diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzigzb" "b/data_all_eng_slimpj/shuffled/split2/finalzzigzb"
new file mode 100644--- /dev/null
+++ "b/data_all_eng_slimpj/shuffled/split2/finalzzigzb"
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\nMuch of the progress in our understanding of the capacity limits of wireless networks over the past decade has come from the pursuit of progressively refined capacity approximations. Generalized degrees of freedom  (GDoF) characterizations represent a most significant step along this path because of their ability to capture arbitrary channel strength and channel uncertainty levels.  The GDoF framework may seem counter-intuitive at first because it allows exponential scaling of signal strengths with various exponents. An intuitive justification for the GDoF framework is as follows. It is important to remember that the goal behind GDoF is to seek capacity approximations for a given wireless network with its arbitrary \\emph{finite} signal strengths and channel uncertainty levels. Unlike the degrees of freedom (DoF) metric which linearly scales all signal strengths and loses the distinction of different channel strengths (every non-zero channel carries $1$ DoF), the GDoF formulation takes a more sophisticated approach.  The key to GDoF is the intuition that if the capacity of every link in a network is scaled by the same factor, then the capacity region of the network should scale by approximately the same factor as well. Normalizing the capacity of the network by the scaling factor then yields a capacity approximation for the original network. Following this intuition, one allows the scaling factor to approach infinity, while guaranteeing that the capacity is always normalized by the scaling factor. The asymptotic behavior of normalized capacity is potentially easier to characterize than a direct approximation of the capacity of the original network. Let $\\alpha_i$ represent the capacity of the $i^{th}$ link in the original network (in isolation from all other links), and let $\\log(P)$ be the scaling factor applied to every link capacity. Then we obtain channels whose capacity scales as $\\alpha_i\\log(P)$, i.e., channels whose strength scales as $P^{\\alpha_i}$, and, according to this intuitive reasoning, normalization of network capacity by $\\log(P)$ in the limit $P\\rightarrow\\infty$ presents the approximation of the capacity of the original network. This approximation is what is known as the GDoF characterization, and along with its abstractions into deterministic channel models, over the past decade it has been the key to finding capacity approximations for many networks whose exact capacity remains intractable. Thus, the linear scaling of capacity naturally corresponds to an exponential scaling of signal strengths in the GDoF model.\n\nGDoF studies started with settings where perfect CSIT is available \\cite{Etkin_Tse_Wang, Jafar_Vishwanath_GDOF, Karmakar_Varanasi}. The opposite extreme of no CSIT was also explored under strong assumptions of statistical equivalence between users \\cite{Huang_Jafar_Shamai_Vishwanath, Varanasi_noCSIT, Guo_noCSIT}. Lately, however, the focus has shifted to the broader assumption of finite precision CSIT  \\cite{Arash_Jafar_TC}, \\cite{Arash_Bofeng_Jafar_BC}. Some of the more sophisticated concepts such as interference alignment \\cite{Jafar_FnT} have turned out to be too fragile to be useful with finite precision CSIT, so that conventional achievable schemes are usually optimal. As such the  main challenge for GDoF studies under finite precision CSIT tends to be the proof of optimality, i.e., the converse, or the GDoF outer bound.  Finding tight GDoF outer bounds under finite precision CSIT is generally a hard problem, as exemplified by the conjecture of Lapidoth et al. \\cite{Lapidoth_Shamai_Wigger_BC} which remained unresolved for nearly a decade. The main idea for these outer bounds is the Aligned Image Sets (AIS) argument that was introduced in  \\cite{Arash_Jafar_PN} in order to settle the conjecture of Lapidoth et al. Generalizations of the AIS approach have also helped settle the GDoF in other settings such as the X channel and the $2$ user MISO BC under finite precision CSIT in \\cite{Arash_Jafar_TC}, and the 2 user MISO BC with arbitrary channel strengths and channel uncertainty levels in \\cite{Arash_Bofeng_Jafar_BC}. Of particular relevance to this work is  \\cite{Arash_Jafar_IC} where the sum GDoF of $K$-user symmetric interference channel (IC) is characterized under finite precision CSIT (see Figure \\ref{fig:Kusert}). This work is motivated by the goal of further broadening the scope of the AIS argument, so that the results of \\cite{Arash_Jafar_IC} may be generalized to MIMO settings.\n\n\\begin{figure*}[h]\n\\center\n\\begin{minipage}[c]{0.54\\textwidth}\n\t\\centerline{\\includegraphics[width=3.7in]{p2+.pdf}}\n\\end{minipage}~~~~~\n\\begin{minipage}[c]{0.33\\textwidth}\n\\begin{eqnarray*}\nd(\\alpha) = \\left\\{\n\\begin{array}{ll} \n1-\\alpha,&0\\leq\\alpha\\leq \\frac{1}{2}\\vspace{0.03in}\\\\ \n\\frac{K-2-(K-3)\\alpha}{K-1}, &\\frac{1}{2}<\\alpha\\leq\\frac{K}{K+1}\\vspace{0.03in}\\\\ \n1-\\left(\\frac{K-1}{K}\\right)\\alpha,&\\frac{K}{K+1}<\\alpha\\leq 1\\\\\n\\frac{\\alpha}{K},& 1<\\alpha\\leq K\\\\\n1, &K<\\alpha\n\\end{array}\n\\right.\n\\end{eqnarray*}\n\\end{minipage}\n\\caption{GDoF\/user of the Symmetric $K$ User Interference Channel with Finite Precision CSIT \\cite{Arash_Jafar_IC}.}\\label{fig:Kusert}\n\\end{figure*}\n\n\nIn this paper, we characterize the GDoF for the symmetric  $K$-user MIMO Interference Channel  under the assumption that the CSIT is limited to finite precision. In this symmetric setting, each transmitter is equipped with $M$ antennas, each receiver is equipped with $N$ antennas, each desired channel (i.e., a channel between a transmit antenna and a receive antenna belonging to the same user) has strength $\\sim P$, while each undesired channel has strength $\\sim P^\\alpha$, where $P$ is a nominal SNR parameter.  GDoF per user take the form of a $W$-curve with respect to $\\alpha$ for fixed values of $M$ and $N$. See Figure \\ref{Fig1}. As usual for finite precision CSIT, achievability is fairly straightforward. While ostensibly the main result of this work is the GDoF characterization for the $K$-user symmetric MIMO IC, the deeper significance of this paper resides in a key generalization of the AIS approach that allows comparisons in the GDoF sense of the entropies of  different numbers of linear combinations (finite precision versus perfectly known channels) of random variables under various power-level partitions. The generalization seems broadly useful for GDoF problems related to MIMO wireless networks. \n\n\n\n\n{\\it Notation:} The notation $|A|$ denotes the cardinality of the set $A$ and the notation $[n]$ is defined as $\\{1,2,\\cdots,n\\}$ for any $n\\in\\mathbb{N}$ where $\\mathbb{N}$ is the set of all positive integer numbers. The notations~ $X^{[T]}$ and $X_{i}^{[T]}$ also stand~ for $\\{X(1), X(2), \\cdots X(T)\\}$ and $\\{X_i(t): \\forall t\\in[T]\\}$, respectively. Moreover, we use the Landau $o(\\cdot)$ notation for the functions $f(x), g(x)$ from $\\mathbb{R}$ to $\\mathbb{R}$ as follows. $f(x)=o(g(x))$ denotes that $\\limsup_{x\\rightarrow\\infty}\\frac{|f(x)|}{|g(x)|}=0$. Finally, we define $\\lfloor x\\rfloor$  as the largest integer that is smaller than or equal to $x$  for any positive real number $x$ and  the smallest integer that is larger than or equal to $x$  for any negative real number $x$. $A^\\dagger$ is the transpose of matrix $A$. The support of a random variable $X$ is denoted as supp$(X)$.\n\n\n\n\n\n\n\\section{Definitions}\n\\begin{definition}\\label{bd}[Bounded Density Channel Coefficients \\cite{Arash_Jafar_PN}] Define a set of real valued random variables, $\\mathcal{G}$ such that the magnitude of each random variable $g\\in\\mathcal{G}$ is bounded away from  infinity, $ |g|\\leq\\Delta<\\infty$, for some positive constant $\\Delta$, and there exists a finite positive constant $f_{\\max}$, such that for all finite cardinality disjoint subsets $\\mathcal{G}_1, \\mathcal{G}_2$ of $\\mathcal{G}$, the joint probability density function of all random variables in $\\mathcal{G}_1$, conditioned on all random variables in $\\mathcal{G}_2$, exists and is bounded above by $f_{\\max}^{|\\mathcal{G}_1|}$. Without loss of generality we will assume that $f_{\\max}\\geq 1, \\Delta\\geq 1$.\n\\end{definition}\n\n\n\n\n\\begin{definition}[Power Levels] Consider integer valued random variables $X_i$ over alphabet $\\mathcal{X}_{\\lambda_i}$,\n\\begin{eqnarray}\n\\mathcal{X}_{\\lambda_i}&\\triangleq&\\{0,1,2,\\cdots,\\bar{P}^{\\lambda_i}-1\\}\n\\end{eqnarray}\nwhere $\\bar{P}^{\\lambda_i}$ is a compact notation for $\\left\\lfloor\\sqrt{P^{\\lambda_i}}\\right\\rfloor$ and the constant $\\lambda_i$ is a positive real number denoting the \\emph{power level} of $X_i$. \n\\end{definition}\n\n\n\n\n\n\\begin {definition}\\label{powerlevel} For $X\\in\\mathcal{X}_\\lambda$, and $0\\leq \\lambda_1\\leq\\lambda$, define the random variables $(X)_{\\lambda_1}^{\\lambda}$ as,\n \\begin{eqnarray}\n(X)_{\\lambda_1}^{\\lambda}&\\triangleq&\\left \\lfloor \\frac{X}{\\bar{P}^{\\lambda_1}} \\right \\rfloor\n\\end{eqnarray}\n\\end {definition}\nIn words, $(X)_{\\lambda_1}^{\\lambda}$ retrieves the top $\\lambda-\\lambda_1$ power levels of $X$. Similarly, for  the vector ${\\bf V}=\\begin{bmatrix}v_1&v_2&\\cdots&v_k\\end{bmatrix}^\\dagger$, we define $({\\bf V})^{\\lambda}_{\\lambda_1}$ as,\n \\begin{eqnarray}\n({\\bf V})^{\\lambda}_{\\lambda_1}&\\triangleq&\\begin{bmatrix}(v_1)^{\\lambda}_{\\lambda_1}&(v_2)^{\\lambda}_{\\lambda_1}&\\cdots&(v_k)^{\\lambda}_{\\lambda_1}\\end{bmatrix}^\\dagger\n\\end{eqnarray}\n\n\\begin {definition}\\label{deflc} For {real}  numbers $x_1,x_2,\\cdots,x_k{\\in\\mathcal{X}_{\\eta}}$ define the notation $L_j^b(x_i,1\\le i\\le k)$  to represent,\n\\begin {eqnarray}\nL^b_j(x_1,x_2,\\cdots,x_k )&=&\\sum_{1\\le i\\le k} \\lfloor g_{j_i}x_i\\rfloor\n\\end{eqnarray}\nfor  distinct random variables $g_{j_i}\\in\\mathcal{G}$. The subscript $j$ is used to distinguish among multiple linear combinations, and may be dropped if there is no potential for ambiguity.  For  the vector $V=\\begin{bmatrix}v_1&v_2&\\cdots&v_k\\end{bmatrix}^\\dagger$ define the notation $L^b_j(V)$  to represent,\n\\begin {eqnarray}\nL^b_j(V)=\\sum_{1\\le i\\le k} \\lfloor g_{j_i}v_i\\rfloor\n\\end{eqnarray}\nfor  distinct random variables $g_{j_i}\\in\\mathcal{G}$.\n\\end {definition}\n\n\\begin{definition}\\label{defvec} For the two vectors $V=\\begin{bmatrix}v_1&\\cdots&v_{k_1}\\end{bmatrix}^\\dagger$ and $W=\\begin{bmatrix}w_1&\\cdots&w_{k_2}\\end{bmatrix}^\\dagger$ define the vector $V\\bigtriangledown W$ as $\\begin{bmatrix}v_1&\\cdots&v_{k_1}&w_1&\\cdots&w_{k_2}\\end{bmatrix}^\\dagger$.\n\\end{definition}\n\n\n\n\n\\section{System Model} {\\label{sec-sys}}\nIn this work we consider only the setting where all variables take real values. Extensions to complex settings are cumbersome but conceptually straightforward  as in \\cite{Arash_Jafar_PN}.\n\\subsection{The Channel}\n\\noindent Define  random variables $\\mathbf{X}_{k}(t)$ and $\\mathbf{Y}_{k}(t)$, $\\forall k\\in[K]$ as,\n\\begin{align}\n\\mathbf{X}_{k}(t)=&\\begin{bmatrix}{X}_{k1}(t)&{X}_{k2}(t)&\\cdots&{X}_{kM}(t)\\end{bmatrix}^\\dagger\\\\\n\\mathbf{Y}_{k}(t)=&\\begin{bmatrix}{Y}_{k1}(t)&{Y}_{k2}(t)&\\cdots&{Y}_{kN}(t)\\end{bmatrix}^\\dagger\n\\end{align}\nwhere the channel uses are indexed by $t\\in[T]$. $X_{km}(t), k\\in[K],m\\in[M],t\\in[T]$ are the symbols sent from $m$-th transmit antenna of the $k$-th transmitter and are subject to unit power constraint, while $Y_{kn}(t), k\\in[K],n\\in[N],t\\in[T]$ are the symbols observed by the $n$-th antenna of the $k$-th receiver. Under the GDoF framework, the channel model for the $K$-user MIMO IC is defined by the following input-output equations\n\\begin{align}\n{\\bf Y}_{k}(t)=&\\sqrt{P}{\\bf G}_{kk}(t)\\mathbf{X}_k(t)+\\sqrt{{P}^{\\alpha}}\\sum_{\\hat{k}=1,\\hat{k}\\neq k}^K{\\bf G}_{k\\hat{k}}(t)\\mathbf{X}_{\\hat{k}}(t)+{\\bf \\Gamma}_{k}(t)\n\\end{align}\nfor all $k\\in[K]$ and $t\\in[T]$. The $N\\times M$ matrix ${\\bf G}_{k\\hat{k}}(t)$ is the channel fading coefficient matrix between the $k$-th receiver and the $\\hat{k}$-th transmitter for any $k,\\hat{k}\\in[K]$. The entry in the $n$-th row and $m$-th column of the matrix ${\\bf G}_{k\\hat{k}}(t)$ is ${G}_{k\\hat{k}nm}(t)$. $\\mathbf{\\Gamma}_{k}(t)$ are  $N\\times 1$  matrices whose components are zero mean unit variance additive white Gaussian noise (AWGN) experienced by $k$-th receiver.  Figure \\ref{Fig1rr} illustrates a $3$-user $3\\times 2$ MIMO IC.  $P$ is a nominal SNR parameter that approaches infinity for GDoF characterizations. CSIR is assumed to be perfect. However, CSIT is limited to finite precision. Under finite precision CSIT  we assume that $G_{k\\hat{k}nm}(t)\\in\\mathcal{G}$ for any $k,\\hat{k}\\in[K],n\\in[N],m\\in[M]$ and $t\\in[T]$, and since transmitters only know the probability density but not the realizations of channel coefficients, we assume that all ${\\bf X}_k(t), t\\in[T], k\\in[K]$ are independent of $\\mathcal{G}$.\n\\begin{figure}[tp]\n\\centering \n\\includegraphics[scale =0.28]{p8.pdf}\n\\caption{Three user $3 \\times 2$ MIMO IC.}\n\\label{Fig1rr}\n\\end{figure}\n\\subsection{GDoF}\nThe definitions of achievable rates $R_i(P)$ and capacity region $\\mathcal{C}(P)$ are standard. The GDoF region is defined as\n\\begin{eqnarray}\n\\mathcal{D}&=&\\{(d_1,d_2,\\cdots,d_K): \\exists (R_1(P),R_2(P), \\cdots,R_K(P))\\nonumber\\\\\n&&\\in\\mathcal{C}(P), \\mbox{ s.t. } d_k=\\lim_{P\\rightarrow\\infty}\\frac{R_k(P)}{\\frac{1}{2}\\log(P)}, \\forall k\\in[K]\\}\n\\end{eqnarray}\nThe maximum value of $d_1+d_2+\\cdots+d_K$ over $\\mathcal{D}$ is known as the sum GDoF value.\n\n\n\n\\section{Main Result}\n\n\\begin{theorem}\\label{theorem:GDoF1} The sum GDoF value for the $K$-user  symmetric MIMO IC for $M\\le \\frac{N}{K}$ is $KM$, and for $\\frac{N}{K}\\le M$ is\n\\begin{align}\n{\\sum_{k=1}^Kd_k}=&\\left\\{\n\\begin{array}{ll} \nK\\min(M,N)(1-\\alpha)+\\frac{K(N-M)^+\\alpha}{K-1},&0\\leq\\alpha\\leq \\frac{1}{2}\\vspace{0.03in}\\\\ \n\\min\\left(\\frac{K}{K-1}\\left((K-2)\\min(M,N)(1-\\alpha)+N(\\alpha)\\right)\\right., &\\vspace{0.03in}\\\\ \nN\\alpha+K\\min(M,N)\\left(1-\\alpha\\right)\\big),&\\frac{1}{2}<\\alpha\\leq 1\\\\\n\\min\\left(D(\\alpha),K\\min(M,N)\\right), &1<\\alpha\n\\end{array}\n\\right.\\label {fg<}\n\\end{align}\nwhere $N(\\alpha)$ and $D(\\alpha)$ are defined as,\n\\begin{eqnarray}\nN(\\alpha)&=&\\min((K-1)M,N)\\alpha+(N-(K-1)M)^+(1-\\alpha)\\\\\nD(\\alpha)&=&(N-(K-1)M)^++\\min(N,(K-1)M)\\alpha \n\\end{eqnarray}\n\\end{theorem}\n\\begin{figure}[tp]\n\\centering \n\\includegraphics[scale =0.4]{p2c.pdf}\n\\caption{Sum GDoF of the three user $3 \\times N$ MIMO IC.}\n\\label{Fig1}\n\\end{figure}\n\\begin{remark} The sum GDoF, i.e., \\eqref{fg<} for $N< M$ yields,\n\\begin{align}\n{\\sum_{k=1}^Kd_k}=&{KN}\\times  \\left\\{\n\\begin{array}{ll} \n(1-\\alpha),&0\\leq\\alpha\\leq \\frac{1}{2}\\vspace{0.03in}\\\\ \n\\frac{K-2-(K-3)\\alpha}{K-1}, &\\frac{1}{2}<\\alpha\\leq\\frac{K}{K+1}\\vspace{0.03in}\\\\ \n1-(\\frac{K-1}{K})\\alpha,&\\frac{K}{K+1}<\\alpha\\leq 1\\\\\n\\frac{\\alpha}{K},& 1<\\alpha\\leq K\\\\\n1, &K<\\alpha\n\\end{array}\n\\right.\\label {GDOFM>N}\n\\end{align}\n\\end{remark}\n\n\n\\section{Proof of Theorem \\ref{theorem:GDoF1}: Converse}\n The first step in the converse proof, identical to \\cite{Arash_Jafar_IC},\n is the transformation\ninto a deterministic setting such that a GDoF outer bound\non the deterministic setting is also a GDoF outer bound on\nthe original setting.  We start directly from the deterministic\nmodel.\n\n\\subsection{Deterministic Model}\\label{DM_1}\n\\vspace{-1em}\n\\begin{align}\n\\bar{\\mathbf{Y}}_{k}(t)&=[\\bar{Y}_{k1}(t)\\ \\bar{Y}_{k2}(t)\\ \\cdots\\ \\bar{Y}_{kN}(t)]^\\dagger\\\\\n\\bar{Y}_{kn}(t)&=L_{kn1}^b(t)\\left({(\\bar{\\mathbf{X}}_{k}(t))}^{\\max(1,\\alpha)}_{\\max(1,\\alpha)-1}\\right)+L_{kn2}^b(t)\\left({(\\bar{\\mathbf{X}}_{j}(t))}^{\\max(1,\\alpha)}_{\\max(1,\\alpha)-\\alpha},\\forall j\\in[K],j\\neq k\\right)\\label{dm1}\n\\end{align}\nfor all $k\\in[K],n\\in[N],t\\in[T]$. $\\bar{\\mathbf{X}}_{k}(t)$ are  defined as,\n\\begin{align}\n\\bar{\\mathbf{X}}_{k}(t)=&[\\bar{X}_{k1}(t)\\ \\bar{X}_{k2}(t)\\ \\cdots\\ \\bar{X}_{kM}(t)]^\\dagger\\label{ggf1}\n\\end{align}\nfor any $k\\in[K]$, $t\\in[T]$ where $\\bar{X}_{km}(t)\\in\\{0, 1, \\cdots, {\\bar{P}}^{\\max(1,\\alpha)}-1\\}$, $\\forall k\\in[K],m\\in[M],t\\in[T]$. \n\n\\subsection{ Key Lemma} \nThe following  lemma is the critical generalization of the AIS bound needed for Theorem \\ref{theorem:GDoF1}.\n\\begin{lemma}\\label{lemma} Define the two random variables $\\bar{\\bf U}_1$ and  $\\bar{\\bf U}_2$ as,\n\\begin{eqnarray}\n\\bar{\\bf U}_1&=&\\left({U}_{11}^{[T]},{U}_{12}^{[T]},\\cdots,{U}_{1N_1}^{[T]}\\right)\\label{lemmamimox1}\\\\\n\\bar{\\bf U}_2&=&\\left({U}_{21}^{[T]},{U}_{22}^{[T]},\\cdots,{U}_{2N_2}^{[T]}\\right)\\label{lemmamimox2}\n\\end{eqnarray}\nwhere for any $t\\in[T]$, $U_{1n}(t)$ and $U_{2n}(t)$ are defined as,\n\\begin{eqnarray}\nU_{1n}(t)&=&L_{1n}^b(t)\\left((\\bar{\\mathbf{V}}_1(t))^{\\eta}_{\\eta-\\lambda_{11}}\\bigtriangledown(\\bar{\\mathbf{V}}_2(t))^{\\eta}_{\\eta-\\lambda_{12}}\\bigtriangledown\\cdots\\bigtriangledown(\\bar{\\mathbf{V}}_l(t))^{\\eta}_{\\eta-\\lambda_{1l}}\\right), \\forall n\\in[N_1]\\label{lemmamimox3}\\\\\nU_{2n}(t)&=&L_{2n}^b(t)\\left((\\bar{\\mathbf{V}}_1(t))^{\\eta}_{\\eta-\\lambda_{21}}\\bigtriangledown(\\bar{\\mathbf{V}}_2(t))^{\\eta}_{\\eta-\\lambda_{22}}\\bigtriangledown\\cdots\\bigtriangledown(\\bar{\\mathbf{V}}_l(t))^{\\eta}_{\\eta-\\lambda_{2l}}\\right), \\forall n\\in[N_2]\\label{lemmamimox4}\n\\end{eqnarray}\nwhere $\\bar{\\mathbf{V}}_i(t)=\\begin{bmatrix}\\bar{V}_{i1}(t)&\\cdots&\\bar{V}_{iM_i}(t)\\end{bmatrix}^\\dagger$, $\\bar{V}_{im}(t)\\in\\mathcal{X}_{\\eta}$  are all independent of $\\mathcal{G}$, and $0\\le\\lambda_{1i},\\lambda_{2i}\\le\\eta$ for any $i\\in[l]$.  Without loss of generality,  $(\\lambda_{1i}-\\lambda_{2i})^+$ are sorted in descending order, i.e., $(\\lambda_{1i}-\\lambda_{2i})^+\\ge(\\lambda_{1j}-\\lambda_{2j})^+$ if $1\\le i< j\\le l$.  Then, for any acceptable\\footnote{Let $\\mathcal{G}(Z)\\subset\\mathcal{G}$ denote the set of all bounded density channel coefficients that appear in $\\bar{\\bf U}_1,\\bar{\\bf U}_2$.  $W$ is acceptable if  conditioned on any $\\mathcal{G}_o\\subset (\\mathcal{G}\/\\mathcal{G}(Z))\\cup \\{W\\}$, the channel coefficients $\\mathcal{G}(Z)$ satisfy the bounded density assumption. For instance, any random variable $W$ independent of $\\mathcal{G}$ can be utilized in  Lemma \\ref{lemma}.} random variable ${W}$, if $N_1\\le \\min(N_2, \\sum_{i=1}^lM_i)$ we have,\n\\begin{eqnarray}\n&&H({\\bar{\\bf U}}_1\\mid {W},\\mathcal{G})-H({\\bar{\\bf U}}_2\\mid {W},\\mathcal{G})\\nonumber\\\\\n&\\le&T\\big((N_1-\\sum_{i=1}^sM_i)(\\lambda_{1,s+1}-\\lambda_{2,s+1})^++\\sum_{i=1}^sM_i(\\lambda_{1i}-\\lambda_{2i})^+\\big)\\log{\\bar{P}}+T~o~(\\log{\\bar{P}})\\label{lemmamimox5}\n\\end{eqnarray}\nwhere $s$ must satisfy the condition  $\\sum_{i=1}^{s}M_i\\le N_1< \\sum_{i=1}^{s+1}M_i$.  \n\\end{lemma} \nProof of Lemma \\ref{lemma} is based on the AIS argument and is relegated to Appendix \\ref{lemmap}.\n\n\n\n\\subsection{Some Insights For the Three User $2\\times3$ MIMO IC}\n\\begin{figure}[!h]\n\\centerline{\\includegraphics[width=6in]{p8d.pdf}}\n\\caption{ Three user $2\\times3$ MIMO IC. The network is fully connected but only the channel strength parameters needed for the application of Lemma \\ref{lemma} are shown in this figure.}\\label{fig:intuit-}\n\\end{figure}\nTo gain some insights into the application of Lemma \\ref{lemma}, consider the three user $2\\times3$ MIMO IC illustrated in Figure \\ref{fig:intuit-} for $\\alpha\\le1$. To apply Lemma \\ref{lemma}, the random variables  $\\bar{\\bf U}_1$, $\\bar{\\bf U}_2$, $\\bar{\\bf V}_1^{[T]}$, $\\bar{\\bf V}_2^{[T]}$ and $W$ are interpreted as $\\bar{\\bf Y}_2^{[T]}$, $\\bar{\\bf Y}_1^{[T]}$, $\\bar{\\bf X}_2^{[T]}$, $\\bar{\\bf X}_3^{[T]}$ and $\\bar{\\bf X}_1^{[T]}$, respectively. The first user receives the top $\\alpha$ power levels of $\\bar{\\bf X}_2^{[T]}$ and $\\bar{\\bf X}_3^{[T]}$ while second reciever sees  the top $1$ power levels of $\\bar{\\bf X}_2^{[T]}$ and  the top $\\alpha$ power levels of $\\bar{\\bf X}_3^{[T]}$.  So we have $\\eta=1, \\lambda_{11}=1, \\lambda_{21}=\\alpha, \\lambda_{12}=\\alpha, \\lambda_{22}=\\alpha$. Therefore, $(\\lambda_{11}-\\lambda_{21})^+=1-\\alpha$ and $(\\lambda_{12}-\\lambda_{22})^+=0$. From Lemma \\ref{lemma} we conclude,\n\\begin{align}\n&H(\\bar{\\bf Y}_2^{[T]}\\mid \\bar{\\bf X}_1^{[T]},\\mathcal{G})-H(\\bar{\\bf Y}_1^{[T]}\\mid \\bar{\\bf X}_1^{[T]},\\mathcal{G})\\nonumber\\\\\n\\le&T\\big(1\\times 0+2\\times(1-\\alpha)\\big)\\log{\\bar{P}}+T~o~(\\log{\\bar{P}})\\label{jl0}\n\\end{align}\n  Let us also explain how intuitively we expect \\eqref{jl0} to be true as well. Conditioned on $\\bar{\\bf X}_{1}^{[T]}$, $\\bar{\\bf Y}_{2}(t)$ is a linear combination of $\\bar{\\bf X}_{2}(t)$ and $(\\bar{\\bf X}_{3}(t))^{\\alpha}$ while $\\bar{\\bf Y}_{1}(t)$ is a linear combination of $(\\bar{\\bf X}_{2}(t))^{\\alpha}$ and $(\\bar{\\bf X}_{3}(t))^{\\alpha}$. Consider the channel illustrated in Figure \\ref{fig:intuit-}. First of all, observe that $\\bar{\\bf X}_{2}(t)$ appears in $\\bar{\\bf Y}_{2}(t)$ with the signal strength levels $1$ and appears in $\\bar{\\bf Y}_{1}(t)$ with the signal strength levels $\\alpha$. Thus, due to the bounded density assumption the maximum difference of $2(1-\\alpha)$ is possible in the GDoF sense between the two entropies. Note that, $\\bar{\\bf X}_{3}(t)$ appears in both the received signals  $\\bar{\\bf Y}_{1}(t)$ and  $\\bar{\\bf Y}_{2}(t)$ with the same signal strength levels of $\\alpha$. Therefore, it cannot contribute positive difference of entropies as in the finite precision CSIT no interference alignment is possible.\n  \nSimilarly, from Lemma \\ref{lemma} we have,  \n\\begin{align}\nH(\\bar{\\bf Y}_3^{[T]}\\mid \\bar{\\bf X}_1^{[T]},\\bar{\\bf X}_2^{[T]},\\mathcal{G})-H(\\bar{\\bf Y}_2^{[T]}\\mid \\bar{\\bf X}_1^{[T]},\\bar{\\bf X}_2^{[T]},\\mathcal{G})\\le&2T(1-\\alpha)\\log{\\bar{P}}+T~o~(\\log{\\bar{P}})\\label{jl00}\n\\end{align} \nOn the other hand, writing Fano's inequality for all the three users (and suppressing $o(T)$ terms for simplicity) we obtain the following bounds,\n\\begin{eqnarray}\nTR_1&\\le& H(\\bar{\\bf Y}_{1}^{[T]}\\mid \\mathcal{G})- H(\\bar{\\bf Y}_{1}^{[T]}\\mid \\bar{\\bf X}_{1}^{[T]},\\mathcal{G})\\label{jl1}\\\\\nTR_2&\\le& H(\\bar{\\bf Y}_{2}^{[T]}\\mid \\bar{\\bf X}_{1}^{[T]},\\mathcal{G})- H(\\bar{\\bf Y}_{2}^{[T]}\\mid \\bar{\\bf X}_{1}^{[T]},\\bar{\\bf X}_{2}^{[T]},\\mathcal{G})\\label{jl2}\\\\\nTR_3&\\le& H(\\bar{\\bf Y}_{3}^{[T]}\\mid  \\bar{\\bf X}_{1}^{[T]},\\bar{\\bf X}_{2}^{[T]},\\mathcal{G})\\label{jl3}\n\\end{eqnarray}\nTherefore, for $\\alpha\\le1$, from \\eqref{jl0}-\\eqref{jl3} we have,\n\\begin{eqnarray}\nTR_1+TR_2+TR_3&\\le& H(\\bar{\\bf Y}_{1}^{[T]}\\mid \\mathcal{G})+4T(1-\\alpha)\\log{\\bar{P}}+T~o~(\\log{\\bar{P}})\\\\\n&\\le&T(6-3\\alpha)\\log{\\bar{P}}+T~o~(\\log{\\bar{P}})\\label{oi}\n\\end{eqnarray}\n\\eqref{oi} is true as discrete entropy of any discrete random variable is bounded by logarithm of its cardinality.\n\n\\subsection{Equivalent Bounds}\nTheorem \\ref{theorem:GDoF1} is concluded from the following bounds,\n\\begin{enumerate}\n\\item If $\\alpha\\in\\mathbb{R}^+,\\alpha\\le\\frac{1}{2}$, then \n\\begin{eqnarray}\n&&\\sum_{k=1}^Kd_k\\nonumber\\\\\n&\\le& \\frac{K\\left(\\min(M,N)(1-\\alpha)+(N-M)^+\\alpha\\right)+K(K-2)\\min(M,N)(1-\\alpha)}{K-1}\\label{b1}\n\\end{eqnarray}\n\\item If $\\alpha\\in\\mathbb{R}^+,\\frac{1}{2}\\le\\alpha\\le1$, then \n\\begin{eqnarray}\n&&\\sum_{k=1}^Kd_k\\nonumber\\\\\n&\\le& \\frac{K\\left(\\min((K-1)M,N)\\alpha+(N-(K-1)M)^+(1-\\alpha)\\right)+K(K-2)\\min(M,N)(1-\\alpha)}{K-1}\\nonumber\\\\\n&&\\label{b1+}\n\\end{eqnarray}\n\\item If $\\alpha\\in\\mathbb{R}^+,\\alpha\\le1$, then \n\\begin{eqnarray}\n\\sum_{k=1}^Kd_k&\\le& N\\alpha+K\\min(M,N)(1-\\alpha)\\label{b2}\n\\end{eqnarray}\n\\item If $\\alpha\\in\\mathbb{R}^+,1\\le\\alpha$, then \n\\begin{eqnarray}\n\\sum_{k=1}^Kd_k&\\le& (N-(K-1)M)^++\\min(N,(K-1)M)\\alpha\\label{b2+}\n\\end{eqnarray}\n\\item For any $\\alpha\\in \\mathbb{R}^+$, \n\\begin{eqnarray}\n\\sum_{k=1}^Kd_k&\\le& K\\min(M,N)\\label{b3}\n\\end{eqnarray}\n\\end{enumerate}\nThus, in order to prove Theorem \\ref{theorem:GDoF1}, the Bounds \\eqref{b1}-\\eqref{b3} should be proved.\n\\subsection{Proof of Bounds \\eqref{b1}-\\eqref{b3}}\\label{mt}\nThe last bound,  $\\sum_{k=1}^Kd_k\\le K\\min(M,N)$ is the trivial combination of single user bounds.  Let us prove the other four  bounds, i.e., \\eqref{b1}-\\eqref{b2+}. \n\\begin{enumerate}\n\\item {\\bf Proof of \\eqref{b1} and \\eqref{b1+}}\\\\\n\n Writing Fano's Inequality for the first $K-1$ receivers we have,\n\\begin{eqnarray}\nTR_1&\\le& I(\\bar{\\bf Y}^{[T]}_{1};\\bar{\\bf X}^{[T]}_{1}\\mid \\mathcal{G})\\label{c1}\\\\\nTR_k&\\le& I(\\bar{\\bf Y}^{[T]}_{k};\\bar{\\bf X}^{[T]}_{k}\\mid \\bar{\\bf X}^{[T]}_{1},\\cdots,\\bar{\\bf X}^{[T]}_{k-1},\\mathcal{G}), \\forall k\\in[K-1],k\\neq 1\\label{c2}\n\\end{eqnarray}\n{Summing}  \\eqref{c1} and \\eqref{c2}, we have,\n\\begin{eqnarray}\n&&T\\sum_{k=1}^{K-1}R_k\\nonumber\\\\\n&\\le& I(\\bar{\\bf Y}^{[T]}_{1};\\bar{\\bf X}^{[T]}_{1}\\mid \\mathcal{G})+\\sum_{k=2}^{K-1} I(\\bar{\\bf Y}^{[T]}_{k};\\bar{\\bf X}^{[T]}_{k}\\mid \\bar{\\bf X}^{[T]}_{1},\\cdots,\\bar{\\bf X}^{[T]}_{k-1},\\mathcal{G})\\\\\n&=& H(\\bar{\\bf Y}^{[T]}_{1}\\mid \\mathcal{G})-H\\left(\\bar{\\bf X'}^{[T]}_{K}\\mid \\mathcal{G}\\right)\\nonumber\\\\\n&&+\\sum_{k=2}^{K-1} \\left(H(\\bar{\\bf Y}^{[T]}_{k}\\mid  \\bar{\\bf X}^{[T]}_{1},\\cdots,\\bar{\\bf X}^{[T]}_{k-1},\\mathcal{G})-H(\\bar{\\bf Y}^{[T]}_{k-1}\\mid  \\bar{\\bf X}^{[T]}_{1},\\cdots,\\bar{\\bf X}^{[T]}_{k-1},\\mathcal{G})\\right)\\\\\n&\\le& H(\\bar{\\bf Y}^{[T]}_{1}\\mid \\mathcal{G})-H\\left(\\bar{\\bf X'}^{[T]}_{K}\\mid \\mathcal{G}\\right)+\\sum_{k=2}^{K-1} T\\min(M,N)(1-\\alpha)\\log{\\bar{P}}+T~o(\\log{\\bar{P}})\\label{c3}\n\\end{eqnarray}\nwhere the new random variable,  $\\bar{\\bf X'}_k(t)$ is defined as\\\\\n\\begin{eqnarray}\n\\bar{\\bf X'}_k(t)&=&\\begin{bmatrix}\\bar{X'}_{k1}(t)&\\bar{X'}_{k2}(t)&\\cdots&\\bar{X'}_{kN}(t)\\end{bmatrix}^\\dagger\\\\\n\\bar{X'}_{kn}(t)&=&{L}_{kn3}^b(t)\\left((\\bar{\\bf X}_{k}(t))^{\\alpha}\\right), \\forall n\\in[N]\n\\end{eqnarray}\nLet us explain how Lemma \\ref{lemma} {yields}  \\eqref{c3}. Substitute the random variables $\\bar{\\bf U}_1$, $\\bar{\\bf U}_2$, $\\bar{\\bf V}_1^{[T]}$, $\\bar{\\bf V}_2^{[T]}$ and $W$  in Lemma \\ref{lemma}  with $\\bar{\\bf Y}^{[T]}_{k}$, $\\bar{\\bf Y}^{[T]}_{k-1}$, $\\bar{\\bf X}_k^{[T]}$,  $\\left(\\bar{\\bf X}_{j}^{[T]},j\\in[K],j\\notin[k]\\right)$ and $\\left(\\bar{\\bf X}_{j}^{[T]},j\\in[k-1]\\right)$, respectively. Next,  we set $\\eta=1, \\lambda_{11}=1, \\lambda_{21}=\\alpha, \\lambda_{12}=\\alpha, \\lambda_{22}=\\alpha,M_1=M,M_2=(K-k)M,N_1=N_2=N$. Thus, we have $(\\lambda_{11}-\\lambda_{21})^+=1-\\alpha$ and $(\\lambda_{12}-\\lambda_{22})^+=0$. Therefore, from Lemma \\ref{lemma},  \\eqref{c3} is concluded. Similar to  \\eqref{c3}, by symmetry we have, \n\\begin{eqnarray}\n&&T\\sum_{k\\in[K],k\\neq j}R_k\\nonumber\\\\\n&\\le& H(\\bar{\\bf Y}^{[T]}_{j+1}\\mid \\mathcal{G})-H\\left(\\bar{\\bf X'}^{[T]}_{j}\\mid \\mathcal{G}\\right)\\nonumber\\\\\n&&+(K-2) T\\min(M,N)(1-\\alpha)\\log{\\bar{P}}+T~o(\\log{\\bar{P}})\\label{c4}\n\\end{eqnarray}\nfor all $j\\in[K]$. Summing \\eqref{c4} for all $j\\in[K]$ we have,\n\\begin{eqnarray}\n&&T(K-1)\\sum_{k=1}^KR_k\\nonumber\\\\\n&=&T\\sum_{j=1}^K\\sum_{k\\in[K],k\\neq j}R_k\\nonumber\\\\\n&\\le& \\sum_{k=1}^K\\left(H(\\bar{\\bf Y}^{[T]}_{k}\\mid \\mathcal{G})-H\\left(\\bar{\\bf X'}^{[T]}_{k}\\mid \\mathcal{G}\\right)\\right)+TK(K-2) \\min(M,N)(1-\\alpha)\\log{\\bar{P}}\\nonumber\\\\\n&&+T~o(\\log{\\bar{P}})\\label{xx}\n\\end{eqnarray}\nNow, let us consider the two cases of $\\alpha\\le\\frac{1}{2}$ and $\\frac{1}{2}\\le\\alpha\\le1$ separately.\n\\begin{enumerate}\n\\item{$\\alpha\\le\\frac{1}{2}$}\n\\begin{eqnarray}\n&&H(\\bar{\\bf Y}^{[T]}_{k}\\mid \\mathcal{G})-H\\left(\\bar{\\bf X'}^{[T]}_{k}\\mid \\mathcal{G}\\right)\\nonumber\\\\\n&\\le&  T\\left(\\min(M,N)(1-\\alpha)+(N-M)^+\\alpha\\right)\\log{\\bar{P}}+T~o(\\log{\\bar{P}})\\label{c5}\n\\end{eqnarray}\nLet us explain how \\eqref{c5} follows from Lemma \\ref{lemma}. Substitute the random variables  $\\bar{\\bf U}_1$, $\\bar{\\bf U}_2$, $\\bar{\\bf V}_1^{[T]}$ and $\\bar{\\bf V}_2^{[T]}$  with $\\bar{\\bf Y}^{[T]}_{k}$, $\\bar{\\bf X'}^{[T]}_{k}$, $\\bar{\\bf X}_k^{[T]}$ and $\\left(\\bar{\\bf X}_{j}^{[T]},j\\in[K],j\\neq k\\right)$, respectively. Moreover,  setting $\\eta=1, \\lambda_{11}=1, \\lambda_{21}=\\alpha, \\lambda_{12}=\\alpha, \\lambda_{22}=0$, we have $(\\lambda_{11}-\\lambda_{21})^+=1-\\alpha$ and $(\\lambda_{12}-\\lambda_{22})^+=\\alpha$. Therefore, from Lemma \\ref{lemma} we conclude \\eqref{c5}. From \\eqref{xx} and \\eqref{c5}, \\eqref{b1} is concluded.\n\\item{$\\frac{1}{2}\\le\\alpha\\le1$}\n\\begin{eqnarray}\n&&H(\\bar{\\bf Y}^{[T]}_{k}\\mid \\mathcal{G})-H\\left(\\bar{\\bf X'}^{[T]}_{k}\\mid \\mathcal{G}\\right)\\nonumber\\\\\n&\\le&  T\\left(\\min((K-1)M,N)\\alpha+(N-(K-1)M)^+(1-\\alpha)\\right)\\log{\\bar{P}}+T~o(\\log{\\bar{P}})\\label{c5g}\n\\end{eqnarray}\n \\eqref{c5g} follows from Lemma \\ref{lemma} similar to \\eqref{c5}. Substitute the random variables  $\\bar{\\bf U}_1$, $\\bar{\\bf U}_2$, $\\bar{\\bf V}_1^{[T]}$ and $\\bar{\\bf V}_2^{[T]}$  with $\\bar{\\bf Y}^{[T]}_{k}$, $\\bar{\\bf X'}^{[T]}_{k}$, $\\left(\\bar{\\bf X}_{j}^{[T]},j\\in[K],j\\neq k\\right)$ and $\\bar{\\bf X}_k^{[T]}$, respectively. The rest of the proof is concluded similar to \\eqref{c5}. From \\eqref{xx} and \\eqref{c5g}, \\eqref{b1+} is concluded.\n\\end{enumerate}\n\\item {\\bf Proof of \\eqref{b2}}\\\\\n\n Summing \\eqref{c1} and \\eqref{c2}, we have,\n\\begin{eqnarray}\n&&T\\sum_{k=1}^{K}R_k\\nonumber\\\\\n&\\le& I(\\bar{\\bf Y}^{[T]}_{1};\\bar{\\bf X}^{[T]}_{1}\\mid \\mathcal{G})+\\sum_{k=2}^{K} I(\\bar{\\bf Y}^{[T]}_{k};\\bar{\\bf X}^{[T]}_{k}\\mid \\bar{\\bf X}^{[T]}_{1},\\cdots,\\bar{\\bf X}^{[T]}_{k-1},\\mathcal{G})\\label{c6+}\\\\\n&=& H(\\bar{\\bf Y}^{[T]}_{1}\\mid \\mathcal{G})\\nonumber\\\\\n&&+\\sum_{k=2}^{K} \\left(H(\\bar{\\bf Y}^{[T]}_{k}\\mid  \\bar{\\bf X}^{[T]}_{1},\\cdots,\\bar{\\bf X}^{[T]}_{k-1},\\mathcal{G})-H(\\bar{\\bf Y}^{[T]}_{k-1}\\mid  \\bar{\\bf X}^{[T]}_{1},\\cdots,\\bar{\\bf X}^{[T]}_{k-1},\\mathcal{G})\\right)\\\\\n&\\le& H(\\bar{\\bf Y}^{[T]}_{1}\\mid \\mathcal{G})+\\sum_{k=2}^{K} T\\min(M,N)(1-\\alpha)\\log{\\bar{P}}+T~o(\\log{\\bar{P}})\\label{c6}\\\\\n&\\le& T\\left((N-M)^+\\alpha+\\min(M,N)\\right)\\log{\\bar{P}}+\\sum_{k=2}^{K} T\\min(M,N)(1-\\alpha)\\log{\\bar{P}}\\nonumber\\\\\n&&+T~o(\\log{\\bar{P}})\\label{cv6}\\\\\n&=& T(N\\alpha+K\\min(M,N)(1-\\alpha))\\log{\\bar{P}}+T~o(\\log{\\bar{P}})\\label{c7}\n \\end{eqnarray}\n\\eqref{c6} follows similar to \\eqref{c3} and \\eqref{cv6}  is concluded as the entropy of a discrete random variable is   bounded by logarithm of the cardinality of its support, i.e., \\footnote{\\eqref{cv6} follows from Lemma \\ref{lemma}  by substituting $\\bar{\\bf U}_1$ and $\\bar{\\bf U}_2$ with $\\bar{\\bf Y}^{[T]}_{1}$ and ${ C}^{[T]}$ where ${\\bf C}^{[T]}$ is a $T$-letter constant variable.  Then, substituting $\\bar{\\bf V}_1^{[T]}$ and $\\bar{\\bf V}_2^{[T]}$  with  $\\bar{\\bf X}_1^{[T]}$ and $\\left(\\bar{\\bf X}_{j}^{[T]},j\\in[K],j\\neq 1\\right)$, \\eqref{cv6} is concluded. Here, we assume $\\eta=1, \\lambda_{11}=1, \\lambda_{21}=\\alpha, \\lambda_{12}=0, \\lambda_{22}=0,,M_1=M,M_2=(K-1)M,N_1=N_2=N$.}\n\\begin{eqnarray}\n H(\\bar{\\bf Y}^{[T]}_{1}\\mid \\mathcal{G})&\\le& T\\left((N-M)^+\\alpha+\\min(M,N)\\right)\\log{\\bar{P}}\n \\end{eqnarray}\nDividing \\eqref{c7} by $T\\log{\\bar{P}}$, \\eqref{b2}  is obtained.\n\\item {\\bf Proof of \\eqref{b2+}}\\\\\n\n Similarly, from (\\eqref{c6+}-\\eqref{c6}) we have,\n\\begin{eqnarray}\n&&T\\sum_{k=1}^{K}R_k\\nonumber\\\\\n&\\le& H(\\bar{\\bf Y}^{[T]}_{1}\\mid \\mathcal{G})+\\sum_{k=2}^{K} T\\min(M,N)(1-\\alpha)^+\\log{\\bar{P}}+T~o(\\log{\\bar{P}})\\\\\n&\\le& H(\\bar{\\bf Y}^{[T]}_{1}\\mid \\mathcal{G})+T~o(\\log{\\bar{P}})\\label{c6-}\\\\\n&\\le& (N-(K-1)M)^+T\\log{\\bar{P}}+\\min(N,(K-1)M)\\alpha T\\log{\\bar{P}}+T~o(\\log{\\bar{P}})\\label{c6--}\n \\end{eqnarray}\n\\eqref{c6-} is true as $1\\le\\alpha$ and \\eqref{c6--} follows similar\\footnote{\\eqref{c6--} follows similar to  \\eqref{cv6}  from Lemma \\ref{lemma}. Substitute $\\bar{\\bf U}_1$, $\\bar{\\bf U}_2$, $\\bar{\\bf V}_1^{[T]}$ and $\\bar{\\bf V}_2^{[T]}$ with $\\bar{\\bf Y}^{[T]}_{1}$, ${ C}^{[T]}$,  $\\left(\\bar{\\bf X}_{j}^{[T]},j\\in[K],j\\neq 1\\right)$ and $\\bar{\\bf X}_1^{[T]}$ and  assume $\\eta=\\alpha, \\lambda_{11}=\\alpha, \\lambda_{21}=1, \\lambda_{12}=0, \\lambda_{22}=0,M_1=(K-1)M,M_2=M,N_1=N_2=N$.} to  \\eqref{cv6}. Dividing \\eqref{c6--} by $T\\log{\\bar{P}}$, \\eqref{b2+}  is obtained.\n\n\n\n\\end{enumerate}\n\n\\section{Proof of Theorem \\ref{theorem:GDoF1}: Achievability}\n\\subsection{A Useful Lemma}\nConsider a $(M_1+M_2)$-user multiple access channel (MAC) where each transmitter is equipped with a single antenna, the receiver has $N$ antennas, $N< M_1+M_2$, and the $N\\times 1$ received signal vector ${\\bf Q}$ is represented as,\n\\begin{align}\n{\\bf Q}=&\\sqrt{P}\\sum_{k=1}^{M_1} {\\bf H}_k{ T}_k+\\sqrt{P^{\\alpha}}\\sum_{k=M_1+1}^{M_1+M_2} {\\bf H}_k{ T}_k+\\sum_{n=1}^{N} \\sqrt{P^{\\alpha_n}}{\\bf G}_nZ_n\\label{mac0}\n\\end{align}\nwhere  $T_1, T_2, \\cdots, T_{M_1+M_2}$ are the transmitted signals, and $Z_1,Z_2,\\cdots,Z_N$ are i.i.d. Gaussian zero mean unit variance noise terms. The ${\\bf H}_k, {\\bf G}_n$ are $N\\times 1$ generic vectors, i.e.,  generated from continuous distributions with bounded density, so that any $N$ of them are linearly independent almost surely. The transmit power constraint is  expressed as,\n\\begin {eqnarray}\n\\mbox{E}{|T_{k}|}^2&\\leq&P^{-\\eta_k},~\\forall k\\in[M_1+M_2]\\label{mac1}\n\\end{eqnarray}\nwhere for any $k\\in[M_1+M_2]$,  $\\eta_k$ is a non-negative integer. Further, define $\\gamma_k$ for $k\\in[M_1+M_2]$ as,\n\\begin {eqnarray}\n\\gamma_k&=&\\left\\{\n\\begin{array}{ll} \n{(1-\\eta_k)}^+,&k\\in[M_1]\\\\ \n{(\\alpha-\\eta_k)}^+, &\\text{Otherwise}\n\\end{array}\n\\right.\\label{gamma1}\n\\end{eqnarray}\nThus $\\gamma_k$ is the received power level of user $k$ in the GDoF sense. The GDoF region $\\mathcal{D}'$ is defined as\n\\begin{align}\n\\mathcal{D}'\\triangleq&\\{(d'_1,d'_2,\\cdots,d'_{M_1+M_2}): \\exists (R'_1(P),R'_2(P),\\cdots,R'_{M_1+M_2}(P))\\in\\mathcal{C}'(P),\\nonumber\\\\\n& \\mbox{ s.t. } d'_k=\\lim_{P\\rightarrow\\infty}\\frac{R'_k(P)}{\\frac{1}{2}\\log{(P)}}, \\forall k\\in[M_1+M_2]\\} \\label {region}\n\\end{align}\n where $\\mathcal{C}'(P)$ is the capacity region of MAC described in (\\ref{mac0}).\n \n \n\\begin{lemma}\\label{lemma:mac} The GDoF tuple $(d'_1, d'_2, \\cdots, d'_{M_1+M_2})$ is achievable in the multiple access channel described above if $\\forall k\\in[M_1+M_2]$, and $\\forall S\\subset[M_1+M_2]$ where $|S|=k$,\n\\begin{align}\n\\sum_{i\\in S} d'_i\n&\\le\\max_{S_2\\in S,|S_2|=\\min(k,N)}\\sum_{i\\in S_2} \\gamma_i -\\min_{S_1\\in[N],|S_1|=\\min(k,N)}\\sum_{i\\in S_1} \\alpha_i \\label{mac3}\n\\end{align}\n\\end{lemma} \nFor proof of Lemma \\ref{lemma:mac} see \\cite{Arash_Jafar_MIMOsym_ArXiv}. It is sufficient to derive the achievability for Theorem \\ref{theorem:GDoF1}, as Theorem \\ref{theorem:GDoF1} is automatically concluded from it.\n\n\\subsection{Proof of Achievability in Theorem \\ref{theorem:GDoF1}} \\label{app3}\nNow, let us achieve the bound (\\ref{fg<}). We will suppress the time-index $t$ in this section to simplify the notation. For any $k\\in[K]$ user $k$'s message $W_k$ is split into messages $(W_{kc},W_{kp})$, representing common message and private message, respectively.  Let us consider the three cases of $\\alpha\\le\\frac{1}{2}$, $\\frac{1}{2}\\le\\alpha\\le1$, and $1\\le\\alpha$ separately as,\n\\begin{enumerate}\n\\item {$\\alpha\\le\\frac{1}{2}$.} Our goal here is to achieve $\\min(M,N)(1-\\alpha)+\\frac{(N-M)^+\\alpha}{K-1}$ GDoF per user where results in $K\\min(M,N)(1-\\alpha)+\\frac{K(N-M)^+\\alpha}{K-1}$ GDoF totally. In order to achieve $\\min(M,N)(1-\\alpha)+\\frac{(N-M)^+\\alpha}{K-1}$ GDoF per user, for any $k\\in[K]$ the public message $W_{kc}$ is encoded into Gaussian codebooks $U_{k1c},U_{k2c},\\cdots,U_{kMc}$ with powers $\\mbox{E}{|U_k|}^2=1-P^{-\\alpha}$ each carrying $\\frac{(N-M)^+\\alpha}{(K-1)M}$ GDoF. These codewords are transmitted through $M$ antennas along  $M\\times 1$ generic unit  vectors ${\\bf V}_{k1},{\\bf V}_{k2},\\cdots,{\\bf V}_{k{M}}$.  The private message $W_{kp}$ is encoded into Gaussian codebooks $U_{k1p},U_{k2p},\\cdots,U_{k\\min(M,N)p}$ with powers $\\mbox{E}{|U_{kjp}|}^2=P^{-\\alpha}$ for any $j\\in[\\min(M,N)]$ so that the total power per transmitter is unity. These codewords are transmitted through $\\min(M,N)$ antennas along the $M\\times 1$ generic unit  vectors ${\\bf V}_{k1},{\\bf V}_{k2},\\cdots,{\\bf V}_{k{\\min(M,N)}}$. Each of the  private messages is carrying $1-\\alpha$ GDoF. The transmitted and received signals are,\n\\begin{align}\n\\mathbf{X}_{k}=&\\sum_{j=1}^{M}\\mathbf{V}_{kj}{U}_{kjc}+\\sum_{j=1}^{\\min{M,N}}\\mathbf{V}_{kj}{U}_{kjp}\\label{TR21a-}\\\\\n\\mathbf{Y}_{k}=&\\sqrt{P}{\\bf G}_{kk}\\mathbf{X}_{k}+\\sum_{j=1,j\\neq k}^{K}\\sqrt{P^{\\alpha}}{\\bf {G}}_{kj}\\mathbf{X}_{j}+\\mathbf{\\Gamma}_{k}\\label{TR21a}\n\\end{align}\n Using Lemma \\ref{lemma:mac} we claim that each receiver, e.g., receiver $1$ can decode  all the signals $U_{kjc}$ and $U_{1jp}$ for all $k\\in[K]$ and $j\\in[M]$  treating all the other signals as noise. Set the variables $M_1=M+\\min(M,N)$, $M_2=(K-1)M$ and $\\alpha_n=0$ for all $n\\in[N]$. Moreover, define the codewords $T_{1},\\cdots,T_{KM+\\min(M,N)}$ as \n\\begin {align}\nT_j=&\\left\\{\n\\begin{array}{ll} \nU_{1jc},&1\\le j\\le {M}\\\\ \nU_{2(j-M)c},&M<j\\le 2M\\\\\n\\vdots &\\vdots\\\\\nU_{K(j-(K-1)M)c},&(K-1)M<j\\le KM\\\\\nU_{1(j-KM)p},&KM<j\\le KM+\\min(M,N)\n\\end{array}\n\\right.\n\\end{align}\nFrom \\eqref{gamma1}, $\\gamma_1,\\cdots,\\gamma_{KM+\\min(M,N)}$ are derived as,\n\\begin {align}\n\\gamma_j=&\\left\\{\n\\begin{array}{ll} \n1,&1\\le j\\le M\\\\ \n\\alpha,&M< j\\le KM\\\\ \n1-\\alpha,&KM< j\\le KM+\\min(M,N)\n\\end{array}\n\\right.\\label{ffd2}\n\\end{align}\nNote that $N\\le KM$ and $\\alpha\\le\\frac{1}{2}$. Thus, from the received signal in (\\ref{TR21a}), $T_1,\\cdots,T_{KM+\\min(M,N)}$ are decoded by  first receiver as (\\ref{mac3}) is satisfied for all $k\\in[KM+\\min(M,N)]$. For instance if we set $k=KM+\\min(M,N)$, the condition (\\ref{mac3}) is equivalent to,\n\n\\begin{align}\n\\sum_{i\\in S} d'_i=&\\frac{K(N-M)^+\\alpha}{K-1}+\\min(M,N)(1-\\alpha)\\nonumber\\\\\n\\le&\\min(M,N)+(N-M)^+\\alpha=\\max_{S_2\\in S,|S_2|=\\min(k,N)}\\sum_{i\\in S_2} \\gamma_i .\n\\end{align}\n\n\\item{$\\frac{1}{2}<\\alpha\\leq1$.} Let us achieve $d$ GDoF where $d$  is equal to,\n\\begin{align}\nd=\\min\\left(\\frac{K}{K-1}\\left((K-2)\\min(M,N)(1-\\alpha)+N(\\alpha)\\right),N\\alpha+K\\min(M,N)\\left(1-\\alpha\\right)\\right) \n\\end{align}\nSimilar to the case $\\alpha\\le\\frac{1}{2}$, the public message $W_{kc}$ is encoded into Gaussian codebooks $U_{k1c},U_{k2c},\\cdots,U_{kMc}$ with powers $\\mbox{E}{|U_k|}^2=1-P^{-\\alpha}$ each carrying $\\frac{d-K\\min(M,N)(1-\\alpha)}{KM}$ GDoF. These codewords are transmitted through $M$ antennas along  $M\\times 1$ generic unit  vectors ${\\bf V}_{k1},{\\bf V}_{k2},\\cdots,{\\bf V}_{k{M}}$.  The private ~message $W_{kp}$ ~is encoded~ into Gaussian~ codebooks ~$U_{k1p},U_{k2p},\\cdots,U_{k\\min(M,N)p}$ with powers $\\mbox{E}{|U_{kjp}|}^2=P^{-\\alpha}$ for any $j\\in[\\min(M,N)]$. These codewords are transmitted through $\\min(M,N)$ antennas along the $M\\times 1$ generic unit  vectors ${\\bf V}_{k1},{\\bf V}_{k2},\\cdots,{\\bf V}_{k{\\min(M,N)}}$. Each of the  private messages is carrying $1-\\alpha$ GDoF. The transmitted and received signals follows the same as \\eqref{TR21a-} and \\eqref{TR21a}. From Lemma \\ref{lemma:mac} each receiver, e.g., receiver $1$ can decode all the codewords $U_{kjc}$ and $U_{1jn}$ for all $k\\in[K]$ and $j\\in[M]$  treating all the other signals as noise. The details how receiver $1$ can decode all these codewords follows the same as the case $\\alpha\\le\\frac{1}{2}$.\n\n\\item{$1\\le\\alpha$.} In this case, $\\min\\left(D(\\alpha),K\\min(M,N)\\right)$ is achieved as follows. Recall that $D(\\alpha)$ was defined as $(N-(K-1)M)^++\\min(N,(K-1)M)\\alpha$. All $\\min(M,N)$ messages of each transmitter are\npublic in this case and are encoded into Gaussian codebooks\n$U_{k1c},U_{k2c},\\cdots,U_{k\\min(M,N)c}$ with unit powers each carrying $\\min\\left(\\frac{D(\\alpha)}{K\\min(M,N)},1\\right)$ GDoF. These codewords are transmitted through $\\min(M,N)$ antennas along the $M\\times 1$ generic unit  vectors ${\\bf V}_{k1},{\\bf V}_{k2},\\cdots,{\\bf V}_{k{\\min(M,N)}}$. The transmitted and received signals are concluded similar to \\eqref{TR21a-} and \\eqref{TR21a} as,\n\\begin{align}\n\\mathbf{X}_{k}=&\\sum_{j=1}^{M}\\mathbf{V}_{kj}{U}_{kjc}\\label{TR21a-1}\\\\\n\\mathbf{Y}_{k}=&\\sqrt{P}{\\bf G}_{kk}\\mathbf{X}_{k}+\\sum_{j=1,j\\neq k}^{K}\\sqrt{P^{\\alpha}}{\\bf {G}}_{kj}\\mathbf{X}_{j}+\\mathbf{\\Gamma}_{k}\\label{TR21a1}\n\\end{align}\nEach receiver, e.g., receiver $1$ can decode  all the signals $U_{kjc}$ for all $k\\in[K]$ and $j\\in[\\min(M,N)]$  treating all the other signals as noise. Set the variables $M_1=\\min(M,N)$, $M_2=(K-1)\\min(M,N)$ and $\\alpha_n=0$ for all $n\\in[N]$ in Lemma \\ref{lemma:mac} . Moreover, define the codewords $T_{1},\\cdots,T_{K\\min(M,N)}$ as \n\\begin {align}\nT_j=&\\left\\{\n\\begin{array}{ll} \nU_{1jc},&1\\le j\\le {\\min(M,N)}\\\\ \nU_{2(j-\\min(M,N))c},&\\min(M,N)<j\\le 2\\min(M,N)\\\\\n\\vdots &\\vdots\\\\\nU_{K(j-(K-1)\\min(M,N))c},&(K-1)\\min(M,N)<j\\le K\\min(M,N)\n\\end{array}\n\\right.\n\\end{align}\nFrom \\eqref{gamma1}, $\\gamma_1,\\cdots,\\gamma_{K\\min(M,N)}$ are derived as,\n\\begin {align}\n\\gamma_j=&\\left\\{\n\\begin{array}{ll} \n1,&1\\le j\\le \\min(M,N)\\\\ \n\\alpha,&\\min(M,N)< j\\le K\\min(M,N)\n\\end{array}\n\\right.\\label{ffd2}\n\\end{align}\nSimilar to the case $\\alpha\\le \\frac{1}{2}$, $T_1,\\cdots,T_{K\\min(M,N)}$ are decoded by  first receiver as (\\ref{mac3}) is satisfied for all $k\\in[K\\min(M,N)]$. For instance if we set $k=K\\min(M,N)$, the condition (\\ref{mac3}) is equivalent to,\n\\begin{align}\n\\sum_{i\\in S} d'_i=&K\\min(M,N)\\times\\min\\left(\\frac{D(\\alpha)}{K\\min(M,N)},1\\right)\\nonumber\\\\\n\\le&\\min\\left((K-1)M,N\\right)\\alpha+\\left(N-(K-1)M\\right)^+=\\max_{S_2\\in S,|S_2|=\\min(k,N)}\\sum_{i\\in S_2} \\gamma_i \n\\end{align}\n\n\\end{enumerate}\n\n \n\\section{Conclusion}\nSymmetric $K$-user MIMO IC with $M$ antennas at each transmitter and $N$ antennas at each receiver is considered. Sum GDoF of this channel is derived. The Sum GDoF  is found to be a $W$ curve as a function of $\\alpha$ for fixed  $M$ and $N$ similar to the SISO case. Outer bound proof is obtained with the help of a key lemma that generalizes the AIS argument.  The achievability follows from the achievability of the GDoF region of a MAC, combined with the `treating interference as noise' scheme. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION \\label{sec:intro}}\n\nImages of spiral and irregular galaxies have been defined historically\nby regions of massive star formation, signposts demarcating important\nfeatures such as spiral arms, bars, and starbursts.  They remind us\nthat galaxies really are evolving, with continuous injection of energy\nand processed material into the galaxy.  Massive star-forming regions\n(MSFRs) present us with a microcosm of starburst astrophysics, where\nstellar winds from O and Wolf-Rayet (WR) stars compete with supernovae\n(SNe) to carve up the neutral medium from which they formed, igniting a\nnew generation of stars.  With X-ray observations, we see the sources\nthat shape the larger view of a galaxy, injecting energy and processed\nmaterial into the disk and halo ISM through winds, ionization fronts,\nSNe, superbubbles, and chimneys.  About 80\\% of all SNe occur in\nsuperbubbles, so cosmic rays generated by SNe also may be accelerated\nin superbubbles \\citep[e.g.][]{Higdon98, Parizot04}.  X-ray\nobservations probe different energetic components than traditional\nvisual and infrared (IR) studies, penetrating the obscuring material of\nthe natal cloud while minimizing confusion from foreground and\nbackground objects.\n\nX-ray studies also detect the presence of past SNe through the shocks\nin their extended remnants.  Some superbubbles are well-known bright\nX-ray sources, where multiple SNe from past OB stars produce soft\nX-rays filling bubbles 50--100~pc in size with $L_x \\sim\n10^{35-36}$~ergs~s$^{-1}$ \\citep{Chu90}, while others (not recently\nbrightened by such ``cavity'' SNe) are X-ray faint \\citep{Chu95}.\nMSFRs thus exhibit a complicated mixture of point and extended\nstructures that are easily confused by low-resolution X-ray\ntelescopes.  30~Doradus (30~Dor) is a classic example of this.  The\n{\\em Chandra X-ray Observatory} \\citep[Figure~\\ref{fig:csmoothimage}\nand][hereafter Paper~II]{Townsley05b} and {\\em XMM-Newton}\n\\citep{Dennerl01} share the highest-quality views yet achieved of this\nextraordinary and complicated X-ray field.\n\n30~Dor is a field of superlatives.  Located in the Large Magellanic\nCloud (LMC), it is the most luminous Giant Extragalactic H{\\sc II}\nRegion (GEHR) and ``starburst cluster'' in the Local Group.  It hosts\nthe massive compact star cluster R136 \\citep[][called ``RMC~136'' in\nSIMBAD]{Feast60}, a testbed for understanding recent and ongoing star\nformation in the 30~Dor complex \\citep{Walborn02}.  Although R136, with\na mass of $\\sim 6 \\times 10^4$~M$_{\\odot}$ \\citep{Brandl05}, does not\nquite qualify as a ``super star cluster'' \\citep{Meurer95}, it has more\nthan 50 times the ionizing radiation of the Orion Nebula.  Nearby, one\nfinds the large supernova remnant (SNR) N157B (30~Dor~B), embedded in\nan H{\\sc II} region generated by the OB association LH~99 \\citep{Chu92}\nand containing an X-ray pulsar \\citep{Marshall98}.  {\\em ROSAT} images\nshowed two fainter sources associated with WR stars near the central\nstar cluster, at least five plasma-filled superbubbles, and the N157B\nSNR \\citep{Wang95, Wang99}.  30~Dor's proximity and complexity provide\nus with a unique and important microscope into the starburst phenomenon\nin galaxies.\n\nFigure~\\ref{fig:csmoothimage} illustrates the spatial and spectral\ncomplexity of the X-ray emission from 30~Dor with a smoothed image of\nour {\\em Chandra} observation obtained with the Advanced CCD Imaging\nSpectrometer (ACIS).  The aimpoint of the $17\\arcmin \\times 17\\arcmin$\nACIS Imaging Array (ACIS-I) was the R136 cluster.  At $D=50$~kpc, the\ndistance we assume throughout this paper and Paper~II, $1^{\\prime} \\sim\n14.5$~pc, so ACIS-I covers $\\sim 240$~pc $\\times$ 240~pc.  Two off-axis\nCCDs from the ACIS Spectroscopy Array (ACIS-S) were operated for this\nobservation, adding a $\\sim 120$~pc $\\times$ 240~pc image of nearby\ninteresting structures (see Figure~\\ref{fig:introimage}), albeit with\npoor spatial resolution.  A more finely binned image of the ACIS-I\ndata, highlighting the X-ray point sources and showing a J2000\ncoordinate grid, can be found in Figure~1 of Paper~II.  \n\n30~Dor lies at the confluence of two LMC supergiant shells and just the\nmain nebula is $\\sim 250$~pc in diameter.  Since the LMC is viewed\nnearly face-on, confusion between 30~Dor and other disk structures is\nminimized.  SNe pervade the region but most go undetected due to age\nand environment \\citep{Chu90}.  Nearby are two pulsars (B0540$-$69.3 in\nthe SNR N158A and J0537$-$6910 in the SNR N157B) and SN1987A\n(Figure~\\ref{fig:introimage}).  Wide-field ground-based H$\\alpha$\nimages \\citep[e.g.][]{Meaburn84,Chu90,Wang99}, enhanced by recent {\\em\nHST} \\citep{Walborn02} and {\\em Spitzer}\ndata\\footnote{\\url{http:\/\/www.spitzer.caltech.edu\/Media\/releases\/ssc2004-01\/release.shtml}},\nshow that the combined actions of stellar winds and SNe from several\ngenerations of high-mass stars in 30~Dor have carved its ISM into an\namazing display of arcs, shells, pillars, voids, and bubbles, ranging\nover spatial scales of 1--100~pc \\citep{Brandl05}. \n\nThis field provides a unique bridge between Galactic giant H{\\sc II}\nregions (e.g.\\ W49A, W51A, NGC~3603) and GEHRs \\citep[e.g.\\ NGC~5471\nand other regions in M101,][]{Chen05}.  30~Dor is the result of\nmultiple epochs of high-mass star formation in a vast molecular cloud\ncomplex chemically enriched by many SNe; it has no Galactic counterpart\nin terms of mass or complexity of its star formation history and is not\neven equalled by other GEHRs in the Local Group.  It provides us with a\nunique view of the most fundamental building block of the starburst\nphenomenon in galaxies; currently we are not able to resolve comparable\nstar-forming complexes in starburst galaxies, only those many times\nlarger and more powerful \\citep[e.g.\\ NGC~253,\nNGC~7714\/7715;][]{Strickland02,Smith05}.\n\nIn \\S\\ref{sec:background} we provide a brief summary of X-ray\nobservations of the 30~Dor complex.  Our {\\em Chandra} observations and\ndata analysis are outlined in \\S\\ref{sec:observation}.  An overview of\nthe X-ray morphology and global spectral properties of the complex is\npresented in \\S\\ref{sec:global}.  The rest of the paper provides more\nin-depth studies of specific diffuse X-ray components of the 30~Dor\ncomplex:  the superbubbles centered on R136 (\\S\\ref{sec:superbubbles}),\nthe SNR N157B (\\S\\ref{sec:N157B}), and other diffuse structures, with a\ncomparison to the H$\\alpha$ kinematic study of 30~Dor by \\citet{Chu94}\n(\\S\\ref{sec:otherdiffuse}).  We conclude with a comparison of the X-ray\ndata to recent H$\\alpha$ and IR data and a summary of our results.  The\nX-ray point source population of 30~Dor revealed by this dataset is\ndescribed in Paper~II.\n\n\n\n\\section{PAST X-RAY OBSERVATIONS\n\\label{sec:background}}\n\n\\subsection{The Large Magellanic Cloud and 30~Doradus}\n\nX-rays were first detected from the LMC in 1968 using rocket-launched\nproportional counters \\citep{Mark69}.  An early survey of the LMC with\nthe {\\em Einstein Observatory} \\citep{Long81} found 75 discrete sources\nassociated with the galaxy, including 25 SNRs and about 25 other\nextended sources.  A re-analysis of the {\\em Einstein} LMC survey\nresulted in a point source catalog of 105 sources, ``roughly half''\nassociated with the LMC \\citep{Wang91c}; the other sources are\nforeground stars or background active galactic nuclei (AGN).  These\nauthors noted source confusion associated with the 30~Dor complex.\n\nDetailed studies of 30~Dor and other LMC superbubbles were performed\nwith {\\em Einstein} \\citep[e.g.][]{Chu90,Wang91a,Wang91b}; they\nconcluded that the bright, diffuse X-ray emission filling the 30~Dor\nsuperbubbles was probably created by off-center cavity SNe inside the\nsuperbubbles shocking the superbubble shells.  Extending the work of\n\\citet{Meaburn84}, \\citet{Chu94} performed a detailed echelle study of\nthe kinematics of 30~Dor and compared their findings to the {\\em\nEinstein} data, concluding that 30~Dor is dominated by a hierarchy of\nexpanding structures with spatial scales ranging over 1--100~pc and\nexpansion velocities of 20--200~km~s$^{-1}$.  The diffuse X-ray sources\nare coincident with these expanding shells, with the brightest X-ray\nemission coincident with high-velocity features.  \\citet{Chu94} argued\nthat both the high-velocity features and the X-ray emission are related\nto high-velocity shocks created by fast stellar winds and SNe.\n\nTwo catalogs of LMC point sources emerged from the {\\em ROSAT}\nObservatory; a few sources (often SNRs or SNR candidates) are\nassociated with the 30~Dor complex \\citep{Haberl99,Sasaki00}.  In\nmosaicked {\\em ROSAT} maps of the hot ISM in the LMC, 30~Dor is an\noutstanding feature---hotter, brighter, and at higher pressure than\nmost other parts of the galaxy \\citep{Snowden94,Points01,Sasaki02}.\n{\\em ROSAT} studies of individual LMC superbubbles showed that several\nare X-ray faint \\citep{Chu95}, consistent with earlier conclusions that\nsuperbubbles are only brightened in X-rays when SNe hit the shell\nwalls, while others exhibit ``breakout regions'' and may be venting hot\ngas into the LMC's ISM \\citep{Dunne01}.\n\nA {\\em ROSAT}\/PSPC study of 30~Dor reported 22 point sources within the\n30~Dor nebula, including sources associated with R136 and N157B\n\\citep{Norci95}.  Spectral fits to the diffuse emission yielded $N_H\n\\sim 8 \\times 10^{21}$~cm$^{-2}$ and $kT \\sim 0.4$~keV, with a\nluminosity (0.1--2.4~keV) of $L_X = $3--6$ \\times\n10^{37}$~ergs~s$^{-1}$ depending on choice of background.  Since the\nGalactic absorbing column toward 30~Dor is $7 \\times\n10^{20}$~cm$^{-2}$, most of the absorption found in their spectral fits\nis local to 30~Dor.  They confirm the {\\em Einstein} results\n\\citep{Wang91a} that the diffuse X-ray emission fills the voids in the\n$H\\alpha$ emission and further conclude that stars in 30~Dor contribute\n$\\leq 2$\\% of its total X-ray luminosity.  While Norci and {\\\" O}gelman\nnote that the central cluster R136 is too young to have produced X-ray\nbinaries that could contribute to the X-ray emission in the region, a\n{\\em ROSAT}\/HRI study suggested that the two point sources found by HRI\nwere associated with the WR stars R140a2 (a WN6 binary) and Melnick 34\n(WN4.5) and could be WR\/black hole binaries \\citep{Wang95}.  This\nconjecture was supported by {\\em ASCA} spectral results that indicated\nhard X-ray emission from the central regions of 30~Dor.  Further {\\em\nROSAT}\/HRI and {\\em ASCA} observations led \\citet{Wang99} to conclude\nthat the diffuse emission came from 2--9~MK thermal plasmas and that\nthe primary mechanism to explain the temperature and X-ray luminosity\nof this emission is mass-loading via mixing of the hot wind-generated\ngas with H{\\sc II} gas from the large number of ionization fronts in\nthe region \\citep{Scowen98}.\n\nThe first light image from the {\\em XMM-Newton Observatory} was\nobtained from a 106~ks pointing just west of the main 30~Dor nebula\n\\citep{Dennerl01}.  This observation yielded broad-band (0.1--10~keV)\nspectra for several important objects in the neighborhood of 30~Dor:\nN157B, the Honeycomb Nebula \\citep[a SNR,][]{Chu95b}, and faint diffuse\nemission northwest of 30~Dor \\citep{Dennerl01}, as well as several\nlikely AGN seen through the LMC disk \\citep{Haberl01}.  A 38~ks {\\em\nXMM} observation of N157B (Observation 0113020201, PI Aschenbach)\nimages even more of the main 30~Dor nebula; although it is beyond the\nscope of this effort to include that observation in this paper, future\nefforts to obtain spectra of various features in 30~Dor from this\ndataset would be worthwhile.\n\n\n\\subsection{N157B and PSR~J0537$-$6910}\n\\label{sec:n157bbkgd}\n\nThe N157B SNR has itself been the subject of many X-ray studies.  A\nknown radio source, it was first detected in X-rays by {\\em Einstein}\n\\citep{Long79}.  A later {\\em Einstein} study \\citep{Clark82}\ndemonstrated that its spectrum showed no strong emission lines and was\nnon-thermal, well-fit by a power law with slope $\\Gamma = 2.9$, $N_H =\n1.2 \\times 10^{21}$~cm$^{-2}$, and luminosity (0.5--4.5~keV) of $L_X =\n2.3 \\times 10^{36}$~ergs~s$^{-1}$; they suggested that a search for a\npulsar in N157B should commence.  \\citet{Chu92} determined the boundary\nof the SNR based on kinematic information and suggested that N157B is\nanother example of an off-center SNR expanding into a pre-existing\nsuperbubble and that the SNR may be interacting with a dark cloud on\nits southern edge; this dark cloud may shadow the soft X-ray emission\nfrom the SNR.\n\nA detailed {\\em ROSAT} and {\\em ASCA} study of N157B was presented by\n\\citet{Wang98a}.  Their spectrum is dominated by a non-thermal\ncomponent with power law slope $\\Gamma \\sim 2.5$, although there are\nhints of emission lines below 2~keV that they attributed to a thermal\nplasma from the SNR with $kT = $0.4--0.7~keV.  {\\em ROSAT}\/HRI revealed\na comet-shaped central nebula that dominates the X-ray emission,\nexplained as a synchrotron nebula from a pulsar moving through the ISM\nat a rate of $\\sim 1000$~km~s$^{-1}$, although no pulsed emission was\napparent in these data. \n\nThe long-awaited detection of PSR~J0537$-$6910 in N157B, with a period\nof 16~msec and a power law photon index of $\\Gamma = 2.6$, came\nserendipitously from an {\\em RXTE} observation of SN1987A\n\\citep{Marshall98} and was soon confirmed by {\\em BeppoSAX}\n\\citep{Cusumano98}.  A slightly later {\\em ROSAT}\/HRI detection\nprovided an arcsecond position for the pulsar \\citep{Wang98b}.  This is\nthe shortest period known for a pulsar associated with a SNR.  This\npulsar remains undetected in the radio \\citep{Crawford05} and no\nunambiguous optical counterpart has been found, even with deep {\\em\nHST}\/ACS imaging \\citep{Mignani05}.\n\nThe thermal emission component in N157B's spectrum is confirmed by {\\em\nXMM} \\citep{Dennerl01}, also confirming N157B as a composite SNR,\ncontaining a Crab-like synchrotron core with $\\Gamma = 2.83$\n\\citep{Dennerl01} surrounded by a thermal shell.  {\\em Chandra} has\nobserved N157B directly with both the HRC \\citep{Wang01} and ACIS-S\nusing subarray readout (ObsID 2783, PI Wang).  With the HRC data,\n\\citet{Wang01} were able to pinpoint the position of the pulsar to\n$<1\\arcsec$ and to resolve a compact pulsar wind nebula at the ``head''\nof the comet-shaped nebula seen by {\\em ROSAT} \\citep{Wang98a}.  This\n$\\sim 2\\arcsec \\times 7\\arcsec$ feature is centered on the pulsar and\noriented perpendicular to the larger cometary nebula.  \\citet{Wang01}\nexplain this feature as a toroidal pulsar wind nebula confined by the\nbow shock from the supersonic motion of the pulsar and depositing its\nenergy in the trailing cometary nebula.\n\t\n\n\\subsection{Past Work on This {\\em Chandra} Observation}\n\t\nOur short {\\em Chandra}\/ACIS-I observation of 30~Dor (described in\ndetail below) was one of the first observations made by {\\em Chandra}\nand was part of the ACIS Team's Guaranteed Time Observation program.  We\nhave reported preliminary results from this observation at several\nmeetings \\citep{Townsley99,Townsley00b,Townsley02} and have used it to\nestimate the position of PSR~J0537$-$6910 in the N157B SNR\n\\citep{Mignani05}.  With this study we finally are able to provide a\ndetailed analysis of this observation, after several years' worth of\ncode development and special analysis techniques were implemented.  Our\ncustom tools are also useful for analysis of other {\\em Chandra}\nfields, so we have made them publicly available (see\n\\S\\ref{sec:observation}).\n\nThis dataset has also been studied by other groups.  Recently,\n\\citet{Lazendic03} proposed two new SNR candidates in the 30~Dor nebula\nbased on radio continuum maps obtained with the {\\em Australia\nTelescope Compact Array}, although the {\\em Chandra} data showed no\nX-ray enhancements associated with these regions.  \\citet{Chu04}\nexamined this claim and concluded, based in part on the absence of\nassociated X-ray emission, that these small radio sources are not SNRs;\nrather one is likely a young star-forming region and the other a\nmolecular cloud.\n\n\\citet{PPL02} found 20 mostly pointlike sources in the inner $11\\arcmin\n\\times 11\\arcmin$ of the ACIS-I field of view and studied 11 of those\nin the central region, within $100\\arcsec$ from the core of R136.  They\nmatched these sources with known early-type (WR and O) stars, some\nknown to be binaries, and concluded that they are bright in X-rays\nbecause they are all probably colliding-wind binaries.\n\\citet{Stevens03} included a spectral analysis of unresolved X-ray\nemission around R136 in their study of the properties of super star\nclusters.  Some unspecified number of point sources was removed from\nthe inner $16\\arcsec$ of R136, then the remaining X-ray emission was\nfit with a thermal plasma model, resulting in fit parameters of $N_H =\n4.2 \\times 10^{21}$~cm$^{-2}$, $kT = 2.1$~keV, and absorption-corrected\nluminosity $L_X = 5.5 \\times 10^{34}$~ergs~s$^{-1}$.  In Paper II we\nshow that more X-ray point sources in R136 are discernable in this\ndataset than these two papers considered; their results would change\nsomewhat if these faint sources were included in their analyses.\n\nSince our observation included the two off-axis ACIS-S chips S3 and S4,\nwe were able to image an $8\\arcmin.5 \\times 17\\arcmin$ region to the\nsouthwest of 30~Dor, featuring SN1987A, the large superbubble 30~Dor~C,\nand the Honeycomb Nebula (see Figures~\\ref{fig:csmoothimage} and\n\\ref{fig:introimage}).  As mentioned above, the {\\em XMM-Newton}\nfirst-light image was centered on this field and gives much better\nspatial and spectral information on these structures than this short\nACIS exposure \\citep{Dennerl01}.  The X-ray brightening of SN1987A has\nbeen monitored by {\\em Chandra} since early in the mission; see\n\\citet{Park05} and references therein.  30~Dor~C, an unusual\nnon-thermal X-ray emitter, has been studied with {\\em Chandra}\n\\citep{Bamba04} and {\\em XMM} \\citep{Smith04}.  Given these better\ndatasets and existing studies, we will not include detailed analysis of\nthese sources in this paper.\n\nThe results we present below represent only one step in the\nhigh-resolution characterization of X-rays from the 30~Dor region.  A\n100~ks {\\em Chandra} ACIS-I observation of 30~Dor is scheduled for\nlater this year; it should give a $\\sim 5$-fold improvement in\nsensitivity and signal over the results reported here.\n\n\n\n\\section{{\\em CHANDRA} OBSERVATIONS AND ANALYSIS\n\\label{sec:observation}}\n\n\\subsection{Instrumental Set-up}\n\nWe observed 30~Dor and its environs for $\\sim 26$~ks with ACIS-I on\n1999 September 21--22.  R136 was placed at the aimpoint.  The focal\nplane temperature at this early point in the mission was $-110$C to\nfacilitate water outgassing; it was later reduced to $-120$C to improve\ndetector performance.  This higher focal plane temperature causes the\neffects of radiation damage to the frontside-illuminated CCDs in the\nACIS-I array to be particularly pronounced \\citep{Townsley02a}.  For\nexample, the spectral resolution at 1.5~keV varies across the device\nfrom $\\sim 75$~eV FWHM at low CCD row numbers to $\\sim 170$~eV at high\nrow numbers.  Unfortunately, the aimpoint of the ACIS-I array falls\nnear the top of the I3 CCD, so spectral resolution for the sources in\nR136 is particularly bad.  A summary of the observations is given in\nTable~\\ref{tbl:obslog}.  \n\nThe first $\\sim 1$~ks of the observation (Observation Identification or\n``ObsID'' 22) was made in the usual ``timed event, faint'' mode, with\n$3 \\times 3$ pixel events, 3.2~s CCD frames, and the Spectroscopy Array\n(ACIS-S) CCDs S2 and S3 operating in addition to the ACIS-I devices.\nThe remainder of the observation (ObsID 62520) was taken in ``timed\nevent, very faint'' mode ($5 \\times 5$ pixel events) using the\n``alternating'' readout mode with a 10:1 cycle; every ten CCD frames\nwere exposed for 3.3~s, then every eleventh frame had a short 0.3~s\nexposure.  The data processing pipeline divides these short and long\nframes from ObsID 62520 into separate event lists.  In order to image\ninteresting structures near the 30~Dor field, ACIS-S CCDs S3 and S4\nwere operational for this observation.\n\nWe chose to use the ACIS camera's alternating-exposure mode because the\n30~Dor field contains PSR~J0537$-$6910, a bright 16~ms pulsar\n\\citep{Marshall98}.  Although it is imaged $\\sim 7\\arcmin$ off-axis, we\nknew that it would still suffer from photon pile-up in regular-length\nCCD frames, so we included the short frames to try to mitigate pile-up\non this source.  In hindsight, this was not an ideal approach, as there\nare too few photons in the short frames to provide much information on\nPSR~J0537$-$6910 and the use of alternating mode this early in the\nmission caused problems for the data processing pipelines, resulting in\nabsent or incorrect Good Time Interval (GTI) tables, incorrect keyword\nvalues, and other complications.\n\nFor this analysis, we used Version 3 of the pipeline processing\n(created 26 July 2001), which has correct GTI tables.  Even this third\nversion of the data still contains some errors though; for example, the\nEXPOSURE and LIVETIME keywords in the 0.3~s event list of ObsID 62520\nare incorrect.  After recalculating these quantities for each CCD and\ncorrecting the header keywords in the event list, a total of $\\sim\n200$~s of data taken with 0.3~s CCD frames is obtained.\n\n\n\\subsection{Data Analysis}\n\nStarting with the Level 1 event lists, the data were reduced using our\nstandard methods, as described in \\citet{Townsley03} and\n\\citet{Getman05}, using {\\it CIAO} 3.0.2.  {\\em Chandra} source\npositions were determined from the source-finding routine {\\it\nwavdetect} \\citep{Freeman02} run on a $2 \\times 2$-pixel binned image\nof the hard band (2--7~keV) data from the ``long'' observations in\nObsID 62520; data below 2~keV were excluded to minimize the\ncontamination of source positions by the soft diffuse emission that\npervades the field.  ObsID 62520 was registered to the astrometric\nreference frame of 2MASS using 4 matches.  Offsets were $+0.95\\arcsec$\nin RA, $+1.0\\arcsec$ in Dec.  \n\nThe {\\em Clean55}\nalgorithm\\footnote{\\url{http:\/\/hea-www.harvard.edu\/\\-~alexey\/\\-vf\\_bg\/vfbg.html}}\ndeveloped by Alexey Vikhlinin was used to flag events in ObsID 62520;\nthese flagged events were removed to suppress background for\nsource-finding and image generation, but this flag can also erroneously\nremove events from the cores of point sources so it was not applied in\nspectral analysis.  Due to a strong background flare caused by\nparticles generated by solar activity, ObsID 62520 has some telemetry\nsaturation that only affects the S3 chip, from frame $\\sim 7248$ ($\\sim\n24300$~sec) to the end of the observation.  Background flares affected\nCCD S3 both early and late in the exposure and those times were\nfiltered from the S3 data; other CCDs were unaffected by these flares.\nThe {\\it CIAO} ``destreaking'' algorithm by John\nHouck\\footnote{\\url{http:\/\/asc.harvard.edu\/ciao\/threads\/destreak\/}} was\napplied to CCD8 to suppress noise on that device.\n  \nWe did not adjust the astrometry of ObsID 22 due to only two 2MASS\nmatches and inconsistent, small offsets with those matches.  Although\nwe would normally not remove the event position randomization added in\nthe standard data processing pipeline for such a short observation, it\nwas removed for ObsID 22 in order to combine these data with ObsID\n62520.  S3 was experiencing background flares for the duration of this\nshort observation.  Given that and the short amount of total\nintegration time for the area covered by CCD S2, only the ACIS-I array\ncomponent of this observation was merged with ObsID 62520.\n\nFor ObsID 62520, individual CCD columns were examined in chip\ncoordinates using our {\\it Event Browser} tool\\footnote{Code available\nat \\url{http:\/\/www.astro.psu.edu\/xray\/docs\/TARA\/}} \\citep{Broos00} to\nsearch for additional hot pixels and columns that were not part of the\npre-defined bad pixel file and to establish the energies of those noise\nevents.  At $-110$C, there are many more hot columns (exhibiting extra\nnoise below 1~keV) than at $-120$C (the focal plane temperature used\nfrom January 2000 to the present), especially on CCD S4.  Those events\nwere removed from both ObsID 62520 and ObsID 22 using CIAO 2.3 because\nthe necessary ``exclude'' syntax (to remove only certain energies from\nspecified spatial coordinates) was not working correctly in CIAO\n3.0.2.  Although the effects of these deleted events cannot be included\nin the exposure map (because only events below certain energies, not\nall events from a given column, were removed), we chose to suffer this\nslight calibration error rather than to retain the noisy events because\nthey become especially pronounced and distracting in smoothed soft-band\nimages.\n\nBoth datasets were corrected for charge transfer inefficiency (CTI)\nusing the Penn State CTI corrector, which was originally developed\nspecifically to address the CTI problem for this\ndataset\\footnote{Details, source code, and calibration products for CTI\ncorrection are available at\n\\url{http:\/\/www.astro.psu.edu\/users\/townsley\/cti}.} \\citep{Townsley00,\nTownsley02a}, due to our interest in spectral analysis of the many\npointlike and diffuse structures in this field.  It remains the only\nCTI corrector available for the backside-illuminated CCD S3 and for all\nACIS data taken at a focal plane temperature of $-110$C (where CTI is\nworse than at $-120$C for frontside-illuminated CCDs, as mentioned\nabove).  Custom response matrices (RMFs) and quantum efficiency\nuniformity (QEU) files were generated to facilitate spectral fitting\nfor $-110$C data \\citep{Townsley02b}.  We also refined the event\npositions in both datasets using the subpixel positioning algorithm\ndeveloped by Koji Mori\\footnote{Code available at\n\\url{http:\/\/www.astro.psu.edu\/users\/mori\/chandra\/subpixel\\_resolution.html}}\n\\citep{Mori01, Tsunemi01}.\n\nThe reduced event lists were then merged and the exposure maps added.\nA {\\em status=0} filter was applied to create an event list appropriate\nfor source searching and image generation.  The data were once again\nexamined in {\\it Event Browser}, showing that the best energy range to\nconsider for imaging studies is 350--7300~eV; the lower limit retains\nevents below 500 eV that are clearly from the diffuse superbubble\nstructures in the field while limiting low-energy noise, while the\nupper limit is set just below the instrumental Ni~K$\\alpha$ line, which\nmarks the beginning of enhanced high-energy background.\n\nNext we conducted a multistage process for identifying and extracting\nphotons of unresolved sources that is described in detail in Paper II.\nIt involves source identification using a wavelet transform and a\nmaximum likelihood image restoration algorithm.  Point source events\nfor the final list of 180 sources were then extracted using PSB's\nIDL-based code {\\it ACIS Extract} \\citep{Broos02} developed to treat\ncomplicated fields with multiple point-like and diffuse\ncomponents\\footnote{The {\\it ACIS Extract} code is available at\n\\url{http:\/\/www.astro.psu.edu\/xray\/docs\/TARA\/ae\\_users\\_guide.html}.}.\nFor analyzing the diffuse components in the 30~Dor field, the only\npoint source properties required are their locations and extraction\nregions; these are provided in Tables 1 and 2 of Paper II.  These are\nused to construct a point source mask and a source-free dataset\nsuitable for analysis of diffuse emission.\n\nWe used a mask produced by the {\\it ACIS Extract} tool {\\it\nae\\_optimal\\_masking} which identifies regions where the expected\nsurface brightness from the calibrated point sources is larger than one\nhalf the smoothed observed local background.  These point source masks\nare illustrated in Figure~\\ref{fig:mask_emap}, which shows the I-array\nexposure map for ObsID 62520.  Chip gaps and bad columns are seen as\nareas of reduced exposure, while the small white regions distributed\nacross the field and concentrated in the center show areas of zero\nexposure representing areas that were masked to exclude point sources.\nThis figure also illustrates that the {\\em Chandra} PSF is a strong\nfunction of off-axis angle; most of the X-ray point sources in the field\nare concentrated in R136 (field center), yet the sharp PSF on-axis leads\nto relatively little area removed for diffuse emission studies there.\n\nExtraction regions for a variety of diffuse structures were then\ndefined by hand, based on the apparent surface brightness in a\nsmoothed, sources-removed image.  Spectra, backgrounds, ARFs, and RMFs\nfor all point sources and diffuse structures were constructed by {\\it\nACIS Extract} and used for both automated and interactive spectral\nfitting using Keith Arnaud's {\\it XSPEC} fitting package\n\\citep{Arnaud96} and {\\it XSPEC} scripts provided by Konstantin Getman.\n\n\n\n\\section{GLOBAL PROPERTIES OF THE X-RAY EMISSION\n\\label{sec:global}}\n\nWe start by describing the large-scale features revealed by this ACIS\nobservation.  Figure~\\ref{fig:csmoothimage} shows the soft-band image\nof 30~Dor created with the adaptive-kernel smoothing tool {\\it csmooth}\nin {\\it CIAO} (a translation of the {\\it asmooth} code by Harald\nEbeling).  This smoothed image emphasizes the soft diffuse structures,\nmapping red to the spectral range 500--700~eV, green to 700--1120~eV,\nand blue to 1120--2320~eV.  The reduced, full-field, full-band binned\ndata are shown in Figure~\\ref{fig:introimage}.  The event data were\nbinned into $8\\arcsec \\times 8\\arcsec$ pixels to create this image.\nSome well-known objects are labeled, as are the ACIS CCDs used in this\nobservation.  Areas of reduced exposure between the CCDs are clearly\nvisible, as is the extensive diffuse X-ray emission present in this\nfield.  Several famous structures are imaged far off-axis on the S3 and\nS4 CCDs.\n\n\n\\subsection{X-ray Morphology\n\\label{sec:globalmorph}}\n\nIn these images we see a bright concentration of hard X-ray events\nassociated with the R136 star cluster at the center of the ACIS-I array\nand fainter emission from several other well-studied stellar objects,\nmostly WR stars (see Figure 1 in Paper II).  The bright SNR N157B is a\ndistinct, extended feature to the southwest of the main 30~Dor\ncomplex.  It is the hardest source in the field due to non-thermal\nemission from its central pulsar and synchrotron nebula, but the larger\nSNR softens with distance from the pulsar.  A number of\nwidely-distributed compact X-ray sources are scattered across the\nfield; some were seen in previous X-ray studies\n(\\S\\ref{sec:background}).  The hard sources are probably mostly\nbackground AGN (see Paper II).\n\nBy far the dominant features, however, are the large diffuse structures\nassociated with the superbubbles produced by existing and past OB\nassociations and their supernova events.  These structures were\nwell-known from {\\em Einstein} and {\\em ROSAT} studies, as outlined in\n\\S\\ref{sec:background} \\citep[see especially the HRI data\nin][]{Wang99}, but  higher on-axis spatial resolution of {\\em Chandra}\nand the ACIS camera's intrinsic spectral capabilities reveal a new\nlevel of complexity.  The center of the field is complicated by\ncrossing ACIS-I chip gaps but clearly contains some of the hardest\nemission in the field, due to the R136 cluster, R140, and R145.  Faint,\nsofter diffuse emission appears to pervade the central region.  Just\nnorthwest of R140 (due west of R139) is a bright, medium-soft clump\n$\\sim 1\\arcmin$ in size.\n\nShell 1 in Figure~\\ref{fig:introimage} contains several point sources\nbut is dominated by a bright, medium-energy clump of diffuse emission\nroughly $1\\arcmin.5 \\times 2\\arcmin$ in size.  An ACIS-I chip gap\nbifurcates this feature, so the two-lobed structure it shows in\nFigure~\\ref{fig:csmoothimage} is largely an artifact \\citep[see Figure\n1 in][]{Wang99}.  Shell 2 is shown as a larger structure in\nFigure~\\ref{fig:introimage} than originally defined\n\\citep{Meaburn84,Wang91a} because the kinematic study of \\citet{Chu94}\nshowed that the expanding shell is larger than the morphological Shell\n2 identified in an H$\\alpha$ image \\citep{Meaburn84}.  It is\nedge-brightened in X-rays, with a large central void and a sharp\nboundary along its southeast edge.  There are small clumps of emission\naround its periphery, many of which are consistent with point sources.\nIts faint emission is harder than the other shell structures in the\n30~Dor nebula.  At its upper center is an interesting comet-shaped\nfeature showing substantial spectral complexity, with a soft point\nsource at its head.\n\nShell 3 is also largely edge-brightened, quite faint and soft around\nits periphery but with a harder, nearly-complete, clumpy central ring\nabout $2\\arcmin.5$ in diameter (hereafter called the ``West Ring'').\nThis ring structure also shows spectral complexity, appearing harder on\nits western half.  A faint, small clump is nearly centered on this\nring, but it is too large and diffuse to be a point source.  The $\\sim\n20$~Myr old stellar cluster Hodge~301 \\citep{Grebel00} sits at the\nnortheast edge of this ring, but we do not resolve any X-ray point\nsources in it.  Shell 3 encompasses two dark voids, one on either side\nof its central ring.\n\nShells 4 and 5 are both distinctly soft and bright, and differ from the\nother shells in that they appear center-filled rather than\nedge-brightened.  Shell 5, unfortunately, has a chip gap running all\nthe way through its long dimension, so some of the structure we see may\nbe affected by that.  Both it and Shell 4 are spectrally uniform and\nsimilar to each other.  Shell 5 is quite clumpy, with a bright lower\nclump about $1\\arcmin \\times 2\\arcmin$ in size containing the\nWR star R144 and a string of smaller, fainter clumps populating\nits interior.\n\nThe off-axis ACIS-S CCDs appear to be pervaded by soft, faint diffuse\nemission.  (Small point-like features around the edges of this field\nseen in Figure~\\ref{fig:csmoothimage} are smoothing artifacts.)  The\nmost striking feature here is the superbubble 30~Dor~C, seen as a\nnearly complete ring of clumpy emission spanning S3 and S4; its western\nhalf is very hard in X-rays due to strong non-thermal emission (see\n\\S\\ref{sec:background}).  SN1987A is the bright point source on S4; its\nlight is spread across a region more than an arcminute in size due to\n{\\em Chandra's} broad PSF at this large off-axis angle ($>20\\arcmin$).\nThe bright, soft feature at the lower center of S4 is the Honeycomb\nSNR; superposed on it appears to be a large ring of soft, diffuse\nemission that arcs to the north and east of the Honeycomb.  Such faint,\nfar off-axis features are more clearly seen in the {\\em XMM}\nfirst-light image \\citep{Dennerl01}.\n\n  \n\\subsection{Other Views\n\\label{sec:diffsrcs}}\n\nGiven the great variety of diffuse X-ray structures apparent in this\nobservation, it is worthwhile to find new ways to highlight them.\nFigure~\\ref{fig:diffuseintro} shows a color composite of three\nadaptively-smoothed images \\citep[from PSB's adaptive kernel smoothing\ntool {\\em adaptive\\_density\\_2d},][]{Townsley03} where the 180 point\nsources in the field have been removed and the resulting ``holes''\nsmoothed over.  The images were made in three soft bands, 350--700~eV,\n700--1100~eV, and 1100--2200~eV, similar to the bands used to create\nFigure~\\ref{fig:csmoothimage}.  They were combined to show the spectral\nvariety that must be present in the diffuse structures, even on spatial\nscales too small for us to extract and fit spectra to demonstrate this\nvariety; the intensities were set to emphasize faint diffuse features.\n\nThis smoothing technique has the advantage (over that used in\nFigure~\\ref{fig:csmoothimage}) that it does not create artifacts around\nthe edges of the field, although the overall appearance is more\nblurry.  We think that the best impression of the field comes from\nactively comparing these two renderings.  For example, the West Ring in\nShell 3 is much more clearly portrayed by Figure~\\ref{fig:csmoothimage},\nbut there are two small, faint loops ($< 2\\arcmin$ in diameter) on its\nnorth and east edges that are only visible in\nFigure~\\ref{fig:diffuseintro}.  We do not have enough data to know\nwhether or not these features are real.\n\nWith the point sources gone from Figure~\\ref{fig:diffuseintro}, we are\nfree to concentrate on the diffuse structures.  Here the outer loop in\nShell 3 is more pronounced, as are other faint diffuse features,\nespecially on the S-array CCDs.  Another intriguing small loop not\napparent in Figure~\\ref{fig:csmoothimage} is seen in the eastern half\nof Shell 2, visible as a green arc $\\sim 1.5\\arcmin$ across just above\na bright green knot in the sharp lower edge of Shell 2.  Again, only\nmore data will allow us to understand this feature in detail.\n\n\n\\subsection{Spectra\n\\label{sec:globalspectra}}\n\nFor comparison with more distant GEHRs, Figure~\\ref{fig:globalspectra}\nshows spectra of large-scale regions of X-ray emission on the ACIS-I\narray and {\\it XSPEC} fits.  Fit results are given in\nTable~\\ref{tbl:globalspectable}.  Column 1 lists the various composite\nregions on which spectral fits were performed; rows 1--4 correspond to\nthe fits shown in Figure~\\ref{fig:globalspectra}.  Column 2 gives the\nnumber of counts used in the fit.  Columns 3--8 give the spectral fit\nparameters, while columns 9--12 show the elemental abundances relative\nto solar values from the $\\it vapec$ thermal plasma model\n\\citep{Smith01} for elements that required $Z > 0.3Z_{\\odot}$.  Columns\n13--17 give the X-ray luminosities based on the spectral fits for a\nvariety of wavebands, with and without correction for absorption.\n\nColumn 8 shows that many of these fits are not formally acceptable (the\nreduced $\\chi^2 > 1.1$).  This is due mainly to the large number of\ndegrees of freedom, which is in turn due to the large number of events\nin these spectra; while in principle more events should allow the\nreduced $\\chi^2$ to approach unity as signal-to-noise improves, in\npractice systematic errors in the ARF and RMF become more important and\nthe reduced $\\chi^2$ remains high.  The fits in\nFigure~\\ref{fig:globalspectra} show that the spectral models used in\nTable~\\ref{tbl:globalspectable} characterize the data quite well in\nspite of the formally unacceptable fit; a more complicated model\n(such as an additional thermal plasma component) does not improve the fit.  \n\nFigure~\\ref{fig:globalspectra}a illustrates the global spectral fit for\nall components of the main 30~Dor nebula as outlined in\nFigure~\\ref{fig:introimage}, including point sources, the SNR N157B,\nits pulsar, and its cometary nebula.  The spectrum is fit with a single\nabsorbing column via the {\\it wabs} model \\citep{Morrison83}, which\nassumes solar abundances for the absorbing material, and a spectral\nmodel consisting of two components:  a variable-abundance thermal\nplasma and a power law.  The average absorption is a factor of 2\nsmaller than the {\\em ROSAT} result \\citep{Norci95}, although the\nplasma temperature is similar.  \\citet{Norci95} do not state what\nabundances they assume for the absorbing material and \\citet{Mignani05}\nnote that abundances of $0.4Z_{\\odot}$ give $N_H$ values a factor of\ntwo higher than those for solar abundances when fitting the spectrum of\nPSR~J0537$-$6910, so perhaps different assumptions regarding the\nmetallicity of the intervening material account for this discrepancy in\nthe average absorption.  The ACIS soft-band intrinsic\n(absorption-corrected) luminosity for the main 30~Dor nebula is a\nfactor of 2.5--5 smaller than the {\\em ROSAT} 0.1--2.4~keV estimate\n\\citep{Norci95}. \n\nFigure~\\ref{fig:globalspectra}b covers the same region of X-ray\nemission as Figure~\\ref{fig:globalspectra}a minus the N157B SNR (a\ncomposite spectrum of the SNR is shown in \\S\\ref{sec:n157bspectra}).\nSince N157B is quite bright and contains substantial non-thermal\nemission that would not necessarily be present in a more distant GEHR\nunless it contained a recent supernova, removing it from the overall\nspectrum of 30~Dor might yield a more typical GEHR spectrum.  The {\\it\nwabs*(vapec + powerlaw)} fit is actually quite similar to that in\nFigure~\\ref{fig:globalspectra}a, with nearly the same $N_H$, $kT$, and\nplasma abundances as that fit, but a steeper power law slope.  This\nshows that, although the non-thermal N157B pulsar and cometary nebula\nare bright compared to other point sources in the field, the overall\nspectral shape of the GEHR is still dominated by a soft thermal plasma\nwith fairly simple spatially-averaged spectral properties---no\nprominent lines are present and there is no obvious need for multiple\nplasma components or absorbing columns to fit the composite spectrum.\nThe only indication of the presence of the pulsar is at high energies,\nwhere it flattens the power law component of the fit.\n\nThe same extraction region is used for Figure~\\ref{fig:globalspectra}c\nas was used in Figure~\\ref{fig:globalspectra}b, but now the 158 point\nsources that this region contains have been removed from the spectrum.\nThe composite spectrum of those point sources is shown in\nFigure~\\ref{fig:globalspectra}d.  With only the diffuse emission\ncomponent remaining in Figure~\\ref{fig:globalspectra}c, the power law\nis no longer needed in the spectral model; rather the emission can be\nrepresented by a single, slightly hotter thermal plasma with enhanced\nabundances.  The composite point source spectrum in\nFigure~\\ref{fig:globalspectra}d requires a hotter thermal plasma and a\nflatter power law; it constitutes less than 10\\% of the full-band\nemission in this region but dominates the hard-band (2--8~keV) emission\nby a factor of 3 over the diffuse component.\n\nThese fits suggest that unresolved GEHRs in more distant galaxies might\nappear as soft, moderately luminous ($L_X \\sim 10^{36}$~ergs~s$^{-1}$)\nX-ray sources, quite distinct from harder and often brighter ($L_X \\sim\n10^{37}$--$10^{40}$~ergs~s$^{-1}$) X-ray binaries.  Distinguishing\nGEHRs from individual SNRs, though, is probably only possible for young\nSNRs, which might present hotter thermal plasmas or occasionally\nnon-thermal emission and may have prominent line features.  Even though\nthe N157B SNR is quite young, it does not show prominent lines and,\nwhen averaged with the emission from the rest of 30~Dor, it is not\nbright enough to harden the composite spectrum.  If its non-thermal\ncentral components were absent, it would not be spectrally distinct\nfrom the other diffuse structures in 30~Dor.  This is illustrated in\nthe spectral fitting presented in \\S\\ref{sec:superbubbles} and\n\\S\\ref{sec:N157B} below.\n\nDespite its complex spatial morphology, 30~Dor's integrated thermal\nemission has simple spectral properties:  it is well-fit by a\nsingle-temperature plasma with $T \\simeq 4$~MK, roughly solar line\nstrengths, and a single absorbing column density.  We\nwill see in \\S\\ref{sec:superspectra} that a considerable variety of\nplasma temperatures and absorbing columns are present on small spatial\nscales.  In the ACIS soft band (0.5--2~keV), the X-ray luminosity comes\nprimarily from the diffuse structures, which are brighter by an order\nof magnitude than the point sources.  Above 2~keV, though, the diffuse\nemission disappears and the integrated emission is dominated by the\npoint sources.  Paper~II demonstrates that this point source emission\nis itself dominated by a few bright sources; one third of the counts\nthat compose the spectrum in Figure~\\ref{fig:globalspectra}d come from\na single source \\citep[probably a colliding-wind binary,][]{PPL02}.\n\n\n\n\\section{THE X-RAY SUPERBUBBLES\n\\label{sec:superbubbles}}\n\n\\subsection{Morphology\n\\label{sec:supermorph}}\n\nAs outlined in \\S\\ref{sec:globalmorph}, the five plasma-filled\nsuperbubbles first defined by \\citet{Wang91a} from {\\em Einstein}\nobservations of 30~Dor appear to be much more complicated structures in\nthese ACIS images, exhibiting knots, whisps, loops, and voids with a\nrange of surface brightnesses and X-ray colors.  In some cases the\nX-ray morphology may be affected by the clumpy molecular clouds in the\nregion \\citep{Johansson98}, the shreds of the material from which R136\nformed.  This neutral material may shadow soft X-rays; comparing CO\ncloud locations from \\citet{Johansson98} to our smoothed images, this\neffect may account for some of the X-ray voids close to R136, although\nprobably not the larger, more distant voids seen in Shells 2 and 3.\n\nSome X-ray regions show distinctive sharp edges, such as the long\nnorth-central bubble comprising Shell 5 and the long arc stretching\nacross the southeast edge of Shell 2.  Other regions such as the large\nloop cut off by the northwest edge of the ACIS-I array have softer\nedges that seem to blend into faint, larger-scale diffuse emission.\nThe two bubbles of Shells 4 and 5 running almost due north and south of\nR136 appear quite red in Figures~\\ref{fig:csmoothimage} and\n\\ref{fig:diffuseintro}, perhaps implying that they are less absorbed\nthan other parts of the field or are filled with cooler gas.  They are\nalso distinctly center-filled, while other superbubbles appear as loops\nwith central voids.  The ACIS-S structures 30~Dor~C and the faint loop\nthat arcs to the northeast of the Honeycomb SNR are good examples of\nsuch loops, but the main 30~Dor complex shows them as well.\n\nIn the hot thermal plasma models that we use for spectral fitting of\nthese data (see \\S\\ref{sec:globalspectra}), most of the emission\nbetween 0.5 and 2~keV comes from a large number of blended spectral\nlines, with only a small remainder caused by thermal bremsstrahlung.\nIn order to understand the morphology of 30~Dor's superbubbles and its\nenergy dependence, Figure~\\ref{fig:lineims}\\footnote{These images\n(point sources removed) were made with the {\\it adaptive\\_density\\_2d}\nprocedure in {\\it ACIS Extract} using the smoothing scales from a\nbroadband (350--2000~eV) image, so that all narrow-band images have the\nsame smoothing scales.  To account for the fact that each narrow-band\nimage was made from a different number of events, each image was\nnormalized by its median intensity, then all were scaled the same in\nthe display.} shows narrow-band smoothed images of the full ACIS field\ncentered on some of the most prominent spectral lines in the thermal\nplasma models, specifically those often seen in SNRs, following {\\it\nChandra} studies of Magellanic Cloud diffuse nebulae by \\citet{Behar01}\nand \\citet{Naze02}.  The continuum emission dominates the\nhighest-energy image ($\\sim$65\\%) but contributes only $\\sim$10--30\\%\nto the other narrow-band images.\n\nWhile small-scale changes between narrow-band images should be viewed\nwith caution due to limited photon statistics, it is clear that all of\nthese images hint at the presence of complex fine spatial structure and\nthat the morphology of large-scale structures changes substantially\nwith energy.  A good example of this is 30~Dor~C, which seems to emerge\nout of faint, unorganized swaths of soft emission to coalesce into the\nwell-defined hard loop that we see in broad-band images.  The large,\nsoft superbubble at the south-center of the ACIS-I array (Shell 4) and\nthe loop containing the Honeycomb SNR are prominent in the softer\nimages, but fade above 950~eV.  Conversely, the sharp southeast ridge\nseen on CCD I1 (edge of Shell 2) becomes more prominent at higher\nenergies and the void to its northwest appears to be filling in, with\nno void left in the 1780--1940~eV image.  Smaller-scale knots and\nwhisps near the center and west-center of 30~Dor also come and go with\nenergy, with some prominent in the softer panels and others bright at\nhigher energies.\n\nThese images show that substantial spectral variation exists in the\n30~Dor superbubbles but do not by themselves reveal whether the\nvariations are produced by spatial changes in absorption columns,\nplasma temperatures, abundances, or other phenomena.  To investigate\nthis question, we divide the diffuse emission on the ACIS-I array as\nshown in Figure~\\ref{fig:diffregions}.  These regions were chosen\nprimarily on the basis of apparent surface brightness of the diffuse\nemission and overall morphology of the 30~Dor complex, so they form a\nphenomenological rather than a physical parameterization of the\nX-ray emission.  We emphasize that they were not chosen by considering\nthe morphology of 30~Dor in any other waveband; this will become relevant\nlater in this paper.  \n\nThe three large green polygons labeled ``B'' are the regions used to\nobtain the background spectrum used in all spectral fitting of diffuse\nregions described in this paper.  From {\\em XMM} data of this region\n\\citep{Dennerl01} and {\\em ROSAT} maps of the LMC\n\\citep[e.g.][]{Points01,Sasaki02}, we know that these regions contain\nfaint diffuse emission, but this may be the most appropriate background\nto use, as the whole field may well be pervaded by such faint diffuse\nemission.\n\nThe regions in Figure~\\ref{fig:diffregions} are best understood by\nconsidering them hierarchically by size, starting with the outermost,\nlargest regions and working inward and to smaller sizes.  The large red\ncontour outlines the region used earlier for the global spectral\nproperties of 30~Dor.  The six white regions define large-scale diffuse\nstructures, including the N157B SNR.  Region 1 encompasses the\nbrightest emission from the whole nebula, including regions 3, 12, 16,\n24, n8, and the central polygon that these regions abut.  Region 2 is\nthe perimeter between region 1 and the outer red contour, also\nexcluding region 18.  The five purple contours define ``voids,'' or\nareas of low apparent surface brightness.  Two central green regions (9\nand 10) outline relatively bright central structures.  Contained within\nthese large-scale regions are smaller regions defining locally bright\nareas, in blue, yellow, cyan, and red.  A cyan contour in the far\nnortheastern corner (region 23) defines an area of faint diffuse\nemission known from {\\em ROSAT} observations \\citep{Points01}.  A\nseries of annular regions (n0--n7) is used to explore the spectral\nproperties of the N157B SNR; \\S\\ref{sec:N157B} gives more detail.\n\nAfter choosing these extraction regions based on the X-ray apparent\nsurface brightness in our {\\em Chandra} images, we compared them to the\n$\\sim 100$~ks {\\em XMM} first light image in \\citet{Dennerl01}, which\ncontains the western half of the main 30~Dor nebula.  This image\nclosely matches the {\\em Chandra} data, showing very similar diffuse\nX-ray structures, and justifies our choice of extraction regions even\nfor faint features such as the outer loop in Shell 3 that we delineate\nas region 18, soft features such as regions 3 and 4 that fill Shell 4,\nand the X-ray voids in regions 19, 20, and 21.\n\n\n\\subsection{Spectra\n\\label{sec:superspectra}}\n\nThe spectral variety in these diffuse regions is shown in\nTable~\\ref{tbl:diffspectable}.  Column 1 gives the identifier of the\nregions shown in Figure \\ref{fig:diffregions}.  Column 2 gives the net\nfull-band (0.5--8~keV) counts in each diffuse region.  Spectral fit\nparameters, allowing for both {\\em vapec} thermal plasma and power law\nfits, are given in Columns 3--7.  Column 8 gives the reduced $\\chi^2$\nvalue for the fit.  All elemental abundances were allowed to vary;\nColumns 9-12 show abundances relative to solar that exceeded the\nnominal value $0.3Z_{\\odot}$.  These abundances were not always\nwell-constrained in the fits; in cases where the fit was improved with\nhigher abundances but actual values were not well-determined, they are\nlisted as ``hi'' in the table.  Derived X-ray luminosities are given in\nColumns 13-16; since the diffuse emission is quite soft in most\nextended regions (apart from the center of N157B), we report the\nintrinsic (absorption-corrected) soft-band luminosity as well as the\nusual intrinsic full-band luminosity.  The area subtended by each\ndiffuse region is given in Column 17 and the table is completed by a\nlist of intrinsic soft-band surface brightnesses in Column 18.\n\nUsing methods similar to those developed by Jeremy Sanders for study of\nextragalactic plasmas \\citep[e.g.][]{Sanders05}, we have used thermal\nplasma fits and the regions defined in Figure~\\ref{fig:diffregions} to\nproduce maps of $N_H$, $kT$, and intrinsic X-ray surface brightness in\n30~Dor, as shown in Figure~\\ref{fig:nhktmaps}.  For uniformity, the\nspectral fits used to make these maps were single-temperature thermal\nplasmas, so the results are not necessarily the same as those presented\nin Table~\\ref{tbl:diffspectable}, where power law components were added\nfor some regions.  Characteristics of the largest regions are recorded\nfirst and are overlaid by those of smaller regions.  This approach has\nthe misleading effect of leaving apparent ``rings'' around void\nregions (e.g.\\ regions 15 and 19) where the underlying large-scale fit\nis not masked by an overlying smaller region.\n\nAlthough crude, these maps are useful for understanding the physical\nstate of the plasmas that make up the 30~Dor diffuse emission.  For\nexample, region 4 is not intrinsically more luminous than the\nsurrounding region 3, it is just less absorbed and, since it has the\nsame temperature as region 3, its apparent surface brightness is\nlarger.  The same explanation applies to region 8.  Region 11 is\nparticularly bright, even though it has the same plasma temperature as\nits surrounding region 10, because it is slightly less absorbed plus it\nhas intrinsically higher surface brightness.  Regions 25 and 26 suffer\nmore absorption than the surrounding region 24 and have cooler plasma\ntemperatures, but they stand out because they have higher intrinsic\nsurface brightness.  Regions 18 and 19 suffer similar high obscuration\nand have the same cool plasma; region 19 appears as a void because it\nis intrinsically fainter than its surrounding region 18, which is only\nvisible to us in regular images because it has relatively high\nintrinsic surface brightness.  Region 22 has similar obscuration and\nplasma temperature, but appears as a void because it lacks this\nenhanced intrinsic brightness.\n\nParticularly unusual is region 15, which has high obscuration and low\nintrinsic surface brightness so it appears to us as a void, but it has\nthe highest plasma temperature in the nebula.  Region 5 is also\ninteresting, exhibiting a low temperature and relatively high\nobscuration, but it is very bright in regular images due to its\nintrinsic brightness, one of the largest in the nebula.  Of even higher\nintrinsic brightness, though, is region 7, yet it is much less\nnoticeable because of its large obscuration.  The intrinsically faint\nand soft emission in region 23, probably associated with more eastern\nstructures rather than with 30~Dor itself, is visible only because of\nits minimal obscuration.\n\nIn order to get enough counts for reliable spectral fitting, the\nregions used here average over many tens of square parsecs.  From the\ncomplex energy variations seen in Figures~\\ref{fig:csmoothimage},\n\\ref{fig:diffuseintro}, and \\ref{fig:lineims}, it is quite possible\nthat each region averages over many distinct plasma components, with\ndifferent pressures, densities, temperatures, absorbing columns, and\npossibly even different abundances.  Study of such features and their\nenergetics will be possible when longer ACIS exposures of the region\nare made.\n\n\n\n\\section{THE SUPERNOVA REMNANT N157B AND PSR~J0537$-$6910\n\\label{sec:N157B}}\n\n\\subsection{Morphology\n\\label{sec:n157bmorph}}\n\nThe {\\em Chandra}\/HRC \\citep{Wang01} and ACIS images of the composite\nSNR N157B are incremental improvements to the {\\em ROSAT} data, which\nalso showed a comet-shaped nebula embedded in a diffuse remnant,\nsuggesting that the neutron star received a kick in the supernova blast\nand is plowing through the surrounding ISM \\citep{Wang98a}.  Smoothed,\nsoft-band and hard-band ACIS images of N157B are shown in\nFigure~\\ref{fig:N157B-regs}.  The soft-band image\n(Figure~\\ref{fig:N157B-regs}a) shows the point source extraction\nregions outlined in red; the hard-band image\n(Figure~\\ref{fig:N157B-regs}b) additionally shows the concentric\ndiffuse extraction regions n7--n0 as defined in\nFigure~\\ref{fig:diffregions} and  Table~\\ref{tbl:diffspectable}.  The\nextraction region for the pulsar is shown in white.\n\nThe ACIS soft-band smoothed image shows diffuse emission from N157B\nextending somewhat farther than that shown in the HRC data\n\\citep[][Figure 1]{Wang01}, giving dimensions for the X-ray SNR of\n$\\sim 3\\arcmin.3 \\times 3\\arcmin.5$, or roughly 50~pc in diameter.  The\nsoft X-ray emission might be partially shadowed along the southern edge\nof the SNR by the dark cloud described in \\citet{Chu92}.  The hard\nX-ray emission in Figure~\\ref{fig:N157B-regs}b, which would not be\nshadowed because it penetrates such material, does not appear to\nextend into this region, though.  Thus it does not appear that the\nSNR extends behind the southern dark cloud.   \n  \nFigure~\\ref{fig:imageN157B} shows ACIS images of the embedded cometary\nnebula and pulsar.  The SNR is imaged $6\\arcmin.9$ off-axis, where the\nPSF is $\\sim 5\\arcsec \\times 10\\arcsec$ in size (shown in white in\nFigure~\\ref{fig:imageN157B}a).  The central source is still quite\ncentrally peaked and sharp enough that photon pile-up affects its\nspectrum.  Since it is immersed in its pulsar wind nebula and may have\nfurther corruption from the surrounding cometary nebula and SNR, our\nestimate of the pulsar's power law slope ($\\Gamma = 2.0$) should be\nconsidered qualitative.  \\citet{Mignani05} estimate $\\Gamma = 1.8$ for\nthe pulsar using an ACIS subarray observation that minimizes photon\npile-up for the pulsar and images it on-axis, so the small PSF limits\nthe contamination from diffuse structures around the pulsar.\n\nFigure~\\ref{fig:imageN157B}b shows the maximum likelihood\nreconstruction of Figure~\\ref{fig:imageN157B}a, using the PSF of the\npulsar for the reconstruction of the whole field. We recover an image\nvery similar to the on-axis {\\em Chandra}\/HRC images in\n\\citet{Wang01}:  the pulsar is a circular point source centered on its\npulsar wind nebula and a bright, $\\sim 4\\arcsec \\times 7\\arcsec$ region\nof emission is oriented perpendicular to the larger, trailing cometary\nnebula.  Although this image is not useful for spectral analysis, it is\nreassuring that even a simple reconstruction algorithm can recover\ninformation far off-axis, for sufficiently bright sources.\n\n\n\\subsection{Spectra\n\\label{sec:n157bspectra}}\n\nFigure~\\ref{fig:n157bspectra}a shows the spectral fit for the entire\nN157B SNR (labeled in Figure~\\ref{fig:introimage} and seen as region n8\nin Figure~\\ref{fig:diffregions}), including its pulsar (suffering from\nmoderate photon pile-up with $\\sim 0.9$ counts per frame), its pulsar\nwind nebula, and the surrounding cometary nebula.  Fit parameters are\ngiven on the last line of Table~\\ref{tbl:globalspectable}.  Using the\nsame spectral model as for Figure~\\ref{fig:globalspectra}a and b, we\nobtain the same $N_H$ but a hotter thermal plasma.  Some elements\nrequire abundances $>0.3Z_{\\odot}$ but are not well-constrained.\n\nOur thermal plasma fit results are quite similar to the {\\em ROSAT}\nresults \\citep{Wang98a}, but the power law component is substantially\nflatter than the results given by other X-ray observations of N157B\n(see \\S\\ref{sec:n157bbkgd}).  Since N157B is a young SNR, we also\ncharacterized its spectrum with a variable-abundance non-equilibrium\nionization model, using the {\\it XSPEC} model {\\it vnei + powerlaw}.\nThe fit was essentially the same as the thermal plasma fit described\nabove, with $kT \\sim 0.9$~keV and $\\Gamma = 2.2$, but the fit\nparameters were not as well-constrained as those for the thermal plasma fit.\n\nRemoving all the point sources from this region leaves the diffuse\nemission associated with the SNR (Figure~\\ref{fig:n157bspectra}b).  Its\nspectral fit parameters are listed under region n8 in\nTable~\\ref{tbl:diffspectable}.  The cometary nebula associated with the\npulsar contributes the power law component of the spectrum.  The\nthermal emission averaged over the whole SNR is quite soft and possible\nemission lines of Ne and Mg are seen.  While our fit results are\nconsistent with the {\\em XMM} results for the thermal component and\noverabundances of Ne and Mg are consistent with {\\em XMM's} discovery\nof emission lines \\citep{Dennerl01}, our power law slope is slightly\nflatter than the $\\Gamma = 2.8$ result from {\\em XMM}\n\\citep{Dennerl01}, even though our fit excluded the pulsar.\n\\citet{Dennerl01} note that, if N157B is similar to the Crab Nebula, we\nwould expect a difference between the core power law slope and that for\nthe entire nebula of $\\sim 0.5$ due to synchrotron losses.  This is\nseen in our {\\em Chandra} data, confirming the analogy with the Crab.\n\nThe {\\em XMM} value for the absorption (assuming an abundance of\n$0.5Z_{\\odot}$) was $N_H = 1.9 \\times 10^{22}$~cm$^{-2}$, more than a\nfactor of 6 higher than our result; the highest absorption we see for\nany part of N157B is at its core, where our $N_H$ is still a factor of\n3 lower than the {\\em XMM} value.  As discussed above, our assumption\nof solar abundance for the absorption can account for a factor of $\\sim\n2$ discrepancy in the measured column, but the cause of the remaining\ndifference in $N_H$ estimates is unknown.  One possibility might be\ncalibration uncertainties in the early {\\em XMM} results or \ncross-calibration issues between the two observatories.  \n\nTo study the SNR on finer spatial scales, spectral fits to the annular\nregions n0--n7 defined in Figure~\\ref{fig:N157B-regs} were performed\nand are given in Table~\\ref{tbl:diffspectable}.  Our spectral fitting\nconfirms the results of \\citet{Dennerl01} and \\citet{Wang98a} that\nN157B is a composite nebula, with a thermal plasma showing hints of\nline emission in the outer regions giving over to synchrotron emission\nin the bright core.  The spectral model in these fits allowed for both\na thermal plasma and a power law component, reverting to a single\ncomponent when an adequate fit could be obtained with just the thermal\nplasma or the power law model.\n\nWe have included two point source extraction regions in this table and\nshow their spectra in Figure~\\ref{fig:n157bspectra}:  p1 contains the\npulsar (CXOU~J053747.41$-$691019.8), p2 (CXOU~J053745.61$-$691011.1) is the\npointlike component of the cometary nebula found by {\\em wavdetect}.\nThese sources are included because their large off-axis PSFs make it\ndifficult to isolate the point source emission from the surrounding\nbright diffuse emission, so these point sources include diffuse\nspectral components.  In fact it is not clear in the ACIS subarray\nobservation of N157B that p2 is a point source at all; it is most\nlikely another example of concentrated diffuse emission that is\nconsistent with the ACIS PSF at this large off-axis angle.  The\nspectral fits in Table~\\ref{tbl:diffspectable} are not identical to\nthose in Paper II because the former were performed by hand, while the\nlatter were performed automatically in {\\em ACIS Extract}.  The results\nare consistent to within errors. \n\nComparing the spectrum of p1 (core, Figure~\\ref{fig:n157bspectra}c) to\np2 (cometary nebula bright point, Figure~\\ref{fig:n157bspectra}d) and\nthe surrounding bright cometary nebula (n0, the smallest blue annulus\nin Figure~\\ref{fig:N157B-regs}b), we see that all three regions are\nadequately fit by a simple power law, but its slope steepens rapidly\naway from the pulsar.  As we progress through the larger concentric\nannular regions n1--n7, an additional thermal component is needed for\nan adequate fit and the power law component apparently becomes very\nsteep.  In the outermost annular region (n7), the power law component\nis no longer necessary.  The spectral fits of these annular regions are\nproblematic due to small number counts; there can be interplay between\nthe thermal plasma temperature, the power law slope, and the absorption\nthat can lead to incongruous results, as we see for region n6.  Here\nthe power law slope is quite flat, but its normalization is low.  The\ngeneral trend of these annular spectra, though, seems to indicate\nelectron cooling through the steepening power law slope, a transition\nfrom non-thermal to thermal spectra with distance from the pulsar, and\nhigher abundances in the SNR, possibly hinting at the presence of\nspectral lines.\n\n\n\n\\section{DIFFUSE STRUCTURES IN THE CONTEXT OF 30~DOR KINEMATICS\n\\label{sec:otherdiffuse}}\n\nFrom {\\em Einstein} and {\\em ROSAT} data, it has long been known that\nmany of the X-ray concentrations in the 30~Dor superbubbles are\nspatially associated with high-velocity optical emission line clouds\n\\citep[][hereafter CK94]{Chu94}.  These kinematic data are invaluable\nfor sorting out the many overlapping X-ray features seen in the {\\em\nChandra} data as well.  For example, the large arc that appears to be\nassociated with the Honeycomb SNR is actually not kinematically related\nto it, but kinematic data indicate that the Honeycomb itself is the\nresult of a cavity supernova explosion.  The part of it seen in X-rays\nis due to the collision of this cavity SNR with an intervening porous\ngas sheet that could be associated with its slow-moving giant shell\n\\citep{Chu95b,Redman99}.  \n  \nWe place our bright X-ray features in the context of CK94's echelle\nstudy in Table~\\ref{tbl:kinematics}.  Several prominent X-ray features\n(identified solely on the basis of their apparent X-ray surface\nbrightness in a smoothed ACIS image, see \\S\\ref{sec:supermorph}) show\nhigh velocities in the kinematic data:  our West Ring is adjacent to\nCK94's NW Loop; our region 5 is CK94's R139W; our region 6 is CK94's\nR136E.  This correspondence between high-velocity features and diffuse\nX-ray emission was noticed and explained by CK94 as X-rays produced in\nthe shocks between high-velocity material and the surrounding\nslow-moving shells.  The {\\em Chandra} data allow us to explore this\ngeneral idea in more detail, with spatial resolution  better matched to\nthat of CK94's echelle data.\n\n\n\\subsection{X-rays from High-velocity Features\n\\label{sec:highvel}}\n\nThe {\\em Chandra} data show a more complex spatial distribution of\nX-ray plasmas than that described by \\citet{Wang99}, who inferred from\n{\\em ASCA} data that the diffuse emission was hot in the core of the\nnebula and cooler in its outer regions.  This complex distribution more\nclosely matches the kinematic portrait of the nebula built by CK94,\nthough.  Many of the bright X-ray regions that match high-velocity\nfeatures listed in Table~\\ref{tbl:kinematics} show cooler plasma\ntemperatures than their surroundings (Figure~\\ref{fig:nhktmaps}b).  We\ninterpret this as evidence that these regions are denser due to the\nhigh-velocity shocks, hence they are able to cool faster than the \nsurrounding superbubbles.\n\nThe high velocities measured near this X-ray-emitting material are more\nconsistent with CK94's interpretation of them as supernova shocks\nrather than as regions of high mass-loading, as proposed by\n\\citet{Wang99}, although our data do not exclude mass loading as a\nsecondary mechanism for increasing X-ray luminosity in some regions.\nWe would expect this mechanism to be most important in regions where\nsubstantial neutral material is known to exist, thus mass loading may\ncontribute to the bright diffuse X-rays seen near the center of the\nnebula, which contains the remains of the GMC.\n\nOur bright, soft region 5 is coaligned with CK94's fast shell R139W,\nwhich shows blueshifted emission with velocities up to\n150~km~s$^{-1}$.  The region of bright X-ray emission is $\\sim 15$~pc\nin diameter and fills a prominent, well-known hole in the H$\\alpha$\nemission that also appears in the {\\em Spitzer} data (see\n\\S\\ref{sec:hotcold}).  Nearby is another bright, soft X-ray patch\n(region 6) also $\\sim 15$~pc in diameter that is well-matched to CK94's\nfast shell R136E.  This region exhibits both redshifted and blueshifted\nemission with speeds in excess of 130~km~s$^{-1}$.  The X-ray emission\nis centered on the WR star R145, but CK94 note that it is unlikely to\nbe caused solely by the winds from that one star because the energy\nrequirements to support the X-ray emission and expansion velocities\nseen would require at least 5 average WR stars.  Both X-ray regions 5\nand 6 are cooler and intrinsically brighter than their surrounding\nregion 9 (see Figure~\\ref{fig:nhktmaps}).  They may well be parts of\nthe bubble blown by R136 that have been brightened by off-center SNe\n\\citep{Chu90}.\n\nWe find that not every high-velocity feature shows bright X-ray\nemission.  A good example is region 15, which coincides with a 20~pc\nshell expanding at 110~km~s$^{-1}$ described by CK94.\nFigure~\\ref{fig:nhktmaps} shows that this region is intrinsically X-ray\nfaint and exhibits the hottest plasma in the 30~Dor nebula.  Perhaps\nthis is the site of a recent cavity supernova that occurred far enough\nfrom the edge of Shell 2 that it did not produce an X-ray-bright shock\nbut instead deposited its energy into heating the surrounding plasma\n\\citep{Chu90}.  A more detailed velocity study of this region is\nwarranted to elucidate the cause of this unusual X-ray feature.\n\n\n\\subsection{Is the West Ring a Cavity Supernova Remnant?\n\\label{sec:westring}}\n\nA prominent feature in all X-ray images featured in this paper is a\nnearly-complete, clumpy ring structure in the middle of Shell 3, as\ndescribed in \\S\\ref{sec:globalmorph}.  We called this structure the\n``West Ring'' and find that it sits $\\sim 1\\arcmin$ west of CK94's\nhigh-velocity ``NW Loop,'' which shows expansion velocities of\n200~km~s$^{-1}$.  CK94 note that such high velocities are never found\nin wind-blown bubbles.  In Figure~\\ref{fig:nhktmaps}, the West Ring is\ncontained in region 17, which is indistinguishable from its surrounding\nregion 16 in obscuration and temperature but has slightly higher\nintrinsic surface brightness.  \n\nThe West Ring could be a SNR:  \\citet{Meaburn88} performed an echelle\nstudy of this region and concluded that the large expansion velocities\nmake a supernova origin the most likely explanation.  It resembles the\n``shell'' SNRs catalogued in \\citet{Williams99}, both in size and\nstructure.  Such a ring structure is predicted by \\citet{Velazquez03}\nwhen a SNR hits its cluster's stellar winds.  Recent echelle\nspectroscopy concentrating on this region also finds knots and shells\nconsistent with a supernova interpretation \\citep{Redman03}.\n\nThis structure is also very similar to the central X-ray ring\n\\citep{Leahy85} in the over-sized Galactic SNR HB3 (G132.7+1.3),\nconsidered to be a cavity SNR \\citep{Routledge91}.  The West Ring is\nabout $2\\arcmin.5$ in diameter, or $\\sim 36$~pc.  The X-ray ring in HB3\nis about $35\\arcmin$ in diameter, which at $D = 2.3$~kpc corresponds to\n$\\sim 23$~pc.  HB3's X-ray ring is also very clumpy and not quite\ncomplete \\citep{Landecker87}; it is strikingly similar in appearance to\nthe West Ring.  From Table~\\ref{tbl:diffspectable}, diffuse region 17\nhas 0.5--8~keV intrinsic luminosity $L_{X,corr} = 4.7 \\times\n10^{35}$~ergs~s$^{-1}$ and $kT = 0.5$~keV.  The {\\em Einstein}\n(0.2--4~keV) X-ray luminosity of HB3 totals $1.6 \\times\n10^{35}$~ergs~s$^{-1}$ and shows a hot central region ($kT \\sim 1$~keV)\nevolving to a cooler limb ($kT \\sim 0.3$~keV) \\citep{Leahy85}.  Our\nspectral fit to region 17 shows solar abundances for O and Ne; a recent\nanalysis of a short {\\em XMM} observation of HB3 shows the possibility\nof enhanced abundances of O, Ne, and Mg in its X-ray ring\n\\citep{Lazendic05}.  Just as in the West Ring, HB3 lacks strong X-ray\nemission lines.\n\nExamining the visual and radio images from \\citet{Dickel94}, it is\nclear that there is no prominent visual or radio continuum source at\nthe location of the West Ring.  As noted by \\citet{Chu00} in their\nstudy of the large LMC X-ray ring RX~J050736-6847.8, though, SNRs in\nlow-density media are not expected to show prominent visual or radio\nfeatures.  Given the similarity of the West Ring to HB3 and other LMC\nSNRs, we suspect that it is in fact a cavity SNR, perhaps produced by a\nstar in the Hodge~301 cluster.\n\nAlthough more detailed comparisons of the data from CK94 and other\nkinematic studies to the diffuse X-ray structures revealed by {\\em\nChandra} are beyond the scope of this paper, CK94 noted that such\ncomparisons likely hold the key to understanding the complex ISM in\n30~Dor.  We will search for the parsec-scale high-velocity knots seen\nby CK94 and \\citet{Meaburn84} in the longer {\\em Chandra} observation.\nWe suspect that this dataset will reveal small-scale diffuse X-ray\nfeatures that will merit new kinematic studies as well.\n\n\n\n\\section{DISCUSSION\n\\label{sec:discussion}}\n\n\\subsection{The Relationship between Hot, Warm, and Cool Interstellar \nMaterial in 30~Dor\n\\label{sec:hotcold}}\n\nThe morphology of the hot X-ray emitting plasma is extremely\ncomplicated and bears little resemblance to theoretical calculations of\nindividual SNRs or superbubbles, which generally predict\nquasi-spherical structures.  Insight into the origins of these\nstructures emerges from comparison with the distribution of\ninterstellar material traced by H$\\alpha$ emission from ionized gas and\nIR emission from dust.  The comparison with H$\\alpha$ has been made\nsince the earliest X-ray images of 30~Dor were obtained by {\\em\nEinstein} \\citep[e.g.][]{Chu90,Walborn91,Wang91a}.\n\nFigure~\\ref{fig:Halpha} shows a high-resolution H$\\alpha$ image from\nthe MCELS project \\citep{Smith00} in red, with adaptively-smoothed\nsoft-band ACIS images (point sources removed) in green and blue.  This\nimage is reminiscent of earlier combinations of H$\\alpha$ and X-ray\ndata that showed that diffuse soft X-ray emission was anticorrelated\nwith H$\\alpha$ emission, often filling the cavities outlined by ionized\ngas \\citep[e.g.][]{Wang99}.  These high-resolution images allow us to\nrefine that picture somewhat, giving more detailed information on the\nrelative locations of harder and softer X-rays and H$\\alpha$ emission.\n\nComparing Figure~\\ref{fig:Halpha} to a similar image based on {\\em\nROSAT}\/HRI data in \\citet[][Figure 3]{Wang99}, both images show that\nthe northern Shell 5 and southern Shell 2 are very clearly outlined by\nlarge-scale H$\\alpha$ structures.  X-rays from the northern part of SNR\nN157B coincide with a dark cavity in the H$\\alpha$ emission caused by a\ndust lane \\citep{Chu92}.  Conversely, the outer loop of Shell 3,\nlabeled as X-ray emitting region 18 in Figure~\\ref{fig:diffregions},\ncoincides with a similar loop in H$\\alpha$; in Figure~\\ref{fig:Halpha}\nwe can see that the northern part of this loop is softer than the\nsouthern part.  Both images show that a prominent H$\\alpha$ dark region\nin Shell 2 (southeast of R136) contains faint X-ray emission, while the\nbright H$\\alpha$ filaments that outline it bifurcate our X-ray void\nlabeled region 15.  Figure~\\ref{fig:Halpha} shows that Shell 2 X-rays\nare relatively hard and Figure~\\ref{fig:nhktmaps} reveals that this is\ndue to a combination of intrinsic spectral hardness and heavy\nobscuration in this part of 30~Dor.  The West Ring is now more clearly\nseen to be a clumpy ring of X-ray emission, fainter in its center, with\nsome soft X-ray emission coincident with the Hodge 301 cluster on the\nnortheast edge of the ring.  Small-scale clumps (such as our regions 7\nand 8) are now distinct from the smoother underlying X-ray emission.\nWith {\\em Chandra's} high on-axis spatial resolution, we have been able\nto excise the point source X-ray emission from Figure~\\ref{fig:Halpha},\nso it is clearer that bright diffuse regions 5 and 6 are distinct from\nthe R136 cluster and from the nearby WR stars R139, R140, and R145.\n\nThe 8$\\mu$m {\\em Spitzer} data (Figure~\\ref{fig:spitzer}) highlighting\nwarm dust also add insight into the diffuse X-ray structures.  Hot\nX-ray plasma fills the interiors of superbubbles that are outlined by\nwarm dust and emission from PAHs (Brandl et al.\\ 2006, in\npreparation).  The X-ray morphology and possible confinement are more\nfully appreciated when anchored by these IR data.  Heated dust provides\nan envelope to the base of the cylindrical X-ray Shell 5.  X-ray\nemission could be suppressed in this region or it could be present but\nabsorbed.  An IR-bright V-shaped ridge of emission (reminiscent of the\nCarina Nebula) separating Shell 3 from Shell 5 is devoid of observable\nsoft X-ray plasma, while the bright X-ray spot dominating Shell 1 is\nnearly devoid of IR emission.  The thermal plasma that constitutes the\nouter regions of the N157B SNR fills a large cavity in the warm dust.\nOne of the brightest clumps of diffuse X-rays (our region 5) fills a\ndistinct hole in both the H$\\alpha$ and IR emission.\n\nTo give a more complete view of the wide range of emission from 30~Dor,\nFigure~\\ref{fig:3bands} combines the 8$\\mu$m data tracing PAH emission\nand warm dust (red) from Figure~\\ref{fig:spitzer}, the H$\\alpha$ data\ntracing ionized $10^4$~K gas (green) from Figure~\\ref{fig:Halpha}, and\nthe 1120--2320~eV X-ray data tracing $10^7$~K plasma (blue) from\nFigure~\\ref{fig:csmoothimage}.  Note that here the X-ray image comes\nfrom the ACIS data processed with {\\em csmooth}, with the point sources\nleft in place, not the adaptively-smoothed images used in\nFigures~\\ref{fig:Halpha} and \\ref{fig:spitzer} where the point sources\nwere removed. \n\nFigure~\\ref{fig:3bands} shows the full ACIS-I field of view and is scaled\nto show the full extent of faint diffuse X-ray emission across the\nfield.  The zoomed image in Figure~\\ref{fig:3bandszoom} is scaled to\nshow just the brighter patches of X-ray emission in the center of the\nfield and removes the color saturation in the core.  Although the {\\em\nChandra} data are not deep enough to match the spatial resolution seen\nin the H$\\alpha$ and {\\em Spitzer} data, it is clear that the 30~Dor\ncomplex cannot be understood by visual and IR studies alone, no matter\nhow high the quality of those datasets.  The high-energy emission is a\nnear-perfect complement to the longer-wavelength emission, filling\ncavities in the complex that are outlined by H{\\sc II} regions.  These\nare in turn outlined by warm dust, because ultraviolet radiation in the\nH{\\sc II} regions destroys PAHs.  \n\nX-rays from the northern parts of SNR N157B fill a prominent hole in\nthe 8$\\mu$m emission.  The eastern side of the Shell 5 X-ray\nemission, which appeared unconfined by warm dust in\nFigure~\\ref{fig:spitzer}, is clearly defined by a large region of\nH$\\alpha$ emission, while its western side shows a narrower H{\\sc II}\nregion bordered by dust.  The southwestern side of the 30~Dor complex\nshows more warm dust than the northern or eastern sides.\n\nThere is a notable absence of X-ray emission in the southeastern corner\nof the image, where the stellar cluster SL~639\n\\citep{Shapley63,Melnick87,Bica99} is clearly seen in the H$\\alpha$ and\n8$\\mu$m data.  Some ionized gas is present around the cluster along\nwith substantial amounts of heated dust.  This cluster may be too young\nto have produced a substantial wind-blown bubble or any supernovae, or\nits ionizing stars may not generate enough wind power to blow a\nsubstantial bubble.  Conversely, if it is older than $\\sim$10~Myr it\nmay have dispersed its hot gas.  It will be an interesting site to\nsearch for faint diffuse X-rays in longer observations.\n\nThe famous central arcs north and west of R136 are bright at 8$\\mu$m as\nwell as in H$\\alpha$, illustrating the transition layers from cold\nmolecular material to heated dust to ionized gas that characterize much\nof the 30~Dor complex.  Figure~\\ref{fig:3bandszoom} shows that the\nbright X-ray emission lies in the cavities interior to these ionization\nfronts, but the saturated center of Figure~\\ref{fig:3bands} suggests\nthat some diffuse X-rays are seen superposed on the H$\\alpha$ and IR\nemission.  This is true in other parts of the image as well and implies\nthat the soft X-rays come from regions that lie in front of the denser\nIR-emitting material since they avoided being absorbed.  This is a\nreminder that the complex ISM in 30~Dor may be absorbing similar soft\nX-rays along other lines of sight.  Thus the regions exhibiting hot\nplasma may be connected via tunnels or fissures that are not visible to\nus in these images.  The regions where we do not see soft X-ray\nemission are not necessarily lacking in hot plasma, especially if it is\nclear from the {\\em Spitzer} data that substantial absorbing material\nlies along the line of sight.\n\nThese data thus augment and support the ideas developed over the last\ntwenty years for the evolution of the 30~Dor complex.  Powerful stellar\nwinds from extremely massive stars are carving holes in an extensive\nGMC and filling those holes with hot plasma.  Additional hot plasma is\nadded as those stars explode inside the cavities they created.  In this\nprocess, cold gas is pushed aside and confines the hot plasma into\nshapes otherwise difficult to understand.  Since this plasma appears to\nemit only soft X-rays for most of its lifetime, some of it may be\nabsorbed by intervening molecular material.  Tunnels and fissures could\nexist in the GMC that allow hot plasma to flow between apparently\nunconnected regions, although none of the hot plasma seen by {\\em\nChandra} appears completely unconfined (there are no empty bubbles that\nappear to have vented their X-ray plasma into the surrounding ISM).\nThe H$\\alpha$ emission traces the ionization fronts, H{\\sc II} region\n``sheets'' that mark the transition between the million-degree plasma\nand the neutral material.  This transition layer is heated, evaporated,\nand is accelerating away from the cloud interface.  Evolved stars are\ndistributed across the entire 30~Dor nebula, providing the fuel for the\nSNRs that occasionally brighten the superbubbles in X-rays and the\nenhanced density that allows the plasma to cool.  Neutral clumps may\nalso enhance the X-ray emission, especially in regions close to the\nremains of the GMC.\n\n\n\\subsection{Summary of Findings}\n\nDiffuse emission dominates the morphology of 30~Dor in soft X-rays,\ncaused by hot plasma filling the large superbubbles created by\ngenerations of star formation and subsequent SNe in this region.  This\nis dramatically illustrated by combining the {\\em Chandra} data with\nH$\\alpha$ and {\\em Spitzer} images\n(Figures~\\ref{fig:Halpha}--\\ref{fig:3bandszoom}); the $10^6$--$10^7$~K\nX-ray plasmas are enveloped and probably confined in most cases by the\ncooler gas and warm dust that define the classic picture of 30~Dor.\nThis example illustrates the need for high-quality X-ray observations\nof star-forming regions; new facets of both stellar and diffuse\ncomponents are revealed by high-energy data.\n\nThe combined spatial and spectral resolution afforded by ACIS shows\nthat these superbubbles are not uniformly filled with a\nsingle-temperature gas; great variety is seen, on a range of spatial\nscales, in absorption, plasma temperature, and intrinsic surface\nbrightness (Figure~\\ref{fig:nhktmaps}).  Some bubbles are\ncenter-filled, perhaps revealing interactions with cold gas left in\nshell interiors \\citep{Arthur96,Wang99}, while others are\nedge-brightened with distinct central voids not caused by obscuring\nforeground material.  The faintest regions (voids and the periphery of\nthe main 30~Dor nebula) are a factor of 20 fainter in intrinsic surface\nbrightness than the brightest regions, which have surface brightnesses\n$>1 \\times 10^{33}$~ergs~s$^{-1}$~pc$^{-2}$.  Our diffuse regions range\nin size from $1\\arcmin$ ($\\sim 14.5$~pc) for small surface brightness\nenhancements to $>7\\arcmin$ for the large shells.  For comparison with\nother GEHRs, we have provided integrated spectra and fits for the main\n30~Dor nebula and for the N157B SNR (Table~\\ref{tbl:globalspectable}).\n\nSpectral fits to the diffuse emission show moderate absorption ($N_H =\n1$--$6 \\times 10^{21}$~cm$^{-2}$) and soft thermal plasmas with $kT =\n0.3$--0.8~keV (3--9~MK).  Many diffuse regions exhibit elevated\nabundances (above the nominal value of $0.3Z_{\\odot}$), perhaps\nindicative of line emission, but none show prominent emission lines.\nAlthough not a definitive demonstration of a supernova origin for the\nX-ray emission, these spectral fit results are consistent with that\ninterpretation.  SNR N157B, also possibly the result of an explosion\nwithin a pre-existing cavity, also lacks strong emission lines, as does\nthe Galactic example of a possible cavity SNR discussed here, HB3.\n\nThe brightest point source in the field is PSR~J0537$-$6910 in the SNR\nN157B, imaged almost $7\\arcmin$ off-axis.  Although the large PSF there\nprevents us from performing accurate spectral analysis of the pulsar\nand its pulsar wind nebula, these data clearly show spectral changes as\na function of distance from the pulsar; the pure non-thermal spectrum\nsteepens and combines with a thermal component, finally giving over to\na pure thermal spectrum in the outer regions of the SNR\n(\\S\\ref{sec:N157B}).\n\nSeveral interesting smaller-scale structures emerge with the\nhigh-resolution {\\em Chandra} observations.  One feature (the ``West\nRing'') may be the relic signature of a past cavity supernova that\nexploded inside a pre-existing wind-blown bubble generated by the\nHodge~301 cluster.  As first shown by CK94, bright patches of X-ray\nemission often coincide with high-velocity features known from\nH$\\alpha$ echelle studies, indicating that X-rays are produced in\nshocks in 30~Dor's ISM (\\S\\ref{sec:otherdiffuse}).\n\nThe MCELS H$\\alpha$ image, the {\\em Spitzer} image, and the recent\nmosaic of {\\em HST} data \\citep{Walborn02} are all dominated by\nhighly-structured arcs and shells on 1--10~pc scales; in the central\nregion, these reveal the interfaces between the central cavity created\nby R136 and the surrounding molecular clouds \\citep{Scowen98} and\ndemonstrate how R136 is shredding its natal environment\n(\\S\\ref{sec:hotcold}).  Here and throughout the 30~Dor nebula, an\nappreciation of the X-ray emission is necessary to further our\nunderstanding of the processes working to shape this complex.  As the\nimages in \\S\\ref{sec:hotcold} show, it is imperative that the diffuse\nX-ray emission become part of the ``classic'' picture of 30~Dor.\n\nNew stellar clusters are now forming in the dense knots that remain in\n30~Dor's GMC; their collapse was probably triggered by R136\n\\citep{Walborn02}.  These clusters are not yet resolved in X-rays,\ndemonstrating that there is much more work to be done to achieve a\ncomplete picture of the X-ray emission from 30~Dor.\n\n  \n   \n\\subsection{30~Dor and Other Massive Star-forming Regions\n\\label{sec:othermsfrs}}\n\n30~Dor's diffuse X-ray emission is, overall, substantially different\nthan that seen in Galactic MSFRs that are too young to have produced\nSNe.  M~17 shows strong diffuse soft X-rays probably due to wind-wind\nand\/or wind-cloud collisions \\citep{Townsley03}, but the primary\ncomponent of this emission is hotter (kT = 0.6~keV) than many 30~Dor\nregions and has low surface brightness\n($10^{31.9}$~ergs~s$^{-1}$~pc$^{-2}$).  These quantities are most\nsimilar to the regions we call ``voids'' in 30~Dor (regions 15, 19, 20,\n21, and 22 in Figure~\\ref{fig:diffregions}).  The diffuse X-ray\nemission reported for the massive Galactic clusters NGC~3603\n\\citep{Moffat02} and Arches \\citep{Yusef-Zadeh02} has higher surface\nbrightness more comparable to the brighter regions in 30~Dor\n\\citep[$10^{32.6}$ and $10^{33.1}$~ergs~s$^{-1}$~pc$^{-2}$\nrespectively, calculated from Table 4 of][]{Townsley03} but the plasmas\nare much hotter (kT = 3.1~keV and 5.7~keV respectively).  Either a very\ndifferent mechanism is generating hard diffuse emission in these\nclusters \\citep[e.g. wind collisions,][]{Canto00}, young SNe dominate\nthe diffuse emission, or substantial unresolved point source emission\nis corrupting the measurements.\n\n\\citet{Chu93} proposed that the soft diffuse X-ray emission seen in the\nCarina star-forming complex \\citep{Seward82} could be due to a cavity\nSNR inside a superbubble blown by Carina's many massive stellar\nclusters; thus the Carina complex should be a Galactic analog to one of\nthe superbubbles seen in 30~Dor.  The entire Carina complex shows\nintegrated X-ray emission with $kT \\sim 0.8$~keV and surface brightness\n$\\geq 10^{32.2}$~ergs~s$^{-1}$~pc$^{-2}$ \\citep{Seward82,Townsley03}.\n{\\em Chandra} and {\\em XMM} resolve this X-ray emission into thousands\nof harder but comparatively faint stellar sources\n\\citep[e.g.][]{Albacete03,Evans03} and bright diffuse emission\npervading the complex yet not centered on the O or WR stars or the\nstellar clusters.  No obvious SNR is present, but the complex is old\nenough to have produced SNe, as evidenced by the presence of evolved\nstars.  This bright, soft diffuse emission is quite comparable in\nsurface brightness and extent to the smaller regions sampled in\n30~Dor.  Using new {\\em Chandra} data centered on the Trumpler~14 OB\ncluster in Carina, we show that the brightest regions of diffuse\nemission are well-separated from the massive stars in the Carina\ncomplex and show filamentary morphology, both consistent with a cavity\nsupernova origin \\citep{TownsleyIAUS227}.  Thus we suspect that Carina\nserves as a good microscope for understanding the processes powering\nthe X-ray emission from 30~Dor.\n\nGiven the morphological complexity that 30~Dor displays at all\nwavelengths, it is surprisingly similar to other GEHRs, most notably\nNGC~604 in M33, the second-largest GEHR in the Local Group.  Although\nnot dominated by a massive central star cluster, the large-scale loops\nand voids of NGC~604 are also characteristic of superbubbles and are\nfilled with hot gas emitting soft X-rays \\citep[][Brandl et al.\\ 2006,\nin preparation]{Maiz04}.  The second-largest GEHR in the LMC, N11, also\nexhibits a central massive stellar cluster surrounded by a superbubble\nand containing bright, soft diffuse X-rays \\citep{MacLow98,Naze04}.\nAlthough more distant GEHRs are not resolvable with current technology,\nwe should expect all GEHRs to be made up of a complex mix of pointlike\nand diffuse X-ray components with the hard-band X-ray luminosity\ndominated by massive stars and the soft-band X-ray luminosity dominated\nby the effects of recent cavity SNe expoding near the edges of\nsuperbubbles, as we see in 30~Dor.\n\n\n\\subsection{Concluding Comments}\n\nWe have analyzed an early {\\em Chandra}\/ACIS observation of 30~Doradus,\nconcentrating here on the diffuse X-ray structures and in Paper II on\nthe point sources.  This study documents a wide variety of diffuse\nX-ray-emitting sources:  a complex hierarchy of diffuse structures,\nfrom small-scale knots and wisps to huge superbubbles; a composite SNR\nincluding its pulsar, pulsar wind nebula, and cometary tail; an X-ray\nring possibly due to a cavity SNR; bright X-ray patches associated with\nhigh-velocity H$\\alpha$ structures.  A coherent understanding of the\nstructures is beginning to emerge from a multiwavelength comparison of\nthe X-ray, H$\\alpha$, and mid-IR maps.  We demonstrate a variety of\ndata analysis tools for study of ACIS fields with mixtures of pointlike\nsources and diffuse structures.  All software used in this study is\npublicly available.\n\nFrom this work, we conclude the following:\n\\begin{itemize}\n\\item  30~Dor's integrated diffuse emission is well-fit by a\nsingle-temperature plasma with $T \\simeq 7$~MK, roughly solar line\nstrengths, and a single absorbing column density of $N_H \\simeq 3\n\\times 10^{21}$~cm$^{-2}$.  The soft-band luminosity is dominated by the\ndiffuse structures, but these disappear above 2~keV and the integrated\nemission is dominated by just a few bright point sources\n(\\S\\ref{sec:globalspectra} and Paper~II).\n\\item  ACIS spectra of 30~Dor's diffuse emission regions often suggest\nchemical enrichment, another argument in favor of a SNR origin for the\nX-ray emission (Table~\\ref{tbl:diffspectable}).\n\\item  Annular spectra around the SNR N157B, centered on\nPSR~J0537$-$6910, indicate electron cooling through a steepening power\nlaw slope, a transition from non-thermal to thermal spectra with\ndistance from the pulsar, and chemical enrichment in the SNR\n(\\S\\ref{sec:N157B}).\n\\item  A powerful method for understanding the complex spectral and\nspatial information contained in high-resolution X-ray studies of\nmassive star-forming complexes like 30~Dor is to study maps of column\ndensity, plasma temperature, and intrinsic surface brightness derived\nfrom X-ray spectral fitting (Figure~\\ref{fig:nhktmaps}).\n\\item  The main 30~Dor nebula exhibits several X-ray ``voids'' not\ncaused by obscuration, most notably the hot region 15\n(\\S\\ref{sec:superspectra}).  These resemble the wind collision plasma\nseen in nearby H{\\sc II} regions like M~17 (\\S\\ref{sec:othermsfrs}).\n\\item  Comparing the diffuse X-ray emission with kinematic studies of\nthe warm gas in H{\\sc II} region complexes is essential to disentangle\nthe confusing morphological information.  While the fast shocks found\nin 30~Dor by CK94 often correlate well with regions of high X-ray\nsurface brightness, the correlation is not complete; some fast shells\nlack bright X-rays while some X-ray features are not known to exhibit\nhigh velocities (\\S\\ref{sec:highvel}).\n\\item Some shells do not currently contain massive clusters to supply\nwind-generated X-rays.  The source of diffuse X-rays may be centered\nSNe that heated the interior of these shells without producing fast\nshocks against the shell walls \\citep{Chu90}.\n\\item  The X-ray structure that we call the West Ring (our region 17)\nis adjacent to CK94's NW Loop and has X-ray properties very similar to\nthe Galactic cavity SNR HB3.  We propose that it is a cavity SNR that\nexploded inside 30~Dor's Shell 3 and that its projenitor most likely\ncame from the massive cluster Hodge~301 (\\S\\ref{sec:westring}).\n\\item  Bright X-ray patches often have cooler plasma temperatures than\ntheir surrounding shells, possibly indicating that the increased\ndensities associated with fast shocks allow the associated X-ray plasma\nto cool more efficiently than the plasma associated with the large,\nlow-density shells.\n\\end{itemize}\n\nOur upcoming 100~ks ACIS-I observation of 30~Dor will provide the\nhigh-quality X-ray dataset that is needed to place the X-ray emission\nin context with state-of-the-art observations in other wavebands.  We\nexpect this observation to reveal more of the high-mass stellar\npopulation in 30~Dor and to give even more detailed information on the\ncomplex morphology of the wind-blown bubbles and superbubbles, both\nwith higher spatial resolution imaging and with spectroscopy on finer\nspatial scales.  True progress in understanding, though, will require\ncooperation.  By combining the power of today's Great Observatories and\nhigh-quality ground-based data, we are confident that unique insight\nawaits.\n\n\n\\acknowledgments\n\nSupport for this work was provided to Gordon Garmire, the ACIS\nPrincipal Investigator, by the National Aeronautics and Space\nAdministration (NASA) through NASA Contract NAS8-38252 and {\\em\nChandra} Contract SV4-74018 issued by the {\\em Chandra X-ray\nObservatory} Center, which is operated by the Smithsonian Astrophysical\nObservatory for and on behalf of NASA under contract NAS8-03060.  LKT\nappreciates technical contributions by Jeremy Sanders and Konstantin\nGetman and helpful discussions with Mike Eracleous and Bruce\nElmegreen.  We thank the Magellanic Cloud Emission Line Survey (MCELS)\nteam for the use of their H$\\alpha$ image of 30~Doradus.  We also\nthank our anonymous referee for investing the time to review this\npaper and for several helpful suggestions.\n\nThis work is based in part on observations made with the {\\em Spitzer\nSpace Telescope}, which is operated by the Jet Propulsion Laboratory,\nCalifornia Institute of Technology under NASA Contract 1407.  This\nresearch made use of data products from the Two Micron All Sky Survey,\nwhich is a joint project of the University of Massachusetts and the\nInfrared Processing and Analysis Center\/California Institute of\nTechnology, funded by NASA and the National Science Foundation.  This\nresearch also made use of the SIMBAD database and VizieR catalogue\naccess tool, operated at CDS, Strasbourg, France.  We are grateful for\nthe invaluable tools of NASA's Astrophysics Data System.\n\n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\\section{Acknowledgements}\nThe authors thank the Amazon Mechanical Turk workers that participated in the user study. This work was supported by the DFG \u2013 EXC number 2064\/1 \u2013 project number 390727645, by the DFG: SFB 1233, Robust Vision: Inference Principles and Neural Mechanisms - project number: 276693517, by the ERC (853489 - DEXIM), and by the BMBF (FKZ: 01IS18039A). The authors thank the International Max Planck Research School for Intelligent Systems (IMPRS-IS) for supporting Leonard Salewski.\n\\section{Conclusion}\nWe introduced the novel CLEVR-X\\xspace{} dataset which contains natural language explanations for the VQA task on the CLEVR dataset.\nOur user study confirms that the explanations in the CLEVR-X\\xspace{} dataset are complete and match the questions and images.\nFurthermore, we have provided baseline performances using the PJ-X and FM frameworks on the CLEVR-X\\xspace{} dataset. The structured nature of our proposed dataset allowed the detailed evaluation of the explanation generation quality according to answer and question types. We observed that the generated explanations were of higher quality for easier answer and question categories.\nOne of our findings is, that explanations for counting problems are worse than for other answer types, suggesting that further research into this direction is needed.\nAdditionally, we find that the four NL\\rev{G} metrics used to evaluate the quality of the generated explanations exhibit different convergence patterns depending on the number of available ground-truth references. \n\nSince this work only considered two natural language generation methods for VQA as baselines, the natural next step will be the benchmarking and closer investigation of additional recent frameworks for textual explanations in the context of VQA on the CLEVR-X\\xspace{} dataset.\nWe hope that our proposed CLEVR-X\\xspace{} benchmark will facilitate further research to improve the generation of natural language explanations in the context of vision-language tasks.\n\\section{The CLEVR-X dataset}\\label{sec:dataclevrx}\n\n\nIn this section, we introduce the CLEVR-X dataset that consists of natural language explanations in the context of VQA\\@. The CLEVR-X dataset extends the CLEVR dataset with 3.6 million natural language explanations for 850k question-image pairs.\nIn Section~\\ref{sec:data-clevr}, we briefly describe the CLEVR dataset, which forms the base for our proposed dataset. Next, we present an overview of the CLEVR-X dataset by describing how the natural language explanations were obtained in Section~\\ref{sec:data-generation}, and by providing a comprehensive analysis of the CLEVR-X dataset in Section~\\ref{sec:data-analysis}.\nFinally, in Section~\\ref{sec:data-userstudy}, we present results for a user study on the CLEVR-X dataset.\n\n\n\n\\subsection{The  CLEVR dataset}\\label{sec:data-clevr}\nThe CLEVR dataset consists of images with corresponding full scene graph annotations which contain information about all objects in a given scene (as nodes in the graph) along with spatial relationships for all object pairs. The synthetic images in the CLEVR dataset contain three to ten (at least partially visible) objects in each scene, where each object has the four distinct properties \\texttt{size}, \\texttt{color}, \\texttt{material}, and \\texttt{shape}.\nThere are three shapes (\\texttt{box}, \\texttt{sphere}, \\texttt{cylinder}), eight colors (\\texttt{gray}, \\texttt{red}, \\texttt{blue}, \\texttt{green}, \\texttt{brown}, \\texttt{purple}, \\texttt{cyan}, \\texttt{yellow}), two sizes (\\texttt{large}, \\texttt{small}), and two materials (\\texttt{rubber}, \\texttt{metallic}). This allows for up to 96 different combinations of properties. \n\n\n\n\n\n\nThere are a total of 90 different question families in the dataset which are grouped into 9 different question types. Each type contains questions from between 5 and 28 question families.\nIn the following, we describe the 9 question types in more detail.\n\n\\myparagraph{\\emph{Hop} questions:}\nThe \\emph{zero hop}, \\emph{one hop}, \\emph{two hop}, and \\emph{three hop} question types contain up to three relational reasoning steps, e.g.\\ \\enquote{What color is the cube \\underline{to the left of} the ball?} is a \\emph{one hop} question.\n\n\\myparagraph{\\emph{Compare} and \\emph{relate} questions:}\nThe \\emph{compare integer}, \\emph{same relate}, and \\emph{comparison} question types require the understanding and comparison of multiple objects in a scene.\nQuestions of the \\emph{compare integer} type compare counts corresponding to two independent clauses (e.g.\\ \\enquote{Are there \\underline{more} cubes \\underline{than} red balls?}).\n\\emph{Same relate} questions reason about objects that have the same attribute as another previously specified object (e.g.\\ \\enquote{What is the color of the cube that has \\underline{the same size} as the ball?}).\nIn contrast, \\emph{comparison} question types compare the attributes of two objects (e.g.\\ \\enquote{\\underline{Is the color} of the cube \\underline{the same} as the ball?}).\n\n\\myparagraph{\\emph{Single and\/or} questions:}\n\\emph{Single or} questions identify objects that satisfy an exclusive disjunction condition (e.g.\\ \\enquote{How many objects are \\underline{either} red \\underline{or} blue?}).\nSimilarly, \\emph{single and} questions apply multiple relations and filters to find an object that satisfies all conditions (e.g.\\ \\enquote{How many objects are red \\underline{and} to the left of the cube.}).\\\\\n\n\\rev{Each CLEVR question can be represented by a corresponding functional program and its natural language realization. \nA functional program is composed of basic functions that resemble elementary visual reasoning operations, such as \\emph{filtering} \\rev{objects} by \\rev{one or more} properties, spatially \\emph{relating} objects to each other, or \\emph{querying} object properties.\nFurthermore, logical operations \\rev{like} \\emph{and} and \\emph{or}, as well as counting operations like \\emph{count}, \\emph{less}, \\emph{more}, and \\emph{equal} are used to build complex questions.\nExecuting the functional program associated with the question against the scene graph yields the correct answer to the question.}\nWe can distinguish between three different answer types: Binary answers (\\texttt{yes} or \\texttt{no}), counting answers (integers from \\texttt{0} to \\texttt{10}), and attribute answers (any of the possible values of \\texttt{shape}, \\texttt{color}, \\texttt{size}, or \\texttt{material}).\n\n\n\\subsection{Dataset generation}%\n\\label{sec:data-generation}\n\\begin{figure}[t]\n     \\centering\n        \\centering\n        \\includegraphics[width=\\linewidth,trim=3mm 85mm 7mm 15mm,clip]{figures\/dataset\/PaperFigures_ask.pdf}\n        \\caption{%\n        CLEVR-X dataset generation: Generating a natural language explanation for a sample from the CLEVR dataset. Based on the question, the functional program for answering the question is executed on the scene graph and traced. A language template is used to cast the gathered information into a natural language explanation.\n        }%\n        \\label{fig:generation-and-examples}\n\\end{figure}\n\nHere, we describe the process for generating natural language explanations for the CLEVR-X dataset.\n\\rev{In contrast to image captions, the CLEVR-X explanations only describe image elements that are relevant to a specific input question.}\nThe explanation generation process for a given question-image pair is illustrated in Fig.~\\ref{fig:generation-and-examples}. It consists of three steps: Tracing the functional program, relevance filtering (not shown in the figure), and explanation generation. In the following, we will describe those steps in detail.\n\n\\myparagraph{Tracing the functional program.}\nGiven a question-image pair from the CLEVR dataset, we trace the execution of the functional program (that corresponds to the question) on the scene graph (which is associated with the image). The generation of the CLEVR dataset uses the same step to obtain a question-answer pair. When executing the basic functions that comprise the functional program, we record their outputs in order to collect all the information required for explaining a ground-truth answer.\n\n\n\\rev{In particular, we trace} the \\emph{filter}, \\emph{relate} and \\emph{same-property} functions \\rev{and record the returned objects and their properties, such as, for instance, \\texttt{shape} or \\texttt{size}.}\nAs a result, the tracing omits objects \\rev{in the scene} that are not relevant for the question.\n\\rev{As we are aiming for complete explanations for all question types, each explanation has to mention all the objects that were needed to answer the question, i.e.\\ all the evidence that was obtained during tracing.\nFor example, for \\emph{counting} questions, all objects that match the \\rev{\\emph{filter} function} preceding the \\emph{counting} step are recorded during tracing.\nFor \\emph{and} questions, we merge the tracing results of the preceding \\rev{functions} which results in short and readable explanations.}\n\\rev{In summary, the tracing} produces a \\emph{complete} \\rev{and \\emph{correct}} understanding of the objects and relevant properties which contributed to an answer.\n\n\n\n\\myparagraph{Relevance filtering.}\nTo keep the explanation at a reasonable length, we filter the object attributes that are mentioned in the explanation according to their relevance.\nFor example, the \\texttt{color} of an object is not relevant for a given question that asks about the \\texttt{material} of said object.\n\\rev{We deem all properties that were listed in the question to be relevant.}\nThis makes it easier to recognize the same referenced object in both the question and explanation.\nAs the \\texttt{shape} property also serves as a noun in CLEVR, our explanations always mention the \\texttt{shape} to avoid using generic shape descriptions like \\enquote{object} or \\enquote{thing}.\nWe distinguish between objects which are used to build the question (e.g.\\ {} \\enquote{[\\ldots] that is left of the \\textit{cube}?}) and those that are the subject of the posed question (e.g.\\ {} \\enquote{What color is the \\textit{sphere} that is left of the cube?}).\n\\rev{For the former, we do not mention any additional properties, and for the latter}, we mention the queried property (e.g.\\ \\texttt{color}) for question types yielding attribute answers.\n\n\\myparagraph{Explanation generation.} To obtain the final natural language explanations, each question type is equipped with one or more natural language templates with variations in terms of the wording used.\nEach template contains placeholders which are filled with the output of the previous steps, i.e.\\ the tracing of the functional program and subsequent filtering for relevance.\nAs mentioned above, our explanations use the same property descriptions that appeared in the question. \n\\rev{This is done to ensure that the wording of the explanation is consistent with the given question, e.g.\\ for the question \\enquote{Is there a \\underline{small} object?} we generate the explanation \\enquote{Yes there is a \\underline{small} cube.}%\n\\footnote{%\nThe explanation could have used the synonym \\enquote{box} instead of \\enquote{cube}. In contrast, \\enquote{tiny} and \\enquote{small} are also synonyms in CLEVR, but the explanation would not have been consistent with the question which used \\enquote{small}.}%\n}.\nWe randomly sample synonyms for describing the properties of objects that do not appear in the question.\nIf multiple objects are mentioned in the explanation, we randomize their order.\n\\rev{If the tracing step returned an empty set, e.g.\\ if no object exists that matches the given filtering function for an \\emph{existence} or \\emph{counting} question, we state that no relevant object is contained in the scene (e.g.\\ \\enquote{There is no red cube.}).}\n\nIn order to decrease the overall sentence length and to increase the readability, we aggregate repetitive descriptions (e.g.\\ \\enquote{There is a \\underline{red cube} and a \\underline{red cube}}) using numerals (e.g.\\ \\enquote{There are \\underline{two} red cubes.}).\nIn addition, if \\rev{a function of} the functional program merely restricts the output set of a preceding \\rev{function}, we only mention the outputs of the later \\rev{function}.\nFor instance, if a \\texttt{same-color} \\rev{function} yields a large and a small cube, and a \\rev{subsequent} \\texttt{filter-large} \\rev{function} restricts th\\rev{e output} to only the large cube, we do not mention the output of \\texttt{same-color}, as the output of the following \\texttt{filter-large} causes natural language redundancies%\n\\footnote{\\rev{E.g.\\ for the question: \\enquote{How many large objects have the same color as the cube?}, we do not generate the explanation \\enquote{There are a small and a large cube that have the same color as the red cylinder of which only the large cube is large.} but instead only write \\enquote{There is a large cube that has the same color as the red cylinder.}}}%\n.\n\n\n\\rev{The selection of different language templates, random sampling of synonyms and randomization of the object order (if possible) results in multiple different explanations.} \nWe uniformly sample up to 10 different explanations \\rev{per question }for our dataset.\n\n\\myparagraph{Dataset split.}\nWe provide explanations for the CLEVR training and validation sets, skipping only a negligible subset (less than $0.04\\permil$) of questions due to malformed question programs from the CLEVR dataset\\rev{, e.g.\\ due to disjoint parts of their abstract syntax trees. In total, this affected 25 CLEVR training and 4 validation questions.}\n\nAs the scene graphs and question functional programs are not publicly available for the \\rev{CLEVR} test set, we use the original CLEVR validation subset as the CLEVR-X test set. 20\\% of the CLEVR training set serve as the CLEVR-X validation set. We perform this split on the image-level to avoid any overlap between images in the CLEVR-X training and validation sets.\nFurthermore, we verified that the relative proportion of samples from each question and answer type in the CLEVR-X training and validation sets is similar, such that there are no biases towards specific question or answer types.\n\n\nCode for generating the CLEVR-X dataset and the dataset itself are publicly available at \\url{https:\/\/github.com\/ExplainableML\/CLEVR-X}.\n\n\n\n\n\\subsection{Dataset analysis}\\label{sec:data-analysis}\n\\input{sections\/body\/tables\/dataset_statistics}\n\n\\begin{figure}[t]\n     \\centering\n     \\begin{subfigure}[t]{0.495\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth,trim=3mm 3mm 3mm 4mm, clip]{figures\/dataset\/xlen_hist_train_v0.7.10-recut.pdf}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[t]{0.495\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth,trim=3mm 3mm 3mm 4mm, clip]{figures\/dataset\/dataset_histogram_comparison_vqax_esnlive_clevr_train_v0.7.10-recut.pdf}\n     \\end{subfigure}\n     \\vspace{-2ex}\n    \\caption{%\n        Stacked histogram of the average explanation lengths measured in words for the 9 question types for the CLEVR-X training set (left).\n        Explanation length distribution for the CLEVR-X\\xspace{}, VQA-X\\xspace{}, and e-SNLI-VE\\xspace{} \\rev{training sets} (right).\n        The long tail of the e-SNLI-VE\\xspace{} distribution (125 words) was cropped out \\rev{for better} readability.\n    }%\n    \\label{fig:dataset-statisics}\n    \\vspace{-0.6em}\n\\end{figure}\n\n\n\nWe compare the CLEVR-X dataset to the related VQA-X and e-SNLI-VE datasets in Table~\\ref{tab:new_dataset_statistics}. Similar to CLEVR-X, VQA-X contains natural language explanations for the VQA task. However, different to the natural images and human explanations in VQA-X, CLEVR-X consists of synthetic images and explanations.\nThe e-SNLI-VE\\xspace{} dataset provides explanations for the visual entailment (VE) task. VE consists of classifying an input image-hypothesis pair into entailment \/ neutral \/ contradiction categories. \n\n\n\n\nThe CLEVR-X\\xspace{} dataset is significantly larger than the VQA-X\\xspace{} and e-SNLI-VE\\xspace{} datasets in terms of the number of images, questions, and explanations.\nIn contrast to the two other datasets, CLEVR-X\\xspace{} provides (on average) multiple explanations for each question-image pair in the train set.\nAdditionally, the average number of words per explanation is also higher. Since the explanations are built to explain each component mentioned in the question, long questions require longer explanations than short questions. Nevertheless, by design, there are no unnecessary redundancies. The explanation length in CLEVR-X\\xspace is very strongly correlated with the length of the corresponding question (Spearman's correlation coefficient between the number of words in the explanations and questions is $0.89$). \n\nFigure~\\ref{fig:dataset-statisics} (left) shows the explanation length distribution in the CLEVR-X dataset for the 9 question types.\nThe shortest explanation consists of 7 words, and the longest one has 53 words. On average, the explanations contain \\rev{21.53} words. In Fig.~\\ref{fig:dataset-statisics} (right) and Table~\\ref{tab:new_dataset_statistics}, we can observe that explanations in CLEVR-X tend to be longer than the explanations in the VQA-X dataset. Furthermore, VQA-X\\xspace has significantly fewer samples overall than the CLEVR-X dataset. The e-SNLI-VE dataset also contains longer explanations (that are up to 125 words long), but the CLEVR-X dataset is significantly larger than the e-SNLI-VE dataset.\nHowever, due to the synthetic nature and limited domain of CLEVR, the vocabulary of CLEVR-X is very small with only 96 different words.\nUnfortunately, VQA-X\\xspace{} and e-SNLI-VE\\xspace{} contain spelling errors, resulting in multiple versions of the same words.\nModels trained on CLEVR-X circumvent those aforementioned challenges and can purely focus on visual reasoning and explanations for the same.\nTherefore, Natural Language Generation (NLG) metrics applied to CLEVR-X indeed capture the factual correctness and completeness of an explanation.\n\n\n\n\n\n\\input{sections\/body\/user_study}\n\\section{Experiments}\nWe describe the experimental setup for establishing baselines on our proposed CLEVR-X dataset in Section~\\ref{sec:exp-setup}. \nIn Section~\\ref{sec:clevrx-results}, we present quantitative results on the CLEVR-X dataset. Additionally, we analyze the generated explanations for the CLEVR-X dataset in relation to the question and answer types in Section~\\ref{sec:type-results}.\nFurthermore, we study the behavior of the NL\\rev{G} metrics when using different numbers of ground-truth explanations for testing in Section~\\ref{sec:length-results}.\nFinally, we present qualitative explanation generation results on the CLEVR-X dataset in Section~\\ref{sec:quali-results}.\n\n\n\\subsection{Experimental setup}\\label{sec:exp-setup}\nIn this section, we provide details about the datasets and models used to establish baselines for our CLEVR-X dataset and about their training details. Furthermore, we explain the metrics for evaluating the explanation generation performance.\n\n\\myparagraph{Datasets.}\nIn the following, we summarize the datasets that were used for our experiments. In addition to providing baseline results on CLEVR-X, we also report experimental results on the VQA-X and e-SNLI-VE datasets.\nDetails about our proposed \\textbf{CLEVR-X} dataset can be found in Section~\\ref{sec:dataclevrx}.\nThe \\textbf{VQA-X} dataset~\\cite{hukpark2018MultimodalExplanationsJustifying} is a subset of the VQA~v2 dataset with a single human-generated textual explanation per question-image pair in the training set and $3$ explanations for each sample in the validation and test sets.\nThe \\textbf{e-SNLI-VE} dataset~\\cite{Do2020eSNLIVE20CV,kayser2021vil} is a large-scale dataset with natural language explanations for the visual entailment task.\n\n\\myparagraph{Methods.}\nWe used multiple frameworks to provide baselines on our proposed CLEVR-X dataset.\nFor the \\textbf{random words} baseline, we sample random word sequences of length $w$ for the answer and explanation words for each test sample. The full vocabulary corresponding to a given dataset is used as the sampling pool, and $w$ denotes the average number of words forming an answer and explanation in a given dataset.\nFor the \\textbf{random~explanations} baseline, we randomly sample an answer-explanation pair from the training set and use this as the prediction. \\rev{The explanations from this baseline are well-formed sentences. However, the answers and explanations most likely do not match the question or the image.}\nFor the random-words and random-explanations baselines, we report the NLG metrics for all samples in the test set (instead of only considering the correctly answered samples, since the random sampling of the answer does not influence the explanation).\nThe Pointing and Justification model \\textbf{PJ-X}~\\cite{hukpark2018MultimodalExplanationsJustifying} provides text-based post-hoc justifications for the VQA task. It combines a modified MCB~\\cite{fukui2016MultimodalCompactBilinear} framework, pre-trained on the VQA v2 dataset, with a visual pointing and textual justification module.\nThe Faithful Multimodal (\\textbf{FM}) model~\\cite{wu2019FaithfulMultimodalExplanation} aims at grounding parts of generated explanations in the input image to provide explanations that are \\textit{faithful} to the input image. It is based  on the Up-Down VQA model~\\cite{anderson2018bottom}. In addition, FM contains an explanation module which enforces consistency between the predicted answer, explanation and the attention of the VQA model.\nThe implementations for the PJ-X and FM models are based on those provided by the authors of~\\cite{kayser2021vil}.\n\n\\myparagraph{Implementation and training details.}\nWe extracted 14$\\times$14$\\times$1024 grid features for the images in the CLEVR-X dataset using a ResNet-101~\\cite{he2016deep}, pre-trained on ImageNet~\\cite{deng2009imagenet}. These grid features served as inputs to the FM~\\cite{wu2019FaithfulMultimodalExplanation} and PJ-X~\\cite{hukpark2018MultimodalExplanationsJustifying} frameworks. \nThe CLEVR-X explanations are lower case and punctuation is removed from the sentences.\nWe selected the best model on the CLEVR-X validation set\nbased on the highest \\rev{mean of the four NLG metrics, where explanations for incorrect answers were set to an empty string}. \\rev{This metric accounts for the answering performance as well as for the explanation quality.}\nThe final models were \nevaluated on the CLEVR-X test set.\nFor PJ-X, our best model was trained for \\rev{52} epochs, using the Adam optimizer~\\cite{kingma2014adam} with a learning rate of 0.0002 and a batch size of \\rev{256}.\nWe did not use gradient clipping for PJ-X.\nOur strongest FM model was trained for \\rev{30} epochs, using the Adam optimizer with a learning rate of 0.000\\rev{2}, a batch size of 128, and gradient clipping of 0.1. All other hyperparameters were taken from~\\cite{hukpark2018MultimodalExplanationsJustifying,wu2019FaithfulMultimodalExplanation}.\n\n\n\\myparagraph{Evaluation metrics.}\nTo evaluate the quality of the generated explanations, we use the standard natural language generation metrics BLEU~\\cite{papineni2001BLEUMethodAutomatic}, METEOR~\\cite{banerjee2005METEORAutomaticMetric}, ROUGE-L~\\cite{lin2004ROUGEPackageAutomatic} and CIDEr~\\cite{vedantam2015CIDErConsensusbased}.\nBy design, there is no correct explanation that can justify a wrong answer. \nWe follow~\\cite{kayser2021vil} and \\rev{report the quality of the generated explanations for the subset of correctly answered questions}.\n\n\n\n\\subsection{Evaluating explanations generated by state-of-the-art methods}\\label{sec:clevrx-results}\nIn this section, we present quantitative results for generating explanations for the CLEVR-X dataset (Table~\\ref{tab:clevr-results}).\nThe random words baseline exhibits weak explanation performance for all NLG metrics on CLEVR-X.\nAdditionally, the random answering accuracy is very low at 3.6\\%.\nThe results are similar on VQA-X\\xspace{} and e-SNLI-VE\\xspace{}.\nThe random explanations baseline achieves stronger explanation results on all three datasets, but is still significantly worse than the trained models.\nThis confirms that, even with a medium-sized answer space (28 options) and a small vocabulary (96 words), it is not possible to achieve good scores on our dataset using a trivial approach.\n\nWe observed that the PJ-X model yields a significantly stronger performance on CLEVR-X in terms of the NL\\rev{G} metrics for the generated explanations compared to the FM model, with METEOR scores of \\rev{58.9} and \\rev{52.5} for PJ-X and FM respectively. Across all explanation metrics, the scores on the VQA-X and e-SNLI-VE datasets are in a lower range than those on CLEVR-X. For PJ-X, we obtain a CIDEr score of \\rev{639.8} on CLEVR-X and \\rev{82.7} and \\rev{72.5} on VQA-X and e-SNLI-VE\\@. This can be attributed to the smaller vocabulary and longer sentences, which allow $n$-gram based metrics (e.g.\\ BLEU) to match parts of sentences more easily.\n\n\\rev{In contrast to the explanation generation performance, the FM model is better at answering questions than PJ-X on CLEVR-X with an answering accuracy of 80.3\\% for FM compared to 63.0\\% for PJ-X.}\n\\rev{Compared to recent models tuned to the CLEVR task, the answering performances of PJ-X and FM do not seem very strong. However, the PJ-X backbone MCB~\\cite{fukui2016MultimodalCompactBilinear} (which is crucial for the answering performance) preceded the publication of the CLEVR dataset.\nA version of the MCB backbone (CNN+LSTM+MCB in the CLEVR publication~\\cite{johnson2017CLEVRDiagnosticDataset}) achieved an answering accuracy of 51.4\\% on CLEVR~\\cite{johnson2017CLEVRDiagnosticDataset}, whereas PJ-X is able to correctly answer 63\\% of the questions.\nThe strongest model discussed in the initial CLEVR publication (CNN+LSTM+SA in~\\cite{johnson2017CLEVRDiagnosticDataset}) achieved an answering accuracy of 68.5\\%.\n}\n\n\n\\input{sections\/body\/tables\/clevr_results}\n\n\\subsection{Analyzing results on CLEVR-X by question and answer types}\\label{sec:type-results}\n\nIn Fig.~\\ref{fig:clevrx-answer-question-type-full-results-and-metrics-with-different-gt} (left and middle), we present the performance for PJ-X on CLEVR-X for the 9 question and 3 answer types. The explanation results for samples which require counting abilities (counting answers) are lower than those for attribute answers (57.3 vs.\\ 63.3).\nThis is in line with prior findings that VQA models struggle with counting problems~\\cite{Trott2018InterpretableCF}.\n\\rev{The explanation quality for binary questions is even lower with a METEOR score of only 55.6.}\nThe generated explanations are of higher quality for easier question types; \\textit{zero-hop} questions yield a METEOR score of 64.9 compared to 62.1 for \\textit{three-hop} questions. \nIt can also be seen that \\textit{single-or} questions are harder to explain than \\textit{single-and} questions.\nThese trends can be observed across all NL\\rev{G} explanation metrics.\n\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{subfigure}[t]{0.6775\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{figures\/types\/question_answer_types.pdf}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[t]{0.2825\\textwidth}\n        \\centering\n    \\includegraphics[width=\\linewidth,trim=4mm 4mm 4mm 4mm,clip]{figures\/average_gt_perf\/average_gt_perf.pdf}\n    \\end{subfigure}\n    \\hfill\n    \\caption{%\n    Explanation generation results for PJ-X on the CLEVR-X test set according to question (left) and answer (middle) types \\rev{compared to the overall explanation quality}. Easier types yield higher METEOR scores.\n    NLG metrics using different numbers of ground-truth explanations on the CLEVR-X test set (right). CIDEr converges faster than the other NLG metrics.}%\n    \\label{fig:clevrx-answer-question-type-full-results-and-metrics-with-different-gt}%\n\\end{figure}\n\n\n\\subsection{Influence of using different numbers of ground-truth explanations}\\label{sec:length-results}\nIn this section, we \nstudy the influence of using multiple ground-truth explanations for evaluation on the behavior of the NL\\rev{G} metrics. \nThis gives insights about whether the metrics can correctly rate a model's performance with a limited number of ground-truth explanations.\nWe set an upper bound $k$ on the number of explanations used and randomly sample $k$ explanations if a test sample has more than $k$ explanations for $k \\in\\{1,2,\\dots,10\\}$.\nFigure~\\ref{fig:clevrx-answer-question-type-full-results-and-metrics-with-different-gt} (right) shows the NL\\rev{G} metrics (normalized with the maximum value for each metric on the test set for all ground-truth explanations) for the PJ-X model depending on the average number of ground-truth references used on the test set.\n\nOut of the four metrics, BLEU-4 converges the slowest, requiring close to 3 ground-truth explanations to obtain a relative metric value of 95\\%. Hence, BLEU-4 might not be able to reliably predict the explanation quality on the e-SNLI-VE dataset which has only one explanation for each test sample.\nCIDEr converges faster than ROUGE and METEOR, and achieves 95.7\\% of its final value with only one ground-truth explanation.\n\\rev{This could be caused by the fact, that CIDEr utilizes a tf-idf weighting scheme for different words, which is built from all reference sentences in the subset that the metric is computed on. This allows CIDEr to be more sensitive to important words (e.g.\\ attributes and shapes) and to give less weight, for instance, to stopwords, such as \\enquote{the}.}\nThe VQA-X and e-SNLI-VE datasets contain much lower average numbers of explanations for each dataset sample (1.4 and 1.0). Since there could be many more possible explanations for samples in those datasets that describe different aspects than those mentioned in the ground truth, automated metric may not be able to correctly judge a prediction \\rev{even if it is correct and faithful w.r.t.\\ to the image and question.\n}\n\n\n\n\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{subfigure}[t]{0.31\\textwidth}\n        \\centering\n        \\scriptsize\n        \\caption*{\\textbf{Question}: How many tiny red things are the same material as the big sphere?}\n        \\includegraphics[width=.75\\linewidth]{figures\/qualitative_examples\/CLEVR_val_014997.png}\n        \\caption*{%\n        \\textbf{GT Answer $\\mid$ Explanation}:\\\\\n        \\rev{1} $\\mid$ The tiny red metal block has the same material as a big sphere.\\\\\n        \\textbf{Pred. Answer $\\mid$ Expl.}\\\\\n        \\rev{1} $\\mid$ \\rev{There is the tiny red metal block which has the identical material as a big sphere.}\\\\\n        \\rev{\\textbf{B4 \/ M \/ RL \/ C:}\\\\100.0 \/ 100.0 \/ 100.0 \/ 744.0}\n        \n        }%\n       \\label{fig:clevrx-qualitative-example-1}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[t]{0.31\\textwidth}\n        \\centering\n        \\caption*{\\textbf{Question}: The cylinder has what size?\\\\}\n        \\includegraphics[width=.75\\linewidth]{figures\/qualitative_examples\/CLEVR_val_000079.png}\n        \\caption*{%\n        \\textbf{GT Answer $\\mid$ Explanation}:\\\\\n        Small $\\mid$ The cylinder is small.\\\\\\\\\\\\\n        \\textbf{Pred. Answer $\\mid$ Expl.}\\\\\n        Small $\\mid$ \\rev{The cylinder is tiny}.\\\\\\\\\\\\\n        \\rev{\\textbf{B4 \/ M \/ RL \/ C:}\\\\100.0 \/ 100.0 \/ 100.0 \/ 462.4}\n        }%\n        \\label{ffig:clevrx-qualitative-example-2}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[t]{0.31\\textwidth}\n        \\centering\n        \\caption*{\\textbf{Question}: Are there any small matte cubes?\\\\}\n        \\includegraphics[width=.75\\linewidth]{figures\/qualitative_examples\/CLEVR_val_000100.png}\n        \\caption*{%\n        \\textbf{GT Answer $\\mid$ Explanation}:\\\\\n        No $\\mid$ There are no small matte cubes.\\\\\\\\\n        \\textbf{Pred. Answer $\\mid$ Expl.}\\\\\n        Yes $\\mid$ There is a small matte cube.\\\\\\\\\n        \\rev{\\textbf{B4 \/ M \/ RL \/ C:}\\\\0.0 \/ 76.9 \/ 57.1 \/ 157.1}\n        }%\n        \\label{ffig:clevrx-qualitative-example-3}\n    \\end{subfigure}\n    \\caption{Examples for answers and explanations generated with the PJ-X framework on the CLEVR-X dataset, showing correct answer predictions (left, middle) and a failure case (right). \\rev{The NLG metrics obtained with the explanations for the correctly predicted answers are high compared to those for the explanation corresponding to the wrong answer prediction.}\n    }%\n    \\label{fig:clevrx-qualitative-examples}%\n\\end{figure}\n\n\n\\subsection{Qualitative explanation generation results}\\label{sec:quali-results}\nWe show examples for explanations generated with the PJ-X framework on CLEVR-X in Fig.~\\ref{fig:clevrx-qualitative-examples}. As can be seen across the three examples presented, PJ-X generates high-quality explanations which closely match the ground-truth explanations.\n\nIn the left-most example in Fig.~\\ref{fig:clevrx-qualitative-examples}, we can observe slight variations in grammar when comparing the generated explanation to the ground-truth explanation. However, the content of the generated explanation corresponds to the ground truth. \nFurthermore, some predicted explanations differ from the ground-truth explanation in the use of another synonym for a predicted attribute. For instance, in the middle example in Fig.~\\ref{fig:clevrx-qualitative-examples}, the ground-truth explanation describes the size of the cylinder as ``small'', whereas the predicted explanation uses the equivalent attribute ``tiny''.\nIn contrast to other datasets, the set of ground-truth explanations for each sample in CLEVR-X contains these variations. Therefore, the automated NLG metrics \\rev{do} not decrease when such variations are found in the predictions. \\rev{For the first and second example, PJ-X obtains the highest possible explanation score (100.0) in terms of the BLEU-4, METEOR, and ROUGE-L metrics.}\n\nWe show a failure case where PJ-X predicted the wrong answer in Fig.~\\ref{fig:clevrx-qualitative-examples}~(right). The generated answer-explanation pair shows that the predicted explanation is consistent with the wrong answer prediction and does not match the input question-image pair.\n\\rev{The NLG metrics for this case are significantly weaker with a BLEU-4 score of $0.0$, as there are no matching $4$-grams between the prediction and the ground truth.}\n\n\n\n\\section{Introduction}\n\nExplanations for automatic decisions form a crucial step towards increasing transparency and human trust in deep learning systems. In this work, we focus on natural language explanations in the context of vision-language tasks.\n\\blfootnote{A version of the contribution has been accepted for publication, after peer review but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record will be available online at: \\url{https:\/\/doi.org\/10.1007\/978-3-031-04083-2_5}.}\n\nIn particular, we consider the\nvision-language task of Visual Question Answering (VQA) which consists of answering a question about an image. This requires multiple skills, such as visual perception, text understanding, and cross-modal reasoning in the visual and language domains. A natural language explanation for a given answer allows a better understanding of the reasoning process for answering the question and adds transparency. \nHowever, it is challenging to formulate what comprises a good textual explanation in the context of VQA involving natural images.\n\nExplanation datasets commonly used in the context of VQA, such as the VQA-X dataset~\\cite{hukpark2018MultimodalExplanationsJustifying} or the e-SNLI-VE dataset~\\cite{Do2020eSNLIVE20CV,kayser2021vil} for visual entailment,\ncontain explanations of widely varying quality since \nthey are generated by humans. The ground-truth explanations in VQA-X and e-SNLI-VE can range from statements that merely describe an image to explaining the reasoning about the question and image involving prior information, such as common knowledge. One example for a ground-truth explanation in VQA-X that requires prior knowledge about car designs from the 1950s can be seen in Fig.~\\ref{fig:teaser-dataset-comparison}. The e-SNLI-VE dataset contains numerous explanation samples which consist of repeated statements (``x because x'').\nSince existing explanation datasets for vision-language tasks contain immensely varied explanations, it is challenging to perform a structured analysis of strengths and weaknesses of existing explanation generation methods.\n\n\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{subfigure}[t]{0.31\\textwidth}\n        \\centering\n        \\scriptsize\n        \\caption*{%\n        {\\bf VQA-X} \\\\ \\textbf{Question}: Does this scene look like it could be from the early 1950s?}\n        \\includegraphics[width=0.75\\linewidth,trim={0 3mm 0 1.5mm},clip]{figures\/teaser\/COCO_val2014_000000416220.jpg}\n        \\caption*{\\textbf{Answer $\\mid$ Explanation:}\\\\\n        Yes $\\mid$ The photo is in black and white and the cars are all classic designs from the 1950s\n}%\n       \\label{fig:vqax-example}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[t]{0.31\\textwidth}\n       \\centering\n        \\caption*{{\\bf e-SNLI-VE} \\\\ \\textbf{Hypothesis}: A woman is holding a child.\\\\}\n        \\includegraphics[width=0.75\\linewidth,trim={0 6mm 0 6mm},clip]{figures\/teaser\/3047751696.jpg}\n        \\caption*{\\textbf{Answer $\\mid$ Explanation:}\\\\\n        Entailment $\\mid$ If a woman holds a child she is holding a child.}%\n        \\label{fig:esnlive-examples}\n    \\end{subfigure}\n     \\hfill\n    \\begin{subfigure}[t]{0.31\\textwidth}\n        \\centering\n        \\caption*{{\\bf CLEVR-X} \\\\\\textbf{Question}: There is a purple metallic ball; what number of cyan objects are right of it?}\n        \\includegraphics[width=0.75\\linewidth]{figures\/teaser\/CLEVR_val_005182.png}\n        \\caption*{\\textbf{Answer $\\mid$ Explanation:}\\\\\n        1 $\\mid$ There is a cyan cylinder which is on the right side of the purple metallic ball.}%\n       \\label{fig:clevrx-examples}\n    \\end{subfigure}\n  \n    \\caption{Comparing examples from the VQA-X\\xspace{} (left), e-SNLI-VE\\xspace{} (middle), and CLEVR-X\\xspace{} (right) datasets. The explanation in VQA-X\\xspace{} requires prior knowledge (about cars from the 1950s), e-SNLI-VE\\xspace{} argues with a tautology, and our CLEVR-X\\xspace{} only uses abstract visual reasoning.\n    \\label{fig:teaser-dataset-comparison}\n\\end{figure}\n\nIn order to fill this gap, we propose the novel, diagnostic CLEVR-X dataset for visual reasoning with natural language explanations. It extends the synthetic CLEVR~\\cite{johnson2017CLEVRDiagnosticDataset} dataset through the addition of structured natural language explanations for each question-image pair. An example for our proposed CLEVR-X dataset is shown in Fig.~\\ref{fig:teaser-dataset-comparison}.\nThe synthetic nature of the CLEVR-X dataset results in several advantages over datasets that use human explanations.\nSince the explanations are synthetically constructed from the underlying scene graph, the explanations are \\textit{correct} and do not require auxiliary prior knowledge. The synthetic textual explanations do not suffer from errors that get introduced with human explanations. Nevertheless, the explanations in the CLEVR-X dataset are human parsable as demonstrated in the human user study that we conducted. Furthermore, the explanations contain all the information that is necessary to answer a given question about an image without seeing the image. This means that the explanations are \\textit{complete} with respect to the question about the image.\n\n\nThe CLEVR-X dataset allows for detailed diagnostics of natural language explanation generation methods in the context of VQA\\@. For instance, it contains a wider range of question types than other related datasets. We provide baseline performances on the CLEVR-X dataset using recent frameworks for natural language explanations in the context of VQA\\@. Those frameworks are jointly trained to answer the question and provide a textual explanation. Since the question family, question complexity (number of reasoning steps required), and the answer type (binary, counting, attributes) is known for each question and answer, the results can be analyzed and split according to these groups. In particular, the challenging counting problem~\\cite{Trott2018InterpretableCF}, which is not well-represented in the VQA-X dataset, can be studied in detail on CLEVR-X.\nFurthermore, our dataset contains multiple ground-truth explanations for each image-question pair. These capture a large portion of the space of correct explanations which allows for a thorough analysis of the influence of the number of ground-truth explanations used on the evaluation metrics.\nOur approach of constructing textual explanations from a scene graph yields a great resource which could be extended to other datasets that are based on scene graphs, such as the CLEVR-CoGenT dataset.\n\n\n\nTo summarize, we make the following four contributions:\n\\begin{enumerate*}[label=(\\arabic*)]\n    \\item We introduce the CLEVR-X dataset with natural language explanations for Visual Question Answering; \n    \\item We confirm that the CLEVR-X dataset consists of correct explanations that contain sufficient relevant information to answer a posed question by conducting a user study;\n    \\item We provide baseline performances with two state-of-the-art methods that were proposed for generating textual explanations in the context of VQA;\n    \\item We use the CLEVR-X dataset for a detailed analysis of the explanation generation performance for different subsets of the dataset and to better understand the metrics used for evaluation.\n\\end{enumerate*}\n\n\n\\section{Related work}\nIn this section, we discuss several themes in the literature that relate to our work, namely \\textit{Visual Question Answering}, \\textit{Natural language explanations (for vision-language tasks)}, and the \\textit{CLEVR dataset}. \n\n\\myparagraph{Visual Question Answering (VQA).} The VQA~\\cite{antol2015vqa} task has been addressed by several works that apply attention mechanisms to text and image features~\\cite{yang2016stacked,xu2016ask,zhu2016visual7w,shih2016look,fukui2016MultimodalCompactBilinear}.\nHowever, recent works observed that the question-answer bias in common VQA datasets can be exploited in order to answer questions without leveraging any visual information~\\cite{agrawal2016analyzing,agrawal2018don,johnson2017CLEVRDiagnosticDataset,zhang2016yin}.\nThis has been further investigated in more controlled dataset settings, such as the CLEVR~\\cite{johnson2017CLEVRDiagnosticDataset}, VQA-CP~\\cite{agrawal2018don}, and GQA~\\cite{hudson2019GQANewDataset} datasets. In addition to a controlled dataset setting, our proposed CLEVR-X dataset contains natural language explanations that enable a more detailed analysis of the reasoning in the context of VQA\\@.\n\n\n\n\\myparagraph{Natural language explanations.}\nDecisions made by neural networks can be visually explained with visual attribution that is determined by introspecting trained networks and their features~\\cite{simonyan2013deep,zeiler2014visualizing,selvaraju2019GradCAMVisual,bach2015pixel,zhang2018top}, by using input perturbations~\\cite{petsiuk2018rise,fong2019understanding,fong_iccv_2017}, or by training a probabilistic feature attribution model along with a task-specific CNN~\\cite{kim2021keep}. Complementary to visual explanations methods that tend to not help users distinguish between correct and incorrect predictions~\\cite{kim2021hive}, natural language explanations have been investigated for a variety of tasks, such as fine-grained visual object classification~\\cite{hendricks2018GroundingVisualExplanations,hendricks2018Generatingcounterfactualexplanations}, or self-driving car models~\\cite{kim2018textual}. The requirement to ground language explanations in the input image can prevent shortcuts, such as relying on dataset statistics or referring to instance attributes that are not present in the image.\nFor a comprehensive overview of research on explainability and interpretability, we refer to recent surveys~\\cite{Barredo_Arrieta_XAI_2020,brundage2020toward,gilpin2018explaining}.\n\n\n\\myparagraph{Natural language explanations for vision-language tasks.}\nMultiple datasets for natural language explanations in the context of vision-language tasks have been proposed, such as the VQA-X~\\cite{hukpark2018MultimodalExplanationsJustifying}, VQA-E~\\cite{li2018vqa}, and e-SNLI-VE datasets~\\cite{kayser2021vil}.   VQA-X~\\cite{hukpark2018MultimodalExplanationsJustifying} augments a small subset of the VQA~v2~\\cite{goyal2017MakingVVQA} dataset for the Visual Question Answering task with human explanations.\nSimilarly, the VQA-E dataset~\\cite{li2018vqa} extends the VQA~v2 dataset by sourcing explanations from image captions. However, the VQA-E explanations resemble image descriptions and do not provide satisfactory justifications whenever prior knowledge is required~\\cite{li2018vqa}. The e-SNLI-VE~\\cite{kayser2021vil,Do2020eSNLIVE20CV} dataset combines human explanations from e-SNLI~\\cite{Camburu2018eSNLINL} and the image-sentence pairs for the Visual Entailment task from SNLI-VE~\\cite{Xie2019VisualEA}.\nIn contrast to the VQA-E, VQA-X, and e-SNLI-VE datasets which consist of human explanations or image captions, our proposed dataset contains systematically constructed explanations derived from the associated scene graphs.\nRecently, several works have aimed at generating natural language explanations for vision-language tasks~\\cite{hukpark2018MultimodalExplanationsJustifying,wu2019FaithfulMultimodalExplanation,wu2020ImprovingVQAits,Marasovi2020NaturalLR,patro2020robust,kayser2021vil}.\nIn particular, we use the PJ-X~\\cite{hukpark2018MultimodalExplanationsJustifying} and FM~\\cite{wu2019FaithfulMultimodalExplanation} frameworks to obtain baseline results on our proposed CLEVR-X dataset.\n\n \n\n\\myparagraph{The CLEVR dataset.}\nThe CLEVR dataset~\\cite{johnson2017CLEVRDiagnosticDataset} was proposed as a diagnostic dataset to inspect the visual reasoning of VQA models.\nMultiple frameworks have been proposed to address the CLEVR task~\\cite{hudson2018CompositionalAttentionNetworks,perezFiLMVisualReasoning,hudson2019learning,johnson2017inferring,Suarez2018DDRprogAC,Shi2019ExplainableAE}. \nTo add explainability, the XNM model~\\cite{Shi2019ExplainableAE} adopts the scene graph as an inductive bias which enables the visualization of the reasoning based on the attention on the nodes of the graph.\nThere have been numerous dataset extensions for the CLEVR dataset, for instance to measure the generalization capabilities of models pre-trained on CLEVR (CLOSURE~\\cite{bahdanau2019closure}), to evaluate object detection and segmentation (CLEVR-Ref+~\\cite{liu2019clevr}), or to benchmark visual dialog models (CLEVR dialog~\\cite{kottur2019clevr}).\nThe Compositional Reasoning Under Uncertainty (CURI) benchmark uses the CLEVR renderer to construct a test bed for compositional and relational learning under uncertainty~\\cite{vedantam2021curi}. \\cite{Holzinger2021KANDINSKYPatternsA} provide an extensive survey of further experimental diagnostic benchmarks for analyzing explainable machine learning frameworks along with proposing the KandinskyPATTERNS benchmark that contains synthetic images with simple 2-dimensional objects. It can be used for testing the quality of explanations and concept learning.\nAdditionally,~\\cite{arras2020ground} proposed the CLEVR-XAI-simple and CLEVR-XAI-complex datasets which provide ground-truth segmentation information for heatmap-based visual explanations.\nOur CLEVR-X augments the existing CLEVR dataset with explanations, but in contrast to (heatmap-based) visual explanations, we focus on natural language explanations.\n\n\n\n\n\n\\section{First Section}\n\\subsection{A Subsection Sample}\nPlease note that the first paragraph of a section or subsection is\nnot indented. The first paragraph that follows a table, figure,\nequation etc. does not need an indent, either.\n\nSubsequent paragraphs, however, are indented.\n\n\\subsubsection{Sample Heading (Third Level)} Only two levels of\nheadings should be numbered. Lower level headings remain unnumbered;\nthey are formatted as run-in headings.\n\n\\paragraph{Sample Heading (Fourth Level)}\nThe contribution should contain no more than four levels of\nheadings. Table~\\ref{tab1} gives a summary of all heading levels.\n\n\\begin{table}\n\\caption{Table captions should be placed above the\ntables.}\\label{tab1}\n\\begin{tabular}{|l|l|l|}\n\\hline\nHeading level &  Example & Font size and style\\\\\n\\hline\nTitle (centered) &  {\\Large\\bfseries Lecture Notes} & 14 point, bold\\\\\n1st-level heading &  {\\large\\bfseries 1 Introduction} & 12 point, bold\\\\\n2nd-level heading & {\\bfseries 2.1 Printing Area} & 10 point, bold\\\\\n3rd-level heading & {\\bfseries Run-in Heading in Bold.} Text follows & 10 point, bold\\\\\n4th-level heading & {\\itshape Lowest Level Heading.} Text follows & 10 point, italic\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\noindent Displayed equations are centered and set on a separate\nline.\n\\begin{equation}\nx + y = z\n\\end{equation}\nPlease try to avoid rasterized images for line-art diagrams and\nschemas. Whenever possible, use vector graphics instead (see\nFig.~\\ref{fig1}).\n\n\\begin{figure}\n\\includegraphics[width=\\textwidth]{fig1.eps}\n\\caption{A figure caption is always placed below the illustration.\nPlease note that short captions are centered, while long ones are\njustified by the macro package automatically.} \\label{fig1}\n\\end{figure}\n\n\\begin{theorem}\nThis is a sample theorem. The run-in heading is set in bold, while\nthe following text appears in italics. Definitions, lemmas,\npropositions, and corollaries are styled the same way.\n\\end{theorem}\n\\begin{proof}\nProofs, examples, and remarks have the initial word in italics,\nwhile the following text appears in normal font.\n\\end{proof}\nFor citations of references, we prefer the use of square brackets\nand consecutive numbers. Citations using labels or the author\/year\nconvention are also acceptable. The following bibliography provides\na sample reference list with entries for journal\narticles~\\cite{ref_article1}, an LNCS chapter~\\cite{ref_lncs1}, a\nbook~\\cite{ref_book1}, proceedings without editors~\\cite{ref_proc1},\nand a homepage~\\cite{ref_url1}. Multiple citations are grouped\n\\cite{ref_article1,ref_lncs1,ref_book1},\n\\cite{ref_article1,ref_book1,ref_proc1,ref_url1}.\n\\subsection{User study on explanation completeness and relevance}\\label{sec:data-userstudy}\n\n\nIn this section, we describe our user study for evaluating the completeness and relevance of the generated ground-truth explanations in the CLEVR-X\\xspace{} dataset.\n\\rev{%\nWe wanted to verify whether humans are successfully able to parse the synthetically generated textual explanations and to select complete and relevant explanations.\nWhile this is obvious for easier explanations like \\enquote{There is a blue sphere.}, it is less trivial for more complex explanations such as \\enquote{There are two red cylinders in front of the green cube that is to the right of the tiny ball.}\nThus, strong human performance in the user study indicates that the sentences are parsable by humans.\n}\n\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{subfigure}[t]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[height=8.3cm]{figures\/user_study\/completeness.png}%\n        \\label{fig:user-study-example-completeness}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[t]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[height=8.3cm]{figures\/user_study\/relevance.png}%\n        \\label{fig:user-study-example-relevance}\n    \\end{subfigure}%\n    \\caption{Two examples from our user study to evaluate the completeness (left) and relevance (right) of natural language explanations in the CLEVR-X dataset.}\\label{fig:user-study-example}\n\\end{figure}\n\nWe performed our user study using Amazon Mechanical Turk (MTurk). It consisted of two types of Human Intelligence Tasks (HITs).\nEach HIT was made up of\n\\begin{enumerate*}[label=(\\arabic*)]\n\\item An explanation of the task;\n\\item A non-trivial example, where the correct answers are already selected;\n\\item A CAPTCHA~\\cite{Ahn2003CAPTCHAUH} to verify that the user is human;\n\\item The problem definition consisting of a question and an image;\n\\item A \\rev{user qualification step}, for which the user has to correctly answer a question about an image. \\rev{This ensures that the user is able to answer the question in the first place, a necessary condition to participate in our user study};\n\\item Two explanations from which the user needs to choose one.\n\\end{enumerate*}\nExample screenshots of the user interface for the user study are shown in Fig.~\\ref{fig:user-study-example}.\n\nFor the two different HIT types, we randomly sampled 100 explanations from each of the 9 question types, resulting in a total of 1800 samples for the completeness and relevance tasks. \nFor each task sample, we requested 3 different MTurk workers based in the US (with high acceptance rate of $>95\\%$ and over 5000 accepted HITs).\nA total of 78 workers participated in the completeness HITs. They took on average 144.83 seconds per HIT\\@.\nThe relevance task was carried out by 101 workers which took on average 120.46 seconds per HIT\\@.\nIn total, 134 people participated in our user study.\nIn the following, we describe our findings regarding the completeness and relevance of the CLEVR-X explanations in more detail.\n\n\n\n\n\\myparagraph{Explanation completeness.}\nIn the first part of the user study, we evaluated whether human users are able to determine if the ground-truth explanations in the CLEVR-X\\xspace{} dataset are complete (and also correct).\nWe presented the MTurk workers with an image, a question, and two explanations. \nAs can be seen in Fig.~\\ref{fig:user-study-example} (left), a user had to first select the correct answer (\\textit{yes}) before deciding which of the two given explanations was complete.\nBy design, one of the explanations presented to the user was the complete one from the CLEVR-X dataset and the other one was a modified version for which at least one necessary object had been removed. As simply deleting an object from a textual explanation could lead to grammar errors, we re-generated the explanations after removing objects from the tracing results. This resulted in incomplete, albeit grammatically correct, explanations.\n\n\nTo evaluate the ability to determine the completeness of explanations, we measured the accuracy of selecting the complete explanation.\nThe human participants obtained an average accuracy of \\rev{92.19}\\%, confirming that  complete explanations which mention all objects necessary to answer a given question were preferred over incomplete ones. \nThe performance was weaker for complex question types, such as \\textit{compare-integer} and \\textit{comparison} with accuracies of only 77.00\\% and 83.67\\% respectively, compared to the easier \\textit{zero-hop} and \\textit{one-hop} questions with accuracies of 100\\% and 98.00\\% respectively. \n\nAdditionally, there were huge variations in performance across different participants of the completeness study (Fig.~\\ref{fig:user-study-worker-answer-accuracies-and-work-time}~(top left)), with the majority performing very well ($>$97\\% answering accuracy) for most question types. For the \\emph{compare-integer}, \\emph{comparison} and \\emph{single or} question types, some workers exhibited a much weaker performance with answering accuracies as low as $0\\%$.\nThe average turnaround time shown in Fig.~\\ref{fig:user-study-worker-answer-accuracies-and-work-time}~(bottom left) confirms that easier question types required less time to be solved than more complex question types, such as \\emph{three hop} and \\emph{compare integer} questions. Similar to the performance, the work time varied greatly between different users.\n\n\\input{sections\/body\/tables\/user_study_results}\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth,trim=4mm 4mm 4mm 4mm,clip]{figures\/user_study\/boxplots\/joint_figure.pdf}\n    \\caption{%\n    \\rev{Average answering accuracies for each worker (top) and average work time (bottom) for the user study (left: completeness, right: relevance). The boxes indicate the mean as well as lower and upper quartiles, the lines extend 1.5 interquartile ranges of the lower and upper quartile. All other values are plotted as diamonds.\n   \n    }\n    }%\n    \\label{fig:user-study-worker-answer-accuracies-and-work-time}%\n\\end{figure}\n\n\n\\myparagraph{Explanation relevance.}\nIn the second part of our user study, we analyzed if humans are able to identify explanations which are relevant for a given image. For a given question-image pair, the users had to first select the correct answer. Furthermore, they were provided with a correct explanation and another randomly chosen explanation from the same question family (that did not match the image). The task consisted of selecting the correct explanation that matched the image and question content. Explanation~1 in the example user interface shown in Fig.~\\ref{fig:user-study-example} (right) was the relevant one, since Explanation 2 does not match the question and image. \n\nThe participants of our user study were able to determine which explanation matched the given question-image example with an average accuracy of 92.52\\%. Again, the performance for complex question types was weaker than for easier questions. The difficulty of the question influences the accuracy of detecting the relevant explanation, since this task first requires understanding the question. Furthermore, complex questions tend to be correlated with complex scenes that contain many objects which makes the user's task more challenging. The accuracy for \\textit{three-hop} questions was 89.00\\% compared to 99.67\\% for \\textit{zero-hop} questions. For \\textit{compare-integer} and \\textit{comparison} questions, the users obtained accuracies of 83.67\\% and 87.33\\% respectively, which is significantly lower than the overall average accuracy.\n\n\n\nWe analyzed the answering accuracy per worker in Fig.~\\ref{fig:user-study-worker-answer-accuracies-and-work-time}~(top right).\nThe performance varies greatly between workers, with the majority performing very well ($>$90\\% answering accuracy) for most question types.\nSome workers showed much weaker performance with answering accuracies as low as $0\\%$ (e.g.\\ for \\emph{compare-integer} and \\emph{single or} questions).\nFurthermore, the distribution of work time for the relevance task is shown in Fig.~\\ref{fig:user-study-worker-answer-accuracies-and-work-time}~(bottom right). The turnaround times for each worker exhibit greater variation on the completeness task (bottom left) compared to the relevance task (bottom right).\nThis might be due to the nature of the different tasks. For the completeness task, the users need to check if the explanation contains all the elements that are necessary to answer the given question. The relevance task, on the other hand, can be solved by detecting a single non-relevant object to discard the wrong explanation.\n\nOur user study confirmed that humans are able to parse the synthetically generated natural language explanations in the CLEVR-X dataset. Furthermore, the results have shown that users prefer complete and relevant explanations in our dataset over corrupted samples.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAs is well known, the study of null geodesics at a given \nspacetime is not only \nrelevant\nfrom a theoretical point of view, but also from a practical perspective.\nTheir analysis, in addition to information about the causal structure of the \nspacetime, can also establish observable consequences of various \nastrophysical \nphenomena \\cite{stuchlik,cov,vsoc,fernando12,vo13,vv13,garcia13}.\n\nThe role of so-called circular null geodesics is not trivial. \nIn some situations, these geodesics can allow the existence of \nwhat is known as a photon surface, which, in the context of \nfour-dimensional spacetimes, is a \nthree-dimensional non-spacelike manifold $\\mathcal{P}$ , such \nthat every null geodesic whose\ntangent vector $l^a$ at a given point $p\\in \\mathcal{P}$ is contained in  \nthe tangent space $T_p\\mathcal{P}$ of $\\mathcal{P}$ at $p$, always remains \nin $\\mathcal{P}$. In particular, in \nthe case of spherically symmetric spacetimes,\na photon sphere can be defined as a $SO(3)\\times\\mathcal{R}$-invariant photon surface. \nFor rigorous \ndefinitions see\\cite{Ellis01}.\n\nThe study of photon spheres in the case of spherical \nsymmetry and circular null geodesics \nin axial symmetric spacetimes\n is important in astrophysics for several reasons: \n\\begin{enumerate}\n\\item[a)] They appear explicitly in the study of \nrelativistic images produced by black \nholes in the\nstrong lensing regime\\cite{Bozza02, Bozza10}.\n\\item[b)] They play an important role in the analysis of quasinormal \nmodes in black hole perturbations\\cite{Cardoso2009, Decanini10}.\n\\item[c)] They determine the shadow of black holes, or \n(which is the same), how they \nlook to outside observers\\cite{Atamurotov13}.\n\\item[d)] In the case of hairy black holes, they determine \na lower bound on the size of the hair\\cite{Hod11b}.\n\\item[e)] They allow a link between points a) and \nb), or more precisely between gravitational waves and \nlensing, as recently shown in\n\\cite{Stefanov10, Wei14}.\n\\end{enumerate}\nThey have also been analyzed in situations that do not contain black holes, as \nfor example, in Boson stars\\cite{Horvat}, where the authors showed that \nin certain configurations \nphoton spheres can occur, or in certain classes of regular metrics \nwith non-negative trace of the \nenergy-momentum tensor, where the existence of at least two \nphoton spheres was shown\\cite{Hod14}.\nRecently, a study of the thermodynamics of a quantum \nversion of photon spheres was presented\\cite{Baldiotti14}.\n\n\nAlthough in axially-symmetric spacetimes there is no notion yet of \nthe analog of photon spheres,\nthere are circular null geodesics on the equatorial plane, and they are \nalso useful for the\ndiscussion of the previous points. \n\nFor all these considerations, the characterization \nand localization of these particular surfaces or circular\nnull geodesics is relevant.\n\n\nRecently\\cite{Hod13}, Hod made some interesting observations \nabout this type of null \ngeodesics. In particular, in the framework of four-dimensional \ngeneral relativity (GR), he analyzed a general family of spherically \nsymmetric black holes,  which satisfy some natural asymptotic \nand energy conditions, finding \nan upper bound for the radius of photon spheres in terms of their ADM mass.\nIn the case of axial symmetry, he studied circular null geodesics on the \nequatorial plane of a \nKerr spacetime, finding that they are also the fastest way to circle a \nKerr black hole\\cite{Hod11} and conjecturing an expression for the \nminimum orbital time \nto circle any compact object in GR in terms of its mass. \nAs shown by Pradhan\\cite{Pradhan13}, the conclusion that \ncircular null geodesics are\nthe fastest way to circle black holes remains valid for the $n$-dimensional \nversion of the Kerr-Newman metric, namely, for charged Myers-Perry spacetimes. \n\n\nThe reason for and interest in the study of theories of\ngravity in higher dimensions is motivated by string\ntheory. As a possibility, the Einstein-Gauss-Bonnet (EGB) gravity\ntheory is selected by the low energy limit of the string\ntheory~\\cite{cuerdas1,cuerdas3}. In this theory,\ncorrective terms to Einstein gravity appear, which are quadratic in the\ncurvature of the spacetime. The effect of those Gauss-Bonnet terms is\nnontrivial for higher dimensions, so the theory of gravity, which\nincludes Gauss-Bonnet terms, is called Einstein-Gauss-Bonnet (EGB)\ngravity. \n\nEven without these Gauss-Bonnet corrections,\nthe $n$-dimensional version of GR is usually studied in \nastrophysical and \ntheoretical contexts, and in general it is also the case using \nother alternative gravitational\ntheories. For some works on these topics in EGB or pure GR see\n\\cite{Tsukamoto14, Cardoso2009, Cardoso2009b, Sadeghi14, Man14}.\n\nBecause of the evidence garnered of the importance of \ncircular null geodesics\nin the characterization of spherically and axially-symmetric \nblack holes, \nwe extend and generalize some of the Hod results to higher-dimensional \ngravitational theories. In particular we study photon spheres \nin $n$-dimensional GR and EGB theory and \ncircular null geodesics on the equatorial plane for generic \naxially-symmetric spacetimes.  \n\nThe article is organized as follows. In section II we review EGB theory, \nand prepare\nthe setting for the discussion of photon spheres.\nIn section III, we establish and prove two theorems which state \nupper bounds on the radii of\nphoton spheres for black holes in GR and EGB theories, in \nterms of their ADM mass. \nIn section IV, we analyze some of their implications, giving \na universal upper bound for the real \npart of quasinormal frequencies in the WKB limit and a lower \nbound for the first relativistic \nimage in the strong lensing regime.\nIn section V, we make some general comments on the relation between \ncircular null geodesics in \naxially-symmetric \nspacetimes and the fastest way to circle black holes. We also show, \nby using an explicit \ncounterexample, that a lower bound for the orbital period \nof circular null geodesics conjectured by Hod in the context of \nfour-dimensional GR\\cite{Hod11}, \ncannot be assumed to be valid also in alternative gravitational theories. \n\n\\section{Background and setting}\n\nThe action that describes Einstein-Gauss-Bonnet gravity coupled\nwith matter fields reads:\n\\begin{equation*}\nS = \\frac {1}{16\\pi } \\int d^nx \\sqrt {-g}  \\left[ R - 2\\Lambda +\n\\alpha (R_{a b c d } R^{a b c d } + \\right.\n\\end{equation*}\n\\begin{equation*}\n\\left. + R^2 - 4R_{a b } R^{a b} )\\right]+S_{\\text{matt}},\n\\end{equation*}\nwhere $S_{\\text{matt}}$ is the action associated with the matter\nfields, and $\\alpha $ is the Gauss-Bonnet coupling constant\nassociated in the string models, with the tension of these\nstrings. This constant introduces a length scale. In fact, the\ncorrections that this theory produces to GR, are noted at short\ndistances, given by the scale $l=\\sqrt{4 \\alpha}$.\n\nThe equations of motion resulting from $\\delta S=0$ are\n\\begin{eqnarray*}\n\\kappa T_{a b } &=& \\mathcal{G}_{a b}=G^{(0)}_{a b}+G^{(1)}_{a\nb}+G^{(2)}_{a b},\n\\end{eqnarray*}\nwhere $\\kappa=8\\pi G\/c^4$ is the gravitational constant, $T_{a b}$ \nis the energy-momentum tensor, \nrepresenting the matter-field distribution resulting from the variation \n$\\delta S_{\\text{matt}}\/\\delta g^{a b},$ and\n\\begin{eqnarray*}\nG^{(0)}_{a\nb}&=&\\Lambda g_{a b}  \\\\\n G^{(1)}_{a\nb}&=& R_{a b }-\\frac{1}{2}Rg_{a b}\\\\\nG^{(2)}_{a b}&=&-\\alpha \\left[\\frac {1}{2} g_{a b} (R_{ c j e\nk}R^{c\n j e k }-4R_{c j }R^{c j }+R^2) \\right. -   \\\\\n&-&\\left. 2RR_{a b}+4R_{a c}R^{c}_{b}+4R_{c j} R^{c j}_{ \\ \\ a\nb}-2R_{a c j e}R_{b}^{ \\ c j e} \\right ].\n\\end{eqnarray*}\n\nFrom now on, we will use unit where $c=1$ and we adopt $\\alpha$ \npositive since this\ncondition arises from the string theory. \nWe also assume asymptotically flat spacetimes, and therefore $\\Lambda=0$. \n\n\nLet us consider a spherically symmetric metric given by\n\\begin{equation}\\label{eq:metric}\nds^2=-e^{-2\\delta(r)}\\mu(r)dt^2+\\mu(r)^{-1}dr^2+r^2d\\Omega^2_{n-2},\n\\end{equation}\nwith $d\\Omega^2_{n-2}$\nbeing the metric of the $(n-2)$-sphere\n\\begin{equation}\nd\\Omega^2_{n-2} = d\\theta^2_1 + \\sum^{n-2}_{i=2}\\prod^{i-1}_{j=1}\n\\sin^{2}\\theta_j\\;d\\theta^2_i \\; ,\n\\end{equation}\nsolution to the Einstein-Gauss-Bonnet equations \nin higher dimensions. \nThe area of the unit $(n-2)$-sphere is given by \n\\begin{equation}\nS_{n-2}=\\frac{2\\pi^{\\frac{n-1}{2}}}{\\Gamma\\left(\\frac{n-1}{2}\\right)},\n\\end{equation} \nwith $\\Gamma(x)=\\int^\\infty_0 t^{x-1}e^{-t} dt$, the gamma function.\n\nThe Einstein-Gauss-Bonnet equations in terms of \nthe components of the \nenergy momentum tensor, \n$T^t_t=-\\rho$, $T^r_r=p_r$ and\n$T^{\\theta_i}_{\\theta_i}=p_\\bot$ reads\n\\begin{equation}\n\\begin{split}\\label{eq:rhogb}\n\\kappa\\rho=&-\\frac{1}{2r^2}(n-2)[r\\mu'+(n-3)(\\mu-1)]\\\\\n&+\\frac{\\widehat{\\alpha}}{2r^4}(n-2)(\\mu-1)[2r\\mu'+(n-5)(\\mu-1)],\n\\end{split}\n\\end{equation}\n\\begin{equation}\n\\begin{split}\\label{eq:prgb}\n\\kappa p_r=&-(n-2)\\left\\{\\frac{1}{2r^2}[2r\\mu\\delta'-r\\mu'-(n-3)(\\mu-1)]\\right.\\\\\n&\\left.-\\frac{\\widehat{\\alpha}}{2r^4}(\\mu-1)[-2r\\mu'+4r\\mu\\delta'-(\\mu-1)(n-5)]\\right\\},\n\\end{split}\n\\end{equation}\nwhere $\\widehat{\\alpha}=\\alpha (n-3)(n-4)$.\nWe have not written the angular-angular equations \nbecause we do not need them. Instead, \nwe will use the only nontrivial component\nof the energy-momentum conservation equation $\\nabla_aT^a_r=0$, which reads\n\\begin{equation}\\label{eq:conser}\np'_r=-\\frac{(e^{-2\\delta}\\mu)'}{2e^{-2\\delta}\\mu}(\\rho+p_r)+\\frac{n-2}{r}(p_\\bot-p_r).\n\\end{equation}\nFrom eqs.(\\ref{eq:rhogb}) and (\\ref{eq:prgb}) it follows that\n\\begin{equation}\\label{eq:delta}\n\\mu[2\\alpha(\\mu-1)-r^2]\\delta'=\\kappa\\frac{r^3}{(n-2)}(\\rho+p_r).\n\\end{equation}\nWe are interested in regular black holes.\nIn particular, we assume that at the horizon $r_H$, $\\mu(r)$ and $\\delta(r)$ satisfy\n\\begin{eqnarray}\\label{crh}\n\\mu(r_H)=0,\\;\\;\\;\\;\\;  \\mu'(r_H)\\geq 0,\\\\\n\\delta(r_H)\\leq\\infty,\\;\\;\\;\\;\\;  \\delta'(r_H)\\leq\\infty.\n\\end{eqnarray}\nThese conditions, together with (\\ref{eq:delta}) imply\n\\begin{equation}\n\\rho(r_H)+p_r(r_H)=0.\n\\end{equation}\n\n\n\nBecause of the asymptotic flatness requirement, we also assume\n\\begin{eqnarray}\n\\mu(r\\rightarrow \\infty)=1,\\label{eq:mu}\\\\\n\\delta(r\\rightarrow \\infty)\\rightarrow 0.\n\\end{eqnarray}\nNote that, by requiring a GR limit as $\\widehat{\\alpha}\\rightarrow 0$,\nthe general solution of (\\ref{eq:rhogb}) can be written as\n\\begin{equation}\n\\mu=1+\\frac{r^2}{2\\widehat{\\alpha}}\\left(1-\\sqrt{1+\\frac{8\\kappa \\widehat{\\alpha} M(r)}\n{(n-2)S_{n-2}r^{n-1}}}\\right),\\label{eq:muM}\n\\end{equation}\nwith $M(r)$ given by\n\\begin{equation}\nM(r)=M_H+S_{n-2}\\int^r_{r_H} \\rho r^{n-2}dr;\n\\end{equation}\nwhich can be shown to be the generalized \nMisner-Sharp mass\\cite{Maeda08}. In particular,\n$M_H$ is the horizon mass, and when $r$ goes \nto infinity,  $M(r)$ goes to\nthe ADM mass $\\mathcal{M}$. In order to have a \nfinite ADM mass, $\\rho$ must satisfy\n\\begin{equation}\n\\lim\\limits_{r\\rightarrow\\infty} r^{n-1}\\rho=0.\\label{eq:rholim}\n\\end{equation} \nIn the vacuum case ($T^a_b=0$), we have  \n\\begin{equation}\n\\mu=1+\\frac{r^2}{2\\widehat{\\alpha}}\\left(1-\\sqrt{1+\\frac{8\\kappa \\widehat{\\alpha}\\mathcal{M}}\n{(n-2)S_{n-2}r^{n-1}}}\\right);\n\\end{equation}\nand the associated vacuum metric is known as the \nBoulware-Deser-Wheeler (BDW) black hole.\n\nIf $\\hat{\\alpha}\\rightarrow 0$, Eq.(\\ref{eq:muM}) reduces to its GR \nlimit,\n\\begin{equation}\n\\mu=1-\\frac{2\\kappa M(r)}\n{(n-2)S_{n-2}r^{n-3}}.\\label{eq:muMe}\n\\end{equation}\n \n\n\n\nFinally, \n\\begin{equation}\nT=-\\rho+p_r+(n-2)p_\\bot;\n\\end{equation} \ndenotes the trace of the energy momentum tensor and we assume \nthat matter satisfies the dominant energy condition (DEC) \n$\\rho\\geq 0$, $\\rho\\geq |p_r|,|p_\\bot|$. From\nthe DEC, we see then that $M_H\\leq M(r)\\leq \\mathcal{M}.$\n\n\n\\section{Upper bound on the photon sphere radii}\nWe are now prepared to establish the following theorems:\\\\\n\n\\textbf{Theorem 1. Hod's theorem ($n$-dimensional GR version):} \\textit{Let $(\\tilde{M},g_{ab})$ \nbe a spherically symmetric spacetime, such that: i) it has a regular \nevent horizon, ii) it is asymptotically flat, iii) the dominant \nenergy condition \nis satisfied and the \nenergy-momentum trace is nonpositive, iv) it satisfies the $n$-dimensional \nEinstein equations. \nThen this metric admits at least one photon sphere, which in the coordinates \ngiven by (\\ref{eq:metric}) is characterized by\na radius $r_\\gamma$, which is bounded by the\nfollowing expression in terms of its total ADM mass $\\mathcal{M}$:\n\\begin{equation}\nr_\\gamma\\leq\\left(\\frac{\\kappa(n-1)}{(n-2)S_{n-2}} \\mathcal{M}\\right)^{\\frac{1}{n-3}}.\\label{boundrgr}\n\\end{equation}\n}\n\nThis theorem can be extended to Einstein-Gauss-Bonnet theory \nin $n$-dimensions. In particular, \nin five dimensions we can obtain an upper bound for the photon sphere radius in terms of \nthe ADM mass and the constant $\\hat\\alpha$.\nFor $n>5$, although we cannot give a general expression for the \nphoton sphere radius bound in terms of the ADM mass \n(with the exception of $n=9$), \nwe can state a more general result. These properties \nare summarized in the following theorem.\\\\\n\n\\textbf{Theorem 2. $n$-dimensional EGB version:} \\textit{Let $(\\tilde{M},g_{ab})$ be a \nspherically symmetric spacetime such that: a) it satisfies conditions i), ii) \nand iii) of Theorem $1$, b) it satisfies the $n$-dimensional \nEGB equations. Then this metric admits at least one photon sphere, \nwhose radius $r_\\gamma$ in the coordinates given by (\\ref{eq:metric}) is always \nbounded\nby the photon sphere radius $r^*_\\gamma$ of a vacuum spherically \nsymmetric BDW black hole \nwith the same total ADM mass $\\mathcal{M}$, i.e. $r_\\gamma\\leq r^*_\\gamma$.\nIn particular in five dimensions the photon sphere radius is bounded by the\nfollowing expression in terms of its total ADM mass $\\mathcal{M}$:\n\\begin{equation}\nr_\\gamma\\leq\\frac{\\sqrt{6}[\\kappa\\mathcal{M}\n(\\kappa\\mathcal{M}-3\\widehat{\\alpha}\\pi^2)]^{1\/4}}{3\\pi}.\\label{rg5}\n\\end{equation}\n}\n \nNote that the inequality (\\ref{rg5}) is saturated for a five-dimensional BDW \nblack hole with mass\n$\\mathcal{M}$.\\\\\n\n\n\\textbf{Proof:} \\textit{Common part.} Since the Einstein equations \ncan be recovered from (\\ref{eq:rhogb}) and (\\ref{eq:prgb}) with \n$\\hat{\\alpha}=0$, in the proof \nthere is a common part for both theorems. \nBasically, we follow the same steps as Hod's proof\\cite{Hod13}, although some \nchanges are \nneeded to incorporate the $n$-dimensional case. \nBecause of the spherical symmetry, null geodesics may without \nloss of generality be taken to be \non the equatorial plane. \nFrom the metric (\\ref{eq:metric}) it is easy to show that the equation\nfor circular null geodesics $\\dot r_\\gamma=(\\dot r_\\gamma)'=0$ \n(where a dot means derivative with\nrespect to an affine parameter and $'$ means derivative with respect to $r$)\nreads\\cite{Chandra83,Cardoso2009}\n\\begin{equation}\n\\tilde{N}(r_\\gamma)=0;\n\\end{equation}\nwith\n\\begin{equation}\n\\tilde{N}(r)=2e^{-2\\delta(r)}\\mu(r)-r[e^{-2\\delta(r)}\\mu(r)]'.\n\\end{equation}\nBy defining the function\n\\begin{equation}\nN(r)=e^{2\\delta(r)}\\tilde{N}(r),\\label{eq:No}\n\\end{equation}\nwe similarly obtain\n\\begin{equation}\\label{N1}\nN(r_\\gamma)=2[1+r_\\gamma\\delta'(r_\\gamma)]\\mu(r_\\gamma)-r_\\gamma\\mu'(r_\\gamma)=0.\n\\end{equation}\n\nFrom the trace of the energy-momentum tensor and (\\ref{eq:No}),\nwe can write (\\ref{eq:conser}) as\n\\begin{equation}\np'_r=\\frac{1}{2\\mu r}\\left(N(\\rho+p_r)+2\\mu T-2n\\mu p_r\\right).\\label{eq:prp}\n\\end{equation}\nNow if we define:\n\\begin{equation}\nP=r^np_r,\n\\end{equation}\nwe arrive at\n\\begin{equation}\\label{consef}\nP'(r)=\\frac{r^{n-1}}{2\\mu}\\left(N(\\rho+p_r)+2\\mu T\\right). \n\\end{equation}\nIn terms of the energy-momentum components, $N$ reads\n\\begin{equation}\\label{N2}\n\\begin{split}\nN=&\\frac{1}{2\\widehat{\\alpha}(\\mu-1)-r^2}\\left\\{(n-1)\\widehat{\\alpha}\\mu^2\\right.\\\\\n&\\left.-[(n-1)r^2+2(n-3)\\widehat{\\alpha}]\\mu+(n-5)\\widehat{\\alpha}+(n-3)r^2\\right.\\\\\n&\\left.+\\frac{2}{n-2}\\kappa r^4p_r\\right\\};\n\\end{split}\n\\end{equation}\nwhich in the $n$-dimensional GR\ncase reduces to\n\\begin{equation}\nN_{GR}=(n-1)\\mu-(n-3)-\\frac{2\\kappa}{(n-2)}r^2p_r.\\label{NGR}\n\\end{equation}\nLet us observe that (\\ref{N1}) admits at least one solution as follows \nfrom the fact that \nthe conditions (\\ref{crh}) and (\\ref{eq:No}) imply \n$N(r_H)\\leq 0$, and taking into account that \n$\\lim\\limits_{r\\rightarrow\\infty} r^2p_r=0$ (which follows \nfrom the dominant energy condition and (\\ref{eq:rholim})), we see that\n$N(r\\rightarrow\\infty)\\rightarrow 2$; therefore, there \nis a $r_\\gamma$ where $N(r_\\gamma)=0$.\nWhat is more, because we are interested in the innermost \nnull circular orbit, $N(r)$\n must satisfy \n\\begin{equation}\\label{N3}\nN(r_H\\leq r<r_\\gamma)<0.\n\\end{equation}\nThen, by using the dominant energy condition \nwe obtain\n\\begin{equation}\np_r(r_H)=-\\rho(r_H)\\leq 0, \\;\\; \\Rightarrow p_r(r_H)\\leq 0 \\Leftrightarrow P(r_H)\\leq 0.\\label{prh1}\n\\end{equation}\nSimilarly from (\\ref{consef}), (\\ref{N3}) and the condition of nonpositive trace $T\\leq 0$ , \nwe see that\n $P'(r_H\\leq r<r_\\gamma)\\leq 0.$ \nFrom this last condition and (\\ref{prh1}) we have\n\\begin{equation}\np_r(r_\\gamma)\\leq 0.\\label{eq:prf}\n\\end{equation}\nLet us analyze the implications of this relation in the Einstein \nand Einstein-Gauss-Bonnet cases\nseparately.\\\\\n\n{\\textsl{ Completion of the proof of Theorem 1.}}\nIn the Einstein case, from (\\ref{eq:prf}) and (\\ref{NGR}) we conclude that\n\\begin{equation}\n\\mu(r_\\gamma)\\leq\\frac{n-3}{n-1},\n\\end{equation}\nwhich  by using (\\ref{eq:muMe}) implies\n\\begin{equation}\nr_\\gamma\\leq\\left(\\frac{\\kappa(n-1)}{(n-2)S_{n-2}}{M(r_\\gamma)}\\right)^{\\frac{1}{n-3}},\n\\end{equation}\nor in terms of the total ADM mass $\\mathcal{M}$,\n\\begin{equation}\nr_\\gamma\\leq\\left(\\frac{\\kappa(n-1)}{(n-2)S_{n-2}} \\mathcal{M}\\right)^{\\frac{1}{n-3}}.\n\\end{equation}\n\n\n\\textsl{Completion of the proof of Theorem 2.} Let us start by \nnoting that in (\\ref{N2}) \nthe following expression appears in \nthe denominator\n\\begin{equation}\n2\\widehat{\\alpha}(\\mu-1)-r^2=-\\frac{r^2}{2\\widehat{\\alpha}}\\sqrt{1\n+\\frac{8\\kappa \\widehat{\\alpha} \nM(r)}{(n-2)S_{n-2}r^{n-1}}}<0.\n\\end{equation} \nTherefore from (\\ref{N2}) and (\\ref{eq:prf}) we deduce that at $r_\\gamma$,\n\n\\begin{equation}\\label{Ngb}\n\\begin{split}\n&N^*\\equiv\\left\\{(n-1)\\widehat{\\alpha}\\mu^2-[(n-1)r^2+2(n-3)\\widehat{\\alpha}]\\mu\n+(n-5)\\widehat{\\alpha}\n\\right.\\\\\n&\\left.+(n-3)r^2\\right\\}=-\\frac{2}{n-2}\\kappa r^4p_r\\geq0.\n\\end{split}\n\\end{equation}\nThe implications of this equation for the allowed $r_\\gamma$ \nwill be discussed \nfirst in five dimensions because this case is exactly solvable, \nand after that, we will\nextend the analysis to higher dimensions\n\nBy writing $N^*$  \nas\n\\begin{equation}\nN^*=(n-1)\\widehat{\\alpha}(\\mu-B_{(n)-})(\\mu-B_{(n)+}),\\label{eq:NbmbM}\n\\end{equation}\nwe see that in five dimensions, (\\ref{Ngb}) implies that at $r_\\gamma$, some\nof these conditions hold:\n \\begin{equation}\\label{eq:mup}\n\\mu\\leq\\frac{1}{2\\widehat{\\alpha}}(\\widehat{\\alpha}+r^2\n-\\sqrt{\\widehat{\\alpha}^2+r^4})\\equiv B_{(5)-},\n\\end{equation}\n\nor\n\\begin{equation}\\label{eq:munp}\n\\mu\\geq\\frac{1}{2\\widehat{\\alpha}}(\\widehat{\\alpha}+r^2+\\sqrt{\\widehat{\\alpha}^2+r^4})\\equiv B_{(5)+};\n\\end{equation}\nwhere in order to write these inequalities we use the fact \nthat $B_{(5)-}$ and $B_{(5)+}$ are both \npositive real \nfunctions and that $B_{(5)-}<B_{(5)+}$.\n\nWe also note [using $M_H\\leq M(r)\\leq \\mathcal{M}$] that independently of \nthe dimension,\n$\\mu$ is bounded from below by\n\\begin{equation}\\label{eq:muMadm}\n\\mu\\geq 1+\\frac{r^2}{2\\widehat{\\alpha}}\\left(1-\\sqrt{1+\\frac{8\\kappa \\widehat{\\alpha} \\mathcal{M}}\n{(n-2)S_{n-2}r^{n-1}}}\\right)\\equiv \\mu_{(n)\\mathcal{M}};\n\\end{equation}\nand by \n\\begin{equation}\\label{eq:muMH}\n\\mu\\leq 1+\\frac{r^2}{2\\widehat{\\alpha}}\\left(1-\\sqrt{1+\\frac{8\\kappa \\widehat{\\alpha} {M_H}}\n{(n-2)S_{n-2}r^{n-1}}}\\right)\\equiv \\mu_{(n){M_H}};\n\\end{equation}\nfrom above.\nConsequently, the validity of the conditions (\\ref{eq:mup}) or (\\ref{eq:munp}) implies\n\\begin{equation}\n\\mu_{(5)\\mathcal{M}}(r_\\gamma)\\leq B_{(5)-}(r_\\gamma);\\label{in1}\n\\end{equation}\nor \n\\begin{equation}\n\\mu_{(5)M_H}(r_\\gamma)\\geq B_{(5)+}(r_\\gamma);\\label{in2}\n\\end{equation}\nrespectively.\nMoreover, the four functions $\\mu_{\\mathcal{M}}(r)$, $\\mu_{(5)M_H}$, $B_{(5)+}(r)$ and \n$B_{(5)-}(r)$ are \nall monotonically increasing functions.\nFrom this fact, and observing that $B_{(5)+}(r)\\geq 1$, and $\\mu_{(5)M_H}(r\\rightarrow\\infty)=1$, \nit follows that \n\\begin{equation}\n\\mu(r)\\leq\\mu_{(5)M_H}(r) < B_{(5)+}(r)\\; \\forall r,\n\\end{equation} \nthus,  \nit is not possible for any real $r_\\gamma$ to satisfy the \ninequality (\\ref{eq:munp}). \n\nLet us study now the inequality (\\ref{in1}). \nObserving that for $r\\geq r_H$,\n\\begin{equation}\n0<B_{(5)-}(r)<B_{(5)-}(r\\rightarrow\\infty)=\\frac{1}{2},\n\\end{equation}\nand \n\\begin{equation}\n\\mu_{(5)\\mathcal{M}}(r=r_H)\\leq 0,\\;\\; \\mu_{(5)\\mathcal{M}}(r\\rightarrow\\infty)=1,\n\\end{equation} \nwe see that (\\ref{in1}) will be valid for all $r$ such that \n$r_H\\leq\\, r\\leq r^*_\\gamma$, with\n$r^*_\\gamma$ solution of \n \\begin{equation}\n\\mu_{(5)\\mathcal{M}}(r^*_\\gamma)=B_{(5)-}(r^*_\\gamma).\n\\end{equation}\nThe only positive real solution of this equation is \n\\begin{equation}\nr^*_\\gamma=\\frac{\\sqrt{6}[\\kappa\\mathcal{M}(\\kappa\\mathcal{M}-3\\widehat{\\alpha}\\pi^2)]^{1\/4}}{3\\pi}.\n\\end{equation}\nOn the other hand, as $\\mu(r_H)=0$, $B_{(5)-}(r_H)>0$, and $\\mu(r)\\geq \\mu_{(5)\\mathcal{M}}(r)$, \nwe conclude that  $\\mu(r)> B_{(5)-}(r)$ for all $r> r^*_\\gamma$.\nConsequently, the radius $r_\\gamma$ which satisfies (\\ref{eq:mup}) \nmust necessarily be\nbounded from above by $r^*_\\gamma$, that is\n\\begin{equation}\nr_\\gamma\\leq\\frac{\\sqrt{6}[\\kappa\\mathcal{M}(\\kappa\\mathcal{M}-\n3\\widehat{\\alpha}\\pi^2)]^{1\/4}}{3\\pi}.\\label{eqrg}\n\\end{equation}\n\n\nIn the $n$-dimensional case,  as was mentioned above, \nwe cannot find an explicit upper bound in terms of the ADM mass;\nhowever, we can show that even in these situations, the \nphoton sphere radius will always \nbe bounded by the photon sphere radius of a BDW black \nhole with the same mass.\nIt can be shown as follows. \n\nFirst, in the general case, following a similar analysis \nas in Eqs.(\\ref{eq:NbmbM}), (\\ref{eq:mup}) \nand (\\ref{eq:munp}), we see that (\\ref{Ngb}) is satisfied at $r_\\gamma$ if\n\\begin{equation}\\label{eq:mupN}\n\\mu\\leq\\frac{1}{2\\widehat{\\alpha}}\\left(\\frac{2(n-3)}{n-1}\\widehat{\\alpha}+r^2-\n\\sqrt{\\frac{16}{(n-1)^2}\\widehat{\\alpha}^2\n+r^4}\\right)\\equiv B_{(n)-},\n\\end{equation}\nor if \n\\begin{equation}\\label{eq:mupN2}\n\\mu\\geq\\frac{1}{2\\widehat{\\alpha}}\\left(\\frac{2(n-3)}{n-1}\\widehat{\\alpha}+r^2+\n\\sqrt{\\frac{16}{(n-1)^2}\\widehat{\\alpha}^2\n+r^4}\\right)\\equiv B_{(n)+},\n\\end{equation}\nwhich using (\\ref{eq:muMadm}) and (\\ref{eq:muMH}) implies\n\\begin{equation}\n\\mu_{(n)\\mathcal{M}}(r_\\gamma)\\leq B_{(n)-}(r_\\gamma);\\label{inn1}\n\\end{equation}\nor \n\\begin{equation}\n\\mu_{(n)M_H}(r_\\gamma)\\geq B_{(n)+}(r_\\gamma).\\label{inn2}\n\\end{equation}\nFor the same reasons as in the five-dimensional case [$B_{(n)+}(r)\\geq 1$, \n$\\mu_{(n){M_H}}(r\\rightarrow\\infty)=1$, and both being \nmonotonically increasing functions], \nwe conclude that there is no real $r_\\gamma$ so that (\\ref{inn2}) [and hence \n(\\ref{eq:mupN2})] \ncan be satisfied.\nOn the other hand, from the fact that $B_{(n)-}$ and $\\mu_{(n)\\mathcal{M}}$ are\nmonotonically increasing functions,\nand taking into account that\n\\begin{equation}\n0\\leq B_{(n)-}(r)< B_{(n)-}(r\\rightarrow\\infty)=\\frac{n-3}{n-1}<1\\;\\forall r\\geq r_H;\n\\end{equation} and $\\mu_{(n)\\mathcal{M}}(r_H)\\leq 0$, \n$\\mu_{(n)\\mathcal{M}}(r\\rightarrow\\infty)=1$, we see that (\\ref{inn1}) holds\nfor all $r_H\\leq r<r^*_\\gamma$ with $r^*_\\gamma$ satisfying\n\\begin{equation}\n\\mu_{(n)\\mathcal{M}}(r^*_\\gamma)=B_{(n)-}(r^*_\\gamma).\\label{eq:genAB}\n\\end{equation}\nAs \n$\\mu(r_H)=0, B_{(n)-}(r_H)>0$ and $\\mu(r)\\geq \\mu_{(n)\\mathcal{M}}(r)$; and following the\nsame arguments as in the five-dimensional situation, we conclude that  \nthe radius $r_\\gamma$ where (\\ref{eq:mup}) holds, will also\nsatisfy $r_\\gamma\\leq r^*_\\gamma$.\nFurthermore, from (\\ref{N2}), (\\ref{N3}) and (\\ref{eq:NbmbM}), and observing that\n\\begin{equation}\nN^*=[2\\widehat{\\alpha}(\\mu-1)-r^2]N-\\frac{2}{n-2}\\kappa r^4p_r,\n\\end{equation} \nwe see that (\\ref{eq:genAB}) is equivalent \nto requiring $N(r^*_\\gamma)=0$ for a \nBDW black hole with ADM-mass $\\mathcal{M}$ ($\\rho=p_r=p_\\bot=0$).\nIn other words, $r^*_\\gamma$ can be interpreted as the radius \nof the photon sphere for a \nBDW  black hole with mass $\\mathcal{M}$. As a result, \nthe bound is \nsaturated in these cases. $\\blacksquare$\\\\\n\n\nLet us make some remarks before continuing. In order to see how \nthe bound (\\ref{boundrgr}) varies \nin terms of the dimension, we plot the quotient $r_\\gamma\/(G\\mathcal{M})^{\\frac{1}{n-3}}$ \nin Fig.(1), assuming a \ncontinuous $n$. The physical values \nmust be taken\nonly for natural $n$ in the graphic. We see that in eight dimensions \nthis quotient reaches its \nminimum value. However, as for $n>4$, the dependence in the mass is not linear, \nthe dimension which \nminimizes the value of the upper bound\nfor the radius $r_\\gamma$ at a given fixed $\\mathcal{M}$ will\ndepend on the value of this mass.\n\\begin{figure}[!h]\n\\centering \n\\includegraphics[clip,width=80mm]{rgammaGMfixed5b.eps}\n\\caption{Upper bound for the quotient $r_\\gamma\/(G\\mathcal{M})^{\\frac{1}{n-3}}$ \nin terms of the \ndimension $n$ of the spacetime.\n}\n\\label{fig:distrib}\n\\end{figure}\n\nLet us also observe that if we take $\\alpha\\rightarrow 0$ in (\\ref{eqrg}), \nwe recover the upper \nbound that we found in the GR \ncase.\n\nAs a final comment, we mention that (\\ref{eq:genAB}) can also be \nsolved in nine dimensions, obtaining \nthe result\n\\begin{equation}\nr_\\gamma\\leq 98(14)^{1\/3}\\left[\\frac{h^{2\/3}-686(14)^{2\/3}\\pi^2\\kappa\\hat\\alpha\\mathcal{M}\n}{\\pi^3h^{1\/3}}\\right]^{1\/4},\n\\end{equation}\nwith\n\\begin{equation}\nh=\\pi\\kappa\\mathcal{M}\\left[18\\kappa\\mathcal{M}+\\sqrt{2\\kappa\\mathcal{M}\n(162\\kappa\\mathcal{M}+7\\pi^4\\hat\\alpha^3)}\\right].\n\\end{equation}\n\n\n\\section{Applications: Quasinormal modes and strong lensing}\n\nRecently(\\cite{Cardoso2009}), Cardoso {\\it et al.}, showed a \ncorrespondence between\nthe quasinormal modes associated to a black hole,\nin the eikonal limit, and some properties \nof photon spheres.\nIn particular, in this limit, the quasinormal\nfrequencies $\\omega_{QNM}$ can be computed from \nanalytical WKB \napproximation methods\\cite{Iyer87},\nobtaining\n\\begin{equation}\n\\omega_{QNM}=l\\Omega_\\infty-i|\\lambda|(m+1\/2),\\label{eq:wqnm}\n\\end{equation}\nwhere \n\\begin{equation}\n\\Omega_\\infty=\\frac{[e^{-2\\delta(r_\\gamma)}\\mu(r_\\gamma)]^{1\/2}}{r_\\gamma},\\label{omegainfty}\n\\end{equation} \nis the angular velocity\nof circular null geodesics at $r_\\gamma$ as\nmeasured by asymptotic observers, $\\lambda$ is the \nLyapunov exponent associated with\nthis kind of geodesics, \n$l$ is the angular momentum of the perturbation \n(and it is assumed that $l\\gg 1$) \nand $m$ is the overtone number. \nBased on this fact, Hod in\\cite{Hod13} presented a simple \nand universal bound\nfor the real part of the quasinormal frequencies\nin the WKB limit, expressed in terms of the event horizon \nradius of the associated black hole.\nNow, we generalize this expression to $n$-dimensional \nGR and EGB theory, and we also\ndiscuss other simple bounds for other observables. \n\nFrom (\\ref{eq:wqnm}), we see that the real part of these frequencies is given by\n\\begin{equation}\n\\omega_l\\equiv\\Re{[\\omega_{QNM}]}=l\\Omega_\\infty.\\label{omegalq}\n\\end{equation}\nBy using (\\ref{eq:mupN}), which can be re-written as\n\\begin{equation}\n\\mu(r_\\gamma)\\leq 2\\frac{\\frac{n-5}{n-1}\\hat\\alpha+\\frac{n-3}{n-1}r^2_\\gamma}\n{\\frac{n-3}{n-1}\\alpha+r^2_\\gamma+\\sqrt{\\frac{16\\hat\\alpha^2}{(n-1)^2}+r^4_\\gamma}};\n\\end{equation}\nand using the fact that $e^{-2\\delta}\\leq 1$, [which follows from (\\ref{eq:delta}) and \nthe asymptotic \ncondition for $\\delta(r)$], we see that (\\ref{omegainfty}) implies \n\\begin{equation}\n\\Omega_\\infty\\leq 2\\left[\\frac{\\frac{n-5}{(n-1)r^2_\\gamma}\\hat\\alpha+\\frac{n-3}{n-1}}\n{\\frac{n-3}{n-1}\\alpha+r^2_\\gamma+\\sqrt{\\frac{16\\hat\\alpha^2}{(n-1)^2}+r^4_\\gamma}}\\right]^{1\/2}.\n\\end{equation}\nHence, from relation (\\ref{omegalq}) we obtain\n\\begin{equation}\\label{eq:omegalp}\n{\\omega_l}\\leq 2 l \\left[\\frac{\\frac{n-5}{(n-1)r^2_\\gamma}\\hat\\alpha+\\frac{n-3}{n-1}}\n{\\frac{n-3}{n-1}\\alpha+r^2_\\gamma+\\sqrt{\\frac{16\\hat\\alpha^2}{(n-1)^2}+r^4_\\gamma}}\\right]^{1\/2}.\n\\end{equation}\nIn order to compare this with the Hod bound, we will give a \nweaker inequality, but one \nwhich is simpler and more universal than (\\ref{eq:omegalp})\nin the sense that it does not depend on the \nparticular properties of the\nblack hole.\nLet us note that, in our study cases, $\\hat\\alpha\\geq 0$ and $r_\\gamma\\geq r_H$; \ntherefore,\n\n\\begin{equation}\n\\begin{split}\n{\\Omega}_\\infty & \\leq 2  \\left[\\frac{\\frac{n-5}{(n-1)r^2_\\gamma}\\hat\\alpha+\\frac{n-3}{n-1}}\n{\\frac{n-3}{n-1}\\hat\\alpha+r^2_\\gamma+\\sqrt{\\frac{16\\hat\\alpha^2}{(n-1)^2}+r^4_\\gamma}}\\right]^{1\/2}\\\\\n& \\leq 2  \\left[\\frac{\\frac{n-5}{(n-1)r^2_H}\\hat\\alpha+\\frac{n-3}{n-1}}\n{\\frac{n-3}{n-1}\\hat\\alpha+r^2_H+\\sqrt{\\frac{16\\hat\\alpha^2}{(n-1)^2}+r^4_H}}\\right]^{1\/2}\\\\\n& \\leq   \\left[\\frac{\\frac{n-5}{(n-1)r^2_H}\\hat\\alpha+\\frac{n-3}{n-1}}\n{r^2_H}\\right]^{1\/2}.\\label{omegacot}\n\\end{split}\n\\end{equation}\n\nCombining this inequality with (\\ref{omegalq}), we finally obtain\n \\begin{equation}\n\\begin{split}\n{\\omega}_l \n\\leq l  \\left[\\frac{\\frac{n-5}{(n-1)r^2_H}\\hat\\alpha+\\frac{n-3}{n-1}}\n{r^2_H}\\right]^{1\/2};\n\\end{split}\n\\end{equation}\nwhich implies\n \\begin{equation}\n\\begin{split}\n{\\omega}_l r_H \n& \\leq l  \\left[\\frac{n-5}{(n-1)r^2_H}\\hat\\alpha+\\frac{n-3}{n-1}\\right]^{1\/2}.\\label{eq:omeg}\n\\end{split}\n\\end{equation}\nThis expression generalizes to $n$-dimensional EGB theory, \nthe universal bound found\nby Hod in the framework of four-dimensional GR\\cite{Hod13}.\nIn particular, in the GR case it reduces to \n\\begin{equation}\n{\\omega}_l r_H  \\leq l  \\left[\\frac{n-3}{n-1}\\right]^{1\/2},\n\\end{equation}\nwhich, when $n=4$ agrees with the expression given in \\cite{Hod13}.\nNote also that in five dimensions the first factor of \n(\\ref{eq:omeg}) does not survive,  \nand we get the simple bound\n\\begin{equation}\n{\\omega}_l r_H \\leq \\frac{1}{\\sqrt{2}} l.\n\\end{equation}\n\nAt this point, we can ask if there is a similar \nuniversal bound for the \nLyapunov exponent; \nhowever, the answer is negative.\nThe Lyapunov exponent for circular null geodesics \nis defined by\\cite{Cardoso2009}\n\\begin{equation}\n\\lambda=\\sqrt{\\frac{r^2_\\gamma [e^{-2\\delta(r_\\gamma)}\\mu(r_\\gamma)]}\n{2L^2}V''(r_\\gamma)},\\label{lambda}\n\\end{equation}\nwith \n\\begin{equation}\nV''(r_\\gamma)=\\frac{L^2}{r^4_\\gamma e^{-2\\delta{(r_\\gamma)}}}\\left\\{e^{-2\\delta{(r_\\gamma)}}\n\\mu(r_\\gamma)-r^2_\\gamma [e^{-2\\delta(r_\\gamma)}\\mu(r_\\gamma)]''\\right\\},\n\\end{equation}\nand $L$ the orbital angular momentum of the circular null geodesics.\nAfter some simple computations it can be shown that in $n$-dimensional GR,\n\\begin{equation}\nV''(r_\\gamma)=\\frac{L^2 N'(r_\\gamma)}{r^3_\\gamma}.\n\\end{equation}\nThis makes it possible to write (\\ref{lambda}) as\n\\begin{equation}\n\\lambda=\\sqrt{ [e^{-2\\delta(r_\\gamma)}\\mu(r_\\gamma)]\n\\frac{ N'(r_\\gamma)}{2r_\\gamma}}.\\label{lambda2}\n\\end{equation}\nFrom (\\ref{NGR}) we have \n\\begin{equation}\n\\begin{split}\nN'(r_\\gamma)&=(n-1)\\mu'(r_\\gamma)-\\frac{2\\kappa}{(n-2)}\\left[2r_\\gamma p_r(r_\\gamma)+\nr^2_\\gamma p'_r(r_\\gamma)\\right]\\\\\n&\\geq (n-1)\\mu'(r_\\gamma)+2{\\kappa} r_\\gamma p_r(r_\\gamma),\\label{Npf}\n\\end{split}\n\\end{equation}\nwhere in order to establish the last inequality, we have taken into account that \nat $r_\\gamma$, $p_r$ and $P'$\nare both nonpositive numbers.\nHowever, from the first factor in (\\ref{lambda2}), which is the factor \n$e^{-2\\delta(r_\\gamma)}\\mu(r_\\gamma)$, \nwe  can only ensure that $e^{-2\\delta(r_\\gamma)}\\mu(r_\\gamma)\\leq\\mu(r_\\gamma)$, \nand therefore\nthis inequality and (\\ref{Npf}) impose bounds in opposite \ndirections, thereby preventing \na universal bound (independent of the matter content) \nfrom being written \nfor the Lyapunov exponent.\n\nComing back to the inequality (\\ref{omegacot}), it can also be used to \nimpose a universal bound over the location of the first\nrelativistic image in the strong lensing regime. In general, if we assume \nthat the observer\nis at a distance $D_{ol}$ from the lens, then the first relativistic \nimage subtends an \nangle $\\theta_\\infty$ \\cite{Bozza02,Stefanov10}, which\ncan be expressed in terms of the circular orbital frequency $\\Omega_\\infty$ as\n\\begin{equation}\n\\theta_\\infty=\\frac{1}{D_{ol}\\Omega_\\infty}.\n\\end{equation}\nFrom this relation and using (\\ref{omegacot}),\nwe obtain the universal and simple lower bound\n\\begin{equation}\n\\theta_\\infty\\geq \\frac{1}{D_{ol}}\\left[\\frac{r^2_H}{\\frac{n-5}{(n-1)r^2_H}\\hat\\alpha+\\frac{n-3}{n-1}}\n\\right]^{1\/2}.\n\\end{equation}\nIn the GR limit it reduces to\n\\begin{equation}\n\\theta_\\infty\\geq \\frac{\\, r_H}{D_{ol}}\\left[{\\frac{n-1}{n-3}}\n\\right]^{1\/2},\n\\end{equation}\nwhich, for the 4-dimensional case gives\n\\begin{equation}\n\\theta_\\infty\\geq \\sqrt{3}\\frac{\\, r_H}{D_{ol}}.\n\\end{equation}\n\\section{General comment on the fastest way to circle \naxially-symmetric spacetimes}\nIt would be very interesting to find a generalization of some \nof the previous results\nfor the case of axial symmetry. Even in this case, the \ncircular null geodesics on the\nequatorial plane share some of the properties found \nin the spherical symmetric case. \nFor example, \nWei and Liu (\\cite{Wei14}) found universal relations between \nthe first relativistic image and the quasinormal frequencies by \nanalyzing equatorial circular null \ngeodesics in arbitrary axially-symmetric black holes, thus \nextending some of the \nresults of spherical symmetry.\n\n Recently, Hod\\cite{Hod11} also showed that the fastest way to circle a \nblack hole in general \nrelativity is through circular null geodesics. In particular, \nhe demonstrated \nthat for any spherically \nsymmetric spacetime in four dimensions, the minimum orbital \ntime as measured by an \nasymptotic observer is realized by circular null geodesics, \nindependently of the underlying gravitational theory. Moreover, \nhe showed that a similar \nresult can be obtained in GR for the case of \na Kerr black hole by noting that the equatorial circular \norbit with minimum traveling\ntime coincides with the solution obtained by solving the equation governing \ncircular null geodesics.\nMore recently, in \\cite{Pradhan13}, Pradhan, in his study of charged \nMyers-Perry black holes \nin higher dimensions, made the observation that Hod's \nconclusion regarding the fastest way \nto circle a black hole \nremains valid in this more general family of metrics.\n\n\nAs mentioned, the fact that circular null geodesics minimize \nthe orbital period in \nthe case of spherically symmetric spacetimes is always valid. \nHowever, in\\cite{Hod11}, when Hod analyzed the Kerr metric,  \nhe obtained two different equations (one for the \nfastest circular orbit, and another for the \ncircular\nnull geodesics, see Eqs.(21) and (31) in his paper), and\nhe showed that they admit the same kind of solutions; therefore, \nit was not clear from his \ndiscussion how general these results were. In other words, \none can ask whether the agreement \nbetween equatorial \ncircular null geodesics and the fastest way to circle\nblack holes is a property unique to GR or it is a \ngeometrical property \ngeneral to any axially-symmetric \nmetric, independently of any gravity theory. \n\n\nAdditionally, Hod made the interesting conjecture that\nthere should be a lower bound for the traveling time, given \nin terms of the ADM mass \nof the black hole\\cite{Hod11}. \nMore precisely, he conjectured a lower bound on the orbital periods \n$T_\\infty$ of circular null geodesics around compact objects \nwith mass $\\mathcal{M}$ \n(as seen from far away \nobservers). In mathematical terms he states that (in units with $G=c=1$)\n\\begin{equation}\nT_\\infty\\geq 4\\pi \\mathcal{M}\\label{Tinf}.\n\\end{equation}\nIn particular, the equality is satisfied by an extreme Kerr black hole.\nThis conjecture is physically motivated by taking into \naccount the rotational dragging \nwhich is maximal in the case of an extremal black hole.\nIf this conjecture were correct, it could also be used to \nestablish another universal bound for\nthe possible location of the first relativistic image of an \nobject lensed by a black hole.\nIn fact, if (\\ref{Tinf}) were valid we should obtain \n$\\Omega_\\infty=2\\pi\/T_\\infty\\leq\\frac{1}{2\\mathcal{M}},$ which would imply\n\\begin{equation}\n\\theta_\\infty\\geq\\frac{2\\mathcal{M}}{D_{ol}}.\\label{thetab}\n\\end{equation}\nIn other words, the expression (\\ref{thetab}) would also be valid\nfor a generic rotating black hole, at least in the GR regime.\n\nEven given the reasonableness of conjecture (\\ref{Tinf}), \nit has not yet been proven. \nHowever, it leads one to wonder whether this conjecture \ncan be kept as potentially valid in alternative \ngravitational theories. \nIt is our purpose in this section to answer these questions. \nIn particular, we \nmake the observation \nthat the orbital period for circular orbits {\\it always} \ncoincides with the  circular null geodesics for \nany axially-symmetric spacetime, \nindependently of the gravitational theory.\nWe also answer the second question negatively \non the validity of (\\ref{Tinf}) for alternative\nblack hole candidates \nby giving an explicit counterexample where the bound \nassumed by Hod is not satisfied. \nWhat is more, we will show that the conjecture can be violated \neven in the case of a nonrotating \nblack hole.\nIn order to do that, we will discuss a special family of \nKaluza-Klein black holes.\n\nLet us start with the first question.\nWe begin, with the more general conformally stationary and\naxially-symmetric metric in $n$-dimensions, without making \nreference to any gravitational theory.\n\nThese kinds of metrics are given by\n\\begin{equation}\n\\begin{split}\nds^2&=e^{2\\Phi(r,\\theta,t)}\\left(g_{tt}dt^2+2g_{t\\phi}dtd\\phi+g_{rr}dr^2\\right.\n\\\\&+\\left. g_{\\theta\\theta}d\\theta^2+g_{\\phi\\phi}d\\phi^2+r^2\\cos(\\theta)^2d\\Omega^2_{n-3}\\right),\n\\end{split}\n\\end{equation}\nwith\n\\begin{equation}\nd\\Omega^2_{n-3}=d\\phi^2_2+\\sin^2\\phi_2\\left[d\\phi^2_3+\\sin^2\\phi_3(\\cdots d\\phi^2_{n-4})\\right],\n\\end{equation}\nand with the components of the metric not depending on $\\phi$.\nAs this metric is axially-symmetric, there are equatorial orbits, \nand in particular we\ncan compute from this metric the orbital time for circular null curves, \ni.e., curves \nsuch that $ds^2=0, \\theta=\\pi\/2$ and $r=r_\\gamma$. \nThe orbital period for these curves is:\n\\begin{equation}\\label{eq:T}\nT_\\infty=2\\pi\\frac{-g_{t\\phi}\\pm\\sqrt{\\Delta}}{g_{tt}},\n\\end{equation}\nwhere \n\\begin{equation}\n\\Delta=g_{t\\phi}^2-g_{tt}g_{\\phi\\phi},\n\\end{equation}\nand the $+\/-$ signs correspond to counter-rotating\/co-rotating orbits, \nrespectively.\nIf we compute their extremes we obtain\n\\begin{equation}\n\\frac{dT}{dr}=\\frac{\\pi}{\\sqrt{\\Delta}}G=0,\n\\end{equation}\nwith \n\\begin{equation}\\label{eq:G}\n\\begin{split}\nG_\\pm&=\\frac{1}{g_{tt}^2}\\left[(\\mp 2g_{t\\phi}\\sqrt{\\Delta}\n+\\Delta+g_{t\\phi}^2)\\frac{dg_{tt}}{dr}\\right.\\\\\n&\\left.+(-2g_{tt}g_{t\\phi}\\pm 2g_{tt}\n\\sqrt{\\Delta})\\frac{dg_{t\\phi}}{dr}+g_{tt}^2\\frac{dg_{\\phi\\phi}}{dr}\\right].\n\\end{split}\n\\end{equation}\nSo, if we assume that $\\Delta$ is a regular function and we \nonly consider the region exterior\nto all horizons (where $\\Delta=0$), the minimum orbital \nperiod is obtained for \nthose curves which satisfy $G_\\pm=0$.\n\nOn the other hand, we can study the condition for the \nexistence of circular null geodesics on the\nequatorial plane.\nWe start from the conservation equations\n\\begin{eqnarray}\nE&=&-\\left(g_{tt}\\dot{t}+g_{t\\phi}\\dot{\\phi}\\right),\\\\\nL&=&g_{t\\phi}\\dot{t}+g_{\\phi\\phi}\\dot{\\phi},\\\\\np_r&=&g_{rr}\\dot{r},\n\\end{eqnarray}\nwhich represent the energy of the null particle as measured by an \nasymptotic observer,\nthe orbital angular momentum and the radial component of the linear \nmomentum, respectively.\nThese equations can be solved for $\\dot{t}$, and $\\dot\\phi$ and replaced in\nthe Hamiltonian $H$,\n\\begin{equation}\nH=-E\\dot{t}+L\\dot\\phi+p_r\\dot{r}=0;\n\\end{equation} \nobtaining the following equation for $\\dot{r}$:\n\\begin{equation}\n\\dot{r}^2=V_r;\n\\end{equation}\nwith \n\\begin{equation}\nV_r=-\\frac{E^2}{g_{rr}\\Delta}\\left(g_{\\phi\\phi}+2g_{t\\phi}b\n+g_{tt}b^2\\right),\n\\end{equation}\nand $b=L\/E$, the impact parameter. \nCircular null geodesics must satisfy $\\dot{r}_\\gamma=\\dot{r}'_\\gamma=0$, or equivalently \n$V_r=V'_r=0$.\nFrom  $V_r=0$, we obtain\n\\begin{equation}\nb=\\frac{\\pm\\sqrt{\\Delta}-g_{t\\phi}}{g_{tt}},\\label{eq:b}\n\\end{equation}\nand from $V'_r=0$ it follows that\n\\begin{equation}\n\\frac{d}{dr}g_{\\phi\\phi}+2b\\frac{d}{dr}g_{t\\phi}+b^2\\frac{d}{dr}g_{tt}=0,\n\\end{equation}\nFinally, by replacing the expression for $b$ \ngiven by (\\ref{eq:b}) in the last equation,\nwe conclude that the circular null geodesics, if they exist, \nmust satisfy\n\\begin{equation}\n\\begin{split}\n&\\frac{1}{g_{tt}^2}\\left[(\\mp 2g_{t\\phi}\\sqrt{\\Delta}\n+\\Delta+g_{t\\phi}^2)\\frac{dg_{tt}}{dr}\\right.\n\\\\&\\left.+(-2g_{tt}g_{t\\phi}\\pm 2g_{tt}\\sqrt{\\Delta})\n\\frac{dg_{t\\phi}}{dr}+g_{tt}^2\\frac{dg_{\\phi\\phi}}{dr}\\right]=0,\n\\end{split}\n\\end{equation}\nwhich agrees with the expression for $G_\\pm$ given previously.\nAs a result, its critical points always coincide with \nthe values of the circular geodesics, generalizing in this way \nthe results given by Hod for\nspherically symmetric spacetimes, and the particular cases of Kerr\\cite{Hod11} \nand Myers Perry\\cite{Pradhan13}. Let us remark that \nthere is not inconsistency \nbetween this result\nand the fact that in\\cite{Hod11} Hod obtained\ntwo different algebraic equations [Eqs.(21) and (31) in \\cite{Hod11} for \nthe minimum orbital period  \nand geodesic motion, respectively]. In fact, \nthe circular null geodesics are obtained by solving $\\dot{r}_\\gamma=0$ \nand  $\\dot{r}'_\\gamma=0$, \nboth being functions of $b$. If we first solve $\\dot{r}'_\\gamma=0$ for $b$ \nand replace its value in \n$\\dot{r}_\\gamma=0$, (as the path followed by Hod) we obtain Eq.(31) \nof\\cite{Hod11}. Alternatively, \nif we first solve for $b$ \nfrom $\\dot{r}_\\gamma=0$ and replace it in $\\dot{r}'_\\gamma=0$, \nwe arrive to his Eq.(21).   \n\nAs mentioned above, in \\cite{Hod11}, by comparing the total \norbital period for rotating Kerr\nblack holes with those of Schwarzschild ones, \na lower bound for the orbital \nperiods \n$T_\\infty$ (as measured from far away \nobservers) of circular null geodesics around compact objects with \nmass $\\mathcal{M}$, \nEq.(\\ref{Tinf}) was conjectured. \nNow, by giving an explicit counterexample, we will show that, \nin contrast to the geometrical \nstatus\nof fastest way to circle black holes, this conjecture cannot be \ngenerally valid in other black holes candidates coming from \nalternative gravitational theories.\n\nThe counterexample is based on the Kaluza-Klein \nblack hole\\cite{Aliev13}. Shadows of these kinds of \nblack holes were recently analyzed in\\cite{Eiroa13}.\nLet us consider the following metric, which satisfies \nthe vacuum Einstein equations in five dimensions. \n\\begin{equation}\nds^2=-(1-\\frac{2m}{r})dt^2+\\frac{dr^2}{1-\\frac{2m}{r}}\n+r^2(d\\theta^2+\\sin^2(\\theta)d\\phi^2)+dy^2.\n\\end{equation}\nThen, by doing a compactification of the extra dimension \nand a boost transformation with \nvelocity $v$ in the $y$-direction, \nand by projecting in the $4$-manifold, a new metric is obtained \nrepresenting a charged \nspherically symmetric black hole together with a \ndilaton field\\cite{Aliev13}. \nThe four-dimensional metric reads:\n\\begin{equation}\n\\begin{split}\nds^2&=-\\frac{1-\\frac{2m}{r}}{B}dt^2+\\frac{B}{1-\\frac{2m}{r}}dr^2+\\\\\n& Br^2\\left(d\\theta^2+\\sin(\\theta)^2d\\phi^2\\right),\n\\end{split}\n\\end{equation}\nwith\n\\begin{equation}\nB=\\sqrt{1+\\frac{2mv^2}{r(1-v^2)}}.\n\\end{equation}\nThis metric together with the\nscalar field \n\\begin{equation}\n\\Phi=-\\frac{\\sqrt{3}}{2}\\ln{B},\n\\end{equation}\nand the electromagnetic potential $\\mathcal{A}$\n\\begin{equation}\n\\mathcal{A}=\\frac{v}{2(1-v^2)}\\frac{1-\\frac{2m}{r}}{B^2}dt,\n\\end{equation}\nsatisfies the set of Einstein-Maxwell-dilaton field equations \nthat follow from the action:\n\\begin{equation}\nS=\\int d^4x\\sqrt{-g}\\left[R+2(\\nabla\\Phi)^2\n-e^{2\\sqrt{3}\\Phi}F^2\\right].\n\\end{equation}\nThis metric represents a black hole with ADM mass $\\mathcal{M}$ and\ncharge $Q$ given in terms of the parameter $m$ and the velocity $v$ \nby:\n\\begin{eqnarray}\n\\mathcal{M}&=&m\\left[1+\\frac{v^2}{2(1-v^2)}\\right],\\\\\nQ&=&\\frac{mv}{1-v^2}.\n\\end{eqnarray}\n\nOne of the characteristics of this type of metric is that the ADM mass,\nthe dilation field and the charge depend on the boost parameter.\nNote that the event horizon is located at $r_H=2m$, which \nfor a fixed mass $\\mathcal{M}$, shrinks \nto zero when $v$ goes to $1$.\nLet us compute now the minimum orbital time that a circular null geodesic \ntakes to orbit this black hole.\nIn order to do this, we must solve for the co-rotating orbits, that is $G_{-}=0$, \nwhich, using the expression (\\ref{eq:G}) in this case gives\n\\begin{equation}\n\\begin{split}\n&(v^2-2)^2r^2-2\\mathcal{M}(v^2-2)(4v^2-3)r\\\\\n&-16\\mathcal{M}^2v^2(1-v^2)=0.\n\\end{split}\n\\end{equation}\nThe physical solution of this equation is\n\\begin{equation}\nr_\\gamma=\\frac{3-4v^2+\\sqrt{9-8v^2}}{2-v^2}\\mathcal{M}.\n\\end{equation}\nBy replacing this radius in (\\ref{eq:T}), \nwe obtain\n\\begin{equation}\nT_\\infty=2\\pi\\frac{3-4{v}^{2}+\\sqrt{-8{v}^{2}+9}}{2-{v}^{2}}\\sqrt{\\frac{\n3+\\sqrt{-8{v}^{2}+9}}{-1+\\sqrt{-8{v}^{2}+9}}}\\mathcal{M}.\n\\end{equation}\nA plot of this function is shown in Fig. 2. At $v=0$ (Schwarzschild case)\nit takes the value $T_\\infty=6\\sqrt{3}\\pi\\mathcal{M}$, \nand it goes to zero when $v\\rightarrow 1$.\n\n\\begin{figure}[]\n\\centering\n\\includegraphics[clip,width=80mm]{Tinfinity.eps}\n\\caption{Plot of the quotient $T_\\infty\/\\mathcal{M}$ as \na function of $v$ We assume $G=1$.\n}\n\\label{fig:Tinfi}\n\\end{figure}\nConsequently, the orbital period around a black hole with mass $\\mathcal{M}$\ncan be made arbitrarily small.\nTherefore, there is a value of $v$ from which the Hod conjecture \ncannot be satisfied for this type of black holes. \nNumerically we found that for $v>0.96591$ all the orbital periods $T_\\gamma$ \nare smaller than $4\\pi\\mathcal{M}$.  \n\n\\section{Summary}\nIn this work, we have found upper bounds for the radius of \nphoton spheres in $n$-dimensional\nGR and EGB theories, \nthereby generalizing the previous work of Hod in\\cite{Hod13}.\nIn the general situation of a black hole dressed with matter \nfields, we have seen that the photon\nsphere radius is always smaller than that corresponding \nto a vacuum black hole with the same mass. \nIt would be interesting to know how generic these results \nare, in the sense of\nhow much they depend on gravitational \nfield equations. In particular, it would be natural to study whether these results \nhold for the more general \nLovelock gravitational theory or if they can be extended to other\ntheories like $f(R)$ ones.\n\nWith respect to the study of circular null geodesics in axial \nsymmetry, we have observed that equatorial circular null geodesics \nalways minimize \nthe orbital time around a black hole. It is a geometrical result. Given \nthe astrophysical importance of this type of geodesics, it would \nalso be interesting to attempt a proof of the Hod conjecture (\\ref{Tinf}). \nWe wish to deal with these and other associated problems in the near future.\\\\\n\n\n\n\\section*{Acknowledgments}\nE. Gallo thanks the Instituto de F\\'isica y\nAstronom\\'ia, Universidad de Valpara\\'iso for their kind hospitality, \nwhile working on this paper. \nE. Gallo acknowledge financial support from CONICET and SeCyT-UNC.\nJ. R. Villanueva is supported by Comisi\\'on Nacional de Investigaci\\'on Cient\\'ifica \ny Tecnol\\'ogica through FONDECYT grants No  11130695.\nIn this post-publisher version two typos were corrected: the signature of the metric in eq.(1) and a factor in eq.(102). We thanks Gary Gibbons and Chris Pope for bringing to our attention these typos. \n\n\\begingroup\\raggedright","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sec:intro}\n\nHierarchical structure formation is one of the most profound predictions of the standard model of cosmology --- the Lambda--Cold Dark Matter ($\\Lambda$CDM) cosmology \\citep[e.g.,][]{10.1093\/mnras\/183.3.341,10.1086\/183911,10.1038\/311517a0,2010gfe..book.....M}. In a $\\Lambda$CDM universe, dark matter is attracted to local density peaks of initial fluctuations, laid down during inflation, forming dark matter haloes. These dark matter haloes grow and merge with one another, and are believed to be the nests within which visible galaxies develop. The spatial distribution of dark matter haloes is hence a powerful prediction of the $\\Lambda$CDM model, and can be compared with observations of galaxy distribution to test our understanding of cosmology and galaxy formation physics. \n\nDespite the nonlinearity in the formation and merging of dark matter haloes, we have a good general picture of the distribution of dark matter haloes, thanks to extensive studies using both mathematical approximations and numerical simulations. In particular, the excursion set formalism provides an analytic description of the halo mass function, halo mass assembly histories, and halo clustering properties  (\\citealt{Press1974,1991ApJ...379..440B}; for a review see \\citealt{10.1142\/S0218271807010511} and Chapter 7 of \\citealt{2010gfe..book.....M}). In this context, the clustering properties of haloes are a function of halo mass alone. This is commonly referred to as ``halo bias'' and can be approximated analytically \\citep[e.g.,][]{Kaiser84,Cole1989,Mo1996,Sheth2001} or calibrated against numerical simulations \\citep[e.g.,][]{Tinker2010}. These analytic descriptions of halo bias are powerful tools within galaxy models that rely on halo mass as the link between haloes and the galaxies that they host, such as the Halo Occupation Distribution \\citep[e.g.,][]{Peacock2000,Seljak2000,Berlind2002,Zheng2005,Zehavi2005} and the Conditional Luminosity Function \\citep[e.g.,][]{Yang2003,vandenBosch2013}, to predict galaxy clustering efficiently. \n\nOn the other hand, it has also been shown, within cosmological $N$-body simulations, that halo clustering also depends on halo properties other than mass, amongst which the most notable is halo assembly history  \\citep{2001PhDT.........7W,Gao2005,Wechsler2006,2008MNRAS.389.1419L}. This is commonly referred to as ``halo assembly bias.'' Although halo assembly bias is a smaller effect than mass-dependent halo bias, it has drawn increasing attention from researchers who model the galaxy--halo connection \\citep[e.g.,][]{1507.01948,Hearin2016,Zentner2016,1611.09787,Lehmann2017,2017MNRAS.469.1809R}, because it is very plausible that the galaxy assembly history is to some extent connected to the assembly history of its host halo. If this is the case, one would need to understand halo assembly bias to predict accurately galaxy clustering and to mitigate potential bias in any inference from clustering measurements \\citep{Reddick2013,Zentner2014}. \n\nIn the excursion set formalism, a characteristic mass $M^*$, below which most haloes have already collapsed and formed, is defined to satisfy $\\sigma(M^*)=\\delta_c D(a)$, where $\\delta_c \\simeq 1.686$ is the critical overdensity, $D(a)$ is the linear growth rate, and $\\sigma(M)$ is the squared root of the mass variance with a top-hat filter of mass $M$. \nStudies have found that, below the characteristic mass $M^*$, haloes that form earlier are more strongly clustered \\citep{2001PhDT.........7W,Gao2005,Wechsler2006,2008MNRAS.389.1419L}. However, above the characteristic mass $M^*$, the signal of halo assembly bias is less clean. \\citet{Wechsler2006} found that, above the collapse mass, haloes with lower \\emph{concentration} are more clustered, but detected no clear bias as a function of halo formation time, despite the correlation between concentration and formation time \\citep{Wechsler2002}. Similarly, \\citet{2008MNRAS.389.1419L} inspected several different definitions of halo formation time, but found little assembly bias for high-mass haloes upon splitting haloes by their formation times. Attempts have been made to explain the physical origin of halo assembly bias \\citep{Sandvik2007,10.1142\/S0218271807010511,Desjacques2007,Dalal2008,2009MNRAS.396.2249W}, and while they provided some heuristic insights, they do not explain why different proxies of halo assembly history (e.g., concentration and formation time) exhibit assembly biases of very different magnitudes for massive haloes.\n\nOn the observational front, recent efforts have been made to detect halo assembly bias with galaxy clusters \\citep{Yang2006,Tinker2012,Miyatake2015,More2016,Dvornik2017}. However, results remain inconclusive as various systematic effects can masquerade as observed halo assembly bias \\citep{Lin2016,2016arXiv161100366Z,1702.01682}. One challenge facing observations is that, because halo properties are not directly observable, one has to use some observational proxy to approximate halo assembly history. In the studies of \\citet{Miyatake2015,More2016,Dvornik2017}, they use the average distance of member galaxies in the cluster as a proxy of halo formation time. The motivation for such a proxy is the well-documented correlation between halo concentration and halo formation time \\citep{Wechsler2002} and the assumed correlation between halo concentration and member galaxy distances. Nevertheless, even if a bias signal due to the average distance of member galaxies had been robustly detected observationally, it would not be clear whether or not this bias is of the same physical origin as the halo assembly bias identified in cosmological $N$-body simulations. \n\nIn addition to halo assembly bias, the clustering of haloes also depends on other halo properties, such as spin, shape, and substructure abundance \\citep[e.g.,][]{2001PhDT.........7W,Wechsler2006,Bett2007,Gao2007,2007MNRAS.375..489H,Faltenbacher2010}. {More recently, there are also studies that explore the relationships amongst different kinds of halo assembly bias \\citep[e.g.,][]{1612.04360,1708.08451}.}  In much of the literature, these other \\emph{secondary halo biases} (i.e., dependence of halo clustering on halo properties other than mass) are also commonly referred to as ``assembly bias,'' regardless of whether or not the secondary halo properties have a direct connection to halo assembly history. This nomenclature could result in some confusion as to whether all the different kinds of secondary halo bias have the same physical origin. While halo assembly history is arguably the most important property other than halo mass that characterizes individual haloes, when it comes to the clustering properties, it is still unclear if or how the different secondary halo biases connect with one another. \n\n\n\nIn this study, we join the exploration of halo secondary bias with two specific aims. First, we examine the dependence of halo clustering on a set of secondary halo properties for cluster-size haloes using a modern, large-volume cosmological simulation to validate previous results, some of which may have suffered from limited volume or resolution. Secondly, we inspect the correlations between different halo properties and how they connect to the secondary bias, to better understand the relation between assembly bias and all different kinds of secondary biases. To this end, we present a novel way to characterize secondary bias, which helps us to gain insight into these questions. \n\nWe focus our current study on cluster-size haloes for three reasons. First, in the cluster-mass regime, the behaviour of assembly bias due to different proxies, such as halo concentration and formation time, is poorly understood. Secondly, at the high-mass end of the halo mass spectrum, halo assembly bias is thought not to be due to nonlinear interactions between neighbouring haloes, but reflective of the initial conditions for structure formation \\citep{10.1142\/S0218271807010511,Dalal2008}. Therefore, it seems plausible that assembly bias is simpler at high mass.\nThird, there have been, and likely will soon be more, observational attempts to directly measure halo assembly bias with galaxy clusters. Hence, this is a timely and crucial study to facilitate the interpretation of current measurements and preparation for forthcoming observations.  \n\nThis paper is structured as follows. In \\autoref{sec:methods}, we introduce the simulation used in this study, define the halo properties, and explain how we remove mass dependence in the bias. We present our main results in \\autoref{sec:results}, where we show and compare the secondary halo biases due to different halo properties, with \\autoref{fig:bias} summarizing this result, and also present a novel way to characterize secondary bias. In \\autoref{sec:discussion}, we turn to explore the interplay between correlation and bias, demonstrate that the correlation between different halo properties does not determine the secondary bias they exhibit, and also discuss the implication for galaxy assembly bias. \\autoref{fig:history} highlights the absence of assembly bias (in its strict definition) at this mass scale, despite the existence of other secondary halo biases. We conclude in \\autoref{sec:conclusion}. We also include a summary of the correlations amongst all secondary halo properties used in this study and their dependence on the environment in Appendix~\\ref{sec:appendix}.\n\n\\begin{figure*}\n\\centering\\includegraphics[width=\\textwidth]{marks.pdf}\n\\caption{\\label{fig:marks}%\nEach scatter plot shows the relation between halo mass and one of the secondary halo properties: concentration parameter \\ensuremath{c}, spin parameter \\ensuremath{\\lambda}, half-mass scale \\ensuremath{a_{1\/2}}, accretion rate before peak mass \\ensuremath{\\Gamma_\\text{peak}}, number of subhaloes \\ensuremath{N_\\text{sub}}, and averaged subhalo distance \\ensuremath{R_\\text{mem}}. All distinct haloes in the sample are plotted and coloured by their mark values, which are always between 0 and 1 and are assigned according to the rank of the secondary property within each halo mass bin as explained in \\autoref{sec:marks}.\n}\n\\end{figure*}\n\n\\section{Methods}\n\\label{sec:methods}\n\n\\subsection{Simulations}\n\\label{sec:sims}\n\nIn this study we use the MultiDark Planck 2 (MDPL2) simulation \\citep{Klypin2016}. MDPL2 is a cosmological gravity-only $N$-body simulation, run with the \\textsc{L-Gadget2} code. It has a periodic volume of 1\\,\\ensuremath{h^{-1}}\\,Gpc$^3$, with 3840$^3$ particles. The mass resolution of each particle is $1.51\\times10^9$\\,\\ensuremath{h^{-1}}\\,\\text{M$_\\odot$}, and the physical force resolution ranges from 5 to 13\\,\\ensuremath{h^{-1}}\\,kpc (smaller at lower redshift). \nMDPL2 adopts the Planck 2013 $\\Lambda$CDM cosmology \\citep{Planck2013cosmology}, and the actual values used in MDPL2 are: total matter density $\\Omega_\\text{m} = 0.307115$, dark energy density $\\Omega_\\Lambda = 1 - \\Omega_\\text{m}$, baryon density $\\Omega_\\text{b} = 0.048206$, Hubble parameter $h = 0.6777$, scalar spectral index $n_s = 0.96$, and the amplitude of mass density fluctuation $\\sigma_8 = 0.8228$.\n\nThe MDPL2 simulation has been analysed by the \\textsc{Rockstar} halo finder and the \\textsc{Consistent Trees} merger tree builder \\citep{2013ApJ...762..109B,2013ApJ...763...18B}. \nThe halo mass definition used here is the virial mass; in this cosmology the virial overdensity corresponds to approximately 100 times the critical density \\citep{Bryan1998}.\nA halo is called a ``subhalo'' if its centre is within the virial radius of any larger halo. Any halo that is not a subhalo is called a ``distinct halo.''\n\nIn this study, we use haloes from the present-day ($z=0$) halo catalogue. We select all distinct haloes with a present-day virial mass $\\ensuremath{M_\\text{vir}} \\geq 10^{14}$\\,\\ensuremath{h^{-1}}\\,\\text{M$_\\odot$}{} as our full halo sample. There are 27,029 distinct haloes in this sample, corresponding to a number density of $2.7\\times 10^{-5}\\,h^3\\,\\text{Mpc}^{-3}$.\n\nAlthough the MDPL2 halo catalogues contain information about halo mass assembly histories, we do not have direct access to the full assembly history of each individual halo. Hence we also use the DarkSky-Gpc simulation when the full assembly history is required (\\autoref{fig:history}). DarkSky-Gpc ({\\tt ds14\\_b}) is part of the Dark Sky Simulations \\citep{Skillman2014}, run with the \\textsc{2HOT} code \\citep{Warren2013}. It also has a periodic volume of 1\\,\\ensuremath{h^{-1}}\\,Gpc$^3$, and was run with 10240$^3$ particles and a mass resolution of $7.63\\times10^7$\\,\\ensuremath{h^{-1}}\\,\\text{M$_\\odot$}. \nDarkSky-Gpc has also been analysed by the \\textsc{Rockstar} halo finder; however, the full \\textsc{Rockstar--Consistent Trees} halo catalogues and merger trees have only been constructed on a downsampled version, which has only $10240^3\/32 \\simeq 3225^3$ particles and an effective mass resolution of $2.44\\times10^9$\\,\\ensuremath{h^{-1}}\\,\\text{M$_\\odot$}%\n\\footnote{See also \\citet{2016PhDT........76M,Lehmann2017}. We note that these two studies mistakenly reported the DarkSky-Gpc particle mass to be twice its actual value.}, similar to the mass resolution of MDPL2. In this study we only use the downsampled version of DarkSky-Gpc. DarkSky-Gpc adopts a flat cosmology close to the Planck 2013 $\\Lambda$CDM cosmology, with $h = 0.688$, $\\Omega_\\text{m} = 0.295$, $n_s = 0.968$, $\\sigma_8 = 0.834$. We also use the virial overdensity as the halo mass definition for DarkSky-Gpc.\n\nWhile we present most of our result (all except for \\autoref{fig:history}) using the MDPL2 simulation because it is publicly available and has slightly better mass resolution, we have verified that the DarkSky-Gpc simulation produces very similar result, with no qualitative difference and little quantitative difference. Our result holds in both simulations.\n\n\n\\subsection{Secondary halo properties}\n\\label{sec:properties}\n\nIn this study we select six different halo properties other than halo mass and investigate the secondary halo bias due to these properties. The properties we choose are as follows:\n\n\\begin{enumerate}\n\\itemsep0.5\\baselineskip\n\n\\item Concentration parameter (\\ensuremath{c}), obtained by fitting the dark matter density profile to a Navarro--Frenk--White (NFW) profile \\citep{Navarro1996,Navarro1997}.\n\n\\item Spin parameter (\\ensuremath{\\lambda}), as defined in \\citet{Peebles1969}.\n\n\\item Half-mass scale (\\ensuremath{a_{1\/2}}), defined as the scale factor at which a distinct halo reaches more than or equal to half of its present-day ($z=0$) mass on its main branch.\n\n\\item Accretion rate {of} peak mass (\\ensuremath{\\Gamma_\\text{peak}}), the average halo mass accretion rate (in \\ensuremath{h^{-1}}\\,\\text{M$_\\odot$}\\,yr$^{-1}$) {of peak mass, i.e., \\[\\left[\\ensuremath{M_\\text{peak}}(z=0) - \\ensuremath{M_\\text{peak}}(z=0.5)\\right]\/\\left[t(z=0) - t(z=0.5)\\right].\\]} Since our halo sample contains the most massive haloes in the simulation, the majority (68.7\\%) have $\\ensuremath{z_\\text{peak}} = 0$ {(the redshift when peak mass takes place)}. The median \\ensuremath{z_\\text{peak}}{} for haloes whose present-day mass is not peak mass (i.e., $\\ensuremath{z_\\text{peak}} > 0$) is 0.093. Hence, for our halo sample, \\ensuremath{\\Gamma_\\text{peak}} is basically the average halo mass accretion rate for $0 < z < 0.5$ .\n\n\\item Number of subhaloes (\\ensuremath{N_\\text{sub}}). We count the number of subhaloes (identified by the `upid' value in the \\textsc{Rockstar--Consistent Trees} catalogue) that have a peak maximal circular velocity (\\ensuremath{V_\\text{peak}}) above 135\\,\\text{km\\,s$^{-1}$}. The peak maximal circular velocity is defined as the largest value of the maximal circular velocity on the main branch of the subhalo in consideration. \n\n\\item Average subhalo distance (\\ensuremath{R_\\text{mem}}), defined as the average three-dimensional distance between all subhaloes and the centre of the main halo. The subhalo definition is the same as the definition when we calculate the number of subhaloes for each distinct halo.\n\n\\end{enumerate}\n\nFor \\ensuremath{c}, \\ensuremath{\\lambda}, \\ensuremath{a_{1\/2}}, and \\ensuremath{\\Gamma_\\text{peak}}, we use the values in the \\textsc{Rockstar--Consistent Trees} catalogue directly.\n\n\n\\subsection{Definition of mass-normalized marks}\n\\label{sec:marks}\n\nFormally, we say a secondary halo bias exists if two groups of haloes that have the same halo mass but different values of a secondary propriety cluster differently. However, practically, because it is difficult to measure spatial clustering within an infinitesimally thin mass bin due to the finite number of haloes even within our large-volume simulations, we need to remove any bias that may be induced by halo mass when we measure the secondary bias. \n\nTo do so, for each secondary halo property in consideration, we assign a ``mass-normalized'' mark value to each halo to represent the secondary halo property. We first bin haloes by halo mass, and then in each mass bin, we assign the mark value based on the rank within that mass bin. Hence, within each mass bin and also overall, the mark values always span $[0,1]$ uniformly. The mark values are always dimensionless.\n\nFor secondary halo properties that are discrete, such as the number of subhaloes and the half-mass scale (which is discrete because of the time interval between the simulation snapshots that are saved), if a group of haloes in a particular mass bin share the same exact value, they are randomly assigned different mark values in a way that still preserve the overall ranks. For example, when assigning the mark for the number of subhaloes, we add a random number drawn from a continuous uniform distribution on $[0, 1)$ to the number of subhaloes before the ranking process. In this fashion, haloes with one subhalo would have different mark values, but their mark values will always be smaller than the mark values of haloes with two subhaloes. This procedure hence ensures that the mark is uniformly distributed and that it is not clustered at a certain value, yet it does not introduce noise in the overall ranks.\n\nThe sample of distinct haloes used in this study spans the mass range of $14 \\leq \\log\\,\\left[M\/(\\ensuremath{h^{-1}}\\,\\text{M$_\\odot$})\\right] < 15.55$. We split this mass range into 30 bins. The bin widths increase quadratically in log mass, such that the lowest mass bins do not contain the vast majority of haloes in the sample. \\autoref{fig:marks} shows the relations between the halo mass and each of the secondary halo properties we considered here. The mark value we assigned to each halo is represented by the colour of each point. We can observe the binning effect on the mark values for properties that change more rapidly with halo mass (\\ensuremath{\\Gamma_\\text{peak}}, \\ensuremath{N_\\text{sub}}, and \\ensuremath{R_\\text{mem}}). Nevertheless, we have tested and find our results are insensitive to the binning schemes. When we use different binning schemes, including uniformly spacing in log mass and also different number of bins, the result still holds.\n\n\n\\section{Results}\n\\label{sec:results}\n\n\\subsection{Secondary halo bias manifested in the bias function}\n\\label{sec:bias}\n\n\\begin{figure}\n\\centering\\includegraphics[width=\\columnwidth]{bias.pdf}\n\\caption{\\label{fig:bias}%\nThe ratio between the pair counting function ($n_\\text{pairs}$) of only haloes with high marks and that of only haloes with low marks for each of the six secondary halo properties: (solid lines from top to bottom) \\ensuremath{\\lambda}, \\ensuremath{R_\\text{mem}}, \\ensuremath{N_\\text{sub}}, \\ensuremath{\\Gamma_\\text{peak}}, \\ensuremath{a_{1\/2}}, and \\ensuremath{c}. A high (or low) mark value means a mark value above (or below) 0.5. The pair counting function is calculated by counting all pairs of haloes that have three-dimensional distances between 2 to 40\\, \\ensuremath{h^{-1}}\\,Mpc, and binned in distance ($r$), normalized by the square of number of haloes. For the concentration parameter (blue solid line), we also show the inverse ratio as a blue dashed line to guide an easy comparison with bias due to other secondary halo properties. For comparison, a thin horizontal black line shows a ratio of unity, and the grey band shows the 3$\\sigma$ deviation for a randomly-assigned mark values. The deviation of the coloured lines from the horizontal black line shows a bias due to the secondary halo property.\n}\n\\end{figure}\n\nThe most straightforward way to evaluate halo bias is to calculate the ratio between the pair counts of two samples of haloes. This is sometimes called the bias function. If the two samples have different numbers of total haloes, one needs to normalize the pair count first, usually by dividing out the expected number of pairs of a set of uniformly distributed random points. As for secondary halo bias, one can then calculate the bias function between two halo samples that differ in a secondary halo property, but not in halo mass. \n\nHere we present in \\autoref{fig:bias}, for each of the six secondary halo properties listed in \\autoref{sec:properties}, the bias function (or more precisely, the ratio between the pair counting functions) of two samples of haloes, split by the mark values. For each case, one sample has all haloes with mark values above 0.5, and the other has all haloes with mark values below 0.5. That is, we compare haloes in the upper half of the mark distribution with those in the lower half of the mark distribution at \\emph{fixed} mass (cf., \\autoref{fig:marks}). Note that because the mark values are assigned in mass bins, the split by mark already excludes any clustering dependence upon mass, and the bias we observe is the secondary bias.\n\nThe pair counting function is evaluated in bins of pair distance ($r$). Here we use 11 equally-spaced bins in logarithmic scale between 2 to 40\\,\\ensuremath{h^{-1}}\\,Mpc, and we have verified that our result is insensitive to the binning scheme.\nWe also estimate how much of the bias signal in the ratio of pair counting functions can come from noise. We uniformly at random assign mark values of $1\/N, 2\/N, \\ldots, 1$ to all haloes in the sample, calculate the ratio of pair counting functions for high and low randomly assigned marks, and repeat this procedure until we obtain convergence of the 99.7\\% (3$\\sigma$) distribution of the bias signal; this is  shown by the grey band in \\autoref{fig:bias}. Hence, deviation beyond this grey band indicates a secondary halo bias signal $>3\\sigma$.\n\n\\begin{figure*}\n\\centering\\includegraphics[width=\\textwidth]{markdist.pdf}\n\\caption{\\label{fig:markdist}%\nThe probability distributions of mark values of each of the six secondary halo properties, when only counting paired haloes. A ``paired halo'' is any distinct halo that is neighboured with at least one other distinct halo in the full halo sample within 10\\,\\ensuremath{h^{-1}}\\,Mpc. If a halo has in multiple neighbour haloes, its mark value is counted only once in the distribution shown by an orange solid line, and counted multiple times (as many as the number of its neighbours) in the distribution shown by a red dashed line. \nFor comparison, a thin horizontal black line shows a uniform distribution, which, by construction, is the distribution of the mark values when including all, paired or not, haloes. The deviation of the coloured lines from the horizontal black line shows a bias due to the secondary halo property. The grey band shows a typical 3$\\sigma$ deviation for a uniform random distribution of the same sample size.\n}\n\\end{figure*}\n\n\\autoref{fig:bias} shows that, amongst the six secondary halo properties, the two that directly quantify some aspect of the mass accumulation histories of haloes, namely the half-mass scale $\\ensuremath{a_{1\/2}}$, and the accretion rate before peak mass $\\ensuremath{\\Gamma_\\text{peak}}$, do not exhibit secondary bias; in both cases, the ratio between the the pair counts is consistent with random marks on all scales.\n\nAll other halo properties exhibit clear secondary bias. Haloes with higher concentrations are less clustered, while haloes with higher spins, more subhaloes, or a larger average subhalo distances are more clustered. In terms of the magnitude of the secondary bias, the spin parameter exhibits the strongest amongst these properties. The concentration parameter, number of subhaloes, and the average subhalo distance all produce a secondary bias of the roughly same magnitude. \\autoref{fig:bias} also shows that, the secondary bias due to the properties listed decreases with scale. Nevertheless, between scales of 2 to 40\\,\\ensuremath{h^{-1}}\\,Mpc, this secondary bias is always statistically significant. Moreover, on all of the scales that we have considered, the secondary bias induced by any individual property never crosses the black horizontal line at unity, meaning that the secondary bias is always of the same sense. \n\nThis result is broadly consistent with previous studies. In this mass regime, the inverted concentration bias has been shown in \\citet{Wechsler2006}, the strong spin bias has been shown in \\citet{Gao2007}, the lack of assembly bias when split by formation time has been shown in \\citet{Gao2005,2008MNRAS.389.1419L}, and the bias due to subhalo distance has been shown in \\citet{More2016}. Our study confirms that the secondary bias signals in previous studies can be reproduced with a larger-volume and higher-resolution simulation. {However, we caution the reader that we present the pair counting ratio in \\autoref{fig:bias}, but this quantity \ndoes not directly translate into the ratio of linear biases of the subsamples. We will not quote linear biases in this work because the \nMDPL2 particle data are not publicly available.}\n\nIt is interesting to note that the term ``assembly bias'' has been widely use to refer to any secondary halo bias, despite the fact that {when splitting cluster-size haloes by their half-mass scales or recent accretion rates, the two sample have indistinguishable halo clustering properties}%\n\\footnote{We note that for low-mass haloes, which we do not address in this work, {there exists assembly bias even with this strict definition}.}.\nThe reason for this nomenclature is very likely due to the common understanding that halo assembly history is correlated with many halo properties, including concentration, spin, and so on. Nevertheless, the fact is that {some} direct measures of halo assembly history (e.g., $\\ensuremath{a_{1\/2}}$ and $\\ensuremath{\\Gamma_\\text{peak}}$) result in no significant secondary bias at this mass regime, and this causes the ``assembly bias'' nomenclature to be potentially confusing and misleading. We will discuss this seemingly counter-intuitive result further in \\autoref{sec:discussion}. \n\n\\subsection{Secondary halo bias through the demography of paired haloes}\n\\label{sec:markdist}\n\n\\begin{figure*}\n\\centering\\includegraphics[width=0.9\\textwidth]{scatter.pdf}\n\\caption{\\label{fig:scatter}%\nA matrix of two-dimensional distributions of the mark values for each pair of the six secondary halo properties. In each cell, the $x$- and $y$-axes both go from 0 to 1 and show the mark value of the corresponding labels. The lower triangular cells of the matrix (with blue points) depict the distributions when only including haloes that are \\emph{not} in pairs, while the upper triangular cells (with orange points) exhibit the distributions including only paired haloes. Paired haloes are those with at least one other distinct halo within 10\\,\\ensuremath{h^{-1}}\\,Mpc. The number in each cell is the value of the Spearman correlation coefficient. In the absence of correlations, the absolute value of Spearman correlation coefficient would be smaller than $0.03$ at the $99\\%$ level  ($3\\sigma$) for a sample the size of our halo sample. Diagonal cells are left blank as only a trivial perfect correlation would appear in those cells. For each cell in the upper triangular portion of the matrix, the marginal distribution of the points would match the orange solid lines in \\autoref{fig:markdist}.\n}\n\\end{figure*}\n\nA different, yet arguably more intuitive, way to demonstrate the halo bias is to first split the halo sample by whether or not the halo contributes to the correlation (pair counting) function. Here we define a ``paired'' halo as any distinct host halo that is within 10\\,\\ensuremath{h^{-1}}\\,Mpc of another cluster-size distinct halo, and an ``unpaired'' halo is an isolated halo that has no other cluster-size distinct haloes within 10\\,\\ensuremath{h^{-1}}\\,Mpc.\nThe choice of 10\\,\\ensuremath{h^{-1}}\\,Mpc is, to some extent, arbitrary, but we have verified that our results are insensitive to the choice of this radius in the range of 5--20\\,\\ensuremath{h^{-1}}\\,Mpc. \nOnce we divide the full halo sample into paired and unpaired subsets, we can inspect the difference between the demography of these two samples, and the difference is a manifestation of halo secondary biases.\n\nWe start with \\autoref{fig:markdist}, showing the mark distribution of the six secondary halo properties for paired haloes only. If paired haloes form an unbiased subset of all haloes, the mark distribution of only paired haloes should be identical to the distribution of the full sample, which \\emph{by construction} is uniformly distributed between $[0,1]$. \nConsequently, the deviations from the uniform distributions in \\autoref{fig:markdist} highlight the secondary halo bias. Here we find that paired haloes are strongly biased towards lower concentration and higher spin, somewhat biased towards higher number of subhaloes and larger average subhalo distance, and nearly unbiased in half-mass scale and accretion rate before peak mass. All signals are consistent with the secondary bias signals in \\autoref{fig:bias}. To verify that these signals are robust, we also show the mark distribution for all paired haloes with repeated counting, to account for the fact that some haloes contribute to multiple pairs and those haloes contribute more to the bias function. Nevertheless, in both counting schemes, the results are generally the same.\n\nNote that this ``demography of paired haloes'' approach, demonstrated with \\autoref{fig:markdist}, contains similar, but not identical, information to the bias function approach shown in \\autoref{fig:bias}. For example, the cross-correlation between high- and low-mark haloes does not contribute to \\autoref{fig:bias}, while any halo in pairs contributes to \\autoref{fig:markdist}. Furthermore, \\autoref{fig:markdist} helps us to understand the demography of paired haloes by allowing us to inspect the actual mark distribution for paired haloes. We can see that, for instance, the distributions of marks that exhibit secondary bias, are generally monotonic with respect to the mark values. This feature indicates that the secondary bias is not due to intermediate mark values but mostly due to high- or low-end tails in the mark distribution. In the next section we will further see the usefulness of this ``demography of paired haloes'' approach. \n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\n\\subsection{Correlations between halo properties}\n\\label{sec:scatter}\n\nThe term ``halo assembly bias'' has been widely used to refer to any kind of secondary halo bias. While this is an issue of nomenclature, the extensive use of the term ``halo assembly bias'' has a significant drawback, as it implicitly suggests that all secondary halo bias may have originated from differences amongst the assembly histories of haloes. However, for haloes with the characteristic masses of galaxy clusters, we do not find significant secondary bias for \\ensuremath{a_{1\/2}}{} and \\ensuremath{\\Gamma_\\text{peak}}{}, the two halo properties that we study which are directly related to halo assembly histories. This finding is in broad agreement with the detailed findings in the previous literature, though not always with the physical interpretation of them. In particular, despite the lack of assembly bias in its strict sense, there is still clear secondary halo bias for other properties that are correlated with halo assembly history, such as halo concentration. How can we understand better these counter-intuitive and seemingly contradictory results? \n\nWe start by inspecting the correlations amongst the secondary halo properties. \\autoref{fig:scatter} shows the scatter plot and the Spearman correlation coefficient between the mark values for each pair amongst the six secondary properties. \\autoref{fig:scatter} shows these distributions for two subsets of haloes, namely paired haloes only (upper triangular cells in orange) and unpaired haloes only (lower triangular cells in blue). In order to interpret \\autoref{fig:scatter}, it is useful to consider that in the absence of correlations, the Spearman correlation coefficient for a sample of the size of our halo subsets would be limited to an absolute value less than 0.03 at the 99.7\\% ($3\\sigma$) level. Therefore, for example, the correlation coefficient of $-0.06$ describing the correlation between $\\lambda$ and $\\ensuremath{R_\\text{mem}}$ is a weak, but likely real, correlation. Correlation coefficients with a magnitude larger than this are very highly significant.\n\nAt first glance, most correlations are as expected based upon the previous literature. The half-mass scale \\ensuremath{a_{1\/2}}{} and the accretion rate before peak mass \\ensuremath{\\Gamma_\\text{peak}}{} are strongly correlated, because the assembly histories of haloes are rather universal and in most cases can be well described by a one-parameter function \\citep{Wechsler2002,Wu2013}. \nHalo concentration \\ensuremath{c}{} is well correlated with both \\ensuremath{a_{1\/2}}{} and \\ensuremath{\\Gamma_\\text{peak}}{}, consistent with the finding of \\citet{Wechsler2002}. Halo concentration \\ensuremath{c}{} is also fairly correlated with the number of subhaloes \\ensuremath{N_\\text{sub}}{}, consistent with the finding of \\citet{Zentner2005,Mao2015}. The halo spin parameter \\ensuremath{\\lambda}{} is correlated with \\ensuremath{c}{}, \\ensuremath{a_{1\/2}}{}, and \\ensuremath{\\Gamma_\\text{peak}}{}, also consistent with the finding of \\cite{2007MNRAS.378...55M}. \nThe average subhalo distance does not exhibit strong correlations with the other five halo properties, but correlates somewhat weakly with \\ensuremath{a_{1\/2}}{} and \\ensuremath{\\Gamma_\\text{peak}}{}; this is consistent with the findings of \\citep{More2016}.\n\nA more unexpected feature of \\autoref{fig:scatter} is that the correlations amongst these halo properties \\emph{seem} to be nearly the same for paired and unpaired haloes, in terms of both the features in the scatter plots and the values of the Spearman correlation coefficients. This may be counter-intuitive as one might naively expect that if two properties are well correlated (such as concentration and half-mass scale), then both of them should result in secondary biases of roughly the same magnitude. However, this naive expectation is mathematically unfounded. As we demonstrate next, the distribution of the orange points (paired haloes) differs slightly from the distribution of the blue points (unpaired haloes). Because of this small difference, the correlation between two secondary properties, say concentration and half-mass scale, is far from a guarantee of that these two properties would result in similar secondary biases. \n\n\n\\subsection{Correlation does not imply secondary bias}\n\\label{sec:demo}\n\n\\begin{figure}\n\\centering\\includegraphics[width=\\columnwidth]{demo.pdf}\n\\caption{\\label{fig:demo}%\nAn illustration of two highly correlated variables that result in different biases when a subset is selected. The main panel shows the two-dimensional distribution (corresponding to \\autoref{fig:scatter}) of two fictitious variables $X$ and $Y$, both uniformly distributed between $[0,1]$ for all points. The top and right-hand panels show the marginal probability distributions (corresponding to \\autoref{fig:markdist}) of $X$ and $Y$, respectively. A subset of points (shown in orange, approximately 40\\% of the total points) exhibit bias in $Y$ but not in $X$.  The numbers in parentheses are the Spearman correlation coefficient.\n}\n\\end{figure}\n\n\\begin{figure*}\n\\centering\\includegraphics[width=0.8\\textwidth]{c-ahalf.pdf}\n\\caption{\\label{fig:c-ahalf}%\nLeft panel shows the two-dimensional distribution of the marks on the concentration parameter (\\ensuremath{c}{}) and the half-mass scale (\\ensuremath{a_{1\/2}}{}), for unpaired (blue points) and paired (orange points) haloes. The distributions are the same as those in \\autoref{fig:scatter}, but here we overlay them for a direct comparison. Right panel shows the mean value of the \\ensuremath{c}{}-mark, conditioned on the \\ensuremath{a_{1\/2}}{}-mark; the black dashed, blue solid, and orange solid lines show the sample of all, paired, and unpaired haloes, respectively. \nThe standard error of the mean is shown by error bars, but too small to be seen.\n}\n\\end{figure*}\n\nTo demonstrate this important statement that two highly correlated halo properties can result in distinctly different secondary biases, consider a fictitious sample of 30,000 points that are described by two highly correlated variables $X$ and $Y$. The variables $X$ and $Y$ yield a Spearman correlation coefficient of $-0.90$, far more significant than the correlations amongst any of our halo secondary properties aside from the correlation of $\\ensuremath{a_{1\/2}}$ with $\\ensuremath{\\Gamma_\\text{peak}}$. From amongst these 30,000 points, we select approximately 40\\% of the total points and put them in a subset $S$. \\autoref{fig:demo} shows the two-dimensional and marginal distributions of $X$ and $Y$, for both points in and out of the subset $S$. We can see that the subset $S$ exhibits bias in $Y$ but not in $X$, despite the very strong correlation between $X$ and $Y$. Of course, the selection of the points in $S$ is not random. The subset $S$ is constructed by preferably selecting points with lower $Y$ in bins of $X$, and hence it is by construction that this subset results in a different bias in $X$ and $Y$. Mathematically speaking, for the full sample we have $P(X)=P(Y)=1$ and $P(X,Y)=P(X|Y)=P(Y|X)$. However, for the subset $S$, we alter the conditional distribution $P_{S}(Y|X)$ so that it differs from $P_{S}(X|Y)$, and hence the marginal distributions $P_S(X)$ and $P_S(Y)$ differ.\n\nThe two-dimensional distribution in \\autoref{fig:demo} is an analogy for \\autoref{fig:scatter}, and the marginal distributions in \\autoref{fig:demo} are analogies for \\autoref{fig:markdist}. The subset of points $S$ is analogous to the subset of paired haloes. In fact, the marginal distributions for each cell in the upper triangular part of \\autoref{fig:scatter} exactly corresponds to the mark distributions of paired haloes, shown by the orange solid lines in \\autoref{fig:markdist}. Hence, even though by eye the two-dimensional distributions of two marks for paired and unpaired haloes look very similar, the small difference between them can result in markedly different secondary clustering biases.\n\nTo take an even closer look at the particular case of halo concentration and half-mass scale, we overlay the two-dimensional distributions of the mark values of halo concentration and half-mass scale for paired and unpaired haloes, and show them in the left-hand panel of \\autoref{fig:c-ahalf}. Clearly, $\\ensuremath{a_{1\/2}}$ and $\\ensuremath{c}$ are similarly correlated for both subsamples. Indeed, they have Spearman correlation coefficients that are consistent with each other given the sample size. However, we can already see the small difference between the two-dimensional distributions for paired and unpaired haloes by eye. To quantify this difference, in the right-hand panel of \\autoref{fig:c-ahalf}, we show the mean value of the concentration mark, conditioned on half-mass scale mark, for the sample of all, paired, and unpaired haloes.\nWe find that, although concentration and half-mass scale are similarly correlated for paired and unpaired haloes, paired haloes have a slightly lower concentration at a given half-mass scale than unpaired haloes, and this small difference gives rise to their different behaviours in secondary bias.\n\nAt this point, it may be useful to summarize the phenomenology of \\ensuremath{c}{}- and \\ensuremath{a_{1\/2}}{}-dependent halo clustering. Haloes exhibit secondary bias based on \\ensuremath{c}{} because paired haloes constitute a subset of haloes with preferentially lower concentrations (in the mass range we consider). However, this subset of paired haloes also has the \\emph{same} distribution of \\ensuremath{a_{1\/2}}{} as unpaired haloes (as shown in \\autoref{fig:markdist}). Consequently, high-mass haloes exhibit \\ensuremath{c}{}-dependent secondary bias, but do not exhibit $\\ensuremath{a_{1\/2}}$-dependent (or $\\ensuremath{\\Gamma_\\text{peak}}$-dependent) secondary bias despite \\ensuremath{a_{1\/2}}{} and \\ensuremath{c}{} being strongly correlated. Selecting haloes based upon whether or not they have a neighbour alters the conditional distributions $P(\\ensuremath{c}|\\ensuremath{a_{1\/2}})$ and $P(\\ensuremath{a_{1\/2}}|\\ensuremath{c})$, in a highly non-trivial manner. The underlying physical reasons for this shift in halo properties induced by proximity to neighbour haloes are not immediately apparent, but our work suggests that halo properties are related to environment in a manner that is considerably more complex than is commonly assumed.\n\nInterestingly, the story of $\\ensuremath{R_\\text{mem}}$ is nearly the opposite of that for \\ensuremath{a_{1\/2}}{}. While \\ensuremath{R_\\text{mem}}{} exhibits a secondary bias similar to \\ensuremath{c}, \\ensuremath{\\lambda}, and \\ensuremath{N_\\text{sub}}{} (see \\autoref{fig:markdist}), it is only weakly correlated with those three properties (see \\autoref{fig:scatter}), for both paired and unpaired haloes. The lack of correlation of two variables by no means implies they cannot have similar or even the same marginal distribution, which is indeed what happens here. For example, the mark distributions of \\ensuremath{R_\\text{mem}}{} and \\ensuremath{N_\\text{sub}}{} for paired haloes are very similar, but these two properties are also the least correlated amongst the properties studied here. \n\nAs a consequence, one should be very cautious when studying the secondary bias due to one halo property through the use of a different halo property as a proxy. Even when the two properties, namely the halo property of interest and the proxy, have been shown to be strongly correlated, it is not true that the two properties must exhibit similar secondary biases. Likewise, when two properties both exhibit similar secondary biases, they may still have little correlation with each other.\n\n\n\\subsection{Implication for galaxy assembly bias}\n\\label{sec:galaxy-bias}\n\nThe term ``assembly bias'' is also frequently used to refer to ``galaxy assembly bias,'' which does not have a single clear definition. In most contexts galaxy assembly bias means that the clustering properties of galaxies depend on some galaxy properties at a fixed host halo mass. With what we have learned here, we shall take a closer look at the idea of galaxy assembly bias. Consider a galaxy property $G$, which depends on a halo property $H$. If $G$ and $H$ are highly correlated and their conditional distributions $P(G|H)$ and $P(H|G)$ are not altered by the presence of nearby haloes (e.g., in the context of our examples, $P(G|H)$ and $P(H|G)$ stay the same for both paired and unpaired haloes), then the secondary halo bias due to $H$ induces a ``galaxy assembly bias'' due to $G$. However, if the conditional distributions $P(G|H)$ and $P(H|G)$ are altered for haloes in pairs, such as $(X,Y)$ and $(\\ensuremath{a_{1\/2}},\\ensuremath{c})$ in our examples above, then the secondary halo bias due to $H$ does \\emph{not} guarantee any bias signal due to $G$. Similarly, any bias signal due to $G$ cannot be used to infer the existence of an underlying secondary halo bias due to $H$. Likewise, seeing both biases due to $G$ and $H$ does not guarantee a correlation between the galaxy property $G$ and the halo property $H$.\n\nIn short, the correlation between two variables and the dependence of clustering properties on these two variables do not have a firm connection. This statement is particularly evident for the secondary halo biases for cluster-size haloes, but it is a general, mathematical statement. Hence, using ``assembly bias'' to refer to different kinds of secondary halo bias and even bias in galaxy clustering is potentially misleading. \n\n\\subsection{Physical robustness of the secondary halo bias signals}\n\\label{sec:robust}\n\nDespite some of the counter-intuitive and seemingly contradictory results that we have presented, the secondary halo biases we show in this work are, in fact, physically robust. One may suppose that either \\ensuremath{a_{1\/2}}{} or \\ensuremath{\\Gamma_\\text{peak}}{} or both do not exhibit significant secondary halo clustering bias because they are measured with considerably more noise than the other halo properties that we explore or because these measures do not probe the particularly important epochs of halo formation. As we will show, this is not the case. Moreover, the existence of other secondary halo biases (due to concentration, spin, and subhalo properties) is also physically robust. \n\nTo demonstrate the robustness of the secondary halo biases that we present above we proceed as follows. For each of the six secondary properties, we identify another, similar property that has approximately the same physical meaning but defined differently. We then investigate whether or not this similar halo property exhibits a similar secondary halo bias. This new set of six properties are as follows.\n\n\\begin{enumerate}\n\\itemsep0.5\\baselineskip\n\n\\item Maximal circular velocity (\\ensuremath{V_\\text{max}}). For a halo that follows a NFW profile perfectly, the maximal circular velocity is a simple function of halo mass and concentration. At fixed mass, higher \\ensuremath{V_\\text{max}}{} implies higher halo concentration.\n\n\\item Spin parameter (\\ensuremath{\\lambda_\\text{Bullock}}), as defined in \\citet{Bullock2001}, which has a different normalization compared to the Peebles spin parameter.  \n\n\\item Scale of last major merger(\\ensuremath{a_\\text{LMM}}), defined as the scale factor at which the halo experiences its last major merger on its main branch. A major merger is defined as a merging event of two haloes with a mass ratio that is larger than one-third. \n\n\\item Present-day instantaneous accretion rate (\\ensuremath{\\Gamma_\\text{inst}}), it is defined as the mass change rate between two adjacent snapshot outputs on the main branch. For MDPL2, this rate is calculated between $z=0$ and 0.0224, which corresponds to approximately 300\\,Myr.\n\n\\item Peak maximal circular velocity of the largest subhalo (\\ensuremath{V_\\text{peak}^\\text{(1st sub)}}). Here largest subhalo means the subhalo that has the largest \\ensuremath{V_\\text{peak}}{} values of all subhaloes in that distinct halo. At a fixed distinct halo mass, this value is correlated with the concentration of the host halo and with the number of subhaloes \\citep{Mao2015}. \n\n\\item Average subhalo distance weighted by subhalo mass (\\ensuremath{R_\\text{mem}^\\text{(weighted)}}), defined as the average three-dimensional distance between all subhaloes and the centre of the main halo, with the contribution of each subhalo weighted by the subhalo mass. The subhalo definition is the same as the definition used to calculate the number of subhaloes for each distinct halo.\n\n\\end{enumerate}\n\n\\begin{figure}\n\\centering\\includegraphics[width=\\columnwidth]{bias_more.pdf}\n\\caption{\\label{fig:bias-more}%\nSame as \\autoref{fig:bias} but for six different (but related) secondary halo properties: (solid lines from top to bottom) \\ensuremath{\\lambda_\\text{Bullock}}, \\ensuremath{R_\\text{mem}^\\text{(weighted)}}, \\ensuremath{V_\\text{peak}^\\text{(1st sub)}}, \\ensuremath{a_\\text{LMM}}, \\ensuremath{\\Gamma_\\text{inst}}, and \\ensuremath{V_\\text{max}}.\nFor \\ensuremath{V_\\text{max}} (blue solid line), we also show the inverse ratio as a blue dashed line to guide an easy comparison with bias due to other secondary halo properties.\n}\n\\end{figure}\n\n\n\\autoref{fig:bias-more} shows the secondary bias (as ratio of pair counting functions) for these six secondary properties, and the behaviours of the secondary biases are very similar to the six properties used in \\autoref{fig:bias}. The spin parameter with \\citet{Bullock2001} definition still exhibits the largest bias. The maximal circular velocity exhibits the second most significant secondary bias, similar to, but slightly stronger than, the secondary bias exhibited by the concentration parameter. The two assembly history-related properties (\\ensuremath{a_\\text{LMM}}{} and \\ensuremath{\\Gamma_\\text{inst}}{}) again exhibit clustering that is consistent with random sampling and do not indicate secondary halo bias based upon halo mass assembly history. Lastly, the two subhalo-related properties (\\ensuremath{V_\\text{peak}^\\text{(1st sub)}}{} and \\ensuremath{R_\\text{mem}^\\text{(weighted)}}{}) show moderate secondary bias. \n\nAs our discussion in \\autoref{sec:demo} points out, correlation does not imply similar clustering bias. Hence the similar secondary bias signals we observe in \\autoref{fig:bias} and \\ref{fig:bias-more} can be interpreted to indicate that both properties in each pair affect the clustering properties. It also implies that the two conditional distributions of each pair of properties stay roughly the same for paired and unpaired haloes. In other words, the small change in definitions, in these particular cases, does not alter the joint distribution in a way that would result in different marginal distributions for paired or unpaired haloes (see \\autoref{fig:conditional-mean} for a full comparison of the conditional distributions amongst all properties). \n\n\n\\begin{figure*}\n\\centering\\includegraphics[width=\\textwidth]{history.pdf}\n\\caption{\\label{fig:history}%\nStacked main-branch assembly histories for halo mass $M(a)\/M(a=1)$ (upper row) and for maximal circular velocity $\\ensuremath{V_\\text{max}}(a)\/\\ensuremath{V_\\text{max}}(a=1)$ (lower row).\nIn each panel, the full sample of cluster-size haloes is split to show the difference between the stacked assembly histories of the two subsamples. \nIn the three columns, the sample is split by concentration mark (left column; orange dashed line for the 50\\% low-concentration haloes and blue solid line for the 50\\% high-concentration),\nby whether or not the haloes reside in pairs (middle column; orange dashed line for paired haloes and blue solid line for unpaired), \nand by large-scale density mark (left column; orange dashed line for the 50\\% haloes in high-density regions and blue solid line for the 50\\% haloes in low-density regions).\nHere paired haloes are haloes that have neighbour haloes within 10\\,\\ensuremath{h^{-1}}\\,Mpc), and the large-scale density is calculated by summing the total mass$^\\text{\\ref{footnote:halo-proxy}}$ within a 20\\,\\ensuremath{h^{-1}}\\,Mpc-radius sphere.\nFor each subsample under consideration, the line shows the median mass assembly history, and the corresponding band shows the 16$^\\text{th}$ and 84$^\\text{th}$ percentiles.\nThis plot is made with the DarkSky-Gpc simulation. \n}\n\\end{figure*}\n\n\\subsection{{Full halo assembly histories and bias}}\n\\label{sec:history}\n\nSo far, we have inspected four different summary statistics of halo mass assembly history: \\ensuremath{a_{1\/2}}, \\ensuremath{a_\\text{LMM}}, \\ensuremath{\\Gamma_\\text{peak}}, and \\ensuremath{\\Gamma_\\text{inst}}, and \\emph{none of them} exhibits any statistically significant secondary clustering bias. Given the similarity of halo mass assembly histories, which can be described well by one or two parameters, {one might \\emph{speculate} that, for high-mass haloes, the entire halo mass assembly history is independent of the environment}. With the ``demography for paired haloes'' approach that we introduced in \\autoref{sec:markdist}, we can directly inspect the mass assembly history for haloes {in different environments to further investigate the relationship between bias and assembly history.}\n\nThe upper row of \\autoref{fig:history} shows the stacked (median) mass assembly histories for main-branch progenitors as a function of scale factor, $M(a)\/M(a=1)$, using three different ways to split the full cluster-size halo sample into two subsamples. To produce \\autoref{fig:history}, we used the DarkSky-Gpc simulation to obtain the full mass assembly histories for all distinct haloes that have a present-day mass $\\ensuremath{M_\\text{vir}} \\geq 10^{14}\\,\\ensuremath{h^{-1}}\\,\\text{M$_\\odot$}$. On the left of \\autoref{fig:history}, we split haloes by their concentration mark. In the middle, we split haloes by whether or not they are in pairs; here a ``paired halo'' is again defined as any distinct halo that has at least one other distinct halo closer than 10\\,\\ensuremath{h^{-1}}\\,Mpc. On the right, we split haloes by their {mass-normalized mark values of} large-scale matter density; we calculate the matter density by summing up the masses of all resolved haloes%\n\\footnote{We calculate the mass by summing up halo masses within 20\\,\\ensuremath{h^{-1}}\\,Mpc spheres because we do not have direct access to the full particle snapshot of DarkSky-Gpc. To make this approximation as close as the actual matter distribution, we use all distinct haloes identified in the DarkSky-Gpc halo catalogue, down to a minimal halo mass of $4.88 \\times 10^{10}$\\,\\ensuremath{h^{-1}}\\,\\text{M$_\\odot$}{} (20 particles). These haloes, in total, contain 42.5\\% of the total matter mass in the simulation box. For our purpose of ranking the large-scale matter densities, this method provides good approximation, as we have verified using independent simulations. \\label{footnote:halo-proxy}}\nwithin a 20\\,\\ensuremath{h^{-1}}\\,Mpc sphere around each distinct halo in our sample, and then calculate the mass-normalized mark values using the procedure outlined in \\autoref{sec:marks}.\n\nWe can immediately see that the stacked mass assembly histories for paired and unpaired haloes are \\emph{essentially identical}. The 16$^\\text{th}$ and 84$^\\text{th}$ percentiles also match between the two samples, indicating that the variety of possible assembly histories is quite similar for both paired and unpaired haloes. \nFurthermore, when the halo sample is split by the large-scale matter density around the haloes, haloes in denser regions have nearly identical stacked assembly history as those haloes in less dense regions, as shown in the upper right-hand panel of \\autoref{fig:history}.\nThe lack of difference between the assembly histories of paired and unpaired haloes (or of haloes in high- and low-density regions) is \\emph{not} caused by the stacking procedure. As the upper left-hand panel of \\autoref{fig:history} shows, when the haloes are split by the concentration mark, there is a clear difference in the stacked assembly histories. The trend is consistent with our expectation, that high-concentration haloes form early. {One should also note that these two groups of haloes \\emph{do} have different clustering biases, as we know the concentration bias does exist at this mass scale.}\n\nWe have also verified that the lack of difference in the assembly histories for haloes in different environments is insensitive to the radii used in the definitions of paired haloes and large-scale density (the result holds when using {5, 10, 20, 30, and 40}\\,\\ensuremath{h^{-1}}\\,Mpc), and is also insensitive to how we split the density mark (the result holds when selecting the 25\\% or 10\\% most extreme mark values).\nIn addition, we further inspect the history of the maximal circular velocity (\\ensuremath{V_\\text{max}}) for main-branch progenitors, which represents the mass assembly history for the core of a halo. As the lower row of \\autoref{fig:history} shows, we again find that the stacked \\ensuremath{V_\\text{max}}{} histories of the paired and unpaired haloes, or of haloes in high- and low-density regions, are nearly identical. The concentration-split histories show a difference that is consistent with the difference in mass assembly history. We also find that the probability distributions of the number of major mergers (mass ratio between $1\/3$ and 1) that happened along the main branch are also essentially the same for paired and unpaired haloes.\n\nThese findings are in good agreement with our main results and also with \\citet{Gao2005} and \\citet{2008MNRAS.389.1419L}. {At this mass scale ($\\gtrsim 10^{14}$\\,\\ensuremath{h^{-1}}\\text{M$_\\odot$}), haloes that are in different environments do not have significantly different assembly histories.\nHowever, this does not guarantee that a correlation of any form between large-scale halo clustering and halo assembly history does not exist. Our results demonstrate that any such correlation is not evident either from the summary statistics that have been explored here or in the stacked assembly histories. The reasons for this phenomenon are twofold. First, at this mass scale, the strength of the secondary bias is small, and hence even if the $c-\\ensuremath{a_{1\/2}}$ relation were not biased for haloes in denser environments, the difference in the stacked assembly histories of haloes in different environments would still be modest. Secondly, as we discussed in \\autoref{sec:demo}, the $c-\\ensuremath{a_{1\/2}}$ relation is slighted biased for haloes in denser environments, and it cancels out any remaining difference in the assembly histories for haloes in different environments.}\n\n{It is possible to estimate the relative sizes of these two effects. In \\autoref{fig:history_diff}, we plot the relative differences between the stacked mass assembly histories of subgroups of haloes split by various halo properties. In each case, we split the samples into two equal-sized subgroups about a mark value of 0.5. For example, the green line in \\autoref{fig:history_diff} shows the normalized difference between the blue and orange lines in the upper left-hand panel of \\autoref{fig:history}. Splitting haloes by their \n\\ensuremath{c}{}, \\ensuremath{a_{1\/2}}{}, or \\ensuremath{\\Gamma_\\text{peak}}{} (green solid, orange dash--dotted, and red dotted lines respectively), yields significant differences in mass assembly histories, as one would expect. On the other hand, splitting haloes by large-scale density (blue solid line) yields assembly histories that are extraordinarily similar. This is another representation of our previous results. \n}\n\n{\nWe can now explore what would be expected of the mass assembly difference between haloes selected by density due only to the fact that density is correlated with concentration. To compute this, we place haloes into narrow bins of concentration mark and then randomly shuffle the density marks. We then split haloes based upon the shuffled density mark. In this manner, the selection upon density becomes meaningless, but the two sub-populations have concentration mark distributions that are identical to the subgroups split on actual density. The mass accretion history difference constructed in this way is shown by the purple dashed line in \\autoref{fig:history_diff}. This line shows the slightly different mass accretion histories of haloes due to the fact that selecting upon density also introduces small differences in the concentration distributions of the two halo subgroups. While this difference is small relative to selecting upon concentration or formation time directly, it is interesting that this difference is significantly larger than the difference in mass accretion histories induced by selecting on {\\em actual} density. Despite the fact that selecting upon density also selects populations with slightly different concentration distributions, selecting upon {\\em actual} density yields a nearly undetectable difference in mass accretion histories. The relation between concentration and mass accretion history is density dependent in such a way as to render the different mass accretion histories nearly identical. The difference in mass accretion histories exhibited by the purple dashed line in \\autoref{fig:history_diff} is also what we would have observed if the secondary bias due to \\ensuremath{a_{1\/2}}{} or \\ensuremath{\\Gamma_\\text{peak}}{} were as significant as the concentration bias.\n}\n\n{A common assumption has been that the secondary bias due to halo concentration is merely a consequence of the simple halo assembly bias. In other words, it has been commonly believed that secondary bias due to concentration was induced by the combination of assembly bias and the formation time--concentration relation. Given what we have learned here, the concentration bias seems more intriguing at this mass scale. In fact, one extremely interesting feature in \\autoref{fig:history_diff} is that when the haloes are split by the concentration mark, the difference in the mass assembly history {is similar to} the difference when the haloes are split by half-mass scale or accretion rate {only at early time ($a < 0.35$)}. This suggests that, at this mass scale, halo concentration, as a summary statistic, {captures only early-time assembly history and is more correlated with large-scale bias.} We leave this intriguing phenomenon for future study. }\n\n\n\\begin{figure}\n\\centering\\includegraphics[width=\\columnwidth]{history_diff.pdf}\n\\caption{\\label{fig:history_diff}%\n{Similar to \\autoref{fig:history} but showing the relative difference in the stacked (median) main-branch mass assembly histories for two groups of haloes, for several different ways to split the haloes: by mark values of large-scale (20 \\ensuremath{h^{-1}}\\,Mpc) density (blue solid, bottom-most), concentration (green solid, top-most), half-mass scale (orange dash--dotted), or accretion rate after peak mass (red dotted lines), and shuffled density (purple dashed line); all splits are made at the mark value of 0.5. The case when the haloes are split by shuffled density, each of the two groups has the same concentration distribution as in the case of split by large-scale density. \nThis plot is made with the DarkSky-Gpc simulation. }\n}\n\\end{figure}\n\n\n\\subsection{Secondary biases due to subhalo properties}\n\nIn \\autoref{fig:bias} and \\ref{fig:bias-more} we see that the number of subhaloes and the average subhalo distance (both weighted and unweighted) give significant secondary bias, and that the peak maximal circular velocity of the largest subhalo, which at a fixed distinct halo mass represents the gap between the first subhalo and the distinct halo, also exhibits a modest secondary bias signal.\nSimilar to halo concentration, both the number of subhaloes and the average subhalo distance are correlated with the assembly history of the parent distinct halo, and both of them exhibit secondary bias despite the lack of assembly bias. This result is in good agreement with the findings in \\citet{More2016}. \n\nThe secondary halo biases due to these subhalo properties are particularly interesting because they are more likely to be directly observable; at this mass scale, most large subhaloes would host galaxies.\nFor example, if the way galaxies \\emph{populate} subhaloes is not influenced by large-scale environment (although there is no clear evidence for such a dependence, it could exist), then the secondary bias due to subhalo occupation would directly translate to observable galaxy occupation bias (i.e., the dependence of central galaxy clustering on the member occupation or richness). Similarly, the average galaxy member distance will exhibit a bias signal if the galaxy--subhalo connection is not influenced by large-scale environment.\n\nThere are, however, complications to these potential observables. First, as we have discussed in \\autoref{sec:galaxy-bias}, the galaxy occupation bias or the average galaxy distance bias does not directly translate to the halo concentration bias or the halo assembly bias (in fact, we have already shown the latter does not exist for cluster-size haloes).\nSecondly, observationally, projection effects and redshift-space distortions can contaminate member assignment, and potentially produce artificial bias signal \\citep{2016arXiv161100366Z,1702.01682}. \nHence, one should be cautious when interpreting the galaxy occupation bias or the average galaxy distance bias.\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nIn this study, we have revisited the complex phenomenon of secondary halo bias, which is commonly referred to as ``halo assembly bias,'' for cluster-size haloes ($\\ensuremath{M_\\text{vir}} \\geq 10^{14}\\,\\ensuremath{h^{-1}}\\,\\text{M$_\\odot$}$) using large-volume simulations. As part of this investigation, we presented a novel approach to highlight secondary halo bias. Our approach was to study the demographics of paired and unpaired haloes, enabling us to determine whether or not the distributions of halo properties are different for haloes in pairs compared to the full halo sample. \n\nUsing both halo two-point functions and paired halo demographics, we found that halo concentration, halo spin, number of subhaloes, and average subhalo distance all exhibit significant secondary halo biases at the cluster mass scale. Amongst these properties, halo spin exhibits the strongest secondary halo bias, in the sense that high-spin haloes cluster more strongly. Low-concentration haloes cluster more weakly than high-concentration haloes, a dependence that is the converse of concentration-dependent secondary halo bias at lower masses. Cluster-size haloes cluster more strongly as a function of both subhalo number and the average distance between subhaloes and the halo centre.\n\nWe have identified no statistically significant secondary halo bias at this mass scale for any of four halo properties that directly measure the mass assembly history: half-mass scale, accretion rate before peak mass, instantaneous accretion rate, and time of last major merger. We have found that the entire main-branch mass assembly histories of paired and unpaired haloes (or haloes in different large-scale densities) are statistically identical. {This suggests that the assembly histories of massive, cluster-size haloes do not correlate with their environments in a simple fashion. This is not equivalent to the statement that there exists no features of halo assembly histories that do correlate with environment and\/or clustering strength. Indeed, some particular features of the halo assembly histories, such as those captured by halo concentration, may still correlate with the large-scale environment.}\n\nWith our halo demographic approach, we further investigated the seemingly contradictory result of the lack of secondary halo bias due to assembly history-related properties, given the clear correlation between halo concentration and halo assembly history. We have demonstrated that the correlation between two variables is, in general, not relevant to the question of whether or not the two variables will result in similar secondary halo clustering biases. If the conditional distributions of the two variables are altered, even only slightly, for paired haloes, then the two variables can easily result in very different secondary clustering biases. For instance, halo concentration and half-mass scale are similarly correlated for paired and unpaired haloes, yet paired haloes have a slightly lower concentration at a given half-mass scale, which results in their different secondary bias signals. Likewise, the fact that two variables (e.g., halo concentration and average subhalo distance) yield similar secondary clustering biases by no means implies  a correlation between the two variables. These statements have the following important consequence: Even when the presence of a galaxy in a halo is a function of a halo property that exhibits secondary halo bias, this does \\emph{not} necessarily imply galaxy ``assembly bias.''\n\nOur study has provided a comprehensive view of secondary halo biases for cluster-size haloes, and leads to a different perspective on secondary halo biases. In particular, we caution against use of the term ``assembly bias,'' particularly for cluster-size haloes.  The term ``assembly bias,'' when used to refer to secondary biases other than those due to the assembly history (e.g., concentration-dependent halo clustering), implies that all such biases have a common origin rooted in some aspect of the mass assembly histories of haloes. \n{While these different secondary biases may still all have some connections with the halo assembly history, those connections are more complex than simple correlations amongst different halo properties. \nIn particular, the halo secondary bias due to concentration is not a direct consequence of the difference in bias between early- and late-forming haloes.}\n\nOur results regarding halo assembly histories have an important consequence for the interpretation of halo clustering. At face value, our result is in qualitative agreement with early analytic studies of halo abundance and clustering using excursion set theory, which predicted no assembly bias (specifically, \\citealt{Kaiser84,Cole1989,1991ApJ...379..440B,Mo1996,Sheth2001}; see \\citealt{10.1142\/S0218271807010511} for a review and subsequent developments). Yet, these predictions stem from ad hoc assumptions adopted for computational ease, rather than for any well-established physical reasons. Several authors have proposed more physically-motivated analytic interpretations of secondary halo bias based upon the assembly histories of haloes \\citep{10.1142\/S0218271807010511,Desjacques2007,Dalal2008}. {However, given our findings, these analytic interpretations of halo assembly bias do not manifest in all different kinds of secondary halo biases in a straightforward manner. Indeed, these interpretations contradict our finding that high-mass haloes cluster independently of halo formation time and other simple metrics of halo age. The explicit relations amongst large-scale environment, halo assembly history, and sundry internal halo properties, such as concentration and spin, remain unclear.}\n\nWhile this study has demonstrated that a very complex phenomenology of secondary halo bias is mathematically possible, it has yet to provide a solid physical explanation for the existence of the concentration bias, the spin bias, the subhalo abundance bias, and the average subhalo distance bias. Previous attempts to explain these secondary biases that rely upon the existence of halo assembly bias in its restricted definition are not valid in this mass regime. It is also important to understand why the correlations between particular halo properties (e.g., between subhalo abundance and assembly history) depend on large-scale environment. We have not yet identified a plausible working theory that is able to explain all of the correlations. We hence leave this interesting problem for future study, with the hope that what we have laid out in this study will provide useful insights. \n\n\n\n\\section*{Acknowledgements}\n\nThe authors thank Andreas Berlind, Philipp Busch, Neal Dalal, \\mbox{Andrew} Hearin, Ari Maller, Surhud More, Rita Tojeiro, \\mbox{Li-Cheng} Tsai, Frank \\mbox{van den Bosch}, Antonio \\mbox{Villarreal}, and Kuan Wang for useful discussions.\nThis research made use of the MDPL2 simulation; the authors gratefully acknowledge the Gauss Centre for Supercomputing e.V.\\ (\\http{www.gauss-centre.eu}) and the Partnership for Advanced Supercomputing in Europe (PRACE, \\http{www.prace-ri.eu}) for funding the MultiDark simulation project by providing computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing Centre (LRZ, \\http{www.lrz.de}), and also thank Peter Behroozi for the direct access to the MDPL2 halo catalogues on SLAC servers.\nThis research made use of the DarkSky-Gpc (\\texttt{ds14\\_b}) simulation, which was part of the Dark Sky Simulations (\\http{darksky.slac.stanford.edu}) produced using an INCITE 2014 allocation on the Oak Ridge Leadership Computing Facility at Oak Ridge National Laboratory, a U.S.\\ Department of Energy Office; the authors thank Sam Skillman, Mike Warren, Matt Turk, and other Dark Sky collaborators for their efforts in creating these simulations and for providing access to them.\nThis research made use of computational resources at SLAC National Accelerator Laboratory, a U.S.\\ Department of Energy Office; YYM and RHW thank the support of the SLAC computational team.\nYYM is supported by the Samuel P.\\ Langley PITT PACC Postdoctoral Fellowship.\nARZ is supported, in part, by grants AST 1516266 and AST 1517563 from the U.S.\\ National Science Foundation as well as by the Pittsburgh Particle physics, Astrophysics, and Cosmology Center (PITT PACC) at the University of Pittsburgh.\nRHW received partial support from the U.S.\\ Department of Energy contract to SLAC No.\\ DE-AC02-76SF00515.\nThis work was completed at the Kavli Institute for Theoretical Physics at the program ``The Galaxy--Halo Connection Across Cosmic Time,'' and supported in part by the National Science Foundation under Grant No.\\ NSF PHY-1125915.\n\nThis research made use of Python, along with many community-developed or maintained software packages, including\nIPython \\citep{ipython},\nJupyter (\\http{jupyter.org}),\nMatplotlib \\citep{matplotlib},\nNumPy \\citep{numpy},\nPandas \\citep{pandas},\nand SciPy \\citep{scipy}.\nThis research made use of NASA's Astrophysics Data System for bibliographic information.\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}