diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzbjlg" "b/data_all_eng_slimpj/shuffled/split2/finalzzbjlg" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzbjlg" @@ -0,0 +1,5 @@ +{"text":"\\section{\\label{sec:level0}INTRODUCTION}\nRecently, the interest in relativistic dissipative fluids in astrophysics and nuclear physics \nhas increased. \nRelativistic dissipative fluid equations have many features that do not appear in the case of nonrelativistic fluid.\nThe most basic difference is the fact that in the presence of heat flux, \nthe fluid velocity cannot be defined uniquely. \nThere are two well-known definitions of fluid velocity; \nEckart velocity~\\cite{Eckart(1940)} that is parallel to particle flow\nand Landau-Lifshitz velocity~\\cite{Landau & Lifshitz(1959)} that is parallel to energy flow.\nIn addition, \nit is well known that standard first-order relativistic Navier-Stokes hydrodynamics exhibits fatal problems regarding causality and stability, \nthat is, small perturbations to the uniform static states grow exponentially~\\cite{Hiscock & Lindblom(1983),Hiscock & Lindblom(1985)}. \nCurrently, \nthe most widely accepted and studied theory is the second-order Israel-Stewart (IS) approach~\\cite{Israel & Stewart(1979)}\nbased on the 14-moment method~\\cite{Cerbook}.\nUnfortunately, \nthis theory is inconvenient for practical use \nbecause we have to restore so many terms that are second-order in deviations from equilibrium, \nthat is, the time derivative of dissipation terms \nand the products of gradients of dissipative quantities. \nHowever, because of the recent finding of the strongly coupled \nquark-gluon plasma (sQGP) in the Relativistic Heavy-Ion Collider (RHIC), \ndescription by relativistic hydrodynamics equations have been vigorously studied in the context of nuclear physics~\\cite{Baier et al.(2008)}, \nand application of IS theory has just begun~\\cite{Molnar et al.(2009)}.\nRecently, a new approach to relativistic dissipative fluid equation has been shown by Tsumura, Kunihiro, and Ohnishi~\n\\cite{Tsumura et al.(2007),Tsumura & Kunihiro(2008),Tsumura & Kunihiro(2009)}.\nThey use the renormalization-group method for obtaining fluid equation from the Boltzmann equation, \nand obtained equation is different from both Eckart and Landau-Lifshitz equation.\n\nMicroscopic phenomena are accurately described by the Boltzmann equation.\nHowever, it is very difficult to solve \nsince its collision term depends on the product of the distribution functions.\nConsequently, a simpler approximation for the collision term has been proposed; \nthe most widely used relativistic kinetic model equations are \nthose of Marle~\\cite{Marle} and Anderson-Witting~\\cite{Anderson & Witting(1974)}.\nThe Marle model is an extension of the nonrelativistic \nBhatnagar-Gross-Krook (BGK) model~\\cite{Bhatnagar et al.(1954)} to the relativistic case \nand is described in the Eckart frame~\\cite{Eckart(1940)}. \nThe Anderson-Witting model is another extension \nand is described in the Landau-Lifshitz frame~\\cite{Landau & Lifshitz(1959)}.\nOf the two, the Anderson-Witting model is widely used~\\cite{Struchtrup(1998),Yano et al.(2007)}\nbecause the Marle model has undesirable properties; for example, \nthe transport coefficients obtained by the Marle model \ndo not agree with those obtained by the full Boltzmann equation~\\cite{Cerbook}.\n\nIn this paper,\nwe compare the dynamics described by the 14-moment theory with that of the kinetic model equation \nand test the applicability of the IS approach. \nTo do numerical simulation of relativistic dissipative fluid, \nwe should know how to treat the small second-order terms \nand how to determine appropriate values of new coefficients, \nwhich urges us to check how important these terms are.\nTo make the problem tractable, \nwe study linear perturbation and compare the solutions of the dispersion relation.\nThe dispersion relations of the relativistic kinetic equations have been studied as a boundary value problem \nby Cercignani and Majorana~\\cite{cer1984,Cercignani & Majorana(1985)}. \nTo understand the dynamics as a Cauchy problem, \nwe solve the dispersion relations with respect to $\\omega$.\nIn addition, \nwe modify the problematic properties of the Marle model \nand use the modified model equation to analyze the Eckart description.\n\nThis paper is organized as follows: \nin Sec.~\\ref{sec:level1}, \nwe introduce the kinetic models of Marle and Anderson-Witting. \nThen, we modify the Marle model \nand obtain the dispersion relations.\nIn Sec.~\\ref{sec:level2}, \nwe solve the dispersion relations numerically with respect to $\\omega$ \nand present our results. \nIn Sec.~\\ref{sec:level3}, \nwe discuss the properties of the Marle and Anderson-Witting models. \nIn addition, \nwe analyze the asymptotic behavior of the dispersion relations. \nFirst, we study the long wavelength limit \nand then we study the short wavelength and high frequency limits.\nWe solve the dispersion relation of the 14-moment theory \nand compare it with the dispersion relations of the kinetic model equation.\n\n\n\\section{\\label{sec:level1}THE LINEARIZED KINETIC EQUATIONS AND THE DISPERSION RELATIONS}\nIn this section, we derive the dispersion relations of the relativistic kinetic models of Marle and Anderson-Witting.\nThroughout this paper, we use the units\n\\begin{equation}\nc = 1, \\quad k_B = 1\n,\n\\end{equation}\nwhere $c$ is the velocity of light, and $k_B$ the Boltzmann constant.\n\nIn Cartesian coordinates, the Minkowski metric tensor $\\eta_{\\mu \\nu}$ is given by\n\\begin{equation}\n\\eta_{\\mu \\nu} = \\mathrm{diag}(1, -1, -1, -1)\n.\n\\end{equation}\nVariables indicated by Greek letters take values from $0$ to $3$, \nand those indicated by Roman letters take values from $1$ to $3$.\n\n\\subsection{\\label{sec:mlevel0}MODIFICATION OF THE BGK MODEL OF MARLE}\nMarle~\\cite{Marle} has proposed the following form of the kinetic model equation,\n\\begin{equation}\n\\left(\\frac{\\partial f}{\\partial t} \\right)_{coll} = \n- \\frac{m}{\\tau_M} (f(t,{\\bf x},{\\bf p})-f_{eq}(t,{\\bf x},{\\bf p})) \n,\n\\label{eq:mBGK}\n\\end{equation}\nwhere $\\tau_M$ is a characteristic time on the order of the mean flight time \n(see below for its physical interpretation), \n$m$ is the rest mass of a particle of the relativistic gas, and \n$f_{eq}$ is the local equilibrium distribution function.\n\nUsing Eq.~(\\ref{eq:mBGK}), \nwe obtain the following form of the kinetic equation\n\\begin{align}\np^{\\mu} \\partial_{\\mu} f &= p^0 \\left(\\frac{\\partial}{\\partial t} + {\\bf v} \\cdot \\nabla \\right) f \n= - \\frac{m}{\\tau_M} (f-f_{eq}) \n,\n\\label{eq:mBGKbol} \\\\\n{\\bf v} &= \\frac{{\\bf p}}{p^0}\n.\n\\end{align}\n\nThe Marle model is an extension of the nonrelativistic BGK model to the relativistic case.\nThe transport coefficients for the Marle model equation reproduce the nonrelativistic results \nin the limiting case of low temperature.\nIt is, however, well known that in the limiting case of high temperature, \nthe transport coefficients of the Marle model\ndiffer from those found for hard-sphere particles obtained by the full Boltzmann equation~\\cite{Cerbook}.\nMore precisely, if we express the transport coefficients ($\\propto \\tau_M$) as a function of $\\zeta = m \/ T$, \nthe transport coefficients of the Marle model behave as $1 \/ \\zeta$ of those found \nby the Boltzmann equation for hard-sphere particles in the limit of high temperature.\nFor this problem, we should recall that the transport coefficients are generally proportional to the relaxation time $\\tau_M$, \nand Eq.~(\\ref{eq:mBGK}) contains $\\tau_M$ as a parameter of the BGK model.\nThis indicates that \nthe appropriate value of $\\tau_M$ is different from the physical relaxation timescale $\\tau_{relax}$ by a factor \nthat becomes unity in the low temperature \nlimit and becomes $\\zeta$ in the high temperature limit. \nWe discuss this new interpretation of $\\tau_M$.\n\nFirst, \nwe clarify the meaning of the parameter $\\tau$ in the BGK model.\nIn the nonrelativistic BGK model, \nthe parameter $\\tau$ is equivalent to the relaxation time.\nThe nonrelativistic kinetic equation of the BGK model is\n\\begin{equation}\n\\left(\\frac{\\partial}{\\partial t} + {\\bf v} \\cdot \\nabla \\right) f \n= - \\frac{1}{\\tau} (f(t,{\\bf x},{\\bf v})-f_{eq}(t,{\\bf x},{\\bf v})) \n.\n\\label{eq:BGKbol}\n\\end{equation}\nIf the one-particle distribution function $f$ does not depend on the spatial coordinates, \nEq.~(\\ref{eq:BGKbol}) reduces to the ordinary first-order differential equation, \nand we can obtain the formal solution \n\\begin{equation}\nf(t) = \\left[f(0) + \\frac{1}{\\tau} \\int^t_0 e^{t' \/ \\tau} f_{eq}(t') dt' \\right] e^{-t\/\\tau}\n.\n\\end{equation}\nThis equation indicates that $\\tau$ is the relaxation time of the distribution function.\n\nNext, \nwe consider the relativistic BGK model of Marle. \nThe kinetic equation of the Marle model is\n\\begin{align}\n&p^{\\mu} \\partial_{\\mu} f = p^0 \\left(\\frac{\\partial}{\\partial t} + {\\bf v} \\cdot \\nabla \\right) f\n\\nonumber\n\\\\\n&= - \\frac{m}{\\tau_M} \\left(f(t,{\\bf x, v}) - f_{eq}(t, {\\bf x, v}) \\right)\n.\n\\label{eq:mBGK2}\n\\end{align}\n\nIf we assume that the one-particle distribution function $f$ does not depend on the spatial coordinates, \nthe formal solution of Eq.~(\\ref{eq:mBGK2}) is\n\\begin{align}\nf(t) &= \\left[f(0) + \\frac{1}{\\tau_{M*}} \\int^t_0 e^{t'\/\\tau_{M*}} f_{eq}(t') dt' \\right] e^{-t\/\\tau_{M*}}\n\\label{eq:Mrelax}\n,\n\\\\\n\\tau_{M*} &= \\frac{p^0}{m} \\tau_M\n.\n\\end{align}\nThis indicates that in a general inertial frame, \nthe relaxation time is not $\\tau_M$ but $\\tau_{M*}$, and \n$\\tau_M$ is the relaxation time in the rest frame where the momentum of particles is ${\\bf p = 0}$.\nMore precisely, \nif we employ the particle's rest frame where {\\bf p = 0}, \nEq.~(\\ref{eq:mBGK2}) becomes\n\\begin{equation}\n\\frac{\\partial}{\\partial t} f(t,{\\bf x,0}) = - \\frac{m}{\\tau_M} (f(t,{\\bf x},{\\bf 0})-f_{eq}(t,{\\bf x},{\\bf 0}))\n.\n\\end{equation}\nThis is the same equation as in the nonrelativistic BGK model, \nindicating that only in this frame does $\\tau_M$ become the relaxation time.\n\nAlthough the transport coefficients of the Marle model are expressed in a form proportional to $\\tau_M$ in the literature~\\cite{Cerbook},\nthe above explanation shows that we should use $\\tau_{M*}$ as the relaxation time instead of $\\tau_M$.\nHowever, \n$\\tau_{M*}$ depends on the momentum $p^0$, so $\\tau_{M*}$ cannot appear in macroscopic descriptions, \nsuch as transport coefficients.\nFor this reason, we have to consider the true relaxation time $\\tau_{relax}$, \nto which the transport coefficients should be proportional, and\nrelate it to the BGK parameter of the Marle model $\\tau_M$.\nThe above discussion suggests that we may regard $1 \/ \\tau_{relax}$ as $\\left\\langle 1 \/ \\tau_{M*} \\right\\rangle$, \nand we can consider $\\tau_{relax}$ as the effective relaxation time in general frames. \nUsing the local equilibrium distribution function, \n$\\tau_M$ is\n\\begin{equation}\n\\tau_M = \\frac{m}{n} \\int \\frac{d^3 p}{p^0} f_{eq} \\tau_{relax} = \\frac{K_1(\\zeta)}{K_2(\\zeta)} \\tau_{relax} \n\\label{eq:taum}\n,\n\\end{equation}\nwhere $K_n$ is the second kind modified Bessel function of order n.\nThe correction $K_1(\\zeta)\/K_2(\\zeta)$ becomes $1$ in the limit of large $\\zeta$ and \n$\\zeta\/ 2$ when $\\zeta$ is nearly $0$.\nThis indicates that this function has the desired properties.\nIn the following, we use this $\\tau_M$ as the BGK parameter of the Marle model.\n\nIn above discussion, \nwe assume that the physical system is not far from equilibrium state, and \ncalculate the average of $\\tau_M$ with respect to the local equilibrium distribution function $f_{eq}$.\nThough this cannot give the correct $\\tau_M$ in the general case,\nit is a good approximation for linear perturbation about the local equilibrium distribution function.\n\n\n\\subsection{\\label{sec:mlevel1}THE LINEARIZED KINETIC EQUATION AND \nDISPERSION RELATION OF THE MODIFIED MARLE MODEL}\nIn this section, \nwe derive the dispersion relation of the modified relativistic kinetic model of Marle.\nTo obtain the dispersion relation, \nwe apply an approach similar to that in the work of Cercignani and Majorana~\\cite{cer1984}.\n\nWhen there is no external field, the equation of the modified Marle model is given by\n\\begin{eqnarray}\n&& \\frac{D}{D s} f = - \\frac{m}{\\tau_M} (f-f_{eq}) \n,\n\\label{eq:mBGKbolkai}\n\\\\\n&& \\frac{D}{D s} f = p^{\\mu} \\partial_{\\mu} f = p^0 \\left(\\frac{\\partial}{\\partial t} + {\\bf v} \\cdot \\nabla \\right) f\n.\n\\end{eqnarray}\n\nIn Eq.~(\\ref{eq:mBGKbolkai}), $\\tau_M$ is the relaxation time modified in Sec.~\\ref{sec:mlevel0}, and \n$f_{eq}$ represents the local Maxwell-J$\\ddot{\\mathrm{u}}$ttner distribution function\n\\begin{eqnarray}\nf_{eq}(t,{\\bf x, p}) &=& \\frac{n(t,{\\bf x})}{4 \\pi m^2 T(t,{\\bf x}) K_2(\\zeta(t,{\\bf x}))} \n\\nonumber\n\\\\\n&\\times&\\exp \\left[- \\frac{p_{\\mu} u^{\\mu}(t,{\\bf x})} {T(t,{\\bf x})}\\right] \n\\label{eq:mjeq}\n,\n\\\\\n\\zeta &=& \\frac{m}{T}\n,\n\\end{eqnarray}\nwhere $m$ is the mass of the particle, and $T$ is the temperature.\n\nEq.~(\\ref{eq:mBGKbolkai}) is a nonlinear equation for $f(t,{\\bf x, p})$ because of \nthe nonlinear dependence of $f_{eq}$ on $f$ through the following conditions called the matching conditions:\n\\begin{eqnarray}\n&&\\int \\left( f_{eq} - f \\right) \\psi \\frac{d^3 p}{p^0} = 0 \n\\label{eq:mrmat}\n,\n\\\\\n&&\\psi = (1, ~p^{\\mu})\n\\label{eq:psi}\n.\n\\end{eqnarray}\n\nTo obtain the dispersion relation, \nwe start by expanding the distribution function around a global equilibrium state $f_0({\\bf p})$, \n\\begin{eqnarray}\n\\delta f = f- f_0, \\quad \\delta f_{eq} = f_{eq} - f_0\n\\end{eqnarray}\nThen, the linearized kinetic equation of the modified Marle model is given by\n\\begin{eqnarray}\n\\left( \\frac{\\partial}{\\partial t} + {\\bf v} \\cdot \\nabla \\right) \\delta f \n&=& - \\frac{\\delta f - \\delta f_{eq}}{\\tau_{M*}}\n,\n \\label{eq:mlinbol} \\\\\n\\tau_{M*} &=& \\frac{p^0}{m} \\tau_M\n.\n\\end{eqnarray}\n\nWe assume a solution in the following form:\n\\begin{equation}\n\\delta f = \\delta \\tilde{f} e^{- i k_{\\mu} x^{\\mu}} = \\delta \\tilde{f} e^{- i \\omega(t-t_0)+i {\\bf k \\cdot x}} \n.\n\\label{eq:mome}\n\\end{equation}\nThen, Eq.~(\\ref{eq:mlinbol}) reduces to\n\\begin{equation}\n\\left( \\frac{1}{\\tau_{M*}} -i \\omega + i {\\bf k \\cdot v} \\right) \\delta f = \\frac{1}{\\tau_{M*}} \\delta f_{eq}\n\\label{eq:mlinbolf}\n.\n\\end{equation}\nWe consider an equilibrium background state in which the fluid is at rest, so that $u^{\\mu} = (1, {\\bf 0})$ and \n$\\delta u^{\\mu} = (0, \\delta {\\bf u})$ owing to the relation $u^{\\mu} \\delta u_{\\mu} = 0$.\nThen, $\\delta f_{eq}$ is given by\n\\begin{align}\n\\delta f_{eq} &= f_0 \n\\nonumber\n\\\\\n&\\times \\left[ \\frac{\\delta n}{n} + \\left(-1 + \\frac{p^0}{T} + \\frac{K_2'}{K_2} \\zeta \\right) \\frac{\\delta T}{T}\n - \\frac{{\\bf p} \\cdot \\delta {\\bf u}}{T} \\right] \n,\n\\\\\nf_0 &= \\frac{n}{4 \\pi m^2 T K_2(\\zeta)} \\exp \\left[- \\frac{p^0}{T} \\right]\n,\n\\end{align}\nwhere $\\delta {\\bf u}$ is the space component of the Eckart velocity, as explained in Sec.~\\ref{sec:level31}, and \n$K_n'$ is the derivative of $K_n$ with respect to $\\zeta$.\n\nUsing the matching conditions, we can rewrite $\\delta \\rho, \\delta {\\bf u}$, and $\\delta T$ as the integrals of $\\delta f$:\n\\begin{align}\n\\delta n(t,{\\bf x}) &= \\int \\frac{d^3 p}{p^0} p^0 \\delta f\n, \\\\\n\\delta {\\bf u}(t,{\\bf x}) &= - \\frac{1}{n} \\int \\frac{d^3 p}{p^0} {\\bf p} \\delta f\n, \\\\\n\\delta T(t,{\\bf x}) &= \\int \\frac{d^3 p}{p^0} p^0 \n\\\\\n\\nonumber\n&\\times\n\\frac{- 1 + K_2' \\zeta \/ K_2 + p^0 \/ T}{\\left(1 - K_1 \\zeta \/ K_2 \\right)\n\\left(3 + \\zeta^2 + K_1 \\zeta \/ K_2 \\right)} \\delta f\n.\n\\end{align}\n\nEq.~(\\ref{eq:mlinbolf}) becomes\n\\begin{align}\n& \\left( \\frac{1}{\\tau_{M*}} -i \\omega + i {\\bf k \\cdot v} \\right) \\delta f({\\bf p}) \n\\\\ \\nonumber\n&= \\int \\frac{d^3 p'}{p'^0} \\frac{f_0({\\bf p})}{\\tau_{M*}} \n\\left[\\frac{p'^0}{n} - \\frac{{\\bf p} \\cdot {\\bf p'}}{T}\n\\right.\n\\\\ \\nonumber\n&+ \\left.\n \\frac{T}{n} \\frac{p'^0}{\\left(1-K_1 \\zeta \/ K_2\\right) \\left(3 + \\zeta^2 + K_1 \\zeta \/ K_2 \\right)} \n\\right.\n\\\\ \\nonumber\n&\\times \\left. \\left(-1 + \\frac{p^0}{T} + \\frac{K_2'}{K_2} \\zeta \\right) \n \\left( - 1 + \\frac{p'^0}{T} + \\frac{K_2'}{K_2} \\zeta \\right) \\right] \\delta f({\\bf p'})\n.\n\\end{align}\n\nIn the following, \nwe take $\\tau_{relax}$ as a unit of time: \n\\begin{equation}\n\\omega \\tau_{relax} \\rightarrow \\omega, \\quad \\tau_{relax} k \\rightarrow k.\n\\end{equation}\n\nFinally, the linearized equation of the BGK model of Marle is\n\\begin{align}\n\\delta f({\\bf p}) &= \\int \\frac{d^3 p'}{p'^0} K({\\bf p,p'}) \\delta f({\\bf p'}) \n\\label{eq:mlinbolk}\n,\n\\\\\nK({\\bf p,p'}) &\\equiv \\frac{f_0({\\bf p})}{1 - \\left(i \\omega - i {\\bf k} \\cdot \\frac{{\\bf p}}{p^0} \\right)\n \\frac{K_1 z}{K_2 \\zeta}}\n\\left[\\frac{p'^0}{n} - \\frac{{\\bf p} \\cdot {\\bf p'}}{T}\n\\right.\n\\\\ \\nonumber\n&+ \\left. \\frac{T}{n} \\frac{p'^0}{\\left(1-K_1 \\zeta \/ K_2\\right) \\left(3 + \\zeta^2 + K_1 \\zeta \/ K_2 \\right)} \n\\right.\n\\\\ \\nonumber\n&\\times \\left. \\left(-1 + z + \\frac{K_2'}{K_2} \\zeta \\right) \n \\left( - 1 + z' + \\frac{K_2'}{K_2} \\zeta \\right) \\right] \\delta f({\\bf p'})\n,\n\\end{align}\nwhere $z = p^0 \/ T$. \nThis equation make sense only when \n$1 - \\left(i \\omega - i {\\bf k} \\cdot \\frac{{\\bf p}}{p^0} \\right) \\frac{K_1 z}{K_2 \\zeta} \\neq 0$; \nwe explain the case where \n$1 - \\left(i \\omega - i {\\bf k} \\cdot \\frac{{\\bf p}}{p^0} \\right) \\frac{K_1 z}{K_2 \\zeta} = 0$ \nlater.\n\nEq.~(\\ref{eq:mlinbolk}) is the homogeneous Fredholm integral equation of the second kind.\nIn particular, the kernel function $K({\\bf p, p'})$ can be separated with respect to the variables ${\\bf p}$ and ${\\bf p'}$. \nThus, this equation can be solved according to a general procedure.\n\nFirst, we integrate Eq.~(\\ref{eq:mlinbolk}) with respect to ${\\bf p}$.\nThen, we multiply by $K_1 k \/ \\zeta$\nand the equation reduces to\n\\begin{equation}\nI_{11} \\frac{\\delta n}{n} + I_{12} {\\bf k} \\cdot \\delta {\\bf u} + I_{13} \\frac{\\delta T}{T} = 0\n,\n\\label{eq:mcont}\n\\end{equation}\nwhere\n\\begin{align}\nI_{11} &= K_1^2 k - \\zeta K_2 P(0) - \\pi K_2 e^{-d}\n,\n\\\\\nI_{12} &= -\\frac{i K_2 \\zeta}{k} \\left\\{\\frac{\\zeta}{k} \\left(\\frac{K_2}{K_1} P(0) - i \\omega P(1) \\right) - K_1 \\right\\}\n\\\\ \\nonumber\n&- \\pi K_2 e^{-d} \\frac{i}{k^2} \\left\\{-i \\omega (1 + d) + \\frac{K_2}{K_1} \\zeta \\right\\}\n,\n\\\\\nI_{13} &= \\left(-3-\\frac{K_1}{K_2} \\zeta \\right) \\left(k K_1^2 - \\zeta K_2 P(0) \\right) \n\\\\ \\nonumber\n&+ \n\\zeta K_2 \\left(k K_1 - \\zeta P(1) \\right) \n- \\pi K_2 e^{-d} \\left(d - 2 - \\frac{K_1}{K_2} \\zeta \\right)\n,\n\\end{align}\n$P(n)$ is\n\\begin{align}\nP(n) &= \\int^{\\infty}_{1} dy \\; e^{-\\zeta y} y^n \\arctan \\frac{\\zeta \\sqrt{y^2 - 1}}{b}\n,\n \\\\\nb &= \\frac{\\zeta}{k} \\left( \\frac{K_2}{K_1} - i \\omega y \\right)\n,\n\\end{align}\nand $d$ is\n\\begin{align}\nd &= - \\frac{K_2 \\zeta}{K_1 \\mathrm{Im}(\\omega)} \\qquad \\mathrm{if} \\quad \\mathrm{Im}(\\omega) < - \\frac{K_2}{K_1}\n,\n\\\\ \\nonumber\nd &= \\zeta \\qquad \\qquad \\qquad \\mathrm{if} \\quad \\mathrm{Im}(\\omega) > - \\frac{K_2}{K_1} \n.\n\\end{align}\nThe derivation of the correction term $c$ is explained in Sec.~\\ref{sec:mlevela2}.\n\nNext, we multiply Eq.~(\\ref{eq:mlinbolk}) by ${\\bf k \\cdot p} \\equiv k p^x$ and \nintegrate with respect to ${\\bf p}$.\nThen, we multiply by $\\zeta K_1$ \nand the equation reduces to\n\\begin{equation}\nI_{21} \\frac{\\delta n}{n} + I_{22} {\\bf k} \\cdot {\\bf \\delta u} + I_{23} \\frac{\\delta T}{T} = 0\n\\label{eq:mmom}\n,\n\\end{equation}\nwhere\n\\begin{align}\nI_{21} &= \\zeta \\left(- i K_1 + i \\frac{K_2 \\zeta}{K_1 k} P(0) + \\frac{\\omega \\zeta}{k} P(1) \\right) \n\\\\ \\nonumber\n&+ \\frac{\\pi}{k} e^{-d} \n\\left\\{ \\omega (1 + d) + i \\frac{K_2}{K_1} \\zeta \\right\\}\n,\n\\\\\nI_{22} &= \\frac{\\zeta^2}{k^2} \\left\\{K_2 (1 - i \\omega) - \\frac{K_2 \\zeta}{K_1 k} \\left(\\frac{K_2}{K_1} P(0) - i \\omega P(1) \\right) \\right.\n\\\\ \\nonumber\n&+ \\left. \\frac{i \\omega \\zeta}{k} \\left(\\frac{K_2}{K_1} P(1) - i \\omega P(2) \\right) - \\zeta K_1 \\right\\}\n\\\\ \\nonumber\n&+ \\frac{\\pi}{k} e^{-d} \\frac{1}{k^2} \\left\\{\\omega^2 (d^2 + 2 d + 2) \n\\right.\n\\\\ \\nonumber\n&+ \\left. \n2 i \\omega \\frac{K_2}{K_1} \\zeta (1 + d) - \n\\left(\\frac{K_2 \\zeta}{K_1} \\right)^2 \\right\\}\n,\n\\\\\nI_{23} &= - i \\zeta \\left(- 3 K_1 - \\frac{K_1^2}{K_2} \\zeta + \\zeta K_2 \\right) \n\\\\ \\nonumber\n&+ i \\frac{K_2 \\zeta^2}{K_1 k} \\left(- 3 P(0) - \\frac{K_1 \\zeta}\n{K_2} P(0) + \\zeta P(1) \\right)\n\\\\ \\nonumber\n&+ \\frac{\\omega \\zeta^2}{k} \\left(- 3 P(1) - \\frac{K_1 \\zeta}{K_2} P(1) + \\zeta P(2) \\right)\n\\\\ \\nonumber\n&+ \\frac{\\pi}{k} e^{-d} \\left[\\omega \\left\\{(1 + d) \\frac{K_2'}{K_2} \\zeta + d^2 + d + 1 \\right\\} \n\\right.\n\\\\ \\nonumber\n&+ \\left. i \\frac{K_2}{K_1} \\zeta \\left(\nd + \\frac{K_2'}{K_2} \\zeta \\right) \\right]\n. \n\\end{align}\n\nNext, we multiply Eq.~(\\ref{eq:mlinbolk}) by ${\\bf k \\times p} \\equiv k p^{\\perp}$ and \nintegrate with respect to ${\\bf p}$.\nThen we multiply by $K_1 k$\nand the equation reduces to\n\\begin{equation}\nI_{\\perp \\perp} \\delta u_{\\perp} = 0\n,\n\\label{eq:mshear} \n\\end{equation}\nwhere\n\\begin{align}\nI_{\\perp \\perp} &= 2 k K_1 - \\int^{\\infty}_1 dy \\; e^{-\\zeta y} \\left[ -b \\zeta \\sqrt{y^2 - 1} \n\\right.\n\\\\ \n\\nonumber\n&+ \\left. \\left\\{ b^2 + \\zeta^2 (y^2 - 1) \\right\\} \n\\arctan \\frac{\\zeta \\sqrt{y^2 - 1}}{b} \\right]\n\\\\ \n\\nonumber\n&- \\frac{\\pi}{\\zeta} e^{-d} \\left\\{(d^2 + 2 d + 2) \\left(1 - \\frac{\\omega^2}{k^2} \\right) \n\\right.\n\\\\\n\\nonumber\n&- \\left. \\frac{2 i K_2}{k^2 K_1} \\zeta \\omega (1 + d)\n+ \\left(\\frac{K_2^2 \\zeta^2}{k^2 K_1^2} - \\zeta^2 \\right) \\right\\}\n,\n\\end{align}\nEq.~(\\ref{eq:mmom}) corresponds to the longitudinal mode ($\\delta u_x \\neq 0, \\delta u_{\\perp} = 0$), \nand Eq.~(\\ref{eq:mshear}) corresponds to the transverse mode ($\\delta u_x = 0, \\delta u_{\\perp} \\neq 0$).\n\nFinally, we multiply Eq.~(\\ref{eq:mlinbolk}) by $p^0$ and \nintegrate with respect to ${\\bf p}$.\nWe multiply by $K_1 k$ and \nthe equation reduces to\n\\begin{equation}\nI_{31} \\frac{\\delta n}{n} + I_{32} {\\bf k} \\cdot \\delta {\\bf u} + I_{33} \\frac{\\delta T}{T} = 0\n,\n\\label{eq:men}\n\\end{equation}\nwhere\n\\begin{align}\nI_{31} &= \\zeta P(1) - K_1 k + \\frac{\\pi}{\\zeta} e^{-d} (1 + d)\n,\n\\\\\nI_{32} &= - \\frac{i \\zeta}{k} \\left\\{K_2 - \\frac{\\zeta}{k} \\left(\\frac{K_2}{K_1} P(1) - i \\omega P(2) \\right) \\right\\}\n\\\\\n\\nonumber\n&+ \\frac{\\pi}{\\zeta} e^{-d} \\frac{i}{k} \\left\\{- i \\omega (d^2 + 2 d + 2) + \\frac{K_2}{K_1} \\zeta (1 + d) \\right\\}\n,\n\\\\\nI_{33} &= \\zeta \\left(- 3 P(1) - \\frac{K_1}{K_2} \\zeta P(1) + \\zeta P(2) \\right) \n\\\\\n\\nonumber\n&+ \\frac{\\pi}{\\zeta} e^{-d} \\left\\{(1 + d) \\frac{K_2'}{K_2} \\zeta + d^2 + d + 1 \\right\\}\n.\n\\end{align}\n\nIf the determinant of the above homogeneous system is set to equal zero,\nthe following dispersion relation is obtained:\n\\begin{eqnarray}\n \\begin{vmatrix}\n I_{11} & I_{12} & I_{13} & 0 & 0 \n \\\\\n I_{21} & I_{22} & I_{23} & 0 & 0 \n \\\\\n I_{31} & I_{32} & I_{33} & 0 & 0\n \\\\\n 0 & 0 & 0 & I_{\\perp \\perp} & 0 \n \\\\\n 0 & 0 & 0 & 0 & I_{\\perp \\perp} \n \\end{vmatrix}\n= 0\n.\n\\label{eq:det1}\n\\end{eqnarray}\nThis condition implies either\n\\begin{eqnarray}\n \\begin{vmatrix}\n I_{11} & I_{12} & I_{13}\n \\\\\n I_{21} & I_{22} & I_{23}\n \\\\\n I_{31} & I_{32} & I_{33}\n \\end{vmatrix}\n= 0\n, \n\\label{eq:det2}\n\\end{eqnarray}\nor\n\\begin{equation}\nI_{\\perp \\perp} =0\n.\n\\label{eq:det3}\n\\end{equation}\n\nUsing this dispersion relation, \nwe can obtain $\\delta f$ in the form \n\\begin{align}\n\\delta f({\\bf v}) &= \\sum_n \\frac{C_n f_0({\\bf v})}{1 - \\left(i \\omega - i {\\bf k} \\cdot \\frac{{\\bf p}}{p^0} \\right)\n \\frac{K_1 z}{K_2 \\zeta}} \n\\label{eq:mgenesol}\n\\\\\n\\nonumber\n&\\times\n\\left[ \\frac{\\delta n_{\\omega_n}}{n} + \\left(-1 + \\frac{p^0}{T} + \\frac{K_2'}{K_2} \\zeta \\right) \\frac{\\delta T_{\\omega_n}}{T}\n\\right.\n\\\\\n\\nonumber\n&- \\left. \\frac{{\\bf p} \\cdot \\delta {\\bf u}_{\\omega_n}}{T} \\right] e^{- i (\\omega_n t + {\\bf k \\cdot x})} \n,\n\\end{align}\nwhere $C_n$ is a constant coefficient, and \n$\\delta n_{\\omega_n}, \\delta T_{\\omega_n}$, and $\\delta {\\bf u}_{\\omega_n}$ are eigenfunctions obtained from the dispersion relations.\n\nIf $1 \/ \\tau_{M*} - i \\omega + i {\\bf k \\cdot v} = 0$,\nthe mode becomes continuous~\\cite{book,Takamoto & Inutsuka (2010)}.\nAccording to Eq.~(\\ref{eq:mlinbolf}), \nthe eigenfunction for this mode satisfies the equation\n\\begin{eqnarray}\n0 = \\delta f_{eq}\n.\n\\end{eqnarray}\nThis mode represents the decay of the moments of $f$ with vanishing $\\delta f_{eq}$, i.e.,\n$\\delta n = \\delta T = 0$, $\\delta {\\bf u = 0}$.\n\nUnlike the case of the nonrelativistic BGK model and Anderson-Witting model, \nthe decay rate of this continuous spectrum is not constant but depends on $p^0$~\\cite{cer1984}.\n\n\\subsection{\\label{sec:AWlevel1}THE LINEARIZED KINETIC EQUATION AND \nDISPERSION RELATION OF THE ANDERSON-WITTING MODEL}\nIn this section, we derive the dispersion relation of the relativistic kinetic model of Anderson-Witting.\nTo obtain the dispersion relation, \nwe apply an approach similar to that in the work of Cercignani and Majorana~\\cite{Cercignani & Majorana(1985)}.\n\nWhen there is no external field, \nthe equation of the Anderson-Witting model~\\cite{Anderson & Witting(1974)} is given by\n\\begin{align}\n&\\frac{D}{D s} f = p^{\\mu} \\partial_{\\mu} f \n\\label{eq:AWBGKbol}\n\\\\ \\nonumber\n&= p^0 \\left(\\frac{\\partial}{\\partial t} + {\\bf v} \\cdot \\nabla \\right) f \n= - u_{\\nu} p^{\\nu} \\frac{f-f_{eq}}{\\tau} \n,\n\\end{align}\nwhere $f_{eq}$ is the local equilibrium distribution function defined by Eq.~(\\ref{eq:mjeq}).\n\nAs in the Marle model, \nEq.~(\\ref{eq:AWBGKbol}) is a nonlinear equation for $f(t,{\\bf x, p})$ \nbecause of the nonlinear dependence of $f_{eq}$ on $f$ through the following matching conditions:\n\\begin{equation}\nu_{\\nu} \\int \\frac{d^3 p}{p^0} p^{\\nu} \\psi \\left( f_{eq} - f \\right) = 0 \n\\label{eq:AWmat}\n,\n\\end{equation}\nwhere $\\psi$ is defined by Eq.~(\\ref{eq:psi}).\n\nTo obtain the dispersion relation, \nwe start by expanding the distribution function around a global equilibrium state $f_0({\\bf p})$, \n\\begin{equation}\n\\delta f = f - f_0, \\quad \\delta f_{eq} = f_{eq} - f_0.\n\\end{equation}\nThe kinetic equation of the Anderson-Witting model Eq.~(\\ref{eq:AWBGKbol}) reduces to\n\\begin{equation}\n\\left( \\frac{\\partial}{\\partial t} + {\\bf v} \\cdot \\nabla \\right) \\delta f \n= - \\frac{\\delta f - \\delta f_{eq}}{\\tau}\n,\n\\label{eq:AWlinbol}\n\\end{equation}\nin a linear approximation.\n\nWe assume a solution of the form\n\\begin{equation}\n\\delta f = \\delta \\tilde{f} e^{- i k_{\\mu} x^{\\mu}} = \\delta \\tilde{f} e^{- i \\omega(t-t_0)+i {\\bf k \\cdot x}}\n.\n\\label{eq:AWome}\n\\end{equation}\n\nThen, Eq.~(\\ref{eq:AWlinbol}) reduces to\n\\begin{equation}\n\\left( \\frac{1}{\\tau} -i \\omega + i {\\bf k \\cdot v} \\right) \\delta f = \\frac{1}{\\tau} \\delta f_{eq}\n.\n\\end{equation}\n\nWe consider an equilibrium background state, \nin which the fluid is at rest, so that $u^{\\mu} = (1, {\\bf 0})$ and \n$\\delta u^{\\mu} = (0, \\delta {\\bf u})$ due to the relation $u^{\\mu} \\delta u_{\\mu} = 0$.\nThen, $\\delta f_{eq}$ is\n\\begin{align}\n\\delta f_{eq} &= f_0 \\left[ \\frac{\\delta n}{n} + \\left(-1 + \\frac{p^0}{T} + \\frac{K_2'}{K_2} \\zeta \\right) \\frac{\\delta T}{T}\n - \\frac{{\\bf p} \\cdot \\delta {\\bf u}}{T} \\right] \n,\n\\\\\nf_0 &= \\frac{n}{4 \\pi m^2 T K_2(\\zeta)} \\exp \\left[- \\frac{p^0}{T} \\right]\n,\n\\end{align}\nwhere $\\delta {\\bf u}$ is the space component of the Landau-Lifshitz velocity, as explained in Sec.~\\ref{sec:level31}.\n\nUsing the matching conditions, we can rewrite $\\delta \\rho, \\delta {\\bf u}$, and $\\delta T$ as integrals of $\\delta f$:\n\\begin{align}\n\\delta n(t,{\\bf x}) &= \\int \\frac{d^3 p}{p^0} p^0 \\delta f\n, \\\\\n\\delta {\\bf u}(t,{\\bf x}) &= - \\frac{1}{n} \\int \\frac{d^3 p}{p^0} {\\bf p} \\delta f\n, \\\\\n\\delta T(t,{\\bf x}) &= \\int \\frac{d^3 p}{p^0} p^0\n\\\\ \\nonumber\n&\\times \\frac{- 1 + K_2' \\zeta \/ K_2 + p^0 \/ T}{\\left(1 - K_1 \\zeta \/ K_2 \\right)\n\\left(3 + \\zeta^2 + K_1 \\zeta \/ K_2 \\right)} \\delta f\n.\n\\end{align}\n\nThen, Eq.~(\\ref{eq:AWlinbol}) becomes\n\\begin{align}\n&\\left( \\frac{1}{\\tau} -i \\omega + i {\\bf k \\cdot v} \\right) \\delta f({\\bf p}) \n\\\\\n\\nonumber\n&= \\int \\frac{d^3 p'}{p'^0} \\frac{f_0({\\bf p})}{\\tau} \n\\left[\\frac{p'^0}{n} - \\frac{{\\bf p} \\cdot {\\bf p'}}{T}\n\\right.\n\\\\ \\nonumber\n&+ \\left. \\frac{T}{n} \\frac{p'^0}{\\left(1-K_1 \\zeta \/ K_2\\right) \\left(3 + \\zeta^2 + K_1 \\zeta \/ K_2 \\right)} \n\\right.\n\\\\ \\nonumber\n&\\times\n\\left.\n\\left(-1 + \\frac{p^0}{T} + \\frac{K_2'}{K_2} \\zeta \\right) \n \\left( - 1 + \\frac{p'^0}{T} + \\frac{K_2'}{K_2} \\zeta \\right) \\right] \\delta f({\\bf p'})\n.\n\\end{align}\n\nIn the following, \nwe take $\\tau$ as a unit of time:\n\\begin{equation}\n\\omega \\tau \\rightarrow \\omega, \\quad \\tau k \\rightarrow k.\n\\end{equation}\n\nFinally, the linearized equation of the BGK model of Anderson-Witting is\n\\begin{align}\n\\delta f({\\bf p}) &= \\int \\frac{d^3 p'}{p'^0} K({\\bf p,p'}) \\delta f({\\bf p'}) \n\\label{eq:AWK}\n,\n\\\\\nK({\\bf p,p'}) &\\equiv \\frac{f_0({\\bf p})}{1 - i \\omega + i {\\bf k} \\cdot \\frac{{\\bf p}}{p^0} }\n\\left[\\frac{p'^0}{n} - \\frac{{\\bf p} \\cdot {\\bf p'}}{T}\n\\right.\n\\\\ \\nonumber\n&+ \\left. \\frac{T}{n} \\frac{p'^0}{\\left(1-K_1 \\zeta \/ K_2\\right) \\left(3 + \\zeta^2 + K_1 \\zeta \/ K_2 \\right)} \n\\right.\n\\\\ \\nonumber\n&\\times \\left.\n\\left(-1 + z + \\frac{K_2'}{K_2} \\zeta \\right) \n \\left( - 1 + z' + \\frac{K_2'}{K_2} \\zeta \\right) \\right] \\delta f({\\bf p'})\n,\n\\end{align}\nwhere $z = p^0 \/ T$. \nThis equation make sense only when \n$1 - i \\omega + i {\\bf k} \\cdot \\frac{{\\bf p}}{p^0} \\neq 0$\n.\nWhen\n$1 - i \\omega + i {\\bf k} \\cdot \\frac{{\\bf p}}{p^0} = 0$\n,\nthe mode becomes continuous, \nas in the Marle model.\n\nEq.~(\\ref{eq:AWK}) is a homogeneous Fredholm integral equation of the second kind.\nIn particular, the kernel function $K({\\bf p, p'})$ can be separated with respect to the variables ${\\bf p}$\nand ${\\bf p'}$, and \nthis equation can be solved according to a general procedure.\n\nFirst, we multiply Eq.~(\\ref{eq:AWK}) by $p^0$ and \nintegrate with respect to ${\\bf p}$.\nThen, we multiply by $K_2 k$ \nand the equation reduces to\n\\begin{equation}\nI_{11} \\frac{\\delta n}{n} + I_{12} {\\bf k} \\cdot \\delta {\\bf u} + I_{13} \\frac{\\delta T}{T} = 0\n,\n\\label{eq:AWcont} \n\\end{equation}\nwhere\n\\begin{align}\nI_{11} &= \\zeta Q(2) - K_2 k\n,\n\\\\\nI_{12} &= - \\frac{i}{k} \\left(3 K_2 + \\zeta K_1 - b \\zeta^2 Q(3) \\right) \n,\n\\\\\nI_{13} &= \\left(- 3 - \\frac{K_1 \\zeta}{K_2} \\right) \\zeta Q(2) + \\zeta^2 Q(3)\n,\n\\end{align}\nand $Q(n)$ and $b$ are defined as follows:\n\\begin{align}\nQ(n) &= \\int^{\\infty}_{1} dy \\; e^{-\\zeta y} y^n \\arctan \\frac{\\sqrt{y^2 - 1}}{b \\:y}\n,\n \\\\\nb &= \\frac{1 - i \\omega }{k}\n.\n\\end{align}\n\nNext, we multiply Eq.~(\\ref{eq:AWK}) by $p^0 \\; {\\bf k \\cdot p}$ and \nintegrate with respect to ${\\bf p}$.\nThen, we multiply by $K_2$, \nand the equation reduces to\n\\begin{equation}\nI_{21} \\frac{\\delta n}{n} + I_{22} {\\bf k} \\cdot \\delta {\\bf u} + I_{23} \\frac{\\delta T}{T} = 0 \n,\n\\label{eq:AWmom} \n\\end{equation}\nwhere\n\\begin{align}\nI_{21} &= 3 K_2 + \\zeta K_1 - b \\zeta^2 Q(3)\n,\n\\\\\nI_{22} &= - i \\left[\\zeta K_3 - \\frac{b}{k} \\left\\{(12 + \\zeta^2) K_2 \n\\right. \\right.\n\\\\ \\nonumber\n&+ \\left. \\left. 3 \\zeta K_1 - b \\zeta^3 Q(4) \\right\\} \\right] \n,\n\\\\\nI_{23} &= - K_1 \\zeta \\left(3 + \\frac{K_1 \\zeta}{K_2} \\right) + \n(3 + \\zeta^2) K_2 \n\\\\ \\nonumber\n&- b \\zeta^2 \\left(- 3 Q(3) - \\frac{K_1}{K_2} \\zeta Q(3) + \\zeta Q(4) \\right)\n,\n\\end{align}\n\nNext, we multiply Eq.~(\\ref{eq:AWK}) by $p^0 \\; {\\bf k \\times p} \\equiv k p^0 p^{\\perp}$ and \nintegrate with respect to ${\\bf p}$.\nThen, we multiply by $2 k K_2$, \nand the equation reduces to\n\\begin{equation}\nI_{\\perp \\perp} \\delta u_{\\perp} = 0\n,\n\\label{eq:AWshear}\n\\end{equation}\nwhere\n\\begin{align}\nI_{\\perp \\perp} &= 2 k \\zeta K_3 - \\zeta^3 \\left\\{(b^2 + 1) Q(4) - Q(2) \\right\\} \n\\\\ \\nonumber\n&+ b \\left\\{(12 + \\zeta^2) K_2 + 3 \\zeta K_2 \\right\\}\n,\n\\end{align}\n\nFinally, we multiply Eq.~(\\ref{eq:AWK}) by $(p^0)^2$ and \nintegrate with respect to ${\\bf p}$.\nThen we multiply by $K_2 k$, \nand the equation reduces to\n\\begin{equation}\nI_{31} \\frac{\\delta n}{n} + I_{32} {\\bf k} \\cdot \\delta {\\bf u} + I_{33} \\frac{\\delta T}{T} = 0\n,\n\\label{eq:AWen} \n\\end{equation}\nwhere\n\\begin{align}\nI_{31} &= k (3 K_2 + \\zeta K_1) - \\zeta^2 Q(3)\n,\n\\\\\nI_{32} &= - \\frac{i}{k} \\left[b \\zeta^3 Q(4) - (\\zeta^2 + 12) K_2 - 3 \\zeta K_1 \\right] \n,\n\\\\\nI_{33} &= k \\left\\{(3 + \\zeta^2) K_2 - \\zeta K_1 \\left(3 + \\frac{K_1 \\zeta}{K_2} \\right) \\right\\} \n\\\\ \\nonumber\n&- \\zeta^2 \\left(- 3 Q(3) - \\frac{K_1}{K_2} Q(3) \\zeta + \\zeta Q(4) \\right)\n,\n\\end{align}\n\nIf the determinant of the above homogeneous system is set to zero,\nwe can obtain dispersion relation the same as Eqs. (\\ref{eq:det1}), (\\ref{eq:det2}), and (\\ref{eq:det3}).\n\nUsing this dispersion relation, we can obtain $\\delta f$ in the form\n\\begin{align}\n\\delta f({\\bf v}) &= \\sum_n \\frac{C_n f_0({\\bf v})}{1 - i \\omega + i {\\bf k} \\cdot \\frac{{\\bf p}}{p^0}} \n\\\\ \\nonumber\n&\\times\n\\left[ \\frac{\\delta n_{\\omega_n}}{n} + \\left(-1 + \\frac{p^0}{T} + \\frac{K_2'}{K_2} \\zeta \\right) \\frac{\\delta T_{\\omega_n}}{T}\n\\right.\n\\\\ \\nonumber\n&- \\left. \\frac{{\\bf p} \\cdot \\delta {\\bf u}_{\\omega_n}}{T} \\right] e^{- i (\\omega_n t + {\\bf k \\cdot x})} \n,\n\\label{eq:AWgenesol}\n\\end{align}\nwhere $C_n$ is a constant coefficient, and \n$\\delta n_{\\omega_n}, \\delta T_{\\omega_n}$, and $\\delta {\\bf u}_{\\omega_n}$ are eigenfunctions obtained from the dispersion relations.\n\n\\section{\\label{sec:level2}RESULTS}\n\\subsection{\\label{sec:mlevel2}MARLE MODEL}\nIn this section, \nwe show the dispersion relations of the modified Marle model obtained in the previous sections.\nWe solve the dispersion relations numerically; \nthe results are shown below.\nFirst, we show the thermal conduction mode in Figs.~{\\ref{fig:1}}, {\\ref{fig:2}}, and {\\ref{fig:3}}.\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=7cm,clip]{relatz100_new.eps}\n \\caption{Decay rate of the thermal conduction mode in the nonrelativistic case: $m \/ T =100$.\n The black curve corresponds to the Anderson-Witting model \n while the thick gray curve corresponds to the modified Marle model.}\n \\label{fig:1}\n\\end{figure}\n\\begin{figure}[here]\n \\centering\n \\includegraphics[width=7cm,clip]{relatz13.eps}\n \\caption{Decay rate of the thermal conduction mode of the modified Marle model \n in the relativistic case: $m \/ T =1$.\n The gray zone represents the region in which the BGK approximation is expected to be worse; \n the black zone represents the region in which we cannot use the BGK approximation.\n }\n \\label{fig:2}\n\\end{figure}\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=7cm,clip]{relatz0013.eps}\n \\caption{Decay rate of the thermal conduction mode of the modified Marle model \n in the ultra-relativistic case: $m \/ T =0.01$.}\n \\label{fig:3}\n\\end{figure}\nAt long wavelengths, \nthe decay rate is proportional to $k^2$ \nand reproduces the result obtained by the first-order Chapman-Enskog approximation.\nNote that the decay rate of the thermal conduction mode diverges at finite wavelengths in the relativistic cases.\nThis may be equivalent to the critical frequency of the thermal wave mode \npredicted in the work of Cercignani and Majorana~\\cite{Cercignani & Majorana(1985)}.\nWe will return to this problem later.\nSecond, we show the sound wave mode in Figs.~{\\ref{fig:4}}, {\\ref{fig:5}}, and {\\ref{fig:6}}.\n\\begin{figure}[here]\n \\centering\n \\includegraphics[width=7cm,clip]{relaxcz100_new.eps}\n \\caption{Dispersion relation of the sound wave mode \n in the nonrelativistic case: $m \/ T = 100$.\n Solid curve represents the decay rate ($-$Im~$\\omega$); \n dotted curve represents the frequency Re~$\\omega$.\n The black curve corresponds to the Anderson-Witting model \n while the thick gray curve corresponds to the modified Marle model.}\n \\label{fig:4}\n\\end{figure}\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=7cm,clip]{relaxcz13.eps}\n \\caption{Dispersion relation of the sound wave mode of the modified Marle model \n in the relativistic case: $m \/ T = 1$.\n Solid curve represents the decay rate ($-$Im~$\\omega$); \n dotted curve represents the frequency Re~$\\omega$.}\n \\label{fig:5}\n\\end{figure}\n\\begin{figure}[here]\n \\centering\n \\includegraphics[width=7cm,clip]{relaxcz0013.eps}\n \\caption{Dispersion relation of the sound wave mode of the modified Marle model \n in the ultra-relativistic case: $m \/ T = 0.01$.\n Solid curve represents the decay rate ($-$Im~$\\omega$); \n dotted curve represents the frequency Re~$\\omega$.}\n \\label{fig:6}\n\\end{figure}\nAs in the thermal conduction mode, \nat the long wavelengths the decay rate is proportional to $k^2$.\nIn the relativistic and ultra-relativistic cases, \nthe phase velocity becomes larger than light velocity at some wavelength, \nand we stop the calculation \nbecause the physical collision term produces a phase speed less than light velocity~\\cite{cer1983}.\nAs in the case of thermal conduction mode, \nthis may be equivalent to the critical frequency predicted in the work of Cercignani and Majorana~\\cite{Cercignani & Majorana(1985)}.\nNumerically we obtain that $k_{max} \\simeq 80$ when $\\zeta = 5$ and $k_{max} > 100$ when $\\zeta = 10$ \nwhere $k$ is the maximum wavelength of applicability of BGK model. \n\nFinally, we show the shear flow mode in Figs.~{\\ref{fig:7}}, {\\ref{fig:8}}, and {\\ref{fig:9}}.\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=7cm,clip]{relasz100_new.eps}\n \\caption{Decay rate of the shear flow mode \n in the nonrelativistic case: $m \/ T =100$.\n The black curve corresponds to the Anderson-Witting model \n while the thick gray curve corresponds to the modified Marle model.}\n \\label{fig:7}\n\\end{figure}\n\\begin{figure}[here]\n \\centering\n \\includegraphics[width=7cm,clip]{relasz1.eps}\n \\caption{Decay rate of the shear flow mode of the modified Marle model \n in the relativistic case: $m \/ T =1$.}\n \\label{fig:8}\n\\end{figure}\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=7cm,clip]{relasz001.eps}\n \\caption{Decay rate of the shear flow mode of the modified Marle model \n in the ultra-relativistic case: $m \/ T = 0.01$.}\n \\label{fig:9}\n\\end{figure}\nAs in the thermal conduction mode, \nat the long wavelengths the decay rate is proportional to $k^2$.\nThe dispersion relation for shear flow has only a decay rate, \nindicating that in rarefied gas shear flow cannot propagate.\n\nIn nonrelativistic case, \nthe relevancy of the adopted equation can be checked by comparing its dispersion relations to \nexperimental data of attenuation rate and phase velocity of sound wave. \nUnfortunately, corresponding experiments are very difficult in relativistic regime, \nand we cannot compare our results to experimental data. \nHowever, our results of nonrelativistic case ($\\zeta = 100$) reproduce the dispersion relation of \nnonrelativistic BGK~\\cite{Bhatnagar et al.(1954),Takamoto & Inutsuka (2010)}\nthat agrees with experimental data~\\cite{Meyer & Sessler(1957)} even in short-wavelength regime.\nFor this reason, we expect that our relativistic dispersion relations should be correct even in relativistic regime, at least qualitatively.\n\n\\subsection{\\label{sec:AWlevel2}ANDERSON AND WITTING MODEL}\nIn this section, \nwe show the dispersion relations of the Anderson-Witting model obtained in previous sections.\nFirst, we show the thermal conduction mode in Figs.~{\\ref{fig:11}} and {\\ref{fig:12}}. \nThe nonrelativistic limit $m \/ T = 100$ is given in Fig. {\\ref{fig:1}}.\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=7cm,clip]{relatAWz13.eps}\n \\caption{Decay rate of the thermal conduction mode of the Anderson-Witting model \n in the relativistic case: $m \/ T =1$.}\n \\label{fig:11}\n\\end{figure}\n\\begin{figure}[here]\n \\centering\n \\includegraphics[width=7cm,clip]{relatAWz0013.eps}\n \\caption{Decay rate of the thermal conduction mode of the Anderson-Witting model \n in the ultra-relativistic case: $m \/ T =0.01$.}\n \\label{fig:12}\n\\end{figure}\nAt long wavelengths, \nthe decay rate is proportional to $k^2$ \nand reproduces the result obtained by the first-order Chapman-Enskog approximation.\nAs in the Marle model, the decay rate diverges at finite wavelengths.\n\nSecond, we show the sound wave mode in Figs.~{\\ref{fig:14}} and {\\ref{fig:15}}.\nThe nonrelativistic limit $m \/ T = 100$ is given in Fig. {\\ref{fig:4}}.\n\\begin{figure}[here]\n \\centering\n \\includegraphics[width=7cm,clip]{relaxcAWz13.eps}\n \\caption{Dispersion relation of the sound wave mode of the Anderson-Witting model \n in the relativistic case: $m \/ T = 1$.\n Solid curve represents the decay rate ($-$Im~$\\omega$); \n dotted curve represents the frequency Re~$\\omega$.}\n \\label{fig:14}\n\\end{figure}\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=7cm,clip]{relaxcAWz0013.eps}\n \\caption{Dispersion relation of the sound wave mode of the Anderson-Witting model \n in the ultra-relativistic case: $m \/ T = 0.01$.\n Solid curve represents the decay rate ($-$Im~$\\omega$); \n dotted curve represents the frequency Re~$\\omega$.}\n \\label{fig:15}\n\\end{figure}\nAs in the thermal conduction mode, \nat the long wavelengths the decay rate is proportional to $k^2$.\nIn the relativistic and ultra-relativistic case, the phase velocity becomes larger than light velocity at some wavelength, \nand we stop the calculation \nas in the Marle's model.\nIn this case, the wave number $k_{max}$ at which phase velocity becomes faster than light is \n $k_{max} \\simeq 64$ when $\\zeta = 5$ and $k_{max} > 100$ when $\\zeta = 10$.\n\nFinally, we show the shear flow mode in Figs.~{\\ref{fig:17}} and {\\ref{fig:18}}.\nThe nonrelativistic limit $m \/ T = 100$ is given in Fig. {\\ref{fig:7}}. \n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=7cm,clip]{relasAWz1.eps}\n \\caption{Decay rate of the shear flow mode of the Anderson-Witting model \n in the relativistic case: $m \/ T =1$.}\n \\label{fig:17}\n\\end{figure}\n\\begin{figure}[here]\n \\centering\n \\includegraphics[width=7cm,clip]{relasAWz001.eps}\n \\caption{Decay rate of the shear flow mode of the Anderson-Witting model \n in the ultra-relativistic case: $m \/ T = 0.01$.}\n \\label{fig:18}\n\\end{figure}\nAs in the thermal conduction mode, \nat the long wavelengths the decay rate is proportional to $k^2$.\nThe dispersion relation for shear flow has only a decay rate, \nindicating that in rarefied gas shear flow cannot propagate.\n\nIn the Anderson-Witting model, \nwe find the kinetic mode. \nFigs.~\\ref{fig:19} and \\ref{fig:20} show the transverse kinetic mode, \nand Figs.~\\ref{fig:21} and \\ref{fig:22} show the longitudinal kinetic mode.\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=7cm,clip]{relaskinAWz100.eps}\n \\caption{Dispersion relation of the transverse kinetic mode of the Anderson-Witting model \n in the nonrelativistic case:\n $m \/ T = 100$.\n Solid curve represents the decay rate ($-$Im~$\\omega$); \n dotted curve represents the frequency Re~$\\omega$.}\n \\label{fig:19}\n\\end{figure}\n\\begin{figure}[here]\n \\centering\n \\includegraphics[width=7cm,clip]{relaskinAWz1.eps}\n \\caption{Dispersion relation of the transverse kinetic mode of the Anderson-Witting model \n in the relativistic case:\n $m \/ T = 1$.\n Solid curve represents the decay rate ($-$Im~$\\omega$); \n dotted curve represents the frequency Re~$\\omega$.}\n \\label{fig:20}\n\\end{figure}\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=7cm,clip]{relaxckinAWz100.eps}\n \\caption{Dispersion relation of the longitudinal kinetic mode of the Anderson-Witting model \n in the nonrelativistic case:\n $m \/ T = 100$.\n Solid curve represents the decay rate ($-$Im~$\\omega$); \n dotted curve represents the frequency Re~$\\omega$.}\n \\label{fig:21}\n\\end{figure}\n\\begin{figure}[here]\n \\centering\n \\includegraphics[width=7cm,clip]{relaxckinAWz1.eps}\n \\caption{Dispersion relation of the longitudinal kinetic mode of the Anderson-Witting model \n in the relativistic case:\n $m \/ T = 1$.\n Solid curve represents the decay rate ($-$Im~$\\omega$); \n dotted curve represents the frequency Re~$\\omega$.}\n \\label{fig:22}\n\\end{figure}\nDue to the numerical difficulties, \nwe cannot find the kinetic mode in the ultra-relativistic case: \nbecause the phase velocity of the longitudinal kinetic mode in the relativistic case is a little \nfaster than that of light, \nthis mode is physically incorrect.\nWe expect that this problem results from the accuracy of the BGK model \nand does not show that there is any longitudinal kinetic mode of in the relativistic case.\n\nAs for the Marle model, \nwe cannot compare these dispersion relations and experiment; \nhowever, we expect that our relativistic dispersion relations is correct even in relativistic regime, \nat least qualitatively for the same reason of Marle model. \n\n\\section{\\label{sec:level3}DISCUSSION}\n\n\\subsection{\\label{sec:level322}ANALYSIS IN THE SHORT WAVE LENGTH AND HIGH FREQUENCY LIMIT}\nIn this section, \nwe analyze the dispersion relation for the short wavelength limit of the shear flow mode \nand the high frequency limit of the thermal conduction mode.\nIn particular, \ndivergence of the decay rate of the thermal conduction mode is not observed in the analysis of the nonrelativistic BGK model,\nso we have to check this divergence analytically.\nFor simplicity we analyze the Anderson-Witting model.\n\nFirst, we analyze the dispersion relation for large wave number in the shear flow mode.\nFrom Sec.~\\ref{sec:AWlevel1}, the dispersion relation of the shear flow mode is\n\\begin{align}\nn \\delta u_{\\perp} &= \\int \\frac{d^3 {\\bf p}}{p^0} p_{\\perp} \\delta f \n\\\\\n\\nonumber\n&= \\int \\frac{d^3 {\\bf p}}{p^0} p_{\\perp} \\frac{f_{eq}}{1 - i \\omega + i k v_x} \n\\\\\n\\nonumber\n&\\times\n\\left[ \\frac{\\delta n}{n} + \\left(-1 + \\frac{p^0}{T} + \\frac{K_2'}{K_2} \\zeta \\right) \\frac{\\delta T}{T} + {\\bf p \\cdot \\delta u} \\right]\n\\\\\n\\nonumber\n&+ \\frac{2 \\pi}{k} \\int^{\\infty}_{\\zeta \/ \\sqrt{1+b^2}} d z~z^2 [(1 + b^2) z^2 - \\zeta^2] f_{eq} \\delta u_{\\perp}\n\\\\\n\\nonumber\n&= \\int \\frac{d^3 {\\bf p}}{p^0} \\frac{p_{\\perp} f_{eq}}{1 - i \\omega + i k v_x} p_{\\perp} \\delta u_{\\perp}\n\\\\\n\\nonumber\n&+ \\frac{2 \\pi}{k} \\int^{\\infty}_{\\zeta \/ \\sqrt{1+b^2}} d z~z^2 [(1 + b^2) z^2 - \\zeta^2] f_{eq} \\delta u_{\\perp}\n,\n\\label{eq:slwl}\n\\\\\nb &= \\frac{1 - i \\omega}{k}\n.\n\\end{align}\nIn the above equation, \nwe add correction terms of analytical continuation, \nbecause the decay rate $-\\mathrm{Im}~\\omega$ is larger than $1 \/ \\tau$ in the short wavelength limit.\n\nIn the high frequency limit, \nwe use the following approximations:\n\\begin{align}\n\\frac{1}{1 - i \\omega + i k v_x} &\\simeq - \\frac{1}{i \\omega} \\left( 1 + \\frac{1 + i k v_x}{i \\omega} \\right)\n,\n\\\\\n\\frac{\\zeta}{\\sqrt{1 + b^2}} &\\simeq \\frac{\\zeta}{b^2}\n,\n\\\\\nb &\\simeq - \\frac{i \\omega}{k}\n,\n\\end{align}\nUsing the above approximations and neglecting the terms higher than the third order of $|1 \/ \\omega|$, \nwe reduce Eq.~(\\ref{eq:slwl}) to\n\\begin{align}\n&1 - \\frac{2 \\pi}{k} \\frac{(i \\omega)^2}{k^2} \\alpha = 0\n,\n\\\\\n&\\alpha = \\int^{\\infty}_{\\zeta\/b^2} dz~z^4 f_{eq}\n\\end{align}\nFinally, we obtain the dispersion relation of the shear flow mode in the high frequency limit, \n\\begin{equation}\n\\omega = - i \\sqrt{2 \\pi \\alpha} k^{3\/2}\n,\n\\end{equation}\nwhere we take the sign representing the decaying mode.\nThis reproduces the results in Sec.~\\ref{sec:AWlevel2}.\nIn the nonrelativistic regime, \nwe cannot neglect the $\\omega \/ k$ dependence in $\\alpha$, \nand the behavior of the shear flow mode in the large wave number limit becomes different from $k^{3\/2}$.\n\nNext, we consider the high frequency limit of the thermal conduction mode.\nSince the decay rate diverges, \nwe study only the highest-order terms in $\\omega$.\n\nFrom Sec.~\\ref{sec:AWlevel1}, \nthe conservation law for particle number Eq.~(\\ref{eq:AWcont}) is\n\\begin{align}\n\\frac{\\delta n}{n} &= \\int \\frac{d^3 {\\bf p}}{p^0} p^0 \\frac{f_{eq}}{1 - i \\omega i k v_x} \n\\\\ \n\\nonumber\n&\\times\n\\left[ \n\\frac{\\delta n}{n} + \\left( - 1 + \\frac{p^0}{T} + \\frac{K_2'}{K_2} \\zeta \\right) \\frac{\\delta T}{T} + {\\bf p \\cdot \\delta u}\n\\right]\n\\\\\n\\nonumber\n& + \\frac{\\pi}{K_2 k \\zeta^2} e^{-d} \n\\left[ (d^2 +2 d + 2) \\frac{\\delta n}{n} + i b (d^3 + 3 d^2 + 6 d) \\delta u_x \n\\right.\n\\\\\n\\nonumber\n&\\left.\n+ \\left\\{\n- \\left( 3 + \\frac{K_1}{K_2} \\zeta \\right) (d^2 + 2 d + 2) \n\\right. \\right.\n\\\\\n\\nonumber\n+& \\left. \\left. d^3 + 3 d^2 + 6 d + 6 \n\\right\\} \\frac{\\delta T}{T}\n\\right]\n,\n\\\\\nd &\\equiv \\frac{\\zeta}{\\sqrt{1 + b^2}}\n.\n\\end{align}\nIn the above equation, \nwe add the correction terms of analytical continuation, \nbecause the decay rate $-\\mathrm{Im}~\\omega$ is larger than $1 \/ \\tau$ in the large wave number limit.\n\nAs in the shear flow mode,\nwe expand the integrand in powers of $\\omega$ and neglect terms higher than second order on the right-hand side.\nThen, the above equation reduces to\n\\begin{align}\n&\\left( - 1 + \\frac{2 \\pi}{K_2 k \\zeta^2} \\right) \\frac{\\delta n}{n} \n\\\\\n\\nonumber\n&+ \\frac{6 \\pi i b}{K_2 k \\zeta^2} \\delta u_x \n- \\frac{12 \\pi}{ K_2 k \\zeta^2} \\frac{K_1}{K_2} \\zeta \\frac{\\delta T}{T} = 0\n,\n\\end{align}\nwhere we consider the relativistic limit $\\zeta \\ll 1$, \nso that we approximate $\\exp [- d] \\simeq 1$.\n\nSimilarly, the conservation of energy Eq.~(\\ref{eq:AWen}) reduces to \n\\begin{align}\n&\\left( - c_v T + \\frac{6 \\pi}{K_2 k \\zeta^2} \\right) \\frac{\\delta n}{n} \n\\\\\n\\nonumber\n&+ \\frac{24 \\pi i b}{K_2 k \\zeta^2} \\delta u_x \n\\left[\n- n c_v + \\frac{6 \\pi}{ K_2 k \\zeta^2} \\left( 1 - \\frac{K_1}{K_2} \\zeta \\right)\n\\right] \\frac{\\delta T}{T} = 0\n,\n\\end{align}\n\nFrom Sec.~\\ref{sec:level31}, \nthe dispersion relation of the Anderson-Witting model includes the conservation of energy Eq.~(\\ref{eq:eflow}), \nso we use the conservation of energy instead of the conservation of momentum Eq.~(\\ref{eq:AWmom}).\n\\begin{equation}\n- i \\omega (c_v T \\delta n + n c_v \\delta T) + i n h k \\delta u_x = 0\n,\n\\end{equation}\nFrom the above equations, \nwe can obtain the dispersion relation in the form\n\\begin{align}\n&\\frac{i \\omega^2}{k^3 K_2^3 \\zeta^4}\n\\left[24 c_v k K_2^2 \\pi \\zeta^2 \n\\right.\n\\\\\n\\nonumber\n&- \\left. 12 c_v \\pi^2 \\left\\{4 K_2 + 3 K_1 \\zeta (- 1 + 8 \\zeta) \\right\\} \\right] + O (\\omega) = 0,\n\\end{align}\nIn the high frequency limit, \nterms of lower order than $\\omega^2$ can be neglected. \nFor the left-hand side to vanish, \nthe coefficient of $\\omega^2$ should be $0$.\nThen we obtain\n\\begin{equation}\nk = \\frac{4 K_2 - 3 K_1 \\zeta + 24 K_1 \\zeta^2}{2 K_2^2 \\zeta^2} \\pi\n.\n\\end{equation}\nIf we insert $\\zeta = 0.01$, we obtain $k \\simeq 3.141\\cdots$.\nThis reproduces the critical wave number for thermal conduction in the Anderson-Witting model accurately.\nSimilarly, if we insert $\\zeta = 100$, we obtain $k \\sim 10^{45}$.\nThis indicates that in the nonrelativistic regime the Anderson-Witting model does not effectively yield the critical wave number. \nIn deriving the above equation, \nwe assume $\\exp[- d] \\sim 1$, \nso we can not reproduce the critical wave number of $\\zeta = 1$ very well.\nMore accurate analysis reproduces the critical wave number of $\\zeta = 1$ as $k \\simeq 4.958$.\n\n\n\\subsection{\\label{sec:level33}COMPARISON TO 14-MOMENT EXPANSION}\nIn this section, we analyze the dispersion relation of the 14-moment theory\nand compare it with that of the BGK model.\nWe assume that the relativistic gas is at rest, \nand we consider only longitudinal waves.\nIn this case, the dispersion relation is given in the work of Cercignani and Kremer\\cite{Cerbook,CercignaniKremer(2001)}.\nThe IS equation is based on 14-moment theory; \nthus, results obtained in this section can be applied to IS equation as well.\n\nIn the nonrelativistic limiting case $\\zeta \\gg 1$, \nthe dispersion relation reduces to\n\\begin{align}\n&\\left( \\frac{k c_s}{\\omega} \\right)^4 \\left[\n\\frac{567}{100} i \\omega^3 - \\frac{477}{100} \\omega^2 - \\frac{9}{10} i \\omega\n\\right]\n\\\\ \\nonumber\n&- \\left( \\frac{k c_s}{\\omega} \\right)^2 \\left[\n\\frac{441}{50} i \\omega^3 - \\frac{342}{25} \\omega^2 - \\frac{63}{10} i \\omega + 1\n\\right]\n\\\\ \\nonumber\n&+ \\frac{9}{4} i \\omega^3 - \\frac{21}{4} \\omega^2 - 4 i \\omega + 1 = 0\n,\n\\end{align}\nwhere\n\\begin{align}\nc_s &= \\sqrt{\\frac{c_p T}{c_v h}}\n,\n\\\\\nh &= m G(\\zeta)\n,\n\\\\\nc_v &= \\zeta^2 + 5 G \\zeta - G^2 \\zeta^2\n,\n\\\\\nc_p &= c_v + 1\n,\n\\\\\nG &= \\frac{K_3(\\zeta)}{K_2(\\zeta)}\n.\n\\end{align}\nWe are interested in the Cauchy problem, \nso we solve the above equation with respect to $\\omega$.\nThe results are illustrated in Figs.~\\ref{fig:23} and \\ref{fig:24} in the case of $\\zeta = m \/ T = 100$.\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=7cm,clip]{IStz100.eps}\n \\caption{Decay rate of the thermal conduction and kinetic modes \n of the 14-moment theory\n in the nonrelativistic case: $m \/ T =100$.}\n \\label{fig:23}\n\\end{figure}\n\\begin{figure}[here]\n \\centering\n \\includegraphics[width=7cm,clip]{ISxcz100.eps}\n \\caption{Dispersion relation of the sound wave mode of the 14-moment theory\n in the nonrelativistic case: $m \/ T = 100$.\n Solid curve represents the decay rate ($-$Im~$\\omega$); \n dotted curve represents the frequency Re~$\\omega$.}\n \\label{fig:24}\n\\end{figure}\nFig.~\\ref{fig:23} shows the heat conduction mode and its accompanying kinetic mode. \nThis figure shows that at the long wavelengths, \nthe decay rate of the heat conduction mode of the 14-moment theory \nis proportional to $k^2$ and reproduces the result of the first-order Chapman-Enskog approximation. \nAt short wavelengths, the decay rate of the heat conduction mode has an upper limit \nand approaches the limit asymptotically.\nIn addition, 14-moment theory reproduces the kinetic mode\nFig.~\\ref{fig:24} shows that the decay rate of the sound wave mode has similar features.\nIn comparison to the kinetic model equation and experiment~\\cite{Meyer & Sessler(1957)}, \nwe find that in the nonrelativistic limit, \nthe behavior of the decay rate of the fluid mode of 14-moment theory\nis consistent with the kinetic equation at long wavelengths \nbut inconsistent at short wavelengths.\nIn contrast to 14-moment theory, \nthe BGK approximation reproduces the result of experiment~\\cite{st65} qualitatively. \nIn addition, the kinetic mode obtained by 14-moment theory decrease with $k$ in contrast to kinetic modes of BGK equation. \n\nNext, we consider the ultra-relativistic limit $\\zeta \\ll 1$.\nIn this case, the dispersion relation reduces to\n\\begin{align}\n&\\left( \\frac{k c_s}{\\omega} \\right)^4 \\left[\n\\frac{225}{16} i \\omega^3 - \\frac{35}{4} \\omega^2 - \\frac{5}{4} i \\omega\n\\right]\n\\\\ \\nonumber\n&- \\left( \\frac{k c_s}{\\omega} \\right)^2 \\left[\n\\frac{175}{8} i \\omega^3 - \\frac{145}{6} \\omega^2 - \\frac{25}{3} i \\omega + 1\n\\right]\n\\\\ \\nonumber\n&+ \\frac{125}{16} i \\omega^3 - \\frac{145}{12} \\omega^2 - \\frac{73}{12} i \\omega + 1 = 0\n.\n\\end{align}\nThe results are illustrated in Fig.~\\ref{fig:25} in the case of $\\zeta = m \/ T = 0.01$.\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=7cm,clip]{IStz001.eps}\n \\caption{Decay rate of the thermal conduction and kinetic modes \n of the 14-moment theory\n in the ultra-relativistic case: $m \/ T = 0.01$.}\n \\label{fig:25}\n\\end{figure}\nFig.~\\ref{fig:25} shows the heat conduction mode and its accompanying kinetic mode.\nWe could not calculate the sound wave mode very accurately \nbecause of the numerical difficulty in the complex Newton-Raphson method, \nbut the behavior of the decay rate of the sound wave mode seems to be similar to that in the nonrelativistic case.\nThese results indicate that at short wavelengths, \nthe dispersion relation of the 14-moment theory is qualitatively different from that of the kinetic equation in the ultra-relativistic limit.\n\nIn conclusion, \nthe 14-moment theory is better than the first-order Chapman-Enskog approximation \nin the sense that this theory is causal and can describe the kinetic mode.\nHowever, the 14-moment theory cannot reproduce the result of the kinetic equation at short wavelengths \nor high frequencies, \neven in the nonrelativistic limit\nin contrast to the kinetic model equations. \nThis indicates that \n the second-order dissipation terms do not reproduce kinetically correct \n results, and keeping these second-order terms may not necessarily improve the physical description \n of the fluid phenomena; \n but just make the mathematical form of fluid equations \n hyperbolic. \n\n\n\\section{\\label{sec:level4}CONCLUSION}\nIn this paper, we have solved the dispersion relation of the kinetic equations of the Marle \nand Anderson-Witting models with respect to $\\omega$ as a function of $k$\nsince we are interested in the Cauchy problem.\nTo obtain the dispersion relation, \nan approach similar to that in the work of Cercignani and Majorana~\\cite{cer1984,Cercignani & Majorana(1985)} is applied.\nTo obtain an acceptable dispersion relation, \nwe have modified the Marle model \nsince it cannot reproduce correct results in the relativistic case.\nOur dispersion relation indicates that \nboth kinetic model equations have a critical wavelength for the sound wave and thermal conduction modes;\nfor the sound wave mode, \nthe phase velocity exceeds the speed of light at that wavelength~\\cite{cer1983}, \nand for the thermal conduction mode, \nthe decay rate diverges at that wavelength.\n\nWe have solved the dispersion relation of the 14-moment theory~\\cite{Cerbook,CercignaniKremer(2001)}\nwith respect to $\\omega$ as a function of $k$\nand compared it with that of the kinetic model equations.\nThe results show that \nthe 14-moment theory reproduces the first-order Chapman-Enskog approximation in the long wavelength region, \nbut does not reproduce the result of the kinetic equation at short wavelengths \neven in the nonrelativistic limit.\nThis indicates that \nthe second-order terms other than the time derivative of the dissipation terms are not useful \nfor physical description of relativistic dissipative fluid. \n\n\\section*{acknowledgments}\nWe thank Takayuki Muto and Takayuki Muranushi for fruitful discussions.\nWe also thank referees for advices.\nThis work was supported by the Grant-in-Aid for the Global COE Program \n\"The Next Generation of Physics, Spun from Universality and Emergence\" \nfrom the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} Observations with the Experiment to Detect the\nGlobal Epoch of Reionization Signature (EDGES) have recently resulted\nin the detection of an absorption profile with full width half maxima\n(FWHM) $19 \\, {\\rm MHz}$ centred at $78 \\, {\\rm MHz}$ in the sky\naveraged spectrum of the background radiation in the frequency range\n$50-100 \\, {\\rm MHz}$ \\citep{Bowman2018}. If confirmed by other\nsimilar experiments like the Large-Aperture Experiment to Detect the\nDark Ages (LEDA; \\citealt{Bernardi2016}) , the Sonda Cosmol\\'ogica de\nlas Islas para la Detecci\\'on de Hidr\\'ogeno Neutro (SCI-HI;\n\\citealt{Voytek2014}), the Probing Radio Intensity at high z from\nMarion (PRIZM; \\citealt{Philip2018}) and the Shaped Antenna measurement of the background\nRadio Spectrum 2 (SARAS 2; \\citealt{Singh2017}), this can be\ninterpreted as the neutral Hydrogen ({H{\\sc I}~}) 21-cm absorption profile\nresulting from the Lyman-$\\alpha$ coupling due to the formation of the\nfirst stars in the early universe \\citep{Pritchard2012}. However, the\nobservation indicates a dip with amplitude $0.5 \\, {\\rm K}$ which is\nmore than a factor of two larger than the largest predictions\n\\citep{Cohen2017}. \\citet{Barkana2018} have proposed that it is\npossible to explain this enhanced dip through a possible interaction\nbetween the baryons and dark matter particles (b-DM interaction). Such\nmodels \\citep{Barkana2018, Fialkov2018} have predicted a $10$ fold\nenhancement of the spatial fluctuations of the redshifted {H{\\sc I}~} 21-cm\nbrightness temperature $\\delta T_b(x)$. We note that other alternative\nexplanations have also been proposed \\citep{Ewall-Wice2018, Feng2018}\nto explain the enhanced dip. The latter models incorporate an\nenhancement in the radio background and they do not predict such\nenhancement in $\\delta T_b(x)$.\n\n\\citet{Dvorkin2014},\\citet{Tashiro2014},\\citet{Munoz2015} and \\citet{Xu2018}\n have considered b-DM interaction in the context of cosmology and\n large-scale structure formation. \n \\citet{Barkana2018} have assumed a non-standard\n Coulomb-like interaction between dark-matter particles and baryons\n that does not depend on whether the baryons are free or bound within\n atoms. By combining this with the radiation\n emitted by the first stars during cosmic dawn, they find a strong 21-cm absorption that can\n explain the feature measured by EDGES. These models \\citep{Barkana2018, Fialkov2018} also \n predict a 10 fold enhancement of the spatial fluctuations of the\n redshifted HI 21-cm brightness temperature $\\delta T_b(x)$.\n \\citet{Munoz2018a} explore whether dark-matter\n particles with an electric ``minicharge\" can significantly alter the\n baryonic temperature and affect the global 21-cm signal. They\n find that the entire dark matter cannot be minicharged at\n a significant level, the constraints coming from Galactic and\n extragalactic magnetic fields. However, minicharged particles that \n comprise a subpercent fraction of the dark matter and have a charge\n $\\sim 10^{-6}$ in units of the electron charge and masses $m_{\\chi}\n \\sim 1 - 60 \\, {\\rm MeV}$ can significantly cool down the\n baryons and explain the EDGES result while remaining consistent\n with other observational constraints. In a recent paper,\n subsequent to the submission of the present {\\em paper}, \n \\citet{Munoz2018b} have analysied the 21-cm brightness temperature \n fluctuations for the minicharge model. Their study confirms a significant\n enhancement in the predicted 21-cm brightness temperature \n fluctuations, including an increase in the amplitude\n of the baryon acoustic oscillation. \n\nUpcoming experiments such as the Hydrogen Epoch of Reionization Array\n(HERA; \\citealt{DeBoer2017}) and the Square Kilometre Array (SKA;\n\\citealt{Koopmans2015}) have the potential of measuring the {H{\\sc I}~} 21-cm\npower spectrum from Cosmic Dawn $(50 - 100 \\, {\\rm MHz})$. Both of\nthese experiments should easily be able to measure the corresponding\nenhanced {H{\\sc I}~} 21-cm power spectrum predicted by the b-DM interaction\nmodels \\citep{Barkana2018}.\n\n\nThe Giant Meterwave Radio Telescope (GMRT; \\citealt{Swarup1991}) is\none of the largest and most sensitive fully operational low-frequency\nradio telescopes in the world today. The array configuration of $30$\nantennas (each of $45 \\, {\\rm m}$ diameter) spanning over $25 \\, {\\rm\n km}$, provides a total collecting area of about $30,000 \\, {\\rm\n sq. \\, m}$ at metre wavelengths. The GMRT is being upgraded (uGMRT,\n\\citealt{Gupta2017}) to have seamless frequency coverage, as far as\npossible, from $50$ to $1500 \\, {\\rm MHz}$. Band-1 of uGMRT, which is\nyet to be implemented, is expected to cover the frequency range $50 -\n80 \\, {\\rm MHz}$. Earlier \\citet{Shankar2009} envisaged a $50 \\, {\\rm\n MHz}$ system developed for GMRT to provide imaging capability in the\nfrequency range $30-90 \\, {\\rm MHz}$.\n\n\nIn this paper, we investigate the prospects of detecting the\nredshifted {H{\\sc I}~} 21-cm signal power spectrum from Cosmic Dawn using the\nuGMRT. For the purpose of this analysis we have considered a\nfunctional bandwidth of $B = 20 \\, {\\rm MHz}$ centred at $\\nu_c = 78\n\\, {\\rm MHz}$, consistent with the frequency coverage described in\n\\citet{Shankar2009}. Frequencies above $90 \\, {\\rm MHz}$ are used for\nFM transmission which restricts the allowed frequency range.\nWe have also carried out a similar \nanalysis for the future SKA-Low, and we present a comparison of \nthe predictions for the uGMRT with those for the future \nSKA-Low.\n\n\\section{Methodology} We have simulated the uGMRT baseline configuration\nfor $8 \\, {\\rm hours}$ of observation targeted on a field at\n$+60^{\\circ}$ DEC with $16 \\, {\\rm s}$ integration time.\nThe entire analysis has been restricted to \n baselines with antenna separations within $2\\, {\\rm km}$,\n which contains the bulk of the cosmological signal. We assume\nthat the bandwidth $B = 20 \\, {\\rm MHz}$ is divided into $N_c = 200$\nspectral channels of $\\Delta \\nu_c = 100 \\, {\\rm KHz}$. Note that the\nvalues of $B$, $N_c$ and $\\Delta \\nu_c$ assumed here are only\nrepresentative values, and the actual values in the final\nimplementation of the telescope may be somewhat different.\nThe $20 \\, {\\rm MHz}$ bandwidth \nspans the redshift range $z=15$ to $20$, and the {H{\\sc I}~} 21-cm power spectrum \nmay evolve significantly within this redshift range. Consequently \nwe have considered three bands each of \nwidth $6 \\, {\\rm MHz}$ centred at \n$72 \\, {\\rm MHz} $, $78 \\, {\\rm MHz}$ and $84 \\, {\\rm MHz}$ which\ncorrespond to $z = 18.7, 17.2$ and $15.9$ respectively. We assume that\nthe measurements from these three bands are combined to enhance the\nsignal-to-noise ratio (SNR).\n\n\nConsidering the simulated baseline $u-v$ distribution, we use\n$\\mathbfit{k}_{\\perp}=2\\pi\\mathbfit{U}\/r$ and $k_{\\parallel m}=2\\pi\nm\/r^{'} B$ to estimate the Fourier modes at which the brightness\ntemperature fluctuations $\\Delta T(\\mathbfit{k})$ will be measured by\nthis observation. Here $\\mathbfit{U}$ refer to different baselines, $0\n\\leq m \\leq N_c\/2$, $r$ is the co-moving distance corresponding to\n$\\nu_c$ and $r^{'} = dr\/d\\nu$ evaluated at $\\nu = \\nu_c$. \nWe assume that the measured $\\Delta T(\\mathbfit{k})$ values\nare gridded in $(\\mathbfit{k}_{\\perp},k_{\\parallel})$ space \nwith a grid spacing $\\Delta k_{\\perp}=2\\pi D\/ \\lambda_c r$ and \n$\\Delta k_{\\parallel}=2\\pi \/r^{'} B$, and the gridded values \nare used to estimate the power spectrum $P(\\mathbfit{k}_g)$\nat each grid point $\\mathbfit{k}_g$. Here we have used the \nsimulation to estimate \nthe sampling function $\\tau(\\mathbfit{k}_g)$ which refers to the number of \n distinct $\\Delta T(\\mathbfit{k})$\nmeasurements that contribute to each grid point $\\mathbfit{k}_g$.\n\n\n \nFor the Cosmic Dawn {H{\\sc I}~} 21-cm signal we have used the\n value of the dimensionless {H{\\sc I}~} 21-cm power spectrum $\\Delta^2_{\\rm\n {H{\\sc I}~}}(k) = k^3 P_{{\\rm {H{\\sc I}~}}}(k)\/2 \\pi^2$ from\n earler works \\citep{Santos2010,Mellema2013}, where $P_{{\\rm {H{\\sc I}~}}}(k)$ refers \n to the {H{\\sc I}~} 21-cm power spectrum.\n\\citep{Barkana2018,Fialkov2018}. We have used the \n $z=17$ {H{\\sc I}~} 21-cm power spectrum \npredictions from \\citet{Santos2010} as the fiducial model for\n all the three bands which we have considered here.\nThese values correspond to the standard scenario, we expect a $10$\nfold enhancement in the brightness temperature fluctuations {\\it i.e.}\na dimensionless power spectrum of $100\\Delta^2_{\\rm {H{\\sc I}~}}$ in the\npresence of the b-DM interaction\n\nIn addition to the {H{\\sc I}~} 21-cm power spectrum, \nwe also have the noise power spectrum $P_N(\\ppk, k_{\\parallel})$ which can be \nestimated as follows. The measured power spectrum is related to the observed \nvisibilities $V(\\mathbfit{U},\\nu)$ as \n\\begin{equation}\nP(\\ppk, k_{\\parallel}) = \\frac{r^{'}}{\\tilde{Q}} \\int \\langle \nV(\\mathbfit{U}, \\nu) V^{*}(\\mathbfit{U}, \\nu + \\Delta \\nu) \\rangle d(\\Delta\\nu)\n\\label{eq:pk}\n\\end{equation}\nwhich can be obtained from (eq.~15.) of \\citet{Bharadwaj2005},\nwhere $\\tilde{Q} = (\\partial B\/\\partial T)^2 r^{-2} \\int A(\\theta)^2 d^2\\theta$, $A(\\theta)$ is the primary beam pattern of the telescope, $(\\partial B\/\\partial T) = 2k_B\/\\lambda^2$,\nConsidering the noise contribution $N(\\mathbfit{U},\\nu)$\nto the observed visibilities $V(\\mathbfit{U},\\nu)$, and assuming that the noise at two \ndifferent frequency channels is uncorrelated, \nwe have the noise power spectrum contribution from the visibilities \nmeasured at a single baseline $\\mathbfit{U}$ to be \n\\begin{equation}\nP_N(\\ppk, k_{\\parallel})=\\frac{r^{'} \\langle \\mid N(\\mathbfit{U},\\nu) \\mid^2 \\rangle \\, \n\\Delta \\nu_c}{\\tilde{Q}}\n\\label{eq:x1}\n\\end{equation}\nwhich is independent of $k_{\\parallel}$. The real and imaginary \nparts of $N(\\mathbfit{U},\\nu)$ both have equal variance $\\sigma_N^2$ \n\\citep{Chengalur2007} with \n\\begin{equation}\n\\sigma_N^2 = \\frac{2}{N_p \\Delta \\nu_c \\, \\Delta t}\n\\left(\\frac{k_B T_{sys}}{ \\eta A_g} \\right)^2 \n\\label{eq:x2}\n\\end{equation}\n where $T_{sys}$ is the system temperature, \n$\\Delta t$ is the integration time, $N_p$ is the number of \npolarizations and the antenna efficiency $\\eta$ is defined through \n $\\lambda^2\/ \\eta \\, A_g = \n\\int A(\\theta) d^2 \\theta $ where $A_g$ is the geometrical \narea of the antennas. \n\nIn our analysis we have gridded \nthe simulated baseline distribution on to the $(\\ppk,k_{\\parallel})$ grid \nintroduced earlier. Figure 5 of \\citealt{Choudhuri2014} shows the \nbaseline distribution corresponding to the uGMRT observations\n considered here. Note that the $\\mathbfit{U}$ values need to be multiplied \n by a factor of $\\approx 2$ to scale them from $150 \\, {\\rm MHz}$\n \\citep{Choudhuri2014} to the central frequency $78 \\, {\\rm MHz}$\n considered here.\n We see that the simulated baselines do not uniformly \nsample the $u-v$ (or equivalently $\\ppk$) plane, and we have used the \ngrid sampling function $\\tau(\\mathbfit{k}_g)$ to quantify the number \nof baselines which contribute to each grid point $\\mathbfit{k}_g$.\nIncorporating \nthis, the noise power spectrum at each grid point can be expressed as \n\\begin{equation}\nP_N(\\mathbfit{k}_g) = \\frac{T_{sys}^2 \\, r^{'} \\, r^2 }\n{\\Delta t \\, N_p \\, \\tilde{\\eta} \\, \\tau(\\mathbfit{k}_g)} \\, ,\n\\label{eq:noise}\n\\end{equation}\nwhere we have defined the dimensionless factor \n $\\tilde{\\eta} = [\\int A^2(\\theta) d^2\\theta]\/[\\int A(\\theta) d^2\\theta]^2$.\n\n\n\nWe have assumed the system temperature $T_{sys} = 3000 \\, {\\rm K}$,\n number of polarizations $N_p = 2$ and integration time $\\Delta t = 16 \\, {\\rm s}$.\nAt the frequencies of our interest the uGMRT primary beam pattern\nis well approximated by a Gaussian\n$A(\\theta)=e^{-(\\theta\/\\theta_0)^2}$ with $\\theta_0=3.1^{\\circ}$\nwhereby $\\tilde{\\eta}=54.4$. Note that it is well justified to use the \nflat sky approximation for the uGMRT. The analysis till now considers $8$ hours \nof observation which roughly corresponds to a single night. For longer observations, \nthe noise power spectrum has been scaled inversely with the number of observing \nnights. \n\n\n \n\n\nWe have binned the $k$-range accessible to uGMRT into $10$\n logarithmic bins. For each bin $a$,we the binned power spectrum \n$P_a=\\sum_g w_g P(\\mathbfit{k}_g)$ is a weighted sum of the \npower spectrum measured at all the grid points within the bin. \nThe weights have been chosen as $w_g= A P_{{\\rm\n {H{\\sc I}~}}}(\\mathbfit{k}_g)\/[P_{{\\rm {H{\\sc I}~}}}(\\mathbfit{k}_g) +\n P_N(\\mathbfit{k}_g)]^2$ (with normalization constant $A$) to\noptimise the SNR. We use the variance $(\\Delta P_a)^2 = {1\/\n \\sum_{g} [P_{{\\rm {H{\\sc I}~}}}(\\mathbfit{k}_g) + P_N(\\mathbfit{k}_g)]^{-2}}$\nto quantify the uncertainty with which it will be possible to measure \npower spectrum in each bin. \n\n \n\n\n\n\nForeground removal (e.g. \\citealt{Ali2008}) is an important \nissue for detecting the cosmological 21-cm power spectrum. \nSeveral studies have \nshown that the foregrounds are expected to be confined to a wedge \nwhich is approximately bounded by \n\\begin{equation}\nk_{\\parallel} \\le \\left[ \\frac{r \\, \\sin(\\theta_l)}{r_{'} \\, \\nu_c} \\right] \n\\,k_{\\perp} \n\\label{eq:a1}\n\\end{equation}\nin the $(k_{\\parallel},k_{\\perp})$ plane \\citep{Datta2010,Vedantham2012,Morales2012,\nParsons2012b, Trott2012} where $\\theta_l$ refers to the largest \nangle (relative to the telescope's pointing direction) \nfrom which we have a significant foreground contamination. \nOnly the $k$-modes outside this \n``foreground wedge\" can be used for power spectrum estimation. \nThe exact extent of this wedge is however still debatable \n(see \\citealt{Pober2014} for a detailed discussion), and \nfor the purpose of this work we consider three different cases \nwhich differ in the extent of the foreground wedge. \n\n\n\\begin{itemize}\n\\item {\\it \\textbf{Case I}:} This is the most optimistic scenario,\n where we assume that the foregrounds have been removed perfectly and\n the whole $\\bf{k}$ space accessible by uGMRT is available for\n measuring the {H{\\sc I}~} 21-cm signal.\n\n\\item {\\it \\textbf{Case II}:} In this moderate scenario we assume that\n the foreground contributions from angles beyond $\\theta_l=18^{\\circ}$ from\n the center of the field of view are highly suppressed by tapering\n the sky response \\citep{Choudhuri2014}. Note that the first null of\n the uGMRT primary beam pattern at $78 \\, {\\rm MHz}$ is expected at\n $\\sim 6^{\\circ} $. In this case the Fourier modes $k_{\\parallel} \\leq 1.813\n \\, |\\ppk|$ are foreground contaminated\n(eq. \\ref{eq:a1}), and only the modes outside \n this foreground wedge are used for measuring the {H{\\sc I}~} 21-cm signal.\n\n\n\\item {\\it \\textbf{Case III}:} In this pessimistic scenario we assume\n that the foreground contribution extends till the horizon \n$(\\theta_l=90^{\\circ})$ , and the\n Fourier modes $k_{\\parallel} \\leq 5.964 \\, |\\ppk|$ are foreground\n contaminated (eq.\\ref{eq:a1}). Only the modes outside this foreground wedge are used \n for measuring the {H{\\sc I}~} 21-cm signal.\n\n\\end{itemize}\n\n\\section{Result}\nWe first consider very large observing times for which \n$P_N \\rightarrow 0$, and the $1-\\sigma$ error $\\Delta P_a$ on the measurement\n of $P_a$ converges to the cosmic variance. We find\n that ${\\rm SNR} > 5$ can be achieved at $k > 0.02 {\\rm Mpc}^{-1}$,\n $0.04 {\\rm Mpc}^{-1}$ and $0.1 {\\rm Mpc}^{-1}$ for Case I, Case II\n and Case III respectively. We only consider these $k$-modes for our\nsubsequent analysis. We see that in all the three cases there is a\nreasonably large $k$-range where a detection is possible provided we\nhave sufficiently deep observations.\n\nThe system noise dominates $\\Delta P_a$ for small observing \ntimes. Figure~\\ref{fig:b} shows a comparison between the dimensionless \n{H{\\sc I}~} 21-cm signal power spectrum and the corresponding $1-\\sigma$ error. \n For Case I\nwe find that $100\\Delta^2_{\\rm {H{\\sc I}~}}$ and $10\\Delta^2_{\\rm {H{\\sc I}~}}$ can be\ndetected with $100$ and $500$ hours of observation for Fourier modes \n$0.06 < k < 0.5 \\, {\\rm Mpc}^{-1}$ and $0.06 < k< 0.25 \\, {\\rm\n Mpc}^{-1}$ respectively. However, a detection of $\\Delta^2_{\\rm\n {H{\\sc I}~}}$ will require more than $1000$ hours of observation. In Case\nII, it is possible to detect $100\\Delta^2_{\\rm {H{\\sc I}~}}$ and\n$10\\Delta^2_{\\rm {H{\\sc I}~}}$ in the $k$-range, $0.1 < k < 0.45 \\, {\\rm\n Mpc}^{-1}$ and $0.1 < k< 0.25 \\, {\\rm Mpc}^{-1}$ in $100$ and $500$\nhours of observation respectively. For the pessimistic scenario, {\\it\n i.e.} Case III, we find that $100\\Delta^2_{\\rm {H{\\sc I}~}}$ can be detected in $100$\n hours of observation in the $k$-range $0.1 < k < 0.4 \\, {\\rm Mpc}^{-1}$ and \nthe detection of $10\\Delta^2_{\\rm {H{\\sc I}~}}$ will require more than $1000$ hours of\nobservation.\n\n\n\\begin{figure*}\n\\psfrag{optimistic}{Case \\, I} \\psfrag{Modarate}{Case \\, II}\n\\psfrag{pessimistic}{Case \\, III} \\psfrag{100DHI2}{$100\\Delta^2_{\\rm\n {H{\\sc I}~}}$} \\psfrag{10DHI2}{$10\\Delta^2_{\\rm {H{\\sc I}~}}$}\n\\psfrag{DHI2}{$\\Delta^2_{\\rm {H{\\sc I}~}}$} \\psfrag{HI signal}{\\qquad {H{\\sc I}~}\n signal} \\psfrag{100 Hr}{100 Hours} \\psfrag{500 Hr}{500 Hours}\n\\psfrag{1000 Hr}{1000 Hours} \\psfrag{D2kk-mK2}{$\\Delta^2(k)\\, {\\rm\n mK}^2$} \\psfrag{k-Mpc-1}{$k \\, \\rm{Mpc}^{-1}$} \\centering\n\\includegraphics[scale=0.75, angle = 270]{cov_comp_6MHz.eps}\n\\caption{This shows a comparison of the dimensionless \n{H{\\sc I}~} 21-cm signal power\n spectrum and the corresponding $1-\\sigma$ error for an uGMRT observation. The\n left, middle and right panels show results for Case I, II and III\n respectively. The {H{\\sc I}~} signal is shown in dashed lines (as mentioned\n in the figure). The error on the measurement of the power spectrum for $100$, $500$ and $1000$\n hours is shown in solid, dotted and fine-dotted lines respectively.}\n\\label{fig:b}\n\\end{figure*}\n\n We finally consider the situation where all the\n available $k$-modes are combined for a detection of the {H{\\sc I}~} 21-cm\n signal. Here we have a single parameter $A_{{\\rm {H{\\sc I}~}}}$ which is\n the amplitude of the {H{\\sc I}~} 21-cm power spectrum. We have estimated the\n ${\\rm SNR}$ for the measurement of $A_{{\\rm {H{\\sc I}~}}}$ using\n\\begin{equation}\n{\\rm SNR}^2 = \\frac{1}{2} \\sum_{\\mathbfit{k}_g} \\left[\\frac{\\partial\n P_{{\\rm {H{\\sc I}~}}}(\\mathbfit{k}_g)}{\\partial ln A_{{\\rm {H{\\sc I}~}}}}\\right]^2\n[P_{{\\rm {H{\\sc I}~}}}(\\mathbfit{k}_g)+ P_N(\\mathbfit{k}_g)]^{-2} \\, .\n\\label{eq:snr}\n\\end{equation}\n\nFigure~\\ref{fig:c} shows the predicted SNR as a function of the observing time.\n The\nhorizontal solid and the dot-dashed lines mark the SNR value of $10$\nand $5$ respectively. For Case I we find that a $10\\sigma$ detection\nof $100\\Delta^2_{\\rm {H{\\sc I}~}}$, $10\\Delta^2_{\\rm {H{\\sc I}~}}$ and $\\Delta^2_{\\rm\n {H{\\sc I}~}}$ is possible in $\\sim 70,\\, 700 \\, {\\rm and} \\, 6000$ hours of\nobservation respectively. A $5\\sigma$ detection of $\\Delta^2_{\\rm\n {H{\\sc I}~}}$ can be achieved in $~3000$ hours of observation. In Case II\nand Case III, it takes $\\sim 140, 1400$ hours and $\\sim 400, 4000$\nhours for a $10\\sigma$ detection of $100\\Delta^2_{\\rm {H{\\sc I}~}}$ and\n$10\\Delta^2_{\\rm {H{\\sc I}~}}$ respectively. It is not possible to detect\n$\\Delta^2_{\\rm {H{\\sc I}~}}$ within reasonable observation time when we\nconsider the pessimistic scenario.\n\n\\begin{figure*}\n\\psfrag{optimistic}{Case \\, I} \\psfrag{moderate}{Case \\, II}\n\\psfrag{pessimistic}{Case \\, III} \\psfrag{100DHI2}{$100\\Delta^2_{\\rm\n {H{\\sc I}~}}$} \\psfrag{10DHI2}{$10\\Delta^2_{\\rm {H{\\sc I}~}}$}\n\\psfrag{DHI2}{$\\Delta^2_{\\rm {H{\\sc I}~}}$} \\psfrag{SNR}{SNR}\n\\psfrag{Observation Hours}{Observation Hours} \\centering\n\\includegraphics[scale=0.75, angle = 270]{fisher_snr_6MHz.eps}\n\\caption{The left, middle and right panels show the predictions for\n Case I, II and III respectively when all the available $k$-modes are\n combined. The dashed, dotted and fine-dotted lines show the\n predictions for the {H{\\sc I}~} 21-cm signal power spectrum $\\Delta^2_{\\rm\n {H{\\sc I}~}}$, $10\\Delta^2_{\\rm {H{\\sc I}~}}$ and $100\\Delta^2_{\\rm {H{\\sc I}~}}$\n respectively . The horizontal dot-dashed and solid lines mark the\n SNR of $5$ and $10$ respectively.}\n\\label{fig:c}\n\\end{figure*}\n\n\n\nFor comparison, we consider the upcoming SKA-Low which will operate \nwithin a frequency range of $50 - 350 \\, {\\rm MHz}$ and investigate the prospects \nof detecting the redshifted {H{\\sc I}~} 21-cm signal power spectrum from Cosmic Dawn.\nSKA-Low is expected to be an array of $\\sim 513$ \nstations, each of diameter $D = 35 \\, {\\rm m}$ . Modelling each station\nas having a circular aperture of diameter $35 \\, {\\rm m}$ we estimate\nthe primary beam pattern to have a full-width at half-maxima (FWHM) of\n$6.44^{\\circ}$, and we model the primary beam pattern as a Gaussian\n$A(\\theta) = e^{-(\\theta\/\\theta_0)^2}$ with $\\theta_0 = 0.62 \\,\n\\theta_{\\rm FWHM} = 4^{\\circ}$ \\citep{Choudhuri2014}. Using this we have\ncalculated $\\tilde{\\eta} = 39.44$ at the frequency of our interest.\nWe have simulated the SKA-Low baseline distribution corresponding to\n$8 \\, {\\rm hrs}$ of observation towards a fixed pointing direction at\ndeclination $\\delta = -30^{\\circ}$. We have used the proposed SKA-Low\\footnote{\\href {https:\/\/astronomers.skatelescope.org\/wp-content\/uploads\/2016\/09\/SKA-TEL-SKO-0000422_02_SKA1_LowConfigurationCoordinates-1.pdf}{SKA1-\nLowConfigurationCoordinates-1.pdf}} antenna\nconfiguration for our simulation. The rest of the\nanalysis was carried out along exactly the same lines as that for the\nuGMRT. Note that the grid spacing $\\Delta \\ppk$ was scaled appropriately\nto account for the different value of the diameter $D$, whereas the\nvalue of $\\Delta k_{\\parallel}$ was maintained the same as for the uGMRT.\nFigure~\\ref{fig:d} shows a comparison between the dimensionless \n{H{\\sc I}~} 21-cm signal power spectrum and the corresponding $1-\\sigma$ error. \nFor Case I we find that $\\Delta^2_{\\rm {H{\\sc I}~}}$ can be\ndetected with $100$ hours of observation for Fourier modes \n$0.02 < k < 1.0 \\, {\\rm Mpc}^{-1}$. However, a detection of $10\\Delta^2_{\\rm \n{H{\\sc I}~}}$ and $100\\Delta^2_{\\rm {H{\\sc I}~}}$ will require less observation time. In Case\nII, it is possible to detect $\\Delta^2_{\\rm {H{\\sc I}~}}$ in the $k$-range,\n$0.1 < k < 1.0 \\, {\\rm \nMpc}^{-1}$ and for the pessimistic scenario, {\\it i.e.} Case III, we find that \n$\\Delta^2_{\\rm {H{\\sc I}~}}$ can be detected in $100$ hours of observation in the \n$k$-range $0.2 < k < 1.0 \\, {\\rm Mpc}^{-1}$. This results are significantly\npromising when compared to the uGMRT. It is a direct consequence of the\nfact that SKA-Low has larger number of antennas and better $uv$-coverage\nwhen compared to uGMRT. It is interesting to note that for SKA-Low the\n$1-\\sigma$ errors\ndo not decrease very much as the observing time is increased from $100$ to\n$1000 \\, {\\rm hrs}$, whereas we find a pretty significant decrease for\nuGMRT (Figure~\\ref{fig:b}). This indicates that the $1-\\sigma$ errors for\nthe SKA-Low are largely cosmic variance dominated whereas these are the\nsystem noise dominated for the uGMRT.\n\\begin{figure*}\n\\psfrag{optimistic}{Case \\, I} \\psfrag{Modarate}{Case \\, II}\n\\psfrag{pessimistic}{Case \\, III} \\psfrag{100DHI2}{$100\\Delta^2_{\\rm\n {H{\\sc I}~}}$} \\psfrag{10DHI2}{$10\\Delta^2_{\\rm {H{\\sc I}~}}$}\n\\psfrag{DHI2}{$\\Delta^2_{\\rm {H{\\sc I}~}}$} \\psfrag{HI signal}{\\qquad {H{\\sc I}~}\n signal} \\psfrag{100 Hr}{100 Hours} \\psfrag{500 Hr}{500 Hours}\n\\psfrag{1000 Hr}{1000 Hours} \\psfrag{D2kk-mK2}{$\\Delta^2(k)\\, {\\rm\n mK}^2$} \\psfrag{k-Mpc-1}{$k \\, \\rm{Mpc}^{-1}$} \\centering\n\\includegraphics[scale=0.75, angle = 270]{ska_cov_comp_6MHz.eps}\n\\caption{This shows a comparison of the dimensionless \n{H{\\sc I}~} 21-cm signal power spectrum and the corresponding $1-\\sigma$ error for an SKA-Low observation. The\n left, middle and right panels show results for Case I, II and III\n respectively. The {H{\\sc I}~} signal is shown in dashed lines (as mentioned\n in the figure). The error on the measurement of the power spectrum for $100$, $500$ and $1000$\n hours is shown in solid, dotted and fine-dotted lines respectively.}\n\\label{fig:d}\n\\end{figure*}\nFigure~\\ref{fig:e} shows the predicted SNR as a function of the observing time for SKA-Low.\n The horizontal solid and the dot-dashed lines mark the SNR value of $20$\nand $10$ respectively. For Case I, II and III we find that a $20\\sigma$ detection\n$\\Delta^2_{\\rm {H{\\sc I}~}}$ is possible in $\\sim 40,\\, 60 \\, {\\rm and} \\, 200$ hours of\nobservation respectively. Here again it is interesting to note that for\nSKA-Low the SNR does not increase as rapidly with the observing time\nas for the uGMRT (Figure~\\ref{fig:c}). As noted earlier, this is a\nconsequence of the fact that for SKA-Low the cosmic variance makes a larger\ncontribution to the total error budget as compared to the uGMRT. Our\npredictions for SKA-Low are roughly consistent with the earlier predictions\nof \\citet{Koopmans2015}.\n\\begin{figure}\n\\psfrag{opt}{Case \\, I} \n\\psfrag{mod}{Case \\, II}\n\\psfrag{pess}{Case \\, III} \n\\psfrag{100DHI2}{$100\\Delta^2_{\\rm{H{\\sc I}~}}$} \n\\psfrag{10DHI2}{$10\\Delta^2_{\\rm {H{\\sc I}~}}$}\n\\psfrag{DHI2}{$\\Delta^2_{\\rm {H{\\sc I}~}}$} \n\\psfrag{SNR}{SNR}\n\\psfrag{Observation Hours}{Observation Hours} \n\\psfrag{SKA-low}{ }\n\\centering\n\\includegraphics[scale=0.45, angle = 270]{ska_fisher_snr_6MHz.eps}\n\\caption{The dashed, dotted and fine-dotted lines show the predictions for\n Case I, II and III respectively when all the available $k$-modes are\n combined. The horizontal dot-dashed and solid lines mark the\n SNR of $10$ and $20$ respectively.}\n\\label{fig:e}\n\\end{figure}\n\n\nThe upcoming experiment, the Hydrogen Epoch of \n Reionization Array (HERA; \\citealt{DeBoer2017}) a 350-element\n interferometer is also expected to operate from $50$ to $250\n \\, {\\rm MHz}$. Note that HERA is a drift scan instrument unlike\n uGMRT and SKA-Low which can track a fixed pointing direction\n on the sky. Considering $1000 \\, {\\rm hrs} $ of observation, \nat $z = 17$ the HERA sensitivity is about $8$ times better for \nthe cosmic dawn {H{\\sc I}~} 21-cm power spectrum measurement\n\\citep{DeBoer2017} as compared to uGMRT.\n\n\n\\section{Conclusion} If the proposed b-DM interaction\n\\citep{Barkana2018,Fialkov2018} enhances the Cosmic Dawn {H{\\sc I}~} 21-cm\npower spectrum, it can be detected with the Band-1 of uGMRT within\nreasonable hours of observation. Such a detection would be an\nindependent confirmation of the enhanced dip reported by\n\\citet{Bowman2018}. \nThe b-DM scattering model depends upon \ntwo additional parameters: the mass of the DM particles, $0.0032 < m_{\\chi} \n< 100 \\, {\\rm GeV}$ and the cross-section $10^{-30} < \\sigma_1 < 3.16 \\times \n10^{-18} \\, {\\rm cm}^2$. This interaction model enhances the Cosmic Dawn {H{\\sc I}~} 21-cm \nbrightness temperature fluctuations and the maximal fluctuation amplitude \n(due to b-DM scattering only) is predicted to be between $0$ and $850 \n\\, {\\rm mK}$, while the maximal fluctuation amplitude without b-DM scattering \nhas been predicted to be anywhere between $1.5$ and $90 \\, {\\rm mK}$ \\citep{Fialkov2018}.\nWith the measurement of the Cosmic Dawn {H{\\sc I}~} 21-cm power spectrum,\none expects to constrain the $m_{\\chi} - \\sigma_1$ parameter space.\nObservations with the Band-1 of uGMRT hold the\nprospect of being an interesting probe of the b-DM interaction in the\nearly universe. Even upper limits from a non-detection of this power\nspectrum would impose useful constraints on the mass of the DM particles, \nthe scattering cross-section and the proposed b-DM interaction. We also note that the \nobservations with the Band-1 of uGMRT hold the possibility to constraint \nthe ``minicharged\" dark matter models.\nEven an upper limit from a non-detection of the \nthe Cosmic Dawn {H{\\sc I}~} 21-cm power spectrum can put an upper limit on such \n``minicharged\" dark matter models. For example, considering $1000 \\, {\\rm hrs}$ of observation\nwith Case I, uGMRT would be sensitive enough to rule out models with\nthe fraction of ``minicharged\" dark matter particles in the range\n$f_{dm} \\geq 0.03$ based on the {H{\\sc I}~} 21-cm power spectrum predictions of \n\\citet{Munoz2018b}. \n\n\n\n\n{\\it Acknowledgement:} The authors would to thank Ravi Subrahmanyan\nfor drawing their attention to the possibility of observing the Cosmic\nDawn redshifted 21-cm signal using Band-1 of uGMRT. The authors would\nalso like to thank Abinash K. Shaw and Anjan K. Sarkar for useful\ndiscussions. SC acknowledges the University Grants Commission, India\nfor providing financial support through Senior Research Fellowship.\n\n\n\\bibliographystyle{mnras} ","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe goal of string phenomenology is to construct realistic\nstandard-like string models with all moduli stabilized. In the\nearly days, string model building was mainly concentrated on the\nweakly coupled heterotic string theory. After the second string\nrevolution, consistent four-dimensional chiral models with\nnon-Abelian gauge symmetry on Type II orientifolds were able to be\nconstructed due to the advent of D-branes~\\cite{JPEW}. In\nparticular, Type II orientifolds with intersecting D-branes, where\nthe chiral fermions arise from the intersections of D-branes in\nthe internal space~\\cite{bdl} with T-dual description in terms of\nmagnetized D-branes~\\cite{bachas}, have played an important role\nin string model building during the last few years.\n\n\nOn Type IIA orientifolds with intersecting D6-branes,\nmany non-supersymmetric three-family\nstandard-like models and Grand Unified Theories (GUTs) were\nconstructed~\\cite{Blumenhagen:2000wh,Angelantonj:2000hi,Blumenhagen:2005mu}.\nAlthough these models were globally consistent, there generically existed\nuncancelled Neveu-Schwarz-Neveu-Schwarz (NSNS) tadpoles\nas well as the gauge hierarchy problem. To solve these two problems,\nsemi-realistic supersymmetric standard-like models,\nPati-Salam models, $SU(5)$ models as well as\n flipped $SU(5)$ models have been constructed in\nType IIA theory on $\\mathbf{T^6\/({\\mathbb Z}_2\\times\n{\\mathbb Z}_2)}$~\\cite{CSU1,CSU2,Cvetic:2002pj,CP,CLL,\nCvetic:2004nk,Chen:2005ab,Chen:2005mj} and\n$\\mathbf{T^6\/({\\mathbb Z}_2\\times\n{\\mathbb Z}_2')}$~\\cite{Blumenhagen:2005tn,Chen:2006sd} orientifolds with\nintersecting D6-branes, and some of their phenomenological\nconsequences have been studied~\\cite{CLS1,CLW}. Moreover, the\nsupersymmetric constructions in Type IIA theory on other\norientifolds were also discussed~\\cite{ListSUSYOthers}. There are\ntwo main constraints on supersymmetric D6-brane model building: RR\ntadpole cancellation conditions and four-dimensional $N=1$\nsupersymmetric D6-brane configurations. Also, K-theory conditions\nprovide minor constraints. In addition, to stabilize the\nclosed-string moduli via supergravity fluxes, the flux models on\nType II orientifolds have also been\nconstructed~\\cite{Blumenhagen:2003vr,Cascales:2003zp,MS,CL,Cvetic:2005bn,\nKumar:2005hf,Chen:2005cf,Villadoro:2005cu,Camara:2005dc,Chen:2006gd,Chen:2006ip}.\n\n\n\nIt is well known that there are two serious problems in almost all\nthe supersymmetric D-brane models: no gauge coupling unification\nat the string scale, and the rank one problem in the Standard\nModel (SM) fermion Yukawa matrices. Although these problems can be\nsolved in the flux models of Ref.~\\cite{Chen:2006gd} where the RR\ntadpole cancellation conditions are relaxed, these models are in\nthe AdS vacua and the question of how to lift these AdS vacua to\nthe Minkowski vacua or dS vacua correctly is still a big\nchallenge. Recently, we found that there is one and only one\nintersecting D6-brane model on Type IIA $\\mathbf{T^6\/({\\mathbb Z}_2\\times\n{\\mathbb Z}_2)}$ orientifold where the above problems can be\nsolved~\\cite{CLL,Chen:2006gd}. Moreover, this model may has a\nrealistic low energy phenomenology~\\cite{Chen:2007px}. Although\nits observable sector has unique phenomological properties, it is\npossible to have different stacks of the D6-branes in the hidden\nsector.\n\n\n\nIn this paper, we discuss three non-equivalent variations of the\nhidden sector where the RR tadpoles are cancelled, the\nfour-dimensional $N=1$ supersymmetry is perserved, and the\nK-theory conditions are satisfied. These three variations seem to\nbe the only possibilities. In the original\nmodel~\\cite{CLL,Chen:2006gd}, the gauge symmetry in the hidden\nsector is $USp(2)_1\\times USp(2)_2\\times USp(2)_3 \\times\nUSp(2)_4$. Interestingly, we can replace the $USp(2)_1\\times\nUSp(2)_2$ gauge symmetry by an $U(2)_{12}$ gauge symmetry, and\/or\nthe $USp(2)_3\\times USp(2)_4$ gauge symmetry by an $U(2)_{34}$\ngauge symmetry since the contributions to the RR tadpoles from the\n$USp(2)^2$ stacks of D6-branes are the same as those of the $U(2)$\nstacks. Thus, there are three non-equivalent variations, and the\ncorresponding gauge symmetries in the hidden sector are $U(2)_{12}\n\\times USp(2)_3 \\times USp(2)_4$, $U(2)_{34} \\times USp(2)_1\n\\times USp(2)_2$, and $U(2)_{12} \\times U(2)_{34}$, respectively.\nMoreover, we discuss the hidden sector gauge symmetry breaking,\nand consider how to decouple the additional exotic particles.\nBecause the observable sector is the same, the discussions on\nphenomenological consequences, for example, the gauge coupling\nunification, supersymmetry breaking soft terms, low energy\nsupersymmetric particle spectrum, dark matter density, and the SM\nfermion masses and mixings, are the same as those in\nRef.~\\cite{Chen:2007px,CLMN-L}.\n\n\nThis paper is organized as follows. We briefly review the\nintersecting D6-brane model building on Type IIA\n$\\mathbf{T^6\/({\\mathbb Z}_2\\times {\\mathbb Z}_2)}$ orientifold in Section II and\nthe realistic intersecting D6-brane model in Section III. We study\nthe three variations of the hidden sector in Section IV.\nDiscussion and conclusions are given in Section V.\n\n\n\\section{Intersecting D6-Brane Model Building in\nType IIA Theory on $\\mathbf{T^6\/({\\mathbb Z}_2\\times {\\mathbb Z}_2)}$\nOrientifold}\n\n\nWe briefly review the intersecting D6-brane model\nbuilding in Type IIA theory on\n$\\mathbf{T^6\/({\\mathbb Z}_2\\times {\\mathbb Z}_2)}$ orientifold~\\cite{CSU1,CSU2}.\n We consider $\\mathbf{T^{6}}$ to be a\nsix torus factorized as\n$\\mathbf{T^{6}} = {\\bf T}^{2} \\times {\\bf T}^{2} \\times {\\bf T}^{2}$\nwhose complex coordinates are $z_i$, $i=1,\\; 2,\\; 3$ for the\n$i$-th two torus, respectively. The $\\theta$ and $\\omega$\ngenerators for the orbifold group ${\\mathbb Z}_{2} \\times {\\mathbb Z}_{2}$\n act on the complex coordinates as following\n\\begin{eqnarray}\n& \\theta: & (z_1,z_2,z_3) \\to (-z_1,-z_2,z_3)~,~ \\nonumber \\\\\n& \\omega: & (z_1,z_2,z_3) \\to (z_1,-z_2,-z_3)~.~\\,\n\\label{Z2Z2}\n\\end{eqnarray}\nWe implement an orientifold projection $\\Omega R$, where $\\Omega$\nis the world-sheet parity, and $R$ acts on the complex coordinates as\n\\begin{equation}\nR:(z_1,z_2,z_3)\\rightarrow(\\overline{z}_1,\\overline{z}_2,\\overline{z}_3)~.~\\,\n\\end{equation}\nSo, there are four kinds of\norientifold 6-planes (O6-planes) for the actions of $\\Omega R$,\n$\\Omega R\\theta$, $\\Omega R \\omega$, and $\\Omega R\\theta\\omega$,\nrespectively. Also, we have two kinds of complex structures\nconsistent with orientifold projection for a two torus --\nrectangular and tilted~\\cite{LUII}. If we denote the\nhomology classes of the three cycles wrapped by the D6-brane\nstacks as $n_a^i[a_i]+m_a^i[b_i]$ and $n_a^i[a'_i]+m_a^i[b_i]$\nwith $[a_i']=[a_i]+\\frac{1}{2}[b_i]$ for the rectangular and\ntilted tori respectively, we can label a generic one cycle by\n$(n_a^i,l_a^i)$ in either case, where in terms of the wrapping\nnumbers $l_{a}^{i}\\equiv m_{a}^{i}$ for a rectangular two torus\nand $l_{a}^{i}\\equiv 2\\tilde{m}_{a}^{i}=2m_{a}^{i}+n_{a}^{i}$ for\na tilted two torus. So, the homology three-cycles for stack $a$\nof $N_a$ D6-branes and its orientifold image $a'$ take the form\n\\begin{equation}\n[\\Pi_a]=\\prod_{i=1}^{3}\\left(n_{a}^{i}[a_i]+2^{-\\beta_i}l_{a}^{i}[b_i]\\right),\\;\\;\\;\n\\left[\\Pi_{a'}\\right]=\\prod_{i=1}^{3}\n\\left(n_{a}^{i}[a_i]-2^{-\\beta_i}l_{a}^{i}[b_i]\\right)~,~\\, \\end{equation}\nwhere $\\beta_i=0$ if the $i$-th two torus is rectangular and\n$\\beta_i=1$ if it is tilted. Also, we define\n$k\\equiv\\beta_1+\\beta_2+\\beta_3$.\n\n\n\\begin{table}[t]\n\\caption{General spectrum for intersecting D6-branes at generic\nangles, where $I_{aa'}=-2^{3-k}\\prod_{i=1}^3(n_a^il_a^i)$,\nand $I_{aO6}=2^{3-k}(-l_a^1l_a^2l_a^3\n+l_a^1n_a^2n_a^3+n_a^1l_a^2n_a^3+n_a^1n_a^2l_a^3)$.\nMoreover,\n${\\cal M}$ is the multiplicity, and $a_S$ and $a_A$ denote\n the symmetric and anti-symmetric representations of\n$U(N_a\/2)$, respectively.}\n\\renewcommand{\\arraystretch}{1.25}\n\\begin{center}\n\\begin{tabular}{|c|c|}\n\\hline {\\bf Sector} & \\phantom{more space inside this box}{\\bf\nRepresentation}\n\\phantom{more space inside this box} \\\\\n\\hline\\hline\n$aa$ & $U(N_a\/2)$ vector multiplet and 3 adjoint chiral multiplets \\\\\n\\hline\n$ab+ba$ & $ {\\cal M}(\\frac{N_a}{2},\n\\frac{\\overline{N_b}}{2})=\nI_{ab}=2^{-k}\\prod_{i=1}^3(n_a^il_b^i-n_b^il_a^i)$ \\\\\n\\hline\n$ab'+b'a$ & $ {\\cal M}(\\frac{N_a}{2},\n\\frac{N_b}{2})=I_{ab'}=-2^{-k}\\prod_{i=1}^3(n_{a}^il_b^i+n_b^il_a^i)$ \\\\\n\\hline $aa'+a'a$ & ${\\cal M} (a_S)=\n\\frac 12 (I_{aa'} - \\frac 12 I_{aO6})$~;~~ ${\\cal M} (a_A)=\n\\frac 12 (I_{aa'} + \\frac 12 I_{aO6}) $ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{spectrum}\n\\end{table}\n\n\nFor a stack of $N$ D6-branes that do not lie on the top of any\nO6-plane, we obtain the $U(N\/2)$ gauge symmetry with three adjoint\nchiral superfields due to the orbifold projections. While for a\nstack of $N$ D6-branes on the top of an O6-plane, we obtain the\n$USp(N)$ gauge symmetry with three anti-symmetric chiral\nsuperfields. The bifundamental chiral superfields arise from the\nintersections of two different stacks of D6-branes or\n one stack of D6-branes and its $\\Omega R$ image~\\cite{CSU1,CSU2}.\nIn short, the general spectrum for intersecting D6-branes at\ngeneric angles, which is valid for both rectangular and tilted two\ntori, is given in Table \\ref{spectrum}. Moreover, a model may\ncontain additional non-chiral (vector-like) multiplet pairs from\n$ab+ba$, $ab'+b'a$, and $aa'+a'a$ sectors if two stacks of the\ncorresponding D-branes are parallel and on the top of each other\non one two torus. The multiplicity of the non-chiral multiplet\npairs is given by the product of the intersection numbers on the\nother two two-tori.\n\n\nBefore further discussions, let us define the products of wrapping\nnumbers\n\\begin{equation}\n\\begin{array}{rrrr}\nA_a \\equiv -n_a^1n_a^2n_a^3, & B_a \\equiv n_a^1l_a^2l_a^3,\n& C_a \\equiv l_a^1n_a^2l_a^3, & D_a \\equiv l_a^1l_a^2n_a^3, \\\\\n\\tilde{A}_a \\equiv -l_a^1l_a^2l_a^3, & \\tilde{B}_a \\equiv\nl_a^1n_a^2n_a^3, & \\tilde{C}_a \\equiv n_a^1l_a^2n_a^3, &\n\\tilde{D}_a \\equiv n_a^1n_a^2l_a^3.\\,\n\\end{array}\n\\label{variables}\n\\end{equation}\n\nThe four-dimensional $N=1$ supersymmetric models from Type IIA\norientifolds with intersecting D6-branes are mainly constrained by\nthe RR tadpole cancellation conditions and the four-dimensional\n$N=1$ supersymmetric D6-brane configurations, and also\nconstrained by the K-theory conditions: \\\\\n\n(1) RR Tadpole Cancellation Conditions \\\\\n\nThe total RR charges of D6-branes and O6-planes must vanish since\nthe RR field flux lines are conserved. And then we obtain\n the RR tadpole cancellation conditions as follows\n\\begin{eqnarray}\n -2^k N^{(1)}+\\sum_a N_a A_a=-2^k N^{(2)}+\\sum_a N_a\nB_a= \\nonumber\\\\ -2^k N^{(3)}+\\sum_a N_a C_a=-2^k N^{(4)}+\\sum_a\nN_a D_a=-16,\\,\n\\end{eqnarray}\nwhere $2 N^{(i)}$ are the number of D6-branes wrapping along\nthe $i$-th O6-plane which is defined in Table \\ref{orientifold}. \\\\\n\n\\renewcommand{\\arraystretch}{1.4}\n\\begin{table}[t]\n\\caption{Wrapping numbers of the four O6-planes.} \\vspace{0.4cm}\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline\n Orientifold Action & O6-Plane & $(n^1,l^1)\\times (n^2,l^2)\\times\n(n^3,l^3)$\\\\\n\\hline\n $\\Omega R$& 1 & $(2^{\\beta_1},0)\\times (2^{\\beta_2},0)\\times\n(2^{\\beta_3},0)$ \\\\\n\\hline\n $\\Omega R\\omega$& 2& $(2^{\\beta_1},0)\\times (0,-2^{\\beta_2})\\times\n(0,2^{\\beta_3})$ \\\\\n\\hline\n $\\Omega R\\theta\\omega$& 3 & $(0,-2^{\\beta_1})\\times\n(2^{\\beta_2},0)\\times\n(0,2^{\\beta_3})$ \\\\\n\\hline\n $\\Omega R\\theta$& 4 & $(0,-2^{\\beta_1})\\times (0,2^{\\beta_2})\\times\n (2^{\\beta_3},0)$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{orientifold}\n\\end{table}\n\n\n(2) Four-Dimensional $N = 1$ Supersymmetric D6-Brane Configurations \\\\\n\n\n The four-dimensional $N=1$ supersymmetry can be preserved by the\norientation projection if and only if the rotation angle of any\nD6-brane with respect to the O6-plane is an element of\n$SU(3)$~\\cite{bdl}, or in other words,\n$\\theta_1+\\theta_2+\\theta_3=0$ mod $2\\pi$, where $\\theta_i$ is the\nangle between the D6-brane and the O6-plane in the $i$-th two\ntorus. This supersymmetry conditions can be rewritten\nas~\\cite{Cvetic:2002pj}\n\\begin{eqnarray}\nx_A\\tilde{A}_a+x_B\\tilde{B}_a+x_C\\tilde{C}_a+x_D\\tilde{D}_a=0,\n\\nonumber\\\\\\nonumber \\\\ A_a\/x_A+B_a\/x_B+C_a\/x_C+D_a\/x_D<0,\n\\label{susyconditions}\n\\end{eqnarray} where $x_A=\\lambda,\\;\nx_B=\\lambda 2^{\\beta_2+\\beta3}\/\\chi_2\\chi_3,\\; x_C=\\lambda\n2^{\\beta_1+\\beta3}\/\\chi_1\\chi_3,\\; x_D=\\lambda\n2^{\\beta_1+\\beta2}\/\\chi_1\\chi_2$, and $\\chi_i=R^2_i\/R^1_i$ are the\ncomplex structure parameters. The positive parameter $\\lambda$ has\nbeen introduced to put all the variables $A,\\,B,\\,C,~{\\rm and}~D$ on an\nequal footing. \\\\\n\n(3) K-theory Conditions \\\\\n\nThe discrete D-brane RR charges classified by\nthe $\\mathbf{{\\mathbb Z}_2}$ K-theory groups in the\npresence of orientifolds, which are subtle and invisible by the\nordinary homology~\\cite{MS,Witten9810188},\nshould also be taken into account~\\cite{Cascales:2003zp}.\nThe K-theory conditions are\n\\begin{eqnarray}\n\\sum_a 2^{-k}\\tilde{A}_a = \\sum_a 2^{-\\beta_1} N_a \\tilde{B}_a\n= \\sum_a 2^{-\\beta_2} N_a \\tilde{C}_a =\n\\sum_a 2^{-\\beta_3} N_a \\tilde{D}_a\n = 0 \\textrm{ mod }4 \\label{K-charges}~.~\\,\n\\end{eqnarray}\n\n\n\\section{The Realistic Intersecting D6-Brane Model}\n\nThere may be one and only one intersecting D6-brane model in\nType IIA theory on $\\mathbf{T^6\/({\\mathbb Z}_2\\times {\\mathbb Z}_2)}$ orientifold\nwith a realistic phenomenology~\\cite{CLL,Chen:2006gd,Chen:2007px}.\nLet us briefly review it. We present the D6-brane\nconfigurations and intersection numbers in\nTable~\\ref{MI-Numbers}, and its spectrum in Table~\\ref{Spectrum}.\nWe put the $a'$, $b$, and $c$ stacks of D6-branes on the top of\neach other on the third two torus, and then we have the additional\nvector-like particles from $N=2$ subsectors.\n\n\n\\begin{table}[h]\n\n\\begin{center}\n\n\\footnotesize\n\n\\begin{tabular}{|@{}c@{}|c||@{}c@{}c@{}c@{}||c|c||c|@{}c@{}|@{}c@{}|\n@{}c@{}||@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|} \\hline\n\n\nstack & $N$ & ($n_1$,$l_1$) & ($n_2$,$l_2$) & ($n_3$,$l_3$) & A &\nS & $b$ & $b'$ & $c$ & $c'$ & $O6^{1}$ & $O6^{2}$ & $O6^{3}$ &\n$O6^{4}$ \\\\ \\hline \\hline\n\n$a$ & 8 & ( 0,-1) & ( 1, 1) & ( 1, 1) & 0 & 0 & 3 & 0(3) & -3 &\n0(3) & 1 & -1 & 0 & 0 \\\\ \\hline\n\n$b$ & 4 & ( 3, 1) & ( 1, 0) & ( 1,-1) & -2 & 2 & - & - & 0(6) &\n0(1) & 0 & 1 & 0 & -3 \\\\ \\hline\n\n$c$ & 4 & ( 3,-1) & ( 0, 1) & ( 1,-1) & 2 & -2 & - & - & - & - &\n-1 & 0 & 3 & 0 \\\\ \\hline \\hline\n\n$O6^{1}$ & 2 & ( 1, 0) & ( 1, 0) & ( 2, 0) & - & - & - & - & - & -\n& - & - & - & - \\\\ \\hline\n\n$O6^{2}$ & 2 & ( 1, 0) & ( 0,-1) & ( 0, 2) & - & - & - & - & - & -\n& - & - & - & - \\\\ \\hline\n\n$O6^{3}$ & 2 & ( 0, -1) & ( 1, 0) & ( 0, 2) & - & - & - & - & - &\n- & - & - & - & - \\\\ \\hline\n\n$O6^{4}$ & 2 & ( 0, -1) & ( 0, 1) & ( 2, 0) & - & - & - & - & - &\n- & - & - & - & - \\\\ \\hline\n\\end{tabular}\n\\caption{The D6-brane configurations and intersection numbers on Type\nIIA $\\mathbf{T}^6 \/ {\\mathbb Z}_2 \\times {\\mathbb Z}_2$ orientifold. The gauge\nsymmetry is $[U(4)_C \\times U(2)_L \\times\nU(2)_R]_{\\rm Observable}\\times [USp(2)_1 \\times USp(2)_2\n\\times USp(2)_3 \\times USp(2)_4]_{\\rm Hidden}$, the SM fermions\nand Higgs fields arise from the intersections on the first\ntwo torus, and the complex structure parameters are\n$2\\chi_1=6\\chi_2=3\\chi_3 =6$. Also, the beta functions for all\n$USp(2)_i$ gauge symmetries are $-3$.}\n\\label{MI-Numbers}\n\\end{center}\n\\end{table}\n\n\n\n\\begin{table}\n[htb] \\footnotesize\n\\renewcommand{\\arraystretch}{1.0}\n\\caption{The chiral and vector-like superfields,\n and their quantum numbers\nunder the gauge symmetry $SU(4)_C\\times SU(2)_L\\times SU(2)_R\n\\times USp(2)_1 \\times USp(2)_2 \\times USp(2)_3 \\times USp(2)_4$.}\n\\label{Spectrum}\n\\begin{center}\n\\begin{tabular}{|c||c||c|c|c||c|c|c|}\\hline\n & Quantum Number\n& $Q_4$ & $Q_{2L}$ & $Q_{2R}$ & Field \\\\\n\\hline\\hline\n$ab$ & $3 \\times (4,\\overline{2},1,1,1,1,1)$ & 1 & -1 & 0 & $F_L(Q_L, L_L)$\\\\\n$ac$ & $3\\times (\\overline{4},1,2,1,1,1,1)$ & -1 & 0 & $1$ & $F_R(Q_R, L_R)$\\\\\n$a1$ & $1\\times (4,1,1,2,1,1,1)$ & $1$ & 0 & 0 & $X_{a1}$ \\\\\n$a2$ & $1\\times (\\overline{4},1,1,1,2,1,1)$ & -1 & 0 & 0 & $X_{a2}$ \\\\\n$b2$ & $1\\times(1,2,1,1,2,1,1)$ & 0 & 1 & 0 & $X_{b2}$ \\\\\n$b4$ & $3\\times(1,\\overline{2},1,1,1,1,2)$ & 0 & -1 & 0 & $X_{b4}^i$ \\\\\n$c1$ & $1\\times(1,1,\\overline{2},2,1,1,1)$ & 0 & 0 & -1 & $X_{c1}$ \\\\\n$c3$ & $3\\times(1,1,2,1,1,2,1)$ & 0 & 0 & 1 & $X_{c3}^i$ \\\\\n$b_{S}$ & $2\\times(1,3,1,1,1,1,1)$ & 0 & 2 & 0 & $T_L^i$ \\\\\n$b_{A}$ & $2\\times(1,\\overline{1},1,1,1,1,1)$ & 0 & -2 & 0 & $S_L^i$ \\\\\n$c_{S}$ & $2\\times(1,1,\\overline{3},1,1,1,1)$ & 0 & 0 & -2 & $T_R^i$ \\\\\n$c_{A}$ & $2\\times(1,1,1,1,1,1,1)$ & 0 & 0 & 2 & $S_R^i$ \\\\\n\\hline\\hline\n$ab'$ & $3 \\times (4,2,1,1,1,1,1)$ & 1 & 1 & 0 & \\\\\n& $3 \\times (\\overline{4},\\overline{2},1,1,1,1,1)$ & -1 & -1 & 0 & \\\\\n\\hline\n$ac'$ & $3 \\times (4,1,2,1,1,1,1)$ & 1 & & 1 & $\\Phi_i$ \\\\\n& $3 \\times (\\overline{4}, 1, \\overline{2},1,1,1,1)$ & -1 & 0 & -1 &\n$\\overline{\\Phi}_i$\\\\\n\\hline\n$bc$ & $6 \\times (1,2,\\overline{2},1,1,1,1)$ & 0 & 1 & -1 & $H_u^i$, $H_d^i$\\\\\n& $6 \\times (1,\\overline{2},2,1,1,1,1)$ & 0 & -1 & 1 & \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nWe have shown that the gauge symmetry in the observable sector can\nbe broken down to the SM gauge symmetry via the Green-Schwarz mechanism,\n D6-brane splittings and supersymmtry preserving Higgs mechanism.\nThe gauge couplings for $SU(4)_C$, $SU(2)_L$ and $SU(2)_R$ are\nunified at the string scale, and the additional exotic particles\nmay be decoupled around the string scale. Also, we calculated the\nsupersymmetry breaking soft terms, and the corresponding low\nenergy supersymmetric particle spectrum that can be tested at the\nLarge Hadron Collider (LHC). The observed dark matter density can\nalso be generated. In addition, we can explain the SM quark masses\nand mixings, and the tau lepton mass. The neutrino masses and\nmixings may be generated via seesaw mechanism as well. Similar to\nthe GUTs~\\cite{Nanopoulos:1982zm}, we have roughly the wrong\nfermion mass relation $m_e\/m_{\\mu} \\simeq m_{d}\/m_s$, and the\ncorrect electron and muon masses can be generated via\nhigh-dimensional operators~\\cite{CLMN-L}. Furthermore, all the\n$USp(2)_i$ gauge symmetries will become strong around the string\nscale~\\cite{CLMN-L}.\n\n\\section{Three variations of the Hidden Sector}\n\nIn the realistic intersecting D6-brane model~\\cite{CLL,Chen:2006gd},\nthe observable sector is unique.\nInterestingly, we find three non-equivalent variations of\nthe hidden sector where we can cancel the RR tadpoles,\n preserve the four-dimensional $N=1$ supersymmetry,\nand satisfy the K-theory conditions. And it seems to us that there\nis no other variation. In the original\nmodel~\\cite{CLL,Chen:2006gd}, the gauge symmetry in the hidden\nsector is $USp(2)_1\\times USp(2)_2\\times USp(2)_3 \\times\nUSp(2)_4$. We notice that the $USp(2)_1\\times USp(2)_2$ gauge\nsymmetry can be replaced by an $U(2)_{12}$ gauge symmetry, and\/or\nthe $USp(2)_3\\times USp(2)_4$ gauge symmetry by an $U(2)_{34}$\ngauge symmetry because the contributions to the RR tadpoles from\nthe $USp(2)^2$ stacks of D6-branes are the same as those of the\n$U(2)$ stacks. Thus, there are three non-equivalent variations,\nand the corresponding gauge symmetries in the hidden sector are\n$U(2)_{12} \\times USp(2)_3 \\times USp(2)_4$, $U(2)_{34} \\times\nUSp(2)_1 \\times USp(2)_2$, and $U(2)_{12} \\times U(2)_{34}$,\nrespectively. Let us present them one by one in the following\nsubsections.\n\n\n\\subsection{$U(2)_{12}\\times USp(2)_3 \\times USp(2)_4$ Hidden Sector}\n\n\n\n\n\n\\begin{table}[h]\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{|@{}c@{}|c||@{}c@{}c@{}c@{}||c|c||c|@{}c@{}|@{}c@{}|\n@{}c@{}||@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|} \\hline\n\nstack & $N$ & ($n_1$,$l_1$) & ($n_2$,$l_2$) & ($n_3$,$l_3$) & A &\nS & $b$ & $b'$ & $c$ & $c'$ & $d$ & $d'$ & $O6^{3}$ & $O6^{4}$\n\\\\ \\hline \\hline\n\n$a$ & 8 & ( 0,-1) & ( 1, 1) & ( 1, 1) & 0 & 0 & 3 & 0(3) & -3 &\n0(3) & 0(2) & 0(1) & 0 & 0 \\\\ \\hline\n\n$b$ & 4 & ( 3, 1) & ( 1, 0) & ( 1,-1) & -2 & 2 & - & - & 0(6) &\n0(1) & 1 & 0(1) & 0 & -3 \\\\ \\hline\n\n$c$ & 4 & ( 3,-1) & ( 0, 1) & ( 1,-1) & 2 & -2 & - & - & - & - &\n-1 & 0(1) & 3 & 0 \\\\ \\hline \\hline\n\n$d$ & 4 & ( 1, 0) & ( 1,-1) & ( 1, 1) & 0 & 0 & - & - & - & - & -\n& - & -1 & 1 \\\\ \\hline\n\n$O6^{3}$ & 2 & ( 0, -1) & ( 1, 0) & ( 0, 2) & - & - & - & - & - &\n- & - & - & - & - \\\\ \\hline\n\n$O6^{4}$ & 2 & ( 0, -1) & ( 0, 1) & ( 2, 0) & - & - & - & - & - &\n- & - & - & - & - \\\\ \\hline\n\\end{tabular}\n\\caption{The D6-brane configurations and intersection numbers on Type\nIIA $\\mathbf{T}^6 \/ {\\mathbb Z}_2 \\times {\\mathbb Z}_2$ orientifold. The complete\ngauge symmetry is $[U(4)_C \\times U(2)_L \\times\nU(2)_R]_{\\rm Observable}\\times [U(2)_{12} \\times\nUSp(2)_3\\times USp(2)_4]_{\\rm Hidden}$.}\n\\label{HA-MI-Numbers}\n\\end{center}\n\\end{table}\n\n\n\n\\begin{table}[htb]\n\\footnotesize\n\\renewcommand{\\arraystretch}{1.0}\n\\caption{The new chiral superfields and their quantum numbers\nunder the gauge symmetry $SU(4)_C\\times SU(2)_L\\times\nSU(2)_R \\times U(2)_{12} \\times USp(2)_3 \\times USp(2)_4 $.}\n\n\\label{HA-Spectrum}\n\\begin{center}\n\\begin{tabular}{|c||c||c|c|c|c||c|c|c|}\\hline\n\n& Representation\n\n& $Q_4$ & $Q_{2L}$ & $Q_{2R}$ & $Q_{12} $ & Field \\\\\n\n\\hline\\hline\n\n$bd$ & $1\\times(1,2,1,\\overline{2},1,1)$ & 0 & 1 & 0 & -1 & $X_{bd}$ \\\\\n\n$cd$ & $1\\times(1,1,\\overline{2},2,1,1)$ & 0 & 0 & -1 & 1 & $X_{cd}$ \\\\\n\n$d3$ & $1\\times (1,1,1,\\overline{2},2,1)$ & 0 & 0 & 0 & -1 & $X_{d3}$ \\\\\n\n$d4$ & $1\\times (1,1,1,2,1,2)$ & 0 & 0 & 0 & 1 & $X_{d4}$ \\\\\n\\hline\n\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nIn the first variation of the hidden sector, we replace\nthe $USp(2)_1\\times USp(2)_2$ gauge symmetry by an\n$U(2)_{12}$ gauge symmetry. We present the D6-brane\nconfigurations and intersection numbers in\nTable~\\ref{HA-MI-Numbers}. Moreover, the particle spectrum has two parts:\n(1) the spectrum for old particles is given\nin Table~\\ref{Spectrum} by removing all the particles that\nare charged under $USp(2)_1\\times USp(2)_2$; (2) the spectrum\nfor the new particles is given in Table~\\ref{HA-Spectrum}.\n\n\nThe anomalies from the global $U(1)$ of $U(2)_{12}$ are cancelled\nby the Green-Schwarz mechanism, and its gauge field obtains mass\nvia the linear $B\\wedge F$ couplings. Then, the effective gauge\nsymmetry is $SU(2)_{12}$. The $SU(2)_{12}$ gauge symmetry can be\nbroken down to $U(1)_{12}$ via D6-brane splitting. Interestingly,\nwe do not have any additional chiral exotic particles that are\ncharged under $SU(4)_C$. The simple way to give masses to the\nextra exotic particles $X_{bd}$ and $X_{cd}$ is instanton\neffects~\\cite{Blumenhagen:2006xt,Ibanez:2006da,Cvetic:2007ku,Ibanez:2007rs}.\nHowever, we do not have the suitable three-cycles wrapped by E2\ninstantons~\\footnote{Note that the E2 branes must also wrap rigid\ncycles.}, and thus the instanton effects are not available.\nSimilar results hold for the next two subsections.\n In addition, the $USp(2)_3$ and $USp(2)_4$ will become strong\nat about the string scale~\\cite{CLMN-L}, and then we will have\nsome composite particles in the $U(2)_{12}$ anti-symmetric and\nsymmetric representations, $\\overline{S}'_d$ and\n$\\overline{T}'_d$ from $X_{d3}$, and $S'_d$ and $T'_d$ from\n$X_{d4}$, respectively. So we can break the $U(1)_{12}$ by giving\nsuitable string-scale vacuum expectation values (VEVs) to\n$\\overline{T}'_d$ and $T'_d$, and we can give the string-scale\nVEVs to $\\overline{S}'_d$ and $S'_d$. Note that we give the\nTeV-scale VEVs to $S_L^i$ and the string-scale VEVs to\n$S_R^i$~\\cite{Chen:2007px}, we can give the GUT-scale masses to\n$X^i_{c3}$ and $X_{cd}$ and the TeV-scale masses to the $X^i_{b4}$\nand $X_{bd}$ via the high-dimensional operators~\\cite{CLMN-L}.\nFurthermore, if we could give the string-scale masses to the\nthree $U(2)_{12}$ adjoint chiral superfields and we do not break\nthe $SU(2)_{12}$ via D6-brane splitting, the $SU(2)_{12}$ gauge\nsymmetry will become strong around the string scale. Then we can\nhave the singlet composite field $S'_L$ in the $U(2)_L$\nanti-symmetric representation with charge $+2$ under $U(1)_L$ from\n$X_{bd}$. And we can give the string-scale VEVs to $S_L^i$ and\n$S'_L$ while keeping the D-flatness of $U(1)_L$. Therefore, we may\nalso give the GUT-scale masses to the $X^i_{b4}$ and $X_{bd}$ via\nthe high-dimensional operators~\\cite{CLMN-L}.\n\n\n\n\\subsection{$U(2)_{34}\\times USp(2)_1 \\times USp(2)_2$ Hidden Sector}\n\n\n\n\\begin{table}[h]\n\n\\begin{center}\n\n\\footnotesize\n\n\\begin{tabular}{|@{}c@{}|c||@{}c@{}c@{}c@{}||c|c||c|@{}c@{}|@{}c@{}|\n@{}c@{}||@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|} \\hline\n\n\nstack & $N$ & ($n_1$,$l_1$) & ($n_2$,$l_2$) & ($n_3$,$l_3$) & A &\nS & $b$ & $b'$ & $c$ & $c'$ & $e$ & $e'$ & $O6^{1}$ & $O6^{2}$\n\\\\ \\hline \\hline\n\n$a$ & 8 & ( 0,-1) & ( 1, 1) & ( 1, 1) & 0 & 0 & 3 & 0(3) & -3 &\n0(3) & 0(2) & 0(0) & 1 & -1 \\\\ \\hline\n\n$b$ & 4 & ( 3, 1) & ( 1, 0) & ( 1,-1) & -2 & 2 & - & - & 0(6) &\n0(1) & 0(3) & -3 & 0 & 1 \\\\ \\hline\n\n$c$ & 4 & ( 3,-1) & ( 0, 1) & ( 1,-1) & 2 & -2 & - & - & - & - &\n0(3) & 3 & -1 & 0 \\\\ \\hline \\hline\n\n$e$ & 4 & ( 0, 1) & (-1, 1) & (-1, 1) & 0 & 0 & - & - & - & - & -\n& - & -1 & 1 \\\\ \\hline\n\n$O6^{1}$ & 2 & ( 1, 0) & ( 1, 0) & ( 2, 0) & - & - & - & - & - & -\n& - & - & - & - \\\\ \\hline\n\n$O6^{2}$ & 2 & ( 1, 0) & ( 0,-1) & ( 0, 2) & - & - & - & - & - & -\n& - & - & - & - \\\\ \\hline\n\n\n\\end{tabular}\n\\caption{The D6-brane configurations and intersection numbers on Type\nIIA $\\mathbf{T}^6 \/ {\\mathbb Z}_2 \\times {\\mathbb Z}_2$ orientifold.\nThe gauge symmetry is $[U(4)_C \\times\nU(2)_L \\times U(2)_R]_{\\rm Observable}\\times [U(2)_{34}\\times\nUSp(2)_1\\times USp(2)_2]_{\\rm Hidden}$. }\n\\label{HB-MI-Numbers}\n\\end{center}\n\\end{table}\n\n\n\n\n\\begin{table}[htb]\n\\footnotesize\n\\renewcommand{\\arraystretch}{1.0}\n\\caption{The new chiral superfields and their quantum numbers\nunder the gauge symmetry $SU(4)_C\\times SU(2)_L\\times\nSU(2)_R \\times U(2)_{34} \\times USp(2)_1 \\times USp(2)_2 $.}\n\n\\label{HB-Spectrum}\n\\begin{center}\n\\begin{tabular}{|c||c||c|c|c|c||c|c|c|}\\hline\n\n& Representation\n\n& $Q_4$ & $Q_{2L}$ & $Q_{2R}$ & $Q_{34}$ & Field \\\\\n\n\\hline\\hline\n\n$be'$ & $3\\times(1,\\overline{2},1,\\overline{2},1,1)$ & 0 & -1 & 0 & -1 & $X^i_{be'}$ \\\\\n\n$ce'$ & $3\\times(1,1,2,2,1,1)$ & 0 & 0 & 1 & 1 & $X^i_{ce'}$ \\\\\n\n$e1$ & $1\\times (1,1,1,\\overline{2},2,1)$ & 0 & 0 & 0 & -1 & $X_{e1}$ \\\\\n\n$e2$ & $1\\times (1,1,1,2,1,2)$ & 0 & 0 & 0 & 1 & $X_{e2}$ \\\\\n\n\\hline\n\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nIn the second variation of the hidden sector, we replace the\n$USp(2)_3\\times USp(2)_4$ gauge symmetry by an $U(2)_{34}$ gauge\nsymmetry. We present the D6-brane configurations and intersection\nnumbers in Table~\\ref{HB-MI-Numbers}. The particle spectrum also\nhas two parts: (1) the spectrum for old particles is given in\nTable~\\ref{Spectrum} by removing all the particles that are\ncharged under $USp(2)_3\\times USp(2)_4$; (2) the spectrum for the\nnew particles is given in Table~\\ref{HB-Spectrum}.\n\nNote that the wrapping numbers for the $d$ stack of D6-branes are\nequivalent to those of the $a$ stack by T duality and orientifold\naction, we can think that we have an $U(6)$ gauge symmetry in the\nbegining. Only the global $U(1)$ of $U(6)$ is anomalous $U(1)$ symmetry,\nand its gauge field obtains mass via the linear $B\\wedge F$ couplings.\nAfter we put four D6-branes on the place with equivalent wrapping\nnumbers (just like the D6-brane splittings), we break the $SU(6)$ down to the\n$SU(4)_C \\times SU(2)_{34} \\times U(1)'$ where the $U(1)'$ generator\nin $SU(6)$ is\n\\begin{eqnarray}\n T_{U(1)'}\n\\equiv ~ {1\\over {2{\\sqrt 6}}} \\, {\\rm diag}\\left(1, 1, 1, 1, -2, -2 \\right) ~.~\\,\n\\label{SU6-GU1A}\n\\end{eqnarray}\nThus, the left-handed and right-handed SM fermions have $U(1)'$ charges\n$+1\/2{\\sqrt 6}$ and $-1\/2{\\sqrt 6}$, respectively.\nIn order to keep the gauge coupling\nunification, we have to break the $U(1)'$ so that it will not become part\nof the $U(1)_Y$. In short, we have to break $U(2)_{34}$\ncompletely.\n\n\nBecause the $USp(2)_1$ and $USp(2)_2$ will become strong at about\nthe string scale~\\cite{CLMN-L}, we will have some composite\nparticles in the $U(2)_{34}$ anti-symmetric and symmetric\nrepresentations, $\\overline{S}'_e$ and $\\overline{T}'_e$ from\n$X_{e1}$, and $S'_e$ and $T'_e$ from $X_{e2}$, respectively. So we\ncan break the $U(2)_{12}$ completely by giving suitable\nstring-scale VEVs to $\\overline{S}'_e$, $\\overline{T}'_e$, $S'_e$,\nand $T'_e$. Moreover, we can have the singlet composite particle\n$S'_L$ in the $U(2)_L $ anti-symmetric representation with charge\n$+2$ under $U(1)_L$ from $X_{b2}$. And then we can give the\nstring-scale VEVs to $S_L^i$ and $S'_L$ while keeping the\nD-flatness of $U(1)_L$. Note that $S_R^i$ also have string-scale\nVEVs, we may give the GUT-scale masses to $X_{b2}$, $X_{c1}$,\n$X^i_{be'}$, and $X^i_{ce'}$ via the high-dimensional\noperators~\\cite{CLMN-L}. Moreover, $X_{a1}$ and $X_{a2}$ may form\nthe vector-like particles if we break the $USp(2)_1$ and\n$USp(2)_2$ down to the diagonal $USp(2)_{D12}$~\\cite{Chen:2007px}.\n\n\n\n\n\\subsection{$U(2)_{12}\\times U(2)_{34}$ Hidden Sector}\n\n\n\n\n\\begin{table}[h]\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{|@{}c@{}|c||@{}c@{}c@{}c@{}||c|c||c|@{}c@{}|@{}c@{}|\n@{}c@{}||@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|} \\hline\n\nstack & $N$ & ($n_1$,$l_1$) & ($n_2$,$l_2$) & ($n_3$,$l_3$) & A &\nS & $b$ & $b'$ & $c$ & $c'$ & $d$ & $d'$ & $e$ & $e'$\n\\\\ \\hline \\hline\n\n$a$ & 8 & ( 0,-1) & ( 1, 1) & ( 1, 1) & 0 & 0 & 3 & 0(3) & -3 &\n0(3) & 0(2) & 0(1) & 0(2) & 0(0) \\\\ \\hline\n\n$b$ & 4 & ( 3, 1) & ( 1, 0) & ( 1,-1) & -2 & 2 & - & - & 0(6) &\n0(1) & 1 & 0(1) & 0(3) & -3 \\\\ \\hline\n\n$c$ & 4 & ( 3,-1) & ( 0, 1) & ( 1,-1) & 2 & -2 & - & - & - & - &\n-1 & 0(1) & 0(3) & 3 \\\\ \\hline \\hline\n\n$d$ & 4 & ( 1, 0) & ( 1,-1) & ( 1, 1) & 0 & 0 & - & - & - & - & -\n& - & 0(1) & 0(2) \\\\ \\hline\n\n$e$ & 4 & ( 0, 1) & (-1, 1) & (-1, 1) & 0 & 0 & - & - & - & - & -\n& - & - & - \\\\ \\hline\n\n\\end{tabular}\n\\caption{The D6-brane configurations and intersection numbers on\nType IIA $\\mathbf{T}^6 \/ {\\mathbb Z}_2 \\times {\\mathbb Z}_2$ orientifold. The\ncomplete gauge symmetry is $[U(4)_C \\times U(2)_L \\times\nU(2)_R]_{\\rm Observable}\\times [U(2)_{12}\\times U(2)_{34}]_{\\rm Hidden}$. }\n\\label{HC-MI-Numbers}\n\\end{center}\n\\end{table}\n\n\n\n\\begin{table}[htb]\n\\footnotesize\n\\renewcommand{\\arraystretch}{1.0}\n\\caption{The chiral and vector-like superfields,\n and their quantum numbers under the gauge symmetry\n $SU(4)_C\\times SU(2)_L\\times\nSU(2)_R \\times U(2)_{12}\\times U(2)_{34}$.}\n\\label{HC-Spectrum}\n\\begin{center}\n\\begin{tabular}{|c||c||c|c|c|c|c||c|c|c|}\\hline\n\n& Representation\n\n& $Q_4$ & $Q_{2L}$ & $Q_{2R}$ & $Q_{12}$ & $Q_{34}$ & Field \\\\\n\n\\hline\\hline\n\n$bd$ & $1\\times(1,2,1,\\overline{2},1)$ & 0 & 1 & 0 & -1 & 0 & $X_{bd}$ \\\\\n\n$be'$ & $3\\times(1,\\overline{2},1,1,\\overline{2})$ & 0 & -1 & 0 & 0 & -1 & $X_{be'}^i$ \\\\\n\n$cd$ & $1\\times(1,1,\\overline{2},2,1)$ & 0 & 0 & -1 & 1 & 0 & $X_{cd}$ \\\\\n\n$ce'$ & $3\\times(1,1,2,1,2)$ & 0 & 0 & 1 & 0 & 1 & $X_{ce'}^i$ \\\\\n\n\\hline\\hline\n\n$de$ & $1 \\times (1,1,1,2,\\overline{2})$ & 0 & 0 & 0 & 1 & -1 & $X_{de}$ \\\\\n& $1 \\times (1,1,1,\\overline{2},2)$ & 0 & 0 & 0 & -1 & 1 & $\\overline{X}_{de}$ \\\\\n\n\\hline\n\n$de'$ & $2 \\times (1,1,1,2,2)$ & 0 & 0 & 0 & 1 & 1 & $X_{de'}^i$ \\\\\n& $2 \\times (1,1,1,\\overline{2},\\overline{2})$ & 0 & 0 & 0 & -1 & -1 &\n$\\overline{X}_{de'}^i$ \\\\\n\n\\hline\n\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nIn the third variation of the hidden sector, we replace the\n$USp(2)_1\\times USp(2)_2$ gauge symmetry by $U(2)_{12}$, and\nreplace the $USp(2)_3\\times USp(2)_4$ gauge symmetry by\n$U(2)_{34}$. We present the D6-brane configurations and\nintersection numbers in Table~\\ref{HC-MI-Numbers}. The particle\nspectrum also has two parts: (1) the spectrum for old particles is\ngiven in Table~\\ref{Spectrum} by removing all the particles that\nare charged under $USp(2)_1\\times USp(2)_2\\times USp(2)_3\\times\nUSp(2)_4$; (2) the spectrum for the new particles is given in\nTable~\\ref{HC-Spectrum}.\n\n\nAs discussed in above subsections, we can break the $U(2)_{12}$\ndown to the $U(1)_{12}$ gauge symmetry via Green-Schwarz mechanism\nand D6-brane splitting, and we have to break the $U(2)_{34}$\ngauge symmetry completely. In order to break the $U(1)_{12}$ and\n$U(2)_{34}$ gauge symmetries, we put the $d$ and $e$ stacks of\nD6-branes on the top of each other on the second two torus, and\nput the $d$ and $e'$ stacks on the top of each other on the third\ntwo torus. Then, we have additional vector-like particles\n$X_{de}$, $\\overline{X}_{de}$, $X^i_{de'}$, and\n$\\overline{X}^i_{de'}$, as given in Table~\\ref{HC-Spectrum}. And\nthere exist the following Yukawa couplings\n\\begin{eqnarray}\nW \\supset && y^A_{ij} X_{bd} X_{be'}^i X_{de'}^j +\ny^B_{ij} X_{cd} X_{ce'}^i \\overline{X}_{de'}^j~,~\\,\n\\end{eqnarray}\nwhere $y^A_{ij}$ and $y^B_{ij}$ are Yukawa couplings. If we give\nthe diagonal string-scale VEVs to $X_{de'}^j$ and\n$\\overline{X}_{de'}^j$, we break the $U(2)_{12}\\times U(2)_{34}$\ndown to the diagonal $U(2)_D$. Moreover, the $X_{bd}$ and one\nlinear combination of $X_{be'}^i$, and the $X_{cd}$ and one linear\ncombination of $X_{ce'}^i$ can have vector-like masses close to\nthe string scale. Note that we can give the TeV-scale VEVs to\n$S_L^i$ and the string-scale VEVs to $S_R^i$~\\cite{Chen:2007px},\nwe can give the GUT-scale masses to $X_{cd}$ and the other two\nlinear combinaions of $X_{ce'}^i$, and the TeV-scale masses to $X_{bd}$\nand the other two linear combinations of $X_{be'}^i$ via the\nhigh-dimensional operators~\\cite{CLMN-L}. Similar to the\ndiscussions in the above subsection A, if we can give the\nstring-scale masses to the three $U(2)_{12}$ adjoint chiral\nsuperfields and do not break the $SU(2)_{12}$ gauge symmetry via\nD6-brane splitting, the $SU(2)_{12}$ gauge symmetry will become\nstrong around the string scale. Then we can have the singlet\ncomposite field $S'_L$ in the $U(2)_L$ anti-symmetric\nrepresentation with charge $+2$ under $U(1)_L$ from $X_{bd}$, and\nwe can give the string-scale VEVs to $S_L^i$ and $S'_L$ while\nkeeping the D-flatness of $U(1)_L$. Therefore, we may also give\nthe GUT-scale masses to $X_{bd}$ and the other two linear combinations of\n$X_{be'}^i$ via the high-dimensional operators~\\cite{CLMN-L}.\n\n\n\n\\section{Discussion and Conclusions}\n\n\nAt present, there is only one known example of an intersecting\nD6-brane model with a realistic observable sector. Interestingly,\nthere are three non-equivalent variations of the hidden sector in\nwhich the theoretical constraints on model building can be\nsatisfied. There does not seem to be any other possible variation\nin the original model~\\cite{CLL,Chen:2006gd}, and the gauge symmetry\nin the hidden sector is $USp(2)_1\\times USp(2)_2\\times USp(2)_3\n\\times USp(2)_4$. We noticed that the $USp(2)_1\\times USp(2)_2$\ngauge symmetry can be replaced by an $U(2)_{12}$ gauge symmetry,\nand\/or the $USp(2)_3\\times USp(2)_4$ gauge symmetry can be\nreplaced by an $U(2)_{34}$ gauge symmetry because the $USp(2)^2$\nstacks of D6-branes contribute to the same RR tadpoles as those of\nthe $U(2)$ stacks. Thus, we obtained three non-equivalent\nvariations, and the corresponding gauge symmetries in the hidden\nsector are $U(2)_{12} \\times USp(2)_3 \\times USp(2)_4$, $U(2)_{34}\n\\times USp(2)_1 \\times USp(2)_2$, and $U(2)_{12} \\times\nU(2)_{34}$, respectively. In addition, we studied the hidden\nsector gauge symmetry breaking, and discussed how to decouple the\nadditional exotic particles. Because the observable sector is the\nsame, the phenomenological discussions in the observable sector\nare the same as those in Ref.~\\cite{Chen:2007px,CLMN-L}.\n\n\n\n\\section*{Acknowledgments}\nThis research was supported in part\nby the Mitchell-Heep Chair in High Energy Physics (CMC),\nby the Cambridge-Mitchell Collaboration in Theoretical Cosmology (TL),\nand by the DOE grant DE-FG03-95-Er-40917 (DVN).\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\nBook embeddings have a long history and arise in various application\nareas such as VLSI design, parallel computing, design of\nfault-tolerant systems~\\cite{Chung87embeddinggraphs}. In a\n\\emph{book embedding} the placement of nodes is restricted to a\nline, the \\emph{spine} of the book. The edges are assigned to\ndifferent \\emph{pages} of the book. A page can be thought of as a\nhalf-plane bounded by the spine where the edges are drawn as\ncircular arcs between their endpoints. We say that a graph admits a\n\\emph{$k$-page book embedding} or is \\emph{k-page embeddable} if one\ncan assign the edges to $k$ pages and there exists a linear ordering\nof the nodes on the spine such that no two edges of the same page\ncross. The minimum number of pages required to construct such an\nembedding is the \\emph{book thickness} or \\emph{page number} of a\ngraph. The book thickness of planar graphs has received much\nattention in the past. Yannakakis~\\cite{JCSS::Yannakakis1989}\ndescribes a linear-time algorithm to embed every planar graph into a\nbook of four pages. We study the problem of embedding 4-planar\ngraphs, i.e., planar graphs with maximum degree four, into books\nwith two pages. Bernhart et al.~\\cite{bk-btg-79} show that a graph\nis two-page embeddable iff it is subhamiltonian. A\n\\emph{subhamiltonian graph} is a subgraph of a planar hamiltonian\ngraph. It is \\textit{NP}-complete to determine whether a graph is\nsubhamiltonian~\\cite{wigderson82}. Often referred to as\n\\emph{augmented hamiltonian cycle}, a \\emph{subhamiltonian cycle} is\na cyclic sequence of nodes in a graph that would form a hamiltonian\ncycle when adding the missing edges without destroying planarity.\nThe relation between subhamiltonian cycles and two-page book\nembeddings is quite intuitive. The order of the nodes on the spine\nis equivalent to the cyclic order of the subhamiltonian cycle. The\nedges are partitioned by whether they lie in the interior of the\ncycle or not.\n\nAn early important result is due to Whitney~\\cite{Whi}, who proves\nthat every maximal planar graph with no separating triangles is\nhamiltonian (recall that a \\emph{separating triangle} is a 3-cycle\nwhose removal disconnects the graph). Tutte~\\cite{Tut56} shows that\nevery 4-connected planar graph has a hamiltonian cycle.\nChiba~et~al.~\\cite{chibanishizeki89} provide a linear-time algorithm\nto find a hamiltonian cycle in a 4-connected planar graph.\nChen~\\cite{Chen:2003fk} gives a proof that every maximal planar\ngraph with at least five vertices and no separating triangles is\n4-connected. Sanders~\\cite{sanders-97} generalizes a theorem of\nThomassen and shows that any 4-connected planar graph has a\nhamiltonian cycle that contains two arbitrarily chosen edges of the\ngraph. Based on Whitney's theorem,\nKainen~et~al.~\\cite{Kainen2007835} show that every planar graph with\nno separating triangles is subhamiltonian. Another result is by\nChen~\\cite{Chen:2003fk} who shows that if a maximal planar graph\ncontains only one such triangle, then it is hamiltonian.\nHelden~\\cite{Helden20071833} improves this result further to two\ntriangles. The aforementioned results are all related to the problem\nof embedding planar graphs into two pages. However, there is an\nextensive amount of literature on embedding various types of graphs\ninto books; for an overview see e.g.~\\cite{DujmovicW04}. One result\nthat is interesting in our context is that of\nHeath~\\cite{heath-thesis-85}. In his thesis, he describes a\nlinear-time algorithm to embed any 3-planar graph into two pages and\nconcludes that it would be interesting to know if a higher degree\nbound is possible.\n\nWe tackle the 4-planar case from two sides. The first approach based\non the subhamiltonicity is restricted to triconnected graphs\n(Section~\\ref{sec:triconnected-planar}) but builds on existent\nresults and is therefore of a simple nature compared to the second\napproach. Extending it to biconnected graphs is not straightforward,\nthough. The algorithm of Section~\\ref{sec:general-planar} --which is\nless efficient in terms of time complexity-- exploits the degree\nrestriction to construct a two-page book embedding.\n\n\\section{Subhamiltonicity of Triconnected 4-Planar Graphs}\n\\label{sec:triconnected-planar}\nIn this section we restrict ourselves to triconnected 4-planar\ngraphs. To state the main result of this section, we proceed in a\nstep-by-step manner. First we investigate the special properties of\nseparating triangles in 4-planar graphs, then we use those to derive\na solution for a single separating triangle. Unlike\nChen~\\cite{Chen:2003fk} and Helden~\\cite{Helden20071833}, we are\nable to extend our approach to an unbounded number of triangles by\nexploiting the degree restriction. We say a subhamiltonian cycle $H$\n\\emph{crosses} a face if there are two consecutive vertices in $H$\nthat are incident to the face but not adjacent to each other.\n\n\\begin{figure}[t]\n \\centering\n \\begin{minipage}[b]{0.4\\textwidth}\n \\centering\n \\includegraphics[width=0.65\\textwidth,page=1]{images\/sep_tri}\n \\caption{Triangle $\\mathcal{T}$ separating $\\inGraph{\\mathcal{T}}$ and $\\outGraph{\\mathcal{T}}$ on removal.}\n \\label{fig:sep_tri_4planar}\n \\end{minipage}\n \\hfill\n \\begin{minipage}[b]{0.5\\textwidth}\n \\centering\n \\includegraphics[width=0.65\\textwidth,page=2]{images\/sep_tri}\n \\caption{Merging $\\inHamilton{\\mathcal{T}}$ (dotted) and $\\outHamilton{\\mathcal{T}}$ (dashed) into $H$ (bold gray).}\n \\label{fig:merging_cycles}\n \\end{minipage}\n\\end{figure}\n\n\\begin{lemma}\n\\label{lemma:linear_time_one_crossing} Every triconnected planar\ngraph with no separating triangles has a subhamiltonian cycle that\ncrosses every face at most once and it can be computed in linear\ntime.\n\\end{lemma}\n\\begin{proof}\nIn the triconnected case, Kainen~et~al.~\\cite{Kainen2007835}\nconstruct a new maximal planar graph $G'=(V',E')$ by inserting a\nvertex into each non-triangular face of $G$ and connect it to the\nvertices of that face. Clearly this takes linear time. $G'$ is\nmaximal planar, free of separating triangles, hence, 4-connected. We\ncan use the linear-time algorithm of\nChiba~et~al.~\\cite{chibanishizeki89} to obtain a hamiltonian cycle\n$H'$ for $G'$. Deleting the newly inserted vertices $V' - V$ yields\na subhamiltonian cycle $H$ for $G$ that crosses each face at most\nonce.\n\\end{proof}\n\nBefore investigating the properties of separating triangles, we\nintroduce some notation. Given an embedded triconnected 4-planar\ngraph $G$ with a fixed outerface and a separating triangle\n$\\mathcal{T}$ with vertices $V(\\mathcal{T})= \\{A, B, \\Gamma\\} $, we\ndenote the subgraph of $G$ contained in $\\mathcal{T}$ by\n$\\inGraph{\\mathcal{T}}$ and the subgraph of $G$ outside\n$\\mathcal{T}$ by $\\outGraph{\\mathcal{T}}$. We also denote\n$\\inGraphInc{\\mathcal{T}} = G - \\outGraph{\\mathcal{T}}$ and\n$\\outGraphInc{\\mathcal{T}} = G - \\inGraph{\\mathcal{T}}$. Since $G$\nis triconnected and 4-planar, every vertex of $\\mathcal{T}$ has\ndegree four and is adjacent to exactly one vertex in\n$\\inGraph{\\mathcal{T}}$ and $\\outGraph{\\mathcal{T}}$, respectively.\nWe denote these with $A_{in}, B_{in}, \\Gamma_{in}$ and $A_{out},\nB_{out}, \\Gamma_{out}$, respectively (see\nFig.~\\ref{fig:sep_tri_4planar}).\n\n\\begin{lemma}\\label{lemma:abc_distinct}\nGiven a 4-planar triconnected graph $G$ and a separating triangle\n$\\mathcal{T} = \\{A,B,\\Gamma\\}$, then $A_{in}, B_{in}, \\Gamma_{in}\n(A_{out}, B_{out}, \\Gamma_{out})$ are pairwise distinct or all\nrepresent the same vertex.\n\\end{lemma}\n\\begin{proof}\nIn the other case, where {w.l.o.g. } $A_{in} = B_{in} = v$ and\n$\\Gamma_{in} \\neq v$, there exists a separation pair $(v,\\Gamma)$\ncontradicting the triconnectivity of $G$. A symmetric argument\napplies to $A_{out}, B_{out}, \\Gamma_{out}$.\n\\end{proof}\n\n\n\\begin{lemma}\\label{lemma:triangle_disjoint}\nIn a 4-planar triconnected graph, every pair of distinct separating\ntriangles $\\mathcal{T}$ and $\\mathcal{T}'$ is vertex disjoint, i.e.\n$V(\\mathcal{T}) \\cap V(\\mathcal{T}') = \\emptyset$.\n\\end{lemma}\n\\begin{proof}\nAssume to the contrary that $\\mathcal{T}$ and $\\mathcal{T}'$ share\nan edge or a vertex. In the first case, let {w.l.o.g. } $e = (u,v)$ be the\ncommon edge. The degree of both $u$ and $v$ is at least five, since\nthree edges are required for $\\mathcal{T}, \\mathcal{T}'$ and two\nadditional edges to connect $\\inGraph{\\mathcal{T}}$ and\n$\\inGraph{\\mathcal{T}'}$ to $\\mathcal{T}$ and $\\mathcal{T}'$,\nrespectively. In the second case, let $v$ denote the common vertex.\nSince $v$ is part of two edge disjoint cycles and connected to\n$\\inGraph{\\mathcal{T}}$ and $\\inGraph{\\mathcal{T}'}$, it follows\nthat $deg(v) \\geq 6$.\n\\end{proof}\n\nConsider now a 4-planar triconnected graph with a single separating\ntriangle $\\mathcal{T}$. Similar to Chen~\\cite{Chen:2003fk}, the idea\nis to compute two cycles $\\inHamilton{\\mathcal{T}}$ and\n$\\outHamilton{\\mathcal{T}} $ for $\\inGraphInc{\\mathcal{T}}$ and\n$\\outGraphInc{\\mathcal{T}}$ and link them via the separating\ntriangle together. The crucial observation is that if two cycles\nintersect as illustrated in Fig.~\\ref{fig:merging_cycles}, i.e.,\nthey contain two edges of the triangle but have only one of them in\ncommon, then we can always merge them into one cycle.\n\\begin{lemma}\n\\label{lemma:cycle_merging} Let $G$ be a triconnected 4-planar\ngraph, $\\mathcal{T}$ a separating triangle, and\n$\\inHamilton{\\mathcal{T}}$ and $\\outHamilton{\\mathcal{T}} $ two\nsubhamiltonian cycles for $\\inGraphInc{\\mathcal{T}}$ and\n$\\outGraphInc{\\mathcal{T}}$, resp. If $E(\\inHamilton{\\mathcal{T}})\n\\cap E(\\mathcal{T})=\\{e_{in},e\\}$ and $E(\\outHamilton{\\mathcal{T}})\n\\cap E(\\mathcal{T})=\\{e_{out},e\\} $ where $\\{e, e_{in} ,e_{out}\\}$\nare the edges of $\\mathcal{T}$, then $G$ is subhamiltonian.\n\\end{lemma}\n\\begin{proof}\nLet {w.l.o.g. } $e = (A,B)$, $e_{in}=(B,\\Gamma)$ and $e_{out}=(A,\\Gamma) $\nas illustrated in Fig.~\\ref{fig:merging_cycles}. The result of\nremoving the edges of $\\mathcal{T}$ from both cycles are two paths\n$P_{out} = B \\rightsquigarrow \\Gamma$ and $P_{in} = \\Gamma\n\\rightsquigarrow A$. Joining them at $\\Gamma$ and inserting $e$\nyields a subhamiltonian cycle.\n\\end{proof}\n\n\\begin{figure}[t]\n\\centering\n \\begin{minipage}[b]{.20\\textwidth}\n \\centering\n \\subfloat[\\label{fig:sep_tri_dummy}{$v_\\mathcal{T}$ in $\\outGraphDummy{\\mathcal{T}}$}]\n {\\includegraphics[width=\\linewidth,page=3]{images\/sep_tri}}\n \\end{minipage}\n \\hfill\n \\begin{minipage}[b]{.20\\textwidth}\n \\centering\n \\subfloat[\\label{fig:sep_tri_dummy_T}{$\\mathcal{T}$ in $\\outGraphInc{\\mathcal{T}}$}]\n {\\includegraphics[width=\\linewidth,page=4]{images\/sep_tri}}\n \\end{minipage}\n \\hfill\n \\begin{minipage}[b]{.15\\textwidth}\n \\centering\n \\subfloat[\\label{fig:sep_tri_inside_cycle}{$v'_\\mathcal{T}$ in $\\inGraphDummy{\\mathcal{T}}$}]\n {\\includegraphics[width=\\linewidth,page=5]{images\/sep_tri}}\n \\end{minipage}\n \\hfill\n \\begin{minipage}[b]{.20\\textwidth}\n \\centering\n \\subfloat[\\label{fig:sep_tri_result_cycle}{$\\mathcal{T}$ in $\\inGraphInc{\\mathcal{T}}$}]\n {\\includegraphics[width=\\linewidth,page=6]{images\/sep_tri}}\n \\end{minipage}\n \\begin{minipage}[b]{.20\\textwidth}\n \\centering\n \\subfloat[\\label{fig:sep_tri_result_cycle}{$G$ with $\\mathcal{T}$ and $H$}]\n {\\includegraphics[width=\\linewidth,page=7]{images\/sep_tri}}\n \\end{minipage}\n \\caption{\n (a)~Subhamiltonian cycle $\\outHamiltonDummy{\\mathcal{T}}$ in $\\outGraphDummy{\\mathcal{T}}$ containing $v_\\mathcal{T}$.\n (b)~Augmenting $\\outHamiltonDummy{\\mathcal{T}}$ yields $\\outHamilton{\\mathcal{T}}$ containing edges $e_1 = (\\Gamma,A)$ and $e_2 = (A,B)$.\n (c)~Dummy vertex $v'_\\mathcal{T}$ as replacement for $\\mathcal{T}$ in $\\inGraphDummy{\\mathcal{T}}$ and a cycle $\\inHamiltonDummy{\\mathcal{T}}$.\n (d)~Rerouting $\\inHamiltonDummy{\\mathcal{T}}$ through $\\mathcal{T}$ resulting in $\\inHamilton{\\mathcal{T}}$ with edges $e'_1 = (\\Gamma,B)$ and $e_2 = (A,B)$.\n (d)~The result of merging $\\inHamilton{\\mathcal{T}}$ and $\\outHamilton{\\mathcal{T}}$ into a cycle $H$ for $G$.}\n\\end{figure}\n\nIt remains to show that we can always find two cycles that satisfy\nthe requirements of Lemma~\\ref{lemma:cycle_merging}. In the\nfollowing, we neglect the degenerated case of\nLemma~\\ref{lemma:abc_distinct}, where $\\outGraph{\\mathcal{T}}$ or\n$\\inGraph{\\mathcal{T}}$ is a single vertex, because finding a cycle\nin that case is trivial. Consider for example\n$\\outGraphInc{\\mathcal{T}}$, for $\\inGraphInc{\\mathcal{T}}$ a\nsymmetric argument holds. To obtain $\\outHamilton{\\mathcal{T}}$, we\ntemporarily replace $\\mathcal{T}$ in $\\outGraphInc{\\mathcal{T}}$\nwith a single vertex $v_\\mathcal{T}$ as depicted in\nFig.~\\ref{fig:sep_tri_dummy}. The resulting graph\n$\\outGraphDummy{\\mathcal{T}}$ remains 4-planar and triconnected,\nbecause $deg(v_\\mathcal{T}) = 3$ by construction and any path via\n$\\mathcal{T}$ can use $v_\\mathcal{T}$ instead. One may argue that\nthis operation may introduce additional separating triangles.\nHowever, such a triangle must contain $v_\\mathcal{T}$ and,\ntherefore, $deg(v_\\mathcal{T}) = 4$, a contradiction. Now let us\nassume that $\\outHamiltonDummy{\\mathcal{T}}$ is a subhamiltonian\ncycle for $\\outGraphDummy{\\mathcal{T}}$. The idea is to reinsert\n$\\mathcal{T}$ and reroute $\\outHamiltonDummy{\\mathcal{T}}$ through\n$\\mathcal{T}$ such that the resulting cycle\n$\\outHamilton{\\mathcal{T}}$ contains two edges $e_1,e_2 \\in\nE(\\mathcal{T})$.\n\n\\begin{lemma}\n\\label{lemma:cycle_rerouting} Let $G$ be a triconnected 4-planar\ngraph, $\\mathcal{T}$ a separating triangle. Furthermore, let\n$\\outGraphDummy{\\mathcal{T}}$ denote the graph resulting from\nreplacing $\\mathcal{T}$ by a vertex $v_\\mathcal{T}$ in\n$\\outGraphInc{\\mathcal{T}}$. A subhamiltonian cycle\n$\\outHamiltonDummy{\\mathcal{T}}$ for $\\outGraphDummy{\\mathcal{T}}$\ncan be augmented to a subhamiltonian cycle\n$\\outHamilton{\\mathcal{T}}$ for $\\outGraphInc{\\mathcal{T}}$ such\nthat it contains two edges of $\\mathcal{T}$, i.e.,\n$E(\\outHamilton{\\mathcal{T}}) \\cap E(\\mathcal{T}) = \\{e_1, e_2\\}$.\nIf $\\outHamiltonDummy{\\mathcal{T}}$ crosses every face of\n$\\outGraphDummy{\\mathcal{T}}$ at most once, one may choose any pair\n$e_1, e_2 \\in E(\\mathcal{T})$ to lie on $\\outHamilton{\\mathcal{T}}$.\n\\end{lemma}\n\\begin{proof}\nTo prove the claim, it is sufficient to consider every combination\nof $e_1, e_2$ and the location of the predecessor and successor of\n$v_\\mathcal{T}$ in $\\outHamiltonDummy{\\mathcal{T}}$. \nIn the following, we enumerate and describe in detail all possible\ncases that occur when augmenting $\\outHamiltonDummy{\\mathcal{T}}$\nsuch that the resulting cycle $\\outHamilton{\\mathcal{T}}$ contains\ntwo edges $e_1, e_2$ of $\\mathcal{T}$. \nTo avoid any redundancies, we omit\nsymmetric cases and consider for the same reason a directed cycle.\nWe distinguish between three main cases depending on the location of\nthe predecessor and successor of $v_\\mathcal{T}$ in\n$\\outHamiltonDummy{\\mathcal{T}}$.\n\n\\begin{figure}[h!]\n\\centering\n \\begin{minipage}[b]{.32\\textwidth}\n \\centering\n \\subfloat[\\label{fig:dummy_edge_edge}{Case 1}]\n {\\includegraphics[page=1]{images\/sep_tri_reroute_cases}}\n \\end{minipage}\\hfill\n \\begin{minipage}[b]{.32\\textwidth}\n \\centering\n \\subfloat[\\label{fig:dummy_edge_face}{Case 2}]\n {\\includegraphics[page=2]{images\/sep_tri_reroute_cases}}\n \\end{minipage} \\hfill\n \\begin{minipage}[b]{.32\\textwidth}\n \\centering\n \\subfloat[\\label{fig:dummy_face_face}{Case 3}]\n {\\includegraphics[page=3]{images\/sep_tri_reroute_cases}}\n \\end{minipage}\\hfill\n \\caption{The three main cases at $v_\\mathcal{T}$: (a)~The cycle uses two of the three edges incident to $v_\\mathcal{T}$.\n (b)~The cycle enters via an edge and leaves through a face.\n (c)~Predecessor and successor are not adjacent to $v_\\mathcal{T}$.}\n \\label{fig:sep_tri_reroute_cases}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centering\n \\begin{minipage}[b]{.32\\textwidth}\n \\centering\n \\subfloat[\\label{fig:reroute_edge_edge_ACB}{$(A,\\Gamma)$ and $(B,\\Gamma)$}]\n {\\includegraphics[page=4]{images\/sep_tri_reroute_cases}}\n \\end{minipage}\n \\hfill\n \\begin{minipage}[b]{.32\\textwidth}\n \\centering\n \\subfloat[\\label{fig:reroute_edge_edge_CAB}{$(A,B)$ and $(A,\\Gamma)$}]\n {\\includegraphics[page=5]{images\/sep_tri_reroute_cases}}\n \\end{minipage} \\hfill\n \\begin{minipage}[b]{.32\\textwidth}\n \\centering\n \\subfloat[\\label{fig:reroute_edge_edge_ABC}{$(A,B)$ and $(B,\\Gamma)$}]\n {\\includegraphics[page=6]{images\/sep_tri_reroute_cases}}\n \\end{minipage}\n \\hfill\n \\caption{\n (a)~Dummy vertex $v_T$ is replaced by the sequence $A,\\Gamma,B$ to obtain a cycle with the edges $(A,\\Gamma)$ and $(B,\\Gamma)$.\n (b)~Sequence $\\Gamma,A,B$ yields a cycle with $(A,B)$ and $(A,\\Gamma)$ where $(A_{out},\\Gamma)$ requires it to cross a face.\n (c)~Augmenting with $A,B,\\Gamma$ results in a cycle containing $(A,B)$ and $(B,\\Gamma)$.}\n \\label{fig:sep_tri_reroute_case_1}\n \\begin{minipage}[b]{.32\\textwidth}\n \\centering\n \\subfloat[\\label{fig:reroute_edge_face_ACB}{$(A,\\Gamma)$ and $(B,\\Gamma)$}]\n {\\includegraphics[page=7]{images\/sep_tri_reroute_cases}}\n \\end{minipage}\n \\hfill\n \\begin{minipage}[b]{.32\\textwidth}\n \\centering\n \\subfloat[\\label{fig:reroute_edge_face_CAB}{$(A,B)$ and $(A,\\Gamma)$}]\n {\\includegraphics[page=8]{images\/sep_tri_reroute_cases}}\n \\end{minipage}\n \\hfill\n \\begin{minipage}[b]{.32\\textwidth}\n \\centering\n \\subfloat[\\label{fig:reroute_edge_face_ABC}{$(A,B)$ and $(B,\\Gamma)$}]\n {\\includegraphics[page=9]{images\/sep_tri_reroute_cases}}\n \\end{minipage}\n \\hfill\n \\caption{\n In (a)~and~(b) the same sequences as before are used to obtain a cycle containing $(A,\\Gamma), (B,\\Gamma)$ and $(A,B), (A,\\Gamma)$, respectively.\n Subcase-specific links are drawn in blue and red. (c) A more complicated case requiring one additional crossing of a face from $A_{out}$ to $\\Gamma$.}\n \\label{fig:sep_tri_reroute_case_2}\n \\begin{minipage}[b]{.32\\textwidth}\n \\centering\n \\subfloat[\\label{fig:reroute_face_face_ACB}{$(A,\\Gamma)$ and $(B,\\Gamma)$}]\n {\\includegraphics[page=10]{images\/sep_tri_reroute_cases}}\n \\end{minipage}\\hfill\n \\begin{minipage}[b]{.32\\textwidth}\n \\centering\n \\subfloat[\\label{fig:reroute_face_face_CAB}{$(A,B)$ and $(A,\\Gamma)$}]\n {\\includegraphics[page=11]{images\/sep_tri_reroute_cases}}\n \\end{minipage} \\hfill\n \\begin{minipage}[b]{.32\\textwidth}\n \\centering\n \\subfloat[\\label{fig:reroute_face_face_ABC}{$(A,B)$ and $(B,\\Gamma)$}]\n {\\includegraphics[page=12]{images\/sep_tri_reroute_cases}}\n \\end{minipage}\\hfill\n \\caption{(a)~Both subcases have a solution. (b,c) When the cycle uses two distinct faces ($f_1\\rightsquigarrow f_3$) a solution for both pairs of edges can be found. If only one face is used ($f_1\\rightsquigarrow f_2$), then no solution exists for the edges $(A,B), (A,\\Gamma)$ and $(A,B), (B,\\Gamma)$.}\n \\label{fig:sep_tri_reroute_case_3}\n\\end{figure}\n\n\\begin{description}\n\\item[Case 1] ($\\text{Edge} \\rightsquigarrow v_\\mathcal{T} \\rightsquigarrow\n\\text{Edge}$): Both the predecessor and successor of $v_\\mathcal{T}$\nin $\\outHamiltonDummy{\\mathcal{T}}$ are adjacent to $v_\\mathcal{T}$,\nhence, the cycle $\\outHamiltonDummy{\\mathcal{T}}$ contains two edges\nincident to $v_\\mathcal{T}$, let us say $(A_{out}, v_\\mathcal{T}),\n(v_\\mathcal{T}, B_{out})$ as illustrated in\nFig.~\\ref{fig:dummy_edge_edge}.\nFig.~\\ref{fig:sep_tri_reroute_case_1} depicts how\n$\\outHamiltonDummy{\\mathcal{T}}$ can be augmented such that every\npair of edges of $T$ is contained in $\\outHamilton{\\mathcal{T}}$.\nNotice that while for the pair $(A,\\Gamma),(B,\\Gamma)$ in\nFig~\\ref{fig:reroute_edge_edge_ACB} no face crossing is required,\nfor the two other pairs one additional face crossing is introduced\n(Fig.~\\ref{fig:reroute_edge_edge_CAB} and\n\\ref{fig:reroute_edge_edge_ABC}).\n\n\\item[Case 2] ($\\text{Edge} \\rightsquigarrow v_\\mathcal{T} \\rightsquigarrow\n\\text{Face}$): In this case, the predecessor, say $A_{out}$, is\nadjacent to $v_\\mathcal{T}$, while the successor is not. Since\n$\\outHamiltonDummy{\\mathcal{T}}$ is a subhamiltonian cycle, the\nsuccessor is incident to one of the three faces incident to\n$v_\\mathcal{T}$. To cover all possible combinations, we distinguish\nbetween whether (i) the predecessor $A_{out}$ is incident to that\nface or (ii) not. Fig.~\\ref{fig:dummy_edge_face} illustrates both\nconfigurations, where $f_1$ denotes the successor located at a face\nof type (i), and $f_2$ the successor that is incident to the face at\nthe opposite side (ii). For both subcases, the rerouting rules for\nthe first two edge pairs are relatively simple, since they follow\nthe basic principle of the first case, see\nFig.~\\ref{fig:reroute_edge_face_ACB}\nand~\\ref{fig:reroute_edge_face_CAB}. However, the third pair is more\ncomplicated. For (i) the sequence $A_{out}, v_\\mathcal{T} , f_1$ is\nreplaced by $A_{out}, \\Gamma, B, A, f_1$, whereas for (ii) $A_{out},\nv_\\mathcal{T} , f_1$ is substituted by $A_{out}, A, B, \\Gamma, f_2$\n(Fig.~\\ref{fig:reroute_edge_face_ABC}).\n\n\\item[Case 3] $\\text{Face} \\rightsquigarrow v_\\mathcal{T} \\rightsquigarrow\n\\text{Face}$. Both predecessor and successor of $v_\\mathcal{T}$ in\n$\\outHamiltonDummy{\\mathcal{T}}$ are not adjacent. Hence, the cycle\nenters and leaves $v_T$ through a face. Again to cover all\npossibilities, we have to deal with two subcases: (i) the two faces\nare distinct or (ii) the cycle $\\outHamiltonDummy{\\mathcal{T}}$\nleaves through the same face as it enters. Rerouting\n$\\outHamiltonDummy{\\mathcal{T}}$ in the first subcase (i) works for\nall three different edge pairs, even without introducing any new\nface crossings. The three solutions for (i) are displayed in\nFig.~\\ref{fig:sep_tri_reroute_case_3}, where the predecessor is\nlabeled by $f_1$ and the successor by $f_3$. So far we have been\nable to resolve every configuration such that any pair of edges can\nbe selected to be part of $\\outHamilton{\\mathcal{T}}$. However, the\ninteresting case is subcase (ii), where the predecessor $f_1$ and\nsuccessor $f_2$ are incident to the same face. While there is a\nsolution for the edge pair $(A,\\Gamma), (B,\\Gamma)$ as displayed in\nFig.~\\ref{fig:reroute_face_face_ACB}, the two remaining edge pairs\ncreate unresolvable configurations, see\nFig.~\\ref{fig:reroute_face_face_CAB}\nand~\\ref{fig:reroute_face_face_ABC}, respectively. This dilemma is\ncaused by the fact that $\\outHamilton{\\mathcal{T}}$ has to either\nenter or leave $\\mathcal{T}$ via $\\Gamma$. However, $\\Gamma$ is not\naccessible from neither $f_1$ nor $f_2$ without destroying\nplanarity.\n\\end{description}\n\nWe may summarize the solutions for the different cases as follows:\nAs long as the cycle does not enter and leave $v_\\mathcal{T}$ via\nthe same face, we can always choose two edges of $\\mathcal{T}$ in\nadvance and reroute the cycle such that these two edges become part\nof $H$. \n\\end{proof}\nAt this point it is tempting to show that we can always find\na cycle that avoids crossing a face twice. By using\nLemma~\\ref{lemma:linear_time_one_crossing}, we may obtain such a\ncycle in a triconnected graph with no separating triangles. This\nraises the question if we can use it and apply the described rules\nto obtain a cycle through multiple triangles for which we may\nspecify two edges in advance. We answer this question negatively\nwith a small counterexample.\n\n\\begin{figure}[t]\n \\centering\n \\begin{minipage}[b]{.32\\textwidth}\n \\centering\n \\subfloat[\\label{fig:sep_tri_reroute_counterexample_0}{$G$ with $\\mathcal{T}$ and $\\mathcal{T}'$}]\n {\\includegraphics[page=1]{images\/sep_tri_reroute_counterexample}}\n \\end{minipage}\\hfill\n \\begin{minipage}[b]{.32\\textwidth}\n \\centering\n \\subfloat[\\label{fig:sep_tri_reroute_counterexample_1}{Cycle from Lemma~\\ref{lemma:linear_time_one_crossing}}]\n {\\includegraphics[page=2]{images\/sep_tri_reroute_counterexample}}\n \\end{minipage} \\hfill\n \\begin{minipage}[b]{.32\\textwidth}\n \\centering\n \\subfloat[\\label{fig:sep_tri_reroute_counterexample_2}{After rerouting at $\\mathcal{T}$ }]\n {\\includegraphics[page=3]{images\/sep_tri_reroute_counterexample}}\n \\end{minipage}\\hfill\n \\caption{\n (a)~Two separating triangles $\\mathcal{T}$ and $\\mathcal{T}'$ with vertices $V(\\mathcal{T}) = \\{A,B,\\Gamma\\}$ and $V(\\mathcal{T}') = \\{A',B',\\Gamma'\\}$ and for each two prescribed edges (bold, blue).\n (b)~$\\mathcal{T}$ and $\\mathcal{T}'$ replaced by $v_\\mathcal{\\mathcal{T}}$ and $v_{\\mathcal{T}'}$, every non triangular face is stellated by inserting additional vertices (squares) and\n edges (dashed), and a (sub)hamiltonian cycle $H'$ (bold, red).\n (c)~Result of applying the corresponding rule to $\\mathcal{T}$ creating an unresolvable configuration for $\\mathcal{T}'$.}\n \\label{fig:sep_tri_reroute_counterexample}\n\\end{figure}\n\nConsider the triconnected 4-planar graph $G$ shown in\nFig.~\\ref{fig:sep_tri_reroute_counterexample_0}. It contains two\nseparating triangles $\\mathcal{T}$ and $\\mathcal{T}'$ with vertices\n$V(\\mathcal{T}) = \\{A,B,\\Gamma\\}$ and $V(\\mathcal{T}') =\n\\{A',B',\\Gamma'\\}$, respectively. In every triangle two edges (bold,\nblue) are prescribed to lie on the augmented subhamiltonian cycle\n$H$. We proceed as described; both triangles are replaced by a dummy\nvertex $v_{\\mathcal{T}}$ and $v_{\\mathcal{T}'}$, respectively. The\nresulting graph (Fig.~\\ref{fig:sep_tri_reroute_counterexample_1}) is\ntriconnected 4-planar and free of separating triangles. The squares\nand dashed lines correspond to the dummy vertices and edges\ninserted by the technique of Kainen et al.~\\cite{Kainen2007835} as\ndescribed in Lemma~\\ref{lemma:linear_time_one_crossing}. We may now\ncompute a hamiltonian cycle $H'$ by applying the linear-time\nalgorithm of Chiba et al.~\\cite{chibanishizeki89}. Assume the result\nis the bold cycle in\nFig.~\\ref{fig:sep_tri_reroute_counterexample_1}. Clearly the cycle\ncrosses every face at most once after we remove the dummy vertices\ninserted by the technique of Kainen et al.~\\cite{Kainen2007835}. We\nreinsert $\\mathcal{T}$ and apply the corresponding rule, i.e., the\naugmentation displayed in Fig.~\\ref{fig:reroute_edge_face_CAB}. The\nresult of augmenting such that the two marked edges of\n$\\mathcal{T}$, namely $(A,\\Gamma), (B,\\Gamma)$, lie on the cycle is\ndisplayed in Fig.~\\ref{fig:sep_tri_reroute_counterexample_2}. Notice\nthat we are forced to enter $\\mathcal{T}$ via $A$ and exit by $B$.\nAs a result, the cycle crosses one face twice. Moreover,\n$\\mathcal{T}'$ must be entered and left through the same face. The\ncorresponding rule, illustrated in\nFig.~\\ref{fig:reroute_face_face_CAB}, implies that we cannot reroute\nthe cycle such that it contains the edges $(A',\\Gamma'),\n(B',\\Gamma')$. However, we may lift the restriction, use the only\nrule applicable in this case (Fig.~\\ref{fig:reroute_face_face_ACB}),\nand obtain a cycle with edges $(A',\\Gamma'), (A',B')$ instead.\nNotice that the graph in this example has even a hamiltonian cycle $H$ through\nthe requested edges. However, the purpose of the example is to\ndemonstrate that for an arbitrary chosen subhamiltonian cycle, the\ndescribed rules cannot always be applied. We may conclude that when\nusing Lemma~\\ref{lemma:linear_time_one_crossing}, we may choose for\none (the first) triangle two edges because the initial cycle visits\nevery face at most once. From there on, we can only guarantee that\ntwo unknown edges are part of the final cycle. In the following we will benefit from this observation.\n\nRecall the aforementioned single-separating-triangle scenario. Both\n$\\outGraphInc{\\mathcal{T}}$ and $\\inGraphInc{\\mathcal{T}}$ are free\nof separating triangles. Therefore, we may construct two graphs\n$\\outGraphDummy{\\mathcal{T}}, \\inGraphDummy{\\mathcal{T}}$ by\nreplacing $\\mathcal{T}$ with dummy vertices. Applying\nLemma~\\ref{lemma:linear_time_one_crossing} to them yields two\nsubhamiltonian cycles $\\outHamiltonDummy{\\mathcal{T}}$ and\n$\\inHamiltonDummy{\\mathcal{T}}$, both crossing every face of\n$\\outGraphDummy{\\mathcal{T}}$ and $\\inGraphDummy{\\mathcal{T}}$ at\nmost once. Hence, we may augment them with the aid of\nLemma~\\ref{lemma:cycle_rerouting} such that they contain each two\nedges of $\\mathcal{T}$. By choosing the combination of the edges such\nthat $\\outHamilton{\\mathcal{T}}$ and $\\inHamilton{\\mathcal{T}}$ meet\nthe requirements of Lemma~\\ref{lemma:cycle_merging}, we can\nmerge them into a single subhamiltonian cycle $H$ for $G$.\n\nWhile the property that $\\outGraphDummy{\\mathcal{T}}$ and\n$\\inGraphDummy{\\mathcal{T}}$ are both free of separating triangles\nenables us to conveniently choose two edges for each cycle\n$\\outHamilton{\\mathcal{T}}, \\inHamilton{\\mathcal{T}}$, this only\nworks for a single separating triangle. However, a closer look\nreveals that it is sufficient to have a choice for either\n$\\outHamilton{\\mathcal{T}}$ or $\\inHamilton{\\mathcal{T}}$, not\nnecessarily both of them. The idea is to first augment the cycle for\nwhich we do not have a choice to see which edges of $\\mathcal{T}$\nare part of it, then we choose the edges for the second cycle\naccordingly. We summarize the idea as the main result of this\nsection and describe it in a more formal manner in form of a proof.\n\n\\begin{theorem}\\label{theorem:triconnected}\nEvery triconnected 4-planar graph is subhamiltonian.\n\\end{theorem}\n\\begin{proof}\nLet $G$ denote a triconnected 4-planar graph and $\\tau(G)$ the\nnumber of separating triangles in $G$. We prove by induction and\nclaim that for any $\\tau(G) \\geq 0$ we can compute a subhamiltonian\ncycle $H$ for $G$. \\emph{Base case}: Since $\\tau(G) = 0$, we can\ndirectly apply Lemma~\\ref{lemma:linear_time_one_crossing}.\n\\emph{Inductive case:} For $\\tau(G) > 0$, we pick a separating\ntriangle $\\mathcal{T}$ such that $\\tau(\\inGraphInc{\\mathcal{T}}) =\n0$. Let $\\outGraphDummy{\\mathcal{T}}$ be the result of replacing\n$\\mathcal{T}$ by $v_\\mathcal{T}$ in $\\outGraphInc{\\mathcal{T}}$.\nNotice that $\\tau(\\outGraphDummy{\\mathcal{T}}) = \\tau(G) - 1$ holds.\nHence, by induction hypothesis, $\\outGraphDummy{\\mathcal{T}}$ has a\nsubhamiltonian cycle $\\outHamiltonDummy{\\mathcal{T}}$. We reinsert\n$\\mathcal{T}$ and augment $\\outHamiltonDummy{\\mathcal{T}}$ such that\nthe result $\\outHamilton{\\mathcal{T}}$ contains two (arbitrary)\nedges $e_1, e_2$ of $\\mathcal{T}$. In a similar way, we replace\n$\\mathcal{T}$ in $\\inGraphInc{\\mathcal{T}}$ by $v'_\\mathcal{T}$ to\nobtain $\\inGraphDummy{\\mathcal{T}}$. Since\n$\\tau(\\inGraphInc{\\mathcal{T}}) = \\tau(\\inGraphDummy{\\mathcal{T}}) =\n0$ holds, we can apply Lemma~\\ref{lemma:linear_time_one_crossing} to\n$\\inGraphDummy{\\mathcal{T}}$ and compute a cycle\n$\\inHamiltonDummy{\\mathcal{T}}$ that crosses each face at most once.\nWith Lemma~\\ref{lemma:cycle_rerouting} we may obtain a cycle\n$\\inHamilton{\\mathcal{T}}$ for $\\inGraphInc{\\mathcal{T}}$ with two\nedges $e'_1, e'_2 \\in E(\\mathcal{T})$ of our choice. Choosing $e'_1\n= e_1$ and $e'_2 \\neq e_2$ yields two cycles\n$\\outHamilton{\\mathcal{T}}, \\inHamilton{\\mathcal{T}}$ that meet the\nrequirements of Lemma~\\ref{lemma:cycle_merging} and we can merge\nthem into one cycle $H$ for $G$.\n\\end{proof}\n\nThe proof of Theorem~\\ref{theorem:triconnected} is constructive.\nEmbedding $G$ and identifying all separating triangles in $G$ can be\ndone in linear time. Augmenting a cycle and merging two of them\ntakes constant time. Disjointness of separating triangles yields a\nlinear number of subproblems and every edge occurs in at most one\nsuch subproblem. Hence, the total time spent for the subroutine of\nLemma~\\ref{lemma:linear_time_one_crossing} is linear in the size of\n$G$.\n\n\\begin{corollary}\nA subhamiltonian cycle of a triconnected 4-planar graph can be found\nin linear time.\n\\end{corollary}\n\nIn this section, we have shown that in the triconnected case a\nrather simple technique can be used to efficiently compute a\nsubhamiltonian cycle in a 4-planar graph. However, the property that\nG is triconnected has been used extensively throughout this section,\nthus, a relaxation to biconnectivity is not straightforward.\n\n\n\\section{Two-Page Book Embeddings of General 4-Planar Graphs}\n\\label{sec:general-planar}\nIn this section, we prove that any planar graph of maximum degree\nfour admits a two-page book embedding. The proof is given by a\nrecursive combinatorial construction, which determines the order of\nthe vertices along the spine and the page in which each edge is\ndrawn. W.l.o.g. we assume that the input graph $G$ is biconnected,\nsince it is known that the page number of a graph equals the maximum\nof the page number of its biconnected components \\cite{bk-btg-79}.\nNote that one can neglect the exact geometry, as two edges that are\ndrawn on the same page cross if and only if their endpoints\nalternate along the spine. We say that an edge $e$ \\emph{nests a\nvertex} $v$ iff one endpoint of $e$ is to the left of $v$ along the\nspine and the other endpoint of $e$ to its right. We also say that\nan edge $e$ \\emph{nests an edge} $e'$ iff both $e$ and $e'$ are\ndrawn on the same page and both endpoints of $e'$ are nested by $e$.\nObserve that nested edges do not cross.\n\nThe general idea of our algorithm is as follows: First remove from\n$G$ cycle $C_{out}$ delimiting the outerface of $G$ and\n\\emph{contract} each bridge-block\\footnote{The \\emph{bridge-blocks}\nof a connected graph $G$ are the connected components formed by\ndeleting all bridges of $G$. The bridge-blocks and the bridges of\n$G$ have a natural tree structure, called \\emph{bridge-block tree}.}\nof the remaining graph into a single vertex. Let $F$ be the implied\ngraph, which is a forest in general, since $G-C_{out}$ is not\nnecessarily connected. Cycle $C_{out}$ is embedded, s.t.: (i)~the\norder of the vertices of $C_{out}$ along the spine is fixed (and\nfollows the one in which the vertices of $C_{out}$ appear along\n$C_{out}$), and, (ii)~all edges of $C_{out}$ are on the same page,\nexcept for the one that connects its outermost vertices. Then, we\ndescribe how to embed without crossings: (i)~the chords of\n$C_{out}$, (ii)~forest $F$, and, (iii)~the edges between $C_{out}$\nand $F$. To obtain a two-page book embedding of $G$, we replace each\nvertex of $F$ with a cycle (embedded similarly to $C_{out}$), whose\nlength equals to the length of the cycle delimiting the outerface of\nthe bridge-block it corresponds to in $G-C_{out}$, and recursively\nembed its interior.\n\nMore formally, consider an arbitrary simple cycle $C: v_1\n\\rightarrow v_2 \\rightarrow \\ldots \\rightarrow v_k \\rightarrow v_1$\nof $G$. The removal of $C$ results in two planar subgraphs\n$\\inGraph{C}$ and $\\outGraph{C}$ of $G$ that are the components of\n$G-C$ that lie in the interior and exterior of $C$ in $G$, resp.\nNote that $\\inGraph{C}$ and $\\outGraph{C}$ are not necessarily\nconnected. Let $\\inGraphInc{C}$ ($\\outGraphInc{C}$, resp.) be the\nsubgraph of $G$ induced by $C$ and $\\inGraph{C}$ ($\\outGraph{C}$,\nresp.). For the recursive step, we assume the following invariant\nproperties:\n\n\\begin{enumerate}[{I}P-1:]\n\\item \\label{ip:1} The order of the vertices of $\\outGraphInc{C}$ along the spine $\\ell$ is fixed and the page in which each edge of $\\outGraphInc{C}$ is drawn (i.e., top or bottom) is determined s.t. the book embedding of $\\outGraphInc{C}$ is planar. In other words, we assume that we have already produced a two-page book embedding for $\\outGraphInc{C}$, in which no edge crosses the spine.\n\n\\item \\label{ip:extra} The combinatorial embedding of $\\outGraphInc{C}$ is consistent with a given planar combinatorial embedding of $G$.\n\n\\item \\label{ip:2} The vertices of $C$ occupy consecutive positions along $\\ell$, s.t. $v_1$ ($v_k$, resp.) is the leftmost (rightmost, resp.) along $\\ell$. Moreover, all edges of $C$ are on the same page, except for the one that connects $v_1$ and $v_k$. Say w.l.o.g. that $(v_1,v_k)$ is on the top-page (or \\emph{top-drawn}), while the remaining edges of $C$, namely edges $(v_i,v_{i+1})$ for $1\\leq i