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+{"text":"\\section{Introduction and Background}\n\nSince their first prediction about four decades ago, the interest in axions has been continually growing \\cite{review,amoreview,casper,irastorza}. Light axions or axion-like-particles with a mass in the $10^{-6} $~eV $< m<$ $10^{-2} $~eV range form a compelling candidate for the dark matter in the universe. The existence of axions would also solve one of the longstanding theoretical problems in the standard model of particle physics; the so-called strong CP problem \\cite{wilczek,pq1,pq2}. Not surprisingly, there has been a large number of experimental efforts to detect this elusive particle. In this paper, we will be focusing on experimental detection schemes that rely on the interaction of the axions with electromagnetic waves. The theoretical proposals exploring axion-photon coupling date back to 80's \\cite{sikivie,krauss,maiani}, spurring much experimental work. ADMX (Axion Dark Matter eXperiment) and related experiments  \\cite{admx1,admx2,admx3,admx4,admx5} rely on converting low-energy axions that form the background dark matter to microwave photons using an intense DC magnetic field. The detection sensitivity of the microwave photons can be enhanced by many orders of magnitude using microwave cavities with a high quality factor (high-$Q$). Solar helioscopes such as CAST (Cern Axion Solar Telescope) \\cite{cast1,cast2} and IAXO (International AXion Observatory) \\cite{iaxo} use a similar mechanism, but instead search for the conversion of high energy solar axions into x-ray photons. The strategy in these experiments is to search for a change in the x-ray flux as the detector points to and away from the sun, or as earth rotates around the sun. Detecting axionic dark matter using possible resonance effects in superconducting Josephson junctions has also been suggested \\cite{josephson1,josephson2}. \n\nThe experiments mentioned above aim to detect axions that are naturally present in the environment. Another set of experiments work towards generating and detecting axions in the lab, and have greater control of their experimental parameters since they do not rely on an external source of axions. One idea is to produce axions using the interaction of an intense laser inside a Fabry-Perot cavity with a DC magnetic field, and detect the generated axions at a different location by back-converting them into photons. This set of experiments are cordially referred to as {\\it light shining through a wall (LSW)}, and their sensitivity has been steadily increasing over the last few decades \\cite{lsw1,lsw_new}. Other experiments along similar lines have aimed to observe a change in the phase, intensity, or the polarization of a laser, as the beam propagates through a region of space and generates axions through its interaction with a strong DC magnetic field \\cite{lsw2,lsw_new2}. Currently, these laboratory-based experiments do not have detection sensitivities comparable to microwave cavity experiments. Detecting axion-like particles using interaction of multiple high-peak-power pulsed laser beams in vacuum has also been discussed \\cite{mendonca,dobrich,homma1,homma2,nobuhiro}. \n\nIn this letter, we suggest a laser-based scheme that belongs to this second group of experiments (i.e., generating and detecting axions in the lab), but also has quite high sensitivity over a broad axion mass range. As we discuss below, the scheme relies on using laser-based four-wave mixing, which is mediated by the hypothetical axion field. Intense pump and Stokes laser beams with appropriately chosen frequencies resonantly drive axion generation. The central idea of our technique is that with the pump and Stokes lasers confined to a waveguide, for example an optical fiber, the generated axions can also be guided. The spatial profiles of the driving laser beams confine the axion generation, producing propagating axion modes, which we refer to as axitons.  The physics of confinement for axitons is similar to electromagnetic modes of a fiber; instead of the radial refractive index profile, the confinement relies on the electric and magnetic beam profiles of the driving pump and Stokes lasers. These confined axitons can then couple and mix with another laser (which we term as the mixing beam), affecting the propagation of a probe laser. In the spirit of LSW experiments, the detection of the axion using mixing and probe lasers can be accomplished in a separate fiber. We predict that our scheme will be able to detect axions with a sensitivity at the level of $10^{-12}$~GeV$^{-1}$ for the axion-photon coupling constant over an axion mass range $10^{-6} $~eV $< m<$ $10^{-2} $~eV. \n\nCompared to traditional LSW experiments, there are four unique advantages of our scheme: (1) The mass range of the hypothetical axions can be scanned by appropriately tuning the frequencies of the pump and the Stokes laser beams. (2) Because we utilize guided modes of a fiber, the interaction length can be as much as hundreds of kilometers, which is much longer than what can be achieved using other approaches. (3) We point out the use of near-zero refractive index for the probe laser beam, which may possibly be achieved using an engineered (metamaterial) waveguide, to further enhance the detection sensitivity \\cite{caspani,liberal,lawson}. To our knowledge, this is the first suggestion of using metamaterials for axion detection. (4) The scheme is purely laser based and does not rely on interaction using a Tesla-level DC magnetic field. These four advantages may prove to be critical in generating and detecting this elusive particle in the near future. \n  \n\\section{Formalism}\n\nWe proceed with a detailed description of our scheme. Were it to exist, a scalar axion field $\\phi (\\vec{r},t)$  would interact with electric and magnetic fields of electromagnetism through the Lagrangian \\cite{gasperini,visinelli}:\n\\begin{eqnarray}\n\\Delta \\mathcal{L} = \\frac{g_{a \\gamma \\gamma}} {\\mu_0c} \\phi \\vec{E}  \\cdot \\vec{B} \\quad , \n\\end{eqnarray}\n\n\\noindent where the quantity $g_{a \\gamma \\gamma}$ is the axion-EM coupling constant. The factor $1\/(\\mu_0 c)$ assures that the product $g_{a \\gamma \\gamma} \\phi$ is dimensionless.  In order to make the physics more explicit, we will use mks units in our analysis (except when we present the bounds on the coupling constant  $g_{a \\gamma \\gamma} $  in unit of GeV$^{-1}$, which is the tradition in the discipline). With the Lagrangian of Eq.~(1), the modified Maxwell's equations in a medium with current density $\\vec{J}$ can be found through the Euler-Lagrange equations and they are:\n\\begin{eqnarray}\n\\nabla \\cdot \\vec{E} & = & - g_{a \\gamma \\gamma} c \\nabla \\phi \\cdot \\vec{B} \\quad , \\nonumber \\\\\n\\nabla \\cdot \\vec{B} & = &  0 \\quad , \\nonumber \\\\\n\\nabla \\times \\vec{E} & = & -\\frac{ \\partial \\vec{B}}{\\partial t}  \\quad , \\nonumber \\\\\n\\nabla \\times \\vec{B} & = &  \\frac{1}{c^2} \\frac{ \\partial \\vec{E}}{\\partial t}  + \\mu_0 \\vec{J} + \\frac{g_{a \\gamma \\gamma}}{c} \\left( \\frac{\\partial \\phi}{\\partial t} \\vec{B} + \\nabla \\phi \\times \\vec{E} \\right) \\quad . \n\\end{eqnarray}\n\n\\noindent The interaction with the axion field produces additional charge and current densities, and modifies the Maxwell's equations for $\\nabla \\cdot \\vec{E}$ and $\\nabla \\times \\vec{B}$, while leaving the other two equations unchanged. As a result of the electromagnetic interaction, the Klein-Gordon equation for the axion field  $\\phi (\\vec{r},t)$  is also modified with a driving term that involves the dot product of the electric and magnetic fields:\n\\begin{eqnarray}\n \\nabla^2 \\phi - \\frac{1}{c^2} \\frac{ \\partial^2 \\phi}{\\partial t^2}- \\left( \\frac{m c} {\\hbar} \\right)^2  \\phi =  \\frac{g_{a \\gamma \\gamma}} {\\mu_0 c} \\vec{E} \\cdot \\vec{B} \\quad .\n\\end{eqnarray}\n\n\\noindent Here $m$ is the mass of the hypothetical axion and $\\nabla^2$ is the Laplacian operator. The $ \\vec{E} \\cdot \\vec{B} $ driving term produces an effective second-order nonlinearity for the vacuum and in our scheme we borrow many ideas from nonlinear optics, specifically from four-wave mixing processes \\cite{boyd}. \n\n\n\\begin{figure}[tbh]\n\\vspace{-0cm}\n\\begin{center}\n\\includegraphics[width=15cm]{axion_generation_scheme.pdf}\n\\vspace{-0.5cm} \n\\caption{\\label{eit_scheme} \\small   Energy level diagram and simplified schematic for producing guided axitons. Pump and Stokes laser beams whose frequency difference, $\\omega_P - \\omega_S$, is tuned close to the rest mass energy of the axion, resonantly drive axion generation. The two lasers are confined to an optical fiber and the solid curves are cartoon schematics for the radial profiles of the two lasers. The spatial profiles of the two beams then confine axion generation, producing guided axitons, which are shown in dashed curve.}\n\\end{center}\n\\vspace{-0cm}\n\\end{figure}\n\n\n\\section{Guided axion waves: axitons}\n\nWe first discuss the generation of axions that are confined by the spatial profiles of the pump and Stokes laser beams through the Klein-Gordon equation of Eq.~(3). Figure~1 shows the relevant energy level diagram and cartoon schematic for producing guided axion waves, which we refer to as axitons. We consider pump and Stokes laser beams propagating through a wave-guide, for example, an optical fiber. Through the $ \\vec{E} \\cdot \\vec{B} $ term, the electric field of the pump  and the magnetic field of the Stokes will drive the axion excitation. We take the fiber to be cylindrically symmetric and assume the pump and Stokes lasers to be in specific modes of the fiber. We work in cylindrical coordinates $(r, \\varphi, z)$, and take the two lasers to be of the form:\n\\begin{eqnarray}\n\\vec{E}_P (r, \\varphi, z) & = & E_P u_P(r) \\exp{ \\left(i l_P \\varphi \\right)} \\exp{ \\left( i \\beta_P z - i \\omega_P t \\right) } \\hat{e} + c. c.  \\quad , \\nonumber \\\\\n\\vec{B}_S (r, \\varphi, z) & = & B_S u_S(r) \\exp{ \\left(i l_S \\varphi \\right)} \\exp{ \\left( i \\beta_S z - i \\omega_S t \\right) } \\hat{e} + c. c.  \\quad . \n\\end{eqnarray}\n\n\\noindent Here $c. c.$ refers to the complex conjugate of the whole expression that is written before. $E_P$ and $B_S$ are the electric and magnetic field amplitudes for the pump and Stokes laser beams, and $\\hat{e}$ denotes the common polarization direction for the two vectors (which is any direction orthogonal to the propagation direction $\\hat{z}$). The quantities $u_P(r)$ and $u_S(r)$ are the radial mode functions of the corresponding lasers, and the integers $l_P$ and $l_S$ are typically referred to as orbital angular momentum numbers for the associated photons \\cite{boyd}. $\\beta_P$ and $\\beta_S$ are the propagation constants of the modes along the fiber $z$ axis (i. e., the longitudinal $k$-vector) and the quantities $\\omega_P$ and $\\omega_S$ are the angular frequencies. Noting the energy level diagram of Fig.~1, the frequency difference of the pump and Stokes laser beams, $\\omega_P-\\omega_S$ is tuned close to the rest-mass energy of the hypothetical axion. By tuning this frequency difference, axions at different masses can be searched for. In general, there is another contribution to the generation of the axion wave, which is driven by the magnetic field of the pump beam and the electric field of the Stokes laser. For plane-wave like lasers propagating inside a bulk material, this second contribution (which is proportional to $B_P E_S^*$), would interfere destructively with the main contribution that we consider below (which is proportional to $E_P B_S^*$), reducing the produced axion amplitude. Here, we take that the relevant modes of the fiber are chosen appropriately so that this second contribution can be ignored compared to the first one. For example, one could use a TE (transverse electric) mode for the pump, whereas a TM (transverse magnetic) mode for the Stokes laser. This way, the vectors $E_P$ (electric field of the pump) and $B_S$ (magnetic field of the Stokes) can be aligned, maximizing the dot product. However, an angle would be present between $B_P$ and $E_S$, which can be tuned minimizing their dot product using specific choice of the modes.  \n\nBecause the axion generation is driven by the  $\\vec{E} \\cdot \\vec{B} $ term, we are interested in the axion field solutions of the form:\n\\begin{eqnarray}\n\\phi (r, \\varphi, z)  =  u_{\\phi} (r) \\exp{ \\left[i (l_P -l_S) \\varphi \\right]} \\exp{ \\left[ i (\\beta_P- \\beta_S) z - i (\\omega_P-\\omega_S) t \\right] } + c. c.  \\quad . \n\\end{eqnarray}\n\n\\noindent If the ansatz solution of the form Eq.~(5) exists, it would indicate confined axions propagating without any change in their profile along the fiber. Using Eqs.~(4) and (5), the Klein-Gordon equation for the axion field can be reduced to a single ordinary differential equation for the radial axion mode function $u_{\\phi}(r)$:\n\\begin{eqnarray}\n\\frac{d^2 u_{\\phi}}{d r^2} + \\frac{1}{r} \\frac{d u_{\\phi}}{d r} - \\frac{(l_P - l_S)^2}{r^2} u_{\\phi}  + \\left[ \\frac{(\\omega_P - \\omega_S)^2} {c^2} - (\\beta_P - \\beta_S)^2 - \\left(\\frac{m c }{\\hbar} \\right)^2 \\right] u_{\\phi} = \\frac{g_{a \\gamma \\gamma}} {\\mu_0c} E_P B_S^* u_P(r) u_S^*(r) \\quad . \\nonumber \\\\\n\\end{eqnarray}\n\n\\noindent We next define the quantity $ \\Delta k ^2 \\equiv \\frac{(\\omega_P - \\omega_S)^2} {c^2} - (\\beta_P - \\beta_S)^2 - \\left(\\frac{m c }{\\hbar} \\right)^2$, which can be thought of as an ``energy detuning\": i.e., the detuning (difference) between the pump and Stokes laser energy difference (i.e., $\\omega_P -\\omega_S$) and the total energy (kinetic  + rest mass) of the excited axions. To elucidate the physics, we normalize the above equation through the definitions $\\tilde{r} \\equiv \\kappa_{axion} r $  and $ \\tilde{ \\Delta k } \\equiv \\Delta k \/ \\kappa_{axion}$. Here, the quantity $\\kappa_{axion}$ is the $k$-vector of the axion particles if their kinetic energy equaled their rest-mass energy, $\\kappa_{axion} = m c \/ \\hbar$. With these definitions, Eq.~(6) reads:\n\\begin{eqnarray}\n\\frac{d^2 u_{\\phi}}{d \\tilde{r}^2} + \\frac{1}{\\tilde{r}} \\frac{d u_{\\phi}}{d \\tilde{r}} - \\frac{(l_P - l_S)^2}{\\tilde{r}^2} u_{\\phi}  + \\tilde{ \\Delta k}^2 u_{\\phi} = \\frac{1}{\\kappa_{axion}^2} \\frac{g_{a \\gamma \\gamma}} {\\mu_0c} E_P B_S^* u_P(\\tilde{r}) u_S^*(\\tilde{r}) \\quad . \n\\end{eqnarray}\n\n\\noindent Given the parameters of the system, Eq.~(7) can now be numerically integrated to find the radial profile of the axion excitation, $u_{\\phi} (\\tilde{r})$. We look for physical solutions where the axion field is a maximum at $ \\tilde{r} =0$, which gives the following initial condition for the first derivative:  $ \\frac{d u_{\\phi}}{d \\tilde{r}} (\\tilde{r} = 0 ) =0$. We then numerically integrate Eq.~(7), with a trial initial condition $ u_{\\phi} (\\tilde{r} = 0) $. For a given initial condition, the integration will typically not result in a bounded axion field, i.e., $u_{\\phi}  (\\tilde{r} \\rightarrow \\infty ) \\neq 0 $, which is not physical. Given the parameters for the system, we vary the initial condition $ u_{\\phi} (\\tilde{r} = 0) $ until a bounded solution with $u_{\\phi}  (\\tilde{r} \\rightarrow \\infty ) = 0$ is found. \n\n\\begin{figure}[tbh]\n\\vspace{-0cm}\n\\begin{center}\n\\includegraphics[width=15cm]{axiton_mode_profiles.pdf}\n\\vspace{-0.5cm} \n\\caption{\\label{eit_scheme} \\small   Numerically calculated normalized axiton profiles, $u_{\\phi}  (\\tilde{r})$, for $ \\tilde{ \\Delta k }^ 2 = -1 $ (solid black),   $ \\tilde{ \\Delta k }^ 2 = -0.1 $ (solid red), and $ \\tilde{ \\Delta k }^ 2 = -0.01 $ (solid green), respectively. For comparison, the dashed blue line shows the mode profiles for the driving laser beams, $ u_P (\\tilde{r}) $ and $ u_S (\\tilde{r}) $. For a bounded physical solution such that $u_{\\phi}  (\\tilde{r} \\rightarrow \\infty ) = 0$, we require  $ \\tilde{ \\Delta k }^ 2 <0 $. As the quantity  $ \\tilde{ \\Delta k }^ 2 $ gets closer to 0, the axion mode profile gets broader and extends significantly beyond the confinement of the driving lasers.  }\n\\end{center}\n\\vspace{-0.5cm}\n\\end{figure}\n\n\nFigure 2 shows numerically calculated normalized axiton profiles, $u_{\\phi}  (\\tilde{r} )$, for $ \\tilde{ \\Delta k }^ 2 = -1 $ (solid black),   $ \\tilde{ \\Delta k }^ 2 = -0.1 $ (solid red), and $ \\tilde{ \\Delta k }^ 2 = -0.01 $ (solid green), respectively. For comparison, the dashed blue line shows the mode profiles for the driving laser beams, $ u_P (\\tilde{r}) $ and $ u_S (\\tilde{r})  $. Here, for simplicity, we take the profiles for the pump and Stokes laser beams to be Gaussian with unity width, $ u_P (\\tilde{r}) = u_S^* (\\tilde{r}) = \\exp{(-\\tilde{r}^2})$. We also take the angular momentum numbers for the pump and Stokes fields to be the same, $l_P=l_S$.  Due to well-known Bessel function solutions to differential equations of the from Eq.~(7), for a physical bounded solution such that $u_{\\phi}  (\\tilde{r} \\rightarrow \\infty ) = 0$, we require  $ \\tilde{ \\Delta k }^ 2 <0 $. As the quantity  $ \\tilde{ \\Delta k }^ 2 $ gets closer to 0, the axiton radial mode profile gets broader and extends significantly beyond the confinement of the driving lasers. This is well-illustrated in the solid green curve in Fig.~2. The quantity $ \\tilde{ \\Delta k }^ 2$  will likely be important parameter to tune in future experiments, since the broader the axion profile, the more it will leak to the second detection fiber as we will discuss in the detection scheme below. The numerically found initial amplitudes for the axitons, each multiplied by the scaling factor in the right hand side of Eq.~(7) (i.e., $\\frac{1}{\\kappa_{axion}^2} \\frac{g_{a \\gamma \\gamma}} {\\mu_0c} E_P B_S^*$) are, $ u_{\\phi} (\\tilde{r} = 0)= 0.23$ (for  $ \\tilde{ \\Delta k }^ 2 = -1 $), $ u_{\\phi} (\\tilde{r} = 0)= 0.48$ (for  $ \\tilde{ \\Delta k }^ 2 = -0.1 $), and $ u_{\\phi} (\\tilde{r} = 0)= 0.76$ (for  $ \\tilde{ \\Delta k }^ 2 = -0.01 $), respectively. Once the radial mode profile for the axion field is found numerically as shown in Fig.~2, the solution of Eq.~(5) gives the full description of the axiton mode. \n\n\\section{Detection of axions using the guided-wave geometry}\n\nWe have so far focused on generation of confined axitons that propagate along the fiber. Now, we discuss the second half of the problem, namely how to detect these guided axitons and prove their existence. The vision that we have for a future experiment is shown in Fig. 3. Inspired by the LSW experiments, we use  a separate fiber, which we refer to as the detection fiber. This is necessary, since any material will posses a four-wave mixing nonlinearity, which would completely overwhelm the four-wave mixing interaction mediated by the axion field. We, therefore, need to make sure that the four involved laser beams do not spatially overlap. The central idea in Fig.~3 is that the axiton mode produced in the generation fiber overlaps with the detection fiber, while the pump and Stokes laser beam profiles do not. If necessary, the extinction of the pump and Stokes lasers at the detection fiber can be guaranteed by putting a metal shield between the two fibers. In the detection fiber, the axion field mixes with the magnetic field of another laser that we refer to as the mixing laser. Through the axion interaction, the mixing laser then affects the propagation (both phase and intensity) of a probe laser beam. The search for the axion relies on detecting this change on the probe laser. \n\n\\begin{figure}[tbh]\n\\vspace{-0cm}\n\\begin{center}\n\\includegraphics[width=15cm]{axion_generation_and_detection_scheme.pdf}\n\\vspace{-0.5cm} \n\\caption{\\label{eit_scheme} \\small  The left panel shows the energy level diagram for the four-wave mixing scheme for generating and detecting the axions. The axion field, $\\phi$, produced by the pump and Stokes laser beams mix with the mixing laser, affecting the propagation of the probe laser at frequency $\\omega_0$. The four-wave mixing interaction forms a closed loop: $\\omega_P-\\omega_S+\\omega_M = \\omega_0$. The mixing and the probe lasers propagate along a separate fiber, which we refer to as the detection fiber. Inspired by the LSW experiments, the axion field produced in the generation fiber (by the pump and Stokes lasers), overlaps with the detection fiber and mediates the interaction between the mixing and probe lasers.  }\n\\end{center}\n\\vspace{-0cm}\n\\end{figure}\n\n\nSimilar to above, we will now take mixing and probe lasers to be modes of the detection fiber and assume the following forms for the two waves:\n\\begin{eqnarray}\n\\vec{E}_{probe} (r, \\varphi, z) & = & E_0(z) u_0(r) \\exp{ \\left(i l_0 \\varphi \\right)} \\exp{ \\left( i \\beta_0 z - i \\omega_0 t \\right) } \\hat{e} + c. c.  \\quad , \\nonumber \\\\\n\\vec{B}_M (r, \\varphi, z) & = & B_M u_M(r) \\exp{ \\left(i l_M \\varphi \\right)} \\exp{ \\left( i \\beta_M z - i \\omega_M t \\right) } \\hat{e} + c. c.  \\quad . \n\\end{eqnarray}\n\nHere, the quantity $B_M$ is the magnetic field amplitude for the mixing laser beam. To allow for a change in the phase and the intensity of the probe wave, we have explicitly made its electric field amplitude to be a function of $z$, $E_0(z)$. Without the axion interaction, the amplitude for the probe mode would be independent of distance, i.e., $E_0(z)=E_0(z=0)$. This amplitude gets modified due to axion interaction and the basic idea of the scheme is to infer the existence of axions using this amplitude modification. In above, the quantities $u_0(r)$ and $u_M(r)$ are the radial mode functions of the corresponding lasers (now in the detection fiber) and the integers $l_0$ and $l_M$ are the corresponding angular momentum numbers. $\\beta_0$ and $\\beta_M$ are the propagation constants of the modes along the fiber $z$ axis  and the quantities $\\omega_0$ and $\\omega_M$ are the angular frequencies. \n\nWe next solve for the propagation of the probe laser field in the presence of the axion field $\\phi$ of Eq.~(5). The detailed procedure is outlined in the Appendix~A. Briefly, we first reduce Maxwell's equations for the probe laser beam [Eq.~(2)] into a single wave equation. We then solve this wave equation in the detection fiber, while taking into account the interaction of the probe laser with the axion field and the mixing laser beam. We make several reasonable simplifications and three key assumptions: (i) {\\it Energy conservation:} we take the frequencies of the interacting laser beams in the four-wave mixing process to form a closed loop: $\\omega_P- \\omega_S + \\omega_M = \\omega_0$. (ii) {\\it Angular momentum conservation:} we take the angular momentum numbers for the four laser beams to form a closed loop as well: $l_P - l_S + l_M = l_0$. (iii) We make the Slowly Varying Envelope Approximation (SVEA) for the probe laser field, $ \\vert \\frac{d E_0} { d z } \\vert << \\beta_0 E_0$. Under these assumptions and simplifications, the propagation of the probe laser electric field amplitude, $E_0(z)$, can be reduced to a single differential equation:\n\\begin{eqnarray}\n 2 i \\beta_0 \\frac{ d E_0} {d z} = \\frac{g_{a \\gamma \\gamma}^2} {c} \\frac{1}{\\kappa_{axion}^2}  (\\omega_P - \\omega_S) \\omega_0  \\left(\\frac{1}{2 \\mu_0} B_M B_S^* \\right) E_P \\exp{[i (\\beta_P - \\beta_S + \\beta_M - \\beta_0) z]} \\quad . \n\\end{eqnarray}\n\n\\noindent Here, the quantity $\\beta_P - \\beta_S + \\beta_M - \\beta_0$ is the phase-mismatch of the four-wave mixing interaction. We next define $\\Delta k _{FWM} \\equiv \\beta_P - \\beta_S + \\beta_M - \\beta_0$ to simplify the notation in Eq.~(9) further. We also define $n_{eff} \\equiv \\frac{\\beta_0} {\\omega_0\/c}$, which can be thought as the effective refractive index for the probe laser mode as it is propagating through the detection fiber. With these definitions, the differential equation that describes the propagation of the probe laser reduces to: \n\\begin{eqnarray}\n\\frac{d E_0}{d z} =  i \\xi E_P  \\exp{(i \\Delta k_{FWM} z)} \\quad , \n\\end{eqnarray} \n\n\\noindent where the quantity $\\xi$ essentially summarizes the whole interaction and is given by:\n\\begin{eqnarray}\n\\xi= g_{a \\gamma \\gamma}^2 \\frac{1}{n_{eff}} \\frac{1}{c}  \\frac{1}{\\kappa_{axion}^2} \\left( \\frac{1}{2 \\mu_0} B_M B_S^* \\right) (\\omega_P - \\omega_S)    \\quad . \n\\end{eqnarray} \n\n\nWe next focus on the ideal phase-matched case where we assume that the mode propagation constants can be adjusted such that $\\Delta k_{FWM} \\rightarrow 0$ (the case of finite $\\Delta k_{FWM}$ is discussed in Appendix~B). In this limit, the probe propagation equation has a particularly simple form and can immediately be solved:\n\\begin{eqnarray}\n\\frac{d E_0}{d z} & = &  i \\xi E_P   \\nonumber \\\\\n\\Rightarrow E_0(L) & = & E_0(0) + i  \\xi E_P L  \\quad . \n\\end{eqnarray} \n\n\\noindent Here, $L$ is the total length of each fiber, which can be viewed as the interaction length. Equation~(12) describes the change in the electric field of the probe laser beam due to the axion interaction. Using this change in the electric field, we can also find the corresponding fractional change in the intensity of the probe laser beam:\n\\begin{eqnarray}\n\\frac{I(L)}{I(0)} = 1 + 2 \\xi L \\frac{\\Im{(E_P)}}{E_0(0)} \\quad .  \n\\end{eqnarray}\n\n\\noindent Here, for simplicity, we have taken the quantities $\\xi$ and $E_0(0)$ to be real. In Eq.~(13), the symbol $\\Im$ stands taking the imaginary part of the quantity inside the brackets. Note that by changing the phase of the pump beam, $E_P$, we can change the sign of the quantity $\\Im{(E_P)}$, and thereby control whether the probe beam will experience absorption or amplification due to axion-mediated four-wave mixing interaction. \n\n\\section{Bounds on the coupling constant}\n\nWe next discuss the bounds on the axion coupling constant that such an experiment can place, given specific experimental conditions. The sensitivity of such an experiment will critically depend on to what precision we can measure the change in the intensity of the probe laser beam. For this purpose, to first order, we assume shot-noise limited detection, with fluctuations of order $\\sqrt{N_{photon}}$, for a total detected number of probe photons of $N_{photon}$. Under these assumptions and simplifications, given an experimental set of parameters, the bound on the square of axion-photon coupling constant from Eq.~(13) is:\n\\begin{eqnarray}\ng_{a \\gamma \\gamma}^2 = \\frac{ n_{eff} \\kappa_{axion} }{2 L  \\left( \\frac{1}{2 \\mu_0} B_M B_S^* \\right) \\sqrt{N_{photon}} \\sqrt{\\frac{P_P}{P_0}}} \\quad .\n\\end{eqnarray}\n\n\\noindent Here, the quantity  $\\frac{1}{2 \\mu_0} B_M B_S^*$ can be thought as the magnetic energy density, $P_P$ is the optical power in the pump laser and $P_0$ is the optical power in the probe beam. The details of deriving Eq.~(14) is given in Appendix~B.  The dependence of the coupling constant on the probe refractive index, $n_{eff}$, is in the spirit of enhanced nonlinearities in near-zero index materials \\cite{caspani,liberal}.  \n\n We next evaluate the bounds on the coupling constant for four envisioned phases of the experiment. The parameters that are used in these four phases are given below, in Table-I. The envisioned parameters for the lasers are well within the current the state of the art of high-power fiber lasers \\cite{laser_review}. We envision that the parameters that are used in the first two phases of the experiment ({\\it phase-1} and {\\it phase-2}) can be achieved in a few year time-scale, while the last two phases of the experiment ({\\it phase-3} and {\\it phase-4}) can be performed within the next 5 to 10 years. There will likely be two main systematics in the experiment: (1) Optical four-wave mixing interaction due to possible residual leakage of the pump and Stokes laser beams to the detection fiber. (2) The change in the intensity of the probe laser beam due to off-resonant Raman scattering in the detection fiber.  To put the parameters that are listed in Table~1 into perspective, the ALPS~II project, which is the next generation LSW experiment currently at construction at DESY, will use a cavity with a finesse of $F=4 \\times 10^4$, and a cavity length of $L=88$~m, with a quite stringent DC magnetic-field of 5.3~T to be maintained within the whole length using dipolar magnets \\cite{review}. The first results from the ALPS~II project is expected in 2022. \n\n\n\\vspace{0.5cm}\n\n\\begin{table}\n\n\\begin{tabular}{ || c | c | c | c | c ||}\n\\hline\nParameter & Phase-1 & Phase-2 & Phase-3 & Phase-4 \\\\\n\\hline\nLength of the fiber ($L$) & $1$~km & $10$~km & $100$~km & $1000$~km \\\\ \n\\hline\nOptical power in probe ($P_0$) & $1$~mW & $10$~mW & $100$~mW & $1$~W  \\\\\n\\hline\nIntegration time ($T$) & $100$~s & $10^3$~s & $10^4$~s & $10^6$~s \\\\\n\\hline\nOptical power in pump ($P_P$) & $1$~W & $10$~W & $100$~W & $10$~kW \\\\\n\\hline\nOptical intensity of Stokes  ($I_S$) & $0.1$~GW\/cm$^2$ & $1$~GW\/cm$^2$ &  $10$~GW\/cm$^2$ &  $100$~GW\/cm$^2$ \\\\\n\\hline \nOptical intensity of mixing ($I_M$) & $0.1$~GW\/cm$^2$ & $1$~GW\/cm$^2$ &  $10$~GW\/cm$^2$ &  $100$~GW\/cm$^2$ \\\\\n\\hline\nThe refractive index for probe ($n_{eff}$) & $1$ & $10^{-1}$ & $10^{-2}$ & $10^{-4}$ \\\\\n\\hline\n\\end{tabular}\n\n\\caption{The set of parameters that are used for the four envisoned phases of the experiment.}\n\n\\end{table}\n\n\\vspace{0.5cm}\n\nThe calculated bounds for the axion-photon coupling for these four envisioned phases of the experiment are shown in Fig.~4. For comparison the predicted sensitivities of the next generation LSW experiment (ALPS~II), as well  the next generation solar helioscope (IAXO) are also plotted (dashed brown lines) \\cite{review}. Solar helioscope IAXO is proposed to utilize 25-m-long, 5.2-m-diameter toroid magnet assembly. The envisioned {\\it phase-4} experiment of our scheme (solid black line) is quite competitive with planned ALPS~II and IAXO \\cite{review}. \n\n\\begin{figure}[tbh]\n\\vspace{0cm}\n\\begin{center}\n\\includegraphics[width=19cm]{bounds_for_four_phases.pdf}\n\\vspace{-1cm} \n\\caption{\\label{eit_scheme} \\small  The calculated detection sensitivity for the axion-photon coupling constant for four different phases of the experiment. The {\\it phase-4} experiment is quite competitive with several planned experiments, such as the next generation LSW experiment (ALPS~II) and the next generation solar helioscope (IAXO).   }\n\\end{center}\n\\vspace{-0cm}\n\\end{figure}\n\n\n\\section{Conclusions and future directions}\n\nIn summary, we have suggested a new approach for generating and detecting axions using lasers in a guided wave geometry. As we mentioned above, there are important advantages of our scheme including the ability to scan a wide mass-range for the axion, and the long interaction length. Furthermore, our scheme does not rely on interaction of the axions with a Tesla level DC magnetic field. The calculations of Fig.~4 predict that our scheme can achieve sensitivities quite competitive to several planned experiments in the near future. While the parameters that are used for the {\\it phase-4} calculation in Fig.~4 are stringent, they are within the state of the art. \n\nWe believe there are several promising avenues for future research which may result in significant enhancement of the detection sensitivity in our scheme: (1) The interaction length can further be enhanced by integrating a high-finesse cavity with the detection and generation fibers. Using a distributed Bragg reflector at each end of the fiber (which may be achieved by modulating the refractive index over a section near the beginning and the end each fiber), the lasers can be made to bounce back and forth along the fiber, thereby significantly increasing the interaction length. For this case, the sensitivity bounds for the axion-photon coupling constant $g_{a \\gamma \\gamma}$ would reduce by a factor $\\sqrt{F}$, where $F$ is the cavity finesse. (2) For the calculation of Fig.~4, we assumed shot noise limited detection for the probe laser beam. If squeezed light for the probe laser beam is used, the fluctuations in either the phase or the intensity of the probe laser can be reduced substantially below the shot-noise limit, thereby resulting in improved sensitivity for axion detection. (3) To our knowledge, our work is the first to suggest using a low refractive index for the probe laser beam to increase the detection sensitivity. It may also be possible to utilize metamaterials in other ways to increase the detection sensitivity. For example, waveguides with a high effective magnetic susceptibility may be used to increase the magnetic field values for the Stokes and Mixing lasers [i.e., it may be possible to enhance the values of the quantities $B_S$ and $B_M$ in Eq.~(14)] \\cite{caspani,liberal}. \n\n\nWe thank Ben Lemberger for many helpful discussions. This work was supported by the University of Wisconsin-Madison through the Vilas Associates Award. \n\n\n\n\\newpage\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThere has been significant interest, especially in recent years, to develop\nspatially periodic band gap materials and structures, based upon Floquet-Bloch\ntheory \\citep{floquet1883, bloch1929}. Recent developments in the field of\narchitectured materials aimed at achieving novel mechanical properties often\nrely on enhancements that include effects neglected by classical theories.\nContinuum models with local microstructural interactions have become\nincreasingly popular after the advance and growth in the field of metamaterials,\nas summarized in the monograph by  \\cite{book:banerjee_metamaterials}. A family\nof models that has regained popularity in the last few years is the so-called\nCosserat-based theories, which are mainly founded on the formulation by\n\\cite{Cosserat1909}. In a wide sense, these material models consider\nmicrostructural effects through a generalization of Cauchy's postulate to\ninclude additional mechanical interactions involving couples per unit surface\nor couple-stresses. In the present work, we focus on a pure continuum mechanics\nrepresentation by widening the modeling capabilities of consistent-couple stress\ntheory (C-CST), originally formulated in \\cite{hadjesfandiari2011couple}. In\nparticular, we establish for the first time a principle of stationary correlated\naction for the corresponding reduced wave equation of elastodynamics and extend\nthe theory to spatially periodic materials, thus providing an objective physical\nbasis to characterize material through its dispersive behaviour.\n\nThe entire family of Cosserat elasticity models depart from the classical Cauchy\nmodels in the consideration of microstructural effects, which are unavoidably\nexpected to occur once the specimen dimensions become comparable to the material\nmicrostructural features. These effects cannot be addressed in classical\ntheories. On the other hand, microstructural effects are introduced through the\nextension of dynamic and kinematic descriptors from classical continuum mechanics\non a range of alternative models. \\cite{Voigt1910} was probably the first to\npostulate a model with asymmetric mechanical interactions in terms of\ncouple-stresses: the interaction between two material points in this continuum\nencompassed couples per unit contact surface in addition to the classical Cauchy\nforces per unit surface. In a landmark contribution, the Cosserat brothers\n\\citep{Cosserat1909} formulated a mathematical theory involving couple-stresses\nin which new kinematic variables were introduced in the form independent\nmicro-rotations. Various later extensions from these theories were also\ndeveloped by \\cite{Eringen1966, book:nowacki1986, Mindlin1964,\neringen1964nonlinear} in micropolar, microstretch and micromorphic theories. \nAlongside, a different branch of developments resulted in a set of couple stress\ntheories in the work by \\cite{Toupin1962}, \\cite{MindlinAndTiersten1962} and\n\\cite{Koiter1964}, who used the gradients of the true continuum rotation field\nto provide the required kinematic enrichment.\n\nDevelopments from \\cite{hadjesfandiari2011couple} have resulted in a\nconsistent version of the models by \\cite{Toupin1962},\n\\cite{MindlinAndTiersten1962} and \\cite{Koiter1964} in terms of couple-stresses.\nThe consistency of this model is reflected in the determinacy of all the\nforce-stress and couple-stress components, the identification of the necessary\nand sufficient set of natural and essential boundary conditions and the\nelimination of redundant force components. An approach for evaluating the\nusefulness and robustness of a continuum mechanics model is through the\ndetermination of its band structure in terms of its dispersion relationships.\nThese indicate the kinematic response of the material through an identification\nof the wave propagation modes that can exist within the model and the frequency\ndependency of the group and phase velocities of these potential waves. An\neffective technique, relying on the assumption of spatial periodicity, is based\non Bloch's theorem from solid state physics \\citep{book:brillouin}, where the\nproblem of finding the band structure reduces to solving a series of generalized\neigenvalue problems for a variation of the wave vector in the reciprocal space.\nIn the case of the C-CST model, this problem poses several computational\nchallenges. First, since the enriched kinematic variables are now curvatures,\ncorresponding to particular second order gradients of the displacement field,\nthe displacement-based finite element formulation now would require \\(C^1\\)\ninterelement continuity. As shown by \\cite{darrall2014}, this numerical issue\ncan be resolved by introducing Lagrange multiplier techniques, however it is\nnot obvious how to incorporate these within Bloch analysis. Second, as a result\nof enforcing the kinematic constraint in terms of Lagrange multipliers, the\ncomputational framework lacks inertial components associated with the rotational\ninteractions. Since there is only a mass matrix associated with the\ntranslational degrees of freedom, special attention is needed in solving the\neigenproblem. Both of these issues are resolved in the present\nwork.\n\nThe characterization of the bulk properties of periodic materials is commonly\ndone finding the band structure or dispersion relations \\citep{hussein2014dynamics}.\nCommonly, this band structure is obtained using a numerical method such as \nthe Boundary Element Method \\citep{li2013bandgap, li2013boundary}, the Finite\nDifference Method \\citep{tanaka2000band, su2010postprocessing,\nisakari2016periodic}, the Finite Element Method \\citep{langlet95, guarin2015, \nvalencia_uel_2019, mazzotti2019modeling, guarin2020,chin2021spectral},  or\nthe Plane Wave Expansions \\citep{cao2004convergence, xie2017improved,\ndal2020elastic}. We favor the use of the Finite Element Method because of its\nmaturity and versatily to represent arbitrary geometries and boundary\nconditions. In this work, we find the dispersion relations modeling\na single unit cell of the material and using Bloch's theorem. There have been\nfew works on periodic materials involving generalized continua and these have\nbeen related to micropolar elasticity \\citep{zhang2018, guarin2020}. To the best\nof our knowledge, this is the first work using a higher-order elasticity model\nfor phononic crystals.\n\nHere we establish a new variational principle in the temporal frequency domain\nfor reduced couple stress elastodynamics and then extend the finite element\nalgorithm from \\cite{darrall2014} to the case of spatially periodic material\ncells with Bloch boundary conditions. We examine first the closed form\ndispersion relationships for the homogeneous version of the model. This\nhomogeneous model already involves micromechanical effects through a length\nscale material parameter, however additional effects can be considered in terms\nof explicit representations of geometric features at the fundamental material\ncell level. We then formulate a variational statement together with the\nimposition of an extended version of the usual Bloch periodic boundary\nconditions that satisfies Hermiticity and positive definiteness for C-CST.\nSubsequently, this statement is modified by introducing an artificial\nindependent rotation field tied to the continuum displacement field through the\nenforcement of a Lagrange multiplier field that is shown to equal the\nskew-symmetric part of the force-stresses. The resulting numerical framework is\ntested by comparing its results with those obtained in closed form\nfor the homogeneous case and by applying it to a porous periodic material cell\ndesign, which displays interesting bandgap behavior that has\nnot been resported previously.\n\n\n\\section{Governing equations}\n\n\\subsection{Forces and moments in the C-CST solid}\nThe fundamental signature of the extended continuum model considered in this\nwork is the presence of rotational mechanical interaction, in addition to the\nclassical translational interaction between material points in the continuum.\nFollowing a generalized Cauchy's postulate \\citep{MindlinAndTiersten1962,\nKoiter1964} we define force and couple traction vectors \\(t_i^{(\\hat{n})}\\)\nand \\(m_i^{(\\hat{n})}\\) respectively as\n\\begin{subequations}\n\\label{eq:tracts}\n\\begin{align}\n        &t_i^{(\\hat{n})} = \\lim_{\\Delta S(\\hat{n}) \\rightarrow 0} \\frac{\\Delta \n        R_i}{\\Delta S(\\hat{n})}\\label{eq:for_T} \\\\\n        &m_i^{(\\hat{n})} = \\lim_{\\Delta S(\\hat{n}) \\rightarrow 0} \\frac{\\Delta \n        M_i}{\\Delta S(\\hat{n})} \\, , \\label{eq:for_M}       \n\\end{align}\n\\end{subequations}\nand where \\(\\Delta S(\\hat{n})\\) is a small element of area oriented with unit\nnormal \\(\\hat{n}\\) while \\(\\Delta R_i\\) and \\(\\Delta M_i\\) are the resultant\nforce and couple moment, respectively. However, only the tangential components of\n\\(m_i^{(\\hat{n})}\\) exist as independent bending couple tractions\n(\\cref{fig:cauchy_principle}).\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=5 cm]{img\/cauchy_principle}\n\\caption{In the C-CST model mechanical effects are described through linear and\nrotational interactions in terms of resultant forces and moments at the material\npoint. These resultants act on the surface element \\(\\Delta S\\) over a plane\nwhose unit outward normal is \\(\\hat{n}\\).}\n\\label{fig:cauchy_principle}\n\\end{figure}\n\nNote that while the force-tractions vector \\(t_i^{(\\hat{n})}\\) is a polar vector,\nthe couple-tractions vector \\(m_i^{(\\hat{n})}\\) is an axial vector.\nForce-tractions and couple-tractions are also described by projections of the\nnon-symmetric force-stress tensor \\(\\sigma_{ij}\\) and the couple-stress\n\\(\\mu_{ij}\\) tensors according to:\n\\begin{subequations}\n\\label{eq:tensions}\n\\begin{align}\n        &t_i^{(\\hat{n})} = \\sigma_{ji} n_j\\, ,\\label{eq:for_sigma}\\\\\n        &m_i^{(\\hat{n})} = \\mu_{ji} n_j=\\epsilon_{ijk}\\mu_k n_j\\, , \\label{eq:cou_sigma}       \n\\end{align}\n\\end{subequations}\nwhere \\(\\mu_{ij}\\) is skew-symmetric. Thus, \\(\\mu_{ij} = -\\mu_{ji}\\) and the\ncouple-stress tensor can be written as a polar vector with\n\\[\\mu_k = \\frac{1}{2}\\epsilon_{kji}\\mu_{ji}\\, ,\\]\nwhere \\(\\epsilon_{ijk}\\) is the Levi-Civita permutation symbol leading to the\nlast form in \\eqref{eq:cou_sigma}, which clearly shows that \\(m_i^{(\\hat{n})}\\)\nis tangential to the surface.\n\nConsideration of the linear and angular balance equations for an arbitrary part\nof the material continuum of volume \\(V\\), bounded by external surface \\(S\\)\nleads to the following force-stress and couple-stress equilibrium equations for\nthe C-CST model:\n\\begin{equation}\n\\begin{split}\n&\\sigma_{ji,j} + f_i = \\rho \\ddot{u}_i\\, ,\\\\\n&\\mu_{ji,j} + \\epsilon_{ijk}\\sigma_{jk} = 0\\, ,\n\\end{split}\n\\label{eq:balance_eqs}\n\\end{equation}\nwhere \\(f_i\\) are forces per unit volume, and \\(\\rho\\) is the mass density.\nNotice that, in contrast to micropolar models \\citep{guarin2020} where there is\na rotational inertial density and a body couple term, in this model the balance\nequations already include those contributions. This particular aspect of the\nC-CST model is discussed in the original paper by \\cite{hadjesfandiari2011couple}\nwhere it is also proved that from\n\\begin{equation}\n\\epsilon_{ijk}(\\mu_{k,j} + \\sigma_{jk}) = 0\\, ,\n\\label{eq:skew_symmetry}\n\\end{equation}\nit follows that \\(\\mu_{k,j} + \\sigma_{jk}\\) is symmetric and as a result its\nskew-symmetric part is zero leading to\n\\[\\sigma_{[ji]} = - \\mu_{[i,j]}\\, .\\]\nThis gives the skew-symmetric part of the force-stress tensor in terms of the\ncouple-stress vector, which also can be described by its dual vector\nrepresentation\n\\[s_i = \\frac{1}{2} \\epsilon_{ijk} \\mu_{k, j}\\, .\\]\n\n\n\\subsection{Kinematics and constitutive relations}\nIn the linear C-CST model, kinematics is described by the classical infinitesimal\nstrain (\\(e_{ij}\\)) and rotation (\\(\\theta_{ij}\\)) tensors\n\\begin{subequations}\n\\label{eq:kinematics}\n\\begin{align}\ne_{ij} = \\frac{1}{2}(u_{i,j} + u_{j,i})\\, , \\label{eq:strain}\\\\\n\\theta_{ij} = \\frac{1}{2}(u_{i,j} - u_{j,i})\\, , \\label{eq:rotation}\n\\end{align}\n\\end{subequations}\nand by the mean curvature tensor\n\\begin{equation}\n\\kappa_{ij} = \\frac{1}{2}(\\theta_{i,j} - \\theta_{j,i})\\, ,\n\\label{eq:curv_tensor}\n\\end{equation}\nwhere\n\\[\\theta_i = \\frac{1}{2}\\epsilon_{ijk}\\theta_{kj}\\, .\\]\n\nEquation \\eqref{eq:curv_tensor} can also be written in polar form, as an\nengineering curvature vector \\citep{darrall2014}\n\\begin{equation}\n\\kappa_i = \\epsilon_{ijk} \\theta_{j,k} =\\frac{1}{2}(u_{i,kk} - \nu_{k,ik})\n\\label{eq:curv}\n\\end{equation}\nsince\n\\[\\kappa_i = \\epsilon_{ijk}\\kappa_{jk}\\, .\\]\n\nFor a linear elastic centrosymmetric C-CST continuum, the constitutive equations\ncan be written as\n\\begin{equation}\n\\begin{split}\n&\\sigma_{(ij)} = C_{ijkl} e_{kl}\\, ,\\\\\n&\\mu_i = D_{ij} \\kappa_j\\, ,\n\\end{split}\n\\label{eq:constitutive_aniso}\n\\end{equation}\nwhere \\(C_{ijjkl}\\) is the stiffness tensor as in classical (anisotropic)\nelasticity, and \\(D_{ij}\\) is an additional material tensor that accounts for \ncouple-stress effects. In the expressions above, parentheses as subindices are\nused to indicate the symmetric part of the tensor. In the case of a linear\nisotropic elastic C-CST continuum,\n\\begin{equation}\n\\begin{split}\n&C_{ijkl} = \\lambda \\delta_{ij} \\delta_{kl} + \\mu ( \\delta_{ik} \\delta_{jl}\n   + \\delta_{il} \\delta_{jk} ), \\\\\n&D_{ij} = 4\\eta \\delta_{ij},\n\\end{split}\n\\label{eq:isotropic_tensors}\n\\end{equation}\nwhere \\(\\mu\\) and \\(\\lambda\\) are the Lam\u00e9 parameters as in classical \nelasticity, while \\(\\eta\\) is the additional material coefficient that accounts\nfor couple-stress effects. Then, the constitutive equations for isotropy can be\nsimplified to\n\\begin{equation}\n\\begin{split}\n&\\sigma_{(ij)} = \\lambda e_{kk} \\delta_{ij} + 2\\mu e_{ij}\\, ,\\\\\n&\\mu_i = 4\\eta \\kappa_i\\, .\n\\end{split}\n\\label{eq:constitutive}\n\\end{equation}\n\n\n\\subsection{Displacement equations of motion}\nAt this point it may be convenient to alternate between index and explicit\nvector notation. In the latter, the gradient operator reads\n\\(\\nabla = \\frac\\partial{\\partial x_i}\\) in Cartesian coordinates. In these\nterms, the time domain displacement equations of motion are obtained after\nusing the constitutive relations \\eqref{eq:constitutive} in the equilibrium\nequations \\eqref{eq:balance_eqs} yielding\n\\begin{equation}\n(\\lambda + 2\\mu)\\nabla (\\nabla \\cdot \\mathbf{u}) - \\mu\\nabla\\times \\nabla \n\\times \\mathbf{u} + \\eta \\nabla^2 \\nabla \\times \\nabla \\times \\mathbf{u} = \n\\rho\\ddot{\\mathbf{u}}\\, .\n\\label{eq:wave_eq}\n\\end{equation}\n\n\nDefining the phase\/group speed for the longitudinal (P) wave \\(c_1\\) (which is\nnot dispersive), the low-frequency (\\(k\\rightarrow 0\\)) phase\/group speed for\nthe transverse wave (S) \\(c_2\\) (which is dispersive) and the intrinsic material\nlength scale parameter \\(l\\) (which is not present in classical elasticity),\nsuch that\n\\begin{equation}\nc_1^2 = \\frac{\\lambda + 2\\mu}{\\rho}\\, ,\\qquad\nc_2^2 = \\frac{\\mu}{\\rho}\\, ,\\qquad\nl^2 = \\frac{\\eta}{\\mu}\\, ,\n\\label{eq:c1c2l2}\n\\end{equation}\nallows us to write \\eqref{eq:wave_eq} in the form\n\\begin{equation}\nc_1^2 \\nabla (\\nabla \\cdot \\mathbf{u}) - c_2^2 (1-l^2 \\nabla^2) \\nabla\\times \\nabla \\times \n\\mathbf{u} = \\ddot{\\mathbf{u}}\\, .\n\\label{eq:wave_eq_speeds}\n\\end{equation}\n\n\n\\subsection{Dispersion relations for unbounded domains}\nUsing a Helmholtz decomposition,  the displacement field can be written in terms\nof the scalar and vector potentials \\(\\varphi\\) and \\(\\mathbf{H}\\)\n\\citep{book:arfken} as\n\\[\\mathbf{u} = \\nabla\\varphi + \\nabla\\times\\mathbf{H}\\, ,\\quad \n\\nabla\\cdot\\mathbf{H} = 0\\, ,\\]\nand replacing this in \\eqref{eq:wave_eq_speeds} gives the following set of\nuncoupled wave equations\n\\begin{align}\n&c_1^2\\nabla^2\\varphi = \\ddot{\\varphi}\\, ,\\\\\n&c_2^2(1 - l^2\\nabla^2)\\mathbf{H} = \\ddot{\\mathbf{H}}\\, ,\n\\end{align}\nwhere it is observed that the equation for the rotational potential follows a\nhigher-order wave equation that is inherently dispersive.  This becomes evident\nafter assuming a solution of the form \\(\\mathbf{u} = \\mathbf{\\tilde u}e^{ik x - \ni\\omega t}\\) which gives the dispersion relations\n\\begin{align}\\label{eq:dispersion}\n&\\omega_P^2 = c_1^2 k^2\\, ,\\\\\n&\\omega_S^2 = c_2^2 k^2 (1 + k^2 l^2)\\, . \\label{eq:dispersion_SV}\n\\end{align}\n\nSolving the above for \\(k\\), we have in each case\n\\begin{align*}\nk_P^2 &= \\frac{\\omega^2}{c_1^2}\\, , &k_S^2 &= \\frac{1}{2 l^2}\\left[\\pm\\sqrt{1 \n+ \\frac{4\\omega^2 l^2}{c_2^2}} - \n1\\right]\\, .\n\\end{align*}\n\nNoticing that the quantity inside the square root is always greater than 1\nindicates that we should consider only the positive root, while the negative\nroot corresponds to an evanescent wave that should arise under certain boundary\nconditions. The phase and group speeds are now given by\n\\begin{equation}\n\\begin{aligned}\nv_P &= c_1\\, , & g_P &= c_1\\, ,\\\\\nv_S(k) &= c_2\\sqrt{1 + k^2 l^2}\\, , & g_S &= c_2 \\frac{1 + 2k^2 l^2}{\\sqrt{1 + \nk^2 l^2}}\\, .\n\\end{aligned}\n\\label{eq:speeds}\n\\end{equation}\n\nTaking the low and high frequency limits \\(k \\rightarrow 0\\) and\n\\(k \\rightarrow \\infty\\) gives\n\\begin{align*}\n&\\lim_{k \\rightarrow 0} v_S = \\lim_{k \\rightarrow 0} g_S = \nc_2\\, ,\\\\\n&\\lim_{k \\rightarrow \\infty} v_S = \\lim_{k \\rightarrow \\infty}g_S \n\\rightarrow  \\infty\\, ,\n\\end{align*}\nwhich shows how the speed of energy flow increases with frequency. All of these \nrelations are displayed in \\cref{fig:dispersion_analytic}.\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[height=1.8 in]{img\/disp_cst_freqs.pdf}\n    \\includegraphics[height=1.8 in]{img\/disp_cst_phase_speed.pdf}\n    \\includegraphics[height=1.8 in]{img\/disp_cst_group_speed.pdf}\n    \\caption{Dispersion relations for a homogeneous C-CST material \n        with properties: \\(\\rho = 1\\times 10^5\\), \\(\\lambda=2.8 \\times 10^{10}\\),\n        \\(\\eta=1.62 \\times 10^9\\), \\(\\mu = 4\\times 10^9\\). The plot  on \n        the left shows the frequency-wave number relation for the \n        non-dispersive P-wave (continuous line) and the dispersive SV (dashed\n        lines). The plots in the middle and right part of the  figure show the\n        phase and group speeds for the dispersive modes.}\n    \\label{fig:dispersion_analytic}\n\\end{figure}\n\n\n\\subsection{Frequency domain equations}\nBloch analysis considering spatial periodicity of the material is naturally\nconducted in the Fourier domain, involving both the temporal frequencies and\nspatial wave numbers.  After performing a Fourier transform of the linear and\nangular momentum equations  \\eqref{eq:balance_eqs} to the temporal frequency\ndomain, these become\n\\begin{equation}\n\\begin{split}\n&\\tilde\\sigma_{ji,j} + \\tilde f_i = - \\rho \\omega^2 \\tilde{u}_i\\, ,\\\\\n&\\tilde\\mu_{ji,j} + \\epsilon_{ijk}\\tilde\\sigma_{jk} = 0\\, ,\n\\end{split}\n\\label{eq:fourier_balance_eqs}\n\\end{equation}\nwhere the superposed tilde denotes a complex Fourier amplitude.\n\nAfter introducing the constitutive equations \\eqref{eq:constitutive_aniso} for\ncentrosymmetric materials into \\eqref{eq:fourier_balance_eqs} and then combining\nthe angular momentum and linear momentum balance laws into a single set in terms\nof displacement, one finds:\n\\begin{equation}\n( C_{ijkl} \\tilde u_{k,l} )_{,j} + \\frac{1}{4}\\epsilon_{pij}\\epsilon_{pmn}\\{D_{nk} (\\tilde u_{k,ll}-\\tilde u_{l,kl})\\}_{,mj} + f_i = -\\rho \\omega^2 \\tilde u_i\n\\label{eq:fourier_disp_aniso}\n\\end{equation}\n\nSubstituting \\eqref{eq:isotropic_tensors} and \\eqref{eq:c1c2l2} for isotropic\nmaterials into \\eqref{eq:fourier_disp_aniso} provides the corresponding Fourier\ndomain reduced wave equations in the absence of body forces, which can be written\n\\begin{equation}\nc_1^2 \\nabla (\\nabla \\cdot \\mathbf{\\tilde u}) - c_2^2 (1-l^2 \\nabla^2) \\nabla\\times \\nabla \\times\n\\mathbf{\\tilde u}  = -\\omega^2 \\mathbf{\\tilde u}\\, .\n\\label{eq:wave_eq_freq}\n\\end{equation}\nNotice that \\eqref{eq:wave_eq_freq} is the temporal Fourier transform of\n\\eqref{eq:wave_eq_speeds}.\n\n\\section{Variational principles}\nWe will describe next a variational formulation for the elastodynamic C-CST\nmodel. An inherent complexity is the presence of second order displacement\ngradients arising in the curvatures \\eqref{eq:curv}, which requires \\(C^1\\)\ncontinuity of the displacement field.\n\nLet us consider a volume \\(V\\) with boundary \\(S\\), having specified body forces\n\\(f_i\\), force-tractions \\(t_i\\), and couple-tractions \\(m_i\\) (see\n\\cref{fig:domain}). The boundary \\(S\\) is split into different segments, where\n\\(S_u\\) represents the portion of \\(S\\) with specified displacements, \\(S_t\\)\nrepresents the surface with prescribed tractions, \\(S_\\theta\\) represents the\nsegment with enforced rotations, and \\(S_m\\) the boundary with prescribed\ncouple-tractions. Additionally, \\(S = S_u \\cup S_t = S_\\theta \\cup S_m\\) and\n\\(S_u \\cap S_t = S_\\theta \\cap S_m = \\emptyset\\). In general, \\(S_u\\) and \\(S_t\\)\nmight overlap with \\(S_\\theta\\) and \\(S_m\\). This is an important aspect of the\nC-CST model that is relevant in the solution of boundary value problems, as we\nshall see later.\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=4 in]{img\/domain.pdf}\n    \\caption{Schematic representation of the domain and boundary conditions for the C-CST model.}\n    \\label{fig:domain}\n\\end{figure}\n\nWe begin with the following couple stress elastodynamic action functional in the\nfrequency domain, as an extension of the elastostatic formulations introduced\nby \\cite{hadjesfandiari2011couple} and \\cite{darrall2014}:\n\\begin{equation}\n\\mathcal{A}[u;\\omega] = \\mathcal{U}[u;\\omega]+ \\mathcal{T}[u;\\omega]+\\mathcal{V}[u;\\omega],\n\\label{eq:Lagrangian_functional}\n\\end{equation}\nHere, and in the remainder of this paper, the superposed tilde has been\nsuppressed for notational convenience.  Meanwhile, the elastic, kinetic and\napplied load actions can be written in explicit form, respectively, as\n\\begin{equation}\n\\mathcal{U}[u;\\omega] = \\frac{1}{2}\\int\\limits_{V} e_{ij}^* C_{ijkl} e_{kl} \\dd{V}\n+ \\frac{1}{2}\\int\\limits_{V} \\kappa_{i}^* D_{ij} \\kappa_{j} \\dd{V},\n\\label{eq:delta_functional_U}\n\\end{equation}\n\\begin{equation}\n\\mathcal{T}[u;\\omega] = -\\frac{\\omega^2}{2} \\int\\limits_{V} u_i^* \\rho u_i \\dd{V},\n\\label{eq:delta_functional_T}\n\\end{equation}\n\\begin{equation}\n\\mathcal{V}[u;\\omega] =- \\int\\limits_{V} u_i^* f_i \\dd{V}\n- \\int\\limits_{S_t}  u_i^* t_i^{(\\hat{n})} \\dd{S}  - \\int\\limits_{S_m} \\theta_i^* m_i^{(\\hat{n})} \\dd{S}. \n\\label{eq:delta_functional_V}\n\\end{equation}\nwith the asterisk denoting complex conjugate.\n\nThe stationarity of this action becomes\n\\begin{equation}\n\\delta \\mathcal{A}[u;\\omega] = \\delta\\mathcal{U}[u;\\omega]\n+ \\delta\\mathcal{T}[u;\\omega]+\\delta\\mathcal{V}[u,\\omega]) = 0,\n\\label{eq:action_variation}\n\\end{equation}\nor\n\\begin{equation}\n\\begin{split}\n&\\delta\\mathcal{A}[u;\\omega] = \n \\int\\limits_{V} \\delta e_{ij}^* C_{ijkl} e_{kl} \\dd{V}\n + \\int\\limits_{V} \\delta\\kappa_{i}^* D_{ij} \\kappa_{j} \\dd{V}\n -\\omega^2 \\int\\limits_{V} \\delta u_i^* \\rho u_i \\dd{V} \\\\\n&- \\int\\limits_{V} \\delta u_i^* f_i \\dd{V}\n- \\int\\limits_{S_t}  \\delta u_i^* t_i^{(\\hat{n})} \\dd{S}\n- \\int\\limits_{S_m} \\delta\\theta_i^* m_i^{(\\hat{n})} \\dd{S} = 0, \\,\n\\label{eq:delta_action_functional}\n\\end{split}\n\\end{equation}\nwhich can serve as the weak form for a finite element formulation in reduced\nelastodynamics.  With the appearance of mean curvature in\n\\eqref{eq:delta_action_functional}, this would require \\(C^1\\) spatial continuity\nof displacements \\(u_i\\).\n\nNext, let us derive the Euler-Lagrange equations associated with the functional\n\\(\\mathcal{A}[u; \\omega]\\). Starting from the first variation in\n\\eqref{eq:delta_action_functional}, we repeatedly apply integration-by-parts\noperations and the divergence theorem to shift all of the spatial derivatives\nfrom the variations to the true fields. This leads to the following\nstatement:\n\\begin{equation}\n\\begin{split}\n&\\int\\limits_{V} \\delta u_i^* [ ( C_{ijkl} u_{k,l} )_{,j}\n  + \\frac{1}{4}\\epsilon_{pij}\\epsilon_{pmn}\\{D_{nk} (u_{k,ll}-u_{l,kl})\\}_{,mj}\n  + f_i + \\rho \\omega^2 u_i] \\dd{V} \\\\\n&+ \\int\\limits_{S_t}  \\delta u_i^* [ t_i^{(\\hat{n})} - \\sigma_{ji} n_j ] \\dd{S} \\\\\n&+ \\int\\limits_{S_m} \\delta\\theta_i^* [ m_i^{(\\hat{n})} -  \\epsilon_{ijk}\\mu_k n_j ] \\dd{S} = 0. \\, \n\\label{eq:delta_action_E-L_functional}\n\\end{split}\n\\end{equation}\nFor arbitrary variations, each set of terms inside the square brackets must be\nzero. Thus, the Euler-Lagrange equations can be written:\n\\begin{equation}\n( C_{ijkl} u_{k,l} )_{,j} + \\frac{1}{4}\\epsilon_{pij}\\epsilon_{pmn}\\{D_{nk} (u_{k,ll}-u_{l,kl})\\}_{,mj}\n  + f_i = -\\rho \\omega^2 u_i \\quad {\\rm in} \\ V \\\\\n\\label{eq:action_E-L_aniso_momentum}\n\\end{equation}\n\\begin{equation}\n\\begin{split}\n&t_i^{(\\hat{n})} = \\sigma_{ji} n_j \\quad {\\rm on} \\ S_t \\\\\n&m_i^{(\\hat{n})} = \\epsilon_{ijk}\\mu_k n_j \\quad {\\rm on} \\ S_m \\,\n\\label{eq:action_E-L_aniso_traction}\n\\end{split}\n\\end{equation}\nNotice that \\eqref{eq:action_E-L_aniso_momentum} are the reduced wave equations\nfrom \\eqref{eq:fourier_disp_aniso}, while \\eqref{eq:action_E-L_aniso_traction}\nrepresent the corresponding natural boundary conditions for C-CST.  In the\nisotropic case, substituting \\eqref{eq:isotropic_tensors} into\n\\eqref{eq:action_E-L_aniso_momentum} produces\n\\begin{equation}\nc_1^2 u_{j,ji}-c_2^2 \\epsilon_{ijk} \\epsilon_{kmn} (u_{n,mj}-l^2 u_{n,mjll})\n+ f_i = -\\omega^2 u_i\\quad {\\rm in} \\ V\n\\label{eq:action_E-L_iso_momentum}\n\\end{equation}\nwhich is the equivalent of \\eqref{eq:wave_eq_freq} in index notation.\n\nPerforming an inverse Fourier transform of individual terms in\n\\eqref{eq:Lagrangian_functional}-\\eqref{eq:delta_functional_V} back to the time domain, one finds\n\\begin{equation}\n\\mathcal{F}^{-1}[u^* v] = (u \\star v)(t)\n\\label{eq:time_domain_functional}\n\\end{equation}\nwhere the $\\star$ operator denotes correlation over time, such that\n\\begin{equation}\n(u \\star v)(t) = \\int_{-\\infty}^\\infty u(\\tau) v(t+\\tau) d\\tau.\n\\label{eq:correlation}\n\\end{equation}\n\nConsequently, we have established the following stationary {\\it Principle of\nCorrelated Action} for couple stress elastodynamics: Of all the possible\ndisplacement fields in \\(V\\) that satisfy the frequency domain kinematic boundary\nconditions on \\(S_u\\) and \\(S_\\theta\\), the one that renders the action\n\\(\\mathcal{A}[u; \\omega]\\) in \\eqref{eq:Lagrangian_functional} stationary\ncorresponds to the solution of the reduced wave equations\n\\eqref{eq:action_E-L_aniso_momentum} and traction boundary conditions\n\\eqref{eq:action_E-L_aniso_traction}.\n\nWe should emphasize that this stationary Principle of Correlated Action also\nholds for classical theory, if one neglects contributions from mean curvature\nand moment tractions.  Thus, the classical correlated action for reduced\nelastodynamics can be written:\n\\begin{equation}\n\\begin{split}\n&\\mathcal{A}_{\\rm cl}[u;\\omega] = \\frac{1}{2}\n \\int\\limits_{V} e_{ij}^* C_{ijkl} e_{kl} \\dd{V}\n -\\frac{\\omega^2}{2} \\int\\limits_{V} u_i^* \\rho u_i \\dd{V} \\\\\n&\\qquad\\qquad- \\int\\limits_{V} u_i^* f_i \\dd{V}\n- \\int\\limits_{S_t} u_i^* t_i^{(\\hat{n})} \\dd{S} . \\,\n\\label{eq:classical_action_functional}\n\\end{split}\n\\end{equation}\n\n\n\\section{Response of Periodic Materials}\nThis section summarizes the most relevant theoretical aspects for the numerical\nanalysis of periodic materials. An in-depth treatment of the subject can be\nfound in classical textbooks, such as \\cite{book:brillouin} and \\cite{book:kittel},\nwhile a comprehensive review is provided in \\cite{hussein2014dynamics}. In our\ndiscussion we will use a generalized form of the reduced wave equation, however\nwe will provide the Bloch-Floquet boundary conditions for the particular case of\nthe C-CST model.\n\n\\subsection{Bloch's theorem}\nConsider a reduced elastodynamic wave equation in the frequency domain of the form\n\\begin{equation}\n  \\mathcal{L} \\mathbf{u}(\\mathbf{x}) = -\\rho \\omega^2 \\mathbf{u}(\\mathbf{x})\\ \n  \\label{eq:reduced}\n\\end{equation}\nvalid for a  field \\(\\mathbf{u}\\) at a spatial point \\(\\mathbf{x}\\). Here\n\\(\\mathcal{L}\\) is a positive definite linear differential operator\n\\citep{book:reddy_functional, book:kreyszig_functional, johnson2007waves},\nwhile \\(\\rho\\) is the mass density and \\(\\omega\\) the corresponding angular\nfrequency. Bloch's theorem \\citep{book:brillouin} establishes that solutions to\n\\eqref{eq:reduced} are of the form\n\\begin{equation}\n  \\mathbf{u}(\\mathbf{x}) = \\mathbf{w}(\\mathbf{x}) e^{i\\mathbf{k}\\cdot \\mathbf{x}}\\\n  \\label{eq:bloch}\n\\end{equation}\nwhere $\\mathbf{w}(\\mathbf{x})$ is a Bloch function carrying with it the same\nperiodicity as the material. Since the spatial period in\n\\(\\mathbf{w}(\\mathbf{x})\\) is the lattice parameter \\(\\mathbf{a}\\), it follows that\n\\[\\mathbf{w}(\\mathbf{x} + \\mathbf{a}) = \\mathbf{w}(\\mathbf{x}).\\]\n\nAccordingly, \\eqref{eq:bloch} is the product of a spatially periodic function\n\\(\\mathbf{w}(\\mathbf{x})\\), with the periodicity of the lattice, and a plane\nwave (of wave vector \\(\\textbf{k}\\)), which is also periodic. As a result, field\nvariables \\(\\mathbf{\\Phi}\\) at opposite sides of the unit cell and separated by\nthe lattice vector \\(\\mathbf{a}\\) are related through\n\\begin{equation}\n\\mathbf{\\Phi}(\\mathbf{x} + \\mathbf{a}) = \\mathbf{\\Phi}(\\mathbf{x})e^{i\\mathbf{k}\\cdot\\mathbf{a}} .\n\\label{eq:gen_bloch}\n\\end{equation}\n\nIn this case, $\\mathbf{\\Phi}$ refers to the principal variable involved in the\nphysical problem, or to any of its spatial derivatives. From a physical point\nof view, \\eqref{eq:gen_bloch} means that a field variable \\(\\mathbf{\\Phi}\\) at\npoints \\(\\mathbf{x}\\) and \\(\\mathbf{x} + \\mathbf{a}\\) differ only by the phase\nshift \\(e^{i\\mathbf{k}\\cdot\\mathbf{a}}\\).\n\nIn the classical elastodynamic case in which \\(\\mathcal{L}\\) is the Navier\noperator of order 2, the generalized Boundary Value Problem (BVP) considering\nBloch boundary conditions (BBCs) takes the form:\n\\begin{subequations}\n\\label{eq:bvp_bloch}\n\\begin{align}\n&\\mathcal{L} \\mathbf{u}(\\mathbf{x}) = -\\rho \\omega^2 \\mathbf{u}(\\mathbf{x})\\, ,\\\\\n&\\mathbf{u}(\\mathbf{x} + \\mathbf{a}) = \\mathbf{u}(\\mathbf{x})e^{i\\mathbf{k}\\cdot\\mathbf{a}}\\, ,\\\\\n&\\mathbf{\\sigma}(\\mathbf{x} + \\mathbf{a}) \\cdot \\hat{\\mathbf{n}}\n  = - \\mathbf{\\sigma}(\\mathbf{x}) \\cdot \\hat{\\mathbf{n}}\\,  e^{i\\mathbf{k}\\cdot\\mathbf{a}}\\, ,\n\\end{align}\n\\end{subequations}\nwhere \\(\\mathbf{u}(\\mathbf{x} + \\mathbf{a})\\) and \\(\\mathbf{u}(\\mathbf{x})\\)\ngive the field at \\(\\mathbf{x} + \\mathbf{a}\\) and \\(\\mathbf{x}\\), respectively,\nand \\(\\mathbf{\\sigma}(\\mathbf{x})\\) is the corresponding stress.  Meanwhile, \n\\(\\mathbf{a} = \\mathbf{a}_1 n_1 + \\mathbf{a}_2 n_2 + \\mathbf{a}_3 n_3\\) is the\nlattice translation vector and \\(n_i\\) are the lattice normal parameters.\n\nNote that the BVP encompassed by \\eqref{eq:bvp_bloch} simultaneously describes\nthe space-time periodicity of the solutions in the cellular material. Time\nperiodicity is present in the frequency-domain nature of the reduced wave\nequation, while space periodicity explicitly appears in the wave number\nrepresentation of the boundary conditions. The periodic relationship between\nopposite sides of the fundamental cell, appearing in the boundary terms, allows\ncharacterization of the fundamental properties of the material with the analysis\nof a single cell. At the same time the wave vector \\(\\textbf{k}\\) in\n\\eqref{eq:bvp_bloch} simultaneously describes: (i) the propagation direction of\na plane wave traveling through the unit cell and (ii) the spatial periodicity of\nthe plane wave. In consequence, finding solutions to the Bloch-BVP amounts to\nfinding those tuples \\((\\omega, \\mathbf{k}, \\mathbf{u})\\) satisfying\n\\eqref{eq:bvp_bloch} when \\(\\mathbf{k}\\) is varied in the dual Fourier based\nrepresentation of the fundamental material cell. This dual space corresponds to\nthe \\textit{reciprocal space} and since it carries with it the periodic\ncharacter of the physical space it suffices to consider values (and directions)\nof \\(\\mathbf{k}\\) within this reciprocal space representation of the unit cell.\n\nIn the case of the C-CST medium, Bloch's theorem states that the eigenfunctions\nof \\eqref{eq:wave_eq_freq} can be expressed in the form\n\\[\\mathbf{u}(\\vb{x}) = \\mathbf{u}(\\vb{x}+\\vb{a})e^{i\\vb{k}\\cdot\\vb{a}} \\]\nwhere $\\vb{a}$ is a vector that represents the periodicity of the material.\nThat is, the solution is the same at opposite sides of the unit cell, except \nfor a phase shift factor $e^{i\\vb{k}\\cdot\\vb{a}}$. Due to the linearity of the\ndifferential equations we also have Bloch-periodic boundary \nconditions for the corresponding rotation and traction vectors.\nThus, in the case of the C-CST elastic solid, Bloch's theorem reduces to the\nfollowing set of boundary conditions for displacements, rotations,\nforce-tractions and couple-tractions in index notation:\n\\begin{subequations}\n\\begin{align}\n    &u_i(\\vb{x}) = u_i(\\vb{x}+\\vb{a})e^{i\\vb{k}\\cdot\\vb{a}}\\, , \\\\ \n    &\\theta_i(\\vb{x}) = \\theta_i(\\vb{x}+\\vb{a})e^{i\\vb{k}\\cdot\\vb{a}}\\, ,\\\\\n    &t_i(\\vb{x}) = -t_i(\\vb{x}+\\vb{a})e^{i\\vb{k}\\cdot\\vb{a}}\\, , \\label{eq:bloch_bcs.3}\\\\ \n    &m_i(\\vb{x}) = -m_i(\\vb{x}+\\vb{a})e^{i\\vb{k}\\cdot\\vb{a}}\\, .\n    \\label{eq:bloch_bcs.4}\n\\end{align}\n\\label{eq:bloch_bcs}\n\\end{subequations}\n\n\\noindent The set of conditions summarized in \\eqref{eq:bloch_bcs} will be\nsatisfied in a variational sense using a finite element formulation, where the\nfirst two are essential boundary conditions and the other two natural boundary\nconditions. Subsequently, a numerical model of the unit cell resulting in a\ngeneralized eigenvalue problem will be solved for various specifications of the\nwave vector.\n\n\\subsection{Hermiticity}\nOur finite element algorithm follows from the action functional formulated in\n\\eqref{eq:Lagrangian_functional}. As discussed previously this amounts to the\nsolution of the weak form of the frequency domain reduced wave equations subject\nto Bloch-periodic boundary conditions, as given by \\eqref{eq:bloch_bcs}.\nNeglecting body forces in \\eqref{eq:Lagrangian_functional}, we have:\n\\begin{equation}\n\\begin{split}\n\\mathcal{A}[u; \\omega] = &\\frac{1}{2}\\int\\limits_{V} e_{ij}^* C_{ijkl} e_{ij} \\dd{V}\n  + \\frac{1}{2}\\int\\limits_{V} \\kappa_{i}^* D_{ij} \\kappa_{i } \\dd{V}\n  -\\frac{\\omega^2}{2}\\int\\limits_{V} u_i^* \\rho u_i \\dd{V}\\\\\n&- \\int\\limits_{S_t}  u_i^* t_i \\dd{S}  - \\int\\limits_{S_m} \\theta_i^* m_i \\dd{S}\\, ,\n\\end{split}\n\\label{eq:hamilton_freq_free}\n\\end{equation}\n\nTo obtain real eigenvalues that correspond to propagating waves in the band\nstructure of the material, the matrices resulting from the finite element\ndiscretization must be Hermitic. Equivalently, we must prove  Hermiticity\n(self-adjointness) in the action functional. This amounts to showing that the\nboundary terms in \\eqref{eq:hamilton_freq_free} vanish under Bloch periodic\nboundary conditions.\n\nSubstitution of \\eqref{eq:bloch_bcs} into surface integral terms of\n\\eqref{eq:hamilton_freq_free} yields\n\\begin{equation}\n\\begin{split}\n&\\int\\limits_S u_i^{*}(\\vb{x})t_i(\\vb{x}) \\dd{S} + \\int\\limits_S \n\\theta_i^{*}(\\vb{x}) m_i(\\vb{x}) \\dd{S} = \\\\\n&\\sum\\limits_q \\left\\lbrace \\int\\limits_{S_q}\\left[u_i^{*}(\\vb{x})t_i(\\vb{x}) + \nu_i^{*}(\\vb{x} + \\vb{a}_q)t_i(\\vb{x} + \\vb{a}_q)\\right]\\dd{S}_q + \\right.\\\\\n&\\left. \\int\\limits_{S_q}\\left[\\theta_i^{*}(\\vb{x}) m_i(\\vb{x}) + \n\\theta_i^{*}(\\vb{x} + \\vb{a}_q) m_i(\\vb{x} + \\vb{a}_q)\\right]\\dd{S}_q \n\\right\\rbrace \\, ,\n\\end{split}\n\\end{equation}\nwith the index $q$ referring to each pair of opposite sides of the boundary.\nIntroducing the phase shifts and pulling out the common factors give:\n\\begin{equation}\n\\begin{split}\n&\\int\\limits_S u_i^{*}(\\vb{x})t_i(\\vb{x}) \\dd{S} + \\int\\limits_S \n\\theta_i^{*}(\\vb{x}) m_i(\\vb{x}) \\dd{S} = \\\\\n&\\sum\\limits_q \\left\\lbrace \\int\\limits_{S_q} u_i^*(\\vb{x})\\left[t_i(\\vb{x}) \n+ e^{i\\vb{k}\\cdot\\vb{a}}t_i(\\vb{x} + \\vb{a}_q)\\right]\\dd{S}_q + \\right.\\\\\n&\\left. \\int\\limits_{S_q}\\theta_i^*(\\vb{x})\\left[m_i(\\vb{x}) + \ne^{i\\vb{k}\\cdot\\vb{a}}m_i(\\vb{x} + \\vb{a}_q)\\right]\\dd{S}_q \\right\\rbrace \n\\, ,\n\\end{split}\n\\end{equation}\nwhich after substituting \n\\eqref{eq:bloch_bcs.3} and \\eqref{eq:bloch_bcs.4}\nleads to the vanishing of the boundary terms, thus proving the Hermiticity\ncondition.\n\n\\subsection{Positive definiteness}\nSimilarly, the proof for positive (semi)-definiteness reduces to showing that\nthe action functionals are related in such a way that:\n\\begin{equation}\n\\omega^2 = \n\\frac{\\mathcal{U}[u;\\omega]}{\\mathcal{\\tilde T}[u;\\omega]}  \\geq 0 \\, ,\n\\label{eq:definiteness}\n\\end{equation}\nwhere\n\\[\\mathcal{U}[u;\\omega] = \\frac{1}{2}\\int\\limits_{V} e_{ij}^* \nC_{ijkl}e_{kl}\\dd{V} + \\frac{1}{2}\\int\\limits_{V} \\kappa_{i}^* D_{ij} \\kappa_{j}\\dd{V} \n\\, \\]\nand\n\\[\\mathcal{\\tilde T}[u;\\omega] = \\frac{1}{2}\\int\\limits_V u_i^* \\rho u_i \\dd{V} \\,  , \\]\nwith the latter deriving directly from $\\mathcal{T}[u;\\omega]$.\n\nNote that we have used the general representation \\(C_{ijkl}\\) and \\(D_{ij}\\) for\nthe constitutive tensors. The functional \\(\\mathcal{U}[u;\\omega]\\) is positive as\nlong as these constitutive tensors are positive definite, which holds true if\nthey satisfy\n\\begin{align*}\nC_{ijkl}  e_{ij} e_{kl} \\geq 0\\quad \\forall \\ e_{mn}\\, ,\\\\\nD_{ij}  \\kappa_{i} \\kappa_{j} \\geq 0\\quad \\forall \\ \\kappa_{m}\\, .\n\\end{align*}\n\nFor isotropic materials, this implies the following constraints for the material\nparameters:\n\\[\\mu >0\\, ,\\quad   3\\lambda + 2\\mu > 0\\, ,\\quad \\eta > 0\\, .\\]\nOn the other hand, the condition  \\(u_i \\neq 0\\), requires \\(\\mathcal{\\tilde T}\\)\nto be different from zero and thus the condition required by\n\\eqref{eq:definiteness}. In the case of rigid body motion, \\(\\mathcal{U}\\)\ncould be zero implying that the form  is positive semi-definite, while the form\n\\(\\mathcal{\\tilde T}\\) is positive definite.\n\n\\section{Finite element formulation}\nIn this section, we derive a consistent finite element formulation for periodic\ncouple stress elastodynamics, as an extension of those formulated by\n\\cite{darrall2014} for the corresponding quasistatic problem and by\n\\cite{guarin2020} for periodic micropolar Bloch analysis.  In particular,\nthe \\(C^1\\) displacement continuity requirement is avoided by using a Lagrange\nmultiplier approach. Other finite element solutions in C-CST include a penalty\nmethod for isotropic elastostatics \\citep{chakravarty2017}, Lagrange multipliers\nfor centrosymmetric anisotropic elastostatics \\citep{pedgaonkar2021} and mixed\nvariable methods for isotropic elastodynamics \\citep{deng2016, deng2017}.\n\n\\subsection{Lagrange multiplier reformulation}\nConsider now a modification of the action given in\n\\eqref{eq:hamilton_freq_free} to include Lagrange multipliers \\(\\lambda_i\\) that\nenforce compatibility between the displacement field \\(u_i\\) and an assumed\nindependent rotation field \\(\\theta_i\\).  Thus, the modified action becomes\n\\begin{equation}\n\\begin{split}\n\\mathcal{\\hat A}[u; \\omega] = &\\frac{1}{2}\\int\\limits_{V} e_{ij}^* C_{ijkl} e_{ij} \\dd{V}\n  + \\frac{1}{2}\\int\\limits_{V} \\kappa_{i}^* D_{ij} \\kappa_{i } \\dd{V}\n  -\\frac{\\omega^2}{2}\\int\\limits_{V} u_i^* \\rho u_i \\dd{V}\\\\\n&- \\int\\limits_{S_t}  u_i^* t_i \\dd{S}  - \\int\\limits_{S_m} \\theta_i^* m_i \\dd{S}\\\\\n&+ \\int\\limits_{V}\\lambda_i^* (\\epsilon_{ijk} u_{k,j} - 2\\theta_i )\\dd{V}\\, .\n\\end{split}\n\\label{eq:hamilton_freq_with_multipliers}\n\\end{equation}\n\nFor stationarity, we require\n\\[\\var{\\hat{\\mathcal{A}}} = \\pdv{\\hat{\\mathcal{A}}}{u_i} \\var{u_i}\n+ \\pdv{\\hat{\\mathcal{A}}}{\\theta_i}\\var{\\theta_i}\n+ \\pdv{\\hat{\\mathcal{A}}}{\\lambda_i} \\var{\\lambda_i} = 0\\, , \\]\nwhich is equivalent to\n\\begin{equation}\n\\begin{split}\n&\\int\\limits_{V} \\delta e_{ij}^* C_{ijkl} e_{ij} \\dd{V}\n  + \\int\\limits_{V} \\delta \\kappa_{i}^* D_{ij} \\kappa_{i } \\dd{V}\n  -\\omega^2 \\int\\limits_{V} \\delta u_i^* \\rho u_i \\dd{V}\\\\\n&\\qquad- \\int\\limits_{S_t}  \\delta u_i^* t_i \\dd{S}  - \\int\\limits_{S_m} \\delta \\theta_i^* m_i \\dd{S}\\\\\n&\\qquad+ \\int\\limits_{V}\\delta \\lambda_i^* (\\epsilon_{ijk} u_{k,j} - 2\\theta_i )\\dd{V}\\\\\n&\\qquad+ \\int\\limits_{V} (\\epsilon_{ijk} \\delta u_{k,j}^* - 2\\delta \\theta_i^* ) \\lambda_i \\dd{V}\\, .\n\\end{split}\n\\label{eq:weak_form_with_multipliers}\n\\end{equation}\n\n\\Cref{eq:weak_form_with_multipliers} is the modified weak form that will be used\nhere as the basis for the finite element Bloch analysis of an elastic\ncouple-stress solid. The Lagrange multiplier terms enforce the required\nkinematic constraint between the continuum rotations \\(\\epsilon_{ijk} u_{k,j}\\)\nof the material point and the independent rotational variables \\(\\theta_i\\).\n\nFrom \\eqref{eq:weak_form_with_multipliers}, we obtain the following\nEuler-Lagrange equations\n\\begin{equation}\n\\begin{split}\n&(C_{ijkl} e_{kl} + \\epsilon_{ijk} \\lambda_k)_{,j} = -\\rho \n\\omega^2u_i\\quad \\text{in } V,\\\\\n&\\epsilon_{ijk} (D_{kl} \\kappa_{l})_{,j} - 2\\lambda_i = 0\\quad \\text{in } V,\\\\\n&\\theta_i = \\frac{1}{2}\\epsilon_{ijk} u_{k,j}\\quad \\text{in } V,\\\\\n&t_i = (C_{ijkl} e_{kl} + \\epsilon_{ijk}\\lambda_k)n_j\\quad   \\text{on } S_t,\\\\\n&m_i = \\epsilon_{ijk} D_{kl} \\kappa_{l} n_j\\quad   \\text{on } S_m\\, ,\n\\end{split}\n\\label{eq:bvp}\n\\end{equation}\nComparing this with \\eqref{eq:skew_symmetry}, we can conclude that the Lagrange\nmultipliers equal the skew-symmetric part of the force-stress tensor, i.e.,\n\\[\\lambda_i = s_i\\, .\\]\n\n\\subsection{Discretization}\nTo discretize \\eqref{eq:weak_form_with_multipliers}, we use for the\nelement-based shape functions second-order Lagrange interpolation for the\ndisplacements and rotations and constant skew-symmetric stresses. This\ntranslates into \\(C^0\\) inter-element displacement and rotation continuity, and\nskew-symmetric stresses that are constant within the element but discontinuous\nbetween elements. \\Cref{fig:element} depicts a typical element for the\ndiscretization and the degrees of freedom used in two-dimensional idealizations.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=2 in]{img\/element.pdf}\n\\caption{Finite element used for the finite element discretization of the\nC-CST material model. A second-order Lagrange interpolation is used for\ndisplacements and rotations while a constant is used for the skew-symmetric\nstresses. The nodal degrees of freedom are depicted as black disks while the\nwhite disk describes the element skew-symmetric force-stress degree of freedom.}\n\\label{fig:element}\n\\end{figure}\n\nTo write the discretized equations, we will use a combined index notation. In\nthis context subscripts will still make reference to scalar components of\ntensors while capital superscripts will indicate interpolation operations. For\ninstance in the expression\n\\[u_i = _{u}\\hspace{-4pt}N_{i}^{Q}u^{Q}\\]\nsubscripts $i$ indicate the scalar components of the vector \\(u\\). To facilitate\nfurther operations this subscript is also placed in the shape function resulting\nin terms like \\(_{u}N_{i}^{Q}\\) and where the term \\(u^{Q}\\)  represents the\nnodal point displacement associated to the $Q$th nodal point. This nodal vector\nimplicitly considers horizontal and vertical rectangular components. To clarify,\nthe displacement interpolation scheme written here as\n\\(u_i = _{u}\\hspace{-4pt}N_{i}^{Q}u^{Q}\\) takes the following explicit form for\nthe single nodal point Q:\n\\begin{equation}\n\\begin{bmatrix}u_x\\\\ u_y\\end{bmatrix}=\n\\left[\\cdots\\begin{array}{cc}N^Q &0\\\\ 0 &N^Q\\end{array}\\cdots\\right]\n\\begin{Bmatrix}\\vdots\\\\ u_x^Q\\\\ u_y^Q\\\\\\vdots\\end{Bmatrix} \\, .\n\\label{eq:interp_nota}\n\\end{equation}\n\nWith this notation we write for the primary variables\n\\((u_i, \\theta_i, s_i )\\) the following interpolated versions\n\\begin{equation}\nu_i = _{u}\\hspace{-4pt}N_{i}^{Q}u^{Q},\\quad \\theta_i\n  = _{\\theta}\\hspace{-4pt}N_i^{Q}\\theta^{Q} ,\\quad s_i\n  = _{s}\\hspace{-4pt}N_{i}^{Q} s^{Q} \\, ,\n\\label{eq:interp_fun_prim}\n\\end{equation}\nand similarly for the secondary kinematic descriptors\n\\(e_{ij} , \\epsilon_{ijk} u_{i,j}\\) and \\(\\kappa_{i}\\) \n\\begin{equation}\ne_{ij} = _{e}\\hspace{-4pt}B_{ij}^{Q}u^{Q} ,\\quad \\epsilon_{ijk} u_{i,j}\n  = _{\\nabla}\\hspace{-4pt}B_{k}^{Q}u^{Q} ,\\quad \\kappa_{i}\n  = _{\\kappa}\\hspace{-4pt}B_{i}^Q \\theta^Q \\, ,\n\\label{eq:interp_fun_second}\n\\end{equation}\ntogether with the constitutive equations\n\\begin{equation}\n\\begin{aligned}\n  &\\sigma_{ij} = C_{ijkl}e_{kl}\\, , \\\\\n  &\\mu_{i} = D_{ij}\\kappa_{j}\\, .\n\\end{aligned}\n\\label{eq:const}\n\\end{equation}\n\n\nSubstitution of the above relations in \\eqref{eq:weak_form_with_multipliers}\ngives the discrete version of the first variation of the modified correlated\naction;\n\\begin{equation}\n\\begin{split}\n&\\var{\\hat{\\mathcal{A}}} = \\delta u^{Q*} \\int\\limits_{V} (_{e}B_{ij}^{Q}) (C_{ijkl}) (_{e}B_{kl}^{P}) \\dd{V} u^{P} - \\rho \\omega ^2 \\delta u^{Q*}  \\int\\limits_{V} (_{u}N_{i}^{Q}) (_{u}N_{i}^{P}) \\dd{V} u^{P} \\\\\n&- \\delta u^{Q*}\\int\\limits_{V}\\ _{u}N_{i}^Q f_i \\dd{V} - \\delta u^{Q*}\\int\\limits_{S}\\ _{u}N_{i}^Q t_i \\dd{S} \n+ \\delta \\theta^{Q*} \\int\\limits_{V} (_{\\kappa}B_{i}^{Q}) (D_{ij}) (_{\\kappa}B_{j}^{P}) \\dd{V} \\theta^{P}  \\\\\n&-\\delta \\theta^{Q*}\\int\\limits_{S}\\ _{\\theta}N_i^Q m_i \\dd{S} + \\delta s^{Q*} \\int\\limits_{V} (_{s}N_{k}^{Q}) (_{\\nabla}B_{k}^{P}) \\dd{V}  u^{P}\n+ \\delta u^{Q*} \\int\\limits_{V} (_{\\nabla}B_{k}^{Q}) (_{s}N_{k}^{P})  \\dd{V} s^{P} \\\\\n&-\\delta s^{Q*} \\int\\limits_{V} 2 (_{s}N_{k}^{Q}) (_{\\theta}N_{k}^{P}) \\dd{V} \\theta^{P} -\\delta \\theta^{Q*} \\int\\limits_{V} 2(_{\\theta}N_{k}^{Q}) (_{s}N_{k}^{P}) \\dd{V} s^{P} = 0 \\, .\n\\end{split}\n\\label{eq:discrete_PVW}\n\\end{equation}\n\nThe explicit form of the interpolators defined above is given in the appendix.\n\n\\subsection{Discrete equilibrium equations}\nFrom the arbitrariness in the variations \\(\\delta u^Q\\) , \\(\\delta \\theta^Q\\)\nand \\(\\delta s^Q\\) in \\eqref{eq:discrete_PVW} it follows that:\n\\begin{equation*}\n\\begin{split}\n&\\int\\limits_{V} (_{e}B_{ij}^{Q}) (C_{ijkl}) (_{e}B_{kl}^{P}) \\dd{V} u^P\n  - \\int\\limits_{V} (_{u}N_{i}^{Q}) (_{u}N_{i}^{P}) \\dd{V} u^P\n  - \\int\\limits_{V}\\ _{u}N_{i}^Q f_i \\dd{V}\n  - \\int\\limits_{S}\\ _{u}N_{i}^Q t_i \\dd{S} = 0\\,, \\\\\n&\\int\\limits_{V} (_{\\kappa}B_{i}^{Q}) (D_{ij}) (_{\\kappa}B_{j}^{P}) \\dd{V} \\theta^P \n  - \\int\\limits_{S}\\ _{\\theta}N_i^Q m_i \\dd{S}\n  - \\int\\limits_{V} 2(_{\\theta}N_{k}^{Q}) (_{s}N_{k}^{P}) \\dd{V} s^P=0\\, , \\\\\n&\\int\\limits_{V} (_{s}N_{k}^{Q}) (_{\\nabla}B_{k}^{P}) \\dd{V}  u^P\n  - \\int\\limits_{V} 2 (_{s}N_{k}^{Q}) (_{\\theta}N_{k}^{P}) \\dd{V} \\theta^P = 0\\, ,\n\\end{split}\n\\end{equation*}\nwhich can be written in the standard finite element form for dynamic equilibrium\n\\begin{equation}\n\\begin{bmatrix}\nK_{uu}^{QP} &0 &K_{us}^{QP}\\\\\n0 &K_{\\theta\\theta}^{QP} &-K_{\\theta s}^{QP}\\\\\nK_{s u}^{QP} &-K_{s\\theta}^{QP} &0\n\\end{bmatrix}\n\\begin{Bmatrix}\nu^P\\\\\n\\theta^P\\\\\ns^P\n\\end{Bmatrix}\n=\n\\omega^2\\begin{bmatrix}\n M_{uu}^{QP} &0 &0\\\\\n0 &0 &0\\\\\n0 &0 &0\n\\end{bmatrix}\n\\begin{Bmatrix}\nu^P\\\\\n\\theta^P\\\\\ns^P\n\\end{Bmatrix}\n+ \\begin{Bmatrix}\nF_u^Q\\\\\nm_\\theta^Q\\,\\\\\n0\\end{Bmatrix}\n\\label{eq:mat_fem}\n\\end{equation}\nwhere the individual terms are defined as\n\\begin{align*}\nK_{uu}^{QP} &= \\int\\limits_{V} (_{e}B_{ij}^{Q}) (C_{ijkl}) (_{e}B_{kl}^{P}) \\dd{V}\\, ,\n&M_{uu}^{QP} &= \\rho \\omega ^2 \\int\\limits_{V} (_{u}N_{i}^{Q}) (_{u}N_{i}^{p}) \\dd{V}\\, ,\\\\\nK_{u s}^{QP} &= \\int\\limits_{V} (_{\\nabla}B_{k}^{Q}) (_{s}N_{k}^{P})  \\dd{V}\\, ,\n&F_{u }^{Q} &= \\int\\limits_{V}\\ _{u}N_{i}^Q f_i \\dd{V} + \\int\\limits_{S}\\ _{u}N_{i}^Q t_i \\dd{S}\\, ,\\\\\nK_{\\theta \\theta}^{QP} &= \\int\\limits_{V} (_{\\kappa}B_{i}^{Q}) (D_{ij}) (_{\\kappa}B_{j}^{P}) \\dd{V}\\, ,\n&K_{\\theta s}^{QP} &= \\int\\limits_{V} 2(_{\\theta}N_{k}^{Q}) (_{s}N_{k}^{P}) \\dd{V}\\, ,\\\\\nm_{\\theta}^{Q} &= \\int\\limits_{S}\\ _{\\theta}N_i^Q m_i \\dd{S}\\, ,\n&K_{s u}^{QP} &= \\int\\limits_{V} (_{s}N_{k}^{Q}) (_{\\nabla}B_{k}^{P}) \\dd{V}\\, ,\\\\\nK_{s \\theta}^{QP} &= \\int\\limits_{V} 2 (_{s}N_{k}^{Q}) (_{\\theta}N_{k}^{P}) \\dd{V}\\, .\n\\end{align*}\n\n\\Cref{eq:mat_fem} can be rewritten in the following set of equilibrium equations\nin terms of nodal forces and couples\n\\begin{equation}\n\\begin{aligned}\nf_{(\\sigma)}^Q + f_{s}^Q - f_I^Q - T^Q &= 0\\, ,\\\\\nm_{\\mu}^Q + m_{s}^Q - q^Q &= 0\\, ,\\\\\ns(\\theta - \\hat{\\theta}) = 0\\, ,\n\\end{aligned}\n\\label{eq:discrete_balance}\n\\end{equation}\nwhere the subindex $(\\sigma)$ refers to the symmetric part of the stress tensor,\nand \\(I\\) to inertial forces. Notice that we do not have an\ninertial term for the second equation as is the case for the micropolar model\n\\citep{guarin2020}. We also have a third equation reflecting the kinematic\nrestriction, between the rotation \\(\\theta\\) and the introduced degree of\nfreedom \\(\\hat{\\theta}\\),  imposed via the Lagrange-multiplier term \\(s\\) in\neach element.\n\nWhen using a Lagrange multiplier formulation as in\n\\eqref{eq:weak_form_with_multipliers} the equations are still self-adjoint, as\ncan be seen in the structure of \\eqref{eq:mat_fem}. Nevertheless, the stiffness\nmatrix is indefinite and the solution of the problem represents a saddle-point\ninstead of a minimum \\citep{arnold_mixed_1990, darrall2014}.\n\n\\subsection{Eigenvalue problem}\nIn finding the dispersion relations, we are interested in the free wave motion\nin the media. This leads to the following eigenvalue problem\n\\begin{equation}\n[K]\\{U\\} = \\omega^2 [M]\\{U\\}\n\\label{eq:eigenvalue_fem}\n\\end{equation}\nwith\n\\[[K] = \\begin{bmatrix}\nK_{uu}^{QP} &0 &K_{us}^{QP}\\\\\n0 &K_{\\theta\\theta}^{QP} &-K_{\\theta s}^{QP}\\\\\nK_{s u}^{QP} &-K_{s\\theta}^{QP} &0\n\\end{bmatrix} ,\\,\n[M] = \\begin{bmatrix}\nM_{uu}^{QP} &0 &0\\\\\n0 &0 &0\\\\\n0 &0 &0\n\\end{bmatrix}, \\,\n\\{U\\} = \n\\begin{Bmatrix}\nu^P\\\\\n\\theta^P\\\\\ns^P\n\\end{Bmatrix}\\, .\\]\n\nIn \\eqref{eq:eigenvalue_fem} Bloch-periodic boundary  conditions are yet to be\nimposed. This can be done in two ways \\citep{valencia_uel_2019}: (i) modifying\nthe connectivity of the elements; and (ii) assembling the matrices without\nconsidering boundary conditions and impose the Bloch-periodicity through\nrow\/column operations. In this work, we follow the second approach as it\nrequires the stiffness and mass matrices to be assembled once and the\ntransformation matrices are computed for every wavenumber in the first\nBrillouin zone. This process results in the following eigenvalue problem\n\\begin{equation}\n[K_R(\\vb{k})]\\{U\\} = \\omega^2 [M_R(\\vb{k})]\\{U\\}\n\\label{eq:reduced_eigenvalue_fem}\n\\end{equation}\nwith\n\\[[K_R(\\vb{k})] = [T(\\vb{k})^H K T(\\vb{k})]\\, ,\\quad [M_R(\\vb{k})] = [T(\\vb{k})^H M T(\\vb{k})]\\, ,\\]\nwhere \\([T(\\vb{k})]\\) represents the transformation matrix for a given \\(\\vb{k}\\),\nand the \\([T^H]\\) refers to the Hermitian transpose of \\([T]\\). For an explicit\nform for the matrices \\([T]\\) refer to \\cite{hussein2014dynamics} or\n\\cite{guarin2012_msc}.\n\nWe conducted the implementation on top of the in-house finite element code\nSolidsPy \\citep{solidspy} and used SciPy to solve the eigenvalue problem\n\\citep{scipy}. To take advantage of the sparsity of the matrices the problem\nshould be written as matrix-vector multiplications, such as\n\\begin{align*}\n\\{x\\} = [T]\\{U\\}\\, ,\\\\\n\\{y\\} = [K]\\{x\\}\\, ,\\\\\n\\{z\\} = [T^H] \\{y\\}\\, ,\n\\end{align*}\nwith $\\{z\\}$ representing the image of the linear operator \\([K_R]\\) over\n\\(\\{U\\}\\). The same procedure can be applied for the right-hand side of\n\\eqref{eq:reduced_eigenvalue_fem}.\n\nThe Lagrange-multiplier approach represents a saddle-point instead of a minimization\nproblem \\citep{arnold_mixed_1990}. This can be seen in the structure of the stiffness\nmatrix obtained in equation \\eqref{eq:eigenvalue_fem}. Furthermore the mass matrix\nis not positive definite anymore. This structure for the eigenvalue problem requires\nthe use of a specific solver such as the LOBPCG method \\citep{knyazev2001}\ninstead of the classical Arnoldi method \\citep{book:arpack}.\n\n\\section{Results: Dispersion relations for C-CST cellular materials}\nIn this section we conduct a series of dispersion analyses intended to show the\neffectiveness of our mixed finite element implementation of the C-CST material\nmodel in predicting the correct wave propagation properties of the material. All\nthe dispersion graphs use the dimensionless frequency\n\\begin{equation}\n\\Omega = \\frac{2d\\omega}{c_2}\\, ,\n\\end{equation}\nfor the vertical axis, where \\(2d\\) is the dimension of the unit cell and\n\\(c_2^2 = \\mu\/\\rho\\) is the speed of the shear wave for a classical elastic material.\nThe Poisson ratio for all the simulations is \\(\\nu = 1\/4\\).\n\nAs a first instance we find the response of a homogeneous periodic material\nwhich has also a closed form solution. We will then continue to study a second\nprototypical example corresponding to a homogeneous material with a circular\npore. These two problems exhibit two different levels of dispersive behavior. In\nthe homogeneous material cell, dispersion is due to the kinematic enrichment of\nthe model associated to the length scale parameter, while in the porous material\nmodel additional dispersion arises due to the explicit microstructural feature.\n\n\\subsection{Homogeneous material}\nAs a test of accuracy and effectiveness of our implementation we consider the\ncase of a homogeneous material cell with the same mechanical properties of the\nmaterial reported previously and with closed form dispersion relations from\n\\eqref{eq:dispersion} and \\eqref{eq:dispersion_SV}. In this model microstructural\neffects are introduced through the material length parameter \\(\\ell\\).\nRecall that \\(\\ell^2\\) is defined by the ratio \\(\\frac{\\eta}{\\mu}\\) where\n\\(\\eta\\) is the curvature-couple-stress module while \\(\\mu\\) is the shear\nmodulus from Cauchy elasticity. The results in terms of the resulting band\nstructure are shown in \\cref{fig:homogeneous}, where we used a\n\\(16\\times16\\) mesh and \\(\\ell^2\/d^2 = 3\/8\\). For a conceptual description of\nthe reciprocal space and a guide on how to interpret the results in a Bloch\nanalysis the reader is referred to \\cite{Valencia_periodic}. Note that this set\nof results is directly comparable with the curves from the closed form solutions\nfrom \\cref{fig:dispersion_analytic}. Since the material is isotropic there are\nno directional effects and, as discussed previously, the only difference between\nthis model and the result from classical elasticity is the dispersive behavior\nof the shear wave. In contrast with the micropolar model \\citep{guarin2020},\nthe present C-CST model does not exhibit additional rotational waves.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[height=2.8 in]{img\/comparison_analytical_mod.pdf}\n\\caption{Dispersion relations for a homogeneous material model. Solid lines\nrepresent FEM results while markers correspond to the analytic solution.\nTriangular and filled-dots describe the P and SV wave modes, respectively.}\n\\label{fig:homogeneous}\n\\end{figure}\n\n\\Cref{fig:homog_lengths} shows the results for the same\nmaterial cell but now we have considered 4 different values of the length scale\nparameter corresponding to \\(\\ell\/d \\in [0.01, 0.1, 1, 10]\\). The mesh in this\ncase is \\(16\\times16\\). Notice that, as expected, the increasing value of this\nparameter only affects the dispersive response of the shear waves while the\nP-waves retain their classical non-dispersive behavior. As seen in\n\\eqref{eq:dispersion_SV} the dispersion increases for higher values of \\(\\ell\\)\ndue to the factor \\(\\sqrt{1 + k^2 \\ell^2}\\) in the dispersion relation. This\nbehavior is closely followed by the numerical results presented in\n\\cref{fig:homog_lengths}.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=5.5 in]{img\/homogeneous_lengths.pdf}\n\\caption{Dispersion relations for a homogeneous material model with varying\nlength scales, \\(\\ell\/d \\in [0.01, 0.1, 1, 10]\\). Solid lines represent FEM\nresults while markers correspond to analytic solutions. As expected, for\nincreasing \\(\\ell\/d\\) the S-wave presents more dispersion due to the factor\n\\(\\sqrt{1 + k^2 \\ell^2}\\), as presented in \\eqref{eq:dispersion_SV}.}\n\\label{fig:homog_lengths}\n\\end{figure}\n\nAs an additional verification, we also tested the convergence\nin the  calculation of the dispersion relations after considering the first 8\nmodes  for a sequence of meshes of \\(1\\times1\\),  \\(2\\times2\\), \\(4\\times4\\),\nand \\(8\\times8\\) elements for \\(\\ell^2\/d^2 = 3\/8\\). The error in the eigenvalue\ncomputation was measured according to\n\\[e = \\frac{\\Vert\\vb*{\\omega}_\\text{ref} - \n\\vb*{\\omega}_h\\Vert_2}{\\Vert\\vb*{\\omega}_\\text{ref}\\Vert_2}\\, ,\\]\nwhere \\(\\vb*{\\omega}_h\\) is the set of eigenvalues (dispersion relation) for a \nmesh of characteristic element size \\(h\\) and \\(\\vb*{\\omega}_\\text{ref}\\) is the \nsolution corresponding to the finer \\(16\\times16\\) elements mesh, which has been\ntaken as reference. The results for this sequence, together with the variation\nin the  error parameter, are displayed in \\cref{fig:convergence}. The estimated \nconvergence rate for the eigenvalues is 2.32. We see that when\nwe refine the mesh it can reproduce the dispersion curves better for higher\nfrequencies. There are still some differences between the \\(8\\times8\\) and\n\\(16\\times16\\) meshes around the dimensionless frequency of 15 but these\ndifferences will disappear with further refinement. Nevertheless, opposed to\nwhat happens in classical continua we would need more points per wavelength\nevery time that we want to increase the maximum frequency. This is due to the\ninherent dispersive behavior of S-waves as can be seen in equation\n\\eqref{eq:dispersion_SV}. Thus, we would expect to need more than 10 points per\nwavelength, customary for finite element methods, or 5, customary for spectral\nelement methods \\citep{komatitsch1999introduction, ainsworth2009dispersive,\nguarin2015}. Again, we should emphasize the dispersive nature of the SV-waves,\nwhile the P-waves remain non-dispersive.\n\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=5.75 in]{img\/convergence.pdf}\n\\caption{Convergence of the first 8 modes in the dispersion relations at\n\\(\\ell^2\/d^2 = 3\/8\\) for a sequence of meshes with: \\(1\\times1\\), \\(2\\times2\\),\n\\(4\\times4\\), and \\(8\\times8\\) elements --- presented as solid blue lines in the \nbackground. The results are compared with a mesh that has \\(16\\times16\\)\nelements --- presented as dots in the foreground. The estimated convergence rate\nfor the eigenvalues in the 2-norm is 2.32.}\n\\label{fig:convergence}\n\\end{figure}\n\n\n\\subsection{Dispersion in cellular material with a circular pore}\nNow, we consider a C-CST composite material cell configured by a circular pore\nembedded in a homogeneous matrix. The presence of the pore provides the model\nwith a second length-scale due to the microstructure, in addition to the one\nintroduced by the material length-scale parameter \\(\\ell\\). For\nillustration, we assume a pore diameter \\(a\\) that is half the cell length\n(i.e., \\(a=d\\)). Thi is equivalent to a porosity of \\(\\pi\/16\\) or approximately\n0.196, which is kept fixed as we modify the size of the unit cell to control\n\\(\\ell\/a\\).\n\nThe resulting dispersion curves for this cellular material\nwith four different length scale ratios are shown in \\cref{fig:pore}\nwith \\(\\ell\/a = [0.01, 0.1, 1, 10]\\). In contrast to the results from the fully\nhomogeneous material cell, the presence of the circular pore introduces\nscattering effects inside each cell and the composite shows much more\ncomplicated elastodynamic behavior.  Most importantly, however, the dispersion\ncurves become more regular with increased \\(\\ell\/a\\) and partial bandgaps\nopen up along the \\(\\Gamma M\\) and \\(X\\Gamma\\) directions, especially for\n\\(\\ell\/a = 1\\) and \\(\\ell\/a = 10\\). This type of band structure is not\nobserved for classical elastodynamic cells with a similar geometric periodicity,\nwhich would exhibit behavior close to that obtained here with \\(\\ell\/a = 0.01\\).\nIn fact, as \\(\\ell\/a \\rightarrow 0\\), C-CST theory recovers the classical\nresult.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=5.5 in]{img\/cpore_lengths_mod.pdf}\n\\caption{Dispersion for a cellular material with circular pores for varying \nlength scales, $\\ell\/a = [0.01, 0.1, 1, 10]$.}\n\\label{fig:pore}\n\\end{figure}\n\nFurthermore, it is important to note that this interesting band structure is\nobtained with a consistent continuum mechanics formulation, which requires only\na single additional material parameter, \\(\\ell\\), beyond those needed in the\nclassical elastic case. When this intrinsic length scale is on the order of the hole\ndiameter, the dispersive SV wave has a fundamental group velocity for the cell\nwith dimension \\(2d\\) approximately equal to the group (and phase) velocity of\nthe non-dispersive P wave, which allows the SV and P branches to follow a\nsimilar path, causing band gaps to open. This behavior, which is not seen in\nclassical elastodynamics, occurs in the regions near to where the two branches\nintersect in \\cref{fig:dispersion_analytic}. Consequently, under C-CST,\nthese band gaps will originate whenever the size of the cellular structure is\ntuned to the material length scale, a potentially significant phenomenon that\nhas not been recognized previously. With further tuning of the porosity level\nand cell size, it may be possible to achieve even a complete bandgap at\nrelatively low non-dimensional frequency \\(\\Omega\\).\n\n\n\\section{Conclusions}\n\nThe present work incorporates several innovative aspects. First of all, we\nhave developed a novel frequency domain correlated action principle for the\nconsistent couple stress theory (C-CST) of \\cite{hadjesfandiari2011couple} and\nused that to extend the Lagrange multiplier finite element algorithm of\n\\cite{darrall2014} to study periodic cellular materials through Floquet-Bloch\ntheory from solid state physics. Particularly, we have addressed the imposition\nof extended Bloch boundary conditions for this material model where in\naddition to force tractions and displacements there are also couple tractions\nand rotations. Secondly, we also discussed numerical aspects related to the\nsolution of the wavenumber dependent generalized eigenvalue problem resulting\nfrom the imposition of the Bloch periodic boundary conditions, overcoming\ncomplications arising from the inclusion of Lagrange multipliers and a\nnon-positive definite mass matrix. The implementation was\nshown to give accurate results for homogeneous and porous unit cells and for\nvarying couple stress material length-scale parameters.\n\nThe analysis of the first cell was used mainly to test the correctness of our\nimplementation as this material has a closed-form dispersion relation. The\nalgorithm was shown to correctly capture the non-dispersive P-wave as well as\nthe dispersive SV-wave. This analysis was complemented by a convergence analysis\nwith four different meshes of increasing refinement for the material cell. The\nobserved convergence rate shows that the Lagrange multiplier algorithm is\neffective in maintaining continuity by imposing the newly introduced kinematic\nconstraint implicit in the mean curvature tensor definition. As the final\ncontribution, we have discovered the interesting bandgap structure of a material\ncell with a circular pore embedded in a homogeneous matrix, which reveals the\nappearance of bandgaps introduced by the kinematic features of C-CST and the\ndispersive behavior of the SV-waves defined in terms of the microstructural\nlength scale parameter.\n\nFrom a general perspective, C-CST is a true size-dependent continuum theory,\nwhich is intended here for periodic elastic material cells at scales for which a\ncontinuum representation is appropriate.  From the results shown in the paper,\nC-CST becomes important when the size of the cell in on the order of the\nintrinsic length scale parameter or smaller.  For larger cells, the classical\ntheory can be used instead. On the other hand, micropolar theory disconnects the\nrotational field from the displacements, which can lead to approximations that\nmay or may not be physical.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\tConvolutional Neural Networks (CNNs) have been successfully used in various tasks with Euclidean structure data in recent years. For non-Euclidean structure datasets, graph convolutional networks (GCNs) use the same idea to extract the topological structure information.\n\t\n\t\\cite{bruna2014spectral} firstly proposed two GCN models with spectral and spatial construction respectively. The spectral model using a Laplacian matrix to aggregate neighborhood information of each node in a graph. The spatial model partition graph into clustering and update them by aggregating function. In order to extract deeper features, model based on CNNs usually deepen the model's layers. While in GCNs, Deepening layers will cause a lot of problems. In spectral construction method, too many layers lead to \\textbf{``over smoothing''}\\cite{li2018deeper,zhou2018graph}: the features of nodes in graph will tend to be the same. In spatial construction method, it will cause exponential growth of the number of sampled neighbor nodes, called \\textbf{``neighbor explosion''}. Node sampling and layer sampling\\cite{hamilton2018inductive,chen2018fastgcn,gao2018large,huang2018adaptive,ying2018hierarchical} were proposed to handle this problem, but due to incomplete sampling, the inaccuracy of nodes' representation accumulates errors between layers.\n\t\n\tTo address these two problems, in this paper, we propose a model called PathSAGE, which can learn high-order topological information by expanding the receptive field.\n\tFirstly, We design a path sampling technique based on random walk to sample paths starting from central nodes with different lengths, then the sequences of paths are fed into Transformer encoder\\cite{vaswani2017attention}, which can extract the semantic and distance information in sequence effectively. As shown in Fig. 1, We view the sequences of paths from the tail nodes to the central node as the central node' neighbors. Secondly, following this idea, we take the average of paths of the same length as the representation of the central node in this level of reception field. Finally, after concatenating the aggregated features of paths with different lengths, the final representation of the central node are used for downstream tasks.\n\t\n\tFor the two problems mentioned earlier, on the one hand, the aggregation of the central node only perform once in training of a sample, which never cased ``over-smoothing''. On the other hand, all the paths were sampled with a fixed number for representing the central node in our model, instead of recursively sampling exponentially growing neighbor nodes. And each path only contributes to the central node, we do not need to calculate and store the temporary representations of nodes from the middle layer. Furthermore, it prevents the error propagation caused by incomplete sampling. \n\t\n\tOur contribution can be summarized in three points:\n\t\n\t\\begin{itemize}\n\t\t\n\t\t\\item We utilize the path sampling to take place of the node sampling to avoid error accumulation caused by incomplete node sampling.\n\t\t\n\t\t\\item We propose and evaluate our model Path-SAGE to solve the existing ``neighour explosion'' and ``over-smoothing'' problems. the model can capture richer and more diverse patterns around the central node with only one layer of structure to the central node.\n\t\t\n\t\t\\item We evaluate our model on three inductive learning tasks, and it reaches the state-of-the-art performance on two of them. We analyze the attention weights of Transformer encoder and detect some patterns in the attention mechanism, which can further illustrate how the model works.\n\t\t\n\t\\end{itemize}\n\t\\section{Related work}\n\tGNNs model was initially proposed by \\cite{bruna2014spectral}, and the convolution operation in the traditional Euclidean structure was introduced into the graph network with the non-Euclidean structure in this article. They \\cite{bruna2014spectral} divided GNN into two construction methods: spectral construction and spatial construction. Subsequently, many studies are carried out around these two aspects. \n\t\n\tIn spectral construction, \\cite{defferrard2017convolutional} used Chebyshev polynomials with learnable parameters to approximate a smooth filter in the spectral domain, which improves computation efficiency significantly. \\cite{kipf2017semisupervised} further reduced the computational cost through local first-order approximation. In spatial construction, MoNet was proposed in \\cite{monti2017geometric}, developing the GCN model by defining a different weighted sum of the convolution operation, using the weighted sum of the nodes as the central node feature instead of the average value. \\cite{hamilton2018inductive} attempted various aggregator to gather the features of neighbor nodes. \\cite{velivckovic2017graph,zhang2018gaan} defined the convolution operation with a self-attention mechanism between the central node and neighbor nodes, \\cite{liu2019geniepath} brought the LSTM(Long Short Term Memory networks)\\cite{hochreiter1997long} from NLP (Natural Language Processing) to GNN, and built an adaptive depth structure by applying the memory gates of LSTM. \\cite{yang2021spagan} sampled the shortest path based on their attention score to the central node and combined them in a mixed way. PPNP and APPNP\\cite{klicpera2018predict} used the relationship between GCN and PageRank to derive an improved communication scheme based on personalized PageRank. \n\t\n\tWith the scale of graph data increasing, the full-batch algorithm is no longer applicable, and the mini-batch algorithm using stochastic gradient descent is applied. In recent years, some studies based on different sampling strategies have been proposed. \\cite{hamilton2018inductive} tried random neighbor node sampling for the first time to limit large amounts of nodes caused by recursive node sampling. Layer-wise sampling techniques was applied in \\cite{chen2018fastgcn,gao2018large,huang2018adaptive,ying2018hierarchical}, which only consider fixed number of neighbors in the graph to avoid ``neighbor explosion''. \\cite{chen2018fastgcn,chen2018stochastic} used a control variate-based algorithm that can achieve good convergence by reducing the approximate variance. Besides, subgraph sampling was first introduced by \\cite{zeng2019accurate}, which used a probability distribution based on degree. \\cite{chiang2019cluster} performed clustering decomposition on the graph before the training phase and randomly sampled a subgraph as a training batch at every step. \\cite{zeng2020graphsaint} sampled subgraph during training based on node and edge importance. All these sampling techniques are based on GCNs, how to extract deeper topology information from large graph datasets is still a problem.\n\t\\section{Proposed method}\n\tIn this section, we present the PathSAGE. Firstly, the sampling algorithm is introduced in 3.1. Secondly, we detail the aggregator in 3.2. Finally, we discuss the difference between our model and the related models.\n\t\\begin{algorithm} \n\t\t\\SetAlgoLined\n\t\t\\DontPrintSemicolon\n\t\t\\caption{Random Path Sampling}  \n\t\t\t\\KwIn{garph $G(V,E)$\\\n\t\t\tcentral node $c$\\\\\n\t\t\tsample depth $s$\\\\\n\t\t\tsample num each length $L =\\{n_1, n_2, \u2026, n_s\\}$\\} }\n\t\t\t\\KwOut{path sequcences with different length\\\\ \n\t\t\t$\\{\\bm{P}_1, \\bm{P}_2, ..., \\bm{P}_s\\}$}\n\t\t\t\\SetKwFunction{Fun}{Ramdom Path Sampling}\n\t\t\t\\Fun{$G$, $c$, $s$, $L$}{\n\t\t\t\t\n\t\t\t\\ForEach{$l = 0 \\to s$}{\n\t\t\t$i\\gets0$;\n\t\t\t\n\t\t\t\\While{$i < n_l$}{\n\t\t\t $u\\gets c$;\n\t\t\t \n\t\t\t $P\\gets \\{u\\}$;\n\t\t\t \n\t\t\t\\For{$j = 0 \\to l$}{\n\t\t\t $u \\gets$ Node randomly selected from $u$'s neighbors;\n\t\t\t \n\t\t\t $P\\gets P \\cup \\{u\\}$}\n\t\t \n\t\t\t $\\bm{P}_l \\gets \\bm{P}_l \\cup \\{P\\}$;\n\t\t\t \n\t\t\t $i \\gets i+1$;}\n\t\t \n\t\t\t}\n\t\t\\Return $\\{\\bm{P}_1, \\bm{P}_2, ..., \\bm{P}_s\\}$\n\t\t}\n\t\\end{algorithm}\n\t\\subsection{Path Sampling}\n\tExcept deepening model, another way to expand receptive field of CNNs is to increase the size of convolution kernel. Following this idea, we sample the node sequences starting from central node and regard these paths as the context of the central node. Therefore, the receptive field can be expanded by extending these paths.\n\t\n\tWe use a straightforward random sampling algorithm based on random walk, shown in \\textbf{Algorithm 1}: for a central node, a sampling starts from it and randomly selects a neighbor node each time until reaching the preset path length, and multiple path sequences for the corresponding central node of various lengths can be obtained in this way, constituting a training sample.\n\n\t\n\t\\begin{figure*}[htb]\n\t\t\\begin{center}\n\t\t\t\\includegraphics[height=4.5cm]{.\/fig1.pdf}\n\t\t\\end{center}\n\t\t\\caption{Mechanism of sampling and aggregation. (a) and (b) are two different possible training samples with the same central node. Paths with same colors are with same lengths same length and share the same aggregators. }\n\t\n\t\t\\label{fig1}\n\t\\end{figure*}\n\t\n\t\\begin{figure}[htbp]\n\t\t\\begin{center}\n\t\t\t\\includegraphics[height=4.5cm]{.\/fig2.pdf}\n\t\t\\end{center}\n\t\t\\caption{Structure of first aggregation in aggregator. A specific sampled path's features adds the position embedding, through Transformer encoder layers to fuse the information.\n\t\t}\n\t\t\\label{fig2}\n\t\\end{figure}\n\t\n\t\\subsection{Aggregator}\n\tThere are two aggregations in the aggregator: the first one aggregate path sequences; the second one aggregate different paths as the final representation of the central node. \t\n\tFor the first aggregation, We formulate each path sequence as:\n\t\\begin{equation}\n\t\t{\\bm{P}}_{ij}^{}\\; = \\;\\{ {\\vec a_{ij}^1,\\;\\vec a_{ij}^2,\\;\\vec a_{ij}^3,\\;...,\\;\\vec a_{ij}^j\\;} \\}\n\t\\end{equation}\n\n\t\n\tTo utilize the position information of path sequence, we define positions of nodes in paths as their distances to the central nodes. As same as Transformer, we add nodes' features and their positional vectors together as the input of the structure:\n\t\\begin{equation}\\label{eq6}\n\t\t\\bm{\\tilde{P}}_{ij}\\; = \\;\\{ {a_{ij}^1\\; + \\;pos\\_emb\\left( 1 \\right),...,\\;a_{ij}^j\\; + \\;pos\\_emb\\left( j \\right)} \\}\n\t\\end{equation}\n\tFor the $pos\\_emb(\\cdot)$, we use $sine$ and $cosine$ functions of different frequencies to generate the positional vectors. In each dimension, the positional embedding is:\n\t\n\t\\begin{equation}\\label{eq7}\n\t\tpos\\_emb(p)_{2i} = sin(\\frac{{p}}{{{{10000}^{\\frac{{2i}}{{{d}}}}}}})\n\t\\end{equation}\n\t\n\t\\begin{equation}\\label{eq8}\n\t\tpos\\_emb(p)_{2i + 1} = cos(\\frac{{p}}{{{{10000}^{\\frac{{2i}}{{d}}}}}})\n\t\\end{equation}\n\tWhere $p$ is the position in the sequences, $2i$ and $2i+1$ mean the odd and even dimensions of position embedding, and $d$ is the number of features. \n\t\n\tAfter that, we apply the Transformer encoder on each path: \n\t\n\t\\begin{equation}\\label{eq9}\n\t\t\\bm{\\tilde{P}}_{ij}^{k} = transformer\\_bloc{k^k}(\\bm{\\tilde{P}}_{ij}^{k-1})\n\t\\end{equation}\n\tWhere $k$ means $k$-th Transformer encoder layer. Noted that, in a $m$-layer Transformer encoder, the output of the last layer $\\bm{\\tilde{P}}_{ij}^{m}$ is a sequence of features. We only take the output at position 0 as the final representation of the path sequence.\n\t\n\t\\begin{equation}\\label{eq10}\n\t\t\\tilde{P}_{ij}' = \\left[\\bm{\\tilde{P}}_{ij}^{m}\\right]_{0}\n\t\\end{equation}\t\n\t\n\tFollowing these equations (\\ref{eq6})-(\\ref{eq10}), we can obtain representations of paths with different length to the central node, as shown in Fig. 1. In the second aggregation, we apply an average pooling layer to aggregate the central node paths. Then we concatenate all the path representation and apply a feed-forward layer with non-linearity to fuse these features. The final output of a central node $C'$ is computed as following:\n\t\n\t\\begin{equation}\n\t\t\\begin{array}{c}\n\t\t\tC = \\;concat\\left( {{C_1},\\;...,\\;{C_s}} \\right)\\\\\n\t\t\t\\\\\n\t\t\twhere\\;{C_i}\\; = \\;Average\\left( \\tilde{P}_{i1}',\\;...,\\;\\tilde{P}_{in}' \\right)\n\t\t\\end{array}\n\t\\end{equation}\n\t\n\t\\begin{equation}\n\t\t{C'}\\; = \\;max\\left( {0,\\;C{\\bm{W}_1}\\; + \\,{b_1}} \\right){\\bm{W}_2}\\; + \\;{b_2}\n\t\\end{equation}\n\twhere $s$ denotes the sample depth, $n$ denote the number of paths sampled in\n\t\n\t\\subsection{Comparisons to related work}\n\n\t\\begin{itemize}\n\t\t\\item We introduce a sequence transduction model into our structure, but it is distinct from related work based on these models. LSTM\\cite{hochreiter1997long} was also used in GraphSAGE\\cite{zeng2020graphsaint} to aggregate node's features, it is very sensitive to the order of the input sequence. In contrast, neighbor nodes are disordered, and the authors have rectified it by consistently feeding randomly-ordered sequences to the LSTM. In our model, paths to the central node already have orders, which are naturally suitable for sequence processing model. \n\t\t\\item The difference between our attention mechanism and GAT is that we collect all the mutual attention information of each node in the path. The output of our model is integrated information of entire sequence, instead of only attend to the central node.\n\t\t\\item Path sampling method is also used in SPAGAN\\cite{yang2021spagan}. But each sampled path in SPAGAN is the shortest one to the central node, this sample technique will leads to a high computational overhead and limits the ability of the model to be applied to large graph datasets. By contrast, we sample paths randomly, which greatly save the computation overhead. Besides, in our sample algorithm, the same node may have a different path to the central node, which may help the model acquire more diverse patterns around the central node while saving the computational overhead. \n\t\\end{itemize}\n\t\\section{Experiments}\n\tIn this section, we introduce the datasets and the experiment setting in 4.1 and 4.2 respectively. we present the results of evaluation in 4.3. \n\t\\begin{table}[htbp]\n\t\t\\caption{Summary of inductive learning tasks' statistics.}\n\t\t\\begin{center}\n\t\t\n\t\t\t\\centering\n\t\t\t\\renewcommand{\\arraystretch}{2.2}\n\t\t\n\t\t\t\\setlength{\\tabcolsep}{4mm}{\n\t\t\t\t\\begin{tabular}{|c|c|c|c|}\n\t\t\t\t\t\\hline\n\t\t\t\t\t&     \\textbf{Reddit} &       \\textbf{Yelp} &     \\textbf{Flickr} \\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\ttype & single-label & multi-label & multi-label \\\\\n\t\t\t\t\t\n\t\t\t\t\t\\#Node &    232,965 &    716,847 &     89,250 \\\\\n\t\t\t\t\t\n\t\t\t\t\t\\#Edges & 11,606,919 &  6,977,410 &    899,756 \\\\\n\t\t\t\t\t\n\t\t\t\t\t\\#Features &        602 &        300 &        500 \\\\\n\t\t\t\t\t\n\t\t\t\t\t\\#Classes &         41 &        100 &          7 \\\\\n\t\t\t\t\t\n\t\t\t\t\t\\makecell[c]{Train \/ Val \\\\\/ Test} & \\makecell[c]{66\\% \/ 10\\% \\\\\/ 24\\%} & \t\\makecell[c]{75\\% \/ 10\\% \\\\\/ 15\\%} & \\makecell[c]{50\\% \/ 25\\% \\\\\/ 25\\%} \\\\\n\t\t\t\t\t\\hline\n\t\t\t\\end{tabular} }\n\t\t\\end{center}\n\t\t\\label{tab2}\n\t\\end{table}\n\t\\subsection{Dataset}\n\tThree large-scale datasets are used to evaluate our model: 1) \\textbf{Reddit}\\cite{hamilton2018inductive}: a collection of monthly user interaction networks from the year 2014 for 2046 sub-reddit communities from Reddit that connect users when one has replied to the other, 2) \\textbf{Flickr}: a social network built by forming links between images sharing common metadata from Flickr. 3) \\textbf{Yelp}\\cite{zeng2020graphsaint}: a network that links up businesses with the interaction of its customers. Reddit is a multiclass node-wise classification task; Flickr and Yelp are multilabel classification tasks. \n\tThe detail statistics of these datasets are shown in Tab 1.\n\t\n\t\\begin{table}[t]\n\t\t\n\t\t\\caption{Perforamnce on inductive learning tasks(Micro-F1).}\n\t\t\\begin{center}\n\t\t\n\t\t\t\\centering\n\t\t\t\n\t\t\t\\renewcommand{\\arraystretch}{2}\n\t\t\t\\setlength{\\tabcolsep}{4mm}{\n\t\t\t\t\\begin{tabular}{|c|c|c|c|}\n\t\t\t\t\t\\hline\n\t\t\t\t\t\\textbf{Model} &     \\textbf{Reddit} &       \\textbf{Yelp} &     \\textbf{Flickr} \\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\tGCN & 0.933$\\pm$0.000 & 0.378$\\pm$0.001 & 0.492$\\pm$0.003 \\\\\n\t\t\t\t\t\n\t\t\t\t\tGraphSAGE & 0.953$\\pm$0.001 & 0.634$\\pm$0.006 & 0.501$\\pm$0.013 \\\\\n\t\t\t\t\t\n\t\t\t\t\tFastGCN & 0.924$\\pm$0.001 & 0.265$\\pm$0.053 & 0.504$\\pm$0.001 \\\\\n\t\t\t\t\t\n\t\t\t\t\tS-GCN & 0.964$\\pm$0.001 & 0.640$\\pm$0.002 & 0.482$\\pm$0.003 \\\\\n\t\t\t\t\t\n\t\t\t\t\tAS-GCN & 0.958$\\pm$0.001 &     -      & 0.504$\\pm$0.002 \\\\\n\t\t\t\t\t\n\t\t\t\t\tClusterGCN & 0.954$\\pm$0.001 & 0.609$\\pm$0.005 & 0.481$\\pm$0.005 \\\\\n\t\t\t\t\t\n\t\t\t\t\tGraphSAINT & 0.966$\\pm$0.001 & \\bf{0.653$\\pm$0.003} & 0.511$\\pm$0.001 \\\\\n\t\t\t\t\t\n\t\t\t\t\tours & \\bf{0.969$\\pm$0.002} & 0.642$\\pm$0.005 & \\bf{0.511$\\pm$0.003} \\\\\n\t\t\t\t\t\\hline\n\t\t\t\\end{tabular} }\n\t\t\t\\label{tab4}\n\t\t\\end{center}\n\t\\end{table}\n\t\\subsection{Experiment setup}\n\t\n\tWe build our model on Pytorch framework\\cite{paszke2017automatic} and construct the Transformer encoder based on UER\\cite{zhao2019uer}. For all tasks, We train the model with Adam SGD optimizer\\cite{kingma2014adam} with learning rate 1e-3 and a learning rate scheduler with 0.1 warmup ratio. We use two layers Transformer, each of which has 8 attention heads to gather the features of paths. The batch size is 32. The dropout layers are applied between each sub-layer in Transformer layer. We use different dropout rates in the output layer and Transformer encoder, which are 0.3 and 0.1 respectively. We set the sampling length of the path ranging from 1 to 8 (depth $s = 8$), and the number of paths sampled in each length are [5, 5, 5, 5, 5, 10, 10, 10]. For multi-label classification task Flickr and Yelp, the final output is obtained through a $sigmoid$ activation, and in Reddit, the final output is obtained through a $softmax$ activation. The hidden dimension is 128 for Reddit and Flickr, 512 for Yelp. \n\t\n\t\\subsection{Result}\n\tWe compare our model with seven state-of-the-art model:  GCN\\cite{kipf2017semisupervised}, GraphSAGE\\cite{hamilton2018inductive}, FastGCN\\cite{chen2018fastgcn}, S-GCN\\cite{chen2018stochastic}, AS-GCN\\cite{huang2018adaptive}, ClusterGCN\\cite{chiang2019cluster}, GraphSAINT\\cite{zeng2020graphsaint}. GraphSAGE uses a random node sampling and LSTM aggregator. FastGCN and S-GCN use a control variate-based algorithm that can achieve good convergence by reducing the sampling variance. ClusterGCN performs clustering decomposition on the graph before the training phase and randomly sampling a subgraph as a training-batch at every step. GraphSAINT samples subgraph while training with several sampling methods based on node and edge importance. The evaluation on these tasks uses the $Micro-F1$ metric, and report the mean and confidence interval of the metrics by five runs.\n\t\n\tThe results of inductive learning experiments are shown in Tab 2. As we can see from the table, For Reddit, our model outperforms all the baseline models. For Flickr, we achieve a comparable F1-score with the top-performing model. For Yelp, we surpass most GCN models, second only to GraphSAINT. One hypothesis to explain the difference of the results is: Reddit and Flickr have more training samples and number of nodes features, which makes the attention mechanism have enough data to capture the relationships between nodes.\n\t\n\t\\begin{figure*}[t]\n\t\t\\begin{center}\n\t\t\t\\includegraphics[height=4.7cm]{.\/emp.pdf}\n\t\t\\end{center}\n\t\t\\caption{Detection of attention weights in Transformer that node have the same label in the path sequence may receive higher weights at some attention heads. The heatmap in the upper-left corner is the weight of position 4 of different layers attend to position 2 and attention heads, the right one is the complete heatmap of the selected head. The lower-left picture shows the labels of the nodes in the sequence. }\n\t\t\\label{fig3}\n\t\\end{figure*}\n\t\n\t\\section{Attention Analysis}\n\tWe observed the attention weight of the trained model in the PubMed test set. At some attention heads, we find that nodes with the same labels get very high attention scores to each other. We visualize an example in Fig 3. Central node 0 and node 4 have the same label and receive extremely high attention scores on second attention heads of the first layer. By observing the attention scores for the whole sequence, we can see that this score also occupies a significant share in the sequence. \n\t\n\tThis observation proves that the attention mechanism can successfully capture the information that is helpful to the downstream tasks in the aggregation of paths.\n\t\n\t\\section{Conclusion}\n\tIn this paper, we propose a model---PathSAGE, which can expand the size of receptive field without stacking layers which solved the problem of ``neighbor explosion'' and ``over smoothing''. Regarding all paths as the neighbor nodes of the central node, PathSAGE samples paths starting from the central node based on random walk and aggregates these features by a strong encoder---Transformer. Consequently, the model can obtain more features of neighbor nodes and more patterns around the central node. In our experiment, PathSAGE achieves the state-of-the-art on inductive learning tasks. Our model provides a novel idea to handle large-scale graph data.\n\t\n\t\\section*{Acknowledgments}.\n\tThis work is supported by National Key R\\&D Program of China (No.2017YFB1402405-5), and the Fundamental Research Funds for the Central Universities (No.2020CDCGJSJ0042). The authors thank all anonymous reviewers for their constructive comments.\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Conclusion}\nIn this paper, we propose a Spatio-Temporal feature auto-selective (STAS) approach that can automatically extract optimal Spatio-Temporal (ST) features hidden in EC for Bias Correcting on Precipitation (BCoP). Experiments on EC benchmark datasets in Eastern China indicate that STAS achieves the highest threat score (TS) on BCop than other 8 algorithms, and has a strong correcting ability in dealing with different degree of precipitation, especially for heavy precipitation. In the future, we will study how to employ ST mechanisms on more complex precipitation scenarios such as squall line, severe convection and thunderstorm.   \n\n\n\n\\section{Experiments}\nWe conduct all experiments on time-series EC benchmarks collected from 1) the high-resolution version of the public European Centre (EC) dataset and 2) Meteorological Information Comprehensive Analysis and Process System (MICAPS)~\\cite{luo2006introduction} that can provide the labels of 5 Meteorological Elements (MEs) including precipitation, temperature, pressure, wind, and dew. Our experiments on Bias Correcting on Precipitation (BCoP) mainly contain two parts. The first part compares the Spatio-Temporal feature Auto-Selective (STAS) model with 8 published machine learning (ML) methods. The second part is a set of ablation experiments on STAS. \n\n\\begin{table}[htp]\n\\LARGE\n\\caption{The details of STAS. $N \\bigtimes C \\bigtimes \\ell \\bigtimes(h \\bigtimes w)_{29\\sim3}$ are multi-scale (from 29*29 to 3*3) dimensions of inputs. $(*)_{[.]}$ represent a operator layer and its parameters setting such as filter size from CNN and output scale from Adaptive Pooling (ADP) or Up-Sampling (UpSp). Specifically, the last parameter of Deformable CNN (D-CNN) is spatial dilation rate. Besides, $\\oplus Noisy$ is addition operation of Gaussian noise, and $(\\cdot)$ is shape of outputs in current module along with pipeline $\\rightarrow$.}\n\\smallskip\n\\centering\n\\resizebox{230pt}{25mm}{\n\\smallskip\\begin{tabular}{l|l|l}\n\\hline\n\\multicolumn{3}{c}{{\\bf Inputs:} $N \\bigtimes C \\bigtimes \\ell \\bigtimes(h \\bigtimes w)_{29\\sim3}$} \\\\\n\\hline \n\\multirow{2}*{SFM} & $CNN_{[1\\times1,0]\\rightarrow[3\\times3,0]}$ & $(N\\bigtimes C\\bigtimes (\\ell-1)\\bigtimes (h\\bigtimes w))$ \\\\\n~ & $D-CNN_{[3\\times3,1,0.8]\\rightarrow[3\\times3,1,0.6]} \\rightarrow ADPooling_{[1\\times1]} \\rightarrow FC$ & $\\rightarrow (N\\bigtimes C \\bigtimes(\\ell-1)\\bigtimes(1\\bigtimes1)) \\rightarrow(N)\\bigtimes(\\ell-1) \\Rightarrow \\vct{s}^*$\\\\\n\\hline\n\\hline\n\\multirow{2}*{Encoder} & $CNN_{[1\\times1,0]\\rightarrow[3\\times3,0]} \\rightarrow ADP_{[16\\times16]}$ & $ \\rightarrow (N\\bigtimes C \\bigtimes \\ell \\bigtimes16\\bigtimes16)$ \\\\\n~ & $\\oplus Noisy$ & $\\oplus \\rightarrow (N\\bigtimes C \\bigtimes \\ell \\bigtimes18\\bigtimes18)$ \\\\\n\\hline\n\\hline\n\\multirow{2}*{TFM} & $3DCNN_{[3\\times3,1]} \\rightarrow (ADP_{[1\\times1]}+FC) \\bigtimes 3$ & $\\rightarrow(N\\bigtimes C\\bigtimes18\\bigtimes18) \\rightarrow(N\\bigtimes6) \\bigtimes 3$  \\\\\n~ & $Concat\\rightarrow FC\\bigtimes2$ & $\\rightarrow (N\\bigtimes18) \\rightarrow (N) \\Rightarrow l^{*}$\\\\\n\\hline\n\\hline\nDecoder & $UpSp_{[18\\times18]}$ & $\\rightarrow (N \\bigtimes C\\bigtimes \\ell \\bigtimes 18 \\bigtimes 18)$ \\\\\n\\hline\n\\hline\nRC & $ CNN_{[3\\times3,1]}\\rightarrow ADP_{[1\\times1]} \\rightarrow FC$ & $(N\\bigtimes C\\bigtimes29\\bigtimes29) \\rightarrow (N\\bigtimes C\\bigtimes 1\\bigtimes1) \\rightarrow (N)$ \\\\\n\\hline\n\\multicolumn{3}{c}{{\\bf Output:} $N$} \\\\\n\\hline\n\\end{tabular}\n}\n\\label{details}\n\\end{table}  \n\n\\subsection{Datasets and Training Details}\n\\noindent {\\bf EC benchmarks (ECb)} are sliced from a high-resolution version of the public EC dataset~\\cite{berrisford2009era}, only covering Eastern China between ranging from June 1st to August 31st for three years (2016-2018). Concretely, ECb consists of 57 weather features (channels) worldwide selected from 601 meteorological factors by Pearson correlation analysis~\\cite{benesty2009pearson}. Every feature stems from a grid where each pixel in the grid means a specific location. The spatial and temporal resolutions of ECb are about $111km$ per pixel and 6h per time-level respectively. Then ECb is divided into four datasets for different experiments. The first three datasets are 1) ECb only including Tiny rainfall (ECbT), 2) ECb only including Moderate rainfall (ECbM) and 3) ECb only including Heavy rainfall (ECbH) separated into half-open range of precipitation intensity interval ($[0,1mm)$, $[1-10mm)$, $[10mm,+\\infty)$). The last one is ECb Mixed 3 rainfall above (ECbMi) and its mixture ratio of samples is $T:M:H = 9:3:1$.\n\n\\noindent {\\bf Labels} are observations of 5 Meteorological Elements (MEs) from MICAPS in the specific locations in Eastern China every $6h$.\n\n\\noindent {\\bf Training details} are shown in Tab.~\\ref{details}. The table includes the main structure of layers and initial settings of modules in STAS. Besides, the batch size of training is 256 and the testing is 64. We employ Adam as optimizers for all modules in STAS and the learning rate is $1e-4$. Besides, the weight ratios on 5MEs for spatio-temporal losses are set as $2:1:1:1:1$ in which rainfall is 2 and other elements are 1. The uniform scale after upsampling and downsampling is set as $16*16$. For ordinal regression and C3AE, the ranking intervals for precipitation value are set as 0.5 and 1.5 respectively. Specifically, we set a constant standard $1e-3$ as white Gaussian noise. We test our model in every 6 epochs on training and the max epoch of training is 80. Finally, all experiments are conducted in 8 NVIDIA GPUs.\n\\begin{table}[htp]\n\\caption{The 5 criteria between 8 machine learning methods and STAS on ECbMi as in Tab.~\\ref{contrastiveI}. $TS_{0.1}$ is TS score in $PI>0.1$. $TS_{1} | TS_{10}$ is $PI>1$ and $PI>10$ respectively. The ECb forecasts are results of predictions from EC benchmarks themselves. SVR is support vector regression, LR is linear regression, MLP is multilayer perceptron, FCN is full convolutional network, FPN is feature pyramid network, LSTM long short-term memory, OBA is ordinal boosting auto-encoder.}\n\\smallskip\n\\centering\n\\resizebox{240pt}{20mm}{\n\\smallskip\\begin{tabular}{c|c|c|c|c|c}\n\\hline \n       &\\multicolumn{5}{|c}{Criteria} \\\\\n\\hline \nMethods& MAE & MAPE & $TS_{0.1}$ & $TS_{1}$ & $TS_{10}$ \\\\\n\\hline \nECb forecasts~\\cite{ran2018evaluation}  & 1.76 & 17.09 & 0.41 & 0.31 & 0.19 \\\\\n\\hline\nSVR~\\cite{srivastava2015wrf} & 1.67 & 15.81 & 0.48 & 0.37 & 0.1 \\\\\n\\hline\nLR~\\cite{hamill2012verification} & 1.73 & 16.90 & 0.35 & 0.35 & 0.21 \\\\\n\\hline\nMLP~\\cite{yuan2007calibration} & 1.59 & 15.13 & 0.46 & 0.39 & 0.21 \\\\\n\\hline\nFCN~\\cite{xu2019towards} & 1.26 & 12.30 & 0.49 & 0.48 & 0.24 \\\\\n\\hline\nFPN~\\cite{lin2017feature} & 1.15 & 7.38 & 0.56 & 0.51 & 0.27 \\\\\n\\hline\nLSTM~\\cite{xingjian2015convolutional} & 1.21 & 9.8 & 0.52 & 0.48 & 0.24 \\\\\n\\hline\nOBA~\\cite{xu2019towards} & 1.01 & 8.96 & 0.58 & 0.53 & 0.25 \\\\\n\\hline\nSTAS(ours) & {\\bf 0.98} & {\\bf 5.84} & {\\bf 0.75} & {\\bf 0.69} & {\\bf 0.38} \\\\\n\\hline\n\\end{tabular}\n}\n\\label{contrastiveI}\n\\end{table} \n\\begin{table*}[t]\n\\centering\n\\caption{Ablation experiments conducted on ECbMi using $TS_{1}$ and $MAPE$. $\\surd$ is defined as an existing component in current ablated STAS for every line of Table. Instead, $\\times$ is no this component in the ablation. SFM-MSMs and TFM-MTMs are spatial and temporal meteorological elements modules severally. Besides, D-CNN is deformable CNN and R, T, P, W, and D represents the module of Rainfall, Temperature, Pressure, Wind and Dew respectively.}\n\\fontsize{8}{9}\\selectfont\n    \\begin{tabularx}{15.5cm}{@{\\extracolsep{\\fill}}|c|c|ccccc|ccccc|c|c||c|c|}\n    \\hline\n    \\multirow{2}{*}{SFM} & \\multirow{2}{*}{TFM} & \\multicolumn{5}{c|}{SFM-MSMs} & \\multicolumn{5}{c|}{TFM-MTMs} & \\multirow{2}{*}{D-CNN} & \\multirow{2}{*}{C3AE} & \\multirow{2}{*}{$TS_1$} & \\multirow{2}{*}{$MAPE$} \\\\\n\n\n   \n          &  & R & T & P & W & D & R & T & P & W & D &  &  &  &  \\\\\n    \\hline\n    $\\surd$ & $\\times$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & \\multicolumn{5}{c|}{N\/A} &$\\surd$ & N\/A & 0.60 & 8.58 \\\\\n    \\hline\n    $\\times$ & $\\surd$ & \\multicolumn{5}{c|}{N\/A} & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ &N\/A & $\\surd$ & 0.64 & 8.01 \\\\\n    \\hline\n    $\\surd$ & $\\surd$ & $\\surd$ & $\\times$ & $\\times$ & $\\times$ & $\\times$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & 0.62 & 7.97 \\\\\n    \\hline        \n    $\\surd$ & $\\surd$ & $\\times$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & 0.65 & 8.15 \\\\\n    \\hline        \n    $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\times$ & $\\times$ & $\\times$ & $\\times$ & $\\surd$ & $\\surd$ & 0.65 & 7.03 \\\\    \n    \\hline        \n    $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\times$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & 0.66 & 7.94 \\\\   \n    \\hline        \n    $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\times$ & $\\surd$ & 0.67 & 6.08 \\\\\n    \\hline        \n    $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\times$ & 0.69 & 6.81 \\\\    \n    \\hline        \n    $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & 0.70 & 5.96 \\\\ \n     \\hline           \n    \\end{tabularx}\n  \\label{ablation}\n\\end{table*}\n\\subsection{Evaluation Metrics}\nMAE and MAPE are regarded as two evaluation criteria for training model.  Here MAE~\\cite{willmott2005advantages} is defined as the Mean Absolute Error of corrected precipitation, and MAPE is a variant of MAE without rainless sampless ($<1mm$). In numerical weather prediction, threat score (TS) is a standard criterion for evaluating the accuracy of forecast~\\cite{mesinger2008bias} as follows:\n\\begin{equation} \\label{reconsLoss}\nTS_{\\rho} = H_{\\rho}\/(H_{\\rho} + M_{\\rho} +FA_{\\rho})\n\\end{equation}where $H_{.}$ is Hit (correction = 1, truth = 1), $M_{.}$ is Miss (correction = 0, truth = 1), and $FA_{.}$ is False Alarm (correction = 1, truth = 0), in which 1 is rainfall and 0 is rainless. Specifically, $\\rho$ is a threshold for splitting the range of Precipitation Intensity (PI) into two intervals and set $[0.1,1,10]$ for three different rainfall cases.\n\\subsection{Contrastive Experiments on ECbMi}\nWe list the assessment results of 8 methods and our models on the ECbMi show in Tab.~\\ref{contrastiveI}. The reported results are the average of 20 repetitions, each of which is the mean of predicted results on all batches. STAS outperforms all the other methods on five criteria and its $TS_{10}$ is $28.94\\%$ higher than the second highest result from OBA in this case. Meanwhile, the performance of FPN and LSTM can extract the spatial and temporal features severally beyond the traditional methods from third line to seventh line. Furthermore, FPN has preferable performance than LSTM. There are mainly two reasons for these phenomena above. 1) the performance of BCoP can be promoted either by learning temporal features or spatial features. 2) As for BCop, the spatial features are more important than temporal features. Furthermore, one possible reason is that FPN can predict rainfall utilized adaptive feature layer that has maximum likelihood~\\cite{lin2017feature}, but only learning temporal features in a fixed time scale for LSTM. Besides, the machine learning methods from the fourth to sixth line have somewhat better performance than original ECb forecasts because of utilizing more information from EC data.\n\\begin{table}[htp]\n\\caption{The 4 criteria between 3 machine learning methods and STAS on ECbT\/ECbM\/ECbH divided by precipitation intensity. N\/A(Not Applicable) is none of the samples in the current condition.}\n\\smallskip\n\\centering\n\\resizebox{150pt}{20mm}{\n\\smallskip\\begin{tabular}{c|c|c|c|c|c}\n\\hline \n       &\\multicolumn{1}{|c}{Methods} &\\multicolumn{4}{|c}{Criteria} \\\\\n\\hline \nEcb & & MAPE & $TS_{0.1}$ & $TS_{1}$ & $TS_{10}$ \\\\\n\\hline \n\\multirow{4}*{EcbT} & OBA & 3.81 & 0.65 & N\/A & N\/A \\\\\n~ & LSTM  & 3.79 & 0.55 & N\/A & N\/A \\\\\n~ & FPN  & 2.14 & 0.60 & N\/A & N\/A \\\\\n~ & STAS & {\\bf 2.01} & 0.78 & N\/A & N\/A \\\\\n\\hline\n\\multirow{4}*{EcbM} & OBA & 8.39 & 0.53 & 0.49 & N\/A \\\\\n~ & LSTM  & 8.01 & 0.51 & 0.47 & N\/A \\\\\n~ & FPN  & 6.93 & 0.54 & 0.50 & N\/A  \\\\\n~ & STAS & {\\bf 4.43} & 0.61 & {\\bf 0.59} & N\/A  \\\\\n\\hline\n\\multirow{4}*{EcbH} & OBA & 13.44 & 0.21 & 0.21 & 0.09 \\\\\n~ & LSTM  & 12.81 & 0.24 & 0.24 & 0.16 \\\\\n~ & FPN  & 10.79 & 0.28 & 0.28 & 0.20  \\\\\n~ & STAS & {\\bf 7.05} & 0.38 & 0.38 & {\\bf 0.28}  \\\\\n\\hline\n\\end{tabular}\n}\n\\label{contrastiveII}\n\\end{table}\n\\subsection{Contrastive Experiments on ECbT\/ECbM\/ECbH}\nFor investigating the influence of ST representation in different precipitation intensity, we compare the performance of the 3 targeted methods and STAS on ECBT, ECBM and ECbH as shown in Tab.~\\ref{contrastiveII}.\n\nWe prefer to select $MAPE$ instead of $MAE$ because correcting rainfall samples are our main purpose. Overall, The performance of all methods on $TS_{1}$ and $TS_{10}$ decreases, compared with the same $TS$s of these methods in Tab.~\\ref{contrastiveI}. The possible reason is that the large rainfall value is hard to correct because of the distribution of longspan and the few limited numbers of samples. Both FPN and LSTM have better performance in $TS_{10}$ than OBA because FPN can capture richer multi-scale spatial features and LSTM can obtain temporal dependency in EC. However, OBA only encodes deep representation. Furthermore, the $TS_{10}$ of LSTM on EcbH is $0.04$ lower than FPN in the same case. One possible explanation is that FPN can automatically select a scale layer with the largest confidence level as the predictive layer in the testing phase, whereas LSTM is a fixed ST scale before testing. Finally, It is worth noting that the $MAPE$s of all methods sharply rise with an increment of precipitation intensity. The reason is that the more samples heavy rainfall has, the bigger contribution MAPE according to its equation. Nevertheless, STAS nearly obtains all SOTA results on 3 Ecbs owing to the learning ability of optimized ST representation.\n\n\\subsection{Ablation Experiments}\nHere we perform ablation experiments to verify the effectiveness of each new-introduced components in STAS.\n\n\\noindent {\\bf Impact of SFM and TFM} \\quad The first two lines of Tab.~\\ref{ablation} show the effectiveness of SFM and TFM. It is obvious that the $TS_{1}$ sharply decreases after removing SFM or TFM, and the $TS_{1}$ of without ($w\/o$) SFM is $0.4$ lower than TFM in the same case. One possible reason is that the role of spatial scale is more important in BCoP than that of temporal scale. \n\n\\noindent {\\bf SFM vs TFM for ME modules}\\quad A similar conclusion can be obtained when we evaluate the SFM\/TFM impacts in 5ME modules if we see the results shown in the third line to the sixth line of Tab.~\\ref{contrastiveII}. Furthermore, it is obvious that the rainfall (R) module has a greater impact on $MAPE$ than other modules since it utilizes the historical observations of precipitation.     \n  \n\\noindent {\\bf Impact of deformable CNN and C3AE}\\quad The influences of Deformable CNN (D-CNN) and C3AE are shown in the seventh and eighth lines. MAE of $w\/o$ C3AE increases because C3AE is a regression method, quite like ordinal regression for solving the longspan range of precipitation distribution. Besides, the decrement of $TS_{1}$ on $w\/o$ D-CNN indicates that D-CNN does work in selecting optimized pixels in the process of learning spatial features.        \n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\columnwidth]{figure7.png} \n\\caption{ The visualized comparisons between predictions of precipitation and corresponding ground-truth in the same regions of specific stations. From left to right: the predicted precipitation on OBA and STAS respectively, and observed precipitation. The color-patch bar on the right of Figure is used for distinguishing precipitation intensity by changing light color to high color.}\n\\label{EC_vs}\n\\end{figure}   \n\n\\subsection{Qualitative Analysis}\nWe visualize predictive results from 2 methods and corresponding observations in several examples shown in Fig.~\\ref{EC_vs}. Among the visualizations, the 2 red ovals on the left and middle pictures show that STAS can almost correct heavy rain (mazarine) accurately while OBA cannot correct exactly. Besides, the 2 orange ovals on the 2 same pictures above reveal that OBA has some errors on calibrating moderate rain (green), but STAS is successful for correcting it. All in all, STAS has better prediction performance than OBA in forecasting mesoscale or large-scale precipitation.   \n\n\n\n\n\n\n\n\\iffalse\n\\noindent {\\bf Typhoon dataset} is the real-time observed typhoon data from MICAPS and records typhoon features such as maximum wind velocity and typhoon radius every $6h$. Specifically, we manually label these data according to maximum wind velocity~\\cite{cao2009wind}.\n\\fi\n\n\\iffalse\n\\noindent {\\bf Pre-processing} is divided into 3 procedures including feature selecting, slicing time-series ECMWF dataset in the given stations, and slicing the ECMWF data with typhoon flags. First, we utilize the Pearson Correlation to select the top 57 features that have the highest coefficients on BCoP from 601features in ECMWF. Second, we fix precipitation stations from MICAPS, and empirically slice the effective EC features around one station that are composed of the refined grid points in the geographical range of $\\mbox{[lat-2, lon-2]}$ to $\\mbox{[lat+2, lon+2]}$, then perform the time-slider in the time dimension to require time-series ECMWF dataset. Third, we select the ECMWF data that have the same time with typhoon and these ECMWF data are also included the $\\pm4$ range around the Lat-Lon of this typhoon radius. Meanwhile, we save the label and radius of the typhoon to guide the selection of ECMWF data and flag this typhoon data.\n\\fi\n\n\\iffalse\n\\begin{table}[htp]\n\\caption{The 5 criteria between 8 machine learning methods and STAS on ECbMi. $TS_{0.1}$ is TS score in $PI>0.1$. $TS_{1} | TS_{10}$ is $PI>1$ and $PI>10$ respectively. SVR is support vector regression, LR is linear regression, MLP is Multilayer Perceptron, LSTM is long short-term memory, and FCN is full convolutional network.}\n\\smallskip\n\\centering\n\\resizebox{240pt}{12mm}{\n\\smallskip\\begin{tabular}{c|c|c|c|c|c}\n\\hline \n       &\\multicolumn{5}{|c}{Criterion($TS_{20\/30\/40}$regarding)} \\\\\n\\hline \nMethods& MAE & MAPE & $TS_{0.1}$ & $TS_{1}$ & $TS_{10}$ \\\\\n\\hline \nBCoP Benchmark Method~\\cite{ran2018evaluation}  & 1.76$\\pm$0.13 & 3.18$\\pm$0.21 & 0.41$\\pm$0.04 & 0.35$\\pm$0.03 & 0.19$\\pm$0.04 \\\\\n\\hline\nSVR~\\cite{srivastava2015wrf} & 1.67$\\pm$0.12 & 3.04$\\pm$0.19 & 0.48$\\pm$0.04 & 0.41$\\pm$0.03 & 0.0$\\pm$0.0 \\\\\n\\hline\nLR~\\cite{hamill2012verification} & 1.73$\\pm$0.12 & 2.77$\\pm$0.15 & 0.35$\\pm$0.04 & 0.23$\\pm$0.03 & 0.21$\\pm$0.04 \\\\\n\\hline\nMLP~\\cite{yuan2007calibration} & 1.59$\\pm$0.11 & 2.54$\\pm$0.15 & 0.51$\\pm$0.04 & 0.41$\\pm$0.04 & 0.21$\\pm$0.04 \\\\\n\\hline\nLSTM~\\cite{yu2018coordinate} & (1.39$\\pm$0.08) & 2.37$\\pm$0.12 & 0.58$\\pm$0.04 & 0.48$\\pm$0.02 & 0.24$\\pm$0.02 \\\\\n\\hline\nFCN~\\cite{hu2018soil} & 1.26$\\pm$0.11 & 2.29$\\pm$0.14 & 0.61$\\pm$0.07 & 0.54$\\pm$0.04 & 0.27$\\pm$0.03 \\\\\n\\hline\nOBA\\iffalse~\\cite{hu2018soil}\\fi & 1.01$\\pm$0.05 & 2.21$\\pm$0.14 & 0.58$\\pm$0.03 & 0.53$\\pm$0.01 & 0.29$\\pm$0.01 \\\\\n\\hline\nSTAS(ours) & {\\bf 0.98$\\pm$0.03} & {\\bf 1.98$\\pm$0.10} & {\\bf 0.67$\\pm$0.03} & {\\bf 0.58$\\pm$0.02} & {\\bf 0.35$\\pm$0.02} \\\\\n\\hline\n\\end{tabular}\n}\n\\label{contrastive}\n\\end{table}\n\n\\fi\n\n\\iffalse The form of mean$\\pm$standard deviation. \\fi\n\\iffalse and changing curves on key hyperparameters respectively.      \\fi\n\n\n\n\n\n\\section{Introduction}\n\nWeather forecast plays a crucial role in disaster monitoring and emergency disposal. Numerical Weather Prediction (NWP) based on the equations set of kinetic and thermodynamics~\\cite{liu2017kinetic} is often used to cope with sudden climate change and extreme weather beforehand. One of the representatives of progressive NWP in the global is the European Centre for medium-range weather forecasts (EC)~\\cite{ran2018evaluation}. However, the predictions of precipitation from EC suffer from some uncertainty intrinsic mechanisms of rainfall, e.g., the elusive physical process of rainfall. Therefore, Bias Correcting on Precipitation (BCoP) is the need to improve the forecast level of EC around a local area. \n\\begin{figure}[htbp]\n\\centering\n\\subfigure[$t-2$]{\n\\begin{minipage}[t]{0.3\\linewidth}\n\\centering\n\\includegraphics[width=1\\columnwidth,height=1.2\\textwidth]{figure1}\n\n\\end{minipage}%\n}%\n\\subfigure[$t-1$]{\n\\begin{minipage}[t]{0.3\\linewidth}\n\\centering\n\\includegraphics[width=1\\columnwidth,height=1.2\\textwidth]{figure2}\n\\end{minipage}%\n}%\n\\subfigure[$t$]{\n\\begin{minipage}[t]{0.3\\linewidth}\n\\centering\n\\includegraphics[width=1\\columnwidth,height=1.2\\textwidth]{figure3}\n\\end{minipage}\n}%\n\\centering\n\\caption{The visualization of EC precipitation of Eastern China in the 3 continuous timestamps with region segmentation, which can embody the spatiality and temporal granularity of precipitation.}\n\\label{ST}\n\\end{figure}\n\nClassical BCoP can be roughly divided into two categories: regression and parameter estimation. Regression methods can be regarded as a probabilistic model obeying a credible distribution judged by historical prior information such as expert experience~\\cite{hamill2008probabilistic}. And the parameter estimating methods figure out a set of optimal parameters in some functions such as Kalman filter~\\cite{monache2008kalman} via historical observation for better correcting. Note that in the era of restricted prior information or history observations, these two correcting methods less utilize the spatio-temporal weather regularity in EC~\\cite{hamill2012verification}.    \n\nA feasible way to refine the performance of BCoP is to continuously learn the weather dynamic features from EC data and boosting the correcting ability with machine learning algorithms~\\cite{srivastava2015wrf}.\n\nNevertheless, shallow or low-level dynamic representation is not enough to significantly improve the performance of BCoP. Therefore, it is necessary to capture the high-level representation such as spatial and temporal-dependencies from EC data. To clarify this point, we visualize the colored EC precipitation region in the 3 continuous timestamps shown in Fig.~\\ref{ST}, and employ a Simple Linear Iterative Clustering (SLIC) is employed for segmenting different precipitation subregions by clustering the pixels in similar semantic information~\\cite{achanta2012slic}. From Fig.~\\ref{ST}, we can observe that pixels with the same color are naturally segmented into the same region, separated by the purple line, which reflects the spatiality of precipitation. Besides, an obvious movement for the positions of precipitation regions over time reflects varied temporal granularity. \n\nWe thus propose a deep ST Feature Auto-Selective (STAS) Model for learning ST representation. Further, we add two pre-trained modules termed Spatial Feature-selective Mechanism (SFM) and Temporal Feature-selective Mechanism (TFM) to STAS, in which five observations of meteorological elements (precipitation\/temperature\/pressure\/wind\/dew) are used to guide the selection of optimal ST scales shown in Fig.~\\ref{pipeline} for better extracting ST features. Besides, we integrate a binary classifier so that the precipitation prediction can be more precise. The final prediction is thus achieved by multiplying regression and classified results. The contributions of STAS are summarized as follows:\n\n\\begin{itemize}\n\\item {\\bf Spatial Adaptivity} The Spatial Feature-selective Mechanism (SFM)  adaptively selects an optimal spatial scale of specific EC data for capturing the richer spatial representation, which can refine the performance of correcting, especially in the heavy rainfall, indicating its practicability in forecasting mesoscale or large-scale precipitation.  \n\\item {\\bf Temporal Adaptivity} The Temporal Feature-selective Mechanism (TFM) is utilized for automatically choosing the optimal time-lagging sequence of time-series features of EC in line with the minimal loss value for acquiring the better temporal representation.      \n\\item {\\bf Effective} Experiments indicate that our model achieved better prediction performance than the other 8 published methods on a BCOP benchmark dataset, especially for correcting the large-scale precipitation.\n\\end{itemize}\n\n\n\n\\section{STAS: A Spatio-Temporal Feature Auto-Selective Model}\n\nIn this section, we will introduce our proposed STAS for automatically selecting the spatial and temporal scales of meteorological features from European Centre for Medium-Range Weather Forecasts (EC) in detail. For better illustration, an overall pipeline of STAS is shown in Fig.~\\ref{pipeline}.\n\n\\begin{figure*}[!thbp] \n\\centering\n\\includegraphics[width=0.8\\textwidth,height=0.28\\textwidth]{figure4.png}\n\\caption{The framework is STAS. SFM and TFM is the mechanism of spatial and temporal feature-selective respectively. $s^{*}$ is adaptive spatial size of an EC data in one batch and $\\ell^{*}$ is adaptive time-lagging length of the encoded time-series features. $s^{*}$ and $\\ell^{*}$ are adaptively adjusted by SFM and TFM respectively. $\\mathcal{L}$s are main loss functions in STAS and RC is rainfall classifier. $\\mathcal{L}_{min}$s are minimal losses from the process of STAS.}\n\\label{pipeline}\n\\end{figure*}\n\n\\subsection{Notations and Methodology}\nFirst, we define several basic symbols for our method. Assuming that we have a total of $N$ surface observation stations from Eastern China. The EC data around one station can be set as the combination of the refined grid points in the geographical range from \\mbox{[$\\emph{da} - \\omega$, $\\emph{do} - \\omega$]} to \\mbox{[$\\emph{da} + \\omega$, $\\emph{do} + \\omega$]}, where \\emph{da} and \\emph{do} are the latitude and longitude of this station respectively, and $\\omega$ is the degree. Furthermore, the time-series EC data from the $i$th station at time level $t$ are defined as $(\\mat{X}^{i}_{t})_{\\ell} = [\\mat{X}_{t}^{u}, \\mat{X}_{t-1}, \\ldots, \\mat{X}_{t-\\tau}]_{\\ell}$, where $i \\in N$, and $\\ell$ is the length of time sequence and $\\ell = \\tau + 1$. $u$ is uniform spatial scale. In this study, the interval of the sequence $\\ell$ is 6h. Therefore, $(\\mat{X}^{i}_{t})_{\\ell}$ can be regarded as a four-dimensional tensor-form input including the dimensions of the channels of features, the length of the sequence, the height and width of features.\n\nWith these notations, we roughly build 3 subdivisions for performing Bias Correcting on Precipitation (BCoP) in order as follows:\n\\begin{eqnarray}\\label{eq_pipeline}\n \\hat{y}_{tp} &= & OR \\left \\{LSTM\\left ( E\\left ([\\mat{X}^{u}_{t}, \\ldots, \\mat{X}^{s^{*}_{\\tau}}_{t-\\tau}]\\right)_{\\ell^{*}}\\right)  \\right\\}  \\\\\n \\hat{y}_{rc} &= & RC(\\mat{X}^{u}_{t})  \\\\\n \\hat{y}_{t} &= & \\hat{y}_{tp} \\otimes \\hat{y}_{rc}\n\\end{eqnarray}\nwhere $s^{*}_{\\tau}$ denotes the optimized size (height $\\times$ width) of the EC data $\\mat{X}_{t-\\tau}$ through spatial feature-selective mechanism (SFM), which will be introduced in Sec.3.2. Parameter $\\ell^{*}$ denotes the refined time-lagging length for time-series features $[(\\mat{X}^{i}_{t})_{\\ell}]^{u}$ that are encoded (Sec.3.3) to a uniform ($u$) size via temporal feature-selective mechanism (TFM), which will be discussed in Sec.3.4. $E(\\cdot)$ is an encoder backbone and $LSTM(\\cdot)$ is a stacked ConvLSTM~\\cite{xingjian2015convolutional}. The acronym $OR(\\cdot)$ means an ordinal regression model~\\cite{zhu2018facial} is used for regressing corrected precipitation value in the end. Besides, we utilize the precipitation binary classifier $RC(\\cdot)$ for classifying raining or rainless samples. Finally, predicted result $\\hat{y}_{t}$ is obtained by multiplying the predicted precipitation value $\\hat{y}_{tp}$ and the classified result $\\hat{y}_{rc}$.\n\n\\subsection{Spatial Feature-Selective Mechanism}\n\\vspace{-0.1em}\nWhen lacking an instructive spatial scale, the EC features centered on all stations are empirically set to be a fixed spatial scale for prediction. However, these rules-of-thumb may impair predictive accuracy when there is a strong connection between features scale and precipitation intensity~\\cite{mu2003conditional}. Therefore, we propose a Spatial Feature-selective Mechanism (SFM) to adaptively search the optimal spatial scales of specific EC data based on observations of 5 Meteorological Elements (MEs) including precipitation, temperature, pressure, wind, dew. Concretely, we can find the optimal spatial scales by minimizing the total spatial losses in different scales, which are the summation of 5 spatial MSE losses between predictive MEs and their observations shown in Fig.~\\ref{frameSm}. The selection is formulated as:\n\\begin{eqnarray}\\label{spatialMe}\ns^{*}  &= & \\mathop{\\arg\\min}_{s}\\mathcal{L}_{s} \\\\ \n\\mathcal{L}_{s}  & = & \\sum_{i=0}^{n(MEs)}\\mathcal{L}_{MSE}(MSM_i(\\mat{X'}^{s}_{t}), y_{t}^{i})\n\\end{eqnarray}where $\\mathcal{L}_{s}$ is the spatial total loss in the scale $s$. $n(MEs)$ is the number of MEs. $MSM_i(\\cdot)$ is $i$-th ME Spatial Module and $\\mat{X'}^{s}_{t}$ is a given EC data that is scale $s$ in timestamp $t$. $y_{t}^{i}$ are the labels of $i$-th ME in timestamp $t$. Therefore, the goal is to search an optimal scale $s^{*}$ shown in Eq \\eqref{spatialMe}. Specifically, a deformable CNN layer is introduced for boosting the rep- resentational ability of 5 modules via learning the offsets of filters that are appropriate for capturing better features~\\cite{dai2017deformable}. \n\n\\subsection{Backbone with Denoising}\n\\vspace{-0.1em}\nThe EC features contain numerous noises~\\cite{xu2019towards} that can produce negative side effects for correcting. To solve this issue, we introduce a denoised Encoder-Decoder (E-D) as backbone shown in Fig.~\\ref{pipeline}. In the process of encoder, the Gaussian white noise is added to the encoded features that have uniform size $u$ via upsampling and downsampling. We also introduce a reconstruction loss utilized for calculating difference between encoded features and reconstructed features by decoder to optimize denoised ability of encoder as follows:\n\\begin{equation} \\label{reconsLoss}\n\\mat{W}_{E}^*, \\mat{W}_{D}^* = \\mathop{\\arg\\min}_{\\mat{W}_{E},\\mat{W}_{D}}{\\Vert E([\\mat{X}^{u}_{t}, \\ldots, \\mat{X}^{s^{*}_{\\tau}}_{t-\\tau}], \\mat{\\epsilon}) - D(Z_{t}) \\Vert}\n\\end{equation}where $\\mat{W}_{E}$ and $\\mat{W}_{D}$ are parameter matrices from encoder $E(\\cdot)$ and decoder $D(\\cdot)$ respectively. $[\\mat{X}^{u}_{t}, \\ldots, \\mat{X}^{s^{*}_{\\tau}}_{t-\\tau}]$ is time-series features with optimized spatial scale $\\vct{s^{*}}$ through SFM. $Z_{t}$ is hidden features by $E(\\cdot)$. Moreover, $\\mat{\\epsilon}$ is Gaussian noise.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=1.4\\columnwidth, height=0.28\\textwidth]{figure5.png} \n\\caption{The structure of Spatial Feature-selective Mechanism (SFM), the visualizations of 5 Meteorological Elements (ME) are temperature, pressure, wind, dew and precipitation. $\\{ \\mathcal{L}_{s1}, \\ldots, \\mathcal{L}_{sw} \\}$ are MSE spatial total losses that are the sum ($\\oplus$) of 5 spatial MSE losses from ME spatial modules in different scales. SFM can select the adaptive spatial scale $s^{*}$ of specific EC data in $\\mat{X'}_{t}$. The deformable CNN is utilized for helping filters to operate the given pixels that can capture the better representation.}\n\\label{frameSm}\n\\end{figure}    \n\\subsection{Temporal Feature-Selective Mechanism}\n\\vspace{-0.1em}\nIt is obvious that rainfall patterns in one station are not only related to EC features around this station, but also closely connected with historical features~\\cite{ciach2006analysis}. Therefore, the temporal Feature-selective Mechanism (TFM) is proposed for adaptively acquiring an optimized encoded features $[(\\mat{X}^{i}_{t})_{\\ell^*}]^{u} = [\\mat{X}^{u}_{t}, \\ldots, \\mat{X}^{u}_{t-\\tau}]_{\\ell^*}$, , from which more useful temporal representation can be learned. Formally, it is defined as: \n\\begin{eqnarray}\\label{temporalMe}\nl^{*}  &= & \\mathop{\\arg\\min}_{\\ell}\\mathcal{L}_{T} \\\\\n\\mathcal{L}_{T}  & = & \\sum_{i=0}^{n(MEs)}\\mathcal{L}_{MAE}(MTM_i([(\\mat{X}^{i}_{t})_{\\ell}]^{u}), y_{t}^{i})\n\\end{eqnarray}where $\\ell^{*}$ is the optimal time-lagging length and $y_{t}^{i}$ are labels of $i$-th MEs in timestamp $t$. $MTM_i(\\cdot)$ is $i$-th ME Temporal Module. Same as SFM, TFM can select optimal time-lagging sequence $\\ell^{*}$ by finding out minimum temporal total $MAE$ loss, which are the sum of 5 temporal $MAE$ losses between predictions of $MTM(\\cdot)$ and labels of MEs in different time-lagging sequences. The detailed structure of C3AE~\\cite{zhang2019c3ae} and 3DCNN~\\cite{zhang2017learning} are shown in Fig.~\\ref{frameTm}. In consideration of maldistribution of precipitation values~\\cite{xu2019towards}, an ordinal regression (OR) method~\\cite{zhu2018facial} is utilized for outputing regressing value of precipitation $\\hat{y}$ shown in Fig.~\\ref{pipeline}. OR may solve the problem of longspan range of precipitation values and convert a regression task into a multi-binary classification one to reduce the complexity of regression.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=1.3\\columnwidth,height=0.28\\textwidth]{figure6.png} \n\\caption{ The structure of Temporal Feature-selective Mechanism (TFM). TFM can select adaptive time-lagging length $\\ell^*$ of one time-series features $(\\mat{X}^{i}_{t})_{\\ell}$. $\\{ \\mathcal{L}_{T1}, \\ldots, \\mathcal{L}_{Th} \\}$ are MAE temporal total losses that are the sum ($\\oplus$) of 5 MAE temporal losses from ME temporal modules in different time-lagging sequence. 3D CNN is utilized for capturing the patial-temporal dependency. Specifically, C3AE is a lightweight rank learning module and suited for regression distribution that has longspan range.}\n\\label{frameTm}\n\\end{figure}   \n\n\\subsection{Training and Testing}\n{\\bf Training}\\quad In the training phase, we firstly pre-train the Spatial Feature-selective Module (SFM) and Temporal Feature-selective Module (TFM) so that these two modules can predict the MEs accurately. Secondly, we integrate SFM into our framework shown in Fig.~\\ref{pipeline} for predicting the MEs from historical EC data $\\mat{X'}_{t}^{s}$ and selecting the optimal scale $s^{*}$ that has minimal MSE loss. Similarly, we begin to train Temporal Feature-selective Module (TFM) and Encoder-Decoder (E-D) together, and use a specific ADAM optimizer for BP when adaptive time-lagging length $\\ell^{*}$ in one batch is selected by TFM. Meanwhile, the rainy classifier (RC) is cross-trained along with TFM\/SFM and E-D.\n\n\\noindent{\\bf Testing}\\quad In the testing phase, SFM plays a role in selecting the optimal scale $s^{*}$ and TFM selects the adaptive time-lagging length $\\ell^{*}$, then SFM is forward to calculate probabilities of all classifiers from ordinal regression. We select specific classifiers with large probabilities according to initialized interval $\\xi$ and transform these probabilities into regression value of precipitation $\\hat{y}_{tp}$ formulated as:\n\\begin{equation} \\label{temporalMe}\n\\hat{y}_{tp} = \\xi * \\sum\\limits_{v=0}^{c-1}(p_{v}\\ge\\xi)\n\\end{equation} where $p_{v}$ is classified probability of the $v$-th binary classifier. Besides, RC is forward to obtain the classified result $\\hat{y}_{rc}$. Final predicted result $\\hat{y}_{t}$ is required by multiplying SFM result $\\hat{y}_{tp}$ and RC result $\\hat{y}_{rc}$ as shown in the third equation of Eq \\eqref{eq_pipeline} in Sec.3.1.\n\n\n\n\n\n\\iffalse\nThe conditional nonlinear optimal perturbation (CNOP)~\\cite{mu2003conditional} is employed for searching a sensitive area in time dimension of history time level that can has an important effect of current prediction of precipitation. Some methods based on CNOP such as Ensemble Transforming Kalman Filter (ETKF) have some effect on finding sensitive area. However, these methods embrace excess intervention from experts and manual features, so increasing the overhead. \n\\fi  \n\n\n\\iffalse\nTheoretically, the classified accuracy of RC is higher than OR because of only having one binary label. \n\\fi\n\n\\iffalse\n~\\cite{ciach2006analysis}\n\\fi\n\n\\iffalse\nSpecifically, the key role of RC is to enhance error-tolerant rate of SFM for improving performance of our model.\n\\fi\n\n\\section{Related Work}\n\n\\subsection{Bias Correcting on Precipitation} \nIn this section, we will give a brief survey on Bias Correcting on Precipitation (BCoP) and spatio-temporal (ST) pattern selection. Classical BCoP can be grouped into regression and parameter estimating methods. Regression methods~\\cite{hamill2008probabilistic} heavily depend on expert experience, which requires manually setting a threshold for the generation of probability, losing their flexibility and adaptivity. Meanwhile, the parameter estimating methods heuristically assess the key parameters set by trial and error in specific models. Nevertheless, traditional BCoP methods suffer from limited available priors and historical observations, and less utilize the clue of possible motion or dynamics existed in European Centre for medium-range weather forecasts (EC) data. To address these issues, some recently proposed models attempt to capture complex climate patterns from the large-scale EC data and optimize the model in line with observations~\\cite{srivastava2015wrf}. However, these methods neglect the potential dependency among variables in EC data, especially the ST dependency.      \n\n\\subsection{Spatio-Temporal Pattern on EC Precipitation}\nExcept for qualitative analysis for reflecting the ST dependencies shown in Fig.~\\ref{ST}, it is proved that rainfall value in one location correlates with weather indicators such as temperature, pressure, wind and dew~\\cite{yapp1982model}. However, we do not know which scale around this location has a strong connection with the precipitation. Moreover, we know that the system on atmospheric dynamics is a  spatio-temporal evolution system, in which physical field changes over time~\\cite{mu2003conditional}. Hence, capturing the ST representation would be a feasible way to improve the performance of BCoP. We consider employing an end-to-end deep-learning model based on multi-scale EC feature~\\cite{lin2017feature}, and adaptively select the features that have an optimal scale and take advantage of these features for correcting precipitation more accurately. To our knowledge, there is no report on how to adaptively extract the ST features based deep-learning model for BCoP in literature. This is the first time to adaptively extract ST representation end-to-end.\n      \n\n\n\\iffalse \n\nIn deep learning domain, as we know, ST Long Short-Term Memory (ST-LSTM) is applied for action recognition by capturing the spatial features from joints of the trunk and temporal features from consecutive frames of action~\\cite{liu2016spatio}. Besides, a ST residual network is employed for traffic flow prediction~\\cite{zhang2017deep}.\n\nMoreover, conditional nonlinear optimal perturbation (CNOP), is utilized for identifying a sensitive time area that has a key influence on specific precipitation along the time dimension of this precipitation~\\cite{mu2003conditional}. \n\\fi \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{Sec. Introduction}\n\nIn his pioneering 1968 Ph.D. thesis, Leonard Parker discovered the surprising phenomenon that evolving cosmological spacetimes can produce quantum particles \\cite{Parker69, Parker71, ZeldovichStarobinsky77}. This work laid much of the theoretical framework for our current understanding of quantum fields in curved spacetimes (see, e.g., the textbooks \\cite{ParkerToms09, BirrellDavies82}). The phenomenon has since been investigated in  astrophysical and cosmological contexts, leading to fundamental theoretical results including the emission of Hawking radiation by blackholes \\cite{Hawking75, Unruh76, Davies76, Page82, Brown86, Frolov87, Anderson93, Anderson94, Anderson95} and the generation of primordial fluctuations during inflation \\cite{Starobinsky79, Allen88, Sahni90, Mukhanov92, Souradeep92, Glenz09, Agullo11}.\n\nCosmological quantum particle production results from the shifting quantum vacuum state due to spacetime expansion. A given mode of a quantum field can begin in a state with no particles, but at a later time have a non-zero particle number expectation value. Parker showed that while this does not happen for massless conformally-coupled fields, it is generic for massive scalar fields of arbitrary coupling to the spacetime curvature. Typically, the particle production is significant when the particle mass $m$ is of the order of the expansion rate $H$, or when $m^{2} \\simeq \\dot{H}$. Simple dimensional arguments show that in a Universe with critical density $\\rho_c \\simeq 3H^{2}\/8\\pi G$, quantum particle production can contribute significantly to the energy density of the Universe at early epochs when $H$ is not too far below the Planck scale. \n\nThis effect naturally suggests that quantum particle production may have a significant impact on the expansion rate of the Universe at early times. The back-reaction problem consists in understanding how the energy density generated through this process alters the evolution of the background spacetime. This seemingly straightforward calculation is actually subtle due to two technical challenges of quantum fields in curved spacetimes. First, the definition of the quantum vacuum and particle are not formally well defined in time-varying spacetimes. Second, the energy-momentum tensor of a quantized field propagating on a curved background possesses a formally infinite expectation value, and must be regularized to yield physically sensible results. In homogeneous and isotropic spacetimes, both problems can be addressed via the method of adiabatic regularization \\cite{ParkerFulling74, FullingParker74, Bunch80, Fulling74, Christensen78, AndersonParker87}, which is particularly useful for numerical computations. The adiabatic notion of particle offers a clear way to track particle number in a spacetime which is expanding sufficiently slowly. This representation has the special property of defining a vacuum state which comes closest to the Minkowski vacuum when the background expansion is sufficiently slow. However, adiabatic regularization introduces several technical complications to the back-reaction problem. As a consequence, ambiguities arise in specifying initial conditions, and computationally the problem becomes very complicated with potential numerical instabilities \\cite{Anderson83, Anderson84, Anderson85, Anderson86, SuenAnderson87}. In practice, the ambiguities and complexities together have prevented any general numerical solution, although the test-field limit in which back-reaction effects are neglected has been investigated in several studies \\cite{Birrell78, Anderson00, Anderson05, Bates10, Habib99}.\n\nFormal developments clarified aspects of the adiabatic regularization approach, offering a clear separation between the energy density due to the field particle content and the divergent contributions from the zero-point energy \\cite{Habib99, AndersonMottola14, AndersonMottola14b, Anderson18}. However, the notion of adiabatic particle appearing in the energy density suffers from ambiguities. For a typical mode of a quantum field, its associated particle number during times of significant particle production depends on the perturbative order of adiabatic regularization employed.  This is obviously an unphysical result, since the dominant term in the energy density is often just a simple function of the particle number density. Typically, successive orders of adiabatic regularization gives the particle number in a given mode as a divergent asymptotic series. Recent important papers by Dabrowski and Dunne \\cite{DabrowskiDunne14, DabrowskiDunne16} employed a remarkable result of asymptotic analysis \\cite{Dingle73, Berry90, Berry82, Berry89, Berry88} to sum the divergent series and provide a sensible notion of particle which is valid at all times. Since any remaining physical ambiguity is then removed from the problem, this result points the way to a general numerical solution.\n\nWe combine these recent results into an approach which numerically solves the quantum back-reaction problem in regimes with field energy density dominated by particle production. We then apply this technique to a toy cosmological model, that of a positive-curvature spacetime with a constant energy density (the closed de Sitter model). In the absence of any quantum fields or other particle content, this spacetime exhibits a bounce behavior, contracting to a minimum scale factor and then expanding again. Here we show explicitly that the existence of a massive scalar field in a particular mass range will cause large changes in the spacetime evolution: even if the contracting spacetime initially contains a quantum field in its adiabatic vacuum state, quantum particle production can create enough energy density to push the spacetime into a radiation crunch. Special values of the field mass can also delay but not eliminate the bounce. This appears to be the first general solution for the quantum back-reaction problem in cosmology.\n\nIn Section II we review standard results for quantized scalar fields propagating in cosmological spacetimes, while Section III discusses the adiabatic field representation and the semi-classical notion of adiabatic particle number. Section IV recasts quantum particle production in terms of the Stokes phenomenon of the complex-plane wave equation for specific modes, including interference between different modes. Section V formulates the back-reaction problem for scenarios in which the field particle content or particle production dominates the field energy density. Section VI outlines our numerical implementation of the mathematical results in Sections IV and V. Physical results for a closed de Sitter model are presented in Section VII. Finally, in Section VIII we discuss the prospects for more general situations, including quantum fields with spin and interacting quantum fields, and the possible relevance of quantum particle production to early-Universe models, including inflationary and bounce scenarios. Salient technical details are summarized in the Appendix. Natural units with $\\hbar = c = 1$ are adopted throughout.\n\n\\section{Scalar Fields in FLRW Spacetimes}\\label{Sec. QFT in FLRW}\n\nWe first summarize basic results for scalar fields in spatially isotropic and homogeneous spacetimes (see, e.g., \\cite{ParkerToms09, BirrellDavies82}). \nConsider a Universe described by the Friedmann-Lema\\^{i}tre-Robertson-Walker (FLRW) metric\n\\begin{equation}\\label{Eq. FLRW Metric}\n\t\\mathrm{d}s^{2} = g_{ab}\\mathrm{d}x^{a}\\mathrm{d}x^{b} = -\\mathrm{d}t^{2} + a^{2}(t) g_{ij}\\mathrm{d}x^{i}\\mathrm{d}x^{j} \\, ,\n\\end{equation}\nwith\n\\begin{equation}\\label{Eq. FLRW Spatial Metric}\n\tg_{ij}\\mathrm{d}x^{i}\\mathrm{d}x^{j} =\\frac{\\mathrm{d}r^{2}}{1 - K r^{2}} + r^{2} \\mathrm{d}\\theta^{2} + r^{2} \\sin^{2}\\theta \\, \\mathrm{d}\\varphi^{2} \\, .\n\\end{equation}\nHere $a(t)$ is the scale factor which describes the cosmological expansion history, and $K = -1, \\, 0, \\, +1$ corresponds to the curvature parameter of an open, flat, and closed Universe, respectively.\nThe non-vanishing components of the Ricci tensor $R_{ab}$ are\n\\begin{subequations}\\label{Eq. Ricci Tensor}\n\\begin{align}\n\tR_{00}(t) &= 3\\Big[ \\dot{H}(t) + H^{2}(t) \\Big] g_{00} \\, ,\\\\[8pt]\n\tR_{ij}(t) &= \\Big[ \\dot{H}(t) + 3 H^{2}(t) + 2K\/a^{2}(t) \\Big] g_{ij} \\, ,\n\\end{align}\n\\end{subequations}\nand the Ricci scalar $R = g^{ab}R_{ab}$ is\n\\begin{equation}\\label{Eq. Ricci Scalar}\n\tR(t) = 6 \\Big[ \\dot{H}(t) + 2 H^{2}(t) + K\/a^{2}(t) \\Big] \\, \n\\end{equation}\nwhere\n\\begin{equation}\\label{Eq. Hubble Parameter}\n\tH(t) \\equiv \\frac{\\dot{a}(t)}{a(t)} \n\\end{equation}\nis the Hubble parameter. Overdots indicate differentiation with respect to proper time $t$.\n\nWe are interested in the evolution of a free scalar field $\\Phi(t,\\, \\mathbf{x})$ of arbitrary mass and curvature coupling in this spacetime. The action for such a field can be expressed generically as\n\\begin{align}\\label{Eq. Field Action}\n\tS = -\\frac{1}{2} \\int \\sqrt{-g} \\, \\mathrm{d}^{4}x \\Big[ \\big( \\nabla_{a} \\Phi \\big) g^{ab} \\big( \\nabla_{b} \\Phi \\big) + \\Big. & \\\\\n\t \\Big. + m^{2} \\Phi^{2} + \\xi R \\Phi^{2} \\Big] & \\, , \\nonumber\n\\end{align}\nwhere $\\nabla_{a}$ is the covariant derivative, $g = \\det{(g_{ab})}$, $m$ is the field mass, and $\\xi$ is the field coupling to the spacetime curvature. Applying the variational principle to this action yields the equation of motion \n\\begin{equation}\\label{Eq. Field Equation of Motion}\n\t\\Big[ \\Box - m^{2} - \\xi R(t) \\Big]\\Phi(t,\\, \\mathbf{x}) = 0 \\, ,\n\\end{equation}\nwhere $\\Box = g^{ab}\\nabla_{a}\\nabla_{b}$ is the d'Alembert operator associated with the spacetime.\n\nDue to the homogeneity and isotropy of the background metric, the solutions of Eq.~(\\ref{Eq. Field Equation of Motion}) \ncan be separated into purely temporal and spatial parts. As a consequence, the quantized field operator can be written as\n\\begin{align}\\label{Eq. Field}\n\t\\hat{\\Phi}(t,\\, \\mathbf{x})=a^{-3\/2}(t) \\! \\int \\mathrm{d}\\mu(k) \\Big[ a_{\\mathbf{k}}^{\\phantom{\\dagger}}f_{k}(t)Y_{\\mathbf{k}}(\\mathbf{x}) + \\Big. & \\\\\n\t\\Big. + a_{\\mathbf{k}}^{\\dagger}f_{k}^{\\ast}(t)Y_{\\mathbf{k}}^{\\ast}(\\mathbf{x}) \\Big] & \\, , \\nonumber\n\\end{align}\nwhere the raising and lowering operators $a_{\\mathbf{k}}^{\\dagger}$ and $a_{\\mathbf{k}}^{\\phantom{\\dagger}}$ satisfy the canonical commutation relations\n\\begin{equation}\\label{Eq. Commutation Relations}\n\\left[ a_{\\mathbf{k}}^{\\phantom{\\dagger}} \\, , a_{\\mathbf{k^{\\prime}}}^{\\dagger} \\right] = \\delta_{\\mathbf{k},\\, \\mathbf{k^{\\prime}}} \\, ,\n\\end{equation}\nand $\\mathrm{d}\\mu(k)$ is a geometry-dependent integration measure given by \n\\begin{equation}\\label{Eq. Integration Measure}\n\t\\int \\mathrm{d}\\mu(k) =\n\t\\left\\{\\begin{aligned}\n\t\t\t&\\sum_{k=1}^{\\infty} k^2\\, , && \\text{for } K = +1 \\\\\n\t\t\t&\\int_{0}^{\\infty} k^2 \\,\\mathrm{d}k \\, , && \\text{for } K = 0,\\,-1 \\, .\n\t\t\\end{aligned}\\right.\n\\end{equation}\nThe functions $Y_\\mathbf{k}(\\mathbf{x})$ and $f_k(t)$ contain the spatial and temporal dependence of each $\\mathbf{k}$-mode. The harmonic functions $Y_\\mathbf{k}(\\mathbf{x})$ are eigenfunctions of the Laplace-Beltrami operator associated with the geometry of spatial hypersurfaces, while the mode functions $f_k(t)$ obey the harmnonic oscillator equation\n\\begin{equation}\\label{Eq. Mode Equation}\n\\ddot{f}_{k}(t) + \\Omega_{k}^{2}(t) f_{k}(t) = 0 \\, \n\\end{equation}\nwith the time-dependent frequency function \n\\begin{equation}\\label{Eq. Mode Frequency}\n\\Omega_{k}^{2}(t) = \\omega_{k}^{2}(t) + \\bigg( \\xi - \\frac{1}{6} \\bigg) R(t) - \\Bigg[ \\frac{\\dot{H}(t)}{2} + \\frac{H^{2}(t)}{4} \\Bigg] \\, ,\n\\end{equation}\nwhere\n\\begin{equation}\\label{Eq. Minkowski Frequency}\n\t \\omega_{k}(t) = \\Bigg[\\frac{k^{2}}{a^{2}(t)} + m^{2}\\Bigg]^{\\!1\/2} \\, .\n\\end{equation}\nThe complex mode functions $f_{k}(t)$ and $f^{\\ast}_{k}(t)$ also satisfy the Wronskian condition\n\\begin{equation}\\label{Eq. Wronskian Condition}\nf_{k}(t)\\dot{f}^{\\ast}_{k}(t) - \\dot{f}_{k}(t) f^{\\ast}_{k}(t) = i \\, .\n\\end{equation}\nIf Eq.~(\\ref{Eq. Wronskian Condition}) holds at some particular time $t$, then Eq.~(\\ref{Eq. Mode Equation}) guarantees it will also hold at all future\ntimes.\n\nThe quantization procedure outlined above naturally leads to the construction of the Fock space of field states. The base element of this space is the vacuum state, which is defined as the normalized state that is annihilated by all lowering operators:\n\\begin{equation}\\label{Eq. Vacuum}\na_{\\mathbf{k}}^{\\phantom{\\dagger}} \\left| 0 \\right\\rangle = 0 \\ \\ \\mathrm{and} \\ \\ \\left\\langle 0 | 0 \\right\\rangle = 1 \\, .\n\\end{equation}\nAll remaining states are generated from the vacuum by the successive application of raising operators, such as\n\\begin{equation}\n\\left| \\mathbf{k}_1 ,\\, \\mathbf{k}_2 ,\\, \\dots \\right\\rangle = a_{\\mathbf{k}_1}^{\\dagger} a_{\\mathbf{k}_2}^{\\dagger} \\dots \\left| 0 \\right\\rangle \\, ,\n\\end{equation}\nand normalized by the requirement of mutual orthonormality. In what follows, we will be interested in the family of field states which are spatially isotropic and homogeneous, as these constitute viable sources of the FLRW metric.\n\nThe field operator $\\hat{\\Phi}(t,\\, \\mathbf{x})$ admits numerous representations of the form shown in Eq.~(\\ref{Eq. Field}), each of which is associated with a different mode function pertaining to the set of solutions of Eq.~(\\ref{Eq. Mode Equation}). These representations are related: the complex mode functions $f_{k}(t)$ and $h_{k}(t)$ belonging to any two different representations can be expressed in terms of one another through the Bogolyubov transformations\n\\begin{subequations}\\label{Eq. Bogolyubov Modes}\n\\begin{align}\n\t{f}_{k}(t) &= \\alpha_{k}h_{k}(t) + \\beta_{k}h^{\\ast}_{k}(t) \\, ,\\\\[8pt]\n\t{f}_{k}^{\\ast}(t) &= \\beta_{k}^{\\ast}h_{k}(t) + \\alpha_{k}^{\\ast}h^{\\ast}_{k}(t) \\, ,\n\\end{align}\n\\end{subequations}\nwhere $\\alpha_{k}$ and $\\beta_{k}$ are known as Bogolyubov coefficients. Due to homogeneity and isotropy, these coefficients depend only on ${k = \\left| \\mathbf{k} \\right|}$. Substituting these expressions into Eq.~(\\ref{Eq. Field}) leads to similar transformations relating the raising and lowering operators belonging to these representations:\n\\begin{subequations}\\label{Eq. Bogolyubov Operators}\n\\begin{align}\na_{\\mathbf{k}}^{\\phantom{\\dagger}} &= \\alpha_{k}^{\\ast} b_{\\mathbf{k}}^{\\phantom{\\dagger}} - \\beta_{k}^{\\ast} b_{\\mathbf{k}}^\\dagger\\, , \\\\[8pt]\na_{\\mathbf{k}}^\\dagger &= \\alpha_{k} b_{\\mathbf{k}}^\\dagger - \\beta_{k} b_{\\mathbf{k}}^{\\phantom{\\dagger}} \\, ,\n\\end{align}\n\\end{subequations}\nfrom which it follows that the Bogolyubov coefficients must satisfy\n\\begin{equation}\\label{Eq. Bogolyubov Constraint}\n\t\\big| \\alpha_{k} \\big|^{2} - \\big| \\beta_{k} \\big|^{2} = 1 \\,\n\\end{equation}\nin order to guarantee that the commutation relations of Eq.~(\\ref{Eq. Commutation Relations}) are valid across all representations.\n\nA direct consequence of Eqs.~(\\ref{Eq. Bogolyubov Operators}) is that the notion of vacuum is not unique for a quantized field defined on a FLRW spacetime \\cite{BirrellDavies82, ParkerToms09}. This is evident from the following simple calculation, which shows that the vacuum defined in Eq.~(\\ref{Eq. Vacuum}) is not necessarily devoid of particles according to the number operator belonging to a different field representation:\n\\begin{align}\n\t\\mathcal{N}_{k} &= \\big\\langle 0 \\big| b_{\\mathbf{k}}^{\\dagger} b_{\\mathbf{k}}^{\\phantom{\\dagger}} \\big| 0 \\big\\rangle \\nonumber \\\\\n\t&= \\big| \\alpha_{k} \\big|^{2} \\big\\langle 0 \\big| a_{\\mathbf{k}}^{\\dagger} a_{\\mathbf{k}}^{\\phantom{\\dagger}}  \\big| 0 \\big\\rangle + \\big| \\beta_{k} \\big|^{2} \\big\\langle 0 \\big| a_{\\mathbf{k}}^{\\phantom{\\dagger}} a_{\\mathbf{k}}^\\dagger \\big| 0 \\big\\rangle \\\\\n\t&= \\big| \\beta_{k} \\big|^{2} \\nonumber \\, .\n\\end{align}\nTherefore, different choices of representation inevitably lead to distinct notions of vacuum and, consequently, to distinct notions of particle. This result is a quite general feature of quantum field theory defined on curved spacetimes, and although it initially seems troublesome, it actually becomes useful in numerical back-reaction calculations. To that end, we introduce in the next section a particularly useful representation which defines the most physical notion of particle in a FLRW spacetime.\n\n\\section{Adiabatic Representation}\\label{Sec. Adiabatic Representation}\n\nDespite the multitude of available representations for a quantized scalar field defined on a FLRW spacetime, one particular choice referred to as the adiabatic representation stands out. This representation has the special property of defining a vacuum state which comes closest to the Minkowski vacuum when the background expansion is sufficiently slow. As a consequence, the adiabatic representation provides the most meaningful notion of physical particle in an expanding homogeneous and isotropic Universe. Here we discuss this representation \nclosely following Ref.~\\cite{Habib99}.\n\nThe adiabatic representation is characterized by mode functions which are the phase-integral solutions \\cite{Parker69, Parker71, ZeldovichStarobinsky77} of Eq.~(\\ref{Eq. Mode Equation}):\n\\begin{equation}\\label{Eq. Adiabatic Modes}\n\th_{k}(t) = \\frac{1}{\\sqrt{2 W_{k}(t)}} \\exp{\\!\\bigg( \\! -i \\! \\int^{t} W_{k}(s) \\, \\mathrm{d}s \\bigg)} \\, ,\n\\end{equation}\nwhere the integral in the exponent can be computed from any convenient reference time, and the function $W_{k}(t)$ is given by the formal asymptotic series\n\\begin{equation}\\label{Eq. Adiabatic W}\n\t W_{k}(t) \\equiv \\Omega_{k}(t) \\sum_{n=0}^{\\infty} \\varphi_{k,\\,2n}(t) \\, .\n\\end{equation}\nThe terms $\\varphi_{k,\\,2n}(t)$ are obtained by substituting Eqs.~(\\ref{Eq. Adiabatic W}) and (\\ref{Eq. Adiabatic Modes}) into Eq.~(\\ref{Eq. Mode Equation}). The expressions which ensue from these substitutions are standard results of the phase-integral method \\cite{FromanFroman96, FromanFroman02}; up to fourth order they are\n\\begin{subequations}\n\\begin{align}\n\t&\\varphi_{k,\\,0}(t) = 1 \\, , \\\\[8pt]\n\t&\\varphi_{k,\\,2}(t) = \\frac{1}{2} \\varepsilon_{k,\\,0}(t) \\, , \\\\[8pt]\n\t&\\varphi_{k,\\,4}(t) = -\\frac{1}{8}\\Big[ \\varepsilon_{k,\\,0}^{2}(t) + \\varepsilon_{k,\\,2}(t) \\Big] \\, ,\n\\end{align}\n\\end{subequations}\nfor which the quantities appearing on the right-hand sides are given by\n\\begin{subequations}\n\\begin{align}\n\t\\varepsilon_{k,\\,0}(t) &\\equiv \\Omega_{k}^{-3\/2}(t) \\, \\frac{d^{2}}{dt^{2}} \\! \\bigg[ \\Omega_{k}^{-1\/2}(t) \\bigg]\\, , \\\\[8pt]\n\t\\varepsilon_{k,\\,m}(t) &\\equiv \\bigg[ \\Omega_{k}^{-1}(t) \\, \\frac{d}{dt} \\bigg]^{\\!m} \\!\\!\\varepsilon_{k,\\,0}(t) \\, .\n\\end{align}\n\\end{subequations}\n\nIn a sense, the functions $W_{k}(t)$ capture the overall time dependence of each $\\mathbf{k}$-mode due the evolving FLRW metric, leaving behind only the Minkowski-like mode oscillations which take place on top of this background \\cite{DabrowskiDunne14, DabrowskiDunne16}. It is this property that makes the adiabatic mode functions $h_{k}(t)$ and $h^{\\ast}_{k}(t)$ such good templates for probing the particle content of fields evolving in cosmological spacetimes. This template role is made precise by the following time-dependent generalization of Eqs.~(\\ref{Eq. Bogolyubov Modes}) \\cite{Parker69, Parker71, ZeldovichStarobinsky77}, which expresses the field modes $f_{k}(t)$ as linear combinations of the adiabatic mode functions:\n\\begin{equation}\\label{Eq. Adiabatic Bogolyubov Mode}\n\t{f}_{k}(t) = \\alpha_{k}(t)h_{k}(t) + \\beta_{k}(t)h^{\\ast}_{k}(t) \\, ,\n\\end{equation}\nwhere the Bogolyubov coefficients $\\alpha_{k}(t)$ and $\\beta_{k}(t)$ are analogous to those appearing in Eqs.~(\\ref{Eq. Bogolyubov Modes})~and~(\\ref{Eq. Bogolyubov Operators}), but are here regarded as time-dependent quantities due to the fact that $h_{k}(t)$ and $h_{k}^{\\ast}(t)$ are merely approximate solutions of Eq.~(\\ref{Eq. Mode Equation}). In order to completely specify these coefficient functions, an additional expression must be provided. For that purpose, it is common to introduce a condition on the time derivative of the mode function which preserves the Wronskian relation of Eq.~(\\ref{Eq. Wronskian Condition}). In its most general form, this condition can be stated as \\cite{Habib99}\n\\begin{align}\\label{Eq. Adiabatic Bogolyubov Mode Derivative}\n\t\\dot{f}_{k}(t) = \\bigg[ -iW_{k}(t) + \\frac{V_{k}(t)}{2}  \\bigg] \\alpha_{k}(t)h_{k}(t) \\,\\, & \\\\\n\t + \\, \\bigg[ iW_{k}(t) + \\frac{V_{k}(t)}{2}  \\bigg] \\beta_{k}(t)h^{\\ast}_{k}(t)& \\, . \\nonumber\n\\end{align}\nHere the arbitrary function $V_{k}(t)$  contains the residual freedom in the definition of the adiabatic vacuum. In this work we will choose this function to be\n\\begin{equation}\\label{Eq. Adiabatic V}\n\tV_{k}(t) = -\\frac{\\dot{W}_{k}(t)}{W_{k}(t)} \\, ,\n\\end{equation}\nas this choice leads to important simplifications in the back-reaction problem. \n\nGathering Eqs.~(\\ref{Eq. Field}) and (\\ref{Eq. Adiabatic Bogolyubov Mode}), we find that the ladder operators associated with the adiabatic representation satisfy the transformations\n\\begin{subequations}\\label{Eq. Bogolyubov Adiabatic Operators}\n\\begin{align}\na_{\\mathbf{k}}^{\\phantom{\\dagger}} &= \\alpha_{k}^{\\ast}(t) b_{\\mathbf{k}}^{\\phantom{\\dagger}}(t) - \\beta_{k}^{\\ast}(t) b_{\\mathbf{k}}^{\\dagger}(t)\\, , \\\\[8pt]\na_{\\mathbf{k}}^\\dagger &= \\alpha_{k}(t) b_{\\mathbf{k}}^{\\dagger}(t) - \\beta_{k}(t) b_{\\mathbf{k}}^{\\phantom{\\dagger}}(t) \\, ,\n\\end{align}\n\\end{subequations}\nwhich in turn imply a time-dependent version of Eq.~(\\ref{Eq. Bogolyubov Constraint}), \n\\begin{equation}\\label{Eq. Adiabatic Bogolyubov Constraint}\n\t\\big| \\alpha_{k}(t) \\big|^{2} - \\big| \\beta_{k}(t) \\big|^{2} = 1 \\, .\n\\end{equation}\n\nFinally, it is useful to characterize field states according to the values of the non-trivial adiabatic bilinears $\\big\\langle b_{\\mathbf{k}}^{\\dagger}(t) b_{\\mathbf{k}}(t) \\big\\rangle$ and $\\big\\langle b_{\\mathbf{k}}(t) b_{\\mathbf{k}}(t) \\big\\rangle$. The first of these bilinears tracks the adiabatic particle content per comoving volume in the $\\mathbf{k}$-mode under consideration. Using the transformations established above by Eqs.~(\\ref{Eq. Bogolyubov Adiabatic Operators}), it follows that\n\\begin{align}\\label{Eq. Adiabatic Particles}\n\t\\mathcal{N}_{k}(t) &= \\big\\langle b_{\\mathbf{k}}^{\\dagger}(t) b_{\\mathbf{k}}(t) \\big\\rangle \\nonumber \\\\\n\t&= \\big| \\alpha_{k}(t) \\big|^{2} \\big\\langle a_{\\mathbf{k}}^{\\dagger} a_{\\mathbf{k}}^{\\phantom{\\dagger}} \\big\\rangle + \\big| \\beta_{k}(t) \\big|^{2} \\big\\langle a_{\\mathbf{k}}^{\\phantom{\\dagger}} a_{\\mathbf{k}}^\\dagger \\big\\rangle \\\\\n\t&= N_{k} + \\sigma_{k} \\, \\big| \\beta_{k}(t) \\big|^{2} \\nonumber \\, ,\n\\end{align}\nwhere $N_{k} = \\big\\langle a_{\\mathbf{k}}^{\\dagger} a_{\\mathbf{k}}^{\\phantom{\\dagger}} \\big\\rangle$ is a constant of motion which can be understood as the initial number of adiabatic particles per comoving volume populating the field mode of wavenumber $k$, and $\\sigma_{k} = 1 + 2N_{k}$ is the Bose-Einstein parameter responsible for stimulated particle production. The second bilinear can be expressed as\n\\begin{align}\\label{Eq. Adiabatic Interference}\n\t\\mathcal{M}_{k}(t) &= \\big\\langle b_{\\mathbf{k}}(t) b_{\\mathbf{k}}(t) \\big\\rangle \\nonumber \\\\\n\t&= \\alpha_{k}(t) \\beta^{\\ast}_{k}(t) \\big\\langle a_{\\mathbf{k}}^{\\dagger} a_{\\mathbf{k}}^{\\phantom{\\dagger}} \\big\\rangle + \\alpha_{k}(t) \\beta^{\\ast}_{k}(t) \\big\\langle a_{\\mathbf{k}}^{\\phantom{\\dagger}} a_{\\mathbf{k}}^\\dagger \\big\\rangle \\\\\n\t&= \\sigma_{k} \\, \\alpha_{k}(t) \\beta^{\\ast}_{k}(t) \\nonumber \\, .\n\\end{align}\n\nIn principle, these bilinears contain all the required information to track the field evolution and, consequently, the time dependence of the field energy density and pressure. In practice, however, these quantities suffer from an irreducible ambiguity which is particularly pronounced when $\\mathcal{N}_{k}(t)$ and $\\mathcal{M}_{k}(t)$ incur rapid changes, such as when particle production occurs. \nThe root of this issue can be traced back to the asymptotic representation of $W_{k}(t)$, which is usually handled by simply truncating the series in Eq.~(\\ref{Eq. Adiabatic W}) at a finite order. \nHowever, the values of the bilinears depend strongly on where the series is truncated if they are rapidly changing (see Ref.~\\cite{DabrowskiDunne14} for striking graphical representations).\nIn the next Section, we discuss a technique for finding the exact universal evolution for both adiabatic bilinears which the asymptotic series represents.\n\n\n\\section{Particle Production and the Stokes Phenomenon}\\label{Sec. Stokes}\n\nThe adiabatic representation introduced in the previous section provides an accurate description of the bilinears $\\mathcal{N}_{k}(t)$ and $\\mathcal{M}_{k}(t)$ whenever $\\left| \\varepsilon_{k,\\,0} \\right| \\ll 1$. The more severely this condition is violated, the more unreliable these adiabatic quantities become. Adiabatic particle production, for instance, coincides with the momentary violation of this condition, implying that the notion of particle remains uncertain until particle production ceases. Nonetheless, a universal notion of particle can be restored for all times when particle production events are understood in terms of the Stokes phenomenon.\n\nThe sharp transitions between asymptotic solutions of a given differential equation which are valid in different regions of the complex plane are termed the Stokes phenomenon. These regions are bounded by the so-called Stokes and anti-Stokes lines. In the context of a scalar field evolving in a FLRW spacetime, the differential equation of interest is the equation of motion for a given field mode extended to a complex time variable $z$:\n\\begin{equation}\\label{Eq. Complex Mode Equation}\n{f}^{\\prime\\prime}_{k}(z) + \\Omega_{k}^{2}(z) f_{k}(z) = 0 \\, ,\n\\end{equation}\nin which the primes stand for differentiation with respect to $z$, the proper time is given by $t \\equiv \\mathrm{Re}\\, z$, and $\\Omega_{k}(z)$ represents the analytic continuation of the time-dependent frequency of Eq.~(\\ref{Eq. Mode Frequency}). The Stokes lines associated with Eq.~(\\ref{Eq. Complex Mode Equation}) are those lines which emanate from the zeros (also known as turning points) and poles of $\\Omega_{k}(z)$ and along which $\\mathrm{Re}\\big[\\Omega_{k}\\mathrm{d}z\\big]~\\!\\!=~\\!\\!0$. An illustration of such a line is shown in Fig.~\\ref{Fig. Stokes Line}. The asymptotic solutions susceptible to the Stokes phenomenon are given by\n\\begin{equation}\\label{Eq. Phase-Integral Solution}\n\tf_{k}(z) = \\alpha_{k}(z) h_{k}(z) + \\beta_{k}(z) h^{\\ast}_{k}(z) \\, ,\n\\end{equation}\nwhere $h_{k}(z)$ and $h_{k}^{\\ast}(z)$ are the complex extensions of the adiabatic mode functions defined in the previous section. As this solution evolves across a Stokes line, the values of the Bogolyubov coefficients $\\alpha_{k}(z)$ and $\\beta_{k}(z)$ change abruptly. By Eqs.~(\\ref{Eq. Adiabatic Particles}) and (\\ref{Eq. Adiabatic Interference}), this implies a sudden change in the adiabatic bilinears and, in particular, the production of adiabatic particles. Remarkably, a result from asymptotic analysis guarantees the existence of a smooth universal form for this rapid transition between different asymptotic regimes. Below we outline the derivation of this important result and summarize the quantities which determine the functional form of such smooth Stokes jumps.\n\n\\begin{figure}[t!]\n    \\includegraphics[width=0.49\\textwidth]{StokesLine.pdf}\n    \\caption{A depiction of the Stokes line sourced by a conjugate pair of simple turning points $\\big( z_{0},\\, z_{0}^{\\ast}\\big)$ of a frequency function $\\Omega_{k}(z)$. Here the Stokes line crosses the real axis at the point $s_{0}$, which corresponds to the time at which particle production occurs for the mode of wavenumber $k$. The guiding lines show the directions for which the condition $\\mathrm{Re}\\big[\\Omega_{k}\\mathrm{d}z\\big] = 0$ is locally satisfied.}\n    \\label{Fig. Stokes Line}\n\\end{figure}\n\nWe start by defining Dingle's singulant variable \\cite{Dingle73} anchored at $z_{0}$:\n\\begin{equation}\\label{Eq. Singulant}\nF_{k}^{(0)}(z) = 2i \\! \\int_{z_0}^{z} \\Omega_{k}(w) \\, \\mathrm{d}w \\, ,\n\\end{equation}\nwhere $z_0$ is a solution of $\\Omega_{k}(z) = 0$ which sources the Stokes line of interest, is closest to the real axis, and is located in the upper half-plane. The singulant is a convenient variable for tracking the change incurred by the Bogolyubov coefficients $\\alpha_{k}(z)$ and $\\beta_{k}(z)$ across a Stokes line. Indeed, it was shown by Berry \\cite{Berry90, Berry89, Berry88, Berry82} that these coefficients satisfy the following differential equations in the vicinity of a Stokes line:\n\\begin{subequations}\\label{Eq. Bogolyubov Couplers}\n\\begin{align}\n\t\\frac{d\\beta_{k}}{dF_{k}^{(0)}} &= C_{\\beta,\\,k}^{(0)}\\,\\alpha_{k}\\, , \\\\[8pt]\n\t\\frac{d\\alpha_{k}}{dF_{k}^{(0)}} &= C_{\\alpha,\\,k}^{(0)}\\,\\beta_{k}  \\, ,\n\\end{align}\n\\end{subequations}\nwhere $C_{\\beta,\\,k}^{(0)}$ and $C_{\\alpha,\\,k}^{(0)}$ are coupling functions which depend on the order at which the series representation of $W_{k}(z)$ is truncated. A remarkable discovery by Dingle \\cite{Dingle73} states that the large-order terms in the asymptotic series of Eq.~(\\ref{Eq. Adiabatic W}) have a closed form given by\n\\begin{equation}\n\\varphi_{k,\\, 2n}(z) \\sim -\\frac{(2n - 1)!}{\\pi F_{k}^{(0)\\,2n}} \\quad \\text{for} \\quad n \\gg 1 \\, .\n\\end{equation}\nIt is clear from this result that the smallest term in such a series corresponds to $n \\approx \\big| F_{k}^{(0)} \\big|$. Terminating the series at this order leads to optimal closed form expressions for $C_{\\beta,\\,k}^{(0)}$ and $C_{\\alpha,\\,k}^{(0)}$, which can be substituted in Eqs.~(\\ref{Eq. Bogolyubov Couplers}) to yield the following universal behaviors for $\\beta_{k}(t)$ and $\\alpha_{k}(t)$ along the real axis and across the Stokes line under consideration:\n\\begin{subequations}\\label{Eq. Smooth Bogolyubov}\n\\begin{align}\n\t\\beta_{k}(t) &\\approx \\frac{i}{2} \\, \\mathrm{Erfc}\\Big( \\! -\\vartheta_{k}^{(0)}(t) \\Big) \\delta_{k}^{(0)}\\, , \\\\[8pt]\n\t\\alpha_{k}(t) &\\approx \\sqrt{1 + \\big| \\beta_{k}(t) \\big|^{2}} \\, ,\n\\end{align}\n\\end{subequations}\nwhere $\\vartheta_{k}^{(0)}(t)$ is a natural time evolution parameter which determines the sharpness of the Stokes jump, and $\\delta_{k}^{(0)}$ corresponds to the jump's amplitude. Both of these parameters are expressible in terms of the singulant variable evaluated over the real axis:\n\\begin{subequations}\n\\begin{align}\n\t&\\vartheta_{k}^{(0)}(t) = \\frac{\\phantom{\\sqrt{2 \\,\\,}}\\mathrm{Im} \\Big[ F_{k}^{(0)}(t) \\Big]}{\\sqrt{2 \\, \\mathrm{Re} \\Big[ F_{k}^{(0)}(t) \\Big]}}\\, , \\\\[8pt]\n\t&\\delta_{k}^{(0)} = \\exp\\Big( \\! -F_{k}^{(0)}(s_{0}) \\Big) \\, .\n\\end{align}\n\\end{subequations}\nHere $F_{k}^{(0)}(s_{0})$ is simply the singulant computed at the point $z=s_{0}$, where the Stokes line sourced by $z_{0}$ intersects the real axis, i.e., \n\\begin{equation}\\label{Eq. Amplitude}\nF_{k}^{(0)}(s_{0}) = 2i \\! \\int_{z_0}^{s_{0}} \\Omega_{k}(w) \\, \\mathrm{d}w = i \\! \\int_{z_0}^{z_0^{\\ast}} \\Omega_{k}(w)\\,\\mathrm{d}w \\, ,\n\\end{equation}\nwhere the last equality follows from the reality of $\\Omega_{k}(z)$ over the real axis. Putting together Eqs.~(\\ref{Eq. Adiabatic Particles}), (\\ref{Eq. Adiabatic Interference}), and (\\ref{Eq. Smooth Bogolyubov}) yields a universal functional form which describes the time evolution of the adiabatic bilinears associated with the field mode of wavenumber $k$:\n\\begin{subequations}\\label{Eq. Smooth Bilinears}\n\\begin{align}\n\t\\mathcal{N}_{k} &\\approx N_{k} + \\frac{\\sigma_{k}}{4} \\Big| \\mathrm{Erfc}\\Big( \\! -\\vartheta_{k}^{(0)} \\Big) \\delta_{k}^{(0)} \\Big|^{2}\\, , \\\\[8pt]\n\t\\mathcal{M}_{k} &\\approx -i \\, \\frac{\\sigma_{k}}{2} \\Big[ \\mathrm{Erfc}\\Big( \\! -\\vartheta_{k}^{(0)} \\Big) \\delta_{k}^{(0)} \\Big] \\Big[ 1 + \\mathcal{N}_{k} \\Big]^{\\!1\/2}\n\\, .\n\\end{align}\n\\end{subequations}\n\nThese results can be further generalized to account for multiple Stokes line crossings, as well as the interference effects between them \\cite{DabrowskiDunne14}. Define the accumulated phase between the first and the $p$-th pair of zeros of $\\Omega_{k}(z)$ as\n\\begin{equation}\\label{Eq. Phase}\n\\theta_{k}^{(p)} = \\int_{s_0}^{s_p} \\Omega_{k}(w)\\,\\mathrm{d}w \n\\end{equation}\nwhere $s_p$ corresponds to the point where the Stokes line associated with the $p$-th conjugate pair of zeros crosses the real axis. The functions which describe both adiabatic bilinears are then given by\n\\begin{subequations}\\label{Eq. Bilinears Multiple Crossings}\n\\begin{align}\n\t\\mathcal{N}_{k} &\\approx N_{k} + \\frac{\\sigma_{k}}{4} \\bigg| \\sum_{p} \\mathrm{Erfc}\\Big( \\! -\\vartheta_{k}^{(p)} \\Big) \\delta_{k}^{(p)} \\exp\\Big(2i\\theta_{k}^{(p)} \\Big) \\bigg|^{2}\\, , \\\\[8pt]\n\t\\mathcal{M}_{k} &\\approx -i \\, \\frac{\\sigma_{k}}{2} \\bigg[ \\sum_{p} \\mathrm{Erfc}\\Big( \\! -\\vartheta_{k}^{(p)} \\Big) \\delta_{k}^{(p)} \\exp\\Big( \\! -2i\\theta_{k}^{(p)} \\Big) \\bigg] \\times \\nonumber \\\\\n\t & \\,\\,\\;\\; \\quad \\quad \\times \\! \\bigg[ 1 + \\mathcal{N}_{k} \\bigg]^{\\!1\/2}\n\\end{align}\n\\end{subequations}\nwith $\\delta_{k}^{(p)}$ and $\\vartheta_{k}^{(p)}(t)$ being the amplitude and time evolution parameter associated with the $p$-th Stokes line.\n\nTherefore, by monitoring the turning points and Stokes lines which accompany each mode's frequency function on the complex plane, we can track the evolution of the adiabatic bilinears related to any physically acceptable field state. In the next section we examine how this evolution affects the Universe's scale factor through the semi-classical Einstein equations.\n\n\\onecolumngrid\n\\section{The Semi-Classical Einstein Equations}\\label{Sec. Semi-Classical Einstein}\n\nIf cosmological quantum particle production occurs at a sufficiently high rate, it can in principle back-react on the cosmic evolution through the semi-classical Einstein equations\n\\begin{equation}\\label{Eq. Einstein Equations}\n\tR_{ab} - \\frac{1}{2}R \\, g_{ab} + \\Lambda \\, g_{ab} = M^{-2} \\big\\langle \\hat{T}_{ab}\\big\\rangle \n\\end{equation}\nwhere $\\Lambda$ represents the cosmological constant, ${M = ({8 \\pi G})^{-1\/2}}$ stands for the reduced Planck mass, and $\\big\\langle \\hat{T}_{ab}\\big\\rangle$ corresponds to the expectation value of the energy-momentum tensor operator, including contributions both from the scalar field we are considering plus any other stress-energy sources. The canonical expression for $\\hat{T}_{ab}$ due to the scalar field is constructed by varying the action in Eq.~(\\ref{Eq. Field Action}) with respect to the metric $g_{ab}$, and subsequently substituting the field operator $\\hat{\\Phi}$ from Eq.~(\\ref{Eq. Field}) into the resulting expression:\n\\begin{equation}\\label{Eq. Energy-Momentum Tensor}\n\t\\hat{T}_{ab} = \\big( \\nabla_{a} \\hat{\\Phi} \\big) \\big( \\nabla_{b} \\hat{\\Phi} \\big) - \\frac{1}{2}g_{ab} \\big( \\nabla^{c} \\hat{\\Phi} \\big) \\big( \\nabla_{c} \\hat{\\Phi} \\big) + \\xi \\bigg[  g_{ab} \\, \\Box - \\nabla_{a}\\nabla_{b} + R_{ab} - \\frac{1}{2}R \\, g_{ab} - \\frac{m^{2}}{2}g_{ab} \\bigg] \\hat{\\Phi}^{2} \\, .\n\\end{equation}\nFor a statistically homogeneous and isotropic field state, $\\big\\langle \\hat{T}_{ab}\\big\\rangle$ is equivalent to the energy-momentum tensor of a perfect fluid for which the field energy density and pressure are given, respectively, by $\\rho(t) = \\big\\langle \\hat{T}_{00} \\big\\rangle$ and $P(t) = \\frac{1}{3}g^{ij}\\big\\langle \\hat{T}_{ij} \\big\\rangle$. As a consequence, such a state naturally sources an FLRW metric, reducing Eq.~(\\ref{Eq. Einstein Equations}) to the usual Friedmann equations:\n\\begin{subequations}\\label{Eq. Friedmann Equations}\n\\begin{align}\n\t& H^{2}(t) = \\frac{1}{3} M^{-2} \\rho(t) + \\frac{\\Lambda}{3} - \\frac{K}{a^{2}(t)} \\label{Eq. First Friedmann Equation} \\\\[8pt]\n\t&\\dot{H}(t) + H^{2}(t) = -\\frac{1}{6} M^{-2} \\Big[ \\rho(t) + 3P(t) \\Big] + \\frac{\\Lambda}{3} \\, .\n\\end{align}\n\\end{subequations}\nFurthermore, it can be shown that $\\big\\langle \\hat{T}_{ab}\\big\\rangle$ is covariantly conserved, resulting in the cosmological continuity equation\n\\begin{equation}\\label{Eq. Cosmological Continuity}\n\t\\dot{\\rho}(t) + 3H(t)\\Big[ \\rho(t) + P(t) \\Big] = 0 \\, .\n\\end{equation}\n\nHowever, at this stage these equations are merely formal, because both the pressure and energy density of a quantized field are in general divergent and need to be regularized. \nIn a FLRW spacetime, these divergencies can be partially isolated by expressing $\\rho(t)$ and $P(t)$ in terms of the adiabatic bilinears $\\mathcal{N}_{k}(t)$ and $\\mathcal{M}_{k}(t)$ \\cite{Habib99}. Substituting Eqs.~(\\ref{Eq. Field}), (\\ref{Eq. Adiabatic Bogolyubov Mode}), and (\\ref{Eq. Adiabatic Bogolyubov Mode Derivative}) into the expectation value of Eq.~(\\ref{Eq. Energy-Momentum Tensor}) and collecting terms with the same adiabatic factor gives\n\\begin{subequations}\\label{Eq. Energy Density and Pressure}\n\\begin{align}\n\t&\\rho(t) = \\big\\langle \\hat{T}_{00} \\big\\rangle = \\frac{1}{4 \\pi a^{3}(t)} \\! \\int d\\mu(k) \\Bigg\\{  \\rho^{\\mathcal{N}}_{k}\\!(t) \\bigg[ \\mathcal{N}_{k}(t) + \\frac{1}{2} \\bigg] + \\rho^{\\mathcal{R}}_{k}(t)\\,\\mathcal{R}_{k}(t) + \\rho^{\\mathcal{I}}_{k}(t) \\, \\mathcal{I}_{k}(t) \\Bigg\\} \\label{Eq. Energy Density} \\\\[8pt] \n\t& P(t) = \\frac{1}{3} g^{ij} \\big\\langle \\hat{T}_{ij} \\big\\rangle = \\frac{1}{4 \\pi a^{3}(t)} \\! \\int d\\mu(k) \\Bigg\\{  P^{\\mathcal{N}}_{k}\\!(t) \\bigg[ \\mathcal{N}_{k}(t) + \\frac{1}{2} \\bigg] + P^{\\mathcal{R}}_{k}(t)\\,\\mathcal{R}_{k}(t) + P^{\\mathcal{I}}_{k}(t) \\, \\mathcal{I}_{k}(t) \\Bigg\\} \\label{Eq. Pressure} \\, .\n\\end{align}\n\\end{subequations}\nThe terms proportional to $\\mathcal{N}_{k}(t)$ capture the contribution to the energy density and pressure due to the evolving distribution of adiabatic particles populating the field modes, while the quantum interference terms contain ${\\mathcal{R}_{k}(t) = \\mathrm{Re}\\big[ \\mathcal{M}_{k}(t) \\big]}$ and ${\\mathcal{I}_{k}(t) = \\mathrm{Im}\\big[ \\mathcal{M}_{k}(t) \\big]}$ \\footnote{Here the definitions for $\\mathcal{R}_{k}(t)$ and $\\mathcal{I}_{k}(t)$ might seem to differ from those found in the literature by a phase factor, but this factor is implicit in our definitions for $\\alpha_{k}(t)$ and $\\beta_{k}(t)$ obtained from asymptotic analysis}. The prefactors in each of these terms are defined as\n\\begin{subequations}\\label{Eq. Adiabatic Factors}\n\\begin{align}\n\t&\\rho^{\\mathcal{N}}_{k}\\!(t) \\equiv \\frac{1}{W_{k}(t)} \\Bigg\\{ W^{2}_{k}(t) + \\omega^{2}_{k}(t) + \\frac{1}{4} \\Big[ V_{k}(t) - H(t) \\Big]^{\\!2} + \\big(6\\xi - 1\\big) \\bigg[ H(t) V_{k}(t) - 2H^{2}(t) + \\frac{K}{a^{2}(t)} \\bigg] \\Bigg\\}\\, , \\\\[8pt]\n\t&\\rho^{\\mathcal{R}}_{k}(t) \\equiv \\frac{1}{W_{k}(t)} \\Bigg\\{ - W^{2}_{k}(t) + \\omega^{2}_{k}(t) + \\frac{1}{4} \\Big[ V_{k}(t) - H(t) \\Big]^{\\!2} + \\big(6\\xi - 1\\big) \\bigg[ H(t) V_{k}(t) - 2H^{2}(t) + \\frac{K}{a^{2}(t)} \\bigg] \\Bigg\\}\\, , \\\\[8pt]\n\t&\\rho^{\\mathcal{I}}_{k}(t) \\equiv  V_{k}(t) - H(t) +2H(t)\\big(6\\xi - 1\\big)\\, , \\\\[8pt]\n\t&P^{\\mathcal{N}}_{k}\\!(t) \\equiv \\frac{1}{3W_{k}(t)} \\Bigg\\{ W^{2}_{k}(t) + \\omega^{2}_{k}(t) - 2m^{2} + \\frac{1}{4} \\Big[ V_{k}(t) - H(t) \\Big]^{\\!2} + \\frac{1}{3}\\big(6\\xi - 1\\big)^{\\!2}R(t) + \\Bigg. \\, \\\\\n\t & \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\Bigg. + \\big(6\\xi - 1\\big) \\bigg[ -2W^{2}_{k}(t) - \\frac{1}{2}V^{2}_{k}(t) + 4H(t)V_{k}(t) + 2\\omega^{2}_{k}(t) + 2\\dot{H}(t) + \\frac{K}{a^{2}(t)} - \\frac{5}{2}H^{2}(t) \\bigg] \\Bigg\\}\\, , \\nonumber \\\\[8pt]\n\t&P^{\\mathcal{R}}_{k}(t) \\equiv \\frac{1}{3W_{k}(t)} \\Bigg\\{ -W^{2}_{k}(t) + \\omega^{2}_{k}(t) - 2m^{2} + \\frac{1}{4} \\Big[ V_{k}(t) - H(t) \\Big]^{\\!2} + \\frac{1}{3}\\big(6\\xi - 1\\big)^{\\!2}R(t) + \\Bigg.\\,  \\\\\n\t & \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\Bigg. + \\big(6\\xi - 1\\big) \\bigg[ 2W^{2}_{k}(t) - \\frac{1}{2}V^{2}_{k}(t) + 4H(t)V_{k}(t) + 2\\omega^{2}_{k}(t) + 2\\dot{H}(t) + \\frac{K}{a^{2}(t)} - \\frac{5}{2}H^{2}(t) \\bigg] \\Bigg\\}\\, , \\nonumber \\\\[8pt]\n\t&P^{\\mathcal{I}}_{k}(t) \\equiv \\frac{1}{3}\\Big[ V_{k}(t) - H(t) \\Big] + \\frac{2}{3} \\big(6\\xi - 1\\big) \\Big[ 4H(t) - V_{k}(t) \\Big] \\, .\n\\end{align}\n\\end{subequations}\nFor adiabatic field states, the contributions to the energy density and pressure due to the real bilinears $\\mathcal{N}_{k}(t)$, $\\mathcal{R}_{k}(t)$, and $\\mathcal{I}_{k}(t)$ are always finite. This implies that the divergencies in Eqs.~(\\ref{Eq. Energy Density}) and (\\ref{Eq. Pressure}) are isolated in the vacuum-like terms characterized by the $\\frac{1}{2}$ factors, henceforth identified as\n\\begin{subequations}\\label{Eq. Energy and Pressure Vacuum}\n\\begin{align}\n\t\\rho_{\\mathrm{vac}}(t) &\\equiv \\frac{1}{8 \\pi a^{3}(t)} \\! \\int d\\mu(k)\\,\\rho^{\\mathcal{N}}_{k}\\!(t) \\label{Eq. Vacuum Energy} \\\\[8pt]\n\tP_{\\mathrm{vac}}(t) &\\equiv \\frac{1}{8 \\pi a^{3}(t)} \\! \\int d\\mu(k)\\,P^{\\mathcal{N}}_{k}\\!(t) \\label{Eq. Vacuum Pressure} \\, .\n\\end{align}\n\\end{subequations}\n\nRegularization consists precisely in controlling the divergent behavior of $\\rho_{\\mathrm{vac}}(t)$ and $P_{\\mathrm{vac}}(t)$ so as to obtain finite expressions for  $\\rho(t)$ and $P(t)$ which still satisfy the cosmological continuity equation. Adiabatic regularization achieves this result by subtracting the fourth-order phase-integral expansions of $\\rho^{\\mathcal{N}}_{k}\\!(t)$ and $P^{\\mathcal{N}}_{k}\\!(t)$ from the integrands of Eqs.~(\\ref{Eq. Vacuum Energy}) and (\\ref{Eq. Vacuum Pressure}), respectively \\cite{ParkerFulling74, AndersonParker87}. From a technical perspective, however, this procedure introduces significant challenges to the numerical implementation of the semi-classical Friedmann equations. Chief among these is the appearance of higher-order time derivatives of $H(t)$ in the integrands of Eqs.~(\\ref{Eq. Energy Density and Pressure}), turning the semi-classical Friedmann equations into a system of integro-differential equations which is not amenable to standard numerical treatments. We circumvent this difficulty by employing an alternative regularization scheme which, albeit cruder, yields a good approximation to $\\rho(t)$ and $P(t)$ in regimes dominated by particle production.\n\nCentral to the regularization approach adopted here is the realization that $\\rho_{\\mathrm{vac}}(t)$ and $P_{\\mathrm{vac}}(t)$ independently satisfy the cosmological continuity equation as long as the function $V_{k}(t)$ has the form established in Eq.~(\\ref{Eq. Adiabatic V}) (see the Appendix). It follows that the vacuum contributions to $\\rho(t)$ and $P(t)$ can be discarded in their entirety while still ensuring that Eq.~(\\ref{Eq. Cosmological Continuity}) remains valid. Despite its simplicity, this procedure yields a good approximation to the field energy density and pressure provided the adiabatically regularized integrands of Eqs.~(\\ref{Eq. Energy Density and Pressure}) are dominated by the real adiabatic bilinears $\\mathcal{N}_{k}(t)$, $\\mathcal{R}_{k}(t)$, and $\\mathcal{I}_{k}(t)$. Therefore, in what follows we take the regularized expressions for the energy density and pressure to be\n\\begin{subequations}\\label{Eq. Reg. Energy Density and Pressure}\n\\begin{align}\n\t\\rho(t) &\\approx \\frac{1}{4 \\pi a^{3}(t)} \\! \\int d\\mu(k) \\Bigg\\{  \\rho^{\\mathcal{N}}_{k}\\!(t) \\, \\mathcal{N}_{k}(t) + \\rho^{\\mathcal{R}}_{k}(t)\\,\\mathcal{R}_{k}(t) + \\rho^{\\mathcal{I}}_{k}(t) \\, \\mathcal{I}_{k}(t) \\Bigg\\} \\label{Eq. Reg. Energy Density} \\\\[8pt] \n\tP(t) &\\approx \\frac{1}{4 \\pi a^{3}(t)} \\! \\int d\\mu(k) \\Bigg\\{  P^{\\mathcal{N}}_{k}\\!(t) \\, \\mathcal{N}_{k}(t) + P^{\\mathcal{R}}_{k}(t)\\,\\mathcal{R}_{k}(t) + P^{\\mathcal{I}}_{k}(t) \\, \\mathcal{I}_{k}(t) \\Bigg\\} \\label{Eq. Reg. Pressure} \\, ,\n\\end{align}\n\\end{subequations}\nwhere the factors $\\rho^{\\mathcal{N}}_{k}\\!(t)$, $\\rho^{\\mathcal{R}}_{k}(t)$, and $\\rho^{\\mathcal{I}}_{k}(t)$, $P^{\\mathcal{N}}_{k}\\!(t)$, $P^{\\mathcal{R}}_{k}(t)$, and $P^{\\mathcal{I}}_{k}(t)$ are computed by truncating the asymptotic series Eq.~(\\ref{Eq. Adiabatic W}) for $W_{k}(t)$ at its optimal order.\n\nFinally, the regularization of $\\big\\langle \\hat{T}_{ab}\\big\\rangle$ also induces the renormalization of the gravitational coupling constants $G$ and $\\Lambda$. Moreover, self-consistency demands the  introduction of a covariantly conserved tensor composed of fourth-order derivatives of the metric into the semi-classical Einstein equations \\cite{FullingParker74, Bunch80}. This tensor is accompanied by a new unknown coupling constant whose renormalization assimilates the ultra-violet divergence in the field energy-momentum tensor. For simplicity, in this work we assume this new coupling constant to be renormalized to zero, thus preserving the form of Eq.~(\\ref{Eq. Einstein Equations}). Non-zero values for this coupling constant will be considered elsewhere.\n\nTaken together, Eqs.~(\\ref{Eq. Bilinears Multiple Crossings}), (\\ref{Eq. Friedmann Equations}), and (\\ref{Eq. Reg. Energy Density and Pressure}) describe the coupled field evolution and cosmic evolution in regimes dominated by particle production. In the next section we present an algorithm which numerically solves this system of equations.\\\\\n\\twocolumngrid\n\n\\section{Numerical Implementation}\\label{Sec. Numerical Implementation}\n\nThe semi-classical Friedmann equations can be formulated as a discretized initial value problem. We take the domain of numerical integration to be a band of the complex plane which is bisected by the real $t$ axis. As illustrated in Figure \\ref{Fig. Numerical Analytic Continuation}, this band is discretized by a uniformly spaced grid where the real-valued entries $t_{j}$ label the physical time. Initial conditions are set by an appropriately chosen functional form for the scale factor $a(t)$ which not only admits an adiabatic field state at the initial time $t_{0}$, but which is also consistent with our choice for the initial distribution of adiabatic particles $\\mathcal{N}_{k}(t_{0}) = N_{k}$ populating the field modes. In addition, we require that\n\\begin{equation}\\label{Eq. Particle Distribution Constraint}\n\tN_{k} < \\mathcal{O}(k^{-3}) \\quad \\text{as} \\quad k \\rightarrow \\infty\n\\end{equation}\nin order to ensure that both the energy density and pressure associated with the initial particle distribution are finite. \n\nWe use a standard finite-difference scheme to step $a(t)$, $H(t)$, and $\\dot{H}(t)$ along the real axis, and employ B-splines to scan the Stokes geometry on the complex plane. The latter is accomplished by generating a numerical sample of $\\Omega_{k}(t)$ through Eq.~(\\ref{Eq. Mode Frequency}), and subsequently performing high-order B-spline interpolations to construct a truncated Taylor polynomial for this function over the real line up to the value of $t$ in the current time step. Due to the analyticity of $\\Omega_{k}(t)$, this series representation is also valid on the complex plane, and thus encodes the analytical continuation of the frequency function. Explicitly, given a grid point $z_{ij}$ on the discretized plane, we compute $\\Omega_{k}(z_{ij})$ through the expression\n\\begin{equation}\\label{Eq. Taylor}\n\\Omega_{k}(z_{ij}) \\approx \\sum_{n = 0}^{T} \\frac{1}{{n!}} \\big(z_{ij} - t_{j} \\big)^{n} \\, \\Omega_{k}^{(n)}(t_{j}) \\, ,\n\\end{equation}\nwhere $t_{j} = \\mathrm{Re}\\,z_{ij}$, as depicted in Figure \\ref{Fig. Numerical Analytic Continuation}. The numerical derivatives $\\Omega_{k}^{(n)}$ are extracted from B-spline interpolations over the real axis, and $T$ corresponds to a truncation order which depends on the density of grid points lying over the real axis. In addition, we feed Eq.~(\\ref{Eq. Taylor}) to a Pad\\'{e} approximant \\cite{Bender13} routine to accelerate its convergence and improve its accuracy. Once this approximate representation of the frequency function has been computed over the discretized plane, it can be interpolated and used in the monitoring of turning points and Stokes lines.\n\n\\begin{figure}[t!]\n    \\includegraphics[width=0.49\\textwidth]{NumericalAnalyticContinuation.pdf}\n    \\caption{A grid of uniformly spaced points covering a band of the complex plane. The grid points lying over the real axis mark the discretization of physical time. Numerically constructing the Taylor polynomial associated with the frequency function around the point $t_{j}$ allows for the optimal evaluation of $\\Omega_{k}(z_{ij})$ at grid points $z_{ij}$ for which $\\mathrm{Re}\\,z_{ij} = t_{j}$.}\n    \\label{Fig. Numerical Analytic Continuation}\n\\end{figure}\n\nWhile the turning points of $\\Omega_{k}(z)$ can be located with the aid of root-finding algorithms designed for multi-valued functions, the problem of determining the Stokes lines sourced by these points requires the numerical integration of an ordinary differential equation. This is evident from the Stokes lines definition ${\\mathrm{Re}\\big[\\Omega_{k}\\mathrm{d}z\\big]=0}$, which implies that, locally, its line element must satisfy ${\\mathrm{d}z \\propto i\/\\Omega_{k}(z)}$. Defining ${t=\\mathrm{Re} \\, z}$ and ${\\tau=\\mathrm{Im} \\, z}$, this condition can be rewritten as\n\\begin{equation}\n\t\\mathrm{d}z = \\mathrm{d}t + i\\,\\mathrm{d}\\tau \\propto \\frac{i}{\\Omega_{k}(z)} \\, .\n\\end{equation}\nTaking the ratio between the matched real and imaginary parts of this proportionality relation leads to the differential equation \n\\begin{equation}\\label{Eq. Stokes ODE}\n\t\\frac{dt}{d\\tau} = \\frac{ \\mathrm{Im} \\, \\Omega_{k}(z) }{ \\mathrm{Re} \\, \\Omega_{k}(z) } \\, \n\\end{equation}\nfor the Stokes line,\nwhich can be numerically integrated from the turning point of interest to yield $t(\\tau)$.\n\nHere is a summary of the minimal set of tasks performed by our algorithm while evolving the physical quantities of interest by one time step:\n\\begin{itemize}\n\t\\item[1.] Take samples of $a(t)$, $H(t)$, and $\\dot{H}(t)$ describing the metric along an interval of the real axis. Over this same interval, sample and interpolate the field energy density $\\rho(t)$ and pressure $P(t)$.\n\t\n\t\\item[2.] Numerically integrate the semi-classical Friedmann equations so as to enlarge the  input metric samples $a(t)$, $H(t)$, and $\\dot{H}(t)$ by a time step $\\Delta t$.\n\n\t\\item[3.] For each field mode, generate a sample of the frequency function $\\Omega_{k}(t)$ over the real axis, and numerically extend this function onto the complex plane to obtain $\\Omega_{k}(z)$.\n\t\n\t\\item[4.] Search for complex turning points of each frequency function $\\Omega_{k}(z)$, and numerically trace their corresponding Stokes lines.\n\n\t\\item[5.] If a Stokes line associated with a mode of wavenumber $k$ is found to intersect the real axis, update the real bilinears $\\mathcal{N}_{k}(t)$, $\\mathcal{R}_{k}(t)$ and $\\mathcal{I}_{k}(t)$ accordingly.\n\t\n\t\\item[6.] For each field mode, compute $W_{k}(t)$ and $V_{k}(t)$ up to the optimal truncation order set by the last Stokes line crossing.\n\t\n\t\\item[7.] Gather the results from all previous steps to evolve the input samples for the field energy density $\\rho(t)$ and pressure $P(t)$ by a time step $\\Delta t$.\n\\end{itemize}\n\nIn general, the Stokes lines associated with field modes of comparable wavenumber will cross the real axis within close proximity of one another, giving rise to overlapping particle production events. In order to correctly capture the influence that such events might have on each other, we apply the stepping algorithm outlined above in an iterative fashion. In other words, once the quantities of interest have been forward-stepped up to $t_{j}$, the following iteration backtracks to $t_{0}$ and then proceeds to step the problem up to $t_{j+1} = t_{j} + \\Delta t$ using as sources for the semi-classical Friedmann equations the field energy density and pressure obtained in the previous iteration.\n\nIn summary, our numerical implementation allows for the scale factor and the Stokes geometry to reconfigure themselves with each iteration and thereby construct a self-consistent solution to the back-reaction problem. \n\n\\section{Numerical Results}\\label{Sec. Results}\n\nTo assess the accuracy of our numerical approach, we first neglect back-reaction effects and compare numerical results to known analytic solutions for a quantized scalar field evolving in a closed de Sitter spacetime \\cite{Mottola85}. This case is characterized by a positive cosmological constant $\\Lambda$ and a curvature parameter of $K = 1$, which together lead to a bouncing scale factor evolution\n\\begin{equation}\\label{Eq. de Sitter}\n\ta(t) = \\bar{H}^{-1} \\cosh{\\left(\\bar{H} t\\right)} \\quad \\text{with} \\quad \\bar{H} = \\sqrt{\\Lambda\/3} \\, .\n\\end{equation}\nHere $\\bar{H}$ is the asymptotic value of the Hubble parameter in the infinite future,\n\\begin{equation}\\label{Eq. Hubble de Sitter}\n\t\\lim_{t \\to \\pm \\infty}{H(t)} = \\pm \\bar{H} \\, .\n\\end{equation}\nThis model Universe contracts for $t < 0$, reaches its minimum size at $t = 0$, and subsequently expands for the $t > 0$.\n\nSubstituting Eq.~(\\ref{Eq. de Sitter}) into Eq.~(\\ref{Eq. Mode Frequency}) yields \n\\begin{equation}\\label{Eq. de Sitter Mode Frequency}\n\\Omega_{k}^{2}(t) = \\bar{H}^{2} \\Bigg[ \\bigg(k^{2} - \\frac{1}{4} \\bigg) \\, \\text{sech}^{2} \\! \\left(\\bar{H} t\\right) + \\frac{m^{2}}{\\bar{H}^{2}} + 12\\,\\xi - \\frac{9}{4} \\Bigg] \\, \n\\end{equation}\nfor the mode frequency function.\nAnalytically extending this function to the complex plane, locating its turning points, and tracing its Stokes lines are straightforward. We verify our numerical calculations against these analytic results. For definiteness, we choose a scalar field of mass $m = 0.1\\,M$ which is conformally coupled to the scalar curvature, $\\xi = \\frac{1}{6}$. We set the cosmological constant to $\\Lambda = 3 \\, m^{2}$, so that $\\bar{H} = 1 \\, m$. All dimensional quantities are thus expressed in terms of the field mass.\n\nA comparison between the analytic extension of Eq.~(\\ref{Eq. de Sitter Mode Frequency}) and the  numerical analytic continuation produced by our algorithm is displayed in Figure \\ref{Fig. Continuation Error} for the field mode of wavenumber $k = 5 \\, m$. The left panel shows the absolute value of the numerically obtained frequency function, while the right panel exhibits how this result deviates from the analytic expression for $\\Omega_{k}(z)$. In addition to correctly reproducing the function's conjugate pair of zeroes $\\big( z_{0},\\, z_{0}^{\\ast}\\big)$ located in this region, the numerical analytic continuation differs from the analytic value by at most 2\\% in the vicinity of these points. As a result, the Stokes lines which occupy this area of the complex plane can be traced with high fidelity. This is demonstrated in the left panel of Figure \\ref{Fig. Frequency Particle}, where the Stokes lines sourced by the pairs of turning points $\\big( z_{0},\\, z_{0}^{\\ast}\\big)$ and $\\big( z_{1},\\, z_{1}^{\\ast}\\big)$ are superimposed over the numerically obtained frequency function. The effects of each Stokes line on the adiabatic bilinear $\\mathcal{N}_{k}(t)$ are displayed in the right panel of Figure \\ref{Fig. Frequency Particle}, wherein this quantity is tracked as a function of time. Each burst of particle production is prompted by a Stokes line crossing, the first of which occurs as the Universe contracts and the field mode under consideration becomes sub-horizon; while the second burst happens after the bounce, when the mode reverts back to being super-horizon due to the Universe's expansion \\cite{Habib99}. Despite the symmetry between these events, constructive interference expressed by Eq.~(\\ref{Eq. Bilinears Multiple Crossings}) causes more particles to be produced in the second burst. The expected values for the particle number plateaus are indicated by the square markers on the vertical axis, both of which agree well with the numerical curve.\n\n\\begin{figure}[t!]\n    \\includegraphics[width=0.49\\textwidth]{ContinuationError.pdf}\n    \\caption{A comparison between the numerical analytic continuation of $\\Omega_{k}(z)$ produced by our algorithm and the expected analytic expression for this function in a closed de Sitter spacetime. The field parameters are $m = 0.1 \\, M$, $\\xi = \\frac{1}{6}$, and $k = 5 \\, m$, while the spacetime is characterized by $\\Lambda = 3 \\, m^{2}$ and $K = 1$. The left panel shows the absolute value of the numerically produced frequency function in the vicinity of the pair of conjugate turning points $\\big( z_{0},\\, z_{0}^{\\ast}\\big)$, while the right panel exhibits the relative difference between the analytic and numerical results.}\n    \\label{Fig. Continuation Error}\n\\end{figure}\n\n\\begin{figure*}[hbtp]\n    \\includegraphics[width=1.0\\textwidth]{FrequencyParticle.pdf}\n    \\caption{The numerically traced Stokes geometry associated with the frequency function $\\Omega_{k}(z)$, and the adiabatic particle number evolution $\\mathcal{N}_{k}(t)$ extracted from it. The field parameters are set to $m = 0.1 \\, M$, $\\xi = \\frac{1}{6}$, $k = 5 \\, m$, and $N_{k} = 0$, while the spacetime is characterized by $\\Lambda = 3 \\, m^{2}$ and $K = 1$. The left panel shows the Stokes lines sourced by the pairs of turning points $\\big( z_{0},\\, z_{0}^{\\ast}\\big)$ and $\\big( z_{1},\\, z_{1}^{\\ast}\\big)$ superimposed over the absolute value of the numerically obtained frequency function. The real axis corresponds to the central dashed line. The effects of each Stokes line on the adiabatic particle number $\\mathcal{N}_{k}(t)$ are illustrated on the right panel, wherein this quantity is tracked as a function of time. Each burst of particle production is prompted by a Stokes line crossing, indicated here by the circular markers on the horizontal axis. The expected values for the particle number plateaus featuring in this image are indicated by the square markers on the vertical axis, both of which show very good agreement with the numerically produced curve for $\\mathcal{N}_{k}(t)$. Constructive interference causes more particles to be produced in the second burst.}\n    \\label{Fig. Frequency Particle}\n\\end{figure*}\n\\begin{figure*}[hbtp]\n    \\includegraphics[width=1.0\\textwidth]{EnergyDensity_NoBackReaction.pdf}\n    \\caption{The evolution of every term appearing on the right-hand side of the semi-classical Friedmann Eq.~(\\ref{Eq. First Friedmann Equation}) in a closed de Sitter spacetime evolution, as well as the quantities describing the metric for this spacetime in the absence of back reaction. The field parameters are set to $m = 0.1 \\, M$ and $\\xi = \\frac{1}{6}$, while the spacetime is characterized by $\\Lambda = 3 \\, m^{2}$ and $K = 1$. The bounce starts at $t_{0} = -5 \\, m^{-1}$ with an initial particle distribution given by $\\mathcal{N}_{k}(t_{0}) = 0$. The left panel follows the evolution of $H^{2}_{\\mathcal{N},\\,\\mathcal{R},\\,\\mathcal{I}}$ (solid line), $H^{2}_{\\mathcal{R},\\,\\mathcal{I}}$ (dot-dashed line), $H^{2}_{\\Lambda}$ (dotted line), and $H^{2}_{K}$ (dashed line). While $H^{2}_{\\mathcal{R},\\,\\mathcal{I}}$ remains negligible throughout, $H^{2}_{\\mathcal{N},\\,\\mathcal{R},\\,\\mathcal{I}}$ grows exponentially and eventually comes to dominate over all other contributions. The right panels illustrate the scale factor $a(t)$ and Hubble parameter $H(t)$ which describe the de Sitter bounce. Because back-reaction effects are being neglected, the Hubble parameter is just $H^{2} = H^{2}_{\\Lambda} + H^{2}_{K}$.} \n    \\label{Fig. Contributions to H2 - No back-reaction}\n\\end{figure*}\n\nBy tracing the Stokes geometry of every field mode, we can also track the evolution of the field energy density as the spacetime evolves. Even though back-reaction effects are being neglected, this quantity shows whether the effects of particle production will eventually become comparable to the contributions from $\\Lambda$ and $K$ which source the background de Sitter spacetime. To that end, we track every term appearing on the right-hand side of  the semi-classical Friedmann Eq.~(\\ref{Eq. First Friedmann Equation}), identifying each contribution according to the notation\n\\begin{subequations}\\label{Eq. Contributions to H2}\n\\begin{align*}\n\tH^{2}_{\\mathcal{N},\\,\\mathcal{R},\\,\\mathcal{I}} \\equiv \\frac{\\rho}{3M^2}\\,\\,, \\quad H^{2}_{\\Lambda} \\equiv \\frac{\\Lambda}{3} \\,\\,, \\quad \\text{and} \\quad H^{2}_{K} \\equiv - \\frac{K}{a^{2}} \\, .\n\\end{align*}\n\\end{subequations}\nAdditionally, we define $H^{2}_{\\mathcal{R},\\,\\mathcal{I}}$ as the contribution to the right-hand side of Eq.~(\\ref{Eq. First Friedmann Equation}) which stems solely from terms proportional to the real bilinears $\\mathcal{R}_{k}$ and $\\mathcal{I}_{k}$. The left panel of Figure~\\ref{Fig. Contributions to H2 - No back-reaction} displays the evolution of the above-defined quantities for a bounce that starts at $t_{0} = -5 \\, m^{-1}$ with an initial particle distribution given by $\\mathcal{N}_{k}(t_{0}) = 0$. Being the only true sources in this case, $H^{2}_{\\Lambda}$ and $H^{2}_{K}$ behave in the standard way, acting in concert to produce the de Sitter bounce. Because back-reaction effects are neglected, the Hubble parameter $H$ shown on the right panel of Figure~\\ref{Fig. Contributions to H2 - No back-reaction} is entirely characterized by these two quantities, i.e., $H^{2} = H^{2}_{\\Lambda} + H^{2}_{K}$. On the other hand, the field-related quantities $H^{2}_{\\mathcal{N},\\,\\mathcal{R},\\,\\mathcal{I}}$ and $H^{2}_{\\mathcal{R},\\,\\mathcal{I}}$ display an interesting behavior which mirrors the result found in Ref.~\\cite{AndersonMottola14}. While $H^{2}_{\\mathcal{R},\\,\\mathcal{I}}$ remains negligible throughout, $H^{2}_{\\mathcal{N},\\,\\mathcal{R},\\,\\mathcal{I}}$ grows exponentially as the Universe progresses toward the bounce. In other words, the field energy density eventually becomes dominated by $\\mathcal{N}_{k}$ -- the field particle content. Physically, the soaring field energy density is due to the blueshift experienced by particles produced in the contracting phase. As a result, the Universe is filled with relativistic particles which effectively behave as radiation, making the field energy density grow as $\\rho \\propto a^{-4}$. This trend is then reversed in the ensuing expanding phase, during which the field energy density drops rapidly as particles are continuously redshifted.\n\\begin{figure*}[hbtp]\n    \\includegraphics[width=1.0\\textwidth]{EnergyDensity_BackReaction.pdf}\n    \\caption{The evolution of every term appearing on the right-hand side of the semi-classical Friedmann Eq.~(\\ref{Eq. First Friedmann Equation}) and the quantities describing the metric evolution in a full back-reacting calculation. The field parameters are $m = 0.1 \\, M$ and $\\xi = \\frac{1}{6}$, while the cosmological constant and curvature parameter are $\\Lambda = 3 \\, m^{2}$ and $K = 1$. The closed de Sitter initial conditions are set at $t_{0} = -5 \\, m^{-1}$, along with an initial particle distribution given by $\\mathcal{N}_{k}(t_{0}) = 0$. The left panel follows the evolution of $H^{2}_{\\mathcal{N},\\,\\mathcal{R},\\,\\mathcal{I}}$ (solid line), $H^{2}_{\\mathcal{R},\\,\\mathcal{I}}$ (dot-dashed line), $H^{2}_{\\Lambda}$ (dotted line), and $H^{2}_{K}$ (dashed line). The exponential growth of $H^{2}_{\\mathcal{N},\\,\\mathcal{R},\\,\\mathcal{I}}$ effectively fills the Universe with relativistic particles, introducing an instability to the initial de Sitter phase. The right panels illustrate the scale factor $a(t)$ and Hubble parameter $H(t)$ transitioning from a Sitter bounce to a radiation dominated phase. Here the solid lines represent the solutions to the back-reaction problem, while the dashed lines trace the pure de Sitter bounce.} \n    \\label{Fig. Contributions to H2 - Back-reaction}\n\\end{figure*}\n\nThe preceding calculations demonstrate that back-reaction effects due to particle production can become dynamically significant in an initially closed de Sitter spacetime. A full account of these effects is shown in Figure~\\ref{Fig. Contributions to H2 - Back-reaction}, using the algorithm for computing back-reaction effects described in the previous Section. In this case, the metric evolution initially matches that of a closed de Sitter spacetime at $t_{0} = -5 \\, m^{-1}$, while the initial particle distribution is given by $\\mathcal{N}_{k}(t_{0}) = 0$. These initial conditions self-consistently satisfy the semi-classical Friedmann equations at the initial time $t_{0}$ within our approximations. As in the case without back reaction, the quantity $H^{2}_{\\mathcal{R},\\,\\mathcal{I}}$ remains sub-dominant throughout the evolution, while $H^{2}_{\\mathcal{N},\\,\\mathcal{R},\\,\\mathcal{I}}$ grows exponentially as newly-created particles are continuously blueshifted. Since they quickly become relativistic, these particles behave as an additional radiation-like component, destabilizing the initial de Sitter phase. This is illustrated in the right panels of Figure~\\ref{Fig. Contributions to H2 - Back-reaction}, where the scale factor and Hubble parameter can be seen transitioning from a de Sitter bounce to a radiation-dominated behavior. The contributions from the regularized vacuum terms discarded in our approximations remain negligible at all times. We stop the numerical integration at $t = -1.3 \\, m^{-1}$, since beyond this time the Hubble parameter becomes of order $H \\simeq M^{-1}$, invalidating the semi-classical picture of gravity on which our calculations rely.\n\nThe de Sitter bounce is not always disrupted by particle production. For sufficiently low values of the field mass, the bounce is merely delayed. Figure~\\ref{Fig. Contributions to H2 - Back-reaction 2} illustrates a near-limiting case with ${m = 0.0145 \\, M}$ for which an initial de Sitter evolution is still driven toward a radiation dominated phase. The contributions due to particle production $H^{2}_{\\mathcal{N},\\,\\mathcal{R},\\,\\mathcal{I}}$ only come to dominate over the combined $H^{2}_{\\Lambda}$ and $H^{2}_{K}$ near the bounce at $t = 0 \\, m^{-1}$. For field masses $m \\lesssim 0.0142 \\, M$, the negative curvature contributions $H^{2}_{K}$ neutralize the growth of $H^{2}_{\\mathcal{N},\\,\\mathcal{R},\\,\\mathcal{I}}$ for long enough to preserve the bounce. The resulting bounce is pushed to a slightly later time and occurs at a smaller value of the scale factor.\n\n\\begin{figure*}[hbtp]\n    \\includegraphics[width=1.0\\textwidth]{EnergyDensity_BackReaction_2.pdf}\n    \\caption{The evolution of every source term featuring on the right-hand side of the semi-classical Friedmann Eq.~(\\ref{Eq. First Friedmann Equation}) and the quantities describing the metric evolution in a full back-reacting calculation. The field parameters are set to $m = 0.0145 \\, M$ and $\\xi = \\frac{1}{6}$, while the cosmological constant and curvature parameter are characterized by $\\Lambda = 3 \\, m^{2}$ and $K = 1$. The closed de Sitter initial conditions are set at $t_{0} = -5 \\, m^{-1}$, along with an initial particle distribution given by $\\mathcal{N}_{k}(t_{0}) = 0$. The left panel follows the evolution of $H^{2}_{\\mathcal{N},\\,\\mathcal{R},\\,\\mathcal{I}}$ (solid line), $H^{2}_{\\mathcal{R},\\,\\mathcal{I}}$ (dot-dashed line), $H^{2}_{\\Lambda}$ (dotted line), and $H^{2}_{K}$ (dashed line). The growth of $H^{2}_{\\mathcal{N},\\,\\mathcal{R},\\,\\mathcal{I}}$ fills the Universe with just enough relativistic particles to destabilize the initial de Sitter phase. The right panels illustrate the scale factor $a(t)$ and Hubble parameter $H(t)$ transitioning from a de Sitter bounce to a radiation dominated phase. Here the solid lines represent the solutions to the back-reaction problem, while the dashed lines trace the pure de Sitter bounce. Had the field mass been set to a value $m \\lesssim 0.0142 \\, M$, a bounce would still take place, albeit at a slightly later time and for a smaller value of the scale factor.}\n    \\label{Fig. Contributions to H2 - Back-reaction 2}\n\\end{figure*}\n\n\\section{Discussion}\\label{Sec. Discussion}\n\nThe back-reaction problem addressed in this work imposes several technical hurdles which have resisted a satisfactory solution for decades. These difficulties stem primarily from the necessity to control the divergent nature of the vacuum energy. Adiabatic regularization accomplishes this at the cost of increasing the problem's complexity. As a result, ambiguities arise in the specification of initial conditions and in the value of physical quantities when particle production is rapid, and computationally the problem becomes susceptible to potential numerical instabilities. In this work we have shown that these issues can be circumvented in scenarios dominated by particle production. Our approach relies on a particular choice of adiabatic mode functions which isolate the vacuum contributions into a separate covariantly conserved component of the total stress-energy. In regimes dominated by particle production, this vacuum component is sub-dominant and can be discarded in its entirety. By definition, the remaining covariantly conserved portion of the stress-energy dominates, as it encapsulates the effects of particle production. This component can be expressed in terms of the particle number density as described by Berry's universal form, resolving the ambiguity in physical quantities, and computed from the analytic continuation of each mode's frequency function onto the complex plane (Figures \\ref{Fig. Frequency Particle} and \\ref{Fig. Contributions to H2 - No back-reaction}). The resulting stress-energy is a calculable source term for the semi-classical Friedmann equations, and can be used to obtain a numerical solution to the back-reaction problem. We have performed this calculation for an initially closed de Sitter spacetime, demonstrating that the effects of particle production in this scenario can become strong enough to drive the cosmic evolution into a radiation-dominated phase (Figures \\ref{Fig. Contributions to H2 - Back-reaction} and \\ref{Fig. Contributions to H2 - Back-reaction 2}). Our results illustrate the reliability of our numerical implementation, and open the possibility of the systematic investigation of cosmological scenarios dominated by quantum particle production.\n\nOn a technical level, our method relies on some previous knowledge of the Stokes geometry associated with the spacetime evolution. For the case studied in this work, all Stokes lines are sufficiently separated from each other so that Berry's universal form for particle production applies without corrections. In general, however, the spacetime evolution might result in near-lying Stokes lines for which higher-order Stokes corrections are required for an accurate description of particle production. Although not included in this work, such corrections are well-documented in the literature \\cite{BerryHowls94, HowlsLongmanDaalhuis04, HowlsDaalhuis12} and could in principle be added to our numerical implementation. More fundamentally, our method is based on a well-defined semi-classical notion of particle. Mathematically, this notion is tied to the existence of a phase-integral expansion for the field mode functions. Such a representation can always be constructed as long as $\\left| \\varepsilon_{k,\\,0} \\right| \\ll 1$. Physically, this requirement typically translates to an approximate bound on the Hubble rate $H \\lesssim m$ set by the mass of the field under consideration. Nonetheless, some scenarios exist for which $\\left| \\varepsilon_{k,\\,0} \\right| \\ll 1$ is satisfied even when $H > m$.\n\nQuantum backreaction  is potentially important in models of the very early Universe. Quantum particle production is actually quite familiar in the context of inflation, as it provides the standard mechanism for the generation of perturbations in an inflating spacetime \\cite{Starobinsky79, Allen88, Sahni90, Mukhanov92, Souradeep92, Glenz09, Agullo11}.  Interestingly, it has been suggested that these same ideas could be applied to the problem of driving inflation itself \\cite{Prigogine89, Calvao92, Lima96, Abramo96, Gunzig98, Lima14, HaroPan16}. Indeed, a phase of accelerated expansion can result if particles are produced at a high enough rate. A time derivative of the usual Friedmann Equation $H^{2}(t) = \\frac{1}{3}M^{-2} \\rho(t)$ shows that an accelerating expansion ${\\ddot a} > 0$ occurs when\n\\begin{equation}\\label{Eq. Accelerated Condition}\n\t\\dot{\\rho}(t) > -\\frac{2}{\\sqrt{3}} \\, M^{-1} \\rho^{3\/2}(t) \\, .\n\\end{equation}\nSuch a scenario has the potential to sidestep some of the conceptual problems of the standard inflationary paradigm. For instance, it has been argued that standard inflation cannot generically start in a patch which is smaller than the cosmological horizon without violating either causality or the weak energy condition \\cite{VachaspatiTrodden99, BereraGordon01}. However, if inflation is initially driven by an increasing energy density due to particle production, the weak energy condition {\\it is} effectively violated. Therefore, inflation driven by such a mechanism could generically start in small patches contained within the cosmological horizon without violating causality. Inflation driven by particle production would also clarify the meaning of the inflaton effective potential by making manifest the high mass-scale physics it represents.\n\nThe same conditions which lead to quantum particle production can also result in particle annihilation. If sufficiently pronounced, this effect can drive a contracting spacetime toward a bounce phase. Indeed, it follows from the cosmological continuity equation that $\\rho(t) + P(t) < 0$ provided the particle annihilation rates are high enough to cause the field energy density to decrease as the Universe contracts. In other words, the null energy condition is effectively violated, making $\\dot{H}(t) > 0$ according to the Friedmann equations \\cite{IjjasSteinhardt18, Ijjas16}. Thus, a classical bounce can emerge provided enough energy density is sequestered by quantum particle annihilation during a phase of cosmological contraction. If realized, such a mechanism could provide a natural description for cosmological bounce scenarios which does not require new physics. Also, successful bounces require constraints on high mass-scale quantum fields, so that quantum back-reaction does not push the contracting phase into a radiation crunch, as with the example solved in this paper.\n\nAnother possibly interesting effect is the production of a relativistic condensate in the early universe. Under certain circumstances, quantum particle production can lead to large occupation numbers for some scalar field modes, representing condensate formation. This phenomenon could lead to additional interesting phenomenology \\cite{AragaoRosa80, ParkerZhang91, ParkerZhang93}. \n\nA number of technical questions remain to be answered. Fermion fields require more complex calculations than scalar fields, and may present some different physics \\cite{Landete13, Landete14}. How to handle interacting fields remains an open question, and multiple fields offer additional possibilities \\cite{Ringwald87, CooperMottola87, PazMazzitelli88, Habib96, Cooper97, MolinaParis00}. We have a long road to travel before the range of interesting early-Universe dynamical scenarios driven by quantum particle production has been fully explored. \n\n\n\\begin{acknowledgments}\nF.Z. thanks D. Boyanovsky, S. Habib, P. Anderson, and E. Mottola for helpful discussions. F.Z. acknowledges support from the Andrew Mellon Predoctoral Fellowship and the A\\&S PITT PACC Fellowship. The authors have been partly supported by the National Science Foundation under the grant AST-1312380. This work made use of many community-developed or community-maintained software packages, including (in alphabetical order): Matplotlib \\cite{Matplotlib}, NumPy \\cite{NumPy}, and SciPy \\cite{SciPy}. Bibliographic information was obtained from the NASA Astrophysical Data System.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}