diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzznvjh" "b/data_all_eng_slimpj/shuffled/split2/finalzznvjh"
new file mode 100644--- /dev/null
+++ "b/data_all_eng_slimpj/shuffled/split2/finalzznvjh"
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nHigh performance codes are becoming increasingly difficult to program,\ndespite a proliferation of successful (but incremental) efforts to\nincrease programmability and productivity for high performance\ncomputing (HPC) systems. The reasons for this range over several\nlayers, beginning with the need for large, international\ncollaborations to combine expertise from many different fields of\nscience,\nto the need to address a wide variety of systems and hardware\narchitectures to ensure efficiency and performance.\n\nAs heterogeneous and hybrid systems are becoming common in HPC\nsystems, additional levels of parallelism need to be addressed, and\nthe bar for attaining efficiency is being raised. Three out of ten,\nand 62 of the top 500 of the fastest computers in the world use\naccelerators of some kind to achieve their performance~\\cite{top500}.\nMore large heterogeneous systems are scheduled to be set up, especially\nincluding new Intel Xeon Phi and Nvidia K20x co-processors.\n\\todo{ES: Should we update this statement? Stampede at TACC (with Xeon\n  Phis) is open to the public, and Blue Waters (with Nvidia GK110\n  coprocessors) is open for testing.}\n\nIn this paper we present \\emph{Chemora\\xspace}, using an integrated approach\naddressing programmability and performance at all levels, from\nenabling large-scale collaborations, to separating physics, numerical\nanalysis, and computer science portions, to\ndisentangling kernel implementations from performance optimization\nannotations. Chemora\\xspace is based on the \\emph{Cactus}\nframework~\\cite{Goodale02a, cactusweb}, a well-known tool used in\nseveral scientific communities for developing HPC applications. Cactus\nis a component-based framework providing key abstractions to \nsignificantly simplify parallel programming for a large class of problems, in\nparticular solving systems of partial differential equations (PDEs) on\nblock-structured grids -- i.e.\\ adaptive mesh refinement (AMR) and\nmulti-block systems (see section \\ref{sec:cactus} below).\n\nChemora\\xspace enables existing Cactus-based applications to continue scaling\ntheir scientific codes and make efficient use of new hybrid systems,\nwithout requiring costly re-writes of application kernels or adopting\nnew programming paradigms. At the same time, it also provides a\nhigh-level path for newly developed applications to efficiently employ\ncutting-edge hardware architectures, without having to target a\nspecific architecture.\n\nWe wish to emphasize that the present work is merely the next step in\nthe currently fifteen year-long history of the Cactus framework. While\nfinding ways to exploit the power of accelerators is perhaps the\nlargest current challenge to increased code performance, it is really\nonly the latest advance in an ever-changing evolution of computer\narchitectures. Suport for new architectures is typically added to the\nlower-level components of frameworks (such as Cactus) by the framework\ndevelopers, allowing the application scientist to take advantage of\nthem without having to significantly rewrite code.\n\nTo create the Chemora\\xspace framework, we have built on top of a number of\nexisting modules that have not been written specifically for this\nproject, as well as creating new modules and abstractions. The main\nresearch and development effort has been the integration of these\nmodules, especially as regards accelerator interfaces, their\nadaptation for production codes as well as automatic optimizations to\nhandle complicated Numerical Relativity codes. The result is that this\nframework allows the use of accelerator hardware in a transparent and\nefficient manner, fully integrated with the existing Cactus framework,\nwhere this was not possible before.  The full contribution to the\ndescribed research work has been described in the section\n\\ref{sec:contribution}. The framework, along with introductory\ndocumentation, will be made publicly available \\cite{chemoracode}.\n\n\\subsection{Scientific Motivation}\n\\label{sec:science}\n\nPartial differential equations are ubiquitous throughout the fields of\nscience and engineering, and their numerical solution is a challenge\nat the forefront of modern computational science.  In particular, our\napplication is that of \\emph{relativistic astrophysics}. Some of the\nmost extreme physics in the universe is characterised by small regions\nof space containing a large amount of mass, and Newton's theory of\ngravity is no longer sufficient; Einstein's theory of General\nRelativity (GR) is required.  For example, black holes, neutron stars,\nand supernovae are fundamentally relativistic objects, and\nunderstanding these objects is essential to our understanding of the\nmodern universe. Their\naccurate description is only possible using GR\\@. The solution of\nEinstein's equations of GR using computational techniques is known as\n\\emph{numerical relativity} (NR\\@). See \\cite{Pfeiffer:2012pc} for a recent\nreview, and see \\cite{Loffler:2011ay} for a detailed description of an open-source\nframework for performing NR simulations.\n\nOne of the most challenging applications of NR is the inspiral and\nmerger of a pair of orbiting black holes.  GR predicts the existence\nof gravitational waves: ripples in spacetime that propagate away from\nheavy, fast-moving objects.  Although there is indirect evidence,\nthese waves have not yet been directly detected due to their low\nsignal strength. The strongest expected sources of gravitational waves are\nbinary black hole and neutron star mergers, and supernova explosions--\nprecisely those objects for which GR is required for accurate\nmodeling.  Several gravitational wave detectors \\cite{Fritschel:2003qw} are\npresently under construction and they are expected to see a signal within\nthe next few years.  The detection of gravitational waves will lead\nto an entirely new view of the universe, complementary to existing\nelectromagnetic and particle observations.  The existence and\nproperties of expected gravitational wave sources will dramatically\nextend our knowledge of astronomy and astrophysics.\n\nNR models the orbits of the black holes, the waveforms they produce,\nand their interaction with these waves\nusing the Einstein equations. Typically, these equations are split\ninto a 3+1 form, breaking the four dimensional character of the\nequations and enabling the problem to be expressed as a time evolution\nof gravitational fields in three spatial dimensions.\nThe Einstein equations in the BSSN formulation~\\cite{Nakamura:1987zz,\n  Shibata:1995we, Baumgarte:1998te}\nare a set of coupled nonlinear\npartial differential equations with 25 variables~\\cite{Alcubierre99d, Alcubierre02a},\nusually written for compactness in abstract index form.\nWhen fully expanded, they contain thousands of terms, and the right\nhand side requires about 7900\nfloating point operations per grid point to evaluate once, if using\neigth order finite differences.\n\nThe simulations are characterised by the black hole mass, $M$,\na length, $G M\/c^2$, and a time, $G M\/c^3$. Usually one \nuses units in which $G = c = 1$, allowing both time and distance to be\nmeasured by $M$. Typical simulations of the type listed above\nhave gravitational waves of\nsize $\\sim 10 M$, and the domain to be simulated is $\\sim\n100$--$1000 M$ in radius.  For this reason, Adaptive Mesh Refinement\n(AMR) or multi-block methods are required to perform long-term BBH\nsimulations.\n\nOver 30 years of research in NR culminated in a major breakthrough in\n2005~\\cite{pretorius2005evolution,Baker:2005vv,Campanelli:2005dd},\nwhen the first successful long-term stable binary black hole\nevolutions were performed.  Since then, the NR community has refined\nand optimized their codes and techniques, and now routinely runs\nbinary black hole simulations, each employing hundreds or thousands of\nCPU cores simultaneously of\nthe world's fastest supercomputers.\nPerformance of the codes is a critical issue, as the\nscientific need for long waveforms with high accuracy is compelling.\nOne of the motivations of the Chemora\\xspace project was taking the NR\ncodes into the era of computing with the use of accelerators (in particular\nGPUs) and improving their performance by an order of magnitude, thus enabling\nnew science.\n\n\\subsection{Related Work}\n\nTo achieve sustained performance on hybrid supercomputers and reduce\nprogramming cost, various programming frameworks and tools have been developed, e.g.,\nMerge~\\cite{Linderman:2008:MPM:1353536.1346318} (a library based framework\nfor heterogeneous multi-core systems), Zippy~\\cite{CGF:CGF1131} (a framework\nfor parallel execution of codes on multiple GPUs),\nBSGP~\\cite{Hou:2008:BBG:1360612.1360618} (a new programming language for\ngeneral purpose computation on the GPU), and\nCUDA-lite~\\cite{springerlink:10.1007\/978-3-540-89740-8_1} (an enhancement to\nCUDA that transforms code based on annotations).\nEfforts are also underway to improve compiler tools\nfor automatic parallelization and optimization of affine loop\nnests for GPUs~\\cite{Baskaran:2008:CFO:1375527.1375562} and for automatic\ntranslation of OpenMP parallelized codes to\nCUDA~\\cite{Lee:2009:OGC:1594835.1504194}.\nFinally, OpenACC is slated to provide OpenMP-like annotations for C and\nFortran code.\n\nStencil computations form the kernel of many scientific applications that \nuse structured grids to solve partial differential equations.\nThis numerical problem can be characterised as the {\\em structured grids} \"Berkeley\nDwarf\" \\cite{berkeleydwarfs2006}, one of a set of algorithmic patterns identified as important\nfor current and near-future computation.\nIn particular, stencil computations parallelized using hybrid architectures\n(especially multi-GPU) are\nof particular interest to many researchers who want to leverage the emerging hybrid\nsystems to speed up scientific discoveries.\nMicik~\\cite{Micik2009} proposed an optimal 3D finite difference\ndiscretization of the wave equation in a CUDA environment, and\nalso proposed a way to minimize the latency of inter-node communication\nby overlapping slow PCI-Express (interconnecting the GPU with the\nhost) data exchange with computations. This may be achieved by\ndividing the computational domain along the slowest varying dimension.\nThibault \\cite{Thibault2009} followed the idea of a domain division pattern and implemented\na 3D CFD model based on finite-difference discretization of the Navier-Stokes equations parallelized\non a single computational node with 4 GPUs.\n\nJacobsen \\cite{Jacobsen2010} extended this model by adding inter-node communication via\nMPI\\@. They followed the approach described in Micik~\\cite{Micik2009} and overlapped the communication with\ncomputations as well as GPU-host with host-host data exchange. However, they did not take\nadvantage of the full-duplex nature of the PCI-Express bus, which would have decreased the\ntime spent for communication. Their computational model also divides the domain along the slowest\nvarying dimension only, and this approach is not suitable for all numerical problems. For example, for large computational\ndomains, the size of the ghost zone becomes noticeable in comparison to the computed part\nof the domain, and the communication cost becomes larger than the computational cost, which can\nbe observed in the non-linear scaling of their model.\n\nNotable work on an example stencil application was selected as a finalist of the Gordon Bell Prize in \nSC 2011 as the first peta-scale result \\cite{Shimokawabe2011}. Shimokawabe et al.\\ demonstrated very high \nperformance of 1.017 PFlop\/s in single precision using 4,000 GPUs along with 16,000 CPU cores on TSUBAME 2.0. \nNevertheless, a set of new and more advanced optimization techniques introduced in the Chemora\\xspace framework as \nwell as its capabilities to generate highly efficient multi-GPU stencil computing codes from a high-level \nproblem description make this framework even more attractive for users of large-scale hybrid systems.\n\nPhysis \\cite{Physis} addresses the problem of dividing the domain in\nall dimensions, and is these days\nseen as one of the most efficient frameworks for stencil\ncomputations over regular multidimensional Cartesian\ngrids in distributed memory environments.\nThe framework in its current state, however, does not divide\nthe domain automatically; this has to be done manually\nat launch time.\nNevertheless, Physis achieves very good scaling by taking\nadvantage of memory transfers overlapped with computations.\nStencil computations are defined in the form of C-based functions (or \\emph{kernels})\nwith the addition of a few special macros that allow accessing values at grid points.\nThe framework also uses CUDA streams that allow for parallel execution\nof multiple kernels at the same time; e.g.\\ regular and boundary kernels\nmay be executed in parallel.\nData dependencies between stencil points are resolved statically,\nhence must be known beforehand, at compile time.\nThe authors put a special emphasis on ease of use, and\nindeed the time needed to write an application in Physis is relatively short.\nThis framework was evaluated using three benchmark programs running on \nthe TSUBAME~2.0 supercomputer, and proved\nto generate scalable code for up to 256 GPUs.\nBelow, we compare Chemora\\xspace with its dynamic compilation and \nauto-tuning methods to Physis, and show that Chemora\\xspace outperforms Physis\nin the area of automatically generated\ncode for GPU clusters.\n\n\\subsection{Contributions}\n\\label{sec:contribution}\nThis paper makes the following contributions:\n\n\\begin{itemize}\n\\setlength{\\itemsep}{-2pt}\n\\item An overview of the Chemora\\xspace framework for generating hybrid\n  CPU\\slash GPU cluster code from PDE descriptions is presented and\n  its performance is characterized.\n\n\\item A language for expressing differential equation models of\n  physical systems suitable for generating hybrid cluster simulation\n  code (based on the existing \\term{Kranc} code-generation package), was developed.\n\n\\item Model-based GPU tile\\slash thread configuration optimization\n  techniques were developed, enabling the exploration of a large\n  search space and the use of dynamic compilation (performed once on\n  the chosen configuration).\n\n\\item Automatic hybrid execution GPU\\slash CPU data staging techniques\n  were developed (the \\term{accelerator} module).\n\n\\item GPU tuning techniques were developed for large kernel codes,\n  such as register-pressure sensitive configuration.\n\n\\item The first demonstration binary black hole simulations using GPUs in full GR\n  were presented.  Since Chemora has not yet been applied to the\n  Carpet AMR driver, these are not suitable for production physics,\n  but prove that existing codes used in numerical relativity can be\n  adapted to Chemora.\n\\end{itemize}\n\n\\section{Chemora\\xspace Framework}\n\nChemora\\xspace takes a physics model described in a high level \n\\emph{Equation Description Language} (EDL) and generates highly optimized code suitable \nfor parallel execution on heterogeneous systems.\nThere are three major components in Chemora\\xspace:\nthe Cactus-Carpet computational infrastructure, \nCaKernel programming abstractions, and the Kranc\ncode generator. Chemora\\xspace is portable to many\noperating systems, and adopts widely-used parallel programming \nstandards (MPI, OpenMP and OpenCL) and models (vectorization and CUDA\\@).\nAn architectural view of the Chemora\\xspace framework is shown in\nFigure~\\ref{fig:chemora_arch}. We describe the individual components below.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{figs\/Chemora}\n\\caption{An architectural view of Chemora\\xspace. Chemora\\xspace consists of three major\ncomponents: The Cactus-Carpet computational infrastructure, CaKernel\nprogramming abstractions, and the Kranc code generator. Chemora\\xspace takes a physics\nmodel described in a high level Equation Description Language and \nproduces highly optimized code suitable for parallel execution on \nheterogeneous systems.}\n\\label{fig:chemora_arch}\n\\end{figure}\n\n\\subsection{Cactus-Carpet Computational Infrastructure}\n\\label{sec:cactus}\n\nThe Cactus computational framework is the foundation of Chemora\\xspace.\nCactus~\\cite{Goodale02a, cactusweb} is an open-source, modular,\nhighly-portable programming environment for collaborative research\nusing high-performance computing. Cactus is distributed with a generic\ncomputational toolkit providing parallelization, domain decomposition,\ncoordinates, boundary conditions, interpolators, reduction operators,\nand efficient I\/O in different data formats. More than 30\ngroups worldwide are using Cactus for their research work in cosmology,\nastrophysics, computational fluid dynamics, coastal modeling, quantum\ngravity, etc. The Cactus framework is a vital part of the Einstein\nToolkit~\\cite{Loffler:2011ay, EinsteinToolkit:web},\nan NSF-funded collaboration enabling a\nlarge part of the world-wide research in numerical relativity by\nproviding necessary core computational tools as well as a common\nplatform for exchanging physics modules. Cactus is part of the\nsoftware development effort for Blue Waters, and in particular the\nCactus team is working with NCSA to produce development interfaces and\nparadigms for large scale simulation development.\n\nOne of the features of Cactus relevant in this context is that it\nexternalizes parallelism and memory management into a module (called\na \\emph{driver}) instead of providing it itself,\nallowing application modules (called \\emph{thorns}) to function mostly\nindependently of the system architecture. Here we employ the\n\\emph{Carpet} driver~\\cite{Schnetter-etal-03b, Schnetter06a,\n  carpetweb} for MPI-based parallelism via spatial domain\ndecomposition. Carpet provides adaptive mesh refinement (AMR) and\nmulti-block capabilities\\footnote{We do not use these capabilities in\n  the examples below.}, and has been shown to scale to more than\n16,000 cores on current NERSC and XSEDE systems.\n\nIn the typical Cactus programming style for application modules, these\nmodules consist either of \\emph{global} routines (e.g.\\ reduction or\ninterpolation routines), or \\emph{local} routines (e.g.\\ finite\ndifferencing kernels). Local routines are provided in the form of\nkernels that are mapped by the driver onto the available resources.\nAt run time, a schedule is constructed, where Cactus orchestrates the\nexecution of routines as well as the necessary data movement\n(e.g.\\ between different MPI processes).\nThis execution model is both easy to understand for application\nscientists, and can lead to highly efficient simulations on large\nsystems. Below, we refine this model to include accelerators\n(e.g.\\ GPUs) with separate execution cores and memory systems.\n\n\\subsection{CaKernel Programming Abstractions}\n\\label{sec:cakernel}\nThe Chemora\\xspace programming framework uses the CaKernel\n\\cite{parco11, ppopp11, sciprog11}, a set of high level programming\nabstractions, and the corresponding implementations.\nBased on the Cactus-Carpet computational infrastructure,\nCaKernel provides two major sets of programming abstractions:\n(1) \\emph{Grid Abstractions} that represent the dynamically \ndistributed adaptive grid\nhierarchy and help to separate the application development from the\ndistributed computational domain;\n(2) \\emph{Kernel Abstractions} that enable automatic generation of numerical\nkernels from a set of highly optimized templates and help to separate the\ndevelopment, scheduling, and execution of numerical kernels.\n\n\\subsubsection{Grid Abstractions}\nThe Cactus flesh and the Cactus computational toolkit contain a collection\nof data structures and functions that \ncan be categorized into the following three grid abstractions, which commonly appear\nin high level programming frameworks for parallel block-structured\napplications~\\cite{parabrow96}:\n\\begin{itemize}\n\\setlength{\\itemsep}{-2pt}\n\\item The \\emph{Grid Hierarchy (GH)} represents the distributed adaptive GH\\@.\nThe abstraction enables application developers to create, operate and destroy\nhierarchical grid structures. The regridding and partitioning operations on a grid\nstructure are done automatically whenever necessary. In Cactus, grid operations\nare handled by a driver thorn which is a special module in Cactus.\n\\item A \\emph{Grid Function (GF)} represents a distributed data structure\ncontaining one of the variables in an application. Storage, synchronization, arithmetic,\nand reduction operations are implemented for the GF by standard thorns. The\napplication developers are responsible for providing routines for\ninitialization, boundary updates, etc.\n\\item The \\emph{Grid Geometry (GG)} represents the coordinates, bounding boxes,\nand bounding box lists of the computational domain. Operations on the GG, such\nas union, intersection, refine, and coarsen are usually implemented in a driver\nthorn as well.\n\\end{itemize}\n\n\\subsubsection{Kernel Abstractions}\nThe kernel abstractions enable automatic code generation with a set of highly optimized\ntemplates to simplify code construction. The definition of a kernel requires\nthe following three components:\n\\begin{itemize}\n\\setlength{\\itemsep}{-2pt}\n\\item A \\emph{CaKernel Descriptor} describes one or more numerical\n  kernels,\n \n  dependencies, such as grid functions and parameters\n  required by the kernel, and grid point relations with its neighbors.\n  the information provided in the descriptor is then used to generate\n  a kernel frame (macros) that performs automatic data fetching,\n  caching and synchronization with the host.\n\\item A \\emph{Numerical Kernel} uses kernel-specific auto-generated\n  macros. The function may be generated via other packages (such as\n  Kranc), and operates point-wise.\n\\item The \\emph{CaKernel Scheduler} schedules CaKernel launchers and\n  other CaKernel functions in exactly the same way as other Cactus\n  functions. Data dependencies are evaluated and an optimal strategy\n  for transferring data and performing computation is selected\n  automatically.\n\\end{itemize}\nThese kernel abstractions not only enable a simple way to write and execute\nnumerical kernels in a heterogeneous environment, but also enable lower-level\noptimizations without modifying the kernel code itself.\n\n\\subsubsection{Hardware Abstraction}\nCaKernel provides an abstraction of the hardware architecture, and\nChemora code is generated on top of this abstraction. The high level\nproblem specification in the Chemora framework may thus remain\nindependent of the architecture. The support for new architectures is\nthe responsibility of the Chemora developers, and thus it is\ntransparent to the end-user, who should not need to significantly\nmodify their code once the underlying CaKernel implementation has been\nmodified.\n\n\\subsection{Describing a Physics Model}\nProgramming languages such as C or Fortran offer a very low level of\nabstraction compared to the usual mathematical notation. Instead of\nrequiring physicists to write equations describing PDEs at this level,\nwe introduce EDL, a\ndomain-specific language for specifying systems of PDEs as well as\nrelated information (initial and boundary conditions, constraints,\nanalysis quantities, etc.) EDL allows equations to be specified independent\nof their discretization, allows abstract index notation to be used as a\ncompact way to write vectors and tensors, and does not limit\nthe options for memory layout or looping order. For Chemora\\xspace, we designed EDL\nfrom scratch instead of piggybacking it onto an existing language\nsuch as Mathematica, Haskell, or C++ so that we could choose a syntax\nthat is easily understood by domain scientists, i.e.\\ physicists and\nengineers.\n\nEDL has a very simple syntax, similar to C, but extended with a\nLaTeX-like syntax for abstract index notation for vectors and tensors.\nSample \\ref{fig:edl} shows as an example the main part of specifying the\nscalar wave equation in a fully first order form (assuming, for\nsimplicity, the propagation speed is $1$.) In addition to specifying\nthe equations themselves, EDL supports constants, parameters,\ncoordinates, auxiliary fields, and conditional expressions.\n\n\\begin{lstlisting}[escapechar=!, caption={Example showing (part of) \nthe scalar wave equation written in \\emph{EDL}, a language designed to describe PDEs. A LaTeX-like\n  syntax allows a compact notation for vectors and tensors. Additional\n  annotations (not shown here) are needed to complete the\n  description.}, label=fig:edl]\n\n!\\color{ForestGreen}{begin calculation}! !\\color{blue}{Init}!\n  u   = !\\color{cyan}{0}!\n  rho = A exp(!\\color{cyan}{-1\/2}! (r\/W)**!\\color{cyan}{2}!)\n  v_i = !\\color{cyan}{0}!\n!\\color{ForestGreen}{end calculation}!\n\n!\\color{ForestGreen}{begin calculation}! !\\color{blue}{RHS}!\n  D_t u   = rho\n  D_t rho = delta^ij D_i v_j\n  D_t v_i = D_i rho\n!\\color{ForestGreen}{end calculation}!\n\n!\\color{ForestGreen}{begin calculation}! !\\color{blue}{Energy}!\n  eps = !\\color{cyan}{1\/2}! (rho**!\\color{cyan}{2}! + delta^ij v_i v_j)\n!\\color{ForestGreen}{end calculation}!\n...\n\\end{lstlisting}\n\nIn addition to describing the system of equations, EDL makes it possible\nto specify a particular discretization by specifying sets of finite\ndifferencing stencils.\nThese\nstencil definitions remain independent of the equations themselves.\n\nThe \\emph{Kranc} code-generation package (see section\n\\ref{sec:kranc}), written in Mathematica and\ndescribed below, has been enhanced in Chemora\\xspace to accept EDL as its input\nlanguage.  Via a J\/Link interface to the Piraha PEG \\cite{brandt2010piraha} Java\nparsing library, the EDL is parsed into Mathematica expressions\nequivalent to those traditionally used as input to Kranc.  The formal\ngrammar which defines the syntax of the language is available as part\nof the Kranc distribution, should other tools need to parse EDL files.\n\nIn spite of its apparent simplicity, the high-level description in EDL\ncaptures everything that is needed to create a complete Cactus module.\nMetadata such as variable declarations, schedule items, and parameter\ndefinitions are extracted from EDL, and implementation choices such as\nmemory layout and loop traversal order are made automatically or even\ndynamically at run time (see below).\n\nKranc is written in Mathematica, and prior to Chemora\\xspace was used by\nwriting a script in the Mathematica language to set up data structures\ncontaining equations and then call Kranc Mathematica functions to\ngenerate the Cactus module.  This allowed great flexibility, but at\nthe same time required users to know the Mathematica language, which\nin several ways is idiosyncratic and is unfamiliar to many users.\nAdditionally, the use of an imperative language meant that Kranc was\nunable to reason about the input script in any useful manner (for\nexample for the purpose of reporting line numbers where errors were\nfound).  A new, simple, declarative domain-specific language was\ntherefore created which allowed a concise expression of exactly the\ninformation needed by Kranc.  Existing languages familiar to the\nmajority of scientists (C, Fortran, Perl, Python) introduce a wide\nvariety of features and semantics unnecessary for our application, and\nnone of these are suitable for expressing equations in a convenient\nmanner.  The block structure of EDL was inspired by Fortran, the\nexpression syntax by C, and the index notation for tensors by LaTeX.\nWe feel that the language is simple enough that it can be learned very\nquickly by reference to examples alone, and that there is not a steep\nlearning curve.\n\nBy providing a high-level abstraction for an application scientist,\nthe use of EDL substantially reduce the time-to-solution, which includes:\nlearning the software syntax, development time from a given system of\nequations to machine code, its parallelization on a heterogeneous\narchitecture, and finally its deployment on production clusters.  It\nalso eliminates many potential sources of errors introduced by low\nlevel language properties, and thus reduces testing time. For further\ninformation about the total time-to-solution, see\n\\cite{hochstein2005parallel}.  \n\n\\subsection{Automated Code Generation with Kranc}\n\\label{sec:kranc}\n\nTranslating equations from a high-level mathematical notation into C\nor Fortran and discretizing them manually is a tedious, error-prone\ntask. While it is straightforward to do for simple algorithms, this\nbecomes prohibitively expensive for complex systems.\nWe identify two levels\nof abstraction.  The first is between the continuum equations and the\napproximate numerical algorithm (discretization), and the second is\nbetween the numerical algorithm and the computational implementation.\n\nWe employ \\emph{Kranc}~\\cite{Husa:2004ip, Lechner:2004cs, Kranc:web}\nas a code-generation package which implements these abstractions. The\nuser of Kranc provides a \\emph{Kranc script} containing a section\ndescribing the partial differential equations to solve, and a section\ndescribing the numerical algorithm to use. Kranc translates this\nhigh-level description into a complete Cactus module, including C++ code\nimplementing the equations using the specified numerical method, as\nwell as code and metadata for integrating this into the Cactus\nframework.\n\nBy separating mathematical, numerical, and computational aspects,\nKranc allows users to focus on each of these aspects separately\naccording to their specialization. Although users can write Kranc\nscripts directly in Mathematica, making use of the EDL \nshields them from\nthe (sometimes arcane) Mathematica syntax (because they are required to follow a\nstrict pattern for specifying PDEs) and provides them with much more\ninformative (high-level) error messages.  Either the traditional Mathematica language, or the new EDL language, can be used\nwith Chemora for GPU code generation.\n\nKranc is able to:\n\\begin{itemize}\n\\setlength{\\itemsep}{-2pt}\n\\item accept input with equations in abstract index notation;\n\\item generate customized finite differencing operators;\n\\item generate codes compatible with advanced Cactus features such as\n  adaptive mesh refinement or multi-block systems;\n\\item check the consistency with non-Kranc generated parts of the\n  user's simulation;\n\\item apply coordinate transformations, in particular of derivative\n  operators, suitable for multi-block systems\n  (e.g.~\\cite{Pollney:2009yz});\n\\item use symbolic algebra based on the high-level description of the\n  physics system to perform optimizations that are inaccessible to the\n  compiler of a low-level language;\n\\item implement transparent OpenMP parallelization;\n\\item explicitly vectorize loops for SIMD architectures (using\n  compiler-specific syntaxes);\n\\item generate OpenCL code (even independent of the CaKernel framework\n  described below);\n\\item apply various transformations and optimizations (e.g.\\ loop\n  blocking, loop fission, multi-threading, loop unrolling) as\n  necessary for the target architecture.\n\\end{itemize}\n\n\\subsubsection{Optimization}\n\nIt is important to note that Kranc does not simply generate the source code for\na specific architecture that\ncorresponds $1:1$ to its input. Kranc has many of the features of a traditional compiler, including\na front-end, optimizer, and code generator, but the code generated is C++\/CaKernel\/CUDA rather than\nmachine code.\n\nThe high-level optimizations currently implemented act on discretized\nsystems of equations, and include the following:\n\\begin{itemize}\n\\setlength{\\itemsep}{-2pt}\n\\item Removing unused variables and expressions;\n\\item Transforming expressions to a normal form according to\n  mathematical equivalences and performing \\emph{constant folding};\n \n \n\\item\n \n   \n    Introducing temporaries to perform \\emph{common subexpression elimination};\n\\item Splitting calculations into several independent calculations\n \n \n \n  to reduce the instruction cache footprint and data cache pressure\n  \\emph{(loop fission)};\n\\item Splitting calculations into two, the first evaluating all\n  derivative operators (using stencils) storing the result into\n  arrays, the second evaluating the actual RHS terms but not using any\n  stencils. This allows different loop optimizations to be applied to\n  each calculation, but requires more memory bandwidth \\emph{(loop\n    fission)}.\n\\end{itemize}\n\nNote in the above that a\n\\emph{calculation} is applied to all grid points, and thus either\nloops over or uses multiple threads to traverse all grid points.\nAlso note that both the high-level and the low-level optimizations could in principle\nalso be performed by an optimizing compiler. However, none of the\ncurrently available compilers for HPC systems are able to do so,\nexcept for very simple kernels. We surmise that the reason for this is\nthat it is very difficult for a compiler to abstract out sufficient\nhigh-level information from code written in low-level languages\nto prove that these\ntransformations are allowed by the language standard. A programmer is\nforced to make many (ad-hoc) decisions when implementing a system of\nequations in a low-level language such as C or C++, and the compiler\nis then unable to revert these decisions and fails to optimize the\ncode.\n\nIt is surprising to see that these optimizations -- which are in\nprinciple standard transformations among compiler builders -- are (1)\nable to significantly improve performance, are (2) nevertheless not\napplied by current optimizing compilers, and are yet (3) so easily\nimplemented in Mathematica's language, often requiring less than a\nhundred lines of code.\n\nKranc is a developed and mature package. Since its conception in 2002,\nit has been continually developed to adapt to changing computational\nparadigms.\nKranc is not just a theoretical tool. In the Einstein\nToolkit~\\cite{Loffler:2011ay},\nKranc is used to generate a highly efficient\nopen-source implementation of the Einstein equations as well as\nseveral analysis modules.\nAll of the above features are used heavily by users of the Toolkit,\nand hence have been well-tested on many production architectures,\nincluding most systems at NERSC or in XSEDE\\@.\n\n\\subsubsection{Debugging the Numerical Code}\nIt is also important to note that Chemora significantly reduces the time\nrequired to debug the application.  The recommended approach for development\nusing Chemora is that the user's Kranc script is considered the canonical\nsource, and only this should be modified during development. The generated code\nshould not be modified, as it will be completely regenerated each time Kranc is\nrun, so any hand-modifications of the generated code will be lost. Unlike when\nwriting a C++ program, every successfully-compiled Kranc script should lead to\ncorrect computational (though not necessarily physical) code. Hence the errors\nare limited to the application domain, for example an incorrect equation is solved.\nSimilarly, use of a source-code level debugger is\nnot typical when working with Kranc, as the ``debugging'' happens at the level of\nthe scientific results (e.g. convergence tests and visualisation) rather than\nat the level of programmatic bugs in the generated code. \nAs such, Kranc is treated as a black box by the application scientist,\nmuch as a compiler would be.\n\n\\subsubsection{Code Generation for CaKernel}\n\nIn order to use Kranc as a component of Chemora\\xspace, the code-generation\nbackend was modified, and\nCaKernel (see section \\ref{sec:cakernel}) was added as an output target. This\nchange is essentially invisible to the application developer; there is merely\nan additional option to generate CaKernel code rather than C++ or OpenCL code.\nEach calculation is then annotated with whether\nit runs on the host (CPU) or the device (GPU\\@).  Kranc\nalso creates all metadata required by CaKernel.\nAdditionally, the new EDL language frontend was added to Kranc.\n\n\\subsubsection{Hybrid Codes}\n\nSince GPU accelerators have to be governed by CPU(s), it is natural to attempt\nto exploit them by employing \\emph{hybrid codes}. In this case, Kranc,\ngenerates both CPU and CaKernel codes from the same script. At\nrun time, each MPI process checks whether to attach itself to a GPU\nand perform its calculations there, or whether to use the CPU for\ncalculations. \n\nThis mechanism works in principle; however, as the Cactus driver\ncurrently assigns the same amount of work to each\nMPI process (uniform load balancing), the large performance disparity between\nCPU and GPU has led to only minimal performance gains so far. We expect this issue\nto be resolved soon.\n\n\\subsection{CaKernel GPU Code Optimization}\n\n\\bitbucket{\nChemora\\xspace optimizes at multiple levels of abstraction, starting with the\nKranc scripts, through domain decomposition and data staging,\nGPU code generation, down to execution configuration and cache\nsettings. Optimizations are performed at the appropriate level, but\ntake advantage of information provided by higher levels.}\n\n\\bitbucket{\nOne goal of Chemora\\xspace is to generate efficient code starting from a\nhigh-level description and without requiring the user to tune for\nefficient execution. CaKernel achieves this by using\nKranc-provided and run-time information to set such important\nparameters as tile shape and GPU cache settings.}\n\nThe CaKernel code generated by Kranc consists of\n\\term{numerical kernels}, routines that operate on a single grid\npoint. The CaKernel parts of Chemora\\xspace use Kranc-provided and\nrun time information to generate efficient GPU executables\nfrom the numerical kernels, without requiring the user to set tuning\nparameters. At build time, numerical kernels are wrapped with\n\\term{kernel frames}, code that implements data staging and iteration,\nproducing a source code package that is compressed and compiled into\nthe Cactus executable. At run time, CaKernel makes use of\ninformation about the kernels provided by Kranc as well as user\nparameters and information on the problem size to choose tiling, etc.\nWith this information, the code package is extracted,\nedited, compiled, loaded to the GPU, and run. This dynamic process\nresults in lightweight GPU code that makes efficient use of GPU\nresources, including caches.\nCaKernel uses several techniques to generate efficient GPU code which\nwe shall elaborate in the following subsections.\n\n\\bitbucket{\n\\term{Numerical kernels} are routines that update a single grid point,\nthey reference their own and neighboring grid points through\n\\term{indexing functions} which are restricted to purely relative\naccesses. To achieve maximum portability, numerical kernels should be\nwritten in a compatible subset of CUDA C, OpenCL, and C++, however\nthis is not enforced and so a kernel intended for CUDA execution can\nuse CUDA-specific features. The routines have Cactus parameters and\nvariables describing grid point location defined in their\nnamespaces. The numerical kernel will ultimately run as a thread on an\naccelerator. Assignment of grid points to threads is a hierarchical\nprocess. At the top, it is the responsibility of a Cactus thorn to\nassign a \\term{local section} of the grid to an accelerator device\n(actually to an MPI processes associated with the accelerator). It is\nthe important responsibility of the kernel frame to assign local grid\npoints to threads, such an assignment will be referred to as a\n\\term{tile selection}.\n\nData staging and iteration over the local grid are performed by the\n\\term{kernel frame} code, which wraps the numerical kernel. The\nparticular type of kernel frame and its options are specified in the\nCaKernel descriptor. There are separate kernel frame types for the\nboundary region and interior points, and for static and dynamic\ncompilation. For the non-boundary types, the descriptor indicates the\nextent of the stencil and tile size and caching hints. The descriptor\nalso identifies the grid functions and other data needed by the\nrespective kernel.}\n\n\\bitbucket{\n\\subsection{Optimization Challenges}\n\nChemora\\xspace was designed to generate efficient hybrid system code, without\nthe need for user tuning, from problems described in terms of\ndifferential equations, in particular those of the complexity of the\nEinstein equations. These are characterized by evolution expressions\nhaving thousands of terms, sometimes rendering standard performance\ntuning guidelines useless. Several innovations needed to achieve\nperformance are described below. They include dynamic tile size\nselection, lightweight kernel generation via dynamic compilation, the\nuse of integrated GPU performance monitoring, and source-level code\ntransformations.}\n\n\\bitbucket{\n\\subsection{Kernel Frame Types}\n\nCaKernel provides manual and automatic means of tile selection; manual\nselection is described briefly below, automatic tile selection, an\nimportant factor in achieving high performance, is described in a\nfollowing section.\n\n\\def\\tile#1,#2,#3;{\\left<#1,#2,#3\\right>}\n\nFor some of the kernel frames tile selection is specified manually by\na three-component \\param{tile} parameter. Tile $\\tile x,y,z;$\nindicates that the thread block should consist of $xy$ threads and\nshould operate on a $x\\times y\\times z$ section of the grid, where\n$x$, $y$, and $z$ are integers. Each thread operates on $z$ grid\npoints. The kernel frame code on the host will launch a kernel\nconsisting of enough such blocks to cover the local section of the\ngrid. The kernel frame code on the device, which wraps the numerical\nkernel, iterates over $z$ and computes the global indices of the grid\nfunction corresponding to the thread at each iteration.\n\nThe CaKernel descriptor is also used to specify which of the grid\nfunctions should be staged in shared memory. These choices do not\naffect the code in the numerical kernel, in particular the same\nindexing function is used whether or not the variable is in shared\nmemory. The kernel frame will load the selected grid functions into\nshared memory and update them as threads iterate. Some kernel frame\ntypes use registers rather than shared memory when stencil patterns\nallow.\n}\n\n\\subsubsection{Stencils and Dynamic Tile Selection}\n\n\\bitbucket{\nClassic CPU loop tiling involves choosing an iteration strategy to maximize\ncache reuse, see for example \\cite{rivera00}. Cache reuse is important\nfor GPU tiling too, however because of latency hiding with\nmultithreading, a primary goal can be minimization of the total data\nrequest size. Many tiling strategies for stencil computations on GPUs\nhave been reported~\\cite{renganarayana07,datta08,meng09,unat11}, the\ncommon goal being to make best use of the limited amount of high-speed\nmemory by taking advantage of the repeated access to data elements.}\n\nCPU and GPU tiling has been extensively studied, though often limited\nto specific stencils,\n\\cite{renganarayana07,datta08,meng09,unat11}. The goal for CaKernel\nwas to develop an automatic tile selection scheme that would work well\nnot just for a few specific stencils, but any stencil pattern the user\nrequested. The tile selection is based not just on the stencil shape\nbut also on the number of grid variables and on the shape of the local\ngrid. The resulting tile makes best use of the cache and potentially\nregisters for minimizing data access. The discussion below provides\nhighlights of the scheme; details will be more fully reported\nelsewhere.\n\nThe following discussion uses CUDA terminology, see \\cite{cuda40,cudatune40}\nfor background. The term \\term{tile} will be used here to mean the portion of\nthe grid assigned to a CUDA block. In GPUs, higher \\term{warp} occupancy means\nbetter latency hiding introduced by common memory access. That can\nbe achieved with multiple blocks, but to maximize L1 cache reuse\nCaKernel will favor a single large block, the maximum block size\ndetermined by a trial compilation of a numerical kernel. Within that\nblock size limit a set of candidate tile shapes are generated using\nsimple heuristics, for example, by dividing the $x$ dimension of the\nlocal grid evenly, (by 1, 2, $3,\\ldots$) and then for each tile $x$\nlength find all pairs of $t_y$ and $t_z$ lengths that fit within the\nblock limit, where $t_x$, $t_y$, and $t_z$ are the tile shape in units\nof grid points.\n\nGiven a candidate tile shape, the number of cache lines requested\nduring the execution of the kernel is computed. Such a \\term{request\n  size} is computed under the \\term{ordering assumption} that memory\naccesses are grouped by grid function and dimension (for stencil\naccesses). As an illustration, if the assumption holds a possible\naccess pattern for grid functions $g$ and $h$ is $g_{0,1,0}$,\n$g_{0,2,0}$, $g_{1,0,0}$, $h_{0,0,0}$, while the pattern $g_{0,1,0}$,\n$h_{0,0,0}$, $g_{1,0,0}$, $g_{0,2,0}$ violates the assumption because\n$h$ is between $g$'s accesses and for $g$ a dimension-$x$ stencil\naccess interrupts dimension-$y$ accesses.\n\nRequest sizes are computed under different cache line \\term{survival\n  assumptions}, and the one or two that most closely match the cache\nare averaged. One survival assumption is that all lines survive (no\nline is evicted) during an iteration in which case the request size is\nthe number of distinct lines the kernel will touch, after accounting\nfor many special cases such as alignment. Another survival assumption\nis that data accessed using stencils along one dimension (say, $x$)\nwill not survive until another dimension access (say, $y$)\n  (e.g., common lines might be evicted). The particular assumption to\nuse is based on the size of the tile and cache.\n\nSkipping details, let $r$ denote the overall request size. An\n\\term{estimated cost} is computed by first normalizing $r$ to the number of\ngrid points, $r\/It_xt_yt_z$, where $I$ is the number of iterations\nperformed by threads in the tile. To account for the lower execution\nefficiency with smaller tiles, a factor determined empirically as\n$1\/(1+256\/t_xt_yt_z)$ is used. The complete expression for the\nestimated cost\nis $\\sigma=(r\/It_xt_yt_z)\/(1+256\/t_xt_yt_z)$.\nThe tile with the lowest estimated cost is selected.\n\nTiles chosen using this method are often much longer in the $x$\ndirection than other dimensions, because the request size includes the\neffect of partially used cache lines. If a stencil extends in all three\ndimensions and there are many grid functions, the tile chosen will be\n``blocky''. If there are fewer grid functions, the tile will be\nplate-shaped, since the request size accounts for cache lines that survive\niterations in the axis orthogonal to the plate. The tile optimization\nis performed for the tile shape, but not for the number of iterations\nwhich so far is chosen empirically.\n\n\\subsubsection{Lightweight Kernel Generation}\n\nA number of techniques are employed to minimize the size of the GPU\nkernels. Dynamic compilation using program parameters and tile shape,\nseen by the compiler as constants, was very effective. Another\nparticularly useful optimization given the large size of the numerical\nkernels is \\term{fixed-offset loads}, in which a single base address\nis used for all grid functions. Normally, the compiler reserves two\n32-bit registers for the base address of each grid function, and uses\ntwo additional registers when performing index arithmetic since the\noverhead for indexing is significant. Fortunately, the Fermi memory\ninstructions have a particularly large offset, at least 26 bits based\non an inspection of Fermi machine code (which is still not well\ndocumented). (An offset is a constant stored in a memory instruction,\nit is added to a base address to compute the memory access address.)\nWith such generous offsets, it is possible to treat all grid\nfunctions (of the same data type) as belonging to one large array.\n\n\\subsubsection{Fat Kernel Detection}\n\nSome numerical kernels are extremely large, and perform very poorly\nusing standard techniques, primarily due to very frequent register\nspill\/reload accesses. \nCaKernel identifies and provides\nspecial treatment for such kernels. The kernels can be automatically\nidentified using CaKernel's integrated performance monitoring code by\nexamining the number of local cache misses. (Currently, they are\nautomatically identified by examining factors such as the number\nof grid functions.)\nSuch fat kernels are handled using two techniques: they are launched\nin small blocks of 128 threads, and source-level code restructuring\ntechniques are applied. Launching in small blocks relieves some\npressure on the L1 cache. (A dummy shared memory request prevents\nother blocks from sharing the multiprocessor.) The source code\nrestructuring rearranges source lines to minimize the number of live\nvariables; it also assigns certain variables to shared memory.\n\n\\subsubsection{Integrated Performance Monitoring}\n\nCaKernel provides performance monitoring using GPU event counters,\nread using the NVIDIA Cupti API\\@. If this option is selected, a\nreport on each kernel is printed at the end of the run. The report\nshows the standard tuning information, such as warp occupancy and\nexecution time, and also cache performance data. To provide some insight\nfor how well the code is performing, the percentage of potential\ninstruction execution and memory bandwidth used by the kernel is\noutput. For example, a 90\\% instruction execution potential would\nindicate that the kernel is close to being instruction bound. We plan\nto use these data for automatic tuning, e.g.\\ to\nbetter identify fat kernels.\n\n\\subsubsection{Effectiveness of Low-Level Optimizations}\n\nMost of the optimizations are highly effective, including dynamic\ncompilation and fixed-offset loads.\nThere are two areas where some\npotential has been left unexploited: tile shape, and the handling of\nfat kernels. \n\nAutomatic tile size selection greatly improves performance over\nmanually chosen tile sizes, however kernels are still running at just\n20\\% of execution utilization while exceeding 50\\% of available memory\nbandwidth, suffering L1 cache miss ratios well above what was\nexpected. The primary weakness in tile selection is assuming an\nordering of memory accesses that does not match what the compiler\nactually generates. (The compiler used was NVIDIA \\code{ptxas} release\n4.1 V0.2.1221.) For example, for a kernel with a $5\\times5\\times5$\nstencil and a $102\\times3\\times3$ tile, the compiler interleaves $n$\naccesses along the $y$ and $z$ axes. The cache can hold all grid\npoints along one axis (273 cache lines would be needed in this\nexample) but not along two (483 cache lines). Several solutions have\nbeen identified, including modifying the model to match compiler\nbehavior, waiting for a better compiler, restructuring the code to\nobtain a\nbetter layout, or rescheduling the loads at the object-file level.\n\nOne of the kernels performing the last step in the time evolution\nhas over 800 floating point\ninstructions in straight-line code. This executes at only 14\\%\ninstruction utilization, suffering primarily from L1 cache misses on\nregister spill\/reload accesses. We address this via fixed offsets and\nother dynamic compilation techniques that reduce register pressure. A\ncombination of source-level scheduling and shared memory use yielded\nfrom 5\\% to 10\\% better performance, and there seems to be a large\npotential for further improvement.\n\n\\subsection{Accelerator Framework}\n\\label{sec:accelerator}\n\nIn large, complex applications based on component frameworks such as\nCactus, GPUs and other accelerators are only useful to those\ncomponents which perform highly parallel arithmetic computations. As\nsuch, it is neither necessary nor useful to port the entire framework\nto run on GPUs -- in fact, much of the code in Cactus-based\napplications is not numerical, but provides support in the form of\norganizing the numerical data.\n\nOne approach to porting a component to run on a GPU is to identify the\nentry and exit points of that component, copy all required data to the\nGPU beforehand, and copy it back after the GPU computation.\nUnfortunately, such data transfer is prohibitively slow, and \nthe performance of this approach is not acceptable.\n\nInstead, we track which data (and which parts of the data) is read and\nwritten by a particular routine, and where this routine executes (host\nor GPU). Data is copied only when necessary, and then only those\nportions that are needed. Note that data is not only accessed for computations,\nbut also by inter-process synchronization and I\/O.\n\nThe metadata available for each Cactus component (or thorn) already contains\nsufficient information in its schedule description for such tracking,\nand during Chemora\\xspace we refined the respective declarations\nto further increase performance. This metadata needs to be provided\nmanually for hand-written thorns, but can be deduced automatically\ne.g.\\ by Kranc in auto-generated thorns.\n\nIn keeping with the Cactus spirit, it is a Cactus component (thorn\n\\term{Accelerator}) that tracks which parts of what grid functions are\nvalid where, and which triggers the necessary host--device copy\noperations that are provided by other, architecture-specific thorns.\n\n\\section{Case studies}\n\n\\subsection{Computing Resources}\nWe tested our framework on different computational systems.\nUnfortunately, clusters available to us at the time this paper was written were\ninsufficient for the purpose of challenging scaling tests.\n\\todo{Steve: Maybe we can we add Tienhe-1 and\/or Super Mike?}\n\n\\subsubsection{Cane}\n\\label{sec:canedesc}\n\n\\emph{Cane} is a heterogeneous cluster located at the Pozna\\'{n} Supercomputing\nand Networking Center.  Although it consists of 334 nodes, at the time we\nperformed the tests only 40 of them were available as\nthe cluster was still being set up.\nEach node is equipped with two AMD Opteron\u2122 6234 2.7GHz processors (with two\nNUMA nodes each; 12 cores per CPU), 64GB of main memory,\nand one NVIDIA M2050 GPU with 3GB of RAM\\@. The computational nodes\nare interconnected by InfiniBand QDR network with the fat-tree topology\n(32Gbit\/s bandwidth). CUDA 4.1 and gcc 4.4.5 were used for GPU and CPU code\ncompilation, respectively.\n\n\\subsubsection{Datura}\n\\label{sec:daturadesc}\n\n\\emph{Datura} is an CPU-only cluster at the Albert-Einstein-Institute in\nPotsdam, Germany. Datura has 200 nodes, each consisting of two Intel\nWestmere 2.666GHz processors with 6 cores and 24GB of memory.\nThe nodes are connected via QDR InfiniBand (40Gbit\/s bandwidth).\nWe used the Intel compilers version 11.1.0.72.\n\n\\subsection{CFD with Chemora\\xspace and Physis}\n\nWe employed a simple CFD (Computational Fluid Dynamics) benchmark\napplication to compare the performance of Chemora\\xspace and Physis.\nThis code solves the Navier-Stokes equations;\nfor details about the problem and\nits discretization see~\\cite{VOF_Hirt79,NASA_VOF2D_Torrey}, and for its\nimplementation in Cactus and CaKernel see~\\cite{parco11,sciprog11,ppopp11}.\nThe setup consists of three stencil kernels:\none that explicitly updates velocity values, one that iteratively solves \nthe conservation of mass (updating velocity and pressure), and one that\nupdates the boundary conditions.\nFor simplicity, we ran 4 iterations of the mass conservation kernel,\nand applied the boundary condition after each iteration. Although the CFD code\nwas written directly in CaKernel native language and its performance was\nalready reported along with our previous\nwork~\\cite{parco11,sciprog11,ppopp11}, we used\nCaKernel's new optimization facilities in this work.\nThese allowed us to obtain improved\nperformance compared to our previous results as well as compared to\nsimilar, publicly\navailable frameworks (e.g.\\ Physis).\n\nTo obtain statistically stable performance results, as many as 1000 iterations\nwere executed in each run. The CFD benchmark uses single-precision\nfloating-point data, which provides sufficient accuracy for this test case.\nBoth frameworks use the GPUs only for computation, and use CPUs only for\ndata transfer and management.\n\nFigure~\\ref{fig:sc12bench_cfd} compares the scalability of the \nframeworks in this CFD benchmark. The problem size of the weak scaling test\nfor each GPU was fixed at $256^3$, and \nthe performance was evaluated using 1 to 36 GPUs with two-dimensional\ndomain decompositions\nalong the $y$ and $z$ directions. We present results for\nthe best domain decompositions for each framework. The performance of both implementations increases\nsignificantly with increasing number of the GPU nodes. Numerous optimizations in Chemora\\xspace such as dynamic \ncompilation and auto-tuning allowed us to find the best GPU block size for\nthe domain size, and execute on the correct number of warps to limit the number of L1 cache \nmisses. As a result, for a single GPU, Chemora\\xspace obtained 90.5\nGFlop\/s, whereas Physis only obtained 43 GFlop\/s.\nThis gap may be also due to the fact that Physis does not make any use of\nshared memory on the GPUs.\n\nFigure~\\ref{fig:sc12bench_cfd} also compares\nthe performance of the two frameworks in a strong scaling test.\nThe problem size for this test was fixed at $656^3$. Both implementations\nscale up very well;\nChemora\\xspace achieved 270 GFlop\/s and 1055 GFlop\/s for 4 and 36 GPUs, respectively, \nwhereas Physis achieved 170 GFlop\/s and 965 GFlop\/s in the same configurations.\nThe parallel efficiency (when increasing the number of GPUs from \n4 to 36) is 43\\% and 63\\% for Chemora\\xspace and Physis, respectively.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.7\\textwidth]{benchmark\/physis\/results\/both-cane-cfd}\n\\caption{Weak and strong scaling test comparing Chemora\\xspace and Physis\n  running on multiple nodes for the same CFD application.\n  Smaller numbers are better, and ideal scaling correspoonds to a\n  horizontal line.\n  Chemora\\xspace\n  achieves a higher per-GPU performance, whereas Physis shows a higher\n  strong scalability. Details in the main text.}\n\\label{fig:sc12bench_cfd}\n\\end{figure*}\n\n\\subsection{Binary Black Hole Simulations with Chemora\\xspace}\n\nWe demonstrate the integration of Chemora\\xspace technologies into our\nproduction-level codes by performing a  Numerical Relativity (NR)\nsimulation. This simulation of a binary black\nhole (BBH) merger event shows that our GPU-accelerated main evolution\ncode can be seamlessly integrated into the pre-existing CPU framework, and that\nit is not necessary to port the entire framework to the GPU\\@.  It also\ndemonstrates the use of the data management aspect of Chemora\\xspace, showing\nhow data is copied between the host and the device on demand.\nAnalysis modules\nrunning on the CPU can make use of data generated on the GPU\nwithout significant modification.\n\nOur production simulations differ from this demonstration only in\ntheir use of adaptive mesh refinement (AMR), which allows a much larger\ncomputational domain for a given computational cost.  This allows the\nsimulation of black hole binaries with larger separations, many more\norbits before merger, and hence longer waveforms when AMR is used.\n\nThe initial condition consists of two black holes on a quasi-circular\norbit about their common center of mass (``QC-0'' configuration). This\nis a benchmark configuration; in a production simulation, the black\nholes would have a much larger separation.\nThis configuration performs approximately one orbit before the energy\nloss due to gravitational wave emission cause the black holes to\nplunge together and form a single, highly-spinning black hole.\n\nGravitational waves are emitted from the orbiting and merging system.\nThese are evaluated on a sphere and decomposed into spherical\nharmonics.\nIt is this waveform which is used in gravitational wave detection.\n\nWe use a 3D Cartesian numerical grid $x^i \\in [-6.75, 6.75]^3$ with\n$270^3$ evolved grid points. To ensure a balanced domain decomposition\nwe run on 27 processes, corresponding to $90^3$ evolved points per\nprocess. This is the largest grid that fits in the 3 GB of GPU memory\non Cane, given the large number of grid variables required.\nAll calculations are performed in double precision.\nWe evolve the system using the \\code{McLachlan} code (see section\n\\ref{sec:science} above), using 8th order finite differencing and a\n3rd order Runge-Kutta time integrator.\n\nAny production Cactus simulation makes use of a large number of\ncoupled thorns; e.g.\\ this simulation contains 42 thorns. Most of\nthese do not need to be aware of the GPU, CaKernel, or the Accelerator\ninfrastructure. In our case, only \\code{McLachlan} and the\n\\code{WeylScal4} gravitational wave extraction thorns were running on a\nGPU\\@. Additional thorns, e.g.\\ tracking the location or shape of the\nblack holes, were run on the CPU\\@.\n\nWe use 27 nodes of the Cane cluster (see section~\\ref{sec:canedesc})\nwith one GPU per node. We do not run any CPU-only processes.\n\nFig.~\\ref{fig:bbh} shows the numerical simulation domain.  On the\n$x-y$ plane we project the $\\Psi_4$ variable which represents\ngravitational waves.  The black hole trajectories are shown as black\ncurves near the center of the grid; they end when the black holes\nmerge into a single black hole located at the center.\nThe sphere on which\nthe multipolar decomposition of the gravitational waves is performed\nis also shown.  In the insets, we show (a) the time evolution of the\n(dominant) $\\ell = 2, m = 2$ mode\nof the gravitational radiation computed on the sphere at $r = 4 M$,\nand (b) the (highly distorted) shape of the common apparent horizon\nformed when\nthe two individual black holes merge.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.50\\textwidth]{figs\/bbhsim}\n\\caption{Visualization of a binary black hole system}\n\\label{fig:bbh}\n\\end{figure}\n\nTable~\\ref{tbl:bbhtimers} shows a break-down of the total run time of\nthe BBH simulation. The routines labeled in bold face run on the\nGPU\\@. The times measured are averaged across all processes. The\n\\emph{Wait} timer measures the time processes wait on each other\nbefore an interprocessor synchronization. This encapsulates the\nvariance across processes for the non-communicating routines.\n\n\\begin{table}\n\\centering\n\\begin{tabular}{lr}\n        & \\hspace{-0.5cm}Percentage of \\\\\n  Timer & \\hspace{-0.5cm}total evolution time \\\\\n\\hline\n \\text{Interprocess synchronization} & 39\\%\\\\\n \\textbf{RHS advection} & 13\\%\\\\\n \\textbf{RHS evaluations} & 12\\%\\\\\n \\text{Wait} & 11\\%\\\\\n \\textbf{RHS derivatives} & 6\\%\\\\\n \\textbf{Compute Psi4} & 5\\%\\\\\n \\text{Multipolar decomposition} & 3\\%\\\\\n \\text{File output} & 3\\%\\\\\n \\text{BH tracking} & 3\\%\\\\\n \\text{Time integrator data copy} & 2\\%\\\\\n \\text{Horizon search} & 2\\%\\\\\n \\textbf{Boundary condition} & 1\\%\\\\\n \\text{BH tracking (data copy)} & 1\\%\\\\\n\\end{tabular}\n\\caption{Timer breakdown for the binary black hole\n  simulation.  Routines in bold face (48\\%) are executed on the GPU\\@.}\n\\label{tbl:bbhtimers}\n\\end{table}\n\nWe see that the interprocess synchronization is a significant portion\n(38\\%) of the total run time on this cluster. One reason for this is\nthat the large number of ghost zones (5) needed for partially-upwinded 8th order stencils\nrequire transmitting a large amount of data. This could likely be\nimproved by using a cluster with more than one GPU or more GPU memory\nper node, as this would reduce the relative cost of inter-process\ncommunication relative to computation.\n\n\\subsection{McLachlan Benchmark}\n\nWe used part of the binary black hole simulation as a weak-scaling\nperformance benchmark. We chose a local problem size that fitted into\nthe GPU memory of Cane (see section~\\ref{sec:canedesc}), corresponding\nto $100^3$ evolved points plus boundary and ghost zones. We ran\nthe benchmark on Cane (on GPUs) and Datura (on CPUs; see\nSec.~\\ref{sec:daturadesc}), using between 1 and 48 nodes.\nFigure~\\ref{fig:s12bench_ml} shows results comparing several\nconfigurations, demonstrating good parallel scalability for these core\ncounts. One of Cane's GPUs achieved about twice the performance\nof one of its CPUs, counting each NUMA node as a single\nCPU\\@.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.7\\textwidth]{figs\/sc12bench_ml_weak_cane}\n\\caption{Weak-scaling test for McLachlan code performed on the Cane and\n  Datura clusters. ($n$)p(($m$)t) stands for $n$ processes per node\n  using $m$ threads each. (no) x-split stands for (not) dividing\n  domain along the $x$ axis. Smaller numbers are better, and ideal\n  weak scaling corresponds to a horizontal line. The benchmark scales\n  well on these platforms.}\n\\label{fig:s12bench_ml}\n\\end{figure*}\n\nAs a measurement unit we use time per grid point update per GPU (or\nCPU\\@). The best performance was achieved for a single GPU: 25\nGFlop\/s, which is 5\\% of the M2050 GPU's peak performance of 515\nGFlop\/s. On 40 nodes, we observed 50\\% scaling efficiency due to\nsynchronization overhead, and achieved a total performance of 500\nGFlop\/s.\n\nCPU performance tests were performed on both Cane and Datura. The\ntotal performance of the parallel OpenMP code, properly vectorized, was\nsimilar to the performance of a single GPU, with similar scaling factor.\n\nWe note that our floating point operation counts consider only those\noperations strictly needed in a sequential physics code, and e.g.~do\nnot include index calculations or redundant computations introduced by\nour parallelization. \\todo{Steve: I'm not quite sure what this\nparagraph is trying to say, it sounds like we're saying our code\nhas inefficiencies and we artificially removed their effects in\nour results.}\n\n\\section{Conclusion}\n\nWe have presented the Chemora\\xspace project, a component-based approach to \nmaking efficient use of current and future accelerator\narchitectures for high-performance scientific codes. \nAlthough the examples we present run on the GPU and use CUDA, \nour work is general and will be applied e.g.\\ to OpenCL and other approaches in\nfuture work. Using Chemora\\xspace, a scientist can \ndescribe a problem in terms of a system of PDEs in our Equation\nDescription Language.\nA module for the Cactus framework is then\ngenerated automatically by Kranc for one or more\ntarget architectures.\nKranc applies many optimizations \nat code-generation time, making use of symbolic algebra,\nand the resulting source code can then be compiled on a\ndiverse range of machines (taking advantage of the established\nportability of Cactus and the availability of CUDA as a uniform GPU\nprogramming environment).  At run-time, the CUDA code is recompiled\ndynamically to enable a range of runtime optimizations.\n\nWe have presented two case studies. The first is a \nComputational Fluid Dynamics (CFD) code, and we demonstrated weak scaling\nusing our infrastructure running on GPUs.  We also used\nthe Physis framework for this same problem and compared the scaling.\nChemora\\xspace has comparable\nor higher performance, a result we attribute to the dynamic\noptimizations that we employ.  The second case study is a Numerical\nRelativity simulation based on the\nMcLachlan code, a part of the freely available open-source (GPL)\nEinstein Toolkit (ET\\@).  McLachlan solves a\nsignificantly more complex set of equations, and integrates with many\nother components of the ET\\@.  We performed a simulation of a binary\nblack hole coalescence using the same codes and techniques as we would\ncurrently use in production CPU simulations, with the omission of\nAdaptive Mesh Refinement (AMR), which is not yet adapted to Chemora\\xspace.\n\nWe plan to implement AMR and multi-block methods next.\nAMR and multi-block are implemented in Cactus in a\nway which is transparent to the application programmer, hence we\nexpect that including AMR in Chemora\\xspace will be straightforward using\nthe Accelerator architecture developed in this work\n(which maintains knowledge of which variables are valid on the host\n(CPU) and which on the device (GPU)).  As with the time integration, we\nwill implement only the basic low-level interpolation operators\nrequired for mesh refinement on the GPU, and the existing AMR code\nCarpet will marshal the required operations to the device.\n\nWith AMR and\/or multi-block methods, Chemora\\xspace will be an even more\ncompelling option for implementing scientific codes, and fields of\nscience (such as Numerical Relativity) requiring the solution of\ncomplex systems of PDEs will be able to reach a new level of\nperformance.\nShould the specifics of accelerator devices change in the future, the\nChemora\\xspace framework, much of which is general, should be easily adaptable to\nthe new technology, and codes built with Chemora\\xspace will have a head start in\nadvancing computational science on the new platform.\n\n\\section*{Acknowledgments}\n\nThe authors would like to thank Gabrielle Allen and Joel E. Tohline at\nthe CCT\nand Krzysztof Kurowski at PSNC for their vision, encouragement, and \ncontinuous support to this project.\n\nThis work was supported by the UCoMS project under award number MNiSW\n(Polish Ministry of Science and Higher Education) Nr 469~1~N~-\nUSA\/2009 in close collaboration with U.S. research institutions\ninvolved in the U.S. Department of Energy (DOE) funded grant under\naward number DE-FG02-04ER46136 and the Board of Regents, State of\nLouisiana, under contract no.\\ DOE\/LEQSF(2004-07)\nand LEQSF(2009-10)-ENH-TR-14.\nThis work was also supported by NSF award 0725070 \\emph{Blue Waters},\nNFS awards 0905046 and 0941653\n\\emph{PetaCactus}, NSF award 0904015 \\emph{CIGR}, \nand NSF award 1010640 \\emph{NG-CHC} to Louisiana State University, and\nby the DFG grant SFB\/Transregio~7 ``Gravitational-Wave Astronomy''.\n\nThis work was performed using computational resources of XSEDE\n(TG-CCR110029, TG-ASC120003), LONI (loni\\_cactus), LSU, and PSNC,\nand on the Datura cluster at the AEI\\@.\n\n\\bibliographystyle{unsrt}\n{\\footnotesize","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\nThe real analytic Eisenstein series is a special function\nthat has been studied classically. It is used in the representation theory of $SL(2,\\mathbb{R})$,\nand in analytic number theory (e.g., cf.\\,\\cite{Kub}).\nIts generalization to the case of many variables was initiated by\nSiegel and later studied more extensively by Langlands \\cite{Lang} and Shimura \\cite{S2}.\n\nLet\n$$\nE_k^{(m)}(z,s)=\\text{det}(y)^s\n                         \\sum_{\\{ c,d\\}}\\text{det}(cz+d)^{-k}\\,|\\text{det}(cz+d)|^{-2s}\n$$\nbe the Eisenstein series of degree $m$ (for a precise definition, see $\\S$ \\ref{sec3-1}).\n\nShimura \\cite{S2} studied the analytic properties of the Eisenstein series, including\nthis type. He reveals the holomorphy of $E_k^{(m)}(z,s)$ in $s$ at $s=0$ by\nanalyzing the Fourier coefficients. The Fourier coefficient essentially consists of\ntwo parts. One is the confluent hypergeometric function, and the other is the Siegl series.\nTherefore, the analytic properties of Fourier coefficients, and the Eisenstein\nseries results in the study of the analytic properties of these two parts. In \\cite{S1},\nShimura established the analytic theory of confluent hypergeometric functions\non tube domains and then applied them to analysis of Eisenstein series.\nThe results of holomorphy of $E_k^{(m)}(z,s)$ studied and extended by Weissauer\n\\cite{W}.\n\nIn Shimura's paper \\cite{S2}, apart from the holomorphy, the residue of Eisenstein\nseries is mentioned. His statement is as follows:\n\\vspace{2mm}\n\\\\\nThe residue of the Eisenstein series $E_{(m-1)\/2}^{(m)}(z,s)$ at $s=1$ can be expressed \nas the product of $\\pi^{-m}$ and a holomorphic modular form of weight $(m-1)\/2$,\nwith rational Fourier coefficients.\n\\vspace{2mm}\n\\\\\n(It is known that the holomorphic modular form stated above is\na (rational) constant multiple of Eisenstein series $E_{(m-1)\/2}^{(m)}(z,0)$.)\n\\vspace{2mm}\n\\\\\nOther than his work, few papers mention concrete forms of the residue for the Eisenstein series,\nexcept for the classical work by Kaufhold \\cite{Kau} (see $\\S$ \\ref{other}).\n\nThis study aims to provide concrete forms of residue $E_0^{(m)}(z,s)$\nat $s=m\/2$. Our results strongly depend on Mizumoto's work \\cite{Mi}, especially his work on\nthe Fourier expansion of $E_k^{(m)}(z,s)$, which is a refinement of Maass's result.\n\\vspace{4mm}\n\\\\\n\\textbf{Theorem} \n\\begin{align*}\n& \\underset{s=m\/2}{{\\rm Res}}E_0^{(m)}(z,s)\\\\\n&=\\mathbb{A}^{(m)}(y)+\\mathbb{B}^{(m)}(y)\n                \\sum_{h\\in\\Lambda_m^{(1)}}\\sigma_0({\\rm cont}(h))\\eta_m(2y,\\pi h;m\/2,m\/2)\n                \\boldsymbol{e}(\\sigma(hx)),\n\\end{align*}\n{\\it where}\n\\begin{align*}\n\\mathbb{A}^{(m)}(y) &=\\frac{1}{2}\\,\\alpha_m(y,m\/2)\\cdot C_{m-1}^{(m)}(y)\n                                                    +\\frac{1}{8}\\,v(m-1)\\text{det}(2y)^{-(m-1)\/2}\\alpha'_m(y,m\/2)\\\\\n                         &+\\frac{1}{8}\\,\\beta'_m(y,m\/2),\\\\\n\\mathbb{B}^{(m)}(y) &=2^{m-2}\\pi^{m\\kappa(m)}\\text{det}(y)^{m\/2}\\Gamma_m(m\/2)^{-1}\\zeta(m)^{-1}\\\\\n                          &\\cdot \\prod_{j=1}^{m-2}\\zeta(m-j)\\,\\prod_{j=1}^{m-1}\\zeta(2m-2j)^{-1}.\n \\end{align*}     \n\\vspace{2mm}\n\\\\\nHere, $C_{m-1}^{(m)}$ is the constant term of the completed Koecher--Maass zeta function\n$\\xi_{m-1}^{(m)}(2y,s)$ at $s=m\/2$ (see (\\ref{DefC})), and $\\alpha_m(y,s)$ and $\\beta_m(y,s)$\nare defined in (\\ref{alpham}) and (\\ref{betam}), respectively,  which are essentially products of the gamma functions\nand zeta functions.\n\\vspace{2mm}\n\\\\\nIn the degree 2 case, the constant $C_{1}^{(2)}(y)$ can be calculated explicitly from the first\nKronecker limit formula.\n\\vspace{2mm}\n\\\\\n\\textbf{Corollary}\n\\\\\n\\begin{align}\n\\underset{s=1}{{\\rm Res}}\\,E_0^{(2)}(z,s) \n& =\\frac{18}{\\pi^2\\sqrt{\\text{det}(y)}}\\left(\\frac{1}{2}\\gamma+\\frac{1}{2}\\log\\frac{v'}{4\\pi}-\\log|\\eta(W_g)|^2 \\right)\n\\nonumber \\\\\n& \\quad +\\frac{36\\,\\text{det}(y)}{\\pi^2}\\sum_{h\\in\\Lambda_2^{(1)}}\\sigma_0(\\text{cont}(h))\\eta_2(2y,\\pi h;1,1)\n             \\boldsymbol{e}(\\sigma(hx)). \\label{A}\n\\end{align}\n(for the notation, see $\\S$ \\ref{Degree2}.)\nIn \\cite{Na}, the author provided a formula for $E_2^{(2)}(z,0)$ (Siegel Eisenstein series of degrees 2 and 2):\n\\begin{align}\nE_2^{(2)}(z,0)=& 1-\\frac{18}{\\pi^2\\sqrt{\\text{det}(y)}}\\left(1+\\frac{1}{2}\\gamma+\\frac{1}{2}\\log\\frac{v'}{4\\pi}-\\log|\\eta(W_g)|^2 \\right) \\nonumber \\\\\n         &-\\frac{72}{\\pi^3}\\sum_{\\substack{0\\ne h\\in\\Lambda_2\\\\ \\text{discr}(h)=\\square}}\\varepsilon_h\n           \\sigma_0(\\text{cont}(h))\\eta_2(2y,\\pi h;2,0)\\boldsymbol{e}(\\sigma(hx))\\nonumber \\\\\n         &+288\\sum_{0\\ne h\\in \\Lambda_2}\\sum_{d\\mid\\text{cont}(h)}d\\,H\\left(\\frac{|\\text{discr}(h)|}{d^2}\\right)\n              \\boldsymbol{e}(\\sigma(hz)). \\label{B}\n\\end{align}\nIt is interesting that the same term appears in each Fourier coefficient in (\\ref{A}) and\n(\\ref{B}).\n\n\n\n\n\\section{Notation}\n\\label{notation}\n$1^\\circ$\\quad\nIf $a$ is an $m\\times m$ matrix, we write it as $a^{(m,n)}$, and as $a^{(m)}$ if $m=n$,\n${}^ta$ denotes the transpose of $a$, and $a_{ij}$ denotes the $(i,j)$-entry of $a$. For a\nmatrix $a$, we write $\\sigma (a)$ as the trace of $a$. If the right-hand side is defined as\nthe identity matrix (resp. zero matrix) of size $m$ and is denoted by $1_m$ (resp. $0_m$).\nFor a commutative ring $R$ with $1$, we denote $R^{(m,n)}$ by the $R$-module of\nall $m\\times n$ matrices with entries $R$. We set $R^{(m)}:=R^{(m,m)}$ and $R^m:=R^{(1,m)}$.\n\\vspace{2mm}\n\\\\\n$2^\\circ$\\quad\nWe put\n\\vspace{1mm}\n\\\\\n$\\bullet$\\quad $\\displaystyle\\kappa (\\nu)=\\frac{\\nu+1}{2}$ for $\\nu\\in\\mathbb{Z}_{\\geqq 0}$.\n\\vspace{2mm}\n\\\\\n$\\bullet$\\quad $\\boldsymbol{e}(z)=\\text{exp}(2\\pi iz)$ for $z\\in\\mathbb{C}$.\n\\vspace{2mm}\n\\\\\n$\\bullet$\\quad $H_m:=\\{\\,z\\in\\mathbb{C}^{(m)}\\,\\mid\\, {}^tz=z,\\;\\text{Im}(z)>0\\,\\}$\\,:\\;upper\nhalf space.\n\\vspace{2mm}\n\\\\\n$\\bullet$\\quad $V_m=\\{\\,x\\in \\mathbb{R}^{(m)}\\,\\mid\\, {}^tx=x\\,\\}$.\n\\vspace{2mm}\n\\\\\n$\\bullet$\\quad $V_m(\\mathbb{C})=V_m\\otimes_{\\mathbb{R}}\\mathbb{C}$.\n\\vspace{2mm}\n\\\\\n$\\bullet$\\quad $P_m:=\\{\\, x\\in V_m\\,\\mid\\,x>0\\,\\}$.\n\\vspace{2mm}\n\\\\\n$\\bullet$\\quad $V_m(p,q,r)$: subset of $V_m$ consisting of the elements with $p$ positive,\\\\\n             \\qquad\\qquad\\qquad\\quad $q$ negative, $r$ zero eigenvalues.\n\\vspace{2mm}\n\\\\\n$3^\\circ$\\quad\nThe function $\\Gamma_m(s)$ is defined by\n$$\n\\Gamma_m(s)=\\pi^{\\frac{m(m-1)}{4}}\\prod_{\\nu=0}^{m-1}\\Gamma\\left(s-\\frac{\\nu}{2}\\right)\n$$\nfor $m>0$, and $\\Gamma_0(s):=1$. \n\\vspace{2mm}\n\\\\\n$4^\\circ$\\quad The set of symmetric half-integral matrices of size $m$ is denoted by\n$\\Lambda_m$. We place\n$$\n\\Lambda_m^{(\\nu)}:=\\{\\,h\\in\\Lambda_m\\,\\mid\\,\\text{rank}(h)=\\nu\\,\\}.\n$$\nFor $\\nu\\in\\mathbb{Z}$ with $1\\leqq \\nu\\leqq m$,\n$$\n\\mathbb{Z}_{\\text{prim}}^{(m,\\nu)}=\\{\\, a\\in \\mathbb{Z}^{(m,\\nu)}\\,\\mid\\,\n                                             a\\; {\\rm is\\; primitive} \\,\\}.\n$$\n$5^\\circ$\\quad Throughout the paper, we understand that the product (resp. sum)\nover an empty set is equal to $1$ (resp. $0$).\n\\section{Preliminary}\n\\label{Pre}\n\\subsection{Eisenstein series}\n\\label{sec3-1}\nFor $m\\in\\mathbb{Z}_{>0}$ and $k\\in 2\\mathbb{Z}_{\\geq 0}$, let\n\\begin{equation}\n\\label{DefEis}\nE_k^{(m)}(z,s)=\\text{det}(y)^s\n                         \\sum_{\\{ c,d\\}}\\text{det}(cz+d)^{-k}\\,|\\text{det}(cz+d)|^{-2s}\n\\end{equation}\nbe the Eisenstein series for $\\Gamma_m=Sp_m(\\mathbb{Z})$ (Siegel modular\ngroups of degrees $m$). Here, $z=x+iy$ is a variable on $H_m$, $s$ is a complex\nvariable, and $\\{ c,d\\}$ runs over a complete system of representatives $\\binom{\\,*\\,\\;*\\,}{c\\;d}$\nof $\\left\\{ \\binom{\\,*\\,*\\,}{\\,0\\;*\\,}\\in \\Gamma_m \\right\\}\\backslash \\Gamma_m$.\nThe right-hand side of (\\ref{DefEis}) converges absolutely, locally, and uniformly on the\n$$\n\\{\\; (z,s)\\in H_m\\times\\mathbb{C}\\;\\mid\\; \\text{Re}(s)>(m+1-k)\/2\\;\\}.\n$$\nAs is well known, the Eisenstein series $E_k^{(m)}(Z,s)$ has a meromorphic continuation\nto the whole $s$-plane (Langlands \\cite{Lang}, Mizumoto \\cite{Mi}).\n\\subsection{Confluent hypergeometric functions}\n\\label{sec3-2}\nShimura studied the confluent hypergeometric functions on the tube domains (\\cite{S1})\nand applied his results to develop the theory of the Eisenstein series (\\cite{S2}). In this section,\nwe summarize some results on the confluent hypergeometric functions that will be used later.\n\nFor $g\\in P_m$, $h\\in V_m$, and $(\\alpha,\\beta)\\in\\mathbb{C}^2$,\n\\begin{equation}\n\\label{xi}\n\\xi_m(g,h;\\alpha,\\beta)=\\int_{V_m}\\boldsymbol{e}^{-\\sigma(hx)}\\text{det}(x+ig)^{-\\alpha}\n                                     \\text{det}(x-ig)^{-\\beta}dx,\n\\end{equation}\nwith $dx=\\prod_{i\\leqq j}dx_{ij}$, which is convergent for $\\text{Re}(\\alpha+\\beta)>m$;\n\\begin{equation}\n\\label{eta}\n\\eta_m(g,h;\\alpha,\\beta)= \\int_{\\substack{V_m\\\\ x\\pm h>0}}\ne^{-\\sigma(gx)}\\text{det}(x+h)^{\\alpha-\\kappa(m)}\n                                     \\text{det}(x-h)^{\\beta-\\kappa(m)}dx,\\\\\n\\end{equation}\nwhich is convergent for $\\text{Re}(\\alpha)>\\kappa (m)-1$, $\\text{Re}(\\beta)>m$.\nWe also use\n\\begin{equation*}\n\\label{etastar}\n\\eta^*_m(g,h;\\alpha,\\beta)=\\text{det}(g)^{\\alpha+\\beta-\\kappa(m)}\\eta_m(g,h;\\alpha,\\beta),\n\\end{equation*}\nwhich satisfies the property\n\\begin{equation*}\n\\eta_m^*(g[a],h[{}^ta^{-1}];\\alpha,\\beta)\n=\\eta^*_m(g,h;\\alpha,\\beta)\n\\quad\n\\text{for all}\n\\quad a\\in GL_m(\\mathbb{R}).\n\\end{equation*}\nBy \\cite{S1}, (1.29),\n\\begin{equation}\n\\label{xieta}\n\\xi_m(g,h;\\alpha,\\beta)= i^{m(\\beta-\\alpha)}\\cdot 2^m\\pi^{m\\kappa(m)}\n                                          \\Gamma_n(\\alpha)^{-1}\\Gamma_n(\\beta)^{-1}\n                                        \\eta_m(2g,\\pi h;\\alpha,\\beta).\n\\end{equation}\nfor $\\text{Re}(\\alpha)>\\kappa(m)-1$, $\\text{Re}(\\beta)>m$.\nWhen $h=0_m$, the following identity holds:\n\\begin{Prop}\n(Shimura \\cite{S1}, (1.31)) {\\it If ${\\rm Re}(\\alpha+\\beta)>2\\kappa(m)-1$, then}\n\\begin{align}\n\\xi_m(g,0_m;\\alpha,\\beta) &= i^{m\\beta-m\\alpha}\\cdot 2^{m(1-\\kappa(m))}(2\\pi)^{m\\kappa(m)}\\nonumber \\\\\n                     & \\cdot \\Gamma_m(\\alpha)^{-1}\\Gamma_m(\\beta)^{-1}\\Gamma_m(\\alpha+\\beta-\\kappa(m)) \n                     \\nonumber \\\\\n                     & \\cdot \\text{det}(2g)^{\\kappa(m)-\\alpha-\\beta}.\n                     \\label{xi0}\n\\end{align}\n\\end{Prop}\nFor $g\\in P_m$, $h\\in V_m(p,q,r)$ with $p+q+r=m$, we put\n\\begin{align*}\n& \\delta_+(hg):=\\text{the product of all positive eigenvalues of} \\;\\;g^{\\frac{1}{2}}hg^{\\frac{1}{2}},\\\\\n& \\delta_{-}(hg):=\\delta_+((-h)g).\n\\end{align*}\nWe then put\n\\begin{align}\n\\omega_m(g,h;\\alpha,\\beta) :=&  2^{-p\\alpha-q\\beta}\\Gamma_p\\left(\\beta-(m-p)\/2\\right)^{-1}\n                                                               \\Gamma_q\\left(\\alpha-(m-q)\/2\\right)^{-1}\\nonumber \\\\\n                                   & \\cdot \\Gamma_r\\left(\\alpha+\\beta-\\kappa(m) \\right)^{-1} \\nonumber\\\\\n                                   & \\cdot \\delta_{+}(hg)^{\\kappa(m)-\\alpha-q\/4}\n                                     \\delta{-}(hg)^{\\kappa(m)-\\beta-p\/4}\\,\n                                     \\eta_n^*(g,h;\\alpha,\\beta), \\label{omegaeta}\n\\end{align}\nOne of the main results in \\cite{S1} is as follows:\n\\begin{Thm}\n\\label{ShimuraMain}\n(Shimura \\cite{S1}, Theorem 4.2)\\;{\\it \nFunction $\\omega_m$ can be continued as a holomorphic function in $(\\alpha,\\beta)$ to\nthe whole $\\mathbb{C}^2$ and satisfies\n$$\n\\omega_m(g,h;\\alpha,\\beta)=\\omega_m\\left(g,h;\\kappa(m)+(r\/2)-\\beta,\\kappa(m)+(r\/2)-\\alpha\\right).\n$$\n}\n\\end{Thm}\n\\subsection{Fourier expansion}\n\\label{FourierEx}\nFor $m\\in\\mathbb{Z}_{>0}$ and $k\\in 2\\mathbb{Z}_{\\geqq 0}$, let $s$ be a complex variable,\nwhere $\\text{Re}(s)>\\kappa (m)$, and let $z=x+iy$ be a variable on $H_m$ with $x\\in V_m$.\nand $y\\in P_m$. Maass (\\cite{Ma}) provided a formula for the Fourier expansion of the\nEisenstein series $E_k^{(m)}(z,s)$:\n\\begin{align}\nE_k^{(m)}(z,s) &= \\text{det}(y)^s+\\text{det}(y)^s\\sum_{\\nu=1}^m\\sum_{h\\in\\Lambda_\\nu}\n                                         \\sum_{q\\in\\mathbb{Z}_{\\text{prim}}^{(m,\\nu)}\/GL_\\nu(\\mathbb{Z})}\\nonumber\\\\\n                 & S_\\nu(h,2s+k)\\xi_{\\nu}(y[q],h;s+k,s)\\boldsymbol{e}(\\sigma(h[{}^tq]x)),  \\label{ExMaass}                 \n\\end{align}\nwhere\n\\begin{equation}\nS_\\nu(h,s)=\\sum_{r\\in V_\\nu\\cap\\, \\mathbb{Q}^\\nu \\text{mod}\\, 1}n(r)^{-s}\\boldsymbol{e}(\\sigma (hr))\n\\end{equation}\nis the singular series (Siegel series), where $n(r)$ is the product of the reduced positive denominators\nof the elementary divisors of $r$, and $\\xi_\\nu$ is the confluent hypergeometric function defined in  (\\ref{xi}).\n\nFrom \\cite{Mi}, Lemma 1.1, we have\n\\begin{Lem}\n{\\it For $\\nu\\in\\mathbb{Z}_{>0}$, each $h\\in\\Lambda_\\nu$ of {\\rm rank} $\\lambda>0$ (that is,\n$h\\in \\Lambda_\\nu^{(\\lambda)}$) is expressed uniquely as\n$$\nh=h_0[{}^tw]\n$$\nwith $h_0\\in\\Lambda_\\lambda^{(\\lambda)}$ and \n$w\\in \\mathbb{Z}_{{\\rm prim}}^{(\\nu,\\lambda)}\/GL_\\lambda(\\mathbb{Z})$.}\n\\end{Lem}\nMizumoto provided a reduced formula for $\\xi_\\nu$ (\\cite{Mi}, Lemma 1.4):\n\\begin{Prop}\n\\label{xi-eta}\n{\\it Let $h=h_0[{}^tw]$ be, as in the above lemma. Suppose that ${\\rm Re}(s)>\\nu$.\nThen, in {\\rm (\\ref{ExMaass})}, we have}\n\\begin{align}\n& \\xi_{\\nu}(y[q],h;s+k,s) \\nonumber \\\\\n& = (-1)^{k\\nu\/2}2^\\nu\\pi^{\\nu\\kappa(\\nu)+\\lambda(\\nu-\\lambda)\/2)}\n                                \\cdot \\Gamma_{\\nu-\\lambda}(2s+k-\\kappa(\\nu))\n                                \\Gamma_\\nu(s)^{-1}\\Gamma_\\nu(s+k)^{-1} \\nonumber \\\\\n                &\\cdot {\\rm det}(2y[q])^{\\kappa(\\nu)-k-2s}\n                \\eta_{\\lambda}^*(2y[qw],\\pi h_0;s+k+(\\lambda-\\nu)\/2,s+(\\lambda-\\nu)\/2).\n\\end{align}\n\\end{Prop}\nLet $m,\\,\\lambda\\in\\mathbb{Z}$ with $m\\geqq \\lambda\\geqq 1$. We define the subgroup\n$\\Delta_\\lambda^{(m)}$ of $GL_m(\\mathbb{Z})$ by\n$$\n\\Delta_\\lambda^{(m)}:=\\left\\{ \\begin{pmatrix} * & * \\\\ 0^{(m-\\lambda,\\lambda)} & * \\end{pmatrix}\\in\nGL_m(\\mathbb{Z}) \\;\\right\\}.\n$$\nFor $r\\in \\mathbb{Z}_{{\\rm prim}}^{(m,\\lambda)}$, $u_r$ is an element of $GL_m(\\mathbb{Z})$\ncorresponding to $r$ under a bijection\n\\begin{align*}\n\\mathbb{Z}_{{\\rm prim}}^{(m,\\lambda)}\/&GL_\\lambda(\\mathbb{Z})\n \\qquad\n\\longleftrightarrow\n\\qquad\nGL_m(\\mathbb{Z})\/\\Delta_\\lambda^{(m)} \\\\\n& r\\qquad\\qquad\\quad  \\longmapsto\\qquad\\qquad u_r\n\\end{align*}\nwhich is determined up to the right action of $\\Delta_\\lambda^{(m)}$.\n\nFor $y\\in P_m$, we write the Jacobi decomposition of $y[u_r]$ as\n\\begin{equation*}\ny[u_r]={\\rm diag}(y[r],g(y,u_r))\\begin{bmatrix} 1_\\lambda & b \\\\ 0 & 1_{m-\\lambda} \\end{bmatrix}.\n\\end{equation*}\nExplicitly, we place $u_r=(r\\,r_1)$ and then\n\\begin{equation}\n\\label{gyur}\ng(y,u_r)=y[r_1]-(y[r])^{-1}[{}^tryr_1].\n\\end{equation}\n\\vspace{3mm}\n\\\\\n\\quad Next, we provide a definition of Koecherer--Maass zeta functions.\nFor $1\\leqq \\nu\\leqq m$ and $g\\in P_m$, we define\n\\begin{equation}\n\\label{KM}\n\\zeta_\\nu^{(m)}(g,s):=\\sum_{a\\in \\mathbb{Z}_{{\\rm prim}}^{(m,\\nu)}\/GL_\\nu(\\mathbb{Z})}\n\\text{det}(g[a])^{-s}\n\\end{equation}\nwhich is convergent for $\\text{Re}(s)>m\/2$. By definition,\n$$\n\\zeta_m^{(m)}(g,s)=\\text{det}(g)^{-s}.\n$$\nFor later purposes, we put\n$$\n\\zeta_0^{(m)}(*,s):=1\\qquad\n\\text{for all}\n\\quad m\\in\\mathbb{Z}_{\\geqq 0}.\n$$\nMizumoto's refinement of Maass' expression is as follows:\n\\begin{Thm} (Mizumoto \\cite{Mi}, Theorem 1.8) \n\\label{FourierMi}\n{\\it\nFor $m\\in\\mathbb{Z}_{>0}$, $k\\in 2\\mathbb{Z}_{\\geqq 0}$, and ${\\rm Re}(s)>m$,\nthe Eisenstein series $E_k^{(m)}(z,s)$ has the following expression:\n\\begin{equation}\n\\label{MiEx}\nE_k^{(m)}(z,s)=\\sum_{\\nu=0}^m\\sum_{\\lambda=0}^\\nu F_{k,\\nu,\\lambda}^{(m)}(z,s)\n\\end{equation}\nwhere\n\\begin{align}\n\\label{F0}\n& F_{k,\\nu,0}^{(m)}(z,s) \\nonumber \\\\\n& =(-1)^{k\\nu\/2}2^\\nu\\pi^{\\nu\\kappa(\\nu)}\\Gamma_\\nu(2s+k-\\kappa(\\nu))\n            \\Gamma_\\nu(s)^{-1}\\Gamma_\\nu(s+k)^{-1} \\nonumber \\\\\n& \\cdot S_\\nu(0_\\nu,2s+k)\\,{\\rm det}(y)^s\\,\\zeta_\\nu^{(m)}(2y,2s+k-\\kappa(\\nu)),\n\\end{align}\nfor $0\\leqq \\nu\\leqq m$, and\n\\begin{equation}\n\\label{Fb}\nF_{k,\\nu,\\lambda}^{(m)}(z,s)\n=\n\\sum_{h\\in\\Lambda_{\\lambda}^{(\\lambda)}}\\sum_{r\\in \\mathbb{Z}_{{\\rm prim}}^{(m,\\lambda)}\/GL_\\lambda(\\mathbb{Z})}\nb_{k,\\nu,\\lambda}^{(m)}(h[{}^tr],y,s)\\boldsymbol{e}(\\sigma(h[{}^tr]x))\n\\end{equation}\nfor $1\\leqq \\lambda\\leqq \\nu\\leqq m$ with\n\\begin{align}\n& b_{k,\\nu,\\lambda}^{(m)}(h[{}^tr],y,s) \\nonumber \\\\\n             &\\quad := (-1)^{k\\nu\/2}2^\\nu\\pi^{\\nu\\kappa(\\nu)+\\lambda(\\nu-\\lambda)\/2}\n              \\Gamma_{\\nu-\\lambda}(2s+k-\\kappa(\\nu))\\Gamma_\\nu(s)^{-1}\\Gamma_\\nu(s+k)^{-1}\n              \\nonumber \\\\\n              &\\qquad \\cdot S_\\nu({\\rm diag}(h,0_{\\nu-\\lambda}),2s+k)){\\rm det}(y)^s{\\rm det}(2y[r])^{\\kappa(\\nu)-k-2s}\n              \\nonumber \\\\\n              &\\qquad \\cdot \\eta_\\lambda^*(2y[r],\\pi h;s+k+(\\lambda-\\nu)\/2,s+(\\lambda-\\nu)\/2)\\nonumber \\\\\n             &\\qquad  \\cdot \\zeta_{\\nu-\\lambda}^{(m-\\lambda)}(2g(y,u_r),2s+k-\\kappa(\\nu)) \\label{Fb1}.\n\\end{align}\nHere, $\\zeta_\\nu^{(m)}(g,s)$ for $0\\leqq \\nu\\leqq m$ is the Koecher--Maass zeta function\ndefined in {\\rm (\\ref{KM})}, and $g(y,u_r)$ is defined by {\\rm (\\ref{gyur})}. Matrix $h[{}^tr]$ runs\nover the set $\\Lambda_m^{(\\lambda)}$ exactly once if $h$ runs over $\\Lambda_\\lambda^{(\\lambda)}$\nand $r$ runs over a complete set of representatives of \n$\\mathbb{Z}_{{\\rm prim}}^{(m,\\lambda)}\/GL_\\lambda(\\mathbb{Z})$.}\n\\end{Thm}\n\\subsection{Siegel series}\n\\label{sec3-4}\nIn this section, we summarize the results of the Siegel series $S_\\nu(h,s)$ that appear in\nthe Fourier expansions (\\ref{ExMaass}) and (\\ref{Fb1}).\n\nFor $h\\in\\Lambda_\\lambda^{(\\lambda)}$, we set\n$$\nd(h):=(-1)^{[\\lambda\/2]}2^{-\\delta((\\lambda-1)\/2)}\\text{det}(2h)\n$$\nwhere\n$$\n\\delta (x):=\\begin{cases} 1  &  x\\in\\mathbb{Z},\\\\\n                                0  &   x\\notin \\mathbb{Z}\n                                \\end{cases}\n$$\nfor $x\\in\\mathbb{Q}$. By \\cite{Mi}, (5.1), \n\\begin{align}\n\\label{Siegelreduce}\nS_\\nu(\\text{diag}(h,0_{\\nu-\\lambda}),s) &=\\zeta(s+\\lambda-\\nu)\\zeta(s)^{-1}\\nonumber\\\\\n                                                  & \\cdot \\prod_{j=1}^{\\nu-\\lambda}(\\zeta(2s-\\nu-j)\\,\\zeta(2s-2j)^{-1})\n                                                  \\nonumber\\\\\n                                                   & \\cdot S_\\lambda(h,s-\\nu+\\lambda)\n\\end{align}\nand\n\\begin{align*}\n& S_\\lambda(h,s)=\\sum_{d\\in A(h)}(\\text{det}(d))^{\\lambda+1-2s}\\widehat{S}_{\\lambda}(h[d^{-1}],s),\\\\\n& \\widehat{S}_{\\lambda}(h,s)=\\zeta(s)^{-1}\\prod_{j=1}^{[\\lambda\/2]}\\zeta(2s-2j)^{-1}\n             L\\left(s-\\lambda\/2,\\left(\\frac{d(h)}{*} \\right)\\right)^{\\delta(\\lambda\/2)}\\prod_p a_p(h,s)\n\\end{align*}\nwhere $L\\left(s,\\left(\\frac{d(h)}{*}\\right)  \\right)$ is Dirichlet $L$-function associated to the quadratic\ncharacter $\\left(\\frac{d(h)}{*}\\right)$, the product of $p$ runs over the prime divisors of $d(h)$,\n$$\nA(h):=GL_\\lambda(\\mathbb{Z})\\backslash\n     \\{\\,d\\in\\mathbb{Z}^{(\\lambda)}\\,\\mid\\, \\text{det}(d)\\ne 0\\;\\;\\text{and}\\;\\; h[d^{-1}]\\in\\Lambda_\\lambda\\,\\},\n$$\nand from \\cite{Bo}, we have\n\\begin{align*}\n& a_p(h,s)=\\\\\n&\n\\begin{cases}\n\\prod_{j=1}^{r\/2}(1-p^{2j-1+\\lambda-2s}) & (\\lambda,r) \\equiv (1,0) \\pmod{2},\\\\\n(1+\\lambda_p(h)p^{(\\lambda+r)\/2-s})\\prod_{j=1}^{(r-1)\/2}(1-p^{2j-1+\\lambda-2s}) \n                                                    & (\\lambda,r) \\equiv (1,1) \\pmod{2},\\\\   \n\\prod_{j=1}^{(r-1)\/2}(1-p^{2j+\\lambda-2s})\n                                                   & (\\lambda,r) \\equiv (0,1) \\pmod{2},\\\\ \n(1+\\lambda_p(h)p^{(\\lambda+r)\/2-s})\\prod_{j=1}^{r\/2-1}(1-p^{2j+\\lambda-2s}) \n                                                  & (\\lambda,r) \\equiv (0,0) \\pmod{2}.\n\\end{cases}\n\\end{align*}\nHere, $r:=r(p)$ is the maximal number, which is the condition \n$h[u] \\equiv \\begin{pmatrix}h^* & 0 \\\\ 0 & 0_r \\end{pmatrix} \\pmod{p}$\nfor some $u\\in\\mathbb{Z}^{(\\lambda)}$ and $\\lambda_p(h):=\\left(\\frac{d(h^*)}{p} \\right)$.\n\\vspace{2mm}\n\\\\\n\\begin{Rem}\n(1)\\quad We understand that $S_0(*,s)=1$.  Therefore, from (\\ref{Siegelreduce}), we obtain the following.\nFormula for $S_\\nu(0_\\nu,s)$:\n\\begin{align}\n\\label{S0}\nS_\\nu(0_\\nu,s) &=\\zeta(s-\\nu)\\zeta(s)^{-1}\\prod_{\\nu=1}^\\nu(\\zeta(2s-\\nu-j)\\,\\zeta(2s-2j)^{-1}) \\nonumber\\\\\n               &= \\zeta(s-\\nu)\\zeta(s)^{-1}\\prod_{\\nu=1}^{[\\nu\/2]}(\\zeta(2s-2\\nu-1+2j)\\,\\zeta(2s-2j)^{-1}).\n\\end{align}\n(2)\\quad In the following discussion, the concrete form of $a_p(h,s)$ is not needed, only its\nholomorphy in $s$.\n\\end{Rem}\n\\subsection{Koecher--Maass zeta function}\n\\label{sec:3-5}\nThe Koecher--Maass zeta function $\\zeta_\\nu^{(m)}(g,s) $ in $ \\S$ \\ref{FourierEx}.\nAnalytic properties of this function are important for the analysis of the Fourier coefficient\n$F_{k,\\nu,\\lambda}^{(m)}(z,s)$. In this section, we recall Arakawa's results for the Koecher--Maass\nzeta function.\n\nFor $1\\leqq \\nu\\leqq m$ and $g\\in P_m$, we define the completed Koecher--Maass zeta\nfunction by\n\\begin{equation}\n\\label{CompKM}\n\\xi_{\\nu}^{(m)}(g,s):=\\prod_{i=0}^{\\nu-1}\\xi(2s-i)\\,\\zeta_\\nu^{(m)}(g,s)\n\\end{equation}\nwhere\n$$\n\\xi(s):=\\pi^{-s\/2}\\Gamma(s\/2)\\,\\zeta(s)\n$$\nand we understand\n$$\n\\xi_0^{(m)}(g,s):=1.\n$$\nThe following result is due to Arakawa, which plays an important role in our investigation.\n\\begin{Prop}\\;(Arakawa \\cite{Ara})\\quad \n\\label{Ara}\n{\\it (1)\\; Suppose $m\\geqq 2\\nu-1$.\nThe function $\\xi_{\\nu}^{(m)}(g,s)$ has simple poles at $s=0,\\,\\frac{1}{2},\\,\\cdots\\,,\\frac{\\nu-1}{2}$\nand $s=\\frac{m-\\nu+1}{2},\\,\\cdots\\,,\\frac{m}{2}$. For $0\\leqq \\mu\\leqq\\nu-1$, the residues of\n$\\xi_{\\nu}^{(m)}(g,s)$ at $s=\\frac{\\mu}{2}$ and $s=\\frac{m-\\mu}{2}$ are given by\n\\begin{align}\n& \n\\underset{s=\\mu\/2}{\\rm Res}\\xi_{\\nu}^{(m)}(g,s)=-\\frac{1}{2}v(\\nu-\\mu)\\xi_{\\mu}^{(m)}(g,\\tfrac{\\nu}{2}),\n\\label{Res1} \\\\\n&\n\\underset{s=(m-\\mu)\/2}{\\rm Res}\\xi_{\\nu}^{(m)}(g,s)=\\frac{1}{2}v(\\nu-\\mu)\n                                               {\\rm det}(g)^{-\\frac{\\nu}{2}}\\xi_{\\mu}^{(m)}(g^{-1},\\tfrac{\\nu}{2}),\n\\label{Res2}\n\\end{align}\nwhere\n$$\nv(\\nu)=\n\\begin{cases}\n\\displaystyle \\prod_{i=2}^\\nu \\xi(i)   &  (\\nu\\geqq 2),\\\\\n1                                             & (\\nu=1).\n\\end{cases}\n$$\n(2)\\; Suppose $\\nu\\leqq m\\leqq 2\\nu-2$. The function \n$\\xi_{\\nu}^{(m)}(g,s)$ has poles at $s=0,\\,\\frac{1}{2},\\,\\cdots\\,,\\frac{m}{2}$ of which\n$s=0,\\,\\frac{1}{2},\\,\\cdots\\,,\\frac{m-\\nu}{2}$ and $s=\\frac{\\nu}{2},\\,\\frac{\\nu+1}{2},\\,\\cdots\\,,\\frac{m}{2}$\nare simples poles. The poles at $s=\\frac{m-\\nu+1}{2},\\,\\frac{m-\\nu+2}{2},\\,\\cdots\\,,\\frac{\\nu-1}{2}$ are\ndouble poles. For $0\\leqq \\mu\\leqq m-\\nu$, the residues of $\\xi_{\\nu}^{(m)}(g,s)$ at $s=\\frac{\\mu}{2}$\nand $s=\\frac{m-\\mu}{2}$ are given by {\\rm (\\ref{Res1}) and (\\ref{Res2})}, respectively.\n}\n\\end{Prop}\n\\begin{Rem}\nWhen $m=2$ and $\\nu=1$, the function $\\zeta_1^{(2)}(g,s)$ appears as a simple\nfactor of Epstein's zeta function for $g$. Therefore, the residue and constant term at\n$s=1$ is explicitly expressed by the Kronecker limit formula (see $\\S$ \\ref{Degree2}).\n\\end{Rem}\n\\section{Residue of Eisenstein series}\n\\label{Main}\nIn the rest of this paper, we assume that $m\\geqq 2$.\nIn this section, we provide an explicit formula for\n$$\n\\underset{s=m\/2}{\\text{Res}}E_0^{(m)}(z,s)\n$$\nwhich is the main result of this paper.\n\n\\subsection{Fourier coefficient of $\\boldsymbol{E_0^{(m)}(z,s)}$}\n\\label{sec:4-1}\nWe recall the Fourier expansion\n$$\nE_0^{(m)}(z,s)=\\sum_{\\nu=0}^m\\sum_{\\lambda=0}^\\nu F_{0,\\nu,\\lambda}^{(m)}(z,s)\n$$\nin Theorem \\ref{FourierMi} and study the analytic property, particularly the singularity\nof $F_{0,\\nu,\\lambda}^{(m)}(z,s)$ and $b_{0,\\nu,\\lambda}^{(m)}(*,y,s)$.\nFor this purpose, we use the results introduced in $\\S$ \\ref{Pre} and consider\nthem dividing into several cases.\n\\vspace{2mm}\n\\\\\n$1^\\circ$\\quad $(\\nu,\\lambda)=(0,0)$:\n$$\nF_{0,0,0}^{(m)}(z,s)=\\text{det}(y)^s.\n$$\n$2^\\circ$\\quad $(\\nu,\\lambda)=(\\nu,0)$,\\;$(0<\\nu<m)$:\n\\begin{align*}\nF_{0,\\nu,0}^{(m)}(z,s) &= c_{\\nu,0}(s)\\Gamma_\\nu(s)^{-2}\\zeta(2s-\\nu)\\zeta(2s)^{-1}\\\\\n                           &\\quad\\cdot \\prod_{j=1}^\\nu\\zeta(4s-2j)^{-1}\\,\\xi_\\nu^{(m)}(2y,2s-\\kappa(\\nu)).\n\\end{align*}\n$3^\\circ$\\quad $(\\nu,\\lambda)=(m,0)$:\n\\begin{align*}\nF_{0,m,0}^{(m)}(z,s) &= c_{m,0}(s)\\Gamma_m(2s-\\kappa(m))\\Gamma_m(s)^{-2}\n\\zeta(2s-m)\\zeta(2s)^{-1}\\\\\n                           &\\quad\\cdot \\prod_{j=1}^m(\\zeta(4s-m-2j)\\zeta(4s-2j)^{-1}).\n\\end{align*}\n$4^\\circ$\\quad $(\\nu,\\lambda)=(\\nu,\\nu)$,\\;$(0<\\nu\\leqq m)$:\n\\begin{align*}\nb_{0,\\nu,\\nu}^{(m)}(*,y,s) &= c_{\\nu,\\nu}(s)\\Gamma_\\nu(s)^{-2}\\zeta(2s)^{-1}\n                                    \\prod_{j=1}^{[\\nu\/2]}\\zeta(4s-2j)^{-1}\n                                    \\cdot \\eta_\\nu(*,*;s,s).\n\\end{align*}\n$5^\\circ$\\quad $(\\nu,\\lambda)$,\\;$(0<\\lambda<\\nu<m)$:\n\\begin{align*}\nb_{0,\\nu,\\lambda}^{(m)}(*,y,s) &= c_{\\nu,\\lambda}(s)\\Gamma_\\nu(s)^{-2}\\zeta(2s)^{-1}\n                                           \\prod_{j=1}^{\\nu-\\lambda}\\zeta(4s-2j)^{-1}\\\\\n                                 & \\cdot \\prod_{j=1}^{[\\lambda\/2]}\\zeta(4s-2\\nu+2\\lambda-2j)^{-1}\\\\\n                                   &\\cdot  \\eta_\\lambda(*,*;s+(\\lambda-\\nu)\/2,s+(\\lambda-\\nu)\/2)\n                                    \\cdot \\xi_{\\nu-\\lambda}^{(m-\\lambda)}(*,2s-\\kappa(\\nu)).\n\\end{align*}\n$6^\\circ$\\quad $(\\nu,\\lambda)=(m,\\lambda)$,\\;$(0<\\lambda<m)$:\n\\begin{align*}\nb_{0,m,\\lambda}^{(m)}(*,y,s) &= c_{m,\\lambda}(s)\\Gamma_{m-\\lambda}(2s-\\kappa(m))\\Gamma_\\nu(s)^{-2}\\\\\n                 &\\cdot \\prod_{j=1}^{m-\\lambda}\\zeta(4s-m-j)\\zeta(4s-2j)^{-1}\n                    \\prod_{j=1}^{[\\lambda\/2]}\\zeta(4s-2m+2\\lambda-2j)^{-1}\\\\\n                 &\\cdot \\eta_{\\lambda}(*,*;s+(\\lambda-m)\/2.s+(\\lambda-m)\/2).\n\\end{align*}\nwhere $c_{\\nu,\\lambda}(s)$ are holomorphic function in $s$.\n\\subsection{Analytic property of Fourier coefficients}\n\\label{sec:4-2}\nWe investigate the analytic property of $F_{0,\\nu,\\lambda}^{(m)}(z,s)$ and $b_{0,\\nu,\\lambda}^{(m)}(*,y,s)$\nat $s=m\/2$, based on the description in $\\S$ \\ref{sec:4-1}.\n\\begin{Prop}\n\\label{Triang}\n{\\it Functions $F_{0,\\nu,\\lambda}^{(m)}(z,s)$ and $b_{0,\\nu,\\lambda}^{(m)}(*,y,s)$ are holomorphic in $s$ at\n$s=m\/2$, except for the following three cases:\n$$\n{\\rm (i)}\\; \\nu=m-1,\\;\\lambda=0\\qquad\n{\\rm (ii)}\\; \\nu=m,\\;\\lambda=0\\qquad\n{\\rm (iii)}\\; \\nu=m, \\;\\lambda=m-1.\n$$\n}\n\\end{Prop}\n\\begin{proof}\nWe use the expressions $1^\\circ-6^\\circ$ given in the previous section.\n\nThe holomorphy for the case $1^\\circ$ is\ntrivial.\n\nFirst, we consider the case $5^\\circ$. The $\\Gamma$-factor $\\Gamma_\\nu(s)^{-2}$,\nand $\\zeta$-factors $\\zeta(2s)^{-1},\\cdots$ are all holomorphic at $s=m\/2$.\nFrom Theorem \\ref{ShimuraMain}, the holomorphy of \n$\\eta_\\lambda(*,*;s+(\\lambda-\\nu)\/2,s+(\\lambda-\\nu)\/2)$ is reduced to that of\n$$\n\\Gamma_p(s+(\\lambda-\\nu)\/2-q\/2)\\quad\n\\Gamma_q(s+(\\lambda-\\nu)\/2-p\/2)\\qquad (p+q=\\lambda),\n$$\nand are both holomorphic at $s=m\/2$. (Note that the factor $\\Gamma_r(*)^{-1}$\ndoes not appear in $\\eta_\\lambda$.)\n\nFunction $\\xi_{\\nu-\\lambda}^{(m-\\lambda)}(*,2s-\\kappa(\\nu))$ is holomorphic at $s=m\/2$\nbecause $\\text{Re}(2s-\\kappa(\\nu))>(m-\\lambda)\/2$.\nConsequently, the functions in the case $5^\\circ$ are holomorphic at $s=m\/2$.\n\nBy a similar argument, we observe that the functions in the case of $4^\\circ$ are holomorphic at $s=m\/2$.\nThe cases we must consider are the cases of $2^\\circ$ and $6^\\circ$.\n\nIn the case of $2^\\circ$, only $F_{0,m-1,0}^{(m)}(z,s)$ and $F_{0,m,0}^{(m)}(z,s)$ are non-holomorphic\nat $s=m\/2$ because $F_{0,m-1,0}^{(m)}(z,s)$ has a factor $\\zeta(2s-m+1)\\xi_{m-1}^{(m)}(*,2s-m\/2)$,\nand $F_{0,m,0}^{(m)}(z,s)$ have the factors $\\Gamma(2s-m)\\zeta(4s-2m+1)$, respectively. In fact,\nthe factors above have double poles at $s=m\/2$.\n\nIn the case $6^\\circ$, only the function $b_{0,m,m-1}^{(m)}(*,y,s)$ is nonholomorphic at $s=m\/2$,\nbecause it contains the factor $\\zeta(4s-2m+1)$, which has a simple pole at $s=m\/2$.\n\nThese facts complete the proof.\n\\end{proof}\n\\begin{Rem}\nThe explicit formulas for $F_{0,m-1,0}^{(m)}(z,s)$, $F_{0,m,0}^{(m)}(z,s)$, \\\\\nand $F_{0,m,1}^{(m)}(z,s)$ will be given in the next sections.\n\\end{Rem}\nHere, we arrange functions $F_{0,\\nu,\\lambda}^{(m)}$ as follows:\n\n{\\small\n\\begin{center}\n\\begin{tabular}{llllll}\n$F_{0,0,0}^{(m)}$  &    {}                    &     {}         &    {}    &     {}    &    {}       \n\\vspace{2mm}\n\\\\\n$F_{0,1,0}^{(m)}$  & $F_{0,1,1}^{(m)}$ &    {}          &   {}     &     {}    &    {}      \n\\vspace{2mm}\n\\\\\n$\\vdots$          &  $\\vdots$         &    $\\ddots$ &  {}    &     {}     &   {},        \n\\vspace{2mm}\n\\\\\n$F_{0,m-2,0}^{(m)}$ &$F_{0,m-2,1}^{(m)}$&  $\\ldots$ & $\\ddots$   &    {}     &    {}       \n\\vspace{2mm}\n\\\\\n$\\boldsymbol{F_{0,m-1,0}^{(m)}}$ &$F_{0,m-1,1}^{(m)}$&$F_{0,m-1,2}^{(m)}$&$ \\ldots$&$F_{0,m-1,m-1}^{(m)}$ &{} \n\\vspace{2mm}\n\\\\\n$\\boldsymbol{F_{0,m,0}^{(m)}}$ &$\\boldsymbol{F_{0,m,1}^{(m)}}$&$F_{0,m,2}^{(m)}$& $\\ldots$&$F_{0,m,m-1}^{(m)}$ &$F_{0,m,m}^{(m)}$\n\n\\end{tabular}\n\\end{center}\n}\nThe proposition asserts that only functions $F_{0,\\nu,\\lambda}^{(m)}$ printed in bold are non-holomorphic\nat $s=m\/2$.\n\\subsection{Residue of the constant term}\n\\label{sec:4-3}\nWe investigate the analytic property of the constant term\n$$\n\\sum_{\\nu=0}^mF_{0,\\nu,0}^{(m)}(z,s)\n$$\nat $s=m\/2$. More specifically, we show that the constant term has a simple pole at $s=m\/2$\nand calculate the residue.\n\nBy Proposition \\ref{Triang}, it is sufficient to investigate only $F_{0,m-1,0}^{(m)}(z,s)$ and\\\\\n$F_{0,m,0}^{(m)}(z,s)$ as far as considering the residue.\n\\vspace{2mm}\n\\\\\n\\textbf{Analysis of} $\\boldsymbol{F_{0,m-1,0}^{(m)}(z,s)}:$\n\\vspace{2mm}\n\\\\\nFrom the definition of $F_{0,\\nu,0}^{(m)}(z,s)$ (see (\\ref{F0})), we have\n\\begin{align}\n\\label{Fm-1}\n F_{0,m-1,0}^{(m)}(z,s) \n & =2^{m-1}\\pi^{2(m-1)s}\\text{det}(y)^s\\Gamma_{m-1}(s)^{-2}\\zeta(2s-m+1)\\zeta(2s)^{-2}\n         \\nonumber \\\\\n&\\quad \\cdot\\prod_{j=1}^{m-1}\\zeta(4s-2j)^{-1}\\cdot\\xi_{m-1}^{(m)}(2y,2s-m\/2). \n\\end{align}\n(We rewrote (\\ref{F0}) with the complete Koecher--Maass zeta function $\\xi_{m-1}^{(m)}$.)\n\nWe separate $F_{0,m-1,0}^{(m)}(z,s)$ into holomorphic and non-holomorphic parts.\nWe define the function $\\alpha_m(y,s)$ by\n$$\nF_{0,m-1,0}^{(m)}(z,s)=\\zeta(2s-m+1)\\xi_{m-1}^{(m)}(2y,2s-m\/2)\\cdot \\alpha_m(y,s).\n$$\nExplicitly,\n\\begin{equation}\n\\label{alpham}\n\\alpha_m(y,s):=2^{m-1}\\pi^{2(m-1)s}\\text{det}(y)^s\\Gamma_{m-1}(s)^{-2}\\zeta(2s)^{-1}\n                  \\prod_{j=1}^{m-1}\\zeta(4s-2j)^{-1}.\n\\end{equation}\nFunctions $\\zeta(2s-m+1)$ and $\\xi_{m-1}^{(m)}(2y,2s-m\/2)$ have a simple pole at\n$s=m\/2$ (for $\\xi_{m-1}^{(m)}$, see Proposition \\ref{Ara}), and $\\alpha_m(y,s)$\nis holomorphic at $s=m\/2$. These facts imply that $F_{0,m-1,0}^{(m)}(z,s)$ has a double pole at\n$s=m\/2$.\n\nWe set\n$$\nF_{0,m-1,0}^{(m)}(z,s)=\\sum_{l=-2}^\\infty A_l^{(m)}(y)(s-m\/2)^l\n\\quad (\\text{Laurent expansion at}\\; s=m\/2)\n$$\nand calculate $A_{-2}^{(m)}(y)$ and $A_{-1}^{(m)}(y)$.\n\nAs a preparation, we investigate the analytic behavior of $\\xi_{m-1}^{(m)}(2y,2s-m\/2)$\nat $s=m\/2$. We consider the completed Koecher--Maass zeta function $\\xi_{m-1}^{(m)}(2y,s)$.\nAccording to Arakawa's result (Proposition \\ref{Ara}), this function has a simple pole with\nresidue\n\\begin{equation}\n\\label{Resm-1}\n\\underset{s=m\/2}{\\text{Res}}\\xi_{m-1}^{(m)}(2y,s)=\\frac{1}{2}v(m-1)\\text{det}(2y)^{-(m-1)\/2}\n\\end{equation}\n\\begin{Def}\n\\label{DefC}\nDefine a constant $C_{m-1}^{(m)}(y)$ by\n$$\nC_{m-1}^{(m)}(y):=\n\\lim_{s\\to m\/2}\\left(\\xi_{m-1}^{(m)}(2y,s)-\\underset{s=m\/2}{\\text{Res}}\\xi_{m-1}^{(m)}(2y,s)(s-m\/2)^{-1}\\right).\n$$\n\\end{Def}\nThat is, $C_{m-1}^{(m)}(y)$ is the constant term of the Laurent expansion of $\\xi_{m-1}^{(m)}(2y,s)$\nat $s=m\/2$.\n\\begin{Rem}\n(1)\\; It should be noted that the constant $C_{m-1}^{(m)}(y)$ is defined from $\\xi_{m-1}^{(m)}(2y,s)$\nnot $\\xi_{m-1}^{(m)}(y,s)$, and the constant term of $\\xi_{m-1}^{(m)}(2y,2s-m\/2)$ at $s=m\/2$\nis equal to that of $\\xi_{m-1}^{(m)}(2y,s)$.\n\\\\\n(2)\\; In the case $m=2$, the constant $C_1^{(2)}(y)$ is explicitly expressed by the Kronecker\nlimit formula (see $\\S$ \\ref{Degree2}).\n\\end{Rem}\n\\begin{Prop} \n{\\it Explicit forms of $A_{-2}(y)$ and $A_{-1}(y)$ are given as follows:\n\\begin{align}\nA_{-2}^{(m)}(y) &= \\frac{1}{8}\\,v(m-1){\\rm det}(2y)^{-(m-1)\/2}\\alpha_m(y,m\/2),\n\\label{EA-2}\n\\\\\nA_{-1}^{(m)}(y) \n&=\\alpha_m(y,m\/2)\\left(\\frac{1}{2}\\,C_{m-1}^{(m)}(y)+\\frac{\\gamma}{4}\\,v(m-1){\\rm det}(2y)^{-(m-1)\/2}\\right)\n        \\nonumber \\\\\n&\\qquad +\\frac{1}{8}\\,v(m-1){\\rm det}(2y)^{-(m-1)\/2}\\cdot \\alpha'_m(y,m\/2),\n\\label{EA-1}\n\\end{align}\nwhere $\\gamma$ is the Euler constant and \n$\\alpha'_m(y,m\/2)=\\left.\\frac{d}{ds}\\alpha_m(y,s)\\right|_{s=m\/2}$.}\n\\end{Prop}\n\\begin{proof}\nThe formulas are derived from the expression\n\\begin{align*}\n\\zeta(2s-m+1)&\\xi_{m-1}^{(m)}(2y,2s-m\/2) \\\\\n                  &= \\frac{1}{8}\\,v(m-1)\\text{det}(2y)^{-(m-1)\/2}(s-m\/2)^{-2}\\\\\n             &+\\left(\\frac{1}{2}\\,C_{m-1}^{(m)}(y)+\\frac{\\gamma}{4}\\,v(m-1)\\text{det}(2y)^{-(m-1)\/2} \\right)(s-m\/2)^{-1}\\\\\n             &+(\\text{a holomorphic function at}\\; s=m\/2). \n\\end{align*}\n\\end{proof}\n\\noindent\n\\textbf{Analysis of} $\\boldsymbol{F_{0,m,0}^{(m)}(z,s)}:$\n\\vspace{2mm}\n\\\\\nBy definition (\\ref{Fb}),\n\\begin{align}\n\\label{Fm}\n F_{0,m,0}^{(m)}(z,s) \n & =2^{-2ms+m(m+3)\/2}\\pi^{m(m+1)\/2}\\text{det}(y)^{-s+(m+1)\/2}\n\\Gamma_m(2s-\\kappa(m)) \\nonumber \\\\\n& \\cdot \\Gamma_{m}(s)^{-2}\\zeta(2s-m)\\zeta(2s)^{-1}\n         \\cdot\\prod_{j=1}^{m}(\\zeta(4s-m-j)\\zeta(4s-2j)^{-1}).\n\\end{align}\nSimilar to that in case $F_{0,m,-1,0}^{(m)}(z,s)$, we define the function $\\beta_m(y,s)$ as\n$\\alpha_m(y,s)$:\n$$\nF_{0,m,0}^{(m)}(z,s)=\\Gamma(2s-m)\\zeta(4s-2m+1)\\,\\beta_m(y,s).\n$$\nExplicitly,\n\\begin{align}\n\\label{betam}\n\\beta_m(y,s):&=2^{-2ms+m(m+3)\/2}\\pi^{(m^2+2m-1)\/2}\\text{det}(y)^{-s+(m+1)\/2}\\nonumber \\\\\n                            & \\cdot \\Gamma_{m-1}(2s-\\kappa(m))\\Gamma_{m}(s)^{-2}\\zeta(2s)^{-1} \\nonumber \\\\\n                            &\\cdot \\prod_{j=1}^{m-2}\\zeta(4s-m-j)\\cdot\\prod_{j=1}^{m-1}\\zeta(4s-2j)^{-1}.\n\\end{align}\nFunctions $\\Gamma(2s-m)$ and $\\zeta(4s-2m+1)$ have a simple pole at $s=m\/2$, respectively,\nand $\\beta_m(y,s)$ is holomorphic at $s=m\/2$. Consequently, we observe that $F_{0,m,0}^{(m)}(z,s)$,\nhas a double pole at $s=m\/2$, as in the previous case.\n\nWe set\n$$\nF_{0,m,0}^{(m)}(z,s)=\\sum_{l=-2}^\\infty B_l^{(m)}(y)(s-m\/2)^l\n$$\nand calculate $B_{-2}^{(m)}(y)$ and $B_{-1}^{(m)}(y)$.\n\\begin{Prop} \n\\label{B-2-1}\n{\\it The explicit forms of $B_{-2}^{(m)}(y)$ and $B_{-1}^{(m)}(y)$ are as follows:\n\\begin{align}\n& B_{-2}^{(m)}(y) = \\frac{1}{8}\\,\\beta_m(y,m\/2),\n\\label{EB-2}\n\\\\\n& B_{-1}^{(m)}(y) \n=\\frac{\\gamma}{4}\\,\\beta_m(y,m\/2)+\\frac{1}{8}\\,\\beta'_m(y,m\/2)\n\\label{EB-1}\n\\end{align}\nwhere\n$\\beta'_m(y,m\/2)=\\left.\\frac{d}{ds}\\beta_m(y,s)\\right|_{s=m\/2}$.}\n\\end{Prop}\n\\begin{proof}\nFunction $\\Gamma(2s-m)\\zeta(4s-2m+1)$ has the Laurent expansion as\n\\begin{align*}\n &\\Gamma(2s-m)\\zeta(4s-2m+1) \\\\\n & = \\frac{1}{8}\\,(s-m\/2)^{-2}+\\frac{\\gamma}{4}\\,(s-m\/2)^{-1}+(\\text{a holomorphic function at}\\; s=m\/2). \n\\end{align*}\nThe formulas for $B_{-2}^{(m)}(y)$ and $B_{-1}^{(m)}(y)$ are obtained from this expression.\n\\end{proof}\nAn important point is the following relationship between $A_{-2}^{(m)}(y)$ and\\\\\n $B_{-2}^{(m)}(y)$.\n\\begin{Prop}\n{\\it The following identity holds.}\n\\begin{equation}\n\\label{A=-B}\nA_{-2}^{(m)}(y)=-B_{-2}^{(m)}(y).\n\\end{equation}\n\\end{Prop}\n\\begin{proof}\nA direct calculation shows that\n\\begin{align}\nA_{-2}^{(m)}(y) &= 2^{(-m^2+4m-8)\/2}\\pi^{(m^3+3m-2)\/4}\\text{det}(2y)^{1\/2}\\zeta(m)^{-1} \\nonumber \\\\\n        & \\cdot \\prod_{i=2}^{m-1}\\Gamma(i\/2)\\prod_{j=0}^{m-2}\\Gamma((m-j)\/2)^{-2}\\prod_{i=2}^{m-1}\\zeta(i)\n        \\prod_{j=1}^{m-1}\\zeta(2m-2j)^{-1}.\\label{ExplicitA-2}\n\\end{align}\nMeanwhile,\n\\begin{align}\nB_{-2}^{(m)}(y) &= -2^{(-m^2+4m-8)\/2}\\pi^{(m^3+3m)\/4}\\text{det}(2y)^{1\/2}\\zeta(m)^{-1} \\nonumber \\\\\n        & \\cdot \\prod_{j=0}^{m-2}\\Gamma((m-1-j)\/2)\\prod_{j=0}^{m-1}\\Gamma((m-j)\/2)^{-2}\\nonumber \\\\\n        & \\cdot  \\prod_{j=1}^{m-2}\\zeta(m-j)  \\prod_{j=1}^{m-1}\\zeta(2m-2j)^{-1}.\\label{ExplicitB-2}\n\\end{align}\nNoting that\n$$\n\\prod_{j=0}^{m-2}\\Gamma((m-1-j)\/2)=\\Gamma(1\/2)\\cdot\\prod_{i=2}^{m-1}\\Gamma(i\/2),\n$$\nwe conclude that $A_{-2}^{(m)}(y)=-B_{-2}^{(m)}(y)$.\n\\end{proof}\nFrom this proposition, we observe that the singularity of function\n$$\nF_{0,m-1,0}^{(m)}(z,s)+F_{0,m,0}^{(m)}(z,s)\n$$\nat $s=m\/2$ is a simple pole.\n\\begin{Thm}\n{\\it The residue of the constant term is as follows: }\n\\begin{align}\n& \\underset{s=m\/2}{{\\rm Res}}\\sum_{\\nu=0}^{m}F_{0,\\nu,0}^{(m)}(z,s)\n  =\\underset{s=m\/2}{{\\rm Res}}(F_{0,m-1,0}^{(m)}(z,s)+F_{0,m,0}^{(m)}(z,s)) \\nonumber \\\\\n&=\\frac{1}{2}\\,\\alpha_m(y,m\/2)\\cdot C_{m-1}^{(m)}(y)\n    +\\frac{1}{8}\\,v(m-1)\\text{det}(2y)^{-(m-1)\/2}\\alpha'_m(y,m\/2)\\nonumber \\\\\n   &\\quad +\\frac{1}{8\\,}\\beta'_m(y,m\/2).\n    \\label{ResConst}\n\\end{align}\n\\end{Thm}\n\\begin{proof}\nWe have\n\\begin{align*}\n& \\underset{s=m\/2}{{\\rm Res}}(F_{0,m-1,0}^{(m)}(z,s)+F_{0,m,0}^{(m)}(z,s))\n   =A_{-1}^{(m)}(y)+B_{-1}^{(m)}(y)\\\\\n&=\\frac{1}{2}\\,\\alpha_m(y,m\/2)\\cdot C_{m-1}^{(m)}(y)+\\frac{\\gamma}{4}\\,v(m-1)\\text{det}(2y)^{-(m-1)\/2}\\alpha_m(y,m\/2)\\\\\n& \\quad +\\frac{1}{8}\\,v(m-1)\\text{det}(2y)^{-(m-1)\/2}\\alpha'_m(y,m\/2)+\\frac{1}{8}\\,\\beta'_m(y,m\/2)+\n      \\frac{\\gamma}{4}\\,\\beta_m(y,m\/2).\n\\end{align*}\nBy the identity (\\ref{A=-B}), the sum of the second and fifth terms in the last formula is equal to zero.\nThis implies (\\ref{ResConst}).\n\\end{proof}\n\\subsection{Calculation of $\\boldsymbol{F_{0,m,1}^{(m)}(z,s)}$}\n\\label{sec:4-4}\nWe have one more non-holomorphic term, that is, $F_{0,m,1}^{(m)}(z,s)$.\n\\begin{Prop}\n{\\it Function $F_{0,m,1}^{(m)}(z,s)$ has the following expression:}\n\\begin{align}\n& F_{0,m,1}^{(m)}(z,s) \\nonumber \\\\\n&  =2^m\\pi^{m\\kappa(m)}\\text{det}(y)^s\\Gamma_m(s)^{-2}\\zeta(2s)^{-1}\n      \\prod_{j=1}^{m-1}(\\zeta(4s-m-j)\\cdot\\zeta(4s-2j)^{-1}) \\nonumber \\\\\n&\\cdot \\sum_{h\\in\\Lambda_m^{(1)}}\\sigma_{m-2s}(\\text{cont}(h))\\cdot \\eta_m(2y,\\pi h;s,s)\n          \\boldsymbol{e}(\\sigma (hx)),\n\\end{align}\n{\\it where $\\sigma_s(a)=\\sum_{0<d|a}d^s$ and for $0\\ne h\\in\\Lambda_m$, \n${\\rm cont}(h):={\\rm max}\\{l\\in\\mathbb{N}\\,\\mid\\,l^{-1}h\\in\\Lambda_m\\}$.}\n\\end{Prop}\n\\begin{proof}\nBy definition (\\ref{Fb1}),\n\\begin{align}\n& F_{0,m,1}^{(m)}(z,s) \\nonumber \\\\\n& =2^m\\pi^{m\\kappa(m)+(m-1)\/2}\\Gamma_{m-1}(2s-\\kappa(m))\\Gamma_m(s)^{-2}\n\\text{det}(y)^s\\text{det}(2y)^{\\kappa(m)-2s} \\nonumber \\\\\n&\\quad \\cdot\\sum_{0\\ne h\\in\\mathbb{Z}}\\sum_{w\\in\\mathbb{Z}_{\\text{prim}}^{(m,1)}\/\\{\\pm 1\\}}\n                   S_m(\\text{diag}(h,0_{m-1}),2s) \\nonumber \\\\\n& \\qquad\\qquad \\cdot (2y[w])^{2s-m}\\eta_1(2y[w],\\pi h;s-(m-1)\/2,s-(m-1)\/2)\\boldsymbol{e}(\\sigma(h[{}^tw]x).\n\\label{DefF0m0}\n\\end{align}\nIt follows from (\\ref{Siegelreduce}) that\n\\begin{align}\n& S_m(\\text{diag}(h,0_{m-1}),2s)\\nonumber \\\\\n& =\\zeta(2s-m+1)\\zeta(2s)^{-1}\\prod_{j=1}^{m-1}(\\zeta(4s-m-j)\\,\\zeta(4s-2j)^{-1})\n        S_1(h,2s-m+1) \\nonumber \\\\\n        &=\\sigma_{m-2s}(h)\\zeta(2s)^{-1}\\prod_{j=1}^{m-1}(\\zeta(4s-m-j)\\,\\zeta(4s-2j)^{-1}).\n        \\label{SiegelS}\n\\end{align}\nIn the above, we used the formula $S_1(h,s)=\\zeta(s)^{-1}\\sigma_{1-s}(h)$ for $h\\in\\mathbb{Z}_{>0}$.\n\nFrom Proposition \\ref{xi-eta}, function $\\eta_1$ is expressed as\n\\begin{align}\n& \\eta_1(2y[w],\\pi h;s-(m-1)\/2,s-(m-1)\/2) \\nonumber \\\\\n& =\\pi^{-(m-1)\/2}\\Gamma_{m-1}(2s-\\kappa(m))^{-1}\\text{det}(2y)^{2s-\\kappa(m)}(2y[w])^{-2s+m} \\nonumber \\\\\n&\\quad \\cdot \\eta_m(2y,\\pi h[{}^tw],s,s) . \\label{eta1}\n\\end{align}\nSubstituting (\\ref{SiegelS}) and (\\ref{eta1}) into (\\ref{DefF0m0}), we obtain the following expression:\n\\end{proof}\nFrom the above proposition, we obtain the following result:\n\\begin{Thm}\n{\\it Function $F_{0,m,1}^{(m)}(z,s)$ has a simple pole at $s=m\/2$. }\n\\begin{align}\n \\underset{s=m\/2}{{\\rm Res}}F_{0,m,1}^{(m)}(z,s)  &=\n2^{m-2}\\pi^{m\\kappa(m)}\\text{det}(y)^{m\/2}\\Gamma_m(m\/2)^{-1}\\zeta(m)^{-1} \\nonumber \\\\\n& \\cdot \\prod_{j=1}^{m-2}\\zeta(m-j)\\prod_{j=1}^{m-1}\\zeta(2m-2j)^{-1} \\nonumber \\\\\n& \\cdot \\sum_{h\\in\\Lambda_m^{(1)}}\\sigma_0({\\rm cont}(h))\\eta_m(2y,\\pi h;m\/2,m\/2)\n\\boldsymbol{e}(\\sigma(hx)). \\label{Main2}\n\\end{align}\n\\end{Thm}\n\\begin{proof}\nIn the expression of $F_{0,m,1}^{(m)}(z,s)$, only the last factor $\\zeta(4s-2m+1)$ in the product\n$\\prod_{j=1}^{m-1}\\zeta(4s-m-j)$ has a simple pole at $s=m\/2$ with residue $1\/4$.\nFrom this fact, we obtain (\\ref{Main2}).\n\\end{proof}\n\\subsection{Conclusion}\n\\label{sec:4-5}\nWe summarize our results in the previous sections.\n\nThe following is a main result of this study.\n\\begin{Thm}\n\\label{Conclusion}\n\\begin{align*}\n& \\underset{s=m\/2}{{\\rm Res}}E_0^{(m)}(z,s)\\\\\n&=\\mathbb{A}^{(m)}(y)+\\mathbb{B}^{(m)}(y)\n                \\sum_{h\\in\\Lambda_m^{(1)}}\\sigma_0({\\rm cont}(h))\\eta_m(2y,\\pi h;m\/2,m\/2)\n                \\boldsymbol{e}(\\sigma(hx)),\n\\end{align*}\n{\\it where}\n\\begin{align*}\n\\mathbb{A}^{(m)}(y) &=\\frac{1}{2}\\,\\alpha_m(y,m\/2)\\cdot C_{m-1}^{(m)}(y)\n                                                    +\\frac{1}{8}\\,v(m-1)\\text{det}(2y)^{-(m-1)\/2}\\alpha'_m(y,m\/2)\\\\\n                         &+\\frac{1}{8}\\,\\beta'_m(y,m\/2),\\\\\n\\mathbb{B}^{(m)}(y) &=2^{m-2}\\pi^{m\\kappa(m)}\\text{det}(y)^{m\/2}\\Gamma_m(m\/2)^{-1}\\zeta(m)^{-1}\\\\\n                          &\\cdot \\prod_{j=1}^{m-2}\\zeta(m-j)\\,\\prod_{j=1}^{m-1}\\zeta(2m-2j)^{-1}.\n \\end{align*}                                                                          \n\\end{Thm}\n\\section{Remarks}\n\\subsection{Low degree cases}\nIn this section, we provide more explicit formulas for $\\text{Res}_{s=m\/2}E_0^{(m)}(z,s)$.\n$m=2,\\,3$. We used the notations in the previous sections as they are.\n\\subsubsection{Case $\\boldsymbol{m=2}$}\n\\label{Degree2}\nIn this case, the constant $C_1^{(2)}(y)$ appearing in the term $\\mathbb{A}^{(2)}(y)$\ncan be expressed more explicitly because we can apply the Kronecker limit formula.\nFor $g\\in P_2$, we consider the Epstein zeta function\n$$\n\\zeta_g(s):=\\sum_{\\boldsymbol{0}\\ne a\\in\\mathbb{Z}^{(2,1)}\/\\{\\pm 1\\}}g[a]^{-s},\n\\qquad\\quad \\text{Re}(s)>1.\n$$\nThe first Kronecker limit formula asserts that $\\zeta_g(s)$ has the following expression:\n\\begin{align}\n& \\zeta_g(s)=\\frac{1}{2}(4\\text{det}(g))^{-s\/2}\\left[\\frac{2\\pi}{s-1}+4\\pi\\beta(g)+O(s-1) \\right]\n\\label{Epstein}\\\\\n& \\beta(g)=\\gamma+\\frac{1}{2}\\log\\frac{v'}{2\\sqrt{\\text{det}(g)}}-\\log|\\eta(W_g)|^2.\n\\end{align}\nHere, for $g=\\begin{pmatrix} v' & w \\\\ w & v \\end{pmatrix}\\in P_2$,\n$$\nW_g:=\\frac{w+i\\sqrt{\\text{det}(g)}}{v'}\\in H_1,\n$$\nand $\\eta(z)$ is the Dedekind eta function:\n$$\n\\eta(z)=\\boldsymbol{e}(z\/24)\\prod_{n=1}^\\infty (1-\\boldsymbol{e}^n(z)),\\qquad z\\in H_1.\n$$\nThe relationship between the complete zeta function $\\xi_1^{(2)}(g,s)$ and $\\zeta_g(s)$\nis expressed as follows:\n$$\n\\xi_1^{(2)}(g,s)=\\xi(2s)\\zeta_1^{(2)}(g,s)=\\pi^{-s}\\Gamma(s)\\zeta_g(s).\n$$\nTherefore, the constant $C_1^{(2)}(y)$, which is the constant term of $\\xi_1^{(2)}(2y,s)$, \ncan be expressed as\n$$\nC_1^{(2)}(y)=\\frac{1}{2}(4\\text{det}(y))^{-1\/2}\\left(\\gamma+\\log\\frac{v'}{8\\pi}-\\log(\\text{det}(y))\n                -2\\log|\\eta(W_g)|^2\\right).\n$$\nConcerning $\\alpha_m(y,m\/2)$, $\\alpha'_m(y,m\/2)$, $\\cdots$ appearing in $\\mathbb{A}^{(m)}(y)$, \nwe can calculate them explicitly as\n\\begin{align*}\n\\alpha_2(y,1) &=72\\pi^{-2}\\text{det}(y)\\,\\zeta(2)^{-2},\\\\\n\\alpha'_2(y,1) &=2\\pi^2\\text{det}(y)\\,\\zeta(2)^{-2}(2\\log\\pi+\\log(\\text{det}(y))+2\\gamma-6\\zeta'(2)\\,\\zeta(2)^{-1}),\\\\\n\\beta_2(y,1) &=-\\pi^2\\text{det}(y)^{1\/2}\\,\\zeta(2)^{-2},\\\\\n\\beta'_2(y,1) &= 2\\pi^2\\text{det}^{1\/2}\\,\\zeta(2)^{-2}\n                     \\left(2\\log 2+\\tfrac{1}{2}\\log(\\text{det}(y))-\\gamma+2\\zeta'(0)+3\\zeta'(2)\\zeta(2)^{-1}\\right).\n\\end{align*}\nFrom these formula,\n\\begin{align*}\n& \\mathbb{A}^{(2)}(y)=\\underset{s=1}{{\\rm Res}}(F_{0,1,0}^{(2)}(z,s)+F_{0,2,0}^{(2)}(z,s))\\\\\n& =\\frac{1}{2}\\,\\alpha_2(y,1)\\cdot C_1^{(2)}(y)+\\frac{1}{8}\\,v(1)\\cdot\\text{det}(y)^{-1\/2}\\alpha'_2(y,1)+\\frac{1}{8}\\,\\beta'_2(y,1)\\\\\n& =\\frac{18}{\\pi^2\\sqrt{\\text{det}(y)}}\\left(\\frac{1}{2}\\gamma+\\frac{1}{2}\\log\\frac{v'}{4\\pi}-\\log|\\eta(W_g)|^2 \\right).\n\\end{align*}\nCombining with $\\text{Res}_{s=1}F_{0,2,1}^{(3)}(z,s)$, we obtain\n\\begin{Prop}\n\\begin{align}\n \\underset{s=1}{{\\rm Res}}\\,E_0^{(2)}(z,s) &\n=\\frac{18}{\\pi^2\\sqrt{\\text{det}(y)}}\\left(\\frac{1}{2}\\gamma+\\frac{1}{2}\\log\\frac{v'}{4\\pi}-\\log|\\eta(W_g)|^2 \\right)\\nonumber\n\\\\\n& \\quad +\\frac{36\\text{det}(y)}{\\pi^2}\\sum_{h\\in\\Lambda_2^{(1)}}\\sigma_0(\\text{cont}(h))\\eta_2(2y,\\pi h;1,1)\n             \\boldsymbol{e}(\\sigma(hx)). \\label{Degrre2Main}\n\\end{align}\n\\end{Prop}\n\\begin{Rem}\nIn \\cite{Na}, the author provided a formula for $E_2^{(2)}(z,0)$ (Siegel Eisenstein series of degree 2 and weight 2):\n\\begin{align}\nE_2^{(2)}(z,0)=& 1-\\frac{18}{\\pi^2\\sqrt{\\text{det}(y)}}\\left(1+\\frac{1}{2}\\gamma+\\frac{1}{2}\\log\\frac{v'}{4\\pi}-\\log|\\eta(W_g)|^2 \\right) \\nonumber\\\\\n         &-\\frac{72}{\\pi^3}\\sum_{\\substack{0\\ne h\\in\\Lambda_2\\\\ \\text{discr}(h)=\\square}}\\varepsilon_h\n           \\sigma_0(\\text{cont}(h))\\eta_2(2y,\\pi h;2,0)\\boldsymbol{e}(\\sigma(hx))\\nonumber \\\\\n         &+288\\sum_{0\\ne h\\in \\Lambda_2}\\sum_{d\\mid\\text{cont}(h)}d\\,H\\left(\\frac{|\\text{discr}(h)|}{d^2}\\right)\n              \\boldsymbol{e}(\\sigma(hz)).\\label{Degree2Weight2}\n\\end{align}\nHere, $H(N)$ is the Kronecker--Hurwitz class number and $\\varepsilon_h=1\/2$ if $\\text{rank}(h)=1;$\n$=1$ if $\\text{rank}(h)=2$.\n\nIt is interesting that the same term appears in each Fourier coefficient in (\\ref{Degrre2Main}) and\n(\\ref{Degree2Weight2}).\n\\end{Rem}\n\\subsubsection{Case $\\boldsymbol{m=3}$}\n\\label{Degree3}\nFrom Theorem \\ref{Conclusion}, we can write \n\\begin{align*}\n& \\underset{s=3\/2}{{\\rm Res}}E_0^{(3)}(z,s)\\\\\n&=\\mathbb{A}^{(3)}(y)+\\mathbb{B}^{(3)}(y)\n                \\sum_{h\\in\\Lambda_3^{(1)}}\\sigma_0({\\rm cont}(h))\\eta_3(2y,\\pi h;3\/2,3\/2)\n                \\boldsymbol{e}(\\sigma(hx)).\n\\end{align*}\nThe quantities $\\mathbb{A}^{(3)}(y)$ and $\\mathbb{B}^{(3)}(y)$ are given as follows:\n\\begin{align*}\n& \\mathbb{A}^{(3)}(y) \\\\\n& = 2^3\\pi^4\\text{det}(y)^{3\/2}\\zeta(2)^{-1}\\zeta(3)^{-1}\\zeta(4)^{-1}\\cdot C_2^{(3)}(y)\n                         +2^{-2}\\pi^3\\text{det}(y)^{1\/2}\\zeta(3)^{-1}\\zeta(4)^{-1}\n\\\\\n& \\quad\\cdot (-2\\Gamma'(1)-4\\zeta'(2)\\zeta(2)^{-1}+4\\zeta'(0)+2\\log(\\text{det}(y))+4\\log\\pi+6\\log 2),\n\\vspace{2mm}\n\\\\\n& \\mathbb{B}^{(3)}(y)\\\\ \n& = 2^2\\pi^{7\/2}\\text{det}(y)^{3\/2}\\zeta(3)^{-1}\\zeta(4)^{-1}.\n\\end{align*}\n\\begin{Rem}\nIn the above formulas, we may substitute\n$$\n\\zeta(2)=\\pi^2\/6,\\quad \\zeta(4)=\\pi^4\/90,\\quad \\zeta'(0)=(-\\log 2\\pi)\/2,\\quad \\Gamma'(1)=-\\gamma.\n$$\n\\end{Rem}\n\\subsection{Residue at the other point}\n\\label{other}\nThe residue we considered above was to $s=m\/2$, and it is represented as a Fourier series.\nThe case $\\text{Res}_{s=(m+1)\/2}E_0^{(m)}(z,s)$ is easier than in the above case. In fact,\nit becomes a constant, explicitly\n\\begin{align*}\n& \\underset{s=(m+1)\/2}{{\\rm Res}}E_0^{(m)}(z,s)\\\\\n& =\\underset{s=(m+1)\/2}{{\\rm Res}}\\xi(2s-m)\\xi(2s)^{-1}\\prod_{j=1}^{[m\/2]}(\\xi(4s-2m-1+2j)\\,\\xi(4s-2j)^{-1}).\n\\end{align*}\n\\begin{Rem}\nKaufhold \\cite{Kau} noted that the residue of\n$$\n\\varPhi_0(s):=E_0^{(2)}(z,s\/2)\n$$\nat $s=3$ is $90\\pi^{-2}$. This is a special case of the above formula because\n$$\n \\underset{s=3\/2}{{\\rm Res}}E_0^{(2)}(z,s)= \\underset{s=3\/2}{{\\rm Res}}\\xi(2s-2)\\xi(2s)^{-1}\\xi(4s-3)\\xi(4s-2)^{-1}\n=\\frac{45}{\\pi^2}.\n$$\n\\end{Rem}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n{\\sc Gist}\\\/ (Game solver from IST) is a tool for (a)~qualitative analysis of \\emph{turn-based probabilistic\ngames ($2\\slopefrac{1}{2}$-player games)} with $\\omega$-regular objectives, and \n(b)~computing environment assumptions for synthesis of unrealizable \nspecifications. \nThe class of $2\\slopefrac{1}{2}$-player games arise in several important applications \nrelated to verification and synthesis of reactive systems. Some key \napplications are: (a) synthesis of stochastic reactive systems; \n(b) verification of probabilistic systems; and (c) synthesis of unrealizable\nspecifications. We believe that our tool will be useful for the above\napplications.\n\n\n\\smallskip\\noindent{\\bf $2\\slopefrac{1}{2}$-player games.} \n$2\\slopefrac{1}{2}$-player games are played on a graph by two players along with \nprobabilistic transitions.\nWe consider $\\omega$-regular objectives over infinite paths specified by \nparity, Rabin and Streett (strong fairness) conditions that can express \nall $\\omega$-regular properties such as safety, reachability, liveness, \nfairness, and most properties commonly used in verification. \nGiven a game and an objective, our tool determines\nwhether the first player has a strategy to ensure that the objective \nis satisfied with probability~1, and if so, it constructs \nsuch a witness strategy. Our tool provides the first implementation of \nqualitative analysis (probability~1 winning) of $2\\slopefrac{1}{2}$-player games \nwith $\\omega$-regular objectives.\n\n\\smallskip\\noindent{\\bf Synthesis of environment assumptions.}\nThe synthesis problem asks to construct a finite-state reactive \nsystem from an $\\omega$-regular specification. In practice, initial specifications \nare often unrealizable, which means that there is no system that implements \nthe specification. A common reason for unrealizability is that assumptions \non the environment of the system are incomplete. The problem of \ncorrecting an unrealizable specification $\\Psi$ by computing an environment \nassumption $\\Phi$ such that the new specification $\\Phi \\to \\Psi$ \nis realizable was studied in~\\cite{CHJ08}.\nThe work~\\cite{CHJ08} constructs an assumption $\\Phi$ that constrains only \nthe environment and is as weak as possible. \nOur tool implements the algorithms of~\\cite{CHJ08}.\nWe believe our implementation will be useful in analysis \nof realizability of specifications and computation of \nassumptions for unrealizable specifications.\n\n\n\n\\section{Definitions}\\label{section:definition}\nWe first present the basic definitions of games and objectives.\n\n\\smallskip\\noindent{\\bf Game graphs.} \nA \\emph{turn-based probabilistic game graph} (\\emph{$2\\slopefrac{1}{2}$-player game graph})\n$G =((S, E), (S_0,S_1,S_{P}),\\delta)$ consists of a directed graph \n$(S,E)$, a partition $(S_0$, $S_1$,$S_{P})$ of the finite set $S$ of states, \nand a probabilistic transition function $\\delta$: $S_{P} \\rightarrow {\\cal D}(S)$, \nwhere ${\\cal D}(S)$ denotes the set of probability distributions over the \nstate space~$S$. \nThe states in $S_0$ are the {\\em player-$0$\\\/} states, where player~$0$\ndecides the successor state; the states in $S_1$ are the {\\em \nplayer-$1$\\\/} states, where player~$1$ decides the successor state; \nand the states in $S_{P}$ are the {\\em probabilistic\\\/} states, where\nthe successor state is chosen according to the probabilistic transition\nfunction~$\\delta$. \nWe assume that for $s \\in S_{P}$ and $t \\in S$, we have $(s,t) \\in E$ \niff $\\delta(s)(t) > 0$.\nThe {\\em turn-based deterministic game graphs} (\\emph{2-player game graphs})\nare the special case of the $2\\slopefrac{1}{2}$-player game graphs with $S_{P} = \\emptyset$.\n\n\\smallskip\\noindent{\\bf Objectives.} We consider the three canonical\nforms of $\\omega$-regular objectives: Streett and its dual Rabin\nobjectives; and parity objectives.\nThe Streett objective consists of $d$ request-response pairs\n$\\set{(Q_1,R_1),(Q_2,R_2),\\ldots, (Q_d,R_d)}$ where $Q_i$ denotes a\nrequest and\n$R_i$ denotes the corresponding response (each $Q_i$ and $R_i$ are subsets of \nthe state space). The objective requires that if a request $Q_i$ happens \ninfinitely often, then the corresponding response must happen infinitely often.\nThe Rabin objective is its dual. \nThe parity (or Rabin-chain objective) is the special case of Streett objectives\nwhen the set of request-responses \n$Q_1 \\subset R_1 \\subset Q_2 \\subset R_2 \\subset  Q_3 \\subset \\cdots \\subset \nQ_d \\subset R_d$ form a chain.\n\n\\smallskip\\noindent{\\bf Qualitative analysis.} The qualitative analysis for \n$2\\slopefrac{1}{2}$-player games is as follows: the input is a \n$2\\slopefrac{1}{2}$-player game graph, and an objective $\\Phi$ (Streett, Rabin or parity \nobjective), and the output is the set of states such that player~0 \ncan ensure $\\Phi$ with probability~1.\nFor detailed description of game graphs, plays, strategies, objectives and \nnotion of winning see~\\cite{KrishThesis}.\nWe focus on qualitative analysis because:\na)~In applications like synthesis the relevant analysis is qualitative\nanalysis: the goal is to synthesize a system that behaves correctly\nwith probability~1; (b)~Qualitative analysis for probabilistic games is independent of \nthe precise probabilities, and thus robust with imprecision in the \nexact probabilities (hence resilient to probabilistic modeling errors). \nThe qualitative analysis can be done with discrete graph theoretic \nalgorithms.\nThus qualitative analysis is more robust and efficient, and our \ntools implements these efficient algorithms.\n\n\\section{Tool Implementation} \nOur tool presents a solution of the following two problems.\n\n\\smallskip\\noindent{\\bf Qualitative analysis of $2\\slopefrac{1}{2}$-player games.} \nOur tool presents the first implementation for the qualitative \nanalysis of $2\\slopefrac{1}{2}$-player games with Streett, Rabin and parity objectives.\nWe have implemented the linear-time reduction for qualitative analysis of \n$2\\slopefrac{1}{2}$-player Rabin and Streett games to $2$-player Rabin and Streett games \nof~\\cite{CdAH05}, and the linear-time reduction for $2\\slopefrac{1}{2}$-player \nparity games to $2$-player parity games of~\\cite{CJH04}.\nThe $2$-player Rabin and Streett games are solved by reducing them to the\n$2$-player parity games using the LAR (latest appearance records) \nconstruction~\\cite{GH82}. The $2$-player parity games are solved using the\ntool PGSolver~\\cite{Lange09}. \n\n\\smallskip\\noindent{\\bf Environment assumptions for synthesis.} \nOur tool implements a two-step algorithm for computing the environment assumptions\nas presented in~\\cite{CHJ08}. \nThe algorithm operates on the game graph that is used to answer the \nrealizability question. \nFirst, a safety assumption that removes a minimal set of \nenvironment edges from the graph is computed. \nSecond, a fairness assumption that puts fairness conditions on some of the\nremaining environment edges is computed. \nThe problem of finding a minimal set of fair edges is \ncomputationally hard~\\cite{CHJ08}, and a reduction to $2\\slopefrac{1}{2}$-player games \nwas presented in~\\cite{CHJ08} to compute a locally minimal fairness assumption.\nThe details of the implementation are as follows: given an LTL formula $\\phi$,\nthe conversion to an equivalent deterministic\nparity automaton is achieved through GOAL~\\cite{Goal}. Our tool then converts\nthe parity automaton into a $2$-player parity game by splitting the states and\ntransitions based on input and output symbols. Our tool then computes the safety\nassumption by solving a safety model-checking problem. \nThe computation of the fairness assumption is achieved in the following steps:\n\\begin{compactitem}\n\\item Convert the parity game with fairness assumption\ninto a $2\\slopefrac{1}{2}$-player game.\n\\item Solve the $2 \\slopefrac{1}{2}$-player game (using our tool) to check whether the\nassumption is sufficient (if so, go to the previous step with a weaker fairness assumption).\n\\end{compactitem}\nThe synthesized system is obtained from a witness strategy of the parity\ngame. \nThe flow is illustrated in Figure~\\ref{fig:example}.\n\n\\begin{figure}\n\\begin{center}\n\\begin{pspicture}[showgrid=false](-1.5,0)(10,2.7)\n\\begin{psmatrix}[rowsep=1.0,colsep=1.5]\n\\psframebox[linearc=0.2,cornersize=absolute,framesep=2pt]{LTL Formula} & \n\\psframebox[linearc=0.2,cornersize=absolute,framesep=2pt]{\\tabular{c} Det. Parity \\\\ Aut. \\endtabular} & \n\\psframebox[linearc=0.2,cornersize=absolute,framesep=2pt]{\\tabular{c} Synthesis \\\\ Game \\endtabular} \\\\\n\\psframebox[linearc=0.2,cornersize=absolute,framesep=2pt]{Synthesized System}  &\n\\psframebox[linearc=0.2,cornersize=absolute,framesep=2pt]{\\tabular{c} $2\\slopefrac{1}{2}$-player \\\\ game \\endtabular} &\n\\psframebox[linearc=0.2,cornersize=absolute,framesep=2pt]{\\tabular{c} Safe \\\\ Synthesis \\\\ Game \\endtabular} \\\\\n\\end{psmatrix}\n\\ncline[linestyle=dashed]{->}{1,1}{1,2}\\naput{GOAL} \\ncline{->}{1,2}{1,3}\n\\ncline{->}{1,3}{2,3} \\ncline{->}{2,3}{2,2} \\ncline{->}{2,2}{2,1}\n\\ncloop[angleA=180,angleB=-270,loopsize=-1,linestyle=dashed]{->}{2,2}{2,2}\\naput{\\tiny{Assumption not locally minimal}}\n\\end{pspicture}\n\\end{center}\n\\caption{An example illustrating the flow of the tool}\n\\label{fig:example}\n\\end{figure}\n\n\\begin{figure}[h]\n\\subfigure[Deterministic Parity Automaton]{\n\\begin{picture}(35,30)(0,0)\n\t\\node[Nmarks=i,Nw=5,Nh=5,Nmr=2.5](0)(5,15){0}\n\t\\node[Nw=5,Nh=5,Nmr=2.5](1)(20,25){1}\n\t\\node[Nw=5,Nh=5,Nmr=2.5,Nmarks=r](2)(20,5){2}\n\t\\drawloop[loopdiam=2,ELdist=0.5](0){$\\neg g$}\n\t\\drawedge[curvedepth=1,ELdist=0.5](0,1){$c \\wedge g$}\n\t\\drawedge[curvedepth=1,ELdist=0](0,2){$\\neg c \\wedge g$}\n\t\\drawloop[loopdiam=2,ELdist=0.5,loopangle=0](1){$T$}\n\t\\drawedge[curvedepth=1,ELdist=0.5](2,0){$\\neg g$}\n\t\\drawedge[curvedepth=-2,ELdist=0.5,ELside=r](2,1){$c \\wedge g$}\n\t\\drawloop[loopdiam=2,ELdist=0.5,loopangle=0](2){$\\neg c \\wedge g$}\n\\end{picture}\n\\label{fig:parity_automaton}\n}\n\\subfigure[$2$-player Parity Game]{\n\\begin{picture}(40,30)(0,0)\n\t\\rpnode[Nmarks=i,fangle=45](0)(5,15)(4,3){0}\n\t\\rpnode[fangle=45,Nmarks=r](1)(25,5)(4,3){1}\n\t\\rpnode[fangle=45](2)(25,25)(4,3){2}\n\t\\rpnode[polyangle=45](3)(15,25)(4,3){3}\n\t\\rpnode[polyangle=45](4)(15,5)(4,3){4}\n\t\\rpnode[polyangle=45](5)(35,25)(4,3){5}\n\n\t\\drawedge[curvedepth=1,ELdist=0.5](0,3){$c$}\n\t\\drawedge[curvedepth=1,ELdist=0.5](3,0){$\\neg g$}\n\t\\drawedge[curvedepth=1,ELdist=0.5](0,4){$\\neg c$}\n\t\\drawedge[curvedepth=1,ELdist=0.5](4,0){$\\neg g$}\n\t\\drawedge(3,2){$g$}\n\t\\drawedge(1,3){$c$}\n\t\\drawedge[curvedepth=1,ELdist=0.5](4,1){$g$}\n\t\\drawedge[curvedepth=1,ELdist=0.5](1,4){$\\neg c$}\n\t\\drawedge[curvedepth=1,ELdist=0.5](5,2){$\\mbox{T}$}\n\t\\drawedge[curvedepth=1,ELdist=0.5](2,5){$\\mbox{T}$}\n\\end{picture}\n\\label{fig:synthesis_game}\n}\n\\subfigure[A $2\\frac{1}{2}$-player game obtained for the fairness\n\tassumption which contains only $(0,4)$]{\n\\begin{picture}(50,30)(0,0)\n\t\\rpnode[Nmarks=i,fangle=45](0)(5,25)(4,3){0}\n\t\\node[Nw=5,Nh=5,Nmr=2.5](0d)(5,5){$\\overline{0}$}\n\t\\rpnode[fangle=45,Nmarks=r](1)(25,5)(4,3){1}\n\t\\rpnode[fangle=45](2)(25,25)(4,3){2}\n\t\\rpnode[polyangle=45](3)(15,25)(4,3){3}\n\t\\rpnode[polyangle=45](4)(15,5)(4,3){4}\n\t\\rpnode[polyangle=45](5)(35,25)(4,3){5}\n\n\t\\drawedge(0,3){}\n\t\\drawedge[curvedepth=1,ELdist=0.5](0d,4){}\n\t\\drawedge(0d,0){}\n\t\\drawedge(3,0d){}\n\t\\drawedge(0,4){}\n\t\\drawedge[curvedepth=1,ELdist=0.5](4,0d){}\n\t\\drawedge(3,2){}\n\t\\drawedge(1,3){}\n\t\\drawedge[curvedepth=1,ELdist=0.5](4,1){}\n\t\\drawedge[curvedepth=1,ELdist=0.5](1,4){}\n\t\\drawedge[curvedepth=1,ELdist=0.5](5,2){}\n\t\\drawedge[curvedepth=1,ELdist=0.5](2,5){}\n\\end{picture}\n\\label{fig:probabilistic_game}\n}\\hfill\n\\subfigure[Environment Assumption]{\n\\begin{picture}(40,30)(0,0)\n\t\\node[Nmarks=i,Nw=5,Nh=5,Nmr=2.5](0)(5,15){0}\n\t\\node[Nw=5,Nh=5,Nmr=2.5,Nmarks=r](1)(20,25){1}\n\t\\node[Nw=5,Nh=5,Nmr=2.5,Nmarks=r](2)(20,5){2}\n\t\\drawloop[loopdiam=2,ELdist=0.5](0){$c \\wedge \\neg g$}\n\t\\drawedge[curvedepth=1,ELdist=0.5,ELside=r](0,1){$c \\wedge g$}\n\t\\drawedge[curvedepth=1,ELdist=0](0,2){$\\neg c$}\n\t\\drawloop[loopdiam=2,ELdist=0.5,loopangle=0](1){$T$}\n\t\\drawedge[curvedepth=1,ELdist=0.5](2,0){$c \\wedge \\neg g$}\n\t\\drawedge[curvedepth=-2,ELdist=0.5,ELside=r](2,1){$c \\wedge g$}\n\t\\drawloop[loopdiam=2,ELdist=0.5,loopangle=0](2){$\\neg c$}\n\\end{picture}\n\\label{fig:environment_assumption}\n}\\hfill\n\\subfigure[Transducer System]{\n\\begin{picture}(20,30)(0,0)\n\t\\node[Nmarks=i,Nw=5,Nh=5,Nmr=2.5](0)(10,15){0}\n\t\\drawloop[loopdiam=2.5,ELdist=0.5](0){$\\neg c \/ g$}\n\t\\drawloop[loopdiam=2.5,ELdist=0.5,loopangle=0](0){$c \/ \\neg g$}\n\\end{picture}\n\\label{fig:transducer_system}\n}\n\\caption{An example that illustrates the tool flow}\n\\label{fig:worked_example}\n\\end{figure}\n\nWe illustrate the working of our tool on a simple example shown in\nFigure~\\ref{fig:worked_example}\nConsider an LTL formula $\\Phi=GF \\mathtt{grant} \\wedge G(\\mathtt{cancel} \\to \\neg \n\\mathtt{grant})$, where $G$ and $F$ denote globally and eventually, respectively.\nThe propositions \\texttt{grant} and \\texttt{cancel} are abbreviated as \\texttt{g} \nand \\texttt{c}, respectively. \nFrom $\\Phi$ our tool obtains a deterministic parity automaton (Figure~\\ref{fig:parity_automaton})\nthat accepts exactly the words that satisfies $\\Phi$. \nThe parity automaton is then converted into a parity game. In Figure~\\ref{fig:synthesis_game},\n$\\Box$ represents player~0 states and $\\Diamond$ represents player~1 states. \nIt can be shown that in this game no safety assumption required.\nWe illustrate how to compute a locally minimal fairness assumption. \nGiven an fairness assumption on edges, our tool reduces the game with the assumption to a $2\\slopefrac{1}{2}$-player parity game\n(see details in~\\cite{CHJ08}). If the initial state in the $2\\slopefrac{1}{2}$-player game is in winning with probability~1 \nfor player~0, then the assumption is sufficient. Figure~\\ref{fig:probabilistic_game} illustrates \nthe $2\\slopefrac{1}{2}$-player game obtained with the fairness assumption on the edge\n$(0,4)$. The $\\bigcirc$ state is \nthe probabilistic state with uniform distribution over its successors.\nThe assumption on this edge is the minimal fairness assumption for the\nexample. Our tool then converts this game back into an automaton to\nobtain the environment assumption as an\nautomaton(Figure~\\ref{fig:environment_assumption}). This assumption\nis equivalent to the formula $G(\\neg (\\mathtt{cancel \\wedge grant})) \\implies GF(\\mathtt{\\neg cancel})$. From a witness\nstrategy in Figure~\\ref{fig:probabilistic_game}\nour tool obtains the system that implements the specification with the\nassumption (Figure~\\ref{fig:transducer_system}).\n\n\\smallskip\\noindent{\\bf Performance of {\\sc Gist}.} Our implementation of reduction \nof $2\\slopefrac{1}{2}$-player games to $2$-player games is linear time and efficient, and\nthe computationally expensive step is solving $2$-player games. \nFor qualitative analysis of $2\\slopefrac{1}{2}$-player games, {\\sc Gist}\\ can handle game \ngraphs of size that can be typically handled by tools solving $2$-player games.\nTypical run-times for qualitative analysis of $2\\slopefrac{1}{2}$-player parity\ngames of various sizes are summarized in Table~\\ref{table:runtimes}. The\ngames used were generated using the benchmark tools of PGSolver and then\nconverting one-tenth of the states into probabilistic states. \n\n\\begin{table}[ht]\n\\begin{center}\n\\label{table:runtimes}\n\\begin{tabular}{|c|c||c|c|c|}\n\\hline\nStates & Edges & \\multicolumn{3}{|c|}{Runtime (sec.)} \\\\\n\\cline{3-5}\n&& Avg. & Best & Worst \\\\\n\\hline\n1000 & 5000 & 1.17 & 0.63 & 1.59 \\\\\n5000 & 25000 & 15.94 & 11.10 & 19.46 \\\\\n10000 & 50000 & 51.43 & 39.38 & 62.61 \\\\\n20000 & 100000 & 282.24 & 267.40 & 310.11 \\\\\n50000 & 250000 & 2513.18 & 2063.40 & 2711.23 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Runtimes for solving $2\\slopefrac{1}{2}$-player parity games}\n\\end{table}\nIn the case of synthesis of environment assumptions, the expensive step is the\nreduction of LTL formula to deterministic parity automata. Our tool can \nhandle formulas that are handled by classical tools for translation of LTL \nformula to deterministic parity automata.\n\n\n\n\n\\smallskip\\noindent{\\bf Other features of {\\sc Gist}.}\nOur tool is compatible with several other game solving and synthesis tools: \n{\\sc Gist}\\\/  is compatible with PGSolver and GOAL. Our tool provides a graphical \ninterface to describe games and automata, and thus can also be used as a \nfront-end to PGSolver to graphically describe games.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nCosmological observations, and in particular Cosmic Microwave\nBackground (CMB) measurements \\cite{COBE,CMB,WMAP}, are\nconsistent with a nearly gaussian and practically scale invariant\nspectrum of primordial perturbations, as predicted by the inflationary\nmodels \\cite{inflation, Linde:1983gd}. An early period of inflation also\naccounts for the inferred flatness of the Universe, and provides a\nsolution to the horizon problem. This makes inflation a robust\ncandidate to account for the early evolution of our\nuniverse. Inflationary predictions are characterized by the spectral\nindex of the primordial spectrum, its tensor contribution, and the\nlevel of non-gaussianity. Present CMB data however sets at\nmost an upper limit on the level of the tensor contribution and\nnon-gaussianity \\cite{WMAP}, and it is not yet able\nto discriminate among the different implementations and models of \ninflation. This situation is expected to change with the next\ngeneration of CMB experiments, like ESA's Planck surveyor satellite\n\\cite{planck}, which will further improve our knowledge of the cosmological\nparameters.      \n\nInflation in brief is no more than an early period of accelerated\nexpansion. In the standard picture of inflation, denoted as cold\ninflation, the universe rapidly \nsupercools, and inflation should be followed by a  \nreheating period, during which the inflationary vacuum energy is\nconverted into radiation. For reheating to take place, the inflaton field has to\ncouple to other degree of freedom, such that it can decay into light,\nrelativistic degrees of freedom that thermalize and provide the\nradiation bath \\cite{reheating}. During cold inflation one assumes that those couplings\nplay no role during the accelerated expansion. The alternative, called\nwarm inflation \\cite{bf1,wi}\n(for earlier related work see \\cite{prewarm}), assumes on the contrary that\nthose coupling can lead to non-negligible dissipative effects, and radiation\nproduction can occur concurrently with the inflationary expansion. \nBoth background evolution and inflaton fluctuations are modified with\nthe inclusion of an extra friction term $\\Upsilon \\dot \\phi$\naccounting for the transfer of energy between the inflation and the radiation. \nThe dynamics of the fluctuations are now governed by a\nLangevin equation including  a noise force term from the influence\nof the radiation fluctuations into the inflaton field \n\\cite{bf1,langevin,warmdeltap,BR1,langevin2}. Thermal fluctuations in\nthe radiation are transfered to the inflaton and become the main\nsource of primordial fluctuations \\cite{bf1,warmdeltap,warmpert}.  \n\nThe dissipative coefficient can be computed from first principles in\nquantum field theory, within an adiabatic approximation. The two-stage\ninteraction configuration proposed in \\cite{BR1} has been\nshown to lead to a large enough dissipative coefficient while keeping\nthe corrections to the inflationary potential under control and\nallowing a period of slow-roll inflation\n\\cite{br05,Hall:2004zr,Moss:2008yb}. \nFor the two-stage mechanism, the inflaton field \ncouples to a heavy catalyst field, and the latter in\nturn couples to light degrees of freedom. During the motion of the\nbackground inflaton, it excites the catalyst fields which then decay\ninto light fields \\cite{Berera:2008ar}. \nUsing this mechanism, the first calculation of the\ndissipative coefficient within the close-to-equilibrium approximation\nwas done in \\cite{mossxiong}, leading to a temperature dependent\ndissipative coefficient. In the low-temperature regime, when the mass\nof the heavy catalyst field is much larger than $T$, one has $\\Upsilon \\propto\nT^3$, while in the high-temperature regime,\nwhen $T$ is above the heavy catalyst field\nmass, depending on type of interaction the  dissipative coefficient\nbecomes linear with $T$ \\cite{BasteroGil:2010pb}\nor goes goes like the inverse of $T$ \\cite{BGR}. All high-$T$ models\nsuffer in general from very large thermal corrections which spoil the\nflatness of the potential \\cite{BGR,Yokoyama:1998ju},\nwith only a few exceptions \\cite{warmdeltap,Berera:1998px}.\nHowever, viable\nmodels of warm inflation  have been studied in the low-$T$ regime \n\\cite{BasteroGil:2009ec,Zhang:2009ge,joao}.  \n\nThe temperature dependence of the dissipative coefficient induces the\ncoupling of the field and radiation fluctuations as shown in\n\\cite{mossgraham}. Previous studies of the primordial spectrum of\nperturbations in warm inflation \\cite{warmpert} did take into account\nthe influence of the thermal fluctuations on the field through the\nnoise term, but  not the coupling through the dissipative term\nitself. In \\cite{mossgraham} it was shown that for positive power of $T$\nin $\\Upsilon$, and when $\\Upsilon$ dominates over the Hubble expansion\nrate, this coupling induces a growing mode in the fluctuations before\nhorizon crossing, enhancing by several order of magnitude the\namplitude of the primordial perturbations with respect to previous\ncalculations. In the calculations of the primordial spectrum\ntypically the radiation bath is treated as a perfect fluid, with an\nequilibrium pressure $p_r \\simeq \\rho_r\/3$, where $\\rho_r$ is the radiation\nenergy density, an approximation valid in the close-to-equilibrium\nregime required for the calculation of the dissipative coefficient to\nhold. However, even in that regime,  the radiation bath is expected to\ndepart from  an ideal fluid as a consequence of the constant\nproduction of radiation particles from  the background field\ndissipation.  Imperfect fluids have dissipative\neffects that can be parameterized in terms of shear and bulk viscosity\ncoefficients, and a heat flow coefficient.  Heat flow happens as a \nconsequence of changes of conserved charges other than the\nstress-energy tensor, but we do not consider this possibility here and\nfocus on the effects of the temperature. Bulk viscous effects, which\ncan be interpreted as a  consequence of the decay of particles within\nthe fluid, have been considered for warm inflation in\n\\cite{delCampo:2010by}, where \nthey studied either a constant bulk viscous pressure or one proportional\nto the radiation energy density.  The bulk\npressure appears at both the background and the perturbation level,\nand being a negative pressure, it will favor warm inflation. For the\namplitude of the spectrum, for the phenomenological model considered\nin \\cite{delCampo:2010by} they found that it could induce a variation in\nthe amplitude of the order of 4\\%. On the other hand, the shear\nviscosity is related to changes in momentum of the  particles of the\nfluid, \nand appears only at the level of the perturbations.     \nShear and bulk viscosity coefficients due to light field have been extensively\ncompute in the literature \\cite{jeon,shear}, leading to power-law\ndependences on $T$ for these coefficients. In addition, the bulk\nviscous coefficient typically ends being much smaller than the shear\nviscosity. Therefore, we will concentrate on the effect of the shear\nviscosity on the spectrum, and do not consider those of the bulk\nviscosity. The shear viscosity, being related to dissipation, appears\nin the radiation fluid equation as a friction term which tends to damp\nthe growth of the fluctuations \\cite{mossgraham}. Eventually, the\ndamping effect will dominate \nover  the enhancement induced by the dissipative source term. The aim\nwill be therefore to quantify, in a model independent way, this effect\non the spectrum, and when it will render the system field-radiation\neffectively decoupled.     \n\nThe outline of the paper is as follow. In section II we review the\nbasic of warm inflation at the background level. In section III we\npresent the equations for the fluctuations of the coupled\nfield-radiation system, when the radiation is taken as an imperfect\nfluid. The numerical solutions for the fluctuations are presented in\nsection IV, with and without the shear viscosity. We  also study\nthe inflationary model dependence of the results on the\nspectrum by considering two generic model of inflation: a chaotic\nmodel with general power $p$, and a standard quadratic hybrid model. \nIn section V we summarize our findings.\n\n\n\n\\section{Warm inflation: background equations}\n\\label{sect2}\n\nIn any particle physics realization of the inflationary framework, the\ninflaton is not an isolated part of the model but it interacts with\nother fields.  These interactions  may \nlead to the dissipation of the inflaton energy into other degrees of\nfreedom, such that a small percent of the inflaton vacuum energy is\ntransferred into other kinds of energy. In the two-stage mechanism for\nwarm inflation, dissipation leads to particle production of light\ndegrees of freedom. When those relativistic particles thermalize fast\nenough, say in less than a Hubble time in an expanding universe, we\ncan model their contribution as that of radiation:\n\\begin{equation}\n\\rho_r \\simeq \\frac{\\pi^2}{30}g_* T^4 \\,,\n\\end{equation}\nwhere $T$ is the temperature of the thermal bath, and $g_*$ the\neffective number of light degrees of freedom\\footnote{If not otherwise\nspecified, we will take $g_* = 228.75$, the effective no. of degrees of\nfreedom for the Minimal Supersymmetric Standard Model, when presenting\nnumerical results.}.  \n\nThe dissipative term appears as an\nextra friction term in the evolution equation for the inflaton field $\\phi$,\n\\begin{equation}\n\\ddot \\phi + ( 3 H + \\Upsilon ) \\dot \\phi + V_\\phi =0\n\\,,\\label{eominf}\n\\end{equation}\n$\\Upsilon$ being the dissipative coefficient, $H=\\dot a\/a$ is the\nHubble rate of expansion, and\n$a$ the scale factor of the Friedmann-Robertson-Walker background\nmetric:\n\\begin{equation}\nds^2 = - dt^2 + a(t)^2 \\delta_{ij}dx^i dx^j \\,.\n\\end{equation} \nEq. (\\ref{eominf})\nis equivalent to the evolution equation for the inflaton energy\ndensity $\\rho_\\phi$:\n\\begin{equation}\n\\dot \\rho_\\phi + 3 H ( \\rho_\\phi + p_\\phi) = - \\Upsilon ( \\rho_\\phi +\np_\\phi) \\,,\n\\label{rhoinf}\n\\end{equation}\nwith pressure $p_\\phi = \\dot \\phi^2\/2 - V(\\phi)$, and $\\rho_\\phi\n+ p_\\phi= \\dot \\phi^2$. Energy conservation then demands that the\nenergy lost of the inflaton field must be gained by the radiation fluid\n$\\rho_r$, with the  RHS of Eq. (\\ref{rhoinf}) acting as\nthe source term:\n\\begin{equation}\n\\dot \\rho_r+ 3 H ( \\rho_r + p_r) = \\Upsilon ( \\rho_\\phi +\np_\\phi) \\,. \\label{eomrad}\n\\end{equation}\nIn warm inflation, radiation  is not\nredshifted away during inflation,\nbecause it is continuously fed by the inflaton \nthrough the dissipation \\cite{wi}. \nInflation happens when $\\rho_R \\ll \\rho_\\phi$, but even if small when\ncompared to the inflaton energy density it can be larger than the\nexpansion rate with $\\rho_R^{1\/4} > H$. Assuming thermalization, this\ntranslates roughly  into $T > H$. Otherwise, when $T < H$ (or\nsimilarly when $\\rho_R^{1\/4} < H$), one just recovers the standard cold\ninflation scenario, where dissipation can be neglected.\n\nDuring warm inflation the motion of the inflaton field has to be\noverdamped in order to have the accelerated expansion, but now this\ncan be achieved due to \nthe extra friction term $\\Upsilon$ instead of that of the Hubble\nrate. And once $\\phi$, $H$, and also $\\Upsilon$, are in this slow-roll\nregime, the\nsame will happen with the radiation energy density, the source term\nnow compensating for the Hubble dilution. In the slow-roll regime, the\nequations of motion reduce to:\n\\begin{align}\n3 H ( 1 + Q ) \\dot \\phi &\\simeq  -V_\\phi    \\,,\\label{eominfsl} \\\\\n4 \\rho_R  &\\simeq 3 Q\\dot \\phi^2\\,, \\label{eomradsl}\n\\end{align}\nwhere we have introduced the dissipative ratio $Q=\\Upsilon\/(3 H)$. \nNotice that\n $Q$ is not necessarily constant. The\ncoefficient  $\\Upsilon$ will depend on  $\\phi$ and $T$, and \ntherefore depending on the model the ratio $Q$ may increase or\ndecrease during inflation \\cite{BasteroGil:2009ec}.\nThe slow-roll conditions are given by \\cite{Moss:2008yb}: \n\\begin{align}\n\\epsilon &= \\frac{m_P^2}{2} \\left ( \\frac{V_{\\phi}} {V}\n\\right)^2 \\frac{1}{1+Q}\\ll 1\\,, \\label{eps}\\\\\n\\eta &= m_P^2 \\left ( \\frac{V_{\\phi \\phi}} {V}\\right) \\frac{1}{1+Q}\n\\ll 1 \\,, \\label{eta} \\\\\n\\beta_\\Upsilon &= m_P^2 \\left ( \\frac{\\Upsilon_\\phi V_\\phi }\n     {\\Upsilon V}\\right) \\frac{1}{1+Q}\\ll 1 \\,, \\label{beta} \\\\\n\\delta &= \\frac{T V_{T\\phi}}{V_\\phi} < 1  \\label{delta}\\,,\n\\end{align}\nwhere the slow-roll parameter $\\beta_\\Upsilon$  takes into account the\nvariation of $\\Upsilon$ with respect to $\\phi$, and \nthe last condition ensures that  thermal corrections to\nthe inflation potential are negligible. Similarly, taking also into\naccount the dependence on $T$ of $\\Upsilon$, one has: \n\\begin{equation}\n\\left|\\frac{d \\ln \\Upsilon}{d \\ln T}\\right| < 4 \\,,\n\\end{equation}\nwhich reflects the fact that radiation has to be produced at a rate\nlarger than the redshift due to the expansion of the universe.   \nThe slow-roll regime ends when any of the above conditions\nEqs. (\\ref{eps})-(\\ref{delta}) is no longer satisfied, such that  either the\nmotion is no longer overdamped and slow-roll ends, or the radiation\nbecomes comparable to the inflaton energy density. Either way,\ninflation will  end shortly  afterwards. \n\nFor warm inflation the first requirement is to have $T>H$, but the\nratio $Q$ can be larger or smaller than unity. In the former case we\nare in the strong dissipative regime (SDR), whereas the latter is\ncalled weak dissipative regime (WDR). In the weak dissipative regime\nthe extra friction added by $\\Upsilon$ is not enough to substantially\nmodify the background inflaton evolution, and it will resemble that\nof cold inflation; still the thermal fluctuations of the\nradiation energy density will modify the field fluctuations, and\naffect the primordial spectrum of perturbations. \n\nThe $T$ and $\\phi$ dependent dissipative coefficient has been computed\nin \\cite{mossxiong}, using the near-equilibrium approximation\nproposed in \\cite{BGR}. The specific field theory models \nconsidered for the inflaton interactions leading to dissipation \nall follow from the two-stage mechanism \\cite{BR1,br,br05}. \nIn this mechanism,  \nthe inflaton field  $\\phi$ \nis coupled to heavy catalyst fields  $\\chi$, which decay into light\nfields  $\\sigma_i$. Consistency of the approximations then demands the\nmicrophysical dynamics determining $\\Upsilon$ to be faster than that\nof the macroscopic motion of the background inflaton and the\nexpansion: \n\\begin{equation}\n\\Gamma_\\chi > \\left|\\frac{\\dot \\phi}{\\phi}\\right|,\\, H, \n\\end{equation}\nwhere $\\Gamma_\\chi$ is the decay width of the heavy fields. \nIn addition, the condition $T \\gg H$ allows  to neglect the expansion of \nthe universe  when computing $\\Upsilon$. In the low $T$ regime, when \n the mass of the catalyst field $m_\\chi$ is larger than $T$, one has:\n\\begin{equation}\n\\Upsilon(\\phi,T) \\propto \\frac{T^3}{m_\\chi^2} \\propto\n\\frac{T^3}{\\phi^2}\\,, \n\\end{equation}\nwhile in the high $T$ regime, where the thermal corrections to the\ncatalyst field mass start to be important,\n\\begin{equation}\n\\Upsilon(\\phi,T) \\propto T \\,.\n\\end{equation}\nAnd in the very high $T$ regime, the dissipative coefficient goes like\nthe inverse of $T$. These are the cases of study we are going to\nconsider in the next section when studying the fluctuations during\nwarm inflation. In general, we will parametrize the dissipative coefficient as:\n\\begin{equation}\n\\Upsilon = C_\\phi \\frac{T^c}{\\phi^{m}} \\,,\n\\end{equation}\nwith $c-m=1$. We will work with $c=$3,1,-1, and also $c=0$, the case\nof no $T$ dependence for the dissipative coefficient. \n\n\n\\section{Fluctuations at linear order: Primordial spectrum}\n\nDuring warm inflation we have a multicomponent\nfluid, a mixture of a \nscalar inflaton field  $\\Phi$ interacting with the radiation fluid.\nBoth components exchange energy and momentum through the dissipative\nterm $\\Upsilon$. Dissipative effects also imply small departures from\nequilibrium, and that the \nradiation fluid will not behave exactly like a perfect fluid during\ninflation. In relativistic \ntheory, these effect can be parametrized\nin terms of a shear viscous tensor $\\pi_{ab}$, an energy flux vector $q_a$\nand a bulk viscous pressure $\\pi_b$, in the stress-energy tensor for the\nradiation fluid~\\cite{weinberg,maartens},\n\\begin{equation} \nT_{ab}^{(r)}= (\\bar \\rho_r + \\bar p_r + \\pi_b) u_a^{(r)}u_b^{(r)} + (\\bar p_r +\n\\pi_b) g_{ab} + q_a^{(r)}u_b^{(r)}+q_b^{(r)}u_a^{(r)}+ \\pi_{ab} \\,,\n\\end{equation}\nwhere $\\bar \\rho_r$ is the energy density, $\\bar p_r$ the adiabatic pressure,\n$u_a^{(r)}$ the four velocity of the radiation fluid, $g_{ab}$ the\nfour-dimensional metric, and  $u_a^{(r)}\n\\pi^{ab}=0=g_{ab}\\pi^{ab}$, $u_a^{(r)} q^{a}=0$. There would be heat \nflow for example in the presence of conserved charges in the system\nother than the stress-energy tensor, but we do not consider such\npossibility in this work, and then set $q_a=0$. The shear viscous\ntensor vanishes in an homogeneous and isotropic background geometry,\nbut at linear order it is given by \\cite{maartens}:\n\\begin{equation}\n\\pi_{ab} \\simeq -2 \\zeta_s \\sigma_{ab} \\,, \\label{shear}\n\\end{equation}\nwhere $\\zeta_s$ is the shear viscosity coefficient and $\\sigma_{ab}$\nthe shear of the radiation fluid:\n\\begin{equation}\n\\sigma_{ab} = \\nabla_{(a}u_{b)} + u_{(a} u^c\\nabla_c u_{b)} -\n\\frac{h_{ab}}{3}\\nabla^c u_c \\,,\n\\end{equation}\n$\\nabla_a$ being the covariant derivative of the metric $g_{ab}$. \nThe bulk viscous pressure can be seen as a non-adiabatic pressure\ncontribution, already present at the background level. Nevertheless,\nthe contribution from the light fields is expected to be small with\nrespect to $p_r$.  Thus, we will also set $\\pi_b=0$ and  focus on studying the\nconsequences of the shear viscosity on the primordial perturbations\nduring warm inflation.   \n\nIn order to study the system of perturbations at linear order, field,\nradiation energy density  and radiation pressure \nare expanded around their background values in a\n{}Friedman-Robertson-Walker metric: \n\\begin{align}\n\\Phi(x,t) &= \\phi(t) + \\delta \\phi(t,x) \\,, \\\\\n\\bar \\rho_r (x,t)&= \\rho_r(t) + \\delta \\rho_r(t,x) \\,, \\\\\n\\bar p_r(x,t) &= p_r(t) + \\delta p_r(t,x) \\,,\n\\end{align}\nand similarly for the dissipative coefficient: $\\bar \\Upsilon(x,t)=\n\\Upsilon(t) + \\delta \\Upsilon(t,x)$.  The  perturbed FRW metric,\nincluding only scalar perturbations,  is given by\\footnote{Latin\n  indexes $i,\\,j,\\,k,\\ldots$ are used for the spatial components, and either\n  $a,\\,b,\\,c,\\ldots$ or Greek letters for space-time indexes. }:  \n\\begin{equation}\nds^2= -(1+2 \\alpha) dt^2 - 2 a \\partial_i \\beta dx^i dt \n     + a^2 [ \\delta_{ij} (1 +2 \\varphi) + 2 \\partial_i \\partial_j \\gamma] dx^i\ndx^j \\,, \\label{metric}\n\\end{equation}\nwhere $\\varphi$ is the intrinsic curvature of a\nconstant time hypersurface. For later use, we introduce the combinations:\n\\begin{align}\n\\chi & = a ( \\beta + a \\dot \\gamma) \\,, \\\\\n\\kappa&= 3 (H \\alpha - \\dot \\varphi) + \\partial_k \\partial^k \\chi \\,, \n\\end{align}\nwhere $\\chi$ is the shear and $-\\kappa$ the\nperturbed expansion scalar of the comoving frame. \n \nThe evolution equations follow from the\nconservation of the energy-momentum tensor: \n\\begin{equation}\n\\nabla^a T_{ab}^{(\\alpha)}= Q_b^{(\\alpha)} \\,,\\;\\;\\; \\sum_\\alpha\nQ_b^{(\\alpha)}=0\\,,\n\\label{DTab}\n\\end{equation}\nwhere $Q_b$ is the four-vector source term  accounting for \nthe exchange of energy and momentum:\n\\begin{equation}\n-Q_b^{(\\phi)}=Q_b^{(r)}=\\Upsilon u_{\\phi}^a \\nabla_a \\Phi \\nabla_b \\Phi \\,, \n\\end{equation}\n$u_\\phi^a$ is now the four-velocity of inflaton fluid:  \n\\begin{equation}\nu_\\phi^a=-\\frac{\\nabla^a \\Phi}{\\sqrt{\\rho_\\phi + p_\\phi}} \\,,\n\\end{equation}\nand then:\n\\begin{equation}\nQ_b^{(r)}=\\Upsilon (\\bar \\rho_\\phi + \\bar p_\\phi)^{1\/2} \\nabla_b \\Phi \\,. \n\\end{equation}\nThe projection of the four-vector source term along the direction of the fluid \ngives the energy density source term, \n\\begin{equation}\nQ^{(r)}=-u_{\\phi}^a Q_a^{(\\phi)}\\,,\n\\end{equation}\nwhich at linear order is given by:\n\\begin{align} \nQ^{(r)}&= Q_r + \\delta Q_r \\,, \\\\\nQ_r & = \\Upsilon \\dot \\phi^2 \\,, \\\\\n\\delta Q_r &= \\delta \\Upsilon \\dot \\phi^2 + 2 \\Upsilon \\dot \\phi \\delta\n\\dot \\phi - 2 \\alpha \\Upsilon \\dot \\phi^2 \\,. \\label{Qr}\n\\end{align}\nThe momentum source term $J_a$ is orthogonal to the fluid velocity:\n\\begin{equation}\nQ^{(\\phi)}_a= Q^{(\\phi)} u^{(\\phi)}_a + J^{(\\phi)}_a\\,, \\;\\;\\;\\;\nu^{(\\phi)\\,a} J_a^{(\\phi)}=0 \\,,\n\\end{equation}\nand vanishes in the FRW geometry; at linear order it reads:\n\\begin{align}\nJ_i^{(r)} &= \\partial_i {\\bf J}_r \\,, \\\\ \n{\\bf J}_r& = - \\Upsilon \\dot \\phi \\delta \\phi \\,.\\label{Jr}\n\\end{align}\nTo complete the specification of the source, we need $\\delta\n\\Upsilon$, which for a general temperature $T$ and field $\\phi$\ndependent dissipative coefficient,   \n$\\Upsilon =  C_\\phi T^c\/\\phi^m$, with $c-m=1$, is given by:  \n\\begin{equation}\n\\delta \\Upsilon =  \\Upsilon \\left(c \\frac{\\delta T}{T} - m\n\\frac{\\delta \\phi}{\\phi} \\right) \n\\,. \\label{dupsilon} \n\\end{equation}\nAlthough dissipation implies departures from thermal equilibrium in\nthe radiation fluid, the system has to be close-to-equilibrium for the\ncalculation of the dissipative coefficient to hold, therefore $p_r \\simeq\n\\rho_r\/3$, $\\rho_r \\propto T^4$ and \n\\begin{equation}\n4 \\frac{\\delta T}{T} \\simeq \\frac{\\delta \\rho_r}{ \\rho_r}\\,.\n\\end{equation}\n\n{}Finally, the evolution equations for the radiation fluctuations are obtained\nexpanding at linear order Eq. (\\ref{DTab}) \\cite{kodama,\n  hwang,hwangnoh,malik}. Working in momentum space, defining the Fourier\ntransform with respect to the comoving coordinates, the equation of\nmotion for the fluctuations with comoving wavenumber $k$ are given\nby\\footnote{For simplicity, we keep the same notation  for the\n  fluctuations $\\delta f({\\bf x},t)$ and their Fourier transform \n$\\delta f({\\bf k},t)$.}:    \n\\begin{align}\n\\delta \\dot \\rho_r + 3 H (\\delta \\rho_r + \\delta\np_r) &= -3 (\\rho_r + p_r) \\dot \\varphi + \\frac{k^2}{a^2} \\left[\n\\Psi_r +(\\rho_r + p_r) \\chi\\right] + \\delta Q_r +\nQ_r \\alpha \\,, \\label{energyalpha}\\\\\n\\dot \\Psi_r + 3 H \\Psi_r &=- (\\rho_r +\np_r)\\alpha  - \\delta p_r+\n\\frac{2 k^2}{3 a^2} \\sigma_r + {\\bf J}_r\n\\,, \\label{momentumalpha} \n\\end{align}\nwhere a ``dot'' denotes the derivative with respect to the metric time\n``$t$'', $\\Psi_r$ is the radiation momentum perturbation, $T^{0\\,(r)}_j=-\n\\partial_j \\Psi_r\/a$, and $\\sigma_r$ the shear viscous pressure at\nlinear order:\n\\begin{equation}\n\\sigma_r \\simeq - 2 \\zeta_s \\left(\\frac{\\Psi_r}{\\rho_r + p_r} + \\chi\\right)\n\\,. \n\\end{equation}\n\nOn the other hand, field fluctuations are described by a stochastic\nsystem whose evolution is determined by a Langevin equation\n\\cite{calzetta,mossgraham}:\n\n\\begin{align}\n\\delta \\ddot \\phi + (3 H + \\Upsilon) \\delta \\dot \\phi +\n\\left(\\frac{k^2}{a^2} + V_{\\phi\\phi}\\right) \\delta \\phi &= \n\\left[2(\\Upsilon + H) T\\right]^{1\/2}\na^{-3\/2}\\xi_k  \\nonumber - \\delta \\Upsilon \\dot \\phi \\\\\n& + \\dot \\phi ( \\kappa + \\dot \\alpha) + (2 \\ddot\n\\phi + 3 H \\dot \\phi) \\alpha \n-\\Upsilon ( \\delta \\dot \\phi - \\alpha \\dot \\phi) \\label{field}\\,,\n\\end{align}\nwhere the stochastic source $\\xi_k$ can be approximated by a\nlocalized gaussian distribution with correlation function:\n\\begin{equation}\n\\langle \\xi(t,x) \\xi(t^\\prime,x^\\prime)\\rangle = \\delta(t -t^\\prime)\n\\delta^{(3)}(x-x^\\prime) \\,. \n\\end{equation}\n\nSo far, the equations for the perturbations at linear\norder are written in a ``gauge ready'' form, without specifying any\nparticular gauge, but the equations can also be written in\nterms of gauge invariant (GI) variables. For any scalar quantity $f$, at\nlinear order one can define a gauge invariant perturbation\n\\cite{kodama,hwang}: \n\\begin{equation}\n\\delta f^{GI} = \\delta f - \\frac{\\dot f}{H} \\varphi \\,, \n\\end{equation}\nwhile similarly the gauge invariant momentum perturbation reads:\n\\begin{equation}\n\\Psi^{GI}= \\Psi - \\frac{\\rho + p}{H} \\varphi \\,,\n\\end{equation}\nand for the metric perturbations one has the gauge invariant combinations:\n\\begin{align}\n{\\cal A} &= \\alpha - \\left[ \\frac{\\varphi}{H} \\right]^\\cdot \\,,\\\\\n\\Phi & = \\varphi - H \\chi \\,.\n\\end{align}\nThe evolution equations then read:\n\\begin{align}\n\\delta \\ddot \\phi^{GI} + 3 H \\delta \\dot \\phi^{GI} + \n\\left(\\frac{k^2}{a^2} + V_{\\phi \\phi}\\right) \\delta \\phi^{GI} &= \n\\left[2 (\\Upsilon+H) T\\right]^{1\/2}a^{-3\/2}  \\xi \n-\\dot \\phi \\delta \\Upsilon^{GI}  -\\Upsilon \\delta \\dot \\phi^{GI}\n\\nonumber  \\\\  \n&+\\Upsilon \\dot \\phi {\\cal A} + \\dot \\phi \\dot {\\cal A}+ 2\n(\\ddot \\phi + 3 H \\dot \\phi) {\\cal A} -\\frac{k^2}{a^2} \\dot \\phi\n\\frac{\\Phi}{H}, \\label{fieldGI} \\\\\n\\delta \\dot \\rho_r^{GI} + 3 H (1 + w_r) \\delta \\rho_r^{GI}  &=\n\\frac{k^2}{a^2} \\Psi_r^{GI}+ \\delta Q_r^{GI} + \\dot \\rho_r {\\cal A} \n\\,, \\label{energyrGI}\\\\\n\\dot \\Psi_r^{GI} + 3 H \\left( 1 + \\frac{k^2}{a^2 H^2}\\bar \\zeta_s\\right) \n\\Psi_r^{GI} &=- w_r \\delta \\rho_r^{GI} -\n\\Upsilon \\dot \\phi \\delta \\phi^{GI} - (\\rho_r + p_r) {\\cal A}\n-\\frac{3 k^2}{ a^2} (\\rho_r+ p_r) \\bar \\zeta_s \\frac{\\Phi}{H} \n\\label{momentumrGI} \\,,\n\\end{align}\nwhere in Eq. (\\ref{momentumrGI}) we have defined: \n\\begin{equation}\n\\bar \\zeta_s= \\frac{4}{9} \\frac{\\zeta_s H}{\\rho_r + p_r} \\label{zetas}\\,.\n\\end{equation} \n{}Finally, from the Einstein equations at linear order, the gauge invariant\nmetric perturbations are given by \\cite{kodama, hwang}:\n\\begin{align}\n{\\cal A}&= -\\frac{\\dot H}{H^2} {\\cal R} \\,, \\label{metricA}\\\\\n\\frac{k^2}{a^2 H^2}\\Phi&= 3 {\\cal A} + \\frac{3}{2} \\frac{\\delta\n  \\rho_T^{GI}}{\\rho_T}  \\label{metricPhi}\\,,\n\\end{align}\nwhere ${\\cal R}$ is the total comoving curvature perturbation,\n\\begin{eqnarray}\n{\\cal R}&=& \\varphi - \\frac{H}{\\rho_T + p_T} \\Psi_T \n\\nonumber \\\\\n&=& -\\frac{H}{\\rho_T\n  +p_T} \\Psi_T^{GI} \\,.\n\\end{eqnarray}\n{}For a multicomponent fluid, the total momentum perturbation is given\nby the sum of the individual components, and ${\\cal R}$ can be written\nas the weighted sum of the individual contributions:\n\\begin{align}\n{\\cal R} &= \n \\sum_{\\alpha=\\phi,\\,r} \\frac{h_\\alpha}{h_T} {\\cal R}_{\\alpha} \\,, \\label{RT}\\\\\n{\\cal R}_\\alpha & = -\\frac{H}{h_\\alpha} \\Psi_\\alpha^{GI}\\,,\n\\end{align}\nwhere we have defined $h_\\alpha= \\rho_\\alpha + p_\\alpha$. \nIn particular, for a scalar field $\\Psi_\\phi^{GI}= - \\dot \\phi \\delta\n\\phi^{GI}$, $h_\\phi = \\dot \\phi^2$,  and\n\\begin{equation}\n{\\cal R}_{\\phi}= \\frac{H}{\\dot \\phi} \\delta \\phi^{GI} \\,, \\label{Rphi}\n\\end{equation}\nfor which the power spectrum would be the form for single\nfield cold inflation. For warm inflation, we shall use the total\ncomoving curvature perturbation to evaluate the primordial spectrum,  \n\\begin{equation}\nP_{\\cal R}(k)= \\frac{k^3}{2 \\pi^2} \\langle |{\\cal R}_k|^2 \\rangle\\,, \\label{PR}\n\\end{equation}\nwhere ``$\\langle \\cdots \\rangle$'' means average over different\nrealizations of the noise term in Eq. (\\ref{field}). In all numerical\nresults shown in this work we have performed averages over 1000 runs. \nThis was found to be more than enough to get convergent numerical results.\".\n \n\nThe largest observable scale in the CMB corresponds to a comoving\nscale $k$ crossing the horizon $N_e$ e-folds before the end of\ninflation, denoted by $k=a_* H_*$. The value of $N_e$ can vary in\ngeneral from 40 to 70 depending on the inflationary model and details on the\nsubsequent reheating process \\cite{reheating}. Although we shall\nconsider different inflationary potentials, we will not consider the\ndetails of reheating period, and simply fix the horizon crossing at\n$N_e\\simeq 60$. As we will see, soon after horizon crossing, the amplitude of\nthe individual curvature perturbations ${\\cal R}_\\phi$ and ${\\cal\n  R}_r$  freezes out, and so does that of the total\ncurvature. This allows to compute the primordial spectrum by\nevaluating Eq. (\\ref{PR}) at horizon crossing, mainly when getting\nanalytical approximations. Nevertheless, when\nshowing numerical results we shall integrate Eqs. (\\ref{fieldGI})-\n(\\ref{momentumrGI}) and evaluate the amplitude of\nthe spectrum say 10 e-folds after horizon crossing. \n\n\n\\section{Equations in the zero-order slow-roll approximation and beyond}\n\nIn order to gain some insight on the evolution of the perturbations,\nwe follow \\cite{mossgraham} and first study the equations for the\nperturbations at zero-order in the slow-roll parameters. That is, \nexpanding the background variables around their slow-roll values and\nneglecting all terms in the equations proportional to slow-roll\nparameters. This eliminates in the equations for the fluctuations the\ndependence on the inflationary potential, and the details on the\nevolution of the dissipative coefficient. For example, the metric\nperturbation  ${\\cal A}$ given in \nEq. (\\ref{metricA}) is proportional to the slow-roll coefficient\n$\\epsilon$, Eq. (\\ref{eps}),    \n$\\epsilon= -\\dot H\/H^2$,\nand can be neglected; \nand the same for the terms proportional to $H \\dot \\phi$, \n{\\it i.e.}, those\nproportional to $\\Phi$, and $\\dot \\phi\/H\\phi$. Defining the\ndimensionless variables:  \n\\begin{align}\ny_k &= \\frac{k^{3\/2}\\delta \\phi^{GI}} {\\left[2 (\\Upsilon + H) T\\right]^{1\/2}} \\,,\\\\\nw_k &= \\frac{k^{3\/2}\\delta \\rho_r^{GI}}{ \\left[2 (\\Upsilon + H) T\\right]^{1\/2} \n(\\Upsilon \\dot \\phi)} \\,, \\\\\nu_k &= \\frac{k^{3\/2}\\Psi_r^{GI}}{ \\left[2 (\\Upsilon + H) T\\right]^{1\/2} \\dot \\phi} \\,,\n\\end{align}\nand using the slow-roll background equations (\\ref{eominfsl}),\n(\\ref{eomradsl}), \nwe have the system:\n\\begin{align}\n\\ddot y_k + 3 H ( 1 +Q) \\dot y_k + H^2 \\left[ z^2 + 3 \\eta (1+Q) - 3m Q\n\\frac{\\dot \\phi}{H\\phi} \n\\right] y_k \n&= \\left(\\frac{k}{a}\\right)^{3\/2}\\xi_k- 3 Q c H^2 w_k \\,, \\label{EOMyk} \\\\\n\\dot w_k + H (4-c) w_k &= \\frac{H}{3 Q}z^2 u_k + 2 \\dot y_k \n\\,, \\label{EOMwk}\\\\ \n\\dot u_k + 3 H(1 + z^2 \\bar \\zeta_s) u_k &= -3 Q H \\left(\\frac{w_k}{3}\n+y_k\\right) \n\\,, \\label{EOMuk}  \n\\end{align}\nwhere $z=k\/(aH)$. Combining Eqs. (\\ref{EOMwk}) and (\\ref{EOMuk}) into\na second order differential equation, we have:\n\\begin{align}\n\\ddot w_k + H ( 9 -c + 3 z^2 \\bar \\zeta_s ) \\dot w_k + H^2 \\left[ 20 - 5 c +\n6 Q c + \\frac{z^2}{3} + 3 z^2 (4-c) \\bar \\zeta_s\\right] w_k \n&=\n2 \\left(\\frac{k}{a}\\right)^{3\/2}\\xi \\nonumber \\\\\n& \n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n+ H ( 4 - 6 Q + 6 z^2 \\bar \\zeta_s) \\dot y_k \n- H^2 \\left[3 z^2 + 6 \\eta (1+Q)- 6 Q m \\frac{\\dot \\phi}{H \\phi}\\right]\ny_k  \\label{fullwk}\\,.  \n\\end{align}\n\n\\begin{figure}\n\\vspace{0.5cm}\n \\includegraphics[width=0.55\\textwidth,\n  angle=0]{spectra_chaot_kk10000_Q100_yy_p.eps}\n\\vspace{0.25cm}\n\\caption{\\label{plot1a} Evolution of the power\n  spectrum of the field \n  $\\langle y_k^2\\rangle$, radiation energy density $\\langle\n  w_k^2\\rangle$, and radiation momentum $\\langle u_k^2\\rangle\/Q^2$,\n  for $Q=100$, and wavenumber \n  $k=10^4 H$. The vertical thin dotted line sets the value of the freeze\n  out scale $k_F$ in warm inflation.     \n  The results are shown for different power dependence on\n  $T$ of the dissipative coefficient: \n  $c=3$ (solid lines), $c=1$ (dashed lines), $c=-1$ (dash-dotted\n  lines), and $c=0$ (dotted lines).\n} \n\\end{figure}\n\n\n\\begin{figure}\n\\vspace{0.5cm}\n\\includegraphics[width=0.55\\textwidth, angle=0]{spectra_chaot_kk10000_Q100_Pr_p.eps}\n\\vspace{0.25cm}\n\\caption{\\label{plot1b} Evolution  of the total curvature perturbation\n  spectrum $P_{\\cal R}^{1\/2}$ (black lines), the \n  radiation $P_{{\\cal R}_r}^{1\/2}$ (red lines), and the field\n  $P_{{\\cal R}_\\phi}^{1\/2}$ (green lines) curvature perturbation spectrum.\n  The results are shown for different power dependence on\n  $T$ of the dissipative coefficient: \n  $c=3$ (solid lines), $c=1$ (dashed lines), $c=-1$ (dash-dotted\n  lines), and $c=0$ (dotted lines).\n} \n\\end{figure}\n\nBefore including shear effects, we study the evolution of the\nperturbations setting $\\bar \\zeta_s=0$. \nThe evolution of the power spectrum of the field $\\langle y_k^2\n\\rangle$, the radiation energy  density $\\langle w_k^2 \\rangle$, and\nthe radiation momentum $\\langle u_k^2 \\rangle$, are shown \nin {} Fig. \\ref{plot1a} as a function of $z^{-1}=aH\/k$. We have\ntaken $Q=100$,  and started the\nintegration at $z_i=10^4$. As mentioned above, quantities as $\\langle\ny_k^2 \\rangle$ denotes the average over 1000 realizations of the\ngaussian noise term,  and by $y_k^2$, $w_k^2$, $u_k^2$ we mean the\nmodulus of the complex variable. We have set the initial conditions\nfor field fluctuations in the vacuum,  while $w_k$ and $u_k$ initially\nvanish, for simplicity. Starting the evolution early enough before horizon\ncrossing, the system is always controlled by the stochastic source\nterm, and the dependence on the initial conditions is quickly erased.  \n\n\nWe have considered different powers of $T$ for the dissipative\ncoefficient $\\Upsilon$, $c=3,\\,1\\,,-1$, and included the\ncase of a constant or \nfield dependent $\\Upsilon$ ( $c=0$, dotted lines) for comparison. \nThe radiation fluctuation $w_k$ acts as a source term for the field,\nbut at early times $z^{-1} \\ll 1$, for subhorizon perturbations, the\nfield evolution is dominated by the stochastic source term, and both\nradiation and field fluctuations evolve like in the case $c=0$. \nIn the latter\ncase, the freeze out of the perturbations takes place before horizon\ncrossing, due to the extra friction term in Eq. (\\ref{EOMyk}), at\naround $k_F\/(aH)\\simeq 3\\sqrt{Q\/2}$ (vertical thin dotted line)\n\\cite{warmdeltap,warmpert, mossgraham}, and\nsoon after field and radiation spectrum level off. The field spectrum\nfor a $T$ independent $\\Upsilon$  can be computed analytically and is\ngiven by \\cite{warmdeltap}:    \n\\begin{equation}\n\\langle y_k^2 \\rangle_0 \\simeq \\frac{\\sqrt{3\\pi}}{4} \n\\frac{\\sqrt{1+Q}}{(1+3 Q)}\\,,  \\label{yy0}\n\\end{equation}\nwhere the subindex ``0'' denotes the value for $c=0$. On the other\nhand, when $c > 0$, field and radiation fluctuations get\neffectively coupled before freeze out at around $z_c^2 \\approx 18 Q c$,\nand both start growing at similar\nrates. Numerically, we get: \n\\begin{equation}\n\\langle y_k^2 \\rangle \\approx \\langle y_k^2 \\rangle_{z_c} \\left( \\frac{z_c}{z}\n\\right)^{5 c} \\,. \\label{yyz} \n\\end{equation}\nThe field spectrum when the fluctuations are still subhorizon and $z >\nz_c$, which is independent of the radiation fluctuation,  can be\nfound in \\cite{mossgraham}: \n\\begin{equation}\n\\langle y_k^2 \\rangle_{z_c} \\approx \\frac{3c}{z_c} \\,, \\label{yyzc}\n\\end{equation}\nand therefore, at horizon crossing: \n\\begin{equation}\n\\langle y_k^2 \\rangle_* \\propto z_c^{5c-1} \\propto Q^{(5c-1)\/2} \\,. \\label{yystar}\n\\end{equation}\nWhen $c< 0$, the effect is the opposite, and effectively the freeze\nout is delayed until practically horizon crossing, which makes the\namplitude of the field spectrum smaller than in the $c=0$ case. \n\n{}Fig. \\ref{plot1b} shows the evolution\nof the comoving curvature power spectrum, $P_{\\cal R}^{1\/2}$, given by\nthe sum of the radiation and the field contributions as in\nEq. (\\ref{RT}), for different values of $c$. Also shown are \nthe power spectra of\nthe radiation, $P_{{\\cal R}_r}^{1\/2}$, \nand the field $P_{{\\cal R}_\\phi}^{1\/2}$. After  \nhorizon crossing they all converge to the same amplitude. During\nslow-roll one has that $h_r = \\rho_r +p_r \\simeq Q h_\\phi$, and from\nEq. (\\ref{EOMuk}) when $z\\ll 1$ the radiation momentum\nbecomes proportional to the field fluctuation:   \n\\begin{equation}    \n\\Psi_r \\simeq Q \\dot \\phi \\delta \\phi \\,,\n\\end{equation}\nand therefore: \n\\begin{equation}\nP_{{\\cal R}_r} = \\frac{H}{h_r} P_{\\Psi_r} \\simeq P_{{\\cal R}_\\phi} \\,.\n\\end{equation}\nOwing to the fact that $h_\\phi \\ll h_r \\simeq h_T$, the main\ncontribution to the primordial spectrum in Eq. (\\ref{PR}) comes from\nthe radiation, before and after horizon crossing. The\nprimordial spectrum is always dominated by the thermal fluctuations. \nBut after horizon crossing one simply has:\n\\begin{equation}\nP_{\\cal R} \\simeq P_{{\\cal R}_r}\\simeq P_{{\\cal R}_\\phi} \\,.\n\\end{equation}  \n Therefore, the amplitude of the primordial spectrum can be written as\n usual in terms of that of the inflaton field:\n\\begin{equation}\nP_{\\cal R} \\simeq \\left( \\frac{H}{\\dot \\phi} \\right)^2 \\frac{(H +\n  \\Upsilon)T}{\\pi^2} \\langle y_k^2 \\rangle_*   \\label{PRyy}\\,,\n\\end{equation}\nevaluated at horizon crossing. \n\n\n\\begin{figure}\n\\vspace{0.55cm}\n\\includegraphics[width=0.55\\textwidth,angle=0]{spectra_chaot_Qiter_kk100_gralT_p.eps}\n\\vspace{0.25cm}\n\\caption{\\label{plot2} The spectrum of the field at horizon crossing \n  $\\langle y_k^2\\rangle_*$ as a function of the\n  dissipative parameter $Q$, at zero order in the slow-roll\n  parameters, for different values of $c$, and no shear $\\bar\n  \\zeta_s=0$. Solid lines are obtained integrating\n  Eqs. (\\ref{EOMyk})-(\\ref{EOMuk}), while dashed lines were obtained\n  with the approximation used in \\cite{mossgraham}.     \n} \n\\end{figure}\n\nIn {}Fig. \\ref{plot2} we have compared the power spectrum of the\nfield, $\\langle y_k^2 \\rangle$ at horizon crossing, as a function of the\ndissipative parameter $Q$ for different values of $c$. The equation\nfor the fluctuations has been integrated keeping the background values\nconstant (solid lines), Eqs. (\\ref{EOMyk})-(\\ref{EOMuk}).  The larger\nthe power $c>0$, the larger the enhancement with $Q$, as the\nfluctuations get coupled earlier. For $c <0$, as mentioned \nbefore, we have the opposite effect, and the spectrum diminished with\nrespect to the case $c=0$. For positive $c$, the curves can be fitted by\na function:\n\\begin{equation}\n\\langle y_k^2 \\rangle_* |_{c>0}\\simeq \\langle y_k^2 \\rangle_0 \n( A_c Q^\\alpha + B_c Q^\\beta) \\,.  \\label{yyapprox}\n\\end{equation}\nbut when $c=-1$, we have found that the curve can be best fitted by: \n\\begin{equation}\n\\langle y_k^2 \\rangle_*|_{c=-1} \\simeq \\frac{ 1+ A_{-1} Q^\\alpha}{ 1+ B_{-1}\n  Q^\\beta} \\,. \\label{yyapproxcn1}\n\\end{equation}\nThe coefficients are given in Table \\ref{table1}. \nFor $c=3,\\,1$, the approximation works well for $Q>50$, while for\n$c=-1$ it is valid for any $Q$. \n\n\n\n\\begin{table}\n\\begin{tabular}{|c|c|c|c|c|}   \n \\hline\n$c$ & $\\alpha$ &  $\\beta$ & $A_c$ & $B_c$ \\\\\n\\hline\n~~~3~~~ & ~~~7.5~~~ & ~~~7.0~~~ & ~~~$1.9\\times 10^{-8}$~~~ & ~~~$3.4 \\times 10^{-6}$~~~ \\\\\n~~~1~~~ & ~~~2.5~~~ & ~~~2.0~~~ & ~~~$2.8 \\times 10^{-2}$~~~ & ~~~$6.8 \\times 10^{-5}$~~~ \\\\\n~~~-1~~~ & ~~~0.2~~~ & ~~~1.4~~~ & ~~~0.78~~~ & ~~~0.088~~~ \\\\\n\\hline\n\\end{tabular} \n\\caption{\\label{table1} Coefficients for the numerical fit of the\n  spectrum, Eq. (\\ref{yyapprox}) and Eq. (\\ref{yyapproxcn1}).}\n\\end{table}\n\n\nWe have also included in {}Fig. \\ref{plot2} the spectrum of the field\nobtained with the approximation used in \\cite{mossgraham} (dashed\nlines) for comparison. We have confirmed the main results about the\npower spectrum obtained in \\cite{mossgraham}; {\\it i.e.}, that for $c >0$,\nthe amplitude of the primordial spectrum in warm inflation is enhanced\nby a factor $\\simeq Q^{\\alpha -1\/2}$. However, while in\n\\cite{mossgraham} they found $\\alpha = 3c$, we have a smaller power\n$\\alpha=5c\/2$, which for $c=3$ can   \nmean a difference of two or 3 orders of magnitude in the amplitude of\nthe spectrum for $Q\\simeq 50 - 100$. The differences can be traced\nback to how the radiation source term is treated. In \\cite{mossgraham}, it\nwas approximated by:\n\\begin{equation}\n\\delta Q_r \\simeq Q_r \\frac{\\delta \\Upsilon}{\\Upsilon} \\simeq c Q_r\n\\frac{\\delta \\rho_r}{4\\rho_r} \\,, \\label{Qrian} \n\\end{equation} \nwhile we have kept the dependence on the field:  \n\\begin{equation}\n\\delta Q_r \\simeq Q_r \\left( c \\frac{\\delta \\rho_r}{4\\rho_r} + 2\n\\frac{\\delta \\dot \\phi}{\\dot \\phi}\\right) \\,. \\label{Qrphi}  \n\\end{equation} \nWith no shear viscosity included, the second order differential\nequation Eq. (\\ref{fullwk}) reads:  \n\\begin{align}\n\\ddot w_k + H ( 9 -c ) \\dot w_k + H^2 \\left( 20 - 5 c  \\frac{z^2}{3}\\right) w_k \n+  H^2 z^2 y_k  \n&= 2 \\left(\\frac{k}{a}\\right)^{3\/2}\\xi  \\nonumber \\\\\n& \n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n- H^2 6 Q c w_k +  H ( 4 - 6 Q ) \\dot y_k \n- H^2 \\left[2 z^2 + 6 \\eta (1+Q) - 6 Q m \\frac{\\dot \\phi}{H \\phi}\\right]\ny_k  \\label{fullwknoshear}\\,,  \n\\end{align}\nwhere we have written the equation such that on the RHS we have the terms\ninduced by the field dependence in Eq. (\\ref{Qrphi}). Thus, setting\nthe RHS to zero one recovers the equation derived with the source term\nas given in Eq. (\\ref{Qrian}). While the term\nproportional to the field perturbation $y_k$ acts as a source term on\nthe radiation that tends to enhance the fluctuation, the extra terms \n$6 H^2 Qc w_k$ and $H(4 -6Q) \\dot y_k$ have the opposite effect, i.e.,\nthat of damping the growth. Although these contributions are not enough\nto avoid the growth of the fluctuations, they have a sizable effect on\ntheir power-law behavior, mainly when $c=3$, as seen in Fig. \\ref{plot2}. \nWhen $c =-1$ the radiation fluctuations do not grow before horizon\ncrossing, so that the effect of the field dependent terms in\nEq.~(\\ref{Qrphi}) is negligible. \n\n\\begin{figure}\n\\vspace{0.55cm}\n\\includegraphics[width=0.55\\textwidth,angle=0]{ianeq_c3_k100_z1_all_p.eps}\n\\vspace{0.25cm}\n\\caption{\\label{plot3} The spectrum of the field \n  $\\langle y_k^2\\rangle$ at horizon crossing as a function of the\n  dissipative parameter $Q_*$, for different inflationary models, $c=3$\n  and no shear $\\bar \\zeta_s=0$. The solid line is the result at zero\n  order in slow-roll; dashed lines are for a chaotic model with\n  $p=$6,4,2, from top to bottom; the dash-dotted line is a quadratic\n  hybrid model. \n} \n\\end{figure}\n\nNeglecting the evolution of the background variables and working at\nzero order in the slow roll parameters,  is a good approximation when\n$Q$ is large enough, and the background parameters hardly vary during\nthe last 60 e-folds of inflation. Indeed the spectrum depends mainly\non the value of the parameters in a smaller interval, 5-6 e-folds\naround horizon crossing, where one may expect the approximation of\nkeeping them constant to be a fairly good one. Still, this is a model\ndependent question. This can be seen in {}Fig. \\ref{plot3}, where we\nshow the field spectrum for some generic inflationary models, and\n$c=3$, $\\bar \\zeta_s=0$. The value of  \n$\\langle y_k^2 \\rangle_*$ has been obtained integrating\nEqs. (\\ref{field})-(\\ref{momentumrGI}), evaluating the amplitude of\nthe comoving curvature spectrum at $N_e=20$ efolds after horizon crossing, and\nusing Eq. (\\ref{PRyy}). We have considered the inflationary models:\n\\begin{align}\nV& = \\frac{V_0}{p} \\left( \\frac{\\phi}{m_P}\\right)^p\\,, \\;\\;\\; & {\\rm\n  (chaotic)} \\,, \\label{chaotic}\\\\ \nV& = V_0 \\left[ 1 + \\frac{\\eta_\\phi}{2} \\left( \\frac{\\phi}{m_P}\\right)^2\n\\right] \\,, \\;\\;\\; & {\\rm (hybrid)}  \\label{hybrid}\\,. \n\\end{align} \n{}For the chaotic model we have run different powers $p=$ 6, 4, 2, and\nset $V_0= 10^{-14} m_P^4$, while for the hybrid $V_0= 10^{-8}m_P^4$\nand $\\eta_\\phi=3$. For each model, the initial value of the inflation\nfield is chosen such that we can have $N_e \\simeq 64$, and from the\nbackground slow-roll equations one derives the initial values of the field\nderivative, $\\rho_r$ and $Q$. We have chosen $k= 100 H_i$, $H_i$ being\nthe initial value of the Hubble parameter. Therefore, horizon crossing\n$k = a_* H_*$ takes places at around 60 e-folds before the end of\ninflation. \n\n\nIn all the examples considered above the dissipative coefficient\nincreases during inflation \\cite{BasteroGil:2009ec}, and the larger the\npower in the potential, the slower the evolution of the background\nvalues.  In the plot, the solid line is the result obtained with constant\n background variables. For a quartic chaotic model or larger power, the\napproximation at zero order in the slow-roll works fine, while for a\nquadratic chaotic it tends to overestimate the spectrum by at least an order of\nmagnitude for $Q \\gtrsim 50 $. {}For the hybrid model, the model\ndependence shows up when $Q \\lesssim 100$. \n\n\nShear effects will further damp the growth of the fluctuations.  \nIn Eq. (\\ref{fullwk}) the shear acts as an extra friction when the\nfluctuations are still subhorizon, suppressing the amplitude of the\nradiation fluctuation before the \nradiation-field system becomes effectively coupled. Whenever the shear\nis large enough, this suppression indeed can prevent altogether the\ngrowth of the field perturbations, as the amplitude of the radiation\nfluctuation is not enough to affect that of the field before horizon\ncrossing. This will happen when $\\bar \\zeta_s= \\zeta_s H\/(3 \\rho_r)\n\\gtrsim 1$ at horizon crossing. During warm inflation, we have the\ncatalyst field \ncoupled to the inflaton field, and to the light degrees of freedom\ngiving rise to the thermal bath. The shear viscosity for light fields\n(light with respect to the temperature $T$ of the thermal bath)\ntypically behaves \nas $\\zeta_s \\propto T^3$ \\cite{shear,Moore:2007ib}, although, depending on the\npattern of interactions, other powers could be\npossible and cannot be excluded. Nevertheless, the damping  is fully\ncontrolled by the value of the dimensionless parameter $\\bar \\zeta_s$ at\nhorizon crossing, and {\\it is independent of  the functional form of the shear\nwith the temperature}, as can be seen in\n{}Fig. \\ref{plot4}. We have integrated the full EOM without\napproximations Eqs. (\\ref{field})-(\\ref{momentumrGI}), for a quartic\nchaotic model, Eq. (\\ref{chaotic}) with $p=4$, and set the initial\nbackground values such that $Q_*\\simeq 40$. We show in the plot  \nthe dependence of the field spectrum with $\\bar \\zeta_s$ at horizon\ncrossing, for different values of $c$, and two \nexamples of the shear viscosity: one proportional to $T^3$ (solid lines), and\nanother linear in $T$ (dashed lines), however the curves lay on top of each\nother. We have checked that this is independent of the inflationary\nmodel considered. We have normalized the field spectrum with the value obtained\nwhen $c=0$. As the shear\nincreases, it does damp the radiation fluctuation enough for the\nfield spectrum to approach the $c=0$ value. Asymptotically, when $\\bar\n\\zeta_s \\gg 1$, for a linear dissipative coefficient with $T$ one\npractically recovers the\n$c=0$ case, for a cubic one the field spectrum is $\\simeq 2 \\langle\ny_k^2\\rangle_0\/5$, while the inverse $T$ case is slightly above $\\simeq\n5 \\langle y_k^2 \\rangle_0\/2$. In all cases, the field spectrum is well\nfitted by a function:\n\\begin{equation}\n\\log_{10} \\frac{\\langle y_k^2 \\rangle}{\\langle y_k^2\\rangle_0 } \\simeq \nA_s - B_s\\left[1+ \\tanh \\left( \\log_{10}\\bar \\zeta_s + \\Delta_s\\right)\\right] \n\\,, \\label{fittingyys}\n\\end{equation}\nwhich interpolates between the result with no shear $\\sim 10^{A_s}$ and \n the  $c=0$ case, modulo a normalization constant $\\sim\n 10^{A_s- 2 B_s}$. As an example, the coefficients $A_s$, $B_s$ and\n $\\Delta_s$ for the  potential  shown in Fig. \\ref{plot4} with\n $Q_*=40$ are given in Table\n   \\ref{table2}.  \n\n\\begin{table}\n\\begin{tabular}{|c|c|c|c|}   \n \\hline\n$c$ & $A_s$ & $B_s$ & $\\Delta_s$ \\\\\n\\hline\n~~~3~~~ & ~~~6.35~~~ & ~~~3.4~~~ & ~~~1.36~~~ \\\\\n~~~1~~~ & ~~~1.9~~~ & ~~~1.2~~~ & ~~~1.33~~~ \\\\\n~~~-1~~~ & ~~~-0.95~~~ & ~~~-0.7~~~ & ~~~-0.66~~~ \\\\\n\\hline\n\\end{tabular} \n\\caption{\\label{table2} Coefficients for the numerical fit of the\n  spectrum of the field with the shear viscosity,\n  Eq. (\\ref{fittingyys}).}\n\\end{table}\n\n\\begin{figure}[t]\n\\vspace{0.5cm}\n\\includegraphics[width=0.55\\textwidth,angle=0]{spectra_chaot_Qiter_kk100_cshear_T_p.eps}\n\\vspace{0.25cm}\n\\caption{\\label{plot4} Field spectrum normalized by the value with\n  $c=0$, as a function of the shear parameter $\\bar \\zeta_s$, for\n  different values of $c$, and $Q=40$. From top to bottom, $c=3,\\,1,\\,-1$. For\n  each curve, we have also considered two different $T$ dependence on\n  the shear viscosity, as indicated in the legend, but both gives the\n  same field spectrum. } \n\\end{figure}\n\nFinally, combining Eq. (\\ref{fittingyys}) with Eq. (\\ref{yyapprox}),\nthe field spectrum when $c>0$ reads:\n\\begin{equation}\n\\langle y_k^2 \\rangle_* \\simeq \\langle y_k^2 \\rangle_0 F_Q[Q]^{F_s[\\bar \\zeta_s]} \\,,\n\\end{equation}\nwhere: \n\\begin{align}\nF_Q[Q] &\\simeq \\left( A_c Q^\\alpha + B_c Q^\\beta \\right) \\,, \\\\\nF_s[\\bar \\zeta_s]& \\simeq \\frac{1}{2}\\left[1- \\tanh \\left( \\log_{10}\\bar \\zeta_s + \n\\Delta_s\\right)\\right] \\,. \n\\end{align}\nTherefore, in the strong dissipative regime when $Q>1$, the primordial\nspectrum of the curvature perturbation can be written as that obtained\nfor a $T$ independent dissipative coefficient, times an enhancement\nfunction $F_Q[Q]$ depending on the dissipative ratio $Q$, but\ncontrolled by a function depending on the shear $F_s[\\bar \\zeta_s]$ :\n\\begin{equation}\nP_{\\cal R} \\simeq \\left( \\frac{H^2}{\\dot \\phi} \\right)^2\n\\frac{\\sqrt{3\\pi}}{4 \\pi^2}\\left[\\frac{(1 +\n  Q)^{3\/2}}{1+3Q}\\right]  \\left(\\frac{T}{H}\\right) \\times F_Q[Q]^{F_s[\\bar \\zeta_s]} \\,, \n\\end{equation}\nand whenever $\\bar \\zeta_s > 1$ one recovers the amplitude of the\nprimordial spectrum obtained when $c=0$. The latter is of a magnitude\ncompatible with the observational value, for model parameters values\ncommon in inflationary model building \\cite{BasteroGil:2009ec,Zhang:2009ge}. For\nexample, without the enhancement, a quartic warm chaotic\nmodel gives rise to the right level of perturbations with a coupling\nconstant $\\lambda \\simeq 10^{-13}-10^{-14}$. From the point of view of\nmodel building, the enhancement produced\nby the backreaction of the radiation fluctuations onto the fields, if\nnot avoided by shear effects, can be compensated by lowering the\nheight of the potential, i.e., by lowering couplings and\nmasses. However, it will also have an impact on the prediction for the\nspectral index: for models with $Q$ increasing at the time of horizon\ncrossing it will render the spectrum too blue-tilted, and the other\nway round, for $Q$ decreasing the spectrum may become too\nred-tilted.  Shear viscosity damps the growth of the fluctuations, and\ntherefore will also affect the spectral index. \n\nEven though the results we have obtained are fairly model independent\nand shown not to depend on the specific dependence on the temperature\nof shear viscosity, it is useful to estimate the\nmagnitude of the ratio $\\zeta_s H\/(3 \\rho_r)$ for typical warm\ninflation models. In kinetic theory, the shear viscosity for\nrelativistic fluids can be expressed parametrically as proportional to\nthe mean free path of quasiparticles in the fluid. Considering as an\nexample a radiation fluid made of relativistic scalar particles\n$\\sigma$, with mass $m_\\sigma\/T \\ll 1$ and self-interaction potential\n$\\lambda_\\sigma \\sigma^4\/4!$, the mean free path of quasiparticles is\ndetermined by the inverse of the thermal width, which is ${\\cal\n  O}(\\lambda_\\sigma^2)$, and the computed value for the shear\nviscosity is \\cite{jeon} $\\zeta_s \\simeq 3 \\times 10^3\nT^3\/\\lambda_\\sigma^2$.  The condition $\\zeta_s H\/(3 \\rho_r) \\gtrsim 1$\ncan then be expressed, for example, as a condition on the magnitude of\nthe radiation bath self-interaction, $\\lambda_\\sigma \\lesssim 55\n\\sqrt{H\/T}$. Since warm inflation requires $T > H$ and in general we\ntypically work with values $T \\gg H$, we find that weakly interacting\nradiation fluids can easily have shear viscosities of sufficient\nmagnitude to counterbalance and suppress completely the growth of\nfluctuations caused by the coupling of the inflaton's fluctuations with\nthose from the radiation.\n\n\nIn supersymmetric warm inflation models however the radiation bath\nself coupling is the same as the coupling to the catalyst field, which\nenters in the calculation of the dissipative coefficient. In general,\nlarge multiplicities of the fields (catalyst $\\chi$ and radiation $\\sigma$)\nare required in order to have enough dissipation. Considering\na model with ${\\cal N}_\\chi$, ${\\cal N}_\\sigma$ copies of the complex fields,\nwith common coupling $h$ among the $\\chi$'s and $\\sigma$'s, the\nself-interaction potential is given by $h^2 \\sum_i |\\sigma|^2\/4$. \nIn the low-$T$ regime,  the dissipative and shear coefficients can be\nwritten as \\cite{BasteroGil:2010pb}:\n\\begin{align}\n\\Upsilon & \\simeq 0.1 h^4 {\\cal N}_\\chi{\\cal N}_\\sigma^2 \\frac{T^3}{\\phi^2}\n\\,, \\label{upslowT}\\\\\n\\zeta_s &\\simeq 127 N_\\sigma (1 + 0.3 h)\\frac{T^3}{h^4} \\,, \\label{zetaslf}\n\\end{align}\nwhere we have included the next-to-leading order correction in the\nshear viscosity \\cite{Moore:2007ib}.  \nThe condition $\\bar \\zeta_s \\geq 1$ then reads:\n\\begin{equation}\nh^4 \\lesssim 128 \\frac{{\\cal N}_\\sigma}{g_*} \\frac{H}{T} \\simeq \n69 \\frac{H}{T} \\,,\n\\end{equation}\nwhere $g_*\\simeq 15 {\\cal N}_\\sigma\/8$. Although \nin a bath of weakly interacting radiation particles, $h \\ll 1$,   \nthe shear viscosity would be rather large, it may not give rise to  enough\ndissipation to sustain a period of warm inflation, unless the weakness of\nthe coupling is compensated by having a large multiplicity for the fields.   \nThis therefore becomes a model dependent question depending on the\nparameters ${\\cal N}_\\chi$, ${\\cal N}_\\sigma$ and the coupling\n$h$. In Fig. \\ref{plot5} we show an example for a quartic chaotic\nmodel, where we compare the value of the field spectrum $\\langle\ny^2_k\\rangle_*$ as a function\nof $Q_*$ for different values of the Yukawa coupling $h$.  \nWe have kept the value ${\\cal\n  N}_\\sigma=10^3$ fixed, and vary the value of ${\\cal N}_\\chi$ to get\ndifferent values of $Q_*$. From top to bottom the value of $h$\ndecreases, starting with the largest possible one $h=\\sqrt{4\\pi}$\n(solid line) for which there is a negligible shear effect. In this case,  \na value $Q_*\\simeq 10$ requires ${\\cal N}_\\chi=35 $, while for $Q_*\\simeq\n100$ we need ${\\cal N}_\\chi \\simeq 110$. As the value of $h$\ndecreases, the multiplicity of the field to get the same value of\n$Q_*$ increases by a factor $(4\\pi\/h^2)^2$.  For $h=1.12$ and $Q_*=100$,\nwe would need ${\\cal N}_\\chi \\simeq 10^4$. \nAs the value of $Q_*$ increases, so it does the\nratio $T\/H$, and therefore the parameter $\\bar \\zeta_s$\ndecreases along the curves. Although  in this example \nviscous effects are not \nenough to completely avoid the growth of the perturbations, \nthey bring  it down to $\\langle y^2_k\\rangle_* \\propto Q^{2.5}$ for\n$h=1,12$ and $\\langle y^2_k\\rangle_* \\propto Q^{4.5}$ for $h=1.67$,\ninstead of $\\langle y^2_k\\rangle_* \\propto Q^7$.      \n\n\\begin{figure}[t]\n\\vspace{0.5cm}\n\\includegraphics[width=0.55\\textwidth,angle=0]{spectra_chaot_Qiter_kk100_pp4_shear_p.eps}\n\\vspace{0.25cm}\n\\caption{\\label{plot5} Field spectrum as a function of $Q_*$, for a\n  quartic chaotic model, with a cubic dissipative coefficient\n  (Eq. \\ref{upslowT}) and $\\zeta_s$ given by Eq. (\\ref{zetaslf}). We\n  have taken ${\\cal N}_\\sigma=10^3$ and $h=\\sqrt{4\\pi}$ (solid line),\n  $h=1.67$ (dashed line) and $h=1.12$ (dot-dashed line). \nWe include for comparison the result with $c=0$. \n} \n\\end{figure}\n\n\n\\section{Conclusions}\n\nDensity perturbations in warm inflation are seeded by thermal\nfluctuations of the inflaton. In warm inflation the inflaton decays into\nradiation through a dissipation term in the inflaton's equation of\nmotion and that originates from the microphysical interactions of the\ninflaton field with other degrees of freedom of the\nmicroscopic Lagrangian describing the complete system. The origin of\nthe dissipation term and its quantum field theoretical treatment has\nbeen extensively discussed in the literature (for a recent review, see\ne.g. Ref. \\cite{Berera:2008ar}).  However, as radiation is produced\nduring the inflaton's evolution, the full treatment of the spectrum of\nperturbations no longer involves only that of the inflaton's\nperturbations but must also account for radiation perturbations.  This\nmakes the treatment of density perturbations in warm inflation \nsimilar to\na multifluid system. Since the larger is the dissipation the larger is\nexpected to be the rate of radiation production, it becomes important\nthe study of how the produced radiation and its perturbations\nbackreacts on the inflaton's evolution and respective\nperturbations. In Ref. \\cite{mossgraham} it was shown that as\na consequence of this backreaction, the infaton's perturbations can\ngrow as the dissipation of the inflaton increases. This increasing of\nthe inflaton's perturbations with increasing dissipation can,\ntherefore, severely constrain the model parameters in warm inflation\nso as to cope with the known measured results for the CMB\nradiation. This backreaction of the produced radiation on the\ninflaton's perturbations is larger the stronger is the coupling\nbetween the radiation and the inflaton; in particular\nthe larger the power of $T$ in the dissipative coefficient,\nthe stronger the $T$ dependence on the perturbations.\nThis is exemplified by the results shown in\n{}Fig. \\ref{plot2}.\n \nIn this paper we have studied how this backreaction of the produced\nradiation, that can lead to this growth mode in the inflaton's\nperturbations, can be counterbalanced by the dissipative effects within\nthe radiation fluid. Dissipative effects in the radiation fluid\nitself are described by viscosity terms.  This is expected when the\nradiation fluid departs from equilibrium, which is the case in any\ndissipative system, where the produced radiation from the system does\nnot immediately equilibrate in the radiation bath and its approach to\nequilibrium is controlled by viscosity coefficients, like the shear\nviscosity, the bulk viscosity and heat transport coefficients.  We\nhave focused on the dissipation effect as coming dominantly from a\nshear viscosity term in the fluctuation equations. We have then shown\nthat the shear viscosity can effectively damp the radiation\nfluctuations so as to avoid altogether the appearance of the growth mode\nin the resulting perturbations. The results we have obtained are model\nindependent and we have shown that the overall effect of compensation\nof the growth mode and its control is determined by the ratio $\\zeta_s\nH\/(3 \\rho_r)$, where $\\zeta_s$ is the shear viscosity coefficient, $H$\nis the Hubble parameter and $\\rho_r$ the radiation energy density.\nWhen the ratio is ${\\cal O}(1)$ or larger, the growth mode disappears\ncompletely.\n\n\nIn this work we have only considered the coupling between the\nfluctuations in the inflaton field with those in \nthe radiation through the\n(temperature dependence on the) dissipation coefficient in the\ninflaton's dynamics. But in a thermal bath, the parameters of the\ninflaton's potential can also adquire temperature corrections. Even\nthough these thermal corrections can be kept under control and small\nin Supersymmetry model building realizations for warm inflation\n\\cite{BR1,Berera:2008ar,BasteroGil:2009ec}, they can still be large enough\nto provide extra sources of couplings between inflaton and radiation\nfluctuations and it should be interesting to analyze their effects in a\nfuture work. Likewise, there can be additional sources of dissipation\nin the radiation fluid, for example as coming from bulk viscosity,\nthat can further help to damp any leftover growing modes as resulting from\nthese additional couplings. In this work we have neglected the effects\nof the bulk viscosity on the grounds that it is in general much\nsmaller than the shear viscosity. {}For example, for the\nself-interacting scalar field radiation discussed in section IV, the ratio of\nthe bulk viscosity, $\\zeta_b$, with the shear viscosity for a high\ntemperature radiation fluid is $\\zeta_b\/\\zeta_s\\sim 10^{-9}\n\\lambda_\\sigma^3$, thus, it is negligible for a weakly interacting\nradiation bath. But there may be other interactions and energy\nregimes for the radiation fluid in which the bulk can be sizable and lead\nto effects in the density perturbation evolution (see\ne.g. Ref. \\cite{giovaninni}). \n\n\nAll these effects, starting with the shear viscosity, will also impact\nthe second order evolution of the perturbations and thus the\ncalculation of the non-gaussinity. Forthcoming cosmological data are\nexpected very soon to set the level of non-gaussinity of the\nprimordial spectrum, which clearly will help to discriminate among inflationary\nmodels. Warm inflation, being a type of \nmulti-fluid model, falls into the category of models with a\nnon-negligible value of the non-linearity parameter $f_{NL}$ for\nnon-gaussinity. This parameter has been computed for a $T$ independent\ndissipative coefficient \\cite{nongauss}, and recently the $T$\ndependence of $\\Upsilon$ has been included \\cite{Moss:2011qc}, which\nprovides an extra \nnon-linear source in the field second order equation. However, if the\ncoupling between field and radiation perturbations at first order is\nsuppressed by viscous effects, qualitatively we expect the same to\nhappen at second order. The question then is whether one simply\nrecover the prediction for a constant dissipative coefficient, or\nnon-linearities are further suppressed by viscous effects. \nThese and other effects mentioned above will be studied\nelsewhere.\n\n\\acknowledgements\n\n\nA.B. acknowledges support from the STFC. R.O.R\nis partially supported by Conselho Nacional de\nDesenvolvimento Cient\\'{\\i}fico e Tecnol\\'ogico (CNPq - Brasil).\nM.B.G. would like to thank the hospitality of the School of \nPhysics and Astronomy at the\nUniversity of Edinburgh which during her visit this work has started.\nMBG is partially supported by MICINN (FIS2010-17395) and ``Junta de\nAndaluc\\'ia\" (FQM101), and by SUPA during the realization of this work\nin the UK.    \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:level1}Introduction }\n\nOne of the predictions of general theory of relativity is the existence of gravitational waves. The sources of  generation of these waves  vary  from   the dynamics of early universe to  massive  astrophysical objects  such as  neutron star binaries and black hole mergers etc. Thus the    waves have a  wide spectrum   of frequencies vary from very low to high ${\\cal{O}}($10$^{-19}$Hz -10$^{10}$Hz). It is possible to discriminates the  relic waves from other sources on the observational point of view also.   The relic gravitational waves are   paramount importance in cosmology because it provides valuable information on the conditions of very early universe.   It is believed  that the relic gravitational waves are mainly generated during inflationary epoch.  And the waves that amplified  during the inflation are low frequency only.  Since the higher frequency  waves are  outside the  ``barrier'' (horizon)\\cite{gh}  \\footnote{ the terminology ``barrier\" is adopted from \\cite{gh}.} the corresponding  amplitudes decreased during the evolution of the universe. \nThe features of  relic gravitational  waves  of  very high frequency range  is interesting   though  the energy scale of conventional inflationary models  are not favoring for it. \nHowever  if  extra dimensions exist (for a review, see \\cite{rex} and motivation for extra dimensions \\cite{ex}) the graviton background  can have a thermal spectrum \\cite{fry}. According to the  extra dimensional models  the  thermal gravitons with very high frequency range also be observed with  a specific peak temperature  today  \\cite{fry} and  \n therefore  the detection of very high  frequency thermal gravitational waves is  an interesting test to   see the possibility of existence of extra dimensions as well. \nThese  thermal gravitational waves   can contribute to the  higher frequency range of the spectrum.    \nThe \nexistence of  thermal graviton background\nwith  the black body type spectrum  is   also discussed in \\cite{24},\\cite{25}. \nIf  the inflation was preceded by a radiation era, then there would be  thermal gravitational waves at the time of inflation \\cite{25}. The  generation of tensor perturbations during inflation  by  the stimulated emission   process leads  to the existence of  thermal  gravitational waves \\cite{29}.  \nDirect detection of the thermal gravitons is challenging\nbut may be possible in the near future with the 21-cm\nemission line of atomic hydrogen. \n  \n  \n  The inflationary scenario \\cite{23} predicts a\nstochastic cosmic background of gravitational waves\n(CGWB) \\cite{24}.\nThe spectrum of  these relic gravitational waves  depends\nnot only on the details  of expansion  during the  inflationary era  but also \n the subsequent stages, including the current  epoch of the universe. \nComputation of  the  spectrum of the  waves   for   matter dominated   universe  is usually  done in  decelerated expanding model \\cite{1}-\\cite{6}. The resulting spectrum is used  for putting constrain on the detection of  gravitational waves  originated from  sources other than  early universe epoch.\nThe result of  astronomical\nobservations on SN Ia \\cite{11}-\\cite{12} shows that the universe is currently under going  accelerated\nexpansion  indicating a non-zero cosmological \nconstant.  According to the $\\Lambda$CDM\nconcordance model, the observed acceleration of the present universe is supposed to be driven by the dark energy. \n Effect of  the current acceleration  on the nature  of  the  spectrum  and  spectral energy density of the relic\ngravitational waves  is studied  \\cite{2},\\cite{15}.  And shown that  the current acceleration phase of the universe  does change  the shape, amplitude and spectrum  of the  waves  \\cite{15}.\n\n\nIn the present work, we   consider   contribution of  very higher frequency  relic  thermal gravitational waves  to its  spectrum and spectral  energy density  for the decelerated as well as  accelerated universe.  The focus of the present work is on the spectrum of  the higher frequency range of the  waves due to extra dimensional effects.\nThe  normalization  of the spectrum is being done with the measured  CMB  anisotropy spectrum of the WMAP.   The inclusion of the higher frequency relic thermal gravitational waves    leads to  enhancement of the spectrum.   This  enhancement  leads to  modification of the amplitude of the spectrum  in the frequency range (10$^{-16}$ -10 $^{8}$ Hz)   as an additional  feature  and is  possible to  compare these with the sensitivity of Advanced.LIGO (Adv.LIGO), Einstein Telescope  (ET) and LISA missions.  The corresponding spectral energy density can be compared with estimated upper bound of various studies.\n Also can check whether the  inclusion of  higher frequency thermal gravitational waves   in  the  total spectral energy density  exceed  the  upper  bound   of  primordial nucleosynthesis rate or not.\n In the present work, we use the unit $c=\\hbar = k_{B} =1$.\n \n  \n \\section{Gravitational waves spectrum in expanding universe}\nThe perturbed metric for a homogeneous  isotropic  flat Friedmann-Robertson-Walker (FRW) universe  can be written   as\n\\begin{equation}\nd s^{2}= S^{2}(\\eta)(d\\eta^{2}-(\\delta_{ij}+h_{ij})dx^{i}dx^{j}),\n\\end{equation}\nwhere $S(\\eta)$ is the cosmological scale factor,  $\\eta$ is the conformal time and $\\delta_{ij}$ is the Kronecker delta  symbol.  The $h_{ij} $ are  metric perturbations  field contain only the pure gravitational waves and is  transverse-traceless i.e; $\\nabla_i h^{ij} =0, \\delta^{ij} h_{ij}=0$.\n\nThe present study mainly deals  with amplitude and spectral energy density of  the  relic gravitational waves   generated by the expanding spacetime  background. Thus  the perturbed matter source is therefore not taken into account. The \n gravitational waves are described with  the  \n linearized field equation given by\n \\begin{equation}\\label{weq}\n \\nabla_{\\mu} \\left( \\sqrt{-g} \\, \\nabla^{\\mu} h_{ij}(\\bf{x}, \\eta)\\right)=0.\n \\end{equation} \nThe tensor perturbations have two independent physical degrees of freedom  and are denotes as $h^{+}$ and $h^{\\times}$, called polarization modes. To compute the spectrum of gravitational waves $h(\\bf{x},\\eta)$ in the thermal states, we express $h^{+}$ and $h^{\\times}$ in terms\nof the creation ($a^{\\dagger}$) and annihilation ($a$) operators,\n\\begin{eqnarray}\\label{1}\n\\nonumber  h_{ij}({\\bf x},\\eta)=\\frac{\\sqrt{16\\pi} l_{pl}}{S(\\eta)} \\sum_{\\bf{p}} \\int\\frac{d^{3}k}{(2\\pi)^{3\/2}} {\\epsilon}_{ij} ^{\\bf {p}}(\\bf {k}) \\\\\n \\times  \\frac{1}{\\sqrt{2 k}} \\Big[a_{\\bf{k}}^{\\bf {p}}h_{\\bf {k}}^{\\bf {p}}(\\eta) e^{i \\bf {k}.\\bf {x}} +a^{\\dagger}_{\\bf {k}} {^{\\bf {p}}} h^{*}_{\\bf {k}}{^{\\bf {p}}} (\\eta)e^{-i\\bf{k}.\\bf{x}}\\Big],\n\\end{eqnarray}\nwhere  $\\bf{k}$ is the comoving wave\nnumber, $k=|\\bf {k}|$, $l_{pl}= \\sqrt{G}$ is the\nPlanck's length and $\\bf{ p}= +, \\times$ are polarization modes. The polarization tensor \n$\\epsilon_{ij} ^{{\\bf p}}({\\bf k})$ is symmetric and transverse-traceless  $ k^{i} \\epsilon_{ij} ^{{\\bf p}}({\\bf k})=0, \\delta^{ij} \\epsilon_{ij} ^{{\\bf p}}({\\bf k})=0$ and \nsatisfy  the conditions $\\epsilon^{ij {\\bf p}}({\\bf k})   \\epsilon_{ij}^{{\\bf p}^{\\prime}}({\\bf k})= 2  \\delta_{ {\\bf p}{{\\bf p}}^{\\prime}} $ and $ \\epsilon^{{\\bf p}}_{ij} ({\\bf -k}) = \\epsilon^{{\\bf p}}_{ij} ({\\bf k}) $, the creation and annihilation operators  satisfy\n$[a_{{\\bf k}}^{{\\bf p}},a^{\\dagger}_{{\\bf k} ^{\\prime}} {{^{{\\bf p}}}^{\\prime}}]= \\delta_{{{\\bf p}} {\\bf {p}}^{\\prime} }\\delta^{3}({\\bf k}-{{\\bf k}}^{\\prime})$, the initial vacuum state is defined  as\n\\begin{equation}\na_{\\bf{k}}^{\\bf{p}}|0\\rangle = 0,\n\\end{equation}\nfor each $\\bf {k}$ and $\\bf {p}$. The energy density of the gravitational waves in vacuum state is $ t_{00}= \\frac{1}{32 \\pi l^2_{pl}} h_{ij,0} h^{ij}_{,0}$.\n\n For a fixed  wave number $\\bf{k} $ and a fixed polarization state $\\bf{p}$ the linearized wave equation (\\ref{weq}) gives\n \\begin{equation}\\label{zz1}\nh^{\\prime \\prime}_{k}+2\\frac{S^{\\prime}}{S}h^{\\prime}_{k}+k^{2}{h}_{k}=0,\n\\end{equation}\nwhere   prime means derivative with respect to the conformal time. Since the  polarization states are  same,   we here onwards denote  $h_{k}(\\eta)$ without the polarization  index.  \n\nNext, we  rescale the filed $h_{k}(\\eta)$ by taking\n$h_{k}(\\eta)=f_{k}(\\eta)\/S(\\eta)$, where the mode functions $f_{k}(\\eta)$ obey the minimally coupled Klein-Gordon equation\n\\begin{equation}\\label{zz}\nf^{\\prime \\prime}_{k}+\\Big(k^{2}-\\frac{S^{\\prime \\prime}}{S} \\Big)f_{k}=0.\n\\end{equation}\n The  general solution of the above equation  is a linear combination of  the Hankel function with a  generic power-law for the scale factor $S= \\eta^{q}$  given by\n\\begin{equation}\nf_k (\\eta)= A_k \\sqrt{ k \\eta} H^{(1)} _{(q-\\frac{1}{2})} (k \\eta)+  B_k \\sqrt{ k \\eta} H^{(2)}_{(q-\\frac{1}{2})} (k \\eta).\\end{equation}\nFor a given  model of the expansion of universe, consisting of a sequence of successive  scale factor with different $q$, we can obtain an exact solution  $f_k (\\eta)$ by matching its value and  derivative at the joining points.\n\n The  approximate computation of the spectrum of  gravitational  waves  is usually performed in two limiting cases depending up on the waves that are within  or outside of the  barrier. For the  gravitational  waves  outside barrier ($k^{2}\\gg S^{\\prime \\prime}\/S$, short wave   approximation)  the corresponding   amplitude  decrease as $h_k \\propto 1\/S(\\eta) $ while for the  waves inside the barrier ($k^{2} \\ll S^{\\prime \\prime}\/S$, long wave  approximation),  $h_k = C_k $ simply a constant. Thus these results can be used to estimate the spectrum for the present epoch of  universe.\n\n\nThe history of  overall expansion of the universe can be modeled as  following sequence\nof successive epochs of power-law expansion.\n\nThe initial stage (inflationary)\n\\begin{equation}\nS(\\eta)=l_{0}|\\eta |^{1+\\beta},\\;\\;\\;\\;\\;\\;-\\infty <\\eta\\leq \\eta_{1},\n\\end{equation}\nwhere $1+\\beta <0$, $\\eta<0$ and $l_{0}$ is a constant.\n\nThe z-stage\n\\begin{equation}\nS(\\eta)=S_{z}(\\eta - \\eta_{p})^{1+\\beta_{s}},\\;\\;\\;\\;\\;\\;\\eta_{1}<\\eta\\leq \\eta_{s},\n\\end{equation}\nwhere $\\beta_{s}+1>0$. Towards the end of inflation, during the reheating, the equation of state of\nenergy in the universe can be quite complicated and is rather model-dependent \\cite{qa}. Hence this\nz-stage is introduced to allow a general reheating epoch, see for details \\cite{3}.\n\nThe radiation-dominated stage\n\\begin{equation}\nS(\\eta)=S_{e}(\\eta-\\eta_{e}),\\;\\;\\;\\;\\;\\;\\eta_{s}\\leq \\eta \\leq \\eta_{2},\n\\end{equation}\n\nThe matter-dominated stage\n\\begin{equation}\nS(\\eta)=S_{m}(\\eta-\\eta_{m})^{2},\\;\\;\\;\\;\\;\\;\\eta_{2}\\leq \\eta \\leq \\eta_{E},\n\\end{equation}\nwhere $\\eta_{E}$ is the time when the dark energy density $\\rho_{\\Lambda}$ is equal to the matter energy density $\\rho_{m}$. Before the discovery of  accelerating expansion of the universe, the current expansion is\nused to take  as decelerating one  because of  the matter-dominated stage. Thus, following\nthe matter-dominated stage,  it reasonable to add an epoch of accelerating stage, which is probably driven\nby either the cosmological constant, or the quintessence, or some other kind of condensate \\cite{sa}. The value of  redshift $z_{E}$ at  $\\eta_{E}$ is  $(1+z_{E})=S(\\eta_{0})\/S(\\eta_{E})$, where $\\eta_{0}$ is the present time.  Since $\\rho_{\\Lambda}$ is constant and $\\rho_{m}(\\eta) \\propto S^{-3}(\\eta)$, we get\n\\begin{equation}\n\\frac{\\rho_{\\Lambda}}{\\rho_{m}(\\eta_{E})}=\\frac{\\rho_{\\Lambda}}{\\rho_{m}(\\eta_{0})(1+z_{E})^3}=1.\n\\end{equation}\nIf  the current value of $\\Omega_{\\Lambda}\\sim0.7$ and $\\Omega_{m}\\sim0.3$, then it follows that\n\\begin{equation}\n1+z_{E}=\\Big(\\frac{\\Omega_{\\Lambda}}{\\Omega_{m}}\\Big)^{1\/3}\\sim 1.33.\n\\end{equation}\n\nThe accelerating stage (up to the present)\n\\begin{equation}\\label{1w}\nS(\\eta)=\\ell_{0}|\\eta- \\eta_{a} |^{-1},\\;\\;\\;\\;\\;\\;\\eta_{E}\\leq \\eta \\leq\\eta_{0}.\n\\end{equation}\nThis stage describes the accelerating expansion of the universe.  And  is a  new feature and hence  its  influence  on  the spectrum of relic gravitational waves is of interesting to study. It is be  noted  that the actual scale factor function $S(\\eta)$ differs from equation (\\ref{1w}), since\nthe matter component exists in the current universe. However, the dark energy is dominant,\ntherefore (\\ref{1w}) is an approximation to the current expansion behaviour.\n\nGiven $S(\\eta)$ for the various epochs, the derivative $S^{\\prime}=dS\/d\\eta$ and  ratio $S^{\\prime}\/S$ follow\nimmediately. Except for  $\\beta_{s}$ which is imposed upon as the model parameter, there are ten\nconstants in the  expressions of $S(\\eta)$. By the continuity conditions of $S(\\eta)$ and $S^{\\prime}(\\eta)$ at\n four given joining points $\\eta_{1}, \\eta_{s}, \\eta_{2},$ and $\\eta_{E}$, one can fix only eight constants. The remaining\ntwo constants can be fixed by the overall normalization of $S$ and  the observed Hubble\nconstant as the expansion rate. Specifically, we put $|\\eta_{0}-\\eta_{a}|=1$ for the normalization of $S$, which fixes the  $\\eta_{a}$, and the constant $\\ell_{0}$ is fixed by the following calculation,\n\\begin{equation}\n\\frac{1}{H}\\equiv \\Big(\\frac{S^{2}}{S^{\\prime}}\\Big)_{\\eta_{0}}=\\ell_{0}.\n\\end{equation}\nwhere $\\ell_{0}$ is  the Hubble radius at present. \n\n In the expanding Friedmann-\nRobertson-Walker spacetime the physical wavelength is related to the comoving wave\nnumber  as\n$\\lambda \\equiv \\frac{2\\pi S(\\eta)}{k},$\nand the wave number $k_{0}$ corresponding to the present Hubble radius is\n$k_{0}=\\frac{2\\pi S(\\eta_{0})}{\\ell_{0}}=2\\pi.$ And\nthere is another wave number\n$k_{E}=\\frac{2\\pi S(\\eta_{E})}{1\/H}=\\frac{k_{0}}{1+z_{E}},$\nwhose corresponding wavelength at the time $\\eta_{E}$ is the Hubble radius $1\/H$.\n\nBy matching $S$ and $S^{\\prime}\/S$ at the joint points, one gets\n\\begin{equation}\\label{kk}\nl_{0}=\\ell_{0}b\\zeta_{E}^{-(2+\\beta)}\\zeta_{2}^{\\frac{\\beta-1}{2}}\\zeta_{s}^{\\beta}\\zeta_{1}^{\\frac{\\beta-\\beta_{s}}{1-\\beta_{s}}},\n\\end{equation}\nwhere $b\\equiv|1+\\beta|^{-(2+\\beta)}$, which is defined differently from  \\cite{p}, $\\zeta_{E}\\equiv\\frac{S(\\eta_{0})}{S(\\eta_{E})}$, $\\zeta_{2}\\equiv\\frac{S(\\eta_{E})}{S(\\eta_{2})}$, $\\zeta_{s}\\equiv\\frac{S(\\eta_{2})}{S(\\eta_{s})}$, and $\\zeta_{1}\\equiv\\frac{S(\\eta_{s})}{S(\\eta_{1})}$. With these specifications, the functions $S(\\eta)$ and $S^{\\prime}(\\eta)\/S(\\eta)$ are fully determined. In particular, $S^{\\prime}(\\eta)\/S(\\eta)$ rises up during the accelerating\nstage, instead of decreasing as in the matter-dominated stage. This causes the modifications to the spectrum of relic gravitational waves.\n\n\\section{Gravitational waves spectrum in thermal vacuum state}\nThe power spectrum  of   gravitational waves is defined  as\n\\begin{equation}\\label{pow}\n\\int_0 ^\\infty h^2 (k,\\eta) \\frac{dk} {k} = \\langle 0 | h^{ij}({\\bf x},\\eta) h_{ij}({\\bf x},\\eta) |0 \\rangle,\n\\end{equation}\n Substituting equation  (\\ref{1}) in (\\ref{pow}) and taking the contribution from each polarization is same, we get\n \\begin{equation}\\label{pp}\n h(k,\\eta)= \\frac{4 l_{pl}}{\\sqrt{\\pi}} k \\mid h(\\eta) \\mid.\n \\end{equation} \nThus once the mode function $h(\\eta)$ is known, the spectrum $h(k,\\eta)$ follows.\n\nThe spectrum at the present time $ h(k,\\eta_0)$ can be obtained, provided the initial  spectrum is specified. The initial condition is taken to be the during the inflationary stage. Thus the initial amplitude of the spectrum is given by\n\\begin{equation}\\label{bet}\nh(k,\\eta_i)= A{\\left(\\frac {k}{k_0}\\right)}^{2+\\beta},\n\\end{equation}\nwhere $A=8\\sqrt{ \\pi} \\frac{l_{pl}}{l_0} $ is a constant. The  power spectrum for the primordial perturbation of energy density  is $P(k)\\propto {\\mid  h(k,\\eta_0)\\mid}^2$ and in terms of  initial spectral index $n$, it is defined as $ P(k) \\propto k^{n-1}$.\nThus the scale invariant spectral index  $n=1$ for the pure de Sitter expansion can be obtained   with the relation $n= 2 \\beta +5 $  for $\\beta $= - 2. \n \n\nAn effective approach to deals with the thermal  vacuum state is the thermo-field dynamics (TFD)\\cite{34}. In this approach  a tilde space  is needed besides the usual\nHilbert space, and the direct product space is made up of the these two spaces. Every operator and state in the Hilbert space has the corresponding counter part in the tilde\nspace \\cite{34}. Therefore a  thermal  vacuum state ($Tv$) can be  defined as\n\\begin{equation}\\label{16}\n|Tv\\rangle ={\\cal T }(\\theta_{k})|0\\; \\tilde{0}\\rangle,\n\\end{equation}\nwhere \n \\begin{equation}\\label{333}\n{\\cal T }(\\theta_{k})=\\mathrm{exp} [-\\theta_{k} (a_{\\bf {k}}\\tilde{a}_{\\bf {k}}-a_{\\bf{k}}^{\\dagger}\\tilde{a}_{\\bf {k}}^{\\dagger})],\n\\end{equation}\nis the thermal operator and $|0\\; \\tilde{0}\\rangle$ is the two mode vacuum state at zero temperature. The  quantity $\\theta_{k}$ is  related to   the average number of the thermal particle, $\\bar{n}_{k}=\\mathrm{sinh}^{2}\\theta_{k}$. The $\\bar{n}_{k}$  for given  temperature T is\nprovided   by the Bose-Einstein distribution \n$\\bar{n}_{k}=[\\mathrm{exp}( k \/T)-1]^{-1}$, \nwhere $\\omega_{k}$ is the resonance frequency of the field. The $a_{{\\bf k}}$, $a_{{\\bf k}}^{\\dagger}$  and  $\\tilde{a}_{{\\bf  k}}$, $\\tilde{a}_{{\\bf k}}^{\\dagger}$, are respectively the annihilation and creation operators in Hilbert and tilde space, satisfy the  usual commutation relations,  \n$[a_{{\\bf k}},a^{\\dagger}_{{\\bf k}^{\\prime}}]=\n[\\tilde{a}_{{\\bf k}},\\tilde{a}^{\\dagger}_{{\\bf k}^{\\prime}}]=\\delta^{3}({\\bf k}-{{\\bf k}^{\\prime}}) \\; $. And all other commutation relations   of these operators are zero.\n By the appropriate action of the operator (\\ref{333})  on $a_{{\\bf k}}$ and $a_{{\\bf k}}^{\\dagger}$, we  get \\cite{35}\n\\begin{eqnarray} \\label{tt}\n\\nonumber {\\cal T }^{\\dagger}a_{\\bf {k}}{\\cal T }=a_{\\bf {k}}\\;\\mathrm{cosh}\\; \\theta_{k} +\\tilde{a}_{\\bf {k}}^{\\dagger}\\; \\mathrm{sinh} \\; \\theta_{k},\\\\\n{\\cal T }^{\\dagger}a^{\\dagger}_{\\bf {k}}{\\cal T }=a^{\\dagger}_{\\bf {k}}\\; \\mathrm{cosh}\\; \\theta_{k} +\\tilde{a}_{\\bf{k}}\\;\\mathrm{sinh}\\; \\theta_{k}.\n\\end{eqnarray} \nHence  the  occupation number   in  thermal vacuum state\n can be written as\n\\begin{equation}\\label{w}\n\\langle a^{\\dagger}_{\\bf {k}} a_{\\bf {k} ^{\\prime}}\\rangle = \\left( \\frac{1}{e^{k\/T} -1} \\right)\\delta^{3}(\\bf {k}-\\bf {k}^{\\prime}).\n\\end{equation}\n Thus, using Eq.(\\ref{1}) and Eqs.(\\ref{16}-\\ref{w}) in Eq.(\\ref{pow}) the power spectrum in thermal vacuum state is obtained as\n  \\begin{equation}\n h^2 _T(k,\\eta)=  \\frac{16 l^2 _{pl} }{\\pi} k^2 {\\mid h(\\eta)\\mid}^2  \\mathrm {coth}\\Big[\\frac{k}{2T}\\Big],\n   \\end{equation}\n\nThus in comparison with Eq.(\\ref{bet}), the spectrum  is expressed as\n\\begin{equation}\nh(k,\\eta_i)= A{\\left(\\frac {k}{k_0}\\right)}^{2+\\beta} \\mathrm {{coth}^{1\/2}\\Big[\\frac{k}{2T}\\Big]}.\n\\end{equation}\n   The  last  term becomes significant when the ratio $k\/(2T)$ is less than unity. The  wave number $k$ and  temperature $T$ are comoving quantity  which are \n    related  to the physical parameters at the time of inflation, see for details \\cite{25}. Thus it is expected an enhancement of the spectrum by a factor $\\mathrm {coth}^{1\/2}[{k}\/{2T}] =\\mathrm {coth}^{1\/2}[{H S_i}\/{2T_i}]$ .\n   \n   It is convenient to consider the amplitude of waves   in different  range of wave numbers \\cite{15}.  Thus the amplitude  of the spectrum in thermal vacuum state  for  different ranges are given by\n   \n       (i) when $k\\leq k_{E}$, the  corresponding wavelength is   greater the present Hubble radius. Thus the amplitude remain as the initial one and can be written as \n\n\\begin{equation}\\label{y}\nh_{T}(k,\\eta_{0})=A\\Big(\\frac{k}{k_{0}}\\Big)^{2+\\beta}\\mathrm {{coth}^{1\/2}\\Big[\\frac{k}{2T}\\Big]},\n\\end{equation}\n\n(ii)   the amplitude remains  approximately same as long as the wave inside the barrier   but begin to  decrease when it leaves the barrier by a factor $1\/S(\\eta)$, depending the value of scale factor at that time. This  process  continue until the barrier becomes higher than $k$ at a time  $\\eta$ earlier than $\\eta_0$, so the  amplitude has decreased by  the ratio of the scale factor at the time of  leaving the barrier $S_b$ to  its value at $\\eta$, $S(\\eta)$. This   is   in the   range  $k_{E}\\leq k\\leq k_{0}$.  \n   \n\\begin{equation}\\label{ke}\nh_{T}(k ,\\eta_{0})=A\\Big(\\frac{k}{k_{0}}\\Big)^{\\beta-1} \\mathrm {{coth}^{1\/2}\\Big[\\frac{k}{2T}\\Big]}\\frac{1}{(1+z_{E})^{3}}.\n\\end{equation}\nNote  that this range is a new feature   on account of the  current acceleration of the universe  which is absent in the decelerating model as pointed out in \\cite{15}.  The amplitude of     the waves  that left  the barrier at $S_b$ with waves numbers $k > k_0$  has  been decreased  up to the present time by a factor $ S_b \/ S( \\eta _0)$. This affect the amplitude of the  present spectrum and is obtained as\n\\begin{equation}\\label{sq}\nh_{T}(k,\\eta_{0})=A\\Big(\\frac{k}{k_{0}}\\Big)^{2+\\beta} \\mathrm {{coth}^{1\/2}\\Big[\\frac{k}{2T}\\Big]} \\frac{S_b}{S(\\eta_0)}.\n\\end{equation}\nThis result  can be  used to obtain  the spectrum of the waves in the remaining range of wave numbers.\n\n(iii) the  wave number that does not hit the barrier in the range $ k_{0}\\leq k\\leq k_{2} $ gives the amplitude as follows\n\\begin{equation}\\label{l}\nh_{T}(k,\\eta_{0})=A\\Big(\\frac{k}{k_{0}}\\Big)^{\\beta} \\mathrm {{coth}^{1\/2}\\Big[\\frac{k}{2T}\\Big]}\\frac{1}{(1+z_{E})^{3}},\n\\end{equation}\n the spectrum  in this   interval  is differ from that of the matter dominated case by a the factor $\\frac{1}{(1+z_{E})^{3}}$. The  wave lengths of the spectrum  in the range  are  long but smaller than the present Hubble radius.\n \n (iv) in  the range of wave number $ k _{2}\\leq  k \\leq k _{s}$ , the amplitude is\n\\begin{equation}\\label{o}\nh(k,\\eta_{0})=A\\Big(\\frac{k}{k _{0}}\\Big)^{1+\\beta}\\Big(\\frac{k_{0}}{k_{2}}\\Big) \\frac{1}{(1+z_{E})^{3}}.\n\\end{equation}\n This is the interesting range on the observational point of view of Adv.LIGO, ET and LISA. Note that the temperature dependent factor in this  range  is negligible hence  the term  is dropped out  because of the low temperature nature of the  relic waves. \n \n (v) for the wave number  range $k _{s}\\leq k \\leq k _{1}$  which is in  the high frequency  case and   gives the corresponding amplitude as\n\\begin{equation*}\nh_{T}(k,\\eta_{0})=A\\Big(\\frac{k}{k _{0}}\\Big)^{1+\\beta-\\beta_{s}}\\Big(\\frac{k _{s}}{k _{0}}\\Big)^{\\beta_{s}}\\Big(\\frac{k _{0}}{k _{2}}\\Big) \\mathrm {{coth}^{1\/2}\\Big[\\frac{k}{2T}\\Big]}\\frac{1}{(1+z_{E})^{3}}.\n\\end{equation*}\n\\begin{equation}\\label{oo}\n\\end{equation}\n \n In the usual case the temperature dependent term can also be  neglected however the extra dimensional scenario  predicts  higher temperature for the  thermal gravitational waves, hence the term again becomes significant. Therefore the contribution from the thermal relic gravitational waves is  expected increase the amplitude of spectrum  particularly in the higher frequency range also. \n \n \n It is to be noted that in  (iv)   the  thermal contribution in $ k _{2}\\leq  k \\leq k _{s}$  range is  negligible   \ndue to  the temperature dependent  term.  Similarly  the thermal effect is  insignificant  in the range $k _{s}\\leq k \\leq k _{1}$ also. However\nby taking into account  the extra dimensional effect,  the spectrum of  relic waves  is   peaked with a  temperature   $T_{*}$=1.19 $\\times$ 10$^{25}$\\;{Mpc}$^{-1}$ \\cite{fry} (See, appendix  A for a brief discussion on $T_{*}$ from extra dimensional scenario.). Therefore it is expected  enhancement  for the amplitude of spectrum (orange lines, Figs.[\\ref{ff1}] and [\\ref{ff2}]) in the range  $ k _{s}\\leq  k \\leq k _{1}$ compared   to $T= 0$ case for the accelerated as well as  decelerated universe. But at the same time, ignoring  the  thermal contribution  to  the amplitude of  spectrum  in the range $ k _{2}\\leq  k \\leq k _{s}$ leads to a discontinuity  at $ k_{s}$, see Fig.[\\ref{ff1}]. \nThis   is evaded  by  fitting a new  line  in the range  $ k _{2}\\leq  k \\leq k _{s}$\nfor  the amplitude $h$  of Eq.(\\ref{o})  as follows. \n \n Let the amplitude of the wave in the range  $k _{0}\\leq  k \\leq k _{2}$ is given by  (\\ref{l}) and can be rewritten as\n\\begin{equation}\\label{1q}\nh_{1T}(k,\\eta_{0})=A\\Big(\\frac{k}{k_{0}}\\Big)^{\\beta} \\mathrm {{coth}^{1\/2}\\Big[\\frac{k}{2T}\\Big]}\\frac{1}{(1+z_{E})^{3}},\n\\end{equation}\nand  the amplitude in the $k _{s}\\leq  k \\leq k _{1}$ is given by (\\ref{oo}) also rewritten as\n\n\\begin{equation}\n\\label{3q}\nh_{2T}(k,\\eta_{0})=A\\Big(\\frac{k}{k _{0}}\\Big)^{1+\\beta-\\beta_{s}}\\Big(\\frac{k _{s}}{k _{0}}\\Big)^{\\beta_{s}}\\Big(\\frac{k _{0}}{k _{2}}\\Big) \\mathrm {{coth}^{1\/2}\\Big[\\frac{k}{2T_{*}}\\Big]}\\frac{1}{(1+z_{E})^{3}}.\n\\end{equation}\nThus the new slope for  Eq.(\\ref{o}), in the range $ k _{2}\\leq  k \\leq k _{s}$, can be obtained \nby taking  $y\\equiv \\log_{10}(h)$ and $x\\equiv \\log_{10}(k)$, then \n\\begin{equation}\\label{poo}\n\\log_{10}(h)-\\log_{10}(h)_{i}=\\frac{\\log_{10}(h)_{f}-\\log_{10}(h)_{i}}{\\log_{10}(k_{f})-\\log_{10}(k_{i})}(\\log_{10}(k)-\\log_{10}(k_{i})),\n\\end{equation}\nwhere the subscribes  $i$ and $f$  are respestively indicating  the first and last points of the straight line.\nBy putting $k_{i}\\equiv k_{2}$ from Eq.(\\ref{1q}) and $k_{f}\\equiv k_{s}$ from Eq.(\\ref{3q}) in   Eq.(\\ref{poo}), we get \\footnote{ here, $\\mathrm {{coth}^{1\/2}\\Big[\\frac{k_{2}}{2T}\\Big]}=1. $}\n\\begin{equation}\\label{p1}\nh=(h_{1T})_{k_{2}}g(k),\n\\end{equation}\nwhere\n\\begin{equation}\ng(k)=\\Big(  \\frac{k}{k_{2}}\\Big)^{\\gamma},\n\\end{equation}\nand\n\\begin{equation}\\label{ss}\n\\gamma=\\frac{\\log_{10}(h_{2T})_{k_{s}}-\\log_{10}(h_{1T})_{k_{2}}}{\\log_{10}(k_{s})-\\log_{10}(k_{2})}=\\frac{\\log_{10}\\Big ( \\Big(\\frac{k_{s}}{k_{2}} \\Big)^{1+\\beta}  \\mathrm {{coth}^{1\/2}\\Big[\\frac{k_{s}}{2T_{*}}\\Big]}\\Big)}{\\log_{10}\\Big( \\frac{k_{s}}{k_{2}}\\Big)},\n\\end{equation}\nis the slope of the line and thus  we find the amplitude, for convenience we  call it as `modified amplitude', given by\n\\begin{equation}\\label{ppp}\nh(k,\\eta_{0})=A\\Big (\\frac{k_{2}}{k_{0}}\\Big)^{\\beta} \\frac{1}{(1+z_{E})^{3}} \\Big( \\frac{k}{k_{2}} \\Big)^{\\gamma}.\n\\end{equation}\nWhen $T_{*}$  becomes  zero    Eq.(\\ref{ss}) leads to $\\gamma=1+\\beta$, and  hence   (\\ref{o}) is recovered  from   (\\ref{ppp})  in the range $k _{2}\\leq  k \\leq k _{s}$.\n\n\n The overall multiplication factor $A$  in all the spectra is determined in  absence of  the temperature dependent  term  with the CMB data of  WMAP  \\cite{15}.  This is  based on the  assumption that the contribution from  gravitational waves  and  the density perturbations  are the same order of magnitude  or if  the CMB anisotropies at low multipole are induced by the gravitational waves,   therefore \n it is possible to write  $\\Delta T \/ T \\simeq h(k,\\eta_{0})$.  The observed CMB anisotropies \\cite{u} at lower multipoles is $\\Delta T \/ T \\simeq0.37\\times10^{-5}$ at $l\\sim2$ which corresponds to the largest\nscale anisotropies that have observed so far. Thus taking this to be the perturbations at the Hubble\nradius  gives\n\\begin{equation}\\label{k}\nh(k_{0},\\eta_{0})=A\\frac{1}{(1+z_{E})^{3}}=0.37 \\times 10^{-5}.\n\\end{equation}\nHowever, there is a subtlety  in the interpretation of $ \\Delta T \/ T$ at low multipoles, whose\ncorresponding scale is very large $ \\sim \\ell_{0}$. At present the Hubble radius is $\\ell_{0}$, and the Hubble\ndiameter is $2\\ell_{0}$. On the other hand, the smallest characteristic wave number is $k_{E}$, \nwhose\ncorresponding physical wave length at present is $2\\pi S (\\eta_{0} )\/k_{E}=\\ell_{0}(1+z_{E})\\simeq1.32 \\ell_{0}$, which\nis within the Hubble diameter $2 \\ell_{0}$, and is theoretically observable. So, instead of Eq.(\\ref{k}), if $ \\Delta T \/ T \\simeq 0.37 \\times 10^{-5}$ at $l\\sim2$ were taken as the amplitude of the spectrum at $\\nu_{E}$, one would\nhave $h_T(k_{E},\\eta_{0})=A\/(1+z_{E})^{2+\\beta}=0.37\\times10^{-5}$, yielding a smaller $A$ than that in Eq. (\\ref{k}) by a factor $(1+z_{}E)^{1-\\beta}\\sim2.3$ \\cite{15}. The allowed values of  $\\beta$ and $\\beta_s$ are obtained and are  respectively give by $\\beta=-1.9$, and $\\beta_{s}=-0.552$ \\cite{15}.\n\n  \\begin{figure}[t]\n {\\includegraphics[scale=0.52]{m1.eps}}\n \\caption{ The  amplitude of  the gravitational waves  for the accelerated (solid lines) and decelerated (dashed lines) universe. }\\label{ff1}\n \\end{figure} \n \n \\begin{figure}[t]\n {\\includegraphics[scale=0.52]{LIG.eps}}\n \\caption{  Comparison  of  the  modified  amplitude of the spectrum for the accelerated (solid black and green lines) and decelerated (dashed black and green lines) universe with  the  sensitivity curves of Adv.LIGO \\cite{alg}   and ET \\cite{et}.  }\\label{f3}\n\\end{figure}    \n\n \\begin{figure}[t]\n {\\includegraphics[scale=0.52]{LIS.eps}}\n \\caption{Comparison  of  the  modified  amplitude of the spectrum for the accelerated (solid black and green lines) and decelerated (dashed black and green lines) universe with   the   LISA sensitivity curve. }\\label{f4}\n\\end{figure}    \n \n Next, we obtain the spectrum in the thermal vacuum state with the following  parameters. By taking   $k=2\\pi\\nu$, $\\nu_{E}=1.5\\times10^{-18}$ Hz, $\\nu_{0}=2\\times10^{-18}$ Hz, $\\nu_{2}=117\\times10^{-18}$ Hz, $\\nu_{s}=10^{8}$ Hz, $\\nu_{1}=3\\times10^{10}$ Hz ( the value of  $\\nu_{1}$  is taken such a way that spectral energy density does not exceed the level of $10^{-6}$ , as required by the rate of  primordial nucleosynthesis). \n The range of frequency is  chosen  in accordance with  generation of gravitational waves  that vary from early universe to  various astrophysical sources. And hence the range  is matching with  the interest   of CMB, Adv.LIGO, ET and LISA  operations for detection of  the gravitational waves.\nThe  spectrum is computed  in the thermal vacuum state  with the chosen values of the parameters  for  the  accelerated as well as decelerated model with  $T=0.001$Mpc$^{-1}$  in the low  range $k < k_{2}$. This temperature is    considered in the context of B mode of  CMB spectrum  in   thermal  state  \\cite{25}. And  $T_{*}$  =1.19$\\times$10$^{25}$Mpc$^{-1}$  \\footnote{here,  $T_{*}$ = 0.905 K=  1.19$\\times$10$^{25}$Mpc$^{-1}.$ } for the high  range $k_{s}\\leqslant k\\leqslant k_{1}$ which is from the   extra dimensional scenario \\cite{fry}.  Since we use the natural unit, the wave number and temperature that appear in the temperature dependent term of the spectrum is computed  numerically in the Mpc$^{-1}$ unit.\nThe obtained spectra are normalized   of  the CMB  anisotropy spectrum of  WMAP data. \n The  amplitude of the spectrum of  the thermal gravitational  waves is  enhanced  compared to its zero temperature case (vacuum case). It is  observed that the  spectrum for  $T=0.001$Mpc$^{-1}$ get maximum enhancement   $\\sim 1.51$  times  than  the  vacuum case, at  $k=k_E$,\nand   it is $\\sim 20$ times for  $T_{*}$=1.19$\\times$10$^{25}$Mpc$^{-1}$ at $k=k_s$. \n\n\n The  plots  for the  amplitude  of   spectrum $h_{T}(k,\\eta_{0})$ versus the frequency $\\nu$ for $\\beta=-1.9$ and $\\beta_{s}=-0.552$  are  given in Fig.[\\ref{ff1}]. The amplitude of the  spectrum get enhanced  in the  frequency ranges, 10$^{-19}$ Hz$ \\leq \\nu <1.49 \\times 10 ^{-17}$Hz, and $\\nu_{s}\\leq \\nu\\leq \\nu_{1}$ ( the lower   value  of this range is  selected  such way that  thermal enhancement of the spectral density does not exceed the upper bound of nucleosynthesis rate.) due to the thermal effect of gravitational waves but  for the  range $ 1.49 \\times 10 ^{-17} $Hz $\\leq \\nu < \\nu_{s}$     there is a suppression because of  the $\\coth^{1\/2}[k\/2T]$   term.  For  comparison, the amplitude of the spectra  are plotted for the decelerated and accelerated  universe, see  Fig.[\\ref{ff1}].\n\n  \n The new enhancement of the gravitational waves spectrum   due to the extra dimensional effect (the  modified amplitude, see  Fig.[\\ref{ff1}], light green lines) can be compared with  the sensitivity of  Adv.LIGO, ET and LISA. An analytical expressions for the Adv.LIGO and ET  interferometers are  discussed in \\cite{bs}. For  Adv.LIGO and ET cases, consider the root mean square amplitude per root Hz which equal to\n \\begin{equation}\n \\frac{h(\\nu)}{\\sqrt{\\nu}}.\n \\end{equation}\n The  comparison of the sensitivity  (10 Hz - 10$^4$ Hz) curve of the ground based interferometer  Adv.LIGO \\cite{alg} with the  gravitational wave spectra  of  $\\beta =-1.9$   for the accelerated and decelerated universe are given in Fig.[\\ref{f3}].  Thus it shows that the Adv. LIGO  is unlikely to   detect  the enhancement   of the spectrum  from the  extra dimensional effect with its current stands but  be possible with  the sensitivity of  ET.\n  \n  \n  \n Next, we compare the enhancement of the spectrum with the sensitivity  (10$^{-7}$ Hz - 10$^0$ Hz) of  space based detector LISA \\cite{ss}. It is assumed that LISA  has one year observation time which corresponds to frequency bin $\\Delta \\nu$ = 3 $\\times$ 10 $^{-8}$Hz ( one cycle\/year) around each frequency.  Hence to make a comparison with the sensitivity curve,  a rescaling  of the spectrum $h(\\nu)$ is required in Eq.(\\ref{pp}) into the root mean square spectrum $h(\\nu, \\Delta \\nu)$ in the band $\\Delta \\nu$, given by\n \\begin{equation}\n h(\\nu, \\Delta \\nu)=  h(\\nu) \\sqrt{\\frac{\\Delta \\nu}{\\nu}}.\n   \\end{equation}\n The plots of the LISA sensitivity  with the  modified amplitude of the spectrum are  given in Fig.[\\ref{f4}] for the accelerated and decelerated universe. This show that the LISA is unlikely to detect the spectrum with the new enhancement feature of the gravitational waves. \n\n\n\nThe spectral energy density\nparameter $\\Omega_{g}(\\nu)$ of gravitational waves is defined through the relation $\\rho_{g}\/\\rho_{c}=\\int\\Omega_{g}(\\nu)\\frac{d\\nu}{\\nu}$, where $\\rho_{g}$ is the energy density of the gravitational waves and $\\rho_{c}$ is the critical energy density.\nThus\n\\begin{equation}\\label{ka}\n\\Omega_{g}(\\nu)=\\frac{\\pi^{2}}{3}h^2_{T}(k,\\eta_{0})\\Big(\\frac{\\nu}{\\nu_{0}}\\Big)^{2}.\n\\end{equation}\n Since the spacetime is assumed to be spatially flat \n$K=0$ with $\\Omega=1$, the fraction density of relic gravitational waves must be less than unity, $\\rho_{g}\/ \\rho_{c}<1$. After  \n  normalization of  the spectrum by using Eq.(\\ref{k}), we  integrate $\\int\\Omega_{g}(\\nu)d\\nu\/ \\nu$ from  $\\nu_{*}=10^{-19}$ Hz up to  $\\nu_{1}=3\\times10^{10}$ Hz,  with $\\beta=-1.9$ and $\\beta_{s}=-0.552$. \nThe integral is  evaluated   for the thermal case  and  zero temperature case, the  obtained results for the accelerated universe are  \n\n(a) $\\nu_{*}\\leq\\nu\\leq\\nu_{E}$,\n\\begin{eqnarray*}\n\\frac{\\rho_{g}}{\\rho_{c}}&=&5.8\\times 10^{-11},\\;\\;\\;\\;T=0,\\\\\n\\frac{\\rho_{g}}{\\rho_{c}}&=&8.8\\times 10^{-11},\\;\\;\\;\\;T=0.001\\;{Mpc}^{-1},\n\\end{eqnarray*}\n\n(b) $\\nu_{E}\\leq\\nu\\leq\\nu_{H}$,\n\\begin{eqnarray*}\n\\frac{\\rho_{g}}{\\rho_{c}}&=&2.3\\times 10^{-11},\\;\\;\\;\\;T=0,\\\\\n\\frac{\\rho_{g}}{\\rho_{c}}&=&3.5\\times 10^{-11},\\;\\;\\;\\;T=0.001\\;{Mpc}^{-1},\n\\end{eqnarray*}\n\n(c) $\\nu_{H}\\leq\\nu\\leq\\nu_{2}$,\n\\begin{eqnarray*}\n\\frac{\\rho_{g}}{\\rho_{c}}&=&2.4\\times 10^{-11},\\;\\;\\;\\;T=0,\\\\\n\\frac{\\rho_{g}}{\\rho_{c}}&=&3.7\\times 10^{-11},\\;\\;\\;\\;T=0.001\\;{Mpc}^{-1},\n\\end{eqnarray*}\n\n(d) $\\nu_{2}\\leq\\nu\\leq\\nu_{s}$,\n\\begin{eqnarray*}\n\\frac{\\rho_{g}}{\\rho_{c}}&=&8.97\\times 10^{-9},\\;\\;\\;\\;T=0,\\\\\n\\end{eqnarray*}\n\n(e) $\\nu_{s}\\leq\\nu\\leq\\nu_{1}$,\n\\begin{eqnarray*}\n\\frac{\\rho_{g}}{\\rho_{c}}&=&2.7\\times 10^{-6},\\;\\;\\;\\;T=0. \\\\\n\\frac{\\rho_{g}}{\\rho_{c}}&=&6.67\\times 10^{-6},\\;\\;\\;\\;T_{*}=1.19 \\times10 ^{25}Mpc^{-1}.\n\\end{eqnarray*}\n\n\\begin{figure*}[t]\n{\\includegraphics[scale=0.52]{m2.eps}}\n\\caption{ The  spectral  energy density of the gravitational waves  for the accelerated (solid lines) and decelerated (dashed lines) universe.}\\label{ff2}\n\\end{figure*}\n\nIt is to be noted that in  (d)  the  thermal case are not shown  because  the  thermal contribution in this frequency range is  negligible   \ndue to  the temperature dependent  term. However\nby taking into the extra dimensional effect, the  upper limit of temperature of the relic waves  is to be $T_{*}$=1.19$\\times$10$^{25}$Mpc$^{-1}$. Thus it is expected an enhancement of the  spectral energy density in range  $\\nu_{s}\\leq\\nu\\leq\\nu_{1}$ compared   to $T= 0$ case for the accelerated as well as  decelerated universe. But at the same time ignoring the thermal contribution  on the spectral density in the range  $\\nu_{2}\\leq\\nu\\leq\\nu_{s}$ leads to a discontinuity  at $ \\nu_{s}$, see Fig.[\\ref{ff2}]. This problem  is solved  by  fitting a new  line  as discussed  in the context  of  estimation of the amplitude of the spectrum and  hence recomputed the spectral density in the range  $\\nu_{2}\\leq\\nu\\leq\\nu_{s}$ which gives the new value $\\frac{\\rho_{g}}{\\rho_{c}}$ = 8.21$\\times 10^{-7}$. This \n changes the slope  indicating enhancement of the spectral energy density of the gravitational waves in the range    $\\nu_{2}\\leq\\nu\\leq\\nu_{s}$, green lines, Fig.[\\ref{ff2}].\n \nThe enhancement spectral energy density $\\Omega_{g}(\\nu)$ in (d) can be  compared with  the estimated upper bound of various  studies and  are given  in Tab.[\\ref{t1}].   Thus $\\Omega^{(dec)}_{g}$ and $\\Omega^{(acc)}_{g}(\\nu)$   are less than the upper bound of  the estimated values  of   the respective frequency range.\n\\begin{table}\n\\caption{\\label{t1}Comparison of  the estimated upper bound  of spectral energy density of  various studies  with   the present work.\nHere $\\Omega^{(dec)}_{g}$ and $\\Omega^{(acc)}_{g} $ are   respectively the   spectral energy density  of the relic gravitational waves in  the decelerated and accelerated universe  of the present study and $\\Omega^{(est)}_{g}$ is the  estimated upper bound of  various studies.\n} \n\n\\footnotesize\\rm\n\\begin{tabular*}{\\textwidth}{@{}l*{15}{@{\\extracolsep{0pt plus12pt}}l}}\n\\br\nFrequency($\\nu$) Hz &$ \\Omega^{(dec)}_{g}(\\nu)$ & $\\Omega^{(acc)}_{g}(\\nu)$&  $\\Omega^{(est)}_{g}(\\nu)$\\\\\n\\mr\n$10^{-9}-10^{-7} $ & $ 4.98 \\times10^{-9}$ & $  1.03 \\times10^{-9}$ & $2 \\times 10^{-8} \\;\\;\\;\\cite{z}$ \\\\\n$69-156$ &  $  34.84 \\times10^{-8}$ & $  7.2 \\times10^{-8}$ & $8.4\\times 10^{-4}\\cite{zz}$ \\\\\n$41.5-169.25$  & $  4.93 \\times10^{-7}$ & $ 1.02 \\times10^{-7}$ & $6.9\\times 10^{-6}\n \\cite{zzz}$\\\\\n\\br\n\\end{tabular*}\n\\end{table}\n\n\n \nFurther  see that  the  contribution  to  $\\rho_{g}\/\\rho_{c}$  from  the low frequency range  is $ {\\cal{O}} (10^{-11}-10^{-10})$ while  from the higher frequency range it is $ {\\cal{O}}(10^{-6})$.  Since the order of  contribution  to   the total $\\rho_{g}\/\\rho_{c}$ from the lower frequency side is   very small  in  contrast with higher frequency side,  we  get  for the accelerated universe as\n \\begin{equation}\n \\rho_{g}\/\\rho_{c}\\simeq 6.67\\times10^{-6}   \\; \\; \\;  \\nu_{*}\\leq\\nu\\leq\\nu_{1},\n\\end{equation}\nand is  the same order as that of the zero temperature case.\n However    $ \\rho_{g}\/\\rho_{c}$  of the gravitational waves with $T\\neq 0$ is higher  than  the zero temperature  case at lower frequency range $\\nu_{*} \\leq\\nu\\leq\\nu_{2}$. Therefore  it is expected an enhancement for   the spectral energy density  in the thermal  vacuum state  in the  frequency range  $\\nu_{*} \\leq\\nu\\leq\\nu_{2}$ and actually  it   is the range  of interest on the observational point of view of the  relic gravitational waves.   The total   estimated value of $ \\rho_{g}\/\\rho_{c}$  by including    the thermal relic   gravitational waves in the very high frequency   does not alter the  upper bound of the nucleosynthesis rate. Thus the relic thermal  gravitons  with very high frequency range  are not ruled out and is   testable with  the upcoming  data of various missions  for detecting gravitational waves.\n \\section{Discussion and conclusion}\n \nGravitational waves are  one of the classical predictions of  Einstein's general theory of relativity. The gravitational waves are generated  during the very early  evolution stages  of the universe as well as from the various astrophysical objects.  Therefore  frequency of the waves are  varying  very widely.  There are many on going  experiments  to detect these waves and the  interested range of frequency is from 10$^{-19}$ Hz to 10$^{10}$Hz.  The existence of gravitational waves with  very high  frequency range is  not  favoring by the energy scale of the conventional  inflationary scenario. However the  very high frequency range gravitational waves are interesting   candidates  in the models with extra dimensions.  The extra dimensional theories predict the existence of thermal gravitons with black body type spectrum.  These relic thermal gravitational waves can also add to the spectrum of the waves thus the corresponding amplitude  also  get enhanced. \nThe nature of spectrum of the waves   to be observed today is   dependents on the  evolution history of the universe. Before the result of SN Ia observations,  the current evolution of the universe  is used to consider as  matter dominated  with decelerated expansion.  But, according  to the $\\Lambda$CDM concordance model the present universe is supposed to be driven by dark energy  resulting an accelerated phase. If this  is the case then the spectral property of the waves  to be studied by taking into account of the current acceleration of the universe.    In the present work we  mainly considered the  very high frequency range  (The low frequency range thermal gravitational case  is carried out   by us without including the  very high frequency thermal waves  that comes from extra dimensional scenario and the   work is under  preparation. The enhancement of the lower frequency range is shown with red lines, see Figs.[\\ref{ff1},\\ref{ff2}]) of relic gravitational waves in the thermal vacuum state and obtained the spectrum for the accelerated as well as decelerated models.  The obtained spectra are normalized with the WMAP data. It is observed that  the inclusion of the very high relic thermal gravitational waves  leads to a discontinuity  in  the  amplitude of the spectrum at  $\\nu_s$ (see Fig.[\\ref{ff1}]). This is  \ndue to the fact that the temperature dependent term is insignificant in the  higher  frequency side of  the  range $\\nu_2 \\leq \\nu \\leq \\nu_s$.  To evade this problem  a new  equation of  line  is derived and thus the amplitude get  enhanced in the range  $\\nu_2  \\leq \\nu \\leq \\nu_s$. This is the new feature of the spectrum and we designates  it as the `modified amplitude' of the spectrum.\n The  modified amplitude of the spectrum can be compared with the sensitivity of  the Adv.LIGO, ET and LISA missions. The comparison of the Adv.LIGO  sensitivity shows that the modified amplitude   is unlikely to   detect    with its current stands of  LISA or   the improved sensitivity of Adv.LIGO. Where  as the  proposed sensitivity of  the ET  is promising to verify the modified amplitude  with its upcoming mission data.\n \n \nThe spectral energy density of the gravitational waves is  estimated   in thermal vacuum state for  the accelerated and decelerated universe.  It is observed that the spectral energy density get enhanced   in  the lower frequency range  $ {\\cal{O}} (10^{-11}-10^{-10})$ and   from the higher frequency range it is $ {\\cal{O}}(10^{-6})$. A comparison of  the estimated upper bound  of spectral energy density of  various studies  with   the present work is done. It shows that\n $\\Omega^{(dec)}_{g}$ and $\\Omega^{(acc)}_{g} $   are less than  the  estimated upper bound of  various studies.  The total   estimated value of $ \\rho_{g}\/\\rho_{c}$  by including    the   very high frequency  thermal relic gravitational waves  does not alter the  upper bound of the nucleosynthesis rate. Thus the relic thermal  gravitons  with very high frequency range  are not ruled out and is   testable with  the upcoming  data of various missions  for detecting gravitational waves.\n\n\\section*{Appendix A}\n \\section*{Extra dimensional Scenario and Thermal Gravitons} \n \n Cosmology with extra dimensions have been motivated since Kaluza  and Klein (KK) showed that classical electromagnetism and general relativity could be combined in a five-dimensional framework. The modern scenarios involving extra dimensions are being explored in particle physics, with most models possessing either a large volume  or a large curvature. Although there exist  different  models of extra dimensions, there are some general features and signals common to all of them.\n\n \n In presence of  $D$ extra spatial dimensions, the 3+D+1- dimensional  action for gravity  for can be written as \n \\begin{equation}\n S = \\int d^4 x \\left [ \\int d^D y \\sqrt{- g_{D}} \\frac{R_{D}}{16 \\pi G_{D}}  + \\sqrt{-g} L_{m} \\right],\n \\end{equation}\n where \n \\begin{equation}\n G_D = G_N \\frac{m_{pl} ^2}{m_D ^{2+D}},\n \\end{equation}\n and $g$ is the four dimensional metric, $G_N$  is  Newton's constant, $g_D, G_N$ and $R_D$ denote the higher dimensional counter parts of the metric, Newton's constant, and the Ricci scalar, respectively. $m_D $ is the fundamental scale of the extra dimensional theory. \n \n    Since the gravitational interactions are not strong enough to produce a thermal gravitons at temperatures below the Planck scale ($m_{pl}  \\sim1.22 \\times 10^{19}$ GeV), the standard  inflationary cosmology  predicted the existence the cosmic gravitational waves background  which are non-thermal in nature. However\n if the universe contains extra dimensions that can generate the thermal gravitational waves and its shape and amplitude of the CGWB may change significantly.  This can happen \n when energies in the universe are higher than the fundamental scale $m_D$, the gravitational coupling strength increases significantly, as the gravitational field spreads out into the full spatial volume. Instead of freezing out at $\\sim { \\cal O}(m_{pl})$, as in 3+1 dimensions, gravitational interactions freeze-out at $\\sim {\\cal O}(m_D)$.  If the gravitational interactions become strong at an energy scale below the reheat temperature ($m_D  < T_{RH}$), gravitons get the opportunity to thermalize, creating a thermal CGWB. The qualitative result, the creation of a thermal CGWB if $m_D < T_{RH}$, is unchanged by the type of extra dimensions chosen \\cite{fry}. \n \n Thus, if extra dimensions do exist, and the fundamental scale of those dimensions is below the reheat temperature, a relic thermal CGWB ought to exist today. Compared to the relic thermal photon background (CMB), a thermal CGWB would have the same shape, statistics, and high degree of isotropy and homogeneity. The energy density ($\\rho_g$) and fractional energy density ($\\Omega_g$ )\tof\ta\tthermal\tCGWB\t are\n \\begin{eqnarray}\n \\rho_g = \\frac{\\pi^2}{15} \\left( { \\frac{3.91}{g_{\\star}}} \\right )^{4\/3} T_{CMB}^4, \\\\\n \\Omega_g = \\frac{\\rho_g}{\\rho_c} \\simeq 3.1 \\times 10 ^{-4} (g_{\\star})^{4\/3},\n \\end{eqnarray}\n where $\\rho_c $ is the critical energy density today, $T_{CMB}$ is the present temperature of the CMB, and $g_{\\star}$ is the number of relativistic degrees of freedom at the scale of $m_D$.  $g_{\\star}$ is dependent on the particle content of the universe, i.e. whether (and at what scale) the universe is supersymmetric, has a KK tower, etc. Other quantities, such as the temperature (T), peak frequency ($\\nu$), number density ($n$), and entropy density ($s$) of the thermal CGWB can be derived from the CMB if $g_{\\star}$ is known, as \n \\begin{eqnarray}\n n_g = n_{CMB} \\left( { \\frac{3.91}{g_{\\star}}} \\right ),\\,\\, \\, \\, \\, \\,  s_g= s_{CMB}  \\left( { \\frac{3.91}{g_{\\star}}} \\right )\\\\\n  T_g = T_{CMB} \\left( { \\frac{3.91}{g_{\\star}}} \\right )^{1\/3},\\,\\, \\, \\, \\, \\,  \\nu_g= \\nu_{CMB}  \\left( { \\frac{3.91}{g_{\\star}}} \\right )^{1\/3}.\n    \\end{eqnarray}\n  These quantities are not dependent on the number of extra dimensions, as the large discrepancy in size between the three large spatial dimensions and the $D$ extra dimensions suppresses those corrections by at least a factor of  $ \\sim 10^{ -29}$. If $m_D$ is just barely above the scale of the standard model, then $ g_{\\star }$= 106.75. The thermal CGWB then has a temperature of 0.905 Kelvin, a peak frequency of 19 GHz \\cite{fry}. \n  \n  \\section*{Acknowledgement}\nAuthors would like to thank S.Hild  for providing the ET sensitivity  data and  also Adv.LIGO and LISA web.\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}