Cosmological fluctuations of a random field and radiation fluid Mar Bastero-Gil,1,∗ Arjun Berera,2,† Ian G. Moss,3,‡ and Rudnei O. Ramos4,§ 1Departamento de F´ısica Te´orica y del Cosmos, Universidad de Granada, Granada-18071, Spain 2SUPA, School of Physics and Astronomy, University of Edinburgh, Edinburgh, EH9 3JZ, United Kingdom 3School of Mathematics and Statistics, Newcastlle University, NE1 7RU, United Kingdom 4Departamento de F´ısica Te´orica, Universidade do Estado do Rio de Janeiro, 20550-013 Rio de Janeiro, RJ, Brazil AgeneralizationoftherandomfluidhydrodynamicfluctuationtheoryduetoLandauandLifshitz isappliedtodescribecosmological fluctuationsinsystemswithradiationandscalarfields. Thevis- couspressures,parametrizedintermsofthebulkandshearviscositycoefficients,andtherespective random fluctuations in the radiation fluid are combined with the stochastic and dissipative scalar 4 evolutionequation. Thisresultsinacompletesetofequationsdescribingtheperturbationsinboth 1 scalarandradiationfluids. Thesederivedequationsarethenstudied,asanexample,inthecontext 0 of warm inflation. Similar treatments can be done for other cosmological early universe scenarios 2 involvingthermal or statistical fluctuations. n PACSnumbers: 98.80.Cq a J 6 ] O I. INTRODUCTION C Fluctuationanddissipationphenomenacouldpotentiallyplayanimportantroleinearlyuniversecosmology. When . h themattercontentoftheuniversecanbesplitintoasubsysteminteractingwithalargeenergyresevoir,thenphysical p processes may be represented through effective dissipation and stochastic noise terms. Various physical systems - o have been proposed for the early universe which are well suited for such a treatment. In particular scenarios where r thermal or any statistical fluctuations seed cosmic structure fit into this category. The role of thermal fluctuations t s in structure formation have been considered since the early work of Peebles [1] and Harrison [2] In recent times, a thermal fluctuations continue to be seen as a possible mechanism for seeding structure formation [3–11]. One of the [ early models with statistical fluctuations seeding structure formation was in the context of inflationary cosmology 1 with the warm inflation paradigm [5]. There are a variety of warm inflation models that have been developed (see v [12, 13] and references therein for severalexamples). Moreover,many models have subsequently been developed that 9 are very similar to the warm inflation picture, with particle production during inflation and non-vaccum density 4 perturbations,suchasnon-commutativeinflation[14],decayingmultifieldinflationmodels[15],trappedinflation[16], 1 cyclic-inflationmodels [11, 17], axionmodels of inflation [18], and effective field theory models of warminflation [19]. 1 . Thermal fluctuations have also been examined as the origin of density perturbations in bouncing universe models 1 [7, 20], string cosmology [21, 22], loop cosmology [23], the near-Milne universe [24], a model of phase transition 0 involvingholography[25],andvaryingspeedoflightmodels[7]. Densityperturbationscreatedfromastatisticalstate 4 1 havebeen examinedinthe radiationdominatedregime [7–11]andother excitedstates [26, 27]. During reheatingand : preheating, particle production has been shown to affect cosmologicalperturbations [28, 29]. v In all these scenarios a common feature is a sizable density of particles in the universe that is pictured in some i X statistical state, usually thermal. In order to thermalize, these particles must interact with one another. The short r scale physics in the early universe is not directly accessible to observation today. Moreover this is a complicated a many-body problem, which, similar to such problems in condensed matter systems, would be effectively intractable to exact calculation. The typical approach to such a problem is for this many-body dynamics to be characterized by one, perhaps many, microphysical scales, within which one can employ a statistical treatment and the dynamics manifests itself through dissipation processes. Associated with such effects will be corresponding stochastic forces. A treatment involvingfluctuation-dissipationdynamics canbe implemented at different levels of coarsegrainingof thedegreesoffreedom. Ideallyoneshouldstartwiththeunderlyingfundamentalquantumfieldtheoryandproceedto coarse grain. This level of sophistication has been realized in some simple cases, most notably the Caldeirra-Leggett modelforcondensedmattersystems[30]andthismodelhasalsobeenexaminedinacosmologicalcontext[31]. Inthis ∗Electronicaddress: [email protected] †Electronicaddress: [email protected] ‡Electronicaddress: [email protected] §Electronicaddress: [email protected] 2 casethequantummodelcanbe coarsegrainedsystematicallyintoastochasticlangevinequationforthe system,with the remaining degrees of freedom represented as a noise force and a dissipative term. In more complicated quantum field theory models, coarse graining has been done at a perturbative level [32–36]. Such approaches treat one part of the quantum field theory as a system and then integrate out the remaining fields into a heat reservoir. In treating cosmological perturbations, the problem is a little more involved than simply deriving the stochastic evolution equation for the system. The heat bath will also have fluctuation-dissipation effects due to the action of the system as well as internal effects from within the heat bath. These effects will play a role in the cosmological perturbationequationsandhaveconsequencesforthedensityperturbations. Thisisamuchharderproblem. Deriving the full system-heat bath dynamics from quantum field theory with all fluctuation and dissipation effects computed to our knowledge has never been achieved. An intermediate approach is to treat the heat bath within a fluid approximation which is then coupled to the system, which is treated from quantum field theory. It is at this level that the study in this paper procedes. In this approximation, the heat bath is treated in terms of quantum fields for computing the transport and noise coefficients of the system and the heat bath itself. However, in treating the cosmological perturbations, the heat bath is then represented as a fluid. The missing step here is showing how the quantum fields that comprise the heat bath can be represented as a fluid. At an intuitive level the correspondence seemsevident,butitisadifficultproblemofcoarsegrainingthatgoeswellbeyondtheconcernsofthespecificproblem being addressed in this paper. This limitation in our approach thus brings some lack of precision in formulating the dynamical problem. Nevertheless, it still captures a great deal of the physics that otherwise is completely ignored in simple mean field treatments. Once the problem is formulated in this way, progress can be made. Hydrodynamics is a macroscopic theory describing the behavior of averaged or mean variables corresponding to the energy density, pressure, fluid velocity and so on. As such, the microscopic physics become manifest in the form of dissipative terms corresponding to the transport coefficients, like bulk and shear viscosities. But from a fluctuation-dissipation stand point, these must also berelatedtostochasticfluctuationsaswell. LandauandLifshitzwerethefirsttoproposeafluctuatinghydrodynamics theory, where random fluxes are added to the usual hydrodynamic equations, with two-point correlation functions related with the transport coefficients through fluctuation-dissipation relations [37]. The Landau-Lifshitz fluctuating hydrodynamics theory was later refined and put in firm theoretical grounds by the work of Fox and Uhlenbeck [38] and extended to the relativistic fluids by Zimdahl [39] (see also [40]). Asideforwarminflation,mostcosmologicalmodelsinvolvingthermalorstatisticalfluctuationshavebeenexamined atonlyameanfieldlevel,wherefluctuation-dissipationeffectshavenotbeentreated. Assuch,importantinformation is ignored about how the short scale physics affects the large scale physics. Calculations in warm inflation have shown that current precision from CMB data demands a treatment beyond a mean field level, and requires account for fluctuation and dissipation effects. This lesson probably also carries over for other scenarios involving thermal fluctuations. In this paper, we will study the density perturbations spectra in terms of the coupled set of radiation equations describing the random radiation fluid equations and the stochastic equation for a scalar field. We believe this is the first study of such a system. The study of cosmological perturbations making use of the relativistic version of the fluctuation hydrodynamics theory of Landau and Lifshitz has already been done before by Zimdahl in [41]. Some papers have treated the density perturbations in a system of a scalar field with dissipation coupled to a radiation fluid [42] as well as affects of viscosity within the radiation fluid [43, 44]. However no work has treated in addition the corresponding noise forces that accompany dissipation and viscosity. Our treatment in this paper includes all these effects andcanbe appliedto problemsincosmologyinvolvingascalarfieldcoupledto othersystems,suchasin inflationary cosmology, cosmic phase transitions, reheating, curvaton decay etc... Often, in addition of including the scalar field dynamics, we also have a mechanism by which the radiation bath is generated and maintained through particle production due to the decay of the scalar field. We analyze in detail not only the interplay of the different dissipation terms, from the scalar field and the bulk and shear viscosities of the radiation fluid, but also the effect of the respectivenoiseterms,connectedwiththedissipationandviscositytermsbythe dissipation-fluctuationrelations. This paper is organized as follows. In Sec. II we introduce the relativistic fluctuating hydrodynamics built from the original version due to Landau and Lifshitz. In Sec. III, guided by the equivalence principle, we extend the relativistic fluctuating hydrodynamic equations for the radiation bath for the cosmological context. The equations are coupled with those of the inflaton field as appropriate in dissipative environments, like in the warm inflation scenario, and the perturbation equations constructed. In Sec. IV we give the general expressions for the dissipative and viscosity coefficients we will be considering along this work. In Sec. V, the full cosmological perturbations are studied numerically and results for the curvature power spectrum for perturbations presented. The effects of both bulk and shear viscosities are analyzed. Finally, in Sec. VI we give our concluding remarks. 3 II. FLUCTUATIONS IN FLAT SPACETIME We are interested primarily in situations with a radiation fluid that is close to being in thermal equilibrium at some local temperature T, and the fluid is hot enough to be treated as classicaland relativistic. Quantum statistical mechanicalfluctuations in such a radiationfluid can be describedusing Landau’s theory of random fluids [37], where the deterministicequationsoffluiddynamicsarereplacedbyasystemofequationswithstochasticsourceterms. The fluid approximation is maintained by microscopic interactions, with small departures from equilibrium which cause both fluctuations and dissipation. The fluctuations of the fluid reach a balance between the effects of the source and the dissipation terms. Fixing the statistical properties of the noise terms to ensure that stochastic averages of fluid variables reproduce the statistical ensemble averages leads to the fundamental fluctuation-dissipation relation. Consider a relativistic fluid with energy density ρ(f) and pressure p(f) in which conserved particle numbers are absent or negligible, and the 4-velocity ua(f) is the velocity of energy transport. Random sources and dissipative stresses are introduced via a stress term Π in the stress-energy tensor, ab T(f) =(p(f)+ρ(f))u (f)u (f)+p(f)g +Π , (2.1) ab a b ab ab where indices a,b... denote spacetime components. In Landau’s theory, dissipation is governed by constitutive relations for shear viscosity η and bulk viscosity η whilst fluctuations are generated by a Gaussian noise term Σ . s b ab In a comoving frame where the spatial components u (f) =0 and the time component u (f) = 1, the non-vanishing i 0 − shear terms are 2 Π = η u (f)+η u (f)+(η η )δ u(f)k Σ , (2.2) ij s i j s j i b s ij k ij − ∇ ∇ − 3 ∇ − (cid:18) (cid:19) where denotes a spatial derivative. The correlation functions of the stochastic noise term Σ are assumed to be i ij ∇ local and determined by the fluctuation-dissipation relation, 2 Σ (x,t)Σ (x′,t′) =2T η δ δ +η δ δ +(η η )δ δ δ(3)(x x′)δ(t t′). (2.3) ij kl s ik jl s il jk b s ij kl h i − 3 − − (cid:18) (cid:19) This will be explored further in Sect. IIC. Landau’s theory can be used reliably for small departures from a stable underlying fluid flow. We shall be concerned mostly with small density, pressure and 3-velocity fluctuations δu(f) of a radiation fluid in an inertial frame with background density ρ(f) and pressure p(f). For example, the momentum conservation equation obtained using the vanishing divergence of the stress-energy tensor outlined above is, 1 (p(f)+ρ(f))δu˙(f)+∇δp+p˙(f)δu(f) =η ∇2δu(f)+ η ∇ δu(f)+η ∇ δu(f)+∇ Σ. (2.4) s s b 3 ∇· ∇· · This can be recognized as the perturbed Navier-Stokes momentum conservation equation with a stochastic source term. The solutions to the stochastic fluid equations can be used to follow the evolution of quantities such as the density perturbations, δρ(f)(x,t)δρ(f)(x′,t) , (2.5) h i bytakingastochasticaverage. Withoutthetheoryofrandomfluids,wewouldonlyhaveknowledgeoftheequilibrium values of the density fluctuations. A. Relativistic fluids coupled to a scalar field Our aimis to couple this radiationfluid to a scalarfield. The behaviorofa relativistic scalarfield inflat spacetime interacting with radiation can be analyzed using non-equilibrium quantum field theory [45]. When the small-scale behavior of the fields is averaged out, the scalar field fluctuations, like the fluid fluctuations, can be described by a stochastic system whose evolutionis determined by a Langevinequation[49]. For a weakly interacting radiationgas, the dissipation and noise terms in the Langevin equation can be approximated by local expressions. This is the case we will consider here. The Langevin equation for a scalar field with thermodynamic potential Ω(φ,T) and damping coefficient Υ(φ,T) is then [34] 4 (cid:3)φ(x,t)+Υφ˙(x,t)+Ω =(2ΥT)1/2ξ(φ)(x,t), (2.6) ,φ − where(cid:3)istheflatspacetimed’Alembertianandξ(φ) isastochasticsource. Theprobabilitydistributionofthesource term will be approximated by a localized gaussian distribution with correlationfunction [34, 35], ξ(φ)(x,t)ξ(φ)(x′,t′) =δ(3)(x x′)δ(t t′). (2.7) h i − − TheLangevinequationapplieswhenthesurroundingradiationisatrest. Forafluidinuniformmotionwith4-velocity ua(f) we wouldneed to choose the Lorentzframe to be the rest frame of the fluid. This canbe expressedin covariant form by replacing φ˙ in the dissipation term by a fluid derivative, Dφ=ua(f) φ. (2.8) a ∇ The dissipation results in a transfer of energy and momentum from the scalar field to the radiation which needs to be included in the fluid equations. Energy and momentum transfer can be tracked by considering the divergence of the stress-energy tensor. We combine the fluid and scalar contributions into a unified stress-energy tensor given by 1 T =Tsu(f)u(f) Ωg + φ φ ( φ)2g +Π , (2.9) ab a b − ab ∇a ∇b − 2 ∇ ab ab whereΠ isorthogonaltothefluidvelocity. Wehaveintroducedtheentropydensitys,definedbythethermodynamic ab relation s= Ω . (2.10) ,T − If s 0, then the thermodynamic potential Ω splits into an effective potential V(φ) depending only on φ and a ,φ ≡ radiation term depending only on T, Ω=V(φ) p(f)(T). (2.11) − In this case, s s(T) and the fundamental thermodynamic relation implies that Ts = ρ(f) +p(f) allowing us to separate off the≡fluid stress-energy tensor T(f) given in Eq. (2.1). This separation into fluid and scalar field terms ab is not possible in general, but a partial separation can be seen in the divergence of the stress-energy tensor, bT = D(Ts)+ ub(f) u (f)+s T + bΠ +((cid:3)φ Ω ) φ. (2.12) ab b a a ab ,φ a ∇ ∇ ∇ ∇ − ∇ (cid:16) (cid:17) The first three terms represent the field equations for the fluid in the absense of the scalar field and they can be separated from the remaining terms by defining fluxes Q(f) and Q(φ) by, a a Q(f) = D(Ts)+ u(f)b u(f) +s T + bΠ , (2.13) a ∇b a ∇a ∇ ab Q(φ) = ((cid:16)(cid:3)φ Ω ) φ. (cid:17) (2.14) a − ,φ ∇a Using the Langevin Eq. (2.6) for the scalar field, we obtain that Q(φ) =Υ(Dφ) φ (2ΥT)1/2ξ(φ) φ. (2.15) a ∇a − ∇a Energy-momentum conservation bT =0 results in a set of fluid equations, ab ∇ D(Ts)+ u(f)b u(f) +s T + bΠ = Q(φ). (2.16) ∇b a ∇a ∇ ab − a (cid:16) (cid:17) 5 Therefore, the flux Q(φ) describes the transfer of energy and momentum to the fluid equations. a As a matter of fact, Eq. (2.15) is not the most general exprsssion which we can obtain for the energy transfer. We might also consider adding a stochastic energy flux term P to the stress energy tensor, rather like the stochastic stress term Σ which we had in Eq. 2.2, so that the stress energy tensor becomes ij 1 T =Tsu(f)u(f) Ωg + φ φ ( φ)2g +Π +2u (f)P . (2.17) ab a b − ab ∇a ∇b − 2 ∇ ab ab (a b) This modifies the energy transfer vector Qa(φ), Q(φ) =Υ(Dφ) φ (2ΥT)1/2ξ(φ) φ+ b 2u (f)P . (2.18) a ∇a − ∇a ∇ (a b) (cid:16) (cid:17) The time component represents energy transfer, Q(φ) =Υ(Dφ)φ˙ (2ΥT)1/2ξ(φ)φ˙ ∇ P. (2.19) 0 − − · The simplest possibility is simply P = 0, but an interesting alternative is to impose the condition that the energy a flux is independent of ξ(φ), by setting ∇ P = (2ΥT)1/2ξ(φ)φ˙. (2.20) · − In this case P has to be included in the momentum flux Q(φ). The calculationsin later sections will consider both of these possibilities. B. Perturbation theory We perturb the fluid quantities and the scalar field, replacing ρ(f) by ρ(f) + δρ(f) and so on, and taking the backgrounds to be homogeneous with vanishing velocity. From this point on we use the indices i,j... to denote the spatial coordinate frame in which the backgroundfluid is at rest. The non-vanishing components of the stress tensor Π aregivenbytheconstitutiverelationsforshearandbulkviscosityaswellastherandomnoisetermΣ generating ab ij the fluctuations, 2 Π = η δu(f)+η δu(f)+(η η )δ δu(f)k Σ . (2.21) ij − s∇i j s∇j i b− 3 s ij∇k − ij (cid:18) (cid:19) The noise term is taken to be gaussian with the correlation function (2.3). The first-order fluid equations obtained from energy-momentum conservation bT =0 using the stress-energy tensor (2.9) are then ab ∇ T δs˙+s˙δT +Ts∇ δu(f) = δQ(φ), (2.22) · − 1 Tsδu(f) ˙ +∇(sδT) η 2δu(f) η + η ∇ δu(f) = δQ(φ)+∇ Σ, (2.23) s b s { } − ∇ − 3 ∇· − · (cid:18) (cid:19) where boldface denotes spatial vectors and δQ(φ) = δQ(φ)0 = δQ(φ). Comparison with the random fluid Eq. (2.4) − 0 suggests that we should identify the fluid density and pressure perturbations as δρ(f) = T δs, (2.24) δp(f) = sδT. (2.25) The fluctuations δρ(f), δp(f) and δφ are obtained from just two thermodynamical degrees of freedom φ and T, so one of the fluctuations is dependent on the other two, the natural choice being the pressure perturbation. By setting δs=s δφ+s δT in (2.24), we arrive at ,φ ,T δp(f) =c2(δρ(f) Ts δφ), (2.26) s − ,φ 6 where the sound speed c2 =s/(Ts ). Differentiating Eqs. (2.24) and (2.25), we also have s ,T Tδs˙+δTs˙ =δρ˙(f)+s δq, (2.27) ,φ where we have defined δq =φ˙δT T˙ δφ. (2.28) − The fluid equations can then be re-written in terms of the density and scalar field fluctuations, δρ˙(f)+(ρ(f)+p(f))∇ δu(f)+s δq = δQ(φ), (2.29) ,φ · − 1 (ρ(f)+p(f))δu(f) ˙ +∇δp(f) η 2δu(f) η + η ∇ δu(f) = δQ(φ)+∇ Σ. (2.30) s b s { } − ∇ − 3 ∇· − · (cid:18) (cid:19) When s 0, then δp(f) = c2δρ(f) and the δq term drops out of the fluid equations. In this case the equations ,φ ≡ s become perturbed versions of the relativistic Navier-Stokes equations with stochastic source terms. Since there are no sources of vorticity at linear order, we can introduce scalar velocity perturbations through δu(f) =∇δv(f), δQ(φ) =∇δJ(φ). (2.31) The fluid perturbations for potential flow satisfy δρ˙(f)+(ρ(f)+p(f)) 2δv(f)+s δq = δQ(φ), (2.32) ,φ ∇ − (ρ(f)+p(f))δv(f) ˙ +δp(f) η′ 2δv(f) = δJ(φ)+(2η′T)1/2ξ(f), (2.33) { } − ∇ − where δp(f)is given by Eq. (2.26) and we have defined η′ as the combination of viscosity coefficients: 4 η′ = η +η . (2.34) s b 3 Using Eq. (2.3), the noise source ξ(f) = −2 i jΣ has correlationfunction ij ∇ ∇ ∇ ξ(f)(x,t)ξ(f)(x′,t′) =δ(3)(x x′)δ(t t′). (2.35) h i − − Thenewfeatureoftheseequationsisthattheycombinetherandomfluidwiththeexchangeofenergyandmomentum to the scalar field, represented by the flux terms δQ(φ) and δQ(φ). For a homogeneous background scalar field, the perturbation of Eq. (2.18) shows that δQ(φ) = δΥφ˙2 2Υφ˙δφ˙+(2ΥT)1/2φ˙ξ(φ)+∇ P (2.36) − − · δJ(φ) = Υφ˙δφ+ −2∇ P˙. (2.37) ∇ · We shall take P = C (2ΥT)1/2φ˙ −2∇ξ(φ). (2.38) P − ∇ ThetwocasesC =0andC =1governwhetherthenoisesourceξ(φ) appearsintheenergyfluxorinthemomentum P P flux. Both cases will be considered in our numerical analysis to be performed in Sec. V. 7 C. Fluctuation-dissipation relations We finish this section with a discussionof the fluctuation-dissipationrelations to verify that the stochastic average δρ(f)(x,t)δρ(f)(x′,t) , (2.39) h i reproduces the quantum-statistical ensemble average on time-independent backgrounds. This is expected on general grounds, but the derivation for relativistic fields is less well known than the non-relativistic case and the density correlations will be useful later. The thermal ensemble averages can be obtained using standard thermodynamical arguments, or by using thermal quantum field theory (see [8] for an example). These thermodynamic results have also been used in cosmologicalsettings, e.g. by [8, 11]. We disconnect the scalar field by setting Q(φ) = s = 0 and take the background density and pressure to be a ,φ constant. This allows Fourier decomposition with δρ(f)(k,ω) = dtd3xδρ(f)(x,t)ei(k·x−ωt), (2.40) Z δv(f)(k,ω) = dtd3xδv(f)(x,t)ei(k·x−ωt). (2.41) Z On substituting these transforms into Eqs. (2.29) and (2.30), the fluctuations satisfy iω (1+c2)ρ(f)k2 δρ(f) 0 − s =(2η′T)1/2 . (2.42) c2 iω(1+c2)ρ(f)+k2η′ δv(f) ξ(f) (cid:18) s s (cid:19)(cid:18) (cid:19) (cid:18) (cid:19) The solution for the density fluctuation is δρ(f)(k,ω)=G(k,ω)k2(2η′T)1/2ξ(f), (2.43) with the Green function G(k,ω)= (γk2 i(ω c k))(γk2 i(ω+c k)) γ2k4 −1, (2.44) s s − − − − and γ =η′/2(1+c2)ρ(f). After using t(cid:2)he noise correlation function (2.35), the dens(cid:3)ity correlation functions become s dω δρ(f)(k,t)δρ(f)(k′,t) = G(k,ω)2(2η′T)k4(2π)2δ(3)(k+k′). (2.45) h i 2π| | Z In the low damping regime γk c , the integration gives s ≪ 1+c2 δρ(f)(k,t)δρ(f)(k′,t) s Tρ(f)(2π)2δ(3)(k+k′). (2.46) h i≈ c2 s For comparison,statisticalmechanicsrelatesthe fluctuations attemperatures largeenoughto ignorequantumeffects to the entropy density s [8], ∂s δρ(f)(k,t)δρ(f)(k′,t) T3 (2π)2δ(3)(k+k′). (2.47) sm h i ≈ ∂T In the case where the density depends only on temperature, we have ρ(f) =aT1+1/c2s, s=a(1+c2)T1/c2s. (2.48) s It follows that 8 1+c2 δρ(f)(k,t)δρ(f)(k′,t) s Tρ(f)(2π)2δ(3)(k+k′). (2.49) h ism ≈ c2 s Equations (2.46) and (2.49) agree, confirming that the coefficient of the noise term was chosen correctly. The fluctuation-dissipation relations for the scalar field can be obtained by following a similar route. We take a constant backgroundscalar field and consider the fluctuations δφ. Their Fourier transforms satisfy δφ(k,ω)=G(k,ω)(2ΥT)1/2ξ(φ), (2.50) where the Green function is G(k,ω)=(k2 ω2 iΥω+m2)−1, (2.51) − − and m2 =V . Following the same steps as above, with Υ k, these give ,φφ ≪ T δφ(k,t)δφ(k′,t) (2π)2δ(3)(k+k′), (2.52) h i≈ ω2 k where ω2 =k2+m2. This is the correctstatisticalmechanicalresult,telling us that the oscillatormodes with energy k ω2δφ2 have an averageenergy T in the classical regime ω T. In the quantum regime ω T, we would have k k ≪ k ≫ 1 δφ(k,t)δφ(k′,t) (2π)2δ(3)(k+k′). (2.53) h i≈ 2ω k This result can be obtained by following the general prescription (see, e.g., [46] where this is explicitly derived) of inserting the factor (ω/2T)cosh(ω/2T) into the Fourier transform of the noise correlation (2.7). III. COSMOLOGICAL PERTURBATIONS In this section we shall describe the effects of fluid and scalar field fluctuations in a cosmological setting where the background spacetime describes a homogeneous, isotropic and spactially flat universe. We assume the fluid to be highly relativistic, such as we might expect in the very early universe. The main dissipative mechanisms are the energyloss by the scalarfield andviscosityin the radiationfluid. Eachof these is associatedwith a stochasticsource term with correlation functions determined by the fluctuation-dissipation relation. We shall take the damping terms and the correlation functions to have a local form, allowing us to apply the equivalence principle. Our gauge-ready formalism for cosmological perturbations follows Hwang and Noh [42]. The spacetime metric for a scalar-type of perturbation is given by ds2 = (1+2α)dt2 2β dtdxi+a2(δ (1+2ϕ)+2γ )dxidxj, (3.1) ,i ij ,ij − − where a(t) is the scale factor and H = a˙/a defines the background expansions rate. Physical combinations of the metric perturbations which will be useful later on are the shear χ and perturbed expansion rate κ, given by χ = a(β+aγ˙), (3.2) κ = 3Hα 3ϕ˙ 2χ. (3.3) − −∇ The backgroundLaplaciandenotes the combination 2 =a−2δij , where is the derivative with respect to xi. i j i Note the factor of a−2 here, and that 2 is the covar∇iantLaplaci∇an∇for the spa∇tial metric g =a2δ ij ij ∇ The stress-energy tensor is conveniently expressed as, T =(ρ+p)n n +pg +n q +n q +Π , (3.4) ab a a ab a b b a ab 9 where q and the trace-free tensor Π are orthogonal to the unit vector n . We shall take na to be the unit normal a ab a to the constant-time surfaces. For scalar perturbations, we replace ρ by ρ+δρ, p by p+δp and define δv and δΠ by 1 q =(ρ+p) δv, δΠ = δΠ g 2δΠ, (3.5) i i ij i j ij ∇ ∇ ∇ − 3 ∇ The perturbed Einstein equations in gauge-readyform are then [42] 2ϕ+Hκ = 4πGδρ, (3.6) ∇ − κ+ 2χ = 12πG(ρ+p)δv, (3.7) ∇ − χ˙ +Hχ α ϕ = 8πGδΠ, (3.8) − − κ˙ +2Hκ+ 2α 3(ρ+p)α = 4πG(δρ+3δp). (3.9) ∇ − Diffeomorphism invariance allows us to fix two of the independent variables. At least two further equations are required,and these come from consideringthe matter sector,which in our case consists of the radiationfluid and the scalar field. A. Fluid and scalar perturbations The stress-energy tensor can be expressed in the velocity frame we used earlier in Eq. (2.9), 1 T =Tsu(f)u(f) Ωg +Π(f)+ φ φ ( φ)2g . (3.10) ab a b − ab ab ∇a ∇b − 2 ∇ ab We have taken the energy flux P =0 to simplify the discussion, but non-vanishing energy flux can easily be accomo- dated. By comparing the two forms of the stress-energy tensor (3.4) and (3.10) on homogeneous backgrounds with vanishing fluid velocity we find the relations, 1 ρ = Ts+ φ˙2+Ω, (3.11) 2 1 p = φ˙2 Ω+Π(f), (3.12) 2 − where Π(f) = Π(f) i. For the fluctuations, comparing the first-order perturbations of the two stress-energy tensors i gives δρ = δρ(f)+φ˙(δφ˙ αφ˙)+Ω δφ, (3.13) ,φ − δp = δp(f)+φ˙(δφ˙ αφ˙) Ω δφ, (3.14) ,φ − − (ρ+p)δv = Tsδv(f) φ˙δφ, (3.15) − where δρ(f) =Tδs and δu(f) = δv(f) as before. (See below for δp(f).) i ∇i B. Fluid equations The fluid equations obtained from the stress-energy tensor (3.10) are T Ds+Ts ua(f)+Π(f)ab u (f) = Q(φ), (3.16) a a b ∇ ∇ − TsDu (f)+sh b T +h Π(f)bc = h Q(φ)c. (3.17) a a b ac b ac ∇ ∇ − whereh =g +u (f)u (f) andD is the comovingderivativeasbefore. Guidedbythe equivalenceprinciple,weadd ab ab a b dissipation and noise sources to the shear stress Π(f) to reproduce the flat-spacetime limit Eq. (2.2), ab 10 Π(f) = 2η σ η h cu (f) Σ . (3.18) ab s ab b ab c ab − − ∇ − The first term relates the shear stress to the rate-of-straintensor σ , ab 1 σ =h ch d u (f) h cu(f). (3.19) ab (a b) ∇c d − 3 ab∇ c Note that the bulk viscosity terms behave like a contribution to the pressure p(f). We are ready to expand these equations to first order in perturbations theory about homogeneous backgrounds. The backgroundfluid equation from Eq. (3.16) is Ts˙+3H(Ts 3Hη )=Υφ˙2. (3.20) b − At first order in perturbation theory, using the metric (3.1), we find the velocity expansion, δ( cu(f))= 2δv(f) κ, (3.21) ∇ c ∇ − and the strain tensor 1 σ = σ g 2σ, σ =δv(f)+χ. (3.22) ij i j ij ∇ ∇ − 3 ∇ We can also modify the pressure to absorb the bulk viscosity, by defining δp(f) =sδT 3Hδη =c2(δρ(f) Ts δφ) 3Hδη , (3.23) − b s − ,φ − b after taking into account Eq. (2.26). The fluid equations (3.16) and (3.17) expanded to first order with the metric (3.1) become δρ˙(f) αTs˙+3H(δρ(f)+δp(f) η κ)+(Ts 3Hη )( 2δv(f) κ)+s δq = δQ(φ), (3.24) b b ,φ − − − ∇ − − a−3 a3(Ts 3Hη )δv(f) ˙ +α(Ts 3Hη )+δp(f) η κ η′ 2(δv(f)+χ) = δJ(φ)+(2η′T)1/2ξ(f).(3.25) b b b { − } − − − ∇ − Theseequationsreducetotheprevioussetofequations(2.32)and(2.33)inflatspaceifwesubstituteTs=ρ(f)+p(f), although it is often advantageous to work with s and T rather than ρ(f) and p(f). The correlationfunction for the stochasticsourceshas to be correctedto accountfor the scalingbetween comoving coordinates xi and inertial frame coordinates axi, resulting in a factor a−3, ξ(f)(x,t)ξ(f)(x′,t′) =a−3δ(3)(x x′)δ(t t′). (3.26) h i − − The energy and momentum transfer terms are given as before by perturbing Eq. (2.15), δQ(φ) = δΥφ˙2 2Υφ˙(δφ˙ αφ˙)+(2ΥT)1/2φ˙ξ(φ)+∇ P, (3.27) − − − · δJ(φ) = Υφ˙δφ+ −2∇ (P˙ +4HP), (3.28) ∇ · where we have allowed for the possibility of modifying the stress energy tensor by including a stochastic energy flux term P. The fluid equations reduce to previously known versions in special cases. The equations agree with other work on random radiation fluids when the scalar field is absent and δQ(φ) = δJ(φ) = s = 0 [41]. The non-viscous case ,φ η = η =0 has been widely discussed in the context warm inflation [8, 48]. A new feature of these equations is the s b noise term in the energy and momentum transfer terms (3.27) and (3.28), and the effect on the amplitude of density perturbationswillbeanalyzedlater. Theviscouscasewithouttherandomfluidsourceshasbeendiscussedin[43,44].