The Effective Fluid Approach to Cosmological Nonlinearities: Applications to Preheating
aa r X i v : . [ a s t r o - ph . C O ] S e p The Effective Fluid Approach to CosmologicalNonlinearities: Applications to Preheating
Hyeyoun Chung
Jefferson Physical Laboratory, Harvard University,17 Oxford St., Cambridge, MA 02138, USA [email protected] 8, 2018
Abstract
In [1], Baumann et al. presented a new formalism for studying cosmologicalsystems where the characteristic scale of non-linearities, k − , is much smallerthan the Hubble scale H − . By integrating out the short-wavelength modes,it is possible to obtain an effective theory of long-wavelength perturbationsthat is described by an imperfect fluid evolving in an FRW background. Asthe long-wavelength perturbations remain small even when the short-scale dy-namics are non-linear, the tools of linear perturbation theory may be applied.The work in [1] deals only with matter in the form of a pressureless perfectfluid with zero anisotropic stress, and also assumes that the short-scale gravi-tational dynamics are Newtonian. In this work we extend this formalism to thecase of a perfect fluid with pressure, and in particular to the case of preheat-ing after inflation, where the matter content of the universe can be modeledby two coupled scalar fields. We discard the assumption that the short-scalegravitational dynamics are Newtonian. We find that our results differ fromBaumann et al.’s even when the pressure is set to zero, which suggests thatrelaxing their assumptions creates appreciable changes in the long-wavelengtheffective theory. We derive equations of motion for the total density pertur-bation and matter velocities during preheating, as well as linearized Einsteinequations for the long-wavelength metric perturbations. We also present theequations governing the effective long-wavelength scalar field dynamics. Introduction
Recently, Baumann et al. have proposed a formalism for analytically studying long-wavelength dynamics even when the density contrast δ of a universe grows large[1].Their method applies when the characteristic scale of non-linearities, k − , is muchsmaller than the Hubble scale H − . This hierarchy allows us to integrate out theshort-wavelength modes by smoothing all perturbations over a scale Λ − that liesbetween the non-linear scale and the scale of the long-wavelength perturbations. Thisprocedure gives an effective theory of long-wavelength perturbations that is modeledto lowest order by an imperfect fluid with effective energy-momentum pseudotensor τ µν , evolving in an FRW universe. The properties of the effective fluid are determinedby the interactions of the short-wavelength modes.The equations of motion in the effective theory can be expressed as a derivativeexpansion of the long-wavelength variables, with higher order terms being suppressedby powers of ( k/ Λ) or ( k/k NL ) . Since the long-wavelength perturbations duringpreheating remain small even when the underlying dynamics have become non-linear,we can then apply perturbation theory techniques. The coefficients of the effectivetheory can either be obtained by matching to the results of numerical simulations, orleft as free parameters to be matched to measurements from experiment.Baumann et al. considered a universe filled with cold dark matter, modeled asa pressureless perfect fluid. In this paper we consider a universe where the mattercontent is a perfect fluid with pressure: in particular, we consider matter in theform of two coupled scalar fields. We also make fewer assumptions than Baumannet al. For example, we discard their assumption that the short-scale gravitationaldynamics are Newtonian. We obtain somewhat different results from those given in[1], even when we set the pressure to zero in our equations, suggesting that relaxingthe underlying assumptions on the short-scale gravitational dynamics significantlyaffects the long-wavelength perturbations. We hope that our approach will thus beapplicable in more general scenarios than a ΛCDM universe.One particular example where our work applies is the period of preheating fol-lowing inflation. Inflation ends when the slow-roll conditions are violated, and theinflaton φ begins to oscillate around its ground state. As the inflaton field is coupledto the standard model (SM) matter fields, these oscillations cause the energy storedin φ to be transferred to the SM fields. This process is known as reheating [3].Reheating was originally analyzed using perturbative quantum field theory, until2t was realized that the coherent nature of the inflaton field at the end of inflationrenders this picture inaccurate. The excitation of SM fields during reheating was thenreformulated as a semi-classical problem, in which the quantum mechanical produc-tion of SM matter particles takes place in the classical background of the inflatonfield. The most common toy model used to describe this situation couples a scalarmatter field χ to the inflaton field φ . In this model of reheating the χ field is excitedvia parametric resonance, and the inflaton decays rapidly as a result, so that thedynamics quickly become non-linear. In this regime the standard tools of linearizedcosmological perturbation theory no longer apply, so most studies of reheating haverelied on numerical simulations[4]. In particular, the occupation numbers of χ and φ are found to be large in the non-linear domain, so that they may be treated as classi-cal fields and studied using lattice simulations. This period of rapid energy transferfrom φ to χ is known as preheating . It results in a highly non-thermal distribution ofmatter fields that then thermalizes to give the initial conditions for a hot big bang.Although computer simulations are useful for providing visual representations ofpreheating dynamics, it would be desirable to have an analytical description of thenon-linear dynamics. In addition to allowing us to make explicit calculations, it couldalso offer a greater understanding of the scalar field interactions. As preheating is ahighly inhomogeneous process, the density contrast δ ≡ ρ/ ¯ ρ − φ and χ soonexceeds 1, so that conventional cosmological perturbation theory (in which we expandin δ , matter velocity v , and metric perturbations Φ) cannot be applied. However,numerical studies have shown that there is a large hierarchy between the non-linearscale and the Hubble scale during preheating, with k − ∼ . H − [4]. Moreover, themetric and velocity perturbations remain small, even while the inhomogeneous partsof φ and χ grow large[5]. Thus it seems that the period of preheating provides asuitable test case for setting up an effective theory of long-wavelength perturbations.This paper is organized as follows. In Section 2 we describe the rules for expandingthe equations of motion in our formalism. In Section 3 we give the basic equationsthat govern the scalar field and metric dynamics during preheating. In Section 4 wederive the effective theory of the long-wavelength modes, and in Section 5 we give theevolution equations for the long-wavelength theory. We conclude in Section 6.3 Cosmological Perturbation Theory: A VelocityExpansion
In cosmological perturbation theory, we usually expand to linear order in the densityperturbation δρ , matter velocity v , and metric perturbations Φ, so that linear per-turbation theory is no longer valid when the density contrast δ >
1. In our work, wefollow the alternative approach outlined in [1], where we expand up to order v invelocity and metric perturbations, and we do not expand in δρ .In perturbation theory, we find that v is related to the Newtonian potential Φand the density contrast δ by v ∼ Φ δ (2.1)In the non-linear regime, when δ ∼
1, we therefore find that v ∼ Φ.On small scales, gradients of the gravitational potential can change the power-counting of standard perturbation theory, as the short-wavelength modes have largemomenta k . The net result is that, at the non-linear scale, each gradient of Φ reducesthe order in v by one. Therefore, when we carry out an expansion to order v , weexpand to linear order in Φ and second order in ∇ Φ.Simulations of the scalar fields during preheating have shown that the metricfluctuations and the gradient energies of the fields remain small, even when the fieldsthemselves (and their kinetic energies) grow large[5]. Therefore, it is valid to applythis expansion when studying preheating dynamics.
We will consider one of the simplest models of preheating, in which a massive inflatonfield φ with inflaton potential V = m φ couples to a massless scalar field χ througha potential V = g φ χ . (Our approach can easily be extended to other models withdifferent interaction potentials.) The full potential is therefore V ( φ, χ ) = V + V = 12 m φ + 12 g φ χ (3.1)At the end of inflation φ is a homogeneous field that approaches φ → M pl √ πmt sin mt, (3.2)4here t is the cosmic time. The spacetime is described by a perturbed FRW metricd s = a ( η ) (cid:0) − e d η + e − d x (cid:1) , (3.3)where η is the conformal time, we have ignored vector and tensor perturbations, andwe use the Poisson gauge. In the case of zero anisotropic stress, we find that Ψ = Φto first order.The basic equations that describe the dynamics during preheating are the Klein-Gordon (KG) equations for φ and χ , and the Einstein equations. The KG equationsare: ✷ φ − V ,φ = 0 (3.4) ✷ χ − V ,χ = 0 (3.5)where V ,φ ≡ ∂V∂φ . The energy-momentum tensor of the scalar fields is given by T µν = ∂ µ φ∂ ν φ + ∂ µ χ∂ ν χ − δ µν (cid:18) ∂ α φ∂ α φ + 12 ∂ α χ∂ α χ + V (cid:19) (3.6)Furthermore, T µν obeys the conservation law ∇ µ T µν = 0 . (3.7)Finally, the Einstein equations are given by G µν = 8 πGT µν . (3.8)In the rest of this section we expand the KG equations and the continuity equationsto order v in the metric perturbations and the matter velocities, but we do not expandin ρ or P . We find that the Einstein equations have a very similar form to those givenin [1], at least to leading order in long-wavelength perturbations. We therefore givethese equations in Appendix B. In order to apply the expansion to the system ofcoupled scalar fields described above, we first consider the ways in which this systemmay be interpreted as a sum of perfect fluids. It is possible to interpret a system of interacting scalar fields as a system of interactingfluids [7, 8, 9, 10]. We can therefore expand the equations of motion in terms of ρ and v , or in terms of φ and χ . This will give us two equivalent descriptions of the5atter fields during preheating, and we can go back and forth between the two usingthe correspondence outlined below. It will be useful to consider both descriptions, asthey offer different physical insights and different calculational advantages. The fluiddescription allows us to consider the energy density of the entire system using onlyone variable, ρ , and gives a simpler formulation of the equations of motion. The scalarfield description allows us to consider the individual field perturbations directly.The energy momentum tensor of a perfect fluid with energy density ρ , pressure P , and zero anisotropic stress is given by T µν = ( ρ + P ) u µ u ν + δ µν P, (3.9)where u µ is the instantaneous 4-velocity of the fluid. The components of the 4-velocityare related to the matter velocity v by the equations u = a − e − Ψ γ ( v ) , u i = a − e Φ v i (3.10) u = − ae Ψ γ ( v ) , u i = ae − Φ v i , (3.11)where γ ( v ) := (1 − v ) − / . Comparing Eq.(3.9) with the form of T µν in Eq.(3.6), wesee that the interacting scalar fields φ and χ can be treated as the sum of two “kineticfluids” with energy density and pressure[8] ρ φ := − ∂ α φ∂ α φ ρ χ := − ∂ α χ∂ α χ (3.12) P φ := − ∂ α φ∂ α φ P χ := − ∂ α χ∂ α χ (3.13)and a single “potential fluid” with energy density and pressure ρ V := V (3.14) P V := − V. (3.15)The instantaneous 4-velocity of each kinetic fluid is is given by u µφ := − ∂ µ φ √− ∂ α φ∂ α φ u µχ := − ∂ µ χ √− ∂ α χ∂ α χ (3.16)and to order v the matter velocity v is given by v iφ = ∂ i φ∂ φ = − ∂ i φ∂ φ , v iχ = ∂ i χ∂ χ = − ∂ i χ∂ χ (3.17)The 4-velocity is not defined for the potential fluid.6he total energy-momentum tensor is given by the sum of the individual energy-momentum tensors of these three fluids. The total veocity perturbation is given by v i = ( ρ φ + P φ ) v iφ + ( ρ χ + P χ ) v iχ ρ + P (3.18)where ρ := ρ φ + ρ χ + ρ V and P := P φ + P χ + P V are the total energy density andtotal pressure respectively.The physical interpretation of the scalar field as a fluid is valid as long as u µ is timelike. In our case, we are assuming that v is small compared to δ , which isequivalent to assuming that the gradient energy of the scalar fields is small comparedto their kinetic energy. Thus it is reasonable to assume that u µ remains timelike duringpreheating. The form of T µν shows that the anisotropic stress is zero. Therefore, wecan set the metric perturbations Φ = Ψ to first order, and will do so for the rest ofthe paper. The components of the energy-momentum tensor calculated to order v in terms of ρ , v are: T = − ( ρ + P ) γ + P (3.19) T i = − v i ( ρ + P ) (3.20) T ij = ( ρ + P ) v i v j + δ ij P (3.21)In terms of φ , χ , the components are: T = − a − − (cid:18) ∂ i φ∂ φ (cid:19) ! ( ∂ φ ) − a − − (cid:18) ∂ i χ∂ χ (cid:19) ! ( ∂ χ ) − V (3.22) T i = 1 a ( ∂ i φ∂ φ + ∂ i χ∂ χ ) (3.23) T ij = 1 a ( ∂ i φ∂ j φ + ∂ i χ∂ j χ ) − δ ij V + 12 a δ ij − − (cid:18) ∂ i φ∂ φ (cid:19) ! ( ∂ φ ) + − − (cid:18) ∂ i χ∂ χ (cid:19) ! ( ∂ χ ) ! (3.24)In [1], the equations of motion of the long-wavelength perturbations are derived us-ing the Euler equations in the Newtonian approximation. We will take a different7pproach and use the full general-relativistic conservation equation (3.7). We canproject this equation along u ν , or along the orthogonal direction: u ν ∇ µ T µν = − √− g ∂ µ (cid:0) √− g ( ρ + P ) u µ (cid:1) + u µ ∂ µ P = 0 (3.25)( g σν + u σ u ν ) ∇ µ T µν = ( ρ + P ) u µ ∇ µ u σ + ∂ σ P + u σ u µ ∂ µ P = 0 (3.26)To order v , Eq.(3.25) is:˙ ρ + ∂ i (( ρ + P ) v i ) − v i ∂ i P + (3 H − v · ˙ v )( ρ + P ) (3.27)To order v , the spatial components of Eq.(3.26) are:( ρ + P ) (cid:0) ˙ v i + v j ∂ j v i + ∂ i Ψ (cid:1) + ∂ i P + v i ∂ P + v i v j ∂ j P = 0 (3.28) To write down the Einstein equations to order v , we decompose the Einstein tensor G µν into a homogeneous background ¯ G µν , a part linear in the perturbations ( G Lµν ),and part non-linear in the perturbations ( G NLµν ). We can then write the non-linearEinstein equations in the form G Lµν = 8 πG ( τ µν − ¯ T µν ) (3.29)where the energy-momentum pseudotensor τ µν is given by τ µν ≡ T µν − G NLµν πG (3.30)The velocity expansion of the Einstein equations is given in Appendix B.1. Expanding the Klein-Gordon equations for φ to order v gives:(1 − ∂ φ − ∂ i φ + (2 H − H − − ˙Ψ) ∂ φ + ∂ i (Φ − Ψ) ∂ i φ + V ,φ = 0 , (3.31)with an analogous equation holding for χ .8 Integrating Out Short-Wavelength Modes
In this section we describe how to integrate out the short-wavelength modes to ob-tain a long-wavelength effective theory. Integrating out the short-wavelength modesamounts to averaging perturbations over a smoothing scale Λ − . Since we are inter-ested in the theory at scales k − much larger than the non-linear scale k − , we choosea smoothing scale Λ − >> k − .The smoothing of perturbations corresponds to a convolution of all fields X ≡{ ρ, Φ , ρ v } with a window function W Λ . We define the long-wavelength mode X l of afield X to be X l ≡ [ X ] Λ ( x ) = Z d x ′ W Λ ( | x − x ′ | ) X ( x ′ ) . (4.1)The short-wavelength mode X s of X is then defined by X ≡ X l + X s . (4.2)We will assume W Λ to be Gaussian for convenience. We also assume that W Λ satisfiesthe following conditions: ∂ j ′ W Λ = − ∂ j W Λ = Λ ( x − x ′ ) j W Λ (4.3) ∂ i ′ ∂ j ′ W Λ = ∂ i ∂ j W Λ = − Λ δ ij W Λ + Λ ( x − x ′ ) i ( x − x ′ ) j W Λ (4.4)If we smooth general bilinear and trilinear quantities using W Λ , we obtain the follow-ing results: [ f g ] Λ = f l g l + [ f s g s ] Λ + 1Λ ∇ f l · ∇ g l + . . . (4.5)[ f gh ] Λ = f l g l h l + [ f s g s h s ] Λ + f l [ g s h s ] Λ + g l [ f s h s ] Λ + h l [ f s g s ] Λ + 1Λ ( f l ∇ g l · ∇ h l + g l ∇ f l · ∇ h l + h l ∇ f l · ∇ g l )+ 1Λ ( ∇ f l · ∇ [ g s h s ] Λ + ∇ g l · ∇ [ f s h s ] Λ + ∇ h l · ∇ [ f s g s ] Λ )+ 12Λ ( ∇ f l [ g s h s ] Λ + ∇ g l [ f s h s ] Λ + ∇ h l [ f s g s ] Λ ) + . . . (4.6)The explicit calculations used to derive Eq.(4.5-4.6) are given in Appendix A. Theexpressions are given up to higher derivative terms of order k / Λ , where k is acharacteristic frequency of a long-wavelength mode. The higher derivative terms aretherefore suppressed, as k << Λ. As mentioned in the Introduction, our expressionfor smoothed trilinear quantities differs from that given in [1], due to the presence9f extra terms that are absent in their formula. It is possible that the discrepancyis due to extra (unspecified) assumptions that have been imposed in their paper toallow these terms to be dropped.
We now use the equations from Section 3 and Section 4 to derive the long-wavelengtheffective theory of preheating dynamics. Smoothing the energy-momentum pseu-dotensor in (3.30), we find that the effective energy-momentum pseudotensor [ τ µν ] Λ has the form [ τ µν ] Λ = τ lµν + τ sµν + τ ∂ µν , (5.1)where τ lµν depends only on the long-wavelength perturbations, τ sµν depends on theshort-wavelength modes, and τ ∂ µν contains higher-derivative corrections that are sup-pressed by powers of k / Λ . Throughout this work, we will drop all such higher-derivative corrections that result from smoothing. We also drop non-linear metriccontributions to τ lµν .The pseudotensor [ τ µν ] Λ describes an imperfect fluid. Thus the effective theory weobtain after smoothing is an imperfect fluid with density perturbation δ l and velocityperturbation v l , evolving in a background FRW metric with scalar metric perturba-tions Φ l and Ψ l . As the scale of non-linearities is much smaller than the smoothingscale, these long-wavelength perturbations remain small even when the small-scale dy-namics have become non-linear. Therefore, we may apply linear perturbation theoryin the effective theory even when δ >> τ µν ] Λ = ( ρ + P ) u µ u ν + ( P − ζ θ ) g µν + Σ µν (5.2)where θ = ∂ i u i , ζ is the bulk viscosity, and Σ µν is the anisotropic stress. The pseu-dotensor is given as a derivative expansion, with higher order terms being suppressedby ( k/k NL ) . All of the quantities in (5.2) are quantities in the effective theory: forexample, ρ := ρ eff = ¯ ρ eff + δρ eff . In order to avoid cluttering the notation, we willomit the “eff” subscript from relevant quantities, with the exception of Section 5.1when we discuss the renormalization of background pressure, energy density, andanisotropic stress due to short-wavelength dynamics. Although we began with a sys-tem of coupled scalar fields that had zero anisotropic stress and zero viscosity, we10ill find that both anisotropic stress and viscosity are induced in the long-wavelengtheffective theory.We use the following ansatz for the anisotropic stress:Σ ij = − ησ ij , σ ij := v ( i,j ) − δ ij v k,k , (5.3)where η is the shear viscosity. To lowest order, the pressure perturbation δP is δP = c s ρδ, (5.4)where c s is the speed of sound squared. We also define the equation of state parameter w = Pρ , and the dimensionless paramter c vis that characterizes the viscosity by c vis := (cid:18) η ζ (cid:19) H ¯ ρ (5.5)The anisotropic stress induced in the effective theory satisfies − ρ k i k j k Σ ij = − c vis θ H (5.6)To leading order in the long-wavelength perturbations, we have1¯ ρ k i k j k [ τ ij ] s Λ = c s δ − c vis θ H (5.7) At super-Hubble scales, with k << H , the short-scale dynamics simply renormalizethe background pressure, energy density, and anisotropic stress. We can determinethe renormalization by evaluating [ τ µν ] Λ as k → ρ eff = lim k → − [ τ ] Λ ¯ P eff = lim k →
13 [ τ ii ] Λ (5.8)The background anisotropic stress is zero. To find the equations governing the dynamics of the effective fluid, we smooth theequations (3.27-3.28). Smoothing the continuity equation gives:˙ ρ l + ∂ i (( ρ l + P l ) v il ) − v il ∂ i P l + 3( H − ˙Φ l )( ρ l + P l ) + 2 v l · ˙ v l ( ρ l + P l )= − ∂ i [( ρ s + P s ) v is ] Λ + [ v is ∂ i P s ] Λ + 3[ ˙Φ s ( ρ s + P s )] Λ − [( ρ + P ) ∂ ( v )] s Λ (5.9)11rom now on we will omit the “ l ” subscript from the variables to avoid cluttering thenotation. Keeping only the terms linear in the perturbations δ l and v l , subtractingthe homogeneous equation and dividing by ¯ ρ gives˙ δ = (1 + w )(3 ˙Φ − ∇ · v ) − H δ (cid:0) c s − w (cid:1) (5.10)To this order we can take w = ¯ P ¯ ρ . Similarly, smoothing Eq.(3.28) gives: ∇ · ˙ v + c a (1 + w ) ∇ · v + ∇ Ψ = − ρ (1 + w ) ∂ i ∂ j [ τ ij ] s Λ (5.11)where w = ¯ P ¯ ρ as before and c a = ˙¯ P ˙¯ ρ is the adiabatic sound speed. (The details of thiscalculation are somewhat involved and are therefore given in Appendix C.) We canintroduce the velocity potential v such that v i = ik i v :˙ v + c a (1 + w ) v − Ψ = − ρ (1 + w ) (cid:18) c s ¯ ρδ − c vis θ H (cid:19) (5.12)Once again, our results differ from those given in [1]. We believe that the differencesare due to extra simplifications that are made in [1]. For example, Baumann et al.assume that the Newtonian approximation holds at scales k − << Λ − .The Einstein equations for the long-wavelength metric perturbations are given inAppendix B.2. Finally, we consider the long-wavelength scalar field perturbations by smoothing theKlein-Gordon equations. In order to simplify our calculations we will ignore themetric perturbations, which is a commonly used approximation when studying fielddynamics during preheating[4, 5, 11, 12]. Separating the fields φ, χ into homogeneousparts ¯ φ, ¯ χ and perturbations δφ, δχ , and assuming that ¯ χ = 0, we obtain the followingequations for the perturbations:¨ δφ − ∇ ( δφ ) + 2 H ˙ δφ + a m δφ + a g ] δχ ( ¯ φ + δφ ) = 0 (5.13)¨ δχ − ∇ ( δχ ) + 2 H ˙ δχ + a g χ ( ¯ φ + δφ ) = 0 (5.14)Smoothing these equations and keeping only the terms linear in the long-wavelengthperturbations gives:¨ δφ l − ∇ ( δφ l ) + 2 H ˙ δφ l + a m δφ l = − a g (cid:0) ¯ φ [ δχ s ] Λ + [ δχ s δφ s ] Λ (cid:1) (5.15)¨ δχ l − ∇ ( δχ l ) + 2 H ˙ δχ l + a g ¯ φ δχ l = − a g (cid:0) φ [ δχ s δφ s ] Λ + [ δχ s δφ s ] Λ (cid:1) (5.16)12e have obtained evolution equations for δφ l and δχ l in terms of the 2 and 3-pointcorrelation functions of the short-wavelength field perturbations. It would be inter-esting to find even approximate analytical forms for the correlation functions, so thatthe equations could be solved explicitly. We have studied field dynamics during preheating in an unusual way, by focusing ourattention on long-wavelength perturbations and integrating out the short-wavelengthmodes. As might be expected, we see that the homogeneous part of the inflatonfield plays an important part in the evolution of both the φ and χ fields, as do thecorrelation functions between the perturbations δφ and δχ . We have also adaptedand extended the formalism in [1] so that it can be applied to a universe whosematter content has pressure, and where the short-scale gravitational dynamics arenot Newtonian. In contrast to the work in [1], the only restriction we placed on ourmodel was that the matter content should be modeled as a perfect fluid.There are several obvious ways in which this work could be extended. Firstly,of course, it would be desirable to find analytic solutions of the equations of motiongiven in this paper, and to compare them with the results of lattice simulations of fielddynamics during preheating. We have also given the linearized Einstein equations,which govern the evolution of large-scale scalar metric perturbations during preheat-ing. Given that data collection is currently underway to find evidence distinguishingbetween various preheating models, it would be interesting to find ways of calculat-ing observable non-Gaussianities in the power spectrum of the metric perturbationsusing this formalism. Also, we have only considered scalar metric perturbations inthis work: the next step might be to consider the evolution of large-scale vector andtensor perturbations. A Smoothing Bilinear and Trilinear Quantities
Here we explicitly outline the calculations that lead to Eq.(4.5-4.6). Smoothing abilinear quantity f g gives the same result as in [1], but we find a different expressionfor [ f gh ] Λ . Therefore we will follow the approach given in [1] to derive [ f g ] Λ , beforeoutlining our calculation for [ f gh ] Λ and emphasizing the differences from the resultin [1]. 13ecall that we assumed the following useful properties for the Gaussian windowfunction W Λ ( | x − x ′ | ): Z x ′ W Λ ( x ′ − x ) i ( x ′ − x ) j = 1Λ δ ij (A.1) ∂ j ′ W Λ = − ∂ j W Λ = Λ ( x − x ′ ) j W Λ (A.2) ∂ i ′ ∂ j ′ W Λ = ∂ i ∂ j W Λ = − Λ δ ij W Λ + Λ ( x − x ′ ) i ( x − x ′ ) j W Λ (A.3)To smooth f g , we take its convolution with the window function:[ f g ] Λ = Z x ′ W Λ f ( x ′ ) g ( x ′ ) . (A.4)Splitting the fields f , g into long-wavelength modes f l , g l and short-wavelength modes f s , g s , we get: [ f g ] Λ = [ f l g l ] Λ + [ f s g s ] Λ + [ f l g s ] Λ + [ f s g l ] Λ . (A.5)Since f l and g l are assumed to be small perturbations, and are long-scale, we canexpand them in a Taylor series about x : f l ( x ′ ) = f l ( x ) + ∂ i f l ( x )( x ′ − x ) i + 12 ∂ i ∂ j f l ( x )( x ′ − x ) i ( x ′ − x ) j + . . . (A.6)and similarly for g l . This gives us[ f l g l ] Λ = f l g l + 1Λ (cid:18) ∇ f l · ∇ g l + 12 f l ∇ g l + 12 g l ∇ f l (cid:19) + . . . (A.7)where the dots indicate higher derivative terms, suppressed by powers of k / Λ . Tosimplify the term [ f l g s ] Λ , we first rewrite it as[ f l g s ] Λ = [ f l g ] Λ − [ f l g l ] Λ , (A.8)where the second term is given by (A.7). To simplify the first term, we again use theTaylor expansion of f l , giving us:[ f l g ] Λ = f l g l − ∂ i f l [( x − x ′ ) i g ( x ′ )] Λ + 12 ∂ i ∂ j f l · [( x − x ′ ) i ( x − x ′ ) j g ( x ′ )] Λ + . . . = f l g l + 1Λ (cid:18) ∇ f l · ∇ g l + 12 g l ∇ f l (cid:19) + . . . (A.9)Interchanging f and g in the above expression gives us [ f s g l ] Λ . Thus we find that[ f g ] Λ = f l g l + [ f s g s ] Λ + 1Λ ∇ f l · ∇ g l + . . . (A.10)14moothing a trilinear term f gh proceeds in the same way. We begin by splittingthe fields f, g, h into long-wavelength modes f l , g l , h l and short-wavelength modes f s , g s , h s , giving:[ f gh ] Λ = [ f l g l h l ] Λ + [ f s g s h s ] Λ + [ f l gh ] Λ + [ f g l h ] Λ + [ f gh l ] Λ − [ f g l h l ] Λ − [ f l gh l ] Λ − [ f l g l h ] Λ (A.11)Expanding f l , g l in Taylor series as before, we can smooth f l g l h :[ f l g l h ] Λ = f l g l h l − g l ∂ i f l [( x − x ′ ) i h ( x ′ )] Λ − f l ∂ i g l [( x − x ′ ) i h ( x ′ )] Λ ∂ i f l ∂ j g l [( x − x ′ ) i ( x − x ′ ) j h ] Λ g l ∂ i ∂ j f l [( x − x ′ ) i ( x − x ′ ) j h ] Λ + 12 ∂ i ∂ j g l [( x − x ′ ) i ( x − x ′ ) j h ] Λ + . . . = f l g l h l + g l ∇ f l · ∇ h l Λ + f l ∇ g l · ∇ h l Λ + h l ∇ f l · ∇ g l Λ + g l h l ∇ f l + f l h l ∇ g l + . . . (A.12)We can also smooth f l gh :[ f l gh ] Λ = f l [ gh ] Λ − ∂ j f l [( x − x ′ ) j gh ] Λ + 12 ∂ i ∂ j f l [( x − x ′ ) i ( x − x ′ ) j gh ] Λ + . . . = f l g l h l + f l [ g s h s ] Λ + f l ∇ g l · ∇ h l Λ + g l ∇ f l · ∇ h l Λ + h l ∇ f l · ∇ g l Λ + g l h l ∇ f l + ∇ f l · ∇ [ g s h s ] Λ Λ + ∇ f l [ g s h s ] Λ + . . . , (A.13)where we have used (A.10). Substituting (A.13) and (A.12) into (A.11), we find[ f gh ] Λ = f l g l h l + [ f s g s h s ] Λ + f l [ g s h s ] Λ + g l [ f s h s ] Λ + h l [ f s g s ] Λ + 1Λ ( f l ∇ g l · ∇ h l + g l ∇ f l · ∇ h l + h l ∇ f l · ∇ g l )+ 1Λ ( ∇ f l · ∇ [ g s h s ] Λ + ∇ g l · ∇ [ f s h s ] Λ + ∇ h l · ∇ [ f s g s ] Λ )+ 12Λ ( ∇ f l [ g s h s ] Λ + ∇ g l [ f s h s ] Λ + ∇ h l [ f s g s ] Λ ) + . . . (A.14)The expression easily generalizes to smoothed polynomial quantities f f · · · f n of anyorder n . 15aumann et al. claim in [1] that smoothing the trilinear quantity ρv i v j gives:[ ρv i v j ] Λ = ρ l v il v jl + [ ρv is v js ] Λ + ρ l ∇ v il · ∇ v jl Λ + . . . = ρ l v il v jl + [ ρ s v is v js ] Λ + [ ρ l v is v js ] Λ + ρ l ∇ v il · ∇ v jl Λ + . . . = ρ l v il v jl + [ ρ s v is v js ] Λ + ρ l [ v is v js ] Λ + ∇ ρ l · ∇ [ v is v js ] Λ Λ + ρ l ∇ v il · ∇ v jl Λ + . . . (A.15)Comparing this to (A.14), we see that our general expression for a smoothed trilinearquantity [ f gh ] Λ does not agree with Baumann et al.’s expression for [ ρv i v j ] Λ upon thesubstitution of ρ, v i , and v j for f, g, and h , due to the presence of extra terms such as v il [ ρ s v js ] Λ in our formula. As [ f gh ] Λ should be symmetrical in f, g, h , we believe thatthese terms should be present unless some additional assumptions or conditions areimposed. In our work, we will make no such additional assumptions. B The Einstein Equations
B.1 Velocity Expansion of the Einstein Equations
As explained in Section 3.3, the energy-momentum pseudotensor τ µν is given by τ µν ≡ T µν − G NLµν πG (B.1)We expand these equations to order v . The expansion of T µν is given by Eq.(3.19-3.21). As explained in Section 2, the metric perturbation Φ is of order v . However,each gradient ∇ Φ lowers the order in v by one, so terms of the form Φ ∇ Φ are oforder v . We therefore work to first order in Φ, with the exception of such gradientterms. We end up with the equations ∇ Φ − H ( ˙Φ + H Ψ) = − πGa ( τ − ¯ T ) (B.2) ∂ i ( ˙Φ + H Ψ) = 4 πGa τ i (B.3)¨Φ + H (2 ˙Φ + ˙Ψ) + ( H + 2 ˙ H )Ψ − ∇ (Φ − Ψ) = 4 πGa τ ii − ¯ T ii ) (B.4) ∂ i ∂ j (cid:20) ∂ i ∂ j (Φ − Ψ) − δ ij ∇ (Φ − Ψ) (cid:21) = 8 πGa ∂ i ∂ j ( τ ij − δ ij τ kk ) (B.5)16here the non-linear parts of the Einstein tensor are − a ( G ) NL ∼ −∇ Φ · ∇ Φ + 4Φ ∇ Φ (B.6) − a ( G i ) NL ∼ − a ( G ij ) NL ∼ δ ij ∇ Φ · ∇ Φ − ∇ i Φ ∇ j Φ (B.8)to order v . The homogeneous Einstein equations are H = − πGa T = − πGa ρ (B.9) H + 2 ˙ H = − πGa T ii = − πGa P (B.10) B.2 The Einstein Equations in the Long-Wavelength Effec-tive Theory
The long-wavelength metric perturbations are given by the Einstein equations, sourcedby the effective energy-momentum pseudotensor. To linear order in the long-wavelengthperturbations, we have ∇ Φ l − H ( ˙Φ l + H Ψ l ) = − πGa ( τ − ¯ τ ) = 4 πGa δ l ¯ ρ (B.11) ∂ i ( ˙Φ l + H Ψ l ) = 4 πGa τ i = 4 πGa ( ¯ ρ + ¯ P ) v il (B.12)¨Φ l + H (2 ˙Φ l + ˙Ψ l ) + ( H + 2 ˙ H )Ψ l − ∇ (Φ l − Ψ l ) = 4 πGa τ ii − ¯ τ ii )= 4 πGa c s ¯ ρδ l (B.13) ∂ i ∂ j (cid:20) ∂ i ∂ j (Φ l − Ψ l ) − δ ij ∇ (Φ l − Ψ l ) (cid:21) = 8 πGa ∂ i ∂ j ( τ ij − δ ij τ kk ) (B.14)= 8 πGa ( ¯ ρ + ¯ P ) c vis θ H (B.15)where the components of τ µν are τ − ¯ τ = − δ l ρ l − [ ρv ] s Λ − [Φ s,k Φ s,k ] Λ − s Φ s,kk ] Λ πGa (B.16) τ i = ( ρ l + P l ) v il + [( ρ s + P s ) v is ] Λ (B.17) τ ij = P l δ ij + [( ρ + P ) v i v j ] s Λ − [Φ s,k Φ s,k ] Λ δ ij − s,i Φ s,j ] Λ πGa (B.18)Equations (B.16-B.18) differ slightly from those given in [1], as we have different ex-pressions for smoothed trilinear quantities. However, as the smoothed short-wavelength17erturbations can be written as a derivative expansion of long-wavelength variables,we find that to leading order in the perturbation variables, the Einstein equations arethe same. C Smoothing the Momentum Conservation Equa-tions
Here we show the explicit steps taken in smoothing the momentum conservationequation ( g σν + u σ u ν ) ∇ µ T µν = ( ρ + P ) u µ ∇ µ u σ + ∂ σ P + u σ u µ ∂ µ P = 0 (C.1)which, when expanded to order v , becomes:( ρ + P ) (cid:0) ˙ v i + v j ∂ j v i + ∂ i Ψ (cid:1) + ∂ i P + v i ∂ P + v i v j ∂ j P = 0 (C.2)We will omit all the higher derivative corrections suppressed by powers of k / Λ thatarise. Smoothing the term ∂ i P is easy:[ ∂ i P ] Λ = ∂ i P l (C.3)Next we smooth the term ( ρ + P ) ∂ i Ψ:[( ρ + P ) ∂ i Ψ] Λ = ( ρ l + P l ) ∂ i Ψ l + [( ρ s + P s ) ∂ i Ψ s ] Λ (C.4)= ( ρ l + P l ) ∂ i Ψ l + 14 πGa [ ∇ Ψ s ∂ i Ψ s ] Λ + [ P s ∂ i Ψ s ] Λ , (C.5)using the Poisson equation, ∇ Ψ s = 4 πGa ρ s . We then integrate by parts and use ∂ j ′ W Λ = − ∂ j W Λ to get[( ρ + P ) ∂ i Ψ] Λ = ( ρ l + P l ) ∂ i Ψ l + [ P s ∂ i Ψ s ] Λ + 18 πGa ∂ j [Ψ ,k Ψ ,k δ ij − ,i Ψ ,j ] Λ (C.6)18inally we smooth the remaining terms, integrating by parts and using the properties(A.1-A.3) of the window function when necessary:[( ρ + P ) (cid:0) ˙ v i + v j ∂ j v i (cid:1) + v i ˙ P + v i v j ∂ j P ] Λ = ∂ [( ρ + P ) v i ] Λ − Z W Λ v i ( ˙ ρ + ˙ P ) + Z W Λ (cid:16) ( ρ + P ) v j ∂ j v i + v i ˙ P + v i v j ∂ j P (cid:17) = ∂ [( ρ + P ) v i ] Λ − Z W Λ v i (cid:16) − ∂ j ′ (( ρ + P ) v j ) − H − ˙Φ)( ρ + P ) − ( ρ + P )˙( v ) (cid:17) + Z W Λ ( ρ + P ) v j ∂ j v i = ∂ [( ρ + P ) v i ] Λ + Z W Λ (cid:0) v i ∂ j ′ (( ρ + P ) v j ) + ( ρ + P ) v j ∂ j v i (cid:1) + Z W Λ v i (cid:16) H − ˙Φ)( ρ + P ) + ( ρ + P ) ˙( v ) (cid:17) = ∂ [( ρ + P ) v i ] Λ + ∂ j Z W Λ ( ρ + P ) v i v j + Z W Λ v i (cid:16) H − ˙Φ)( ρ + P ) + ( ρ + P ) ˙( v ) (cid:17) (C.7)Using the smoothed continuity equation (5.9), we can substitute for ˙ ρ + ˙ P , whichgives:[( ρ + P ) (cid:0) ˙ v i + v j ∂ j v i (cid:1) + v i ˙ P + v i v j ∂ j P ] Λ = ( ρ l + P l ) ˙ v li + ∂ [( ρ s + P s ) v is ] Λ + ∂ j Z W Λ ( ρ + P ) v i v j + v il (cid:16) − ∂ j (( ρ l + P l ) v jl ) − ∂ j [( ρ s + P s ) v is ] Λ + ˙ P l + v jl ∂ j P l + [ v js ∂ j P s ] Λ (cid:17) (C.8)Adding Eq.(C.3-C.8) gives:( ρ l + P l ) ˙ v li + ( ρ l + P l ) v jl ∂ j v li + v li ˙ P l + v il v jl ∂ j P l + ( ρ l + P l ) ∂ i Ψ l + ∂ i P l = ∂ j (cid:20) Ψ ,k Ψ ,k δ ij − ,i Ψ ,j πGa (cid:21) Λ + v il ∂ j [( ρ s + P s ) v is ] Λ − v il [ v js ∂ j P s ] Λ − ∂ [( ρ s + P s ) v is ] Λ − ∂ j [( ρ + P ) v i v j ] s Λ − [ P s ∂ i Ψ s ] Λ (C.9)Keeping only the leading terms linear in the matter and velocity perturbations, weget: (1 + w ) ˙ v li + v il ˙ P l ρ l + (1 + w ) ∂ i Ψ l + ∂ i P l ρ = ∂ j [ τ ij ] s Λ (C.10)19 eferences [1] D. Baumann, A. Nicolis, L. Senatore, and M. Zaldarriaga, “Cosmological Non-Linearities as an Effective Fluid,” [arXiv:1004.2488].[2] K. Enqvist and G. Rigopoulos, “Non-linear mode coupling and the growth ofperturbations in ΛCDM,” [arXiv:1008.2751].[3] B. A. Bassett, S. Tsujikawa, and D. Wands, “Inflation Dynamics and Reheating,” Rev. Mod. Phys. , 537589 (2006), [arXiv:astro-ph/050763v2].[4] A. V. Frolov, “DEFROST: A New Code for Simulating Preheating after Inflation,” JCAP , 009, (2008) [arXiv:0809.4904v2].[5] G. N. Felder and L. Kofman, “Nonlinear Inflaton Fragmentation after Preheat-ing,” Phys. Rev. D , 043518 (2007) [arXiv:hep-ph/0606256].[6] F. Bernardeau, S. Colombi, E. Gazta˜naga, R. Scoccimarro, “Large-Scale Structureof the Universe and Cosmological Perturbation Theory,” Physics Reports , 1-248, (2002) [arXiv:astro-ph/011255].[7] R. Mainini, “Scalar field-perfect fluid correspondence and nonlinear perturbationequations,”
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