Impact of anisotropic stress of free-streaming particles on gravitational waves induced by cosmological density perturbations
aa r X i v : . [ a s t r o - ph . C O ] J a n Impact of anisotropic stress of free-streaming particles on gravitational waves inducedby cosmological density perturbations
Shohei Saga, Kiyotomo Ichiki,
1, 2 and Naoshi Sugiyama
1, 2, 3 Department of Physics and Astrophysics, Nagoya University, Aichi 464-8602, Japan ∗ Kobayashi-Maskawa Institute for the Origin of Particles and the Universe, Nagoya University, Nagoya 464-8602, Japan Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU),The University of Tokyo, Chiba 277-8582, Japan
Gravitational waves (GWs) are inevitably induced at second-order in cosmological perturbationsthrough non-linear couplings with first order scalar perturbations, whose existence is well establishedby recent cosmological observations. So far, the evolution and the spectrum of the secondary inducedGWs have been derived by taking into account the sources of GWs only from the product of firstorder scalar perturbations. Here we newly investigate the effects of purely second-order anisotropicstresses of photons and neutrinos on the evolution of GWs, which have been omitted in the literature.We present a full treatment of the Einstein-Boltzmann system to calculate the spectrum of GWswith anisotropic stress based on the formalism of the cosmological perturbation theory. We findthat photon anisotropic stress amplifies the amplitude of GWs by about 150% whereas neutrinoanisotropic stress suppress that of GWs by about 30% on small scales k & . h Mpc − compared tothe case without anisotropic stress. The second order anisotropic stress does not affect GWs withwavenumbers k . . h Mpc − . The result is in marked contrast with the case at linear order, wherethe effect of anisotropic stress is damping in amplitude of GWs. PACS numbers: 04.30.-w, 98.80.-k
I. INTRODUCTION
The standard cosmological model contains two types of cosmological perturbations, namely, the curvature pertur-bations (i.e. the scalar mode) and the primordial gravitational waves (GWs) (i.e. the tensor mode). Among the two,the existence and the property of the scalar type perturbations have been established by a number of observations,such as fluctuations in the cosmic microwave background (CMB) and large scale structure of the universe [1–5]. Onthe other hand, the observational cosmology through the tensor mode perturbations has just begun with the breakingdiscovery of primordial B-mode polarizations of CMB anisotropies by the BICEP2 experiment [6]. If this B-modesignal is attributed to the primordial GWs produced during inflation, the scalar-to-tensor ratio is as large as r ≈ . r ≈ . ∗ Electronic address: [email protected] interaction between photons and electrons. More recently, the gauge dependence and the invariance are analyzed inRef. [30].The worth of the second-order cosmological perturbation theory is not only in the improvement of accuracy of thetheory but also in the appearance of new effects which do not arise in the first-order cosmological perturbation theory.For example, non-Gaussianity arises from non-linearity at second-order since the non-linear couplings cause the strainof the primordial Gaussian profile of perturbations. In the PLANCK experiment, the non-Gaussianity is characterizedby f NL , which is found to be consistent with zero as f localNL = 2 . ± . SONG [34, 43] and CosmoLib2 nd [44] take into accountthe effects of purely second-order anisotropic stress on the second-order CMB bispectrum and the second-order B-mode polarization spectrum. Therefore, this paper for the first time analyses the full second-order gravitational wavespectrum, while its indirect effect has already been included in the photon bispectrum analysis and in the analysis ofthe second-order B-modes in the above-mentioned works.In the first-order cosmological perturbation theory, anisotropic stress of neutrinos is shown to affect the spectrumof background GWs from inflation at several tens of percent level [45–49]. These studies found that the anisotropicstress pulls out the energy of GWs, which causes the damping of GWs and also CMB anisotropies generated fromthe tensor mode. In preparation for the future experiments to detect cosmological GWs directly, such as DECIGO[50] and atomic gravitational wave interferometric sensors [51], the precise estimation of the amplitude of GWs wouldbe needed. In this paper, we estimate the amplitude of the secondary GWs including not only the product of thefirst-order perturbations but also the purely second-order perturbations. To achieve this, we solve the full system ofthe Einstein-Boltzmann equations at second-order in the tensor mode numerically.The plan of this paper is as follows. In the next section, we expand the Einstein and Boltzmann equationsup to second-order in the cosmological perturbation theory. In section III, we show and discuss results of ournumerical calculation. To set up the numerical calculation, we derive the solution up to the first-order in the tight-coupling approximation, which is adopted as our initial condition. In section IV, we discuss the effects of second-orderanisotropic stress on the gravitational wave spectrum. Section V is devoted to our summary.Throughout this paper, we use the units in which c = ~ = 1 and the metric signature with ( − , + , + , +). We obeythe rule that the Greek indices run from 0 to 3 and the alphabets run from 1 to 3, respectively. II. SECOND-ORDER PERTURBATION THEORY
In this section, we formulate the second-order cosmological perturbation theory. Throughout this paper, we workin the Poisson gauge [24] whose metric is given by ds = a ( η ) (cid:2) − e dη + 2 ω i dηdx i + (cid:0) e − δ ij + χ ij (cid:1) dx i dx j (cid:3) , (1)where the gauge conditions ω i,i = χ ij,j = 0 and the traceless condition χ ii = 0 are imposed on ω i and χ ij . Raisingor lowering indices of perturbations are done by δ ij . Owing to these gauge conditions and the traceless condition, ω i and χ ij contain only the vector and tensor modes, respectively.In the second-order cosmological perturbation theory, scalar, vector and tensor modes must mix due to the non-linearity. Scalar, vector and tensor modes correspond to the perturbations of density or curvature, vorticity, and GWs,respectively. We neglect the first-order vector mode since the vector mode has only a decaying solution in the standardcosmology. We also neglect the first order tensor mode for clarity because our aim here is to precisely estimate theamplitude of the second-order tensor perturbations which are generated from the first order density perturbations. Weexpand cosmological perturbations in the metric as Ψ = Ψ (1) + Ψ (2) , Φ = Φ (1) + Φ (2) , ω i = ω (2) i , and χ ij = χ (2) ij .In Ref. [41], the authors only considered the secondary GWs induced through the convolutions of the first-orderscalar perturbations, in which case the second-order Boltzmann equation is not necessary. In this paper, we needsolve the second-order Boltzmann equation because we consider not only the convolutions but also purely second-ordereffects. A. Boltzmann equation
The Boltzmann equation describes the evolution of distribution functions of particles including microscopic colli-sions. Let us consider the Boltzmann equation for photons, which is given by dfdλ ( x µ , P µ ) = ˜ C [ f ] , (2)where f ( x µ , P µ ) is the distribution function, P µ is the canonical momentum, and λ is the affine parameter and ˜ C [ f ]is the collision term due to the Thomson interaction between photons and electrons. Here we omit the interactionbetween photons and protons because the Thomson cross section of protons is much smaller than that of electrons.In the Boltzmann equation for dark matter or neutrinos, the collision term must vanish.To calculate the perturbed Boltzmann equation, it is useful to change the coordinate system from Poisson gauge( x µ , P µ ) to the local inertial frame ( x µ , p µ ) [21]. Since we consider the cosmological perturbations up to second-order,the distribution function is expanded as f ( η, x , p, ˆ n ) = f (0) ( η, p ) + f (1) ( η, x , p, ˆ n ) + 12 f (2) ( η, x , p, ˆ n ) , (3)where p and ˆ n are the amplitude and the direction of photon’s momentum, respectively. The zeroth order distributionfunction, f (0) ( η, p ), is fixed to the Planck distribution. It is useful to define the brightness function which is given by∆ (1 , ( η, x , ˆ n ) = R dp p f (1 , ( η, x , p, ˆ n ) R dp p f (0) ( η, p ) , (4)where the denominator of the right-hand side is proportional to the mean energy density of photons. The relationsbetween the temperature fluctuation of CMB, Θ ≡ δT /T , and the brightness function are given by ∆ (1) = 4Θ (1) and∆ (2) = 4Θ (2) + 16(Θ (1) ) [33] at first and second order, respectively.The angle dependence of the brightness function is expanded by the spherical harmonics as∆ (1 , ( η, x , ˆ n ) = X ℓ ℓ X m = − ℓ ∆ (1 , ℓ,m ( η, x )( − i ) ℓ r π ℓ + 1 Y ℓ,m (ˆ n ) . (5)The Boltzmann equation of photons in terms of ∆ (2) ℓ,m up to second-order is written by˙∆ (2) ℓ,m + k (cid:20) c ℓ +1 ,m ℓ + 3 ∆ (2) ℓ +1 ,m − c ℓ,m ℓ − (2) ℓ − ,m (cid:21) = S (2) ℓ,m , (6)where c ℓ,m ≡ √ ℓ − m . A dot represents a derivative with respect to the conformal time η . Here we have translatedfrom real space to Fourier space, following the convention of the Fourier transformation as f ( x ) = Z d k (2 π ) f ( k ) e i k · x . (7)The source term S (2) ℓ,m can be expressed as S (2) ℓ,m ( k , η ) = C (2) ℓ,m ( k , η ) + G (2) ℓ,m ( k , η ) . (8)Here C (2) ℓ,m is the collision term that is proportional to the differential optical depth ˙ τ c ≡ − an e σ T , where a , n e , and σ T are the number density of the electron, scale factor, and the Thomson scattering cross-section, respectively, and G (2) ℓ,m denotes the gravitational effects, i.e., the lensing and the redshift terms. In this paper, we call C ℓ,m and G ℓ,m thescattering term and the gravitational term, respectively. The explicit form of C (2) ℓ,m and G (2) ℓ,m are given as C (2) ℓ,m = ˙ τ c ∆ (2) ℓ,m − ˙ τ c (cid:18) ∆ (2)00 δ ℓ, δ m, + 4 v (2)b m δ ℓ, + 110 ∆ (2)2 ,m δ ℓ, (cid:19) + ˙ τ c Z d k (2 π ) h − δ b + Ψ) (1) ( k ) δ (1) γ ( k ) − k · ˆ k ) v (1) γ ( k ) v (1)b 0 ( k ) i δ ℓ, δ m, + ˙ τ c Z d k (2 π ) h − k · ˆ k ) v (1)b 0 ( k ) v (1)b 0 ( k ) i δ ℓ, δ m, + ˙ τ c Z d k (2 π ) h − v (1)b 0 ( k )( δ b + Ψ) (1) ( k ) − v (1)b 0 ( k ) δ (1) γ ( k ) − v (1)b 0 ( k )Π (1) γ ( k ) i r π Y ∗ ,m (ˆ k ) δ ℓ, + ˙ τ c Z d k (2 π ) h − Π (1) γ ( k )( δ b + Ψ) (1) ( k ) i r π Y ∗ ,m (ˆ k ) δ ℓ, + ˙ τ c Z d k (2 π ) h (1) ℓ, ( k )( δ b + Ψ) (1) ( k ) i r π ℓ + 1 Y ∗ ℓ,m (ˆ k )+ ˙ τ c i ( − i ) − ℓ ( − m (2 ℓ + 1) X ℓ X m ,m ( − i ) ℓ (cid:18) ℓ ℓ (cid:19) (cid:18) ℓ ℓm m − m (cid:19) × Z d k (2 π ) h (2 + δ ℓ , )∆ (1) ℓ , ( k ) v (1)b 0 ( k ) i r π ℓ + 1 Y ∗ ℓ ,m (ˆ k ) r π Y ∗ ,m (ˆ k )+ ˙ τ c ( − i ) − ℓ ( − m (2 ℓ + 1) X m ,m (cid:18) ℓ (cid:19) (cid:18) ℓm m − m (cid:19) × Z d k (2 π ) h v (1)b 0 ( k ) v (1)b 0 ( k ) − v (1) γ ( k ) v (1)b 0 ( k ) i r π Y ∗ ,m (ˆ k ) r π Y ∗ ,m (ˆ k ) , (9)and G (2) ℓ,m = 4 ˙Φ (2) δ ℓ, δ m, − X λ = ± ω (2) λ δ ℓ, δ m,λ + 4 k Ψ (2) δ ℓ, δ m, − X σ = ± χ (2) σ δ ℓ, δ m,σ +4 k Z d k (2 π ) h Ψ (1) ( k )Ψ (1) ( k ) i δ ℓ, δ m, + Z d k (2 π ) h (1) ℓ, ( k ) ˙Φ (1) ( k ) i r π ℓ + 1 Y ∗ ℓ,m (ˆ k )+2 i ( − i ) − ℓ (2 ℓ + 1) X ℓ X m ,m ( − m (2 ℓ + 1) (cid:18) ℓ ℓ (cid:19) (cid:18) ℓ ℓm m − m (cid:19) × Z d k (2 π ) h k (Ψ + Φ) (1) ( k ) ˜∆ (1) ℓ ( k ) i r π Y ∗ ,m (ˆ k ) r π ℓ + 1 Y ∗ ℓ ,m (ˆ k )+ Z d k (2 π ) h k Ψ (1) ( k )Φ (1) ( k ) i r π Y ∗ ,m (ˆ k ) δ ℓ, − i ( − i ) − ℓ (2 ℓ + 1) X ℓ ( − i ) ℓ (cid:18) ℓ ℓ (cid:19) (cid:18) ℓ ℓ (cid:19) Z d k (2 π ) h k ∆ (1) ℓ , ( k )(Ψ (1) + Φ (1) )( k ) i r π ℓ + 1 Y ∗ ℓ,m (ˆ k ) − i ( − i ) − ℓ ( − m (2 ℓ + 1) X ℓ X m ,m ( − i ) ℓ (cid:18) ℓ ℓ (cid:19) (cid:18) ℓ ℓm m − m (cid:19) × Z d k (2 π ) h k ∆ (1) ℓ , ( k )Ψ (1) ( k ) i r π ℓ + 1 Y ∗ ℓ ,m (ˆ k ) r π Y ∗ ,m (ˆ k )+2 i ( − i ) ℓ ( − m (2 ℓ + 1) X L,L ′ ,L ′′ X M ′ ,M ′′ (2 L + 1)(2 L ′ + 1)(2 L ′′ + 1) × (cid:18) L ′ (cid:19) (cid:18) L ′ ℓ L (cid:19) (cid:18) L L ′′ (cid:19) (cid:18) L ′′ ℓM ′ M ′′ − m (cid:19) (cid:26) ℓ L ′′ L L ′ (cid:27) × Z d k (2 π ) h k (Ψ + Φ) (1) ( k ) ˜∆ (1) L ( k ) i r π Y ∗ ,M ′ (ˆ k ) r π L ′′ + 1 Y ∗ L ′′ ,M ′′ (ˆ k ) , (10)where Fourier wavevectors k , k , and k satisfy the relation k = k + k . The relations between the distributionfunction and the density perturbation δ , the velocity perturbation v , and anisotropic stress Π γ are defined in Ref. [32].In Eq. (10), we have defined ˜∆ (1) ℓ as˜∆ (1) ℓ ′′ ≡ (2 ℓ ′′ + 3)∆ (1) ℓ ′′ +1 + (2 ℓ ′′ + 7)∆ (1) ℓ ′′ +3 + · · · , (11)which comes from the lensing term [33]. We see that the lensing term contains higher multipole moments. The sourceterm of the first-order Boltzmann equation vanish when m = 0, because we consider only the scalar mode in the firstorder perturbations. However for the second-order perturbations, not only the scalar mode ( m = 0), but also thevector ( m = λ ) and tensor ( m = σ ) modes arise due to non-linear couplings, where λ = ± σ = ±
2, respectively.Note that in the Einstein gravity, there is no source of the modes with | m | ≥ τ c = 0 in the above equations because massless neutrinos interactwith the other fluids only through gravity. We do not write down the hierarchical equation of neutrinos here sinceit is trivial. The distribution function of neutrinos is also expanded by the spherical harmonics and we write theexpansion coefficients as N (1 , ℓ,m in this paper. B. Tensor decomposition of the Einstein equation
Let us write down the second-order Einstein equation. Here we concentrate only on the tensor mode, which isequivalent to GWs. The second-order Einstein and energy-momentum tensors are, respectively [32], a G ij = e (cid:0) Φ ,i,j − Ψ ,i,j (cid:1) + Φ ,i Φ ,j − Ψ ,i Ψ ,j − (cid:0) Φ ,i Ψ ,j + Φ ,j Ψ ,i (cid:1) + H (cid:2) ˙ χ ij − (cid:0) ω i,j + ω j,i (cid:1)(cid:3) + 12 (cid:2) ¨ χ ij − (cid:0) ˙ ω i,j + ˙ ω j,i (cid:1) − χ ij,a,a (cid:3) +(diagonal part) δ ij , (12)and T i r j = ρ r Π i r j + (diagonal part) δ ij , (13) T i m j = ρ m v (1)m i v (1)m j + (diagonal part) δ ij , (14)where T i r j and T i m j denote massless (relativistic) particles such as photons and neutrinos, and massive (non-relativistic)particles such as baryons and dark matter, respectively. As the GWs are equivalent to the traceless and transversepart of the metric perturbations, we do not pick up the diagonal part in the Einstein and the energy momentumtensors for the non-relativistic matters. We decompose the tensor mode by the following expansion, χ (2) ij ( x , η ) = Z d k (2 π ) e i k · x X σ = ± χ (2) σ ( k, η ) e ( σ ) ij (ˆ k ) , (15)where e ( σ ) ij (ˆ k ) is the polarization tensor and is constructed by the polarization vectors as e ( ± ij (ˆ k ) = − r ǫ ( ± i (ˆ k ) ǫ ( ± j (ˆ k ) . (16)This polarization tensor satisfies the traceless and transverse conditions asˆ k i e ( ± ij (ˆ k ) = e ( ± ii (ˆ k ) = 0 . (17)By contracting Eqs. (12) and (14) with e ( − σ ) ij , we can obtain the tensor part of the Einstein equation as¨ χ (2) σ + 2 H ˙ χ (2) σ + k χ (2) σ = 8 πGa ρ (0) γ
415 ∆ (2)2 ,σ + 8 πGa ρ (0) ν N (2)2 ,σ + X s=b , dm πGa ρ (0)s Z d k (2 π ) " r v (1)s 0 ( k ) v (1)s 0 ( k ) π Y ∗ ,λ (ˆ k ) r π Y ∗ ,λ (ˆ k )+ Z d k (2 π ) k h Φ (1) ( k )Φ (1) ( k ) + Ψ (1) ( k )Ψ (1) ( k ) i r π Y ∗ ,σ (ˆ k ) , (18)where subscripts of “b” and “dm” mean baryons and dark matter, respectively, H = ˙ a/a , and ∆ (2)2 ,σ and N (2)2 ,σ areanisotropic stresses of photons and neutrinos, respectively. The third term of r.h.s. in Eq. (18) ∼ v (1)s ( k ) v (1)s ( k ) canbe read as the anisotropic stress for the non-relativistic matters, while the products of the velocity perturbations forthe relativistic particles are included in the first and second terms. These purely second-order anisotropic stresses ofphotons and neutrinos have not yet been considered in the previous work [41]. Here we newly take into account thesecontributions to the amplitude of GWs and find that the effect of the stress is significant, as we shall show below.The energy density spectrum of the GWs is defined as (see e.g., [52])Ω (2)GW ≡ k H (cid:20) k π P (2) χ ( k ) (cid:21) , (19)where P (2) χ ( k ) is the spectrum of the second-order tensor perturbations We present these spectra and their timeevolutions in the next section. C. Structure of the second-order perturbation theory
In the second-order perturbation theory, the transfer function depends on ( k, k , k ) or equivalently ( k, µ , k ), where µ is defined as ˆ k · ˆ k . The equation below is a schematic equation for the evolution of second-order perturbations [58],ˆ L h ∆ (2) ( k , η ) i = Z d k (2 π ) [ S ( k , η ; k , k )] , (20)where k = k + k , ˆ L is a general linear operator and S is a source term which is constructed from the first-orderperturbation variables. The source term S ( k , η ; k , k ) is given by the products of the first order perturbations, whichcan be expressed using the linear transfer functions ∆ T1 ( k , η ) and ∆ T2 ( k , η ), and primordial curvature perturbations ψ ( k ) and ψ ( k ) as ∆ T1 ( k , η ) ψ ( k )∆ T2 ( k , η ) ψ ( k ), where we used the fact that linear transfer functions do notdepend on the direction of the wavevector. The statistics of ψ ( k ) obeys the random Gaussian and characterized bythe primordial power spectrum, as h ψ ( k ) ψ ∗ ( k ) i = (2 π ) P ( k ) δ ( k − k ) , (21)where P ( k ) is the power spectrum.Observationally, the primordial power spectrum is nearly scale-invariant and parameterized as [1, 59] k π P ( k ) = 49 ∆ R ( k ) (cid:18) kk (cid:19) n s − . (22)In this paper, we employ the standard cosmological model, i.e., the Λ-CDM model, and we set ∆ R ( k =0 . h Mpc − ) = 2 . × − [1] and consider a scale-invariant scalar spectrum with n s = 1 for illustration pur-poses. Because we split perturbations into transfer functions and primordial perturbations, the second-order evolutionequation (Eq. 20) is translated to the equation for second order transfer functions in ( k , k , k ) space asˆ L h ∆ (2)T ( k , η ; k , k ) i = S ( k , η ; k , k ) , (23)where the subscript T represents the transfer function. Taking an ensemble average is the final step to derive thepower spectra of the second-order perturbations. We can split any second-order perturbation variable into a transferfunction and the first-order primordial perturbations as∆ (2) ( k , η ; k , k ) = ∆ (2)T ( k , η ; k , k ) × ψ ( k ) ψ ( k ) . (24)From the above expression, we can calculate the spectrum of the second-order variable as D ∆ (2) ( k , η ; k , k )∆ ∗ (2) ( k ′ , η ; k ′ , k ′ ) E = h ∆ (2)T ( k , η ; k , k ) i h ∆ (2)T ( k ′ , η ; k ′ , k ′ ) i × h ψ ( k ) ψ ( k ) ψ ∗ ( k ′ ) ψ ∗ ( k ′ ) i . (25)By using Wick’s theorem, the bracket in the above equation is reduced to h ψ ( k ) ψ ( k ) ψ ∗ ( k ′ ) ψ ∗ ( k ′ ) i = h ψ ( k ) ψ ∗ ( k ′ ) i h ψ ( k ) ψ ∗ ( k ′ ) i + h ψ ( k ) ψ ∗ ( k ′ ) i h ψ ( k ) ψ ∗ ( k ′ ) i = (2 π ) P ( k ) P ( k ) [ δ ( k ′ − k ) + δ ( k ′ − k )] δ ( k − k ′ ) . (26)To proceed the derivation of the power spectrum, we define the second-order spectrum as D ∆ (2) ( k , η ; k , k )∆ ∗ (2) ( k ′ , η ; k ′ , k ′ ) E ≡ (2 π ) P ∆ (2) ( k , η ; k , k , k ′ , k ′ ) δ ( k − k ′ ) . (27)Finally we calculate the convolution as P ∆ (2) ( k , η ) = Z d k (2 π ) Z d k ′ (2 π ) P ∆ (2) ( k , η ; k , k , k ′ , k ′ )= 2 Z d k (2 π ) h ∆ (2)T ( k , η ; k , k ) i P ( k ) P ( k ) . (28)In the second equality in Eq. (28), we assume that the source term S ( k , η ; k , k ) is symmetric with respect to theexchange of k and k , which means that the transfer function ∆ (2)T ( k , η ; k , k ) is also symmetric in k and k . III. NUMERICAL RESULTS
To solve cosmological perturbations up to second-order, we need the time evolutions of the transfer functions ofthe first order perturbations, Φ (1) ( k, η ), Ψ (1) ( k, η ), ∆ (1) ℓ,m ( k, η ), and N (1) ℓ,m ( k, η ) in the Poisson gauge. We obtain thesevariables using the CAMB code [57]. In practice, we store the first-order variables in k -space, whose range is taken as[5 × − h Mpc − , h Mpc − ]. We truncate the first-order hierarchies of photon and neutrino Boltzmann equationsat ℓ = 30 and the second-order hierarchies of them at ℓ = 25. We checked that the results are stable against thesechoices.In our numerical calculation, we store first-order transfer functions and solve the Einstein-Boltzmann system upto second-order and sample second-order transfer functions in the ( k , k ) plane with a fixed real k . Owing to thetriangle condition about k , k , and k , we reduce the sampling area of the ( k , k ) plane, namely, we need solve theequations only in the region where | k − k | ≤ k ≤ k + k . Furthermore, as we stock the first-order transfer functionswith a logarithmic interval, this triangle condition is effective to reduce the cost of numerical calculation. A. Initial conditions
To solve the second-order equations derived above numerically, we should set up the initial condition of eachperturbation variable. Thus we first solve the equations analytically with kη ≪ τ c is large[36, 53–55]. Although the photon and baryon fluids would behave as a single fluid, there is a small difference in motionbetween photon and baryon fluids. For this reason, we can expand the perturbation variables using the tight-couplingparameter which is given by ǫ ≡ (cid:12)(cid:12)(cid:12)(cid:12) k ˙ τ c (cid:12)(cid:12)(cid:12)(cid:12) ∼ − (cid:18) k − (cid:19) (cid:18) z (cid:19) − (cid:18) Ω b h . (cid:19) − , (29)where Ω b is the baryon density normalized by the critical density and h ≡ H /
100 [km s − Mpc − ] is the normalizedHubble parameter with H being the Hubble constant. In what follows we derive the tight-coupling solution up tofirst order to set the initial condition of photon and baryon fluids at second order in cosmological perturbations andto calculate the evolution of perturbations in a numerically stable manner.We expand the cosmological perturbation variables using the tight-coupling parameter up to first order as,∆ (CPT=1 , = ∆ (CPT=1 , , TCA=Ø) + ∆ (CPT=1 , , TCA= I ) , (30)where the Arabic number (1 , I ) represent the order in the cosmological perturbationtheory (CPT) and the tight coupling approximation (TCA), respectively. Note that the tight-coupling expansion isindependent of the order of cosmological perturbations.Let us now focus on the tensor mode ( m = σ ). It is useful to define the function Y ℓ ,ℓ ℓ,m (ˆ k , ˆ k ) as Y ℓ ,ℓ ℓ,m (ˆ k , ˆ k ) ≡ ( − m (2 ℓ + 1) X m ,m (cid:18) ℓ ℓ ℓ (cid:19) (cid:18) ℓ ℓ ℓm m − m (cid:19) r π ℓ + 1 Y ∗ ℓ ,m (ˆ k ) r π ℓ + 1 Y ∗ ℓ ,m (ˆ k ) , (31)where ℓ + ℓ + ℓ must be even because of a property of the Wigner-3j symbol. Note that, for the special case that ℓ = 0 or ℓ = 0, the dependence on ˆ k or ˆ k vanishes as Y ℓ , ℓ,m (ˆ k , ˆ k ) = r π ℓ + 1 Y ∗ ℓ,m (ˆ k ) δ ℓ,ℓ , Y ,ℓ ℓ,m (ˆ k , ˆ k ) = r π ℓ + 1 Y ∗ ℓ,m (ˆ k ) δ ℓ,ℓ . (32)Firstly, using the function defined above we obtain the solution at zeroth order in the tight coupling approximationas, ∆ (2 , Ø)2 ,σ = 20 Z d k (2 π ) h v (1 , Ø) γ ( k ) v (1 , Ø) γ ( k ) i Y , ,σ (ˆ k , ˆ k ) , (33)∆ (2 , Ø) ℓ ≥ ,σ = 0 . (34)It is interesting that at second-order in cosmological perturbations, anisotropic stress of photons arises even at ze-roth order in the tight coupling approximation, while it vanishes at first order in cosmological perturbations. Theanisotropic stress of photons in the second-order survives because of the coupling between velocity perturbations ofphotons in the scalar mode as is shown in Eq. (33). This result is consistent with Refs. [32, 42].Next, we consider the next order in the tight coupling approximation (CPT= 2, TCA= I ). We find the results as910 ∆ (2 ,I )2 ,σ = 2˙ τ c ˙ χ (2 , Ø) σ − Z d k (2 π ) h (1 ,I ) γ ( k )( δ (1 , Ø)b − Φ (1 , Ø) )( k ) i r π Y ∗ ,σ (ˆ k )+4 Z d k (2 π ) h v (1 , Ø) γ ( k ) v (1 ,I ) γ ( k ) + 8 v (1 , Ø) γ ( k ) δv (1 ,I ) γ b 0 ( k ) i Y , ,σ (ˆ k , ˆ k )+ Z d k (2 π ) (cid:20)(cid:18) k ˙ τ c (cid:19) (cid:16) δ (1 , Ø) γ ( k ) − (1 , Ø) (cid:17) ( k ) v (1 , Ø) γ ( k ) (cid:21) Y , ,σ (ˆ k , ˆ k ) , (35)∆ (2 ,I )3 ,σ = − (cid:18) k ˙ τ c (cid:19) √
55 ∆ (2 , Ø)2 ,σ + 15 Z d k (2 π ) h Π (1 ,I ) γ ( k ) v (1 , Ø) γ ( k ) i Y , ,σ (ˆ k , ˆ k ) , (36)∆ (2 ,I ) ℓ ≥ ,σ = 0 . (37)It might be surprising, but at this order, the octupole moment ( ℓ = 3) can survive because of two source terms.One is the anisotropic stress of photons at zeroth order in the tight-coupling approximation, which generates ℓ = 3moment through the streaming effect in the left-hand side of the Boltzmann equation. The other is a convolution ofthe first order cosmological perturbations. This term comes from the collision term. These terms do not appear inthe first order cosmological perturbation theory and the result here is a genuine second-order effect. Note that, highermultipoles than ℓ = 3 are equal to zero because the source is absent.By using the tight-coupling solution, we can analyze the behavior of anisotropic stress of photons in early times.If we adopt the adiabatic initial condition for the first-order variables [56], the velocity perturbation of photons isproportional to η in the Poisson gauge. The anisotropic stress of photons is proportional to η and therefore we findthe initial time dependence of the second-order anisotropic stress of photons must be proportional to η at zeroth orderin the tight coupling approximation. In the next section, we will show that these analytic estimates are consistentwith our numerical calculation.Before analyzing the second-order GWs, we focus on the evolutions of anisotropic stresses of photons and neutrinosat second-order, which are the essential sources of the GWs. To understand the behavior of the power spectra ofanisotropic stresses of photons and neutrinos, let us first investigate the source terms in the evolution equation ofanisotropic stress, S (2)2 ,σ ( k , η ; k , k ), in the subsection below. B. Sources for photon and neutrino anisotropic stresses
Let us investigate the sources for anisotropic stress at second-order to understand its time evolution. First of all,we note two key points in order to understand properties of the second-order power spectrum and source terms.First, in the second-order tensor-mode, the sources of the gravitational waves in Eq. (18) are suppressed on smallscales in the ΛCDM model. Therefore, the sources with wavenumbers k , k . k mainly contribute to the second-orderanisotropic stress. Second, as for the source terms, we note that the difference between the sources of photon andneutrino anisotropic stress is only in the scattering term, C (2) ℓ,m ∝ ˙ τ c , as is shown in Eq. (8). The source of photon -6 -5 -4 -3 -2 -1 -6 -5 -4 -3 -2 -1 S ( ) , σ a k = 10 -3 [ h Mpc -1 ] NeutrinoPhoton -4 -2 -6 -5 -4 -3 -2 -1 S ( ) , σ a k = 10 -1 [ h Mpc -1 ] NeutrinoPhoton -10 -8 -6 -4 -2 -6 -5 -4 -3 -2 -1 S ( ) , σ a k = 10 [ h Mpc -1 ] NeutrinoPhoton
FIG. 1: Time evolutions of the source terms of anisotropic stresses of photons and neutrinos S (2)2 ,σ for three typical scales.Dummy wavenumbers, which are used for convolutions are taken as k ≈ k ≈ k . Blue lines represent the source term ofanisotropic stress of neutrinos and red lines do that of photons. Top left, top right and bottom panels show evolutions at k ∼ − h Mpc − , 10 − h Mpc − , and 10 h Mpc − , respectively. Note that the source term of photon anisotropic stressbefore recombination η < η rec is not used in our numerical calculations because we use the tight-coupling solution for photonanisotropic stress in this epoch. anisotropic stress has two contributions from the gravitational G (2) ℓ,m and the scattering C (2) ℓ,m terms, while that ofneutrinos has only the gravitational term. In Fig. 1, the evolutions of source terms for anisotropic stresses of photonsand neutrinos that satisfy the triangle configuration as k ≈ k ≈ k are depicted on several scales.
1. Photons
Let us first investigate the evolution of sources for the photon anisotropic stress on large scales ( k ≈ − h Mpc − )shown in the top left panel of Fig. 1. Before recombination, the scattering term dominates due to the frequent Comptonscattering, while the term becomes negligible compared to the gravitational term after recombination. Therefore wecan write the source term of photon anisotropic stress as S (2)2 ,σ ≈ ( C (2)2 ,σ ( η < η rec ) G (2)2 ,σ ( η rec < η ) , (38)while the source of neutrino anisotropic stress has only the gravitational term. Therefore, after recombination, thesource term corresponds to the evolutions of the free-streaming particles.On intermediate scales ( k ≈ . h Mpc − , the top right panel of Fig. 1), there is a subtle difference between thesource terms of photons and neutrinos even after recombination. This is because the first order perturbations ofphotons and neutrinos on these scales enter the horizon before recombination and evolve differently. On much smallerscales ( k ≈ h Mpc − , the bottom panel of Fig. 1), the evolutions of source terms of photons and neutrinos aresignificantly different. The source of neutrino anisotropic stress is larger than that of photons in late times becausethe Silk damping effect erases photon perturbations exponentially before recombination. Meanwhile, the source ofneutrino anisotropic stress decays as η − with oscillation inside the sound horizon because of the free-streaming effect.0 -35 -30 -25 -20 -15 -8 -7 -6 -5 -4 -3 -2 -1 P ∆ ( ) , σ ak = 10 -3 [ h Mpc -1 ] k = 10 -1 [ h Mpc -1 ] k = 10 [ h Mpc -1 ] -35 -30 -25 -20 -15 -8 -7 -6 -5 -4 -3 -2 -1 P N ( ) , σ ak = 10 -3 [ h Mpc -1 ] k = 10 -1 [ h Mpc -1 ] k = 10 [ h Mpc -1 ] FIG. 2: Time evolutions of anisotropic stresses of photons ( left ) and neutrinos ( right ) for wavenumbers k = 10 − h Mpc − ,10 − h Mpc − , and 10 h Mpc − as indicated in the panels. Here we set ∆ R ( k ) = 2 . × − and n s = 1 .
0. It is shown thatthere are noisy fluctuations due to numerical errors in late time on small scales. However these fluctuations do not affect onthe final result of the spectrum of GWs since the most of the contribution is coming from the epoch of horizon crossing, e.g., a ∼ − for k = 10 h Mpc − .
2. Neutrinos
Next, let us investigate the evolution of neutrino anisotropic stress. We find that the source term of neutrinoanisotropic stress is proportional to η on superhorizon scales, since the source term contains the combination of theform: S (2)2 ,σ ; ν ∋ N (1) ℓ ≥ , × Ψ (1) . (39)On super horizon scales, this source term evolves as ∝ η , since the time evolutions of N (1) ℓ ≥ , and Ψ (1) are given as N (1)1 , ∝ η and Ψ (1) ∝ η , respectively. Note that the gravitational lensing terms such as in Eq. (39) transport thefirst-order higher multipole moments to the second-order lower ones. On the other hand, gravitational redshift termstransport the first-order lower multipole moments to the second-order higher ones as seen in Eq. (11).The structure of the gravitational term of neutrinos is same as that of photons. At first order, ∆ (1) ℓ,m and N (1) ℓ,m onsuch large scales evolve in the same way after recombination since both of them undergo only free-streaming in acommon gravitational potential. Therefore, we expect that the difference between photons and neutrinos must vanishon large scales after recombination even at the second order, as shown in the figure. C. Photon and neutrino anisotropic stresses
The evolutions of anisotropic stresses of photons and neutrinos at second-order, which are the essential sourcesof the GWs, are depicted in Fig. 2. It is difficult to analyze power spectra of second-order perturbations quantita-tively because the power spectra are obtained after carrying out complicated convolutions. However we can roughlyunderstand the shape of the spectra by counting the most contributing terms.
1. Photons
Let us first consider the evolution of anisotropic stress of photons on small scales, in three characteristic epochsseparately; (i) the tight-coupling epoch, (ii) the epoch from when the tight coupling approximation is broken to theonset of recombination, and (iii) the epoch after recombination.When the tight-coupling approximation is valid, anisotropic stress of photons is given by Eq. (33) and the powerspectrum of the photon anisotropic stress evolves as η , since Eq. (33) indicates that ∆ (2)2 ,σ ∝ ( v (1) γ ) ∝ η . When thezeroth order tight-coupling approximation is broken, the first-order tight-coupling approximation by Eq. (35) gives amore appropriate solution for ∆ (2)2 ,σ . We find from our numerical calculation that the right hand side of Eq. (35) isdominated by the first term given by (2 / ˙ τ c ) ˙ χ (2) σ and also find numerically that ˙ χ (2) σ evolves as η . . Therefore in the1first-order tight-coupling approximation, ∆ (2)2 ,σ ∝ η . . The transition from the zeroth order to the first order solutionscan be seen as a kink in the early time evolution of photon anisotropic stress in the left panel of Fig. 2, for example,at a ≈ × − for k = 10 h Mpc − (the solid black line). After the mode enters the horizon, anisotropic stress issustained by the source in Eq. (35) until the tight-coupling approximation is broken.After the tight coupling approximation is broken but before recombination, the source for the anisotropic stressof photons experiences the Silk damping effect on small scales. Consequently, the second-order anisotropic stressof photons loses its source and also experiences the Silk damping. Note that on the larger scales the first orderperturbations do not experience the Silk damping effect and can sustain the anisotropic stress at second-order.After the recombination epoch, the second-order anisotropic stress of photons on scales smaller than the Silkdamping scale undergoes the free-streaming, since sources of anisotropic stress of photons have already decayed away.However, on larger scales, the source of anisotropic stress can survive owing to the gravitational term (see the modeswith k . . h Mpc − in the left panel of Fig. 2).
2. Neutrinos
Next, let us consider the neutrino anisotropic stress, which is sourced only by the gravitational term. Before thehorizon crossing, neutrino anisotropic stress is generated by the source, which is of the form S (2)2 ,σ ; ν ∋ N (1) ℓ ≥ , × Ψ (1) ∝ η × η (see Eq. (10)). This means that the power spectrum of anisotropic stress of neutrinos evolves as η onsuperhorizon scales. After the horizon crossing, the anisotropic stress of neutrinos on small scales undergoes thefree-streaming. This is because the sources of the anisotropic stress of neutrinos, i.e., N (1) ℓ ≥ , , also undergo the free-streaming and the first order gravitational potentials decay away in the radiation dominated epoch. However, onlarge scales, the anisotropic stress does not decay in the matter dominated epoch because it can be sustained by thesources consist of the scalar gravitational potentials that are constant in time in the matter dominated epoch. D. The power spectrum of secondary GWs
Finally, we show the complete spectrum of the secondary GWs by taking into account all the contributions atsecond order, namely, the products of the first-order scalar-perturbations and purely second-order anisotropic stressesof photons and neutrinos in Fig. 3. As shown in Fig. 3, the anisotropic stresses of photons and neutrinos as awhole amplify the secondary GWs by about 120% on small scales, k & . h Mpc − . However, on large scales, k . . h Mpc − , anisotropic stresses of photons and neutrinos do not affect the secondary GWs. In the followingsection, we discuss the impact of the anisotropic stress on the secondary GWs for photons and neutrinos, separately. IV. DISCUSSION
We discuss effects of second-order anisotropic stress on the gravitational wave spectrum. We depict the spectrumof GWs and the ratio of the spectra with and without the purely second-order anisotropic stress in Fig. 4.First, we focus on small scales, say, k & . h Mpc − . We find that the photon anisotropic stress amplifies GWs about150% but neutrino anisotropic stress suppresses GWs about 30%. The net effect of purely second-order anisotropicstress on small scales is the amplification of GWs by about 120% compared to the case without the photon andneutrino anisotropic stresses.To understand why photon and neutrino give the opposite effects on the GWs, we show the second-order trans-fer function of anisotropic stress at k ≈ k ≈ k , in Fig. 5, which scales mainly contribute to the second-orderpower spectrum (see section III B). As is seen in Fig. 5, before the horizon crossing, the transfer functions for pho-tons and neutrinos have minus sign. However, after the horizon crossing, the transfer functions for photons andneutrinos have different signs. Neutrino anisotropic stress undergoes free-streaming after the horizon crossing andstarts to oscillate. In contrast, from Eq. (33), the photon anisotropic stress is sourced by the term proportional to v (1) γ ( k ) v (1) γ ( k ) Y , ,σ (ˆ k , ˆ k ) due to the scattering term. For the most contributing configuration where k = k = k ,the first terms, v (1) γ v (1) γ do not change the overall sign. On the other hand, the second term, Y , ,σ ( k = k = k )gives a negative sign from Eq. (31). The net sign of the photon anisotropic stress is therefore negative through itsevolution in the tight-coupling regime. After the horizon crossing, photons amplify the GWs since the sign of photonanisotropic stress is same as that of the GWs, while neutrinos gives the opposite result.2 -22 -21 -20 -19 -18 -17 -16 -15 -14 Ω G W w/ Anisotropic stressw/o Anisotropic stress -4 -3 -2 -1 Ω G W t o t / Ω G W k [ h Mpc -1 ] z = 0 FIG. 3: The secondary GWs at z = 0 induced by all the contributions, i.e., the products of the first-order scalar-perturbationsand prey second-order anisotropic stresses of photons and neutrinos (black line in the top panel). The top panel shows thegenerated GWs with and without the contributions from the purely second-order anisotropic stress at z = 0 as indicated. Thebottom panel shows the ratio of the spectra with (Ω GW tot ) and without (Ω
GW0 ) the contributions from the purely second-orderanisotropic stress. For reference, the ratio equal to unity is shown (magenta, short dashed line). The spectrum of GWs for k . . h Mpc − at z = 0 is consistent with the result of Ref. [41] that was derived without contributions from the purelysecond-order anisotropic stress. Second, we focus on GWs on large scales in the matter dominated epoch. On these scales effects from the second-order anisotropic stress should be negligible as the first order gravitational potentials directly sustain the second-order GWs. The transition scales correspond to those above which the sources from the products of first orderperturbations, i.e., the gravitational potentials in the matter dominated epoch, can sustain the GWs. On those scales,the gravitational potentials at first order completely determine the amplitude of GWs and hence the contributionsfrom purely second-order anisotropic stress become negligible. More specifically, from Eq. (18), the GWs are sourcedby 8 πGρ (0) γ ∆ (2)2 ,σ and 8 πGρ (0) ν N (2)2 ,σ . In the matter dominated epoch, the prefactor of anisotropic stress, namely ρ (0) γ and ρ (0) ν , must be negligibly small. Thus, in the matter dominated epoch, the anisotropic stress does not affect the GWs.In the radiation dominated epoch, on the other hand, the anisotropic stress affects the second-order GWs. FromFig. 5, the photon and neutrino anisotropic stresses contribute to the GWs as the negative source. Consequently, inthe radiation dominated epoch, the amplitude of the GWs is suppressed on large scales. However, these suppressionsdo not affect final results since the main contributions are determined around the horizon crossing.To confirm the above analytic investigations, we show the difference between time evolutions of the GWs with andwithout anisotropic stress in Fig. 6, for three typical scales, k = 10 − h Mpc − , 10 − h Mpc − , and 10 h Mpc − .Before the horizon crossing, the GWs are generated by the product of first-order scalar perturbations and the purelysecond-order anisotropic stress. After the horizon crossing, the power spectrum of GWs decay as a − because the3 -26 -24 -22 -20 -18 -16 -14 Ω G W + z = 3400 z = 100 z = 0 -4 -3 -2 -1 Ω G W + / Ω G W k [ h Mpc -1 ] -26 -24 -22 -20 -18 -16 -14 Ω G W + z = 3400 z = 100 z = 0 -4 -3 -2 -1 Ω G W + / Ω G W k [ h Mpc -1 ] FIG. 4: Impact of the anisotropic stresses of photons ( left ) and neutrinos ( right ) on the GWs. Top figures show the generatedGWs including all the contributions at each time as indicated. Bottom figures show the ratio of with (Ω
GW+ ) and without(Ω
GW0 ) the anisotropic stress contributions. For reference, the ratio equal to unity is shown (magenta, short dashed line). -6 -5 -4 -3 -2 -1 -7 -6 -5 -4 -3 T r a n s f e r f un c ti on a GWsPhotonNeutrino
FIG. 5: Transfer functions of anisotropic stresses for photons (red), neutrinos (blue) and GWs without anisotropic stress(black) at k ≈ k ≈ k ≈ . h Mpc − . In this figure, solid lines and dashed lines represent plus and minus signs, respectively. product of the first-order scalar perturbations decay faster after the horizon crossing. However on large scales, i.e., k . − h Mpc − , the product of the first-order scalar perturbations remains constant and sustains the second-orderGWs. Consequently, for large scales where the product of the first-order scalar perturbations dominates to generatethe second-order GWs, the effect of anisotropic stress of free-streaming particles is negligible. These evolutions areconsistent with the spectra of GWs in Fig. 4.The reason why we only consider up to the second order perturbation is as follows. First of all, GWs from densityperturbations can arise only from the second order. Therefore it is essential to consider the second order perturbations.Furthermore, from Fig. 6, we can see that the anisotropic stress affects the GWs only at the horizon crossing where the4 -36 -34 -32 -30 -28 -26 -24 -22 -20 -18 -16 P χ k = 10 -3 [ h Mpc -1 ] k = 10 -1 [ h Mpc -1 ] k = 10 [ h Mpc -1 ] -8 -7 -6 -5 -4 -3 -2 -1 Ω G W + / Ω G W a -36 -34 -32 -30 -28 -26 -24 -22 -20 -18 -16 P χ k = 10 -3 [ h Mpc -1 ] k = 10 -1 [ h Mpc -1 ] k = 10 [ h Mpc -1 ] -8 -7 -6 -5 -4 -3 -2 -1 Ω G W + / Ω G W a FIG. 6: The effects of the anisotropic stresses of photons ( left ) and neutrinos ( right ) on the evolutions of the GWs. Topfigures show evolutions of generated GWs including all contributions at each scale, k = 10 − h Mpc − , 10 − h Mpc − , and10 h Mpc − . Bottom figures show the ratio of the spectra with (Ω GW+ ) and without (Ω
GW0 ) the anisotropic stress. Forreference, the ratio equal to unity is shown (magenta, short dashed line). These evolutions have been discussed in Ref. [41]without the second-order anisotropic stress. perturbation approach is valid even on smaller scales if the power spectrum of the primordial density perturbationsis nearly scale invariant. After the horizon crossing, the second order anisotropic stress should decay away togetherwith its sources, i.e., the first order gravitational potentials in the radiation dominated era. Therefore, the higherorder contributions, e.g., those from the third order, should be small compared with those from the second order onsmall scales.Finally, let us discuss observational implications of our result. Cosmological tensor perturbations are known toinduce curl modes of weak gravitational lensing of background galaxies [60]. It is found that the secondary tensormode generated from density perturbations produces larger signal in the curl modes than the gravitational wavesgenerated by inflation if the tensor-to-scalar ratio is less than 0 . r . − , and becomes relevant in direct detection experiments such as DECIGO [50] andatomic gravitational wave interferometric sensors [51]. In this case, the second-order GWs may be directly observed,and they can be used as yet another probe of standard CDM cosmology. For example, it is claimed that DECIGOmay detect the stochastic GWs background up to the order of Ω GW ∼ − at f ∼ . (2nd)GW ∼ f Π × − on small scales, with f Π being the amplification factor by the second-order anisotropic stress. Therefore, the estimated amplitude of thesecondary GWs is at the same order of magnitude as the noise level assumed for DECIGO [50], and it will becomeimportant for the future GWs experiments to estimate the impact of the anisotropic stress on the secondary GWs.Furthermore, the secondary induced tensor mode has been used to place constraints on the amplitude of primordialscalar perturbations, which can not be probed by other ways [16, 17, 62]. The reason is that the amplitude of thesecondary induced tensor mode is proportional to P , and therefore non-detection of GWs in gravitational waveexperiments places an upper bound on P scalar at observed scales. If we assume that our result applies to much smallerscales k ≫
100 Mpc − , the constraints on P scalar should become tighter by a factor of 1 .
5, because our result showsthat on small scales the secondary induced tensor mode should be larger by a factor of 2 . ∼ /f γ ∼ f γ ≈ .
681 is the energy fraction of photons in the radiation dominated era with temperature . MeV.Other relativistic particles such as electrons and positrons are also expected to amplify the GWs even further for k ≫
100 Mpc − . V. SUMMARY
In this paper, we explored the impact of anisotropic stresses of photons and neutrinos on the secondary GWsgenerated from first order density perturbations. To estimate the spectrum of GWs including the anisotropic stress, wereformulated the cosmological perturbation theory up to second-order, which is based on the Einstein and Boltzmannequations. To solve the second-order equations numerically, we considered the initial conditions of the Boltzmannequation of photons using the tight coupling approximation. Deep in the radiation dominated era, photons and baryonsfrequently interact with each other, which allows us to expand the equations using the tight-coupling parameter k/ ˙ τ c . Under the tight-coupling approximation, we solved the hierarchical Boltzmann equations and showed the tight-coupling solution up to first-order. In the first-order cosmological perturbation theory, the anisotropic stress of photonsmust vanish due to frequent collisions between photons and baryons, as well as the photon’s higher multipoles. Howeverin the second-order cosmological perturbation theory, the anisotropic stress of photons does not vanish because themode coupling of the first-order scalar perturbations, i.e., velocity perturbations of photons, generates anisotropicstress. Moreover, differently from the first order perturbation theory, the octupole moment ( ℓ = 3) of the distributionfunction of photons can arise even in the first-order tight-coupling solution. We found that this is because the octupolemoment is also sourced by the mode coupling of the first-order scalar perturbations and the streaming term. In thenext order of the tight-coupling expansion, we showed that the relative velocity between photons and baryons doesnot vanish, which is the same result as in the first-order perturbation theory. We adopted these solutions as the initialconditions for second order perturbations, and solved the Einstein-Boltzmann system numerically.Photons and massless neutrinos have anisotropic stress in the perturbed universe. We considered effects ofanisotropic stresses of photons and neutrinos on the secondary generated GWs from density perturbations. We foundthat, before the recombination epoch, anisotropic stresses of photons and neutrinos affects the amplitude of GWs onboth small and large scales. On super horizon scales, the anisotropic stress is growing because of the first-order scalarperturbations. Generated anisotropic stress can be a source of the GWs. In the matter dominated epoch, the effectof anisotropic stress on the GWs becomes negligible, because the first-order scalar gravitational potentials directlysustain the GWs. On small scales, on the other hand, the GWs have been affected by the anisotropic stress becausethe scalar gravitational potentials have decayed away on small scales. Photon anisotropic stress, which is sourced bythe square of the first-order velocity perturbations, amplifies the GWs by about 150%. On the other hand, neutrinoanisotropic stress suppresses the GWs by about 30% because the second-order neutrinos undergo the free-streamingwith oscillations around the origin. This difference can be explained by the different signs of the transfer functions.To conclude, the effect of the anisotropic stress at second-order in cosmological perturbations is to amplify the GWsby about 120 % for k & . h Mpc − in the present universe, compared to the case without taking the anisotropicstress into account. Acknowledgments
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