Gauge Independence of Induced Gravitational Waves

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Gauge independence of induced gravitational waves

Keisuke Inomata1,2 and Takahiro Terada 3 1ICRR, University of Tokyo, Kashiwa 277-8582, Japan 2Kavli IPMU (WPI), UTIAS, University of Tokyo, Kashiwa 277-8583, Japan 3Center for Theoretical Physics of the Universe, Institute for Basic Science (IBS), Daejeon 34126, Korea

(Received 10 December 2019; published 28 January 2020)

We study gauge (in)dependence of the gravitational waves (GWs) induced from curvature perturbations. For the GWs produced in a radiation-dominated era, we find that the observable (late-time) GWs in the transverse-traceless (synchronous) gauge and in the Newtonian gauge are the same in contrast to a claim in the literature. We also mention the interpretation of the gauge dependence of the tensor perturbations which appears in the context of the induced GWs.

DOI: 10.1103/PhysRevD.101.023523

I. INTRODUCTION determine or constrain the amplitude of the small-scale perturbations. One of the main predictions of cosmological inflation is In contrast to the linear perturbation theory, the the generation of the cosmic perturbations. So far, there is second-order (induced) tensor perturbations are known only an upper bound on the amplitude of tensor pertur- to have gauge dependence. A large change of the bation, but that of scalar perturbations, namely the power spectrum of the tensor perturbations induced in curvature perturbations, has been well established by a matter-dominated (MD) era was reported in Ref. [35]. the precise observations of the cosmic microwave back- More aspects of the gauge dependence were studied in ground and the large-scale structure [1,2]. Although these Refs. [36–40]. observations provide us information on the large scale, we In the context of the tensor perturbations induced by cannot access much smaller scales due to the Silk damping scaler perturbations, the tensor perturbation can be of the radiation perturbations or the nonlinear growth of divided into two parts. One is the freely propagating the matter perturbations. Such information on the small- tensor perturbation, following the equation of motion scale perturbations is desirable to understand the global without any source. This kind of tensor perturbations picture of the inflaton potential and particle physics is widely known as gravitational waves and the time behind it. dependence can be written as hij ∝ sinðkηÞ=a or Gravitational waves (GWs) are a useful probe of small- ð ηÞ scale curvature perturbations [3–8] since if the latter is cos k =a (a: scale factor), as is well known. enhanced, a sizable amount of GWs is produced at the Although they couple with scalar perturbations at their second order of the cosmological perturbation. We call production, they finally decouple from the scalar per- it the induced GWs1 (see early works [9–12] and recent turbations and freely propagate. Once they decouple developments [13–22]). Such an enhancement of the from the scalar perturbation, they are independent of the curvature perturbations are often considered in the context scalar perturbations and therefore do not depend on the of the primordial black hole (PBH) [23–25] (see also gauge. The other is the tensor perturbation coupling with Refs. [7,26–31]). PBHs have recently been studied by the scalar perturbations, in which the freely propagating many authors mainly because they can be a dark matter tensor perturbations are included only until they decou- (DM) candidate and may explain the binary black hole ple from the scalar perturbations. Since this kind of merger events observed by LIGO/Virgo [32]. If the PBHs tensor perturbation is controlled by the scalar perturba- explain the DM or LIGO/Virgo events, the induced GWs tions, the time dependence of this tensor perturbation can be observed by the future projects, such as LISA [33] inherits those of the scalar perturbations. Note that the and SKA [34], and even if there are few PBHs in the gauge dependence appears in this kind of tensor per- Universe, the future observations of the induced GWs could turbation. This tensor perturbation is also often called a gravitational wave in many references, but in this paper, to discriminate these two kinds of tensor perturbations, 1It is also called the second-order GWs, secondary GWs, or we call only the freely propagating tensor perturbation a scalar-induced GWs, etc. gravitational wave.

2 Recently, it was claimed in Ref. [41] that the power where hij is the freely propagating tensor perturbation, and spectrum of the induced GWs (freely propagating tensor the overline denotes the oscillation average. Thanks to the perturbations) calculated in the TT gauge (also known as oscillation average, we may calculate the spatial derivatives synchronous gauge) is reduced by at least one order of hij;khij;k instead of the time derivatives, which is justified magnitude compared to that calculated in the Newtonian on subhorizon scales. gauge (also called the Poisson gauge, the longitudinal gauge, The current value of the power spectrum of the induced Ω ðη Þ ≡ or the zero-shear gauge), commonly used in the literature for GWs per a logarithmic waveR number bin GW ;k ρ ðη Þ ρ ρ ðηÞ¼ ρ ðη Þ the GWs produced during a radiation-dominated (RD) era. GW ;k = total with GW dlnk GW ;k is obtained However, we do not expect that the induced GWs depend from on the gauge as we mentioned above. The gauge inde- 4 ð Þ 106 75 3 pendence of the induced GWs can also be expected from Ω ðη Þ¼0 387Ω g Tc . Ω ðη Þ GW 0;k . r 106 75 ð Þ GW c;k ; the coincidence of the GWs calculated in the Newtonian . g;s Tc gauge and the uniform curvature gauge (also called the flat ð2Þ gauge), shown in Ref. [40] (see the case of w>0 in the reference). In the present paper, we revisit the calculation in where η ¼ η0 is the current conformal time; a subscript c Ref. [41] and find that the difference between the tensor denotes a time at which the tensor perturbation is domi- 3 perturbations induced during a RD era in the TT gauge and nated by the freely propagating tensor perturbation ; g and in the Newtonian gauge decreases on subhorizon scales, g;s are the numbers of relativistic degrees of freedom for Ω and therefore the late-time (i.e., observable) GWs are the the energy and entropy density, respectively; and r is the same in both gauges, in contrast to Ref. [41]. We also current energy density fraction of radiation. This formula mention the interpretation of the gauge dependence assumes the production of the GWs in a RD era. The Ω reported in Ref. [40] and an important role played by constant value of GW can be calculated as the diffusion damping effect at the end of this paper. Z 1 2 ∞ Ω ðη Þ¼ k GW c;k dv 6 H 0 II. INDUCED GRAVITATIONAL WAVES Z IN TT GAUGE 1þv 4v2 − ð1 þ v2 − u2Þ2 2 × du In this section, we calculate the induced GWs in the j1−vj 4uv TT gauge and compare them with those in the Newtonian 2 × I ðu; v; x ÞPζðukÞPζðvkÞ; ð3Þ gauge known in the literature. We take the convention in c Refs. [14,42]. The comparison of the conventions in the where x ≡ kη is a dimensionless time, H ¼ðda=dηÞ=a is literature is given in Appendix A. To avoid the repetition of the conformal Hubble parameter, PζðkÞ is the power steps of derivation, we write only the essential equations for spectrum of the primordial curvature perturbations, and a review. Details of derivation can be found in Refs. [14,42] the late time limit x → ∞ is practically used for evaluation (see also Ref. [36] for early works on the formulation in ¼ η ð Þ at xc k c. The function I u; v; x is defined as the TT gauge). Z x ðη¯Þ The GW energy density is given by [43] ¯ a ¯ ¯ Iðu; v; xÞ¼ dx kGkðη; ηÞfðu; v; xÞ; ð4Þ 0 aðηÞ 1 ρ ¼ h 0 0 i ð Þ GW h h ; 1 where x¯ ¼ kη¯, and G ðη; η¯Þ is the Green function for 16a2 ij ij k the tensor perturbations, which is given as kGkðη; η¯Þ¼ sinðx − x¯Þ in the production in a RD era. In this expression, 2 One of the main claims in Ref. [41] is that the description of u and v are symmetric under their exchange.4 the detection of GWs with a space-based interferometer is most straightforward in the transverse-traceless (TT) gauge, and they 3 advocate the use of the TT gauge (if the strength of the induced If we define the parameter ΩGW as the energy density GWs is gauge dependent at all). The argument is as follows. The parameter of the total tensor perturbations, as in many references, size of the space-based interferometer is typically not sufficiently the subscript c denotes a time at which ΩGW reaches the large compared to the wavelength of the GWs. In such a situation, asymptotic constant value. one cannot take the proper detector frame (Fermi normal 4The power spectrum of the induced GWs is nothing but a coordinates) for the whole experimental region. The completely 2-point correlation function, which can be written as a 4-point relativistic treatment is required, and they give at least two function of scalar source modes. In the absence of non- reasons that motivate one to use the TT gauge. For one thing, one Gaussianity, the 4-point function can be expressed in terms of has to otherwise add the integral of the gravitational potential two different combinations of 2-point functions (two ways of along the photon path to make the time shift (measured as the Wick contraction). From this and from the (u ↔ v) symmetry phase shift of the laser) gauge invariant in other gauges. Note that of the rest of the integrand as well as the integration region, the the result of the TT gauge reduces to the simplest case where the remaining asymmetry under the exchange of u and v, if any, description in the proper detector frame is possible. For the other, must vanish. Once this is understood, one can freely symmetrize the sensitivity curve of LISA is calculated in the TT gauge. fðu; v; xÞ or equivalently Iðu; v; xÞ as a convention.

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Now, the final missing piece in the above equation is the Also, our definition of the transfer functions is 3 times explicit definition of the source function fðu; v; x¯Þ, which larger than that in Ref. [41] for the transfer function of is different between the TT gauge and the Newtonian (or the gauge invariant Bardeen potential Ψ to be unity for some other) gauge. The difference arises only from this. It η → 0. Therefore, we expect an amplitude and shape of the can be derived from the expression of the scalar-scalar power spectrum of the induced GWs different from those source term in the equation of motion for the second-order of Ref. [41]. tensor perturbation, Eq. (46) of Ref. [39]. In terms of the Similarly to the case of the Newtonian gauge studied in transfer functions Tψ and Tσ of the scalar perturbation Refs. [13,14], we can analytically integrate Eq. (4) using variable ψ and the shear potential σ in the TT gauge, it is the method of Ref. [10]. The result is long, so it is written in given by Appendix B. First of all, it is different from the case of the Newtonian gauge for finite x ¼ kη. The difference between 2 2 ð Þ ≡ − − ð Þ ð Þ the two gauges is given by f u; v; x 3 Tψ ux Tψ vx pffiffiffi pffiffiffi ðNewtonian gaugeÞ ðTT gaugeÞ 3 3 3 3 I ðu; v; xÞ − I ðu; v; xÞ þ v 0 ð Þ ð Þþ u 0 ð Þ ð Þ pffiffiffi pffiffiffi 2 Tψ vx Tσ ux 2 Tψ ux Tσ vx 18 3 3 u v ¼ 1 − puxffiffiffi 1 − pvxffiffiffi 2 2 sin sin 3 2 0 0 u v x ux 3 vx 3 þ TσðuxÞTσðvxÞ − uvx Tψ ðuxÞTψ ðvxÞ ; 2uv x3 18 ≃ min ;x; ; ð7Þ ð5Þ 18 u2v2x pffiffiffi 9 − 9 cosðx= 3Þ where Iðu; v; xÞ ≡ xIðu; v; xÞ, and the superscripts Tψ ðxÞ¼ ; x2 denote in which gauge it is evaluated. We introduce this pffiffiffi pffiffiffi I −3 3 þ 9 ð 3Þ combination as its oscillation average asymptotes to a ð Þ¼ sin x= ð Þ Tσ x 2 ; 6 constant while I decreases by redshift. In the second x equality, we have used the fact that max½u2;v2 ≃ 1 where the prime denotes the differentiation with respect when min½u; v ≪ 1. From the above expression, it is 0ð Þ ≡ dTðuxÞ to the argument, e.g., T ux dðuxÞ . More explanations clear that the gauge difference decreases in time on on the transfer functions and the choice of an integration subhorizon scales, and the asymptotic constant values, constant are given in Appendix A. A difference of an which are relevant for direct observations, coincide I ðTT gaugeÞð Þ¼ overall factor from Ref. [41] is just because of different with each other. That is, limx→∞ u; v; x ðNewtonian gaugeÞ conventions. We find the coefficient of the first term inside limx→∞I ðu; v; xÞ. Therefore, we can use the parenthesis in Eq. (5) −1, but 2 is used in Ref. [41]. the same formula [14] (see also Ref. [13])

1 3ð 2 þ 2 −3Þ 2 3−ð þ Þ2 2 pffiffiffi I2ð Þ¼ u v −4 þð 2 þ 2 −3Þ u v þπ2ð 2 þ 2 −3Þ2Θð þ − 3Þ ð Þ lim u;v;x 3 3 uv u v log 2 u v u v ; 8 x→∞ 2 4u v 3−ðu−vÞ in both the TT gauge and the Newtonian gauge, where Θð·Þ tensor perturbation (k ¼ H)occursatx ¼ 1.Thevaria- is the Heaviside step function. Thus, once the power bles u and v are to be interpreted as the ratio of the length spectrum of the primordial curvature perturbations PζðkÞ scales of the tensor perturbation and the typical or is specified, one can use Eqs. (2), (3), and (8) to compute relevant scalar source. For example, if we consider the the power spectrum of the induced GWs, and the result is monochromatic power spectrum of the curvature pertur- irrespective of the choice of the TT gauge or the Newtonian bation at k, taken in Ref. [41], the relevant values of u gauge. Explicitly, and v are k=k. Let us begin with the small x (superhorizon) limit. The ΩðTT gaugeÞðη Þ¼ΩðNewtonian gaugeÞðη Þ ð Þ 2 GW 0;k GW 0;k ; 9 leading order terms of Iðu; v; xÞ in this limit are ð2=9Þx in the Newtonian gauge5 and ð1=6Þx2 in the TT gauge. In η η and an analogous formula with 0 replaced by c hold. terms of I, the difference is thus ð1=18Þx3. In the remainder of this section, let us study the Ið Þ After the horizon entry of the scale of the would-be behavior of u; v; x in each gauge and their differences GWs, the difference in the two gauges becomes large for finite x. Examples of the time evolution of Iðu; v; xÞ for given sets of u and v in both gauges are shown in Fig. 1.Inthefigure,x ¼ kη can be interpreted as the time 5It is written as ð1=2Þx2 in Ref. [14], but an overall factor was parameter in the unit in which the horizon entry of the missing.

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FIG. 1. Comparison of time evolution of the function jIðu; v; xÞj in the Newtonian gauge (cyan) and in the TT gauge (purple). The values of u and v are set as (u ¼ 1, v ¼ 1) in the left panel, and (u ¼ 1, v ¼ 0.01) in the right panel. The gray dotted lines are x and 18=ðu2v2xÞ. for small u or v (right panel of Fig. 1). This should come III. DISCUSSION from the gauge dependence of the part of the tensor In this paper, we reconsider the results in Ref. [41] and perturbations coupling with scalar perturbations. After a show that the late-time (e.g., observable) GWs calculated I ∝ x−1 while, the difference of decreases slowly as . in the TT gauge and in the Newtonian gauge coincide with Finally, the difference becomes smaller than the asymptotic each other, in contrast to Ref. [41]. Here, the “late time” x ≫ 1 value at a late time , which means the tensor limit means the subhorizon limit: time much after the perturbations in the TT gauge are dominated by the freely horizon entry of the relevant scalar source modes as well propagating tensor perturbations (GWs). Let us see these as the tensor mode. In addition, the calculation in the behaviors quantitatively below. uniform curvature gauge (flat gauge) coincides with the Typically, the Newtonian gauge results (cyan line) reach Newtonian-gauge result too [40]. The main reason for the asymptotic value as early as its horizon entry (both in the agreement in these gauges is because the tensor the left and right panel). TT gauge results (purple line) are perturbations are finally dominated by the freely propa- similar if u and v are not smaller than 1 (left panel), but this gating tensor perturbations (GWs) in all of the three changes if u or v are small, i.e., uv ≲ 1 (right panel).6 In this pffiffiffi gauges. I 3 2 ≲ case, pffiffiffiin the TT gauge once increases as x for On the other hand, the tensor perturbations could be 2 2 xp≲ffiffiffi3 2=ðuvÞ and then decreases as 18=ðu v xÞ for x ≳ different in the comoving gauge even in the subhorizon 3 2=ðuvÞ until it reaches the asymptotic value, whose limit during a RD era, according to Ref. [40]. We can (u, v) dependence ispffiffiffi not simple [see Eq. (8)]. The peak of find that the difference between the Newtonian gauge I, given as x ≃ 3 2=ðuvÞ, roughly corresponds to the and the comoving gauge comes from the termsp whoseffiffiffi horizon entry of the scalar perturbation with the wave oscillations arepffiffiffi of the form cos ððu vÞkη= 3Þ or number min½uk; vk. This means that once both the scalar sin ððu vÞkη= 3Þ, which are not the oscillations of perturbations with their wave numbers being uk and vk gravitational waves, cosðkηÞ or sinðkηÞ; see Eq. (65) in enter the horizon, the gauge dependence of the tensor Ref. [40]. This indicates that the gauge dependence appears perturbations starts to decrease and finally becomes neg- only in the tensor perturbations coupling with scalar ligible. This behavior is similar to the decay of the gauge perturbation, not in the gravitational wave.7 dependence for the energy density perturbation on sub- We also comment on the gauge dependence from horizon scales [44]. In summary, the gauge dependence the viewpoint of the diffusion damping [45], which vanishes well after the tensor mode and the scalar modes occurs on small scales, since many authors focus on the enter the horizon. Note again the difference between the GWs induced by the small-scale scalar perturbations tensor perturbations in the two gauges at a finite time is due (often in the context of PBHs). After inducing GWs, the to the part of those coupling with the scalar perturbations.

7A corollary of this discussion is that the large enhancement of 6Note that u and v cannot be small simultaneously because of the induced GWs found in Ref. [19], which one may suspect a momentum conservation: see the integration region in Eq. (3). gauge artifact, is not a gauge artifact. They are produced in a RD Also, uv ≪ 1 is possible only at the corners of the integration era, just after a sudden reheating transition, and the late-time time region, so the net effect of gauge dependence after integration dependence of Iðu; v; xÞ is that of genuine gravitational waves, over u and v is suppressed. i.e., cosðkηÞ=a or sinðkηÞ=a.

023523-4 GAUGE INDEPENDENCE OF INDUCED GRAVITATIONAL WAVES PHYS. REV. D 101, 023523 (2020) small-scale perturbations finally decay exponentially, Note added.—The results of the latest version of Refs. ∝ ð− 2 2 ð ÞÞ −1ð Þ exp k =kD t , within the diffusion scale kD t [41,48] agree with ours. because the free-streaming length of some species, such as neutrinos or photons, increases as the temperature decreases (see, e.g., Ref. [46] for the time dependence APPENDIX A: CONVENTION of the damping scale). This means that the scalar pertur- We use the reduced Planck unit, c ¼ ℏ ¼ 8πG ¼ 1, bations inducing the GWs during the RD era get exponen- and basically follow the notations and conventions in tially suppressed in most cases, and therefore the gauge Refs. [14,42]. dependence of the tensor perturbations would disappear. The line element is parametrized as Although we leave the analysis of the diffusion damping 2 ¼ − 2ð1 þ 2ϕÞ η2 þ 2 2 η i effect on the evolutions of the tensor perturbations for ds a d a B;id dx future work, we expect that the gauge dependence that 1 þ 2 ð1 − 2ψÞδ þ 2 þ i j ð Þ appears in the comoving gauge [40] would disappear a ij E;ij 2 hij dx dx ; A1 after the exponential damping of the scalar perturbations. This is because if the scalar perturbations are exponentially where we only write the relevant perturbation variables. suppressed and can be negligible, the difference between The quantities ϕ, ψ, B, and E are first-order scalar the tensor perturbations in two different gauges, which perturbations, and hij is the second-order transverse trace- should be written in terms of the scalar perturbations, can less tensor perturbation. The comma denotes differentia- also be negligible. In summary, gauge dependence dis- tion: for example, E;ij ¼ ∂i∂jE. The TT gauge is defined appears due to the diffusion damping, so the late-time GWs as the gauge where ϕ ¼ B ¼ 0. On the other hand, the should be unique (the same as the one calculated in the Newtonian gauge is defined as E ¼ B ¼ 0. Newtonian gauge). The normalizations and signs of ψ, E, and the shear Note that the situation is different for the tensor pertur- potential σ ¼ ∂E=∂η in the TT gauge are the same as those bation induced by the scalar perturbations during the late in Ref. [41]. If we pick up the second-order component MD era. In this case, the scalar perturbations continue to of hij [see their Eq. (A.12g)], it is also the same in the couple with the tensor perturbations on subhorizon scales coordinate space. The convention of the Fourier transform even in the Newtonian gauge because of the growing matter is different, but it does not affect most parts of the perturbations. Therefore, the tensor perturbations are domi- calculation. When there is no anisotropic stress, the nated by the ones coupling with the scalar perturbations. following equations are satisfied in each gauge: That is, this kind of tensor perturbations can be larger than the GWs induced during the RD era on large scales [11]. ϕ ¼ ψ ðNewtonian gaugeÞ; ðA2Þ This is why the tensor perturbation easily depends on the ∂σ gauge during a MD era. For example, in Appendix B2,we ¼ −ðψ þ 2HσÞðTT gaugeÞ: ðA3Þ show that the function I in the TT gauge is different from ∂η that in the Newtonian gauge even at late time. However, we The relation of perturbation variables in the TT gauge should notice that this kind of tensor perturbation is not and in the Newtonian gauge is given by Eqs. (5.2) and (5.3) a gravitational wave. The tensor perturbation is time- in Ref. [41]: independent in the Newtonian gauge and does not behave Z η aðη¯Þ D1ðkÞ as radiation [11,12,14]. Since most observations assume ψ ðTTgaugeÞðη;kÞ¼Ψ þ HðηÞ Ψðη¯;kÞdη¯ − ; that the tensor perturbations are gravitational waves, the 0 aðηÞ aðηÞ observational sensitivities for this kind of tensor perturba- ðA4Þ tion will be different from those for conventional gravita- Z tional waves. Also, the observation of this tensor η aðη¯Þ D1ðkÞ perturbation might require a discussion about the suitable σðTT gaugeÞðη; kÞ¼− Ψðη¯; kÞdη¯ þ ; ðA5Þ 0 aðηÞ aðηÞ gauge for the observation because of its gauge dependence. where Ψ is a gauge invariant Bardeen potential which ACKNOWLEDGMENTS reduces to ψ in the Newtonian gauge, and D1ðkÞ is an η This work was supported in part by World Premier integration constant which does not depend on . We take D ðkÞ¼0 CðkÞ¼−2ζðkÞ 2 International Research Center Initiative (WPI Initiative), 1 , which corresponds to =k in MEXT, Japan, the JSPS Research Fellowship for Young Ref. [41]: for comparison, see also their footnote 6. The ð Þ ð Þ Scientists (K. I.), JSPS KAKENHI Grants No. JP18J12728 transfer functions of ψ ¼ ψ TT gauge and σ ¼ σ TT gauge 8 (K. I.), Advanced Leading Graduate Course for Photon during a RD era in the TT gauge are defined as [41] Science (K. I.), and IBS under the project code, IBS-R018- D1 (T. T.). We used the Mathematica package XPAND [47] 8We take the sign notation for the curvature perturbation which during our work. is opposite to that taken in Ref. [41].

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2 TABLE I. Comparison of notations. In this table, the wave- ψðη; kÞ¼− ζðkÞTψ ðη;kÞ; ðA6Þ 3 number of the tensor and scalar perturbations are denoted by k pffiffiffi and p, respectively, with k ¼jkj and p ¼jpj. η¯ is the conformal 2 3 time as an integration variable. σðη kÞ¼− ζðkÞ ðη Þ ð Þ ; 3 Tσ ;k ; A7 k This paper Ref. [41] Correspondence where ζðkÞ is the primordial curvature perturbations. v ¼ p=k x ¼ p=k v ↔ x Table I summarizes the correspondence of the notations u ¼jk − pj=k y ¼jk −ppffiffiffij=k u ↔ py ffiffiffi in this paper (as well as Refs. [14,18,19,42]) and in x ¼ kη z ¼ kη= 3 x ↔ 3z x¯ ¼ kη¯ u ¼ kη¯ x¯ ↔ u Ref. [41]. See the last part of Appendix D of Ref. [13] I ¼ I þ I 9I ↔ I for complementary information. (The notation of Ref. [41] xI c cos x s sin x c c, ¼ I cos x þ I sin x 9I ↔ I is largely the same as that of Ref. [13].) c s s s

APPENDIX B: ANALYTICAL EXPRESSION OF FUNCTION I(u; v; x) IN TT GAUGE 1. Radiation-dominated era In a RD era, the Eq. (4) is integrated as follows: 3 pffiffiffi ux pffiffiffi vx pffiffiffi ux vx Iðu; v; xÞ¼ −6uvx − uvðu2 þ v2 − 3Þx2 sin x þ 6 3v sin pffiffiffi þ 6 3u sin pffiffiffi − 6 3u cos pffiffiffi sin pffiffiffi u3v3x3 3 3 3 3 pffiffiffi vx ux ux vx ux vx − 6 3v cos pffiffiffi sin pffiffiffi þ 3ðu2 þ v2 − 3Þx sin pffiffiffi sin pffiffiffi þ 6uvx cos pffiffiffi cos pffiffiffi 3 3 3 3 3 3 2 2 2 3ðu þ v − 3Þ ðu þ vÞx ðu þ vÞx ðu − vÞx þ sin x −Ci x þ pffiffiffi − Ci x − pffiffiffi þ Ci x þ pffiffiffi 4u3v3x 3 3 3 2 ð−u þ vÞx 3 − ðu þ vÞ ðu þ vÞx ðu þ vÞx þ Ci x þ pffiffiffi þ log þ cos x Si x þ pffiffiffi þ Si x − pffiffiffi 3 3 − ðu − vÞ2 3 3 ðu − vÞx ð−u þ vÞx − Si x þ pffiffiffi − Si x þ pffiffiffi ; ðB1Þ 3 3 R R ð Þ ≡ x ¯ sin x¯ ð Þ ≡ − ∞ ¯ cos x¯ where Si x 0 dx x¯ and Ci x x dx x¯ . The counterpart in the Newtonian gauge is Eq. (22) of Ref. [14] [see also Ref. [13] ]. This formula can also be expressed in the form

xIðu; v; xÞ¼Icðu; v; xÞ cos x þ I sðu; v; xÞ sin x; ðB2Þ where 1 pffiffiffi vx pffiffiffi ux ux vx I ðu; v; xÞ¼ 9x cos x −2uvx þ 2 3u sin pffiffiffi þ 2 3v sin pffiffiffi þ 2uvx cos pffiffiffi cos pffiffiffi c u3v3x3 3 3 3 3 pffiffiffi ux vx pffiffiffi vx ux ux vx − 2 3u cos pffiffiffi sin pffiffiffi − 2 3v cos pffiffiffi sin pffiffiffi þðu2 þ v2 − 3Þx sin pffiffiffi sin pffiffiffi 3 3 3 3 3 3 ux vx pffiffiffi vx pffiffiffi ux − 3 sin x 6uvx 1 þ cos pffiffiffi þ cos pffiffiffi − 12 3u sin pffiffiffi − 12 3v sin pffiffiffi 3 3 3 3 ux vx ux vx − 18uvx cos pffiffiffi cos pffiffiffi þ 3ðu2 þ v2 þ 3Þx sin pffiffiffi sin pffiffiffi 3 3 3 3 pffiffiffi ux vx þ 3uð12 þðu2 − v2 − 3Þx2Þ cos pffiffiffi sin pffiffiffi þðu ↔ vÞ 3 3 3ðu2 þ v2 − 3Þ2 ðu þ vÞx ðu þ vÞx þ Si x þ pffiffiffi þ Si x − pffiffiffi 4u3v3 3 3 ðu − vÞx ð−u þ vÞx − Si x þ pffiffiffi − Si x þ pffiffiffi ; ðB3Þ 3 3

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1 pffiffiffi ux pffiffiffi vx I ðu; v; xÞ¼ −3uvðu2 þ v2 − 3Þx3 þ 9x sin x −2uvx þ 2 3v sin pffiffiffi þ 2 3u sin pffiffiffi s u3v3x3 3 3 pffiffiffi ux vx ux vx ux vx − 2 3u cos pffiffiffi sin pffiffiffi þðu ↔ vÞþ2uvx cos pffiffiffi cos pffiffiffi þðu2 þ v2 − 3Þx sin pffiffiffi sin pffiffiffi 3 3 3 3 3 3 ux vx pffiffiffi ux pffiffiffi vx þ 3 cos x 6uvx 1 þ cos pffiffiffi þ cos pffiffiffi − 12 3v sin pffiffiffi − 12 3u sin pffiffiffi 3 3 3 3 ux vx ux vx − 18uvx cos pffiffiffi cos pffiffiffi þ 3ðu2 þ v2 þ 3Þx sin pffiffiffi sin pffiffiffi 3 3 3 3 pffiffiffi ux vx þ 3ð12 þðu2 − v2 − 3Þx2Þ cos pffiffiffi sin pffiffiffi þðu ↔ vÞ 3 3 2 2 2 3ðu þ v − 3Þ ðu þ vÞx ðu þ vÞx − Ci x þ pffiffiffi þ Ci x − pffiffiffi 4u3v3 3 3 2 ðu − vÞx ð−u þ vÞx 3 − ðu þ vÞ − Ci x þ pffiffiffi − Ci x þ pffiffiffi − log ; ðB4Þ 3 3 3 − ðu − vÞ2 where (u ↔ v) denotes the term obtained by exchanging u and v in the previous term.

2. Matter-dominated era Taking into account the relation Ψ ¼ −ð3 þ 3wÞ=ð5 þ 3 Þζ ¼ ρ In the case of tensor perturbations induced in a MD era, w on the superhorizon scales where w P= is the equation-of-state parameter with P denoting the pressure, we take the following normalizations for the transfer 2 −2 functions of ψ and σ: we define fðu; v; xÞ in a MD era ð3=5Þ ð2=3Þ ð¼ 81=100Þ times Eq. (5). Strictly speaking, the coefficient of the last 0 3 term should also be modified. However, Tψ vanishes in a ψðη; kÞ¼− ζðkÞTψ ðη;kÞ; ðB5Þ 5 MD era, so it does not matter. The explicit form of f is pffiffiffi fðu; v; xÞ¼1 − x2=50, which grows in time. 3 3 The function Iðu; v; xÞ in the MD era is σðη; kÞ¼− ζðkÞ Tσðη;kÞ: ðB6Þ 5 k − 5 þ 60 3 þ 180 − 180 ð Þ¼ x x x cos x sin x ð Þ Then, the concrete expressions of the transfer functions are I u; v; x 3 : B8 50x given as [41] The late-time limit is Iðu; v; x → ∞Þ¼−x2=50. Note that 5 −x 6 5 Tψ ðxÞ¼ ;TσðxÞ¼ pffiffiffi : ðB7Þ the counterpart in the Newtonian gauge is = [Eq. (36) of 3 3 3 Ref. [14]; see also Ref. [12] ].

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