Gravitational wave cosmology I: high frequency approximation
Jared Fier, Xiongjun Fang, Bowen Li, Shinji Mukohyama, Anzhong Wang, Tao Zhu
YYITP-21-11, IPMU21-0011
Gravitational wave cosmology I: high frequency approximation
Jared Fier , ∗ Xiongjun Fang , † Bowen Li , ‡ Shinji Mukohyama , , § Anzhong Wang , ¶ and Tao Zhu , ∗∗ GCAP-CASPER, Department of Physics, Baylor University,One Bear Place Department of Physics, Key Laboratory of Low DimensionalQuantum Structures and Quantum Control of Ministry of Education,and Synergetic Innovation Center for Quantum Effects and Applications,Hunan Normal University, Changsha, Hunan 410081, P. R. China Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, 606-8502, Kyoto, Japan Kavli Institute for the Physics and Mathematics of the Universe (WPI),The University of Tokyo Institutes for Advanced Study,The University of Tokyo, Kashiwa, Chiba 277-8583, Japan Institute for Theoretical Physics & Cosmology, Zhejiang University of Technology, Hangzhou, 310023, China United Center for Gravitational Wave Physics (UCGWP),Zhejiang University of Technology, Hangzhou, 310023, China (Dated: February 19, 2021)In this paper, we systematically study gravitational waves (GWs) first produced by remote com-pact astrophysical sources and then propagating in our inhomogeneous universe through cosmicdistances, before arriving at detectors. To describe such GWs properly, we introduce three scales, λ, L c and L , denoting, respectively, the typical wavelength of GWs, the scale of the cosmologicalperturbations, and the size of the observable universe. For GWs to be detected by the current andforeseeable detectors, the condition λ (cid:28) L c (cid:28) L holds. Then, such GWs can be approximated ashigh-frequency GWs and be well separated from the background γ µν by averaging the spacetimecurvatures over a scale (cid:96) , where λ (cid:28) (cid:96) (cid:28) L c , and g µν = γ µν + (cid:15)h µν with (cid:15) ≡ λ/L , and h µν denotesthe GWs. In order for the backreaction of the GWs to the background spacetimes to be negligible,we must assume that | h µν | (cid:28)
1, in addition to the condition (cid:15) (cid:28)
1, which are also the conditions forthe linearized Einstein field equations for h µν to be valid. Such studies can be significantly simplifiedby properly choosing gauges, such as the spatial ( χ µ = 0), traceless ( γ µν χ µν = 0), and Lorentz( ∇ ν χ µν = 0) gauges, where χ µν ≡ h µν − hγ µν /
2, and ∇ ν denotes the covariant derivative withrespect to γ µν . We show that these three different gauge conditions can be imposed simultaneously,even when the background is not vacuum, as long as the high-frequency GW approximation is valid.However, to develop the formulas that can be applicable to as many cases as possible, in this paperwe first write down explicitly the linearized Einstein field equations for χ µν by imposing only thespatial gauge. Then, applying these formulas together with the geometrical optics approximationto such GWs, we find that they still move along null geodesics and its polarization bi-vector isparallel-transported, even when both the cosmological scalar and tensor perturbations are present.In addition, we also calculate the gravitational integrated Sachs-Wolfe effects due to these two kindsof perturbations, whereby the dependences of the amplitude, phase and luminosity distance of theGWs on these perturbations are read out explicitly. I. INTRODUCTION
The detection of the first gravitational wave (GW)from the coalescence of two massive black holes (BHs) bythe advanced Laser Interferometer Gravitational-WaveObservatory (aLIGO) marked the beginning of a new era, the GW astronomy [1]. Following this observation, soonmore than 50 GWs were detected by the LIGO/Virgoscientific collaboration [2–4]. The outbreak of intereston GWs and BHs has further gained momenta after the ∗ Electronic address: Jared [email protected] † Electronic address: [email protected] ‡ Electronic address: Bowen [email protected] § Electronic address: [email protected] ¶ Electronic address: Anzhong [email protected]; correspondingauthor ∗∗ Electronic address: [email protected] detection of the shadow of the M87 BH [5–10].One of the remarkable observational results is the dis-covery that the mass of an individual BH in these binarysystems can be much larger than what was previouslyexpected, both theoretically and observationally [11–13],leading to the proposal and refinement of various forma-tion scenarios, see, for example, [14–17], and referencestherein. A consequence of this discovery is that the earlyinspiral phase may also be detectable by space-based ob-servatories, such as LISA [18], TianQin [19], Taiji [20],and DECIGO [21], for several years prior to their coales-cence [22, 23]. Multiple observations with different de-tectors at different frequencies of signals from the samesource can provide excellent opportunities to study theevolution of the binary in detail. Since different detectorsobserve at disjoint frequency bands, together they coverdifferent evolutionary stages of the same binary system.Each stage of the evolution carries information about dif-ferent physical aspects of the source. As a result, multi- a r X i v : . [ a s t r o - ph . C O ] F e b band GW detections will provide an unprecedented op-portunity to test different theories of gravity in the strongfield regime [24].Recently, some of the present authors generalized thepost-Newtonian (PN) formalism to certain modified the-ories of gravity and applied it to the quasi-circular inspi-ral of compact binaries. In particular, we calculated indetail the waveforms, GW polarizations, response func-tions and energy losses due to gravitational radiation inBrans-Dicke (BD) theory [25], screened modified grav-ity (SMG) [26–28], and gravitational theories with par-ity violations [29–32] to the leading PN order, with whichwe then considered projected constraints from the third-generation detectors. Such studies have been further gen-eralized to triple systems [33, 34] in Einstein-aether (æ-)theory [35–37]. When applying such formulas to the firstrelativistic triple system discovered in 2014 [38], we stud-ied the radiation power, and found that quadrupole emis-sion has almost the same amplitude as that in generalrelativity (GR), but the dipole emission can be as largeas the quadrupole emission. This can provide a promis-ing window to place severe constraints on æ-theory withmulti-band GW observations [39, 40].More recently, we revisited the problem of a binarysystem of non-spinning bodies in a quasi-circular inspi-ral within the framework of æ-theory [41–46], and pro-vided the explicit expressions for the time-domain andfrequency-domain waveforms, GW polarizations, and re-sponse functions for both ground- and space-based de-tectors in the PN approximation [47]. In particular,we found that, when going beyond the leading order inthe PN approximation, the non-Einsteinian polarizationmodes contain terms that depend on both the first andsecond harmonics of the orbital phase. With this in mind,we calculated analytically the corresponding parameter-ized post-Einsteinian parameters, generalizing the exist-ing framework to allow for different propagation speedsamong scalar, vector and tensor modes, without assum-ing the magnitude of its coupling parameters, and mean-while allowing the binary system to have relative motionswith respect to the aether field. Such results will partic-ularly allow for the easy construction of Einstein-aethertemplates that could be used in Bayesian tests of GR inthe future.It is remarkable to note that the space-based detectorsmentioned above, together with the current and forth-coming ground-based ones, such as KAGRA [48], Voy-ager [49], the Einstein Telescope (ET) [50] and CosmicExplorer (CE) [51], are able to detect GWs emitted fromsuch systems as far as the redshift is about z (cid:39)
100 [52],which will result in a variety of profound scientific conse-quences. In particular, GWs propagating over such longcosmic distances will carry valuable information not onlyabout their sources, but also about the detail of the cos-mological expansion and inhomogeneities of the universe,whereby a completely new window to explore the uni-verse by using GWs is opened, as so far our understand-ing of the universe almost all comes from observations of electromagnetic waves only (possibly with the importantexceptions of cosmic rays and neutrinos) [53].In this paper, we shall generalize our above studies tothe cases in which the GWs are first generated by remoteastrophysical sources and then propagate in the inhomo-geneous universe through cosmic distances before arriv-ing at detectors, either space- and/or ground-based ones.It should be noted that recently such studies have alreadyattracted lots of attention, see, for example, [54] and ref-erences therein. In particular, using Isaacson’s high fre-quency GW formulas [55, 56], Laguna et al studied thegravitational analogue of the electromagnetic integratedSachs-Wolf (iSW) effects in cosmology, and found thatthe phase, frequency, and amplitude of the GWs expe-rience iSW effects, in addition to the magnifications onthe amplitude from gravitational lensing [57]. More re-cently, Bertacca et al connected the results of Laguna etal obtained in real space frame to the observed frame, byusing the cosmic rulers formulas [58], whereby the correc-tions to the luminosity distance due to velocity, volume,lensing and gravitational potential effects were calculated[59].On the other hand, Bonvin et al [60] studied the effectsof the universe on the gravitational waveform, and foundthat the acceleration of the Universe and the peculiar ac-celeration of the binary with respect to the observer dis-tort the gravitational chirp signals from the simplest GRprediction, not only a mere time independent rescalingof the chirp mass, but also the intrinsic parameter esti-mations for binaries visible by LISA. In particular, theeffect due to the peculiar acceleration can be much largerthan the one due to the Universe acceleration. Moreover,peculiar accelerations can introduce a bias in the estima-tion of parameters such as the time of coalescence andthe individual masses of the binary. An error in the esti-mation of the time of coalescence made by LISA will havean impact on the prediction of the time at which the sig-nal will be visible by ground based interferometers, forsignals spanning both frequency bands.Lately, such studies have been further generalized toscalar-tensor theories [61], including Horndeski [62–64]and SMG [64] theories.However, it should be noted that in all these stud-ies, the cosmological tensor perturbations have been ne-glected. As observing the primordial GWs (the tensorperturbations) is one of the main goals in the currentand forthcoming cosmological observations [65], in thispaper we shall consider the cosmological background thatconsists of both the scalar and tensor perturbations, butrestrict ourselves only to Einstein’s theory, and leave thegeneralizations to other theories of gravity to other occa-sions. What we are planning to do in the current paperare the following: • First, to describe the GWs propagating through theinhomogeneous universe from cosmic distances toobservers properly, we first introduce three scales, λ, L c and L , which denote, respectively, the typicalwavelength of GWs, the scale of the cosmologicalperturbations, and the size of the observable uni-verse. For GWs to be detected by the current andforeseeable detectors, we find that the condition λ (cid:28) L c (cid:28) L, (1.1)always holds. Then, such GWs can be approxi-mated as high-frequency GWs and be well separatedfrom the background γ µν by averaging the space-time curvatures over a scale (cid:96) , where λ (cid:28) (cid:96) (cid:28) L c ,and the total metric of the spacetime is given by g µν = γ µν + (cid:15)h µν , (1.2)where (cid:15) ≡ λ/L , and γ µν denotes the background,while h µν represents the GWs. In order for thebackreaction of the GWs to the background space-times to be negligible, we must assume that | h µν | (cid:28) , (1.3)in addition to the condition (cid:15) (cid:28)
1, which are alsothe conditions for the linearized Einstein field equa-tions for h µν to be valid. • Such studies can be significantly simplified by prop-erly imposing gauge conditions, such as the spatial,traceless, and Lorentz gauges , given, respectively,by χ µ = 0 , (1.4) χ = 0 , (1.5) ∇ ν χ µν = 0 , (1.6)where χ µν ≡ h µν − γ µν h, h ≡ γ µν h µν , (1.7)and ∇ ν denotes the covariant derivative with re-spect to γ µν . We show that these three differ-ent gauge conditions can be imposed simultane-ously, even when the background is not vacuum, aslonger as the high-frequency GW approximationsare valid. • However, to develop the formulas that can be ap-plicable to as many cases as possible, in this pa-per we write down explicitly the linearized Einsteinfield equations for χ µν by imposing only the spatialgauge. Applying these formulas together with thegeometrical optic approximations to such GWs, wefind that they still move along null geodesics and itspolarization bi-vector is parallel-transported, evenwhen both the cosmological scalar and tensor per-turbations are present. In addition, we also calcu-late the gravitational integrated Sachs-Wolfe (iSW)effects due to these two kinds of perturbations,whereby the dependences of the amplitude, phaseand luminosity distance of the GWs on these per-turbations are read off explicitly. The rest of the paper is organized as follows: In Sec.II, after introducing the three different scales, λ, L c , L ,we show that, for the GWs to be detected by the currentand foreseeable both ground- and space-based detectors,such GWs can be well approximated as high frequencyGWs. Then, we derive the Einstein field equations, andfind that, to make the backreaction of the GWs to thebackground negligible, as well as to have the linearizedEinstein field equations for h µν to be valid, the condition(1.3) must hold. In this section, we also provide a verybrief review on the cosmological background that con-sists of both the cosmological and tensor perturbations.In Sec. III, we consider the gauge freedom for GWs,and show that the three different gauge conditions, (1.4)-(1.6), can be still imposed simultaneously, even when thebackground spacetime is not vacuum, as long as the high-frequency approximations are valid. Then, by imposingonly the spatial gauge condition (1.4), we write down thelinearized Einstein field equations for the GWs, so theformulas can be applied to cases with different choices ofgauges. In Sec. IV we study the GWs with the geometri-cal optics approximation, and calculate the effects of thecosmological scalar and tensor perturbations on the am-plitudes and phases of such GWs, and find the explicitexpressions of the iSW effects due to both the cosmologi-cal scalar and tensor perturbations. When applying themto a binary system, we calculate explicitly the effects ofthese two kinds of the cosmological perturbations on theluminosity distance and the chirp mass [cf. Eq.(4.51)].Finally, we summarize our main results in Sec. V, andpresent some concluding remarks.There are also three appendices, A, B and C, in whichsome mathematical computations are presented. In par-ticular, in Appendix A, we give a very brief review overthe inhomogeneous universe, when both the cosmologi-cal scalar and tensor perturbations are present, while inAppendix B, we present the field equations for the GWs χ µν by imposing only the spatial gauge (1.4). In Ap-pendix C, we first decompose χ µν as χ µν = χ (0) µν + (cid:15) c χ (1) µν and then write down explicitly the field equations for χ (1) µν only with the spatial gauge.Before proceeding to the next section, we would liketo note that GWs produced by remote astrophysicalsources and then propagating through the homogeneousand isotropic universe have been systematical studied byAshtekar and his collaborators through a series of papers[66–72], and various subtle issues related to the de Sitterbackground were clarified [73–75] (See also [76–84]).In addition, in this paper we shall adopt the followingconventions, which are different from those adopted in[55, 56], but the same as those used in [85]. In particular,in this paper the signature of the metric is ( − , + , + , +),while the Christoffel symbols, Riemann and Ricci tensors,as well as the Ricci scalar, are defined, respectively, byΓ αµν ≡ g αβ ( g βν,µ + g βµ,ν − g µν,β ) , ( D α D β − D β D α ) X µ = R µναβ X ν , R µν ≡ R αµαν , R ≡ g µν R µν , (1.8)where D α denotes the covariant derivative with respectto metric g µν , g µν,λ ≡ ∂g µν /∂x λ , and R αµνλ = Γ αµλ,ν − Γ αµν,λ + Γ αβν Γ βµλ − Γ αβλ Γ βµν . (1.9)The Einstein field equations read, R µν − g µν R = κT µν , (1.10)where κ ≡ πG/c , with G denoting the Newtonian con-stant, and c the speed of light. In addition to D α and ∇ α , we also introduce the covariant derivative ¯ ∇ α withrespect to the homogeneous metric ¯ γ µν , where γ µν = ¯ γ µν + (cid:15) c ˆ γ µν , (1.11)with (cid:15) c ≡ L c /L (cid:28)
1. We shall also adopt the conven-tions, A ( µν ) ≡ ( A µν + A νµ ) / , A [ µν ] ≡ ( A µν − A νµ ) / II. GRAVITATIONAL WAVES PROPAGATINGIN INHOMOGENEOUS UNIVERSE
In this section, we shall consider GWs first producedby remote astrophysical sources and then propagating incosmic distances through the inhomogeneous Universe,before arriving at detectors. To study such GWs, letus first consider several characteristic lengths that arehighly relevant to their propagations and polarizations.
A. Characteristic Scales of Background
In this paper, we shall consider our inhomogeneous uni-verse as the background, which includes two parts, thehomogeneous and isotropic universe and its inhomoge-neous perturbations, given by ¯ γ µν and ˆ γ µν , respectively,so the background metric γ µν can be written as γ µν = ¯ γ µν + (cid:15) c ˆ γ µν + O (cid:0) (cid:15) c (cid:1) ,γ µν = ¯ γ µν − (cid:15) c ˆ γ µν + O (cid:0) (cid:15) c (cid:1) , (2.1)where (cid:15) c , | ˆ γ | (cid:28) γ µλ γ νλ = δ µν + O (cid:0) (cid:15) c (cid:1) , ¯ γ µλ ¯ γ νλ = δ µν + O (cid:0) (cid:15) c (cid:1) , ˆ γ µν ≡ ¯ γ µα ˆ γ αν , ˆ γ µν ≡ ¯ γ µα ¯ γ νβ ˆ γ αβ , (2.2)and so on.The size of the observational universe is about L (cid:39) . × m . On the other hand, in the momentum spaceof the cosmological perturbations, we have L c (cid:39) /k ,where k denotes the typical wavenumber of the perturba-tions, and L c the length over which the change of the cos-mological perturbations becomes appreciable. When themodes are outside the Hubble horizon, it can be shownthat L c /L (cid:39) − . But, once they re-enter the hori-zon these modes decay suddenly and then are oscillating rapidly about a minimum [86]. In addition, the densityperturbation δρ is the order of δρ/ ¯ ρ (cid:39) − . So, it isquite reasonable to assume that (cid:15) c (cid:39) L c L (cid:28) . (2.3) B. Typical Gravitational Wavelengths
For the second generation of the ground-based detec-tors, such as LIGO, Virgo, and KAGRA, the wavelengthof the detected GWs are λ (cid:39) ∼ m , whilethe wavelength of GWs to be detected by the space-based detectors, such as LISA, TianQin and Taiji, are λ (cid:39) ∼ m . Therefore, for the ground-baseddetectors, we have (cid:15) (cid:39) λ/L ∈ (cid:0) − , − (cid:1) , while forthe space-based detectors, we have (cid:15) ∈ (cid:0) − , − (cid:1) .Thus, for the GWs to be detected by the current andforeseeable detectors, the following is always true, λL c = (cid:15)(cid:15) c (cid:28) . (2.4)Therefore, all such GWs can be well approximated as highfrequency GWs with respect to the distance over which theinhomogeneities of the Universe change significantly . C. Einstein Field Equations
Following the above analyses, we find that λ , L c and L denote, respectively, the characteristic length over which h µν , ˆ γ µν or ¯ γ µν changes significantly. Thus, their deriva-tives are typically of the orders, ∂ ¯ γ ∼ ¯ γL , ∂ ¯ γ ∼ ¯ γL ,∂ ˆ γ ∼ ˆ γL c , ∂ ˆ γ ∼ ˆ γL c ,∂h ∼ hλ , ∂ h ∼ hλ . (2.5)To estimate orders of terms, following Isaacson [55], weregard L as order of unity, and say that the metric (1.2)contains a high-frequency GW, if and only if there existsa family of coordinate systems (related by infinitesimalcoordinate transformations), in which we have (cid:15) (cid:28) (cid:15) c (cid:28) , (2.6)and ¯ γ µν , ¯ γ µν,α , ¯ γ µν,αβ (cid:39) O (1) , The frequencies of GWs detected by the second generation of theground-based detectors is f (cid:39) − f (cid:39) − − Hz. ˆ γ µν (cid:39) O (ˆ γ ) , ˆ γ µν,α (cid:39) O (ˆ γ/(cid:15) ) , ˆ γ µν,αβ (cid:39) O (cid:0) ˆ γ/(cid:15) (cid:1) ,h µν (cid:39) O ( h ) , h µν,α (cid:39) O ( h/(cid:15) ) ,h µν,αβ (cid:39) O (cid:0) h/(cid:15) (cid:1) , (2.7)where γ µν,α ≡ ∂γ µν /∂x α , etc. Note that, in contrast to[55], here we do not assume h µν (cid:39) O (1), in order toneglect the backreaction of the GWs to the backgroundspacetime γ µν , as to be shown below.Expanding the Riemann and Ricci tensors R µναβ ( g µν )and R µν ( g µν ) in terms of (cid:15) , we find [55, 85], R αβγδ ( g µν ) = R αβγδ (0) + (cid:15)R αβγδ (1) + (cid:15) R αβγδ (2) + O (cid:0) (cid:15) (cid:1) ,R αβ ( g µν ) = R αβ (0) + (cid:15)R αβ (1) + (cid:15) R αβ (2) + O (cid:0) (cid:15) (cid:1) , (2.8)where R αβγδ (0) = R αβγδ ( γ µν ) ,R αβγδ (1) = 12 (cid:104) h βγ ; αδ + h αδ ; βγ − h αγ ; βδ − h βδ ; αγ + R ασγδ (0) h σβ − R βσγδ (0) h σα (cid:105) , (2.9) R αβ (0) = R αβ ( γ µν ) ,R αβ (1) = 12 γ ρτ (cid:16) h τα ; βρ + h τβ ; αρ − h ρτ ; αβ − h αβ ; ρτ (cid:17) , (2.10) R αβ (2) = 14 (cid:110) h ρτ ; β h ρτ ; α + 2 h ρτ (cid:0) h τρ ; αβ + h αβ ; τρ − h τα ; βρ − h τβ ; αρ (cid:1) + 2 h τβ ; ρ (cid:0) h τα ; ρ − h ρα ; τ (cid:1) − (2 h ρτ ; ρ − h ; τ ) (cid:0) h τα ; β + h τβ ; α − h αβ ; τ (cid:1)(cid:111) . (2.11)Here the semi-colon “;” denotes the covariant deriva-tive with respect to the background metric γ µν . Forthe sake of convenience, we shall also use ∇ λ to denotethe covariant derivative with respect to γ µν , so we have h µν ; λ ≡ ∇ λ h µν , etc. The background metric γ µν ( γ µν ) isalso used to lower (raise) the indices of h µν , such as h µν ≡ γ µα h αν = γ να h µα , h ≡ h λλ = γ αβ h αβ , (2.12)and so on.The background curvatures R αβγδ (0) ( γ ) and R αβ (0) ( γ )can be further expanded in terms of (cid:15) c , as R αβγδ (0) ( γ ) = ¯ R αβγδ (¯ γ ) + (cid:15) c ˆ R αβγδ (ˆ γ )+ (cid:15) c ˆ R (2) αβγδ (ˆ γ ) + O (cid:0) (cid:15) c (cid:1) ,R αβ (0) ( γ ) = ¯ R αβ (¯ γ ) + (cid:15) c ˆ R αβ (ˆ γ )+ (cid:15) c ˆ R (2) αβ (ˆ γ ) + O (cid:0) (cid:15) c (cid:1) , (2.13)where ˆ R αβγδ (ˆ γ ) = 12 (cid:104) ˆ γ βγ | αδ + ˆ γ αδ | βγ − ˆ γ αγ | βδ − ˆ γ βδ | αγ + ¯ R ασγδ ˆ γ σβ − ¯ R βσγδ ˆ γ σα (cid:105) , (2.14)ˆ R αβ (ˆ γ ) = 12 ¯ γ ρτ (cid:16) ˆ γ τα | βρ + ˆ γ τβ | αρ − ˆ γ ρτ | αβ − ˆ γ αβ | ρτ (cid:17) , (2.15)and ˆ R (2) αβ (ˆ γ ) is given by Eq.(2.11) with the replacement( h αβ , ∇ µ ) → (cid:0) ˆ γ αβ , ¯ ∇ µ (cid:1) . Here the vertical bar “ | ” denotesthe covariant derivative with respect to ¯ γ µν , which is alsodenoted by ¯ ∇ λ , so that ˆ γ ρτ | α ≡ ¯ ∇ α ˆ γ ρτ , etc. Taking L (cid:39) O (1) and considering Eq.(2.7) we find¯ R αβγδ , ¯ R αβ (cid:39) O (1) , (2.16) (cid:15) c ˆ R αβγδ , (cid:15) c ˆ R αβ ∼ O (ˆ γ/(cid:15) c ) ,(cid:15) c ˆ R (2) αβγδ , (cid:15) c ˆ R (2) αβ (cid:39) O (cid:0) ˆ γ (cid:1) , (2.17) (cid:15)R αβγδ (1) , (cid:15)R αβ (1) (cid:39) O ( h/(cid:15) ) ,(cid:15) R αβγδ (2) , (cid:15) R αβ (2) (cid:39) O (cid:0) h (cid:1) . (2.18)To write down the Einstein field equations, let us firstnote that( ∇ α ∇ β − ∇ β ∇ α ) χ γδ = − R σγαβ (0) χ σδ − R σδαβ (0) χ γσ . (2.19)Then, we find that in terms of χ µν , R αβ (1) is given by R αβ (1) = 12 (cid:16) R γαβσ (0) χ γσ + R σα (0) χ βσ + R σβ (0) χ ασ + ∇ α ∇ δ χ βδ + ∇ β ∇ δ χ αδ (cid:17) − (cid:3) χ αβ + 14 γ αβ (cid:3) χ, (2.20)where (cid:50) χ αβ ≡ γ µν χ αβ ; µν , and χ µν ≡ h µν − γ µν h, χ ≡ γ µν χ µν = − h. (2.21)It should be noted that in [55] Isaacson considered thevacuum case, for which we have R αβ (1) = 0, that is, (cid:3) χ αβ − γ αβ (cid:3) χ − ∇ α ∇ δ χ βδ − ∇ β ∇ δ χ αδ + 2 R α γβσ (0) χ γσ − R σα (0) χ βσ − R σβ (0) χ ασ = 0 , (2.22)which is precisely Eq.(5.7) of [55], after the differencebetween the conventions used here and the ones used in[55] is taken into account.However, in the present paper we consider the propaga-tion of GWs through the inhomogeneous universe, whichhas non-zero Riemann and Ricci tensors. So, we expectthat the corresponding Einstein field equations for h µν are different from Eq.(2.22). To see this, we first notethat g µν = γ µν − (cid:15)h µν + (cid:15) h µα h αν + O (cid:0) (cid:15) (cid:1) , R ≡ g µν R µν = R (0) + (cid:15)R (1) + (cid:15) R (2) + O (cid:0) (cid:15) (cid:1) , (2.23)where R (0) ≡ γ µν R (0) µν ,R (1) ≡ γ µν R (1) µν − h µν R (0) µν = ∇ α ∇ β χ αβ − χ αβ R (0) αβ + 12 (cid:16) (cid:3) + R (0) (cid:17) χ,R (2) ≡ γ µν R (2) µν − h µν R (1) µν + h µα h αν R (0) µν . (2.24)Inserting Eqs.(2.8) and (2.23) into the Einstein fieldequations, we find that R (0) µν − γ µν R (0) + (cid:15) (cid:20) R (1) µν − (cid:16) γ µν R (1) + h µν R (0) (cid:17)(cid:21) + (cid:15) (cid:20) R (2) µν − (cid:16) γ µν R (2) + h µν R (1) (cid:17)(cid:21) + O (cid:0) (cid:15) (cid:1) = κ (cid:16) T (0) µν + (cid:15) T µν (cid:17) , (2.25)where T (0) µν denote the energy-momentum tensor that pro-duces the background, while T µν denotes the astrophysi-cal source that produces the GWs. D. Separation of GWs from Background
To separate GWs produced by astrophysical sourcesfrom the inhomogeneous background, we can average thefield equations over a length scale (cid:96) , which is much largerthan the typical wavelength of the GWs but much smallerthan L c , λ (cid:28) (cid:96) (cid:28) L c . (2.26)Then, this process will extract the slowly varying back-ground from GWs, as the latter will vanish when averag-ing over such a scale. In particular, we have (cid:104) γ µν (cid:105) = γ µν , (cid:68) R µναβ (0) (cid:69) = R µναβ (0) , (cid:68) R µν (0) (cid:69) = R µν (0) , (cid:68) T µν (0) (cid:69) = T µν (0) , (2.27) (cid:104) h µν (cid:105) = (cid:68) R µν (1) (cid:69) = (cid:68) R (1) (cid:69) = 0 , (2.28) (cid:68) R µν (2) (cid:69) = (cid:68) R µν (2) (cid:69) (cid:96) , (cid:68) R (2) (cid:69) = (cid:68) R (2) (cid:69) (cid:96) , (cid:68) h µν R (1) (cid:69) = (cid:68) h µν R (1) (cid:69) (cid:96) , (cid:104)T µν (cid:105) = (cid:104)T µν (cid:105) (cid:96) . (2.29)Note that quadratic terms of h µν may survive such anaveraging process, if two modes are almost equal butwith different signs, although each of them representsa high frequency mode. For example, for h µν ∝ e iω x and h αβ ∝ e − iω x , we have h µν h αβ ∝ e iω x , where ω ≡ ω − ω . Thus, although ω , ω (cid:29)
1, we can have ω (cid:28)
1, if ω (cid:39) ω . Therefore, due to the nonlinear in-teractions among different modes, low frequency modescan be produced, which will survive with such averagingprocesses. If we are only interested in the linearized Ein-stein field equations of h µν , such modes must be taken ofcare properly. With this in mind, taking the average ofEq.(2.25) we find that R (0) µν − γ µν R (0) + (cid:15) (cid:68) G µν (2) (cid:69) (cid:96) = κ (cid:16) T (0) µν + (cid:15) (cid:104)T µν (cid:105) (cid:96) (cid:17) , (2.30)where G µν (2) ≡ R (2) µν − (cid:16) γ µν R (2) + h µν R (1) (cid:17) , (2.31)which is a quadratic function of h µν . Then, substitutingEqs.(2.30) and (2.31) back to Eq.(2.25), we find that thehigh-frequency part takes the form, R (1) µν − (cid:16) γ µν R (1) + h µν R (0) (cid:17) + (cid:15) (cid:68) G (2) µν (cid:69) high = κ (cid:104)T µν (cid:105) high , (2.32)where (cid:68) G (2) µν (cid:69) high ≡ G (2) µν − (cid:68) G µν (2) (cid:69) (cid:96) , (cid:104)T µν (cid:105) high ≡ T µν − (cid:104)T µν (cid:105) (cid:96) . (2.33)On the other hand, from Eqs.(2.16)-(2.18) we find that G (0) µν ≡ R (0) µν − γ µν R (0) (cid:39) O (ˆ γ/(cid:15) c ) , (cid:68) G µν (2) (cid:69) (cid:96) (cid:39) O (cid:0) h /(cid:15) (cid:1) , T (0) µν (cid:39) O (cid:0) (cid:15) − c (cid:1) . (2.34)Note that, after introducing the cosmological perturba-tion scale L c , the leading order of G (0) µν becomes G (0) µν (cid:39) (cid:15) c ˆ R µν (cid:39) O (ˆ γ/(cid:15) c ), instead of L − [61]. The same is truefor T (0) µν , as it can be seen from Appendix A. Then, fromEq.(2.30) we find that each term has the following order, O (ˆ γ/(cid:15) c ) + O (cid:0) h (cid:1) = O (ˆ γ/(cid:15) c ) + (cid:15) O (cid:0) (cid:104)T µν (cid:105) (cid:96) (cid:1) . (2.35)Therefore, to have the backreaction of the GWs to thebackground be negligible, so that the background space-time γ µν is uniquely determined by T (0) µν , i.e., R µν (0) − γ µν R (0) = κT µν (0) , (2.36)we must assume that h (cid:28) ˆ γ(cid:15) c , (2.37) (cid:15) · (cid:12)(cid:12) (cid:104)T µν (cid:105) (cid:96) (cid:12)(cid:12) (cid:28) ˆ γ(cid:15) c . (2.38)In addition, from Eq.(2.32) we find that (cid:15) (cid:68) G (2) µν (cid:69) high (cid:39) O (cid:0) h /(cid:15) (cid:1) . (2.39)Therefore, in order for the quadratic terms from G (2) µν notto affect the linear terms of the leading orders h/(cid:15) and h/(cid:15) in Eq.(2.32), we must assume that | h | (cid:28) . (2.40)With the above conditions, we find that Eq.(2.32) canbe written as (cid:3) χ αβ + γ αβ ∇ γ ∇ δ χ γδ − ∇ α ∇ δ χ βδ − ∇ β ∇ δ χ αδ + 2 R αγβσ (0) χ γσ = − κ (cid:104)T αβ (cid:105) high . (2.41)From the above derivations, we can see that the linearizedEinstein field equations (2.41) are valid only to the twoleading orders, (cid:15) − and (cid:15) − . For orders higher than them,these equations are not applicable. This is particularlytrue for the zeroth-order of (cid:15) . In addition, since (cid:15) − c (cid:28) (cid:15) − , we find that in Eq.(2.41) the term2 R αγβσ (0) χ γσ (cid:39) O (ˆ γh/(cid:15) c ) (cid:28) O ( h/(cid:15) ) , (2.42)can also be neglected. However, in order to compare ourresults with the ones obtained in [55–57], we shall keepit, and drop the corresponding terms only at the end ofour calculations. E. The Inhomogeneous Universe
In this subsection, we shall give a very brief introduc-tion over the flat FRW universe with its linear scalar andtensor perturbations, described by the metric (1.11). Interms of the conformal coordinates x µ = (cid:0) η, x i (cid:1) , ( i =1 , , γ µν = a ( η ) η µν , ¯ γ µν = a − ( η ) η µν , (2.43)with η µν = diag ( − , +1 , +1 , +1). The coordinate η isrelated to the cosmic time via the relation, η = (cid:82) dta ( t ) .Following the standard process, we decompose thelinear perturbations ˆ γ µν into scalar, vector and tensormodes,ˆ γ µν = a ( η ) (cid:18) − φ ∂ i B − S i sym − ψδ ij + 2 ∂ ij E + 2 ∂ ( i F j ) + H ij (cid:19) , (2.44)where ∂ i S i = ∂ i F i = 0 , ∂ i H ij = 0 = H ii , (2.45)with ∂ i ≡ δ ij ∂ j and H ij ≡ δ ik H kj . However, the vec-tor mode will decay quickly with the expansion of theuniverse, and can be safely neglected [87, 88]. Then, using the gauge transformations, as shown explicitly inAppendix A, we can always set B = E = 0 , (2.46)in which the gauge is completely fixed. This is often re-ferred to as the Newtonian gauge, under which the gauge-invariant quantities defined in Eq.(A.11) become,Φ = φ, Ψ = ψ, ( B = E = 0) , (2.47)that is, in the Newtonian gauge, the potentials φ and ψ are equal to the gauge-invariant ones, Φ and Ψ. There-fore, with this gauge and ignoring the vector part, wehave ˆ γ µν = a ( η ) (cid:18) − φ H ij − ψδ ij (cid:19) , ˆ γ µν = a − ( η ) (cid:18) − φ H ij − ψδ ij (cid:19) . (2.48)In the rest of this paper, we shall restrict ourselves tothis gauge. III. LINEARIZED FIELD EQUATIONS FORGWS IN INHOMOGENEOUS UNIVERSE
In this section, we shall consider the field equations for χ µν given by Eq.(2.41) in the inhomogeneous cosmolog-ical background of Eq.(1.11) with the Newtonian gauge(2.46), by neglecting the vector perturbations, for whichˆ γ µν and ˆ γ µν are given by Eq.(2.48). A. Gauge Fixings for GWs
Before writing down these linearized field equations ex-plicitly, let us first consider the gauge freedom for χ µν . Atthe end of the last section, we had considered the gaugetransformations for the cosmological perturbations, andhad already used the gauge freedom,˜ x µ = x µ + (cid:15) c ζ µ , (3.1)to set B = E = 0 [cf. Eq.(2.46)], the so-called Newtoniangauge, as shown explicitly in Appendix A. These choicescompletely fix the gauge freedom for the cosmologicalperturbations.In this subsection, we shall consider another kind ofgauge transformations for the GWs, given byˇ x α = x α + (cid:15)ξ α , (3.2)where ξ α (cid:39) O ( (cid:15)h ) , ξ α ; β (cid:39) O ( h ) , ξ α ; β ; γ (cid:39) O ( h/(cid:15) ) . (3.3) In writing down the leading order of ξ α , we had set the slowly-changing part that is of order one to zero, as it is irrelevant tothe high frequency GWs considered here. Since (cid:15) c (cid:29) (cid:15) , we can see that to the first order of (cid:15) c ,the background metric γ µν does not change under thecoordinate transformations (3.2), that is,ˇ γ µν = γ µν + O (cid:0) (cid:15) c (cid:1) , (3.4)a property that is required for the transformations (3.2)to be the gauge transformations only for the GWs. Onthe other hand, under the coordinate transformations(3.2), we haveˇ g µν ≡ ˇ γ µν + (cid:15) ˇ h µν + O (cid:0) (cid:15) (cid:1) = γ µν + (cid:15) ( h µν − ξ µ ; ν − ξ ν ; µ ) + O (cid:0) (cid:15) (cid:1) , (3.5)that is, ˇ h µν = h µν − ξ ( µ ; ν ) . (3.6)Hence, we findˇ R (1) αβγδ − R αβγδ (1) = −L ξ R αβγδ (0) = O ( h ˆ γ/(cid:15) c ) , ˇ R (1) αβ − R αβ (1) = −L ξ R αβ (0) = O ( h ˆ γ/(cid:15) c ) , (3.7)as can be seen from Eqs.(2.16)-(2.18), and (3.3), where L ξ denotes the Lie derivative. Therefore, Eq.(2.41) is gauge-invariant only up to O ( h ˆ γ/(cid:15) c ). However, since (cid:15) − c (cid:28) (cid:15) − , terms that are order of (cid:15) − and (cid:15) − are still gauge-invariant, while the ones of order of (cid:15) are not. Thisis because in the scale λ the spacetime appears locallyflat, and the curvature is locally gauge-invariant. Thus,provided that the following conditions hold, | h | , | ˆ γ | (cid:28) , (cid:15) (cid:28) (cid:15) c (cid:28) , (3.8)the GW produced by an astrophysical source can be con-sidered as a high-frequency GW, and their low-frequencycomponents are negligible, so that the local-flatness be-havior carries over to the case in which the backgroundis even curved.On the other hand, from the field equations (2.41) wecan see that they will be considerably simplified, if wechoose the Lorentz gauge, ∇ ν ˇ χ µν = 0 , (3.9)where ˇ χ µν ≡ ˇ h µν − γ µν ˇ h = χ µν − ∇ ( µ ξ ν ) + γ µν ∇ λ ξ λ , (3.10)as it can be seen from Eq.(3.6), where ξ µ ≡ γ µν ξ ν . Then,we find that the Lorentz gauge (3.9) requires, (cid:50) ξ µ + R (0) νµ ξ ν = ∇ ν χ µν . (3.11)Note that R (0) νµ ξ ν (cid:39) O ( h ˆ γ(cid:15)/(cid:15) c ) (cid:28) O ( h/(cid:15) ), so it can beneglected in the above equation. Clearly, for any given χ µν (with some proper continuous conditions [89], which are normally assumed always to exist.), the above equa-tion in general has non-trivial solutions [55].In addition, Eq.(3.11) does not completely fix thegauge. In fact, the gauge residual,ˇˇ x α = ˇ x α + (cid:15)ς α , (3.12)exists, for which the Lorentz gauge (3.9) still holds, ∇ ν ˇˇ χ µν = 0 , (3.13)as long as ς α satisfies the conditions, (cid:50) ς µ + R (0) νµ ς ν = 0 . (3.14)Again, in this equation the term R (0) νµ ς ν (cid:39) O ( h(cid:15) ˆ γ/(cid:15) c ) isnegligible, in comparing with the one (cid:50) ς µ (cid:39) O ( h/(cid:15) ).An interesting question is that: can we use this gaugeresidual further to set ˇˇ χ µ = 0 . (3.15)To answer this question, we first note that if this is thecase, ς µ must satisfy the additional conditions, ∇ ς ν + ∇ ν ς − γ ν ∇ α ς α = ˇ χ ν . (3.16)Clearly, for any given γ µν and ˇ χ µν (again with certainregular conditions [89]), in general the above equationhas solutions. However, we must remember that ς ν alsoneeds to satisfy Eq.(3.14). To see if these conditions areconsistent or not, let us take the covariant derivative ∇ µ in both sides of Eq.(3.16), which results in ∇ ν ∇ ς ν + (cid:50) ς − ∇ ∇ ν ς ν = (cid:50) ς + R (0)0 α ς α = 0 = ∇ ν ˇ χ ν . (3.17)Therefore, we conclude that it is consistent to impose theLorentz and spatial gauges simultaneously, even when thebackground is curved [55].Finally, we note that the traceless condition χ = 0 , (3.18)was also introduced in [55]. In fact, provided that theLorentz gauge ∇ ν χ µν = 0 holds, from the field equations(2.41) we find (cid:3) χ = − κγ αβ (cid:104)T αβ (cid:105) high . (3.19)Note that in writing down the above equation, we haddropped the term 2 R γσ (0) χ γσ , which is order of h ˆ γ/(cid:15) c , asshow above. Therefore, far from the source ( T αβ = 0), ifthe Lorentz gauge holds, one can also consistently im-pose the traceless gauge. Together with the Lorentzand spatial gauges, it leads to the well-known traceless-transverse (TT) gauge, frequently used when the back-ground is Minkowski [85, 90, 91].It should be noted that in curved backgrounds theabove three different gauge conditions can be imposed si-multaneously only for high frequency GWs, and are validonly up to the order of (cid:15) − [55]. In other situations, whenimposing them, one must pay great cautions, as theseconstraints in general represent much more degrees thanthe four degrees of the gauge freedom that the generalcovariance normally allows. B. Field Equations for GWs
To write down explicitly the field equations (2.41) for χ µν , and to make our expressions as much applicableas possible, in Appendix A, we only impose the spatialgauge, χ µ = 0 , ( µ = 0 , , , , (3.20)and then calculate each term appearing in Eq.(2.41),before putting them together to finally obtain the ex-plicit expressions for each component of the field equa-tions. In particular, the non-vanishing components of theterm 2 R γασβ (0) χ γσ are given by Eq.(B.5), while the onesof (cid:50) χ αβ are given by Eqs.(B.7) and (B.8). The term γ αβ ∇ γ ∇ δ χ γδ is given by Eqs.(B.10) and (B.11), whilethe one ∇ α ∇ δ χ βδ is given by Eq.(B.13). Setting G αβ ≡ (cid:3) χ αβ + γ αβ ∇ γ ∇ δ χ γδ − ∇ α ∇ δ χ βδ − ∇ β ∇ δ χ αδ +2 R αγβσ (0) χ γσ , (3.21)we find that the field equations (2.41) take the form, G αβ = − κ (cid:104)T αβ (cid:105) high , (3.22)where the non-vanishing components of G αβ are given byEqs.(B.15) - (B.17). IV. GEOMETRICAL OPTICSAPPROXIMATION
To study the propagation of GWs in our inhomoge-neous universe, let us first note that, when far away fromthe source that produces the GWs, we have T µν = 0.Then, Eq.(3.22) reduces to, G αβ = 0 , ( T µν = 0) . (4.1)Following Isaacson [55] and Laguna et al [57], we con-sider the geometrical optics approximation, for which wehave χ αβ = A αβ e iϕ/(cid:15) = e αβ A e iϕ/(cid:15) , (4.2)where e αβ denotes the polarization tensor with e αβ e αβ = 1 , (4.3)and A and ϕ characterize, respectively, the amplitudeand phase of the GWs with e αβ ≡ γ αµ γ βν e µν . Notethat in writing the above expression we made the change, ϕ I → ϕ/(cid:15) , by following Laguna et al [57], where ϕ I is thequantity used by Isaacson [55]. With this in mind, we cansee that both the amplitude A and the phase ϕ are slowlychanging functions [55], ∂ α ϕ (cid:39) O (1) , A αβ ; γ (cid:39) O (1) . (4.4) With the gauge (3.20), we must set A β = 0 = e β . (4.5)Moreover, as shown in the last section, in addition tothe spatial gauge, we can consistently impose the Lorentzand traceless gauges, ∇ ν χ µν = 0 , χ = 0 . (4.6)Then, from Eqs.(4.2) and (4.5) we find that the Lorentzgauge yields, ∇ ν A µν + i(cid:15) k ν A µν = 0 , (4.7)where k α ≡ ∇ α ϕ and k α ≡ γ αβ k β . Considering Eq.(4.4)we find that, to the leading order ( (cid:15) − ), we have k ν A µν = 0 ⇒ k ν e µν = 0 . (4.8)Therefore, the propagation direction of the GW is orthog-onal to its polarization plane spanned by the bivector e µν .Note that the first term in Eq.(4.7) is of order (cid:15) , andshould be discarded. Otherwise, it will lead to inconsis-tent results, as mentioned above. Therefore, in the restof this paper we shall ignore such terms without furthernotifications. See [55, 57, 61] for more details.In addition, the traceless condition requires γ αβ e αβ = 0 . (4.9)Plugging Eq.(4.2) into Eq.(3.22) and considering Eq.(4.4)and the Lorentz gauge (4.6), we find that the field equa-tions to the orders of (cid:15) − and (cid:15) − are given, respectively,by (cid:15) − : k µ k µ A αβ = 0 , (4.10) (cid:15) − : k µ ∇ µ e αβ + (cid:18) k µ ∇ µ ln A + 12 ∇ µ k µ (cid:19) e αβ = 0 . (4.11)Since A µν (cid:54) = 0, from Eq.(4.10) we find k λ k λ = 0 . (4.12)Then, for such a null vector k µ , we can always define acurve x µ = x µ ( λ ) by setting dx µ ( λ ) dλ ≡ k µ , (4.13)where λ denotes the affine parameter along the curve. Itis clear that such a defined curve is a null geodesics, k λ ∇ µ k λ = k λ ∇ λ k µ = 0 , (4.14)as now we have ∇ µ k λ = ∇ µ ∇ λ ϕ = ∇ λ ∇ µ ϕ = ∇ λ k µ ,that is, GWs are always propagating along null geodesicsin our inhomogeneous universe, even when both the cos-mological scalar and tensor perturbations are all present,as long as the geometrical optics approximation are valid .0On the other hand, Multiplying e αβ in both sides ofEq.(4.11) and taking Eq.(4.3) into account, we find that ddλ ln A + 12 ∇ µ k µ = 0 , (4.15)where d/dλ ≡ k ν ∇ ν . Introducing the current J µ ≡ A k µ of the gravitons moving along the null geodesics, theabove equation can be written in the form, ∇ µ J µ = 0 . (4.16)Therefore, the current of the gravitons moving along thenull geodesics defined by k µ is conserved, even when theprimordial GWs (or cosmological tensor perturbations)are present ( H ij (cid:54) = 0 ) .Inserting Eq.(4.15) into Eq.(4.11), we find that k µ ∇ µ e αβ = 0 . (4.17)Thus, the polarization bivector e αβ is still parallel-transported along the null geodesics, even when the pri-mordial GWs are present .It is interesting to note that Eqs.(4.7)-(4.17) hold notonly for the inhomogeneous universe, but also for anycurved background, as long as the geometrical optics ap-proximation are applicable to the high frequency GWs .To study them further, we expand χ µν in terms of (cid:15) c as, ˆ χ µν = χ (0) µν + (cid:15) c χ (1) µν + O (cid:0) (cid:15) c (cid:1) , (4.18)and then consider them order by order. A. GWs Propagating in Homogeneous andisotropic Background
To the zeroth-order of (cid:15) c , we have γ µν (cid:39) ¯ γ µν = a η µν ,and χ µν (cid:39) χ (0) µν + O ( (cid:15) c ) , (4.19)where we had set χ (0) µν ≡ A (0) µν e iϕ (0) /(cid:15) = e (0) µν A (0) e iϕ (0) /(cid:15) . (4.20)Then, from Eqs.(4.16) and (4.17) we immediately ob-tain, ¯ ∇ ν (cid:16) A (0)2 k (0) ν (cid:17) = 0 , (4.21) ddλ e (0) ij = 0 , (4.22)where k (0) µ ≡ ¯ ∇ µ ϕ (0) = (cid:0) ϕ (0) ,η , ϕ (0) ,i (cid:1) , and k (0) µ ≡ ¯ γ µν k (0) ν . B. Gravitational iSW Effects
The derivation of the iSW effect in cosmology is basedcrucially on the fact that the electromagnetic radiationpropagating along null geodesics in the inhomogeneousuniverse. Laguna et al [57] took the advantage of thefact that GWs are also propagating along null geodesicsand derived the gravitational iSW effect for GWs whenonly the cosmological scalar perturbations are present( H ij = 0). In this subsection, we shall generalize theirstudies further to the case where both the cosmologicalscalar and tensor perturbations are present. As shown byEq.(4.12), even when both of them are present, the GWsproduced by astrophysical sources are still propagatingalong the null geodesics. Therefore, such a generalizationis straightforward.In particular, let us first introduce the conformal met-ric ˜ γ µν by d ˜ s = ˜ γ µν dx µ dx ν ≡ a − γ µν dx µ dx ν = − (1 + 2 (cid:15) c φ ) dη + (cid:104) (1 − (cid:15) c ψ ) δ ij + H ij (cid:105) dx i dx j . (4.23)Since γ µν and ˜ γ µν are related to each other by a con-formal transformation, so the null geodesics x µ ( λ ) in the γ µν spacetime is the same as ˜ x µ (˜ λ ) in the ˜ γ µν spacetime,where dλ = ad ˜ λ, k µ = 1 a ˜ k µ , (4.24)and ˜ λ is the affine parameter of the null geodesics ˜ x µ inthe spacetime of ˜ γ µν .The advantage of working with the metric ˜ γ µν is thatthe zeroth-order spacetime now becomes the Minkowskispacetime, and the corresponding null geodesics are thestraight lines, given by d ˜ x (0) µ (˜ λ ) d ˜ λ ≡ ˜ k (0) µ . (4.25)Thus, to simplify our calculations, we shall work with˜ γ µν . In particular, to the zeroth-order of (cid:15) c , we have˜ k (0) µ = (cid:0) , − n i (cid:1) , (4.26)where ˜ k (0) i ≡ − n i represents the spatial direction of theGWs from the source propagating to the observer [cf.Fig. 1]. Then, from Eq.(4.21) we find, dd ˜ λ ln (cid:16) a A (0) (cid:17) = −
12 ˜ k (0) ν,ν = 0 , (4.27)which implies that the quantity defined by Q ≡ RA (0) , (4.28)is constant along the GW path, and will be determinedby the local wave-zone source solution, where R ≡ ar de-notes the physical distance between the observer and the1 i zx y0 − n β α FIG. 1: A gravitational wave is propagating along the spatialdirection ˜ k (0) i ≡ − n i to the observer located at the origin. source, while r denotes the comoving distance, given by r ≡ (cid:113) ( x e − x r ) + ( y e − y r ) + ( z e − z r ) , where x ie ≡ ( x e , y e , z e ) and x ir ≡ ( x r , y r , z r ) are the spatial locationsof the source and observer, respectively.In the following, we shall set up the coordinates as fol-lows [57]: The observer is located at the origin with itsproper time denoted by τ and world line x µ ( τ ). Denot-ing the time to receive the GW by τ r , this event willbe recorded as x µ ( τ ) = ( τ r , (cid:126)o ). The emission time ofthe GW by an astrophysical source corresponds to theproper time τ e of the observer with x µ ( τ ) = ( τ e , (cid:126)o ).Then, the GW will move along the null geodesics, de-scribed by ˜ x µ (˜ λ ) = ˜ x (0) µ (˜ λ ) + (cid:15) c ˜ x (1) µ (˜ λ ), which corre-sponds to the wave vector ˜ k µ (˜ λ ) = ˜ k (0) µ (˜ λ ) + (cid:15) c ˜ k (1) µ (˜ λ ),where ˜ x (0) µ (˜ λ ) = (cid:16) ˜ λ, (˜ λ r − ˜ λ ) n i (cid:17) , and ˜ λ r is the momentwhen the GW arrives at the origin with τ (˜ λ r ) = τ r .The effects of the scalar and tensor perturbations aremanifested from the perturbations of the null geodesics.Considering the fact ˜Γ (0) µνλ = 0 in the ˜ γ µν spacetime, wefind that, to the first-order of (cid:15) c , ˜ k (1) µ (˜ λ ) is given by d ˜ k (1) µ d ˜ λ + ˜Γ (1) µαβ ˜ k (0) α ˜ k (0) β = 0 , (4.29)where ˜Γ (1) µαβ denotes the Christoffel symbols of the first-order of (cid:15) c . As mentioned previously, for the scalar per-turbations, we shall not assume that ψ = φ , that is, the trace of the anisotropic stress of the universe does notnecessarily vanish, as shown by Eq.(A.15) in AppendixA. Then, for µ = 0 we find that ddλ ˜ k (1)0 = ∂ τ ( φ + ψ ) − dφdλ − n k n l ∂ τ H kl , (4.30)where d Φ dλ ≡ (cid:0) ∂ τ − n i ∂ i (cid:1) Φ . (4.31)Thus, integrating Eq.(4.30) we find,˜ k (1)0 = − ( φ + ψ ) | λ e + 12 n k n l H kl | λ e − φ | λλ e + I ( s ) iSW − I ( t ) iSW , (4.32)where I ( s ) iSW represents the gravitational iSW effect due tothe cosmological scalar perturbations, and was first cal-culated in [57]. The new term I ( t ) iSW is the gravitationalintegrated effect due to the cosmological tensor pertur-bations. They are given, respectively, by I ( s ) iSW ≡ (cid:90) λλ e ∂ τ ( φ + ψ ) dλ (cid:48) , (4.33) I ( t ) iSW ≡ n k n l (cid:90) λλ e ∂ τ H kl dλ (cid:48) . (4.34)On the other hand, the spatial components of the wave-vector are given by, ddλ ˜ k (1) i (cid:107) = − n i (cid:34) ∂ τ ( φ + ψ ) + ddλ ( φ − ψ ) − n k n l (cid:32) dH kl dλ + ∂ τ H kl (cid:33)(cid:35) , (4.35) ddλ ˜ k (1) i ⊥ = − ⊥ ij (cid:20) ∂ j ( φ + ψ ) − n k dH jk dλ − n k n l ∂ j H kl (cid:21) , (4.36)where we had set ˜ k (1) i = ˜ k (1) i (cid:107) + ˜ k (1) i ⊥ , with the parallelcomponent of the spatial wave-vector being defined by˜ k (1) i (cid:107) = n i n j ˜ k (1) j , and the perpendicular component by˜ k (1) i ⊥ = ⊥ ij ˜ k (1) j . The projection operator ⊥ ij is definedby ⊥ ij = δ ij − n i n j , with n i ≡ δ ik n k . After integrations,the above two equations yield,˜ k (1) i (cid:107) = − n i (cid:34) ( ψ − φ ) | λλ e − n k n l H kl | λλ e + I ( s ) iSW − I ( t ) iSW (cid:35) , (4.37)˜ k (1) i ⊥ = − ⊥ ij (cid:34) (cid:90) λλ e ∂ j ( φ + ψ ) dλ (cid:48) − n k H jk | λλ e − n k n l (cid:90) λλ e ∂ j H kl dλ (cid:48) (cid:35) . (4.38)The GW phase is then given by, dϕdλ = φ + ψ − n k n l (cid:90) λλ e H kl dλ (cid:48) , (4.39)which leads to δϕ = ϕ − ϕ e = (cid:90) λλ e ( φ + ψ ) dλ (cid:48) − n k n l (cid:90) λλ e H kl dλ (cid:48) . (4.40)The frequency of the GW is defined as ω = − u µ k µ , where u µ is the 4-velocity of the fluid of the universe, fromwhich we find that the ratio of receiving and emittingfrequencies is given by ω r ω e = 1 − Υ1 + z , (4.41)where 1 + z ≡ a r /a e , andΥ ≡ φ | λ r λ e + v i n i + 12 n k n l H kl | λ r λ e − I ( s ) iSW | λ r + 12 I ( t ) iSW | λ r . (4.42)In addition, setting A = A (0) (1 + ξ ), from Eq.(4.21)we find − dξdλ = ∂ τ ˜ k (1)0 + ∂ i ˜ k (1) i (cid:107) + ∂ i ˜ k (1) i ⊥ + ˜Γ (1) µµν ˜ k (0) ν , (4.43)where ∂ τ ˜ k (1)0 = ∂ τ (cid:18) − φ + I ( s ) iSW − I ( t ) iSW (cid:19) ,∂ i ˜ k (1) i (cid:107) = ddλ (cid:16) ψ − φ + I ( s ) iSW (cid:17) − ∂ τ (cid:16) ψ − φ + I ( s ) iSW (cid:17) − ddλ (cid:16) n k n l H kl + I ( t ) iSW (cid:17) + 12 ∂ τ (cid:16) n k n l H kl + I ( t ) iSW (cid:17) ,∂ i ˜ k (1) i ⊥ = − ⊥ ij (cid:20) (cid:90) λλ e ∂ i ∂ j ( φ + ψ ) dλ (cid:48) − n k ∂ i H kj − n k n l (cid:90) λλ e ∂ i ∂ j H kl dλ (cid:48) (cid:21) , ˜Γ (1) µµν ˜ k (0) ν = ddλ ( φ − ψ ) . (4.44)Notice that in the last term, there are no contributionsfrom the tensor perturbations. Collecting all of this to-gether, Eq.(4.43) yields, − dξdλ = − ∂ τ ( φ + ψ ) + ddλ (cid:16) − ψ + I ( s ) iSW (cid:17) − ⊥ ij (cid:90) λλ e ∂ i ∂ j ( φ + ψ ) dλ (cid:48) + 12 n k n l ∂ τ H kl − ddλ (cid:16) n k n l H kl + I ( t ) iSW (cid:17) + ⊥ ij n k ∂ i H jk + 12 ⊥ ij n k n l (cid:90) λλ e ∂ i ∂ j H kl dλ (cid:48) , (4.45)which has the general solution, ξ = − ψ | λλ e + 12 ⊥ ij (cid:90) λλ e (cid:90) λ (cid:48) λ e ∂ i ∂ j ( φ + ψ ) dλ (cid:48) dλ (cid:48)(cid:48) − n k (cid:34) − n l H kl | λλ e + ⊥ ij (cid:90) λλ e ∂ i H jk dλ (cid:48) + 12 ⊥ ij n l (cid:90) λλ e (cid:90) λ (cid:48) λ e ∂ i ∂ j H kl dλ (cid:48) dλ (cid:48)(cid:48) (cid:35) . (4.46)In terms of the gravitational tensorial iSW effect definedby Eq.(4.34), the above expression can be written in theform, ξ = (cid:18) ψ − n k n l H kl (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) λλ e + 12 I ( t ) iSW − ⊥ ij (cid:90) λλ e (cid:90) λ (cid:48) λ e ∂ i ∂ j (cid:34) n k n l H kl − φ + ψ ) (cid:35) dλ (cid:48)(cid:48) dλ (cid:48) − n k (cid:90) λλ e ∂ l H kl dλ (cid:48) . (4.47)Combining all of our results together, we are at thepoint to construct the gravitational waveform throughEq.(4.2), from which we find that h µν = χ µν − χγ µν = e µν ˜ h, ˜ h ≡ A e iϕ = (1 + z ) Q d L (1 + ξ ) e i ( ϕ e + δϕ ) , (4.48)where δϕ and ξ are given, respectively, by Eqs.(4.40) and(4.47), and d L ≡ (1 + z ) R is the luminosity distance.Note that in writing the expression for the response func-tion ˜ h we had set (cid:15) = 1.For a binary system, we have [57, 91], Q = M e ( πf e M e ) / ,ϕ e = ϕ c − ( πf e M e ) − / , (4.49)where M e and f e denote, respectively, the intrinsic chirpmass and frequency of the binary, and φ c is the valueof the phase at the merge, at which we have f = ∞ .Therefore, the function ˜ h for a binary system can be castin the form,˜ h = M r D L ( πf r M r ) / e i ( ϕ e + δϕ ) , (4.50)3where the modified luminosity distance D L and the chirpmass M r measured by the observer are given, respec-tively, by D L ≡ d L − Υ − ξ , M r ≡ (cid:18) z − Υ (cid:19) M e , (4.51)where Υ is given by Eq.(4.42). V. CONCLUSIONS
In this paper, we have systematically studied GWs,which are first produced by some remote compact astro-physical sources, and then propagate in our inhomoge-neous universe through cosmic distances before arrivingat the detectors. Such GWs will carry valuable informa-tion of both their sources and the cosmological expan-sion and inhomogeneities of the universe, whereby a com-pletely new window to explore our universe by using GWsis opened. As the third generation (3G) detectors, suchas the space-based ones, LISA [18], TianQin [19], Taiji[20], DECIGO [21], and the ground-based ones, ET [50]and CE [51], are able to detect GWs emitted from suchsources as far as at the redshift z (cid:39)
100 [52], it is veryimportant and timely to carry out such studies system-atically. Such studies were already initiated some yearsago [57, 59, 60] in the framework of Einstein’s theory,and more recently in scalar-tensor theories [61–64, 64].In this paper, in order to characterize effectively suchsystems, we first introduced three scales, λ, L c and L ,which represent, respectively, the typical wavelength ofthe GWs, the scale of the cosmological perturbations,and the size of our observable universe. For GWs to bedetected by the current and foreseeable (both ground-and space-based) detectors, in Sec. II we showed thatthe relation λ (cid:28) L c (cid:28) L, (5.1)is always true, that is, such GWs can be well approxi-mated as high frequency GWs , for which the general for-mulas were already developed by Isaacson more than halfcentury ago [55, 56].However, Isaacson considered only the case where thebackground is vacuum, while in [57, 59, 60] only the cos-mological scalar perturbations were considered. In thispaper, we considered the most general case in which thebackground also includes the cosmological tensor pertur-bations. The inclusion of the latter is important, as nowone of the main goals of cosmological observations is theprimordial GWs (the tensor perturbations) [65]. In thenon-vacuum case, (in Sec. II) we showed explicitly thatthe conditions | h µν | (cid:28) , (cid:15) (cid:28) (cid:15) c (cid:28) , (5.2)must hold, in order for the backreaction of the GWs tothe background to be neglected, and the linearized Ein-stein field equations given by Eq.(2.41) to hold, where the total metric of the spacetime is expanded as g µν = γ µν + (cid:15)h µν , with γ µν ( ≡ ¯ γ µν + (cid:15) c ˆ γ µν ) representing thebackground.In Sec. III, we considered the gauge choices, and foundthat the three different gauge conditions, spatial, trace-less, and Lorentz , given respectively by Eqs.(1.4) - (1.6),can be still imposed simultaneously , even when both thecosmological scalar and tensor perturbations are present,as long as the GWs can be approximated as the high-frequency GWs. However, by imposing only the spatialgauge (1.4), the linearized Einstein field equations (2.41)are explicitly given in Appendix B. If χ µν is decomposedinto two parts, χ µν = χ (0) µν + (cid:15) c χ (1) µν + O (cid:0) (cid:15) c (cid:1) , (5.3)the field equations for χ (1) µν are given explicitly in Ap-pendix C.As an application of our general formulas, developedin Secs. II and III, in Sec. IV we studied the GWs byusing the geometrical optics approximation, χ αβ = e αβ A e iϕ/(cid:15) , (5.4)where e αβ represents the polarization tensor, A and ϕ denote, respectively, the amplitude and phase of theGWs. We showed explicitly that even when both thecosmological scalar and tensor perturbations are present, such GWs are still propagating along null geodesics, andthe current of gravitons moving along the null geodesicsis conserved, and the polarization tensor is parallel-transported , i.e., k λ ∇ λ k µ = 0 , k λ ∇ λ e αβ = 0 , ∇ λ J λ = 0 , (5.5)where k µ ≡ ∇ µ ϕ, J µ ≡ A k µ . In fact, these are truefor any curved background, provided that: (a) the GWscan be considered as high-frequency GWs; and (b) thegeometrical optics approximation are valid .With these remarkable features, we calculated the ef-fects of the cosmological scalar and tensor perturbationson the amplitude A and phase ϕ , given by Eqs.(4.40),(4.47) and (4.48). Restricting to GWs produced by abinary system, the effects of the cosmological perturba-tions, both scalar and tensor, on the luminosity distanceand the chirp mass are given explicitly by Eq.(4.51),which represent a natural generalization of the resultsobtained in [57, 59, 60] to the case in which the cosmo-logical tensor perturbations are also present.The applications of our general formulas developed inthis paper to other studies are immediate, including thegravitational analogue of the electromagnetic Faraday ro-tations [83, 84, 92, 93], and their detections by the space-and ground-based detectors. We wish to return to theseimportant issues in other occasions soon.It would be also very important to extend such stud-ies to include the relations between the GWs and theirsources, high-order corrections to the geometrical opticsapproximations, and more interesting the non-high fre-quency GWs.4 Acknowledgments
We would like very much to thank David Wand andWen Zhao for valuable discussions. This work waspartially supported by the National Key Research andDevelopment Program of China under the Grant No.2020YFC2201503, the National Natural Science Founda-tion of China under the grant Nos. 11675143, 11675145,11705053, 11975203 and 12035005, the Zhejiang Provin-cial Natural Science Foundation of China under GrantNos. LR21A050001, LY20A050002, and the Fundamen-tal Research Funds for the Provincial Universities of Zhe-jiang in China under Grant No. RF-A2019015. Thework of S.M. was supported in part by Japan Societyfor the Promotion of Science Grants-in-Aid for Scien-tific Research No. 17H02890, No. 17H06359, and byWorld Premier International Research Center Initiative,MEXT, Japan. J.F. and B.-W.L. acknowledge the sup-port from Baylor University through the Baylor Univer-sity Physics graduate program.
Appendix A: Decompositions of cosmologicalperturbations and gauge choice
Following [87, 88], the linear perturbations ˆ γ µν can bedecomposed into scalar, vector and tensor modes, andgiven explicitly by Eq.(2.44).The energy-momentum tensor T µν (0) of a fluid takesthe form [87], T µν (0) = ( ρ + p ) u µ u ν + pδ µν + π µν , (A.1)where u µ is the 4-velocity of the fluid, ρ and p are itsenergy density and isotropic pressure, respectively, and π µν is the anisotropic stress tensor, which has only spatialcomponents, i.e., π µ = 0. Setting ρ = ¯ ρ + (cid:15) c δρ, p = ¯ p + (cid:15) c δp,u µ = ¯ u µ + (cid:15) c δu µ , (A.2)where ¯ u µ = a − δ µη is the 4-velocity of the fluid of thehomogenous and isotropic universe, and ¯ ρ and ¯ p are itsenergy density and isotropic pressure, respectively, wefind that δu µ can be decomposed as δu µ = 1 a (cid:0) − φ, ∂ i v + v i (cid:1) , (A.3)where ∂ i v i = 0. Then, from u µ ≡ γ µν u ν = ¯ u µ + (cid:15) c δu µ ,we find that δu µ = a ( − φ, ∂ i v + ∂ i B + v i − S i ) , (A.4)which leads to u µ u µ = − O (cid:0) (cid:15) c (cid:1) , as expected.On the other hand, setting π ji = (cid:15) c ˆ π ji , similar to ˆ γ µν ,the anisotropic stress tensor ˆ π ji can be decomposed intoscalar, vector and tensor modes,ˆ π ji = (cid:18) ∂ j ∂ i − δ ji ∂ (cid:19) Π + 12 (cid:0) ∂ i Π j + ∂ j Π i (cid:1) + Π j i , (A.5)where ∂ i Π i = 0 = Π i i , ∂ j Π j i = 0, Π i ≡ δ ik Π k , Π i j ≡ δ ik Π kj , ∂ ≡ ∂ i ∂ i , etc. Then, we find that T = − ¯ ρ − (cid:15) c δρ,T i (0) = (cid:15) c (¯ ρ + ¯ p ) [ ∂ i ( v + B ) + v i − S i ] ,T i = − (cid:15) c (¯ ρ + ¯ p ) (cid:0) ∂ i v + v i (cid:1) ,T i j (0) = ¯ pδ ij + (cid:15) c (cid:0) δpδ ij + ˆ π ij (cid:1) . (A.6) A. Gauge Transformations of CosmologicalPerturbations
Considering the gauge transformations,˜ η = η + (cid:15) c ζ , ˜ x i = x i + (cid:15) c (cid:0) ∂ i ζ + ζ i (cid:1) , (A.7)where ∂ i ζ i = 0, we find that˜ φ = φ − H ζ − ζ (cid:48) , ˜ ψ = ψ + H ζ , ˜ B = B + ζ − ζ (cid:48) , ˜ E = E − ζ, ˜ δρ = δρ − ζ ¯ ρ (cid:48) , ˜ δp = δp − ζ ¯ p (cid:48) , ˜ v = v + ζ (cid:48) , (A.8)˜ F i = F i − ζ i , ˜ S i = S i + ζ (cid:48) i , ˜ v i = v i + ζ i (cid:48) , (A.9)˜ H ij = H ij , ˜ π ij = π ij , (A.10)where H ≡ a (cid:48) /a with a (cid:48) ≡ da/dη . From the above gaugetransformations we can see that the following quantitiesare gauge-invariant,Φ ≡ φ + H ( B − E (cid:48) ) + ( B − E (cid:48) ) (cid:48) , Ψ ≡ ψ − H ( B − E (cid:48) ) , Φ i ≡ S i + F (cid:48) i . (A.11)On the other hand, if we choose ζ = E, ζ = E (cid:48) − B and ζ i = F i , we have˜ B = ˜ E = 0 , ˜ F i = 0 , (A.12)in which the gauge is completely fixed. This is oftenreferred to as the Newtonian gauge. Then, we are leftwith six scalars, ( φ, ψ, v, δρ, δp, Π), two vectors, ( S i , v i ),and two tensors, ( H ij , Π ij ). However, the vector partdecreases rapidly with the expansion of the universe, sowe can safely set them to zero [87, 88], S i = F i = v i = Π i = 0 . (A.13)Then, for the scalar perturbations, there are six-independent equations, given, respectively, by [87], ψ (cid:48)(cid:48) + 2 H ψ (cid:48) + H φ (cid:48) + (cid:0) H (cid:48) + H (cid:1) φ = 4 πGa (cid:18) δp + 23 ∇ Π (cid:19) , (A.14)5 ψ − φ = 8 πGa Π , (A.15)3 H ( ψ (cid:48) + H φ ) − ∇ ψ = − πGa δρ, (A.16) ψ (cid:48) + H φ = − πGa (¯ ρ + ¯ p ) v, (A.17) δρ (cid:48) + 3 H ( δρ + δp ) = (¯ ρ + ¯ p ) (cid:0) ψ (cid:48) − ∇ v (cid:1) , (A.18)[(¯ ρ + ¯ p ) v ] (cid:48) + δp + 23 ∇ Π= − (¯ ρ + ¯ p ) ( φ + 4 H v ) . (A.19)Note that Eqs.(A.14) and (A.15) are obtained from thelinearized (i, j)-components of the Einstein field equa-tions, and Eqs.(A.16) and (A.17) are the energy and mo-mentum constraints, while Eqs.(A.18) and (A.19) are ob-tained from the conservation of the energy-momentumtensor.For the tensor perturbations, we have H (cid:48)(cid:48) ij + 2 H H (cid:48) ij − ∇ H ij = 16 πGa Π ij , (A.20)which is obtained from the equations δG (0) ij = κδT (0) ij .It must be noted that in writing the linearized fieldequations, (A.14) - (A.20), we had implicitly assumedthat the quadratic terms (cid:15) c ˆ R (2) µν (ˆ γ ) (cid:39) O (ˆ γ ) (cid:28)
1, whichis equivalent to ˆ γ (cid:28) , (A.21) where ˆ R (2) µν (ˆ γ ) is given by Eq.(2.11) with the replacement( h µν , ∇ α ) → (cid:0) ˆ γ µν , ¯ ∇ α (cid:1) . Otherwise, these quadraticterms cannot be neglected from the Einstein field equa-tions for the background spacetimes,¯ G µν (¯ γ ) + (cid:15) c ˆ G µν (ˆ γ ) + (cid:15) c ˆ G (2) µν (ˆ γ ) = κT (0) µν , (A.22)where ¯ G µν (¯ γ ) (cid:39) O (1) , (cid:15) c ˆ G µν (ˆ γ ) (cid:39) O (ˆ γ/(cid:15) c ) ,(cid:15) c ˆ G (2) µν (ˆ γ ) (cid:39) O (ˆ γ ) , (A.23)as can be seen from Eq.(2.17). Appendix B: Field Equations for χ ij In this Appendix, we shall calculate all the componentsof the quantities appearing in the field equations (3.22)for χ αβ , by imposing only the spatial gauge, χ µ = 0 . In particular, to calculate the non-vanishing componentsof the tensor G αβ , we first note that χ ij ≡ γ iµ γ jν χ µν = γ ik γ jl χ kl = 1 a (cid:110) δ ik δ jl + (cid:15) c (cid:2) ψδ ik δ jl − (cid:0) δ ik H jl + δ jl H ik (cid:1)(cid:3) (cid:111) ˆ χ kl ,γ ij χ ij = ˆ χ + (cid:15) c (cid:0) ψ ˆ χ − H kl ˆ χ kl (cid:1) , χ ≡ γ µν χ µν = γ ij χ ij = γ ij χ ij ,γ ij χ ik ˆ π jk = (cid:104) ˆ π kl + (cid:15) c (cid:0) ψ ˆ π kl − ˆ π km H ml (cid:1) (cid:105) ˆ χ kl ,χ γσ T γσ (0) − χT (0) = 12 (¯ ρ − ¯ p ) ˆ χ + 12 (cid:15) c (cid:104) (¯ ρ − ¯ p ) (cid:0) ψ ˆ χ − H kl ˆ χ kl (cid:1) + ( δρ − δp ) ˆ χ + 2ˆ π kl ˆ χ kl (cid:105) , (B.1)where ˆ χ ≡ δ ij ˆ χ ij , χ ij ≡ a ˆ χ ij , ˆ π ij ≡ δ ik ˆ π kj , etc. Then, we find that the non-vanishing (independent) components ofthe Riemann tensor, R µναβ (0) = ¯ R µναβ + (cid:15) c ˆ R µναβ , (B.2)are given, respectively, by¯ R i j = a (cid:18) H − a (cid:48)(cid:48) a (cid:19) δ ij , ¯ R minj = a H ( δ ij δ mn − δ in δ mj ) , (B.3)andˆ R i j = a (cid:40) φ ,ij + H φ (cid:48) δ ij + (cid:34) ( ψ (cid:48)(cid:48) + H ψ (cid:48) ) + 2 (cid:18) a (cid:48)(cid:48) a − H (cid:19) ψ (cid:35) δ ij − (cid:34) (cid:0) H ij (cid:48)(cid:48) + H H ij (cid:48) (cid:1) + 2 (cid:18) a (cid:48)(cid:48) a − H (cid:19) H ij (cid:35)(cid:41) , ˆ R ijk = a (cid:104) H ( φ ,j δ ik − φ ,k δ ij ) + (cid:0) ψ (cid:48) ,j δ ik − ψ (cid:48) ,k δ ij (cid:1) + 12 (cid:0) H (cid:48) ij,k − H (cid:48) ik,j (cid:1) (cid:105) , ˆ R ijkl = − a H φ ( δ ik δ jl − δ il δ jk ) − a (cid:104) ( δ jk ψ ,il + δ il ψ ,jk − δ ik ψ ,jl − δ jl ψ ,ik ) + 2 H ( ψ (cid:48) + 2 H ψ ) ( δ ik δ jl − δ il δ jk ) (cid:105) + 12 a (cid:40) ( H jk,il + H il,jk − H ik,jl − H jl,ik ) − H (cid:104) δ il (cid:0) H (cid:48) jk + 2 H H jk (cid:1) + δ jk ( H (cid:48) il + 2 H H il )6 − δ jl ( H (cid:48) ik + 2 H H ik ) − δ ik (cid:0) H (cid:48) jl + 2 H H jl (cid:1) (cid:105)(cid:41) . (B.4)Hence, we find that2 R i j (0) χ ij = 2 (cid:18) a (cid:48)(cid:48) a − H (cid:19) ˆ χ + (cid:15) c (cid:26) (cid:0) ∂ i ∂ j φ (cid:1) ˆ χ ij + 2 H φ (cid:48) ˆ χ + 2 (cid:0) ψ (cid:48)(cid:48) + H ψ (cid:48) (cid:1) ˆ χ − (cid:18) a (cid:48)(cid:48) a − H (cid:19) ψ ˆ χ − (cid:0) H ij (cid:48)(cid:48) + H H ij (cid:48) (cid:1) ˆ χ ij + 2 (cid:18) a (cid:48)(cid:48) a − H (cid:19) H ij ˆ χ ij (cid:27) , R jik (0) χ jk = 2 (cid:15) c (cid:40) H (cid:2) ( ∂ i φ ) ˆ χ − (cid:0) ∂ k φ (cid:1) ˆ χ ik (cid:3) + ( ∂ i ψ (cid:48) ) ˆ χ − (cid:0) ∂ k ψ (cid:48) (cid:1) ˆ χ ik + 12 (cid:104)(cid:16) ∂ k H ji (cid:48) (cid:17) − (cid:16) ∂ i H jk (cid:48) (cid:17)(cid:105) ˆ χ jk (cid:41) , R ikjl (0) χ kl = 2 H (cid:0) δ ij ˆ χ − ˆ χ ij (cid:1) + (cid:15) c (cid:26) H φ ( ˆ χ ij − ˆ χδ ij ) + 4 H ψ (cid:48) ( ˆ χ ij − ˆ χδ ij )+2 (cid:104) ( ∂ i ∂ j ψ ) ˆ χ + (cid:0) ∂ k ∂ l ψ (cid:1) ˆ χ kl δ ij − (cid:0) ∂ k ∂ i ψ (cid:1) ˆ χ jk − (cid:0) ∂ k ∂ j ψ (cid:1) ˆ χ ik (cid:105) − H H kl ˆ χ kl δ ij + 2 H H ij ˆ χ − H H k ( i (cid:48) ˆ χ j ) k + H H ij (cid:48) ˆ χ + H H kl (cid:48) ˆ χ kl δ ij + (cid:16) ∂ ( i ∂ l H kj ) − ∂ k ∂ l H ij − ∂ i ∂ j H kl (cid:17) ˆ χ kl (cid:27) . (B.5)On the other hand, similar to the above expression, writing (cid:3) χ αβ in the form, (cid:3) χ αβ ≡ ¯ (cid:3) χ αβ + (cid:15) c ˆ (cid:3) χ αβ , (B.6)we find they are given, respectively, by¯ (cid:3) χ = 2 H ˆ χ, ¯ (cid:3) χ i = − H ∂ j ˆ χ ij , ¯ (cid:3) χ ij = − ˆ χ (cid:48)(cid:48) ij − H ˆ χ (cid:48) ij + ∂ ˆ χ ij + 2 H ˆ χ ij , (B.7)and ˆ (cid:3) χ = − H (cid:104) ψ (cid:48) − H ψ ) ˆ χ − (cid:16) H ij (cid:48) − H H ij (cid:17) ˆ χ ij (cid:105) , ˆ (cid:3) χ i = (cid:0) ∂ j φ (cid:48) (cid:1) ˆ χ ij + 2 (cid:0) ∂ j φ (cid:1) (cid:0) ˆ χ (cid:48) ij + H ˆ χ ij (cid:1) + (cid:0) ∂ j ψ (cid:48) (cid:1) ˆ χ ij + 2 ( ψ (cid:48) − H ψ ) ∂ j ˆ χ ij + 2 H (cid:2)(cid:0) ∂ j ψ (cid:1) ˆ χ ij − ( ∂ i ψ ) ˆ χ (cid:3) − (cid:16) H jk (cid:48) − H H jk (cid:17) ∂ k ˆ χ ij + H (cid:0) ∂ i H jk (cid:1) ˆ χ jk , ˆ (cid:3) χ ij = 2 φ ˆ χ (cid:48)(cid:48) ij + ( φ (cid:48) + 4 H φ ) ˆ χ (cid:48) ij + (cid:0) ∂ k φ (cid:1) ∂ k ˆ χ ij − H φ ˆ χ ij +2 ψ∂ ˆ χ ij + 4 ∂ ( i ψ∂ k ˆ χ j ) k + 3 (cid:0) ∂ k ψ (cid:1) ∂ k ˆ χ ij − (cid:0) ∂ k ψ (cid:1) ∂ ( i ˆ χ j ) k + 2 ∂ k ∂ ( i ψ ˆ χ j ) k + 2 (cid:0) ∂ ψ (cid:1) ˆ χ ij − ∂ ( i ∂ k ψ ˆ χ j ) k − ψ (cid:48) ˆ χ (cid:48) ij − ψ (cid:48)(cid:48) + 4 H ψ (cid:48) ) ˆ χ ij − H kl ∂ k ∂ l ˆ χ ij − ∂ l H k ( i ∂ l ˆ χ j ) k − ∂ ( i H kl ∂ l ˆ χ j ) k + 2 ∂ k H ( il ∂ l ˆ χ j ) k + 2 H k ( i (cid:48) ˆ χ (cid:48) j ) k + H k ( i (cid:48)(cid:48) ˆ χ j ) k +4 H H k ( i (cid:48) ˆ χ j ) k − ∂ H k ( i ˆ χ j ) k , (B.8)where 2 ∂ k ∂ ( i ψ ˆ χ j ) k ≡ (cid:0) ∂ k ∂ i ψ (cid:1) ˆ χ jk + (cid:0) ∂ k ∂ j ψ (cid:1) ˆ χ ik , that is, the partial derivative acts only to the first function. Thesame is true for other terms, for example, 2 ∂ l H k ( i ∂ l ˆ χ j ) k ≡ (cid:0) ∂ l H ki (cid:1) ∂ l ˆ χ jk + (cid:0) ∂ l H kj (cid:1) ∂ l ˆ χ ik .On the other hand, defining G (1) αβ ≡ γ αβ ∇ γ ∇ δ χ γδ , (B.9)we find that G (1)00 = −G (1)0 − (cid:15) c (cid:16) φ G (1)0 + G (1)1 (cid:17) , G (1)0 i = G i = 0 , G (1) ij = δ ij G (1)0 + (cid:15) c (cid:104) δ ij (cid:16) G (1)1 − ψ G (1)0 (cid:17) + H ij G (1)0 (cid:105) , (B.10)where G (1)0 ≡ H ˆ χ (cid:48) + (cid:18) a (cid:48)(cid:48) a + H (cid:19) ˆ χ + ∂ i ∂ j ˆ χ ij , G (1)1 ≡ − H φ ˆ χ (cid:48) − (cid:20) (cid:18) a (cid:48)(cid:48) a + H (cid:19) φ + H φ (cid:48) (cid:21) ˆ χ + 2 (cid:0) ∂ i φ (cid:1) (cid:0) ∂ j ˆ χ ij (cid:1) + (cid:0) ∂ i ∂ j φ (cid:1) ˆ χ ij − ( ψ (cid:48) − H ψ ) ˆ χ (cid:48) − (cid:20) ψ (cid:48)(cid:48) + 3 H ψ (cid:48) − ∂ ψ − (cid:18) a (cid:48)(cid:48) a + H (cid:19) ψ (cid:21) ˆ χ + ( ∂ i ψ ) ∂ i ˆ χ + 4 ψ∂ i ∂ j ˆ χ ij − (cid:0) ∂ i ∂ j ψ (cid:1) ˆ χ ij + 12 (cid:40) (cid:16) H ij (cid:48) − H H ij (cid:17) ˆ χ (cid:48) ij + (cid:20) H ij (cid:48)(cid:48) − (cid:18) a (cid:48)(cid:48) a + H (cid:19) H ij − ∂ H ij (cid:21) ˆ χ ij − H ik ∂ k ∂ j ˆ χ ij − (cid:0) ∂ k H ij (cid:1) (cid:0) ∂ k ˆ χ ij (cid:1) − (cid:0) ∂ i H jk (cid:1) ( ∂ k ˆ χ ij ) (cid:41) . (B.11)On the other hand, defining G (2) αβ ≡ ∇ α ∇ δ χ βδ , (B.12)we find that it has the following non-vanishing components, G (2)00 = − a (cid:48) a ˆ χ (cid:48) − (cid:18) a (cid:48)(cid:48) a − H (cid:19) ˆ χ + (cid:15) c (cid:34) H φ (cid:48) ˆ χ − (cid:0) ∂ i φ (cid:1) ∂ k ˆ χ ik + ( ψ (cid:48) − H ψ ) ˆ χ (cid:48) + ( ψ (cid:48)(cid:48) − H ψ (cid:48) ) ˆ χ − (cid:18) a (cid:48)(cid:48) a − H (cid:19) ψ ˆ χ − (cid:16) H ij (cid:48) − H H ij (cid:17) ˆ χ (cid:48) ij − (cid:16) H ij (cid:48)(cid:48) − H H ij (cid:48) (cid:17) ˆ χ ij + (cid:18) a (cid:48)(cid:48) a − H (cid:19) H ij ˆ χ ij (cid:35) , G (2)0 i = ∂ k ˆ χ (cid:48) ik − H ∂ k ˆ χ ik + (cid:15) c (cid:34) (cid:0) ∂ j φ (cid:1) ˆ χ (cid:48) ij + (cid:0) ∂ j φ (cid:48) − H ∂ j φ (cid:1) ˆ χ ij + H ( ∂ i φ ) ˆ χ +2 ψ∂ k ˆ χ (cid:48) ik − (cid:0) ∂ k ψ (cid:1) ˆ χ (cid:48) ik + (3 ψ (cid:48) − H ψ ) ∂ k ˆ χ ik + ( ∂ i ψ ) ˆ χ (cid:48) − (cid:0) ∂ k ψ (cid:48) − H ∂ k ψ (cid:1) ˆ χ ik + ( ∂ i ψ (cid:48) − H ∂ i ψ ) ˆ χ − H jk ∂ k ˆ χ (cid:48) ij − (cid:16) H jk (cid:48) − H H jk (cid:17) ∂ k ˆ χ ij − H ji (cid:48) ∂ k ˆ χ jk − H jk,i ˆ χ (cid:48) jk − (cid:16) H jk (cid:48) − H H jk (cid:17) ,i ˆ χ jk (cid:35) , G (2) i = −H (cid:0) ∂ k ˆ χ ik + ∂ i ˆ χ (cid:1) + (cid:15) c (cid:34) H ( ∂ i φ ) ˆ χ − H (cid:0) ∂ k φ (cid:1) ˆ χ ik + ( ψ (cid:48) − H ψ ) (cid:0) ∂ k ˆ χ ik + ∂ i ˆ χ (cid:1) + H (cid:0) ∂ k ψ (cid:1) ˆ χ ik + ( ψ (cid:48) − H ψ ) ,i ˆ χ − (cid:16) H jk (cid:48) − H H jk (cid:17) ∂ i ˆ χ jk − H ji (cid:48) ∂ k ˆ χ jk + H H jk ∂ k ˆ χ ij − (cid:16) H jk (cid:48) − H H jk (cid:17) ,i ˆ χ jk (cid:35) , G (2) ij = ∂ i ∂ k ˆ χ jk + H ˆ χδ ij + (cid:15) c (cid:34) (cid:0) ∂ k φ (cid:1) ∂ i ˆ χ jk + (cid:0) ∂ i ∂ k φ (cid:1) ˆ χ jk − H φ ˆ χδ ij − ∂ i (cid:2)(cid:0) ∂ k ψ (cid:1) ˆ χ jk (cid:3) + ∂ i [( ∂ j ψ ) ˆ χ ] + 2 ∂ i ( ψ∂ k ˆ χ jk ) + ( ∂ i ψ ) ∂ k ˆ χ jk + ( ∂ j ψ ) ∂ k ˆ χ ik − (cid:0) ∂ k ψ (cid:1) ∂ l ˆ χ kl δ ij − H ψ (cid:48) ˆ χδ ij − ∂ i ( H kl ∂ l ˆ χ jk ) − ∂ i (cid:2)(cid:0) ∂ j H kl (cid:1) ˆ χ kl (cid:3) + 12 (cid:0) ∂ k H ij − H ki,j − H kj,i (cid:1) ∂ l ˆ χ kl + 12 H (cid:16) H kl (cid:48) − H H kl (cid:17) ˆ χ kl δ ij + 12 H (cid:0) H (cid:48) ij + 2 H H ij (cid:1) ˆ χ (cid:35) . (B.13)Note that G (2) αβ is not symmetric, G (2) αβ (cid:54) = G (2) βα , as can be seen from its definition given by Eq.(B.12).Finally, defining G αβ as G αβ ≡ (cid:3) χ αβ + γ αβ ∇ γ ∇ δ χ γδ − ∇ α ∇ δ χ βδ − ∇ β ∇ δ χ αδ + 2 R αγβσ (0) χ γσ , (B.14)we find that its non-vanishing components are given by G = H ˆ χ (cid:48) − (cid:18) a (cid:48)(cid:48) a + H (cid:19) ˆ χ − ∂ i ∂ j ˆ χ ij + (cid:15) c (cid:40) H φ (cid:48) ˆ χ + (cid:0) ∂ i ∂ j φ (cid:1) ˆ χ ij − φ (cid:0) ∂ i ∂ j ˆ χ ij (cid:1) + (cid:20) ψ (cid:48)(cid:48) + 7 H ψ (cid:48) − (cid:18) a (cid:48)(cid:48) a + H (cid:19) ψ − ∂ ψ (cid:21) ˆ χ − (cid:0) ψ (cid:48) − H ψ (cid:1) ˆ χ (cid:48) (cid:0) ∂ i ∂ j ψ (cid:1) ˆ χ ij − ψ (cid:0) ∂ i ∂ j ˆ χ ij (cid:1) − (cid:0) ∂ k ψ (cid:1) ∂ k ˆ χ − (cid:20) (cid:16) H ij (cid:48)(cid:48) − ∂ H ij (cid:17) + 4 H H ij (cid:48) − (cid:18) a (cid:48)(cid:48) a + H (cid:19) H ij (cid:21) ˆ χ ij + 12 (cid:0) H ij (cid:48) − H H ij (cid:1) ˆ χ (cid:48) ij + 2 H ik ( ∂ k ∂ j ˆ χ ij ) + 12 ( ∂ k H ij ) ∂ k ˆ χ ij + ( ∂ i H jk ) ∂ k ˆ χ ij (cid:41) , (B.15) G i = H ∂ i ˆ χ − ∂ j ˆ χ (cid:48) ij + (cid:15) c (cid:40) (cid:0) ∂ j φ (cid:1) ˆ χ (cid:48) ij + 2 H (cid:0) ∂ j φ (cid:1) ˆ χ ij − ψ (cid:48) ( ∂ j ˆ χ ij ) + 2 H ( ∂ i ψ ) ˆ χ − ψ ( ∂ j ˆ χ (cid:48) ij ) + ( ∂ j ψ ) ˆ χ (cid:48) ij − ( ∂ i ψ ) ˆ χ (cid:48) − ( ψ (cid:48) − H ψ ) ∂ i ˆ χ −H ( ∂ i H jk ) ˆ χ jk + (cid:16) ∂ k H ji (cid:48) (cid:17) ˆ χ jk + H ji (cid:48) (cid:0) ∂ k ˆ χ jk (cid:1) + H jk (cid:0) ∂ k ˆ χ (cid:48) ij (cid:1) + 12 ( ∂ i H jk ) ˆ χ (cid:48) jk + 12 H jk (cid:48) ( ∂ i ˆ χ jk ) − H H jk ( ∂ i ˆ χ jk ) (cid:41) , (B.16) G ij = − ˆ χ (cid:48)(cid:48) ij + ∂ ˆ χ ij − H ˆ χ (cid:48) ij + H δ ij ˆ χ (cid:48) + δ ij (cid:18) a (cid:48)(cid:48) a + H (cid:19) ˆ χ + ∂ k ∂ l ˆ χ kl δ ij − ∂ i ∂ k ˆ χ jk − ∂ j ∂ k ˆ χ ik + (cid:15) c (cid:26) φ ˆ χ (cid:48)(cid:48) ij + (cid:0) φ (cid:48) + 4 H φ (cid:1) ˆ χ (cid:48) ij − (cid:20) H φ (cid:48) + 2 (cid:18) a (cid:48)(cid:48) a + H (cid:19) φ (cid:21) δ ij ˆ χ − H φδ ij ˆ χ (cid:48) − ( ∂ j ∂ k φ ) ˆ χ ik − ( ∂ i ∂ k φ ) ˆ χ jk + ( ∂ k φ ) ∂ k ˆ χ ij + 2( ∂ k φ ) ∂ l ˆ χ kl δ ij − ( ∂ k φ ) ∂ i ˆ χ kj − ( ∂ k φ ) ∂ j ˆ χ ik + ( ∂ k ∂ l φ ) ˆ χ kl δ ij +2 (cid:0) ∂ ψ − ψ (cid:48)(cid:48) − H ψ (cid:48) (cid:1) ˆ χ ij + (cid:0) ∂ ψ − ψ (cid:48)(cid:48) − H ψ (cid:48) (cid:1) ˆ χδ ij − ψ (cid:48) (cid:0) ˆ χ (cid:48) ij + ˆ χ (cid:48) δ ij (cid:1) + 2 ψ∂ ˆ χ ij + ∂ k ψ (cid:0) ∂ k ˆ χ ij + ∂ k ˆ χδ ij − ∂ i ˆ χ jk − ∂ j ˆ χ ik + 2 ∂ l ˆ χ kl δ ij (cid:1) + 2 ψ∂ k ∂ l ˆ χ kl δ ij + ( ∂ k ∂ l ψ ) ˆ χ kl δ ij − ∂ ( i ψ∂ k ˆ χ j ) k − ∂ ( i ∂ k ψ ˆ χ j ) k − ψ∂ ( i ∂ k ˆ χ j ) k − ∂ ( i ψ∂ j ) ˆ χ + 12 (cid:0) H kl (cid:48) − H H kl (cid:1) ˆ χ (cid:48) kl δ ij + (cid:20) H kl (cid:48)(cid:48) − (cid:18) a (cid:48)(cid:48) a + H (cid:19) H kl (cid:21) ˆ χ kl δ ij + H k ( i (cid:48)(cid:48) ˆ χ j ) k + 2 H H k ( i (cid:48) ˆ χ j ) k +2 H k ( i (cid:48) ˆ χ (cid:48) j ) k + (cid:20) H ˆ χ (cid:48) + (cid:18) a (cid:48)(cid:48) a + H (cid:19) ˆ χ + ∂ k ∂ l ˆ χ kl (cid:21) H ij − H jk ∂ k ∂ m ˆ χ lm δ ij − ∂ H kl ˆ χ kl δ ij − ∂ m H kl ∂ m ˆ χ kl δ ij − ∂ l H k ( i ∂ l ˆ χ j ) k + 2 ∂ k H l ( i ∂ l ˆ χ j ) k − ∂ H k ( i ˆ χ j ) k + ∂ ( i H kl ∂ j ) ˆ χ kl + ∂ ( i H kj ) ∂ l ˆ χ kl + 2 ∂ ( i ∂ l H kj ) ˆ χ kl − ∂ k ∂ l H ij ˆ χ kl − ∂ k H ij ∂ l ˆ χ kl + 2 H kl ∂ ( i ∂ l ˆ χ j ) k − ∂ k H ml ∂ m ˆ χ kl δ ij − H kl ∂ k ∂ l ˆ χ ij (cid:27) . (B.17) Appendix C: Field Equations to the First-order of (cid:15) c Following Eq.(4.18), we write ˆ χ αβ in the form,ˆ χ αβ (cid:39) ˆ χ (0) αβ + (cid:15) c ˆ χ (1) αβ + O (cid:0) (cid:15) c (cid:1) , (C.1)where to the zeroth-order, the TT gaugeˆ χ (0)0 β = 0 , ˆ χ (0) = 0 , ∂ i ˆ χ (0) ij = 0 , (C.2) will be chosen. But, to the first order, we shall not imposethe traceless and Lorentz gauge conditions. The onlygauge that now we choose isˆ χ (1)0 β = 0 . (C.3)With this gauge choice, to the first-order of (cid:15) c , thenon-vanishing components of the tensor G αβ given byEqs.(B.15)-(B.17) yield, G (1)00 = H ˆ χ (cid:48) (1) − (cid:18) a (cid:48)(cid:48) a + H (cid:19) ˆ χ (1) − ∂ i ∂ j ˆ χ (1) ij + (cid:2) ∂ i ∂ j ( φ + ψ ) (cid:3) ˆ χ (1) ij + ˆ G (1)00 , (C.4) G (1)0 i = H ∂ i ˆ χ (1) − ∂ j ˆ χ (cid:48) (1) ij + ˆ G (1)0 i , (C.5) G (1) ij = − ˆ χ (cid:48)(cid:48) (1) ij + ∂ ˆ χ (1) ij − H ˆ χ (cid:48) (1) ij + H δ ij ˆ χ (cid:48) (1) + δ ij (cid:18) a (cid:48)(cid:48) a + H (cid:19) ˆ χ (1) + ∂ k ∂ l ˆ χ (1) kl δ ij − ∂ i ∂ k ˆ χ (1) jk − ∂ j ∂ k ˆ χ (1) ik + ˆ G (1) ij , (C.6)9whereˆ G (1)00 = − (cid:20) (cid:16) H ij (cid:48)(cid:48) − ∂ H ij (cid:17) + 4 H H ij (cid:48) − (cid:18) a (cid:48)(cid:48) a + H (cid:19) H ij (cid:21) ˆ χ (0) ij + 12 (cid:16) H ij (cid:48) − H H ij (cid:17) ˆ χ (cid:48) (0) ij + 12 (cid:0) ∂ k H ij (cid:1) ∂ k ˆ χ (0) ij + (cid:0) ∂ i H jk (cid:1) ∂ k ˆ χ (0) ij , (C.7)ˆ G (1)0 i = + (cid:0) ∂ j φ (cid:1) ˆ χ (cid:48) (0) ij + 2 H (cid:0) ∂ j φ (cid:1) ˆ χ (0) ij + (cid:0) ∂ j ψ (cid:1) ˆ χ (cid:48) (0) ij −H (cid:0) ∂ i H jk (cid:1) ˆ χ (0) jk + (cid:16) ∂ k H (cid:48) ji (cid:17) ˆ χ (0) jk + H jk (cid:16) ∂ k ˆ χ (cid:48) (0) ij (cid:17) + 12 H (cid:48) jk (cid:16) ∂ i ˆ χ (0) jk (cid:17) − H H jk (cid:16) ∂ i ˆ χ (0) jk (cid:17) , (C.8)ˆ G (1) ij = +2 φ ˆ χ (cid:48)(cid:48) (0) ij + ( φ (cid:48) + 4 H φ ) ˆ χ (cid:48) (0) ij − (cid:0) ∂ j ∂ k φ (cid:1) ˆ χ (0) ik − (cid:0) ∂ i ∂ k φ (cid:1) ˆ χ (0) jk + (cid:0) ∂ k φ (cid:1) ∂ k ˆ χ (0) ij − (cid:0) ∂ k φ (cid:1) ∂ i ˆ χ (0) kj − (cid:0) ∂ k φ (cid:1) ∂ j ˆ χ (0) ik + (cid:0) ∂ k ∂ l φ (cid:1) ˆ χ (0) kl δ ij +2 (cid:0) ∂ ψ − ψ (cid:48)(cid:48) − H ψ (cid:48) (cid:1) ˆ χ (0) ij − ψ (cid:48) ˆ χ (cid:48) (0) ij + 2 ψ∂ ˆ χ (0) ij + ∂ k ψ (cid:16) ∂ k ˆ χ (0) ij − ∂ i ˆ χ (0) jk − ∂ j ˆ χ (0) ik (cid:17) + (cid:0) ∂ k ∂ l ψ (cid:1) ˆ χ (0) kl δ ij + 12 (cid:16) H kl (cid:48) − H H kl (cid:17) ˆ χ (cid:48) (0) kl δ ij + (cid:20) H kl (cid:48)(cid:48) − (cid:18) a (cid:48)(cid:48) a + H (cid:19) H kl (cid:21) ˆ χ (0) kl δ ij + H k ( i (cid:48)(cid:48) ˆ χ (0) j ) k + 2 H H k ( i (cid:48) ˆ χ (0) j ) k +2 H k ( i (cid:48) ˆ χ (cid:48) (0) j ) k − ∂ H kl ˆ χ (0) kl δ ij − ∂ m H kl ∂ m ˆ χ (0) kl δ ij − ∂ l H k ( i ∂ l ˆ χ (0) j ) k + 2 ∂ k H l ( i ∂ l ˆ χ (0) j ) k − ∂ H k ( i ˆ χ (0) j ) k + ∂ ( i H kl ∂ j ) ˆ χ (0) kl + 2 ∂ ( i ∂ l H kj ) ˆ χ (0) kl − ∂ k ∂ l H ij ˆ χ (0) kl + 2 H kl ∂ ( i ∂ l ˆ χ (0) j ) k − ∂ k H ml ∂ m ˆ χ (0) kl δ ij . 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