Late time tails and nonlinear memories in asymptotically de Sitter spacetimes
LLate time tails and nonlinear memories in asymptotically deSitter spacetimes
Yi-Zen Chu , M. Afif Ismail , and Yen-Wei Liu Department of Physics, National Central University, Chungli 32001, Taiwan Center for High Energy and High Field Physics (CHiP),National Central University, Chungli 32001, Taiwan Department of Physics, National Tsing-Hua University, Hsinchu, Taiwan
Abstract
We study the propagation of a massless scalar wave in de Sitter spacetime perturbed by an arbitrarycentral mass. By focusing on the late time limit, this probes the portion of the scalar signal travelinginside the null cone. Unlike in asymptotically flat spacetimes, the scalar field amplitude does not decayback to zero but develops a spacetime constant shift – both at zeroth and first order in the centralmass. This indicates that massless scalar field propagation in asymptotically de Sitter spacetimesexhibits both linear and nonlinear tail-induced memories.
Non-linear perturbation theory about cosmological background is of physical interest. One of the rea-sons is, the asymptotic structure of cosmological spacetimes is quite different from that of flat back-ground. In particular, linear perturbation theory on a de Sitter background is well studied [7] [9]. Forinstance, Ashtekar, Bonga and Kesavan [8] found that adding a cosmological constant to the Einsteinequation introduces new features to the quadrupole formula of gravitational waves [8]. By integratingthe equations-of-motion for the Weyl tensor, Bieri, Garfinkle and Yau [10] found the lightcone part ofgravitational waves give rise to a memory effect similar to its Minkowski counterpart, except a redshift1 a r X i v : . [ g r- q c ] J a n actor. Others [2], [14], [13] have since confirmed and extended the results to more general Friedman-Robertson-Walker cosmologies. (Also, see [11], [12], [13] for studies of the asymptotic symmetriesof cosmological spacetimes; and [15] for memories in anti de Sitter spacetimes.) Being in de Sitterbackground, the gravitational waves do not just propagate strictly on the lightcone, but also inside thelightcone; this is known as the tail effect. One of us [3], [2] discovered that the gravitational wavetail also contributes to the memory effect. Now, about asymptotically flat spacetimes, it is known sinceChristodoulou [5] and Blanchet and Damour [6] that nonlinear corrections from General Relativity pro-duces additional memories that at times might be even larger than the linear ones. In this paper, weinitiate an examination of cosmological nonlinear memories by computing that of the massless scalarinteracting gravitationally with an arbitrary central mass. Furthermore, we will assume the role of anobserver approaching timelike infinity, to extract the tail-induced memories. As elucidated below, ouranalysis probes the scalar-graviton-scalar interaction, and is analogous to the de Sitter graviton -pointself-interactions that would encode any potential nonlinear gravitational wave memories.The late time behavior of radiative fields in asymptotically-flat spacetimes has been well studiedover the past decades. What is now know as “Price’s fall-off theorem" was first discovered by Price [16]:the spin-0 radiative field decays in time according to an inverse power-law, with a power determinedby the angular profile of the initial wave profile. In his work, Price is using the spherically symmet-ric Schwarzschild spacetime. The corresponding falloff properties of radiative fields in cosmologicalbackgrounds is less well understood, although there are a number of numerical studies, such as Brady,Chambers, Krivan and Laguna (BCKL) [18] and Brady, Chambers, Laarakkers and Poisson (BCLP) [19]on exact Schwarzschild-de Sitter and Reissner-Nordström-de Sitter backgrounds.In this paper, we want to investigate the simplest radiative field, a massless scalar field, in a per-turbed de Sitter spacetime by using Poisson’s approach to learn its late-time behavior [17] analytically.Given an initial profile for the scalar wave packet, we found a constant tail signal observed at late times.This effect can be interpreted as tail induced memory effect to the scalar field. This complements andextends the results of BCKL [18] and BCLP [19]. In §(2) we describe our setup; in §(3) we introduce afrequency space method to reproduce Poisson’s work [17] on the late time behavior of a scalar field inperturbed Minkowski background. Following which, in §(4) we work on the scalar field solution in thelate time regime of the perturbed de Sitter background.2 Setup
We consider a massless free scalar field Ψ , satisfying the equation (cid:3) x Ψ[ x ] = 0 , x µ = ( x , (cid:126)x ) , (2.0.1)where (cid:3) is the wave operator (cid:3) ≡ g µν ∇ µ ∇ ν . The geometry would take the following perturbed form , g µν = a ( η µν + χ µν ) , | χ µν | (cid:28) , (2.0.2) η µν ≡ diag [1 , − , − , − (2.0.3)where in the asymptotically Minkowski case x (cid:48) = t (cid:48) ∈ R , x = t ∈ R , and a = 1; (2.0.4)whereas in the asymptotically de Sitter case x (cid:48) = η (cid:48) , x = η, and a [ η ] = − Hη , (2.0.5)with these cosmological conformal times running over the negative real line, −∞ < η, η (cid:48) < , and H > is the Hubble constant.Suppose at some initial time x (cid:48) , the scalar field and its velocity are described by the followingmultipole expansion , Ψ[ (cid:126)x (cid:48) ] = (cid:88) (cid:96) (cid:48) ,m (cid:48) C m (cid:48) (cid:96) (cid:48) [ r (cid:48) ] Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48) ] , ˙Ψ[ (cid:126)x (cid:48) ] = (cid:88) (cid:96) (cid:48) ,m (cid:48) ˙ C m (cid:48) (cid:96) (cid:48) [ r (cid:48) ] Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48) ] , (2.0.6)where Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48) ] are spherical harmonics (which form basis orthonormal functions on the 2-sphere), and C m (cid:48) (cid:96) (cid:48) and ˙ C m (cid:48) (cid:96) (cid:48) [ r (cid:48) ] are arbitrary functions that are non-zero only within some finite radii. This regionwhere the initial conditions are non-zero is illustrated on the initial time hypersurface in Fig. (3.1). Wewill evolve this initial profile to the final time hypersurface x by using Kirchhoff representation of thescalar field, up to first order in the perturbation χ µν : Ψ[ x > x (cid:48) , (cid:126)x ] = (cid:90) R d (cid:126)x (cid:48) (cid:112) | h | (cid:98) n α (cid:48) (cid:0) G [ x, x (cid:48) ] ∂ α (cid:48) Ψ[ x (cid:48) ] − Ψ[ x (cid:48) ] ∂ α (cid:48) G [ x, x (cid:48) ] (cid:1) = (cid:90) R d (cid:126)x (cid:48) a [ η (cid:48) ] (cid:18) η ij χ ij − χ (cid:19) (cid:16) G [ x, x (cid:48) ] ˙Ψ[ x (cid:48) ] − Ψ[ x (cid:48) ] ∂ (cid:48) G [ x, x (cid:48) ] (cid:17) . (2.0.7) Greek indices µ, ν, ...., run from (time) to , while Latin ones i, j, ..., run over only the spatial values to We denote (cid:80) (cid:96),m as the sum over all multipole indices, namely (cid:80) ∞ (cid:96) =0 (cid:80) (cid:96)m = − (cid:96) , unless otherwise explained. | h | is the determinant of induced metric on the constant time hypersurface, which has (cid:98) n α as thenormal vector. h αβ d x α d x β = − a ( δ ij − χ ij ) d x i d x j (2.0.8) (cid:98) n α = a − δ α (cid:18) − χ + O (cid:2) χ (cid:3)(cid:19) (2.0.9)The retarded Green function G [ x, x (cid:48) ] describes the propagation of initial profile located at x (cid:48) to an ob-server at x (see Fig. (2)) that obeys the wave equation (cid:3) G [ x, x (cid:48) ] = δ (4) [ x − x (cid:48) ] (cid:112) g [ x ] g [ x (cid:48) ] . (2.0.10)In §[4], a scheme was devised to solve Green’s functions in a perturbed spacetime g µν = g µν + h µν ,in terms of the Green’s functions in the unperturbed one g µν . In our case, the massless scalar Green’sfunction reads G [ x, x (cid:48) ] = G [ x, x (cid:48) ] + δ G [ x, x (cid:48) ]; (2.0.11)where G solves the massless scalar wave equation with respect to g µν = a η µν , ∂ µ (cid:0) a η µν ∂ ν G (cid:1) a = δ (4) [ x − x (cid:48) ] a [ x ] a [ x (cid:48) ] ; (2.0.12)and, in turn, δ G [ x, x (cid:48) ] = − (cid:90) d x (cid:48)(cid:48) a [ η (cid:48)(cid:48) ] ∂ α (cid:48)(cid:48) G [ x, x (cid:48)(cid:48) ] ¯ χ α (cid:48)(cid:48) β (cid:48)(cid:48) ∂ β (cid:48)(cid:48) G [ x (cid:48)(cid:48) , x (cid:48) ]; (2.0.13)with all double-primed indices denoting evaluation with respect to the integration variable x (cid:48)(cid:48) ; and the‘trace-reversed’ perturbation is defined as ¯ χ µν ≡ χ µν − η µν η αβ χ αβ , (2.0.14)with its indices moved by the flat metric.The metric perturbation χ µν itself is sourced by the presence of an arbitrary quasi-static massdistribution consistent with energy-momentum conservation with respect to the background metric g µν = a η µν , namely T µν [ x (cid:48)(cid:48) ] = ρ [ (cid:126)x (cid:48)(cid:48) ] a δ µ δ ν ; (2.0.15)This allows us to re-express the first order Green’s function in eq. (2.0.13) as a Feynman diagram in Fig.(2) involving scalar-graviton-scalar interactions. In turn, this allows us to assert that the scalar memoryrevealed below is in fact of nonlinear character. 4igure 1: The Feynman diagram representing the massless scalar retarded Green’s function at first orderin the central mass. The dashed lines denote the zeroth order scalar Green’s functions; whereas thewavy line tied to the blob labeled ‘ M ’ describes the metric perturbation sourced by the static but oth-erwise arbitrary central mass distribution. From this depiction, we see that the scalar Green’s functionat first order in the central mass involves the scalar-graviton-scalar nonlinear interactions. (Drawn withJaxoDraw [22].)As an important aside: in a co-moving orthonormal frame, T (cid:98) (cid:98) = ρ/a and therefore the totalmass with respect to the background metric is M ≡ (cid:90) R T (cid:98) (cid:98) a d (cid:126)x (cid:48) = (cid:90) R ρ [ (cid:126)x (cid:48)(cid:48) ] d (cid:126)x (cid:48)(cid:48) . (2.0.16)All our major results below will be expressed in terms of this M .We will focus on the late time limit of the scalar field in both Minkowski and de Sitter back-grounds. This means the difference between observation time x and the initial time x (cid:48) obeys x − x (cid:48) (cid:29) r + r (cid:48) , where r ≡ | (cid:126)x | and r (cid:48) ≡ | (cid:126)x | are respectively the radial coordinates of the observer and an arbitrarypoint within the region where the initial scalar profile is non-trivial. Furthermore, in the Minkowski case,we will take both the null infinity limit, where the retarded time u ≡ t − r is held fixed and the advancedtime v ≡ t + r is sent to infinity; as well as the timelike infinity limit, where the observer time is sentto infinity ( t → ∞ ) while her radial position r is held still. In the de Sitter case, we will only considerthe timelike infinity limit, where η/η (cid:48) → while r is held constant. Furthermore, we shall assume that − η (cid:48) is the largest length scale in our problem: − r (cid:48) /η (cid:48) < − r/η (cid:48) (cid:28) . As we shall witness in some detailbelow, that the observer continues to receive the scalar signal in such a late time regime, is due solelyto the existence of tails – see Fig. (2) – i.e., a portion of the initial scalar field propagates inside the nullcone. 5igure 2: Localized initial scalar field profile and late time observer. Bottom and top planes are, respec-tively, the initial x (cid:48) hypersurface and final (observer) x hypersurface. The observer is located at (cid:126)x andthe cone denotes her past null cone. This illustrates, in the late time regime where the two hypersurfacesgrow in distance, the observer continues to receive a scalar signal due solely to the tail effect. As a warm up to de Sitter calculation, we first discuss the Minkowski background case. The late timebehavior of a scalar field in a weakly curved spacetime was investigated by Poisson in [17]. The calcula-tion was performed by solving the Kirchhoff representation (2.0.7) directly in position space, and boththe timelike and null infinity limits of the scalar solution were extracted.Poisson extended Price’s fall-off theorem of a scalar field by working in a perturbed Minkowskispacetime sourced by a central mass without any particular spatial symmetries, as opposed to the spher-ically symmetric black hole solution used by Price. In detail, Poisson assumed a spatially localized sta-tionary mass distribution with mass density T = ρ [ (cid:126)x ] and mass-current density T i = (cid:126)j [ (cid:126)x ] = ρ(cid:126)v , where (cid:126)v is the velocity within the matter distribution. By parametrizing eq. (2.0.2) asd s = (1 + 2Φ) d t − (1 − d (cid:126)x · d (cid:126)x − (cid:126)A · d (cid:126)x ) d t, (3.0.1)the linearized Einstein’s equations (with zero cosmological constant) reduce to a pair of Poisson’s equa-tions, whose solutions are Φ[ (cid:126)x ] = − πG N (cid:90) R ρ [ (cid:126)x (cid:48)(cid:48) ] | (cid:126)x − (cid:126)x (cid:48)(cid:48) | d (cid:126)x (cid:48)(cid:48) (3.0.2) (cid:126)A [ (cid:126)x ] = − πG N (cid:90) R (cid:126)j [ (cid:126)x (cid:48)(cid:48) ] | (cid:126)x − (cid:126)x (cid:48)(cid:48) | d (cid:126)x (cid:48)(cid:48) . (3.0.3)6e found the scalar field behavior at timelike infinity is given by Ψ[ t (cid:29) r, r, Ω] = 4 G N M ( − (cid:96) +1 (2 (cid:96) + 2)!!(2 (cid:96) + 1)!! r (cid:96) t (cid:96) +3 Y m(cid:96) [ (cid:98) x ] (cid:18) ˙ C (cid:96) − (cid:96) + 3 t C (cid:96) (cid:19) , (3.0.4)and at null infinity is given by r Ψ[ v → ∞ , u, Ω] = 2 G N M ( − (cid:96) +1 ( (cid:96) + 1)!(2 (cid:96) + 1)!! 1 u (cid:96) +2 Y m(cid:96) [ (cid:98) x ] (cid:18) ˙ C (cid:96) − (cid:96) + 2 u C (cid:96) (cid:19) . (3.0.5)where (cid:98) x ≡ (cid:126)x/ | (cid:126)x | are the angular coordinates; and we use retarded time coordinate u = t − r as well asadvanced time coordinate v = t + r . The C (cid:96) and ˙ C (cid:96) are respectively defined to be C (cid:96) ≡ (cid:82) d r r (cid:96) +2 C [ r (cid:48) ] and ˙ C (cid:96) ≡ (cid:82) d r r (cid:96) +2 ˙ C [ r (cid:48) ] .The key finding of Poisson’s results in equations (3.0.4) and (3.0.5) is that, even without as-suming any spatial symmetries of the central mass, the falloff behavior of Ψ is completely governed byits monopole moment in eq. (2.0.16) – i.e., no higher mass multipoles entered the final answer – andthe power law in time depends on the multipole index of the initial data. We now proceed to reproducethe same results by employing a frequency space based method, which will also be employed in the deSitter analysis in §(4) below. The advantage of this approach is, it allows us to use a spherical harmonicbasis from the outset. We will be contend with reproducing Poisson’s result involving the mass density only; for simplicity wewill set (cid:126)j and hence (cid:126)A to zero. The first step is to perform a spherical harmonic decomposition to thesolution to eq. (3.0.2) using the relation π | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48)(cid:48)(cid:48) | = 1 r > ∞ (cid:88) (cid:96) (cid:48)(cid:48) =0 (cid:96) (cid:48)(cid:48) (cid:88) m (cid:48)(cid:48) = − (cid:96) (cid:48)(cid:48) Y m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) [ (cid:98) x (cid:48)(cid:48) ] Y m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) [ (cid:98) x (cid:48)(cid:48)(cid:48) ]2 (cid:96) (cid:48)(cid:48) + 1 (cid:18) r < r > (cid:19) (cid:96) (cid:48)(cid:48) , (3.1.1)where r < = min[ r (cid:48)(cid:48) , r (cid:48)(cid:48)(cid:48) ] and r > = max[ r (cid:48)(cid:48) , r (cid:48)(cid:48)(cid:48) ] . This yields Φ[ (cid:126)x (cid:48)(cid:48) ] = − πG N (cid:88) (cid:96) (cid:48)(cid:48) ,m (cid:48)(cid:48) ( − m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) + 1 Y m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) [ (cid:98) x (cid:48)(cid:48) ] r (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) +1 M m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) , (3.1.2)where M m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) ≡ (cid:90) r m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) d r (cid:48)(cid:48)(cid:48) r (cid:48)(cid:48)(cid:48) (cid:96) (cid:48)(cid:48) (cid:90) S d Ω (cid:48)(cid:48)(cid:48) Y m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) [ (cid:98) x (cid:48)(cid:48)(cid:48) ] ρ m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) [ (cid:126)x (cid:48)(cid:48)(cid:48) ] (3.1.3)7s the mass multipole moment. The mass density is localized inside radius r m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) of degree (cid:96) (cid:48)(cid:48) and azimuthalindex m (cid:48)(cid:48) and (cid:82) d Ω (cid:48)(cid:48)(cid:48) is the solid angle integral over the 2-sphere. Zeroth Order Green Function
In Minkowski spacetime, the propagation of a masslessscalar field is described by the following retarded Green function G [ x, x (cid:48) ] = δ [ t − t (cid:48) − | (cid:126)x − (cid:126)x (cid:48) | ]4 π | (cid:126)x − (cid:126)x (cid:48) | , (3.1.4)that obeys the wave equation (2.0.12), with a = 1 ; namely, ∂ G = δ (4) [ x − x (cid:48) ] . By re-expressing thedelta function using its integral representation, we obtain the frequency space retarded Green function. G [ x, x (cid:48) ] = (cid:90) ∞−∞ d ω π e − iω ( t − t (cid:48) ) (cid:101) G + [ ω ; (cid:126)x − (cid:126)x (cid:48) ] , (3.1.5)where (cid:101) G + [ ω ; (cid:126)x − (cid:126)x (cid:48) ] ≡ exp[ iω | (cid:126)x − (cid:126)x (cid:48) | ]4 π | (cid:126)x − (cid:126)x (cid:48) | . (3.1.6)The spherical harmonic decomposition of eq. (3.1.6) is (cid:101) G + [ ω ; (cid:126)x − (cid:126)x (cid:48) ] = iω (cid:88) (cid:96),m j (cid:96) [ ωr < ] h (1) (cid:96) [ ωr > ] Y m(cid:96) [ (cid:98) x ] Y m(cid:96) [ (cid:98) x (cid:48) ] , (3.1.7)where j (cid:96) [ ωr < ] is the spherical Bessel function; h (1) (cid:96) [ ωr > ] is the Hankel function of the first kind; and theradii denote r < ≡ min[ r, r (cid:48) ] and r > ≡ max[ r, r (cid:48) ] . First Order Green Function
We will now include the perturbation to the Green functiongenerated by the gravitational potential Φ . This potential will scatter the null signals transmitted by thezeroth order Green’s functions, so the perturbed signals can be seen as tail propagation from the per-spective of the observer at (cid:126)x . This is why the signal can be observed at late times, and is to be contrastedagainst the pure light-cone propagation of the signal in the unperturbed Minkowski background.The perturbed retarded Green function can be obtained from eq. (2.0.13) by setting a = 1 andutilizing eq. (3.1.4). Up to the first order in perturbation, δ G [ x, x (cid:48) ] = − (cid:90) R , d x (cid:48)(cid:48) ∂ t (cid:48)(cid:48) δ [ t − t (cid:48)(cid:48) − | (cid:126)x − (cid:126)x (cid:48)(cid:48) | ]4 π | (cid:126)x − (cid:126)x (cid:48)(cid:48) | Φ[ (cid:126)x (cid:48)(cid:48) ] ∂ t (cid:48)(cid:48) δ [ t (cid:48)(cid:48) − t (cid:48) − | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | ]4 π | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | . (3.1.8)Evaluation of the time integral t (cid:48)(cid:48) over one of the Green functions will give us δ G [ x, x (cid:48) ] = − ∂ t ∂ t (cid:48) (cid:90) R d (cid:126)x (cid:48)(cid:48) δ [ t − t (cid:48) − | (cid:126)x − (cid:126)x (cid:48)(cid:48) | − | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | ]4 π | (cid:126)x − (cid:126)x (cid:48)(cid:48) | π | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | Φ[ (cid:126)x (cid:48)(cid:48) ] . (3.1.9)8he delta function in eq. (3.1.8) indicates the null signals actually propagate from the initial point at x (cid:48) , and reflects off the potential Φ at (cid:126)x (cid:48)(cid:48) before reaching the observer at x . The volume integral (cid:126)x (cid:48)(cid:48) mustbe evaluated on the intersection of future lightcone of the initial scalar field x (cid:48) and past lightcone of theobserver at x , which is an ellipsoid with foci at (cid:126)x and (cid:126)x (cid:48) . By aligning the (cid:98) z axis to be parallel to (cid:126)x − (cid:126)x (cid:48) ,the spatial components in a Cartesian basis of an arbitrary point of this ellipsoid reads [17], [4] (cid:126)x (cid:48)(cid:48) [ θ (cid:48)(cid:48) , φ (cid:48)(cid:48) ] = (cid:126)x + (cid:126)x (cid:48) (cid:112) ( t − t (cid:48) ) − | (cid:126)x − (cid:126)x (cid:48) | sin[ θ (cid:48)(cid:48) ] (cid:98) e ⊥ [ φ (cid:48)(cid:48) ] + t − t (cid:48) θ (cid:48)(cid:48) ] (cid:98) z. (3.1.10)The (cid:98) e ⊥ = (cos φ (cid:48)(cid:48) , sin φ (cid:48)(cid:48) , is the unit radial vector lying on the 2-dimensional plane orthogonal to (cid:126)x − (cid:126)x (cid:48) .In Fig. (3.1), we illustrate with a spacetime diagram the causal structure of the scattering of null signalsin eq. (3.1.8).Figure 3: Spacetime diagram of the first order Green’s function δ G . The bottom and top planes are,respectively, the initial and final time hypersurfaces. The cylindrical tube joining the two planes is theworldtube swept out by the central mass, whose proper mass density is ρ [ (cid:126)x (cid:48)(cid:48) ] /a , which we assume iszero outside some small radius. The shaded region on the initial time (lower) plane is where the initial Ψ and its velocity are non-zero, which we too assume is a small region centered around the spatialorigin. The observer on the final time (top) surface is located at (cid:126)x , where we base her backward lightcone. The first order massless scalar Green’s function receives a signal emitted from the shaded regionon the initial time surface, which propagates on the displayed forward null cone before scattering offthe gravitational perturbation ¯ χ [ x (cid:48)(cid:48) ] engendered by the central mass, and taking another null path tothe observer at ( η, (cid:126)x ) . The locus of spacetime points { x (cid:48)(cid:48) } where the scalar signal scatters off ¯ χ is, bycausality, given by the intersection of the forward light cone of x (cid:48) ≡ ( x (cid:48) , (cid:126)x (cid:48) ) and the backward light coneof x ≡ ( x , (cid:126)x ) . Notice, in the late time limit, the size of the ellipsoid will grow very large compared to | (cid:126)x (cid:48) | and the spatial size of the mass distribution. 9o decompose the integrand of (3.1.9) in terms of spherical harmonics, we again replace thedelta function with its integral representation, with ω as the frequency. δ G [ x, x (cid:48) ] = − (cid:90) R d (cid:126)x (cid:48)(cid:48) (cid:90) ∞−∞ d ω ω e − iω ( t − t (cid:48) ) e iω | (cid:126)x − (cid:126)x (cid:48)(cid:48) | π | (cid:126)x − (cid:126)x (cid:48)(cid:48) | e iω | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | π | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | Φ[ (cid:126)x (cid:48)(cid:48) ] , (3.1.11)where ω is due to the ∂ t ∂ t (cid:48) in eq. (3.1.9). Since the expression (3.1.11) is time translation invariant,described by exp[ − iω ( t − t (cid:48) )] , we can set the initial time t (cid:48) of the scalar field to be . Employing thespherical harmonic decompositions of equations (3.1.7) and (3.1.2), we may re-cast eq. (3.1.11) into δ G [ x, x (cid:48) ] = (cid:90) R d (cid:126)x (cid:48)(cid:48) (cid:90) ∞−∞ d ω ω e − iωt πG N (cid:88) (cid:96) (cid:48)(cid:48) ,m (cid:48)(cid:48) ( − m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) + 1 Y m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) [ (cid:98) x (cid:48)(cid:48) ] r (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) +1 M m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) × iω (cid:88) (cid:96),m j (cid:96) [ ωr < ] h (1) (cid:96) [ ωr > ] Y m(cid:96) [ (cid:98) x ] Y m(cid:96) [ (cid:98) x (cid:48)(cid:48) ] × iω (cid:88) (cid:96) (cid:48) ,m (cid:48) j (cid:96) (cid:48) [ ωr (cid:48) < ] h (1) (cid:96) (cid:48) [ ωr (cid:48) > ] Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48)(cid:48) ] Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48) ] , where r < = min[ r, r (cid:48)(cid:48) ] , r > = max[ r, r (cid:48)(cid:48) ] , r (cid:48) < = min[ r (cid:48) , r (cid:48)(cid:48) ] , r (cid:48) > = max[ r (cid:48) , r (cid:48)(cid:48) ] . In the late time limit, r (cid:48) willalways be evaluated inside the boundary of initial scalar field r Ψ which is very small compared to r (cid:48)(cid:48) ,where r (cid:48)(cid:48) is evaluated on the ellipsoid. Therefore, we can directly set r (cid:48) < = r (cid:48) and r (cid:48) > = r (cid:48)(cid:48) . For r < and r > , will be determined by further calculation. Since r (cid:48) is the smallest length scales in our problem, weexpect the final answer to admit a power series in r (cid:48) . Motivated by this observation, we take the smallargument limit of the spherical Bessel functions j (cid:96) (cid:48) [ ωr (cid:48) ] and use integral representation of j (cid:96) [ ωr ] , δ G [ x, x (cid:48) ] = − (cid:90) ∞ d r (cid:48)(cid:48) r (cid:48)(cid:48) (cid:90) S d Ω (cid:90) ∞−∞ d ω ω e − iωt πG N (cid:88) (cid:96) (cid:48)(cid:48) ,m (cid:48)(cid:48) ( − ) m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) + 1 Y m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) [ (cid:98) x (cid:48)(cid:48) ] r (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) +1 M m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) (3.1.12) × (cid:88) (cid:96),m (cid:90) − d c e iωr < c (cid:96) ! (cid:16) − ωr < (cid:17) (cid:96) ( c − (cid:96) ( − i ) (cid:96) +1 e iωr > ωr > (cid:96) (cid:88) s =0 i s s !(2 ωr > ) s ( (cid:96) + s )!( (cid:96) − s )! Y m(cid:96) [ (cid:98) x ] ¯ Y m(cid:96) [ (cid:98) x (cid:48)(cid:48) ] × (cid:88) (cid:96) (cid:48) ,m (cid:48) ( ωr (cid:48) ) (cid:96) (cid:48) (2 (cid:96) (cid:48) + 1)!! (cid:0) O [( ωr (cid:48) ) ] (cid:1) ( − i ) (cid:96) (cid:48) +1 e iωr (cid:48)(cid:48) ωr (cid:48)(cid:48) (cid:96) (cid:48) (cid:88) s (cid:48) =0 i s (cid:48) s (cid:48) !(2 ωr (cid:48)(cid:48) ) s (cid:48) ( (cid:96) (cid:48) + s (cid:48) )!( (cid:96) (cid:48) − s (cid:48) )! Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48)(cid:48) ] ¯ Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48) ] By collecting ω in the integrand, we can see that ω has positive power (cid:96) + (cid:96) (cid:48) − s − s (cid:48) . Therefore,10 (cid:96) + (cid:96) (cid:48) − s − s (cid:48) can simply be replaced with ( i∂ t ) (cid:96) + (cid:96) (cid:48) − s − s (cid:48) , and the integral over ω evaluated. δ G [ x, x (cid:48) ] = − (cid:88) (cid:96),m (cid:18) − (cid:19) (cid:96) (cid:90) − d c (cid:96) ! ( c − (cid:96) ( − i ) (cid:96) +1 (cid:96) (cid:88) s =0 i s s !(2) s ( (cid:96) + s )!( (cid:96) − s )! Y m(cid:96) [ (cid:98) x ]( − ) m (3.1.13) π G N (cid:88) (cid:96) (cid:48)(cid:48) ,m (cid:48)(cid:48) ( − ) m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) + 1 M m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) (cid:88) (cid:96) (cid:48) ,m (cid:48) ( r (cid:48) ) (cid:96) (cid:48) (2 (cid:96) (cid:48) + 1)!! ( − i ) (cid:96) (cid:48) +1 (cid:96) (cid:48) (cid:88) s (cid:48) =0 i s (cid:48) s (cid:48) !(2) s (cid:48) ( (cid:96) (cid:48) + s (cid:48) )!( (cid:96) (cid:48) − s (cid:48) )! ¯ Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48) ] ∂ (cid:96) + (cid:96) (cid:48) − s − s (cid:48) t ( − i ) (cid:96) + (cid:96) (cid:48) − s − s (cid:48) (cid:40) Θ[ t − c r − r ] r (cid:96) ( t − c r ) (cid:96) (cid:48)(cid:48) + s (cid:48) + s +1
12 + Θ (cid:20) r − t − r c (cid:21) (cid:18) t − r c (cid:19) (cid:96) − (cid:96) (cid:48)(cid:48) − s (cid:48) r s +1
11 + c (cid:41)(cid:18) (2 (cid:96) + 1)(2 (cid:96) (cid:48) + 1)(2 (cid:96) (cid:48)(cid:48) + 1)4 π (cid:19) / (cid:96) (cid:96) (cid:48) (cid:96) (cid:48)(cid:48) (cid:96) (cid:96) (cid:48) (cid:96) (cid:48)(cid:48) − m m (cid:48) m (cid:48)(cid:48) (cid:0) O [( r (cid:48) /t ) ] (cid:1) , where we have evaluated the solid angle integral Ω (cid:48)(cid:48) with three spherical harmonics in the integrand,which introduces the Wigner 3j-symbols, and the r (cid:48)(cid:48) integral will collapses the delta functions into stepfunctions.We will consider two possible configurations of the observer. The first: the observer is locatednear time-like infinity, t → ∞ with r fixed. In this case, the largest quantity is the observer time t . Thesecond: the observer is located at null-infinity, where the advanced time grows without bound, v → ∞ but the retarded time u is large but fixed. Time-like infinity
Consider the observer at timelike infinity. For r (cid:48)(cid:48) > r , we have a stepfunction with − − c + t/r as the argument. Since we consider the observer at time-like infinity t → ∞ ,the step function always has a positive value argument, i.e Θ[( t/r − c ) (cid:29)
1] = 1 . However, for r > r (cid:48)(cid:48) as t → ∞ , the argument is always negative, Θ[( c − t/r ) → −∞ ] = 0 , since the maximum value of cosine is . Take the leading contribution of this limit, ( − c r + t ) → t , followed by evaluating the integral, δ G [ x, x (cid:48) ] = (cid:88) (cid:96),m (cid:18) − (cid:19) (cid:96) (cid:96) ! √ π Γ[ (cid:96) + 1]Γ[ (cid:96) + 3 /
2] ( − i ) (cid:96) +1 (cid:96) (cid:88) s =0 i s s !(2) s ( (cid:96) + s )!( (cid:96) − s )! Y m(cid:96) [ (cid:98) x ]( − ) m × π G N (cid:88) (cid:96) (cid:48)(cid:48) ,m (cid:48)(cid:48) ( − ) m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) + 1 M m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) (cid:88) (cid:96) (cid:48) ,m (cid:48) ( r (cid:48) ) (cid:96) (cid:48) (2 (cid:96) (cid:48) + 1)!! ( − i ) (cid:96) (cid:48) +1 (cid:96) (cid:48) (cid:88) s (cid:48) =0 i s (cid:48) s (cid:48) !(2) s (cid:48) ( (cid:96) (cid:48) + s (cid:48) )!( (cid:96) (cid:48) − s (cid:48) )! ¯ Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48) ] × Γ[3 + (cid:96) + (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) ] t − − (cid:96) − (cid:96) (cid:48) − (cid:96) (cid:48)(cid:48) Γ[ (cid:96) (cid:48)(cid:48) + s + s (cid:48) + 1]( i ) (cid:96) + (cid:96) (cid:48) − s − s (cid:48) r (cid:96) (cid:114) (2 (cid:96) + 1)(2 (cid:96) (cid:48) + 1)(2 (cid:96) (cid:48)(cid:48) + 1)4 π (cid:96) (cid:96) (cid:48) (cid:96) (cid:48)(cid:48) (cid:96) (cid:96) (cid:48) (cid:96) (cid:48)(cid:48) − m m (cid:48) m (cid:48)(cid:48) × (cid:0) O (cid:2) ( r (cid:48) /t ) , r/t (cid:3)(cid:1) . (3.1.14)Parity invariance – see Appendix (A) – tells us that only (cid:96) (cid:48)(cid:48) = 0 mode survives and the double summationover s and s (cid:48) is ( − (cid:96) . The usual rules of angular momentum addition (as contained within the Wigner11j-symbols) then informs us the surviving terms are those where (cid:96) = (cid:96) (cid:48) . δ G [ x, x (cid:48) ] = 4 G N M (cid:88) (cid:96),m t − (cid:96) − ( rr (cid:48) ) (cid:96) Y m(cid:96) [ (cid:98) x ] ¯ Y m(cid:96) [ (cid:98) x (cid:48) ] (2 (cid:96) + 2)!!(2 (cid:96) + 1)!! ( − (cid:96) +1 (cid:0) O [( r (cid:48) /t ) , r/t ] (cid:1) . (3.1.15)We are going to use this Green function to describe the behavior of the scalar field. Recalling the Kirchhoffrepresentation in eq. (2.0.7), with the initial data in eq. (2.0.6), we arrive at Ψ[ x ] = 4 G N M (cid:88) (cid:96),m t − (cid:96) − r (cid:96) Y m(cid:96) [ (cid:98) x ] (2 (cid:96) + 2)!!(2 (cid:96) + 1)!! ( − (cid:96) +1 × (cid:90) ∞ d r (cid:48) r (cid:48) (cid:96) +2 (cid:18) C m(cid:96) [ r (cid:48) ] − (cid:96) + 3 t ˙ C m(cid:96) [ r (cid:48) ] (cid:19) . (3.1.16)Notice that we used time translation invariance on the second term, which introduced a minus sign. Thissolution is equivalent to Poisson’s result (3.0.4). Null infinity
Now, let’s consider the scalar field at null infinity. To do that, we introducethe retarded time coordinate u ≡ t − r and advanced time coordinate v ≡ t + r in (3.1.13), and proceedto consider the limits: v → ∞ and u fixed. δ G [ x, x (cid:48) ] = − (cid:88) (cid:96),m (cid:18) − (cid:19) (cid:96) (cid:90) − d c (cid:96) ! ( c − (cid:96) ( − i ) (cid:96) +1 (cid:96) (cid:88) s =0 i s s !(2) s ( (cid:96) + s )!( (cid:96) − s )! Y m(cid:96) [ (cid:98) x ]( − ) m (3.1.17) × π G N (cid:88) (cid:96) (cid:48)(cid:48) ,m (cid:48)(cid:48) ( − ) m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) + 1 M m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) (cid:88) (cid:96) (cid:48) ,m (cid:48) ( r (cid:48) ) (cid:96) (cid:48) (2 (cid:96) (cid:48) + 1)!! ( − i ) (cid:96) (cid:48) +1 (cid:96) (cid:48) (cid:88) s (cid:48) =0 i s (cid:48) s (cid:48) !(2) s (cid:48) ( (cid:96) (cid:48) + s (cid:48) )!( (cid:96) (cid:48) − s (cid:48) )! ¯ Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48) ] × ∂ (cid:96) + (cid:96) (cid:48) − s − s (cid:48) u ( − i ) (cid:96) + (cid:96) (cid:48) − s − s (cid:48) (cid:18) Θ (cid:20) uv − u − − c (cid:21) − (cid:96) − ( v − u ) (cid:96) (1 − c ) (cid:96) (cid:48)(cid:48) + s (cid:48) + s +1 +Θ (cid:20) c + 2 uu − v (cid:21) u (cid:96) − (cid:96) (cid:48)(cid:48) − s (cid:48) s +1 ( v − u ) s +1 (1 + c ) (cid:96) − (cid:96) (cid:48)(cid:48) − s (cid:48) +1 (cid:33) × (cid:18) (2 (cid:96) + 1)(2 (cid:96) (cid:48) + 1)(2 (cid:96) (cid:48)(cid:48) + 1)4 π (cid:19) / (cid:96) (cid:96) (cid:48) (cid:96) (cid:48)(cid:48) (cid:96) (cid:96) (cid:48) (cid:96) (cid:48)(cid:48) − m m (cid:48) m (cid:48)(cid:48) (cid:16) O (cid:104)(cid:0) r (cid:48) /v (cid:1) , u/v (cid:105)(cid:17) . We have already taken the leading contribution at the null-infinity. The c integral is in fact the repre-sentation of the beta function B [ α, β ] . After carrying out the u − derivatives, we found that the first stepfunction term is suppressed by u/v relative to the second step function term. In the latter, we also onlyretain the lowest power of /v . Therefore, we take (cid:96) (cid:48)(cid:48) = 0 and s = 0 which from angular momentumaddition rules imply (cid:96) (cid:48) = (cid:96) . The perturbed Green function now becomes δ G [ x, x (cid:48) ] = (cid:88) (cid:96),m G N M ( − (cid:96) +1 r (cid:48) (cid:96) r u (cid:96) +2 ( (cid:96) + 1)!!(2 (cid:96) + 1)!! Y m(cid:96) [ (cid:98) x ] ¯ Y m(cid:96) [ (cid:98) x (cid:48) ] (cid:16) O (cid:104)(cid:0) r (cid:48) /t (cid:1) , u/v (cid:105)(cid:17) . (3.1.18)12e now use the form of the Green’s function in eq. (3.1.18) in the Kirchhoff representation of (2.0.7)with eq. (2.0.6) as the initial profile of the scalar field. Finally, we have r Ψ[ x ] = 2 G N M (cid:88) (cid:96),m u (cid:96) +2 ( (cid:96) + 1)!(2 (cid:96) + 1)!! Y m(cid:96) [ (cid:98) x ] (cid:90) ∞ d r (cid:48) r (cid:48) (cid:96) +2 (cid:18) C m(cid:96) [ r (cid:48) ] − (cid:96) + 2 u ˙ C m(cid:96) [ r (cid:48) ] (cid:19) . (3.1.19)We have again used time translation invariance on the second term, which introduced a minus sign. Thissolution is equivalent to Poisson’s result (3.0.5). We now turn to the de Sitter case, where a [ η ] = − / ( Hη ) . Within the generalized de Donder gaugecondition η∂ µ χ µν = 2 χ ν − δ ν η ρσ χ ρσ , the metric perturbation χ µν in eq. (2.0.2) (or, equivalently, itstrace-reversed cousin ¯ χ µν in eq. (2.0.14)) satisfying the linearized Einstein’s equations with a positivecosmological constant Λ = 3 H had been solved analytically in [1] for a general source T µν ; namely δ G µν − Λ a χ µν = 8 πG N T µν , (4.1.1)where δ G µν is the Einstein tensor expanded about a de Sitter background and containing precisely onepower of χ µν . For technical simplicity we are considering a stress tensor in eq. (2.0.15) whose onlynon-zero component is its mass density. Hence, the only non-zero component is ¯ χ , which can be foundfrom the so-called pseudo-trace mode solution in [1]. ¯ χ [ η (cid:48)(cid:48) , (cid:126)x (cid:48)(cid:48) ] = − πG N a [ η (cid:48)(cid:48) ] (cid:90) R d (cid:126)x (cid:48)(cid:48)(cid:48) ρ [ (cid:126)x (cid:48)(cid:48)(cid:48) ]4 π | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48)(cid:48)(cid:48) | (4.1.2)In terms of ¯ χ , the metric perturbation components are χ = 12 ¯ χ , χ ij = 12 δ ij ¯ χ , χ i = 0 . (4.1.3)Eq. (4.1.2) tells us, up to the scale factor, ¯ χ is identical to the flat spacetime Newtonian gravitationalpotential, which in turn means the spherical harmonic decomposition in eq. (3.1.1) may continue to beexploited. Specifically, if the ¯ χ is evaluated outside the matter distribution then eq. (4.1.2) becomes ¯ χ [ η (cid:48)(cid:48) , (cid:126)x (cid:48)(cid:48) ] = − πG N a [ η (cid:48)(cid:48) ] (cid:88) (cid:96) (cid:48)(cid:48) ,m (cid:48)(cid:48) ( − ) m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) + 1 M m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) [ r m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) ] r (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) +1 Y (cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) [ (cid:98) x (cid:48)(cid:48) ] (4.1.4)13here the mass multipoles are M m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) [ r m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) ] ≡ (cid:90) r m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) d r (cid:48)(cid:48)(cid:48) (cid:90) S d Ω (cid:48)(cid:48)(cid:48) r (cid:48)(cid:48)(cid:48) (cid:96) (cid:48)(cid:48) +2 Y (cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) [ (cid:98) x (cid:48)(cid:48)(cid:48) ] ρ [ (cid:126)x (cid:48)(cid:48)(cid:48) ] . (4.1.5)On the other hand, if ¯ χ is evaluated inside the matter distribution, the integration needs to be separatedinto two different regions. The first region lies between the origin and the location of metric perturbation r (cid:48)(cid:48) , where r > = r (cid:48)(cid:48) . However, the second region is between r (cid:48)(cid:48) and the boundary of matter distribution r m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) , with r > = r (cid:48)(cid:48)(cid:48) . Then, (4.1.2) becomes ¯ χ [ η (cid:48)(cid:48) , (cid:126)x (cid:48)(cid:48) ] = − πG N a [ η (cid:48)(cid:48) ] (cid:88) (cid:96) (cid:48)(cid:48) ,m (cid:48)(cid:48) ( − ) m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) + 1 Y (cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) [ (cid:98) x (cid:48)(cid:48) ] (cid:32) M m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) [ r (cid:48)(cid:48) ] r (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) +1 + N m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) [ r (cid:48)(cid:48) ] r (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) (cid:33) (4.1.6)where the internal multipoles are now M m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) [ r (cid:48)(cid:48) ] ≡ (cid:90) r (cid:48)(cid:48) d r (cid:48)(cid:48)(cid:48) (cid:90) S d Ω (cid:48)(cid:48)(cid:48) r (cid:48)(cid:48)(cid:48) (cid:96) (cid:48)(cid:48) +2 Y (cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) [ (cid:98) x (cid:48)(cid:48)(cid:48) ] ρ [ (cid:126)x (cid:48)(cid:48)(cid:48) ] ,N m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) [ r (cid:48)(cid:48) ] ≡ (cid:90) r m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) r (cid:48)(cid:48) d r (cid:48)(cid:48)(cid:48) (cid:90) S d Ω (cid:48)(cid:48)(cid:48) r (cid:48)(cid:48)(cid:48)− (cid:96) (cid:48)(cid:48) +1 Y (cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) [ (cid:98) x (cid:48)(cid:48)(cid:48) ] ρ [ (cid:126)x (cid:48)(cid:48)(cid:48) ] . (4.1.7)These M m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) and N m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) contribute, respectively, to the two integration regions < r (cid:48)(cid:48)(cid:48) < r (cid:48)(cid:48) and r (cid:48)(cid:48) < r (cid:48)(cid:48)(cid:48) Now, let us prepare the Kirchhoff representation of the scalarfield in a perturbed de Sitter background. Here, we set the initial hypersurface at η (cid:48) . Therefore, upto first order in χ µν , the square root of induced metric determinant and normal vector of the initialhypersurface are (cid:112) | h | ≈ a [ η (cid:48) ] (cid:18) − 34 ¯ χ (cid:19) , (cid:98) n α ≈ a [ η (cid:48) ] − δ α (cid:18) − 14 ¯ χ (cid:19) . (4.1.8)Up to first order in perturbations, we have from (2.0.7), Ψ[ x ] = (cid:90) R d (cid:126)x (cid:48) a [ η (cid:48) ] (cid:16) G [ x, x (cid:48) ] ˙Ψ[ x (cid:48) ] − Ψ[ x (cid:48) ] ∂ η (cid:48) ( G [ x, x (cid:48) ]) (cid:17) + (cid:90) R d (cid:126)x (cid:48) a [ η (cid:48) ] ( − ¯ χ ) (cid:16) G [ x, x (cid:48) ] ˙Ψ[ x (cid:48) ] − Ψ[ x (cid:48) ] ∂ η (cid:48) ( G [ x, x (cid:48) ]) (cid:17) + (cid:90) R d (cid:126)x (cid:48) a [ η (cid:48) ] (cid:16) ( δ G x,x (cid:48) ) ˙Ψ[ x (cid:48) ] − Ψ[ x (cid:48) ] ∂ η (cid:48) ( δ G x,x (cid:48) ) (cid:17) ≡ Ψ (0) [ η, η (cid:48) , (cid:126)x ] + Ψ (0 ,χ ) [ η, η (cid:48) , (cid:126)x ] + Ψ (1) [ η, η (cid:48) , (cid:126)x ] . (4.1.9)The first, second and third terms after the first equality are defined, respectively, as Ψ (0) [ η, η (cid:48) , (cid:126)x ] , Ψ (0 ,χ ) [ η, η (cid:48) , (cid:126)x ] and Ψ (1) [ η, η (cid:48) , (cid:126)x ] . Ψ (0) and Ψ (0 ,χ ) will be discussed in the next section. Unlike in 4D Minkowski, where14here is no linear tail effect, these two terms do contribute to the observed scalar field at late timesbecause the background de Sitter Green’s function (in eq. (4.1.10) below) contains a tail. Therefore, weneed to include these terms and compare them with the nonlinear signal Ψ (1) [ η, η (cid:48) , (cid:126)x ] associated withscalar-gravity scattering described by δ G . Perturbed Green Function The retarded Green function of a massless scalar field in deSitter spacetime contains two terms. One of them, proportional to a delta function, describes the prop-agation of the field on the lightcone; while the other, proportional to a step function, describes thepropagation of the field inside the future lightcone of the source at x (cid:48) . It is given by G [ x, x (cid:48) ] = H π (cid:18) δ [ η − η (cid:48) − | (cid:126)x − (cid:126)x (cid:48) | ] | (cid:126)x − (cid:126)x (cid:48) | ηη (cid:48) + Θ[ η − η (cid:48) − | (cid:126)x − (cid:126)x (cid:48) | ] (cid:19) . (4.1.10)The first order perturbed Green function is given by eq. (2.0.13). Its late time limit reads δ G [ x, x (cid:48) ] = − πG N (cid:90) ∞ d r (cid:48)(cid:48) r (cid:48)(cid:48) (cid:90) S d Ω (cid:48)(cid:48) H (cid:88) (cid:96) (cid:48)(cid:48) ,m (cid:48)(cid:48) ( − ) m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) + 1 Y (cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) [ (cid:98) x (cid:48)(cid:48) ] r (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) +1 M m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) (4.1.11) × (cid:0) ηη (cid:48) A − − ηη (cid:48) ∂ η (cid:48) A + ηB − ,(cid:126)x − ηη (cid:48) ∂ η A + ηη (cid:48) ∂ η ∂ η (cid:48) A − η∂ η B ,(cid:126)x − η (cid:48) B − ,(cid:126)x (cid:48) + η (cid:48) ∂ η (cid:48) B ,(cid:126)x (cid:48) − C − (cid:1) where we have defined A a ≡ (cid:90) −∞ d η (cid:48)(cid:48) η (cid:48)(cid:48) a δ [ η − η (cid:48)(cid:48) − | (cid:126)x − (cid:126)x (cid:48)(cid:48) | ]4 π | (cid:126)x − (cid:126)x (cid:48)(cid:48) | δ [ η (cid:48)(cid:48) − η (cid:48) − | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | ]4 π | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | , (4.1.12) B b,(cid:126)x o ≡ (cid:90) −∞ d η (cid:48)(cid:48) η (cid:48)(cid:48) b δ [ η − η (cid:48)(cid:48) − | (cid:126)x − (cid:126)x (cid:48)(cid:48) | ] δ [ η (cid:48)(cid:48) − η (cid:48) − | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | ](4 π ) | (cid:126)x o − (cid:126)x (cid:48)(cid:48) | , (cid:126)x o ∈ { (cid:126)x, (cid:126)x (cid:48) } , (4.1.13) C − ≡ (cid:90) −∞ d η (cid:48)(cid:48) π ) η (cid:48)(cid:48) δ [ η − η (cid:48)(cid:48) − | (cid:126)x − (cid:126)x (cid:48)(cid:48) | ] δ [ η (cid:48)(cid:48) − η (cid:48) − | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | ] . (4.1.14)That only delta functions are present in equations (4.1.12)–(4.1.14) is because the derivatives in eq.(2.0.13) have converted the Θ ’s of the tail term in eq. (4.1.10) into δ ’s. Moreover, these delta functionsteach us that, as far as the causal structure of the signal is concerned, the de Sitter massless scalar Green’sfunction at first order in the central mass is similar to its asymptotically flat cousin: the signal at theobserver’s location x arises from the signal emitted at x (cid:48) scattering off the central mass’ gravitationalpotential lying on the ellipsoid that is associated with the intersection of the future light cone of x (cid:48) andthe past light cone of x – recall Fig. (3.1). This also explains why, we have employed in eq. (4.1.11) theform of the perturbation in eq. (4.1.4); for, the ellipsoid of integration at late times always lies outsidethe matter distribution itself.In this work, we will be extracting the leading order terms from expanding in powers of η/η (cid:48) , r/η (cid:48) and r (cid:48) /η (cid:48) . In particular, we shall specialize to an observer approaching timelike infinity, where15 /η (cid:48) → for fixed r . By using that assumption the first six terms from the left subleading relative to theremaining three terms: δ G [ x, x (cid:48) ] = − πG N H (cid:90) ∞ d r (cid:48)(cid:48) (cid:90) S d Ω (cid:48)(cid:48) (cid:88) (cid:96) (cid:48)(cid:48) ,m (cid:48)(cid:48) ( − m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) + 1 Y (cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) [ (cid:98) x (cid:48)(cid:48) ] r (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) − M m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) × (cid:0) − η (cid:48) B − ,(cid:126)x (cid:48) + η (cid:48) ∂ η (cid:48) B ,(cid:126)x (cid:48) − C − (cid:1) . (4.1.15)The calculations of B − ,(cid:126)x (cid:48) , B ,(cid:126)x (cid:48) and C − have been performed in the appendix (B). Here we will discusshow each term contributes. By including the r (cid:48)(cid:48) integration from δ G , B − ,(cid:126)x (cid:48) , B ,(cid:126)x (cid:48) and C − have thefollowing forms (cid:90) ∞ d r (cid:48)(cid:48) B − ,(cid:126)x (cid:48) r (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) − = − (cid:88) (cid:96),m ( r ) (cid:96) (2 (cid:96) + 1)!! Y m(cid:96) [ (cid:98) x ] ¯ Y m(cid:96) [ (cid:98) x (cid:48)(cid:48) ] (cid:88) (cid:96) (cid:48) ,m (cid:48) ( r (cid:48) ) (cid:96) (cid:48) (2 (cid:96) (cid:48) + 1)!! Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48)(cid:48) ] ¯ Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48) ] ( − (cid:96) + (cid:96) (cid:48) ( (cid:96) + (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) )!( η − η (cid:48) ) (cid:96) + (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) +1 × π − (cid:96) (cid:48)(cid:48) Γ[ (cid:96) (cid:48)(cid:48) ]Γ (cid:2) − (cid:96) − (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) (cid:3) Γ (cid:2) (cid:96) − (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) +12 (cid:3) Γ (cid:2) − (cid:96) + (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) +12 (cid:3) Γ (cid:2) (cid:96) + (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) +22 (cid:3) (cid:18) O (cid:20) ηη (cid:48) (cid:21)(cid:19) (4.1.16) (cid:90) ∞ d r (cid:48)(cid:48) B ,(cid:126)x (cid:48) r (cid:48)(cid:48)− (cid:96) (cid:48)(cid:48) +1 = (cid:32) π (cid:88) (cid:96) (cid:48) ,m (cid:48) ( r (cid:48) ) (cid:96) (cid:48) (2 (cid:96) (cid:48) + 1)!! Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48)(cid:48) ] ¯ Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48) ] 2 (cid:96) (cid:48)(cid:48) ( − (cid:96) (cid:48) ( η − η (cid:48) ) − (cid:96) (cid:48) − (cid:96) (cid:48)(cid:48) ( (cid:96) (cid:48)(cid:48) ) − (cid:96) (cid:48) + (cid:88) (cid:96) =1 ,m (cid:88) (cid:96) (cid:48) ,m (cid:48) (cid:96) (cid:48)(cid:48) − r (cid:96) r (cid:48) (cid:96) (cid:48) ( − (cid:96) + (cid:96) (cid:48) ( η − η (cid:48) ) − (cid:96) (cid:48)(cid:48) − (cid:96) − (cid:96) (cid:48) (2 (cid:96) (cid:48) + 1)!! Γ[ (cid:96) (cid:48)(cid:48) + (cid:96) + (cid:96) (cid:48) ](2 (cid:96) + 1)!! (4.1.17) × π − (cid:96) (cid:48)(cid:48) Γ[ (cid:96) (cid:48)(cid:48) − Y m(cid:96) [ (cid:98) x ] ¯ Y m(cid:96) [ (cid:98) x (cid:48)(cid:48) ] Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48)(cid:48) ] ¯ Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48) ]Γ (cid:2) − (cid:96) − (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) (cid:3) Γ (cid:2) (cid:96) − (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) − (cid:3) Γ (cid:2) − (cid:96) + (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) +12 (cid:3) Γ (cid:2) (cid:96) + (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) (cid:3) (cid:33) (cid:32) O (cid:34)(cid:18) rη (cid:48) (cid:19) (cid:35)(cid:33)(cid:90) ∞ d r (cid:48)(cid:48) C − r (cid:48)(cid:48)− (cid:96) (cid:48)(cid:48) +1 = (cid:32) − π (cid:88) (cid:96),m ( r ) (cid:96) (2 (cid:96) + 1)!! Y m(cid:96) [ (cid:98) x (cid:48)(cid:48) ] ¯ Y m(cid:96) [ (cid:98) x (cid:48) ] 2 (cid:96) (cid:48)(cid:48) ( − (cid:96) ( η − η (cid:48) ) − (cid:96) − (cid:96) (cid:48)(cid:48) ( (cid:96) (cid:48)(cid:48) ) − (cid:96) − (cid:88) (cid:96) (cid:48) =1 ,m (cid:48) (cid:88) (cid:96),m (cid:96) (cid:48)(cid:48) − r (cid:96) r (cid:48) (cid:96) (cid:48) ( − (cid:96) + (cid:96) (cid:48) ( η − η (cid:48) ) − (cid:96) (cid:48)(cid:48) − (cid:96) − (cid:96) (cid:48) (2 (cid:96) + 1)!! Γ[ (cid:96) (cid:48)(cid:48) + (cid:96) + (cid:96) (cid:48) ](2 (cid:96) (cid:48) + 1)!! (4.1.18) × π − (cid:96) (cid:48)(cid:48) Γ[ (cid:96) (cid:48)(cid:48) − Y m(cid:96) [ (cid:98) x ] ¯ Y m(cid:96) [ (cid:98) x (cid:48)(cid:48) ] Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48)(cid:48) ] ¯ Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48) ]Γ (cid:2) − (cid:96) − (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) (cid:3) Γ (cid:2) (cid:96) (cid:48) − (cid:96) + (cid:96) (cid:48)(cid:48) − (cid:3) Γ (cid:2) − (cid:96) (cid:48) + (cid:96) + (cid:96) (cid:48)(cid:48) +12 (cid:3) Γ (cid:2) (cid:96) + (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) (cid:3) (cid:33) (cid:32) O (cid:34)(cid:18) r (cid:48) η (cid:48) (cid:19) (cid:35)(cid:33) In the first term of eq. (4.1.17), because the Pochhammer symbol ( (cid:96) (cid:48)(cid:48) ) − (cid:96) (cid:48) will blow up if (cid:96) (cid:48) = (cid:96) (cid:48)(cid:48) > ,only the (cid:96) (cid:48)(cid:48) = (cid:96) = 0 terms survive. This in turn renders the first term of eq. (4.1.17) independent ofboth η and η (cid:48) , and since B ,(cid:126)x (cid:48) shows up in eq. (4.1.15) only as a derivative with respect to η (cid:48) , we maytherefore drop the first term of eq. (4.1.17). Similar arguments related to ( (cid:96) (cid:48)(cid:48) ) − (cid:96) in the first term of eq.(4.1.18) tell us only its (cid:96) = 0 term remains.Additionally, the intermediate summations leading up to equations (4.1.16), (4.1.17) and(4.1.18) have been tackled in appendix (A.1) – see the discussion leading up eq. (A.1.5). Now, theGamma functions in the denominators of (4.1.16), (4.1.17) and (4.1.18) will blow up when their ar-guments are non-positive integers. In particular, parity arguments, spelt out in more detail in appendix16A.2), provide the constraint (cid:96) (cid:48)(cid:48) +2 q = (cid:96) + (cid:96) (cid:48) , where q is an arbitrary integer. This implies the Γ (cid:104) − (cid:96) − (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) (cid:105) in equations (4.1.16), (4.1.17) and (4.1.18) diverges, except when (cid:96) (cid:48)(cid:48) = 0 due to the Γ[ (cid:96) (cid:48)(cid:48) ] in eq. (4.1.16)and Γ[ (cid:96) (cid:48)(cid:48) − in equations (4.1.17) and (4.1.18). Angular momentum addition tells us (cid:96) = (cid:96) (cid:48) when (cid:96) (cid:48)(cid:48) = 0 ;hence, at this point, we simply need to compute the limits lim (cid:96) (cid:48)(cid:48) → + − (cid:96) (cid:48)(cid:48) π Γ[ (cid:96) (cid:48)(cid:48) ]Γ (cid:2) (cid:96) (cid:48)(cid:48) − (cid:96) (cid:3) Γ (cid:2) ( (cid:96) (cid:48)(cid:48) + 1) (cid:3) Γ (cid:2) ( (cid:96) (cid:48)(cid:48) + 1) (cid:3) Γ (cid:2) (cid:96) (cid:48)(cid:48) + ( (cid:96) + 1) (cid:3) = ( − (cid:96) . and lim (cid:96) (cid:48)(cid:48) → + − (cid:96) (cid:48)(cid:48) π Γ[ (cid:96) (cid:48)(cid:48) − (cid:2) (cid:96) (cid:48)(cid:48) − (cid:96) (cid:3) Γ (cid:2) (cid:96) (cid:48)(cid:48) (cid:3) Γ (cid:2) ( (cid:96) (cid:48)(cid:48) + 2) (cid:3) Γ (cid:2) (cid:96) (cid:48)(cid:48) + (cid:96) (cid:3) = 2 (cid:96) ( − (cid:96) . Altogether, we gather δ G [ x, x (cid:48) ] = (cid:0) δ , G + δ , G [ x, x (cid:48) ] (cid:1) (cid:32) O (cid:34) ηη (cid:48) , (cid:18) rη (cid:48) (cid:19) , (cid:18) r (cid:48) η (cid:48) (cid:19) (cid:35)(cid:33) , (4.1.19)where we define the monopole-only term (with (cid:96) = (cid:96) (cid:48) = (cid:96) (cid:48)(cid:48) = 0 ) as δ , G ≡ G N H M π , (4.1.20)and the higher multipole terms as δ , G [ x, x (cid:48) ] = 2 G N M H ∞ (cid:88) (cid:96) =1 (cid:96) (cid:88) m = − (cid:96) ( − (cid:96) (cid:96) !!(2 (cid:96) + 1)!! (cid:18) rr (cid:48) η (cid:48) (cid:19) (cid:96) Y m(cid:96) [ (cid:98) x ] ¯ Y m(cid:96) [ (cid:98) x (cid:48) ] . (4.1.21) Linear Propagation, Ψ (0) Recall the definition of Ψ (0) from (4.1.9), Ψ (0) [ η, η (cid:48) , (cid:126)x ] ≡ (cid:90) R d (cid:126)x (cid:48) a [ η (cid:48) ] (cid:16) G [ x, x (cid:48) ] ˙Ψ[ x (cid:48) ] − Ψ[ x (cid:48) ] ∂ η (cid:48) G [ x, x (cid:48) ] (cid:17) . (4.1.22)For late-times, the delta function term in eq. (4.1.10) will not contribute to the observed scalar field Ψ (0) [ η, η (cid:48) , (cid:126)x ] . Moreover, the time derivative of G [ x, x (cid:48) ] on the second term will turn the tail of the Greenfunction into a null signal, which will also be excluded. Additionally, Θ[∆ η − R ] = 1 at late times. Hence,we are left with Ψ (0) [ η, η (cid:48) , (cid:126)x ] ≡ a [ η (cid:48) ] (cid:90) R d (cid:126)x (cid:48) H π ˙Ψ[ η (cid:48) , (cid:126)x (cid:48) ] . (4.1.23)By using the initial condition of the scalar fields in eq.(2.0.6), Ψ (0) [ η, η (cid:48) , (cid:126)x ] ≡ H π a [ η (cid:48) ] (cid:90) ∞ d r (cid:48) r (cid:48) ˙ C [ r (cid:48) ] . (4.1.24)17 inear Propagation, Ψ (0 ,χ ) The next part is Ψ (0 ,χ ) . From eq. (4.1.9), Ψ (0 ,χ ) [ η, η (cid:48) , (cid:126)x ] = (cid:90) R d (cid:126)x (cid:48) a [ η (cid:48) ]( − ¯ χ ) (cid:16) G [ x, x (cid:48) ] ˙Ψ[ x (cid:48) ] − Ψ[ x (cid:48) ] ∂ η (cid:48) G [ x, x (cid:48) ] (cid:17) . (4.1.25)The same late time considerations for Ψ (0) apply here, since their integral representations employ thesame unperturbed retarded Green function G [ x, x (cid:48) ] . We thus have Ψ (0 ,χ ) [ η, η (cid:48) , (cid:126)x ] = − (cid:90) R d (cid:126)x (cid:48) a [ η (cid:48) ] ¯ χ [ (cid:126)x (cid:48) ] H π (cid:88) (cid:96) (cid:48) ,m (cid:48) ˙ C m (cid:48) (cid:96) (cid:48) [ r (cid:48) ] Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48) ] , (4.1.26)where we use the initial condition of the scalar fields given in eq.(2.0.6). Now, let us recall trace-reversedperturbation metric ¯ χ . Since the r (cid:48) integral runs over all radius, we will have two integration regions:outside of the mass distribution r (cid:48) > r m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) and inside of mass distribution r m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) > r (cid:48) > . We thereforeneed to invoke eq. (4.1.6) to deduce Ψ (0 ,χ ) [ η, η (cid:48) , (cid:126)x ] = 4 G N H (cid:90) ∞ d r (cid:48) r (cid:48) a [ η (cid:48) ] (cid:88) (cid:96) (cid:48) ,m (cid:48) (cid:32) ( − m (cid:48) (cid:96) (cid:48) + 1 1 r (cid:48) (cid:96) (cid:48) +1 M m (cid:48) (cid:96) (cid:48) [ r m (cid:48) (cid:96) (cid:48) ]Θ[ r (cid:48) − r m (cid:48) (cid:96) (cid:48) ]+ ( − m (cid:48) (cid:96) (cid:48) + 1 (cid:40) M m (cid:48) (cid:96) (cid:48) [ r (cid:48) ] r (cid:48) (cid:96) (cid:48) +1 + N m (cid:48) (cid:96) (cid:48) [ r (cid:48) ] r (cid:48) (cid:96) (cid:48) (cid:41) Θ[ r m (cid:48) (cid:96) (cid:48) − r (cid:48) ] (cid:33) ˙ C m (cid:48) (cid:96) (cid:48) [ r (cid:48) ] , (4.1.27)where the first and second terms respectively describe the contribution from the initial scalar field outside(eq. (4.1.4)) and inside (eq. (4.1.6)) the central mass, with M m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) [ r (cid:48) ] and N m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) [ r (cid:48) ] already defined in eq.(4.1.7). While only the initial scalar field velocity’s monopole contributed to the central mass indepen-dent linear propagation result in eq. (4.1.24); we see from eq. (4.1.27), all the initial velocity’s multi-poles do contribute once the gravitational potential of the central mass is included, because the formerare now tied to the mass multipoles in a one-to-one manner. Moreover, both equations (4.1.24) and(4.1.27) describe a final scalar field that does not decay back to zero amplitude, but instead asymptotesto a constant. In addition, we may compare equations (4.1.24) and (4.1.27) to the results obtained byBCLK [18] and BCLP [19], where they dealt with a spherically symmetric black hole in de Sitter space-time. Like them, we find the final scalar field to scale as H ; but unlike them, because of our arbitrarymatter distribution, we uncovered sensitivity to both external and internal multipoles of all orders. Eventhough BCLK and BCLP did not name it as such, they found – as we did here – their massless scalar fieldto develop a memory at late times. Non-Linear Propagation, Ψ (1) Now, let us find how the non-linear parts contribute. Here,the propagation is governed by the first order perturbed retarded Green function. Recall the definition18f Ψ from (4.1.9), Ψ (1) [ η, η (cid:48) , (cid:126)x ] = (cid:90) R d (cid:126)x (cid:48) a [ η (cid:48) ] (cid:16) δ G [ x, x (cid:48) ] ˙Ψ[ x (cid:48) ] − Ψ[ x (cid:48) ] ∂ η (cid:48) δ G [ x, x (cid:48) ] (cid:17) (4.1.28)The perturbed Green function has been computed in eq. (4.1.19). With the initial profile in eq. (2.0.6), Ψ (1) [ η, η (cid:48) , (cid:126)x ] = a [ η (cid:48) ] M G N H (cid:90) ∞ d r (cid:48) r (cid:48) (cid:40) ˙ C [ r (cid:48) ]4 π / (4.1.29) + (cid:88) (cid:96) (cid:48) =1 ,m (cid:48) (cid:18) ˙ C m (cid:48) (cid:96) (cid:48) [ r (cid:48) ] − (cid:96)η (cid:48) C m (cid:48) (cid:96) (cid:48) [ r (cid:48) ] (cid:19) Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x ] (cid:18) rr (cid:48) η (cid:48) (cid:19) (cid:96) (cid:48) (cid:96) (cid:48) )!!( − (cid:96) (cid:48) (2 (cid:96) (cid:48) + 1)!! (cid:41) (cid:32) O (cid:34) ηη (cid:48) , (cid:18) rη (cid:48) (cid:19) , (cid:18) r (cid:48) η (cid:48) (cid:19) (cid:35)(cid:33) Observe that, because only the mass monopole term contributes to δ G in eq. (4.1.19), there is anangular momentum conservation at play here: the initial (cid:96) th multipole of the scalar field at η (cid:48) propagatesforward in time into a pure (cid:96) -multipole at late times η → . Comparison to Static Patch Results The line element of de Sitter in static patch coordi-nates is given by d s = (1 − H r s ) d t s − (1 − H r s ) − d r s − r s d Ω . (4.1.30)The transformation rules from flat slicing coordinates ( η, r ) in eq. (2.0.2) to static patch coordinates ( t s , r s ) is r = r s exp[ − Ht s ] (cid:112) − H r s , and η = − exp[ − Ht s ] H (cid:112) − H r s . (4.1.31)By employing these transformation rules to r (cid:96) in our solution (4.1.29), the (cid:96) th multipole of the scalarfield becomes Ψ (1) ∼ e − (cid:96)Ht s . (4.1.32)This decay law was first obtained by BCLP [19].We also highlight that the late time assumption that allowed us to simplify the Green’s functionexpression to that in eq. (4.1.15), while made in the flat slicing coordinates of eq. (2.0.2), continuesto hold even in the static patch of de Sitter spacetime. This is because η/η (cid:48) ≈ exp[ − H ( t s − t (cid:48) s )] → as t s → ∞ . In the previous section, §(4.1), we saw that at leading order in η/η (cid:48) , r/η (cid:48) , and r (cid:48) /η (cid:48) , the multipolemoments of the central mass distribution did not contribute to the perturbed Green’s function δ G . In19his section, we will show how we may recover this mass-monopole-only result from a position space-time calculation. In the late time limit, we have seen that ¯ χ will never be evaluated inside the massdistribution, and hence the integrand of eq. (4.1.2) may be Taylor expanded as follows ¯ χ [ η (cid:48)(cid:48) , (cid:126)x (cid:48)(cid:48) ] = − G N a [ η (cid:48)(cid:48) ] (cid:32) M | (cid:126)x (cid:48)(cid:48) | + + ∞ (cid:88) (cid:96) (cid:48)(cid:48) =1 ( − ) (cid:96) (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) ! Ξ i ...i (cid:96) ∂ i (cid:48)(cid:48) . . . ∂ i (cid:48)(cid:48) (cid:96) | (cid:126)x (cid:48)(cid:48) | (cid:33) (4.2.1) Ξ i ...i (cid:96) ≡ (cid:90) R d (cid:126)zz i . . . z i (cid:96) (cid:48)(cid:48) ρ [ (cid:126)z ] . (4.2.2)This is equivalent to the spherical harmonic expansion of eq. (4.1.2), with the (cid:96) (cid:48)(cid:48) th term in the summationcorresponding to the (cid:96) (cid:48)(cid:48) th mass multipole moment. Since we only wish to recover the mass monopoleresult of the previous section, we shall discard the summation in eq. (4.2.1) and insert it and eq. (4.1.10)into eq. (2.0.13). By explicitly splitting the de Sitter Green’s function G into the null cone G ( direct ) andtail G ( tail ) pieces, G [ x, x (cid:48) ] = G (direct) [ x, x (cid:48) ] + G (tail) [ x, x (cid:48) ] ,G (direct) [ x, x (cid:48) ] = H π δ [ η − η (cid:48) − | (cid:126)x − (cid:126)x (cid:48) | ] | (cid:126)x − (cid:126)x (cid:48) | ηη (cid:48) ,G (tail) [ x, x (cid:48) ] = H π Θ[ η − η (cid:48) − | (cid:126)x − (cid:126)x (cid:48) | ]; (4.2.3)this leads us to δ G [ x, x (cid:48) ] = (cid:88) ≤ I ≤ δ G ( I ) [ x, x (cid:48) ]; (4.2.4)where δ G (1) [ x, x (cid:48) ] ≡ − η ρµ (cid:48)(cid:48) η σν (cid:48)(cid:48) (cid:90) −∞ d η (cid:48)(cid:48) (cid:90) R d (cid:126)x (cid:48)(cid:48) a [ η (cid:48)(cid:48) ] ∂ µ (cid:48)(cid:48) G ( direct ) [ x, x (cid:48)(cid:48) ] χ ρσ [ x (cid:48)(cid:48) ] ∂ ν (cid:48)(cid:48) G ( direct ) [ x (cid:48)(cid:48) , x (cid:48) ] , = − G N M π H (cid:40) ∂ η ∂ η (cid:48) (cid:18) Θ[ η − η (cid:48) − R ] ηη (cid:48) ( η + η (cid:48) )2 I (cid:19) − ∂ η (cid:18) Θ[ η − η (cid:48) − R ] (cid:18) ηη (cid:48) I + η ( η + η (cid:48) )2 I (cid:19)(cid:19) − ∂ η (cid:48) (cid:18) Θ[ η − η (cid:48) − R ] (cid:18) ηη (cid:48) I + η (cid:48) ( η + η (cid:48) )2 I (cid:19)(cid:19) + Θ[ η − η (cid:48) − R ] (cid:18) ( η + η (cid:48) ) I + η + η (cid:48) I + 2 ηη (cid:48) η + η (cid:48) I (cid:19) (cid:41) , (4.2.5) δ G (2) [ x, x (cid:48) ] ≡ − η ρµ (cid:48)(cid:48) η σν (cid:48)(cid:48) (cid:90) −∞ d η (cid:48)(cid:48) (cid:90) R d (cid:126)x (cid:48)(cid:48) a [ η (cid:48)(cid:48) ] ∂ µ (cid:48)(cid:48) G ( tail ) [ x, x (cid:48)(cid:48) ] χ ρσ [ x (cid:48)(cid:48) ] ∂ ν (cid:48)(cid:48) G ( direct ) [ x (cid:48)(cid:48) , x (cid:48) ] , (4.2.6) = − G N M π H (cid:40) ∂ η (cid:48) (cid:16) Θ[ η − η (cid:48) − R ] η (cid:48) ( η − η (cid:48) ) I − (cid:17) − Θ[ η − η (cid:48) − R ]( η − η (cid:48) ) (cid:18) η (cid:48) η + η (cid:48) I − + I − (cid:19) (cid:41) ,δ G (3) [ x, x (cid:48) ] ≡ − η ρµ (cid:48)(cid:48) η σν (cid:48)(cid:48) (cid:90) −∞ d η (cid:48)(cid:48) (cid:90) R d (cid:126)x (cid:48)(cid:48) a [ η (cid:48)(cid:48) ] ∂ µ (cid:48)(cid:48) G ( direct ) [ x, x (cid:48)(cid:48) ] χ ρσ [ x (cid:48)(cid:48) ] ∂ ν (cid:48)(cid:48) G ( tail ) [ x (cid:48)(cid:48) , x (cid:48) ] , (4.2.7)20 G N M π H (cid:40) ∂ η (cid:16) Θ[ η − η (cid:48) − R ] η ( η − η (cid:48) ) I +4 (cid:17) − Θ[ η − η (cid:48) − R ]( η − η (cid:48) ) (cid:18) I +4 + 2 ηη + η (cid:48) I +5 (cid:19) (cid:41) ,δ G (4) [ x, x (cid:48) ] ≡ − η ρµ (cid:48)(cid:48) η σν (cid:48)(cid:48) (cid:90) −∞ d η (cid:48)(cid:48) (cid:90) R d (cid:126)x (cid:48)(cid:48) a [ η (cid:48)(cid:48) ] ∂ µ (cid:48)(cid:48) G ( tail ) [ x, x (cid:48)(cid:48) ] χ ρσ [ x (cid:48)(cid:48) ] ∂ ν (cid:48)(cid:48) G ( tail ) [ x (cid:48)(cid:48) , x (cid:48) ] , = G N M π H (cid:18) Θ[ η − η (cid:48) − R ] ( η − η (cid:48) ) η + η (cid:48) I (cid:19) , (4.2.8)and, by placing (cid:126)x (cid:48)(cid:48) on the ellipsoidal surface parametrized in eq. (3.1.10) and defining R ≡ | (cid:126)x − (cid:126)x (cid:48) | , theintegrals involved are defined by I ≡ (cid:90) S d Ω (cid:48)(cid:48) π | (cid:126)x (cid:48)(cid:48) | (4.2.9) I ≡ (cid:90) S d Ω (cid:48)(cid:48) π − ξ cos θ (cid:48)(cid:48) | (cid:126)x (cid:48)(cid:48) | , ξ ≡ − Rη + η (cid:48) , < ξ < ,I ≡ (cid:90) S d Ω (cid:48)(cid:48) π | (cid:126)x (cid:48)(cid:48) | (1 − ξ cos θ (cid:48)(cid:48) ) ,I ± ≡ (cid:90) S d Ω (cid:48)(cid:48) π ± ζ cos θ (cid:48)(cid:48) | (cid:126)x (cid:48)(cid:48) | , ζ ≡ Rη − η (cid:48) , < ζ < , η − η (cid:48) > r + r (cid:48) ,I ± ≡ (cid:90) S d Ω (cid:48)(cid:48) π ± ζ cos θ (cid:48)(cid:48) | (cid:126)x (cid:48)(cid:48) | (1 − ξ cos θ (cid:48)(cid:48) ) ,I ≡ (cid:90) S d Ω (cid:48)(cid:48) π (1 + ζ cos θ (cid:48)(cid:48) )(1 − ζ cos θ (cid:48)(cid:48) ) | (cid:126)x (cid:48)(cid:48) | (1 − ξ cos θ (cid:48)(cid:48) ) . The I was evaluated by DeWitt and DeWitt [20]; for η − η (cid:48) > r + r (cid:48) , relevant for our late timecalculations, it reads I = 1 R ln (cid:20) ζ − ζ (cid:21) . (4.2.10)Whereas, in the timelike infinity limit − r/η (cid:48) , − r (cid:48) /η (cid:48) , η/η (cid:48) (cid:28) , both the ξ and ζ tend to zero. As such,the rest of I , , , , are really I up to fractional corrections of order − r/η (cid:48) . I , , , , = I (1 + O [ ζ, ξ ]) (4.2.11)Taking all these into account, and returning to our perturbed Green’s function in eq. (4.2.4), δ G [ x, x (cid:48) ] = − π G N M H (cid:0) η − η (cid:48) (cid:1) (cid:0) ( η − η (cid:48) ) − ηη (cid:48) − R (cid:1)(cid:0) ( η − η (cid:48) ) − R (cid:1) (cid:18) O (cid:20) Rη + η (cid:48) , Rη − η (cid:48) (cid:21)(cid:19) , (4.2.12)Next, we follow Poisson [17] to expand eq. (4.2.12) in powers of cos γ = (cid:98) x · (cid:98) x (cid:48) , via the relation (cos γ ) k = ∞ (cid:88) (cid:96) =0 (cid:96) +1 π (cid:16) − k − (cid:96) (cid:17) Γ[ k + 1]Γ (cid:2) k + (cid:96) + 1 (cid:3) Γ (cid:2) k − (cid:96) + 1 (cid:3) Γ [ k + (cid:96) + 2] (cid:96) (cid:88) m = − (cid:96) Y m(cid:96) [ θ, φ ] Y m(cid:96) [ θ (cid:48) , φ (cid:48) ] . (4.2.13)21t this point, we have recovered δ G of eq. (4.1.19) within the timelike infinity limit: δ G [ x, x (cid:48) ] = 2 G N M H ∞ (cid:88) (cid:96) =0 (cid:96) (cid:88) m = − (cid:96) ( − (cid:96) (cid:96) !!(2 (cid:96) + 1)!! (cid:18) rr (cid:48) η (cid:48) (cid:19) (cid:96) Y m(cid:96) [ θ, φ ] Y m ∗ (cid:96) [ θ (cid:48) , φ (cid:48) ] × (cid:18) O (cid:20) ηη (cid:48) , rη (cid:48) , r (cid:48) η (cid:48) (cid:21)(cid:19) . (4.2.14) In this paper, we have calculated the behavior of a scalar field in the late-time regime of perturbed deSitter spacetime, where the perturbation is generated by a localized mass distribution. We performedthe calculation in a (fictitious) frequency space that amounted to replacing delta functions with theirintegral representations. This allowed us to implement a spherical harmonic decomposition early on inthe computation. We tested the method in §(3.1) by recovering a portion of Poisson’s asymptotically flatresults in [17].By taking the leading order terms of η/η (cid:48) → , − r (cid:48) /η (cid:48) < − r/η (cid:48) (cid:28) expansion at the early stageof calculation, , we recognized three contributions to the observed scalar field at ( η, (cid:126)x ) in the timelikeinfinity limit. The first term is the linear part of the scalar field propagation itself, coming from the tail ofthe de Sitter massless scalar Green’s function. The second is just the linear part with a first order metricperturbation, where the initial scalar field is sensitive to both the internal and external mass multipoles.The first and second terms yield linear scalar memory, as η/η (cid:48) → at fixed r : Ψ does not decay backto zero but to a spacetime constant proportional to H . The third term is the non-linear part of thescalar field propagation, governed by the perturbed Green function. Here, the scalar interacts with thegravitational potential of the central mass through scattering off the intersection of the future lightconeof the initial scalar profile and past lightcone of the observer. There is a nonlinear scalar memory effectlike the linear case, where the scalar decays to a spacetime constant proportional to H M , where M isthe total mass of the matter distribution. However, this nonlinear scalar memory does not appear to besensitive at all to the higher mass multipoles. We also provided an analytic understanding of the decaylaw first uncovered by BCLK [18] for the higher multipoles of the final scalar field: the (cid:96) th momentdecays in time as Ψ (1) ∼ H r (cid:96) .In the future, we might consider the scalar field at null infinity within the late time regime.This point could be quite tricky in de Sitter spacetime, where there exists a cosmological horizon. We22ay also consider more general matter distributions or perhaps an astrophysical systems. It could beof physical interest to extend our late time analysis to that of intermediate times, to understand thefull causal structure of the scalar signal. We also wish to extend the scalar analysis here to that ofgravitational waves – and ask: what sort of nonlinear gravitational memories are there in asymptoticallyde Sitter spacetimes? We were supported by the Ministry of Science and Technology of the R.O.C. under the grant 106-2112-M-008-024-MY3. YWL is currently supported by the Ministry of Science and Technology of the R.O.C.under Project No. MOST 109-2811-M-007-514 A The Multipoles A.1 Evaluation of Double Sum In this paper, we encountered the following double summation that appeared in the spherical harmonicexpansion of the perturbed scalar Green’s function δ G in both Minkowski and de Sitter spacetimes: S [ (cid:96), (cid:96) (cid:48) , (cid:96) (cid:48)(cid:48) ] ≡ (cid:96) (cid:88) n =0 (cid:96) (cid:48) (cid:88) n (cid:48) =0 ( − n + n (cid:48) ( (cid:96) (cid:48)(cid:48) + n + n (cid:48) )! ( (cid:96) + n )! n !( (cid:96) − n )! ( (cid:96) (cid:48) + n (cid:48) )! n (cid:48) !( (cid:96) (cid:48) − n (cid:48) )! , (A.1.1)We may first perform the n (cid:48) summation in eq. (A.1.1) to obtain S [ (cid:96), (cid:96) (cid:48) (cid:96) (cid:48)(cid:48) ] = (cid:96) (cid:88) n =0 ( − n Γ[ (cid:96) + n + 1]Γ[ (cid:96) (cid:48)(cid:48) + n ]Γ[ n + 1]Γ[ (cid:96) − n + 1]Γ[ (cid:96) (cid:48)(cid:48) + (cid:96) (cid:48) + n + 1]Γ[ (cid:96) (cid:48)(cid:48) − (cid:96) (cid:48) + n ] , Re [ (cid:96) (cid:48)(cid:48) ] > , (A.1.2)where we have used the following limit to replace the ( − n (cid:48) factor in eq. (A.1.1), Γ[ − (cid:96) (cid:48) + n (cid:48) ]Γ[ − (cid:96) (cid:48) ] = ( − n (cid:48) Γ[ (cid:96) (cid:48) + 1]Γ[ (cid:96) (cid:48) − n (cid:48) + 1] , ≤ n (cid:48) ≤ (cid:96) (cid:48) . (A.1.3)as well as the identity F [ − (cid:96) (cid:48) , (cid:96) (cid:48) + 1; (cid:96) (cid:48)(cid:48) + n + 1; 1] = Γ[ (cid:96) (cid:48)(cid:48) + n + 1]Γ[ (cid:96) (cid:48)(cid:48) + n ]Γ[ (cid:96) (cid:48)(cid:48) + (cid:96) (cid:48) + n + 1]Γ[ (cid:96) (cid:48)(cid:48) − (cid:96) (cid:48) + n ] , Re [ (cid:96) (cid:48)(cid:48) + n ] > . (A.1.4)23trictly speaking, eq. (A.1.2) does not hold for the monopole (cid:96) (cid:48)(cid:48) = 0 case. Nevertheless, we may proceedto compute the n -summation assuming (cid:96) (cid:48)(cid:48) > , followed by taking the (cid:96) (cid:48)(cid:48) → limit.To perform the n -summation, we may exploit the replacement (A.1.3) again (with n (cid:48) → n and (cid:96) (cid:48) → (cid:96) ) to reach S [ (cid:96), (cid:96) (cid:48) (cid:96) (cid:48)(cid:48) ] = 2 − (cid:96) (cid:48)(cid:48) π Γ[ (cid:96) (cid:48)(cid:48) ]Γ (cid:2) ( (cid:96) (cid:48)(cid:48) − (cid:96) − (cid:96) (cid:48) ) (cid:3) Γ (cid:2) ( (cid:96) (cid:48)(cid:48) + (cid:96) − (cid:96) (cid:48) + 1) (cid:3) Γ (cid:2) ( (cid:96) (cid:48)(cid:48) − (cid:96) + (cid:96) (cid:48) + 1) (cid:3) Γ (cid:2) ( (cid:96) (cid:48)(cid:48) + (cid:96) + (cid:96) (cid:48) + 2) (cid:3) , (A.1.5)for Re [ (cid:96) (cid:48)(cid:48) ] > , where we have employed one of Whipple’s identities (see [DLMF, Eq. 16.4.7][21]): F [ a, − a, c ; d, c − d + 1; 1] (A.1.6) = 2 − c π Γ[ d ]Γ[2 c − d + 1]Γ (cid:2) c + ( a − d + 1) (cid:3) Γ (cid:2) c + 1 − ( a + d ) (cid:3) Γ (cid:2) ( a + d ) (cid:3) Γ (cid:2) ( d − a + 1) (cid:3) , Re [ c ] > . A.2 Constraints on Multipole Indices from Parity Up to this point, the only constraint on the multipole indices occuring within the δ G comes from theusual rules of angular momentum addition. In this section, we will discover another set of constraintsfrom parity considerations.In δ G , we have three multipoles (cid:96) , (cid:96) (cid:48) , and (cid:96) (cid:48)(cid:48) that correspond respectively to the multipoles ofthe observed scalar field, initial profile of the scalar field, and the mass distribution. Let us recall thestructure of δ G , up the scale factor for de Sitter case, δ G [ t, t (cid:48) ; (cid:126)x, (cid:126)x (cid:48) ] ∼ (cid:90) d x (cid:48)(cid:48) ∂ t (cid:48)(cid:48) (cid:18) δ [ t − t (cid:48)(cid:48) − | (cid:126)x − (cid:126)x (cid:48)(cid:48) | ]4 π | (cid:126)x − (cid:126)x (cid:48)(cid:48) | (cid:19) (cid:90) d (cid:126)x (cid:48)(cid:48)(cid:48) ρ [ (cid:126)x (cid:48)(cid:48)(cid:48) ] | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48)(cid:48)(cid:48) | ∂ t (cid:48)(cid:48) (cid:18) δ [ t (cid:48)(cid:48) − t (cid:48) − | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | ]4 π | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | (cid:19) (A.2.1)Upon the parity flip (cid:126)x → − (cid:126)x and (cid:126)x (cid:48) → − (cid:126)x (cid:48) , followed a similar change in the integration variables, δ G [ t, t (cid:48) ; − (cid:126)x, − (cid:126)x (cid:48) ] ∼ (cid:90) d x (cid:48)(cid:48) ∂ t (cid:48)(cid:48) (cid:18) δ [ t − t (cid:48)(cid:48) − | ( − (cid:126)x ) − (cid:126)x (cid:48)(cid:48) | ]4 π | ( − (cid:126)x ) − (cid:126)x (cid:48)(cid:48) | (cid:19) (cid:90) d (cid:126)x (cid:48)(cid:48)(cid:48) ρ [ (cid:126)x (cid:48)(cid:48)(cid:48) ] | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48)(cid:48)(cid:48) | ∂ t (cid:48)(cid:48) (cid:18) δ [ t (cid:48)(cid:48) − t (cid:48) − | (cid:126)x (cid:48)(cid:48) − ( − (cid:126)x (cid:48) ) | ]4 π | (cid:126)x (cid:48)(cid:48) − ( − (cid:126)x (cid:48) ) | (cid:19) (A.2.2) = (cid:90) d x (cid:48)(cid:48) ∂ t (cid:48)(cid:48) (cid:18) δ [ t − t (cid:48)(cid:48) − | (cid:126)x − (cid:126)x (cid:48)(cid:48) | ]4 π | (cid:126)x − (cid:126)x (cid:48)(cid:48) | (cid:19) (cid:90) d (cid:126)x (cid:48)(cid:48)(cid:48) ρ [ − (cid:126)x (cid:48)(cid:48)(cid:48) ] | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48)(cid:48)(cid:48) | ∂ t (cid:48)(cid:48) (cid:18) δ [ t (cid:48)(cid:48) − t (cid:48) − | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | ]4 π | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | (cid:19) (A.2.3)The spherical harmonics dependence of each multipole in eq. (A.2.2) is Y m(cid:96) [ (cid:98) x ] ¯ Y m(cid:96) [ (cid:98) x (cid:48)(cid:48) ] Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48)(cid:48) ] ¯ Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48) ] Y m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) [ (cid:98) x (cid:48)(cid:48)(cid:48) ]( − ) (cid:96) + (cid:96) (cid:48) , Y m(cid:96) [ (cid:98) x ] ¯ Y m(cid:96) [ (cid:98) x (cid:48)(cid:48) ] Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48)(cid:48) ] ¯ Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48) ] Y m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) [ (cid:98) x (cid:48)(cid:48)(cid:48) ]( − ) (cid:96) (cid:48)(cid:48) . Since spherical harmonics are linearly independent, equations (A.2.2) and (A.2.3) should in fact be thesame: (cid:96) + (cid:96) (cid:48) = (cid:96) (cid:48)(cid:48) + 2 q, q ∈ Z . (A.2.4) B Contributions to δ G in de Sitter Background B.1 Spherical Harmonic Decomposition of exp[ iωR ] /R As we shall witness below, the spherical harmonic decomposition of exp[ iωR ] /R is needed in the com-putation of δ G in de Sitter. First re-express it as e iωR πR = i (cid:90) ω d ¯ ω e i ¯ ωR πR + 14 πR . (B.1.1)The first term can be computed by integrating (3.1.7), i (cid:90) ω d ¯ ω e i ¯ ωR πR = (cid:88) (cid:96),m ( − ir ) (cid:96) r (cid:48)(cid:48) (2 (cid:96) + 1)!! (cid:96) (cid:88) s =0 i s s !(2 r (cid:48)(cid:48) ) s ( (cid:96) + s )!( (cid:96) − s )! Y m(cid:96) [ (cid:98) x ] ¯ Y m(cid:96) [ (cid:98) x (cid:48)(cid:48) ] (cid:96) − s (cid:88) k =0 C [ (cid:96) − s, k ] e iωr (cid:48)(cid:48) k !( ω ) − (cid:96) + s + k ( − i ) k r (cid:48)(cid:48)− − k − (cid:88) (cid:96),m (cid:16) rr (cid:48)(cid:48) (cid:17) (cid:96) (cid:96) (cid:96) ! Y m(cid:96) [ (cid:98) x ] ¯ Y m(cid:96) [ (cid:98) x (cid:48)(cid:48) ] r (cid:48)(cid:48) (2 (cid:96) + 1)!! (cid:0) O [(¯ ωr ) ] (cid:1) (B.1.2)For the second term, we refer to Dixon and Lacroix work [23]. They tell us π | (cid:126)x − (cid:126)x (cid:48) | = 14 π ∞ (cid:88) (cid:96) =0 (cid:96) + 12 rr (cid:48)(cid:48) Q (cid:96) (cid:20) r/r (cid:48)(cid:48) ) r/r (cid:48)(cid:48) ) (cid:21) P (cid:96) [ (cid:98) x · (cid:98) x (cid:48)(cid:48) ] (B.1.3)We can approximate ( r/r (cid:48)(cid:48) ) → . In doing so, we need to exploit the behavior of Q (cid:96) (cid:104) r/r (cid:48)(cid:48) ) r/r (cid:48)(cid:48) ) (cid:105) at infinity,which gives us π | (cid:126)x − (cid:126)x (cid:48) | ≈ (cid:88) (cid:96),m (cid:96) ! r (cid:48)(cid:48) (cid:16) rr (cid:48)(cid:48) (cid:17) (cid:96) (cid:96) (2 (cid:96) + 1)!! Y m(cid:96) [ (cid:98) x ] ¯ Y m(cid:96) [ (cid:98) x (cid:48)(cid:48) ] (B.1.4)25y combining equations (B.1.2) and (B.1.4), e iωR πR ≈ (cid:88) (cid:96),m ( − ir ) (cid:96) r (cid:48)(cid:48) (2 (cid:96) + 1)!! (cid:96) (cid:88) s =0 i s s !(2 r (cid:48)(cid:48) ) s ( (cid:96) + s )!( (cid:96) − s )! Y m(cid:96) [ (cid:98) x ] ¯ Y m(cid:96) [ (cid:98) x (cid:48)(cid:48) ] (cid:96) − s (cid:88) k =0 C [ (cid:96) − s, k ] e iωr (cid:48)(cid:48) k !( ω ) − (cid:96) + s + k ( − i ) k r (cid:48)(cid:48)− − k (cid:0) O [(¯ ωr ) ] (cid:1) (B.1.5)Observe that (B.1.4) canceled the second term of (B.1.2). B.2 A a Recall the definition of A a from (4.1.12).We can collapse one delta function by simply integrate η (cid:48)(cid:48) , A a ≡ (cid:90) −∞ d η (cid:48)(cid:48) η (cid:48)(cid:48) a δ [ η − η (cid:48)(cid:48) − | (cid:126)x − (cid:126)x (cid:48)(cid:48) | ]4 π | (cid:126)x − (cid:126)x (cid:48)(cid:48) | δ [ η (cid:48)(cid:48) − η (cid:48) − | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | ]4 π | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | = (cid:0) η − | (cid:126)x − (cid:126)x (cid:48)(cid:48) | (cid:1) a δ [ η − η (cid:48) − | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | − | (cid:126)x − (cid:126)x (cid:48)(cid:48) | ]4 π | (cid:126)x − (cid:126)x (cid:48)(cid:48) | π | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | (B.2.1)where a = − , , . Different value of a may lead to different results. B.2.1 A For a = 0 , we have A ≡ (cid:90) ∞−∞ d ω π e − iω ( η − η (cid:48) ) e iω ( | (cid:126)x − (cid:126)x (cid:48)(cid:48) | ) π | (cid:126)x − (cid:126)x (cid:48)(cid:48) | e iω ( | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | ) π | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | . (B.2.2)We can impose multipole expansions of e iω ( | (cid:126)x − (cid:126)x (cid:48)(cid:48)| ) π | (cid:126)x − (cid:126)x (cid:48)(cid:48) | and e iω ( | (cid:126)x (cid:48)(cid:48)− (cid:126)x (cid:48)| ) π | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | with ω as the frequency andspherical harmonics as basis functions. Here we take the small argument limit of spherical bessel j (cid:96) [ ωr ] and j (cid:96) [ ωr (cid:48) ] since they are small quantities in the time-like infinity case. After we evaluate the ω integral,we have A = i (cid:88) (cid:96),m ( r ) (cid:96) (2 (cid:96) + 1)!! ( − i ) (cid:96) +1 r (cid:48)(cid:48) (cid:96) (cid:88) s =0 i s s !(2 r (cid:48)(cid:48) ) s ( (cid:96) + s )!( (cid:96) − s )! Y m(cid:96) [ (cid:98) x ] ¯ Y m(cid:96) [ (cid:98) x (cid:48)(cid:48) ] (B.2.3) i (cid:88) (cid:96) (cid:48) ,m (cid:48) ( r (cid:48) ) (cid:96) (cid:48) (2 (cid:96) (cid:48) + 1)!! ( − i ) (cid:96) (cid:48) +1 r (cid:48)(cid:48) (cid:96) (cid:48) (cid:88) s (cid:48) =0 i s (cid:48) s (cid:48) !(2 r (cid:48)(cid:48) ) s (cid:48) ( (cid:96) (cid:48) + s (cid:48) )!( (cid:96) (cid:48) − s (cid:48) )! Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48)(cid:48) ] ¯ Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48) ]( i∂ η ) (cid:96) − s + (cid:96) (cid:48) − s (cid:48) δ [( η − η (cid:48) ) − r (cid:48)(cid:48) ] By including the r (cid:48)(cid:48) integral and r (cid:48)(cid:48)− (cid:96) +1 , and referring to the δ G expression in eq.(4.1.15), we can26ollapse the delta function that contains r (cid:48)(cid:48) into η − η (cid:48) and evaluate the η (cid:48) derivative (cid:90) d r (cid:48)(cid:48) r (cid:48)(cid:48)− (cid:96) (cid:48)(cid:48) +1 A = − (cid:88) (cid:96),m ( r ) (cid:96) (2 (cid:96) + 1)!! (cid:96) (cid:88) s =0 s ! ( (cid:96) + s )!( (cid:96) − s )! Y m(cid:96) [ (cid:98) x ] ¯ Y m(cid:96) [ (cid:98) x (cid:48)(cid:48) ] (cid:88) (cid:96) (cid:48) ,m (cid:48) ( r (cid:48) ) (cid:96) (cid:48) (2 (cid:96) (cid:48) + 1)!! (cid:96) (cid:48) (cid:88) s (cid:48) =0 s (cid:48) ! ( (cid:96) (cid:48) + s (cid:48) )!( (cid:96) (cid:48) − s (cid:48) )! Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48)(cid:48) ] ¯ Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48) ]2 (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) ( − − (cid:96) − (cid:96) (cid:48) + s + s (cid:48) ( η − η (cid:48) ) − (cid:96) − (cid:96) (cid:48) − (cid:96) (cid:48)(cid:48) − Γ[ (cid:96) + (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) + 1]Γ[ s + s (cid:48) + (cid:96) (cid:48)(cid:48) + 1] (B.2.4) B.2.2 A For a = 1 , we have A ≡ (cid:90) ∞−∞ d ω π e − iω ( η − η (cid:48) ) ( η − | (cid:126)x − (cid:126)x (cid:48)(cid:48) | ) e iω ( | (cid:126)x − (cid:126)x (cid:48)(cid:48) | ) π | (cid:126)x − (cid:126)x (cid:48)(cid:48) | e iω ( | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | ) π | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | = (cid:90) ∞−∞ d ω π e − iω ( η − η (cid:48) ) η e iω ( | (cid:126)x − (cid:126)x (cid:48)(cid:48) | ) π | (cid:126)x − (cid:126)x (cid:48)(cid:48) | e iω ( | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | ) π | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | − (cid:90) ∞−∞ d ω π e − iω ( η − η (cid:48) ) e iω ( | (cid:126)x − (cid:126)x (cid:48)(cid:48) | ) π e iω ( | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | ) π | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) |≡ ηA + A , (B.2.5)The first term is exactly A multiplied by η ; while the second term A , is unknown at this point.Therefore, we need to compute the latter separately. Note that the spherical harmonic expansion of exp[ iω ( | (cid:126)x − (cid:126)x (cid:48)(cid:48) | )] / π can be obtained from that of exp[ iω ( | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | )] / π | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | (which is a standardresult) via differentiation, namely − i∂ ω exp[ iω ( | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | )]4 π | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | = exp[ iω ( | (cid:126)x − (cid:126)x (cid:48)(cid:48) | )]4 π . (B.2.6)This leads us to A , = − (cid:90) ∞−∞ d ω π e − iω ( η − η (cid:48) ) (cid:88) (cid:96),m r (cid:48)(cid:48) rω (cid:96) + 1 (cid:16) h (1) (cid:96) − [ r (cid:48)(cid:48) ω ] j (cid:96) − [ rω ] − h (1) (cid:96) +1 [ r (cid:48)(cid:48) ω ] j (cid:96) +1 [ rω ] (cid:17) Y m(cid:96) [ (cid:98) x ] ¯ Y m(cid:96) [ (cid:98) x (cid:48)(cid:48) ] iω (cid:88) (cid:96) (cid:48) ,m (cid:48) j (cid:96) (cid:48) [ ωr (cid:48) ] h (1) (cid:96) (cid:48) [ ωr (cid:48)(cid:48) ] Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48)(cid:48) ] ¯ Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48) ] (B.2.7)We cannot take small argument limit of j − [ ωr ] directly. What we should do instead is to do the (cid:96) = 0 separately. By using a certain recursion relation of the spherical Bessel function, we have A (cid:96) =01 , = − (cid:90) ∞−∞ d ω π e − iω ( η − η (cid:48) ) r (cid:48)(cid:48) rω (cid:18)(cid:18) − h (1)1 [ r (cid:48)(cid:48) ω ] + 1 ωr (cid:48)(cid:48) h (1)0 [ r (cid:48)(cid:48) ω ] (cid:19) (cid:18) − j [ rω ] + 1 ωr j [ rω ] (cid:19) − h (1)1 [ r (cid:48)(cid:48) ω ] j [ rω ] (cid:19) Y [ (cid:98) x ] ¯ Y [ (cid:98) x (cid:48)(cid:48) ] iω (cid:88) (cid:96) (cid:48) ,m (cid:48) j (cid:96) (cid:48) [ ωr (cid:48) ] h (1) (cid:96) (cid:48) [ ωr (cid:48)(cid:48) ] Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48)(cid:48) ] ¯ Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48) ] δ (cid:96), (B.2.8)27he same procedure as the previous calculation can be used at this level. Take the small argument limitof j (cid:96) [ ωr ] , evaluate the ω integral, and the include r (cid:48)(cid:48) integral to obtain (cid:90) ∞ d r (cid:48)(cid:48) r (cid:48)(cid:48)− (cid:96) (cid:48)(cid:48) +1 A (cid:96) =01 , = − δ (cid:96), π (cid:88) (cid:96) (cid:48) ,m (cid:48) ( r (cid:48) ) (cid:96) (cid:48) ( η − η (cid:48) ) − (cid:96) (cid:48) − (cid:96) (cid:48)(cid:48) − (2 (cid:96) (cid:48) + 1)!! (cid:96) (cid:48) (cid:88) s (cid:48) =0 s (cid:48) ! ( (cid:96) (cid:48) + s (cid:48) )!( (cid:96) (cid:48) − s (cid:48) )! Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48)(cid:48) ] ¯ Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48) ]2 (cid:96) (cid:48)(cid:48) − ( − − (cid:96) (cid:48) + s (cid:48) ( (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) )!( s (cid:48) + (cid:96) (cid:48)(cid:48) )! (cid:18) r (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) + 1) + ( η − η (cid:48) ) (cid:19) (B.2.9)The first term is supressed by ( r/η (cid:48) ) , while the second term is the leading term which contributes tothe overall power of η (cid:48) .For A , with (cid:96) ≥ in eq.(B.2.7), A (cid:96) ≥ , ≡ − (cid:90) ∞−∞ d ω π e − iω ( η − η (cid:48) ) (cid:88) (cid:96) =1 ,m r (cid:48)(cid:48) rω (cid:96) + 1 (cid:16) h (1) (cid:96) − [ r (cid:48)(cid:48) ω ] j (cid:96) − [ rω ] − h (1) (cid:96) +1 [ r (cid:48)(cid:48) ω ] j (cid:96) +1 [ rω ] (cid:17) Y m(cid:96) [ (cid:98) x ] ¯ Y m(cid:96) [ (cid:98) x (cid:48)(cid:48) ] iω (cid:88) (cid:96) (cid:48) ,m (cid:48) j (cid:96) (cid:48) [ ωr (cid:48) ] h (1) (cid:96) (cid:48) [ ωr (cid:48)(cid:48) ] Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48)(cid:48) ] ¯ Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48) ] . (B.2.10)This can be computed with the same procedure as previous calculation. Here we can directly take smallargument limit of j (cid:96) [ ωr ] . By including the r (cid:48)(cid:48) integral, its solution is (cid:90) ∞ d r (cid:48)(cid:48) r (cid:48)(cid:48)− (cid:96) (cid:48)(cid:48) +1 A (cid:96) ≥ , = − (cid:88) (cid:96) =1 ,m (cid:88) (cid:96) (cid:48) ,m (cid:48) (cid:96) (cid:48) (cid:88) s (cid:48) =0 (cid:96) (cid:48)(cid:48) − r (cid:96) r (cid:48) (cid:96) (cid:48) ( − − s (cid:48) + (cid:96) + (cid:96) (cid:48) ( η − η (cid:48) ) − (cid:96) (cid:48)(cid:48) − (cid:96) − (cid:96) (cid:48) − (2 (cid:96) + 1)(2 (cid:96) (cid:48) + 1)!! s (cid:48) ! ( (cid:96) (cid:48) + s (cid:48) )!( (cid:96) (cid:48) − s (cid:48) )! (B.2.11) (cid:32) (cid:96) − (cid:88) s =0 ( − ) s ( η − η (cid:48) ) Γ( s + (cid:96) )Γ( (cid:96) (cid:48)(cid:48) + (cid:96) + (cid:96) (cid:48) )(2 (cid:96) − !! Γ( (cid:96) − s )Γ[ (cid:96) (cid:48)(cid:48) + s + s (cid:48) ] s ! − (cid:96) +1 (cid:88) s =0 ( − ) s r Γ( s + (cid:96) + 2)Γ( (cid:96) (cid:48)(cid:48) + (cid:96) + (cid:96) (cid:48) + 2)(2 (cid:96) + 3) !! Γ[2 − s + (cid:96) ]Γ[ (cid:96) (cid:48)(cid:48) + s + s (cid:48) ] s ! (cid:33) . B.2.3 A − For a = − , we have A − ≡ (cid:90) ∞−∞ d ω π e − iω ( η − η (cid:48) ) η − | (cid:126)x − (cid:126)x (cid:48)(cid:48) | e iω ( | (cid:126)x − (cid:126)x (cid:48)(cid:48) | ) π | (cid:126)x − (cid:126)x (cid:48)(cid:48) | e iω ( | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | ) π | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | . (B.2.12)At late time η → , as long as | (cid:126)x − (cid:126)x (cid:48)(cid:48) | (cid:54) = 0 , we can expand (1 − η | (cid:126)x − (cid:126)x (cid:48)(cid:48) | ) − around η = 0 . Up to the zerothorder, we have A − ≈ (cid:90) ∞−∞ d ω π e − iω ( η − η (cid:48) ) −| (cid:126)x − (cid:126)x (cid:48)(cid:48) | e iω ( | (cid:126)x − (cid:126)x (cid:48)(cid:48) | ) π | (cid:126)x − (cid:126)x (cid:48)(cid:48) | e iω ( | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | ) π | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | (cid:18) O [ η | (cid:126)x − (cid:126)x (cid:48)(cid:48) | ] (cid:19) (B.2.13)28et’s employ spherical harmonic decomposition on this expression. Notice that the multipole expansionof exp[ iωR ] /R is non-trivial – see appendix (B.1). A − ≈ − (cid:90) ∞−∞ d ω π e − iω ( η − η (cid:48) ) (cid:88) (cid:96),m ( − rω ) (cid:96) (2 (cid:96) + 1)!! e iωr (cid:48)(cid:48) r (cid:48)(cid:48) (cid:96) (cid:88) s =0 i s + (cid:96) s s ! ( (cid:96) + s )!( (cid:96) − s )! (cid:96) − s (cid:88) k =0 ( − i ) − k k !( ωr (cid:48)(cid:48) ) − k − s C [ (cid:96) − s, k ] Y m(cid:96) [ (cid:98) x ] ¯ Y m(cid:96) [ (cid:98) x (cid:48)(cid:48) ] iω (cid:88) (cid:96) (cid:48) ,m (cid:48) ( ωr (cid:48) ) (cid:96) (cid:48) (2 (cid:96) (cid:48) + 1)!! e iωr (cid:48)(cid:48) ωr (cid:48)(cid:48) (cid:96) (cid:48) (cid:88) s (cid:48) =0 i s (cid:48) ( − i ) (cid:96) (cid:48) +1 s (cid:48) !(2 ωr (cid:48)(cid:48) ) s (cid:48) ( (cid:96) (cid:48) + s (cid:48) )!( (cid:96) (cid:48) − s (cid:48) )! Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48)(cid:48) ] ¯ Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48) ] (B.2.14)By following the same scenario as previous calculation, and including the r (cid:48)(cid:48) integral, we have (cid:90) ∞ d r (cid:48)(cid:48) r (cid:48)(cid:48)− (cid:96) (cid:48)(cid:48) +1 A − = − (cid:88) (cid:96),m ( r ) (cid:96) (2 (cid:96) + 1)!! (cid:96) (cid:88) s =0 s ! ( (cid:96) + s )!( (cid:96) − s )! (cid:96) − s (cid:88) k =0 k ! C [ (cid:96) − s, k ] Y m(cid:96) [ (cid:98) x ] ¯ Y m(cid:96) [ (cid:98) x (cid:48)(cid:48) ] (cid:88) (cid:96) (cid:48) ,m (cid:48) ( r (cid:48) ) (cid:96) (cid:48) (2 (cid:96) (cid:48) + 1)!! (cid:96) (cid:48) (cid:88) s (cid:48) =0 s (cid:48) ! ( (cid:96) (cid:48) + s (cid:48) )!( (cid:96) (cid:48) − s (cid:48) )! Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48)(cid:48) ] ¯ Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48) ]2 k + (cid:96) (cid:48)(cid:48) +1 ( − − k − s − s (cid:48) + (cid:96) + (cid:96) (cid:48) Γ[ (cid:96) + (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) + 2]Γ[ k + s + s (cid:48) + (cid:96) (cid:48)(cid:48) + 2] ( η − η (cid:48) ) − (cid:96) − (cid:96) (cid:48) − (cid:96) (cid:48)(cid:48) − (B.2.15) B.3 B b,(cid:126)x o The next part is B b,(cid:126)x o , which defined as B b,(cid:126)x o ≡ (cid:90) −∞ d η (cid:48)(cid:48) η (cid:48)(cid:48) b δ [ η − η (cid:48)(cid:48) − | (cid:126)x − (cid:126)x (cid:48)(cid:48) | ] δ [ η (cid:48)(cid:48) − η (cid:48) − | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | ](4 π ) | (cid:126)x (cid:48)(cid:48) − (cid:126)x o | , (B.3.1)where b = − , and (cid:126)x o = { (cid:126)x, (cid:126)x (cid:48) } . We can collapse one delta function by carrying out the η (cid:48)(cid:48) integral, B b,(cid:126)x o = (cid:0) η − | (cid:126)x − (cid:126)x (cid:48)(cid:48) | (cid:1) b δ [ η − η (cid:48) − | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | − | (cid:126)x − (cid:126)x (cid:48)(cid:48) | ](4 π ) | (cid:126)x (cid:48)(cid:48) − (cid:126)x o | (B.3.2)Now, let use integral representation of delta functions to convert the integrand to frequencyspace, B b,(cid:126)x o = (cid:90) ∞−∞ d ω π (cid:0) η − | (cid:126)x − (cid:126)x (cid:48)(cid:48) | (cid:1) b e − iω ( η − η (cid:48) ) e iω ( | (cid:126)x − (cid:126)x (cid:48)(cid:48) | ) e iω ( | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | ) (4 π ) | (cid:126)x (cid:48)(cid:48) − (cid:126)x o | . (B.3.3)In the next subsection, we will evaluate B b,(cid:126)x o for different values of b and (cid:126)x o . B.3.1 B ,(cid:126)x o For b = 0 , we have B ,(cid:126)x o ≡ (cid:90) ∞−∞ d ω π e − iω ( η − η (cid:48) ) e iω ( | (cid:126)x − (cid:126)x (cid:48)(cid:48) | ) e iω ( | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | ) (4 π ) | (cid:126)x (cid:48)(cid:48) − (cid:126)x o | . (B.3.4)29e recognize that the integrand is identical to that of A , . For (cid:126)x o = (cid:126)x (cid:48) , B ,(cid:126)x (cid:48) = − A , ; (B.3.5)while for (cid:126)x o = (cid:126)x , we can just swap r and r (cid:48) from the A , result (cid:90) ∞ d r (cid:48)(cid:48) r (cid:48)(cid:48)− (cid:96) (cid:48)(cid:48) +1 B (cid:96) (cid:48) =00 ,(cid:126)x = δ (cid:96) (cid:48) , π (cid:88) (cid:96),m ( r ) (cid:96) ( η − η (cid:48) ) − (cid:96) − (cid:96) (cid:48)(cid:48) − (2 (cid:96) + 1)!! (cid:96) (cid:88) s =0 s ! ( (cid:96) + s )!( (cid:96) − s )! Y m(cid:96) [ (cid:98) x (cid:48)(cid:48) ] ¯ Y m(cid:96) [ (cid:98) x ]2 (cid:96) (cid:48)(cid:48) − ( − (cid:96) + s ( (cid:96) + (cid:96) (cid:48)(cid:48) )!( s + (cid:96) (cid:48)(cid:48) )! (cid:18) r (cid:48) (cid:96) + (cid:96) (cid:48)(cid:48) + 1) + ( η − η (cid:48) ) (cid:19) , (B.3.6) (cid:90) ∞ d r (cid:48)(cid:48) r (cid:48)(cid:48)− (cid:96) (cid:48)(cid:48) +1 B (cid:96) (cid:48) ≥ ,(cid:126)x = (cid:88) (cid:96),m (cid:88) (cid:96) (cid:48) =1 ,m (cid:48) (cid:96) (cid:88) s =0 (cid:96) (cid:48)(cid:48) − r (cid:96) r (cid:48) (cid:96) (cid:48) ( − − s + (cid:96) + (cid:96) (cid:48) ( η − η (cid:48) ) − (cid:96) (cid:48)(cid:48) − (cid:96) − (cid:96) (cid:48) − (2 (cid:96) + 1)(2 (cid:96) (cid:48) + 1)!! s ! ( (cid:96) + s )!( (cid:96) − s )! (B.3.7) (cid:32) (cid:96) (cid:48) − (cid:88) s (cid:48) =0 ( − ) s (cid:48) ( η − η (cid:48) ) Γ( s (cid:48) + (cid:96) (cid:48) )Γ( (cid:96) (cid:48)(cid:48) + (cid:96) + (cid:96) (cid:48) )(2 (cid:96) (cid:48) − !! Γ( (cid:96) (cid:48) − s (cid:48) )Γ[ (cid:96) (cid:48)(cid:48) + s + s (cid:48) ] s (cid:48) ! − (cid:96) (cid:48) +1 (cid:88) s (cid:48) =0 ( − ) s (cid:48) r (cid:48) Γ( s (cid:48) + (cid:96) (cid:48) + 2)Γ( (cid:96) (cid:48)(cid:48) + (cid:96) + (cid:96) (cid:48) + 2)(2 (cid:96) (cid:48) + 3) !! Γ[2 − s (cid:48) + (cid:96) (cid:48) ]Γ[ (cid:96) (cid:48)(cid:48) + s + s (cid:48) ] s (cid:48) ! (cid:33) The first term of eq. (B.3.6) and second term of eq.(B.3.7) is supressed by ( r (cid:48) /η (cid:48) ) B.3.2 B − ,(cid:126)x o For b = − , we have B − ,(cid:126)x o ≡ (cid:90) ∞−∞ d ω π e − iω ( η − η (cid:48) ) η − | (cid:126)x − (cid:126)x (cid:48)(cid:48) | e iω ( | (cid:126)x − (cid:126)x (cid:48)(cid:48) | ) e iω ( | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | ) (4 π ) | (cid:126)x (cid:48)(cid:48) − (cid:126)x o | . (B.3.8)At late time η → , as long as | (cid:126)x − (cid:126)x (cid:48)(cid:48) | (cid:54) = 0 , we can expand (1 − η | (cid:126)x − (cid:126)x (cid:48)(cid:48) | ) − around η = 0 . Up to the zerothorder, B − ,(cid:126)x o ≈ (cid:90) ∞−∞ d ω π e − iω ( η − η (cid:48) ) −| (cid:126)x − (cid:126)x (cid:48)(cid:48) | e iω ( | (cid:126)x − (cid:126)x (cid:48)(cid:48) | ) e iω ( | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | ) (4 π ) | (cid:126)x (cid:48)(cid:48) − (cid:126)x o | (cid:18) O [ η | (cid:126)x − (cid:126)x (cid:48)(cid:48) | ] (cid:19) . (B.3.9)For (cid:126)x o = (cid:126)x (cid:48) , B − ,(cid:126)x (cid:48) is exactly A with minus sign difference B − ,(cid:126)x (cid:48) ≈ − A ; (B.3.10)while, for (cid:126)x o = (cid:126)x , it will be quite different B − ,(cid:126)x ≈ − (cid:90) ∞−∞ d ω π e − iω ( η − η (cid:48) ) e iω ( | (cid:126)x − (cid:126)x (cid:48)(cid:48) | ) e iω ( | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | ) (4 π ) | (cid:126)x − (cid:126)x (cid:48)(cid:48) | . (B.3.11)The use of equations (B.2.6) and (B.1.5) hands us B − ,(cid:126)x ≈ (cid:90) ∞−∞ d ω π e − iω ( η − η (cid:48) ) (cid:88) (cid:96),m ( − rω ) (cid:96) (2 (cid:96) + 1)!! e iωr (cid:48)(cid:48) r (cid:48)(cid:48) (cid:96) (cid:88) s =0 i s + (cid:96) s s ! ( (cid:96) + s )!( (cid:96) − s )! (cid:96) − s (cid:88) k =0 ( − i ) − k k !( ωr (cid:48)(cid:48) ) − k − s C [ (cid:96) − s, k ] Y m(cid:96) [ (cid:98) x ] ¯ Y m(cid:96) [ (cid:98) x (cid:48)(cid:48) ] (cid:88) (cid:96) (cid:48) ,m (cid:48) r (cid:48)(cid:48) r (cid:48) ω (cid:96) (cid:48) + 1 (cid:16) h (1) (cid:96) (cid:48) − ( r (cid:48)(cid:48) ω ) j (cid:96) (cid:48) − ( r (cid:48) ω ) − h (1) (cid:96) (cid:48) +1 ( r (cid:48)(cid:48) ω ) j (cid:96) (cid:48) +1 ( r (cid:48) ω ) (cid:17) Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48)(cid:48) ] ¯ Y m (cid:48) (cid:96) (cid:48) [ (cid:98) x (cid:48) ] (B.3.12)30ust as in A , , we cannot take small argument limit of j − [ ωr (cid:48) ] directly when (cid:96) (cid:48) = 0 . Because of that,we separate the calculation for (cid:96) (cid:48) = 0 and (cid:96) (cid:48) ≥ . The procedure is the same as previous calculation. Byincluding r (cid:48)(cid:48) , we found (cid:90) ∞ d r (cid:48)(cid:48) r (cid:48)(cid:48)− (cid:96) (cid:48)(cid:48) +1 B (cid:96) (cid:48) =0 − ,(cid:126)x = δ (cid:96) (cid:48) , π (cid:88) (cid:96),m (cid:96) (cid:88) s =0 (cid:96) − s (cid:88) k =0 k !2 k + (cid:96) (cid:48)(cid:48) r (cid:96) ( s + (cid:96) )!( (cid:96) − s )! ( − k − s + (cid:96) ( (cid:96) + (cid:96) (cid:48)(cid:48) )!( η − η (cid:48) ) − (cid:96) − (cid:96) (cid:48)(cid:48) − C [ (cid:96) − s, k ] s !(2 (cid:96) + 1)!!Γ[ k + s + (cid:96) (cid:48)(cid:48) + 2] (cid:0) η − η (cid:48) ) (1 + k + s + (cid:96) (cid:48)(cid:48) ) + 2 r (cid:48) ( (cid:96) + (cid:96) (cid:48)(cid:48) + 1)( (cid:96) + (cid:96) (cid:48)(cid:48) + 2) (cid:1) Y m(cid:96) [ (cid:98) x ] ¯ Y m(cid:96) [ (cid:98) x (cid:48)(cid:48) ] (B.3.13) (cid:90) ∞ d r (cid:48)(cid:48) r (cid:48)(cid:48)− (cid:96) (cid:48)(cid:48) +1 B (cid:96) (cid:48) ≥ − ,(cid:126)x = (cid:88) (cid:96),m (cid:96) (cid:88) s =0 (cid:96) − s (cid:88) k =0 (cid:88) (cid:96) (cid:48) =1 ,m (cid:48) (cid:32) (cid:96) (cid:48) − (cid:88) s (cid:48) =1 ( η − η (cid:48) ) ( (cid:96) (cid:48) − s (cid:48) ) ( (cid:96) (cid:48) − s (cid:48) + 1)(2 (cid:96) (cid:48) − (cid:96) (cid:48) + s (cid:48) ]Γ[ (cid:96) + (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) + 1] − (cid:96) (cid:48) +1 (cid:88) s (cid:48) =1 r (cid:48) Γ[ (cid:96) (cid:48) + s (cid:48) + 2] Γ[ (cid:96) + (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) + 3](2 (cid:96) (cid:48) + 3)!! (cid:33) ( − i ) (cid:96) k + (cid:96) (cid:48)(cid:48) Γ[ k + 1]( ir ) (cid:96) r (cid:48) (cid:96) (cid:48) Γ[ (cid:96) + s + 1]( − − k + (cid:96) (cid:48) − s − s (cid:48) ( η − η (cid:48) ) − (cid:96) − (cid:96) (cid:48) − (cid:96) (cid:48)(cid:48) − C [ (cid:96) − s, k ](2 (cid:96) (cid:48) + 1) s ! s (cid:48) !(2 (cid:96) + 1)!!( (cid:96) − s )!Γ( (cid:96) (cid:48) − s (cid:48) + 2)Γ[ k + (cid:96) (cid:48) + s + s (cid:48) + 1] (B.3.14) B.4 C − The last one is C − : C − = (cid:90) −∞ d η (cid:48)(cid:48) π ) η (cid:48)(cid:48) δ [ η − η (cid:48)(cid:48) − | (cid:126)x − (cid:126)x (cid:48)(cid:48) | ] δ [ η (cid:48)(cid:48) − η (cid:48) − | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | ] . (B.4.1)We may evaluate the η (cid:48)(cid:48) integral by collapsing one of the delta functions, and use the delta function’sintegral representation to convert the integrand to frequency space, C − = 1(4 π ) (cid:90) ∞−∞ d ω π e − iω ( η − η (cid:48) ) η − | (cid:126)x − (cid:126)x (cid:48)(cid:48) | e iω ( | (cid:126)x − (cid:126)x (cid:48)(cid:48) | ) e iω ( | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | ) (B.4.2)At late time η → , as long as | (cid:126)x − (cid:126)x (cid:48)(cid:48) | (cid:54) = 0 , we can expand (1 − η | (cid:126)x − (cid:126)x (cid:48)(cid:48) | ) − around η = 0 . Up to zerothorder, C − ≈ − (cid:90) ∞−∞ d ω π e − iω ( η − η (cid:48) ) (4 π ) | (cid:126)x − (cid:126)x (cid:48)(cid:48) | e iω ( | (cid:126)x − (cid:126)x (cid:48)(cid:48) | ) e iω ( | (cid:126)x (cid:48)(cid:48) − (cid:126)x (cid:48) | ) (cid:18) O [ η | (cid:126)x − (cid:126)x (cid:48)(cid:48) | ] (cid:19) = − B ,(cid:126)x . (B.4.3) References [1] Y. Z. Chu, “Gravitational Wave Memory In dS n and 4D Cosmology,” Class. Quant. Grav. ,no.3, 035009 (2017) doi:10.1088/1361-6382/34/3/035009 [arXiv:1603.00151 [gr-qc]].312] Y. Z. Chu, “More On Cosmological Gravitational Waves And Their Memories,” Class. Quant. Grav. , no.19, 194001 (2017) doi:10.1088/1361-6382/aa8392 [arXiv:1611.00018 [gr-qc]].[3] Y. Z. Chu, “Transverse traceless gravitational waves in a spatially flat FLRW universe:Causal structure from dimensional reduction,” Phys. Rev. D , no.12, 124038 (2015)doi:10.1103/PhysRevD.92.124038 [arXiv:1504.06337 [gr-qc]].[4] Y. Z. Chu and G. D. Starkman, “Retarded Green’s Functions In Perturbed Spacetimes For Cosmologyand Gravitational Physics,” Phys. Rev. D , 124020 (2011) doi:10.1103/PhysRevD.84.124020[arXiv:1108.1825 [astro-ph.CO]].[5] D. Christodoulou, “Nonlinear nature of gravitation and gravitational wave experiments,” Phys. Rev.Lett. , 1486-1489 (1991) doi:10.1103/PhysRevLett.67.1486[6] L. Blanchet and T. Damour, “Hereditary effects in gravitational radiation,” Phys. Rev. D , 4304-4319 (1992) doi:10.1103/PhysRevD.46.4304[7] H. J. de Vega, J. Ramirez and N. G. Sanchez, “Generation of gravitational waves by generic sourcesin de Sitter space-time,” Phys. Rev. D , 044007 (1999) doi:10.1103/PhysRevD.60.044007[arXiv:astro-ph/9812465 [astro-ph]].[8] A. Ashtekar, B. Bonga and A. Kesavan, “Asymptotics with a positive cosmological constant: III. Thequadrupole formula,” Phys. Rev. D , no.10, 104032 (2015) doi:10.1103/PhysRevD.92.104032[arXiv:1510.05593 [gr-qc]].[9] G. Date and S. J. Hoque, “Gravitational waves from compact sources in a de Sitter background,”Phys. Rev. D , no.6, 064039 (2016) doi:10.1103/PhysRevD.94.064039 [arXiv:1510.07856 [gr-qc]].[10] L. Bieri, D. Garfinkle and S. T. Yau, “Gravitational wave memory in de Sitter spacetime,” Phys. Rev.D , no.6, 064040 (2016) doi:10.1103/PhysRevD.94.064040 [arXiv:1509.01296 [gr-qc]].[11] B. Bonga and K. Prabhu, “BMS-like symmetries in cosmology,” Phys. Rev. D , no.10, 104043(2020) doi:10.1103/PhysRevD.102.104043 [arXiv:2009.01243 [gr-qc]].[12] Y. Hamada, M. S. Seo and G. Shiu, “Memory in de Sitter space and Bondi-Metzner-Sachs-like supertranslations,” Phys. Rev. D , no.2, 023509 (2017) doi:10.1103/PhysRevD.96.023509[arXiv:1702.06928 [hep-th]]. 3213] A. Kehagias and A. Riotto, “BMS in Cosmology,” JCAP , 059 (2016) doi:10.1088/1475-7516/2016/05/059 [arXiv:1602.02653 [hep-th]].[14] A. Tolish and R. M. Wald, “Cosmological memory effect,” Phys. Rev. D , no.4, 044009 (2016)doi:10.1103/PhysRevD.94.044009 [arXiv:1606.04894 [gr-qc]].[15] C. S. Chu and Y. Koyama, “Memory effect in anti–de Sitter spacetime,” Phys. Rev. D , no.10,104034 (2019) doi:10.1103/PhysRevD.100.104034 [arXiv:1906.09361 [hep-th]].[16] R. H. Price, “Nonspherical Perturbations of Relativistic Gravitational Collapse. II. Integer-Spin,Zero-Rest-Mass Fields,” Phys. Rev. D , 2439-2454 (1972) doi:10.1103/PhysRevD.5.2439[17] E. Poisson, “Radiative falloff of a scalar field in a weakly curved space-time without symmetries,”Phys. Rev. D , 044008 (2002) doi:10.1103/PhysRevD.66.044008 [arXiv:gr-qc/0205018 [gr-qc]].[18] P. R. Brady, C. M. Chambers, W. Krivan and P. Laguna, “Telling tails in the presence of a cosmo-logical constant,” Phys. Rev. D , 7538-7545 (1997) doi:10.1103/PhysRevD.55.7538 [arXiv:gr-qc/9611056 [gr-qc]].[19] P. R. Brady, C. M. Chambers, W. G. Laarakkers and E. Poisson, “Radiative falloff in Schwarzschild-deSitter space-time,” Phys. Rev. D , 064003 (1999) doi:10.1103/PhysRevD.60.064003 [arXiv:gr-qc/9902010 [gr-qc]].[20] C. M. DeWitt and B. S. DeWitt, “Falling charges,” Physics Physique Fizika , 3-20 (1964) [erratum:Physics Physique Fizika , 145 (1964)] doi:10.1103/PhysicsPhysiqueFizika.1.3[21] [DLMF] NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.1.0 of2020-12-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W.Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds.[22] D. Binosi, J. Collins, C. Kaufhold and L. Theussl, “JaxoDraw: A Graphical user interface for drawingFeynman diagrams. Version 2.0 release notes,” Comput. Phys. Commun.180