Cosmological memory effect in scalar-tensor theories

The recent detection of gravitational waves (GWs) has opened up a new window to probe different aspects of gravitational interaction that otherwise are impossible to be explored [1, 2]. As one of the observable byproduct of the GWs, the so-called memory effect, which is a permanent and sudden change of the relative distance between two observers through the passage of GWs, has received a lots of attention in recent years. This effect was first indicated by Zel’dovich and Polnarev [3] and later completed by Christodoulou by taking into account nonlinear effects [4] (see also [5, 6]).

In combination with the soft theorem and the asymptotic symmetry, the memory effect constitutes the so-called “infrared triangle” [7]. The soft theorem was first discovered in the context of QED [8] and a few years later generalized by Weinberg to theories involving particles with arbitrary spins including gravitons [9]. In QED, it is implemented to cancel infrared divergences to preserve the consistency of the field theory [10]. On the other hand, the known asymptotic symmetry for gravity is the so-called Bondi, van der Burg, Metzner and Sachs (BMS) symmetry which is the symmetry group of diffeomorphism transformations that do not break asymptotic flatness at conformal infinities [11, 12]. The three corners of the infrared triangle are expected to be equivalent to each other: i) equivalence of BMS symmetry with the soft graviton theorem can be shown by Ward-Takahashi identity [13], ii) the permanent change of the distance between observers caused by the memory effect can be realized through a step functional change of the spacetime metric and this change is indeed the same as the one generated by the BMS transformations [7], iii) the change in an asymptotic metric due to the memory effect has the same form as the coefficient of the scattering amplitude due to the addition of soft gravitons in the soft theorem [13]. It is worth mentioning that these three subjects have been developed independently and these equivalences are quite nontrivial. Moreover, this triangle equivalence is not restricted to the gravitational theory and similar equivalences also show up in some gauge theories [14, 15]. Therefore, infrared triangle may have a deeper origin arising from the infrared consistency of a theory under consideration.

In order to better understand the infrared structure of gravity, it is then quite important to further study each corner of the infrared triangle. The memory effect, which is the subject of this paper, is originally found in Minkowski spacetime and later studied in asymptotically flat spacetime. In these cases, the radiation part of the gravitational field, which includes the memory effect, can be distinguished from the other tidal gravitational effects by looking at the fall off of the gravitational field near the spatial and null infinities. In the case of the cosmological spacetime, which is not asymptotically flat, characterizing the memory effect is more subtle. Recently, this issue was studied by different groups with different approaches [16, 17, 18, 19, 20, 21, 22] and among them, we will focus on the approach adopted by Tolish and Wald [23]. Indeed, using the fact that the spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime is conformally flat, they have developed a general setup to study memory effect which is applicable as far as an idealized particle-like source for GWs is considered. Apart from the fact that we need to clarify the notion of memory in a cosmological background to study the universe, their setup provides a framework to better understand the infrared regime of gravity at cosmological scales. Their analysis of cosmological memory effect is based on general relativity and it will be interesting to explore what happens in modified theories of gravity. In this paper, we focus on scalar-tensor theories. According to the results of Ref. [23] for the linear perturbations, only tensor perturbations contribute to the cosmological memory effect in general relativity. We will show that, in scalar-tensor theories, scalar perturbations also contribute to the cosmological memory.

The rest of the paper is organized as follows. In Sec. II, we present our scalar-tensor model which is coupled to a perfect fluid and particle-like sources, as a source of GWs. We then study the background equations and linear perturbations around a spatially flat FLRW spacetime. In Sec. III, we focus on the scalar perturbations and we find the direct part of the corresponding retarded Green’s function. Using this result, in Sec. IV, we show that the scalar perturbations contribute to the cosmological memory effect in scalar-tensor theories. Sec. V is devoted to the summary of the paper. Moreover, we present our model in the Jordan and the Einstein frames in Appendix A and we present relation between the energy-momentum tensors in different frames in Appendix B. In Appendix C, we show explicit forms of the mass and source matrices for the sake of completeness. In Appendix D, we present scalar perturbations in terms of the gauge-invariant counterpart of the scalar field perturbation to make the minimal coupling and constant scalar background limit of the theory manifest. Finally, in Appendix E, we develop a general framework which can be used to study coupled wave equations in any curved spacetime region which admits a foliation by time slices. The cosmological spacetime, which we deal with in this paper, can be considered as a special subset.

We consider a scalar-tensor theory with a linear kinetic term and without higher derivative terms. The action of the system in the Jordan frame, in which matter fields directly couple to the metric, is given by

	$\displaystyle S_{\rm J}[{\tilde{g}},\phi,\psi]$	$\displaystyle=\int d^{4}x\sqrt{-{\tilde{g}}}\bigg{[}\frac{M_{\rm Pl}^{2}}{2}F(\phi){\tilde{R}}$
		$\displaystyle-\frac{1}{2}{\tilde{K}}(\phi){\tilde{g}}^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-{\tilde{V}}(\phi)\bigg{]}+S_{\rm m}[{\tilde{g}},\psi]\,,$		(1)

where $M_{\rm Pl}=(8\pi{G})^{-1/2}$ is the reduced Planck mass, ${\tilde{R}}$ is the Ricci scalar in the Jordan frame, $F$, ${\tilde{K}}$ and ${\tilde{V}}$ are functions of the scalar field, and $S_{\rm m}$ is the matter action in which $\psi$ collectively represents all matter fields and particles which are present in the system under consideration. The gravitational part of the action (1) is constructed out of the metric in the Jordan frame ${\tilde{g}}_{\mu\nu}$ and scalar field $\phi$. Performing the conformal transformation

$${\tilde{g}}_{\mu\nu}=F(\phi)^{-1}g_{\mu\nu}\,,$$

(2)

where $g_{\mu\nu}$ is the metric in the Einstein frame, the action in the Einstein frame takes the form (see Appendix A)

	$\displaystyle S_{\rm E}[{g},\varphi,\psi]$	$\displaystyle=\int d^{4}x\sqrt{-{g}}\bigg{[}\frac{M_{\rm Pl}^{2}}{2}{R}$
		$\displaystyle-\frac{1}{2}{g}^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi-{V}(\varphi)\bigg{]}+S_{\rm m}[\tilde{g},\psi]\,,$		(3)

where we have defined

$\displaystyle\varphi\equiv\int\sqrt{K(\phi)}\,d\phi,\hskip 5.69046ptK\equiv\frac{{\tilde{K}}}{F}+\frac{3}{2}M_{\rm Pl}^{2}\left(\frac{F_{,\phi}}{F}\right)^{2},\hskip 5.69046ptV\equiv\frac{{\tilde{V}}}{F^{2}},$

(4)

and it is understood that $\tilde{g}$ in (II) is given by (2) and that $\phi$ is considered as a function of $\varphi$.

Comparing the equivalent actions (1) and (II), we find that the form of the action for $g_{\mu\nu}$ is different from that for $\tilde{g}_{\mu\nu}$. This difference is compensated by the fact that the coupling to the matter sources is different: the matter couples only to ${\tilde{g}}_{\mu\nu}$ in the Jordan frame while it couples to both $g_{\mu\nu}$ and $\varphi$ in the Einstein frame. As it is well-known, this is the reason why we consider the scalar-tensor theories as modified gravity theories, despite the fact that $g_{\mu\nu}$ is described by the Einstein-Hilbert action as in general relativity.

As it is easier, we perform all calculations in the Einstein frame and translate only the final results in terms of the Jordan frame quantities. The details of the transformation between the Jordan frame and the Einstein frame are presented in Appendix A. From now on (throughout this and next sections), all calculations are presented in the Einstein frame. The Einstein equations can be deduced by taking the variation of the action (II) with respect to the metric $g^{\mu\nu}$

$\displaystyle M_{\rm Pl}^{2}G_{\mu\nu}=T^{(\varphi)}_{\mu\nu}+T^{({\rm m})}_{\mu\nu}\,;\hskip 14.22636ptT^{({\rm m})}_{\mu\nu}=\frac{-2}{\sqrt{-g}}\frac{\delta{S}_{\rm m}}{\delta{g}^{\mu\nu}}\,,$

(5)

where $G_{\mu\nu}$ is the Einstein tensor and $T^{({\rm m})}_{\mu\nu}$ is the energy-momentum tensor of the matter (see Appendix B) while energy-momentum tensor of the scalar field is given by

$\displaystyle T^{(\varphi)}_{\mu\nu}=\partial_{\mu}\varphi\partial_{\nu}\varphi-g_{\mu\nu}\left[\frac{1}{2}g^{\alpha\beta}\partial_{\alpha}\varphi\partial_{\beta}\varphi+V(\varphi)\right]\,.$

(6)

Taking variation of the action (II) with respect to the scalar field $\varphi$, we find

$\displaystyle\Box{\varphi}-V_{,\varphi}=\frac{F_{,\varphi}}{2F}\,T^{({\rm m})}\,;\hskip 14.22636ptT^{({\rm m})}=g^{\alpha\beta}T^{({\rm m})}_{\alpha\beta}\,.$

(7)

We consider two types of sources for the matter sector: perfect fluid which is responsible for the energy density of the universe and particle-like sources as idealized sources for GWs production. The matter energy-momentum tensor then can be separated to two parts

$\displaystyle T^{({\rm m})}_{\mu\nu}={T}^{({\rm F})}_{\mu\nu}+{T}^{({\rm P})}_{\mu\nu}\,.$

(8)

The conservation equations for the matter [see Eq. (84)] imply

$\displaystyle\nabla_{\mu}{T}^{({\rm F})\mu}{}_{\nu}=-\frac{F_{,\varphi}}{2F}\,{T}^{({\rm F})}\nabla_{\nu}\varphi\,,$

(9)

$\displaystyle\nabla_{\mu}{T}^{({\rm P})\mu}{}_{\nu}=-\frac{F_{,\varphi}}{2F}\,{T}^{({\rm P})}\nabla_{\nu}\varphi\,.$

(10)

The energy-momentum tensor for the perfect fluid is given by

$${T}^{({\rm F})}_{\mu\nu}=(\rho+p){U}_{\mu}{U}_{\nu}+p\,{g}_{\mu\nu}\,,$$

(11)

where $\rho$ and $p$ are the energy density and pressure in the Einstein frame while $U_{\mu}$ is the four-velocity normalized with respect to the Einstein frame metric $g^{\mu\nu}{U}_{\mu}{U}_{\nu}=-1$.

For the particle energy-momentum tensor we have

	$\displaystyle{T}^{({\rm P})}_{\mu\nu}$	$\displaystyle=\sum_{l,{\rm in}}{T}^{(M,l)}_{\mu\nu}+\sum_{n,{\rm in}}{T}^{(N,n)}_{\mu\nu}$
		$\displaystyle+\sum_{l^{\prime},{\rm out}}{T}^{(M,l^{\prime})}_{\mu\nu}+\sum_{n^{\prime},{\rm out}}{T}^{(N,n^{\prime})}_{\mu\nu}\,,$		(12)

in which the upper indices $M$ and $N$ indicate massive and massless particles respectively with

	$\displaystyle T^{(M,{\rm in})}_{\mu\nu}$	$\displaystyle=$	$\displaystyle mu_{\mu}u_{\nu}\delta^{(3)}({\bf x}-{\bf z}(t))\frac{1}{\sqrt{-{g}}}\frac{d\tau}{dt}\Theta(-t)\,,$		(13)
	$\displaystyle T^{(N,{\rm in})}_{\mu\nu}$	$\displaystyle=$	$\displaystyle k_{\mu}k_{\nu}\delta^{(3)}({\bf x}-{\bf y}(t))\frac{1}{\sqrt{-{g}}}\frac{ed\lambda}{dt}\Theta(-t)\,,$		(14)

where $u_{\mu}=\frac{dx^{\mu}}{d\tau}$ is the four-velocity of the massive particles and $k^{\mu}=\frac{dx^{\mu}}{ed\lambda}$ is the four-momentum of the massless particles. Notice that $\tau$ is the proper time while $t$ is the local time coordinate for the particles. The label “in” shows that the particles come from the past lightcone for $t<0$ while the label “out” corresponds to the particles in the future lightcone with $t>0$. Therefore, the energy-momentum tensor for the outgoing massive and massless particles have the same forms as Eqs. (13) and (14) with the replacement of $\Theta(-t)$ with $\Theta(t)$.

Working with the conformal time $\eta=\int[{\bar{N}}(\tau)/a(\tau)]d\tau$, the background line element (15) takes the following conformally flat form

$$ds^{2}=a(\eta)^{2}\left(-d\eta^{2}+\delta_{ij}dx^{i}dx^{j}\right)\,.$$

(43)

For the tensor perturbations, Eq. (26) gives

$\displaystyle\left[\partial_{\eta}^{2}+\frac{2}{a}\frac{da}{d\eta}\partial_{\eta}-\partial^{2}\right]h_{ij}=4\pi a^{2}\mu_{ij}\,,$

(44)

where we have defined normalized source $\mu_{ij}\equiv\frac{1}{2\pi M_{\rm Pl}^{2}}{\cal T}_{ij}$.

For the scalar perturbations, Eq. (39) in the component form yields

	$\displaystyle\left[\partial_{\eta}^{2}+\left(aN_{11}-\frac{1}{a}\frac{da}{d\eta}\right)\partial_{\eta}-\partial^{2}+a^{2}M_{11}\right]\zeta$
	$\displaystyle+\left[aN_{12}\partial_{\eta}+a^{2}M_{12}\right]\delta{Q}=4\pi a^{2}\mu^{(P)}_{\zeta}\,,$		(45)
	$\displaystyle\left[\partial_{\eta}^{2}+\left(aN_{22}-\frac{1}{a}\frac{da}{d\eta}c_{s}^{2}\right)\partial_{\eta}-c_{s}^{2}\partial^{2}+a^{2}M_{22}\right]\delta{Q}$
	$\displaystyle+\left[aN_{21}\partial_{\eta}+a^{2}M_{21}\right]\zeta=4\pi a^{2}\mu^{(P)}_{\delta{Q}}\,.$		(46)

As we have already mentioned, the memory effect for tensor perturbations $h_{ij}$ is studied in Ref. [23]. Moreover, it is shown that the field $\delta{Q}$, which corresponds to the scalar degree of freedom of the perfect fluid, does not develop any memory effect in general relativity [23]. Therefore, we only need to focus on the scalar graviton field $\zeta$. More precisely, it is expected that we only need to obtain the direct part of the retarded Green’s function for $\zeta$ which is the subject of this section and Appendix E. In Appendix E, starting from the first principle and working out the fundamental solutions for the scalar modes $\zeta$ and $\delta{Q}$ in the scalar-tensor theory, we prove that $\delta{Q}$ does not develop any singularity along the lightcone of the mode $\zeta$ [Eq. (168)] while it changes the evolution of $\zeta$ inside the lightcone [Eqs. (169)]. Therefore, $\delta{Q}$ does not contribute to the direct part of the mode $\zeta$. Then, we have systematically calculated the direct part of the corresponding retarded Green’s function in Eq. (190). Thus we can simply use the result Eq. (190) and move to the next section.

Here in this section we obtain the same result by implementing a less rigorous but more intuitive approach which we believe is easier for the readers to follow. The readers who are only interested in the final result may simply move to the next section. On the other hand, the readers who are interested in the rigorous treatment are directed to Appendix E.

As it is shown in Appendix E, the retarded Green’s function for $\delta{Q}$ does not develop any singularities along the lightcone of $\zeta$. We thus treat the term proportional to $\delta{Q}$ in Eq. (45) as a source for $\zeta$ and we rewrite it in the following form

$\displaystyle\left[\partial_{\eta}^{2}+\frac{2}{A_{\zeta}}\frac{dA_{\zeta}}{d\eta}\partial_{\eta}-\partial^{2}+a^{2}M_{11}\right]\zeta=4\pi a^{2}\mu_{\zeta}\,,$

(47)

where we have defined

		$\displaystyle\mu_{\zeta}\equiv{\mu}^{(P)}_{\zeta}-\frac{1}{4\pi}\left(\frac{1}{a}N_{12}\partial_{\eta}+M_{12}\right)\delta{Q}\,,$		(48)
		$\displaystyle A_{\zeta}(\eta)\equiv a(\eta)\exp\left[\frac{1}{2}\int_{\eta}\left(aN_{11}-\frac{3}{a}\frac{da}{d{\bar{\eta}}}\right)d{\bar{\eta}}\right]\,.$		(49)

The corresponding Green’s function then satisfies

		$\displaystyle\left[\partial_{\eta}^{2}+\frac{2}{A_{\zeta}}\frac{dA_{\zeta}}{d\eta}\partial_{\eta}-\partial^{2}+a^{2}M_{11}\right]G_{\zeta}^{\rm ret}(x,x^{\prime})$
		$\displaystyle=\frac{4\pi}{a^{2}}\delta^{(4)}(x-x^{\prime})\,.$		(50)

In general, the retarded solution for the above equation has the Hadamard representation [25, 26]

$\displaystyle G_{\zeta}^{\rm ret}(x,x^{\prime})=\left[U_{\zeta}(x,x^{\prime})\delta(\sigma_{\zeta})+V_{\zeta}(x,x^{\prime})\Theta(-\sigma_{\zeta})\right]\Theta(t-t^{\prime})\,,$

(51)

where $U_{\zeta}$ and $V_{\zeta}$ characterize the direct and tail parts respectively and $\sigma_{\zeta}(x,x^{\prime})$ is the geodetic interval (squared of the geodesic distance) between $x$ and $x^{\prime}$ which satisfies

$\displaystyle{\bar{g}}^{\mu\nu}{\bar{\nabla}}_{\mu}\sigma_{\zeta}{\bar{\nabla}}_{\nu}\sigma_{\zeta}=2\sigma_{\zeta}\,,$

(52)

where a bar denotes that the covariant derivatives are defined in the spirit of the background metric $\bar{g}_{\mu\nu}$. Following the approach implemented in Refs. [27, 28, 29], by substituting

$\displaystyle G_{\zeta}^{\rm ret}(x,x^{\prime})=\frac{A_{\zeta}(\eta^{\prime})}{A_{\zeta}(\eta)a(\eta^{\prime})^{2}}g_{\zeta}^{\rm ret}(x,x^{\prime})\,,$

(53)

in Eq. (50), we find

		$\displaystyle\left[\partial_{\eta}^{2}-\partial^{2}+\left(a^{2}M_{11}-\frac{1}{A_{\zeta}}\frac{d^{2}A_{\zeta}}{d\eta^{2}}\right)\right]g_{\zeta}^{\rm ret}(x,x^{\prime})$
		$\displaystyle=4\pi\delta^{(4)}(x-x^{\prime})\,.$		(54)

The above equation is similar to the equation for the Green’s function in flat spacetime. Indeed the first two terms on the left hand side of (54) correspond to the flat spacetime operator and they determine the direct part. Therefore, considering the relation between $G_{\zeta}^{\rm ret}$ and $g_{\zeta}^{\rm ret}$ in Eq. (53), from Eq. (51) we find

$\displaystyle U_{\zeta}(x,x^{\prime})=\left[\frac{A_{\zeta}(\eta^{\prime})}{A_{\zeta}(\eta)}\right]{\bar{U}}(x,x^{\prime})\,,$

(55)

where ${\bar{U}}$ is the corresponding quantity in the flat spacetime. Using the fact that ${\bar{U}}=1$, we find the direct part of the Green’s function (51) as follows

$\displaystyle U_{\zeta}(\eta,\eta^{\prime})=\frac{A_{\zeta}(\eta^{\prime})}{A_{\zeta}(\eta)}\,.$

(56)

The above result coincides with the result (190) which is obtained from a more rigorous approach.

Apart from the mass term, Eqs. (44) and (47) have the same forms so that $A_{\zeta}$ plays the role of an effective scale factor for the scalar mode $\zeta$. Therefore, with the same approach, we can easily find the direct part for the tensor perturbations $h_{ij}$ as $U_{h}(\eta,\eta^{\prime})=a(\eta^{\prime})/a(\eta)$ [23].

In our idealized case in which GWs are produced due to the particle-like sources with energy-momentum tensor (12), the memory effect is characterized by the “existence of first derivative of the delta functions” in the components of the Riemann tensor [23]. The electric components of the Riemann tensor up to the linear order in tensor and scalar perturbations (22) are given by

		$\displaystyle\delta_{1}R_{i00}{}^{j}=\frac{1}{2}\left(\partial^{2}_{\eta}{h}_{ik}+\partial_{\eta}{h}_{ik}\right)\delta^{kj}$
		$\displaystyle+\left(\partial_{i}\partial^{j}-\frac{1}{3}\delta_{i}{}^{j}\partial^{2}\right)\Phi+\left[\partial_{\eta}^{2}\Psi+\frac{1}{a}\frac{da}{d\eta}\left(\partial_{\eta}\Psi-\partial_{\eta}\Phi\right)\right]\delta_{i}{}^{j}\,,$		(57)

where we have defined the Bardeen potentials [30]

$\displaystyle\Phi\equiv\alpha+\frac{1}{a}\partial_{\eta}\beta\,,$

$\displaystyle\Psi\equiv\zeta+\frac{1}{a^{2}}\frac{da}{d\eta}\beta\,.$

(58)

For the tensor perturbations, the electric components of the Riemann tensor (57) obviously include $\partial^{2}_{\eta}h_{ij}$ which provides first derivative of the delta function and, therefore, tensor memory effect shows up [23]. For the scalar perturbations, (57) include $\partial^{2}_{\eta}\zeta$ while they do not include $\partial_{\eta}^{2}\delta{Q}$. The later does not provide any first derivative of the delta function and our aim is here to show that, similarly to $\partial^{2}_{\eta}h_{ij}$, the former $\partial^{2}_{\eta}\zeta$ does include first derivative of the delta function.

Up to here, we have presented all results in the Einstein frame while to interpret the results we need to go back to the Jordan frame. In the unitary gauge $\delta\varphi=0$, which we have implemented in (28), based on (2), we can simply use conformal transformation ${\tilde{g}}_{\mu\nu}={\bar{F}}^{-1}{g}_{\mu\nu}$ where only background value of conformal factor ${\bar{F}}$ is considered. The line element for the background (43) takes the following form

	$\displaystyle{\widetilde{d}s}^{2}$	$\displaystyle=\tilde{\bar{g}}_{\mu\nu}dx^{\mu}dx^{\nu}$
		$\displaystyle={\tilde{a}}(\eta)^{2}\left(d\tau^{2}+\delta_{ij}dx^{i}dx^{j}\right),\hskip 5.69046pt{\tilde{a}}(\eta)\equiv a(\eta)/\sqrt{{\bar{F}}(\eta)}\,,$		(59)

where ${\tilde{a}}$ denotes the scale factor in the Jordan frame. Taking into account the change in the scale factor, we find that $\zeta$ defined by (22) and (28) does not change by the conformal transformation (see Refs. [31, 32, 33, 34, 35] for the conformal invariance of the scalar perturbations in scalar-tensor theories). More precisely, the linear scalar perturbations do not change as the non-dynamical fields like $\beta$ can be simply redefined. Therefore, in the unitary gauge (28), in order to go back from the Einstein frame to the Jordan frame, we only need to rewrite all results in terms of the scalar factor ${\tilde{a}}$ in the Jordan frame defined in (59). The equation of motion for $\zeta$ presented in Eq. (47) takes the following form in the Jordan frame

$\displaystyle\left[\partial_{\eta}^{2}+\frac{2}{A_{\zeta}}\frac{dA_{\zeta}}{d\eta}\partial_{\eta}-\partial^{2}+{\tilde{a}}^{2}{\tilde{M}}_{11}\right]\zeta=4\pi{\tilde{a}}^{2}{\tilde{\mu}}_{\zeta}\,,$

(60)

where ${\tilde{M}}_{11}\equiv{\bar{F}}{M}_{11}$, ${\tilde{\mu}}_{\zeta}\equiv{\bar{F}}{\mu}_{\zeta}$, and also

$\displaystyle A_{\zeta}=\left(\sqrt{\bar{F}}{\tilde{a}}\right)^{3}\frac{d{\bar{\varphi}}}{d\eta}\exp\left[\frac{3}{2}\int\left(\frac{w{\bar{\rho}}+{\bar{p}}_{\varphi}}{{\bar{\rho}}+{\bar{\rho}}_{\varphi}}-1\right)d\ln\left({\sqrt{\bar{F}}{\tilde{a}}}\right)\right]\,,$

(61)

in which we have used Eqs. (18) and (20) when we substitute the value of $N_{11}$ defined in (41). In the absence of the fluid ${\bar{\rho}}=0$, the mass term vanishes ${\tilde{M}}_{11}=0$ as it can be seen from Eq. (105). We thus find the well-known result $A_{\zeta}\propto{\tilde{a}}$ for the minimally coupled ${F}=1$ massless scalar field with ${\bar{p}}_{\varphi}={\bar{\rho}}_{\varphi}$ and $d{\bar{\varphi}}/d\eta\propto{\tilde{a}}^{-2}$. There is also an apparent subtlety: for the minimal coupling and constant scalar background limit ${F}=1$ and $\dot{\bar{\varphi}}=0$, as it can be seen from Eq. (106), the source does not vanish ${\tilde{\mu}}_{\zeta}={\mu}_{\zeta}\neq 0$. This is simply an artifact of the unitary gauge (28) that we have implemented. From Eq. (27), we see that $\zeta=\psi$ in the unitary gauge (28) when $\delta\varphi=0$ while $\zeta=\frac{H}{\dot{\bar{\varphi}}}\delta\varphi$ if we work in the spatially flat gauge with $\psi=0$ and $E=0$. As it is shown in Appendix D, there will be no source if we work with the gauge-invariant counterpart of $\delta\varphi$, which is given by $\delta\varphi+\dot{\bar{\varphi}}\beta$ for $E=0$, in the limit $F=1$ and $\dot{\bar{\varphi}}=0$. Although the $\dot{\bar{\varphi}}=0$ limit is not manifest in the unitary gauge, the results away from this limit are much simpler in this gauge and that is why we have implemented this gauge.

The retarded solution for $\zeta$ is given by

$\displaystyle\zeta(x)=\int d^{4}x^{\prime}\sqrt{-{\bar{g}}(x^{\prime})}G^{\rm ret}_{\zeta}(x,x^{\prime}){\tilde{\mu}}_{\zeta}(x^{\prime})\,,$

(62)

where the retarded Green’s function $G^{\rm ret}_{\zeta}$ is given by Eq. (51). Substituting Eq. (51) in solution (62) and taking into account the fact that $U_{\zeta}$ and $V_{\zeta}$ are regular functions, it can be shown that the tail part including $V_{\zeta}$ can at most provide singularity proportional to the delta function while the direct part including $U_{\zeta}$ can indeed provide singularity at the level of derivative of the delta function. Therefore, only the direct part $U_{\zeta}$ contributes to the memory effect. This result is completely independent of the explicit functional forms of $U_{\zeta}$ and $V_{\zeta}$. We do not repeat the corresponding analysis here, as we only need the final result in our upcoming analysis and we refer the readers to Ref. [23] for the details.

Instead of performing the integral directly in the retarded solution (62), the fact that the memory effect is encoded only in the direct part and the spatially flat FLRW spacetime is conformally flat, makes it possible to find a universal explicit relation between the memory effect in the FLRW spacetime and its counterpart in Minkowski spacetime. In order to do so, we first note that Eq. (55) immediately implies

$\displaystyle\zeta^{\rm dir}(x)=\left[\frac{A_{\zeta}(\eta_{s})}{A_{\zeta}(\eta_{o})}\right]\bar{\zeta}^{\rm dir}(x)\,,$

(63)

where $\eta_{s}$ is the conformal time at the source, $\eta_{o}$ is the conformal time at which the detector observes the signal, and $\bar{\zeta}^{\rm dir}(x)$ is the flat spacetime counterpart of $\zeta^{\rm dir}(x)$, i.e., scalar mode characterizing the scalar field perturbations in the absence of cosmic expansion. From Eq. (57) we have $\delta_{1}R_{i00}{}^{j}\supset\partial_{\eta}^{2}\zeta$. Taking into account the fact that the time derivatives of $A_{\zeta}$ do not contribute to the direct part, we find

$\displaystyle\delta_{1}R^{\rm dir}_{({\rm S})i00}{}^{j}=\left[\frac{A_{\zeta}(\eta_{s})}{A_{\zeta}(\eta_{o})}\right]\overline{\delta_{1}R}^{\rm dir}_{({\rm S})i00}{}^{j}\,,$

(64)

where subscript ${\rm S}$ shows that only scalar perturbations are taken into account.