††thanks: Equal contribution to this work.††thanks: Equal contribution to this
work.

# Gravitational wave memory signatures in identifying transition redshift

Indranil Chakraborty<EMAIL_ADDRESS>Susmita Jana
<EMAIL_ADDRESS>S. Shankaranarayanan<EMAIL_ADDRESS>Department of
Physics, Indian Institute of Technology Bombay, Mumbai 400076, India

###### Abstract

Many astrophysical and cosmological observations consistently indicate that
the universe is currently accelerating. Despite many possible explanations,
the exact cause of this acceleration remains unknown. Therefore, additional
observational probes are necessary to pinpoint the cause. Gravitational waves
(GWs) have the potential to unravel some of the unresolved mysteries in
cosmology. In this work, we highlight the potential utility of _gravitational
wave memory_ as a tool to identify the cause of this acceleration. We evaluate
cosmological memory as a particular case of the master equation for GW memory
in Locally Rotationally Symmetric type II spacetimes. Unlike the previous
works, the master equation for GW memory contains non-linear dependence of the
background quantities. Hence, even though the successive GWs generated are
smaller than their predecessors, we demonstrate that their cumulative effect
over cosmological time leads to observable signatures, akin to the growth of
density perturbations resulting in large-scale structures. Finally, we show
that the GW memory exhibits distinct signatures between accelerated and
decelerated universes, potentially enabling the identification of the
transition redshift from a matter-dominated to a dark-energy-dominated
universe.

##  I Introduction

The $\Lambda$CDM model has become the cornerstone of modern cosmology due to
its impressive ability to shed light on the structure and evolution of the
Universe [1, 2, 3]. However, it also presents several unanswered questions
[4]. $\Lambda$CDM model addresses the flatness of the Universe and the
uniformity of the cosmic microwave background radiation by incorporating
inflation [5]. However, the specifics of inflation remain unclear [6, 7]. The
second puzzle is dark matter — a particle inferred solely from interaction
with gravity [8, 9, 10]. Third, the current acceleration of the universe [11,
12, 13, 14, 15]. The cosmological constant ($\Lambda$), scalar field and
modified gravity theories can explain late-time acceleration [16, 17, 18, 19,
20, 21, 22, 23], however, each leading to different transition redshift [24,
25].

The direct detection of gravitational waves (GWs) opened an unprecedented
channel to probe some of these open questions in cosmology [26]. For instance,
the first GW detection from merging black holes (BHs) has reignited the
possibility that primordial BHs (PBHs) may constitute most of the dark matter
[27, 28, 29]. The first binary neutron star event (GW170817) has constrained
the speed of the GWs w.r.t. the photon speed to better than one part in
$10^{15}$ [30, 31]. This has ruled out many modified gravity theories of dark
energy [32, 33, 34]. The same event has provided a new probe of Hubble
constant ($H_{0}$) [35, 36]. It will be possible to measure $H_{0}$ with $1\%$
accuracy through GW observations alone [37, 38].

GWs travel unimpeded over cosmological distances due to the weakness of the
gravitational interaction [26]. Hence, one only considers the redshifting of
the GW frequencies and dilution of the GW amplitude. Including these two
effects is not difficult if we understand the background dynamics. However, as
GWs propagate through cosmological distances, they induce subsequent GWs,
creating additional GWs. In Minkowski spacetime, this phenomena leads to
persistent change in the spacetime geometry even after the passage of the GW,
referred to as _gravitational memory_ [39, 40]:

$\displaystyle\Delta
h_{+,\times}^{\mathrm{mem}}=\lim_{t\rightarrow+\infty}h_{+,\times}(t)-\lim_{t\rightarrow-\infty}h_{+,\times}(t),$
(1)

where, $h_{+},\,h_{\times}$ is the plus and cross mode of the GW. This effect
manifests as a permanent change in separation (displacement memory), or change
in relative velocity (velocity memory), between free test masses [41, 42, 43,
40, 44, 45]. This effect has been extensively studied in asymptotically flat
(AF) spacetimes owing to its relation to Bondi-Metzner-Sachs (BMS) symmetries
[46, 47]. Comprising of the low-frequency component of the emitted GWs,
gravitational memory is determined by solving the sourced Einstein’s field
equations [47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60]. While BMS
symmetries and memory in AF spacetimes provide important insights into
infrared physics, they can not be directly applied to cosmological settings,
as one needs to take into account the background curvature. Previous studies
on memory employing perturbative approaches have shown the enhancement
attributable to the redshift factor [61, 62] along with the presence of tail
terms [63, 64, 56].

This leads us to the following questions: Is there a unified treatment of GW
memory for a class of spacetimes? Can such an analysis help us understand
curvature effects on memory? If yes, what are the observational consequences?
Interestingly, in Ref. [65] the authors obtained a master equation for
electromagnetic memory in an arbitrary spacetime, including Kerr. Since
gravity is non-linear, extending the analysis to general relativity is
difficult. However, as we show in this work, it is possible to obtain a
_master equation_ for GW memory for a class of geometries, referred to as
Locally Rotationally Symmetric of type II (LRS-II) spacetimes [66].

LRS symmetry means that spacetime is invariant under rotations around at least
one spatial direction at every point. LRS-II spacetimes are both time and
space-dependent and contain many physically interesting solutions like
spherically-symmetric perfect fluids, Lemaitre–Tolman–Bondi (LTB) cosmologies,
Kantowski–Sachs, Bianchi I and III cosmologies, and the flat and hyperbolic
Friedmann-Lema$\hat{i}$tre-Robertson-Walker (FLRW) models [67, 66]. Using the
covariant approach [68, 69, 70, 71], we obtain gravitational memory for LRS-II
spacetimes. For the Minkowski spacetime, the master equation reduces to the
standard expression derived in the literature. For FLRW spacetime, due to the
background curvature, even though the successive GWs generated have a smaller
magnitude than their predecessors, we demonstrate that that their cumulative
effect over cosmological time leads to observable signatures. This is similar
to growth of density perturbations leading to large scale structures in the
universe. In particular, we show that the memory has distinct features between
accelerated and decelerated universes, thus providing a possibility to
identify the transition redshift from matter-dominated to dark-energy-
dominated universe.

##  II The Set-up

We use the covariant $1+1+2$ or semi-tetrad formalism to derive the master
equation (ME) for GW memory [68, 69, 70, 71]. In this formalism, $4-$D
spacetime is split into a timelike, a spacelike direction, and a $2-$D hyper-
surface, which is orthogonal to both the timelike and spacelike directions.
The timelike direction is the velocity $u^{a}$, obeying $u^{a}u_{a}=-1$, of
the comoving observer, which is tangent to the timelike congruence $\gamma$.
The spacelike direction $n^{a}$, obeying $n^{a}n_{a}=1$, is a preferred
direction depending upon the symmetry of the spacetime. The projection tensor
$N_{ab}$, for the 2-D surface is related to the $4-$D metric $g_{ab}$ [with
signature $(-,+,+,+)$] via:

$\displaystyle g_{ab}=-u_{a}u_{b}+n_{a}n_{b}+N_{ab}\,.$ (2)

In covariant formalism, the $4-$D spacetime is described by the _geometrical
variables_ — expansion, shear, and vorticity associated with $u_{a}$ and
$n_{a}$, and components of the Weyl tensor ($C_{abcd}$) [68, 69, 70, 71].
Specifically, $C_{abcd}$ describes the perturbation due to GWs [72, 73]. The
electric part of Weyl tensor ($E_{ab}$, a tensor in 3-space) contains
information about the tidal deformation of the spacetime, and the magnetic
part of Weyl tensor ($B_{ab}$) contains information about the gravitational
radiation that travels to future infinity from the source [74, 72]. In $1+1+2$
formalism, the components of $B_{ab},\,E_{ab}$ along the 2-D surface carry
information about the passing GWs. This provides a natural representation for
the GWs where the two traceless-transverse DOF lie on the 2-D surface [68, 69,
70, 71].

For LRS spacetimes with spherical symmetry, we can choose $n_{a}$ such that
all the geometrical variables on the $2$-D surface $N_{ab}$ vanish [66]. For
the LRS-II spacetimes, which is the focus of this work, the vorticity
associated with $u^{a}$ and $n^{a}$ also vanish [66]. Due to the vanishing of
the vorticity in LRS-II spacetime, $E_{ab}$ and $B_{ab}$ can be orthogonal,
and $E_{ab}$ can be expressed in terms of $B_{ab}$ [75]. As we show, this
property helps us to derive the master equation for memory in these
spacetimes.

Figure 1: Schematic depiction of GW memory in LRS-II spacetime. The generated
GWs (shown in red) are due to hyperbolic scattering between the two stars
($t=t_{0}$) which alter the shear of the 2-space at $t=t_{1}$. The blue arrow
at the bottom depicts the direction of time ($u^{a}$). The orange arrows
denote the trajectory of the stars.

A schematic depiction of the GW memory in the LRS-II spacetime is shown in
Fig. 1. At $t=t_{0}$, the astrophysical process results in the generation of
GWs. Assuming that the GWs travel from left to right, the generated GWs
perturb the 2-D surface that is orthogonal to the direction of propagation in
the semitetrad formalism. We are interested in the integrated effect of the
GWs passing through the 2-D surface from $t=t_{0}$ to $t=t_{1}$. Before the
generation of the GW ($t<t_{0}$), the LRS-II spacetime is described by the
following geometrical variables [69]:

$\mathcal{D}_{0}=\\{\Theta,\phi,\Sigma,{\mathcal{A}},\mathcal{E}\\}\cup\mbox{matter
components},$ (3)

where, $\Theta$ and $\phi$ represents expansion along $u^{a}$ and $n^{a}$,
respectively, $\Sigma$ is component of shear tensor along $n^{a}$,
$\mathcal{A}$ is the acceleration projected along $n^{a}$ of the comoving
fictitious observer, and $\mathcal{E}$ is projection of electric-Weyl tensor
along $n^{a}$. As mentioned above, all other geometrical variables vanish on
the 2-D surface. This has to be contrasted with the covariant perturbation
theory in cosmological spacetimes where the growth of tensor and scalar
perturbations are generated due to the quantum fluctuations [76, 77].

GWs generated at $t=t_{0}$ permanently alter the 2-D surface, and the
spacetime is no longer LRS-II. However, since the energy carried by the GWs is
small, we can assume that the spacetime is a perturbed LRS-II. The perturbed
spacetime will now contain _non-zero_ geometrical variables on the 2-D
surface. More specifically, it will contain non-zero components of the
electric and magnetic Weyl
$(\mathcal{E}_{a},\,\mathcal{E}_{ab},\,\mathcal{B}_{a},\,\mathcal{B}_{ab})$
and shear tensor $(\Sigma_{ab})$ that carries information about the incoming
GW. Thus, the perturbed LRS-II spacetime is described by the following
variables: $\mathcal{D}_{\rm
n}\equiv\mathcal{D}_{0}\cup\\{\mathcal{E}_{a},\,\mathcal{E}_{ab},\,\mathcal{B}_{a},\,\mathcal{B}_{ab},\,\Sigma_{ab}\\}\cup\text{others}$,
where
$\\{\mathcal{E}_{a},\,\mathcal{E}_{ab},\,\mathcal{B}_{a},\,\mathcal{B}_{ab},\,\Sigma_{ab}\\}$
depend on the properties of the incoming GWs. For details, see the Appendix A.

As mentioned above, the 2-D surface carries information about the passing GWs.
Thus, to evaluate the GW memory of the perturbed LRS-II spacetime, we need to
identify quantities related to the perturbed 2-D surface. In covariant
formalism, the GW memory is related to the change in the geometrical
quantities associated with the 2-D surface w.r.t the timelike direction
$u^{a}$: $\mathcal{X}_{ab}=\mathcal{L}_{u}N_{ab}$. Since the GWs are symmetric
and traceless, we have:

$\displaystyle\mathcal{X}_{\\{ab\\}}=2\,\Sigma_{ab}\,.$ (4)

For details, see Appendix (B). This identification is the key to obtaining the
master equation for memory in LRS-II spacetime and we refer $\Sigma_{ab}$ as
_instantaneous memory tensor_. This has to be contrasted with the cumulative
memory $(\Delta N_{ab})$ which captures the integrated GW memory along the
timelike vector. In this work, the final results are in terms of observables
computed from the cumulative memory tensor.

##  III Master equation for memory in LRS-II spacetimes

In covariant formulation, like in 1+3 formulation, we write the perturbed
Einstein’s equations in LRS-II spacetime into a set of evolution, propagation
and constraint equations of electric
$(\mathcal{E},\,\mathcal{E}_{a},\,\mathcal{E}_{ab})$ and magnetic parts
$(\mathcal{B},\,\mathcal{B}_{a},\,\mathcal{B}_{ab})$ of Weyl tensor and the
memory tensor ($\Sigma_{ab}$) [70]. To the first order in perturbations, the
propagation equations (along the radial direction, denoted by over-hat, and
the 2-space derivatives are given by $\delta$) for $\mathcal{E},\,\mathcal{B}$
are:

$\displaystyle\hat{\mathcal{E}}=-\delta_{a}\mathcal{E}^{a}-3\phi\mathcal{E}/2;~{}~{}\hat{\mathcal{B}}=-\delta_{a}\mathcal{B}^{a}-3\phi\mathcal{B}/2\,.$
(5)

Substituting these in the propagation equation and demanding that the electric
and magnetic Weyl are orthogonal, we obtain the following constraints:

$\displaystyle\mathcal{E}_{a}=\varepsilon_{ab}\mathcal{B}^{b}~{};~{}\mathcal{E}_{ab}=\varepsilon_{ad}\mathcal{B}^{d}\,_{b}$
(6)

where, $\varepsilon_{ab}$ is Levi-Civita tensor in $2-$D surface. Substituting
the above constraint leads to:

$\displaystyle
3\mathcal{E}\,\Sigma_{ab}+\left[2\Theta+3\Sigma+\phi\right]\mathcal{E}_{ab}+\left[\mu+p\right]\Sigma_{ab}=-\mathcal{G}_{ab}.$
(7)

where, $\mathcal{G}_{ab}\equiv
2\,\epsilon_{c\\{a}\delta^{c}\mathcal{B}_{b\\}}$, captures the incoming GWs
and the matter is represented by a perfect fluid with pressure $p$ and energy
density $\mu$. The evolution of the instantaneous memory observable is,

$\displaystyle\dot{\Sigma}_{\\{ab\\}}=-\left(\textstyle\frac{2}{3}\Theta+\textstyle\frac{1}{2}\Sigma\right)\Sigma_{ab}-\mathcal{E}_{ab}.$
(8)

Substituting Eq. (8) in the constraint Eq (7) leads to the following master
equation for GW memory:

$\displaystyle\left(2\Theta+3\Sigma+\phi\right)\left[\dot{\Sigma}_{\\{ab\\}}+\left(\textstyle\frac{2}{3}\Theta+\textstyle\frac{1}{2}\Sigma\right)\Sigma_{ab}\right]$
$\displaystyle-\left[3\mathcal{E}\,+\mu+p\right]\Sigma_{ab}=\mathcal{G}_{ab}$
(9)

Here, dot denotes derivative w.r.t timelike direction $u^{a}$. This is the key
expression of this work, regarding which we want to discuss the following
points: To begin with, the master equation relates the memory tensor and the
geometrical quantities of the background spacetime along with the incoming GW.
This equation is valid for all LRS-II spacetimes, including Minkowski, FLRW,
and LTB cosmologies111In deriving the above expression, we have assumed
acceleration ($\mathcal{A}$) vanishes. In Appendix. (C), we have derived the
expression for nonzero acceleration ($\mathcal{A}$).. However, all the earlier
work on GW memory was restricted to asymptotically flat spacetime described by
Bondi-Metzner-Sachs metric [78, 79, 80].

In addition, it is easy to verify that the above results match with Bondi
shear for Minkowski spacetime [47]. $\phi(=2/r)$ is the only nonzero LRS
scalar. Substituting $\phi$ in Eq. (III) and using the fact that
$\dot{N}_{ab}=2\Sigma_{ab}$, we have:

$\displaystyle
2\,\dot{\Sigma}_{ab}=r\,\mathcal{G}_{ab}~{}~{}\Longrightarrow~{}~{}\ddot{N}_{ab}=r\,\mathcal{G}_{ab}\,.$
(10)

where dot denotes partial derivative w.r.t. the retarted time coordinate $u$.
Since the GW memory depends on the incoming GW, to keep calculations
transparent, we assume the incoming GW to have a burst profile ($h_{ab}^{I}$)
due to hyperbolic scattering between two stars [81]. Simulations of PBHs in
galaxy clusters reveal numerous hyperbolic encounters where two BHs interact,
emitting GW Bremsstrahlung [82]. For close enough encounters, the GWs emitted
may be detectable in the present and future GW detectors [83, 84]. Evaluating
the electric and magnetic parts of the Weyl tensor for the plus polarization
of the incoming GW, we get [85]:

$\displaystyle\ddot{N}_{\vartheta\vartheta}(u)=r^{2}\,\cos\vartheta\,\cos(2\varphi)\,\ddot{h}_{\vartheta\vartheta}^{I}(u)\,.$
(11)

Details of the burst profile are irrelevant to the present discussion. (See
Appendix D.) Integrating the above equation for a finite time $(t_{0},t_{1})$
interval, we get $\Delta N_{\vartheta\vartheta}\sim r^{2}\Delta
h_{\vartheta\vartheta}^{I}$. Since the GW burst falls as $\mathcal{O}(1/r)$,
we find the memory observable in our formalism scale as $\mathcal{O}(r)$.
Thus, it is a _subleading term_ , much akin to the Bondi shear [47]. See also
Appendix D. Thus, our analysis reveals the conventional non-oscillatory
behavior of the memory observable such that the final spacetime is a shifted
Minkowski [40, 47] (See Fig. 3). Furthermore, Eq.(11) resembles a forced
system, where the incoming GW provides the forcing term. Such forced systems
have been discussed earlier in literature in Ref. [39, 58].

Lastly, the master equation (III) is a crucial step towards future
observational implications of GW memory as it explicitly contains the non-
linear dependence of the background quantities. For instance, the earlier work
on GW memory in cosmological settings did not account for the non-linear
dependence [56]. Like in the electromagnetic memory [65], this analysis shows
that the GW memory contains a non-linear contribution of the geometry, which
is a quintessential property of gravity! We now show that cosmological GW
memory provides a distinct signature in identifying cosmological transition
redshift.

##  IV Cosmological GW memory

Consider the following spatially flat FLRW line-element
$ds^{2}=a^{2}(\eta)[-d\eta^{2}+d{\bf x}^{2}]$, where $\eta$ is conformal time,
$a(\eta)$ is the scale factor and $d{\bf x}^{2}$ is 3-space. In FLRW
spacetime, $\Theta=3{\mathcal{H}}/a,{\mathcal{H}}={a^{\prime}}{/a}\equiv d(\ln
a)/d\eta$ and $\Sigma=-\Theta/3$ are non-zero [86]. Substituting these in Eq.
(III) reduces to:

$\displaystyle
2\,\Theta\,\dot{\Sigma}_{ab}+\Theta^{2}\,\Sigma_{ab}-2\,(\mu+p)\,\Sigma_{ab}=2\,\mathcal{G}_{ab}\,,$
(12)

where dot refers to the retarded time $\tilde{u}=\eta-r$ and $\mu,p$ refer to
the energy density and pressure of the cosmological fluid. This is the second
key expression of this work regarding which we want to discuss the following
points. First, unlike Minkowski, $\phi$ contributes to the subleading terms.
Due to homogeneity and isotropic nature of the FLRW, $\Theta,\mu$, and $p$ do
not have any radial dependence. Hence, asymptotic (large $r$) behavior of the
GW memory in FLRW is different from Minkowski as the LRS scalars governing
their evolution have different fall-offs. Thus, in FLRW, $\Sigma_{ab}$ _does
not_ have $\mathcal{O}(1/r)$ fall-off. This feature is unique from our
analysis.

Second, this can be understood from the Lagrangian and Eulerian viewpoint in
fluid mechanics. The asymptotic static observers in Minkowski spacetime
correspond to Eulerian observers, while, the comoving observers in FLRW
spacetime are Lagrangian. Hence, unlike in Minkowski spacetime, we do not have
to transform the above expression to Eulerian observers [65]. This was missed
in the earlier works on memory in cosmological settings. Since cosmological
memory of the comoving observers do not have $\mathcal{O}(1/r)$ fall-off, this
provides unique opportunity to have observable consequences.

To understand the consequence, we consider an incoming (plus polarization) GW
burst produced due to hyperbolic scattering between two stars at time
$\eta_{0}$, and after traversing through the Universe reaches us at
$\eta=\eta_{1}$. Since we are interested in the time evolution of the
amplification factor, keeping $r$ fixed, rewriting $\tilde{u}$ in-terms of
$\eta$, Eq. (12) reduces to:

$\Sigma_{\vartheta\vartheta}^{\prime}\,+\Gamma(\eta)\,\Sigma_{\vartheta\vartheta}=({a(\eta)}/{3{\mathcal{H}}})\,r\cos\vartheta\cos(2\varphi)\,h_{\vartheta\vartheta}^{I^{\prime\prime}}(\eta)$
(13)

where,
$\Gamma(\eta)=-c_{0}{\mathcal{H}}+c_{1}{{\mathcal{H}}^{\prime}}/{{\mathcal{H}}}$,
$c_{0}={1}/{2}+c_{1}$, and $c_{1}={1}/{(12\pi)}$. In terms of
$N_{\vartheta\vartheta}$, Eq. (13) reads as:

$\displaystyle\left[\frac{N_{\vartheta\vartheta}}{a^{2}}\right]^{\prime\prime}\\!\\!+({\mathcal{H}}+\Gamma)\left[\frac{N_{\vartheta\vartheta}}{a^{2}}\right]^{\prime}\\!\\!=\frac{2r}{3{\mathcal{H}}}\cos\vartheta\cos(2\varphi)\,h_{\vartheta\vartheta}^{I^{\prime\prime}}$
(14)

As mentioned above, $N_{\vartheta\vartheta}\sim\mathcal{O}(1)$ fall-off at
asymptopia. Hence, unlike in Minkowski (11), the damping term play a crucial
role in the cosmological memory. Specifically, the form of ${\cal H}$ and
${\cal H}^{\prime}$ determine the cumulative memory at the final time
$\eta_{1}$. Physically, the successive waves generated have a smaller
magnitude than their predecessors, however, integrated over cosmological time
lead to observable signatures.

Since GWs are massless, the energy density scales as
$N_{\vartheta\vartheta}^{2}/a^{4}$. To understand the effect of background
geometry on GW memory, we compute the amplification factor ($A_{C}$) which is
the ratio of GW memory amplitude at the final and initial time. For the
previously discussed GW burst, we have:

$\displaystyle\\!\\!\\!\\!A_{C}=\frac{\left({N_{\vartheta\vartheta}}/{a^{2}}\right)\rvert_{\eta_{1}}}{\left({N_{\vartheta\vartheta}}/{a^{2}}\right)\rvert_{\eta_{0}}}=1+\dfrac{a^{2}(\eta_{0})}{h_{I}(\eta_{0})}\,[F(\eta_{1})-F(\eta_{0})].$
(15)

For details, see Appendix D.2. We are now in a position to understand the
relation between the amplification factor and cosmological epochs. To go about
this, we consider three scenarios — matter dominated (MD), dark energy (DE)
dominated and matter to $\Lambda$ dominated [11].

Fig.(2) is the plot of $A_{C}$ as a function of cosmic time ($t$). For MD and
DE, we have assumed one e-folding of expansion and the final times are chosen
to be identical. In the case of matter to $\Lambda$ dominated, we have used
Sahni and Starobinsky exact solution
$a(t)=\sinh^{2/3}(\textstyle\frac{3}{2}t)$ [11] which transitions between a MD
universe to deSitter (dS) at $t\to 0$ and $t\to+\infty$, respectively. In this
case, we have considered a larger range to show the imprint of transition in
the amplification factor. All the scenarios show a gradual change in $A_{C}$.
While $A_{C}$ decays for both DE and MD, it increases at the transition epoch
for the matter to $\Lambda$ dominated scenario. As a result, we can use
$A_{C}$, the cosmological memory-related amplification factor, to detect the
cosmological transition redshift.

Figure 2: The evolution of the amplification factor ($A_{C})$ for three
scenarios as a function of cosmic time. Left plot: Matter $a(t)=t^{2/3}$ and
Dark energy, $a(t)=t^{2}$. Right Plot: Sahni-Starobinsky model,
$a(t)=\sinh^{2/3}(\textstyle\frac{3}{2}t)$ [11].

##  V Conclusions and Observational Implications

GW memory, yet to be detected from observations, provides an interesting
avenue for understanding features of strong-field gravity. However, prior to
this work, there did not exist a unified formalism for studying GW memory for
a larger class of spacetimes. Our work tries to fill this gap by constructing
a master equation for a class of background geometries known as LRS-II
spacetimes. We showed that the Master equation for Minkowski spacetime leads
to Bondi shear. In the case of FLRW spacetime, we showed that the cosmological
memory does not have a $\mathcal{O}(1/r)$ fall-off, thus providing a unique
opportunity to be used as an observational probe. Additionally, the cumulative
cosmological memory bears similarity to the formation of large-scale
structures in the universe through the growth of density perturbations.

Our analysis shows that the cosmological memory has the potential to identify
the epoch of matter to $\Lambda$ transition. This work gives a novel proof-of-
principle to capture cosmological transition redshift ($z_{t}$) by the
measurement of GW memory. Till date, most prescriptions have either relied
upon the cosmic ladder (SNe) [87, 88] or the cosmic distance approaches (BAO)
[89]. Independent estimates by using expansion rate of passively evolving
galaxies [25] along with Planck missions [90] have constrained the value at
$z_{t}=0.64^{+0.11}_{-0.07}$ for $\Lambda$CDM model. Viable cosmological
models in $f(R)$ and $f(T)$ gravity theories have set $z_{t}$ as
$0.7679^{+0.1831}_{-0.1829}$ and $0.643^{0.034}_{-0.030}$ [91, 92]. Thus,
there exists parameter range where $z_{t}$ can be further constrained.

We outline two avenues where our work can provide crucial pointers in
cosmology and GW science. First, our methodology only employs the measurement
of GW memory. The obtained memory signal would provide a signature of the
source redshift and the corresponding era of the universe. Second, knowing the
source distance from the oscillatory part of the GW signal along with redshift
of the host galaxy from EM data, one can compare our results and provide
better constraints to both cosmological models as well as theory of gravity.
These are currently under investigation.

_Acknowledgement:_ The authors thank R. Goswami, J. P. Johnson, S. Kar, A.
Kushwaha, and S. Mandal for discussions and comments on the earlier draft. The
work is supported by SERB-CRG (RD/0122-SERB000-044). Thanks to Einstein
Telescope team for providing the background for Fig. 1.

##  Appendix A Semitetrad covariant formalism

In this section, we briefly recapitulate the semi-tetrad formalism [93, 69],
which enables us to study GW memory for LRS-II spacetime. A crucial feature of
these semi-tetrad decompositions is their locality, defined on any open set
$\mathcal{S}$. Initially, the properties of spacetime are analyzed relative to
a real or fictitious observer whose velocity aligns with the tangent of a
timelike congruence, splitting the $4-$d spacetime into a timelike direction
and a $3-$space. Subsequently, if the spacetime exhibits certain symmetries
such as local rotational symmetry, a preferred spatial direction emerges. The
spacetime is further decomposed using this preferred spatial congruence. The
field equations are then reformulated in terms of the geometric variables
associated with these congruences and the curvature tensor of the spacetime
(appropriately decomposed using the congruences).

Although this formalism is well-studied in the literature [70], for
completeness, we provide a overview of the same.

###  A.1 Semitetrad 1+3 formalism

Covariant formalism, first proposed in Refs. [94, 95], later were extensively
used in relativistic comlogy [77]. In this formalism the $4-$d spacetime is
deconstructed w.r.t a fictitious co-moving observer, moving with velocity
$u^{a}=dx^{a}/d\tau$($\tau$ is the affine parameter), satisfying
$u_{a}u^{a}=-1$. The spacetime comprises a timelike congruence $\gamma$ and a
$3-$d space orthogonal to $u^{a}$. The $3-$d space is described by the
projection tensor $h_{ab}$ that follows:

$g_{ab}=-u_{a}\hskip 0.99594ptu_{b}+h_{ab},$ (16)

iff the 3-space has no twist or vorticity, $h_{ab}$ becomes metric of the
$3-$space. The covariant time derivative along the observers’ worldlines,
denoted by ‘${\hskip 0.99594pt\hskip 0.99594pt{}^{\cdot}\hskip 0.99594pt\hskip
0.99594pt}$’, is defined using the vector ${u^{a}}$, as

$\dot{Z}^{a...b}{}_{c...d}=u^{e}\hskip 0.99594pt\nabla_{e}\hskip
0.99594ptZ^{a...b}{}_{c...d},$ (17)

for any tensor ${Z^{a...b}{}_{c...d}}$. The fully orthogonally projected
covariant spatial derivative, denoted by ‘ ${D}$ ’, is defined using the
spatial projection tensor ${h_{ab}}$, as

$D_{e}\hskip 0.99594ptZ^{a...b}{}_{c...d}=h^{r}{}_{e}h^{a}{}_{f}\hskip
0.99594pt...\hskip 0.99594pth^{b}{}_{g}\hskip 0.99594pth^{p}{}_{c}\hskip
0.99594pt...\hskip 0.99594pth^{q}{}_{d}\hskip 0.99594pt\nabla_{r}\hskip
0.99594ptZ^{f...g}{}_{p...q},$ (18)

The covariant derivative of the 4-velocity vector ${u^{a}}$ is decomposed
irreducibly as follows

$\displaystyle\nabla_{a}\hskip 0.99594ptu_{b}$ $\displaystyle=$
$\displaystyle-u_{a}\hskip 0.99594ptA_{b}+\frac{1}{3}h_{ab}\hskip
0.99594pt\Theta+\sigma_{ab}+\epsilon_{abc}\hskip 0.99594pt\omega^{c},$ (19)

where ${A_{b}}$ is the acceleration, ${\Theta}$ is the expansion of ${u_{a}}$,
${\sigma_{ab}}$ is the shear tensor, ${\omega^{a}}$ is the vorticity vector
representing rotation and ${\epsilon_{abc}}$ is the effective volume element
in the rest space of the comoving observer. The vorticity vector $\omega^{q}$
is related to vorticity tensor $\omega^{ab}$ as:
$\omega^{a}\equiv(1/2)\,\epsilon^{abc}\,\omega_{bc}$.

Furthermore, the energy-momentum tensor of matter or fields present in the
spacetime, decomposed relative to ${u^{a}}$, is given by

$\displaystyle T_{ab}$ $\displaystyle=$ $\displaystyle\mu\hskip
0.99594ptu_{a}\hskip 0.99594ptu_{b}+p\hskip 0.99594pth_{ab}+q_{a}\hskip
0.99594ptu_{b}+u_{a}\hskip 0.99594ptq_{b}+\pi_{ab},$ (20)

where ${\mu}$ is the effective energy density, ${p}$ is the isotropic
pressure, ${q_{a}}$ is the 3-vector defining the energy-momentum flux and
${\pi_{ab}}$ is the anisotropic stress. The Weyl tensor also is decomposed
into electric part $E_{ab}$ and magnetic part $B_{ab}$ as follows,

$\displaystyle E_{ab}$ $\displaystyle=$ $\displaystyle C_{acbd}\hskip
0.99594ptu^{c}\hskip 0.99594ptu^{d}=E_{<ab>},$ (21) $\displaystyle B_{ab}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\varepsilon_{ade}\hskip
0.99594ptC^{de}{}_{bc}\hskip 0.99594ptu^{c}=B_{<ab>}.$ (22)

Here, the angle brackets denote orthogonal projections of vectors onto the
three space as well as the projected, symmetric and trace-free (PSTF) part of
tensors:

$\displaystyle\dot{V}_{<a>}$ $\displaystyle=$ $\displaystyle h_{a}{}^{b}\hskip
0.99594pt\dot{V}_{b},$ (23) $\displaystyle Z_{<ab>}$ $\displaystyle=$
$\displaystyle\left(h^{c}{}_{(a}\hskip
0.99594pth^{d}{}_{b)}-\frac{1}{3}h_{ab}\hskip 0.99594pth^{cd}\right)\hskip
0.99594ptZ_{cd}.$ (24)

###  A.2 Semitetrad 1+1+2 formalism

The $3-$space mentione din the $1+3$ covariant formalism can be further split
into a one spaclike direction $e_{a}$ satisfying $e_{a}e^{a}=1$ and a $2-$d
surface orthogonal to both $u^{a}$ and $e^{a}$. The 1+1+2 covariantly
decomposed spacetime is expressed in terms of the projection tensor $N_{ab}$
associated to the $2-$d surface as:

$g_{ab}=-u_{a}\hskip 0.99594ptu_{b}+e_{a}\hskip 0.99594pte_{b}+N_{ab},$ (25)

where ${N_{ab}}$ ${\left(e^{a}\hskip 0.99594ptN_{ab}=0=u^{a}\hskip
0.99594ptN_{ab},N^{a}{}_{a}=2\right)}$ projects vectors onto the 2-sheets,
orthogonal to ${u^{a}}$ and ${e^{a}}$. We introduce two new derivatives for
any tensor ${\phi_{a...b}{}^{c...d}}$:

$\displaystyle\hat{\phi}_{a..b}{}^{c..d}$ $\displaystyle\equiv$ $\displaystyle
e^{f}\hskip 0.99594ptD_{f}\hskip 0.99594pt\phi_{a..b}{}^{c..d},$ (26)
$\displaystyle\delta_{f}\phi_{a...b}{}^{c...d}$ $\displaystyle\equiv$
$\displaystyle
N_{f}{}^{j}N_{a}{}^{l}...N_{b}{}^{g}N_{h}{}^{c}...N_{i}{}^{d}D_{j}\phi_{l...g}{}^{h...i}.$

Eq.(26) denotes the derivative along the preferred spacelike direction $e^{a}$
while $\delta_{f}$ in Eq.(A.2) gives the 2-space derivative.

The 1+3 geometrical and dynamical quantities and anisotropic fluid variables
are split irreducibly as

$\displaystyle A^{a}$ $\displaystyle=$ $\displaystyle\mathcal{A}\hskip
0.99594pte^{a}+\mathcal{A}^{a},$ (28) $\displaystyle\omega^{a}$
$\displaystyle=$ $\displaystyle\Omega\hskip 0.99594pte^{a}+\Omega^{a},$ (29)
$\displaystyle\sigma_{ab}$ $\displaystyle=$
$\displaystyle\Sigma\left(e_{a}\hskip 0.99594pte_{b}-\frac{1}{2}\hskip
0.99594ptN_{ab}\right)+2\hskip 0.99594pt\Sigma_{(a}\hskip
0.99594pte_{b)}+\Sigma_{ab},$ (30) $\displaystyle E_{ab}$ $\displaystyle=$
$\displaystyle\mathcal{E}\left(e_{a}\hskip 0.99594pte_{b}-\frac{1}{2}\hskip
0.99594ptN_{ab}\right)+2\hskip 0.99594pt\mathcal{E}_{(a}\hskip
0.99594pte_{b)}+\mathcal{E}_{ab},$ (31) $\displaystyle B_{ab}$
$\displaystyle=$ $\displaystyle\mathcal{H}\left(e_{a}\hskip
0.99594pte_{b}-\frac{1}{2}\hskip 0.99594ptN_{ab}\right)+2\hskip
0.99594pt\mathcal{H}_{(a}\hskip 0.99594pte_{b)}+\mathcal{H}_{ab}\,.$ (32)

The fully projected 3-derivative of ${e^{a}}$ is given by

$\displaystyle D_{a}\hskip 0.99594pte_{b}$ $\displaystyle=$ $\displaystyle
e_{a}\hskip 0.99594pta_{b}+\frac{1}{2}\hskip 0.99594pt\phi\hskip
0.99594ptN_{ab}+\xi\hskip 0.99594pt\varepsilon_{ab}+\zeta_{ab},$ (33)

where traveling along ${e^{a}}$, ${a_{a}}$ is the sheet acceleration, ${\phi}$
is the sheet expansion, ${\xi}$ is the vorticity of ${e^{a}}$ (the twisting of
the sheet) and ${\zeta_{ab}}$ is the shear of ${e^{a}}$.

We can immediately see that the Ricci identities and the doubly contracted
Bianchi identities, which specify the evolution of the complete system, can
now be written as the time evolution, spatial propagation, and spatial
constraints of an irreducible set of geometrical variables:

$\displaystyle\mathcal{D}_{geom}=$ $\displaystyle\\{\Theta,\hskip
0.99594pt\mathcal{A},\hskip 0.99594pt\Omega,\hskip 0.99594pt\Sigma,\hskip
0.99594pt\mathcal{E},\hskip 0.99594pt\mathcal{B},\hskip 0.99594pt\phi,\hskip
0.99594pt\xi,\hskip 0.99594pt\mathcal{A}_{a},\hskip 0.99594pt\Omega_{a},\hskip
0.99594pt\Sigma_{a},\hskip 0.99594pt\alpha_{a},\hskip 0.99594pta_{a},\hskip
0.99594pt\mathcal{E}_{a},$ $\displaystyle\hskip
0.99594pt\mathcal{B}_{a},\hskip 0.99594pt\Sigma_{ab},\hskip
0.99594pt\zeta_{ab},\hskip 0.99594pt\mathcal{E}_{ab},\hskip
0.99594pt\mathcal{B}_{ab}\\}$ (34)

###  A.3 Degrees of freedom of incoming GW

Let us consider the following scenario: As show in Fig. (3), LRS-II spacetime
is perturbed due to the incoming GW. This disturbs the LRS-II spacetime and
leads to GW memory. To identify the GW memory, we need to know the true GW
degrees of freedom (DOF) in this formalism. In this appendix, for
completeness, we identify the true GW DOF.

Figure 3: The presence of GW alters the background LRS spactime.

Let the initial LRS-II spacetime be represented by the geometric variables
$\mathcal{D}_{0}$. Due to the incoming GW, the spacetime is no more LRS-II.
Since the incoming GW is weak, we can assume that the resulting spacetime to
be perturbed LRS-II. Let us represent the geometric variables of this
spacetime as $\mathcal{D}_{\rm n}$.

The GW mimicking variable may consist of one or several vector or tensor
quantities listed in $\mathcal{D}_{\rm n}$. As described in Refs. [72, 73],
the Weyl curvature tensor $C_{abcd}$ characterizes the gravitational
distortion or perturbation caused by GWs. The electric part of the Weyl
tensor, denoted by the 3-tensor $E_{ab}$, provides information about the tidal
deformation in spacetime, while the magnetic part, represented by $H_{ab}$,
presents the gravitational radiation traveling to future infinity from the
source [72, 71]. Both the electric and magnetic parts of the Weyl tensor
follow a closed form of the wave equation in the FLRW spacetime [96, 71].

In the $1+3$ formalism, $H_{ab}$ and $E_{ab}$ represent certain features of
GWs, while in the $1+1+2$ formalism, their components such as
$\mathcal{B},,\mathcal{B}_{a},,\mathcal{B}_{ab}$ and
$\mathcal{E},,\mathcal{E}_{a},,\mathcal{E}_{ab}$ carry information about the
passing GWs during $[\tau_{0},,\tau_{1}]$. As mentioned earlier, GWs perturb
the $2-$dimensional surface $S_{0}$ orthogonal to it. Hence, the vector and
tensor components of $H_{ab}$ and $E_{ab}$ in the $2-$dimensional surface
($\mathcal{B}_{a},,\mathcal{B}_{ab}$) and
($\mathcal{E}_{a},,\mathcal{E}_{ab}$) describe the GW distortion during the
time interval $[\tau_{0},,\tau_{1}]$.

As discussed above, in the $1+1+2$ formalism
$\mathcal{E}_{a},,\mathcal{E}_{ab},,\mathcal{B}_{a},,\mathcal{B}_{ab}$ each
have two components, resulting in a total of 8 DOF. However, considering the 2
constraint equations for the scalar part of electric and magnetic Weyl
(45),(46), the DOF reduces to 6. Furthermore, the relationship among the
components of the electric and magnetic parts of the Weyl tensor, as given by
Eq.(53), (54) in the Appendix (C), reduces the DOF to 2, matching the DOF of
GWs in linearized gravity theory [97, 73]. In principle, any of
$\mathcal{E}_{a},,\mathcal{E}_{ab},,\mathcal{B}_{a},,\mathcal{B}_{ab}$ can
represent the passing gravitational wave. However, to formulate the master
equation for GW memory as done in Appendix (C), we choose $\mathcal{B}_{a}$ to
describe the incoming GW.

##  Appendix B Relation between GW memory and shear

Gravitational wave (GW) memory refers to the change in the metric or spacetime
resulting from the passage of GWs. Mathematically, it is defined as:

$\displaystyle\Delta
h_{+,\times}^{\mathrm{mem}}=\lim_{t\rightarrow+\infty}h_{+,\times}(t)-\lim_{t\rightarrow-\infty}h_{+,\times}(t),$
(35)

where $t$ represents the time of the asymptotic (static) observer, and $h_{+}$
and $h_{\times}$ denote the plus and cross modes of the GW signal,
respectively.

However, in covariant formalism, $u^{a}$ defines the time. The change of any
vector or tensor with respect to time is equivalent to their covariant
derivative projected along the timelike direction $u^{a}$. This change can be
obtained from the Lie derivative of any tensor or vector along $u^{a}$. To
quantify gravitational memory and relate it to the geometrical variables
described in $\mathcal{D}_{\rm n}$, we begin with the Lie derivative of
$N_{ab}$ along $u^{a}$.

$\displaystyle\mathcal{X}_{ab}\equiv\mathcal{L}_{u}N_{ab}=u^{c}\nabla_{c}N_{ab}+N_{cb}\nabla_{a}u^{c}+N_{ca}\nabla_{b}u^{c}$
(36)

Substituting the geometrical variables described in $1+1+2$ formalism we
obtain,

$\displaystyle\mathcal{X}_{ab}=2n_{(a}\left[-\alpha_{b)}+\Sigma_{b)}-\varepsilon_{b)\,c}\,\Omega^{c}\right]+2\,\Sigma_{ab}$
$\displaystyle+N_{ab}\left[\frac{2\Theta}{3}-\Sigma\right]$ (37)

The brackets $()$ in the subscript denote the symmetrization of the terms. In
Ref. [70], the author mentions the existence of a $2-$surface if the Greenberg
vector $\Sigma^{a}+\varepsilon^{ab}\Omega_{b}-\alpha^{a}=0$. Using this
Greenberg vector condition in the previous equation, we obtain:

$\displaystyle\mathcal{X}_{ab}=-4n_{(a}\varepsilon_{b)c}\,\Omega^{c}+2\Sigma_{ab}+N_{ab}\left(\frac{2\Theta}{3}-\Sigma\right)$
(38)

The quantity $\mathcal{X}_{ab}$ above is symmetric but not transverse-
traceless. Its transverse trace-less part is derived as follows:

$\displaystyle\mathcal{X}_{\\{ab\\}}=N^{\alpha}\,_{(a}N^{\beta}\,_{b)}\mathcal{X}_{cd}-\frac{1}{2}N_{ab}N^{cd}\mathcal{X}_{cd}$
(39)

Substituting $\mathcal{X}_{\alpha\beta}$ into the RHS of the above equation,
we finally obtain:

$\displaystyle\mathcal{X}_{\\{ab\\}}=2\,\Sigma_{ab}$ (40)

The $\Sigma_{ab}$ tensor serves as an indicator of memory in this scenario as
it represents the time evolution of the projection tensor onto the $2-$space.
We need to establish the conditions under which the projection tensor $N_{ab}$
reduces to the metric of the $2-$space. These conditions are: 1) the Greenberg
vector $\Sigma^{a}+\varepsilon^{ab}\Omega_{b}-\alpha^{a}=0$, and 2)
$\xi=\Omega=0$).

Similarly, the covariant derivative of $N_{ab}$ projected along the spacelike
direction will be $\zeta_{ab}$, where $\zeta_{ab}$ represents the shear
related to the spacelike direction $n^{a}$. Hence, both $\sigma_{ab}$ and
$\zeta_{ab}$ will contribute to memory.

In the following calculation, we assume that the projection tensor $N_{ab}$
reduces to the metric of the $2-$space if:

$\displaystyle\text{Greenberg
vector:}\quad\Sigma^{a}+\varepsilon^{ab}\Omega_{b}-\alpha^{a}$
$\displaystyle=0,$ (41) $\displaystyle\text{}\xi=\Omega$ $\displaystyle=0$
(42)

are satisfied.

##  Appendix C Formulation of gravitational memory in _LRS-II_ spacetime

In this section we formulate the master equation to obtain GW memory in the
_LRS-II_ spacetime using the evolution, propagation and constraint equations
of electric $\mathcal{E},\,\mathcal{E}_{a},\,\mathcal{E}_{ab}$ and magnetic
parts $\mathcal{B},\,\mathcal{B}_{a},\,\mathcal{B}_{ab}$ of Weyl curvature
tensor. Here, we use the equations derived in Ref. [70] considering terms till
first order w.r.t. the quantities listed in $\mathcal{D}_{\rm n}$. We assume
that the _LRS-II_ background contains only the energy density and pressure in
the energy-momentum tensor that is a function of time along. We ignore the
anisotropic stress and heat flux. The propagation equations for
$\mathcal{E},\,\mathcal{B}$ are:

$\displaystyle\hat{\mathcal{E}}$
$\displaystyle=-\delta_{a}\mathcal{E}^{a}-\frac{3}{2}\phi\mathcal{E}$ (43)
$\displaystyle\hat{\mathcal{B}}$
$\displaystyle=-\delta_{a}\mathcal{B}^{a}-\frac{3}{2}\phi\mathcal{B}$ (44)

As GWs propagate along the null direction, instead of using the time
coordinate $t$’ and radial coordinate $r$’, we utilize the null coordinate
$u\equiv t-r$. In an asymptotically flat spacetime, at the asymptotic limit,
the time and radial derivatives can be expressed as
$\partial_{t}\sim\partial_{u}$ and $\partial_{r}\sim-\partial_{u}$. Using this
notation, we rewrite the propagation equations as follows:

$\displaystyle\partial_{u}\mathcal{E}-\frac{3}{2}\phi,\mathcal{E}$
$\displaystyle=\delta_{a}\mathcal{E}^{a}$ (45)
$\displaystyle\partial_{u}\mathcal{B}-\frac{3}{2}\phi,\mathcal{B}$
$\displaystyle=\delta_{a}\mathcal{B}^{a}$ (46)

These are the constraint equations relating the scalar part
$\mathcal{E}(\mathcal{B})$ to the vector $\mathcal{E}_{a}(\mathcal{B}_{a})$.
Now, we aim to establish if there exists any relation between the electric and
magnetic parts of the Weyl tensor. To proceed, we derive the evolution
equations for $\mathcal{E}_{a}$ and $\mathcal{B}_{a}$ after incorporating
conditions (41) and (42) in the null coordinate.

$\displaystyle\partial_{u}\mathcal{E}_{a}-\frac{1}{2}\varepsilon_{ab}\partial_{u}\mathcal{B}^{b}$
$\displaystyle=\frac{3}{4}\varepsilon_{ab}\delta^{b}\mathcal{B}+\frac{1}{2}\varepsilon_{bc}\delta^{b}\mathcal{B}^{c}\,_{a}-\frac{1}{2}(\mu+p)\,\Sigma_{a}$
$\displaystyle+\left(\frac{3}{4}\Sigma-\Theta\right)\mathcal{E}_{a}-\left(\frac{1}{4}\phi+\mathcal{A}\right)\varepsilon_{ab}\mathcal{B}^{b}$
(47)

and,

$\displaystyle\partial_{u}\mathcal{B}_{a}+\frac{1}{2}\varepsilon_{ab}\partial_{u}\mathcal{E}^{b}$
$\displaystyle=-\frac{3}{4}\varepsilon_{ab}\delta^{b}\mathcal{E}-\frac{1}{2}\varepsilon_{bc}\delta^{b}\mathcal{E}^{c}\,_{a}$
$\displaystyle+\left(\frac{3}{4}\Sigma-\Theta\right)\mathcal{B}_{a}+\left(\frac{1}{4}\phi+\mathcal{A}\right)\varepsilon_{ab}\mathcal{E}^{b}$
(48)

We write the propagation equations of $\mathcal{E}_{a},\,\mathcal{B}_{a}$ in
terms of _null coordinate_ as follows,

$\displaystyle\partial_{u}\mathcal{E}_{a}$
$\displaystyle=-\frac{1}{2}\delta_{a}\mathcal{E}+\delta^{b}\mathcal{E}_{ab}+\frac{3}{2}\phi\mathcal{E}_{a}+\frac{3}{2}\Sigma\,\varepsilon_{ab}\mathcal{B}^{b}$
(49)

and,

$\displaystyle\partial_{u}\mathcal{B}_{a}$
$\displaystyle=-\frac{1}{2}\delta_{a}\mathcal{B}+\delta^{b}\mathcal{B}_{ab}+\frac{3}{2}\phi\mathcal{B}_{a}-\frac{3}{2}\Sigma\varepsilon_{ab}\mathcal{E}^{b}$
(50)

Now, multiplying Eq. (C) with $\varepsilon^{ae}$ and adding it to Eq. (C) (by
setting the same dummy index to $a$) we obtain the following relation,

$\displaystyle\frac{3}{2}\partial_{u}\left(\mathcal{E}_{a-}\varepsilon_{ab}\mathcal{B}^{b}\right)=\frac{3}{4}\left(\varepsilon_{ab}\delta^{b}\mathcal{B}-\delta_{a}\mathcal{E}\right)-\frac{1}{2}\left(\mu+p\right)\Sigma_{a}$
$\displaystyle-\frac{3}{4}\left(\alpha_{a}-2\mathcal{A}_{a}\right)\mathcal{E}+\left(\frac{\phi}{4}+\mathcal{A}+\frac{3}{4}\Sigma-\Theta\right)\left(\mathcal{E}_{a}-\varepsilon_{ab}\mathcal{B}^{b}\right)$
$\displaystyle+\frac{1}{2}\left(\varepsilon_{bc}\delta^{b}\mathcal{B}^{c}\,_{a}-\delta^{b}\mathcal{E}_{ab}\right)$
(51)

Substituting Eqs. (49) and (50) into the above equation and multiplying the
result by $\varepsilon^{ad}$ we obtain,

$\displaystyle
2\delta^{b}\left(\mathcal{E}_{ab}-\varepsilon^{ad}\mathcal{B}^{d}\,_{b}\right)$
$\displaystyle+2(\mu+p)\Sigma_{a}=$
$\displaystyle\left(-2\phi+\mathcal{A}+3\Sigma-\Theta\right)\left(\mathcal{E}_{a}-\varepsilon_{ab}\mathcal{B}^{b}\right)$
(52)

One can check that $\alpha_{a}$=0 for perturbed FLRW and Minkowski spacetimes.
This automatically ensures that $\Sigma_{a}=0$ if $\Omega_{a}=0$ from the
Greenberg vector condition. We choose in rest of our work the following
relation between $\\{\mathcal{E}_{a},\,\mathcal{B}_{a}\\}$ and
$\\{\mathcal{E}_{ab},\,\mathcal{B}_{ab}\\}$.

$\displaystyle\mathcal{E}_{a}=\varepsilon_{ab}\mathcal{B}^{b}$ (53)
$\displaystyle\mathcal{E}_{ab}=\varepsilon_{ad}\mathcal{B}^{d}\,_{b}$ (54)

We have the evolution equation of $\mathcal{E}_{ab}$, tensor part of the
electric part of Weyl tensor in _null coordinate_ as,

$\displaystyle\partial_{u}\mathcal{E}_{\\{ab\\}}+\partial_{u}\left(\varepsilon_{c\\{a}\mathcal{B}_{b\\}}^{~{}~{}c}\right)=-\varepsilon_{c\\{a}\delta^{c}\mathcal{B}_{b\\}}-\textstyle\frac{3}{2}\mathcal{E}\Sigma_{ab}$
$\displaystyle-\textstyle\frac{3}{2}\mathcal{B}\varepsilon_{c\\{a}\zeta_{b\\}}^{~{}~{}c}-\left(\Theta+\textstyle\frac{3}{2}\Sigma\right)\mathcal{E}_{ab}+\left(\textstyle\frac{1}{2}\phi+2{\cal
A}\right)\varepsilon_{c\\{a}\mathcal{B}_{b\\}}^{~{}~{}c}$
$\displaystyle-\textstyle\frac{1}{2}(\mu+p)\Sigma_{ab}$ (55)

We have assumed $\partial_{u}\varepsilon_{ab}=0$. Now substituting Eq. (54)
relation and simplifying the above equation we obtain,

$\displaystyle
3\mathcal{E}\,\Sigma_{ab}+\left(2\Theta+3\Sigma+\phi+4\mathcal{A}\right)\mathcal{E}_{ab}+\left(\mu+p\right)\Sigma_{ab}$
$\displaystyle=-2\varepsilon_{c\\{a}\delta^{c}\,\mathcal{B}_{b\\}}$ (56)

We already mentioned that the shear tensor $\Sigma_{ab}$ holds information
about the alteration of the spacetime or the memory the passing GW. Hence we
want to re-write the above Eq. (C) substituting $\mathcal{E}_{ab}$ in terms of
$\Sigma_{ab}$. To do so, we have the evolution equation of $\Sigma_{ab}$ as,

$\displaystyle\dot{\Sigma}_{\\{ab\\}}=\delta_{\\{a}{\cal
A}_{b\\}}-\left(\textstyle\frac{2}{3}\Theta+\textstyle\frac{1}{2}\Sigma\right)\Sigma_{ab}+\mathcal{A}\zeta_{ab}-\mathcal{E}_{ab}$
(57)

We set $\mathcal{A}=\mathcal{A}_{a}=0$ in the above equation and obtain,

$\displaystyle\mathcal{E}_{ab}=-\dot{\Sigma}_{\\{ab\\}}-\left(\textstyle\frac{2}{3}\Theta+\textstyle\frac{1}{2}\Sigma\right)\Sigma_{ab}$
(58)

Substituting Eq. (58) into Eq. (C) we obtain the final master equation for GW
memory,

$\displaystyle
3\mathcal{E}\,\Sigma_{ab}-\left(2\Theta+3\Sigma+\phi\right)\dot{\Sigma}_{\\{ab\\}}+\left(\mu+p\right)\Sigma_{ab}$
$\displaystyle-\left(2\Theta+3\Sigma+\phi\right)\left(\textstyle\frac{2}{3}\Theta+\textstyle\frac{1}{2}\Sigma\right)\Sigma_{ab}=-2\varepsilon_{c\\{a}\delta^{c}\,\mathcal{B}_{b\\}}$
(59)

##  Appendix D Gravitational wave memory profiles in Minkowski and FLRW
spacetime

In this section, we try to examine the behavior of the memory observable
($N_{\vartheta\vartheta}$) in Minkowski spacetime and the amplification factor
($A_{C}$) in FLRW.

###  D.1 Minkowski spacetime

As previously outlined, the fall-off behavior of the memory observable in
Minkowski spacetime is $1/r$. Assuming the form of the incoming GW profile for
the plus polarization to be [81]:

$\displaystyle h_{+}^{I}=$ $\displaystyle
4,{\dfrac{m_{A}m_{B}}{b,r}},\bigg{[}\frac{1}{4}\left(\frac{1}{\sqrt{l^{2}+1}}+\frac{1}{\left(l^{2}+1\right)^{3/2}}\right)$
(60)
$\displaystyle+\frac{1}{2}\left(\frac{l}{\sqrt{l^{2}+1}}+\frac{l}{\left(l^{2}+1\right)^{3/2}}+1\right)-\frac{1}{4\left(l^{2}+1\right)^{3/2}}\bigg{]}$

where $l=\sqrt{1+u^{2}}$. $m_{A}$ and $m_{B}$ represent the masses of the
stars undergoing hyperbolic scattering, $b$ is the impact parameter, and $r$
is the distance between the source and the observation point (detector). Since
our observable is a subleading term of the 2-sphere metric with radius $r$,
the evolution of the factor $(N_{\vartheta\vartheta}/r^{2})$ is depicted in
Fig.(4), rather than solely $N_{\vartheta\vartheta}$. The profile exhibits the
characteristic signal of a GW burst with non-zero memory.

Figure 4: The evolution of memory observable $N_{\vartheta\vartheta}$ due to
the passage of GW burst.

###  D.2 Amplification factor in Cosmological GW memory

Here, we aim to derive the form of the amplification factor. We begin by
rewriting the relation between instantaneous memory and cumulative memory
given in Eq.(4) for FLRW spacetime:

$\displaystyle\partial_{\vartheta\vartheta}\left({N_{\vartheta\vartheta}}/{a^{2}}\right)={2\Sigma_{\vartheta\vartheta}}/{a}.$
(61)

Considering that the burst profile traverses for a finite interval of
conformal time ($\eta_{0},\eta_{1}$) and integrating Eq.(61) in that region,
we obtain:

$\displaystyle\left({N_{\vartheta\vartheta}}/{a^{2}}\right)\bigg{\rvert}_{\eta_{1}}-\left({N_{\vartheta\vartheta}}/{a^{2}}\right)\bigg{\rvert}_{\eta_{0}}=F(\eta_{1})-F(\eta_{0}),$
(62)

where $F(\eta)=\int\frac{2\Sigma_{\vartheta\vartheta}}{a},d\eta$. Since the GW
are massless, its energy density behaves as $(\frac{N_{+}}{a^{2}})^{2}$. Thus,
to obtain the amplification of the GW memory amplitude, we compute the ratio
of the quantity $(\frac{N_{+}}{a^{2}})$ at initial and final conformal times.
We define amplification as:

$A_{C}=\frac{\left({N_{\vartheta\vartheta}}/{a^{2}}\right)\rvert_{\eta_{1}}}{\left({N_{\vartheta\vartheta}}/{a^{2}}\right)\rvert_{\eta_{0}}}.$

Plugging this relation back to Eq.(62) we get,

$\displaystyle
A_{C}=1+\dfrac{a^{2}(\eta_{0})}{h_{I}(\eta_{0})}\,[F(\eta_{1})-F(\eta_{0})].$
(63)

Note at initial time $\eta_{i}$,
$N_{\vartheta\vartheta}=h_{\vartheta\vartheta}^{I}(\eta_{0})$.

###  D.3 FLRW spacetime

We have previously discussed the amplification factor for matter domination
(MD), dark energy (DE), and matter-to-$\Lambda$ transition. Here, we
demonstrate how the amplification factor behaves for radiation-dominated (RD),
and super-inflation (SI) scenarios [98, 99] for unit $e$-fold expansion.

Figure 5: Evolution of the amplification factor for SI
($a(\eta)=1/(-\eta)^{1/20}$), and RD ($a(t)=t^{1/2}$).

Only in the case of SI do we observe an increase in amplification. In RD, we
observe similar decaying features as were obtained for MD and DE phases.

## References

  * Weinberg [2008] S. Weinberg,  _Cosmology_ , Oxford Graduate Texts (Oxford University Press, 2008).
  * Peter and Uzan [2013] P. Peter and J.-P. Uzan,  _Primordial Cosmology_ , Oxford Graduate Texts (Oxford University Press, 2013).
  * Rees [2022] M. J. Rees, Ann. Rev. Astron. Astrophys. 60, 1 (2022).
  * Melia [2022] F. Melia, Publ. Astron. Soc. Pac. 134, 121001 (2022).
  * Ellis and Wands [2023] J. Ellis and D. Wands, (2023), arXiv:2312.13238 [astro-ph.CO] .
  * Martin  _et al._ [2014] J. Martin, C. Ringeval, and V. Vennin, Phys. Dark Univ. 5-6, 75 (2014), arXiv:1303.3787 [astro-ph.CO] .
  * Odintsov  _et al._ [2023] S. D. Odintsov, V. K. Oikonomou, I. Giannakoudi, F. P. Fronimos, and E. C. Lymperiadou, Symmetry 15, 1701 (2023), arXiv:2307.16308 [gr-qc] .
  * Profumo  _et al._ [2019] S. Profumo, L. Giani, and O. F. Piattella, Universe 5, 213 (2019), arXiv:1910.05610 [hep-ph] .
  * Rajendran [2022] S. Rajendran, SciPost Phys. Lect. Notes 56, 1 (2022), arXiv:2204.03085 [hep-ph] .
  * Carr and Kuhnel [2020] B. Carr and F. Kuhnel, Ann. Rev. Nucl. Part. Sci. 70, 355 (2020), arXiv:2006.02838 [astro-ph.CO] .
  * Sahni and Starobinsky [2000] V. Sahni and A. A. Starobinsky, Int. J. Mod. Phys. D 9, 373 (2000), arXiv:astro-ph/9904398 .
  * Peebles and Ratra [2003] P. J. E. Peebles and B. Ratra, Rev. Mod. Phys. 75, 559 (2003), arXiv:astro-ph/0207347 .
  * Padmanabhan [2003] T. Padmanabhan, Phys. Rept. 380, 235 (2003), arXiv:hep-th/0212290 .
  * Joyce  _et al._ [2016] A. Joyce, L. Lombriser, and F. Schmidt, Ann. Rev. Nucl. Part. Sci. 66, 95 (2016), arXiv:1601.06133 [astro-ph.CO] .
  * Motta  _et al._ [2021] V. Motta, M. A. García-Aspeitia, A. Hernández-Almada, J. Magaña, and T. Verdugo, Universe 7, 163 (2021), arXiv:2104.04642 [astro-ph.CO] .
  * Weinberg [1989] S. Weinberg, Rev. Mod. Phys. 61, 1 (1989).
  * Weinberg [2000] S. Weinberg, in  _4th International Symposium on Sources and Detection of Dark Matter in the Universe (DM 2000)_ (2000) pp. 18–26, arXiv:astro-ph/0005265 .
  * Carroll [2001] S. M. Carroll, Living Rev. Rel. 4, 1 (2001), arXiv:astro-ph/0004075 .
  * Di Valentino  _et al._ [2021] E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri, D. F. Mota, A. G. Riess, and J. Silk, Class. Quant. Grav. 38, 153001 (2021), arXiv:2103.01183 [astro-ph.CO] .
  * Shah  _et al._ [2021] P. Shah, P. Lemos, and O. Lahav, Astron. Astrophys. Rev. 29, 9 (2021), arXiv:2109.01161 [astro-ph.CO] .
  * Schöneberg  _et al._ [2022] N. Schöneberg, G. Franco Abellán, A. Pérez Sánchez, S. J. Witte, V. Poulin, and J. Lesgourgues, Phys. Rept. 984, 1 (2022), arXiv:2107.10291 [astro-ph.CO] .
  * Kamionkowski and Riess [2023] M. Kamionkowski and A. G. Riess, Ann. Rev. Nucl. Part. Sci. 73, 153 (2023), arXiv:2211.04492 [astro-ph.CO] .
  * Shankaranarayanan and Johnson [2022] S. Shankaranarayanan and J. P. Johnson, Gen. Rel. Grav. 54, 44 (2022), arXiv:2204.06533 [gr-qc] .
  * Lima  _et al._ [2012] J. A. S. Lima, J. F. Jesus, R. C. Santos, and M. S. S. Gill, (2012), arXiv:1205.4688 [astro-ph.CO] .
  * Moresco  _et al._ [2016] M. Moresco, L. Pozzetti, A. Cimatti, R. Jimenez, C. Maraston, L. Verde, D. Thomas, A. Citro, R. Tojeiro, and D. Wilkinson, JCAP 05, 014 (2016), arXiv:1601.01701 [astro-ph.CO] .
  * Sathyaprakash and Schutz [2009] B. S. Sathyaprakash and B. F. Schutz, Living Rev. Rel. 12, 2 (2009), arXiv:0903.0338 [gr-qc] .
  * Sasaki  _et al._ [2016] M. Sasaki, T. Suyama, T. Tanaka, and S. Yokoyama, Phys. Rev. Lett. 117, 061101 (2016), [Erratum: Phys.Rev.Lett. 121, 059901 (2018)], arXiv:1603.08338 [astro-ph.CO] .
  * Bird  _et al._ [2016] S. Bird, I. Cholis, J. B. Muñoz, Y. Ali-Haïmoud, M. Kamionkowski, E. D. Kovetz, A. Raccanelli, and A. G. Riess, Phys. Rev. Lett. 116, 201301 (2016), arXiv:1603.00464 [astro-ph.CO] .
  * Hayasaki  _et al._ [2016] K. Hayasaki, K. Takahashi, Y. Sendouda, and S. Nagataki, Publ. Astron. Soc. Jap. 68, 66 (2016), arXiv:0909.1738 [astro-ph.CO] .
  * Abbott  _et al._ [2017a] B. P. Abbott _et al._ (LIGO Scientific, Virgo, Fermi-GBM, INTEGRAL), Astrophys. J. Lett. 848, L13 (2017a), arXiv:1710.05834 [astro-ph.HE] .
  * Abbott  _et al._ [2017b] B. P. Abbott _et al._ (LIGO Scientific, Virgo), Phys. Rev. Lett. 119, 161101 (2017b), arXiv:1710.05832 [gr-qc] .
  * de Rham and Melville [2018] C. de Rham and S. Melville, Phys. Rev. Lett. 121, 221101 (2018), arXiv:1806.09417 [hep-th] .
  * Ezquiaga and Zumalacárregui [2017] J. M. Ezquiaga and M. Zumalacárregui, Phys. Rev. Lett. 119, 251304 (2017), arXiv:1710.05901 [astro-ph.CO] .
  * Sakstein and Jain [2017] J. Sakstein and B. Jain, Phys. Rev. Lett. 119, 251303 (2017), arXiv:1710.05893 [astro-ph.CO] .
  * Schutz [1986] B. F. Schutz, Nature 323, 310 (1986).
  * Abbott  _et al._ [2017c] B. P. Abbott _et al._ (LIGO Scientific, Virgo, 1M2H, Dark Energy Camera GW-E, DES, DLT40, Las Cumbres Observatory, VINROUGE, MASTER), Nature 551, 85 (2017c), arXiv:1710.05835 [astro-ph.CO] .
  * Del Pozzo [2012] W. Del Pozzo, Phys. Rev. D 86, 043011 (2012), arXiv:1108.1317 [astro-ph.CO] .
  * Nissanke  _et al._ [2013] S. Nissanke, D. E. Holz, N. Dalal, S. A. Hughes, J. L. Sievers, and C. M. Hirata, (2013), arXiv:1307.2638 [astro-ph.CO] .
  * Braginsky and Grishchuk [1985] V. B. Braginsky and L. P. Grishchuk, Sov. Phys. JETP 62, 427 (1985).
  * Favata [2010] M. Favata, Class. Quant. Grav. 27, 084036 (2010), arXiv:1003.3486 [gr-qc] .
  * Zel’dovich and Polnarev [1974] Y. B. Zel’dovich and A. G. Polnarev, Sov. Astron. 18, 17 (1974).
  * Christodoulou [1991] D. Christodoulou, Phys. Rev. Lett. 67, 1486 (1991).
  * Braginskii and Thorne [1987] V. B. Braginskii and K. S. Thorne, Nature 327, 123 (1987).
  * Favata [2009] M. Favata, Astrophys. J. Lett. 696, L159 (2009), arXiv:0902.3660 [astro-ph.SR] .
  * Zhang  _et al._ [2017] P. M. Zhang, C. Duval, G. W. Gibbons, and P. A. Horvathy, Phys. Rev. D 96, 064013 (2017), arXiv:1705.01378 [gr-qc] .
  * Strominger and Zhiboedov [2016a] A. Strominger and A. Zhiboedov, JHEP 01, 086 (2016a), arXiv:1411.5745 [hep-th] .
  * Flanagan and Nichols [2017] E. E. Flanagan and D. A. Nichols, Phys. Rev. D 95, 044002 (2017), [Erratum: Phys.Rev.D 108, 069902 (2023)], arXiv:1510.03386 [hep-th] .
  * Nichols [2017] D. A. Nichols, Phys. Rev. D 95, 084048 (2017), arXiv:1702.03300 [gr-qc] .
  * Mitman  _et al._ [2020] K. Mitman, J. Moxon, M. A. Scheel, S. A. Teukolsky, M. Boyle, N. Deppe, L. E. Kidder, and W. Throwe, Phys. Rev. D 102, 104007 (2020), arXiv:2007.11562 [gr-qc] .
  * Strominger [2017] A. Strominger,  _Lectures on the Infrared Structure of Gravity and Gauge Theory_ (2017) arXiv:1703.05448 [hep-th] .
  * Compère and Fiorucci [2018] G. Compère and A. Fiorucci, (2018), arXiv:1801.07064 [hep-th] .
  * Thorne [1992] K. S. Thorne, Phys. Rev. D 45, 520 (1992).
  * Wiseman and Will [1991] A. G. Wiseman and C. M. Will, Phys. Rev. D 44, R2945 (1991).
  * Jenkins and Sakellariadou [2021] A. C. Jenkins and M. Sakellariadou, Class. Quant. Grav. 38, 165004 (2021), arXiv:2102.12487 [gr-qc] .
  * Mukhopadhyay  _et al._ [2021] M. Mukhopadhyay, C. Cardona, and C. Lunardini, JCAP 07, 055 (2021), arXiv:2105.05862 [astro-ph.HE] .
  * Jokela  _et al._ [2022] N. Jokela, K. Kajantie, and M. Sarkkinen, Phys. Rev. D 106, 064022 (2022), arXiv:2204.06981 [gr-qc] .
  * Shore [2018] G. M. Shore, JHEP 12, 133 (2018), arXiv:1811.08827 [gr-qc] .
  * Siddhant  _et al._ [2021] S. Siddhant, I. Chakraborty, and S. Kar, Eur. Phys. J. C 81, 350 (2021), arXiv:2011.12368 [gr-qc] .
  * Chakraborty [2022] I. Chakraborty, Phys. Rev. D 105, 024063 (2022), arXiv:2110.02295 [gr-qc] .
  * Heisenberg  _et al._ [2023] L. Heisenberg, N. Yunes, and J. Zosso, Phys. Rev. D 108, 024010 (2023), arXiv:2303.02021 [gr-qc] .
  * Tolish and Wald [2016] A. Tolish and R. M. Wald, Phys. Rev. D 94, 044009 (2016), arXiv:1606.04894 [gr-qc] .
  * Bieri  _et al._ [2017] L. Bieri, D. Garfinkle, and N. Yunes, Class. Quant. Grav. 34, 215002 (2017), arXiv:1706.02009 [gr-qc] .
  * Chu [2017] Y.-Z. Chu, Class. Quant. Grav. 34, 035009 (2017), arXiv:1603.00151 [gr-qc] .
  * Kehagias and Riotto [2016] A. Kehagias and A. Riotto, JCAP 05, 059 (2016), arXiv:1602.02653 [hep-th] .
  * Jana and Shankaranarayanan [2023] S. Jana and S. Shankaranarayanan, Phys. Rev. D 108, 024044 (2023), arXiv:2301.11772 [gr-qc] .
  * van Elst and Ellis [1996] H. van Elst and G. F. R. Ellis, Class. Quant. Grav. 13, 1099 (1996), arXiv:gr-qc/9510044 .
  * Ellis [1967] G. F. R. Ellis, J. Math. Phys. 8, 1171 (1967).
  * Clarkson  _et al._ [2004] C. A. Clarkson, M. Marklund, G. Betschart, and P. K. S. Dunsby, Astrophys. J. 613, 492 (2004), arXiv:astro-ph/0310323 .
  * Clarkson and Barrett [2003] C. A. Clarkson and R. K. Barrett, Class. Quant. Grav. 20, 3855 (2003), arXiv:gr-qc/0209051 .
  * Clarkson [2007] C. Clarkson, Phys. Rev. D 76, 104034 (2007), arXiv:0708.1398 [gr-qc] .
  * Goswami and Ellis [2021] R. Goswami and G. F. R. Ellis, Class. Quant. Grav. 38, 085023 (2021), arXiv:1912.00591 [gr-qc] .
  * Owen  _et al._ [2011a] R. Owen _et al._ , Phys. Rev. Lett. 106, 151101 (2011a), arXiv:1012.4869 [gr-qc] .
  * Zhang  _et al._ [2012] F. Zhang, A. Zimmerman, D. A. Nichols, Y. Chen, G. Lovelace, K. D. Matthews, R. Owen, and K. S. Thorne, Phys. Rev. D 86, 084049 (2012), arXiv:1208.3034 [gr-qc] .
  * Maartens and Bassett [1998] R. Maartens and B. A. Bassett, Class. Quant. Grav. 15, 705 (1998), arXiv:gr-qc/9704059 .
  * Herrera  _et al._ [2006] L. Herrera, N. O. Santos, and J. Carot, J. Math. Phys. 47, 052502 (2006), arXiv:gr-qc/0511112 .
  * Bruni  _et al._ [1992] M. Bruni, P. K. S. Dunsby, and G. F. R. Ellis, Astrophys. J. 395, 34 (1992).
  * Ellis  _et al._ [2012] G. F. R. Ellis, R. Maartens, and M. A. H. MacCallum,  _Relativistic Cosmology_ (Cambridge University Press, 2012).
  * Bieri and Garfinkle [2014] L. Bieri and D. Garfinkle, Phys. Rev. D 89, 084039 (2014), arXiv:1312.6871 [gr-qc] .
  * Tolish  _et al._ [2014] A. Tolish, L. Bieri, D. Garfinkle, and R. M. Wald, Phys. Rev. D 90, 044060 (2014), arXiv:1405.6396 [gr-qc] .
  * Strominger and Zhiboedov [2016b] A. Strominger and A. Zhiboedov, JHEP 01, 086 (2016b), arXiv:1411.5745 [hep-th] .
  * Kovacs and Thorne [1978] S. J. Kovacs and K. S. Thorne, Astrophys. J. 224, 62 (1978).
  * Trashorras  _et al._ [2021] M. Trashorras, J. García-Bellido, and S. Nesseris, Universe 7, 18 (2021), arXiv:2006.15018 [astro-ph.CO] .
  * De Vittori  _et al._ [2012] L. De Vittori, P. Jetzer, and A. Klein, Phys. Rev. D 86, 044017 (2012), arXiv:1207.5359 [gr-qc] .
  * Mukherjee  _et al._ [2021] S. Mukherjee, S. Mitra, and S. Chatterjee, Mon. Not. Roy. Astron. Soc. 508, 5064 (2021), arXiv:2010.00916 [gr-qc] .
  * Nichols  _et al._ [2011] D. A. Nichols _et al._ , Phys. Rev. D 84, 124014 (2011), arXiv:1108.5486 [gr-qc] .
  * Mavrogiannis and Tsagas [2021] P. Mavrogiannis and C. G. Tsagas, Phys. Rev. D 104, 124053 (2021), arXiv:2110.02489 [gr-qc] .
  * Riess  _et al._ [1998] A. G. Riess _et al._ (Supernova Search Team), Astron. J. 116, 1009 (1998), arXiv:astro-ph/9805201 .
  * Perlmutter  _et al._ [1999] S. Perlmutter _et al._ (Supernova Cosmology Project), Astrophys. J. 517, 565 (1999), arXiv:astro-ph/9812133 .
  * Eisenstein  _et al._ [2005] D. J. Eisenstein _et al._ (SDSS), Astrophys. J. 633, 560 (2005), arXiv:astro-ph/0501171 .
  * Aghanim  _et al._ [2020] N. Aghanim _et al._ (Planck), Astron. Astrophys. 641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO] .
  * Capozziello  _et al._ [2014] S. Capozziello, O. Farooq, O. Luongo, and B. Ratra, Phys. Rev. D 90, 044016 (2014), arXiv:1403.1421 [gr-qc] .
  * Capozziello  _et al._ [2015] S. Capozziello, O. Luongo, and E. N. Saridakis, Phys. Rev. D 91, 124037 (2015), arXiv:1503.02832 [gr-qc] .
  * Ellis and van Elst [1999] G. F. R. Ellis and H. van Elst, NATO Sci. Ser. C 541, 1 (1999), arXiv:gr-qc/9812046 .
  * Heckmann and Schucking [1955] O. Heckmann and E. Schucking, zap 38, 95 (1955).
  * Raychaudhuri [1955] A. Raychaudhuri, Physical Review 98, 1123 (1955).
  * Hawking [1966] S. W. Hawking, Astrophys. J. 145, 544 (1966).
  * Owen  _et al._ [2011b] R. Owen _et al._ , Phys. Rev. Lett. 106, 151101 (2011b), arXiv:1012.4869 [gr-qc] .
  * Biswas and Mazumdar [2014] T. Biswas and A. Mazumdar, Class. Quant. Grav. 31, 025019 (2014), arXiv:1304.3648 [hep-th] .
  * Basak and Shankaranarayanan [2015] A. Basak and S. Shankaranarayanan, JCAP 05, 034 (2015), arXiv:1410.5768 [hep-ph] .