# Inferring fundamental spacetime symmetries with gravitational-wave memory:
from LISA to the Einstein Telescope

Boris Goncharov<EMAIL_ADDRESS>Gran Sasso Science Institute (GSSI),
I-67100 L’Aquila, Italy  INFN, Laboratori Nazionali del Gran Sasso, I-67100
Assergi, Italy  Laura Donnay  International School for Advanced Studies
(SISSA), Via Bonomea 265, 34136 Trieste, Italy  INFN, Sezione di Trieste, Via
Valerio 2, 34127 Trieste, Italy  Jan Harms  Gran Sasso Science Institute
(GSSI), I-67100 L’Aquila, Italy  INFN, Laboratori Nazionali del Gran Sasso,
I-67100 Assergi, Italy

###### Abstract

General Relativity (GR) yields a number of gravitational wave memory effects
that correspond to symmetries of spacetime infinitely far away from
gravitational fields. These symmetries and memory effects hint at the
fundamental mathematical connection between gravity and quantum fields in the
low-energy “infrared” regime. In this study, we propose to shift the paradigm
of memory from merely a target in gravitational wave searches to a unique way
of measuring symmetries of nature. Thus, we extend previous memory detection
efforts to the proof-of-principle parameter estimation and model selection.
Through simulating binary black hole (BBH) mergers in scenarios where
spacetime exhibits certain sets of Bondi-Metzner-Sachs symmetries, we point
out that both the current and the future gravitational wave observatories are
excellent probes of spacetime symmetries. In particular, the design
sensitivity of the proposed Einstein Telescope (ET) allows to constrain the
strain amplitude of the leading-order displacement memory, associated with
superrotational symmetries, to a $2\%$ level in one year. Whereas the weaker
spin memory amplitude can be constrained to a $22\%$ level, providing a
pathway to probe superrotational symmetries. Finally, there is almost no doubt
among GR experts that displacement memory exists in nature, although its
effect on parameter estimation was largely neglected in the predictions of the
science output of future experiments such as LISA and ET. We find that it may
lead to an overestimation of the measurement uncertainties for inferred
parameters of the loudest BBH mergers by the order of $10\%$.

##  I  Introduction

In recent years, observation of gravitational waves (GW) from compact binaries
has provided new means to test General Relativity (GR) and fundamental physics
(see, e.g., Abbott et al. (2021a, 2019); The LIGO Scientific Collaboration et
al. (2021)). A peculiar prediction of GR is that, in addition to the
oscillatory contributions to GW strain, there are also non-oscillatory
contributions which are referred to as gravitational memory. A linear
component of the memory was first predicted by Zel’dovich and Polnarev
Zel’dovich and Polnarev (1974) when considering hyperbolic encounters of
compact objects in galactic nuclei. Later, Christodoulou found an additional
memory effect arising from a full nonlinear analysis of Einstein’s equations
Christodoulou (1991) (see also Blanchet and Damour (1992)). These non-
linearities originate from the cumulative effect of the gravitational waves
themselves to the stress-energy tensor Thorne (1992). The effect is also
referred to as displacement memory because it manifests as if observers or
test masses experiencing this effect remained permanently displaced after the
passage of GW. By Strominger and Zhiboedov Strominger and Zhiboedov (2014), GW
memory has been shown to represent one of the three corners of the so-called
“infrared triangle”. The latter establishes universal relationships between
the memory effect, Weinberg’s soft graviton theorem in quantum field theory
Weinberg (1965) and the Bondi-Metzner-Sachs (BMS) symmetry group of
asymptotically flat spacetime Bondi et al. (1962); Sachs (1962). The three
seemingly unrelated subjects turn out to be three equivalent ways of
characterizing physics at very long distances and the recent exploration of
their connection has shed new light on the fascinating universal features of
gravity; see Strominger (2018) for a review.

The standard BMS group describes the asymptotic symmetries of flat spacetime
at null infinity, i.e. as seen by observers infinitely far away from the
gravitational wave field of interest. The authors of Bondi et al. (1962);
Sachs (1962) initially expected that symmetries of such spacetimes would
represent the Poincaré group, with ten conserved quantities – referred to as
charges – being the total energy, the momentum, the angular momentum, and the
position of the center of mass. However, they ended up discovering a superset
of the Poincaré group which also contains an infinite amount of so-called
supermomentum charges. These charges are associated with the symmetries known
as _supertranslations_ , as they extend the usual group of Poincaré
translations.

BMS charges are conserved in the absence of fluxes of radiation to null
infinity. In the presence of radiation, they rather obey flux-balance laws
(among which one finds the Bondi mass loss formula Bondi et al. (1962)). The
displacement memory arises as the permanent shift of the asymptotic shear
after the passage of gravitational waves. This shift can be equivalently
described as a transition between two different asymptotic BMS frames related
by a supertranslation Ashtekar (2014); Strominger and Zhiboedov (2014).

Later studies proposed new symmetry groups which include the BMS group as a
subset. These further extensions correspond to more relaxed fall-off
conditions for the metric near null infinity and have been shown to lead to
new gravitational wave memory terms. The symmetry group which we refer to as
the extended BMS group includes additional “superrotation” symmetries that
correspond to supermomentum and superspin charges Barnich and Troessaert
(2010a, b); Kapec et al. (2014). It was shown that they can be related to a
new gravitational wave memory known as the spin memory Pasterski et al.
(2016); Nichols (2018); Himwich et al. (2021); Flanagan and Nichols (2017),
which manifests as a relative time delay between two observers in counter-
orbiting trajectories. Additional BMS symmetry groups were later proposed as
well Campiglia and Laddha (2015); Freidel et al. (2021); Godazgar et al.
(2019), and associated additional memory-type effects were discussed Compère
et al. (2018); Seraj and Oblak (2021); Godazgar et al. (2022). Given the
variety of proposals in the literature for what the largest nontrivial
asymptotic symmetry group should be, it is of great interest to determine
which set of BMS symmetries can be accounted for by observational data.
Moreover, deviations from GR may lead to different apparent memory amplitude,
as discussed in Heisenberg et al. (2023). There may also be subtleties related
to the treatment of memory in data processing. Therefore, measurements of the
memory amplitude could be utilized for tests of GR and the robustness of data
analysis.

From the experimental side, previous studies mainly concerned the aspect of
the detection of memory effects. Current ground-based laser interferometers,
the Advanced LIGO111Laser Interferometer Gravitational-wave Observatory
(LIGO). LIGO Scientific Collaboration et al. (2015) and the Advanced Virgo
Acernese et al. (2015), may be able to detect the displacement component of
the memory, as suggested by Lasky et al. Lasky et al. (2016). The search for
displacement memory has been carried out by Hübner et al. with the data from
the first Hübner et al. (2020) and the second Hübner et al. (2021) LIGO-Virgo
gravitational-wave transient catalog. The data suggested no evidence of
memory, and the authors found that the cumulative detection requires between a
few hundred and a few thousand compact binaries detected by a network
comprising the Advanced LIGO and the Advanced Virgo at design sensitivity.
Grant and Nichols Grant and Nichols (2023) developed a framework to forecast
the detection of displacement and spin memory, further evaluating the
prospects of resolving spin memory with the next-generation instruments, the
Einstein Telescope (ET, Punturo et al., 2010) and the Cosmic Explorer (CE,
Reitze et al., 2019). With Pulsar Timing Arrays (PTA), the displacement memory
(Wang et al., 2015; Arzoumanian et al., 2015) may in principle be detectable
at $2\sigma$ as pointed out by Van Haasteren and Levin van Haasteren and Levin
(2010). Prospects of memory detection are brighter Favata (2010); Sun et al.
(2023) with the space-based gravitational wave detector LISA222Laser
Interferometer Space Observatory (LISA). which is to operate at millihertz
frequencies Amaro-Seoane et al. (2017). Note, there are additional potentially
observable effects that have previously been referred to as “memory”, e.g.
“velocity memory” Grishchuk and Polnarev (1989). However, they do not
correspond to asymptotic symmetries, and so the term “persistent observables”
was coined by Flanagan, Grant, Harte, and Nichols to describe this more
general class of phenomena Flanagan et al. (2019). It is also worth a mention
that several publications discussed the merits of displacement memory in
breaking gravitational-wave parameter estimation degeneracies Gasparotto et
al. (2023), identifying low-mass binary black holes Ebersold and Tiwari
(2020), and distinguishing black holes and neutron stars in gravitational wave
observations Yang and Martynov (2018); Tiwari et al. (2021); Lopez et al.
(2023).

In this work, we focus on evaluating the prospects of characterizing memory
signals and performing model selection to determine which set of symmetries
describes the spacetime we live in. We simulate gravitational wave signals
from binary black hole mergers in different scenarios of spacetime at null
infinity. Using the principles of Bayesian model selection and the Fisher
information, we then evaluate the odds of distinguishing between the
aforementioned scenarios with ground-based and space-based gravitational wave
detectors. The rest of the paper is organized as follows. In Section II, we
outline the models of spacetime symmetries and their correspondence to
gravitational wave memory effects. Next, in Section III, we describe the
details of the simulated observations and the results of model selection
between spacetime symmetry groups with ground-based detectors (III.2) and LISA
(III.1). In Section IV, we introduce a test of spacetime symmetries based on
introducing the strain amplitude of gravitational wave memory as a free
parameter. We summarize our results and discuss next steps for inferring
fundamental spacetime symmetries in Section V.

##  II  Models

The boundary of flat spacetimes (null infinity $\mathscr{I}$) is a lightlike
hypersurface parametrized by a null time $u=t-r$ and a two-dimensional sphere
$S^{2}$, called the celestial sphere. The latter has a metric given by
$\gamma_{AB}dx^{A}dx^{B}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$ in terms of the
usual spherical coordinates $x^{A}=(\theta,\phi)$. BMS transformations are
generated by vector fields $\xi$ of the following form333There are also
subleading terms in $1/r$; we only wrote here the restriction at $\mathscr{I}$
of the BMS vector field.:

$\xi=\left[T(x^{A})+\frac{u}{2}D_{A}Y^{A}(x^{B})\right]\partial_{u}+Y^{A}(x^{B})\partial_{A}\,,$
(1)

where $D_{A}$ denotes the covariant derivative associated to $\gamma_{AB}$.
The different models correspond to different choices of the supertranslation
and superrotation generators $T$, and $Y^{A}$, respectively. In both Poincaré
and standard BMS groups, the vector fields $Y^{A}$ are $\ell=1$ vector
spherical harmonics generating the usual 6 Lorentz transformations. The
infinite-dimensional nature of the standard BMS group stems from the fact that
the function $T(x^{A})$ is allowed to be an arbitrary (smooth) function of the
sphere angles (in contrast, in the Poincaré group, $T$ is restricted to be a
linear combination of $\ell=0,1$ spherical harmonics).

In the extended BMS group, supertranslations are supplemented by
superrotations Barnich and Troessaert (2010a), where $Y^{A}$ are taken to be
_local_ conformal Killing vectors (CKV) of the sphere. In order for the
asymptotic symmetry group to include those extra symmetries, the key is to
allow for singularities at isolated points on the celestial sphere444This, is
turn, implies that supertranslations can now also be singular.. While this
might seem an odd thing to do from a conservative GR point of view (see
however Strominger and Zhiboedov (2017) for a physical interpretation of
superrotation symmetries), there is a very strong motivation coming from a
field theorist’s perspective. Indeed, the algebra spanned by local CKV is
nothing but the Virasoro algebra, an infinite-dimensional algebra ubiquitous
in string theory and two-dimensional conformal field theory (CFT). As a matter
of fact, the realization that Virasoro symmetries could arise as part of the
symmetry group acting on the celestial sphere kicked off the ambitious
Celestial Holography program (see Pasterski et al. (2021) and references
therein), whose aim is to provide a holographic description of quantum gravity
in flat spacetime in terms of a CFT living on the celestial sphere.

Finally, let us mention that there exists a fourth option (not analyzed here)
which is the generalized BMS group. The latter extension was first proposed in
Campiglia and Laddha (2015) and is characterized by the fact that $Y^{A}$ can
be chosen to be any smooth vector fields (i.e. they generate the full
diffeomorphisms of the sphere, Diff$(S^{2})$). These generalized symmetries
can now deform the boundary sphere metric and require enlarging the
gravitational phase space; see Compère et al. (2018) for proposed associated
memory effects. The four models are summarized in Table 1.

Table 1: Models. | Asymptotic symmetry groups
---|---
| Poincaré | Standard BMS | Extended BMS | Generalized BMS
Asymptotic symmetries | 4 translations | supertranslations | supertranslations | supertranslations
| 6 Lorentz | 6 Lorentz | local CKV555local conformal Killing vector field, as opposed to global CKV (i.e. Lorentz transformations) | Diff($S^{2}$)
Boundary metric | $\gamma_{AB}$ | $\gamma_{AB}$ |  $\gamma_{AB}$ locally666i.e except at isolated points | not fixed
Memory terms | None | Displacement | Displacement, spin | Refraction, velocity kick

We reduce our attention to the three first models of spacetime symmetries at
null infinity: the Poincaré spacetime, the standard BMS, and the extended BMS.
The Poincaré model corresponds to the observation of no gravitational wave
memory. It can thus be associated with seemingly unlikely scenarios where,
e.g., the cumulative contribution of gravitational waves is omitted from the
stress-energy tensor in Einstein field equations. Nevertheless, let us
speculate that the detection of such a scenario may be a product of data
analysis subtleties such as model misspecification. Thus, we still consider
this model for generality. The standard BMS model corresponds to observing
only the displacement memory term. The relationship between superrotations and
the spin memory is more subtle. Indeed, an important difference between the
displacement and the spin memory is the fact that, while the former can be
understood as a transition from an early time frame to a supertranslated
frame, the spin memory has not an immediate relation to a spacetime
“superrotated” from a specific early time frame Nichols (2017); Compère et al.
(2018). Keeping this in mind, we will nevertheless refer to the extended BMS
as the model related to the observations of both the displacement and the spin
memory. We simulate gravitational wave strain as a function of time $h(t)$
based on a hybridized numerical relativity surrogate waveform approximant
Varma et al. (2019). Hybridization means that early times of the inspiral of a
compact binary are modeled with post-Newtonian and effective-one-body
approaches, whereas at the merger the signal is predicted based on the results
of numerical relativity simulations. The term “surrogate” refers to the fact
the model is based on interpolation between available numerical relativity
simulations which are only available for a limited set of compact binaries.
The numerical relativity simulations underlying the approximant do not exhibit
displacement memory, although they are reported to contain $50\%$ of the
strain of the expected spin memory for an unknown reason Mitman et al. (2020).
The memory contributions to $h(t)$ are calculated based on Mitman et al.
(2021), using BMS balance laws, such that any putative pre-existing spin
memory residual is treated as a memory-free part of the waveform. Because the
artefact is $\sim 10^{-3}$ of the gravitational wave strain, once absorbed to
the spin memory it should manifest as $\sim 10^{-6}$ of the strain, and thus
have a negligible effect on our results. The artefact disappears with the
recent extension of the employed surrogate approximant with built-in memory
contributions Yoo et al. (2023). Note that other numerical relativity
simulation techniques that evolve Einstein equations to future null infinity
do recover the displacement Pollney and Reisswig (2011) and the spin memory
Mitman et al. (2020), under the assumption of the same boundary metric as for
the standard BMS. Thus, keeping in mind subtleties associated with the spin
memory and the extended BMS group, non-detection of the symmetries discussed
in this work may also be interpreted as a deviation from GR.

##  III  Observing spacetime symmetries

With gravitational wave observations, determining which spacetime symmetry
group represents the universe we live in777While we live in a de Sitter
universe, the approximation of an asymptotically flat universe is valid up to
astrophysical scales smaller than cosmological scales Kehagias and Riotto
(2016). reduces to determining which set of memory terms is the best
description of data. In this paper, we simulate measurements with three types
of gravitational wave instruments:

  * $\bullet$

The network of current gravitational wave detectors at design sensitivity, the
Advanced LIGO, and the Advanced Virgo.

  * $\bullet$

Next-generation ground-based detectors proposed for the 2030s, the Einstein
Telescope (ET), and the Cosmic Explorer (CE).

  * $\bullet$

LISA, a space-based instrument planned for the late 2030s. It will measure
gravitational wave strain through time-delay interferometry (TDI) Tinto et al.
(2002).

Figure 1: Demonstration of the gravitational wave memory contribution to
strain from a merger of two non-spinning binary black holes in the extended
BMS scenario,
$(m_{1},m_{2},\theta_{jn},z)=(30~{}M_{\odot},30~{}M_{\odot},\pi/3,0.06)$. The
three top panels show the time-domain strain of the total gravitational wave
signal (left), its displacement memory component (middle), and its spin memory
component (right). Solid lines show $h_{+}$, dashed lines show $h_{\times}$.
The three bottom panels show the respective frequency series of
$|\tilde{h}(f)|$. The effect of windowing on time series data is shown only in
the left top panel, and it corresponds to a decline in gravitational wave
strain at the end of the time series data.

State-of-the-art gravitational wave data analysis is based on constructing and
sampling the likelihood $\mathcal{L}(d|\theta)$ of the data $d$ as a function
of parameters $\theta$ of a model. For ground-based detectors and LISA, the
data is fit to the model in the frequency ($f$) domain, $\tilde{h}(f)$, where
the tilde denotes a Fourier transform. The measurement of parameters is given
by the posterior probability distribution
$\mathcal{P}(\theta|d)=\mathcal{L}(d|\theta)\pi(\theta)/\mathcal{Z}$, where
the equality manifests a Bayes theorem. The function $\pi(\theta)$ is a prior
distribution and $\mathcal{Z}=\int\mathcal{L}(d|\theta)\pi(\theta)d\theta$ is
referred to as the evidence. The ratio of evidences of the two models is
referred to as the Bayes factor, $\mathcal{B}$. It is equal to the Bayesian
odds ratio when prior odds of occurrence of the two models are assumed to be
equal, which is what we consider in this study. We will use these terms
interchangeably throughout the paper. The universality of gravitational wave
memory allows us to probe its presence in multiple measurements by taking a
product of Bayes factors from these measurements Lasky et al. (2016); Hübner
et al. (2020, 2021). In Bayesian inference, the priors are usually chosen such
that they represent the observed distribution of $\theta$. However, this has a
negligible effect on our results. So, we set $\pi(\theta)=1$, such that the
posterior is driven entirely by the likelihood. To be able to study thousands
of signals, we Taylor-expand the likelihood at simulated parameter values
$\theta_{0}$ and approximate it by neglecting terms past the second order. The
resulting posteriors will become a multivariate Gaussian distribution centred
at true parameter values, with measurement uncertainty given by the inverse of
the Fisher information matrix Dupletsa et al. (2023). This is a valid and
sufficiently accurate approximation for the loudest gravitational wave signals
we discuss throughout the paper. By knowing the form of the posterior, we also
evaluate Bayesian evidence analytically to perform model selection.

Following the procedure above, we evaluate $\mathcal{P}(\theta|d)$ for a full
set of parameters that describe a binary black hole merger: binary component
masses $m_{1,2}$, angular momenta $a_{1,2}$, gravitational wave phase $\phi$
and polarization $\psi$, orbital inclination with respect to the line of sight
$\theta_{jn}$, luminosity distance $D_{\text{L}}$, sky position and merger
time. We also introduce additional parameters, multiplicative factors of the
predicted gravitational wave memory strain amplitude, $A_{\text{d}}$ for
displacement memory and $A_{\text{s}}$ for spin memory, for the following
reason. Model selection requires evaluating the Bayesian evidence for both the
correct model and the incorrect model. Fitting an incorrect model to the data
introduces systematic errors and, on average, a decrease in the maximum
likelihood value. Introducing $A_{\text{d},\text{s}}$ allows measuring
posterior probability, as well as the Bayesian evidence, for both the model
where memory is present in the data ($A_{\text{d},\text{s}}=1$) and for the
model where it is not present ($A_{\text{d},\text{s}}=0$), by considering
slices along $A_{\text{d},\text{s}}$ of the full posterior. Thus,
$A_{\text{d},\text{s}}$ manifests as a Bayesian hyper-parameter. For more
details and a discussion of the methods, please refer to Appendix A. The
visualization of a fiducial gravitational wave signal and its memory
components are shown in Figure 1.

###  III.1  LISA

Space-based interferometer LISA is to explore the gravitational wave universe
between $10^{-4}$ and $1$ Hz in frequency, with a peak sensitivity at around
$10^{-2}$ Hz. The loudest expected astrophysical sources for LISA are mergers
of massive binary black holes, which may be detected with a rate of a few per
year Barausse et al. (2020). Frequency-domain strain $\tilde{h}(f)$ is
projected onto the three LISA TDI channels, using detector specifications from
Amaro-Seoane et al. (2017). We simulate the analysis of frequency series from
$f_{\text{min}}=2^{-13}~{}\text{Hz}\approx 1.2\times 10^{-4}~{}\text{Hz}$ to
$f_{\text{max}}=2^{-6}~{}\text{Hz}\approx 1.6\times 10^{-2}~{}\text{Hz}$, with
a spacing $\Delta f=f_{\text{min}}$. The frequency range based on powers of
two is chosen for an exact match with the Fourier transform of $h(t)$ provided
by the waveform approximant.

Figure 2: Results of the simulated model selection between spacetime
symmetries with LISA based on a single non-spinning binary black hole with
component masses of $10^{6}~{}M_{\odot}$. The panels show the inferred log
odds, $\ln\mathcal{B}$, in favour of one symmetry group representing our
universe over the other, with probability distributions outlining the
uncertainty given by drawing gravitational wave parameters other than masses
from the prior. The symmetry scenarios are shown in the pairs of left, middle,
and right panels. The model comparison between the Poincaré group and either
the original or the extended BMS symmetries is shown in the top three panels,
whereas the model comparison between the two BMS groups is shown in the bottom
three panels. Red vertical dashed lines corresponds to $\ln\mathcal{B}=3$
(strong evidence), red solid lines correspond to $\ln\mathcal{B}=5$
(detection, decisive evidence). The left bottom panel shows the case of model
misspecification, where neither of the compared models represents the correct
symmetry in nature.

We consider an observation of a non-spinning equal-mass binary with component
masses of $10^{6}~{}M_{\odot}$, positioned at a luminosity distance between
$1$ and $3$ Gpc. LISA is expected to detect such a signal during the mission
lifetime of 4-6 years. We simulate stochastic realizations of this observation
in universes described by different asymptotic spacetime symmetry groups,
marginalizing over sky position, merger time, $\phi$, $\psi$, and
$\theta_{jn}$. The results are shown in Figure 2, where marginalized Bayesian
odds between symmetry groups are shown against $D_{\text{L}}$. In panels with
log-y-axes, $5$-$95$ % cumulative density levels are shown as horizontal black
lines across the distributions. The log scaling is chosen to focus on the
minority of quieter events in asymmetric distributions. Horizontal solid lines
show the Bayes factors $\ln\mathcal{B}=5$ corresponding to decisive evidence
following the Kass-Raftery scale, whereas dashed lines correspond to positive
evidence with $\ln\mathcal{B}=3$. From the top three panels, it is evident
that LISA should be able to rule out or, possibly, infer that the spacetime at
null infinity is asymptotically flat, i.e. it is described by the Poincaré
group. This outcome is driven entirely by the detection or non-detection of
displacement memory. The bottom three panels correspond to the model selection
between the original and the extended BMS models, which is likewise driven
entirely by the ability of LISA to resolve spin memory. In the Poincaré
scenario, model selection between higher-order symmetries represents the case
of model misspecification. It is shown in the bottom left panel. It stands out
from all other panels, as Bayes factors appear both positive and negative,
with the majority of observing scenarios favouring the simpler symmetry group,
as expected. We suggest that, given a sufficiently large sample of
observations, a comparison of the results between real data and simulations
may be able to indicate a case where neither of the known symmetry groups
describes our universe sufficiently well. Finally, the two bottom right panels
indicate that single-event-based inference of higher-order BMS symmetries is
likely only for $10^{6}~{}M_{\odot}$ binaries within the $2$ Gpc range.

###  III.2  Ground-based detectors

The Advanced LIGO and Virgo are sensitive to gravitational waves with
frequencies between around $10$ and $1000$ Hz. With ongoing efforts to push
these instruments to reach design sensitivity, 94 inspirals-mergers of compact
binaries have already been observed since 2015. It is estimated that on the
order of $10^{5}$ binary black holes and $10^{6}$ neutron stars merge in the
Hubble volume every year. The proposed next-generation instruments, ET and CE,
would observe most of them at target sensitivity. In this work, we consider 1
year of observation of binary black holes with four different detector
networks: (a) LIGO and Virgo, (b) LIGO Voyager and Virgo, (c) ET, (d) ET and
CE. We do not discuss binaries that contain neutron stars as they are
typically more quiet. We simulate frequency series from $f_{\text{min}}=10$ Hz
to $f_{\text{max}}=1024$ Hz, with a spacing $\Delta f=2^{-7}$ Hz.

For ground-based detectors, single-event-based memory inference is not as
powerful as for LISA, thus we focus on combining the information from multiple
observations of compact binary coalescences. Because the observations are
independent, the total log Bayes factor
$\ln\mathcal{B}=\sum_{i}\ln\mathcal{B}_{i}$, where $i$ is the index of a
binary black hole observation. Note, in the frequentist statistics which
operates with the signal-to-noise ratio (SNR) instead,
$\ln\mathcal{L}\propto\text{SNR}^{2}$, so that the sum of log Bayes factors is
similar to the widely-used procedure of adding SNRs in quadrature. Based on
this, $\ln\mathcal{B}$ is expected to grow linearly with the observation time
and the SNR is expected to grow as a square root of the observation time. As
in Grant and Nichols (2023), we find that LIGO and Virgo may be able to detect
displacement memory, yet the detection of spin memory with these instruments
is not realistic. Thus, below we only present model selection between Poincaré
group and BMS groups for (a) LIGO-Virgo and (b) Voyager-Virgo. Whereas the
model selection results between different BMS groups are only shown for (c) ET
and (d) ET-CE. Furthermore, it was pointed out in Lasky et al. (2016) that the
degeneracy between $\phi$ and $\psi$ in the dominant $(2,2)$ mode of a
gravitational wave signal makes it impossible to determine the sign of the
memory. This invalidates the assumption that there is only one maximum-
aposteriori point in the parameter space, so that $\ln\mathcal{B}$ is
overestimated. The good news is that we use the waveform approximant with
high-order modes that break this degenearcy Varma et al. (2019). We only have
to make sure that the high-order mode content of the signal is sufficiently
loud. For this, we check that the optimal SNR of the $m$-odd part of the
waveform is $>2$, following the prescription by Grant and Nichols (2023). For
other signals, one could, in principle, reduce $\ln\mathcal{B}$ accordingly.
However, for simplicity, we assign $\ln\mathcal{B}$ to zero for signals that
do not meet the criterion. This results in a slighly more conservative
prediction.

We simulate two populations of binary black holes (BBHs) up to $z=30$ based on
the results of the population analysis of the second LIGO-Virgo transient
catalog Abbott et al. (2021b). Our main population is based on the median-
posterior local merger rate of $23.9~{}\text{Gpc}^{-3}\text{yr}^{-1}$ and the
auxiliary one is based on the lower limit at 95% credibility,
$15.3~{}\text{Gpc}^{-3}\text{yr}^{-1}$, to demonstrate the impact of the
associated uncertainty and the more conservative rate assumptions. The mass
distribution is described by the best-fit model and best-fit parameters from
Abbott et al. (2021b). Note, our inference of spacetime symmetries does not
suffer from astrophysical selection effects, so in principle, we can choose
any subset of the population for the analysis. Further on, we focus on the
analysis of the best 1000 events which we picked based on maximizing the
optimal signal-to-noise ratio for the most elusive spin memory component. This
number of events is sufficient for resolving spacetime symmetries considered
in this work. The consequences of this choice are that (1) there is more
information that can be extracted from the data, and (2) the selected events
are only sub-optimal for resolving the displacement memory term.

Figure 3: Simulated model selection between spacetime symmetries with ground-
based detectors based on combining evidence from multiple gravitational wave
observations. In all panels, solid lines and dotted lines correspond to the
two cases where each of the two symmetry groups that are tested is the correct
one. The log odds ratio, $\ln\mathcal{B}$, is always calculated in favor of
the correct model over the incorrect model so that the values are positive.
Colors indicate different detector networks and filled regions show the
difference between the main and the auxiliary population described in Section
III.2. Top: selection between the Poincaré group and the original BMS group.
Detection of either symmetry can be achieved with a network of current
detectors at design sensitivity. Bottom: Selection between the original and
the extended BMS model, which we only find conclusive for next-generation
instruments (ET, CE). Evidence in favor of the correct symmetry group over the
incorrect one is shown as a function of the observation time in the left two
panels and as a function of the number of optimal binary black hole
observations in the right two panels.

The results of model selection are shown in Figure 3. As for LISA, we simulate
three observation scenarios where spacetime symmetries at null infinity are
described by the Poincaré, the original BMS, and the extended BMS scenario.
The top two panels illustrate that LIGO and Virgo resolve supertranslational
symmetries of the original BMS group through sensitivity to the displacement
component of the memory. The left top panel shows that it takes around a year
for the LIGO-Virgo network to confidently resolve the correct symmetry group.
Depending on the local black hole merger rate, the detection can be made with
$\approx 100$ optimal signals. It is clear from the top right panel in Figure
3 that adding additional 1000 signals observed the same year does not add to
our measurement. This is because the high-order modes in these signals are not
prominent enough to break the $\phi$-$\psi$ degeneracy, as we mentioned
earlier. Considering LIGO Voyager and an auxiliary Virgo interferometer
significantly reduces times to detection and the number of optimal events
required for the detection. However, we find that even such a network still
can not confidently infersuperrotation symmetries related to the extended BMS
model. The bottom two panels show how enhanced symmetries, means to
distinguish between the original and the extended BMS models, may be inferred
with the ET and the joint ET-CE observations. As previously, we find that the
outcome is due to the sensitivity of detector networks to spin memory. For ET,
it is sufficient to analyze around $100$ optimal events to make such a
conclusion or to operate for $150$-$175$ days. Whereas for an ET-CE network,
this reduces to $10$-$30$ optimal events and $90$ days of observation,
respectively. Note, the Bayes factors we predict for observations with ground-
based detectors are noise-averaged. The effect of Gaussian noise will
introduce a range of the predicted values around our estimates. Precisely, the
linear growth of $\ln\mathcal{B}$ with the observation time will turn out to
be a stochastic random-walk-like process.

Let us also point out a few other notable features in the results. First, we
find that the local black hole merger rate is not an important factor at
observation timescales of $\sim 100$ days required for the detection of
relevant symmetry groups. In these early days, at some observation intervals,
the auxiliary low-rate population even yields higher evidence for a correct
symmetry group compared to the primary population based on the best-fit merger
rate. This is especially prominent for LIGO-Virgo and Voyager-Virgo in the
context of displacement memory. This is partly due to the modest contribution
of the majority of the signals to the measurement. As shown in the right upper
panel in Figure 3, only 2% events contribute 50% of all evidence, whereas the
growth of evidence turns over after around $200$ observed black hole mergers.
Whereas for the case of ET probing higher-order BMS symmetries through spin
memory, the positive impact of the higher merger rate is more clear. There,
10% of the events contribute to 50% of all evidence, as shown in the right
bottom panel in Figure 3. Second, for ET and CE, the evidence for a correct
symmetry group tends to match in value between the cases where the roles of
correct and incorrect symmetry groups are switched. So, the information about
spacetime symmetry group inferred from the presence of the spin memory term is
equal to the information inferred from the absence of spin memory in the
opposite scenario. However, there appears to be a discrepancy in the case of
displacement memory and observations with LIGO, Virgo, and Voyager. The dotted
line, which represents the case of the simulated original BMS symmetry group,
is visually below the solid line, which represents the case of the simulated
Poincaré symmetry group. The offset can be explained by stronger
gravitational-wave parameter estimation degeneracies for LIGO, Virgo, and
Voyager, compared to ET and CE. These degeneracies make it more difficult to
evaluate the Fisher information matrix that governs the posterior,
$\mathcal{P}(\theta|d)$. In turn, this affects the evidence values,
$\mathcal{Z}$, and thus the Bayes factors. Finally, one may notice a visible
leap in evidence between 150 and 175 observation days with ET and CE, compared
to ET only. For ET and CE, the log Bayes factor is increased by $2.3$. This is
due to a significant improvement in parameter estimation thanks to CE, for
binary black hole merger with component masses of $35~{}M_{\odot}$ and
$27~{}M_{\odot}$, with an orbital inclination of $54$ deg. The most prominent
difference for this event was related to sky position and polarization
parameters, which were almost unconstrained for ET. For this event, it is
likely that ET sky position uncertainty is underestimated, as typically Fisher
matrix approximation of the posterior is suboptimal for multi-modal
probability distributions with nonlinear covariance (e.g., Figures 6-9 in
Dupletsa et al., 2023). Nevertheless, this event is an example of the positive
impact of forming a network of next-generation detectors, compared to relying
on the observations of individual detectors.

##  IV  Constraining spacetime symmetries with the Einstein Telescope

Introducing $A_{\text{d},\text{s}}$ in Section III yields posterior
distributions on these parameters. Previously, we only considered information
from these posteriors when evaluating the Bayes factors. However, we suggest
that the measurements of $A_{\text{d},\text{s}}$ could be of interest as well.
Any deviations in $A_{\text{d},\text{s}}$ from the values of $0$ and $1$ could
be indications of model misspecification. This could either indicate new
physics, e.g. a symmetry group that represents our null infinity is neither of
the groups we consider, or unknown systematic errors Scargle et al. (2021).
Similarly to taking the product of Bayes factors, the posterior distribution
given all available data $\mathcal{P}(A_{\text{d},\text{s}}|d)$ factorizes as
a product of posterior distributions from individual gravitational wave
events, marginalized over nuisance parameters. The only requirement is that
the marginalization is not performed over common parameters between the
measurements, such as $A_{\text{d},\text{s}}$. Additionally, unlike in Section
III, here it is important to take Gaussian noise into account. For individual
gravitational wave signals, it does not affect the measurement uncertainty.
Instead, it introduces offsets between simulated (true) parameters and
measured parameters. When combining multiple events, however, the offsets do
lead to increased measurement uncertainty. The offsets due to Gaussian noise
are simulated following Equation 5 in Sharma and Harms (2020).

The results of the combined constraints on $A_{\text{d},\text{s}}$ across the
Poincaré, the original BMS, and the extended BMS, with 1000 optimal BBHs
observed by the Einstein Telescope (ET) in a year are shown in Figure 4. In
all cases, we find that the 1-$\sigma$ uncertainty of the displacement memory
amplitude $A_{\text{d}}$ is $0.03$, whereas the respective uncertainty of the
spin memory amplitude $A_{\text{s}}$ is $0.26$. Furthermore, we show results
for combining the data from all binary black hole mergers at redshifts
$z\lessapprox 1$ with the ET, such that the number of events reaches
approximately $10^{4}$. The additional events further constrain the
uncertainty of $A_{\text{d}}$ to $0.02$ and that of $A_{\text{s}}$ to $0.22$.
The best-fit $A_{\text{d,s}}$ are consistent with the simulated values.

Following Gasparoto et al. Gasparotto et al. (2023), we also investigate if
the presence of gravitational wave memory terms improves the measurement of
gravitational-wave parameters – describing the inspiral and merger of binary
black holes – for ET. The presence of elusive spin memory, as expected, does
not play a role for ET. However, the presence of displacement memory in a
simulated observation scenario yields visible improvements in parameter
estimation. We report our findings for the $4000$ optimally chosen binary
black hole mergers from a year of observations, with caveats that the
imporovements we report are (i) subject to selection effects, (ii) only apply
to the loudest events and thus the improvements for the other $\approx 98\%$
of quieter gravitational wave events are expected to be more modest. Most of
all, the presence of displacement memory improves gravitational-wave phase
$\phi$ and polarization $\psi$. On average, we find the uncertainties on
$\phi$ are reduced by $81\%$ and the unicertainties on $\psi$ \- by $74\%$.
The sky position uncertainty is reduced by $41$-$42$%. The merger time of a
binary black hole is measured by $41\%$ better with the displacement memory.
The luminosity distance uncertainty, $D_{\text{L}}$, is reduced by $37\%$. The
binary inclination angle, $\theta_{jn}$, which is degenerate with respect to
$D_{\text{L}}$, is only measured $4\%$ better with the displacement memory.
The uncertainty in binary component masses is reduced by $32$-$34\%$.
Measurement uncertainties for spin parameters – binary angular momenta – are
reduced by $12\%$. For several loudest binary black hole mergers systematic
errors from assuming incorrect $A_{\text{d,s}}$ reach $1$-$2~{}\sigma$. This
motivates further development of gravitational waveforms that include memory
Yoo et al. (2023). Furthermore, our findings suggest that displacement memory
– which is expected to exist in nature – would allow us to extract more
information about the astrophysics of compact binary mergers.

(a) Poincaré Universe

(b) Original BMS Universe

(c) Extended BMS Universe

Figure 4: Measurement of the relative displacement and spin memory amplitudes,
$A_{\text{d}}$ and $A_{\text{s}}$, from the 1000 optimally-chosen binary black
hole mergers observed by Einstein Telescope in a year. The three panels
correspond to scenarios with different symmetry groups describing the null
infinity of the observer. The memory amplitudes corresponding to these
scenarios – either 0 (memory off) or 1 (memory on) – are indicated by red
dashed lines. The color represents the posterior probability density,
$\mathcal{P}(A_{\text{d,s}}|d)$, whereas three solid lines show
$1,2,3$-$\sigma$ credible levels. The dotted lines show $1$-$\sigma$ credible
levels when we include all black hole mergers up to a redshift $z\approx 1$
($10^{4}$ events).

##  V  Conclusion

The boundary of asymptotically flat spacetimes (null infinity, celestial
sphere) is a possible bridge between General Relativity and Quantum Field
Theory. Gravitational wave memory is a direct probe of edge symmetries of such
spacetimes, which makes it a very interesting phenomenon to study in detail.
The symmetries are set by the choices of the supertranslation and
superrotation generators $T$ and vector fields $Y^{A}$ on the celestial
sphere. A range of gravitational wave memories arise from imposing fewer
restrictions on $T$ and $Y^{A}$. We acknowledged four different symmetry
groups in Table 1. For three of these models we simulated one year of
gravitational wave observations of binary black holes with LIGO-Virgo,
Voyager, ET, CE, as well as realizations of a single binary black hole
observation with LISA. The ability to determine which of the spacetime
symmetry groups describes the universe we live in is based on resolving the
presence or absence of the displacement memory and the spin memory. In
observation scenarios that represent averages over Gaussian noise, we find
that LIGO-Virgo will be able to distinguish between a GR-predicted standard
BMS group and a more exotic Poincaré group without any gravitational wave
memories, in less than a year at design sensitivity. Selection between the
standard BMS group and the extended BMS group is not realistic with LIGO-Virgo
or Voyager. However, ET could achieve it in less than a year at design
sensitivity. Simultaneous gravitational wave observations with ET and CE
further reduce this timescale and increase the likelihood of observing fairly
strong ($\ln\mathcal{B}\approx 2$) evidence for spin memory in individual
signals. Additionally, we find that LISA will almost certainly distinguish
between the standard BMS spacetime and the Poincaré spacetime when observing a
$10^{6}~{}M_{\odot}$ binary at $<3.5~{}\text{Gpc}$. LISA also has a $\approx
50\%$ chance of distinguishing between the standard BMS and the extended BMS
scenario if such a binary is observed at $<1.5~{}\text{Gpc}$. Moreover, we
test the idea of measuring the amplitude of displacement memory,
$A_{\text{d}}$, and of the spin memory, $A_{\text{s}}$. This may be employed
as a test of spacetime symmetries, akin to tests of General Relativity. It may
help to identify issues in data analysis as well as model misspecification,
i.e. the case where the true symmetry group of the asymptotic spacetime we
live in is neither of the symmetry groups we expect. For example, a deviation
from the expected $A_{\text{d,s}}$ may occur for binary black hole mergers at
high redshifts Ng et al. (2023), where the approximation of asymptotically
flat spacetime is no longer valid Kehagias and Riotto (2016). We evaluate
constraints on $A_{\text{d,s}}$ amplitudes of the displacement and the spin
memory, respectively, on a scale of 1 of the prediction given by the extended
BMS model. We find that ET with $\approx 10^{4}$ optimally-chosen events,
which represent $\approx 5\%$ of binary black holes observed in a year, would
constrain $A_{\text{d}}$ to $0.02$ and $A_{\text{s}}$ to $0.22$ at $1\sigma$.
Finally, we find that the displacement memory is an additional feature of the
observational data that may help to improve parameter estimation in the
loudest signals observed by ET and CE, whereas neglect of memory may lead to
systematic errors at $1$-$2\sigma$ level. This motivates further development
of gravitational waveform approximants with memory effects Yoo et al. (2023)
for future observatories such as ET, CE, and LISA.

The measurements we demonstrated with simulated data may be extended to
ongoing memory searches with LIGO-Virgo Hübner et al. (2020, 2021), as well as
to future experiments. The predictive accuracy of our results is subject to a
number of both conservative and optimistic assumptions. The conservative
assumptions are the following. First, we neglect the fact that some black
holes have non-negligible spins, although non-zero spins increase memory
amplitude. Secondly, we only used the subset of loudest events, where in
practice it is easier to control the noise and systematic errors. It is not so
straightforward in the analysis of significant yet relatively memory-quiet
events (e.g., face-on), although including these is likely to improve the
measurement. The use of quieter signals would require a full Bayesian
inference. Among the optimistic assumptions, we note that it is not guaranteed
that the instruments will reach design sensitivity. Additionally, data quality
issues and non-Gaussian noise in LIGO and Virgo are known to limit the ability
to interpret the astrophysical origin of signals Payne et al. (2022). In
principle, it is also possible that the method outlined in Section IV may be
sub-optimal for the identification of certain cases of a misspecified
gravitational wave memory model.

We hope the results and the discussion we presented will motivate future work
in theory and observations. First, further theoretical studies of low-energy
physics in asymptotically flat spacetimes. The positive prospect of
distinguishing between spacetime symmetries and constraining memory would
allow for observational confirmation of the laws of physics at spacetime
boundaries. Second, the development of models of gravitational wave strain
from the refraction and the velocity kick memory of generalized BMS Compère et
al. (2018) and the predictions on memory signatures beyond GR Heisenberg et
al. (2023). Finally, efforts in improving the low-frequency sensitivity of
gravitational wave detectors, which pose significant technological challenges
Hall et al. (2021); Punturo et al. (2010); Kagra Collaboration et al. (2019)
and yet benefit observations of gravitational wave displacement and spin
memory (see Figure 1).

## Acknowledgements

We thank Paul Lasky, Eric Thrane, Geoffrey Compère, Keefe Mitman, and Leo
Stein for useful discussions. We also appreciate early feedback on the
manuscript from Tjonnie Li and Nathan Johnson-McDaniel. We thank the
organizers and speakers of the GW memory workshop at the Queen Mary University
of London for an opportunity to touch base with the cutting-edge gravitational
wave memory research prior to the completion of this work. Finally, we thank
OzGrav, The Australian Research Council Centre of Excellence for Gravitational
Wave Discovery, for providing the opportunity to utilize the OzSTAR Australian
national facility (high-performance computing) at Swinburne University of
Technology. BG is supported by the Italian Ministry of Education, University
and Research within the PRIN 2017 Research Program Framework, n. 2017SYRTCN.
L.D. is supported by the European Research Council (ERC) Project 101076737 –
CeleBH. Views and opinions expressed are however those of the author only and
do not necessarily reflect those of the European Union or the European
Research Council. Neither the European Union nor the granting authority can be
held responsible for them. L.D. is also partially supported by INFN Iniziativa
Specifica ST&FI.

## Data availability

The simulated data utilized in this study will be provided by the
corresponding author upon request.

## Code availability

The response of detectors to gravitational wave signals and the calculations
of the Fisher matrix and the error matrix is performed with gwfish Dupletsa et
al. (2023). To calculate the gravitational wave strain for a given event, we
employ the numerical relativity waveform approximant nrhybsur3dq8 Varma et al.
(2019) available through gwsurrogate.

## References

  * Abbott et al. (2021a) R. Abbott, T. D. Abbott, S. Abraham, F. Acernese, K. Ackley, A. Adams, C. Adams, R. X. Adhikari, V. B. Adya, C. Affeldt, and et al.  Tests of general relativity with binary black holes from the second LIGO-Virgo gravitational-wave transient catalog.  _Phys. Rev. D_ , 103(12):122002, June 2021a.  doi: 10.1103/PhysRevD.103.122002.
  * Abbott et al. (2019) B. P. Abbott, R. Abbott, T. D. Abbott, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, V. B. Adya, and et al.  Tests of General Relativity with GW170817.  _Phys. Rev. Lett._ , 123(1):011102, July 2019.  doi: 10.1103/PhysRevLett.123.011102.
  * The LIGO Scientific Collaboration et al. (2021) The LIGO Scientific Collaboration, the Virgo Collaboration, the KAGRA Collaboration, R. Abbott, H. Abe, F. Acernese, K. Ackley, N. Adhikari, R. X. Adhikari, V. K. Adkins, and et al.  Tests of General Relativity with GWTC-3.  _arXiv e-prints_ , art. arXiv:2112.06861, December 2021.  doi: 10.48550/arXiv.2112.06861.
  * Zel’dovich and Polnarev (1974) Y. B. Zel’dovich and A. G. Polnarev.  Radiation of gravitational waves by a cluster of superdense stars.  _Sov. Astron. Lett_ , 18:17, 1974.
  * Christodoulou (1991) D. Christodoulou.  Nonlinear nature of gravitation and gravitational wave experiments.  _Phys. Rev. Lett._ , 67:1486–1489, 1991.  doi: 10.1103/PhysRevLett.67.1486.
  * Blanchet and Damour (1992) Luc Blanchet and Thibault Damour.  Hereditary effects in gravitational radiation.  _Phys. Rev. D_ , 46:4304–4319, 1992.  doi: 10.1103/PhysRevD.46.4304.
  * Thorne (1992) Kip S. Thorne.  Gravitational-wave bursts with memory: The Christodoulou effect.  _Phys. Rev. D_ , 45(2):520–524, January 1992.  doi: 10.1103/PhysRevD.45.520.
  * Strominger and Zhiboedov (2014) Andrew Strominger and Alexander Zhiboedov.  Gravitational Memory, BMS Supertranslations and Soft Theorems.  _arXiv e-prints_ , art. arXiv:1411.5745, November 2014.  doi: 10.48550/arXiv.1411.5745.
  * Weinberg (1965) Steven Weinberg.  Infrared photons and gravitons.  _Phys. Rev._ , 140:B516–B524, 1965.  doi: 10.1103/PhysRev.140.B516.
  * Bondi et al. (1962) H. Bondi, M. G. J. van der Burg, and A. W. K. Metzner.  Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems.  _Proc. Roy. Soc. Lond._ , A269:21, 1962.  doi: 10.1098/rspa.1962.0161.
  * Sachs (1962) R. K. Sachs.  Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times.  _Proc. Roy. Soc. Lond._ , A270:103–126, 1962.  doi: 10.1098/rspa.1962.0206.
  * Strominger (2018) Andrew Strominger.  _Lectures on the Infrared Structure of Gravity and Gauge Theory_.  Princeton University Press, 2018.
  * Ashtekar (2014) Abhay Ashtekar.  Geometry and Physics of Null Infinity.  9 2014.
  * Barnich and Troessaert (2010a) Glenn Barnich and Cedric Troessaert.  Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited.  _Phys. Rev. Lett._ , 105:111103, 2010a.  doi: 10.1103/PhysRevLett.105.111103.
  * Barnich and Troessaert (2010b) Glenn Barnich and Cedric Troessaert.  Aspects of the BMS/CFT correspondence.  _JHEP_ , 05:062, 2010b.  doi: 10.1007/JHEP05(2010)062.
  * Kapec et al. (2014) Daniel Kapec, Vyacheslav Lysov, Sabrina Pasterski, and Andrew Strominger.  Semiclassical Virasoro symmetry of the quantum gravity $\mathcal{S}$-matrix.  _JHEP_ , 08:058, 2014.  doi: 10.1007/JHEP08(2014)058.
  * Pasterski et al. (2016) Sabrina Pasterski, Andrew Strominger, and Alexander Zhiboedov.  New Gravitational Memories.  _JHEP_ , 12:053, 2016.  doi: 10.1007/JHEP12(2016)053.
  * Nichols (2018) David A. Nichols.  Center-of-mass angular momentum and memory effect in asymptotically flat spacetimes.  _Phys. Rev. D_ , 98(6):064032, 2018.  doi: 10.1103/PhysRevD.98.064032.
  * Himwich et al. (2021) Elizabeth Himwich, Zahra Mirzaiyan, and Sabrina Pasterski.  A Note on the Subleading Soft Graviton.  _JHEP_ , 04:172, 2021.  doi: 10.1007/JHEP04(2021)172.
  * Flanagan and Nichols (2017) Éanna É. Flanagan and David A. Nichols.  Conserved charges of the extended Bondi-Metzner-Sachs algebra.  _Phys. Rev. D_ , 95(4):044002, 2017.  doi: 10.1103/PhysRevD.95.044002.
  * Campiglia and Laddha (2015) Miguel Campiglia and Alok Laddha.  New symmetries for the Gravitational S-matrix.  _JHEP_ , 04:076, 2015.  doi: 10.1007/JHEP04(2015)076.
  * Freidel et al. (2021) Laurent Freidel, Roberto Oliveri, Daniele Pranzetti, and Simone Speziale.  The Weyl BMS group and Einstein’s equations.  _JHEP_ , 07:170, 2021.  doi: 10.1007/JHEP07(2021)170.
  * Godazgar et al. (2019) Hadi Godazgar, Mahdi Godazgar, and C. N. Pope.  New dual gravitational charges.  _Phys. Rev. D_ , 99(2):024013, 2019.  doi: 10.1103/PhysRevD.99.024013.
  * Compère et al. (2018) Geoffrey Compère, Adrien Fiorucci, and Romain Ruzziconi.  Superboost transitions, refraction memory and super-Lorentz charge algebra.  _JHEP_ , 11:200, 2018.  doi: 10.1007/JHEP11(2018)200.  [Erratum: JHEP 04, 172 (2020)].
  * Seraj and Oblak (2021) Ali Seraj and Blagoje Oblak.  Gyroscopic Gravitational Memory.  12 2021.
  * Godazgar et al. (2022) Mahdi Godazgar, George Macaulay, George Long, and Ali Seraj.  Gravitational memory effects and higher derivative actions.  _JHEP_ , 09:150, 2022.  doi: 10.1007/JHEP09(2022)150.
  * Heisenberg et al. (2023) Lavinia Heisenberg, Nicolás Yunes, and Jann Zosso.  Gravitational wave memory beyond general relativity.  _Phys. Rev. D_ , 108(2):024010, July 2023.  doi: 10.1103/PhysRevD.108.024010.
  * LIGO Scientific Collaboration et al. (2015) LIGO Scientific Collaboration, J. Aasi, B. P. Abbott, R. Abbott, T. Abbott, M. R. Abernathy, K. Ackley, C. Adams, T. Adams, P. Addesso, and et al.  Advanced LIGO.  _Classical and Quantum Gravity_ , 32(7):074001, April 2015.  doi: 10.1088/0264-9381/32/7/074001.
  * Acernese et al. (2015) F. Acernese, M. Agathos, K. Agatsuma, D. Aisa, N. Allemandou, A. Allocca, J. Amarni, P. Astone, G. Balestri, G. Ballardin, and et al.  Advanced Virgo: a second-generation interferometric gravitational wave detector.  _Classical and Quantum Gravity_ , 32(2):024001, January 2015.  doi: 10.1088/0264-9381/32/2/024001.
  * Lasky et al. (2016) Paul D. Lasky, Eric Thrane, Yuri Levin, Jonathan Blackman, and Yanbei Chen.  Detecting Gravitational-Wave Memory with LIGO: Implications of GW150914.  _Phys. Rev. Lett._ , 117(6):061102, August 2016.  doi: 10.1103/PhysRevLett.117.061102.
  * Hübner et al. (2020) Moritz Hübner, Colm Talbot, Paul D. Lasky, and Eric Thrane.  Measuring gravitational-wave memory in the first LIGO/Virgo gravitational-wave transient catalog.  _Phys. Rev. D_ , 101(2):023011, January 2020.  doi: 10.1103/PhysRevD.101.023011.
  * Hübner et al. (2021) Moritz Hübner, Paul Lasky, and Eric Thrane.  Memory remains undetected: Updates from the second LIGO/Virgo gravitational-wave transient catalog.  _Phys. Rev. D_ , 104(2):023004, July 2021.  doi: 10.1103/PhysRevD.104.023004.
  * Grant and Nichols (2023) Alexander M. Grant and David A. Nichols.  Outlook for detecting the gravitational-wave displacement and spin memory effects with current and future gravitational-wave detectors.  _Phys. Rev. D_ , 107(6):064056, March 2023.  doi: 10.1103/PhysRevD.107.064056.
  * Punturo et al. (2010) M. Punturo, M. Abernathy, F. Acernese, B. Allen, N. Andersson, K. Arun, F. Barone, B. Barr, M. Barsuglia, M. Beker, N. Beveridge, S. Birindelli, S. Bose, L. Bosi, S. Braccini, C. Bradaschia, T. Bulik, E. Calloni, G. Cella, E. Chassande Mottin, S. Chelkowski, A. Chincarini, J. Clark, E. Coccia, C. Colacino, J. Colas, A. Cumming, L. Cunningham, E. Cuoco, S. Danilishin, K. Danzmann, G. De Luca, R. De Salvo, T. Dent, R. De Rosa, L. Di Fiore, A. Di Virgilio, M. Doets, V. Fafone, P. Falferi, R. Flaminio, J. Franc, F. Frasconi, A. Freise, P. Fulda, J. Gair, G. Gemme, A. Gennai, A. Giazotto, K. Glampedakis, M. Granata, H. Grote, G. Guidi, G. Hammond, M. Hannam, J. Harms, D. Heinert, M. Hendry, I. Heng, E. Hennes, S. Hild, J. Hough, S. Husa, S. Huttner, G. Jones, F. Khalili, K. Kokeyama, K. Kokkotas, B. Krishnan, M. Lorenzini, H. Lück, E. Majorana, I. Mandel, V. Mandic, I. Martin, C. Michel, Y. Minenkov, N. Morgado, S. Mosca, B. Mours, H. Müller–Ebhardt, P. Murray, R. Nawrodt, J. Nelson, R. Oshaughnessy, C. D. Ott, C. Palomba, A. Paoli, G. Parguez, A. Pasqualetti, R. Passaquieti, D. Passuello, L. Pinard, R. Poggiani, P. Popolizio, M. Prato, P. Puppo, D. Rabeling, P. Rapagnani, J. Read, T. Regimbau, H. Rehbein, S. Reid, L. Rezzolla, F. Ricci, F. Richard, A. Rocchi, S. Rowan, A. Rüdiger, B. Sassolas, B. Sathyaprakash, R. Schnabel, C. Schwarz, P. Seidel, A. Sintes, K. Somiya, F. Speirits, K. Strain, S. Strigin, P. Sutton, S. Tarabrin, A. Thüring, J. van den Brand, C. van Leewen, M. van Veggel, C. van den Broeck, A. Vecchio, J. Veitch, F. Vetrano, A. Vicere, S. Vyatchanin, B. Willke, G. Woan, P. Wolfango, and K. Yamamoto.  The Einstein Telescope: a third-generation gravitational wave observatory.  _Classical and Quantum Gravity_ , 27(19):194002, October 2010.  doi: 10.1088/0264-9381/27/19/194002.
  * Reitze et al. (2019) David Reitze, Rana X. Adhikari, Stefan Ballmer, Barry Barish, Lisa Barsotti, GariLynn Billingsley, Duncan A. Brown, Yanbei Chen, Dennis Coyne, Robert Eisenstein, Matthew Evans, Peter Fritschel, Evan D. Hall, Albert Lazzarini, Geoffrey Lovelace, Jocelyn Read, B. S. Sathyaprakash, David Shoemaker, Joshua Smith, Calum Torrie, Salvatore Vitale, Rainer Weiss, Christopher Wipf, and Michael Zucker.  Cosmic Explorer: The U.S. Contribution to Gravitational-Wave Astronomy beyond LIGO.  In _Bulletin of the American Astronomical Society_ , volume 51, page 35, September 2019.  doi: 10.48550/arXiv.1907.04833.
  * Wang et al. (2015) J. B. Wang, G. Hobbs, W. Coles, R. M. Shannon, X. J. Zhu, D. R. Madison, M. Kerr, V. Ravi, M. J. Keith, R. N. Manchester, Y. Levin, M. Bailes, N. D. R. Bhat, S. Burke-Spolaor, S. Dai, S. Osłowski, W. van Straten, L. Toomey, N. Wang, and L. Wen.  Searching for gravitational wave memory bursts with the Parkes Pulsar Timing Array.  _MNRAS_ , 446(2):1657–1671, January 2015.  doi: 10.1093/mnras/stu2137.
  * Arzoumanian et al. (2015) Z. Arzoumanian, A. Brazier, S. Burke-Spolaor, S. J. Chamberlin, S. Chatterjee, B. Christy, J. M. Cordes, N. J. Cornish, P. B. Demorest, X. Deng, T. Dolch, J. A. Ellis, R. D. Ferdman, E. Fonseca, N. Garver-Daniels, F. Jenet, G. Jones, V. M. Kaspi, M. Koop, M. T. Lam, T. J. W. Lazio, L. Levin, A. N. Lommen, D. R. Lorimer, J. Luo, R. S. Lynch, D. R. Madison, M. A. McLaughlin, S. T. McWilliams, D. J. Nice, N. Palliyaguru, T. T. Pennucci, S. M. Ransom, X. Siemens, I. H. Stairs, D. R. Stinebring, K. Stovall, J. Swiggum, M. Vallisneri, R. van Haasteren, Y. Wang, W. W. Zhu, and NANOGrav Collaboration.  NANOGrav Constraints on Gravitational Wave Bursts with Memory.  _ApJ_ , 810(2):150, September 2015.  doi: 10.1088/0004-637X/810/2/150.
  * van Haasteren and Levin (2010) Rutger van Haasteren and Yuri Levin.  Gravitational-wave memory and pulsar timing arrays.  _MNRAS_ , 401(4):2372–2378, February 2010.  doi: 10.1111/j.1365-2966.2009.15885.x.
  * Favata (2010) Marc Favata.  The gravitational-wave memory effect.  _Classical and Quantum Gravity_ , 27(8):084036, April 2010.  doi: 10.1088/0264-9381/27/8/084036.
  * Sun et al. (2023) Shuo Sun, Changfu Shi, Jian-dong Zhang, and Jianwei Mei.  Detecting the gravitational wave memory effect with TianQin.  _Phys. Rev. D_ , 107(4):044023, February 2023.  doi: 10.1103/PhysRevD.107.044023.
  * Amaro-Seoane et al. (2017) Pau Amaro-Seoane, Heather Audley, Stanislav Babak, John Baker, Enrico Barausse, Peter Bender, Emanuele Berti, Pierre Binetruy, Michael Born, Daniele Bortoluzzi, Jordan Camp, Chiara Caprini, Vitor Cardoso, Monica Colpi, John Conklin, Neil Cornish, Curt Cutler, Karsten Danzmann, Rita Dolesi, Luigi Ferraioli, Valerio Ferroni, Ewan Fitzsimons, Jonathan Gair, Lluis Gesa Bote, Domenico Giardini, Ferran Gibert, Catia Grimani, Hubert Halloin, Gerhard Heinzel, Thomas Hertog, Martin Hewitson, Kelly Holley-Bockelmann, Daniel Hollington, Mauro Hueller, Henri Inchauspe, Philippe Jetzer, Nikos Karnesis, Christian Killow, Antoine Klein, Bill Klipstein, Natalia Korsakova, Shane L Larson, Jeffrey Livas, Ivan Lloro, Nary Man, Davor Mance, Joseph Martino, Ignacio Mateos, Kirk McKenzie, Sean T McWilliams, Cole Miller, Guido Mueller, Germano Nardini, Gijs Nelemans, Miquel Nofrarias, Antoine Petiteau, Paolo Pivato, Eric Plagnol, Ed Porter, Jens Reiche, David Robertson, Norna Robertson, Elena Rossi, Giuliana Russano, Bernard Schutz, Alberto Sesana, David Shoemaker, Jacob Slutsky, Carlos F. Sopuerta, Tim Sumner, Nicola Tamanini, Ira Thorpe, Michael Troebs, Michele Vallisneri, Alberto Vecchio, Daniele Vetrugno, Stefano Vitale, Marta Volonteri, Gudrun Wanner, Harry Ward, Peter Wass, William Weber, John Ziemer, and Peter Zweifel.  Laser Interferometer Space Antenna.  _arXiv e-prints_ , art. arXiv:1702.00786, February 2017.  doi: 10.48550/arXiv.1702.00786.
  * Grishchuk and Polnarev (1989) L. P. Grishchuk and A. G. Polnarev.  Gravitational wave pulses with “velocity-coded memory”.  _Soviet Journal of Experimental and Theoretical Physics_ , 69(4):653, October 1989.
  * Flanagan et al. (2019) Éanna É. Flanagan, Alexander M. Grant, Abraham I. Harte, and David A. Nichols.  Persistent gravitational wave observables: General framework.  _Phys. Rev. D_ , 99(8):084044, April 2019.  doi: 10.1103/PhysRevD.99.084044.
  * Gasparotto et al. (2023) Silvia Gasparotto, Rodrigo Vicente, Diego Blas, Alexander C. Jenkins, and Enrico Barausse.  Can gravitational-wave memory help constrain binary black-hole parameters? A LISA case study.  _Phys. Rev. D_ , 107(12):124033, June 2023.  doi: 10.1103/PhysRevD.107.124033.
  * Ebersold and Tiwari (2020) Michael Ebersold and Shubhanshu Tiwari.  Search for nonlinear memory from subsolar mass compact binary mergers.  _Phys. Rev. D_ , 101(10):104041, May 2020.  doi: 10.1103/PhysRevD.101.104041.
  * Yang and Martynov (2018) Huan Yang and Denis Martynov.  Testing Gravitational Memory Generation with Compact Binary Mergers.  _Phys. Rev. Lett._ , 121(7):071102, August 2018.  doi: 10.1103/PhysRevLett.121.071102.
  * Tiwari et al. (2021) Shubhanshu Tiwari, Michael Ebersold, and Eleanor Z. Hamilton.  Leveraging gravitational-wave memory to distinguish neutron star-black hole binaries from black hole binaries.  _Phys. Rev. D_ , 104(12):123024, December 2021.  doi: 10.1103/PhysRevD.104.123024.
  * Lopez et al. (2023) Dixeena Lopez, Shubhanshu Tiwari, and Michael Ebersold.  Memory matters: Gravitational wave memory of compact binary coalescence in the presence of matter effects.  _arXiv e-prints_ , art. arXiv:2305.04761, May 2023.  doi: 10.48550/arXiv.2305.04761.
  * Strominger and Zhiboedov (2017) Andrew Strominger and Alexander Zhiboedov.  Superrotations and Black Hole Pair Creation.  _Class. Quant. Grav._ , 34(6):064002, 2017.  doi: 10.1088/1361-6382/aa5b5f.
  * Pasterski et al. (2021) Sabrina Pasterski, Monica Pate, and Ana-Maria Raclariu.  Celestial Holography.  In _Snowmass 2021_ , 11 2021.
  * Nichols (2017) David A. Nichols.  Spin memory effect for compact binaries in the post-Newtonian approximation.  _Phys. Rev. D_ , 95(8):084048, 2017.  doi: 10.1103/PhysRevD.95.084048.
  * Varma et al. (2019) Vijay Varma, Scott E. Field, Mark A. Scheel, Jonathan Blackman, Lawrence E. Kidder, and Harald P. Pfeiffer.  Surrogate model of hybridized numerical relativity binary black hole waveforms.  _Phys. Rev. D_ , 99(6):064045, March 2019.  doi: 10.1103/PhysRevD.99.064045.
  * Mitman et al. (2020) Keefe Mitman, Jordan Moxon, Mark A. Scheel, Saul A. Teukolsky, Michael Boyle, Nils Deppe, Lawrence E. Kidder, and William Throwe.  Computation of displacement and spin gravitational memory in numerical relativity.  _Phys. Rev. D_ , 102(10):104007, November 2020.  doi: 10.1103/PhysRevD.102.104007.
  * Mitman et al. (2021) Keefe Mitman, Dante A. B. Iozzo, Neev Khera, Michael Boyle, Tommaso De Lorenzo, Nils Deppe, Lawrence E. Kidder, Jordan Moxon, Harald P. Pfeiffer, Mark A. Scheel, Saul A. Teukolsky, and William Throwe.  Adding gravitational memory to waveform catalogs using BMS balance laws.  _Phys. Rev. D_ , 103(2):024031, January 2021.  doi: 10.1103/PhysRevD.103.024031.
  * Yoo et al. (2023) Jooheon Yoo, Keefe Mitman, Vijay Varma, Michael Boyle, Scott E. Field, Nils Deppe, François Hébert, Lawrence E. Kidder, Jordan Moxon, Harald P. Pfeiffer, Mark A. Scheel, Leo C. Stein, Saul A. Teukolsky, William Throwe, and Nils L. Vu.  Numerical relativity surrogate model with memory effects and post-Newtonian hybridization.  _arXiv e-prints_ , art. arXiv:2306.03148, June 2023.  doi: 10.48550/arXiv.2306.03148.
  * Pollney and Reisswig (2011) Denis Pollney and Christian Reisswig.  Gravitational Memory in Binary Black Hole Mergers.  _ApJ_ , 732(1):L13, May 2011.  doi: 10.1088/2041-8205/732/1/L13.
  * Kehagias and Riotto (2016) A. Kehagias and A. Riotto.  BMS in cosmology.  _J. Cosmology Astropart. Phys_ , 2016(5):059, May 2016.  doi: 10.1088/1475-7516/2016/05/059.
  * Tinto et al. (2002) Massimo Tinto, F. B. Estabrook, and J. W. Armstrong.  Time-delay interferometry for LISA.  _Phys. Rev. D_ , 65(8):082003, April 2002.  doi: 10.1103/PhysRevD.65.082003.
  * Dupletsa et al. (2023) U. Dupletsa, J. Harms, B. Banerjee, M. Branchesi, B. Goncharov, A. Maselli, A. C. S. Oliveira, S. Ronchini, and J. Tissino.  GWFISH: A simulation software to evaluate parameter-estimation capabilities of gravitational-wave detector networks.  _Astronomy and Computing_ , 42:100671, January 2023.  doi: 10.1016/j.ascom.2022.100671.
  * Barausse et al. (2020) Enrico Barausse, Irina Dvorkin, Michael Tremmel, Marta Volonteri, and Matteo Bonetti.  Massive Black Hole Merger Rates: The Effect of Kiloparsec Separation Wandering and Supernova Feedback.  _ApJ_ , 904(1):16, November 2020.  doi: 10.3847/1538-4357/abba7f.
  * Abbott et al. (2021b) R. Abbott, T. D. Abbott, S. Abraham, F. Acernese, K. Ackley, A. Adams, C. Adams, R. X. Adhikari, V. B. Adya, C. Affeldt, and et al.  Population Properties of Compact Objects from the Second LIGO-Virgo Gravitational-Wave Transient Catalog.  _ApJ_ , 913(1):L7, May 2021b.  doi: 10.3847/2041-8213/abe949.
  * Scargle et al. (2021) Jeffrey D. Scargle, Zhoujian Cao, and Zhi-Chao Zhao.  Detection of the Permanent Strain Offset Component of Gravitational-Wave Memory in Black Hole Mergers.  _arXiv e-prints_ , art. arXiv:2110.07754, October 2021.  doi: 10.48550/arXiv.2110.07754.
  * Sharma and Harms (2020) Ashish Sharma and Jan Harms.  Searching for cosmological gravitational-wave backgrounds with third-generation detectors in the presence of an astrophysical foreground.  _Phys. Rev. D_ , 102(6):063009, September 2020.  doi: 10.1103/PhysRevD.102.063009.
  * Ng et al. (2023) Ken K. Y. Ng, Boris Goncharov, Shiqi Chen, Ssohrab Borhanian, Ulyana Dupletsa, Gabriele Franciolini, Marica Branchesi, Jan Harms, Michele Maggiore, Antonio Riotto, B. S. Sathyaprakash, and Salvatore Vitale.  Measuring properties of primordial black hole mergers at cosmological distances: Effect of higher order modes in gravitational waves.  _Phys. Rev. D_ , 107(2):024041, January 2023.  doi: 10.1103/PhysRevD.107.024041.
  * Payne et al. (2022) Ethan Payne, Sophie Hourihane, Jacob Golomb, Rhiannon Udall, Derek Davis, and Katerina Chatziioannou.  Curious case of GW200129: Interplay between spin-precession inference and data-quality issues.  _Phys. Rev. D_ , 106(10):104017, November 2022.  doi: 10.1103/PhysRevD.106.104017.
  * Hall et al. (2021) Evan D. Hall, Kevin Kuns, Joshua R. Smith, Yuntao Bai, Christopher Wipf, Sebastien Biscans, Rana X. Adhikari, Koji Arai, Stefan Ballmer, Lisa Barsotti, Yanbei Chen, Matthew Evans, Peter Fritschel, Jan Harms, Brittany Kamai, Jameson Graef Rollins, David Shoemaker, Bram J. J. Slagmolen, Rainer Weiss, and Hiro Yamamoto.  Gravitational-wave physics with Cosmic Explorer: Limits to low-frequency sensitivity.  _Phys. Rev. D_ , 103(12):122004, June 2021.  doi: 10.1103/PhysRevD.103.122004.
  * Kagra Collaboration et al. (2019) Kagra Collaboration, T. Akutsu, M. Ando, K. Arai, Y. Arai, S. Araki, A. Araya, N. Aritomi, H. Asada, Y. Aso, S. Atsuta, K. Awai, S. Bae, L. Baiotti, M. A. Barton, K. Cannon, E. Capocasa, C. S. Chen, T. W. Chiu, K. Cho, Y. K. Chu, K. Craig, W. Creus, K. Doi, K. Eda, Y. Enomoto, R. Flaminio, Y. Fujii, M. K. Fujimoto, M. Fukunaga, M. Fukushima, T. Furuhata, S. Haino, K. Hasegawa, K. Hashino, K. Hayama, S. Hirobayashi, E. Hirose, B. H. Hsieh, C. Z. Huang, B. Ikenoue, Y. Inoue, K. Ioka, Y. Itoh, K. Izumi, T. Kaji, T. Kajita, M. Kakizaki, M. Kamiizumi, S. Kanbara, N. Kanda, S. Kanemura, M. Kaneyama, G. Kang, J. Kasuya, Y. Kataoka, N. Kawai, S. Kawamura, T. Kawasaki, C. Kim, J. Kim, J. C. Kim, W. S. Kim, Y. M. Kim, N. Kimura, T. Kinugawa, S. Kirii, Y. Kitaoka, H. Kitazawa, Y. Kojima, K. Kokeyama, K. Komori, A. K. H. Kong, K. Kotake, R. Kozu, R. Kumar, H. S. Kuo, S. Kuroyanagi, H. K. Lee, H. M. Lee, H. W. Lee, M. Leonardi, C. Y. Lin, F. L. Lin, G. C. Liu, Y. Liu, E. Majorana, S. Mano, M. Marchio, T. Matsui, F. Matsushima, Y. Michimura, N. Mio, O. Miyakawa, A. Miyamoto, T. Miyamoto, K. Miyo, S. Miyoki, W. Morii, S. Morisaki, Y. Moriwaki, T. Morozumi, M. Musha, K. Nagano, S. Nagano, K. Nakamura, T. Nakamura, H. Nakano, M. Nakano, K. Nakao, T. Narikawa, L. Naticchioni, L. Nguyen Quynh, W. T. Ni, A. Nishizawa, Y. Obuchi, T. Ochi, J. J. Oh, S. H. Oh, M. Ohashi, N. Ohishi, M. Ohkawa, K. Okutomi, K. Ono, K. Oohara, C. P. Ooi, S. S. Pan, J. Park, F. E. Peña Arellano, I. Pinto, N. Sago, M. Saijo, S. Saitou, Y. Saito, K. Sakai, Y. Sakai, Y. Sakai, M. Sasai, M. Sasaki, Y. Sasaki, S. Sato, N. Sato, T. Sato, Y. Sekiguchi, N. Seto, M. Shibata, T. Shimoda, H. Shinkai, T. Shishido, A. Shoda, K. Somiya, E. J. Son, A. Suemasa, T. Suzuki, T. Suzuki, H. Tagoshi, H. Tahara, H. Takahashi, R. Takahashi, A. Takamori, H. Takeda, H. Tanaka, K. Tanaka, T. Tanaka, S. Tanioka, E. N. Tapia San Martin, D. Tatsumi, T. Tomaru, T. Tomura, F. Travasso, K. Tsubono, S. Tsuchida, N. Uchikata, T. Uchiyama, T. Uehara, S. Ueki, K. Ueno, F. Uraguchi, T. Ushiba, M. H. P. M. van Putten, H. Vocca, S. Wada, T. Wakamatsu, Y. Watanabe, W. R. Xu, T. Yamada, A. Yamamoto, K. Yamamoto, K. Yamamoto, S. Yamamoto, T. Yamamoto, K. Yokogawa, J. Yokoyama, T. Yokozawa, T. H. Yoon, T. Yoshioka, H. Yuzurihara, S. Zeidler, and Z. H. Zhu.  KAGRA: 2.5 generation interferometric gravitational wave detector.  _Nature Astronomy_ , 3:35–40, January 2019.  doi: 10.1038/s41550-018-0658-y.
  * Vallisneri (2008) Michele Vallisneri.  Use and abuse of the Fisher information matrix in the assessment of gravitational-wave parameter-estimation prospects.  _Phys. Rev. D_ , 77(4):042001, February 2008.  doi: 10.1103/PhysRevD.77.042001.

##  Appendix A  Reduced-order Bayesian inference

###  A.1  Likelihood

Let us provide more details on the use of Fisher matrix approximation of the
likelihood in Bayesian inference from our simulations. We start with Taylor-
expanding the log-likelihood, $\ln\mathcal{L}(d|\bm{\theta})$, with respect to
the vector of parameters $\bm{\theta}$ at the position of true parameters
$\bm{\theta}_{0}$ that describe the data, $d$:

$\ln\mathcal{L}(d|\bm{\theta})\bigg{|}_{\bm{\theta}\xrightarrow{}\bm{\theta}_{0}}=\sum_{n=0}^{\infty}\ln\mathcal{L}^{(n)}(d|\bm{\theta}_{0})(\bm{\theta}-\bm{\theta}_{0})^{n},$
(2)

where $(n)$ denotes the $n$’th derivative of the likelihood with respect to
$\bm{\theta}$. The $n=0$ term corresponds to the maximum (log-)likelihood,
$\mathcal{L}_{0}=\text{const}(\bm{\theta})$. Because the $n=1$ term
corresponds to the first derivative of the likelihood at its maximum point, it
is equal to zero. Note that the $n=2$ term contains

$\ln\mathcal{L}^{(2)}(d|\bm{\theta}_{0})=\langle\frac{\partial^{2}}{\partial\theta_{i}\partial\theta_{j}}\ln\mathcal{L}(d|\bm{\theta}_{0})\rangle\equiv\bm{F},$
(3)

which is equal to the Fisher information matrix Dupletsa et al. [2023],
$\bm{F}$, a Hessian matrix where $i$ and $j$ denote indices of the first and
the second partial derivatives. Neglecting terms past the second order, the
likelihood is approximated as

$\mathcal{L}(d|\bm{\theta})\bigg{|}_{\bm{\theta}\xrightarrow{}\bm{\theta}_{0}}\approx\mathcal{L}_{0}\exp\bigg{(}-\frac{1}{2}(\bm{\theta}-\bm{\theta}_{0})\bm{F}(\bm{\theta}-\bm{\theta}_{0})\bigg{)}.$
(4)

Notice that this approximation represents a multivariate Gaussian function,
with a covariance matrix $\bm{C}=\bm{F}^{-1}$ given by the inverse of the
Fisher information matrix. The dependence of the likelihood on $\theta$ is
what contributes to the posterior distribution,
$\mathcal{P}(\bm{\theta}|d)\propto\mathcal{L}(d|\bm{\theta})\pi(\bm{\theta})$,
representing our simulated measurement uncertainties. In this work, we assume
flat uninformative priors, such that $\pi(\bm{\theta})=1$ and the posterior is
determined only by the likelihood. When true parameters $\bm{\theta}_{0}$ are
not known, e.g. the case of real observations, priors should be chosen such
that they outline the true distributions of the observed parameters in nature.
However, this is not the case in our study.

###  A.2  Model selection and conditional posteriors

One may also evaluate the posterior and the evidence conditioned upon fixing
one or more of the parameters of the model that were previously free. Such a
posterior will remain a multivariate Gaussian function, a slice of the
original multivariate Gaussian posterior. Let us outline how to find
$(\bm{\theta}_{0}^{\prime},\bm{C}^{\prime},\mathcal{L}_{0}^{\prime})$, the new
mean, the new covariance matrix, and the new maximum likelihood position of
such a conditioned posterior, respectively, which fully define it. Let us
split the parameters into the fixed, denoted by an index “f”, and the
remaining ones, denoted by an index “r”, such that
$(\bm{\theta}_{\text{f}},\bm{\theta}_{\text{r}})=\bm{\theta}$. Similarly, we
break the true values into
$(\bm{\theta}_{\text{f,0}},\bm{\theta}_{\text{r,0}})=\bm{\theta}_{0}$. When
fixing parameters $\bm{\theta}_{\text{f}}=\bm{a}=\text{const}$, we reduce the
dimensionality of the posterior. To solve for $\bm{\theta}_{0}^{\prime}$ and
$\bm{C}^{\prime}$, we break the covariance matrix into so-called Schur
complements,

$\bm{C}=\begin{pmatrix}\bm{C}_{\text{rr}}&\bm{C}_{\text{rf}}\\\
\bm{C}_{\text{fr}}&\bm{C}_{\text{ff}}\end{pmatrix},$ (5)

where $\bm{C}_{\text{ff}}$ are the elements where rows and columns correspond
to the fixed parameters $\bm{\theta}_{\text{f}}$, $\bm{C}_{\text{rr}}$ are the
elements where rows and columns correspond to the remaining parameters
$\bm{\theta}_{\text{r}}$, $\bm{C}_{\text{rf}}$ is where rows correspond to the
remaining parameters and columns correspond to the fixed parameters, and
$\bm{C}_{\text{fr}}$ is the opposite to the previous one. Then, it follows
that the mean of a conditioned Gaussian function is

$\bm{\theta}_{0}^{\prime}=\bm{\theta}_{\text{r,0}}+\bm{C}_{\text{rf}}\bm{C}_{\text{ff}}^{-1}(\bm{a}-\bm{\theta}_{\text{f,0}}).$
(6)

The second term in the equation above manifests, in practice, systematic
errors in the remaining free parameters $\bm{\theta}_{\text{r}}$ due to fixing
parameters $\bm{\theta}_{\text{f}}$ at incorrect values. Whereas the variance
of a conditioned Gaussian is

$\bm{C}^{\prime}=\bm{C}_{\text{rr}}-\bm{C}_{\text{rf}}\bm{C}_{\text{ff}}^{-1}\bm{C}_{\text{fr}}.$
(7)

Note that the parameters can be fixed at both the correct values,
$\bm{a}=\bm{\theta}_{\text{f,0}}$, and the incorrect values,
$\bm{a}\neq\bm{\theta}_{\text{f,0}}$. In the former case,
$\mathcal{L}_{0}^{\prime}=\mathcal{L}_{0}$. In the latter case, based on
Equation 4, the new conditional posterior is positioned at
$\bm{\theta}_{\text{r,0}}$ will have a maximum value

$\mathcal{L}_{0}^{\prime}=\mathcal{L}_{0}\exp\bigg{(}-\frac{1}{2}(\bm{\theta}^{\prime\prime}-\bm{\theta}_{0})\bm{F}(\bm{\theta}^{\prime\prime}-\bm{\theta}_{0})\bigg{)}<\mathcal{L}_{0},$
(8)

where $\bm{\theta}^{\prime\prime}=(\bm{a},\bm{\theta}_{\text{r,0}})$.

Considering nested models, the approach above allows one to evaluate the Bayes
factors for one model over the other. For all calculations throughout this
work, we implicitly evaluate the Fisher matrix for $A_{\text{d,s}}$ (see
Section IV) in addition to the standard gravitational wave parameters. Next,
following the previous paragraph, we evaluate the posterior for a set of
hypotheses where $\theta_{\text{f}}=A_{\text{d,s}}$ are fixed to values $0$
and $1$, depending on the spacetime symmetry in question. Because our
approximation of the posterior given by Equation 4 represents a function for
which the integral is known, we evaluate Bayesian evidence analytically,

$\mathcal{Z}\equiv\int\mathcal{L}(d|\bm{\theta})\pi(\bm{\theta})d\bm{\theta}=\mathcal{L}_{0}\sqrt{(2\pi)^{m}\det{\bm{C}}},$
(9)

where $m$ is a number of parameters in vector $\bm{\theta}$, the
dimensionality of a multivariate Gaussian. Similarly, we evaluate the evidence
$\mathcal{Z}^{\prime}$ for the nested models that depend on
$(\bm{\theta}_{0}^{\prime},\bm{C}^{\prime},\mathcal{L}_{0}^{\prime})$.
Finally, we evaluate a Bayes factor in favor of model A against model B as
$\mathcal{B}\equiv\mathcal{Z}_{\text{A}}/\mathcal{Z}_{\text{B}}$.

###  A.3  On the accuracy of the approximation

A number of caveats apply to the use of Fisher matrix approximation of the
likelihood. In particular, the accuracy may be subject to sufficiently high
signal-to-noise ratios (SNRs) for the signals of interest, the inclusion of
priors, as well as the numerical accuracy of the matrix inversion Vallisneri
[2008]. Except for the LIGO-Virgo simulations, for which we point out the
effect of a weak-signal case in Section III.2, we work in the very high SNR
regime as we focus on a sub-population of the loudest binary black hole
mergers. The effect manifests as a $\approx 10\%$ discrepancy between
$\ln\mathcal{B}^{\text{Poincar{\'{e}}}}_{\text{BMS}}|_{\text{Poincar{\'{e}}}}$
and $\ln\mathcal{B}^{\text{BMS}}_{\text{Poincar{\'{e}}}}|_{\text{BMS}}$ in
Figure 3. Note, the effect is still smaller than the uncertainties associated
with the (1) binary black hole merger rate, (2) cosmic variance. At ballpark,
our results are consistent with full Bayesian inference by Hübner et al.
[2020]. They find that LIGO-Virgo require around $1500$ binary black hole
mergers to detect the displacement memory at $\ln\mathcal{B}=5$. Approximately
the same amount of events is to be detected by LIGO-Virgo at design
sensitivity in a year, which is equal to our detection time scale. Note that
it was also found in Hübner et al. [2020] that the result is largely a subject
to cosmic variance, which we did not simulate for ground-based detectors.
Nevertheless, the effect of cosmic variance is apparent in Figure 3 when at
some point during the observation the cumulative evidence in favor of a
correct symmetry group given the pessimistic black hole merger rate exceeds
that obtained with the best-fit merger rate. There are also more optimistic
and more pessimistic prospects in the literature. Shortly after the first
gravitational wave detection, Lasky et al. Lasky et al. [2016] suggested that
LIGO may be able to detect the displacement memory in 90 days at SNR of 5,
assuming all events are like GW150914. Although the assumption is now
considered too optimistic, our detection timescale for the displacement memory
with LIGO-Virgo is larger only by a factor of four. Regarding the prospects of
detection of both the displacement and the spin memory, our detection
timescales are shorter compared to Grant and Nichols [2023] by around a factor
of 3. This may be due to differences in the methodology and partly to due to
consideration of different detector networks. In any case, we argue that the
level of the precision of the Fisher matrix approximation is inferior to other
uncertainties for the purpose of our study.