# Prospects for reconstructing the gravitational-wave signals from core-
collapse supernovae with Advanced LIGO-Virgo and the BayesWave algorithm

Nayyer Raza<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of British Columbia, Vancouver, BC V6T1Z1, Canada  Department of
Physics, McGill University, Montréal, QC H3A2T8, Canada  Jess McIver
<EMAIL_ADDRESS>Department of Physics and Astronomy, University of British
Columbia, Vancouver, BC V6T1Z1, Canada  Gergely Dálya  Institute of Physics,
Eötvös University, 1117 Budapest, Hungary  Department of Physics and
Astronomy, Universiteit Gent, B-9000 Gent, Belgium  Peter Raffai  Institute of
Physics, Eötvös University, 1117 Budapest, Hungary

###### Abstract

Our current understanding of the core-collapse supernova explosion mechanism
is incomplete, with multiple viable models for how the initial shock wave
might be energized enough to lead to a successful explosion. Detection of a
gravitational-wave signal emitted in the initial few seconds after stellar
core-collapse would provide unique and crucial insight into this process. With
the Advanced LIGO and Advanced Virgo detectors expected to approach their
design sensitivities soon, we could potentially detect this signal from a
supernova within our galaxy. In anticipation of such a scenario, we study how
well the BayesWave algorithm can recover the gravitational-wave signal from
core-collapse supernova models in simulated advanced detector noise, and
optimize its ability to accurately reconstruct the signal waveforms. We find
that BayesWave can confidently reconstruct the signal from a range of
supernova explosion models in Advanced LIGO-Virgo for network signal-to-noise
ratios $\gtrsim 30$, reaching maximum reconstruction accuracies of $\sim 90\%$
at SNR $\sim 100$. For low SNR signals that are not confidently recovered, our
optimization efforts result in gains in reconstruction accuracy of up to
$20-40\%$, with typical gains of $\sim 10\%$.

††preprint: APS/123-QED

##  I Introduction

Despite many detailed multi-wavelength electromagnetic observations of core-
collapse supernovae (CCSNe), the exact mechanism powering these explosions is
not yet fully understood (see e.g. Burrows and Vartanyan [1] and references
therein). This is in part because the electromagnetic signal only escapes from
the very outer layers of the star after shock break-out (SBO), on time scales
of hours to days after the core collapses [2]. However, most of the critical
explosion physics occurs in the very central core of the star within the first
few seconds [3]. The thermodynamic state of the explosion in this crucial
period is imprinted on to the escaping neutrino flux, which carries away
almost all of the total explosion energy, $\sim 10^{53}~{}\mathrm{ergs}$ [4].
Non-spherical, accelerated mass motions in the dense supernova core at this
time also produce gravitational waves (GWs), lasting $\lesssim 1$ s after core
bounce, which then probe the internal dynamics of the explosion mechanism (see
e.g. Abdikamalov  _et al._ [5] for a recent review). We can thus use multi-
messenger astronomy to study these highly energetic events and help solve the
mystery of what powers CCSNe.

While Supernova SN1987A marked the first (and so far only) supernova for which
the neutrino signal was also observed [6, 7, 8], GWs from CCSNe have not yet
been detected [9, 10]. This is especially challenging since CCSNe are rare
events, expected to only occur approximately once or twice per century in
large galaxies such as the Milky Way [11].

The Advanced LIGO [12] and Advanced Virgo [13] ground-based interferometric GW
detectors are expected to achieve their design sensitivities for their fourth
observing run [14]. At these sensitivities (along with the addition of new
detectors such as KAGRA [15, 16] and LIGO-India [17] in the future), we could
potentially observe a Galactic CCSN with GWs [18, 19]. In anticipation of such
an event, in this study we follow up on the CCSN sensitivity analyzed in the
all-sky short duration burst search of the most recent LIGO-Virgo observing
run [10], and characterize how well the BayesWave algorithm [20] can
reconstruct CCSN models embedded in Advanced LIGO and Advanced Virgo design
sensitivity detector noise. We also aim to tune BayesWave for CCSN signals,
and produce a set of recommendations to be used for analysis on a candidate
detection that will optimize the reconstruction accuracy.

###  I.1 Core-collapse supernovae

CCSNe are the end stage of the life of massive stars with $M\gtrsim
8~{}\mathrm{M_{\odot}}$. When the mass of the central iron core exceeds the
effective Chandrasekhar mass, runaway collapse of the core begins, marking the
onset of the explosion. This value can vary between $M_{Ch}\sim
1.34-1.8~{}\mathrm{M_{\odot}}$, depending on a range of different properties
of the progenitor star (see e.g. Woosley  _et al._ [21] for a classic review).
A shock forms between the supersonically infalling inner core of $\sim
0.6-0.8~{}\mathrm{M_{\odot}}$ and the subsonically infalling outer core of
$\sim 0.6~{}\mathrm{M_{\odot}}$. As the temperature and density increases, the
collapse is accelerated by photo-dissociation and electron capture, which also
produces a burst of neutrinos that are initially trapped in the ultra-dense
inner core. As the central density reaches nuclear density, the inner core
cannot collapse any further and rebounds, transferring in-falling energy-
momentum by launching a shock wave outwards.

This shock wave loses significant energy as it propagates through the still
in-falling outer core and heats it, dissociating iron nuclei into free protons
and neutrons. Furthermore, as the density behind the shock gets low enough,
$\sim 10^{11}~{}\mathrm{g\cdot cm^{-3}}$, the neutrinos start to escape,
contributing to further energy loss. Eventually, the shock stalls, around
100-200 km from the center (a few tens of milliseconds after bounce), near the
outer iron layer. From this point, to get a successful explosion the shock
must be revived within a few hundred milliseconds before the continued
accretion onto the central proto neutron star (PNS) triggers further collapse
into a black hole.

The exact mechanism behind the re-energization of the shock wave and the
subsequent successful explosion is not fully understood [1]. The delayed
neutrino heating mechanism has emerged as a promising candidate to explain the
shock revival for most slowly rotating progenitors. In this scenario the
intense neutrino flux from the hot PNS deposits a small fraction of its energy
($\sim 1\%$) into the gain region behind the stalled shock (see e.g. Janka
[22] for a review of neutrino-driven explosions). For rapidly rotating stars
with strong magnetic fields (expected to account for $\sim 1\%$ of CCSN
progentiors [23]), magnetohydrodynamic effects can allow the transfer of
energy from the highly magnetized PNS into the outer stellar layers for a
violent explosion, in the magnetorotationally-driven mechanism (e.g. [24]).
While it’s expected that neutrino heating plays a key role in most explosion
models, the exact role of the different potential drivers of the explosion,
which include the neutrino flux, differential rotation of the progenitor star,
and strong magnetic fields, remains to be fully understood. We direct the
reader to the review by Janka [3] for a detailed discussion of the different
possible CCSN mechanisms, and to the recent review by Burrows and Vartanyan
[1] and the references therein for current limitations in our understanding.

###  I.2 Gravitational waves from CCSNe

The GW signal can provide unique insight into the state of the PNS in this
crucial time period, and inform about the explosion mechanism [5]. Some
expected features in a GW signal are common to most CCSN models. A quadrupole
moment from the deformation of the PNS is expected to occur after the core
bounce. Waves from the outer convective region are likely to travel in and
strike the outer core, and waves from convection in the PNS are likely to
internally excite core and surface oscillations. For slowly rotating stars,
the GW signal is then expected to be dominated by the fundamental oscillation
modes of the PNS (f, g, p modes), which increase in frequency with time as the
PNS contracts (e.g. [25, 26, 27, 28]). The exact shape and amplitude of the
signal can then also place constraints on the source properties such as the
PNS mass, size, core compactness and explosion energy (e.g. [29, 30, 31]).

Bulk fluid motion from the advective acoustic cycle, particularly in the gain
region, may also cause global asymmetric perturbations leading to emission in
the low-frequency regime. Linear or spiral oscillation modes (l, m) of the
shock front may arise from the Standing Accretion Shock Instability (SASI)
[32].

In the case that the explosion is powered by strong magnetic fields or
amplified by a high rotation rate, prompt broadband emission can be expected
within $\sim 20$ ms after core bounce (e.g. [33, 34]). A rapidly rotating
stellar core is deformed due to its angular momentum and leads to the
derivatives of the quadrupole moment to change significantly as the core
collapses, and the core bounce prompt convection signal can become a dominant
feature (e.g. [35, 36])

###  I.3 Detecting the GW signal with Advanced LIGO-Virgo

The stochastic nature of the CCSN explosion makes predicting exactly which
features will occur currently impossible. However, the dominant emission
features that are expected from the GW signal of different CCSN models, $\sim$
20 Hz - 2 kHz, lie in the most sensitive frequency regime of the Advanced LIGO
and Virgo detectors [14]. This sensitivity overlap makes CCSN within our
Galaxy a potentially promising target for current detectors. While the
expected event rate of 1-2 per century [11] for such events is low, a
detection of the associated GW signal should a Galactic CCSN happen within the
next observing run would be invaluable, and could allow us to distinguish
between different explosion models and mechanisms (e.g. [37, 38]), as well as
extract astrophysical parameters of the source progenitor and remnant.

GW detection and analysis algorithms have been developed over the years
targeted towards characterizing short duration GW signals in the LIGO and
Virgo detectors, and some of these have been used to study GWs from CCSN. In
particular, previous work by Gossan  _et al._ [18] used the X-Pipeline
algorithm [39] and GW waveform models from multi-dimensional simulations to
study in detail the detectability of these models for the advanced era
detectors. The recent study by Szczepańczyk  _et al._ [19] used the coherent
WaveBurst algorithm [40] to analyze it’s ability to detect and reconstruct
CCSN GW signals, with an expanded and updated set of waveform models, for the
anticipated fourth and fifth observing runs.

Another candidate analysis algorithm is BayesWave, which is designed to
reconstruct the signal from GW burst sources (short duration transient
signals) using a wavelet basis, while making minimal assumptions about the
specific waveform morphology [20]. This un-modeled approach, as opposed to
matched filtering using known waveforms for compact binary coalescence
candidates (e.g. [41, 42, 43]), makes BayesWave a suitable waveform
reconstruction pipeline for CCSN studies.

To prepare for a candidate detection, we aim to study and quantify BayesWave’s
potential ability to reconstruct different types of CCSN signals. Furthermore
we aim to identify which algorithm settings are most sensitive to CCSN
signals, and produce a set of recommendations to be used for analysis on a
real signal. In order to achieve these goals of characterizing BayesWave’s
performance, we study five different recent CCSN explosion models from three
dimensional simulations. These models encompass the same set used in Abbott
_et al._ [10], and span a range of explosion models, including failed
explosions, neutrino driven explosions and magnetorotationally driven
explosions.

The rest of the paper is organized as follows. In Section II we describe the
BayesWave algorithm, the supernova waveform models we employed, the general
procedure for our reconstruction studies, and our optimization efforts. In
Section III we present our results, highlighting BayesWave’s reconstruction
accuracy for CCSN, gains from our optimization, and the recovery of a
realistic distribution of sources in the Milky Way. Section IV follows with a
discussion of BayesWave’s performance between the different models, the
distances to which it can reconstruct CCSN signals, and the limitations of
BayesWave in this work. We summarize and conclude with final remarks and
future work in Section V.

##  II Methods

The aim of our study is to analyze and optimize the reconstruction of CCSN GW
signals with BayesWave. In the next few subsections we detail the methods that
we employ, including a summary of the BayesWave algorithm and its features, a
description of the CCSN waveform models selected, the steps taken to perform
the reconstructions in simulated Advanced LIGO-Virgo detectors, and the
optimization efforts employed in the procedure to yield more reliable and
accurate reconstructions.

###  II.1 BayesWave

BayesWave is a GW signal reconstruction algorithm for short bursts that makes
minimal assumptions about the signal morphology [20]. For each detector
BayesWave models the analyzed data ($d$) as a linear combination of the GW
signal in the detector frame ($s$), the detector’s stationary Gaussian noise
($n$), and transient instrumental noise (glitches, $g$): $d=s+n+g$. The ratio
of the Bayesian evidences for each model determines which has the most
support.

In practice BayesWave uses a trans-dimensional reversible jump Markov chain
Monte Carlo (RJMCMC) to place a variable number of Morlet-Gabor (sine-
Gaussian) wavelets, the linear combination of which forms a reconstructed
signal. For a multiple detector configuration, the wavelets placed for the
signal model must be coherent across detectors, requiring the same set of
extrinsic parameters (sky location, ellipticity, polarization angle) to
properly project the signal onto the detectors. For the glitch model, however,
the wavelets are uncorrelated.

Since the supernova analysis pipeline must make minimal assumptions about the
waveform, BayesWave is then one of the more suitable choices to use in our
study. Because it is agnostic to the type or source of the signal, BayesWave
has been extensively used for un-modeled reconstructions of GW signals from
compact binary coalescences (e.g. [44, 45, 46, 47]) and white-noise bursts
[48], as well as a glitch subtraction tool to ‘clean’ a portion of the
detector data that has an instrumental noise event overlapping with a real
signal (e.g. [49, 50]). It could also potentially be used as a follow-up tool
to cWB to help better discriminate CCSNe events from instrumental noise events
[51].

If the analysis involves recovering a known signal that has been injected
(added) into detector noise (as is the case for our study), then BayesWave
also calculates the overlap (also called match) between the waveform of the
injected signal $h_{i}$ and the BayesWave wavelet reconstruction $h_{r}$. The
overlap is defined as

$O=\frac{(h_{i}|h_{r})}{\sqrt{(h_{i}|h_{i})(h_{r}|h_{r})}},$ (1)

where the notation $(x|y)$ denotes a noise-weighted inner product using the
detector’s one-sided noise power spectral density $S_{h}(f)$:

$(x|y)=2\int_{0}^{\infty}\frac{x(f)y^{*}(f)+x^{*}(f)y(f)}{S_{h}(f)}df$ (2)

Thus the overlap characterizes the accuracy of the waveform reconstruction,
measuring the similarity of the injected and recovered waveforms. This value
ranges from -1 to 1, with $O=1$ meaning a perfect match between the two
waveforms, $O=0$ meaning no match, and $O=-1$ indicating a perfect anti-
correlation. In our work we quote the combined weighted network overlap for
all $N=3$ detectors, which is computed as

$O_{net}=\frac{\sum_{j=1}^{N}(h_{i}^{(j)}|h_{r}^{(j)})}{\sqrt{\sum_{j=1}^{N}(h_{i}^{(j)}|h_{i}^{(j)})\sum_{j=1}^{N}(h_{r}^{(j)}|h_{r}^{(j)})}},$
(3)

where $j$ denotes the $j$-th detector.

In the context of our study, this means that if the BayesWave reconstructed
CCSN waveform has an overlap of $O=0.5$, for example, then that reconstructed
waveform accurately matches 50% of the total time-frequency and phase
signature of the true waveform. In the rest of this paper, we then often say
that the reconstruction accuracy is $X\%$ to reflect the fact that the overlap
between the reconstructed and injected signals is $X/100$. Furthermore, as
outlined in Section II.3, in our studies BayesWave produces 20000
reconstructed waveforms for each injected signal, and calculates the overlap
for each of these. The result is a distribution of overlap values, from which
we select the median value as the characteristic overlap, and compute the
standard deviation of the distribution as an estimate of the overlap
uncertainty.

###  II.2 Supernova waveform models

Here we briefly describe the supernova waveform models from 3D simulations
used in our study and the GW features they exhibit (as shown in the first row
in Fig. 1). We refer the interested reader to their source papers for more
detailed information about the details of their simulation procedure and
modeling techniques.

  * •

The O’Connor and Couch [26] mesa20-3D-pert model (hereafter referred to as
m20-3Dp) has a $\mathrm{20\ M_{\odot}}$ solar metallicity non-rotating
progenitor star, and fails to explode within $\sim 500$ ms after core bounce,
the extent of the simulation. However, the model does emit a GW signal that is
dominated by the g-mode oscillation of the PNS surface, which begins at $\sim
300$ Hz and grows with time, peaking at $\sim 600-700$ Hz at $\sim 200$ ms
after core bounce. A subdominant signal due to the spiral SASI modes is also
present at $\sim 50-200$ Hz.

  * •

The Radice  _et al._ [27] s9 model has a $\mathrm{9\ M_{\odot}}$ solar
metallicity non-rotating progenitor, and represents a lower mass neutrino-
driven explosion. A broadband prompt convection signal develops $\sim 50$ ms
after bounce in conjunction with the neutrino shock breakout. After a short
quiescent period, the PNS g-mode oscillation begins, rising to frequencies of
$\sim 700$ Hz. The shock is successfully revived at $\sim 200$ ms after core
bounce, and no significant GW emission occurs after $\sim 300$ ms.

  * •

The Powell and Müller [28] s18 model has a $\mathrm{18\ M_{\odot}}$ solar
metallicity non-rotating progenitor, and represents a typical neutrino driven
explosion. GW emission from g-mode surface oscillations of the PNS begins
$\sim 100$ ms after core bounce, and peaks at $\sim 800-1000$ Hz just after
shock revival and onset of the explosion $\sim 300$ ms after bounce.
Significant GW emission continues up to $\sim 600$ ms.

  * •

The Powell and Müller [52] m39 model has a rapidly rotating Wolf-Rayet star as
its progenitor with a $\mathrm{39\ M_{\odot}}$ helium star mass, 2% solar
metallicity, and an initial surface rotation velocity of
$600~{}\mathrm{km\cdot s^{-1}}$. It is a strong neutrino driven explosion,
with GW features amplified by the rapid rotation. Strong GW emission from
prompt convection at low frequencies occurs shortly after bounce time, mainly
visible in observer angles towards the equator. There is also a strong core
bounce signal at frequencies $>500$ Hz. This is followed by emission
associated with f-mode oscillations of the PNS, which peaks at a frequency of
$\sim 750$ Hz about 400 ms after core bounce. The shock revival occurs at
$\sim 200$ ms, and GW emission continues up to $\sim 600$ ms.

  * •

The Obergaulinger and Aloy [24] 35OC-RO model has a $\mathrm{35\ M_{\odot}}$
rapidly rotating progenitor with strong magnetic fields, with a sub-solar
metallicity and an equatorial surface rotation velocity of
$380~{}\mathrm{km\cdot s^{-1}}$. It represents a strong magneto-rotationally
driven explosion. A bounce signal is quickly followed by an evolving PNS
oscillation track. After the shock is revived $\sim 200-300$ ms after bounce,
the strong explosion rapidly inhibits further accretion onto the PNS, beyond
which the GW signal is somewhat broadband and lasts until $\sim 800$ ms. Most
of the emission occurs near frequencies of $\sim 300-500$ Hz.

###  II.3 General procedure

Figure 1: The combined Q-transform spectrograms for the three detectors for
the injected signal waveform (top row), the data that BayesWave analyzes
(injected signal in simulated gaussian detector noise, middle row), and the
BayesWave recovered waveform (bottom row). Out of the 100 injections for each
waveform model, one representative case with a relatively high network SNR
$\sim 60$ is shown. We can see that for such loud signals, in each case
BayesWave manages to recover all dominant features visible in the spectrogram,
with overlap accuracies of $\sim 80\%$ for all waveforms. This includes the
high-frequency evolving g or f mode track from the PNS oscillation, which is
most easily identifiable in the s18 and m39 models, as well as the bounce and
prompt convection signal which is visible in the s9 and m39 models. The 35OC-
RO waveform is the most distinct with a relatively broadband track since the
explosion is magneto-rotationally driven instead of due to neutrino heating.
Note that the time-scale for the m20-3Dp and s9 waveforms is shorter than the
rest ($\sim 0.5$ s compared to $\sim 0.8$ s).

For each waveform model we first take the GW signal as generated by the
corresponding simulations in the source frame and project them into the
detector frame based on a given set of extrinsic parameters. This set of
parameters includes the sky position of the supernova, i.e. its right
ascension and declination, the distance to the source, the orientation of the
source relative to the line of sight, the polarization angle of the signal,
and the time of observation, which influences the detector antenna response
pattern in relation to the sky position.

We generate two population sets for these parameters for each model: in the
first set the sky positions follow a random uniform distribution across the
whole sphere, the source orientations and polarization angles are uniform
randomly distributed across all physically possible values, the time of
observation is set to be 10 seconds apart (for a uniform-in-sky population the
observation time makes no practical difference, and we choose this time for
convenience), and the distances are then determined such that the resulting
signal in the detector frame is log-uniform randomly distributed in SNR from
10 to 100. We produce 100 distinct signals to analyze in this way. In the
second population set we change the sky positions and distances to model after
a realistic spatial distribution of stars in the Milky Way galaxy, as done in
Abbott  _et al._ [10]. The Galaxy is modeled as a combination of the bulge,
thin disk and thick disk with parameters from the best-fitting contracted NFW
halo model in Cautun  _et al._ [53]. The time interval between signals is
increased to 10 minutes, and a total of 150 detector signals are thus
generated spanning approximately 24 hours. This is done since the detector sky
antenna sensitivity pattern changes on the time scale of a day due to the
rotation of the Earth, and we must account for the fact that the sources are
not uniformly distributed on the sky, but mostly lie within a narrow band that
traces the Galactic disk.

These selected signals are then injected (added) into simulated Gaussian noise
that is generated based on the expected design spectral noise sensitivity of
the Advanced LIGO [12] and Advanced Virgo [13] detectors, taking into account
the relative positions and orientations of the individual Hanford, Livingston,
and Virgo detectors. The end result is a GW data file for each injected signal
that has three separate channels for the strain data in each detector,
mirroring what we would expect to analyze for a real signal during an
observing run.

Finally, we analyze each of these signal-plus-noise data files using the
BayesWave algorithm, performing waveform reconstruction of the supernova GW
signal within the simulated detector noise (the injection-reconstruction
process is illustrated in Fig. 1). Since BayesWave is based on a Bayesian
approach to model the signal waveform and it samples the posterior
distribution of the waveform parameters, for each run the output is a
distribution of recovered waveforms, where each waveform represents one step
in the sampler. For the runs in our study BayesWave performs a total of 4
million RJMCMC sampler iterations. To keep computational costs reasonable we
then sub-select and save every hundredth sample in the latter half of the run
(i.e. once the distribution is steady). The BayesWave output products are then
calculated based on these 20000 waveforms, and the median recovered waveform
(as described in Cornish  _et al._ [54]) is used as the representative final
waveform. Similarly, for the network overlap we quote the median network
overlap value of the 20000 waveforms, and show the $1\sigma$ standard
deviation of that distribution as a measure of uncertainty on the final value
wherever possible.

###  II.4 Optimizations

We aim to optimize the BayesWave run settings to maximize waveform recovery
from supernovae for the different waveform models considered and across the
range of possible extrinsic signal parameters. We characterize differences in
performance between different run settings using the network overlap
parameter, where an increase in the overlap indicates a positive gain. We
consider this to be a more robust measure of whether the changes we introduce
have a positive or negative impact, as compared to measuring the SNR of the
recovered waveform. This is because the recovered SNR increases as BayesWave
places more wavelets, regardless of whether those wavelets are capturing
features appropriately. The overlap, however, appropriately penalizes any
frivolously placed wavelets, focusing not on simply the excess power in the
recovered signal, but how accurately that excess power matches the shape of
the true injected signal.

Since in the case of a real detection we will not know a priori what kind of
explosion model is the underlying cause of the signal, any optimization
efforts we make must be robust across all five different waveforms considered,
and for the entire SNR range studied. From the potential optimizations we
consider, we find two significant improvements based on: i) knowing accurate
sky localization information from electromagnetic and neutrino counterparts
(e.g. [55, 56]), and ii) increasing the allowed wavelet quality factor $Q$,
which is the number of cycles of the wavelet over one e-folding of the
Gaussian envelope (i.e. describes how spread out in time it is). Thus our
optimized runs correspond to setting a fixed sky location and $Q_{max}=100$,
compared to the initial un-optimized runs with default settings in which the
sky location is not fixed (uniform prior over the whole sky) and $Q_{max}=40$.

We also explored using frequency-evolving ‘chirplets’ [57] instead of the
fixed-frequency wavelets. However, this approach did not yield consistently
positive results, and we discuss this in more detail in Section IV. We also
note that a recent update to the BayesWave algorithm enables relaxing
polarization constraints from elliptical polarization to generic polarization
[54]. We were unable to fully explore this new feature for our present work,
and leave a more robust analysis of its potential for CCSN signals to future
studies.

Along with this we find that in order for the comparison to be appropriate and
for the BayesWave runs to converge in all cases, we have to be careful of a
few other run parameters. The analyzed segment length should not be less than
4 seconds, as below that BayesLine [58] can not determine the spectral density
of the noise accurately. The analysis window where BayesWave is allowed to
place signal wavelets should be kept to approximately 1 second to make sure no
part of the potential signal is being missed, and be centered on the duration
of the signal (in a real scenario the ‘trigger’ time of the event can be
estimated either from low-latency GW search algorithms like coherent WaveBurst
[40]) or from electromagnetic or neutrino counterpart detections of CCSNe. The
maximum number of allowed wavelets should scale with the SNR of the signal,
where we had to set this limit to 150 wavelets for the highest SNR signals in
this study. The prior on the number of wavelets should also be specified to be
uniform across the allowed window of {1, 150}. The number of iterations in the
RJMCMC must also not be less than 4 million, as the sampler can sometimes take
at least this many steps to converge to a steady distribution, especially for
higher SNR CCSN signals. Finally, we also recommend keeping the allowed
frequency range wide, between $\sim 16-32$ Hz at the lower end to $\sim 2$ kHz
(i.e. a maximum sampling rate of 4 kHz) at the higher end, to account for most
of the features we can expect to observe from CCSN GW waveform models with
current detectors.

##  III Results

In all subsequent sections we present our main results and figures based on
the optimized BayesWave run settings (fixed sky location, $Q_{max}=100$). We
first show how accurate the BayesWave reconstructions are for the different
types of CCSN waveform models and how this varies based on the source
parameters. We then quantify and describe the increase in overlap accuracy we
are able to achieve when using the optimized runs compared to the initial un-
optimized runs (uniform sky prior, $Q_{max}=100$). This is followed by the
BayesWave CCSN signal reconstruction prospects presented in the context of a
population of sources modeled after the Milky Way stellar distribution.

###  III.1 Accuracy of BayesWave reconstructions

To show the potential capability of BayesWave in recovering a CCSN waveform
and all its features if the signal is loud, one instance of the injection-
reconstruction process and the median recovered waveform for each model is
shown in Fig. 1. For each model we consider all the injections in the SNR
range 55-65, and then select the one which has the median overlap value within
that sample as a representative example. The features are most clearly
distinguished in time-frequency spectrograms, which are plotted in Fig. 1
using a constant-Q transform [59] ($Q=16$). Since our reconstructions are
based on the three detector HLV network and BayesWave simultaneously computes
the recovered waveform in all three detectors, we sum the power in the
individual spectrograms after accounting for the appropriate time delay shift,
and show the HLV combined spectrograms (noting that this coherent sum
emphasizes the signal more over the background noise when compared to single
detector signal+noise spectrograms). At such high SNRs, we can see that
BayesWave is recovering all previously identified GW emission features,
including the rising g or f mode PNS oscillation track visible in all models,
the early broadband prompt convection in e.g. m39, and the low-frequency SASI
in m20-3Dp.

Figure 2: A scatter plot of the BayesWave reconstructions for all the
injections in the study, showing the median accuracy of the recovered waveform
as measured by the network overlap (match) parameter plotted against the
network SNR of the injected signal. Uncertainty error bars are omitted here
for readability (see Fig. 3). The dashed black line indicates the 50% accuracy
threshold. The reconstruction accuracy between models varies greatly at low
SNRs, with a network SNR of $\gtrsim 30$ required to exceed the 50% threshold
for all models. The accuracy converges to $\sim 90\%$ for the highest SNR
signals in this study. The differences between models can be explained by the
differences in the waveform features and how compact they are in time-
frequency. The scatter in the overlap accuracy for a given model near a
specific SNR is a result of the small changes in the waveform morphology given
varying extrinsic parameters (such as inclination angle).

Fig. 2 shows the overlap accuracy of the BayesWave reconstructed waveform for
all the supernova GW signal injections analyzed in our study. The median
network overlap (match) for each reconstruction in the optimized runs is
plotted against the injected signal SNR for the 3 detector HLV network (root-
sum-square of the individual detector SNRs). As observed in previous studies
(e.g. [19]), the accuracy of the reconstruction generally increases as the
signal SNR increases. Table 1 lists the network SNR required for each model
for the network overlap to exceed 0.5 and the majority of the waveform be
recovered by BayesWave. Overall, when the SNR $\gtrsim 30$, all explosion
models are confidently recovered.

Waveform Model | $\mathbf{~{}SNR_{net}~{}}$ |  Distance (kpc)
---|---|---
m20-3Dp | 25 | 1.2
s9 | 25 | 0.6
s18 | 30 | 5
m39 | 20 | 40
35OC-RO | 30 | 45
Table 1: The approximate GW signal network SNR value or source distance
required for the BayesWave reconstruction network overlap value to be
$O_{net}\approx 0.5$ for each model. For signals with higher SNRs or lower
source distances than those listed the reconstruction will have $O_{net}>0.5$,
i.e. the majority of the waveform will be accurately recovered by BayesWave.

However, similar to what prior work by Gossan  _et al._ [18] and Szczepańczyk
_et al._ [19] found for other analysis pipelines, there is a difference in
BayesWave’s reconstruction performance between the different explosion models,
especially at lower SNRs. We can see that the s18 and 35OC-RO models have the
lowest accuracy, the m20-3Dp and s9 models somewhat higher, and the m39 model
has the highest reconstruction accuracy (these trends are also true for the
un-optimized run results which aren’t shown). At higher SNRs these differences
become less pronounced, with reconstruction accuracies reaching $\sim 90\%$ at
SNRs $\sim 100$. The differences can be explained by the different waveform
morphologies. How the power is distributed in time-frequency governs how well
the features can be captured by BayesWave’s Morlet-Gabor wavelet basis, with
well-localized signals expected to be better reconstructed.

For each explosion model we also see that the points near a given SNR for that
model have a small scatter due to different extrinsic parameters. Waveform
features that BayesWave can more accurately reconstruct are a little bit more
pronounced for some combination of extrinsic parameters than others that still
give a similar SNR. For example, for the m39 model we know that the prompt
convection signal right after core bounce is most readily observed when the
line of sight is aligned with the star’s equatorial plane (inclination angle
$=90^{\circ}$) as opposed to the pole [52]. And since this feature is more
compact in time-frequency as compared to the f-mode PNS oscillation track,
BayesWave can recover the equatorial signals better than the polar ones.

Figure 3: The accuracy of the BayesWave recovered waveform as a function of
the source distance (and explosion model) with $1\sigma$ uncertainty error
bars. The dashed black line indicates the 50% accuracy threshold. Also shown
for reference are three benchmark distances: Betelgeuse, the Milky Way center,
and the center of the Large Magellanic Cloud. The distances at which the
waveform is accurately recovered by BayesWave and passes the 50% threshold
varies by almost two orders of magnitude between the different waveform
models. For the low mass and relatively low explosion energy models of m20-3Dp
and s9 this occurs at $\sim 1~{}\mathrm{kpc}$, for the typical neutrino-driven
explosion model of s18 this is closer to the Milky Way center, and for the
high explosion energy models with rapidly rotating progenitors the recoverable
distances are up to the LMC at $\sim 50~{}\mathrm{kpc}$.

Following the approach in Abbott  _et al._ [10], to find out how far away a
supernova source can be for BayesWave to be able to detect and reconstruct the
signal in Advanced LIGO-Virgo, we show the reconstructed overlap accuracy for
the injections analyzed plotted against the source distance in Fig. 3. Since
each waveform model is based on very different progenitor stars and uses
different modeling techniques, the total amount of energy released in the
explosion, and thus in GWs, varies over a few orders of magnitude. The source
distance required to have a high amplitude GW signal reach the LIGO-Virgo
detectors is then significantly dependent on the waveform model considered (as
was previously found [18, 19, 10]). Table 1 also summarizes the source
distances at which the BayesWave reconstructed signal has an overlap accuracy
of $\sim 0.5$ for each waveform model, showing an almost two orders of
magnitude difference between the low mass neutrino-driven explosion of the s9
model and the magnetorotationally driven explosion of the 35OC-RO model.

###  III.2 Optimization gains

Figure 4: The gain in the network overlap of the recovered waveform when using
the optimized BayesWave run settings as compared to the initial un-optimized
runs, plotted against the injected signal SNR. The scatter plot on the left
shows the gains for each individual injection-reconstruction, along with the
$1\sigma$ uncertainty error bars on the gain. The gains are either positive or
consistent with zero. The plot on the right summarizes this information
showing the median overlap gain (solid lines) and the maximum gains observed
(dashed lines) near a given SNR. The median gains are on the order of $\sim
10\%$ in the low SNR regime of $\sim 15-20$, and consistent with zero for SNR
$\gtrsim 30$. The maximum observed gains can be quite large and significant,
reaching an increase in overlap accuracy of about 40% for the m20-3Dp and m39
models, and more than 20% for the other three models.

In order to quantify the effects of our BayesWave optimization efforts, and to
see in which cases they make the biggest impact, in Fig. 4 we plot the gain in
reconstruction accuracy of the waveform between the initial and optimized
BayesWave runs against the network SNR of the injected signal. The gains are
most pronounced in the low SNR regime of $\sim 15-20$, where they are
typically on the order of $\sim 10\%$. However, the maximum gains observed
across our study can be very high, from $\sim 20\%$ for the 35OC-RO model to
$\sim 40\%$ for the m20-3Dp and m39 models. This SNR regime corresponds to the
transition between cases where the signal is too low in amplitude for
BayesWave to distinguish much of it from the detector noise, and cases where
the signal is loud enough that BayesWave is starting to confidently recover
most of the waveform. Thus the optimizations are most impactful for the
marginal cases, and have no significant impact for signals with
$\mathrm{SNR}\gtrsim 30$.

Figure 5: The supernova waveform signals and their reconstructions in the time
domain, showing the comparison between the initial BayesWave reconstructions
and the optimized version. Plotted in darkest purple is the median
reconstructed waveform, while the lighter shaded bands are the 50% and 90%
credible intervals. These are overlaid on top of the true injected signal,
plotted in teal. To get a sense of how this signal would look like in one of
the LIGO detectors, the combined signal plus detector noise is shown in gray
in the background. Out of the 100 injections analyzed for each waveform model,
one representative case in the low network SNR regime ($\sim 15-20$) is shown.
This is the SNR range where the optimization leads to the highest gains in
reconstruction accuracy, as seen in Fig. 4. We can see that the gains in
reconstruction can generally be attributed to BayesWave’s ability to more
confidently and accurately capture the high-amplitude features. Note that the
time-scale for the m20-3Dp and s9 waveforms is shorter than the rest ($\sim
0.5$ s compared to $\sim 0.8$ s).

A more qualitative picture of how these optimizations manifest is shown in a
comparison of the time series reconstructions of the waveforms in Fig. 5. For
signals with $\mathrm{SNR}\sim 15-20$ we choose the case which has the median
gain in overlap within that SNR range as a representative example for each
model, and show the injected and BayesWave reconstructed signals in one of the
LIGO detectors (Hanford and Livingston have the same expected design
sensitivity [12], so they are qualitatively interchangeable in our study). We
can see that the gains in reconstruction for these examples can be attributed
to a combination of BayesWave being able to: i) find the highest amplitude
part of the signal when it wasn’t able to recover anything before (35OC-RO),
ii) recover the same part of the signal it was before but more accurately (m39
and s9), and iii) recover additional high-amplitude parts of the signal
compared to only the highest amplitude portion before (s18 and m20-3Dp).

###  III.3 Prospects for a Milky Way source distribution

Figure 6: The histogram distribution of the optimized BayesWave reconstructed
network overlap for each supernova model, for a population of injections with
extrinsic parameters modeled according to a realistic distribution of stars in
the Milky Way. We see that for such a distribution, we do not expect to be
able to recover the GW waveform with BayesWave from low energy m20-3Dp- or
s9-like supernova models. For more typical neutrino-driven explosion models
like s18, BayesWave can recover a non-zero overlap in many cases, but high
accuracy reconstructions with overlap $>0.5$ occur only in $\sim 15\%$ of
cases. For stronger explosions in the Galaxy from rapidly rotating progenitors
(m39, 35OC-RO), almost all of them can be expected to be confidently and
accurately recovered.

We show the prospects for accurate recovery of each model waveform using a
realistic distribution of CCSN sources within the Milky Way galaxy, modeled
according to its stellar distribution, in Fig. 6. This figure shows the
histogram distribution of the network overlap in the optimized BayesWave runs
for Milky Way distributed signals.

We see that for the low explosion energy models of m20-3Dp and s9 we can not
expect to reconstruct the GW signal in almost all cases (the overlap is non-
zero for $<5\%$ of sources), consistent with what Abbott  _et al._ [10] find.
This is primarily because the stellar distribution in the Galaxy is most
concentrated towards the central bulge near 8 kpc, while we see in Fig. 3 that
these models are not confidently recovered beyond $\sim 1-2$ kpc.

For the s18 model we have a somewhat broader distribution, with a non-zero
overlap in $\sim 65\%$ of cases, and a confident reconstruction with overlap
$>0.5$ for $\sim 15\%$ of the sources. The most likely (non-zero) overlap
value, coincident with sources at the Galactic center, is $\sim 0.3$. Due to
the fact that our optimizations were largely effective at lower SNRs, and this
SNR regime is coincident with near the Galactic center for the s18 model, it
is also the only model for which we see a measurable increase in the
percentage of cases that are recovered between the un-optimized and optimized
runs: a 5% increase from 11% to 16% for confident reconstructions, and a 15%
increase from 52% to 67% for non-zero overlap.

For the two high explosion energy models we see that $>95\%$ of cases have
overlap $>0.5$. The most likely overlap, coincident with sources at the center
of the Milky Way, is $\sim 0.85$ for m39 and $\sim 0.95$ for 35OC-RO. In Fig.
7 we also show spectrograms for the most likely reconstruction for sources at
the Galactic center for each waveform model.

##  IV Discussion

###  IV.1 Reconstruction accuracy model dependence

The duration of the GW emission plays a central role in influencing the
quality of the BayesWave reconstruction. Since the SNR is an integrated
measure of the signal power across the analyzed segment, when comparing
different model injections at a fixed SNR, their total power is the same.
However, the injected signals in Fig. 1 show that significant GW emissions
last for about 0.5 seconds for s18 and about 0.7 seconds for the 35OC-RO
model, making them more spread out in time compared to the $\sim 0.3$ seconds
emissions from m20-3Dp and s9. On average, this effectively distributes the
signal power so that analyzing a similar 0.3 second segment in the longer
models would have about half the power as compared to the shorter ones. Since
BayesWave places wavelets in time-frequency trying to capture this localized
power wherever it confidently can [20], the wavelet basis better captures
signals that have their excess power concentrated in a shorter time (following
the same reasoning why high mass compact binary coalescences are better
reconstructed compared to low mass ones in Ghonge  _et al._ [46]). For the m39
model, even though it is one of the longer duration models with significant GW
emissions lasting about 0.6 seconds, depending on the source orientation it
often produces much of that excess power in the short duration early
rotational bounce signal (seen in Fig. 1), which BayesWave easily recovers.

Compactness in frequency of the GW emission further explains differences in
reconstruction accuracy. For example, in the m39 model the f-mode PNS
oscillation track is very narrow in time-frequency. Compare this to the s18
model where no high amplitude prompt convection mechanism contributes and the
signal energy is distributed throughout a relatively broader PNS g-mode
oscillation track (as well as what seems to be a low-frequency persistent
background component), making it harder for BayesWave to recover. The other
three models are similarly more spread out in frequency than m39, negatively
affecting the ability of the compact Morlet-Gabor wavelets to capture this
more broadband power.

At lower SNRs these differences in the reconstruction accuracy are more
pronounced because BayesWave needs the localized SNR in each segment of the
signal to be above a certain threshold to justify placing a wavelet there
(since it takes a Bayesian approach, the increase in signal likelihood must
exceed the Occam factor penalty for increasing the signal model complexity
[20, 60, 61]). Capturing localized excess is then only possible close to where
the signal amplitude peaks for lower SNR signals (as seen in Fig. 5), which
varies significantly from model to model. On the other hand, for high SNR
signals the signal amplitude is high enough to be distinguished from the noise
across the whole waveform, and so BayesWave can place wavelets across the
signal relatively unaffected by the specific distribution of the power.

In future work we plan to do a deeper investigation on the possibility of
distinguishing between different model features and extracting physical
parameters of the progenitor using machine learning techniques, based on the
waveform reconstructions that BayesWave outputs.

###  IV.2 Reconstruction distances in context

In Fig. 3 along with presenting the distance dependence of the BayesWave
reconstructions for the optimized runs, we also include distances to some
common benchmarks for comparison, similar to Figure 7 in Abbott  _et al._
[10]. Betelgeuse is the nearest red supergiant to Earth, expected to end its
life in a CCSN [62]. The Milky Way center is the region with the highest
density of massive stars in the Galaxy, and so is the most likely source for
the next Galactic supernova. The Large Magellanic Cloud is the host galaxy to
supernova 1987A, one of the most recent and closest CCSN observed, and the
only one also detected with neutrinos [7, 6, 8]. We see that for the low
explosion energy models of s9 and m20-3Dp, the GW waveform is very well
recovered by BayesWave at distances close to Betelgeuse, but the accuracy
falls below the 50% threshold very quickly beyond $\sim 1-2$ kpc. The typical
neutrino driven explosion for s18 is recovered well at galactic distances, up
to the center of the Milky Way, but can not be recovered for sources beyond
our galaxy. For the very highest explosion energy models of m39 and 35OC-RO
however, the BayesWave reconstruction has accuracies of $\sim 90\%$ close to
the Milky Way center, and is well recovered even up to the LMC. If Betelgeuse
goes supernova during the advanced detector era then we can see that BayesWave
will accurately recover and reconstruct almost all of the signal, regardless
of waveform model (even if the explosion is not successful, as in the m20-3Dp
model).

These distances are also significantly larger than the detection distances
found for the same set of waveform models studied with BayesWave in Abbott
_et al._ [10], as shown in Figure 7 of their paper. This is expected as the
LVK study is based on the actual detector noise curves recorded for the LIGO
and Virgo detectors during the third observing run, which is roughly a factor
of 1.5 less sensitive than the design Advanced LIGO-Virgo configuration that
we use in this study, in anticipation of the fourth observing run [14]. Note
that there is also a difference in the distance measure analyzed: the LVK
study quotes the distance at which 50% (and 10%) of the total injections are
detected, whereas in our study we look at the distance at which the recovered
waveform accuracy is on average 50%. We also find that the BayesWave
reconstruction overlap accuracies at given distances (and SNRs) for the
advanced LIGO-Virgo detectors are broadly consistent with those reported by
Szczepańczyk  _et al._ [19] using the coherent WaveBurst algorithm.

As was done for the third observing run sensitivity in Abbott  _et al._ [10],
for a realistic distribution of CCSN progenitors within the Milky Way we have
also quantified the prospects for BayesWave to confidently reconstruct the
emitted GW signal for the expected sensitivities in the fourth observing run
(Fig. 6). This also demonstrates how the distribution of sources within the
Galaxy modulates the detectability of each supernova model, as most stars are
concentrated near the center of the Galaxy $\sim 8$ kpc away from the Sun.
Should the next Galactic CCSN be a low energy or failed explosion, we do not
anticipate being able to detect it in Advanced LIGO-Virgo with BayesWave
(unless we are very lucky and the source is $\lesssim 1$ kpc away). If it is a
typical neutrino driven explosion like the s18 model however, we expect that
there is a $\sim 65\%$ chance BayesWave will recover a part of the GW signal,
and a $\sim 15\%$ chance it will be confidently reconstructed. Finally, if the
explosion follows from a rapidly rotating progenitor, then BayesWave will
almost certainly recover the GW features confidently ($>95\%$ probability).
However, such rapid rotation is only expected to occur in $\sim 1\%$ of all
CCSN progenitors [23].

###  IV.3 Limits to BayesWave reconstructions

The work by Pannarale  _et al._ [63] characterizes BayesWave’s performance in
general from a theoretical approach, and suggests a formulation for the
maximum possible overlap (match) in Equation 8 of their paper. We test this
prediction against our CCSN reconstructions for all five of the waveform
models studied, and plot the difference between the two for our optimized runs
in Fig. 8 (included in the appendix). What we see is similar to what Pannarale
_et al._ [63] find, which is that more complicated and longer waveforms (such
as the 35OC-RO in our study) show a larger disparity with the predictions and
have a harder time reaching that theoretical maximum as compared to waveforms
that are more compact in time-frequency (such as the m39 waveform in our
study). This also explains why BayesWave has a harder time recovering the
signal accurately from all of the supernova models in our study when compared
to it’s performance in recovering the signal from coalescing binary black
holes as studied in Pannarale  _et al._ [63].

The disparity between the predicted overlaps and the CCSN overlaps in our
study also show an SNR dependence, where the predictions overestimate much
more at lower SNRs than at higher. This happens because Pannarale  _et al._
[63] state Equation 8 is valid in the limit that the squared SNR recovered per
pixel (wavelet) is much greater than unity, which is a requirement that holds
true for reconstruction of high SNR signals in our study but becomes more
tenuous for lower SNR signal reconstructions. Theoretically we are expecting
that there should be some useful information, but practically at such SNRs the
signal is buried in Gaussian noise and BayesWave has a very hard time
extracting any part of it, giving very little or no information.

We also consider the limits of BayesWave to understand why our run
optimization efforts work well at relatively lower SNRs. When the signals are
not very loud, BayesWave’s certainty in determining the time delay of certain
signal features between the three different detectors decreases, and so the
reconstructed sky localization is not accurate, with the 90% confidence sky
area not constrained to better than $\sim 50~{}\mathrm{deg^{2}}$ for SNR
$\lesssim 30$. This increase in the posterior distribution for the source sky
position parameter is associated with a decrease in the calculated likelihood
when placing a wavelet to capture that signal feature, and so in the RJMCMC
BayesWave places a wavelet to capture that power less often, resulting in a
lower median overlap. When the true sky position is known and fixed however,
the parameter space is significantly reduced, and so BayesWave places wavelets
at the corresponding signal feature locations much more confidently and thus
much more often, increasing the median overlap.

The increase in overlap due to a higher wavelet quality factor is a direct
result of the fact that supernova waveforms generally have high amplitude
features that are spread out in time, and so longer duration wavelets are
required to capture the power not just at the precise time where the amplitude
is large, but also in its immediate vicinity (as illustrated in Fig. 5). At
higher SNRs this makes little difference because instead of having to capture
the excess power using only one wavelet, the signal is loud enough that
BayesWave can confidently place many wavelets to capture the same power.

In our optimization efforts during our study we had also considered using
frequency-evolving ‘chirplets’ [57] instead of the fixed-frequency wavelets,
i.e. increasing the allowed waveform complexity in the bid to capture the
nuances in the signal morphologies more accurately. However, this was not
included in the final optimized BayesWave runs as it did not yield positive
gains for all the different waveform models. While using chirplets increased
the overlap accuracy by $\sim 5\%$ for low SNR signals for the more broadband
35OC-RO model, it also resulted in a slight net negative overlap at low SNRs
for the m20-3Dp and m39 models (while having no measurable impact on the other
two models). Combined with the fact that the computing time is almost twice as
long for the chirplet model as compared to the wavelet, we recommend that the
wavelet model be used for supernova analysis unless the signal is known to be
broadband from other analyses.

##  V Conclusion

Core-collapse supernova explosion models predict the production of
gravitational waves within the first second after core-collapse, exhibiting
dominant emission features with frequencies that lie in the most sensitive
regime of current interferometric ground based detectors. Galactic CCSNe
represent a rare but promising astrophysical source that could be detected by
the Advanced LIGO and Advanced Virgo detectors in one of their upcoming
observing runs.

In order to prepare for such a detection, in this paper we study how well the
BayesWave algorithm can reconstruct the GW signal emitted by a range of
different explosion models with different waveform morphologies, if such a
signal is detected by a design sensitivity LIGO-Virgo network configuration.
We also optimize the BayesWave run settings for CCSN, focusing on maximizing
the reconstruction overlap accuracy, and provide a set of recommendations for
the run settings for analysis on a real candidate signal.

We find that BayesWave’s reconstruction performance varies depending on the
complexity of the waveform, with models that exhibit features that are more
compact in time-frequency being better recovered than models that have their
power distributed over a longer time and have broadband emission, consistent
with previous work [19]. That said, BayesWave can confidently recover the GW
signal from a range of supernova explosion models in Advanced LIGO-Virgo for
network $\mathrm{SNR\gtrsim 30}$, with maximum accuracies of $\sim 90\%$ for
SNR $\sim 100$, as shown in Fig. 2.

The corresponding distances, as seen in Fig. 3, are from as low as
$\mathrm{\sim 1~{}kpc}$ for low mass neutrino driven explosions, to up to
$\mathrm{\sim 50~{}kpc}$ for high mass magneto-rotationally driven explosions.
We put this into context of what percentage of CCSN within our Galaxy we could
confidently recover with BayesWave by also studying the reconstructions for a
realistic distribution of stellar sources in the Miky Way (Fig. 6). We find
that should the next Galactic CCSN be a low mass or failed explosion,
BayesWave is likely not to detect it. However, there is a $\sim 15\%$ chance
the source is close enough that for typical neutrino driven explosions like
the s18 model, BayesWave will be able to confidently reconstruct the GW
emission. Should a high energy CCSN from a rapidly rotating progenitor be the
source, BayesWave will confidently recover the GW signal in $>95\%$ of cases.

A key result of our work is that optimizing the BayesWave run settings
increases the reconstructed accuracy significantly for low SNR signals, with
gains of up to $\sim 20-40\%$ for lower SNR signals, and typically $\sim 10\%$
in the SNR $\sim 15-20$ regime, as shown in Fig. 4. These gains are based on:
i) exploiting the multi-messenger nature of CCSN so that the sky location is
known from the electromagnetic and neutrino counterpart observations, allowing
BayesWave to more confidently recover waveform features, and ii) increasing
the allowed wavelet quality factor (to $Q_{max}=100$) so that high amplitude
features that are more spread out in time for supernova signals are more
accurately reconstructed. For typical neutrino-driven explosions like the s18
model, this also translates into being able to recover the signal from $\sim
15\%$ more sources in our Galaxy.

Our results show that if there is a CCSN signal detection in Advanced LIGO-
Virgo, for loud enough signals the BayesWave algorithm will be able to
accurately reconstruct the dominant GW emission features, regardless of the
specific model. This will help us distinguish between different supernova
explosion models and mechanisms, and allow crucial insight into the dynamics
of the very first second of the explosion.

###### Acknowledgements.

The authors gratefully acknowledge Jade Powell, Meg Millhouse, Katerina
Chatziioannou, Tyson Littenberg, Bence Bécsy, Sudarshan Ghonge and Evan Goetz
for useful discussions related to this work. The manuscript has benefited from
internal LVK collaboration review under the document ID P2100404. This
material is based upon work supported by NSF’s LIGO Laboratory which is a
major facility fully funded by the National Science Foundation. The authors
gratefully acknowledge the support of the NSF for provision of computational
resources. This research is funded by an NSERC Discovery Grant. GD is
supported through the iBOF-project BOF20/IBF/124.

*

## Appendix A

To supplement the discussion in the main text, in Fig. 7 we show spectrograms
for the most likely BayesWave reconstruction for sources near the Galactic
center for each waveform model. For each model we consider all the injections
from the Milky Way distributed population, in the distance range 7-9 kpc, and
then select the one which has the median overlap value within that sample as a
representative example.

Figure 7: The combined Q-transform spectrograms for the three detectors for
the injected signal waveform (top row), the data that BayesWave analyzes
(injected signal in simulated gaussian detector noise, middle row), and the
BayesWave recovered waveform (bottom row). Out of the 150 Milky Way
distributed injections for each waveform model, one representative case at the
Galactic center at a distance of $\sim 8$ kpc is shown. The m20-3Dp and s9
waveforms are not reconstructed at all, the s18 model has an overlap
reconstruction accuracy of $\sim 30\%$, the m39 model has accuracy of $\sim
85\%$, and the 35OC-RO model has accuracy of $\sim 95\%$. The large
differences in accuracy of reconstruction is due to the difference in GW
amplitude and energy between models.

We also test the predictions made in the work by Pannarale  _et al._ [63]
characterizing BayesWave’s performance from a theoretical approach, against
our CCSN reconstructions for all five of the waveform models studied, and the
difference between the two for our optimized runs is shown in Fig. 8.

Figure 8: Comparison of the theoretical predictions of the BayesWave overlap
according to Equation 8 in Pannarale  _et al._ [63], and the measured overlap
in our study. Our measurements are worse than the predicted values (as
expected), and quantify how the prediction breaks down for longer duration
signals like CCSNe. The mismatch is larger for lower SNR signals, and
generally larger for models that have more complex features in time-frequency
such as the 35OC-RO model.

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