Detecting Gravitational Wave Memory in the Next Galactic Core-Collapse Supernova

Detecting Gravitational Wave Memory in the Next Galactic Core-Collapse Supernova
Colter J. Richardson,1, ∗
Haakon Andresen,2
Anthony Mezzacappa,1
Michele Zanolin,3
Michael G. Benjamin,1
Pedro Marronetti,4
Eric J. Lentz,1, 5
and Marek J. Szczepańczyk6
1
Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA
2
The Oskar Klein Centre, Department of Astronomy, AlbaNova, SE-106 91 Stockholm, Sweden
3
Embry-Riddle Aeronautical University, 3700 Willow Creek Road, Prescott, Arizona 86301, USA
4
Physics Division, National Science Foundation, Alexandria, Virginia 22314, USA
5
Physics Division, Oak Ridge National Laboratory,
P.O. Box 2008, Oak Ridge, Tennessee 37831-6354, USA
6
Faculty of Physics, University of Warsaw, Ludwika Pasteura 5, 02-093 Warszawa, Poland
(Dated: April 3, 2024)
We present an approach to detecting (linear) gravitational wave memory in a Galactic core-collapse
supernova using current interferometers. Gravitational wave memory is an important prediction of
general relativity that has yet to be confirmed. Our approach uses a combination of Linear Prediction
Filtering and Matched-Filtering. We present the results of our approach on data from core-collapse
supernova simulations that span a range of progenitor mass and metallicity. We are able to detect
gravitational wave memory out to 10 kpc. We also present the False Alarm Probabilities assuming
an On-Source Window compatible with the presence of a neutrino detection.
Introduction The deaths of massive stars in core-
collapse supernovae (CCSNe) are promising sources of
gravitational waves (GWs). Stellar core collapse, core
bounce at super-nuclear densities, fluid instabilities in
the newly-formed proto-neutron star and in the cavity
between the proto-neutron star surface and the super-
nova shock wave, believed to be vital to the explosive
central engine, as well as explosion itself and anisotropic
neutrino emission, are all expected to generate GWs [1–
3]. The fluid instabilities, as well as the turbulence they
induce, are expected to excite GW emission at frequen-
cies between 50 Hz and a few kHz [4–64].
Detection strategies for CCSN GWs until now relied
on excess-energy methods because the stochastic nature
of the signals impeded the use of matched filtering. How-
ever, it has been pointed out recently that matched filter-
ing alongside multi-messenger observations can improve
the detection efficiency of nearby events [55]. Besides
emissions above 50 Hz, a slowly evolving signal compo-
nent, associated with the GW (linear) memory, is ex-
pected below a few 10’s of Hz [6, 10, 11, 13, 15, 16, 18,
21, 28, 36, 41, 58–60, 65–73]. The memory in a CCSN
stems from asymmetric emission of neutrinos during the
explosion and the non-spherical expansion of the super-
nova blast wave. Although this low-frequency compo-
nent contributes minimally to the total energy emitted,
its amplitude can be several times larger than that of
the emission above 50 Hz. Strictly speaking, the mem-
ory only refers to a constant offset in the strain after the
GW pulse has passed. However, in our discussion of the
memory we include the secular ramp-up to the satura-
tion value. In terms of detectability, the GW memory
from CCSNe has been largely overlooked due to the lim-
ited sensitivity of current GW detectors below 10 Hz.
Moreover, even if the peak of the frequency band of the
memory (including the secular ramp-up) is below 10 Hz,
there may be a detectable strain (or energy) present at
and above 10 Hz.
In this Letter, we demonstrate that the slow and reg-
ular time evolution of the memory is uniquely suited
to matched-filter techniques. We show how matched-
filtering can be utilized to detect the GW memory from
CCSNe in current interferometers. Observing the mem-
ory, or signs of it, would confirm an important prediction
of general relativity that has yet to be confirmed.
Models We study the memory from three state-of-
the-art, three-dimensional core-collapse supernova sim-
ulations. The simulations were carried out with the
Chimera [74] code, initiated from three non-rotating
progenitors with zero-age main sequence masses of 9.6,
15, and 25 Solar masses, and zero and Solar metallicity
[58]. The models are labeled by a “D” (for Chimera
D-series simulations) followed by the mass of the progen-
itor from which the simulation in the series was initiated.
Rapid shock expansion sets in at ∼125 ms, ∼250 ms, and
∼500 ms for D9.6, D25, and D15, respectively.
Gravitational Wave Signals The solid lines in the top
panel of Fig. 1 show the plus polarization mode of the
combined (matter and neutrino) GW strains (h+) from
our three models: the blue, orange, and green curves
represent D25, D15, and D9.6, respectively. Within a
spherical coordinate system centered on the simulations,
the models are observed at randomly-chosen directions:
D9.6 at ϕ = −35◦
, θ = 90◦
; D15 at ϕ = 60◦
, θ = 70◦
;
and D25 at ϕ = 35◦
, θ = 0◦
. The GW signals of all
three models show the slow ramp-up to a non-zero strain
value that is characteristic of the memory. (Note, the
D9.6 model is representative of low-mass CCSNe, which
typically have low ejecta asymmetry.)
arXiv:2404.02131v1
[astro-ph.HE]
2
Apr
2024

2
20
10
0
10
20
Dh
+
[cm]
0.0 0.2 0.4 0.6 0.8 1.0
Time [s]
10
5
0
5
10
Filtered
Dh
+
[cm]
D25
D15
D9.6 ×10
Fit
Fit
Fit
FIG. 1. Gravitational wave signals from all three models.
Solid lines show the signals from the simulations. Dashed
lines show the fit of the waveforms to a logistic function with
a tapering (see Eq. 1 and Table I). Tapering reduces the high-
frequency noise induced by the abrupt end of the simulations.
We taper with a frequency of 1/10 Hz. The top panel displays
unfiltered signals; the bottom panel shows the same signals
with an 8 Hz high-pass Butterworth filter applied. The D9.6
signal is scaled up by a factor of 10.
Analytical Fit To isolate the memory, we fit the signal
to a tapered logistic function defined as follows:
hfit
(t) =
L
1 + e−k(t−t0)
(1 − H(t − ts))
+
L
2
1 + cos (2πft(t − ts))

H(t − ts). (1)
Here t0 is the center of the rise time, k is the inverse of
the typical rise time, L is the memory saturation value,
ts is the time of saturation (or the end time of the simula-
tion), and H(t) is the Heaviside step function. The taper-
ing is characterized by the tapering frequency, ft, which
we chose based on the noise characteristics of the LIGO-
Virgo-Kagra (LVK) detectors. Current ground-based de-
tectors have a sharp increase in their characteristic noise
at approximately 10 Hz. As long as the tapering is longer
than 0.1 s, the small amount of energy added by the ta-
pering is negligible compared to the noise. In reality the
signals are expected to saturate at some non-zero value,
but we taper the signals to avoid inducing high-frequency
noise in our Fourier analysis. The dashed curves in the
top panel of Fig. 1 show the fits. See Table 1 for the fit
parameters. For the D9.6 and D15 models, we start the
tapering right as the simulations end. On the other hand,
we extrapolate the D25 signal until it saturates (details
regarding extrapolating the signals can be found in [72]).
We extrapolate the signal from D25 because applying the
tapering directly after the end of the simulation led to a
discontinuity in the signal’s derivative. The extrapola-
tion is conservative and is not instrumental to detecting
the actual signal.
TABLE I. The parameters used for the fit and tapering of
the GW signals from the simulations (see Eq.1). Each row
corresponds to a particular model.
Model t0 [s] L [cm] k [Hz] ts [s] ft [Hz]
D9.6 0.28 -1.30 29.45 0.300 0.1
D15 0.60 17.73 22.60 0.7414 0.1
D25 0.41 24.23 18.73 0.472 0.1
GWOSC
Noise
Signal
from
Simulation
Tapering Fit
Extension
Train LPF
(Without
Signal)
+
Apply LPF
Subtract Predicted Data
Correlation
Matched-
filter
Template
FIG. 2. Flow chart outlining the procedure presented in this
Letter. The end node labeled “Correlation” corresponds to
the final result of our analysis and is what we show in Fig. 4.
Matched Filtering The procedure implemented in this
Letter, starting from the GWOSC noise data and the
waveforms predicted by our simulations, is outlined in
Fig.2.
We inject the tapered signals into a sample of LVK
data obtained from the Gravitational Wave Open Science
Center (GWOSC) [75]—specifically, a 4096 s segment of
data from the O3b run of the Livingston and Hanford de-
tectors, with an initial GPS time of 1262178304. Due to
the nature of the publicly available data from GWOSC,
which has a high-pass filter already applied to it, after
injection we apply to the strain a high-pass Butterworth
filter with a cut-off of 8 Hz. The second panel of Fig. 1
shows the signals and the fits after the filter has been
applied. For all of the models, the secular ramp-up is
reduced, but not erased.
After injecting the signal, we train a Linear Prediction
Filter (LPF) [76, 77] with 16384 trained parameters on a
2048 s segment of the data that does not contain the sig-
nal. We then subtract the portion of the signal predicted

3
10 26
10 25
10 24
10 23
10 22
10 21
10 20
ASD
[Hz
1/2
]
10 20 40 60 80 100
Frequency [Hz]
10 26
10 25
10 24
10 23
10 22
10 21
ASD
[Hz
1/2
]
Noise D25 D15 D9.6
FIG. 3. Top: The Amplitude Spectral Density of: 1) the
noise data from the Livingston detector (gray) and 2) of the
three signals (colored lines). Bottom: The Amplitude Spec-
tral Density of: 1) the detector noise with the part predicted
by the LPF subtracted (gray) and 2) the whitened signals.
Signals were scaled to a source distance of 1kpc. (colored
lines)
by the LPF and define
Ŝ = S − SLPF, (2)
where S is the strain from the detector, including the
injected signal, and SLPF is the output of the LPF. In
Fig. 3 we show the amplitude spectral density (ASD) of
the noise and the signals. The top (bottom) panel shows
the data before (after) applying the LPF. The data in the
bottom panel are defined in Eq. 2, and, for the purpose of
the plots only, we assume a source distance of 1 kpc. The
filtered data do not represent the actual detector strain,
but by predicting, and then removing the predicted por-
tion of the noise, we are able to better locate the memory
component. The LPF below 200 Hz improves detectabil-
ity metrics (like the SNR or cross-correlations with tem-
plates) for the memory by several orders of magnitude.
It does affect the signal as well, but to a far lesser degree.
Therefore, Ŝ is a better starting point for our matched
filtering than the actual strain data.
We then calculate the discrete correlation—i.e., the
match—between Ŝ and hfit
, which is defined as follows:
⟨Ŝ, hfit
⟩(tn) =
X
m
Ŝ(tm)hfit
(tm−n). (3)
Here we sum over all tm in our data, and tn refers to the
n-th sample time. We simulate a two-detector network
by evaluating
⟨Ŝ, hfit
⟩N
= ⟨Ŝ, hfit
⟩H
· ⟨Ŝ, hfit
⟩L
, (4)
where ⟨Ŝ, hfit
⟩H
and ⟨Ŝ, hfit
⟩L
represent ⟨Ŝ, hfit
⟩ calcu-
lated with data from the LIGO Hanford and LIGO Liv-
ingston detectors, respectively. A detector network re-
duces the number of false alarms by enabling coincident
analysis.
For a Galactic CCSN, timing information based on a
detected neutrino event would enable us to significantly
narrow our search window. Assuming that the on-source
window can be reduced to two seconds, we split the 4096
s window into two-second segments and define the FAP as
the ratio between the number of segments with triggers
above some threshold versus the total number. We leave
a more complete exploration of the impact of detector
networks to future work.
When performing matched filtering, the sharp edges
of noise segments at the beginning and the end of a
data stream can lead to large and nonphysical correla-
tions (edge effects). Therefore, we apply a window to the
noise data before our matched search. We used a Tukey
window of the same length as our noise, with a shape
parameter α = 0.2.
Lastly, we note that the signals are injected at a
randomly-chosen time and that choosing a different injec-
tion time does not change the general conclusions of this
work. However, for the noise data we use, there are three
clusters where the amplitude of the correlation, which is
at its base an inner product, is large, at roughly 750 s,
894 s, and 1350 s. Injecting the signals near the noise
clusters slightly decreases the efficiency of our approach.
For a source distance of 10 kpc, the False Alarm Proba-
bility (FAP) increases by a few percent, but signals can
still be clearly identified.
Results The matched-filter results for all three mod-
els, with a source distance of 10 kpc and using a two-
detector network at O3b sensitivity (for the expected
design sensitivity of O5, we expect at least a 50% in-
crease), are presented in Fig. 4. The top, middle, and
bottom panels correspond to models D9.6, D15, and D25,
respectively. The signals were injected at one randomly-
chosen time, indicated by gray dashed lines. The signals
from D15 and D25 are identifiable, but the weak sig-
nal of D9.6 is not visible in the detector noise. In addi-
tion to the signal, our matched-filter approach picks out
several other noise events (for example around 750 and
2656 s, see Fig.4). However, the correlations between
the filtered template and noise events are smaller than

4
the correlation between the template and the actual sig-
nal (except for D9.6). To calculate a FAP we select a
match threshold and only consider events with a cor-
relation, ⟨Ŝ, hfit
⟩N
(tn)/⟨Ŝ, hfit
⟩N
(tinj), higher than the
chosen threshold as potential detection candidates. The
threshold is chosen to achieve a desired FAP (for brevity
we leave for future studies the discussion of varying the
template parameters while performing the matched fil-
tering, even if our tests indicate that this will have a
small impact on the FAP curves).
For the D15 signal injected at 10 kpc, the correlation
has two distinct peaks, one at the injection site and one
at a glitch. This glitch is not present in either the D9.6
or the D25 case, indicating that this noise event only cor-
relates with the D15 template. Using a threshold of 0.8,
we see that the signal is one of two triggers, resulting in
a FAP of 50%. While we apply a mask that removes the
categorized glitches (provided by the GWOSC), a more
in-depth analysis of the noise may remove these features.
Lowering the threshold to 0.5 results in six triggers, lead-
ing to a FAP of 83.33%. For the D25 signal, applying the
same 0.8 and 0.5 thresholds, we detect no false triggers
at the higher threshold and one false trigger at the lower
threshold, achieving FAPs of ≤ 1/2048 (0.05%) and 50%,
respectively.
Given a coincident neutrino detection (or a search with
a two-second temporal window), Fig. 5 shows how the
FAP depends on the chosen match threshold and the
distance to the source. The blue, orange, and green
curves correspond to models D25, D15, and D9.6, re-
spectively. Different markers indicate different distances:
1 kpc (dots), 10 kpc (triangles), and 100 kpc (squares).
At a distance of 1 kpc, the memory in the D15 and D25
models is detectable with a FAP less than 0.05% for any
match threshold, while for the D9.6 model, a large match
threshold is required in order to obtain similar results.
(N.B. Our results at O(1) kpc reflect the results we can
expect at O(10) kpc given next-generation detectors.) At
10 kpc, both D15 and D25 have a FAP of less than 5% for
a match threshold of 0.3. For D9.6, the FAP curve does
not change relative to the 1 kpc case. This occurs when,
at the injection distance, the signal becomes dominated
by the noise, rendering the quantity plotted in Fig. 4 in-
dependent of the distance. This would eventually happen
for the D15 and D25 models, as well, at some distance
above 100 kpc. At 100 kpc, a FAP for models D15 and
D25 below 5-10% is possible, but only at match thresh-
olds approaching 1.0. At a threshold of 1.0, the FAP for
both models is greater than 0.05%, which means at this
distance there are always triggers stronger than the in-
jected signal. Note, the FAP curves for the D9.6 model
and at large distances for models D15 and D25 (e.g., at
100 kpc) will vary with injection times, and will require
potentially larger match thresholds at those distances.
Conclusions In this Letter, we have shown that, given
the secular ramp-up of the linear GW memory in a
0.0
0.5
1.0 D9.6
0.0
0.5
1.0
S,
h
fit
N
(t
n
)/
S,
h
fit
N
(t
inj
)
D15
0 1000 2000 3000 4000
Time [s]
0.0
0.5
1.0 D25
FIG. 4. The two-detector correlation (Eq. 4) between the fil-
tered templates (bottom panel of Fig 1) and the whitened de-
tector data. The top, middle, and bottom panels correspond
to signals from the D9.6, D15, and D25 models, respectively,
all at a distance of 10 kpc. The vertical gray dashed lines
indicate the location of injection.
0.0 0.2 0.4 0.6 0.8 1.0
Match Threshold
0.0
0.2
0.4
0.6
0.8
1.0
False
Alarm
Probability
D25
D15
D9.6
1 kpc
10 kpc
100 kpc
FIG. 5. The False Alarm Probability as a function of the
correlation threshold used to classify a trigger. Different line
markers indicate different source distances, and the colors cor-
respond to different models.
CCSN and given the use of a Linear Prediction Filter,
a matched-template search (similar to the current detec-
tion strategy for binary mergers) can be performed to
detect the memory using current interferometers and for
the first time confirm an important prediction of general
relativity.
With a focus on detecting CCSNe, we have shown that,
in the absence of a multimessenger detection, our ap-
proach would be effective out to a distance of 10 kpc.
Of course, at these distances a multimessenger detection
is expected. The detection range afforded by our ap-
proach is perhaps best discussed in the context of next-

5
generation interferometers. For the Einstein Telescope
and Cosmic Explorer, the combination of their reduced
noise floor across all frequencies, which is projected to
be approximately one order of magnitude in magnitude
across the sensitivity band, and the reduction of the low-
frequency wall from 10 Hz to below 10 Hz, may enable
the detection range of CCSNe—specifically, through the
detection of GW memory—out to Mpc distance scales,
necessarily without a concurrent neutrino detection.
While this does not impact our main conclusions, in fu-
ture publications we will discuss the variability of the re-
sults using different saturation levels for the memory (as
a detection and parameter estimation template search).
Recent findings suggest that we can have saturation val-
ues up to 60 times larger than our signals [60], and even
larger for GRBs [78], potentially extending the detection
range to several Mpc in current detectors. Additionally,
it has been found that asymmetric neutrino emission can
lead to large neutron star kicks, with potential amplifi-
cation through neutrino flavor conversion [79, 80]. Such
large, neutrino-induced kicks would imply a substantial
GW memory, detectable at far greater distances than our
current signals suggest.
Acknowledgements H.A. is supported by the Swedish
Research Council (Project No. 2020-00452). A.M. ac-
knowledges support from the National Science Founda-
tion’s Gravitational Physics Theory Program through
grants PHY-1806692 and PHY 2110177. M.Z. is sup-
ported by the National Science Foundation Gravita-
tional Physics Experimental and Data Analysis Program
through award PHY-2110555. P.M. is supported by the
National Science Foundation through its employee IR/D
program.
This research has made use of data or software ob-
tained from the Gravitational Wave Open Science Center
(gwosc.org), a service of the LIGO Scientific Collabora-
tion, the Virgo Collaboration, and KAGRA. This ma-
terial is based upon work supported by NSF’s LIGO
Laboratory, which is a major facility fully funded by
the National Science Foundation, as well as the Sci-
ence and Technology Facilities Council (STFC) of the
United Kingdom, the Max-Planck-Society (MPS), and
the State of Niedersachsen/Germany for support of the
construction of Advanced LIGO and construction and
operation of the GEO600 detector. Additional support
for Advanced LIGO was provided by the Australian Re-
search Council. Virgo is funded, through the European
Gravitational Observatory (EGO), by the French Cen-
tre National de Recherche Scientifique (CNRS), the Ital-
ian Istituto Nazionale di Fisica Nucleare (INFN) and the
Dutch Nikhef, with contributions by institutions from
Belgium, Germany, Greece, Hungary, Ireland, Japan,
Monaco, Poland, Portugal, and Spain. KAGRA is sup-
ported by the Ministry of Education, Culture, Sports,
Science, and Technology (MEXT), the Japan Society for
the Promotion of Science (JSPS) in Japan, the National
Research Foundation (NRF) and Ministry of Science and
ICT (MSIT) in Korea, and the Academia Sinica (AS)
and National Science and Technology Council (NSTC)
in Taiwan.
∗
cricha80@vols.utk.edu
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