# The Gravitational-Wave Signature of Core-Collapse Supernovae

David Vartanyan  0000-0003-1938-9282<EMAIL_ADDRESS>Carnegie
Observatories, 813 Santa Barbara St, Pasadena, CA 91101, USA; NASA Hubble
Fellow  Adam Burrows  0000-0002-3099-5024 Department of Astrophysical
Sciences, 4 Ivy Lane, Princeton University, Princeton, NJ 08544, USA  Tianshu
Wang  0000-0002-0042-9873 Department of Astrophysical Sciences, 4 Ivy Lane,
Princeton University, Princeton, NJ 08544, USA  Matthew S.B. Coleman
0000-0001-5939-5957 Department of Astrophysical Sciences, 4 Ivy Lane,
Princeton University, Princeton, NJ 08544, USA  Department of Physics and
Engineering Physics, Stevens Institute of Technology, Castle Point on the
Hudson, Hoboken, NJ 07030, USA  Christopher J. White  0000-0002-3099-5024
Department of Astrophysical Sciences, 4 Ivy Lane, Princeton University,
Princeton, NJ 08544, USA  Center for Computational Astrophysics, Flatiron
Institute, 162 5th Ave., New York, NY 10010, USA

###### Abstract

We calculate the gravitational-wave (GW) signatures of detailed 3D core-
collapse supernova simulations spanning a range of massive stars. Most of the
simulations are carried out to times late enough to capture more than 95% of
the total GW emission. We find that the f/g-mode and f-mode of proto-neutron
star oscillations carry away most of the GW power. The f-mode frequency
inexorably rises as the proto-neutron star (PNS) core shrinks. We demonstrate
that the GW emission is excited mostly by accretion plumes onto the PNS that
energize modal oscillations and also high-frequency (“haze”) emission
correlated with the phase of violent accretion. The duration of the major
phase of emission varies with exploding progenitor and there is a strong
correlation between the total GW energy radiated and the compactness of the
progenitor. Moreover, the total GW emissions vary by as much as three orders
of magnitude from star to star. For black-hole formation, the GW signal tapers
off slowly and does not manifest the haze seen for the exploding models. For
such failed models, we also witness the emergence of a spiral shock motion
that modulates the GW emission at a frequency near $\sim$100 Hertz that slowly
increases as the stalled shock sinks. We find significant angular anisotropy
of both the high- and low-frequency (memory) GW emissions, though the latter
have very little power.

The theory of core-collapse supernova (CCSN) explosions has been developed
over the last six decades and is now a mature field at the interface of
gravitational, particle, nuclear, statistical, and numerical physics. The
majority of explosions are thought to be driven by neutrino heating behind a
shock wave formed upon the collision of the rebounding inner core with the
infalling mantle of the Chandrasekhar mass birthed in the center of stars more
massive than $\sim$8 M⊙[1, 2, 3, 4, 5]. After implosion ensues, this inner
white dwarf core, with a mass near $\sim$ 1.5M⊙ and a radius of only a few
thousand kilometers, requires only hundreds of milliseconds of implosion to
achieve a central density above that of the atomic nucleus. At this point, the
inner core stiffens, rebounds, and collides with the outer core, thereby
generating a shock wave that should be the supernova explosion in its infancy.
However, detailed 3D simulations [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
18, 19, 20, 21, 22] and physical understanding dictate that this shock
generally stalls into accretion, but is often reenergized into explosion by
heating via the neutrinos emerging from the hot, dense, accreting proto-
neutron star (PNS), aided by the effects of vigorous neutrino-driven turbulent
convection behind that shock [23, 24, 25, 15, 26, 22]. The delay to explosion
can last another few hundred milliseconds, after which the explosion is driven
to an asymptotic state in a period of from $\sim$a few to $\sim$10 seconds. An
extended period of neutrino heating seems required [5, 15, 27]. The shock wave
then takes a minute to a day to emerge from the massive star, and this
emergence inaugurates the brilliant electromagnetic display that is the
supernova. The outcomes and timescales depend upon the progenitor core density
and thermal structure at the time of collapse [28, 29], which itself is an
important function of progenitor mass, metallicity, and rotational profile. If
a black hole eventually forms, the core must still go through the PNS stage,
and it is still possible to launch an explosion, even when a black hole is the
residue. There is never a direct collapse to a black hole. A small fraction of
supernova (hypernovae?) may be driven by magnetic jets from the PNS if the
cores are rotating at millisecond periods. Otherwise, magnetic effects are
generally subdominant, but of persistent interest in the context of pulsar and
magnetar birth [30, 31, 32, 33, 34, 35, 36].

Though this scenario is buttressed by extensive simulation and theory, and
most 3D (and 2D) models now explode without artifice [37, 8, 13, 12, 38, 17,
15, 20, 26, 21, 22], direct verification of the details and the timeline
articulated above are difficult to come by. However, the neutrino and
gravitational-wave (GW) signatures of this dynamical event would allow one to
follow the theoretically expected sequence of events in real time. The
detection of 19 neutrinos from SN1987A was a landmark [39, 40], but little was
learned, other than that a copious burst of neutrinos, whose properties are
roughly in line with theory, attends CCSN and the birth of a compact object.
The real-time witnessing of the events described as they unfold is the promise
of GW detection from a supernova explosion. The detection of these GWs and the
simultaneous detection of the neutrinos overlapping in time is the holy grail
of the discipline.

The CCSN GW signal [41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54]
from bounce through supernova explosion and into the late-time (many-second)
proto-neutron star (PNS) [55, 56, 32] cooling phase (or black hole formation)
can be decomposed into various stages with characteristic features, frequency
spectra, strains and polarizations [57]. GWs are generated immediately around
bounce in CCSN by time-dependent rotational flattening (if initially rotating,
see e.g. [58]) and prompt post-shock overturning convection (lasting tens of
milliseconds) [24, 59], then during a low phase (lasting perhaps
$\sim$50$-$200 milliseconds) during which post-shock neutrino-driven
turbulence builds, followed by vigorous accretion-plume-energized PNS modal
oscillations (predominantly a mixed f/g-mode early, then a pure f-mode later,
see §II.1) [41, 51]. If an explosion ensues, these components are accompanied
by low-frequency ($\sim$1$-$25 Hertz (Hz)) GW “memory” (see §II.6) due to
asymmetric emission of neutrinos and aspherical explosive mass motions [60,
61, 62, 63]. The duration of the more quiescent phase between prompt
overturning convection and vigorous turbulence depends upon the seed
perturbations in the progenitor core [15], of which the duration of this phase
is diagnostic. Relevant for the GW signature is the equation of state (EOS)
[64, 59, 65, 66], the rate of neutronization and neutrino cooling of the core
[67, 56], and the stellar core’s initial angular momentum and mass density
distributions. Measurable GW signatures of rotation require particular
progenitors that rotate fast, while all other phases/phenomena are expected to
be operative in any core-collapse context, except matter memory, which
requires an asymmetric explosion. The rotational signature is primarily
through its dependence upon the ratio of rotational to gravitational energy
($T/W$) [44, 50, 18], and, for fast rotating cores, to the degree of initial
differential rotation. Black hole formation will approximately recapitulate
the sequence followed by neutron star formation, except that when the black
hole forms long after collapse the GW signal ceases abruptly [68] and that due
to the shrinking stalled shock radius a spiral mode is excited that quasi-
periodically modulates its GW signal (see §II.2). The termination of the
neutrino emission is correspondingly abrupt. Though these phases are generic,
their duration, strain magnitudes, and degree of stochastic variation and
episodic bursting (due in part to episodic accretion and fallback) is a
function of massive star progenitor density structure, degree of rotation, and
the chaoticity of the turbulence.

Though neutrino-driven convection generally overwhelms manifestations of the
“SASI” (Standing Accretion Shock Instability [69, 70]), the SASI is sometimes
discernible as a near 100$-$200 Hz subdominant component. However, when the
explosion aborts and the average shock radius sinks deeper below $\sim$100 km,
the so-called “spiral SASI” [71, 46, 72] emerges (§II.2). This spiral
rotational mode has a frequency of $\sim$100-200 Hz, is interestingly
polarized, and if present clearly modulates both the neutrino and GW
signatures until the general-relativistic instability that leads to black hole
formation.

Therefore, each phase of a supernova has a range of characteristic signatures
in GWs that can provide diagnostic constraints on the evolution and physical
parameters of a CCSN and on the dynamics of the nascent PNS. Core bounce and
rotation, the excitation of core oscillatory modes, neutrino-driven
convection, explosion onset, explosion asymmetries, the magnitude and geometry
of mass accretion, and black hole formation all have unique signatures that,
if measured, would speak volumes about the supernova phenomenon in real time.

For this paper, we have run a broad set of 3D simulations (11 progenitors in
total, with progenitors ranging from 9 to 23 M⊙) out to the late post-bounce
times (up to $\sim$6 seconds post-bounce). These late-time 3D simulations
illustrate the sustained multi-second GW signal and are the longest 3D core-
collapse simulations with sophisticated neutrino transport performed to date.
Simulations out to one second or less do not capture the entire time evolution
of the GW signal. We find that the GW signal persists out to late times for
all models, is strongly correlated with the turbulence interior to the shock
(see §II.7), and is not correlated with proto-neutron star convection
(§II.7.1). We see a memory signature at $\leq$25 Hz associated with large-
scale ejecta for our exploding models, and a spiral SASI signature at
$\sim$100 Hz for the non-exploding models. All our models show an early
prompt-convection phase at $\sim$50 milliseconds (ms) associated with a
negative entropy gradient interior to the stalled shock front at $\sim$150 km.
For most models, this is followed by a quiescent phase of duration $\leq$50
ms, after which the strain grows in association with turbulent motion interior
to the stalled shock. The 23-M${\odot}$ model shows an interesting exception,
as the strain illustrates another feature lasting from $\sim$175 to 350 ms,
coincident with the shock receding before reviving. As the accretion rate and
turbulence diminish, so too does the strain. However, at late times, we see a
consistent offset in the strain associated with matter and neutrino memory in
the exploding models (§II.6).

We now proceed to a more detailed discussion of our new results that span a
broad progenitor mass range, capture for the first time and for most models
the entire GW signal of CCSN, and do not suffer from Nyquist sampling
problems. In this paper, we focus on initially non-rotating progenitors, whose
general behavior should also encompass models for slowly rotating initial
cores. We note that even initially non-rotating 3D CCSN models experience core
spin up due to stochastic fallback [73], and this effect is a normal by
product of sophisticated 3D simulations. In §I, we provide information on the
simulation suite and its characteristics and briefly describe the various
models’ hydrodynamic developments. Then, in §II we present our comprehensive
set of findings concerning the complete GW signature of initially non-rotating
CCSN, partitioned into subsections that each focus on a different aspect of
this signature and its import. In subsection §II.1, we lay out the basic
signal behaviors as a function of progenitor. This section contains our major
results and then is followed in §II.2 by a digression into the GW signature of
black hole formation. In §II.3, we present an interesting finding concerning
the dependence of the total radiated GW energy on compactness [74] and in
§II.4 we note the avoided crossing that is universally manifest in all CCSN GW
spectra and seems a consequence of the presence of inner proto-neutron star
(PNS) convection and its growth with time. In §II.5 we discuss the solid-angle
dependence of the matter-sourced GW emission, both at high and low (matter
“memory”) frequencies, and in §II.6 we present our results concerning neutrino
memory at low frequencies. Then, we transition in §II.7 to a discussion of the
predominant excitation mechanism. Finally, in §III we recapitulate our basic
findings and wrap up with some observations.

##  I Setup and Hydrodynamics Summary

We present in this paper a theoretical study of the GW emission of eleven CCSN
progenitors, from 9 to 23 M⊙, evolved in three dimensions using the radiation-
hydrodynamic code Fornax [75]. To calculate the quadrupole tensor and the GW
strains we employ the formalisms of references [76] and [77] (see also §A) and
we dump these data at high cadences near the LIGO sampling rate (Table 1). The
progenitor models were selected from [78] for the 14-, 15.01-, and 23-M⊙
models and from [79] for models between 9 and 12.25 M⊙. The radial extent of
the models spans 20,000 kilometers (km) to 100,000 km, generally increasing
with progenitor mass. All the models (except the 9-M⊙ model on Blue Waters,
which had 648 radial zones) were run with 1024$\times$128$\times$256 cells in
radius, $\theta$, and $\phi$. We employ 12 neutrino energy groups for each of
the $\nu_{e}$, $\bar{\nu}_{e}$, and “$\nu_{\mu}$”s followed (see [21, 73]) and
the SFHo equation of state [80]. The progenitor models are non-rotating,
though some degree of rotation is naturally induced due to fallback [73].
These simulations include two of the longest 3D CCSN simulations run to date,
a 11-M⊙ model evolved past 4.5 seconds post-bounce, and a 23-M⊙ model evolved
to $\sim$6.2 seconds post-bounce. All of our models explode except the 12.25-
and 14-M⊙ progenitors, and all models besides the 14-M⊙ are evolved beyond one
second post-bounce. We include four simulations of the 9-M⊙ model on different
high-performance clusters (Frontera, Theta, and Blue Waters) at various stages
of code evolution. The low-mass 9-M⊙ models asymptote early on in diagnostic
quantities such as explosion energy and residual mass, and one iteration has
been evolved past two seconds post-bounce. Our models, run times, and
explosion outcome are summarized in Table 1. Several of the models have been
published before, including three of the four 9-M⊙ iterations (the fourth, the
longest simulation, is new) in references [81], [21], and [73].

In Figure 1, we plot the density profiles against enclosed mass for all eleven
models studied here. Note the association of the silicon-oxygen (Si/O)
interface density drop (see, e.g. [82, 14, 38, 21, 83, 5, 29, 28]) with the
onset of successful shock revival. The 14-M⊙ and 12.25-M⊙ progenitors lack
such a strong interface and do not explode. Low-mass progenitors (e.g. 9-9.5
M⊙) have a steep density profile and explode easily (though still with the aid
of turbulent convection). For instance, compared to models 11-M⊙, models
15.01-, and 23-M⊙ have Si/O interfaces successively further out, and explode
successively later.

We show the angle-averaged shock radii at early and late-times in Figure 2.
All models except the 12- and 14-M⊙ models explode, with an approximate
correlation between progenitor compactness [74] and explosion time. The two
non-exploding models experience $\sim$10 ms oscillations in the shock radii
due to a spiral SASI that manifests itself after $\sim$350 ms. We also see a
longer secular timescale oscillation of $\sim$70 ms in the 12.25-M⊙ and 14-M⊙
black-hole formers. We summarize the eleven models in Table 1. The more
massive 15.01- and 23-M⊙ progenitors explode later, with the latter showing
shock revival only after $\sim$0.5 seconds post-bounce. A later shock revival
time, again $\sim$0.5 s, was also seen for the 25-M⊙ progenitor in [21]. After
the first $\sim$500 ms, the shock velocities settle into approximately
asymptotic values that range from 7000 to 16000 km s-1, inversely correlated
crudely with the progenitor mass (see also [84]).

##  II Results

###  II.1 General Gravitational-Wave Signal Systematics of Core-Collapse
Supernovae

As stated, we highlight for this study of the GW signatures of CCSN eleven of
our recent initially non-rotating 3D Fornax simulations. Care has been taken
to calculate the quadrupole tensor with a high enough cadence to avoid Nyquist
sampling problems, and we have been able to simulate to late enough times to
capture what is effectively the entire GW signal after bounce for a large
subset of the models. Table 1 provides the duration of each simulation and the
minimum Nyquist frequency achieved during each run. We will discuss both
matter and neutrino contributions to GW energy, and the relevant equations are
summarized in Appendix A.

In Figure 3, we plot the plus (black) and cross (blue) polarizations in the
x-direction of our computational grid of the strain multiplied by the distance
to the source. Other orientations yield qualitatively similar numbers for the
higher frequency components that dominate the GW power. However, there is a
large variation at low frequencies ($\leq$25 Hz) of the matter and neutrino
memories with solid angle (see §II.6). Note that the x- and y-axes cover
different ranges for each model. A red star on the panels indicates the rough
time of explosion, defined loosely. As Figure 3 demonstrates, all the
exploding models transition through similar phases. During the first $\sim$50
ms there is a burst of emission due to prompt overturning convection driven by
the negative entropy gradient produced behind the shock wave as it stalls. The
detailed time behavior of this overturn will depend on the initial accreted
perturbations, which will set the number of e-folds to the non-linear phase.
However, the basic behavior and timescales are broadly similar. Figure 4
focuses on this early first 0.25 seconds. For the lower-mass progenitors,
explosion (the red star) ensues towards the end, or not long after, the prompt
signal, and this is followed by the early growth of the second phase. For the
more massive progenitors (such as the 23-M⊙ model), the onset of explosion can
be much later. As Figure 3 shows, the growth phase of the GW emission
continues beyond what is shown in Figure 4 to a strong peak. That peak phase
is powered by the accretion of the infalling plumes during explosion. A core
aspect of 3D core-collapse explosions is the breaking of spherical symmetry
that allows simultaneous accretion in one direction and explosion in another
[72, 21, 5, 28]. For exploding models, the infalling plumes that strike the
surface of the PNS can achieve supersonic speeds before impact. For the black
hole formers (the 14-M⊙ and 12.25 M⊙ models here), the accretion is
maintained, but impinges upon the PNS core subsonically. This will have
interesting consequences we discuss in §II.2.

Figures 3 show that the lower-mass models have smaller strains and that the
phase of high strain lasts for a shorter time. For the 9-M⊙ through 9.5-M⊙
models, much of the GW emission subsides by $\sim$0.25-0.5 seconds, while the
high phase lasts $\sim$1.2 seconds for the 23-M⊙ model and continues beyond
$\sim$1.0 and $\sim$1.5 seconds for the 11-M⊙ and 15.01-M⊙ models,
respectively. These differences reflect the differences in the initial density
profiles (Figure 1) and the compactness (see also Figure 11).

After this vigorous phase, the pounding of the accretion plumes subsides, but
the signal continues at a low amplitude. Though as much as $\sim$95% of the GW
energy emission has already occurred, the f-mode continues to the latest times
we have simulated as a low hum of progressively increasing
frequency111Sonifications of the signals are available upon request.. Hence,
we see universally for the exploding models a transition from a high-
amplitude, lower-frequency stage ($\leq$0.3-1.5 seconds, depending upon the
progenitor) to a lower-amplitude high-frequency stage ($\geq$1.5 seconds). As
Figure 3 indicates, for the exploding models a very-low frequency memory is
superposed that represents a permanent metric strain. There is no such matter
memory signal for the black-hole formers (§II.6), but the accretion phase
continues for them to very late times, abating only slowly as the mantle
continues to accrete the mass of the outer mantle until the general-
relativistic instability that leads to a black hole ensues.

It has too often been thought that strain signals such as are depicted in
Figures 3 and 4 are too noisy to be templated cleanly, and this to a degree is
true. There is a lot of stochasticity due to chaotic turbulence. However, the
frequency content of these signals tells a different story. In Figures 5 and
6, we plot spectrograms of the GW power versus time after bounce for our 3D
models and see distinct structures. The most obvious feature is the f/g-mode
[41, 51, 85, 86, 87] from $\sim$400 Hz early, rising to $\sim$1000-3000 Hz
after $\sim$0.8 seconds after bounce. It is in this band that most of the
emitted GW power of supernovae resides (see Table 1). This is a natural
consequence of the fact that the peak in the eigenfunction of the f-mode is in
the PNS periphery where the collisions of the accreta with the core are
occurring. Hence, the excitation and the fundamental f-mode eigenfunction
overlap nearly optimally. Associated with this feature in the earlier phases
is a dark band near $\sim$1000-1300 Hz. This has been interpreted as a
manifestation of an avoided crossing [51] between a trapped $\ell=2$ g-mode
and the $\ell=2$ f-mode. All the spectrograms for all our models show the same
modal interaction, though at slightly different frequencies. For instance, at
two seconds after bounce, the f-mode frequency is $\sim$1.75 kHz, 1.8 kHz, 2
kHz, and 2.5 kHz for the 9-, 11-, 12.25-, and 23-M⊙ models, respectively,
reflecting the variation in model PNS masses. Early on power is in the lower
frequency component (mostly a trapped g-mode, mixed with the f-mode), and then
it jumps to the higher frequency component (mostly the f-mode). This modal
repulsion, or “bumping,” is a common feature in asteroseismology [88] and
seems generic in core-collapse seismology.

All the models show the early prompt convection phase, with power from
$\sim$300 to $\sim$2000 Hz. After this, all the models manifest a “haze” of
emission that extends above the f/g-mode to frequencies up to $\sim$2000 to
$\sim$5000 Hz. The duration of this haze is from $\sim$0.25 to $\sim$1.5
seconds and tracks the phase of vigorous accretion (see §II.7). Individual PNS
pulsation modes, likely the $\ell=2;n=1$ p-mode, can also be seen superposed
in this “haze” and extending beyond it to later times. This is particularly
the case for the 12.25-, 15.01-, and 23-M⊙ models. It is only for the models
with the most vigorous accretion onto the core that this mode is clearly seen
at later times.

The origin of this haze is still a bit unclear, though it is definitely
excited by the pummeling accreta (§II.7). The p-mode frequencies of the PNS
for radial node numbers from 1 to 10 reside in this space and we could be
seeing an overlapping and unresolved superposition of these modes. However,
exploding models experience simultaneous explosion and accretion, the latter
through infalling funnels that are few in number, can achieve supersonic
speeds, and dance over the PNS surface. It is likely that the time-changing
quadrupole moment of these funnels as they impinge upon the PNS surface is the
source of this power. The timescales of their deceleration are about right,
those timescales have a spread which could translate into a broad feature, and
at any particular time they represent a low angular order perturbation.
Importantly, however, we don’t see this haze for the black-hole formers
12.25-M⊙ and 14-M⊙. It is only for the exploding models that there is a
breaking of spherical symmetry that results in simultaneous explosion and
supersonic funnel infall. Figure 7 portrays two snapshots of the Mach-number
distribution of the inner 100 km of the residue of the 23-M⊙ model, clearly
showing such funnel collisions. The accretion onto the proto-black-hole is
always subsonic. Though the haze constitutes at most only a few 10’s of
percent of the total emission, its origin is clearly an interesting topic for
future scrutiny.

In Figure 8, we plot spectrograms for two representative models of the
effective strain versus time after bounce. The effective strain is defined as
the average of both strain polarizations,

$h_{\mathrm{eff}}=0.5\left(h_{+}+h_{\times}\right)\,.$ (1)

This figure provides a focus on the low-frequency regions. We see for the
14-M⊙ black-hole former some power near $\sim$100 Hz, which we identify with
the spiral-SASI [71] (see §II.2). Such a mode emerges in our model set only
for the proto-black-holes (see §II.2). In the red blot in the lower left-hand
corner of the left panel may be a signature of the traditional SASI [69]. We
generally see little evidence in the GW signature of this SASI, but always see
the spiral SASI when the explosion is aborted and the stalled shock recedes.
In the right panel of Figure 8, the red band is a signature of the matter
memory associated with the asymmetrical explosion of the 23-M⊙ model. Whether
the traditional SASI is seen to the left of this band is unclear, but the
early recession of its shock before explosion might be conducive to its brief
appearance.

###  II.2 Signatures of Failed Explosions - The Prelude to Black Hole
Formation

If a black hole forms by late-time fallback after many, many seconds to hours
after the launching of a stalled shock that seemed to herald a successful
explosion (but didn’t), then the GW signal will be similar to those seen in
the context of successful explosions. If, however, the stalled shock is never
“reignited,” it will slowly settle to progressively smaller radii and the
mantle of the progenitor core will continue to accrete through it onto the
PNS. Eventually, the fattening PNS will experience the general-relativistic
instability to a black hole, at which time the GW emission abruptly terminates
within less than a millisecond. This latter phase could take many seconds to
many minutes to reach222For the 14-M⊙ model, we estimate a black hole
formation time (using a maximum baryon mass of 2.477 M⊙ from Steiner et al.
[80] at the onset of collapse) of $\sim$500 seconds.. The GW signature of this
modality of black hole formation, representatives of which are our 14- and
12.25-M⊙ models, has particular diagnostic features that set these evolutions
apart. First, the breaking of symmetry that results in the simultaneous
accretion of lower-entropy plumes with the explosion of high-entropy bubbles
does not occur. The result is that for this channel of black hole formation
the infalling plumes do not dance over the PNS, do have high Mach numbers, and
don’t excite the higher-frequency “haze” that we have identified for the
exploding models seen in the associated spectrograms (Figures 5 and 6). We do
see in Figure 6 power not only in the dominant f-mode, but weakly in an
overtone p-mode as well. However, as shown in Table 1 for the 12.25 M⊙ black-
hole former, the fraction of the total GW energy radiated in the f-mode is
correspondingly higher, as much as $\sim$95% of the total, than for the
exploding models that also generate power in the haze.

This channel of black hole formation also experiences the emergence of what we
identify as the spiral-SASI [71]. This is seen in Figure 2 in the clear
$\sim$100-200 Hz periodicity of the late-time mean shock position of both the
14- and 12.25-M⊙ models after $\sim$300$-$400 milliseconds after bounce and
very clearly in the spectrogram of the shock dipole depicted in Figure 9.
Generally, this feature emerges after the mean stalled shock radius sinks
below $\sim$100 km and is not seen in exploding models. The timescale of the
periodicity scales roughly with $\Delta{R_{s}}/c_{s}+\Delta{R_{s}}/v_{acc}$,
where $R_{s}$, $c_{s}$, and $v_{acc}$ are the shock radius, speed of sound,
and post-shock accretion speed [70].

Another feature seen most clearly in Figure 2 in the context of these black-
hole formers is a much longer-timescale modulation of the mean shock position
with a period near $\sim$70 ms. Not seen clearly in the GW spectrograms or
strain plots (though there may be a hint in the strain plot for the 14-M⊙
model), this oscillation may be due to a global pulsation mode associated with
the neutrino heating, cooling, and transport of the mantle, but this
speculation remains to be verified. Nevertheless, this feature has never
before been identified in studies of 3D CCSN and is interesting in itself.
Finally, as Figure 3 suggests for the 14- and 12.25-M⊙ models, since those
cores that form black holes by this channel do not explode, they are expected
to have no net low-frequency matter memory component.

###  II.3 Total Gravitational-Wave Energy Radiated

In Figure 10, we plot versus time the integrated radiated GW energy due to
matter motions. Model 23-M⊙ radiates the most GW energy ($\sim$3.0$\times$1046
erg, or $\sim$2$\times$10-8 M⊙ c2 after $\sim$5 seconds), while the collection
of 9-M⊙ models radiates the least. There are a few important features of this
plot. First, we see that we have captured what is basically the entire GW
signal for many of the models (the 15.01-M⊙, 14-M⊙, and 12.25-M⊙ model
emissions are still climbing). Within $\sim$1.5 seconds, most exploding models
have radiated $\geq$95% of the total energy to be radiated, and after $\sim$2
seconds they have radiated $\geq$98%. Table 1 provides the total energy
radiated via the f/g-mode, as well as the fraction of this total radiated in
the f-mode after 1.5 seconds. Not shown in Table 1 is the fact that more than
95% of the total GW energy radiated after 1.5 seconds is via the f-mode. As
indicated for the 12.25-M⊙ model in Table 1 and Figure 10, due to continuing
accretion the black hole formers radiate to later times than the exploding
models, and this mostly in the f-mode.

Figure 10 shows that the various phases described in §II.1 are recapitulated
via stair steps until finally asymptoting. Moreover, the continuum of models
highlighted in this paper demonstrate collectively that the radiated GW signal
energies vary by as much as three orders of magnitude from the lowest-mass to
the higher-mass models. This is a consequence of the differences in their
initial density profiles (see Figure 1), and directly from the resulting mass
accretion histories. Even more directly, as Figure 11 demonstrates, there is a
strong monotonic relation for exploding models between the total GW energy
radiated and the compactness [74] of the initial progenitor “Chandrasekhar”
core333We define the compactness here as
$\xi_{M}=\frac{M/M_{\odot}}{R(M)/1000\,\mathrm{km}}$, where $M=1.75$ M⊙..
Though compactness does not correlate with explodability [21, 83, 5, 28, 29],
it does seem to correlate with residual neutron star mass, radiated neutrino
energy, and, as now indicated in Figure 11, the total GW energy radiated. In
fact, we derive a power-law with index $\sim$0.73 between the two.

Finally, we note that the collection of 9-M⊙ models don’t behave exactly the
same. This is due to the fact that these models were simulated with slightly
different code variants (as we continued to update and upgrade Fornax); one
(the 9a model) had small artificial imposed perturbations in the initial model
and three different supercomputers were used. The natural chaos in the flow
and in the simulations will pick up on any slight variations and amplify them,
with the result that the evolution can be slightly different. Though the
qualitative behavior of all four of these 9-M⊙ models is the same, they
exploded at slightly different times (see Figure 3), with the resulting
different developments of their GW signals. As a consequence, the radiated
energy varies by about a factor of two. In a crude sense, we might view this
spread as an imperfect indicator of the likely spread in Nature due to the
chaos of turbulence for the “same” progenitor, but we have certainly not
demonstrated this.

###  II.4 Avoided Crossing and Trapped g-mode

A distinctive feature seen clearly in the spectrograms for all the models (see
Figures 5, 6) is a dark band near $\sim$1000$-$1300 Hz during the first
$\sim$0.3$-$1.0 seconds of the post-bounce evolution. This is most likely due
to an avoided crossing [88] of interfering $\ell=2$ PNS pulsation modes that
are coupled and mixed [51]. The best current thinking is that the interfering
modes are a trapped g-mode and the $\ell=2$ f-mode [51, 86], though much work
remains to be done to determine the details and the nature of the couplings.
Non-linear mode coupling may be involved. There is also evidence for some GW
power in the “bumped” g-mode (that thereafter trends to lower frequencies),
seen just after the mode repulsion in Figures 5 and 6, but most clearly in
Figure 17 below. This is a qualitatively similar feature to that highlighted
recently in [89]. Nevertheless, within $\sim$0.3 to $\sim$1.0 seconds
(depending upon the progenitor) most power is clearly in the f-mode, where it
persists thereafter.

G-modes, however, are generally at low frequencies below $\sim$500 Hz and
don’t contribute much to the GW signature. Moreover, the presence of lepton-
gradient-driven PNS convection [55, 32] introduces a region in the PNS for
which g-modes are evanescent and non-propagating. Figure 12 depicts the
evolution of PNS convection for most of the models presented in this work. One
sees clearly that the extent of PNS convection starts in a narrow shell, but
grows wider with time. By $\sim$1.6 to $\sim$4.0 seconds PNS convection has
grown to encompass the center and most of the residual PNS and will persist
beyond the simulation times of this study [56]. During the early post-bounce
phase, though most g-modes have frequencies too low to couple with the f- and
p-modes, at an early stage before the region of PNS convection has grown too
thick it is possible for a g-mode trapped mostly interior to PNS convection to
couple with them. With time, the coupling will be broken by the evolving
thickness of the convective shell; it is this growth that eventually severs
the coupling with the outer regions where the impinging plumes are providing
the excitation and that leads to the jump to the pure f-mode. However, much
work remains to be done to fully demonstrate the details of this coupling and
“bumping” transition. Nevertheless, the manifest presence of this avoided
crossing in the GW spectrograms and in the GW signature of core collapse
universally is an interesting direct marker of the presence of PNS convection
that deserves further study.

###  II.5 The Angular Anisotropy of the Matter Gravitational-Wave Emissions

In Figure 13, we plot the matter strain for the 9b, 12.25-, and 23-M⊙ models
as a function of time after bounce to illustrate the anisotropy with various,
arbitrarily chosen viewing angles. Note that significant anisotropy manifests
at late times in the (low-frequency) memory component, which captures the
large-scale asymmetry in the explosion ejecta. The non-exploding 12.25-M⊙
model, by comparison, has virtually no anisotropy. The early high-frequency
component is stochastic, whereas the low-frequency late-time memory show
secular time-evolution that does not average to zero and indicates a metric
shift, reaching values of $\sim$5 cm for the various massive models (showing a
general trend with the progenitor mass/explosion asymmetry). We find similar
significant anisotropy in the (low-frequency) neutrino memory component
(discussed in §II.6). Importantly, however, when calculating the total
inferred “isotropically-equivalent” radiated GW energy, which is dominated by
the higher-frequency component in and near the LIGO band, as a function of
angle we find that it varies by $\sim$10 to $\sim$15% around an angle-averaged
mean. This implies that, though the higher-frequency emissions are indeed
anisotropic, the integrated high-frequency signals are only weakly dependent
on angle.

###  II.6 The Neutrino Memory Component

In addition to the matter memory, asymmetries in the emission of neutrinos
produce another low-frequency memory component, the neutrino memory [90, 60,
91, 61, 62, 63]. In Figure 14, we plot the GW strain due to anisotropic
neutrino emission as a function of time after bounce for the models studied
here. The neutrino strain is significantly larger in magnitude than the matter
contribution, reaching over 1000 cm for the most massive progenitors. There is
generally a hierarchy of strain amplitude with progenitor mass, reflecting the
sustained turbulent accretion in more massive progenitors, which results in
higher neutrino luminosities and generally more anisotropic explosions. The
11-M⊙ model is an exception and fields the highest strain amplitude. In
addition, the neutrino memory shows much lower frequency evolution and more
secular time-evolution than the matter component, which is fundamentally
because it is a cumulative time-integral of the anisotropy-weighted neutrino
luminosity (see Appendix A as well as [72]). The difference in the mean
frequencies of the neutrino and matter memories may, therefore, provide a
means someday to distinguish them observationally.

In Figure 15, we plot the GW energy due to neutrino emissions as a function of
time after bounce for our various 3D models. There are several key and
distinguishable features. First, like the GW energy due to matter motions
displayed in Figure 10, we see growth by over two orders of magnitude (but not
three, as in the former) in the neutrino memory. Additionally, we generally
see a hierarchy with progenitor mass, however with the 11-M⊙ surpassing the
23-M⊙ until $\sim$4.5 seconds. Unlike the matter component of the GW energy,
the neutrino component shows sustained growth for our longest duration model
(the 23-M⊙ model). In comparison with Figure 5 from [92], this emphasizes
again the need to carry simulations out to late times to capture the entire
signal. Note that, despite the higher strains seen in the neutrino component
of the GW signature and its sustained growth, due to the much smaller
frequencies it is still more than two orders of magnitude less energetic than
the matter-sourced GW energy. In addition, though both components capture the
development of turbulence, the neutrino component does not show a prompt
convective phase and begins to develop $\sim$100 to 200 ms later than the
matter component.

As with the matter component, the neutrino component is most pronounced for
delayed explosions of models with higher compactness reflecting their more
vigorous turbulent accretion history and more anisotropic explosions.

###  II.7 Turbulent Accretion Excites Gravitational-Wave Emission

The major excitation mechanism of GW from CCSN and black hole formers is the
pounding accretion onto the PNS core [41, 48, 43, 49, 51, 52]. As shown in
§II.1, much of the GW power comes out at the frequencies associated with the
pulsational modes of the PNS. To demonstrate the correlation of the
gravitational strain with the matter accretion, we first remove the matter
memory by applying a high-pass Butterworth filter below 15 Hz to the strains.
Then, in Figure 16 we plot the turbulent hydrodynamic luminosity evolution
interior to the shock with this filtered strain timeline.

The turbulent hydrodynamic flux is defined following [35] as

$F_{conv}=\Bigl{\langle}\left(\frac{1}{2}\rho
v_{turb}^{2}+u+p\right)v_{turb}^{r}\Bigr{\rangle}_{d\Omega}\,,$ (2)

including the turbulent kinetic energy, the internal energy $u$, and the
pressure $p$. The turbulent velocity is defined here as the radial component
of the turbulent velocity,

$v_{conv}^{r}=\langle\left(v^{r}-v_{ave}^{r}\right)^{2}\rangle_{d\Omega}^{1/2}\,,$
(3)

where $v_{ave}^{r}$ is the density-weighted angle average of the radial
velocity. This is calculated at 110 km for all models except the non-exploding
12.25- and 14-M⊙ progenitors, whose shocks early on sink below this radius;
for these models we calculate the turbulent hydrodynamic flux at 2.5 times the
PNS radius (defined here as the density cutoff at 1010 g cm-3).

The strong correlation throughout their evolution (even in detail) between the
turbulent hydrodynamic flux impinging onto the core and the GW strains
demonstrates that turbulent accretion through the shock and onto the PNS core
is the major agency of GW excitation and emission in CCSN. We note that this
correlation was demonstrated even though the flux was angle-averaged and the
strain was for emission along the x-axis. No attempt was made to break the
flux into $\ell=2;m=[-2,-1,0,1,2]$ components, yet the correlation is clear.

To provide another perspective, in Figure 17, we show the relation between the
GW energy spectrogram and the accretion rate “power” in frequency components
with $f>25$ Hz (orange lines) for the 9.5- and 23-M⊙ models. As the panels
demonstrate, after the explosion and the period of heavy infall subsides
(during which the effects of individual accretion events overlap), the
accretion rate power shows clear correlations with excursions in the GW energy
spectrogram. Spikes and gaps on the spectrogram coincide directly with the
peaks and troughs on the curve, meaning that the GWs are excited by, or at
least correlated with, the short-period variations in accretion rate onto the
core. Such mass accretion rate variations directly tie both episodic fallback
and outflow events with the GW emission.

We note that the colormap used for these plots reveals, particularly for the
23-M⊙ model, some power in the g-mode “bumped” by the f-mode and the repulsion
between the two modes around $\sim$0.8 seconds (see §II.4). Though weak, for
the 23-M⊙ model this signature continues almost to $\sim$2.0 seconds, and
perhaps beyond.

####  II.7.1 Possible Secondary Role of Proto-Neutron-Star Convection

Some have suggested that inner PNS convection itself excites much of the GW
emission from CCSN [47]. In Figures 18 and 19, we overplot the angle-averaged
convective hydrodynamic luminosity at its peak inside the PNS (see also
Figures 12) against the Butterworth filtered strain. We follow Eq. 3, but
limit the radius to that of the maximum convective luminosity within the PNS
(positive outwards), which lies near $\sim$15 km for the various models. As
Figures 18 and 19 show there seems to be no correlation between the two. This
is particularly clear at late times when the PNS convective flux is still
large while the GW emissions have all but subsided. If PNS convection were the
agency of excitation at all phases, GW emission would not have subsided to
such a degree at later times (from 0.3 to 1.5 seconds after bounce). This
comparison demonstrates the importance of simulating to late times to capture
the entire GW signal.

However, as Figures 5 and 6 themselves show, the f-mode persists to the latest
times and manifests an episodically modulated (see Figure 17), though
continuous, signal. We have yet to identify the major excitation mechanism for
this component at late times. Anisotropic winds and neutrino emissions from
the residual core could be causes, but inner PNS convection may be also a
factor here. It is also the case that the mode should ring down over a period
given by the dominant damping mechanism. Such mechanisms include sound
generation, the back reaction of anisotropic winds, neutrino emission coupling
and viscosity, non-linear parent-daughter mode coupling, and numerical
dissipation. We reiterate, however, that the f-mode continues to ring and
produce a weak GW signal for the duration of all our simulations. Clearly,
this topic deserves more detailed scrutiny in the future. Nevertheless, this
later phase amounts to only a few percent of the total energy emitted.

##  III Conclusions

In this paper, we have presented and analyzed the GW signatures of an
extensive suite of detailed initially non-rotating 3D core-collapse supernova
simulations spanning a wide range of massive-star progenitors. For the first
time, most of the published simulations were carried out to late enough times
to capture more than 99% of the total GW emission from such events. Moreover,
we have endeavored to dump the relevant quadrupole data at a rate sufficient
to effectively eliminate Nyquist sampling errors. We see that the f/g-mode and
f-mode oscillation modes of the PNS core carry away most of the GW power and
that generically there are avoided crossings and modal interactions likely
associated with the evolution, extent, and character of lepton-driven PNS
convection. The f-mode frequency inexorably rises as the proto-neutron star
core shrinks during its Kelvin-Helmholtz contraction phase, driven by neutrino
loses, and its power and frequency behavior are central features of the GW
emissions from the core-collapse event. Other modes are also seen in the GW
spectra, in particular a $\ell=2;n=1$ p-mode and, perhaps directly, a trapped
g-mode, though most g-modes are not excited. Whether other p-modes are in
evidence is to be determined.

We demonstrate that the GW emission is powered mostly by accretion plumes onto
the PNS that excite its modal oscillations and also produce a “haze” of higher
frequency emission also correlated with the phase of violent accretion, after
which the signal subsides to be dominated by the chirp of the f-mode signal at
low power that nevertheless continues beyond the duration of even these
simulations, albeit weakly. The duration of the major phase of emission varies
with exploding progenitor and is generally shorter for the lower-mass
progenitors ($\sim$0.3-0.5 seconds) and longer for the higher-mass progenitors
($\sim$1.5 seconds). We find a strong correlation between the total GW energy
radiated and the compactness of the progenitor whose mantle explodes as a
supernova. Furthermore, we find that the total GW energy emissions can vary by
as much as three orders of magnitude from star to star. Hence, there is a
severe progenitor dependence that must be factored into any discussion of
detectability. For the black-hole forming models, since accretion is not
reversed at any time or at any solid angle, their GW signal lasts until the
black hole forms, tapering off only slowly until then. In addition, they do
not manifest the high-frequency haze seen for the exploding models. For these
black-hole formers, we also witness the emergence of a spiral shock motion
that modulates the GW emission at a frequency near $\sim$100 Hz that slowly
increases as the stalled shock sinks.

In Figure 20, we plot the sensitivity curves with the amplitude spectral
densities at 10 kiloparsecs of our 3D models. More massive models generally
leave larger footprints, with the 11 M⊙ studied being the exception. Current
and next-generation detectors can observe, for galactic events, a signature
spanning orders of magnitude, from subHz to $\sim$3000 Hz. While Advanced
LIGO/Virgo/Kagra (LVK, [93, 94, 95]) can detect $\sim$30$-$3000 Hz signals for
the more massive progenitors, upcoming detectors, including the Einstein
Telescope [96, 97], the Cosmic Explorer [98], BBO [99], and Decigo [100, 101]
(at lower frequencies, see also [102]) should be able to detect galactic
events for all progenitor masses studied here through almost three orders of
magnitude in both frequency and total energy radiated. Note that neutrino GWs
dominate at lower frequencies (from sub-Hz to 10s of Hz) and matter GWs
dominate at higher frequencies (at several hundreds of Hz). However, a
detailed retrieval analysis, informed by the best signal-processing
approaches, has yet to be performed and is an important topic for future work.

Though we have endeavored here to provide a comprehensive look at the
gravitational-wave signatures of core collapse, there remains much yet to
understand. Topics unaddressed here are the nuclear equation-of-state
dependencies, the role of rapid rotation, the possible signatures of strong
magnetic fields [89, 36], and the results for other progenitor massive stars
and from other stellar evolution codes. Importantly, the analysis of a
detected GW signal would be significantly aided if done in concert with a
corresponding analysis of the simultaneous neutrino signal. Optimal
methodologies with which to extract physical information from such an analysis
have yet to be designed. Nevertheless, we have in the GW signal of core-
collapse supernovae a direct and real-time window into the supernova mechanism
and PNS evolution. Therefore, such a methodology would likely pay rich
scientific dividends when astronomy is finally presented with the opportunity
to employ it.

## Data Availability

The numerical data associated with this article and sonifications of the GW
strains will be shared upon reasonable request to the corresponding author.
The GW strains as well as the quadrupole data are available publicly at
https://dvartany.github.io/data/ and
https://www.astro.princeton.edu/~burrows/gw.3d.new/.

## Acknowledgments

We thank Jeremy Goodman, Eliot Quataert, David Radice, Viktoriya Morozova,
Hiroki Nagakura, and Benny Tsang for insights and advice during the
germination and execution of this project. DV acknowledges support from the
NASA Hubble Fellowship Program grant HST-HF2-51520. We acknowledge support
from the U. S. Department of Energy Office of Science and the Office of
Advanced Scientific Computing Research via the Scientific Discovery through
Advanced Computing (SciDAC4) program and Grant DE-SC0018297 (subaward
00009650), support from the U. S. National Science Foundation (NSF) under
Grants AST-1714267 and PHY-1804048 (the latter via the Max-Planck/Princeton
Center (MPPC) for Plasma Physics), and support from NASA under award JWST-
GO-01947.011-A. A generous award of computer time was provided by the INCITE
program, using resources of the Argonne Leadership Computing Facility, a DOE
Office of Science User Facility supported under Contract DE-AC02-06CH11357. We
also acknowledge access to the Frontera cluster (under awards AST20020 and
AST21003); this research is part of the Frontera computing project at the
Texas Advanced Computing Center [103] under NSF award OAC-1818253. In
addition, one earlier simulation was performed on Blue Waters under the
sustained-petascale computing project, which was supported by the National
Science Foundation (awards OCI-0725070 and ACI-1238993) and the state of
Illinois. Blue Waters was a joint effort of the University of Illinois at
Urbana–Champaign and its National Center for Supercomputing Applications.
Finally, the authors acknowledge computational resources provided by the high-
performance computer center at Princeton University, which is jointly
supported by the Princeton Institute for Computational Science and Engineering
(PICSciE) and the Princeton University Office of Information Technology, and
our continuing allocation at the National Energy Research Scientific Computing
Center (NERSC), which is supported by the Office of Science of the U. S.
Department of Energy under contract DE-AC03-76SF00098.

##  Appendix A General Equations

We follow the method in ([76], see also [77, 47, 104, 72]) to calculate the
gravitational-wave strain tensor. We calculate the first time derivative of
the mass quadrupole with the following formula:

$q_{ij}=\frac{d}{dt}Q_{ij}=\int
d^{3}x\,(v_{i}x_{j}+v_{j}x_{i}-\frac{2}{3}v_{r}r),$ (4)

where $v$ is the velocity, $x$ is the Cartesian coordinate, $Q_{ij}$ is the
transverse-traceless quadrupole tensor, and $r$ is the radius. The transverse-
traceless GW strain tensor $h^{TT}_{ij}$ is calculated by taking the numerical
time derivative of $q_{ij}$, i.e.,

$h^{TT}_{ij}=\frac{2G}{c^{4}D}\frac{dq_{ij}}{dt}\,,$ (5)

where $D$ is the distance to the source. Hereafter, we drop the superscript
“TT”. We also calculate and dump the quadrupole $Q_{ij}=\int
d^{3}x\,\rho(x_{i}x_{j}-\frac{1}{3}r^{2}\delta_{i}j)$, and its numerical
derivatives are consistent with the values calculated by the above equation.
However, taking numerical derivatives can be viewed as convolving the signal
with a window function, thus it will introduce a bit of extra noise at high
frequencies.

The “plus” and “cross” polarized strains along ($\theta,\phi$) direction are
given by

$h_{+}=\frac{G}{c^{4}D}\left(\frac{dq_{\theta\theta}}{dt}-\frac{dq_{\phi\phi}}{dt}\right)$
(6)

and

$h_{\times}=\frac{2G}{c^{4}D}\frac{dq_{\theta\phi}}{dt}$ (7)

where (from [77])

$\begin{split}q_{\theta\theta}=&\,\,(q_{xx}\cos^{2}\phi+q_{yy}\sin^{2}\phi+2\,q_{xy}\sin\phi\cos\phi)\cos^{2}\theta\\\
&+q_{zz}\sin^{2}\theta-2\,(q_{xz}\cos\phi+q_{yz}\sin\phi)\sin\theta\cos\theta\end{split}$
(8)
$\begin{split}q_{\phi\phi}=&\,\,q_{xx}\sin^{2}\phi+q_{yy}\cos^{2}\phi-2\,q_{xy}\sin\phi\cos\phi\\\
\end{split}$ (9)
$\begin{split}q_{\theta\phi}=&\,\,(q_{yy}-q_{xx})\cos\theta\sin\phi\cos\phi+q_{xy}\cos\theta\,(\cos^{2}\phi-\sin^{2}\phi)\\\
&+q_{xz}\sin\theta\sin\phi-q_{yz}\sin\theta\cos\phi\,.\end{split}$ (10)

The total energy emitted in GWs is

$\begin{split}E_{GW}&=\int_{0}^{t}\sum_{ij}\frac{G}{5c^{5}}\left(\frac{d^{3}Q_{ij}}{dt^{3}}\right)^{2}\\\
\end{split}$ (11)

To calculate GWs from neutrino asymmetries, we follow the prescription of
([91], see also [92]). We include angle-dependence of the observer through the
viewing angles $\alpha\in[-\pi,\pi]$ and $\beta\in[0,\pi]$. The time-dependent
neutrino emission anisotropy parameter for each polarization is defined as

$\alpha_{S}(t,\alpha,\beta)=\frac{1}{\Lambda(t)}\int_{4\pi}d\Omega^{\prime}\,W_{S}(\Omega^{\prime},\alpha,\beta)\frac{d\Lambda}{d\Omega^{\prime}}(\Omega^{\prime},t)\,,$
(12)

where the subscript is $S\in\\{+,\times\\}$ and the GW strain from neutrinos
is defined as:

$h_{S}(t,\alpha,\beta)=\frac{2G}{c^{4}D}\int_{0}^{t}dt^{\prime}\,\Lambda(t^{\prime})\,\alpha_{S}(t^{\prime},\alpha,\beta)\,,$
(13)

where $\Lambda(t)$ is the angle-integrated neutrino luminosity as a function
of time, $D$ is the distance to the source, and WS
($\mathrm{\Omega}^{\prime},\alpha,\beta)$ is the geometric weight for the
anisotropy parameter given by

$W_{\mathrm{S}}=\frac{D_{\mathrm{S}}(\theta^{\prime},\phi^{\prime},\alpha,\beta)}{N(\theta^{\prime},\phi^{\prime},\alpha,\beta)}\,,$
(14)

where (from [91])

$\displaystyle\begin{split}D_{+}&=[1+(\cos(\phi^{\prime})\cos(\alpha)+\sin(\phi^{\prime})\sin(\alpha))\sin(\theta^{\prime})\sin(\beta)+\cos(\theta^{\prime})\\\
&\cos(\beta)]\\{[(\cos(\phi^{\prime})\cos(\alpha)+\sin(\phi^{\prime})\sin(\alpha))\sin(\theta^{\prime})\cos(\beta)\\\
&-\cos(\theta^{\prime})\sin(\beta)]^{2}-\sin^{2}(\theta^{\prime})(\sin(\phi^{\prime})\cos(\alpha)-\cos(\phi^{\prime})\sin(\alpha))^{2}\\}\end{split}$
(15a)
$\displaystyle\begin{split}D_{\times}&=[1+(\cos(\phi^{\prime})\cos(\alpha)+\sin(\phi^{\prime})\sin(\alpha))\sin(\theta^{\prime})\sin(\beta)+\cos(\theta^{\prime})\\\
&\cos(\beta)]2[(\cos(\phi^{\prime})\cos(\alpha)+\sin(\phi^{\prime})\sin(\alpha))\sin(\theta^{\prime})\cos(\beta)\\\
&-\cos(\theta^{\prime})\sin(\beta)]\sin(\theta^{\prime})(\sin(\phi^{\prime})\cos(\alpha)-\cos(\phi^{\prime})\sin(\alpha))^{2}\end{split}$
(15b)
$\displaystyle\begin{split}N&=[(\cos(\phi^{\prime})\cos(\alpha)+\sin(\phi^{\prime})\sin(\alpha))\sin(\theta^{\prime})\cos(\beta)-\cos(\theta^{\prime})\\\
&\sin(\beta)]^{2}+\sin^{2}(\theta^{\prime})(\sin(\phi^{\prime})\cos(\alpha)-\cos(\phi^{\prime})\sin(\alpha))^{2}\,.\end{split}$
(15c)

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Table 1: Model Properties

Model |  Run Time |  Explosion? |  Shock Velocity |  f/g-mode |  Nyquist
---|---|---|---|---|---
|  |  |  |  Energy Fraction |  Frequency
(M⊙) |  (s, pb) |  |  (km s-1) |  Total (Late) |  (Hz)
9$\mathrm{\Theta}$ |  1.47 |  ✓ |  16,000 |  56.08% (N/A) |  5000
9BW |  1.1 |  ✓ |  14,000 |  68.11% (N/A) |  8000
9F |  1.6 |  ✓ |  16,000 |  60.92% (5.75%) |  3000
9F, l |  2.1 |  ✓ |  16,000 |  76.68% (6.63%) |  5000
9.25 |  2.7 |  ✓ |  13,000 |  72.55% (3.06%) |  8000
9.5 |  2.1 |  ✓ |  11,000 |  76.35% (0.72%) |  8000
11 |  4.5 |  ✓ |  11,000 |  85.02% (13.55%) |  6000
12.25 |  2.0 |  ✗ |  |  94.55% (25.71%) |  8000
14 |  1.0 |  ✗ |  |  75.82% (N/A) |  8000
15.01 |  2.0 |  ✓ |  6,000 |  84.46% (5.12%) |  8000
23 |  6.2 |  ✓ |  6,000 |  85.74% (8.50%) |  8000

Table of some the features of the 3D simulation. We include the simulation
time, in seconds post-bounce, the explosion outcome, the asymptotic shock
velocity, and the fraction of the GW energy radiated via the f/g-mode. Models
with a checkmark explode, and models with an ✗ do not explode. The various 9
solar mass models were run on Theta (ALCC), Blue Waters (NCSA), and Frontera
(TACC), respectively. 9F,l is a longer simulation of the 9-M⊙ progenitor, also
done on Frontera. The f/g-mode energy is the gravitational-wave energy within
$\pm 100$ Hz of its central frequency. “Total” means for all time, while
“Late” means after 1.5 seconds after bounce. After the mode repulsion, almost
all the power is the an unmixed f-mode. Also in the table are the lowest
Nyquist sampling frequencies for each run. The behavior at frequencies below
these should not be compromised by Nyquist errors.

Figure 1: Mass density profiles for the various progenitors studied here, from
9.0 (reddish) to 23 (blue) M⊙. The accretion of the Si/O interface frequently
corresponds with the onset of shock revival. Note that the 12.25- and 14-M⊙
models do not explode and are destined to produce a stellar-mass black hole.
Figure 2: Early post-bounce behavior of the mean shock radii (in km) for all
the models as a function of time (in seconds after core bounce). All models
explode except for the 12.25- and 14-M⊙ progenitors, which show evidence for a
spiral SASI after $\sim$400 and $\sim$300 ms post-bounce, respectively, in the
form of $\sim$10 ms oscillations in the shock radii. Note that the 12.25- and
14-M⊙ (non-exploding) progenitors also shows a secular $\sim$70 ms oscillation
in the shock radii. Figure 3: The gravitational-wave strain, from matter
motions, times distance (product in cm) for all the 3D models studied in this
paper as a function of time after bounce (in seconds) observed along the
positive x-axis in the simulation frame. Both cross (blue) and plus (black)
polarizations are illustrated. The red star indicates the approximate time of
the onset of explosion, defined here as when the stalled shock radius changes
concavity at $\sim$150 km and accelerates. See Figure 2.  Figure 4: Same as
Figure 3, but now focusing on the matter strain during the first 250 ms after
core bounce to illustrate the prompt convection. Note the stately oscillation
present in most models of $\sim$10 ms periods visible in the $+$ polarization
and absent in the $\times$ polarization. This is a generic feature indicative
of the strain polarization geometry. Note that only model 9a has any
perturbations on bounce.

Figure 5: In this figure and the next, we provide the gravitational-wave
energy spectrogram (to a normalization constant of order unity) at 10
kiloparsecs (kpc) as a function of time after bounce (in seconds) for the 3D
models highlighted in this paper. Note that the frequency ranges and durations
differ from panel to panel. The colormap provides the spectral density in
units of Bethes (1051 egs) per Hertz, where we have integrated over a running
40-millisecond bin around each time. We have found that the results do not
depend significantly upon the width of this window. Though the models in this
first set are for 9-, 9.25-, and 9.5 M⊙, the description in this caption also
applies to the 11-, 12.25-, 15.01-, and 23-M⊙ models in the accompanying
figure (Figure 6). All models show a strong feature out to $\sim$50 ms and up
to $\sim$2000 Hz associated with prompt convection, followed by a short
quiescent phase, during which the turbulence between the shock and the proto-
neutron star grows. Thereafter, all models illustrate high frequency behavior
before $\sim$one second (even earlier for the rapidly exploding lower
progenitor mass models), which truncates in power as the turbulent accretion
which excites it subsides. This “emission haze” is either a superposition of
core pulsational $\ell=2$ p-modes with a range of node numbers or the GW
emission from the supersonic plumes themselves impinging upon the PNS core, or
some combination of both. The fundamental f/g-mode (early), then the f-mode
(later), is manifest and significant in all models for the duration of the
simulations. We note that the 15.01-M⊙ model (next figure), which has a
relatively lower ejecta mass, sustains turbulent accretion and high frequency
haze emission until the end of our simulation, $\sim$2.0 seconds post-bounce.
We also note that we see higher harmonics/overtones after the first second and
after the earlier emission haze abates, most visibly in the 15.01- and 23-M⊙
models. These later-time PNS pulsational modes are likely the $\ell=2,n=1$
p-mode. The fundamental f-mode frequency grows roughly quadratically with time
([104]) and is associated with the shrinkage of the PNS as it executes its
Kelvin-Helmholtz contraction phase, driven by neutrino losses. Except for the
very early phase after bounce, most g-modes are suppressed due both to the
presence and growth of PNS convection in the deep core and to their lower
frequencies ($\sim$a few hundred Hz). Early on, all models show a dark band
near $\sim$1000 Hz which may be a signature of an avoided crossing and
interference with a trapped g-mode. See text in §II.1 and §II.4 for a
discussion.

Figure 6: Same as Figure 5, but for models 11-, 12.25-, 15.01-, and 23-M⊙.

Figure 7: Plots (2D slices of 3D data) of the Mach number distribution in the
inner 100 kilometers of the 23-M⊙ simulation at 574 and 632 milliseconds after
bounce. Seen are representative supersonic plumes crashing onto the PNS core.
Such plumes not only excite the GW emission, but themselves emit a component
of the GW signal. See text in §II.1 for a discussion.

Figure 8: Plots of the effective strains as a function of time after bounce
(in seconds) for two representative 3D models (the black-hole former 14-M⊙ and
late-exploding 23-M⊙ model). Note that we see a dark red band at low ($\leq$50
Hz) frequencies for the exploding model associated with the asymmetric ejecta
at later times (matter “memory”). The non-exploding model shows a spiral SASI
signature at $\sim$100 Hz. This quasi-periodic feature is also seen after
$\sim$0.4 seconds in Figure 2 for both the 14-M⊙ and 12.25 M⊙ black-hole
formers. Such a feature shows up only for non-exploding models when the shock
radius has shrunken below $\sim$100 km. These features are much less prominent
in the gravitational-wave energy spectra because they are low-frequency, and,
hence, low-power components. There may be a hint of the SASI itself in the
lower left-hand corner of the panel to the right. See text in §II.1 for a
discussion.

Figure 9: In this figure, we show the spectrogram of the z-component of shock
dipole position for four models. Note the focus on lower frequencies. In the
non-exploding 12.25- and 14-M⊙ models, the spiral-SASI modes are clearly seen
and manifest gradually increasing frequencies as the PNS core shrinks. The
exploding models (for example, the 15.01 and 23 M⊙ models depicted here) don’t
show this feature, but do manifest the even lower frequency signals due to the
matter memory caused by explosion anisotropy (see also Figure 8). We note that
the 23-M⊙ model may show a weak SASI signature (analogous with that found in
the 25-M⊙ model in reference [72]) that vanishes at $\sim 0.55$ seconds and is
perhaps indicative of its delayed explosion time. The onset of explosion
corresponds with the rise of the low-frequency signal memory in the bright
yellow band. Figure 10: The gravitational-wave energy from matter motions (in
M⊙ c2) for all the 3D models highlighted in this paper as a function of time
after bounce (in seconds). Note that the total GW energy radiated differs by
$\sim$three orders of magnitude from the least massive, 9-M⊙ to the most
massive, 23-M⊙ progenitor. This energy grows by three orders of magnitude for
the most massive progenitors over the first $\sim$2 seconds of simulation, but
is already asymptoting shortly after one second post-bounce. All models show
rapid growth in the first $\sim$50 ms, associated with the onset of prompt
convection driven by the overturn of the shocked mantle dynamically generated
at and after bounce as the shock stalls initially into accretion. After this
phase, the neutrino-driven turbulence between the shock and the proto-neutron
star core grows in vigor over a period of $\sim$100 milliseconds and excites a
spectrum of core pulsational f- and some p-modes and likely generates a GW
component due to the impinging of the plumes onto the PNS that all together
constitute the bulk of the gravitational radiation issuing from the supernova.
This phase can last from hundreds of milliseconds to $\leq$1.5 seconds,
depending upon progenitor, after which the strains subside to a hum dominated
by the fundamental $\ell=2$ f-mode and (weakly) overtones. This last phase can
last for many seconds. However, the signals from those models destined to
leave black holes are still vigorous for a longer period of time, since
accretion persists for these models until the black hole forms, after which
the signal ceases abruptly. See the text for a discussion. Figure 11: The
gravitational-wave energy from matter motions (in M⊙c2) for all the 3D models
in this study versus the model compactness at 1.75 M⊙ [74]. Note the clear
linear correlation on the log-log scale, with a power-law slope of $\sim$0.73.
The outliers are the non-exploding models (12.25- and 14-M⊙, shown with
triangle markers) and these models generally take longer to radiate a given
amount of GW energy (see Figure 10). The black-hole formers don’t as a rule
manifest turbulence that is as vigorous as those models that eventually
explode. All exploding models show a steep growth in the GW energy associated
with the onset of vigorous turbulence. The 23-M⊙, which explodes the latest,
has the latest growth spurt in gravitational energy. By contrast, non-
exploding models show a slower rate of of increase of GW emission, associated
with a slower growth rate of turbulence. However, due to the persistently
stalled shock and continued accretion, the black-hole formers will generate
GWs for a longer time and will saturate their total GW emission more slowly.
Finally, a spiral SASI mode emerges for the black-hole formers when the
stalled shock wave retreats significantly (below $\sim$100 km). This mode
modulates the stalled accretion shock radius and the GW emissions of these
black-hole progenitors quasi-periodically.

Figure 12: Spacetime diagrams illustrating the convective regions in the PNS
for eight representative models, identified by the convective luminosity, as a
function of time after bounce. Note that convection reaches the PNS core
within $\sim$1.6 to $\sim$4.0 seconds, earlier for the low-mass progenitors
and later for the 23-M⊙ model. For all models, PNS convection starts in a
shell and then grows to encompass much of the core. The fact that g-modes are
evanescent in convective regions bears on the suppression of g-mode excitation
in general, but also on the eventual severing of the interaction/coupling
between a trapped g-mode interior to it and the PNS surface where the plumes
are exciting much of the GW emission. See text for a discussion (§II.4).

Figure 13: The gravitational-wave strain h+D (cm) as a function of time after
bounce (in seconds) for the 9b model, the non-exploding model 12.25-M⊙, and
model 23-M⊙ for various, arbitrary, viewing angles illustrated by color. These
plots do not include the neutrino memory (§II.6). Note the large anisotropy by
viewing angle, which is dominated at late times by the matter memory, which is
sourced by the large-scale explosion asymmetry. In the black hole model
(12.25-M⊙), we see no memory, and little anisotropy, at late times. Though we
show only the $+$ polarization, the results are consistent for both
polarizations. We see that in principle some measure of the degree and angle
of the matter explosion asymmetry can be discerned by the sign and magnitude
of this memory. However, the neutrino memory, if only weakly correlated with
the matter memory, will complicate the extraction of such a feature.
Nevertheless, the difference in the frequency spectra of the matter and
neutrino memories generically, with the former at a higher average frequency,
may allow one someday to disentangle the two. Figure 14: Same as Figure 3, but
for the gravitational-wave memory due to neutrino emission anisotropy. Because
of the cumulative nature of the neutrino GW memory, the strain is both larger
and shows less variation on small timescales when compared to the matter-
driven strain. We reach strains as high as 1000 cm for the 11- and 23-M⊙
models. Note that this is consistent with earlier work [92] showing continued
growth with time of the memory strain. The strain growth rate also
accelerates, showing upwards of 600 cm growth from $\sim$2 to $\sim$3 seconds
for the 11-M⊙ progenitor. This emphasizes our essential point that models
should be carried out to late times to fully capture not only the explosion
dynamics, but the emerging GW signatures. The 11-M⊙ progenitor has the highest
neutrino memory strain among those models we present in this paper. Otherwise,
we see a general correlation with mass, with the low-mass progenitors having
the smallest neutrino memory. The non-exploding 12.25- and 14-M⊙ models show
significant evolution and magnitude in their memory signatures. Figure 15:
Same as Figure 10, but for the radiated gravitational-wave energy due to the
neutrino anisotropy. As seen in Figure 14, unlike the matter contribution, the
neutrino anisotropy continues to contribute to the gravitational signal past
six seconds (for our longest simulation), emphasizing the presence of
sustained neutrino emission asymmetry. We see again at least two orders of
magnitude of growth with time after $\sim$50 ms in the associated GW energy,
this time in the neutrino memory component. The 23-M⊙ progenitor has neutrino
GW memory energies only slightly higher than the matter GW components for the
least energetic, lowest-mass progenitors. Importantly, for the same progenitor
model, the late time neutrino-sourced GW energies, albeit still growing, are
more than two orders of magnitude smaller than the corresponding matter
components.

Figure 16: Turbulent luminosity (blue, in 1051 erg s-1) at 110 km (except for
model 12.25-M${}_{\odot},$ where it is evaluated at 2.5 times the PNS radius),
overplotted with both filtered GW strain polarizations (green and black)
multiplied by distance (product in cm) as a function of time after bounce (in
seconds). We emphasize the clear correlation shown here $-$ the outer
turbulent luminosity closely follows the strain. It reproduces both the global
time behavior, rising and following at coincident times with the strain, and
the episodic ‘packets’ of accretion (for instance, reproducing the bump in the
15.01-M⊙ strain at $\sim$1.35 s and for the 23-M⊙ at $\sim$3.5 s). From this
figure, it is clear that it is the outer downward turbulent convective flux
onto the PNS that drives the gravitational-wave signature after the prompt
convection phase.

Figure 17: In this figure, we show the correlation between the gravitational-
wave energy spectrogram and the accretion rate power in frequency components
with $f>25$ Hz for two representative models. The background spectra in this
figure are the same as those plotted in Figure 5 and 6, but with different
color-bar ranges so that the late time structures can be more clearly seen.
The orange curve shows the accretion rate “power” in frequency components with
$f>25$ Hz. The 25-Hz filter is chosen here because the spectrogram is
calculated using a 0.04s-wide sliding window. The accretion rates are measured
at a radius of 25 km, and the PNS radii of the two models fall below this
value at $\sim$0.9 seconds and $\sim$1.0 seconds after bounce, respectively.
Note also that this colormap for the 23-M⊙ model spectrogram reveals a weak
signal below the f-mode associated with the avoided crossing that may be the
bumped g-mode and that seems to continue beyond $\sim$1.5 seconds. See text in
§II.7 for a discussion.

Figure 18: Convective hydrodynamic luminosity within the PNS (blue, in 1051
erg s-1) at its peak value in the inner PNS, overplotted with both
gravitational-wave strain polarizations (black and green) multiplied by
distance (product in cm) as a function of time after bounce (in seconds) for
models 9a, 9b, 9.25, and 9.5. Figure 19 depicts the same quantities, but for
models 11-, 12.25-, 15.01-, and 23-M⊙. Note that PNS convection first
increases in strength over time and then stays approximately constant (with
some minor excursions) or stays high with a secular evolution for the duration
of the simulations. Moreover, PNS convection grows in extent over time to
encompass more and more of the core, expanding deeper in radius and reaching
the center of the PNS after from $\sim$1.7 seconds (the 9-M⊙ model) to $\sim$4
seconds (the 23-M⊙ model, see Figure 19), all the while maintaining its vigor.
This behavior shows little or no correlation with the GW strain, which
decreases in magnitude with time from an earlier peak and nearly flattens at
late time (dominated then by a humming fundamental f-mode (see Figure 5). In
particular, at late times, when the strain has subsided substantially, the
inner PNS convective luminosity is always still strong, indicating that PNS
convection is not the dominant excitation mechanism of CCSN GW emission.

Figure 19: Same as Figure 18, but for models 11-, 12.25-, 15.01-, and 23-M⊙.
Figure 20: Gravitational-wave sensitivity curve for all the models as a
function of time (in seconds after core bounce), compared with the CCSN GW
signals at 10 kpc, including both matter (high frequency, dominant above
several hundred Hz) and neutrino contributions to GW emissions (low frequency,
dominant from sub-Hz to several tens of Hz). Note that DECIGO, the Einstein
Telescope, and the Cosmic Explorer can probe down to the lowest-mass
progenitors across three decades in frequency ($\sim$10 to $\sim$5000 Hz),
whereas aLIGO is sensitive between $\sim$30 to $\sim$3000 Hz only to the most
massive progenitors. Our lower limit in frequency is set by the duration of
the CCSN simulation. Since our longest simulation is out to 6.2 seconds,
simulations are generally unable to constrain the spectrogram data between
0.1$-$1 Hz. Our upper limit in frequency is set by the Nyquist limit in Table
1.