# Gravitational Waves from Neutrino-Driven Core Collapse Supernovae:
Predictions, Detection, and Parameter Estimation

A. Mezzacappa  Department of Physics and Astronomy, The University of
Tennessee, Knoxville, 1408 Circle Drive, Knoxville, TN 37996 USA  M. Zanolin
Physics and Astronomy Department, Embry-Riddle University, 3700 Willow Creek
Road, Prescott, AZ 86301 USA

###### Contents

  1. I Introduction
  2. II The Physics of Neutrino-Driven Core Collapse Supernovae
  3. III The Status of Core Collapse Supernova Models
  4. IV Sources of Core Collapse Supernova Gravitational Waves
  5. V Calculating Core Collapse Supernova Gravitational Wave Emission: Formalism
  6. VI Predictions of Core Collapse Supernova Gravitational Wave Emission from Multi-Physics Simulations
    1. VI.1 General Characteristics of Core Collapse Supernova Gravitational Wave Strains
    2. VI.2 General Characteristics of Core Collapse Supernova Spectrograms
    3. VI.3 Sourcing Gravitational Wave Emission in Core Collapse Supernovae
    4. VI.4 Culling Information about the Proto-Neutron Star through a Gravitational Wave Detection
  7. VII Detection and Parameter Estimation of Gravitational Waves from Core Collapse Supernovae
    1. VII.1 Detection Strategies
    2. VII.2 Parameter Estimation Strategies
  8. VIII Role of Multimessenger Identification of an Observation Window
  9. IX Conclusions and Outlook
  10. X Acknowledgments

##  I Introduction

In what will hopefully be the not too distant future, Mother Nature will
supply us with a Galactic or near-extra-Galactic core collapse supernova
(CCSN). This will be the multi-messenger astrophysics event of the century,
possibly literally, for which we have all been waiting, and will grace us with
all three astrophysical messengers: gravitational waves (GWs), neutrinos, and
photons across the electromagnetic spectrum. The volumes of information that
will be brought to us by these cosmic couriers will enable us to validate and,
no doubt, greatly improve our models of the supernova central engine, to probe
the high-density, neutron-rich physics of the newly forming neutron star, and
to probe neutrino physics that may manifest only in such a unique, to say the
least, neutrino laboratory.

These are exciting times for both CCSN theory and CCSN observations. For the
first time in the nearly sixty years that core collapse supernovae (CCSNe)
have been modeled numerically, a consensus across leading CCSN theory groups
worldwide is developing: CCSNe can in fact be driven by neutrinos, as
originally proposed by Colgate and White Colgate and White (1966) and, then,
in the version of the story on which all modern CCSN theory is based, by Bethe
and Wilson Bethe and Wilson (1985). Across numerical methods implemented, CCSN
codes that instantiate them, and progenitor characteristics, such as mass,
rotation, and metallicity, explosions are now obtained by these groups
routinely. The CCSN theory community has entered a new phase of exploration.
The challenge now is to obtain results that are in quantitative agreement with
observations (e.g., explosion energies commensurate with observed explosion
energies), not qualitative agreement (e.g., explosion versus the lack of an
explosion).

The momentum the worldwide CCSN modeling community now has, and the promise of
a multimessenger detection, which could happen at any time, now provide the
impetus to the modeling community to address the significant challenges that
remain to develop models sufficiently accurate to be able to predict
successfully the outcomes for a given progenitor and, in particular, the
progenitor that will be associated with the next Galactic or near-extra-
Galactic event. This remains a tall order, and one that relies not only on the
CCSN modeling community to fill. CCSN theory is still limited by the lack of
three-dimensional progenitor models, although important progress has been made
on at least the late stages of stellar evolution. And uncertainties in
neutrino weak interaction physics, both theoretical and experimental, and in
the high-density, neutron-rich, hot nuclear equation of state must be
addressed by others. Nonetheless, decades of progress toward realizing the
goal of understanding the CCSN explosion mechanism has taught us one thing:
Progress is made, sometimes in increments, sometimes in leaps, by improving
the models in each sector of the macroscopic physics (e.g., the models of
gravity, hydrodynamics, magnetohydrodynamics, and neutrino kinetics
(transport) implemented) and the microscopic physics (e.g., the neutrino weak
interaction rates and nuclear equation of state), one by one. Certainly, the
development of high-fidelity models of the macroscopic physics together with
the implementation of the state of the art in weak interaction and equation of
state physics is the CCSN modeling community’s job.

To this end, and duly noting the significant progress that has been made and
all of the hard work that went into accomplishing it, it is important to note
that only recently have we seen the first published general relativistic CCSN
model with state-of-the-art microphysics (see Section III). And to be clear,
by general relativistic we mean the implementation of general relativistic
hydrodynamics or magnetohydrodynamics, general relativistic gravity, and
general relativistic neutrino kinetics, and their coupling. As we will discuss
in Section III, some of the leading models are general relativistic but lack
critical microphysics, while others that include the microphysical state of
the art implement approximations to general relativity across the major
sectors of gravity, hydrodynamics, and neutrino kinetics. Overcoming the
shortcomings on either side of this modeling frontier, thereby bridging the
two, is the primary challenge CCSN modelers face now. But progress continues
to be rapid, and we should have every expectation this gap will be bridged in
the not too distant future.

Even with this gap bridged, modelers must face some longer-term challenges
that may effect CCSN model outcomes and, with that, our ability to derive the
maximal benefit from a Galactic or near-extra-Galactic CCSN detection. In
order to get these benefits we also need to have the best possible instruments
to record the electromagnetic (EM) emissions, GWs, and neutrinos from these
extraordinary events.

The detection of the first GWs from binary-black-hole mergers in 2015 enabled
new comparisons between theoretical predictions and observations. The benefits
from similar comparisons in the case of CCSNe would be enormous, even with the
observation of a single CCSN. While other reviews already discuss the value of
CCSN EM and neutrino observations, here we focus on the idiosyncrasies of the
detection and parameter estimation (PE) from the gravitational wave (GW)
channel, using ground-based laser interferometers. Several valuable resources
describe the detection and PE software for compact-binary coalescences (CBCs),
where the key tool is cross correlation between analytical or numerical
deterministic templates and laser interferometric data. The development of
algorithms for the detection and PE of CCSN signals on the other hand is more
complex both because of the stochastic aspects of the waveforms, as well as
the presence of, as we will see, what can only be termed emergent parameters.
It is important to notice that what was initially perceived as only a
stochastic GW time series has revealed deterministic features both in the time
domain and in the time–frequency domain. Regarding the algorithms for
detection and PE, the landscape of methods of which they are composed, both
developed and under development, is vast and still growing: likelihood
methods, wavelet decomposition, machine learning (ML), and frequentist and
Bayesian approaches are some of the more common methods finding application in
this setting. We review the current status of the field, including a
discussion of current research directions where hypothetical EM and neutrino
observations are paired with GW observations in multi-messenger strategies.

##  II The Physics of Neutrino-Driven Core Collapse Supernovae

CCSNe mark the death of massive stars ($M>8-10$ M⊙). The iron core of such
stars is supported primarily by degenerate relativistic electron pressure, for
which a maximum mass, the Chandrasekhar mass, can be supported. As fusion
continues in the silicon layer above the iron core, the iron core mass comes
to exceed the Chandrasekhar mass, and the core collapses, initiating the
supernova. Sound speed decreases with density away from the center and infall
velocity increases with distance away from the center. The formation of a
sonic point is inevitable, which will set the stage for the formation of the
supernova shock wave that will ultimately be responsible for the demise of the
progenitor star. The inner, subsonic core proceeds through nuclear matter
densities, between $1-2\times 10^{14}$ gcm-3, as it collapses and undergoes a
phase transition from nucleons and nuclei to nuclear pasta to bulk nuclear
matter. At sufficiently high densities, the hard core in the nucleon–nucleon
interaction potential, which renders the nucleon–nucleon interaction
repulsive, and the Pauli exclusion principle, which also repels like nucleons,
lead to a “stiffening” of the nuclear equation of state (the pressure
increases more rapidly with an increase in density). This stiffening halts the
collapse and, moreso, leads to a rebound of the inner core. At the sonic
point, information carried by sound about this rebound can no longer propagate
outward, and a discontinuity in the flow, with outflow meeting inflow, forms –
i.e., the supernova shock wave.

The so-called “prompt mechanism” for CCSNe, where the newly formed shock wave
propagates outward, as a dynamical shock, through the outer iron core and the
stellar layers above it, reversing the infall of the stellar matter and
generating, instead, an outflow, fails. There are two culprits: dissociation
and neutrino losses. As the shock propagates through the outer iron core, the
nuclei are dissociated by shock compression and heating, costing the shock
energy to the tune of $1\times 10^{51}$ ergs per $0.1M_{\odot}$ of mass
dissociated. Then, as the shock passes the electron neutrinosphere, the sphere
of last scattering of the electron neutrinos produced through electron capture
on nuclei and nucleons during collapse, above which the electron neutrinos are
free to escape, carrying energy with them, the shock loses sufficient energy,
loses its status as a dynamical shock, and “stalls,” becoming instead a
standing accretion shock.

The stage is then set for the “delayed shock mechanism,” whereby the shock is
reenergized to become a dynamical shock. The core becomes stratified. The
region below the neutrinospheres (at this point, all three flavors of
neutrinos are present) constitutes the proto-neutron star (PNS). Above it, in
the cavity between the neutrinospheres and the shock, the matter is heated and
cooled by charged-current electron-neutrino interactions involving the
nucleons in this cavity, released from nuclei during dissociation. Cooling
dominates at higher densities at the base of the cavity, and heating dominates
above it, in the region just below the accretion shock. A cooling layer and a
heating layer are formed. Neutrino energy deposition in the heating layer is
responsible for the revival of the shock.

At core bounce and shock formation and its initial propagation through the
core, entropy and lepton gradients are formed, which in turn drive a period of
“prompt (Ledoux) convection” in the PNS. Heating of the region below the
supernova shock wave by neutrinos emerging from the PNS drives “neutrino-
driven (Schwarzchild) convection” in the heating region. These instabilities
assist the neutrinos in generating an explosion by boosting the neutrino
luminosities at the neutrinospheres and by boosting the neutrino heating
efficiency of the material below the shock. They also generate GWs. These
“local” instabilities are joined by a “global” instability of the supernova
shock wave, the so-called Standing Accretion Shock Instability (SASI),” the
latter of which assists neutrino heating by increasing the size of the heating
region. The SASI, too, generates GWs. Thus, the very ingredients that lead to
explosion by neutrino heating, each have gravitational wave (GW) fingerprints.
This will be critical in determining whether or not a CCSN has been detected
in GWs, as well as necessary in order to use such a detection to validate CCSN
models.

##  III The Status of Core Collapse Supernova Models

Fortunately, we are now in possession of dozens of three-dimensional CCSN
models that cut across progenitor characteristics of mass, rotation, and
metallicity, and that have implemented different nuclear equations of state.
All of these models are quite sophisticated, though all possess major
weaknesses. These weaknesses stem from three shortcomings: (1) The use of an
effective gravitational potential that is intended to capture the stronger
gravitational fields expected from general relativity in an otherwise
Newtonian treatment. (2) The use of the so-called “ray-by-ray” approximation
for neutrino kinetics. (3) The lack of a complete (or nearly complete) set of
neutrino interaction physics. By complete, we mean the inclusion of all weak
interactions known to be important, or at least nonnegligible, and the
implementation of state of the art treatments of them.

The use of an effective potential in the context of CCSNe was first proposed
by Rampp and Janka (2002). In this approximation, Newtonian hydrodynamics is
deployed and the Newtonian gravitational potential is modified. Specifically,
the monopole term is replaced by a spherically symmetric potential derived by
comparing the Newtonian equation for hydrostatic equilibrium with the
Tolman–Oppenheimer–Volkoff equation for general relativistic hydrostatic
equilibrium. In a multidimensional context, input to the general relativistic
potential is obtained by spherical averaging of the needed quantities, such as
the rest mass density, internal energy density, pressure, etc. Liebendoerfer
et al. Liebendörfer  _et al._ (2005) pointed out that the use of an effective
potential, at least as originally formulated, overestimates general
relativistic effects. As a result, a number of modifications to the procedure
were developed and tested (Cases A through N in Marek  _et al._ (2006)), with
one option, Case A, considered to reproduce best the results of general
relativistic simulations. However, a later study by Mueller, Janka, and Marek
Müller  _et al._ (2012), in the context of two-dimensional CCSN simulations
beginning with the same 15 $M_{\odot}$ progenitor, yielded qualitatively
different outcomes for the simulation that was general relativistic and the
simulation that relied on the effective potential approach (Case A), with the
former yielding an explosion and the latter not. The effective potential
approach is used by four separate groups: the Max Planck group, the
MSU–Stockholm collaboration, the Princeton group, and the UT–ORNL group. These
four groups have produced many of the most sophisticated three-dimensional
models published to date. Obviously, this approximation to general relativity
has enabled the community to make progress, but no one would argue that we
should be satisfied with such an approach and not move toward general
relativistic treatments of gravity. Decades of supernova modeling have
revealed that model outcomes are sensitive even to small variations in the
input physics or the treatment of the major physics components in the models.

The ray-by-ray approximation to neutrino transport was also first proposed by
Rampp and Janka (2002). In this approximation, the solution of the three-
dimensional neutrino kinetics equations is replaced by the solution of a set
of spherically symmetric neutrino kinetics equations, one for each radial ray
through each $(\theta,\phi)$ point on the computational grid. This approach
was motivated in part by the fact that sophisticated neutrino kinetics solvers
had been developed for spherically symmetric simulations of CCSNe. The ray-by-
ray approximation is exact, though redundant, for spherically symmetric
problems. Thus, the efficacy of the approximation depends on how close to
spherical symmetry the problem – more precisely, the PNS – remains during the
evolution. In axisymmetry, massive accretion funnels persist along the plain
perpendicular to the symmetry axis, exciting neutrino emission from the PNS
surface in a manner far from spherical symmetry. In these cases, the
approximation is not ideal, as discussed by Skinner, Burrows, and Dolence
Skinner  _et al._ (2016). In three dimensions, without the imposition of a
symmetry axis and, therefore, the formation of a small number of massive
accretion funnels pinned to equatorial regions of the PNS, the approximation
is more effective, as demonstrated by Glas et al. Glas  _et al._ (2019). Two
of the above four groups use the ray-by-ray approximation: the Max Planck
group and the UT–ORNL group. Again, while the ray-by-ray approximation has
enabled important advancements in CCSN theory, particularly in three-
dimensional CCSN modeling, no one would argue that we should not move toward
three-dimensional solutions to the neutrino kinetics equations.

The history of the deployment of increasingly realistic neutrino weak
interaction physics in CCSN models has followed the development of the theory
of weak interaction physics itself, especially in the years immediately
following the publication of the electroweak theory. In particular, neutral-
current weak interactions, which allow coherent scattering of neutrinos on
nuclei, fundamentally altered the dynamics of stellar core collapse Wilson
(1974). The later introduction of new weak interactions – e.g., neutrino pair
production by nucleon–nucleon bremsstrahlung Hannestad and Raffelt (1998) –
opened up new channels for the production of neutrino pairs of all three
flavors, and more recent work factoring in nucleon correlations in nuclei Hix
_et al._ (2003); Langanke  _et al._ (2003) and nuclear matter Burrows and
Sawyer (1998); Reddy  _et al._ (1998), all proved to be essential improvements
to CCSN models. Thus, a minimal set of neutrino weak interactions must include
(1) electron capture on nuclei, with nucleon correlations taken into account,
(2) electron and positron capture on nucleons, including degeneracy and
relativity, (3) coherent, isoenergetic scattering on nuclei, (4)
nonisoenergetic scattering on nucleons, factoring in the small energy
exchange, (5) nonisoenergetic scattering on electrons and positrons, (6)
neutrino pair production from electron–positron pairs, and (7) neutrino pair
production from nucleon–nucleon bremsstrahlung. Prior to near the turn of the
millennium, all models included (1)–(6) but without the effects of nucleon
correlations, degeneracy, relativity, and small-energy scattering – the so-
called “Bruenn85” opacity set Bruenn (1985). Some of today’s leading three-
dimensional models Lentz  _et al._ (2015); Melson  _et al._ (2015); Burrows
_et al._ (2019); Vartanyan  _et al._ (2019), at least from a microphysical
perspective, include the minimal set and, in some cases more (e.g., the
inclusion of muons and their associated weak interactions Bollig  _et al._
(2017)). Until recently, other of today’s leading models Roberts  _et al._
(2016); Kuroda  _et al._ (2016b), from a macrophysical perspective [i.e., were
three-dimensional and general relativistic (or nearly so)], did not include
the minimal set (they include the Bruenn85 set or the Bruenn85 set plus pair
production from nucleon–nucleon bremsstrahlung).

Kuroda (2021) finally advanced the frontier to perform the first three-
dimensional, general relativistic CCSN simulations with an extensive set of
neutrino interactions.

##  IV Sources of Core Collapse Supernova Gravitational Waves

At this point in time, a number of studies of CCSN GW emission have been
conducted based on simulation data from three-dimensional models Afle  _et
al._ (2023); Andresen  _et al._ (2017, 2019); Bruel  _et al._ (2023); Drago
_et al._ (2023); Hayama  _et al._ (2015, 2018, 2016); Kawahara  _et al._
(2018); Kotake  _et al._ (2011, 2009); Kuroda  _et al._ (2022, 2017, 2016a,
2018); Lin  _et al._ (2023); Matsumoto  _et al._ (2022); Mezzacappa  _et al._
(2020, 2023); Nakamura  _et al._ (2022); O’Connor and Couch (2018); Ott  _et
al._ (2013); Pajkos  _et al._ (2023, 2021); Pan  _et al._ (2018, 2021); Powell
and Müller (2019, 2020, 2022); Radice  _et al._ (2019); Richardson  _et al._
(2022); Richers  _et al._ (2017); Scheidegger  _et al._ (2010); Shibagaki  _et
al._ (2020, 2021); Srivastava  _et al._ (2019); Takiwaki and Kotake (2018);
Vartanyan and Burrows (2020); Vartanyan  _et al._ (2019, 2023); Warren  _et
al._ (2020). Given the results of these studies, and the conclusions reached,
the sources of GWs in CCSNe can be broken down as follows:

Figure 1: Sources of CCSN GW emission (taken from Andresen  _et al._ (2017)),
beginning with sustained Ledoux convection in Region A1 deep within the PNS,
moving outward to convective overshoot in Region A2, then on to neutrino-
driven convection in the gain layer, Region C, which generates GWs per se but
also leads to accretion plumes impinging on the PNS surface. These plumes in
turn excite GW emission from the PNS. Finally, the SASI modulates the flow in
Region C, as well as the accretion onto the PNS surface, indicated here by the
$\rho=10^{10}$ gcm-3 contour, generating low-frequency GW emission.

Core Bounce: We expect the majority of CCSN progenitors to be rotating, albeit
slowly. In the presence of such rotation, stellar core bounce results in a
significant time-dependent quadrupole moment, which in turn produces the first
significant feature in the GW strain. This feature comprises a prominent spike
in the strain amplitude, together with secondary peaks associated with bounce-
induced oscillations of the PNS (see, for example, Richers  _et al._ (2017).
In most of the studies cited above, the bounce signature is not present in the
waveforms given that rotation was not included in the associated models. In
the absence of three-dimensional progenitors, it is difficult to know what the
differential core rotation is across the progenitors considered.

Prompt Convection in the Proto-Neutron Star: This is convection that occurs
immediately after bounce, due to the gradients in entropy and lepton fraction
left behind by (i) the weakening shock and (ii) the core deleptonization
during collapse and the escape of copious electron neutrinos during the
electron-neutrino burst within the first $\sim$5 ms after core bounce,
respectively. It is short-lived, normally occurring between 5 and 20 ms after
bounce. The convection flattens the gradients that give rise to it. Prompt
convection gives rise to high-frequency ($>500$ Hz) GW emission.

Sustained Proto-Neutron Star Ledoux Convection: Unlike prompt convection, PNS
Ledoux convection is sustained by lepton gradients that are sustained by the
continuing deleptonization of the PNS by neutrino diffusion out of it. It
begins at about 100 ms after bounce and is sustained for the duration of the
simulations. Like prompt convection, PNS convection gives rise to high-
frequency GW emission.

Convective Overshoot from Sustained Proto-Neutron Star Ledoux Convection: This
results from the convective overshoot into the convectively stable layer above
the Ledoux convection layer. Its contribution to the high-frequency GW
emission is typically larger than the emission emanating from the convective
region itself. Here too, the GW emission is of high frequency.

Neutrino-Driven Convection in the Gain Layer: Shock revival is mediated by
heating of the material below the stalled shock wave by electron-neutrino and
electron-antineutrino absorption on the dissociation-liberated neutrons and
protons, respectively. Because of the $r^{-2}$ dependence of the neutrino flux
emanating from the PNS surface (the neutrinospheres), the heating is greatest
at the base of the heating or “gain” layer. As a result, an entropy gradient
develops in this layer and Schwarzchild convection follows. The convection
becomes turbulent. Turbulent neutrino-driven convection gives rise to low-
frequency ($<250$ Hz) GW emission.

The Standing Accretion Shock Instability (SASI): The stalled supernova shock
wave is unstable to non-spherical perturbations. The perturbations grow,
leading to “sloshing” Blondin  _et al._ (2003) and “spiraling” Blondin and
Mezzacappa (2007) motions beneath it, which in turn are reflected in the shape
of the shock wave itself as the explosion develops. Like turbulent neutrino-
driven convection, the SASI gives rise to low-frequency GW emission.

Accretion onto the Proto-Neutron Star Surface: Given turbulent neutrino-driven
convection and the SASI, the accretion onto the PNS, which occurs via
accretion plumes that penetrate the gain radius separating the gain layer and
the neutrino cooling layer below it, impinging on the PNS surface and exciting
high-frequency GW emission.

Explosion: The non-spherical explosion of material gives rise to a low-
frequency time-dependent quadrupole moment and, consequently, very-low-
frequency ($<50$ Hz) GW emission.

Most of the eight sources listed above are illustrated in Figure 1.

As we will discuss in Section VI, predictions for high- and low-frequency GW
emission across the studies conducted to date by various groups are
qualitatively very similar. On the other hand, conclusions regarding the
sourcing of this GW emission have varied considerably across the same studies.

##  V Calculating Core Collapse Supernova Gravitational Wave Emission:
Formalism

The Einstein equations for the spacetime metric, $g_{\mu\nu}$, in the line
element

$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}$ (1)

are given in natural units by

$G_{\mu\nu}=8\pi T_{\mu\nu},$ (2)

where the Einstein tensor, $G_{\mu\nu}$, is

$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R,$ (3)

and the Ricci tensor, $R_{\mu\nu}$, and the Ricci scalar, $R$, are

$R_{\mu\nu}=R^{\lambda}_{\mu\lambda\nu}$ (4)

and

$R=g^{\mu\nu}R_{\mu\nu}.$ (5)

The dependence on the metric implicit in the quantities above is made explicit
by considering the Riemann curvature tensor, $R^{\lambda}_{\mu\beta\nu}$,

$R^{\lambda}_{\mu\beta\nu}=\partial_{\beta}\Gamma^{\lambda}_{\mu\nu}-\partial_{\nu}\Gamma^{\lambda}_{\mu\beta}+\Gamma^{\lambda}_{\alpha\beta}\Gamma^{\alpha}_{\mu\nu}-\Gamma^{\lambda}_{\alpha\beta}\Gamma^{\alpha}_{\mu\beta},$
(6)

which is expressed in terms of the connection coefficients,
$\Gamma^{\lambda}_{\mu\nu}$,

$\Gamma^{\lambda}_{\mu\nu}=\frac{1}{2}g^{\lambda\rho}\left(\partial_{\nu}g_{\mu\rho}+\partial_{\mu}g_{\nu\rho}-\partial_{\rho}g_{\mu\rho}\right).$
(7)

In the case of CCSNe, GWs can be considered as linear perturbations of the
background spacetime geometry. Choosing the background geometry to be the
geometry far from the source, we write

$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}.$ (8)

Upon substitution of this metric in the Einstein equations, Eqn. (2), we
obtain a wave equation for the metric perturbation, $h_{\mu\nu}$,

$\Box\bar{h}_{\mu\nu}=-16\pi T_{\mu\nu},$ (9)

where

$\bar{h}_{\mu\nu}=h_{\mu\nu}-\frac{1}{2}hg_{\mu\nu},$ (10)

and where $h$ is the trace of $h_{\mu\nu}$,

$h=g^{\mu\nu}h_{\mu\nu}.$ (11)

In the vacuum case, Eqn. (9) becomes (from now on we remove the bar notation)

$\Box h_{\mu\nu}=0,$ (12)

with solutions of the form

$h_{\mu\nu}=e_{\mu\nu}e^{ik\cdot x},$ (13)

where $e_{\mu\nu}$ is the polarization tensor, which for a GW propagating
along the $\hat{\mathbf{e}}_{z}$ direction is

$e_{\mu\nu}=\left(\begin{matrix}h_{+}&h_{\times}&0\\\ h_{\times}&-h_{+}&0\\\
0&0&0\end{matrix}\right).$ (14)

$h_{+}$ and $h_{\times}$ represent the amplitudes of the two polarizations of
the GW, the so-called “plus” and “cross” polarizations, respectively.

In the presence of a source, the solution to Eqn. (2) can be written as

$\bar{h}_{\mu\nu}=-16\pi\int
d^{4}x^{\prime}G(x-x^{\prime})T_{\mu\nu}(x^{\prime}),$ (15)

where the Green’s function, $G(x-x^{\prime})$, satisfies

$\Box G(x-x^{\prime})=\delta^{(4)}(x-x^{\prime}),$ (16)

whose solution is

$G(x-x^{\prime})=-\frac{1}{4\pi|\mathbf{x}-\mathbf{x}^{\prime}|}\delta(t-|\mathbf{x}-\mathbf{x}^{\prime}|-t^{\prime}).$
(17)

Substituting Eqn. (17) in Eqn. (15), we obtain

$\bar{h}_{\mu\nu}(\mathbf{x},t)=4\int
d^{3}x^{\prime}\frac{T_{\mu\nu}(\mathbf{x}^{\prime},t-|\mathbf{x}-\mathbf{x}^{\prime}|)}{|\mathbf{x}-\mathbf{x}^{\prime}|},$
(18)

where the integration is over the source volume.

We now adopt the slow motion approximation and expand $T_{\mu\nu}$ in Eqn.
(18) in powers of $v/c$. In addition, we perform a transverse traceless
decomposition of the metric to arrive at (putting back the factors of $c$)

$h^{TT}_{ij}=\frac{G}{c^{4}}\frac{1}{r}\sum_{m=-2}^{+2}\frac{d^{2}I_{2m}}{dt^{2}}(t-\frac{r}{c})f^{2m}_{ij},$
(19)

where

$I_{2m}=\frac{16\sqrt{3}\pi}{15}\int\rho Y^{*}_{2m}r^{2}dV$ (20)

is the mass quadrupole moment and

$f^{lm}_{ij}=r^{2}\left(\begin{matrix}0&0&0\\\ 0&W_{lm}&X_{lm}\\\
0&X_{lm}&-\sin^{2}\theta\,W_{lm}\end{matrix}\right)$ (21)

are the tensor spherical harmonics, with

$X_{lm}=2\frac{\partial}{\partial\phi}\left(\frac{\partial}{\partial\theta}-\cot\theta\right)Y_{lm}(\theta,\phi)$
(22)

and

$W_{lm}=\left(\frac{\partial^{2}}{\partial\theta^{2}}-\cot\theta\frac{\partial}{\partial\theta}-\frac{1}{\sin^{2}\theta}\frac{\partial^{2}}{\partial\phi^{2}}\right)Y_{lm}(\theta,\phi).$
(23)

The gravitational wave “strain” amplitudes of the plus and cross polarizations
are then given by

$h_{+}=\frac{h_{\theta\theta}}{r^{2}}$ (24)

and

$h_{\times}=\frac{h_{\theta\phi}}{r^{2}\sin\theta}.$ (25)

For additional details, we refer the reader to Kotake  _et al._ (2006).

##  VI Predictions of Core Collapse Supernova Gravitational Wave Emission from
Multi-Physics Simulations

###  VI.1 General Characteristics of Core Collapse Supernova Gravitational
Wave Strains

Figure 2: CCSN GW strains (at the source) for both polarizations for a
progenitor of mass 15 M⊙, from Mezzacappa  _et al._ (2023).

Figure 2 shows the plus- and cross-polarization GW strains (at the source) for
a 15 M⊙ model from Mezzacappa  _et al._ (2023). Given the lack of rotation,
there is no bounce signal and, with the exception of some low-amplitude
emission from prompt convection, GW emission does not begin until $\sim$100 ms
after bounce, when several excitation mechanisms set in. By this time,
sustained Ledoux convection within the PNS has begun, and neutrino-driven
convection and the SASI have developed, all three of which in turn excite
high-frequency GW emission from the PNS. This period of significant high-
frequency emission is supplemented by an overall offset of the strain after
$\sim$500 ms post-bounce. This offset, or “memory” results from explosion.

###  VI.2 General Characteristics of Core Collapse Supernova Spectrograms

Figure 3 shows the spectrogram for the model considered above. Focusing first
on the time period $t>100$ ms after bounce, we see that the spectrogram
exhibits two primary features: The first, a component above $\sim$500 Hz that
rises in frequency with post-bounce time. The second, a component below
$\sim$250 Hz that remains within a constant range in frequency with post-
bounce time. The high-frequency component of the spectrogram originates in the
PNS. It is excited by prompt convection, sustained Ledoux convection, and
sustained Ledoux convection overshoot in the PNS, as well as accretion onto
the PNS. The low-frequency component can originate from the gain layer due to
the SASI’s impact on the flow in this region, and/or from low-frequency
modulation of the accretion onto the PNS by the SASI. At later times, at very
low frequencies, the GW emission originates from explosion itself.

The situation is dramatically different for $t<100$ ms, where little GW
emission is seen. This is in part an artifact of the use of non-rotating and
non-perturbed progenitors (i.e., spherically symmetric progenitors).
Mezzacappa  _et al._ (2023) document a dramatically different case, initiated
from a 9.6 M$\odot$ progenitor. During collapse, this model exhibits nuclear
burning as a result of its composition, which perturbs the core prior to
bounce, yielding significant GW emission early after bounce. This, together
with earlier work by Couch  _et al._ (2015), Müller  _et al._ (2017), and
Vartanyan  _et al._ (2022), highlights the need for three-dimensional
progenitor models for both explosion modeling and predicting accurately this
early phase of GW emission. Until these models are available, confidence in
our predictions of CCSN GW strains and spectrograms will be higher at later
times after bounce, where agreement across model predictions by different
groups has been demonstrated.

###  VI.3 Sourcing Gravitational Wave Emission in Core Collapse Supernovae

Figure 3: Spectrogram for the model considered in Figure 2. Figure 4: Region
R5 spectrogram for the model considered in Figure 2. Figure 5: GW strain by
region for the model considered in Figure 2.

GW emission from neutrino-driven CCSNe is directly linked to the components of
the central engine driving such supernovae. The construction of a detailed GW
strain from a detection of the next Galactic or near-extra-Galactic event will
then provide an opportunity to probe these components and to validate our
models of them. However, this will rely on our ability to source spatially the
GW emission from the center of the PNS outward. Thus far, both indirect and
direct methods have been developed to do so.

In the context of three-dimensional CCSN models, direct methods to source
spatially the GW emission have been developed by Andresen  _et al._ (2017) and
Mezzacappa  _et al._ (2020, 2023). Figure 5 shows the strain amplitude for the
15 M⊙ model of Mezzacappa  _et al._ (2023). The strain is computed for five
distinct regions, R1 through R5, corresponding to A1 (R1), A2 (R2), B (R3),
and C (R4 and R5) in Figure 1. The evolution of the high-frequency emission
and its sources is made clear by this spatial decomposition. Over the first
several hundred milliseconds after bounce, it emanates initially from R4. The
emission then moves inward to region R3 – i.e., the PNS surface layer – and
R2, excited by accretion onto the PNS. However, at later post-bounce times,
the high-frequency emission emanates largely from deep within the PNS, from
regions R1 and R2 – i.e., the region of Ledoux convection and the convective
overshoot region above it. This one example illustrates how complex and time
dependent the emission and sourcing in a model can be. In this model, there is
no single source or region that is responsible for the high-frequency GW
emission throughout the evolution.

Focusing now on the low-frequency emission in this model, Figure 4 shows the
spectrogram for the same 15 M⊙ model but for region R5 only. Comparing Figures
3 and 4, it is evident that in this model low-frequency emission stems from
region R5, the gain layer. For Andresen et al., on the other hand, the low-
frequency emission stemmed from the PNS as the result of the low-frequency
modulation of accretion onto the PNS by the SASI.

Moving now to indirect methods that have been developed to source CCSN GW
emission, we consider the example provided by Radice  _et al._ (2019).
Specifically, these authors computed the energy radiated in GWs versus the
turbulent energy accreted onto the PNS for four models of 9, 10, 11, and 12 M⊙
they considered. The strong correlation between the two, shown in Figure 6,
led Radice  _et al._ (2019) to conclude that the GW emission is sourced in
these models primarily by accretion onto the PNS by turbulent-convection– and
SASI–induced funnels.

Figure 6: Total GW energy released as a function of the energy (kinetic and
thermal) accreted by the PNS, for the models considered in Figure 6 of Radice
_et al._ (2019).

###  VI.4 Culling Information about the Proto-Neutron Star through a
Gravitational Wave Detection

In light of the discussion in the previous section, the detection of GW
emission from a CCSN will enable us to observe the fluid instabilities at play
that aid the neutrinos in powering an explosion. The hope, too, is that a
detection would also enable us to say something about the PNS remnant – e.g.,
its mass and radius. For this, the GW astrophysics community has turned to
asteroseismology.

Applications of asteroseismology to CCSNe, more specifically the PNS, were
first carried out by Ferrari  _et al._ (2003) and later by Fuller  _et al._
(2015), Sotani and collaborators Sotani and Takiwaki (2016); Sotani  _et al._
(2017, 2019); Sotani and Takiwaki (2020); Sotani  _et al._ (2021), Torres-
Forne’ and collaborators Torres-Forné  _et al._ (2018, 2019a, 2019b), Morozova
and collaborators Morozova  _et al._ (2018); Radice  _et al._ (2019),
Westernacher-Schneider and collaborators Westernacher-Schneider  _et al._
(2019); Westernacher-Schneider (2020), and others (e.g., most recently by Afle
_et al._ (2023) and Mori  _et al._ (2023)). This work has led to the
association of specific features of the CCSN spectrograms computed by various
groups with specific modes: $g$-modes, mediated by gravity, $p$-modes,
mediated by pressure, and the fundamental or $f$-mode, also mediated by
pressure. Arguably the prominent high-frequency feature observed in all CCSN
spectrograms computed to date is a focus of CCSN GW detection and PE, in the
latter case particularly with regard to estimating the mass and radius of the
PNS.

It is well known from asteroseismology that the peak frequency of $g$-modes
depend on the gravity at the source – i.e.,

$f_{\rm peak}\sim\frac{GM}{R^{2}},$ (26)

where $G$ is Newton’s gravitational constant, $M$ is the enclosed mass, and
$R$ is the radius, at which the modes are excited. On the other hand, the peak
frequency of an $f$-mode depends on the square root of the mean density of the
star – i.e.,

$f_{\rm peak}\sim\sqrt{\frac{M}{R^{3}}}\,,$ (27)

where in this case $M$ and $R$ are the stellar mass and radius, respectively.
If we are to use a CCSN GW detection – specifically the peak frequency of the
detected GWs and its time evolution – to determine the mass and/or radius of
the PNS and, in turn, to learn something about its equation of state, we must
first know what modes are giving rise to the peak frequency emission.

The results from two-dimensional CCSN models pointed to surface $g$-modes
excited by accretion funnels from the gain layer impinging on the PNS surface.
Three-dimensional models on the other hand have painted a more complex and
varied picture, as discussed in Section VI.3. $g$-modes can be excited by
accretion onto the PNS and by convection (specifically, convective overshoot)
deep within the PNS. In both cases, the relationship to the peak frequency,
Eqn. (26), remains the same, but $M$ and $R$ are obviously not the PNS mass
and radius in both cases. With regard to the fundamental mode, this ambiguity
does not arise given that $M$ and $R$ in Eqn. (27) are in fact the PNS mass
and radius. However, up until now we have assumed that the peak frequency
emission seen in CCSN spectrograms is due to a single mode, $g$ or $f$, but
given the results of past analyses, this emission results from a combination
of $g$\- and $f$-modes, with $g$-mode emission dominating early and $f$-mode
emission dominating late. (See Figure 7, which is taken from Morozova  _et
al._ (2018).)

Thus, our ability to cull information about the PNS from a CCSN GW detection
is predicated on (i) capturing the evolution of the peak frequency of the GW
emission, (ii) determining which modes are present in the peak emission, and
(iii) sourcing the peak emission.

Figure 7: Spectrogram for a 10 M⊙ non-rotating progenitor derived from an
axisymmetric simulation using the FORNAX code and the SFHo nuclear equation of
state, taken from Morozova  _et al._ (2018). Superimposed on the spectrogram
are the results of a modal analysis for $l=2$ modes. The peak frequency
emission is tracked early on by a $g^{2}_{2}$ mode and later on by an $f_{2}$
mode. The results were obtained by relaxing the Cowling approximation (i.e.,
$\delta\alpha\neq 0$).

Of course, the surface of the PNS provides the natural boundary for any modal
analysis, but it is important to note here that different groups define this
surface to be at different densities – e.g., $\rho=10^{10}$ or $10^{11}$
gcm-3. To further complicate matters, Sotani  _et al._ (2019) have
demonstrated that the outcomes of the modal analyses are sensitive to the
choice of outer boundary. And regarding (ii), Rodriguez  _et al._ (2023) have
recently shown that the classification of modes varies as the method of
classification varies, as well. That is, in one classification a mode may be
classified as a $g$-mode, whereas in another, it may be classified as an
$f$-mode.

Other difficulties include the fact that modal analyses of PNS GW emission
have been confined to spherical symmetry (i.e., the analyses are conducted on
spherical averages of three-dimensional simulation data). Aside from the
obvious implications of averaging the simulation data, the inclusion of
rotation in the analysis is not possible without breaking the assumed
symmetry, which would in turn render the modal analyses far more challenging.
As demonstrated and discussed by Powell and Müller (2020), one might then
expect that analytic expressions for the evolution of the peak frequency
derived under the assumption of spherical symmetry may no longer accurately
track the predicted evolution of the peak frequency from CCSN simulations,
particularly as the progenitor rotation increases.

Finally, a more accurate determination of the modes that may exist in the PNS
and their frequencies is made possible when the equations that are linearized
are the equations that are used in the CCSN simulations providing the modal
analyses backgrounds. A number of the models that have been used for such
modal analyses, including the models discussed here, deploy the pseudo-
Newtonian treatment of gravity discussed in Section III. On the other hand,
the majority of the modal analysis tools that have been developed for the
purpose of PNS asteroseismology are based on the conformally flat condition
(CFC) implementation of general relativity. This may lead to over-estimation
of the mode frequencies, as discussed extensively by Westernacher-Schneider
(2020). Moreover, the results shown in Figure 7 were obtained by relaxing the
so-called Cowling approximation, wherein the metric variables (the lapse
function, $\alpha$, the shift vector, $\beta$, and the conformal factor,
$\psi$) are not varied in the linear analysis. Results using the Cowling
approximation are included in Morozova  _et al._ (2018), as well, and yield
different results not in sync with the peak frequency evolution, for any mode.
Further discussion on the validity of the Cowling approximation can be found
in Torres-Forné  _et al._ (2019b) and Sotani and Takiwaki (2020).

##  VII Detection and Parameter Estimation of Gravitational Waves from Core
Collapse Supernovae

Up to this point we have reviewed (a) the status of the numerical modeling of
CCSNe and the challenges of achieving accurate modeling of the explosions and
(b) the relationship between the physics of the central engine and the GW
production. The extraction of physical information from GWs requires that
consider we investigate (c) the response of a laser interferometer to the GWs
from CCSNe, as well as multimessenger opportunities/challenges. This sector of
CCSN research could be divided into (a) detection strategies, (b) PE
strategies, and (c) the role of the temporal observation window on (a) and
(b).

###  VII.1 Detection Strategies

In this section we describe the key conceptual elements involved in the
detection process for GWs from CCSNe and the subsequent PE. In order to
understand the performance and mathematical structure of the current detection
and PE algorithms, it is useful to review the time-frequency properties of the
data recorded by laser interferometers and their relationship with related
algorithms for CBCs.

Laser interferometers are sensitive to a linear combination of the two GW
polarizations with additive noise Misner  _et al._ (1973):

$x_{i}(t)=h_{i}(t)+n_{i}(t),$ (28)

with

$\displaystyle h_{i}(t)$ $\displaystyle=$ $\displaystyle
H_{i+}(\theta,\phi,\psi)h_{+}(t-\tau_{i})$ (29) $\displaystyle+$
$\displaystyle H_{i\times}(\theta,\phi,\psi)h_{\times}(t-\tau_{i}),$

where $\tau_{i}$ is the delay of the interferometer $i$ with respect to a
fixed, Earth location, usually the arrival time at the center of the Earth,
and $\theta$, $\phi$ and $\psi$ are Euler angles in the interferometer
reference frame having the two arms located as its $x$ and $y$ axis. See
Vitale and Zanolin (2011) for a review of the relationships between the three
types of reference frames involved in the interferometer responses: the
interferometer frames, the Earth frame, and the GW frames. $H_{+}$ and
$H_{\times}$ are the interferometer responses (or antenna patterns)

$\displaystyle H_{i+}(\theta,\phi,\psi)$ $\displaystyle=$
$\displaystyle\frac{1}{2}(1+\cos^{2}(\theta))\cos 2\phi\cos 2\psi$ (30)
$\displaystyle+$ $\displaystyle\cos\theta\sin 2\phi\sin 2\psi,$

$\displaystyle H_{i\times}(\theta,\phi,\psi)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}(1+\cos^{2}(\theta))\cos 2\phi\cos 2\psi$ (31)
$\displaystyle+$ $\displaystyle\cos\theta\sin 2\phi\cos 2\psi,$

and $\psi$ is a gauge-freedom angle determining in the Cartesian frame whose
$z$ axis is the direction of propagation of the GW how to fix the GW $x$ and
$y$ axis and consequently $h_{+}(t)$ and $h_{\times}(t)$. $h_{i}(t)$, which is
gauge invariant, varies, however, depending on the interferometer location,
orientation, and hardware specifics. It is worth specifying that the exact
interferometer response is frequency dependent and the generally-used equation
above is a long wavelength approximation carrying errors of at most 1 to 2
percent for frequencies up to a thousand Hz Rakhmanov  _et al._ (2008).

Figure 8: Interferometer diagram and examples of noise spectra. The four main
features defining the noise spectra are (a) sharp increase of the noise at
low-frequency due to ground vibrations (b) the increase of the noise at high
frequencies because of quantum effects like photon shot noise (controlled by
Posson statistics) (c) the lower limits imposed by the coating vibrations in
the most sensitive frequency and (d) the spikes related to resonant
frequencies of the apparatus (especially mirror supporting strings),
calibration frequencies ”lines”, and power lines disturbances (60Hz in the
US). The interferometer geometry has a natural choice of reference frame where
the two arms are labelled x and y arms.

The detection process involves computing detection metrics from the data and
declaring a detection if the values are above particular thresholds chosen to
limit the False Alarm Probability (FAP; usually for GW CBC signals a 5-sigma
FAP is used). We want algorithms that maximize the detection probability for a
desired maximum tolerable FAP. In other words, we want the best possible
Receiver Operating Curve (ROC). It is also useful to mention the concept of
statistical bias in designing an algorithm. While there is a large amount of
freedom in defining the strategy, one is not allowed to choose the detection
architecture and related thresholds after looking at the GW candidate events
(commonly described as “opening the box”). The reason is, one could make
decisions that minimize the FAP for an event of interest, which would not have
been made just looking at noise events (the so-called time-lag procedure) or
GWs from injected CCSN (see for example Szczepańczyk  _et al._ (2023) and
references therein).

In the following we review the issues involved in choosing the best algorithms
for the detection and PE of CCSNe. The guiding principle is to incorporate as
much as possible the idiosyncrasies of the instrument response to GWs, noise,
and GW morphological features, to maximize the ROCs and PE accuracy.

According to the Neyman-Pearson lemma, the metric producing the best ROC for a
scalar detection problem where a single measurement x is performed in the
presence of additive Gaussian noise is the so-called likelihood ratio J.Neyman
(1933). This is the ratio of the Probability Distribution Function (PDF) of
the measurement if the signal is present to the PDF if it is not:

$\Lambda=\frac{p(x(t);h(t,\xi))}{p(x(t))}\,,$ (32)

where $h(\xi)$ represents the dependence of the measurement on a function of
astrophysical parameters $\xi$. Such ratios can also be calculated for
measurements of time series in Gaussian noise and a network of laser
interferometers. Unfortunately there is no theorem indicating the optimal
metric for a realistic network of laser interferometers (and in general, non-
Gaussian noise). With the assumption that even in the absence of a general
theorem supporting the usage of the likelihood ratio, which is generally given
by

$\Lambda=\frac{\Pi_{i=1}^{K}p_{i}(x_{i}(t);h_{i}(t,\xi))}{\Pi_{i=1}^{K}p_{i}(x_{i}(t))},$
(33)

can still be used as a starting point in designing a detection algorithm,
although with some customizations, which we review in this section. The index
$i$ runs through the different interferometers of the network, $p_{i}$ are the
PDFs of the data at different interferometers, which, for non co-located
interferometers, can be assumed to be statistically independent (this is the
reason why the numerator and denominator are written as products of the
probability distributions at different sites).

The PDF of each time series at the interferometer $i$, when non-Gaussian noise
disturbances (glitches) are not present, can be written as a time-invariant,
correlated-Gaussian process

$\displaystyle p(x,h)=$ (34) $\displaystyle
Me^{-\int_{-\infty}^{+\infty}(x(t_{1})-h(t_{1}))R(t_{1}-t_{2})(x(t_{2})-h(t_{2}))dt_{1}dt_{2}},$

where $R(t_{1}-t_{2})=\langle n(t_{1})n(t_{2})\rangle$ is the noise
correlation function (here we denote by $\langle.\rangle$ the process of
statistical averaging) and the normalization M is signal independent. If we
discretize the data and perform a discrete Fourier transform of the data (to
illustrate the concepts we assume that we collect N temporal samples at each
inteferometer):

$\tilde{x}(m)=\sum_{l=0}^{N-1}e^{-\frac{2\pi
lm}{N}}x(t_{l})\equiv\sum_{l=0}^{N-1}F_{ml}x(t_{l}),$ (35)

Eqn. (VII.1) becomes

$p(\tilde{x},h)=\tilde{M}e^{-(\tilde{x}-\tilde{h})^{\dagger}C(\tilde{x}-\tilde{h})},$
(36)

where the covariance matrix, whose elements are given by
$C_{lp}=(FRF^{-1})_{lp}$ in the frequency domain, is diagonal, with
eigenvalues equal to 1 over the square of the noise spectra at the specific
frequency (see the curves in the left panel of Figure 8, and Eqn. (2.4) of
Sutton  _et al._ (2010) for a detailed proof). If the data are whitened (see
the discussion later in this section), $C_{ll}$ become identical for all
values of $l$. Applying thresholds on $\Lambda$ or $L=\log(\Lambda)$ is
equivalent, with the benefit that $L$ can be written as

$L=\frac{1}{2}\sum_{i=1}^{K}-(\tilde{x_{i}}-\tilde{h_{i}})^{\dagger}C^{i}(\tilde{x_{i}}-\tilde{h_{i}}),$
(37)

where we sum over the $K$ interferometers of the network ($C^{i}$ is the
covariance matrix of interferometer $i$), and we omit the additive signal-
independent terms in $\Lambda$, like the logarithm of the normalizations,
given they are constants. Calculating $\Lambda$ from observed data runs into
the problem that $h_{i}(t)$ are unknown. In the case of CCSNe we do not have
analytical templates describing the expected GWs. The calculation of
$h_{+}(t)$ and $h_{\times}(t)$ can be performed by imposing stationarity of
$L$ (similar to the derivation of Lagrange’s equations from a classical
action) either in the time domain or in the frequency domain:

$0=\frac{\delta L}{\delta h}|_{h=h_{\rm stationary}}.$ (38)

For CBC signals, solving Eqn. (38) and plugging the solution into $\Lambda$ is
equivalent to computing the so-called matched-filter signal-to-noise ratio
(SNR) Creighton and Anderson (2011) at each interferometer,

${\rm SNR}_{\rm
MF}=\sqrt{\int_{-\infty}^{+\infty}\frac{h(f)^{2}}{n(f)^{2}}df},$ (39)

where $h(f)$ is the strain spectrum recorded at the interferometer and $n(f)$
is the noise spectrum at the same interferometer. This SNR is an important
metric for detectability because it measures the ratio of the squared spectrum
of the signal with respect to the squared spectrum of the noise, at each
frequency (this expression suggests that the low-frequency seismic noise in
ground interferometers will pose a challenge in achieving large SNR values
with the low-frequency GW memory from CCSNe). For template searches,
calculating the ${\rm SNR}_{\rm MF}$ involves performing cross correlations of
waveform templates with the interferometric data and choosing the template
that (on average among different interferometers) best correlates with the
data (assuming that the data contains a GW). The waveform templates depend on
three types of parameters: physical (such as the mass of the two compact
objects), geometrical (such as the direction of arrival), and those
corresponding to testing GR (such as effects due to possible new
polarizations). The calculation of the cross correlation for CBC signals is
performed by varying all the parameters at the same time and choosing the
values that maximize the correlation both for detection and PE. The dimension
of the parameter space is constrained to up to a few tens of parameters.

For burst signals, and CCSNe in particular, we treat each frequency (or time)
sample of $h_{+}(t)$ and $h_{\times}(t)$ as free parameters, and, if there are
no constraints, there are approximately 16384 (number of samples per second),
times 2, degrees of freedom for each second of signal. The direction of
arrival can be unknown as well (for example in the case where we have a failed
CCSN without an EM counterpart and without a neutrino signature as well).

According to Eqn. (29), $h_{+,\times}(t)$ have identical temporal shape for
all of the interferometers, but a specific time delay at each interferometer
(see for example the expressions in Markowitz  _et al._ (2008)) for a fixed
direction of arrival with respect to a fixed frame of reference whose origin
is at the center of the Earth. In general, these delays are unknown. Recalling
that a shift in time in the time domain corresponds to multiplication by an
exponential in the frequency domain: If we know the direction of arrival we
can evaluate the $\tau_{i}$ of Eqn. (29), remove the effect of the delay by
shifting the data accordingly in the time domain, or by multiplying by
$e^{-2\pi\tau_{i}f_{l}}$ in the frequency domain, and then carry on with the
rest of the procedure described below. If we do not know the direction, we can
repeat the process below for a grid of sky directions and use the values of
the detection metric to estimate the direction. We can also encounter the
scenario where we know the direction but the network has poor directional
resolution. In this instance, for detection purposes we can employ a grid of
sky locations, as well.

For a given direction of arrival, the solution can be obtained at one
frequency at a time, of index $l$, to obtain $(\tilde{h}_{\rm
stationary}(l))^{T}=((\tilde{h}_{+})_{\rm
stationary}(l),(\tilde{h}_{\times})_{\rm stationary}(l))$. Then, an inverse
Fourier transformation to the time domain can be performed to obtain
$((h_{+})_{\rm stationary}(t),(h_{\times})_{\rm stationary}(t))$. The solution
is usually presented for the scenario where the recorded additive noise has
been whitened (see a discussion of the whitening procedure later in this
section). Whitening is equivalent to dividing
$(\tilde{x}(l))^{T}=(\tilde{x}_{1}(l),\ldots,\tilde{x}_{K}(l))$ by the
standard deviation of the noise, $\sqrt{C_{ll}}$, at the corresponding
frequency and interferometer, which yields

$(\tilde{d}(l))^{T}=(\frac{\tilde{x}_{1}(l)}{\sqrt{C_{ll}^{1}}},\ldots,\frac{\tilde{x}_{K}(l)}{\sqrt{C_{ll}^{K}}}).$
(40)

The corresponding solutions of Eqn. (37) at a fixed direction of arrival (see
for example Sutton  _et al._ (2010) and Klimenko  _et al._ (2016)) are

$\tilde{h}_{\rm stationary}=(\tilde{H}\tilde{H}^{T})^{-1}\tilde{H}\tilde{d},$
(41)

where

$\tilde{H}_{+}=(\frac{H_{1+}}{\sqrt{C^{1}_{ll}}},..,\frac{H_{K+}}{\sqrt{C^{K}_{ll}}}),$
(42)
$\tilde{H}_{\times}=(\frac{H_{1\times}}{\sqrt{C^{1}_{ll}}},..,\frac{H_{K\times}}{\sqrt{C^{K}_{ll}}}),$
(43)

and

${\tilde{H}}=\begin{bmatrix}{\tilde{H}_{+}}\\\
\tilde{H}_{\times}\end{bmatrix},$ (44)

and where, in the antenna patterns, the $\theta,\phi$ angles have to be
individually calculated, for the chosen direction of arrival, with respect to
a fixed, Earth reference frame.

Inspection of the inversion in Eqn. (41), meaning that
$(\tilde{H}\tilde{H}^{T})^{-1}$ is real and symmetric and that if we increment
$\psi$ by an amount $\gamma$ the following relations hold:

$\tilde{H}_{+}^{{}^{\prime}}=\cos\gamma\tilde{H}_{+}+\sin\gamma\tilde{H}_{\times}$
(45)

and

$\tilde{H}_{\times}^{{}^{\prime}}=-\sin\gamma\tilde{H}_{+}+\cos\gamma\tilde{H}_{\times},$
(46)

reveals that at each frequency there is a choice of $\psi$, and consequently
$h_{+,x}(t)$, where $(\tilde{H}\tilde{H}^{T})^{-1}$ is diagonal, the network
has the maximum antenna response to one of the two polarizations, and, in an
orthogonal direction, the minimum antenna response. In the reference frame
corresponding to the value of $\psi$ where $(\tilde{H}\tilde{H}^{T})^{-1}$ is
diagonal, the inversion is performed by two operators
$\frac{(H^{{}^{\prime}}_{+})_{l}}{|H^{{}^{\prime}}_{+}|_{l}}$ and
$\frac{(H^{{}^{\prime}}_{\times})_{l}}{|H^{{}^{\prime}}_{\times}|_{l}}$ that
project and rescale the data of the interferometers. The full inversion is not
possible in the case of aligned interferometers that are blind to one of the
two polarizations. See Eqn. (2.31) of Sutton  _et al._ (2010) and the
corresponding discussion in Klimenko  _et al._ (2016) for the use of the ratio
$\frac{|H^{{}^{\prime}}_{+}|}{|H^{{}^{\prime}}_{\times}|}$ as a tool to assess
the relative sensitivity of a network to the two polarizations in the primed
frame and the discussion on constraints (sometimes termed regulators) later in
this section for how this information can be used in the design of the
detection/reconstruction algorithms. This frame is named the Dominant
Polarization Frame (DPF) Klimenko  _et al._ (2005, 2016); Sutton  _et al._
(2010). It is important to stress that the exact value of $\psi$ that
diagonalizes $(\tilde{H}\tilde{H}^{T})^{-1}$ depends on the orientation of the
interferometers, as well as on the relative amplitude of the noise spectra
between different interferometers (if the amplitude of the spectra at all
interferometers is the same, the quantitites $\sqrt{C^{i}_{ll}}$ factor out in
Eqns. (40) to (44) making the $\psi$ of the DPF frequency independent). When
this relative amplitude varies with frequency, so does $\psi$.

Interferometric data is available for periods of time that can reach days at a
time. However, the data is divided into shorter intervals (candidate events)
of up to a few seconds, processed with wavelet decomposition and clustering,
described later in this section. The resulting reconstructed signals from
applying the procedure above to the candidate events are then plugged into L
giving:

$L_{\rm
stationary}=\sum_{l}\left[\frac{(H^{{}^{\prime}}_{+}\tilde{d})_{l}^{2}}{|H^{{}^{\prime}}_{+}|_{l}^{2}}+\frac{(H^{{}^{\prime}}_{\times}\tilde{d})_{l}^{2}}{|H^{{}^{\prime}}_{\times}|_{l}^{2}}\right]=\sum_{l}{\rm
SNR}^{2}_{l}.$ (47)

This quantity can be calculated for an arbitrary direction of arrival, and
only the largest value of the values obtained from all directions is retained
as the representative value for an event. Histograms of the $L_{\rm
stationary}$ values for all events can be produced for the data that are
processed in a specific CCSN search or observation temporal window (see VIII).
If only the most energetic event is considered for detection, we refer to the
method as the Loudest Event statistical method.

If the Loudest Event method produces a FAP that is well above what is needed
to declare a detection, other approaches must be considered. In the course of
a GW search in which we will assume for the moment that the detection of GWs
from a CCSN has occurred, there will be many noise-induced events as well.
Moreover, with regard to the physical signals detected, there could be more
than one – e.g., in the case of black-hole formation in a failed supernova.
The CCSN-associated signal (successful or failed) could be the loudest event,
the weakest event, or neither. In the case of multiple candidate events,
“population studies methods” can be triggered, in which one looks for evidence
of the CCSN given the distribution of the number of events as a function of
SNR. These techniques, already employed for the search of GWs from gamma-ray
bursts (see for example Abbott  _et al._ (2022)) focus on detecting the
presence of GWs in a population of events that can also contain noise events.
Population studies have the potential to establish detection using events that
would not be detectable with the Loudest Event statistics approach. It is
worth noting, this approach is similar to the approach used for the detection
of the Higgs Boson. In the energy distribution of the recorded events at CERN,
the Higgs was detected not because it produced the most energetic events, but
because the observed distribution profile versus energy would not be expected
in the absence of the Higgs.

In either scenario it is necessary to produce populations of noise-only
events, by either processing data outside the designed observation window
(off-source data) or by changing the time stamp of the data in one or more of
the interferometers by an amount much larger than the distance between
interferometer sites divided by the speed of light (producing so-called time-
lagged data). These noise populations allow us to establish the FAP of the
candidate events (see the discussion in section VIII).

A recent application of this procedure for CCSNe can be found in Szczepańczyk
_et al._ (2023) (this reference also summarizes the current detection ranges
for CCSNe, as well as provides some conservative projections for the near
future). However, the orders of magnitude more degrees of freedom relative to
CBC signals and the intrinsic weaker GW emissions affect, in comparison to CBC
signals, the detection range, as well as the ROCs. Years of trials and code
development demonstrated that the detection performance is much better if the
projectors are not applied directly to the interferometric data but, instead,
to pre-processed data, with modules that isolate the more relevant degrees of
freedom for CCSNe and, in general, burst searches. The key ingredients we
describe below in optimizing the usage of the likelihood ratio for CCSNe are
the pre-processing modules (1) whitening, to remove predictable components of
the noise, (2) wavelet decomposition and clustering, to consider only
connected time-frequency structures of realistic duration for GWs from CCSNe,
(3) network constraints, to include the relative sensitivity to the two GW
polarizations and the relative sensitivity in different sky locations (this
and the next category have the goal of reducing the rate of false alarm
events), and (4) morphological constraints, such as duration constraints or
time-frequency contraints. Furthermore, in the search and PE we also apply (5)
vetoes and data quality, to eliminate events with strong statistical
correlation to Earth-induced noise sources, and in this review briefly discuss
the difference between (6) frequentist versus Bayesian approaches to
incorporating constraints.

The gap in detection performance between CBC and CCSN algorithms and signals
with equal ${\rm SNR}_{\rm MF}$ is decreasing with time through these
improvements. Several algorithms have been used in this effort over the years
(see for example TFClusters Sylvestre (2003), Q-pipeline Chatterji  _et al._
(2004), X-pipelines Adams  _et al._ (2013), and cWB Klimenko  _et al._ (2005);
Mukherjee  _et al._ (2021)), and more are under development. New trends also
involve investigating pure ML algorithms for the detection of GWs from CCSNe.

(1) Whitening: It can be shown that the noise of the laser interferometers has
components that are predictable given previous temporal samples. It has also
been shown that it is easier to detect GWs if these components are subtracted
before performing the subsequent steps of the analysis. These components can
be estimated with linear predictor filters (LPF; for example with the Levinson
algorthm described in Press  _et al._ (1991)), where the coefficients are
determined or “trained” with past test data:

$x_{i,{\rm predicted}}(t_{n})=\sum_{l=1}^{P}a_{l}x_{i}(t_{n-l}),\\\ $ (48)

where the $a_{l}$ are the P coefficients of the LPF. The data actually
analyzed becomes

$x_{i,{\rm analyzed}}(t_{n})=x_{i}(t_{n})-x_{i,{\rm predicted}}(t_{n}).$ (49)

This subtraction process also has the effect of whitening the resulting data,
from which the name originated. There are several implementations of these
LPFs in GW data analysis algorithms. Two that deserve special mention for
CCSNe are the implementation in the wavelet domain (see the subsequent
discussion of wavelets) performed by cWB Klimenko  _et al._ (2005), where the
wavelet family is chosen to reduce as much as possible the frequency leakage
described below, and the implementation by MULASEC Mukherjee  _et al._ (2021),
where the values of the algorithm’s free signal-processing parameters (like
those of the adopted Wiener filters and convolutional neural network) were
specifically optimized by testing the performance on GWs from CCSNe. In both
situations the whitening has been shown to increase the SNR of the candidate
events, because the LPF subtracts energy from the noise but not from from the
signal (consistent with the fact that the LPF training is performed on noise
and not GWs). These LPFs have to be optimized to account for the fact that the
noise spectrum is time dependent and has a drastic frequency variability,
especially below 20 Hz. This last fact makes it prone to energy leakage
between frequencies. For example, when we analyze a finite-duration signal
extracted from a time series, it is equivalent to multiplying an infinite time
series with a time window of finite duration. In the frequency domain this is
equivalent to a convolution between the spectrum of the original signal and
the spectrum of the time window. Every time window of finite duration has a
finite nonzero bandwidth, which automatically broadens the band of each of the
frequencies in the original signal. If someone is not careful the broadening
of the low-frequency noise amplitude could “fill” the “noise spectral well”
around 100 Hz.

(2) Wavelet Decomposition: Instead of applying the inversion equations for
$h_{+,\times}(t)$ on just recorded time series or whitened time series, it has
been shown to be useful to also reduce the degrees of freedom with methods
developed for time-frequency representations of the data. The usage of
spectrograms has a long history in signal detection and characterization (see
for example Chan  _et al._ (2020) and references therein). Spectrograms are a
special case of wavelet decomposition. The detectability of a GW signal
increases when the signal can be described with the least possible number of
wavelet components (see for example Sutton (2013) and references therein). The
ultimate performance is achieved when a signal is matched by a single wavelet,
rendering the search a template search. If someone would have to take
inspiration from CBC GW signals and choose an appropriate wavelet
decomposition the wavelets would be temporally long and narrowband at lower
frequencies and temporally short/broadband at higher frequencies (because the
frequency is slowly evolving at the beginning and rapidly evolving near the
merger). Having narrowband wavelet components for low frequencies is also
useful for controlling frequency leakage from the so-called “seismic wall” at
low frequencies.

The general approach used for CCSN signal processing is to use CBC-inspired
wavelet decomposition with, however, larger dimensionality than the
dimensionality of the time series – for example by using multiple copies of
the same wavelet temporally translated by a fraction of the wavelet duration.
To illustrate how this could be relevant for GWs from CCSNe, let us assume
that we have in the GW a SASI component made of a single frequency of a
duration of 100 milliseconds and that we have a wavelet basis containing
wavelet components having exactly the same frequency and duration, but in
performing the wavelet decomposition, the wavelets are obtained by translating
the mother wavelet exactly 100 milliseconds. It is quite likely that none of
the wavelets have start and end times exactly identical with the SASI
component. The choice of temporally translated wavelets by a fraction of the
duration can aid the detection because the probability of perfect alignment
increases and a wavelet perfectly alligned with the GW could produce a wavelet
coefficient amplitude that passes internal algorithm thresholds, whereas
partially-correlated wavelets might not. See for example the discussions on
“black-pixel” thresholds in Klimenko  _et al._ (2005), where only a small
percentage of more energetic wavelet components is retained, as well as only
those that form clusters. This last strategy is motivated by the canonical
spectrograms from CCSNe – e.g., see Figure 3. The fraction of data retained
and the clustering procedure used are some of the tunable parameters of the
algorithm. The tuning process is usually based on maximizing ROCs and
detection ranges for the average of trials with a representative population of
GW signals.

It was shown that using multiple wavelet decompositions in parallel is
beneficial Chatterji  _et al._ (2004), and now it is common to have this
feature in detection algorithms (see for example Adams  _et al._ (2013) and
Klimenko  _et al._ (2005)). We foresee more research being conducted on this
front, as the progress has been driven by testing the assumptions on evolving
data and CCSN GW databases.

(3) Network Constraints: These involve (a) metrics that quantify the
similarity of the reconstructed signal between different interferometers and
(b) constraints on polarization (if a configuration of the laser
interferometer network is mostly sensitive to a single polarization, we do not
try to reconstruct the other because it would produce mostly noise-induced
events – for example by choosing in the middle expression of Eqn. (47) only
the contribution from the $+$ polarization). (c) A network may also have sky
locations where it is particularly insensitive, and events reconstructed in
these directions might also be mostly noise-induced events.

(4) Constraints on signal time-frequency structure: In CCSN searches, we
assume that the frequency content is limited between $\sim$20 Hz and
$\sim$2000 Hz because the detectors are much less sensitive outside this
frequency band. We also do not include events that are longer than a few
seconds. In recent years it has proven useful to use different versions of ML
algorithms that distinguish noise-induced events from events of astrophysical
origin Cavaglia  _et al._ (2020); Morales  _et al._ (2021); Antelis  _et al._
(2022); Lopez  _et al._ (2022).

(5) Veto and Data Quality: For all algorithms devoted to detection and PE, it
is critical to identify stretches of data that have been damaged by the
coupling with non-astrophysical disturbances. Currently there is no CCSN-
specialized approach to this. It is worth mentioning that different stretches
of data are sometimes used for different purposes. For example if we have an
exceptionally interesting GW candidate or an exceptional astronomical event
such as a Galactic CCSN we would use the data even if the time associated with
the candidate or event is in a period of time when many known correlations
exist with Earth-based sources of strong glitches. However, where no event is
deemed a GW event, astrophysical statements can still be made – for example on
the maximum energy emitted in GWs for the source of interest (the most recent
of such statements for CCSNe were produced in Szczepańczyk  _et al._ (2023)).
These statements are based on the fact that no events louder than the loudest
observed noise event were present in the data set. For this usage, it is
usually better to discard intervals of the data with transients correlated to
known noise disturbances so that astrophysical statements are stronger.

The noise can also be divided into an ever-present Gaussian component (which
emerges as a product of many small disturbances blending, as described by the
central limit theorem) and glitches – i.e., sporadic, sub-second transients
induced by environmental disturbances (a falling tree, a train, a fluctuation
in the power supply, etc.). These glitches, which affect more strongly the
detection confidence than PE algorithms, are studied and classified on a
continuous basis by different laser interferometer collaborations (see for
example D. Davis (2021) for LIGO, Yuzurihara (2023) for Kagra, and Acernese
_et al._ (2023) for Virgo).

(6) Frequentist versus Bayesian Approaches: The use of a frequentist versus a
Bayesian approach is one aspect of the algorithms that has both practical and
conceptual ramifications. In the frequentist approach the probability depends
on the physical value of the parameters treated as deterministic quantities,
while in the Bayesian approach the parameters are random variables as well. In
this second scenario the joint PDF can be factorized according to Bayes’
theorem

$p(x,h)=p(x;h)p(h),$ (50)

where $p(x;h)$ is the frequentist PDF and $p(h)$ is the prior, describing the
data-independent assumptions someone would like to make regarding the GW,
including the direction of arrival. The Bayesian prior needs to be provided. A
Bayesian detection process could be implemented in Eqn. (33) by replacing the
frequentist PDFs with Eqn. (50). It may be problematic to make prior
assumptions about GW properties before we have detections from CCSNe. In
practice though, it should be possible to implement with Bayesian priors the
network and morphology constraints discussed earlier in this section. While we
do not describe in detail the Bayesian implementation, Pannarale  _et al._
(2019), Sutton  _et al._ (2010), and Cornish  _et al._ (2021), and references
cited therein, provide a good overview of current efforts.

The strategies in this section have the goal of improving the detection range
for a broad range of CCSN morphologies and sky locations. They are constantly
re-evaluated because of theoretical and signal processing advancements, as
well as the evolution of the instruments, which produce noise spectral
properties that are slightly different at every data collection run, including
different types of transient disturbances. For a given value of the SNR a
matched-filtering approach can still provide a detection range a few times
larger than a pure coincident excess-energy approach. However, the techniques
in this section are reducing the performance gap. Of course, reducing the
number of degrees of freedom of CCSN searches is essential. One important
aspect of the entire enterprise we do not cover in this review has to do with
the computational cost associated with any given approach adopted. For
searches performed specifially for GWs from CCSNe in laser interferometric
data, see Szczepańczyk  _et al._ (2023) and Abbott  _et al._ (2020b, 2016,
2009).

###  VII.2 Parameter Estimation Strategies

We now review the main issues involved in PE for CCSNe. A whole range of PE
activities have been (or are being) developed for the GW sector and its role
in MMA PE (for activities included in CCSN search papers see Szczepańczyk  _et
al._ (2023) and Abbott  _et al._ (2020b, 2009) and for activities documented
in methodological papers see Lin  _et al._ (2023); Gossan  _et al._ (2016);
Lagos  _et al._ (2023); Villegas  _et al._ (2023); Szczepańczyk and Zanolin
(2022); Gill  _et al._ (2022); Antelis  _et al._ (2022); Richardson  _et al._
(2022); Abbott  _et al._ (2021b); Szczepanczyk  _et al._ (2021); Cavaglia  _et
al._ (2020); Lin  _et al._ (2020); Abbott  _et al._ (2020a); Powell and Müller
(2022); Roma  _et al._ (2019); Suvorova  _et al._ (2019); Pastor-Marcos  _et
al._ (2023); Drago  _et al._ (2023); more references are discussed in the rest
of this section).

The main starting point is the appreciation of the basic features of GWs from
CCSNe (with their striking differences relative to other GW transient sources
such as CBC sources). Even if the time series display a marked stochasticity,
there are deterministic features that are visible either in the time domain,
frequency domain, or time–frequency domain. The polarization state is expected
to be mostly randomly polarized, with the exception of (a) the core bounce
component for rapidly rotating progenitors and (b) spiral SASI effects, which
are expected to be strongly and somehow elliptically polarized. We summarize
in the following these features and the efforts to both extract them (detect
them in interferometric recordings, with algorithms that use minimal
assumptions about their morphologies) and quantify them (estimate
deterministic parameters, their intrinsic uncertainties, and the expected
estimation uncertainties given interferometric data):

(1) Core Bounce: The estimation of physical parameters from the GW core-bounce
phase of rotating CCSN progenitors aims to both estimate information about the
degree of rotation in the progenitor and about the PNS equation of state. It
can complement the results of the PE performed on other CCSN GW features (like
the evolution of the high-frequency GW emission from the PNS).

The core-bounce GW component is amenable to be modeled by templates Richers
_et al._ (2017), a special feature, and, accordingly, by techniques already
used for modeled searches and their PE, as in the case of matched filtering
implemented for GWs from CBCs.

There is active research in choosing the GW templates to match with the GW
signal embedded in realistic interferometric noise Pastor-Marcos  _et al._
(2023); Villegas  _et al._ (2023). These templates could be sourced directly
out of numerical simulations, whose accuracy is evolving over time, or by
identifying phenomenological templates. In the second scenario, optimal
template construction (using a matched-filtering fitting factor between the
templates and numerical simulations as a metric of “goodness”) can be done in
terms of (a) the size of the parameter space, (b) the types of parameters:
phenomenological or physical, such as (i) the ratio of kinetic to
gravitational potential energy, $\beta$ (there are indications that the
parametrization for $\beta\leq 0.06$ and above 0.06 are different Richers  _et
al._ (2017)) or (ii) source orientation, and (c) the GW polarization state.
Furthermore, Richers  _et al._ (2017) suggest that $\beta$ is more closely
related to the core bounce amplitude and $\sqrt{G\bar{\rho}_{c}}$ to the
duration (where $\bar{\rho}_{c}$ is the average PNS density), with further
corrections from different EOS and radial rotational profiles, as suggested
for example in Abdikamalov  _et al._ (2014).

Performing a search and PE in already-detected GW signals from CCSNe could
allow us to detect the presence of the core-bounce component, as well as
estimate its physical parameters such as $\beta$. Identification ROCs (meaning
assessing the probability to detect the presence of the core bounce for a
given false identification probability) can also be produced similarly to what
has been developed for the SASI in Lin  _et al._ (2023). The PE of $\beta$
from the core-bounce component could also allow us to establish the presence
of rotation if the accuracy is sufficient to establish that $\beta$ is
different than zero.

The usage of numerical/phenomenological templates also allows us to study
theoretical minima (Cramer-Rao Lower Bounds; CRLB) in the errors any algorithm
could achieve for a given signal, interferometer network, and noise floor, as
well as higher-order corrections to them when non-linearities in the
relationship between the observed data and the estimators of the physical
parameters make the CRLB an underestimation of the actual error. The
calculation of the CRLB and higher-order asymptotic expansions involve the
evaluation of mean values of the log-likelihood derivatives with respect to
the parameters. This methodology has been previously applied to GWs from
binary black holes (see Vitale and Zanolin (2011) and references therein) to
quantify the accuracy in estimating physical parameters such as total mass,
reduced mass, and phase.

(2) Memory: The first evidence of GW memory from CCSNe was seen with two-
dimensional numerical simulations Murphy  _et al._ (2009); Müeller  _et al._
(2013); Yakunin  _et al._ (2010, 2015), even though it was understood that the
imposition of axial symmetry would potentially maximize the asphericity of the
explosion and, consequently, the memory, given that explosion can occur only
along the symmetry axis, in a pronounced prolate fashion. Indeed, in O’Connor
and Couch (2018) and Mezzacappa  _et al._ (2020) the GW strain amplitudes for
CCSN simulations initiated from the same progenitor in two- and three-
dimensional simulations were compared, with the amplitudes for all phases of
the strain in the latter case being much reduced. The key point is, the exact
characteristics and asymptotic values of the signal can be determined
accurately only from three-dimensional numerical simulations, although it is
important to stress that the axial symmetry of rapidly rotating progenitors
has the potential to develop much larger memory amplitudes.

The amplitude of the strain for the low-frequency emission from GW memory can
be significantly larger than the strain caused by other sources, such as the
high-frequency emission from PNS oscillations or the low-frequency emission
from the SASI. However, the energy contained in the low-frequency component is
much less because the energy density in the frequency domain is proportional
to the square of the frequency.

In the low-frequency limit, GW memory is expected to be linearly polarized.
More explicitly, the GW spacetime metric being symmetric and real can always
be diagonalized. However the frame where this happens is in general constantly
evolving. Because this is not the case for the memory, we can say that it is
linearly polarized. The sensitivity of the network to the memory depends on
the relative orientation of this frame with respect to the DPF at low
frequencies, unless we have a network equally sensitive to both polarizations.

A discussion of the key issues associated with GW memory from CCSNe can be
found in Richardson  _et al._ (2022), where they discuss how the GW memory has
a peak frequency of approximately 1 Hz, related to the typical time scale for
its development, which is approximately one second. The memory dominates the
GW emission from CCSNe below a few tens of Hz. The noise floor of ground-based
interferometers is currently too high to be able to see the signal above the
noise, at 1 Hz. However, the spectrum of the memory decreases with increasing
frequency at a lower rate than the noise spectrum. In general, for any ground-
based interferometer configuration, a critical feature for memory
detectability is the rate of frequency variability of the seismic noise. This
rate will evolve as vibration control systems evolve.

Low-frequency GWs from neutrino emission asymmetries in CCSNe were studied in
the context of three-dimensional models by Vartanyan and Burrows (2020). They
found that the neutrino-generated GW signal from a Galactic supernova will be
detectable by AdvancedLIGO, the Einstein Telescope, and proposed space-based
detectors. However, this conclusion is for the total signal and not
specifically the memory. It is also unclear whether or not the sudden
termination of the GW signal at a non-zero value introduced artefacts in the
Fourier analysis, or in detectability estimates for present and future ground-
based interferometers. The issue of how continuation of truncated simulations
should be managed in sensitivity studies performed with real interferometric
noise is discussed in Richardson  _et al._ (2022), as well as the angular
dependence of the matter-induced memory with respect to source orientation.
More research is needed for the study of the angular dependence of the
neutrino-induced memory amplitude with respect to source orientation. More
generally, research is needed to develop a CCSN memory detection pipeline.

It is worth noting that future space interferometers have strong potential for
the detection of GW memory. An initial quantitative analysis is presented in
Richardson  _et al._ (2022). In addition, Mukhopadhyay  _et al._ (2022)
describe how improving the noise of DECIGO by one order of magnitude will
allow robust time triggers for CCSNe at distances D $\sim$40–300 Mpc. And
Branchesi  _et al._ (2023) discuss how lunar ranging experiments can be used
to detect GW memory, as well.

(3) High-Frequency Feature: This feature is recognizable in a time-frequency
spectrogram, as in Figure 3, as a continuous, increasing, approximately linear
feature (for the currently visible part of the spectrum, below 1000 Hz),
starting at $O(100)$ Hz and increasing up to $\sim 1-2$ kHz with time after
bounce. The rate of increase of the frequency is believed to be strongly
correlated with progenitor characteristics, such as the progenitor mass and
degree of rotation (for studies in the context of three-dimensional CCSN
modeling, see for example Powell and Müller (2020), Pan  _et al._ (2021), and
Pajkos  _et al._ (2021)), and with characteristics of the PNS, such as its EOS
(again, for studies in the context of three-dimensional CCSN modeling, see for
example Kuroda  _et al._ (2017)).

At later stages the frequency evolution is expected to plateau or disappear
depending on whether the PNS mass and radius stabilize or a black hole forms.
Given current laser interferometers it is expected to be difficult to estimate
more than the initial linear growth of this feature because the frequencies
above 500 Hz are expected to be mostly buried in the interferometer noise. A
first attempt to estimate the slope of the feature with real interferometric
noise was performed in Lin  _et al._ (2023), applying a chi-squared method to
a low-order polynomial evolution of the resonant frequency. The authors
applied the procedure to CCSN events that would be identified by cWB, the
flagship algorithm for the detection of GW bursts. The study in Bizouard  _et
al._ (2021) proposed an approach involving the modal decomposition described
in Section VI.4, along with a polynomial interpolation and simulated Gaussian
noise, to infer the time evolution of a combination of the mass and radius of
the compact remnant, as discussed in that section. Currently the best
performance in estimating the initial slope with real laser interferometric
data is documented in Lagos  _et al._ (2023), where an optimized neural
network approach was used.

The amount of information that could be extracted from the slope estimates and
the corresponding errors (which depend on the signals, the distances, and the
interferometer network) in real interferometric data is the focus of current
research.

(4) SASI: Among the deterministic features in a CCSN GW spectrogram is the
feature induced by the SASI. It imparts a distinctive feature to the neutrino
luminosities as well, in the form of quasi-periodic fluctuations. This feature
makes it a natural candidate for multi-messenger studies. The SASI descriptive
parameters include its frequency (in both GW and neutrino channels), duration,
amplitude, and GW polarization. The average SASI GW frequency contains
information about the average radius of the stalled shock front and the
coupling mechanism between the shock wave and the PNS Blondin and Mezzacappa
(2006); Walk  _et al._ (2020). Longer SASI duration could be a distinct
feature of failed supernovae Walk  _et al._ (2020).

It is of interest to analyze the statistical conditions needed to detect the
presence of the SASI and to estimate its parameters for realistic detectors,
where noise is present. The noise, as well as the signal processing artifacts,
of neutrino and GW detectors are different. However, in both the neutrino and
GW channels, noise can induce some energy in the SASI time-frequency regions,
thus complicating the analyses.

Spectral properties of the SASI-induced features in the neutrino luminosity
were described in Lund  _et al._ (2010, 2012) and Müller  _et al._ (2013) for
a specific set of progenitors. The question of the detectability of the SASI
was discussed, not with respect to a specific algorithm, but in terms of the
spectral amplitude relative to a Cherenkov detector’s shot noise. In Lund  _et
al._ (2010, 2012), the shot noise was estimated by Fourier transforming a
neutrino time series. The estimated shot noise became independent of
frequency. Note, however, that when the statistical fluctuations of the
neutrino signals themselves dominate over the noise of the detector’s
background, the frequency-independence assumption may only serve as a rough
approximation. (The neutrino time series used by Lund et al. never dominated
the noise.) The extension to a full SASI detection methodology in the neutrino
channel was performed in Lin  _et al._ (2020), where the “SASI-meter” was
proposed in order to detect the presence of the SASI with a desired
statistical confidence, as well as obtain an estimate of the frequency for the
SASI candidates that pass a desired confidence threshold. In that work, it was
also pointed out, there is an intrinsic uncertainty in the frequency of the
SASI, both in the GW channel and in the neutrino channel, related to the
finite duration of the instability.

For the SASI GW emission a few approaches have been explored, each with
specific idiosyncrasies. In Roma  _et al._ (2019) a Bayesian method that uses
a training process on an existing database of GWs was proposed in order to
identify the presence of the SASI. In this study, magnetorotational emission
models were assumed not to contain the SASI, and PE and false identification
probabilities were not involved. In Takeda  _et al._ (2021), they explored the
usefulness of an application of the Hilbert-Huang Transform (HHT). While this
is an interesting new direction, the role of noise was not part of the study.
In Lin  _et al._ (2023), the work of Lin  _et al._ (2020) on SASI
detectability and PE was extended to real interferometric noise. The
probability that the presence of the SASI can be established was analyzed, as
well as the intrinsic uncertainties in determining the SASI frequency and the
SASI false identification probability. The method used a frequentist
inference, which does not apply prior information of the SASI from any
specific numerical simulation. The method used theoretical knowledge to
identify conservative boundaries of the time-frequency region of a GW signal
where SASI contributions would be present. The scenario adopted is when a CCSN
detection has been established in both the (time-coincident) neutrino and GW
channels. This work also extends the neutrino analysis in Lin  _et al._ (2020)
with an estimate of the duration of the SASI. The wavelet decomposition of the
GW data recorded at laser interferometers KAGRA LIGO Virgo Collaborations
(2018) was performed using cWB. The main metric that was introduced to detect
the presence of the SASI in a GW event was the relative SASI SNR. Explicitly,
given a GW event wavelet decomposition, the relative SASI SNR is the sum of
the SNR of individual wavelet components in the SASI time-frequency region
versus the sum of the SNR of all the wavelet components in the event. This
metric is a quantitative measure of the predominance of the SASI
time–frequency region with respect to the whole GW event. It is not an overall
detection metric like Eqn. (47).

##  VIII Role of Multimessenger Identification of an Observation Window

When searching for GWs from CCSNe it is beneficial for the searches to have
information about the time when the GW emission might have reached the Earth.
The on-source window (OSW) is a time interval during which we search for the
GW transient. It is defined as $[t_{1},t_{2}]$, where $t_{1}$ and $t_{2}$ are
the beginning and end times, respectively. The main benefit of having a short
OSW, $t_{2}-t_{1}$, is that the FAP of a candidate event (the main statistical
metric for detection) is given by

${\rm FAP}=1-e^{-(t_{2}-t_{1}){\rm FAR}}\simeq(t_{2}-t_{1}){\rm FAR},$ (51)

where the first equality is the Poisson probability to have more than zero
noise events and the second equality follows from the fact we usually operate
when $(t_{2}-t_{1}){\rm FAR}<<1$. An all-sky search and a targeted search are
the two types of GW searches we can perform for CCSN GW signals. In an all-sky
search the amount of data to be searched corresponds to a duration of the
order of one or more years (the whole duration of a scientific data collection
run). In a targeted search we can use an EM and/or neutrino observation to
search data corresponding to a significantly shorter period of time.

The reduction in the OSW can be used to either (a) lower the FAP at a fixed
FAR, (b) increase the FAR at a fixed FAP, or (c) affect a combination of both.
The second option makes the search more sensitive because an increase in FAR
is achieved by using smaller thresholds in SNR and, therefore, by detecting
weaker signals (the SNR of an event is inversely proportional to the distance
of the source). The choice of strategy among (a),(b), or (c) depends on
whether or not it is more pressing to decrease the FAP, or detect signals at a
larger distance, as well as on the properties of the data.

Figure 9: The change in the SNR thresholds for a given change in FAR can be
different depending on the region of the FAR vs SNR curve, the all-sky
threshold (AST), and the targeted-search thresholds (TST). On the vertical
axis the two black arrows signify the increase of FAR that we can tolerate
while maintaining the FAP fixed because of the reduction of the temporal
observation window.

The FAR versus SNR distribution, how many noise events per unit time are
expected above a given SNR (see for example the cartoon in Figure 9 or Figure
3 of Abbott  _et al._ (2008) and Figure 5 of Abadie  _et al._ (2010) for real
data distributions), might make it easier to get the benefits in (a) rather
than the benefits in (b), or vice versa. Histograms of FAR versus SNR can
exhibit two types of behavior. At low SNR, the FAR changes very rapidly, by
orders of magnitude, for small changes in SNR. At large SNR, when vetoes or
other measures fail to remove some of the loud glitches, small changes in the
FAR are connected to large variations in the SNR. As a consequence, if the
all-sky SNR threshold is in a glitch region, the tolerable increase of the FAR
of a targeted search is better used to increase the range, given that the
tolerable increase corresponds to a significant decrease in SNR and,
consequently, a significant increase in the range. On the other hand, at low
SNR, it is better to use the reduced OSW duration to decrease the FAP while
keeping the FAR fixed.

The methodology to compute $t_{1}$ and $t_{2}$ should also be a compromise
between achieving a high probability (see Gill  _et al._ (2022) for some of
the limitations in achieving a 100% probability) to contain the possible GW
emission from the CCSN and, at the same time, not lose the benefits of having
a shorter OSW relative to all-sky searches (see for example Abbott  _et al._
(2021a)).

Before we discuss the OSW options in more detail, it is useful to compare
detection and PE scenarios. Detection algorithms in general have multiple
internal thresholds that are chosen to identify a GW while discarding with a
desired FAP Earth-generated disturbances. Regarding the OSW impact on PE, the
situation is different. PE can be performed either on all candidate events
(possibly as a way to reduce further the FAP if we use the feedback of PE to
disregard events with morphology incompatible with a CCSN) or only on events
that have been detected as GW events. While in the first scenario the duration
of the OSW will affect how many events will be processed and a
characterization of the PE output on the corresponding population of noise
events will be needed, in the second scenario the OSW has no direct impact on
the PE results. This means, if PE routines are performed in a follow-up mode
of already detected events, we can investigate algorithm configurations that
would not be viable for detection purposes as long as they legitimately
improve the reconstruction performance (see the discussion on SASI
reconstruction in the previous section).

Neutrino OSW: The electron-neutrino burst and the rapid rise of the three-
flavor neutrino and antineutrino luminosities are expected to happen within
$O(10)$ ms of core bounce. The OSW duration can be set to be of the order of a
few seconds to accommodate all phases of the anticipated GW emission and
possible biases on the GW estimated arrival time, as well as signal start and
end times due to performing wavelet decompositions and clustering in the
presence of interferometric noise. In this scenario there are about 7 orders
of magnitude improvement that can be achieved in the FAP and/or the FAR (and
according to the discussion of this section, sensitivity) with respect to an
all-sky search. Studies for extragalactic CCSNe also exist – e.g., see Aartsen
_et al._ (2014) and Mukhopadhyay  _et al._ (2022), the latter of whom discuss
how a nearly background-free sample of $\sim$3–70 neutrino events per Mt per
decade of operation will be available.

EM OSW: If neutrinos are not available, measurements of EM emission could be
employed as well, as described below. The OSW estimation from EM observations
requires two steps: estimating the time of shock breakout (SBO) – i.e., the
time when the supernova shock wave reaches the progenitor’s surface – and
estimating the delay between the GW emission and the time of SBO.

Delay between Collapse and Shock Breakout: Published work estimating the time
delay between the iron core collapse and SBO for a given CCSN progenitor can
be found in Barker  _et al._ (2022). In particular, their Figure 6 relates the
mass of the progenitor to this delay. In addition, an extensive study of CCSN
progenitor masses in the local universe was performed in Smartt (2009) (within
30 Mpc, where this study is feasible). As summarized in their Figure 6, in the
local universe there is an apparent upper limit of CCSN progenitor masses of
16.5 M⊙. In the absence of progenitor-mass information, the mass–delay
relationship in Barker  _et al._ (2022) would indicate that $\Delta t_{GW-
SBO}=3$ days would be an appropriate upper limit for the collapse-to-SBO
delay. Clearly then, one discussion that goes beyond the scope of this review
but is important for the OSW estimation is the degree of trust we have in the
estimate of the CCSN progenitor mass, both for the collapse-to-SBO delay and
for the CCSN population characterization. To illustrate the difficulties, in
the case of SN2023ixf Pledger and Shara (2023) suggested that the progenitor
mass was 8 to 10 $M_{\odot}$, Kilpatrick  _et al._ (2023) predicted the mass
to be 11 $M_{\odot}$, and Soraisam  _et al._ (2023) instead predicted the mass
to be $20\pm 4$ $M_{\odot}$.

Estimating the Time of Shock Break Out: There are three different types of
methods that can be used to calculate the time of SBO. The choice depends on
the availability of early observational (pre-peak luminosity) data and the
availability of published tools to model the light curves.

(i) The Early Observation Method: In this method, the initial supernova
observation provided by EM facilities (discovery time, $t_{\rm disc}$ ) is
used as the time of SBO and the end of the OSW, $t_{2}$. The beginning of the
OSW is calculated from the last EM null observation of a CCSN in the host
galaxy, $t_{\rm null}$. However, we should not use $t_{\rm null}$ as $t_{1}$,
because even if it is before SBO, it may be after core bounce and the
initiation of the GW emission, which in turn would have left the exploding
progenitor before SBO and arrived at the Earth before $t_{\rm null}$ (it is
also important to asses that the sensitivity of the available telescope
observations is not impairing the capability to ensure that $t_{\rm null}$ is
really before the SBO Gill  _et al._ (2022)). $t_{1}$ can be chosen to be
$t_{\rm null}$ minus the delay between collapse and SBO. An upper limit for
the delay between collapse and SBO could be inferred from Barker  _et al._
(2022) given the progenitor-mass estimate. If someone is sufficiently
confident in the progenitor-mass estimate and mass–delay conversion, $t_{2}$
can be moved closer to $t_{1}$ by using the lower limit in the delay from
Barker  _et al._ (2022). Overall the decision is a trade off between making
sure that the OSW contains the GW emission and maximizing the detection
capabilities.

(ii) Quartic Interpolation: If the light curve is discovered in the homologous
phase of the shock expansion we can use the fact that it has in this phase a
high degree of universality among observed CCSNe. In Szczepańczyk  _et al._
(2023) the use of the simplest polynomial fit was investigated (where the
coefficients of different polynomials were estimated with a chi-squared
minimization between the data and the polynomials). Such a fit was used, as a
testing ground, on a detailed light curve (from the moment of observed SBO)
obtained for KSN2011a by Kepler Garnavich  _et al._ (2016) and Astropy
(Robitaille  _et al._ (2013); Price-Whelan  _et al._ (2018, 2022)). Figure 3
of Szczepańczyk  _et al._ (2023) shows possible interpolations in cases where
a potential first observation can be up to a few days after SBO. A simple
quadratic polynomial fit revealed biases. In this case, the estimated time of
SBO was earlier than the actual time of SBO because the slope of the light
curve around its peak is smaller than in its early stages. A quartic
polynomial fit is instead able to more accurately estimate SBO even when the
first few days of data are missing.

(iii) Physics-Based: Another approach used for some of the candidates in
Szczepańczyk  _et al._ (2023) and described in more detail in Gill  _et al._
(2022) is to estimate the time of SBO with light curves based on either
physical modelling or matched filtering with template banks of recorded light
curves. This approach also provides posterior distributions for the possible
time of arrival of the GW at Earth.

For targeted CCSN searches with neutrino triggers versus EM triggers, the data
associated with the seconds-long OSWs will be available for the whole time
interval, and it is straightforward to choose an OSW that is both short and
has a $100\%$ probability of containing the GW emission. On the other hand if
EM triggers are used (as well as for all-sky searches) a sizable fraction of
the days-long OSW will not be available in at least one of the
interferometers. The so-called duty cycle of available science-quality
coincident data can vary from $20\%$ to close to $90\%$ (see examples in
Szczepańczyk  _et al._ (2023)). Furthermore it is currently hard to quantify
precisely all the uncertainties associated with the OSW from an EM
observation. Therefore for an EM-targeted search, having an OSW that has a
probability between $90\%$ and $100\%$ to contain the GW emission and increase
the detection chances is preferable to a very long OSW with no detection
benefit, and in fact such a strategy was adopted in the most recent search
Szczepańczyk  _et al._ (2023) (also because the data outside the OSW is
processed in parallel, with lower sensitivity, by the all-sky searches).
Finally, the OSW discussion is also relevant for all-sky searches in order to
evaluate the possible correlation of candidate events with known CCSNe.

##  IX Conclusions and Outlook

In this review, we hope we have conveyed the status of CCSN modeling and GW
prediction, efforts to detect a Galactic or near-extra-Galactic event, and
efforts to infer defining characteristics of the progenitor and supernova
central engine given a detection. Three-dimensional models have reached a
significant level of sophistication, though important work remains, especially
to elevate all of the leading models across groups to full three-
dimensionality and full general relativity. Supercomputing architectures are
enabling these advances. Our understanding of CCSN GW emission has grown
dramatically in recent years, in sync with the increase in the sophistication
of explosion modeling and, most important, full physics three-dimensional
modeling. Lessons from GW detection and PE of CBCs have provided a foundation
on which detection and PE methods specifically designed for CCSNe have been
developed. The advances in modeling mentioned above have aided this
development, providing significant incite into the GW features associated with
stellar core bounce, PNS convection, neutrino-driven convection, the SASI, and
explosion. Fortunately, in some cases, the expected spectral characteristics
of GW emission from these central engine components lie within the most
sensitive parts of the interferometer sensitivity curves. And the development
of GW data analysis has helped hone CCSN simulation, as well, in order to
provide the targeted, optimal output for GW detection and PE.

GW astronomy is just beginning and, even more so, CCSN GW astronomy. The
challenges we face, in light of the weakness of the GW emission from CCSNe and
the paucity of such events in the Galaxy, are significant. But the scientific
payload is too great to withdraw from such challenges. Thus far, the community
has not, and we hope we have conveyed the significant progress that has been
made along all three necessary fronts: modeling, detection, and PE. Given this
progress, and given that we may be overdue for a Galactic event, these are
truly exciting times to be modeling CCSNe and their GW emission, and to be
investing in methods, tools, and instruments designed to see them in ways only
GWs can enable, to the very center of the supernova central engine.

##  X Acknowledgments

A.M. was supported in part by the National Science Foundation Gravitational
Physics Theory Program through awards PHY 1806692 and PHY 2110177. M.Z. was
supported by the National Science Foundation Gravitational Physics
Experimental and Data Analysis Program through award PHY 2110555. The authors
would like to express their thanks to Haakon Andresen, Viktoriya Morozova, and
David Radice for kindly letting us include their figures in this review.

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