Rotational or fragmentation instabilities encountered by the collapsing star will also produce GWs. Neutron star remnants of collapse may emit GWs due to the growth of rotational or rmode instabilities. Black hole remnants will also be sources of GWs if they experience accretion induced ringing. All of these phenomena have the potential of being detected by gravitational wave observatories because they involve the rapid change of dense matter distributions.
Observation of gravitational collapse by gravitational wave detectors will provide unique information, complementary to that derived from electromagnetic and neutrino detectors. Gravitational radiation arises from the coherent superposition of mass motion, whereas electromagnetic emission is produced by the incoherent superposition of radiation from electrons, atoms, and molecules. Thus, GWs carry different kinds of information than other types of radiation. Furthermore, electromagnetic radiation interacts strongly with matter and thus gives a view of the collapse only from lower density regions near the surface of the star, and it is weakened by absorption as it travels to the detector. Neutrinos can escape from much further within the collapsing star, but even they are scattered by the highest density regions in the core. By contrast, gravitational waves can propagate from the innermost parts of the stellar core to detectors without attenuation by intervening matter.
The characteristics of the GW emission from gravitational collapse have been the subject of much study. Core collapse supernovae, in particular, have been investigated as sources of gravitational radiation for more than three decades (see,
e.g., [
165,
198,
166,
50,
148,
139,
188,
65,
137,
217,
160,
71,
72]
).
However, during this time research has produced estimates of GW strength that vary over orders of magnitude. This is due to the complex nature of core collapse. Important theoretical and numerical issues include

∙
construction of accurate progenitor models, including realistic angular momentum distributions,

∙
proper treatment of microphysics, including the use of realistic equations of state and neutrino transport,

∙
simulation in threedimensions to study nonaxisymmetric effects,

∙
inclusion of general relativistic effects,

∙
inclusion of magnetic field effects, and

∙
study of the effect of an envelope on core behavior.
To date, collapse simulations generally include stateoftheart treatments of only one or two of the above physics issues (often because of numerical constraints). For example, those studies that include advanced microphysics have often been run with Newtonian gravity (and approximate evaluation of the GW emission; see Section 2.4 ). A 3D, general relativistic collapse simulation that includes all significant physics effects is not feasible at present. However, good progress has been made on the majority of the issues listed above; the more recent work will be reviewed in some detail here.
The remainder of this article is structured as follows. Each category of gravitational collapse will be discussed in a separate section (AIC in Section
2 , collapse of massive stars in Section 3 , collapse of Population III stars in Section 4 , and collapse of SMSs in Section 5 ). Each of these sections ( 2 , 3 , 4 , 5 ) is divided into subsection topics: collapse scenario, formation rate, GW emission mechanisms, and numerical predictions of GW emission. In the subsections on numerical predictions, the detectability of the GW emission from various phenomena associated with collapse is examined.
In particular, the predicted characteristics of GW emission are compared to the sensitivities of LIGO (for sources with frequencies of
$1$
to
${10}^{4}Hz$
) and LISA (for sources with lower frequencies in the range of
${10}^{4}$
to
$1Hz$
).
2 Accretion Induced Collapse
2.1 Collapse scenario
As a white dwarf accretes matter from a binary companion, its density will increase. If the rate of the resulting compressional heating exceeds the white dwarf 's cooling rate, the star's temperature will increase as well. When the mass of the white dwarf exceeds the Chandrasekhar stability limit, it will begin to collapse.
If the temperature of the core is high enough, nuclear burning will begin and the stellar pressure will increase. However, electron capture will act to reduce both the temperature and pressure behind the burn front. The collapse outcome is determined by the relative strength of the nuclear burning and electron capture. If nuclear burning is strong enough, the white dwarf will explode as a Type Ia supernova (SN). If electron capture wins out, the collapse will continue and result in neutron star formation. Electron capture will dominate if the density at which nuclear ignition occurs exceeds a critical density
${\rho}_{crit}$
. For CO white dwarfs,
${\rho}_{crit}$
is in the range
$6\times {10}^{9}\text{\u2013}{10}^{10}gc{m}^{3}$
[
26]
. For ONeMg white dwarfs, electron capture may be stronger than nuclear burning under most conditions (if a RayleighTaylor instability does not produce a turbulent burn front) [
155,
98]
. The collapse of an ONeMg white dwarf begins when its central density reaches
$4\times {10}^{9}gc{m}^{3}$
. (For more details about the conditions under which AIC occurs, see [
155,
98,
26,
25,
125]
.) The dynamics of the collapse itself are somewhat similar to the dynamics of core collapse SNe (without the resulting explosive disruption of the star). The collapse is halted when nuclear densities are reached in the core. The core bounces and sends a bounceshock outward through the star. After several milliseconds, the shock stalls and the remainder of the star collapses through the shock front (see, e.g., [
211]
). Less than
${10}^{1}{M}_{\odot}$
will likely be ejected by the star (due to the bounce itself or due to neutrino absorption/wind mechanisms) [
69]
.
2.2 Formation rate
The AIC occurrence rate is difficult to determine for a number of reasons. These include incomplete understanding of binary star evolution and accretion processes [
40,
111]
. Another uncertainty is whether the collapse of an accreting Chandrasekhar mass white dwarf results in a supernovae explosion or a complete AIC (with accompanying neutron star formation).
The AIC rate can be indirectly inferred from the observed amount of rare, neutron rich isotopes present in the Galaxy. These isotopes (formed via electron capture) are present in the portion (
$\sim 0.1{M}_{\odot}$
) of the outer envelope ejected by the star during an AIC. If all of these isotopes present in the Galaxy are assumed to have originated in AICs, an upper limit of
$\sim {10}^{5}y{r}^{1}$
can be set for the galactic AIC rate [
69]
.
Binary population synthesis analysis can be used to determine which accreting white dwarfs will undergo AIC. The results of Yungelson and Livio [
214]
predict that the galactic AIC rate is between
$8\times {10}^{7}$
and
$8\times {10}^{5}y{r}^{1}$
. Thus, a reasonable occurrence rate can be found for an observation distance of
$100Mpc$
.
2.3 GW emission mechanisms
During AIC, emission of GWs will occur if the infall of matter is aspherical. GWs will also be produced if the collapsing star or neutron star remnant encounters nonaxisymmetric instabilities.
These include global rotational mode,
$r$
mode, and fragmentation instabilities.
Global rotational instabilities in fluids arise from nonaxisymmetric modes
${e}^{\pm im\phi}$
, where
$m=2$
is known as the “barmode” [
192,
6]
. It is convenient to parameterize a system's susceptibility to these modes by the stability parameter
$\beta ={T}_{rot}/\leftW\right$
. Here,
${T}_{rot}$
is the rotational kinetic energy and
$W$
is the gravitational potential energy. Dynamical rotational instabilities, driven by Newtonian hydrodynamics and gravity, develop on the order of the rotation period of the object. For the uniformdensity, incompressible, uniformly rotating MacLaurin spheroids, the dynamical barmode instability sets in at
${\beta}_{d}\approx 0.27$
. For differentially rotating fluids with a polytropic equation of state, numerical simulations have determined that the stability limit
${\beta}_{d}\approx 0.27$
is valid for initial angular momentum distributions that are similar to those of MacLaurin spheroids [
180,
54,
130,
97,
157,
92,
201]
. If the object has an offcenter density maximum,
${\beta}_{d}$
could be as low as
$0.10$
[
200,
210,
157,
41]
. Furthermore, recent work by Shibata, Kariono, and Eriguchi [
184]
suggests that
${\beta}_{d}$
could be as low as
$\sim 0.01$
for stars with a large degree of differential rotation. General relativity may enhance the dynamical barmode instability by slightly reducing
${\beta}_{d}$
[
182,
170]
. Secular rotational instabilities are driven by dissipative processes such as gravitational radiation reaction and viscosity. When this type of instability arises, it develops on the timescale of the relevant dissipative mechanism, which can be much longer than the rotation period. The secular barmode instability limit for MacLaurin spheroids is
${\beta}_{s}\approx 0.14$
.
In rotating stars, gravitational radiation reaction drives the
$r$
modes toward unstable growth [
5,
67]
.
In hot, rapidly rotating neutron stars, this instability may not be suppressed by internal dissipative mechanisms (such as viscosity and magnetic fields) [
121]
. If not limited, the dimensionless amplitude
$\alpha $
of the dominant (
$m=2$
)
$r$
mode will grow to order unity within ten minutes of the formation of a neutron star rotating with a millisecond period. The emitted GWs carry away angular momentum, and will cause the newly formed neutron star to spin down over time. The spindown timescale and the strength of the GWs themselves are directly dependent on the maximum value
${\alpha}_{max}$
to which the amplitude is allowed to grow [
122,
123]
. Originally, it was thought that
${\alpha}_{max}\sim 1$
. Later work indicated that
${\alpha}_{max}$
may be
$\ge 3$
[
122,
189,
173,
123]
. Some research suggests that magnetic fields, hyperon cooling, and hyperon bulk viscosity may limit the growth of the
$r$
mode instability, even in nascent neutron stars [
110,
109,
163,
164,
123,
120,
80,
6]
(significant uncertainties remain regarding the efficacy of these dissipative mechanisms). Furthermore, a recent study of a simple barotropic neutron star model by Arras et al. [
8]
suggests that multimode couplings could limit
${\alpha}_{max}$
to values
$\ll 1$
. If
${\alpha}_{max}$
is indeed
$\ll 1$
, GW emission from
$r$
modes in collapsed remnants is likely undetectable. For the sake of completeness, an analysis of GW emission from
$r$
modes (which assumes
${\alpha}_{max}\sim 1$
) is presented in the remainder of this paper. However, because it is quite doubtful that
${\alpha}_{max}$
is sizeable,
$r$
mode sources are omitted from figures comparing source strengths and detector sensitivities and from discussions of likely detectable sources in the concluding section.
There is some numerical evidence that a collapsing star may fragment into two or more orbiting clumps [
75]
. If this does indeed occur, the orbiting fragments would be a strong GW source.
2.4 Numerical predictions of GW emission
2.4.1 Full collapse simulations
The accretioninduced collapse of white dwarfs has been simulated by a number of groups [
13,
133,
211,
69]
.
The majority of these simulations have been Newtonian and have focused on mass ejection and neutrino and
$\gamma $
ray emission during the collapse and its aftermath (note that neglecting relativistic effects likely introduces an error of order
$(v/c{)}^{2}\sim 10\%$
for the neutron star remnants of AIC; see below). The most sophisticated are those carried out by Fryer et al. [
69]
, as they include realistic equations of state, neutrino transport, and rotating progenitors.
As a part of their general evaluation of upper limits to GW emission from gravitational collapse, Fryer, Holz, and Hughes (hereafter, FHH) [
71]
examined an AIC simulation (model 3) of Fryer et al. [
69]
. FHH used both numerical and analytical techniques to estimate the peak amplitude
${h}_{pk}$
, energy
${E}_{GW}$
, and frequency
${f}_{GW}$
of the gravitational radiation emitted during the collapse simulations they studied.
For direct numerical computation of the GWs emitted in these simulations, FHH used the quadrupole approximation, valid for nearly Newtonian sources [
136]
. This approximation is standardly used to compute the GW emission in Newtonian simulations. The reduced or traceless quadrupole moment of the source can be expressed as
$$\begin{array}{c}I{}_{ij}=\int \rho \left({x}_{i}{x}_{j}\frac{1}{3}{\delta}_{ij}{r}^{2}\right){d}^{3}r,\end{array}$$ 
(1)

where
$i,j=1,2,3$
are spatial indices and
$r=\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}$
is the distance to the source. The two polarizations of the gravitational wave field,
${h}_{+}$
and
${h}_{\times}$
, can be computed in terms of
$\stackrel{\text{}...\text{}}{I}ij$
, where an overdot indicates a time derivative
$d/dt$
. The energy
${E}_{GW}$
is a function of
$\stackrel{\text{}...\text{}}{I}ij$
. Equation ( 1 ) can be used to calculate
$I{}_{ij}$
from the results of a numerical simulation by direct summation over the computational grid. Numerical time derivatives of
$I{}_{ij}$
can then be taken to compute
${h}_{+}$
,
${h}_{\times}$
, and
${E}_{GW}$
. However, successive application of numerical derivatives generally introduces artificial noise. Methods for computing
$\stackrel{\text{}..\text{}}{I}ij$
and
$\stackrel{\text{}...\text{}}{I}ij$
without taking numerical time derivatives have been developed [
65,
20,
137]
. These methods recast the time derivatives of
$I{}_{ij}$
as spatial derivatives of hydrodynamic quantities computed in the collapse simulation (including the density, velocities, and gravitational potential). Thus, instantaneous values for
$\stackrel{\text{}..\text{}}{I}ij$
and
$\stackrel{\text{}...\text{}}{I}ij$
can be computed on a single numerical time slice (see [
137,
217]
for details). Note that FHH define
${h}_{pk}$
as the maximum value of the rms strain
$h=\sqrt{\langle {h}_{+}^{2}+{h}_{\times}^{2}\rangle}$
, where angular brackets indicate that averages have been taken over both wavelength and viewing angle on the sky.
Errors resulting from the neglect of general relativistic effects (in collapse evolutions as a whole and in GW emission estimations like the quadrupole approximation) are of order
$(v/c{)}^{2}$
. These errors are typically
$\sim 10\%$
for neutron star remnants of AIC,
$<30\%$
for neutron star remnants of massive stellar collapse, and
$>30\%$
for black hole remnants. Neglect of general relativity in rotational collapse studies is of special concern because relativistic effects counteract the stabilizing effects of rotation (see Section 3.4 ).
Because the code used in the collapse simulations examined by FHH [
69]
was axisymmetric, their use of the numerical quadrupole approximation discussed above does not account for GW emission that may occur due to nonaxisymmetric mass flow. The GWs computed directly from their simulations come only from polar oscillations (which are significant when the mass flow during collapse [or explosion] is largely aspherical).
In order to predict the GW emission produced by nonaxisymmetric instabilities, FHH employed rough analytical estimates. The expressions they used to approximate the rms strain
$h$
and the power
$P=d{E}_{GW}/dt$
of the GWs emitted by a star that has encountered the barmode instability are
$$\begin{array}{c}{h}_{bar}=\sqrt{\frac{32}{45}}\frac{G}{{c}^{4}}\frac{m{r}^{2}{\omega}^{2}}{d}\end{array}$$ 
(2)

and
$$\begin{array}{c}{P}_{bar}=\frac{32}{45}\frac{G}{{c}^{5}}{m}^{2}{r}^{4}{\omega}^{6}.\end{array}$$ 
(3)

Here
$m$
,
$2r$
, and
$\omega $
are the mass, length, and angular frequency of the bar and
$d$
is the distance to the source. FHH vary the mass
$m$
assumed to be enclosed by the bar (which has a corresponding length
$2r$
) and compute the characteristics of the GW emission as a function of this enclosed mass.
For simplicity, FHH assumed that a fragmentation instability will cause a star to break into two clumps (although more clumps could certainly be produced). Their estimates for the rms strain and power radiated by the orbiting binary fragments are
$$\begin{array}{c}{h}_{bin}=\sqrt{\frac{128}{5}}\frac{G}{{c}^{4}}\frac{m{r}^{2}{\omega}^{2}}{d}\end{array}$$ 
(4)

and
$$\begin{array}{c}{P}_{bin}=\frac{128}{5}\frac{G}{{c}^{5}}{m}^{2}{r}^{4}{\omega}^{6}.\end{array}$$ 
(5)

Here
$m$
is the mass of a single fragment,
$2r$
is the separation of the fragments, and
$\omega $
is their orbital frequency.
For their computation of the GWs radiated via rmodes, FHH used the method of Ho and Lai [
89]
(which assumes
${\alpha}_{max}=1$
) and calculated only the emission from the dominant
$m=2$
mode. This approach is detailed in FHH. If the neutron star mass and initial radius are taken to be
$1.4{M}_{\odot}$
and
$12.53km$
, respectively, the resulting formula for the average GW strain is
$$\begin{array}{c}h\left(t\right)=1.8\times {10}^{24}\alpha \left(\frac{{\nu}_{s}}{1kHz}\right)\left(\frac{20Mpc}{d}\right),\end{array}$$ 
(6)

where
$\alpha $
is the mode amplitude and
${\nu}_{s}$
is the spin frequency.
FHH's numerical quadrupole estimate of the GWs from polar oscillations in the AIC simulation of Fryer
et al. [
69]
predicts a peak dimensionless amplitude
${h}_{pk}=5.9\times {10}^{24}$
(for
$d=100Mpc$
).
The energy
${E}_{GW}=3\times {10}^{45}erg$
is emitted at a frequency of
${f}_{GW}\approx 50Hz$
. This amplitude is about an order of magnitude too small to be observed by the advanced LIGOII detector. The sensitivity curve for the broadband configuration of the LIGOII detector is shown in Figure 1 (see Appendix A of [
71]
for details on the computation of this curve). Note that the characteristic strain
$h$
is plotted along the vertical axis in Figure 1 (and in LISA's sensitivity curve, shown in Figure 12 ). For burst sources,
$h={h}_{pk}$
. For sources that persist for
$N$
cycles,
$h=\sqrt{N}{h}_{pk}$
.
Figure 1
: A comparison between the GW amplitude
$h\left(f\right)$
for various sources and the LIGOII sensitivity curve. See the text for details regarding the computations of
$h$
. The AIC sources are assumed to be located at a distance of
$100Mpc$
; the SNe sources at
$10Mpc$
; and the Population III sources at a luminosity distance of
$\sim 50Gpc$
. Secular barmode sources are identified with an (s), dynamical barmodes with a (d).
The simulation of Fryer et al. [
69]
considered by FHH does produce an object with an offcenter density maximum. However, because the maximum
$\beta $
reached in the AIC simulation of Fryer et al.
was
$<0.06$
(and because the degree of differential rotation present in the remnant was low [
68]
), the collapsing object is not likely to encounter dynamical rotational instabilities. The choice of initial angular momentum
$J$
for the progenitor in an AIC simulation can affect this outcome. Fryer et al.'s choice of
$J={10}^{49}gc{m}^{2}{s}^{1}$
is such that the period of the white dwarf progenitor is
$10s$
less than the shortest observed period
$30s$
of a cataclysmic variable white dwarf [
112]
. If higher values of
$J$
exist in accreting white dwarfs, barmode instabilities may be more likely to occur (see the discussion of work by Liu and Lindblom below).
According to FHH, the remnant of this AIC simulation will be susceptible to
$r$
mode growth.
Assuming
${\alpha}_{max}\sim 1$
(which is likely not physical; see Section 2.3 ), they predict
${E}_{GW}>{10}^{52}erg$
.
FHH compute
$h\left({f}_{GW}\right)$
for coherent observation of the neutron star as it spins down over the course of a year. For a neutron star located at a distance of
$100Mpc$
, this track is always below the LIGOII noise curve. The point on this track with the maximum
$h$
, which corresponds to the beginning of
$r$
mode evolution, is shown in Figure 1 . The track moves down and to the left (i.e.,
$h$
and
$f$
decrease) in this figure as the
$r$
mode evolution continues.
2.4.2 Analyses of equilibrium representations of collapse
In addition to full hydrodynamics collapse simulations, many studies of gravitational collapse have used hydrostatic equilibrium models to represent stars at various stages in the collapse process. Some investigators use sequences of equilibrium models to represent snapshots of the phases of collapse (e.g., [
14,
153,
125]
). Others use individual equilibrium models as initial conditions for hydrodynamical simulations(e.g., [
187,
157,
152,
41]
). Such simulations represent the approximate evolution of a model beginning at some intermediate phase during collapse or the evolution of a collapsed remnant. These studies do not typically follow the intricate details of the collapse itself. Instead, their goals include determining the stability of models against the development of nonaxisymmetric modes and estimation of the characteristics of any resulting GW emission.
Liu and Lindblom [
125,
124]
have applied this equilibrium approach to AIC. Their investigation began with a study of equilibrium models built to represent neutron stars formed from AIC [
125]
.
These neutron star models were created via a twostep process, using a Newtonian version of Hachisu's selfconsistent field method [
76]
. Hachisu's method ensures that the forces due to the centrifugal and gravitational potentials and the pressure are in balance in the equilibrium configuration.
Liu and Lindblom's process of building the nascent neutron stars began with the construction of rapidly rotating, precollapse white dwarf models. Their Models I and II are CO white dwarfs with central densities
${\rho}_{c}={10}^{10}$
and
$6\times {10}^{9}gc{m}^{3}$
, respectively (recall this is the range of densities for which AIC is likely for CO white dwarfs). Their Model III is an ONeMg white dwarf that has
${\rho}_{c}=4\times {10}^{9}gc{m}^{3}$
(recall this is the density at which collapse is induced by electron capture). All three models are uniformly rotating, with the maximum allowed angular velocities. The models' values of total angular momentum are roughly 3 – 4 times that of Fryer et al.'s AIC progenitor Model 3 [
69]
. The realistic equation of state used to construct the white dwarfs is a Coulomb corrected, zero temperature, degenerate gas equation of state [
171,
44]
.
In the second step of their process, Liu and Lindblom [
125]
built equilibrium models of the collapsed neutron stars themselves. The mass, total angular momentum, and specific angular momentum distribution of each neutron star remnant is identical to that of its white dwarf progenitor (see Section 3 of [
125]
for justification of the specific angular momentum conservation assumption). These models were built with two different realistic neutron star equations of state.
Liu and Lindblom's cold neutron star remnants had values of the stability parameter
$\beta $
ranging from
$0.23\text{\u2013}0.26$
. It is interesting to compare these results with those of Zwerger and Müller [
217]
.
Zwerger and Müller performed axisymmetric hydrodynamics simulations of stars with polytropic equations of state (
$P\propto {\rho}^{\Gamma}$
). Their initial models were
$\Gamma =4/3$
polytropes, representative of massive white dwarfs. All of their models started with
${\rho}_{c}={10}^{10}gc{m}^{3}$
. Their model that was closest to being in uniform rotation (A1B3) had 22% less total angular momentum than Liu and Lindblom's Model I. The collapse simulations of Zwerger and Müller that started with model A1B3 all resulted in remnants with values of
$\beta <0.07$
. Comparison of the results of these two studies could indicate that the equation of state may play a significant role in determining the structure of collapsed remnants. Or it could suggest that the assumptions employed in the simplified investigation of Liu and Lindblom are not fully appropriate. Zwerger and Müller's work will be discussed in much more detail in Section 3 , as it was performed in the context of core collapse supernovae.
In a continuation of the work of Liu and Lindblom, Liu [
124]
used linearized hydrodynamics to perform a stability analysis of the cold neutron star AIC remnants of Liu and Lindblom [
125]
.
He found that only the remnant of the ONeMg white dwarf (Liu and Lindblom's Model III) developed the dynamical barmode (
$m=2$
) instability. This model had an initial
$\beta =0.26$
. Note that the
$m=1$
mode, observed by others to be the dominant mode in unstable models with values of
$\beta $
much lower than
$0.27$
[
200,
210,
157,
41]
, did not grow in his simulation. Because Liu and Lindblom's Models I and II had lower values of
$\beta $
, Liu identified the onset of instability for neutron stars formed via AIC as
${\beta}_{d}\approx 0.25$
. Liu estimated the peak amplitude of the GWs emitted by the Model III remnant to be
${h}_{pk}\approx 1.4\times {10}^{24}$
and the LIGOII signaltonoise ratio (for a persistent signal like that seen in the work of [
152]
and [
28]
) to be
$S/N\le 3$
(for
${f}_{GW}\approx 450Hz$
). These values are for a source located at
$100Mpc$
. He also predicted that the timescale for gravitational radiation to carry away enough angular momentum to eliminate the barmode is
${\tau}_{GW}\sim 7s$
(
$\sim 3\times {10}^{3}$
cycles). Thus,
$h\sim 8\times {10}^{23}$
.
(Note that this value for
$h$
is merely an upper limit as it assumes that the amplitude and frequency of the GWs do not change over the
$7s$
during which they are emitted. Of course, they will change as angular momentum is carried away from the object via GW emission.) Such a signal may be marginally detectable with LIGOII (see Figure 1 ). Details of the approximations on which these estimates are based can be found in [
124]
.
Liu cautions that his results hold if the magnetic field of the protoneutron star is
$B\le {10}^{12}G$
. If the magnetic field is larger, then it may have time to suppress some of the neutron star's differential rotation before it cools. This would make bar formation less likely. Such a large field could only result if the white dwarf progenitor's
$B$
field was
$\ge {10}^{8}G$
. Observationbased estimates suggest that about
$25\%$
of white dwarfs in interacting close binaries (cataclysmic variables) are magnetic and that the field strengths for these stars are
$\sim {10}^{7}\text{\u2013}3\times {10}^{8}G$
[
206]
.
2.5 Going further
The AIC scenario is generally discussed in terms of the collapse of an accreting white dwarf to a neutron star. However, Shibata, Baumgarte, and Shapiro have examined the collapse of a rotating, supramassive neutron star to a black hole [
183]
. Such supramassive neutron stars (with masses greater than the maximum mass for a nonrotating neutron star) could be formed and pushed to collapse via accretion from a binary companion. They performed 3D, fully general relativistic hydrodynamics simulations of uniformly rotating neutron stars. Dynamical nonaxisymmetric instabilities (such as the barmode) did not have time to grow in their simulations prior to black hole formation. Differentially rotating neutron star progenitors could have higher values of
$\beta $
than the uniformly rotating models used in this study and may be susceptible to nonaxisymmetric instabilities on a shorter timescale.
3 Collapse of Massive Stars
3.1 Collapse scenario
Stars with mass greater than
$\sim 8{M}_{\odot}$
will undergo core collapse at the end of their thermonuclear burning life cycles. The collapse will trigger a supernova explosion if the star's mass is
$<40\text{\u2013}50{M}_{\odot}$
(for reviews, see [
7,
18,
32]
). Core collapse SNe include Types II and Ib/Ic. SNe Ib/Ic are distinguished from Type II SNe by the absence of hydrogen in their spectra. SNe Ib/Ic are thought to result from the collapse of the cores of massive stars that have lost their hydrogen envelopes (and possibly part of their helium envelopes) by stellar winds or by mass transfer. The SN Ib/Ic progenitors that lose their outer envelopes via stellar winds are known as Wolf–Rayet stars and have initial masses
$\stackrel{>}{\sim}30{M}_{\odot}$
; those that undergo mass loss via mass transfer in binaries have progenitor masses in the range
$12{M}_{\odot}\text{\u2013}18{M}_{\odot}$
[
143]
.
It is believed that the remnant of a core collapse SN is a neutron star if the mass of the progenitor is less than
$\sim 20\text{\u2013}25{M}_{\odot}$
[
73,
71]
. If the progenitor's mass is in the range
$2025\stackrel{<}{\sim}M\stackrel{<}{\sim}40\text{\u2013}50{M}_{\odot}$
, not the entire star is ejected in the SN explosion. More than
$2{M}_{\odot}$
will fall back onto the nascent neutron star and lead to black hole formation. If the progenitor's mass exceeds roughly
$40\text{\u2013}50{M}_{\odot}$
, then no explosion will occur and the star will collapse directly to a black hole. These objects are known as collapsars. However, it is unclear if high metallicity stars with
$M\stackrel{>}{\sim}40\text{\u2013}50{M}_{\odot}$
actually end their lives in collapse or are prevented from doing so by stellar wind driven mass loss [
71]
. Note that the limits on the progenitor masses quoted in this paragraph (especially the
$40\text{\u2013}50{M}_{\odot}$
lower limit for direct black hole formation) are uncertain because the progenitor mass dependence of the neutrino explosion mechanism (see below) is unknown [
81,
147]
.
The massive iron cores of SN II/Ib/Ic progenitors are supported by both thermal and electron degeneracy pressures. The density and temperature of such a core will eventually rise, due to the build up of matter consumed by thermonuclear burning, to the point where electron capture and photodissociation of nuclei begin. Dissociation lowers the photon and electron temperatures and thereby reduces the core's thermal support [
62]
. Electron capture reduces the electron degeneracy pressure. One or both of these processes will trigger the collapse of the core. The relative importance of dissociation and electron capture in instigating collapse is determined by the mass of the star [
62]
. The more massive the core, the bigger is the role played by dissociation.
Approximately
$70\%$
of the inner portion of the core collapses homologously and subsonically.
The outer core collapses at supersonic speeds [
62,
137]
. The maximum velocity of the outer regions of the core reaches
$\sim 7\times {10}^{4}km{s}^{1}$
. It takes just 1 s for an earthsized core to collapse to a radius of 50 km [
7]
.
The inward collapse of the core is halted by nuclear forces when its central density
${\rho}_{c}$
is 2 – 10 times the density of nuclear material [
12,
11]
. The core overshoots its equilibrium position and bounces. A shock wave is formed when the supersonically infalling outer layers hit the rebounding inner core. If the inner core pushes the shock outwards with energy
$E>{10}^{51}erg$
(supplied by the binding energy of the nascent neutron star), then the remainder of the star can be ejected in about 20 ms [
7]
. This socalled “prompt explosion” mechanism will only succeed if a very soft supranuclear equation of state is used in conjunction with a relatively small core (
$M\stackrel{<}{\sim}1.35{M}_{\odot}$
, derived from a very low mass progenitor) and a large portion of the collapse proceeds homologously [
18,
143]
.
Inclusion of general relativistic effects in collapse simulations can increase the success of the prompt mechanism in some cases [
12,
190]
.
Both dissociation of nuclei and electron capture can reduce the ejection energy, causing the prompt mechanism to fail. The shock will then stall at a radius in the range 100 – 200 km. Colgate and White [
47]
were the first to suggest that energy from neutrinos emitted by the collapsed core could revive the stalled shock. (See Burrows and Thompson [
36]
for a recent review of core collapse neutrino processes.) However, their simulations did not include the physics necessary to accurately model this “delayed explosion” mechanism. Wilson and collaborators were the first to perform collapse simulations with successful delayed ejections [
24,
23,
207,
19,
209,
208]
. However, their simulations and those of others had difficulty producing energies high enough to match observations [
45,
30,
104]
. Observations of SN 1987a show that significant mixing occurred during this supernovae (see Arnett et al. [
7]
for a review). Such mixing can be attributed to nonradial motion resulting from fluid instabilities. Convective instabilities play a significant role in the current picture of the delayed explosion mechanism. The outer regions of the nascent neutron star are convectively unstable after the shock stalls (for an interval of 10 – 100 ms after bounce) due to the presence of negative lepton and energy gradients [
143]
. This has been confirmed by both 2D and 3D simulations [
33,
34,
104,
140,
141,
105,
143,
134,
100,
102,
159]
. Convective motion is more effective at transporting neutrinos out of the protoneutron star than is diffusion. Less than
$10\%$
of the neutrinos emitted by the neutron star need to be absorbed and converted to kinetic energy for the shock to be revived [
143]
. The “hot bubble” region above the surface of the neutron star also has been shown to be convectively unstable [
45,
18,
46,
74]
. Janka and Müller have demonstrated that convection in this region only aids the explosion if the neutrino luminosity is in a narrow region [
106]
. Some simulations that include advanced neutrino transport methods have cast doubt on the ability of convection to ensure the success of the delayed explosion mechanism [
135]
. However, Rampp and Janka [
102,
159]
have recently performed collapse simulations with sophisticated neutrino physics (including Boltzmann transport and stateoftheart neutrinomatter interactions) that are very close to producing explosions. Their results indicate that successful explosions could be very sensitive to the details of the neutrino physics. Multidimensional simulations that include the full effects of general relativity, rotation, and an improved treatment of neutrino interactions will likely be necessary to properly model the delayed explosion mechanism [
103,
102]
.
In addition to the mixing seen in SN 1987a, observations of (i) polarization in the spectra of several core collapse SNe, (ii) jets in the Cas A remnant, and (iii) kicks in neutron stars suggest that supernovae are inherently aspherical (see [
7,
3,
90]
and references therein). Note that these asphericities could originate in the central explosion mechanism itself and/or the mechanism(s) for energy transfer between the core and ejecta [
102]
. These observations partly motivated the multidimensional studies of convection in the delayed explosion mechanism discussed above and have also spurred investigations of the role jets may play in SN explosions [
128,
205,
212]
. Recent 3D radiation hydrodynamics simulations performed by Höflich et al. [
90]
indicate that low velocity jets stalled inside SN envelopes can account for the observed asymmetries. Possible sources of such asymmetries include magnetic fields [
15,
3]
, anisotropic neutrino emission, and convection.
3.2 Formation rate
Type II/Ib/Ic supernovae are observed to occur in only spiral and irregular galaxies. The most thorough computation of SN rates is that of Cappellaro et al. [
38]
. Their sample includes 137 SNe from five different SN searches. They determined that the core collapse SN rate in the Galaxy is
$0.6\times {10}^{3}\text{\u2013}1.6\times {10}^{2}y{r}^{1}$
. Thus, a reasonable occurrence rate can be found for an observation distance of 10 Mpc. (Note that a recent infrared survey estimates that the rate may actually be an order of magnitude higher [
129]
.) Approximately 5 – 40% of these SNe will leave behind black hole remnants [
73]
. The formation rate of collapsars, the massive cores that collapse to black holes without an accompanying SNe explosion, is unknown. This is because of the uncertainty in stellar winddriven massloss rates [
73]
.
3.3 GW emission mechanisms
Gravitational radiation will be emitted during the collapse/explosion of a core collapse SN due to the star's changing quadrupole moment. A rough description of the possible evolution of the quadrupole moment is given in the remainder of this paragraph. During the first 100 – 250 ms of the collapse, as the core contracts and flattens, the magnitude of the quadrupole moment
$I{}_{ij}$
will increase. The contraction speeds up over the next 20 ms and the density distribution becomes a centrally condensed torus [
137]
. In this phase the core's shrinking size dominates its increasing deformation and the magnitude of
$I{}_{ij}$
decreases. As the core bounces,
$I{}_{ij}$
changes rapidly due to the deceleration and rebound. If the bounce occurs because of nuclear pressure, its timescale will be
$<1ms$
. If centrifugal forces play a role in halting the collapse, the bounce can last up to several
$ms$
[
137]
. The magnitude of
$I{}_{ij}$
will increase due to the core's expansion after bounce. As the resulting shock moves outwards, the unshocked portion of the core will undergo oscillations, causing
$I{}_{ij}$
to oscillate as well. The shape of the core, the depth of the bounce, the bounce timescale, and the rotational energy of the core all strongly affect the GW emission. For further details see [
62,
137]
.
Convectively driven inhomogeneities in the density distribution of the outer regions of the nascent neutron star and anisotropic neutrino emission are other sources of GW emission during the collapse/explosion [
35,
146]
.
As discussed in the case of AIC, global rotational instabilities (such as the
$m=2$
barmode) may develop during the collapse itself or in a neutron star remnant. A neutron star remnant will likely also be susceptible to the radiation reaction driven
$r$
modes. Both of these types of instabilities will emit GWs, as will a fragmentation instability if one occurs. See Section 2.3 for further details regarding these instabilities.
If the collapsed remnant is a black hole, GWs will be radiated as the infall of the remaining stellar matter distorts the black hole's geometry. This “ringdown phase” will end when gravitational radiation has dissipated all of the black hole's accretioninduced distortion. Zanotti, Rezzolla, and Font [
215]
have recently suggested that the torus of matter surrounding the black hole may be an even stronger source of GWs than the collapse itself (see also [
113,
204]
). When perturbed, such “toroidal neutron stars” may undergo regular oscillations and thus emit copious GWs.
3.4 Numerical predictions of GW emission
3.4.1 Historical investigations
The collapse of the progenitors of core collapse supernovae has been investigated as a source of gravitational radiation for more than three decades. In an early study published in 1971, Ruffini and Wheeler [
165]
identified mechanisms related to core collapse that could produce GWs and provided orderofmagnitude estimates of the characteristics of such emission.
Quantitative computations of GW emission during the infall phase of collapse were performed by Thuan and Ostriker [
198]
and Epstein and Wagoner [
57,
56]
, who simulated the collapse of oblate dust spheroids. Thuan and Ostriker used Newtonian gravity and computed the emitted radiation in the quadrupole approximation. Epstein and Wagoner discovered that postNewtonian effects prolonged the collapse and thus lowered the GW luminosity. Subsequently, Novikov [
156]
and Shapiro and Saenz [
177,
166]
included internal pressure in their collapse simulations and were thus able to examine the GWs emitted as collapsing cores bounced at nuclear densities. The quadrupole GWs from the ringdown of the collapse remnant were initially investigated by the perturbation study of Turner and Wagoner [
202]
and later by Saenz and Shapiro [
167,
168]
.
Müller [
139]
calculated the quadrupole GW emission from 2D axisymmetric collapse based on the Newtonian simulations of Müller and Hillebrandt [
144]
(these simulations used a realistic equation of state and included differential rotation). He found that differential rotation enhanced the efficiency of the GW emission.
The first fully general relativistic investigations of stellar core collapse were Nakamura's 2D simulations of rotating collapse [
148,
149]
. However, because of the limits of his numerical formalism and computational resources, he was unable to compute the emitted gravitational radiation (the energy of this emission is quite small compared to the rest mass energy and thus was difficult to extract numerically). The results of this work indicate that collapse does not lead to black hole formation if the parameter
$q=J/{M}^{2}$
exceeds unity (here
$J$
and
$M$
are the angular momentum and gravitational mass of the remnant).
Stark and Piran [
188,
158]
were the first to compute the GW emission from fully relativistic collapse simulations, using the groundbreaking formalism of Bardeen and Piran [
10]
. They followed the (pressurecut induced) collapse of rotating polytropes in 2D. Their work focused in part on the conditions for black hole formation and the nature of the resulting ringdown waveform, which they found could be described by the quasinormal modes of a rotating black hole. In each of their simulations, less than
$1\%$
of the gravitational mass was converted to GW energy.
Seidel and collaborators also studied the effects of general relativity on the GW emission during collapse and bounce [
175,
176]
. They employed a perturbative approach, valid only in the slowly rotating regime.
The gravitational radiation from nonaxisymmetric collapse was investigated by Detweiler and Lindblom, who used a sequence of nonaxisymmetric ellipsoids to represent the collapse evolution [
50]
. They found that the radiation from their analysis of nonaxisymmetric collapse was emitted over a more narrow range of frequency than in previous studies of axisymmetric collapse.
For further discussion of the first two decades of study of the GW emission from stellar collapse see [
63]
. In the remainder of Section 3.4 , more recent investigations will be discussed.
3.4.2 Axisymmetric simulations
The core collapse simulations of Mönchmeyer et al. began with better iron core models and a more realistic microphysical treatment (including a realistic equation of state, electron capture, and a simple neutrino transport scheme) than any previous study of GW emission from axisymmetric stellar core collapse [
137]
. The shortcomings of their investigation included initial models that were not in rotational equilibrium, an equation of state that was somewhat stiff in the subnuclear regime, and the use of Newtonian gravity. Each of their four models had a different initial angular momentum profile. The rotational energies of the models ranged from 0.1 – 0.45 of the maximum possible rotational energy.
The collapses of three of the four models of Mönchmeyer
et al. were halted by centrifugal forces at subnuclear densities. This type of low
${\rho}_{c}$
bounce had been predicted by Shapiro and Lightman [
178]
and Tohline [
199]
(in the context of the “fizzler” scenario for failed supernovae; see also [
82,
83,
96]
), and had been observed in earlier collapse simulations [
145,
191]
. Mönchmeyer and collaborators found that a bounce caused by centrifugal forces would last for several
$ms$
, whereas a bounce at nuclear densities would occur in
$<1ms$
. They also determined that a subnuclear bounce produced larger amplitude oscillations in density and radius, with larger oscillation periods, than a bounce initiated by nuclear forces alone. They pointed out that these differences in timescale and oscillatory behavior should affect the GW signal. Therefore, the GW emission could indicate whether the bounce was a result of centrifugal or nuclear forces.
Mönchmeyer
et al. identified two different types of waveforms in their models (computed using the numerical quadrupole approximation discussed in Section 2.4 ). The waveforms they categorized as Type I (similar to those observed in previous collapse simulations [
139,
65]
) are distinguished by a large amplitude peak at bounce and subsequent damped ringdown oscillations. They noted that Type I signals were produced by cores that bounced at nuclear densities (or bounced at subnuclear densities if the cores had small ratios of radial kinetic to rotational kinetic energies).
The quadrupole gravitational wave amplitude
${A}_{20}^{E2}$
for a Type I waveform is shown in Figure 2 (see [
194,
217]
for expressions relating
${A}_{20}^{E2}$
to
$h$
). The waveforms identified as Type II exhibit several maxima, which result from multiple bounces (see Figure 3 for an example of a Type II waveform).
Note that the waveforms displayed in Figures
2 and 3 are from the study of Zwerger and Müller [
217]
, discussed below.
Figure 2
: Type I waveform (quadrupole amplitude
${A}_{20}^{E2}$
as a function of time) from one of Zwerger and Müller's [
217]
simulations of a collapsing polytrope. The vertical dotted line marks the time at which the first bounce occurred. (Figure 5d of [
217]
; used with permission.)
Figure 3
: Type II waveform (quadrupole amplitude
${A}_{20}^{E2}$
as a function of time) from one of Zwerger and Müller's [
217]
simulations of a collapsing polytrope. The vertical dotted line marks the time at which bounce occurred. (Figure 5a of [
217]
; used with permission.)
The model of Mönchmeyer et al. that bounced due to nuclear forces had the highest GW amplitude of all of their models,
${h}_{pk}\sim {10}^{23}$
for a source distance
$d=10Mpc$
, and the largest emitted energy
${E}_{GW}\sim {10}^{47}erg$
. The accompanying power spectrum peaked in the frequency range
$5\times {10}^{2}\text{\u2013}{10}^{3}Hz$
.
The most extensive Newtonian survey of the parameter space of axisymmetric, rotational core collapse is that of Zwerger and Müller [
217]
. They simulated the collapse of 78 initial models with varying amounts of rotational kinetic energy (reflected in the initial value of the stability parameter
${\beta}_{i}$
), differential rotation, and equation of state stiffness. In order to make this large survey tractable, they used a simplified equation of state and did not explicitly account for electron capture or neutrino transport. Their initial models were constructed in rotational equilibrium via the method of Eriguchi and Müller [
58]
. The models had a polytropic equation of state, with initial adiabatic index
${\Gamma}_{i}=4/3$
. Collapse was induced by reducing the adiabatic index to a value
${\Gamma}_{r}$
in the range 1.28 – 1.325. The equation of state used during the collapse evolution had both polytropic and thermal contributions.
Figure 4
: Type III waveform (quadrupole amplitude
${A}_{20}^{E2}$
as a function of time) from one of Zwerger and Müller's [
217]
simulations of a collapsing polytrope. The vertical dotted line marks the time at which bounce occurred. (Figure 5e of [
217]
; used with permission).
The major result of Zwerger and Müller's investigation was that the signal type of the emitted gravitational waveform in their runs was determined by the stiffness of the equation of state of the collapsing core (i.e., the value of
${\Gamma}_{r}$
). In their simulations, Type I signals (as labelled by Mönchmeyer et al. [
137]
) were produced by models with relatively soft equations of state,
${\Gamma}_{r}\stackrel{<}{\sim}1.31$
. Type II signals were produced by the models with stiffer equations of state,
${\Gamma}_{r}\stackrel{>}{\sim}1.32$
. They found a smooth transition between these signal types if
${\Gamma}_{r}$
was increased while all other parameters were held fixed. They also observed another class of signal, Type III, for their models with the lowest
${\Gamma}_{r}$
(
$=1.28$
). Type III waveforms have a large positive peak just prior to bounce, a smaller negative peak just after bounce, and smaller subsequent oscillations with very short periods (see Figure 4 ). Type III signals were not observed in the evolution of strongly differentially rotating
${\Gamma}_{r}=1.28$
models. Their waveforms were computed with the same technique used in [
137]
.
In contrast to the results of Mönchmeyer
et al. [
137]
, in Zwerger and Müller's investigation the value of
${\rho}_{c}$
at bounce did not determine the signal type. Instead, the only effect on the waveform due to
${\rho}_{c}$
was a decrease in
${h}_{pk}$
in models that bounced at subnuclear densities. The effect of the initial value of
${\beta}_{i}$
on
${h}_{pk}$
was nonmonotonic. For models with
${\beta}_{i}\stackrel{<}{\sim}0.1$
,
${h}_{pk}$
increased with increasing
${\beta}_{i}$
. This is because the deformation of the core is larger for faster rotators. However, for models with larger
${\beta}_{i}$
,
${h}_{pk}$
decreases as
${\beta}_{i}$
increases. These models bounce at subnuclear densities. Thus, the resulting acceleration at bounce and the GW amplitude are smaller. Zwerger and Müller found that the maximum value of
${h}_{pk}$
for a given sequence was reached when
${\rho}_{c}$
at bounce was just less than
${\rho}_{nuc}$
. The degree of differential rotation did not have a large effect on the emitted waveforms computed by Zwerger and Müller. However, they did find that models with soft equations of state emitted stronger signals as the degree of differential rotation increased.
The models of Zwerger and Müller that produced the largest GW signals fell into two categories:
those with stiff equations of state and
${\beta}_{i}<0.01$
; and those with soft equations of state,
${\beta}_{i}\ge 0.018$
, and large degrees of differential rotation. The GW amplitudes emitted during their simulations were in the range
$4\times {10}^{25}\stackrel{<}{\sim}h\stackrel{<}{\sim}4\times {10}^{23}$
, for
$d=10Mpc$
(the model with the highest
$h$
is identified in Figure 1 ). The corresponding energies ranged from
${10}^{44}\stackrel{<}{\sim}{E}_{GW}\stackrel{<}{\sim}{10}^{47}erg$
.
The peaks of their power spectra were between 500 Hz and 1 kHz. Such signals would fall just outside of the range of LIGOII. Yamada and Sato [
213]
used techniques very similar to those of Zwerger and Müller [
217]
in their core collapse study. Their investigation revealed that the
${h}_{pk}$
for Type I signals became saturated when the dimensionless angular momentum of the collapsing core,
$q=J/(2GM/c)$
, reached
$\sim 0.5$
.
They also found that
${h}_{pk}$
was sensitive to the stiffness of the equation of state for densities just below
${\rho}_{nuc}$
. The characteristics of the GW emission from their models were similar to those of Zwerger and Müller.
3.4.3 Nonaxisymmetric simulations
The GW emission from nonaxisymmetric hydrodynamics simulations of stellar collapse was first studied by Bonazzola and Marck [
131,
22]
. They used a Newtonian, pseudospectral hydrodynamics code to follow the collapse of polytropic models. Their simulations covered only the prebounce phase of the collapse. They found that the magnitudes of
${h}_{pk}$
in their 3D simulations were within a factor of two of those from equivalent 2D simulations and that the gravitational radiation efficiency did not depend on the equation of state.
The first 3D hydrodynamics collapse simulations to study the GW emission well beyond the core bounce phase were performed by Rampp, Müller, and Ruffert [
160]
. These authors started their Newtonian simulations with the only model (A4B5G5) of Zwerger and Müller [
217]
that had a postbounce value for the stability parameter
$\beta =0.35$
that significantly exceeded
$0.27$
(recall this is the value at which the dynamical bar instability sets in for MacLaurin spheroidlike models).
This model had the softest EOS (
${\Gamma}_{r}=1.28$
), highest
${\beta}_{i}=0.04$
, and largest degree of differential rotation of all of Zwerger and Müller's models. The model's initial density distribution had an offcenter density maximum (and therefore a toruslike structure). Rampp, Müller, and Ruffert evolved this model with a 2D hydrodynamics code until its
$\beta $
reached
$\approx 0.1$
. At that point, 2.5 ms prior to bounce, the configuration was mapped onto a 3D nested cubical grid structure and evolved with a 3D hydrodynamics code.
Before the 3D simulations started, nonaxisymmetric density perturbations were imposed to seed the growth of any nonaxisymmetric modes to which the configuration was unstable. When the imposed perturbation was random (5% in magnitude), the dominant mode that arose was
$m=4$
.
The growth of this particular mode was instigated by the cubical nature of the computational grid.
When an
$m=3$
perturbation was imposed (10% in magnitude), three clumps developed during the postbounce evolution and produced three spiral arms. These arms carried mass and angular momentum away from the center of the core. The arms eventually merged into a barlike structure (evidence of the presence of the
$m=2$
mode). Significant nonaxisymmetric structure was visible only within the inner 40 km of the core. Their simulations were carried out to
$\sim 14ms$
after bounce.
The amplitudes of the emitted gravitational radiation (computed in the quadrupole approximation) were only
$\sim 2\%$
different from those observed in the 2D simulation of Zwerger and Müller. Because of low angular resolution in the 3D runs, the energy emitted was only
$65\%$
of that emitted in the corresponding 2D simulation.
The findings of Centrella
et al. [
41]
indicate it is possible that some of the postbounce configurations of Zwerger and Müller, which have lower values of
$\beta $
than the model studied by Rampp, Müller, and Ruffert [
160]
, may also be susceptible to nonaxisymmetric instabilities. Centrella et al. have performed 3D hydrodynamics simulations of
$\Gamma =1.3$
polytropes to test the stability of configurations with offcenter density maxima (as are present in many of the models of Zwerger and Müller [
217]
).
The simulations carried out by Centrella and collaborators were not full collapse simulations, but rather began with differentially rotating equilibrium models. These simulations tracked the growth of any unstable nonaxisymmetric modes that arose from the initial
$1\%$
random density perturbations that were imposed. Their results indicate that such models can become dynamically unstable at values of
$\beta \stackrel{>}{\sim}0.14$
. The observed instability had a dominant
$m=1$
mode.
Centrella
et al. estimate that if a stellar core of mass
$M\sim 1.4{M}_{\odot}$
and radius
$R\sim 200km$
encountered this instability, the values of
${h}_{pk}$
from their models would be
$\sim 2\times {10}^{24}\text{\u2013}2\times {10}^{23}$
, for
$d=10Mpc$
. The frequency at which
${h}_{pk}$
occurred in their simulations was
$\sim 200Hz$
. This instability would have to persist for at least
$\sim 15$
cycles to be detected with LIGOII. Brown [
29]
carried out an investigation of the growth of nonaxisymmetric modes in postbounce cores that was similar in many respects to that of Rampp, Müller, and Ruffert [
160]
. He performed 3D hydrodynamical simulations of the postbounce configurations resulting from 2D simulations of core collapse. His precollapse initial models are
$\Gamma =4/3$
polytropes in rotational equilibrium. The differential rotation laws used to construct Brown's initial models were motivated by the stellar evolution study of Heger, Langer, and Woosley [
86]
. The angular velocity profiles of their precollapse progenitors were broad and Gaussianlike. Brown's initial models had peak angular velocities ranging from 0.8 – 2.4 times those of [
86]
. The model evolved by Rampp, Müller, and Ruffert [
160]
had much stronger differential rotation than any of Brown's models. To induce collapse, Brown reduced the adiabatic index of his models to
$\Gamma =1.28$
, the same value used by [
160]
.
Brown found that
$\beta $
increased by a factor
$\stackrel{<}{\sim}2$
during his 2D collapse simulations. This is much less than the factor of
$\sim 9$
observed in the model studied by Rampp, Müller, and Ruffert [
160]
.
This is likely a result of the larger degree of differential rotation in the model of Rampp
et al.
Brown performed 3D simulations of the two most rapidly rotating of his postbounce models (models
$\Omega 24$
and
$\Omega 20$
, both of which had
$\beta >0.27$
after bounce) and of the model of Rampp et al.
(which, although it starts out with
$\beta =0.35$
, has a sustained
$\beta <0.2$
). Brown refers to the Rampp et al. model as model RMR. Because Brown's models do not have offcenter density maxima, they are not expected to be unstable to the
$m=1$
mode observed by Centrella et al. [
41]
. He imposed random
$1\%$
density perturbations at the start of all three of these 3D simulations (note that this perturbation was of a much smaller amplitude than those imposed by [
160]
).
Brown's simulations determined that both his most rapidly rotating model
$\Omega 24$
(with postbounce
$\beta >0.35$
) and model RMR are unstable to growth of the
$m=2$
barmode. However, his model
$\Omega 20$
(with postbounce
$\beta >0.3$
) was stable. Brown observed no dominant
$m=3$
or
$m=4$
modes growing in model RMR at the times at which they were seen in the simulations of Rampp et al.
This suggests that the mode growth in their simulations was a result of the large perturbations they imposed. The
$m=2$
mode begins to grow in model RMR at about the same time as Rampp et al. stopped their evolutions. No substantial
$m=1$
growth was observed.
The results of Brown's study indicate that the overall
$\beta $
of the postbounce core may not be a good diagnostic for the onset of instability. He found, as did Rampp, Müller, and Ruffert [
160]
, that only the innermost portion of the core (with
$\rho >{10}^{10}gc{m}^{3}$
) is susceptible to the barmode.
This is evident in the stability of his model
$\Omega 20$
. This model had an overall
$\beta >0.3$
, but an inner core with
${\beta}_{ic}=0.15$
. Brown also observed that the
$\beta $
of the inner core does not have to exceed
$0.27$
for the model to encounter the barmode. Models
$\Omega 24$
and RMR had
${\beta}_{ic}\approx 0.19$
. He speculates that the inner cores of these later two models may be barunstable because interaction with their outer envelopes feeds the instability or because
${\beta}_{d}<0.27$
for such configurations.
Fryer and Warren [
74]
have recently performed the first 3D collapse simulations to follow the entire collapse through explosion. They used a smoothed particle hydrodynamics code, a realistic equation of state, the fluxlimited diffusion approximation for neutrino transport, and Newtonian spherical gravity. Their initial model was nonrotating. Thus, no barmode instabilities could develop during their simulations. The only GW emitting mechanism present in their models was convection in the core. The maximum amplitude
$h$
of this emission, computed in the quadrupole approximation, was
$\sim 3\times {10}^{26}$
, for
$d=10Mpc$
[
72]
. Preliminary results from rotating simulations performed with this same 3D code do not show the development of bar or fragmentation instabilities [
72]
.
The GW emission from nonradial quasinormal mode oscillations in protoneutron stars has recently been examined by Ferrari, Miniutti, and Pons [
61]
. They found that the frequencies of emission
${f}_{GW}$
during the first second after formation (600 – 1100 Hz for the first fundamental and gravity modes) are significantly lower than the corresponding frequencies for cold neutron stars and thus reside in the bandwidths of terrestrial interferometers. However, for first generation interferometers to detect the GW emission from an oscillating protoneutron star located at 10 Mpc, with a signaltonoise ratio of 5,
${E}_{GW}$
must be
$\sim {10}^{3}\text{\u2013}{10}^{2}{M}_{\odot}{c}^{2}$
. It is unlikely that this much energy is stored in these modes (the collapse itself may only emit
$\sim {10}^{7}{M}_{\odot}{c}^{2}$
in gravitational waves [
53]
).
3.4.4 General relativistic simulations
General relativistic effects oppose the stabilizing influence of rotation in precollapse cores.
Thus, stars that might be prevented from collapsing due to rotational support in the Newtonian limit may collapse when general relativistic effects are considered. Furthermore, general relativity will cause rotating stars undergoing collapse to bounce at higher densities than in the Newtonian case [
192,
217,
160,
31]
.
The full collapse simulations of Fryer and Heger [
70]
are the most sophisticated axisymmetric simulations from which the resultant GW emission has been studied [
71,
72]
. Fryer and Heger include the effects of general relativity, but assume (for the purposes of their gravity treatment only) that the mass distribution is spherical. The GW emission from these simulations was evaluated with either the quadrupole approximation or simpler estimates (see below).
The work of Fryer and Heger [
70]
is an improvement over past collapse investigations because it starts with rotating progenitors evolved to collapse with a stellar evolution code (which incorporates angular momentum transport via an approximate diffusion scheme) [
84]
, incorporates realistic equations of state and neutrino transport, and follows the collapse to late times. The values of total angular momentum of the inner cores of Fryer and Heger (
$0.95\text{\u2013}1.9\times {10}^{49}gc{m}^{2}{s}^{1}$
) are lower than has often been assumed in studies of the GW emission from core collapse. Note that the total specific angular momentum of these core models may be lower by about a factor of 10 if magnetic fields were included in the evolution of the progenitors [
3,
85,
87]
.
FHH's [
71]
numerical quadrupole estimate of the GWs from polar oscillations in the collapse simulations of Fryer and Heger [
70]
predicts a peak dimensionless amplitude
${h}_{pk}=4.1\times {10}^{23}$
(for
$d=10Mpc$
), emitted at
${f}_{GW}\approx 20Hz$
. The radiated energy
${E}_{GW}\sim 2\times {10}^{44}erg$
. This signal would be just out of the detectability range of the LIGOII detector.
The cores in the simulations of Fryer and Heger [
70]
are not compact enough (or rotating rapidly enough) to develop bar instabilities during the collapse and initial bounce phases. However, the explosion phase ejects a good deal of low angular momentum material along the poles in their evolutions. Therefore, about 1 s after the collapse,
$\beta $
becomes high enough in their models to exceed the secular bar instability limit. The
$\beta $
of their model with the least angular momentum actually exceeds the dynamical bar instability limit as well (it contracts to a smaller radius and thus has a higher spin rate than the model with higher angular momentum). FHH (and [
72]
) compute an upper limit (via Equation ( 2 )) to the emitted amplitude from their dynamically unstable model of
$h\sim 3\times {10}^{22}$
(if coherent emission from a bar located at 10 Mpc persists for 100 cycles).
The corresponding frequency and maximum power are
${f}_{GW}\approx {10}^{3}Hz$
and
${P}_{GW}={10}^{53}erg{s}^{1}$
.
LIGOII should be able to detect such a signal (see Figure
1 , where FHH's upper limit to
$h$
for this dynamical barmode is identified).
As mentioned above, the protoneutron stars of Fryer and Heger are likely to be unstable to the development of secular bar instabilities. The GW emission from protoneutron stars that are secularly unstable to the barmode has been examined by Lai and Shapiro [
117,
115]
. Because the timescale for secular evolution is so long, 3D hydrodynamics simulations of the nonlinear development of a secular bar can be impractical. To bypass this difficulty, Lai [
115]
considers only incompressible fluids, for which there are exact solutions for (Dedekind and Jacobilike) bar development. He predicts that such a bar located at 10 Mpc would emit GWs with a peak characteristic amplitude
$h\sim {10}^{21}$
, if the bar persists for
${10}^{2}\text{\u2013}{10}^{4}$
cycles. The maximum
${f}_{GW}$
of the emitted radiation is in the range
${10}^{2}\text{\u2013}{10}^{3}Hz$
. This type of signal should be easily detected by LIGOI (although detection may require a technique like the fast chirp transform method of Jenet and Prince [
107]
due to the complicated phase evolution of the emission).
FHH predict that a fragmentation instability is unlikely to develop during core collapse SNe because the cores have central density maxima (see also [
72]
). However, they do give estimates (calculated via Equations ( 4 , 5 )) for the amplitude, power, and frequency of the emission from such an instability:
${h}_{pk}\sim 2\times {10}^{22}$
,
${P}_{GW}={10}^{54}erg{s}^{1}$
,
${f}_{GW}\approx 2\times {10}^{3}Hz$
. Again, this signal would fall just beyond the upper limit of LIGOII's frequency range. Drawing on analogy with fragmentation observed in star formation simulations, Davies et al. [
49]
have recently suggested that the fragmentation of collapsing massive stellar cores and the subsequent gravitational radiationdriven coalescence of the fragments may power gammaray bursts (see also [
113]
). They say that due to kick velocity constraints, this would be possible only for supernovae that leave behind black hole remnants with
$M\stackrel{>}{\sim}12{M}_{\odot}$
(formed via the merger of the fragments). Under the assumption that the collapsing core splits into two fragments (each with
$M=0.7{M}_{\odot}$
), they estimate that LIGOII should detect the merger of
$\sim 400$
core collapse fragments per year. Note that the signal strength would be even greater if the total mass of the black hole remnant were indeed
$12{M}_{\odot}$
(however the formation rate for these more massive objects is likely lower than that of lower mass cores).
The GW emission from
$r$
mode unstable neutron star remnants of core collapse SNe would be easily detectable if
${\alpha}_{max}\sim 1$
(which is likely not physical; see Section 2.3 ). Multiple GW bursts will occur as material falls back onto the neutron star and results in repeat episodes of
$r$
mode growth (note that a single
$r$
mode episode can have multiple amplitude peaks [
122]
). FHH calculate that the characteristic amplitude of the GW emission from this
$r$
mode evolution tracks from
$6\text{\u2013}1\times {10}^{22}$
, over a frequency range of
${10}^{3}\text{\u2013}{10}^{2}Hz$
(see Section 2.4 for details). They estimate the emitted energy to exceed
${10}^{52}erg$
.
If the collapse remnant is a black hole, GWs will be emitted as the accreting black hole “rings down”. Even with very optimistic accretion scenarios, FHH conclude that such radiation will be of very low amplitude and beyond the upper frequency reach of LIGOII (see [
71]
for details).
General relativity has been more fully accounted for in the core collapse studies of Dimmelmeier, Font, and Müller [
51,
52,
53]
, which build on the Newtonian, axisymmetric collapse simulations of Zwerger and Müller [
217]
. In all, they have followed the collapse evolution of 26 different models, with both Newtonian and general relativistic simulations. As in the work of Zwerger and Müller, the different models are characterized by varying degrees of differential rotation, initial rotation rates, and adiabatic indices. They use the conformally flat metric to approximate the space time geometry [
48]
in their relativistic hydrodynamics simulations. This approximation gives the exact solution to Einstein's equations in the case of spherical symmetry. Thus, as long as the collapse is not significantly aspherical, the approximation is relatively accurate. However, the conformally flat condition does eliminate GW emission from the spacetime. Because of this, Dimmelmeier, Font, and Müller used the quadrupole approximation to compute the characteristics of the emitted GW signal (see [
217]
for details).
The general relativistic simulations of Dimmelmeier
et al. showed the three different types of collapse evolution (and corresponding gravitational radiation signal) seen in the Newtonian simulations of Zwerger and Müller (regular collapse – Type I signal; multiple bounce collapse – Type II signal; and rapid collapse – Type III signal). However, relativistic effects sometimes led to a different collapse type than in the Newtonian case. This is because general relativity did indeed counteract the stabilizing effects of rotation and led to much higher bounce densities (up to 700% higher). They found that multiple bounce collapse is much rarer in general relativistic simulations (occurring in only two of their models). When multiple bounce does occur, relativistic effects shorten the time interval between bounces by up to a factor of four. Movies of the simulations of four models from Dimmelmeier et al. [
53]
are shown in Figures 5 , 6 , 7 , and 8 . The four evolutions shown include a regular collapse (Movie 5 ), a rapid collapse (Movie 6 ), a multiple bounce collapse (Movie 7 ), and a very rapidly and differentially rotating collapse (Movie 8 ). The left frames of each movie contain the 2D evolution of the logarithmic density. The upper and lower right frames display the evolutions of the gravitational wave amplitude and the maximum density, respectively.
These movies can also be viewed at [
132]
.
Figure 5
: Movie of the evolution of the regular collapse model A3B2G4 of Dimmelmeier et al. [
53]
. The left frame contains the 2D evolution of the logarithmic density. The upper and lower right frames display the evolutions of the gravitational wave amplitude and the maximum density, respectively.
Figure 6
: Same as Movie 5 , but for rapid collapse model A3B2G5 of Dimmelmeier et al. [
53]
.
Figure 7
: Same as Movie 5 , but for multiple collapse model A2B4G1 of Dimmelmeier et al. [
53]
.
Figure 8
: Same as Movie 5 , but for rapid, differentially rotating collapse model A4B5G5 of Dimmelmeier et al. [
53]
.
Dimmelmeier et al. found that models for which the collapse type was the same in both Newtonian and relativistic simulations had lower GW amplitudes
${h}_{pk}$
in the relativistic case. This is because the Newtonian models were less compact at bounce and thus had material with higher densities and velocities at larger radii. Both higher and lower values of
${h}_{pk}$
were observed in models for which the collapse type changed. Overall, the range of
${h}_{pk}$
(
$4\times {10}^{24}\text{\u2013}3\times {10}^{23}$
, for a source located at 10 Mpc) seen in the relativistic simulations was quite close to the corresponding Newtonian range. The average
${E}_{GW}$
was somewhat higher in the relativistic case (
$1.5\times {10}^{47}erg$
compared to the Newtonian value of
$6.4\times {10}^{46}erg$
). The overall range of GW frequencies observed in their relativistic simulations (60 – 1000 Hz) was close to the Newtonian range. They did note that relativistic effects always caused the characteristic frequency of emission,
${f}_{GW}$
, to increase (up to fivefold). For most of their models, this increase in
${f}_{GW}$
was not accompanied by an increase in
${h}_{pk}$
. This means that relativistic effects could decrease the detectability of GW signals from some core collapses. However, the GW emission from the models of Dimmelmeier et al. could be detected by the first generation of groundbased interferometric detectors if the sources were fortuitously located in the Local Group of galaxies. A catalog containing the signals and spectra of the GW emission from all of their models can be found at [
138]
.
Fully general relativistic collapse simulations (
i.e., without the conformally flat approximation) have been performed by Shibata [
181]
. He used an axisymmetric code that solves the Einstein equations in Cartesian coordinates and the hydrodynamics equations in cylindrical coordinates.
The use of the Cartesian grid eliminates the presence of singularities and allows for stable, longduration axisymmetric simulations [
4]
. The focus of this work was the effect of rotation on the criteria for prompt black hole formation. Shibata found that if the parameter
$q=J/{M}^{2}$
is less than
$0.5$
, black hole formation occurred for rest masses slightly greater than the maximum mass of spherical stars. However, for
$0.5<q<1$
, the maximum stable rest mass is increased by
$\sim 70\text{\u2013}80\%$
. The results are only weakly dependent on the initial rotation profile. Shibata did not compute the GW emission in his collapse simulations.
The new general relativistic hydrodynamics simulations of Zanotti, Rezzolla, and Font [
215]
suggest that a torus of neutron star matter surrounding a black hole remnant may be a stronger source of GWs than the collapse itself. They used a high resolution shockcapturing hydrodynamics method in conjunction with a static (Schwarzschild) spacetime to follow the evolution of “toroidal neutron stars”. Their results indicate that if a toroidal neutron star (with constant specific angular momentum) is perturbed, it could undergo regular oscillations. They estimate that the resulting GW emission would have a characteristic amplitude
${h}_{c}$
ranging from
$6\times {10}^{24}\text{\u2013}5\times {10}^{23}$
, for ratios of torus mass to black hole mass in the range 0.1 – 0.5. (These amplitude values are likely underestimated because the simulations of Zanotti et al. are axisymmetric.) The corresponding frequency of emission is
${f}_{GW}\sim 200Hz$
. The values of
${h}_{c}$
and
${f}_{GW}$
quoted here are for a source located at 10 Mpc. This emission would be just outside the range of LIGOII (see Figure 1 ).
Further numerical investigations, which study tori with nonconstant angular momenta and include the effects of selfgravity and black hole rotation, are needed to confirm these predictions. Movies from the simulations of Zanotti
et al. can be viewed at [
162]
.
Magnetized tori around rapidly spinning black holes (formed via either core collapse or neutron starblack hole coalescence) have recently been examined in the theoretical study of van Putten and Levinson [
204]
. They find that such a torus–black hole system can exist in a suspended state of accretion if the ratio of poloidal magnetic field energy to kinetic energy
${E}_{B}/{E}_{k}$
is less than
$0.1$
.
They estimate that
$\sim 10\%$
of the spin energy of the black hole will be converted to gravitational radiation energy through multipole mass moment instabilities that develop in the torus. If a magnetized torus–black hole system located at 10 Mpc is observed for
$2\times {10}^{4}$
rotation periods, the characteristic amplitude of the GW emission is
$\sim 6\times {10}^{20}$
. It is possible that this emission could take place at several frequencies. Observations of xray lines from gammaray bursts (which are possibly produced by these types of systems) could constrain these frequencies by providing information regarding the angular velocities of the tori (preliminary estimates from observations suggest
${f}_{GW}\sim 500Hz$
, placing the radiation into a range detectable by LIGOI).
3.4.5 Simulations of convective instabilities
Convectively driven inhomogeneities in the density distribution of the outer regions of the nascent neutron star and anisotropic neutrino emission are other sources of GW emission during the collapse/explosion [
35,
143]
. GW emission from these processes results from smallscale asphericities, unlike the largescale motions responsible for GW emission from aspherical collapse and nonaxisymmetric global instabilities. Note that Rayleigh–Taylor instabilities also induce timedependent quadrupole moments at composition interfaces in the stellar envelope. However, the resultant GW emission is too weak to be detected because the Rayleigh–Taylor instabilities occur at very large radii [
143]
.
Burrows and Hayes [
35]
have suggested that the large proper motions observed in pulsars may result from asymmetrical core collapse SNe. They have computed the resultant GW emission under the assumption that such asymmetries were caused by convection and anisotropic neutrino emission during the collapse and explosion. Hydrodynamic simulations and theoretical investigations suggest that asymmetries present in the star prior to collapse (in part due to convection during silicon and oxygen burning) will be amplified during the collapse [
16,
116]
. Based on these arguments, Burrows and Hayes induced a mass dipole anisotropy of
$<0.1\%$
in the precollapse core used in their “exploratory” collapse simulation. The initial quadrupole anisotropy present in their model was
$0.012$
. They imposed no aspherical perturbations on the initial velocities. The collapse was followed in 1D inside a radius of 15 km and in 2D (with azimuthal symmetry) to larger radii. To reduce the computational burden of this exploratory simulation, they artificially hardened the neutrino spectrum to drive the explosion at a faster rate (note that this practice was observed to minimize the GW emission in the simulations of Müller and Janka [
146]
; see below).
The initial asymmetries were observed to grow in Burrows and Hayes' calculation. The nascent neutron star developed a recoil speed of
$\sim 530km{s}^{1}$
. Aspherical mass motion was responsible for the vast majority of the recoil. Only 16% of the recoil velocity could be attributed to anisotropic neutrino emission. The gravitational waveform from this simulation (including separate matter and neutrino contributions) is shown in Figure 9 . The contributions of matter motion and neutrino emission to the GW amplitude
$h$
were of opposite sign. Anisotropic neutrino radiation was basically the sole contributor to
$h$
for the first 20 ms, after which the contribution from mass motion became significant. The peak amplitude calculated was
${h}_{pk}\sim 3\times {10}^{24}$
, for a source located at 10 Mpc. Most (
$\sim 70\%$
) of the emitted
${E}_{GW}\sim 2.0\times {10}^{45}erg$
came from mass motion (because the mass quadrupole was changing more rapidly as the bounce occurred). The frequency of the GWs emitted was in the range
$10<{f}_{GW}<100Hz$
. Burrows and Hayes predicted LIGOII should be able to detect this emission, with a signaltonoise ratio of 10, from a core collapse that fortuitously happened to occur in our Galaxy (with
$d=10kpc$
).
Figure 9
: The gravitational waveform (including separate matter and neutrino contributions) from the collapse simulations of Burrows and Hayes [
35]
. The curves plot the product of the gravitational wave amplitude and distance to the source as a function of time. (Figure 3 of [
35]
; used with permission.)
The study by Nazin and Postnov [
151]
predicts a lower limit for
${E}_{GW}$
emitted during an asymmetric core collapse SN (where such asymmetries could be induced by both aspherical mass motion and neutrino emission). They assume that observed pulsar kicks are solely due to asymmetric collapse. They suggest that the energy associated with the kick (
$M{v}^{2}/2$
, where
$M$
and
$v$
are the mass and velocity of the neutron star) can be set as a lower limit for
${E}_{GW}$
(which can be computed without having to know the mechanism behind the asymmetric collapse). From observed pulsar proper motions, they estimate the degree of asymmetry
$\epsilon $
present in the collapse and the corresponding characteristic GW amplitude (
$h\propto \sqrt{\epsilon}$
). This amplitude is
$3\times {10}^{25}$
for a source located at 10 Mpc and emitting at
${f}_{GW}=1kHz$
. Müller and Janka performed both 2D and 3D simulations of convective instabilities in the protoneutron star and hot bubble regions during the first second of the explosion phase of a Type II SN [
146]
. They numerically computed the GW emission from the convectioninduced aspherical mass motion and neutrino emission in the quadrupole approximation (for details, see Section 3 of their paper).
Figure 10
: Convective instabilities inside the protoneutron star in the 2D simulation of Müller and Janka [
146]
. The evolutions of the temperature (left panels) and logarithmic density (right panels) distributions are shown for the radial region
$15\text{\u2013}95km$
. The upper and lower panels correspond to times
$12$
and
$21ms$
, respectively, after the start of the simulation. The temperature values range from
$2.5\times {10}^{10}$
to
$1.8\times {10}^{11}K$
. The values of the logarithm of the density range from
$10.5$
to
$13.3gc{m}^{3}$
. The temperature and density both increase as the colors change from blue to green, yellow, and red. (Figure 7 of [
146]
; used with permission.)
For typical iron core masses, the convectively unstable region in the protoneutron star extends over the inner
$0.7\text{\u2013}1.20{M}_{\odot}$
of the core mass (this corresponds to a radial range of
$\sim 10\text{\u2013}50km$
).
The convection in this region, which begins approximately 10 – 20 ms after the shock forms and may last for
$\approx 20ms\text{\u2013}1s$
, is caused by unstable gradients in entropy and/or lepton number resulting from the stalling of the prompt shock and deleptonization outside the neutrino sphere.
Müller and Janka's simulations of convection in this region began with the 1D, nonrotating, 12 ms postbounce model of Hillebrandt [
88]
. This model included general relativistic corrections that had to be relaxed away prior to the start of the Newtonian simulations. Neutrino transport was neglected in these runs (see Section 2.1 of [
146]
for justification); however, a sophisticated equation of state was utilized. Figure 10 shows the evolution of the temperature and density distributions in the 2D simulation of Müller and Janka. The peak GW amplitude resulting from convective mass motions in these simulations of the protoneutron star was
$\approx 3\times {10}^{24}$
in 2D and
$\approx 2\times {10}^{25}$
in 3D, for
$d=10Mpc$
. The emitted energy was
$9.8\times {10}^{44}erg$
in 2D and
$1.3\times {10}^{42}erg$
in 3D. The power spectrum peaked at frequencies of 200 – 600 Hz in 2D and 100 – 200 Hz in 3D. Such signals would not be detectable with LIGOII. The reasons for the differences between the 2D and 3D results include smaller convective elements and less underand overshooting in 3D. The relatively low angular resolution of the 3D simulations may have also played a role. The quadrupole gravitational wave amplitude
${A}_{20}^{E2}$
from the 2D simulation is shown in the upper left panel of Figure 11 (see [
217,
194]
for expressions relating
${A}_{20}^{E2}$
to
$h$
).
Figure 11
: Quadrupole amplitudes
${A}_{20}^{E2}[cm]$
from convective instabilities in various models of [
146]
. The upper left panel is the amplitude from a 2D simulation of protoneutron star convection. The other three panels are amplitudes from 2D simulations of hot bubble convection. The imposed neutrino flux in the hot bubble simulations increases from the top right model through the bottom right model. (Figure 18 of [
146]
; used with permission.)
Convection in the hot bubble region between the shock and neutrino sphere arises because of an unstable entropy gradient resulting from neutrino heating. This unstable region extends over the inner mass range
$1.25\text{\u2013}1.40{M}_{\odot}$
(corresponding to a radial range of
$\approx 100\text{\u2013}1000km$
). Convection in the hot bubble begins
$\approx 50\text{\u2013}80ms$
after shock formation and lasts for
$\approx 100\text{\u2013}500ms$
.
Only 2D simulations were performed in this case. These runs started with a 25 ms postbounce model provided by Bruenn. A simple neutrino transport scheme was used in the runs and an imposed neutrino flux was located inside the neutrino sphere. Due to computational constraints,
the computational domain did not include the entire convectively unstable region inside the protoneutron star (thus this set of simulations only accurately models the convection in the hot bubble region, not in the protoneutron star).
The peak GW amplitude resulting from these 2D simulations of convective mass motions in the hot bubble region was
${h}_{pk}\approx 5\times {10}^{25}$
, for
$d=10Mpc$
. The emitted energy was
$\stackrel{<}{\sim}2\times {10}^{42}erg$
. The energy spectrum peaked at frequencies of 50 – 200 Hz. As the explosion energy was increased (by increasing the imposed neutrino flux), the violent convective motions turn into simple rapid expansion. The resultant frequencies drop to
${f}_{GW}\sim 10Hz$
. The amplitude of such a signal would be too low to be detectable with LIGOII. Müller and Janka estimated the GW emission from the convection induced anisotropic neutrino radiation in their simulations (see [
146]
for details). They could not directly compute the characteristics of the GW emission from neutrinos because their protoneutron star simulations did not include neutrino transport and the neutrino transport in their hot bubble simulations was only 1D. They found that the amplitude of the GWs emitted can be a factor of 5 – 10 higher than the GW amplitudes resulting from convective mass motion. However, their results indicate that the
${E}_{GW}$
coming from anisotropic neutrino emission is only a small fraction of that emitted by convective mass motion.
Note that the qualitative results (including
$h$
and
${E}_{GW}$
magnitudes) of Burrows and Hayes' [
35]
“exploratory” 2D calculation are in rough agreement with the more detailed 2D simulations of Müller and Janka. However, both investigations have shortcomings. Necessary improvements would include more realistic initial models (which include rotation and are produced with multidimensional collapse simulations to better estimate initial asphericities), use of sophisticated neutrino transport schemes, inclusion of general relativistic effects, and higherresolution 3D runs.
3.5 Going further
The background of GW emission from a population of core collapse SNe at cosmological distances may be detectable by LIGOII, according to Ferrari, Matarrese, and Schneider [
60]
. They determined the SN rate as a function of redshift using observations to determine the evolution of the star formation rate. Only collapses that lead to black hole formation were considered. This simplified the study because the GW emission from such collapses is generally a function of just the black hole mass and angular momentum. They found that the stochastic background from these sources is not continuous and suggest that this could be used to optimize detection strategies. The maximum GW spectral strain amplitude they computed was in the range
${10}^{28}\text{\u2013}{10}^{27}Hz$
, at frequencies of a few times
${10}^{2}Hz$
. Such a signal may be detected by a pair of LIGOII detectors.
4 Collapse of Population III Stars
4.1 Collapse scenario
The first generation of stars to form in the early universe are known as Population III stars (formed at redshifts
$z\stackrel{>}{\sim}5$
). Theoretical and computational evidence suggests that Population III stars may have had masses
$\stackrel{>}{\sim}100{M}_{\odot}$
[
2,
1,
75]
. Since these massive stars contained no metals, it was possible for them to form directly and then evolve with very low stellar winds and thus very little mass loss. If the mass of a nonrotating Population III star is
$\stackrel{>}{\sim}260{M}_{\odot}$
, its fate is to collapse directly to a black hole at the end of its life [
75]
. If rotational support prevents the star from direct collapse to a black hole, explosive thermonuclear burning will cause the star to undergo a giant hypernova explosion. Prior to black hole formation, the rotating, collapsed core will have a mass of
$50\text{\u2013}70{M}_{\odot}$
and a radius of 1000 – 2000 km. Note that because these massive stars evolve so quickly (in a few million years [
9]
), the events associated with their deaths will take place at roughly the redshifts of their births.
4.2 Formation rate
The formation rate of Population III stars can be indirectly estimated from the reionization fraction of the early universe, which was reionized by light from these stars [
91,
39]
. Using estimates of the ultraviolet light emission from Population III stars, their ionization efficiency, and the reionization fraction of the early universe, one can determine that about 0.01% – 1% of the universe's baryonic matter was found in these very massive stars. This corresponds to
$\sim {10}^{4}\text{\u2013}{10}^{7}$
Population III stars in a
${10}^{11}{M}_{\odot}$
galaxy and thus a collapse rate that is
$\stackrel{<}{\sim}{10}^{3}y{r}^{1}$
. Thus, a reasonable occurrence rate can be found for an observation (luminosity) distance of
$\sim 50Gpc$
(which corresponds to a redshift of
$z=5$
, in the cosmology used by [
71]
). However, uncertainties in the assumptions make this formation rate uncertain by a few orders of magnitude [
1,
75,
2,
71]
. It is unknown how many Population III stars will collapse directly to black holes.
4.3 GW emission mechanisms
The GW emission mechanisms related to the collapse of Population III stars are a subset of those discussed in the sections on AIC and SNe/collapsars. These include aspherical collapse, global rotational and fragmentation instabilities that may arise during the collapse/explosion and in the collapse remnant (prior to black hole formation), and the “ringdown” of the remnant black hole.
4.4 Numerical predictions of GW emission
The GW emission from the collapse of Population III stars has recently been investigated by Fryer and collaborators (Fryer, Woosley, and Heger [
75]
, FHH [
71]
, and Fryer, Holz, Hughes, and Warren [
72]
). The collapse simulations of Fryer, Woosley, and Heger again started with rotating collapse progenitors that had been evolved with a stellar evolution code [
84]
. The initial models used by the evolution code were in rigid rotation with a surface ratio of centrifugal to gravitational forces of 20% (this ratio is seen in current observations of O stars).
The results of Fryer, Woosley, and Heger suggest that the collapse remnant (prior to black hole formation) is susceptible to the development of a secular barmode instability. However, at
$z>5$
, the GW emission would be redshifted out of LIGOII's frequency range. At
$z=5$
,
${h}_{pk}=8\times {10}^{23}$
, with a corresponding frequency of 10 Hz [
71,
72]
. Even if such a signal persists for a hundred cycles, it probably would be undetectable by LIGOII. Note that these signal strengths are orders of magnitude lower than the qualitative estimates of signal strength given in Carr, Bond, and Arnett [
39]
.
LIGOII may be able to detect the GW emission from binary clumps formed via a fragmentation instability. If such a signal is emitted at