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  identified mechanisms related to core collapse that could produce GWs and provided order-of-magnitude estimates of the characteristics of such emission.
Quantitative computations of GW emission during the infall phase of collapse were performed by Thuan and Ostriker  and Epstein and Wagoner [66, 65], 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 post-Newtonian effects prolonged the collapse and thus lowered the GW luminosity. Subsequently, Novikov  and Shapiro and Saenz [218, 204] 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  and later by Saenz and Shapiro [205, 206].
Müller  calculated the quadrupole GW emission from 2D axisymmetric collapse based on the Newtonian simulations of Müller and Hillebrandt  (these simulations used a realistic equation of state and included differential rotation). He found that differential rotation enhanced the efficiency of the GW emission. Update
Stark and Piran [233, 193] were the first to compute the GW emission from fully relativistic collapse simulations, using the ground-breaking formalism of Bardeen and Piran . They followed the (pressure-cut 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 quasi-normal 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 [215, 216]. They employed a perturbative approach, valid only in the slowly rotating regime.
The gravitational radiation from non-axisymmetric collapse was investigated by Detweiler and Lindblom, who used a sequence of non-axisymmetric ellipsoids to represent the collapse evolution . They found that the radiation from their analysis of non-axisymmetric 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 . In the remainder of Section 3.4, more recent investigations will be discussed.
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 . 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 bounce had been predicted by Shapiro and Lightman  and Tohline  (in the context of the “fizzler” scenario for failed supernovae; see also [105, 106, 124]), and had been observed in earlier collapse simulations [175, 238]. Mönchmeyer and collaborators found that a bounce caused by centrifugal forces would last for several , whereas a bounce at nuclear densities would occur in . 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 [171, 74]) 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 for a Type I waveform is shown in Figure 3 (see [241, 271] for expressions relating to ). The waveforms identified as Type II exhibit several maxima, which result from multiple bounces (see Figure 4 for an example of a Type II waveform). Note that the waveforms displayed in Figures 3 and 4 are from the study of Zwerger and Müller , discussed below.The model of Mönchmeyer et al. that bounced due to nuclear forces had the highest GW amplitude of all of their models, for a source distance , and the largest emitted energy . The accompanying power spectrum peaked in the frequency range .
The most extensive Newtonian survey of the parameter space of axisymmetric, rotational core collapse is that of Zwerger and Müller . They simulated the collapse of 78 initial models with varying amounts of rotational kinetic energy (reflected in the initial value of the stability parameter ), 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 . The models had a polytropic equation of state, with initial adiabatic index . Collapse was induced by reducing the adiabatic index to a value in the range 1.28 - 1.325. The equation of state used during the collapse evolution had both polytropic and thermal contributions (note that simulations using more sophisticated equations of state get similar results ).UpdateThe 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 ). In their simulations, Type I signals (as labelled by Mönchmeyer et al. ) were produced by models with relatively soft equations of state, . Type II signals were produced by the models with stiffer equations of state, . They found a smooth transition between these signal types if was increased while all other parameters were held fixed. They also observed another class of signal, Type III, for their models with the lowest (). 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 5). Type III signals were not observed in the evolution of strongly differentially rotating models and were also not seen in subsequent investigations [144, 190].Update Their waveforms were computed with the same technique used in .
In contrast to the results of Mönchmeyer et al. , in Zwerger and Müller’s investigation the value of at bounce did not determine the signal type. Instead, the only effect on the waveform due to was a decrease in in models that bounced at subnuclear densities. The effect of the initial value of on was non-monotonic. For models with , increased with increasing . This is because the deformation of the core is larger for faster rotators. However, for models with larger , decreases as 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 for a given sequence was reached when at bounce was just less than . 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 ; and those with soft equations of state, , and large degrees of differential rotation. The GW amplitudes emitted during their simulations were in the range , for (the model with the highest is identified in Figure 2). The corresponding energies ranged from . The peaks of their power spectra were between and . Such signals would fall just outside of the range of LIGO-II. Magnetic fields lower these amplitudes by 10% , but realistic stellar profiles can lower the amplitudes by a factor of [190, 177], restricting the detectability of supernovae to within our Galaxy (). Update
Yamada and Sato  used techniques very similar to those of Zwerger and Müller  in their core collapse study. Their investigation revealed that the for Type I signals became saturated when the dimensionless angular momentum of the collapsing core, , reached . They also found that was sensitive to the stiffness of the equation of state for densities just below . The characteristics of the GW emission from their models were similar to those of Zwerger and Müller.
The GW emission from non-axisymmetric hydrodynamics simulations of stellar collapse was first studied by Bonazzola and Marck [163, 28]. They used a Newtonian, pseudo-spectral hydrodynamics code to follow the collapse of polytropic models. Their simulations covered only the pre-bounce phase of the collapse. They found that the magnitudes of 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 . These authors started their Newtonian simulations with the only model (A4B5G5) of Zwerger and Müller  that had a post-bounce value for the stability parameter that significantly exceeded (recall this is the value at which the dynamical bar instability sets in for MacLaurin spheroid-like models). This model had the softest EOS (), highest , and largest degree of differential rotation of all of Zwerger and Müller’s models. The model’s initial density distribution had an off-center density maximum (and therefore a torus-like structure). Rampp, Müller, and Ruffert evolved this model with a 2D hydrodynamics code until its reached . At that point, 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, non-axisymmetric density perturbations were imposed to seed the growth of any non-axisymmetric modes to which the configuration was unstable. When the imposed perturbation was random (5% in magnitude), the dominant mode that arose was . The growth of this particular mode was instigated by the cubical nature of the computational grid. When an perturbation was imposed (10% in magnitude), three clumps developed during the post-bounce 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 bar-like structure (evidence of the presence of the mode). Significant non-axisymmetric structure was visible only within the inner of the core. Their simulations were carried out to after bounce.
The amplitudes of the emitted gravitational radiation (computed in the quadrupole approximation) were only 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.  indicate it is possible that some of the post-bounce configurations of Zwerger and Müller, which have lower values of than the model studied by Rampp, Müller, and Ruffert , may also be susceptible to non-axisymmetric instabilities. Centrella et al. have performed 3D hydrodynamics simulations of polytropes to test the stability of configurations with off-center density maxima (as are present in many of the models of Zwerger and Müller ). 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 non-axisymmetric 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 . The observed instability had a dominant mode. Centrella et al. estimate that if a stellar core of mass and radius encountered this instability, the values of from their models would be , for . The frequency at which occurred in their simulations was . This instability would have to persist for at least cycles to be detected with LIGO-II.
Brown  carried out an investigation of the growth of non-axisymmetric modes in post-bounce cores that was similar in many respects to that of Rampp, Müller, and Ruffert . He performed 3D hydrodynamical simulations of the post-bounce configurations resulting from 2D simulations of core collapse. His pre-collapse initial models are 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 . The angular velocity profiles of their pre-collapse progenitors were broad and Gaussian-like. Brown’s initial models had peak angular velocities ranging from 0.8 - 2.4 times those of . The model evolved by Rampp, Müller, and Ruffert  had much stronger differential rotation than any of Brown’s models. To induce collapse, Brown reduced the adiabatic index of his models to , the same value used by .
Brown found that increased by a factor during his 2D collapse simulations. This is much less than the factor of observed in the model studied by Rampp, Müller, and Ruffert . 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 post-bounce models (models and , both of which had after bounce) and of the model of Rampp et al. (which, although it starts out with , has a sustained ). Brown refers to the Rampp et al. model as model RMR. Because Brown’s models do not have off-center density maxima, they are not expected to be unstable to the mode observed by Centrella et al. . 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 ).
Brown’s simulations determined that both his most rapidly rotating model (with post-bounce ) and model RMR are unstable to growth of the bar-mode. However, his model (with post-bounce ) was stable. Brown observed no dominant or 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 mode begins to grow in model RMR at about the same time as Rampp et al. stopped their evolutions. No substantial growth was observed.
The results of Brown’s study indicate that the overall of the post-bounce core may not be a good diagnostic for the onset of instability. He found, as did Rampp, Müller, and Ruffert , that only the innermost portion of the core (with ) is susceptible to the bar-mode. This is evident in the stability of his model . This model had an overall , but an inner core with . Brown also observed that the of the inner core does not have to exceed for the model to encounter the bar-mode. Models and RMR had . He speculates that the inner cores of these later two models may be bar-unstable because interaction with their outer envelopes feeds the instability or because for such configurations.
Fryer and Warren  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 flux-limited diffusion approximation for neutrino transport, and Newtonian spherical gravity. Their initial model was nonrotating. Thus, no bar-mode instabilities could develop during their simulations. The only GW emitting mechanism present in their models was convection in the core. The maximum amplitude of this emission, computed in the quadrupole approximation, was , for . In later work, Fryer & Warren  included full Newtonian gravity through a tree algorithm and studied the rotating progenitors from Fryer & Heger . By the launch of the explosion, no bar instabilities had developed. This was because of several effects: they used slowly rotating, but presumably realistic, progenitors , the explosion occured quickly for their models () and, finally, because much of the high angular momentum material did not make it into the inner core. These models have been further studied for the GW signals . The fastest rotating models achieved a signal of for and characteristic frequencies of . For supernovae occuring within the Galaxy, such a signal is detectable by LIGO-II. Update
Fryer and collaborators have also modeled asymmetric collapse and asymmetric explosion calculations in 3 dimensions [80, 90]. These calculations will be discussed in Section 3.4.5.
The GW emission from nonradial quasinormal mode oscillations in proto-neutron stars has been examined by Ferrari, Miniutti, and Pons . They found that the frequencies of emission during the first second after formation ( 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 proto-neutron star located at , with a signal-to-noise ratio of 5, must be . It is unlikely that this much energy is stored in these modes (the collapse itself may only emit in gravitational waves ).
General relativistic effects oppose the stabilizing influence of rotation in pre-collapse 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 [239, 271, 198, 37].
The full collapse simulations of Fryer and Heger  are the most sophisticated axisymmetric simulations from which the resultant GW emission has been studied [86, 88]. 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  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) , 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 () 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, 232, 110].Update
FHH’s  numerical quadrupole estimate of the GWs from polar oscillations in the collapse simulations of Fryer and Heger  predicts a peak dimensionless amplitude (for ), emitted at . The radiated energy . This signal would be just out of the detectability range of the LIGO-II detector.
The cores in the simulations of Fryer and Heger  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 after the collapse, becomes high enough in their models to exceed the secular bar instability limit. The 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 ) compute an upper limit (via Equation (2)) to the emitted amplitude from their dynamically unstable model of (if coherent emission from a bar located at persists for 100 cycles). The corresponding frequency and maximum power are and . LIGO-II should be able to detect such a signal (see Figure 2, where FHH’s upper limit to for this dynamical bar-mode is identified).
As mentioned above, the proto-neutron stars of Fryer and Heger are likely to be unstable to the development of secular bar instabilities. The GW emission from proto-neutron stars that are secularly unstable to the bar-mode has been examined by Lai and Shapiro [148, 146]. 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  considers only incompressible fluids, for which there are exact solutions for (Dedekind and Jacobi-like) bar development. He predicts that such a bar located at would emit GWs with a peak characteristic amplitude , if the bar persists for cycles. The maximum of the emitted radiation is in the range . This type of signal should be easily detected by LIGO-I (although detection may require a technique like the fast chirp transform method of Jenet and Prince  due to the complicated phase evolution of the emission).
Alternatively, Ou et al.  bypassed the long secular timescale by increasing the driving force of the instability. They found that a bar instability was maintained for several orbits before sheer flows, producing GW emission that would have a signal-to-noise ratio greater than 8 for LIGO-II out to . A movie of this simulation is shown in Figure 6. Update
The GW emission from -mode unstable neutron star remnants of core collapse SNe would be easily detectable if (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 -mode growth (note that a single -mode episode can have multiple amplitude peaks ). FHH calculate that the characteristic amplitude of the GW emission from this -mode evolution tracks from , over a frequency range of (see Section 2.4 for details). They estimate the emitted energy to exceed .
General relativity has been more fully accounted for in the core collapse studies of Dimmelmeier, Font, and Müller [58, 59, 60] and Shibata and collaborators ,Update which build on the Newtonian, axisymmetric collapse simulations of Zwerger and Müller . 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  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  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.  are shown in Figures 7, 8, 9, and 10. The four evolutions shown include a regular collapse (Movie 7), a rapid collapse (Movie 8), a multiple bounce collapse (Movie 9), and a very rapidly and differentially rotating collapse (Movie 10). 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 .
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 [41, 173]. GW emission from these processes results from small-scale asphericities, unlike the large-scale motions responsible for GW emission from aspherical collapse and non-axisymmetric global instabilities. Note that Rayleigh-Taylor instabilities also induce time-dependent 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 .
Since convection was suggested as a key ingredient into the explosion, it has been postulated that asymmetries in the convection can produce the large proper motions observed in the pulsar population . Convection asymmetries can either be produced by asymmetries in the progenitor star that grow during collapse or by instabilities in the convection itself. Burrows and Hayes  proposed that asymmetries in the collapse could produce the pulsar velocities. The idea behind this work was 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 [20, 147]. These asymmetries will then drive asymmetries in the convection and ultimately, the supernova explosion. Burrows and Hayes  found that not only could they produce strong motions in the nascent neutron star, but detectable gravitational wave signals. The peak amplitude calculated was , for a source located at .
Fryer  was unable to produce the large neutron star velocities seen by Burrows and Hayes  even after significantly increasing the level of asymmetry in the initial star in excess of 25%. This discrepancy is now known to be due to the crude 2-dimensional model and gravity scheme used by Burrows and Hayes . However, the gravitational wave signal produced by both simulations is comparable. Figure 11 shows the gravitational waveform from the Burrows & Hayes simulation (including separate matter and neutrino contributions). Figures 12, 13 show the matter and neutrino contributions respectively to the gravitational wave forms for the Fryer results. The gravitational wave amplitude is dominated by the neutrino component and can exceed , for a source located at in Fryer’s most extreme example.The study by Nazin and Postnov  predicts a lower limit for 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 (, where and are the mass and velocity of the neutron star) can be set as a lower limit for (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 present in the collapse and the corresponding characteristic GW amplitude (). This amplitude is for a source located at and emitting at .
Müller and Janka performed both 2D and 3D simulations of convective instabilities in the proto-neutron star and hot bubble regions during the first second of the explosion phase of a Type II SN . They numerically computed the GW emission from the convection-induced aspherical mass motion and neutrino emission in the quadrupole approximation (for details, see Section 3 of their paper).
The peak GW amplitude resulting from convective mass motions in these simulations of the proto-neutron star was in 2D and in 3D, for . More recent calculations get amplitudes of in 2D  and in 3D .Update The emitted energy was in 2D and in 3D. The power spectrum peaked at frequencies of in 2D and in 3D. Such signals would not be detectable with LIGO-II. The reasons for the differences between the 2D and 3D results include smaller convective elements and less under- and overshooting in 3D. The relatively low angular resolution of the 3D simulations may have also played a role. The quadrupole gravitational wave amplitude from the 2D simulation is shown in the upper left panel of Figure 15 (see [271, 241] for expressions relating to ).
The case for GWs from convection induced asymmetric neutrino emission has also varied with time. Müller and Janka estimated the GW emission from the convection induced anisotropic neutrino radiation in their simulations (see  for details). 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. Müller et al. (2004)  argue now that the GWs produced by asymmetric neutrino emission is less than that of the convective motions.
Our understanding of the convective engine is evolving with time. Scheck et al.  found that the convective cells could merge with time, producing a single lobe convective instability, fulfilling the prediction by Herant . Blondin et al.  argue that a standing shock instability could develop to drive low-mode convection and Burrows et al.  argue that it is the shocks produced in this convection that truly drives the supernova explosion. This convection can drive oscillations in the neutron star which may also be a source for GWs (see Fig. 17). Update
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