Gravitational Waves from Gravitational Collapse
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Abstract
Gravitationalwave emission from stellar collapse has been studied for nearly four decades. Current stateoftheart numerical investigations of collapse include those that use progenitors with more realistic angular momentum profiles, properly treat microphysics issues, account for general relativity, and examine nonaxisymmetric effects in three dimensions. Such simulations predict that gravitational waves from various phenomena associated with gravitational collapse could be detectable with groundbased and spacebased interferometric observatories. This review covers the entire range of stellar collapse sources of gravitational waves: from the accretioninduced collapse of a white dwarf through the collapse down to neutron stars or black holes of massive stars to the collapse of supermassive stars.
Keywords
Black Hole Neutron Star Massive Star White Dwarf Quasinormal Mode1 Introduction
The field of gravitationalwave (GW) astronomy will soon become a reality. The first generation of groundbased interferometric detectors (LIGO [177], VIRGO [324], GEO600 [122], TAMA300 [301]) are beginning their search for GWs. Upgrades for two of these detectors (LIGO, VIRGO) have already begun and in the next few years, the sensitivity of both these detectors will be increased. In addition, these two detectors have begun working more closely together, improving their net sensitivity. After these detectors are complete, they will have sensitivities necessary to regularly detect emissions from astrophysical sources. New detectors, such as the LargeScale Cryogenic Gravitationalwave Telescope (LCGT) as well as technology developing sites such as the Australian Interferometric Gravitational Observatory (AIGO) are also contributing to the progress in GW astronomy (see [145] for a review). A spacebased interferometric detector, LISA [182], could be launched in the next decade. One important class of sources for these observatories is stellar gravitational collapse. This class covers an entire spectrum of stellar masses, from the accretioninduced collapse (AIC) of a white dwarf and the collapse of lowmass stars, including electroncapture supernovae (M_{star} < 10 M_{⊙}), through the collapse of massive stars (M < 10 M_{⊙}) that produce and the even more massive stars (M < 20 M_{⊙}) that produce the “collapsar” engine believed to power longduration gammaray bursts [333], massive and very massive Population III stars (M = 20–500 M_{⊙}), and supermassive stars (SMSs, M > 10^{6} M_{⊙}). Some of these collapses result in explosions (Type II, Ib/c supernovae and hypernovae) and all leave behind neutronstar or blackhole remnants^{1}.
Strong GWs can be emitted during a gravitational collapse/explosion and, following the collapse, by the resulting compact remnant [309, 205, 206, 88, 269, 108, 150]. GW emission during the collapse itself may result if the collapse or explosion involves aspherical bulk mass motion or convection. Rotational or fragmentation instabilities encountered by the collapsing star will also produce GWs. Pulsations and instabilities in the newly formed neutron star (a.k.a. proto neutron star) may also produce observable GWs. Asymmetric neutrino emission can also produce a strong GW signature. Neutronstar remnants of collapse may emit GWs due to the growth of rotational or rmode instabilities. Blackhole remnants will also be sources of GWs if they experience accretioninduced ringing or if the disks around the black hole develop instabilities. All of these phenomena have the potential of being detected by GW observatories because they involve the rapid change of dense matter distributions.
Observation of gravitational collapse by GW 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. In contrast, gravitational waves can propagate from the innermost parts of the stellar core to detectors without attenuation by intervening matter. With their weak interaction crosssections, neutrinos can probe the same region probed by GWs. But whereas neutrinos are extremely sensitive to details in the microphysics (equation of state and crosssections), GWs are most sensitive to physics driving the mass motions (e.g., rotation). Combined, the neutrino and the GW signals can teach us much about the conditions in the collapsing core and ultimately the physics that governs stellar collapse (e.g., [7, 107, 167, 228]).

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 generalrelativistic effects,

inclusion of magneticfield 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, e.g., Section 4.1). Very few, if any simulations, have reached any convergence in spatial resolution. Many of the codes have not been tested to see if their algorithm implementations guard against standard numerical artifacts. For example, very few codes used have tested the effects of the nonconservation of angular momentum and the numerical transport of this angular momentum. 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. We first review the basic modes of GW emission in stellar collapse presenting, where they exist, analytic formulae that have been used to estimate these GWs (Section 2). The latter half of Section 2 presents many of the numerical approximations used to calculate GW emission. Section 3 covers the various collapse scenarios and their GW sources: normal corecollapse supernovae (Section 3.1), the accretion induced collapse (AIC) of a white dwarf (Section 3.2), the collapse of low mass stars and electron capture supernovae (Section 3.3), and the collapse of massive (Section 3.4) and supermassive (Section 3.5) stars. For each, we review the current understanding of the occurrence rate, collapse evolution, and the specific causes behind GW emission. Section 4 then discusses the current state of the calculations for sources arising from: bounce (Section 4.1), convection (Section 4.2), bar modes (Section 4.3), neutrinos (Section 4.4), rmodes (Section 4.5), fragmentation (Section 4.6), and ringing (Section 4.7). We conclude by tying together all of these sources with their emission mechanisms to predict a complete GW sky.
One final word of warning. In many cases, the total GW signal from stellar collapse can be tuned by altering key initial conditions (such as the rotation rate of the collapsing star). Many of the strongest GW estimates in the literature tend to use rotation rates that are orders of magnitude higher than that predicted for most stars. These more optimistic results often predict that the current set of detectors should observe GWs from astrophysical sources. In many cases, studies of these extreme conditions provide insight into possible mechanisms for GW emission. To include these new insights, we will discuss these results in this review. The exact nature of the initial conditions may make certain GW signals undetectable. For each of the these scenarios, we strive to distinguish academic studies with more realistic estimates of the signal. Our summary is based on what we judge to be the more realistic signal predictions.
2 Gravitational Wave Emission
Most GW estimates are based on Equation (1). When bulk mass motions dominate the dynamics, the first term describes the radiation. For example, this term gives the wellknown “chirp” associated with binary inspiral. It can be used to model barmode and fragmentation instabilities. At least conceptually, this term also applies to black hole ringing, provided one interprets \({{\mathcal I}_{jk}}\) as a moment of the spacetime rather than as a mass moment [310, 162]). In practice, ringing waves are computed by finding solutions to the wave equation for gravitational radiation [303] with appropriate boundary conditions (radiation purely ingoing at the hole’s event horizon, purely outgoing at infinity). The second term in Equation (1) gives radiation from mass currents, and is used to calculate GW emission due to the rmode instability.
There are several phases during the collapse (and resultant explosion) of a star that produce rapidly changing quadrupole moments: both in the baryonic matter and in the neutrino emission. We will break these phases into four separate aspects of the corecollapse explosion: the bounce of the core, the convective phase above the proto neutron star, cooling in the neutron star, and formation phase of a black hole. Scientists have focused on different phases at different times, arguing that different phases dominated the GW emission. These differences are primarily due to different initial conditions. A specific stellar collapse will only pass through a subset of these phases and the magnitude of the GW emission from each phase will also vary with each stellar collapse. Fryer, Holz & Hughes (hereafter FHH) [106] reviewed some of these phases and calculated upper limits to the GWs produced during each phase [106]. Recent studies have confirmed these upper limits, typically predicting results between 5–50 times lower than the secure FHH upper limits [165, 228]. Before we discuss simulations of GWs, we review these FHH estimates and construct a few estimates for phases missed by FHH.
FHH focused on three emission sources after the bounce. In the convective regime, they studied the development of two different instabilities: fragmentation and bar modes. Bar modes can also develop in the neutron star as it cools. FHH [106] also estimated the rmode strength in neutron stars. Let’s review these estimates.
Considerably more effort has been focused on bar modes, or more generally, nonaxisymmetric instabilities. Classical, high β (where β is the ratio of the rotational to potential energy ≡ T/W), bar modes can be separated into two classes: dynamical instabilities and secular instabilities. The classical analysis predicts that β ≳ 0.27 is needed for dynamical instabilities and (β ≳ 0.14) is required for secular instabilities. Collapse calculations of many modern progenitors (e.g., [138]) followed through collapse calculations [245, 113, 69] suggest that such “classical” instabilities do not occur in nature.
2.1 Making numerical estimates
In core collapse, estimates of the GW emission from simulations are usually done outside of the actual hydrodynamics calculation — that is, there is no feedback from the loss of energy from GWs from the system. Of course, where the GWs drive or damp an unstable mode (e.g., bar modes, Section 4.3), this approximation will lead to erroneous results. Such an assumption is almost always justified, as the GW emission is many orders of magnitude less than the total energy in the system. Calculating the GW emission from a hydrodynamic simulation requires the discretization of Equation (1) and the discretization used depends upon the numerical technique and the dimension of the calculation. Here we present some common discretizations.
The signal arising from neutrino emission is an example of a GW burst “with memory”, where the amplitude rises from a zero point and ultimately settles down to a value offset from this initial value. The noise sources and optimal data analysis techniques for these signals will differ from, for example, signals from bounce that do not have memory [25, 80].
3 Astrophysical Sources of Gravitational Wave Emission

The collapse of OxygenMagnesiumNeon and Carbon Oxygen White Dwarfs pushed beyond the Chandrasekhar limit

The collapse of Iron or OxygenMagnesiumNeon Stellar Cores of massive stars that become too large to support themselves (also roughly at the Chandrasekhar limit).
For both classes of stellar collapse, the core is supported by a combination of thermal and degeneracy pressures. When the mass is too great for these pressures to support the star/core, it begins to compress and heat up. The compression leads to electron capture, neutrino emission, and ultimately, dissociation of the elements. Electron capture reduces the support from degeneracy pressure while Urca processes and dissociation of elements remove thermal support. With less support, the core compresses further, accelerating the rate of electron capture and dissociation, ultimately leading to a runaway implosion. This collapse continues until the matter reaches nuclear densities and the formation of a proto neutron star, or, in the case of massive stars above 100 M_{⊙}, a proto black hole [115]^{4}.
In stellar collapse, it is the structure (density, entropy, rotation) of the core that determines whether a black hole or neutron star is formed^{5} For those stellar collapses that form neutron stars, the structure of the star just beyond the collapsed core is critical in defining the fate of the system. As the collapsed material reaches nuclear densities, nuclear forces and neutron degeneracy pressure halt the collapse and send a bounce shock out through the star. For nearly all systems, this bounce shock stalls. The shock can be revived by neutrinos leaked from the core depositing their energy above the proto neutron star. This is the basis of the neutrinodriven supernova mechanism [56, 19].
This picture does not change even if the standing accretion shock instability contributes to the convective instabilities. Many of the recent simulations have focused on the SASI and it is worth reviewing how it effects the basic convective picture of supernovae. This instability was originally discussed in the context of whitedwarf and neutronstar accretion in scenarios under the assumption that the accretion envelope was stable to RayleighTaylor instabilities [144]. Asymmetries and strong RayleighTaylor convection in actual models [101] coupled with the long predicted growth times for the SASI (∼ 3 s [144]) in neutron star accretion, led the accretion community to limit such instabilities to late times. Blondin et al. [21] introduced this instability into the corecollapse supernovae by setting up conditions that were stable to RayleighTaylor instabilities to study the SASI. In this case, SASI dominates and, as Blondin et al. [21] found, can produce lowmode convection.
There are two dominant questions currently under discussion with regard to this instability. First, how dominant is SASI when entropy gradients do exist? Recall, the analytic estimates of the SASI were originally derived under the assumption that the envelope is RayleighTaylor stable [144] and in neutronstar fallback; simulations argue that RayleighTaylor instabilities dominate^{6}. Lowmode convection is not a sure indicator of the SASI, as RayleighTaylor instabilities also predict a growth toward low modes [139]). Second, is the SASI driven by an advectiveacoustic instability or is it simply an acoustic instability? Blondin and collaborators, who first mentioned the possibility of advected vortices in the supernova context, later argued for an acoustic scenario [20]. Answering these question has spawned a great deal of both simulation and analytic theory work [38, 39, 82, 81, 94, 93, 92, 116, 195, 196, 229, 264, 263, 164, 336].
Our analysis of explosion energy holds regardless of the dominant instability or the cause of that instability if the assumption that the energy is stored in the convective region remains true. In such a case, it is difficult to construct a strong explosion from a mechanism that has a long delay. But other sources of energy exist. The neutron star itself can store energy in oscillations to add to the explosion energy [40, 39]. Fallback may also drive additional explosions [101, 100]. We will study both of these in more detail below. This has repercussions on the explosion engine and the resultant GW mechanisms.
Alternative mechanisms for the corecollapse supernova mechanism exist, most notably the magneticfield mechanism [175, 299, 2, 3, 6, 35, 61, 62, 265]. The idea here is that magnetic fields strengthened in the collapse and subsequent convection (especially if the star is rotating rapidly) will drive an explosion. How the magnetic fields affect the GW signal will depend on how and when the magnetic field develops. For most magneticfield generation schemes, the field grows after the bounce of the core, and signal from the collapse/bounce phase (and indeed some of the convective phase) from supernovae will be the same whether the mechanism is this alternate magneticfield driven mechanism or the standard convective engine. For some magnetar models, the magnetic field develops after the launch of a weak, convectivelydriven explosion [304]. In such a case, the GW signal will be identical to the convectivedriven mechanism.
Finally, if the core collapses to a black hole and the infalling material has enough angular momentum to prevent its infall directly into the event horizon, explosions might be produced. An accretion disk forms and, either through the windup of magnetic fields or neutrino annihilation above the disk, an explosion may result. This type of explosion has been posited as the engine behind GRBs [219, 333]. The association of GRBs with supernovalike outbursts has argued strongly for a massivestar origin for at least longduration GRBs [335, 117].
In this section, we review the various progenitors of core collapse, describing their physical properties, its occurrence rate, evolution, and likely GW emission mechanisms. Rather than follow the order of progenitor mass, we first discuss the fate of stars in the 12–20 M_{⊙} range, the likely progenitors for corecollapse supernovae. These are the moststudied gravitationalcollapse systems. We then discuss the AIC of white dwarfs and the very similar collapse of stars in the ∼ 9–12 M_{⊙} range. We then move upwards in mass studying massive, very massive (20–500 M_{⊙}) and supermassive (> 10^{5} M_{⊙}) stars.
3.1 Corecollapse supernovae
Stars more massive than ∼ 10–12 M_{⊙} and less massive than ∼ 20 M_{⊙} are the primary candidates for what is considered the standard formation scenario behind corecollapse supernovae. For these stars, the current belief is that the collapse proceeds through bounce and proto neutron star formation. Convection, or perhaps an alternate mechanism like magnetic fields, then deposits enough energy above the proto neutron star to drive an explosion. Scientists presently think that this process is responsible for most type Ib/c and type II supernovae^{7}. This is the best studied, and probably bestunderstood, corecollapse scenario in astronomy.
3.1.1 Corecollapse supernovae rate
The rate of corecollapse supernovae can be determined by simply multiplying the starformation rate times the fraction of stars in the ∼ 10–12 and ∼ 20 M_{⊙} mass range. The uncertainties in such a calculation lie in determining the limits on either side of this range (determined by theory), the power in the initial mass function and the star formation rate (both beyond theory at this point). The latter two quantities, both set by observations, dominate the errors in this rate estimate.
3.1.2 Corecollapse evolution
The evolution of a corecollapse supernova passes through a number of phases: collapse and bounce, postbounce and convection and neutronstar cooling. The runaway collapse of the core continues until the core reaches nuclear densities. At this point, nuclear forces and neutron degeneracy pressure sharply increase the core pressure, causing the core to bounce and sending a shock wave throughout the star. The nature of the bounce depends upon the equation of state: a stiffer equation of state causes the star to bounce more quickly but also more weakly than a softer equation of state. Within the current uncertainties, the bounce still occurs at an enclosed mass (mass zone measured from the center of the star) of 0. 55 ± 0. 2 M_{⊙}. By the time the bounce shock stalls, the mass of the core is closer to 0.9 ± 0.2 M_{⊙}. Including the effects of general relativity acts to effectively soften the equation of state. Rotation can cause the bounce to occur at lower densities (as centrifugal support contributes to the bounce), but for the current fastestrotating cores produced in stellar models, this effect is less than 20% [113, 69].
As the proto neutron star cools, it may also develop strong convection and this could be another source of GWs [160, 34]. But the extent of this convection ranges from growing throughout the entire proto neutron star [160], to select regions in the star [60], to small regions in the star [140, 112, 113, 116]. Only when the convection is in the entire proto neutron star [160] is it considered to play a large role in the supernova explosion. Although a small contribution to the GW signal, this convection may not be negligible in calculating the full GW signal from core collapse [228].
Another possible source of GWs is the oscillations in the proto neutron star [40]. But, like the strong protoneutronstar convection, there are arguments against such strong oscillations [116, 85, 343, 195, 327].
In strongly magnetized neutron stars, when neutrino oscillations into sterile neutrinos in the core occur, we also expect asymmetries in the neutrino emission to lead to asymmetries in the neutrinos escaping the core, producing GWs [170, 120].
But, by far, the most studied source of GWs from proto neutron stars are the rotationallyinduced barmode instabilities. All of these can contribute to the outflows from the cooling neutron star. If magnetic fields develop or there is a high massinfall rate [305], these outflows can add significantly to the explosion energy and at least one supernovae seems to have experienced such a secondary explosion [191].
3.1.3 Core collapse: Sources of gravitational wave emission
In Section 2, we reviewed the wide variety of mechanisms that lead to GW emission, i.e., time varying quadrupole moments in the matter or the radiation. The complex dynamics and physics within the evolution of a standard corecollapse supernova allow for a variety of scenarios for GW emission at different phases in the collapse.
Bounce: At core bounce, when the infalling material reaches nuclear densities and the collapse halts, the matter reaches its peak acceleration. If the collapse phase is asymmetric, either by asymmetries in the stellar structure or through rotation, this phase can lead to the strongest GW emission. The primary source of this emission is the rapidly changing quadrupole moment in the matter as the asymmetries evolve (Section 4.1). But the asymmetries in the bounce also produce the initial asymmetry in the neutrino emission and this asymmetry may lead to a strong GW signal as well (Section 4.4).
Postbounce and Convection: The convection above the proto neutron star can also develop strong asymmetries as the convective cells merge into lowmode convection. This can lead to rapidly varying quadrupole moments in both the matter (Section 4.2) and the core (Section 4.4).
In the Neutron Star: Convection in the cooling neutron star could produce strong GW emission. So can asymmetric neutrino emission produced by neutrino oscillations to stellar neutrinos in the core^{9}. Pulsations in the newly formed proto neutron star may also produce a strong GW signal [85, 229]. But the moststudied GW source arises from barmode instabilities, in part because, if they develop, they may produce a GW signal that rivals both the bounce and convective GW signals.
3.2 Accretioninduced collapse
Stars with masses in the range ∼ 1−8 M_{⊙}, through burning flashes in an Asymptotic Giant Branch (AGB) and postAGB phase, ultimately lose their envelopes, leaving behind their Carbon/Oxygen (CO) cores (now CO White Dwarfs) and a planetary nebula (for a review of AGB evolution, see Herwig [141] or recent papers [286, 287, 241]). If it were not for the fact that many of these newly formed white dwarfs are in binary systems, such objects would be unimportant for stellar collapse. But there is growing evidence that perhaps most planetary nebulae are formed in binaries [200]. If a white dwarf accretes enough matter through binary mass transfer, its mass can approach the Chandrasekhar limit, causing it to collapse and produce a supernova explosion.

high whitedwarf mass: If the white dwarf is massive (above ∼ 1.2 M_{⊙}), the heat inflow to the core is minimal and the white dwarf heats up primarily due to adiabatic compression. In this case, by the time the white dwarf ignites, it is very dense and the URCA process can cause significant cooling, causing the star to collapse instead of explode.

high accretion rates: If the accretion rate is very high, the accretion process will ignite carbon at the edge of the white dwarf, burning it into an Oxygen/Neon (ONe) white dwarf. In this case, Urca processes provide a cooling mechanism causing the star to collapse as it reaches the Chandrasekhar limit (see [127] for review and counterexamples).
At this time, stellar modelers do not expect that Nature produces that many CO white dwarfs with initial masses above 1.2 M_{ ⊙ } [287, 241]. The most massive white dwarfs are probably formed as ONe white dwarfs from stars above 8 M_{ ⊙ } — see Section 3.3. So this class of AIC formation is likely to be rare. But high accretion rates are expected in a very important, and perhaps common, event: the merger of two white dwarfs (generally a He or CO white dwarf merging with a CO white dwarf). Most current simulations predict that this merger process is very rapid, arguing that the moremassive white dwarf will tidally disrupt and accrete its companion in a very short time. Such a rapid process suggests high accretion rates, arguing that most mergers produce AICs. However, there are a number of uncertainties in such calculations and no other mechanism produces a sufficiently high rate to explain the occurrence of type Ia supernovae, so we cannot rule out that these mergers produce thermonuclear supernovae instead of AICs.
3.2.1 AIC rates
Thus far, no outburst from the AIC of a white dwarf has been observed. Given that the outburst is expected to be very dim because the shock heating is negligible and the predicted ^{56}Ni yields are all low (< 0.05 M_{ ⊙ } [102, 161, 61]), the lack of observed AICs does not place firm upper limits on the AIC rate.
Theoretical estimates of the rate of AICs are also quite uncertain. If we accept the conclusions of Nomoto & Kondo [224], the AIC rate may be well above the type Ia supernova rate (∼ 10^{−2} y^{−1} in a MilkyWaysized galaxy). This result depends upon a number of assumptions in the accretion process in these binary systems and the true rate of AICs could be many orders of magnitude lower than this value. Studies of binary mass transfer [198, 199, 16, 270, 126, 203, 72, 342] and white dwarf accretion [91] are both becoming more accurate. As they are coupled with stellar evolution models of these systems, our understanding of the rate of AICs should become more accurate as well.
Alternatively, one can use observed features of AIC explosions to constrain the AIC rate. Fryer [102] argued that the neutronrich ejecta from an AIC limits their rate to ∼ 10^{−4} y^{−1}. More recent results, which eject a smaller fraction of neutronrich material, may loosen this limit by 1 order of magnitude [161, 61].
3.2.2 Evolution of AIC
As a white dwarf accretes material up to the Chandrasekhar limit, it will begin to compress. As it does, it heats up and, in principle, can ignite explosive burning. But the URCA process cooling (electron capture followed by a beta decay, which leads to the emission of two neutrinos, which escape and cool the system, see Couch & Arnett [58] for a summary) on nuclei can allow the white dwarf to cool enough that the burning does not generate enough energy to disrupt the white dwarf. When the white dwarf becomes sufficiently dense, the material is dissociated and the collapse proceeds in a very similar manner to that of a “standard” corecollapse supernova. The core proceeds through a runaway collapse that ends when nuclear forces and neutron degeneracy pressure halt the collapse, causing a “bounce”. Especially in the case of a white dwarf/white dwarf merger, the accretion phase just prior to collapse is likely to spin up the core and these are almost certainly the most rapidlyrotating corecollapse models.
The structure of AIC cores is very similar to the structure of lowmass stars, and a large set of simulations of both have been conducted in the past few decades [142, 11, 334, 102, 63, 61]. The latest simulations [63, 61] use progenitors including rotation [341, 339]. The shock stalls in AICs, just as in standard corecollapse supernovae, but there is no stellar envelope with which to prevent further accretion and the explosion is quickly revived (hence there will be little convective overturn). The cooling neutron star could well develop instabilities (both to convection and to bar modes). If magnetic fields develop, additional outflows can occur.
3.2.3 AIC: Sources of gravitational wave emission
Due to the lack of convection, AICs may only have a subset of the GW sources that we see in corecollapse supernovae. However, AICs have the potential to be much faster rotating, which may make them strong GW sources.
Bounce: The potentially highrotation in AICs mean that the bounce signal in these systems could be quite large. The primary source of this emission is the rapidly changing quadrupole moment in the matter as the asymmetries evolve (Section 4.1), although we cannot rule out a strong signal from asymmetric neutrino emission (Section 4.4).
In the Neutron Star: Convection in the cooling neutron star could produce GW emission. But more likely, due to the strong rotation, barmode instabilities can become important in these collapsed systems (Section 4.3).
3.3 Electroncapture supernovae and corecollapse supernovae below 10–12 solar masses: Low mass stars
Stars below about 10–12 M_{ ⊙ } ([286, 287, 241], Poelarends et al., in preparation) have very diffuse envelopes surrounding their cores when they collapse. The lowest mass stars in this class lie on the border between white dwarf (and planetary nebula) formation and the formation of an iron core (see Herwig [141] for a review). These stars tend to form ONe white dwarfs, ejecting most of their envelope. They are supported by electron degeneracy pressure and electron capture causes them to collapse down to neutron stars. Their resultant explosions are termed Electron Capture Supernovae [152]. The more massive stars in this class do form iron cores, but the envelopes surrounding these core have low densities (when compared to normal supernovae progenitors) at the time of collapse.
3.3.1 Rate of supernovae from lowmass stars
The rate of core collapse from low mass stars has the same uncertainties of star formation rate and initial mass function of stars that made it difficult to estimate the normal supernova rate. But it also has an additional uncertainty: understanding the lower limit at which these collapses occur [197, 286, 287, 241]. This lower limit marks the dividing line between white dwarf and neutron star formation and the explosive ejection of the star’s envelope (part of the white dwarf formation process) has proven very difficult for traditional stellar evolution codes to model. The actual position of this lower limit has been a matter of debate for over three decades (see Iben & Renzini [152] for a review). The metallicity dependence on this limit is even more controversial. Heger et al. [136] argued that the metallicity dependence was negligible and that the lower limit was roughly at 9 M_{⊙} at all metallicities. Poelarends et al. [241] have studied this matter in more detail and have a range of solutions for the metallicity dependence. For neutron stars forming through electron capture supernovae, their “preferred” model predicts that the limit drops from 9 M_{⊙} at solar metallicity down to 6.3 M_{⊙} at z_{metal} = 10^{−3}z_{ ⊙ }. This means that for most initial mass functions, these low mass corecollapse progenitors dominate corecollapse at low metallicities.
Unfortunately, these lowmass supernovae do not produce very much radioactive Ni, the power source for supernova lightcurves. Although their envelopes are likely to be denser than the material surrounding AICs, the tenuous envelopes certainly affect the explosion and may limit the shockpowered lightcurve. But surveys are now catching these supernovae [288]. These surveys will ultimately place reasonable estimates on the rate of these supernovae. Based on current estimates of the mass limits and the initial mass function in this mass range, we expect the rate of these supernovae to be roughly equal to that of the corecollapse supernova rate.
3.3.2 Lowmass star collapse: Evolution
The structure of the core of lowmass stars is very similar to the structure of AIC cores, and a large set of simulations of both have been conducted in the past few decades [142, 11, 334, 102, 63, 61]. The core collapses and bounces, but the envelope is not massive enough to prevent a quick revival. A convective region is unlikely to develop above the proto neutron star. But these stars are probably not rotating as fast as most AICs, and their subsequent GW emission is likely to be weaker (e.g., smaller chance of barmode instabilities, etc.).
3.3.3 Lowmass star collapse: Gravitational wave emission mechanisms
The GW emission mechanisms at bounce and in the neutron star are the same as AICs but the likelihood of the development of instabilities (and the strength of the GW emission from these instabilities) will be affected by slower rotation.
3.4 Corecollapse supernovae from stars above 20 solar masses: Massive star collapse
In the absence of winds, as the stellar mass approaches 20 M_{⊙}, the explosion energy predicted by engine models decreases while the binding energy of these stars increases (recall Figure 3). As this happens, more and more material from the stellar mantle falls back onto the newly formed neutron star. Roughly at 20 M_{⊙}, so much material falls back that the mass of the compact remnant exceeds the maximum neutron star mass and ultimately collapses to form a black hole. However, note that Heger et al. [136] argue that, at solar metalliciety, many massive stars lose so much mass in their winds that they will form neutron stars, not black holes (recall Figure 4).
3.4.1 Massive star collapse: Evolution
The evolution of a massive star collapse begins identically to lesser mass stars. In general, it is likely that the core bounces just as in normal corecollapse supernovae [90, 97, 296, 295].^{10} The bounce shock stalls as in normal supernovae. The primary exception are the low metallicity (< 10^{−4} Z_{⊙}) stars in the 140–260 M_{⊙} range that produce pairinstability supernovae [114]. These stars explode completely leaving no remnant behind. We will ignore these stars in the rest of our discussion.
These massive stars are much less likely to revive the stalled shock. Although the region above the proto neutron star is convective, the convective engine is unable to quickly drive an explosion (recall Section 3.1). In such cases, lowmode instabilities (e.g., SASI, see Section 3.1.2) are more likely to dominate the convection. If the shock is revived at all, the explosion is weaker than normal supernovae. Considerable material begins to fall back within 1–2 s of the explosion with accretion rates nearly at 1 M_{⊙} s^{−1} [190, 347, 100]. This accretion may lead to additional convection and possible further outbursts [105, 100]. With both the SASI and the accretion convection, these stars are likely to have more asymmetric convection than normal corecollapse supernovae (beneficial for GW emission).
Ultimately, the mass accretion causes the proto neutron star to collapse down to a black hole [55, 54]. After the collapse, a disk forms around the black hole if the star’s angular momentum is sufficiently high. Magnetic dynamos in the disk, or neutrino emission, might lead to an additional explosion. Indeed, this is the mechanism behind the collapsar GRB [219, 333].
Massive stars might avoid the collapse to a black hole if strong magnetarlike fields can be produced in the dense environment produced when the above engine fails [35, 175, 2, 3, 6, 329].
3.4.2 Rates of massive star collapse
The rate of massive star collapse can be determined by the same method used to determine the normal supernova rate: multiplying the star formation rate times the fraction of stars above ∼ 20 M_{⊙}. The uncertainties in such a calculation lie in determining the lower limit for black hole formation (determined by theory, and bolstered by observations), the power in the initial mass function and the star formation rate (both beyond theory at this point). As with core collapse, the latter two quantities dominate the errors in this rate estimate. Fryer & Kalogera [109] argue that between 10–40% of all massive stars above 10 M_{⊙} (those that are likely to form bright corecollapse supernovae and contribute to the observed supernova rate) form black holes. This answer depends primarily on their prescription for winds and the initial mass function. The fraction is likely to be lower at lowredshift but increase with increasing redshift as winds remove less mass from the star, allowing more stars to form black holes.
Unfortunately, there are no direct observational estimates for the blackholeformation rate. If we assume this collapse is the primary scenario for the production of longduration GRBs we can estimate a lower limit for the blackholeformation rate (and estimate the formation rate of fastrotating systems). This value is roughly 0.1–0.01% the total corecollapse supernova rate.
3.4.3 Gravitational waves from massive stars
For most massive stars, the GW signal will be similar to normal corecollapse supernovae. Massive stars above 300 M_{⊙} are likely to have much stronger signals than normal corecollapse supernovae, but they will only occur at high redshift.
Bounce: For the lowermass range of these stars, the signal is unlikely to be demonstrably different than normal corecollapse supernovae.
PostBounce and Convection: Because the convective phase in these massive stars persists longer than normal supernovae, it can develop strong lowmode activity. Such convective cycles produce the strongest GWs, so we might expect stronger GWs from this convective phase for massive stars than for any other stellar collapse scenario. Fryer et al. [115] also found that bar instabilities could develop in ∼ 300 M_{⊙} stars.
Black Hole Formation: The initial blackhole formation and the subsequent accretion leads to a perturbed geometry of the black hole, initially distorting it. This distortion causes the hole to “ring” in distinct harmonics as gravitational radiation removes the perturbation and the black hole settles into a stationary Kerr state.
Disk Fragmentation: Depending upon the angular momentum profile, the mass in the disk can be large enough that selfgravity can drive instabilities and induce fragmentation.
3.5 Supermassive stars
There is a large body of observational evidence that supermassive black holes (SMBHs, M ≳ 10^{6} M_{⊙}) exist in the centers of many, if not most galaxies (see, e.g., the reviews of Rees [246] and Macchetto [188]). The masses of SMBHs in the centers of more than 45 galaxies have been estimated from observations [83] and there are more than 30 galaxies in which the presence of a SMBH has been confirmed [163].
3.5.1 Supermassive stars: Evolution
One of the possible formation mechanisms for SMBHs involves the gravitational collapse of SMSs. The timescale for this formation channel is short enough to account for the presence of SMBHs at redshifts z > 6 [155]. SMSs may contract directly out of the primordial gas, if radiation and/or magneticfield pressure prevent fragmentation [131, 75, 130, 186, 29, 1, 348]. Alternatively, they may build up from fragments of stellar collisions in clusters [262, 14]. Supermassive stars are radiation dominated, isentropic and convective [276, 345, 186]. Thus, they are well represented by an n = 3 polytrope. If the star’s mass exceeds 10^{6} M_{⊙}, nuclear burning and electron/positron annihilation are not important.
After formation, an SMS will evolve through a phase of quasistationary cooling and contraction. If the SMS is rotating when it forms, conservation of angular momentum requires that it spins up as it contracts. There are two possible evolutionary regimes for a cooling SMS. The path taken by an SMS depends on the strength of its viscosity and magnetic fields and on the nature of its angular momentum distribution.
In the first regime, viscosity or magnetic fields are strong enough to enforce uniform rotation throughout the star as it contracts. Baumgarte and Shapiro [12] have studied the evolution of a uniformlyrotating SMS up to the onset of relativistic instability. They demonstrated that a uniformlyrotating, cooling SMS will eventually spin up to its mass shedding limit. The mass shedding limit is encountered when matter at the star’s equator rotates with the Keplerian velocity. The limit can be represented as β_{shed} = (T/W)_{shed}. In this case, β_{shed} = 9 × 10^{−3}. The star will then evolve along a massshedding sequence, losing both mass and angular momentum. It will eventually contract to the onset of relativistic instability [151, 49, 50, 276, 155, 185].
The SMS formation scenario for SMBHs is just one of many. It fits into a broad class of scenarios invoking the collapse of supermassive objects formed in halos of dense gas, e.g., [15]. Although the structure used in many of these SMS calculations may not be appropriate for this broad class of supermassive objects, many of the basic features studied in these SMS simulations will persist.
Baumgarte and Shapiro used both a secondorder, postNewtonian approximation and a fully generalrelativistic numerical code to determine that the onset of relativistic instability occurs at a ratio of R/M ∼ 450, where R is the star’s radius and G = c = 1 in the remainder of this section. Note that a secondorder, postNewtonian approximation was needed because rotation stabilizes the destabilizing role of nonlinear gravity at the first postNewtonian level. If the mass of the star exceeds 10^{6} M_{⊙}, the star will then collapse and possibly form a SMBH. If the star is less massive, nuclear reactions may lead to explosion instead of collapse.
The major result of Baumgarte and Shapiro’s work is that the universal values of the following ratios exist for the critical configuration at the onset of relativistic instability: T/W, R/M, and J/M^{2}. These ratios are completely independent of the mass of the star or its prior evolution. Because uniformlyrotating SMSs will begin to collapse from a universal configuration, the subsequent collapse and the resulting gravitational waveform will be unique.
In the opposite evolutionary regime, neither viscosity nor magnetic fields are strong enough to enforce uniform rotation throughout the cooling SMS as it contracts. In this case, it has been shown that the angular momentum distribution is conserved on cylinders during contraction [22]. Because viscosity and magnetic fields are weak, there is no means of redistributing angular momentum in the star. So, even if the star starts out rotating uniformly, it cannot remain so.
The star will then rotate differentially as it cools and contracts. In this case, the subsequent evolution depends on the star’s initial angular momentum distribution, which is largely unknown. One possible outcome is that the star will spin up to massshedding (at a different value of β_{shed} than a uniformlyrotating star) and then follow an evolutionary path that may be similar to that described by Baumgarte and Shapiro [12]. The alternative outcome is that the star will encounter the dynamical bar instability prior to reaching the massshedding limit. New and Shapiro [222, 223] have demonstrated that a barmode phase is likely to be encountered by differentiallyrotating SMSs with a wide range of initial angularmomentum distributions. This mode will transport mass and angular momentum outward and thus may hasten the onset of collapse.
3.5.2 Rates of supermassive stars
An estimate of the rate of the collapse of SMSs can be derived from the quasar luminosity function. Haehnelt [129] has used the quasar luminosity function to compute the rate of GW bursts from SMBHs, assuming that each quasar emits one such burst during its lifetime (and that each quasar is a supermassive black hole). If it is assumed that each of these bursts is due to the formation of a supermassive black hole via the collapse of an SMS, then Haehnelt’s rate estimates can be used as estimates of the rate of SMS collapse. This rate is likely an overestimate of the SMS collapse rate because many SMBHs may have been formed via merger. Haehnelt predicts that the integrated event rate through redshift z = 4.5 ranges from ∼ 10^{−6} yr^{−1} for M = 10^{8} M_{⊙} objects to ∼ 1 yr^{−1} for M = 10^{6} M_{⊙} objects. Thus, as in the case of Population III stars, a reasonable occurrence rate can be determined for an observation (luminosity) distance of 50 Gpc.
3.5.3 Gravitationalwave emission mechanisms of supermassive stars
The GW emission mechanisms related to the collapse of SMSs are a subset of those discussed in the sections on AIC, SNe/collapsars, and Population III stellar collapse.
Bounce: Like very massive stars [97], it is likely that these SMSs will first form a proto black hole before collapsing to form a black hole. If so, they can emit GWs from the bounce that occurs when the proto black hole is formed. Because of the mass of the SMS, an aspherical collapse can produce a much stronger signal than any normal supernovae.
PostBounce: GWs may be emitted due to global rotational and fragmentation instabilities that may arise during the collapse/explosion and in the collapsed remnant (prior to blackhole formation).
Black Hole: Ringing in the black hole as it forms and accretes matter may also drive a strong signal.
4 Gravitational Wave Emission Mechanisms
The collapse of the progenitors of corecollapse 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 [251] 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 [311] and Epstein and Wagoner [78, 77], 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 [225] and Shapiro and Saenz [273, 253] 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 [318] and later by Saenz and Shapiro [254, 255].
Müller [204] calculated the quadrupole GW emission from 2D axisymmetric collapse based on the Newtonian simulations of Müller and Hillebrandt [207] (these simulations used a realistic equation of state and included differential rotation). He found that differential rotation enhanced the efficiency of the GW emission.
Stark and Piran [291, 238] were the first to compute the GW emission from fullyrelativistic 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 blackhole 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 GW emission during collapse and bounce [271, 272]. They employed a perturbative approach, valid only in the slowlyrotating 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 [64]. 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 GW emission from stellar collapse see [87, 167, 228]. In the remainder of this section we will discuss more recent investigations.
4.1 Bounce
Most of the original GW studies of corecollapse supernovae focused on the bounce phase. Gravitational radiation will be emitted during the collapse/explosion of a corecollapse 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 \({{\mathcal I}_{jk}}\) will increase. The contraction speeds up over the next 20 ms and the density distribution becomes more centrally condensed [201]. In this phase the core’s shrinking size dominates its increasing deformation and the magnitude of \({{\mathcal I}_{jk}}\) decreases. As the core bounces, \({{\mathcal I}_{jk}}\) changes rapidly due to deceleration and rebound. If the bounce occurs because of nuclear pressure, its timescale will be < 1 ms. If centrifugal forces play a role in halting the collapse, the bounce can last up to several ms [201]. The magnitude of \({{\mathcal I}_{jk}}\) will increase due to the core’s expansion after bounce. As the resulting shock moves outward, the unshocked portion of the core will undergo oscillations, causing \({{\mathcal I}_{jk}}\) 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 [86, 201, 167, 228].
The core collapse simulations of Mönchmeyer et al. [201] 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 stellarcore 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 ρ_{c} bounce had been predicted by Shapiro and Lightman [274] and Tohline [312] (in the context of the “fizzler” scenario for failed supernovae; see also [133, 134, 153]), and had been observed in earlier collapse simulations [210, 299]. 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 < 1 ms. 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.
The model of Mönchmeyer et al. [201] that bounced due to nuclear forces had the highest GW amplitude of all of their models, h_{pk} ∼ 10^{−23} for a source distance d =10 Mpc, and the largest emitted energy E_{ GW } ∼ 10^{47} erg. The accompanying power spectrum peaked in the frequency range 5 × 10^{2} −10^{3} Hz.
The most extensive Newtonian survey of the parameter space of axisymmetric, rotational core collapse is that of Zwerger and Müller [350]. 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 [79]. The models had a polytropic equation of state, with initial adiabatic index Γ_{i} =4/3. Collapse was induced by reducing the adiabatic index to a value Γ_{r} 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 [168]).
In contrast to the results of Mönchmeyer et al. [201], in Zwerger and Müller’s investigation the value of ρ_{c} at bounce did not determine the signal type. Instead, the only effect on the waveform due to ρ_{c} was a decrease in h_{pk} in models that bounced at subnuclear densities. The effect of the initial value of β_{i} on h_{pk} was nonmonotonic. For models with β_{i} ≲ 0.1, h_{pk} increased with increasing β_{i}. This is because the deformation of the core is larger for faster rotators. However, for models with larger β_{i}, h_{pk} decreases as β_{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 ρ_{c} at bounce was just less than ρ_{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 β_{i} < 0.01; and those with soft equations of state, β_{i} ≥ 0.018, and large degrees of differential rotation. The GW amplitudes emitted during their simulations were in the range 4 × 10^{−25} ≲ h ≲ 4 × 10^{−23}, for d =10 Mpc. The corresponding energies ranged from 10^{44} ≲ E_{GW} ≲ 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 [337] used techniques very similar to those of Zwerger and Müller [350] 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 ∼ 0.5. They also found that h_{pk} was sensitive to the stiffness of the equation of state for densities just below ρ_{nuc}. The characteristics of the GW emission from their models were similar to those of Zwerger and Müller. A series of new results (e.g., [168, 169, 232, 233, 68, 69]) have essentially confirmed these results.
Kotake et al. [169] found that magnetic fields can lower these amplitudes (they lower the rotation rate) by ∼ 10%, but realistic stellar profiles can lower the amplitudes by a factor of ≳ 4–10 [230, 209], restricting the detectability of supernovae to within our galaxy (≲ 10 kpc). Obviously, this effect depends on the strength of the magnetic fields and Obergaulinger et al. [227, 226] studied this effect by using a variety of magneticfield strengths. Generalrelativistic magnetohydrodynamic calculations of the bounce phase have confirmed these results [48, 47]. Magnetohydrodynamic calculations are becoming more common, gradually increasing our intuition about this important piece of physics [267, 266, 300].
Fryer and Warren [112] 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 gray diffusion approximation for neutrino transport, and Newtonian spherical gravity. The gray approximation is a limiting assumption and a multigroup calculation can produce different results [34, 325]. Their initial model was nonrotating. Thus, no barmode instabilities could develop during their simulations. The only GWemitting mechanism present in their models was convection in the core. The maximum amplitude h of this emission, computed in the quadrupole approximation, was ∼ 3 × 10^{−26}, for d =10 Mpc [108]. In later work, Fryer & Warren [113] included full Newtonian gravity through a tree algorithm and studied the rotating progenitors from Fryer & Heger [103]. 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 [103], the explosion occured quickly for their models (≲ 100 ms) 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 [107]. The fastestrotating models achieved a signal of h ∼ 2 × 10^{−24} for d =10 Mpc and characteristic frequencies of f_{GW} ∼ 1000 Hz. For supernovae occuring within the galaxy, such a signal is detectable by LIGOII. Dimmelmeier et al. [68, 69] and Ott et al. [232, 233] have studied these rotating progenitors, producing amplitudes on par with these results, but providing much more detailed information, including spectra. These models included full general relativity and are discussed in Section 4.1.1.
Fryer and collaborators have also modeled asymmetric collapse and asymmetric explosion calculations in three dimensions [98, 110]. These calculations will be discussed in Sections 4.2 and 4.4.
4.1.1 General relativistic calculations
Generalrelativistic 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 generalrelativistic effects are considered. Furthermore, general relativity will cause rotating stars undergoing collapse to bounce at higher densities than in the Newtonian case [302, 350, 245, 33].
The full collapse simulations of Fryer and Heger [103] followed the axisymmetric evolution of the collapse, bounce and explosion including equation of state and neutrino transport from which the resultant GW emission has been studied [106, 108]. Fryer and Heger only include the effects of general relativity in the monopole approximation. This ignores many feedback mechanisms. The GW emission from these simulations was evaluated with either the quadrupole approximation or simpler estimates (see below).
The work of Fryer and Heger [103] was 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) [135], 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 (0.95–1.9 × 10^{49} g cm^{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 [2, 290, 138].
FHH’s [106] numerical quadrupole estimate of the GWs from oscillations observed in the collapse simulations of Fryer and Heger [103] predicts an upper limit of the peak dimensionless amplitude h_{pk} = 4.1 × 10^{−23} (for d =10 Mpc), emitted at f_{GW} ≈ 20 Hz. The radiated energy E_{GW} ∼ 2 × 10^{44} erg. This signal would be just out of the detectability range of the LIGOII detector. However, most simulations predict lower amplitudes peaking at different frequencies.
Dimmelmeier and collaborators [68, 69] have now completed a very extensive study of axisymmetric collapse including modern equations of state and a recipe for the evolution of the electron fraction. The range of their results was consistent with the upper limits placed by FHH, with their strongest signal have a peak dimensionless amplitude h_{pk} = 1.0 × 10^{−23} (for d =10 Mpc).
General relativity has been more fully accounted for in the core collapse studies of Dimmelmeier, Font, and Müller [65, 66, 67] and Shibata and collaborators [281, 283], which build on the Newtonian, axisymmetric collapse simulations of Zwerger and Müller [350]. In all, they have followed the collapse evolution of 26 different models, with both Newtonian and generalrelativistic 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 conformallyflat metric to approximate the spacetime geometry [57] 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 conformallyflat 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 [350] for details).
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 × 10^{−24} − 3 × 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 × 10^{47} erg compared to the Newtonian value of 6.4 × 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). Studies of nonlinear pulsations in neutron stars expect high frequencies between 1.8–3.6 kHz [293]. 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 [292].
Ott et al. [232, 233] argue that the GW signal from the collapse, bounce, and early postbounce phases of the core collapse evolution is much more generic than many of these past results show, arguing that variations in the rotation (within the limits studied in their calculations) do not alter the signal significantly. If these results are confirmed, firm estimates can be provided to the GW detector community.
4.2 Convection
As we discussed at the beginning of Section 3, it is becoming increasingly accepted that convection above the proto neutron star plays a major role in the supernova explosion mechanism (e.g., Figure 1). Convectivelydriven 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 [37, 206]. Inside the proto neutron star, GW emission from these processes results from smallscale asphericities, unlike the largescale motions responsible for GW emission from aspherical collapse and nonaxisymmetric global instabilities. But the SASI and lowmode convection do produce largescale accelerated mass motions [165, 213]. Note that RayleighTaylor instabilities in the exploding star 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 RayleighTaylor instabilities occur at very large radii (with low velocities) [206].
4.2.1 Asymmetric collapse
Since convection was suggested as a key ingredient in the explosion, it has been postulated that asymmetries in the convection can produce the large proper motions observed in the pulsar population [139]. 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 [37] proposed that asymmetries in the collapse could produce 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 [13, 172]. These asymmetries will then drive asymmetries in the convection and ultimately, the supernova explosion. Burrows and Hayes [37] found that not only could they produce strong motions in the nascent neutron star, but detectable GW signals. The peak amplitude calculated was h_{pk} ∼ 3 × 10^{−24}, for a source located at 10 Mpc.
The study by Nazin and Postnov [220] predicts a lower limit for E_{GW} emitted during an asymmetric corecollapse 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 ϵ present in the collapse and the corresponding characteristic GW amplitude \((h \propto \sqrt \epsilon)\). This amplitude is 3 × 10^{−25} for a source located at 10 Mpc and emitting at f_{GW} = 1 kHz.
4.2.2 Proto neutron star convection
Müller and Janka [208] 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 convectioninduced aspherical mass motion and neutrino emission in the quadrupole approximation (for details, see Section 3 of their paper [208]).
4.2.3 Convection above the proto neutron star
The peak GW amplitude resulting from these 2D simulations of convective mass motions in the hot bubble region was h_{pk} ≈ 5 × 10^{−25}, for d =10 Mpc. The emitted energy was ≲ 2 × 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} ∼ 10 Hz. The amplitude of such a signal would be too low to be detectable with LIGOII.
The case for GWs from convectioninduced asymmetric neutrino emission has also varied with time. Müller and Janka estimated the GW emission from the convectioninduced anisotropic neutrino radiation in their simulations (see [208] for details). They found that the amplitude of the GWs emitted can be a factor of 5–10 times higher than the GW amplitudes resulting from convective mass motion. More recent simulations by Müller et al. [209] argue that the GWs produced by asymmetric neutrino emission is weaker than that of the convective motions. But Marek et al. [196] have argued strongly that calculating the GW signal from asymmetries in neutrinos is extremely difficult and detailed neutrino transport is required to determine the GW signal from neutrinos.
4.2.4 Lowmode convection and the standing accretionshock instability
Our understanding of the convective engine is deepening with time. The latest focus of attention has been the SASI instability, which produces, at late times, extremely low mode convection. This topic is currently a matter of heated debate. Whether this instability dominates at late times, or whether the latetime, lowmode convection is simply the merger of convective cells [139] is, in the opinion of these authors, yet to be conclusively determined. Convection is very difficult to simulate and has been studied for many decades on a variety of applications from the combustion engine to astrophysics, with no accepted resolution. But many groups are now finding lowmode convection above the proto neutron star (which can heighten the GW emission) and all agree that this occurs at late times (more than a few hundred milliseconds after bounce). As we have shown in Figure 5, such late explosions will be weak and, if the assumptions of that analysis are correct, these late explosion mechanisms cannot explain the observed corecollapse supernovae. Marek & Janka [195] believe otherwise. Certainly, if material can continue to accrete onto the neutron star after the launch of the explosion, which it does in some of the recent results of the Mezzacappa team (in preparation), stronger explosions may be produced.
4.3 Bar modes
Rotational instabilities in proto neutron stars, if they exist, could be very powerful GW sources. Global rotational instabilities in a collapsed core lead to rapid variations in the quadrupole moment, and hence, strong GW emission. Global rotational instabilities in fluids arise from nonaxisymmetric modes e^{±imϕ}, where m = 2 is known as the “bar mode” [302, 5]. It is convenient to parameterize a system’s susceptibility to these modes by the stability parameter β = T_{rot}/W. 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, uniformlyrotating MacLaurin spheroids, the dynamical barmode instability sets in at β_{d} ≈ 0.27. These instabilities have been confirmed by a number of studies [276, 192, 74, 313, 331, 314, 147, 289, 148, 146, 237, 315, 154, 221, 30, 184, 232, 233], and we will discuss some of these results here. At lower rotation rates, a secular instability may develop that produces bar modes if viscous or gravitational radiation reaction forces can redistribute the angular momentum [52, 173, 171, 236].
First, scientists have learned that dynamical instabilities can excite m = 1 modes as well as the wellstudied, m = 2 bar modes [46, 258, 234, 235, 260]. Second, scientists have discovered that nonaxisymmetric instabilities can occur at much lower values of β when the differential rotation is very high [278, 279, 282, 326, 257, 260]. For some models, cores with high differential rotation have exhibited nonaxisymmetric instabilities for values of β ≈ 0.01. These results may drastically change the importance of these modes in astrophysical observations.
4.3.1 Equilibrium models to study instabilities
One way to determine whether a specific star collapses to develop bar modes is through equilibrium models as initial conditions for hydrodynamical simulations (e.g., [289, 237, 221, 46]). 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 [184, 183] have applied this equilibrium approach to AIC. Their investigation began with a study of equilibrium models built to represent neutron stars formed from AIC [184]. These neutron star models were created via a twostep process, using a Newtonian version of Hachisu’s selfconsistent field method [128]. 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 rapidlyrotating, precollapse white dwarf models. Their Models I and II are CO white dwarfs with central densities ρ_{c} = 10^{10} and 6 × 10^{9} g cm^{−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 ρ_{c} = 4 × 10^{9} g cm^{−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 [102]. The realistic equation of state used to construct the white dwarfs is a Coulombcorrected, zero temperature, degenerate gas equation of state [261, 51].
In the second step of their process, Liu and Lindblom [184] built equilibrium models of the collapsed neutron stars themselves. The mass, total angular momentum, and specific angular momentum distribution of each neutronstar remnant is identical to that of its white dwarf progenitor (see Section 3 of [184] for justification of the specific angular momentum conservation assumption). These models were built with two different realistic neutronstar equations of state.
Liu and Lindblom’s cold neutronstar remnants had values of the stability parameter β ranging from 0.23–0.26. It is interesting to compare these results with those of Villain et al. [323] or of Zwerger and Müller [350]. Villain et al. [323] found maximum β values of 0.2 for differentiallyrotating models and 0.11 for rigidlyrotating models. Zwerger and Müller performed axisymmetric hydrodynamics simulations of stars with polytropic equations of state (P ∝ ρ^{Γ}). Their initial models were Γ = 4/3 polytropes, representative of massive white dwarfs. All of their models started with ρ_{c} = 10^{10} g cm^{−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 β < 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.
In a continuation of the work of Liu and Lindblom, Liu [183] used linearized hydrodynamics to perform a stability analysis of the cold neutronstar AIC remnants of Liu and Lindblom [184]. He found that only the remnant of the ONeMg white dwarf (Liu and Lindblom’s Model III) developed the dynamical bar mode (m = 2) instability. This model had an initial β = 0.26. Note that the m =1 mode, observed by others to be the dominant mode in unstable models with values of β much lower than 0.27 [314, 332, 237, 46], did not grow in this simulation. Because Liu and Lindblom’s Models I and II had lower values of β, Liu identified the onset of instability for neutron stars formed via AIC as β_{d} ≈ 0.25.
Liu estimated the peak amplitude of the GWs emitted by the Model III remnant to be h_{pk} ≈ 1.4 × 10^{−24} and the LIGOII signaltonoise ratio (for a persistent signal like that seen in the work of [221] and [30]) to be S/N ≤ 3 (for f_{GW} ≈ 450 Hz). These values are for a source located at 100 Mpc. He also predicted that the timescale for gravitational radiation to carry away enough angular momentum to eliminate the bar mode is τ_{GW} ≈ 7 s (∼ 3 × 10^{3} cycles). Thus, h ≈ 8 × 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 7 s 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 23). Details of the approximations on which these estimates are based can be found in [183].
Liu cautions that his results hold if the magnetic field of the proto neutron star is B ≤ 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 ≥ 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 ∼ 0^{7} − 3 × 10^{8} G [330]. Strong toroidal magnetic fields of B ≥ 10^{14} G will also suppress the bar instabilities [119].
4.3.2 Hydrodynamic models
The GW emission from nonaxisymmetric hydrodynamics simulations of stellar collapse was first studied by Bonazzola and Marck [194, 23]. 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 use of 3D hydrodynamics collapse simulations to study the GW emission well beyond the core bounce phase was performed by Rampp, Müller, and Ruffert [245]. These authors started their Newtonian simulations with the only model (A4B5G5) of Zwerger and Müller [350] that had a postbounce value for the stability parameter β = 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 equation of state (Γ_{r} = 1.28), highest β_{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 β reached ∼ 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 ∼ 14 ms 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. [46] indicate it is possible that some of the postbounce configurations of Zwerger and Müller, which have lower values of β than the model studied by Rampp, Müller, and Ruffert [245], may also be susceptible to nonaxisymmetric instabilities. Centrella et al. have performed 3D hydrodynamics simulations of Γ = 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 [350]). The simulations carried out by Centrella and collaborators were not full collapse simulations, but rather began with differentiallyrotating 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 β ≳ 0.14. The observed instability had a dominant m =1 mode. Centrella et al. estimate that if a stellar core of mass M ∼ 1.4 M_{⊙} and radius R ∼ 200 km encountered this instability, the values of h_{pk} from their models would be ∼ 2 × 10^{−24} − 2 × 10^{−23}, for d =10 Mpc. The frequency at which h_{pk} occurred in their simulations was ∼ 200 Hz. This instability would have to persist for at least ∼ 15 cycles to be detected with LIGOII.
Brown [31] 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 [245]. He performed 3D hydrodynamical simulations of the postbounce configurations resulting from 2D simulations of core collapse. His precollapse initial models are Γ = 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 [137]. 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 [137]. The model evolved by Rampp, Müller, and Ruffert [245] had much stronger differential rotation than any of Brown’s models. To induce collapse, Brown reduced the adiabatic index of his models to Γ = 1.28, the same value used by [245].
Brown found that β increased by a factor ≲ 2 during his 2D collapse simulations. This is much less than the factor of ∼ 9 observed in the model studied by Rampp, Müller, and Ruffert [245]. 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 rapidlyrotating of his postbounce models (models Ω24 and Ω20, both of which had β > 0.27 after bounce) and of the model of Rampp et al. (which, although it starts out with β = 0.35, has a sustained β < 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. [46]. 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 [245]).
Brown’s simulations determined that both his most rapidlyrotating model Ω24 (with postbounce β > 0.35) and model RMR are unstable to growth of the m = 2 bar mode. However, his model Ω20 (with postbounce β > 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 β of the postbounce core may not be a good diagnostic for the onset of instability. He found, as did Rampp, Müller, and Ruffert [245], that only the innermost portion of the core (with ρ > 10^{10} g cm^{−3}) is susceptible to the bar mode. This is evident in the stability of his model Ω20. This model had an overall β > 0.3, but an inner core with β_{ic} = 0.15. Brown also observed that the β of the inner core does not have to exceed 0.27 for the model to encounter the bar mode. Models Ω24 and RMR had β_{ic} ≈ 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 β_{d} < 0.27 for such configurations.
The GW emission from nonradial quasinormal mode oscillations in proto neutron stars has been examined by Ferrari, Miniutti, and Pons [85]. 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 proto neutron star located at 10 Mpc, with a signaltonoise ratio of 5, E_{GW} must be ∼ 10^{−3} − 10^{−2} M_{⊙}c^{2}. It is unlikely that this much energy is stored in these modes (the collapse itself may only emit ∼ 10^{−7} M_{⊙}c^{2} in gravitational waves [67]).
Shibata et al. [278, 279] found that, with extremely differentiallyrotating cores, a bar mode instability can occur at β values of 0.01. They found that such an instability was weakly dependent on the polytropic index describing the equation of state and on the velocity profile (as long as the differential rotation is high). They predict an effective amplitude of roughly 10^{−22} at a distance of 100 Mpc.
Studies of systems with extreme differential rotation have also discovered the development of onearm (m = 1) instabilities. The work of Centrella et al. [46] has been followed by a large set of results, varying the density and angular velocity profiles [46, 258, 234, 235, 260]. Ou & Tohline [235] argued that these instabilities are akin to the Rossby wave instabilities studied in blackhole accretion disks [176]. These onearmed spirals could also produce a considerable GW signal.
Watts et al. [326] argue that these lowβ instabilities can be explained if the corotating fmode develops a dynamical shear instability. This occurs when the corotating fmode enters the corotation band and when the degree of differential rotation exceeds a threshold value (that is within those produced in some collapse progenitors). These new instabilities are drastically changing our view of nonaxisymmetric modes.
4.4 Neutrinos
Up until now, we have focused on the GWs from a changing quadrupole moment of the baryonic matter. Anisotropic neutrino emission may also produce GWs. Although less studied than baryonic motions, GW emission from aniostropic neutrino emission [76, 317] has been investigated in proto neutron star convection, convection above the proto neutron star, and in asymmetric collapse (caused either by asymmetries in the collapse or by rotation).
The asymmetric collapse simulations discussed in Section 4.2 are one way to increase the GW signal from anisotropic neutrino emission. Recall from Figure 17 that Burrows & Hayes found that the neutrinoinduced term dominated the GW signal for their asymmetric collapse simulations. Fryer et al. [107] found the same result. The GW amplitude is dominated by the neutrino component and can exceed h_{pk} ∼ 6 × 10^{−24}, for a source located at 10 Mpc in Fryer’s most extreme example. A caveat in this result is that Fryer et al. were artificially increasing the level of the asymmetry in the collapse in an attempt to obtain strong neutron star kicks and it is unlikely that any stellar system will have such large asymmetries in nature. This signal estimate should be seen as an extreme upper limit.
4.5 rModes
rmodes are quasitoroidal oscillations in rotating fluids that occur because of the Coriolis effect (akin to the Rossby modes studied in planetary “spots”). In GWs, these modes are driven unstable by gravitational radiation reaction. The primary questions surrounding these instabilities arise in calculating the saturation amplitude of rmodes. The GW emission from rmode unstable neutronstar remnants of corecollapse SNe would be easily detectable if α_{max} ∼ 1. In rotating stars, gravitational radiation reaction drives the rmodes toward unstable growth [4, 96]. In hot, rapidlyrotating neutron stars, this instability may not be suppressed by internal dissipative mechanisms (such as viscosity and magnetic fields) [179]. If not limited, the dimensionless amplitude α of the dominant (m = 2) rmode 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 newlyformed neutron star to spin down over time. The spindown timescale and the strength of the GWs themselves are directly dependent on the maximum value α_{max} to which the amplitude is allowed to grow [180, 181].
Originally, it was thought that α_{max} ∼ 1. In such a case, we can estimate the GW signal from stellar collapse. We expect multiple GW bursts to occur as material falls back onto the neutron star and results in repeat episodes of rmode growth (note that a single rmode episode can have multiple amplitude peaks [180]). Using Equation (10), FHH [106] calculate that the characteristic amplitude of the GW emission from this rmode evolution tracks from 6 − 1 × 10^{−22}, over a frequency range of 10^{3} − 10^{2} Hz for a source driven by fallback at 10 Mpc. They estimate the emitted energy to exceed 10^{52} erg.
Later work indicated that α_{max} may be ≥ 3 [180, 294, 268, 181]. But most recent research suggests that magnetic fields, hyperon cooling, and hyperon bulk viscosity may limit the growth of the rmode instability, even in nascent neutron stars [159, 158, 248, 249, 181, 178, 132, 5] (significant uncertainties remain regarding the efficacy of these dissipative mechanisms). One way to reduce this viscosity is to invoke nonstandard physics in the dense equation of state, e.g., quark material, antikaon core [53, 71]. Even with this more exotic physics, the reduction in viscosity is limited to specific regions in the neutron star and in the spin/temperature phase space. More work is needed to determine if such modifications can allow rmodes to make a detectable signal.
In addition, a study of a simple barotropic neutron star model by Arras et al. [8] argue that multimode couplings could limit α_{max} to values ≪ 1. If α_{max} is indeed ≪ 1 (see also [125, 27, 26, 28, 252, 24]), GW emission from rmodes in collapsed remnants is likely undetectable. From Equation (10), we can see that the GW signal is proportional to the mode amplitude, so a decrease in the maximum amplitude by an order of magnitude corresponds to an order of magnitude decrease in the GW strain. If correct, and much of the community believes the maximum of the mode amplitude may be even smaller than 0.1, the GW signal from rmodes is much lower than any other GW source and will not contribute significantly to the observed signal in stellar collapse. Because of this, rmode sources are omitted from figures comparing source strengths and detector sensitivities and from discussions of likely detectable sources in the concluding section.
rmodes may still produce signals in accreting systems such as lowmass Xray binaries. Work continues in this subject for these systems, but this is beyond the subject of this review.
4.6 Fragmentation
Fragmentation requires even more extreme spin rates than barmode instabilities. It is unlikely that any neutronstarforming systems will have enough rotation to produce fragmentation. For most stellar models, the angular momentum increases with radius (or enclosed mass). When black holes form, material further out in the star (with higher angular momentum) becomes part of the compact remnant and it is possible to form proto black holes with higher spin rates than proto neutron stars. These proto black holes may be more likely to fragment. Even so, fragmentation requires extreme levels of rotation. As we shall see, some cases of GW emission from fragmentation is intimately related to blackhole ringing and some of the GW sources discussed here will also be discussed in Section 4.7.
Duez et al. [73] found that if a black hole does form, but the disk is spinning rapidly, that the disk will fragment and its subsequent accretion will be in spurts, causing a “splash” onto the black hole, producing ringing and GW emission. Their result implies very strong GW amplitudes ≳ 10^{−21} at distances of 10 Mpc. Black hole ringing was also estimated by FHH [106], where they too assumed discrete accretion events. They found that, even with very optimistic accretion scenarios, such radiation will be of very low amplitude and beyond the upper frequency reach of LIGOII (see [106] for details).
The generalrelativistic hydrodynamics simulations of Zanotti, Rezzolla, and Font [344] suggest that a torus of neutronstar matter surrounding a blackhole remnant may be a stronger source of GWs than the collapse itself. They used a highresolution 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 quasiperiodic oscillations. They estimate that the resulting GW emission would have a characteristic amplitude h_{c} ranging from 6 × 10^{−24} − 5 × 10^{−23}, for ratios of torus mass to blackhole 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} ∼ 200 Hz. 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 23). 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 [247].
Magnetized tori around rapidlyspinning black holes (formed either via core collapse or neutronstarblackhole coalescence) have been examined in the theoretical study of van Putten and Levinson [321]. They find that such a torusblackhole system can exist in a suspended state of accretion if the ratio of poloidal magneticfield energy to kinetic energy E_{ B }/E_{ k } is less than 0.1. They estimate that ∼ 10% of the spin energy of the black hole will be converted to gravitationalradiation energy through multipole massmoment instabilities that develop in the torus. If a magnetized torusblackhole system located at 10 Mpc is observed for 2 × 10^{4} rotation periods, the characteristic amplitude of the GW emission is ∼ 6 × 10^{−20}. It is possible that this emission could take place at several frequencies. Observations of xray lines from GRBs (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} ≈ 500 Hz, placing the radiation into a range detectable by LIGOI [321].
By studying the results of current stellarcollapse models, FHH [106] predict that a fragmentation instability is unlikely to develop during corecollapse SNe because the cores have central density maxima (see also [108]). However, they do give estimates [calculated via Equations (7) and (6)] for the amplitude, power, and frequency of the emission from such an instability: hpk ∼ 2 × 10^{−22}, P_{GW} = 10^{54} erg s^{−1}, f_{GW} ≈ 2 × 10^{3} Hz. Again, this signal would fall just beyond the upper limit of LIGOII’s frequency range.
4.7 Ringing
Most studies of quasinormal modes in black holes (blackhole ringing) have focused on the collapse of SMSs. A number of reviews already exist on this topic [162, 18, 17, 84] and we refer the reader to these texts for more details. For stellarmassed black holes, ringing is intimately connected to fragmentation. As we discussed in Section 4.6, although Duez et al. [73] found that fragmentation could lead to blackhole ringing with very strong GW amplitudes ≳ 10^{−21} at distances of 10 Mpc, FHH found that even with very optimistic accretion scenarios, such radiation will be of very low amplitude and beyond the upper frequency reach of LIGOII (see [106] for details). In numerical studies, Nagar et al. [215, 216] argue that this accretion is not a superposition of quasinormal modes (although see [9], indicating that the signal would be even lower than simple analytic estimates might predict). The strongest blackhole ringing signal will occur in SMBHs and the rest of this section will focus on the GW signal from these objects.
Shibata and Shapiro [284] have published a fully generalrelativistic, axisymmetric simulation of the collapse of a rapidly, rigidlyrotating SMS. They found that the collapse remained homologous during the early part of the evolution. An apparent horizon does appear in their simulation, indicating the formation of a black hole. Because of the symmetry condition used in their run, nonaxisymmetric instabilities did not develop.
The collapse of a uniformlyrotating SMS has been investigated with postNewtonian hydrodynamics, in 3+1 dimensions, by Saijo, Baumgarte, Shapiro, and Shibata [259]. Their numerical scheme used a postNewtonian approximation to the Einstein equations, but solved the fully relativistic hydrodynamics equations. Their initial model was an n = 3 polytrope.
Saijo et al. also consider the GW emission from the ringdown of the blackhole remnant. For the l = m = 2 quasinormal mode of a Kerr black hole with a/M = 0.9, they estimate the characteristic amplitude of emission to be h ≈ 1.2 × 10^{−20} [ΔE_{GW}/M)/10^{−4}]^{1/2} at f_{GW} ∼ 2 × 10^{−2} Hz for an M = 10^{6} M_{⊙} source located at a luminosity distance of 50 Gpc (see [174, 307, 285] for details). Here, ΔE_{GW}/M is the radiated energy efficiency and may be ≲ 7 × 10^{−4} [291]. This GW signal is within LISA’s range of sensitivity (see Figure 30).
5 Summary
How likely are we to detect GWs from stellar collapse? And if we detect them, will we be able to use the observation to constrain our understanding of their source? The former question has been at the head of almost every investigation of GWs from gravitational collapse. But with the almost certain detection of neutronstar mergers with advanced LIGO^{12}, the question of detection is gradually being superseded by the question of what we will learn from the detection. To answer either question, a key first step is to determine how nearby the stellar collapse must be to be detectable by current or upandcoming instruments. Improvements on both the LIGO and VIRGO detectors as well as new observatories, such as LCGT, will soon make GW astronomy a reality. For SMSs, lowfrequency detectors like LISA will also play a role. For our discussion, we will use LIGO and LISA as our guides, but bear in mind that these other detectors have comparable detection limits and will be crucial for the success of GW astronomy.
5.1 Detection of collapse GW signals
In general, the signal at bounce is strongest for the rotating (and most asymmetric) explosions. The strong bounce signal is possible in AICs, lowmass collapse and normal corecollapse supernovae. The rest of the signals in Figures 32 and 33 are only appropriate for normal corecollapse supernovae. Except in the fastestrotating cases, the signals from AICs, lowmass collapse and normal supernovae are only observable if they occur within local group galaxies. We are unable to observe stellar collapse in the nearby Virgo cluster. A detection of a stellar collapse in this cluster would argue that either extreme rotation does occur in stars or that bar modes exist in these cores.
If bar modes do develop in the proto neutron star, the GW signal may be orders of magnitude higher amplitude than that produced in the bounce or convective phase. Figure 23 shows the signal for barmode sources assuming the source is at 10 Mpc instead of the 10 kpc. Dynamical bar modes could be easily detected out to the Virgo cluster. Even secular bar modes should be detected out to Virgo. If such modes are produced in even 1/10th of all stellar collapses, advanced LIGO should detect multiple GW outbursts per year as soon as it reaches its design specifications. Nondetections place limits on the spin rate of stars.
Black hole forming systems have the potential to form stronger GWs. But the fragmentation claimed in many results did not occur in the 3dimensional models by Rockefeller et al. [250]. Their predicted GW signal was on par with the upper limits to rotational collapse shown in Figure 32. Like normal stellar collapse, blackholeforming objects are unlikely to be observed as far out as the Virgo cluster.
Although the signal for very massive stars (∼ 300 M_{⊙}) is expected to be much larger than other blackholeforming systems, these objects are believed to only form in the early universe (modest metallicities cause these stars to lose most of their mass in winds, in case they behave more like lowermass blackholeforming systems). At such distances, these collapses will be difficult to observe. However, if the rate of these objects is high, it may be that future detectors such as the DECihertz Interferometer Gravitational wave Observatory (DECIGO) and the Big Bang Observer (BBO) may be able to detect GW emission from these objects [298].
The possible exception for detectable GW emission from blackhole formation is the formation of SMBHs. SMBHs exist. If they are formed in the collapse of an SMS, a number of observational sources should allow us to observe these objects with LISA (Figure 30).
5.2 Using GWs to study core collapse
Beyond the detection of GWs, GW observations of stellar collapse allow us to probe the mechanism behind corecollapse explosions. Along with neutrinos, GWs provide one of the few windows into the heart of the corecollapse supernova explosion mechanism. As we have already discussed, the detection (or lack thereof) of a GW signal can tell us something about the spin rate of the core. For instance, a strong signal from a supernova at the Virgo cluster would require barmode instabilities and, hence, a rotation rate in the core that is much faster than we currently believe. Likewise, a strong GRB GW signal would imply fragmentation in the disk that we, as yet, do not observe. This would lead to a major rethinking in the GRB engine.
Scientists also argue that the strength and form of the GW signal in a galactic supernova would allow us to distinguish between asymmetric and rotating collapse and determine the magnitude of rotation (or degree of asymmetry) in these collapse models (e.g., [107, 228, 297]). If we knew the theoretical signal unambiguously, scientists could determine the progenitor rate and degree of differential rotation [297]. Unfortunately, at this time, the range of results discussed in this review is too large for such a study; it points out the potential for GW to address very specific astronomy questions if collapse models reach some agreement on the exact GW signal.
But precision measurements in gravitational collapse have been hampered by the lack of true verification and validation studies of the existing codes. Although the amount of angular momentum in the core is critical to determining the exact wave signal, very few solid tests of angular momentum conservation (and artificial angularmomentum transport) have been made on most codes. This issue is even more important for threedimensional calculations. Until this source of error can be quantified and minimized, the errors in the GW signal will remain large, stagnating this field and limiting what we can learn from an observed signal.
The detection of GWs from weak or dim supernovae will provide clues to these rarelyobserved classes of supernovae. Dim supernovae can be produced both by the collapse of lowmass stars (electroncapture supernovae) or the collapse of massive stars that have considerable fallback. Models argue that these explosions are difficult to observe and such supernovae may occur nearby without detection in the optical. GWs may provide the only direct means to study these supernovae. Similarly, different GRB models predict very different GW signals and the GW detection of a GRB will place strong constraints on the current engine models.
Finally, detectors such as LIGO will build up a sample of merging systems consisting of neutron stars or black holes. From these detections, we can derive strong observational constraints on the birth mass distributions of both neutron stars and stellarmassed black holes. This set of observations will indirectly constrain the explosion mechanisms for core collapse.
Current studies have begun to probe the tip of the iceberg of what we will be able to study in core collapse with GWs and we expect the number of papers probing this topic to increase dramatically in the next few years.
Footnotes
 1.
Note that some stars in M = 100–500 M_{⊙} mass range may not collapse at all, but rather explode in a pairinstability supernovae [136].
 2.
Currently, there are major discussions on the details of the drive in the convection above the proto neutron star. Instabilities can be driven by the postbounceshock entropy profile, neutrino heating, or the standing accretion shock instability (SASI). Similarly, debate surrounds the magnitude of the entropy/lepton driven convection within the proto neutron star.
 3.
Müller & Janka [208] did not specifically describe what they used, but any standard derivative discretization would work, e.g., ∂Φ/∂r = (Φ^{k+1} − Φ^{k−1})/(r^{k+1} − r^{k−1}).
 4.
The term “proto black hole” is used to describe the large (above 50 M_{⊙}) core produced in the collapse of a massive star. Because of high entropies, the core does not collapse down to nuclear densities but collapses straight to a black hole, never producing a compact proto neutron star at nuclear densities.
 5.
Our models of these collapsing cores are also sensitive to the numerical modeling of the physics as well as uncertainties in the physics, both leading to the current variety in results between different simulations.
 6.
 7.
These different supernova types are thought to differ primarily based on the envelope left on the star at collapse, not based on a different collapse mechanism: type II supernovae still have a hydrogen envelope, whereas type Ib/c SNe have lost their hydrogen, and for the type Ic SNe, most of its helium, envelopes.
 8.
Fryer & Young [116] have pointed out that current simulations are likely to be underresolved and predict less vigorous convection than what should happen in nature: see also [41, 70]. In general, models of convective instabilities suggest that high resolution (above 512^{3} zones) across a convective cell is required to resolve convection accurately (e.g., [243]). If such constraints from the turbulence community are correct, the current gridbased simulations in core collapse (e.g., Marek & Janka [195]) remain underresolved.
 9.
 10.
The possible exception is extremely massive stars above ∼ 300 M_{⊙}. Fryer et al. [114, 217, 218] found that such stars may have enough entropy and angular momentum to alter the “bounce” density (and mass) considerably. These stars form proto black holes and collapse to a black hole shortly after the initial bounce with no supernova explosion whatsoever.
 11.
This “softening” of the equation of state has been studied for three decades [322].
 12.
Even the more pessimistic/realistic estimates of the rate of neutronstar mergers predict that such a detection will occur within a few years of operation.
Notes
Acknowledgements
It is a pleasure to thank John Blondin, Adam Burrows, Thierry Foglizzo, Scott Hughes, Kei Kotake, Paolo Mazzali, Ewald Müller, Ken’ichi Nomoto, Shangli Ou, Gabriel Rockefeller, and Stuart Shapiro for helpful conversations and/or permission to reprint figures/movies from their published works. This work was carried out in part under the auspices of the National Nuclear Security Administration of the Department of Energy (DOE) at Los Alamos National Laboratory and supported by a LANL/LDRD program.
Supplementary material
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