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Gravitational Waves from Core-Collapse Supernovae

Gravitational waves are a potential direct probe for the multi-dimensional flow during the first second of core-collapse supernova explosions. Here we outline the structure of the predicted gravitational wave signal from neutrino-driven supernovae of non-rotating progenitors from recent 2D and 3D simulations. We sketch some quantitative dependencies that govern the amplitudes of this signal and its evolution in the time-frequency domain.


Introduction
Core-collapse supernovae [1] are the explosions of stars that begin their life with a mass of 8M ⊙ . Such stars go through successive nuclear burning stages, synthesizing progressively heavier elements up to the iron group. Eventually, they build up a degenerate iron core at the centre. As the core comes close to its effective Chandrasekhar mass, electron capture reactions and photodisintegration of heavy nuclei trigger its collapse. After reaching supranuclear densities, the core rebounds, and a shock wave is launched into the infalling outer shells. The shock initially stalls, and is later revived on a time scale of hundreds of milliseconds, likely by neutrino heating, or in rare cases of rapidly rotating progenitors by magnetohydrodynamic effects. Hours later, the supernova outburst becomes visible across the electromagnetic spectrum when the shock reaches the stellar surface.
Both observations and theory have shown that core-collapse supernova explosions typically exhibit strong asymmetries [2]. In slowly spinning progenitors, these asymmetries arise naturally already during the first second as neutrino heating drives convective overturn behind the shock [3,4] or from the large-scale "standing accretion shock instability" (SASI) [5]. In fact, these instabilities are thought to play an important supporting role in the neutrino-driven mechanism. Bipolar asymmetry also develops naturally in magnetorotationally-driven explosions [6,7].
The electromagnetic signatures from corecollapse supernovae are determined at later times, and thus only provide indirect clues about the multi-dimensional flow during the operation of the supernova "engine" for the first second(s) after collapse. Gravitational waves (GWs) potentially provide a more direct way to probe the dynamics in the supernova core. Several works have already addressed the potential to discriminate between different supernova mechanisms or at least facilitate GW detection by using predicted waveforms from simulations in conjunction with sophisticated data analysis methods [8][9][10][11]; and this prompts the question whether a prospective GW detection could reveal even more of the physics at play in supernova cores. Focusing on GW emission by convection and the SASI in neutrino-driven explosions, we here review the predicted GW signatures and sketch some of the physics that determines important signal properties.

Gravitational Wave Emission in Core-Collapse Supernovae
The link between non-spherical fluid flow in the supernova core and GW emission is provided by the Einstein quadrupole formula, which relates (to first order in the far-field limit) the transverse-traceless components h TT ij of the metric perturbations at a distance D from the source to the symmetric trace-free (STF) component of the second time derivative of the mass quadrupole moment Q ij of a system. Dimensional analysis shows that the GW amplitude scales with the kinetic energy E kin of a system and an overlap factor α Q that quantifies the overlap with the quadrupole component that enters in Equation (1),

WAVES 2017, Minneapolis
Historically, the classical scenario for GW emission in supernovae has been that of the collapse and bounce of a rotating iron core (see [12,13] for reviews). Here, α Q is non-zero from the outset due to the rotational deformation of the core. As the shape of the bounce signal is highly generic [14], the amplitude and frequency (which is set by the fundamental quadrupolar oscillation mode of the young proto-neutron star (PNS) [15]) of the signal are suitable probes for inferring the bulk parameters of the collapsing iron core: For a Galactic supernova from a rapidly rotating progenitor, the angular momentum of the core may be measured within 20% accuracy with Advanced LIGO [16].

The Stochastic Signal from SASI and Convection
Even in the absence of rotation, aspherical mass motions in the neutrino heating layer (or "gain layer") also lead to GW emission in the later post-bounce phase due to temporal variations in the mass quadrupole moment. Recent 2D and 3D simulations [17][18][19][20][21][22] typically show several distinct phases: Shock ringing after prompt convection leads to a low-frequency signal around 100 Hz for about 50 ms, followed by a signal at several hundred Hz with strong stochastic amplitude modulations, and possibly a "tail" due to asymmetric shock expansion in the explosion phase. There are still few 3D models, which show lower amplitudes by a factor of ∼10 [19,22], but the 3D models have also revealed a new distinct low-frequency component from the SASI at 100-200 Hz [22,23]. Inferring physics from the GW signal of nonrotating progenitors is less straightforward because of its stochastic character. Without a clear relation to physical parameters of the accretion flow onto the PNS, it is also not obvious what range of variations should be expected in the waveforms for different supernova progenitors.

The Signal in the Time-Frequency Domain
However, a closer analysis of the waveforms in the time-frequency domain and of the hydrodynamical processes behind GW emission reveal a more ordered picture as we shall sketch here for the high-frequency component of the signal. 2D simulations showed that this signal compo-nent originates from the deceleration of downflows at the interface between the heating and cooling region [18]. This can be understood as the stochastic excitation of an ℓ = 2 surface gmode, which leaves a clearly defined narrow frequency band in wavelet spectrograms [20]. The time-dependent frequency f g of the GW emission from this surface g-mode can be related to the PNS mass M and radius R, to the electron antineutrino mean energy Eν e (which is a potentially measurable proxy for the PNS surface temperature), and the nucleon mass m n as including relativistic correction terms [20]. In 3D, the picture is modified [22]: The excitation of the surface g-mode is less efficient because downflows in the heating region are strongly decelerated before striking the PNS surface and (in contrast to the 2D case) lack the rapid time variability needed for resonant excitation of the surface g-mode. The high-frequency emission still follows f g , but GW emission is now mainly due to the overshooting of plumes from the convective region inside the PNS. Thus, there is a still a well-defined structure in the time-frequency domain, but the distance-independent quadrupole amplitudes A = hD are only 1-5 cm, i.e. much smaller than in 2D. Mass motions in the gain region instead excite low-frequency PNS surface oscillations non-resonantly.

Estimate of Gravitational Wave Amplitudes
Understanding the GW amplitudes due to the excitation of PNS surface oscillations by mass motions in the gain region or PNS convection zone is less straightforward. At present, we can only formulate a crude physical model that roughly explains the amplitudes found in simulations and makes their dependence on the parameters of the accretion flow transparent without aiming for quantitative accuracy.
Using the stress formula of [24], the GW amplitude from oscillatory motions in the PNS surface layer can be estimated from the quadrupole component ρ 2 of the density perturbations in this region, and dimensional analysis suggests a relation to the potential energy E g stored in the surface g-mode: We expect the energy flux L g of excited waves in the convectively stable region region to scale with the convective luminosity L conv and Mach number Ma in the adjacent convective region [25]. For stochastic excitation of the surface gmode, coherence of the forcing holds roughly over one turnover time scale τ so that we can relate E g to the kinetic energy E conv in convection in the heating region or inside the PNS as follows, where we have allowed for an additional factor α 1 quantifying the overlap of the forcing with the spatial dependence and frequency of the ℓ = 2 mode visible in GWs.
For excitation by convection in the gain region, we can estimate the maximum amplitude A max around the onset of the explosion by expressing E conv in terms of the mass M gain of the gain region and the typical convective velocity. If we express the post-shock sound speed c s in terms of the shock radius r sh as c s = (GM/3r sh ) 1/2 [26], we find (6) Relating the mass in the gain region to the explosion energy E expl via the typical residual recombination energy ǫ rec per baryon [27] of around 5-6 MeV [28], we further obtain or by using r sh ≈ 200 km and a typical value of Ma 2 = 0.3 at shock revival [29]: This estimate yields values in the ballpark of the GW emission from non-resonant excitation of low-frequency surface oscillations by convection and SASI in the 3D models of [22]. It qualitatively reproduces the trend towards stronger GW signals from more energetic explosions seen in 2D models [20].
The explosion energy is likely the dominant parameter in Equation (7), and there is little reason to expect that the other parameters are strongly anticorrelated to it. With both observations of Type IIP supernovae [30] to a large spread in explosion energies in ordinary (likely neutrino-powered) supernovae by more than an order of magnitude, we can expect at least a corresponding spread in wave amplitudes. This suggests there is still room for somewhat stronger 3D signals from SASI and convection in the gain region than those predicted by [22].
By contrast, the work of [22] suggests that variations in the GW signal from PNS convection stem primarily from the overlap parameter α since one does not expect strong variations in E conv in this region. Estimates of the kinetic energy E conv of convective motions in this region can be formed based on dimensional analysis along the lines of mixing-length theory. Equating the convective luminosity to the diffusive core neutrino luminosity L core yields in terms of the typical mass ∆M ≈ 1M ⊙ and width ∆R ≈ 10 km of the PNS convection zone during the pre-explosion phase. L core does not vary considerably across progenitors as reflected by modest variations in the (individual) luminosity L µ/τ of the heavy flavour species [31], which is a good proxy for the core luminosity (L core ≈ 6L µ/τ ). With a typical value of L core = 10 53 erg s −1 and Ma = 0.05, we obtain A ∼ 1 cm × α, again roughly in line with [22].

Outlook
Here we have attempted to qualitatively outline some of the relevant physical dependencies that govern the structure of the GW signal from convection and SASI during the accretion phase in non-rotating supernovae, underscoring recent findings that the predicted GW signal from non-rotating neutrino-driven supernovae contains significant physical information despite its stochastic character. The identification of clearly defined structures of the GW signal in the time-frequency domain has seen further progress than can be discussed here in detail with the identification of SASI activity via GW spectrograms [22,23] and of a host of modes in the PNS in long-time simulations of potential collapsar progenitors [32]. It now remains to be seen whether these findings can be exploited to infer physics from the prospective detection of GWs from a supernovae, or a least facilitate such a detection with the help of advancements in data analysis techniques.