The gravitational wave burst signal from core collapse of rotating stars
We present results from detailed general relativistic simulations of stellar core collapse to a proto-neutron star, using two different microphysical nonzero-temperature nuclear equations of state as well as an approximate description of deleptonization during the collapse phase. Investigating a wide variety of rotation rates and profiles as well as masses of the progenitor stars and both equations of state, we confirm in this very general setup the recent finding that a generic gravitational wave burst signal is associated with core bounce, already known as type I in the literature. The previously suggested type II (or “multiple-bounce”) waveform morphology does not occur. Despite this reduction to a single waveform type, we demonstrate that it is still possible to constrain the progenitor and postbounce rotation based on a combination of the maximum signal amplitude and the peak frequency of the emitted gravitational wave burst. Our models include to sufficient accuracy the currently known necessary physics for the collapse and bounce phase of core-collapse supernovae, yielding accurate and reliable gravitational wave signal templates for gravitational wave data analysis. In addition, we assess the possiblity of nonaxisymmetric instabilities in rotating nascent proto-neutron stars. We find strong evidence that in an iron core-collapse event the postbounce core cannot reach sufficiently rapid rotation to become subject to a classical bar-mode instability. However, many of our postbounce core models exhibit sufficiently rapid and differential rotation to become subject to the recently discovered dynamical instability at low rotation rates.
pacs:04.25.D-, 04.30.Db, 97.60.Bw, 02.70.Bf, 02.70.Hm
The final event in the life of a massive star is the catastrophic collapse of its central, electron-degenerate core composed of iron-peak nuclei. When silicon shell burning pushes the iron core over its effective Chandrasekhar mass, collapse is initiated by a combination of electron capture and photo-disintegration of heavy nuclei, both leading to a depletion of central pressure support. Massive stars in the approximate mass range of about to solar masses () experience such a collapse phase until their homologously contracting (1); (2) inner core reaches densities near and above nuclear saturation density where the nuclear equation of state (EoS) stiffens, leading to an almost instantaneous rebound of the inner core (core bounce) into the still supersonically infalling outer core. The hydrodynamic supernova shock is born, travels outward in radius and mass, but rapidly loses its kinetic energy to the dissociation of infalling iron-group nuclei and to neutrinos that deleptonize the immediate postshock material and stream off from these regions quasi-freely. The shock stalls, turns into an accretion shock and must be revived to produce the observable explosion associated with a core-collapse supernova. Mechanisms of shock revival are still under debate (a recent review is presented in (3), but see also (4); (5); (6)) and may involve heating of the postshock region by neutrinos, multi-dimensional hydrodynamic instabilities of the accretion shock, in the postshock region, and/or in the proto-neutron star, rotation, magnetic fields, and nuclear burning. If the shock is not revived, black-hole formation (on a timescale of (7)) is inevitable and the stellar collapse event may remain undetected by conventional astronomy or, perhaps, appear as a gamma-ray burst if the progenitor star has a compact enough envelope and sufficiently rapid rotation in its central regions (8); (9).
Conventional astronomy can constrain core-collapse supernova theory and the supernova explosion mechanism via secondary observables only, e.g., the explosion energy, ejecta morphology, nucleosynthesis yields, residue neutron star or black hole mass and proper motion, and pulsar magnetic fields. Neutrinos and gravitational waves, on the other hand, are emitted deep inside the supernova core and travel to observers on Earth practically unscathed by intervening material. They can act as messengers to provide first-hand and live dynamical information on the intricate multi-dimensional dynamics of the proto-neutron star and postshock region and may constrain directly the core-collapse supernova mechanism. Importantly, core-collapse events that do not produce the canonical observational astronomical signature or whose observational display is shrouded from view can still be observed in neutrinos and gravitational waves if occurring sufficiently close to Earth.
Gravitational waves, in contrast to neutrinos, have not yet been observed directly, but an international array of gravitational wave observatories (see, e.g., (10)) is active and taking data. Since gravitational waves from astrophysical sources are expected to be weak, their detection is notoriously difficult and involves extensive signal processing and detailed analysis of the detector output. Chances for the detection of an astrophysical event of gravitational wave emission are significantly enhanced if accurate theoretical knowledge of the expected gravitational wave signature from such an event is at hand.
Theoretical predictions of the gravitational wave signature from a core-collapse supernova are complicated, since the emission mechanisms are very diverse. While the prospective gravitational wave burst signal from the collapse, bounce, and the very early postbounce phase is present only when the core rotates (11); (12); (13); (14); (15); (16); (17); (18), gravitational wave signals with sizeable amplitudes can also be expected from convective motions at postbounce times, instabilities of the standing accretion shock, anisotropic neutrino emission, excitation of various oscillations in the proto-neutron star, or nonaxisymmetric rotational instabilities (19); (20); (21); (22); (23); (17).
In the observational search for gravitational waves from merging black hole or neutron star binaries, powerful data analysis algorithms such as matched filtering are applicable, as the waveform from the inspiral phase can be modeled with high accuracy (see, e.g., (24)) and gravitational wave data analysts already have access to robust template waveforms that depend only on a limited number of macroscopic parameters. In contrast, the complete gravitational wave signature of a core-collapse supernova cannot be predicted with template-level accuracy as the postbounce dynamics involve chaotic processes (turbulence, [magneto-] hydrodynamic instabilities) that are sensitive not only to a multitude of precollapse parameters, but also to small-scale perturbations of any of the hydrodynamic variables.
While the complete supernova gravitational wave signature may remain inaccessible to template-based data analysis, a number of individual constituent emission processes, in particular those involving coherent global bulk dynamics and/or rotation, allow, in principle, for accurate and robust waveform predictions that may be applied to template-based searches. Rotating core collapse and core bounce as well as pulsations or nonaxisymmetric rotational deformations of a proto-neutron star constitute this group of processes. Among them, rotating collapse and bounce is the historically most extensively studied case (see, e.g., (25) for a historical review) and may be the most promising for becoming robustly predictable in its gravitational wave emission. Yet, to date, the gravitational wave signal from rotating stellar core collapse and bounce has not been predicted with the desired accuracy and robustness.
These deficiencies of previous simulations result from the fact that the physically realistic modeling of core collapse requires a general relativistic description of consistently coupled gravity and hydrodynamics in conjunction with a microphysical treatment of the sub- and supernuclear EoS, electron capture on heavy nuclei and free protons, and neutrino radiation transport. Only very few multi-dimensional general relativistic codes have recently begun to approach these requirements (17); (18). In addition, the properties of the EoS around and above nuclear density are not very well constrained by theory or experiments. The same applies to the rotation rate and angular velocity profile of the progenitor core, which are also not directly accessible by observation and very difficult to model numerically in stellar evolution codes. Furthermore, variations with progenitor structure and mass are to be expected. Therefore, the influence of rotation and progenitor structure on the collapse and bounce dynamics and thus the gravitational wave burst signal must be investigated by extensive and computationally expensive parameter studies.
Previous parameter studies have considered a large variety of rotation rates and progenitor core configurations, but generally ignored important microphysical aspects and/or the influence of general relativity. Mönchmeyer et al. (12) performed axisymmetric Newtonian calculations with progenitor models from stellar evolutionary studies. They employed the microphysical nuclear EoS of Hillebrandt and Wolff (26) and included deleptonization via a neutrino leakage scheme and electron capture on free protons. Capture on heavy nuclei was neglected, which resulted in a too high electron fraction at core bounce and a consequently overestimated inner core mass (2); (27). In that study a limited set of four calculations was computed and two qualitatively and quantitatively different types of gravitational wave burst signals were identified. Their morphology can be classified alongside with the collapse and bounce dynamics: Type I signals are emitted when the collapse of the quasi-homologously contracting inner core is not strongly influenced by rotation, but stopped by a pressure-dominated bounce due to the stiffening of the EoS near nuclear density , where the adiabatic index rises above . This leads to a bounce with a maximum core density . Type II signals occur when centrifugal forces, which grow during contraction owing to angular momentum conservation, are sufficiently strong to halt the collapse, resulting in consecutive (typically multiple) centrifugal bounces with intermediate coherent re-expansion of the inner core, seen as density drops by sometimes more than an order of magnitude; thus here after bounce. Type I and II dynamics and waveforms were also found in the more recent Newtonian studies by Kotake et al. (15), who employed a more complete leakage/capture scheme, but still obtained too high at bounce, and by Ott et al. (16), who performed an extensive parameter study and for the first time also considered variations in progenitor star structure, but neglected deleptonization during collapse.
Zwerger and Müller (13) carried out an extensive two-dimensional Newtonian study of rotating collapse of idealized polytropes in rotational equilibrium (28) with a simplified hybrid EoS, consisting of a polytropic and a thermal component (29). Electron capture during collapse was mimicked by an instantaneous lowering of the adiabatic index from its initial value of to trigger the onset of collapse. At , the adiabatic index was raised to to qualitatively model the stiffening of the nuclear EoS. Zwerger and Müller also obtained the previously suggested signal types and introduced type III signals that appear in a pressure-dominated bounce when the inner core has a very small mass due to very efficient electron capture (approximated in (13) via a in their hybrid EoS). Obergaulinger et al. (30) also employed the hybrid EoS, but included magnetic fields. They introduced the additional dynamics/signal type IV, which occurs only in the case of very strong precollapse core magnetization. They found that weak to moderate core magnetization in agreement with predictions from stellar evolution theory (see, e.g., (31)) has little effect on the collapse and bounce dynamics and the resulting gravitational wave signal. This finding is in agreement with (32) (see also (5); (33)), where magneto-rotational collapse simulations were performed, a smaller model set was considered, but the neutrino leakage scheme of (15) was employed and it was made use of two different microphysical EoSs to study the EoS dependence of the collapse dynamics and gravitational wave signal.
The first extensive set of general relativistic simulations of rotating iron core collapse to a proto-neutron star were presented by Dimmelmeier et al. (14), who employed an analytic hybrid EoS and polytropic precollapse models in rotational equilibrium as initial data (but see also the pioneering early work of (34)). These simulations were subsequently confirmed in (35); (36); (25); (37). Dimmelmeier et al. studied a subset of the models in (13) in the same parameter space of rotation rate and degree of differential rotation, and found that general relativistic effects counteract centrifugal support and shift the occurrence of type II dynamics and wave signals to a higher precollapse rotation rate at a fixed degree of differential rotation.
Recently, new general relativistic simulations of rotating core collapse in two and three dimensions were carried out by Ott et al. (25); (17); (38) who included the microphysical EoS of Shen et al. (39), precollapse models from stellar evolutionary calculations as well as an approximate deleptonization scheme (40). The results of these calculations indicate that the gravitational wave burst signal associated with rotating core collapse is exclusively of type I. In addition, the simulations showed that rotating stellar iron cores stay axisymmetric throughout collapse and bounce, and only at postbounce times develop nonaxisymmetric features.
In a general relativistic two-dimensional follow-up study, Dimmelmeier et al. (18); (41) considerably extended the number of models and comprehensively explored a wide parameter space of precollapse rotational configurations. Even for this more general setup they found gravitational wave signals solely of type I form, although for rapid precollapse rotation some of their models experience a core bounce due to centrifugal forces only, which however is always a single centrifugal bounce rather than the multiple ones observed in earlier work (see, e.g., (13); (14); (16)). They identified the physical conditions that lead to the emergence of this generic gravitational wave signal type and quantified their relative influence. These results strongly suggest that the waveform of the gravitational wave burst signal from the collapse of rotating iron cores in a core-collapse event is much more generic than previously anticipated.
In this work, we extend the above study of the gravitational wave signal from rotating core collapse and consider not only variations in the precollapse rotational configuration, but also in progenitor structure and nuclear EoS. In this way, we carry out the to-date largest and most complete parameter study of rotating stellar core collapse that includes all the (known) necessary physics to produce reliable predictions of the gravitational wave signal associated with rotating collapse and bounce. All our computed gravitational wave signals are made available to the detector data analysis community in a freely accessible waveform catalog (42).
We perform a large number of two-dimensional simulations with our general relativistic core-collapse code CoCoNuT and employ , , , and (masses at zero-age main sequence) precollapse stellar models from the stellar evolutionary studies of Heger et al. (43); (31). In addition to the EoS by Shen et al. (39) used in our previous studies, we also calculate models with the EoS by Lattimer and Swesty (44). We describe in detail and explain comprehensively the qualitative and quantitative aspects of the collapse and bounce dynamics and the resultant gravitational wave signal. We lay out the individual effects of general relativity, deleptonization, precollapse stellar structure and rotational configuration, and nuclear EoS on the gravitational wave signature from rotating core collapse. We study the prospects for nonaxisymmetric rotational instabilities in our postbounce cores, which could lead to an enhancement of the gravitational wave signature. Furthermore, we set our model gravitational radiation waveforms in context with present and future detector technology and assess their detectability.
This paper is organized as follows: In Section II we introduce our treatment of the general relativistic spacetime curvature and hydrodynamics equations. Furthermore, we introduce our variants of the two microphysical EoS we employ, the scheme for deleptonization and neutrino pressure contributions, our precollapse model set, and the gravitational wave extraction technique employed. Section III discusses the numerical methods used in the CoCoNuT code and the computational grid setup for the simulations presented in this paper. In Section IV we present the collapse dynamics and waveform morphology of our simulated models, while in Section V we investigate the stratification of the postbounce core and its impact on the gravitational wave signal. The detection prospects for the gravitational wave burst from core bounce are discussed in Section VI, while the rotational configuration of the proto-neutron star and its susceptibility to nonaxisymmetric rotational instabilities are examined in Section VII. Finally, in Section VIII, we summarize and discuss our results.
Throughout the paper we use a spacelike signature and units in which . Greek indices run from 0 to 3, Latin indices from 1 to 3, and we adopt the standard Einstein summation convention.
Ii Physical model and equations
ii.1 General relativistic hydrodynamics
We adopt the Arnowitt–Deser–Misner (ADM) formalism of general relativity to foliate the spacetime endowed with a four-metric into spacelike hypersurfaces (45). In this approach the line element reads
where is the lapse function, is the shift vector, and is the spatial three-metric induced in each hypersurface.
The hydrodynamic evolution of a perfect fluid in general relativity with four-velocity , rest-mass current , and stress-energy tensor is determined by a system of local conservation equations,
where denotes the covariant derivative with respect to the four-metric. Here is the rest-mass density, is the specific enthalpy, is the fluid pressure, and the three-velocity with respect to an Eulerian observer moving orthogonally to the spacelike hypersurfaces is given by . We define a set of conserved variables as
In the above expressions is the Lorentz factor, which satisfies the relation .
Here , and and are the determinant of and , respectively, with . are the four-Christoffel symbols. Since we use a microphysical EoS that requires information on the local electron fraction per baryon , we add an advection equation for the quantity to the standard form of the conservation equations (4). The radiation stress due to the neutrino pressure (as defined in Section II.4), is included in the form of an additive term in the source of both the momentum and energy equations. Note also that here we use an analytically equivalent reformulation of the energy source term in contrast to the one presented in (14).
ii.2 Metric equations in the conformal flatness approximation
Using the ADM formalism, the Einstein equations split into a coupled set of first-order evolution equations for the three-metric and the extrinsic curvature ,
and constraint equations,
In the above equations, is the covariant derivative with respect to the three-metric , is the three-Ricci tensor and is the scalar three-curvature. The projection of the stress-energy tensor onto the spatial hypersurface is , the ADM energy density is given by , and is the momentum density. In addition, we have chosen the maximal slicing condition for which the trace of the extrinsic curvature vanishes: .
In order to simplify the ADM metric equations and to ameliorate the stability properties when numerically solving those equations, we employ the conformal flatness condition (CFC) introduced in (47) and first used in a pseudo-evolutionary context in (48). In this approximation the spatial three-metric is replaced by the conformally flat three-metric, , where is the flat-space metric and is the conformal factor. Then the metric equations (8–11) reduce to a set of elliptic equations for , , and ,
where and are the Laplace and covariant derivative operators associated with the flat three-metric, and . The CFC metric equations (12–14) do not contain explicit time derivatives, and thus the metric components are evaluated in a fully constrained approach.
Imposing CFC in a spherically symmetric spacetime is equivalent to solving the exact Einstein equations. For nonspherical configurations the CFC approximation may be roughly regarded as full general relativity without the dynamical degrees of freedom of the gravitational field that correspond to the gravitational wave content (49). However, even spacetimes that do not contain gravitational waves can be not conformally flat. A prime example are the spacetime of a Kerr black hole (50) or rotating fluids in equilibrium. For rapidly rotating models of stationary neutron stars the deviation of certain metric components from conformal flatness has been shown to reach up to in extreme cases (51), while the oscillation frequencies of such models typically deviate even less from the corresponding values obtained in full general relativistic simulations (52). In the context of rotating stellar core collapse the excellent quality of the CFC approximation has been demonstrated extensively (35); (36); (17).
Due to its fully constrained nature, the CFC approximation permits a straightforward and numerically more robust implementation of the metric equations in coordinate systems containing coordinate singularities (e.g., spherical polar coordinates) compared to a Cauchy free-evolution scheme. Furthermore, by definition it allows no constraint violations, which is a significant benefit in cases where a perturbation is added to the initial data. More details on the CFC equations can be found in, e.g., (14).
ii.3 Equations of state
In our simulations we employ two tabulated nonzero-temperature equations of state, the one by Shen et al. (39); (53) (Shen EoS), and the one by Lattimer and Swesty (44) (LS EoS). The LS EoS is based on a compressible liquid-drop model (54). The transition from inhomogeneous to homogeneous matter is established by a Maxwell construction, and the nucleon-nucleon interactions are expressed by a Skyrme force. In our version of this EoS, the incompressibility modulus of bulk nuclear matter is taken to be and the symmetry energy parameter has a value of . In contrast, the Shen EoS is based on a relativistic mean field model and is extended with the Thomas–Fermi approximation to describe the homogeneous phase of matter as well as the inhomogeneous matter composition. The parameter for the incompressibility of nuclear matter is and the symmetry energy has a value of .
Both EoSs employed in this study are the same as in Marek et al. (55) and include contributions of baryons, electrons, positrons, and photons. Furthermore, in this study the LS EoS has been extended to densities below by a smooth transition to the Shen EoS which is tabulated down to .
The microphysical EoS returns the fluid pressure (and additional thermodynamic quantities) as a function of , where is the temperature. Since the hydrodynamic equations (4) operate on the specific internal energy , we determine the corresponding temperature iteratively with a Newton–Raphson scheme and the EoS table. All interpolations are carried out in tri-linear fashion and the tables are sufficiently densely spaced to lead to an artificial entropy increase in an adiabatic collapse by not more than .
ii.4 Deleptonization and neutrino pressure
Electron capture on free protons and heavy nuclei during collapse reduces (i.e., “deleptonizes” the collapsing core) and consequently decreases the size of the homologously collapsing inner core that depends on the average value of in a roughly quadratic way (see, e.g., (56)). The material of the inner core is in sonic contact and determines the dynamics and the gravitational wave signal at core bounce and in the early postbounce phases. Hence, deleptonization has a direct influence on the collapse dynamics and the gravitational wave signal, and thus it is essential to include deleptonization during collapse.
Since multi-dimensional radiation-hydrodynamics calculations in general relativity are not yet computationally feasible, in our simulations we make use of a recently proposed approximative scheme (40) where deleptonization is parametrized based on data from detailed spherically symmetric calculations with Boltzmann neutrino transport, for which (as in (18)) we take the latest available electron capture rates (57). Following the main assumption in (40) that the local electron fraction for each fluid element during the contraction phase can be modeled rather accurately by a dependence on the density only, these simulations yield a universal relation . Furthermore, we find that this relation varies only slightly with progenitor mass, as shown in Fig. 1, where models with identical progenitor but different EoS have the same color, but different hues (e.g., dark green versus light green for the s20 progenitor). Consequently, we utilize the progenitor to create such a profile for each of the two EoSs. This profile is then used to correct the value of obtained from the advection by an amount
after each time integration step. This procedure assures that approaches the phenomenological input profile with the constraint that must be negative. Accordingly, in order to model the entropy loss by neutrinos escaping the collapsing core, for densities below an adopted neutrino trapping density the internal specific energy is re-adjusted at constant and such that the specific entropy per baryon is changed by
where is an average escape energy for the neutrinos, is the Boltzmann constant and where , , and are the proton, neutron, and electron chemical potentials, respectively. Note that when equilibrium between neutrinos and matter (i.e., -equilibrium) is established, this balance requires for the neutrino chemical potential .
We stop deleptonization at the time of core bounce (i.e., as soon as the specific entropy per baryon exceeds at the outer boundary of the inner core). After core bounce, for lack of a simple yet accurate approximation scheme for treating the further deleptonization in the nascent proto-neutron star, we advect only passively according to the conservation equation (4), although this effectively prevents the factual cooling and contraction of the proto-neutron star.
In all collapse phases, however, as in (40) we approximate the pressure contribution of the neutrinos by that of an ideal Fermi gas,
with being the Fermi–Dirac function of order 3. The neutrino pressure is included only in the regime which is optically thick to neutrinos, which we define for densities above .
ii.5 Initial models
All presupernova stellar models available to-date are end products of Newtonian spherically-symmetric stellar evolutionary calculations from hydrogen burning on the main sequence to the onset of core collapse by photo-dissociation of heavy nuclei and electron captures (see, e.g., (58)). Here, we employ various nonrotating models of (58) with zero-age main sequence masses (core model s11.2, here for simplicity labeled s11), (core model s15), (core model s20), and (core model s40). Recently, the first presupernova models that include rotation in a one-dimensional approximate fashion have become available (43); (31), and of these we select ones with (models e15a and e15b) as well as (core models e20a and e20b). All progenitors have solar-metallicity (at zero-age main sequence) and we generate our initial models by taking the data obtained from stellar evolution out to a radius where the density drops to a value that equals of the initial precollapse central density . Selected quantities that describe the properties of these stellar cores are summarized in Table 1.
We set those cores that are initially nonrotating (core models s11, s15, s20, and s40) artificially into rotation according to the rotation law specified in (28), where the specific angular momentum is given by
Here the length parametrizes the degree of differential rotation (stronger differentiality with decreasing ) and is the precollapse value of the angular velocity at the center. In the Newtonian limit, this reduces to
with being the distance to the rotation axis.
In order to determine the influence of different angular momentum distributions on the collapse dynamics, we parameterize the precollapse rotation of our models in terms of (A1: , almost uniform; A2: , moderately differential; A3: , strongly differential) and . The model nomenclature for the precollapse rotation parameters is shown in Table 2. We have selected the rotational configuration of the models in such a way that for the s20 progenitor they are a representative subset of the models investigated in (18); (41). They reflect different properties of the collapse dynamics and the gravitational radiation waveform discussed in that work, namely pressure-dominated bounce with or without significant postbounce convective overturn as well as single centrifugal bounce.
|core model||||||[%]||core model||||||[%]||core model||||||[%]|
Note that models with the same rotation specification (but different progenitor mass or EoS) have an identical angular velocity profile, while the precollapse rotation rate , which is the precollapse ratio of rotational energy to gravitational energy, varies. We have decided to compare models with identical initial angular velocity and not precollapse rotation rate , as the latter quantity is rather sensitive to the chosen core radius in the case of (almost) uniform rotation.
The models that are based on progenitor calculations including rotation (core models e15a, e15b, e20a, and e20b) are mapped onto our computational grids under the assumption of constant rotation on cylindrical shells of constant distance to the rotation axis. We also point out that due to the one-dimensional nature, none of the considered models are in rotational equilibrium. Still, in slowly rotating initial models this effect is small and thus negligible. For more rapidly rotating models, which exhibits the strongest deviation from rotational equilibrium, the collapse proceeds more slowly due to stabilizing centrifugal forces, and hence the star has more time for the adjustment to the appropriate angular stratifications for its rate of rotation.
In this study, we focus on the collapse of massive presupernova iron cores with at most moderate differential rotation and precollapse rotation rates that except for the most slowly rotating models lead to proto-neutron stars that are probably spinning too fast to yield spin periods of cold neutron stars in agreement with observationally inferred injection periods of young pulsars into the diagram (31); (59). However, they may be highly relevant in the collapsar-type gamma-ray burst scenario (59); (60); (9).
ii.6 Gravitational wave extraction
We employ the Newtonian quadrupole formula in the first-moment of momentum density formulation as discussed, e.g., in (14); (61); (62) to extract the gravitational waves generated by nonspherical accelerated fluid motions. It yields the quadrupole wave amplitude as the lowest order term in a multipole expansion of the radiation field into pure-spin tensor harmonics (63). The wave amplitude is related to the dimensionless gravitational wave strain in the equatorial plane by
where is the distance to the emitting source.
We point out that although the quadrupole formula is not gauge invariant and is only valid in the Newtonian slow-motion limit, for gravitational waves emitted by pulsations of rotating NSs (i.e., in astrophysical situations comparable to collapsing stellar cores at bounce in terms of compactness and rotation rates) it yields results that agree very well in phase and to in amplitude with more sophisticated methods (61); (64).
In order to assess the prospects for detection by current and planned interferometer detectors and to specifically address the issue of detection range and expected event rates, we calculate the dimensionless characteristic gravitational wave strain of the signal according to (65). We first perform a Fourier transform of the gravitational wave strain ,
To obtain the (detector dependent) integrated characteristic signal frequency
and the integrated characteristic signal strain
the power spectral density of the detector is needed (with ). We approximate the average over randomly distributed angles by , assuming optimal orientation of the interferometer detector. From Eqs. (22, 23) the signal-to-noise ratio can be computed as , where is the value of the rms strain noise (i.e., the theoretical sensitivity window) for the detector.
Iii Numerical methods
We perform all simulations using the CoCoNuT code described in detail in (14); (62). The equations of general relativistic hydrodynamics are solved in semi-discrete fashion. The spatial discretization is performed by means of a high-resolution shock-capturing (HRSC) scheme employing a second-order accurate finite-volume discretization. We make use of the Harten–Lax–van Leer–Einfeldt (HLLE) flux formula for the local Riemann problems and piecewise-parabolic reconstruction (PPM) of the primitive variables at cell interfaces. For a review of such methods in general relativistic hydrodynamics, see (66). The time integration and coupling with curvature are carried out with the method of lines (67) in combination with a second-order accurate explicit Runge–Kutta scheme. Once the state vector is updated in time, the primitive variables are recovered from the conserved ones given in Eq. (3) through an iterative Newton–Raphson method. Note that the component associated to in the system (4) of hydrodynamic evolution equations is treated as a passive advection equation, which does not contribute to the characteristic structure in the form of eigenvalues and eigenvectors required by some flux solvers.
To numerically solve the metric equations we utilize an iterative nonlinear solver based on spectral methods. The spectral grid of the metric solver is split into 6 radial domains with 33 radial and 17 angular collocation points each. The combination of HRSC methods for the hydrodynamics and spectral methods for the metric equations (the Mariage des Maillages or “grid wedding” approach) in a multidimensional numerical code has been presented in detail in (62). Even when using spectral methods the calculation of the spacetime metric from the system (12–14) of elliptic equations is computationally expensive. Hence, in our simulations the metric is updated only once every 100/10/50 hydrodynamic time steps before/during/after core bounce, and extrapolated in between. The numerical adequacy of this procedure has been tested and discussed in detail in (14).
In this study we focus on the gravitational wave signal associated with core bounce. As demonstrated by (17); (25), effects that may break rotational symmetry are most likely unimportant in this context. Hence, we assume axisymmetry and in addition impose symmetry with respect to the equatorial plane.
The CoCoNuT code utilizes Eulerian spherical coordinates , and for the computational grid we choose 250 logarithmically-spaced, centrally-condensed radial zones with a central resolution of and 45 equidistant angular zones covering . A small part of the grid is covered by an artificial low-density atmosphere extending beyond the core’s outer boundary defined where .
We also note that we have performed extensive resolution tests with different grid resolutions to ascertain that the grid setup specified above is appropriate for our simulations.
Iv Collapse dynamics and waveform morphology
iv.1 Generic waveform type
We begin our discussion with an analysis of the gravitational radiation waveform emitted during core bounce as an indicator for the influence of the EoS, the progenitor structure, and the precollapse rotational configuration on the collapse and bounce dynamics. In Fig. 2, we present example waveforms for representative collapsing cores selected from the investigated parameter space of models (i.e., less or more massive progenitors with slow or rapid precollapse rotation, varying degree of differential rotation, and using either the Shen EoS or LS EoS). The waveforms of all models are of type I, hence exhibit a positive prebounce rise and a large negative peak, followed by a ring-down. In the light of a considerably extended parameter space in terms of EoS and progenitor mass of the rotating core collapse models investigated in this work, we thus confirm the observation presented in (17); (38); (18); (41) that in general relativistic gravity all models with microphysics exhibit gravitational wave burst signals of type I.
For a slowly rotating core, prompt convective overturn at early postbounce times after the pressure-dominated bounce adds a significant low-frequency contribution to the regular ring-down signal (see, e.g., model s11A1O07).
In the case of moderately rapid rotation which still leads to a pressure-dominated bounce, this convection is effectively suppressed due to the growing influence of angular momentum gradients (68); (33) and does not strongly stand out in the postbounce ring-down signal (see, e.g., models s11A3O13, s15A2O05, s20A2O09, s40A1O07, or e15a).
If rotation is sufficiently rapid, the core bounces at subnuclear or only slightly supernuclear densities due to the increased effects of centrifugal forces, reflected by a significant widening of the bounce peak of the waveform and an overall lower frequency of the signal (see, e.g., models s40A3O13 or e20b).
Fig. 2 also demonstrates that for comparable precollapse rotational configuration (as specified by the parameters and ) the impact of the EoS on the collapse dynamics and, hence, the gravitational wave signal is small. In Table 3 we mark each model with its type of collapse dynamics, and in Fig. 3 we encode that type in the parameter space spanned by rotational configuration, progenitor mass/model, and EoS.
For our set of collapse models, only in four cases (models s11A1O13, s15A1O07, s20A1O09, and s40A1O05) the LS EoS yields a signal with dominant convective contribution while the Shen EoS does not, and only a single model (s15A2O15) changes its collapse type from a centrifugal bounce to a pressure-dominated bounce when replacing the Shen EoS with the LS EoS. However, Fig. 3 shows that the transition between the three different collapse and waveform subtypes occurs at different precollapse rotational configurations for the various progenitor masses. This is a consequence of differences in the mass of the inner core at bounce as discussed in Section IV.3.
The growth of the strong prompt early postbounce convection in slowly rotating models depends sensitively on the seed perturbations resulting from the numerical scheme/grid. In nature, prompt convection will be influenced by random small-scale to large-scale variations in the final stages of silicon burning that are frozen in during collapse. We point out that the duration of the prompt postbounce convection is most likely overestimated in our approach, since in full postbounce radiation-hydrodynamics calculations, energy deposition by neutrinos in the immediate postshock region rapidly smoothes out the negative entropy gradient left behind by the shock (see, e.g., (20); (69)) and quickly damps this early convective instability.