Gravitational wave signals from 3D neutrino hydrodynamics simulations of core-collapse supernovae

TLDR: In this article, a 3D multi-group neutrino hydrodynamics simulation of core-collapse supernovae of progenitors with 11.2 µun, 20 µun and 27 µun was presented.

Abstract: We present gravitational wave (GW) signal predictions from four 3D multi-group neutrino hydrodynamics simulations of core-collapse supernovae of progenitors with 11.2 Msun, 20 Msun, and 27 Msun. GW emission in the pre-explosion phase strongly depends on whether the post-shock flow is dominated by the standing accretion shock instability (SASI) or convection and differs considerably from 2D models. SASI activity produces a strong signal component below 250 Hz through asymmetric mass motions in the gain layer and a non-resonant coupling to the proto-neutron star (PNS). Both convection- and SASI-dominated models show GW emission above 250 Hz, but with considerably lower amplitudes than in 2D. This is due to a different excitation mechanism for high-frequency l=2 motions in the PNS surface, which are predominantly excited by PNS convection in 3D. Resonant excitation of high-frequency surface g-modes in 3D by mass motions in the gain layer is suppressed compared to 2D because of smaller downflow velocities and a lack of high-frequency variability in the downflows. In the exploding 20 Msun model, shock revival results in enhanced low-frequency emission due to a change of the preferred scale of the convective eddies in the PNS convection zone. Estimates of the expected excess power in two frequency bands suggests that second-generation detectors will only be able to detect very nearby events, but that third-generation detectors could distinguish SASI- and convection-dominated models at distances of ~10 kpc.

Figures

Figure 7. Frequency of strongest GW emission above 250Hz in the spectrogram of model s27 as a function of time (blue crosses). We also plot the expected characteristic frequency of GW emission excited by buoyancy effects in the PNS surface layer B and the overshooting layer A2 (red curve, see definition in equation 19). The exact value of N depends on the radius where equation (19) is evaluated, but we find similar numerical values within layers B and A2. Note that the trends seen for model s27 are common to all our models.

Figure 13. Squared Fourier amplitudes, in logarithmic scale, for the l = 1 and l = 2 components of the expansion of the shock position into spherical harmonics. The Fourier amplitudes have been calculated for the timewindow between 100 and 350ms after core bounce. The upper panel shows model s27 and the bottom panel shows the results for model s20. The curves in both panels have been normalized by the same factor.

Figure 1. GW amplitudes A+ and A× as functions of time after core bounce. From the top: s27, s20, s20s and s11, respectively. The two columns show the amplitudes for two different viewing angles: an observer situated along the z-axis (pole; left) and an other observer along the x-axis (equator; right) of the computational grid, respectively. Episodes of strong SASI activity occur between the vertical red lines; dashed and solid lines are used for model s27 to distinguish between two different SASI episodes.

Figure 5. Turbulent mass flux fm (blue curve) for model s27, calculated 192ms after core bounce. The shaded region indicates the convectively unstable region and the overshooting layer, which are lumped together as layer A (see Fig. 4).

Figure 6. Squared Fourier amplitudes of the total volume-integrated GW signal and of the signal contributions arising from the three different layers of the simulation volume. From the top: total signal (T), the PNS convective region and the overshooting layer (A), the PNS surface layer (B), and the volume between the PNS surface and the outer grid boundary (C). Top row: the left and right columns show the results for models s27 and s20, respectively. Bottom row: the left and right columns show the results for models s20s and s11.2. See Fig. 4 for a sketch of the three regions used for this analysis. The Fourier amplitudes are calculated according to equations (14), (4) and (5).

Figure 11. Amplitude spectrograms, for a sliding window of 50 ms, of the low-frequency GW signal arising from the three different layers, summed over the two polarization modes (|Ã+|2 + |Ã×|2). From the top: total signal (T), layers A, B and C. Columns are ordered by progenitor (left: s27, middle: s20, right: s20s). See Fig. 4 for a sketch of the three regions. The observer is chosen to be the north pole in the computational grid. As in Figs 1 and 3, strong SASI episodes are bracketed by vertical lines. The colour scale is logarithmic and all panels have been normalized by the same global factor that has also been used for Fig. 3.

Full text

Gravitational wave signals from 3D neutrino hydrodynamics
simulations of core-collapse supernovae
Andresen, H., Mueller, B., Mueller, E., & Janka, H-T. (2017). Gravitational wave signals from 3D neutrino
hydrodynamics simulations of core-collapse supernovae.
Monthly Notices of the Royal Astronomical Society
,
468
(2), 2032-2051. https://doi.org/10.1093/mnras/stx618
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MNRAS 468, 2032–2051 (2017) doi:10.1093/mnras/stx618
Advance Access publication 2017 March 14
Gravitational wave signals from 3D neutrino hydrodynamics simulations
of core-collapse supernovae
H. Andresen,
1,2‹
B. M
¨
uller,
3,4
E. M
¨
uller
1‹
and H.-Th. Janka
1
1
Max-Planck-Institut f
¨
ur Astrophysik, Karl-Schwarzschild-Str. 1, D-85748 Garching, Germany
2
Physik Department, Technische Universit
¨
at M
¨
unchen, James-Franck-Str. 1, D-85748 Garching, Germany
3
Astrophysics Research Centre, School of Mathematics and Physics, Queen’s University Belfast, Belfast BT7 1NN, UK
4
Monash Centre for Astrophysics, School of Physics and Astronomy, Building 79P, Monash University, VIC 3800, Australia
Accepted 2017 March 10. Received 2017 March 10; in original form 2016 July 15
ABSTRACT
We present gravitational wave (GW) signal predictions from four 3D multigroup neutrino
hydrodynamics simulations of core-collapse supernovae of progenitors with 11.2, 20 and
27 M

. GW emission in the pre-explosion phase strongly depends on whether the post-shock
flow is dominated by the standing accretion shock instability (SASI) or convection and differs
considerably from 2D models. SASI activity produces a strong signal component below 250 Hz
through asymmetric mass motions in the gain layer and a non-resonant coupling to the proto-
neutron star (PNS). Both convection- and SASI-dominated models show GW emission above
250 Hz, but with considerably lower amplitudes than in 2D. This is due to a different excitation
mechanism for high-frequency l = 2 motions in the PNS surface, which are predominantly
excited by PNS convection in 3D. Resonant excitation of high-frequency surface g modes
in 3D by mass motions in the gain layer is suppressed compared to 2D because of smaller
downflow velocities and a lack of high-frequency variability in the downflows. In the exploding
20 M

model, shock revival results in enhanced low-frequency emission due to a change of
the preferred scale of the convective eddies in the PNS convection zone. Estimates of the
expected excess power in two f requency bands suggest t hat second-generation detectors will
only be able to detect very nearby events, but that third-generation detectors could distinguish
SASI- and convection-dominated models at distances of ∼10 kpc.
Key words: gravitational waves – hydrodynamics – instabilities – supernovae: general.
1 INTRODUCTION
Despite impressive progress during recent years, the explosion
mechanism powering core-collapse supernovae is still not fully
understood. For ordinary supernovae with explosion energies up
to ∼10
51
erg, the prevailing theory is the delayed neutrino-driven
mechanism (see Janka 2012; Burrows 2013 for current reviews). In
this scenario, the shock wave formed during the rebound (bounce)
of the inner core initially stalls and only propagates out to a radius of
∼150 km. The energy needed to revitalize the shock is provided by
the partial re-absorption of neutrinos emitted from the proto-neutron
star (PNS).
Hydrodynamical instabilities operating behind the stalled shock
front have been found to be crucial for the success of this sce-
nario as they help to push the shock further out by generating
large Reynolds stresses (or ‘turbulent pressure’, see Burrows, Hayes

E-mail: haakoan@mpa-garching.mpg.de (HA);
ewald@MPA-Garching.MPG.DE (EM)
&Fryxell1995; Murphy, Dolence & Burrows 2013; Couch &
Ott 2015;M
¨
uller & Janka 2015) and transporting neutrino-heated
material out from the gain radius, which then allows the material
to be exposed to neutrino heating over a longer ‘dwell time’ (Buras
et al. 2006b; Murphy & Burrows 2008b). Moreover, if the instabili-
ties lead to the formation of sufficiently large high-entropy bubbles,
the buoyancy of these bubbles can become high enough to allow
them to rise and expand continuously (Thompson 2000; Dolence
et al. 2013;Fern
´
andez 2015).
Two such instabilities have been identified in simulations, namely
the more familiar phenomenon of convection driven by the unsta-
ble entropy gradient arising due to neutrino heating (Bethe 1990;
Herant et al. 1994;Burrows et al. 1995; Janka&M
¨
uller 1996;M
¨
uller
& Janka 1997), and the so-called standing accretion shock instabil-
ity (SASI), which manifests itself in large-scale sloshing and spiral
motions of the shock (Blondin, Mezzacappa & DeMarino 2003;
Blondin & Mezzacappa 2006; Ohnishi, Kotake & Yamada 2006;
Foglizzo et al. 2007, 2015;Ohnishietal.2008; Scheck et al. 2008;
Guilet & Foglizzo 2012). After initial setbacks in three-dimensional
(3D) supernova modelling, we are now starting to see the emergence
C

2017 The Authors
Published by Oxford University Press on behalf of the Royal Astronomical Society

Gravitational waves 2033
of the first generation of successful 3D simulations of explosions
with three-flavour multigroup neutrino transport, culminating in
the recent models of the Garching and Oak Ridge groups (Lentz
et al. 2015; Melson, Janka & Marek 2015a;Melsonetal.2015b)
with their rigorous treatment of the transport and neutrino micro-
physics in addition to many more obtained with more approximate
transport schemes, as for example the studies of Takiwaki, Kotake
&Suwa(2012, 2014), M
¨
uller (2015) and Roberts et al. (2016).
1
Our means to validate these numerical models by observations
are limited. Classical photon-based observations of supernovae and
their remnants (e.g. mixing in the envelope, see Wongwathanarat,
M
¨
uller & Janka 2015 and references therein; pulsar kicks, Scheck
et al. 2006; Wongwathanarat, Janka & M
¨
uller 2010; Nordhaus
et al. 2012; Wongwathanarat, Janka & M
¨
uller 2013) provide only in-
direct constraints on the workings of the hydrodynamic instabilities
in the inner engine of a supernova. For a nearby, Galactic super-
nova event, messengers from the core in the form of neutrinos and
gravitational waves (GWs) could furnish us with a direct glimpse
at the engine. Neutrinos, for example, could provide a smoking gun
for SASI activity through fast temporal variations (Marek, Janka
&M
¨
uller 2009; Lund et al. 2010; Brandt et al. 2011; Tamborra
et al. 2013, 2014a;M
¨
uller & Janka 2014) and could even allow
a time-dependent reconstruction of the shock trajectory (M
¨
uller &
Janka 2014).
Likewise, a detection of GWs could potentially help to unveil
the multidimensional effects operating in the core of a supernova.
The signal from the collapse and bounce of rapidly rotating iron
cores and triaxial instabilities in the early post-bounce phase has
long been studied in 2D (i.e. under the assumption of axisymme-
try) and 3D (e.g. Dimmelmeier et al. 2007, 2008; Ott et al. 2007;
Scheidegger et al. 2008; Abdikamalov et al. 2010). Understanding
the GW signal generated by convection and the SASI in the more
generic case of slow or negligible rotation has proved more difficult
due to a more stochastic nature of the signal. During the recent
years, however, a coherent picture of GW emission has emerged
from parametrized models (Murphy, Ott & Burrows 2009)and
first-principle simulations of supernova explosions in 2D (Marek,
Janka & M
¨
uller 2009;M
¨
uller, Janka & Marek 2013): the models
typically show an early, low-frequency signal with typical frequen-
cies of ∼100 Hz arising from shock oscillations that are seeded by
prompt convection (Marek et al. 2009; Murphy et al. 2009; Yakunin
et al. 2010;M
¨
uller et al. 2013; Yakunin et al. 2015). This signal
component is followed by a high-frequency signal with stochastic
amplitude modulations that is generated by forced oscillatory mo-
tions in the convectively stable neutron star surface layer (Marek
et al. 2009; Murphy et al. 2009;M
¨
uller et al. 2013) with typical
frequencies of 300. . . 1000 Hz that closely trace the Brunt–V
¨
ais
¨
ala
frequency in this region (M
¨
uller et al. 2013). Prior to the explosion,
these oscillations, tentatively identified as l = 2 surface g modes by
M
¨
uller et al. (2013), are primarily driven by the downflows imping-
1
Takiwaki et al. (2012, 2014) employ the isotropic diffusion source ap-
proximation (Liebend
¨
orfer, Whitehouse & Fischer 2009) and use further
approximations to treat heavy lepton neutrinos. Takiwaki et al. (2014)em-
ploy a leakage scheme to account for heavy lepton neutrinos and Takiwaki
et al. (2012) neglect the effect of these neutrinos altogether. M
¨
uller (2015)
utilizes the stationary fast multigroup transport scheme of M
¨
uller & Janka
(2015), which at high optical depths solves the Boltzmann equation in a
two-stream approximation and matches the solution to an analytic variable
Eddington factor closure at low optical depths. Roberts et al. (2016)employ
a full 3D two-moment (M1) solver in general relativistic simulations, but
ignore velocity-dependent terms.
ing on to the neutron star, whereas PNS convection takes over as
the forcing agent a few hundred milliseconds after shock revival as
accretion dies down. This high-frequency contribution dominates
the energy spectrum and the total energy emitted in GWs can reach
∼10
46
erg (M
¨
uller et al. 2013; Yakunin et al. 2015).
Since 3D supernova models have proved fundamentally different
to 2D models in many respects, it stands to reason that much of
what we have learned about GW emission from first-principle 2D
models will need to be revised. In 2D, the inverse turbulent cas-
cade (Kraichnan 1967) facilitates the emergence of large-scale flow
structures also in convectively dominated models and helps to in-
crease the kinetic energy in turbulent fluid motions in the post-shock
region (Hanke et al. 2012). Furthermore, accretion downflows im-
pact the PNS with much higher velocities in 2D than in 3D (Melson
et al. 2015a) due to the inverse turbulent cascade and the stronger in-
hibition of Kelvin–Helmholtz instabilities at the interface of super-
sonic accretion downflows (M
¨
uller 2015). In the SASI-dominated
regime, on the other hand, the additional dimension allows the
development of the spiral mode (Blondin & Mezzacappa 2007;
Blondin & Shaw 2007;Fern
´
andez 2010) in 3D, which can store
more non-radial kinetic energy than pure sloshing motions in 2D
(Hanke et al. 2013;Fern
´
andez 2015), contrary to earlier findings of
Iwakami et al. (2008). Such far-reaching differences between 2D
and 3D cannot fail to have a significant impact on the GW signal.
While the impact of 3D effects on the GW signals from the post-
bounce phase has been investigated before, all available studies
have relied on a rather approximate treatment of neutrino heating
and cooling such as simple light-bulb models (M
¨
uller & Janka 1997;
Kotake et al. 2009; Kotake, Iwakami-Nakano & Ohnishi 2011), grey
neutrino transport (Fryer, Holz & Hughes 2004;M
¨
uller, Janka &
Wongwathanarat 2012a) or a partial implementation of the isotropic
diffusion source approximation of Liebend
¨
orfer et al. (2009)inthe
works of Scheidegger et al. (2008, 2010), which were also limited
to the early post-bounce phase. Arguably, none of these previous
studies have as yet probed precisely the regimes encountered by the
best current 3D simulations (e.g. the emergence of a strong SASI
spiral mode) and therefore cannot be relied upon for quantitative
predictions of GW amplitudes and spectra, which are extremely
sensitive to the nature of hydrodynamic instabilities, the neutrino
heating and the contraction of the PNS.
In this paper, we present GW waveforms of the first few hun-
dred milliseconds of the post-bounce phase computed from 3D
models with multigroup neutrino transport. Waveforms have been
analysed for four supernova models of progenitors with zero-age
main sequence (ZAMS) masses of 11.2, 20 (for which we study an
exploding and a non-exploding simulation) and 27 M

. With four
simulations based on these three different progenitors, we cover
both the convective regime (11.2 M

) and the SASI-dominated
regime (20 and 27 M

). Our aim in studying waveforms from
these progenitors is twofold: on the one hand, we shall attempt to
unearth the underlying hydrodynamical phenomena responsible for
the GW emission in different regions of the frequency spectrum
during different phases of the evolution. We shall also compare the
GW emission in 3D and 2D models, which will further illuminate
dynamical differences between 2D and 3D. Furthermore, with 3D
models now at hand, we are in a position to better assess the de-
tectability of GWs from the post-bounce phase in present and future
instruments than with 2D models affected by the artificial constraint
of axisymmetry.
One of our key findings is that the GW signal from SASI-
dominated models is clearly differentiated from convection-
dominated model by strong emission in a low-frequency band
MNRAS 468, 2032–2051 (2017)

2034 H. Andresen et al.
around 100. . . 200 Hz. Very recently, Kuroda, Kotake & Takiwaki
(2016) also studied the GW signal features (in models using grey
neutrino transport) during phases of SASI activity for a 15 M

star,
comparing results for three different nuclear equations of state. Go-
ing beyond Kuroda et al. (2016), we clarify why this signature has
not been seen in 2D models and point out that the hydrodynamic
processes underlying this low-frequency signal are quite complex
and seem to require a coupling of SASI motions to deeper layers
inside the PNS. Moreover, we show that broad-band low-frequency
GW emission can also occur after the onset of the explosion and is
therefore not an unambiguous signature of the SASI. We also pro-
vide a more critical assessment of the detectability of this new signal
component, suggesting that it may only be detectable with second-
generation instruments like Advanced LIGO for a very nearby event
at a distance of 2 kpc or less.
Our paper is structured as follows: we first give a brief descrip-
tion of the numerical setup and the extraction of GWs in Section 2.
In Section 3, we present a short overview of the GW waveforms
and then analyse the hydrodynamical processes contributing to dif-
ferent parts of the spectrum in detail. We also compare our results
to recent studies based on 2D first-principle models. In Section 4,
we discuss the detectability of the predicted GW signal from our
three progenitors by Advanced LIGO (The LIGO Scientific Collab-
oration et al. 2015) and by the Einstein Telescope (Sathyaprakash
et al. 2012) as next-generation instrument. We also comment on pos-
sible inferences from a prospective GW detection. Our conclusions
and a summary of open questions for future research are presented
in Section 5.
2 SIMULATION SETUP
2.1 Numerical methods
The simulations were performed with the
PROMETHEUS-VERTEX code
(Rampp & Janka 2002; Buras et al. 2006a). The Newtonian hy-
drodynamics module
PROMETHEUS (Fryxell, Arnett & M
¨
uller 1991;
M
¨
uller, Fryxell & Arnett 1991) features a dimensionally split
implementation of the piecewise parabolic method of Colella &
Woodward (1984) in spherical polar coordinates (r, θ, ϕ). Self-
gravity is treated using the monopole approximation, and the ef-
fects of general relativity are accounted for in an approximate fash-
ion by means of a pseudo-relativistic effective potential (case A of
Marek et al. 2006). The neutrino transport module
VERTEX (Rampp
& Janka 2002) solves the energy-dependent two-moment equations
for three neutrino species (ν
e
,¯ν
e
, and a species ν
X
representing all
heavy flavour neutrinos) using a variable Eddington factor tech-
nique. The ‘ray-by-ray-plus’ approximation of Buras et al. (2006a)
is applied to make the multi-D transport problem tractable. In the
high-density regime, the nuclear equation of state (EoS) of Lattimer
& Swesty (1991) with a bulk incompressibility modulus of nuclear
matter of K = 220 MeV has been used in all cases.
2.2 Supernova models
2.2.1 3D models
We study four 3D models based on three solar-metallicity progenitor
stars with ZAMS masses of 11.2 (Woosley, Heger & Weaver 2002),
20 (Woosley & Heger 2007) and 27 M

(Woosley et al. 2002). An
initial grid resolution of 400 × 88 × 176 zones in r, θ and ϕ was
used for the 3D models, and more radial grid zones were added
during the simulations to maintain sufficient resolution around the
PNS surface. The innermost 10 km were simulated in spherical
symmetry to avoid excessive limitations on the time step when
applying a spherical polar grid.
(i) s11.2: model s11.2 (Tamborra et al. 2014b) is based on the
solar-metallicity 11.2 M

progenitor of Woosley et al. (2002). This
model exhibits transient shock expansion after the infall of the Si/O
shell interface, but falls slightly short of an explosive runaway. After
the average shock radius reaches a maximum of ≈250 km at a time
of ≈200 ms after bounce, the shock recedes and shock revival is not
achieved by the end of the simulation 352 ms after core bounce. The
post-shock region is dominated by buoyancy-driven convection;
because of the large shock radius no growth of the SASI is observed.
The convective bubbles remain of moderate scale: even during the
phase of strongest shock expansion around ∼200 ms after bounce
when the shock deformation is most pronounced and the kinetic
energy in convection motions reaches its peak value, the bubbles
subtend angles of no more than 60
◦
.
(ii) s20: model s20 is based on the 20 M

solar-metallicity pro-
genitor of Woosley & Heger (2007) and has been discussed in
greater detail in Tamborra et al. (2013, 2014a), where quasi-periodic
modulations of the neutrino emission were analysed and traced back
to SASI-induced variations of the mass-accretion flow to the PNS.
No explosion is observed by the end of the simulation 421 ms
post-bounce. There is an extended phase of strong SASI activity
(dominated by the spiral mode) between 120 and 280 ms after core
bounce. After a period of transient shock expansion following the
infall of the Si/O shell interface, SASI activity continues, but the ki-
netic energy in the SASI remains considerably smaller than during
its peak between 200 and 250 ms.
(iii) s20s: this model is based on the same 20 M

progenitor
as s20, but a non-zero contribution from strange quarks to the
axial-vector coupling constant, g
s
a
=−0.2, from neutral-current
neutrino–nucleon scattering was assumed (Melson et al. 2015b).
This modification of the neutrino interaction rates results in a suc-
cessful explosion (Melson et al. 2015b). Shock revival sets in around
300 ms after bounce. Prior to shock revival, the post-shock flow is
dominated by large-scale SASI sloshing motions between 120 and
280 ms post-bounce. By the end of the simulation 528 ms post-
bounce, the average shock radius is ≈1000 km, and a strong global
asymmetry stemming from earlier SASI activity remains imprinted
on to the post-shock flow. Asymmetric accretion on to the PNS
still continues, but the accretion rate is reduced by a factor of ∼2
compared to model s20.
(iv) s27: our most massive model is based on the 27 M

solar-
metallicity progenitor of Woosley et al. (2002) and has been dis-
cussed in greater detail in Hanke et al. (2013) and, for SASI-induced
neutrino emission variations, by Tamborra et al. (2013, 2014a).
Shock revival did not occur by the end of the simulation 575 ms
after bounce. There are two episodes of pronounced SASI activity
that are interrupted by a phase of transient shock expansion follow-
ing the infall of the Si/O interface. The first SASI phase takes place
between 120 and 260 ms post-bounce and the second period sets in
around 410 ms post-bounce and lasts until the end of the simulation.
2.2.2 2D models
In addition to the 3D models, we also analyse two 2D models based
on the same progenitor as s27.
(i) s27-2D: model s27-2D wassimulatedwiththesame numerical
setup as s27 (see Hanke et al. 2013), with an initial grid resolution
MNRAS 468, 2032–2051 (2017)

Gravitational waves 2035
of 400 × 88 zones in r and θ and the innermost 10 km being
simulated in spherical symmetry to allow for optimal comparison
with the 3D model. SASI activity sets in about 150 ms after core
bounce. Between 220 and 240 ms after bounce the accretion rate
drops significantly after the Si/O shell interface has crossed the
shock. The decreasing accretion rate leads to shock expansion, and
shock revival occurs around 300 ms post-bounce.
(ii) G27-2D: in order to compare our results to those of a rel-
ativistic 2D simulation of t he SASI-dominated s27 model, we
also reanalyse the 2D model G27-2D presented by M
¨
uller et al.
(2013), which was simulated with
COCONUT-VERTEX (M
¨
uller, Janka &
Dimmelmeier 2010).
COCONUT (Dimmelmeier, Font & M
¨
uller 2002;
Dimmelmeier et al. 2005) uses a directionally unsplit implementa-
tion of the piecewise parabolic method (with an approximate Rie-
mann solver) for general relativistic hydrodynamics in spherical
polar coordinates. The metric equations are solved in the extended
conformal flatness approximation (Cordero-Carri
´
on et al. 2009).
The model was simulated with an initial grid resolution of 400 ×128
zones in r and θ , with the innermost 1.6 km being simulated in spher-
ical symmetry to reduce time-step limitations. For consistency, we
recompute the GW amplitudes for this model based on the rela-
tivistic stress formula (appendix A of M
¨
uller et al. 2013) instead of
the time-integrated quadrupole formula with centred differences as
originally used by M
¨
uller et al. (2013). The stress formula leads to
somewhat larger amplitudes particularly at late times when central
differencing is no longer fully adequate due to the increasing signal
frequency.
The model is characterized by strong post-shock convection for the
first 50 ms after core bounce, which is followed by a phase of strong
SASI activity. Around 120 ms after core bounce the average shock
radius starts to increase steadily. The criterion for runaway shock
expansion is met approximately 180 ms after bounce and the shock
is successfully revived at ∼210 ms post bounce.
The evolution of models G27-2D and s27-2D differs significantly
during the pre-explosion phase: in G27-2D large-scale deforma-
tion of the shock already occurs ∼50 ms after bounce, without a
preceding phase of hot-bubble convection (M
¨
uller et al. 2013). s27-
2D develops SASI activity later: since the average shock radius is
∼20–30 km larger in model s27-2D than in model G27-2D, condi-
tions favour the development of neutrino-driven convection. Conse-
quently, s27-2D shows an initial phase of convection before SASI
activity sets in when the shock starts to retract ∼100–150 ms after
bounce. Due to the early development of SASI activity in model
G27-2D at a time when the accretion rate is high, particularly strong
supersonic downflows on to the PNS develop.
A possible reason for stronger and earlier SASI activity in model
G27-2D is that, in contrast to model s27-2D, model G27-2D exhibits
a phase of strong prompt post-shock convection (between a few ms
after bounce and about 50 ms after bounce), which leaves the shock
appreciably deformed with |a
1
|/a
0
∼ 0.01–0.02 as shown in fig. 7
of M
¨
uller, Janka & Heger (2012b). Therefore, the SASI amplitude
only needs to growby a factor of ∼30 to reach the non-linear regime.
In Hanke et al. (2013, fig. 2), the l = 1 amplitude is much smaller at
early times. Such differences in the post-bounce evolution can have
a variety of reasons. Besides the pure stochasticity of simulations,
the initial perturbations may also play a role: model G27-2D was
simulated in 2D from the onset of core collapse, while model s27-2D
was started from a spherical model with seed perturbations imposed
15 ms after core bounce. The presence or absence of strong prompt
post-shock convection also depends on the details of the entropy
and electron fraction profiles, which are determined by the exact
shock dynamics during the first milliseconds after core bounce.
Without a very careful analysis of all the differences between the two
simulations, we are not able to localize the origin of the differences
between model G27-2D and model s27-2D in the different gravity
treatment or any of the other aforementioned aspects.
Despite (or because of) the differences in the dynamics of the post-
shock flow to s27-2D, model G27-2D is useful for illustrating the
differences of the 3D model to the extreme end of the spectrum of
recent 2D models in terms of peak GW amplitude and illustrates the
mechanism of GW emission by stochastic surface g-mode excitation
due to overshooting plumes from the gain region (Marek et al. 2009;
Murphy et al. 2009;M
¨
uller et al. 2013) in the clearest form.
3 STRUCTURE AND ORIGIN OF THE
GRAVITATIONAL WAVE SIGNAL
The different 3D models used in our analysis probe distinctly dif-
ferent regimes that can be encountered in supernova cores. In this
section, we will investigate how these dynamical differences are
reflected in the GW signals. We also compare our 3D models to
the two 2D models and investigate how and why the GW signal
changes when going from 2D to 3D.
3.1 Gravitational wave extraction
In order to extract the GW signal from the hydrodynamical simula-
tions, we post-process our simulations using the quadrupole stress
formula (Finn 1989; Nakamura & Oohara 1989; Blanchet, Damour
& Schaefer 1990). Here, we only give a concise description of the
formalism and refer the reader to M
¨
uller et al. (2012a)
2
for a full
explanation.
In the transverse traceless (TT) gauge and the far-field limit, the
metric perturbation, h
TT
, can be expressed in terms of the amplitudes
of the two independent polarization modes in the following way,
h
TT
(X ,t) =
1
D
[
A
+
e
+
+ A
×
e
×
]
. (1)
Here, D denotes the distance between the source and the observer,
A
+
denotes the wave amplitude of the plus-polarized mode, A
×
is
the wave amplitude of the cross-polarized mode and e
×
and e
+
denote the unit polarization tensors. The unit polarization tensors
are given by
e
+
= e
θ
⊗ e
θ
− e
φ
⊗ e
φ
, (2)
e
×
= e
θ
⊗ e
φ
+ e
φ
⊗ e
θ
, (3)
where e
θ
and e
φ
are the unit vectors in the θ and φ direction of
a spherical coordinate system and ⊗ denotes the tensor product.
Using the quadrupole approximation in the slow-motion limit, the
amplitudes A
×
and A
+
can be computed from the second time deriva-
tive of the symmetric trace-free (STF) part of the mass quadrupole
tensor Q (Oohara, Nakamura & Shibata 1997),
A
+
=
¨
Q
θθ
−
¨
Q
φφ
, (4)
A
×
= 2
¨
Q
θφ
. (5)
The components of Q in the orthonormal basis associated with
spherical polar coordinates used in this formula can be obtained
2
Note, however,that the description of the formalism in M
¨
uller et al. (2012a)
contains some typos: their equation 24 is incomplete. The superscript TT is
missing from
¨
Q
ij
, as is also the case in equations (22) and (23), and, more
importantly, the trace term is missing.
MNRAS 468, 2032–2051 (2017)

Frequently asked questions

Q1. What have the authors contributed in "Gravitational wave signals from 3d neutrino hydrodynamics simulations of core-collapse supernovae" ?

The authors present gravitational wave ( GW ) signal predictions from four 3D multigroup neutrino hydrodynamics simulations of core-collapse supernovae of progenitors with 11. Estimates of the expected excess power in two frequency bands suggest that second-generation detectors will only be able to detect very nearby events, but that third-generation detectors could distinguish SASIand convection-dominated models at distances of ∼10 kpc. 

Q2. What is the effect of the forward turbulent cascade on the surface g-mode?

Braking of downflows by the forward turbulent cascade and fragmentation into smaller eddies strongly suppress surface g-mode excitation in 3D. 

Q3. Why does the model develop strong supersonic downflows?

Due to the early development of SASI activity in model G27-2D at a time when the accretion rate is high, particularly strong supersonic downflows on to the PNS develop. 

Q4. What is the effect of the resonant excitation of the l = 2 surface?

the spectrum of turbulent motions does not extend to high frequencies in 3D both in SASI-dominated and convection-dominated models so that the resonant excitation of the l = 2 surface g mode at its eigenfrequency becomes ineffective.(iii) 

Q5. What is the metric perturbation in the transverse traceless gauge?

In the transverse traceless (TT) gauge and the far-field limit, the metric perturbation, hTT, can be expressed in terms of the amplitudes of the two independent polarization modes in the following way,hTT(X, t) = 1 D [A+e+ + A×e×] . 

Q6. What is the reason for the weak low frequency emission component in model s11.2?

The lack of large-scale motions with a significant l = 2 component also explains the weak lowfrequency GW activity in model s11.2, where the post-shock flow is dominated by smaller convective bubbles and the kinetic energy in non-radial fluid motions is typically smaller than for the more massive progenitors. 

Q7. How long after the core bounce is the shock revived?

The criterion for runaway shock expansion is met approximately 180 ms after bounce and the shock is successfully revived at ∼210 ms post bounce. 

Q8. What is the l = 2 mode in the exploding model?

In the exploding model, the l = 2 mode is generally stronger than in the non-exploding model and it remains strong throughout the simulation in contrast to the non-exploding model, where the l = 2 mode decreases in strength after the SASI-dominated phase ends. 

Q9. What is the reason why the excitation of g modes in the surface layer is suppresse?

while the excitation of g modes in the surface layer is strongly suppressed in 3D, there is still some residual g-mode activity (Melson et al. 2015a; Müller 2015). 

Q10. What are the reasons why the excitation of oscillations in the PNS surface layer is?

There are presumably several reasons why the excitation of oscillations in the PNS surface layer is found to be more efficient in 2D than in 3D. 

Q11. What is the SNR for matched filtering?

Assuming an optimally orientated detector and a roughly isotropic frequency spectrum for different observer directions, the SNR for matched filtering is formally given by (Flanagan & Hughes 1998, cp. their equation 5.2 for the second form),(SNR)2 = 4 ∫ ∞0 df |h̃(f )|2 S(f )= ∫ ∞0 df h2c f 2S(f ) , (27)where hc = √ 2Gπ2c3D2 dEGW df (28)is the characteristic strain, S(f) is the power-spectral density of the detector noise as a function of frequency f and dEGW/df is the spectral energy density of the GWs. 

Q12. How do the authors calculate the SNR for the low frequency band?

Using equation (33), the authors obtain a detection threshold of SNRlow 11 for the low-frequency band and SNRhigh 15 for the high-frequency band assuming t = 0.5 s. Since the critical SNR depends weakly on t, these fiducial values can be used for all models. 

Q13. How do the authors compute the SNRs for the low frequency band?

In order to better assess the detectability and possible inferences from the signal structure, the authors compute SNRs quantifying the excess power in a low-frequency band (SNRlow for 20 Hz ≤ f < 250 Hz, i.e. δf = 230 Hz) and a high-frequency band (SNRhigh for 250 Hz ≤ f < 1200 Hz, i.e. δf = 950 Hz). 

Q14. What is the main contribution to the high-frequency signal?

The decomposition of the integration volume into three layers (left-hand panel of Fig. 9) reveals that the main contribution to the high-frequency signal stems from the PNS surface (layer B).