GRhydro

1
PRL 94, 131101 (2005)
Luca Baiotti
Albert-Einstein-Institut, Golm, Germany
In collaboration with
L. Rezzolla, E. Schnetter, I. Hawke
2
Plan of the talk

• Whisky a general, 3D, finite-difference,
  GR-hydro code
• Waveforms from 3D rotating neutron-star collapse
  to a Kerr black hole
• Conclusions and future work

3
The EU-network and the need of a Whisky
Over the last three and a half years, also funded
through the European Network on Sources of
Gravitational Waves, we have developed Whisky a
3D, finite-difference, parallel code in Cartesian
coordinates, solving the relativistic
hydrodynamics equations on a generic and
time-varying spacetime.
Whisky works within the framework of Cactus
(www.cactuscode.org) from which it inherits
parallelization, input/output infrastructures,
portability on different platforms and other
facilities.
Cactus
Whisky
4
Highlights of the Whisky code

• High-Resolution Shock-Capturing techniques.
• These guarantee very-high-accuracy correct
  treatment of shocks, with very small numerical
  dissipation
• and require the equations to be written in
  conservative form so the primitive variables
  are transformed into the conserved variables
• Method of Lines.
• Any stable ODE integrator can be used and the
  control of the truncation error is transparent
  the coupling between different treatments in the
  hydro and spacetime is minimized
• Excision of matter and fields.

5
Excising parts of the spacetime with singularities
The region of spacetime inside a horizon (yellow
region) is causally disconnected from the outside
(blue region).
apparent horizon
So a region inside a horizon may be excised from
the numerical domain. This is successfully done
in pure spacetime evolutions since the work of
Nadëzhin Novikov Polnarev (1978). Our group PRD
71, 104006, (2005) and other groups have
recently shown that it can be done also in
non-vacuum simulations.
In practice, the actual excision region is
alegosphere (black region) and is placed well
inside the apparent horizon (which is found at
every time step) and is allowed to move on the
grid.
6
Highlights of the Whisky code

• High-Resolution Shock-Capturing techniques.
• These guarantee very-high-accuracy correct
  treatment of shocks, with very small numerical
  dissipation
• and require the equations to be written in
  conservative form so the primitive variables
  are transformed into the conserved variables
• Method of Lines.
• Any stable ODE integrator can be used and the
  control of the truncation error is transparent
  the coupling between different treatments in the
  hydro and spacetime is minimized
• Excision of matter and fields.
• Fixed or Progressive Mesh Refinement.

7
Carpet progressive fixed mesh-refinement
Initial meshes
Final meshes

• Carpet is an important technical achievement
  (developed by E. Schnetter CQG 21, 1465
  (2004)), which has removed the limitation of
  using uniform 3D grids.
• Carpet is an infrastru-cture that allows for
  fixed refined meshes to be set up. While the
  meshes are fixed in location, they can be
  activated during the evolution. We have used 7
  levels of refinement in the present simulation.

4 levels
7 levels
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3D collapse to a black hole
The gravitational collapse is probably among the
most common phenomena in astrophysics and
cosmology and still remains among the most
challenging processes to be investigated in full
General Relativity.

• Important milestone calculations have been those
  of
• May and White (67) 1D, Lagrangian
  hydrodynamics, artificial viscosity
• Stark and Piran (85) 2D, Eulerian
  hydrodynamics, artificial viscosity
• Shibata and Shibata, Shapiro Baumgarte
  (00-05), 2D or 3D, Eulerian hydrodynamics
  artificial viscosity (or HRSC in 2D)

• However, important questions remained with
  approximate answers
• How massive is the formed black hole?
• How rapidly is it spinning?
• How much gravitational radiation is produced?
• What waveforms are expected?

9
Initial Data uniformly rotating polytropes
We have built sequences of uniformly rotating
polytropes with constant angular momentum and
evolved them with
or
D1 M 1.67 M? rc 3.3x1014 g/cm3 M/R
0.22 rp/re 0.95 J/M 2 0.21 T/W 0.012
D4 M 1.86 M? rc 1.8x1014 g/cm3 M/R
0.19 rp/re 0.65 J/M 2 0.54
0.077
with
With this EoS, the fastest possible spinning star
has aJ/M2 0.54
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Wave-extraction techniques
There exist several ways of extracting
gravitational waves from relativity codes.
We use a gauge-invariant extraction of
perturbations of a Schwarzschild spacetime.
Observers placed on 2-spheres at rE decompose
the metric into tensor spherical-harmonics to
calculate the odd and even-parity perturbations
Qlm of a Schwarzschild black hole.
All of this is repeated on several nested
2-spheres.
The functions Qlm are then used to reconstruct
the asymptotic gravitational-wave quantities h,
h?, E, J, etc.
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Results with uniform grids
The same but for Q40 .

Note that the high-order modes converge more
rapidly
Asymptotic form for Q20 from the simulation
discussed before (typical unigrid extraction)
Re /M5.1
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Waveforms at last!

• Overlapping of the waveforms extracted at
  different distances
• Even further boundary location makes no
  additional improvement.

Most of the signal is in the lowest multipole
Q20
Stark Piran (1985)
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Waveforms at last!
Once computed, the different multipoles Qlm and
Q?lm can be combined to obtain the asymptotic
gravitational-wave amplitudes in the TT-gauge (we
use up to lm5).

• h h? (the collapse is essentially
  axisymmetric)
• the computed amplitudes are one order of
  magnitude smaller than those of Stark Piran
  (pressure is fundamental)

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Frequencies
There is consistency with the expected
frequencies.
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More waveforms
Waves from initial models with different angular
momenta.
Waves from initial models with different pressure
depletions.
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GW amplitude and energetics
Following Thorne (1987) the characteristic
frequency is defined as
with h(f) the FFT of the waveform and Sh(f) the
PSD of the instrument. The characteristic
amplitude is then
These are the numbers we find for a star rotating
at the mass-shedding limit and collapsing at 10
kpc
M/M?
M/M?
LIGO1
VIRGO
NOTE h(f) has maxima at 6 kHz (l2) and at 14
kHz (l4)
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Energy losses
From the wave amplitudes we calculate the energy
lost to gravitational waves as
whose integral yields
DE 1.5x10-6 M/M?
Smaller efficiency than calculated by Stark
Piran (i.e. DE10-4 -10-5 M/M?), but larger than
that obtained in core-collapse to proto-neutron
stars (i.e. DE10-9 M/M?) (Müller et al, ApJ 603,
221, 2004).
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Detectability
Again, for a star uniformly rotating near the
mass-shedding limit at 10kpc, the signal-to-noise
ratio is
0.25 for VIRGO/LIGO I 4 for Advanced
NOTE A significantly stronger
gravitational-wave emission is expected form the
collapse of initially differentially rotating
models, with supra-Kerr angular-momentum
parameter (a1).
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Conclusions

• After having built and tested Whisky, our
  parallel, 3D code for hydrodynamical simulations
  in full General Relativity, implementing HRSC
  techniques, excision of singularities, method of
  lines, progressive mesh refinement
• with Whisky we have been able to produce the
  most accurate description of NS collapse to
  rotating black holes and the first calculations
  of gravitational-wave emission in 3D.
• The detectability of the gravitational-wave
  signal from these sources is promising (though
  not the most promising).
• Intense work is in progress on different source
  modelling (differential rotation, realistic EoSs)
  for the collapse and on different systems
  (binaries) in both of which cases even stronger
  gravitational-wave emission is expected.