Unambiguous determination of gravitational waveforms from binary black hole mergers.
ABSTRACT Gravitational radiation is properly defined only at future null infinity (J+), but in practice it is estimated from data calculated at a finite radius. We have used characteristic extraction to calculate gravitational radiation at J+ for the inspiral and merger of two equal-mass nonspinning black holes. Thus we have determined the first unambiguous merger waveforms for this problem. The implementation is general purpose and can be applied to calculate the gravitational radiation, at J+, given data at a finite radius calculated in another computation.
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arXiv:0907.2637v3 [gr-qc] 8 Jan 2010
Unambiguous determination of gravitational waveforms from binary black hole mergers
C. Reisswig,1N. T. Bishop,2,3D. Pollney,1and B. Szil´ agyi4
1Max-Planck-Institut f¨ ur Gravitationsphysik, Albert-Einstein-Institut, 14476 Golm, Germany
2Department of Mathematics, Rhodes University, Grahamstown 6140, South Africa
3Department of Mathematical Sciences, University of South Africa, Unisa 0003, South Africa
4Theoretical Astrophysics, California Institute of Technology, Pasadena, CA 91125, USA
Gravitational radiation isproperly defined only at future null infinity (J+), but in practice it is estimated from
data calculated at a finite radius. We have used characteristic extraction to calculate gravitational radiation at
J+for the inspiral and merger of two equal mass non-spinning black holes. Thus we have determined the first
unambiguous merger waveforms for this problem. The implementation is general purpose, and can be applied
to calculate the gravitational radiation, at J+, given data at a finite radius calculated in another computation.
PACS numbers: 04.25.dg, 04.30.Db, 04.20.Ha, 04.30.Nk
diation from black hole merger events has attracted consider-
able attention, since the pioneeringworkby Smarrand collab-
orators [1–3]. With the advent of ground-based laser interfer-
ometric gravitational wave detectors, as well as the prospect
of the Laser InterferometerSpace Antenna (LISA), interest in
the problem has considerably increased. The measurement of
gravitational waves will soon provide an important probe of
strong-field nonlinear gravity, the domain of many fundamen-
tal questions in astrophysics. The sensitivity of LISA, and of
the upcoming advanced ground-based detectors AdLIGO and
AdVirgo, is so high that even an error in the waveform cal-
culation of 0.1% (in a sense made precise later) could lead to
an incorrect interpretation of the astrophysical properties of a
source, or of a test of general relativity. Nowadays, there are
several codes that can produce a stable and convergent sim-
ulation of a black hole spacetime. However, a particular dif-
ficulty with measuring gravitational radiation arises from the
fact that, in general relativity, it cannot be defined locally but
is defined only at future null infinity (J+), which, physically,
is thelimit that is approachedbyradiationmovingat thespeed
of light away from an isolated source. Since numerical evolu-
tions are normallycarriedout onfinite domains,there is a sys-
tematic error caused by estimating the gravitational radiation
from fields on a worldtube at finite radius and the uncertainty
in how it relates to measurement at J+. Even if this er-
roris small, theexpectedsensitivity of AdLIGO,AdVirgoand
LISA implies that it is important to obtain an accurate result.
A rigorous formalism for the global measurement of grav-
itational energy at null infinity has been in place since the
pioneering work of Bondi, Penrose and collaborators in the
1960s [5, 6]; and subsequently, techniques for calculating
gravitational radiation at J+have been developed. The idea
which we pursue here is to combine a Cauchy or “3 + 1” nu-
merical relativity code with a characteristic code . Given
astrophysical initial data, such a method has only discretiza-
tion error , and a complete mathematical specification has
been developed. There have been efforts to implementthis
method, often called Cauchy characteristic extraction (CCE),
or characteristic extraction [10, 11]. Previous work has con-
sidered test problems rather than that of the inspiral and
merger of two black holes or neutron stars. Also, earlier ef-
forts have combined the Cauchy and characteristic algorithms
within the same code.
Here, we describe the implementation of a CCE code as
well as results obtained from the code for the inspiral and
merger waveform of two equal mass, non-spinning, black
holes. The waveforms are calculated at J+, and are thus
the first unambiguous waveforms which have been obtained
for this problem, in the sense of being free of gauge or finite-
radius effects. Further, the code is generalpurpose, in that it is
independent of the details of the Cauchy code, requiring only
that it prescribes the required geometrical data on a world-
tube. Thus its applicationto other astrophysicalproblems will
For the specific problem of a binary black hole (BBH)
merger, we show that the waveform obtained at J+contains
only numerical error and is gauge-invariant. We demonstrate
second-order convergence to zero in the amplitude and phase
differencesbetween two CCE runs usingboundarydata at dif-
ferent radii. We comparethe waveformobtainedat J+with a
finite-radius extrapolated waveform, and find that the correc-
tions introducedby CCE are visible in the groundbaseddetec-
tors AdLIGO and AdVirgo, as well as the space-based LISA
Cauchy Evolution. The scenario we envision is an isolated
system (perturbed single body, or gravitationally bound bi-
nary), in a region on which the Einstein (and possibly hydro-
dynamic) equations must be solved. A standard procedurefor
doingthis is to formulatethe equationsas an initial-boundary-
value, or Cauchy, problem, in which data for the 3-metric and
its embedding is prescribed at a given time on a closed re-
gion of the spacetime, Σt. These are evolved according to
the Einstein equations on the interior of the domain, and ar-
tificial conditions on the timelike boundary, ∂Σt. The first
stable evolutions of a binary black hole system were carried
out by Pretorius . Two approaches to the evolution of
the interior equations are in use: (a) the harmonic formu-
lation of the Einstein equations with excised black hole in-
teriors [12, 13]; (b) the BSSNOK (see  and references
therein) evolution system with the black holes specified as
moving “punctures” [15, 16].
For the Cauchy evolutions used here, we have followed the
latter approach using the formulation outlined in . The
spacetime is discretised using finite differences on Cartesian
grids and Berger-Oliger mesh refinement in the neighbour-
hood of the black holes . The wave zone is discretised by
six overlapping coordinate patches with spherical topology.
Interior boundary data between adjacent patches are commu-
nicated by interpolation [19, 20].
The outer boundarycondition on the exterior of the domain
is given by a linear outgoing wave condition on each of the
evolved tensor components. Importantly, through the use of
spherical grids in the wave zone, a sufficient resolution can
be maintained even to a distant outer boundary, reducing the
effect of grid reflections common in mesh-refinement codes.
The size of the evolution domain is chosen according to the
amount of time required, T, and the location of the outer-
most measurement sphere, ri. Since physical as well as con-
straint violating modes propagate with the speed of light, an
outer boundary located at r∂Σt> T + riensures that mea-
surements are causally disconnected from the influence of the
is based on the mathematical prescription given in , and
here we provide only an outline. The process is illustrated
schematically in Fig. 1. Within a Cauchy simulation that uses
Minkowski-likecoordinates(t,x,y,z), wedefinea worldtube
Γ by x2+ y2+ z2= r2
βiand the 3-metric γijon Γ, as well as their first time and ra-
dial derivatives. This data is then decomposed into spherical
harmonics. The Cauchy code writes this spherical harmonic
coefficient data to file, and later the CCE code postprocesses
the data to reconstruct the 4-metric on the inner world tube.
In this way, the CCE code is general purpose, as it runs inde-
pendently of whatever Cauchy code was used to generate the
The CCE code defines angular coordinates φAas well as
a time coordinate u (=t) on Γ, and constructs outgoing null
geodesics with affine parameter λ.
Cauchy 4-metric to (u,λ,φA) coordinates, and calculates a
formation to (u,rS,φA) coordinates, in order to obtain the
Bondi-Sachs metric data in a neighbourhood of Γ. This pro-
vides the inner boundary data for the characteristic evolution,
usinga Cactus  implementationof the PITTnullevolution
code with square stereographic coordinate patches . The
characteristic code uses coordinates based on outgoing null
cones, and so the equations remain regular when the radial
coordinate is compactified (by rS → z = rS/(rS+ rSΓ)),
and in this way J+is included on the computational grid.
The code computes the gravitational radiation at J+as the
Weyl componentψ4. The coordinate-independentquantityψ4
is commonlyused in numericalrelativity; in appropriatecoor-
dinates, it is the second time derivative of the strain measured
by a detector.
Binary black hole evolution. We have carried out fully rela-
tivistic evolutionsof an equal-mass non-spinningbinaryblack
hole inspiral and merger. The initial data parameters for the
Newtonian evolution from large separation in order to deter-
mine the momentafor low-eccentricity(quasi-circular)trajec-
tories . The subsequent full nonlinear numerical relativity
evolution proceeds for approximately 8 orbits (1350M), fol-
Our implementation of CCE
Γ, and compute the lapse α, the shift
It then transforms the
u = const.
FIG.1: Schematicof theCCEalgorithmwithtwospatial dimensions
suppressed. Spacelike slices, Σtareevolved according totheCauchy
evolution scheme (horizontal lines). Geometrical data is recorded on
a world-tube, Γ, which is used as interior boundary data for a char-
acteristic evolution along u = constant null surfaces, transporting
the data to J+. The outer boundary of the Cauchy domain, ∂Σt is
chosen so that it is causally disconnected from Γ over the course of
lowed by the merger and ringdown lasting another 100M.
The evolutions have been carried out at two different grid
resolutionsin orderto verifythe convergenceof the numerical
scheme. The grid settings for the Cauchy code are: The cen-
tral Cartesian grid consists of 6 levels of 2:1 mesh-refinement,
with coarse grid spacings of h = 0.96M and h = 0.64M,
respectively. A grid of spherical topology covers the far field,
r ∈ [35M,3600M]; so that, during the time period of in-
terest, the outer boundary is causally disconnected from any
extraction sphere (see Fig. 1). The radial spacing is commen-
surate with the coarse Cartesian grid at the interface, and (for
reasons of efficiency) is gradually scaled to h = 3.84M and
h = 2.56M at the outer boundary for the two runs. We use a
corresponding Nang = 21 and Nang = 31 points in each of
the angular directions per patch.
Characteristic boundary data were interpolated onto world-
tubes located at r = 100M and r = 200M, and stored in the
form of spherical harmonic coefficients, up to ℓ = 8, which
was found to be the highest resolved mode. The resolutions
of the characteristic evolutions are set up according to the re-
spective resolutions of the Cauchy run. We use Nr = 321
and Nr= 481 radial points, with Nang= 51 and Nang= 76
angular points per angular patch. The dominant ℓ = 2, m = 2
modeof the gravitationalwaveformresulting fromthe numer-
ical evolution is plotted in Fig. 2, to be described in more de-
Invariancewithrespect tothe worldtubelocationis demon-
strated in Fig 3. We have considered the differences between
waveforms at J+resulting from two independent character-
istic evolutions using boundary data at rΓ= 100M and rΓ=
200M, respectively, and for two resolutions, h = 0.96M
and h = 0.64M. The difference between the results should
be entirely due to the discretisation error, and indeed this is
what we find. The differences converge to zero with approx-
imately second-order accuracy, as expected for the null evo-
FIG.2: Inspiral, merger and ringdown phase of ψ4(ℓ = 2,m = 2) as
obtained from finite radius extrapolation (red) and at J+(blue). The
waveforms are aligned at their peaks. There isa maximum difference
of 1.08% in the amplitude and a dephasing of 0.019 radians between
the two waves. These differences can introduce systematic errors to
parameter estimation of events detected at high SNR.
FIG. 3: Differences ∆ψ4 in the amplitude of ψ4(ℓ = 2,m = 2)
between two characteristic runs using boundary data from RΓ =
100M and RΓ = 200M. The red curve shows the difference at res-
olution h = 0.96M while the blue curve shows the difference for
h = 0.64M, scaled so as to line up for second-order convergence.
The expected second order convergence of our code is thus demon-
lution code. The figure displays the differences ∆ψ4in the
amplitude of the wave mode ψ4(ℓ = 2,m = 2) for resolu-
tion h = 0.96M and h = 0.64M scaled for second-order
convergence. The same order of convergence is also obtained
for higher order modes such as ψ4(ℓ = 4,m = 4) (not dis-
played). The differences between the waveforms at J+for
resolution h = 0.64M are of order of 0.03% in amplitude
with a dephasing of 0.002 radians.
The CCE waves can be used to evaluate the quality of stan-
dard finite radius measurements, extrapolated to r → ∞; pre-
viously, this was the most accurate option available. We do
so by finding the Weyl component ψ4relative to a radially
oriented null tetrad  (we prefer ψ4to gauge-invariantper-
turbative methods [24–26]). We have evaluated ψ4at six radii
(r = 280,300,400,500,600,1000M) and extrapolated. De-
tails are given in , and the error is estimated as 0.03% in
amplitude and 0.003 radians in phase.
In Fig. 2, we compare the extrapolated waveform with
that calculated at J+via CCE. The differences between the
two waveforms have maximum and mean values of 1.08%
and 0.166% in amplitude, and −0.019 and −0.004 radians
in phase, respectively. That is, for the resolutions used, the
numerical error in the characteristic evolution (see Fig. 3) is
smaller by one order of magnitude than the error between the
extrapolated and characteristic waveforms in both amplitude
and phase. Further, we note that the estimated error in the ex-
characteristic waveform and extrapolated waveform. This in-
dicates that the systematic error in extrapolation has, previ-
ously,been underestimated. The correctionis towards slightly
larger amplitudes and frequencies. CCE post-processes data
producedby a Cauchy code, and as such there is an additional
cost. However, it is relatively small: the Cauchy run reported
here took ∼ 336 hours, and then CCE required ∼ 10 hours.
Will the small correction to waveforms introduced by CCE
dependon the signal-to-noiseratio(SNR) ofthe event. At low
SNR, whether CCE or extrapolated waveforms are used as a
template will not affect physical interpretation. This is partic-
ularly relevant, as numericalwaveforms are being constructed
with the intention of evaluating and parametrising detector
templates and search algorithms , and to constrain ana-
lytic models [28–31]. Our results indicate that extrapolations
froma finite radiuscanbeused toconstructdetectortemplates
well within the accuracy standards required by matched filter-
However,at largeSNR, the differencesare significantto the
determinationof thephysicalparametersofa modelmeasured
in detector data. To demonstrate this, we follow methods de-
scribed in [32, 33] to determinethe minimumSNR neededfor
a detected signal from a merger event to lead to different pa-
rameter estimates depending on which waveform is used as a
template. TableIdisplaystheresultsforselectedmasses, indi-
cating the maximum distance at which the differencebetween
the waveforms will be relevant for the given merger event.
The differencebetween the waveforms is unlikely to be rel-
evant for LIGO, (e)LIGO and Virgo. Reasonable stellar mass
black hole merger rates are expected only for a volume en-
compassingsourcesupto a distanceof at least 100Mpc. Thus,
there may well be events detected by AdLIGO and AdVirgo
for which the difference is important. Finally, the differences
will certainly be relevant for LISA as they will be applicable
to any supermassive black hole merger event throughout the
visible universe (cH−1is the Hubble radius).
Acknowledgments. The authors thank Stanislav Babak,
Luciano Rezzolla and Jeffrey Winicour for their helpful
input. CR, DP and BS thank the University of South
Africa, and NTB thanks Max-Planck-Institut f¨ ur Gravita-
tionsphysik, for hospitality. This work was supported by
(e)LIGO 50M⊙+ 50M⊙
AdLIGO 50M⊙+ 50M⊙
AdVirgo 50M⊙+ 50M⊙
107M⊙+ 107M⊙ > cH−1
TABLE I: Maximum distance at which the difference between the
extrapolated waveform and that at J+would be significant for a
black hole merger event.
National Research Foundation, South Africa; Budesminis-
terium f¨ ur Bildung und Forschung, Germany; and DFG grant
SFB/Transregio 7 “Gravitational Wave Astronomy”. DP was
supported by a grant from the VESF. BS was supported by
grants from the Sherman Fairchild Foundation,by NSF grants
DMS-0553302, PHY-0601459, PHY-0652995, and by NASA
grant NNX09AF97G. Computations were performed at the
AEI, at LRZ-M¨ unchen, on Teragrid clusters (allocation TG-
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