We present a detailed descriptive analysis of the gravitational radiation from binary mergers of non-spinning black holes, based on numerical relativity simulations of systems varying from equal-mass to a 6:1 mass ratio. Our analysis covers amplitude and phase characteristics of the radiation, suggesting a unified picture of the waveforms' dominant features in terms of an implicit rotating source, applying uniformly to the full wavetrain, from inspiral through ringdown. We construct a model of the late-stage frequency evolution that fits the l = m modes, and identify late-time relationships between waveform frequency and amplitude. These relationships allow us to construct a predictive model for the late-time waveforms, an alternative to the common practice of modelling by a sum of quasinormal mode overtones. We demonstrate an application of this in a new effective-one-body-based analytic waveform model

Baker, J. G.

Boggs, W. D.

Centrella, J. M.

Kelly, B. J.

McWilliams, S. T.

vanMeter, J. R.

NASA Technical Reports Server

Gravitational radiation characteristics of nonspinning
black-hole binaries
B J Kelly, J G Baker, W D Boggs, J M Centrella, J R van Meter and
S T McWilliams
NASA Goddard Space Flight Center, Greenbelt MD 20771, U.S.A.
E-mail: bernard.j.kelly@nasa.gov, john.g.baker@nasa.gov, william.d.boggs@nasa.gov,
joan.m.centrella@nasa.gov, james.r.vanmeter@nasa.gov, sean.t.mcwilliams@nasa.gov
Abstract. We present a detailed descriptive analysis of the gravitational radiation from binary
mergers of non-spinning black holes, based on numerical relativity simulations of systems varying
from equal-mass to a 6:1 mass ratio. Our analysis covers amplitude and phase characteristics
of the radiation, suggesting a unified picture of the waveforms’ dominant features in terms of
an implicit rotating source, applying uniformly to the full wavetrain, from inspiral through
ringdown. We construct a model of the late-stage frequency evolution that fits the ℓ = m
modes, and identify late-time relationships between waveform frequency and amplitude. These
relationships allow us to construct a predictive model for the late-time waveforms, an alternative
to the common practice of modelling by a sum of quasinormal mode overtones. We demonstrate
an application of this in a new effective-one-body-based analytic waveform model.
1. Outline
Since the discovery of the moving puncture recipe, numerical research into the merger of black-
hole binaries (BHBs) has exploded [1, 2, 3, 4]. Moving past the canonical case of the equal-mass,
nonspinning binary, assuming zero eccentricity, and taking advantage of the overall mass-scaling
nature of the problem, BHBs occupy a seven-parameter space (mass ratio, spins of each hole).
With LIGO being retooled as Enhanced LIGO, and LISA anticipated to fly in the next decade,
it is imperative that we explore this parameter space as quickly and efficiently as possible,
to supply the gravitational-wave detector community with the information they need for data
analysis and detector tuning.
In this paper, we outline recent work investigating the one-dimensional parameter space of
nonspinning binaries of varying mass ratio, with a focus on the relationships between harmonic
modes and between mass ratios, somewhat complementary to work by [5]. The material in this
brief article is a summary of numerical experiments and results more fully described in [6].
2. Numerical Methods
We evolved BHBs with mass ratios of 1:1, 2:1, 4:1 and 6:1 using the Goddard Hahndol code [7],
which solves Einstein’s equations on a Cartesian spatial grid, with adaptive mesh refinement
supplied by the Paramesh infrastructure [8]. Initial data is of the “puncture” type [9], with
momenta determined from a 2PN-accurate circular-orbit approximation [10], with constraints
solved using the AMRMG multigrid elliptic solver [11]. Our evolution uses a moified form of the
https://ntrs.nasa.gov/search.jsp?R=20080047958 2019-08-30T05:45:12+00:00Z
-400 -300 -200 -100 0 100
t (M)
-1
-0.5
0
0.5
1
R
h +
/η
1:1
2:1
4:1
6:1
Figure 1. Gravitational-wave signals along polar axis from mass ratios 1:1, 2:1, 4:1 and 6:1.
[Reprinted from [6].]
BSSN equations [12, 13], with 1+log slicing and Gamma-driver shift conditions appropriate for
the “moving puncture” recipe [14].
3. Numerical Relativity Waveforms for Unequal-Mass BHBs
The first thing we note about the waveforms of the different mass ratios is that they show
remarkable consistency, if viewed along the polar axis. Fig. 3 shows the “plus” polarisation
of the waveform strain, h+, as measured along the orbital axis, with amplitudes scaled by the
symmetric mass ratio η ≡ m1m1/M2. The waveform shape is remarkably coherent across mass
ratios from 200M before merger to the merger itself (taken here as the time of peak radiation
amplitude). After the merger, the waveform is dominated by the fundamental quasinormal mode
(QNM) ringdown frequency of the end-state hole, which depends strongly on the spin of the
hole.
It is not surprising, perhaps, that the waveforms look so similar when they are all dominated
by the quadrupole (ℓ,m) = (2,±2) mode, which is insensitive to mass asymmetries. Viewed far
off-axis, where the (2,2) mode no longer dominates, the signals can look very different. However
we retain much simplicity if we decompose the signal into its harmonic modes:
h ≡
∑
ℓm
hℓm −2Y
ℓ,m(θ, ϕ). (1)
On performing this decomposition, we note that, after an initial pulse of high-frequency junk
radiation, the strongest modes are circularly polarized. For instance, we can write the strain-rate
– the first time-derivative of the waveform strain – in the form h˙ℓm = Aℓme
imφ.
4. The Implicit Rotating Source Picture
Looking at gravitational-wave modes across harmonics reveals several general trends. For
example, looking at the power emitted, mode by mode, we find that it is partitioned by modes
according to leading-post-Newtonian (PN) order. The left panel of Fig. 2 shows the power for
several leading modes for the 4:1 merger. The solid lines are the raw numerical data, while the
dashed lines are the leading-order post-Newtonian predictions for the emitted power, as found
in [15], calculated using the orbital frequency extracted from the (2,2) mode.
-500 -400 -300 -200 -100 0 100 200
t (M)
1×10-4
1×10-3
1×10-2
E. (
l,m
}
(2,2)
(2,1)
(3,3)
(4,4)
(3,2)
1×1056
1×1057
1×1058
(erg s-1)
-500 -400 -300 -200 -100 0
t (M)
0
10
20
30
40
50
60
Φ l
m
(2,2)
(2,1)
(3,2)
(3,3)
(4,4)
(5,5)
tracks
Figure 2. Left panel: Power in each mode for 4:1 mass ratio. Right panel: Rotational phase
of each mode for 4:1 mass ratio. [Reprinted from [6].]
Similarly, we find that the phases of different modes are simply related through merger. While
each mode accumulates phase at a different rate, we can define a “rotational phase” for the
system:
Φℓm = φℓm/m+ const. (2)
The right panel of Fig. 2 shows this rotational phase calculated for all the dominant modes of
the 4:1 run. After initial noise (due to an initial burst of junk radiation, which is not circularly
polarised), the different phase estimates agree very well, up to a common “elbow”, where the
binary merges. The result is a picture of BHB as a slowly changing rigid rotator – an “implicit
rotating source”– generating gravitational waves all through inspiral, merger & ringdown such
that each mode is tied to the corresponding multipole moment of the radiating source [16].
5. Full Late-Merger Frequency Model
One common feature of the different modes is that each (ℓ,m) mode’s frequency ω increases
monotonically until it plateaus after the peak amplitude, associated with the merger. The
characteristic shape of the frequency-time curve is very similar for the strongest modes. It can
be well modelled through merger by a smooth step-like function:
Ω(t) = Ωi + (Ωf − Ωi)
(
1 + tanh[ln
√
κ+ (t− t0)/b]
2
)κ
, (3)
where t0 is the time at which the rotational frequency growth rate (chirp rate) is maximum: Ω˙0.
The left panel of Fig. 3 shows the leading modes in the 4:1 mass ratio case, from 60M before
merger to 40M after merger, with the frequency fit (3) overlaid. This fitting works well over all
mass ratios examined, and appears to be valid for more extreme mass ratios as well. The right
panel of Fig. 3 shows the rotational frequency Ω22 as derived from the dominant (2,2) modes for
the 1:1, 2:1, 4:1 & 6:1 numerical evolutions, as well as for the extreme 100:1 mass ratio1. The
dashed horizontal line indicates the QNM ringdown frequency for an end-state Schwarzschild
black hole. As might be expected, as the mass ratio increases, the final hole’s spin decreases
toward the Schwarzschild limit.
1 100:1 data supplied by T. Damour and A. Nagar
-40 -20 0 20 40
t (M)
0
0.2
0.4
0.6
0.8
1
M
ω
(2,1)
(3,3)
(4,4)
-40 -20 0 20
t (M)
0.1
0.15
0.2
0.25
0.3
M
Ω
1:1
2:1
4:1
6:1
100:1 (D&N)
Figure 3. Left panel: fitting Ωℓm for leading modes in 4:1 mass ratio to functional form (3).
Right panel: fitting Ω22 for all mass ratios. [Reprinted from [6].]
-20 0 20
t - t0 (M)
0
0.05
0.1
dJ
, d
E,
 d
Ω
dJ
dE
dΩ
dΩh 
-20 0 20
t - t0 (M)
0
0.1
0.2
dJ
,d
E,
dΩ
dJ
dE
dΩ
dΩh 
Figure 4. Decay of Eℓm, Jℓm, & Ωℓm for 1:1 (2,2) mode (left), 4:1 (2,1) mode (right). Both
time axes are shifted relative to the peak chirp time t0. [Reprinted from [6].]
6. Late-Merger E(Ω) & J(Ω) Relations
A consequence of the IRS picture is that both the energy and angular momentum flux assume
simple forms near merger:
dEℓm
dt
∝ ΩdΩ
dt
,
dJℓm
dt
∝ dΩ
dt
. (4)
This can be seen in Fig. 4, which shows the decay of the modal contributions to radiated E and
Jz towards their final values for the equal-mass (left panel) and 4:1 (right panel) cases, as well
as the decay of two measures of the rotational frequency, Ω from the strain-rate, and Ωh from
the strain (these should differ only at 2.5 PN order).
As a consistency check of this picture, if we assume that dJ/dΩ ∼ constant near merger, we can
use the frequency fit (3) to derive the approximate delay between peak chirp (rate of increase
of Ω) and peak radiation amplitude:
tpeak − t0 ≈ −
Ω˙20
Ω0
...
Ω0
> 0. (5)
-400 -300 -200 -100 0
t (M)
0
0.2
0.4
ω
 
,
 
|dh
/dt
|
ω numerical
ω model 1
ω model 2
3 |dh/dt| numerical
3 |dh/dt| model 1
3 |dh/dt| model 2
Figure 5. Frequency and strain-rate amplitude for the IRS modelled (2,2) waveform (dotted
and dashed lines) compared with the numerical waveform (solid lines), for the 4:1 merger. t = 0
is the time of peak amplitude. [Reprinted from [6].]
Inserting values of Ω0 and its derivatives from the frequency fit yield values of this difference of
between ∼ 2− 3M , in good agreement with the fitted values for t0 itself.
7. Relevance to LISA: New BHB Waveform Templates
New insights into gravitational waveforms from mergers can lead to better templates for data
analysis. Although pure numerical waveforms are rarely long enough to cover the full visible
lifetime of a LISA observation, they can be combined with post-Newtonian pre-merger waveforms
for a hybrid template.
The simplest way to achieve this is to “stitch” the two segments together in such a way as
to maintain an everywhere smooth waveform. The point or interval over which the waveforms
can be stitched should be one where both post-Newtonian and numerical results can be trusted.
An early example of this for the equal-mass case was supplied by Baker et al. [17].
The drawback with this direct approach is that each point in parameter space (mass ratio
here) requires its own numerical waveform of limited duration to be attached to the post-
Newtonian pre-merger waveform. As it is impractical to carry out a full evolution for every
mass ratio of interest, we must develop instead a better analytic insight.
Several semi-analytical methods have been proposed involving the combination of effective-
one-body (EOB) theory [18] and multiple quasinormal ringdown modes (QNMs) [19, 15], with
flexibility parameters tuned using numerical merger data. The current work, with the empirical
functional frequency (3), has allowed an alternative to QNMs, with a late-time strain-rate
amplitude given in terms of the rotational frequency:
Aℓm ≈
√
16πξℓmΩℓmΩ˙ℓm, (6)
where the constant ξ can be fixed by demanding continuity of amplitude at a matching frequency.
Fig. 7 shows the modelled frequency and strain-rate amplitude (dotted lines) compared with
the numerical results (solid lines) for the 4:1 evolution. We see that this new model encodes
both frequency and amplitude well.
8. Summary
At the time of writing, fully numerical evolution of black-hole-binary spacetimes is possible to
high accuracy. Nevertheless, LISA data analysis demands waveforms of much greater duration
than is numerically feasible; additionally an understanding is required of waveforms from systems
over a seven-parameter configuration space (more is eccentricity is considered crucial). To fill
this gap in computational practicality, we are drawn to the development of analytical models of
the gravitational-wave signals.
In this paper, we have introduced a set of numerical evolutions probing one dimension of
parameter space – mass ratio. We have noted common characteristics of the dominant waveform
modes, and suggested a useful heuristic picture of the radiation as being produced by an “implicit
rotating source” through inspiral, merger, and ringdown. We have shown how this picture can be
used to develop an alternative model to a sum of quasinormal modes for the transition through
merger and ringdown. This model can then be combined with effective-one-body theory to
produce analytic waveform templates that seem to capture well the important characteristics of
the merger. A fuller discussion of these issues can be found in [6].
Work is underway to assess the utility of these templates in data analysis, and to extending
our analysis to other parts of parameter space, involving spinning black holes.
Acknowledgments
We thank Emmanuele Berti and Alessandra Buonanno for interesting discussions, Luciano
Rezzolla for useful comments, and Thibault Damour and Alessandro Nagar for supplying data
from the 100:1 case. This work was supported in part by NASA grant 05-BEFS-05-0044 and 06-
BEFS06-19. The simulations were carried out using Project Columbia at the NASA Advanced
Supercomputing Division (Ames Research Center) and at the NASA Center for Computational
Sciences (Goddard Space Flight Center). B.J.K. was supported by the NASA Postdoctoral
Program at the Oak Ridge Associated Universities.
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Gravitational Radiation Characteristics of Nonspinning Black-Hole Binaries

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