Gravity@Malta 2018: WG2 - Numerical Relativity in Astrophysics (vacuum) - Patricia Schmidt (Radboud University) Valletta, 23.1.2018 - Numerical ...
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Gravity@Malta 2018:
WG2 - Numerical Relativity in
Astrophysics (vacuum)
+ a bit of waveform modelling
Patricia Schmidt (Radboud University)
!
Valletta, 23.1.2018
Binary Black Holes 2
‣ Advanced LIGO and Virgo have observed 5.9 binary black holes
!
‣ Probes of the highly dynamical non-linear regime of General Relativity
high mass
low mass
Binary Black Holes & GWs 3
‣ The characteristic „chirp“
!
‣ Signal „sweeps“ through the
detector’s sensitivity band
! 1 ¥ 10-21
‣ Depending on the total mass 5 ¥ 10-22
of the BBH, the merger
signal&noise@Hz-1ê2 D
1 ¥ 10-22
regime is visible 5 ¥ 10-23
‣ Inspiral-merger-ringdown
1 ¥ 10-23
(IMR) waveforms are key 5 ¥ 10-24
‣ BBH science only as good
as our models 1 ¥ 10-24
10 50 100 500 1000 5000
f@HzD
Numerical Relativity 4
‣ Only very few exact solutions of the
Einstein field equations (i.e. the metric) are
known
!
[Pretorius05]
‣ Many astrophysical space times require
numerical solutions, incl. the binary black
hole problem
‣ Numerical relativity is a key ingredient
to understand LIGO’s and Virgo’s
observations!
!
‣ Breakthrough in 2005 [Pretorius; Baker+;
Campanelli+]
‣ Simulation of the final inspiral, merger
& ringdown plus extraction of the
gravitational-signal signal FIG. 3: A sample of the gravitational waves emitted during
the merger, as estimated by the Newman-Penrose scalar Ψ4
(from the medium resolution simulation). Here, the real com-
ponent of Ψ4 multiplied by the coordinate distance r from the
center of the grid is shown at a fixed angular location, though
several distances r. The waveform has also been shifted in
time by amounts shown in the plot, so that the oscillations
Numerical Relativity - a brief timeline 5
2000-04 2011 Lousto+
1952 Choquet-Bruhat 1992,3 Choptiuk
AEI/UTB/NASA 2005 Pretorius
IVP Abrahams, Evans q=100
Lazarus IMR w harmonic
critical phenomena 1999-00
1962 ADM 2011 Lovelace+
AEI/PSU 2005-06 Campanelli+,
formulation a=0.97
Grazing Collisions Baker+; IMR w BSSN &
1997
1964 Hahn-Linquist Brandt&Brügmann ~2000 Choptuik, moving punctures 2015 Szilagyi+
2 wormholes puncture data Brügmann, Schnetter 175 orbits
mesh refinement 2006-08 SXS
1984 Unruh 1994-98 2014-17 Ruchlin+
2005 Gundlach+ IMR w spectral
excision BBH grand challenge high spin puncture ID
constraint damping
1950s 2005 2015+
2000-02
1975-77 Smarr-Eppley 1994 Cook 1999
Alcubierre 2006,07 Baker+;
head on collisions Bowen-York ID BSSN 2009-11
gauge conditions Gonzalez+
Bishop+
1999 York 2004 non-spinning BBH
1979 York 94-95 NCSA/WSU CCE
CTS ID Brügmann+ kicks
Kinematics & improved head-on
one orbit 2007-11 2010
dynamics of GR 1999-2005
2003-08 RIT; Jena; AEI;... Bernuzzi+
JW York,
1989-95 Cook, Pfeiffer ea BBH superkicks C4z
Cornell, Caltech, LSU
Bona-Masso improved ID
hyperbolic formulations 2008
Modified ADM 2000 Ashtekar NINJA
Adapted from slides by C. Lousto & H. Pfeiffer isolated horizons
Two main approaches to BBH evolutions 6
‣ Puncture initial data ‣ Quasi-equilibrium excision initial
! data
‣ BSSN or C4z with moving ‣ Generalised harmonic (GH) with
punctures constraint damping
! !
‣ 1+log, Gamma-driver shift ‣ Damped harmonic gauge
condition
!
!
!
‣ Sommerfeld outer boundary
‣ Constraint preserving, minimally
condition
! reflective outer boundary
! condition
‣ Finite differencing (FD) with !
adaptive mesh refinement (AMR) ‣ Multi-domain spectral methods
! !
‣ BAM, MayaKranc, LazEv, Einstein !
Toolkit, Lean, Goddard, Perimeter, ‣ SXS Collaboration (SpEC)
GRChombo, …
Pretorius:
FD, GH, AMR
Modelling gravitational waves 7
Newtonian dynamics
orbital
separation
post-Newtonian theory
black hole test
path to merger
particle
perturbation limit
theory
Numerical
WG2 topic „source modelling“
relativity T. Hinderer
merger dynamics & GW signal mass ratio
mass & spin of the final BH Waveform models aim to combine
recoil velocity the regimes and the techniques!
gravitational luminosity
Effective-One-Body (EOB) Formalism 8
‣ The basics in a nutshell:
! Effective description
Binary
! problem
! MAP effective particle
! m1 m2
µ=
! m1 + m2
m1 ! [Buonanno, Damour
1999, 2000]
! m2 effective spacetime
!
M = m1 + m2 ⌫ = µ/M 2 [0, 1/4]
Courtesy:
! T. Hinderer
‣ Recipe: EOB Hamiltonian + GW dissipation + wave generation + merger-
ringdown
Numerical relativity
EOB, analytical knowledge only
EOB, calibrated
Courtesy: A. Taracchini
GW cycles
Phenomenological Waveform Models 9
‣ Model IMR signal (amplitude & phase) in the frequency domain
! i (f ; )
!
h̃ 22 (f ; ) = A(f ; ) e
‣ Modular approach to separately model inspiral, merger & ringdown
!
!
!
!
!
!
Khan+ 2015
!
!
‣ Aligned-spin waveforms forms basis for precessing waveform model:
X̀
hP ' R`m0 m hA ⇥ hRD Jˆ
`m (q, ~ 1 , ~ 2 ) `m0 (q, 1L , 2L ) `m
m0 = ` L̂
Encodes the precession of the orbital plane
[Schmidt+10, Schmidt+12, Hannam+13]
NR Surrogates 10
‣ NR simulations across the binary parameter
space are expensive
!
‣ Are pure NR waveform models, i.e. no
analytical approximations, achievable?
‣ One proposal: surrogate models
!
‣ Surrogate: Continuous interpolation between
discrete waveforms [Field+, Galley+]
‣ 5D NR surrogate in hypercube around
GW150914 parameters [Blackman+16]
‣ ~ 270 NR simulations spanning ~20
orbits
‣ 7D NR surrogate for precessing binaries
with mass ratios q=1-2 [Blackman+17]
‣ ~ 744 NR simulations
[Blackman+16]
!
‣ Not yet feasible to replace IMR models by pure
NR surrogates to perform GW data analysis!Kicks 11
‣ GWs carry energy, linear & angular momentum
‣ Anisotropic GW emissions leads to the build-up of net
momentum
‣ After the merger, the GW emission „switches off“ forcing the
remnant black hole to recoil
‣ Kick velocities up to 4000 km/s [Baker+, Gonzalez+,
Campanelli+]
‣ Larger than escape velocity of galaxies
‣ Could explain a population of extra-galactic BHs
!
‣ If the kick is along the line-of-sight, a Doppler shift may be
observable in the GW signal [Moore+17] „relativistic
water sprinkler“
[Moore+]Precession 12
6
‣ Occurs when spins are misaligned with the
far enough away that boundary effects do not interfere
orbital
with angular
the orbital momentum
dynamics of the system. In addition, we
0.3 [Campanelli+07]
Induces
‣ evolved
also the amplitude & phase
SP4 configuration modulations
with a central resolu- 0.2
tion of h = M/25, a grid size of 6402 × 320, and outer 0.1
! at 200M .
boundary 0
! Figures 1 and 2 show the puncture trajectory and -0.1
horizon-spin direction along this track for the SP3 con-
!
figuration (the latter suppressing the z-direction). Note
-0.2
! the scale of the z-axis in Fig. 1 is 1/10th that of
that -0.3
the x and y axes. From the plots one can clearly see the -3 -2
‣ First simulations
orbital plane precess outperformed as early
of the equatorial plane, as 2006
as well
-1 0 1 2 3 1 0 -1 -2 -3
3 2
[Campanelli+]
as the spin axis rotating by approximately 90◦ in the xy
plane during the course of the merger. The spins are ini- FIG. 1: The puncture trajectories along with spin direction
! aligned along the y-axis, but at merger they show
tially (every 4M ) for the SP3 configuration for the M/30 resolution
‣ 7D parameter
both space (q,and
a significant z-component ~a1an, ~aapproximate
2 ) 90◦ run. The spins are initially aligned along[Schmidt+11]
the y-axis, but rotate
rotation to the −x-axis. The individual horizon spins by ∼ 90 during the 1.25 last orbits and also acquire a non-
◦
Difficult
at‣ the merger areto sample
S⃗coord = (−0.121 ± 0.002, −0.007 ± negligible z-component. Note that the z-scale is 1/10th the x
and y scale.
‣ NR
0.003, simulations
0.037 ± 0.003) (we concentrated
use the coordinatearoundbased mea-
sure of the spin at the merger because the calculation of 4
SIH isq=1-3
not accurate when the black holes are this close 2
‣ Long
together; seesimulations
comments below).required
Hence the to total
resolve a
preces-
while Fig. 05 shows the value of the z-component of the
sion angle for the SP3 configuration is Θp = 98◦ . Note 5
that complete precession cyclebetween the ori-
z
specific spin Sz /m2 (where m is the horizon mass) based
there is no discernible correlation -2
on the z-component of S⃗IH for the three resolutions. In
entation of the projected horizon and the projected spin 0
direction. torb t prec t insp
-4
this latter figure
-5
y
the curves have been translated. (A
⃗coord ) and Killing convergence plot of S ⃗IH would not be meaningful be-
In Fig. 3 we show the coordinate (S 0 -5
⃗IH ) calculation of the spin components cause the size of the step discontinuities in S
x ⃗IH are larger
vector based (S 5
⃗IH displays a step-function-like behavior than the differences in the spin direction with resolution.)
versus time. S
For the z-component of the spin, we expect that, given
due to the difficulty in finding the poles (i.e. the ze-Long simulations 13
‣ Long simulations desirable to probe the consistency between early inspiral
models (e.g. EOB or PN)
‣ Resolve precession cycle?
!
‣ Stable evolutions longer than ~20 orbits are difficult & expensive
‣ Gauge drifts
‣ Build-up of numerical errors
!
‣ 2015: ~175 orbit long non-spinning simulation with q=7
[Szilagyi+; Courtesy: D. Hemberger]
100000 80000 60000 40000 20000 0
t/Mmerger, and ringdown have been limited by an apparently insurmountable barrier: the merging
holes’ spins could not exceed 0.93, which is still a long way from the maximum possible value in
terms of the physical effects of the spin. In this paper, we surpass this limit for the first time, opening
arXiv:1010.2777v3 [gr-qc] 11 Jan 2011 Extreme Spins & Final Spin
the way to explore numerically the behavior of merging, nearly extremal black holes. Specifically,
using an improved initial-data method suitable for binary black holes with nearly extremal spins, 14
we simulate the inspiral (through 12.5 orbits), merger and ringdown of two equal-mass black holes
with equal spins of magnitude 0.95 antialigned with the orbital angular momentum.
‣ Extreme spin: First breakthroughs in 2011
PACS numbers: 04.25.dg, 04.30.-w
for quasi-equilibrium initial data [Lovelace
I. INTRODUCTION Rotational energy / rotational energy if extremal
+11]
1.1 1.1
1 1
‣ Fundamental
Although there is spin limit
considerable for Bowen-York
uncertainty, it is pos- 0.9 SKS initial data 0.9
sible that astrophysical black holes exist with nearly 0.8 Not accessible with (this paper): 0.8
(BY) initial
extremal spins data
(i.e., in dimensionless units spins close Bowen-York (B.Y.) first BBH merger
Erot / Erot,extremal
0.7 beyond B.Y. limit 0.7
to 1, the theoretical upper limit for a stationary black initial data
! hole). Binary black hole (BBH) [Dain+]a 0.93
mergers in vacuum typ-
0.6 0.6
max 0.5 0.5
! ically lead to remnant holes with dimensionless spins 0.4 0.4
χ ∼ 0.7 − 0.8 [1–3], although if the merging holes are
‣ 2014 onwards:
surrounded incorporation
by matter the remnant’s spinof typically
non- could 0.3 0.3
be higher than χ ∼ 0.9 [1, 3]. Black holes can reach 0.2 0.2
conformally
higher spins viaflatprolonged
initial data into
accretion [4, the moving
5]: thin accre- 0.1 0.1
[Lovelace+10] 0
punctures framework
tion disks (with [Ruchlin+14,
magnetohydrodynamic effects neglected) 0
-0.1 -0.1
lead to spins as large as χ ∼ 0.998 [6], while thick-disk ac- -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
Zlochower+17] to go beyond
cretion with magnetohydrodynamic theincluded
effects BY limit can Spin / (Mass)
2
lead to spins as large as χ ∼ 0.95 [7, 8]. Even without ac-
cretion, at very high mass ratios with spins aligned with FIG. 1. The rotational energy of a Kerr black hole as a
the orbital angular momentum, binary black hole mergers function of the hole’s dimensionless spin parameter χ :=
can also lead to holes with nearly extremal spins [9–11].
‣ Final spin: most accurately fit to NR data for non-spinning Spin/(Mass)2 . The & aligned-spin
thick red line indicates the Bowen-York
There is observational evidence suggesting the existence limit: standard Bowen-York puncture initial data—used in
binaries [Rezzolla+,
of black holes with nearlyHemberger+, Lousto+,
extremal spins in quasars [12], Healy+,
almost all Hofmann+]
numerical binary-black-hole calculations to date—
and some efforts to infer the spin of the black hole in mi- cannot yield rotational energies more than 60% of the way to
No NRGRS
‣ croquasar data1915+105
used in precessing
from mergers
its x-ray spectra suggest [Barausse+]
extremality. By using instead initial data based on two su-
a spin larger than 0.98, though other analyses suggest perposed Kerr-Schild holes (“SKS initial data”), in this paper
the spin may be much lower [13–15]. we surpass the Bowen-York limit (green circle), opening the
way for numerical studies of merging, nearly extremal black
Merging BBHs—possibly with nearly extremal spins— holes.
are among the most promising sources of gravitational
waves for current and future detectors. Numerical simu-Higher-order modes 15
‣ Current waveform models describe h2,±2
!
‣ Highly asymmetric systems show strong excitations of higher-order modes
‣ Individually resolvable at high SNRs
‣ Break parameter degeneracies
‣ Tests of Kerrness possible
{2, 2}
‣ Higher modes are often {2, 1} [BAM, Husa+, Courtesy: G. Pratten]
difficult to resolve {3, 3}
{3, 2}
accurately in NR {4, 4}
simulations {4, 3}
! -
‣ Recent progress: First
aligned-spin IMR waveform -
models with higher modes
[London+17]; EOB model
[Cotesta+ in prep] -
- - -High mass ratios 16
4.5
‣ Very few points beyond mass ratio
q=8 2.5
‣ non-spinning q=100 [Lousto+]
(y1 − y2)/M
0.5
‣ non-spinning q=10 [SXS]
h=h0*1.2
‣ aligned-spin q=18 [BAM, Husa h=h0
−1.5 h=h0/1.2
+15] h=h0/1.2
2
! −3.5
‣ Waveform models are poorly
−5.5
calibrated in the high mass regime −5.5 −3.5 −1.5 0.5 2.5 4.5
(x1 − x2)/M
due to the lack of NR simulations! [Lousto+]
‣ Test particle limit information
Extreme mass ra*os [Lousto+]
q18, {0,0.0}
included
! q18, {0,+0.4}
‣ Problems:
‣ Large difference in horizon size
‣ Difficult to resolve the smaller
BH
‣ Spins additionally distort the -
horizon
‣ Computationally very - [BAM, Husa+, Courtesy: G. Pratten]
expensive! - - -Eccentricity 17
‣ Circularisation of the orbit during inspiral
[Peters&Matthews+63]
!
‣ Some astrophysical systems may have non-
negligible eccentricity at small separation
‣ e.g. BBH formed through dynamical [Gold+13]
capture in globular clusters
!
‣ Similar to precession, eccentricity leaves a
visible imprint in the waveform
‣ Chirp augmented with burst-like
structures
‣ Zoom-whirl behavior [Healy+,
Sperhake+, Gold+]
!
‣ Ongoing effort to model the IMR waveforms
from eccentric binaries [Huerta+, Haney+]
! Maria Haney’s talk
‣ Lack of calibration to eccentric NR
simulationsProbing waveform models - systematics 18
‣ Independent NR simulations used to test waveform models and quantify
modelling errors („bias“)
‣ Inject NR waveforms as „mock“ signals [Schmidt+16]
‣ Exemplified on GW150914
Tests the physics present in NR
Overall 1.00but neglected in waveform models
Prior
IMRPhenom IMRPhenom
35
EOBNR
0.75
0.50
30
0.25
/M
e↵
msource
0.00
25
2
0.25
20 0.50
0.75
15
1.00
25 30 35 40 45 50 0.00 0.25 0.50 0.75 1.00
msource
1 /M p
[LVC, CQG 34 (2017) no.10]Conclusions & Outlook 19
‣ Tremendous progress in the last decade
‣ O(1000s) BBH simulations
!
‣ Remaining challenges for BBH
simulations are mostly of technical
nature
!
‣ Matter simulations become increasingly
important
‣ Binary neutron stars, neutron star -
black hole binaries, supernova [Blackman+16]
explosion, BBH + accretion disks,
BBH + stellar environment … Albino Perego’s talk
‣ Difficult to get the microphysics right
!
!
‣ Simulations of BBH mergers in
alternative theories of gravity (see e.g.
Okounkova+17) Uli Sperhake’s talk [NR data: Tim Dietrich]You can also read
