LIGO

GWastrophysicsof compact binaries

Ilya Mandel(University of Birmingham)

January 20, 2014UKZN, Durban

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Gravitational Waves

Ripples in spacetime:

Caused by time-varying mass quadrupole moment; GW frequency is twice the orbital frequency for a circular, non-spinning binary

Huge amounts of energy released: GW energy output of SMBH binary greater than EM radiation from entire galaxy over a Hubble time

3

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Indirect observations of GWs PSR 1913+16 Discovered in 1974 GR precession of 4.2 deg/

yr (vs. 43 arcsec/century for Mercury, out of 5600)

5J0737-3039A: [Kramer et al., 2005]

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Opportunity and ChallengeGWs carry a lot of energy, but interact weakly: can pass through everything, including detectors!

Michelson-type interferometers

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Detection Challenges

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Detection Challenges

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LIGO Noise Spectrum

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A few initial LIGO/Virgo highlights GRB070201 overlapped

Andromeda (at 770 Mpc); binary coalescence in M31 excluded at >99% confidence [LSC+Hurley, 2008, ApJ 681 1419]

Best upper limit (Ω0 < 6.9 x 10-6) on stochastic background energy density at 100 Hz [LVC, 2009, Nature 460 990]

Beat spindown limit on emission from Crab (<0.02 E) and Vela (<0.45 E) pulsars, [LVC, 2008, ApJL 683 45; 2011, ApJ 737 93]

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M31, GRB 070201

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Types of GW sources Continuous sources [sources with a slowly

evolving frequency]: e.g., non-axisymmetric neutron stars, slowly evolving binaries

Coalescence sources [known waveforms, matched filtering]: compact object binaries

Burst events [unmodeled waveforms]: e.g., asymmetric SN collapse, cosmic string cusps

Stochastic GW background [early universe]

??? [expect the unexpected]

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Predicting merger rates

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Method Strength WeaknessDirect extrapolation from observed Galactic binaries

Most direct available probe; ~10 known (~5 merging) Galactic binary pulsars

Low statistics, poorly known selection effects, only relevant for BNS systems

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Predicting merger rates

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Method Strength WeaknessDirect extrapolation from observed Galactic binaries

Most direct available probe; ~10 known (~5 merging) Galactic binary pulsars

Low statistics, poorly known selection effects, only relevant for BNS systems

Extrapolation from short GRB rates

Potentially direct probe of mergers involving NS out to large distances (z~2)

Uncertain provenance, ill-constrained beaming factors and selection effects

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Predicting merger rates

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Method Strength WeaknessDirect extrapolation from observed Galactic binaries

Most direct available probe; ~10 known (~5 merging) Galactic binary pulsars

Low statistics, poorly known selection effects, only relevant for BNS systems

Extrapolation from short GRB rates

Potentially direct probe of mergers involving NS out to large distances (z~2)

Uncertain provenance, ill-constrained beaming factors and selection effects

Population synthesis of isolated binaries

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Population synthesis models No observed NS-BH or BH-BH binaries Predictions based on population-synthesis

models for isolated binary evolution with StarTrack [Belczynski et al., 2005, astro-ph/0511811] or similar codes

Many poorly constrained parameters: SN kicks, stellar winds, mass transfer efficiency, common envelope physics [O’Shaughnessy et al., 2005 ApJ 633 1076; 2008 ApJ 672 479]

Also: metallicity variations, SN mechanism/fallback, fate of Hertzsprung gap donor in common envelope... [Dominik et al., 2012 ApJ, 759, 52]

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Population synthesis predictions

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[Dominik et al., 2012 ApJ, 759, 52]

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Predictions of component mass distributions

17[Dominik et al., 2012 ApJ, 759, 52]

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Predicting merger rates

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Method Strength WeaknessDirect extrapolation from observed Galactic binaries

Most direct available probe; ~10 known (~5 merging) Galactic binary pulsars

Low statistics, poorly known selection effects, only relevant for BNS systems

Extrapolation from short GRB rates

Potentially direct probe of mergers involving NS out to large distances (z~2)

Uncertain provenance, ill-constrained beaming factors and selection effects

Population synthesis of isolated binaries

Applies to all binary types, creates models for future astrophysical inference

A number of poorly known input parameters (SNe kicks, winds, common envelope)

Source Rlow

Rhigh

NS-NS (MWEG�1 Myr�1) 1 1000NS-BH (MWEG�1 Myr�1) 0.05 100BH-BH (MWEG�1 Myr�1) 0.01 30

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Merger Rate Predictions

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S6 / VSR2,3 Upper Limits

Predicted rates

[Abadie et al., 2011]

[Abadie et al., CQG 27:173001,2010]

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Predicting merger rates

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Method Strength WeaknessDirect extrapolation from observed Galactic binaries

Most direct available probe; ~10 known (~5 merging) Galactic binary pulsars

Low statistics, poorly known selection effects, only relevant for BNS systems

Extrapolation from short GRB rates

Potentially direct probe of mergers involving NS out to large distances (z~2)

Uncertain provenance, ill-constrained beaming factors and selection effects

Population synthesis of isolated binaries

Applies to all binary types, creates models for future astrophysical inference

A number of poorly known input parameters (SNe kicks, winds, common envelope)

Forward evolution of observed X-ray binaries

Combination of observations and population synthesis

Uncertain selection effects, mass measurements, and modeling assumptions

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Predicting merger rates

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Method Strength WeaknessDirect extrapolation from observed Galactic binaries

Most direct available probe; ~10 known (~5 merging) Galactic binary pulsars

Low statistics, poorly known selection effects, only relevant for BNS systems

Extrapolation from short GRB rates

Potentially direct probe of mergers involving NS out to large distances (z~2)

Uncertain provenance, ill-constrained beaming factors and selection effects

Population synthesis of isolated binaries

Applies to all binary types, creates models for future astrophysical inference

A number of poorly known input parameters (SNe kicks, winds, common envelope)

Forward evolution of observed X-ray binaries

Combination of observations and population synthesis

Uncertain selection effects, mass measurements, and modeling assumptions

Dynamical formation in dense environments

Independent scenario, less sensitive to binary evolution

Poorly known dynamics of globular and nuclear clusters

NG (L10) =43�

�Dhorizon

Mpc

⇥3

(2.26)�3(0.02)

N = R�NG

�(Dhorizon) � 8

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LIGO sensitivity

|h(f)| = 2/D � (5µ/96)1/2(M/⇥2)1/3f�7/6

(merger rate) = (merger rate per L10) * (NG in L10's)

� �

⇥

4� fISCO

0

|h(f)|2Sn(f)

df

1/2.26 -- sky and orientation averaging; 0.02 L10 per Mpc3

S4 S5 aLIGO

[plot from Kopparapu et al., 2008 ApJ 675 1459 ]

22

101

102

103

10−23

10−22

10−21

10−20

10−19

f, Hz

!S

n(f

),1/

!H

z

Initial LIGO

Initial Virgo

Advanced LIGO

Advanced Virgo

IFO Source Nlow

Nhigh

yr�1 yr�1

NS-NS 2⇥ 10�4 0.2Initial NS-BH 7⇥ 10�5 0.1

BH-BH 2⇥ 10�4 0.5NS-NS 0.4 400

Advanced NS-BH 0.2 300BH-BH 0.4 1000UKZN: Jan 20, 2014

Merger and Detection Rates

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[IM & O’Shaughnessy, 2010, CQG 27 114007;Abadie et al., CQG 27:173001, 2010, arXiv:1003.2480]

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Advanced LIGO overviewWhat is Advanced?

EOMLaser

Parameter Initial LIGO Advanced LIGO

Input Laser Power

10 W (10 kW arm)

180 W (>700 kW arm)

Mirror Mass 10 kg 40 kg

Interferometer Topology

Power-recycled

Fabry-Perot arm cavity Michelson

Dual-recycled Fabry-Perot arm cavity Michelson

(stable recycling cavities)

GW Readout Method

RF heterodyne DC homodyne

Optimal Strain Sensitivity

3 x 10-23 / rHz Tunable, better than 5 x 10-24 / rHz in broadband

Seismic Isolation Performance

flow ~ 50 Hz flow ~ 13 Hz

Mirror Suspensions

Single Pendulum

Quadruple pendulum slide courtesy

of Dave Reitze

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Advanced detector prospects

25[Aasi+ (LSC+Virgo), arXiv:1304.0670]

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The case of the buried signal

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Spectrograms

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Burst searches• Coherent WaveBurst: An algorithm which is used for unmodeled searches in a

wide parameter space that could detect all parts of the coalescence event

• Omega pipeline: Multi-resolution time-frequency search, equivalent to a template-based search for sinusoidal Gaussians in whitened data

Laura CadonatiG070801-00 28

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The case of the resurrected signal

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+

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Matched filteringFilter

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slide courtesy of Damir Buskulic

M � 50M�

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Waveform families Typical frequency scales

as 1/Mass For massive systems

( for LIGO), merger and ringdown contribute significantly to signal-to-noise ratio (SNR)

Spins add complications

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INSPIRAL:post-Newtonian

approximate waveforms

RINGDOWN:perturbative

solutionsMERGER:

need Numerical Relativity!

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Background estimation

tsignal?

Hanford 1Virgo

background coincidence32

Background is highly non-Gaussian

Can be estimated with time slides between detectors

Gaussian

Non-Gaussian tails

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False Alarm estimation

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Candidate GW100916 recovered with estimated False Alarm Rate < 1 in 7,000 years! [Abadie et al., Phys. Rev. D 85, 082002 (2012)]

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The Big Dog

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Astrophysics: the Inverse Problem Comparing predicted rates of binary mergers with model

predictions can allow us to constrain the input (astro)physics» Even non-trivial upper limits can do the trick:

37[IM & O’Shaughnessy, 2010, CQG 27 114007]

UKZN: Jan 20, 2014

Astrophysics: the Inverse Problem Comparing predicted rates of binary mergers with model

predictions can allow us to constrain the input (astro)physics

Can learn a lot more by comparing distributions of observed parameters (masses, spins) with model predictions

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Predictions of component mass distributions

39[Dominik et al., 2012 ApJ, 759, 52]

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Astrophysics: the Inverse Problem Comparing predicted rates of binary mergers with model

predictions can allow us to constrain the input (astro)physics

Can learn a lot more by comparing distributions of observed parameters (masses, spins) with model predictions» Requires accurate parameter estimation on individual sources

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Parameter estimation: challenges

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Explore a large physical parameter space: 9 to 15 dimensions

Analyze data streams coherently Make use of a priori information Infer posterior distribution on signal parameters Could be multi-modal:

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Solution: Bayesian inference & stochastic sampling

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[van der Sluys et al., 2008, 2008, 2009; Veitch & Vecchio, 2008, 2008; Raymond et al., 2009,2010; Farr and IM, 2011; Veitch et al., 2012...]

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Accurate Parameter Estimation

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van der Sluys, IM, Raymond, et al., 2009, CQG 26, 204010

UKZN: Jan 20, 2014

Astrophysics: the Inverse Problem Comparing predicted rates of binary mergers with model

predictions can allow us to constrain the input (astro)physics

Can learn a lot more by comparing distributions of observed parameters (masses, spins) with model predictions» Requires accurate parameter estimation on individual sources» Requires combining information from multiple events to construct a

statement about population distribution (accounting for selection bias, etc.)» Requires a library of catalogs of simulations based on different assumed

astrophysical parameters» Requires a pipeline for comparing observations and catalogs» We need to be able to test population synthesis models themselves: need

to over-determine the parameters... how many detections will this require? what will be the correlations/degeneracies in the astrophysical parameter space?

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UKZN: Jan 20, 2014

Astrophysics: the Inverse Problem Comparing predicted rates of binary mergers with model

predictions can allow us to constrain the input (astro)physics

Can learn a lot more by comparing distributions of observed parameters (masses, spins) with model predictions

(Almost) Model-independent inference

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» Evidence for a mass gap? [Dominik, IM, Belczynski, in prep.]

UKZN: Jan 20, 2014

Astrophysics: the Inverse Problem Comparing predicted rates of binary mergers with model

predictions can allow us to constrain the input (astro)physics

Can learn a lot more by comparing distributions of observed parameters (masses, spins) with model predictions

More model-independent inference

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» Search for subpopulations (e.g., distinguish isolated and dynamically formed BH-BH binaries based on spin-orbit alignment)

» Directly measure time delays by observing dependence of merger rate on redshift [IM+, in prep.]

UKZN: Jan 20, 2014

Astrophysics: the Inverse Problem Comparing predicted rates of binary mergers with model

predictions can allow us to constrain the input (astro)physics

Can learn a lot more by comparing distributions of observed parameters (masses, spins) with model predictions

More model-independent inference

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» Search for subpopulations (e.g., distinguish isolated and dynamically formed BH-BH binaries based on spin-orbit alignment)

» Measure binary kick velocities from GWs without EM counterparts [L. Kelley et al., 2010, ApJL 725 L91]

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Multimessenger astronomy “Holy grail of GW astronomy”

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Targeted archival search

GWs from binary merger

EM transient

Rapid(?) followup

GW candidate

EM counterpart

survey

deep pointing

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Cosmology with Advanced Detectors GWs as standard candles [Schutz, Nature, 1986]

Mass-redshift degeneracy can be broken by:» Direct EM associations [e.g. Nissanke et al., 2010]» Statistical EM associations [Schutz, 1986 and others]» Tidal effects [Messenger & Read, 2011] » Knowledge of mass distribution [Taylor, Gair, IM, 2011]

Could independently measure Hubble constant:

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Testing GR with extreme-mass-ratio inspirals

LIGO sensitive @ a few hundred Hz» NS-NS, NS-BH, BH-BH binaries » and intermediate-mass-ratio

inspirals of NSs or BHs into IMBHs– could observe up to tens per year

[IM+, 2008, ApJ 681 1431]

LISA sensitive @ a few mHz» massive black-hole binaries» galactic white dwarf (and compact

object) binaries» extreme-mass-ratio inspirals of

WDs/NSs/BHs into SMBHs– could observe tens to hundreds to

z~1 [e.g., Amaro-Seoane et al., 2007, CQG 24 R113]

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Extreme Mass Ratio Inspirals

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Exploring the spacetime...

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... taking lots of pictures

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Testing the “no-hair” theorem

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Testing the no-hair theorem

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Testing the no-hair theorem?

Stationary, vacuum, asymptotically flat spacetimes in which the singularity is fully enclosed by a horizon with no closed timelike curves outside the horizon are described by the Kerr metric

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Do black holes have hair?

Ryan’s theorem [1995]: GWs from nearly circular, nearly equatorial orbits in stationary, axisymmetric spacetimes encode all of the spacetime multipole moments... in principle

Manko-Novikov spacetime, an exact solution of Einstein’s equations:

Search for observable imprints of a “bumpy” spacetime, such as deviations from the full set of isolating integrals (energy, angular momentum, Carter constant) in Kerr [Gair, Li, IM, 2009, PRD 77:024035]

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The emergence of chaos

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Solve the geodesic equation and study Poincare maps:

- Plot dρ/dt vs. ρ for z=z0 crossings- Phase space plots should be closed curves for all z0 iff there is a third isolating integral [Carter constant]

Newtonian+hexadecapole:

M2=10 M0; M4=400 M0

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Order and Chaos, side by side

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integral.

0 2 4 6 8 10 12 14−5

−4

−3

−2

−1

0

1

2

3

4

5

ρ/Μ

z/M

FIG. 4: E!ective potential for geodesic motion around a bumpy black hole with ! = 0.9, q = 0.95,

E = 0.95, and Lz = 3M . The thick dotted curves indicate zeros of the e!ective potential. The

trajectory of a typical geodesic in the outer region is shown by a thin curve. The regular pattern of

self-intersections of the geodesic projection onto the "!z plane indicates (nearly) regular dynamics.

If |q| is increased from the value shown in Figure 4, the two regions of bound motioneventually merge. When this first occurs, the “neck” joining the regions is extremely narrow.Geodesics exist which can pass through the neck, but this requires extreme fine tuning. As|q| is further increased, the neck gradually widens and eventually disappears. At that stage,the single allowed region for bound orbits has a similar shape to the outer region of Figure 4.

These general properties of the e!ective potential seem to be common to all spacetimeswith q > 0 and ! "= 0. More relevant for the EMRI problem is to fix q and ! and to varyE and Lz . For E = 1 and su"ciently large Lz, there are two regions of allowed motionbounded away from the origin, in addition to the plunging zone connected to the singularityat " = 0, |z| # 1. The outermost of the allowed regions stretches to infinity and containsparabolic orbits. The inner region of bounded motion is the analogue of the inner boundregion described above and lies very close to the central object. If the angular momentumis decreased, while keeping E = 1, the two non-plunging regions get closer together andeventually merge to leave one allowed region that stretches to infinity. For fixed E < 1the behavior is qualitatively the same, except that for Lz $ M there is no outer region(there is a maximum allowed angular momentum for bound orbits of a given energy, as inthe Kerr spacetime). As Lz is decreased, the outer region for bound motion appears andthen eventually merges with the inner region. Decreasing Lz further eventually causes the

11

f

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Other signs of non-Kerr spacetimes Location and character of ISCO

Periapsis and orbital-plane precession

61

1

2

3

4

5

6

7

8

-4 -2 0 2 4

ρIS

CO

/M

q

Radially unstable - first branchRadially unstable - second branch

Vertically unstable

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

-4 -2 0 2 4

(MΩφ) I

SC

O

q

Radially unstable - first branchRadially unstable - second branch

Vertically unstable

FIG. 11: Properties of the equatorial ISCO in spacetimes with ! = 0, as a function of q. We show

the " coordinate of the ISCO (left panel) and the dimensionless frequency of the orbit at the ISCO

(right panel). As described in the text, the ISCO radius has three branches, depending on whether

it is determined by one of the two branches of radial instability or the branch of vertical instability.

These branches are indicated separately in the diagram. For values of q where all three branches

are present, the dashed line denotes the “OSCO” and the dotted line denotes "ISCO as discussed

in the text. Allowed orbits lie above the curve in the left panel, and below the curve in the right

panel.

comparatively small deviations from Kerr.

V. PERIAPSIS AND ORBITAL-PLANE PRECESSIONS

In Section III we saw that astrophysically relevant orbits in the Manko-Novikov space-time are multi-periodic to high precision. In such cases, there is no smoking-gun signaturethat indicates the presence of “bumpiness” in the spacetime. Instead, the imprint of thespacetime bumpiness will be observationally apparent in the location of the last stable orbit,as discussed in the previous section, and in the following ways: (1) in the three fundamen-tal frequencies of the gravitational waves generated while the inspiraling object is on aninstantaneous geodesic orbit; (2) in the harmonic structure of the gravitational-wave emis-sion, i.e., the relative amplitudes and phases of the various harmonics of the fundamentalfrequencies; and (3) in the evolution of these frequencies and amplitudes with time as theobject inspirals. A full analysis of the accuracies that could be achieved in observationswould involve computing gravitational waveforms in the bumpy spacetimes, performing aFisher-Matrix analysis to account for parameter correlations, and comparing to a similaranalysis for Kerr. That is beyond the scope of this paper. However, we can examine thefirst of these observational consequences by comparing the fundamental frequencies betweenthe bumpy and Kerr spacetimes.

The complication in such an analysis is to identify orbits between di!erent spacetimes.

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-4

-2

0

2

4

-0.06 -0.04 -0.02 0 0.02 0.04 0.06

Δ pρ

MΩφ

q = -0.5q = -1

q = -5q = 0.5

q = 1q = 5

FIG. 17: Di!erence between periapsis precessions in a bumpy spacetime with ! = 0 and the

Schwarzschild spacetime, "p!(#", q) = p!(#", q) ! p!(#", q = 0).

the precession (see Eq. (B15) in the Appendix) then gives

p! =

!

3 (M!0)2

3 ! 4! (M!0) +3

2

"

9 + !2 + q#

(M!0)4

3 + · · ·$

+%

2 (M!0)2

3 ! 4! (M!0) + 2"

9 + !2 + q#

(M!0)4

3 + · · ·& !" ! !0

!0

+

!

!1

3(M!0)

2

3 +1

3

"

9 + !2 + q#

(M!0)4

3 + · · ·$ !

!" ! !0

!0

$2

+ · · ·

= b0 + b1!" ! !0

!0+ b2

!

!" ! !0

!0

$2

+ · · · (24)

In this kind of expansion the multipole moments again contribute at all orders. However,provided the initial frequency !0 " 1, the dominant piece of the constant term, b0, is(M!0)

2

3 , so this term can be used to estimate M . Similarly, the dominant piece of 2b0 ! 3b1

is 4! (M!0), so this can be used to estimate !, and that estimate of ! can be used to improve

the estimate of M from b0. The dominant piece of b0 ! b1 + 3b2 is (9 + !2 + q) /2 (M!0)4

3 ,so this can be used to estimate the excess quadrupole moment q and so on. In the sameway, if an eccentric inspiral is observed in a regime where the initial frequency is small (andhence the frequency at capture was also small), we can use the same type of expansion anduse combinations of the coe"cients to successively extract each multipole moment and theinitial eccentricity. To do this requires an expansion of e2 ! e2

0 as a function of !"/!0 ! 1.The necessary derivatives de2/d(M!") are known in the weak-field, and to lowest orderin the multipoles (see, for example, reference [22]). However, this calculation is somewhatinvolved, so we leave it for a future paper.

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-1

-0.5

0

0.5

1

1.5

2

-0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25

Δ p

z

MΩφ

q = -0.5q = -1

q = -5q = 0.5

q = 1q = 5

FIG. 19: Di!erence between orbital-plane precessions in a bumpy spacetime with ! = 0.9 and the

Kerr spacetime with ! = 0.9, "pz(#!, q) = pz(#!, q) ! pz(#!, q = 0).

from large distances. Most astrophysically relevant orbits are regular and appear to possessan approximate fourth integral of the motion, and the orbits are tri-periodic to high accu-racy. The deviations of the central body from Kerr then manifest themselves only in thechanges in the three fundamental frequencies of the motion and the relative amplitude ofthe di!erent harmonics of these frequencies present in the gravitational waves. For nearlycircular, nearly equatorial orbits, the dependence of the precession frequencies on the orbitalfrequency is well fit by a combination of a weak field expansion that encodes the multipolemoments at di!erent orders, plus a term that diverges as the innermost stable circular orbitis approached. The frequency of the ISCO and its nature (whether it is defined by a radialor vertical instability) is another observable signature of a non-Kerr central object.

To derive these results, we have focussed on a particular family of spacetimes due toManko and Novikov [8]. However, we expect the generic features of the results in the weakfield and as the ISCO is approached to be true for a wide range of spacetimes. Chaos has beenfound for geodesic motion in several di!erent metrics by various authors [9, 10, 11, 12, 13].In all cases, however, the onset of chaos was qualitatively similar to what we found here— it occurred only very close to the central object, and for a very limited range of orbitalparameters. The conclusion that gravitational waves from ergodic EMRIs are unlikely to beobserved is thus probably quite robust.

Precessions for spacetimes that deviate from the Kerr metric have also been consideredby several authors [2, 6, 7]. Our results agree with this previous work in the weak-field as itshould. However, the results in the present paper are the first that are valid in the strong-field since previous work was either based on a weak-field expansion [2] or a perturbativespacetime [6, 7]. The main feature of the precessions in the strong-field — the divergence of

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10−6 10−5 10−4 10−3 10−2 10−110−1

100

101

102

103

104

105

106

d

Accu

mul

ated

cyc

le d

iffer

ence

3.5 PN vs 3 PN, M=1e4 Msun3.5 PN vs 3 PN, M=1e5 Msun3.5 PN vs 3 PN, M=1e6 Msun3.5 PN vs EMRI, M=1e4 Msun3.5 PN vs EMRI, M=1e5 Msun3.5 PN vs EMRI, M=1e6 Msun

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Measuring the mass quadrupoleCan measure mass quadrupole moment to around 20% of Kerr value with Advanced LIGO [Brown et al., 2007, PRL 99, 201102]

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Waveforms are a problem: both post-Newtonian and self-force waveforms currently fail in the intermediate regime [IM and Gair, 2009, PRD 72 084025]

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IMRI: null-hypothesis test of Kerrness

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[Rodriguez, IM, Gair, PRD 85, 062002]

UKZN: Jan 20, 2014

Summary Advanced LIGO is likely to observe GWs from NS-NS,

NS-BH, BH-BH coalescences; tens or more coalescences may be observed according to some models, including signatures of dynamical formation

GW detections and upper limits for compact-object coalescences will allow us to constrain astrophysical parameters through comparisons with model predictions

Extreme- or intermediate- mass-ratio inspirals can serve as precise tests of General Relativity

There’s lots of work to be done in order to make true GWastrophysics a reality!

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