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Scott A. Hughes, MIT
Using gravitational-wave observations of black hole mergers to understand the growth of structure
Gravitational waves from the coalescence of massive black holes as cosmological tools
and the large-scale geometry of our universe...
Cosmological Frontiers of Fundamental Physics, 14 June 2011
... and how well we expect to be able to do this as plans for a space-based
GW detector evolve.
Scott A. Hughes, MIT
Gravity grows overdensities: A slight overdensity at z = 1100 will become progressively more dense as
that region attracts more matter to itself.Initial overdensity is tiny: δρ/ρ ~ 10-6. Evolution accurately described using simple linear theory
(Zeldovich 1970, A&A, 5, 84)
Cosmic microwave background gives first glimpse of the universe’s
largest structures
A brief history of galaxies
Cosmological Frontiers of Fundamental Physics, 14 June 2011
Scott A. Hughes, MIT
Linear evolution not accurate when δρ/ρ ~ 1!
http://www.mpa-garching.mpg.de/galform/virgo
Further evolution: Large-scale
N-body simulations
A brief history of galaxies
Credit:The VIRGO
Cosmological N-body Project
Cosmological Frontiers of Fundamental Physics, 14 June 2011
Scott A. Hughes, MIT
Zoom through matter field: Find that structure
is hierarchical. Big structures built from the
repeated merger of smaller structures.
A brief history of galaxies
http://www.mpa-garching.mpg.de/galform/virgo
Galaxies and their host halos merge often,
especially at moderate to high redshift.
Cosmological Frontiers of Fundamental Physics, 14 June 2011
Scott A. Hughes, MIT
Black holes rapidly grew: We see quasars at redshift z > 6.
Most extreme example known: CFHQS J2329-0301 at z = 6.43 ... luminosity
implies MBH ~ 5 x 108 Msun.
Comparable to largest black holes at z = 0 ... but universe was only 700 million years old.
Mystery: How did these black holes get so massive so quickly?
High redshift quasars: Early structure hosts massive black holes
Cosmological Frontiers of Fundamental Physics, 14 June 2011
Scott A. Hughes, MIT
Early rapid growth of black holesVery first black holes: Poorly understood. Models suggest they are remnants of Pop III
stars (roughly 100 Mo) or due to direct collapse of metal-poor, low angular momentum gas which falls into an extreme
overdensity (roughly 105 Mo).
Cosmological Frontiers of Fundamental Physics, 14 June 2011
Image: Tom Abel, Physics Today, April 2011
These need to grow to ~109 Mo in roughly 108 years in
order to fit observations of high redshift quasars.
Simulation of first star at z ~ 15. Likely to leave ~100 Mo
remnant following SN.
Scott A. Hughes, MIT
Multiple mergers allow holes to grow from “seed”
to quasar scale
Early rapid growth of black holesSolution: Mergers.
Cosmological Frontiers of Fundamental Physics, 14 June 2011
Merging of structures which host first black holes brings in a great deal of gas: Catalyzes fierce star formation,
fuels accretion, promotes rapid black hole growth.
Also very naturally leads to formation and merger of binary black holes ...
Li et al 2006, Astrophysical Journal 665, 187
Gravitational waves and MBBH
Scott A. Hughes, MIT
Model the evolution of black hole binaries formed through the
mergers of dark matter halos via merger tree.
Each curve: Different assumptions about seed holes and their growth. All are consistent with AGN optical luminosity on the range 1 < z < 6.
Cosmological Frontiers of Fundamental Physics, 14 June 2011
Sesana, Volonteri, & Haardt 2007 MNRAS 377, 1711
Gravitational waves and MBBH
Scott A. Hughes, MIT
Sesana, Volonteri, & Haardt 2007 MNRAS 377, 1711
Model rates peak at z > 2 for mBH ~ 105
Msun and smaller, at z < 2 for mBH ~ 106 Msun and bigger.
Key point: Even in conservative models,
there are many events per year ... but need instrument which can reach to z ~ 5 — 10.
Cosmological Frontiers of Fundamental Physics, 14 June 2011
Character of BBH waves
Scott A. Hughes, MIT
Waves sweep across band from low (set by detector) to a high “merger” frequency (corresponds to binary’s members merging into a single body):
Sweep rate set by energy loss due to gravitational waves; depends on total mass and mass ratio:
f =48
5!
c5
G5/3µM
5/3
tot (2!f)11/3
For binaries in 105 Mo < Mtot < 107 Mo, takes weeks to a year to sweep from 10-4 Hz to merger.
fmerge !c3
GMtot
[2 " 6]!3/2
!= (0.02 " 0.004) Hz
!
106 M"
Mtot
"
Cosmological Frontiers of Fundamental Physics, 14 June 2011
Scott A. Hughes, MIT
“Classic” LISA band and sensitivity
Sensitive band runs from about 10-4 Hz to 0.1 Hz.
3 spacecraft in passive heliocentric orbits, hold
roughly equilateral triangle lagging earth by 20º, tilted
to ecliptic by 60º.
Cosmological Frontiers of Fundamental Physics, 14 June 2011
Scott A. Hughes, MIT
The eventual successor to LISA ...As Schutz described in the opening talk, the design for LISA is in flux! The design which is likely to be the focus for the next decade or so will:
Cosmological Frontiers of Fundamental Physics, 14 June 2011
•Have shorter arms (will not go as low in frequency — implications on the mass of black holes it can “hear”)
•Not have links between all pairs of spacecraft (cannot measure both polarizations at once)
•Have smaller telescopes (reducing GW sensitivity)•Have different orbits (probably reduced mission
duration; different sensitivity to GW polarization).Need detailed studies to see how this impacts binary
black hole measurements ... now underway.Rest of talk: Results based on “classic” LISA, highlighting what is likely to change and how.
Scott A. Hughes, MIT
Want to assess how well we can learn about the system generating GWs.
Recipe: 1. Build model for GWs; 2. Run model through detector response; 3. Use result to see what we can learn by measuring waves.
Past work focussing on LISA: Cutler (PRD 1998); Hughes (MNRAS 2002); Vecchio
(PRD 2004); Berti, Buonanno, & Will (PRD 2005), Kocsis et al (PRD 2007, ApJ 2008).
How well can we measurethese waves?
Results presented here taken from Lang & Hughes[PRD 74, 122001 (2006); ApJ 677, 1184 (2008)]
Cosmological Frontiers of Fundamental Physics, 14 June 2011
Scott A. Hughes, MIT
Break coalescence into 3 epochs:
Main focus of this talk. LISA measured inspirals will last months to years — very rich structure, measured with high signal-to-noise, makes it possible to study
source characteristics with great precision.
Inspiral: Slow evolution driven by GW loss of orbital energy and
angular momentum.
Wave model
Cosmological Frontiers of Fundamental Physics, 14 June 2011
Wave model
Scott A. Hughes, MIT
Break coalescence into 3 epochs:
Merger: Extremely violent dynamics of
spacetime: Two black holes smash together, leaving one behind.Numerical relativity can now directly model the merger and
its waves ... find that the effective one-body approach gives an outstanding analytic tool for describing these waves.
Cosmological Frontiers of Fundamental Physics, 14 June 2011
T. Damour, Int. J. Mod. Phys. A 23, 1130 (2008)T. Damour and A. Nagar, Phys. Rev. D 79, 081503 (2009)Y. Pan et al, Phys. Rev. D 81, 084041 (2010); also arXiv:1106.1021.
Wave model
Scott A. Hughes, MIT
Break coalescence into 3 epochs:Ringdown: Last
wiggles of the merger, takes the form of a damped sinusoid.Simply described using black hole perturbation theory.
Expect mix of modes; each mode’s frequency and damping time set by final mass and spin.
Measure mixture of modes — measure final mass and spin. With “Classic” LISA, can be done with excellent
precision [Berti, Cardoso, Will, PRD 74, 061503 (2006)].Cosmological Frontiers of Fundamental Physics, 14 June 2011
Scott A. Hughes, MIT
Break coalescence into 3 epochs:
Remainder: Dissection of the inspiral waves for binaries whose members are comparable
mass. Focus on how characteristics of binary are imprinted on the waveform.
Inspiral: Slow evolution driven by GW loss of orbital energy and
angular momentum.
Wave model
Cosmological Frontiers of Fundamental Physics, 14 June 2011
Pieces of inspiral waveform
Scott A. Hughes, MIT
1. Phase. Depends on how rapidly the orbit evolves. Rate is controlled by binary’s masses and spins.
Measure the phase, measure masses and spins.
Cosmological Frontiers of Fundamental Physics, 14 June 2011
Scott A. Hughes, MIT
To get that, need relativistic equations of motion. Post-Newtonian expansion of general
relativity gives us a good form for inspiral:
Lowest order piece: Newtonian gravity
Phase: Comes from integrating up the (relativistic analog of) Kepler’s law
Post-Newton gives corrections in v/c.
Cosmological Frontiers of Fundamental Physics, 14 June 2011
Scott A. Hughes, MIT
... and more corrections ...
Cosmological Frontiers of Fundamental Physics, 14 June 2011
Scott A. Hughes, MIT
[Blanchet 2006, Liv Rev Rel 9, 4, Eq. (168)]
... and a few more.
Cosmological Frontiers of Fundamental Physics, 14 June 2011
Scott A. Hughes, MIT
Integrate up this motion: 103 - 105 radians of phase accumulate over measurement.
Pieces of inspiral waveform
Cosmological Frontiers of Fundamental Physics, 14 June 2011
Key feature: The phase depends on — and thus encodes — masses & spins of the binary’s members.
Measure phase: Measure masses and spins.
Dynamical inclination
Scott A. Hughes, MIT
Relativistic effect: a “magnetic-type” coupling of mass currents to spacetime.
Creates new “forces”, modifying orbit acceleration; also causes spins of binary’s members to precess.
“Gravitomagnetic” field due to
orbital motion
“Gravitomagnetic” field due to other
body’s spin
Cosmological Frontiers of Fundamental Physics, 14 June 2011
Scott A. Hughes, MIT
Creates new “forces”, modifying orbit acceleration; also causes spins of binary’s members to precess.
Angular momentum is globally conserved:
J = L + S1 + S2 = constantMeans that the orbital plane precesses to compensate.
(Known as Lense-Thirring precession in weak-field.)
Relativistic effect: a “magnetic-type” coupling of mass currents to spacetime.
Dynamical inclination
Cosmological Frontiers of Fundamental Physics, 14 June 2011
Scott A. Hughes, MIT
Precession of angular momentum vectors in a binary black hole system.
Dynamical inclination
Cosmological Frontiers of Fundamental Physics, 14 June 2011
Scott A. Hughes, MIT
Inspiral measurements
Example waveform: Both black holes non-
spinning.
Smooth chirp from low to high frequencies.
Cosmological Frontiers of Fundamental Physics, 14 June 2011
Scott A. Hughes, MIT
Inspiral measurements
Spins cranked up!Spin 1 = Spin 2 = 99% maximum
Strong frequency and amplitude modulation gives spin precision.
Cosmological Frontiers of Fundamental Physics, 14 June 2011
Scott A. Hughes, MIT
Survey how well parameters can be measured using Monte Carlo simulations:
Binaries at z = 1
m1 = 106 Mo
m2 = 3 x 105 Mo
Results for “Classic” LISA
Precision of mass determination: Peaks at ~0.1% relative error, distribution confined to
< 2 - 3%.
Similar results at higher redshift,
degrading ~1/DL.Cosmological Frontiers of Fundamental Physics, 14 June 2011
Scott A. Hughes, MIT
Binaries at z = 1
m1 = 106 Mo
m2 = 3 x 105 Mo
Precision of spin determination: Peaks at < 1%
absolute error, distribution confined to
< 10%.Degrades as 1/DL as we go to higher
redshift.Cosmological Frontiers of Fundamental Physics, 14 June 2011
Survey how well parameters can be measured using Monte Carlo simulations:
Results for “Classic” LISA
Scott A. Hughes, MIT Cosmological Frontiers of Fundamental Physics, 14 June 2011
Deep survey of many plausible binaries over range of z:Results for “Classic” LISA
NOTE: Results are robust because these parameters are encoded by the phase. ANY LISA mission will measure these parameters well — the “Classic” results are a good guide to what will eventually be accomplished.
Δm/m = 10-4 — several x 10-2 at z ≤ 3; degrades to percent to several tens of percent at high z.Similar results for Kerr spin parameter a = |S|/m2.
Gravitational-wave observations will be able to track with high precision the evolution of black
hole masses and spins from the earliest moments of cosmic history.
Pieces of inspiral waveform
Scott A. Hughes, MIT
2. Inclination of orbital plane to line of sight.Measure both polarizations, you measure this angle.
1. Phase. Depends on how rapidly the orbit evolves. Rate is controlled by binary’s masses and spins.
Measure the phase, measure masses and spins.
Cosmological Frontiers of Fundamental Physics, 14 June 2011
Naively very hard to do!
Scott A. Hughes, MIT
Orbital inclination is degenerate with sky position! Must break degeneracy to measure location of binary.
Problem is that what we measure is a linear combination of the two polarizations:
Solution in multiple parts:1. Motion of LISA makes θ, ϕ, ψ time dependent. Modulation untangles these angles.2. Inclination angle ι oscillates when spin effects are taken into account.
Cosmological Frontiers of Fundamental Physics, 14 June 2011
hmeas = F+(!, ", #)h+($) + F!(!, ", #)h!($)
Breaking degeneracy
Scott A. Hughes, MIT
hmeas = F+(!, ", #)h+ + F!(!, ", #)h!
1. Motion-induced modulation of waveform.
2. Spin-precession-induced modulation of waveform.3. “Higher harmonics”: Sketch of waveform given before is only the leading quadrupole harmonic. Other harmonics also contribute:
h = hl=2 +
!
v
c
"
hl=3 +
!
v
c
"2
hl=4 + . . .
v/c can be large, SNR is large — “corrections” crucial!These terms encode inclination differently than leading
harmonic ... helps to pin down position.
Cosmological Frontiers of Fundamental Physics, 14 June 2011
Scott A. Hughes, MIT
Example localizations, selected from Monte Carlo sample. Shows
time evolution of localization for a z = 1 source. Typically find
degree-scale accuracy 1 month before merger, several to several tens
of arcminutes at merger.
Cosmological Frontiers of Fundamental Physics, 14 June 2011
Survey how well parameters can be measured using Monte Carlo simulations:
Results for “Classic” LISA
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Scott A. Hughes, MIT Cosmological Frontiers of Fundamental Physics, 14 June 2011
Results for “Classic” LISADeep survey of many plausible binaries over range of z:
NOTE: These results are NOT robust — sky position parameters are encoded in amplitude, which is
far more delicate to measure than phase.
Our ability to determine the position of merging binaries is very likely to be highly degraded relative to “classic” LISA results.
z ~ 1: binaries localized to several to several tens of arcminutes at merger ... and can be localized to a degree or two a few weeks prior to merger.z ~ 5: binaries localized to a few degrees at merger ... localization quite poor prior to merger.
Pieces of inspiral waveform
Scott A. Hughes, MIT
2. Inclination of orbital plane to line of sight.Measure both polarizations, you measure this angle.
3. Luminosity distance. Sets amplitude, once masses and inclination are determined.
1. Phase. Depends on how rapidly the orbit evolves. Rate is controlled by binary’s masses and spins.
Measure the phase, measure masses and spins.
Cosmological Frontiers of Fundamental Physics, 14 June 2011
Pieces of inspiral waveform
Scott A. Hughes, MIT
By measuring GW phase and the amplitudes of both polarizations, the luminosity distance to cosmic events
can be directly determined.
Binary GWs are standard candles (“sirens”)... standardized by general relativity.
Cosmological Frontiers of Fundamental Physics, 14 June 2011
Distance directly measured ...but redshift is not.
Scott A. Hughes, MIT
Consider nearby source: Phase measures timescale for orbit change; tells us mass scale:
Consider cosmological source: Now measure a redshifted timescale; infer redshifted mass:
Redshift is degenerate with masses!!True when taken to higher order as well ... cannot infer redshift from GW measurables.
Cosmological Frontiers of Fundamental Physics, 14 June 2011
Opposite of “normal” astronomy
Scott A. Hughes, MIT
Usual situation: Redshift is direct measurable (assuming you measure and identify lines);
distance must be inferred indirectly.
GWs: Distance is direct measurable; redshift must be measured some other way.
Two options:1. Assume an underlying cosmology. Can then invert distance/redshift relation, infer z.
2. Measure an “electromagnetic” counterpart to the GW event. Directly measure both redshift and distance to event.
Cosmological Frontiers of Fundamental Physics, 14 June 2011
Standard “siren”
Scott A. Hughes, MIT
If we can simultaneously measure distance (with GWs) and redshift (with photons), then we can
independently probe the distance-redshift relation: A calibration-free standard “siren.”
Accuracy in principle quite high ... 2 binary black hole sources could constrain cosmological geometry as well as 3000 supernovae (Holz & Hughes 2005).
Cosmological Frontiers of Fundamental Physics, 14 June 2011
In practice, gravitational lensing degrades sirens ... main utility with LISA likely independent systematic check on (more numerous!) standard standard candles.
Conclusion
Scott A. Hughes, MIT
Massive black hole binaries: Tracers of the growth of cosmic structure.
Hierarchical growth of structure probably guarantees an interesting
astrophysical event rate
Cosmological Frontiers of Fundamental Physics, 14 June 2011
Even a variant of LISA which is degraded relative to the“classic” design will provide precision data about the masses and spins of the first black holes which grew in our universe.