coalescence of massive black holes as cosmological tools · Cosmological Frontiers of Fundamental...

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.


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



Linear evolution not accurate when δρ/ρ ~ 1!

http://www.mpa-garching.mpg.de/galform/virgo

Further evolution: Large-scale

N-body simulations


Credit:The VIRGO

Cosmological N-body Project





Zoom through matter field: Find that structure

is hierarchical. Big structures built from the

repeated merger of smaller structures.



Galaxies and their host halos merge often,

especially at moderate to high redshift.





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



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).


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.


Multiple mergers allow holes to grow from “seed”

to quasar scale

Early rapid growth of black holesSolution: Mergers.


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


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.


Sesana, Volonteri, & Haardt 2007 MNRAS 377, 1711

Gravitational waves and MBBH


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.


Character of BBH waves


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

"



“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.



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:


•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.


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)]



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


Wave model



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.


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


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



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


Pieces of inspiral waveform


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.



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.



... and more corrections ...



[Blanchet 2006, Liv Rev Rel 9, 4, Eq. (168)]

... and a few more.



Integrate up this motion: 103 - 105 radians of phase accumulate over measurement.



Key feature: The phase depends on — and thus encodes — masses & spins of the binary’s members.

Measure phase: Measure masses and spins.

Dynamical inclination


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



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.




Precession of angular momentum vectors in a binary black hole system.




Inspiral measurements

Example waveform: Both black holes non-

spinning.

Smooth chirp from low to high frequencies.



Inspiral measurements

Spins cranked up!Spin 1 = Spin 2 = 99% maximum

Strong frequency and amplitude modulation gives spin precision.



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


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



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.



2. Inclination of orbital plane to line of sight.Measure both polarizations, you measure this angle.




Naively very hard to do!


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.


hmeas = F+(!, ", #)h+($) + F!(!, ", #)h!($)

Breaking degeneracy


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.



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.




−100 0 100

−100

0

100

−50

50

(a)

−100 0 100

−100

0

100

−50

50

(b)

−50 0 50−50

0

50

−25

25

(c)

−100 0 100−100

0

100

−50

50

(d)

−50 0 50

−50

0

50

(e)

−50 0 50

−50

0

50

(f )

−200 0 200−200

0

200

−100

100

(g)

−50 0 50 −50

−25

0

25

50(h)

−100 0 100−100

0

100

−50

50

(i)

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.



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.






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.


Distance directly measured ...but redshift is not.


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.


Opposite of “normal” astronomy


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.


Standard “siren”


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).


In practice, gravitational lensing degrades sirens ... main utility with LISA likely independent systematic check on (more numerous!) standard standard candles.

Conclusion


Massive black hole binaries: Tracers of the growth of cosmic structure.

Hierarchical growth of structure probably guarantees an interesting

astrophysical event rate


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.

coalescence of massive black holes as cosmological tools · Cosmological Frontiers of Fundamental...

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