Gravitational Waves from Massive Black-Hole Binaries

Stuart Wyithe (U. Melb)

NGC 6420

Outline

• The black-hole - galaxy relations.

• Regulation of growth during quasar phase.

• The quasar luminosity function.

• Evolution of the BH mass function.

• Rate of gravity wave detection (LISA).

• The gravity wave back-ground.

• The occupation fraction of SMBHs in halos and GW predictions.

Black Hole - Galaxy Relations

Ferrarese (2002)

5c

5/3halobh vMM ∝∝

• Quasar hosts at high z are smaller than at z=0 (Croom et al. 2004).

The Black Hole-Bulge Relationship

The Black Hole-Bulge Relationship

• Radio quiet QSOs conform to the Mbh-* with little dependence on z (Shields et al. 2002).

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Φ(L B, )z = dΔM halo

35ε

tdyn

5.7 ×103

dnps

(d Mhalo

−ΔMhalo)

d2Nmerge

dΔM halodt(M

halo−ΔM

halo)

0

0.5M halo∫

Three assumptions:• One quasar episode per major merger.

• Accretion at Eddington Rate with median spectrum.

• Hypothesis: Black-Hole growth is regulated by feedback over the dynamical time.

Model Quasar Luminosity Function

Wyithe & Loeb (ApJ 2003)

This hypothesis provides a physical origin for the Black-Hole mass scaling.

The dynamical time is identified as the quasar lifetime.

( )5/25/3halo

5cirbh z1MvM +∝∝⇒

Wyithe & Loeb (ApJ 2003;2004)

Model Quasar Luminosity Function.

clustering of quasars

• The black-hole -- dark matter halo mass relation agrees with the evolution of clustering.

• The galaxy dynamical time reproduces the correct number of high redshift quasars.

Properties of Massive BHs• Ubiquitous in galaxies >1011Msolar at z~0.

• Tight relation between Mbh and * (or vc, Mhalo).

• Little redshift evolution of Mbh~f(*) to z~3.

• Feedback limited growth describes the evolution of quasars from z~2-6.

• Massive BHs (Mbh>109Msolar) at z>6.

• Is formation via seed BHs at high z or through continuous formation triggered by gas cooling?

• What is the expected GW signal?

Evolution of Massive BHs

• Were the seeds of super-massive BHs the remnant stellar mass BHs from an initial episode of metal free star formation at z~20?

• The BH seeds move into larger halos through hierachical merging.

Evolution of Massive BHs

• Is super-massive BH formation ongoing and triggered by gas cooling inside collapsing dark-matter halos?

BH Evolution Triggered by Gas Cooling

• Prior to reionization, cooling of gas inside dark-matter halos is limited by the gas cooling thresh-hold (104K for H).

• Following reionization the infall of gas into dark-matter halos is limited by the Jeans Mass.

•

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≈108 1+ z20 ⎛ ⎝ ⎜

⎞ ⎠ ⎟-3/2

M solar solar5vir

-3/210 M

K10

T

10

z110 ⎟

⎠

⎞⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛ +≈

High z Reionisation Low z

• Reionization may affect BH formation in low mass galaxies as it does star formation.

Merging Massive BHs• Satellite in a virialized halo sinks on a timescale (Colpi et al.

1999)

• Allow at most one coalescence per tsink.

• BBHs in some galaxies will converge within H-1

• Coalescence more rapid in triaxial galaxies.

• Brownian motion of a binary black hole results in a more rapid coalescence.

• We parameterise the hard binary coalescence efficiency by εmrg.

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tsink ≈0.25H−1 M +ΔMΔM

LISA GW Event Rate (hc>10-22 at fc=10-3Hz)

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d2Ngw

dtdz= dM dΔMΘ ,M Δ ,M fc,hc,z( )×S ,z M bh,ΔM bh( )

0

M∫0

∞∫

×dnbh

dM× d2N

dΔ MdtM

dnbhdΔ M

dnps

dΔ M

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

εmrg

1+ z4π d2V

dzdΩ

• An event requires the satellite galaxy to sink, rapid evolution through hard binary stage, and a detectable GW signal.

Number counts resulting from BH seeds

Number counts resulting from continuous BH formation

Characteristic Strain Spectrum

• hspec<10-14 (current)

• hspec<10-15.5 (PPTA)

€

Sh( )f = dh0

∞∫ dz0

∞∫ h2dΦ

dhdf( )4z π d2V

dzdΩ

(f)fS(f)h hspec =

Jenet et al. (2006)

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M bh

M halo

=1.2ε0M halo

1012

⎛ ⎝ ⎜

⎞ ⎠ ⎟

23

Ferrarese (2002):ε0=10-5.0 =5.5

WL (2002):ε0=10-5.4 =5.0

hspec is Sensitive to the Mbh-vc Relation

Sesna et al. (2004)

Massive Black-Holes at low z Dominate GW Back Ground

Black-Hole Mass-Function

• The halo mass-function over predicts the density of local SMBHs.

• Most GWBG power comes from z<1-2.

Model Over-Predicts Low-z Quasar Counts at High Luminosities

Galaxy Occupation Fraction• The occupation

fraction is the galaxy LF / halo MF

• Assume 1 BH/galaxy

Reduced GW Background

• Inclusion of the occupation fraction lowers the predicted GW background by 2 orders of magnitude.

Conclusions• The most optimistic limits on the spectrum of strain of

the GW back-ground are close to expected values. Tighter limits or detection of the back-ground may limit the fraction of binary BHs.

• Allowance should be made for the occupation of SMBHs in halos, which lower estimates of the GW background based on the halo mass function by 2 orders of magnitude.

• Models are very uncertain! PTAs will probe the evolution of the most massive SMBHs at low z.

Limits on the GW Back-Ground

• Pulsar Timing arrays limit the energy density in GW.

Ωgwh2<2x10-9

(Lommen 2002)

dlnf

dñ

ñ

1(f)Ù GW

critGW =

• Atomic hydrogen cooling provides the mechanism for energy loss that allows collapse to high densities.

• This yields a minimum mass in a neutral IGM.

Minimum Halo Mass for Star formation

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Mmin =108 1+ z

10

⎛

⎝ ⎜

⎞

⎠ ⎟−

3

2Msolar

• Assume gas settles into hydrostatic equilibrium after collapse into a DM halo from an adiabatically expanding IGM.

• This yields a minimum mass in an ionized IGM.

Minimum Halo Mass for Baryonic Collapse

€

δb =ρ b

ρ b−1= 1+

6

5

Tvir

T

⎛

⎝ ⎜

⎞

⎠ ⎟

3

2−1 ⇒ Tvir >17.2T (δb >100)

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Mmin = 5 ×109 1+ z

10

⎛

⎝ ⎜

⎞

⎠ ⎟−

3

2Msolar

(Dijkstra et al. 2004)

• A minimum mass is also seen in simulations. The minimum mass is reduced at high redshift.

Minimum Halo Mass for Baryonic Collapse

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z=11

z=2

Median Quasar Spectral Energy DistributionElvis et al. (1994); Haiman & Loeb (1999)

• The median SED can be used to compute number counts.

• The SED can also be used to convert low luminosity X-ray quasar densities to low luminosity optical densities.

Binary BH Detection by LISA

104

107

106

105

10-1.5Hz10-3.5Hz

Black-holes at high z accrete near their Eddington Rate

)(fSf3H

2∂(f)Ù h

320

2

GW =

A BBH in a pair of Merging Galaxies (NGC 6420; Komossa et al. 2003)

Gravitational Waves from BBHs• The observable is a strain amplitude

• In-spiral due to gravitational radiation.

( )1620

1/3bhbh

bhbh2/3

c 1010ÄMM

ÄMM

R(z)

fh −− −≈

+∝

( )5/3

5

1sec

PP10Pt ⎟

⎠

⎞⎜⎝

⎛≈

Merger Rates for DM Halos

(M)Mdtd

Nd2

Δ

k

δ

δcrit(z)

Large M Small M

Time

Lacey & Cole (1993)

The Press-Schechter Mass Function

Z=30Z=0

• Reionization may affect BH formation in low mass galaxies as it does starformation.

Binary Evolution Timescales (Yu 2002)

• BBHs in some galaxies will converge within H-1

• Coalescence more rapid in triaxial galaxies.

• Residual massive BH binaries have P>20yrs and a>0.01pc.

Merging Massive BHs• Satellite in a virialized halo sinks on a timescale (Colpi et

al. 1999)

• Allow at most one coalescence during the decay plus coalescence times.

ÄM

ÄMM0.25H

å

ÄMÄMM

lne

ÄMÄMM

v

r1.2t

1

0.4

c

virdecay

+≈

⎟⎠⎞

⎜⎝⎛ +

⎥⎦⎤

⎢⎣⎡

+⎟⎟⎠

⎞⎜⎜⎝

⎛≈

−

Reduced Event Rate

• Inclusion of the occupation fraction lowers the predicted event rate by an order of magnitude.