Evolution of binarysupermassive black holes:

the final parsec problem is solved

Eugene Vasiliev

Lebedev Physical Institute, Moscow

Based on Vasiliev, Antonini & Merritt 2014, 2015 (in prep.)

29 April 2015

Plan of the talk

Why binary supermassive black holes?

Evolutionary stages of binary black holes

Loss cone theory in spherical and non-spherical systems

N-body simulations and their limitations

The novel Monte Carlo simulation method

Results and analysis of Monte Carlo simulations

Conclusions

Origin of binary supermassive black holes (SBH)

I Most galaxies are believed to host central massive black holes

I In the hierarchical merger paradigm, galaxies in the Universehave typically 1–3 major and multiple minor mergers in theirlifetime

I Every such merger brings two central black holes from parentgalaxies together to form a binary system

I We dont see much evidence for widespread binary SBH (tosay the least) – therefore they need to merge rather efficiently

I Merger is a natural way of producing huge black holes fromsmaller seeds

Evolutionary track of binary SBH

I Merger of two galaxies creates a common nucleus; dynamicalfriction rapidly brings two black holes together to form abinary (distance: a ∼ 10 pc)

I Three-body interaction of binary with stars of galactic nucleusejects most stars from the vicinity of the binary by theslingshot effect; a “mass deficit” is created and the binarybecomes “hard” (a ∼ 1 pc)

I The binary further shrinks by scattering off stars that continueto flow into the “loss cone”, due to two-body relaxation orother factors

I As the separation reaches ∼ 10−2 pc, gravitational wave(GW) emission becomes the dominant mechanism that carriesaway the energy

I Reaching a few Schwarzschild radii (∼ 10−5 pc), the binaryfinally merges

Evolutionary stages and timescales

10-3

10-2

10-1

100

101

102

103

104

105

0 500 1000 1500 2000 2500 3000

1/a

(pc-1

)

T (Myr)

a−1hard

a−1GW

Loss cone evolution

Initial hardeningGW emission

inve

rse

sem

ima

jor

axi

s

[from Khan+ 2012]

Gravitational slingshot and binary hardening

A star passing at a distance . 2a from the binaryexperiences a complex 3-body interaction whichresults in ejection of the star with velocity

vej ∼√

m1m2

(m1 + m2)2vbin.

These stars carry away energy and angular momentumfrom the binary, so that its semimajor axis a decreases:

d

dt

(1

a

)≈ 16

G ρ

σ≡ Sfull [Quinlan 1996]

Thus, if density of field stars ρ remains constant, the binaryhardens with a constant rate.However, the reservoir of low angular momentum stars which canbe ejected is finite and may be depleted quickly, so that the binarystalls at a radius astall ∼ (0.1− 0.4)ah.

Loss cone theory

The region of phase space with angular momentumL2 < L2

LC ≡ 2G (m1 + m2) a is called the loss cone.Gravitational slingshot eliminates stars from the loss cone in oneorbital period Torb. The crucial parameter for the evolution is thetimescale for repopulation of the loss cone.In the absence of other processes, the repopulation time is

Trep ∼ TrelL2

LC

L2circ

, where Trel =0.34σ3

G 2 m? ρ? ln Λis the relaxation time.

If Trep . Torb, the loss cone is full (refilled faster than orbitalperiod). In real galaxies, however, the opposite regime applies –the empty loss cone. In this case the hardening rate

S ≡ d

dt(a−1) ' Torb

TrepSfull.

Relaxation is too slow for an efficient repopulation of the loss cone:in the absense of other processes the binary would not merge ina Hubble time.This is the “final-parsec problem” [Milosavljevic&Merritt 2003]

N-scaling in the empty loss cone regimein

vers

ese

mim

ajo

ra

xis

a−1hard

short-term long-term evolution

Hardening rate S ≡ d

dt(a−1) ∝ T−1

rel ∝ N−1

[from Merritt+ 2007]

Possible ways to enhance the loss cone repopulation

I Brownian motion of the binary (enables interaction with largernumber of stars) [Milosavljevic&Merritt 2001; Chatterjee+ 2003]

I Non-stationary solution for the loss cone repopulation rate[Milosavljevic&Merritt 2003]

I Secondary slingshot (stars may interact with binary severaltimes) [MM03]

I Gas physics – under special circumstances[Lodato+ 2009, Roskar+ 2014]

I Perturbations to the stellar distribution arising from transientevents (such as infall of large molecular clouds, additionalminor mergers and massive black holes, ...)

I Effects of non-sphericity on the orbits of stars in the nucleus[Berczik+ 2006; Preto+ 2011; Khan+ 2011,2012,2013; Vasiliev+ 2014]

Loss cone in non-spherical stellar systems

Angular momentum L of any star is not conserved, but experiencesoscillations due to torques from non-spherical distribution of stars.

Therefore, much larger number of stars can attain low values of Land enter the loss cone at some point in their (collisionless)evolution, regardless of two-body relaxation.

This has led to a conclusion that the loss cone should remain fullin axisymmetric and especially triaxial systems.

0

0.1

0.2

0.3

0.4

0.5

0 2 4 6 8 10 12 14

Axisymmetric

L2

time

loss cone

TubeSaucer

0

0.1

0.2

0.3

0.4

0.5

0 2 4 6 8 10 12 14

Triaxial

L2

time

TubePyramid

Loss cone in non-spherical stellar systems

0.0 0.2 0.4 0.6 0.8 1.0

Lz

0.0

0.2

0.4

0.6

0.8

1.0

L

centrophobic

centrophilic

loss cone

particles that can arrive into the loss cone

not in loss region

spherical

axisymmetric

triaxial

Merger simulations hint for a full loss cone

isolated

merger

isolated

merger

[Preto+ 2011] [Khan+ 2011]

Hardening rates in merger simulations

were found to be N-independent

Evolution of isolated systems in different geometries

We have performed simulations of binaryblack hole evolution in three sets of models:spherical, axisymmetric and triaxial.

In all three cases the hardening rate appears todrop with N in the range 105 . N . 2× 106,but it drops slower in non-spherical cases.

Moreover, this rate is several times lower thanthe rate that would be expected in the full losscone regime.

[Vasiliev, Antonini & Merritt 2014]

I Is there a convergence in the limit N →∞?

I If yes, why the limiting hardening rate seems to bemuch smaller than the full loss cone value?

I Does it stay constant with time, after all?

I Why the results of merger simulations are different?

1

2

5

10

20

104 105 106

har

den

ing

rate

N

N−0.5

Full loss cone

sphericalaxisymmetric

triaxial

0

200

400

600

800

1000

1200

0 20 40 60 80 100

spherical

0 20 40 60 80 100

axisymmetric

8k16k32k64k

125k250k500k

1000k2000k

0 20 40 60 80 100

triaxial

inve

rse

sem

ima

jor

axi

s

time

Evolution of isolated systems in different geometries

We have performed simulations of binaryblack hole evolution in three sets of models:spherical, axisymmetric and triaxial.

In all three cases the hardening rate appears todrop with N in the range 105 . N . 2× 106,but it drops slower in non-spherical cases.

Moreover, this rate is several times lower thanthe rate that would be expected in the full losscone regime.

[Vasiliev, Antonini & Merritt 2014]

I Is there a convergence in the limit N →∞?

I If yes, why the limiting hardening rate seems to bemuch smaller than the full loss cone value?

I Does it stay constant with time, after all?

I Why the results of merger simulations are different?

1

2

5

10

20

104 105 106

har

den

ing

rate

N

N−0.5

Full loss cone

sphericalaxisymmetric

triaxial

0

200

400

600

800

1000

1200

0 20 40 60 80 100

spherical

0 20 40 60 80 100

axisymmetric

8k16k32k64k

125k250k500k

1000k2000k

0 20 40 60 80 100

triaxial

inve

rse

sem

ima

jor

axi

s

time

Problems with direct N-body simulations

I We model galaxies with N? ∼ 1010−12, but our simulations arefeasible only for N ∼ 106;

I therefore, one needs to extrapolate our findings to much higher N;

I but one cannot simply scale the hardening rate to different N:

I the contribution of collisional relaxation to loss cone repopulationscales as N−1, but the collisionless processes are independent of N;

I we cannot afford having much larger N even withthe best hardware and improved algorithms;

I need a simulation method in which we may adjustthe relaxation rate independently of particle number.

I the way to go: combine scattering experiments,evolution of the orbits in a smooth field,and an approximate treatment of relaxation.

[GRAPE cluster in RIT]

Scattering experiments

The interaction between the binary MBH and the field stars maybe described as a sum of individual 3-body scattering events.In each such event, one tracks the changes in energy and angularmomentum of the binary [e.g. Sesana+ 2006,2007], or records thesechanges for the star and adjusts the binary orbit parameters usingthe conservation laws [Meiron&Laor 2012].The population of stars that are able to interact with the binarydoes not stay constant – it needs to be tracked self-consistently:

I stars with low angular momentum are depleted;

I the density cusp around the binary is eroded;

I stars may re-enter the loss cone due to two-body relaxationand non-spherical torques;

I the geometry of non-spherical cusp changes with time.

To address these issues, a model for the evolution of the stellardistribution must be developed.

Self-consistent field methodThis method is used for the study of collisionless systems withmoderate deviation from spherical shape.The idea is to expand both density and potential of the systemusing a suitable (usually orthogonal) set of basis functions whichare themselves solutions of Poisson equation (Hernquist&Ostriker 1992):

ρ(x) =∑

n

Cnρn(x) , Φ(x) =∑

n

CnΦn(x) , ∇2Φn(x) = 4πρn(x) for ∀n.

Usually one takes the basis functions to be products of somefunction in radius and spherical harmonics:Φn(r) = Φn,l(r)Ym

l (θ, φ); n ≡ {n, l ,m}.The coefficients of expansion Cn are computed from the positionsof all N particles as Cn =

∫d3x Φn(x) ρ(x) =

∑Ni=1 Φn(xi )mi .

The simulation workflow is:

1. Compute the coefficients of potential from particle positions;

2. Move particles according to forces obtained by differentiatingthe potential, with a timestep ∆t � Tdyn.

Self-consistent field method

I This method works rather accurately for systems that have awell-defined center and are well approximated by a moderatenumber (∼ few dozen) of expansion terms.

I Since particles do not interact with each other explicitly, but theirmotion is mediated by a smooth potential which represents themean field, this method is well suited for collisionless simulations.

I However, it is not entirely free of numerical relaxation, since thediscreteness noise in the expansion coefficients lead totime-dependent fluctuations in the potential. In fact, themagnitude of numerical relaxation is only a factor of few lower thatfor other methods with the same N.

I A possible way to reduce fluctuations:

1. use longer time intervals between updating the potential expansioncoefficients (but keep short enough timesteps for integrating the orbits);

2. during each update interval, store several sampling points for eachparticle, to increase the effective number of points used in computingcoefficients and hence to reduce discreteness noise.

Temporal smoothing in the SCF method

Using a longer interval between potential recomputation andincreasing the number of sampling points per particle does help toreduce the artificial relaxation rate by 1− 2 orders of magnitude.

−1.0 −0.8 −0.6 −0.4 −0.2 0.0

E

10-8

10-7

10-6

10-5

10-4

⟨ ∆E2⟩ /E

2

−1.0 −0.8 −0.6 −0.4 −0.2 0.0

E

10-8

10-7

10-6

10-5

10-4

⟨ ∆L2⟩

analyticN-body, ǫ=0N-body, ǫ=0.05SCF, no symSCF, triaxRaga, dt=1Raga, dt=10Raga, dt=10,Nsamp=10

Monte Carlo method for collisional stellar systems

Introduced in early 1970s by Spitzer and Henon as an approximateway of treating two-body relaxation phenomena.In the formulation of Spitzer&Hart(1971), the equations of motionof each particle in the system are integrated numerically, and aftereach step a perturbation is applied to the particle’s velocity, usingfirst- and second-order diffusion coefficients calculated from thestandard two-body relaxation theory:

∆v‖ = 〈∆v‖〉∆t + ζ1

√〈∆v2

‖ 〉∆t ,

∆v⊥ = ζ2

√〈∆v2

⊥〉∆t ,

where ζ1, ζ2 are two independent normally distributed random numbers.The potential of the system is then recomputed using the newpositions of all particles. In the original formulation, this has beenused in a spherical geometry only, but it is easy to generalize tonon-spherical cases, using the SCF approach.

The treatment of two-body relaxation

Local (position-dependent) velocity diffusion coefficients:

v〈∆v‖〉 = −(

1 + mm?

)I1/2 ,

〈∆v2‖ 〉 = 2

3

(I0 + I3/2

),

〈∆v2⊥〉 = 2

3

(2I0 + 3I1/2 − I3/2

),

here m and m? are masses of the test and field stars, and

I0 ≡ Γ

∫ 0

EdE ′ f (E ′),

In/2 ≡ Γ

∫ E

Φ(r)dE ′ f (E ′)

(E ′ − Φ(r)

E − Φ(r)

)n/2

,

Γ ≡ 16π2G 2m? ln Λ = 16π2G 2Mtot × (N−1? ln Λ).

distribution function of stars

gravitational potential

scalable amplitude of perturbation

Implementations of the Monte Carlo method

Name Reference relaxation treatment timestep 1:11 BH2 remarks

Princeton Spitzer&Hart(1971),Spitzer&Thuan(1972)

local dif.coefs. in velocity,Maxwellian background f (r, v)

∝ Tdyn − −

Cornell Marchant&Shapiro(1980)

dif.coef. in E , L, self-consistentbackground f (E)

indiv., Tdyn − + particle cloning

Henon Henon(1971) local pairwise interaction, self-consistent bkgr. f (r, v‖, v⊥)

∝ Trel − −

Stodo lkiewicz(1982) Henon’s block, Trel (r) − − mass spectrum, disc shocksStodo lkiewicz(1986) binaries, stellar evolution

Giersz(1998) same same + − 3-body scattering (analyt.)Mocca Hypki&Giersz(2013) same same + − single/binary stellar evol.,

few-body scattering (num.)

Joshi+(2000) same ∝ Trel (r = 0) + − partially parallelizedCmc Umbreit+(2012),

Pattabiraman+(2013)(shared) + + fewbody interaction, single/

binary stellar evol., GPU

Me(ssy)2 Freitag&Benz(2002) same indiv.∝ Trel − + cloning, SPH physical collis.

Raga Vasiliev(2015) local dif.coef. in velocity, self-consistent background f (E)

indiv.∝ Tdyn − + arbitrary geometry

1One-to-one correspondence between particles and stars in the system

2Massive black hole in the center, loss-cone effects

The novel Monte Carlo method for arbitrary geometry

The combination of the temporally-smoothed SCF approach forrepresenting arbitrary non-spherical potential with the Spitzer’sformulation of two-body relaxation using local velocity diffusioncoefficients leads to a new implementation of the Monte Carlomethod suitable for arbitrary geometry.

I Gravitational potential:particles move in a smooth potential represented by a basis-set expansion.

I Orbit integration:variable timestep Runge-Kutta; orbits are computed in parallel, independently

from each other, during each update interval.

I Two-body relaxation:apply perturbation to particle velocity using local diffusion coefficients.

I Potential and distribution function update:update interval � dynamical time ⇒ temporal smoothing; use many sampling

points per particle during each update interval ⇒ reduce discreteness noise.

An example of orbit in a triaxial potentialO

rigi

nal

orb

itP

ertu

rbed

orb

it(N

?=

106)

An

gular

mom

entu

mE

nergy

time

Application to the final-parsec problem

I Follow the merger and initial hardening by a conventional N-bodycode;

I after the formation of hard binary, switch to Monte Carlo method:

I during each episode, evolve particles in a time-dependent potentialof binary MBH moving on a Keplerian orbit with fixed parameters;

I at the end of episode, record the changes of energy and angularmomentum of each particle during each close encounter with thebinary, sum them up and adjust the orbit of the binary usingconservation laws [e.g. Sesana 2010, Meiron&Laor 2012];

I this automatically accounts for depletion and repopulation of theloss cone, secondary slingshot, and change of shape of thegravitational potential; does not account for brownian motion;

I may also include two-body relaxation in addition to non-sphericaltorques ⇒ naturally interpolate between finite N and N =∞.

Calibration of Monte Carlo simulations

I Monte Carlo simulations are in qualitative agreement with directN-body simulations for all combinations of parameters that weexplored (N, mass ratio, eccentricity, geometry, ...)

I There are no free parameters in the Monte Carlo method(apart from the pre-factor η in the Coulomb logarithm log Λ = log ηN, η ∼ 0.02

calibrated by measuring energy diffusion rate in simulations without binary).

0 20 40 60 80 100

time

0

200

400

600

800

1000

1200

1/a

spherical, N=250k, NBMCspherical, N=250k, e=0.9, NBMCaxisymm, N=500k, q=1/9, NBMCtriaxial, N=2000kMC

0 50 100 150 200 250

time

0

200

400

600

800

1000

1/a

128k256k512k1024kN-bodyMonte Carlo

Isolated models Merger simulations

Long-term binary evolution in Monte Carlo simulations

I Hardening rate decreases with time in all three geometries.

I In the absense of relaxation (N? =∞), it drops to zero in sphericaland axisymmetric cases, but stays high enough in triaxial case.

I Systems with relaxation eventually settle at a constant hardeningrate for large enough time.

I There is little difference between axisymmetric and triaxial systemseven for N? as large as 5× 106, but in the collisionless limit theirevolution is qualitatively different!

0 200 400 600 800 1000 1200 1400

time

0

10

20

30

40

50

ah/a

S N =2×106

S N =5×106

S N =∞

A N=5×

106

A N =108

A N =∞

T N

=5×

106

T N=∞

1 2 5 10 20 50ah/a

0.2

0.5

1

2

5

10

hard

enin

g r

ate

S N =2×106

S N =5×106

S N

=∞

A N =5×106

A N =108

A N

=∞

T N =5×106

T N =∞

Qualitative analysis of long-term collisionless evolution

I To shrink the binary by a factor of two, one needs to eject starswith total mass ∼ M•; thus one needs to supply a few×M• worthof stars into the loss cone over the entire evolution.

I Stars on centrophilic orbits in the extended loss region caneventually enter the loss cone; but in the axisymmetric case thevolume of loss region shrinks as the binary hardens.

I The population of loss region is gradually depleted, faster at highbinding energies; so the hardening rate S decreases with time.

0.0 0.2 0.4 0.6 0.8 1.0

Lz

0.0

0.2

0.4

0.6

0.8

1.0

L

time

regularchaotic

(centrophobic)(centrophilic)

LLC − loss cone

Lz=L

LC

f(E

)ξ(E

)

S∝∫f(E) ξ(E) dE

1.0 0.8 0.6 0.4 0.2 0.0

E

0

1

ξ(E

)

time

Estimates of the coalescence time

The joint evolution of binary semimajor axis a and eccentricity ein a collisionless triaxial system is given by

S ≡ d(1/a)

dt= S?(a) + SGW(a, e) ,

de

dt=

de

dt

∣∣∣∣?

+de

dt

∣∣∣∣GW

,

S? ≈ µ Sfull LC (a/ah)ν , µ . 1, ν ∼ 0.3− 0.6,

de

dt

∣∣∣∣?

≈ AS?a e (1− e2)b(

1 +a

a0

)−1

, b = 0.6, a0 = 0.2ah, A = 0.3,

SGW =64

5

G 3M3•

c5a5

q

(1 + q)2

1 + 7324e

2 + 3796e

4

(1− e2)7/2,

de

dt

∣∣∣∣GW

= −G 3M3•

c5a4

q

(1 + q)2

e(304 + 121e2)

15(1− e2)5/2[Peters 1964].

Estimates of the coalescence time

If one ignores the eccentricity change, the evolution equation for ais solved analytically. The coalescence time for e = 0 estimated byequating S? = SGW is

Tcoal,est. ' 1.7× 108 yr×(

rinfl

30 pc

) 10+4ν5+ν

(M•

108 M�

)− 5+3ν5+ν

µ−4

5+ν 20ν .

The numerical solution of the full ODE system agrees well with theMonte Carlo simulations.

0.0 0.2 0.4 0.6 0.8 1.0

eccentricity

0.0

0.2

0.4

0.6

0.8

1.0

Tco

al, G

yr

evol.trackS =const

no ecc.growth

simple estimate

103

104

105

1/a

MC, with GWapprox.MC, no GW

0 500 1000 1500 2000

time

0.5

0.6

0.7

0.8

0.9

1.0

e

Summary

I The final-parsec problem in the binary MBH evolution is relatedto the efficiency of repopulation of the loss cone.

I This repopulation occurs faster in non-spherical geometry,because the loss cone is fed from a much larger “loss region” dueto collisionless processes.

I The loss cone never stays “full”, even in triaxial geometry; as theloss region is gradually depleted, the hardening rate slows down.

I It is difficult to disentangle collisional and collisionless effects(suppress 2-body relaxation) in conventional N-body simulations.

I A novel Monte Carlo method with arbitrary geometry andadjustable relaxation rate is applied to the binary evolution.

I In the collisionless limit, the final parsec problem may be solvedonly in triaxial systems (axisymmetry is not enough).

Thank you!