The Final Parsec: Orbital Decay of Massive Black Holes in Galactic Stellar Cusps A. Sesana 1, F. Haardt 1, P. Madau 2 1 Universita` dell'Insubria, via Valleggio 11, Como, Italy 2 University of California, 1156 High Street, Santa Cruz, CA Como, 20 September 2005

OUTLINE >Merging History of Massive Black Holes >MBHBs Dynamics: the “Final Parsec Problem” >Scattering Experiments: Model Description >Results: Binary Decay in a Time-Evolvig Cuspy Background: the Study Case of the SIS > Effects on the Stellar Population > Returning Stars > Tidal Disruption Rates > Implication for SMBH Coalescence >Summary

MERGING HISTORY OF SMBHs Z=0 Z=20 (Volonteri, Haardt & Madau 2003) Galaxy formation proceeds as a series of subsequent halo mergers MBH assemby follow the galaxy evolution starting from seed BHs with mass ~100M ⊙ forming in minihalos at z~20 During mergers, MBHBs will inevitably form!!

SMBHs DYNAMICS 1. dynamical friction (Lacey & Cole 1993, Colpi et al. 2000) ● from the interaction between the DM halos to the formation of the BH binary ● determined by the global distribution of matter ● efficient only for major mergers against mass stripping 2. hardening of the binary (Quinlan 1996, Merritt 1999, Miloslavljevic & Merritt 2001) ● 3 bodies interactions between the binary and the surrounding stars ● the binding energy of the BHs is larger than the thermal energy of the stars ● the SMBHs create a stellar density core ejecting the background stars 3. emission of gravitational waves (Peters 1964) ● takes over at subparsec scales ● leads the binary to coalescence

DESCRIPTION OF THE PROBLEM We want MBHBs to coalesce after a major merger Dynamical friction is efficient in driving the two BHs to a separation of the order The ratio can be written as we need a physical mechanism able to shrink the binary separation of about two orders of magnitude! GW emission takes over at separation of the order

GRAVITATIONAL SLINGSHOT Extraction of binary binding energy via three body interactions with stars Scattering experiments (e.g. Mikkola & Valtonen 1992, Quinlan 1996) N-body simulations (e.g. Milosavljevic & Merritt 2001) resolution problem > More feasibles > need a large amount of data for significative statistics (eccentricity problem) > warning: connection with real galaxies! > initial conditions > loss cone depletion > contribution of returning stars > presence of bound stellar cusps

SCATTERING EXPERIMENTS Y X Z > MBHB M 1 >M 2 on a Keplerian orbit with semimajor axis a and eccentricity e > incoming star with m * <<M 2 and velocity v > The initial condition is a point in a nine dimensional parameter space: > q=M 2 /M 1, e, m * /M 2 > v, b, , , ,  Our choices: > In the limit m * <<M 2 : results are indipendent on m * we set m * = M (M=M 1 +M 2 ) > we sampled six values of q: 1, 1/3, 1/9, 1/27, 1/81, 1/243 and seven values of e: 0.01, 0.15, 0.3, 0.45, 0.6, 0.75, 0.9 for each q > we sampled 80 values of v in the range 3x (M 2 /M) 1/2 < v/Vc < 3x10 2 (M 2 /M) 1/2 > we sampled b and the four angles in order to reproduce a spherical distribution of incoming stars

> Tolerance is settled so that the energy conservation for each orbit is of the order E * > Integration is stopped when: > the star leave r i with positive total energy > the integration needs more than 10 6 steps > the physical integration time is >10 10 yrs > the star is tidally disrupted We integrate the nine coupled second order, differential equations using the explicit Runge-Kutta integrator DOPRI5 (Hairer & Wanner 2002) > At the end of each run the program records: > the position and velocity of each star > the quantities B and C defined as:

C and B-C distributions vs. x, a rescaled impact parameter defined as M 2 /M 1 =1 e=0

SEMIANALITICAL MODEL We consider: > a MBHB with a semimajor axis a and eccentricity e > a spherically simmetric stellar background >  (r) =  0 (r/r 0 ) -  is the power law density profile. (  0 is the density at the reference distance r 0 from the centre) > f(v,  ) is the stellar velocity distribution.  is the 1- D velocity dispersion (in the following we will always consider a Maxwellian distribution)

C and B can be used to compute the MBHB evolution Writing d 2 N(b,t)/dbdt=2  b  (b,t)v/m * and  (b,t)=  0 F(b a x,t) we find: Weighting over a velocity distribution f(v,  ) we finally get H is the HARDENING RATE Similarly we find the equation for the eccentricity evolution K is the ECCENTRICITY GROWTH RATE Starting from the energy exchange during a single scattering event we can write:

F(b a x,t) is a function, to be determined, of the rescaled impact parameter x and of the time t and depends on the density profile of the stellar distribution Early studies (Mikkola & Valtonen 1992, Quinlan 1996) assumed F(b a x,t) =1 i.e. they studied the hardening problem in a flat core of density  0 constant in time!! Warning: connection with real galaxies! 1- Almost all galaxies show cuspy density profiles in their inner regions   r -  0<  <2.5 (n.b. faint early type galaxies show steeper cusps that giants ellipticals) 2- In real galaxies there is a finite supply of stars to the hardening process LOSS CONE PROBLEM

1-HARDENING IN A CUSPY PROFILE We consider a density profile   r -  where  =  - 1 > If  >1, then > The hardening rate is: Hard binaries hardens at a constant rate only in a flat stellar background!

Eccentricity Growth K is typically small: eccentricity evolution will be modest

2-MODELLING THE LOSS CONE CONTENT Definition: the loss cone is the portion of the space E, J constituded by those stars that are allowed to approach the MBHB as close as  x a, where  is a constant (we choose  = 5) Given  (r ) we can evaluate the mass in the unperturbed loss cone as and the interacting mass integrating where M 2 /M 1 =1 e=0

THE SINGULAR ISOTHERMAL SPHERE (SIS) > we can factorize F(b a x,t)  F 0 (b a x) x  (t) > The umperturbed loss cone mass content is M lc ~ 3/2  M 2 > We model, as a studing case, the stellar distribution as a SIS with density profile r is related to t simply as dr/dt=3 1/2  > The MBHB mass is chosen to satisfy the M-  relation (Tremaine et al. 2002)

1- MBHB Shrinking

2-Distribution of Scattered Stars

The loss of low angular momentum stars Partial loss cone depletion ~20% of the interacting stars returns in the new loss cone of the shrinked binary

Stellar distribution flattening and corotation with the MBHB Interacting star distribution tends to flatten and corotate with the MBHB Ejected mass The ejected mass is of the order M ej ≈0.7M

3-The Role of Returning Stars Total shrinking The shrinking factor  scales as (M 2 /M) 1/2 and is weakly dependent on e Total loss cone depletion The inner density profile flatten significatively

Final Velocity Distribution

4-Tidal Disruption Rates A star is tidally disrupted if it approaches one of the holes as close as the tidal disruption radius r td,i ~(m * / M i ) 1/3 r * We can then derive the mean TD rate as: N TD stars / hardening time > The TD rate is extremely high during the hardening phase (respect to TD rates due to a single BH ~ star/yr) > The high TD rate phase is extremely short Hard to detect a MBHB via TD stars

5-Binary Coalescence As the shrinking factor  is proportional to (M 1 /M) 1/2, writing a f =  x a h, we finally get

e=0 e=0.9 e=0.6 LISA binaries ( M ⊙ ) may need extra help to coalesce within an Hubble time!!!

What can help ? > MBHB random walk ( e.g. Quinlan & Hernquist 1997, Chatterjee et al. 2003) > Star diffusion in the loss cone via two body relaxation (Milosavljevic & Merritt 2001) > Loss cone amplification (loss wedge) in axisimmetric and triaxial potentials (Yu 2002, Merritt & Poon 2004) > Torques exerted on the MBHB by a gaseous disk (Armitage & Natarajan 2002, Escala et al. 2005, Dotti et al. in preparation) M <10 5 M ⊙

Summary > We have studied the interaction MBHB-stars in detail using scattering experiments coupled with a semianalitical model for MBHB and steller background evolution including: >a cuspy time-evolving stellar background >the effect of returning stars > H in the hard stage is proportional to a -  /2 > K is typically positive, but the eccentricity evoution of the binary is modest >Interacting stars typically corotate with the MBHB >MBHB-star interactions flatten the stellar distribution >A mass of the order of 0.7M is ejected from the bulge on nearly radial corotating orbits in the MBHB plane >LISA binaries may need the support of other mechanisms to reach coalescence within an Hubble time Results

Future Prospects Investigate the contribution of other mechanisms to the binary hardening Evaluate the eventual role of bound stellar cusps Include this treatment of MBHB dynamics in a merger tree model to give realistic estimations for the number counts of “LISA coalescences”