4.2 Evolution in an evolving background

The scattering experiments summarized above treat the binary’s environment as fixed and homogeneous. In reality, the binary is embedded at the center of an inhomogeneous and evolving galaxy, and the supply of stars that can interact with it is limited.

In a fixed spherical galaxy, stars can interact with the binary only if their pericenters lie within $~ R × a$ , where $R$ is of order unity. Let $L = RaV ~ 2[E----P(Ra)]-- ~~ V~ 2Gm---Ra- lc 12$ , the angular momentum of a star with pericenter $Ra$ . The “loss cone” is the region in phase space defined by $L < Llc$ . The mass of stars in the loss cone is

integral integral Llc M (a) = m dE dL N (E, L2) lc * 0 integral integral L2lc = m* dE dL24p2f (E,L2)P (E, L2) 0 integral ~~ 8p2Gm12m*Ra dEf (E)Prad(E). (25)

Here $P$ is the orbital period, $f$ is the number density of stars in phase space, and $N (E,L2)dEdL$ is the number of stars in the integral-space volume defined by $dE$ and $dL$ . In the final line, $f$ is assumed isotropic and $P$ has been approximated by the period of a radial orbit of energy $E$ . An upper limit to the mass that is available to interact with the binary is $~ Mlc(ah)$ , the mass within the loss cone when the binary first becomes hard; this is an upper limit since some stars that are initially within the loss cone will “fall out” as the binary shrinks. Assuming a singular isothermal sphere for the stellar distribution, $r oc r -2$ , and taking the lower limit of the energy integral to be $P(ah)$ , Equation (25

) implies

We can compute the change in $a$ that would result if the binary interacted with this entire mass, by using the fact the mean energy change of a star interacting with a hard binary is $~ 3Gm/2a$ [177 ]. Equating the energy carried away by stars with the change in the binary’s binding energy gives

if $DM$ is equated with $Mlc$ . Only for very low mass ratios ( $q <~ 10-3$ ) is this decay factor large enough to give $tgr < 1010yr$ (Equation 8

), but the time required for such a small black hole to reach the nucleus is likely to exceed a Hubble time [132 ]. Hence even under the most favorable assumptions, the binary would not be able to interact with enough mass to reach gravity-wave coalescence.

But the situation is even worse than this, since not all of the mass in the loss cone will find its way into the binary. The time scale for the binary to shrink is comparable with stellar orbital periods, and some of the stars with $rperi ~~ ah$ will only reach the binary after $a$ has fallen below $~ ah$ . We can account for the changing size of the loss cone by writing

integral oo dM-- = --1--dMlc-dE dt E0(t) P(E) dE integral oo = 8p2Gm12m*Ra(t) fi(E)dE, (29) E0(t)

where $M (t)$ is the mass in stars interacting with the binary and $fi(E)$ is the initial distribution function; setting $P (E0) = t$ reflects the fact that stars on orbits with periods less than $t$ have already interacted with the binary and been ejected. Combining Equations (27

) and (29

Solutions to Equation (30

) show that a binary in a singular isothermal sphere galaxy stalls at $ah/a ~~ 2.5$ for $m2 = m1$ , compared with $ah/a ~~ 10$ if the full loss cone were depleted (Equation 28

). In galaxies with shallower central cusps, decay of the binary would stall at even greater separations.