Spin distribution following minor mergers and the effect of spin on the detection range for low-mass-ratio inspirals

We compute the probability distribution for the spin of a black hole following a series of minor mergers with isotropically distributed, non-spinning, inspiraling compact objects. By solving the Fokker-Planck equation governing this stochastic process, we obtain accurate analytical fits for the evolution of the mean and standard deviation of the spin distribution in several parameter regimes. We complement these analytical fits with numerical Monte-Carlo simulations in situations when the Fokker-Planck analysis is not applicable. We find that a ~150 solar-mass intermediate-mass black hole that gained half of its mass through minor mergers with neutron stars will have dimensionless spin parameter chi=a/M~0.2 \pm 0.08. We estimate the effect of the spin of the central black hole on the detection range for intermediate-mass-ratio inspiral (IMRI) detections by Advanced LIGO and extreme-mass-ratio inspiral(EMRI) detections by LISA. We find that for realistic black hole spins, the inclination-averaged Advanced-LIGO IMRI detection range may be increased by up to 10% relative to the range for IMRIs into non-spinning intermediate-mass black holes. For LISA, we find that the detection range for EMRIs into 10^5 solar-mass massive black holes (MBHs) is not significantly affected by MBH spin, the range for EMRIs into 10^6 solar-mass MBHs is affected at the ~ 10% level, and EMRIs into maximally spinning 10^7 solar-mass MBHs are detectable to a distance ~25 times greater than EMRIs into non-spinning black holes. The resulting bias in favor of detecting EMRIs into rapidly spinning MBHs will play a role when extracting the MBH spin distribution from EMRI statistics.