LISA extreme-mass-ratio inspiral events as probes of the black hole mass function

One of the sources of gravitational waves for the proposed space-based gravitational wave detector, the Laser Interferometer Space Antenna (LISA), are the inspirals of compact objects into supermassive black holes in the centers of galaxies—extreme-mass-ratio inspirals (EMRIs). Using LISA observations, we will be able to measure the parameters of each EMRI system detected to very high precision. However, the statistics of the set of EMRI events observed by LISA will be more important in constraining astrophysical models than extremely precise measurements for individual systems. The black holes to which LISA is most sensitive are in a mass range that is difficult to probe using other techniques, so LISA provides an almost unique window onto these objects. In this paper we explore, using Bayesian techniques, the constraints that LISA EMRI observations can place on the mass function of black holes at low redshift. We describe a general framework for approaching inference of this type—using multiple observations in combination to constrain a parametrized source population. Assuming that the scaling of the EMRI rate with the black-hole mass is known and taking a black-hole distribution given by a simple power law, dn/dln⁡M=A₀(M/M_*)^α₀, we find that LISA could measure the parameters to a precision of Δ(ln⁡A₀)∼0.08, and Δ(α₀)∼0.03 for a reference model that predicts ∼1000 events. Even with as few as 10 events, LISA should constrain the slope to a precision ∼0.3, which is the current level of observational uncertainty in the low-mass slope of the black-hole mass function. We also consider a model in which A₀ and α₀ evolve with redshift, but find that EMRI observations alone do not have much power to probe such an evolution.