Supermassive Black Hole Merger Rates: Uncertainties from Halo Merger Theory

The merger of two supermassive black holes is expected to produce a gravitational-wave signal detectable by the satellite LISA. The rate of supermassive-black-hole mergers is intimately connected to the halo merger rate, and the extended Press-Schechter formalism is often employed when calculating the rate at which these events will be observed by LISA. This merger theory is flawed and provides two rates for the merging of the same pair of haloes. We show that the two predictions for the LISA supermassive-black-hole-merger event rate from extended Press-Schechter merger theory are nearly equal because mergers between haloes of similar masses dominate the event rate. An alternative merger rate may be obtained by inverting the Smoluchowski coagulation equation to find the merger rate that preserves the Press-Schechter halo abundance, but these rates are only available for power-law power spectra. We compare the LISA event rates derived from the extended Press-Schechter merger formalism to those derived from the merger rates obtained from the coagulation equation and find that the extended Press-Schechter LISA event rates are thirty percent higher for a power spectrum spectral index that approximates the full Lambda-CDM result of the extended Press-Schechter theory.