Implications of pulsar timing array observations for LISA detections of massive black hole binaries

Pulsar timing arrays (PTAs) and the Laser Interferometer Space Antenna (LISA) will open complementary observational windows on massive black hole binaries (MBHBs), i.e. with masses in the range similar to 10(6) -10(10) M-circle dot. While PTAs may detect a stochastic gravitational wave background from a population of MBHBs, during operation LISA will detect individual merging MBHBs. To demonstrate the profound interplay between LISA and PTAs, we estimate the number of MBHB mergers that one can expect to observe with LISA by extrapolating direct observational constraints on the MBHB merger rate inferred from PTA data. For this, we postulate that the common signal observed by PTAs (and consistent with the increased evidence recently reported) is an astrophysical background sourced by a single MBHB population. We then constrain the LISA detection rate, R, in the mass-redshift space by combining our Bayesian-inferred merger rate with LISA's sensitivity to spin-aligned, inspiral-merger-ringdown waveforms. Using an astrophysically informed formation model, we predict a 95 per cent upper limit on the detection rate of R < 134 yr(-1) for binaries with total masses in the range 10(7)-10(8) M-circle dot. For higher masses, i.e. >108 M-circle dot, we find R < 2 (1) yr(-1) using an astrophysically informed (agnostic) formation model, rising to 11 (6) yr(-1) if the LISA sensitivity bandwidth extends down to 10(-5) Hz. Forecasts of LISA science potential with PTA background measurements should improve as PTAs continue their search.