The nightmare scenario: measuring the stochastic gravitational-wave background from stalling massive black-hole binaries with pulsar-timing arrays

Massive black-hole binaries, formed when galaxies merge, are among the primary sources of gravitational waves targeted by ongoing Pulsar Timing Array (PTA) experiments and the upcoming space-based LISA interferometer. However, their formation and merger rates are still highly uncertain. Recent upper limits on the stochastic gravitational-wave background obtained by PTAs are starting being in marginal tension with theoretical models for the pairing and orbital evolution of these systems. This tension can be resolved by assuming that these binaries are more eccentric or interact more strongly with the environment (gas and stars) than expected, or by accounting for possible selection biases in the construction of the theoretical models. However, another (pessimistic) possibility is that these binaries do not merge at all, but stall at large ($\sim$ pc) separations. We explore this extreme scenario by using a galaxy-formation semi-analytic model including massive black holes (isolated and in binaries), and show that future generations of PTAs will detect the stochastic gravitational-wave background from the massive black-hole binary population within $10-15$ years of observations, even in the "nightmare scenario" in which all binaries stall at the hardening radius. Moreover, we argue that this scenario is too pessimistic, because our model predicts the existence of a sub-population of binaries with small mass ratios ($q \lesssim 10^{-3}$) that should merge within a Hubble time simply as a result of gravitational-wave emission. This sub-population will be observable with large signal-to-noise ratios by future PTAs thanks to next-generation radio telescopes such as SKA or FAST, and possibly by LISA.