Connecting low-redshift LISA massive black hole mergers to the nHz stochastic gravitational wave background

Pulsar Timing Array (PTA) experiments worldwide recently reported evidence of a nHz stochastic gravitational wave background (sGWB) compatible with the existence of slowly inspiralling massive black hole (MBH) binaries (MBHBs). The shape of the signal contains valuable information about the evolution of $z<1$ MBHs above $\rm 10^8 M_{\odot}$, suggesting a faster dynamical evolution of MBHBs towards the gravitational-wave-driven inspiral or a larger MBH growth than usually assumed. In this work, we investigate if the nHz sGWB could also provide constraints on the population of merging lower-mass MBHBs ($\rm {<} 10^7 \, M_{\odot}$) detectable by LISA. To this end, we use the $\texttt{L-Galaxies}$ semi-analytical model applied to the $\texttt{Millennium}$ suite of simulations. We generate a population of MBHs compatible simultaneously with current electromagnetic and nHz sGWB constraints by including the possibility that, in favourable environments, MBHs can accrete gas beyond the Eddington limit. The predictions of the model show that the global (integrated up to high-$z$) LISA detection rate is {\it not} significantly affected when compared to a fiducial model whose nHz sGWB signal is ${\sim}\,2$ times smaller. In both cases, the global rate yields ${\sim}\,12 \rm yr^{-1}$ and is dominated by systems of $\rm 10^{5-6} M_{\odot}$. The main differences are limited to low-$z$ ($z<3$), high-mass (${>}\rm 10^6\, M_{\odot}$) LISA MBHBs. The model compatible with the latest PTA results predicts up to ${\sim}\,1.6$ times more detections, with a rate of ${\sim}1\rm yr^{-1}$. We find that these LISA MBHB systems have 50\% probability of shining with bolometric luminosities $>10^{43}\rm erg/s$. Hence, in case PTA results are confirmed and given the current MBH modelling, our findings suggest there will be higher chances to perform multimessenger studies with LISA MBHB than previously expected.