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We estimate the amplitude of the nano-Hz stochastic gravitational wavebackground (GWB) resulting from an unresolved population of inspirallingmassive black hole binaries (MBHBs). To this aim, we use the L-Galaxiessemi-analytical model applied on top of the Millennium merger trees. Thedynamical evolution of MBHBs includes dynamical friction, stellar and gasbinary hardening, and gravitational wave feedback. At the frequencies proved bythe \textit{Pulsar Timing Array} experiments, our model predicts an amplitudeof ${\sim}1.2\,{\times}\,10^{-15}$ at ${\sim}\,3\,{\times}\,10^{-8}\, \rm Hz$in agreement with current estimations. The contribution to the background comesprimarily from equal mass binaries with chirp masses above $\rm 10^{8}\,M_{\odot}$. We then consider the recently detected common red noise inNANOGrav, PPTA, and EPTA data, working under the hypothesis that it is indeed astochastic GWB coming from MBHBs. By boosting the massive black hole growth viagas accretion, we show that our model can produce a signal with an amplitude$A\approx 2-3 {\times}\,10^{-15}$. There are, however, difficulties inpredicting this background level without mismatching key observationalconstraints such as the quasar bolometric luminosity functions or the localblack hole mass function. This highlights how current and forthcominggravitational wave observations can, for the first time, confront galaxy andblack hole evolution models.

Massive black hole evolution models confronting the n-Hz amplitude of the stochastic gravitational wave background