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The massive black holes we observe in galaxies today are the naturalend-product of a complex evolutionary path, in which black holes seeded inproto-galaxies at high redshift grow through cosmic history via a sequence ofmergers and accretion episodes. Electromagnetic observations probe a smallsubset of the population of massive black holes (namely, those that are activeor those that are very close to us), but planned space-based gravitational-waveobservatories such as the Laser Interferometer Space Antenna (LISA) can measurethe parameters of ``electromagnetically invisible'' massive black holes out tohigh redshift. In this paper we introduce a Bayesian framework to analyze theinformation that can be gathered from a set of such measurements. Our goal isto connect a set of massive black hole binary merger observations to theunderlying model of massive black hole formation. In other words, given a setof observed massive black hole coalescences, we assess what information can beextracted about the underlying massive black hole population model. Forconcreteness we consider ten specific models of massive black hole formation,chosen to probe four important (and largely unconstrained) aspects of the inputphysics used in structure formation simulations: seed formation, metallicity``feedback'', accretion efficiency and accretion geometry. For the first timewe allow for the possibility of ``model mixing'', by drawing the observedpopulation from some combination of the ``pure'' models that have beensimulated. A Bayesian analysis allows us to recover a posterior probabilitydistribution for the ``mixing parameters'' that characterize the fractions ofeach model represented in the observed distribution. Our work shows that LISAhas enormous potential to probe the underlying physics of structure formation.

Reconstructing the massive black hole cosmic history through gravitational waves