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Stellar and Black Hole Mass Densities as Empirical Tracers of Co-evolution Show Lock-step Growth since $z{\sim}3$

At redshifts beyond $z{\sim}1$ measuring the black hole galaxy relations proves to be a difficult task. The bright light of the AGN aggravates deconvolution of black hole and galaxy properties. On the other hand high redshift data on these relations is vital to understand in what ways galaxies and black holes co-evolve and in what ways they don't. In this work we use black hole (BHMDs) and stellar mass densities (SMDs) to constrain the possible co-evolution of black holes with their host galaxies since $z{\sim}5$. The BHMDs are calculated from quasar luminosity functions (QLF) using the Soltan argument, while we use integrals over stellar mass functions (SMFs) or the star formation rate density to obtain values for the stellar mass density. We find that both quantities grow in lock-step below redshifts of $z{\sim}3$ with a non-evolving BHMD to SMD ratio. A fit to the data assuming a power law relation between the BHMD and the SMD yields exponents around unity ($1.0{-}1.5$). Up to $z{\sim}5$ the BHMD to SMD ratio doesn't show a strong evolution given the larger uncertainty in the completeness of high-redshift datasets. Our results, always applying the same analysis technique, seem to be consistent across all adopted data sets.