A simple model to link the properties of quasars to the properties of dark matter haloes out to high redshift

We present a simple model of how quasars occupy dark matter haloes from z= 0 to 5 using the observed mBHâ-σ relation and quasar luminosity functions. This provides a way for observers to statistically infer host halo masses for quasar observations using luminosity and redshift alone. Our model is deliberately simple and sidesteps any need to explicitly describe the physics. In spite of its simplicity, the model reproduces many key observations and has predictive power: (i) model quasars have the correct luminosity function (by construction) and spatial clustering (by consequence); (ii) we predict high-redshift quasars of a given luminosity live in less massive dark matter haloes than the same luminosity quasars at low redshifts; (iii) we predict a factor of ~5 more 108.5 M⊙ black holes at z~ 2 than is currently observed; (iv) we predict a factor of ~20 evolution in the amplitude of the mBHâ-Mhalo relation between z= 5 and the present day; (v) we expect luminosity-dependent quasar lifetimes of between tQ~ 107 and 108 yr , but which may become as short as 105-6 yr for quasars brighter than L* and (vi) while little luminosity-dependent clustering evolution is expected at z≲ 1 , increasingly strong evolution is predicted for L > L* quasars at higher redshifts. These last two results arise from the narrowing distribution of halo masses that quasars occupy as the Universe ages. We also deconstruct both 'downsizing' and 'upsizing' trends predicted by the model at different redshifts and space densities. Importantly, this work illustrates how current observations cannot distinguish between more complicated physically motivated quasar models and our simple phenomenological approach. It highlights the opportunities such methodologies provide.