Non-linear density–velocity divergence relation from phase space dynamics

We obtain the non-linear relation between cosmological density and velocity perturbations by examining their joint dynamics in a two-dimensional density–velocity divergence phase space. We restrict to spatially flat cosmologies consisting of pressureless matter and non-clustering dark energy characterized by a constant equation of state w. Using the spherical top-hat model, we derive the coupled equations that govern the joint evolution of density and velocity perturbations and examine the flow generated by this system. In general, the initial density and velocity are independent, but requiring that the perturbations vanish at the big bang time sets a relation between the two. This traces out a curve in the instantaneous phase space, which we call the ‘Zel'dovich curve’. We show that this curve acts like an attracting solution for the phase space dynamics and is the desired non-linear extension of the density–velocity divergence relation. We obtain a fitting formula which is a combination of the formulae by Bernardeau and Bilicki & Chodorowski, generalized to include the dependence on w. We find that as in the linear regime, the explicit dependence on the dark energy equation of state stays weak even in the non-linear regime. Although the result itself is somewhat expected, the new feature of this work is the interpretation of the relation in the phase space picture and the generality of the method. Finally, as an observational implication, we examine the evolution of galaxy cluster profiles using the spherical infall model for different values of w. We demonstrate that using only the density or the velocity information to constrain w is subject to degeneracies in other parameters such as σ8 but plotting observations on to the joint density–velocity phase space can help lift this degeneracy.