Statistical Decoupling of a Lagrangian Fluid Parcel in Newtonian Cosmology

Abstract

The Lagrangian dynamics of a single fluid element within a self-gravitational matter field is intrinsically non-local due to the presence of the tidal force. This complicates the theoretical investigation of the nonlinear evolution of various cosmic objects, e.g., dark matter halos, in the context of Lagrangian fluid dynamics, since fluid parcels with given initial density and shape may evolve differently depending on their environments. In this paper, we provide a statistical solution that could decouple this environmental dependence. After deriving the evolution equation for the probability distribution of the matter field, our method produces a set of closed ordinary differential equations whose solution is uniquely determined by the initial condition of the fluid element. Mathematically, it corresponds to the projected characteristic curve of the transport equation of the density-weighted probability density function (ρPDF). Consequently it is guaranteed that the one-point ρPDF would be preserved by evolving these local, yet nonlinear, curves with the same set of initial data as the real system. Physically, these trajectories describe the mean evolution averaged over all environments by substituting the tidal tensor with its conditional average. For Gaussian distributed dynamical variables, this mean tidal tensor is simply proportional to the velocity shear tensor, and the dynamical system would recover the prediction of the Zel'dovich approximation (ZA) with the further assumption of the linearized continuity equation. For a weakly non-Gaussian field, the averaged tidal tensor could be expanded perturbatively as a function of all relevant dynamical variables whose coefficients are determined by the statistics of the field.

On the statistical evolution of the large-scale structure and the effective dynamics

“…Similar equations can be found e.g. in [36][37][38] in cosmological context as well. Moreover, this framework describes the effective terms as the ensemble average of small-scale interactions conditional on large-scale modes.…”

“…However, that does not mean we will not learn anything valuable. Following the PDF-based technique developed by [35] in turbulence (see also its cosmology application in [38]), one could formally write down the time evolution equation of P(Ψ Λ , η) as…”

“…Now let us consider an ensemble of systems (2.3). There are certain freedom in choosing the ensemble, for example in [38], it is a sample of Lagrangian fluid elements, or equivalently the density-weighted field in Eulerian space, but here we assume it is different realizations of our Universe. By definition, the PDF of the ensemble could be obtained by taking average of the fined-grained PDF [35], i.e.…”

“…density PDF, topologies and covariance matrices etc. For example, [38] applied a similar method to study the evolution of a given Lagrangian fluid parcel. There, the effective parameter is described by the conditional average of the tidal tensor.…”