Coarse-Grained Cosmological Perturbation Theory

TL;DR

The paper develops a coarse-grained perturbation theory for cosmological structure formation by introducing a finite smoothing scale L that separates long- and short-distance physics. Long-wavelength perturbations are evolved perturbatively while short-distance effects are encapsulated in external source terms derived from a Vlasov-fluid framework and calibrated with N-body simulations. This approach yields a self-consistent generation of macroscopic velocity dispersion and recovers standard one-loop results while revealing additional L-dependent contributions to the power spectrum and cross-spectra, evidenced by comparison with GADGET-2 simulations. The method offers a pathway to incorporate short-scale physics into perturbation theory in a cosmology-dependent way, potentially enabling faster predictions across parameter spaces and suggesting an RG-like interpretation with L as a flow parameter.

Abstract

Semi-analytical methods, based on Eulerian perturbation theory, are a promising tool to follow the time evolution of cosmological perturbations at small redshifts and at mildly nonlinear scales. All these schemes are based on two approximations: the existence of a smoothing scale and the single-stream approximation, where velocity dispersion of the dark matter fluid, as well as higher moments of the particle distributions, are neglected. Despite being widely recognized, these two assumptions are, in principle, incompatible, since any finite smoothing scale gives rise to velocity dispersion and higher moments at larger scales. We describe a new approach to perturbation theory, where the Vlasov and fluid equations are derived in presence of a finite coarse-graining scale: this allows a clear separation between long and short distance modes and leads to a hybrid approach where the former are treated perturbatively and the effect of the latter is encoded in external source terms for velocity, velocity dispersion, and all the higher order moments, which can be computed from N-body simulations. We apply the coarse-grained perturbation theory to the computation of the power spectrum and the cross-spectrum between density and velocity dispersion, and compare the results with N-body simulations, finding good agreement.

Figures (5)

• Figure 1: Snapshots of various quantities obtained from the simulation described in the text. The left column is the density field, the central one is the trace of the velocity dispersion tensor, and the right one the ratio $\bar{\sigma}_{11}/\bar{v}_1^2$. The different lines are obtained by taking grids of spacing $L=4,\,8,$ and $16\,\mathrm{Mpc}/h$, from top to bottom.
• Figure 2: On the left column the same density field as in Figure \ref{['3Slices']} is plotted. The right column represents the macroscopic contribution to velocity dispersion, as defined by the second line of eq. (\ref{['sigbar']}).
• Figure 3: Comparison between the perturbative computation described in Section \ref{['pertsourc']} and simulations. $P_{11}$ represents the density field PS, whereas $P_{13}$ is the cross-correlator between density and the macroscopic component of the trace of the velocity dispersion tensor. The fields have been rescaled according to eq. (\ref{['fielddef']}). The continuos (dashed) lines represent the results obtained with the smooth (sharp) cut-off. The dotted line is the standard 1-loop result, while the dots are obtained from N-body simulations.
• Figure 4: Ratios between $P_{13}$ and $P_{11}$. The continuos (dashed) lines represent the results obtained with the smooth (sharp) cut-off. The dots are obtained from N-body simulations.
• Figure 5: The continuous lines and the dots are the results for $P_{11}$ with the same symbols as in Figure \ref{['Ptutti']}. The dash-dotted line and the open squares represent the short-distance contribution to the perturbative result and from simulations, respectively.