Analytical Description of Baryonic Matter Fluctuations Using Jeans Filtering Functions in Second-Order Cosmological Perturbation Theory

Abstract

Cosmological perturbation theory provides the fundamental framework for describing the evolution of the matter-energy density field in an expanding Universe and serves as the basis for understanding the formation of large-scale structures within the $\Lambda$CDM paradigm. We present an analytical approach to describe the evolution of fluctuations in a mixed fluid composed of cold dark matter (CDM) and baryonic matter. Assuming that the Universe is governed by General Relativity, we employ the Vlasov equation to derive the general equations of motion for this mixed cosmological fluid, incorporating baryonic effects through the stress tensor by considering only the contributions from baryonic pressure. We introduce the Jeans filtering functions as a biasing tool that allows us to describe baryonic fluctuations with CDM as a tracer, and we obtain an analytical description of the fluctuations -- a novel and uncommon approach compared to the accepted computational advances currently available in this field. First- and second-order solutions are obtained through a single iteration of the equations of motion, with the aim of identifying how the filtering scale behaves in a second-order theory compared to the linear one, as well as some of its impacts on the matter power spectrum without the need to compute it explicitly. For the first time, these kind of solutions are derived entirely through an analytical method. Finally, we obtain analytical expressions for baryonic fluctuations in the density and velocity fields, which can be readily evaluated and provide valuable insights into the role of baryons in the Large-scale structure of the Universe. Consequently, these results reveal how pressure effects shift the filtering scale and how including this component could influence parameters such as the filtering mass and the temperature of the pressure-supported components.