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

TL;DR

This paper develops an analytical framework for the nonlinear evolution of baryonic fluctuations in a mixed CDM–baryon cosmology within GR, using the Vlasov equation and Jeans filtering functions to bias baryons with CDM as a tracer. By applying Makino’s second-order approach to the two-fluid system, it derives explicit first- and second-order solutions for the Jeans filtering functions, revealing how pressure shifts the filtering scale and modulates the density and velocity fields. The key result is a scale-dependent baryon bias $g_2^{(0)}(k,\tau)$ that produces a ~30% nonlinear correction around the Jeans scale and shifts the effective filtering scale to $k_F \approx 1.3\,k_J$, with significant implications for the inferred baryon temperature and filtering mass. Overall, the work provides a self-contained analytic method to assess baryonic effects on large-scale structure without explicit power-spectrum computation, highlighting the importance of nonlinear terms in shaping small-scale fluctuations and the velocity field.

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 $Λ$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.

Figures (5)

• Figure 1: Spherical Coordinate System.
• Figure 2: Second-order solution for the evolution of the JFF with zero iteration. The black line shows the evolution of baryonic fluctuations, while the orange dashed line marks the scale $k = k_{\hbox{\tiny J}}$, and the light green band represents the region where fluctuations in the density field exhibit competition between baryonic and CDM fluctuations.
• Figure 3: Left panel: First- and second-order solutions for the evolution of the JFF with zero iteration. The red line shows the evolution of baryonic fluctuations at first order —denoted by the subscript $(1)$—, while the black line shows the second-order solution —denoted by the subscript $(2)$—. The orange dashed line marks the scale $k = k_{\hbox{\tiny J}}$. Right panel: Difference between the first- and second-order solutions, highlighting the nonlinear correction.
• Figure 4: Second-order solution for the divergence of the peculiar velocity field, $h^{(0)}_{_2}(\boldsymbol{k})$, The black solid line shows the behavior of baryonic velocity fluctuations as a function of scale. On large scales $k \ll k_{\hbox{\tiny J}}$, baryons follow the dark matter flow. As the scale approaches $k_{\hbox{\tiny J}}$, the baryonic pressure starts to oppose the gravitational infall, reducing the amplitude of velocity divergences. The shaded blue region indicates the damping regime, where the transition between the coupled and pressure-dominated behavior occurs.
• Figure 5: Left panel: First- and second-order solutions for the evolution of the JFF with zero iteration. The red line shows the evolution of the velocity field of baryonic fluctuations at first order —denoted by the subscript $(1)$—, while the black line shows the second-order solution —denoted by the subscript $(2)$—. The orange dashed line marks the scale $k = k_{\hbox{\tiny J}}$. Right panel: Difference between the first- and second-order solutions, highlighting the nonlinear correction that appears well before the Jeans scale is reached.