Jeans Filtering Functions in Cosmology

• Jeans filtering functions are mathematical tools that quantify how baryonic pressure suppresses small-scale density fluctuations in the universe.
• They are derived from cosmological perturbation theory, incorporating both linear and nonlinear corrections to map baryon-CDM density bias.
• Applications include refining intergalactic medium models and interpreting Lyα forest data, which impacts our understanding of reionization and structure formation.

Jeans filtering functions quantify how baryonic pressure in the Universe suppresses small-scale fluctuations in the baryonic matter density field, differentiating it from the cold dark matter (CDM) component. These functions, emerging from the linear and nonlinear cosmological perturbation theory, mathematically encapsulate the pressure-induced bias between baryons and CDM, and define the so-called Jeans (or filtering) scale—below which baryonic structures are smoothed out. Analytical approaches based on the Vlasov equation and perturbative expansions provide explicit formulae for these filtering functions and clarify their impact on the density, velocity, and power spectra of cosmic structures, as well as cosmological observables such as the Lyman α forest.

1. Theoretical Foundations: From the Vlasov Equation to Filtering Functions

The starting point for understanding Jeans filtering functions is the collisionless Boltzmann (Vlasov) equation for the baryon phase-space distribution: $\frac{d f}{dt} = 0$ Taking successive moments yields the continuity and Euler equations in comoving coordinates. For baryons, a barotropic equation of state is assumed, introducing a pressure term proportional to the density fluctuation: $$-\frac{\nabla_x P_B}{\rho_B} \longrightarrow -c_s^2(\tau) \nabla_x\,\delta_B(\tau,\mathbf x)$$ where $c_s^2 = \partial P / \partial \rho$ is the squared sound speed. In the linear regime and Einstein–de Sitter cosmology, the wavenumber where pressure balances gravity is the Jeans wavenumber: $k_J(\tau) = \frac{\sqrt{6}}{c_s(\tau)\,\tau}$ Modes with $k \gg k_J$ are pressure-supported and oscillate, suppressing baryonic growth; for $k \ll k_J$, gravitational collapse proceeds.

2. Linear and Nonlinear Jeans Filtering Functions

The linear Jeans filtering function, $F_1(k,\tau)$, is defined as the ratio of first-order baryon and CDM density perturbations: $$g_1(k,\tau) = \frac{\tilde\delta_B^{(1)}(k,\tau)}{\tilde\delta_C^{(1)}(k,\tau)} \equiv F_1(k,\tau)$$ with the growing-mode solution: $F_1(k,\tau) = [1 + (k/k_J)^2]^{-1}$ In second-order cosmological perturbation theory, the baryonic density perturbation is expanded as: $$\tilde\delta_B(k,\tau) = a(\tau) F_1(k,\tau) \tilde\delta_{1,C}(k) + a^2(\tau) F_2(k)\tilde\delta_{2,C}(k) + \cdots$$ where

$$F_2(k) = \frac{10/3 - (7/3)[1 - \delta'_{2,C}(k)/\delta_{2,C}(k)]}{10/3 + (k/k_J)^2}$$

The quantities $\delta_{2,C}(k)$ and $\delta'_{2,C}(k)$ are convolution integrals over first-order CDM fields modulated by symmetric second-order kernels and the linear filter $F_1$.

3. Application to Cosmic Density and Velocity Fields

The Jeans filtering functions act as k-dependent bias factors mapping CDM to baryonic fluctuations up to the second perturbative order in both density and velocity, under the expansion: $$\begin{aligned} \tilde\delta_C(k,\tau) &= \sum_{n\ge1} a^n(\tau) \tilde\delta_{n,C}(k) \ \tilde\delta_B(k,\tau) &= \sum_{n\ge1} a^n(\tau) g_n(k,\tau) \tilde\delta_{n,C}(k) \end{aligned}$$ The velocity divergence expansion is analogous: $$\tilde\theta_B(k,\tau) = \dot a\,a^0\,F_1(k,\tau)\,\tilde\theta_{1,C}(k) + \dot a \, a^1 \, h_2(k) \tilde\theta_{2,C}(k) + \cdots$$ with $h_1 = F_1$ and $h_2(k)$ as the second-order velocity filter.

4. Nonlinear Shift of the Filtering Scale, Mass, and Temperature

The filtering scale, at which baryon fluctuations are significantly suppressed, is shifted to higher wavenumbers in nonlinear theory: $k_F \approx 1.3\,k_J \quad \text{(second order);}$

$k_F \sim 1.4\,k_J\ \text{(third order)}$

Consequently, the effective filtering mass

$M_F \sim (1.3)^{-3} M_J \approx 0.45 M_J$

is lower by a factor of about 2.2 relative to linear predictions. Since $k_J \propto T_B^{-1/2}$, the inferred baryon temperature from a linear fit would be systematically higher than in the nonlinear case: $$T_{B,\rm nl} \sim (1.3)^{-2} T_{B,\rm lin} \approx 0.6\,T_{B,\rm lin}$$ indicating up to $\sim$40% underestimation of temperature when nonlinear effects are neglected (Fonseca et al., 11 Nov 2025).

5. Implementation in Semi-Analytic IGM Models and Lyα Forest

Jeans filtering in semi-analytic models (e.g., Rorai et al. (Rorai et al., 2013)) is realized by smoothing the underlying dark matter density field either via a Gaussian kernel in Fourier space: $$\delta_b(k) = W_J(k;\lambda_J) \delta_\mathrm{DM}(k),\quad W_J(k; \lambda_J) = \exp [-\tfrac{1}{2} k^2 \lambda_J^2]$$ or a cubic-spline kernel of finite support $R_J$ in real space, with $\lambda_J = R_J/3.25$. The filtered baryon field sets the temperature–density relation and neutral fraction for the fluctuating Gunn–Peterson approximation within Lyα forest calculations. Both Jeans filtering and 1D thermal broadening suppress small-scale power in mock and observed Lyα forest transmission spectra, though the geometrical nature and $k$-dependence differ:

• Thermal broadening operates along the line of sight as a velocity-space convolution,
• Jeans filtering modulates the full 3D field prior to projection.

Transverse coherence of Lyα absorption in close quasar pairs provides a direct constraint on $\lambda_J$, largely decoupled from thermal Doppler broadening, via the correlation of phase angles of homologous Fourier modes: $$P(\Delta\theta) = \frac{1}{2\pi} \frac{1-\zeta^2}{1+\zeta^2-2\zeta\cos \Delta\theta}$$ with inference of $\zeta(k,r_\perp|\lambda_J)$ leading to a measurement of $\lambda_J$.

6. Impact on the Matter Power Spectrum and Clumping

In both linear and nonlinear (second-order) theory, the effect of Jeans filtering is a scale-dependent suppression of small-scale baryon power relative to CDM: $P_b(k) = F_1^2(k)\,P_C(k)$ In second-order, nonlinear couplings introduce corrections: $$P_b(k) \simeq F_1^2(k)\,P_C(k) + 2 F_1(k)F_2(k) P_{1\otimes2}(k) + \cdots$$ At wavenumbers $k \sim k_J$, the difference between $F_1$ and $F_2$ reaches ~30%, corresponding to up to a 70% change in baryon power. The IGM clumping factor, controlling recombination rates, depends explicitly on the small-scale baryonic power spectrum and is thus sensitive to the adopted filter: $$C = 1 + \sigma_b^2,\quad \sigma_b^2 = \int d\ln k\, [k^3 P_b(k)/(2\pi^2)]$$ A plausible implication is that accurate modeling of reionization and intergalactic chemistry necessitates precise characterization of the Jeans filtering function and its dependence on baryonic temperature and nonlinear physics.

7. Observational Measurement and Inference

Direct measurement of the Jeans (filtering) scale in the IGM via the coherence of Lyα forest absorption across close QSO pairs has been achieved by analyzing the phase difference distribution of longitudinal Fourier modes. Bayesian inference on the wrapped Cauchy distribution parameter $\zeta$ at various separations $r_\perp$ and Fourier modes $k$ enables the extraction of $\lambda_J$ to ~5% precision with only ~20 quasar pairs, robust to continuum fitting, instrumental noise, and metal-line systematics (Rorai et al., 2013). The method exploits the fact that Jeans filtering, in contrast to thermal Doppler broadening, determines how rapidly phase coherence is lost as separation increases.

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In summary, Jeans filtering functions $F_1$ and $F_2$ (and the corresponding real-space or Fourier-space smoothing kernels) provide the analytic framework for describing baryonic fluctuation bias, the suppression of small-scale power, and the shift in effective filtering mass and temperature stemming from pressure effects and nonlinear coupling. Both analytical derivations (Fonseca et al., 11 Nov 2025) and semi-analytic modeling (Rorai et al., 2013) demonstrate the central role of Jeans filtering in setting the thermal and morphological properties of the low-density IGM, the minimum halo mass for collapse, and cosmological observables such as the Lyα forest.