Third-order Perturbation Theory With Non-linear Pressure

TL;DR

This work extends perturbation theory to third order by including a pressure gradient for a pressured component in a two-component matter system, enabling a self-consistent analysis of non-linear pressure effects on the matter power spectrum. The authors derive linear and non-linear Jeans-filtering functions, show that non-linearities reduce the effective filtering scale relative to linear theory, and quantify implications for baryons and neutrinos within a simplified toy model with constant $k_J$. They provide explicit expressions for the non-linear power spectrum and compare to the STT neutrino approximation, finding that neutrinos exhibit substantial non-linearity below their free-streaming scale and that linear treatments can misestimate the power spectrum and filtering properties. The results imply that, in practice, inferences about IGM temperature and filtering mass from Ly$$ forest data could be biased if linear filtering scales are assumed, highlighting the need for time-dependent $k_J$ treatments and full Boltzmann modeling for neutrinos. Overall, the paper demonstrates that non-linear pressure effects are important for accurate small-scale clustering and cosmological inferences.

Abstract

We calculate the non-linear matter power spectrum using the 3rd-order perturbation theory without ignoring the pressure gradient term. We consider a semi-realistic system consisting of two matter components with and without pressure, and both are expanded into the 3rd order in perturbations in a self-consistent manner, for the first time. While the pressured component may be identified with baryons or neutrinos, in this paper we mainly explore the physics of the non-linear pressure effect using a toy model in which the Jeans length does not depend on time, i.e., the sound speed decreases as 1/a^{1/2}, where a is the scale factor. The linear analysis shows that the power spectrum below the so-called filtering scale is suppressed relative to the power spectrum of the cold dark matter. Our non-linear calculation shows that the actual filtering scale for a given sound speed is smaller than the linear filtering scale by a factor depending on the redshift and the Jeans length. A ~40% change is common, and our results suggest that, when applied to baryons, the temperature of the Inter-galactic Medium inferred from the filtering scale observed in the flux power spectrum of Lyman-alpha forests would be underestimated by a factor of two, if one used the linear filtering scale to interpret the data. The filtering mass, which is proportional to the filtering scale cubed, can also be significantly smaller than the linear theory prediction especially at low redshift, where the actual filtering mass can be smaller than the linear prediction by a factor of three. Finally, when applied to neutrinos, we find that neutrino perturbations deviate significantly from linear perturbations even below the free-streaming scales, and thus neutrinos cannot be treated as linear perturbations.

Figures (5)

• Figure 1: Decaying mode solution for the linear filtering function at the zeroth-order iteration ($f_c\rightarrow 1$), $\Delta g_1^{(0)}(k,\tau)\equiv g_1^{(0)}(k,\tau)-1/(1+k^2/k_J^2)$, where $g_1^{(0)}(k,\tau)$ is the numerical solution of eq. (\ref{['eq:zerothgeq']}), with the initial conditions given by $g_1^{(0)}(k,\tau_*)=1$ and $\dot{g}_1^{(0)}(k,\tau_*)=0$ where $\tau_*$ is the conformal time at $z_*=10$. The top and bottom lines at $k/k_J\sim 1$ are at $z=8$ and 0, respectively, and the other lines correspond to the intermediate redshifts.
• Figure 2: Ratio of the total matter power spectrum, $P_{tot}(k,z)$, to the CDM part, $P_c(k,z)$, at $z=0.1$ (top), 1, 3, 5, 10, and 30 (bottom). (Left) The input Jeans wavenumber of $k_J=1~h~{\rm Mpc}^{-1}$. (Right) $k_J=3~h~{\rm Mpc}^{-1}$. The dashed lines show the ratios calculated from the linear theory, whereas the dot-dashed lines show the linear calculations with $k_J=2$ and $6~h~{\rm Mpc}^{-1}$ for the left and right panels respectively, to show that the actual filtering wavenumbers, predicted by the 3PT calculations, can be $\sim$40% as large as the linear filtering wavenumber at low redshift.
• Figure 3: Fractional difference between our full calculation and the approximation used by saito/etal:2008 (STT), $[P_{tot}(k)-P^{\rm STT}_{tot}(k)]/P_{tot}(k)$, for $\Omega_\nu/\Omega_m=1/100$ (top), $1/20$ (middle), and $1/10$ (bottom), which corresponds to $\sum m_\nu\simeq 0.13$, $0.64$, and $1.3$ eV, respectively.
• Figure 4: Fractional difference between the non-linear neutrino power spectrum, $P_\nu(k)$, and the linear power spectrum, $P_\nu^{lin}(k)$, $[P_{\nu}(k)-P^{\rm lin}_{\nu}(k)]/P_{\nu}(k)$, for $\Omega_\nu/\Omega_m=1/100$ (top), $1/20$ (middle), and $1/10$ (bottom), which corresponds to $\sum m_\nu\simeq 0.13$, $0.64$, and $1.3$ eV, respectively.
• Figure 5: The dimensionless power spectra, $\Delta^2(k)\equiv k^3P(k)/(2\pi^2)$, for a matter component with pressure (i.e., baryon, neutrino, etc) are shown for several redshifts ($z=0.1$, $1.0$, $3.0$, $5.0$, $10$ and $30$). We show the non-linear calculations with 3PT in the solid and dotted lines for $k_J=1.0$ and $3.0~h~{\rm Mpc^{-1}}$, respectively. We also show the linear calculations in the dashed and dot-dashed lines for $k_J=1.0$ and $3.0~h~{\rm Mpc^{-1}}$, respectively.