Higher order corrections to the large scale matter power spectrum in the presence of massive neutrinos

TL;DR

This work provides the first systematic one-loop correction to the large-scale matter power spectrum in a cosmology with subdominant massive neutrinos, deriving the perturbative kernels from the full equations of motion and incorporating neutrino free-streaming effects. By diagonalising the linear operator and formulating recursion relations for the interaction kernels, the authors obtain corrected $P_{cb}$ and a new cross-term $P_{cb\nu}$, enabling a consistent one-loop computation of the total power spectrum $P(k,\tau)$. They find that nonlinear corrections can enhance neutrino-induced suppression of small-scale power beyond linear expectations, aligning with multi-component N-body simulations, and provide detailed regions of validity (roughly up to $k \sim 0.2$–$0.4\,h\,\mathrm{Mpc}^{-1}$ depending on redshift) for 1% accuracy. The study also assesses practical approximations, showing that certain simplifications preserve accuracy at the 1% level for typical neutrino fractions, while highlighting the need to include nonlinear neutrino evolution for a fully complete treatment. These results improve modeling fidelity for upcoming cosmological surveys aiming to constrain neutrino masses through small-scale structure.

Abstract

We present the first systematic derivation of the one-loop correction to the large scale matter power spectrum in a mixed cold+hot dark matter cosmology with subdominant massive neutrino hot dark matter. Starting with the equations of motion for the density and velocity fields, we derive perturbative solutions to these quantities and construct recursion relations for the interaction kernels, noting and justifying all approximations along the way. We find interaction kernels similar to those for a cold dark matter-only universe, but with additional dependences on the neutrino energy density fraction f_nu and the linear growth functions of the incoming wavevectors. Compared with the f_nu=0 case, the one-loop corrected matter power spectrum for a mixed dark matter cosmology exhibits a decrease in small scale power exceeding the canonical ~8 f_nu suppression predicted by linear theory, a feature also seen in multi-component N-body simulations.

Figures (5)

• Figure 1: Contributions to the total matter power spectrum (units: $h^3 \ {\rm Mpc}^{-3}$) at $z=1$ (left) and $z=3$ (right) for a $\Lambda$CHDM cosmology with $f_\nu=0.1$. Top: CDM+baryon auto-correlation $P_{cb}(k,\tau)$. Middle: CDM+baryon--neutrino cross-correlation $P_{cb\nu}(k,\tau)$. Bottom: Total matter power spectrum $P(k,\tau)$. In all cases the linear contribution is shown in red/thin solid, the one-loop correction in green/thick solid (green/short dash when negative), and their sum in blue/long dash (blue/dotted when negative). All spectra have been divided by $a^2$ to facilitate comparison. The three vertical lines demarcate, from left to right, the upper limits in $k$ for which the linear matter power spectrum is accurate to 1% or less, the one-loop corrected spectrum to 1% or less, and the one-loop corrected spectrum to 5% or less.
• Figure 2: Relative differences between the total matter power spectra for a pure $\Lambda$CDM cosmology and three $\Lambda$CHDM models with $f_\nu=0.1$ (red/solid), 0.05 (blue/dotted), and 0.01 (green/dash) at $z=1$ (left) and $z=3$ (right). Thick lines indicate results including the one-loop correction, while the linear results are represented by the thin lines. The three vertical lines indicate the maximum $k$ values at which the linear and the one-loop corrected matter power spectra are accurate to better than 1% and 5%.
• Figure 3: One-loop corrections to the CDM+baryon auto-correlation (top) and the CDM+baryon--neutrino cross-correlation (middle) at $z=1$ (left) and $z=3$ (right) for a $\Lambda$CHDM cosmology with $f_\nu=0.1$. In the top and middle rows, the red/thin solid lines indicate the linear contribution, while the green/thick solid lines denote the one-loop correction (green/short dash when negative). The blue/long dash lines represent an approximation to the one-loop correction (blue/dotted when negative), computed by setting $S_{1,2,4}=T_{1,\cdots,8}=1$ and $S_3=0$. The fractional error incurred in the total matter power spectrum by this approximation is shown in the bottom row in red/solid. Also plotted in the bottom row in blue/long dash is the fractional error incurred by neglecting the one-loop correction to the CDM+baryon--neutrino cross-correlation.
• Figure 4: Same as figure \ref{['fig:compare']}, but for $f_\nu=0.05$.
• Figure 5: Same as figure \ref{['fig:compare']}, but for $f_\nu=0.01$.