Efficient analytic approximation for small-scale non-cold relic perturbations

TL;DR

The paper addresses the computational bottleneck in modeling non-cold relic perturbations by deriving a fully analytic small-scale quasi-stationary approximation to the collisionless Boltzmann equation and implementing it in CLASSIER. This approach yields runtime reductions of roughly a factor of two to six compared with existing methods, while preserving high accuracy in observables. It demonstrates sub-0.1% accuracy in the present-day matter power spectrum up to k around 100 Mpc^-1 across several neutrino masses. The method provides a practical path to faster, high-precision cosmological analyses and can be extended to non-standard dark-matter models.

Abstract

We develop a highly accurate analytic approximation for small-scale non-cold relic perturbations by solving the collisionless Boltzmann equation in the quasi-stationary regime. The approximation is implemented in CLASSIER (CLASS Integral Equation Revision), a modified version of the Boltzmann solver CLASS that replaces the traditional truncated Boltzmann hierarchy of non-cold relic multipoles with a small set of integral equations solved iteratively. Applying it to massive neutrinos yields a factor-of-two reduction in total runtime relative to CLASSIER without the approximation. Compared to standard CLASS runs (with $\ell_{\rm max}^{\rm NCDM}=40$ and no late-time massive neutrino fluid approximation) under the same precision setting, CLASSIER with this approximation is faster by a factor of 3-6. The approximation faithfully reproduces the late-time behavior of massive neutrino perturbations and preserves sub-$0.1\%$ accuracy in the matter power spectrum today up to comoving wavenumber $k=100\,{\rm Mpc}^{-1}$. With this approximation, massive-neutrino perturbations are no longer the computational bottleneck on small scales for linear-theory predictions. The approach can be readily extendable to non-standard dark-matter models, and offers prospects for further efficiency gains in high-precision cosmological analyses.

Figures (6)

• Figure 1: Source terms in the Boltzmann equation in the synchronous gauge: $\dot{\eta}$ (top) which decays quickly and $(\dot{h}+6\dot{\eta})/2$ (bottom) which determines the smooth late-time behavior of neutrino perturbations. The relatively small rapid oscillations on top of a smooth behavior in $(\dot{h}+6\dot{\eta})/2$ allow us to find an accurate quasi-stationary analytic approximation for NCDM perturbations.
• Figure 2: The first three multipole moments of massive neutrino phase-space distribution function perturbations, $\Psi_0$, $\Psi_1$, and $\Psi_2$, for comoving wavenumber $k = 10$ Mpc$^{-1}$ and comoving momentum $q = 12.64 ~T_0$, where $T_0$ is the neutrino temperature today. Blue: numerical solutions from CLASSIER; Red: small-scale approximations from Eqs. \ref{['eq:Psi0-ss']}--\ref{['eq:Delta_l']}.
• Figure 3: Regions in $(k,q)$ space that satisfy the small-scale approximation criterion of Eq. \ref{['eq:criterion']}. Different colors correspond to different neutrino masses. Vertical dashed lines indicate the $q$-grid used. Here $q$ is the comoving momentum in units of $T_0$, the neutrino temperature today.
• Figure 4: Runtime distribution for evolving perturbations of three representative $k$-modes in CLASSIER (left: $k=100\;{\rm Mpc}^{-1}$, middle: $k=10\;{\rm Mpc}^{-1}$, right: $k=1\;{\rm Mpc}^{-1}$), shown without (left bar) and with (right bar) the small-scale approximation. Each bar is split into contributions from the zeroth-iteration step that solves the truncated Boltzmann hierarchy (BH) with the late-time fluid approximation (blue), from the integral-equation evaluation of NCDM perturbations (red hatched), and from the rest of the system (orange). The small-scale approximation substantially reduces the time spent on the NCDM integral-equation step, which otherwise accounts for a non-negligible share of the runtime at high $k$. At small $k$, fewer momenta qualify, but these modes contribute negligibly to the overall runtime. When the approximation is used, we reduce the resolution of $\tau$-grid for convolution calculations and this additionally saves the small amount of runtime.
• Figure 5: Absolute fractional differences of the $z=0$ matter power spectrum with respect to CLASS with $\ell^{\rm NCDM}_{\rm max}=500$. The solid lines are with the neutrino mass $m_\nu=0.06$eV, and the dashed and the dot-dashed lines are with $m_\nu=0.01$eV and $0.3$eV, respectively.