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

TL;DR

This paper introduces a fast and accurate analytic approximation for small-scale non-cold relic perturbations, significantly reducing computational time in cosmological simulations while maintaining high precision.

Contribution

The authors develop a novel integral equation-based approximation implemented in CLASSIER, improving efficiency and accuracy in modeling massive neutrino perturbations compared to standard methods.

Findings

Reduces total runtime by a factor of two in neutrino calculations.

Achieves 0.1% accuracy in matter power spectrum up to k=100 Mpc^{-1}.

Faster by a factor of 3-6 compared to standard CLASS with similar precision.

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.