The Cosmic Linear Anisotropy Solving System (CLASS) II: Approximation schemes

TL;DR

This paper introduces and validates three approximation schemes—tight-coupling approximation (TCA), ultra-relativistic fluid approximation (UFA), and radiation streaming approximation (RSA)—to accelerate Boltzmann-code calculations in LCDM cosmology without sacrificing precision. It provides detailed derivations, implementation strategies, and cross-scheme comparisons within CLASS, including multiple TCA variants and UFA/RSA options, highlighting the default compromise_TCA and ufa_class implementations. The study demonstrates substantial computational savings (often factors of a few to orders of magnitude) while maintaining sub-percent accuracy in CMB and matter power spectra, aided by a stiff integrator (ndf15). Collectively, these approximations enable efficient, accurate, and flexible cosmological parameter estimation, with clear guidelines for extending to non-minimal models.

Abstract

Boltzmann codes are used extensively by several groups for constraining cosmological parameters with Cosmic Microwave Background and Large Scale Structure data. This activity is computationally expensive, since a typical project requires from 10'000 to 100'000 Boltzmann code executions. The newly released code CLASS (Cosmic Linear Anisotropy Solving System) incorporates improved approximation schemes leading to a simultaneous gain in speed and precision. We describe here the three approximations used by CLASS for basic LambdaCDM models, namely: a baryon-photon tight-coupling approximation which can be set to first order, second order or to a compromise between the two; an ultra-relativistic fluid approximation which had not been implemented in public distributions before; and finally a radiation streaming approximation taking reionisation into account.