Machine Learning improved fits of the sound horizon at the baryon drag epoch

The baryon acoustic oscillations (BAO) have proven to be an invaluable tool in constraining the expansion history of the Universe at late times and are characterized by the comoving sound horizon at the baryon drag epoch $r_\mathrm{s}(z_\mathrm{d})$. The latter quantity can be calculated either numerically using recombination codes or via fitting functions, such as the one by Eisenstein and Hu (EH), made via grids of parameters of the recombination history. Here we quantify the accuracy of these expressions and show that they can strongly bias the derived constraints on the cosmological parameters using BAO data. Then, using a machine learning approach, called the genetic algorithms, we proceed to derive new analytic expressions for $r_\mathrm{s}(z_\mathrm{d})$ which are accurate at the $\sim0.003\%$ level in a range of $10\sigma$ around the Planck 2018 best-fit or $\sim0.018\%$ in a much broader range, compared to $\sim 2-4\%$ for the EH expression, thus obtaining an improvement of two to three orders of magnitude. Moreover, we also provide fits that include the effects of massive neutrinos and an extension to the concordance cosmological model assuming variations of the fine structure constant. Finally, we note that our expressions can be used to ease the computational cost required to compute $r_\mathrm{s}(z_\mathrm{d})$ with a Boltzmann code when deriving cosmological constraints using BAO data from current and upcoming surveys.