SYREN-HALOFIT: A fast, interpretable, high-precision formula for the ΛCDM nonlinear matter power spectrum

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Context.Rapid and accurate evaluation of the nonlinear matter power spectrum,P(k), as a function of cosmological parameters and redshift is of fundamental importance in cosmology. Analytic approximations provide an interpretable solution, yet current approximations are neither fast nor accurate relative to numerical emulators.Aims.We aim to accelerate symbolic approximations toP(k) by removing the requirement to perform integrals, instead using short symbolic expressions to compute all variables of interest. We also wish to make such expressions more accurate by re-optimising the parameters of these models (using a larger number of cosmologies and focussing on cosmological parameters of more interest for present-day studies) and providing correction terms.Methods.We use symbolic regression to obtain simple analytic approximations to the nonlinear scale,kσ, the effective spectral index,neff, and the curvature,C, which are required for theHALOFITmodel. We then re-optimise the coefficients ofHALOFITto fit a wide range of cosmologies and redshifts. We then again exploit symbolic regression to explore the space of analytic expressions to fit the residuals betweenP(k) and the optimised predictions ofHALOFIT. Our results are designed to match the predictions ofEUCLIDEMULATOR2, but we validate our methods againstN-body simulations.Results.We find symbolic expressions forkσ,neffandCwhich have root mean squared fractional errors of 0.8%, 0.2% and 0.3%, respectively, for redshifts below 3 and a wide range of cosmologies. We provide re-optimisedHALOFITparameters, which reduce the root mean squared fractional error (compared toEUCLIDEMULATOR2) from 3% to below 2% for wavenumbersk = 9 × 10−3 − 9 h Mpc−1. We introduceSYREN-HALOFIT(symbolic-regression-enhancedHALOFIT), an extension toHALOFITcontaining a short symbolic correction which improves this error to 1%. Our method is 2350 and 3170 times faster than currentHALOFITandHMCODEimplementations, respectively, and 2680 and 64 times faster thanEUCLIDEMULATOR2 (which requires runningCLASS) and theBACCOemulator. We obtain comparable accuracy toEUCLIDEMULATOR2 and theBACCOemulator when tested onN-body simulations.Conclusions.Our work greatly increases the speed and accuracy of symbolic approximations toP(k), making them significantly faster than their numerical counterparts without loss of accuracy.