syren-halofit: A fast, interpretable, high-precision formula for the Lambda CDM nonlinear matter power spectrum

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. We aim to accelerate symbolic approximations to $P(k)$ 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. We use symbolic regression to obtain simple analytic approximations to the nonlinear scale, $k_ the effective spectral index eff $, and the curvature, $C$, which are required for the model. We then re-optimise the coefficients of to 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 between $P(k)$ and the optimised predictions of Our results are designed to match the predictions of but we validate our methods against $N$-body simulations. We find symbolic expressions for $k_ eff $ and $C$ which 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-optimised parameters, which reduce the root mean squared fractional error (compared to ) from 3 to below 2 for wavenumbers $k=9 $. We introduce (symbolic-regression-enhanced an extension to containing a short symbolic correction which improves this error to 1. Our method is 2350 and 3170 times faster than current and implementations, respectively, and 2680 and 64 times faster than (which requires running and the emulator. We obtain comparable accuracy to and the emulator when tested on $N$-body simulations. Our work greatly increases the speed and accuracy of symbolic approximations to $P(k)$, making them significantly faster than their numerical counterparts without loss of accuracy.