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

Abstract

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 use symbolic regression to obtain simple analytic approximations to the nonlinear scale, $k_{\sigma }$, the effective spectral index, $n_{\mathrm{eff}}$, and the curvature, $C$, which are required for the halofit model. We then re-optimise the coefficients of halofit to fit a wide range of cosmologies and redshifts. We explore the space of analytic expressions to fit the residuals between $P(k)$ and the optimised predictions of halofit. Our results are designed to match the predictions of EuclidEmulator2, but are validated against $N$-body simulations. Our symbolic expressions for $k_{\sigma }$, $n_{\mathrm{eff}}$ and $C$ 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. The re-optimised halofit parameters reduce the root mean squared fractional error (compared to EuclidEmulator2) from 3% to below 2% for wavenumbers $k=9\times 10^{-3}-9h\mathrm{Mp}{\mathrm{c}}^{-1}$. We introduce syren-halofit (symbolic-regression-enhanced halofit), an extension to halofit containing a short symbolic correction which improves this error to 1%. Our method is 2350 and 3170 times faster than current halofit and hmcode implementations, respectively, and 2680 and 64 times faster than EuclidEmulator2 (which requires running class) and the BACCO emulator. We obtain comparable accuracy to EuclidEmulator2 and BACCO 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.

11 pages, 8 figures. Accepted for publication in A&A

References in corpus (10)

• Wilkinson Microwave Anisotropy Probe (WMAP) Three Year Results: Implications for Cosmology
• Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation
• Efficient Cosmological Parameter Estimation with Hamiltonian Monte Carlo
• Machine learning constraints on deviations from general relativity from the large scale structure of the Universe
• A precise symbolic emulator of the linear matter power spectrum
• Machine learning cosmic backreaction and its effects on observations
• Cosmic backreaction and the mean redshift drift from symbolic regression
• Priors for symbolic regression
• Optimal Inflationary Potentials
• Cosmological parameter constraints using phenomenological symbolic expressions: On the significance of symbolic expression complexity and accuracy

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