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

Deaglan J. Bartlett*, Benjamin D. Wandelt, Matteo Zennaro, Pedro G. Ferreira, Harry Desmond

*Corresponding author for this work

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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 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. 

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 the halofit model. We then re-optimise the coefficients of halofit 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 halofit. Our results are designed to match the predictions of euclidemu-lator2, but we validate our methods against N-body simulations. 

Results: We find symbolic expressions for Kσ, neff 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 halofit parameters, which reduce the root mean squared fractional error (compared to euclidemulator2) from 3% to below 2% for wavenumbers k = 9×10-3-9 h Mpc-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 the bacco emulator when tested on N-body simulations. 

Conclusions: 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.

Original languageEnglish
Article numberA150
Number of pages11
JournalAstronomy and Astrophysics
Publication statusPublished - 5 Jun 2024


  • Cosmological parameters
  • Cosmology: theory
  • Large-scale structure of Universe
  • Methods: numerical
  • UKRI
  • STFC

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