syren-new: Precise formulae for the linear and nonlinear matter power spectra with massive neutrinos and dynamical dark energy

TL;DR

This work introduces syren-new, a symbolic-regression–based emulator for both linear and nonlinear matter power spectra in extended $\Lambda$CDM cosmologies that include massive neutrinos and a CPL dark energy equation of state. By building on prior syren-halofit methods, it provides compact analytic expressions for $P_{lin}$, $P_{nl}$, and $\sigma_8$ that rival numerical emulators in accuracy (RMSEs around $0.1\%$ for $\sigma_8$, $0.3\%$ for $P_{lin}$, and $1.3\%$ for $P_{nl}$) but are orders of magnitude faster to evaluate, even on GPUs. The linear spectrum is modeled as a corrected growth-factor and shape of a $\Lambda$CDM template, while the nonlinear spectrum is learned directly from simulations via a regression on the ratio to the linear spectrum, achieving sub-percent accuracy across a wide parameter range and redshifts. The authors demonstrate competitive performance against existing emulators and $N$-body results, show favorable error behavior in regions near current observational constraints, and assess the impact on weak-lensing forecasts for LSST-like surveys, concluding that the symbolic models are sufficiently accurate for cosmological inference while offering interpretable, portable expressions. The work also notes limitations related to baryonic physics and suggests future enhancements to incorporate baryonic corrections or direct simulation fitting.

Abstract

Current and future large scale structure surveys aim to constrain the neutrino mass and the equation of state of dark energy. We aim to construct accurate and interpretable symbolic approximations to the linear and nonlinear matter power spectra as a function of cosmological parameters in extended $Λ$CDM models which contain massive neutrinos and non-constant equations of state for dark energy. This constitutes an extension of the syren-halofit emulators to incorporate these two effects, which we call syren-new (SYmbolic-Regression-ENhanced power spectrum emulator with NEutrinos and $W_0-w_a$). We also obtain a simple approximation to the derived parameter $σ_8$ as a function of the cosmological parameters for these models. Our results for the linear power spectrum are designed to emulate CLASS, whereas for the nonlinear case we aim to match the results of EuclidEmulator2. We compare our results to existing emulators and $N$-body simulations. Our analytic emulators for $σ_8$, the linear and nonlinear power spectra achieve root mean squared errors of 0.1%, 0.3% and 1.3%, respectively, across a wide range of cosmological parameters, redshifts and wavenumbers. We verify that emulator-related discrepancies are subdominant compared to observational errors and other modelling uncertainties when computing shear power spectra for LSST-like surveys. Our expressions have similar accuracy to existing (numerical) emulators, but are at least an order of magnitude faster, both on a CPU and GPU. Our work greatly improves the accuracy, speed and range of applicability of current symbolic approximations to the linear and nonlinear matter power spectra. We provide publicly available code for all symbolic approximations found.

Figures (15)

• Figure 1: Pareto front of solutions found by operon for $\sigma_8/\sqrt{10^9 A_{\rm s}}$ as a function of cosmological parameters. We plot the curves for our training and validation sets separately, and indicate the chosen model (\ref{['eq:sigma8_result']}) with a dotted vertical line.
• Figure 2: Predicted values (upper) and fractional errors (lower) for the $\sigma_8$ emulator (\ref{['eq:sigma8_result']}) across the training and validation sets. The root mean squared error is 0.1%, with a maximum error of 1% for very small $\sigma_8$ values.
• Figure 3: Fractional error on the $\sigma_8$ prediction for a test set which is restricted to $\Lambda$CDM models only. We compare the results of the emulator given here to that of Bartlett_2024_linear. Our emulator has slightly weaker performance given its extended validity, but the error is almost always within 0.5%.
• Figure 4: Pareto front of solutions for the correction to the growth factor, $R(a,\bm{\theta})$, (left), and the correction to the redshift-zero matter power spectrum, $10 \log_{10} S(k,\bm{\theta})$, (right) as found by operon. The chosen models are indicated by the vertical lines, and we plot the root mean squared errors on the predictions for the training and validation sets separately.
• Figure 5: Distribution of fractional errors on $P_{\rm lin}(k, a, \bm{\theta})$ as a function of $k$ for the extended cosmological models considered in this work (left) and for $\Lambda$CDM (right). The bands given the 1 and $2\sigma$ values across the LH (\ref{['tab:cosmo_par_prior']}), with the dashed lines indicating 1% error. When averaged over $k$, we find that our expressions have a root mean squared error of 0.28% for the extended cosmology, and 0.41% for $\Lambda$CDM.