Symbolic Emulators for Cosmology: Accelerating Cosmological Analyses Without Sacrificing Precision

TL;DR

This work addresses the computational bottleneck in cosmological parameter inference by developing symbolic emulators for key $\Lambda$CDM quantities. Using a genetic-programming based symbolic regression framework, it extends prior ranges and delivers analytic, differentiable expressions for the radial comoving distance, linear growth factor, $A_s\leftrightarrow\sigma_8$ conversion, linear power spectrum, and halofit variables, achieving sub-percent accuracy across relevant redshifts and parameter space. When embedded in a DES-Y1–like $3\times2$pt analysis, these emulators yield cosmological posteriors in excellent agreement with camb, while delivering large speedups (≈$60\times$) and reduced memory usage; they also reveal biases when coupled with the Eisenstein & Hu + halofit approach. The results demonstrate that symbolic emulators are a viable, efficient, and interpretable alternative for Stage-III cosmology and have strong potential for scaling to Stage-IV analyses, with code and models made publicly available.

Abstract

In cosmology, emulators play a crucial role by providing fast and accurate predictions of complex physical models, enabling efficient exploration of high-dimensional parameter spaces that would be computationally prohibitive with direct numerical simulations. Symbolic emulators have emerged as promising alternatives to numerical approaches, delivering comparable accuracy with significantly faster evaluation times. While previous symbolic emulators were limited to relatively narrow prior ranges, we expand these to cover the parameter space relevant for current cosmological analyses. We introduce approximations to hypergeometric functions used for the $Λ$CDM comoving distance and linear growth factor which are accurate to better than 0.001% and 0.05%, respectively, for all redshifts and for $Ω_{\rm m} \in [0.1, 0.5]$. We show that integrating symbolic emulators into a Dark Energy Survey-like $3\times2$pt analysis produces cosmological constraints consistent with those obtained using standard numerical methods. Our symbolic emulators offer substantial improvements in speed and memory usage, demonstrating their practical potential for scalable, likelihood-based inference.

Figures (6)

• Figure 1: Fractional errors on our approximations to the hypergeometric functions required to evaluate the radial comoving distance (left, \ref{['eq:chi_approx']}) and linear growth factor (right, \ref{['eq:D_approx_lcdm']}) for a $\Lambda$CDM cosmology. In both cases, $x \equiv a^3 (\Omega_{\rm m} - 1) / \Omega_{\rm m}$, for scale factor $a$ and present-day matter density parameter $\Omega_{\rm m}$. The approximations were obtained for $x$ in the grey shaded region (corresponding to the prior range in \ref{['tab:parameter_prior']}), and they thus show very good extrapolation behaviour, particularly for values of $x$ relevant for cosmology (red hatched region).
• Figure 2: Left: Fractional error on symbolic fits to the redshift-zero linear matter power spectrum when compared against camb. We plot the 68% error distributions when the cosmological parameters are varied across the range given in \ref{['tab:parameter_prior']}. Right: The various contributions to the prediction for $\log F$ at the Planck 2018 cosmology. The high frequency oscillations of $\cos(f_3)$ are a plotting artefact from using a logarithmic $x$ scale: for large $k$, $f_3$ is linear in $k$.
• Figure 3: Comparison between the true and symbolic fits for the halofit variables: the nonlinear scale ($k_\sigma$; left), the effective slope ($n_{\rm eff}$; centre), and curvature ($C$; right). We plot the predicted vs true values in the upper panels and the distribution of their fractional errors in the lower panels.
• Figure 4: Fractional error on symbolic fits to the halofit approximation to the nonlinear matter power spectrum when compared against camb's implementation of halofit. We plot the 68% and 95% error distributions when the cosmological parameters are varied across the range given in \ref{['tab:parameter_prior']}, and in each panel we choose randomly sampled redshifts in the ranges given by the titles. Our approximation utilises the symbolic approximation for the linear matter power spectrum (\ref{['eq:pk_lin_fit']}), the growth factor (\ref{['eq:D_approx_lcdm']}) and the halofit variables (\ref{['eq:ksigma_fit', 'eq:neff_fit', 'eq:C_fit']}).
• Figure 5: One- and two-dimensional posterior distributions of the cosmological parameters for our mock cosmological DES-Y1 like analysis. The red contours are obtained using an 'exact' model (camb) and are compared to those obtained using the symbolic approximations produced in this work (blue contours), and to the combination of E&H and halofit, as implemented in jax-cosmo (green contours). The grey dashed lines indicate the true parameters. In the left panel we consider the sampled cosmological parameters (which all have uniform priors) whereas in the right panel we also consider a derived parameter, $S_8$. The high level of consistency between our contours and the exact model indicate that the emulators are sufficiently accurate for such an analysis. This is not true for the E&H+halofit model.