syren-halofit: A fast, interpretable, high-precision formula for the $Λ$CDM nonlinear matter power spectrum (2402.17492v2)

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, $n_{\rm eff}$, 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 $k_\sigma$, $n_{\rm eff}$ 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 $k=9\times10^{-3}-9 \, h{\rm Mpc^{-1}}$. 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 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.

• The paper introduces syren-halofit, a symbolic regression-based model that dramatically accelerates ΛCDM power spectrum computations by bypassing intensive integrations.
• It develops concise analytic expressions and optimizes halofit parameters to reduce prediction errors to below 1% compared to leading numerical emulators.
• The advancement enhances computation efficiency by over 2000 times, offering a practical, interpretable tool for precise cosmological analyses.

Advancements in Symbolic Regression Enhancements for ΛCDM Nonlinear Matter Power Spectrum Modeling

Enhancing Speed and Accuracy in Cosmological Analyses

The nonlinear matter power spectrum, $P(k)$, underpins much of contemporary cosmology, encapsulating the distribution of cosmic structures across the universe. Traditional modeling techniques either rely on time-intensive N-body simulations or adopt symbolic approximations like the halofit model for computational efficiency. We discuss recent breakthroughs spearheaded by Bartlett et al. in refining symbolic approximations, significantly outpacing the computational speed of numerical emulators like euclidemulator2 and bacco, without compromising the accuracy required for cutting-edge cosmological inference.

Symbolic Regression to the Rescue

By employing symbolic regression, the team has managed to eliminate performance bottlenecks common in earlier symbolic models, specifically those related to the halofit model. The innovation lies in generating concise symbolic expressions that bypass computationally intensive steps such as integrations and root-finding algorithms. Significant upgrades have been made in three key areas:

1. Introduction of Analytic Expressions for Halofit Variables: Crafting simple yet precise analytic forms for key variables within the halofit model negates the need for complex integrals or external Boltzmann solvers, propelling the model's execution speed by factors exceeding 2000 compared to its numerical counterparts.
2. Optimization of Halofit Parameters: By recalibrating halofit's parameters against a broader and more contemporary set of cosmologies, the paper enhances the model's precision, ensuring it aligns with modern cosmological expectations and observations. Adjustments have culminated in the halofit+ model, showcasing reduced errors and improved fidelity across a spectrum of cosmological scenarios.
3. Correction Equation to Halofit Predictions: A novel symbolic expression systematically corrects predictions from the modified halofit model, boosting accuracy to within 1% of state-of-the-art numerical emulators, an achievement marking a significant milestone for symbolic approximation methods in cosmology.

Implications for Future Cosmological Studies

The work introduces syren-halofit, an evolved version of the halofit model incorporating these enhancements, signaling a pivotal shift toward more efficient and accurate symbolic approaches in cosmological analysis. This leap forward not only establishes a new benchmark for precision but also assures a drastic reduction in computational overhead, enabling more extensive and detailed cosmological studies without the traditional trade-off between accuracy and speed.

Reflecting on the broader impact, the advancements advocate for a resurgence in improving and trusting symbolic models, augmented by machine learning techniques like symbolic regression. This approach doesn't just offer a practical alternative to heavier computational methods but also ensures interpretability and adaptability—a crucial advantage for exploring novel cosmological paradigms.

Looking Forward

While the current focus has been on the standard ΛCDM model with simplifying assumptions about dark energy and neutrino masses, future extensions will likely encompass more complex cosmologies. Moreover, the prospect of refining the halofit model's underlying formalism through symbolic regression opens new avenues for research, promising even more substantial improvements in both the accuracy and computational efficiency of cosmological power spectrum predictions.

In conclusion, the development and validation of syren-halofit mark a significant stride in cosmological modeling, bridging the gap between symbolic approximation's convenience and numerical emulation's accuracy. It underscores the untapped potential of symbolic regression in cosmology, setting the stage for a new era of efficient, accurate cosmos studies that are both computationally feasible and rich in scientific insight.