Hintze Centre for Astrophysical Surveys

Abstract:

Context. Computing the matter power spectrum, P(k), as a function of cosmological parameters can be prohibitively slow in cosmological analyses, hence emulating this calculation is desirable. Previous analytic approximations are insufficiently accurate for modern applications, so black-box, uninterpretable emulators are often used.

Aims. We aim to construct an efficient, differentiable, interpretable, symbolic emulator for the redshift zero linear matter power spectrum which achieves sub-percent level accuracy. We also wish to obtain a simple analytic expression to convert A_s to σ₈ given the other cosmological parameters.

Methods. We utilise an efficient genetic programming based symbolic regression framework to explore the space of potential mathematical expressions which can approximate the power spectrum and σ₈. We learn the ratio between an existing low-accuracy fitting function for P(k) and that obtained by solving the Boltzmann equations and thus still incorporate the physics which motivated this earlier approximation.

Results. We obtain an analytic approximation to the linear power spectrum with a root mean squared fractional error of 0.2% between k = 9 × 10⁻³ − 9 h Mpc⁻¹ and across a wide range of cosmological parameters, and we provide physical interpretations for various terms in the expression. Our analytic approximation is 950 times faster to evaluate than CAMB and 36 times faster than the neural network based matter power spectrum emulator BACCO. We also provide a simple analytic approximation for σ₈ with a similar accuracy, with a root mean squared fractional error of just 0.1% when evaluated across the same range of cosmologies. This function is easily invertible to obtain A_s as a function of σ₈ and the other cosmological parameters, if preferred.

Conclusions. It is possible to obtain symbolic approximations to a seemingly complex function at a precision required for current and future cosmological analyses without resorting to deep-learning techniques, thus avoiding their black-box nature and large number of parameters. Our emulator will be usable long after the codes on which numerical approximations are built become outdated.

Abstract:

<jats:p><jats:italic>Context.</jats:italic>Rapid and accurate evaluation of the nonlinear matter power spectrum,<jats:italic>P</jats:italic>(<jats:italic>k</jats:italic>), 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.</jats:p><jats:p><jats:italic>Aims.</jats:italic>We aim to accelerate symbolic approximations to<jats:italic>P</jats:italic>(<jats:italic>k</jats:italic>) 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.</jats:p><jats:p><jats:italic>Methods.</jats:italic>We use symbolic regression to obtain simple analytic approximations to the nonlinear scale,<jats:italic>k</jats:italic><jats:sub><jats:italic>σ</jats:italic></jats:sub>, the effective spectral index,<jats:italic>n</jats:italic><jats:sub>eff</jats:sub>, and the curvature,<jats:italic>C</jats:italic>, which are required for the<jats:sc>HALOFIT</jats:sc>model. We then re-optimise the coefficients of<jats:sc>HALOFIT</jats:sc>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<jats:italic>P</jats:italic>(<jats:italic>k</jats:italic>) and the optimised predictions of<jats:sc>HALOFIT</jats:sc>. Our results are designed to match the predictions of<jats:sc>EUCLIDEMULATOR</jats:sc>2, but we validate our methods against<jats:italic>N</jats:italic>-body simulations.</jats:p><jats:p><jats:italic>Results.</jats:italic>We find symbolic expressions for<jats:italic>k</jats:italic><jats:sub><jats:italic>σ</jats:italic></jats:sub>,<jats:italic>n</jats:italic><jats:sub>eff</jats:sub>and<jats:italic>C</jats:italic>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<jats:sc>HALOFIT</jats:sc>parameters, which reduce the root mean squared fractional error (compared to<jats:sc>EUCLIDEMULATOR</jats:sc>2) from 3% to below 2% for wavenumbers<jats:italic>k</jats:italic> = 9 × 10<jats:sup>−3</jats:sup> − 9 <jats:italic>h</jats:italic> Mpc<jats:sup>−1</jats:sup>. We introduce<jats:sc>SYREN-HALOFIT</jats:sc>(symbolic-regression-enhanced<jats:sc>HALOFIT</jats:sc>), an extension to<jats:sc>HALOFIT</jats:sc>containing a short symbolic correction which improves this error to 1%. Our method is 2350 and 3170 times faster than current<jats:sc>HALOFIT</jats:sc>and<jats:sc>HMCODE</jats:sc>implementations, respectively, and 2680 and 64 times faster than<jats:sc>EUCLIDEMULATOR</jats:sc>2 (which requires running<jats:sc>CLASS</jats:sc>) and the<jats:sc>BACCO</jats:sc>emulator. We obtain comparable accuracy to<jats:sc>EUCLIDEMULATOR</jats:sc>2 and the<jats:sc>BACCO</jats:sc>emulator when tested on<jats:italic>N</jats:italic>-body simulations.</jats:p><jats:p><jats:italic>Conclusions.</jats:italic>Our work greatly increases the speed and accuracy of symbolic approximations to<jats:italic>P</jats:italic>(<jats:italic>k</jats:italic>), making them significantly faster than their numerical counterparts without loss of accuracy.</jats:p>