Dr Harry Desmond

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>

Abstract:

Inflation is a highly favored theory for the early Universe. It is compatible with current observations of the cosmic microwave background and large scale structure and is a driver in the quest to detect primordial gravitational waves. It is also, given the current quality of the data, highly underdetermined with a large number of candidate implementations. We use a new method in symbolic regression to generate all possible simple scalar field potentials for one of two possible basis sets of operators. Treating these as single-field, slow-roll inflationary models we then score them with an information-theoretic metric ("minimum description length") that quantifies their efficiency in compressing the information in current data. We explore two possible priors on the parameter space of potentials, one related to the functions' structural complexity and one that uses a Katz back-off language model to prefer functions that may be theoretically motivated. This enables us to identify the inflaton potentials that optimally balance simplicity with accuracy at explaining current data, which may subsequently find theoretical motivation. Our exploratory study opens the door to extraction of fundamental physics directly from data, and may be augmented with more refined theoretical priors in the quest for a complete understanding of the early Universe.