Aletheia: Emulating the non-linear matter power spectrum in the context of evolution mapping

TL;DR

Aletheia tackles the challenge of predicting the non-linear matter power spectrum across a wide cosmological parameter space by adopting an evolution-mapping framework that decouples shape and evolution, compressing redshift dependence into σ_{12} and a growth-history descriptor ̃{x}. It implements a two-stage Gaussian Process emulation: first predicting the non-linear boost factor B(k) as a function of Θ_s and σ_{12}, then correcting for growth history with a derivative emulator ∂R/∂ ̃{x}, and finally applying a resolution-correction factor to extend accuracy to high k. Validation against independent N-body runs shows sub-percent accuracy, with 0.2% variance across dynamic DE models, outperforming EuclidEmulator2 in stability and generalization, and even handling DESI-best-fit cosmologies outside standard training ranges. The results demonstrate that evolution mapping yields a robust, extensible tool for precision cosmology, and the publicly available Aletheia package enables broader applications to non-linear large-scale structure statistics.

Abstract

We present Aletheia, a new emulator of the non-linear matter power spectrum, $P(k)$, built upon the evolution mapping framework. This framework addresses the limitations of traditional emulation by focusing on $h$-independent cosmological parameters, which can be separated into those defining the linear power spectrum shape ($\mathbfΘ_{\mathrm{s}}$) and those affecting only its amplitude evolution ($\mathbfΘ_{\mathrm{e}}$). The combined impact of evolution parameters and redshift is compressed into a single amplitude parameter, $σ_{12}$. Aletheia uses a two-stage Gaussian Process emulation: a primary emulator predicts the non-linear boost factor as a function of ($\mathbfΘ_{\mathrm{s}}$) and $σ_{12}$ for fixed evolution parameters, while a second one applies a small linear correction based on the integrated growth history. The emulator is trained on shape parameters spanning $\pm$5$σ$ of Planck constraints and a wide clustering range $0.2 < σ_{12} < 1.0$, providing predictions for $0.006\,{\rm Mpc}^{-1} < k < 2\,{\rm Mpc}^{-1}$. We validate Aletheia against N-body simulations, demonstrating sub-percent accuracy. When tested on a suite of dynamic dark energy models, the full emulator's predictions show a variance of approximately 0.2%, a factor of five smaller than that of the state-of-the-art EuclidEmulator2 (around 1% variance). Furthermore, Aletheia maintains sub-percent accuracy for the best-fit dynamic dark energy cosmology from recent DESI data, a model whose parameters lie outside the training ranges of most conventional emulators. This demonstrates the power of the evolution mapping approach, providing a robust and extensible tool for precision cosmology.

Figures (10)

• Figure 1: The left panel shows the ratio of non-linear power spectra for a set of simulations with identical shape parameters but widely varying evolution parameters, all evaluated at redshifts that correspond to the same values of $\sigma_{12}$. The specific cosmological parameters for each model are detailed in Table 2 of Esposito2024_VelEvoMap. The right panel shows the ratio of the true $P(k)$ to the reference prediction (using $\tilde{x}_0$) and the resulting derivative $\partial R / \partial \tilde{x}$ measured from the simulations, illustrating the accuracy of the first-order Taylor expansion in equation (\ref{['eq:pk_taylor_xtilde']}).
• Figure 2: The ratio of power spectra, $R(k)$, as a function of the integrated growth history parameter $\tilde{x}$ for four choices of $k$ and $\sigma_{12}$, as indicated in the legend. The points show the measurements from the Aletheia simulations, where each colour represents a different cosmology Esposito2024_VelEvoMap, all evaluated at a fixed $\sigma_{12}$. The dashed lines show the best-fitting linear relation for each case. The remarkable linearity of the response validates the first-order approximation in equation (\ref{['eq:pk_taylor_xtilde']}) and allows for a direct measurement of the derivative $\partial R / \partial \tilde{x}$ from the simulations.
• Figure 3: Distribution of cosmological parameters for the training (blue points) and testing (orange points) sets of simulations used for $\mathcal{E}_B(k)$. The panels show 2D projections of the parameter space $(\omega_{\mathrm{b}}, \omega_{\mathrm{c}}, n_{\mathrm{s}}, \sigma_{12})$. The grey ellipses represent the $1\sigma$ and $2\sigma$ confidence regions derived from Planck 2018 data for the shape parameters $(\omega_{\mathrm{b}}, \omega_{\mathrm{c}}, n_{\mathrm{s}})$. Our sampling strategy uses the eigenvector directions of the Planck covariance matrix to broadly cover the relevant parameter space. The sampling for $\sigma_{12}$ covers the range from $0.2$ to $1.0$.
• Figure 4: Raw training data for the Aletheia emulators. The left panel shows the measured boost factor, $B(k)$, for the 100 simulations to train $\mathcal{E}_B(k)$, colour-coded by the clustering amplitude $\sigma_{12}$ at which they were evaluated. This panel illustrates the strong, smooth dependence of the boost factor on $\sigma_{12}$. The right panel displays the measured derivative term $\partial R/\partial \tilde{x}$ for the same cosmologies, determined from the derivative simulations described in Section \ref{['sssec:sims_dRdx']}. The smooth dependency of both quantities on the cosmological parameters and $k$ makes them well-suited for Gaussian Process emulation.
• Figure 5: Correction of the Aletheia emulator for resolution effects. The solid lines show the measured ratio of the uncorrected emulator prediction, $\mathcal{E}_P(k)$, to the high-resolution power spectrum, $P_{\mathrm{HR}}(k)$, derived from the AletheiaMass simulations. The lines are colour-coded by the value of $\sigma_{12}$. The fluctuations are caused by cosmic variance due to the smaller box size of the high-resolution runs. The dashed lines represent the corresponding two-dimensional smoothing spline interpolation, $\mathcal{C}(k, \sigma_{12})$, which is used to apply the correction to the final emulator prediction.