Accelerating Large-Scale-Structure data analyses by emulating Boltzmann solvers and Lagrangian Perturbation Theory

TL;DR

This work tackles the computational bottleneck in large-scale-structure analyses by introducing neural-network emulators for the linear matter power spectrum that run in roughly 1 ms and achieve subpercent accuracy across a broad cosmological space, including massive neutrinos and dynamical dark energy. It also develops 15 LPT-based emulators to rapidly supply cross-spectra needed for second-order galaxy bias models. The authors validate the linear emulator against Boltzmann solvers and demonstrate unbiased cosmological constraints from mock Euclid-like weak-lensing data, while the LPT emulators provide subpercent accuracy for galaxy-related spectra and remain useful within perturbative regimes. Collected under the baccoemu project, these tools, together with nonlinear and baryonic emulators, offer a fast, accurate, and publicly available framework for next-generation LSS analyses.

Abstract

The linear matter power spectrum is an essential ingredient in all theoretical models for interpreting large-scale-structure observables. Although Boltzmann codes such as CLASS or CAMB are very efficient at computing the linear spectrum, the analysis of data usually requires $10^4$-$10^6$ evaluations, which means this task can be the most computationally expensive aspect of data analysis. Here, we address this problem by building a neural network emulator that provides the linear theory (total and cold) matter power spectrum in about one millisecond with 0.2% (0.5%) accuracy over redshifts $z \le 3$ ($z \le 9$), and scales $10^{-4} \le k \, [h {\rm Mpc^{-1}}] < 50$. We train this emulator with more than 200,000 measurements, spanning a broad cosmological parameter space that includes massive neutrinos and dynamical dark energy. We show that the parameter range and accuracy of our emulator is enough to get unbiased cosmological constraints in the analysis of a Euclid-like weak lensing survey. Complementing this emulator, we train 15 other emulators for the cross-spectra of various linear fields in Eulerian space, as predicted by 2nd-order Lagrangian Perturbation theory, which can be used to accelerate perturbative bias descriptions of galaxy clustering. Our emulators are specially designed to be used in combination with emulators for the nonlinear matter power spectrum and for baryonic effects, all of which are publicly available at http://www.dipc.org/bacco.

Figures (7)

• Figure 1: The mean absolute fractional error of our neural network, $\lambda = \left \langle \frac{|P_{\rm NN}-P_{\textsc{\tt CLASS}\xspace}|}{P_{\textsc{\tt CLASS}\xspace}} \right \rangle$, as a function of the number of epochs employed for its training. Black circles and red diamonds show the results when $\lambda$ is evaluated on the training and the validation set, respectively. The vertical blue line marks the minimum of $\lambda$ in the validation set and thus the training we will adopt thereafter in this paper.
• Figure 2: The accuracy of our neural network predictions for the linear matter power spectrum for multiple cosmologies and redshifts. We display the ratio of the power spectra computed by our emulator, $P_{\rm NN}$, to that of the Boltzmann solver CLASS, $P_{\textsc{\tt CLASS}\xspace}$, in the standard (left panel) and in the extended (right panel) cosmological parameter spaces (see \ref{['tab:cosmologies']}). In the extended space, all the cosmological parameters have values not included in the standard space. The shaded regions enclose 68%, 95%, and 99.7% of the cosmologies in our validation set, and the mean is shown as a thick black line. As an example, thin coloured lines show the results for 100 randomly selected cosmologies.
• Figure 3: Validation of the predictions of our neural-network emulator for the linear matter power spectrum. Left panel: Growth factor at $k=0.2\, h\,{\rm Mpc}^{-1}$ as a function of expansion factor. Middle panel: The ratio of the linear power spectrum over its smooth (or de-wiggled) counterpart which isolates the contribution of baryonic acoustic oscillations to the power spectrum. In both of these panels we show 10 randomly-selected cosmological models within the standard space. Right panel: The ratio of the power spectrum computed including neutrinos of a various mass, as specified in the legend, over its respective neutrino massless case. In all three cases, we show the results obtained with CLASS as solid lines, and with our emulator as dashed lines. In the bottom panels, we display the compare these predictions indicating differences of 0.1% and 0.2% as shaded regions.
• Figure 4: The accuracy of our emulators for the cross-spectrum of linear fields in Eulerian coordinates predicted by Lagrangian Perturbation Theory. The two fields defining the cross-spectra are indicated in the legend of each panel, where $1$ is an homogeneous Lagrangian field; $\delta$ and $\delta^2$ are the linear density field and its square, respectively; $s^2$ is the shear field; and $\nabla^2\delta$ is the Laplacian of the linear density field. In each panel we display the ratio of the emulator prediction over the same quantity computed by directly solving the relevant LPT expression. Shaded regions enclose 68% and 95% of the measurements in our validation set, and the mean is is marked by the thick black line. For comparison we show a randomly-selected set of cosmologies as coloured lines.
• Figure 5: The accuracy of our emulators for the galaxy auto power spectrum (left panel) and galaxy - matter cross-spectrum predicted by Lagrangian Perturbation Theory. We used bias factors randomly drawn from the priors computed in Zennaro2021. In each panel we display the ratio of the emulator prediction over the same quantity computed by directly solving the relevant LPT expression. Shaded regions enclose 68% and 95% of the measurements in our validation set, and the mean is is marked by the thick black line. For comparison we show a randomly-selected set of cosmologies as coloured lines.