Bayesian deep learning for cosmic volumes with modified gravity

TL;DR

This work addresses cosmological parameter inference under modified gravity using Bayesian neural networks with multiplicative normalising flows, trained on MG-PICOLA simulations for 3D overdensity fields and power spectra. By comparing Bayesian last-layer (BLL) and fully Bayesian (FullB) architectures, the authors demonstrate accurate recovery of $Ω_m$ and $σ_8$ with well-calibrated uncertainties, while the MG parameter $f_{R0}$ remains degenerate with $σ_8$ in the current small-volume, nonlinear regime. Both voxel-grid 3D convolutional representations and two-point statistics (power spectra) yield consistent results, with FullB giving slight gains at the cost of longer inference times, and BLL often offering a practical speed-up by about a factor of two. The study highlights the potential and challenges of using Bayesian deep learning to extract cosmological information from nonlinear cosmic volumes, pointing to higher-resolution simulations and additional data (e.g., velocity information) as avenues for improved MG constraints.

Abstract

The new generation of galaxy surveys will provide unprecedented data allowing us to test gravity at cosmological scales. A robust cosmological analysis of the large-scale structure demands exploiting the nonlinear information encoded in the cosmic web. Machine Learning techniques provide such tools, however, do not provide a priori assessment of uncertainties. This study aims at extracting cosmological parameters from modified gravity (MG) simulations through deep neural networks endowed with uncertainty estimations. We implement Bayesian neural networks (BNNs) with an enriched approximate posterior distribution considering two cases: one with a single Bayesian last layer (BLL), and another one with Bayesian layers at all levels (FullB). We train both BNNs with real-space density fields and power-spectra from a suite of 2000 dark matter only particle mesh $N$-body simulations including modified gravity models relying on MG-PICOLA covering 256 $h^{-1}$ Mpc side cubical volumes with 128$^3$ particles. BNNs excel in accurately predicting parameters for $Ω_m$ and $σ_8$ and their respective correlation with the MG parameter. We find out that BNNs yield well-calibrated uncertainty estimates overcoming the over- and under-estimation issues in traditional neural networks. We observe that the presence of MG parameter leads to a significant degeneracy with $σ_8$ being one of the possible explanations of the poor MG predictions. Ignoring MG, we obtain a deviation of the relative errors in $Ω_m$ and $σ_8$ by at least $30\%$. Moreover, we report consistent results from the density field and power spectra analysis, and comparable results between BLL and FullB experiments which permits us to save computing time by a factor of two. This work contributes in setting the path to extract cosmological parameters from complete small cosmic volumes towards the highly nonlinear regime.

Figures (7)

• Figure 1: Multidimensional parameter space variations. Each line represents the parameter values of a data point. The four parameters $\Omega_m,\, h,\,\sigma_8$, and $0.1\log_{10}|f_{R0}|$ are visualised along separate axes.
• Figure 2: Projected overdensity field at redshift $z=0$ derived from an arbitrarily chosen simulation within the ensemble of 2500 MG simulations. The normalised density field was calculated using a CIC mass-assignment scheme.
• Figure 3: Projected density field of dark matter in a region of $256\times256\times20$ ($h^{-1}Mpc$)${}^3$ from 100 out of 2500 arbitrarily chosen simulations of MG. The snapshots are taken at $z=0$, and the legend displays the set of cosmological parameters of {$\Omega_m$, $h$, $\sigma_8$, $f_{R_0}$}. The cuts in the density field highlight the broad coverage of the parameter space of the MG simulations. Different features can be observed by naked eye, such as variations in the filament structure of the cosmic web.
• Figure 4: Matter power spectrum at $z=0$ of the MG simulation suit. The variations in the spectrum correspond to changes in each of the four parameters that were varied, $\Omega_m$, $h$, $\sigma_8$ and $|0.1\log f_{R0}|$. The corresponding range of each of parameter is shown in the label.
• Figure 5: Resblock schema depending on the architecture used. Top: Resblock when SeResNet18 is employed. The dashed orange rectangle shows the skip SE-connection schema used in the SeResNet18 resblock. Bottom: Resblock when ResNet is employed.