Deep learning reconstruction of the large scale structure of the Universe from luminosity distance observations

Supernovae Ia (SNe) can provide a unique window on the large scale structure (LSS) of the Universe at redshifts where few other observations are available, by solving the inversion problem (IP) consisting in reconstructing the LSS from its effects on the observed luminosity distance. So far the IP was solved assuming some restrictions about space-time, such as spherical symmetry for example, while we obtain for the first time solutions of the IP problem for arbitrary space-time geometries using deep learning. The method is based on the use of convolutional neural networks (CNN) trained on simulated data. The training data set is obtained by first generating random density and velocity fields, and then computing their effects on the luminosity distance. The CNN, based on an appriately modified version of U-Net to account for the tridimensionality of the data, is then trained to reconstruct the density and velocity fields from the luminosity distance. We find that the velocity field inversion is more accurate than the density field, because the effects of the velocity on the luminosity distance only depend on the source velocity, while in the case of the density it is an integrated effect along the line of sight, giving rise to more degeneracy in the solution of the IP. Improved versions of these neural networks, modified to accommodate the non uniform distribution of the SNe, can be applied to observational data to reconstruct the large scale structure of the Universe at redshifts at which few other observations are available.