Joint reconstruction of $H(z)$ and $fσ_8(z)$ with physics informed neural networks

Abstract

We present a proof of concept for the joint reconstruction of the Hubble parameter $H(z)$ that assumes no dark energy equation of state and the growth rate of large scale structure $fσ_8(z)$ using a physics informed neural network. Rather than fitting these two observables separately and checking their consistency post hoc, we couple them through the linear growth equation of general relativity directly during training, using the equation residual evaluated at collocation points via automatic differentiation as an additional loss term. The network employs a shared backbone feeding two independent output heads, one per observable. We train an ensemble of 100 independently seeded networks on a compilation of 50 $H(z)$ measurements from Cosmic Chronometers and Baryon Acoustic Oscillations and 63 $fσ_8(z)$ measurements from Redshift Space Distortions, and study four values of the physics coupling weight $λ\in \{0,\,0.01,\,0.1,\,1.0\}$. We then anchor the $H_0$ normalization using two independent local distance scale determinations: the SH0ES result $H_0 = 73.04 \pm 1.04$\,km\,s$^{-1}$\,Mpc$^{-1}$ and the Local Distance Network consensus $H_0 = 73.50 \pm 0.81$\,km\,s$^{-1}$\,Mpc$^{-1}$. With either prior the Hubble constant is recovered exactly at the prior value, and the two reconstructions are indistinguishable in $fσ_8(z)$. The reconstructed $fσ_8(z)$ sits systematically below the $Λ$CDM prediction at all redshifts, consistent with the $σ_8$ tension, while the $\mathrm{Om}(z)$ null test shows a marked departure from the flat $Λ$CDM expectation at low redshift. The results establish that coupling the two observables through the growth equation during training is both feasible and beneficial, and that the reconstruction is robust to the choice between the two local $H_0$ determinations.