Translation and Rotation Equivariant Normalizing Flow (TRENF) for OptimalCosmological Analysis

Our universe is homogeneous and isotropic, and its perturbations are translation and rotation equivariant. We develop a generative Normalizing Flow(NF) architecture which explicitly incorporates these symmetries, defining the data likelihood via a sequence of Fourier space based convolutions and non linear transforms, evaluating their Jacobian at each layer. This allows to train the NF by maximizing the data likelihood p(x|y) as a function of the labels y, such as cosmological parameters. In contrast to other generative models the NF approach has no loss of information since it preserves the full dimensionality of the data, and gives direct access to the data likelihoodp(x|y). We apply this to outputs of cosmological N-body simulations and show that the summary statistics of the generated samples agree well with the simulations. We show that the reverse mapping is visually indistinguishable from a Gaussian white noise: when this is perfectly achieved the resulting p(x|y) likelihood analysis becomes optimal. On simple Gaussian examples we show that this approach maximizes the information in the data and saturates the Fisher information content in the labels y. On N-body simulation outputs we show that this leads to significant improvements in constraining power over the standard summary statistics such as the power spectrum. Finally, we develop a generalization of this NF that can handle effects that break the symmetry of the data, such as the survey mask.