Perturbative Likelihoods for Large-Scale Structure of the Universe

Abstract: This work presents a formalism for deriving likelihoods of the cosmological density field directly from first principles within Perturbation Theory (PT). By assuming a perturbative expansion around the Gaussian initial density field and additional stochastic components, we analytically compute two forms of the likelihood. Full marginalization over all underlying fields yields the likelihood of the observed density field, expressed in terms of its summary statistics (such as the power spectrum and bispectrum), which are naturally given by the formalism, and conditioned on model parameters. Marginalizing only over the stochastic fields results in the field-level likelihood. A key strength of this method is its ability to automatically specify the precise combinations of initial field covariances and PT expansion kernels required at each perturbative order (e.g., tree-level power spectrum and bispectrum, and the 1-loop power spectrum). This guarantees that the resulting likelihoods are fully consistent with PT at the chosen order of accuracy, avoiding ad-hoc choices in constructing the statistical model.