  Cosmological simulations play a crucial role in elucidating the effect of
physical parameters on the statistics of fields and on constraining parameters
given information on density fields. We leverage diffusion generative models to
address two tasks of importance to cosmology -- as an emulator for cold dark
matter density fields conditional on input cosmological parameters $\Omega_m$
and $\sigma_8$, and as a parameter inference model that can return constraints
on the cosmological parameters of an input field. We show that the model is
able to generate fields with power spectra that are consistent with those of
the simulated target distribution, and capture the subtle effect of each
parameter on modulations in the power spectrum. We additionally explore their
utility as parameter inference models and find that we can obtain tight
constraints on cosmological parameters.
