  We present a Bayesian hierarchical modelling approach to infer the cosmic
matter density field, and the lensing and the matter power spectra, from cosmic
shear data. This method uses a physical model of cosmic structure formation to
infer physically plausible cosmic structures, which accounts for the
non-Gaussian features of the gravitationally evolved matter distribution and
light-cone effects. We test and validate our framework with realistic simulated
shear data, demonstrating that the method recovers the unbiased matter
distribution and the correct lensing and matter power spectrum. While the
cosmology is fixed in this test, and the method employs a prior power spectrum,
we demonstrate that the lensing results are sensitive to the true power
spectrum when this differs from the prior. In this case, the density field
samples are generated with a power spectrum that deviates from the prior, and
the method recovers the true lensing power spectrum. The method also recovers
the matter power spectrum across the sky, but as currently implemented, it
cannot determine the radial power since isotropy is not imposed. In summary,
our method provides physically plausible inference of the dark matter
distribution from cosmic shear data, allowing us to extract information beyond
the two-point statistics and exploiting the full information content of the
cosmological fields.
