NumCosmoGalaxyShapeFactorVarAdd

Variance-add evaluation of the intrinsic-ellipticity marginal.

This is the classic (approximate) Gaussian treatment: instead of marginalizing over $\chi_I$, the observed ellipticity is pulled back through the inverse shear map, $\chi_I^\ast = f_{\tilde g}^{-1}(\epsilon_\mathrm{obs})$, and evaluated on a Gaussian whose variance is the sum of the intrinsic and noise variances, $$P(\epsilon_\mathrm{obs} \mid g) \approx \frac{e^{-|\chi_I^\ast|^2 / 2(\sigma^2 + \sigma_\mathrm{noise}^2)}} {2\pi\,(\sigma^2 + \sigma_\mathrm{noise}^2)}\, \left|\det J_{f^{-1}}(\epsilon_\mathrm{obs})\right|.$$ The variance-add combination, as opposed to the exact truncated convolution, is exact only where the shear map is expanded jointly to first order in $g$ and $\chi_I$ — not in $g$ alone, since the dropped terms mix the two — and the intrinsic integral is extended from the physical unit disc $|\chi_I|<1$ to the whole plane, which is the only way two independent Gaussians combine in closed form.

This implementation removes the map linearization by using the exact inverse map and its exact Jacobian at a single pulled-back point. It keeps the plane-instead-of-disc approximation, that is the untruncated Gaussian normalization. See the Variance-Add Approximation section of the theory page for the derivation.

The variance addition is defined only for a population parameterized by an untruncated Gaussian width, so the population resolved from the NcmMSet must support nc_galaxy_shape_pop_get_sigma(): currently NcGalaxyShapePopGauss or NcGalaxyShapePopGaussLocal, global or per-galaxy.