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 itself (rather than the exact truncated convolution) is exact only in the doubly-linear regime where the shear map is expanded jointly to first order in $g$ AND $\chi_I$ (not just 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 (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, but keeps the plane-instead-of-disc approximation (the plain un-truncated Gaussian normalization). See the Variance-Add Approximation section of the theory page for the full derivation.

The variance addition is only meaningful for a population parameterized by an (untruncated) Gaussian width sigma, so this method requires the population resolved from the NcmMSet to support nc_galaxy_shape_pop_get_sigma() (currently NcGalaxyShapePopGauss or NcGalaxyShapePopGaussLocal, Global or per-galaxy).