NumCosmoGalaxyShapeFactorCGF

CGF-expansion evaluation of the intrinsic-ellipticity marginal.

Approximates $$P(\epsilon_\mathrm{obs} \mid g) = \int_{|\chi_I|<1} \mathrm{d}^2\chi_I\, P_\mathrm{pop}(\chi_I)\, N_2\big(\epsilon_\mathrm{obs} - f_g(\chi_I); \sigma_\mathrm{noise}^2\big)$$ by a single Gaussian in $\epsilon_\mathrm{obs}$ obtained from a cumulant-generating-function (moment) expansion of the pushforward of the intrinsic distribution through the forward shear map $S(g,\cdot) \equiv f_g$. Writing the analytic low-order response moments of $S$ around $\chi_I=0$ — the value $S_0 = S(g,0)$, the (real $2\times2$) Jacobian $A = \partial S/ \partial\chi_I|0$, and the Laplacian $\nabla^2 S|_0$ — the observed ellipticity is modelled as $\mathcal{N}(\mu, C)$ with $$\mu = S_0 + \tfrac12 V\,\nabla^2 S|_0, \qquad C = V\,A A^{\mathsf T} + \sigma\mathrm{noise}^2\, I_2,$$ where $V$ is the per-component variance of the intrinsic ellipticity.

Unlike NcGalaxyShapeFactorVarAdd (which pulls $\epsilon_\mathrm{obs}$ back through the inverse map and adds scalar variances), CGF keeps the map’s curvature: the mean picks up the Laplacian correction and the covariance is the full $A A^{\mathsf T}$ pushforward of the intrinsic variance, not an isotropic sum. It is therefore more accurate than VarAdd, while remaining a single closed-form arithmetic evaluation — cheaper than NcGalaxyShapeFactorLaplace, which needs a per-galaxy Newton mode search and a Hessian.

The variance $V$ used here is the truncated per-component second moment of the intrinsic ellipticity, i.e. $V = e_\mathrm{rms}^2$ with $e_\mathrm{rms}$ from nc_galaxy_shape_pop_e_rms(), NOT the square of the untruncated Gaussian width nc_galaxy_shape_pop_get_sigma() (which always exceeds it and would bias the recovered shear high).

The moment expansion keeps only the intrinsic second moment, so it is a Gaussian-population method: like NcGalaxyShapeFactorVarAdd it requires the population resolved from the NcmMSet to support nc_galaxy_shape_pop_get_sigma() (currently NcGalaxyShapePopGauss or NcGalaxyShapePopGaussLocal, Global or per-galaxy).

The approximation is exact in the doubly-linear ($g\to0$ or $V\to0$) limit and degrades as the reduced shear or the intrinsic width grows (the dropped higher moments and higher map derivatives then matter). See the accuracy envelope measured in tests/python/nc/lss/galaxy/test_galaxy_shape_factor_cgf.py.