NumCosmoGalaxyShapeFactorFixedQuad

Fixed-node lens-domain quadrature evaluation of the intrinsic-ellipticity marginal.

Like NcGalaxyShapeFactorQuad, evaluates $$P(\epsilon_\mathrm{obs} \mid g) = \int_{|\chi_L|<1} \mathrm{d}^2\chi_L\, P_\mathrm{pop}\big(f_g^{-1}(\chi_L)\big)\, \left|\det J_{f_g^{-1}}(\chi_L)\right|\, N_2\big(\epsilon_\mathrm{obs} - \chi_L; \sigma_\mathrm{noise}^2\big) $$ exactly (no series truncation in $g$), with a FIXED node count over the INTERSECTION of two discs instead of Quad’s adaptive Divonne cubature over a generic box: the noise kernel is supported near $\epsilon_\mathrm{obs}$ (radius $\sim n_\sigma\sigma_\mathrm{noise}$), $P_\mathrm{pop}\circ f_g^{-1}$ only over the unit disc, and their overlap is a two-circle “lens” in general, a plain disc when one contains the other.

The noise kernel lives in $\chi_L$-space and does not depend on $g$, so the whole quadrature domain (node positions, weights, and the noise-kernel value at each node) is $g$-INDEPENDENT: it depends only on $(R,\phi,\sigma_\mathrm{noise})=(\lvert\epsilon_\mathrm{obs}\rvert, \arg\epsilon_\mathrm{obs},\sigma_\mathrm{noise})$, cached per galaxy and reused across every $g$ a fit tries. Validated end to end against an independent scipy oracle and against NcGalaxyShapeFactorQuad itself (see the test suite and Fixed-Node Lens-Domain Quadrature).

Every term summed is manifestly non-negative (quadrature weights, population density, $\lvert\det J\rvert$, and the two-arc domain’s Jacobian are all non-negative), so unlike NcGalaxyShapeFactorSeriesLensed there is no truncated polynomial that can cross zero: this class stays accurate at any physical $g$, real or complex, through $\lvert g\rvert=0.99$.

Works for ANY population (not just Gaussian): each node evaluates nc_galaxy_shape_pop_eval_p() directly at $x_i=\lvert\chi_I\rvert^2$, the same convention NcGalaxyShapeFactorQuad uses (not eval_p_rho2(), whose $\rho^2$ argument is the pre-compactification radius of the $(u,v)$-plane substitution this class does not use), so unlike NcGalaxyShapeFactorSeriesLensed there is no Gaussian-only guard.

Limitation: a fixed grid cannot resolve a population much narrower than its node spacing ($\sigma_\mathrm{pop}\lesssim0.05$, or a sharply concentrated Beta population); use NcGalaxyShapeFactorQuad for narrower or more exotic populations. Production only uses Gaussian populations with $\sigma_\mathrm{pop}\in(0.2,0.4)$, comfortably inside this class’s validated regime. See docs/theory/wl_shape_factor_history.md for why an adaptive alternative was tried and rejected for the narrow-population case, and for the design history of this class more generally.

Cost cliff: at $\sigma_\mathrm{noise}$ where the noise disk is comparable in size to the unit disc (roughly $\sigma_\mathrm{noise}\in(0.05,0.2)$, see docs/theory/wl_shape_marginalization_fixed_quad.qmd’s own “cost cliff” section), nearly every galaxy lands in the genuine-lens branch ($\mathtt{n_lens}^2$ nodes, 1681 at the default 41) rather than the cheaper contained branches — expensive, though this project’s actual production regime ($\sigma_\mathrm{noise}\sim0.3$) is unaffected either side of it. NcGalaxyShapeFactorFixedQuad:auto-lens-nodes (default FALSE, opt-in) calibrates a per-galaxy lens-branch node count instead of always using the configured NcGalaxyShapeFactorFixedQuad:n-lens, cutting cost in that regime (~2x fewer nodes typical, more in the expensive middle) with no change to shipped behavior unless explicitly enabled — see _calibrate_n_lens()’s own docs for the calibration strategy.