Galaxy Shapes and Ellipticity Frames

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Galaxy Shapes and Ellipticity Frames

This page describes the weak-lensing forward model for galaxy shapes — how an intrinsic ellipticity is sheared, calibrated, and observed — and the two conventions that NumCosmo must keep track of: the ellipticity convention (how a shape is turned into a complex number) and the handedness frame (which basis the two ellipticity components are projected onto). The forward model is implemented by the shape distributions NcGalaxySDShape (concretely NcGalaxySDShapeHSMGauss and NcGalaxySDShapeHSMGaussGlobal); the reduced-shear transforms live in NcWLEllipticityFrame and the surrounding nc_wl_ellipticity functions. The reduced shear itself comes from NcWLSurfaceMassDensity (see Weak-Lensing Surface Mass Density) and the lens geometry from NcHaloPosition.

Complex Ellipticity and the Two Conventions

A galaxy image is summarized by its second-brightness-moment (quadrupole) matrix, which defines an ellipse with semi-axes \(a \ge b\) and orientation \(\theta\). The shape is encoded as a single spin-2 complex number. Two normalizations are in common use:

\[ \chi = \frac{a^2 - b^2}{a^2 + b^2}\, e^{2 i \theta}, \qquad \epsilon = \frac{a - b}{a + b}\, e^{2 i \theta} . \]

The trace normalization \(\chi\) (the distortion) and the trace–determinant normalization \(\epsilon\) (the axis-ratio ellipticity) are selected by NcGalaxyWLObsEllipConv (NC_GALAXY_WL_OBS_ELLIP_CONV_TRACE and NC_GALAXY_WL_OBS_ELLIP_CONV_TRACE_DET). They carry the same orientation information but transform differently under shear and have different intrinsic distributions; the spin-2 factor \(e^{2 i \theta}\) is common to both.

The Reduced Shear

For a source at projected radius \(R\) and redshift \(z\) behind a lens at \(z_l\), the lensing distortion is set by the convergence \(\kappa\) and the (tangential) shear \(\gamma\) through the complex reduced shear

\[ g = \frac{\gamma}{1 - \kappa}. \]

NumCosmo computes the real tangential reduced shear \(g_t(R,z) = \gamma_t/(1-\kappa)\) from NcWLSurfaceMassDensity and orients it with the source position angle (see below),

\[ g = g_t\, e^{2 i \phi}. \]

Sources in front of the lens (\(z \le z_l\)) are unlensed, \(g_t = 0\).

Shear Transformation of the Ellipticity

An intrinsic shape \(\epsilon^{(s)}\) is mapped to the lensed (observed) shape by a Möbius transformation whose form depends on the convention and on whether the lensing is weak (\(|g| \le 1\)) or strong (\(|g| > 1\)). For the axis-ratio ellipticity \(\epsilon\),

\[ \epsilon^{(o)} = \begin{cases} \dfrac{\epsilon^{(s)} + g}{1 + g^{*}\,\epsilon^{(s)}}, & |g| \le 1, \\[2ex] \dfrac{1 + g\,\epsilon^{(s)*}}{\epsilon^{(s)*} + g^{*}}, & |g| > 1, \end{cases} \]

with the inverse map (observed \(\to\) intrinsic) obtained by \(g \to -g\). For the distortion \(\chi\) the corresponding transformation is

\[ \chi^{(o)} = \frac{\chi^{(s)} + g\,(g\,\chi^{(s)*} + 2)} {1 + |g|^2 + 2\,\mathrm{Re}(g\,\chi^{(s)*})} . \]

These are the standard reduced-shear ellipticity transforms (see e.g. Bartelmann and Schneider (2001)). They are implemented as inlined kernels (apply_shear, apply_shear_inv) selected once per convention; the public introspectable entry points are nc_galaxy_sd_shape_apply_shear and nc_galaxy_sd_shape_apply_shear_inv.

Bartelmann, M., and P. Schneider. 2001. “Weak Gravitational Lensing.” Phys. Rep. 340 (January): 291–472. https://doi.org/10.1016/S0370-1573(00)00082-X.

Calibration

Real catalogs carry shape-measurement biases. NumCosmo applies a multiplicative bias \(m\) and an additive bias \(c = c_1 + i c_2\) to the reduced shear before the transformation,

\[ g \;\longrightarrow\; (1 + m)\,g + c , \]

and adds a complex measurement noise \(n\) to the result, \(\epsilon^{(o)} \to \epsilon^{(o)} + n\). The additive bias \(c\) is a per-catalog calibration and is taken to be already expressed in the catalog’s storage frame (see below); it is therefore never re-projected.

Handedness Frames: Celestial vs. Euclidean

Every sky position in NumCosmo is given as right ascension and declination, so the position angle is always measured in the celestial convention: building a halo-centred coordinate system with nc_halo_position_polar_angles returns the polar angle \(\theta\) (the lens–source separation) and the azimuth \(\phi_C\), which increases eastward (with increasing RA).

What the catalog chooses is the orthonormal basis onto which the two ellipticity components are projected. The “celestial” / “Euclidean” nomenclature for these two handedness frames follows the GalSim image-simulation toolkit (Rowe et al. 2015); the Subaru Hyper Suprime-Cam shape catalogs, for instance, adopt the celestial convention throughout (Mandelbaum et al. 2018; Li et al. 2022). This is the handedness frame NcWLEllipticityFrame:

Rowe, B. T. P., M. Jarvis, R. Mandelbaum, et al. 2015. “GalSim: The Modular Galaxy Image Simulation Toolkit.” Astronomy and Computing 10: 121–50. https://doi.org/10.1016/j.ascom.2015.02.002.
Mandelbaum, R., H. Miyatake, T. Hamana, et al. 2018. “The First-Year Shear Catalog of the Subaru Hyper Suprime-Cam Subaru Strategic Program Survey.” Publications of the Astronomical Society of Japan 70: S25. https://doi.org/10.1093/pasj/psx130.
Li, Xiangchong, Hironao Miyatake, Wentao Luo, et al. 2022. “The Three-Year Shear Catalog of the Subaru Hyper Suprime-Cam SSP Survey.” Publications of the Astronomical Society of Japan 74: 421–59. https://doi.org/10.1093/pasj/psac006.

  • Celestial (CELESTIAL): the holonomic frame of the celestial convention. The position angle is \(\phi_C\) and the complex ellipticity is \(\epsilon_C = |\epsilon|\,e^{2 i \phi_C}\). This is the reference frame.

  • Euclidean / image (CARTESIAN): the frame an observer on the ground sees when facing the source with their head toward the North celestial pole. The right-hand direction then points East, so the image \(x\)-axis increases toward the West (decreasing RA). This is the opposite handedness (the legacy name was EUCLIDEAN).

The two frames differ by a parity (handedness) flip. The position angle reverses sense,

\[ \phi_E = \pi - \phi_C \quad (\equiv -\phi_C \bmod \pi), \]

and, because the ellipticity is spin-2, this is equivalent to complex conjugation of the shape (its cross component flips sign),

\[ \boxed{\;\epsilon_E = \epsilon_C^{*}\;} \]

The equivalence follows from \(e^{2 i \phi_E} = e^{2 i (\pi - \phi_C)} = e^{-2 i \phi_C}\). Conjugating the ellipticity and flipping \(\phi\) are thus the same operation, and the parity is its own inverse — a single map converts celestial \(\to\) Euclidean and Euclidean \(\to\) celestial alike. The two helpers [[numcosmo|nc_wl_ellipticity_celestial_to_frame_c]] (conjugates \(\epsilon\) for CARTESIAN, identity for CELESTIAL) and [[numcosmo|nc_wl_ellipticity_celestial_to_frame_angle]] (\(\phi_C \to \phi_E\)) encode exactly this, and are no-ops for the celestial frame.

Generation is frame-symmetric

When a mock galaxy is simulated, the intrinsic shape and the measurement noise are drawn from isotropic distributions in the celestial frame, the reduced shear is oriented with \(\phi_C\), and the resulting observed ellipticity is then mapped into the catalog’s storage frame so that it is stored consistently. Because the intrinsic shape and noise are isotropic, conjugating them leaves their distribution unchanged: generating in celestial-and-conjugating is statistically identical to generating directly in the Euclidean frame. The parity flip therefore affects only the deterministic pieces — the position angle and hence the lensed shape — while the calibration bias \(c\), assumed to live in the storage frame, is added without any flip.

The Shape Likelihood

For inference the model needs \(P(\epsilon^{(o)} \mid z, \text{lens})\). The observed shape is mapped back to the intrinsic one, \(\epsilon^{(s)} = \texttt{apply\_shear\_inv}\big((1+m)g + c,\ \epsilon^{(o)}\big)\), and the intrinsic distribution is evaluated there, weighted by the Jacobian of the change of variables. For the Gaussian shape models the intrinsic shape and noise combine into a single isotropic Gaussian of total variance \(\sigma_{\rm tot}^2 = \sigma^2 + \sigma_{\rm noise}^2\), giving

\[ P\!\left(\epsilon^{(o)} \mid z\right) = \frac{1}{2\pi\,\sigma_{\rm tot}^2}\, \exp\!\left[ -\frac{\lvert\epsilon^{(s)}\rvert^2}{2\,\sigma_{\rm tot}^2} + \ln\left\lvert\det\frac{\partial\epsilon^{(s)}}{\partial\epsilon^{(o)}}\right\rvert \right]. \]

For the \(\epsilon\) convention in the weak regime the log-Jacobian is

\[ \ln\left\lvert\det\frac{\partial\epsilon^{(s)}}{\partial\epsilon^{(o)}}\right\rvert = 2\ln\!\left(1 - |g|^2\right) - 2\ln\left\lvert 1 - g^{*}\epsilon^{(o)}\right\rvert^2 , \]

with analogous expressions for the \(\chi\) convention and for the strong-lensing branch. These are provided by lndet_jac (e.g. nc_galaxy_sd_shape_lndet_jac).

Because the likelihood is rotation invariant, the implementation works in the tangential/cross frame: the stored observed ellipticity is rotated by \(e^{-2 i \phi}\) (with \(\phi = \phi_E\) in the catalog frame), which makes the reduced shear real (\(g \to g_t\)) and the calibration bias is pre-rotated consistently. The full observed likelihood marginalizes this expression over the source redshift distribution, handled by NcDataClusterWL.

API Reference

The forward model and the frame machinery are spread over a few classes: