Data vector and systematics

DSF models the excess surface density \(\Delta\Sigma(R)\), the primary observable of galaxy-galaxy lensing. The prediction is constructed from a three-dimensional galaxy-matter power spectrum and can include additional physical contributions and observational systematics.

Excess surface density

The excess surface density is defined as

\[\Delta\Sigma(R,z) = \bar{\Sigma}(<R,z)-\Sigma(R,z),\]

where \(\Sigma(R,z)\) is the projected surface density and \(\bar{\Sigma}(<R,z)\) is its mean within projected radius \(R\).

In Fourier space, the galaxy-matter contribution can be expressed as a second-order Hankel transform,

\[\Delta\Sigma(R,z) = \bar{\rho}_{m,0} \int_0^\infty \frac{k\,\mathrm{d}k}{2\pi} P_{gm}(k,z)J_2(kR),\]

where \(P_{gm}(k,z)\) is the galaxy-matter power spectrum, \(J_2\) is the second-order Bessel function, and \(\bar{\rho}_{m,0}\) is the mean comoving matter density.

DSF evaluates this projection using a CCL Pk2D object, allowing different physical models to be supplied through the same interface.

Tomographic data vector

For a lens tomographic bin \(i\) with redshift distribution \(n_l^i(z)\), DSF computes the lens-averaged profile,

\[\Delta\Sigma^i(R) = \frac{ \int \mathrm{d}z\, n_l^i(z)\Delta\Sigma(R,z) }{ \int \mathrm{d}z\,n_l^i(z) }.\]

The tomographic data vector is formed by concatenating the profiles for the selected lens-source bin pairs,

\[\boldsymbol{\mu} = \left( \Delta\Sigma^{i_1}(R_1),\ldots, \Delta\Sigma^{i_1}(R_{N_R}), \Delta\Sigma^{i_2}(R_1),\ldots \right)^{\mathsf T}.\]

The ordering follows the selected tomographic bin pairs. For the baseline prediction, pairs sharing the same lens bin have identical \(\Delta\Sigma\) profiles; source-bin dependence enters through the lensing geometry and covariance.

DSF uses projected comoving radii and reports \(\Delta\Sigma\) in \(M_\odot/\mathrm{pc}^2\).

Physical modelling and systematics

The galaxy-matter power spectrum can be constructed using different physical prescriptions without changing the projection or data-vector interface.

Halo occupation distribution (HOD)

The HOD model describes the occupation of dark matter halos by galaxies and determines the galaxy-matter clustering signal through the adopted halo-model prescription.

Intrinsic alignments

Intrinsic alignments correlate galaxy shapes with the surrounding tidal field. DSF provides nonlinear alignment (NLA) combinations, controlled by the intrinsic-alignment amplitude \(A_{\mathrm{IA}}\).

Baryonic effects

Baryonic processes redistribute matter within halos, modifying the small-scale density profile. DSF provides a baryonified HOD prescription in which the halo concentration-mass relation is rescaled through the parameter \(f_c\).

Lens magnification

Magnification changes the observed lens-galaxy number density through flux magnification and area distortion. DSF provides an additive lens-magnification correction,

\[\Delta\Sigma_{\mathrm{obs}}(R) = \Delta\Sigma_{\mathrm{base}}(R) + \Delta\Sigma_{\mathrm{mag}}(R),\]

whose amplitude is proportional to \(\alpha_{\mathrm{lens}}-1\), where \(\alpha_{\mathrm{lens}}\) characterizes the magnification response of the lens sample.

Stellar point mass

The stellar mass of the central lens galaxy contributes

\[\Delta\Sigma_\star(R) = \frac{M_\star}{\pi R^2}.\]

This contribution is most relevant at small projected separations.

These ingredients can be used to compare alternative physical predictions or construct systematic corrections to the baseline data vector.

Further examples

See Data vector for the basic \(\Delta\Sigma\) calculation and Systematics modelling for comparisons of HOD, intrinsic-alignment, baryonic, and magnification models.