Covariance modelling

DSF constructs Gaussian covariance matrices for projected galaxy-galaxy lensing and clustering observables.

The covariance combines cosmological power spectra, survey geometry, tomographic redshift distributions, shape noise, and galaxy shot noise. Fourier-space covariance contributions are projected into radial bins using Hankel transforms.

Gaussian covariance

For a data vector \(\boldsymbol{d}\), the covariance is

\[C_{ab} = \left\langle (d_a-\langle d_a\rangle) (d_b-\langle d_b\rangle) \right\rangle.\]

Under the Gaussian approximation, the covariance of two power-spectrum estimators follows from Wick contractions,

\[\operatorname{Cov} \left[ \hat P_{ab}(k),\hat P_{cd}(k) \right] \propto \left(P_{ac}+N_{ac}\right) \left(P_{bd}+N_{bd}\right) + \left(P_{ad}+N_{ad}\right) \left(P_{bc}+N_{bc}\right),\]

where \(P_{ab}\) denotes a signal power spectrum and \(N_{ab}\) denotes the corresponding noise contribution. The proportionality includes the survey-volume and mode-counting normalization.

DSF uses the galaxy-galaxy, galaxy-matter, and matter-matter power spectra,

\[P_{gg}(k,z), \qquad P_{gm}(k,z), \qquad P_{mm}(k,z),\]

together with the appropriate geometric and noise factors to construct the projected covariance.

Covariance components

DSF supports three covariance contributions:

• gm x gm: covariance of the galaxy-matter excess surface density.

• gg x gg: covariance of the projected galaxy-clustering contribution.

• gm x gg: cross-covariance between the two observables.

For a joint data vector, these form the block matrix

\[\begin{split}\mathbf{C}_{\mathrm{joint}} = \begin{pmatrix} \mathbf{C}_{gm,gm} & \mathbf{C}_{gm,gg} \\ \mathbf{C}_{gm,gg}^{\mathsf T} & \mathbf{C}_{gg,gg} \end{pmatrix}.\end{split}\]

The covariance includes the relevant combinations of power spectra, with galaxy shot noise and source shape noise entering the corresponding terms.

Survey geometry and noise

The covariance depends on the survey area and the comoving volume of each lens tomographic bin.

Galaxy shot noise is determined by the three-dimensional lens number density,

\[N_g = \frac{1}{\bar n_g}.\]

Source shape noise depends on the intrinsic ellipticity dispersion \(\sigma_e\) and effective source number density \(n_{\mathrm{eff}}\).

For galaxy-galaxy lensing, the conversion between shear and excess surface density introduces critical surface density factors,

\[\Sigma_{\mathrm{crit}}(z_l,z_s) = \frac{c^2}{4\pi G} \frac{D_s}{D_l D_{ls}},\]

in the proper-distance convention. DSF evaluates the corresponding effective factors using the lens and source redshift distributions, with the appropriate conversion to its comoving convention.

Line-of-sight projection windows and lens-bin geometry also enter the projected covariance.

Tomographic covariance

Covariance ingredients are evaluated for each selected lens-source tomographic bin pair.

DSF averages the relevant power spectra over the lens redshift distribution and evaluates source-dependent quantities, including shape noise and critical surface density factors, for each source bin.

The current block-diagonal construction assumes that different selected tomographic bin pairs are independent,

\[\mathbf{C} = \operatorname{diag} \left( \mathbf{C}^{(1)}, \mathbf{C}^{(2)}, \ldots, \mathbf{C}^{(N_{\mathrm{pairs}})} \right).\]

Within each bin-pair block, correlations between radial bins are retained. Cross-covariances between different tomographic bin pairs are not included in this approximation.

Radial projection

DSF projects Fourier-space covariance contributions into real-space radial bins using Hankel transforms.

Schematically,

\[C(R,R') = \int \frac{k\,\mathrm{d}k}{2\pi} \int \frac{k'\,\mathrm{d}k'}{2\pi} J_n(kR)\, C(k,k')\, J_m(k'R').\]

The implementation supports radial bin averaging, configurable Bessel orders, and numerical tapering.

For transform conventions, numerical backends, and implementation details, see Hankel transforms.

Implementation

The DeltaSigmaCovarianceBuilder assembles the covariance from the tomographic samples and cosmological inputs.

By default, the galaxy spectra are constructed using a linear galaxy-bias prescription,

\[P_{gm}(k,z) = b_g(z)P_{mm}(k,z),\]

\[P_{gg}(k,z) = b_g^2(z)P_{mm}(k,z).\]

Alternatively, custom Pk2D objects can be supplied for both \(P_{gm}\) and \(P_{gg}\).

The builder provides individual covariance blocks, joint covariance matrices, and block-diagonal tomographic covariances.

For a practical calculation, see Analytical covariance.