Simultaneous measurement of cosmology and intrinsic alignments using joint cosmic shear and galaxy number density correlations

TL;DR

This work tackles the IA contamination in cosmic shear by proposing a joint analysis of tomographic galaxy ellipticity, galaxy number density, and ellipticity-density cross-correlations. It introduces a flexible two-dimensional grid parametrization for intrinsic alignments and galaxy bias, and uses a Fisher-matrix framework to show that including density and cross-correlations can self-calibrate these systematics while preserving cosmological information. The results indicate that even with hundreds of nuisance parameters, the combined signal substantially recovers the lensing information, especially for Euclid-like surveys, and that cross-correlations between density and ellipticity are particularly valuable for breaking degeneracies. The study highlights important dependencies on redshift distribution, priors, and magnification slopes, suggesting this joint approach could become standard for future large-area surveys.

Abstract

Cosmic shear is a powerful method to constrain cosmology, provided that any systematic effects are under control. The intrinsic alignment of galaxies is expected to severely bias parameter estimates if not taken into account. We explore the potential of a joint analysis of tomographic galaxy ellipticity, galaxy number density, and ellipticity-number density cross-correlations to simultaneously constrain cosmology and self-calibrate unknown intrinsic alignment and galaxy bias contributions. We treat intrinsic alignments and galaxy biasing as free functions of scale and redshift and marginalise over the resulting parameter sets. Constraints on cosmology are calculated by combining the likelihoods from all two-point correlations between galaxy ellipticity and galaxy number density. The information required for these calculations is already available in a standard cosmic shear dataset. We include contributions to these functions from cosmic shear, intrinsic alignments, galaxy clustering and magnification effects. In a Fisher matrix analysis we compare our constraints with those from cosmic shear alone in the absence of intrinsic alignments. For a potential future large area survey, such as Euclid, the extra information from the additional correlation functions can make up for the additional free parameters in the intrinsic alignment and galaxy bias terms, depending on the flexibility in the models. For example, the Dark Energy Task Force figure of merit is recovered even when more than 100 free parameters are marginalised over. We find that the redshift quality requirements are similar to those calculated in the absence of intrinsic alignments.

Figures (10)

• Figure 1: Fiducial power spectra for all considered correlations. The upper right panels depict the contributions to $\epsilon\epsilon$ (in black) and $nn$ (in magenta) correlations. The lower left panels show the contributions to correlations between number density fluctuations and ellipticity. Since we only show correlations $C_{\alpha \beta}^{(ij)}(\ell)$ with $i \leq j$, we make in this plot a distinction between $n \epsilon$ (in red; number density contribution in the foreground, e.g. gG) and $\epsilon n$ (in blue; number density contribution in the background, e.g. Gg) correlations. In each sub-panel a different tomographic redshift bin correlation is shown. For clarity only odd bins are displayed. In the upper right panels the usual cosmic shear signal (GG) is shown as a black solid lines; the intrinsic alignment GI term is shown by the black dashed lines; the intrinsic alignment II term is shown by the dotted black line; the usual galaxy clustering signal (gg) is shown by the magenta solid line; the cross correlation between galaxy clustering and lensing magnification (gm) is shown by the magenta dashed line; the lensing magnification correlation functions (mm) are shown by the magenta dotted line. In the lower left panels the solid blue line shows the correlation between lensing shear and galaxy clustering (Gg); the blue dashed line shows the correlation between lensing shear and lensing magnification (gm); the blue dot-dashed line shows the correlation between intrinsic alignment and galaxy clustering (Ig or equivalently gI); the red solid line shows the correlation between galaxy clustering and lensing shear (gG), which is equivalent to the blue solid line with redshift bin indices $i$ and $j$ reversed; similarly the red dashed line shows the correlation between lensing magnification and lensing shear (mG), for cases where the magnification occurs at lower redshift than the shear ($i<j$); finally the dotted line shows the correlation between lensing magnification and intrinsic alignment (mI).
• Figure 2: Left panels: Figures of merit as a function of the number of free parameters as a function of wave vector $N_K$ in the bias terms For each line type, the upper curve is obtained for a number of free bias parameters as a function of redshift $N_Z=2$, the lower is for $N_Z=4$. Right panels: Same as on the left, but as a function of $N_Z$, i.e. the number of redshift parameters in the bias terms. The upper curves for each set correspond now to $N_K=2$ and the lower ones to $N_K=4$, respectively. Upper panels: Figure of merit taking into account the full cosmological parameter space, ${\rm FoM}_{\rm TOT}$, see (\ref{['eq:fomtot']}). Lower panels: Dark Energy figure of merit from the Dark Energy Task Force ${\rm FoM}_{\rm DETF}$, see (\ref{['eq:fomdetf']}). Dashed curves correspond to results using galaxy ellipticity correlations ($\epsilon\epsilon$) only, dotted black curves to galaxy number density correlations ($nn$) only, and solid black curves to results using all correlations ($\epsilon\epsilon$, $nn$ and $\epsilon n$). The grey dotted lines show results for $nn$ correlations without imposing cuts in angular frequency. The constant grey line marks the FoM computed for the pure lensing, i.e. GG, signal, assuming intrinsic alignments do not exist. In addition we show the resulting figures of merit when using our most flexible parametrisation with $N_K=N_Z=7$ as filled symbols. Circles correspond to $\epsilon\epsilon$, triangles to $nn$, and diamonds to all correlations.
• Figure 3: $1\,\sigma$-contours for all pairs of cosmological parameters considered, marginalised over all other parameters. We have used a photometric redshift uncertainty parameter value $\sigma_{\rm ph}=0.05$, ten photometric redshift bins for tomography $N_{\rm zbin}=10$, and the most flexible intrinsic alignment and bias model considered in this paper, with over two hundred free parameters ($N_K=7$, and $N_Z=7$). Orange (light hatched) confidence regions result from using galaxy number density correlations ($nn$) (excluding the non-linear regime) only, red (dark hatched) regions use ellipticity correlations ($\epsilon \epsilon$) alone, and blue (filled) regions correspond to using all available information including density-ellipticity cross-correlations. For reference, the contours obtained from a pure lensing signal are shown as black lines. Flat priors on cosmological parameters have been applied.
• Figure 4: Upper panel: The Figure of Merit for all cosmological parameter space ${\rm FoM}_{\rm TOT}$ as a function of the number of photo-z bins used for tomography $N_{\rm zbin}$, shown for $\epsilon \epsilon$ (dashed line), $nn$ (dotted line), and all (solid) correlations. The grey line corresponds to results for lensing only (GG). Throughout, $N_K=N_Z=5$ nuisance parameters for the bias terms are used. These results are obtained for the standard set of parameters and $\sigma_{\rm ph}=0.05$. Lower panel: Same as above, but in terms of the dark energy figure of merit ${\rm FoM}_{\rm DETF}$.
• Figure 5: Upper panel: The difference $d_{\rm FoM}$, defined in (\ref{['eq:fomrelative']}), as a function of the photo-z dispersion $\sigma_{\rm ph}$, shown for $\epsilon \epsilon$ correlations (dashed line), all correlations (black solid line), and the lensing only signal (grey solid line). Throughout, nuisance parameters $N_K=N_Z=5$ are used. These results are obtained for the standard set of parameters and $N_{\rm zbin}=10$. Lower panel: Same as above, but in terms of the ratio $r_{\rm FoM}$, given in (\ref{['eq:fomrelative2']}).