Algorithm for the direct reconstruction of the dark matter correlation function from weak lensing and galaxy clustering

The clustering of matter on cosmological scales is an essential probe for studying the physical origin and composition of our Universe. To date, most of the direct studies have focused on shear-shear weak lensing correlations, but it is also possible to extract the dark matter clustering by combining galaxy-clustering and galaxy-galaxy-lensing measurements. In order to extract the required information, one must relate the observable galaxy distribution to the underlying dark matter distribution. In this study we develop in detail a method that can constrain the dark matter correlation function from galaxy clustering and galaxy-galaxy-lensing measurements, by focusing on the correlation coefficient between the galaxy and matter overdensity fields. Our goal is to develop an estimator that maximally correlates the two. To generate a mock galaxy catalogue for testing purposes, we use the halo occupation distribution approach applied to a large ensemble of $N$-body simulations to model preexisting SDSS luminous red galaxy sample observations. Using this mock catalogue, we show that a direct comparison between the excess surface mass density measured by lensing and its corresponding galaxy clustering quantity is not optimal. We develop a new statistic that suppresses the small-scale contributions to these observations and show that this new statistic leads to a cross-correlation coefficient that is within a few percent of unity down to $5h^{-1}\mathrm{Mpc}$. Furthermore, the residual incoherence between the galaxy and matter fields can be explained using a theoretical model for scale-dependent galaxy bias, giving us a final estimator that is unbiased to within 1%, so that we can reconstruct the dark matter clustering power spectrum at this accuracy up to $k\sim 1h{\mathrm{Mpc}}^{-1}$. We also perform a comprehensive study of other physical effects that can affect the analysis, such as redshift space distortions and differences in radial windows between galaxy clustering and weak lensing observations. We apply the method to a range of cosmological models and explicitly show the viability of our new statistic to distinguish between cosmological models.

Figure 1

Top panel: Excess surface mass density $\Delta \Sigma (R)$ (solid) and ADSD ${\rm Y}(R;R_0)$ (dashed) with $R_0=3h^{-1}\mathrm{Mpc}$ for our fiducial cosmological model and the luminosity-threshold LRG sample. We show the statistics for the galaxy autocorrelation (top red), the galaxy-matter cross-correlation (central green) and the matter autocorrelation (bottom blue). The upturn of the cross-correlation towards small scales leads to a cross-correlation coefficient in excess of unity as we will see later. Bottom panel: Cross-correlation coefficient of the clustering statistics shown in the top panel. The bare excess surface mass density (solid) leads to strong deviations from unity, whereas the ADSD with $R_0=3h^{-1}\mathrm{Mpc}$ (dashed) recovers a cross correlation close to unity.

Figure 2

Left panel: Window functions for ${\rm Y}(R;R_0)$ with $R_0=3h^{-1}\mathrm{Mpc}$ in real space. We show the window function for $R=5$, 10, $20h^{-1}\mathrm{Mpc}$ as green solid, orange dashed and red dash-dotted lines, respectively. Right panel: Window functions for ${\rm Y}(R;R_0)$ with $R_0=3h^{-1}\mathrm{Mpc}$ in $k$ space. We show the products of window function and power spectrum for $R=5$, 10, $20h^{-1}\mathrm{Mpc}$ as green solid, orange dashed, and red dashed-dotted line, respectively. For reference we also plot the window for the correlation function as thin lines. Note that we multiplied with $x$ and $k$, respectively, to account for the logarithmic scale on the ordinate axis.

Figure 3

Left panel: Nonlinear corrections to the real space correlation function as function of radial separation calculated for redshift $z=0.23$. We show the linear (black dotted) and nonlinear matter correlation function (thick blue solid) as well as the $B$ (red dashed-dotted) and $A$ (green solid and dashed) correction terms. The dashed portion of the graph of $A(r)$ denotes the range where it is negative. Right panel: Perturbation theory fit (dashed) over scales $6h^{-1}\mathrm{Mpc}\le r\le 80h^{-1}\mathrm{Mpc}$ to the measured galaxy-correlation functions (solid) of the luminosity-threshold sample. The fits to ${\xi }_{\mathrm{gg}}$ (upper green), ${\xi }_{\mathrm{gm}}$ (central blue) and a joint fit provide consistent results. We are not expecting a good agreement on scales below $r\approx 3h^{-1}\mathrm{Mpc}$, where the correlation function is dominated by nonlinear clustering. For reference, we also plot the matter correlation function ${\xi }_{\mathrm{mm}}$ as the bottom solid red line.

Figure 4

Left panel: Perturbation terms for the projected surface mass density $w$. We show the nonlinear matter correlation function (black dotted) as well as the $w_{\mathrm{B}}$ (red dashed-dotted) and $w_{\mathrm{A}}$ (green dashed and solid) terms. Note that the dashed part of the latter curve is negative. Right panel: Perturbation terms for the ADSD and $R_0=3h^{-1}\mathrm{Mpc}$. Again we show the nonlinear matter statistic (black dotted) together with the ${{\rm Y}}_{\mathrm{A}}$ (green dashed and solid) and ${{\rm Y}}_{\mathrm{B}}$ (red dashed-dotted) terms. Note that the scale at which the ${{\rm Y}}_{\mathrm{B}}$ term becomes comparable to the nonlinear correlation is shifted further out to $R\approx 9h^{-1}\mathrm{Mpc}$.

Figure 5

Mean galaxy number per halo as a function of halo mass for the luminosity-threshold sample and our best fit parameters from Table . We show the number of central galaxies (blue dashed-dotted), satellite galaxies (red dashed) and the total number of galaxies (black solid). Note that the satellite number exceeds unity only for haloes with $M>6\times {10}^{14}{h}^{-1}{M}_{\odot }$, corresponding to virial radii exceeding $r_{\mathrm{vir}}\gtrsim 2h^{-1}\mathrm{Mpc}$.

Figure 6

Cross-correlation coefficient of the ADSD ${\rm Y}$ for the luminosity-threshold (left panel) and luminosity-bin sample (right panel). The trivial case $R_0=0$ (black with error bars), corresponding to bare $\Delta \Sigma$, leads to a cross-correlation coefficient that is far from unity and furthermore strongly scale dependent. If we instead choose $R_0=3h^{-1}\mathrm{Mpc}$ (blue with error bars), inspired by the virial radii of the haloes under consideration, we restore a cross-correlation coefficient close to unity on the 4% level for all scales $R>R_0$. Furthermore, we can model the residual scale dependence reasonably well using the perturbation theory expression of Eq. (35) [solid blue line], whereas the bare $\Delta \Sigma$ deviates from the corresponding perturbation theory result (black dashed-dotted). Since the Taylor expansion is no longer justified for scales below $R=8h^{-1}\mathrm{Mpc}$, we also plot the full expression according to Eq. (43) [red dashed] for ${\rm Y}$.

Figure 7

Reconstructed matter ADSD for the luminosity-threshold (left panel) and luminosity bin sample (right panel). We plot the inferred value from the simulations including corrections for $r_{\mathrm{cc}}\ne 1$ as well as the nonlinear prediction derived from the measured matter-correlation of the simulations (red solid line) and the linear theory value (black dashed line). The nonlinear correlation is well reproduced by the reconstruction, whereas there are remarkable deviations from linear theory on small scales.

Figure 8

Left panel: Residual effect of redshift space distortions on the projected galaxy-galaxy autocorrelation function $w_{\mathrm{gg}}$. We show the simulation measurements for the bright sample as crosses with error bars for ${\chi }_{\max }=50h^{-1}\mathrm{Mpc}$ (upper blue) and ${\chi }_{\max }=100h^{-1}\mathrm{Mpc}$ (lower red) and the corresponding linear theory predictions. Right panel: Residual effect on the annular differential surface density ${{\rm Y}}_{\mathrm{gg}}(R;R_0=3h^{-1}\mathrm{Mpc})$ for the same integration lengths. As on the left panel, the upper blue curve and points correspond to ${\chi }_{\max }=50h^{-1}\mathrm{Mpc}$ whereas the lower red curve and points correspond to ${\chi }_{\max }=100h^{-1}\mathrm{Mpc}$.

Figure 9

Correction factors that must be applied to the galaxy clustering measurements in order to remove redshift space distortions and to construct a quantity equivalent to the lensing signal. The total correction (red solid) is a product of a window correction (blue dashed) and a redshift correction (green dashed-dotted). Left panel: Projected galaxy-galaxy autocorrelation function $w_{\mathrm{gg}}$. The size of the corrections is quite remarkable on the largest scales. The apparent effects on scales below $10h^{-1}\mathrm{Mpc}$ arise from the flattening of the linear power spectrum on these scales. Furthermore the correlation on these scales would be affected by the finger-of-god effects not included in our analysis. Right panel: Annular differential surface density for $R_0=3h^{-1}\mathrm{Mpc}$. The residual correction is much smaller, 3% on the typical scales probed by galaxy-galaxy lensing.

Figure 10

Columns extracted from the correlation matrix of the bright LRG sample Left panel: $\mathrm{Corr}(w)$ Central panel: $\mathrm{Corr}(\Delta \Sigma )$ Right panel: $\mathrm{Corr}({\rm Y})$, where the vertical dashed line shows the cutoff radius $R_0$. Comparing the correlation matrices for $w$ and $\Delta \Sigma$ it is clear that the width of the off-diagonal contributions is larger for the projected correlation function than for the excess surface mass density. The same remains true if one compares the correlation of the ADSD and the projected correlation function.

Figure 11

Cross-correlation coefficient of the ADSD ${\rm Y}$ for the full galaxy sample. The panels show simulation output as circles with error bars and theoretical predictions of Eq. (35) for $R_0=5h^{-1}\mathrm{Mpc}$ (red solid lines). Top left panel: Reduced spectral index $n_{\mathrm{s}}=0.95$ Top right panel: Increased normalization ${\sigma }_8=0.9$ Bottom left panel: Reduced matter density ${\Omega }_{\mathrm{m}}=0.2$, ${\Omega }_{\Lambda }=0.8$ Bottom right panel: Increased matter density ${\Omega }_{\mathrm{m}}=0.3$, ${\Omega }_{\Lambda }=0.7$ We see that the increased number of high mass haloes in the ${\Omega }_{\mathrm{m}}=0.3$ and ${\sigma }_8=0.9$ models leads to a higher number of satellite galaxies and thus partially compensates the drop of the cross-correlation coefficient for haloes on small scales.

Figure 12

Same as Fig. 11 but for the central LRGs. The central LRGs are a cleaner representation of a halo sample and thus better reproduce our theoretical expectations.

Figure 13

Same as Fig. 11, but for $R_0=3h^{-1}\mathrm{Mpc}$ and for LRG sample from which the clusters with mass exceeding $M=3\times {10}^{14}{h}^{-1}{M}_{\odot }$ and all satellites were removed. Because of the cluster subtraction the bias ratio changes to $\alpha =0.41$. In addition to the usual correction (red solid line), we also plot the phenomenological correction $r_{\mathrm{cc}}\approx 1-{\alpha }^2\xi (R/2)/4$ (blue dashed line) The cluster subtraction also allowed us to reduce the cutoff radius to $R_0=3h^{-1}\mathrm{Mpc}$.

Figure 14

Top panel: Reconstructed matter ADSD of the variant cosmologies for $R_0=5h^{-1}\mathrm{Mpc}$. The points with error bars show the simulation results for the four variant cosmologies as measured in the simulations, whereas the solid lines show the corresponding nonlinear matter correlation function. From top to bottom: ${\Omega }_{\mathrm{m}}=0.3$ (black stars), ${\sigma }_8=0.9$ (green squares), $n_{\mathrm{s}}=0.95$ (red circles) and ${\Omega }_{\mathrm{m}}=0.2$ (blue diamonds). The thick black line is the nonlinear matter correlation function of our fiducial model plotted here for reference. Bottom panel: Fractional difference of the reconstructed matter statistics with respect to the fiducial model. From top to bottom we show ${\Omega }_{\mathrm{m}}=0.3$ (solid black), ${\sigma }_8=0.9$ (green dotted), $n_{\mathrm{s}}=0.95$ (red dashed-dotted) and ${\Omega }_{\mathrm{m}}=0.2$ (blue dashed). For the ${\Omega }_{\mathrm{m}}=0.2$ and ${\Omega }_{\mathrm{m}}=0.3$ cosmologies we also include the effect of wrong a priori cosmology as thin lines with corresponding line style (for further discussion see text).