Joint analysis of galaxy-galaxy lensing and galaxy clustering: Methodology and forecasts for Dark Energy Survey

The joint analysis of galaxy-galaxy lensing and galaxy clustering is a promising method for inferring the growth function of large-scale structure. Anticipating a near future application of this analysis to Dark Energy Survey (DES) measurements of galaxy positions and shapes, we develop a practical approach to modeling the assumptions and systematic effects affecting the joint analysis of small-scale galaxy-galaxy lensing and large-scale galaxy clustering. Introducing parameters that characterize the halo occupation distribution (HOD), photometric redshift uncertainties, and shear measurement errors, we study how external priors on different subsets of these parameters affect our growth constraints. Degeneracies within the HOD model, as well as between the HOD and the growth function, are identified as the dominant source of complication, with other systematic effects being subdominant. The impact of HOD parameters and their degeneracies necessitate the detailed joint modeling of the galaxy sample that we employ. We conclude that DES data will provide powerful constraints on the evolution of structure growth in the Universe, conservatively/optimistically constraining the growth function to 7.9%/4.8% with its first-year data that cover over 1000 square degrees, and to 3.9%/2.3% with its full five-year data that will survey 5000 square degrees, including both statistical and systematic uncertainties.

Figure 1

An example of the average number of central/satellite galaxies, $⟨N_{\mathrm{c}/\mathrm{s}}(M_{\mathrm{h}})⟩$, calculated from Eq. (3) with parameter settings $\log M_{\min }=12.36$, $\log M_1=13.69$, ${\sigma }_{\log M}=0.32$, $\alpha =1.28$. These parameter values are selected to match our fiducial default values presented in Table 1. The dashed and dotted black lines respectively represent the central and satellite galaxy counts, with the solid black line showing their sum, i.e. the total number of galaxies in a halo of mass $M_{\mathrm{h}}$. The solid and dashed red lines respectively represent the satellite and total counts using $\log M_0=8.35$ in addition, i.e. Eq. (3) before our simplification. For the galaxy sample under consideration, the effect of the satellite cutoff mass scale $M_0$ is negligible.

Figure 2

The lens redshift distributions (top) and source redshift distributions (bottom) are presented. For the lenses, we show the measured redshift distribution (dashed) and the deduced true redshift distribution (solid) for the two lens bins of our mock catalog. For the sources, we show the Y1 and Y5 redshift distributions, with color-filled regions indicating the three source bins used.

Figure 3

Left: Correlation matrix of galaxy clustering and galaxy-galaxy lensing with a single source bin for illustrative purposes. A large range of scales ($1^{\prime }–100^{\prime }$) is shown with a larger lens sample to reduce shot/shape noise and highlight the correlations of density modes. The black box indicates the range of scales considered in this analysis. Right: Correlation matrix of galaxy clustering and tomographic galaxy-galaxy lensing for the DES Y5 $0.3<z<0.4$ lens sample and range of scales considered in this analysis (cf. Sec. 3c). The panels marked as $z_j$ correspond to the tangential shear measurement vector at the jth source redshift bin.

Figure 4

Forecast constraints on the HOD and growth scaling parameters from the conservative (blue) and optimistic (red) Y1 analyses. Diagonal blocks represent marginalized one-dimensional parameter constraints, and off-diagonal blocks represent two-dimensional $1\sigma$ confidence ellipses for corresponding parameters.

Figure 5

Forecast constraints on the HOD and growth scaling parameters from the conservative (blue) and optimistic (red) Y5 analyses. Diagonal blocks represent marginalized one-dimensional parameter constraints, and off-diagonal blocks represent two-dimensional $1\sigma$ confidence ellipses for corresponding parameters.

Figure 6

Forecast constraints on the HOD and growth scaling parameters from the default conservative (red) and relaxed HOD priors (blue) Y1 analyses. The relaxed HOD priors are a factor of 2.5 weaker than the default priors.

Figure 7

Fractional $1\sigma$ uncertainty on $A_i$ for different Y1 analyses, plotted against HOD prior widths as a factor of the default conservative widths.

Figure 8

Forecast constraints on the systematic effects and growth scaling parameters from the conservative (blue) and optimistic (red) Y1 analyses. Diagonal blocks represent marginalized one-dimensional parameter constraints, and off-diagonal blocks represent two-dimensional $1\sigma$ confidence ellipses for corresponding parameters.

Figure 9

Forecast constraints on the systematic effects and growth scaling parameters from the conservative (blue) and optimistic (red) Y5 analyses. Diagonal blocks represent marginalized one-dimensional parameter constraints, and off-diagonal blocks represent two-dimensional $1\sigma$ confidence ellipses for corresponding parameters.

Figure 10

Forecast constraints on the systematic effects and growth scaling parameters from the default conservative (red) and relaxed systematic effects priors (blue) Y1 analyses. The relaxed systematic effects priors are a factor of 2.5 weaker than the default priors.

Figure 11

Fractional $1\sigma$ uncertainty on $A_i$ for different simulated Y5 analyses, plotted against systematic effects prior widths as a factor of the default widths.

Figure 12

Forecasts for the DES $1\sigma$ bounds on the growth function $D(z)$ in two different redshift bins, presented for different assumptions on data stage and parameter priors. Points are offset for visibility.

Figure 13

Plot of 10,000 random HOD configurations with mean halo masses within 1% of the fiducial default value for lens bin 1, presented with the galaxy biases derived from these configurations. The plot ranges are set to match the range of parameter values used to generate random HOD configurations. Light blue squares and lines mark the fiducial default HOD for lens bin 1, as presented in Table 1.

Figure 14

Marginalized one-dimensional parameter constraints with dotted vertical lines at $\pm 1\sigma$ (diagonal) and two-dimensional $1\sigma$ confidence ellipses (off diagonal), representing the conservative (blue) and optimistic (red) Y1 parameter constraint forecasts for the first lens bin. Vertical and horizontal axes represent the four HOD parameters, five systematic effects parameters, and the growth scaling parameter, respectively. Light blue lines and squares correspond to the true values used in generating the simulated measurement vector.

Figure 15

Marginalized one-dimensional parameter constraints with dotted vertical lines at $\pm 1\sigma$ (diagonal) and two-dimensional $1\sigma$ confidence ellipses (off-diagonal), representing the conservative (blue) and optimistic (red) Y5 parameter constraint forecasts for the first lens bin. Vertical and horizontal axes represent the four HOD parameters, five systematic effects parameters, and the growth scaling parameter, respectively. Light blue lines and squares correspond to the true values used in generating the simulated measurement vector.