Testing Gravity with the Stacked Phase Space around Galaxy Clusters

In general relativity, the average velocity field of dark matter around galaxy clusters is uniquely determined by the mass profile. The latter can be measured through weak lensing. We propose a new method of measuring the velocity field (phase space density) by stacking redshifts of surrounding galaxies from a spectroscopic sample. In combination with lensing, this yields a direct test of gravity on scales of 1–30 Mpc. Using $N$-body simulations, we show that this method can improve upon current constraints on $f(R)$ and Dvali-Gabadadze-Porrati model parameters by several orders of magnitude when applied to upcoming imaging and redshift surveys.

Figure 1

Lower panel: the $v_{\mathrm{los}}-r_p$ phase space distribution (in logarithmic scale) as measured using halo catalogs constructed from $N$-body simulations in $\Lambda \mathrm{CDM}$ (see also 7); we considered primary halos (“clusters”) with masses $\ge {10}^{14}{M}_{\odot }/h$ and secondary halos (“galaxies”) in the range $3\times {10}^{13}\le M\le {10}^{14}{M}_{\odot }/h$. Upper panel: the dispersion of the line-of-sight velocity distribution ${\sigma }_{v_{\mathrm{los}}}$ as function of $r_p$. The data points with error bars are computed from the simulation results in the lower panel, while the solid curve is our analytical model prediction. The error bars are scaled to mimic the measurement accuracies for a spectroscopic survey of 2000 sq degrees over $0.2<z<0.4$.

Figure 2

Upper panel: ratio of the velocity dispersion ${\sigma }_v$ along the line of sight measured around halos with $M_{300}>{10}^{14}{M}_{\odot }/h$ in $f(R)$ simulations to that measured around halos of the same mass in $\Lambda \mathrm{CDM}$ simulations. The error bars are estimated from the simulations, as in Fig. 1, for a spectroscopic survey of 2000 sq degrees. Lower panel: ratio of the enclosed projected mass profiles of the same halos in $f(R)$ and $\Lambda \mathrm{CDM}$ simulations. This is approximately what stacked lensing would measure. The shaded region indicates the range of statistical uncertainties for an imaging survey of the same area (see text).

Figure 3

Same as Fig. 2, but for DGP models (see text).