Cluster density profiles as a test of modified gravity

We present a new test of gravitational interactions at the $r\simeq (0.2–20)\mathrm{Mpc}$ scale, around the virial radius of dark matter halos measured through cluster-galaxy lensing of maxBCG clusters from the Sloan Digital Sky Survey (SDSS). We employ predictions from self-consistent simulations of $f(R)$ gravity to find an upper bound on the background field amplitude of $|f_{R0}|<3.5\times {10}^{-3}$ at the 1D-marginalized 95% confidence level. As a model-independent assessment of the constraining power of cluster profiles measured through weak gravitational lensing, we also constrain the amplitude $F_0$ of a phenomenological modification based on the profile enhancement induced by $f(R)$ gravity when not including effects from the increased cluster abundance in $f(R)$. In both scenarios, dark-matter-only simulations of the concordance model corresponding to $|f_{R0}|=0$ and $F_0=0$ are consistent with the lensing measurements, i.e., at the 68% and 95% confidence level, respectively.

Figure 1

The shape $g(r)$ of the relative enhancement of ${\xi }_{\mathrm{hm}}(r)$ in $f(R)$ gravity simulations, for $|f_{R0}|={10}^{-3}$ in the abundance- (left panel) and threshold-matched case with scatter $\sigma =0$ (middle panel) and $\sigma =0.6$ (right panel), respectively. The middle panel shows the best-fit Gaussian function, Eq. (20), to the simulation output.

Figure 2

Left: Simulation measurements (points) and model predictions (lines) for the peak enhancement of ${\xi }_{\mathrm{hm}}(r)$, i.e., $\alpha A(|f_{R0}|,{\sigma }_8^{\Lambda \mathrm{CDM}})$, as a function of $|f_{R0}|$. Note the approximately logarithmic dependence of $\alpha A$ on $|f_{R0}|$. Middle: The peak enhancement of ${\xi }_{\mathrm{hm}}(r)$ in the abundance-matched case as a function of $|f_{R0}|$ for different values of the power spectrum normalization ${\sigma }_8^{\Lambda \mathrm{CDM}}$. Right: Same for the threshold-matched case, including the dependence on the scatter $\sigma$. Effects from scatter are negligible in the abundance-matched case.

Figure 3

Effects on the halo density profile ${\xi }_{\mathrm{hm}}$ from varying the cosmological parameters with respect to the fiducial case. Upper left: Different parameter values for ${\sigma }_8^{\Lambda \mathrm{CDM}}$ (dashed), $n_{\mathrm{s}}$ (dot-dashed), and ${\Omega }_{\mathrm{m}}$ (dotted). Upper right: $|f_{R0}|={10}^{-3}$ for the abundance-matched case (dashed) and for the fiducial $\Lambda \mathrm{CDM}$ cosmology with different values of scatter $\sigma$ (dotted). Lower left: $|f_{R0}|={10}^{-3}$ for the threshold-matched case with $\sigma =0$ (dashed) and $\sigma =0.6$ (dotted) (with the fiducial case corrected for scatter). Lower right: $F_0=1$ for the phenomenological scenario (see Sec. 4b).

Figure 4

Effects on the excess surface mass density $\Delta \Sigma$ from varying the cosmological parameters with respect to the fiducial case. The lensing data has been rebinned for illustrative purposes (cf. Fig. 5) Top left: ${\sigma }_8^{\Lambda \mathrm{CDM}}$ (dashed), $n_{\mathrm{s}}$ (dot-dashed), ${\Omega }_{\mathrm{m}}$ (dotted). Top right: $|f_{R0}|={10}^{-3}$ for the abundance-matched case (dashed) and for the fiducial $\Lambda \mathrm{CDM}$ cosmology with different values of scatter $\sigma$ (dotted). Bottom left: $|f_{R0}|={10}^{-3}$ for the threshold-matched case with $\sigma =0$ (dashed) and $\sigma =0.6$ (dotted) (with corresponding values of $\sigma$ used in the fiducial $\Delta \Sigma$ for each case). Bottom right: $F_0=1$ for the phenomenological scenario.

Figure 5

Excess surface mass density, Eq. (26), as a function of the comoving transverse separation to the cluster center (BCG), $r_{\perp }$, measured in the maxBCG sample (points). The lines show the predictions from the fiducial and best-fit $\Lambda \mathrm{CDM}$ models as well as for the best-fit phenomenological scenario (see Sec. 2b). Note that the best-fit abundance-matched $f(R)$ model is indistinguishable from the best-fit concordance model (see Table ) and is therefore not shown separately.

Figure 6

2D-marginalized contour plots for the abundance-matched (top row) and the phenomenological enhancement case (bottom row), showing 68%, 95%, and 99% confidence levels. The dashed line corresponds to $\Lambda \mathrm{CDM}$ predictions from DMO simulations.

Figure 7

One- and two-tail 1D-marginalized likelihood. The dotted lines indicate the 68%, 95%, and 99% confidence levels, the dashed line corresponds to the $\Lambda \mathrm{CDM}$ prediction from DMO simulations. Left: $|f_{R0}|$ in the abundance-matched case. Right: $F_0$ in the phenomenological scenario with a Gaussian fit in $\ln r$ to the enhancement in the TM case.

Figure 8

Left: Best-fit prediction for ${\xi }_{\mathrm{hm}}$ with respect to the best-fit $\Lambda \mathrm{CDM}$ model prediction for the phenomenological scenario (dashed) and fiducial $\Lambda \mathrm{CDM}$ cosmology (dotted), respectively. Right: Best-fit prediction for the excess surface mass density $\Delta \Sigma$ in the phenomenological scenario (dashed) and the fiducial $\Lambda \mathrm{CDM}$ model (dotted) with respect to the best-fit concordance model. Note that the lensing data has been rebinned for illustrative purposes (cf. Fig. 5). The shaded areas indicate regions in the abundance-matched case bounded by the best-fit model from below, and $|f_{R0}|=3.5\times {10}^{-3}$ (corresponding to the 68% CL bound) with otherwise identical parameter values from above. The best-fit model for the abundance-matched case is essentially identical to its $\Lambda \mathrm{CDM}$ counterpart and is therefore not shown separately.

Figure 9

Current constraints on $f(R)$ gravity. On linear scales, the strongest bound on $|f_{R0}|$ is obtained through the comparison of predicted to observed cross correlations of the ISW with foreground galaxies. In the nonlinear regime, enhancements of the abundance of clusters and the cluster density profile due to the $f(R)$ modification are incompatible with observations unless $|f_{R0}|$ is smaller than ${10}^{-4}$ and ${10}^{-3}$, respectively. The currently strongest bounds on $|f_{R0}|$, however, are inferred from requiring the modification to be suppressed by the chameleon mechanism within the solar system and the dark matter halo as well as from strong gravitational lenses.

Figure 10

Left: Halo model prediction scaled by the overall factor $\alpha =0.52$ (Sec. 3c) in comparison with simulation measurements for the abundance-matched case and $|f_{R0}|={10}^{-3}$. Middle: Same as left panel, but for the threshold-matched case without scatter (halo model scaled by $\alpha =0.73$). Right: Same as middle panel, but with a scatter of $\sigma =0.6$ (halo model scaled by $\alpha =0.77$).