Constraints on modified gravity from Sunyaev-Zeldovich cluster surveys

We investigate the constraining power of current and future Sunyaev-Zeldovich cluster surveys on the $f(R)$ gravity model. We use a Fisher matrix approach, adopt self-calibration for the mass-observable scaling relation, and evaluate constraints for the South Pole Telescope (SPT), Planck, SPT polarimeter (SPTpol), and Atacama Cosmology Telescope polarimeter (ACTpol) surveys. The modified gravity effects on the mass function, halo bias, matter power spectrum, and mass-observable relation are taken into account. We show that, relying on number counts only, the Planck cluster catalog is expected to reduce current upper limits by about a factor of 4, to ${\sigma }_{f_{R0}}=2\times {10}^{-5}$ (68% confidence level) while SPT, SPTpol, and ACTpol yield about $3\times {10}^{-5}$. Adding the cluster power spectrum further improves the constraints to ${\sigma }_{f_{R0}}=5\times {10}^{-6}$ for Planck and ${\sigma }_{f_{R0}}=2\times {10}^{-5}$ for SPTpol, pushing cluster constraints significantly beyond the limit where number counts have no constraining power due to the chameleon screening mechanism. Further, the combination of both observables breaks degeneracies, especially with the expansion history (effective dark energy density and equation of state). The constraints are only mildly worsened by the use of self-calibration but depend on the mass threshold and redshift coverage of the cluster samples.

Figure 1

Mass limit of cluster surveys in $\Lambda \mathrm{CDM}$ (solid) and $f(R)$ gravity (dashed) with $f_{R0}={10}^{-4}$ (left) and $f_{R0}=3\times {10}^{-5}$ (right). The mass limits in $f(R)$ are reduced due to the effect on dynamical mass measurements (Sec. 3a).

Figure 2

Upper left. The redshift distribution of clusters in the Planck, ACTpol, SPT, and SPTpol survey in the fiducial $\Lambda \mathrm{CDM}$ cosmology. Upper right, bottom left, bottom right. The fractional deviation of the number density between $f(R)$ and $\Lambda \mathrm{CDM}$ models due to all effects of $f(R)$ (dotted lines), $\Delta n_{\ln }$ effect only (dashed lines), and dynamical mass effect only (dash-dotted lines) evaluated at $f_{R0}={10}^{-4}$, $f_{R0}=5\times {10}^{-5}$, and $f_{R0}=2\times {10}^{-5}$, respectively. For the largest field value, the effect on the mass function dominates the enhancement of the cluster abundance at low $z$, while the dynamical mass effect dominates at $z\gtrsim 0.3$. The Planck and SPTpol surveys have the lowest mass threshold at $z<0.2$ and $z>0.2$, respectively, and hence are most sensitive to the $f(R)$ effects for small-field values.

Figure 3

Relative deviations in the $f(R)$ halo power spectrum from $\Lambda \mathrm{CDM}$, i.e., $\Delta P_h/P_h$ for the Planck survey, with $|f_{R0}|={10}^{-5}$. Upper left. Total deviation. Upper right. Deviation due to $P_L(k)$ only. Bottom. Deviation due to halo bias $b_L$ only. For this value of $f_{R0}$, the dynamical mass effect on the power spectrum is negligible and therefore we do not show it here. The redshift and scale dependence in the relative deviations from other cluster surveys are similar to the ones shown here.

Figure 4

The dependence on redshift (top) and wave number (bottom) of the effective volume [Eq. (26)] for a single mass bin and each survey: Planck (black), SPT (green), SPTpol (magenta), and ACTpol (blue). The effective volume is a weak function of wave number $k$ but strongly depends on the redshift.

Figure 5

Fully marginalized 68% confidence level (CL) constraints on $f_{R0}$ from the number count of clusters only (using Planck CMB priors), as a function of the step size $\Delta f_{R0}$, for the surveys considered in this paper. The dotted (red) line indicates ${\sigma }_{f_{R0}}=\Delta f_{R0}$. For a given survey, the intersection of this line with the predicted constraints yields the final expected constraint (Sec. 4f). Solid (dashed) lines represent the case when dynamical mass is (is not) considered. The sharp upturn at $\Delta f_{R0}\lesssim 5\times {10}^{-5}$ is due to the chameleon mechanism.

Figure 6

Same as Fig. 5, but from the power spectrum of clusters only (using Planck CMB priors).

Figure 7

Fully marginalized constraints on $f_{R0}$ from the power spectrum of clusters only, as a function of maximum cluster redshift $z_{\max }$. $\Delta f_{R0}=7\times {10}^{-6}$ was used for all values shown here.

Figure 8

Combined $dN/dz+P(k)$ 68% CL marginalized constraints on $f_{R0}$ as a function of the step size $\Delta f_{R0}$ for the different surveys. As in Fig. 5, the dotted (red) line shows the identity ${\sigma }_{f_{R0}}=\Delta f_{R0}$.

Figure 9

Joint constraints on the Compton wavelength ${\lambda }_{C0}$ [in Mpc; see Eq. (28)] and (counterclockwise from top left) ${\Omega }_{\Lambda }$, ${\sigma }_8$, $w_o$, and $w_a$. All curves denote 68% confidence level, and are for number counts only (blue), power spectrum only (cyan), and a combination of the two (green). The results are shown for the Planck survey with $\Delta f_{R0}=5\times {10}^{-6}$.