Consistent modified gravity analysis of anisotropic galaxy clustering using BOSS DR11

We analyze the clustering of a cosmic large scale structure using a consistent modified gravity perturbation theory, accounting for anisotropic effects along and transverse to the line of sight. The growth factor has a particular scale dependence in $f(R)$ gravity and we fit for the shape parameter $f_{R0}$ simultaneously with the distance and the large scale (general relativity) limit of the growth function. Using more than 690,000 galaxies in the baryon oscillation spectroscopy survey data release 11, we find no evidence for extra scale dependence, with the 95% confidence upper limit $|f_{R0}|<8\times 10^{-4}$. Future clustering data, such as from the dark energy spectroscopic instrument, can use this consistent methodology to impose tighter constraints.

Figure 1

(Left panel) The linear power spectra $P_{\delta \delta }(k)$, $P_{\delta \mathrm{\Theta }}(k)$ and $P_{\mathrm{\Theta }\mathrm{\Theta }}(k)$ from top to bottom. The power spectra of the $\mathrm{\Lambda }\mathrm{CDM}$ model are presented as black solid curves, and the power spectra of $f(R)$ gravity models with $|f_{R0}|=3.2\times 10^{-5}$ and $|f_{R0}|=3.0\times 10^{-4}$ are presented as blue dashed and dotted curves, respectively. The results are evaluated at redshift $z=0.57$. (Right panel) The growth functions $D_{+}^{\delta }$ and $D_{+}^{\mathrm{\Theta }}$ have a scale independent amplitude, depending on ${\mathrm{\Omega }}_{\mathrm{\Lambda }}$ (taking fixed primordial amplitude), and scale dependent shape, depending on $|f_{R0}|$. The results in each subpanel are shown at $z=0.57$ for $|f_{R0}|=3.2\times 10^{-5}$, and ${\mathrm{\Omega }}_{\mathrm{\Lambda }}=0.68$ (black solid curves) and 0.71 (blue dashed curves).

Figure 2

(Left panel) The correlation function $\xi (\sigma ,\pi )$ of the $\mathrm{\Lambda }\mathrm{CDM}$ model is shown as black solid contours. Blue dashed and dotted contours represent $\xi (\sigma ,\pi )$ of the $\mathrm{\Lambda }\mathrm{CDM}$ model in which $D_{+}^{\delta }$ is replaced by that of $f(R)$ gravity models with $|f_{R0}|=3.2\times 10^{-5}$ and $|f_{R0}|=3.0\times 10^{-4}$, respectively. The thick solid circle represents the BAO ring. The levels of contours are given by $(-0.005,-0.0025,-0.004,0.003,0.005,0.016,0.05)$ from outer to inner contours. (Right panel) Black solid contours represent $\xi (\sigma ,\pi )$ of the $\mathrm{\Lambda }\mathrm{CDM}$ model, and blue dashed and dotted contours represent $\xi (\sigma ,\pi )$ of the $\mathrm{\Lambda }\mathrm{CDM}$ model in which $D_{+}^{\mathrm{\Theta }}$ is replaced by that of $f(R)$ gravity models of $|f_{R0}|=3.2\times 10^{-5}$ and $|f_{R0}|=3.0\times 10^{-4}$, respectively.

Figure 3

The best fit correlation function $\xi (\sigma ,\pi )$ of $\mathrm{\Lambda }\mathrm{CDM}$ (black solid unfilled contours) and the correlation function of $f(R)$ gravity models with $|f_{R0}|=3.2\times 10^{-5}$ (black dashed unfilled contours) and $3.0\times 10^{-4}$ (dotted unfilled contours). The blue filled contours represent the measured $\xi (\sigma ,\pi )$ from the DR11 CMASS data. The levels of contours are given by $(-0.001,0.002,0.005,0.016,0.05)$ from the outer to inner contours.

Figure 4

The measured constraints on $f_{R0}$, and their robustness to various tests, are presented. The measured likelihood function appears in the top panels and the measured difference of ${\chi }^2$ is in the bottom panels. (Left panel) Results marginalizing over the scale independent growth rate $G_{\mathrm{\Theta }}$ are shown by the black solid curve, while the constraints fixing $G_{\mathrm{\Theta }}=0.46$, given by the Planck concordance $\mathrm{\Lambda }\mathrm{CDM}$ model, are blue dashed curves. The results for $f_{R0}$ do not depend appreciably on the scale independent behavior. (Right panel) The results also do not depend significantly on whether the initial power spectrum $P(k)$ used matches the Planck (black solid) or WMAP9 (black dashed) model. The blue dotted curve represents the results from analyzing galaxy clustering from $\mathrm{\Lambda }\mathrm{CDM}$ mock catalogues, verifying that $|f_{R0}|\to 0$ is recovered in this case.

Figure 5

The 2D joint likelihood contours at 68% and 95% CL measured for $D_A$, $H^{-1}$, $G_b$ and $G_{\mathrm{\Theta }}$ are shown, using $s_{\text{cut}}=50h{\mathrm{Mpc}}^{-1}$ and ${\sigma }_{\text{cut}}=40h{\mathrm{Mpc}}^{-1}$. The x denotes the Planck best-fit $\mathrm{\Lambda }\mathrm{CDM}$ predictions (with $G_b$ taking the value from fixing $f_{R0}=0$).

Figure 6

The 2D joint likelihood contours at 68% and 95% CL measured for $\log |f_{R0}|$ and $G_{\mathrm{\Theta }}$ are shown in the left panel, and for $\log |f_{R0}|$ and ${\sigma }_p$ are shown in the right panel, using $s_{\text{cut}}=50h^{-1}\mathrm{Mpc}$ and ${\sigma }_{\text{cut}}=40h^{-1}\mathrm{Mpc}$.