Difficulties distinguishing dark energy from modified gravity via redshift distortions

The bulk motion of galaxies induced by the growth of cosmic structure offers a rare opportunity to test the validity of general relativity across cosmological scales. However, modified gravity can be degenerate in its effect with the unknown values of cosmological parameters. More seriously, even the “observed” value of the redshift-space distortions used to measure the fluctuation growth rate depends on the assumed cosmological parameters (the Alcock-Paczynski effect). We give a full analysis of these issues, showing how to combine redshift-space distortions with baryon acoustic oscillations and CMB data, in order to obtain joint constraints on deviations from general relativity and on the equation of state of dark energy while allowing for factors such as nonzero curvature. In particular we note that the evolution of ${\Omega }_m(z)$, along with the Alcock-Paczynski effect, produces a degeneracy between the equation of state $w$ and the modified growth parameter $\gamma$. Typically, the total marginalized error on either of these parameters will be larger by a factor $\simeq 2$ compared to the conditional error where one or the other is held fixed. We argue that future missions should be judged by their figure of merit as defined in the $w_p-\gamma$ plane, and note that the inclusion of spatial curvature can degrade this value by an order of magnitude.

Figure 1

Left panel: Joint constraints on modified gravity and dark energy from a combination of the cosmic microwave background and large-scale structure. The solid contours represent the 1- and $2\mathrm{\text{−}}\sigma$ constraints for a $10h^{-3}{\mathrm{Gpc}}^3$ redshift survey at $z=1$, with $\overline{n}={10}^{-4}{h}^3{\mathrm{Mpc}}^{-3}$, and combined with the DETF Planck Fisher matrix. If we had chosen to neglect the Alcock-Paczynski effect from the redshift distortions, but leaving the BAO information intact as discussed in the text, we would arrive at the dashed contours. In both cases, a weak prior is applied to ${\sigma }_p$; on the basis it may be well measured on scales much smaller than those considered here. Right panel: Varying the redshift bin of the survey from $z=0.5$, 1, and 2, shown as dashed, solid, and dotted lines, respectively.

Figure 2

The same survey specifications as in Fig. 1, but now assuming a flat universe. The dashed contours illustrate the bias induced by an actual value of ${\Omega }_k=-0.01$.

Figure 3

The substantial error in $\gamma$ that can arise from a relatively small uncertainty in $\beta$. The data set is the same as Fig. 1, and the dashed line is for a fixed $w=-1$.

Figure 4

The dash-dotted lines highlight the degeneracy directions for redshift distortions at $z=0.5$, 1, and 2, as predicted by (21). The solid contours correspond to the same data set as in Fig. 1, and we have fixed $w_a=0$ to provide a consistent comparison.