How to optimally parametrize deviations from general relativity in the evolution of cosmological perturbations

The next generation of weak lensing surveys will trace the growth of large scale perturbations through a sequence of epochs, offering an opportunity to test general relativity on cosmological scales. We review in detail the parametrization used in MGCAMB to describe the modified growth expected in alternative theories of gravity and generalized dark energy models. We highlight its advantages and examine several theoretical aspects. In particular, we show that the same set of equations can be consistently used on superhorizon and subhorizon linear scales. We also emphasize the sensitivity of data to scale-dependent features in the growth pattern, and propose using principal component analysis to converge on a practical set of parameters which is most likely to detect departures from general relativity. The connection with other parametrizations is also discussed.

Figure 1

The growth factor, $(\Delta (k,a)/a)/(\Delta (k,a_i)/a_i)$, for the wave numbers: $k=0.1$ (dotted line), 0.01 (long-dashed line), ${10}^{-3}$ (short-dashed line), and ${10}^{-4}$ (dot-dashed line) h/Mpc as a function of $a$ for $\mu (z)$ given by (22). The red solid line is the $\Lambda \mathrm{CDM}$ solution, identical for each $k$. Note the approximately scale-independent enhancement for subhorizon modes ($k=0.1$ and $0.01\mathrm{h}/\mathrm{Mpc}$) at $z<1$ due to a rescaling of Newton’s constant by $\mu$. The long wavelength modes ($k={10}^{-3}$ and $k={10}^{-4}\mathrm{h}/\mathrm{Mpc}$) experience a suppression as expected from (16) for ${\mu }^{\prime }>0$. However, because $\Delta$ is $\mathcal{O}(p^2)$ for small $k$, this suppression is concealed by cosmic variance as illustrated in Fig. 2.

Figure 2

The matter power spectrum at $z=0$ for $\Lambda \mathrm{CDM}$ (black solid curve) along with the associated cosmic variance (blue shaded region), and for the modified gravity (MG) example of Sec. 2d (red dashed curve). Note the potentially observable enhancement on subhorizon scales, while the suppression due to the superhorizon variation of $\mu$ is completely hidden in cosmic variance.

Figure 3

The evolution of $\Psi (k,a)/\Psi (k,a_i)$ for the wave numbers: $k=0.1$ (dotted line), 0.01 (long-dashed line), ${10}^{-3}$ (short-dashed line), and ${10}^{-4}$ (dot-dashed line) h/Mpc as a function of $a$ for $\mu (z)$ given by (22), (as in Fig. 1). The red solid line is the $\Lambda \mathrm{CDM}$ solution, which is scale-independent. Modifying $\mu$ affects only the modes that cross the horizon. The superhorizon gravitational potential, as expected, does not depend on the Poisson equation and, hence, the choice of $\mu$.

Figure 4

The three best constrained eigenmodes of $\mu$ for DES, adapted from Ref. 21.

Figure 5

The best measured eigenmodes of w from SNe (black solid line), CMB (red dash line), galaxy counts (green dash-dot line), weak lensing (blue dash-dot-dot line) and combined (magenta short dash line). This forecast result is adapted from Ref. 54.