Multiparameter investigation of gravitational slip

A detailed analysis of gravitational slip, a new post-general relativity cosmological parameter characterizing the degree of departure of the laws of gravitation from general relativity on cosmological scales, is presented. This phenomenological approach assumes that cosmic acceleration is due to new gravitational effects; the amount of spacetime curvature produced per unit mass is changed in such a way that a universe containing only matter and radiation begins to accelerate as if under the influence of a cosmological constant. Changes in the law of gravitation are further manifest in the behavior of the inhomogeneous gravitational field, as reflected in the cosmic microwave background, weak lensing, and evolution of large-scale structure. The new parameter ${\varpi }_0$ is naively expected to be of order unity. However, a multiparameter analysis, allowing for variation of all of the standard cosmological parameters, finds that ${\varpi }_0={0.09}_{-0.59}^{+0.74}(2\sigma )$, where ${\varpi }_0=0$ corresponds to a cosmological constant plus cold dark matter universe under general relativity. Future probes of the cosmic microwave background (Planck) and large-scale structure (Euclid) may improve the limits by a factor of 4.

Figure 1

The potentials $\varphi$ and $\psi$ are shown versus the scale factor, for different values of ${\varpi }_0$. The blue, dark curves are $\varphi$, whereas the green, light curves are $\psi$. Note that they behave oppositely; when $\varphi$ becomes shallower, $\psi$ becomes deeper, and vice versa.

Figure 2

The matter density contrast is shown versus the scale factor, for different values of ${\varpi }_0$. The time evolution is obtained by integrating Eqs. (7, 8), with initial conditions $\varphi =-{10}^{-5}$, $\dot{\varphi }=0.0$ for $k=0.01{\mathrm{Mpc}}^{-1}$. Positive (negative) values of ${\varpi }_0$ enhance (slow) the growth of density perturbations.

Figure 3

The degree of deviation from the Poisson equation versus scale factor is shown for different values of ${\varpi }_0$. Because $\varpi$ is scale independent, so too is the ratio $-k^2\varphi /(4\pi G{a}^2\delta \rho )$. For positive (negative) ${\varpi }_0$, a given $\varphi$ corresponds to a larger (smaller) density contrast than in GR.

Figure 4

The CMB quadrupole moment is shown versus ${\varpi }_0$. As explained in the text, the quadratic dependence can be understood in terms of the influence of ${\varpi }_0$ on the ISW effect. [Reproduced from Ref. 22 with our new normalization Eq. (3).]

Figure 5

The conformal-time derivative of the gravitational potential $\varphi$ is shown versus the scale factor, for different values of ${\varpi }_0$. Initial conditions are the same as in Fig. 2. The potential well is decaying when $\frac{d\varphi }{d\tau }/|{\varphi }_i{H}_0|>0$ and is deepening when negative.

Figure 6

The 68% and 95% likelihood contours in the ${\varpi }_0-{\Omega }_m$ parameter space are shown. The broad, blue contours are based on CMB data alone. The smaller, red contours add weak lensing, type 1a supernovae and galaxy-CMB cross-correlation data.

Figure 7

The 68% and 95% likelihood contours in the ${\varpi }_0-{\sigma }_8$ parameter space are shown. Shading is the same as in Fig. 6. Note that the addition of large-scale structure data breaks the degeneracy in ${\varpi }_0-{\sigma }_8$.

Figure 8

The projected 68% and 95% likelihood contours in the ${\varpi }_0-{\Omega }_m$ parameter space are shown. The long and narrow, yellow contours are based on mock Planck data. The tiny, green contours add mock weak lensing data. The underlying model is assumed to be ${\varpi }_0=0$ with ${\Omega }_m=0.26$. The current constraints are shown for reference.

Figure 9

The projected 68% and 95% likelihood contours in the ${\varpi }_0-{\sigma }_8$ parameter space are shown. Shading is the same as in Fig. 8.