Consistent cosmological modifications to the Einstein equations

General relativity is a phenomenologically successful theory that rests on firm foundations, but has not been tested on cosmological scales. The deep mystery of dark energy (and possibly even the requirement of cold dark matter), has increased the need for testing modifications to general relativity, as the inference of such otherwise undetected fluids, depends crucially on the theory of gravity. In this work I outline a general scheme for constructing consistent and covariant modifications to the Einstein equations. This framework is such that there is a clear connection between the modification and the underlying field content that produces it. I argue that this is mandatory for distinguishing modifications of gravity from conventional fluids. I give two nontrivial examples, the first of which is a simple metric-based modification of the fluctuation equations for which the background is exact $\Lambda \mathrm{CDM}$ and the second has a Dvali-Gabadadze-Porrati background but differs from it in the perturbations. I present their impact on observations of the cosmic microwave background radiation.

Figure 1

The CMB spectrum for the simple modified gravity model in the text. The solid curve is the plain $\Lambda \mathrm{CMD}$ model ($\beta =0$), while the dotted, dashed, and dotted-dashed curves are with $\beta =\{0.1,0.5,1\}$, respectively.

Figure 2

The time evolution of $\widehat{\Phi }-\widehat{\Psi }$ at $k={10}^{-3}{\mathrm{Mpc}}^{-1}$ for $\beta =0$ (solid), $\beta =0.1$ (dotted), $\beta =0.5$ (dashed), and $\beta =1$ (dotted-dashed).

Figure 3

The phase portrait of $\Phi (t,k)$ and $\Psi (t,k)$ at $k=\{0.001,0.005,0.01,0.05\}{\mathrm{Mpc}}^{-1}$ for $\beta =0$ (solid), $\beta =0.1$ (dotted), $\beta =0.25$(dashed), $\beta =0.5$ (long-dashed), and $\beta =1$ (dotted-dashed). On small scales the phase portrait is squeezed close to a line as it would be for the case of plain $\Lambda \mathrm{CDM}$.

Figure 4

The phase portrait of parameters $Q(t,k)$ and ${\eta }_A(t,k)$ at $k=\{0.001,0.005,0.01,0.05\}{\mathrm{Mpc}}^{-1}$ for $\beta =0$ (solid), $\beta =0.1$ (dotted), $\beta =0.25$ (dashed), $\beta =0.5$ (long-dashed), and $\beta =1$ (dotted-dashed). Notice that this model can cover a wide range of the $\{Q,{\eta }_A\}$ plane rather being effectively one dimensional, as, for example, in the case of DGP [17] or clustering dark energy [44] models [48].

Figure 5

The CMB spectrum for the minimal DGP-like model. The solid curve is the plain $\Lambda \mathrm{CDM}$ model with ${\Omega }_{\Lambda }=0.721$, while the dashed curve is the DGP-like model with ${\Omega }_{\Lambda }=0$ and $r_c=8681\mathrm{Mpc}$. Both models have the same distance to recombination, however, the DGP-like model lowers the power on large scales. This is in contrast to the proper DGP model [51]. The data points are from the 5 yr data release of WMAP [50].