Nonlocal gravity with a Weyl-square term

Recent work has shown that modifications of general relativity based on the addition to the action of a nonlocal term $R{\square }^{-2}R$, or on the addition to the equations of motion of a term involving $(g_{\mu \nu }{\square }^{-1}R)$, produce dynamical models of dark energy which are cosmologically viable both at the background level and at the level of cosmological perturbations. We explore a more general class of models based on the addition to the action of terms proportional to $R_{\mu \nu }{\square }^{-2}{R}^{\mu \nu }$ and $C_{\mu \nu \rho \sigma }{\square }^{-2}{C}^{\mu \nu \rho \sigma }$, where $C_{\mu \nu \rho \sigma }$ is the Weyl tensor. We find that the term $R_{\mu \nu }{\square }^{-2}{R}^{\mu \nu }$ does not give a viable background evolution. The nonlocal Weyl-square term, in contrast, does not contribute to the background evolution but we find that, at the level of cosmological perturbations, it gives instabilities in the tensor sector. Thus, only nonlocal terms which depend just on the Ricci scalar $R$ appear to be cosmologically viable. We discuss how these results can provide a hint for the mechanism that might generate these effective nonlocal terms from a fundamental local theory.

Figure 1

Scalar perturbations for three different comoving reduced wavelengths $ƛ=1/k$. For each wavelength we plot the result for $\mathrm{\Lambda }\mathrm{CDM}$ (black solid line), the pure $R{\square }^{-2}R$ model with ${\mu }_1=0.014H_0^2$ and ${\mu }_2=0$ (blue, dashed), and the model with ${\mu }_1={\mu }_2=0.014H_0^2$ (red, dotted). For $ƛ=20\mathrm{Mpc}$ and for $ƛ=93\mathrm{Mpc}$ the blue and the red curves are basically indistinguishable on this scale, while for $ƛ=1\mathrm{Gpc}$ the blue and black lines are almost indistinguishable.

Figure 2

The evolution of tensor perturbations, for three different comoving reduced wavelengths. For each wavelength we plot the result for $\mathrm{\Lambda }$CDM (black solid line) and for the model with ${\mu }_1={\mu }_2=0.014H_0^2$ (red, dotted).

Figure 3

As in Fig. 2, on a logarithmic vertical scale, and extending the horizontal axis to $\log a=1$. Observe the different vertical scale among the figures.

Figure 4

The value of $h_0^2{\mathrm{\Omega }}_{\mathrm{gw}}(f)$ for a primordial inflationary spectrum further amplified during the evolution to the present time by the nonlocal Weyl-square term.

Figure 5

Numerical evolution of the potentials $\mathrm{\Phi }$ (blue, solid line) and $\mathrm{\Psi }$ (red, dashed line), with $k=10^{-3}{\mathcal{H}}_0$ and ${\gamma }_2={\gamma }_1=0.014H_0^2$. Left: The background evolution is driven by standard matter and radiation only. The analytical solution for $\mathrm{\Psi }$ in matter dominance is the green, dotted line. Right: The background evolution includes dark energy. When the latter becomes relevant at the background level, the solution follows the de Sitter growing mode.

Figure 6

Numerical evolution of the amplitude of the metric tensor mode $h_g$ for $k={10}^3{H}_0$ and ${\gamma }_2={\gamma }_1=0.014H_0$. The numerical solution (blue, solid line) is compared to the analytic one valid for subhorizon modes at late times (green, dotted line).