Conformal symmetry and nonlinear extensions of nonlocal gravity

We study two nonlinear extensions of the nonlocal $R{\square }^{-2}R$ gravity theory. We extend this theory in two different ways suggested by conformal symmetry, either replacing ${\square }^{-2}$ with $(-\square +R/6{)}^{-2}$, which is the operator that enters the action for a conformally-coupled scalar field, or replacing ${\square }^{-2}$ with the inverse of the Paneitz operator, which is a four-derivative operator that enters in the effective action induced by the conformal anomaly. We show that the former modification gives an interesting and viable cosmological model, with a dark energy equation of state today $w_{\mathrm{DE}}\simeq -1.01$, which very closely mimics $\mathrm{\Lambda }\mathrm{CDM}$ and evolves asymptotically into a de Sitter solution. The model based on the Paneitz operator seems instead excluded by the comparison with observations. We also review some issues about the causality of nonlocal theories, and we point out that these nonlocal models can be modified so to nicely interpolate between Starobinski inflation in the primordial universe and accelerated expansion in the recent epoch.

Figure 1

Top left panel: the dark energy density ${\rho }_{\mathrm{DE}}(x)/{\rho }_0$. Top right panel: the energy fractions of radiation (dot-dashed), matter (dashed) and dark energy (solid line). Bottom panel: the Hubble parameter $h(x)$.

Figure 2

The EOS parameter of dark energy, $w_{\mathrm{DE}}$, as a function of $x=\ln a$ (top left), of redshift $z$ (top right) and of scale factor $a$ (bottom).

Figure 3

Left: the dark energy density ${\rho }_{\mathrm{DE}}(x)/{\rho }_0$. Right: $h(x)$.

Figure 4

The EOS parameter of dark energy, $w_{\mathrm{DE}}$, as a function of $x=\ln a$ (top left), of redshift $z$ (top right) and of scale factor $a$ (bottom).