Dynamical analysis of $R\frac{1}{{\square }^2}R$ cosmology: Impact of initial conditions and constraints from supernovae

We discuss the cosmological implications of the $R{\square }^{-2}R$ nonlocal modification to standard gravity. We relax the assumption of special initial conditions in the local formulation of the theory, perform a full phase-space analysis of the system, and show that the late-time cosmology of the model exhibits two distinct evolution paths, on which a large range of values for the present equation of state can be reached. We then compare the general solutions to supernovae data and place constraints on the parameters of the model. In particular, we find that the mass parameter of the theory should be smaller than 1.2 in Hubble units.

Figure 1

The two evolution paths A and B for the background cosmology of $R{\square }^{-2}R$ gravity, in terms of the initial value $U_0$ of the auxiliary field $U$ and the value of $\gamma \equiv \frac{m^2}{9H_0^2}$. The diagonal line depicts the critical value $\overline{U}$ as a function of $\gamma$. The green (red) region corresponds to the realizations of path A (B).

Figure 2

(Left) Evolution of the effective equation of state $w_{\mathrm{eff}}$ as a function of $N\equiv \ln a$ for path A. (Right) Evolution of the density parameters ${\mathrm{\Omega }}_{\mathrm{M}},{\mathrm{\Omega }}_{\mathrm{R}}$, and ${\mathrm{\Omega }}_{\mathrm{DE}}$ for the same path with $U_0=0$. In both plots we have fixed $V_0=0$.

Figure 3

Evolution of the auxiliary fields $U$ and $V$, and their derivatives with respect to $N\equiv \ln a$, $U^{\prime }$ and $V^{\prime }$, for path A. In the plots of $U$ and $U^{\prime }$ we have fixed $V_0=0$, and in the plots of $V$ and $V^{\prime }$ we have fixed $U_0=0$.

Figure 4

(Left) Evolution of the effective equation of state $w_{\mathrm{eff}}$ as a function of $N\equiv \ln a$ for path B. (Right) Evolution of the density parameters ${\mathrm{\Omega }}_{\mathrm{M}}$, ${\mathrm{\Omega }}_{\mathrm{R}}$, and ${\mathrm{\Omega }}_{\mathrm{DE}}$ for the same path with $U_0=-60$. In both plots we have fixed $V_0=0$.

Figure 5

Evolution of the auxiliary fields $U$ and $V$, and their derivatives with respect to $N\equiv \ln a$, $U^{\prime }$ and $V^{\prime }$, for path B. In the plots of $U$ and $U^{\prime }$ we have fixed $V_0=0$, and in the plots of $V$ and $V^{\prime }$ we have fixed $U_0=-20$.

Figure 6

Value of $U_0$ needed to reach path A (blue curve) or path B (red dotted curve) when ${\mathrm{\Omega }}_{\mathrm{M}}^0=0.3$. For large $m$, the two curves converge to $\overline{U}$, represented by the intermediate dashed curve, which follows Eq. (28).

Figure 7

Supernovae likelihood contours at $2\text{−}\sigma$ level for path A (left panel) and path B (right panel). The associated values of $U_0$ are also displayed.

Figure 8

Supernovae likelihood contours at $2\text{−}\sigma$ level for path A (upper panels) and path B (lower panels). The associated values of $w_0$ and $w_1$ in the standard parametrization $w_{\mathrm{eff}}=w_0+(1-a)w_1$ are also displayed.