Dynamics of nonlocal cosmology

Nonlocal quantum corrections to gravity have been recently proposed as a possible solution to the cosmological fine-tuning problems. We study the dynamics of a class of nonlocal actions defined by the function of the inverse d’Alembertian of the Ricci scalar. Power-law and exponential functions are considered in detail, but we also show a method to reconstruct a nonlocal correction that generates a given background expansion. We find that even the simplest terms can, while involving only Planck scale constants, drive the late time acceleration without changing early cosmology. This leads to a sudden future singularity which, however, may be avoided by regularizing the d’Alembertian. We also consider the Einstein frame versions of these models.

Figure 1

Evolution of the scalar fields and comparison with the analytic approximations. The left panel is the power-law model with $n=3$ (row 3 in Table ) and the right panel is the exponential model with $\alpha =-0.5$. The lower solid (blue) line is the numerical solution for $\varphi$ (divided by ten to fit into the figure), and the dashed (black) line is the approximation (41). The upper solid (cyan) line is the numerical solution for $\Psi$, and the dashed (red) line the approximation (51). We note that even after the Universe has begun to accelerate, the matter-dominated solutions for the fields remain reasonable approximations.

Figure 2

The cosmological evolution leading to (nonaccelerating) effects from nonlocal corrections. Left panel: The power-law model with $\alpha =6.0$ and $f_0={10}^{-6}$. The solid (magenta) line is ${\Omega }_m/(1+\Psi )$, and the dashed (red) line is the $w_{\mathrm{eff}}$. One notes that the effect of $f$ slowly decays and the Universe asymptotically returns to the standard Einstein-de Sitter stage. For curiosity, we plot also the evolution in the Einstein frame. The dotted (black) line is ${\tilde{\Omega }}_m$, and the dash-dotted (blue) line is the ${\tilde{w}}_{\mathrm{eff}}$; these are understood as functions of the logarithm of the Einstein frame scale factor $\tilde{a}$. In this frame ${\tilde{\Omega }}_m$ becomes negligible, but the expansion is asymptotically identical to the dust dominated universe. Right panel: The exponential model with $\alpha =-1.0$ and ${\widehat{f}}_0={10}^{-6}$. The solid (magenta) line is ${\Omega }_m/(1+\Psi )$, and the dashed (red) line is the $w_{\mathrm{eff}}$. This is a scaling solution where we may consistently have ${\Omega }_m/(1+\Psi )>1$ because there can be negative effective energy sources. The evolution is plotted also in the Einstein frame. The dotted (black) line is ${\tilde{\Omega }}_m$, and the dash-dotted (blue) line is the ${\tilde{w}}_{\mathrm{eff}}$, and these are understood as functions of the logarithm of the Einstein frame scale factor $\tilde{a}$. In both frames we find asymptotic expansion with a nonzero constant effective equation state for the exponential model.

Figure 3

The cosmological evolution leading to the acceleration and singularity in various cases. Left panel: The power-law model with $n=3$ and $n=6$ (rows 3 and 6 in Table ). The solid (magenta) line is ${\Omega }_m$, and the dashed (red) line is the $w_{\mathrm{eff}}$ for $n=3$. The dotted (black) line is ${\Omega }_m$, and the dash-dotted (blue) line is the $w_{\mathrm{eff}}$ for $n=6$. The onset of acceleration leads to a singularity in the effective equation of state. The evolution is steeper for larger $n$. With smaller $n$, the relative amount of matter ${\Omega }_m$ begins to decrease earlier and $w_{\mathrm{eff}}$ drops later to minus infinity. Right panel: The exponential model with $\alpha =-0.1$ and $\alpha =-0.5$. The solid (magenta) line is ${\Omega }_m$, and the dashed (red) line is the $w_{\mathrm{eff}}$ for $\alpha =0.1$. The dotted (black) line is ${\Omega }_m$, and the dash-dotted (blue) line is the $w_{\mathrm{eff}}$ for $\alpha =0.5$. The onset of acceleration leads to a singularity in the effective equation of state. The evolution gets steeper for more negative $\alpha$. To set ${\Omega }_m=0.3$ today fixes $f_0=-0.296$ for $\alpha =-0.1$ and $f_0=-0.140$ for $\alpha =-0.5$.