Cosmology with Eddington-inspired gravity

We study the dynamics of homogeneous, isotropic universes which are governed by the Eddington-inspired alternative theory of gravity which has a single extra parameter, $\kappa$. Previous results showing singularity-avoiding behavior for $\kappa >0$ are found to be upheld in the case of domination by a perfect fluid with equation of state parameter $w>0$. The range $-\frac{1}{3}<w<0$ is found to lead to universes which experience unbounded expansion rates while still at a finite density. In the case $\kappa <0$ the addition of spatial curvature is shown to lead to the possibility of oscillation between two finite densities. Domination by a scalar field with an exponential potential is found to also lead to singularity-avoiding behavior when $\kappa >0$. Certain values of the parameters governing the potential lead to behavior in which the expansion rate of the universe changes sign several times before transitioning to regular general relativity-like behavior.

Figure 1

Left: Expansion rate of the universe against density [normalized by the maximum density ${\rho }_B=(\kappa w{)}^{-1}$] for $\kappa =1$, $w=1/3$, $k=0$. Right top: Scale factor (normalized by the minimum scale factor $a_B$) against time (in reduced Planck units) for the full numerical solution to the Friedman equation, and approximations in two phases: loitering, as in Sec. 2b1, and GR-like low-density limit. Right bottom: Behavior of the parameters describing the auxiliary metric (for the same universe).

Figure 2

Left: Expansion rate of the universe against density (normalized by the critical density ${\rho }_{*}$) for $\kappa =1$, $w=-1/6$, $k=0$. Right top: Scale factor (normalized by the critical scale factor $a_{*}$) against time (in reduced Planck units) for the full numerical solution to the Friedman equation, and an approximation close to the critical density. Right bottom: Behavior of the parameters describing the auxiliary metric (for the same universe).

Figure 3

Left: Expansion rate of the universe against density (normalized by the maximum density ${\rho }_B=-{\kappa }^{-1}$) for $\kappa =-1$, $w=1/3$, $k/a_B^2=1/10$. Right top: Scale factor (normalized by the minimum scale factor $a_B$) against time (in reduced Planck units) for the full numerical solution to the Friedman equation, and approximations in two phases: close to the maximum density, and GR-like low-density limit. Right bottom: Behavior of the parameters describing the auxiliary metric (for the same universe).

Figure 4

Top: Amplitude (minimum density, normalized by maximum density) of the oscillations described in Sec. 2d against spatial curvature of the universe for $\kappa =-1$ and various values for the equation of state parameter. Full numerical solutions, as well as approximations in the GR regime from Eq. (41). Bottom: Period of the oscillations (in reduced Planck units) for the same universes.

Figure 5

Scale factor (top left), scalar field (bottom left), energy density (top right) and equation of state parameter ($w\equiv P/\rho$) (bottom right) as a function of time (in reduced Planck units) for various values of the free parameters.

Figure 6

Energy density as a function of scale factor for various values of the free parameters.

Figure 7

Hubble expansion parameter (top), scale factor (middle) and energy density (bottom) as a function of time (in reduced Planck units) for various values of the free parameters, showing universes which expand-contract-expand.

Figure 8

Hatched regions are those forbidden due to the square root in $G$; solid lines indicate the solutions to $F=0$ and so represent the locations of discontinuities in $H^2$; dashed lines indicate the asymptotes to which the solid lines tend.