Higher powers in gravitation

We consider the Friedmann-Robertson-Walker cosmologies of theories of gravity that generalize the Einstein-Hilbert action by replacing the Ricci scalar $R$ with some function $f(R)$. The general asymptotic behavior of these cosmologies is found, at both early and late times, and the effects of adding higher and lower powers of $R$ to the Einstein-Hilbert action is investigated. The assumption that the highest powers of $R$ should dominate the Universe’s early history, and that the lowest powers should dominate its future is found to be inaccurate. The behavior of the general solution is complicated, and while it can be the case that single powers of $R$ dominate the dynamics at late times, it can be either the higher or lower powers that do so. It is also shown that it is often the lowest powers of $R$ that dominate at early times, when approach to a bounce or a Tolman solution are generic possibilities. Various examples are considered, and both vacuum and perfect fluid solutions are investigated.

Figure 1

The $(x,y)$ plane in the phase space of vacuum solutions, defined by Eqs. (14, 15, 16). The arrows indicate the trajectories of solutions, confined to the circle in each plot by the constraint equation (17). Stationary points are indicated by dots, squares, and triangles, and divide the phase space into the four subspaces i to iv, between which trajectories cannot pass. Dots correspond to the points 1 and 2, and do not move. Squares correspond to points 3 and 4, and also do not move. The triangles correspond to the critical points defined by (38), which can move. They approach the stationary points 5 and 6 in the limit $B\to B_0=\mathrm{\text{constant}}$.

Figure 2

The $(x,y)$ plane in the phase space of solutions defined by Eqs. (46, 47, 48, 49, 50). Here the infinite $(x,y)$ plane has been compacted to a unit disk. Dots, squares, triangles, and stars indicate the position of stationary points. Squares correspond to points 3, 4, 7, and 8, and, hence, the Tolman solution (13). Stars correspond to points 9 and 10, and the matter dominated solution (12). Triangles are at points 5 and 6, corresponding to the vacuum dominated solution (10). Dots are at points 1 and 2, where $a^{\prime }/a=0$. The phase plane is separated into three different regions by the invariant submanifolds, where $x^2+xy-S=0$. Gray regions have $R<0$, and white regions have $R>0$. In these plots, we take $B=2/3$ and $\gamma =1$, corresponding to pressureless dust and a gravitational Lagrangian dominated by a term $\sim R^3$.

Figure 3

These plots show the $(x,y)$ plane in the phase space of solutions when $B=2$ and $\gamma =1$, corresponding to pressureless dust and a gravitational Lagrangian dominated by a term $\sim R^{-1}$. See the caption of Fig. 2 for further details.

Figure 4

Plots showing the $(x,y)$ plane in the phase space of solutions when $B=1/10$ and $\gamma =1$, corresponding to pressureless dust and a gravitational Lagrangian dominated by a term $\sim R^{10/9}$. See the caption of Fig. 2 for further details.

Figure 5

Plots showing the $(x,y)$ plane in the phase space of solutions when $B=-1/2$ and $\gamma =1$, corresponding to pressureless dust and a gravitational Lagrangian dominated by a term $\sim R^{2/3}$. See the caption of Fig. 2 for further details.

Figure 6

Plots showing the $(x,y)$ plane in the phase space of solutions when $B=-1/2$, $S=1$, and $\gamma =2$. This situation corresponds to a stiff fluid and a gravitational Lagrangian dominated by a term $\sim R^{2/3}$. The left-hand white region in this plot has only a single repellor and no attractor. Trajectories in this region can then be seen to originate from the repellor, but have no stable critical point to attract them in the future. They therefore spiral outwards forever, away from the repellor. The right-hand white region can similarly be seen to have an attractor but no repellor. See the caption of Fig. 2 for further details.

Figure 7

The functions $A(z)$ and $B(z)$ for the theory $f(R)=R+{10}^3{R}^2+0.1R^3$. The dashed line corresponds to $A$, and the solid line to $B$.

Figure 8

The functions $A(z)$ and $B(z)$ for the theory $f(R)=R-{10}^{-5}/R+{10}^{-5}{R}^3$. The dashed line corresponds to $A$, and the solid line to $B$.

Figure 9

The functions $A(z)$ and $B(z)$ for the theory $f(R)=(R+100R^3)e^{{10}^{-5}R}$. The dashed line corresponds to $A$, and the solid line to $B$.