Geodesic completeness and homogeneity condition for cosmic inflation

There are two disjointed problems in cosmology within general relativity (GR), which can be addressed simultaneously by studying the nature of geodesics around $t\to 0$, where $t$ is the physical time. One is related to the past geodesic completeness of the inflationary trajectory due to the presence of a cosmological singularity, and the other one is related to the homogeneity condition required to inflate a local space-time patch of the Universe. We will show that both the problems have a common origin, arising from how the causal structure of null and timelike geodesics are structured within GR. In particular, we will show how a nonlocal extension of GR can address both problems, while satisfying the null energy condition for the matter sources.

Figure 1

A conformal diagram depicting a bouncing cosmology in an isotropic and homogenous FRW metric. Shaded regions are antitrapped and unshaded regions are normal. A patch begins to inflate at later times, $t$, from O to Q with inflationary size $x_{\text{inf}}$, where the line OP borders the apparent inflationary horizon. The arrow, pictured, describes an ingoing null ray entering an antitrapped surface from a normal surface. All notations are as in [11].