String excitation by initial singularity of inflation

Abstract

We discuss excitation of string oscillation modes by an initial singularity of inflation. The initial singularity of inflation is known to occur with a finite Hubble parameter, which is generally lower than the string scale, and hence it is not clear that stringy effects become significant around it. With the help of Penrose limit, we find that infinitely heavy oscillation modes get excited when a singularity is strong in the sense of Krolak’s classification. We demonstrate that the initial singularities of Starobinsky and hill top inflation, assuming the slow roll inflation to the past infinity, are strong. Hence stringy corrections are inevitable in the very early stage of these inflation models. We also find that the initial singularity of the hill top inflation could be weak for non-slow roll case.

Implications of the singularity theorem for the size of a nonsingular universe

“…Even for flat Friedmann-Lemaître-Robertson-Walker (FLRW) geometry, one can construct a non-singular universe by considering the maximal extension of an inflationary universe as studied in Refs. [28][29][30]. What we will show in this paper is that any such non-singular universe has a compact Cauchy surface and the size of the universe, defined later through the affine length of null geodesics, is smaller than the exact de Sitter space.…”

“…This does not necessarily mean the presence of singularity [37] and as studied in Refs. [28][29][30], the spacetime can be extended beyond the endpoint of incomplete null geodesics if lim t→−∞ ∂ t H/a 2 is finite. Actually, our example satisfies this condition because…”

On the initial singularity and extendibility of flat quasi-de Sitter spacetimes

“…It is generally unknown whether the past geodesic incompleteness of inflation always translates into a 'big bang-like' initial singularity or whether the spacetime is actually extendible beyond its past boundary. A first step in this direction was given by [27] (further developed and applied in [28][29][30][31][32]), where the nature of the singularity in quasi-de Sitter spacetimes was explored. Building on mathematical results relating spacetime extendibility to parallelly propagated curvature singularities [33][34][35][36][37], it was shown in [27,29] that the universe had to approach dS sufficiently quickly in order to avoid parallelly propagated curvature singularities.…”

“…One may say that a p. p. singularity is weaker than a scalar curvature singularity, but it may nevertheless be physically relevant. In fact, the relation between p. p. singularities and spacetime extendibility was the crux of the work by Clarke [33,36,37], which was applied in cosmology in [27][28][29][30][31][32]. For instance, the approach of [27] was precisely to evaluate the curvature tensor in a null parallelly propagated basis within a flat FLRW spacetime and finding the criterion that determined the presence/absence of a p. p. curvature singularity.…”

“…Either way, it still remains crucial to correctly determine whether or not a given classical solution is singular and/or (in)extendible 22. First steps in that direction may be found in[30].…”