Time-dependent Orbifolds and String Cosmology

TL;DR

This work surveys time-dependent orbifolds in string theory, focusing on three-dimensional Minkowski space to model cosmological singularities and horizons. It develops a detailed framework for propagating free and interacting fields on these orbifolds, analyzes stability and backreaction via two-dimensional dilaton gravity, and demonstrates that perturbative divergences can be tempered through eikonal resummation in suitable regimes. The authors further connect these time-dependent geometries to orientifold cosmology, establishing dualities with O8/O8 systems and constructing four-dimensional cosmologies exhibiting transient or cyclic acceleration. Collectively, the paper positions time-dependent orbifolds as a tractable laboratory for studying quantum gravity effects near cosmological singularities and for exploring string-inspired resolutions involving orientifolds and holographic-like dualities.

Abstract

In these lectures, we review the physics of time-dependent orbifolds of string theory, with particular attention to orbifolds of three-dimensional Minkowski space. We discuss the propagation of free particles in the orbifold geometries, together with their interactions. We address the issue of stability of these string vacua and the difficulties in defining a consistent perturbation theory, pointing to possible solutions. In particular, it is shown that resumming part of the perturbative expansion gives finite amplitudes. Finally we discuss the duality of some orbifold models with the physics of orientifold planes, and we describe cosmological models based on the dynamics of these orientifolds.

Figures (20)

• Figure 1: The different space--time regions for the shifted--boost orbifold in $X^\pm$--plane. Image points are displaced in the $X$--direction by $2{ \if@compatibility \mathchar"0119 {} \mathchar"0119 } Rn$. All CTC's must cross region III and none goes into region I. The dashed lines correspond to closed time--like geodesics.
• Figure 2: CP diagram for the shifted--boost orbifold quotient space. The surface where the Killing vector ${ \if@compatibility \mathchar"0114 {} \mathchar"0114 }$ is null becomes, in the compactified geometry, a time--like singularity. The future horizon of the contracting region ${\rm I}_{in}$ is a Cauchy Horizon. Region III is excluded from the diagram.
• Figure 3: Effective temperature plot for $m=0$. This curve is the cosmological analogue of the Hawking radiation grey body factor for black holes. For massive particles the maximum of the curve is shifted.
• Figure 4: The fundamental domain for the boost--orbifold. There are CTC's in the whiskers and light--cone points have images arbitrarily close to the origin.
• Figure 5: CP diagram for the boost--orbifold quotient space. Here we consider the covering space $\hbox{M}^3$, but the generalization to higher dimensions is trivial. Only the contracting and expanding regions are represented in this diagram. The remaining regions are the whiskers, which end at the singularity.