[Paper Review] Mirage resolution of cosmological singularities

Frédéric Leblond|ArXiv.org|Mar 22, 2004

Black Holes and Theoretical Physics77 references3 citations

TL;DR

This paper investigates the resolution of cosmological singularities in string theory and supergravity by analyzing time-dependent backgrounds and compactifications. It proposes that while lower-dimensional effective theories may exhibit non-singular bounces or positive acceleration, these are 'mirage' solutions—apparent regularity masked by unavoidable singularities when uplifted to higher dimensions, as dictated by generalized Hawking-Penrose theorems in higher-dimensional spacetimes.

ABSTRACT

We study time-dependent backgrounds in the low energy regimes of string theories. In particular the emphasis is on the general study of exotic phenomena such as positive acceleration and gravitational bounces. We generalize the usual Hawking-Penrose cosmological singularity theorems to higher-dimensional spacetimes and discuss their implications for time-dependent solutions in supergravity theories. The explicit examples we consider fall in two categories. First we consider effective lower-dimensional gravitational theories obtained from compactifications of ten and eleven-dimensional supergravity. We argue and explain why non-singular solutions (e.g., with positive acceleration and possibly a bounce) can in principle be obtained. However we show that their uplift to higher dimensions is always singular as predicted by the theorems. Secondly we revisit the issue of supergravity s-branes. Our main result is to propose a generic mechanism by which the usual singularities can be resolved.

Motivation & Objective

• To investigate whether non-singular bouncing cosmologies or positive acceleration can be realized in effective lower-dimensional theories derived from ten- and eleven-dimensional supergravity.
• To examine the role of compactifications and flux compactifications in embedding such lower-dimensional solutions while preserving consistency with higher-dimensional singularity theorems.
• To analyze supergravity s-branes as a framework for probing singularity resolution, particularly through the interplay of dilaton and breathing modes.
• To determine whether the inclusion of dynamical dilaton fields can resolve curvature singularities in bouncing spacetimes, using numerical and analytical methods.
• To clarify the distinction between apparent regularity in lower-dimensional observers' frames and actual singularities in the full higher-dimensional geometry.

Proposed method

• Generalizes the Hawking-Penrose singularity theorems to higher-dimensional spacetimes to establish conditions under which singularities are unavoidable in time-dependent solutions.
• Analyzes (p+1)-dimensional Einstein gravity coupled to a scalar field with positive exponential potential, deriving analytic solutions for flat and positively curved spatial foliations.
• Applies flux compactification of 10D and 11D supergravity on maximally symmetric spaces to embed lower-dimensional cosmological models and study their higher-dimensional consistency.
• Uses a metric ansatz with gauge choice $ A = pB $ and derives equations of motion for the scale factor $ B $, dilaton $ heta $, and scalar field $ ho $, incorporating Friedmann constraints.
• Performs numerical analysis on systems with non-zero kinetic energy in the dilaton field ($ c_1 \neq 0 $) to test whether the dilaton can stabilize bouncing solutions.
• Compares solutions with and without the dilaton field, using transformed parameters $ \bar{\alpha} $ and $ \bar{C}^2 $ to assess the impact of dilaton dynamics on singularity resolution.

Experimental results

Research questions

• RQ1 Can non-singular bouncing cosmologies be realized in effective lower-dimensional theories derived from supergravity, even if they appear regular in the reduced dimensionality?
• RQ2 To what extent do higher-dimensional singularity theorems rule out non-singular time-dependent solutions, especially when only a submanifold exhibits bouncing behavior?
• RQ3 Can the inclusion of a dynamical dilaton field resolve curvature singularities in bouncing spacetimes that are otherwise singular in the absence of dilaton dynamics?
• RQ4 How do flux compactifications of 10D and 11D supergravity preserve or break the apparent regularity of lower-dimensional cosmological solutions?
• RQ5 What is the role of the dS/CFT correspondence and Euclidean path integral duality in understanding the nature of singularities in time-dependent backgrounds?

Key findings

• Non-singular bouncing solutions in lower-dimensional effective theories (e.g., with flat or positively curved spatial foliations) are found to exist analytically for scalar field potentials with appropriate parameters.
• However, when uplifted to higher-dimensional spacetimes, these solutions always develop curvature singularities, as predicted by generalized Hawking-Penrose theorems.
• The presence of positive acceleration in the lower-dimensional models is linked to violation of the strong energy condition, but this does not prevent future or past curvature singularities.
• For s-brane solutions, the inclusion of a dynamical dilaton field does not resolve singularities; numerical experiments show that singularities persist even when the dilaton has non-zero kinetic energy at the bounce.
• The effective potential in the compactified model includes terms dependent on $ \bar{\alpha} $ and $ \bar{C}^2 $, but the condition $ \bar{\alpha} > \alpha $ ensures that the same singularity constraints apply as in the single-scalar case.
• The key result is that apparent regularity in lower-dimensional observers' frames is a 'mirage'—a geometric illusion—since the full higher-dimensional geometry remains singular.