[Paper Review] Implementing Markov's Limiting Curvature Hypothesis

Robert Brandenberger|ArXiv.org|Oct 1, 1995

Cosmology and Gravitation Theories3 citations

TL;DR

This paper proposes a class of higher-derivative gravitational actions that implement Markov's limiting curvature hypothesis, ensuring all symmetric spacetimes remain nonsingular with bounded curvature. By constructing effective actions where gravity becomes asymptotically free at high curvature, the theory avoids singularities and suppresses matter-gravity coupling near the limiting curvature scale.

ABSTRACT

The effective action for gravity at high curvatures is likely to contain higher derivative terms. These corrections may have profound consequences for the singularity structure of space-time and for early Universe cosmology. In this contribution, recent work is reviewed which demonstrates that it is possible to construct a class of effective gravitational actions for which all solutions with sufficient symmetries have limited curvature and are nonsingular. Near the limiting curvature, the coupling between matter and gravity goes to zero and in this sense the theory is asymptotically free.

Motivation & Objective

• To realize Markov's limiting curvature hypothesis in a consistent gravitational theory.
• To resolve spacetime singularities in cosmological models using higher-derivative corrections.
• To construct effective gravitational actions that yield finite curvature in symmetric spacetimes.
• To ensure gravity becomes asymptotically free at high curvature, suppressing matter coupling.
• To provide a framework for nonsingular early Universe cosmology via modified gravity.

Proposed method

• Constructing effective gravitational actions with higher-derivative terms that modify Einstein-Hilbert gravity at high curvatures.
• Imposing symmetry constraints (e.g., spatial homogeneity and isotropy) to analyze curvature behavior in cosmological solutions.
• Using variational principles to derive field equations from the modified action and study their solutions.
• Analyzing the coupling between matter and gravity in the high-curvature regime to show it vanishes near the limiting curvature.
• Demonstrating that curvature invariants remain bounded in all symmetric solutions, preventing singularities.
• Employing the framework of effective field theory to justify the inclusion of higher-derivative terms.

Experimental results

Research questions

• RQ1 Can higher-derivative gravity actions be constructed such that all symmetric spacetimes have bounded curvature?
• RQ2 Does the coupling between matter and gravity vanish at high curvature, implying asymptotic freedom?
• RQ3 Can the limiting curvature hypothesis be realized in a consistent quantum gravity-compatible framework?
• RQ4 What are the implications of such a theory for the initial singularity in cosmology?
• RQ5 How do the modified field equations prevent curvature blow-up in homogeneous and isotropic models?

Key findings

• All solutions with sufficient symmetries in the proposed gravitational actions exhibit limited curvature, avoiding singularities.
• The coupling between matter and gravity vanishes near the limiting curvature, indicating asymptotic freedom.
• The effective action contains higher-derivative terms that stabilize curvature inhomogeneities and prevent divergence.
• The theory remains nonsingular even in the early Universe, offering a potential resolution to the big bang singularity.
• The limiting curvature scale acts as a natural ultraviolet cutoff, suppressing quantum divergences.
• The model is consistent with the principles of effective field theory and provides a viable alternative to standard general relativity at high energies.