[Paper Review] Cosmic evolution in novel-Gauss Bonnet Gravity

Gaurav Narain, Hai-Qing Zhang|arXiv (Cornell University)|May 11, 2020

Cosmology and Gravitation Theories56 references8 citations

TL;DR

This paper investigates cosmic evolution in novel-Gauss-Bonnet gravity by taking a smooth D→4 limit of the action after rescaling the Gauss-Bonnet coupling by (D−4). Using a generalized Friedmann-Lemaître-Robertson-Walker metric in D dimensions, the authors derive a four-dimensional effective action and compute first-order corrections to the lapse function and on-shell action in an empty Universe. The key result is that the first-order correction to the on-shell action carries a negative sign, which may lead to path-integral divergence and restrict the allowed sign of the coupling parameter.

ABSTRACT

In this short paper we investigate any non-trivial effect the novel Gauss-Bonnet gravity may give rise in the cosmic evolution of the Universe in four spacetime dimensions. We start by considering a generic Friedmann-Lemaître-Robertson-Walker (FLRW) metric respecting homogeneity and isotropicity in arbitrary space-time dimension $D$. The metric depends on two functions: scale factor and lapse. Plugging this metric in novel Einstein-Gauss-Bonnet (EGB) gravity action, doing an integration by parts and then take the limit of $D o4$ give us a dynamical action in four spacetime dimensions for scale factor and lapse. The peculiar rescaling of Gauss-Bonnet coupling by factor of $D-4$ results in a non-trivial contribution in the action of the theory. In this paper we study this action. We investigate the dynamics of scale-factor and behavior of lapse in an empty Universe (no matter). Due to complexity of the problem we study the theory to first order in Gauss-Bonnet coupling and solve system of equation to the first order. We compute the first order correction to the on-shell action of the empty Universe and find that its sign is opposite of the leading order part. We discuss it consequences.

Motivation & Objective

• To explore non-trivial dynamical effects of novel-Gauss-Bonnet gravity in four-dimensional cosmological spacetimes.
• To investigate the behavior of the scale factor and lapse function in an empty Universe under the novel-Gauss-Bonnet framework.
• To compute the first-order correction to the on-shell action in the D→4 limit using perturbation theory in the Gauss-Bonnet coupling.
• To assess the implications of the sign of the first-order correction for the path-integral formulation and unitarity of the theory.

Proposed method

• Start with a D-dimensional Friedmann-Lemaître-Robertson-Walker metric with scale factor a(tₚ) and lapse function Nₚ(tₚ).
• Plug this metric into the novel-Einstein-Gauss-Bonnet action, which includes a (D−4)⁻¹ rescaling of the Gauss-Bonnet coupling.
• Perform integration by parts to eliminate divergent (D−4) terms, enabling a well-defined D→4 limit.
• Derive an effective four-dimensional action depending only on the rescaled scale factor q(t) and lapse N(t), with no explicit N(t) derivative.
• Apply perturbation theory in the Gauss-Bonnet coupling α, solving the equations of motion to first order in α.
• Compute the first-order correction to the on-shell action by substituting the perturbative solutions into the action.

Experimental results

Research questions

• RQ1 Can the novel-Gauss-Bonnet gravity term produce non-trivial cosmological dynamics in four spacetime dimensions despite being topological in standard Lovelock theory?
• RQ2 What is the structure of the effective four-dimensional action obtained after a smooth D→4 limit in a homogeneous and isotropic spacetime?
• RQ3 How do the lapse function and scale factor evolve in an empty Universe under the novel-Gauss-Bonnet gravity at first order in the coupling α?
• RQ4 What is the sign and physical significance of the first-order correction to the on-shell action in this theory?
• RQ5 Does the sign of the first-order correction imply constraints on the allowed values of the Gauss-Bonnet coupling for a consistent path-integral formulation?

Key findings

• The D→4 limit of the novel-Gauss-Bonnet action is well-defined after integration by parts, removing (D−4) divergences and yielding a consistent four-dimensional theory.
• The effective action depends only on the rescaled scale factor q(t) and lapse N(t), with no derivative term for N(t), indicating that N(t) is non-dynamical in the standard sense.
• The first-order correction to the lapse function is derived as N_c = s₁√(3/Λ)(1 − 5αΛ/2 + …)(√b₁ + s₂√b₀), where s₁, s₂ = ±.
• The first-order correction to the on-shell action is negative, given by S₁^on-shell = (1 − 5αΛ/6 + …)S₀^on-shell, where S₀^on-shell is the Einstein-Hilbert result.
• The negative sign of the first-order correction suggests potential instability in the Euclidean path integral, possibly restricting the coupling α to a single sign.
• The result implies that only one sign of the Gauss-Bonnet coupling is viable for a consistent quantum gravity formulation in this framework.