[Paper Review] Lorentzian quantum cosmology in novel Gauss-Bonnet gravity from Picard-Lefschetz methods

Gaurav Narain, Hai-Qing Zhang|arXiv (Cornell University)|Jun 3, 2020

Black Holes and Theoretical Physics43 references4 citations

TL;DR

This paper investigates Lorentzian quantum cosmology in novel Gauss-Bonnet gravity using Picard-Lefschetz theory to compute transition amplitudes directly in Lorentzian signature without Wick rotation. By formulating a mini-superspace action in D dimensions and taking a smooth D→4 limit via integration by parts, the authors derive a non-trivial 4D action with finite Gauss-Bonnet coupling. The key result is a non-perturbative-looking correction to the transition amplitude at first order in the Gauss-Bonnet coupling, arising from complex saddle points and topological contributions in the path integral.

ABSTRACT

In this paper we study some aspects of classical and quantum cosmology in the novel-Gauss-Bonnet (nGB) gravity in four space-time dimensions. Starting with a generalised Friedmann-Lemaître-Robertson Walker (FLRW) metric respecting homogeneity and isotropicity in arbitrary space-time dimension $D$, we find the action of theory in four spacetime dimension where the limit $D o4$ is smoothly obtained after an integration by parts. The peculiar rescaling of Gauss-Bonnet coupling by factor of $D-4$ results in a non-trivial contribution to the action. We study the system of equation of motion to first order nGB coupling. We then go on to compute the transition probability from one $3$-geometry to another directly in Lorentzian signature. We make use of combination of WKB approximation and Picard-Lefschetz (PL) theory to achieve our aim. PL theory allows to analyse the path-integral directly in Lorentzian signature without doing Wick rotation. Due to complication caused by non-linear nature of action, we compute the transition amplitude to first order in nGB coupling. We find non-trivial correction coming from the nGB coupling to the transition amplitude, even if the analysis was done perturbatively. We use this result to investigate the case of classical boundary conditions.

Motivation & Objective

• To explore quantum cosmology in a novel formulation of Gauss-Bonnet gravity that allows a finite D→4 limit.
• To compute transition amplitudes between 3-geometries directly in Lorentzian signature, avoiding the ambiguities of Wick rotation.
• To investigate the role of the Gauss-Bonnet term in quantum gravity using a mini-superspace approximation and Picard-Lefschetz theory.
• To determine whether non-trivial quantum corrections emerge from the Gauss-Bonnet coupling in 4D despite perturbative treatment.

Proposed method

• Formulate a generalized Friedmann-Lemaître-Robertson-Walker metric in D spacetime dimensions with time-dependent lapse and scale factor.
• Derive the mini-superspace action for novel Gauss-Bonnet gravity by integrating by parts, ensuring a well-defined D→4 limit without Kaluza-Klein reduction.
• Apply the Picard-Lefschetz method to analyze the Lorentzian path integral directly, identifying relevant complex saddle points.
• Use the WKB approximation to evaluate the path integral to first order in the Gauss-Bonnet coupling α.
• Compute the transition amplitude by evaluating the action and Hessian at saddle points corresponding to different classical solutions.
• Extract quantum corrections from the imaginary and real parts of the action at complex saddle points, including logarithmic and arctangent terms.

Experimental results

Research questions

• RQ1 Can a finite and non-trivial 4D action be derived from novel Gauss-Bonnet gravity without Kaluza-Klein dimensional reduction?
• RQ2 Does the Picard-Lefschetz method enable a consistent computation of Lorentzian transition amplitudes without Wick rotation?
• RQ3 What is the structure of quantum corrections to the transition amplitude in novel Gauss-Bonnet gravity at first order in α?
• RQ4 How do classical boundary conditions affect the quantum transition amplitude in this framework?
• RQ5 What role do complex saddle points and topological terms (e.g., arctangent contributions) play in the final amplitude?

Key findings

• The D→4 limit of the novel Gauss-Bonnet action is well-defined after integration by parts, yielding a finite 4D theory without extra scalar degrees of freedom.
• Non-trivial quantum corrections to the transition amplitude arise even at first order in the Gauss-Bonnet coupling α, indicating non-perturbative effects in the path integral.
• The transition amplitude contains complex exponential terms involving π²/4ℏΛ and arctangent functions of the scale factor parameters u and v.
• The saddle point contributions at the two relevant complex solutions (denoted by □ and ○) yield distinct expressions involving rational, polynomial, and logarithmic terms in u and v.
• The order-α correction to the amplitude includes terms proportional to tan⁻¹(u) and tan⁻¹(v), as well as rational functions of (u±v), showing non-trivial dependence on the boundary 3-geometry parameters.
• The final amplitude exhibits interference-like behavior due to the superposition of two saddle point contributions with opposite phase factors, suggesting quantum coherence in the transition process.