Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Existence and bounds of nonlinear singularity-free cosmological solutions in a string-inspired gravity

Chihang He, Chao Liu

TL;DR

<3-5 sentence high-level summary>The paper proves the existence of globally non-singular FLRW cosmological solutions in Einstein-dilaton-Gauss-Bonnet gravity with exponential coupling, addressing the Big Bang singularity problem in a string-inspired context. A novel power identity method is developed to manage the strong nonlinearities and is coupled with a first-hit contradiction argument to obtain global bounds on the Hubble parameter. The authors establish a unique, globally defined solution for all time t, with H(t) staying positive and vanishing asymptotically, and the dilaton-like field φ evolving monotonically, in agreement with numerical simulations. This work, together with its companion paper on quadratic coupling, provides a rigorous mathematical foundation for singularity-free cosmology in EdGB/EsGB theories and highlights the viability of exponential dilaton couplings in string-inspired gravity.

Abstract

We provide a rigorous proof for the existence of homogeneous, isotropic and globally singularity-free cosmological solutions in Einstein-dilaton-Gauss-Bonnet (EdGB) gravity with exponential coupling. While numerical studies suggested such solutions exist, a formal proof remained elusive. By employing a novel ``power identity method'' and overcoming significant challenges posed by the strong nonlinearities of the exponential coupling, which are not present in the quadratic coupling analyzed in our companion paper \cite{he2025proofssingularityfreesolutionsscalarization}, we establish a FLRW solution valid for all time $t\in(-\infty,+\infty)$, where the Hubble parameter remains positive and vanishes asymptotically, while the scalar field evolves monotonically. This result align with numerical simulations and offer a firm mathematical foundation for singularity-free cosmology in a string-inspired setting.

Existence and bounds of nonlinear singularity-free cosmological solutions in a string-inspired gravity

TL;DR

<3-5 sentence high-level summary>The paper proves the existence of globally non-singular FLRW cosmological solutions in Einstein-dilaton-Gauss-Bonnet gravity with exponential coupling, addressing the Big Bang singularity problem in a string-inspired context. A novel power identity method is developed to manage the strong nonlinearities and is coupled with a first-hit contradiction argument to obtain global bounds on the Hubble parameter. The authors establish a unique, globally defined solution for all time t, with H(t) staying positive and vanishing asymptotically, and the dilaton-like field φ evolving monotonically, in agreement with numerical simulations. This work, together with its companion paper on quadratic coupling, provides a rigorous mathematical foundation for singularity-free cosmology in EdGB/EsGB theories and highlights the viability of exponential dilaton couplings in string-inspired gravity.

Abstract

We provide a rigorous proof for the existence of homogeneous, isotropic and globally singularity-free cosmological solutions in Einstein-dilaton-Gauss-Bonnet (EdGB) gravity with exponential coupling. While numerical studies suggested such solutions exist, a formal proof remained elusive. By employing a novel ``power identity method'' and overcoming significant challenges posed by the strong nonlinearities of the exponential coupling, which are not present in the quadratic coupling analyzed in our companion paper \cite{he2025proofssingularityfreesolutionsscalarization}, we establish a FLRW solution valid for all time , where the Hubble parameter remains positive and vanishes asymptotically, while the scalar field evolves monotonically. This result align with numerical simulations and offer a firm mathematical foundation for singularity-free cosmology in a string-inspired setting.

Paper Structure

This paper contains 18 sections, 27 theorems, 126 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main Theorem
  3. Related Works
  4. Outlines and Overview of Methods
  5. Core Strategies and Key Techniques
  6. The Two Regions of the Proof
  7. Global Existence
  8. Additional Remarks
  9. Summary
  10. Constraints and local existence
  11. Initial data and the constraint
  12. Local existence
  13. Estimates of the FLRW Solution in the EsGB System
  14. Bounds of $H$ for $t \in (0,\mathcal{T}_+)$
  15. Bounds of $H$ for $t \in (\mathcal{T}_-,0)$
  16. ...and 3 more sections

Key Result

Theorem 1.1

Suppose the initial data satisfy and the conditionNote that eq:2.5, known as the Hamiltonian constraint, is quadratic in $\dot \phi$, giving two algebraic solution branches of $\dot\phi$. Only the negative branch eq:2.11!! leads to singularity-free solutions. then there exists a unique globally singularity-free, homogeneous, and isotropic FLRW solution$(g,\phi)\in C^2((-\infty, +\infty))$, where

Figures (2)

  • Figure 1: Bounds for $H$ and $\phi$
  • Figure 2: Hierarchical Estimates

Theorems & Definitions (51)

  • Theorem 1.1
  • Remark 1.1
  • Lemma 2.1
  • proof
  • Proposition 2.2: Local Existence
  • proof
  • Remark 3.1
  • Proposition 3.1
  • proof
  • Corollary 3.1
  • ...and 41 more