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Bianchi Cosmologies in a Thurston-Based Theory of Gravity

Quentin Vigneron, Hamed Barzegar

TL;DR

This work advances a topology-aware modification of gravity (topo-GR) by introducing a topology-dependent reference Ricci tensor ${\boldsymbol{\bar{Ric}}}$ that modifies the Einstein equations without adding new parameters. It systematically derives the 3+1 and orthonormal formulations for locally spatially homogeneous BKS spacetimes across all Thurston geometries, revealing that shear-free perfect-fluid and static-vacuum solutions exist for all topologies, and that a positive cosmological constant enforces late-time isotropization in all but a Nil non-LRS case. The key results include non-fine-tuned avoidance of recollapse under weak energy conditions, Wald-like isotropization theorems extended to topo-GR, and a unifying framework where inflation can be modeled simply in any topology. The study highlights both the unifying potential of topo-GR across topologies and the peculiarities of Nil geometry, suggesting broad implications for cosmology and quantum inflation initial conditions without introducing extra parameters.

Abstract

The strong interplay between Bianchi--Kantowski--Sachs (BKS) spacetimes and Thurston geometries motivates the exploration of the role of topology in our understanding of gravity. As such, we study non-tilted BKS solutions of a theory of gravity that explicitly depends on Thurston geometries. We show that shear-free solutions with perfect fluid, as well as static vacuum solutions, exist for all topologies. Moreover, we prove that, aside from non-rotationally-symmetric Bianchi II models, all BKS metrics isotropize in the presence of a positive cosmological constant, and that recollapse is never possible when the weak energy condition is satisfied. This contrasts with General Relativity (GR), where these two properties fail for Bianchi IX and KS metrics. No additional parameters compared to GR are required for these results. We discuss, in particular, how this framework might allow for simple inflationary models in any topology.

Bianchi Cosmologies in a Thurston-Based Theory of Gravity

TL;DR

This work advances a topology-aware modification of gravity (topo-GR) by introducing a topology-dependent reference Ricci tensor that modifies the Einstein equations without adding new parameters. It systematically derives the 3+1 and orthonormal formulations for locally spatially homogeneous BKS spacetimes across all Thurston geometries, revealing that shear-free perfect-fluid and static-vacuum solutions exist for all topologies, and that a positive cosmological constant enforces late-time isotropization in all but a Nil non-LRS case. The key results include non-fine-tuned avoidance of recollapse under weak energy conditions, Wald-like isotropization theorems extended to topo-GR, and a unifying framework where inflation can be modeled simply in any topology. The study highlights both the unifying potential of topo-GR across topologies and the peculiarities of Nil geometry, suggesting broad implications for cosmology and quantum inflation initial conditions without introducing extra parameters.

Abstract

The strong interplay between Bianchi--Kantowski--Sachs (BKS) spacetimes and Thurston geometries motivates the exploration of the role of topology in our understanding of gravity. As such, we study non-tilted BKS solutions of a theory of gravity that explicitly depends on Thurston geometries. We show that shear-free solutions with perfect fluid, as well as static vacuum solutions, exist for all topologies. Moreover, we prove that, aside from non-rotationally-symmetric Bianchi II models, all BKS metrics isotropize in the presence of a positive cosmological constant, and that recollapse is never possible when the weak energy condition is satisfied. This contrasts with General Relativity (GR), where these two properties fail for Bianchi IX and KS metrics. No additional parameters compared to GR are required for these results. We discuss, in particular, how this framework might allow for simple inflationary models in any topology.

Paper Structure

This paper contains 55 sections, 14 theorems, 124 equations, 7 tables.

Table of Contents

  1. Introduction
  2. Notations, definitions, and conventions
  3. Maximal and minimal geometries with their metrics
  4. Maximal geometries
  5. Minimal geometries
  6. Minimal metrics as left-invariant metrics on Bianchi groups
  7. Isometry groups of minimal metrics
  8. Spatial isometry group
  9. Spacetime isometry group
  10. Ricci tensors of the maximal metrics
  11. Topo-GR
  12. Field equations
  13. Reference Ricci tensor
  14. Discussions
  15. Motivations for topo-GR and the form of the reference Ricci tensor
  16. ...and 40 more sections

Key Result

Theorem 2.1

The eight maximal, simply connected $3$-dimen-sional geometries $(X, G_{\rm max})$ admitting compact quotient manifolds are as follows:

Theorems & Definitions (38)

  • Theorem 2.1: Thurston, 1982 1982_Thurston1997_Thurston_threegeometry
  • Definition 2.2: Geometric manifold
  • Theorem 2.3: Thurston--Hamilton--Perelman, 2003 1982_Thurston1982_Hamilton_Ricci2002_Perelman2003_Perelman_a
  • Remark 2.4
  • Definition 2.5: Maximal metric
  • Theorem 2.6: 1985_Fagundes1994_Koike_et_al1997_Tanimoto_et_al
  • Remark 2.7
  • Definition 2.8: Minimal metric
  • Remark 2.9
  • Remark 2.10
  • ...and 28 more