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Cosmological viability of anisotropic inflation in Thurston spacetimes

Devika J. S., Tanay Gupta, Sukanta Panda

TL;DR

This work investigates inflation in homogeneous but anisotropic Thurston spacetimes by introducing an axisymmetric vector field coupled to the inflaton. Through a dynamical-systems approach, the authors derive geometry-specific autonomous systems in phase space and identify a unique stable anisotropic fixed point A that attracts trajectories for $Q\gg1$, indicating a robust violation of the cosmic no-hair theorem across several Thurston geometries. The analysis shows that anisotropic hair can persist during inflation with a residual shear determined by the potential form (power-law vs. exponential) and coupling strength, offering a unified picture of anisotropic inflation in curved spatial geometries and motivating further study of tensor perturbations and observational imprints.

Abstract

The introduction of anisotropy into the large-scale geometry of the universe has led to the development of exotic spacetimes possessing an inherent anisotropic curvature in them. Studies related to these anisotropic spacetimes have revealed how they could explain the evolutionary dynamics and light propagation in the universe. Here, we consider one such interesting set of spacetimes that preserve homogeneity but place no constraint on isotropy during the inflationary epoch, to examine whether we can address the possibility of anisotropic inflation in the universe. Researchers have proposed inflationary models in which a vector field coupled to the inflaton is found to violate the cosmic no-hair theorem for the anisotropic Bianchi type I spacetime, due to the existence of a stable anisotropically inflationary fixed point. Lately, this study has been extended to axisymmetric spacetimes of Bianchi type II, III, and the Kantowski-Sachs metric, and it has been inferred that the entire family of spacetimes is attracted to the anisotropic Bianchi I fixed point. By fabricating Thurston spacetimes as inflationary models with an explicit vector field sourced effectively by their inherent eccentricity coupled to inflation, we perform dynamical stability and phase space analyses to confer the feasibility of anisotropic inflation. The results obtained for the considered set of Thurston geometries showed the presence and convergence to a unique, stable inflationary fixed point, which was found to be similar to those obtained in Bianchi spacetimes, thereby indicating the cosmological viability of inflation with anisotropic hair.

Cosmological viability of anisotropic inflation in Thurston spacetimes

TL;DR

This work investigates inflation in homogeneous but anisotropic Thurston spacetimes by introducing an axisymmetric vector field coupled to the inflaton. Through a dynamical-systems approach, the authors derive geometry-specific autonomous systems in phase space and identify a unique stable anisotropic fixed point A that attracts trajectories for , indicating a robust violation of the cosmic no-hair theorem across several Thurston geometries. The analysis shows that anisotropic hair can persist during inflation with a residual shear determined by the potential form (power-law vs. exponential) and coupling strength, offering a unified picture of anisotropic inflation in curved spatial geometries and motivating further study of tensor perturbations and observational imprints.

Abstract

The introduction of anisotropy into the large-scale geometry of the universe has led to the development of exotic spacetimes possessing an inherent anisotropic curvature in them. Studies related to these anisotropic spacetimes have revealed how they could explain the evolutionary dynamics and light propagation in the universe. Here, we consider one such interesting set of spacetimes that preserve homogeneity but place no constraint on isotropy during the inflationary epoch, to examine whether we can address the possibility of anisotropic inflation in the universe. Researchers have proposed inflationary models in which a vector field coupled to the inflaton is found to violate the cosmic no-hair theorem for the anisotropic Bianchi type I spacetime, due to the existence of a stable anisotropically inflationary fixed point. Lately, this study has been extended to axisymmetric spacetimes of Bianchi type II, III, and the Kantowski-Sachs metric, and it has been inferred that the entire family of spacetimes is attracted to the anisotropic Bianchi I fixed point. By fabricating Thurston spacetimes as inflationary models with an explicit vector field sourced effectively by their inherent eccentricity coupled to inflation, we perform dynamical stability and phase space analyses to confer the feasibility of anisotropic inflation. The results obtained for the considered set of Thurston geometries showed the presence and convergence to a unique, stable inflationary fixed point, which was found to be similar to those obtained in Bianchi spacetimes, thereby indicating the cosmological viability of inflation with anisotropic hair.

Paper Structure

This paper contains 21 sections, 111 equations, 3 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Thurston spacetimes
  3. The inflationary model
  4. On Thurston geometries
  5. Field equations and dynamic system formulation
  6. $\mathbb{E}\times \mathbb{H}^2 / S^2$ geometries
  7. Nil geometry
  8. Solv geometry
  9. Dynamical stability analysis
  10. $\mathbb{E}\times \mathbb{H}^2 / S^2$ geometries
  11. Nil geometry
  12. Solv geometry
  13. Phase flow analysis
  14. Anisotropic phase break
  15. Results & Analysis
  16. ...and 6 more sections

Figures (3)

  • Figure 1: Phase flow plots for (X, Z, $\Omega_\kappa$) (left) and (X, Y, Z) (right) for $\lambda$ = 0.1 and Q = 50 using the initial conditions $X_0$ = 0.01, $Y_0$ = $Z_0$ = 0.25 & $\Omega_{\kappa,0}$ = 0.1
  • Figure 2: Phase space (left) and anisotropy evolution (right) plots for potential $V(\phi) = \frac{1}{2}m^2 \phi^2$
  • Figure 3: Fig. \ref{['powerlaw']} for potential $V(\phi) = V_0 e^\frac{\lambda \phi}{M_{(Pl)}}$