[Paper Review] From inflation to dark energy in scalar-tensor cosmology

Jibitesh Dutta, Laur Järv+2 more|arXiv (Cornell University)|Jul 13, 2020

Cosmology and Gravitation Theories99 references4 citations

TL;DR

This paper develops a unified dynamical systems framework for scalar-tensor gravity that captures the entire cosmic evolution—from inflation to dark energy—by identifying fixed points corresponding to inflation, radiation domination, matter domination, and dark energy domination. It derives conditions on the scalar potential and nonminimal coupling that ensure a viable, generic cosmic history with stable transitions between eras, particularly emphasizing the challenge of achieving a graceful exit from inflation via a saddle-point de Sitter fixed point.

ABSTRACT

The methods of dynamical systems have found wide applications in cosmology, with focus either upon inflation or upon the passage into dark energy era. In this paper, we endeavor to capture the whole history of the universe into a dynamical system by considering generic scalar tensor gravity with radiation and dust matter fluids in flat Friedmann-Lemaitre-Robertson-Walker spacetime. We construct the dynamical variables in such a way that the main stages of the cosmic evolution, viz. inflation, radiation domination, matter domination, and dark energy domination can be represented by the respective fixed points in the phase space. As the evolution of solutions is ruled by the sequence of these fixed points with appropriate properties, we can determine the conditions that the scalar potential and nonminimal coupling must satisfy for the model to deliver viable cosmic history in a generic manner. We illustrate the construction by a scalar field with quartic potential, with and without quadratic nonminimal coupling to curvature.

Motivation & Objective

• To develop a unified dynamical systems framework that describes the full cosmic evolution from inflation to dark energy in scalar-tensor gravity.
• To identify the necessary conditions on the scalar potential and nonminimal coupling for fixed points corresponding to inflation, radiation domination, matter domination, and dark energy domination.
• To ensure that trajectories in phase space naturally evolve through all eras without skipping or getting trapped, particularly focusing on the transition from inflation to radiation domination.
• To analyze the stability of de Sitter fixed points using center manifold theory when higher-order derivatives of the potential vanish at the critical point.

Proposed method

• Construct a set of dynamical variables in flat FLRW spacetime that map the main cosmic eras to fixed points in phase space.
• Formulate the cosmological equations as a system of autonomous ordinary differential equations in terms of dimensionless variables representing energy density fractions and scalar field dynamics.
• Identify fixed points corresponding to inflation (de Sitter), radiation domination, matter domination, and dark energy domination, analyzing their stability via linearization and center manifold techniques.
• Apply center manifold theory to analyze non-hyperbolic de Sitter fixed points where the third and fourth derivatives of the potential vanish at the critical field value.
• Use perturbative expansions in new variables to derive the reduced dynamics on the center manifold and determine stability based on the sign of the third and fourth derivatives of the potential.
• Illustrate the framework with a quartic potential and quadratic nonminimal coupling, verifying the existence of the required fixed points and their stability properties.

Experimental results

Research questions

• RQ1 What conditions on the scalar potential and nonminimal coupling ensure the existence of fixed points corresponding to inflation, radiation domination, matter domination, and dark energy domination in scalar-tensor cosmology?
• RQ2 How can a dynamical systems framework be constructed to describe the full cosmic evolution in a single unified model, rather than in isolated early or late-time analyses?
• RQ3 What stability properties must the de Sitter fixed point in the early universe possess to allow a graceful exit into radiation domination, and how is this governed by the potential's higher-order derivatives?
• RQ4 Under what conditions does the center manifold theory predict stability or instability of a non-hyperbolic de Sitter fixed point when the third and fourth derivatives of the potential vanish at the critical point?
• RQ5 Can a model with a quartic potential and quadratic nonminimal coupling realize all four cosmic eras as stable or saddle fixed points in the phase space?

Key findings

• The de Sitter fixed point in the early universe must be a saddle with a repulsive eigendirection pointing toward the radiation domination fixed point to ensure a viable exit from inflation.
• The radiation and matter domination fixed points naturally emerge as saddles, possibly connected by a heteroclinic orbit that guides trajectories through these eras.
• An attractive de Sitter fixed point is required to collect all trajectories in the late-time dark energy era, ensuring cosmic acceleration.
• For non-hyperbolic de Sitter fixed points where the third derivative of the potential vanishes, stability is determined by the sign of the fourth derivative: stable if positive, saddle if negative.
• The center manifold analysis shows that the stability of the de Sitter point depends on the ratio of the third and fourth derivatives of the potential at the critical field value, with explicit expressions derived for the coefficients of the reduced dynamics.
• The model with a quartic potential and quadratic nonminimal coupling successfully realizes all four cosmic eras as fixed points with appropriate stability, demonstrating the framework's viability.