[Paper Review] Generalized modified gravity models: the stability issue

Guido Cognola, Sergio Zerbini|ArXiv.org|Nov 17, 2008

Cosmology and Gravitation Theories17 references3 citations

TL;DR

This paper investigates the stability of spatially flat homogeneous cosmologies in generalized modified gravity models using dynamical systems methods. It derives critical points for various F(R,P,Q,Q₃) models, showing that de Sitter and Minkowski solutions can be stable under specific parameter conditions, with analytical results for models including quadratic and cubic curvature invariants, demonstrating the method's power for complex gravity theories beyond standard F(R) gravity.

ABSTRACT

A brief introduction on the issue of stability in generalized modified gravity is presented and the dynamical system methods are used in the investigation of the stability of spatially flat homogeneous cosmologies within a large class of generalized modified gravity models in the presence of a relativistic matter-radiation fluid.

Motivation & Objective

• To address the stability of cosmological solutions in generalized modified gravity models, particularly those involving higher-order curvature invariants.
• To extend the dynamical systems approach beyond standard F(R) gravity to models built from multiple geometric invariants like P, Q, and Q₃.
• To determine the conditions under which de Sitter and Minkowski solutions are stable, ensuring compatibility with both cosmological observations and local gravity tests.
• To provide analytical and numerical criteria for stability in models including quadratic and cubic curvature terms, such as those arising from two-loop quantum gravity corrections.
• To demonstrate the applicability of the dynamical system method to previously unexplored models, such as those involving the cubic invariant Q₃.

Proposed method

• Formulate an autonomous first-order system of differential equations classically equivalent to the equations of motion for generalized F(R,P,Q,Q₃) gravity models in Friedmann-Robertson-Walker spacetime.
• Apply the dynamical systems approach to analyze critical points corresponding to de Sitter, Minkowski, and cosmological constant-dominated solutions.
• Use the trace equation and perturbation analysis around constant curvature solutions to derive effective scalaron mass and stability conditions.
• Introduce compact notation for critical points: P ≡ (X, Y, Z, Ωρ, weff), with P₀ and PΛ representing de Sitter and de Sitter-with-cosmological-constant solutions.
• Analyze specific models analytically, including F = R − μ⁴/R, F = R + aR² + bP + cQ, F = R − d²Q₃, and F = R + aR² + bP + cQ − d²Q₃.
• Establish stability criteria based on the sign of the effective mass squared M² and positivity of the effective Newton constant, requiring 1 + f′(R₀) > 0.

Experimental results

Research questions

• RQ1 Under what conditions is the de Sitter solution stable in generalized modified gravity models with arbitrary curvature invariants?
• RQ2 How do higher-order curvature invariants like Q₃ affect the stability of cosmological solutions compared to standard F(R) models?
• RQ3 Can the dynamical systems method be systematically applied to generalized F(R,P,Q,Q₃) gravity models to identify and classify critical points?
• RQ4 What are the stability conditions for Minkowski and de Sitter solutions in models including quadratic and cubic curvature terms?
• RQ5 How do the stability properties of F(R) models compare with those of more complex models involving multiple invariants, particularly in the context of quantum gravity corrections?

Key findings

• For the F = R − μ⁴/R model, the de Sitter point P₀ with R₀ = √3 μ² is unstable, while a critical point at (−1/2, −1, −2, 0, −2/3) is stable with H₀ = 0.
• The model F = R + aR² + bP + cQ has a stable Minkowski solution (P₀, R₀ = 0) if 3a + b + c > 0, and a stable de Sitter solution PΛ if the same condition holds.
• The F = R − d²Q₃ model features a stable Minkowski solution at R₀ = 0 and a stable non-de Sitter critical point at (0.05, 0.60, −3.60, 0, −1.03), with H₀ = 0.
• For the generalized model F = R + aR² + bP + cQ − d²Q₃, the de Sitter solution P₀ with R₀ = 6/d is stable if 3a + b + c + 3d > 0.
• The Minkowski solution P₀ with R₀ = 0 is stable if 3a + b + c > 0, and the de Sitter point PΛ exists and is stable under appropriate parameter constraints.
• The method successfully identifies stable solutions in models involving the cubic invariant Q₃, which to our knowledge have not been studied analytically before, demonstrating the method's broad applicability.