[Paper Review] Cosmography in F(G) modified gravity

M. R. Setare, Naser Mohammadipour|arXiv (Cornell University)|Jun 1, 2012

Cosmology and Gravitation Theories10 references3 citations

TL;DR

This paper develops a cosmographic approach to constrain F(G) modified gravity by expressing the present-day values of F(G) and its derivatives in terms of cosmographic parameters (q₀, j₀, s₀, l₀) and H₀, without solving field equations. It derives model-independent constraints on F(G) and its derivatives using fiducial data, showing that viable F(G) models can be reconstructed directly from observational cosmography, with stability conditions dependent on F''(G) > 0 and H₀⁶F''(G₀) < 0.0017.

ABSTRACT

Investigating the accelerated expansion of the universe with cosmography is a best method to constraint cosmological models. In this work, in the $F(G)$ modified gravity framework, we obtain equations of motion in a flat FRW metric. Then we reconstruct the present day values of $F(G)$ and its derivatives with the cosmographic parameters on the only assumption that the universe is homogenous and isotropic on large scale. Also we investigate the conditions of cosmologically viable $F(G)$ gravity models with the fiducial data set values.

Motivation & Objective

• To derive model-independent constraints on F(G) gravity using cosmographic parameters instead of solving Friedmann equations.
• To reconstruct present-day values of F(G) and its derivatives (F', F'', F''') from cosmographic observables (q₀, j₀, s₀, l₀, H₀).
• To assess cosmological viability of F(G) models using fiducial data and stability conditions.
• To eliminate the need for assuming f′(0) = 1 (as in f(R) gravity) by directly relating F(G) to cosmographic parameters.

Proposed method

• Derives field equations for F(G) gravity in a spatially flat FRW metric, expressing curvature invariants R and G in terms of Hubble parameter H and its time derivatives.
• Expresses F(G), F′(G), F′′(G), and F′′′(G) at z=0 via a third-order Taylor expansion around G₀.
• Uses cosmographic parameters (q₀, j₀, s₀, l₀) and H₀ to express F(G) and its derivatives, with Ωₘ₀ as an additional input.
• Applies observational constraints on (H₀, q₀, j₀, s₀, l₀, Ωₘ₀) to derive bounds on F(G) derivatives, summarized in Table 2.
• Tests viability of a power-law model F(G) = αGⁿ + βG ln G by deriving α, β, and n in terms of f₀, f₂, f₃, and G₀.
• Establishes stability conditions: F''(G) > 0 and H₀⁶F''(G₀) < 1/600 ≈ 0.0017 for a stable spiral attractor.

Experimental results

Research questions

• RQ1 Can the present-day values of F(G) and its derivatives be reconstructed directly from cosmographic parameters without solving the field equations?
• RQ2 How do the cosmographic parameters (q₀, j₀, s₀, l₀, H₀) constrain the functional form of F(G) in modified Gauss-Bonnet gravity?
• RQ3 What are the model-independent constraints on F(G), F′(G), F′′(G), and F′′′(G) at z=0 using fiducial data?
• RQ4 Under what conditions is an F(G) model cosmologically viable, and how do these conditions relate to observable cosmographic parameters?
• RQ5 Can the ΛCDM model be recovered within F(G) gravity under the assumption of a third-order Taylor expansion of F(G)?

Key findings

• The present-day value of F(G) is expressed as f₀ = F(G₀) = α(n−1)G₀ⁿ + βG₀, with α and β derived from f₀, f₂, and f₃.
• The second derivative F′′(G₀) is given by f₂ = αn(n−1)G₀ⁿ⁻² + βG₀⁻¹, which must be positive for stability.
• The third derivative F′′′(G₀) is f₃ = αn(n−1)(n−2)G₀ⁿ⁻³ − βG₀⁻², used to constrain the power-law index n.
• A closed-form solution for n is derived as n = [P ± √Q]/T, where P, Q, and T are functions of f₀, f₂, f₃, and G₀, enabling model testing.
• The model-independent reconstruction shows that F(G) derivatives depend on H₀, q₀, j₀, s₀, l₀, and Ωₘ₀, with no need for f′(0) = 1.
• Stability requires F′′(G₀) > 0 and H₀⁶F′′(G₀) < 0.0017, consistent with a damped oscillatory attractor in phase space.