Construction of cosmologically viable f(G) gravity models (0810.5712v2)

Abstract: We derive conditions under which f(G) gravity models, whose Lagrangian densities f are written in terms of a Gauss-Bonnet term G, are cosmologically viable. The most crucial condition to be satisfied is that f_GG, the second derivative of f with respect to G, must be positive, which is required to ensure the stability of a late-time de-Sitter solution as well as the existence of standard radiation/matter dominated epochs. We present a number of explicit f(G) models in which a cosmic acceleration is followed by the matter era. We find that the equation of state of dark energy can cross the phantom divide before reaching the present Universe. The viable models have asymptotic behavior f_GG goes to +0 when |G| goes to infinity, in which case a rapid oscillation of perturbations occurs unless such an oscillating degree of freedom is suppressed relative to a homogeneous mode in the early universe. We also introduce an iterative method to avoid numerical instabilities associated with a large mass of the oscillating mode.

• The paper establishes stability with f,GG > 0 and a critical range (0 < H1^6 f,GG < 1/384) to support a stable late-time de-Sitter epoch.
• It introduces several functional forms and a fixed-point iterative method to manage high-redshift oscillations while mimicking ΛCDM dynamics.
• The findings provide a rigorous framework for modified gravity models, opening new avenues for testing dark energy alternatives through observations.

Overview of Cosmologically Viable $f(G)$ Gravity Models

The research paper addresses the construction of $f(G)$ gravity models formulated to be cosmologically viable, presenting an alternative to the conventional Einstein gravity to account for the observed cosmic acceleration. The $f(G)$ gravity models modify the gravitational action by introducing a function $f$ that depends on the Gauss-Bonnet invariant $G$. The authors, Antonio De Felice and Shinji Tsujikawa, provide conditions under which these models can offer a stable late-time de-Sitter solution while maintaining consistency with radiation and matter-dominated epochs.

Key Findings

1. Stability Conditions: The paper delineates specific conditions necessary for $f(G)$ gravity models to be cosmologically viable. A critical requirement is $f_{,GG} > 0$, ensuring the stability of cosmological perturbations and the absence of ghost-like behaviors. The stability of the de-Sitter point—a haLLMark of late-time cosmic acceleration—demands that $0 < H_1^6 f_{,GG} < 1/384$.
2. Model Proposals: Several functional forms of $f_{,GG}$ are introduced:

• Model A: $$f_{,GG} = \frac{\lambda}{G_*^{3/2}}\left[ 1 + \left(G^2/G_*^2\right)^n \right]$$
• Model B: $$f_{,GG} = \frac{2\lambda}{G_*^{3/2}}\left(1 + G^2 / G_*^2 \right)^n$$
• Model C: $$f_{,GG} = \frac{\lambda}{G_*^{3/2}} \left[1 - \tanh^2(G/G_*)\right]$$

These models are engineered to closely approximate the $\Lambda$CDM behavior during high curvature (early universe) epochs, with deviations becoming significant at lower curvatures.

1. Numerical Solutions and Constraints: The paper emphasizes solving the cosmological equations numerically to determine the behavior of these models over time. Notably, the mass squared of perturbations, $M^2$, exhibits rapid oscillations at high redshift, requiring careful control to prevent numerical instability.
2. Iterative Solution for High-Redshifts: To address the challenges of solving equations with high-redshift oscillations, a fixed-point iterative method is proposed. This approach enables accurate approximations of the cosmological evolution, akin to the $\Lambda$CDM model in the early universe.

Implications and Future Directions

The paper's exploration of $f(G)$ models contributes significantly to the field of modified gravity theories by illustrating an alternative mechanism for dark energy without requiring an exotic matter component. These models, mimicking the $\Lambda$CDM model in the deep matter era, propose subtle deviations that could potentially be tested with future cosmological observations, such as gravitational lensing and large-scale structure formations.

One of the theoretical limitations highlighted is the reliance on a cosmological constant term, complicating the motivation behind purely geometrical explanations of dark energy. The explicit construction of these models, along with their numerical tractability, offers fertile ground for further investigations into compatibility with local gravity constraints and extensions toward other metrics or higher-dimensional theories.

In summary, the paper by De Felice and Tsujikawa provides a rigorous framework to construct and analyze $f(G)$ gravity models, broadening our understanding of the late-time cosmic acceleration through the lens of modified gravitation. Future developments in observational precision and theoretical advancements will be essential in validating or refining these models as viable alternatives to traditional dark energy paradigms.