Matter density perturbations and effective gravitational constant in modified gravity models of dark energy (0705.1032v4)

Abstract: We derive the equation of matter density perturbations on sub-horizon scales for a general Lagrangian density f(R, phi, X) that is a function of a Ricci scalar R, a scalar field phi and a kinetic term X=-(nabla phi)^2/2. This is useful to constrain modified gravity dark energy models from observations of large-scale structure and weak lensing. We obtain the solutions for the matter perturbation delta_m as well as the gravitational potential Phi for some analytically solvable models. In a f(R) dark energy model with the Lagrangian density f(R)=alpha R^{1+m}-Lambda, the growth rates of perturbations exhibit notable differences from those in the standard Einstein gravity unless m is very close to 0. In scalar-tensor models with the Lagrangian density f=F(phi)R+2p(phi,X) we relate the models with coupled dark energy scenarios in the Einstein frame and reproduce the equations of perturbations known in the current literature by making a conformal transformation. We also estimate the evolution of perturbations in both Jordan and Einstein frames when the energy fraction of dark energy is constant during the matter-dominated epoch.

• The paper derives a general linear matter density perturbation equation for modified gravity models, including f(R) and scalar-tensor theories.
• It reveals that changes in the effective gravitational constant lead to noticeably different growth rates compared to the standard ΛCDM predictions.
• The study refines weak lensing parameters and offers observational criteria to constrain dark energy within modified gravity frameworks.

An Analysis of Matter Density Perturbations in Modified Gravity Models of Dark Energy

This paper by Tsujikawa explores the field of modified gravity models as a theoretical approach to understanding dark energy. Specifically, the paper focuses on the derivation of equations for matter density perturbations under a general Lagrangian framework that encompasses various modified gravity models. By constructing and analyzing perturbation equations in these non-standard frameworks, the research aims to elucidate criteria for distinguishing modified gravity models from the conventional Einstein gravity paradigm, based on observational data.

Key Highlights and Findings

1. General Framework for Perturbation Equations:
   • The paper provides an intricate derivation of the linear matter density perturbation equation on sub-horizon scales for a general Lagrangian density f(R,φ,X), where R denotes the Ricci scalar, φ is a scalar field, and X represents a kinetic term.
   • This model encompasses a wide array of modified gravity theories, including f(R) gravity and scalar-tensor theories, which are often proposed to dynamically account for dark energy.
2. Differences from Standard Model Predictions:
   • In the context of f(R) gravity models, the research reveals that deviation from standard Einstein gravity arises from the modification of the effective gravitational constant. This leads to distinct growth rates for density perturbations compared to the ΛCDM model, especially noticeable when the parameter m deviates from zero.
   • Scalar-tensor models show equivalence to coupled dark energy scenarios after conformal transformations, ensuring the reproduction of well-known perturbation equations in literature.
3. Effective Gravitational Constant and Weak Lensing:
   • The paper introduces an effective gravitational constant, Geff, which assimilates modifications from the f(R, φ, X) gravity framework, predicting varying growth of structure.
   • It also refines parameters η and Σ, which quantify the anisotropic stress and deviation from traditional gravity. These aspects become crucial for cosmic surveys, particularly in the context of weak lensing.
4. Potential Constraints on Modified Gravity:
   • The analysis extends to practical constraints on these models, demonstrating that models where the condition k²a≤Rm holds, remain compliant with local gravity constraints. This results in challenging bounds for such models, especially when cosmological observations and local experiments are considered.
   • Models featuring variable m can transition from a matter-dominated epoch fit to standard cosmology into patterns aligning more with dark energy behavior as the universe evolves.

Implications and Future Work

The research conducted by Tsujikawa opens pathways by which precision cosmology can potentially differentiate between modified gravity models and Einsteinian gravity. These developments are instrumental in enhancing our understanding of the nature of dark energy, guiding the interpretation of observational data from large-scale universe surveys such as those of the Cosmic Microwave Background (CMB), galaxy clustering, and weak gravitational lensing.

The capability to parameterize deviations through η and Σ offers a methodical framework to contrast gravitational theories rigorously against empirical data. Looking forward, future studies might expand this analysis to include the influence of higher-order curvature corrections not considered in this paper. For instance, Gauss-Bonnet terms, prevalent in certain string theories, necessitate further investigation to explore their impact on cosmic evolution and perturbations.

Overall, this paper contributes substantially to the ongoing discourse on modified gravity theories and their implications for dark energy research. It lays the groundwork for robust quantitative analyses that are expected to guide the interpretation of data collated in the next generation of cosmological observations.