[Paper Review] Stability of Palatini-f(R) cosmology

Seokcheon Lee|ArXiv.org|Oct 12, 2007

Cosmology and Gravitation Theories3 references3 citations

TL;DR

This paper investigates the stability of linear cosmological perturbations in Palatini f(R) gravity, focusing on the physical frame where matter is minimally coupled. It shows that for $ f(\hat{R}) \sim \hat{R}^n $ with $ n \neq 0,2,3 $, superhorizon metric fluctuations remain stable, and matter density perturbations evolve identically to general relativity at superhorizon scales, while subhorizon dynamics differ, offering a probe for modified gravity.

ABSTRACT

The evolution of linear cosmological perturbations in modified theories of gravity is investigated assuming the Palatini formalism. It has been discussed about the stability problem in this model based on the equivalence between f(R) gravity and the scalar tensor theory. However, we study this problem in the physical frame where the matter is minimally coupled. In general, the stability of the superhorizon metric evolution depends on models. We show that the deviation from the superhorizon metric evolution is null for a specific choice for the nonlinear Einstein-Hilbert action, $f(\hat{R}) \sim \hat{R}^{n}$, where $n eq 0,2,3$. Thus the stability of metric fluctuation is guaranteed in these models. We also study the matter density fluctuation in the general gauge and show the differential equations in super and sub-horizon scales.

Motivation & Objective

• To analyze the stability of linear cosmological perturbations in Palatini f(R) gravity within the physical frame where matter is minimally coupled.
• To resolve inconsistencies in Newtonian limit and stability results from previous scalar-tensor equivalence approaches.
• To investigate the evolution of metric and matter density fluctuations across superhorizon and subhorizon scales.
• To determine conditions under which high-curvature limit stability is preserved, especially for viable f(R) models.
• To provide a framework for testing modified gravity through subhorizon perturbation behavior distinct from general relativity.

Proposed method

• Uses the Palatini formalism, treating metric and connection as independent variables, leading to second-order field equations.
• Derives the modified Einstein equation in the form $ FG_{\mu\nu} = \kappa^2 T_{\mu\nu} - \frac{3}{2} \frac{1}{F} \nabla_\mu F \nabla_\nu F + \nabla_\mu \nabla_\nu F + \cdots $, where $ F = f' $.
• Applies the conformal Newtonian gauge for metric perturbations: $ ds^2 = a^2[ -(1+2\Psi)d\tau^2 + (1-2\Phi)dx^i dx_i ] $.
• Derives perturbed field equations for $ \delta G^\mu_\nu $, $ \delta \hat{R}^\mu_\nu $, and $ \delta \hat{R} $ using the Ricci tensor and scalar curvature relations.
• Solves the differential equations for metric potentials $ \Phi, \Psi $ and density contrast $ \delta $ in both superhorizon and subhorizon limits.
• Analyzes the high-curvature limit to assess stability of metric fluctuations, particularly for power-law models $ f(\hat{R}) \sim \hat{R}^n $.

Experimental results

Research questions

• RQ1 Under what conditions is the superhorizon evolution of metric perturbations stable in Palatini f(R) gravity?
• RQ2 How do matter density fluctuations evolve in Palatini f(R) gravity compared to general relativity across different scales?
• RQ3 What is the behavior of metric and density perturbations in the high-curvature limit for power-law f(R) models?
• RQ4 Can the subhorizon dynamics of perturbations in Palatini f(R) gravity distinguish it from general relativity?
• RQ5 How does the physical frame (matter minimally coupled) affect the stability and evolution of cosmological perturbations in f(R) gravity?

Key findings

• For $ f(\hat{R}) \sim \hat{R}^n $ with $ n \neq 0,2,3 $, the deviation from superhorizon metric evolution is null, guaranteeing stability of metric fluctuations.
• At superhorizon scales, the evolution of matter density fluctuation $ \delta $ and metric potentials $ \Phi, \Psi $ matches that of general relativity.
• Subhorizon evolutions of metric and density perturbations differ significantly from general relativity, providing a potential observational signature for modified gravity.
• The differential equations for $ \Phi, \Psi, \delta $ are derived in general gauge without fixing the gauge, allowing for broader applicability.
• The high-curvature limit stability is preserved only for specific $ f(\hat{R}) $ models, particularly power-law forms with $ n \neq 0,2,3 $, due to cancellation of destabilizing terms.
• The analysis confirms that the physical frame approach avoids inconsistencies in the Newtonian limit and provides a consistent framework for perturbation stability.