[Paper Review] Novel aspects of C-theories in Cosmology

Tomi Koivisto, David F. Mota+1 more|arXiv (Cornell University)|May 21, 2013

Cosmology and Gravitation Theories39 references6 citations

TL;DR

This paper investigates C-theories, a class of modified gravity models where the Ricci scalar is replaced by a conformally scaled version dependent on the curvature scalar itself. It derives field equations in FRW cosmology and finds that linearized perturbations around general relativity are generally unstable or inconsistent, severely limiting the viability of such theories as late-time cosmological models unless they deviate significantly from GR.

ABSTRACT

The field equations in FRW background for the so called C-theories are presented and investigated. In these theories the usual Ricci scalar is substituted with $f(\mathcal{R})$ where $\mathcal{R}$ is a Ricci scalar related to a conformally scaled metric $\hat{g}_{μν} = \mathcal{C}(\mathcal{R})g_{μν}$, where the conformal factor itself depends on $\mathcal{R}$. It is shown that homogeneous perturbations of this Ricci scalar around general relativity FRW background of a large class of these theories are either inconsistent or unstable.

Motivation & Objective

• To investigate the cosmological viability of C-theories, a class of modified gravity models unifying Einstein and Palatini f(R) gravity.
• To derive and reformulate the field equations of C-theories in a Friedmann-Robertson-Walker (FRW) spacetime background.
• To analyze the stability and consistency of linearized perturbations around general relativity in this framework.
• To determine whether C-theories can serve as viable alternatives to ΛCDM in explaining cosmic acceleration.
• To clarify the relationship between C-theories and standard f(R) gravity, especially in the limit of constant conformal factor.

Proposed method

• The theory introduces a conformal factor 𝒞(ℛ) that depends on the Ricci scalar ℛ of a conformally related metric, leading to a generalized action principle.
• Field equations are derived using a variational principle that interpolates between metric and Palatini formalisms.
• The equations are reformulated in FRW geometry to facilitate numerical and linearized analysis.
• Linearized perturbation theory is applied to subclasses of C-theories near general relativity, focusing on scalar modes.
• The stability and consistency of the resulting equations are assessed by examining constraints and dynamical behavior of perturbations.
• The limit of constant 𝒞 is analyzed to recover standard f(R) gravity and compare consistency conditions.

Experimental results

Research questions

• RQ1 Are linearized perturbations in C-theories around a general relativity FRW background stable and consistent?
• RQ2 What constraints do the field equations impose on the matter content in C-theories near GR?
• RQ3 How do the field equations of C-theories differ from those of standard f(R) gravity, especially in the constant-conformal-factor limit?
• RQ4 Can viable cosmological solutions exist in C-theories without violating stability or consistency conditions near GR?
• RQ5 Does the instability of perturbations near GR imply a fundamental flaw in the C-theory framework for late-time cosmology?

Key findings

• Linearized perturbations in a broad class of C-theories around general relativity are found to be unstable or inconsistent, particularly when the conformal factor 𝒞 is not constant.
• The instability arises from constraints that couple the perturbation of the Ricci scalar to the matter content, violating the expected freedom in cosmological evolution.
• When the conformal factor 𝒞 is constant, the theory reduces to standard f(R) gravity, and the inconsistencies vanish, confirming consistency with known f(R) results.
• The zeroth-order equation for the Ricci scalar perturbation δφ shows that δφ must compensate for shifts in f(R), indicating no independent dynamical constraint.
• The first-order perturbation equation for δφ contains terms involving δf′′′/δf′′ and δf′, which can lead to runaway growth or unphysical behavior unless higher-order terms are fine-tuned.
• Numerical studies of cosmological evolution are deemed difficult due to the challenge of finding stable and physically meaningful initial conditions in the viable parameter space.