[Paper Review] Cosmological Models in Modified f(R) Gravity Theories

Alejandro Guarnizo|arXiv (Cornell University)|Nov 11, 2012

Cosmology and Gravitation Theories93 references3 citations

TL;DR

This master's thesis investigates cosmological models in metric $f(R)$ gravity as an alternative to dark energy for explaining the universe's accelerated expansion. Using the Robertson-Walker metric and dynamical systems analysis, it derives viability criteria for $f(R)$ functions that allow a transition from matter domination to accelerated expansion, and derives general expressions for cosmological distances via the Geodesic Deviation Equation, providing a framework for testing $f(R)$ gravity against observational data.

ABSTRACT

The actual accelerated expansion of the universe continues being a mystery in physics. Some models had been proposed for this explanations, among them the dark energy, which however has problems of experimental character as well as theoretical. Other approximations, like modified gravity theories are an interesting alternative for this problem. Motivated in this approach we study cosmological models in f(R) theories which are natural extension of General Relativity with arbitrary functions of the Ricci scalar. One chapter has dedicated to obtain the modified field equations in the metric formalism of f(R) theories, including the discussion about boundary terms in the action. Later, we apply these equations in order to describe the dynamics of the universe, using for this as space-time, the FLRW universe. We focus our study in the problem of cosmological distances in f(R) theories. From the study of the Geodesic Deviation Equation (GDE) in this modified scenario, we obtain differential equations for the angular diameter distance, and as an extension, the Dyer-Roeder like equation in f(R) gravity.

Motivation & Objective

• To explore $f(R)$ gravity theories as a viable alternative to dark energy in explaining the universe's accelerated expansion.
• To derive the modified field equations in the metric formalism of $f(R)$ gravity, including boundary term considerations.
• To analyze cosmological dynamics using the Robertson-Walker metric and dynamical systems approach to identify viable $f(R)$ functions.
• To develop general expressions for cosmological distances in $f(R)$ gravity using the Geodesic Deviation Equation.
• To establish criteria for $f(R)$ functions that ensure a physical transition from matter-dominated to accelerated expansion epochs.

Proposed method

• Derives the modified Einstein field equations in the metric formalism of $f(R)$ gravity, including careful treatment of boundary terms in the action.
• Applies the field equations to a Robertson-Walker spacetime to obtain the dynamical equations governing the scale factor and Ricci scalar evolution.
• Employs a dynamical systems approach to analyze the phase space of $f(R)$ cosmologies, identifying fixed points corresponding to matter-dominated and accelerated expansion epochs.
• Uses the Geodesic Deviation Equation (GDE) to derive general expressions for cosmological distance measures in $f(R)$ gravity, accounting for curvature and higher-order corrections.
• Performs explicit calculations of curvature and derivative terms involving $f'(R)$, $f''(R)$, and $f'''(R)$ along geodesics to isolate the correction terms to the tidal acceleration.
• Derives the final contribution to the geodesic deviation as proportional to $\epsilon \bigl(5Hf''(R)\dot{R} + f''(R)\ddot{R} + f'''(R)\dot{R}^2\bigr)f'(R)\eta^\alpha$, where $\epsilon = V_\alpha V^\alpha$.

Experimental results

Research questions

• RQ1 What conditions must $f(R)$ functions satisfy to allow a transition from a matter-dominated era to a late-time accelerated expansion in $f(R)$ gravity?
• RQ2 How do the modified field equations in metric $f(R)$ gravity differ from those in General Relativity, particularly regarding boundary terms?
• RQ3 What are the general expressions for cosmological distances in $f(R)$ gravity, derived from the Geodesic Deviation Equation?
• RQ4 How do higher-order derivatives of $f(R)$—$f''(R)$ and $f'''(R)$—affect the tidal forces and geodesic deviation in $f(R)$ cosmologies?
• RQ5 Can the dynamical systems approach identify viable $f(R)$ models that reproduce the observed cosmic expansion history?

Key findings

• The paper derives a general expression for the geodesic deviation in $f(R)$ gravity, showing that the correction to tidal forces is proportional to $\epsilon \bigl(5Hf''(R)\dot{R} + f''(R)\ddot{R} + f'''(R)\dot{R}^2\bigr)f'(R)\eta^\alpha$, where $\epsilon = V_\alpha V^\alpha$.
• The analysis confirms that the geodesic deviation correction is independent of the direction of the deviation vector $\eta^\alpha$, and no cross-terms arise between time and spatial components.
• The dynamical systems approach yields viability criteria for $f(R)$ functions, requiring them to support a transition from matter domination to accelerated expansion, which constrains the behavior of $f'(R)$, $f''(R)$, and $f'''(R)$.
• The derivation shows that the contribution from the $f(R)$-dependent terms in the geodesic deviation equation is fully determined by the Ricci scalar's time evolution and its derivatives along the geodesic.
• The paper establishes that $f(R)$ gravity can reproduce the standard cosmological distance relations in the limit $f(R) = R$, and provides corrections that depend on the functional form of $f(R)$.
• The work provides a systematic framework for computing cosmological observables such as luminosity and angular diameter distances in $f(R)$ gravity, essential for confronting the theory with observational data.