f(R) Theories Of Gravity (0805.1726v4)

Abstract: Modified gravity theories have received increased attention lately due to combined motivation coming from high-energy physics, cosmology and astrophysics. Among numerous alternatives to Einstein's theory of gravity, theories which include higher order curvature invariants, and specifically the particular class of f(R) theories, have a long history. In the last five years there has been a new stimulus for their study, leading to a number of interesting results. We review here f(R) theories of gravity in an attempt to comprehensively present their most important aspects and cover the largest possible portion of the relevant literature. All known formalisms are presented -- metric, Palatini and metric-affine -- and the following topics are discussed: motivation; actions, field equations and theoretical aspects; equivalence with other theories; cosmological aspects and constraints; viability criteria; astrophysical applications.

• The paper details f(R) gravity’s extension of General Relativity by using a function of the Ricci scalar to address dark energy and dark matter issues.
• It systematically compares metric, Palatini, and metric-affine formalisms, outlining their differential orders and implications for stability.
• The study underscores the equivalence to Brans-Dicke theories, exploring cosmological applications and key constraints like the Dolgov-Kawasaki instability.

Overview of $f(R)$ Theories of Gravity

The paper "$f(R)$ Theories of Gravity" by Thomas P. Sotiriou and Valerio Faraoni offers an extensive review of modified gravity theories, with a particular emphasis on $f(R)$ gravity. This class of theories is a generalization of Einstein's General Relativity (GR), where the gravitational Lagrangian is extended to be a function of the Ricci scalar $R$.

Motivation and Historical Context

The motivation for investigating $f(R)$ theories arises from the limitations of GR, including its non-renormalizability and challenges in explaining cosmological observations like dark energy and dark matter without hypothetical forms of matter or energy. The authors explore the historical context of $f(R)$ theories, tracing their development from early modifications of GR by Weyl and Eddington to contemporary attempts to incorporate quantum corrections and string theoretical insights.

Formalism and Field Equations

The review presents all known formalisms of $f(R)$ gravity, including the metric, Palatini, and metric-affine approaches. The authors meticulously derive the field equations for each formalism and highlight their differences. In the metric formalism, the field equations are of fourth order, due to the higher-order derivatives of the metric tensor introduced by the Ricci scalar. In contrast, the Palatini approach results in second-order equations but with a nontrivial relation between the Ricci scalar and the matter content. The metric-affine formalism is the most general and allows for independent variations with respect to the metric and connection, leading to equations involving both curvature and non-metricity.

Equivalence with Brans-Dicke Theory

The paper explores the equivalence of $f(R)$ theories to Brans-Dicke scalar-tensor theories with specific Brans-Dicke parameters. This equivalence is instrumental in understanding the dynamics and stability properties of $f(R)$ gravity, providing insights into its scalar degree of freedom, which can be seen as an effective scalar field.

Cosmological and Astrophysical Applications

$f(R)$ theories have significant implications for cosmology and astrophysics. The paper reviews how these theories can explain the accelerated expansion of the universe without invoking dark energy and discusses their consequences for the evolution of cosmological perturbations. It also examines how $f(R)$ gravity might account for galactic rotation curves without dark matter, although this application faces numerous challenges and constraints from Solar System tests.

Stability and Viability Criteria

The authors analyze various stability conditions essential for the viability of $f(R)$ models. They address the Dolgov-Kawasaki instability in the metric formalism, which imposes constraints on the functional form of $f(R)$. For de Sitter solutions to be stable, conditions related to the second derivative of the function at de Sitter points must be satisfied. The absence of ghost fields and the well-posedness of the Cauchy problem are also pivotal for these theories to be physically acceptable.

Problems with Palatini $f(R)$ Gravity

A significant portion of the discussion is devoted to the shortcomings of Palatini $f(R)$ gravity. While it simplifies some mathematical aspects, the authors reveal several conceptual and physical issues, such as the appearance of surface singularities in polytropic stars and incompatibilities with the Standard Model of particle physics due to non-perturbative corrections.

Conclusions and Future Directions

The paper concludes by recognizing $f(R)$ gravity as a valuable theoretical framework that provides a deeper understanding of gravitational interactions beyond standard GR. However, despite its potential, several $f(R)$ models face serious challenges in satisfying empirical data and theoretical consistency. The authors suggest that further extensions or modifications might be necessary to reconcile $f(R)$ gravity with observations and theoretical principles. Future research may consider coupling $f(R)$ gravity with other fields or exploring entirely new gravitational frameworks, possibly inspired by alternative interpretations of higher-order gravity theories.