Extended Theories of Gravity (1108.6266v2)

Abstract: Extended Theories of Gravity can be considered a new paradigm to cure shortcomings of General Relativity at infrared and ultraviolet scales. They are an approach that, by preserving the undoubtedly positive results of Einstein's Theory, is aimed to address conceptual and experimental problems recently emerged in Astrophysics, Cosmology and High Energy Physics. In particular, the goal is to encompass, in a self-consistent scheme, problems like Inflation, Dark Energy, Dark Matter, Large Scale Structure and, first of all, to give at least an effective description of Quantum Gravity. We review the basic principles that any gravitational theory has to follow. The geometrical interpretation is discussed in a broad perspective in order to highlight the basic assumptions of General Relativity and its possible extensions in the general framework of gauge theories. Principles of such modifications are presented, focusing on specific classes of theories like f (R)-gravity and scalar-tensor gravity in the metric and Palatini approaches. The special role of torsion is also discussed. The conceptual features of these theories are fully explored and attention is payed to the issues of dynamical and conformal equivalence between them considering also the initial value problem. A number of viability criteria are presented considering the post-Newtonian and the post-Minkowskian limits. In particular, we discuss the problems of neutrino oscillations and gravitational waves in Extended Gravity. Finally, future perspectives of Extended Gravity are considered with possibility to go beyond a trial and error approach.

• The paper introduces extended gravity models using higher-order curvature and scalar-tensor theories to explain cosmic acceleration and galactic dynamics.
• The paper employs both metric and Palatini formalisms to derive modified field equations that address issues like singularities and dark matter phenomena.
• The paper connects theoretical predictions with observational tests, highlighting its potential to refine our understanding of inflation and strong-field gravity.

Overview of "Extended Theories of Gravity"

The paper "Extended Theories of Gravity" explores a comprehensive exploration of gravitational theories that extend Einstein's General Relativity (GR). The aim of these extensions is to address some of the unresolved issues in astrophysics and cosmology, such as dark energy, dark matter, and quantum gravity, without discarding the successful aspects of GR at solar system scales.

Core Concepts and Theoretical Foundations

1. Motivations for Extended Theories:
   • Infrared and Ultraviolet Corrections: GR is successful at cosmological scales but faces challenges at extreme large-scale (infrared) and small-scale (ultraviolet) phenomena such as galactic rotation curves, cosmic acceleration, and the singularity problem in black holes.
   • Quantum Gravity and Unification: Extended theories aim to incorporate features from quantum field theory (QFT) and general gauge theories to potentially unify gravity with other fundamental forces.
2. Forms of Extensions:
   • Higher-Order Gravity: Theories such as $f(R)$-gravity introduce higher-order curvature invariants to the Einstein-Hilbert action, providing a richer dynamical behavior.
   • Scalar-Tensor Theories: Inspired by the Brans-Dicke theory, these incorporate scalar fields coupled to the metric, offering alternative explanations for cosmic acceleration and varying gravitational 'constants'.
   • Non-Minimal Couplings: In these models, matter fields couple directly to curvature, affecting the geometry dynamically and potentially explaining dark energy enigmas.
3. Palatini Formalism:
   • Unlike the metric approach where the metric tensor alone is the dynamical variable, the Palatini formalism treats the metric and connection independently. This approach provides a bi-metric structure and can align more naturally with gauge theory perspectives. It results in field equations differing from the metric approach, potentially offering alternative gravitational dynamics and resolving curvature singularities.

Numerical Results and Predictions

The paper reviews theories like $f(R)$-gravity and scalar-tensor theories, emphasizing their ability to provide effective models for inflation and possibly explain the late-time accelerated expansion of the Universe. These models can lead to distinct cosmological predictions such as:

• Inflation and Cosmic Acceleration: Certain forms of $f(R)$ and non-minimally coupled scalar theories naturally predict inflationary behavior and match observable parameters of the large-scale structure of the Universe.
• Modified Gravity at Galactic and Solar Scales: These theories attempt to modify the Newtonian potential, potentially addressing the flat rotation curves of galaxies without invoking dark matter.

Implications and Future Directions

1. Theoretical Implications:
   • Conformal Transformations: Extended theories often involve transformations that redefine the physical frame irrelevant of differing equations of motion in varying frames, raising questions about which frame corresponds to 'physical reality.'
   • Reduction to Scalar-Tensor Forms: The ability to recast extended theories into scalar-tensor frames simplifies their paper, connecting them to more familiar physics and facilitating numerical solutions.
2. Experimental and Observational Prospects:
   • The search for deviations from GR predictions in the strong field regime (e.g., near black holes) and through gravitational wave observations offers potential avenues to test these theories.
   • The discrepancy in cosmological scales provided by these theories may inspire new designs for infrastructural experiments and observational strategies.
3. Challenges:
   • Quantum and Semiclassical Integrations: Integrating these theories with quantum mechanics remains a challenge, crucial for a complete quantum gravity theory.
   • Assessment and Viability Tests: Establishing viability criteria based on experimental results at different scales (e.g., solar system tests vs. cosmic scale behaviors) is ongoing.

Conclusion

Extended Theories of Gravity represent a vibrant field of research, striving to expand the limitations of GR. They hold the potential to bridge gaps between astronomical observations and fundamental physics principles. The work featured in this paper constitutes an essential step towards resolving cosmological puzzles and pursuing a coherent unification of gravitational and quantum realms. Future work seeks to refine these theories to produce models with stronger predictive power, ushering in an era of novel gravitational insights.