# Hybrid Metric-Palatini Gravity

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## Abstract

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## 1. Introduction

## 2. Hybrid Metric-Palatini Gravity: The General Formalism

#### 2.1. Action and Gravitational Field Equations

#### 2.2. Scalar-Tensor Representation

#### 2.3. The Cauchy Problem

#### 2.4. More General Hybrid Metric-Palatini Theories

#### 2.4.1. Metric $f\left(R\right)$ Models

#### 2.4.2. Palatini $f\left(\mathcal{R}\right)$ Models

#### 2.4.3. Hybrid $f\left(X\right)$ Models

#### 2.4.4. The Hybrid $f(R,\mathcal{R})$ Models

_{+}> 0 and r

_{−}> 0. The second condition would require that

#### 2.4.5. The hybrid Ricci-Squared $f(\mathcal{R},\widehat{Q})$ Theories

## 3. Hybrid-Gravity Cosmology

#### 3.1. Background Expansion

#### 3.1.1. The Friedmann Equations

#### 3.1.2. Dynamical System Analysis

- The matter dominated fixed point should be a saddle point, the de Sitter fixed point an attractor. Then we naturally obtain a transition to acceleration following standard cosmological evolution.
- At the present epoch the field value should be sufficiently close to zero. Then we avoid conflict with the Solar System tests of gravity (this will be clarified in Section 4.1).

#### 3.1.3. On Cosmological Solutions

#### 3.2. Cosmological Perturbations

#### 3.2.1. Field Equations and Conservation Laws

#### 3.2.2. Matter Dominated Cosmology

#### 3.2.3. Vacuum Fluctuations

## 4. Astrophysical Applications

#### 4.1. The Weak Field Limit

#### 4.2. Galactic Phenomenology: Stable Circular Orbits of Test Particles around Galaxies

#### 4.2.1. Galactic Geometry and Tangential Velocity Curves in Hybrid Metric-Palatini Gravity

#### 4.2.2. On Astrophysical Tests of Hybrid Metric-Palatini Gravity at the Galactic Level

#### 4.3. Galactic Clusters: The Generalized Virial Theorem in Hybrid Metric-Palatini Gravity

#### 4.3.1. Galaxy Cluster As a System of Identical and Collisionless Point Particles

#### 4.3.2. The Relativistic Boltzmann Equation

#### 4.3.3. Geometrical Quantities Characterizing Galactic Clusters

#### 4.3.4. The Generalized Virial Theorem in Hybrid Metric-Palatini Gravity

#### 4.3.5. On Astrophysical Tests of Hybrid Metric-Palatini Gravity at the Cluster Level

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Capozziello, S.; Harko, T.; Koivisto, T.S.; Lobo, F.S.N.; Olmo, G.J.
Hybrid Metric-Palatini Gravity. *Universe* **2015**, *1*, 199-238.
https://doi.org/10.3390/universe1020199

**AMA Style**

Capozziello S, Harko T, Koivisto TS, Lobo FSN, Olmo GJ.
Hybrid Metric-Palatini Gravity. *Universe*. 2015; 1(2):199-238.
https://doi.org/10.3390/universe1020199

**Chicago/Turabian Style**

Capozziello, Salvatore, Tiberiu Harko, Tomi S. Koivisto, Francisco S. N. Lobo, and Gonzalo J. Olmo.
2015. "Hybrid Metric-Palatini Gravity" *Universe* 1, no. 2: 199-238.
https://doi.org/10.3390/universe1020199