# Constraining ƒ(R) Gravity by the Large-Scale Structure

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## Abstract

**:**

## 1. Introduction

## 2. f(R) Gravity

#### 2.1. Chameleon Models

#### 2.2. Analytical f(R) Gravity Models and Yukawa-Like Gravitational Potentials

## 3. Constraining f(R) Gravity Models Using Clusters of Galaxies

Model | ${c}_{\mathbf{500}}$ | ${\mathbf{\alpha}}_{a}$ | ${\mathbf{\beta}}_{a}$ | ${\mathbf{\gamma}}_{a}$ | ${P}_{\mathbf{0}}$ | Reference |
---|---|---|---|---|---|---|

Arnaud et al. 2010 | 1.177 | 1.051 | 5.4905 | 0.3081 | 8.403 ${h}_{70}^{3/2}$ | [114] |

Sayers et al. 2013 | 1.18 | 0.86 | 3.67 | 0.67 | 4.29 | [115] |

Planck et al. 2013 | 1.81 | 1.33 | 4.13 | 0.31 | 6.41 | [116] |

#### 3.1. Pressure Profile from Yukawa-Like Gravitational Potential

#### 3.1.1. Data and Results

**Figure 1.**Comptonization parameter for Comacluster (z = 0.023). The pressure profile is integrated along the line of sight for: the three universal profiles (dashed, solid and dash-dotted lines; the model parameters are quoted in Table 1); β = 2/3 model (long dashed line); and the f(R) model (red solid line, [δ, L, γ] = [-0.98, 0.1, 1.2]).

Parameterization | δ | γ | L | ζ |
---|---|---|---|---|

(A) | $[-0.99,1.0]$ | $[1.0,1.6]$ | - | $[0.1,4]$ |

(B) | $[-0.99,1.0]$ | $[1.0,1.6]$ | $[0.1,20]$ | - |

68% CL | 95% CL | 68% CL | 95% CL | |
---|---|---|---|---|

δ | < $-0.46$ | < $-0.10$ | < $-0.43$ | < $-0.08$ |

γ | > $1.35$ | > $1.12$ | > $1.45$ | > $1.2$ |

$L\left(or\phantom{\rule{0.166667em}{0ex}}\zeta \right)$ | < $2.5$ | < $3.7$ | < 12 | < 19 |

**Figure 2.**2D contours at the 68% (dark green) and 95% (light green) confidence levels of the marginalized likelihoods for Parameterization (A). In panel (

**a**), (

**b**), and (

**c**) there are shown the 2D contours for the parameters (ζ, L), (δ, L), and (δ, ζ), respectively. Since the contours are opened, only upper limits on the parameters can be given.

**Figure 3.**2D contours from the marginalized likelihoods for Parameterization (B). Contours follow the same convention of Figure 2. This parameterization also provides opened contours. Therefore, also in this case, one can only give upper limits on the parameters and can not distinguish which parameterization is the best one.

#### 3.2. Chameleon Gravity: Hydrostatic and Weak Lensing Mass Profile of Galaxy Cluster

#### 3.2.1. Data and Results

**Figure 4.**The confidence contours for the renormalized parameters (ϕ

_{∞,2}, β

_{2}) The 95% and 99% confidence levels are plotted in light gray and medium gray, respectively. The results are from [53]. The 95% and 99% confidence contours from [52] are over-plotted also in red and blue, respectively. The vertical line corresponds to |f,

_{R0}| < 6 × 10

^{−5}.

## 4. N-Body Hydrodynamical Simulations in f(R) Gravity

## 5. Constraining the Expansion History of the Universe in f(R) Gravity

**Figure 5.**The 1σ and 2σ marginalized contours for the parameters γ

_{0}and γ

_{1}in the γ-parameterization. Shown is the reference case (shaded yellow regions), with the optimistic error bars (green long-dashed ellipses) and the pessimistic ones (black dotted ellipses). Red circle represent the ΛCDM model (γ = 0.545), while triangles represent the f(R) model [54].

## 6. Testing Gravity Using the Cosmic Microwave Background Data

**Figure 6.**The $68\%$ and $95\%$ contour plots for the two parameters, $\{{\mathrm{Log}}_{10}\left({\mathrm{B}}_{0}\right),\tau \}$. There is a degeneracy between the two parameters for Planck temperature power spectrum (TT) + BSH (the combination of BAO, SNIa, and ${H}_{0}$ datasets). Adding lensing will break the degeneracy between the two. Here, Planck indicates Planck TT.

## 7. Discussion and Future Perspectives

## Acknowledgments

## Conflicts of Interest

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De Martino, I.; De Laurentis, M.; Capozziello, S.
Constraining ƒ(R) Gravity by the Large-Scale Structure. *Universe* **2015**, *1*, 123-157.
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De Martino I, De Laurentis M, Capozziello S.
Constraining ƒ(R) Gravity by the Large-Scale Structure. *Universe*. 2015; 1(2):123-157.
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