# Inflationary f (R) Cosmologies

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Matter Description

## 3. Exponential Expansion

#### 3.1. Reconstruction of $f(R)$ for Exponential Expansion Models

#### 3.2. Potential $V(\varphi )$ for Exponential Expansion Models

## 4. Linear Expansion

#### 4.1. Reconstruction of $f(R)$ for Linear Inflation Models

#### 4.2. Potential $V(\varphi )$ for Linear Inflation Models

## 5. Application to Inflation Epoch

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Guth, A.H. Inflationary universe: A possible solution to the horizon and flatness problems. Phys. Rev. D
**1981**, 23, 347–356. [Google Scholar] - Starobinsky, A.A. A new type of isotropic cosmological models without singularity. Phys. Lett. B
**1980**, 91, 99–102. [Google Scholar] - Barrow, J.D.; Paliathanasis, A. Reconstructions of the dark-energy equation of state and the inflationary potential. arXiv, 2016; arXiv:1611.06680. [Google Scholar]
- Nojiri, S.; Odintsov, S.D.; Oikonomou, V.K. Modified Gravity Theories on a Nutshell: Inflation, Bounce and Late-time Evolution. Phys. Rept.
**2017**, 692, 1–104. [Google Scholar] - Nojiri, S.; Odintsov, S.D. Modified gravity with negative and positive powers of curvature: Unification of inflation and cosmic acceleration. Phys. Rev. D
**2003**, 68, 123512. [Google Scholar] - Nojiri, S.; Odintsov, S.D. Unifying inflation with ΛCDM epoch in modified f (R) gravity consistent with Solar System tests. Phys. Lett. B
**2007**, 657, 238–245. [Google Scholar] - Nojiri, S.; Odintsov, S.D. Future evolution and finite-time singularities in f (R) gravity unifying inflation and cosmic acceleration. Phys. Rev. D
**2008**, 78, 046006. [Google Scholar] - Nojiri, S.; Odintsov, S.D. Unified cosmic history in modified gravity: From f (R) theory to Lorentz non-invariant models. Phys. Rep.
**2011**, 505, 59–144. [Google Scholar] - Bamba, K.; Nojiri, S.; Odintsov, S.D.; Sáez-Gómez, D. Inflationary universe from perfect fluid and f (R) gravity and its comparison with observational data. Phys. Rev. D
**2014**, 90, 124061. [Google Scholar] - Amin, M.; Khalil, S.; Salah, M. A viable logarithmic f (R) model for inflation. J. Cosmol. Astropart. Phys.
**2016**, 2016, 043. [Google Scholar] - Li, B.; Barrow, J.D. Cosmology of f (R) gravity in the metric variational approach. Phys. Rev. D
**2007**, 75, 084010. [Google Scholar] [CrossRef] - Chakraborty, S.; SenGupta, S. Solving higher curvature gravity theories. Eur. Phys. J. C
**2016**, 76, 552. [Google Scholar] [CrossRef] - Sáez-Gómez, D. Modified f (R) gravity from scalar-tensor theory and inhomogeneous EoS dark energy. Gen. Relativ. Gravit.
**2009**, 41, 1527–1538. [Google Scholar] [CrossRef] - Nojiri, S.; Odintsov, S.D.; Sáez-Gómez, D. Cosmological reconstruction of realistic modified f (R) gravities. Phys. Lett. B
**2009**, 681, 74–80. [Google Scholar] [CrossRef] - Ntahompagaze, J.; Abebe, A.; Mbonye, M. On f (R) gravity in scalar-tensor theories. Int. J. Geom. Methods Mod. Phys.
**2017**, 14, 1750107. [Google Scholar] [CrossRef] - Sami, H.; Namane, N.; Ntahompagaze, J.; Elmardi, M.; Abebe, A. Reconstructing f (R) Gravity from a Chaplygin Scalar Field in de Sitter Spacetimes. arXiv, 2017; arXiv:1706.07790. [Google Scholar]
- Faraoni, V. De Sitter space and the equivalence between f (R) and scalar-tensor gravity. Phys. Rev. D
**2007**, 75, 067302. [Google Scholar] [CrossRef] - Paliathanasis, A. f (R)-gravity from Killing tensors. Class. Quantum Gravity
**2016**, 33, 075012. [Google Scholar] [CrossRef] - Paliathanasis, A.; Tsamparlis, M.; Basilakos, S. Constraints and analytical solutions of f (R) theories of gravity using Noether symmetries. Phys. Rev. D
**2011**, 84, 123514. [Google Scholar] [CrossRef] - Guth, A.H. Inflation. Proc. Natl. Acad. Sci. USA
**1993**, 90, 4871–4877. [Google Scholar] [CrossRef] [PubMed] - Linde, A. Inflationary cosmology after Planck. In Post-Planck Cosmology: Lecture Notes of the Les Houches Summer School: Volume 100, July 2013; Oxford University Press: Oxford, UK, 2015; Volume 100, pp. 231–303. [Google Scholar]
- Bassett, B.A.; Tsujikawa, S.; Wands, D. Inflation dynamics and reheating. Rev. Mod. Phys.
**2006**, 78, 537–589. [Google Scholar] [CrossRef] [Green Version] - Huang, Q.G. A polynomial f (R) inflation model. J. Cosmol. Astropart. Phys.
**2014**, 2014, 035. [Google Scholar] [CrossRef] - Ellis, G.; Madsen, M. Exact scalar field cosmologies. Class. Quantum Gravity
**1991**, 8, 667–676. [Google Scholar] [CrossRef] - Gorini, V.; Kamenshchik, A.; Moschella, U.; Pasquier, V. The Chaplygin Gas as a Model for Dark Energy. In Proceedings of the Tenth Marcel Grossmann Meeting, Rio de Janeiro, Brazil, 16–18 July 2003; pp. 840–859. [Google Scholar]
- Frolov, A.V. Singularity problem with f (R) models for dark energy. Phys. Rev. Lett.
**2008**, 101, 061103. [Google Scholar] [CrossRef] [PubMed] - Liddle, A.R.; Parsons, P.; Barrow, J.D. Formalizing the slow-roll approximation in inflation. Phys. Rev. D
**1994**, 50, 7222–7232. [Google Scholar] [CrossRef] - Liddle, A.R.; Lyth, D.H. COBE, gravitational waves, inflation and extended inflation. Phys. Lett. B
**1992**, 291, 391–398. [Google Scholar] [CrossRef] - Ade, P.; Aghanim, N.; Arnaud, M.; Arroja, F.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Ballardini, M.; Banday, A.; Barreiro, R.; et al. Planck 2015 results-XX. Constraints on inflation. Astron. Astrophys.
**2016**, 594, A20. [Google Scholar]

**Figure 1.**Plot of $\frac{f(R)}{R}$ versus R from Equation (28) for $M=1$, $w=0.14$, ${\varphi}_{0}=1$ and ${C}_{1}=0$ for the exponential expansion, where the green and blue lines refer to the positive and negative Lagrangian for $f(R)$, while the red line refers to the Lagrangian for general relativity (GR). One can see that the deviations from GR occur at higher-curvature regimes.

**Figure 2.**Plot of $V(\varphi )$ versus $\varphi $ from Equation (34) for $M=1$, $A=0.9$, $w=0.12$ and ${\varphi}_{0}=0.5$.

**Figure 3.**Plot of $\frac{f(R)}{R}$ versus R from Equation (42) for $M=1$, ${C}_{1}=-100$, ${\varphi}_{0}=50$, $k=1$ and $A=0.11$ for linear expansion, where the blue and green lines refer to the positive and negative Lagrangian for $f(R)$, while the red line refers to the Lagrangian for general relativity (GR).

**Figure 4.**(

**a**) Plot of $r(w)$ for $M=1$, $A=15$, ${\varphi}_{0}=1$ and $\varphi =1.2$. For exponential expansion, the red line refers to the Planck data. (

**b**) Plot of ${n}_{s}(w)$ for $M=1$, $A=15$, ${\varphi}_{0}=1$ and $\varphi =1.2$. For exponential expansion, the red and green lines refer to the upper and lower bounds of Planck data, respectively.

**Figure 5.**(

**a**) Plot of $r(A)$ for $M=1$, $w=0.15$, ${\varphi}_{0}=1$ and $\varphi =1.5$. For exponential expansion, the red line refers to the Planck data. (

**b**) Plot of ${n}_{s}(A)$ for $M=1$, $w=0.15$, ${\varphi}_{0}=1$ and $\varphi =1.5$. For exponential expansion, the red and green lines refer to the upper and lower bounds of Planck data, respectively.

**Figure 6.**(

**a**) Plot of $r({\varphi}_{0})$ for $M=1$, $w=0.15$, $A=15$ and $\varphi =1.15$. For exponential expansion, the red line refers to the Planck data. (

**b**) Plot of ${n}_{s}({\varphi}_{0})$ for $M=1$, $w=0.15$, $A=15$ and $\varphi =1.15$. For exponential expansion, the red and green lines refer to the upper and lower bounds of Planck data, respectively.

**Figure 7.**(

**a**) Plot of $r(\varphi )$ for $M=1$, $w=0.15$, $A=15$ and ${\varphi}_{0}=1.15$. For exponential expansion, the red line refers to the Planck data. (

**b**) Plot of ${n}_{s}(\varphi )$ for $M=1$, $w=0.15$, $A=15$ and ${\varphi}_{0}=1.15$. For exponential expansion, the red and green lines refer to the upper and lower bounds of Planck data, respectively.

**Figure 8.**(

**a**) Plot of $r(A)$ for $M=1$, ${\varphi}_{0}=750$ and $\varphi =1.05$. For linear expansion, the red line refers to the Planck data. (

**b**) Plot of ${n}_{s}(A)$ for $M=1$, ${\varphi}_{0}=750$ and $\varphi =1.05$. For linear expansion, the red and green lines refer to the upper and lower bounds of Planck data, respectively.

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sami, H.; Ntahompagaze, J.; Abebe, A.
Inflationary *f* (*R*) Cosmologies. *Universe* **2017**, *3*, 73.
https://doi.org/10.3390/universe3040073

**AMA Style**

Sami H, Ntahompagaze J, Abebe A.
Inflationary *f* (*R*) Cosmologies. *Universe*. 2017; 3(4):73.
https://doi.org/10.3390/universe3040073

**Chicago/Turabian Style**

Sami, Heba, Joseph Ntahompagaze, and Amare Abebe.
2017. "Inflationary *f* (*R*) Cosmologies" *Universe* 3, no. 4: 73.
https://doi.org/10.3390/universe3040073