#
Unified Inflation and Late-Time Accelerated Expansion with Exponential and R^{2} Corrections in Modified Gravity

## Abstract

**:**

## 1. Introduction

## 2. Field Equations

## 3. The Models

**Local Gravity Constraints.**

**Model 2.**

## 4. Discussion

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Copeland, E.J.; Sami, M.; Tsujikawa, S. Dynamics of dark energy. Int. J. Mod. Phys. D
**2006**, 15, 753–1936. [Google Scholar] [CrossRef] [Green Version] - Sahni, V. Dark Matter and Dark Energy. Lect. Notes Phys.
**2004**, 653, 141–180. [Google Scholar] - Padmanabhan, T. Cosmological Constant-the Weight of the Vacuum. Phys. Rep.
**2003**, 380, 235–320. [Google Scholar] [CrossRef] [Green Version] - Bamba, K.; Capozziello, S.; Nojiri, S.; Odintsov, S.D. Dark energy cosmology: The equivalent description via different theoretical models and cosmography tests. Astrophys. Space Sci.
**2012**, 342, 155–228. [Google Scholar] [CrossRef] [Green Version] - Dolgov, A.D.; Kawasaki, M. Can modified gravity explain accelerated cosmic expansion? Phys. Lett. B
**2003**, 573, 1–4. [Google Scholar] [CrossRef] [Green Version] - Nojiri, S.; Odintsov, S.D. Introduction to Modified Gravity and Gravitational Alternative for Dark Energy. Int. J. Geom. Meth. Mod. Phys.
**2007**, 4, 115–146. [Google Scholar] [CrossRef] [Green Version] - Amendola, L.; Polarski, D.; Tsujikawa, S. Are f(R) dark energy models cosmologically viable? Phys. Rev. Lett.
**2007**, 98, 131302. [Google Scholar] [CrossRef] [Green Version] - Capozziello, S.; Tsujikawa, S. Solar system and equivalence principle constraints on f(R) gravity by chameleon approach. Phys. Rev. D
**2008**, 77, 107501. [Google Scholar] [CrossRef] [Green Version] - Sotiriou, T.P.; Faraoni, V. f(R) Theories Of Gravity. Rev. Mod. Phys.
**2010**, 82, 451–497. [Google Scholar] [CrossRef] [Green Version] - Tsujikawa, S. Modified gravity models of dark energy. Lect. Notes Phys.
**2010**, 800, 99–145. [Google Scholar] - Nojiri, S.; Odintsov, S.D. Accelerating cosmology in modified gravity: From convenient F(R) or string-inspired theory to bimetric F(R) gravity. Int. J. Geom. Meth. Mod. Phys.
**2014**, 11, 1460006. [Google Scholar] [CrossRef] - Capozziello, S. Curvature Quintessence. Int. J. Mod. Phys. D
**2002**, 11, 483–492. [Google Scholar] [CrossRef] [Green Version] - Capozziello, S.; Carloni, S.; Troisi, A. Quintessence without scalar fields. Recent Res. Dev. Astron. Astrophys.
**2003**, 1, 625. [Google Scholar] - Nojiri, S.; Odintsov, S.D. Where new gravitational physics comes from: M-theory? Phys. Lett. B
**2003**, 576, 5–11. [Google Scholar] [CrossRef] [Green Version] - Nojiri, S.; Odintsov, S.D. Modified gravity with negative and positive powers of the curvature: Unification of the inflation and of the cosmic acceleration. Phys. Rev. D
**2003**, 68, 123512. [Google Scholar] [CrossRef] [Green Version] - Carroll, S.M.; Duvvuri, V.; Trodden, M.; Turner, M.S. Is Cosmic Speed-Up Due to New Gravitational Physics? Phys. Rev. D
**2004**, 70, 043528. [Google Scholar] [CrossRef] [Green Version] - Faraoni, V. The stability of modified gravity models. Phys. Rev. D
**2005**, 72, 124005. [Google Scholar] [CrossRef] [Green Version] - De la Cruz-Dombriz, A.; Dobado, A. A f(R) gravity without cosmological constant. Phys. Rev. D
**2006**, 74, 087501. [Google Scholar] [CrossRef] [Green Version] - Brookfield, A.W.; van de Bruck, C.; Hall, L.M.H. Viability of f (R) Theories with Additional Powers of Curvature. Phys. Rev. D
**2006**, 74, 064028. [Google Scholar] [CrossRef] [Green Version] - Li, B.; Barrow, J.D. The Cosmology of f(R) Gravity in the Metric Variational Approach. Phys. Rev. D
**2007**, 75, 084010. [Google Scholar] [CrossRef] [Green Version] - Abdalla, M.C.B.; Nojiri, S.; Odintsov, S.D. Consistent modified gravity: Dark energy, acceleration and the absence of cosmic doomsday. Class. Quant. Grav.
**2005**, 22, L35. [Google Scholar] [CrossRef] [Green Version] - Cognola, G.; Elizalde, E.; Nojiri, S.; Odintsov, S.D.; Zerbini, S. One-loop f(R) gravity in de Sitter universe. JCAP
**2005**, 502, 010. [Google Scholar] [CrossRef] [Green Version] - Capozziello, S.; Cardone, V.F.; Troisi, A. Reconciling dark energy models with f(R) theories. Phys. Rev. D
**2005**, 71, 043503. [Google Scholar] [CrossRef] - Allemandi, G.; Borowiec, A.; Francaviglia, M.; Odintsov, S.D. Dark Energy Dominance and Cosmic Acceleration in First Order Formalism. Phys. Rev. D
**2005**, 72, 063505. [Google Scholar] [CrossRef] [Green Version] - Koivisto, T.; Kurki-Suonio, H. Cosmological perturbations in the Palatini formulation of modified gravity. Class. Quant. Grav.
**2006**, 23, 2355–2369. [Google Scholar] [CrossRef] - Brevik, I. Crossing of the w = −1 barrier in viscous modified gravity. Int. J. Mod. Phys. D
**2006**, 15, 767–776. [Google Scholar] [CrossRef] [Green Version] - Nojiri, S.; Odintsov, S.D. Modified f(R) gravity consistent with realistic cosmology: From matter dominated epoch to dark energy universe. Phys. Rev. D
**2006**, 74, 086005. [Google Scholar] [CrossRef] [Green Version] - Capozziello, S.; Nojiri, S.; Odintsov, S.D.; Troisi, A. Cosmological viability of f(R)-gravity as an ideal fluid and its compatibility with a matter dominated phase. Phys. Lett. B
**2006**, 639, 135–143. [Google Scholar] [CrossRef] [Green Version] - Nojiri, S.; Odintsov, S.D. Modified gravity as an alternative for Lambda-CDM cosmology. J. Phys. A
**2007**, 40, 6725–6732. [Google Scholar] [CrossRef] - Olmo, G.J. Limit to General Relativity in f(R) theories of gravity. Phys. Rev. D
**2007**, 75, 023511. [Google Scholar] [CrossRef] [Green Version] - Hu, W.; Sawicki, I. Models of f(R) Cosmic Acceleration that Evade Solar-System Tests. Phys. Rev. D
**2007**, 76, 064004. [Google Scholar] [CrossRef] [Green Version] - Starobinsky, A.A. A new type of isotropic cosmological models without singularity. Phys. Lett. B
**1980**, 91, 99–102. [Google Scholar] [CrossRef] - Starobinsky, A.A. Disappearing cosmological constant in f(R) gravity. JETP Lett.
**2007**, 86, 157–163. [Google Scholar] [CrossRef] [Green Version] - Appleby, S.A.; Battye, R.A. Do consistent F(R) models mimic General Relativity plus Λ? Phys. Lett. B
**2007**, 654, 7–12. [Google Scholar] [CrossRef] [Green Version] - Nojiri, S.; Odintsov, S.D. Newton law corrections and instabilities in f(R) gravity with the effective cosmological constant epoch. Phys. Lett. B
**2007**, 652, 343–348. [Google Scholar] [CrossRef] [Green Version] - Cognola, G.; Elizalde, E.; Nojiri, S.; Odintsov, S.D.; Sebastiani, L.; Zerbini, S. Class of viable modified f(R) gravities describing inflation and the onset of accelerated expansion. Phys. Rev. D
**2008**, 77, 046009. [Google Scholar] [CrossRef] [Green Version] - Elizalde, E.; Nojiri, S.; Odintsov, S.D.; Sebastiani, L.; Zerbini, S. Non-singular exponential gravity: A simple theory for early- and late-time accelerated expansion. Phys. Rev. D
**2011**, 83, 086006. [Google Scholar] [CrossRef] [Green Version] - Bamba, K.; Nojiri, S.; Odintsov, S.D. Future of the universe in modified gravitational theories: Approaching to the finite-time future singularity. J. Cosmol. Astropart. Phys.
**2008**, 810, 045. [Google Scholar] [CrossRef] - Barrow, J.D.; Clifton, T. Exact cosmological solutions of scale-invariant gravity theories. Class. Quant. Grav.
**2005**, 23, L1. [Google Scholar] [CrossRef] - Clifton, T.; Barrow, J.D. Further Exact Cosmological Solutions to Higher-Order Gravity Theories. Class. Quant. Grav.
**2006**, 23, 2951. [Google Scholar] [CrossRef] - Capozziello, S.; de Felice, A. f(R) cosmology by Noether’s symmetry. J. Cosmol. Astropart. Phys.
**2008**, 808, 016. [Google Scholar] [CrossRef] [Green Version] - Capozziello, S.; Stabile, A.; Troisi, A. Spherically symmetric solutions in f(R)-gravity via Noether Symmetry Approach. Class. Quant. Grav.
**2007**, 24, 2153–2166. [Google Scholar] [CrossRef] [Green Version] - Capozziello, S.; Stabile, A.; Troisi, A. Spherical symmetry in f(R)-gravity. Class. Quant. Grav.
**2008**, 25, 085004. [Google Scholar] [CrossRef] [Green Version] - Barrow, J.D.; Cotsakis, S. Inflation and the conformal structure of higher-order gravity theories. Phys. Lett. B
**1988**, 214, 515–518. [Google Scholar] [CrossRef] - Nojiri, S.; Odintsov, S.D. Modified Gauss-Bonnet theory as gravitational alternative for dark energy. Phys. Lett. B
**2005**, 631, 1–6. [Google Scholar] [CrossRef] [Green Version] - Nojiri, S.; Odintsov, S.D.; Gorbunova, O.G. Dark energy problem: From phantom theory to modified Gauss-Bonnet gravity. J. Phys. A
**2006**, 39, 6627–6634. [Google Scholar] [CrossRef] - Cognola, G.; Elizalde, E.; Nojiri, S.; Odintsov, S.D.; Zerbini, S. Dark energy in modified Gauss-Bonnet gravity: Late-time acceleration and the hierarchy problem. Phys. Rev. D
**2006**, 73, 084007. [Google Scholar] [CrossRef] [Green Version] - Nojiri, S.; Odintsov, S.D. Unifying inflation with LambdaCDM epoch in modified f(R) gravity consistent with Solar System tests. Phys. Lett. B
**2007**, 657, 238. [Google Scholar] [CrossRef] [Green Version] - Nojiri, S.; Odintsov, S.D. Modified f(R) gravity unifying R
^{m}inflation with ΛCDM epoch. Phys. Rev. D**2008**, 77, 026007. [Google Scholar] [CrossRef] [Green Version] - Odintsov, S.D.; Saez-Chillon, D.; Sharov, G.S. Is exponential gravity a viable description for the whole cosmological history? Eur. Phys. J. C
**2017**, 77, 862. [Google Scholar] [CrossRef] [Green Version] - Odintsov, S.D.; Saez-Chillon, D.; Sharov, G.S. Testing logarithmic corrections on R
^{2}-exponential gravity by observational data. Phys. Rev. D**2019**, 99, 024003. [Google Scholar] [CrossRef] [Green Version] - Odintsov, S.D.; Oikonomou, V.K. Unification of Constant-roll Inflation and Dark Energy with Logarithmic R
^{2}-corrected and Exponential F(R) Gravity. Nucl. Phys. B**2017**, 923, 608–632. [Google Scholar] [CrossRef] - Li, B.; Barrow, J.D.; Mota, D.F. The Cosmology of Modified Gauss-Bonnet Gravity. Phys. Rev. D
**2007**, 76, 044027. [Google Scholar] [CrossRef] [Green Version] - Oikonomou, V.K. Singular Bouncing Cosmology from Gauss-Bonnet Modified Gravity. Phys. Rev. D
**2015**, 92, 124027. [Google Scholar] [CrossRef] [Green Version] - Amendola, L.; Gannouji, R.; Polarski, D.; Tsujikawa, S. Conditions for the cosmological viability of f(R) dark energy models. Phys. Rev. D
**2007**, 75, 083504. [Google Scholar] [CrossRef] [Green Version] - Tsujikawa, S. Observational signatures of f(R) dark energy models that satisfy cosmological and local gravity constraints. Phys. Rev. D
**2008**, 77, 023507. [Google Scholar] [CrossRef] [Green Version] - Rausan, R.; Chaubey, R. Finsler-Randers cosmology in the framework of a particle creation mechanism: A dynamical systems perspective. Eur. Phys. J. Plus
**2020**, 135, 228. [Google Scholar] [CrossRef] - Papagiannopoulos, G.; Basilakos, S.; Paliathanasis, A.; Savvidou, S.; Stavrinos, P.C. Finsler-Randers cosmology: Dynamical analysis and growth of matter perturbations. Class. Quantum Gravity
**2017**, 34, 22. [Google Scholar] [CrossRef] - Granda, L.N. Modified gravity with an exponential function of curvature. arXiv
**2020**, arXiv:2003.09006. [Google Scholar]

**Figure 1.**Trajectories in the $(r,m)$-plane for four different scenarios with ${\lambda}_{2}={10}^{-7}$ and $(\eta ,p)=(0.1,10)$ (solid line),$(0.05,20)$ (dashed),$(0.02,50)$ (dotdashed),$(0.01,100)$ (dotted). In all cases $p\eta =1$, but for smaller $\eta $ and larger p the trajectories become closer to $\mathsf{\Lambda}$CDM. All trajectories connect the matter dominated saddle point ${P}_{M}$ with the late time de Sitter attractor at $r=-2$ with $0<m<1$.

**Figure 2.**The cosmic evolution of the density parameters for matter, radiation and dark energy for the model (31). In this example we take the path of Figure 1 for the parameters $\eta =0.01$, $p=100$ and ${\lambda}_{2}={10}^{-7}$, using the numerical fit for $m\left(r\right)$ given by Equation (64), with initial conditions $x(-5)=0$, $y(-5)=-0.5$, $z(-5)=0.5000134$ and $w(-5)=0.05$. The behavior is compatible with the current cosmic observations on the evolution of density parameters. The obtained current densities are ${\mathsf{\Omega}}_{m}\left(0\right)\simeq 0.3$, ${\mathsf{\Omega}}_{DE}\left(0\right)\simeq 0.7$ and ${\mathsf{\Omega}}_{r}\left(0\right)\simeq {10}^{-4}$.

**Figure 3.**The effective equation of state ${w}_{eff}$ and the equation of state associated with the geometric dark energy ${w}_{DE}$ for the cosmological evolution of the density parameters described in Figure 2. The initial conditions lead to a scenario very close to the $\mathsf{\Lambda}$CDM.

**Figure 4.**The scalar field potential in the Einstein frame for the model (68) for $\eta =0.03$, ${\lambda}_{2}=1$. It can be observed that the potential has the appropriate behavior to develop slow-roll inflation.

© 2020 by the author. 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**

Granda, L.
Unified Inflation and Late-Time Accelerated Expansion with Exponential and *R*^{2} Corrections in Modified Gravity. *Symmetry* **2020**, *12*, 794.
https://doi.org/10.3390/sym12050794

**AMA Style**

Granda L.
Unified Inflation and Late-Time Accelerated Expansion with Exponential and *R*^{2} Corrections in Modified Gravity. *Symmetry*. 2020; 12(5):794.
https://doi.org/10.3390/sym12050794

**Chicago/Turabian Style**

Granda, Luis.
2020. "Unified Inflation and Late-Time Accelerated Expansion with Exponential and *R*^{2} Corrections in Modified Gravity" *Symmetry* 12, no. 5: 794.
https://doi.org/10.3390/sym12050794