# Some Cosmological Solutions of a New Nonlocal Gravity Model

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. New Nonlocal Gravity Model

#### 2.1. Equations of Motion

#### 2.2. Ghost-Free Condition

## 3. Cosmological Solutions

#### 3.1. Cosmological Solution $a\left(t\right)=A\phantom{\rule{0.166667em}{0ex}}\sqrt{t}\phantom{\rule{0.166667em}{0ex}}{e}^{\frac{\Lambda}{4}{t}^{2}}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}k=0$

#### 3.2. Cosmological Solution $a\left(t\right)=A\phantom{\rule{0.166667em}{0ex}}{e}^{\Lambda {t}^{2}}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}k=0$

#### 3.3. Other Vacuum Solutions: $R\left(t\right)$ = const

## 4. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Wald, R.M. General Relativity; University of Chicago Press: Chicago, IL, USA, 1984. [Google Scholar]
- Sotiriou, T.P.; Faraoni, V. f(R) theories of gravity. Rev. Mod. Phys.
**2010**, 82, 451–497. [Google Scholar] [CrossRef][Green Version] - Clifton, T.; Ferreira, P.G.; Padilla, A.; Skordis, C. Modified gravity and cosmology. Phys. Rep.
**2012**, 513, 1–189. [Google Scholar] [CrossRef][Green Version] - 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] [CrossRef][Green Version] - Nojiri, S.; Odintsov, S.D.; Oikonomou, V.K. Modified gravity theories on a nutshell: Inflation, bounce and late-time evolution. Phys. Rep.
**2017**, 692, 1–104. [Google Scholar] [CrossRef][Green Version] - Novello, M.; Bergliaffa, S.E.P. Bouncing cosmologies. Phys. Rep.
**2008**, 463, 127–213. [Google Scholar] [CrossRef] - Deser, S.; Woodard, R. Nonlocal cosmology. Phys. Rev. Lett.
**2007**, 99, 111–301. [Google Scholar] [CrossRef][Green Version] - Woodard, R.P. Nonlocal models of cosmic acceleration. Found. Phys.
**2014**, 44, 213–233. [Google Scholar] [CrossRef][Green Version] - Belgacem, E.; Dirian, Y.; Foffa, S.; Maggiore, M. Nonlocal gravity. Conceptual aspects and cosmological predictions. J. Cosmol. Astropart. Phys.
**2018**, 2018, 2. [Google Scholar] [CrossRef][Green Version] - Biswas, T.; Mazumdar, A.; Siegel, W. Bouncing universes in string-inspired gravity. J. Cosmol. Astropart. Phys.
**2006**, 603, 9. [Google Scholar] [CrossRef] - Biswas, T.; Koivisto, T.; Mazumdar, A. Towards a resolution of the cosmological singularity in non-local higher derivative theories of gravity. J. Cosmol. Astropart. Phys.
**2010**, 1011, 8. [Google Scholar] [CrossRef][Green Version] - Biswas, T.; Gerwick, E.; Koivisto, T.; Mazumdar, A. Towards singularity and ghost free theories of gravity. Phys. Rev. Lett.
**2012**, 108, 31–101. [Google Scholar] [CrossRef] [PubMed][Green Version] - Biswas, T.; Koshelev, A.S.; Mazumdar, A.; Vernov, S.Y. Stable bounce and inflation in non-local higher derivative cosmology. J. Cosmol. Astropart. Phys.
**2012**, 8, 24. [Google Scholar] [CrossRef] - Biswas, T.; Conroy, A.; Koshelev, A.S.; Mazumdar, A. Generalized gost-free quadratic curvature gravity. Class. Quantum Grav.
**2014**, 31, 159501. [Google Scholar] [CrossRef][Green Version] - Biswas, T.; Koshelev, A.S.; Mazumdar, A. Consistent higher derivative gravitational theories with stable de Sitter and Anti-de Sitter backgrounds. Phys. Rev.
**2017**, 95, 043533. [Google Scholar] [CrossRef][Green Version] - Dragovich, B. On Nonlocal modified gravity and cosmology. Springer Proc. Math. Stat.
**2014**, 111, 251–262. [Google Scholar] - Koshelev, A.S.; Vernov, S.Y. On bouncing solutions in non-local gravity. Phys. Part. Nuclei
**2012**, 43, 666–668. [Google Scholar] [CrossRef][Green Version] - Koshelev, A.S. Stable analytic bounce in non-local Einstein-Gauss-Bonnet cosmology. Class. Quantum Grav.
**2013**, 30, 155001. [Google Scholar] [CrossRef][Green Version] - Koshelev, A.S.; Kumar, K.S.; Starobinsky, A.A. R
^{2}inflation to probe non-perturbative quantum gravity. JHEP**2018**, 1803, 71. [Google Scholar] [CrossRef][Green Version] - Koshelev, A.S.; Modesto, L.; Rachwal, L.; Starobinsky, A.A. Occurrence of exact R
^{2}inflation in non-local UV-complete gravity. JHEP**2016**, 11, 1–41. [Google Scholar] - Buoninfante, L.; Koshelev, A.S.; Lambiase, G.; Mazumdar, A. Classical properties of non-local, ghost- and singularity-free gravity. JCAP
**2018**, 2018, 34. [Google Scholar] [CrossRef][Green Version] - Koshelev, A.S.; Marto, J.; Mazumdar, A. Towards conformally flat, non-Kasner vacuum solution in infinite derivative gravity. JCAP
**2019**, 2019, 20. [Google Scholar] [CrossRef][Green Version] - Elizalde, E.; Pozdeeva, E.O.; Vernov, S.Y. Stability of de Sitter solutions in non-local cosmological models. arXiv
**2013**, arXiv:1202.0178. [Google Scholar] - Conroy, A.; Koivisto, T.; Mazumdar, A.; Teimouri, A. Generalised quadratic curvature, non-local infrared modifications of gravity and Newtonian potentials. Clas. Quantum Grav.
**2015**, 32, 015024. [Google Scholar] [CrossRef][Green Version] - Dragovich, B.; Khrennikov, A.Y.; Kozyrev, S.V.; Volovich, I.V.; Zelenov, E.I. p-Adic mathematical physics: The first 30 years. p-Adic Numbers Ultrametric Anal. Appl.
**2017**, 9, 87–121. [Google Scholar] [CrossRef] - Modesto, L. Super-renormalizable quantum gravity. Phys. Rev. D
**2012**, 86, 044005. [Google Scholar] [CrossRef][Green Version] - Modesto, L.; Rachwal, L. Super-renormalizable and finite gravitational theories. Nucl. Phys. B
**2014**, 889, 228. [Google Scholar] [CrossRef][Green Version] - Stelle, K.S. Renormalization of higher derivative quantum gravity. Phys. Rev. D
**1977**, 16, 953. [Google Scholar] [CrossRef] - Dimitrijevic, I.; Dragovich, B.; Grujic, J.; Rakic, Z. On modified gravity. Springer Proc. Math. Stat.
**2013**, 36, 251–259. [Google Scholar] - Dimitrijevic, I.; Dragovich, B.; Grujic, J.; Rakic, Z. New cosmological solutions in nonlocal modified gravity. Rom. Journ. Phys.
**2013**, 58, 550–559. [Google Scholar] - Dimitrijevic, I.; Dragovich, B.; Grujic, J.; Rakic, Z. Some power-law cosmological solutions in nonlocal modified gravity. Springer Proc. Math. Stat.
**2014**, 111, 241–250. [Google Scholar] - Dimitrijevic, I.; Dragovich, B.; Grujic, J.; Koshelev, A.S.; Rakic, Z. Cosmology of non-local f(R) gravity. Filomat
**2019**, 33, 1163–1178. [Google Scholar] [CrossRef] - Dimitrijevic, I.; Dragovich, B.; Stankovic, J.; Koshelev, A.S.; Rakic, Z. On nonlocal modified gravity and its cosmological solutions. Springer Proc. Math. Stat.
**2016**, 191, 35–51. [Google Scholar] - Dimitrijevic, I.; Dragovich, B.; Grujic, J.; Rakic, Z. Some cosmological solutions of a nonlocal modified gravity. Filomat
**2015**, 29, 619–628. [Google Scholar] [CrossRef][Green Version] - Dimitrijevic, I. Cosmological solutions in modified gravity with monomial nonlocality. Appl. Math. Comput.
**2016**, 285, 195–203. [Google Scholar] [CrossRef][Green Version] - Dimitrijevic, I.; Dragovich, B.; Rakic, Z.; Stankovic, J. On nonlocal gravity with constant scalar curvature. Publ. De L’Institut Math. Nouv. Série
**2018**, 103, 53–59. [Google Scholar] [CrossRef] - Dimitrijevic, I.; Dragovich, B.; Koshelev, A.S.; Rakic, Z.; Stankovic, J. Cosmological solutions of a nonlocal square root gravity. Phys. Lett. B
**2019**, 797, 134848. [Google Scholar] [CrossRef] - Dimitrijevic, I.; Dragovich, B.; Rakic, Z.; Stankovic, J. Variations of infinite derivative modified gravity. Springer Proc. Math. Stat.
**2018**, 263, 91–111. [Google Scholar] - Aghanim, N.; Akrami, Y.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Ballardini, M. Planck 2018 results. VI. Cosmological parameters, Planck collaboration. arXiv
**2018**, arXiv:1807.06209. [Google Scholar]

© 2020 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**

Dimitrijevic, I.; Dragovich, B.; Koshelev, A.S.; Rakic, Z.; Stankovic, J.
Some Cosmological Solutions of a New Nonlocal Gravity Model. *Symmetry* **2020**, *12*, 917.
https://doi.org/10.3390/sym12060917

**AMA Style**

Dimitrijevic I, Dragovich B, Koshelev AS, Rakic Z, Stankovic J.
Some Cosmological Solutions of a New Nonlocal Gravity Model. *Symmetry*. 2020; 12(6):917.
https://doi.org/10.3390/sym12060917

**Chicago/Turabian Style**

Dimitrijevic, Ivan, Branko Dragovich, Alexey S. Koshelev, Zoran Rakic, and Jelena Stankovic.
2020. "Some Cosmological Solutions of a New Nonlocal Gravity Model" *Symmetry* 12, no. 6: 917.
https://doi.org/10.3390/sym12060917