# Bouncing Cosmology in Modified Gravity with Higher-Order Gauss–Bonnet Curvature Term

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. $\mathit{F}(\mathit{R},\mathcal{G})$ Gravity and Field Equations

## 3. Bouncing Model I

## 4. Bouncing Model II

## 5. Scalar Perturbations

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

- Riess, A.G.; Filippenko, A.V.; Clocchiatti, A.; Diercks, A.; Garnavich, P.M.; Gilliland, R.L.; Hogan, C.J.; Jha, S.; Kirshner, R.P.; Leibundgut, B.; et al. Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant. Astron. J.
**1998**, 116, 1009. [Google Scholar] [CrossRef] [Green Version] - Perlmutter, S.; Aldering, G.; Goldhaber, G.; Knop, R.A.; Nugent, P.; Castro, P.G.; Deustua, S.; Fabbro, S.; Goobar, A.; Groom, D.E.; et al. Measurements of Ω and Λ from 42 High-Redshift Supernovae. Astrophys. J.
**1999**, 517, 565. [Google Scholar] [CrossRef] - Bennett, C.L.; Hill, R.S.; Hinshaw, G.; Nolta, M.R.; Odegard, N.; Page, L.; Spergel, D.N.; Weiland, J.L.; Wright, E.L.; Halpern, M.; et al. First-Year Wilkinson Microwave Anisotropy Probe (WMAP)* Observations: Foreground Emission. Astrophys. J. Suppl. Ser.
**2003**, 148, 1. [Google Scholar] [CrossRef] [Green Version] - Spergel, D.N.; Verde, L.; Peiris, H.V.; Komatsu, E.; Nolta, M.R.; Bennett, C.L.; Halpern, M.; Hinshaw, G.; Jarosik, N.; Kogut, A.; et al. First-Year Wilkinson Microwave Anisotropy Probe (WMAP)* Observations: Determination of Cosmological Parameters. Astrophys. J.
**2003**, 148, 175. [Google Scholar] [CrossRef] [Green Version] - Spergel, D.N.; Bean, R.; Doré, O.; Nolta, M.R.; Bennett, C.L.; Dunkley, J.; Hinshaw, G.; Jarosik, N.; Komatsu, E.; Page, L.; et al. Three-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Implications for Cosmology. Astrophys. J.
**2007**, 170, 377. [Google Scholar] [CrossRef] [Green Version] - Percival, W.J.; Reid, B.A.; Eisenstein, D.J.; Bahcall, N.A.; Budavari, T.; Frieman, J.A.; Fukugita, M.; Gunn, J.E.; Ivezić, Z.; Knapp, G.R.; et al. Baryon acoustic oscillations in the Sloan Digital Sky Survey Data Release 7 galaxy sample. Mon. Not. R. Astron. Soc.
**2010**, 401, 2148. [Google Scholar] [CrossRef] [Green Version] - Ade, P.A.R.; Aikin, R.W.; Barkats, D.; Benton, S.J.; Bischoff, C.A.; Bock, J.J.; Brevik, J.A.; Buder, I.; Bullock, E.; Dowell, C.D.; et al. Detection of B-Mode Polarization at Degree Angular Scales by BICEP2. Phys. Rev. Lett.
**2014**, 112, 241101. [Google Scholar] [CrossRef] [Green Version] - 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] [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] - Barragán, C.; Olmo, G.J.; Sanchis-Alepuz, H.; Olmo, G.J.; Sanchis-Alepuz, H. Bouncing cosmologies in Palatini f(R) gravity. Phys. Rev. D
**2019**, 80, 024016. [Google Scholar] [CrossRef] - Harko, T.; Francisco Lobo, S.N.; Nojiri, S.; Odintsov, S.D. f(R,T) gravity. Phys. Rev. D
**2011**, 84, 024020. [Google Scholar] [CrossRef] [Green Version] - Cai, Y.F.; Chen, S.H.; Dent, J.B.; Dutta, S.; Saridakis, E.N. Matter bounce cosmology with the f(T) gravity. Class. Quant. Grav.
**2011**, 28, 215011. [Google Scholar] [CrossRef] [Green Version] - Abedi, H.; Capozziello, S.; D’Agostino, R.; Luongo, O. Effective gravitational coupling in modified teleparallel theories. Phys. Rev. D
**2018**, 97, 084008. [Google Scholar] [CrossRef] [Green Version] - Xu, Y.; Li, G.; Harko, T.; Liang, S.D. f(Q,T) gravity. Eur. Phys. J. C
**2019**, 79, 708. [Google Scholar] [CrossRef] [Green Version] - Nojiri, S.; Odintsov, S.D. Modified Gauss–Bonnet theory as gravitational alternative for dark energy. Phys. Lett. B
**2005**, 631, 6. [Google Scholar] [CrossRef] [Green Version] - Cai, Y.F.; Qiu, T.; Zhang, X.; Piao, Y.S.; Li, M. Bouncing universe with Quintom matter. J. High Energy Phys.
**2007**, 10, 071. [Google Scholar] [CrossRef] [Green Version] - Peter, P.; Pinto-Neto, N. Primordial perturbations in a nonsingular bouncing universe model. Phys. Rev. D
**2002**, 66, 063509. [Google Scholar] [CrossRef] [Green Version] - Lin, C.; Brandenberger, R.H.; Perreault, L.L. A matter bounce by means of ghost condensation. J. Cosmol. Astropart. Phys.
**2011**, 04, 019. [Google Scholar] [CrossRef] [Green Version] - Fabris, J.C.; Furtado, R.G.; Peter, P.; Pinto-Neto, N. Regular cosmological bouncing solutions in low energy effective action from string theories. Phys. Rev. D
**2003**, 67, 124003. [Google Scholar] [CrossRef] [Green Version] - Qiu, T.; Evslin, J.; Cai, Y.F.; Li, M.; Zhang, X. Bouncing Galileon cosmologies. J. Cosmol. Astropart. Phys.
**2011**, 10, 036. [Google Scholar] [CrossRef] - Kounnas, C.; Partouche, H.; Toumbas, N. Thermal duality and non-singular cosmology in d-dimensional superstrings. Nucl. Phys. B
**2012**, 855, 280. [Google Scholar] [CrossRef] [Green Version] - Novello, M.; Perez Bergliaffa, S.E. Bouncing cosmologies. Phys. Rep.
**2008**, 463, 127. [Google Scholar] [CrossRef] - Elizalde, E.; Haro, J.; Odintsov, S.D. Quasimatter domination parameters in bouncing cosmologies. Phys. Rev. D
**2015**, 90, 063522. [Google Scholar] [CrossRef] [Green Version] - Peebles, P.J.E.; Yu, J.T. Primeval Adiabatic Perturbation in an Expanding Universe. Astrophys. J.
**1970**, 162, 815. [Google Scholar] [CrossRef] - Sunyaev, R.A.; Zeldovich, Y.B. Small-scale fluctuations of relic radiation. Astrophys. Space Sci.
**1970**, 7, 3. [Google Scholar] [CrossRef] - Battefeld, D.; Peter, P. A critical review of classical bouncing cosmologies. Phys. Rep.
**2015**, 571, 1. [Google Scholar] [CrossRef] [Green Version] - Barrau, A.; Bolliet, B.; Schutten, M.; Vidotto, F. Bouncing black holes in quantum gravity and the Fermi gamma-ray excess. Phys. Lett. B
**2017**, 772, 58. [Google Scholar] [CrossRef] - Bojowald, M. Absence of a Singularity in Loop Quantum Cosmology. Phys. Rev. Lett.
**2011**, 86, 5227. [Google Scholar] [CrossRef] [Green Version] - Odintsov, S.D.; Oikonomou, V.K. Matter Bounce Loop Quantum Cosmology from F(R) Gravity. Phys. Rev. D
**2014**, 90, 124083. [Google Scholar] [CrossRef] [Green Version] - Khoury, J.; Ovrut, B.A.; Steinhardt, P.J.; Turok, N. Ekpyrotic universe: Colliding branes and the origin of the hot big bang. Phys. Rev. D
**2001**, 64, 123522. [Google Scholar] [CrossRef] - Gasperini, M.; Giovannini, M.; Veneziano, G. Perturbations in a non-singular bouncing Universe. Phys. Lett. B
**2003**, 569, 113. [Google Scholar] [CrossRef] [Green Version] - Nojiri, S.; Saridakis, E.N. Phantom without ghost. Astrophys. Space Sci.
**2003**, 347, 221. [Google Scholar] [CrossRef] [Green Version] - Saridakis, E.N. Cyclic universes from general collisionless braneworld models. Nucl. Phys. B
**2009**, 808, 224. [Google Scholar] [CrossRef] [Green Version] - Brandenberger, R. Matter bounce in Hořava-Lifshitz cosmology. Phys. Rev. D
**2009**, 80, 043516. [Google Scholar] [CrossRef] [Green Version] - Cai, Y.F.; Saridakis, E.N. Non-singular cosmology in a model of non-relativistic gravity. J. Cosmol. Astropart. Phys.
**2009**, 10, 020. [Google Scholar] [CrossRef] - Agrawal, A.S.; Pati, L.; Tripathy, S.K.; Mishra, B. Matter bounce scenario and the dynamical aspects in f(Q,T) gravity. Phys. Dark Universe
**2021**, 33, 100863. [Google Scholar] [CrossRef] - Cai, Y.F.; Duplessis, F.; Easson, D.A.; Wang, D.G. Searching for a matter bounce cosmology with low redshift observations. Phys. Rev. D
**2016**, 93, 043546. [Google Scholar] [CrossRef] [Green Version] - Shabani, H.; Ziaie, A.H. Bouncing cosmological solutions from f(R,T) gravity. Eur. Phys. J. C
**2018**, 78, 397. [Google Scholar] [CrossRef] [Green Version] - Mishra, B.; Ribeiro, G.; Moraes, P.H.R.S. De Sitter and bounce solutions from anisotropy in extended gravity cosmology. Mod. Phys. Lett. A
**2019**, 34, 1950321. [Google Scholar] [CrossRef] - Tripathy, S.K.; Mishra, B.; Ray, S.; Sengupta, R. Bouncing universe models in an extended gravity theory. Chin. J. Phys.
**2021**, 71, 610. [Google Scholar] [CrossRef] - Agrawal, A.S.; Tripathy, S.K.; Pal, S.; Mishra, B. Role of extended gravity theory in matter bounce dynamics. Phys. Scr.
**2022**, 97, 025002. [Google Scholar] [CrossRef] - Agrawal, A.S.; Tello-Ortiz, F.; Mishra, B.; Tripathy, S.K. Bouncing Cosmology in Extended Gravity and Its Reconstruction as Dark Energy Model. Fortschr. Phys.
**2022**, 70, 2100065. [Google Scholar] [CrossRef] - Amani, A.R. The bouncing cosmology with F(R) gravity and its reconstructing. Int. J. Mod. Phys. D
**2016**, 25, 1650071. [Google Scholar] [CrossRef] [Green Version] - Nojiri, S.; Odintsov, S.D.; Oikonomou, V.K. Bounce universe history from unimodular F(R) gravity. Phys. Rev. D
**2016**, 93, 084050. [Google Scholar] [CrossRef] [Green Version] - Ilyas, M.; Rahman, W.U. Bounce cosmology in f(R) gravity. Eur. Phys. J. C
**2021**, 81, 160. [Google Scholar] [CrossRef] - Amoros, J.; de Haro, J.; Odintsov, S.D. Bouncing loop quantum cosmology from F(T) gravity. Phys. Rev. D
**2013**, 87, 104037. [Google Scholar] [CrossRef] [Green Version] - Caruana, M.; Farrugia, G.; Levi Said, J. Cosmological bouncing solutions in f(T,B) gravity. Eur. Phys. J. C
**2020**, 80, 640. [Google Scholar] [CrossRef] - Odintsov, S.D.; Oikonomou, V.K.; Paul, T. Bottom-up reconstruction of non-singular bounce in F(R) gravity from observational indices. Nucl. Phys. B
**2020**, 959, 115159. [Google Scholar] [CrossRef] - Karimzadeh, S.; Shojaee, R. Phantom-Like Behavior in Modified Teleparallel Gravity. Adv. High Energy Phys.
**2019**, 8, 4026856. [Google Scholar] [CrossRef] [Green Version] - Duchaniya, L.K.; Lohakare, S.V.; Mishra, B.; Tripathy, S.K. Dynamical stability analysis of accelerating f(T) gravity models. Eur. Phys. J. C
**2022**, 82, 448. [Google Scholar] [CrossRef] - Barrow, J.D.; Cotsakis, S. Inflation and the conformal structure of higher-order gravity theories. Phys. Lett. B
**1988**, 214, 515. [Google Scholar] [CrossRef] - Capozziello, S.; Cardone, V.F.; Carloni, S.; Troisi, A. Can higher order curvature theories explain rotation curves of galaxies? Phys. Lett. A
**2004**, 326, 292. [Google Scholar] [CrossRef] [Green Version] - Elizalde, E.; Myrzakulov, R.; Obukhov, V.V.; Sáez-Gómez, D. ΛCDM epoch reconstruction from F(R,G) and modified Gauss–Bonnet gravities. Class. Quantum Grav.
**2010**, 27, 095007. [Google Scholar] [CrossRef] [Green Version] - De la Cruz-Dombriz, A.; Saez-Gomez, D. On the stability of the cosmological solutions in f(R,G) gravity. Class. Quantum Grav.
**2012**, 29, 245014. [Google Scholar] [CrossRef] [Green Version] - 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] - De Felice, A.; Suyama, T. Vacuum structure for scalar cosmological perturbations in modified gravity models. J. Cosmol. Astropart. Phys.
**2009**, 0906, 034. [Google Scholar] [CrossRef] - De Felice, A.; Gerard, J.M.; Suyama, T. Cosmological perturbation in f(R,G) theories with a perfect fluid. Phys. Rev. D
**2010**, 82, 063526. [Google Scholar] [CrossRef] [Green Version] - De Felice, A.; Suyama, T.; Tanaka, T. Stability of Schwarzschild-like solutions in f(R,G) gravity models. Phys. Rev. D
**2011**, 83, 104035. [Google Scholar] [CrossRef] [Green Version] - Makarenko, A.N.; Obukhov, V.V.; Kirnos, I.V. From Big to Little Rip in modified F(R,G) gravity. Astrophys. Space Sci.
**2013**, 343, 481. [Google Scholar] [CrossRef] [Green Version] - De Laurentis, M.; Paolella, M.; Capozziello, S. Cosmological inflation in F(R,G) gravity. Phys. Rev. D
**2015**, 91, 083531. [Google Scholar] [CrossRef] - De Martino, I.; de Laurentis, M.; Capozziello, S. Tracing the cosmic history by Gauss–Bonnet gravity. Phys. Rev. D
**2020**, 102, 063508. [Google Scholar] [CrossRef] - Haro, J. Future singularity avoidance in phantom dark energy models. J. Cosmol. Astropart. Phys.
**2012**, 07, 007. [Google Scholar] [CrossRef] - Haro, J.; Amorós, J. Viability of the matter bounce scenario in F(T) gravity and Loop Quantum Cosmology for general potentials. J. Cosmol. Astropart. Phys.
**2014**, 12, 031. [Google Scholar] [CrossRef] [Green Version] - Cai, Y.F.; Saridakis, E.N.; Setare, M.R.; Xia, J.Q. Quintom cosmology: Theoretical implications and observations. Phys. Rep.
**2010**, 493, 1. [Google Scholar] [CrossRef] [Green Version] - Caldwell, R. A phantom menace? Cosmological consequences of a dark energy component with super-negative equation of state. Phys. Lett. B
**2002**, 545, 23. [Google Scholar] [CrossRef] [Green Version] - Steinhardt, P.; Wang, L.; Zlatev, I. Cosmological tracking solutions. Phys. Rev. D
**1999**, 59, 123504. [Google Scholar] [CrossRef] [Green Version] - Wang, B.; Gong, Y.; Abdalla, E. Thermodynamics of an accelerated expanding universe. Phys. Rev. D
**2006**, 74, 083520. [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. [Google Scholar] [CrossRef] [Green Version] - Capozziello, S.; de Laurentis, M. Extended Theories of Gravity. Phys. Rep.
**2011**, 509, 167. [Google Scholar] [CrossRef] [Green Version] - Sahni, V.; Saini, T.D.; Starobinsky, A.A.; Alam, U. Statefinder—A new geometrical diagnostic of dark energy. JETP Lett.
**2003**, 77, 201. [Google Scholar] [CrossRef] - Alam, U.; Sahni, V.; Saini, T.D.; Starobinsky, A.A. Exploring the expanding Universe and dark energy using the statefinder diagnostic. Mon. Not. R. Astron. Soc.
**2003**, 344, 1057. [Google Scholar] [CrossRef] - Abdussattar; Prajapati, S.R. Role of deceleration parameter and interacting dark energy in singularity avoidance. Astrophys. Space Sci.
**2011**, 331, 657. [Google Scholar] [CrossRef]

**Figure 1.**Behavior of Hubble parameter (

**left panel**) and deceleration parameter (

**right panel**) versus cosmic time t.

**Figure 2.**Behavior of energy density (

**left panel**) and EoS parameter (

**right panel**) versus cosmic time t for $\alpha =-0.30,\beta =0.15$.

**Figure 3.**Behavior of Gauss–Bonnet invariant (

**left panel**) and energy conditions (

**right panel**) versus cosmic time t. The parameters values are $\alpha =-0.30$, $\beta =0.15$.

**Figure 4.**Symmetric behavior of the jerk parameter (

**above left panel**) and snap parameter (

**above right panel**) versus cosmic time t. The jerk (

**below left panel**) and snap (

**below right panel**) parameter in positive time domain.

**Figure 5.**Behavior of the Hubble parameter (

**left panel**) and deceleration parameter (

**right panel**) versus cosmic time t with $\lambda =1.01$.

**Figure 6.**Behavior of energy density (

**left panel**) and EoS parameter (

**right panel**) versus cosmic time t, $\lambda =1.01$.

**Figure 7.**Behavior of Gauss–Bonnet invariant (

**left panel**) and energy conditions (

**right panel**) versus cosmic time t, $\alpha =-0.30$, $\beta =0.15$ and $\lambda =1.01$.

**Figure 8.**Symmetric behavior of the jerk parameter (

**above left panel**) and snap parameter (

**above right panel**) versus cosmic time t. The jerk (

**below left panel**) and snap (

**below right panel**) parameter in positive time domain with $\lambda =1.01$.

**Figure 9.**$\delta \left(t\right)$ (

**left panel**) and ${\delta}_{m}\left(t\right)$ (

**right panel**) versus cosmic time for Model I. The parameter scheme: $\alpha =-0.30,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\beta =0.15,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\chi =0.15,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\rho}_{0}=1.2$.

**Figure 10.**$\delta \left(t\right)$ (

**left panel**) and ${\delta}_{m}\left(t\right)$ (

**right panel**) versus cosmic time for Model II. The parameter scheme: $\alpha =-0.30,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\beta =0.15,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\lambda =1.01,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\rho}_{0}=1.2$.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lohakare, S.V.; Tello-Ortiz, F.; Tripathy, S.K.; Mishra, B.
Bouncing Cosmology in Modified Gravity with Higher-Order Gauss–Bonnet Curvature Term. *Universe* **2022**, *8*, 636.
https://doi.org/10.3390/universe8120636

**AMA Style**

Lohakare SV, Tello-Ortiz F, Tripathy SK, Mishra B.
Bouncing Cosmology in Modified Gravity with Higher-Order Gauss–Bonnet Curvature Term. *Universe*. 2022; 8(12):636.
https://doi.org/10.3390/universe8120636

**Chicago/Turabian Style**

Lohakare, Santosh V., Francisco Tello-Ortiz, S. K. Tripathy, and B. Mishra.
2022. "Bouncing Cosmology in Modified Gravity with Higher-Order Gauss–Bonnet Curvature Term" *Universe* 8, no. 12: 636.
https://doi.org/10.3390/universe8120636