# Viability of Baryon to Entropy Ratio in Modified Hořava–Lifshitz Gravity

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Modified Hořava-Lifshitz Gravity

## 3. Gravitational Baryogenesis

## 4. Model: I

## 5. Model: II

## 6. Conclusions and Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- 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. Suppl.
**2003**, 148, 175. [Google Scholar] [CrossRef] [Green Version] - Burles, S.; Nollett, K.M.; Turner, M.S. What is the big-bang-nucleosynthesis prediction for the baryon density and how reliable is it? Phys. Rev. D
**2001**, 63, 063512. [Google Scholar] [CrossRef] - Davoudiasl, H.; Kitano, R.; Kribs, G.D.; Murayama, H.; Steinhardt, P.J. Gravitational baryogenesis. Phys. Rev. Lett.
**2004**, 93, 201301. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Majumdar, A.S.; Gupta, P.D.; Saxena, R.P. Baryogenesis from black hole evaporation. Int. J. Mod. Phys. D
**1995**, 4, 517. [Google Scholar] [CrossRef] - Stewart, E.D.; Kawasaki, M.; Yanagida, T. Affleck-Dine baryogenesis after thermal inflation. Phys. Rev. D
**1996**, 54, 6032. [Google Scholar] [CrossRef] [Green Version] - Kolb, E.W.; Linde, A.; Riotto, A. Grand-unified-theory baryogenesis after preheating. Phys. Rev. Lett.
**1996**, 77, 4290. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Trodden, M. Electroweak baryogenesis. Rev. Mod. Phys.
**1999**, 71, 1463. [Google Scholar] [CrossRef] [Green Version] - Brandenberger, R.H.; Yamaguchi, M. Spontaneous baryogenesis in warm inflation. Phys. Rev. D
**2003**, 68, 023505. [Google Scholar] [CrossRef] [Green Version] - Takahashi, F.; Yamaguchi, M. Spontaneous baryogenesis in flat directions. Phys. Rev. D
**2004**, 69, 083506. [Google Scholar] [CrossRef] [Green Version] - Odintsov, S.D.; Oikonomou, V.K. Gauss–Bonnet gravitational baryogenesis. Phys. Lett. B
**2016**, 760, 259. [Google Scholar] [CrossRef] [Green Version] - Oikonomou, V.K.; Saridakis, E.N. f(T) gravitational baryogenesis. Phys. Rev. D
**2016**, 94, 124005. [Google Scholar] [CrossRef] [Green Version] - Ramos, M.P.L.P.; Paramos, J. Baryogenesis in nonminimally coupled f(R) theories. Phys. Rev. D
**2017**, 96, 104024. [Google Scholar] [CrossRef] [Green Version] - Sahoo, P.K.; Bhattacharjee, S. Gravitational Baryogenesis in Non-Minimal Coupled f(R,T) Gravity. Int. J. Theor. Phys.
**2020**, 59, 1451. [Google Scholar] [CrossRef] [Green Version] - Sakharov, A.D. Violation of CP-invariance, C-asymmetry, and Baryon Asymmetry of the Universe. JETP Lett.
**1967**, 5, 24. [Google Scholar] - Lambiase, G.; Scarpetta, G. Baryogenesis in f(R) theories of gravity. Phys. Rev. D
**2006**, 74, 087504. [Google Scholar] [CrossRef] - Bhattacharjee, S.; Sahoo, P.K. Baryogenesis in f(Q,τ) gravity. Eur. Phys. J. C
**2020**, 80, 289. [Google Scholar] [CrossRef] [Green Version] - Nojiri, S.; Odintsov, S.D. Unifying phantom inflation with late-time acceleration: Scalar phantom–non-phantom transition model and generalized holographic dark energy. Gen. Rel. Grav.
**2006**, 38, 1285. [Google Scholar] [CrossRef] [Green Version] - Nojiri, S.; Odintsov, S.D. Covariant generalized holographic dark energy and accelerating universe. Eur. Phys. J. C
**2017**, 77, 528. [Google Scholar] [CrossRef] - Nojiri, S.; Odintsov, S.D.; Saridakis, E.N. Modified cosmology from extended entropy with varying exponent. Eur. Phys. J. C
**2019**, 79, 242. [Google Scholar] [CrossRef] - Nojiri, S.; Odintsov, S.D.; Paul, T. Barrow entropic dark energy: A member of generalized holographic dark energy family. Phys. Lett. B
**2022**, 825, 136844. [Google Scholar] [CrossRef] - Jawad, A.; Sultan, A.M. Cosmic Consequences of Kaniadakis and Generalized Tsallis Holographic Dark Energy Models in the Fractal Universe. Adv. High Energy Phys.
**2021**, 5519028, 1. [Google Scholar] [CrossRef] - Zhu, T.; Zhao, W.; Huang, Y.; Wang, A.; Wu, Q. Effects of parity violation on non-Gaussianity of primordial gravitational waves in Hořava-Lifshitz gravity. Phys. Rev. D
**2013**, 88, 063508. [Google Scholar] [CrossRef] [Green Version] - Huang, Y.; Wang, A.; Yousefi, R.; Zhu, T. Primordial non-Gaussianity of gravitational waves in Hořava-Lifshitz gravity. Phys. Rev. D
**2013**, 88, 023523. [Google Scholar] [CrossRef] [Green Version] - Greenwald, J.; Lenells, J.; Satheeshkumar, V.H.; Wang, A. Gravitational collapse in Hořava-Lifshitz theory. Phys. Rev. D
**2013**, 88, 024044. [Google Scholar] [CrossRef] [Green Version] - Wang, A. Stationary axisymmetric and slowly rotating spacetimes in Hořava-lifshitz gravity. Phys. Rev. Lett.
**2013**, 110, 091101. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Sultan, A.M.; Jawad, A. Cosmic and thermodynamic study of non-canonical scalar field in parameterized modified gravity. Phys. Scrip.
**2022**, 97, 065004. [Google Scholar] [CrossRef] - Wang, A.; Wands, D.; Maartens, R. Scalar field perturbations in Hořava-Lifshitz cosmology. JCAP
**2010**, 03, 013. [Google Scholar] [CrossRef] [Green Version] - Wang, A.; Maartens, R. Linear perturbations of cosmological models in the Hořava-Lifshitz theory of gravity without detailed balance. Phys. Rev. D
**2010**, 81, 024009. [Google Scholar] [CrossRef] [Green Version] - Wang, A.; Wu, Y. Thermodynamics and classification of cosmological models in the Hořava-Lifshitz theory of gravity. JCAP
**2009**, 07, 012. [Google Scholar] [CrossRef] [Green Version] - Bogdanos, C.; Saridakis, E.N. Perturbative instabilities in Horava gravity. Class. Quant. Grav.
**2010**, 27, 075005. [Google Scholar] [CrossRef] [Green Version] - Rani, S.; Jawad, A.; Sultan, A.M.; Shad, M. Cosmographic and thermodynamic analysis of Kaniadakis holographic dark energy. Int. J. Mod. Phys. D
**2022**, 31, 2250078. [Google Scholar] [CrossRef] - Saridakis, E.N. Hořava-Lifshitz Dark Energy. Eur. Phys. J. C
**2010**, 67, 229. [Google Scholar] [CrossRef] - Saridakis, E.N.; Gonzalez-Diaz, P.F.; Siguenza, C.L. Unified dark energy thermodynamics: Varying w and the −1-crossing. Class. Quant. Grav.
**2009**, 26, 165003. [Google Scholar] [CrossRef] [Green Version] - Nojiri, S.; Odintsov, S.D. Introduction to modified gravity and gravitational alternative for dark energy. Int. J. Geom. Methods Mod. Phys.
**2007**, 4, 115. [Google Scholar] [CrossRef] [Green Version] - Felice, A.D.; Tsujikawa, S. f(R) Theories. Living Rev. Relativ.
**2010**, 13, 1. [Google Scholar] [CrossRef] [PubMed] [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] - 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**, 345, 155. [Google Scholar] [CrossRef] [Green Version] - Bento, M.C.; Felipe, R.G.; Santos, N.M.C. Gravitational baryogenesis in Gauss-Bonnet braneworld cosmology. Phys. Rev. D
**2005**, 71, 123517. [Google Scholar] [CrossRef] [Green Version] - Oikonomou, V.K. Constraints on singular evolution from gravitational baryogenesis. Int. J. Geom. Methods Mod. Phys.
**2016**, 13, 1650033. [Google Scholar] [CrossRef] [Green Version] - Odintsov, S.D.; Oikonomou, V.K. Loop quantum cosmology gravitational baryogenesis. Europhys. Lett.
**2016**, 116, 49001. [Google Scholar] [CrossRef] [Green Version] - Baffou, E.H.; Houndjo, M.J.S.; Kanfon, D.A.; Salako, I.G. f(R,T) models applied to baryogenesis. Eur. Phys. J. C
**2019**, 79, 112. [Google Scholar] [CrossRef] - Bhattacharjee, S. Gravitational baryogenesis in extended teleparallel theories of gravity. Phys. Dark Univ.
**2020**, 30, 100612. [Google Scholar] [CrossRef] - Azhar, N.; Jawad, A.; Rani, S. Generalized gravitational baryogenesis of well-known f(T,T
_{G}) and f(T,B) models. Phys. Dark Univ.**2020**, 30, 100724. [Google Scholar] [CrossRef] - Agrawal, A.S.; Tripathy, S.K.; Mishra, B. Gravitational baryogenesis models comparison in f(R) Gravity. Chin. J. Phys.
**2021**, 71, 333. [Google Scholar] [CrossRef] - Azhar, N.; Jawad, A.; Rani, S. Impact of f(G,T) and f(R,G) on gravitational baryogenesis and observational bounds. Phys. Dark Univ.
**2021**, 32, 100815. [Google Scholar] [CrossRef] - Mavromatos, N.E. Matter-antimatter asymmetry in the universe via string-inspired CPT violation at early eras. J. Phys. Conf. Ser.
**2018**, 952, 012006. [Google Scholar] [CrossRef] - Jawad, A.; Sultan, A.M. Analysis of baryon to entropy ratio in Ricci inverse gravity. EPL
**2022**, 138, 29001. [Google Scholar] [CrossRef] - Hořava, P. Quantum Gravity at a Lifshitz Point. Phys. Rev. D
**2009**, 79, 084008. [Google Scholar] [CrossRef] [Green Version] - Calcagni, G. Cosmology of the Lifshitz universe. JHEP
**2009**, 0909, 112. [Google Scholar] [CrossRef] [Green Version] - Brandenberger, R. Matter bounce in Hořava-Lifshitz cosmology. Phys. Rev. D
**2009**, 80, 043516. [Google Scholar] [CrossRef] [Green Version] - Kiritsis, E.; Kofinas, G. Hořava–Lifshitz cosmology. Nucl. Phys. B
**2009**, 821, 467. [Google Scholar] [CrossRef] [Green Version] - Mukohyama, S. Scale-invariant cosmological perturbations from Hořava-Lifshitz gravity without inflation. JCAP
**2009**, 0906, 001. [Google Scholar] [CrossRef] [Green Version] - Takahashi, T.; Soda, J. Chiral primordial gravitational waves from a Lifshitz point. Phys. Rev. Lett.
**2009**, 102, 231301. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Mukohyama, S. Hořava–Lifshitz cosmology: A review. Class. Quantum Grav.
**2010**, 27, 223101. [Google Scholar] [CrossRef] [Green Version] - Carloni, S.; Chaichian, M.; Nojiri, S.; Odintsov, S.D. Modified first-order Hořava-Lifshitz gravity: Hamiltonian analysis of the general theory and accelerating FRW cosmology in a power-law F(R) model. Phys. Rev. D
**2010**, 82, 065020. [Google Scholar] [CrossRef] [Green Version] - Sultan, A.M.; Jawad, A. Compatibility of big bang nucleosynthesis in some modified gravities. Eur. Phys. J. C
**2022**, 82, 905. [Google Scholar] [CrossRef] - Jawad, A.; Sultan, A.M. Analyzing stability of five-dimensional Einstein Chern–Simons gravity through dynamical systems. Phys. Dark Univ.
**2022**, 38, 101127. [Google Scholar] [CrossRef] - Kluson, J. New models of f(R) theories of gravity. Phys. Rev. D
**2010**, 81, 064028. [Google Scholar] [CrossRef] [Green Version] - Mukohyama, S. Dark matter as integration constant in Hořava-Lifshitz gravity. Phys. Rev. D
**2009**, 80, 064005. [Google Scholar] [CrossRef] [Green Version] - Cohen, A.G.; Rujula, A.D.; Glashow, S.L. A matter-antimatter universe? Astrophys. J.
**1998**, 495, 539. [Google Scholar] [CrossRef] [Green Version] - Jawad, A.; Rani, S.; Saleem, M. Cosmological study of reconstructed f(T) models. Astrophys. Space Sci.
**2017**, 362, 63. [Google Scholar] [CrossRef]

**Figure 1.**Plot of $\frac{{\eta}_{B}}{S}$ versus parameter $\alpha $, for $\mu =(0.60,0.75,0.90$), while other fixed parameters are ${T}_{D}=2\times {10}^{16}$ GeV, ${a}_{0}=1,\phantom{\rule{3.33333pt}{0ex}}c=0.3,\phantom{\rule{3.33333pt}{0ex}}{g}_{{}_{b}}=1,\phantom{\rule{3.33333pt}{0ex}}{g}_{*s}=106,\phantom{\rule{3.33333pt}{0ex}}\lambda =0.7$, ${M}_{*}={10}^{12}$ GeV, and $n=0.5$.

**Figure 2.**Variation of $\frac{{\eta}_{B}}{S}$ against the parameter n for the model $F\left(\tilde{R}\right)=\tilde{R}+\alpha {\tilde{R}}^{2}$.

**Figure 3.**Variation of baryon number to entropy ratio against parameter $\alpha $ for various values of parameter $\mu $ mentioned in the panel. The other parameters are ${a}_{o}=1$, ${g}_{b}=1$, ${g}_{*s}=106$, $\lambda =0.7$, ${M}_{*}={10}^{12}$ GeV, $n=0.5$, and ${T}_{D}=2\times {10}^{16}$ GeV.

**Figure 4.**Plot of $\frac{{\eta}_{B}}{S}$ against parameter n for the model $F\left(\tilde{R}\right)=\tilde{R}+\alpha {\tilde{R}}^{2}$.

**Table 1.**Baryogenesis for $F\left(\tilde{R}\right)=\tilde{R}+\alpha {\tilde{R}}^{2}+\beta {\tilde{R}}^{m}$ when $m=3$.

Sr. No | $\mathit{\mu}$ | $\frac{{\mathit{\eta}}_{{}_{\mathit{B}}}}{\mathit{S}}$ |
---|---|---|

1 | $-0.90$ | $9.453\times {10}^{-23}$ |

2 | $-0.85$ | $9.442\times {10}^{-23}$ |

3 | $-0.80$ | $9.433\times {10}^{-23}$ |

4 | $-0.75$ | $9.426\times {10}^{-23}$ |

5 | $-0.70$ | $9.422\times {10}^{-23}$ |

**Table 2.**Baryogenesis for $F\left(\tilde{R}\right)=\tilde{R}+\alpha {\tilde{R}}^{2}+\beta {\tilde{R}}^{m}$ when $m=4$.

S. No | $\mathit{\mu}$ | $\frac{{\mathit{\eta}}_{{}_{\mathit{B}}}}{\mathit{S}}$ |
---|---|---|

1 | $0.20$ | $1.176\times {10}^{-26}$ |

2 | $0.18$ | $1.437\times {10}^{-26}$ |

3 | $0.16$ | $1.637\times {10}^{-26}$ |

4 | $0.14$ | $1.819\times {10}^{-26}$ |

5 | $0.12$ | $1.999\times {10}^{-26}$ |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 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**

Jawad, A.; Sultan, A.M.; Rani, S.
Viability of Baryon to Entropy Ratio in Modified Hořava–Lifshitz Gravity. *Symmetry* **2023**, *15*, 824.
https://doi.org/10.3390/sym15040824

**AMA Style**

Jawad A, Sultan AM, Rani S.
Viability of Baryon to Entropy Ratio in Modified Hořava–Lifshitz Gravity. *Symmetry*. 2023; 15(4):824.
https://doi.org/10.3390/sym15040824

**Chicago/Turabian Style**

Jawad, Abdul, Abdul Malik Sultan, and Shamaila Rani.
2023. "Viability of Baryon to Entropy Ratio in Modified Hořava–Lifshitz Gravity" *Symmetry* 15, no. 4: 824.
https://doi.org/10.3390/sym15040824