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Study of baryogenesis in the framework of Hořava–Lifshitz cosmology with Starobinsky potential

  • Gargee Chakraborty and Surajit Chattopadhyay ORCID logo EMAIL logo
Published/Copyright: August 19, 2022
Zeitschrift für Naturforschung A
From the journal Zeitschrift für Naturforschung A Volume 77 Issue 11

Abstract

Motivated by the work of Paliathanasis et al. (A. Paliathanasis and G. Leon, “Cosmological solutions in Hořava–Lifshitz scalar field theory,” ZnA, vol. 75, p. 523, 2020), this work reports the baryogenesis in Hořava–Lifshitz cosmology by taking the background evolution as modified Chaplygin gas and modified holographic dark energy. The Starobinsky potential has been selected to initiate the study. The scalar field and its potential have been reconstructed and found to be consistent with the universe’s expansion. The quintessence behaviour of equation of state parameters has been observed for both cases. Finally, baryogenesis has been studied in both cases. The baryon entropy ratio attained the observed value. It is also well explained that either the model will achieve an equal number of baryon and antibaryon densities or will satisfy the Generalized Second Law of Thermodynamics.

Keywords: baryogenesis; EoS parameter; Hořava–Lifshitz cosmology; Starobinsky potential; statefinder parameter
Corresponding author: Surajit Chattopadhyay, Department of Mathematics, Amity University, Major Arterial Road, Action Area II, Rajarhat, New Town, Kolkata 700135, India, E-mail:

Funding source: Council of Scientific and Industrial Research

Award Identifier / Grant number: 03(1420)/18/EMR-II

Acknowledgement

The authors are thankful to the anonymous reviewers for their valuable suggestions. Surajit Chattopadhyay acknowledges financial support from the Council of Scientific and Industrial Research (Government of India) with Grant No. 03(1420)/18/EMR-II.

  1. Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix A

A.1 Reconstructed density and pressure for Starobinsky-type potential in HL gravity

Using ϕ and V from Eqs. (23) and (24), respectively, we get ρ ϕ,S and p ϕ,S and the expressions are presented in Eqs. (46) and (47).

(46) ρ ϕ , S = 1 8 ( − 1 + 3 λ ) 4 − 1 + e − t 1 2 − 3 n 2 γ C 1 BesselJ 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + C 2 BesselY 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) m 2 V 0 + 1 4 t − 1 − 3 n ( − 1 + 3 λ ) 2 C 1 t V 0 γ 2 m 2 ( − 1 + 3 λ ) BesselJ 3 2 ( − 1 + n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) − BesselJ 1 2 + 3 n 2 , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + C 1 ( 1 − 3 n ) BesselJ 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + 2 C 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) BesselY 3 2 ( − 1 + n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) − BesselY 1 2 + 3 n 2 , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + C 2 ( 1 − 3 n ) BesselY 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) 2

and

(47) p ϕ , S = 1 8 ( − 1 + 3 λ ) − 4 − 1 + e − t 1 2 − 3 n 2 γ C 1 BesselJ 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + C 2 BesselY 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) m 2 V 0 + 1 4 t − 1 − 3 n ( − 1 + 3 λ ) 2 C 1 t V 0 γ 2 m 2 ( − 1 + 3 λ ) BesselJ 3 2 ( − 1 + n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) − BesselJ 1 2 + 3 n 2 , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + C 1 ( 1 − 3 n ) BesselJ 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + 2 C 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) BesselY 3 2 ( − 1 + n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) − BesselY 1 2 + 3 n 2 , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + C 2 ( 1 − 3 n ) BesselY 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) 2

A.2 MCG and MHDE under the purview of HL gravity

Here, we get reconstructed Hubble parameter H MCG,rec of HL gravity with background evolution as MCG and the reconstructed H is presented in Eq. (48)

(48) H MCG,rec = 1 3 B 1 + A 1 1 + α + 1 2 ( − 1 + 3 λ ) 1 − e − γ C 1 t 1 2 ( 1 − 3 n ) BesselJ 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + C 2 t 1 2 ( 1 − 3 n ) BesselY 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) m 2 V 0 + 1 4 ( − 1 + 3 λ ) 1 2 C 1 ( 1 − 3 n ) t − 1 + 1 2 ( 1 − 3 n ) BesselJ 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + C 1 t 1 2 ( 1 − 3 n ) V 0 γ 2 m 2 ( − 1 + 3 λ ) BesselJ − 1 + 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) − BesselJ 1 + 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + 1 2 C 2 ( 1 − 3 n ) t − 1 + 1 2 ( 1 − 3 n ) BesselY 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + C 2 t 1 2 ( 1 − 3 n ) V 0 γ 2 m 2 ( − 1 + 3 λ ) BesselY − 1 + 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) − BesselY 1 + 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) 2

Using the density of HL gravity as a background evolution as MHDE ρ HL,MHDE from Eq. (36) and ρ ϕ,S from Eq. (46) in the first Friedmann equation H 2 = 1 3 ( ρ H L , M H D E + ρ ϕ , S ) , we get reconstructed Hubble parameter H MHDE,rec as

(49) H MHDE,rec = 1 3 2 − n t 2 + 3 n 2 η 2 t 2 ( 1 − 6 λ ) ( − β + η ) ( − 1 + 3 λ ) 2 + 1 2 ( − 1 + 3 λ ) × 1 − e − γ C 1 t 1 2 ( 1 − 3 n ) BesselJ 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + C 2 t 1 2 ( 1 − 3 n ) BesselY 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) m 2 V 0 + 1 4 ( − 1 + 3 λ ) 1 2 C 1 ( 1 − 3 n ) t − 1 + 1 2 ( 1 − 3 n ) BesselJ 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + C 1 t 1 2 ( 1 − 3 n ) V 0 γ 2 m 2 ( − 1 + 3 λ ) BesselJ − 1 + 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) − BesselJ 1 + 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + 1 2 C 2 ( 1 − 3 n ) t − 1 + 1 2 ( 1 − 3 n ) BesselY 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + C 2 t 1 2 ( 1 − 3 n ) V 0 γ 2 m 2 ( − 1 + 3 λ ) BesselY − 1 + 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) − BesselY 1 + 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) 2

A.3 Entropy and number density of baryon for MCG and MHDE for baryogenesis in HL gravity

(50) S MCG = 1 C 3 t n 1 2 ( − 1 + 3 λ ) − 1 − e − γ C 1 t 1 2 ( 1 − 3 n ) BesselJ 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + C 2 t 1 2 ( 1 − 3 n ) BesselY 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) m 2 V 0 + 1 4 ( − 1 + 3 λ ) 1 2 C 1 ( 1 − 3 n ) t − 1 + 1 2 ( 1 − 3 n ) BesselJ 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + C 1 t 1 2 ( 1 − 3 n ) V 0 γ 2 m 2 ( − 1 + 3 λ ) BesselJ − 1 + 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) − BesselJ 1 + 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + 1 2 C 2 ( 1 − 3 n ) t − 1 + 1 2 ( 1 − 3 n ) BesselY 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + C 2 t 1 2 ( 1 − 3 n ) V 0 γ 2 m 2 ( − 1 + 3 λ ) BesselY − 1 + 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) − BesselY 1 + 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) 2 + 1 2 ( − 1 + 3 λ ) × 1 − e − γ C 1 t 1 2 ( 1 − 3 n ) BesselJ 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + C 2 t 1 2 ( 1 − 3 n ) BesselY 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) m 2 V 0 + 1 4 ( − 1 + 3 λ ) 1 2 C 1 ( 1 − 3 n ) t − 1 + 1 2 ( 1 − 3 n ) BesselJ 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + C 1 t 1 2 ( 1 − 3 n ) V 0 γ 2 m 2 ( − 1 + 3 λ ) BesselJ − 1 + 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) − − BesselJ 1 + 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + 1 2 C 2 ( 1 − 3 n ) t − 1 + 1 2 ( 1 − 3 n ) BesselY 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + C 2 t 1 2 ( 1 − 3 n ) V 0 γ 2 m 2 ( − 1 + 3 λ ) BesselY − 1 + 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) − BesselY 1 + 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) 2

(51) S MHDE = 1 C 3 t n 1 2 ( − 1 + 3 λ ) − 1 − e − γ C 1 t 1 2 ( 1 − 3 n ) BesselJ 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + C 2 t 1 2 ( 1 − 3 n ) BesselY 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) m 2 V 0 + 1 4 ( − 1 + 3 λ ) 1 2 C 1 ( 1 − 3 n ) t − 1 + 1 2 ( 1 − 3 n ) BesselJ 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + C 1 t 1 2 ( 1 − 3 n ) V 0 γ 2 m 2 ( − 1 + 3 λ ) BesselJ − 1 + 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ )   − BesselJ 1 + 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + 1 2 C 2 ( 1 − 3 n ) t − 1 + 1 2 ( 1 − 3 n ) BesselY 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + C 2 t 1 2 ( 1 − 3 n ) V 0 γ 2 m 2 ( − 1 + 3 λ ) BesselY − 1 + 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ )   − BesselY 1 + 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) 2 + 1 2 ( − 1 + 3 λ ) × 1 − e − γ C 1 t 1 2 ( 1 − 3 n ) BesselJ 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + C 2 t 1 2 ( 1 − 3 n ) BesselY 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) m 2 V 0 + 1 4 ( − 1 + 3 λ ) 1 2 C 1 ( 1 − 3 n ) t − 1 + 1 2 ( 1 − 3 n ) BesselJ 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + C 1 t 1 2 ( 1 − 3 n ) V 0 γ 2 m 2 ( − 1 + 3 λ ) BesselJ − 1 + 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ )   − BesselJ 1 + 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + 1 2 C 2 ( 1 − 3 n ) t − 1 + 1 2 ( 1 − 3 n ) BesselY 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + C 2 t 1 2 ( 1 − 3 n ) V 0 γ 2 m 2 ( − 1 + 3 λ ) BesselY − 1 + 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ )   − BesselY 1 + 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) 2

(52) n B , M C G = 1.35185 × 1 0 27 B 1 + A 1 1 + α + 1 2 ( − 1 + 3 λ ) 1 − e − γ C 1 t 1 2 ( 1 − 3 n ) BesselJ 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + C 2 t 1 2 ( 1 − 3 n ) BesselY 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) m 2 V 0 + 1 4 ( − 1 + 3 λ ) 1 2 C 1 ( 1 − 3 n ) t − 1 + 1 2 ( 1 − 3 n ) BesselJ 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + C 1 t 1 2 ( 1 − 3 n ) V 0 γ 2 m 2 ( − 1 + 3 λ ) BesselJ − 1 + 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) − BesselJ 1 + 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + 1 2 C 2 ( 1 − 3 n ) t − 1 + 1 2 ( 1 − 3 n ) BesselY 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + C 2 t 1 2 ( 1 − 3 n ) V 0 γ 2 m 2 ( − 1 + 3 λ ) BesselY − 1 + 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) − BesselY 1 + 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) 2

(53) n B , M H D E = 1.35185 × 1 0 27 2 − n t 2 + 3 n 2 η 2 t 2 ( 1 − 6 λ ) ( − β + η ) ( − 1 + 3 λ ) 2 + 1 2 ( − 1 + 3 λ ) × 1 − e − γ C 1 t 1 2 ( 1 − 3 n ) BesselJ 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + C 2 t 1 2 ( 1 − 3 n ) BesselY 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) m 2 V 0 + 1 4 ( − 1 + 3 λ ) 1 2 C 1 ( 1 − 3 n ) t − 1 + 1 2 ( 1 − 3 n ) BesselJ 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + C 1 t 1 2 ( 1 − 3 n ) V 0 γ 2 m 2 ( − 1 + 3 λ ) BesselJ − 1 + 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ )   − BesselJ 1 + 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + 1 2 C 2 ( 1 − 3 n ) t − 1 + 1 2 ( 1 − 3 n ) BesselY 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) + C 2 t 1 2 ( 1 − 3 n ) V 0 γ 2 m 2 ( − 1 + 3 λ ) BesselY − 1 + 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) − BesselY 1 + 1 2 ( − 1 + 3 n ) , 2 t V 0 γ 2 m 2 ( − 1 + 3 λ ) 2

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Received: 2022-05-09
Accepted: 2022-07-22
Published Online: 2022-08-19
Published in Print: 2022-11-25

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Dynamical Systems & Nonlinear Phenomena
  3. Qualitative behavior of a discrete predator–prey system under fear effects
  4. Nonlinear behaviour of ion acoustic shock waves in a two-electron temperature nonthermal complex plasma
  5. Gravitation & Cosmology
  6. Study of baryogenesis in the framework of Hořava–Lifshitz cosmology with Starobinsky potential
  7. Solid State Physics & Materials Science
  8. Role of graphene-oxide and reduced-graphene-oxide on the performance of lead-free double perovskite solar cell
  9. Thermodynamics & Statistical Physics
  10. Thermodynamics of the classical spin triangle
https://doi.org/10.1515/zna-2022-0130
Keywords for this article
baryogenesis; EoS parameter; Hořava–Lifshitz cosmology; Starobinsky potential; statefinder parameter

Articles in the same Issue

  1. Frontmatter
  2. Dynamical Systems & Nonlinear Phenomena
  3. Qualitative behavior of a discrete predator–prey system under fear effects
  4. Nonlinear behaviour of ion acoustic shock waves in a two-electron temperature nonthermal complex plasma
  5. Gravitation & Cosmology
  6. Study of baryogenesis in the framework of Hořava–Lifshitz cosmology with Starobinsky potential
  7. Solid State Physics & Materials Science
  8. Role of graphene-oxide and reduced-graphene-oxide on the performance of lead-free double perovskite solar cell
  9. Thermodynamics & Statistical Physics
  10. Thermodynamics of the classical spin triangle
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