# Inflation via logarithmic entropy-corrected holographic dark energy model

## Abstract

We study the inflation in terms of the logarithmic entropy-corrected holographic dark energy (LECHDE) model with future event horizon, particle horizon, and Hubble horizon cut-offs, and we compare the results with those obtained in the study of inflation by the holographic dark energy HDE model. In comparison, the spectrum of primordial scalar power spectrum in the LECHDE model becomes redder than the spectrum in the HDE model. Moreover, the consistency with the observational data in the LECHDE model of inflation constrains the reheating temperature and Hubble parameter by one parameter of holographic dark energy and two new parameters of logarithmic corrections.

## Keywords

Dark Energy Spectral Index Cosmic Microwave Background Friedmann Equation Holographic Dark Energy## 1 Introduction

The recent cosmological and astrophysical data from cosmic microwave background (CMB) radiation, the observations of type Ia supernovae and large scale structure (LSS) persuasively express that the universe experiences an accelerated expansion phase [1, 2, 3, 4]. The accelerated expansion phase is derived by an energy component with negative pressure, the so-called dark energy (DE). The most simple candidate for dark energy is the cosmological constant, \(\Lambda \). However, the cosmological constant candidate suffers from the fine-tuning and the cosmic coincidence problems [5, 6]. Therefore, cosmologists suggested some different models for DE, including tachyon, quintessence, phantom, k-essence, chaplygin gas, holographic, and new agegraphic models [7, 8, 9, 10, 11, 12, 13].

*c*is a numerical constant to be determined by observational data.

*L*and \(M_\mathrm{P}\) are the cut-off radius and the reduced Planck mass, respectively.

*L*takes a very small amount. This means that the correction terms are important in early universe and when the universe becomes large, the second and third terms are ignorable and the logarithmic entropy-corrected holographic dark energy model reduces to the ordinary holographic dark energy model. The fractional energy density of LECHDE is given by

We emphasize that the study of holographic dark energy model, considering the cosmological constant problem, leads to the fact that the Hubble horizon and particle horizon cut-offs contradict observations, and only the one with the future event horizon cut-off is consistent with observations [9]. However, for the sake of generality, in this work we intend to study inflation with logarithmic entropy-corrected holographic dark energy model considering the future event horizon, particle horizon, and Hubble horizon cut-offs.

## 2 Inflation and perturbational analysis

*B*is the constant value. The power spectrum of \(\mathcal {R}\) is given by [26]

## 3 LECHDE model with future event horizon cut-off in inflation

*a*is the scale factor. Now, by assuming \(V=M_\mathrm{P}^{4}\) [24] and inserting in Eq. (29), we obtain

*c*increases, the energy density \(\Omega _{\Lambda }\) becomes more dominant at earlier times.

*x*, \(\alpha , \beta \), and

*c*by solving the following equation:

## 4 LECHDE model with particle horizon cut-off in inflation

*c*increases, the energy density \(\Omega _{\Lambda }\) becomes more dominant at earlier times.

*x*, \(\alpha , \beta \), and

*c*by solving the following equation:

## 5 LECHDE model with Hubble cut-off in inflation

*M*is a constant with the dimension of energy. We compare Eqs. (75), (76), and (77) with Eqs. (42), (43) and (44) in Ref. [23]. Then we can write the above equations as follows:

## 6 Concluding remarks

In this work, we have investigated the inflation by logarithmic entropy-corrected holographic dark energy LECHDE model for different cut-offs. We have assumed that the inflaton field does not couple to the logarithmic entropy-corrected holographic dark energy and hence it is not affected by the existence of the logarithmic entropy-corrected holographic dark energy. Also, we have assumed that the LECHDE model depends on the background and it does not create the perturbations. Therefore, the standard perturbation equations remain unchanged. We have also assumed that the reheating period occurs immediately after the inflation period. Considering these assumptions, we have compared our results for the LECHDE model with the results of the HDE model obtained in [23]. We have found that, for the future event horizon cut-off (see Fig. 1), the evolution of \(\Omega _{\Lambda }\) with respect to the scale factor for the LECHDE model is faster than that of the HDE model. Also, in the evolution of \(\Omega _{\Lambda }\) with respect to \(x=\ln (a)\), we found that, as *c* increases, the energy density \(\Omega _{\Lambda }\) becomes more dominant at earlier times for the LECHDE model compared with the HDE model. For the particle horizon cut-off (see Fig. 4), we have found that the evolution of \(\Omega _{\Lambda }\) with respect to the scale factor for the LECHDE model is slower than that of the HDE model. Also, in the evolution of \(\Omega _{\Lambda }\) with respect to \(x=\ln (a)\), we found that as *c* increases, the energy density \(\Omega _{\Lambda }\) becomes more dominant at earlier times for the LECHDE model compared with the HDE model.

We have derived the corrections to the spectral index produced by the LECHDE model with the event future horizon, the particle horizon, and the Hubble horizon cut-offs, and we found that the effect of the LECHDE model for all three cut-offs is making the spectrum redder than the HDE model. The requirement of consistency with the observational data in the LECHDE model of inflation constrains the reheating temperature and Hubble parameter by one parameter of holographic dark energy and two new parameters of logarithmic corrections, compared to the HDE model.

## References

- 1.A.G. Riess et al., Astron. J.
**116**, 1009 (1998)ADSCrossRefGoogle Scholar - 2.S. Perlmutter et al., Astrophys. J.
**517**, 565 (1999)ADSCrossRefGoogle Scholar - 3.P. de Bernardis et al., Nature
**404**, 955 (2000)ADSCrossRefGoogle Scholar - 4.S. perlmutter et al., Astrophys. J. 598, 102 (2003)Google Scholar
- 5.E.J. Copeland, M. Sami, S. Tsujikawa, IJMPD
**15**, 1753 (2066)Google Scholar - 6.S. Weinberg, Rev. Modern Phys.
**61**, 1 (1989)ADSMathSciNetCrossRefGoogle Scholar - 7.T. Padmanabhan, Phys. Rept.
**380**, 235 (2006)ADSCrossRefGoogle Scholar - 8.A.G. Cohen, D.B. Kaplan, A.E. Nelson, Phys. Rev. Lett.
**82**, 4971 (1999)ADSMathSciNetCrossRefGoogle Scholar - 9.S.D.H. Hsu, Phys. Lett. B
**594**, 13 (2004)ADSCrossRefGoogle Scholar - 10.H. Wei, R.G. Cai, Phys. Lett. B
**660**, 113 (2008)ADSCrossRefGoogle Scholar - 11.Y.F. Cai, E.N. Saridakis, M.R. Setare, J.Q. Xia, Phys. Rept.
**493**, 1 (2010)ADSCrossRefGoogle Scholar - 12.M.R. Setare, Phys. Lett. B
**653**, 116 (2007)ADSMathSciNetCrossRefGoogle Scholar - 13.M.R. Setare, J. Sadeghi, A.R. Amani, Phys. Lett. B
**673**, 241 (2009)ADSCrossRefGoogle Scholar - 14.L. Susskind, J. Math. Phys.
**36**, 6377 (1995)ADSMathSciNetCrossRefGoogle Scholar - 15.S. Nojiri, S.D. Odintsov, Gen. Rel. Grav.
**38**, 1285 (2006)ADSCrossRefGoogle Scholar - 16.K. Bamba, S. Capozziello, S.D. Odintsov, Astrophys. Space Sci.
**342**, 155 (2012)ADSCrossRefGoogle Scholar - 17.R.M. Wald, Phys. Rev. D
**48**, 3427 (1993)ADSMathSciNetCrossRefGoogle Scholar - 18.R. Banerjee, B.R. Majhi, Phys. Lett. B
**662**, 62 (2008)ADSMathSciNetCrossRefGoogle Scholar - 19.R. Banerjee, B.R. Majhi, JHEP
**06**, 095 (2008)ADSCrossRefGoogle Scholar - 20.R. Banerjee, B.R. Majhi, J. Zhang. Phys. Lett. B
**668**, 353 (2008)CrossRefGoogle Scholar - 21.Y.F. Cai, J. Liu, H. Li, Phys. Lett. B
**690**, 213 (2010)ADSCrossRefGoogle Scholar - 22.H. Wei, Commun. Theor. Phys.
**52**, 743 (2009)ADSCrossRefGoogle Scholar - 23.B. Chen, M. Li, Y. Wang, Nucl. Phys. B
**774**, 256 (2007)ADSCrossRefGoogle Scholar - 24.D.H. Lyth, A. Riotto, Phys. Rept.
**314**, 1 (1999)ADSCrossRefGoogle Scholar - 25.V.F. Mukhanov, H.A. Feldman, R.H. Brandenberger, Phys. Rept.
**215**, 203 (1992)ADSCrossRefGoogle Scholar - 26.B. Chen, M. Li, T. Wang, Y. Wang, Mod. Phys. Lett. A
**22**, 1987 (2007)ADSCrossRefGoogle Scholar - 27.Q.G. Huang, Y. Gong, JCAP
**0408**, 006 (2004)ADSCrossRefGoogle Scholar - 28.H.C. Kao, W.L. Lee, F.L. Lin, Phys. Rev. D
**71**, 123518 (2005)ADSCrossRefGoogle Scholar - 29.X. Zhang, Int. J. Mod. Phys. D
**14**, 1597 (2005)ADSCrossRefGoogle Scholar - 30.X. Zhang, F.Q. Wu, Phys. Rev. D
**72**, 043524 (2005)ADSCrossRefGoogle Scholar - 31.Z. Chang, F.Q. Wu, X. Zhang, Phys. Lett. B
**633**, 14 (2006)ADSCrossRefGoogle Scholar - 32.X. Zhang, Int. J. Mod. Phys. D
**74**, 103505 (2006)Google Scholar - 33.P.A.R. Ade et al., Astron. Astrophys.
**571**(A), 22 (2014)Google Scholar

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