Abstract
The effects of generalized uncertainty principle (GUP) on the inflationary dynamics and the thermodynamics of the early universe are studied. Using the GUP approach, the tensorial and scalar density fluctuations in the inflation era are evaluated and compared with the standard case. We find a good agreement with the Wilkinson Microwave Anisotropy Probe data. Assuming that a quantum gas of scalar particles is confined within a thin layer near the apparent horizon of the Friedmann-Lemaitre-Robertson-Walker universe which satisfies the boundary condition, the number and entropy densities and the free energy arising form the quantum states are calculated using the GUP approach. A qualitative estimation for effects of the quantum gravity on all these thermodynamic quantities is introduced.
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Acknowledgments
The research of AT has been partly supported by the German–Egyptian Scientific Projects (GESP ID: 1378). AT likes to thank Prof. Antonino Zichichi for his kind invitation to attend the twenty-ninth World Laboratory Meeting at the “Ettore Majorana Foundation and Centre for Scientific Culture” in Erice-Italy, where the present work is completed. The research of AFA is supported by Benha University. The authors gratefully thank the anonymous referee for useful comments and suggestions which helped to improve the paper.
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Appendix A: Entropy and free energy
Appendix A: Entropy and free energy
At temperature \(T=1/\beta \), the entropy can be deduced from the number of quantum states, Eq. (46), [50]
We notice that the expression contains complex terms. The results at upper and lower bounds of \(\alpha \) are given in Fig. 6 (shown in the left and middle panels, respectively). The difference between \(s(\omega )\) at upper and lower bound of \(\alpha \) is given in the right panel. Only real values are drawn.
Left and middle panels in Fig. 6 show the results at upper and lower bounds of \(\alpha \), respectively. Only real values are taken into consideration. The absolute values in the latter case are nearly three orders of magnitude larger than that in the earlier case. In the earlier case \(s(\omega )\) diverges at small \(\omega \). It shows a kind of saturated plateau up to \(\omega \sim 2\) GeV. This is almost the same as it will be shown in Fig. 7. One of the apparent differences is the sign. Here, \(s(\omega )\) is negative. With increasing \(\omega \), it decreases almost exponentially and flips its sign to negative one. At lower bound of \(\alpha \), we find that \(s(\omega )\) behaves almost contrarily. We notice that \(s(\omega )\) remains positive and it decreases almost exponentially. The difference between upper- and lower-bound-results is shown in the right panel. This difference is assumed to approximately give a qualitative estimation for the effects of the quantum gravity on the entropy density. It is obvious that \(s(\omega )\) remains negative. Increasing \(\omega \) results in decrease in the absolute values of the entropy density. It is apparent that negative entropy contradicts the laws of thermodynamics.
Also, the free energy can be deduced, directly, from the number of quantum states, Eq. (46),
Again, we notice that the expression contains complex terms. The results at the upper and lower bounds of \(\alpha \) are given in Fig. 7: left and middle panel, respectively. The difference between \(s(\omega )\) at upper and lower bound of \(\alpha \) is given in the right panel.
The dependence of free energy on \(\omega \) is illustrated in Fig. 7. In the left panel, we show the results at the upper bound of \(\alpha \). In doing this, we take into consideration the real values, only. We notice that the free energy diverges to negative values at very small values of \(\omega \). Then, \(F(\omega )\) makes a plateau up to \(\omega \sim 2\) GeV. With increasing \(\omega \), the free energy arising from the quantum states switches to positive values. Afterwards, it increase, nearly exponentially. The middle panel shows the result at the lower bound of \(\alpha \). We notice that the absolute value of \(\alpha \) is about three orders of magnitude larger than in the case of upper bound (left panel). Also, we notice that lower-bound-values remain negative although they exponentially decay with increasing \(\omega \). The difference between upper- and lower-bound-results is shown in the right panel. It give a qualitative estimation for the effects of the quantum gravity i.e., GUP on the free energy when taking into consideration the quantum gravity i.e., applying the GUP approach.
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Tawfik, A., Magdy, H. & Farag Ali, A. Effects of quantum gravity on the inflationary parameters and thermodynamics of the early universe. Gen Relativ Gravit 45, 1227–1246 (2013). https://doi.org/10.1007/s10714-013-1522-0
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DOI: https://doi.org/10.1007/s10714-013-1522-0