# A Possible Cosmological Application of Some Thermodynamic Properties of the Black Body Radiation in n-Dimensional Euclidean Spaces

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Generalization to n Dimensions

^{4}-law for n = 3. Equation (4) was found also by Barrow and Hawthorne [9]. The Helmholtz function F (n, V, T) is given by the following expression [35],

_{V}can be calculated as usual,

## 3. Adiabatic Processes

^{n}as the black body system, where R is the length of its sides. By using Equation (4) and the expression for p in Equation (9), the adiabatic condition for two different states is:

_{0}and T

_{0}are some initial conditions, and since we are dealing with a region close to the Planck scale, it could be considered that the initial conditions are the temperature and length of Planck T

_{P}and l

_{P}.

## 4. Carnot Cycle

_{1}and Q

_{2}are the input and output heats, respectively. Let us consider a heat engine working between two reservoirs with temperatures T

_{1}and T

_{2}(T

_{1}> T

_{2}) with black body radiation as the working substance. As we know, a Carnot cycle consists of two isothermal processes at temperatures T

_{1}and T

_{2}and two adiabatic processes. The calculation for this efficiency is given in Appendix B, where one obtains the already familiar expression for the Carnot efficiency,

## 5. Critical Points of the Thermodynamic Potentials

_{P}= 1.42 × 10

^{32}K, and the energy of Planck is E

_{P}= 1.9561 × 10

^{9}J. In each plot, the most relevant region is the one at high temperatures and low dimensions, as can be seen in Figures 1–3.

## 6. A Possible Simple Application to Cosmology

^{32}K are only possible in a very primeval time in the evolution of the universe. It is known that the period dominated by radiation is from t = 10

^{−44}s to t ≈ 10

^{10}s [46], that means that in this period, the universe could have been well described by a spatially-flat, radiation-only model [47]. Thus, considering the whole primeval universe as a black body system in a Euclidean space is in principle a reasonable approach.

_{P}was about u ≈ 1E

_{P}/V

_{P}(V

_{P}is a Planck volume) and the temperature approximately T ≈ 1T

_{P}. The energy density and temperature obtained when the maximum of the energy density is located at n = 3 (see Figure 2) are surprisingly near the values referred to above.

_{P}, the difference between these energies is larger. For T > 0.72T

_{P}, the energy does not have any critical point, and the energy difference grows considerably for each initial dimensionality scenario.

^{3}/3, with a = 4σ/c) for calculating S

_{0}at the Planck scale and S

_{f}at the present time, immediately, one obtains S

_{0}≈ 10

^{−23}J/K and S

_{f}≈ 10

^{76}J/K, respectively. These values in Planck units are S

_{0}≈ 1 and S

_{f}≈ 10

^{99}, which are within the range of other estimates [51]. It should be noted that, in fact, if we consider the cosmological evolution of the universe, the entropy is conserved in the radiation-dominated era. However, the photons’ entropy grows due to the additional influence in the evolution of the universe, taking into account inflation, dark matter and dark energy components.

_{n}, (κ = 4π

^{n}

^{/2}Gn/n(n − 2) Γ (n/2); see [52]) or to the Plank length in 3-dim $\left(\kappa =16\pi {l}_{p}^{n+1}/n(n-1\right)$; see [53]).

_{0}= (4/κU

_{0}(n+1)

^{2})

^{1/2}is the present time, and the evolution of the energy density is described by:

_{max}∼ d

_{h}, the minimum χ value is:

_{P}, the original integral should be reduced by a factor of order 10

^{−3}when the value of χ

_{min}is considered. However, for T = 0.08T

_{P}, the two integrals are different by approximately 5%. For higher values of n, the difference between the integrals is even smaller.

_{P}, the number of particles is N ≈ 100; meanwhile, for n = 9, N ≈ 10

^{6}. That is, in low dimensionality, the statistical behavior is not accurate. In fact, as T → T

_{P}, the number of photons tends to unity. The latter should be corrected if one pursues including the effects of the evolution of the universe near the Planck scale and the behavior of the black body radiation in the universe. The former issue is a very complicated problem, because the speed of light depends on the energy, and the quantum effects have to be incorporated by means of loop quantum cosmology, for example. One of the possible modifications consists of a different form of the dependence of the energy density on the scale factor, ε = ε

_{0}α(a)a

^{−4}(see [54]), where α is calculated from loop quantum cosmology.

## 7. LQG Corrections

_{P}is the Planck length and it depends on the dimensionality of the space-time [57–59], E is the energy and α and α′ take different values depending on the details of the quantum gravity candidates. Finally, the spectral energy density in the interval ν to ν + dν shown in [31] is,

_{P}must be fulfilled. In this case, by dividing the total internal energy U between the number of particles N, it is possible to obtain a mean energy per photon E

_{mean}, which is E

_{mean}≤ 0.3E

_{P}. In the Taylor expansion (Equation (24)), the terms of order higher than ${L}_{P}^{6}{E}^{8}$ are neglected, which corresponds to contributions on the order of 6.5 × 10

^{−5}. In this way, the qualitative behavior of the model can be justified. In regards to the number of photons, when the classical cosmological approach is considered with no LQG modifications, the number of photons N is N ≈ 100 at T = 0.65T

_{P}, then statistical physics is not accurate. With the same classical cosmology approach and LQG modifications included, the number of photons at T = .08 is such that N > 5 × 10

^{5}for n = 3 and N > 10

^{34}for n = 9. As mentioned before, this certainly must be corrected by quantum effects and a theory of quantum cosmology

## 8. Conclusions

_{P}.

## Appendix

## A. Riemann’s Zeta Function

## B. Carnot Efficiency

_{2}= p

_{3}:

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 4.**First derivative in the dimension for the energy density u (n, T) given by Equation (4). The vertical dashed line marks the location of the saddle point, located at n < 5. The horizontal dashed line represents an isothermal process, in which by increasing the dimension from n = 1 to n = 25, the black body radiation (BBR) system has an increasing energy density u (positive derivative); then, it reaches a local maximum and decreases (negative derivative) until it reaches a local minimum, and then, u increases again. The saddle point is located at approximately n ≈ 4.8. If n is considered as an integer number, the maxima appear at n ≤ 4 and the minima at n > 4, as summarized in Table 1.

**Figure 8.**First derivative in the dimension for the energy density u (n, T) given by Equation (26). The values of α = 0.1 and α′ = 2.28 are such that the curves shown in [31] are reproduced.

**Table 1.**The locations of the maxima and minima of energy, Helmholtz free energy, entropy and enthalpy densities are restricted to certain dimensionalities.

u (n, T) | f (n, T) | s (n, T) | h (n, T) | |
---|---|---|---|---|

Maxima in | n ≲ 4 | n > 3 | n ≲ 4 | n ≲ 4 |

Minima in | n > 4 | n ≲ 3 | n > 4 | n > 4 |

**Table 2.**Energy (Equation (4)) at T = 0.65T

_{p}within a Planck volume. T = 0.65T

_{p}is an arbitrary value close to T

_{P}.

n | u [E_{p}/V_{P}] | E (GeV/V_{P}) |
---|---|---|

2 | 0.11 | 1.3E18 |

3 | 0.12 | 1.4E18 |

4 | 0.11 | 1.3E18 |

9 | 0.08 | 0.9E19 |

25 | 1.35 | 1.65E19 |

n_{initial} | n_{final} | ΔE_{p} | %E_{missing} |
---|---|---|---|

2 | 3 | −0.01 | −11.8 |

4 | 3 | −0.008 | −7 |

9 | 3 | −0.04 | −54.3 |

10 | 3 | −0.04 | −54.5 |

25 | 3 | 1.24 | 91.3 |

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Gonzalez-Ayala, J.; Perez-Oregon, J.; Cordero, R.; Angulo-Brown, F.
A Possible Cosmological Application of Some Thermodynamic Properties of the Black Body Radiation in n-Dimensional Euclidean Spaces. *Entropy* **2015**, *17*, 4563-4581.
https://doi.org/10.3390/e17074563

**AMA Style**

Gonzalez-Ayala J, Perez-Oregon J, Cordero R, Angulo-Brown F.
A Possible Cosmological Application of Some Thermodynamic Properties of the Black Body Radiation in n-Dimensional Euclidean Spaces. *Entropy*. 2015; 17(7):4563-4581.
https://doi.org/10.3390/e17074563

**Chicago/Turabian Style**

Gonzalez-Ayala, Julian, Jennifer Perez-Oregon, Rubén Cordero, and Fernando Angulo-Brown.
2015. "A Possible Cosmological Application of Some Thermodynamic Properties of the Black Body Radiation in n-Dimensional Euclidean Spaces" *Entropy* 17, no. 7: 4563-4581.
https://doi.org/10.3390/e17074563