Is the (3 + 1)-d nature of the universe a thermodynamic necessity?

It is well established that at early times, long before the time of radiation-matter density equality, the universe could have been well described by a spatially flat, radiation only model. In this article we consider the whole primeval universe, as a first approach, as a black-body radiation system in an n-dimensional Euclidean space. We propose that the (3 + 1)-dimensional nature of the universe could be the result of a thermodynamic selection principle stemming from the second law of thermodynamics. In regard to the three spatial dimensions we suggest that they were chosen by means of the minimization of the Helmholtz free energy per hypervolume unit following possibly a kind of broken symmetry process, while the time dimension, as is well known, is related with the principle of increment of entropy for closed systems: the so-called arrow of time. Copyright c © EPLA, 2016 Introduction. – The question of why space is 3-dimensional goes back to ancient Greece [1]. In modern times this question was first raised by Kant in 1746 [2]. Later, Ehrenfest in 1917 by means of the stable orbits argument showed that n = 3 [3]. In 1983, Barrow brought forward a very interesting approach to the dimensionality problem [1]. Since then, many authors have worked on this problem extended to the case (3 + 1). Such is the case of Brandenberger and Vafa [4], that in 1989 proposed a natural mechanism for explaining why there are 3 large space dimensions in the context of string gas cosmology [4,5]. Regarding the (3+1) problem, Tegmark published an enlightening article summarized through his fig. 1 [6]. For a deeper discussion on this issue one can also see among others the following works [6–9]. Since the works by Kaluza and Klein [10,11], many proposals about universe models with dimensionality different from three have been published [4,5,10–25]. Remarkably this has been the case of results stemming from the string, D-branes and gauge theories [4,5,18–24]. However, nowadays we only have evidence for a universe with three space and one time dimensions. In 1989, Landsberg and De Vos [12] proposed a spatial n-dimensional generalization of the Planck distribution, the Wien displacement, and the Stefan-Boltzmann laws for black-body radiation (BBR) for a zero curvature space. Later, Menon and Agrawal [13] modified the n-dimensional Stefan-Boltzmann constant found by Landsberg and De Vos by using the appropriate spindegeneracy factor of the photon without affecting the normalized Planck spectrum given by Landsberg and De Vos. Shortly thereafter, Barrow and Hawthorne investigated the behavior of matter and radiation in thermal equilibrium in an n-dimensional space in the early universe, in particular they calculated the number of particles N , the pressure p and the energy density u [14]. More recently, Gonzalez-Ayala et al. [26] calculated several thermodynamic potentials for BBR such as the Helmholtz potential F , the enthalpy H, the Gibbs potential G and the entropy S by means of the generalized Planck distribution for an n-dimensional Euclidean space. Moreover, they calculated the corresponding densities per hypervolume unit for these potentials; that is f , h, g and s, respectively. They display in the region of very high temperatures (of the order of Planck’s temperature TP ) a convex behavior [26]. Temperatures as high as 10 K are only possible in very early times in the evolution of the universe. It is known that this period was dominated by energy in the form of