This idea is useful to link the matter era to the earlier radiation era. It will be emphasized in the next subsection 3.2 that the radiation field, almost mono-chromatic at the beginning of the radiation era, turned into a more complex spectrum of wavelengths because of the concurrent expansion of the universe; so quantum fluctuations and possible events of constructive interference, statistically allowed to occur anywhere in the radiation field, promoted favorable conditions to form local couples of virtual particles uniformly distributed in the available volume of the early universe.
over all the elementary volumes [V.sub.0], it will be shown later that an effective entropy driven mechanism in fact marked the transition from the radiation era to the matter era; so the sum of (3,15) reads actually
On the one hand these considerations are interesting because [P.sub.in] controls the expansion of the universe, as it will be shown below; on the other hand the idea of [V.sub.0] bulk states allowed to protons and antiprotons, although suggested by the numerical values of (3,12) only, is attracting because it links radiation era and matter era, at the beginning of which couples of matter/antimatter particles were in fact formed.
The previous ways to estimate T refer to the time where early hadrons began to form everywhere in the radiation field of such universe and indicate a temperature corresponding to a uniform distribution of virtual couples occupying the available states at the end of the radiation era. The same equations could in principle estimate the local T even during the subsequent matter era, when the bombardment with energetic neutrons allowed forming heavy elements; yet the concurrent clustering of matter determined a structure of the universe locally inhomogeneous, so at that later time a unique average T does no longer make sense.
So radiation density, radiation energy density and pressure during the radiation era read
At the beginning of the radiation era, therefore, [DELTA]r = [square root of hG/[c.sup.3]] with [lambda] = [DELTA]r and n = 1 has the expected order of the Planck length with which in effect has been calculated the Planck volume [V'.sub.0].
So, trying to understand the physical meaning of these results beyond the numerical estimates, the radiation era was just after the very early time step of the creation of radiation just concerned; this initial step can be therefore nothing else but the Planck era.
Actually particles and antiparticles with the same [m.sub.p] concurrently formed after the radiation era have statistically the same probability of being found in the boundary state; if so, the initial configuration of coexisting protons and antiprotons uniformly occupying all available bulk states generates subsequently a boundary halo of virtual couples plus possible annihilation photons along with corresponding vacuum states and matter states in the bulk universe.
This section describes a mechanism really possible soon after the end of the radiation era; the couples proton/antiproton just formed from the very hot radiation field have actual physical meaning, instead of being mere statistical entities suggested by (3,12).
It has been already estimated that just after the radiation era T was of the order of [10.sup.32]/[10.sup.33] K; this range of values seems high enough to account for a Dirac-like process.