  We investigate how the universal constants, including the fine structure
constant, have varied since the early universe close to the Planck energy scale
($E_P\sim 10^{19}$GeV) and, thus, how they have evoluted over the cosmological
time related to the temperature of the expanding universe. We have already
shown that the speed of light was much higher close to the Planck scale. In the
present work, we will go further, first by showing that both the Planck
constant and the electron charge were also too large in the early universe.
However, we conclude that the fine structure constant ($\alpha\cong 1/137$) has
remained invariant with the age and temperature of the universe, which is in
agreement with laboratory tests and some observational data. Furthermore, we
will obtain the divergence of the electron (or proton) mass and also the
gravitational constant ($G$) at the Planck scale. Thus, we will be able to
verify the veracity of Dirac's belief about the existence of "coincidences"
between dimensionless ratios of sub-atomic and cosmological quantities, leading
to a variation of $G$ with time, i.e., the ratio of the electrostatic to
gravitational force between an electron and a proton ($\sim 10^{41}$) is
roughly equal to the age of the universe divided by an elementary time
constant, so that the strength of gravity, as determined by $G$, must vary
inversely with time just in the approximation of lower temperature or for times
very far from the early period, in order to compensate for the time-variation
of the Hubble parameter ($H\sim t^{-1}$). In short, we will show the validity
of Dirac's hypothesis only for times very far from the early period or $T\sim
T_P\sim 10^{32}$K.
