Variation of the fundamental constants over the cosmological time: veracity of Dirac's intriguing hypothesis

Abstract

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.

arXiv comment6 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1211.1936, arXiv:2009.05397, arXiv:1205.2298

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