The Age of the Universe Predicted by a Time-Varying G Model?

Based on previous work, it is shown how a time varying gravitational constant can account for the apparent tension between Hubble’s constant and a newly predicted age of the universe. The rate of expansion, about nine percent greater than previously estimated, can be accommodated by two specific models, treating the gravitational constant as an order parameter. The deviations from ΛCDM are slight except in the very early universe, and the two time varying parametrizations for G lead to precisely the standard cosmological model in the limit where, 0 G G → (cid:31) , as well as offering a possible explanation for the observed tension. It is estimated that in the current epoch, 0 0.06 G G H = − (cid:31) , where 0 H is Hubble’s parameter, a value within current observational bounds.

example, we refer the reader to reference [4]. In particular, see Figure/Graph 6a, in Appendix D, in that reference, which we reproduce here as Figure 1. This figure replicates exactly the newest age without any modifications or revisions of existing cosmological parameters as determined by the Planck XIII collaboration [5]. Figure 1 gives the "look back" times for three specific models, the ΛCDM model, model A, and, model B, as a function of the cosmic scale parameter, "a".
Models, A, and B, are two specific time-varying G models. Specifically this figure graphs the age correction factors, F, for the various models under consideration.
The specific values, as determined in reference [4] The two models, A and B, were introduced in reference [4] in order to explain the quintessence parameter, 0.98 w = − , versus 1 w = − , which holds in the ΛCDM model. We also sought to explain the cosmological constant fine tuning problem, the discrepancy between the present day very weak value of the cosmological constant, and the much greater vacuum energy found in earlier epochs. We assumed a connection exists, and that the cosmological constant is a characteristic of the vacuum. The present-day observed value is, In order to derive, means of two separate functions, which we distinguished as model A, and model B. The cosmic scale parameter, "a" is related to CMB temperature, T, by is a constant to be determined, having units of degrees Kelvin, 0 2.275 K T = , and "a" is our scale parameter. In the present epoch, "a" = 1, and thus, 0 is the saturation value of 1 G − , applicable in the limit where the CMB temperature approaches zero, or equivalently, when "a" approaches infinity.
In model B, we have correspondingly, In Equation, (2), ( ) L x is the Langevin function, defined by the equation be determined, having units of degrees Kelvin, and, 0 Thus, for any scale parameter, "a", Equations (1) and (2) Beyond this temperature, gravity, as we currently know it, did not exist.
Therefore gravity is considered to be an emergent "low energy" phenomenon, which came into being at very high temperatures. We also claimed that, if G is not fundamental, then there is nothing fundamental about the Planck scale. It is to be noted that even though models, A, and B, are underpinned by two different functions, the inception of gravity is roughly at the same temperature. We consider this to be more than a coincidence. The temperatures indicated by Equations, (5), and, (6), are well below the Planck temperature of, 1.42 × 10 32 K, but well above the temperature of The details are presented in reference [4].
With these two time varying G models, we can make a case for the quintessence parameter, 0.98 w = − . We can also show that the cosmological constant, Λ , scales. In fact, The subscript 0 refers to current epoch values, and ρ Λ is the dark energy mass density. In the earliest times, G is proportional to temperature, T, in both models, A, and B. We can furthermore show that, at inception, Both values are similar and quite large. We believe, nevertheless, that there could very well be a twenty order of magnitude increase in G at the extremely high temperatures, indicated by Equations (5), and (6). We substitute Equations, (12), and, (13), into relation, (11). This gives We have instead a more modest 41 order of magnitude increase as indicated by Equations, (14), and, (15).
With these models A, and B, it now appears, that in addition to deriving value, then we obtain the ages as indicated above in our figure /graph. This may provide the solution to the "tension" problem currently being discussed. We went into great detail in reference [4]  replaced by a scalar field, as first suggested by Jordan [16]. In fact, we find that in this instance, here G M is the mass of the hypothetical massive particle representing gravity, The scale factor, s, can be found using Equations, (9), and (10)

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.