We use the ideas of a million black holes, at the boundary of contribution to the shift from Pre - Planckian to Planckian physics, as a summed up contribution from one million primordial black holes. I.e. this is assuming a quantum bounce . This is an extension of work done by the author as to explain the nature of a transition from being tiny to when becomes 1 in value. Taking this into account, this article is a way to delineate the physics, inherent in the transition from to which puts a premium upon the growth of the inflaton, due to , with but with changing from , an 10^{255} increase in magnitude. This increase in magnitude may be the driver of subsequent inflation. When we have a pre quantum, especially if the inequality becomes an equality, and then the transition to marks the start of quantum gravity, whereas our black hole entropy model used to obtain a non zero entropy contribution from 1 million primordial relic black holes, as referenced, comes from Dr. Sen in an October 10 Run Run Shaw lecture in Stonybrook University.
Dr. Sen, in 2016 [
N, in this case, is a counting mechanism, for “particles” leaving the event horizon of a black hole and we will have more to say about an alleged counting mechanism later, while r, in this case, is a radial “distance” which is assuming a nonsingular treatment with r, in this case equivalent to an event horizon [
The idea of a 2^{nd} order transition in cosmology can be looked up in [
Take about 1 million black holes behaving as given in Equation (3) and also assume, [
And we will be using in Equation (2)
In addition, from [
Furthermore, Sciama, in 1982 [
Here, if the time is about 10^{−44} seconds (Planck time), then
This would mean then 1 primordial black hole would produce, if the mass of a graviton is 10^{−62} grams [
Or, for a million black holes about 10^{58} gravitons and we would, do the following for change in energy, namely write, from [
Furthermore, we will be assuming, using for Graviton production, that
For the remainder of this document we will be working with
We will be working with Equation (13) to isolate out what we can extract from this, in terms of early universe conditions. The approximation for Gravitons and entropy is based upon, Ng, namely we will, as a start, incorporate Ng’s infinite quantum statistics idea, of entropy being equivalent to a count of particles, i.e. by [
All this will be elaborated upon in the main analysis leading to the change in inflaton values, next.
Given the above, we can write, if we do the math, that we need to do a basic re normalization via Planck units of the above in terms of
Then if
Now if the frequency, initially was of the order of
We get, then that
i.e. the inflaton, nearly zero, in the Pre-Planckian regime, becomes enormously large, right after the phase transition, and we are assuming that the scale factor,
No one knows. It is a seminal question, but Equation (2) is a good imbedding of inflation. i.e. if one uses the Penrose Cyclic conformal cosmology as given in [
The final question to ask, is about the N in the right hand side of Equation (1). It can be viewed, as say the number of operations, for the Universe. i.e. in this sense is a counter point to the [
This work is supported in part by National Nature Science Foundation of China grant No. 11375279.
Beckwith, A.W. (2017) Calculating dg_{tt} at Boundary of Start of Planckian Physics Due to 1 Million Relic Black Holes. Journal of High Energy Physics, Gravitation and Cosmology, 3, 29-33. http://dx.doi.org/10.4236/jhepgc.2017.31005