Calculating δgtt at Boundary of Start of Planckian Physics Due to 1 Million Relic Black Holes

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1. Introduction

Dr. Sen, in 2016 [1] makes use of a simple black hole generation of entropy analogy which we write as, using Planck units for 3 + 1 dimensional geometry

(1)

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 [2] [3] . We will though for the sake of a model, state that we are fixing say 10^{6} (a million) relic black holes, at the boundary of Pre Planckian to Planckian physics. And that we are when doing that, making the following transformation, as given by [4]

(2)

The idea of a 2^{nd} order transition in cosmology can be looked up in [5] [6] [7] but in fact what we are examining is due to [3] , namely if we are looking at the generation of gravitational waves/gravitons from decay of the following mass via

(3)

Take about 1 million black holes behaving as given in Equation (3) and also assume, [8] , i.e. a quantum bounce, with [8]

(4)

And we will be using in Equation (2)

(5)

In addition, from [9] we will be using the following for the inflaton, if, then

(6)

(7)

Furthermore, Sciama, in 1982 [10] allows us to write the following, namely Sciama [10] in 1982 argued for the lifetime of a black hole, of mass M, that the following holds

(8)

Here, if the time is about 10^{−44} seconds (Planck time), then. If so, then, according to [2] , Calmert, et al. about 0.1% of the energy emitted, in the traditional 4 dimensional black hole (3 + 1 dimensions) would be gravitons. Then, becomes linked to Gravitons according to

(9)

This would mean then 1 primordial black hole would produce, if the mass of a graviton is 10^{−62} grams [11]

(10)

Or, for a million black holes about 10^{58} gravitons and we would, do the following for change in energy, namely write, from [2] , and using [4]

(11)

Furthermore, we will be assuming, using for Graviton production, that, i.e. the Planck length is approximately the same as the event Horizon of the Black hole, that then we will use Equation (1) directly with the result that for 3 + 1 dimensions, we are using if we use Planck length, that

(12)

For the remainder of this document we will be working with

(13)

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 [12]

(14)

All this will be elaborated upon in the main analysis leading to the change in inflaton values, next.

2. Isolation of the Value of the Inflaton, Using Equation (13), Equation (14)

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, if so then we have that we rewrite Equation (13) via

(15)

Then if we can rewrite the Equation (13). To read as follows. If the mass of a graviton is 10^{−}^{62} g, and the value of Planck mass is about 10^{−5} g with Planck mass renormalized by Planck scaling to be 1, then in the Planck rescaling we have

(16)

Now if the frequency, initially was of the order of

(17)

We get, then that

(18)

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, is invariant, in Equation (18). If so then there is a 10^{255} increase in the inflaton, according to Equation (18).

3. Conclusion: Is the Increase of 10^{255} for the Inflaton, a Driver of Inflation?

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 [19] of Seth Lloyd which has a power relationship of the entropy being 3/4^{th} the power of the computational bits. i.e. our suggestion is that perhaps there are many more N computations than was supposed in Seth Lloyds [19] reference.

Acknowledgements

This work is supported in part by National Nature Science Foundation of China grant No. 11375279.

References

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