Gedankenexperiment for fluctuation of mass of a Graviton , based on the trace of a GR stress energy tensor-PrePlanckian conditions lead to gaining of Graviton mass , and Planckian Conditions lead to Graviton mass shrinking to 10 ^-62 grams

Abstract. We will be looking at the energy of a graviton, based upon the Stress energy tensor, and from there ascertaining how fluctuations in early universe conditions impact the mass of a graviton. Physically the mass of the graviton would be shrinking right after Planck time and presumably it would be going to its equilibrium value of about 10^-62 grams, for its present day value. It, graviton mass, would increase up to the Plank time of about 10^44 seconds.

1. Introduction, setting up for calculation of using the results of initial energy as due to 2 and comparing it to a more general energy expression given below Start off with looking at from [1], a generalized energy expression with momentum also obeying, if m is for graviton mass.Begin with from [1] Next, from Giovannini [2], if T is the trace of the Stress-Energy tensor, we have that If so, then, the fluctuation of energy would be represented, if Then If we go to look at what [1] 2 ( ) then is saying, the above is then rendered as 2. Utilizing Eq. ( 6) in terms of the initial fluctuation of the graviton mass.
From [4,5,6,7] use We follow what to expect from as a way to quantify energy density when we have what is coming from Weinberg [6] on initial energy density and then from there to say something about initial time step and also potential energy as given Padmanbhan [7] .Doing so will isolate out values of the Potential energy, as in [6] which will then be compared to [7]'s potential energy value, which in turn gets a value of time, which we will set by first considering the following evolution equation.From [6] 3 ( ) 0 Then, look at () V  from [6] as having the value of, if M is related to mass, with a variable parameter 4 () So, then the  is given by [6]     And also look at Padmanabhan's generalized inflaton potential [7], of comparing Eq.( 2) with Eq.( 12) below We Then, we could get the following variance in time, tt

Finding how to use this value of tt  in order to estimate a relic GW frequency
If so, then, up to a point, in the Pre Plankian regime of space time, according to the signs on Eq.( 13) and Eq.( 14) and [4,5] for the change in Set then, in early universe conditions, let us set, if we are considering gravitons, that we will set, say that the expression below would be for pre Planckian times, with t < 10^ -44 seconds. .The upshot would be that there would be a GW frequency, in many cases, as a result of pre Planckian physics of greater than or equal 10 ^ 32 Hz, which would be red shifted down to about 10 ^ 10 Hz, i.e. a 22 order of magnitude drop, in the present era.This is assuming 2 110 ( ) ~10 a initial  , as well as we are assuming N ~ 10^ 37, as seen in [4,5] The M as given in this would correspond to the Mass value of the universe, which is roughly 3 x 10 55 g ( where g is for grams.).[8] .

Conclusion: Putting Eq. (15) into Eq.(7) . What it says, physically
Note that time in Eq.( 14) remains finite but very small, as it came out less than 10 to the minus 44 power seconds, less than Planck time, with the parameter  usually larger than 2. Time, in Eq. ( 14) as estimate is actually negative, unless we have that we chose in Eq. ( 14) the Pre Planckian option, which is saying that likely Planck time may not be the earliest sub division of time as we know it.This last point above will be important in our future research.As well as entropy production models due to discussions in [9,10,11,12] in terms of entropy generation in the Pre Planckian era.The entropy values will influence the N used in Eq. (15) above.After this is set, for Eq. ( 15) we put Eq.(15) into Eq.( 7) and thereby obtain The first term of Eq.( 16) roughly cancels with the number of gravitons, which approximately leaves  The change in graviton mass is not so much affected by N, entropy count , as this is partly neutralized by the near speed of light conditions, for massive gravitons.What is left though is the variation in total mass, M is divided by Physically what this is saying is that the mass of the graviton would be shrinking right after Planck time and presumably it would be going to its equilibrium value of about 10^-62 grams, for its present day value.It, graviton mass, would increase up to the Plank time of about 10^-44 seconds.
expands during the Pre Planckian space-time regime, and which shrinks right after Planckian time is breached, in the Planckian era ( the Universe begins a massive deceleration.The term  would usually be expected to be less than 2.
have the Hubble parameter, if before Planck time, during Plank time