Examination of Schrodinger equation in pre planckian space-time early universe

We look at Viutilli (1999) write up of a generalized schrodinger equation with its Ricci scalar inclusion, in curved space-time. This has a simplified version in Pre Planckian regime, which leads to comparing a resultant admissible wave function with Bohmian reformulations of quantum physics. As was done earlier, we compare this result with a formulation of a modified ‘Poisson’ equation from Poissons and Will from 2014, and then use inflaton physics . The resuting inflaton is then compared to the wavefunctional in the first part of this document. Key words, Ricci tensor, Schrodinger equation, Modified Poisson Equation , massive gravity, inflaton physics.


Viutilli treatment of Schrodinger equation for curved space-time
Here, we bring up [1], and a re set of the Schrodinger equation in early space-time, with curvature.
In [1] we start with Then, after more derivation [1]

Simplifying Eq. (3) in Pre Planckian space-time
We will re write Eq.
(3) to read as follows, with the result that in Pre Planckian-space-time. 2 We are assuming that the term () k flatness approaches zero in Pre Planckian space-time. Obviously if it did not, the last term 2 6 ( ) initial k flatness a would be dominant.
Having said this, with an evolutionary equation statement as to the phase value of  , it is time to look at the initial initial  and to try to learn some physics from it. In order to do this, it may be useful to look at the classical degeneracy argument for forming initial  , and the reference by Rubakov may be useful for this [4] i.e. using a false vacuum analogy, we can write, if q is a generalized space-time unit of "length", and we examine a quartic potential, i.e. look at

Defining the initial value
Now, use the usual given If using Ng infinite quantum statistics, [5], and a non zero massive graviton mass ( massive gravity) [6]   And if we have here that q + is in some sense proportional to length less than or equal to Planck length, the astonishing conclusion is that Eq. (8) would probably be biased toward a low ( nonzero) entropy count, which would mean for finite initial entropy, connected with information transfer, that we would through Eq. (8) and Eq. (9) have a bias toward initially low, say 10^5 or so initial entropy, as a way to quantify the input into the formation of an initial wavefunction, using entropy as equivalent to information as given by Lloyd [7]

Comparing the inputs into Eq(5), Eq.(6), Eq(8) and
Eq. (9) against force against individual 'gravitons' in the Pre Planckian space-time We will go to the [3] reference, page 85, in order to look at a change in the stress energy tensor, 00 0 (0) , Using this, and stating that in the Pre Planckian regime of space-time due to [8], that 0 , 0 j j T  , then if so, using [3] and [8] ( ) Here, we have that the initial volume would be less than the cube power of a Planck length [9] , but not zero, whereas the initial time would be less than, Planck time, but not equal to zero, [9]. For our analysis in the Pre Planckian regime, we will specify 2 initial a as the square of a nonzero initial scale factor for a nonsingular regime of space-time for General relativity with a value as given by [10], and elaborated upon in [11]. In addition we have that the inflaton, as given in Eq. (11) is explained in context by [12] , and we will use an argument below as to how a nonzero graviton mass is linkable to a non zero initial radii, which in turn will state that it is highly unlikely that which is proportional to a radial distance, cubed, goes to zero.. Note that Eq. (11) is also an argument as to why there would be a finite, not almost "infinite" initial value of entropy, so then that Eq. (8) would not go to zero.
Having said that, let us use a semi classical argument as to why the radii would not go to zero, even in Pre Planckian space-time. This 8 would be to insure that ( ) 3 V  would not go to zero, even in Pre Planckian space-time

5.
What is important about the modified Poissons equation [12]? Getting a non zero initial We will first of all refer to two necessary and sufficient conditions for the onset of a massive graviton given in [13] and combined with Padmanablan's reference [12].
I.e., what we will be doing is to re do the reference calculations given in [13] Here, we will be using in the Pre Planckian potential the inputs from the data usually associated with [12] ( ) In other words, we will be using the inflation given by If so, then ( ) Then, after algebra, we have the following , from [14] ( ) The quadratic Equation this engenders is, how to say.
A candidate for the density functional will come next, with the way of obtaining a critical value for r is given by [14] as follows, i.e. if As far as applications to: [1] ( ) This would, lend itself to a quadratic equation for r, and the cube of r would be proportional to What we are examining if our qualitative argument which in sum yields a Pre Planckian wavefunction as compared against the construction given in [15] which is in spirit comparable, up to a point with [16] According to [15] (20) Here, ( S action ) would be the same as Eq. (7), whereas we have V as given by Eq. (18), and then we have Here, M would be given by Eq. (9) , i.e. and V were given by Eq. (18) we would find to a point that Furthermore, we have that analysis of Eq. (24) may be in tandem with analysis of the Corda paper [20] as to if gravity is possibly scalartensor, an extension of GR (possibly with some semi classical treatment of presumed quantum gravity formulations), or something else Finally, since our paper is with respect to relic conditions, if so, then if we are to use a variant of interferometer methods, reference [21] and [22] if relic conditions are observable, via some form of space bound system, may allow us to with refinements get enough control of stochastic noise contamination of GW and the foot print of massive gravity to come up with confirmable data sets as to early universe conditions. With luck, with considerable refinement of instrumentation, we may also be able to get experimental confirmation of [23] and its predictions as to inflaton physics and possibly massive gravity

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