1. Basic Idea, Can Two First Integrals Give Equivalent Information?
Our supposition is that if we wish to make an equivalence between two action integrals, i.e., first integrals that we need to have a 1 to 1 and onto linkage between the integrands, in the two cases so referenced.
To do this, we are making several assumptions.
1) The two mentioned integrals are evaluated from a Pre Planckian to Planckian space-time domain, i.e. in the same specified integral of space-time.
2) In the process of doing so, the Universe is assumed to avoid the so called cosmic singularity. In doing so, assuming a finite “Pre Planckian to Planckian” regime of space time is similar to that given in   .
2. Specifying the Particulars of the Two First Integrals in Pre-Planckian to Planckian Space-Time
Before proceeding, it is advisable to define some of the symbols which will be used in the integrals and the integrands in our document.
First of all, we have what is known as a scale factor. Which is nearly zero, in the Pre Planckian regime of space-time if we assume  does not hold, and that is 1 in the present era. A good reference as to the physics behind how we set up is  . In addition we will define, for the purpose of analysis, of the integrals, the following symbols as given in  , for the Quantum paths sensitive first integral, with
These are the purported volume elements of the  first integral. The second first integral is using the usual GR inputs as defined by Padmanbhan in  . To review what is meant by first integrals we refer the readers to     .
Roughly put, according to    a Lagrangian multiplier invokes a constraint of how a “minimal surface” is obtained by constraining a physical process so as to use the idea of    which invokes the idea of minimization of a physical processes. In the case of  , the minimization process is implicitly that, if were a scale factor as defined by Roos,  and if were a time com- ponent of a metric tensor, which we will later define via   .
Here, the subscripts 3 and 4 in the volume refer to 3 and 4 dimensional spatial dimensions, and this will lead to us writing, via  a 1st integral as defined by   , in the form, if G is the gravitational constant, that if we have following  , a first integral defined by
This should be compared against the Padmabhan 1st integral  of the form, with the third entry of Equation (3) having a Ricci scalar defined via  and usually the curvature set as extremely small, with the general relativity version of
Also, the variation of as given by   will have an inflaton, given by 
Leading to 
Here, we have that is a minimum value of the scale factor presumably given by  as a tiny but non zero value. Or at least a quantum bounce as given by  .
The innovation we will be looking at will be in comparing a 1-1 and onto equivalence, i.e. an information based isomorphism between 1st integrals with a nod to 
We will be making a simple equivalence between the two first integrals via Equation (6) assuming that even in the Pre Planck-Planck regime that curvature will be a very small part of Ricci scalar and that to first approximation even in the Plank time regime, that to first order  has a value altered to be
This last approximation will make a statement as to applying Equation (6) far easier may not be defensible, but we will use it for the time being.
2.1. Comparison between Equations ((2) and (3) with (5)-(7))
In order to obtain maximum results, we will be stating that the following will be assumed to be equivalent.
If the term is indeed a constant (i.e. we avoid Quinessence, and the vacuum energy is invariant), then Equation (10) puts a profound restriction upon which will be elaborated upon in the next section. i.e. for the sake of Argument we will make the following assumptions which may be debatable, i.e.
is approximately a constant. (11)
For extremely small time intervals (in the boundary between Pre Planckian to Planckian physics boundary regime).
The next section will be investigating the physical implications of such assumptions.
2.2. What We Can Extract in Physics, If Equations (9)-(12) Hold?
Simply put a relationship of the Lagrangian multiplier giving us the following:
If the following is true, i.e. in a Pre Plankian to Planckian regime of space- time
Then what has been done is to conflate the Lagrangian as equivalent to which if is also a constant is implying that the cosmological constant is obtaining for us the consomological constant value chosen as a precursor for (DE?) expansion of the universe, as given in the scale factors as of Equation (9) and Equation (8). i.e. what we are inferring then is similar to a result assumed by Padmanabhan, in  .
But what is noticeable is that the inflaton equation as given by Padmanabhan  hopefully will not be incommensurate with the physics of the Corda Criteria given in the Gravity’s breath document  . Keep in mind the importance of the result from reference  below which forms the core of Equation (15) below
Furthermore, we should keep in mind the physics incorporated in   , i.e. as to the work of LIGO. i.e. it is important to keep in mind that in addition,  has confirmed that a subsequent analysis of the event GW150914 by the LSC constrained the graviton Compton wavelength of those alternative theories of gravity in which the graviton is massive and placed a level of 90% confidence on the lower bound of 1013 km for a Compton wavelength of the graviton. Doing this sort of vetting protocols in line with being consistent with investigation as to a real investigation as to the fundamental nature of gravity. This is a way of confirming and showing via experimental data sets if general relativity is the final theory of gravitation. i.e., if massive gravity is confirmed, as given in  , then GR is perhaps to be replaced by a scalar-tensor theory, as has been shown by Corda.
This work is supported in part by National Nature Science Foundation of China grant No. 11375279.
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