**Authors:** Andrew Beckwith

We are looking at comparison of two action integrals and we identify the Lagrangian multiplier as setting up a constraint equation (on cosmological expansion). This is a direct result of the fourth equation of our manuscript which unconventionally compares the action integral of General relativity with the second derived action integral, which then permits equation 5, which is a bound on the Cosmological constant. What we have done is to replace the Hamber Quantum gravity reference-based action integral with a result from John Klauder’s “Enhanced Quantization”. In doing so, with Padamabhan’s treatment of the inflaton, we then initiate an explicit bound upon the cosmological constant. The other approximation is to use the inflaton results and conflate them with John Klauder’s Action principle for a way to, if we have the idea of a potential well, generalized by Klauder, with a wall of space time in the Pre Planckian-regime to ask what bounds the Cosmological constant prior to inflation. And, get an upper bound on the mass of a graviton. We conclude with a redo of a multiverse version of the Penrose cyclic conformal cosmology to show how this mass of a heavy graviton is consistent from cycle to cycle. All this is possible due to equation 4. And we compare all this with results of reference [1] in the conclusion. While showing its relevance to early universe production of black holes, and the volume of space producing 100 black holes of say 10^2 times Planck Mass. Initially in radii of 10^3 Planck length, of space-time for say entropy of about 1000 initially speaking. Key words: Inflaton, action integral, Cosmological Constant, Penrose cyclic cosmology, black holes, massive gravitons, enhanced quantization

**Comments:** 19 Pages. Update for the purpose of being a physics source document for 2 articles in Marcel Grossman 15, and 2 articles in DICE 2018, plus a description of gravity to be cherry picked and used in experimental gravity, Marcel Grossman 15

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[v1] 2018-05-22 15:56:30

[v2] 2018-06-04 15:46:27

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