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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 a radii of 10^3 Planck length, of space-time for say entropy of about 1000 initially speaking.

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Review of arguments in opposition to Dyson’s alleged prohibition against use of device physics as to determining if Gravitons can be determined to exist is followed up by use of a hot Plasma within a Tokamak in a re do of the modeled generation of amplitude of alleged Gravitational waves. This overlaps with Gravitons, and we follow up with an analysis of the pertinent form of Gravitons, In addition we also obtain Gravitational waves of amplitude as low as (having a strain value) of 10^-26 or 10^- 27 five meters above the center of a Tokamak, as say of the design specifications given in Hefei , China. The Author compares this modeling with specifics as to fusion device physics the author observed in a recent visit to Los Alamos Laboratory, where Dr. Scott Hsu showed him a new form of fusion device, as well as discussions of the onset of turbulence which would greatly enhance device stress and Strain above the limits of the 1975 Grishkuk calculations for GW strain amplitudes for a cold Toroidal geometry of simple E and M fields. The limits of the experimental detection system for GW via Gravitons assumes the use of Chongqing University's Photon Perturbative flux system which is an extension of the Gertsenshtein effect as outlined as of M. E. Gertsenshtein, 1961. \Wave Resonance of Light and Gravitational Waves", JETP,41, 113-114, English translation in Soviet Physics JETP, 14, 84-85 (1962).

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We conflate the existence of a bound to the cosmological constant with Dark Energy, and this formation at the initial phases of inflation, from Pre Planckian to Planckian physics is for later speeding up of the rate of expansion of the universe which is observed a billion years ago. Our attention here, is in use by Klauder's enhanced quantization for a protocol for creating a bound to the cosmological constant, which we say is identical to creation of dark energy. The existence of a cosmological constant, of bounded value, is immediately linked to a small graviton mass (heavy gravity). We end this with an appeal to how the graviton mass, as so formed by the cosmological constant (Dark energy) is linked to the publication, "Can Massive Gravitons be an Alternative to Dark Energy?" Marcio E. S. Alves, Oswaldo D. Miranda, Jose C. N. de Araujo, where we from first principles appeal to a mix of the Alves results, with production of a bound to the cosmological constant, as a link to heavy gravity.

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We find that having the scale factor close to zero due to a given magnetic field value in, an early universe magnetic field affects how we would interpret Mukhanov’s chapter on ‘self reproduction of the universe’ in in his reference “Physical foundations of cosmology” terms of production of inhomogeneity during inflation and its aftermath. The stronger an early universe magnetic field is, the greater the likelihood of production of about 20 new domains of size 1/ H, with H early universe Hubble’s constant, per Planck time interval in evolution. One final caveat to consider. What may happen is that the Camara (2004) density and Quintessential density (Corda et al.) are both simultaneously satisfied, which would put additional restrictions on the magnetic field which in turn affects structure formation. In time, once Eq.(16) of this paper is refined further, the author hopes that some of the issues raised by Kobayashi and Seto as to allowed inflation models may be addressed, once further refinement of these preliminary results commences . We mention fluctuations in the Hubble expansion parameter, H, as given below may affect structure as given in reference [10] below. We close with statements as to the value of alpha in a gravitational potential proportional 1/r^(alpha) and how this adjustment affects the 3 body problem

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