Using “Enhanced Quantization” to Bound the Cosmological Constant, (for a Bound-on Graviton Mass), by Comparing Two Action Integrals (One Being from General Relativity) at the Start of Inflation

Abstract

The first result from 2018 is looking at two action integrals and also a Lagrangian multiplier as a constraint equation (on cosmological expansion). In doing so, with Padmanabhan’s version version of an inflaton, we then have a bound upon the cosmological constant. For the record, this is in fidelity with the author’s publication, in JHEPGC, entitled “Using ‘Enhanced Quantization’ to Bound the Cosmological Constant, and Computing Quantum Number n for Production of 100 Relic Mini Black Holes in a Spherical Region of Emergent Space-Time” which was in 2018. And was the genesis of the two integral comparison idea. The second result from 2018 is to use the inflaton results and conflate them with John Klauder’s action principle for a way to 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. The third result from 2018 and the first cited reference is a redo of a multiverse version of the Penrose cyclic conformal cosmology to show how this mass of a heavy graviton inconsistent from cycle to cycle. The fourth result from 2020 is to ask if we can, using an idea from a publication by Diosi, in the Dice 2018 physics conference use a high energy comparison of Planck Length, and a De Broglie wavelength, to find out if we can extract from our estimate of Planck mass a statement as to entropy of the early universe, and the fifth result from 2020 is to comment upon a comparison of the power of the entropy result so obtained with the number of e foldings arising in inflation. This last question we view as essential for answering if there is a foundation of inflation which is linked to quantum gravity. We wish to avoid the anthropic principle in setting initial conditions for the massive graviton, which is why we referenced the modification of the Penrose CCC theory in part of our manuscript.

Appendices

Appendix A: Infinite Quantum Statistics as Given by Jack Ng

Jack Ng changes conventional statistics: he outlines how to get S ≈ N, which with additional arguments we refine to be S ≈  < n> (where <n> is graviton density). Begin with a partition function (Beckwith 2014, 2018; Ng 2008)

$$ {Z}_N\sim \left(\frac{1}{N!}\right)\cdot {\left(\frac{V}{\lambda^3}\right)}^N $$

(3.24)

This, according to Ng, leads to entropy of the limiting value of, if S = (log[Z_N])

$$ S\approx N\cdot \left(\log \left[V/N{\lambda}^3\right]+5/2\right)\underset{\mathrm{Ng}\hbox{-} \operatorname{inf}\;\mathrm{inite}\hbox{-} \mathrm{Quantum}\hbox{-} \mathrm{Statistics}}{\to }N\cdot \left(\log \left[V/{\lambda}^3\right]+5/2\right)\approx N $$

(3.25)

Appendix B: Micro Black Hole, at the Start of the Universe and Their Contribution to Early Universe GW Generation Via an Entropy Count

This is in partial fidelity to Beckwith (2018) and may be added as a factor in future entropy contributions as a secondary effect which may affect future analysis of this phenomenon. We begin first with what would transpire for micro black holes, at the start of the space-time regime. That is the first level of analysis we have done in the main document purports to create graviton type disturbances at or before the electroweak regime of space-time. The addition of entropy so given here is meant to be included, pending an evaluation as to how many primordial black holes may be considered to be created. If the number is large (many micro black holes), then what we have below will significantly add to entropy. Or it may not be a decisive factor. The analysis is included for this document as a secondary effect for entropy generation. The idea would be that we would have, for a quantum level, n, specified for a black hole, due to what Corda developed the following temperature distribution which would ALSO add into more entropy. We include it in as a future works project. The first term below comes from Dyson (1966) and is linkable to a way to also, in addition to the mechanism so brought up a way to also add more early universe entropy, which is linkable to gravitons.

$$ {\displaystyle \begin{array}{l}{T}_{\mathrm{H}}={m}_D\cdot {\left(\frac{m_D\cdot \left(n+2\right)}{m_{bh}\cdot 8\cdot \Gamma \left(\frac{n+3}{2}\right)}\right)}^{\left(1/n+1\right)}\cdot \left(\frac{n+1}{4\sqrt{\pi }}\right)\\ {}{m}_D\doteq 1\;\mathrm{TeV}\equiv {10}^{12}\kern0.37em \mathrm{eV}=1.783\times {10}^{-30}\;\mathrm{g}\\ {}{m}_{bh}\doteq 1.22\times {10}^{21}\;\mathrm{TeV}=2.175\times {10}^{-9}\ \mathrm{g}\\ {}{T}_{\mathrm{H}}\propto 1\;\mathrm{TeV}\cdot {\left(\frac{\left(n+2\right)}{1.22\times {10}^{21}\cdot 8\cdot \Gamma \left(\frac{n+3}{2}\right)}\right)}^{\left(1/n+1\right)}\cdot \left(\frac{n+1}{4\sqrt{\pi }}\right)\end{array}} $$

(3.26)

We look at the initial state of created gravitons, and use the physics given in Hawking (1974) with the specified Hawking temperature, as the main physics phenomenon of interest to our analysis. We then can, if we have this Hawkings temperature, as given in Eq. (3.7) consider the question of first, if the black holes have classical or quantum behavior as well as Γ being a gamma function, i.e., look at what is given in Beckwith (2018), in its conclusion which we will cite here. That is, the idea is based upon the formation of a finite number of black holes, which decay. Quoting Beckwith (2018) we have that we will be looking at the following:

Quote, Beckwith (2018)

“Our physics is simplified if we change Planck length to be scaled as 1 and we look at a ‘unit’ evaluated space-time volume.” Then we can set n of Eq. (3.26) equal to zero initially and obtain the following from Corda (2018) and Beckwith (2018)

$$ {\displaystyle \begin{array}{l}16\pi \left(\frac{{\tilde{n}}_{\mathrm{qm}}}{4\times {10}^4}\right)\cdot \left|\sqrt{1-\left(\frac{{\tilde{n}}_{\mathrm{qm}}-1}{{\tilde{n}}_{\mathrm{qm}}}\right)}\right|\approx {10}^6\\ {}{m}_{bh}\approx {10}^2\times {m}_{\mathrm{planck}}\left(4-\dim \right)\\ {}\Delta {V}_{\mathrm{total}}\simeq {10}^2\times \Delta {V}_{{\tilde{n}}_{\mathrm{qm}}-1\to {\tilde{n}}_{\mathrm{qm}}}\approx {10}^2\times 16\pi \left(\frac{{\tilde{n}}_{\mathrm{qm}}}{4\times {10}^4}\right)\cdot \left|\sqrt{1-\left(\frac{{\tilde{n}}_{\mathrm{qm}}-1}{{\tilde{n}}_{\mathrm{qm}}}\right)}\right|\approx {10}^8\end{array}} $$

(3.27)

This puts a serious restriction on the number of allowed quantization levels ${\tilde{n}}_{\mathrm{qm}}$, but it also means that within this horizon space we may be seeing mini black holes created which could release gravitons. We will then discuss what may be pertinent to characterizing if the black holes are behaving classically or quantum mechanically. Note, if n in Eqs. (3.26) and (3.31) is set equal to zero, we have that if we literally interpreted Eq. (3.26), with n equal zero (4 dimensions) with a 10² plank mass (in four dimensions) black hole, that we would have

$$ {T}_{\mathrm{H}}\propto 1\;\mathrm{TeV}\cdot \left(\frac{1}{1.22\times {10}^{21}\cdot 2\cdot \sqrt{\pi }}\right)\cdot \left(\frac{1}{4\sqrt{\pi }}\right) $$

(3.28)

Noticeably we have a set of reality problems to attend to. Hawking radiation and the Ng (2008) paradigm alter our process so that we obtain the following results as given in Eq. (3.29) for Hawking blackbody style results.

Diosi (2019) and Calmet (n.d.) will add further refinement as to the physics of Eq. (3.29) and hopefully alter them to reflect more of the known observational physics, since Eq. (3.29) is a greatly over simplified version of the gravitational physics input into observational gravitational astronomy. We hope in doing so we obtain data sets as to confirm or falsify our hypothesis as given in this document.

$$ {\displaystyle \begin{array}{l}{T}_{\mathrm{H}}\left(\mathrm{normal}-4\dim \right)=\frac{\mathrm{\hslash}{c}^3}{8\pi {Gm}_{bh}{k}_{\mathrm{B}}}=6.17\times {10}^{-8}\;\mathrm{K}-\mathrm{if}\;{m}_{bh}=1-\mathrm{solar}\hbox{-} \mathrm{mass}\\ {}{T}_{\mathrm{H}}\left(\mathrm{normal}-4\dim \right)=\frac{\mathrm{\hslash}{c}^3}{8\pi {Gm}_{bh}{k}_{\mathrm{B}}}\simeq 6.17\times {10}^{28}\;\mathrm{K}-\mathrm{if}\;{m}_{bh}={10}^2\times {m}_{\mathrm{planck}}\left(4-\dim \right)\\ {}{T}_{\mathrm{H}}\left(\mathrm{normal}-4\dim \right)=\frac{6.17\times {10}^{28}}{11,604}\;\mathrm{eV}-\mathrm{if}\;{m}_{bh}={10}^2\times {m}_{\mathrm{planck}}\left(4-\dim \right)\\ {}{T}_{\mathrm{H}}\left(\mathrm{normal}-4\dim \right)=5.317\times {10}^{12}\;\mathrm{TeV}-\mathrm{if}\;{m}_{bh}={10}^2\times {m}_{\mathrm{planck}}\left(4-\dim \right)\end{array}} $$

(3.29)

The sun is 9.8 times 10³⁷ Planck masses, so this means that the Hawkings temperature of a 10² times four-dimensional Planck mass black hole, as postulated here would be then about 10³⁶ times higher, 1 eV in Kelvin is 11,604 K, hence if we have only n = 0 we really would prefer not to use Eq. (3.28). Hence, using Eq. (3.29). we can then go to Hawking (1974) for entropy, i.e., we have then that

$$ {\displaystyle \begin{array}{l}{S}_{bh}=\frac{1+n}{2+n}\cdot \frac{m_{bh}}{T_{bh}}=\frac{1+n}{2+n}\cdot \frac{10^2\times {m}_{\mathrm{planck}}\left(4-\dim \right)}{5.317\times {10}^{12}\;\mathrm{TeV}}\approx \frac{1+n}{2+n}\cdot \frac{10^2\times 1.22\times {10}^{16}\;\mathrm{TeV}}{5.317\times {10}^{12}\;\mathrm{TeV}}\\ {}{S}_{bh}\approx \frac{1+n}{4+2n}\cdot {10}^4\underset{n\to 0}{\to }{10}^3\end{array}} $$

(3.30)

That is, if we had 10² black holes, of mass about 10² times a four-dimensional black hole, we probably would be looking at an initial entropy of about 10⁵. Then using Kolb and Turner (1991), we would see

$$ s\left(\mathrm{entropy}\hbox{-} \mathrm{density}\right)=\frac{2{\pi}^2}{45}{g}_{\ast}\cdot {\left({T}_{\mathrm{universe}}/{T}_{\mathrm{Planck}}\right)}^2 $$

(3.31)

This postulates that relic, initially formed black holes would be formed, say in a one-meter radius ball about 10⁻²⁷ s, after the onset of inflation and that we would see the rapid decay of these micro-sized black holes in less than 10⁻²⁷ s.

End of quote from Beckwith (2018)

This construction above, with suitable work later on, will be useful in removing the anthropic principle from serious consideration in cosmology (Barrow and Tipler 1988) Once again, the relevance to the analysis given in the main text is heavily flavored as to precisely how many primordial black holes are created. If it is just 10², the number of primordial black holes, then this is a very minor addition to the entropy. If the number of primordial black holes is significantly higher, then the contribution is different and then alters the calculations in potentially significant ways. See Ruutu et al. (1996) as to another model of what may transpire as yet another effect if there are many primordial black holes, creating early universe turbulence which may contribute (the turbulence) to entropy generation, i.e., in a cosmological scale replicating (Ruutu et al. 1996) and cosmic string representations of black holes (t’Hooft n.d.) in the early universe.