# A Rough Calculation Relating the Extremely Small Cosmological Constant to the Extremely Large QFT Vacuum Energy Density

Version 2 : Received: 11 January 2019 / Approved: 14 January 2019 / Online: 14 January 2019 (12:23:21 CET)

Version 3 : Received: 28 July 2021 / Approved: 29 July 2021 / Online: 29 July 2021 (14:25:14 CEST)

Version 4 : Received: 6 August 2021 / Approved: 9 August 2021 / Online: 9 August 2021 (12:38:30 CEST)

Version 5 : Received: 30 March 2022 / Approved: 31 March 2022 / Online: 31 March 2022 (14:01:14 CEST)

How to cite:
Cornwall, R. A Rough Calculation Relating the Extremely Small Cosmological Constant to the Extremely Large QFT Vacuum Energy Density. *Preprints* **2019**, 2019010113. https://doi.org/10.20944/preprints201901.0113.v2
Cornwall, R. A Rough Calculation Relating the Extremely Small Cosmological Constant to the Extremely Large QFT Vacuum Energy Density. Preprints 2019, 2019010113. https://doi.org/10.20944/preprints201901.0113.v2

## Abstract

^{st}order removes the “embarrassing” QFT vacuum constant from the Einstein tensor and then covers nearly all of the 120 orders of magnitude difference between the Cosmological Constant and Vacuum Energy by introducing it as an higher order correction in (G/c

^{4})

^{3}. There is a proviso for further work, that the difference of a few orders we calculate, might be made up by considering fluctuations or running constants in the QFT vacuum and Cosmic Inflation.

## Keywords

## Subject

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## Comments (2)

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Commenter:

Commenter's Conflict of Interests: I am the author leaving comments for people to see my thinking and working for a next version of the paper.

A few notes on the work in progress and a guess by dimensional analysis of the result we aim for:

We want to integrate the third term and get something like k^3.Tuv but this is dimensionally incorrect, we would need a factor k^3.Tuv.Factor with units (J/m)^2 or N^2 or (m kg / s^2)^2

Perhaps we might write N^2 as (Nm / m)^2 and seek three factors, one N/m^2, Nm and m.

What would they be? Perhaps the 1st would be pressure or energy density, the 2nd a moment or Energy, (what do we do with the final m?).

What does it all mean? What can it be? The easiest way would be to have (kTuv)^3 suitably contracted but we would find pvac cubed and the whole point of the hunch is to get k^3 * pvac.QFT -> 1 (still a billion off but much better and discussed in the paper as to how we might be missing some vacuum energy)

I suggested the 1st and 2nd factors but their magnitudes (contracted down) would have to be 1 < magnitude < 10^9. It can't be p from the stress energy tensor for the vacuum energy, modelled as an ideal fluid as pvac = p. Perhaps when we contract down/take the trace of the SE of the ideal fluid it isn't pvac - 3p = 0 when taken in global co-ords of the problem (see comment below about integration path and that space is curving along the path)?

What could the 2nd suggestion of the factor be as an energy? Well we are dealing in tensor densities, so that would be odd. Perhaps, as by the paper, I perform an integration path from a volume with a classical vacuum to a real vacuum (alright, not real but a mathematical trick), so it could be the total energy over this (infinitesimal) integration path. Perhaps I could start thinking about a flow of momenergy into the region from the classical space to the real vacuum as modelled...

All very hunchy, very intuitive at the moment. Of course, the whole approach could be wrong... Cubing k^3 gets close but it's still 10^9 out...

Creative guessing = heuristics.

Commenter:

Commenter's Conflict of Interests: I am the author leaving a comment for my thought processes on what is wrong with the paper. Please leave the comments.

I am trying to link vacuum energy with the cosmological constant by the trick of moving from a space with a purely classical vacuum (no QFT vacuum) to one with it. Imagine you are an observer and pass into a region with this ideal fluid... The aim is to perturbatively model this and have the large vacuum constant enter as third order in k (as in k^3.Tuv.Factors) by doing an expansion in Tuv about a point with x = k.deltaTuv. I might then "integrate up" (hence the Factors) and have higher order expressions for Tuv. It gets "close" and suggests that vacuum energy might be higher than calculated from QFT for extra degrees of freedom.

I might be able to get some factor to k^3 by considering (from the point of view of the observer) the flow of momenergy into his environment for the QFT vacuum modelled as an ideal fluid.