Fixing the Vacuum Catastrophe

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Stuart LaForge

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Jul 22, 2022, 9:05:03 PM7/22/22

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So I have figured out a method that uses Quantum Harmonic Oscillators to fix the vacuum catastrophe. Basically QFT models the quantum vacuum as a field of an infinite number of quantum harmonic oscillators (QHO) of every possible frequency located at every point in space.

Then when they do the math, they either get an infinite energy density for the vacuum, or when they try to constrain the problem, a finite energy density that is still 122 orders larger than the empirically determined value of about a nanojoule per cubic meter.

Here are some references for the vacuum catastrophe and why it is a problem:

Adler, R.J., Casey, B. and Jacob, O.C. (1995) Vacuum Catastrophe: An Elementary Exposition of the Cosmological Constant. American Journal of Physics, 63, 620-626.

Persinger, M. (2014) A Possible Explanation for the Vacuum Catastrophe. International Journal of Astronomy and Astrophysics, 4, 178-180.

https://www.scirp.org/pdf/IJAA_2014031211294319.pdf

https://www.youtube.com/watch?v=n6jAOV7bZ3Y

The fix requires several new assumptions that act as additional boundary conditions on the problem:

Assumptions

1. Our causally connected patch of the universe has a fundamental frequency $\omega_0 = 2 \pi H$ that changes in time in relation to the causal volume's change in size.

2. The Planck frequency, $\omega_P = 2 \pi / t_P$ , where t_P is the Planck time, is the maximum possible frequency in the universe that does not get censored by event horizons. It serves as the upper bound of our calculation.

3. Not all vibrational frequencies are possible. The only allowed frequencies are the harmonics, i.e. integer multiples, of the fundamental frequency $\omega_k = 2 \pi k \omega_0$ . It is presumed that the other infinite vibrations of the field cancel each other out by destructive interference as per Feynmann. Or maybe it is the result of something analogous to Pauli's exclusion principle but for virtual particles.

4. The universe is a very large hypersphere with a very small positive curvature k. According to the final analysis of the Planck satellite data, the team is 99% confident that the universe is closed with $-0.095 < \Omega_k < -0.0007$ .

Here is a supporting reference for this geometry for the universe:

Di Valentino, Eleonora & Melchiorri, Alessandro & Silk, Joseph. (2020). Planck evidence for a closed Universe and a possible crisis for cosmology. Nature Astronomy. 4. 1-8. 10.1038/s41550-019-0906-9.

https://www.nature.com/articles/s41550-019-0906-9.epdf?sharing_token=YcJkhzluLn0ILDMuy2MTzdRgN0jAjWel9jnR3ZoTv0NqA1r1vCcOgNn4LDpENPnwoeHNKdiccffHhZYAzAfWiraW4OXgPFDNmN-1T_5jg5x-qeRUqoEyxmGP-5JmiD4yVerC96a1oeUikU4IRQ0E1gfAlqExIriLQStA5l3rf8Q_KNuv3me3U7m2yiWBIOJEUA8g95wO2NwZXRrVL_NGcg3RL7pOO6larXFp2QMQC6_aX2ac9g3dkOeaERiMuX81YeIiWXsvproalIsufMz_Ag%3D%3D&tracking_referrer=www.space.com

Derivation

Now using those four assumptions or boundary conditions, the equation for the quantum harmonic oscillator (QHO) is:

$E_n = (n + \frac{1}{2}) \hbar \omega$

setting n to zero gives us the formula for the ground state of a single QHO then adding the ground state of all possible frequencies gives us:

$E_0 = \sum \limits_{k} \frac{\hbar \omega_k}{2}$

Since k represents all possible wave vectors in the field, previously this led to unmanageable infinities. Transforming the equation into a continuous integral did not help matters much, allowing for finite but nonsensical numbers 122 orders of magnitude too large..

However, applying my previously stated boundary conditions to the equation gives us a new formula that does not generate infinite energies from fields enclosed by event horizons. Note that k has gone from being the index of a wave vector to being the k-th harmonic of the fundamental frequency.

$E_0 = \sum \limits_{k = 0}^{k_{max}} \frac{\hbar \omega_k}{2} = \sum \limits_{k = 0}^{k_{max}} \frac{\hbar k \omega_0}{2}$

Since \omega_0 can be considered constant during the time frames associated with measurement, one can remove it and the other constant factors from the sum. We can also substitute $k_{max} = \frac{\omega_P}{\omega_0}$

into our upper bound. Since k is an integer ranging from 0 to the highest allowed harmonic of the QHO, the sum reduces to nth triangular number.

$T_n = \sum \limits_{k = 0}^{n} k = \frac{n (n + 1)}{2} = \frac{n^2 + n}{2}$

$E_0 = \frac{\hbar \omega_0}{2} \sum \limits_{k = 0}^{\frac{\omega_P}{\omega_0}} k$

$E_0 = \frac{\hbar \omega_0}{2} \left[ \frac{\left(\frac{\omega_P}{\omega_0} \right)^2 + \frac{\omega_P}{\omega_0}}{2} \right]$

$E_0 = \frac{\hbar}{4} \left (\frac{\omega_P^2}{\omega_0} + \omega_P \right)$

At this point, we can apply our initial boundary conditions and definitions, $\omega_0 = 2 \pi H$ and $\omega_P = \frac{2 \pi}{t_P} = 2 \pi \sqrt{\frac{c^5}{\hbar G}}$ and substitute them into our energy equation and then simplifying yields the following expression for the total zero-point energy for the vacuum of our causal space:

$E_0 = \frac{\hbar}{4} \left (\frac{2 \pi c^5}{H \hbar G} + 2 \pi \sqrt{ \frac{c^5}{\hbar G}} \right ) = \frac{\pi}{2} \left ( \frac{c^5}{H G} + \sqrt{\frac{\hbar c^5}{G}} \right ) \approx 2.6 \times 10^{70} J$

So now we have the zero point energy of our causal space, we still need to figure out $\varepsilon$ which is the energy density of our causal space. For that, we need to know the volume of our causal space. Since from the Nature article referenced above, there is a 99% chance the universe is closed, we will posit a 4-D hypersphere for the shape of a causal space. We assume that the radius of the hypersphere is the same as the radius of our Hubble sphere which defines our causal space.

The Hubble radius is $R_H = c / H \approx 1.4 \times 10^{10} ly$ or about 14 billion light years.Since the current Euclidean 3-D Hubble volume only encompasses our plane of simultaneity, our now, it is only meaningful in the present moment. If we want to know our entire causal volume, past, present and future then we need to use the 3-D surface area of the 4-D hypersphere as our volume. The equation for this "surface volume" is $S_{3D} = 2 \pi^{2} r^{3}$ .

We plug the expression for the Hubble radius into our volume expression to give us the total volume of our causal space:

$V_{CS} = S_{3D}(R_H) = 2 \pi^{2} \frac{c^3}{H^3}$

Now we can determine the energy density of the vacuum $\varepsilon$ as follows:

$\varepsilon = \frac{E_0}{V_{CS}} = \frac{\frac{\pi}{2} \left (\frac{c^5}{H G} + \sqrt{\frac{\hbar c^5}{G}} \right )}{2 \pi^{2} \frac{c^3}{H^3}}$

Cancelling terms and simplifying gives us our final expression for the energy density of our causally connected vacuum:

$\varepsilon = \frac{1}{4 \pi} \left (\frac{H^{2} c^{2}}{G} + H^{3} \sqrt{\frac{\hbar}{G c}} \right ) \approx 5.2 \times 10^{-10} J$

This expression gives us a value very close to experimental observation. We can perform a final sanity check on our result by finding out what fraction of the total mass-energy of our causal space this number represents. We do this by calculating omega $\Omega$ , the unitless cosmic density parameter which is the ratio of the calculated energy density of our vacuum over the critical energy density of the universe for closure.

$\Omega = \frac{\varepsilon}{\varepsilon_{c}} = \frac{\varepsilon}{\rho_{c} c^2}$

Here $\rho_c = \frac{3 H^2}{8 \pi G}$ is the cosmological critical mass density.

Plugging in our known expression for the energy density and simplifying gives gives us:

$\Omega = \frac{\frac{1}{4 \pi} \left (\frac{H^{2} c^{2}}{G} + H^{3} \sqrt{\frac{\hbar}{G c}} \right )}{\frac{3 H^2 c^2}{8 \pi G}} = \frac{2}{3} \left (1 + H \sqrt{\frac{G \hbar}{c^5}} \right )$

We can simplify omega even further by noticing that amazingly we have recovered the Planck time on the right hand side of the equation:

$\Omega = \frac{2}{3} \left (1 + H \sqrt{\frac{G \hbar}{c^5}} \right ) = \frac{2}{3} \left (1 + H t_P \right )$

This results in a dark energy component that remains relatively stable at approximately 2/3 or 67% of the critical density for a wide range of values for the Hubble parameter H and behaves very much like a cosmological constant.

So there you go, that is how to fix the vacuum catastrophe. You are welcome. :)

Stuart LaForge

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Jul 22, 2022, 9:13:09 PM7/22/22

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This is in supposed to be J/m^3 I just didn't correctly type the units. Sorry. The math still works out.

Stathis Papaioannou

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Jul 22, 2022, 9:28:16 PM7/22/22

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“So I have figured out a method that uses Quantum Harmonic Oscillators to fix the vacuum catastrophe.”

I’d like to use that as a conversation starter one day!

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Stathis Papaioannou

Stuart LaForge

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Jul 22, 2022, 10:26:29 PM7/22/22

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There is definitely something to be said for having friends that can appreciate an opener like that. :)

Brent Allsop

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Jul 22, 2022, 11:27:14 PM7/22/22

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Yes, good group of friends!!

To view this discussion on the web visit https://groups.google.com/d/msgid/extropolis/a7a3366b-c8cc-4e5e-afe9-6dedaf187c35n%40googlegroups.com.

John Clark

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Jul 23, 2022, 11:20:53 AM7/23/22

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On Fri, Jul 22, 2022 at 9:05 PM Stuart LaForge <stuart....@gmail.com> wrote:

> The universe is a very large hypersphere with a very small positive curvature k. According to the final analysis of the Planck satellite data, the team is 99% confident that the universe is closed with $-0.095 < \Omega_k < -0.0007$

I certainly don't think that is the final analysis of a huge data set like we got from the Planck Satellite, and most cosmologists would disagree with the idea that the universe is closed; the likelihood that it is just a statistical fluke is about the same as the likelihood that a coin is fair if you flip it and get heads 11 times in a row, however if you flip the coin hundreds of thousands of times it's not so unlikely that on one of those occasions you'll get heads 11 times in a row.

But who knows maybe it is closed, time will tell.

What Shape Is the Universe?

John K Clark

Stuart LaForge

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Jul 25, 2022, 1:54:12 AM7/25/22

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On Fri, Jul 22, 2022 at 6:04 PM Stuart LaForge <stuart....@gmail.com> wrote:

So I have figured out a method that uses Quantum Harmonic Oscillators to fix the vacuum catastrophe. Basically QFT models the quantum vacuum as a field of an infinite number of quantum harmonic oscillators (QHO) of every possible frequency located at every point in space.

Then when they do the math, they either get an infinite energy density for the vacuum, or when they try to constrain the problem, a finite energy density that is still 122 orders larger than the empirically determined value of about a nanojoule per cubic meter.

Here are some references for the vacuum catastrophe and why it is a problem:

Adler, R.J., Casey, B. and Jacob, O.C. (1995) Vacuum Catastrophe: An Elementary Exposition of the Cosmological Constant. American Journal of Physics, 63, 620-626.

Persinger, M. (2014) A Possible Explanation for the Vacuum Catastrophe. International Journal of Astronomy and Astrophysics, 4, 178-180.
https://www.scirp.org/pdf/IJAA_2014031211294319.pdf

https://www.youtube.com/watch?v=n6jAOV7bZ3Y

The fix requires several new assumptions that act as additional boundary conditions on the problem:

Assumptions

1. Our causally connected patch of the universe has a fundamental frequency $\omega_0 = 2 \pi H$ that changes in time in relation to the causal volume's change in size.

2. The Planck frequency, $\omega_P = 2 \pi / t_P$ , where t_P is the Planck time, is the maximum possible frequency in the universe that does not get censored by event horizons. It serves as the upper bound of our calculation.

3. Not all vibrational frequencies are possible. The only allowed frequencies are the harmonics, i.e. integer multiples, of the fundamental frequency $\omega_k = 2 \pi k \omega_0$ .

This is a mistake. This should read as $\omega_k = k \omega_0 = 2 \pi H k$ .

Stuart LaForge

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Jul 25, 2022, 3:11:42 AM7/25/22

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On Saturday, July 23, 2022 at 8:20:53 AM UTC-7 johnk...@gmail.com wrote:

On Fri, Jul 22, 2022 at 9:05 PM Stuart LaForge <stuart....@gmail.com> wrote:

> The universe is a very large hypersphere with a very small positive curvature k. According to the final analysis of the Planck satellite data, the team is 99% confident that the universe is closed with $-0.095 < \Omega_k < -0.0007$

I certainly don't think that is the final analysis of a huge data set like we got from the Planck Satellite, and most cosmologists would disagree with the idea that the universe is closed; the likelihood that it is just a statistical fluke is about the same as the likelihood that a coin is fair if you flip it and get heads 11 times in a row, however if you flip the coin hundreds of thousands of times it's not so unlikely that on one of those occasions you'll get heads 11 times in a row.

You still probably shouldn't bet on it, unless you are the one that flips the coins. Multiverse theory, and the anthropic principle that tags along with it, seems to warrant universes being finite. I mean if no part of a whole is finite, then how do you understand its relation to the whole? What makes your reality special? If boring everyday (every nanosecond?) quantum fluctuations big-banged all of causal nature into fruition, then how could those fluctuations have been both commonplace and special?

Cosmology seems to me an esoteric science where everybody argues over the "correct" value of something like 6 parameters from the same equations, all the likelihood margins of whom all fall within the bounds of measurement error. Like all we need are a few more decimal places and then all will be made clear. Any way, I thought you didn't like infinity?

Stuart LaForge

Lawrence Crowell

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Jul 25, 2022, 5:56:09 AM7/25/22

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I do not see anything wrong in particular with the algebra of this. However, by assuming the critical density and the value of H you sort of put this into the derivation.

John Clark

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Jul 25, 2022, 7:15:58 AM7/25/22

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On Mon, Jul 25, 2022 at 3:11 AM Stuart LaForge <stuart....@gmail.com> wrote:

> Multiverse theory, and the anthropic principle that tags along with it, seems to warrant universes being finite.

Multiverse theory is consistent with infinity but does not demand it; it just says that everything that can happen does happen, it doesn't say if the number of things that can happen is finite or infinite; basically that depends on if time or space is continuous or not.

> What makes your reality special?

Nothing, one world is as real as another.

>I mean if no part of a whole is finite, then how do you understand its relation to the whole?

We know for a fact the speed of light is finite, so we know for a fact the time it takes to communicate between any 2 points must be greater than zero, and we know for a fact the size of the observable universe is not infinitely small, and we know for a fact the observable universe is not infinitely old. So we don't have to understand everything about the entire universe to have a pretty good understanding of a small part of it because we can safely ignore large parts of the whole.

> Cosmology seems to me an esoteric science where everybody argues over the "correct" value of something like 6 parameters from the same equations, all the likelihood margins of whom all fall within the bounds of measurement error. Like all we need are a few more decimal places and then all will be made clear.

If we knew those 6 numbers with sufficient accuracy we still would not know everything, or even most things, of physical interest, but we would know the shape, size and texture of the observable universe and, most important of all, how those things change with time.

> Any way, I thought you didn't like infinity?

I don't, but my tastes are not important because it is not known if nature has the same likes and dislikes that I do.