# The end of the quantum vacuum catastrophe ?

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I did a lot with the equation of state in my own investigations, especially in the non-zero value context which was required for the non-conservation of a dynamically expanding fluid. The zero context of it was the first assumption of Friedmann (maybe influenced by Noether) and it produces the constancy of energy in his theorem. This is why the Friedmann equation is considered by some physicists, as a statement of conservation. Motz has argued, the constancy of energy as spacetime expands is an unfounded assumption.

I think the equation of state though is a reasonable description for late cosmology, not too sure when the universe was young and curvature dominated. It is these sistuations that can easily lead to non-conservation, in at least two different ways I know about, both involving on-shell and off-shell matter.

Carrol seems to believe the universe does not globally conserve energy but for different reasons, such as no global time means no translation with energy and so no conservation.

Edited by Dubbelosix

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I begin to think that the way by the pulsation no need more proof to be validate than I have already said before with this :

$\omega_\text{planck}=\frac{m_\text{Planck}*c^2}{\hbar}\text{ in } s^{-1}$

$\omega_\Lambda= \Lambda^{1/2}\text{ in } s^{-1}$

$(\omega_\text{planck}/\omega_\Lambda)^2*(8\pi)=$

$\omega_\text{planck}^2/\Lambda*(8\pi)$= exact  value of vacuum catastrophe in 10^122 from this presentation :

https://arxiv.org/ftp/physics/papers/0611/0611115.pdf and for $\Lambda = 10^{-36}s^{-2}$ , even if it's not the exact value of $\Lambda$

$\omega_\text{planck}$ : pulsation of Planck.

$\Lambda$ : cosmological constant

What is your opinion Mordred (and of course someone else) please ?

Edited by stephaneww
latex

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oh my god, I didn't see that:

On 17/10/2017 at 0:20 AM, Mordred said:

The equation of state w=-1 for the cosmological constant is valid and agrees with observational evidence thus far.

What is not mentioned is that it is the same equation of state for an incomparable fluid which can be mathematically defined under the Euler hydrostatic equations. ( Though few textbooks will mention this either).

An equation of state in essence gives us a dimensionless value for the energy density to in essence, pressure influence. Via the ideal gas laws.

Later tonight I will try to find the mathematical proof for the W=-1 with regards to Lambda. They can be tricky to find.

of course my last thread have not matter...

did you progress, Mordred ?

Edited by stephaneww

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not yet work has been keeping me hopping. Might have time this weekend

when you can

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For the w=-1 equation of state this scholarpedia article has sufficient coverage to get you started

It is the same you will find in most intro textbooks of cosmology

sorry it took so long busy RL schedule

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thank you very much for the link

I'm not abble to understand most of the page but I try something . You'll say me where  I'm wrong....

$\omega=p/ \rho$ [under  eq(7) ]

so if $\omega_{\Lambda}=-1$ then $p_{\Lambda}=- \rho_{\Lambda}$

so $\rho_{\Lambda}+3p_{\Lambda}=-2\rho_{\Lambda}$ and" in deriving Eq. (11)" ... [find under  eq (12)], so " cosmological constant has " $\omega_{\Lambda}=-1$  is true

but it really seems too simple for me to be the solution. I would be surprised if it was enough ...

Edited by stephaneww

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ooops :

read $w$ and no $\omega$

sorry

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Well keep in mind the FRW metric is a simplification of GR. You essentially have the above correct but the reason I suspect you don't follow why is that math your not following.

Don't feel bad on that as some of the key relations above is mentioned but not fully described.

So I need some more feedback, Is it the stress tensor equations that your not understanding ie the Minkowskii Stress equation given as equation 3 which this article defines as the perfect fluid energy tensor.?

Secondly have you worked much with statistical mechanics ? ie in this case an adiabatic and istropic fluid? (ie are you familiar with how to mathematically model such using statistical mechanics.

thirdly equation, 9 has a mathematical proof that can be heuristically described via Newtonian physics involving the shell theorem. If I recall Liddle has it described in one of his books. I can dig it up as it better explains how the critical density formula works in which equation 9 works with.

(how familiar are you with the FLRW metric? GR? statistical mechanics and shell theorem?) I need a guideline to what level of treatment to provide.

Edit I think I may have posted the Liddle proof of the above previously will check

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2 hours ago, Mordred said:

...You essentially have the above correct but the reason I suspect you don't follow why is that math your not following.

...

(how familiar are you with the FLRW metric? GR? statistical mechanics and shell theorem?) I need a guideline to what level of treatment to provide.

...

You are right and that's the problem : I know nothing of this. That's why I tried a Boolean approach with already written relationships.

Edited by stephaneww

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Fair kk no prob but will take a bit to keep the explanation as Newtonian as possible. Ok lets start with a homogeneous and isotropic state of a given radius. ( no preferred location nor direction)

In essence a uniform distribution, this is how Lambda would be modelled as a scalar field with no preferred direction..

Now ti understand the $w=\frac{p}{\rho}$ relation one must understand that pressure is force per unit volume. This obviously requires momentum, however in the case of Lambda one of no preferred direction ie random Brownian motion.

Matter fields with low momentum exert no pressure, w=0. Radiation exerts higher pressure due to high kinetic energy.

Now in the ideal gas laws under statistical mechanics, you often see pressure as described as particles bouncing of walls,

in this case the treatment is number of particles that will cross a given coordinate axis in a given unit of time.

w is just a dimensionless constant for the ratio of proprtionality of the last descriptive. The FLRW metric itself primarily concerns itself with how the volume of the universe evolves. (as per) the bold above but we can simply use the change in radius and apply another dimensionless constant for how the volume itself evolves via Hubbles parameter (careful it is only constant everywhere at a given moment in time)

That being the scale factor. a(t) This change in volume will naturally be modelled in a vector like fashion as were concerned if it is expanding or contracting.

As matter causes collapse its natural to keep this positive with w being negative being expanding. (negative pressure, contracting volume of gas naturally increases pressure of a vacuum).

if were good with the above, we can show this under math to explain equation 9 better.

(ps this treatment above is essentially similar to particles in a box under QM)

Edited by Mordred

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Here this will save some time on critical density with regards to Equations of state.

I also have my own article on universe geometry that details more into the FRW metric.

Page two covers how the line metric of the FLRW metric works but doesn't cover the fluid equations themself as per the first link.

The FLRW is probably the easiest approach to modelling in Cosmology and to understanding the more complex aporoaches under other field theories.

this is a good overview descriptive of how the FLRW metric works.

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Just to add to Mordred, the density parameter from the Friedmann equations measures the ratio of the observed vacuum density to the critical density. Only when these two quantities are [exactly] the same does the Friedmann equation allow a geometry which would fit Euclidean flat spacetime. It's not difficult mathematics,  consider an equation of state

$\dot{\rho} = \frac{\dot{R}}{R}(\rho + P)$

I have left a constant out of the equation. Ordinary density goes down in an expanding universe and the critical density is when $\rho \rightarrow \frac{\Lambda c^2}{8 \pi G}$ and that results in the equation

$(\frac{\dot{a}}{a})^2 = \frac{8 \pi G}{3}\rho + \frac{kc^2}{a} \rightarrow \frac{8 \pi G}{3}\frac{\Lambda c^2}{8 \pi G} + \frac{kc^2}{a}$

Note, there appears to be something wrong with the equation since, in absence of any non-luminous matter, the critical energy (a tool used to explain possible collapse models) is worked out to be five atoms of hydrogen per cubic metre of space which is (actually) far denser than what is observed. Our universe appears to be well under the critical energy.

Notice when you plug in the critical density, a factor of $\frac{\Lambda c^2}{3}$ appears - this is just the standard definition of how the cosmological constant which drives acceleration enters the equation for critical density - the cosmological constant is believed to only get significant when a universe get's large enough. It seems strange that we might measure a different measure of density for the vacuum than that predicted by the Friedmann equation. Of course, things like vacuum density becomes obscured under relativity anyway since moving observers will not agree on things like density, but I feel like this is a different issue entirely.

Personally, I have likened this lack of density being linked to acceleration of the universe. Something rings true about the prediction, just not the model we are using..

Edited by Dubbelosix

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Let me get back to you on this, I'd rather see how well Stephaneww picks up on the above first hehe as my reply will step into a very useful equation.

55 minutes ago, Dubbelosix said:

Note, there appears to be something wrong with the equation since, in absence of any non-luminous matter, the critical energy (a tool used to explain possible collapse models) is worked out to be five atoms of hydrogen per cubic metre of space which is (actually) far denser than what is observed. Our universe appears to be well under the critical energy.

Notice when you plug in the critical density, a factor of Λc23 appears - this is just the standard definition of how the cosmological constant which drives acceleration enters the equation for critical density - the cosmological constant is believed to only get significant when a universe get's large enough. It seems strange that we might measure a different measure of density for the vacuum than that predicted by the Friedmann equation. Of course, things like vacuum density becomes obscured under relativity anyway since moving observers will not agree on things like density, but I feel like this is a different issue entirely.

No problem.

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I feel like going back to school lol

question: in https://arxiv.org/pdf/astro-ph/0409426.pdf, page 16 and page 17,

what is the unity of $\theta$ and $\phi$ please ?

what its that function of time a(t) also ?

Edit: uh, if the two previous questions are only moderately useful I start to understand again from (1.15)

sorry I need an answer for : what its that function of time a²(t) also ? please

Edited by stephaneww

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The first is just your spherical coordinates see

The a(t) is that other dimensionless constant called the scale factor.

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Sorry forgot to answer your first question. The FRW metric uses normalized units $c=g=\hbar=1$

unity under math means 1 in normalized units and De Moivre numbers as per

Edited by Mordred

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You wanted me to find paper on why energy causes spacetime to distort, but you are asking why it distorts, yes?

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Well , I still do not master $a$ and $z$ of this document (https://arxiv.org/abs/astro-ph/0409426), even if I have a little advanced and revised my maths (including integrales).

I stopped my study just before the end of Chapter 1. Question:  Will the rest be useful to me for this document ?

I will continue to review the rest of the submitted documents and revise or learn the maths needed but it will take time for this question of w = -1. If anyone wants to give the answer, I would be very happy.

I have, however, turned my attention to the equalities (1.41) page 26. Indeed they resume almost my approach with the pulsations but allow an additional hypothesis on the topology of the universe, always trying to solve the vacuum catastrophe :

"...For ultra–relativistic matter like photons, the energy of each-particule is not given by the rest mass but by the frequency $\nu$ or the wavelength $\lambda$

$E = h \nu = h c / \lambda$ (1.41)..."

Novelty : by necessity we will consider a topology of the universe of spherical 3D space, or another topology that needs to transform $h$ into  $h / (2 \pi) = \hbar$ for the "wavelength of the cosmological constant" $\lambda_ \Lambda=1/ \Lambda^{1/2}$. We will use the same method as for the pulsation.

So the average energy densities per unit volume for Planck's and then for the "quantum cosmic constant" are :

$m_pc/l_p^3$ Joules per m^3

and

$\hbar c \Lambda^{1/2} /l_p^3$ Joules per m^3 (a)

We have we want find $\Lambda$ with (a)  :

$( \frac{ \hbar c \Lambda^{1/2} ) }{ l_p^3 } ) ^2/\frac{ m_p c^2 }{ l_p^3 } / 8 \pi=\rho_\Lambda c^2=\frac{ F_p \Lambda}{8 \pi}$ = value of average energy density per unit volume of cosmological constant in $\Lambda CDM$ model

so

$\frac{ \hbar^2 c^2 \Lambda }{ l_p^3 m_p c^2 }= F_p \Lambda$

as $F_p=c^4/G$,  $l_p= (\hbar G /c^3)^{1/2}$ and  $m_p=(\hbar c/ G)^{1/2}$, we have :

$l_p^3 m_p = G \hbar^2 / c^4$ and this equality is true.  I leave you to check it.

Keep in mind that this presentation seems to constrain the possibilities of topology of 3D spaces because the geodesic of the straight line "of a dark energy photon" describes a circle if we do not take into account the time

Now, I'm going back to my hard studies

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On 10/11/2017 at 6:30 AM, Mordred said:

Here this will save some time on critical density with regards to Equations of state.

...

Where can we find one or more examples of numerical computations with $a$ , $\dot{a}$ , and $\ddot{a}$ please ?

I think it will help me a lot to understand  $a$ .

I unterstand the text (end of page 5 and page 6)  but I haven't the maths keys to demonstrate w=-1  for now

Edited by stephaneww

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On 10/11/2017 at 6:30 AM, Mordred said:

Here this will save some time on critical density with regards to Equations of state.

Eh eh. not bad pages 1,2,3,4 .... starting with page 1
I may be able to get there for $a$ and $z$

Edited by stephaneww

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It takes time to fully understand but your not doing too bad. Hopefully you picked up the $\dot{a}$ is a time derivative while $\ddot{a}$ is the second time derivative.

now the scale factor

Hubble parameter is defined as

[

$H=\frac{\dot{a}(t)}{a(t)}$

recessive velocity



$v_{rec}=\frac{\dot{a}(t)}{a}$

so here is some examples, note a(t) today is 1.00 see stretch 1.00 column. a(t) at z=1090 s 0.001 this is the CMB surface of last scattering. the a with values greater than 1 is future expansion. (in this case roughly 88 Gyrs into the future lol.

here perhaps this will help understand scale factor.

taken from the user guide for the cosmological calculator example given below though the calc has numerous more columns and rows when fully set up.

"The LightCone tabular cosmological calculator is a versatile tool for understanding the expansion history of the universe: past, present and future. Stages in expansion history are designated by the corresponding scale factor a, which is the size of a generic distance compared with its size at present. For instance, a=1 denotes the present and a=0.5 the stage when cosmic distances were half their present size. In the same way, a=2 refers to a future stage when distances will be twice what they are now."

Marcus. Jorrie and I wewnt through a lot of effort to simplify how we wrote the guides as we encountered numerous posters where the textbook descriptives were difficult for the average poster to understand (hope this helps)

here is an example table.

${\small\begin{array}{|c|c|c|c|c|c|}\hline T_{Ho} (Gy) & T_{H\infty} (Gy) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.9&0.693&0.307\\ \hline \end{array}}$ ${\small\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&S&z&T (Gy)&R (Gly)&D_{now} (Gly)&D_{then}(Gly)&D_{hor}(Gly)&H/Ho \\ \hline 0.001&1090.000&1089.000&0.000373&0.000628&45.331596&0.041589&0.056714&22915.263\\ \hline 0.001&683.804&682.804&0.000810&0.001326&44.962398&0.065753&0.089864&10859.192\\ \hline 0.002&428.979&427.979&0.001722&0.002756&44.477683&0.103683&0.142116&5224.758\\ \hline 0.004&269.117&268.117&0.003606&0.005666&43.849475&0.162938&0.224202&2541.361\\ \hline 0.006&168.829&167.829&0.007463&0.011563&43.042568&0.254948&0.352603&1245.393\\ \hline 0.009&105.913&104.913&0.015309&0.023478&42.012463&0.396668&0.552333&613.344\\ \hline 0.015&66.444&65.444&0.031211&0.047518&40.702622&0.612585&0.860719&303.042\\ \hline 0.024&41.683&40.683&0.063355&0.095974&39.041469&0.936624&1.332155&150.041\\ \hline 0.038&26.150&25.150&0.128224&0.193578&36.938267&1.412573&2.043059&74.389\\ \hline 0.061&16.405&15.405&0.258995&0.390062&34.278330&2.089532&3.094542&36.917\\ \hline 0.097&10.291&9.291&0.522342&0.785104&30.917756&3.004225&4.606237&18.342\\ \hline 0.155&6.456&5.456&1.051751&1.575989&26.679131&4.132295&6.685941&9.137\\ \hline 0.247&4.050&3.050&2.109877&3.133394&21.362526&5.274330&9.344906&4.596\\ \hline 0.394&2.541&1.541&4.180384&6.013592&14.827243&5.835394&12.323993&2.395\\ \hline 0.627&1.594&0.594&7.955449&10.346218&7.320583&4.592515&14.935503&1.392\\ \hline 1.000&1.000&-0.000&13.787206&14.399932&0.000000&0.000000&16.472274&1.000\\ \hline 1.585&0.631&-0.369&20.956083&16.410335&5.731185&9.083316&17.046787&0.877\\ \hline 2.512&0.398&-0.602&28.694196&17.063037&9.638020&24.209612&17.203810&0.844\\ \hline 3.981&0.251&-0.749&36.601471&17.239540&12.159687&48.408586&17.239540&0.835\\ \hline 6.310&0.158&-0.842&44.553231&17.284732&13.760162&86.820752&17.284732&0.833\\ \hline 10.000&0.100&-0.900&52.516301&17.296130&14.771503&147.715032&17.296130&0.833\\ \hline 15.849&0.063&-0.937&60.482221&17.298988&15.409856&244.229762&17.298988&0.832\\ \hline 25.119&0.040&-0.960&68.448857&17.299697&15.812667&397.196249&17.299697&0.832\\ \hline 39.811&0.025&-0.975&76.415673&17.299867&16.066830&639.632027&17.299867&0.832\\ \hline 63.096&0.016&-0.984&84.382534&17.299901&16.227197&1023.866895&17.299901&0.832\\ \hline 100.000&0.010&-0.990&92.349407&17.299900&16.328381&1632.838131&17.299900&0.832\\ \hline \end{array}}$

This will give you some useful relations to compare wiki has not too bad a coverage for equations of state
remember the cosmological constant is constant over time...

Hopefully it is enough to show that the scale factor conpares the radius of the sphere today with the radius of the sphere at the time being examined. It is simply a ratio of change between two measurement events.

Edited by Mordred

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