Dark Energy: What is the expected form of the equation?

[69911e]

What is the expected equation for total dark energy in universe as a function of size of the universe?
ie
size of universe=D
Dark Energy f(D)= (D^n)*constant ; where n=-2,-1,-.5,0,.5,1,2

Dark Energy f(D)= D*constant
or
Dark Energy f(D)= (1/D)*constant
or
Dark Energy = constant
or
Dark Energy f(D)= D^2*constant

or other?

 

[Jorrie]

or other?

Yes, other. Firstly, the "size of universe" is not known or even sensible. We presently only have a size estimate for the currently observable universe.
Secondly, if you insist on calling it an "energy" (which it is likely not), it will change with the volume of the space that you consider. The best plan is to talk of the energy density of empty space, which according to our best observations is around 6.3 x 10^-10 Joule per cubic meter.

This energy density (of empty space) is thought to be constant over time, despite the expansion. It can be thought as an intrinsic constant spacetime curvature of space (after inflation). As radiation and matter are diluted by the expansion, this remains and will eventually cause mildly exponential expansion of distances between clusters.

 

[69911e]

Jorrie: Thanks for the response. I have no background in this subject, but am working on it with my kids.

Sorry, I omitted, but intended D=size observable universe; my mistake.

"Secondly, if you insist on calling it an "energy" (which it is likely not)"
I did not intend to imply what it is (energy or not), I tried to use what I thought was the industry standard term "dark energy" as I thought it conveys the question best. How should I have phrased the question without using the words "Dark Energy"?

Do we know the value of "energy density of empty space" you referred to is constant over time?
If so, it seems the equation would be (please correct if this is wrong)
Dark Energy f(D)= constant * D^3
as volume increase with D^3

It seems any value of n >= 0 would lead to a Dark energy dominated observable universe at large Time/observable universe size if the formula below would be applicable.
Dark Energy f(D)= (D^n)*constant

If we don't know the value of "energy density of empty space" is constant over time, do we have reasonably defined bounds for it (either values or a function) from CMB or other data?

Thanks

 

[phyzguy]

Whether the dark energy term is constant or whether it varies in space and time is a question that literally thousands of people are working on trying to answer. If you look at this paper on the Planck data of measurements of the CMB, this is exactly the question they are trying to address in Section 6.3. Figure 28 shows how the Planck measurements of the CMB combined with other measurements compare to possible models where the dark energy varies. The values of wa=0 and w0=-1 correspond to the cosmological constant, meaning that the dark energy does not vary, and is constant in space and time. So the cosmological constant hypothesis is fully consistent with the data, but the error bars are still large. Many people are working to try and shrink the error bars on the graph of Figure 28 by measuring the CMB and other astronomical measurements.

 

[69911e]

phyzguy: Thanks for the link. It will take quite a while to read.
[QUOTE="The values of wa=0 and w0=-1 correspond to the cosmological constant, meaning that the dark energy does not vary, and is constant in space and time. .[/QUOTE]

If wa=0 and w0=-1, would this rule out any value other than n = 3 (as this seems to lead to a constant value/unit volume)?
Dark Energy f(D)= (D^n)*constant

Am I oversimplifying by attempting to create this equation for the dark energy contained within a defined volume of space? I understand it can't be looked at as standard energy, but I have seen many references to the ratio of dark energy/mass being very roughly 2.

Thanks

 

[phyzguy]

As Jorrie said, I think you are looking at it wrong. Dark Energy is an energy density in empty space, which we can call ρDE. So any volume of space will have a dark energy content equal to ρDE * Volume. The question is just whether ρDE is constant everywhere in space and at all times, or whether it varies with time or across space.

 

[69911e]

The question is just whether ρDE is constant everywhere in space and at all times, or whether it varies with time or across space.

Thanks phyzguy, that appears to be a better way of phrasing my question.

Does present data show best fit to ρDE being a simple constant or a function of(time,space,other)?

 

[Jorrie]

Do we know the value of "energy density of empty space" you referred to is constant over time?

As phyzguy said, we cannot be sure, due to the difficulty of measuring it and the error bands of the respective independent methods. The simplest interpretation of dark energy is still to take it as the effect of Einstein's cosmological constant and until observations rule this out, it remains the preferred interpretation for modeling the expansion (Lambda-Cold-Dark-Matter or LCDM modeling). But it is not necessarily the "best fit" for all observations - there exist some "tensions" between some measurements and LCDM predictions, but nothing conclusive yet.

 

[69911e]

This is tangentially related to the topic of this post:

I found this graph which appears to show the Hubble constant transitioning from decreasing to increasing.

I have found on this site from marcus:
"Hubble's "constant" has been decreasing since the start of expansion and it is expected to continue decreasing but to level out in the long term."

This graph and this statement seem contradictory. I am likely misinterpreting something. Does the graph not show the Hubble constant?

 

[Jorrie]

I found this graph which appears to show the Hubble constant transitioning from decreasing to increasing.

I have found on this site from marcus:
"Hubble's "constant" has been decreasing since the start of expansion and it is expected to continue decreasing but to level out in the long term."

This graph and this statement seem contradictory. I am likely misinterpreting something. Does the graph not show the Hubble constant?

The graph does not plot the Hubble constant over cosmological time, but rather against lookback time, as observed today. In essence, the horizontal axis does represent cosmological time increasing from left to right, but it has a different basis.
Marcus' statement refers to the Hubble constant as observers living at time T would have measured it. Here is what that same curve would look like against cosmological time.

 

[Helios]

The simplest interpretation of dark energy is still to take it as the effect of Einstein's cosmological constant and until observations rule this out, it remains the preferred interpretation for modeling the expansion (Lambda-Cold-Dark-Matter or LCDM modeling).

Why is this not a rhetorical tautology? What is the cosmological constant ? We think that it is a measure of dark energy. What is dark energy? We think that it is an effect of Einstein's cosmological constant.

 

[Chronos]

The cosmological constant is the intrinsic curvature of spacetime under the einstein field equations. It is basically the k variable in the Friedmann equation.

 

[Jorrie]

Why is this not a rhetorical tautology? What is the cosmological constant ? We think that it is a measure of dark energy. What is dark energy? We think that it is an effect of Einstein's cosmological constant.

Further to what Chronos said, the cosmological constant represents one special case of dark energy, the one with constant energy density over space and time. Dark energy has much wider possibilities.

 

[Chronos]

The cosmological constant would be ruled out by a confirmed finding of any time dependent variation in dark energy density. Studies are underway to test this idea- which would really challenge our cosmological models. Thus far, no such luck.

 

[Haelfix]

The cosmological constant is the intrinsic curvature of spacetime under the einstein field equations. It is basically the k variable in the Friedmann equation.

Err, I think you have this confused. The cosmological constant is NOT the k variable in the Friedmann equation. The k variable is constrained to be -1, 0, 1 (closed, flat, open respectively) and is physically interpreted to be the normalized expression for spatial curvature at a specific scale factor. The associated curvature density parameter has very different evolution dynamics than the cosmological constant (spatial curvature density dilutes as the square of the scale factor for instance, whereas the cosmological constant is well constant).

 

[69911e]

The graph does not plot the Hubble constant over cosmological time, but rather against lookback time, as observed today. In essence, the horizontal axis does represent cosmological time increasing from left to right, but it has a different basis.
Marcus' statement refers to the Hubble constant as observers living at time T would have measured it. Here is what that same curve would look like against cosmological time.
View attachment 93166

That lightcone graph you posted is the one I expected when graphing the Hubble constant , yet I found this other one in a peer review paper. I have seen both graphs with equal frequency on random websites which one obviously has to be careful of.
I understand the horizontal axis difference. The confusion I have is the vertical axis label is the Hubble constant (I could be wrong) but the curve looks like it is an acceleration of the Hubble constant. It shows the Hubble constant lower 6GY ago.
The lightcone graph should (I would think) show the same data from 0 to 14GYear (ie ignore data after 14GY for comparison). But they don't seem to as the first one has the minimum at 6GY ago; the lightcone graph does not match this curve and has no minimum. The graph I posted looks like the derivative of the lightcone graph.

I must be missing something as a number of people have replied, and none appear to see the point which creates my confusion.

 

[Jorrie]

There is a bit of confusion, because the term 'cosmic expansion rate' sometimes means different things in different papers. Some see the the first derivative of the scale factor as 'expansion rate', while others view the Hubble value H(t) as 'expansion rate'. I think the paper that you referred to falls into the first category, while here we have agreed that the second view is more appropriate. In Marcus' parlance, the red curve below is the recession rate of a generic galaxy (in this case one that is presently on the Hubble sphere). I think it has the same form as the "expansion rate" of the paper that you refer to.

Does this make more sense?
It's a pity Marcus is not around at present, because he has a succulent way of explaining all of this.

 

[69911e]

Jorrie, the graph with the 3 curves are of the form I am familiar with. The project I have been working on with my kids yielded almost identical (within 15%) graphs directly from quantum effects. This project is what led to my original question; we currently trying to test the results of the project to the known data. I know this site does not allow (for very good reasons as most are wrong and/or misleading) personal theories, so I will not post any graphs. We have started this project from an Engineering (EE) perspective, thus, my questions and terminology are way off from physics perspective/terminology. I will show them the Lightcone calculator in your signature; it looks awesome. Thanks!
The Lightcone graphs you have posted, your comments on the graph, and Marcus' explanation I quoted all make complete sense.

Unfortunately, I am still confused how the first graph (not originally found on this website) is correct.
The vertical axis units are labeled "Km/S/Mpc". How can this be a derivative of the Hubble constant? I thought Km/S/Mpc (effectively; 1/s) was the the units of the Hubble constant. The numeric values also fall into the proper range of the Hubble constant's expected value, yet it follows the shape of the red line on the lightcone graph which, as far as I can tell, can't have those units.
I still am missing something...any help appreciated.

 

[Bandersnatch]

I must say, I also find it confusing. Can you post a link to the paper from which the graph came from? Maybe some context will help.

 

[Jorrie]

It's a pity Marcus is not around at present, because he has a succulent way of explaining all of this.

In some previous thread Marcus and I agreed that we should call the red curve the 'recession rate' and the gold curve the 'expansion rate'. The latter is like an interest rate, a certain % growth, independent of the amount of money. One could say that this is the "expansion rate of your investment". In our cosmos, it is today about 1/144 % per million years.

In the case of the red curve, we look at the recession rate of one galaxy that was originally very close to us and was receding from us a high rate per unit distance (Mpc). During decelerating expansion, the rate dropped and during the late accelerated expansion, the rate increased. Hence the minimum at around year 8 billion.

Edit: In our case, "the bank dropped the interest rate faster than our money could grow to compensate". However, lately the amount of money became large enough so that it compensated for the drop in interest rate and the interest rate is stabilizing towards a more constant value. Our money is starting to show proper exponential growth, as one would expect from compounded interest at a constant interest rate.

I hope this helps.

 

[69911e]

What is the units of the red curve? Does this match the units , 1/s, of the graph I don't understand?

By my understanding the first graph should have units of 1/s². Any chance it is labeled wrong and this is the cause of my confusion?

 

[69911e]

I must say, I also find it confusing. Can you post a link to the paper from which the graph came from? Maybe some context will help.

http://www.scholarpedia.org/article...om_baryonic_acoustic_oscillation_measurements
Sponsored by: Dr. Lydia Roos, CNRS/IN2P3/LPNHE, Paris, France
Reviewed by: http://www.scholarpedia.org/article/User:Anze_Slosar
Reviewed by: http://www.scholarpedia.org/article/User:Nicolas_G._Busca

 

[69911e]

The values of wa=0 and w0=-1 correspond to the cosmological constant, meaning that the dark energy does not vary, and is constant in space and time.

Is the lightcone graph on page1 consistent with: wa=0 and w0=-1?

 

[Jorrie]

What is the units of the red curve? Does this match the units , 1/s, of the graph I don't understand?

By my understanding the first graph should have units of 1/s². Any chance it is labeled wrong and this is the cause of my confusion?

The curves of the LightCone graphs on page 1 are highly normalized (hence all equal 1 at the present time) and all three are dimensionless. But we can laboriously un-normalize them.

More to the point, V_gen/c is dimensionless, so V_gen has the same units as c, i.e. km/s as used for Ho. The "generic" subscript was defined to be for a galaxy presently at the Hubble radius, so the units effectively become km/s/(Hubble radius). Now one Hubble radius presently equals 14400 Mly, or 4390 Mpc. So, the red curve can be converted to Km/s/Mpc by multiplying it by c (3e5 km/s) and dividing by 4390 Mpc. This simply changes the scale, not the curve itself.

The units of H is the same as Ho (traditionally km/s/Mpc), so essentially the two curves have the same units. This does not mean they represent the same function though. I think the graph in the paper is labeled for convenience and is correct.

 

[phyzguy]

It's a pity Marcus is not around at present, because he has a succulent way of explaining all of this.

Perhaps you mean succinct??

Is the lightcone graph on page1 consistent with: wa=0 and w0=-1?

Sorry which light cone graph are you referring to?

 

[Jorrie]

Perhaps you mean succinct??

Yup - even after a virtual lifetime of learning English, a non-native speaker still stumbles ;)

I think 69911e referred to the LighCone graphs that I posted before and repeated above. The LightCone program only considers the LCDM model with wa=0 and w0=-1.

 

[69911e]

So, the red curve can be converted to Km/s/Mpc by multiplying it by c (3e5 km/s) and dividing by 4390 Mpc. This simply changes the scale, not the curve itself.
The units of H is the same as Ho (traditionally km/s/Mpc), so essentially the two curves have the same units. This does not mean they represent the same function though.

Now I understand. The same units, with the same name for 2 associated but different things is what I missed. Now we get back to applying this...
Thanks!