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Relativity and Cosmology

This is a Blog on relativity and cosmology for engineers and the like. My website "Relativity-4-Engineers" has more in-depth stuff.

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Cosmology Equations Part 3

Posted October 26, 2006 2:13 PM by Jorrie

The cosmological density parameter (Ω) is one of the most used parameters in cosmology. It is dimensionless and expresses the ratio between the actual energy density of the universe and the so-called critical energy density.

Critical energy density is the borderline case between enough energy density to make the universe eventually contract again and too little energy density, causing infinite expansion. The density parameter is therefore:

where ρ is the actual observed density and ρc is the critical density. Ω has the value that we observe today. In an expanding (or contracting) universe, the density cannot remain constant, e.g., if we consider mass density, it must have been higher in the past of an expanding universe.

Further, energy density does not refer only to mass density - it refers to all forms of energy. The main contributors (in a cosmological sense) are: mass energy, radiation energy and vacuum energy. These components add up in pretty logical way, which is best expressed in terms of the universal expansion factor a, as defined in Part 2 of this mini-series. This gives the density parameter as a function of the expansion factor as:

Does this look logical? Maybe not at a first glance, but actually, it is! Here's why. Mass density varies inversely with the volume of space, i.e., a3 - straight engineer's logic.

Radiation density also varies inversely with the volume of space and, in addition to that, it varies inversely with another factor of a, due to the redshift caused by the expansion of the universe - straight cosmologist's logic.

Vacuum energy density does not care about expansion - the more vacuum, the more vacuum energy in a linear fashion - straight quantum theorist's logic.

The scary thing is, according to latest observations, the quantum theorists have got a stake in this, but they cannot agree on the size of their stake!

Relativity 4 Engineers has got some more readily digestible info on all of this weird stuff…

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Anonymous Poster
#1

Re: Cosmology Equations Part 3

10/27/2006 3:20 PM

Jorrie, I have a problem with: "The cosmological density parameter (Ω) is one of the most used parameters in cosmology. It is dimensionless and expresses the ratio between the actual energy density of the universe and the..."

How do they (cosmologists, or whoever), observe the density? As it seems from your article, they include all sorts of weird things in the energy density, so how do they measure this lot???

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#2
In reply to #1

Re: Cosmology Equations Part 3

10/28/2006 4:49 AM

Guest asked: "How do they (cosmologists, or whoever), observe the density?"

OK, maybe I overstated it a bit in my OP, especially where I said "where ρ is the actual observed density". Ordinary matter and radiation energy densities can in fact be observed directly, but dark matter and vacuum energy (aka dark energy) density can only be inferred indirectly.

For ordinary, visible matter, you simply estimate the mass of objects in a fair volume of space and you have density. The mass estimates are made increasingly more accurate by employing various techniques for the same thing.

For radiation energy density, on can simply measure the average temperature of the background radiation (all frequencies, not only microwaves) of the observable universe and you have the radiation density.

For dark matter, one observes the effect that it has on visible stuff (especially when it bends light) and you know the density. For dark energy, one must infer it from the 'shape' of the cosmological expansion curve.

Hope this clears it up a bit...

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Anonymous Poster
#3
In reply to #2

Re: Cosmology Equations Part 3

10/29/2006 1:10 PM

"For dark energy, one must infer it from the 'shape' of the cosmological expansion curve."

Thanks for the reply, but the last one (quoted) appears a bit shaky to me! How does one measure density from the shape of a curve????

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#4

Re: Cosmology Equations Part 3

04/13/2007 7:42 PM

Hi Jorrie,

"Vacuum energy density does not care about expansion - the more vacuum, the more vacuum energy..."

How does this work? It seems that it violates the conservation of energy law. Would not there have to be matter destroyed to compensate for it? If you are talking about the gravitational potential energy with the assigned minus sign, then wouldn't matter have to be created to balance?

S

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#5
In reply to #4

Re: Cosmology Equations Part 3

04/14/2007 1:37 AM

Hi SG,

Yes, this is a rather scary part of cosmology, because it brings in quantum physics. Vacuum energy has a characteristic that works both as expansion energy and as gravitational potential energy at the same time!

Right or wrong, I keep my sanity by viewing it as follows: the inflation epoch of the Big Bang 'borrowed' energy from the vacuum, leaving the vacuum with negative energy. This is the negative potential energy of gravity. It is as if the vacuum is trying to get it's loan back by pulling things together.

That negative energy was balanced by the positive kinetic energy of the expansion rate, hence Ω=1 (exactly) and the total energy of the universe was zero. Inflation stopped when the "false vacuum" went through a "supercooled phase transition" and in the process some of the energy of the false vacuum was transformed into matter and radiation. (This is the scary part, which I do not quite grasp!)

At that point the total energy of that matter and radiation was balanced by the total energy loan from the vacuum and there was no (or very little) further extra energy being borrowed from the vacuum. Hence the extreme expansion rate started to slow down under the total gravity, much like a probe shot away from Earth at precisely escape velocity - always traveling slower, but never quite stopping to fall back.

However, if there was a tiny amount of energy still coming from every square meter of space after inflation, that energy would grow as space gets larger due to expansion. At some point, (it looks like some 8 billion years ago), space has expanded so much that the tiny energy of the vacuum added up to more than the total mass and radiation energies of the universe.

This is when accelerated expansion started - the vacuum producing significant positive energy of expansion and, because it is a "loan", the vacuum has more negative potential energy balancing the books, so Ω=1 and remains so forever. New expansion energy is created on the fly, as space expands, but it is perfectly balanced by negative potential energy, with the total energy remaining zero!

This is, in brief, my view. It may be off the mark here and there, but in general it follows mainstream thinking. At least it makes me sleep better...

Regards, Jorrie

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#6
In reply to #5

Re: Cosmology Equations Part 3

04/14/2007 12:15 PM

Hi Jorrie,

Thanks for the explanation. The book I am reading has something about a false vacuum, but I am not there yet. It's an amazing theory, and getting more amazing as time goes on.

S

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#7
In reply to #5

Re: Cosmology Equations Part 3

04/17/2007 4:13 PM

I goofed twice in one paragraph in my last post, saying: "However, if there was a tiny amount of energy still coming from every square meter of space after inflation, that energy would grow as space gets larger due to expansion. At some point, (it looks like some 8 billion years ago), space has expanded so much that the tiny energy of the vacuum added up to more than the total mass and radiation energies of the universe."

It should have read "every cubic meter" and "some 5 billion years ago". The 8 billion years refer to the age of the universe at the time of equal mass energy and vacuum (dark) energy, with the mass energy including that of dark matter.

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Anonymous Poster
#8
In reply to #5

Re: Cosmology Equations Part 3

05/08/2007 6:56 AM

With all this borrowing and payback, perhaps we should have an accountant review the books...double entry and all...sounds like an interest free loan to me, probably the only one in existence..... :) HTRN

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#9
In reply to #8

Re: Cosmology Equations Part 3

05/08/2007 7:42 AM

Hi HTRN, yep.

In my pdf on the Friedmann Equation I use the terms "density accounting" and "balancing the books".

One problem with the "interest free loan" is that while there is accelerated expansion, the loan's size grows! In essence, due to entropy, the loan can never be paid back, because that would demand a decrease in entropy...

-J

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