Cosmo Const. and Dark Energy

[Martin]

there is some misleading posts at SFN about what the cosmological constant is, and what dark energy is. the topic needs clearing up

 

COSMOLOGICAL CONSTANT

 

the clearest source on CC I have at the moment is a brief 2-page online thing by Sean Carroll who is one of the world's top people in cosmology.

 

http://relativity.livingreviews.org/Articles/lrr-2001-1/node3.html

 

Sean is at the University of Chicago and it is significant that the LivingReviews editors at Max Planck Geselschaft asked him to do their article on the Cosmological Constant. Each LivingReviews article is written by someone they think is a top expert on the topic. the articles are supposed to be recent or else periodically revised to keep up to date.

 

Also Sean is an entertaining writer on non-technical subjects, you might enjoy his blog "Preposterous Universe"

 

I recently saw someone refer to the cosmological constant as "Einstein's Antigravitational Force" which is wrong. It is not a force and if someone were to read that, and think that it is a force, then they could get confused, so that they might go through life never understanding the cosmological constant, or else it might delay them getting a clear picture.

 

If you look at Sean's brief LivingReview essay on the CC you will see that the CC is conventionally denoted by Lambda and that it is the reciprocal of an area. that is, [math]\inline\sqrt{\Lambda}[/math] is the reciprocal of a length.

If one plugs in the standard estimates for things which cosmologists have nowadays one can say what that length is, either in meters or lightyears or whatever unit one pleases.

The length [math]\inline\frac{1}{\sqrt{\Lambda}}[/math] is 9.5 billion LY.

 

So if you want to picture the Cosmological Constant, think of a square which is 9.5 billion LY on a side, think of the area of that square, and the Cosmological Constant, Lambda, is one over that area

 

In spacetime geometry curvature is measured in units of inverse length squared (could also be inverse time squared but time and distance should be measured with the same unit anyway). So a small curvature is written as the reciprocal of a large area. This small extra curvature (which A.E. originally stashed into his equation for his own misguided reasons.) is what we are talking. Hence this large area.

 

It is a good idea to avoid mixing up 9.5 billion lightyears with some other familiar large distance like the Hubble distance, and to avoid confusing 9.5 billion years with the age of the universe! There is no direct connection---by itself, one does not determine the other. Current estimates, using the same data from which I get the 9.5, put the age at 13.7 billion years and the Hubble distance at somewhat over 13.8 billion lightyears (nor are 13.7 and 13,8 simply related). Moral is it's simple but not quite as simple as you might like.

 

DARK ENERGY

 

If you look at an reputable cosmologist's equations they give the same story as Sean's with minor variations of notation---Michael Turner is another worldclass cosmologist who has done some exposition like this. And one thing they very commonly do is introduce an energy density

rho-sub-Lamda like this.

 

[math]\rho_{\Lambda} = \frac{c^4}{8 \pi G}\Lambda[/math]

 

here G is Newton's constant and you often do not see the c⁴ because by that stage of the exposition c has usually been set equal to one, but it is there.

 

the reason for defining this energy density is that this is the quantity which in the Friedmann equations, if you add it in with the energy densities you already have, will have just the same effect as the tiny fudge curvature Lambda. I put the Friedmann equations down at the bottom.

 

If you like to believe things (as we all love to do) then it is a fine idea to believe that this rho sub Lambda is a real (but not understood) form of energy distributed in an absolutely constant uniform way thru space and time. If it is a constant uniform vacuum energy then it can be shown rather easily that it exerts a negative pressure. I may have explained this already at SFN or someone else may have---it is easy and fun, one of the better parts of the story. Anyway the pressure is minus one times the density---they are measured in the same units.

 

[math]p_{\Lambda} = - \rho_{\Lambda}[/math]

 

cosmologists call this "the dark energy equation of state" and they play around with the minus one factor in order to get pathological stuff to happen like the notorious "Big Rip". But this is largely just them having fun.

Physicists who want to get in on the action love to speculate as to what the energy density could be and whether or not the equation of state ratio is really minus one. Observations have narrowed it down now so that it has to be close to minus one if not exactly.

 

the important thing to remember is that what is observed is the slight extra curvature in spacetime which can be described as an acceleration in the rate of expansion. this curvature could have come into the geometry of spacetime some other way

 

so it is very convenient to rewrite the Friedmann equations----the basic model used in cosmology----using the extra energy density rho sub Lambda, instead of the original curvature Lambda. Very convenient and everybody does it! and it helps keep track of the percentage effects of things because everything is causing curvature, all energy densities contribute, so why not put Lambda in in that disguise.

 

But even tho everybody does it and all the articles say rho-Lambda is 0.73 of the total energy density, and all that. Even so, it may not be right. We have to be prepared for the eventuality that Lambda will turn out to be a bit of extra curvature pure and simple! Maybe a tiny quantum blink or twitch just before the big bang set it up and it became intrinsic in the particular universe we got, a kind of built-in scale for large distances, or a long-wavelength cut-off. Warning, highly speculative! Nobody has anything very satisfactory to say about this except "don't read any cosmology books from before 1998"

 

It was in 1998 that people first encountered the data on accelerating expansion and were forced to confront this distance scale of 9.5 billion lightyears for the first time.

 

Sean Carroll was among the first people to publish about this in 1998 and I guess he has various books or essays you can read about it. but watch out for popularizations no matter who writes them as if the treatment is overly verbal instead of mathematical it can damage your brain.

 

 

here are the Friedmann equations out of the Sean Carroll LivingReviews article I put the link to. see also SFN post 69672

Here the cosmological constant appears separately as Lamda, the inverse distance squared term, and rho is all the other energy density, not counting dark energy:

 

[math](\frac{a'}{a})^2 = \frac{8\pi G}{3}\rho - \frac{k}{a^2} + \frac{\Lambda}{3}[/math]

 

[math]\frac{a''}{a}= -\frac{4\pi G}{3}(\rho + 3p)+\frac{\Lambda}{3}[/math]

 

a = a(t) is the cosmological scale factor, which varies with time, getting larger as spatial distances increase. the acceleration everybody talks about shows up in a'', the second derivative wrt. time. for simplicty c = 1 otherwise there would be c all over the place in these equations.

 

http://www.scienceforums.net/forums/showthread.php?p=69672#post69672

[TheProphet]

Thanks Martin. Articles, books and all i have reade haven't put any good math in it just said that it's 0,73... Im not shure i get u right away.. (Math skill sucks over here) But some atleast wnt in my brain! Thanks for pointing out the difference in such explicit terms =D