Hubble's law and cosmological density

[Naty1]

While reading I came across this chart in Wikipedia

http://en.wikipedia.org/wiki/Metric_expansion_of_space

Anybody know the source or a similar plot with some explanations? What do you make of it?? What does it purport to show? Is it in the correct section [Theoretical basis and first evidence] of the article...If so which is it: do you think the plot represents a 'theoretial basis', a prediction, or does it suggest 'first evidence' , experimental confirmation..?

The y-axis is p(a) and the x-axis a(t)...is (a) cosmological scale factor??...so where "today" on the chart is about 3ct₀...or 42B years or so?

I can see that radiation density falls off most rapidly (proportional to 1/a⁴, matter density falls off with volume, curvature as 1/a², and dark energy density remains constant...is it obvious the first three decline linearly?

So far I haven't found a title, explanation, legend, or anything else...Apparently it was posted March 12, 2012... so it seems pretty new. Maybe Muhammad the author will add more..

Why are matter and radiation shown at a common starting point with equal densities...?
Why that point was selected? I'm guessing it's because nobody knows the radiation density at the big bang, so its convenient..or is that where linear change can be approximated for those two variables??

thanks...

 

[Nabeshin]

I think what you're missing is that this is meant to be a log plot. In the radiation dominated universe, $a \propto \sqrt{t}$, so $\log a \propto 1/2 \log t$, which is of course a straight line with slope 1/2 in log-log space. Similarly for matter dominated, the curvature term, and of course DE dominated is constant. The plot is usually used to show the various epochs in our universe's history, such as the matter-radiation equality, along with the transition into a de-Sitter universe.

That should clear it up, I think.

 

[marcus]

Here's the direct link to the chart you are discussing in case anyone is wondering which it is:
http://en.wikipedia.org/wiki/File:Density_evolution_in_a_curved_universe.png

I was intrigued by this one instead:
http://en.wikipedia.org/wiki/File:Embedded_LambdaCDM_geometry.png#Mathematical_details

 

[Chalnoth]

I can see that radiation density falls off most rapidly (proportional to 1/a⁴, matter density falls off with volume, curvature as 1/a², and dark energy density remains constant...is it obvious the first three decline linearly?

As others have mentioned, this is a logarithmic plot. Variables with power-law relationships look linear on a log-log plot.

Why are matter and radiation shown at a common starting point with equal densities...?
Why that point was selected? I'm guessing it's because nobody knows the radiation density at the big bang, so its convenient..or is that where linear change can be approximated for those two variables??

They don't start off at equal densities. But because radiation is decreasing in density faster, at some point they do have equal densities.

And by the way, the radiation density at the emission of the CMB is extremely well-known, and we can extrapolate the previous radiation density back in time from there. At some point that extrapolation obviously has to break down.

 

[Naty1]

Apparently I am 'out of touch' with with 'new' graphing techniques of no title, no axis labels, no legend, ...well, not even any text... You guys and girls will have to excuse me for being unable to start from scratch...next time I plot something, I think I'll just plot the variables: why bother with axis, labels, etc...makes stuff more 'mysterious'...and maybe I'll rotate 1/4 turn from standard plots for additional 'coolness'...I DON"T THINK SO!

I think what you're missing is that this is meant to be a log plot.

I AM missing a lot more than that! but that insight DOES help! [Can I get 'half credit' for recognizing that all those linear declines would be really, really unusual??]

And by the way, the radiation density at the emission of the CMB is extremely well-known, and we can extrapolate the previous radiation density back in time from there. At some point that extrapolation obviously has to break down.

nicely stated...
but I still don't know why equal matter and radiation densities were picked...although it does look rather nice that way!...that didn't coincidentally happen at the first CMB emission, right?? It appears a specific scale factor was picked, a_eq which if 'eq' means equal I'll take a stab in the dark an guess it means when matter and radiation density were equal...??

this #$*&^% confounded graph got me all distracted from the scale factor and metric expansion I was trying to read more about...now somebody admit it: doesn't a graph need something besides greek letters!

 

[Chalnoth]

Apparently I am 'out of touch' with with 'new' graphing techniques of no title, no axis labels, no legend, ...well, not even any text...

No need to get snarky. Obviously it's a badly-composed plot. We only know what it means because we're familiar with this sort of thing.

but I still don't know why equal matter and radiation densities were picked...although it does look rather nice that way!...

It's just a plotting choice. Nothing special about it. They simply chose to start the plot just before matter and radiation reached equality. There was a fair amount of time before that.

that didn't coincidentally happen at the first CMB emission, right??

No, no. Matter/radiation equality was quite a bit earlier.

It appears a specific scale factor was picked, a_eq which if 'eq' means equal I'll take a stab in the dark an guess it means when matter and radiation density were equal...??

Yes, that's the usual label for the particular time when matter and radiation had equal densities.

 

[Chalnoth]

Oh, interesting. Just looked at the plot again. This plot also doesn't conform to reality. That's probably part of why it's so vague: it's simply a plot of what the various energy densities do over time. But the particular energy densities just don't match reality at all.

How do I know this? Well, it lists a substantial "energy density" for curvature at the present time. This just isn't the case.

So it's just a heuristic plot to show how the different sorts of energy densities change over time.

Note: I put energy density in quotes for curvature because it isn't really an energy density. But it acts much like one when considering its effect on the expansion.

 

[Naty1]

Chalnoth..ha,ha...same thing happened to you as to me... that darn graph has me distracted!

But the particular energy densities just don't match reality at all.
How do I know this? Well, it lists a substantial "energy density" for curvature at the present time. This just isn't the case.

You may be right, but I'm not so sure ...take a look at the defined intersection points to the right where some x-axis points are labelled [like ρ_c =ρ _λ...where the curvature line seems anchored]...I don't know what those mean but somebody seems to have had something specific in mind...?

 

[Chalnoth]

You may be right, but I'm not so sure ...take a look at the defined intersection points to the right where some x-axis points are labelled [like ρ_c =ρ _λ...where the curvature line seems anchored]...I don't know what those mean but somebody seems to have had something specific in mind...?

Look at the point that says "present day." It shows $\rho_c > \rho_\Lambda$, which is definitely not the case.

 

[Naty1]

My whole issue is that I don't know what those symbols mean...I can infer from the chart what ρ_λ means...cause dark matter density supposedly remains constant but that's about it...

 

[Chalnoth]

My whole issue is that I don't know what those symbols mean...I can infer from the chart what ρ_λ means...cause dark matter density supposedly remains constant but that's about it...

Okay, so, $a$ is the scale factor, and each $\rho$ is a different energy density. Specifically, $\rho_r$ is the radiation energy density, $\rho_m$ is the matter energy density, $\rho_\Lambda$ is the dark energy density, and $\rho_c$ is the effective curvature density.

Does that help?

 

[Naty1]

ρ c is the effective curvature density...

DUH!

I kept thinking "c" for lightspeed ...but I am laboring from the disadvantage of never having heard of 'curvature density'...What is that??

Anything to do with any of this:

http://en.wikipedia.org/wiki/Scalar_curvature

 

[lostprophets]

..cause dark matter density supposedly remains constant ...

really ?
hmmmm

 

[Nabeshin]

DUH!

I kept thinking "c" for lightspeed ...but I am laboring from the disadvantage of never having heard of 'curvature density'...What is that??

Anything to do with any of this:

http://en.wikipedia.org/wiki/Scalar_curvature

Indeed they are related, see here:
http://en.wikipedia.org/wiki/Friedmann_equations#The_equations

 

[Nabeshin]

really ?
hmmmm

Yeah, that's basically the hallmark of Dark Energy, and why it connects back to Einstein's cosmological constant rather than a standard matter density.

 

[Chalnoth]

DUH!

I kept thinking "c" for lightspeed ...but I am laboring from the disadvantage of never having heard of 'curvature density'...What is that??

Anything to do with any of this:

http://en.wikipedia.org/wiki/Scalar_curvature

Not really. At least, not exactly. The scalar curvature in the FRW metric has two components: one stemming from the expansion itself, the other coming from the spatial curvature. This term, $\rho_c$ is only related to the spatial curvature.

Taking the first Friedmann equation, for example:

$$H^2 = {8\pi G \over 3}\rho - {kc^2 \over a^2}$$

The parameter $\rho_c$ is simply defined so that it contains the information of the curvature parameter $k$, but enters this equation like an energy density:

$$\rho_c = {-3kc^2 \over 8\pi Ga^2}$$

So that now we can write the first Friedmann equation as:

$$H^2 = {8 \pi G \over 3} \tilde{\rho}$$

With:

$$\tilde{\rho} = \rho + \rho_c$$

 

[cepheid]

Yeah, that's basically the hallmark of Dark Energy, and why it connects back to Einstein's cosmological constant rather than a standard matter density.

Naty1 actually said that the dark *matter* density was constant, which may have been why lostprophets questioned him.

Also, if w != -1 then the dark energy density is not constant and can't be described simply as a cosmological constant term.

 

[Nabeshin]

Naty1 actually said that the dark *matter* density was constant, which may have been why lostprophets questioned him.

Also, if w != -1 then the dark energy density is not constant and can't be described simply as a cosmological constant term.

Oops! Of course, you can have w != -1, but in the standard Lambda-CDM it's taken to be -1.

 

[Naty1]

ρ

Originally Posted by Naty1

..cause dark matter density supposedly remains constant ...

really ?

oops for me too..I meant to say 'energy' density...
Nabeshin: thank's for the reference

tp://en.wikipedia.org/wiki/Friedma...#The_equations

The density parameter,Ω , is defined as the ratio of the actual ρ (or observed) density to the critical density ρ_c of the Friedmann universe. The relation between the actual density and the critical density determines the overall geometry of the universe. In earlier models, which did not include a cosmological constant term, critical density was regarded also as the watershed between an expanding and a contracting Universe.

so the ρ_c is CRITICAL density... Now I understand Chalnoth's comment:

it lists a substantial "energy density" for curvature at the present time. This just isn't the case.

I've only seen critical density referred to as an 'energy density' rather than a 'spatial curvature' as well...

I sure have to keep a CLOSE watch on you guys! Thanks.

 

[Naty1]

Can someone explain what 'w' is in posts #17 and #18. Is this an easy way to write
Ω, that is actual/critical density...??

All I think I know is that 'new' space causes cosmological expansion, 'arrives' with it's own energy density, same as old space, and so with more space we get more cosmological total energy...we call it negative vacuum energy pressure...

[Now that I have noticed 'quick symbols' to the right of my when posting [is that new?] screen I could get even more 'dangerous' posting stuff like Ω...cool!]

 

[cepheid]

Can someone explain what 'w' is in posts #17 and #18. Is this an easy way to write
Ω, that is actual/critical density...??

All I think I know is that 'new' space causes cosmological expansion, 'arrives' with it's own energy density, same as old space, and so with more space we get more cosmological total energy...we call it negative vacuum energy pressure...

[Now that I have noticed 'quick symbols' to the right of my when posting [is that new?] screen I could get even more 'dangerous' posting stuff like Ω...cool!]

The parameter w appears in the equation of state of dark energy. The equation of state relates the pressure P to the energy density rho. For ordinary (non-relativistic) matter we assume it has negligible pressure and P = 0. For radiation, the pressure is given by P = (1/3)$\rho$, which is something you can derive for an isotopic radiation field.

For dark energy, the eqn of state is unknown, and we parameterize our ignorance with w by writing$$P = w\rho$$.

One thing we know is that in order for the dark energy to produce an acceleration, it must have negative pressure. If you look at the Friedmann acceleration equation, you'll see that in order for the second derivative of the scale factor to be positive, it must be true that P < -(1/3)rho.

The simplest case is if w = -1 (ie P =-rho) in which case you can show that the energy density of dark energy is constant with time.