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Critical Density

by nick1o2
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nick1o2
#1
Nov7-13, 04:40 AM
P: 14
I was reading up on critical density, and found the "current" number for it, but can't fine any past records or graphs to show how they have changed over time. Any help?
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dauto
#2
Nov7-13, 09:48 AM
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What critical density? If you care enough to actually tell us what you're talking about you might eventually even get an answer. Just saying...
nick1o2
#3
Nov12-13, 03:11 AM
P: 14
The Average Critical Density of the universe. Sorry should of made that clear.

jtbell
#4
Nov12-13, 05:56 AM
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Critical Density

Moved to the Cosmology forum because this looks like a cosmology topic. If I'm mistaken, say so and I'll move it somewhere else.
marcus
#5
Nov12-13, 10:48 AM
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Quote Quote by nick1o2 View Post
I was reading up on critical density, and found the "current" number for it, but can't fine any past records or graphs to show how they have changed over time. Any help?
Quote Quote by nick1o2 View Post
The Average Critical Density of the universe. Sorry should of made that clear.
ρcrit(t) is just something you calculate from H(t). It changes as the Hubble expansion rate changes. It is proportionate (by a constant factor) to H2.

So you can track it by looking at a record of H(t) over time. I'll try to think of how to get a graph or table.
marcus
#6
Nov12-13, 11:18 AM
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If you're familiar with Friedman eqn. then you remember that
H2 = (8πG/3c2

with ρ expressed as an energy density (if you like it as a mass density then omit the c2)

So solve for ρ:

ρ = (3c2/8πG)H2

Are you familiar with the Hubble time? It is simply the reciprocal of the rate:
THubble = 1/H
Let's denote it by Θ so we don't have to write the subscript
Θ(t) = 1/H(t)
So if I can show you a table of the past history of the time Θ(t) you can calculate ρcrit!

ρcrit = 3c2/(8πG Θ2)

Here's a table of past values of the Hubble time Θ listed in billions of years.

[tex]{\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline T (Gy)&\Theta (Gy) \\ \hline 0.473&0.7105\\ \hline 0.566&0.8504\\ \hline 0.678&1.0176\\ \hline 0.811&1.2173\\ \hline 0.971&1.4558\\ \hline 1.162&1.7401\\ \hline 1.390&2.0787\\ \hline 1.663&2.4807\\ \hline 1.988&2.9566\\ \hline 2.375&3.5172\\ \hline 2.835&4.1732\\ \hline 3.380&4.9340\\ \hline 4.023&5.8050\\ \hline 4.777&6.7856\\ \hline 5.654&7.8652\\ \hline 6.666&9.0202\\ \hline 7.819&10.2134\\ \hline 9.114&11.3964\\ \hline 10.549&12.5168\\ \hline 12.111&13.5285\\ \hline 13.787&14.3999\\ \hline \end{array}}[/tex]

This takes you from around year 470 million (first stars and galaxies were forming) up to around year 13.8 billion (the present).

You can use google calculator to convert the Thetas to nanojoules per cubic meter. for example to get the present rho_crit just paste this into the google box:

3c^2/(8 pi G (14.4 billion years)^2))

Google will say 0.778 nanopascal which is the same as 0.778 nanojoule per cubic meter (when you sort the units out.)

Or if you want the density when the first stars were forming just paste this into google box:

3c^2/(8 pi G (0.7105 billion years)^2))

Google will say 319.8 nanopascal which is equivalent to 319.8 nanojoule per cubic meter.
nick1o2
#7
Nov16-13, 08:39 PM
P: 14
Thankyou Markus!
Is there anyway to find out the density of different parts of the universe? For example is the density in the northern hemisphere of the universe bigger or less than the southern hemisphere? I would think they would be different because the universe isn't uniform, because we have the big bang model not steady state model, but i there any resources to show the total density in these area's and how they have changed?
Calimero
#8
Nov17-13, 03:32 AM
P: 256
No, cosmology operates on premise of homogeneity and isotropy, meaning that average density of sufficiently big volume is the same throughout the universe.
Calimero
#9
Nov17-13, 03:50 AM
P: 256
You can google "wmap" and look at the picture of CMB radiation to see how amazingly universe is uniform. There are tiny fluctuations, roughly 1 part in 100 000, which served as seeds for later structure formation.
marcus
#10
Jan19-14, 09:07 AM
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The question came up as to how I got the information in the short table in this post:
Quote Quote by marcus View Post
If you're familiar with Friedman eqn. then you remember that
H2 = (8πG/3c2

with ρ expressed as an energy density (if you like it as a mass density then omit the c2)

So solve for ρ:

ρ = (3c2/8πG)H2

Are you familiar with the Hubble time? It is simply the reciprocal of the rate:
THubble = 1/H
Let's denote it by Θ so we don't have to write the subscript
Θ(t) = 1/H(t)
So if I can show you a table of the past history of the time Θ(t) you can calculate ρcrit!

ρcrit = 3c2/(8πG Θ2)

Here's a table of past values of the Hubble time Θ listed in billions of years.

[tex]{\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline T (Gy)&\Theta (Gy) \\ \hline 0.473&0.7105\\ \hline 0.566&0.8504\\ \hline 0.678&1.0176\\ \hline 0.811&1.2173\\ \hline 0.971&1.4558\\ \hline 1.162&1.7401\\ \hline 1.390&2.0787\\ \hline 1.663&2.4807\\ \hline 1.988&2.9566\\ \hline 2.375&3.5172\\ \hline 2.835&4.1732\\ \hline 3.380&4.9340\\ \hline 4.023&5.8050\\ \hline 4.777&6.7856\\ \hline 5.654&7.8652\\ \hline 6.666&9.0202\\ \hline 7.819&10.2134\\ \hline 9.114&11.3964\\ \hline 10.549&12.5168\\ \hline 12.111&13.5285\\ \hline 13.787&14.3999\\ \hline \end{array}}[/tex]

This takes you from around year 470 million (first stars and galaxies were forming) up to around year 13.8 billion (the present).

You can use google calculator to convert the Thetas to nanojoules per cubic meter. for example to get the present rho_crit just paste this into the google box:

3c^2/(8 pi G (14.4 billion years)^2))

Google will say 0.778 nanopascal which is the same as 0.778 nanojoule per cubic meter (when you sort the units out.)

Or if you want the density when the first stars were forming just paste this into google box:

3c^2/(8 pi G (0.7105 billion years)^2))

Google will say 319.8 nanopascal which is equivalent to 319.8 nanojoule per cubic meter.
I was using a temporary nonstandard symbol Theta for the Hubble time THubble because I didn't want to bother with writing subscript, should go back to more conventional notation and say THubble.

The HUBBLE RADIUS is just the Hubble time multiplied by the speed of light. So if the time is 14.4 Gy (billion years) then the radius is 14.4 Gly (billion lightyears).

Jorrie's "Lightcone" calculator gives the Hubble radius RH in Gly, so I just relabeled the R numbers. That's where the table came from. I'll show this in next post. You can easily learn how to use the Lightcone calculator to make your own tables with however many rows you want covering whatever range of expansion you want.
marcus
#11
Jan19-14, 09:23 AM
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Here's the link to Lightcone

Here's the table it prints if you select the range to be from S=11 to S=1, with 20 steps.
That means it will compute and tabulate the universe's history from a time when distances were 1/11 present size up to the present, when distances are their current size i.e. S=1.
[tex]{\scriptsize\begin{array}{|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{\infty} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.9&0.693&0.307\\ \hline \end{array}}[/tex] [tex]{\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&S&T (Gy)&R (Gly)&D_{now} (Gly)&D_{then}(Gly)&D_{hor}(Gly)&V_{now} (c)&V_{then} (c) \\ \hline 0.091&11.000&0.4726&0.7105&31.447&2.859&4.358&2.18&4.02\\ \hline 0.102&9.757&0.5659&0.8504&30.481&3.124&4.814&2.12&3.67\\ \hline 0.116&8.655&0.6776&1.0176&29.456&3.403&5.308&2.05&3.34\\ \hline 0.130&7.677&0.8112&1.2173&28.368&3.695&5.843&1.97&3.04\\ \hline 0.147&6.809&0.9710&1.4558&27.214&3.997&6.418&1.89&2.75\\ \hline 0.166&6.040&1.1621&1.7401&25.990&4.303&7.032&1.80&2.47\\ \hline 0.187&5.358&1.3905&2.0787&24.693&4.609&7.686&1.71&2.22\\ \hline 0.210&4.752&1.6631&2.4807&23.319&4.907&8.376&1.62&1.98\\ \hline 0.237&4.215&1.9883&2.9566&21.865&5.187&9.098&1.52&1.75\\ \hline 0.267&3.739&2.3755&3.5172&20.330&5.437&9.846&1.41&1.55\\ \hline 0.302&3.317&2.8355&4.1732&18.711&5.642&10.613&1.30&1.35\\ \hline 0.340&2.942&3.3803&4.9340&17.011&5.782&11.387&1.18&1.17\\ \hline 0.383&2.609&4.0230&5.8050&15.233&5.837&12.155&1.06&1.01\\ \hline 0.432&2.315&4.7767&6.7856&13.382&5.781&12.904&0.93&0.85\\ \hline 0.487&2.053&5.6541&7.8652&11.471&5.587&13.617&0.80&0.71\\ \hline 0.549&1.821&6.6657&9.0202&9.516&5.225&14.278&0.66&0.58\\ \hline 0.619&1.615&7.8185&10.2134&7.540&4.668&14.874&0.52&0.46\\ \hline 0.698&1.433&9.1144&11.3964&5.570&3.887&15.393&0.39&0.34\\ \hline 0.787&1.271&10.5488&12.5168&3.635&2.860&15.832&0.25&0.23\\ \hline 0.887&1.127&12.1114&13.5285&1.765&1.565&16.189&0.12&0.12\\ \hline 1.000&1.000&13.7872&14.3999&0.000&0.000&16.472&0.00&0.00\\ \hline \end{array}}[/tex]

To find out what the rows mean, click on the link, you will see a sample table, hover the mouse over the blue dots. Then click on "column selection" and you will get more blue info dots telling what the columns mean. And also the "column selection" menu will allow you to select which columns to show. To make that table with only TWO COLUMNS I just selected only the T and the R columns to be shown. The time (in billions of years Gy) and the Hubble radius (in Gly)

The only other thing I did was to set the S upper limit to 11 and the S lower limit to 1 (i.e. to present day) and tell it to cover that range from 11 down to 1 in 20 steps.
You replace the default values of Supper and Slower and STEPS by typing 11, 1, and 20 in those three boxes, and press "calculate".


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