The critical density of the universe is(adsbygoogle = window.adsbygoogle || []).push({});

rho_{crit}= (3c^{2}H_{0}^{2})/8piG

which is the average energy density required for flatness.

Many people are put off by formulas this complicated and can't say just by looking at it what the density is in familiar terms---how many BTU per cubic yard or whatever makes sense to you.

1. why does it matter what it is?

2. what's a simpler formula for it?

3. what does it actually work out to be?

1. It turns out that as far as we can tell our universe is actually flat, and so what you get from the formula (which takes into account the cosmological constant or dark energy) is our best estimate of the density of the world. Cosmologists are in the habit of giving other densities (like matter, radiation, dark matter, etc) as FRACTIONS of rho crit. So if you know rho crit you can interpret what they are saying and if you dont you cant. So it is good to know.

2. In natural units (c=G=hbar=1) the critical density is simply

rho_{crit}= (3/8pi) divided by the square of the Hubble time.

The Hubble time is 1/H_{0}. It works out to around 8E60 in natural units or about 13.8 billion years. When you see E60 in natural units that is the same order of magnitude as a billion years---same ballpark timescale. The square of the Hubble time is about 64E120.

The number (3/8pi) is roughly 1/8 so rho crit works out to

around 1/8 divided by 64E120. It is a very small density which is good because if it were bigger the universe would go crunch.

If you need more precision in line with presentday accurate measurements of the Hubble parameter, use 8.06E60 for the time.

rho_{crit}= (3/8pi) t_{H}^{-2}

Note that the Hubble time is not generally equal to the age of the universe although in some models it may be fairly close to it. The Hubble parameter is something that is directly measureable from data on observations----whereas the age is something people theorize about and infer from models. Different beasts.

3. To take a for instance: a density which is 2 joules per cubic mile works out to be E-123 in natural units.

So if some density happens to be 1.8E-123 in planck, and you want to interpret it in everyday language, you can call it 3.6 joules per cubic mile. This will be considered perverse by metric purists who abhor miles----therefore, so that the purists may rejoice, we say that E-123 is half a joule per cubic kilometer. Then 1.8E-123 comes out to be around 0.9 joule per cubic km. None of this appears to matter much because it is just translation into some arbitrary conventional terms. I find it's more useful to know in planck.

4. In case you like differential equations the two Friedmann equations are what the Einstein GR equation boils down to assuming a nice homogeneous isotropic universe and the second

Friedmann says (in the zero curvature case):

H_{0}^{2}= (8piG/3c^{2}) rho

that tells what rho has to be in the zero curvature case and

it's easy to rearrange that equation so as to solve for rho,

and it gives the definition of rho crit quoted at the beginning.

This may be why rho crit is so useful----its definition is a disguised

form of one of the two favorite equations of cosmology.

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# Tutorial on rho crit

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