What is the critical density of the universe?

[marcus]

The critical density of the universe is

(3c2H02)/8piG

which is the average energy density required for flatness.
And in fact recent measurements imply that our universe is indeed flat and therefore has this energy density.

In addition, the Hubble parameter H0 has been
determined in the past 5 years with remarkable accuracy
and is 71 km/s per Mpc plusminus some modest uncertainty.
So what is the density of our universe? We live here
and ought to have some idea what it is.

Presumably you know c, and G, and pi, and 71, so
it should be a snap for you to guesstimate the rough size.

Knowing this density can be handy because it lets one
compare other energy densities to it----like the energy lost
from the CMB thru expansion----and figure out what various
fractions of it amount to, like the part of the energy that is visible matter and the part contributed by radiation and so on. But it's
good as well because it gives a feel for how thinly or thickly space is occupied.

[chroot]

cf. Carroll and Ostlie's "Introduction to Modern Astrophysics."

- Warren

[marcus]

Originally posted by chroot
cf. Carroll and Ostlie's "Introduction to Modern Astrophysics."

- Warren

Which of the four answers is closest to what Carroll and Ostlie say?

I have high expectations of PF people----I expect them either to know the rough order of magnitude of rho crit, or else to
be able to make a quick guesstimate using what they know
about H0. Besides which it can be useful to know.

[chroot]

What's with you and all this "I'm going to test PF" attitude? You've made it clear now in several threads that you really don't know as much as you think you know.

- Warren

[schwarzchildradius]

I disagree. The critical density depends on the geometry of the entire universe.

[marcus]

Originally posted by schwarzschildradius
I disagree. The critical density depends on the geometry of the entire universe.

This is a possible objection. Bravo.
You are objecting to the formula

(3c2 H02)/8piG

as a way to calculate rho crit.

It is the standard Astro 1 formula for rho crit (not even Astro 101 [:)]) and comes right
out of the second Friedmann equation.

Because of that there are some underlying assumptions built into it of homogeneous isotropic.

Everybody is allowed to doubt that on the large scale the U
is homog and isotropic. And also to doubt that the Einstein
equation of GR (which the Friedmann eqs derive from) is correct.

But there is a widespread belief that these things are pretty darn near right. The people who are investigating the possibility that the universe looks like a donut are pretty far out on the margin. It is not really a hot research topic.

So why not accept GR and Friedmann eqs and flatness and this resulting formula at least provsionally and see what it says about your universe?

In which case, which of the four answers is the energy density
that comes out of the formula?

[schwarzchildradius]

That value comes right out of specific solution to the field equations; the critical density depends on the behavior of the cosmological constant.

[schwarzchildradius]

Ok your super new fancy version of H = 2.3e-18 s-1
c=3e8
G=6.67e-11
GMm/r=mv2/2 {v=c}
2GM/c2=r
M = (4/3)πr3ρ
2G(4/3)πr2ρ/c2=1
ρ=3c2/(8πGr2)
r = c/H
ρ=3H2/8πG
=9.5e-27 kg/m3

[marcus]

Originally posted by schwarzchildradius
Ok your super new fancy version of H = 2.3e-18 s-1
c=3e8
G=6.67e-11
GMm/r=mv2/2 {v=c}
2GM/c2=r
M = (4/3)πr3ρ
2G(4/3)πr2ρ/c2=1
ρ=3c2/(8πGr2)
r = c/H
ρ=3H2/8πG
=9.5e-27 kg/m3

I will check your answer if you wish. Your figure for the Hubble time is 1/2.3E-18 seconds which works out to 13.8 billion years, so your figure for H is correct.

Now let's look at the bottom line. You have it expressed as a mass density so I have to convert it to energy density, multiplying by 9E16 the square of the speed of light and that turns 9.5E-27 kg/m3 into 0.855E-9 joules per cubic meter.

That is 0.855 joules per cubic kilometer!

YIPPPEEEE!, now a cubic mile is about 4.2 cubic km so indeed your figure converts to 3.6 joules per cubic mile. This is one of the options on the poll.

I mixed the units----joules per cubic mile----in the poll so that one could not simply copy something out of a book or off the web. Your value of around 0.86E-9 joules per cubic meter is a standard mainstream estimate of rho crit. Bravo and congratulations!!!!

Especially since you did not merely copy it from somewhere but obviously went out on a limb and calculated it yourself.
This shows that Foxans have the pioneer spirit or chutzpah or something.

Put a click beside 3.6 joules per cubic mile, if you want.

[marcus]

Originally posted by schwarzchildradius
Ok your super new fancy version of H = 2.3e-18 s-1
c=3e8
G=6.67e-11
...
ρ=3H2/8πG
=9.5e-27 kg/m3

SchwaR. your suggestion that our form of government is a
kleptocracy reminds me of Swift's way of expressing his rage,
which was to turn it into very funny satire. I think his best
piece is not Gulliver but the short essay A Modest Proposal,
which explains how to solve the problem of overpopulation
and is even more relevant today than in the 18th. Have you
read Swift by any chance? there is a kind of Irishness about
his way of expressing outrage.

I will match you in natural units

For a metric user the Hubble time is 4.35E17 seconds
(I have simply taken the reciprocal of your
"2.3e-18 s-1")

and for a natural user the same time is 8.06E60.

A user of natural units simply divides (3/8pi) by the square of 8.06E60, and is done.

A user of metric units would proceed rather much as you did.
Divide (3/8pi) by 6.673E-11, multiply by the square of 3E8, divide by the square of 4.35E17----that gives 0.85 nanojoules per cubic meter, which is the right answer.

[schwarzchildradius]

your equation for ρ is incorrect

[marcus]

Originally posted by schwarzchildradius
your equation for ρ is incorrect

Please be more specific. In fact I did give an equation for
the critical energy density rho crit.

And the formula I gave was

(3c2 H02)/8piG

Are you saying that formula is incorrect?

There is no other formula in cosmology for the critical
energy density, that I know of.

[schwarzchildradius]

ahh. alright, just dont get it confused with vacuum energy density.