Critical Density: What is the Energy Density of the Universe?

[Olivia Grace]

Do we know what the critical density of the universe is? I always thought we did know, but apparently (according to friend) we don't...

 

[marcus]

Do we know what the critical density of the universe is? I always thought we did know, but apparently (according to friend) we don't...

I calculated it to be around 0.8 joules per cubic kilometer

 

[marcus]

you can check my work if you want, Olivia.

there is a standard equation all cosmology is based on called Friedmann equation, and the standard formula for rho_crit comes right out of that

3H²/(8 pi G)

So you just have to plug in the measured values of Hubble parameter H and Newton constant G

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what I actually got as an answer was 0.83 joules per cubic kilometer-----but it is just a rough estimate so I rounded it to 0.8

there is an easy way to do the calculation, if you want to try it: the usual figure for the Hubble parameter, at present, is 71 km/second per Mpc which is a bit messy but it boils down to a reciprocal time. It boils down to one over 13.8 billion years.

you can convert 13.8 billion years to T (the time in seconds).

Then you just have to compute 3/(8 pi G T²)

If you use the usual figure for G in SI metric units, that gives a mass density expressed in kilogram per cubic meter. I prefer to multiply by c² to get it in energy terms, joules per cubic meter or actually cubic kilometer----easier to remember.

 

[Olivia Grace]

Thanks very much, that's really helpful!
One thing,
"the usual figure for the Hubble parameter, at present, is 71 km/second per Mpc which is a bit messy "
How did you boil this down to the 1/13.8 billion years? I'm having trouble converting the units to SI ones...
Your help is muchly appreciated!

 

[marcus]

Thanks very much, that's really helpful!
One thing,
"the usual figure for the Hubble parameter, at present, is 71 km/second per Mpc which is a bit messy "
How did you boil this down to the 1/13.8 billion years? I'm having trouble converting the units to SI ones...
Your help is muchly appreciated!

my handbook table of equivalents says that a parsec is 3.0857 x 10^13 kilometers

so I guess there are several ways you could simplify the fraction, one being like this
put the Megaparsec into kilometers, so it is 3.0857 x 10^19 km

divide that by 71 km/s

that gives you the Hubble time expressed in seconds (which you still must convert to years :-( sorry about that)

===========================
perhaps there are smarter ways to do it, but here is what i am proposing, in gory detail, as one method

the inverted Hubble parameter is One Mpc/ (71 km/s)

so that is 3.0857 x 10^18 km/(71 km/s) and kilometers cancel so it's ( (3.0857 x 10^19 )/71) seconds

 

[Chronos]

The critical density is difficult to calculate in an expanding universe. The density 'now' required to produce a static universe is the usual definition. The measurable amount of matter in the universe is nowhere near enough to explain the current rate of expansion. Nor is it sufficient to explain the rotation curves of nearby galaxies. That leads to the dark matter conjecture - which nicely accounts for both effects.

 

[marcus]

the inverted Hubble parameter is One Mpc/ (71 km/s)

so that is 3.0857 x 10^18 km/(71 km/s) and kilometers cancel so it's ( (3.0857 x 10^19 )/71) seconds

anyway 308.57/71 = 4.346

so the Hubble time, which Olivia Grace seemed to want, is
4.3 x 10^17 seconds

maybe one should postpone rounding off and say 4.346 x 10^17 seconds.

that is the thing everybody calls "13.8 billion years" but it is actually more useful sometimes to know it in terms of seconds, like we have here!

 

[marcus]

Olivia seems to have vanished
I was hoping she would finish calculating the critical density

to get it in joules per cubic kilometer all we needed was to
calculate 3 c^2/(8 pi G T^2)

and just looking at mantissas we have
c = 2.9979...
G = 6.6742...
pi = 3.1415926...
hubble time T = 4.346...

when I plug those mantissas in I get 16.074...divided by the square of 4.346

that is where the POINT EIGHT joules comes from. You see it is
16.074... divided by 18.8877...
which is 0.851...

OK what I remembered was 0.83 joules and using today's numbers it is 0.85.

Anyway if you believe dark energy is energy and you average out all the energy in the universe on a per cubic kilometer basis then if Omega is exactly one then it should come to 0.85 joules per each cube.
================

Just for a lark let's take NED WRIGHT'S BEST FIT OMEGA of 1.011.

Then the REAL energy density of the universe equals 1.011 times crit. and crit is 0.851
and that gives 0.860 joules per cubic kilometer.
================

so if Olivia were still around here, we would say to her that based on 71 for the hubble,
the CRIT is 0.85 joules per cubic km
and based on Ned Wright best fit estimate, the REAL energy density of the universe is 1.011 times crit
so the REAL is 0.86 joules per cubic km