Can two galaxies receded from one another faster than C?

[marcus]

... I assume that it represents the effect of radiation. It's a bit puzzling why this isn't an input parameter into the model in the first place.
...

I think you are right. I think it is approximately zero as fraction of total energy density and therefore usually ignored or lumped in with matter.

I guess I'm concerned that the radiation effects may not be being modeled, because we may not have good information on them - i.e. we may be implicitly assuming that \Omega_{ro} = 0 when we use either Morgan's or Ned Wright's calculator

in the usual percentage breakdown dark energy or the cosm. const is 73% and matter is 27%
and radiation is a fraction of a percent, so it is "in the noise". One is not expressing the 27% accurately enough for something as slight as density of radiant energy to register. So instead of writing down "~0%", which would seem a bit pedantic, one just ignores it.

However if you are especially interested I have seen detailed energy breakdowns of the contents of the universe with estimates carried out to several decimal places. I might be able to find a link.

A possibly interesting detail is that almost all the radiation energy (which is already small on a per unit volume basis) is in the CMB!
Compared with the CMB the rest of the radiant energy-----like starlight for example----is negligible. Note that the temperature of space is the temperature of the CMB radiation (the starlight contributes almost nothing to the temperature). And Planck's cavity radiation density formula can provide the estimate---using the measured 2.75 Kelvin as input.

Let me know if you are especially interested in the average density of EM radiation in space and I will take time to look it up or do a back-of-the-envelope.

but for the moment, just call it zero. After all baryonic matter is only around 4%!

 

[pervect]

I think you are right. I think it is approximately zero as fraction of total energy density and therefore usually ignored or lumped in with matter.
in the usual percentage breakdown dark energy or the cosm. const is 73% and matter is 27%
and radiation is a fraction of a percent, so it is "in the noise". One is not expressing the 27% accurately enough for something as slight as density of radiant energy to register. So instead of writing down "~0%", which would seem a bit pedantic, one just ignores it.

However if you are especially interested I have seen detailed energy breakdowns of the contents of the universe with estimates carried out to several decimal places. I might be able to find a link.

Well, I'm mainly interested in how far back we can go with the online calculators. Mostly because it was in your example, but it also seems like a reasonable thing to want to know about the results (the realm of their validity).

The effect of radiation becomes more important the earlier one goes.

If we look at the generalized Friedmann eq, Ned Wright version, we find that

<br /> \dot{a} = H_0 \sqrt{\frac{\Omega_m}{a} + \frac{\Omega_r}{a^2} + \Omega_v a^2 + (1-\Omega_t)}<br />

http://www.astro.ucla.edu/~wright/Distances_details.gif, eq 4

a is defined such that it is now unity, H_0 is the current value of the Hubble constant, and the various \Omega represent the proportion of energy in mass, radiation, and vacuum energy.

Early in the history of the universe, a was much smaller. Since a=1/z+1, for z=1000, a was about .001. We can see that this amplifies the importance of the \Omega_r term by a factor of 1000.

Thus in order for the above equation to be accurate to this era, we must have \Omega_r &lt;&lt; .001 \Omega_m.

 

[marcus]

The effect of radiation becomes more important the earlier one goes.
...

good observation!
That is quite true. In fact in the very early universe radiation is dominant.

so perhaps you do not care what modest fraction of a percent it is now

Instead, perhaps you would like to know the epoch, or the redshift, at which radiation ceases to be dominant and matter takes over?

my guess is that it is well before 300,000 years old but that is merely a guess. It would take some looking up. Would you like to do that?

My sense is you are very capable of searching out that kind of information, so if you would like to track it down, please do! (I will be interested to know the result.)

 

[SpaceTiger]

The epoch of matter-radiation equality is around z~10000 and \frac{\Omega_r}{\Omega_m}\sim 10^{-5}. The radiation content of the universe is presently dominated by the CMB, so the previous relation can be derived by considering the energy density of a blackbody (at 2.7 K):

u=aT^4

and comparing it to the energy equivalent of the critical density:

\rho_c=\frac{3H_0^2}{8\pi G}

 

[marcus]

Excellent! Thanks SpaceTiger. See also:

A possibly interesting detail is that almost all the radiation energy (which is already small on a per unit volume basis) is in the CMB!
Compared with the CMB the rest of the radiant energy-----like starlight for example----is negligible. Note that the temperature of space is the temperature of the CMB radiation (the starlight contributes almost nothing to the temperature). And Planck's cavity radiation density formula can provide the estimate---using the measured 2.75 Kelvin as input.

Let me know if you are especially interested in the average density of EM radiation in space and I will take time to look it up or do a back-of-the-envelope.

so maybe we should use the radiation density formula----involving the fourth power of temp----that Nick has written for us, and do a back-of-envelope calculation?

If I remember correctly the critical density (which one assumes is THE REAL density in a flat model) is around one joule per cubic kilometer. Nick provides us with a formula which we could use to check that.
the rest should be easy. would anyone like to try?

=========================

EDIT: I am delighted with how the thread is going and will reply to your next post here, pervect, so as not to add a superfluous post. I don't think you really need to worry about radiation in what you are doing since it is not close enough to z = 10,000. but if you happen to want the constant "a" in the fourthpower energy density law it is

the same as the stefanboltzmann constant but multiplied by 4/c
so stefanboltzmann is 5.67E-8 W/sq meter per K to the fourth
and the "a" in what Nick wrote is
7.56E-16 Joule/cubic meter per K to the fourth

What that means for example is that if you have a cubic kilometer of space at temp 100 Kelvin then the thermal radiation in that cubic kilometer is 75.6 Joules
but if the temp is only 1 Kelvin then the radiation in the cubic kilometer is only 75.6 E-8 Joules
which is to say a bit under a microJoule

the critical density of the universe, at present, is around one joule per cubic km
I calculated it one time and seem to remember something like 0.86 but anyway something like 1.
You can calculate it from the Hubble parameter using the formula Nick mentioned.
And the critical density corresponds pretty well with the actual real density (the galaxies the dark energy etc. plus a little light)
And so in an average cubic km most of that Joule is made up of dark energy and dark matter and a little baryonic.

things seem to be OK and it is my bedtime so buenas noches.

 

[pervect]

OK, Ned Wright's calculator does not give us some of the information in Morgan's output, but we can compute the additional information from what Ned Wright gives us and some standard equations.

If we add the following eq's

<br /> D_{then} =\frac{D_{now}}{z+1}<br />
<br /> H = \frac{\dot{a}}{a}<br />
clarification: H_{then} = H evaluated at a = \frac{1}{z+1}
<br /> V_{then} = H_{then} D_{then}<br />

to Ned Wright's form of the Friedmann eq, i.e.

\dot{a} = H_0 \sqrt{\frac{\Omega_m}{a} + \frac{\Omega_r}{a^2} + \Omega_v a^2 + (1-\Omega_t)}<br />

we have everything we need to calculate the additional information. (I have not repeated the information on how to calculate D_{now}, just added the information needed to calculate the 'extra' quantities that Morgan's calculator gives and Ned Wright's does not.)

We can safely set \Omega_r=0 as discussed for anything we can see (i.e. the equations will work to z=1100 and as an added bonus a fair ways beyond).

The results we get by doing this appear to agree with Morgan's results for the test case.

[add]The main motive for writing this out is that Morgan's calculator gives answers, but does not have an explanatory page for the formulas used. Ned Wright's calculator has such an explanatory page, but does not calculate everything that Morgan's calculator calculates. Thus it is useful to document the formulas needed to calculate the extra quantities.