# Redshift at the epoch of matter-radiation equality

#### jackal0206

I have a homework question I am not sure how to approach:

We are told that
1) CMB provides about 10^9 photons/baryons.
2) For matter dominated era, assume flat university with H_o = 73km/s/Mpc
3) Stefan-Boltzman law for energy density: ρ_rad = αT^4 where α = 7.56x10^-15 ergs / cm^3 K^4

I read Marcus' explaination on
https://www.physicsforums.com/showthread.php?t=163941
but it is really difficult for me to understand as I am taking a 1st year AST course.

Any help will be appreciated, thx.

#### marcus

Science Advisor
Gold Member
Dearly Missed
I have a homework question I am not sure how to approach:

We are told that
1) CMB provides about 10^9 photons/baryons.
2) For matter dominated era, assume flat university with H_o = 73km/s/Mpc
3) Stefan-Boltzman law for energy density: ρ_rad = αT^4 where α = 7.56x10^-15 ergs / cm^3 K^4

I read Marcus' explaination on
https://www.physicsforums.com/showthread.php?t=163941
but it is really difficult for me to understand as I am taking a 1st year AST course.

Any help will be appreciated, thx.
If it is a homework question you need to get it moved to homework forum, or start a new thread.

Several ways to do it; one way is to calculate rho_crit. The critical density.
Do you have a figure for that, as an energy density?
The matter contribution is estimated at about 27 percent of the total. So you can find out rho_matter.

Going that way you need to know the energy density corresponding to matter (both ordinary and dark matter combined)

the redshift is going to turn out to be the ratio

today's matter energy-equiv density divided by today's CMB energy density

that is the ratio

rho_matter/rho_rad

If you do it that way you had better learn how to calculate both---the numerator and the denominator.

Maybe someone else, in homework forum, will suggest an alternate way.

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#### jackal0206

If it is a homework question you need to get it moved to homework forum, or start a new thread.

Several ways to do it; one way is to calculate rho_crit. The critical density.
Do you have a figure for that, as an energy density?
The matter contribution is estimated at about 27 percent of the total. So you can find out rho_matter.

Going that way you need to know the energy density corresponding to matter (both ordinary and dark matter combined)

the redshift is going to turn out to be the ratio

today's matter energy-equiv density divided by today's CMB energy density

that is the ratio

rho_matter/rho_rad

If you do it that way you had better learn how to calculate both---the numerator and the denominator.

Maybe someone else, in homework forum, will suggest an alternate way.

Thanks Marcus you're smart. Sorry I didn't know there is a homework forum, I searched up the thread on google and thought this would be the right place to ask. I will keep that in mind and post in the correct forum next time.

One question I want to know though, isn't rho_matter/rho_rad = Scale factor and not redshift?

#### Chronos

Science Advisor
Gold Member
A thought to consider - can photons remain homgogenously distributed throughout the universe without redshift?

#### marcus

Science Advisor
Gold Member
Dearly Missed
One question I want to know though, isn't rho_matter/rho_rad = Scale factor and not redshift?
I think yes, you're right, if I understand you. To paraphrase what I think you mean:
rho_matter/rho_rad
is the ratio of scalefactor now and scalefactor then (when matter and radiation were in balance).
In other words the ratio by which distances have expanded since that time.

So instead of being exactly the redshift z, it is 1+z.

Because the answer is large there is not much percentage difference if add or take away one, but you're right to keep the concepts clear like that and make the distinction. Good luck with your astro course!

A thought to consider - can photons remain homogenously distributed throughout the universe without redshift?
I think the answer might depend somewhat on the model, Chronos. If I understand you, it is a kind of "what if" question. What if, after year 380,000, there had been no expansion? Suppose some genii had frozen the geometry of the universe at that moment, so that distances could neither expand further or begin to re-collapse. Imagine you are the light. What would happen to you?

I think eventually you would be scattered. Re-absorbed and your energy re-emitted. Because the density of year 380,000 is just too high.

The reason that, for us, year 380,000 (approx.) works as a "moment of last scattering" of the great majority of CMB light is that by that time the mean-free-path or the expected free flight time for a photon is long enough, that by the time it would have been absorbed or scattered (barring expansion) there will have been enough additional expansion to extend its flight still more. When you solve the equation that gives 380,000 it takes into account that kind of feedback (or synergy or cooperation among the variables).

Part of the reason 380,000 works IOW is precisely because distances are expanding fast enough---density is declining fast enough--- so that the photon can "recalculate its own mean free path as it goes along", continually getting a new lease on life, and make it to mathematical infinity. It is the point where the half-life blows up so to speak. I hope this sloppy explanation is not making things worse. I looked at the derivation a couple of decades ago and wouldn't be able to re-produce it, just retain the vague idea. Popular accounts simplify by saying that 380,000 was the moment that the medium became transparent. But it didn't become perfectly transparent---just transparent enough that, allowing for future expansion, you had infinite path expectancy.

Like. Density determines your life expectancy as a photon. If density is decreasing fast enough it's like medical science progressing fast enough. We come to a point where you can live forever. Because you will almost surely make it for another five years and by that time medical science will have improved and given you another ten years and so on.

So the whole microwave background thing depends on there being expansion. Not just the homogeneity but even having all that ancient light around at all.

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#### Chronos

Science Advisor
Gold Member
It is a 'what if' question, marcus. In my worldview, expansion smears all photons across the entire universe. I view that as evidence the universe is finite. Were it not, there would be no CMB.

#### sylas

Science Advisor
It is a 'what if' question, marcus. In my worldview, expansion smears all photons across the entire universe. I view that as evidence the universe is finite. Were it not, there would be no CMB.
This is inconsistent with conventional physics, which is what a student is meant to be learning.

What expansion does is expand space. The simplest models are for a universe which is uniformly filled with photons at every point, without limit and with no boundaries. That's the CMB. It comes to us from every direction in the sky; and as far as we can tell space is filled with this radiation. In the simplest models, there's no limit or edge to it; it fills all of space, whether space is finite or infinite. As space expands, the number of background photons per unit volume falls. But they remain always uniformly distributed through all of space.

Cheers -- Sylas

#### jackal0206

I think yes, you're right, if I understand you. To paraphrase what I think you mean:
rho_matter/rho_rad
is the ratio of scalefactor now and scalefactor then (when matter and radiation were in balance).....
This is how I do it and I got a huge number for redshift, please let me know if I am doing anything wrong.

TODAY:
rho_matter = rho_critical = 3/8pi * H^2/G = 1e-14 [kg/m^3]
rho_rad = alpha T^4 = 7.56e-16 * 2.74^4 = 4.26e-14 [J/m^3]
Do a unit conversion
rho_rad = 4.26e-14 / c^2 = 4.74e-31 [kg/m^3]

Since
rho_matter is propotional to R^-3
rho_rad is propotional to R^-4
rho_matter / rho_rad = c * R = c / (1+z)

We know rho_matter, rho_rad, R and z for today, as well as c, so we found c to be 2.11e16.

then I use rho_matter / rho_rad = c / (1+z), where rho_matter / rho_rad = 1 to find z.
Turns out z = c-1, which like you said, it's the ratio between rho_matter and rho_rad.

But I am not sure if it is correct as I calculated z = 2.11e16, which is very large.

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