Is the supply of the observable CMB radiation limited?

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About 0.38 Myr after the Big Bang, the universe cooled to about 3,000 K and went through the recombination era. Electrons and protons combined to form neutral hydrogen atoms and the photons became free to travel through space.

The freed photons can be considered as photon sources of a small duration, a one-time event happened everywhere in the universe simultaneously. If the space is divided into concentric shells centered at our location, as illustrated in the figure below, one can imagine that the freed photons originated from the nearby shells would reach and pass our location first and those in shells farther away would follow as time goes forward. The shell labelled as event horizon is a boundary resulted from the expansion of space. The freed photons originated from regions outside this boundary will never reach our location.
1628068758307.png

Entering z = 1090 into Jorrie’s calculator for the redshift, one finds that the CMB received by us now were originated from a shell with a radius of R = 41.6 Mly at t = 0.372 Myr during the recombination era and that the event horizon had a radius of 56.7 Mly at that time. Because of the expansion of space, it took the CMB photons 13.8 Gyr to reach our location.

If the freed photons that can reach our location are limited to those originated inside the event horizon at that time, does that mean the supply of the observable CMB photons is limited? What are the implications of this?
 
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  • #2
Ibix
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does that mean the supply of the observable CMB photons is limited?
One should be very wary of photon counting arguments since photon number isn't always conserved. That said, in this case, you can certainly count emission events inside the event horizon. Since that's a finite volume there were a finite number of emission events too. But remember that the red shift applies to the emission event rate as well as the light frequency. When the temperature of the CMB was twice what it is today we would also have observed twice as many emission events per second as we do now. In the future the rate of emissions will fall as the temperature falls - asymptotic to zero. This means that one day (in trillions upon trillions of years, I would imagine) we will receive our last CMB light. However, we would never know that there wasn't one more emission event even closer to the event horizon whose light still hasn't reached us.

Incidentally, the same applies to a light source dropped into a black hole. Classically the continuous emission is ever more red shifted and fainter, but in principle is always visible. But in a discrete emission model there is a last bit of light, but without following it in you can never know that there isn't one more emission struggling its way out from a bit closer to the horizon.
 
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  • #3
Orodruin
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If the freed photons that can reach our location are limited to those originated inside the event horizon at that time, does that mean the supply of the observable CMB photons is limited?
No. The CMB light that reaches us today originated on that horizon. The CMB light that reached us yesterday was emitted slightly closer and that which will reach us tomorrow slightly further away and so on. Hence, the CMB will not disappear.
 
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  • #4
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One should be very wary of photon counting arguments since photon number isn't always conserved. That said, in this case, you can certainly count emission events inside the event horizon. Since that's a finite volume there were a finite number of emission events too. But remember that the red shift applies to the emission event rate as well as the light frequency. When the temperature of the CMB was twice what it is today we would also have observed twice as many emission events per second as we do now. In the future the rate of emissions will fall as the temperature falls - asymptotic to zero. This means that one day (in trillions upon trillions of years, I would imagine) we will receive our last CMB light. However, we would never know that there wasn't one more emission event even closer to the event horizon whose light still hasn't reached us.

According to the Big Bang model, the CMB photons, unlike lights emitted by objects in space, are photons existed during the early universe and freed during the recombination era. Around t = 0.38 Mly, the photons had a blackbody distribution at a temperature around 3000 K. It is a photon gas, not a photon emitter. Its blackbody temperature changes according to,
1628163728756.png

where a(t) is the scale factor at cosmological time t. The wavelengths of the CMB photons change with expansion of space,
1628163807735.png

All CMB photons we observe now have the same light-travel-time to our location (from the same shell in the figure), about 13.8 Gyr - 0.38 Myr = 13.8 Gyr. These photons are the same photons freed during the recombination era, but with wavelengths redshifted by a factor of 1090. The article, Last Scattering Surface, has an interesting analogy to the case discuss here: ‘a large field filled with people screaming’.
 
  • #5
Bandersnatch
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No. The CMB light that reaches us today originated on that horizon.
I don't think you have this right. The conformal diagrams show clearly that the light emitted on the event horizon never gets to reach us. The currently observed CMB light was emitted significantly inside the EH (in terms of comoving distance).
 
  • #6
Orodruin
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I don't think you have this right. The conformal diagrams show clearly that the light emitted on the event horizon never gets to reach us. The currently observed CMB light was emitted significantly inside the EH (in terms of comoving distance).
I may have read the OP quite fast. My impression was that they were not referring to the cosmological horizon.
 
  • #7
Ibix
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According to the Big Bang model, the CMB photons, unlike lights emitted by objects in space, are photons existed during the early universe and freed during the recombination era.
I'm not sure what your point is here. That I should have said "last scattering events" instead of "emission events"?
 
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  • #8
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I'm not sure what your point is here. That I should have said "last scattering events" instead of "emission events"?

My point is that, according to the Big Bang model, the CMB photons were not emitted by objects in space, they do not emit photons, they exist in space as a blackbody photon gas, and they do not change besides redshifted and diluted due to space expansion and tiny fraction of them being absorbed by objects in space.
 
  • #9
Buzz Bloom
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No. The CMB light that reaches us today originated on that horizon. The CMB light that reached us yesterday was emitted slightly closer and that which will reach us tomorrow slightly further away and so on. Hence, the CMB will not disappear.
The CMB photons described in Post #1 are all from the recombination region. As time passes, the comoving distance between this region and observation of the photons from Earth (or any arbitrary comoving stationary area) gets bigger as time passes. This region is at a fixed comoving distance from the Planck region (or from the discontinuity region at time zero). If the universe is an FIU flat-infinite (or hyperbolic-infinite) universe , what you say in the quote is correct. But if the universe is an HSF, hyper-spherical-finite universe, it is wrong because at some time the recombination region will no longer be on any part of the HSF universe at that time. In general, the region on the boundary of the observable universe (not the HSF) at the most distance from Earth at a given time will be more recent than the states of the universe at any earlier time. Therefore at such a time, the CMB will not exist in the radius of observable universe which will be larger than the half-circumference the HSF universe.
 
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Ibix
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My point is that, according to the Big Bang model, the CMB photons were not emitted by objects in space
But they scattered off matter in space, and it's that process that thermalises them. And we can certainly talk about the rate of last scattering events.
 
  • #11
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I am not sure why we talk of event horizons here because unlike a black hole which is a part of space where conditions change between the surrounding space and whatever happens beyond the event horizon, in the early universe the Big bang model predicts that the whole universe was at a homogeneous density and temperature, so after recombination the CMB photons were produced everywhere in space at once as space was expanding also everywhere not just at some parts.
So this being a "one off" event would imply that the CMB was "born" everywhere in space and the only thing that has happened to it is redshift and some absorption. I fail to see how something that happens everywhere can have a "horizon" because the word horizon implies a boundary between two different states.
No. The CMB light that reaches us today originated on that horizon. The CMB light that reached us yesterday was emitted slightly closer and that which will reach us tomorrow slightly further away and so on. Hence, the CMB will not disappear.
If what I said above in my post is correct I fail to see what meaning is there to concepts like "slightly closer and further". If the whole world would only consist of a homogeneous ocean (resembling the CMB) then what would it matter where in the ocean one is at a given moment ?
 
  • #12
Ibix
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I am not sure why we talk of event horizons here
Due to the curvature of FLRW spacetime there exist regions that can never send a light signal to us. The boundary between the regions that can send us a light signal and those that cannot is the event horizon. It's not the same as a black hole event horizon, no, in that it's observer dependent. The cosmologcal event horizon associated with Earth is not the same as the cosmological event horizon associated with, for example, a star in Andromeda.
So this being a "one off" event would imply that the CMB was "born" everywhere in space and the only thing that has happened to it is redshift and some absorption.
Yes, but the red shift goes to infinity at a finite distance from us.
 
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  • #13
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If the freed photons that can reach our location are limited to those originated inside the event horizon at that time, does that mean the supply of the observable CMB photons is limited?

Let’s look at some of the properties of the event horizon using calculation results. We will use the following Planck 2015 data for all the calculations here:
1628683061458.png

Similar to the results mentioned in Post #1, the CMB photons received by us now were originated from a shell with a radius of R = 41.57 Mly at t = 0.372 Myr during the recombination era and the event horizon had a radius of 56.72 Mly at the time. According to Rindler (1956), Visual Horizons in World Models, the CMB photons originated outside the event horizon will never reach our location. This is because those photons are carried away from our location due to space expansion in the Big Bang model.

How about the CMB photons originated inside of and close to the event horizon? They should be seriously affected by the space expansion too. We may ask: When do the CMB photons, for example, originated in the shell of radius of 30 Mly, 50 Mly, or 56.55 Mly arrive at our location? The arrival times can be calculated with the calculator post by Gnedin or that by Jorrie using the equations posted by Bandersnatch in his thread, Distances between observers using the Lightone7 calculator. The calculation results are summarized here.

The following shows the relation between the radius of the shell in which the CMB photons were originated and the arrival time of these photons to our location:

1628682756760.png

The figure below shows the graphic form of these results:

1628682848407.png


Here are some observations from the above results:
  • All CMB photons reaching our location are originated in shells inside the event horizon, consistent with the statement that no photons outside the event horizon at t = 0.372 Mly will reach our location.
  • Journeys of the CMB photons inside of and close to the event horizon during the recombination era have prolonged journeys to reach our location. For example, R = 41.57 Mly, t = 13.8 Gyr; R = 50 Mly, t = 27.62 Gyr; and R = 56.55 Mly, t = 90.15 Gyr.
  • These results indicate that the supply of the observable CMB radiation is limited in the ΛCDM (Big Bang) model due to space expansion.
  • The supply of the observable CMB radiation, however, does not end abruptly. It is a dwindling process; when the radius of the source shell getting closer and closer to the event horizon, the available CMB photons become less and less (see figure).
  • It is interesting to note that in an infinite non-expanding universe, there is no cosmological event horizon.
Question:

Does this mean the Big Bang model cannot explain the blackbody nature of the CMB photon gas because of the expansion of space?​
 
  • #14
Bandersnatch
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Does this mean the Big Bang model cannot explain the blackbody nature of the CMB photon gas because of the expansion of space?
You may need to expand on the question as it's not clear how it follows from the rest of the post. Why do you think it can't? There is at least one sense in which the answer is indeed 'no, it can't' (the horizon problem). But I'm not sure if that's what you meant.

It is interesting to note that in an infinite non-expanding universe, there is no cosmological event horizon.
It's not just the static universes, but any non-accelerating ones too. I.e. a steadily expanding, or decelerating, universe won't have an event horizon either.
 
  • #15
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You may need to expand on the question as it's not clear how it follows from the rest of the post. Why do you think it can't? There is at least one sense in which the answer is indeed 'no, it can't' (the horizon problem). But I'm not sure if that's what you meant.
I will answer my question by demonstrate that based on the results presented in this thread, the Big Bang model does not guarantee the blackbody nature of the CMB photon gas. But first, would you please expand your comment, “There is at least one sense in which the answer is indeed 'no, it can't' (the horizon problem)”.
 
  • #16
PeterDonis
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These results indicate that the supply of the observable CMB radiation is limited in the ΛCDM (Big Bang) model due to space expansion.
You are using the word "limited", but you have not defined what you mean by it.

Here is a meaning that makes the sentence above valid: "limited" means that the total energy that would be received by a comoving observer, for all future time, from the CMB is a finite value. This is true; it's basically what your calculation is showing. Is this what you mean by "limited"?

The supply of the observable CMB radiation, however, does not end abruptly. It is a dwindling process; when the radius of the source shell getting closer and closer to the event horizon, the available CMB photons become less and less (see figure).
Yes, this is also true; it's just another way of saying that the CMB redshifts as the universe expands, or, to put it another way, the CMB's energy density decreases as the universe expands.

Does this mean the Big Bang model cannot explain the blackbody nature of the CMB photon gas because of the expansion of space?
No. Nothing that has been said above has anything whatever to do with whether the CMB is blackbody radiation. All your calculation is looking at is the energy density of the CMB. Your calculation says nothing whatever about the spectrum of the CMB, which is what "black body" refers to.
 
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  • #17
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Yes, this is also true; it's just another way of saying that the CMB redshifts as the universe expands, or, to put it another way, the CMB's energy density decreases as the universe expands.
But isn't redshifting and energy density (per cubic meter of space for example) two different things?
Redshift is a lowering of frequency or increase in wavelength but energy density I assume would be the amount of photons present per given area , I think even if the expansion of space wouldn't cause redshift it would still cause lowering of energy density simply because the same number of photons now have to fill a larger volume of space.
The way I see they are related with regards to space is that both happens at the same time, both redshift and expansion with the same number of particles present.
 
  • #18
Bandersnatch
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But first, would you please expand your comment, “There is at least one sense in which the answer is indeed 'no, it can't' (the horizon problem)”.
You should be able to find more detailed discussions of the horizon problem just by looking it up. Especially in the context of introducing inflation. But any cosmology textbook should have at least a short section on the problems with BB.
In short, in the standard big bang model the causally connected regions of the universe monotonically shrink as one rolls back the time, approaching zero at the limit of the singularity. I.e. the comoving extent of the particle horizon shrinks towards 0 at t=0.
This immediately poses a problem. Since the currently (or at any other moment) observed CMB was emitted at the current maximum comoving extent of the particle horizon, those regions are only now coming into causal contact with us. And they have never been in causal contact with the CMB at the opposite side of the sky. And yet, the whole CMB looks like a black body spectrum, i.e. the CMB emitting regions appear to have been thermalised at some point in the past.
The vanilla BB model doesn't address this problem. But, again, it doesn't have much to do with the topic at hand (i.e. the problem persists whether the event horizon exists or not and the CMB supply is 'limited' or not).
 
  • #19
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Since the currently (or at any other moment) observed CMB was emitted at the current maximum comoving extent of the particle horizon, those regions are only now coming into causal contact with us. And they have never been in causal contact with the CMB at the opposite side of the sky. And yet, the whole CMB looks like a black body spectrum, i.e. the CMB emitting regions appear to have been thermalised at some point in the past.
But wasn't the CMB said to be already thermalized back when it was first created due to space being much smaller after recombination as well as the plasma before it being rather homogeneous?
 
  • #20
PeterDonis
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isn't redshifting and energy density (per cubic meter of space for example) two different things?
No.

Redshift is a lowering of frequency
Which is energy for light. And in the case of cosmological expansion, since the redshifting is due to the increase in the scale factor, there is a direct relationship between the redshift and the decrease in energy density of the light.

I think even if the expansion of space wouldn't cause redshift it would still cause lowering of energy density simply because the same number of photons now have to fill a larger volume of space.
"The same number of photons now have to fill a larger volume of space" is the cosmological redshift. They're the same thing.
 
  • #21
Bandersnatch
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But wasn't the CMB said to be already thermalized back when it was first created due to space being much smaller after recombination
In the standard BB model, the scale factor (i.e. 'space') being smaller at recombination (or any other early moment) doesn't help, as the size of the past light cones at that time was also smaller. It is generally true of the expanding universe that the further back you go, the less of the 'contents' of the universe you are in causal contact with.
On the conformal diagram, it looks like this (taken from Ned Wright's tutorial):
1628778927533.png

The large red triangle being the current light cone extending into the past. I.e. it's all the events that we currently observe. The extent of it's base marks the furthest comoving regions we are in causal contact with. The blue line marks some early event (such as recombination). The yellow triangles mark all the events that were in causal contact with what we observe of that event. If you were to push the blue line even further down into the past, the yellow triangles would shrink even further. The two triangles are not overlapping, so they couldn't have exchanged any information with each other.
So, observing two opposite regions of the sky, we see the CMB as thermalised, even though there was no way for them to 'know' what temperature they should equalize to.
as well as the plasma before it being rather homogeneous?
Sure, but you have to assume it was homogeneous, for whatever reason. The BB model doesn't provide any reason for it to be so.
Remember that the BB model doesn't purport to describe any creation event - rather, it's looking back from the present time to as far back as we can see, and trying to model the behaviour of the universe by extrapolating the observed motions of its content.
We can observe that the current somewhat-clumpy distribution of matter smooths out as we look back in time. But in the BB model there's no physical reason for why it is approaching this state.
Or, to put it yet another way, if you were to choose the initial conditions to be that of a perfectly homogeneous, hot and dense plasma, then running the model will produce something like what we observe. But the model doesn't tell you why such initial conditions should be there in the first place.

Inflationary theory attempts to provide such reasons, but it's not strictly speaking a part of the standard BB model. More like an extension intended to address certain problems (such as the one discussed here).
 
  • #22
Bandersnatch
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Which is energy for light. And in the case of cosmological expansion, since the redshifting is due to the increase in the scale factor, there is a direct relationship between the redshift and the decrease in energy density of the light.
I think what was meant was that the energy density of radiation has a component of changing frequency, as well as dilution. I.e. the extra ##a^{-1}## in the ##a^{-4}## factor for radiation density as opposed to ##a^{-3}## for matter density is from frequency change. If radiation didn't have the effects of the frequency shift applied, it'd behave just like matter.
So I think it's fair to say that they're not the same thing.
 
  • #23
PeterDonis
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  • #24
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You are using the word "limited", but you have not defined what you mean by it.

The calculation results in this thread illustrate that no CMB photons freed outside the event horizon (R = 56.72 Mly) during the recombination era will ever reach our location. The number of the CMB photons inside the event horizon is limited and among them, only those traveling toward us will reach our location.

For an infinite, non-expanding universe in a similar situation, there is no event horizon and such a limit does not exist.
 
  • #25
PeterDonis
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The calculation results in this thread illustrate that no CMB photons freed outside the event horizon (R = 56.72 Mly) during the recombination era will ever reach our location. The number of the CMB photons inside the event horizon is limited and among them, only those traveling toward us will reach our location.
This is equivalent to the definition I gave in post #16. So the other things I said in that post apply.

For an infinite, non-expanding universe in a similar situation, there is no event horizon and such a limit does not exist.
There is also no event horizon in any expanding universe that does not have a cosmological constant.
 

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