CMb Photons: Finite or Infinite?

[skydivephil]

Is there a known quantity of CMb phtons? Presumably we can only receive a finite amount in our detectors, but is that due to the fininite distance to our comsic horizon?
But if the universe were infinite would it have emitted an infinite amount of photons from the CMb/ Would everyplace in the an infinite universe receive CMb photons?
If there is a finite amount, is there any way to estimate that number?

 

[bapowell]

Sure. The number density, $n_i$, of particles with $g_i$ internal degrees of freedom has the statistical mechanical definition:
$$n_i(T) = g_i\int f_i(p,T)\frac{d^3p}{2\pi^3}$$
where $f_i(p,T)$ is the phase space distribution function, which depends on temperature $T$ and momentum $p$. For bosons, we use the Bose-Einstein distribution:
$$f_i(p,T) = \frac{1}{e^{E_i/T} - 1}$$
where the chemical potentials have been taken to vanish for simplicity. Computing this integral for photons (m=0) gives
$$n(T) = \frac{\zeta(3)}{\pi^2}gT^3$$
where $\zeta$ is the Riemann zeta function. Lastly, just plug in the temperature of the cosmic microwave background, T = 2.7K, and you've got your answer.

 

[skydivephil]

Thanks for your answer, not sure that asnwer was entriely lucid, I am asking about the number of phtotons emitted by the CMB, is it a finite number? Not asking about the density. If so could we conclude that if the universe is infinite there would be some places in the universe that wouldn't receive any CMb photons?

 

[bapowell]

We would need to know the size of the universe in order to calculate the actual number of photons. If the universe is infinite, and the entire thing experienced a big bang with sufficiently similar conditions, then there would be an infinite number of photons. The only quantity sensitive to experiment is the number density of photons, and this density applies only to the observable universe.

 

[skydivephil]

That was pretty much what I was thinking but wanted to get a expert opinion. Many thanks Bapowell.

 

[marcus]

Hello Skydive!

Brian Powell explained the basic theory behind it. The simple answer to your question is that there are an estimated 413 CMB photons in every cubic centimeter.

Look at the bottom of page 28 here
http://www.astronomy.ohio-state.edu/~dhw/A873/notes4.pdf
where it says:
"The CMB temperature determines the number density of CMB photons, n_γ = 413 photons cm⁻³ ."

That 5-page PDF (pages 26-30 of some Ohio State course notes) has more info about the CMB which could be helpful. Like there are about a billion CMB photons per baryon (i.e. per proton or neutron of ordinary matter)

The author of the notes, Prof. David Weinberg,
http://www.astronomy.ohio-state.edu/~dhw/
has good credentials and seems interested in informing the public as well as teaching and research. His email is there on his homepage.

If communicating with wide audience it might work better to convert the CMB photon number density to "413 million photons per cubic meter." I'm not sure which is more visually evocative.

 

[marcus]

BTW Skydive, belated congratulations on the success of your latest video!

PZ Meyers devoting a post to it at Pharyngula blog, and embedding it there, calling it a "wonderful video" etc. High praise!

I've been distracted with other things and your question slipped my mind. I've been thinking about Rovelli's new book on the very beginnings of the scientific worldview, called The First Scientist.
https://www.amazon.com/dp/1594161313/?tag=pfamazon01-20

It is roughly half about the history of the ideas of natural law and natural (non-god) explanations for things, going back to 600 BC in Ionia---the Greek cities on the Eastern shore of the Aegean.
And roughly half about the present and the issues we face in the present including science/religion and science/anti-science issues.

 

[skydivephil]

Thanks MArcus and thanks for checking our work, really appreciate it. The bools looks great will order that one for sure, Have a great weekend

 

[Cosmo Novice]

I enjoyed the video, thanks for the link Marcus.

Really nice video and the lady commentator was great.