Quick question on CMB anisotropy in Earth frame

[WannabeNewton]

Hi guys. Consider the mean occupation number and specific intensity of the CMB photons in the CMB frame as given by the blackbody formulas: $\eta = \frac{1}{e^{h\nu/k_{B}T_0} - 1}$ and $I_{\nu} =\frac{2h\nu^3}{e^{h\nu/k_{B}T_0} - 1}$ with $T_0$ the thermal bath temperature in the CMB frame and $c = 1$.

Now we consider the Earth as a Lorentz frame moving relative to the CMB frame with some velocity $v$ relative to the $x$-axis of the CMB frame and a telescope in the Earth frame oriented at some angle $\theta$. Note that the mean occupation number can be put in the frame-independent form $\eta = \frac{1}{e^{-p_{\mu}u^{\mu}/k_{B}T_0} - 1}$ where $u^{\mu}$ is the 4-velocity of the CMB frame and $p^{\mu}$ the 4-momentum of the photons.

In the Earth frame, $u^{\mu} = \gamma(1,-v)$ and $p^{\mu} = h(\nu', -\nu' \cos\theta, -\nu' \sin\theta)$ so $\eta = \frac{1}{e^{\gamma h\nu'(1 - v\cos\theta )/k_{B}T_0} - 1} = \frac{1}{e^{h\nu'/k_{B}T} - 1}$ where $T = T_0 \frac{\sqrt{1-v^2}}{1 - v\cos\theta}$. Then the specific intensity in this frame would be $I_{\nu'} = \frac{2h\nu'^3}{e^{h\nu'/k_{B}T} - 1}$. Here $\nu = \gamma \nu' (1 - v\cos\theta)$ so $\nu' = \frac{\sqrt{1 - v^2}}{1 - v\cos\theta}\nu$ is the doppler shifted frequency in the Earth frame.

However, in p.16 of http://www.astro.princeton.edu/~jeremy/heap.pdf, one is given the result $\eta = \frac{1}{e^{h\nu/k_{B}T} - 1}$ instead (the paper's $\hat{T}_{\text{CMB}}$ is the $T$ above) so where did I go wrong?

On the other hand in p.20 of http://www.staff.science.uu.nl/~proko101/JildouBaarsmaCMB.pdf, one is given $\eta = \frac{1}{e^{|p'|/k_{B}T} - 1} = \frac{1}{e^{h\nu'/k_{B}T} - 1}$ (the paper's $T'$ is the $T$ above) which seems to agree with what I have so I'm confused.

Thanks in advance.

 

[George Jones]

However, in p.16 of http://www.astro.princeton.edu/~jeremy/heap.pdf, one is given the result $\eta = \frac{1}{e^{h\nu/k_{B}T} - 1}$ instead (the paper's $\hat{T}_{\text{CMB}}$ is the $T$ above)

At a quick glance, it looks the same to me. Note carefully the notation for the frames given in the paragraph immediately after 1.29.

 

[Mordred]

you might want to also check out this handy article

"Physics of the intergalactic medium." I'm still studying it myself as its fairly intense lol. However you may find it useful for your research

http://arxiv.org/abs/0711.3358

by the way nice articles I'll be adding them to my collection to study myself

 

[WannabeNewton]

At a quick glance, it looks the same to me. Note carefully the notation for the frames given in the paragraph immediately after 1.29.

Oh haha I totally missed the difference in notation for the frames. Well that's embarrassing. Thanks George!

you might want to also check out this handy article

"Physics of the intergalactic medium." I'm still studying it myself as its fairly intense lol. However you may find it useful for your research

http://arxiv.org/abs/0711.3358

by the way nice articles I'll be adding them to my collection to study myself

Awesome, thanks Mordred!

 

[sardar]

Hello,

Thank you for your question. It appears that there may be some confusion regarding the equations and variables used in the papers you have referenced. It is important to carefully define and understand the terms and equations being used in order to accurately interpret and compare results.

One possible explanation for the discrepancy in your calculations could be the use of different definitions for the variables $T$ and $T_0$. In the first paper, $T$ is defined as the thermal bath temperature in the CMB frame, while in the second paper, $T$ is defined as the temperature in the Earth frame. It is possible that this difference in definitions could lead to the discrepancy in your results.

Additionally, it is important to consider the assumptions and approximations made in the papers, as well as any possible simplifications or corrections that may have been applied. It may be helpful to carefully read through the papers and compare the methods and assumptions used in each to better understand the results.

I hope this helps clarify the discrepancy you have encountered. If you have any further questions or need clarification on any specific equations or variables, please do not hesitate to reach out. Thank you.