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##.(adsbygoogle = window.adsbygoogle || []).push({});

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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Quick question on CMB anisotropy in Earth frame

Loading...

Similar Threads for Quick question anisotropy |
---|

I Question about accelerating expansion |

I Shape of the universe, and a few other questions... |

B Question about gravity |

B Black hole gravitational pull questions |

I Question about an equation |

**Physics Forums | Science Articles, Homework Help, Discussion**