Wednesday, April 11, 2012

CMB dipole - fact or fiction?

the cosmic microwave background has a dipolar structure which is interpreted in terms of the Milky Way moving relative to the CMB, inducing Doppler-shifts in the photon energies and boosing the flux. would it be possible to distinguish between a motion dipole and a cosmological dipole on the last-scattering surface? why is the Doppler-boosted photon spectrum still a Planck-spectrum with the correct Wien- and Stefan-Boltzmann-laws?

bonus question: what's $\int\mathrm{d}x\:x\ln(x)$?

1. The bonus question: Using partial integration, one gets $\int\mathrm{d}x\,x\ln x=\frac{1}{2}x^2\ln x-\int\mathrm{d}x\,\frac{1}{2}x=\frac{1}{2}x^2\ln x-\frac{1}{4}x^2=\frac{1}{2}x^2\left(\ln x-\frac{1}{2}\right)$.

1. the answer checks out! :)

2. We can't tell the difference. Otherwise we'd be able to distinguish a special system of reference.

But I have trouble seeing why the spectrum shouldn't change. The invariant should be the occupation number
$n_{\nu} = \frac{1}{\exp{\left( \frac{KT}{h\nu} \right) -1}}$.
That is proportional to
$\frac{I_{nu}}{\nu^3}$.
If I boost, frequancies will multiply by a factor $\Gamma$ depending on the boost parameters $\gamma,\theta$.So my spectrum will be multiplied by $\Gamma^3$.

3. Apparently your question led to a paper a year later at arXiv from Kanpur India's Institute of Technology.

http://arxiv.org/pdf/1308.0924.pdf