Wednesday, October 29, 2014

correlation coefficient

can you provide a proof of the Cauchy-Schwarz inequality and show with it that the correlation coefficient
\begin{equation}
r = \frac{\langle xy\rangle}{\sqrt{\langle x^2\rangle\:\langle y^2\rangle}}
\end{equation}
always ranges between $-1\leq r\leq +1$?

bonus question: can you compute the normalisation, variance and kurtosis of $p(x)\mathrm{d}x\propto\exp(-x^4)\mathrm{d}x$?

Wednesday, October 22, 2014

Eddington temperature

Astrophysical objects are limited in their luminosity by the Eddington luminosity, where radiation pressure exceeds gravity. Would it be sensible to define a limiting Eddington-temperature for Planckian spectra?

Wednesday, October 15, 2014

Boltzmann-tail of the Planck-spectrum

can you estimate the time how long would you have to wait until
1. the Hubble space telescope would see a CMB photon at visible energies?
2. XMM-Newton registers a CMB photon at $X$-ray energies?
3. Fermi-LAT would register a CMB photon at MeV-energies?

Wednesday, October 8, 2014

natural value of the Hubble constant

a motivation for investigating dark energy is that the vacuum energy density is 120 orders of magnitude smaller than expected (which btw applies to the density of everything, dark matter, photons and neutrinos alike): could one not simply say that we don't understand the value of the Hubble constant? what would be a natural value, and could you make an example of units which can be easily imagined?

Wednesday, October 1, 2014

Boltzmann-photons

can you estimate what fraction of photons in a Planck-spectrum are described by classical Boltzmann-statistics?