## Wednesday, November 28, 2012

### Hubble expansion and superluminality

why do we observe light from galaxies if their recession velocity exceeds the speed of light? at what redshift in a $\Lambda$CDM cosmology would a galaxy move as fast as light? why isn't that a problem with special relativity?

bonus question: can you show the property $\Gamma(x+1) = x\Gamma(x)$ of the $\Gamma$-function which is defined by $\Gamma(x)\equiv \int_0^\infty\mathrm{d}t\:t^{x-1}\mathrm{e}^{-t}$?

2. bonus question: it's a matter of integration by parts, with the integrand vanishing at the integration boundaries: $$\Gamma(x+1) = \int_0^\infty\mathrm{d}t\:t^x\mathrm{e}^{-t} = \left.-t^x\mathrm{e}^{-t}\right|_0^\infty + x\int_0^\infty\mathrm{d}t\:t^{x-1}\mathrm{e}^{-t} = x\Gamma(x).$$