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 $\Gamma(x)\equiv \int_0^\infty\mathrm{d}t\:t^{x-1}\mathrm{e}^{-t}$?

Wednesday, November 21, 2012

redundancy in the Friedmann equations

under which circumstances is the $\dot{a}$-Friedmann equation the integral of the $\ddot{a}$-equation? what acts as the integration constant? why would that particular variable not be able to introduce de/ac-celeration?

bonus question: why is (in contrast to real analysis) the derivative of a complex differentiable function always again complex differentiable?

Wednesday, November 14, 2012

mechanical similarity and planetary orbits

what would be Kepler's law of planetary motion (the cube of the orbit's major axis is proportional to the square of the orbital period) if the Newtonian gravitational potential was $\propto 1/r^2$ instead of $\propto 1/r$? can you generalise Kepler's law to arbitrary integer power laws for the potential, $\propto 1/r^n$?

would the two other laws by Kepler be affected (elliptical orbits and angular momentum conservation)?

bonus question: what's $\int_0^\infty\mathrm{d}x\:x^3\exp(-\alpha x^2)$?

Wednesday, November 7, 2012

cold core (aka cooling flow)

this week's question was suggested by R. Schmidt (a fellow cosmologist):
$X$-ray observations suggest that the centres of clusters of galaxies are sometimes colder than the rest of the cluster. what's the physical reason for these cold cores? Would cold cores show up in the thermal Sunyaev-Zel'dovich effect?

bonus question: can you show that $\ln\det(A) = \mathrm{tr}\ln(A)$, or equivalently $\exp\mathrm{tr}(A) = \mathrm{det}\exp(A)$ for any non-singular square matrix A?

bonus question 2: what do you think, do you have more or less friends than your friends have on average?