Wednesday, January 30, 2013

gravity in 2d and 3d

why can't gravity rotate galaxy images in gravitational lensing, but introduce physical rotation of galaxies?

bonus question: where's the global minimum of $x^x$, $x>0$?

Wednesday, January 23, 2013

homogeneous and anisotropic cosmology

can a cosmological model homogeneous but anisotropic? what would that construction imply? are there observable consequences, and which ones would that be?

(question courtesy of R. Schmidt, Centre for Astronomy, Heidelberg)

bonus question: what's the angle between an edge and the space diagonal in an $n$-dimensional unit hypercube? what's the limit of this angle in the case $n\rightarrow\infty$?

for the solution CQW shows here the angle in question (in units of $\pi$) as a function of dimension $n$:

Wednesday, January 16, 2013

destruction of gravitationally bound systems

is it possible that gravitationally bound systems (clusters, galaxies, globular clusters, solar systems, planets with moons) are taken apart by the Hubble-expansion? if yes, under which circumstances and on what time scale?

bonus question: can you show that the integral over spherical harmonics almost always vanishes,
\begin{equation}
\int\mathrm{d}\Omega\:Y_{\ell m}(\theta,\phi) = 0
\end{equation}
except of course for the monopole $\ell=m=0$?

Wednesday, January 9, 2013

conformal time in $\Lambda$-driven expansion

CQW wishes all readers a successful new year 2013, in particular with PLANCK's upcoming data release!

what's the relation between conformal time and cosmic time (or equivalently between comoving and proper distance) during epochs of exponential expansion? can you write the relation as a power-law?

physics bonus question: what's the reason why magnetic fields play such an important role in astrophysics, and electrical fields so little?

maths bonus question: can you determine the derivative of $\exp(-1/(1-x^\alpha))$ both at $x=0$ and $x=1$ for arbitrary $\alpha>0$?

i've added a plot of that function to CQW: $\alpha=2,3,4,5,6$ (green solid lines), $\alpha=1$ (red solid line), $\alpha>1$ in steps of $0.1$ (red dashed lines), $\alpha<1$ in steps of $0.1$ (blue dashed lines):