Wednesday, April 24, 2013

invariants of the lensing Jacobian

the lensing Jacobian $\mathcal{A}=\partial\vec{\beta}/\partial\vec{\theta}$ describes how the lensing deflection angle $\beta$ changes with position $\theta$. how many invariants (under orthogonal transformations) of $\mathcal{A}$ can you construct? what are they and what is their physical interpretation?

bonus question: can you show that $\exp(x)^{\sin(x)}$ is an even function and that the derivative is an odd function?

Wednesday, April 17, 2013

$\Lambda$ as a source of energy

this week, CQW has a guest question by U. Bastian:
is it possible to tap dark energy as a source of (mechanical) energy? could you set up a machine that would accomplish this?

bonus question: assume you're trying to compute the expectation value $\langle x\rangle$ of a random distribution $p(x)\mathrm{d}x$, which has the cumulative distribution $P(x)$. you could write:
\begin{equation}
\langle x\rangle = \int\mathrm{d}x\:xp(x) = \int\mathrm{d}x\:x\frac{\mathrm{d}}{\mathrm{d}x}P(x) = xP(x) - \int\mathrm{d}x\: P(x)
\end{equation}
by integration by parts, where all integration boundaries are taken to be $-\infty\ldots+\infty$. both expressions in the final expression are not finite. where's the mistake?

Wednesday, April 10, 2013

thermal equilibrium and the CMB

why does the CMB have a Planckian spectrum? after all, it has been generated over a range in redshift ($\Delta z\simeq 100$) over which the temperature changed, and there has been energy exchange with the electron plasma.

bonus question: can you show that the asymptote of the integral sine, $\int_0^x\mathrm{d}\ln(t)\:\sin(t)$, is $\pi/2$ for $x\rightarrow\infty$?

Wednesday, April 3, 2013

limitations of the NFW-profile shape

the NFW-profile is an approximation to the density distribution inside cold dark matter structures. where does the NFW-approximation break down and for what reason? if there was annihilation of dark matter, what could you say about the central brightness?

bonus question: where is the first minimum of $x^{\sin(x)}$, $x>0$?