Wednesday, March 27, 2013

decaying vorticity

can you show that in the course of linear structure formation during matter domination the vorticity $\omega=\mathrm{rot}\upsilon$ decays $\propto 1/a$ with the scale factor $a$? what's the relation during radiation domination? by how much has the vorticity decreased from the end of inflation (when vorticity modes could have been seeded) until today?

bonus question: what's the value of the integral
\begin{equation}
\int_0^\pi\mathrm{d}x\:\sin(x)^{x^x}
\end{equation}
CQW accepts numerical values and we'd be really interested in an analytical solution, because the numerical value is somewhat...suggestive.

Wednesday, March 20, 2013

Higgs mechanism

what fraction of your body weight is due to the Higgs mechanism? in what way is the remaining mass being produced?

bonus question: can you show that parity transformation $\vec{r}\rightarrow -\vec{r}$ introduces a factor $(-1)^\ell$ in the spherical harmonics $Y_{\ell m}(\theta,\phi)$?

Wednesday, March 13, 2013

electroweak phase transition

before the electroweak phase transition of the Universe, all weak bosons are massless just like the photons, and with the temperatures well above TeV, the Higgs-particle hasn't given any mass to the charged leptons yet. would there be a way of distinguishing electrons, muons and taus?

physics bonus question: in quantum mechanics, what property of the Hamiltonian operator is responsible for the conservation of probability?

maths bonus question: can you show that $\exp(A)$ is unitary if $A$ is anti-Hermitean, $A^+=-A$, where $A^+$ is the complex-conjugated, transposed matrix $A$?

Wednesday, March 6, 2013

thinking outside the box

we can observe the cosmos on the past light cone and naturally, we see finite amounts of objects and of fluctuations which sets statistical limits on inferences from cosmological data (aka cosmic variance). are there possibilities of looking outside the past light cone and what type of information would that provide?

bonus question: can you estimate the value of
\begin{equation}
\log\int_0^\infty\mathrm{d}t\:t^x\exp(-t)
\end{equation}
for large $x$?