Thursday, March 29, 2012

CMB temperature

if the ionisation temperature of hydrogen is $T=10^4$K, the CMB decoupled at a redshift of $z=10^3$, and if for photons the temperature drops proportional to $1/a$, shouldn't the CMB have a temperature of $10$K instead of $3$K?

bonus question: would it be possible to generalise the hyperbolic functions to a basis $a\neq e=\exp(1)$ in a way that $\mathrm{asinh}(x)$ behaves asymptotically as $\log_{a}(x)$ for $x\gg 1$?

Thursday, March 22, 2012

age of the Universe

imagine you're stuck in a discussion with a creationist: what evidence is there that the Universe is more than ten billion years old? of course there are multiple answers, try to think of as many as you can.

bonus question (a tough one, this time): what's the derivative $\frac{\mathrm{d}}{\mathrm{d}x}x^{x^x}$?

Thursday, March 15, 2012

time-dependence of cosmological constants

in cosmology, we define all quantities by their values today. would we need to revise their numerical values in the future? if yes, on what time scale?

bonus question: what's larger, $|\pi^i|$ or $|i^\pi|$, with $i=\sqrt{-1}$ (principal values only)?

Wednesday, March 14, 2012

Mercury's orbit

can one explain the perihelion precession of Mercury's orbit with Newtonian gravity? what would be needed for this explanation? please keep in mind that at the time of Einstein's derivation (happy birthday, Prof. Einstein!) there was no real observational need for general relativity.

for bonus points: what's the order of magnitude of the perinigricon (point of closest approach to the black hole) precession of the stars orbiting the black hole at the centre of the Milky Way?

Wednesday, March 7, 2012

precision cosmology

a question for the era of precision cosmology... future surveys such as EUCLID and PLANCK are designed to yield almost $10^{-4}$ precision on the cosmological parameters $\Omega_m$ and $H_0$ (in a reasonably simple model). at the same time table-top experiments constrain the gravitational constant $G$ to a similar precision (it's a bit scary to realise that $G$ is the least known constant of Nature with only 4 significant digits!). would it make sense to improve cosmological measurements without improving on the gravitational constant?

bonus question: what's the asymptotic behaviour of $\frac{d}{dx}\left(\frac{1}{x}\right)^\frac{1}{x}$ for large $x$?

Thursday, March 1, 2012

lensing on an expanding metric

in relation to the last question: why is it possible to treat gravitational lensing on an expanding background metric as if it was Minkowskian, weakly perturbed by the gravitational potential? if this is related to a property of photons, to which other particle trajectories would this apply?

bonus question: what's $\sqrt[i]{i}$ with $i=\sqrt{-1}$?