Wednesday, May 29, 2013

dark energy takes over...

what possibilities are there for defining the moment where the Universe switches from matter domination to dark energy domination? the two most obvious choices would be the instant $a$ where the density parameters are equal, $\Omega_m(a) = \Omega_\Lambda(a)$, or perhaps when the deceleration parameter changes sign, $q(a)=0$... how would you define it and what numerical values for $a$ would you get with the two above definitions?

math bonus question: can you derive the Taylor expansion of the logarithm using the geometric series of $1/(1+x)$?

Wednesday, May 22, 2013

size of the Universe

can you estimate a minimum size of the Universe $\chi_K$ given current bounds on the curvature paramter $\Omega_K$ and compare this length scale to the size $\chi_H=c/H_0$ of the observable Universe?

maths bonus question: what's the slope of $x^{\sin(x)^x}$ at $x=0$ and where's the first minimum in the positive $x$-range?

physics bonus question: why do neutrons in neutron stars not decay? with neutron decay half-time of about a quarter of an hour, neutron stars should decay rather quickly!

Wednesday, May 15, 2013

tests for FLRW-cosmologies

is there an observational test of verifying that we live in a FLRW-universe, i.e. of all (which?) symmetry assumptions behind choosing the Robertson-Walker-metric? is this a sufficient condition?

physics bonus question: can we test if the cosmological fluids are ideal?

maths bonus question: can you prove (by induction) the Bernoulli inequality, namely that
\begin{equation}
(1+x)^n\geq 1+nx
\end{equation}
for all real numbers $x\geq-1$?

Wednesday, May 8, 2013

energy input into the CMB

the CMB is produced in thermal equilibrium.... would it be possible to notice if there was some extra energy put into it? what consequences could this have, depending on the way it was done?

maths bonus question: what are the asymptotes of $\exp(\tanh(x))$ for $x\rightarrow\pm\infty$ and what is the slope at $x=0$?

physics bonus question: can you derive the energy-time-uncertainty from the momentum-position-uncertainty in quantum mechanics?

Wednesday, May 1, 2013

inflationary perturbations

guest post by Shaun Hotchkiss (from the blog trenches of discovery):

if, through the CMB and large scale structure, we can study the primordial density perturbation over a range of scales corresponding to five orders of magnitude (i.e. $k_\mathrm{max} = 10^5 k_\mathrm{min}$), what fraction $\Delta \phi/\phi$ of the inflationary potential does this probe (if inflation is true and the inflationary potential is $V(\phi)=m^2\phi^2$)?

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