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!


  1. $\Omega_K$ is given by $K\chi_H^2$ with the Hubble distance $\chi_h=c/H_0$, and at the same time $\Omega_K=1-(\Omega_m+\Omega_\Lambda)$. substituting $\Omega_K\simeq10^{-2}$ implies $1/\sqrt{K}=10\chi_H$ for the curvature radius.

  2. physics bonus question: the neutrons in a neutron star are gravitationally bound and the energy released by the neutron decay is not enough for the decay products to leave the system. so there are constant transformations from neutrons into protons and electrons (and vice versa, at a rate given by the temperature and the energy difference).