## 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.