## Wednesday, March 25, 2015

### constant radius of the Hubble sphere

can you set up a FLRW-model in which the Hubble sphere has a constant comoving radius? what would be needed for such a construction?

bonus question: for which $n$ is the equality $\mathrm{i}^{n^2}=\pm\mathrm{i}^n$ with $\mathrm{i^2}=-1$ fulfilled?

2. answer to the bonus question: like the last week's case, it's best to split up the proof into 4 cases: $n=4m+q$, $q=0,1,2,3$ with the respective squares following from the binomial formula: $n^2=16m^2+8q+q^2$. then, $\mathrm{i}^{n^2}=+\mathrm{i}^n$ is fulfilled for $q=0,1$ and $\mathrm{i}^{n^2}=-\mathrm{i}^n$ for $q=2,3$.