decaying vorticity

Can you show that in the course of linear structure formation during matter domination the vorticity \(\omega=\mathrm{rot}\upsilon\) decays \(\propto 1/a\) with the scale factor \(a\)? What’s the relation during radiation domination? By how much has the vorticity decreased from the end of inflation (when vorticity modes could have been seeded) until today?

bonus question: What’s the value of the integral \(\int_0^\pi\mathrm{d}x\:\sin(x)^{x^x}\)

CQW accepts numerical values and we’d be really interested in an analytical solution, because the numerical value is somewhat…suggestive.