Closed-form solutions for some metric perturbations from a closed Friedmann model and the distortion of the universe

We consider perturbations of a particular type (homogeneous of Bianchi type IX) from a closed Friedmann-Robertson-Walker model. By considering a perfect fluid, we obtain closed-form solutions for the perturbations in three interesting cases: stiff matter, radiation, and dust. As a cosmological application, we obtain an upper limit on the relative shear, ${\left(\frac{\sigma }{\theta }\right)}_0<b_{0\exp }$, if the microwave background radiation was last scattered at a mean red-shift $〈z〉\in [7,{10}^3]$ and the density parameter is $\Omega \in \mathrm{}(1,10)\mathrm{}$, where $b_{0\exp }$ is the relative root-mean-square deviation of the temperature over the whole sky.