Motivation for using $\sigma_{8}$? Press-Schechter formalism allows one to universally quantify the number of massive dark matter halos in a given volume -- The Halo Mass Function. The density contrast is given by the rms mass variance in Fourier space, which is the dependent on the linear power spectrum $P_{\rm lin}(k,z)$ in a smooth window $\widetilde{W}(R;k)$
\begin{align}
\sigma^{2}(R,z) 
= \frac{1}{2\pi^{2}} 
\int_{0}^{\infty} dk\ k^{2} P_{\rm lin}(k,z) \left| \widetilde{W}(k;R)\right|^{2}
\end{align}

The halo count can be scaled via observations in a defined window region of $R = 8\ h^{-1}\rm Mpc$. We can solve for this analytical/numerically and measure it from CMB fluctuations, given by the parameter $\sigma_{8}$. The power spectrum, as well as the halo mass function, is then scaled through the normalization of the analytical variance and the observed variance.

My question is then, why do we choose a smoothing radius of $R = 8\ h^{-1}\rm Mpc$? Is it arbitrarily chosen to compute the number variance in a cluster, or is their more too that? What prevents me from defining the window radius of $R = 6\ h^{-1}\rm Mpc$ (just ignore the fact that the  WMAP and PLANCK results give $\sigma_{8}$ in the radius of $8$)?
The specific radius of $8 h^{−1} \text{Mpc}$ is used because the value of $\sigma_8$ turned out to be close to unity. To quote e.g. from Amendola &amp; Tsujikawa (2010), p. 39:


  If the cells have a radius of $8 h^{−1} \text{Mpc}$, it turns out that $\sigma_R$ is close to unity. Conventionally the normalization of the power spectrum is therefore given by quoting $\sigma_8$.