Pivot scale in cosmology and CMB What is a pivot scale, pivot frequency in general, and particularly in the cosmology? I don't get the idea.
Almost everywhere I found the same the same sentence "we take the standard value of a pivot scale $k=0.05$ Mpc$^{−1}$ (unit of wavenumber) for Planck and pivot frequency as $fcmb=(c/2π)k$". What does it mean?
For example in https://www.cosmos.esa.int/documents/387566/387653/Planck_2018_results_L10.pdf
 A: The power spectrum of primordial density perturbations is often modeled as a power law, $$P(k) = A \left(\frac{k}{k_0}\right)^{n-1}.$$ This is the form predicted by slow roll inflation and it fits current CMB and large scale structure data well.  Here, $k_0$ is the pivot scale.  It is the wavenumber (scale) were the amplitude, $A$, is measured.
It is possible to include so-called "running", in which the spectral index, $n$, itself has a $k$-dependence, $\alpha = dn/d\ln k$, so that $$\ln P(k) = \ln A+(n(k_0)-1)\ln\left(\frac{k}{k_0}\right)+ \frac{1}{2}\alpha\ln\left(\frac{k}{k_0}\right)^2.$$ Here $k_0$ serves as the scale on which the spectral index is measured.
The choice of pivot scale is not arbitrary in models with running, as certain scales minimize the correlation between $n$ and $\alpha$ (see https://inspirehep.net/literature/744039) which generally reduces errors.
EDIT: For the case of power-law spectra, it is easy to understand why this is called a pivot scale, since the spectra rotate about this amplitude as one varies the spectral index,

