Is the CMB shift parameter connected with the growth of cosmological perturbations?

We verify numerically that in the context of general relativity (GR), flat models which have the same ${\mathrm{\Omega }}_{\mathrm{m}}$ and CMB shift parameter $R$ but different $H(a)$ and $w(a)$ also have very similar (within less than 8%) growth of perturbations even though the dark energy density evolution is quite different. This provides a direct connection between geometrical and dynamical tests of dark energy and may be used as a cosmological test of general relativity.