  Future precision measurements of CMB polarizations can shed new light on the
problem so called Hubble tension. The Hubble tension comes from the difference
of the evolutions of the Hubble parameter which are determined with two
different distance ladders. The standard distance ladder with the observation
of Cepheid variables and type Ia supernovae gives larger values of the Hubble
constant, and the inverse distance ladder with the observation of the baryon
acoustic oscillations both in the CMB and in the clustering of galaxies gives
smaller values of the Hubble constant. These different evolutions of the Hubble
parameter indicate different evolutions of the free electron density in the
process of the reionization of the universe and different magnitudes of low-l
polarizations of the CMB, since these polarizations are mainly produced through
the Thomson scattering of CMB photons off these free electrons. We investigate
the effect on CMB E-mode and B-mode polarizations of l < 12 assuming
non-trivially time-dependent equation of state of dark energy. We find that the
case of the standard distance ladder gives higher power of polarizations than
the prediction in the LambdaCDM model.
