The effect of inhomogeneities on the distance to the last scattering
  surface and the accuracy of the CMB analysis
  The standard analysis of the CMB data assumes that the distance to the last
scattering surface can be calculated using the distance-redshift relation as in
the Friedmann model. However, in the inhomogeneous universe, even if
<\delta\rho> =0, the distance relation is not the same as in the unperturbed
universe. This can be of serious consequences as a change of distance affects
the mapping of CMB temperature fluctuations into the angular power spectrum. In
addition, if the change of distance is relatively uniform no new temperature
fluctuations are generated. It is therefore a different effect than the lensing
or ISW effects which introduce additional CMB anisotropies. This paper shows
that the accuracy of the CMB analysis can be impaired by the accuracy of
calculation of the distance within the cosmological models. Since this effect
has not been fully explored before, to test how the inhomogeneities affect the
distance-redshift relation, several methods are examined: the Dyer-Roeder
relation, lensing approximation, and non-linear Swiss-Cheese model. In all
cases, the distance to the last scattering surface is different than when
homogeneity is assumed. The difference can be as low as 1% and as high as 80%.
Excluding extreme cases, the distance changes by about 20-30%. Since the
distance to the last scattering surface is set by the position of the CMB
peaks, in order to have a good fit, the distance needs to be adjusted. After
correcting the distance, the cosmological parameters change. Therefore, a not
properly estimated distance to the last scattering surface can be a major
source of systematics. This paper shows that if inhomogeneities are taken into
account when calculating the distance then models with positive spatial
curvature and with \Omega_\Lambda ~ 0.8-0.9 are preferred. The \Lambda CDM
model in most cases, is at odds with the current data.
