A discrete estimation of the uncertainty of the background redshift and its effects on cosmological parameters

We develop a discrete model to account for the effects of inhomogeneities on the redshift of photons. Using this model we compute the probability distribution of the observed redshift respect to the background value, obtaining that its dispersion around the central value is proportional the square root of the comoving distance. This implies that data analysis should include a contribution to the total error budget which depends on the distance of the source. Assuming large scale inhomogeneities with a power spectrum given by primordial curvature perturbations the effect is expected to be small at low red-shift, and to become important only at very high redshift. We then consider what are the general implications for the estimation of background cosmological parameters, giving some example for the case of quantities related to the luminosity distance such as the Hubble parameter and the cosmological constant. The model correctly reproduces the expected stochastic properties of the propagation of photons in inhomogeneous media and due to its computational simplicity it could be particularly suitable for the numerical estimation of the effects of inhomogeneities on cosmological observables using Montecarlo methods.