We consider future cosmological tests based on observations of Fast Radio Bursts (FRBs). We use Illustris Simulation to realistically estimate the scatter in the dispersion measure (DM) of FRBs caused by the inhomogeneous distribution of ionized gas in the Intergalactic Medium (IGM). We find ~13% scatter in DM to a source at z=1 and ~7% at z=3 (one sigma). The distribution of DM is close to Gaussian. We simulate samples of FRBs and examine their applicability to simple cosmological tests. Our calculations show that using a sample of 100 FRBs and fixing cosmological model one can find the redshift and sample averaged fraction of ionized gas with ~1% uncertainty. Finding the ionized fraction with ~1% accuracy at few different epochs would require ~10 000 FRBs with known redshifts. Because DM is proportional to the product of ionized fraction, baryon density and the Hubble constant it is impossible to constrain these parameters separately with FRBs. Constraints on cosmological densities are possible in a flat LCDM model but give uninterestingly low accuracy. Using FRBs with other type of data improves the constraints, but the role of FRBs is not crucial. Thus constraints on the distribution of ionized gas are probably the most promising application of FRBs which allow for "tomography" if sources redshifts are known, as opposed to measuring electron scattering optical depth or S-Z y parameters with CMB observations.

8 pages, 5 figures, accepted by MNRAS