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We present a general analytical formalism to calculate accurately several statistics related to underdense regions in the Universe. The statistics are computed for dark matter halo and galaxy distributions both in real space and redshift space at any redshift. Using this formalism, we found that void statistics for galaxy distributions can be obtained, to a very good approximation, assuming galaxies to have the same clustering properties as haloes above a certain mass. We deduced a relationship between this mass and that of haloes with the same accumulated number density as the galaxies.  We also found that the dependence of void statistics on redshift is small. For instance, the number of voids larger than  13  h  −1  Mpc  (defined to not contain galaxies brighter than   M r  =−20.4 + 5 log  h  ) change less than 20 per cent between   z = 1  and 0. However, the dependence of void statistics on σ 8  and Γ is considerably larger, making them appropriate to develop tests to measure these parameters. We have shown how to efficiently construct several of these tests and discussed in detail the treatment of several observational effects. The formalism presented here along with the observed statistics extracted from current and future large galaxy redshift surveys will provide an independent measurement of the relevant cosmological parameters. Combining these measurements with those found using other methods will contribute to reduce their uncertainties.

The statistics of voids as a tool to constrain cosmological parameters: σ 8 and Γ