We study the reionization history of the Universe in cosmological models with non-Gaussian density fluctuations , taking them to have a renormalized \chi ^ { 2 } probability distribution function parametrized by the number of degrees of freedom , \nu .
We compute the ionization history using a simple semi-analytical model , considering various possibilities for the astrophysics of reionization .
In all our models we require that reionization is completed prior to z = 6 , as required by the measurement of the Gunn–Peterson optical depth from the spectra of high-redshift quasars .
We confirm previous results demonstrating that such a non-Gaussian distribution leads to a slower reionization as compared to the Gaussian case .
We further show that the recent WMAP three-year measurement of the optical depth due to electron scattering , \tau = 0.09 \pm 0.03 , weakly constrains the allowed deviations from Gaussianity on the small scales relevant to reionization if a constant spectral index is assumed .
We also confirm the need for a significant suppression of star formation in mini-halos , which increases dramatically as we decrease \nu .