The earliest epoch of re-ionization in the standard Λ cold dark matter model

We show that the earliest possible re-ionization of the Universe is approximately at z≃ 13.5 and the optical depth is τ≃ 0.17 in the conventionally accepted Λ cold dark matter (ΛCDM) model with adiabatic fluctuations of a flat spectrum normalized to the cosmic microwave background. This is consistent with the re-ionization found by the Wilkinson Microwave Anisotropy Probe (WMAP) (the apparently earlier re-ionization epoch of the WMAP is ascribed to the adoption of the instantaneous re-ionization approximation), i.e. the WMAP result is realized only if re-ionization of the Universe takes place nearly at the maximal efficiency in the ΛCDM model.