An Analytic Formulation of the 21 cm Signal from the Early Phase of the Epoch of Reionization

Abstract

We present an analytic formulation to model the fluctuating component of the H i signal from the epoch of reionization during the phase of partial heating. During this phase, we assume self-ionized regions, whose size distribution can be computed using excursion set formalism, to be surrounded by heated regions. We model the evolution of the heating profile around these regions (near zone) and their merger into the time-dependent background (far zone). We develop a formalism to compute the two-point correlation function for this topology, taking into account the heating autocorrelation and heating-ionization cross-correlation. We model the ionization and X-ray heating using four parameters: efficiency of ionization, ζ ; number of X-ray photons per stellar baryon, N heat ; spectral index of X-ray photons, α ; and minimum frequency of X-ray photons, ν min . We compute the H i signal in the redshift range 10 <  z  < 20 for the ΛCDM model for a set of these parameters. We show that the H i signal for a range of scales 1–8 Mpc shows a peak strength of 100–1000 (mK) 2 during the partially heated era. The redshift at which the signal makes a transition to a uniformly heated universe depends on the modeling parameters; e.g., if ν min is changed from 100 eV to 1 keV, this transition moves from z  ≃ 15 to z  ≃ 12. This result, along with the dependence of the H i signal on the modeling parameters, is in reasonable agreement with existing results from N -body simulations.