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

Abstract

We present an analytic formulation to model the fluctuating component of the HI 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 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 auto-correlation and heating-ionization cross-correlation. We model the ionization and X-ray heating using four parameters: efficiency of ionization, $\zeta$, number of X-ray photons per stellar baryon, $N_{\rm heat}$, the spectral index of X-ray photons, $\alpha$, and the minimum frequency of X-ray photons, $\nu_{\rm min}$. We compute the HI signal in the redshift range $10 < z < 20$ for the $\Lambda$CDM model for a set of these parameters. We show that the HI signal for a range of scales $1\hbox{-}8 \, \rm Mpc$ show a peak strength $100\hbox{-}1000 \, \rm (mK)^2$ during the partially heated era. The redshift at which the signal makes a transition to uniformly heated universe depends on modelling parameters, e.g. if $\nu_{\rm min}$ is changed from $100 \, \rm eV$ to $1 \, \rm keV$, this transition moves from $z \simeq 15$ to $z \simeq 12$. This result, along with the dependence of the HI signal on modelling parameters, is in reasonable agreement with existing results from N-body simulations.Comment: 35 pages, 17 figures, published in Ap