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