Analytic Formulation of 21 cm Signal from Cosmic Dawn: Ly$\alpha$ Fluctuations

We present an analytic formalism to compute the fluctuating component of the \ion{H}{1} signal and extend it to take into account the effects of partial Lyman-$\alpha$ coupling during the era of cosmic dawn. We use excursion set formalism to calculate the size distribution of randomly distributed self-ionized regions. These ionization bubbles are surrounded by partially heated and Lyman-$\alpha$ coupled regions, which create spin temperature $T_S$ fluctuations. We use the ratio of number of Lyman-$\alpha$ to ionizing photon ($f_L$) and number of X-ray photons emitted per stellar baryons ($N_{\rm heat}$) as modeling parameters. Using our formalism, we compute the global \ion{H}{1} signal, its auto-correlation and power spectrum in the redshift range $10 \le z \le 30$ for the $\Lambda$CDM model. We check the validity of this formalism for various limits and simplified cases. Our results agree reasonably well with existing results from N-body simulations, in spite of following a different approach and requiring orders of magnitude less computation power and time. We further apply our formalism to study the fluctuating component corresponding to the recent EDGES observation that shows an unexpectedly deep absorption trough in global \ion{H}{1} signal in the redshift range $15 <z< 19$. We show that, generically, the EDGES observation predicts larger signal in this redshift range but smaller signal at higher redshifts. We also explore the possibility of negative real-space auto-correlation of spin temperature and show it can be achieved for partial Lyman-$\alpha$ coupling in many cases corresponding to simplified models and complete model without density perturbations.