Thermal Evolution of the IGM due to Lyman-{\alpha} photons during the Cosmic Dawn

The first star-forming objects which formed at high redshifts during the cosmic dawn (CD) also emitted photons between Lyman-$\alpha$ and Lyman-limit frequencies. These photons are instrumental in coupling the spin temperature of the neutral hydrogen (HI) atoms with the kinetic temperature of the intergalactic medium (IGM). Along with this coupling effect, these photons also impact the kinetic temperature by exchanging energy with the HI atoms. The injected Lyman-$\alpha$ photons in general cool the medium, while the continuum photons heat the medium. While studying this effect in the literature, quasi-static profile around the Lyman-$\alpha$ frequency is assumed. In this paper, we solve the time-dependent coupled dynamics of the photon intensity profile along with the evolution of the thermal state of the IGM and HI spin temperature. It is expected that, during the CD era, the IGM has a mix of continuum photons with 10-20% of injected photons. For this case, we show that the system reaches thermal equilibrium in around 1 Myr, with final temperature in the range 50-100 K. This time scale is comparable to the source lifetime of PopIII stars at high redshifts. One impact of switching off short-lived sources is that it can keep the system heated above the temperature of the quasi-static state. We also show that the quasi-static equilibrium for the continuum photons is only achieved on time scales of 100 Myr at $z\simeq 20$, comparable to the age of the Universe. We also briefly discuss how the Lyman-$\alpha$ induced heating can impact the 21 cm signal from CD.