Time-Dependent Behavior of Lyman$\alpha$ Photon Transfer in High Redshift Optically Thick Medium

With Monte Carlo simulation method, we investigate the time dependent behavior of Ly$\alpha$ photon transfer in optically thick medium of the concordance $\Lambda$CDM universe. At high redshift, the Ly$\alpha$ photon escaping from optically thick medium has a time scale as long as the age of the luminous object, or even comparable to the age of the universe. In this case, time-independent, or stationary solutions of the Ly$\alpha$ photon transfer with resonant scattering will overlook important features of the escaped Ly$\alpha$ photons in physical and frequency spaces. More seriously, the expansion of the universe leads to that the time-independent solutions of the Ly$\alpha$ photon transfer may not exist. We show that time-dependent solutions sometimes are essential for understanding the Ly$\alpha$ emission and absorption at high redshifts. For Ly\alpha photons from sources at redshift 1+z=10 and being surrounded by neutral hydrogen IGM of the $\Lambda$CDM universe, the escape coefficient is found to be always less, or much less than one, regardless of the age or life time of the sources. Under such environment, we also find that even when the Ly$\alpha$ photon luminosity of the sources is stable, the mean surface brightness is gradually increasing in the first 10^6 years, and then decreasing with a power law of time, but never approaches a stable, time-independent state. That is, all 1+z=10 sources in a neutral Hubble expanding IGM with Ly$\alpha$ luminosity L have their maximum of mean surface brightness ~ 10^{-21}(L/(10^{43}erg/s)) erg s^{-1} cm^{-2} arcsec^{-2} at the age of about 10^6 years. The time-dependent effects on the red damping wing profile are also addressed.