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.
