  In recent years, the possibility of measuring the cosmological constant
$\Omega_\Lambda$ through the application of the Alcock-Paczynski test to the
Lyman Alpha (Ly$\alpha$) forest has been suggested (McDonald et al. 1999; Hui
et al. 1999). Despite the theoretical uncertainties due to a few other
cosmological parameters, some of the greatest difficulties we encounter concern
the huge uncertainties due to cosmic variance and noise. In this paper, we
propose a maximum likelihood estimation (MLE) method to deal with cosmic
variance and noise using synthetic spectra of quasistellar objects (QSOs) from
our cosmological hydrodynamic simulations. We demonstrate that the MLE method
can overcome the cosmic variance problem. Applying the MLE method, we find that
we have more than 90% probability to determine $\Omega_\Lambda$ within 20%
error and approximately of 66% probability to determine $\Omega_\Lambda$ within
10% error by using 30 pairs QSO spectra when other cosmological parameters are
assumed. Another important source of error is from noise in the flux spectra,
and we have modeled the corresponding effect by studying artificial spectra
with different kinds of noise added. We discover that the noise distribution
does not have significant effect on the final cross-correlation functions as
long as the signal-to-noise ratio (S/N) is fixed. Finally, a preliminary test
and discussion about the sensitivities to other cosmological parameters are
included in this paper as well.
