Probing the Intergalactic Medium with the Lyman alpha forest along multiple lines of sight to distant QSOs
(abridged) We present an effective implementation of analytical calculations of the Lyalpha opacity distribution of the Intergalactic Medium (IGM) along multiple lines of sight (LOS) to distant quasars in a cosmological setting. This method fully accounts for the expected correlations between LOS and the cosmic variance in the large-scale modes of the dark matter distribution. Strong correlations extending up to ~300 kpc (proper) and more are found at redshift z~2->3, in agreement with observations. These correlations are investigated using the cross-correlation coefficient and the cross-power spectrum of the flux distribution along different LOS and by identifying coincident absorption features as fitted with a Voigt profile fitting routine. The cross-correlation coefficient between the LOS can be used to constrain the shape-parameter Gamma of the power spectrum if the temperature and the temperature density relation of the IGM can be determined indepedently. We also propose a new technique to recover the 3D linear dark matter power spectrum by integrating over 1D flux cross-spectra which is complementary to the usual `differentiation' of 1D auto-spectra. The cross-power spectrum suffers much less from errors uncorrelated in different LOS, like those introduced by continuum fitting. Investigations of the flux correlations in adjacent LOS should thus allow to extend studies of the dark matter power spectrum with the Lyalpha forest to significantly larger scales than is possible with flux auto-power spectra. 30 pairs with separation of 1-2 arcmin should be sufficient to determine the 1D cross-spectrum at scales of 60 Mpc/h with an accuracy of about 30% if the error is dominated by cosmic variance.