Absorption effects in the expanding Universe: spectral transmittance functions of intergalactic medium for distant sources

We analyse the formation of broad absorption troughs in the spectra of high-redshift sources in the redshift range $5 \le z \le 15$ for two observationally motivated reionization histories inferred from distant galaxy spectra and CMB polarization measurements. {\color{blue} The calculations assume that neutral hydrogen and helium in a homogeneous intergalactic medium reside predominantly in their ground states and absorb radiation through the Lyman-series lines and continua of HI, HeI, and HeII. The wavelength-dependent optical depths are computed for the first 40 Lyman-series lines of HI and HeII, the first 10 lines of HeI, and the corresponding continua, and are used to derive spectral transmittance functions of the intergalactic medium, $S(\lambda; z)$. The results show that spectral features in the continuous spectra of sources at $5 \lesssim z \lesssim 7$ are particularly sensitive to the reionization histories of both hydrogen and helium. We present a compact analytic prescription for the effective intergalactic spectral transmittance over the redshift range $0 \le z \le 15$, providing closed-form expressions for wavelength-dependent transmission calibrated to observational constraints. As an illustrative application, we compute the spectra of starless virialized halos of the Cloud9 type at different redshifts, demonstrating how intergalactic absorption reshapes their intrinsic emission.