Line intensity mapping (LIM) provides a unique and powerful means to probe cosmic structures by measuring the aggregate line emission from all galaxies across redshift. The method is complementary to conventional galaxy redshift surveys that are object-based and demand exquisite point-source sensitivity. The Tomographic Ionized-carbon Mapping Experiment (TIME) will measure the star formation rate (SFR) during cosmic reionization by observing the redshifted [CII] 158$mu$m line ($6 lesssim z lesssim 9$) in the LIM regime. TIME will simultaneously study the abundance of molecular gas during the era of peak star formation by observing the rotational CO lines emitted by galaxies at $0.5 lesssim z lesssim 2$. We present the modeling framework that predicts the constraining power of TIME on a number of observables, including the line luminosity function, and the auto- and cross-correlation power spectra, including synergies with external galaxy tracers. Based on an optimized survey strategy and fiducial model parameters informed by existing observations, we forecast constraints on physical quantities relevant to reionization and galaxy evolution, such as the escape fraction of ionizing photons during reionization, the faint-end slope of the galaxy luminosity function at high redshift, and the cosmic molecular gas density at cosmic noon. We discuss how these constraints can advance our understanding of cosmological galaxy evolution at the two distinct cosmic epochs for TIME, starting in 2021, and how they could be improved in future phases of the experiment.
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