This document summarizes the science cases related to cosmology studies with the MaunaKea Spectroscopic Explorer (MSE), a highly-multiplexed (4332 fibers), wide FOV (1.5 sq deg), large aperture (11.25 m in diameter), optical/NIR (360nm to 1300nm) facility. The MSE High-z Cosmology Survey is designed to probe a large volume of the Universe with a galaxy density sufficient to measure the extremely-large-scale density fluctuations required to explore primordial non-Gaussianity and therefore inflation. We expect a measurement of the local parameter $f_{NL}$ to a precision $sigma(f_{NL}) = 1.8$. Combining the MSE High-z Cosmology Survey data with data from a next generation CMB stage 4 experiment and existing DESI data will provide the first $5sigma$ confirmation of the neutrino mass hierarchy from astronomical observations. In addition, the Baryonic Acoustic Oscillations (BAO) observed within the sample will provide measurements of the distance-redshift relationship in six different redshift bins between $z=1.6$ and 4.0, each with an accuracy of $sim0.6%$. The simultaneous measurements of Redshift Space Distortions (RSD) constrain the amplitude of the fluctuations, at a level ranging from $1.9%$ to $3.6%$. The proposed survey covers 10,000 ${rm deg}^2$, measuring redshifts for three classes of target objects: Emission Line Galaxies (ELGs) with $1.6<z<2.4$, Lyman Break Galaxies (LBGs) with $2.4<z<4.0$, and quasars $2.1<z<3.5$. The ELGs and LBGs will be used as direct tracers of the underlying density field, while the Lyman-$alpha$ forests in the quasar spectra will be utilized to probe structure. Exposures of duration 1,800sec will guarantee a redshift determination efficiency of $90%$ for ELGS and at least $50%$ for LBGs. The survey will represent 100 nights per year for a 5-year MSE program. Finally, three ideas for additional projects of cosmological interest are proposed.
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