  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 $5\sigma$ 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.
