We present a science forecast for the eBOSS survey, part of the SDSS-IV project, which is a spectroscopic survey using multiple tracers of large-scale structure, including luminous red galaxies (LRGs), emission line galaxies (ELGs) and quasars (both as a direct probe of structure and through the Ly-$alpha$ forest). Focusing on discrete tracers, we forecast the expected accuracy of the baryonic acoustic oscillation (BAO), the redshift-space distortion (RSD) measurements, the $f_{rm NL}$ parameter quantifying the primordial non-Gaussianity, the dark energy and modified gravity parameters. We also use the line-of-sight clustering in the Ly-$alpha$ forest to constrain the total neutrino mass. We find that eBOSS LRGs ($0.6<z<1.0$) (combined with the BOSS LRGs at $z>0.6$), ELGs ($0.6<z<1.2$) and Clustering Quasars (CQs) ($0.6<z<2.2$) can achieve a precision of 1%, 2.2% and 1.6% precisions, respectively, for spherically averaged BAO distance measurements. Using the same samples, the constraint on $fsigma_8$ is expected to be 2.5%, 3.3% and 2.8% respectively. For primordial non-Gaussianity, eBOSS alone can reach an accuracy of $sigma(f_{rm NL})sim10-15$, depending on the external measurement of the galaxy bias and our ability to model large-scale systematic errors. eBOSS can at most improve the dark energy Figure of Merit (FoM) by a factor of $3$ for the Chevallier-Polarski-Linder (CPL) parametrisation, and can well constrain three eigenmodes for the general equation-of-state parameter (Abridged).
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