High-redshift cosmology with oxygen lines from H$\alpha$ surveys
  A new generation of cosmological experiments will spectroscopically detect
the H$\alpha$ line from ELGs at optical/near-infrared frequencies. Other
emission lines will also be present, which may come from the same H$\alpha$
sample or constitute a new galaxy sample altogether. Our goal is to assess the
value, for cosmological investigation, of galaxies at $z\gtrsim2$ present in
H$\alpha$ galaxy surveys and identifiable by the highly redshifted ultra-violet
and optical lines - namely the OII line and the OIII doublet in combination
with the H$\beta$ line. We use state-of-the-art luminosity functions to
estimate the number density of OIIIb and OII ELGs. We study the constraining
power of these high-redshift galaxy samples on cosmological parameters such as
the BAO amplitude, $H(z)$, $D_{\rm A}(z)$, $f\sigma_8(z)$, and $b\sigma_8(z)$
for different survey designs. We present a strong science case for extracting
the OIIIb sample, which we consider as an independent probe of the Universe in
the redshift range $2-3$. Moreover, we show that the OII sample can be used to
measure the BAO and growth of structure above $z=3$; albeit it may be
shot-noise dominated, it will nonetheless provide valuable tomographic
information. We discuss the scientific potential of a sample of galaxies which,
so far, has been mainly considered as a contaminant in H$\alpha$ galaxy
surveys. Our findings indicate that planned H$\alpha$ surveys should include
the extraction of these oxygen-line samples in their pipeline, to enhance their
scientific impact on cosmology.
