Flatness without CMB - the Entanglement of Spatial Curvature and Dark
  Energy Equation of State
  The cosmic spatial curvature parameter $\Omega_k$ is constrained, primarily
by cosmic microwave background (CMB) data, to be very small. Observations of
the cosmic distance ladder and the large scale structure can provide
independent checks of the cosmic flatness. Such late-universe constraints on
$\Omega_k$, however, are sensitive to the assumptions of the nature of dark
energy. For minimally coupled scalar-field models of dark energy, the equation
of state $w$ has nontrivial dependence on the cosmic spatial curvature
$\Omega_k$ (Miao and Huang, 2018). Such dependence has not been taken into
account in previous studies of future observational projects. In this paper we
use the $w$ parameterization proposed by Miao and Huang, where the dependence
of $w$ on $\Omega_k$ is encoded, and perform Fisher forecast on mock data of
three benchmark projects: a WFIRST-like Type Ia supernovae survey, an
EUCLID-like spectroscopic redshift survey, and an LSST-like photometric
redshift survey. We find that the correlation between $\Omega_k$ and $w$ is
primarily determined by the data rather than by the theoretical prior. We thus
validate the standard approaches of treating $\Omega_k$ and $w$ as independent
quantities.
