# Flatness without CMB - the Entanglement of Spatial Curvature and Dark Energy Equation of State

## TL;DR

This study investigates how the correlation between cosmic spatial curvature and dark energy's equation of state affects observational constraints, emphasizing the importance of considering their entanglement in future cosmological surveys.

## Contribution

The paper introduces a Fisher forecast analysis incorporating the dependence of dark energy's equation of state on spatial curvature, validating the independence assumption in observational constraints.

## Key findings

- Correlation between $\
- The data primarily determines the $\
- Standard approaches treating $\

## Abstract

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.