Constrain Spatial Curvature and Dark Energy with Strong Lenses and Complementary Probes: a Forecast for Next-Generation Surveys

Inferring spatial curvature of the Universe with high-fidelity is a longstanding interest in cosmology. However, the strong degeneracy between dark energy equation-of-state parameter $w$ and curvature density parameter $\Omega_{\rm K}$ has always been a hurdle for precision measurements of curvature from late-universe probes. With the imminent commissioning of Vera C. Rubin Observatory's Legacy Survey of Space and Time (LSST), we demonstrate for the first time, using simulations of stage-IV surveys, the crucial role of time-delay distances from strong gravitational lenses in breaking this degeneracy. Our findings suggest that in non-flat $ow$CDM model, while strong lensing data alone only yield a $\Omega_{\rm K}$ constraint at $\sim O(10^{-1})$ level, the integration with SNe Ia and BAO data breaks the $w$-$\Omega_{\rm K}$ degeneracy and refines the $\Omega_{\rm K}$ constraint to $\sim O(10^{-2})$. This surpasses the constraints typically derived from SNe Ia Hubble diagrams and BAO data and is comparable to the constraints obtained from \textit{Planck} Primary CMB data. Additionally, we present a non-parametric approach using Gaussian Process to avoid parameter-dependency of the expansion history $H(z)$ and achieve similar $O(10^{-2})$ level constraint on $\Omega_{\rm K}$. This study demonstrates the significant potential of strong gravitational lenses and Stage-IV surveys like LSST to achieve high-fidelity, independent constraints on $\Omega_{\rm K}$, contributing to our understanding of the Universe's geometry and the dynamics of dark energy.