Constraining the spatial curvature with cosmic expansion history in a cosmological model with a non-standard sound horizon

Spatial curvature is one of the most fundamental parameters in our current concordance flat $\Lambda$CDM model of the Universe. The goal of this work is to investigate how the constraint on the spatial curvature is affected by an assumption on the sound horizon scale. The sound horizon is an essential quantity to use the standard ruler from the Cosmic Microwave Background (CMB) and Baryon Acoustic Oscillations (BAOs). As an example, we study the curvature constraint in an axion-like Early Dark Energy (EDE) model in light of recent cosmological datasets from Planck, the South Pole Telescope (SPT), and the Atacama Cosmology Telescope (ACT), as well as BAO data compiled in Sloan Digital Sky Survey Data Release 16. We find that, independent of the CMB datasets, the EDE model parameters are constrained only by the CMB power spectra as precisely and consistently as the flat case in previous work, even with the spatial curvature. We also demonstrate that combining CMB with BAO is extremely powerful to constrain the curvature parameter even with a reduction of the sound-horizon scale in an EDE model, resulting in $\Omega_K=-0.0056\pm 0.0031$ in the case of ACT+BAO after marginalizing over the parameters of the EDE model. This constraint is as competitive as the Planck+BAO result in a $\Lambda$CDM model, $\Omega_{K}=-0.0001\pm 0.0018$.