Towards constraining cosmological parameters with SPT-3G observations of 25% of the sky

The South Pole Telescope (SPT), using its third-generation camera, SPT-3G, is conducting observations of the cosmic microwave background (CMB) in temperature and polarization across approximately 10 000 deg$^2$ of the sky at 95, 150, and 220 GHz. This comprehensive dataset should yield stringent constraints on cosmological parameters. In this work, we explore its potential to address the Hubble tension by forecasting constraints from temperature, polarization, and CMB lensing on Early Dark Energy (EDE) and the variation in electron mass in spatially flat and curved universes. For this purpose, we investigate first whether analyzing the distinct SPT-3G observation fields independently, as opposed to as a single, unified region, results in a loss of information relevant to cosmological parameter estimation. We develop a realistic temperature and polarization likelihood pipeline capable of analyzing these fields in these two ways, and subsequently forecast constraints on cosmological parameters. Our findings indicate that any loss of constraining power from analyzing the fields separately is primarily concentrated at low multipoles ($\ell$ < 50) and the overall impact on the relative uncertainty on standard $Λ$CDM parameters is minimal (< 3%). Our forecasts suggest that SPT-3G data should improve by more than a factor of 300 and 3000 the Figure of Merit (FoM) of the EDE and the varying electron mass models, respectively, when combined with Planck data. The likelihood pipeline developed and used in this work is made publicly available online.