A thorough investigation of the prospects of eLISA in addressing the Hubble tension: Fisher Forecast, MCMC and Machine Learning

We carry out an in-depth analysis of the capability of the upcoming space-based gravitational wave mission eLISA in addressing the Hubble tension, with a primary focus on observations at intermediate redshifts ($3<z<8$). We consider six different parametrizations representing different classes of cosmological models, which we constrain using the latest datasets of cosmic microwave background (CMB), baryon acoustic oscillations (BAO), and type Ia supernovae (SNIa) observations, in order to find out the up-to-date tensions with direct measurement data. Subsequently, these constraints are used as fiducials to construct mock catalogs for eLISA. We then employ Fisher analysis to forecast the future performance of each model in the context of eLISA. We further implement traditional Markov Chain Monte Carlo (MCMC) to estimate the parameters from the simulated catalogs. Finally, we utilize Gaussian Processes (GP), a machine learning algorithm, for reconstructing the Hubble parameter directly from simulated data. Based on our analysis, we present a thorough comparison of the three methods as forecasting tools. Our Fisher analysis confirms that eLISA would constrain the Hubble constant ($H_0$) at the sub-percent level. MCMC/GP results predict reduced tensions for models/fiducials which are currently harder to reconcile with direct measurements of $H_0$, whereas no significant change occurs for models/fiducials at lesser tensions with the latter. This feature warrants further investigation in this direction.