Multi-messenger cosmology: A route to accurate inference of dark energy beyond CPL parametrization from XG detectors

One of the central challenges in modern cosmology is understanding the nature of dark energy and its evolution throughout the history of the Universe. Dark energy is commonly modeled as a perfect fluid with a time-varying equation-of-state parameter, w(z), often modeled under CPL parametrization using two parameters w0 and wa. In this study, we explore both parametric and non-parametric methods to reconstruct the dark energy Equation of State (EoS) using Gravitational Wave (GW) sources, with and without electromagnetic (EM) counterparts called bright sirens and dark sirens respectively. In the parametric approach, we extend the widely used w0-wa model by introducing an additional term, wb, to better capture the evolving dynamics of dark energy up to high redshift which is accessible from GW sources. This extension provides increased flexibility in modeling the EoS and enables a more detailed investigation of dark energy's evolution. Our analysis indicates that, with five years of observation time and a 75% duty cycle using Cosmic Explorer and the Einstein Telescope, it will be possible to measure the dark energy EoS with remarkable precision, better than any other cosmological probe in the coming years, from bright standard sirens using a multi-messenger approach. These findings highlight the potential of GW observations in synergy with EM telescopes to offer valuable insights into the nature of dark energy, overcoming the current limitations in cosmological measurements.