# Complementary constraints on dark energy equation of state from strongly lensed gravitational wave

## TL;DR

This paper explores how measuring time delays in strongly lensed gravitational wave systems can significantly improve constraints on dark energy's equation of state, especially when combined with other cosmological probes.

## Contribution

It demonstrates that incorporating time delay measurements from lensed GWs enhances dark energy constraints, using the CPL parametrization and future detector capabilities.

## Key findings

- Adding 30 lensed GW systems improves dark energy figure of merit by a factor of 2.
- Third-generation detectors can achieve constraints comparable to millions of standard siren events.
- Time delays in lensed GW systems provide a powerful complementary probe for dark energy studies.

## Abstract

It has been shown that time delay of strong gravitational lensing is not only an effective cosmological probe to constrain the Hubble constant, the matter density and the curvature of the universe but also useful for breaking the degeneracy of the dark energy equation of state and thus provide complementarity with other popular probes, such as type Ia supernovae, baryon acoustic oscillations, and cosmic microwave background radiation. Interestingly, compared to traditional strong lensing systems where quasars act as sources, strongly lensed gravitational waves (GWs) from compact binary coalescence and their electromagnetic (EM) counterparts systems have been recently proposed to be more powerful for studying cosmology since GWs and their EM counterparts are transients. Short durations of GWs and their EM counterparts are very advantageous to time delay measurement and lens profile modeling. Here, in the framework of Chevalier-Polarski-Linder (CPL) parametrization, we investigate improvement of constraining power on dark energy equation of state due to including time delay measurements of strong lensed GW systems. It is suggested that, on the basis of the third generation ground-based detector, e.g. Einstein Telescope, adding time delay of only 30 strong lensed GW systems to type Ia supernovae and cosmic microwave background radiation can improve the dark energy figure of merit by a factor 2. For the traditional standard siren method where the uncertainties of luminosity distances of GWs are $\sim10\%$, a few $\times~10^4$ events are expected to present similar constraints. In the precision cosmology era, this progress is of great significance for studying the nature of dark energy.