# Testing the Etherington's distance duality relation at higher redshifts: the combination of radio quasars and gravitational waves

## TL;DR

This study tests Etherington's distance duality relation at high redshifts by combining simulated gravitational wave data from the Einstein Telescope with VLBI radio quasar observations, achieving high-precision verification.

## Contribution

It introduces a novel method of testing the distance duality relation using gravitational waves and radio quasars, extending tests to higher redshifts with improved accuracy.

## Key findings

- Current radio quasar data verify DDR at 1% precision.
- Future ET data could verify DDR at 0.1% precision.
- Combining GW and EM signals enhances cosmological tests.

## Abstract

In this paper we analyse the implications of the latest cosmological data sets to test the Etherington's distance duality relation (DDR), which connects the luminosity distance $D_L$ and angular diameter distance $D_A$ at the same redshift. For $D_L$ we consider the simulated data of gravitational waves from the third-generation gravitational wave detector (the Einstein Telescope, ET), which can be considered as standard candles (or standard siren), while the angular diameter distances $D_A$ are derived from the newly-compiled sample of compact radio quasars observed by very-long-baseline interferometry (VLBI), which represents a type of new cosmological standard ruler. Alleviating the absorption and scattering effects of dust in the Universe, this will create a valuable opportunity to directly test DDR at much higher precision with the combination of gravitational wave (GW) and electromagnetic (EM) signals. Our results show that, with the combination of the current radio quasar observations, the duality-distance relation can be verified at the precision of $10^{-2}$. Moreover, the Einstein Telescope ET would produce more robust constraints on the validity of such distance duality relation (at the precision of $10^{-3}$), with a larger sample of compact milliarcsecond radio quasars detected in future VLBI surveys.