# Model-Independent Determination of $H_0$ and $\Omega_{K,0}$ using Time-Delay Galaxy Lenses and Gamma-Ray Bursts

## TL;DR

This paper presents a model-independent method to determine the Hubble constant and spatial curvature using time-delay galaxy lensing and gamma-ray bursts, extending redshift coverage and testing cosmological models.

## Contribution

It introduces a novel combination of lensing and gamma-ray burst distances to constrain $H_0$ and $\

## Key findings

- $H_0=71.5^{+4.4}_{-3.0}$ km/s/Mpc from lensing+GRB
- $ ext{Omega}_{K,0}=-0.07^{+0.13}_{-0.06}$ indicating flatness
- Combining lensing, GRB, and SN Ia reduces $H_0$ uncertainty by 25%

## Abstract

Combining the `time-delay distance' ($D_{\Delta t}$) measurements from galaxy lenses and other distance indicators provides model-independent determinations of the Hubble constant ($H_0$) and spatial curvature ($\Omega_{K,0}$), only based on the validity of the Friedmann-Lema\^itre-Robertson-Walker (FLRW) metric and geometrical optics. To take the full merit of combining $D_{\Delta t}$ measurements in constraining $H_0$, we use gamma-ray burst (GRB) distances to extend the redshift coverage of lensing systems much higher than that of Type Ia Supernovae (SNe Ia) and even higher than quasars, whilst the general cosmography with a curvature component is implemented for the GRB distance parametrizations. Combining Lensing+GRB yields $H_0=71.5^{+4.4}_{-3.0}$~km s$^{-1}$Mpc$^{-1}$ and $\Omega_{K,0} = -0.07^{+0.13}_{-0.06}$ (1$\sigma$). A flat-universe prior gives slightly an improved $H_0 = 70.9^{+4.2}_{-2.9}$~km s$^{-1}$Mpc$^{-1}$. When combining Lensing+GRB+SN Ia, the error bar $\Delta H_0$ falls by 25\%, whereas $\Omega_{K,0}$ is not improved due to the degeneracy between SN Ia absolute magnitude, $M_B$, and $H_0$ along with the mismatch between the SN Ia and GRB Hubble diagrams at $z\gtrsim 1.4$. Future increment of GRB observations can help to moderately eliminate the $M_B-H_0$ degeneracy in SN Ia distances and ameliorate the restrictions on cosmographic parameters along with $\Omega_{K,0}$ when combining Lensing+SN Ia+GRB. We conclude that there is no evidence of significant deviation from a (an) flat (accelerating) universe and $H_0$ is currently determined at 3\% precision. The measurements show great potential to arbitrate the $H_0$ tension between the local distance ladder and cosmic microwave background measurements and provide a relevant consistency test of the FLRW metric.