Inverse distance ladder method for determining $$H_0$$ from angular diameter distances of time-delay lenses and supernova observations

Abstract Time-delay measurements from strong lensing systems combined with spectroscopic measurements of stellar kinematics in deflecting galaxies provide a natural way to infer absolute distances (lensing distances). This means that it can be used to anchor the relative distances of Type Ia supernovae (SNe Ia), and further infer the Hubble constant $$H_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> in a cosmological model-independent way, while avoiding the assumptions of curvature and the equation of state of dark energy. Indeed, observations based on gravitational lensing time delays can measure $$H_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> directly, but usually require assumptions about the specific cosmological models. Meanwhile, this method suffers the mass-sheet degeneracy obstacle. These factors may induce the bias on determination of $$H_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> . However, the inverse distance ladder method we use avoids these assumptions altogether. In this study, we seek for the Pantheon and Pantheon plus datasets and use Gaussian process regression to reconstruct the unanchored distance to match the distance at the redshift of the lens to determine $$H_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> , respectively. Based on the four H0LiCOW lenses, the unanchored distances reconstructed by combining the Pantheon and Pantheon plus datasets yielded $$H_0=80.1^{+7.0}_{-6.9} \mathrm {~km/s/Mpc}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>80</mml:mn> <mml:mo>.</mml:mo> <mml:msubsup> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>6.9</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> <mml:mn>7.0</mml:mn> </mml:mrow> </mml:msubsup> <mml:mrow> <mml:mspace/> <mml:mi>km</mml:mi> <mml:mo>/</mml:mo> <mml:mi>s</mml:mi> <mml:mo>/</mml:mo> <mml:mi>Mpc</mml:mi> </mml:mrow> </mml:mrow> </mml:math> and $$H_0=81.2^{+7.1}_{-7.0} \mathrm {~km/s/Mpc}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>81</mml:mn> <mml:mo>.</mml:mo> <mml:msubsup> <mml:mn>2</mml:mn> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>7.0</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> <mml:mn>7.1</mml:mn> </mml:mrow> </mml:msubsup> <mml:mrow> <mml:mspace/> <mml:mi>km</mml:mi> <mml:mo>/</mml:mo> <mml:mi>s</mml:mi> <mml:mo>/</mml:mo> <mml:mi>Mpc</mml:mi> </mml:mrow> </mml:mrow> </mml:math> with $$1 \sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mi>σ</mml:mi> </mml:mrow> </mml:math> observational uncertainty, respectively. All the lenses show the measured $$H_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> is in good agreement with the local measurement results reported by the SH0ES collaboration within $$\sim $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>∼</mml:mo> </mml:math> $$1.3 \sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1.3</mml:mn> <mml:mi>σ</mml:mi> </mml:mrow> </mml:math> confidence level.