Extragalactic test of General Relativity with time-delay gravitational lenses

Technical Summary: Extragalactic Test of General Relativity with Time-Delay Gravitational Lenses

Problem Statement
General Relativity (GR) serves as the foundation of modern cosmology, yet the precision era has revealed tensions (e.g., $H_0$ and $S_8$) that may suggest unidentified systematic effects or the need for new physics beyond the standard $\Lambda$CDM model. While GR has been rigorously tested within the solar system (e.g., Cassini mission), theoretical mechanisms like screening effects or chameleon mechanisms could suppress deviations in local environments, necessitating tests in the non-linear regime on kiloparsec (kpc) scales. Previous extragalactic tests using strong gravitational lensing (SGL) have been limited by their dependence on specific cosmological models (typically $\Lambda$CDM) or by the need to convert luminosity distances ($D_L$) to angular diameter distances ($D_A$) using the Etherington reciprocity formula, which can introduce biases. There is a critical need for a model-independent approach to constrain the parameterized post-Newtonian (PPN) parameter $\gamma_{PPN}$ (where $\gamma_{PPN}=1$ in GR) and the sound horizon scale ($r_d$) simultaneously without assuming the contents of the universe or the theory of gravity.

Methodology
The authors employ a model-independent framework combining time-delay measurements from the H0LiCOW strong lensing systems with reconstructed angular diameter distances derived from Baryon Acoustic Oscillation (BAO) data.

1. Theoretical Framework: The study utilizes the relationship between the time-delay distance ($D_{\Delta t}$) and the angular diameter distance to the lens ($D_l$). In modified gravity, the Weyl potential differs from the Newtonian potential, introducing $\gamma_{PPN}$. The authors derive a relation where the inferred lens distance $D'_l$ is related to the GR distance $D_l$ by $D'_l = \frac{1+\gamma_{PPN}}{2} D_l$.
2. Data Reconstruction (GPR): To avoid cosmological model bias, the authors reconstruct the angular BAO scale $\theta(z)$ using Gaussian Process Regression (GPR). This reconstruction utilizes the latest BAO measurements from the Dark Energy Spectroscopic Instrument (DESI) Data Release 2 (DR2), supplemented by data from SDSS DR16 (including LRGs, ELGs, quasars, and Ly$\alpha$ forests) and previous SDSS/DES transverse BAO measurements. This dataset covers the redshift range of the H0LiCOW lenses ($0.4 < z < 4.2$), eliminating the need for extrapolation.
3. Lensing Sample: The analysis focuses on four H0LiCOW lens systems (RXJ1131-1231, PG1115+080, B1608+656, and J1206+4332) which provide measurements for both $D_{\Delta t}$ and $D_l$. The authors utilize the posterior distributions (MCMC chains or parametric PDFs) provided by H0LiCOW.
4. Analysis Strategy:
   • Simultaneous Estimation: By combining the reconstructed $\theta(z)$ (which depends on $r_d$) with the lensing observables, the authors simultaneously estimate $\gamma_{PPN}$ and $r_d$.
   • Distance Ratio Method: To further mitigate bias induced by $r_d$, a second analysis is performed using the distance ratio $D_{\Delta t}/D'_l$. This ratio depends on the ratio of angular diameter distances ($D_s/D_{ls}$), which is derived directly from the reconstructed $\theta(z)$, effectively removing the explicit dependence on the sound horizon scale in the constraint of $\gamma_{PPN}$.

Key Contributions

• First Simultaneous Model-Independent Measurement: The paper presents the first simultaneous measurement of the post-Newtonian parameter $\gamma_{PPN}$ and the sound horizon scale $r_d$ without assuming the contents of the universe or a specific theory of gravity.
• Direct $D_A$ Reconstruction: Unlike previous studies that relied on converting $D_L$ (from Supernovae) to $D_A$, this work directly reconstructs the angular BAO scale $\theta$ using GPR on BAO data, providing a more robust and direct path to $D_A$.
• Bias Reduction: The introduction of the distance ratio method ($D_{\Delta t}/D'_l$) offers a refined approach to constrain $\gamma_{PPN}$ by avoiding the degeneracy and bias introduced by the sound horizon scale $r_d$.

Results

• Primary Constraints: Based on the four H0LiCOW lenses and the GPR-reconstructed distances, the authors estimate:
  • $\gamma_{PPN} = 0.93^{+0.16}_{-0.17}$
  • $r_d = 136.36^{+5.14}_{-3.20}$ Mpc
    These results are consistent with the GR prediction ($\gamma_{PPN}=1$) within $1\sigma$.
• Refined Constraints: Using the distance ratio method to eliminate $r_d$ bias, the constraint on the PPN parameter improves to:
  • $\gamma_{PPN} = 0.89^{+0.19}_{-0.15}$
    This result also aligns well with GR predictions.
• Sound Horizon Tension: The derived value of $r_d$ shows a tension (within $2.7\sigma$) with the Planck 2018 result ($147.09 \pm 0.26$ Mpc), a finding consistent with other low-redshift observations and potentially indicative of the Hubble tension.

Significance
The paper claims to provide a direct test of General Relativity at extragalactic scales. The results demonstrate that GR holds true within $1\sigma$ on kpc scales, consistent with previous solar system tests and other cosmological probes (such as SLACS and CMB analyses). By achieving a model-independent measurement of $\gamma_{PPN}$ and $r_d$, the study offers a robust verification of GR that is free from the assumptions of the $\Lambda$CDM model. The authors note that while current constraints are consistent with GR, future data from the China Space Station Telescope (CSST), which is expected to detect 160,000 galaxy-scale strong lensing systems, could provide even more stringent tests, particularly regarding screening effects.