Impact of Large-Scale Structure along Line-of-Sight on Time-Delay Cosmography

TL;DR

The paper addresses how line-of-sight large-scale structure biases time-delay cosmography. It integrates high-resolution ELUCID N-body simulations with multi-plane ray tracing to quantify deviations from single-plane models, showing that LoS matter within ~$2\ \mathrm{arcmin}$ induces arcsecond-scale fluctuations that cannot be fully captured by external shear alone. Even after partial corrections, the time-delay distance $D_{\rm dt}$ shows a residual bias of about $6.2\times10^{-3}$ with ~${2.5}\%$ scatter, underscoring the necessity of multi-plane modeling for precise Hubble-parameter constraints. The work demonstrates that external flexion improves image reconstructions but does not eliminate time-delay biases, highlighting the need for comprehensive multi-plane analyses in time-delay cosmography.

Abstract

Time-delay cosmography, by monitoring the multiply imaged gravitational lenses in the time domain, offers a promising and independent method for measuring cosmological distances. However, in addition to the main deflector that produces the multiple images, the large-scale structure along the line-of-sight (LoS) will also deflect the traveling light rays, known as weak lensing (WL). Due to resolution limitations, accurately measuring WL on arcsecond scales is highly challenging. In this work, we evaluate the LoS effects on both lensing images and time-delay measurements using a more straightforward, high-resolution N-body simulation that provides a more realistic matter distribution compared to the traditional, computationally cheaper halo rendering method. We employ the multi-plane ray tracing technique, which is traditionally utilized to compute WL effects at the arcminute scale, extending its application to the strong lensing regime at the arcsecond scale. We focus on the quadruple-image system and present the following findings: 1. In addition to a constant external convergence, large-scale structures within a region approximately 2 arcminutes in angular size act as external perturbers, inducing inhomogeneous fluctuations on the arcsecond scale; 2. These fluctuations cannot be fully accounted for by external shear alone, necessitating the inclusion of external flexion; 3. While incorporating flexion provides a reasonably good fit to the lensing image, the time-delay distance still exhibits a $6.2$\textperthousand~bias and a $2.5\%$ uncertainty. This underscores the limitations of the single-plane approximation, as time-delay errors accumulate along the LoS.

Figures (12)

• Figure 1: The distribution of the line-of-sight (LoS) lensing potential differences in the central region for 200 random cases, comparing different calculation region sizes. The yellow, green, and blue lines represent the normalized differences in lensing potential between light cone sizes of 25.6, 51.2, 102.4, and 204.8 arcsec, respectively.
• Figure 2: The lensing perturbation impact on lensing image. Figure (a) shows the LoS convergence map and shear map for one sample case from the 100 mock datasets. The black line indicates the critical curve of the main lens. Figure (b) presents the best-fit reduced $\chi^2$ values from the reconstruction of the image data, with yellow representing the SIE+Shear model and blue representing the SIE+Shear+Flexion model.
• Figure 3: A showcase of the reconstructed time delay distribution and the posterior of the time delay distance. The upper-right panel presents the posterior distribution of the time delay distance, with the red line indicating from the theoretical prediction of single main lens plane, $(1+z_d)\frac{D_lD_s}{D_{ls}}$. The triangle contour plots display the reconstructed time delays derived from the posterior of the model parameters, while the gray lines indicate the injected values obtained from ray-tracing. In both panels, the yellow lines correspond to the fit from the SIE+Shear model, and the blue lines represent the fit from the SIE+Shear+Flexion model.
• Figure 4: The distribution of the modeling bias on the time delay distance $D_{\rm dt}$ for 100 mock systems. Each half of the violin plot shows the distribution of model-predicted biases for two cases: using only shear, and using shear plus flexion. The left and right sides of each violin compare results with and without the constant external $\kappa_{\rm ext}$ correction. The outer contour of each violin shows the kernel density estimate, while the background histograms reveal the underlying distribution. The central vertical black error bars in each half-violin plot indicate the median (dot) and the 1$\sigma$ range (horizontal caps). A horizontal reference gray line at y = 0 is included for alignment.
• Figure 5: Raytracing cumulative time delays from observer to source normalized by the time delay in the source plane. Solid lines represent the normalized time delays from the multi-lens simulation, while dash-dotted lines indicate the case with only the main lens.