For the uncertainty from the Fermat potential, which is determined by lens modeling, source host galaxies without dazzling AGNs are conducive to the reconstruction of the mass distribution of the lens galaxies in lensed FRB systems. Simulations show that the Fermat potential component contributes approximately ∼0.8% uncertainty on DΔt (Li et al. 2018b). Actually, in lens modeling, except for the contamination from the light of the dazzling AGN in the source, mass-sheet degeneracy (MSD; a family of mass density profiles that could reproduce the same lensing observables, e.g., image positions and relative fluxes, but yields different measured values of H0) also plays an important role in leading to the loss of precision and accuracy. In recent years, this issue has been intensively investigated. Generally, there are two ways, which include applying theoretical priors (make radial-mass density profile assumptions) and appealing nonlensing data (e.g., stellar kinematics) to break the MSD. For instance, the TDCOSMO collaboration (Millon et al. 2020) has achieved ∼2% in the inference of H0 from time-delay cosmography under the assumption that the radial-mass distribution of the lens can be described by a power-law mass profile or a composite of a dark-matter halo (Navarro et al. 1997) and baryon matter. Meanwhile, Gomer & Williams (2020) tested the effects of the power-law assumption and found that the power-law assumption would introduce significant bias in the recovery of H0. In practice, as suggested in Li et al. (2018b), a high-quality optical/IR image of the source–lens system could be very helpful to avoid choosing the wrong models. That is, the model we choose to characterize the mass distribution of the lens according to the high-resolution image might be more complex than the simple power-law one, and thus could significantly reduce the bias. However, it is difficult to choose an exactly right model for the lens, thus the uncertainty of the Fermat potential might be larger than 0.8% in real observations. For example, using eight time-delay galaxy lenses and more flexible modeling methods, a precision of 4.97% on H0 was achieved (Denzel et al. 2021). With spatially resolved kinematics and external observations to break MSD, a 5% precision of H0 was inferred by assuming that the deflector of TDCOSMO and the Sloan Lens ACS (SLACS) lenses are drawn from the same population (Birrer et al. 2020). It is promising that by increasing the size of samples and applying the hierarchical framework introduced by Birrer et al. (2020), a precision of 1.5% or 1.2% on H0 will be achieved without assumptions on the radial-mass profile of lens galaxies (Birrer & Treu 2021). More recently, Ding et al. (2021) implemented realistic simulations in lens modeling based on HST WFC3 observations from transient sources (e.g., supernovae, gamma-ray bursts, FRBs, and GWs) to compare the precision of H0 inferred from the transient case and the lensed AGN case. They found that, compared with traditional lensed quasars, the precision for inferring the Shapiro delay and the geometric delay could be improved by a factor of 3.8 and 4.7 for lensed transient systems, respectively. It means that the precision of the Fermat potential reconstruction could be improved by a factor of about 6. Furthermore, the lensed transient system also facilitates the determination of higher signal-to-noise stellar kinematics of the main deflector, and thus its mass density profile, which in turn plays a key role in breaking the mass-sheet degeneracy. With the abovementioned investigations taken into consideration, we first take 0.8% relative uncertainty on the Fermat potential on the basis of simulations presented in Ding et al. (2021) and Li et al. (2018b). In addition, we also consider a more conservative 1.5% relative uncertainty on Fermat potential for comparison.
Citation: Millon et al. 2020
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