Hierarchical cosmological constraints through strong lensing distance ratio

Strong gravitational lensing provides an independent and powerful probe of cosmic expansion by directly linking observables to cosmological distances. Upcoming surveys such as LSST will discover large number of galaxy-galaxy strong lensing systems, offering a new route to precise cosmological constraints. In this paper, we propose a Fisher-like sensitivity factor to map how the cosmological information of strong-lensing distances changes across the lens-source redshift plane. Applying such factor to the distance ratio $D_{ls}/D_s$, the time-delay distance $D_{Δt}$, and the double-source-plane ratio, we determine the ``sensitivity valleys’’ where an observable becomes insensitive to a given parameter. The realistically simulated LSST lens population, which largely lies outside the distance-ratio valleys, covers the most sensitive region for $(w_0,w_a)$ parameter space. We then develop a new hierarchical framework, which could calibrate the redshift evolution of lens mass-density slopes and constrain cosmological parameters simultaneously. Focusing on the LSST mock data, we demonstrate that ignoring mass-profile evolution can bias $Ω_m$ by up to $\sim 10σ$, while modeling the lens evolution could perfectly recovers the fiducial cosmology and yield stringent cosmological constraints (e.g., $ΔΩ_m \simeq 0.01$ and $Δw \simeq 0.1$ for $\sim 10^4$ lenses).

💡 Research Summary

The paper presents a novel hierarchical framework for extracting cosmological information from strong gravitational lensing observations, focusing on distance ratios such as the lens‑source over source angular‑diameter distance (D_{ls}/D_s), the time‑delay distance (D_{\Delta t}), and the double‑source‑plane (DSP) ratio. The authors first introduce a “Fisher‑like sensitivity factor” that quantifies how the information content of these distance combinations varies across the lens‑source redshift plane ((z_l, z_s)). By applying this factor to the three distance observables, they map “sensitivity valleys” where a given observable becomes nearly insensitive to a specific cosmological parameter (e.g., (\Omega_m), (w_0), (w_a)). They show that the realistic LSST mock lens population largely avoids these valleys, meaning that LSST will probe the most informative region of the ((w_0,w_a)) parameter space.

The theoretical backbone consists of three cosmological models: the standard flat (\Lambda)CDM, a flat constant‑(w) extension (wCDM), and the Chevallier‑Polarski‑Linder (CPL) parametrization (w(a)=w_0+w_a(1-a)). For the lenses, they adopt an extended power‑law mass‑density model where the total mass density (\rho_{\rm tot}\propto r^{-\gamma}) and the luminous component (\rho_{\rm lum}\propto r^{-\delta}). Both slopes are allowed to evolve linearly with redshift: (\gamma(z)=\gamma_0+\gamma_s(z-z_{\rm med})) and similarly for (\delta). Anisotropy in the stellar velocity dispersion is encoded by (\beta). Solving the spherical Jeans equation yields a dynamical mass expression that links the observable Einstein radius (\theta_E) and the aperture‑corrected velocity dispersion (\sigma_v) to the distance ratio (D). Propagation of measurement uncertainties gives an analytic error model for (D).

The data set comprises 161 galaxy‑scale lenses drawn from several surveys (SLACS, SL2S, BELLS, etc.). For each system the authors use spectroscopic redshifts, Einstein radius (assumed 5 % fractional error), effective radius, and velocity dispersion corrected to a standard aperture (half the effective radius). The total velocity‑dispersion error includes statistical, aperture‑correction, and a 3 % systematic component. They also generate a realistic LSST mock catalog containing of order (10^4) lenses.

The hierarchical Bayesian analysis proceeds in two tiers. The first tier calibrates the redshift evolution of the lens mass‑density slopes without assuming any specific dark‑energy model, using unanchored Type Ia supernova distances reconstructed by an artificial neural network. The second tier incorporates these calibrated slope evolutions into the lens model, computes the theoretical distance ratios for a given cosmology, and compares them with the observed ratios. A joint posterior over the lens‑profile parameters ((\gamma_0,\gamma_s,\delta_0,\delta_s,\beta)) and cosmological parameters ((\Omega_m,w_0,w_a)) is sampled via Markov Chain Monte Carlo.

Results demonstrate a strong degeneracy between cosmology and lens‑profile evolution. Ignoring the evolution of (\gamma) and (\delta) biases (\Omega_m) by up to (\sim10\sigma) and significantly distorts the inferred dark‑energy equation‑of‑state parameters. When the evolution is modeled, the input (fiducial) cosmology is recovered without bias. For a LSST‑like sample of (\sim10^4) lenses, the hierarchical method yields uncertainties of (\Delta\Omega_m\simeq0.01) and (\Delta w\simeq0.1), a substantial improvement over current analyses based on a few hundred lenses.

The authors discuss limitations, notably the neglect of covariance between velocity dispersion and Einstein radius, and the simplified treatment of environmental effects, external convergence, and selection biases. They argue that future work should incorporate these systematics, but the present study establishes that strong‑lensing distance ratios, when combined with a hierarchical treatment of lens‑profile evolution, can provide competitive and independent constraints on the expansion history of the Universe, complementing CMB, BAO, and supernova probes.