Hierarchical cosmological constraints through strong lensing distance ratio

Abstract

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).

Figures (10)

• Figure 1: Redshift distribution of lens galaxies and background sources in the combined sample used in this work. Blue squares represent lenses from SL2S, green circles represent lenses from BELLS, red down triangles represent lenses from SLACS, purple pentagons represent lenses from LSD, black triangles represent lenses from S4TM, and yellow diamonds represent lenses from BELLS GALLERY.
• Figure 2: Sensitivity maps for the three cosmological parameters $\Omega_m$, $w_0$, and $w_a$. Top row: sensitivity factor of the distance ratio as a function of lens redshift $z_l$ and source redshift $z_s$. Middle row: sensitivity factor of the time-delay distance as a function of $z_l$ and $z_s$. Bottom row: sensitivity factor of the double–source–plane distance ratio with the higher-redshift source fixed at $z_{s2}=3.0$, shown as a function of $z_l$ and the lower-redshift source $z_{s1}$. Larger values correspond to stronger sensitivity of a given distance combination to the cosmological parameters, although the absolute normalization is not directly comparable between different distance combinations. The orange arrows indicate how the locations of the low-sensitivity valleys shift when a given parameter is increased.
• Figure 3: Sensitivity maps for $w_0$ and $w_a$ assessed using the distance ratio. Blue circles indicate the redshift distribution of the simulation data based on collett2015population.
• Figure 4: These three subplots quantify the response of the observables to perturbations of the input cosmological parameters $\Omega_m$, $w_0$, and $w_a$ away from a fiducial model. Specifically, they show how the distance ratio $\mathcal{D}$, the time-delay distance $D_{\Delta t}$, and the double–source–plane distance ratio $\mathcal{D}_{\mathrm{DSP}}$ vary under parameter shifts. Curves are expressed as fractional changes relative to the fiducial value, $\mathcal{R}/\mathcal{R}_{\mathrm{fid}}$, so larger departures potentially indicate higher sensitivity.
• Figure 5: Joint constraints on $\Omega_m$ and $w$. Blue solid contours show constraints from strong-lensing distance ratios alone. Green dashed contours show Planck-only constraints. Orange solid contours show the joint constraints from strong-lensing distance ratios + Planck.