Curved Arc Basis (CAB) in Lensing

• CAB is a local gravitational lensing formalism that describes arc distortions using eigenvalues and eigenvectors of the local Jacobian.
• It seamlessly bridges weak, semi-linear, and highly non-linear regimes with a continuous, local expansion that avoids rigid global modeling.
• CAB has practical applications in cluster and quasar lensing, enabling robust substructure searches while minimizing false-positive detections.

Curved Arc Basis (CAB) is a local gravitational-lensing formalism that parameterizes the distortion of extended lensed arcs through the eigenvectors and eigenvalues of the local lensing Jacobian and their directional differentials, rather than through a single globally parameterized deflector. In its foundational formulation, it was introduced as a continuous description of observables and degeneracies from the weak to the strong lensing regime, with a parameterization tightly linked to observable features in extended sources and applicable from the weak linear regime, the semi-linear regime, and up to the highly non-linear regime of highly magnified arcs of multiple images (Birrer, 2021).

1. Origin and conceptual scope

CAB was formulated to address a central problem in lensing inference: gravitationally lensed curved arcs contain substantial information about the underlying lensing distortions, but extracting that information from extended sources requires a parameterization that remains accurate as one moves from weak distortions to strongly magnified, highly curved images. The original formulation identifies a non-local and non-linear extended deflector basis that inherits the local properties of the lensing Jacobian, and it is explicitly designed to allow accurate extraction of lensing information from extended images without imposing an explicit global deflector model (Birrer, 2021).

This positioning distinguishes CAB from standard macromodeling. A global macromodel constrains the deflection field everywhere through a single parametric mass profile, whereas CAB only describes the lens mapping in the vicinity of a given arc. Later applications make this interpretation explicit: in cluster lenses, CAB is used as a local smooth model against which subhalo-induced residual distortions are sought, and in quadruply imaged quasars it is used as a macromodel-free description of the deflection field around each resolved host-galaxy arc (Şengül et al., 2023).

The scope of CAB is therefore dual. At the formal level, it is a continuous local description of lensing observables and degeneracies. At the methodological level, it is a practical basis for pixel-level modeling in regimes where extended arcs are informative but a global deflector parameterization is either too rigid or too computationally burdensome.

2. Mathematical structure

CAB is built on the standard thin-lens relation

$$\vec{\beta} = \vec{\theta} - \vec{\alpha}(\vec{\theta}),$$

with local behavior encoded by the Jacobian

$$A_{ij} \equiv \frac{\partial \beta_i}{\partial \theta_j} = \delta_{ij} - \frac{\partial^2 \psi}{\partial \theta_i \partial \theta_j}.$$

In the CAB description, the Jacobian is diagonalized into approximately radial and tangential eigendirections, $(\hat e_{\rm rad},\hat e_{\rm tan})$, defined by

$$\mathbf A \cdot \hat e_{\rm rad} = \lambda_{\rm rad}^{-1}\hat e_{\rm rad}, \qquad \mathbf A \cdot \hat e_{\rm tan} = \lambda_{\rm tan}^{-1}\hat e_{\rm tan}.$$

The corresponding stretch factors $\lambda_{\rm rad}$ and $\lambda_{\rm tan}$ immediately give the local magnification,

$\mu = \lambda_{\rm rad}\lambda_{\rm tan},$

and the tangential curvature is defined by

$$s_{\rm tan} \equiv \partial_{\rm tan}\phi_{\rm tan},$$

where $\phi_{\rm tan}$ is the local angle of $\hat e_{\rm tan}$ relative to a reference direction (Paugnat et al., 5 Sep 2025).

In the simplest CAB model, the local data consist of a reference position $$A_{ij} \equiv \frac{\partial \beta_i}{\partial \theta_j} = \delta_{ij} - \frac{\partial^2 \psi}{\partial \theta_i \partial \theta_j}.$$0, an orientation $$A_{ij} \equiv \frac{\partial \beta_i}{\partial \theta_j} = \delta_{ij} - \frac{\partial^2 \psi}{\partial \theta_i \partial \theta_j}.$$1, local stretch factors $$A_{ij} \equiv \frac{\partial \beta_i}{\partial \theta_j} = \delta_{ij} - \frac{\partial^2 \psi}{\partial \theta_i \partial \theta_j}.$$2 and $$A_{ij} \equiv \frac{\partial \beta_i}{\partial \theta_j} = \delta_{ij} - \frac{\partial^2 \psi}{\partial \theta_i \partial \theta_j}.$$3, and a curvature $$A_{ij} \equiv \frac{\partial \beta_i}{\partial \theta_j} = \delta_{ij} - \frac{\partial^2 \psi}{\partial \theta_i \partial \theta_j}.$$4. The construction assumes that $$A_{ij} \equiv \frac{\partial \beta_i}{\partial \theta_j} = \delta_{ij} - \frac{\partial^2 \psi}{\partial \theta_i \partial \theta_j}.$$5 is constant, so $$A_{ij} \equiv \frac{\partial \beta_i}{\partial \theta_j} = \delta_{ij} - \frac{\partial^2 \psi}{\partial \theta_i \partial \theta_j}.$$6 follows a circle of radius $$A_{ij} \equiv \frac{\partial \beta_i}{\partial \theta_j} = \delta_{ij} - \frac{\partial^2 \psi}{\partial \theta_i \partial \theta_j}.$$7; that $$A_{ij} \equiv \frac{\partial \beta_i}{\partial \theta_j} = \delta_{ij} - \frac{\partial^2 \psi}{\partial \theta_i \partial \theta_j}.$$8 and $$A_{ij} \equiv \frac{\partial \beta_i}{\partial \theta_j} = \delta_{ij} - \frac{\partial^2 \psi}{\partial \theta_i \partial \theta_j}.$$9 are constant along that circle; and that $(\hat e_{\rm rad},\hat e_{\rm tan})$0. Under these assumptions, the resulting local deflector is equivalent to a singular isothermal sphere plus a mass-sheet transformation. An extended version relaxes the assumption that $(\hat e_{\rm rad},\hat e_{\rm tan})$1 is constant along the arc by allowing $(\hat e_{\rm rad},\hat e_{\rm tan})$2, producing a local singular isothermal ellipsoid plus mass-sheet transformation (Paugnat et al., 5 Sep 2025).

This mathematical structure is important for two reasons. First, it expresses arc morphology directly in terms of local radial stretching, tangential stretching, curvature, and orientation. Second, it recasts higher-order local distortions into directional derivatives aligned with physically meaningful eigendirections, rather than into a generic Cartesian Taylor expansion.

3. Observables, degeneracies, and local inference

A central claim of the foundational CAB formalism is that its parameterization is tightly linked to observable features in extended sources, which permits model-independent extraction of lensing information and quantification of what degeneracies can be broken under assumptions about the local lensing nature and the intrinsic source shape (Birrer, 2021). CAB is thus not merely a compact parameterization; it is also a framework for reasoning about which combinations of lensing distortions are observationally identifiable from a given arc.

Because CAB is local, its degeneracy structure differs from that of global lens models. In cluster-arc applications, a source-size/magnification degeneracy appears: if the source is rescaled by a constant $(\hat e_{\rm rad},\hat e_{\rm tan})$3 and the stretches are rescaled by $(\hat e_{\rm rad},\hat e_{\rm tan})$4, the image remains unchanged. One practical choice used to break this degeneracy is to fix $(\hat e_{\rm rad},\hat e_{\rm tan})$5 for the first image, so that the remaining smooth parameters describe distortions relative to that image (Şengül et al., 2023). In quasar applications, CAB models also include a constant deflection shift per cutout, but by the prismatic degeneracy this shift has no observable effect (Paugnat et al., 5 Sep 2025).

The same locality that makes CAB flexible also imposes a validity scale. Later applications emphasize that CAB is a local expansion or local basis for the smooth distortion field. It works best when the image is a distinct arc image; it is less suitable where images straddle critical curves or are image mergers, and merged fold or cusp images are specifically identified as problematic for a single CAB component (Şengül et al., 2023). This locality is not a defect of the formalism so much as its defining assumption: CAB trades global physical closure for local descriptive power.

4. CAB in cluster strong lensing and perturber searches

A major operational use of CAB is the search for low-mass perturbers near highly magnified cluster arcs. In this setting, the smooth cluster field is represented locally by CAB, while subhalos, line-of-sight halos, or wandering supermassive black holes are added as perturbations. The smooth arc-forming distortion is parameterized by four principal quantities: inverse curvature radius $(\hat e_{\rm rad},\hat e_{\rm tan})$6, tangential stretch $(\hat e_{\rm rad},\hat e_{\rm tan})$7, radial stretch $(\hat e_{\rm rad},\hat e_{\rm tan})$8, and orientation angle $(\hat e_{\rm rad},\hat e_{\rm tan})$9. In the limit $$\mathbf A \cdot \hat e_{\rm rad} = \lambda_{\rm rad}^{-1}\hat e_{\rm rad}, \qquad \mathbf A \cdot \hat e_{\rm tan} = \lambda_{\rm tan}^{-1}\hat e_{\rm tan}.$$0, the arc becomes locally straight and CAB reduces to a constant magnification plus shear description (Şengül et al., 2023).

The first detailed cluster application studied three lensed images of a background galaxy in SMACS J0723, taken by the James Webb Space Telescope, and used CAB together with a source-light model based on two shapelet sets. The source complexity was controlled by shapelet order $$\mathbf A \cdot \hat e_{\rm rad} = \lambda_{\rm rad}^{-1}\hat e_{\rm rad}, \qquad \mathbf A \cdot \hat e_{\rm tan} = \lambda_{\rm tan}^{-1}\hat e_{\rm tan}.$$1, with

$$\mathbf A \cdot \hat e_{\rm rad} = \lambda_{\rm rad}^{-1}\hat e_{\rm rad}, \qquad \mathbf A \cdot \hat e_{\rm tan} = \lambda_{\rm tan}^{-1}\hat e_{\rm tan}.$$2

basis coefficients, and the real-data application adopted $$\mathbf A \cdot \hat e_{\rm rad} = \lambda_{\rm rad}^{-1}\hat e_{\rm rad}, \qquad \mathbf A \cdot \hat e_{\rm tan} = \lambda_{\rm tan}^{-1}\hat e_{\rm tan}.$$3 because that gave the lowest BIC. The paper then injected perturbers into realistic JWST-like mock data and reported practical sensitivity to NFW-like perturbers down to about $$\mathbf A \cdot \hat e_{\rm rad} = \lambda_{\rm rad}^{-1}\hat e_{\rm rad}, \qquad \mathbf A \cdot \hat e_{\rm tan} = \lambda_{\rm tan}^{-1}\hat e_{\rm tan}.$$4, with the optimal-location threshold lying between $$\mathbf A \cdot \hat e_{\rm rad} = \lambda_{\rm rad}^{-1}\hat e_{\rm rad}, \qquad \mathbf A \cdot \hat e_{\rm tan} = \lambda_{\rm tan}^{-1}\hat e_{\rm tan}.$$5 and $$\mathbf A \cdot \hat e_{\rm rad} = \lambda_{\rm rad}^{-1}\hat e_{\rm rad}, \qquad \mathbf A \cdot \hat e_{\rm tan} = \lambda_{\rm tan}^{-1}\hat e_{\rm tan}.$$6, and sensitivity to point-mass wandering SMBHs down to about $$\mathbf A \cdot \hat e_{\rm rad} = \lambda_{\rm rad}^{-1}\hat e_{\rm rad}, \qquad \mathbf A \cdot \hat e_{\rm tan} = \lambda_{\rm tan}^{-1}\hat e_{\rm tan}.$$7 (Şengül et al., 2023).

The same study showed that CAB can constrain more than perturber mass in favorable cases. For $$\mathbf A \cdot \hat e_{\rm rad} = \lambda_{\rm rad}^{-1}\hat e_{\rm rad}, \qquad \mathbf A \cdot \hat e_{\rm tan} = \lambda_{\rm tan}^{-1}\hat e_{\rm tan}.$$8 NFW perturbers, it recovered perturber redshift for a foreground line-of-sight halo at $$\mathbf A \cdot \hat e_{\rm rad} = \lambda_{\rm rad}^{-1}\hat e_{\rm rad}, \qquad \mathbf A \cdot \hat e_{\rm tan} = \lambda_{\rm tan}^{-1}\hat e_{\rm tan}.$$9, a cluster subhalo at $\lambda_{\rm rad}$0, and a background line-of-sight halo at $\lambda_{\rm rad}$1; it also found that concentration and ellipticity could be measured for sufficiently massive perturbers, whereas cored NFW core size was not recovered at JWST resolution and noise (Şengül et al., 2023).

These results position CAB as a local macro-lensing basis for cluster arcs. Rather than reconstructing the entire cluster potential at pixel level, it uses the resolved morphology of giant arcs to infer the smooth local mapping and then tests whether residual image-level distortions require an additional perturber.

5. Macromodel-free flux-ratio prediction in quads

CAB has also been extended to quadruply imaged quasars with resolved host-galaxy arcs. In that setting, the lensing field is represented by four independent local CAB models, one per cutout and one per quasar image, while the source is reconstructed jointly across all four cutouts. Because the reference location $\lambda_{\rm rad}$2 is anchored at the fitted quasar image position, the fitted local stretch factors directly predict the smooth magnification at that image through

$\lambda_{\rm rad}$3

The analysis then works in log magnifications and log flux ratios,

$\lambda_{\rm rad}$4

to assess flux-ratio anomalies without imposing a global macromodel (Paugnat et al., 5 Sep 2025).

In a mock study of 27 systems—9 baseline, 9 complex, and 9 subhalo—CAB model-predicted flux ratios reproduced the expected values with a typical precision of $\lambda_{\rm rad}$5. The paper calibrated a systematic uncertainty floor of $\lambda_{\rm rad}$6 for default $\lambda_{\rm rad}$7 cutouts, rising to $\lambda_{\rm rad}$8 for $\lambda_{\rm rad}$9 cutouts. In the baseline case, where the mock lens truly followed the same elliptical power-law plus shear family used for the global fit, the global macromodel had smaller uncertainties, as expected. In the complex case, however, the oversimplified macromodel frequently produced false-positive anomalies: 8 out of 9 complex mocks showed false-positive flux-ratio anomalies $\lambda_{\rm tan}$0, all 9 showed $\lambda_{\rm tan}$1 discrepancies, and 5 out of 9 showed $\lambda_{\rm tan}$2 discrepancies. CAB remained consistent with the true smooth flux ratios for all 9 complex mocks, within its uncertainties (Paugnat et al., 5 Sep 2025).

The same work tested whether CAB might simply absorb true subhalo perturbations. It did not. In the subhalo mocks, CAB still predicted the unperturbed smooth-lens flux ratios rather than the perturbed observed ones, so localized dark-matter-induced anomalies remained visible. This establishes CAB as a conservative complement to global macromodeling: less precise when the macromodel is exactly correct, but more robust to angular complexity in the main deflector and less prone to false-positive anomaly claims (Paugnat et al., 5 Sep 2025).

6. Source-model systematics and validation in real cluster data

Subsequent work on the cluster Abell S1063 emphasized that CAB-based substructure claims are highly sensitive to source modeling. Using JWST/NIRCam short-wavelength imaging in filters F115W, F150W, and F200W, the analysis modeled system 4, a triply imaged source at $\lambda_{\rm tan}$3, with CAB or CAB+NFW and compared two source reconstructions: shapelets and pixel-based source reconstruction based on Delaunay triangulation. Cross-filter stability was quantified through the pairwise tension

$\lambda_{\rm tan}$4

on the premise that the same lens should yield the same CAB parameters in all filters (Ephremidze et al., 25 Feb 2025).

The empirical pattern was sharp. With Large Masks, both source models showed significant disagreement across filters, with average tension $\lambda_{\rm tan}$5, indicating that CAB validity was breaking down when extended too far from the arc centers. With Small Masks, cross-filter consistency improved, but Delaunay remained substantially better than shapelets: shapelets gave average $\lambda_{\rm tan}$6, Delaunay gave $\lambda_{\rm tan}$7, and denser Delaunay with $\lambda_{\rm tan}$8 tessellation points gave $\lambda_{\rm tan}$9. After doubling the pixel errors, shapelets still showed average $\mu = \lambda_{\rm rad}\lambda_{\rm tan},$0, while Delaunay improved to $\mu = \lambda_{\rm rad}\lambda_{\rm tan},$1, bringing all Delaunay-inferred CAB parameters into agreement within $\mu = \lambda_{\rm rad}\lambda_{\rm tan},$2 across filters (Ephremidze et al., 25 Feb 2025).

The same case study also produced a cautionary result on subhalo detection. Shapelets yielded a formally strong $\mu = \lambda_{\rm rad}\lambda_{\rm tan},$3 candidate near image #2, with $\mu = \lambda_{\rm rad}\lambda_{\rm tan},$4, but the signal was not stable across source models or filter diagnostics. When the source was reconstructed with unregularized Delaunay triangulation, the evidence for substructure disappeared, with $\mu = \lambda_{\rm rad}\lambda_{\rm tan},$5 in all filters and $\mu = \lambda_{\rm rad}\lambda_{\rm tan},$6. Reintroducing source regularization into Delaunay could itself recreate spurious perturbers, reaching $\mu = \lambda_{\rm rad}\lambda_{\rm tan},$7 in one F200W configuration (Ephremidze et al., 25 Feb 2025).

This established a methodological constraint on CAB usage. CAB can support local substructure searches, but multi-band analysis, local-mask control, and sufficiently flexible source reconstruction are necessary to prevent convincing but spurious detections driven by source-model inadequacy rather than by dark matter.

7. Interpretation, limitations, and current status

CAB is best understood as a local arc-based lensing formalism rather than as a replacement for global physical mass models. The foundational work already framed it as a model-independent way to extract lensing information and to guide inference efforts in the right choices in complexity based on the data at hand (Birrer, 2021). Later applications preserved that framing: cluster-lens studies used CAB as a local smooth model for arc neighborhoods, and quasar studies used it as a macromodel-free baseline against which flux-ratio anomalies could be assessed (Şengül et al., 2023).

The limitations are correspondingly well defined. CAB is local rather than global; it has a systematic floor associated with local extrapolation assumptions; its performance depends on informative, resolved arcs; it is sensitive to cutout size and source reconstruction; and it can admit unphysical local solutions if unconstrained. In the quasar application, the method was explicitly described as not a full replacement for macromodels, because it does not recover a global physical mass distribution and has larger statistical uncertainties than a correct global model. In the cluster case study, Large Masks, restrictive source models, and regularization were all shown to compromise robustness (Paugnat et al., 5 Sep 2025).

At the same time, the later applications clarify why CAB remains significant. It directly exploits resolved arc morphology, packages the local Jacobian geometry into physically interpretable stretch and curvature variables, and can operate in situations where global macromodel rigidity itself becomes a source of false-positive dark-matter claims. This suggests a durable role for CAB as a complementary formalism: a local, arc-centered description of the deflection field that is particularly valuable when the scientific objective is to isolate small-scale perturbations without overcommitting to a global deflector family.