Flux Ratio Anomalies in Lensing

• Flux ratio anomalies are discrepancies between observed and predicted brightness ratios in lensed systems, indicating perturbations from dark matter subhaloes, baryonic structures, or line-of-sight haloes.
• They are most pronounced in quadruply imaged quasar lenses where cusp and fold configurations near critical curves amplify the impact of small potential variations.
• Multi-wavelength observations, from radio to CO emission, provide a robust, multi-tracer approach to diagnosing lens model inadequacies and dark matter microphysics.

Flux ratio anomalies are discrepancies between the observed brightness ratios of multiple images in a strong gravitational lens and the ratios predicted by a smooth lens model. In quasar lenses, they arise most conspicuously in quadruply imaged systems, especially fold and cusp configurations near critical curves, where magnifications are large and locally sensitive to small perturbations in the lensing potential. The phenomenon is central to small-scale structure studies because the same observational signature can be produced by dark matter subhaloes, line-of-sight haloes, baryonic structures in the main deflector, stellar microlensing, or source-structure effects, so its interpretation is intrinsically diagnostic but not unique (Stacey et al., 2018, Zelko et al., 2023).

1. Formal definitions and anomaly diagnostics

For compact emission, the basic observable is an image-to-image flux ratio,

$R_{ij} = \frac{S_i}{S_j},$

which, in a smooth macrolens, should approximately trace the ratio of magnifications,

$R_{ij} \approx \frac{\mu_i}{\mu_j}.$

This approximation is explicitly conditioned on the source being compact and not being altered by microlensing, extinction, or structure-dependent lensing effects. In the common local description of strong lensing, the magnification is

$\mu = \frac{1}{\det A},$

or, in the scalar convergence–shear form,

$\mu = \frac{1}{(1-\kappa)^2-\gamma^2}.$

Flux-ratio anomalies are therefore magnification anomalies: small perturbations in the local Jacobian can produce large fractional changes in flux while leaving image positions comparatively less affected (Stacey et al., 2018, Hsueh et al., 2016).

Two asymptotic diagnostics dominate the literature. For a fold pair with magnifications $\mu_A,\mu_B$,

$$R_{\rm fold} = \frac{\mu_A + \mu_B}{|\mu_A| + |\mu_B|},$$

and for a cusp triplet with $\mu_A,\mu_B,\mu_C$,

$$R_{\rm cusp} = \frac{|\mu_A+\mu_B+\mu_C|}{|\mu_A|+|\mu_B|+|\mu_C|}.$$

In the smooth limit, both tend to zero as the relevant opening angle becomes small. The fold opening angle is commonly denoted $\phi_1$, and the cusp opening angle $\Delta\phi$ or $R_{ij} \approx \frac{\mu_i}{\mu_j}.$0. Large $R_{ij} \approx \frac{\mu_i}{\mu_j}.$1 or $R_{ij} \approx \frac{\mu_i}{\mu_j}.$2 therefore indicates a violation of the smooth catastrophe relation, with sensitivity controlled by how close the source lies to the fold or cusp caustic (Hsueh et al., 2017, Hou et al., 23 Jan 2026).

This sensitivity is fundamentally local. One recent formulation emphasizes that flux-ratio anomalies respond to the second derivatives of the projected lensing potential, which is why sub-kpc perturbations can produce strong magnification shifts without proportionally large astrometric displacements. A plausible implication is that flux ratios and astrometric anomalies are complementary rather than redundant observables (Zelko et al., 2023).

2. Physical origins and interpretive ambiguity

The canonical interpretation invokes dark matter substructure in the lens halo. In that picture, low-mass haloes or subhaloes perturb the local convergence and shear near one image, producing a flux anomaly that is strongest for highly magnified cusp and fold configurations. Several simulation-based studies support the general sensitivity of flux ratios to cold-dark-matter subhaloes, but they also show that intrinsic substructure alone does not generically explain every observed anomaly. One analysis of radio quads found that MG0414+0534 could be reproduced by adding CDM subhalos with a probability of 5%–20%, whereas B0712+472, B1422+231, B1555+375, and B2045+265 were reproduced only at the level of a few percent (Xu et al., 2014). A related study with larger halo samples found that substructures significantly affect small-separation cusp and fold systems, but not large-separation systems, where discrepancies are attributed to simplifications inherent in smooth halo models (Xu et al., 2013).

Line-of-sight structure provides a second major class of explanation. In a $R_{ij} \approx \frac{\mu_i}{\mu_j}.$3CDM universe, haloes and subhaloes projected anywhere between source and observer can perturb the magnifications. For typical lensing geometry with lens at redshift 0.6 and source at redshift 2, background haloes are more likely to cause a cusp violation than foreground haloes, and the combined effect of perturbing structures within the lens and along the line of sight yields a cusp-violation probability of 20–30% (Xu et al., 2011). In a complementary weak-lensing treatment, structures with $R_{ij} \approx \frac{\mu_i}{\mu_j}.$4, or $R_{ij} \approx \frac{\mu_i}{\mu_j}.$5, were shown to significantly affect magnification ratios, with the ratio of magnification perturbation due to intervening halos to that of a primary lens typically $R_{ij} \approx \frac{\mu_i}{\mu_j}.$6 per cent (Inoue et al., 2012). MG0414+0534 has also been modeled with a minifilament or minivoid along the line of sight, each reproducing $R_{ij} \approx \frac{\mu_i}{\mu_j}.$7 with $R_{ij} \approx \frac{\mu_i}{\mu_j}.$8 and $R_{ij} \approx \frac{\mu_i}{\mu_j}.$9, without invoking any subhalo in the lens galaxy (Inoue, 2014).

Baryonic structure in the main deflector is an equally important contaminant and, in some systems, the dominant explanation. Ray tracing through the Illustris simulation, with subhalos explicitly excluded, showed that baryonic components alone can be a major contribution to flux-ratio anomalies: the probability of strong anomalies increases by about 8% in early-type lenses and by about 10–20% in disc lenses, with edge-on discs producing the strongest effect and 13% of mock lenses showing astrometric anomalies $\mu = \frac{1}{\det A},$0 mas (Hsueh et al., 2017). In CLASS B1555+375 and CLASS B0712+472, high-resolution imaging revealed edge-on disc components crossing the anomalous image pairs, and adding an exponential disc to the lens model reproduced the radio anomalies without requiring dark matter substructure; the inferred projected disc mass fractions within the Einstein radius were about 15% and 16%, respectively (Hsueh et al., 2016, Hsueh et al., 2017). More recently, non-clumpy departures from elliptical symmetry—$\mu = \frac{1}{\det A},$1, $\mu = \frac{1}{\det A},$2, radial variation in $\mu = \frac{1}{\det A},$3, and position-angle twist—were shown to rescue macro-model fits to B1422+231, demonstrating that flux-ratio anomalies can appear or disappear depending on model flexibility and photometric precision (Feng et al., 13 Jul 2026). In barred spiral lenses, the anomaly magnitude correlates strongly with higher-order even Fourier modes of the bar, with Spearman coefficients $\mu = \frac{1}{\det A},$4 for the boxy/peanut component and $\mu = \frac{1}{\det A},$5 for the hexapole component (Shan et al., 3 Oct 2025).

Microlensing and source-size effects complicate wavelength-dependent interpretations. Near-IR and optical fluxes are classically vulnerable to stellar microlensing and extinction. Even mid-infrared measurements are not always immune: inverse ray-shooting simulations of a composite accretion-disc plus dusty-torus source found that microlensing variations as large as 0.1 mag are very common at 11 microns (observer-frame), that a disc flux fraction as small as $\mu = \frac{1}{\det A},$6 can matter, and that significant deviations from $\mu = \frac{1}{\det A},$7 may therefore be induced without dark substructure (Sluse et al., 2013). By contrast, narrow emission lines from the narrow-line region, warm dust, and mm/sub-mm emission are used as progressively cleaner probes because their emitting regions are physically larger and much less affected by stellar microlensing (Zelko et al., 2023).

3. Observational manifestations across wavebands

MG J0414+0534 is a benchmark case because the anomaly persists across multiple tracers. ALMA observations at 340–342 GHz show four compact quasar images, A1, A2, B, and C, together with an Einstein ring from diffuse thermal dust emission in the host galaxy. The total 340 GHz continuum flux density is $\mu = \frac{1}{\det A},$8 mJy, with about half in the compact quasar images. The continuum flux ratios are

$\mu = \frac{1}{\det A},$9

showing that A2 is suppressed relative to A1, as already known from radio and mid-infrared measurements. The major new result in the same system was the detection of CO (11–10) line emission at $\mu = \frac{1}{(1-\kappa)^2-\gamma^2}.$0 GHz, tracing compact, warm, dense gas very close to the quasar nucleus. A single Gaussian fit gives $\mu = \frac{1}{(1-\kappa)^2-\gamma^2}.$1, and using the blue-shifted component $\mu = \frac{1}{(1-\kappa)^2-\gamma^2}.$2 to $\mu = \frac{1}{(1-\kappa)^2-\gamma^2}.$3 yields

$\mu = \frac{1}{(1-\kappa)^2-\gamma^2}.$4

The CO anomaly is consistent with the radio, mid-IR, and sub-mm continuum anomalies, suggesting that high-excitation CO can probe substructure while avoiding microlensing and dust extinction. The same reanalysis rejected the previously claimed dusty dwarf “object Y”: a $\mu = \frac{1}{(1-\kappa)^2-\gamma^2}.$5 peak in the pipeline image vanished after proper self-calibration, and the final image noise was $\mu = \frac{1}{(1-\kappa)^2-\gamma^2}.$6 rather than $\mu = \frac{1}{(1-\kappa)^2-\gamma^2}.$7, leading to the conclusion that object Y was a noise artefact. The cause of the anomaly therefore remained unknown (Stacey et al., 2018).

PSJ2107-1611 illustrates extreme wavelength dependence. It is a fold-configuration quad with a $\mu = \frac{1}{(1-\kappa)^2-\gamma^2}.$8 separation and a bright lensed arc. A smooth SIE plus external shear fit to the astrometry predicts that the fold pair should be among the brightest images, yet the observed optical-to-near-IR flux ratios are grossly inconsistent with that expectation: the fold pair is reported to be $\mu = \frac{1}{(1-\kappa)^2-\gamma^2}.$9 times too faint. Variability, time delay effects, and reddening are ruled out through multiple-epoch imaging and color information, whereas the VLASS S-band radio flux ratios are compatible with the smooth mass macromodel. The system therefore points to a wavelength-dependent combination of microlensing, source structure, and possibly substructure, rather than a purely achromatic lens-potential perturbation (Dux et al., 2023).

RXJ1131-1231 exemplifies the current move toward microlensing-resistant spectroscopy. JWST/NIRSpec IFU data were used to obtain the first narrow-line flux ratios for a lensed quasar with JWST, using the [S III] 9071/9533 Å doublet. A full lens-model reconstruction separated unresolved nuclear emission from extended narrow-line emission, and the spectra were jointly modeled with lensqso-specfit. The resulting narrow-line flux ratios have $\mu_A,\mu_B$0 uncertainties, comparable to the precision of JWST/MIRI warm-dust measurements, and show a clear anomaly in the cusp images relative to a standard smooth lens model. The narrow-line ratios are broadly consistent with previous narrow-line and warm-dust results, but marginal $\mu_A,\mu_B$1 deviations remain; small $\mu_A,\mu_B$2 pc spatial offsets between differently sized emission regions were shown to be sufficient to enhance such shifts (Paugnat et al., 22 Jun 2026).

4. Macromodeling, local reconstruction, and measurement precision

A recurring empirical fact is that lens models often reproduce image positions well but fail on fluxes. In a statistical study of time-delay lenses, model-predicted anomaly histograms were compared to direct microlensing measurements. The average value of the model anomalies was $\mu_A,\mu_B$3 magnitudes, substantially larger than the mean impact of microlensing, $\mu_A,\mu_B$4 magnitudes, and the model histograms contained a fat tail of high anomalies: 36% of image pairs in the Shajib sample had $\mu_A,\mu_B$5 mag, whereas only one case in the direct microlensing histogram, about 2.2%, reached $\mu_A,\mu_B$6 mag. Epps-Singleton, Anderson-Darling, and Cramér–von Mises tests rejected a common origin, while Kolmogorov-Smirnov was not always conclusive because it is less sensitive to the tail. The conclusion was that microlensing cannot statistically explain the bulk of model flux-ratio anomalies, and that model inadequacy is a major systematic (Mediavilla et al., 2024).

The degree to which an anomaly exists can depend strongly on how flexible the macro-model is allowed to be. In B1422+231, SIE+$\mu_A,\mu_B$7 and EPL+$\mu_A,\mu_B$8 models fit image positions within $\mu_A,\mu_B$9 when $$R_{\rm fold} = \frac{\mu_A + \mu_B}{|\mu_A| + |\mu_B|},$$0 mas, but at $$R_{\rm fold} = \frac{\mu_A + \mu_B}{|\mu_A| + |\mu_B|},$$1 mas astrometric anomalies appear if smooth macro-models alone are adopted. When positions and flux ratios are fitted jointly with $$R_{\rm fold} = \frac{\mu_A + \mu_B}{|\mu_A| + |\mu_B|},$$2 mas, the SIE+$$R_{\rm fold} = \frac{\mu_A + \mu_B}{|\mu_A| + |\mu_B|},$$3 family already shows flux-ratio anomalies at photometric uncertainty $$R_{\rm fold} = \frac{\mu_A + \mu_B}{|\mu_A| + |\mu_B|},$$4, whereas EPL+$$R_{\rm fold} = \frac{\mu_A + \mu_B}{|\mu_A| + |\mu_B|},$$5 can fit both positions and flux ratios at $$R_{\rm fold} = \frac{\mu_A + \mu_B}{|\mu_A| + |\mu_B|},$$6 and $$R_{\rm fold} = \frac{\mu_A + \mu_B}{|\mu_A| + |\mu_B|},$$7, but not at $$R_{\rm fold} = \frac{\mu_A + \mu_B}{|\mu_A| + |\mu_B|},$$8. Adding the extracted perturbations $$R_{\rm fold} = \frac{\mu_A + \mu_B}{|\mu_A| + |\mu_B|},$$9, $\mu_A,\mu_B,\mu_C$0, $\mu_A,\mu_B,\mu_C$1, and $\mu_A,\mu_B,\mu_C$2 can rescue the macro-models and explain the observed anomalous flux ratios. This suggests that anomaly statements are precision- and parametrization-dependent (Feng et al., 13 Jul 2026).

Macromodel-free local methods have been proposed to reduce false positives. In the Curved Arc Basis framework, the resolved host-galaxy arcs are used to reconstruct the local lensing Jacobian independently around each quasar image, rather than imposing a global mass profile. On simulated quads, CAB-predicted flux ratios reach a typical precision of $\mu_A,\mu_B,\mu_C$3, with a systematic floor $\mu_A,\mu_B,\mu_C$4. In “complex” lenses with $\mu_A,\mu_B,\mu_C$5 multipoles, a simplistic EPL+shear macromodel produced false-positive flux-ratio anomalies $\mu_A,\mu_B,\mu_C$6 in 8 out of 9 mocks, all 9 discrepancies exceeded $\mu_A,\mu_B,\mu_C$7, and 5 out of 9 exceeded $\mu_A,\mu_B,\mu_C$8; CAB remained consistent with the true flux ratios for all 9 complex mocks. In subhalo-injected mocks, CAB did not absorb the local perturbation, so the genuine subhalo-induced discrepancy remained visible (Paugnat et al., 5 Sep 2025).

Measurement precision itself is becoming a design parameter for anomaly science. Simulations of upcoming adaptive-optics facilities indicate that narrow-line flux ratio errors below $\mu_A,\mu_B,\mu_C$9 and submilliarcsecond astrometric precision will be attainable for typical quadruply imaged quasars. The same study found that, with TMT/IRIS + NFIRAOS, only a few minutes per system are required, and that to reach the same uncertainties as LIGER+KAPA in 600 s, IRIS+NFIRAOS needs only 84 s. This precision directly affects the ability to discriminate smooth structure, baryonic complexity, and genuine dark perturbers (Zelko et al., 2023).

5. Dark matter inference and competing microphysical interpretations

Flux-ratio anomalies are widely used as a gravity-only probe of non-luminous structure on sub-galactic scales. In CDM, low-mass subhaloes are abundant and can perturb magnifications without strongly shifting image positions. Simulation campaigns that combine realistic smooth lenses with subhaloes generally find that substructure broadens the flux-ratio distributions most strongly for close pairs and close triplets, but the predicted anomaly rate depends on host-halo mass, lens morphology, line-of-sight structure, and the adopted smooth-lens family (Xu et al., 2013, Xu et al., 2014). A plausible implication is that no single anomaly statistic directly maps to a unique subhalo abundance unless these nuisance sectors are controlled.

Population-level inference has therefore shifted toward distributional statistics. Harvey et al. proposed doubly imaged quasars as a statistically efficient probe because they are 20 times more abundant than quadruply imaged quasars. Using zoom-in simulations with baryonic feedback, line-of-sight structures, and quasar variability, they found that warm-dark-matter cosmologies predict a $$R_{\rm cusp} = \frac{|\mu_A+\mu_B+\mu_C|}{|\mu_A|+|\mu_B|+|\mu_C|}.$$0 difference in the cumulative distribution functions of flux ratios relative to CDM, with CDM predicting many more small ratios, and estimated that $$R_{\rm cusp} = \frac{|\mu_A+\mu_B+\mu_C|}{|\mu_A|+|\mu_B|+|\mu_C|}.$$1 doubly imaged quasars would be needed to unambiguously distinguish CDM from the two sterile-neutrino WDM models studied there (Harvey et al., 2019).

Recent work has pushed anomaly statistics into model discrimination beyond CDM. A study of the largest microlensing-free sample of 17 quadruply imaged cusp systems combined mid-IR, radio, and narrow-line flux ratios with Monte Carlo mock lenses under CDM, SIDM, and FDM. It identified minor-axis cusp lenses and narrow major-axis cusp lenses with $$R_{\rm cusp} = \frac{|\mu_A+\mu_B+\mu_C|}{|\mu_A|+|\mu_B|+|\mu_C|}.$$2 as a particularly discriminating subset, and found J1042+1641 to be $$R_{\rm cusp} = \frac{|\mu_A+\mu_B+\mu_C|}{|\mu_A|+|\mu_B|+|\mu_C|}.$$3 incompatible with both CDM and SIDM in that region, with a Bayes factor exceeding 100 in favor of FDM over even optimistic CDM and SIDM scenarios. The same study emphasized that only 11 cusp lenses currently lie in the discriminating subset, so the statistical generality remains unresolved (Hou et al., 23 Jan 2026).

An independent fuzzy-dark-matter lensing calculation provides a different route to similar phenomenology. In $$R_{\rm cusp} = \frac{|\mu_A+\mu_B+\mu_C|}{|\mu_A|+|\mu_B|+|\mu_C|}.$$4DM halos, order-unity density granules, $$R_{\rm cusp} = \frac{|\mu_A+\mu_B+\mu_C|}{|\mu_A|+|\mu_B|+|\mu_C|}.$$5, perturb the projected surface density and generate flux ratio anomalies of order a few tens of percent. Reported mean anomaly amplitudes are about 10–30%, comparable to observed quasar anomalies, and the same framework predicts rare hexad and octad image configurations from spiky caustic structure (Chan et al., 2020). These results do not by themselves settle the dark-matter question, but they demonstrate that the anomaly observable is sensitive to dark-sector microphysics as well as to ordinary lens complexity.

6. Clean tracers, future surveys, and unresolved issues

The current observational strategy is to use emission regions that are as free as possible from stellar microlensing and dust extinction while remaining compact enough to be sensitive to substructure. Radio fluxes long served this role, but propagation effects and scatter broadening can remain problematic in individual systems. Mid-infrared warm dust is useful but not perfectly microlensing-free. High-excitation CO in the mm regime, narrow-line flux ratios from the narrow-line region, and resolved host-galaxy arcs now form the most actively developed toolkit. In MG J0414+0534, the combination of compact AGN-related continuum, CO (11–10), and an Einstein ring from host-galaxy dust motivated the broader claim that lensed quasar-starbursts are excellent targets for detecting dark sub-haloes and testing models for dark matter (Stacey et al., 2018).

The instrumental outlook is correspondingly expansive. AO-assisted integral-field spectroscopy is expected to make follow-up of 100–1000 systems from Rubin, Euclid, and Roman practical, while JWST/NIRSpec IFU methods are already generalizable to additional quads with suitable narrow lines (Zelko et al., 2023, Paugnat et al., 22 Jun 2026). Local-arc approaches such as CAB are being developed as complementary priors to traditional macromodels, precisely because overly simplistic parametrizations can generate false-positive anomaly detections, whereas overly flexible global models can absorb genuine perturbations (Paugnat et al., 5 Sep 2025).

Several controversies remain open. One concerns attribution: baryons, line-of-sight structure, microlensing, and source offsets can all mimic or modulate the same signal. Another concerns statistics: small samples of microlensing-free cusp or fold systems limit model selection even when individual anomalies are striking. A third concerns model adequacy: the same system can exhibit or lose an “anomaly” as observational precision improves or the macro-model family is generalized. The field therefore increasingly treats flux-ratio anomalies not as a single diagnostic but as a multi-tracer inference problem in which wavelength, source size, astrometry, arc morphology, and model flexibility jointly determine what can be claimed about dark matter (Hou et al., 23 Jan 2026, Mediavilla et al., 2024).