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arxiv: 2605.03905 · v1 · submitted 2026-05-05 · 🌌 astro-ph.GA · astro-ph.CO

Microlensing time-scales and flux magnification probabilities of a sample of 204 lensed quasars

F. \'Avila-Vera , V. Motta , E. Mediavilla This is my paper

Pith reviewed 2026-05-07 14:59 UTC · model grok-4.3

classification 🌌 astro-ph.GA astro-ph.CO
keywords quasar microlensinglensed quasarsaccretion disk sizemicrolens mass fractionmagnification histogramssource crossing timeEinstein radius crossing time
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The pith

Modeling microlensing across 204 lensed quasars estimates an average quasar half-light radius of 5.4 light-days and requires at least 15 percent of lens mass in compact objects.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper applies automatic lens modeling to every image in a sample of 204 lensed quasars to generate microlensing magnification maps that incorporate realistic optical depths and the gravity of the lens galaxy. Using thin-disk source sizes scaled to recent measurements, it computes mean source crossing times of 2.59 years and Einstein-radius crossing times of 11.29 years. The resulting histogram of mean magnifications is compared with the observed distribution; a good match occurs when 20 percent of the mass is placed in microlenses. This match supplies an average half-light radius for the quasar sources of 5.4 light-days together with a lower limit of 0.15 on the microlens mass fraction. The maps also show that any single lensed image has roughly a 9 percent chance of undergoing a high-magnification event.

Core claim

By generating microlensing magnification maps for each image in 204 lensed quasar systems and comparing the histograms of mean microlensing magnifications to the experimental distribution, while adopting thin-disk source sizes scaled to recent measurements, the work finds that a microlens mass fraction α = 0.2 produces a good match and yields an average source half-light radius R_{1/2} = 5.4 ± 2.7 light-days with α ≥ 0.15 as a lower limit. The same modeling gives a mean source crossing time of 2.59 ± 0.07 years and an Einstein radius crossing time of 11.29 ± 0.05 years, plus an average 9 percent probability that any given image experiences a high-magnification event.

What carries the argument

Microlensing magnification maps and histograms generated by automatic lens modeling of each of the 204 systems, which encode the combined effects of source size, optical depth, and microlens mass fraction for direct statistical comparison with observed magnifications.

If this is right

  • A microlens mass fraction of 0.2 produces a modeled magnification histogram that matches the observed one.
  • The average quasar source half-light radius is 5.4 ± 2.7 light-days.
  • Any given lensed image has approximately a 9 percent probability of a high-magnification event with Δm ≤ -0.32.
  • A subset of images with the largest high-magnification probabilities and smallest crossing times can be identified for targeted monitoring.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Future larger samples from wide-field surveys could tighten the constraints on both average source size and the microlens mass fraction.
  • The reported crossing times imply that multi-year monitoring campaigns are needed to capture typical microlensing variability in most systems.
  • If the thin-disk scaling holds across the sample, it supports standard accretion-disk models for quasars at a wide range of redshifts.
  • The 9 percent high-magnification probability indicates that a non-negligible fraction of known lensed images are currently experiencing significant flux changes.

Load-bearing premise

Automatic lens modeling applied uniformly to all 204 systems produces accurate magnification maps, and scaling thin-disk source sizes to recent measurements correctly represents the emitting region in every quasar.

What would settle it

An independent size measurement, such as from reverberation mapping or interferometry, that yields a mean half-light radius for these quasars differing by more than a few light-days from 5.4 would falsify the reported average.

Figures

Figures reproduced from arXiv: 2605.03905 by E. Mediavilla, F. \'Avila-Vera, V. Motta.

Figure 1
Figure 1. Figure 1 view at source ↗
Figure 2
Figure 2. Figure 2: Microlensing magnification patterns produced by stars in the lensing galaxy corresponding to image A of the system DES2158−5812. Left: Original magnification map. The color scale encodes different magnification levels, while the diamond-shaped curves trace the caustics.Right: The same map after convolution with a Gaussian profile with a sigma size of 3.72 [pix], showing the effective magnification pattern.… view at source ↗
Figure 4
Figure 4. Figure 4: Values of κ and γ for our sample (red filled diamonds), compared with literature samples: unfilled black diamonds Neira et al. (2025), un￾filled circles Ertl et al. (2023), and unfilled black squares Schmidt et al. (2023). imately 1.1 light-days per pixel. This resolution is sufficient to avoid introducing additional biases in our results, as demon￾strated by Vernardos & Fluke (2013). Magnification maps ar… view at source ↗
Figure 2
Figure 2. Figure 2: To account for the magnification of a finite-size source, maps were convolved with a Gaussian source profile of standard devi￾ation rs , I(r) ∝ exp (−r 2 /2r 2 s ). As shown by Mortonson et al. (2005), the specific shape of the brightness profile does not strongly affect the statistical magnification properties, but just its half-light radius, which for a Gaussian profile is R1/2 = 1.18rs and R1/2 = 2.44RS… view at source ↗
Figure 5
Figure 5. Figure 5: Histogram of the Einstein radii at the source plane for the sys￾tems, in blue the distribution of all the objects, and grey the distribution for the systems with the redshift of the lens known tal of 3120 convolved maps for all the pairs (α, rs). These maps are then used to study the effects of α and rs on the magnifica￾tion probability distributions and, combined with observational microlensing constraint… view at source ↗
Figure 7
Figure 7. Figure 7: The median value of the histogram is 807 view at source ↗
Figure 6
Figure 6. Figure 6: Histogram of the S&S based sizes of the source, RS , before and after applying microlensing size correction (204 systems). 1000 2000 3000 4000 5000 6000 Effective velocity (v) [km/s] 0 20 40 60 80 100 120 Number of systems All objects (204) Known lens redshift (119) view at source ↗
Figure 8
Figure 8. Figure 8: Top: Distribution of Einstein time scales. Bottom: Distribution of source time scales. and 2.59 ± 0.09 years for known redshift lens systems and all the sample, respectively, in contrast with the 0.61 years calcu￾lated by M&K. In this case, the discrepancy arises from both, the re-scaling of the theoretical sizes and the updated effective velocity. 4.3. Microlensing magnification statistics To explore the … view at source ↗
Figure 7
Figure 7. Figure 7: Histogram of the distribution of the effective velocities (Equa￾tion 10). these histograms, we can derive, with 1 σ uncertainties, that the mean Einstein ring crossing time is 11.29 ± 0.05 years for the entire sample and 10.82±0.20 years for the systems with known lens redshifts, whereas M&K obtained a value of 20.6 years. The difference between the two estimates is mainly related to using the updated valu… view at source ↗
Figure 9
Figure 9. Figure 9: Probability distribution of unsigned microlensing magnification for three different scaling factor values for 0.3 × {rs}, 1.0 × {rs}, and 2.0 × {rs}, with two different stellar densities (α). these means (520 maps each (α, rs) pair), which give us the prob￾ability of obtaining an (unsigned) value of microlensing magni￾fication if we randomly select one image from the sample. The mean values of these histog… view at source ↗
Figure 10
Figure 10. Figure 10: Histogram of unsigned microlensing magnification for three different scaling factors values for rs , with two different stellar densities (α), comparing the results from Mediavilla et al. (2024) and this work with all histograms normalized to the same total count. 2 4 6 8 10 Half light radius [lt-day] 0.10 0.12 0.14 0.16 0.18 0.20 Fraction of total mass in stars 0.0050 0.0100 0.0150 0.0155 0.0200 0.0250 0… view at source ↗
Figure 11
Figure 11. Figure 11: Relative probability distribution (∝ e −χ 2 /2 ) in the (half light radius [light days], fraction of total mass in stars) parameter space. The red contour marks the 68% confidence region, enclosing the highest￾probability values. distinctive feature of these cumulative distributions is the size dependence of the probability of high magnifications. The inte￾grated probability of events with ∆m < −0.40 for … view at source ↗
Figure 12
Figure 12. Figure 12: Figure: Cumulative distribution functions of magnitude changes during microlensing events. The left column shows the probability of observing a demagnification event with ∆mag > 0, while the right column shows a magnification event with ∆mag < 0. The top row corresponds to a stellar fraction of α = 0.1, and the bottom row to α = 0.2. 0.00 0.05 0.10 0.15 0.20 0.25 Probability mag < 0.32 0 10 20 30 40 Frequ… view at source ↗
Figure 13
Figure 13. Figure 13: Top: histogram of the probabilities of obtaining a magnification less than ∆mag = −0.32 for different stellar densities and values of rs . Bottom: density histogram of the probabilities of obtaining a demagnification greater than ∆mag = 0.32 for different stellar densities and values of rs . α ≥ 0.15, also in good agreement with previous studies (see, e.g., Esteban-Gutiérrez et al. 2022, 2023). To establi… view at source ↗
Figure 14
Figure 14. Figure 14: Same convolved microlensing magnification map as in view at source ↗
Figure 15
Figure 15. Figure 15: Top: probability of ∆mag ≤ −0.32 versus Einstein radius crossing time (tE [years]), with blue (orange) representing values obtained for a stellar density of 0.1 (0.2) and a 1.0 × {rs}, the red stars represent the object that shows the smaller Einstein radius crossing time and high magnification probability that correspond to DESJ2038-4008 and Q2237+030. Bottom: probability or ∆mag ≤ −0.32 versus source cr… view at source ↗
Figure 16
Figure 16. Figure 16: Correlation between the values of κ and γ in our sample, with the color bar representing the HME probability. The panels are separated by α and by multiples of rs . 4. We found a median time scale tE = 11.29 ± 0.05 years for the Einstein radius crossing time and tS = 2.59 ± 0.07 years for the source size crossing time after accounting for uncertain￾ties through Monte Carlo simulations. These values differ… view at source ↗
read the original abstract

Quasar microlensing is both a very useful tool in cosmology and astrophysics, and a source of uncertainty in some studies like the determination of the Hubble constant from lensed quasars. Microlensing probability and time-scales have been statistically studied using as a reference scale the Einstein ring crossing time of an isolated mass. Our goal is to extend the statistical analysis of microlensing to all currently known lensed quasars with available data, considering realistic optical depths and the gravitational effect of the lens galaxy. We take into account new observational results about quasar sizes and peculiar velocities of lens galaxies. We apply automatic lens modeling to the 204 systems available. For each image, we compute microlensing magnification maps and histograms. Using thin disk source sizes scaled to take into account recent measurements of accretion disk sizes, we find a mean source crossing time of $2.59\pm 0.07$ years. The mean Einstein radius crossing time is $ 11.29 \pm 0.05$ years. When a fraction of mass in microlenses $\alpha=0.2$ is adopted, we find a good matching between the modeled histogram of mean microlensing magnifications for the images in our sample and the experimental histogram of microlensing magnifications. From the modeling of microlensing magnification histograms, we estimate the average half-light radius of the quasar source, $R_{1/2}=5.4\pm 2.7$ light-days, and a lower limit to the mass fraction in microlenses, $\alpha\ge 0.15$. From the microlensing magnification maps, we find that a lensed quasar image has a mean probability of approximately 9% of being involved in a high-magnification event ($\Delta m \le -0.32$). We select a group of images with the largest probabilities and the smallest crossing times.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper performs a statistical study of microlensing in 204 lensed quasars by applying automatic lens modeling to produce magnification maps and histograms for each image. Incorporating realistic optical depths, galaxy effects, and thin-disk source sizes scaled from recent measurements, it reports a mean source crossing time of 2.59 ± 0.07 years and Einstein radius crossing time of 11.29 ± 0.05 years. Adopting α = 0.2 yields a match between modeled and observed mean-magnification histograms, from which the authors derive an average quasar half-light radius R_{1/2} = 5.4 ± 2.7 light-days and a lower limit α ≥ 0.15, plus a ~9% mean probability of high-magnification events (Δm ≤ -0.32) and a selected subsample of high-probability images.

Significance. If the automatic modeling and source scaling hold, the work supplies the largest-sample constraints to date on quasar accretion-disk sizes and the microlens mass fraction, with direct utility for time-delay cosmography and accretion physics. The uniform treatment of 204 systems and the concrete, uncertainty-quantified time-scale and probability estimates constitute a clear advance over smaller prior studies.

major comments (3)
  1. [Abstract] Abstract: The choice of α = 0.2 is stated to produce a good match to the experimental histogram, after which the same modeling is used to report a lower limit α ≥ 0.15; this procedure reduces the independence of the lower-limit claim and requires explicit justification of how the threshold was set without circularity.
  2. [Automatic lens modeling] Automatic lens modeling description: No external validation (e.g., comparison of automatic vs. published manual models on a subset of lenses) or per-system goodness-of-fit metrics are reported; because the aggregate histogram comparison underpins both the R_{1/2} estimate and the α ≥ 0.15 limit, systematic offsets in even a modest fraction of magnification maps would propagate directly into the headline results.
  3. [Source modeling] Source-size scaling procedure: The thin-disk sizes are scaled uniformly to external measurements, yet the scaling factor is listed among the free parameters; without a clear, object-by-object justification or sensitivity test, this choice affects the convolution step that produces the modeled histogram and the derived R_{1/2} value.
minor comments (2)
  1. [Abstract] The abstract would benefit from stating the total number of images analyzed in addition to the 204 systems.
  2. [Introduction] Notation for crossing times and R_{1/2} should be defined at first use with explicit units and reference to the thin-disk model employed.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments, which help improve the clarity and robustness of our analysis. We address each major comment below and indicate where revisions will be made.

read point-by-point responses

  1. Referee: [Abstract] The choice of α = 0.2 is stated to produce a good match to the experimental histogram, after which the same modeling is used to report a lower limit α ≥ 0.15; this procedure reduces the independence of the lower-limit claim and requires explicit justification of how the threshold was set without circularity.

    Authors: We agree that the abstract and main text could better separate the representative value from the limit derivation to avoid any appearance of circularity. α = 0.2 was selected as a fiducial value yielding a close match to the observed mean-magnification histogram. The lower limit α ≥ 0.15 is obtained independently by scanning α downward until the modeled histogram deviates from the data beyond the reported uncertainties. We will revise the abstract and add an explicit paragraph in Section 3 describing the threshold procedure and confirming the independence of the limit. revision: yes

  2. Referee: [Automatic lens modeling] No external validation (e.g., comparison of automatic vs. published manual models on a subset of lenses) or per-system goodness-of-fit metrics are reported; because the aggregate histogram comparison underpins both the R_{1/2} estimate and the α ≥ 0.15 limit, systematic offsets in even a modest fraction of magnification maps would propagate directly into the headline results.

    Authors: The automatic modeling employs the same established pipeline used in our prior publications on smaller samples, with parameters constrained by the observed image positions and flux ratios. We did not include a dedicated external validation subset or per-system χ² metrics in the present work. We will add a new subsection discussing the method's internal consistency checks, quoting the typical residuals from the lens equation solutions, and noting that the statistical results are robust to modest per-system errors given the sample size of 204. A full manual re-modeling of a validation subset lies beyond the scope of this statistical study. revision: partial

  3. Referee: [Source modeling] Source-size scaling procedure: The thin-disk sizes are scaled uniformly to external measurements, yet the scaling factor is listed among the free parameters; without a clear, object-by-object justification or sensitivity test, this choice affects the convolution step that produces the modeled histogram and the derived R_{1/2} value.

    Authors: Because individual size measurements exist for only a small fraction of the 204 systems, we apply a uniform scaling factor anchored to the recent observational compilation cited in the paper. This factor is varied within the modeling to produce the best match to the observed histogram, directly yielding the reported R_{1/2}. We will add a sensitivity test (new figure or table) showing how R_{1/2} and the α limit shift when the scaling factor is changed by ±30 % around the adopted value, thereby quantifying the impact on the final results. revision: yes

Circularity Check

1 steps flagged

α=0.2 adopted to match histograms then used to derive α≥0.15 lower limit from same models

specific steps
  1. fitted input called prediction [Abstract]
    "When a fraction of mass in microlenses α=0.2 is adopted, we find a good matching between the modeled histogram of mean microlensing magnifications for the images in our sample and the experimental histogram of microlensing magnifications. From the modeling of microlensing magnification histograms, we estimate the average half-light radius of the quasar source, R_{1/2}=5.4±2.7 light-days, and a lower limit to the mass fraction in microlenses, α≥0.15."

    α=0.2 is explicitly chosen to achieve the histogram match; the subsequent lower-limit estimate α≥0.15 is then extracted from the identical set of models, so the reported bound is not an independent prediction but a direct consequence of the fitting choice.

full rationale

The central estimates for source size and microlens mass fraction are obtained after tuning α to produce a match between modeled and observed magnification histograms. This matches the fitted-input-called-prediction pattern with partial circularity, but the thin-disk scaling draws on external measurements and crossing-time statistics appear independent of the α fit. No self-definitional equations, self-citation load-bearing steps, or ansatz smuggling are present in the provided text. The automatic modeling procedure itself is described without reducing to prior self-citations.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

Relies on standard microlensing theory plus fitted α and scaled source sizes plus domain assumptions on disk geometry and modeling accuracy; no new entities postulated.

free parameters (2)
  • microlens mass fraction α
    Set to 0.2 to match modeled and observed histograms; later used for lower bound α≥0.15.
  • thin-disk source-size scaling factor
    Applied to scale sizes from recent measurements before computing crossing times.
axioms (2)
  • domain assumption Thin-disk geometry describes the quasar accretion disk for all systems
    Used when scaling source sizes for crossing times and histograms.
  • domain assumption Automatic lens modeling produces statistically accurate magnification maps for the sample
    Required to generate histograms and probabilities without per-system checks.

pith-pipeline@v0.9.0 · 9575 in / 1447 out tokens · 83538 ms · 2026-05-07T14:59:53.678732+00:00 · methodology

discussion (0)

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