The Lens Made of Gravity: Gravitational Lensing and the Mapping of Dark Matter

The Hook: A Telescope Nobody Built

In 2022 a team published an image of a single star that shone when the universe was roughly 900 million years old. Seeing individual stars from that epoch is supposed to be impossible: they are far too faint to be resolved as point sources even by the James Webb Space Telescope. The star, now called "Earendel," was nevertheless visible — because a galaxy cluster sits between it and us, and that cluster's mass does not merely bend the light but focuses and amplifies it. The magnification is of order a few thousand.

Nobody built this telescope. It consists of nothing but mass and spacetime curvature, its lens is invisible, and its focus cannot be adjusted. And it is only the most spectacular application of an effect that has gone, over the last forty years, from curiosity to central tool of cosmology.

The reason is a property that gravitational lensing shares with no other method in astronomy: it measures mass without having to assume anything about the state of that mass. Every classical mass determination in the cosmos is indirect and assumption-laden. Infer a cluster's mass from the velocities of its member galaxies, and you must assume the cluster is in dynamical equilibrium. Compute it from the X-ray emission of the hot gas, and you must assume hydrostatic equilibrium. Estimate it from luminosity, and you must assume a mass-to-light ratio. Lensing assumes none of this. Light is deflected by mass whether that mass is at rest or rotating, luminous or dark, made of protons or of something we cannot yet name. The deflection is also achromatic: it is identical for red and blue light, which distinguishes it from any kind of scattering by dust.

That makes lensing the most direct instrument we have for answering a question open since the 1930s: exactly where does the mass we cannot see sit in the universe — and how much of it is there?

This article traces how a footnote to general relativity became a surveying technique, which three very different measurement regimes it produced, how it delivered the most immediate evidence for dark matter to date in a single galaxy cluster — and why the largest controversy of recent years, the so-called \(S_8\) tension, now looks less like new physics than like a lesson in measurement systematics.

Part 1: The Factor of Two

Soldner, Einstein, and the Half-Truth

The idea that light might be deflected by mass is older than relativity. Johann Georg von Soldner worked it out in a Newtonian framework in 1801 and published it in 1804: treat light as a stream of massive particles, and it follows a hyperbolic path as it passes a celestial body, with a calculable deflection angle.

Einstein arrived at exactly the same value in 1911 — by an entirely different route. He derived it from the equivalence principle alone, that is, from the indistinguishability of a gravitational field and an accelerated frame of reference, without the fully worked-out field equations. Only in 1915, with complete general relativity in hand, did he notice that this result is half the correct value. The reason: light deflection has two contributions of equal size. One comes from time dilation (what the equivalence principle alone delivers), the other from the curvature of spatial geometry — an effect a Newtonian calculation knows nothing about.

The correct deflection angle for a light ray passing a point mass \(M\) at impact parameter \(b\) is

$$\alpha = \frac{4GM}{c^2 b}$$

exactly twice the Newtonian value. For a ray grazing the solar limb, that comes to about 1.75 arcseconds.

1919: The Measurement That Made a Theory Famous

That factor of two was a gift to experimental physics. The two competing theories made predictions that differed not qualitatively but quantitatively by exactly a factor of two — the rare case of an experiment that can not only confirm a theory but cleanly discriminate between two.

On 29 May 1919 expeditions under Arthur Eddington and Frank Watson Dyson used a total solar eclipse to photograph the positions of stars near the occulted solar disk and compare them with night-time images of the same star fields. The result favored the relativistic value. By today's standards the precision was modest, the selection of plates used was later debated, and the finding was only brought to the per-mille level and beyond by radio interferometry and ultimately by astrometry missions. But the direction was clear, and it made Einstein world-famous overnight.

For the subject of this article, what matters is what nobody then regarded as a tool: if mass deflects light, then every mass distribution in the universe distorts the image of everything behind it. The deflection is not only a test — it is a measurable quantity.

Part 2: The Lens Equation and Its Three Regimes

Einstein's 1936 Note and Zwicky's Correction

The idea that one star could act as a lens for a star behind it was published by Einstein in Science in 1936 — a two-page note titled "Lens-Like Action of a Star by the Deviation of Light in the Gravitational Field," written at the urging of the Czech engineer Rudi W. Mandl, who kept pressing him on the point. Einstein worked the case through and closed with an assessment now famous for having been wrong: there is "no great chance" of ever observing this phenomenon, because the angular separations are far too small.

He had assumed a star as the lens. Fritz Zwicky showed as early as 1937, in two short Physical Review papers, that the calculation looks entirely different with a galaxy as the lens: the masses are many orders of magnitude larger, the angular separations therefore measurable, and the probability of finding such a system not negligible. Zwicky also named the application that remains the most important one — it would allow the masses of nebulae to be determined directly.

The Einstein Radius as a Characteristic Scale

When source, lens, and observer lie exactly on one line, a rotationally symmetric lens produces not an image but a ring. Its angular radius is the Einstein radius:

$$\theta_E = \sqrt{\frac{4GM}{c^2}\cdot\frac{D_{LS}}{D_L D_S}}$$

where \(D_L\), \(D_S\), and \(D_{LS}\) are the angular-diameter distances to the lens, to the source, and between the two. This quantity is the field's measuring tape. Inside \(\theta_E\) the distortions are dramatic; outside, they fall off quickly. And because \(\theta_E\) scales as the square root of mass, nature divides the effect into three practically separate regimes:

This three-way split is not merely didactic. The three regimes use different instruments, have different error sources, and are pursued by different communities — and they answer different questions. That they must agree in the end is one of cosmology's most powerful consistency arguments.

Part 3: Strong Lensing — From the Twin Quasar to the Clock of the Universe

1979: Two Quasars That Are the Same One

Forty years after Einstein's note and forty-two after Zwicky's correction, the observation happened. Dennis Walsh, Robert Carswell, and Ray Weymann reported in Nature in 1979 on the object pair 0957+561: two quasars just six arcseconds apart, with nearly identical spectra and identical redshift. Two coincidentally adjacent quasars with the same spectrum are infinitely improbable. This was one object whose light reached us by two paths, deflected by a foreground galaxy.

Nine years later a group led by Jacqueline Hewitt found in radio data the object MG 1131+0456 — the first Einstein ring, the almost perfectly axisymmetric configuration Einstein had written could practically not be expected. Today Einstein rings are routine; the most striking come from the Hubble and Webb telescopes, and some of the first Euclid images show them in objects previously thought well studied.

Refsdal's Idea: Time Delay as a Distance Measure

The conceptually most interesting application of strong lensing had been formulated by Sjur Refsdal as early as 1964, fifteen years before the first discovery. His observation: if the light of a variable source reaches us along several paths, those paths have different lengths — geometrically, and additionally because they traverse the lens's gravitational potential to different depths (Shapiro delay). If the source flickers, the same flicker appears in the different images offset in time. That time delay is measurable in days or months.

And, unlike almost everything in cosmology, it is an absolute quantity with the dimension of a time. From it follows a distance — and thus the Hubble constant, with no ladder of calibrated standard candles. That is a fully independent route to the very number at issue in The Cosmic Tension: Why the Universe Seems to Have Two Rates of Expansion.

The Mass-Sheet Degeneracy — and Why It Is the Problem

As elegant as the idea is, its catch is equally stubborn, and it has a name: the mass-sheet degeneracy. One can add a uniform sheet of mass to the lens model while simultaneously rescaling the lens's mass profile, and not a single observed image position or image shape changes. All imaging observations stay identical — but the inferred time delays, and hence \(H_0\), shift.

This degeneracy is not a measurement error but an exact mathematical symmetry of the lensing map. It cannot be broken by better images, only by additional information of a different kind. In practice that means stellar velocities in the lens galaxy: the stars' velocity dispersion depends on the actual mass profile and is unaffected by the degeneracy.

This is precisely the core of the TDCOSMO program. In the 2025 analysis (Birrer, Buckley-Geer, Cappellari, Courbin et al., Astronomy & Astrophysics 704, 2025), eight lens systems with measured time delays were analyzed jointly with eleven SLACS and four SL2S lenses without time delays in a hierarchical Bayesian framework; new spectroscopy from JWST, Keck, and the VLT supplied the kinematic constraints. The result for a flat \(\Lambda\)CDM model combined with Pantheon+ supernova data is

$$H_0 = 71.6^{+3.9}_{-3.3}\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$$

about 4.6% precision — and what is remarkable about it is less the number than the methodological decision behind it. The collaboration deliberately maximized the uncertainty from the mass-sheet degeneracy, parametrizing the mass profile so permissively that no tacit assumption could artificially shrink the error bar. The result is a larger uncertainty, but a more honest one.

Supernova Refsdal: A Prediction That Came True

In 1964 Refsdal had argued using a lensed supernova. In 2014 one was found — in a galaxy cluster, imaged four times over. The models predicted that a further image of the same explosion must appear about a year later at a different position. It appeared. In a field full of statistical arguments, this is one of the rare testable point predictions: a model of the invisible mass distribution of a galaxy cluster predicted the place and time of a future point of light, and nature delivered it.

Part 4: Microlensing — How You Rule Dark Matter Out

The Obvious Hypothesis

When it became clear in the 1970s and 1980s that galaxies are held together by more mass than their stars provide, the most obvious explanation was not a new particle but ordinary matter in an inconspicuous form: burnt-out white dwarfs, brown dwarfs, neutron stars, black holes. These objects acquired the collective name MACHOs (Massive Astrophysical Compact Halo Objects) — a deliberate counterpoint to the particle physicists' WIMPs.

Here lensing shows its strength as an instrument of exclusion. A compact object that happens to drift in front of a background star briefly amplifies its light. The images are far too close together to separate, but the brightness change is measurable: a symmetric, achromatic rise and fall over days to months. Both properties distinguish it from every known form of stellar variability.

If the Milky Way's halo consists of such objects, then observing millions of stars in the Magellanic Clouds must yield a calculable rate of such events. That is a quantitative prediction, and it was tested.

The Result: At Most a Side Contribution

In 5.7 years of observing the Large Magellanic Cloud, the MACHO project found 13 to 17 events — more than expected from the known stellar population, but far fewer than required for a halo made entirely of MACHOs. The inferred halo fraction was about 20% (95% interval 8% to 50%) for typical lens masses between 0.15 and 0.9 solar masses; a halo made entirely of MACHOs was excluded (Alcock et al., The Astrophysical Journal 542, 2000).

The EROS-2 survey tightened the limit considerably. With far greater monitoring statistics, Tisserand et al. (Astronomy & Astrophysics 469, 2007, 387–404) found so few candidates that compact objects across a broad mass range were limited to a single-digit percentage of the halo.

The significance of this finding is hard to overstate. The conservative, physically comfortable explanation of dark matter — ordinary matter in dark packaging — was dispatched not by a theoretical argument but by a count of non-events. A null result decided more here than most of the field's positive measurements. As a by-product, the same technique now provides the coldest method of exoplanet detection: a planet orbiting a lensing star produces a characteristic brief extra spike in the light curve.

Part 5: Weak Lensing — One Percent, a Million Times Over

The Measurement Problem

The cosmologically most powerful regime is the least spectacular. When light from a distant galaxy crosses the irregular gravitational field of large-scale structure, its image is not multiplied but only minimally distorted — a circular outline becomes slightly elliptical. The typical amplitude of this cosmic shear is a few percent, in many fields about 2%.

The problem: galaxies are intrinsically elliptical, with a scatter of roughly 30% — about fifteen times larger than the signal. For a single galaxy the measurement is therefore worthless. It becomes meaningful only statistically: average over many galaxies in neighboring directions on the sky, and the random intrinsic shapes average out, leaving the coherent distortion caused by the same foreground mass. One does not measure shapes but shape correlations.

The first detection of such a coherent distortion was delivered by J. Anthony Tyson, Francisco Valdes, and Richard Wenk in 1990 (The Astrophysical Journal 349, L1) for galaxy clusters: a systematic tangential alignment of background galaxies around the cluster center. Measuring the signal of general large-scale structure — with no conspicuous lens in the foreground — took another ten years. In 2000 four independent groups achieved it within a few weeks at three different 4-meter telescopes: Bacon, Refregier & Ellis; Kaiser, Wilson & Luppino; van Waerbeke et al.; Wittman et al. That simultaneity was no coincidence but the consequence of large-format CCD cameras having just become available.

The Four Systematics

Extracting a two-percent distortion from images in which every other error source is larger means having a systematics problem rather than a statistics problem. Four disturbances dominate:

1. The point spread function (PSF). Every optical system and every atmosphere distorts images in its own right, typically by more than the signal being measured. The PSF must be reconstructed from the stars in the image and removed — across the whole field, with its spatial variation.
2. Photometric redshifts. Shear depends on the distances to lens and source. For millions of faint galaxies there are no spectra, only colors. A systematic error in the mean redshift of a sample propagates directly into the cosmological result.
3. Intrinsic alignments. The assumption that "intrinsic shapes are randomly oriented" is false. Neighboring galaxies form in a shared tidal field and align physically with each other. This effect mimics shear and must be modeled — KiDS-Legacy uses a model that depends on halo mass and spectral type.
4. Baryonic feedback. On small angular scales, star formation and active galactic nuclei alter the matter distribution relative to pure dark matter. This affects precisely the scales with the highest signal strength, which is why many analyses deliberately discard them.

To this is added a null test of remarkable elegance: pure gravitational lensing produces, to leading order, only a so-called E-mode distortion pattern, never a B-mode pattern — a curl-like component is not physically expected. If an analysis finds B-modes, it has a systematic, and it knows this before a single cosmological parameter has been computed. That is a diagnostic criterion that works independently of the expected result.

\(S_8\): What Is Actually Measured

Cosmic shear is primarily sensitive to the product of two quantities: the matter density \(\Omega_m\) and the amplitude of matter fluctuations \(\sigma_8\). Pinning both down separately is hard; their combination is not. Hence the parameter

$$S_8 = \sigma_8 \sqrt{\frac{\Omega_m}{0.3}}$$

has become the standard quantity: it is what weak lensing actually measures well. In substance, \(S_8\) describes how lumpy the universe is today.

This is exactly where the controversy arose. The microwave background surveys the universe about 380,000 years after the Big Bang and predicts from that how lumpy it should be today. Planck gives \(\Omega_m = 0.315 \pm 0.007\) and \(\sigma_8 = 0.811 \pm 0.006\) (Astronomy & Astrophysics 641, A6, 2020), hence \(S_8 \approx 0.83\). The lensing surveys of the 2010s, by contrast, systematically measured less structure. The Dark Energy Survey's third-year analysis gave \(S_8 = 0.759^{+0.025}_{-0.023}\) — a value \(2.3\sigma\) below the CMB expectation (DES Collaboration, Physical Review D 105, 2022).

Two to three sigma is not a discovery, but several surveys pointed independently in the same direction. Proposed explanations ranged from decaying dark matter through massive neutrinos to modified gravity.

The Resolution: KiDS-Legacy 2025

The complete analysis of the Kilo-Degree Survey has changed the picture. Wright, Stölzner, Asgari et al. (Astronomy & Astrophysics 703, 2025) analyzed 1,347 square degrees in nine optical and near-infrared bands, with improved redshift estimation, extended calibration data, multi-band simulations, and the physically motivated intrinsic-alignment model. The result:

$$S_8 = 0.81^{+0.016}_{-0.021}$$

and thus agreement with Planck at the \(0.73\sigma\) level. The tension has, at least in this data set, vanished — not through new physics but through better calibration, while statistical constraining power rose by about 32%.

I am of the opinion that this trajectory is the real lesson of the chapter, regardless of how the question is finally decided: a two-to-three-sigma deviation that several independent analyses find in the same direction is not automatically a physical result. It can also mean that several groups treat the same systematic in similar ways — using the same redshift calibrations, say, or the same alignment model. Correlated systematics look like a reproduced finding. At the same time, caution cuts both ways: a single data set that resolves a tension is also only a single data set.

Part 6: The Bullet Cluster — The Argument That Is Hard to Get Around

All the findings so far share one weakness: they show that more gravity is present than visible mass. That leaves two ways out. Either there is invisible mass — or gravity itself follows a different law at small accelerations, as modified theories in the MOND tradition propose. Both account for flat rotation curves.

The galaxy cluster 1E 0657−558 at redshift \(z = 0.296\), known as the Bullet Cluster, separates the two possibilities. It is a collision of two clusters, and it physically separates their components, because they collide differently:

• The hot X-ray gas is a fluid. During the passage it collides with itself, is decelerated, heated, and left behind in the middle.
• The galaxies themselves behave like a collisionless gas of point masses. They fly through largely undisturbed and now sit ahead of the gas.
• Dark matter, if it is a collisionless particle, should travel with the galaxies, not with the gas.

The decisive measurement is where lensing sees the mass. Clowe, Bradač, Gonzalez, Markevitch, Randall, Jones, and Zaritsky published the answer in 2006 under the deliberately pointed title "A Direct Empirical Proof of the Existence of Dark Matter" (The Astrophysical Journal Letters 648, L109, 2006): the centroids of the lensing-reconstructed mass lie not with the gas but with the galaxies. The two mass peaks are detected at 8$\sigma$ and 12$\sigma$ significance respectively, and each is offset from its X-ray centroid by roughly 8$\sigma$.

This is the point at which the argument bites. In a cluster, most of the baryonic mass resides in the gas, not in the stars. Any theory in which gravity is generated by baryonic mass alone — whatever the force law — must find the center of mass where the gas is. It is demonstrably somewhere else. The Bullet Cluster does not formally rule out modified gravity; theories with additional fields can be adjusted. But it shifts the burden of proof: you need a substance that travels with the galaxies and not with the gas.

What makes the case so methodologically clean is the separation of the measurements. Gas position from X-rays, galaxy position from optical photometry, mass position from shape distortion — three different instruments, three different physical processes, no shared systematic error channel. The finding is not a number with an error bar but a spatial offset. Such geometric statements are far more robust against calibration errors than amplitude measurements.

Part 7: Substructure — How Cold Is Dark Matter?

The question "is there dark matter" has largely been settled by lensing. The more interesting question is now: what properties does it have? Here, too, lensing supplies a measurable quantity.

The difference between "cold" and "warm" dark matter is its velocity in the early universe. Warm particles move fast, stream out of small density perturbations, and thereby erase structure below a characteristic mass — the free-streaming length corresponds to a half-mode mass \(m_{hm}\), below which the number of halos collapses. Cold dark matter, by contrast, forms halos down to very small masses.

These small halos contain no stars. They are not merely dark; they are also impossible to weigh dynamically — there is nothing inside whose velocity could be measured. Lensing can nevertheless notice them, by a detour: in quadruply imaged quasars, the four images lie very close to the Einstein radius. The relative brightnesses of the four images are predicted by the smooth mass model. If a small subhalo happens to sit near one of the images, it shifts that image's magnification. Evaluated statistically across many systems, these flux-ratio anomalies become a census of halos that are never seen directly.

Gilman and colleagues carried out this analysis for eight quadruply imaged quasars (Monthly Notices of the Royal Astronomical Society, 2020) and obtained, at 95% confidence, \(m_{hm} < 10^{7.8}\,M_\odot\), corresponding to a lower bound of \(m_{DM} > 5.2\) keV for a thermal relic; particle masses of 4.0 keV and 3.0 keV were disfavored at likelihood ratios of 7:1 and 30:1. The analysis probed projected substructure in the range \(10^6\) to \(10^9\) solar masses.

The character of this statement deserves precision. No mass is weighed and no object is seen. From a deviation in brightness ratios, the existence of a population of objects that are unobservable by construction is inferred, and from the abundance of that population, a particle property. The chain is long, and it is model-dependent at every link. Accordingly, part of the interpretation is that such limits can move when assumptions about the lens galaxy or about halos along the line of sight change.

Part 8: The Present — Two Machines, One Method

Two projects are currently turning lensing from a specialist discipline into a mass survey.

Euclid, in space since July 2023, is purpose-built for weak lensing: a stable, sharp point spread function that is well characterized across the field and free of atmosphere, plus near-infrared photometry for redshifts. The first taste is instructive. In the Quick Data Release Q1, covering merely about 0.45% of the planned survey area, Lines, Collett, Nightingale, Walmsley, and colleagues identified roughly 500 high-quality strong lens candidates (Nature Astronomy 9, 2025). Extrapolated to the full mission, they expect over 100,000 strong lenses — an increase of two orders of magnitude over all systems previously known. The pipeline itself is notable: fine-tuned image classification models, then crowdsourcing by volunteers, then visual inspection by experts. None of the three stages would have sufficed alone.

On the schedule, precision is worth the effort, because it is often reported in abbreviated form. The first regular data release comes in two stages. In November 2026, "DR1-Foundation" delivers raw data, calibrated images, catalogues, and spectra over roughly 1,900 square degrees. The higher-level products needed for cosmology — galaxy catalogues and shear maps — follow only in mid-2027. Official cosmological results from Euclid DR1 are therefore not to be expected before mid-2027.

The Vera C. Rubin Observatory works from the ground, but with a different lever: area and repetition. The Legacy Survey of Space and Time officially began on 30 June 2026 and runs for ten years; the southern sky is covered every few nights, each position on the sky roughly 800 times, a new image about every 40 seconds, some 10 terabytes per night. For strong lensing, the repetition means something specific: time delays can only be measured by observing the same source again and again over years. Rubin is thus the first machine capable of running Refsdal's idea at industrial scale.

The two projects are complementary rather than redundant: Euclid supplies the image quality, Rubin the time series. That they use different instruments, sites, and analysis pipelines is not duplicated effort but the only way to detect correlated systematics — exactly the problem that kept the \(S_8\) story open for so long.

The Methods Compared

The last row is why the Bullet Cluster is so convincing: it measures not an amplitude but a position.

The Central Takeaway

The transferable lesson of this topic concerns not dark matter but the structure of measurement errors.

Over the last two decades, gravitational lensing has delivered two different kinds of results, and they have held up very differently. The amplitude measurements — how lumpy is the universe, how large is \(H_0\) — have shifted repeatedly, and the shifts almost never came from more data but from better-understood systematics: from redshift calibration, from intrinsic alignments, from a mathematical degeneracy made explicit. The geometric measurements, by contrast — does the mass sit with the gas or with the galaxies, does the predicted image appear at the predicted place — have held.

The difference is systematic, not accidental. An amplitude is vulnerable to every multiplicative error anywhere in the chain; a position or an offset is not. This is why a null test like the B-mode check is so valuable: it does not ask "is the result plausible?" but "does my setup show a signal that physically cannot exist?" — a question you can answer without already knowing the answer, and which therefore cannot be bent by expectation.

The practical prompt: next time you build a measurement — a benchmark, a latency measurement, a monitoring dashboard, a model evaluation — ask two questions that are rarely asked in this order. First: is there a quantity my setup would measure even though it must be zero? If so, measure it first and publish it alongside. A benchmark that also measures a known null load; an A/B test with an A/A arm; an evaluation set containing items whose correct answer is "don't know." Second: can my core claim be stated as a comparison or an offset rather than an absolute value? "Variant B is X faster than A, measured in the same run" is nearly immune to calibration error; "Variant B handles Y requests per second" is not.

And third, as a warning: when several independent teams obtain the same surprising result, check whether they are really independent — or whether they share a library, a calibration file, a preprocessed data set. Reproducibility across shared infrastructure is not reproducibility. The \(S_8\) tension occupied groups for ten years, and the best answer to it so far was not a new theory but a new calibration.

Cross-References in the Vault

This topic is closely interlocked with several articles already covered. Time-delay cosmography offers an independent route to exactly the dispute developed in The Cosmic Tension: Why the Universe Seems to Have Two Rates of Expansion; the standard model's other great tension, over the equation of state of dark energy, is treated in The Accelerating Cosmos: Type Ia Supernovae, Dark Energy, and Whether Einstein's Constant Is Wavering. The Bullet Cluster and the mass maps of galaxy clusters belong to the same program of surveying the invisible as the imaging of an event horizon in The Shadow of the Invisible: Black Holes, the Event Horizon, and the Event Horizon Telescope, where light deflection in strong fields is likewise the actual instrument. The idea of extracting a cosmic quantity from an absolutely measured time rather than from a calibration ladder links Refsdal's time delay to the methods in The Clocks Made of Neutrons: Pulsars, Pulsar Timing Arrays, and the Hum of Spacetime. How to extract a distance from a light signal without a standard candle is shown by the dispersion measure in Milliseconds from the Cosmos: Fast Radio Bursts and the Universe's Missing Matter, and another messenger that reveals matter otherwise invisible is the subject of Messengers from the Hidden: Neutrinos, the Ghost Particles, and the Birth of a New Astronomy. Where the heavy elements come from whose luminosity we measure in lensed images is settled in Cosmic Gold: Neutron Star Collisions, the r-Process, and the Origin of the Heavy Elements. That discovering 100,000 lens systems is now a job for image classification models leads directly to The Error That Learns Backward: Backpropagation, the Chain Rule, and the Algorithm Behind Deep Learning and to the question of how well such models generalize in More Parameters, Fewer Errors: Double Descent, Grokking, and the Riddle of Generalization.

Something to Think About

The Bullet Cluster convinces because it measures a position and not an amplitude; the \(S_8\) tension occupied theorists for a decade because it was an amplitude. Where in your own work do you rest an important claim on an absolute value, when you could just as well state it as a difference, an offset, or a ranking within the same measurement run? And which signal in your setup ought to be exactly zero — are you measuring it?

Sources

• A. Einstein, "Lens-Like Action of a Star by the Deviation of Light in the Gravitational Field," Science 84 (1936), 2188, 506–507: https://www.science.org/doi/10.1126/science.84.2188.506
• F. Zwicky, "Nebulae as Gravitational Lenses," Physical Review 51 (1937), 290: https://journals.aps.org/pr/abstract/10.1103/PhysRev.51.290
• D. Walsh, R. F. Carswell, R. J. Weymann, "0957 + 561 A, B: twin quasistellar objects or gravitational lens?," Nature 279 (1979), 381–384: https://www.nature.com/articles/279381a0
• J. A. Tyson, F. Valdes, R. A. Wenk, "Detection of Systematic Gravitational Lens Galaxy Image Alignments: Mapping Dark Matter in Galaxy Clusters," The Astrophysical Journal 349 (1990), L1: https://ui.adsabs.harvard.edu/abs/1990ApJ...349L...1T/abstract
• A. Refregier, "Weak Gravitational Lensing by Large-Scale Structure," Annual Review of Astronomy and Astrophysics 41 (2003), 645–668 (review containing the four independent first detections of cosmic shear in 2000): https://www.annualreviews.org/doi/10.1146/annurev.astro.41.111302.102207
• C. Alcock et al. (MACHO Collaboration), "The MACHO Project: Microlensing Results from 5.7 Years of Large Magellanic Cloud Observations," The Astrophysical Journal 542 (2000), 281–307: https://arxiv.org/abs/astro-ph/0001272
• P. Tisserand et al. (EROS-2 Collaboration), "Limits on the Macho content of the Galactic Halo from the EROS-2 Survey of the Magellanic Clouds," Astronomy & Astrophysics 469 (2007), 387–404, DOI 10.1051/0004-6361:20066017: https://ui.adsabs.harvard.edu/abs/2007A&A...469..387T
• D. Clowe, M. Bradač, A. H. Gonzalez, M. Markevitch, S. W. Randall, C. Jones, D. Zaritsky, "A Direct Empirical Proof of the Existence of Dark Matter," The Astrophysical Journal Letters 648 (2006), L109–L113, DOI 10.1086/508162: https://chandra.harvard.edu/photo/2006/1e0657/media/paper.pdf
• Planck Collaboration, "Planck 2018 results. VI. Cosmological parameters," Astronomy & Astrophysics 641 (2020), A6, DOI 10.1051/0004-6361/201833910: https://www.aanda.org/articles/aa/full_html/2020/09/aa33910-18/aa33910-18.html
• DES Collaboration (A. Amon, L. F. Secco et al.), "Dark Energy Survey Year 3 Results: Cosmology from Cosmic Shear and Robustness to Modeling Uncertainty," Physical Review D 105 (2022), 023515: https://arxiv.org/abs/2105.13544
• A. H. Wright, B. Stölzner, M. Asgari et al., "KiDS-Legacy: Cosmological constraints from cosmic shear with the complete Kilo-Degree Survey," Astronomy & Astrophysics 703 (2025), DOI 10.1051/0004-6361/202554908: https://www.aanda.org/articles/aa/full_html/2025/11/aa54908-25/aa54908-25.html
• S. Birrer, E. Buckley-Geer, M. Cappellari, F. Courbin et al. (TDCOSMO Collaboration), "TDCOSMO 2025: Cosmological constraints from strong lensing time delays," Astronomy & Astrophysics 704 (2025), DOI 10.1051/0004-6361/202555801: https://www.aanda.org/articles/aa/full_html/2025/12/aa55801-25/aa55801-25.html
• D. Gilman, S. Birrer, A. Nierenberg, T. Treu, X. Du, A. Benson, "Warm dark matter chills out: constraints on the halo mass function and the free-streaming length of dark matter with eight quadruple-image strong gravitational lenses," Monthly Notices of the Royal Astronomical Society (2020): https://arxiv.org/abs/1908.06983
• N. E. P. Lines, T. E. Collett, J. W. Nightingale, M. Walmsley et al., "The revolution in strong lensing discoveries from Euclid," Nature Astronomy 9 (2025), DOI 10.1038/s41550-025-02616-5: https://www.nature.com/articles/s41550-025-02616-5
• ESA Euclid, "DR1 Timeline" (two-stage release: DR1-Foundation November 2026 over ~1,900 deg², higher-level products mid-2027): https://www.cosmos.esa.int/web/euclid/dr1-timeline
• NSF–DOE Vera C. Rubin Observatory, "Action! Rubin LSST Begins" (official start of the Legacy Survey of Space and Time on 30 June 2026): https://rubinobservatory.org/news/action-rubin-lsst-begins
• B. Welch et al., "JWST Imaging of Earendel, the Extremely Magnified Star at Redshift z = 6.2," The Astrophysical Journal Letters 940 (2022), L1, DOI 10.3847/2041-8213/ac9d39; discovery: B. Welch et al., Nature 603 (2022), 815–818: https://www.nature.com/articles/s41586-022-04449-y

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