680:, detection of a galaxy-galaxy shear signal requires one to measure the shapes of background source galaxies, and then look for statistical shape correlations (specifically, source galaxy shapes should be aligned tangentially, relative to the lens center.) In principle, this signal could be measured around any individual foreground lens. In practice, however, due to the relatively low mass of field lenses and the inherent randomness in intrinsic shape of background sources (the "shape noise"), the signal is impossible to measure on a galaxy-by-galaxy basis. However, by combining the signals of many individual lens measurements together (a technique known as "stacking"), the signal-to-noise ratio will improve, allowing one to determine a statistically significant signal, averaged over the entire lens set.
278:(PSF) due to instrumental and atmospheric effects, which causes the observed images to be smeared relative to the "true sky". This smearing tends to make small objects more round, destroying some of the information about their true ellipticity. As a further complication, the PSF typically adds a small level of ellipticity to objects in the image, which is not at all random, and can in fact mimic a true lensing signal. Even for the most modern telescopes, this effect is usually at least the same order of magnitude as the gravitational lensing shear, and is often much larger. Correcting for the PSF requires building for the telescope a model for how it varies across the field. Stars in our own galaxy provide a direct measurement of the PSF, and these can be used to construct such a model, usually by
33:
315:
567:
2817:", makes it possible to map out the 3D distribution of mass. Because the third dimension involves not only distance but cosmic time, tomographic weak lensing is sensitive not only to the matter power spectrum today, but also to its evolution over the history of the universe, and the expansion history of the universe during that time. This is a much more valuable cosmological probe, and many proposed experiments to measure the properties of
2735:. Detecting the extremely faint cosmic shear signal requires averaging over many background galaxies, so surveys must be both deep and wide, and because these background galaxies are small, the image quality must be very good. Measuring the shear correlations at small scales also requires a high density of background objects (again requiring deep, high quality data), while measurements at large scales push for wider surveys.
2848:. Even though there are no distinct resolved sources, perturbations on the origining surface are sheared in a similar way to galaxy weak lensing, resulting in changes to the power spectrum and statistics of the observed signal. Since the source plane for the CMB and high-redshift diffuse 21 cm are at higher redshift than resolved galaxies, the lensing effect probes cosmology at higher redshifts than galaxy lensing.
244:
271:
intrinsic ellipticity is much greater than the shear (by a factor of 3-300, depending on the foreground mass). The measurements of many background galaxies must be combined to average down this "shape noise". The orientation of intrinsic ellipticities of galaxies should be almost entirely random, so any systematic alignment between multiple galaxies can generally be assumed to be caused by lensing.
2283:
3653:; Schneider, Donald P.; Smith, J. Allyn; Stoughton, Chris; Szalay, Alexander S.; Szokoly, Gyula P.; Thakar, Aniruddha R.; Vogeley, Michael S.; Waddell, Patrick; Weinberg, David H.; York, Donald G.; The SDSS Collaboration (September 2000). "Weak Lensing with Sloan Digital Sky Survey Commissioning Data: The Galaxy-Mass Correlation Function to 1 H Mpc".
738:) mass to light ratio – again due to the insensitivity of lensing to matter type. Assuming that luminous matter can trace dark matter, this quantity is of particular importance, since measuring the ratio of luminous (baryonic) matter to total matter can provide information regarding the overall ratio of baryonic to dark matter in the universe.
768:, which act as tracers of (among other things) stellar population, galaxy age, and local mass environment. By separating lens galaxies based on these properties, and then further segregating samples based on redshift, it is possible to use galaxy-galaxy lensing to see how several different types of galaxies evolve through time.
1690:
1056:
1976:
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342:, this effect can cause dramatic distortions of a background source object detectable by eye such as multiple images, arcs, and rings (cluster strong lensing). More generally, the effect causes small, but statistically coherent, distortions of background sources on the order of 10% (cluster weak lensing).
699:
Using techniques similar to those in cluster-scale lensing, galaxy-galaxy lensing can provide information about the shape of mass density profiles, though these profiles correspond to galaxy-sized objects instead of larger clusters or groups. Given a high enough number density of background sources,
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Instead of running statistical analysis on the distortion of galaxies based on the assumption of a positive weak lensing that usually reveals locations of positive mass "dark clusters", these researchers propose to locate "negative mass clumps" using negative weak lensing, i.e. where the deformation
659:
have greatly increased the observed number density of both background source and foreground lens galaxies, allowing for a much more robust statistical sample of galaxies, making the lensing signal much easier to detect. Today, measuring the shear signal due to galaxy-galaxy lensing is a widely used
578:
of the cluster in question. Lensing mass maps can also potentially reveal "dark clusters," clusters containing overdense concentrations of dark matter but relatively insignificant amounts of baryonic matter. Comparison of the dark matter distribution mapped using lensing with the distribution of the
4374:
Van
Waerbeke, L.; Mellier, Y.; Erben, T.; Cuillandre, J.C.; Bernardeau, F.; Maoli, R.; Bertin, E.; McCracken, H.J.; Le Fèvre, O.; Fort, B.; Dantel-Fort, M.; Jain, B.; Schneider, P. (June 2000). "Detection of correlated galaxy ellipticities from CFHT data: first evidence for gravitational lensing by
3648:
Fischer, Philippe; McKay, Timothy A.; Sheldon, Erin; Connolly, Andrew; Stebbins, Albert; Frieman, Joshua A.; Jain, Bhuvnesh; Joffre, Michael; Johnston, David; Bernstein, Gary; Annis, James; Bahcall, Neta A.; Brinkmann, J.; Carr, Michael A.; Csabai, István; Gunn, James E.; Hennessy, G. S.; Hindsley,
270:
of the background galaxies and construct a statistical estimate of their systematic alignment. The fundamental problem is that galaxies are not intrinsically circular, so their measured ellipticity is a combination of their intrinsic ellipticity and the gravitational lensing shear. Typically, the
297:
are unavailable. Redshift information is also important in separating the background source population from other galaxies in the foreground, or those associated with the mass responsible for the lensing. With no redshift information, the foreground and background populations can be split by an
748:
is finite, an observer on the Earth will see distant galaxies not as they look today, but rather as they appeared at some earlier time. By restricting the lens sample of a galaxy-galaxy lensing study to lie at only one particular redshift, it is possible to understand the mass properties of the
654:
J.A. Tyson and collaborators first postulated the concept of galaxy-galaxy lensing in 1984, though the observational results of their study were inconclusive. It was not until 1996 that evidence of such distortion was tentatively discovered, with the first statistically significant results not
318:
The effects of foreground galaxy cluster mass on background galaxy shapes. The upper left panel shows (projected onto the plane of the sky) the shapes of cluster members (in yellow) and background galaxies (in white), ignoring the effects of weak lensing. The lower right panel shows this same
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Sheldon, Erin S.; Johnston, David E.; Frieman, Joshua A.; Scranton, Ryan; McKay, Timothy A.; Connolly, A. J.; Budavári, Tamás; Zehavi, Idit; Bahcall, Neta A.; Brinkmann, J.; Fukugita, Masataka (May 2004). "The Galaxy-Mass
Correlation Function Measured from Weak Lensing in the Sloan Digital Sky
426:
and collaborators published the first sample of galaxy clusters detected via their lensing signals, completely independent of their baryon content. Clusters discovered through lensing are subject to mass selection effects because the more massive clusters produce lensing signals with higher
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between the points where stars appear on the image. This model can then be used to reconstruct the "true" ellipticities from the smeared ones. Ground-based and space-based data typically undergo distinct reduction procedures due to the differences in instruments and observing conditions.
1406:
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Gavazzi, Raphaël; Treu, Tommaso; Rhodes, Jason D.; Koopmans, Léon V. E.; Bolton, Adam S.; Burles, Scott; Massey, Richard J.; Moustakas, Leonidas A. (September 2007). "The Sloan Lens ACS Survey. IV. The Mass
Density Profile of Early-Type Galaxies out to 100 Effective Radii".
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field galaxies that existed during this earlier time. Comparing the results of several such redshift-restricted lensing studies (with each study encompassing a different redshift), one can begin to observe changes in the mass features of galaxies over a period of several
759:
Lens redshift is not the only quantity of interest that can be varied when studying mass differences between galaxy populations, and often there are several parameters used when segregating objects into galaxy-galaxy lens stacks. Two widely used criteria are galaxy
1964:
correlation functions. In analogy with electric and magnetic fields, the E-mode field is curl-free and the B-mode field is divergence-free. Because gravitational lensing can only produce an E-mode field, the B-mode provides yet another test for systematic errors.
2278:{\displaystyle \langle M_{ap}^{2}\rangle (\theta )=\int _{0}^{2\theta }{\frac {\phi d\phi }{\theta ^{2}}}\leftT_{+}\left({\frac {\phi }{\theta }}\right)=\int _{0}^{2\theta }{\frac {\phi d\phi }{\theta ^{2}}}\leftT_{-}\left({\frac {\phi }{\theta }}\right)}
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scenario, but includes the effects of lensing. The middle panel shows a 3-d representation of the positions of cluster and source galaxies, relative to the observer. Note that the background galaxies appear stretched tangentially around the cluster.
2813:) to divide the survey into multiple redshift bins. The low-redshift bins will only be lensed by structures very near to us, while the high-redshift bins will be lensed by structures over a wide range of redshift. This technique, dubbed "cosmic
2742:
cameras enabled surveys of the necessary size and quality. In 2000, four independent groups published the first detections of cosmic shear, and subsequent observations have started to put constraints on cosmological parameters (particularly the
1223:
785:
usually used in cluster and galaxy lensing does not always work in this regime, because structures can be elongated along the line of sight. Instead, the distortion can be derived by assuming that the deflection angle is always small (see
646:). Of the three typical mass regimes in weak lensing, galaxy-galaxy lensing produces a "mid-range" signal (shear correlations of ~1%) that is weaker than the signal due to cluster lensing, but stronger than the signal due to cosmic shear.
3168:
Wittman, D.; Dell'Antonio, I.P.; Hughes, J.P.; Margoniner, V.E.; Tyson, J.A.; Cohen, J.G.; Norman, D. (May 2006). "First
Results on Shear-selected Clusters from the Deep Lens Survey: Optical Imaging, Spectroscopy, and X-Ray Follow-up".
1412:
520:
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is impossible to detect in a single background source. However, even in these cases, the presence of the foreground mass can be detected, by way of a systematic alignment of background sources around the lensing mass.
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can be related to projections (integrals with certain weight functions) of the dark matter density correlation function, which can be predicted from theory for a cosmological model through its
Fourier transform, the
622:. In order for lensing clusters to be a precision probe of cosmology in the future, the projection effects and the scatter in the lensing mass-observable relation need to be thoroughly characterized and modeled.
370:
who discovered giant luminous arcs in a survey of galaxy clusters in the late 1970s. Lynds and
Petrosian published their findings in 1986 without knowing the origin of the arcs. In 1987, Genevieve Soucail of the
3819:
Parker, Laura C.; Hoekstra, Henk; Hudson, Michael J.; van
Waerbeke, Ludovic; Mellier, Yannick (November 2007). "The Masses and Shapes of Dark Matter Halos from Galaxy-Galaxy Lensing in the CFHT Legacy Survey".
1284:
1279:, which measure the mean product of the shear at two points as a function of the distance between those points. Because there are two components of shear, three different correlation functions can be defined:
247:
Distortions of the type produced by lensing, acting on circles and a distribution of ellipses similar to that of real galaxies. The distortion shown here is greatly exaggerated relative to real astronomical
533:
for the cluster, which can be determined by using a reconstructed mass distribution or optical or X-ray data, a model can be fit to the shear profile as a function of clustrocentric radius. For example, the
4232:
Wittman, David; Tyson, J. A.; Kirkman, David; Dell'Antonio, Ian; Bernstein, Gary (May 2000). "Detection of weak gravitational lensing distortions of distant galaxies by cosmic dark matter at large scales".
4059:
Samuroff, S.; Mandelbaum, R.; Blazek, J.; Campos, A.; MacCrann, N.; Zacharegkas, G.; Amon, A.; Prat, J.; Singh, S.; Elvin-Poole, J.; Ross, A. J.; Alarcon, A.; Baxter, E.; Bechtol, K.; Becker, M. R. (2023).
704:). Since the effects of lensing are insensitive to the matter type, a galaxy-galaxy mass density profile can be used to probe a wide range of matter environments: from the central cores of galaxies where
781:
also produces intrinsic alignment (IA) - an observable pattern of alignments in background galaxies. This distortion is only ~0.1%-1% - much more subtle than cluster or galaxy-galaxy lensing. The
2622:
An exact decomposition thus requires knowledge of the shear correlation functions at zero separation, but an approximate decomposition is fairly insensitive to these values because the filters
235:
is thus an intrinsically statistical measurement, but it provides a way to measure the masses of astronomical objects without requiring assumptions about their composition or dynamical state.
558:. Individual mass estimates from weak lensing can only be derived for the most massive clusters, and the accuracy of these mass estimates are limited by projections along the line of sight.
1685:{\displaystyle \xi _{\times +}(\Delta \theta )=\xi _{+\times }(\Delta \theta )=\langle \gamma _{+}({\vec {\theta }})\gamma _{\times }({\vec {\theta }}+{\vec {\Delta \theta }})\rangle }
4696:
Kitamura, T.; Izumi, K.; Nakajima, K.; Hagiwara, C.; Asada, H. (April 2014). "Microlensed image centroid motions by an exotic lens object with negative convergence or negative mass".
1051:{\displaystyle {\frac {\partial \beta _{i}}{\partial \theta _{j}}}=\delta _{ij}+\int _{0}^{r_{\infty }}drg(r){\frac {\partial ^{2}\Phi ({\vec {x}}(r))}{\partial x^{i}\partial x^{j}}}}
3400:
Clowe, D.; Gonzalez, A. H.; Markevitch, M. (April 2004). "Weak-Lensing Mass
Reconstruction of the Interacting Cluster 1E 0657-558: Direct Evidence for the Existence of Dark Matter".
1756:
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Robert B.; Hull, Charles; Ivezić, Željko; Knapp, G. R.; Limmongkol, Siriluk; Lupton, Robert H.; Munn, Jeffrey A.; Nash, Thomas; Newberg, Heidi Jo; Owen, Russell; Pier, Jeffrey R.;
1954:
1883:
2738:
While weak lensing of large-scale structure was discussed as early as 1967, due to the challenges mentioned above, it was not detected until more than 30 years later when large
550:
and the redshift distribution of the background galaxies is also necessary for estimation of the mass and size from a model fit; these redshifts can be measured precisely using
443:
can be recovered from the measurement of the ellipticities of the lensed background galaxies through techniques that can be classified into two types: direct reconstruction and
1783:
4643:
Izumi, K.; Hagiwara, C.; Nakajima, K.; Kitamura, T.; Asada, H. (July 2013). "Gravitational lensing shear by an exotic lens object with negative convergence or negative mass".
846:
1813:
817:
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121:
3929:"Galaxy halo masses and satellite fractions from galaxy-galaxy lensing in the Sloan Digital Sky Survey: stellar mass, luminosity, morphology and environment dependencies"
2709:
1924:
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to the lenses and background sources are important for converting the lensing observables to physically meaningful quantities. These distances are often estimated using
3310:
Schneider, P.; Seitz, C. (February 1995). "Steps towards nonlinear cluster inversion through gravitational distortions. 1: Basic considerations and circular clusters".
2680:
2650:
2613:
2583:
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Because large-scale cosmological structures do not have a well-defined location, detecting cosmological gravitational lensing typically involves the computation of
1529:{\displaystyle \xi _{\times \times }(\Delta \theta )=\langle \gamma _{\times }({\vec {\theta }})\gamma _{\times }({\vec {\theta }}+{\vec {\Delta \theta }})\rangle }
1109:
870:
1258:
522:
where λ is an arbitrary constant. This degeneracy can be broken if an independent measurement of the magnification is available because the magnification is not
1078:
655:
published until the year 2000. Since those initial discoveries, the construction of larger, high resolution telescopes and the advent of dedicated wide field
263:. The convergence term magnifies the background objects by increasing their size, and the shear term stretches them tangentially around the foreground mass.
1785:
is the component at 45°. These correlation functions are typically computed by averaging over many pairs of galaxies. The last correlation function,
618:
the mass-observable relation via a stacked weak lensing signal around an ensemble of clusters, although this relation is expected to have an intrinsic
465:
222:
While the presence of any mass bends the path of light passing near it, this effect rarely produces the giant arcs and multiple images associated with
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2423:
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570:
Image of the Bullet
Cluster from the Hubble Space Telescope with total mass contours (dominated by dark matter) from a lensing analysis overlaid.
209:
396:
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that distorts the images of background objects (usually galaxies) near a foreground mass. The transformation can be split into two terms, the
1815:, is not affected at all by lensing, so measuring a value for this function that is inconsistent with zero is often interpreted as a sign of
1401:{\displaystyle \xi _{++}(\Delta \theta )=\langle \gamma _{+}({\vec {\theta }})\gamma _{+}({\vec {\theta }}+{\vec {\Delta \theta }})\rangle }
379:
and proposed a gravitational lensing interpretation. The first cluster weak lensing analysis was conducted in 1990 by J. Anthony Tyson of
4511:. 17th International Conference on Supersymmetry and the Unification of Fundamental Interactions. Boston: American Institute of Physics.
2957:"Intrinsic galaxy alignments from the 2SLAQ and SDSS surveys: luminosity and redshift scalings and implications for weak lensing surveys"
782:
3565:
Tyson, J. A.; Valdes, F.; Jarvis, J. F.; Mills, A. P. Jr. (June 1984). "Galaxy mass distribution from gravitational light deflection".
418:
surveys). The sample of galaxy clusters studied with lensing was thus subject to various selection effects; for example, only the most
3096:
Soucail, G.; Mellier, Y.; Fort, B.; Mathez, G.; Hammer, F. (October 1987). "Further data on the blue ring-like structure in A 370".
4144:
Schneider, P.; van
Waerbekere, L.; Kilbinger, M.; Mellier, Y. (December 2002). "Analysis of two-point statistics of cosmic shear".
2911:). Such negative mass clumps would be located elsewhere than assumed dark clusters, as they would reside at the center of observed
141:
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260:
574:
Cluster mass estimates determined by lensing are valuable because the method requires no assumption about the dynamical state or
126:
2723:
makes it a potentially powerful probe of cosmological parameters, especially when combined with other observations such as the
452:
363:
778:
688:
Galaxy-galaxy lensing (like all other types of gravitational lensing) is used to measure several quantities pertaining to
610:
should be able to constrain cosmological parameters. In practice, however, projections along the line of sight cause many
643:
202:
2955:
Hirata, C.M.; Mandelbaum, R.; Ishak, M.; Seljak, U.; Nichol, R.; Pimbblet, K.A.; Ross, N.P.; Wake, D. (November 2007).
2887:
As the presence of exotic matter would bend spacetime and light differently than positive mass, a Japanese team at the
535:
4062:"The Dark Energy Survey Year 3 and eBOSS: Constraining galaxy intrinsic alignments across luminosity and colour space"
3347:
Metzler, C.A.; White, M.; Norman, M.; Loken, C. (July 1999). "Weak Gravitational Lensing and Cluster Mass Estimates".
2809:
For current and future surveys, one goal is to use the redshifts of the background galaxies (often approximated using
634:, in which the foreground object responsible for distorting the shapes of background galaxies is itself an individual
602:
In principle, since the number density of clusters as a function of mass and redshift is sensitive to the underlying
423:
4353:
Kaiser, Nick; Wilson, Gillian; Luppino, Gerard (March 2000). "Large-Scale Cosmic Shear Measurements". p. 3338.
4749:
Nakajima, K.; Izumi, K.; Asada, H. (October 2014). "Negative time delay of light by a gravitational concave lens".
591:. The Bullet Cluster data provide constraints on models relating light, gas, and dark matter distributions such as
2936:
2895:
of galaxies is interpreted as being due to a diverging lensing effect producing radial distortions (similar to a
849:
787:
668:, often used in parallel with other measurements in determining physical characteristics of foreground galaxies.
171:
44:
3927:
Mandelbaum, Rachel; Seljak, Uroš; Kauffmann, Guinevere; Hirata, Christopher M.; Brinkmann, Jonathan (May 2006).
3257:
Bartelmann, M.; Narayan, R.; Seitz, S.; Schneider, P. (June 1996). "Maximum-likelihood Cluster Reconstruction".
4877:
2920:
195:
4197:
Gunn, James E. (December 1967). "On the Propagation of Light in Inhomogeneous Cosmologies. I. Mean Effects".
3127:"Detection of systematic gravitational lens galaxy image alignments - Mapping dark matter in galaxy clusters"
1727:
1929:
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48:
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790:). As in the thin lens case, the effect can be written as a mapping from the unlensed angular position
3098:
1761:
52:
822:
753:, leading towards a better understanding of the evolution of mass on the smallest cosmological scales.
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in cosmology as an alternative interpretation to dark matter, which classically has a positive mass.
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1788:
793:
460:
286:
256:
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Kaiser, N.; Squires, G. (February 1993). "Mapping the dark matter with weak gravitational lensing".
2904:
2896:
3987:
3453:
Hoekstra, H.; Jain, B. (May 2008). "Weak Gravitational Lensing and its Cosmological Applications".
3016:
Diaferio, A.; Schindler, S.; Dolag, K. (February 2008). "Clusters of Galaxies: Setting the Stage".
1218:{\displaystyle g(r)=2r\int _{r}^{r_{\infty }}\left(1-{\frac {r^{\prime }}{r}}\right)W(r^{\prime })}
750:
456:
131:
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2778:
2748:
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a typical galaxy-galaxy mass density profile can cover a wide range of distances (from ~1 to ~100
338:. The gravitational fields of these clusters deflect light-rays traveling near them. As seen from
661:
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411:
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1896:
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Mbarek, S.; Paranjape, M. B. (November 2014). "Negative mass bubbles in de Sitter space-time".
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2826:
765:
226:. Most lines of sight in the universe are thoroughly in the weak lensing regime, in which the
3508:"Improved optical mass tracer for galaxy clusters calibrated using weak lensing measurements"
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3603:; Blanford, Roger D.; Smail, Ian (August 1996). "Weak Gravitational Lensing by Galaxies".
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2927:
proposing exotic matter of negative mass as an alternative interpretation to dark matter.
1234:
8:
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of field galaxies. Specifically, the quantity measured through lensing is the total (or
731:
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baryons using optical and X-ray data reveals the interplay of the dark matter with the
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Lynds, R.; Petrosian, V. (September 1986). "Giant Luminous Arcs in Galaxy Clusters".
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proposed to use "negative" weak gravitational lensing related to such negative mass.
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448:
399:. Lensing has been used as a tool to investigate a tiny fraction of the thousands of
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4105:"The Correlation Function of Galaxy Ellipticities Produced By Gravitational Lensing"
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Historically, lensing analyses were conducted on galaxy clusters detected via their
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Hoekstra, H.; Franx, M.; Kuijken, K.; Carlberg, R. G.; Yee, H. K. C. (April 2003).
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515:{\displaystyle \kappa \rightarrow \kappa ^{\prime }=\lambda \kappa +(1-\lambda )}
444:
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323:
93:
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Hossenfelder, S. (15 August 2008). "A Bi-Metric Theory with Exchange Symmetry".
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Reyes, R.; Mandelbaum, R.; Hirata, C.; Bahcall, N.; Seljak, U. (February 2008).
852:
of the transform can be written as an integral over the gravitational potential
362:
The effects of cluster strong lensing were first detected by Roger Lynds of the
4780:
4727:
4674:
4571:
4485:
2869:
2546:{\displaystyle T_{-}(x)=576\int _{0}^{\infty }{\frac {dt}{t^{3}}}J_{4}(xt)^{2}}
2412:{\displaystyle T_{+}(x)=576\int _{0}^{\infty }{\frac {dt}{t^{3}}}J_{0}(xt)^{2}}
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3759:
3256:
2908:
2857:
635:
630:
Galaxy-galaxy lensing is a specific type of weak (and occasionally strong)
551:
314:
294:
459:, where the cluster surface mass density κ can be determined only up to a
4389:
4359:
4312:
4247:
4158:
3945:
3888:
3778:
3721:
3667:
3617:
3414:
3361:
3324:
3271:
3183:
2975:
2822:
2818:
761:
713:
566:
335:
327:
303:
102:
4594:
3505:
2833:, and the Legacy Survey of Space and Time (LSST) to be conducted by the
1231:, which defines the efficiency of lensing for a distribution of sources
587:
components. A notable example of such a joint analysis is the so-called
4293:
2954:
2830:
2814:
1263:
As in the thin-lens approximation, the Jacobian can be decomposed into
723:
392:
347:
343:
267:
156:
97:
79:
75:
70:
4862:
4816:
4526:
4413:
375:
and her collaborators presented data of a blue ring-like structure in
4595:"Negative mass hypothesis in cosmology and the nature of dark energy"
4264:
665:
615:
383:
and collaborators. Tyson et al. detected a coherent alignment of the
376:
266:
To measure this tangential alignment, it is necessary to measure the
4352:
3599:
3095:
4440:
4294:
Bacon, David; Refregier, Alexandre; Ellis, Richard (October 2000).
4218:
4129:
4104:
4078:
3905:
3851:
3738:
3684:
3634:
3586:
3431:
3378:
3346:
3288:
3243:
3200:
3153:
3126:
619:
547:
530:
331:
4763:
4710:
4657:
4554:
4517:
4468:
4296:"Detection of weak gravitational lensing by large-scale structure"
4004:
3834:
3524:
3467:
3399:
3030:
2923:. Such test based on negative weak lensing could help to falsify
2900:
440:
384:
735:
705:
407:
274:
Another major challenge for weak lensing is correction for the
4058:
3015:
4695:
4642:
1956:
are not independent, and they can be decomposed further into
339:
243:
4802:
Piran, Tsvi (November 1997). "On Gravitational Repulsion".
1893:
Because they both depend on a single scalar density field,
689:
592:
580:
354:
are among the most prominent examples of lensing clusters.
161:
730:, galaxy-galaxy lensing can also provide insight into the
422:
clusters were investigated. In 2006, David Wittman of the
3124:
677:
334:
with approximately 80% of cluster content in the form of
2806:) that are competitive with other cosmological probes.
309:
726:(averaged over the entire galaxy stack) in a specific
2781:
2751:
2688:
2658:
2628:
2591:
2561:
2426:
2292:
1979:
1968:
The E-mode correlation function is also known as the
1932:
1899:
1861:
1828:
1791:
1764:
1730:
1700:
1543:
1415:
1287:
1237:
1119:
1090:
1066:
880:
858:
825:
796:
468:
3988:"Cosmology with cosmic shear observations: a review"
3125:
Tyson, J.A.; Valdes, F.; Wenk, R.A. (January 1990).
526:
under the aforementioned degeneracy transformation.
4592:
4748:
3309:
2797:
2767:
2703:
2674:
2644:
2607:
2577:
2545:
2411:
2277:
1948:
1918:
1877:
1847:
1807:
1777:
1750:
1716:
1684:
1528:
1400:
1252:
1217:
1103:
1072:
1050:
864:
840:
811:
514:
455:suffers from a limitation known as the mass sheet
4500:
4447:
4066:Monthly Notices of the Royal Astronomical Society
3933:Monthly Notices of the Royal Astronomical Society
3766:Monthly Notices of the Royal Astronomical Society
3221:
2962:Monthly Notices of the Royal Astronomical Society
2840:Weak lensing also has an important effect on the
606:, cluster counts derived from large weak lensing
4869:
3452:
4539:
708:dominate the total mass fraction, to the outer
4593:Petit, J.-P.; d'Agostini, G. (December 2014).
4102:
3068:
1270:
4742:
4689:
3455:Annual Review of Nuclear and Particle Science
3071:Bulletin of the American Astronomical Society
2719:The ability of weak lensing to constrain the
2714:
203:
4506:
4453:
4405:
2856:Minimal coupling of general relativity with
2001:
1980:
1679:
1600:
1523:
1444:
1395:
1316:
4533:
1724:is the component along or perpendicular to
4795:
4636:
4586:
2825:have focused on weak lensing, such as the
683:
561:
210:
196:
31:
4815:
4762:
4709:
4656:
4553:
4516:
4467:
4388:
4358:
4329:
4311:
4246:
4157:
4128:
4087:
4077:
4003:
3985:
3962:
3944:
3887:
3833:
3795:
3777:
3720:
3666:
3616:
3541:
3523:
3466:
3413:
3360:
3323:
3270:
3182:
3152:
3029:
2992:
2974:
434:
2851:
625:
565:
536:singular isothermal sphere (SIS) profile
364:National Optical Astronomy Observatories
313:
242:
4858:Weak gravitational lensing on arxiv.org
4411:
1751:{\displaystyle {\vec {\Delta \theta }}}
451:reconstructed without knowledge of the
4870:
4103:Miralda-Escudé, Jordi (October 1991).
4801:
3762:"Lensing by galaxies in CNOC2 fields"
3485:10.1146/annurev.nucl.58.110707.171151
2921:large-scale structure of the universe
1949:{\displaystyle \xi _{\times \times }}
1878:{\displaystyle \xi _{\times \times }}
306:cut, but this is much less accurate.
4196:
310:Weak lensing by clusters of galaxies
2907:similar to the image produced by a
722:Comparing the measured mass to the
644:large-scale structure of the cosmos
614:. Weak lensing can also be used to
546:. Knowledge of the lensing cluster
13:
4804:General Relativity and Gravitation
4507:Hossenfelder, Sabine (June 2009).
4414:"Can the hidden mass be negative?"
2753:
2462:
2328:
1735:
1663:
1588:
1560:
1507:
1432:
1379:
1304:
1207:
1181:
1156:
1111:are the transverse distances, and
1032:
1019:
987:
978:
949:
899:
884:
859:
593:Modified Newtonian dynamics (MOND)
480:
19:Part of a series of articles about
14:
4889:
4851:
1778:{\displaystyle \gamma _{\times }}
540:Navarro-Frenk-White (NFW) profile
424:University of California at Davis
4331:10.1046/j.1365-8711.2000.03851.x
3986:Kilbinger, Martin (2015-07-01).
3964:10.1111/j.1365-2966.2006.10156.x
3797:10.1046/j.1365-8711.2003.06350.x
3543:10.1111/j.1365-2966.2008.13818.x
2994:10.1111/j.1365-2966.2007.12312.x
841:{\displaystyle {\vec {\theta }}}
4367:
4346:
4287:
4225:
4190:
4137:
4096:
4052:
3979:
3920:
3866:
3812:
3753:
3699:
3641:
3593:
3558:
3499:
3446:
3393:
3340:
2937:Large Synoptic Survey Telescope
1808:{\displaystyle \xi _{\times +}}
812:{\displaystyle {\vec {\beta }}}
788:Gravitational Lensing Formalism
772:
4602:Astrophysics and Space Science
3992:Reports on Progress in Physics
3303:
3250:
3215:
3161:
3118:
3089:
3062:
3009:
2948:
2534:
2530:
2524:
2511:
2508:
2499:
2443:
2437:
2400:
2396:
2390:
2377:
2374:
2365:
2309:
2303:
2239:
2233:
2214:
2208:
2108:
2102:
2083:
2077:
2010:
2004:
1742:
1676:
1670:
1650:
1641:
1628:
1622:
1613:
1594:
1585:
1566:
1557:
1520:
1514:
1494:
1485:
1472:
1466:
1457:
1438:
1429:
1392:
1386:
1366:
1357:
1344:
1338:
1329:
1310:
1301:
1247:
1241:
1212:
1199:
1129:
1123:
1014:
1011:
1005:
999:
990:
971:
965:
832:
803:
509:
497:
472:
238:
1:
4022:10.1088/0034-4885/78/8/086901
3259:Astrophysical Journal Letters
2942:
2919:within the lacunar, web-like
2775:and power spectrum amplitude
777:The gravitational lensing by
2878:bimetric theories of gravity
2798:{\displaystyle \sigma _{8}~}
2768:{\displaystyle \Omega _{m}~}
1717:{\displaystyle \gamma _{+~}}
649:
224:strong gravitational lensing
7:
2930:
2874:Modified Newtonian Dynamics
2842:Cosmic Microwave Background
2725:cosmic microwave background
1277:shear correlation functions
1271:Shear correlation functions
1265:shear and convergence terms
671:
10:
4894:
4781:10.1103/PhysRevD.90.084026
4728:10.1103/PhysRevD.89.084020
4675:10.1103/PhysRevD.88.024049
4572:10.1103/PhysRevD.90.101502
4486:10.1103/PhysRevD.78.044015
4377:Astronomy and Astrophysics
4176:10.1051/0004-6361:20021341
4146:Astronomy and Astrophysics
3312:Astronomy and Astrophysics
3099:Astronomy and Astrophysics
2715:Weak lensing and cosmology
2704:{\displaystyle \theta =0~}
1919:{\displaystyle \xi _{++~}}
1848:{\displaystyle \xi _{++~}}
678:cluster-scale weak lensing
597:Λ-Cold Dark Matter (Λ-CDM)
556:estimated using photometry
357:
287:Angular diameter distances
233:Weak gravitational lensing
4622:10.1007/s10509-014-2106-5
4412:Milgrom, M. (July 1986).
4375:large-scale structures".
3822:The Astrophysical Journal
3709:The Astrophysical Journal
3605:The Astrophysical Journal
3402:The Astrophysical Journal
3349:The Astrophysical Journal
3171:The Astrophysical Journal
3132:The Astrophysical Journal
3048:10.1007/s11214-008-9324-5
2899:instead of the classical
2835:Vera C. Rubin Observatory
257:coordinate transformation
3876:The Astronomical Journal
3655:The Astronomical Journal
872:along the line of sight
330:bound structures in the
4834:10.1023/A:1018877928270
4614:2014Ap&SS.354..611P
4399:2000A&A...358...30V
4168:2002A&A...396....1S
3334:1995A&A...294..411S
3112:1987A&A...184L...7S
819:to the lensed position
783:thin lens approximation
684:Scientific applications
662:observational astronomy
562:Scientific implications
295:spectroscopic redshifts
4089:10.1093/mnras/stad2013
2860:allows solutions like
2799:
2769:
2705:
2676:
2675:{\displaystyle T_{-}~}
2646:
2645:{\displaystyle T_{+}~}
2609:
2608:{\displaystyle J_{4}~}
2579:
2578:{\displaystyle J_{0}~}
2547:
2413:
2279:
1970:aperture mass variance
1950:
1920:
1879:
1849:
1809:
1779:
1752:
1718:
1686:
1530:
1402:
1254:
1219:
1105:
1074:
1052:
866:
842:
813:
576:star formation history
571:
542:are two commonly used
516:
435:Observational products
366:and Vahe Petrosian of
320:
249:
4878:Gravitational lensing
4863:Observing Dark Worlds
4421:Astrophysical Journal
4199:Astrophysical Journal
4109:Astrophysical Journal
3651:Rockosi, Constance M.
3567:Astrophysical Journal
3224:Astrophysical Journal
3018:Space Science Reviews
2862:traversable wormholes
2852:Negative weak lensing
2811:photometric redshifts
2800:
2770:
2721:matter power spectrum
2706:
2677:
2647:
2610:
2580:
2548:
2414:
2280:
1951:
1921:
1888:matter power spectrum
1880:
1850:
1810:
1780:
1753:
1719:
1687:
1531:
1403:
1255:
1220:
1106:
1104:{\displaystyle x^{i}}
1075:
1053:
867:
865:{\displaystyle \Phi }
843:
814:
779:large-scale structure
741:Galaxy mass evolution
696:Mass density profiles
632:gravitational lensing
626:Galaxy-galaxy lensing
569:
517:
429:signal-to-noise ratio
401:known galaxy clusters
317:
291:photometric redshifts
276:point spread function
261:convergence and shear
253:Gravitational lensing
246:
25:Gravitational lensing
3601:Brainerd, Tereasa G.
2779:
2749:
2745:dark matter density
2686:
2656:
2626:
2589:
2559:
2424:
2290:
1977:
1930:
1897:
1859:
1826:
1789:
1762:
1728:
1698:
1541:
1413:
1285:
1253:{\displaystyle W(r)}
1235:
1117:
1088:
1064:
878:
856:
823:
794:
732:mass to light ratios
719:Mass-to-light ratios
466:
373:Toulouse Observatory
4826:1997GReGr..29.1363P
4773:2014PhRvD..90h4026N
4720:2014PhRvD..89h4020K
4667:2013PhRvD..88b4049I
4564:2014PhRvD..90j1502M
4478:2008PhRvD..78d4015H
4433:1986ApJ...306....9M
4322:2000MNRAS.318..625B
4257:2000Natur.405..143W
4211:1967ApJ...150..737G
4121:1991ApJ...380....1M
4014:2015RPPh...78h6901K
3955:2006MNRAS.368..715M
3898:2004AJ....127.2544S
3844:2007ApJ...669...21P
3788:2003MNRAS.340..609H
3731:2007ApJ...667..176G
3677:2000AJ....120.1198F
3627:1996ApJ...466..623B
3579:1984ApJ...281L..59T
3534:2008MNRAS.390.1157R
3477:2008ARNPS..58...99H
3424:2004ApJ...604..596C
3371:1999ApJ...520L...9M
3281:1996ApJ...464L.115B
3236:1993ApJ...404..441K
3193:2006ApJ...643..128W
3145:1990ApJ...349L...1T
3083:1986BAAS...18R1014L
3040:2008SSRv..134....7D
2985:2007MNRAS.381.1197H
2925:cosmological models
2889:Hirosaki University
2880:consider invisible
2846:21cm line radiation
2466:
2332:
2164:
2033:
2000:
1162:
955:
410:content (e.g. from
389:faint blue galaxies
368:Stanford University
65:Strong lens systems
2827:Dark Energy Survey
2795:
2765:
2701:
2672:
2642:
2605:
2575:
2543:
2452:
2409:
2318:
2275:
2147:
2016:
1983:
1946:
1916:
1875:
1845:
1805:
1775:
1748:
1714:
1682:
1526:
1398:
1250:
1215:
1141:
1101:
1070:
1048:
934:
862:
838:
809:
716:is more prevalent.
572:
512:
321:
300:apparent magnitude
250:
4810:(11): 1363–1370.
4751:Physical Review D
4698:Physical Review D
4645:Physical Review D
4542:Physical Review D
4527:10.1063/1.3327545
4456:Physical Review D
4241:(6783): 143–148.
2794:
2764:
2700:
2671:
2641:
2604:
2574:
2487:
2353:
2269:
2188:
2138:
2057:
1913:
1842:
1745:
1711:
1673:
1653:
1625:
1517:
1497:
1469:
1389:
1369:
1341:
1189:
1082:comoving distance
1073:{\displaystyle r}
1046:
1002:
913:
835:
806:
756:Other mass trends
638:(as opposed to a
544:parametric models
449:mass distribution
381:Bell Laboratories
220:
219:
4885:
4846:
4845:
4819:
4799:
4793:
4792:
4766:
4746:
4740:
4739:
4713:
4693:
4687:
4686:
4660:
4640:
4634:
4633:
4599:
4590:
4584:
4583:
4557:
4537:
4531:
4530:
4520:
4504:
4498:
4497:
4471:
4451:
4445:
4444:
4418:
4409:
4403:
4402:
4392:
4390:astro-ph/0002500
4371:
4365:
4364:
4362:
4360:astro-ph/0003338
4350:
4344:
4343:
4333:
4315:
4313:astro-ph/0003008
4291:
4285:
4284:
4265:10.1038/35012001
4250:
4248:astro-ph/0003014
4229:
4223:
4222:
4194:
4188:
4187:
4161:
4159:astro-ph/0206182
4141:
4135:
4134:
4132:
4100:
4094:
4093:
4091:
4081:
4072:(2): 2195–2223.
4056:
4050:
4049:
4007:
3983:
3977:
3976:
3966:
3948:
3946:astro-ph/0511164
3924:
3918:
3917:
3891:
3889:astro-ph/0312036
3882:(5): 2544–2564.
3870:
3864:
3863:
3837:
3816:
3810:
3809:
3799:
3781:
3779:astro-ph/0211633
3757:
3751:
3750:
3724:
3722:astro-ph/0701589
3703:
3697:
3696:
3670:
3668:astro-ph/9912119
3661:(3): 1198–1208.
3645:
3639:
3638:
3620:
3618:astro-ph/9503073
3597:
3591:
3590:
3562:
3556:
3555:
3545:
3527:
3518:(3): 1157–1169.
3503:
3497:
3496:
3470:
3450:
3444:
3443:
3417:
3415:astro-ph/0312273
3397:
3391:
3390:
3364:
3362:astro-ph/9904156
3344:
3338:
3337:
3327:
3325:astro-ph/9407032
3307:
3301:
3300:
3274:
3272:astro-ph/9601011
3254:
3248:
3247:
3219:
3213:
3212:
3186:
3184:astro-ph/0507606
3165:
3159:
3158:
3156:
3122:
3116:
3115:
3093:
3087:
3086:
3066:
3060:
3059:
3033:
3013:
3007:
3006:
2996:
2978:
2976:astro-ph/0701671
2969:(3): 1197–1218.
2952:
2917:galaxy filaments
2915:located between
2876:as well as some
2804:
2802:
2801:
2796:
2792:
2791:
2790:
2774:
2772:
2771:
2766:
2762:
2761:
2760:
2710:
2708:
2707:
2702:
2698:
2681:
2679:
2678:
2673:
2669:
2668:
2667:
2651:
2649:
2648:
2643:
2639:
2638:
2637:
2617:Bessel Functions
2614:
2612:
2611:
2606:
2602:
2601:
2600:
2584:
2582:
2581:
2576:
2572:
2571:
2570:
2552:
2550:
2549:
2544:
2542:
2541:
2523:
2522:
2498:
2497:
2488:
2486:
2485:
2476:
2468:
2465:
2460:
2436:
2435:
2418:
2416:
2415:
2410:
2408:
2407:
2389:
2388:
2364:
2363:
2354:
2352:
2351:
2342:
2334:
2331:
2326:
2302:
2301:
2284:
2282:
2281:
2276:
2274:
2270:
2262:
2256:
2255:
2246:
2242:
2232:
2231:
2207:
2206:
2189:
2187:
2186:
2177:
2166:
2163:
2155:
2143:
2139:
2131:
2125:
2124:
2115:
2111:
2101:
2100:
2076:
2075:
2058:
2056:
2055:
2046:
2035:
2032:
2024:
1999:
1994:
1955:
1953:
1952:
1947:
1945:
1944:
1925:
1923:
1922:
1917:
1915:
1914:
1911:
1884:
1882:
1881:
1876:
1874:
1873:
1854:
1852:
1851:
1846:
1844:
1843:
1840:
1817:systematic error
1814:
1812:
1811:
1806:
1804:
1803:
1784:
1782:
1781:
1776:
1774:
1773:
1757:
1755:
1754:
1749:
1747:
1746:
1741:
1733:
1723:
1721:
1720:
1715:
1713:
1712:
1709:
1691:
1689:
1688:
1683:
1675:
1674:
1669:
1661:
1655:
1654:
1646:
1640:
1639:
1627:
1626:
1618:
1612:
1611:
1584:
1583:
1556:
1555:
1535:
1533:
1532:
1527:
1519:
1518:
1513:
1505:
1499:
1498:
1490:
1484:
1483:
1471:
1470:
1462:
1456:
1455:
1428:
1427:
1407:
1405:
1404:
1399:
1391:
1390:
1385:
1377:
1371:
1370:
1362:
1356:
1355:
1343:
1342:
1334:
1328:
1327:
1300:
1299:
1259:
1257:
1256:
1251:
1224:
1222:
1221:
1216:
1211:
1210:
1195:
1191:
1190:
1185:
1184:
1175:
1161:
1160:
1159:
1149:
1110:
1108:
1107:
1102:
1100:
1099:
1079:
1077:
1076:
1071:
1057:
1055:
1054:
1049:
1047:
1045:
1044:
1043:
1031:
1030:
1017:
1004:
1003:
995:
986:
985:
975:
954:
953:
952:
942:
930:
929:
914:
912:
911:
910:
897:
896:
895:
882:
871:
869:
868:
863:
847:
845:
844:
839:
837:
836:
828:
818:
816:
815:
810:
808:
807:
799:
521:
519:
518:
513:
484:
483:
326:are the largest
212:
205:
198:
35:
16:
15:
4893:
4892:
4888:
4887:
4886:
4884:
4883:
4882:
4868:
4867:
4854:
4849:
4800:
4796:
4747:
4743:
4694:
4690:
4641:
4637:
4597:
4591:
4587:
4538:
4534:
4509:Antigravitation
4505:
4501:
4452:
4448:
4416:
4410:
4406:
4372:
4368:
4351:
4347:
4292:
4288:
4230:
4226:
4195:
4191:
4142:
4138:
4101:
4097:
4057:
4053:
3984:
3980:
3925:
3921:
3871:
3867:
3817:
3813:
3758:
3754:
3704:
3700:
3646:
3642:
3598:
3594:
3563:
3559:
3504:
3500:
3451:
3447:
3398:
3394:
3345:
3341:
3308:
3304:
3255:
3251:
3220:
3216:
3166:
3162:
3123:
3119:
3094:
3090:
3067:
3063:
3014:
3010:
2953:
2949:
2945:
2933:
2903:distortions of
2854:
2786:
2782:
2780:
2777:
2776:
2756:
2752:
2750:
2747:
2746:
2717:
2687:
2684:
2683:
2682:are small near
2663:
2659:
2657:
2654:
2653:
2633:
2629:
2627:
2624:
2623:
2596:
2592:
2590:
2587:
2586:
2566:
2562:
2560:
2557:
2556:
2537:
2533:
2518:
2514:
2493:
2489:
2481:
2477:
2469:
2467:
2461:
2456:
2431:
2427:
2425:
2422:
2421:
2403:
2399:
2384:
2380:
2359:
2355:
2347:
2343:
2335:
2333:
2327:
2322:
2297:
2293:
2291:
2288:
2287:
2261:
2257:
2251:
2247:
2224:
2220:
2199:
2195:
2194:
2190:
2182:
2178:
2167:
2165:
2156:
2151:
2130:
2126:
2120:
2116:
2093:
2089:
2068:
2064:
2063:
2059:
2051:
2047:
2036:
2034:
2025:
2020:
1995:
1987:
1978:
1975:
1974:
1937:
1933:
1931:
1928:
1927:
1904:
1900:
1898:
1895:
1894:
1866:
1862:
1860:
1857:
1856:
1833:
1829:
1827:
1824:
1823:
1796:
1792:
1790:
1787:
1786:
1769:
1765:
1763:
1760:
1759:
1734:
1732:
1731:
1729:
1726:
1725:
1705:
1701:
1699:
1696:
1695:
1662:
1660:
1659:
1645:
1644:
1635:
1631:
1617:
1616:
1607:
1603:
1576:
1572:
1548:
1544:
1542:
1539:
1538:
1506:
1504:
1503:
1489:
1488:
1479:
1475:
1461:
1460:
1451:
1447:
1420:
1416:
1414:
1411:
1410:
1378:
1376:
1375:
1361:
1360:
1351:
1347:
1333:
1332:
1323:
1319:
1292:
1288:
1286:
1283:
1282:
1273:
1236:
1233:
1232:
1206:
1202:
1180:
1176:
1174:
1167:
1163:
1155:
1151:
1150:
1145:
1118:
1115:
1114:
1095:
1091:
1089:
1086:
1085:
1065:
1062:
1061:
1039:
1035:
1026:
1022:
1018:
994:
993:
981:
977:
976:
974:
948:
944:
943:
938:
922:
918:
906:
902:
898:
891:
887:
883:
881:
879:
876:
875:
857:
854:
853:
827:
826:
824:
821:
820:
798:
797:
795:
792:
791:
775:
702:effective radii
686:
674:
652:
628:
612:false positives
564:
479:
475:
467:
464:
463:
437:
360:
328:gravitationally
324:Galaxy clusters
312:
241:
216:
187:
186:
149:
116:
108:
107:
96:
87:
78:
66:
55:
51:
47:
43:
12:
11:
5:
4891:
4881:
4880:
4866:
4865:
4860:
4853:
4852:External links
4850:
4848:
4847:
4794:
4741:
4688:
4635:
4608:(2): 611–615.
4585:
4548:(10): 101502.
4532:
4499:
4446:
4441:10.1086/164314
4404:
4366:
4345:
4306:(2): 625–640.
4286:
4224:
4219:10.1086/149378
4189:
4136:
4130:10.1086/170555
4095:
4051:
3978:
3939:(2): 715–731.
3919:
3906:10.1086/383293
3865:
3852:10.1086/521541
3811:
3772:(2): 609–622.
3752:
3739:10.1086/519237
3715:(1): 176–190.
3698:
3685:10.1086/301540
3640:
3635:10.1086/177537
3592:
3587:10.1086/184285
3557:
3498:
3445:
3432:10.1086/381970
3408:(2): 596–603.
3392:
3379:10.1086/312144
3339:
3318:(2): 411–431.
3302:
3289:10.1086/310114
3249:
3244:10.1086/172297
3230:(2): 441–450.
3214:
3201:10.1086/502621
3177:(1): 128–143.
3160:
3154:10.1086/185636
3117:
3106:(1–2): L7–L9.
3088:
3061:
3008:
2946:
2944:
2941:
2940:
2939:
2932:
2929:
2870:energy density
2864:stabilized by
2853:
2850:
2789:
2785:
2759:
2755:
2733:galaxy surveys
2716:
2713:
2697:
2694:
2691:
2666:
2662:
2636:
2632:
2599:
2595:
2569:
2565:
2540:
2536:
2532:
2529:
2526:
2521:
2517:
2513:
2510:
2507:
2504:
2501:
2496:
2492:
2484:
2480:
2475:
2472:
2464:
2459:
2455:
2451:
2448:
2445:
2442:
2439:
2434:
2430:
2406:
2402:
2398:
2395:
2392:
2387:
2383:
2379:
2376:
2373:
2370:
2367:
2362:
2358:
2350:
2346:
2341:
2338:
2330:
2325:
2321:
2317:
2314:
2311:
2308:
2305:
2300:
2296:
2273:
2268:
2265:
2260:
2254:
2250:
2245:
2241:
2238:
2235:
2230:
2227:
2223:
2219:
2216:
2213:
2210:
2205:
2202:
2198:
2193:
2185:
2181:
2176:
2173:
2170:
2162:
2159:
2154:
2150:
2146:
2142:
2137:
2134:
2129:
2123:
2119:
2114:
2110:
2107:
2104:
2099:
2096:
2092:
2088:
2085:
2082:
2079:
2074:
2071:
2067:
2062:
2054:
2050:
2045:
2042:
2039:
2031:
2028:
2023:
2019:
2015:
2012:
2009:
2006:
2003:
1998:
1993:
1990:
1986:
1982:
1943:
1940:
1936:
1910:
1907:
1903:
1872:
1869:
1865:
1839:
1836:
1832:
1822:The functions
1802:
1799:
1795:
1772:
1768:
1744:
1740:
1737:
1708:
1704:
1681:
1678:
1672:
1668:
1665:
1658:
1652:
1649:
1643:
1638:
1634:
1630:
1624:
1621:
1615:
1610:
1606:
1602:
1599:
1596:
1593:
1590:
1587:
1582:
1579:
1575:
1571:
1568:
1565:
1562:
1559:
1554:
1551:
1547:
1525:
1522:
1516:
1512:
1509:
1502:
1496:
1493:
1487:
1482:
1478:
1474:
1468:
1465:
1459:
1454:
1450:
1446:
1443:
1440:
1437:
1434:
1431:
1426:
1423:
1419:
1397:
1394:
1388:
1384:
1381:
1374:
1368:
1365:
1359:
1354:
1350:
1346:
1340:
1337:
1331:
1326:
1322:
1318:
1315:
1312:
1309:
1306:
1303:
1298:
1295:
1291:
1272:
1269:
1249:
1246:
1243:
1240:
1229:lensing kernel
1214:
1209:
1205:
1201:
1198:
1194:
1188:
1183:
1179:
1173:
1170:
1166:
1158:
1154:
1148:
1144:
1140:
1137:
1134:
1131:
1128:
1125:
1122:
1098:
1094:
1069:
1042:
1038:
1034:
1029:
1025:
1021:
1016:
1013:
1010:
1007:
1001:
998:
992:
989:
984:
980:
973:
970:
967:
964:
961:
958:
951:
947:
941:
937:
933:
928:
925:
921:
917:
909:
905:
901:
894:
890:
886:
861:
834:
831:
805:
802:
774:
771:
770:
769:
757:
754:
746:speed of light
742:
739:
720:
717:
697:
685:
682:
673:
670:
657:galaxy surveys
651:
648:
640:galaxy cluster
627:
624:
589:Bullet Cluster
563:
560:
511:
508:
505:
502:
499:
496:
493:
490:
487:
482:
478:
474:
471:
461:transformation
439:The projected
436:
433:
359:
356:
352:Bullet Cluster
311:
308:
240:
237:
218:
217:
215:
214:
207:
200:
192:
189:
188:
185:
184:
179:
174:
169:
164:
159:
154:
144:
137:
136:
135:
134:
124:
117:
114:
113:
110:
109:
106:
105:
100:
94:SDSSJ0946+1006
91:
85:Bullet Cluster
82:
73:
67:
64:
63:
60:
59:
49:Strong lensing
37:
36:
28:
27:
21:
20:
9:
6:
4:
3:
2:
4890:
4879:
4876:
4875:
4873:
4864:
4861:
4859:
4856:
4855:
4843:
4839:
4835:
4831:
4827:
4823:
4818:
4817:gr-qc/9706049
4813:
4809:
4805:
4798:
4790:
4786:
4782:
4778:
4774:
4770:
4765:
4760:
4757:(8): 084026.
4756:
4752:
4745:
4737:
4733:
4729:
4725:
4721:
4717:
4712:
4707:
4704:(8): 084020.
4703:
4699:
4692:
4684:
4680:
4676:
4672:
4668:
4664:
4659:
4654:
4651:(2): 024049.
4650:
4646:
4639:
4631:
4627:
4623:
4619:
4615:
4611:
4607:
4603:
4596:
4589:
4581:
4577:
4573:
4569:
4565:
4561:
4556:
4551:
4547:
4543:
4536:
4528:
4524:
4519:
4514:
4510:
4503:
4495:
4491:
4487:
4483:
4479:
4475:
4470:
4465:
4462:(4): 044015.
4461:
4457:
4450:
4442:
4438:
4434:
4430:
4426:
4422:
4415:
4408:
4400:
4396:
4391:
4386:
4382:
4378:
4370:
4361:
4356:
4349:
4341:
4337:
4332:
4327:
4323:
4319:
4314:
4309:
4305:
4301:
4297:
4290:
4282:
4278:
4274:
4270:
4266:
4262:
4258:
4254:
4249:
4244:
4240:
4236:
4228:
4220:
4216:
4212:
4208:
4204:
4200:
4193:
4185:
4181:
4177:
4173:
4169:
4165:
4160:
4155:
4151:
4147:
4140:
4131:
4126:
4122:
4118:
4114:
4110:
4106:
4099:
4090:
4085:
4080:
4075:
4071:
4067:
4063:
4055:
4047:
4043:
4039:
4035:
4031:
4027:
4023:
4019:
4015:
4011:
4006:
4001:
3998:(8): 086901.
3997:
3993:
3989:
3982:
3974:
3970:
3965:
3960:
3956:
3952:
3947:
3942:
3938:
3934:
3930:
3923:
3915:
3911:
3907:
3903:
3899:
3895:
3890:
3885:
3881:
3877:
3869:
3861:
3857:
3853:
3849:
3845:
3841:
3836:
3831:
3827:
3823:
3815:
3807:
3803:
3798:
3793:
3789:
3785:
3780:
3775:
3771:
3767:
3763:
3756:
3748:
3744:
3740:
3736:
3732:
3728:
3723:
3718:
3714:
3710:
3702:
3694:
3690:
3686:
3682:
3678:
3674:
3669:
3664:
3660:
3656:
3652:
3644:
3636:
3632:
3628:
3624:
3619:
3614:
3610:
3606:
3602:
3596:
3588:
3584:
3580:
3576:
3572:
3568:
3561:
3553:
3549:
3544:
3539:
3535:
3531:
3526:
3521:
3517:
3513:
3509:
3502:
3494:
3490:
3486:
3482:
3478:
3474:
3469:
3464:
3461:(1): 99–123.
3460:
3456:
3449:
3441:
3437:
3433:
3429:
3425:
3421:
3416:
3411:
3407:
3403:
3396:
3388:
3384:
3380:
3376:
3372:
3368:
3363:
3358:
3355:(1): L9–L12.
3354:
3350:
3343:
3335:
3331:
3326:
3321:
3317:
3313:
3306:
3298:
3294:
3290:
3286:
3282:
3278:
3273:
3268:
3264:
3260:
3253:
3245:
3241:
3237:
3233:
3229:
3225:
3218:
3210:
3206:
3202:
3198:
3194:
3190:
3185:
3180:
3176:
3172:
3164:
3155:
3150:
3146:
3142:
3138:
3134:
3133:
3128:
3121:
3113:
3109:
3105:
3101:
3100:
3092:
3084:
3080:
3076:
3072:
3065:
3057:
3053:
3049:
3045:
3041:
3037:
3032:
3027:
3024:(1–4): 7–24.
3023:
3019:
3012:
3004:
3000:
2995:
2990:
2986:
2982:
2977:
2972:
2968:
2964:
2963:
2958:
2951:
2947:
2938:
2935:
2934:
2928:
2926:
2922:
2918:
2914:
2910:
2906:
2905:convex lenses
2902:
2898:
2892:
2890:
2885:
2883:
2882:negative mass
2879:
2875:
2871:
2867:
2866:exotic matter
2863:
2859:
2858:scalar fields
2849:
2847:
2843:
2838:
2836:
2832:
2828:
2824:
2820:
2816:
2812:
2807:
2805:
2787:
2783:
2757:
2741:
2736:
2734:
2730:
2726:
2722:
2712:
2695:
2692:
2689:
2664:
2660:
2634:
2630:
2620:
2618:
2597:
2593:
2567:
2563:
2553:
2538:
2527:
2519:
2515:
2505:
2502:
2494:
2490:
2482:
2478:
2473:
2470:
2457:
2453:
2449:
2446:
2440:
2432:
2428:
2419:
2404:
2393:
2385:
2381:
2371:
2368:
2360:
2356:
2348:
2344:
2339:
2336:
2323:
2319:
2315:
2312:
2306:
2298:
2294:
2285:
2271:
2266:
2263:
2258:
2252:
2248:
2243:
2236:
2228:
2225:
2221:
2217:
2211:
2203:
2200:
2196:
2191:
2183:
2179:
2174:
2171:
2168:
2160:
2157:
2152:
2148:
2144:
2140:
2135:
2132:
2127:
2121:
2117:
2112:
2105:
2097:
2094:
2090:
2086:
2080:
2072:
2069:
2065:
2060:
2052:
2048:
2043:
2040:
2037:
2029:
2026:
2021:
2017:
2013:
2007:
1996:
1991:
1988:
1984:
1972:
1971:
1966:
1963:
1959:
1941:
1938:
1934:
1908:
1905:
1901:
1891:
1889:
1870:
1867:
1863:
1837:
1834:
1830:
1820:
1818:
1800:
1797:
1793:
1770:
1766:
1738:
1706:
1702:
1692:
1666:
1656:
1647:
1636:
1632:
1619:
1608:
1604:
1597:
1591:
1580:
1577:
1573:
1569:
1563:
1552:
1549:
1545:
1536:
1510:
1500:
1491:
1480:
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3573:: L59–L62.
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2823:dark matter
2819:dark energy
714:dark matter
397:CL 1409+524
336:dark matter
239:Methodology
4079:2212.11319
2943:References
2831:Pan-STARRS
2815:tomography
2729:supernovae
766:morphology
744:Since the
724:luminosity
457:degeneracy
393:Abell 1689
350:, and the
344:Abell 1689
255:acts as a
228:deflection
157:Pan-STARRS
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616:calibrate
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650:History
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620:scatter
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