Weak gravitational lensing

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 strong gravitational lensing. Most lines of sight in the universe are thoroughly in the weak lensing regime, in which the deflection 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. Weak gravitational lensing 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.

Gravitational lensing acts as a coordinate transformation that distorts the images of background objects (usually galaxies) near a foreground mass. The transformation can be split into two terms, the convergence and shear. The convergence term magnifies the background objects by increasing their size, and the shear term stretches them tangentially around the foreground mass.

To measure this tangential alignment, it is necessary to measure the ellipticities 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 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.

Another major challenge for weak lensing is correction for the point spread function (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 interpolating 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.

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