2. Gravitational Lensing 25
2.2. Weak Gravitational Lensing
Hereκis the dimensionless surface mass density defined as κ(~θ) =Σ(Ddθ)~
Σcrit , (2.11)
where Σcrit= 4πGc2 DDs
dDds. The reduced deflection angle can also be expressed as the gradient of a deflection potentialψ
~
α=∇ψ(~θ), (2.12)
where
ψ(~θ) = 1 π
Z
R2
d2θ0κ(~θ0) ln|~θ−~θ0|. (2.13) Finally, we can say that
∇2ψ= 2κ , (2.14)
while making use of∇2ln|~θ|= 2πδD(~θ), where in this caseδDis the Dirac delta function.
2.2. Weak Gravitational Lensing
So far, we only considered point sources or single light rays, but what happens with extended sources? Images of extended sources will get distorted due to the differential deflection of every light ray belonging to the image. The distortion of such an image is described by the Jacobian of the lens equation
A(~θ) = ∂ ~β
∂~θ =δij−∂2ψ(θ)~
∂θi∂θj
=
1−κ−γ1 −γ2
−γ2 1−κ+γ1
, (2.15)
whereγis the so-called shear. It is a complex number and thus has two components γ1 andγ2
γ=γ1+ iγ2. (2.16)
The inverse of the determinant ofAis the magnification µ(θ) =~ 1
det(A(~θ)) = 1
(1−κ)2− |γ|2 . (2.17)
In extreme cases, lensing can lead to giant arcs (see Section 2.3). However, when κ and |γ| are both much smaller than unity, we are in the weak gravitational lensing regime, where these distortions are only small and the effect is subtle. In order to describe the change in galaxy shapes we first need to be able to quantify the shape. We do this by introducing the complex ellipticity
=||e2iφ , (2.18)
whereφin this case is the position angle and
||= 1−r
1 +r , (2.19)
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CHAPTER 2. GRAVITATIONAL LENSING
withr being the axis ratio of the light distribution of the image of a galaxy. It was shown that the ellipticity under lensing changes as follows (Schneider & Seitz, 1995)
len = int+g 1 +g∗int
,|g| ≤1 or len=1 +g∗int
∗int+g∗,|g|>1, (2.20) whereg is the reduced shear
g= γ
1−κ=g1+ ig2=|g|e2iφ. (2.21) The change in ellipticity depends only on the reduced shear, not onκand γ separately, thus g is the only quantity that can be inferred from measuring galaxy ellipticities.
In practice one does not know the intrinsic ellipticities of background galaxies and thus it is not possible to judge if a single galaxy has been lensed or not. For a large enough ensemble of galaxies, however, we can assume that due to the cosmological principlehinti= 0. If those galaxies were lensed this would not be the case. So for an ensemble of galaxies it is possible to measure the weak gravitational lensing signal. Typical lenses here can be galaxies, galaxy clusters or even the large scale structure of the universe. We will describe these applications in the next sections.
2.2.1. Cluster Weak Lensing
The gravitational potential of a single galaxy cluster can act as a gravitational lens in the strong (see Section 2.3) as well as the weak lensing regime. In order to find the weak lensing signal around a galaxy cluster, one usually uses the tangential shear,
γt=−<[γe−2iφ], (2.22) where φ is the angle that describes the position of the source galaxy with respect to the lens, which is in the centre of the coordinate frame. Astronomical lenses like galaxy clusters or galaxies should introduce a distortion to the sources that is tangential with respect to the lens centre.
Thus, we use the tangential part of the shear whereas the cross shear,
γx =−=[γe−2iφ], (2.23)
will be zero in such a case. If a non-zeroγx is measured, this normally points to systematics in the data. Usually,γtis measured in annular bins around the lens. So instead of just having one source galaxy, people make use of all background galaxies around the cluster and thus measure the average γt as a function of separation to the lens. When assuming a mass profile for the lens, this signal can be predicted and using this, a mass for the cluster can be estimated. When redshift information is available one can also use the annular differential excess surface mass density
∆Σ(R) =hγtiΣcrit (2.24)
as the lensing observable, where R is the projected separation between lens and source. A statistically complete sample of clusters can even be used to measure the so-called halo mass function and thus to constrain cosmological parameters. Recent results of cluster lensing studies can for example be found in Applegate et al. (2014) and Hoekstra et al. (2015).
2.2. WEAK GRAVITATIONAL LENSING
2.2.2. Galaxy-Galaxy Lensing
Instead of a cluster of galaxies, one can also measure the average tangential shear as a function of separation around galaxies. The signal of a single galaxy is normally not strong enough to be detected, which can be overcome by stacking the signal of many lens galaxies. The resulting signal can then be used to learn about the average properties of the lens population. Recent results in this field are for example van Uitert et al. (2011), van Uitert et al. (2012), Mandelbaum et al. (2013), or Velander et al. (2014). An actual application of galaxy-galaxy lensing can be found in Chapter 6.
2.2.3. Cosmic Shear
Light rays emitted by a high-redshift source get deflected many times on their way to the observer.
The lens in this case is the large scale structure of the universe. This deflection again changes the shapes of galaxies and can be measured in a statistical sense. The tools being used here are two-point statistics of the shear, namely the correlation functions
ξ+(ϑ) = hγtγti(ϑ) +hγxγxi(ϑ), (2.25) ξ−(ϑ) = hγtγti(ϑ)− hγxγxi(ϑ), (2.26)
ξx(ϑ) = hγtγxi(ϑ). (2.27)
Here we use pairs of galaxies, each with a measured ellipticity, to estimate these functions, which is why we can again define a tangential as well as a cross part of the shear for each pair. Those functions are directly connected to the convergence power spectrum and thus to the matter power spectrum. This makes cosmic shear a powerful tool to constrain cosmological parameters. The first detections of cosmic shear happened about 15 years ago. Recent results are for example Schrabback et al. (2010) or Heymans et al. (2013).
2.2.4. The Aperture Mass
A special estimator for weak gravitational lensing is the aperture mass (Schneider 1996; Schneider et al. 1998). It was initially designed to overcome the mass-sheet degeneracy, which describes the problem that κfor a given lens can only be constrained up to a constant λ, which means that we cannot observe a difference betweenκand aκ0, where
κ0(θ) =~ λκ(~θ) + (1−λ). (2.28) The aperture mass Map is now defined in such a way that it is insensitive to the mass sheet degeneracy. We define it as
Map= Z
dφ φU(φ)κ(φ), (2.29)
whereU is a compensated filter function andφthe aperture radius. In terms of the tangential shearγtinstead of the convergenceκ, it turns into
Map= Z
dφ φQ(φ)γt(φ), (2.30)
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CHAPTER 2. GRAVITATIONAL LENSING
whereQ is related toU via
Q(φ) = 2 φ2
Z φ 0
dφ0φ0U(φ0)− U(φ). (2.31) Several possible filters have been suggested and used in measurements, most of them polynomials.
In Chapter 6 we will present a new set of filters forMapand apply these new estimators to data.