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Cosmic Lensing

Page history last edited by Adam Amara 15 years, 6 months ago

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Galaxy Distortion


In the strong lensing regime, the gravitational bending of light has a distinct effect on the images of background objects that can be easily seen by eye (see strong lensing for examples). When we look at what appears to be an empty region of sky, the effects of gravitational lensing are not so obvious, but they are present, nevertheless. Light from all distant galaxies must travel to us over vast distances. On this journey, the light will inevitably interact with matter structure in the Universe. This interaction distorts the images of the galaxies and, therefore, leaves an imprint of the Universe that we can measure. This study, however, needs to done statistically by carefully measuring and comparing the shapes of millions of galaxies, since cosmic lensing (often called cosmic shear) cannot be detected on individual galaxies.  


Figure CL1 is an illustration that demonstrates the way that the images of galaxies get distorted. On the left, we see a regular grid of pink blobs, where each blob represents a galaxy. These 'galaxies' have not undergone any gravitational lensing, so their images are the same as their inherent shapes, which, in this case, are circular. On the right, we see the same galaxies after they have been lensed by a typical distribution of dark matter that would lie between us and the galaxies. In fact, to be able to visualise the effect we have exaggerated the lensing effect by a factor of 10, so that it can be see by eye. On inspection we see that gravitational lensing has introduced a complex pattern of swirls and alignments. These patterns have distinct properties that depend on both the properties of the matter distribution in the Universe, as well as its cosmic geometry. 



Figure CL1: This figure shows the effect of image distortion. On the left we see a regular grid that represents an unlensed image of galaxies.  For this simple illustration galaxies are represented by circular Gaussian blobs that are distributed on a regular grid.  On the left we see what happens when lensing is applied.  The lensing effect here has been calculated using a dark matter density that is a Gaussian random field.  Furthermore it should be noted that we have exaggerated the lensing effect (by a factor of 10) to show the effect.


Convergence and Shear 


In general lensing, we introduced the concept of convergence (roughly the 2D mass distribution) for the thin lens approximation. For cosmic lensing, the mass along the entire line of sight needs to be considered since there is not one single lens. Instead, lensing is caused by all mass between us and the galaxy. In this case the converges is given by 




where Ωm is the density of matter in the Universe, H0 is the Hubble constant, c is the speed of light, D is the comoving angular diameter, χ is the comoving radial distance, z redshift and χs is the radial distance to the background galaxies (i.e. the ones being lensed). δ is the matter over density, defined as




where ρ is the matter density and <ρ> is the average density. The left panel of Figure CL2 shows the convergence map that was used to lens the 'galaxies' shown in CL1. In an analogous way to what is discussed in general lensing, there is also a corresponding shear field (γ). The right panel of Figure CL2 shows the shear from the convergence in the left panel. This shear is what caused the 'galaxies' in CL1 to go from circles to ellipses.  



Figure CL2: On the left we see the convergence map used to produce the lensing effect shown in Figure CL1. light/white corresponds to regions with high mass along the line of sight, while dark/black regions have low matter density.  On the right we show the corresponding shear field. 


Over plotting the convergence and the shear fields (Figure CL3) shows the link between shear and convergence. We see that the peaks in the convergence maps (light blue -> white) are surrounded by a circular (tangential) shear pattern. Regions with no matter (dark blue -> black) are surrounded by a radial shear pattern. These characteristic patterns are discussed in more detail in the section E-B mode decomposition below.  


Figure CL3: Here we overlay the convergence and shear shown in Figure CL2.  Here we see that the shear field and the convergence field are strongly linked.  White regions (which are high convergence and high mass) tend to be surrounded by a ring like shear pattern while under dense regions (black) tend to have radial shear patterns.

Between the Galaxy and Us 


The pink blobs shown in Figure CL1 represent an idealised case where galaxies have a simple initial shape and that the only observable distortion is caused by lensing. However, things are not this simple in reality. Figure CL4 shows some of the significant effects that must be considered when performing a cosmic shear measurements. Let us begin by following the steps that affect galaxies. On the left, we see that the initial shapes of galaxies is complex. Typically, galaxies will have an intrinsic ellipticity, as well as further structure.   Gravitational lensing by mass between us and the galaxies causes the image of the galaxy to be distorted. To first order this is well described a simple shearing, which for example, changed circles into ellipses. Note that the shearing shown in figure CL4 is exaggerated and that a cosmic shear would is much smaller, causing ~1% change in ellipticity.  The next effect shown in figure CL4 (centre panels) shows what happens to the image of the galaxy its light travels through the atmosphere and our telescopes. The blurring we see comes from the convolution with the Point-Spread-Function (PSF) of the experiment. To measure the galaxy image we use a detector (e.g. a CCD) which pixelate the image.  Finally we must bare in mind that, as with any measurement, there will me noise.  The exact properties of the noise will depend on the their origin.  


Astronomers interesting cosmic shear begin with the image shown on the far right and try to recover the shear signal introduced in step 2.  This might seem like an impossible task but two factors allow us to measure cosmic shear.  The first is that as well as the galaxies we also measure stars, and the second is that the original shapes of galaxies are random.


As we can see in the bottom panels of figure CL4 stars images go through many of the same steps as galaxies.  However, the crucial stars do not respond to shear.  This is is because stars are so small that they act as point sources.  Measuring stars therefore allows us to measure and characterize the contaminants.  We can then correct for these effects in  our galaxy images.  This takes care of the last three steps in the forward process. The challenge then is to disentangle the shear signal from the (unknown) original shape of the galaxy.  We able to do this because the light from two galaxies that are close to each other on the sky (small angular separation)  will go through similar mass structures.  This means that the lensing signals from these two galaxies will be correlated.  To first order we can assume that the intrinsic shapes of the two galaxies are random and are therefore uncorrelated.  By looking at the correlation between of ellipticity of galaxies we are therefore able to extract the lensing signal and discard the intrinsic shape. There is a small caveat that it is know that there are effect which can correlate intrinsic galaxy shapes. This is known as intrinsic alignment and can contaminate the lensing signal if it is not corrected.




Observable Statistics





E & B Modes









Five Easy Steps


  1. Exercise 1
  2. Exercise 2
  3. Exercise 3
  4. Exercise 4
  5. Exercise 5


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