Weak gravitational lensing is a powerful probe of both the dark matter and the dark energy content of the universe.


What Causes Gravitational Lensing?


Gravitational lensing can be explained by Einstein's general theory of relativity. Galaxies, which are very massive objects, act on spacetime by causing it to become curved. This curved spacetime then influences particles in its vicinity, affecting their motion. For example, photons travel the path of minimum spacetime. What appears to be the shortest path in flat spacetime is not the shortest path in curved spacetime. For this reason, photons passing a massive object take curved paths. Differential curvature results in distorted and magnified background images. In typical weak lensing, these effects are on the order of a few percent. Since galaxies have random rotation and inclination, shear effects must be determined by statistically preferred orientations.

 



 

 

What Effects Should We See

 


Unlensed

 


Lensed

 


Sources of Error

The primary source of error in lensing measurements is due to the convolution of a point spread function (PSF) with your lensed image. A PSF is a distortion due to several factors, primarily imperfect optics, as well as atmospheric smearing in ground-based studies or thermal breathing in space based observations. There are also numerous other contributions due to physical limitations. There are other sources of error removed from the optics as well, such as centroid determination for ellipticity calculations, Poisson, and statistical bias/errors due to orientation assumptions and limited galaxy magnitudes.

 


What Do Noisy Images Look Like?

 


Method I: Shapelets

"Shapelets," developed by Refregier (2001), is a method which has been applied to analyzing images and accurately measuring weak shear. This is accomplished by decomposing individual galaxy images into a complete orthonormal basis set consisting of weightier Hermite Polynomials. The basis functions are perturbations around a circular, 2-dimensional Gaussian distribution, and are eigenstates of the 2-dimensional Quantum Harmonic Oscillator (QHO). The shapelets method has notable advantages over other methods. First, it is invariant under Fourier transformations, which allows accurate Correction of the PSF through a deconvolution process. It is also possible to determine the total flux, centroid, and the radius of a lensed object, of which the accurate measurements provide a significant advantage in measuring shear.


Method II: Kaiser, Squires, and Broadhurst (KSB)

The KSB method measures the ellipticity of a galaxy image. In the weak lensing regime, the shear is proportional to the ellipticity, which can be measured by determining the quadropole moments of an image. To reduce noise, the signal is convoluted with a weighting, typically a gaussian. KSB's primary advantages are its mathematical ease and relatively simple implementation. However, KSB also requires an independent correction for the PSF, generally difficult, and since KSB measures ellipticity, it is necessary to introduce a statistical correction factor based upon assumed average orientations to obtain the shear.



References
Kirsten Howley, Javier Guedes, Daniel Raburn, Chris Sitzer, Prof. Jodi Lamoureux(Cal Poly San Luis Obispo), & Prof. George Smoot(UC Berkeley),"Weak Gravitational Lensing by Clusters of Galaxies"
Lectures on Gravitaional Lensing-Bartlemann & Narayah; 2004,astro-ph/9606001
Bartelmann, M.2002, astro-ph/0207032
Kaiser, N.,Squires,G.,Broadhurst, T., 1995, astro-ph/9411005
Refreiger, A, 2001, astro-ph/0105178
Refregier, A., Bacon, D., 2001, astro-ph/0105179