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My 2008 Summer REU Paper Siu Cheung Wong Department of Physics, The Chinese University of Hong Kong Advisor: Prof. Jon Thaler Title: Image Quality and Weak Gravitational Lensing ABSTRACT Since the discovery of the accelerating expansion of the universe in 1998, there have been many speculations about the nature of dark energy which is hypothesized to explain this observation. To give a more precise study of the dark energy, a large national and international collaboration formed in 2003 proposed the Dark Energy Survey which will build a new 550-megapixel CCD camera for the existing Blanco 4-m telescope in Chile. Four independent but complementary techniques will be employed in this single large experiment, in which weak lensing study is one of them. The weak lensing technique measures the gravitational lensing effect by a mass distribution on a cluster of background galaxies. It makes a powerful tool in mapping the spatial distribution and time evolution of large-scale structure. As the accuracy of the measurement depends on the image quality affected by such factors as atmospheric condition, my work was on the relation between image quality and the estimation of lens mass and position. It was found that the more massive the lens object is, the smaller relative uncertainty it has in its mass estimation. I. Background and Introduction gravitational attraction of normal matter In 1998, observations of type Ia decelerates the expansion, this acceleration supernovae by the Supernova requires something new and strange: dark 1 Cosmology Project at the Lawrence energy. Since then, these observations have Berkeley National Laboratory and the been corroborated by several independent High-z Supernova Search Team2 sources. Measurements of the cosmic suggested that the expansion of the microwave background, gravitational universe is accelerating. Because the lensing, and the large scale structure of the cosmos as well as improved spectrum measurement within redshift measurements of supernovae have been shells to z ~ 1.1 using 300 million galaxies, consistent with the Lambda-CDM model, and (4) a survey to measure ~ 2000 which indicates that the dark energy supernova Ia distances. Each method component dominates and accounts for constrains a different combination of about 70% of the total mass-energy of properties of the universe (i.e. dark energy the universe. However, little more is density Ω Λ , matter density Ω M , equation of known about it. A key objective in state of dark energy w, etc.). Bringing these cosmology and high-energy physics four different techniques together in one today is to understand the nature of this experiment will give a far more precise dark energy. High-precision measurement of the dark energy. To measurements of the expansion of the accomplish this experiment, a new Survey universe are required to understand how Instrument, consisting of a wide field the expansion rate changes over time. In corrector and a 3 degree 2 CCD camera will general relativity, the evolution of the be built, which will be used to carry out a expansion rate is parameterized by the deep four band optical photometric survey cosmological equation of state. on the Blanco 4-m telescope, currently Measuring the equation of state of dark operated by the National Optical Astronomy energy is one of the biggest efforts in Observatory (NOAO) in the southern observational cosmology today. hemisphere. The observation phase will The Dark Energy Survey (DES)3 extend from 2011 through 2015. DES will focuses on this mystery. Its collaboration constitute the most precise study of the dark consisting of scientists from Fermi energy and pave the way for the longer National Accelerator Laboratory timescale, complementary experiments like (Fermilab), the University of Illinois, the the Supernova Acceleration Probe (SNAP)4 University of Chicago, Lawrence and the Large Synoptic Survey Telescope Berkeley National Laboratory (LBNL) (LSST).5 and the Cerro Tololo Inter-American Among the four methods to be used in Observatory (CTIO) was formed in 2003. DES, weak lensing study is for mapping the DES will use four independent methods spatial distribution of large-scale structures to study the nature of the dark energy: (1) and their evolution in time. It can be used to a galaxy cluster survey extending to a study the cosmic shear, which is a measure redshift z ~ 1.1 and detecting ~ 30000 of the gravitational lensing effect of the clusters, (2) a weak lensing study of the matter distribution in the nearby universe on cosmic shear extending to large angular distant galaxies. Gravitational lensing was scales, (3) a galaxy angular power first studied by Einstein in 1912, well before the 1919 eclipse verification of from these information. In addition, how the Einstein’s formula for light deflection. uncertainties in image shape and orientation One of the consequences of Einstein’s propagate into the errors in the lens’ general theory of relativity is that light position and mass is studied. rays are deflected by gravity. The deflection angle is actually twice the II. Method result from Newtonian mechanics, the The propagation of light in arbitrary factor of two arising because of the curved spacetimes is in general a curvature of the metric. complicated theoretical problem. However, Because matter distorts spacetime, for almost all cases of relevance to the path of light from distant galaxies is gravitational lensing, it can be assumed that altered by concentrations of matter, the overall geometry of the universe is well much like a lens focuses light. In general described by the Friedmann-Lemaître- these lensing path changes are quite Robertson-Walker metric and that the small, and the net effect is to slightly matter inhomogeneities which cause the stretch or distort the image of a lensing are no more than local perturbations. background galaxy. By mapping galaxy Light paths propagating from the source shapes and orientations over large past the lens to the observer can then be regions of the sky, it is possible to infer broken up into three distinct zones. In the the matter distribution, be it ordinary or first zone, light travels from the source to a dark matter, in the nearby universe. point close to the lens through unperturbed However, it is not without problems, spacetime. In the second zone, near the lens, with earth-bound observation in light is deflected. Finally, in the third zone, particular. Atmospheric turbulence, light again travels through unperturbed tracking errors, and jitters in the spacetime. In this project, point-mass lens telescope all affect the image quality and of mass M lying in a lens plane is in turn generate uncertainties in mass considered. For a position in the lens calculation. Only with smaller CCD pixels, improved seeing conditions at the plane, its light deflection angle close to the telecope sites and improved image lens is also a two-component vector. In this quality of the telescope optics, the weak special case of a circularly symmetric lens, lensing effect could ultimately be the coordinate origin can be shifted to the measured. In this paper, I will look into centre of symmetry and so light deflection is the weak lensing effect on image shape reduced to a one-dimensional problem. The and orientation, and also how to deflection angle then points toward the ˆ calculate the mass of a point-mass lens centre of symmetry, and its magnitude is 4GM The distance measures are angular- ˆ , (1) c 2 diameter distances, which is where ξ is the distance from the lens, G c 1 z 1 (4) D d z . universal gravitational constant, and c H 1 z 0 M 1 z M 11 z 3 2 speed of light. The geometry of the As D is a function of Hubble’s parameter H, gravitational lens system is shown in Ω Λ , Ω M and z, distance of the lens from the Figure 1. observer D d depends on H 0 , Ω Λ0 , Ω M0 , the corresponding current values and the lens’ redshift z d , distance of background galaxy D s on H 0 , Ω Λ0 , Ω M0 and its redshift z s , and distance between the source and the lens D ds on H d , Ω Λd , Ω Md , the corresponding values at the lens and the redshift of the source as observed from the lens z ds . H d is determined by Hd H0 0 M 0 1 zd 0 M 0 11 zd 3 2 , (5) Ω Λd by 2 H d 0 0 , (6) Hd Figure 1 Illustration of a gravitational lens and Ω Md by system. 2 H0 Md M 0 1 zd 3 The angles are related by a lens . (7) Hd equation: In this study, weak lensing regime is 2 E , (2) considered, where shear << 1 with just a slight distortion of image. There is no where β and θ are respectively the multiple image formation which is usually angular separations of the source and the found in the strong lensing effect. Besides, image from the optic axis as seen by the standard cosmological model – ΛCDM observer. θ E is the Einstein radius given model is used, in which universe is flat, w = by -1, Ω Λ0 ≈ 0.7, and Ω M0 ≈ 0.3. A point mass 4GM Dds is assumed in the lens system. As single E . (3) c 2 Dd Ds image gives no information on the lens characteristics, I apply statistical analysis to the images of a cluster of distributed according to that found by computer-simulated background sources. Yannick Mellier,6 depicted in Figure 3. Physical quantities to be measured are ellipticity ε and orientation ϕ of an 40 image. Ellipticity is defined as b 1 , (8) 30 a where a and b are the semi-major and semi-minor axes of an elliptical image. 20 Orientation of an image is the angle between its major axis and a line joining the lens and its centre, and π < ϕ ≤ 0. 10 Initially, background galaxies are assumed to be intrinsically circular. By 0 0.0 0.2 0.4 0.6 0.8 1.0 using the lens equation, a distribution of Figure 3 Distribution of source intrinsic ellipticities. image ellipticities ε i versus their angular separations d i from the lens is found, as An ellipticity matrix S is found for each shown in Figure 2. source 1 Image ellipticity, ec cos s sin s 0 cos s sin s S 1 s sin sin s cos s cos s 1 2 2 0 s 0.5 (9) 0.4 5M 1 s sin s 2 s cos s sin s 1 s 1 s 0.3 M =1014 MŸ cos sin 1 s cos 2 s s s s 0.2 1 s 1 s 0.1 0.1M In the weak lensing limit, a matrix L for the Image distance, dic arcsec 50 100 150 200 lens is given by Figure 2 Image ellipticity versus distance. 1 d sin 2 d d cos d sin d In the graph, M = 1014 M ʘ , and the 1 d 1 d L , (10) 22 cos sin 1 d cos 2 d effect of varying M is included. d d d 1 d 1 d In the second part, sources are generated with intrinsic ellipticities and where random orientations such that the average ellipticity of the cluster is close to zero. The source ellipticities ε s are 2 the weak lensing regime. More sources and E d , (11) their images are simulated in Figure 5. 100 and y d tan 1 . (12) 2 x 50 Image ellipticity matrix I is then calculated by M =1014 MŸ y arcsec 0 I = LS. (13) ε i and ϕ i are found from I’s eigenvalues and eigenvectors respectively. -50 III. Results and Discussion -100 Circular background galaxies are - 100 - 50 0 x arcsec 50 100 generated randomly around a point-mass Figure 5 Simulated circular sources and images. lens of M = 1014 M ʘ . Their image positions and shapes are calculated by A more informative plot is a shear map the lens equation. The resulting graph is which represents shapes by line segments of shown in Figure 4. 100 e=0.1 20 50 10 M=1014 MŸ y arcsec 0 y arcsec 0 M =1014 MŸ -50 -10 -20 -100 - 100 - 50 0 50 100 x arcsec - 20 - 10 0 10 20 x arcsec Figure 6 Shear map corresponding to Figure 5. Figure 4 Circular sources and their images. different lengths scaled according to their The images with banana-shape are not ellipticities. The shear map corresponding to considered in my study as they are not in Figure 5 is given in Figure 6. When sources’ intrinsic ellipticities are taken into However, it is time-consuming and account, their shear map is illustrated in technically difficult to do analysis on so Figure 7. many data points. Therefore, the data are binned and also ϕ is incorporated into my Simulated Elliptical Background Galaxies and Their Images 2nd y arcsec calculation, i.e. I work with the ellipticity 60 matrices S, L, and I. The results are shown 40 in Figure 9. 20 Ellipticity x arcsec M 0.6 - 60 - 40 - 20 20 40 60 0.5 -20 0.4 -40 0.3 -60 0.2 0.1 Figure 7 Shear map when sources have intrinsic Distance 50 100 150 200 250 ellipticities. Figure 9 Binned data of ε against d for both sources and images. 1000 galaxies are simulated with random positions and orientations, and χ2-fitting is used to do data analysis. It is distributed intrinsic ellipticities. The given by information on ε i and ϕ i for each image 2 is found. First image orientation ϕ i is not N i ,observed i ,theory . 2 (14) taken into consideration, a scatter plot of i 1 i ε against d for both sources and images χ2 for varying position (x M , y M ) and M of is drawn in Figure 8. the lens are shown in Figures 10, 11 and 12 Ellipticity respectively. 0.7 c2 0.6 250 0.5 200 0.4 0.3 150 0.2 100 0.1 Distance 50 50 100 150 200 Figure 8 Scatter plot of ε versus d for both xm arcsec -20 - 15 -10 -5 5 10 sources and images. Figure 10 χ2 versus x M . c2 known as the lens plane is divided into 20.5 equal square regions for binning data. The 20.0 size is 100 by 100 arcsec2. In the simulation, 19.5 sources with position |x| < 20 arcsec or |y| < 19.0 20 arcsec are not included. The resulting ym arcsec plot is depicted in Figure 13. -10 -5 5 10 Figure 11 χ2 versus y M . c2 105 c2 100 350 95 300 90 250 85 200 150 80 100 xm arcsec -20 - 10 10 20 M MŸ 1. μ 1014 1.5 μ 1014 2. μ 1014 Figure 14 χ2 versus x M (2D fitting). Figure 12 χ2 versus M. The set of χ2 plots for lens position is given In Figure 10, the graph is skewed and in Figures 14 and 15, not a perfect parabola because the c2 images are assumed to be distributed in one-dimensional positive x-axis. In view 105 of this problem, a two-dimensional 100 fitting is employed. The x-y plane, 95 90 85 80 ym arcsec -20 - 10 10 20 Figure 15 χ2 versus y M (2D fitting). and the corresponding contour plot in Figure 16. The minimum χ2 values for x M and y M are respectively 74.6143 and 74.4879, which are larger than the expected Figure 13 Scatter plot for 2D distribution of both value of 35. In Figure 16, the lens position sources and images. is estimated to be a bit off the origin. For lens solar masses, the mass of a galaxy cluster, to 1015 solar masses, that of a galaxy supercluster. The relation between % uncertainty in lens mass estimation and M is shown in Figure 18. As M becomes larger, the corresponding % uncertainty decreases. % uncertainty æ 60 50 40 æ 30 20 æ 10 M MŸ Figure 16 χ2 contour plot for lens position. 2. μ 1014 4. μ 1014 6. μ 1014 8. μ 1014 1. μ 1015 Figure 18 % uncertainty against lens mass. 2 mass, Figure 17 is a plot showing χ versus M. The minimum χ2 is 74.0731, In my simulation, the data did not give the best-fit mass 1.0954 x 1014 solar a good fitting (large χ2 value) to the masses, and σ M 0.6125 x 1014 solar theoretical ellipticity surface. The reason for masses. this can be that the bin size of 100 by 100 c2 arcsec2 is not small enough. To improve this, 105 smaller size, for example 50 by 50 arcsec2, 100 is used in a follow-up simulation, for the 95 bins around the best-fit lens position in 90 particular. Also my results showed that more 85 massive lens gave a better estimation of its 80 M MŸ mass. This is because there is a bigger 1. μ 1014 1.5 μ 1014 systematic change in image distortion on Figure 17 χ2 versus M (2D fitting). top of the shape noise mainly contributed by the presence of intrinsic ellipticities in Percentage uncertainty is calculated by background galaxies. This constitutes a M more easily and accurately detectable signal % uncertainty 100% . (15) M in studying weak lensing effect. Its value for M = 1014 solar masses is My project can be extended to study 61.25%. the weak lensing by a continuous mass I varied the lens mass from 1014 distribution like galaxy clusters and large-scale structure. Further work can thank The Chinese University of Hong be done on the effect of atmospheric Kong for providing me with this research condition, CCD pixel size and opportunity and financial support. telescope’s tracking error on image quality and in turn on the measurement VI. References of weak lensing effect. In addition, how 1. S. Perlmutter et al., Astrophys. J. 517, errors in redshift measurement propagate 565 (1999). to the uncertainties in lens mass and 2. Adam G. Riess et al., Astron. J. 116, position estimation can by studied. 1009 (1998). 3. James Annis. DES Plone - The Dark IV. Conclusions Energy Survey [Internet]. Batavia (IL): The background galaxies have a Fermi National Accelerator Laboratory, distribution of intrinsic ellipticities Experimental Astrophysics Group; which constitutes a shape noise and 2004 Aug 31 [modified 2008 May 19; should be taken into account in studying cited 2008 Jul 24]. Available from: weak lensing. The shape or distortion of http://www.darkenergysurvey.org/. an image is described by its ellipticity 4. SNAP [Internet]. Berkeley (CA): and orientation, both of which are Lawrence Berkeley National important in doing statistical analysis of Laboratory; [cited 2008 Jul 24]. a cluster of images. Available from: http://snap.lbl.gov/. In my study, the data did not give a 5. LSST – Home [Internet]. Tucson (AZ): reasonable χ2 fit. This problem can LSST Corporation; [cited 2008 Jul 24]. probably be solved by using a smaller Available from: http://www.lsst.org/lsst bin size, especially for the bins near the 6. Yannick Mellier, Annu. Rev. Astron. expected lens position. The study also Astrophys. 37, 127 (1999). showed that heavier lens had smaller % uncertainty in its mass estimation. V. Acknowledgments I am sincerely grateful to Prof. Jon Thaler for his guidance and patience. His advice was full of insights into the subject of my research. The REU program is supported by National Science Foundation Grant PHY-0243675. I would also like to

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