My 2008 Summer REU Paper

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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  11  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 11  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)
                                                                                              22   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