Ast 8110 Problem Set #6 

Ast 8110
Problem Set #6 
1. Weak shear lensing.
In this problem we will construct a shear map of idealized background cosmic wallpaper consisting of a grid of regularly spaced, initially circular sources, of radius 0.01. Let the grid span
0 → 1 in both x and y directions on the sky. Place your sources on a grid, spaced 0.05 apart, so
that there are 21 sources along each axis. Ignore the deflection of the images w.r.t. the source
positions, i.e., assume that images appear at the locations of corresponding sources.
Let the foreground mass distribution be represented by a sum of 3 singular isothermal spheres
at these locations: (0.44, 0.44), (0.51, 0.51), (0.58, 0.58). The combination of these 3 circular lenses
approximates an elongated lens. Recall that lensing potentials from different mass distributions
add up linearly.
We derived all the necessary lensing quantities for a singular isothermal sphere in class:
Ψ = b[x
2 + y
2
]
1/2
, and κ =
b
2R
, γ1 =
b[y
2 − x
2
]
2R3
, γ2 =
−bxy
R3
,
where R =
p
(x
2 + y
2) is the distance from the lens, on the plane of the sky. Assume b = 0.07, and
use expressions for reduced shear derived in class.
Plot a grid of lensed source shapes (sheared ellipses) due to this mass distibution. Note that in
the central region of your image plane some shear ellipses may become very elongated.
2. Weak magnification lensing.
Weak magnification by a factor µ stretches the whole area behind the lens by µ. As we discussed
in class, this has two effects on a population of background sources: it magnifies each source by µ,
and at the same time stretches the area behind the lens by the same factor, thereby diluting the
sky projected number density of sources.
Let the intrinsic (unlensed) luminosity function (LF) of sources, all located at some redshift
behind the lens, be given by
nU (f) = (f
s1 + f
s2
)
−1
,
where nU (f) is the unlensed number density of sources with fluxes in a small interval around f,
and s1 and s2 the slopes of the faint and bright end of the LF. Assume s1 = 0.3 and s2 = 3.0, and
the LF applies between f = 0.05 and f = 10.0.
Suppose these sources are behind a lens of magnification probability distribution of a Gaussian
form:
p(µ) dµ =
1
q
πσ2
µ
exp(−[µ − µ0]
2
/σ2
µ
) dµ.
The lensed LF is a convolution of the unlensed LF, nU (f) with magnification PDF, p(µ), and
also includes the area dilution effect:
nL(f) = Z
nU (f /µ) p(µ)
µ
dµ.
Numerically calculate the lensed LFs and plot them on the same plot with the corresponding
unlensed LF:
(a) µ0 = 2.5, σµ = 0.5;
(b) µ0 = 0.7, σµ = 0.3.
Make sure that the break in the LF is roughly in the middle of the plotted flux range, and the flux
range extends over at least 2 decades in f. Please make your plots log-log.