2. 2
This thesis titled
Effect of Wavelength Dependent Point Spread Function on Shear Measurements
by
RIFFAT MUNIR
has been approved for
the Department of Physics and Astronomy
and the College of Arts and Science by
Douglas Clowe
Associate Professor of Physics and Astronomy
Robert Frank
Dean,College of Arts and Science
3. 3
Abstract
MUNIR, RIFFAT, M.S., August 2016, Physics
Effect of Wavelength Dependent Point Spread Function on Shear Measurements (54 pp.)
Director of Thesis: Douglas Clowe
Weak lensing is one of the powerful tools for measuring mass energy content of
galaxies and galaxy cluster. Light is deflected by gravity when passing by massive
objects(galaxy,galaxy cluster). Shear measurement due to distortion of light depends upon
the Point spread Function(PSF) which is caused by telescope optics and atmosphere.
Previous work had been done on how monochromatic PSF affects shear measurements.
But each star has different spectral energy distribution and galaxy has internal colour
gradient. We have taken these two effects into consideration in our study. To find the
dependence upon wavelength of PSF we took different stellar spectra. The AB magnitude
was calculated using two infrared regions(1.24-1.57)µm and (1.57-2.00)µm. We found
that the size of PSf does not vary linearly with wavelength. This implies that based upon
stellar spectra we can not estimate the PSF for a galaxy. In the simulation we considered
one wavelength range in the infrared region(1.57-2.00)µm and divided the region into 21
parts . We took galaxies which have internal colour gradients.The convolution was done
with wavelength dependent PSF. We found that the ratio of monochromatic and chromatic
shear vs radius from the center of lens, the ratio varies around 0.1 but needs to be below
0.001. This implies that the colour dependence of PSF and galaxy internal colour
gradients needs to be studied more.
5. 5
List of Tables
Table Page
3.1 Logarithmic Flux and Gaussian Radius for Different Stars . . . . . . . . . . . . 43
6. 6
List of Figures
Figure Page
1.1 Deflection of light of an object from source to observational plane source. The
source is at angular diameter distance Ds and the lens is at distance Dl. The
deflection angle is α(θ) Rebecca Santana [9] . . . . . . . . . . . . . . . . . . 11
1.2 Transformation of a circular source to an elliptical source , convergence causes
transform it in to a circle where shear causes different axis ratio from the
original source . image from Sneider [8]. . . . . . . . . . . . . . . . . . . . . . 15
1.3 Change in shape for different values of ellipticity , the x and y axis denote
ellipticity e1 and e2 respectively source Sneider [8] . . . . . . . . . . . . . . . 18
1.4 SED of a G5V star and SA galaxy Figure From. The solid black curve shows
the spectra whereas the rainbow coloured region shows the SED when it’s taken
in LSST filters(r and i band) Meyers and Burchat [3] . . . . . . . . . . . . . . 21
1.5 Various contribution on psf from different term in the optical region. As can
be seen from figure the diffraction limited term would more more dominated
in the infrared regions. Figure from Cypriano et al. [2] . . . . . . . . . . . . . 22
1.6 Full width half maxima of stars and galaxies at diffrerent filter where F1 is the
broadband filter Y1 is the space based filter in infrared region and r1 is the filter
in optical region and Cypriano et al. [2]. The inset shows the residuals between
galaxy and stellar polynomial FWHM values at a given color. . . . . . . . . . . 23
1.7 Disc(blue colour) and bulge(red colour) spectra for two different filter. Figure
from Voigt et al. [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.8 Variation of multiplicative and additive bias as a function of filter width source
Voigt et al. [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.9 Variation of multiplicative bias as a function of different galaxy parameters
Voigt et al. [1]. Here ns,b is galaxy sersic index,B/T is bulge to total flux ratio,eg
is the ellipticity of galaxy and y0 is the position of peak intensity relative to the
center of the postage stamp . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.1 A sample postage stamp(upper panel) and postage stamp transformed accord-
ing to the parameters in catalog(lower panel) . . . . . . . . . . . . . . . . . . . 32
2.2 Convolved and distorted galaxy image . . . . . . . . . . . . . . . . . . . . . . 37
3.1 Integrating flux from SED files for different wavelength . . . . . . . . . . . . . 41
3.2 Variation of gaussian radius of the Psf for different flux . . . . . . . . . . . . . 44
3.3 Spectral energy Distribution of a typical G and L star . . . . . . . . . . . . . . 45
3.4 Shear and smear polarisibility vs color. The color is defined as the logarithmic
ratio of flux calculated in two different regions. . . . . . . . . . . . . . . . . . 46
3.5 Ratio of shear and smear polarisibility vs color. The color is defined as the
logarithmic ratio of flux calculated in two different regions. . . . . . . . . . . 47
3.6 Ratio of chromatic and monochromatic ellipticity vs radius(pixels) . . . . . . . 49
7. 7
3.7 Variation of chromatic and monochromatic Pγ in different bins . . . . . . . . . 51
3.8 Ratio of chromatic and monochromatic shear as a function of radius(pixels) . . 52
8. 8
1 Introduction
According to the standard model of cosmology dark matter is one of the dominant
components of the total mass energy content of the Universe. Dark matter is mainly
dominant in large scale structure(galaxy and galaxy clusters ). Anisotropy in the cosmic
microwave background,cosmic structure formation,galaxy formation and evolution
suggest the presence of dark matter. As it does not interact with electromagnetic radiation
and visible matter the method it can be detected is by measuring it’s effect on ordinary
baryonic matter. Gravitational lensing provides us a way to see how dark matter along
with visible matter is distributed in a galaxy or galaxy cluster. This is predicted by
Einstein’s general theory of relativity which predicts the deflection of light in a
gravitational field produced by a massive object .
1.1 Gravitational lensing
The function of a lens is to bend light when the light passes through lens. In
gravitational lensing light from the background galaxy sources is deflected by the tidal
gravitational field of a foreground object .
Gravitational lensing is divided into three classes:
Strong lensing: If the background source is aligned with the foreground object and if
the foreground galaxy is massive enough,it can produce multiple images of background
objects which is called strong lensing. The strength of a lens depend upon the critical
surface mass density. Examples of strong lensing are Giant luminous arc and Einstein ring
Weak lensing: The primary focus of this paper is on weak lensing . So how can we
distinguish between strong and weak lensing? The answer lies in the critical mass density.
When the background source is not aligned with the lens , the foreground lens it is not
able to distort light strongly to produce multiple images. In weak lensing , only one image
of background source is produced. Weak lensing is mainly statistical in nature. Observing
9. 9
distortion of one source we can’t strongly determine the mass distribution of a given
galaxy.The galaxies have intrinsic ellipticity which is also one of the main sources of
noise. A large sample of background sources is needed to observe the net distortion
caused by lens .
Microlensing: When the lens object is a star instead of a galaxy the microlensing
effect is observed . Here the multiple images are so close that it cant be observed as
separate image . The lensing effect can be detected by observing the lens position at the
different times on the sky.Microlensing was mainly used to detect exoplanet,brown
dwarf,neutron stars . It was also used to see whether the compact object makes the amount
of dark matter sufficient to explain the flat rotation curve for Milky way galaxy .
Use of lensing: Lensing is related to the gravitational field. From Poisson’s equation
we see that the gravitational field depends on the mass distribution . As lensing does not
depend on the nature of the matter (luminous or dark), it is an ideal tool for measuring
mass distribution of an object. Due to the bending of space , light travels different distance
for different images. This time delay can be used to measure the Hubble parameter by
strong lensing .The sensitivity of weak lensing to different variables( for example density
fluctuations) enables constraints on different cosmological parameters . The probability of
a lensing event depends on the number density of lenses. So observing total number of
lensing effects enable us to estimate compact objects in dark matter halo , redshift
evolution of the galaxies producing strong lensing . As the object appears brighter due to
magnification after lensing spectroscopic properties of the lensed galaxies can be inferred
from the image . Thus lens can act as a natural telescope .
10. 10
1.2 Weak lensing formalism
Figure 1.1 shows the example of lensing. All the distances measured are angular
diameter distance which is defined as ratio of an object’s original size to its observed
angular size. The source galaxy is located at distance Ds or at redshift zs .The lens plane is
at a distance Dl from the observer. The distance between source and lens is Dls .
In an expanding universe
Dl + Dls Ds (1.1)
Light from the source plane is deflected when passing through the lens plane by a
deflection angle ˆα(ξ). This angle is predicted by Einstein’s general theory of Relativity
ˆα(ξ) =
4GM
c2ξ
. (1.2)
Where M is the mass of the lens deflecting the light, G is Newton’s Gravitational constant,
c is the speed of light and ξ is the impact parameter.
Now from figure 1.1 we see that
η + δη
Ds
= θ (1.3)
ξ
Dl
= θ (1.4)
Here δη is distance between the source and the image and θ is the angular position of the
image.
For the same value of θ we can equate eq.(1.3) and eq(1.4)
η + δη
Ds
=
ξ
Dl
(1.5)
The angle δη is given by
δη = Dls ˆα(ξ) (1.6)
11. 11
Figure 1.1: Deflection of light of an object from source to observational plane source. The
source is at angular diameter distance Ds and the lens is at distance Dl. The deflection angle
is α(θ) Rebecca Santana [9]
If we now substitute the value of δη from eq.(1.6) to eq.(1.5) we get
η =
Dsξ
Dl
− Dls ˆα(ξ) (1.7)
The angular position of the source β with respect to the observer is related to η by
η = Dsβ (1.8)
12. 12
The parameter ξ is related to the angular position of the image θ by
ξ = Dlθ (1.9)
If we substitute the value of η and ξ from eq.(1.8) and eq.(1.9) to eq.(1.7) we get
β = θ −
Dls
Ds
ˆα(Dlθ) (1.10)
So from the above equation we see that if the source is at angular position β we see an
image at angular position θ. In the strong lensing case eqn.(1.10) has multiple solution θ
that can result same β i.e. we see multiple image for the same source.
The strength of a lens depend on the dimensionless surface mass density or
convergence which is defined by
κ(θ) =
(Dlθ)
cr
(1.11)
Here cr is the critical surface mass density . If (Dlθ) > cr then κ(θ) > 1
In terms of κ(θ) the scaled deflection angle α(θ) reads
α(θ) =
1
π
d2
θ κ(θ )
θ − θ
|θ − θ |2
(1.12)
α(θ) can be expressed as deflection of gravitational potential ψ(θ) as
α = ψ (1.13)
so ψ would be
ψ(θ) =
1
π
d2
θ κ(θ ) ln |θ − θ | (1.14)
where ψ ,the gravitational potential is derived from Poisson’s equation
2
ψ(θ) = 2κ(θ) (1.15)
13. 13
1.3 Magnification and distortion
The shape of the image differs from that of the original galaxy . The change in the
shape is described by the shear
γ = γ1 + iγ2 = |γ| exp(2iϕ) (1.16)
where γ1 and γ2 are its two components and ϕ is the phase .
As there is no absorption or emission of photons the surface brightness of the image
would be same as that of source
I(θ) = Is
[β(θ)] (1.17)
If θ0 is a point within an image corresponding to the point β0 = β(θ0) then the Taylor
expansion around θ0 gives
β(θ) = β0 + (θ − θ0)
∂β
∂θ
(1.18)
We can define the distortion matrix A(θ) in terms of ∂β
∂θ
as
A(θ) =
∂β
∂θ
= (δij −
∂2
ψ(θ)
∂θi∂θj
)
=
1 − ψ,11 −ψ,12
−ψ,21 1 − ψ,22
We define our shear component as
γ1 =
1
2
(ψ,11 − ψ,22)
γ2 = ψ,12
From eqn.(1.15) we can write the surface mass density κ as
κ =
1
2
(ψ,11 + ψ,22) (1.19)
14. 14
So the distortion matrix interms of γ1,γ2 and κ
A(θ) =
1 − κ − γ1 −γ2
−γ2 1 − κ + γ1
(1.20)
In terms of distortion matrix A(θ) eqn.(1.17) can be written as
I(θ) = Is
[β0 + A(θ)(θ − θ0)] (1.21)
This equation represents an ellipse. The ratio of the radius of the circular source to
that of semi axes of lensed images are
a =
R
1 − κ − |g|
(1.22)
b =
R
1 − κ + |g|
(1.23)
Where 1 − κ ± |g| are the eigenvalues of the matrix A.
The magnification tensor is determined by the inverse of the Jacobian of A
µ(θ) = A−1
(1.24)
We can see from figure 1.2 that how a circular source transforms into an ellipse .
1.4 Reduced shear
We consider our galaxy image as separated (i.e. it’s not influenced by other objects) ,
so the surface brightness at a given point on the galaxy depends only on that position.
Suppose I(θ) is the surface brightness at angular position (θ). Blanford et al. [4]
defined the center of the image as
¯θ =
d2
(θ)qI[I(θ)]θ
d2(θ)qI[I(θ)]
(1.25)
15. 15
Figure 1.2: Transformation of a circular source to an elliptical source , convergence causes
transform it in to a circle where shear causes different axis ratio from the original source .
image from Sneider [8].
Here qI[I(θ)] is suitably chosen weight function. Once the weight function is chosen
the tensor of second brightness moment is defined as
Qij =
d2
(θ)qI[I(θ)](θi − ¯θi)(θj − ¯θj)
d2(θ)qI[I(θ)]
(1.26)
From the definition of tensor of second brightness moment Schneider and Seitz [7].
defined the complex ellipticity as
χ =
Q11 − Q22 + 2iQ12
Q11 + Q22
(1.27)
The complex ellipticity can also be written in terms of the axis ratio r and position angle ν
of elliptical isophote
χ =
1 − r2
1 + r2
e2iν
(1.28)
So we can see that if the image is rotated by π the complex ellipticity remains unchanged .
Center of the source ¯βs
and tensor of second brightness moment Qs
ij can be similarly
defined as that of image.
16. 16
Qs
and Q are related by
Qs
= AQAT
(1.29)
Where A is the Jacobian matrix of the lens equation at position θ .
Schneider and Seitz [7] defined the complex ellipticity of the source χs
in terms of
the complex ellipticity of the image χ as
χs
=
χ − 2g + g2
χ∗
1 + |g|2 − 2R(gχ∗)
(1.30)
In eqn(1.30) g is the reduced shear
g(θ) =
γ(θ)
1 − κ(θ)
(1.31)
When g is sufficiently small that O(g2
) term can be neglected, then equation (1.30)
gives
χs
= χ − 2g (1.32)
Now from equation(1.20) the distortion matrix A(θ) is
A(θ) =
1 − κ − γ1 −γ2
−γ2 1 − κ + γ1
= (1 − κ)
1 − γ1
1−κ
− γ2
1−κ
− γ2
1−κ
1+γ1
1−κ
= (1 − κ)
1 − g1 −g2
−g2 1 + g1
(1.33)
17. 17
Bonnet And Miller [5] defined another ellipticity parameter when the value of g becomes
comparitively large so ellipticity χ can’t be used
ε =
Q11 − Q22 + 2iQ12
Q11 + Q22 + 2(Q11 + Q22 + 2 [Q11Q22 − Q2
12])
(1.34)
ε and χ are related through
ε =
χ
1 + [1 − |χ|2]
1
2
(1.35)
χ =
2ε
1 + |ε|2
(1.36)
The transformation between source and image ellipticity is then given by
εs
=
ε − g
1 − g∗ε
if|g| ≤ 1
=
1 − g∗
ε
ε∗ − g∗
if|g| > 1
(1.37)
We can get an idea how the shape of an object changes with their two component
ellipticity by figure-1.3.
From equation (1.37) we see that the expectation value is
E(ε) = g if|g| ≤ 1
= 1/g ∗ if|g| > 1
(1.38)
Thus ellipticity provides the information about local shear but it is greatly affected by
the noise which comes from the intrinsic elliptical shape of galaxy images. The noise σ
in the intrinsic ellipticity dispersion is given by
σ =
√
< s s∗ > (1.39)
Schneider and Seitz [7] showed that the error can be written as
σ = σε
[1 − min(|g|2
, |g|−2
)]
√
N
(1.40)
18. 18
Figure 1.3: Change in shape for different values of ellipticity , the x and y axis denote
ellipticity e1 and e2 respectively source Sneider [8]
As the total number of galaxy images N increases the noise becomes smaller. So
during the observation it’s necessary to take as large number of galaxy images as possible
so that noise can be reduced.
In general in the weak field regime
γ ≈ g ≈< ε >≈
< χ >
2
(1.41)
1.5 PSF correction
If we observe a point through telescope it looks a like smeared object. This happens
due to effects of telescope optics , Atmosphere etc. This effect is described by the function
known as the point spread function(PSF).
19. 19
So if I(ϑ) is the original brightness profile then the observed brightness profile would
be
Is
(ϑ) = dϑI(ϑ)P(θ − ϑ) (1.42)
In general the PSF is a bell shaped function whose full width half maxima is called the
seeing of the image. In weak lensing, we mainly consider a large number of faint galaxy
sources. If the sources are smaller than the ’seeing’ size of PSF their shape is dominated
by the PSF .
The effect of the PSF primarily is to smear and make an elliptical source rounder. So
a small source with large ellipticity would appear as a round object if its size is smaller
than the PSF. If the PSF has anisotropy i.e. it’s not a smooth function then due to this
anisotropy a round object would look more elliptical which mimics shear. This anisotropy
is large for small object.So if it’s not corrected properly then shear can be misestimated
In KSB approximation of distortion by PSF can be described by a small but highly
anisotropic kernel convolved with a large circular symmetric seeing disc . The ellipticity
of a PSF corrected galaxy is then given by
εcor
α = εobs
α − Psm
αβ Pβ (1.43)
Where Pβ is the vector which measures PSF anisotropy and Psm
is the smear polarisability
tensor which depends upon weighted moments
As the star has zero ellipticity εcor
= 0 then the above equation gives us
Pβ = Psm∗−1
αβ ε∗obs
α (1.44)
This gives the estimate of anisotropy p(θ) at stellar position θ .
Lupino and Kaiser [6] Proposed shear polarisability tensor Pγ
which is related
through the shear with ellipticity
20. 20
εcor
α = εs
α + Pγ
αβγβ (1.45)
Where εs
α is intrinsic ellipticity and γ is pre seeing gravitational shear .
Lupino and Kaiser [6] have shown that in terms of galactic shear polarisability tensor
Psh
Pγ
αβ = Psh
αβ −
Psh∗
δβ
Psm∗
µδ
Psm
αµ (1.46)
Where Psh∗
and Psm∗
denote the stellar smear and shear polarisability.
If we substitute the value of Pγ
from equation (1.39) to eq.(1.38) and use (1.36) we
get the estimate for shear
ˆγβ = Pγ
αβ
−1
[εobs
α − Psm
αβ Pβ] (1.47)
1.6 Dependence on wavelength of Psf
We defined the second order moments in section 1.0.4 which depends on the surface
brightness profile or intensity. Intensity is determined from spectral energy
distribution(SED). SED of stars and galaxy have different shapes and profiles which we
can see from Figure 1.4. The top figure shows the SED of a typical G5V star and bottom
one shows the SED of a spiral galaxy. So if we use the SED of a star instead of a galaxy to
estimate PSF that varies with wavelength we will get wrong shear measurement when this
PSF is applied.
Cypriano et al. [2] discussed the dependence upon wavelength of the PSF. They
considered the PSF to be made up of several components
F2
psf (λ) = F2
D(λ) + F2
MTF(λ) + F2
J (1.48)
where the terms which contributes are
21. 21
Figure 1.4: SED of a G5V star and SA galaxy Figure From. The solid black curve shows
the spectra whereas the rainbow coloured region shows the SED when it’s taken in LSST
filters(r and i band) Meyers and Burchat [3]
1)FD(λ) → nearly diffraction limited telescope optics
2)FMTF(λ) → the CCD modulation transfer function
3)FJ → wavelength independent part such as telescope jitter
As UCLID was a space based telescope they neglected atmospheric effect.
They plotted the various terms and total contribution as a function of wavelength
which can be seen in figure 1.5.
As we can see as the wavelength increases the diffraction term becomes more and
more dominant. Here the wavelength range is in optical region. Our main aim is to seee
how this terms go to infrared regions .
22. 22
Figure 1.5: Various contribution on psf from different term in the optical region. As can be
seen from figure the diffraction limited term would more more dominated in the infrared
regions. Figure from Cypriano et al. [2]
Cypriano et al. [2] assumed that the PSF contributions from different wavelegths have
the same centroid. Based upon this assumption, they calculated FWHM(Full width Half
Maxima) of the composite PSF from the FWHM of each component and the transmitted
flux S (λ)T(λ) where S (λ) is the spectral energy distribution of object and T(λ) is
instrumental plus filter response.
F2
psf =
S (λ)T(λ)F2
psf (λ)dλ
S (λ)T(λ)dλ
(1.49)
Cypriano et al. [2] also observed the FWHM of stars and galaxies in different
filters.They took one broadband filter F1(wavelength range 5500-9200 A◦
),one space
based filter Y1 in infrared region and one ground based filter r1 in optical region.
As we see from figure there is a strong correlation between FWHM and color. We
also see that this correlation is stronger in r − F1 than Y − F1. Comaparing fig 1 and 2 in
23. 23
Figure 1.6: Full width half maxima of stars and galaxies at diffrerent filter where F1 is the
broadband filter Y1 is the space based filter in infrared region and r1 is the filter in optical
region and Cypriano et al. [2]. The inset shows the residuals between galaxy and stellar
polynomial FWHM values at a given color.
the top panel we also see that the bias on PSF size reduces when using the ground based
photometry(r − F1) filter.
Paulin and Henrikson [10] have shown that if the Psf size and Psf ellipticity is
misestimated the systematic bias on a galaxy ellipticity component δ sys
gal,i would be
δ sys
gal,i = (
Rpsf
Rgal
)2
(2( gal,i − psf,i)
δRpsf
Rpsf
− δ psf,i) (1.50)
24. 24
Where gal,i are the original ellipticity component for galaxy and Rgal is the galaxy size.
δRpsf is the misestimation on PSF size and δ psf,i is PSF ellipticity. The misestimated
ellipticity then effects in the measurement of shear.
The observed shear can be written in terms of true shear as
ˆγi = mγi + c (1.51)
Where m is the multiplicative bias and c is the additive bias .
The shear γi is defined as
γi =
gal,i
Pγ
(1.52)
So the change in shear can be written as
δγi =
δ sys
gal,i
Pγ
(1.53)
So the observed shear ˆγi can be written interms of true shear γi
ˆγi = γi + δγi (1.54)
The full width half maxima(F) is related to object size by the relation
F = 2
√
ln2R (1.55)
Here R is the 1 − σ width of Gaussian.
If we substitue eq.(1.51) and (1.52) in equation(1.53) we get
ˆγi =
gal,i
Pγ
+
δ sys
gal,i
Pγ
= (1 + 2(
Rpsf
Rgal
)2 δFpsf
Fpsf
) gal,i −
1
Pγ
(
Rpsf
Rgal
)2
(2 psf
δFpsf
Fpsf
+ δ psf,i)
(1.56)
25. 25
If we compare equation (1.50) with equation (1.55) we get
m = 1 + 2(
Rpsf
Rgal
)2 δFpsf
Fpsf
(1.57)
c =
1
Pγ
(
Rpsf
Rgal
)2
(2 psf
δFpsf
Fpsf
+ δ psf,i) (1.58)
Amara and Refregier [11] showed that for a full sky survery(2 ∗ 104
square degrees
of extragalactic sky) the shear multiplicative error must be below 10−3
to keep systematic
bias on cosmological parameters below random uncertainties.
Voigt et al. [1] mainly discussed how the multiplicative bias changes with respect to
different parameters. They also investigated the impact of galaxy colour gradients on
shear measurements .Galaxy model was chosen so that the disc follows the exponential
profile and bulge follows De vaucouler’s profile. They choose one broadband filter
(550-900 nm) and a narrow filter (725-900 nm). The SED of bulge and disc in these two
filters can be seen in the fig 1.4.
Figure 1.7: Disc(blue colour) and bulge(red colour) spectra for two different filter. Figure
from Voigt et al. [1]
After fixing the parameters Voigt et al. [1] observed how the multiplicative bias
changes as function of filter width. The change can be seen from fig 1.8 .
26. 26
Figure 1.8: Variation of multiplicative and additive bias as a function of filter width source
Voigt et al. [1]
In this figure the red solid line is for fiducial galaxy model where the parameters are
redshift z = 0.9, bulge to disc ratio
re,b
re,d
= 1.1 bulge to total flux B
t
= 0.25. The green dot
dashed line is the one for which width of bulge to disc ratio
re,b
re,d
= 0.4. The blue dashed
one is z = 1.4 As can be seen for a smaller filter width the bias decreases because bulge
and disc spectra are similar when the filter width is small. From the figure we see that the
bias increases when bulge to disc half light radius decreases (the slope of red solid line is
less than that of green dot dashed line). This indicates that the galaxy internal color
gradient strongly affects the bias.
27. 27
Then Voigt et al. [1] investigate how much multiplicative bias can change when some
parameters are varied which can be seen in Figure 1.9 .
Figure 1.9: Variation of multiplicative bias as a function of different galaxy parameters
Voigt et al. [1]. Here ns,b is galaxy sersic index,B/T is bulge to total flux ratio,eg is the
ellipticity of galaxy and y0 is the position of peak intensity relative to the center of the
postage stamp
28. 28
From the figure 1.9 we see that when the bulge sersic parameter ns,b is varied the
multiplicative bias changes at most by a factor of 3.Similarly we see that when the galaxy
ellipticty eg and position of the peak intensity as a relative to the center of the postage
stamp y0 is varied the multiplicative bias does not change that much. But when bulge to
total flux ratio is varied we see a lot of variation in the multiplicative bias.
Voigt et al. [1] discussed mainly how bias varies for different parameters. We want to
see how well we can measure the shear when the PSF is dependent upon wavelength. As
galaxy and stellar spectra are different we need to first correct the stellar psf then apply it
for galactic one to check how the bias decreases. Previous work has been done on bias
estimation but our main focus is now on shear measurement. Voigt et al. [1] took a galaxy
model which follows mainly sersic profile , but we want to see how it behaves for real
galaxies from HST images.
29. 29
2 Implementation Details
From the previous section, we saw that neglecting colour gradient affects shear
measurements . The main target of our simulation is to create a realistic lensing effect.
Galaxies were specified by different parameters and a particular distance from the lens.
We chose comparatively simple model ”singular isothermal sphere”(page 31) for the lens
in the simulation. Initially we considered only monochromatic simulation, i.e. our
simulation ran only once for one wavelength and convolution was done using one PSF.
Then we modified our programme so that instead of running for one wavelength the
simulation now runs for multiple wavelength range. We took the infrared wavelength
range 1.57µm-2.00µm and divided this range into 20 parts. During this step galaxies were
taken which have different flux on two different bands ”F606w”(blue) and ”F818”(red).In
our simulations we mentioned them as ”blue” and ”red” galaxies.Linear interpolation
according to specific wavelength was done by using these galaxies to create galaxy
samples which have a certain percentage of red and certain percentage of blue flux. These
galaxies were then convolved with the PSF of that particular range. We ran the simulation
for 90◦
rotated angle also so that when combined the two output the net intrinsic ellipticity
of the galaxy shape can be reversed.
Our aim is to see how the PSF affects shear measurements . To do so, I used the
programme ”jedisim” written by Ian Dell’ antonio.In this programme we run a python
script named ”jedimaster.py” . This script reads from a text file named ”config” . This text
file contains the information about different parameters. ”jedimaster.py” runs six C
programmes which creates the simulation step by step. These steps are described in the
next pages.
30. 30
2.1 Postage stamps
The images were obtained from an HST UDF image. From that image individual
galaxies are converted into 600 by 600 pixel postage stamps. In the postage stamps the
galaxy is isolated on a blank background. The resolution of the images are 0.03 arc
second per pixel . We took total 128 postage stamps so that our sample contains a diverse
set of galaxy shapes and orientations.
2.2 Making the catalog
The C programme “jedicatalog.c” creates a realistic galaxy catalog . This catalog
contains the information of galaxy images to be created. There were some parameters
which were specified for each galaxy to be created. These parameters are discussed below.
1. Magnitude
We choose the magnitude of each galaxy to be in the range 22≤M≤28 The
distribution is given by Power law distribution:
P(M + dM) ∝ 10BM
(2.1)
where M is the magnitude and B is the constant B = 0.33 We take zero point magnitude to
be M=30 which is the lowest magnitude . For example a galaxy having a magnitude
M=22 would be brighter than a galaxy having magnitude M=25.
2. Radius
HdF catalogs contains database of r50 galaxies. That database was binned by integer
part of the magnitude and a list of radii is made for each magnitude bin . As we have
already assigned a magnitude for our galaxy this magnitude corresponds to a particular
bin and a r50 radius was randomly chosen from that bin.
3. Redshift
31. 31
In general different galaxies have different redshift but for our simulation we take all
the galaxies to be at a single redshift. All the background galaxies are in redshift z = 1.5
and lens is at z = 0.3.
4. Position
The center of the postage stamps were selected from the range [301,40960] in both x
and y direction so that the whole galaxy fits within image and edge effect can be avoided.
5. Angle
As the universe is homogenous and isotropic galaxies are in general randomly
oriented in the sky. The ’jedicatalog.c’ programme randomly choose an angle from the
range[0,2π]. In general the orientation has 3 degrees of freedom but as we are dealing
with 2D projection of galaxies,we can make the orientation random in only 1 degrees of
freedom.
So in general ’Jedicatalog.c’ creates a catalog which contains the name of source
postage stamps,the radius,magnitude,redshift,angle and the galaxy image name which to
be created in the next step.
2.3 Transforming the galaxy according to the catalog
’Jeditransform.c’ takes the catalog and source postage stamps. The postage stamps
are then scaled down to corrected radius, the flux is adjusted for image according to the
magnitude which was specified in the catalog and each galaxy is rotated through the
assinged angle.The postage stamps are then cut out so that the final image contains all the
non zero pixel of the galaxy. In Figure 2.1 we see a sample postage stamp and the same
stamp (lower figure) transformed according to parameter.
32. 32
Figure 2.1: A sample postage stamp(upper panel) and postage stamp transformed according
to the parameters in catalog(lower panel)
33. 33
2.4 Distorting the galaxies
As we have seen before in the weak field limit for a point mass, the lens equation is
β(θ) = θ − α(θ)
= θ −
Dls
Ds
ˆα(θ)
(2.2)
We are considering here the lens as a symmetric mass distribution with the center at
arbitrary position and redshift at z=0.3 . The deflection term in the lens equation becomes
ˆα(θ) = α(r)ˆr (2.3)
Where α(r) is the radial deflection which depends upon the mass distribution.
We are considering the Singular Isothermal Sphere(SIS) profile in the simulation. For
this profile the density is given by
ρ =
σ2
v
2πGr2
(2.4)
Here σv is the velocity dispersion and G is universal gravitational constant. When r→ 0
the density ρ → ∞. So we see that at r=0 this is not a physical situation. But as long as it
is finitely bounded it constitutes a possible physical distribution and can be used as lens.
The deflection in pixels due to an SIS profile is given by
α(r, σv)
r
=
4π
r
(
σv
c
)2
S (2.5)
Where S is the conversion factor between pixels and radians given by
S =
π
180
3600
resolution
(2.6)
So we have seen that the amount of distortion depends upon the distance between the lens
and galaxy. The lens position was selected to be (6144,6144).
34. 34
If we take a galaxy at position(5754,7909) then the distance from the lens to the
galaxy is
r1 = (6144 − 5744)2 + (6144 − 7909)2
= 1809.75pixels
(2.7)
Similarly if we take another galaxy which is at position(690,5217) the distance from
the lens for this second galaxy would be
r1 = (6144 − 690)2 + (6144 − 5217)2
= 5532.21pixel
(2.8)
From equation (2.5) we have seen that deflection is inversely proportional to r ,i.e the
galaxies are at larger position from lens would be deflected less than the galaxies at nearby
position. So according to the value of r1 and r2 galaxy 1 would be distorted more than
galaxy 2.
2.5 Embedding all the unconvolved image
’Jedidistort.c’ simulated the distortion effect in all the galaxies. It also specified two
keywords ’xembed’ and ’yembed’ in the fits image which are the x and y cordinate of
lower left pixel if the individual image is to be embedded on a larger image . ’Jedipaste’
takes all these 12000 images and embeds them on to a larger image which is 12288 by
12288 pixel. Due to computer memory allocation problem ”jedipaste.c” can’t embed all
these large number of images into a single huge image at once . Instead it divides the
larger image(12288,12288) into two equal rectangular parts which was called ’bands’ in
our programme. So band1 has cordinates where x range is (0,12288) and y range is
(0,6144) . Band2 has xrange same as band 1 but y range is from(6144,12288).
35. 35
”jedipaste.c” then takes all these image and embeds images according to their ’xembedd’
and ’yembed’ co-ordinate position in first in band 0 and then in band 1.
2.6 Convolution with PSF
The light from the distant galaxies not only is deflected by the intervening
gravitational field of the foreground sources but also is affected by the telescope optics .
This effect is known as the PSF which we discussed above(Section 1.4).
In general, we can write the convolution of two functions f(x) and g(x) as
f(x) ∗ g(x) = f(y)g(x − y)dy (2.9)
If there are n data points then from the above equation we see that we need to
calculate n2
terms . For an image with size 12288 pixel by 12288 pixel this is not so
practical . Instead we take the Fourier transform of the image and PSF and use the
convolution theorem. The convolution theorem states that
F( f ∗ g) = F(f(x)) F(g(x)) (2.10)
The programme ’Jediconvolve.c’ uses the ’fftw3’ library to implement this idea. This
is a very fast procedure to obtain the convolved image.
In the programme ’jediconvolve.c’ it reads the pixels of the image by using the array
”pImg” and pixels of the PSF using the array ”pPsf”. Under the ’fftw3’ there are some
plans for example we used fftw plan dft r2c 2d. When these plans are executed we get
the Fourier transform of the object.For example we used fftwf execute(pPIMG).This
transforms the image array. Similarly it also does the same procedure for ’PSF’ array.
The image after the convolution is still in the frequency space,so the inverse
transform was taken to get it in real space.
36. 36
As the image ’HST0.fits’ is very large(12288 by 12288 pixel) the programme can not
transform it at once . The programme divides the whole image into 6 bands. so in each
band there are 2048 pixels. Each of these bands is Fourier transformed and then convolved
with the Fourier transform of the PSF and the output of this image was inverse Fourier
transformed. As there are total 6 bands we get 6 images .
2.7 Making convolved image
The”jediconvolve.c” creates 6 images. The jedipaste programme once again takes
these 6 images and reads the x and y co ordinate of the lower left pixel. It then divides the
whole image into 2 rectangular parts ’band 0’ and ’band 1’ and embeds these images
respectively in those two bands and creates one final image “HST convolved.fits” . From
figure 2.2 we can get an idea how the galaxies are embedded in large image.
2.8 Rescaling accroding to the resolution of WFIRST
The resolution of the image “Jedipaste.c” creates is 0.03 arcsecond.But the WFIRST
has resolution 0.11 arcsecond. So “jedirescale.c” mainly scales down this image into a
larger pixelscale and trims off the border.It finds the box which each pixel makes on the
image,integrates over the area under that box,averages and assigns that value to the new
pixel.
Adding noise to the image Finally there is some random noise associated with each
image. ’Jedinoise.c’ added the poisson noise of mean value 10 to the final image
“LSST convolved noise.fits”. We tried to keep noise as low as possible in our simulation
so that it does not effect in shear measurement.
2.9 Changes made to original programme
Implementing wavelength dependence To test the color dependency we took
wavelength at 1.57-2.00 µm range.Then we divided this range into 20 parts . I edited the
37. 37
Figure 2.2: Convolved and distorted galaxy image
Jedisim programme in a way so that unlike the first part instead of running for only one
time now it runs for 21 times.
I wrote one programme ’color.c’ which would take these blue and red images and
creates the final image by linear interpolation
In terms of equation for any run n
38. 38
outputimage = (1 −
n
20
) ∗ blueimage +
n
20
∗ redimage (2.11)
We wrote the formula in this way because the wavelength is going from blue to red.
So when n = 0 i.e. the wavelength is 1.57µm the blue image would contribute most and
when n = 21 (wavelength 2.00 µm)the red images would be dominant .
Previously during the convolution the PSF was monochromatic. But now we need the
PSF which would depend upon the wavelenth. We generated 21 wavelength dependent
PSF from the website.
http : //localhost : 8888/notebooks/WebbPS FWFIRS T Tutorial.ipynb
The PSF’s are wavelength dependent, so as the wavelength increases the FWHM also
increases . It follows the same step as before but instead of one output after “jedirescale.c”
the programme produces 21 outputs. I wrote one programme “avg20.c” which takes these
21 output. It averages them and create one image “LSST convolved.fits”. In the final step
the noise was added using the programme “Jedinoise.c” as before
Similarly the programme also runs for the 90 degree rotated case as before .
39. 39
3 Results
3.1 Weighted average of PSF
One of the main purposes of the study was to observe how the PSF size changes as a
function of wavelength.PSFs are in general estimated using stellar images. For a given
wavelength range different stars have different spectral energy distributions.Finding flux
for a particular PSF in that range enbles us to find flux weighted average of the PSF. This
in turn helps us to compare the size of the PSF for different interval energy distributions .
I downloaded 15 stellar SED text file from the website
http ://irtfweb.ifa.hawaii.edu/ spex/IRTF S pectral Library/. The SED’s were taken for
F,G,M,K,L stars .
I wrote a programme ”newprac.py” to find flux for each of the 21 PSF files. This
programme reads from the spectral energy distribution(SED) file . For example we need
flux at particular wavelength xi . So the programme finds from SED file two nearby values
xi−1 and xi+1 and the corresponding flux value yi−1 and yi+1 . Then it linearly interpolates
between those two points.
So the slope m is
m =
yi+1 − yi−1
xi+1 − xi−1
(3.1)
Once we got the slope , the y intercept for that straight line is
c = yi+1 − mxi+1 (3.2)
So if we now subsitute the m and c in the straight line for a particular wavelength x
the flux would be
y = mx + c (3.3)
40. 40
The programme ”newprac.py” which takes the midpoint of each regions. For a
particular value it integrates the flux between those two midpoint values. For example if
we need to know the flux at 1.5915 µm(for psf1.fits file) the programme first find the value
at 1.58075 µm and 1.60225 µm by linear interpolation. Then it takes all the values
between these points from SED files and integrate over this range.
After making these changes the flux for SED file and PSF files were plotted which
we can see from Figure 3.1.
I then wrote one programme ”pyfits2.py” which finds the weighted average for the 21
PSF files. For example for a star if f(xi) denotes the data value of the fits file and xi
denotes the corresponding flux then the weighted average would be
¯x =
21
n=1 f(xi)xi
21
n=1 xi
(3.4)
42. 42
One of the main purpose is to study how the Psf changes with wavelength . To test
this I wrote one programme ”inte.py”. this programme finds the flux ratio for two
wavelength range one from (1.24-1.57 )µm and the other one is (1.57-2.00) µm. Based
upon the SED files it integrates over the wavelength ranges. Then it finds the absolutte
magnitude which was defined as a variable ”color” in the programme.
The color in magnitudes is given by
m = −2.5log(
f1
f2
) (3.5)
Where f1 is the flux for first wavelength range and f2 is for the second wavelength range .
I used one programme written by Douglas Clowe ”psf riffat.pl” to fit a gaussian
around my weighted average PSF. This programme uses the software ”IMCAT” . The
IMCAT software mainly finds peak for a particular image,finds the second order moments
from the light distribution,fits a Gaussian ,measures ellipticity and shear. The radius of the
gaussian is known as rg. For each 15 file the programme found the radius rg that we can
compare with ”color”.
The data is given in table 3.1.After plotting rg vs color figure 3.2 was produced.
45. 45
In general L and M stars are comparatively red,so we would expect the radius to
increase. But from figure 3.2 we see that the radius decreases. To understand let’s
compare the SED of a G star and L star in figure 3.3. The color was defined as the
logarithmic flux ratio between two range (1.24-1.57 ) µm and (1.57-2.00) µm. From G star
SED we see that there is comparatively more flux on the left hand region(1.24-1.57)µm.
But for L star there is comparatively more flux on the right hand region(1.57-2.00)µm. So
in that region L star is internally a ’blue’ star although for allover the wavelength region
it’s a red star. That is why the radius decreases for L and M star.
Figure 3.3: Spectral energy Distribution of a typical G and L star
46. 46
The shear and smear polarisibility tensor were also calculated using the same
programme . Shear polarisibility tensor Psh depends inversely on the area of the image
and area depends inversely upon the radius. We can see from figure 3.4.
Figure 3.4: Shear and smear polarisibility vs color. The color is defined as the logarithmic
ratio of flux calculated in two different regions.
We can see how the ratio changes Psh
Psm
from figure 3.5.
47. 47
Figure 3.5: Ratio of shear and smear polarisibility vs color. The color is defined as the
logarithmic ratio of flux calculated in two different regions.
As we can see from figure 3.5 the ratio varies in the opposite way compared to Psh or
Psm. This is due to the fact that Psm varies more than Psh. So when we take the ratio it
varies opposite way.
48. 48
3.2 Adjusting flux from galaxy internal colour gradient
As we saw from the last section that the size of PSF does not vary proportionally
with wavelength. It depends upon the spectral energy distribution of stars. So if we want
to use P∗sh
P∗sm
for shear calculation in KSB we will get a wrong estimate over shear. That’s
why we first found the red end flux and blue end flux of the galaxy and took that for
wavelength 1.57 and 2.00 respectively. To find the flux for other 19 wavelength between
this two regions we did linear interpolation between those two fluxes. After we get flux
for each 21 wavelength we took a weighted average of the PSF.
To measure how the ellipticity changes with wavelength I used two programmes
”cc rif fat.pl” and ”cc2 rif fat.pl” written by Douglas Clowe. I created 10 simulations
for the wavelength dependent run. The first programme ”cc rif fat.pl” creates 4 catalog
One simulation was for the middle outfut file ”aout11”,one for final output
”LS S T convolvednoise” . the other two are for 90◦
rotated files ”90aout11”
,”90 LS S T convolvednoise” The middle outputs denoted the monochromatic run while
the final outputs denoted the chromatic run. All the four catalogs were merged by
”cc2 rif fat.pl”. So for 10 simulations we got 10 catalog. At the final stage all this 10
catalogs were merged and we final catalog.From that we got the chromatic and
monochromatic ellipticity and plotted the ratio with radius from the center.
As we can see from figure 3.6 there is a sharp decrease in the ratio when the radius
decreases. This is due to the fact that we have different PSF for bulge and disk. So for
chromatic case the bulge is convolved with a larger psf and disk is convolved with smaller
PSF. So for when the radius is smaller from the center the minor axis stretches more than
the major axis . Thats why the chromatic ellipticity is less than the monochromatic ones
for smaller psf. But as the radius increases these two ellipticity nearly become equal
which we can see from the figure 3.6.
49. 49
Figure 3.6: Ratio of chromatic and monochromatic ellipticity vs radius(pixels)
50. 50
As discussed in section 1.4 the term Pγ
is given by
Pγ
αβ = Psh
αβ −
Psh∗
δβ
Psm∗
µδ
Psm
αµ (3.6)
To get the pγ
I divided the radius into 19 bins. To find the stellar estimate P∗sh
P∗sm
we
used the weighted average which we obtained by galaxies internal colour gradient . To
find Psh and psm I used the midpoint out put file ”aout11” and ”90aout11” for
monochromatic and chromatic files.we plot the ratio of chromatic and monochromatic pγ
vs the radius bin in Figure 3.7.
Finally we plot the ratio of chromatic and monochromatic shear vs radius in Figure
3.8.
We see that although ellipticity increases as the radius increases the shear decreases.
The shear is related to the ellipticity by
γ =
ε
Pγ
(3.7)
From the Pγ
plot we see that the slope is less than for Pγ
than ε vs radial plot. That’s
why the shear decreases with increasing radius .
52. 52
Figure 3.8: Ratio of chromatic and monochromatic shear as a function of radius(pixels)
53. 53
4 Conclusion
Our main target was to create a realistic weak lensing simulation. That’s why we
constrained some parameters. The redshift was kept fixed which is not so realistic . We
need to study more how varying redshift affects the flux measurements which in turn
affects the estimation of shear. We used Singular Isothermal Sphere for lens mass profile.
Whether the Navarro Frenk White profile is one of the more practical. The noise level was
kept small ,next target would be increasing the noise to a realistic sky level and see how it
does affect the shear measurements.
The major purpose of the study was to analyse the wavelength dependence of PSF.
Initially we took the Spectral Energy Distribution of different stars. We found that the size
of the PSF varies according to different star’s SED. It does not increase linearly with
wavelength. That’s why we could not use SEDs of star’s to get an estimate for galaxy
shear measurements.
As we could not use the SED of stars we used flux of galaxies and did linear
interpolation to adjust the flux for weighted average. The main drawback of this procedure
galactic SED does not vary linealy with wavelength. Like stars there are different types of
galaxies as well which have their own spectra. So the major next step is to take into
account the analysis of galactic spectra to find flux for the PSF.
We also saw that the ratio of chromatic and monochromatic shear γc
γm
is near 0.01 for
smaller radius. while it needs to be below 0.001 if we want try to reduce the multiplicative
bias for a space based telescope like WFIRST .
From the figure 3.5,3.6,3.7 we see that there is some noise present in the data . We
can reduce the noise by increasing the number of simulations.
54. 54
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