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Loay Khalifa, Jesus Pando
DePaul University, Chicago, IL
DECAM-NFC, June 2018, Chicago, IL
Probing the Non-Linearity in Galaxy Clusters Through theAnalysis of the Fractal Dimension
6/28/18DCAM_NFC 2018 1
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What is covered?1. Goals and motivations for the project2. Previous work.3. The Data: Baryonic Oscillations Spectroscopic Survey
(BOSS) Galaxies and Mock Galaxy Catalogs4. Our work (Algorithm and Results)5. Conclusion and future research
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Goals and motivations
• Testing whether the galaxy clusters distributions can be described as fractal system or not, i.e. self affine processes, by calculating the fractal dimension.
• The fractal dimension of a system describes how “irregular” or “homogenous/inhomogenous” is the distribution.
• In the context of galaxy distribution the fractal dimension indicates the galaxy clustering density and the dominance of voids.
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Previous work • Different authors reported different values of D according to
the methods and assumptions.• Davies and Pebbles reported D=1.2 using the correlation
function at 5 Mpc.• Irribarem et.al reported D=0.5 and D=1.4 using the galaxy
volume number densities.• Our method: Estimate the angular fractal dimension using the
wavelet methods.
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Data• SDSS• SDSS DR 13 BOSS:
spatial distribution of galaxies.
• Mock Galaxy Catalogs were created using Particle Mesh Method with ⌦m = 0.274 (1)
⌦⇤ = 0.726 (2)
⌦b = 0.046 (3)
1
Source: SDSS6/28/18DCAM_NFC 2018 5
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Galaxy clusters distribution
The number of galaxies decreased as Z increased
Highest density
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Work flow2-D Signal Matrix
WPT and 2-d Power Spectrum
Radial Averaging
Log Plot of the Power Spectrum vs. Frequency
Hurst Exponent
SDSS Raw Data RA and DEC
Slope
Fractal Dimension
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Fractal Dimension estimation• Find the power spectrum
• The Hurst Exponent is estimated using the relation
• The fractal dimension is calculated using the relation
The power spectrum is calculated using
var(dj, k) =1
2j
2j�1X
k=0
|dj, k|2 (1)
The Hurst exponentH is estimated by
H =↵� 1
2(2)
where ↵ is the slope of the straight line
The fractal dimension is calculated by
D = d + 1�H (3)
1
var(dj, k) =1
2j
2j�1X
k=0
|dj, k|2 (1)
1
The power spectrum is calculated using
var(dj, k) =1
2j
2j�1X
k=0
|dj, k|2 (1)
The Hurst exponentH is estimated by
H =↵ + 3
2(2)
where ↵ is the slope of the straight line
The fractal dimension is calculated by
D = d + 1�H (3)
1
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Creating the 2-d signal matrix
Raw SDSS data.RA, DEC, and Z
Pick Z with redshift bin width (Z=0.04)
Extract the RA and DEC coordinates
Count the number of galaxies
Populate a (256*256) matrix with galaxies
1
2
3
4
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HealPix and SDSSPix• Divide the sky into pixels of equal areas and equal sizes.• Depending on the resolution, the higher the resolution, the
larger the pixels. • Using indexing schemes, each pixel is assigned 2 indices i
and j.• The two indices i and j are converted into row and column
indices, then populated into the two dimensional matrix.• Populate the two dimensional matrix with galaxies at
different redshifts 28.06.18 г.DCAM_NFC 2018 10
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HealPix and SDSSPix
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Wavelets• Wavelets are a tool which is used in signal analysis.• Localized in the time as well as the frequency domain.• Operates by partitioning the signal into different sub-signals
with frequency components mapped to coefficients having different energies.
• The transform operates like a microscope for detail examination.
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Wavelets
Wavelet Transform:
Types:1- Discrete Wavelet Transform2- Wavelet Packet Transform
Discrete Wavelet Transform
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2NX
j=1
s
j
�(x) =NX
j=1
a
j
�
0(x) +NX
j=1
d
j
0(x) (1)
1
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Wavelet Packet Transform
Source: Mathworks
Low pass filter High pass filter
Sweep
Signal
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Spectrum estimation
-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0
log (Average Frequency)
2
3
4
5
6
7
8
9
10
log
(Rad
ial A
vera
ge P
ower
Spe
ctru
m)
Power Spectrum at Z=0.55
data1 linear
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
(Average Frequency)
0
500
1000
1500
2000
2500
3000
(Ra
dial
Ave
rage
Pow
er S
pect
rum
)
Power Spectrum at Z=0.55
Fitting a straight line to the log-log plotPower law distribution
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Hurst Exponent H• Values between (0-1).• (H= 0) implies surfaces of extreme irregularity.• (H= 0.5) implies random process, i.e., Brownian motion.• (H> 0.5 to H< 1) surfaces getting smoother.• (H=1) smooth surface, i.e. homogenous or regular
distribution.
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Results
As H decrease, the surface is more irregular.
0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75
Redshift (Z)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Hurst
expo
nent
(H)
Redshift (Z) vs. Hurst exponent (H)
Mock Galaxies CatalogsReal Distribution of Galaxies
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Results
•
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0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75
Redshift (Z)
0
0.5
1
1.5
2
2.5
3
Frac
tal D
imen
sion (
D)
Redshift (Z) vs. Fractal Dimension (D)
Mock Galaxies CatalogsReal Distribution of Galaxies D is constant around
1.5, single fractal system.
Inhomogenousdistribution, D < 3.
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Conclusion
• We developed a wavelet based algorithm to test whether the large scale structure of the universe can be modeled as fractal systems or not, by calculating the angular fractal dimension.
• Galaxy clusters behave as a power law against cosmological distances.
• The fractal dimension is constant, and the galaxy clusters distribution is inhomogenous.
• The large scale structure is dominated by voids.
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Thank You!• Q&A
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References
1. Y. Nievergelt. Wavelets Made Easy. Boster: Birkh auser, 1999.
2. Steven Bradley Lowen, Malvin Carl Teich. Fractal Based Point Processes. John Wiley & Sons, Inc, 2005
3. G. Conde-Saavedraa, A. Iribarrema, Marcelo B. Ribeiro: Fractal analysis of the galaxy distribution in the redshift range 0.45 ≤ z ≤ 5.0. Physica A 417: 332-344, 2015. doi:10.1016/j.physa.2014.09.044
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References4. L.J. Rangel Lemos and M.B. Riberio: Spatial and observational homogeneities of the galaxy distribution in standard cosmologies. Astronomy and Astrophysics, arxiv:0805.3336v35. Marie-Noelle Celerier and Reuben Thieberger: Fractal dimensions of the galaxy distribution varying by steps? arxiv:astro-ph/0504442v1
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Additional slides
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Data: SDSS
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Source: SDSSNorthern Galactic Cap Southern Galactic Cap
Using SDSS IV DR 13
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SDSS Instrumentation• SDSS makes use of ‘plug plates.’ These are aluminum plates,
32 inches in diameter, in which more than 1000 holes are strategically drilled.
• Each hole corresponds to a target object, and when the plate is assembled to the SDSS 2.5 meter telescope, 62ʹʹ fiber-optic cables are plugged into each hole in the plate.
• During telescope operation, these fiber-optic cables then collect light from the target source and carry it to a spectroscopic analyzer.
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SDSS Instrumentation
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Plates