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
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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
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
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
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.
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
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