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