Galaxies distribution and inhomogeneities in the Universe Umberto Esposito Relatore: Francesco Sylos...

Galaxies distribution and inhomogeneities in the Universe

Umberto Esposito

Relatore: Francesco Sylos Labini

Fisica dei Sistemi Complessi Anno Accademico 2010-2011

Main statistical properties – PDF

• Stationarity: probability density function is translational and rotational invariant (translational and rotational invariance)

Copernican Principle

• Ergodicity: for any generic observable F, ensemble average over different realizations of the stochastic process and spatial average in a finite volume

V become equal in the infinite volume limit.

Galaxies distribution is a realization of a stochastic point process: at any microscopic density function is associated a probability .( )r ( )P r

Main statistical properties – PDF

• Uniformity (spatial homogeneity): ensemble average density is strictly positive

Cosmological Principle

• Self-averaging: for any generic observable F, fluctuations from ensemble

average become small in the infinite volume limit. A single large enough

system is sufficient to represent the whole ensemble

0

30 0;

0

1lim 0

;R V R xr d r x

V R x

22 0L L Llim F F

Galaxies distribution is a realization of a stochastic point process: at any microscopic density function is associated a probability .( )r ( )P r

Plus Copernican Principle

Main statistical properties – Fluctuations

Fluctuations from the ensemble average values in a single realization of the stochastic process are usually studied through the two-point correlation function

1 2r r ��

1 2

120

p

r rr

��

Unconditional Conditional

Homogeneous field: 0 0

12 12

12 20 0

1 1pr r

r

Depends on the amplitude A

Depends only on the rate of decay

0 1r

Determines the typical size of fluctuation structures

Marks the crossover from large to small fluctuations

Reduced two point correlation function

exp cr A r r

Homogeneous field: 0 0

Main statistical properties – Fluctuations

Mass fluctuations 12r

22

2 3 31 2 122 2

1

V R V R

M R M RR d r d r r

V RM R

1. for -3<n<1

2. for n=1

3. for n>1

32 nR R

3 12 logR R R

3 12 R R

n=0 Substantially-Poisson: no correlations (purely-Poisson) or short range correlations

with finite exp cr A r r cr

n<0 Super-Poisson: long-range correlations with infinite r r

cr

Sub-Poisson or Super-homogeneous: fastest possible decay

General constraint: 2lim 0R R

Large scales behavior:

Standard Model in Cosmology

• Copernican Principle: no special points or directions

• Cosmological Principle

Stationarity

Friedmann-Robertson-Walker

1. The scale factor describes the geometry of the Universe obeying to the Friedmann equations

2. Matter density is constant in a spatial hyper-surface

Unifomity

Cosmological Principle is a special case of Copernican one: it can be proved that the latter implies the former when taken together with the hypothesis that matter distribution is a smooth function of position.


• Super-homogeneous fluctuations with n=1 3

0

0d r r

1. In the early universe the homogeneity scale is of the order of inter-particle distance while it grows during the process of structure formation

2. Due to small initial dispersion velocity fluctuations remain of small amplitude at large scales while they acquire a large relative amplitude at small scales

3. Evolution of fluctuations into non-linear structures is not considered to have sensible effect on the evolution of the space-time which is driven by the uniform

mean field

4. At small scales the two point correlation function is positive and goes to zero in a way depending on the model while at large scales must be negative (anti-

correlation) with tale 4r


Two additional ingredients are necessary to satisfy the constraints obtained from observations

• Dark energy: dominant repulsive component

1. Cannot be inferred from a-priori principles

2. Can be modeled by a positive cosmological constant

3. It’s amplitude in Cosmology seems absurdly small in the context of quantum physics

4. Abundance: ¾ of total sources

• Dark matter: non-baryonic component

1. Weak interaction with radiation

2. Abundance: ¼ of total sources

These two components play a crucial role in Cosmology ΛCDM model

Standard Model in CosmologyThe ΛCDM model

Density fluctuations in the early Universe are coupled with radiation. From

1. Information about initial conditions by studying CMBR

2. Linear perturbation analysis of self-gravitating fluid in an expanding Universe

It is possible to predict what we expect to see today about scales of density fluctuations

0 10 /nowt Mpc h 100 /cr Mpc h

Standard Model in CosmologyPredictions

•There can be large amplitude density fluctuations only up to

•Small amplitude density fluctuations with positive correlations are present up to

•For larger scales there are anti-correlations and the correlation function has to goes to zero as

0 10 /nowt Mpc h

100 /cr Mpc h

4r

Results in observations

•1985: distribution is anything but random, with a single large structure limited only by sample size: the Great Wall

•In subsequently survey more galaxy large scale structures were identified, with a detection of the Taurus void: a large circular void with diameter of about

•Few years ago, in the SDSS, it has been discovered the Sloan Great Wall, three times longer than the Great Wall

30 /Mpc h

Statistical analysis of the catalogs data has identified

A charcteristic scale, defined to be the one at which fluctuations in the galaxy density field are about twice the value of the sample density

Fluctuations in galaxy counts in different regions of the order of on scales of the order of

0 5 15 /r Mpc h

30%200 /Mpc h

Large scale structures and wide fluctuations at scales of the order ofseem to be in contrast with both the small value of and

100 /Mpc h0r cr

More general statistical methods

Large scale structures and wide fluctuations at scales of the order ofseem to be in contrast with both the small value of and0r

100 /Mpc h

It’s necessary to review our hypotheses in a critical way, developing statistical tests to verify their validity

In doing this we cannot use usual statistical methods on which standard models are constructed, but we have to elaborate much more general ones

Cosmological Principle is a special case of Copernican one: it can be proved that the latter implies the former when taken together with the

hypothesis that matter distribution is smooth function of position.

cr


Conditional quantities: inhomogeneous distributions have 0 0

They are substantially empty: if we randomly take a finite volume it typically contains no points

Unconditional quantities are not well defined, but only conditional ones: 1 2

120

p

r rr

��

Estimators: inhomogeneous distributions have 12lim 0r pr

Quantities such as two point correlation function are meaningless

We have to construct quantities averaged over a finite volume: estimators of the statistical ones:

12

1p

s

rr

s s p

R where


Self-averaging: in inhomogeneous distributions measurements in different sub-regions can show systematic difference, making estimators meaningless

Estimators are meaningful when self-averaging is verified: , ; , ;i jP G L S P G L S i j

By virtue of this properties of inhomogeneous distributions, we consider the statistical properties of the stochastic variable defined by number of points contained in a sphere of radius centered on the point; this depends on the scale and on the spatial position of the sphere’s centre, i.e. its radial

distance from a given origin and its angular coordinates . Integrating over for fixed radial distance we obtain

;i iN r N r R

thir

iRi

r thii

1 1

1 1M r M ri

ipi i

N rn r n r

M r V r M r

22 2

1

1 M r

p i pi

r n r n rM r

,P N r

Testing the standard model - stationarity

•Lack of self-averaging , ; , ;i jP G L S P G L S

Lack of stationarity:

There is a center breaking overall translational invariance

Validity of stationarity:

there is a finite-size effect related to the presence of long-range correlated

fluctuations: sizes of sample not large enough

Distribution is inhomogeneousEstimators are meaningless

How to distinguish the two possibilities?

One has to change the scale where is measured, determining whether the PDF is self-averaging

r ,P N r

Testing the standard model - homogeneity

•Validity of self-averaging , ; , ;i jP G L S P G L S i j Estimators are meaningful

We can ask about homogeneity, studying scaling properties of estimators

Lack of homogeneity:

as long as presents a scaling behavior as a function of spatial

separation

Validity of homogeneity:

when pn r

r

pn r const

This constant furnish an estimation of the ensemble average density

The scale where the transition to the constant behavior occurs marks the

homogeneity scale

0

Results in the data

Conditional number of galaxies contained in the sphere of radius

Sloan Digital Sky Survey (SDSS)

10 /r Mpc h iN r


Results in the data

Self-averaging test

Left panel: 10 /r Mpc h Right panel: 80 /r Mpc h

7 62DR DR 125,400 /R Mpc h

Holds in both cases Lack in DR6

Hold in DR7

Finite size effects at

80 /r Mpc h

Results in the data

Distribution of conditional density : shape of PDF


in r

Gumbel distribution is a clear sign of inhomogeneity

Away from criticality correlations are small- ranged and any global observable has

Gaussian fluctuations: all homogeneous point distributions lead to Gaussian fluctuations.At criticality correlations are long-ranged and long-tailed distributions are found

exp expP x x x

Gumbel distribution

Free parameter Gumbel distribution

exp expy y

P y

yx

Results in the data

Conditional average density


1

1 M ri

pi

N rn r

M r V r

: scaling behavior with exponent close to one

: scaling behavior with change of slope

20 /r Mpc h

20 /r Mpc h

0.290.011n r r 0.0133

logn r

r or

This can be interpreted as a signature of inhomogeneity and of fractality of galaxy distribution at these scales

Moreover the density does not saturate to a constant up to the largest scales probed in this sample, , for which the statistics is sufficiently robust80 /r Mpc h 410M r

From spheres to cylinders

By virtue of geometrical constraints, we have an upper bound to the sphere’s radius, given by , while in the SDSS we have80 /r Mpc h 500 /R Mpc h

maxR

maxr

We are not able to completely investigate the changing of slope in n r


max max2R r

y

x

z

h R

R1

With cylinders we can reach 250 /r Mpc h

From spheres to cylindersWe want to verify if with cylinders analysis we are able to distinguish the

homogeneous distribution from inhomogeneous one in the region of slope changing

We calculate with r

a rn r

r

For homogeneous

For inhomogeneous

3

CV

N R d rn r

For :0 20 /r Mpc h 0.0133a r a 1 1

For :0 20 /r Mpc h 1

0a r b a

0.750a r b a

2 0

2 0.25

•Polar coordinates in the plain

• We can perform both integrals in polar coordinates

,y z 0,2 , 0,h

0h R

From spheres to cylindersWe want to verify if with cylinders analysis we are able to distinguish the

homogeneous distribution from inhomogeneous one in the region of slope changing

2

2 2 22 2 10 01 0 0 0 0 02 ln 1N R a R R R R

R R

.2 0


0

23 42 2 22 2 0.75 5 2 5 2 0.75 20 0

2 0 0 0 0 0 0

8 42 ln 1

15 3

R

N R a R R R dx x RR R

We want to verify if with cylinders analysis we are able to distinguish the homogeneous distribution from inhomogeneous one in the region of slope changing

2 0.25 .


The difference between the two behaviors is

much more evident in cylinders analysis

Conclusions

•Up to :

Distribution of conditional density seems to be Gumbel-like

shows a power-law behavior with fractal dimension

Self-averaging holds

in r

in r

80 /r Mpc h

Copernican principle seems to hold, instead of the Cosmological one

2.7

•The limit of is given by the maximum value of sphere’s radius which we can inscribe in the sample. To improve this limit, up to , we can use cylinders, from which we can investigate the slope changing zone and the difference between homogeneous and not homogeneous behavior seems much more evident.

250 /r Mpc h80 /r Mpc h

Galaxies distribution and inhomogeneities in the Universe Umberto Esposito Relatore: Francesco Sylos...

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