1. Galaxies distribution and inhomogeneities in the Universe Umberto Esposito Relatore: Francesco Sylos Labini Fisica dei Sistemi ComplessiAnno Accademico 2010-2011

2. 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.

3. 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 Galaxies distribution is a realization of a stochastic point process: at any microscopic density function is associated a probability. Plus Copernican Principle

4. 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 UnconditionalConditional Homogeneous field: Depends on the amplitude A Depends only on the rate of decay Determines the typical size of fluctuation structures Marks the crossover from large to small fluctuations Reduced two point correlation function

5. Homogeneous field: Main statistical properties – Fluctuations Mass fluctuations 1. for -3<n<1 2. for n=1 3. for n>1 n=0 Substantially-Poisson: no correlations (purely-Poisson) or short range correlations with finite n<0 Super-Poisson: long-range correlations with infinite Sub-Poisson or Super-homogeneous: fastest possible decay General constraint: Large scales behavior:

6. 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.

7. Standard Model in Cosmology Super-homogeneous fluctuations with n=1 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

8. Standard Model in Cosmology 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

9. Standard Model in Cosmology The 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

10. Standard Model in Cosmology Predictions 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

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

12. More general statistical methods Large scale structures and wide fluctuations at scales of the order of seem to be in contrast with both the small value of and 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.

13. More general statistical methods  Conditional quantities: inhomogeneous distributions have 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:  Estimators: inhomogeneous distributions have Quantities such as two point correlation function are meaningless We have to construct quantities averaged over a finite volume: estimators of the statistical ones: where

14. More general statistical methods  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: 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

15. Testing the standard model - stationarity Lack of self-averaging 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 inhomogeneous  Estimators are meaningless How to distinguish the two possibilities? One has to change the scale where is measured, determining whether the PDF is self-averaging

16. A toy model Homogeneous, short-range positive correlationsHomogeneous, long-range positive correlations Inhomogeneous, long-range positive correlationsInhomogeneous, long-range positive correlations, not self-averaging

17. Testing the standard model - homogeneity Validity of self-averaging  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  This constant furnish an estimation of the ensemble average density  The scale where the transition to the constant behavior occurs marks the homogeneity scale

18. Results in the data Conditional number of galaxies contained in the sphere of radius Sloan Digital Sky Survey (SDSS)

19. Results in the data Self-averaging test Left panel:Right panel: Holds in both cases Lack in DR6 Hold in DR7 Finite size effects at

20. Results in the data Distribution of conditional density : shape of PDF Sloan Digital Sky Survey (SDSS) 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 Gumbel distribution Free parameter Gumbel distribution

21. Results in the data Conditional average density Sloan Digital Sky Survey (SDSS) : scaling behavior with exponent close to one : scaling behavior with change of slope 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 robust

22. 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 have We are not able to completely investigate the changing of slope in

23. From spheres to cylinders With cylinders we can reach

24. From spheres to cylinders 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 We calculate with  For homogeneous  For inhomogeneous For : Polar coordinates in the plain We can perform both integrals in polar coordinates

25. From spheres to cylinders 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 ..

26. From spheres to cylinders 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 ..

27. From spheres to cylinders The difference between the two behaviors is much more evident in cylinders analysis

28. Conclusions Up to :  Distribution of conditional density seems to be Gumbel-like  shows a power-law behavior with fractal dimension  Self-averaging holds Copernican principle seems to hold, instead of the Cosmological one 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.