Statistical physics for cosmic structures / A. Gabrielli ... [and others].

Format:

Book

Contributor:

Gabrielli, A. (Andrea)

Language:

English

Subjects (All):

Cosmology--Statistical methods.

Cosmology.

Statistical physics.

Statistics.

Physical Description:

xiii, 424 pages : illustrations ; 24 cm

Place of Publication:

Berlin ; New York : Springer, [2005]

Summary:

The physics of scale-invariant and complex systems is a novel interdisciplinary field. Its ideas allow us to look at natural phenomena in a radically new and original way, eventually leading to unifying concepts independent of the detailed structure of the systems. The objective is the study of complex, scale-invariant, and more general stochastic structures that appear both in space and time in a vast variety of natural phenomena, which exhibit new types of collective behaviors, and the fostering of their understanding. This book has been conceived as a methodological monograph in which the main methods of modern statistical physics for cosmological structures and density fields (galaxies, Cosmic Microwave Background Radiation, etc.) are presented in detail. The main purpose is to present clearly, to a workable level, these methods, with a certain mathematical accuracy, providing also some paradigmatic examples of applications. This should result in a new and more general framework for the statistical analysis of the many new data concerning the different cosmic structures which characterize the large scale Universe and for their theoretical interpretation and modeling.

Contents:

1.1 Motivations and Purpose of the Book 1

1.2 Structures in Statistical Physics: A New Perspective 2

1.3 Structures in Statistical Physics: The Methods 8

1.4 Applications to Cosmology 11

1.5 Perspectives for the Future 22

Part I Statistical Methods

2 Uniform and Correlated Mass Density Fields 27

2.2 Basic Statistical Properties and Concepts 31

2.2.1 Spatial Averages and Ergodicity 34

2.2.2 Homogeneity and Homogeneity Scale 34

2.3 Correlation Functions 35

2.3.1 Characteristic Function and Cumulants Expansion 36

2.3.2 Correlation Length 39

2.3.3 Other Properties of the Reduced Two-Point Correlation Function 40

2.3.4 Mass Variance 41

2.4 Poisson Point Process 44

2.5 Stochastic Point Processes with Spatial Correlations 46

2.5.1 Conditional Properties 48

2.5.2 Integrated Conditional Properties 50

2.5.3 Detection of the Homogeneity Scale of a Discrete SPP 50

2.6 Nearest Neighbor Probability Density in Point Processes 52

2.6.1 Poisson Case 52

2.6.2 Particle Distributions with Spatial Correlations 54

2.7 Gaussian Continuous Stochastic Fields 55

2.8 Power-Laws and Self-Similarity 58

2.9 Mass Function and Probability Distribution 61

2.10 The Random Walk and the Central Limit Theorem 64

2.10.1 Probability Distribution of Mass Fluctuations in Large Volumes 68

2.11 Gaussian Distribution as the Most Probable Probability Distribution 69

3 The Power Spectrum and the Classification of Stationary Stochastic Fields 73

3.2 General Properties 73

3.2.1 Mathematical Definitions 73

3.2.2 Limit Conditions 76

3.3 The Power Spectrum for the Poisson Point Process and Other SPP 77

3.4 The Power Spectrum and the Mass Variance: A Complete Classification 78

3.4.1 The Complete Classification of Mass Fluctuations versus Power Spectrum 83

3.5 Super-Homogeneous Mass Density Fields 84

3.5.1 The Lattice Particle Distribution 85

3.5.2 The One Component Plasma 88

3.6 Further Analysis of Gaussian Fields 91

3.6.1 Real Space Composition of Gaussian Fields, Correlation Length and Size of Structures 95

4 Fractals 101

4.2 The Metric Dimension 102

4.3 Conditional Density 107

4.3.1 Conditional Density and Smooth Radial Particle Distributions 109

4.3.2 Statistically Homogeneous and Isotropic Distribution of Radial Density Profiles 113

4.3.3 Nearest Neighbor Probability Density for Radial and Fractal Point-Particle Distributions 113

4.4 The Two-Point Conditional Density 116

4.5 The Conditional Variance in Spheres 118

4.6 Corrections to Scaling 119

4.6.1 Correction to Scaling: Deterministic Fractals 120

4.6.2 Correction to Scaling: Random Fractals 124

4.7 Fractal with a Crossover to Homogeneity 127

4.8 Correlation, Fractals and Clustering 127

4.9 Probability Distribution of Mass Fluctuations in a Fractal 130

4.10 Intersection of Fractals 132

4.11 Morphology and Voids 134

4.12 Angular and Orthogonal Projection of Fractal Sets 134

4.12.1 On the Uniformity of the Angular Projection 137

5 Multifractals and Mass Distributions 143

5.3 Deterministic Multifractals 145

5.4 The Multifractal Spectrum 149

5.5 Random Multifractals 151

5.6 Self-Similarity of Fluctuations and Multifractality in Temporal Multiplicative Processes 154

5.7 Spatial Correlation in Multifractals 158

5.8 Multifractals and "Mass" Distributions 159

Part II Applications to Cosmology

6 Fluctuations in Standard Cosmological Models: A Real Space View 167

6.2 Basic Properties of Cosmological Density Fields 167

6.3 The Cosmological Origin of the HZ Spectrum 171

6.4 The Real Space Correlation Function of CDM/HDM Models 173

6.5 P(0) = 0 and Constraints in a Finite Sample 177

6.6 CMBR Anisotropies in Direct Space 179

6.6.1 CMBR Anisotropies and the Matter Power Spectrum 180

6.6.2 The Origin of Oscillations in the Power Spectrum 183

6.6.3 A Simple Example of k-Oscillations 184

6.6.4 Oscillations in the CDM PS 185

6.6.5 Oscillations in the CMBR Anisotropies 187

7 Discrete Representation of Fluctuations in Cosmological Models 193

7.2 Discrete versus Continuous Density Fields 194

7.3 Super-Homogeneous Systems in Statistical Physics 196

7.4 HZ as Equilibrium of a Modified OCP 197

7.5 A First Approximation to the Effect of Displacement Fields 199

7.6 Displacement Fields: Formulation of the Problem 200

7.7 Effects of Displacements on One and Two-Point Properties of the Particle Distribution 203

7.7.1 Uncorrelated Displacements 206

7.7.2 Asymptotic Behavior of P(k) for Small k 208

7.7.3 The Shuffled Lattice with Uncorrelated Displacements 209

7.8 Correlated Displacements 212

7.8.1 Correlated Gaussian Displacement Field 214

8 Galaxy Surveys: An Introduction to Their Analysis 219

8.2 Basic Assumptions and Definitions 220

8.3 Galaxy Catalogs and Redshift 221

8.4 Volume Limited Samples 224

8.5 The Discovery of Large Scale Structure in Galaxy Catalogs 227

8.6 Standard Characterization of Galaxy Correlations and the Assumption of Homogeneity 228

9 Characterizing the Observed Distribution of Visible Matter I: The Conditional Average Density in Galaxy Catalogs 235

9.2 The Conditional Average Density in Finite Samples 236

9.3 Sample Size Smaller than the Homogeneity Scale 240

9.3.1 The Reduced Correlation Function for a Particle Distribution with Fractal Behavior in the Sample 240

9.4 Sample Size Greater Than the Homogeneity Scale 242

9.4.1 Critical Case 243

9.4.2 Substantially Poisson Case 244

9.4.3 Super-Homogeneous Case 245

9.4.4 Some Remarks 245

9.5 Estimating the Average Conditional Density in a Finite Sample 246

9.5.1 Estimators of the Average Conditional Density 247

9.5.2 Effective Depth of Samples 250

9.6 The Average Conditional Density (FS) in Real Galaxy Catalogs 250

9.6.1 Normalization of the Average Conditional in Different VL Samples 257

9.6.2 Estimation of the Conditional Average Luminosity Density 259

9.6.3 Measuring the Average Mass Density [Omega] from Redshift Surveys 260

10 Characterizing the Observed Distribution of Visible Matter II: Number Counts and Their Fluctuations 265

10.2 Number Counts in Real Space 266

10.3 Number Counts as a Function of Apparent Magnitude 268

10.3.1 Poisson Distribution 268

10.3.2 Simple Fractal Distribution 271

10.3.3 Effect of Long-Ranged Correlations in Homogeneous Distributions 273

10.4 Normalization of the Magnitude Counts to Real Space Properties in Euclidean Space 276

10.4.1 Average Distance 276

10.4.2 Normalization of Distance to Magnitude Counts 277

10.5 Galaxy Counts in Real Catalogs 278

10.5.1 Real Space Counts 279

10.5.2 Magnitude Space Counts 283

11 Luminosity in Galaxy Correlations 291

11.2 Standard Methods for the Estimation of the Luminosity Function 292

11.3 Multifractality, Luminosity and Space Distributions 293

12 The Distribution of Galaxy Clusters 299

12.2 Cluster Correlations and Multifractality 300

12.3 Galaxy Cluster Correlations 303

12.3.1 The Average Conditional Density for Galaxy Clusters 306

12.3.2 Galaxy-Cluster Mismatch 306

12.4 Luminosity Bias and the Richness-Clustering Relation 308

13 Biasing a Gaussian Random Field and the Problem of Galaxy Correlations 313

13.2 Biasing of Gaussian Random Fields 314

13.3 Biasing and Real Space Correlation Properties 318

13.4 Biasing and the Power Spectrum 325

14 The Gravitational Field in Stochastic Particle Distributions 335

14.2 Nearest Neighbor Force Distribution 336

14.3 Gravitational Force PDF in a Poisson Particle Distribution 338

14.4 Gravitational Force in Weakly Correlated Particle Distributions: the Gauss-Poisson Case 342

14.5 Generalization of the Holtzmark Distribution to the Gauss-Poisson Case 343

14.5.1 Large F Expansion 344

14.5.2 Small F Expansion 347

14.5.3 Comparison with Simulations 347

14.5.4 Nearest-Neighbor Approximation for the Gauss-Poisson Case 348

14.6 Gravitational Force in Fractal Point Distributions 350

14.7 An Upper Limit in the Fractal Case 351

14.8 Average Quadratic Force in a Fractal 354

14.9 The General Importance of the Force-Force Correlation 358

A Scaling Behavior of the Characteristic Function for Asymptotically Small Values of k 365

B Fractal Algorithms 369

B.1 Cantor Set and Random Cantor Set 369

B.2 Levy Flight 372

B.3 Random Trema Dust 372

C Cosmological Models: Basic Relations 375

C.1 Cosmological Parameters 376

C.1.1 Comoving (Radial) Distance 376

C.1.2 Comoving (Transverse) Distance 377

C.1.3 Luminosity Distance 377

C.1.4 Magnitude 377

C.2 Cosmological Corrections in the Analysis of Redshift Surveys 378

C.2.1 Flat Cosmologies: FMD and FLD 378

C.2.2 Open Model: OBD 380

D Cosmological and k-Corrections to Number Counts 381

D.1 k-Corrections 381

D.2 k-Corrections and the Radial Number Counts 382

D.3 Dependence on the Cosmological Model 383

E Fractal Matter in an Open FRW Universe 385

E.2 Friedmann Solution in an Empty Universe 386

E.3 Curvature Dominated Phase 387

E.4 Radiation Dominated Era 390

E.5 Fluctuations in the CMBR 391

E.6 Other Remarks 392

F Errors in Full Shell Estimators 395

F.1 Bias and Variance of Estimators 395

F.2 Unconditional Average Density 396

F.3 Conditional Number of Points in a Sphere 397

F.4 Integrated Conditional Density 398

F.5 Conditional Average Density in Shells 399

F.6 Reduced Two-Point Correlation Function 402

G Non Full-Shell Estimation of Two Point Correlation Properties 405

G.1 Estimators with Simple Weightings 406

G.2 Other Pair Counting Estimators 407

G.3 Estimation of the Conditional Density Beyond R[subscript s] 409

H Estimation of the Power Spectrum 411.

Notes:

Includes bibliographical references (pages [413]-420) and index.

ISBN:

3540407456

OCLC:

57170791