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.
Author(s): A. Gabrielli, F. Sylos Labini, M. Joyce, L. Pietronero,
Edition: 1
Publisher: Springer
Year: 2004
Language: English
Pages: 420
Contents......Page 7
1.1 Motivations and Purpose of the Book......Page 15
1.2 Structures in Statistical Physics: A New Perspective......Page 16
1.3 Structures in Statistical Physics: The Methods......Page 22
1.4 Applications to Cosmology......Page 25
1.5 Perspectives for the Future......Page 36
Part I Statistical Methods......Page 38
2.1 Introduction......Page 39
2.2 Basic Statistical Properties and Concepts......Page 43
2.3 Correlation Functions......Page 47
2.4 Poisson Point Process......Page 56
2.5 Stochastic Point Processes with Spatial Correlations......Page 58
2.6 Nearest Neighbor Probability Density in Point Processes......Page 64
2.7 Gaussian Continuous Stochastic Fields......Page 67
2.8 Power-Laws and Self-Similarity......Page 70
2.9 Mass Function and Probability Distribution......Page 73
2.10 The Random Walk and the Central Limit Theorem......Page 76
2.11 Gaussian Distribution as the Most Probable Probability Distribution......Page 81
2.12 Summary and Discussion......Page 83
3.2 General Properties......Page 85
3.3 The Power Spectrum for the Poisson Point Process and Other SPP......Page 89
3.4 The Power Spectrum and the Mass Variance: A Complete Classification......Page 90
3.5 Super-Homogeneous Mass Density Fields......Page 96
3.6 Further Analysis of Gaussian Fields......Page 103
3.7 Summary and Discussion......Page 108
4.1 Introduction......Page 112
4.2 The Metric Dimension......Page 113
4.3 Conditional Density......Page 118
4.4 The Two-Point Conditional Density......Page 127
4.5 The Conditional Variance in Spheres......Page 129
4.6 Corrections to Scaling......Page 130
4.8 Correlation, Fractals and Clustering......Page 138
4.9 Probability Distribution of Mass Fluctuations in a Fractal......Page 141
4.10 Intersection of Fractals......Page 143
4.12 Angular and Orthogonal Projection of Fractal Sets......Page 145
4.13 Summary and Discussion......Page 152
5.1 Introduction......Page 154
5.2 Basic Definitions......Page 155
5.3 Deterministic Multifractals......Page 156
5.4 The Multifractal Spectrum......Page 160
5.5 Random Multifractals......Page 162
5.6 Self-Similarity of Fluctuations and Multifractality in Temporal Multiplicative Processes......Page 165
5.7 Spatial Correlation in Multifractals......Page 169
5.8 Multifractals and "Mass" Distributions......Page 170
5.9 Summary and Discussion......Page 172
Part II Applications to Cosmology......Page 175
6.2 Basic Properties of Cosmological Density Fields......Page 176
6.3 The Cosmological Origin of the HZ Spectrum......Page 180
6.4 The Real Space Correlation Function of CDM/HDM Models......Page 182
6.5 P(0) = 0 and Constraints in a Finite Sample......Page 186
6.6 CMBR Anisotropies in Direct Space......Page 188
6.7 Summary and Discussion......Page 198
7.1 Introduction......Page 201
7.2 Discrete versus Continuous Density Fields......Page 202
7.3 Super-Homogeneous Systems in Statistical Physics......Page 204
7.4 HZ as Equilibrium of a Modified OCP......Page 205
7.5 A First Approximation to the Effect of Displacement Fields......Page 207
7.6 Displacement Fields: Formulation of the Problem......Page 208
7.7 Effects of Displacements on One and Two-Point Properties of the Particle Distribution......Page 211
7.8 Correlated Displacements......Page 220
7.9 Summary and Discussion......Page 225
8.1 Introduction......Page 226
8.2 Basic Assumptions and Definitions......Page 227
8.3 Galaxy Catalogs and Redshift......Page 228
8.4 Volume Limited Samples......Page 231
8.5 The Discovery of Large Scale Structure in Galaxy Catalogs......Page 234
8.6 Standard Characterization of Galaxy Correlations and the Assumption of Homogeneity......Page 235
8.7 Summary and Discussion......Page 240
9.1 Introduction......Page 241
9.2 The Conditional Average Density in Finite Samples......Page 242
9.3 Sample Size Smaller than the Homogeneity Scale......Page 246
9.4 Sample Size Greater Than the Homogeneity Scale......Page 248
9.5 Estimating the Average Conditional Density in a Finite Sample......Page 252
9.6 The Average Conditional Density (FS) in Real Galaxy Catalogs......Page 256
9.7 Summary and Discussion......Page 269
10.1 Introduction......Page 270
10.2 Number Counts in Real Space......Page 271
10.3 Number Counts as a Function of Apparent Magnitude......Page 273
10.4 Normalization of the Magnitude Counts to Real Space Properties in Euclidean Space......Page 281
10.5 Galaxy Counts in Real Catalogs......Page 283
10.6 Summary and Discussion......Page 293
11.1 Introduction......Page 295
11.2 Standard Methods for the Estimation of the Luminosity Function......Page 296
11.3 Multifractality, Luminosity and Space Distributions......Page 297
11.4 Summary and Discussion......Page 301
12.1 Introduction......Page 302
12.2 Cluster Correlations and Multifractality......Page 303
12.3 Galaxy Cluster Correlations......Page 306
12.4 Luminosity Bias and the Richness-Clustering Relation......Page 311
12.5 Summary and Discussion......Page 314
13.1 Introduction......Page 315
13.2 Biasing of Gaussian Random Fields......Page 316
13.3 Biasing and Real Space Correlation Properties......Page 320
13.4 Biasing and the Power Spectrum......Page 327
13.5 Summary and Discussion......Page 332
14.1 Introduction......Page 336
14.2 Nearest Neighbor Force Distribution......Page 337
14.3 Gravitational Force PDF in a Poisson Particle Distribution......Page 339
14.4 Gravitational Force in Weakly Correlated Particle Distributions: the Gauss-Poisson Case......Page 343
14.5 Generalization of the Holtzmark Distribution to the Gauss-Poisson Case......Page 344
14.6 Gravitational Force in Fractal Point Distributions......Page 351
14.7 An Upper Limit in the Fractal Case......Page 352
14.8 Average Quadratic Force in a Fractal......Page 355
14.9 The General Importance of the Force-Force Correlation......Page 359
14.10 Summary and Discussion......Page 361
Part III Appendixes......Page 363
A. Scaling Behavior of the Characteristic Function for Asymptotically Small Values of k......Page 364
B.1 Cantor Set and Random Cantor Set......Page 367
B.3 Random Trema Dust......Page 370
C. Cosmological Models: Basic Relations......Page 372
C.1 Cosmological Parameters......Page 373
C.2 Cosmological Corrections in the Analysis of Redshift Surveys......Page 375
D.1 k-Corrections......Page 378
D.2 k-Corrections and the Radial Number Counts......Page 379
D.3 Dependence on the Cosmological Model......Page 380
E.1 Introduction......Page 382
E.2 Friedmann Solution in an Empty Universe......Page 383
E.3 Curvature Dominated Phase......Page 384
E.4 Radiation Dominated Era......Page 387
E.5 Fluctuations in the CMBR......Page 388
E.6 Other Remarks......Page 389
F.1 Bias and Variance of Estimators......Page 391
F.2 Unconditional Average Density......Page 392
F.3 Conditional Number of Points in a Sphere......Page 393
F.4 Integrated Conditional Density......Page 394
F.5 Conditional Average Density in Shells......Page 395
F.6 Reduced Two-Point Correlation Function......Page 398
G. Non Full-Shell Estimation of Two Point Correlation Properties......Page 401
G.1 Estimators with Simple Weightings......Page 402
G.2 Other Pair Counting Estimators......Page 403
G.3 Estimation of the Conditional Density Beyond R[sub(s)]......Page 405
H. Estimation of the Power Spectrum......Page 407
References......Page 409
C......Page 417
H......Page 418
N......Page 419
Z......Page 420
