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El. knyga: Statistical Physics for Cosmic Structures

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Andrea Gabrielli, Luciano Pietronero, F. Sylos Labini, Michael Joyce

Formatas: PDF+DRM
Išleidimo metai: 23-Feb-2006
Leidėjas: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Kalba: eng
ISBN-13: 9783540269991

Kitos knygos pagal šią temą:

Statistical physics

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Formatas: PDF+DRM
Išleidimo metai: 23-Feb-2006
Leidėjas: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Kalba: eng
ISBN-13: 9783540269991

Kitos knygos pagal šią temą:

Statistical physics

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This book has its roots in a series of collaborations in the last decade at the interface between statistical physics and cosmology. The speci c problem which initiated this research was the study of the clustering properties of galaxies as revealed by large redshift surveys, a context in which concepts of modern statistical physics (e. g. scale-invariance, fractality. . ) ?nd ready application. In recent years we have considerably broadened the range of problems in cosmology which we have addressed, treating in particular more theoretical issues about the statistical properties of standard cosmological models. What is common to all this research, however, is that it is informed by a perspective and methodology which is that of statistical physics. We can say that, beyond its speci c scienti c content, this book has an underlying thesis: such interdisciplinary research is an exciting playground for statistical physics, and one which can bring new and useful insights into cosmology. Thebook does not represent a ?nal point, but in our view, a marker in the development of this kind of research, which we believe can go very much further in the future. Indeed as we complete this book, new developments - which unfortunately we have not been able to include here - have been made on some of the themes described here. Our focus in this book is on the problem of structure in cosmology.

Recenzijos

From the reviews:

"Any book on an interdisciplinary subject must build up a self-consistent theory able to explain a great number of observations. The concepts introduced must be pertinent for the phenomena in question and sufficiently elaborated in order to extract solid conclusions in an clear and concise way. These requirements have been perfectly fulfilled in the material presented in this book. the book is highly recommendable due to the insight it gives into the field of cosmic structures, providing sound basis for further research." (J. M. Rubi, Journal of Statistical Physics, 2006)

Introduction

(26)

Motivations and Purpose of the Book

(1)

Structures in Statistical Physics: A New Perspective

(6)

Structures in Statistical Physics: The Methods

(3)

Applications to Cosmology

(11)

Perspectives for the Future

(5)

Part I Statistical Methods

Uniform and Correlated Mass Density Fields

(46)

Introduction

(4)

Basic Statistical Properties and Concepts

(4)

Spatial Averages and Ergodicity

(1)

Homogeneity and Homogeneity Scale

(1)

Correlation Functions

(9)

Characteristic Function and Cumulants Expansion

(3)

Correlation Length

(1)

Other Properties of the Reduced Two-Point Correlation Function

(1)

Mass Variance

(3)

Poisson Point Process

(2)

Stochastic Point Processes with Spatial Correlations

(6)

Conditional Properties

(2)

Integrated Conditional Properties

(1)

Detection of the Homogeneity Scale of a Discrete SPP

(2)

Nearest Neighbor Probability Density in Point Processes

(3)

Poisson Case

(2)

Particle Distributions with Spatial Correlations

(1)

Gaussian Continuous Stochastic Fields

(3)

Power-Laws and Self-Similarity

(3)

Mass Function and Probability Distribution

(3)

The Random Walk and the Central Limit Theorem

(5)

Probability Distribution of Mass Fluctuations in Large Volumes

(1)

Gaussian Distribution as the Most Probable Probability Distribution

(2)

Summary and Discussion

(2)

The Power Spectrum and the Classification of Stationary Stochastic Fields

(28)

Introduction

(1)

General Properties

(4)

Mathematical Definitions

(3)

Limit Conditions

(1)

The Power Spectrum for the Poisson Point Process and Other SPP

(1)

The Power Spectrum and the Mass Variance: A Complete Classification

(6)

The Complete Classification of Mass Fluctuations versus Power Spectrum

(1)

Super-Homogeneous Mass Density Fields

(7)

The Lattice Particle Distribution

(3)

The One Component Plasma

(3)

Further Analysis of Gaussian Fields

(5)

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

(1)

Summary and Discussion

(5)

Fractals

101

(42)

Introduction

101

(1)

The Metric Dimension

102

(5)

Conditional Density

107

(9)

Conditional Density and Smooth Radial Particle Distributions

109

(4)

Statistically Homogeneous and Isotropic Distribution of Radial Density Profiles

113

(1)

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

113

(3)

The Two-Point Conditional Density

116

(2)

The Conditional Variance in Spheres

118

(1)

Corrections to Scaling

119

(8)

Correction to Scaling: Deterministic Fractals

120

(4)

Correction to Scaling: Random Fractals

124

(3)

Fractal with a Crossover to Homogeneity

127

(1)

Correlation, Fractals and Clustering

127

(3)

Probability Distribution of Mass Fluctuations in a Fractal

130

(2)

Intersection of Fractals

132

(2)

Morphology and Voids

134

(1)

Angular and Orthogonal Projection of Fractal Sets

134

(7)

On the Uniformity of the Angular Projection

137

(4)

Summary and Discussion

141

(2)

Multifractals and Mass Distributions

143

(24)

Introduction

143

(1)

Basic Definitions

144

(1)

Deterministic Multifractals

145

(4)

The Multifractal Spectrum

149

(2)

Random Multifractals

151

(3)

Self-Similarity of Fluctuations and Multifractality in Temporal Multiplicative Processes

154

(4)

Spatial Correlation in Multifractals

158

(1)

Multifractals and ``Mass'' Distributions

159

(2)

Summary and Discussion

161

(6)

Part II Applications to Cosmology

Fluctuations in Standard Cosmological Models: A Real Space View

167

(26)

Introduction

167

(1)

Basic Properties of Cosmological Density Fields

167

(4)

The Cosmological Origin of the HZ Spectrum

171

(2)

The Real Space Correlation Function of CDM/HDM Models

173

(4)

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

177

(2)

CMBR Anisotropies in Direct Space

179

(10)

CMBR Anisotropies and the Matter Power Spectrum

180

(3)

The Origin of Oscillations in the Power Spectrum

183

(1)

A Simple Example of k-Oscillations

184

(1)

Oscillations in the CDM PS

185

(2)

Oscillations in the CMBR Anisotropies

187

(2)

Summary and Discussion

189

(4)

Discrete Representation of Fluctuations in Cosmological Models

193

(26)

Introduction

193

(1)

Discrete versus Continuous Density Fields

194

(2)

Super-Homogeneous Systems in Statistical Physics

196

(1)

HZ as Equilibrium of a Modified OCP

197

(2)

A First Approximation to the Effect of Displacement Fields

199

(1)

Displacement Fields: Formulation of the Problem

200

(3)

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

203

(9)

Uncorrelated Displacements

206

(2)

Asymptotic Behavior of P(k) for Small k

208

(1)

The Shuffled Lattice with Uncorrelated Displacements

209

(3)

Correlated Displacements

212

(5)

Correlated Gaussian Displacement Field

214

(3)

Summary and Discussion

217

(2)

Galaxy Surveys: An Introduction to Their Analysis

219

(16)

Introduction

219

(1)

Basic Assumptions and Definitions

220

(1)

Galaxy Catalogs and Redshift

221

(3)

Volume Limited Samples

224

(3)

The Discovery of Large Scale Structure in Galaxy Catalogs

227

(1)

Standard Characterization of Galaxy Correlations and the Assumption of Homogeneity

228

(5)

Summary and Discussion

233

(2)

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

235

(30)

Introduction

235

(1)

The Conditional Average Density in Finite Samples

236

(4)

Sample Size Smaller than the Homogeneity Scale

240

(2)

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

240

(2)

Sample Size Greater Than the Homogeneity Scale

242

(4)

Critical Case

243

(1)

Substantially Poisson Case

244

(1)

Super-Homogeneous Case

245

(1)

Some Remarks

245

(1)

Estimating the Average Conditional Density in a Finite Sample

246

(4)

Estimators of the Average Conditional Density

247

(3)

Effective Depth of Samples

250

(1)

The Average Conditional Density (FS) in Real Galaxy Catalogs

250

(13)

Normalization of the Average Conditional in Different VL Samples

257

(2)

Estimation of the Conditional Average Luminosity Density

259

(1)

Measuring the Average Mass Density Ω from Redshift Surveys

260

(3)

Summary and Discussion

263

(2)

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

265

(26)

Introduction

265

(1)

Number Counts in Real Space

266

(2)

Number Counts as a Function of Apparent Magnitude

268

(8)

Poisson Distribution

268

(3)

Simple Fractal Distribution

271

(2)

Effect of Long-Ranged Correlations in Homogeneous Distributions

273

(3)

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

276

(2)

Average Distance

276

(1)

Normalization of Distance to Magnitude Counts

277

(1)

Galaxy Counts in Real Catalogs

278

(10)

Real Space Counts

279

(4)

Magnitude Space Counts

283

(5)

Summary and Discussion

288

(3)

Luminosity in Galaxy Correlations

291

(8)

Introduction

291

(1)

Standard Methods for the Estimation of the Luminosity Function

292

(1)

Multifractality, Luminosity and Space Distributions

293

(4)

Summary and Discussion

297

(2)

The Distribution of Galaxy Clusters

299

(14)

Introduction

299

(1)

Cluster Correlations and Multifractality

300

(3)

Galaxy Cluster Correlations

303

(5)

The Average Conditional Density for Galaxy Clusters

306

(1)

Galaxy-Cluster Mismatch

306

(2)

Luminosity Bias and the Richness-Clustering Relation

308

(3)

Summary and Discussion

311

(2)

Biasing a Gaussian Random Field and the Problem of Galaxy Correlations

313

(22)

Introduction

313

(1)

Biasing of Gaussian Random Fields

314

(4)

Biasing and Real Space Correlation Properties

318

(7)

Biasing and the Power Spectrum

325

(5)

Summary and Discussion

330

(5)

The Gravitational Field in Stochastic Particle Distributions

335

(78)

Introduction

335

(1)

Nearest Neighbor Force Distribution

336

(2)

Gravitational Force PDF in a Poisson Particle Distribution

338

(4)

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

342

(1)

Generalization of the Holtzmark Distribution to the Gauss-Poisson Case

343

(7)

Large F Expansion

344

(3)

Small F Expansion

347

(1)

Comparison with Simulations

347

(1)

Nearest-Neighbor Approximation for the Gauss-Poisson Case

348

(2)

Gravitational Force in Fractal Point Distributions

350

(1)

An Upper Limit in the Fractal Case

351

(3)

Average Quadratic Force in a Fractal

354

(4)

The General Importance of the Force-Force Correlation

358

(2)

Summary and Discussion

360

(5)

Part III Appendixes

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

365

(4)

B Fractal Algorithms

369

(6)

B.1 Cantor Set and Random Cantor Set

369

(3)

B.2 Levy Flight

372

(1)

B.3 Random Trema Dust

372

(3)

C Cosmological Models: Basic Relations

375

(6)

C.1 Cosmological Parameters

376

(1)

C.1.1 Comoving (Radial) Distance

376

(1)

C.1.2 Comoving (Transverse) Distance

377

(1)

C.1.3 Luminosity Distance

377

(1)

C.1.4 Magnitude

377

(1)

C.2 Cosmological Corrections in the Analysis of Redshift Surveys

378

(1)

C.2.1 Flat Cosmologies: FMD and FLD

378

(2)

C.2.2 Open Model: OBD

380

(1)

D Cosmological and k-Corrections to Number Counts

381

(4)

D.1 k-Corrections

381

(1)

D.2 k-Corrections and the Radial Number Counts

382

(1)

D.3 Dependence on the Cosmological Model

383

(2)

E Fractal Matter in an Open FRW Universe

385

(10)

E.1 Introduction

385

(1)

E.2 Friedmann Solution in an Empty Universe

386

(1)

E.3 Curvature Dominated Phase

387

(3)

E.4 Radiation Dominated Era

390

(1)

E.5 Fluctuations in the CMBR

391

(1)

E.6 Other Remarks

392

(3)

F. Errors in Full Shell Estimators

395

(10)

F.1 Bias and Variance of Estimators

395

(1)

F.2 Unconditional Average Density

396

(1)

F.3 Conditional Number of Points in a Sphere

397

(1)

F.4 Integrated Conditional Density

398

(1)

F.5 Conditional Average Density in Shells

399

(3)

F.6 Reduced Two-Point Correlation Function

402

(3)

G Non Full-Shell Estimation of Two Point Correlation Properties

405

(6)

G.1 Estimators with Simple Weightings

406

(1)

G.2 Other Pair Counting Estimators

407

(2)

G.3 Estimation of the Conditional Density Beyond Rs

409

(2)

H Estimation of the Power Spectrum

411

(2)

References

413

(8)

Index

421

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