(Gaussian) Random Fields

Transcription

1 23/01/2017 (Gaussian) Random Fields Echo of the Big Bang: Cosmic Microwave Background Planck (2013) Earliest view of the Universe: yrs. after Big Bang, 13.8 Gyr ago. 1

2 CMB Temperature Perturbations Cosmic Structure at Edge Visible Universe Planck (2013) CMB temperature map: T K T T 10 5 Fluctuation Field (almost) perfectly Gaussian Origin: inflationary era, t = sec. Gaussian Random Field P N 1 exp 2 N N 1 f i 1 j 1 i M f ij j N N 2 detm 1/2 k 1 df k 2

3 Gaussian Random Field: Multiscale Structure dk ˆ ik x f x f 3 ke 2 i ˆ ˆ ˆ ˆ k f k f k i f k f k e r i Gaussian Random Field: Density 3

4 Gaussian Random Field: Gravity Gaussian Random Field: Gravity Vectors 4

5 Gaussian Random Field: Potential Power Spectrum 5

6 Power Spectrum dk ˆ ik x f x f 3 ke 2 fˆ k fˆ k i fˆ k fˆ k e i k r i The key characteristic of Gaussian fields is that their structure is FULLY, COMPLETELY and EXCLUSIVELY determined by the second order moment of the Gaussian distribution. P(k) specifies the relative contribution of different scales to the density fluctuation field. It entails a wealth of cosmological information. 2 dk P( k) P( k) fˆ( k ) fˆ k 3 2 Power Spectrum P(k) specifies the relative contribution of different scales to the density fluctuation field. It entails a wealth of cosmological information. 2 dk P( k) P( k) fˆ( k) fˆ k 3 2 Formal definition: 3 ˆ ˆ 2 Pk ( 1) D k1k2 f k1 f k2 Pk ( ) fˆ k f k ˆ 1 2 6

7 Power Spectrum Correlation Function Gaussian random field fully described by 2 nd order moment: - in Fourier space: power spectrum - in Configuration (spatial) space: 2-pt correlation function 3 ˆ ˆ 2 Pk ( 1) D k1k2 f k1 f k2 3 ik r Pk ( ) dr r e r, r r r f r f r r 3 dk 2 3 P( k) e ik r Primordial Gaussian Field 1 exp 2 P N N N 1 f i 1 j 1 i M f ij j N N 2 detm 1/2 k 1 df k Key aspects of Gaussian fields: solely & uniquely dependent on 2nd order moment all Fourier modes mutually independent & Gaussian distributed 2 2 fˆk ˆ i f k i PN exp exp i 2Pki i 2Pki ˆ ˆ ˆ ˆ 1 ˆ f k f k d f k P f k d f k exp 2 Pk ( ) Pk ( ) 2 7

8 Power Spectrum P(k) specifies the relative contribution of different scales to the density fluctuation field. It entails a wealth of cosmological information. 2 dk P( k) P( k) fˆ( k) fˆ k 3 2 Power Law Power Spectrum: Pk ( ) k n as index n lower, density field increasingly dominated by large scale modes. For an arbitrary spectrum, dlog P( k) nk ( ) dlog k 8

9 Power Spectrum Physical & Observed CDM Power Spectrum P(k) 9

10 Power Spectrum P(k) 10

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12 Phases & Patterns 12

13 Random Field Phases dk f( x) fˆ ( k) e 2 3 ik x ˆ( ) ˆ ( ) ˆ( ) ˆ i f k f k i f k f( k) e r i ( k) When a field is a Random Gaussian Field, its phases q(k) are uniformly distributed over the interval [0,2p]: ( k) U[0,2 ] As a result of nonlinear gravitational evolution, we see the phases acquire a distinct non-uniform distribution. 13

14 Power Spectrum: Pattern Information & Phases 14

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18 Ergodic Theorem Statistical Cosmological Principle Cosmological Principle: Universe is Isotropic and Homogeneous ( x) p[ ( x a)] p[ ( x )] p[ ( x y)] p[ ( x y )] Homogeneous & Isotropic Random Field : Homogenous Isotropic Within Universe one particular realization : ( x) ( x) Observations: only spatial distribution in that one particular Theory: p[ ( x)] 18

19 Ergodic Theorem Ensemble Averages Spatial Averages over one realization of random field Basis for statistical analysis cosmological large scale structure In statistical mechanics Ergodic Hypothesis usually refers to time evolution of system, in cosmological applications to spatial distribution at one fixed time Ergodic Theorem Validity Ergodic Theorem: Proven for Gaussian random fields with continuous power spectrum Requirement: spatial correlations decay sufficiently rapidly with separation such that many statistically independent volumes in one realization All information present in complete distribution function from single sample ( p[ ( x)] x) over all space available 19

20 Fair Sample Hypothesis Statistical Cosmological Principle Weak cosmological principle (small fluctuations initially and today over Hubble scale) Ergodic Hypothesis + + fair sample hypothesis (Peebles 1980) 20