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Galaxy Auto-Correlations

This version was saved 13 years, 3 months ago View current version     Page history
Saved by Anais Rassat
on August 14, 2008 at 5:33:14 pm
 

 

Outline 

 


 


 


 

 

Statistical Observables

 

The following statistical observables can all be used to describe fluctuations in the galaxy density field.

 

Fourier space power spectrum Formula:

 

 

The Fourier space matter power spectrum is defined by: 

Formula,

where Formula represents the Fourier transform of the matter overdensities Formula and the mean density of the Universe is Formula. The term Formula represents the Dirac delta function.

 

 

Projected spherical harmonic power spectrum Formula

 

One can decompose the projected galaxy density field of an all-sky survey using an angular decomposition in spherical harmonics:

 

Formula,

where Formula represent the set of orthonormal spherical harmonics which span the sphere and the Formula's are the spherical harmonic coefficients.  The spherical coordinates

Formula and Formula are related to the Galactic corrdinates by Formula and Formula.

 

For a given discrete density distribution, the coefficients can be determined by direct summation over the galaxy angular positions:

 

Formula,

where the sum is over all galaxies in the survey, i.e. Formula correspond to the angular positionof each galaxy, and Formula denotes the density (as opposed to overdensity).  The above harmonic coefficients are related to the galaxy overdensity coefficients by: 

 

Formula, where Formula denotes the number of galaxies per steradian.

 

The harmonic coefficients are not rotationally invariant quantities.  It is the spherical harmonic overdiensity power spectrum that one can compare with the linear theory predicitions (Peebles 1980).  The power spectrum is related to the harmonic coefficients by:

 

Formula

 

 

 

 

 

 

 

Real space correlation function Formula

 

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Configuration space correlation function  Formula

 

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Building Blocks of Galaxy Correlations

 

 

Features in the galaxy power spectrum

 

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Features in the projected spherical harmonic power spectrum

 

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Focussing on the the BAO wiggles

 

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