



Galaxy Auto-Correlations

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Saved by Anais Rassat
on August 18, 2008 at 11:55:09 am








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: 




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.


The power spectrum is sometimes represented in its dimensionless form: 





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:




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:




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: 




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:





Building Blocks of the linear power spectrum P(k)


Here we focus on the features of the linear Fourier space galaxy power spectrum P(k).



Primordial Power Spectrum 


First we start with the initial dark matter power spectrum, which is assumed to be of the form: Formula, where n is the spectral index and controls the tilt of the primordial power spectrum.


We recall that Poisson's equation relates the gravitational field to the dark matter density: 



or in Fourier space: 


so that:



The power spectrum of perturbations can then be expressed by: 


so that the dimensionless power spectrum of perturbations is given by:



Considering the primordial power spectrum: 





With this last equation, we see that the Harrison-Zel'dovich value of the spectral index, Formula, corresponds to a scale invariant (i.e., independent of k) power spectrum of perturbations.


Transfer Function


The evolution of the dark matter perturbations - and therefore the dark matter power spectrum P(k) - depend on all possible interactions between the dark matter, baryons, neutrinos and photons.  The effect of these interactions are encoded in the Boltzman equations, and there exists several  public codes which solve these (see CAMB, or CMBFAST).


The resulting power spectrum at redshift z can be written: 



where T(k) is the transfer function which is a solution to Boltzman equations.




The linear galaxy bias


Baryon Acoustic Oscillations


Linear redshift distortions



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