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Galaxy Auto-Correlations

This version was saved 13 years, 2 months ago View current version     Page history
Saved by Anais Rassat
on August 18, 2008 at 5:27:30 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.

 

The power spectrum is sometimes represented in its dimensionless form: 

 

Formula.

 

 

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.

 

 

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: 

 

Formula,

or in Fourier space: 

Formula,

so that:

Formula.

 

The power spectrum of perturbations can then be expressed by: 

Formula,

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

Formula.

 

Considering the primordial power spectrum: 

Formula,

 

Formula.

 

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: 

 

Formula,

where T(k) is the transfer function which is a solution to Boltzman equations.  The linear growth factor D(z) quantifies the evolution fo the linear growth of structure with redshift, and is usually normalized to equal 1 today.

 

 

The linear galaxy bias

 

In a galaxy survey, the observable is the galaxy overdensity Formula, which is assumed to trace the underlying matter distribution Formula following: 

 

Formula,

where b(z, k) is the galaxy bias, which can be both redshift and scale dependent.  The galaxy power spectrum is then given by:

 

Formula.

 

On linear scales, it is often assumed that the scale dependence of the galaxy bias can be dropped so that Formula.  In this case, the bias only modulates the overall amplitude of the galaxy power spectrum. 

 

 

Linear redshift distortions

 

An observer can only measure the galaxy power spectrum in redshift space, which is distorted compared to the popwer spectrum in real space.  Redshift distortions arise because of peculiar velocities of galaxies, i.e. the component of a galaxy's velocity which is not due to the Hubble recession.  In particuliar it is the radial component of the peculiar velocity which will generate distortions in the observed galaxy field.

 

The reason for this is because the observed redshift of a galaxy is always the sum of two quantities: 

 

Formula,

 

Here, Formula represents the redshift due to the global cosmological expansion and Formula is the redshift due to the radial component of the galaxy's peculiar velocity.  The transverse component does not induce any distortions, as it does not affect our estimate of the galaxy's redshift.

 

This means only structures along the line of sight will appear distorted.  Imagine a structure which is isotropic in real space, then an observer will measure something like this

 

 

 

Baryon Acoustic Oscillations

 

In the early Universe, just before recombination, fluctuations in the coupled baryon-photon fluid were subject to two competing effects: attractive gravity and repulsive pressure.  These two effects created a characteristic scale in the galaxy distribution, which manifests itself as a

baryon peak in the correlation function Formula, and a series of acoustic peaks - or Baryon Acoustic Oscillations - in the galaxy power spectrum P(k).  The acoustic peaks are also observed in the temperature-temperature power spectrum Formula of the CMB.

 

 

The scale of these oscillations provides a standard ruler which can be used to constain dark energy parameters.

 

 

 

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