Outline


Statistical Observables
The following statistical observables can all be used to describe fluctuations in the galaxy density field.
Fourier space power spectrum :
The Fourier space matter power spectrum is defined by:
,
where represents the Fourier transform of the matter overdensities and the mean density of the Universe is . The term represents the Dirac delta function.
The power spectrum is sometimes represented in its dimensionless form:
.
Projected spherical harmonic power spectrum :
One can decompose the projected galaxy density field of an allsky survey using an angular decomposition in spherical harmonics:
,
where represent the set of orthonormal spherical harmonics which span the sphere and the 's are the spherical harmonic coefficients. The spherical coordinates
and are related to the Galactic corrdinates by and .
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. correspond to the angular positionof each galaxy, and denotes the density (as opposed to overdensity). The above harmonic coefficients are related to the galaxy overdensity coefficients by:
,
where 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: , 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 HarrisonZel'dovich value of the spectral index, , 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 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
Baryon Acoustic Oscillations
Linear redshift distortions
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