Outline    Outline  Statistical Observables Fourier space power spectrum : Projected spherical harmonic power spectrum : Spatial Correlation Function : . Configuration Space Correlation Function : Building Blocks of the linear power spectrum P(k) Primordial Power Spectrum  Transfer Function The linear galaxy bias Linear redshift distortions Baryon Acoustic Oscillations

 

 

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 all-sky 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:

 

.

 

Spatial Correlation Function :

 

The spatial correlation function  is defined as the excess probability of finding a pair of galaxies at a seperation r, and is related to the Fourier space power spectrum by:

 

.

 

Configuration Space Correlation Function :

 

This quantity is sometimes referred to as the angular correlation fucntion, which we will not use to avoid confusion with its spherical harmonic counterpart.  Both correlation functions and are equivalent in describing the galaxy distribution, they are related by:

 

,

 or equivalently: 

 

 

,

where in both cases is the Legendre polynomial.

 

 

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 Harrison-Zel'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

 

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

 

,

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

 

.

 

On linear scales, it is often assumed that the scale dependence of the galaxy bias can be dropped so that .  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: 

 

,

 

Here, represents the redshift due to the global cosmological expansion and 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

 

 

Here the x-axis corresponds to distances transverse to the line of site, and the y-axis to radial distances.  The general squashing of the structure along the line of sight is due to linear redshift distortions.   On smaller scales, the structure is elongated along the line of sight due to non-linear structures.

 

The strenght of the redshift distortion depends on the distortion parameter .

 

 

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 , 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 of the CMB.

 

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

 

Measurement of the BAO scale will also be subject to redshift distortions (see above).  This means that although in real space the radial and transverse BAO scale should be the same, they will be observed as different - and the change in the radial direction will be modulated by the cosmology dependent distortion parameter defined above.   This means both the radial and transverse scale can be used to constrain our cosmological model, and that the constraint is more powerful when they are used together.