Home Page  Overview  Site Map  Index  Appendix  Illustration  About  Contact  Update  FAQ 
The twopoint correlation function in statistics can be defined as the probability of relating a function f(x) to another function f(y) at the spatial vector r in excess of the random distribution (Figure 01a1). This statement is valid only when it is averaged over a large number of such configurations in the space. It is expressed as < f(x)f(y) > in mathematical notation and sometimes denoted as _{}(x,y) or (r). The coefficient P(k) in the Fourier transform to kspace (k = 2/) of the twopoint correlation function is called Power Spectrum. Following is a brief sketch of the mathematics involved. BTW, the formulas become simpler in cosmology, which assumes an isotopic (rotational invariance) and homogeneous (translational invariance) massenergy distribution so that the formulas depend only on the distance r and the absolute value of k, i.e., k = k.  
Figure 01a1 _{} Correlation Function 
Figure 01a2 Bessel Function, 1st Kind [view large image] 
Figure 01a3 Si Function 
Figure 01a4 Correlation Examples [view large image] 
Figure 01a5 _{} Power Spectrum, Modification 
The conventional definition of the variance is _{} as an indicator for fluctuation, uncertainty, spread, ... of the measurements on certain variable x, 
Theoretical physicists use the power spectrum from the observational data to determine the cosmic parameters. Essentially, the power spectrum (now denoted by C_{}) is measured by taking the temperature fluctuations (n) between two points on the 2D celestial sphere with angular separation (,_{}) = (_{2},_{}_{2})  (_{1},_{}_{1}) = n_{2}  n_{1}, e.g., (n_{2}) = [(T_{2}  T_{0})/T_{0}] and (n_{1}) = [(T_{1}  T_{0})/T_{0}], with the correlation (,_{}) = < (n_{1})(n_{2}) > (where T_{0} = 2.726^{o}K is the average temperature). The power spectrum at that particular angular separation (,_{}) is calculated by averaging the Fourier transform of (,_{}) over the whole sky similar to Eqs.(1a, 1b). Figure 01a6 shows three views of the temperature variation at different angular scales ~ 2/. The wavy curve is a theoretical model of the power spectrum C_{} based on several parameters such as the total cosmic density, the baryon density (luminous matter) and the Hubble's constant as explained in more details below. There are literally millions of such models. The task is to obtain one that is best fit to the observational data.  
Figure 01a6 Power Spectrum 
The shape of the power spectrum in Figure 01a6 can be separated into sections corresponding to different underlying physical processes (since the matterradiation decoupling) as summarized below: 
Figure 01b shows the strength of density fluctuations for the CMBR and other astronomical objects of various size as explained in the topic of "Superclusters". The corresponding power spectrum is depicted in Figure 01c. Actually, this is not the kind of sound wave we hear on Earth. Its wavelength is very long in the order of 1  1000 Mpc, and its medium is not the air but hot plasma with a mixture of photons and other elementary particles.  
Figure 01b Density Fluctuation _{} 
Figure 01c Power Spectrum 


Figure 02 Acoustic Oscillations _{} 
Figure 03a Gaussian Distribution _{} 
Figure 03b Recombination [view large image] 
_{}  
Figure 03c Legendre Polynomials [view large image] 
Thus, in layman's language, the Power Spectrum can be defined as : "The amount of fluctuation (in term of variance) per small interval of k (in logarithmic scale)" . 
Any map drawn on a sphere, whether it be the CMBR's temperature or the topography of the earth, can be broken down into multipoles. The lowest multipoles are the largestarea, continent and oceansize undulations on the temperature map. Higher multipoles are like successively smallerarea plateaus, mountains and hills (and trenches and valleys) inserted on top of the larger features. The entire complicated topography is the sum of the individual multipoles. The lowest mode ( = 0) is the monopole  the entire sphere pulses as one. This is the average temperature (2.726^{o}K) of the CMBR. The next lowest mode ( = 1) is the dipole, in which the temperature goes up in one hemisphere and down in the other. In the CMBR mapping, the dipole is dominated by the Doppler shift of the solar system's motion relative to the CMBR; the sky appears slightly hotter in the direction the sun is traveling (see Figure 0205 in Topic 02, Observable Universe). The CMBR power spectrum begins at C_{=2} because the real information about cosmic fluctuations begins with the quadrupole ( = 2). Note that the peak variation occurs at about = 200 corresponding to an angular size of about 1 degree (Figures 06, and 01a6). Figure 04 shows the multipoles with = 0, 1, 2. The red color represents variation above the average (green); while the blue color denots less.  
Figure 04 Multipoles 
In the photon fluid approximation, the medium for sound propagation is a fluid of pure photons without taking into account the matter and expansion effects. Figure 05 is a plot of the displacement (red) and its square (blue) of the sound wave at the moment of recombination as a function of k, i.e., it is a much simplified version of the power spectrum. It shows many differences when compares to the observed power spectrum in Figure 06. See a different derivation of this primordial power spectrum in "Quantum Fluctuations and Cosmic Structures". 

Figure 05 Simplified Power Spectrum _{} 
Figure 06 Observed Power Spectrum [view large image] 
The relationship between the sound wave and the Hubble horizon is crucial to understand the differences between the simplified and observed power spectrum. Figure 07 plots the inverse of the Hubble horizon (in a comoving frame) against the conformal time in the inflationary era (blue), the radiation era (orange), and the matter era (red). For those values of k under the colored curves, the corresponding wavelength is greater than the Hubble horizon. These kinds of sound wave are frozen and cannot oscillate. As time progresses beyond the inflationary era, sound wave with longer and longer wavelength can reappear first into the radiation era then to the matter era.  
Figure 07 Hubble Horizon and Sound Wave _{} 
As the amplitude and position of the primary and secondary peaks are intrinsically determined by the number of electron scatterers (density) and by the geometry of the Universe, they can be used to calculate the density of baryons and dark matter, as well as other cosmological constants. Specifically, the first and second peaks yield information about the total density, baryon density and the Hubble's constant. Figure 08 shows the different theoretical models  low Hubble's constant H_{0}, dominant cosmologic repulsion, neutrino with mass (Hot Dark Matter), high baryon density, open universe, and early universe with textures (which is a theory different from the inflationary model and based on topological defects^{1}).  
Figure 08 Power Spectrum Models [view large image] 
There are altogether 10 parameters in these equations, including the densities of CDM, baryons, neutrinos, vacuum energy and curvature, the reionization optical depth, and the normalization and tilt for both scalar (unpolarized) and tensor (polarized) fluctuations, etc. Usually, numerical computation is used to construct models with various values of the parameters. NASA has provided an online computer program "Build A Universe" to crank out power spectrum with various input parameters. Figure 09 is another one called "Max's Cosmic Cinema" by Max Tegmark of MIT. It shows the effects of varying the parameters on the theoretical curves. The graph on the top is the CMBR power spectrum, while the one below shows the power spectrum of the large scale structures. Click the STOP button on the toolbar (the _{}) to view a stationary graph.  
Figure 09 Power Spectrum Animation [view animation] 