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The topic of "Power Spectrum" involves many intermingling concepts. This "Introduction" is an attempt to clarify the ideas before going into the details. It can also serve as a brief summary for those who do not want to go into the nitty gritty of the subject. It covers 2 different cosmic structures, namely, the CMB (Cosmic Microwave Background) and CDF (Cosmic Density Fluctuation). They may look different, but basically from the same "Quantum Fluctuations" at the very beginning of the Universe. Their theoretical foundations also share the same kind of formalism. 

Table 00 Introduction to Power Spectrum and Beyond [view large image] 
looks homogenous (almost the same color at every point) showing low correlation between the color of any 2 points; while the CDF image displays marked difference between the wall and void of the galactic clusters meaning high correlation (or fluctuation). This is somewhat similar to the Pearson Correlation between 2 points (x and y in Figure 00a) with the coefficient R = 1 for prefect correlation and R = 0 for no correlation.  
Figure 00a Pearson Correlation 
P_{}(cos) the phase. 

Figure 00b Fourier Transform [view large image] 
For zero separation, the correlation becomes variance of the function. Then P_{}(cos) = 1, and sin(kr)/kr = 1 for CMB and CDF respectively. It represents the statistical variance of each point in the sky map. 


Figure 00c CMB 
Figure 00d CDF Simulation 
Figure 00d is a computer simulation, see Figure 10 or column 4, Table 00 for measurement in density fluctuation (r). See Figure 10a for an illustration of the relationship between density fluctuation and twopoint correlation function. 
The twopoint correlation function in statistics can be defined as the probability of relating (multiplying) 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 space. It is expressed as < f(x)f(y) > in mathematical notation and sometimes denoted as _{}(x,y) or (r). The amplitude f(k) in the Fourier transform to kspace (k = 2/) of the twopoint correlation function with r = 0 (or x = y) is related to the 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, 
where j _{0}(kr) = sin(kr)/kr is the Spherical Bessel Function of the First Kind (Figure 01a2). 
Figure 01a4 Power Spectrum Evolution [view large image] 
directly from observation of the fluctuations. In fitting observational data, it is often assumed a particular form of the Power Spectrum P(k) = Ak^{n}. The power index n = 1 is found to be suitable for CMB data. Such form can also prevent the divergence of the gravitational potential fluctuations on both large and small scales. The primordial fluctuation is heavily modified on the short wave portion once the matter and radiation decoupled at cosmic age of about 0.3 Myr corresponding to the Hubble horizon d_{H} ~ 10 Mpc (in comoving distance scale to remove the effect of cosmic expansion). Examples of the CMB and density power spectrums, and correlation function are illustrated in Figure 01a5,a,b.  
Figure 01a5 Power Spectrum 


Figure 01a6 _{} Frequency Bands 
The 22 LFI horns are arranged around the edge of the HFI instrument, as shown by the insert in upper left of Figure 01a6, the insert on the upper right is the PLanck satellite in action at L2. 
The pixels are then arranged to match the elliptical shape of the map similar to the TV screen pixel format such as 1920x1080 (width x height), but with a varying width for the CMB map. With the monopole, dipole, and Milkyway foregrounds removed (see "CMBR Fluctuations"), the 9 frequency bands are merged into one map, in which the data for each pixel would contain contributions from all 9 frequency bands (Figure 01a7,b). It is the task in the next step to plot the data as a function of frequency (or its equivalance of wavelength , wave number k=2/, or multipole number ).  
Figure 01a7 CMB Data Reduction _{} 
The very large CMB data sets requires innovative tools of analysis. The followings introduce a noval techniques that has been used since the early days of CMB researches. 


Figure 01a8 Bayesian Probability _{} 
In application of the theorem to pixelize the CMB time ordered data (TOD), "H" is identified to the TOD d_{t} and "D" to the signal s_{p}. 
For example, in the Bernoulli trial (Figure 01a9, k = x), if the probability for the success of tossing a biased coin is p (if k=1), then the probability of failure (for k=0) is certainly 1p. Its corresponding likelihood function is shown in Figure 01a10, in which p = becomes a running variable with k = 0 and 1 to be the parameters. Interpretation of the graph is now very different. It shows that the most likely case for failure (k = 0) is to bias the coin to such shape that = 0; while the opposite case for complete success (k = 1) is to make = 1.  
Figure 01a9 Bernoulli Distribution _{} 
Figure 01a10 _{} Likelihood Function 
Note the similarity between the Bayesian probabilities P(AB), P(BA) and the interchanging role between f(p,x) and f(x,p) in defining the likelihood function () = f(x,). 
Figure 01a11 _{} Gaissian Distribution 
Figure 01a12 CMB Data Likelihood, Multivariate Gaussian _{} 
In notation of the likelihood function with = (, ), and x = (x_{1}, ...., x_{n}), () = f(x_{1}, ...., x_{n}, ), n ~ 10^{10}. 
translate a musical signal into fundamental and harmonics. It demonstrates the virtue of FT to turn a mumblejumble signal into something tangible. However, this is not exactly the process to generate a power spectrum as the FT can generate negative coefficients. A power spectrum is closely related to the optical spectrum but in unit of power instead of power/area. Its mathematical derivation is shown in the previous section on "correlation". It is actually the  
Figure 01a13 Spectrum from Prism [view large image] 
Figure 01a14 Fourier Transform of Musical Signal _{} 
Fourier transfrom of the "variance" and also shows the presence of positively defined fundametal and harmonics (see Figure 01a7,c). 
Actually, the CMB power spectrum (now denoted as C_{} instead of P(k)) is a function of the multipole number instead of k (see "Observational Data"). Figure 01a15 shows three views of the temperature variation at different angular scales ~ 180^{o}/. 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. The shape of the power spectrum in Figure 01a15 can be separated into sections corresponding to different underlying physical processes (since the matterradiation decoupling) as summarized below:  
Figure 01a15 CMB Power Spectrum [view Large Image] 
In actual practice, the power spectrum is derived by the likelihood function as described earlier. It is also based on the assumption of Gaussian distribution, but the various terms carry different meaning as shown below. Starting from the temperature variation as the sum of Spherical Harmonics, the likelihood function for this particular case is expressed as:  
Figure 01a16 Galactic Coordinates [view large image] 
Figure 01a17 Band Power Spectrum [view large image] 
See Figure 01a16 for illustration of the Galactic corordinates and its relationship to various CMB temperature definitions, and Figure 01a17 for a spectrum produced by Band Power from the observations of Planck and SPT. The likelihood function in Eq.(1j) now considers the temperature variation _{i} as parameter and the variance (_{i})^{2} becomes the running variable to determine the maximum . Figure 01a18 shows a series of likelihood variations as a function of (+1)C_{}/2 and the location of the maximum by numerical computation. 

Figure 01a18 Likelihood Power Spectrum [view large image] 
Note that (2+1)C_{}/4 ~ (+1)C_{}/2 per for > > 1. The (2+1) is related to the number of m states for each and it is also part of the normalization constant for Y_{m}; while  (+1) is the eigenvalue for r^{2}_{}^{2}Y_{m} =  (+1)Y_{m}. See "Spherical Harmonics". 
The temperature T was about 3000 K at the time of the CMB formation (Figure 01a19). 

Figure 01a19 Black Body Spectrum [view large image] 
A fullfledged General Relativity treatment would involve the spacetime metric g_{} determined by the GR field equation, and the geodesic equation (the GR version of the equation of motion, i.e., F = ma). The collision term C[f] is evaluated from the Compton scattering process (p) + e^{}(q) (p') + e^{}(q'). See "Boltzmann Equation" for details. 
Figure 01a20 Optical Depth Effect 
Essentially, the lhs of Eq.(1k) is related to gravitational interaction, while those terms on the rhs modify its appearance through e^{} Compton scattering in the intervening space. 
Figure 01a21 CMB Formation 
The acoustic oscillations (or Doppler peaks) determine 6 cosmic parameters : _{k}, _{}, _{cd}, _{b}, f_{}, and the Hubble constant H_{0}. See a list of the 10 parameters, also "Baryon Acoustic Oscillations (BAO)" and "Generation of CMBR". 
The likelihood function is computed numerically for different values of a given parameter. A particular value corresponding to the likelihood maximum is chosen (see Figure 01a7,d). 

Figure 01a22 Cosmic Parameters [view large image] 
Figure 01a22 illustrates the variation of the power spectrum model with the increasing value of the parameter. 


Figure 02 Acoustic Oscillations _{} 
Figure 03a Gaussian Distribution _{} 
Figure 03b Recombination _{} 
_{}  
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 n()". 
Any map drawn on the surface of 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 01a15). 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. It should occur in the era of recombination. 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 CMB Power Spectrum Models _{} 
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 or esc button to view a stationary graph.  
Figure 09 Power Spectrum Animation _{} 
The density fluctuationn of the cosmic structure is different from the CMBR temperature fluctuation since there is insufficient repulsive force to counteract the gravitational attraction. The plot (see Figure 10) does not show the "up and down" variation as in Figure 0208. It displays a smooth curve for the variation of galaxy counts on different scale. The measurements have been taken by both the 2dF and SDSS (Sloan Digital Sky Survey) teams with consistent results. Essentially, the measurements were performed with a series of spheres of a given radius at random in the universe and counting the number of galaxies in each one and compute the average difference. The procedure was repeated with spheres of various radii to produce the plot in Figure 10, which is in broad agreement with CDM theory.  
Figure 10 Density Fluctuation in Size Scale [view large image] 
Thus, the galaxy is the basic unit in determining the density of certain spherical volume with radius r by counting the number of such unit within. Density fluctuation is defined by the averaged density variation from one volume (of fixed radius r) to another. Figure 10a illustrates the relationship with the twopoint correlation function, where is a referencedensity. For example, the observable universe with radius r ~ 13.8 Glys and total number of galaxies ~ 2x10^{12} within is used to define ~ 0.1 galaxy/Mly^{3}.  
Figure 10a _{} 2Point Correlation 


Figure 11 Cosmic Era 
Figure 12 Density Power Spectrum, Measured _{} 
Figure 13 Density Power Spectrum, Components _{} 
CMB data, giving a value . 
In effect, the spectral index n in Eq.(3b) varies with a range of 2 1 0 from ancient to modern. The normalization constant A also depends on time as shown in Figure 14. It also shows that in early universe, the P(k) curve is almost a straight line extended to high k value (short wave length) enabling the portion of power spectrum in the form of Ak and then evolves by shifting the maximum toward lower value of k (longer wave length). Just a casual inspection at the power spectrum suggests that it could be simulated by projectile formula of the form : 

Figure 14 Power Spectrum Evolution [view large image] 
Figure 15 Power Spectrum Model [view large image] 
(k) = P(k)/k = ak^{2} + bk + c where a, b, c are constants to be determined by observational data. 
(k) = P(k)/k = ak^{2} + bk + c + e(k)  (6). Further investigation reveals that it is possible to fit the entire power spectrum by adding one term to the power spectrum (k) after the maximum as shown in Figure 16 (in red). It means that the effect is turned on only at the moment of recombination. The damping produced by the origin projectile formula is too much, it has to be reduced by an extra term as shown. In Figure 16, the scale is compressed while the k scale is expanded in plotting the graph. 

Figure 16 Power Spectrum, Simulated [view large image] 
Figure 17 Simulated Density Fluctuation _{} 
The correctional term e(k) = 0.95x10^{8}x(k  0.02)^{2} has to be very precise. It would not work for just a little deviation from the value of 0.95x10^{8}. It has to be turned on only after the era of recombination. 