## Power Spectrum

### Contents

Introduction
Two-Point Correlation Function (Theoretical Base of Power Spectrum)
CMB Data Reduction (2019 Edition)
Generation of the CMBR Power Spectrum
Observational Data
Photon Fluid Approximation
CMB Theoretical Models
Density Power Spectrum

### Introduction

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]

Here's the "Introduction" (see Table 00 for very brief explanation, click underlining subject to go into more detail):

• Sky Map - This is the measurements of some physical parameter over part or whole of the sky.

1. CMB - This one measures the very small temperature variation over the foreground of 2.726oK blackbody radiation in unit of K. It is plotted in galactic coordinates which presents a two dimensional view of the whole sky.
2. CDF - This is a segment of a density map from "sky surveys" with a range of distance (depth) 100 - 1000 Mpc from Earth.

• Two-Point Correlation Function - This is a mathematical tool to quantify the relationship of certain identical variable between 2 spatial points against the " average". A higher value signifies more contrast between them. The CMB image in the first column of Table 00
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 [view large image]

In general, correlation is a statistical measure that quantifies the difference of certain property between 2 variables (at 2 different locations) against a common background (e.g., T1 - Tavg relating to T2 - Tavg, where Tk is the temperature at point k). In other words, it indicates how closely the values of two variables tend to change together. A positive correlation indicates that as one variable increases, the other tends to increase as well. A negative correlation indicates that as one variable increases, the other tends to decrease. For examples, a correlation of +1 indicates a perfect positive correlation, where the variables move in perfect unison in the same direction; a correlation of -1 indicates a perfect negative correlation, where the variables move in perfect unison but in opposite directions; while a correlation of 0 indicates no linear correlation between the variables.

1. CMB - Since the observation is on a 2-dimensional surface, the spatial separation is expressed in the angle . The corresponding formula for the 2-point correlation function in Table 00 is the equivalence of Fourier transform (Figure 00b) but in terms of the multipole moment . The C is essentially the amplitude of the Fourier component and
P(cos) the phase.
2. CDF - It can also be expressed in Fourier transform, the power spectrum (k) is the amplitude and sin(kr)/kr is the phase.
3. #### 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.

• Power Spectrum - It is the amplitude of the Fourier component, which is easier to discern in data processing. The form coded in column 3, Table 00 is the most "likely" one. The power spectrum in column 2 would vary from one point to another in the sky map.

1. CMB - The data reduction begins with collection of microwave signals, which are then translated into tiny temperature (squared) deviation from the average. This process produces the temperature variation for all spatial point in the sky (Figure 00c,b). The production of CMB power spectrum has to go through another round of elaborate process called "Likelihood Estimate" to obtain a most probable value for a given multipole moment from the huge volume of data.
2. CDF - The power spectrum for this case is an ad hoc formula derived by matching the observational data.

• Observation - The CMB observation is usually expressed in term of the Power Spectrum and sky map of tiny temperature variation. The CDF observation is in density fluctuation; r is identified to the size of the astronomical system (e.g., cluster of galaxies, the galaxy is the basic unit in this case).

1. CMB - Figure 00c shows 3 different views of the CMB :
• (a) This is the cosmic foreground (blackbody) radiation in galactic coordinate. It was emitted about 380,000 years after the Big Bang. The original temperature is about 4000oK, it has been reduced to the present 2.726oK by cosmic expansion.
• (b) It is the very small temperature variation (from the average) of 1 part in 100000 as measured by WMAP's observation.
• (c) This is the power spectrum to show the temperature variation (squared) as a function of multipole moment. It is from this graph that many of the cosmic parameters are derived.
2. #### Figure 00c CMB

3. CDF - Figure 00d shows the variation of cosmic density fluctuation from z = 10 (about 12 Glys away) to distance of only a few 100 million light years. It should be viewed as from a fixed solid angle with progressively larger visual field at further distance. It is obvious that the fluctuation evolves from near homogeneous in very large size to marked contrast in smaller space.
4. #### Figure 00d CDF Simulation [view large image]

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 two-point correlation function.

### Two-Point Correlation Function (Theoretical Base of Power Spectrum)

The two-point 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 k-space (k = 2/) of the two-point 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) mass-energy distribution so that the formulas depend only on the distance r and the absolute value of k, i.e., k = |k|.

#### Figure 01a2 Bessel Function, 1st Kind [view large image]

where j 0(kr) = sin(kr)/kr is the Spherical Bessel Function of the First Kind (Figure 01a2).
Thus, the power spectrum would have the dimension of f(x) or f(k) squared and is related to the amplitude f(k) of the k-waves only.

#### Figure 01a4 Power Spectrum Evolution [view large image]

in which, n = ns and A = As.

The case of P(k) = Ak is referred to as primordial spectrum (Figure 01a4,a). It is actually, the portion in the range of k (the long wave) which is outside the Hubble horizon in early universe (Figure 01a5,c, see "Quantum Fluctuation"). This range changes with the expansion of the Hubble horizon allowing the re-entrance of longer wave (corresponding to larger cosmic structure). The P(k) curve also shifts upward with cosmic age (Figure 01a4,b).

The conventional definition of the variance is as an indicator for fluctuation, uncertainty, spread, ... of the measurements on certain variable x, where is the mean (average) value, and N the total number of measurements. See a graphic illustration in Figure 03a in which f(x)dx = (ni/N)dx where ni is the number of measurements within the range dx at xi.
See "Large-scale Structure Formation" for the mathematical detail on the derivation of variance from correlation.

In cosmological applications, the function f(x) is usually designated by the fluctuation of measurements with respect to an averaged value, i.e., (x) = [d(x) - d0] / d0, where d0 is the average value. Then the correlation (r) = <(x)(y)>. The fluctuation (when r = 0) can be the matter density , the temperature T, or the gravitational potential . For the cases of and T, the correlation can be measured
directly from observation of the fluctuations. In fitting observational data, it is often assumed a particular form of the Power Spectrum P(k) = Akn. 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 de-coupled at cosmic age of about 0.3 Myr corresponding to the Hubble horizon dH ~ 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.

### CMB Data Reduction (2019 Edition)

CMB discovery via 1% of the TV static, circa 1964.
• Time Ordered Data - The Planck detectors were designed to detect microwaves at 9 frequency bands (Figure 01a6) from the radio to far-infrared (~ 1 - 0.03 cm) while the expected CMB is at the range of (~ 0.6 - 0.06 cm). The wider coverage was designed to collect more information for the removal of noise from other astronomical objects. See "Planck Instruments". There are 74 detectors in total, 22 and 52 for the Low and High FI (Frequency Instrument) respectively. As the spacecraft has operated for 4 years from 2009 to 2013, it would have collected data from every point in the sky many many times.
• #### 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.

• Frequency Bands - LFI : (30, 44, 70), HFI : (100, 143, 217, 353, 545, 857) GHz corresponding to wave length range from 1 to 0.035 cm. The band width / is 0.2 for LFI and 0.33 for HFI.

• Data - The data stream (ultimately in the form of nano-volt, Figure 01a7,a) consists of the measurements of the sky intensity obtained by the Planck spacecraft as it spun at 1 revolution per minute. Each measurement is tagged by the location on the sky that was pointed to, the time of the observation, and whether or not a solar system object was within the beam (this information is used for noise correct).

• Data Volume - There are about 1 TB data collected in the CMB measurements. Thus, many schemes are devised to minimize the processing effort.

• Maps - Map-making is the process of turning time-ordered data (TOD) into an image of the sky. Since the TOD also tagged the location (on the sky), the data can be pixelized and mathematically represented by the equation :

dt = (Atp)sp + nt ---------- (1g) ,

where dt is a vector for the time ordered data, another vector sp denotes the signal from a location at pixel p, (Atp) represents a matrix, which has elements relating the TOD to specific pixel, while nt is the noise vector (time ordered). The processing is to link each of the pixel p to a signal, which is interpreted as temperature in the CMB maps (color coded). For the case of no noise, the unknown sp is determined by the inverse Apt, i.e.,

sp = (Apt)dt.

The case with noise is more complicated, given by the formula (see other alternatives in "Map-making Methods for CMB"):

sp = [ATN-1A]-1ATN-1dt

where N is the matrix (in boldface) of the noise in the time-line: N = < nnT >, and AT, nT are the transpose of A, n respectively.

The pointing matrix A has a huge dimension of Nt X Np ~ 100 billion entries. Typically it is a very sparse matrix, where each row has a single non-zero entry for a temperature observation at corresponding pixel. Since a pixel would have been scanned many times, each column would have many non-zero entries. It becomes more complicated with the additional measurements of polarization. The examples below are a much reduced pointing matrix and signal vector with Nt = 4, Np = 5 for a total of 20 entries.

,     and a 1080p HD TV screen shot     .

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.

• Bayes' Theorem - It introduces a new kind of probability P(H|D), which relates the probability of "event H" occurring given "event D" has taken place. As shown in Figure 01a8, P(H|D) = [P(D|H) P(H)] / P(D); P(H) and P(D) are the probabilities of observing H and D independently of each other.
• #### Figure 01a8 Bayesian Probability

In application of the theorem to pixelize the CMB time ordered data (TOD), "H" is identified to the TOD dt and "D" to the signal sp (see "Bayesian Monte Carol" for more details).

See "Mapmaking" for detail.

• Likelihood Function - For a probability function f(p,x), where p is the parameter(s) and x is the running variable, the likelihood function (x) = f(x,) applies the Bayesian methodology to treat p = as running variable with x at a fixed value.
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 1-p. 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 01a10 Likelihood Function

Note the similarity between the Bayesian probabilities P(A|B), P(B|A) and the interchanging role between f(p,x) and f(x,p) in defining the likelihood function () = f(x,).

• CMB Map Making - The process involves determination of the signal at one spatial point (the pixel) sp from a collection of time ordered data (TOD) dt's including noise nt's over a period of time as described by Eq.(1g). It is suggested by observations that the distribution of the TOD is in the form of Gaussian (or Normal, see Figure 01a11). This is the "prior" in Bayesian inference, which causes so much distress for Calvin the comic character (see insert in Figure 01a8). Anyway, if we identify xi = dt, and = sp then it is "believed" that the noise nt would take the role of variance . However, since the noise is random, it has to be treated as part of the data such that i = nt. The formulation is more complicated, in term of the likelihood function with assuming the role of independent variable, the equation can be written in multivariate Gaussian :

#### Figure 01a12 CMB Data Likelihood, Multivariate Gaussian

In notation of the likelihood function with = (, ), and x = (x1, ...., xn), () = f(x1, ...., xn|, ), n ~ 1010.

• Power Spectrum - Spectrum is produced by dispersing the components in a source into the individual parts. For example, the optical spectrum is the decomposition of white light into a range of wavelength by a prism as shown in Figure 01a13. It is expressed in unit of intensity = power/area. A similar idea in mathematics is the Fourier transform (FT), which acts like a mathematical prism, to turn a signal into components on a range of wave number (k = 2/). Figure 01a14 is an example to
translate a musical signal into fundamental and harmonics. It demonstrates the virtue of FT to turn a mumble-jumble 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 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).

BTW, in the above derivation, the power spectrum P(k) is dimensionless in term of the fluctuation f(x) = (x) = [y(x) - y] / y, where y is the average. The CMB power spectrum is defined somewhat differently with f(x) = (x) = [T(x) - T] as the millionth temperature difference at point x to its average. Thus, the power spectrum P(k) has the dimansion of millionth temperature squared, i.e., (Ko)2. Even though it is positively defined and temperature is proportional to energy as E ~ kBT, such unit is only remotely related to power (= energy/sec) creating lot of confusion for perplexing novice: "where the heck is the power"? A more appropriate designation would be something like "Temperature Variance Spectrum" (or "TV Spectrum", no pun intended).

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 ~ 180o/. 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 matter-radiation de-coupling) as summarized below:

#### Figure 01a15 CMB Power Spectrum [view Large Image]

1. ISW (Integrated Sachs-Wolfe Effect) Rise - This effect arose from the time-dependent perturbations of the gravitational field. The effect is the sum from contributions along the path of the photons. It has been confirmed through correlations between the large-angle anisotropies (temperature fluctuations) and large-scale structure.
2. Sachs-Wolfe Plateau - Perturbation of the gravitation field at large scale is responsible for this near constant appearance at lower s. Anisotropies at this scale have not evolved significantly, and hence directly reflect the "initial conditions".
3. Acoustic (Doppler) peaks - The rich structure in this region is the consequence of the acoustic oscillation driven by repulsive radiation pressure and attractive gravity (as explained in more details later). The main peak is the oscillatory mode that went through 1/4 of a period (reaching maximal compression) at the time of recombination (between electrons and protons to form neutral atoms). The lower peaks correspond to the harmonic series of the main peak frequency. An additional effect comes from geometrical projection such that the angular position of the peaks is sensitive to the spatial curvature of the universe.
4. Damping Tail (Doppler Foothills) - The recombination process is not instantaneous, giving a thickness to the last scattering surface. This leads to a damping of the anisotropies at the high s, corresponding to scales smaller than that subtended by this thickness. The damping cuts off the anisotropies at multipoles above ~ 2000.

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 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 Ym; while - (+1) is the eigenvalue for r22Ym = - (+1)Ym. See "Spherical Harmonics".

• Cosmological Parameters - The Boltzmann equation for photon distribution in 4-D phase space (x, p) is the methematical tool to derive the theoretical power spectrum of CMB. The following provides a very brief summery of the formation.

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 full-fledged General Relativity treatment would involve the space-time 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.

The temperature variation can be defined as the small difference between 2 photon distributions integrating over the frequency :

#### Figure 01a20 Optical Depth Effect [view large image]

Essentially, the l-h-s of Eq.(1k) is related to gravitational interaction, while those terms on the r-h-s modify its appearance through -e- Compton scattering in the intervening space.

Here's a few comments on Eq.(1k) as illustrated by Figure 01a21:

• The optical depth is the dominant factor for the collison effect on the r-h-s of Eq.(1k). It is defined by :

The process occurs during the transmission of the CMB signal. The optical depth = 0 for no scattering; otherwise Rt < Ri, the power spectrum is distorted or flattened as shown in Figure 01a20. In the absence of gravitational variations and other effects, Eq.(1k) shows that = 0 e-.

• The second term on the r-h-s of Eq.(1k) is related to the Doppler shift, while the quadrupole moment in the third term has something to do with polarization of the signals and hence gravitational wave. This effect is related to the tensor tilt parameter nt. See the failed attempt to link them in "B-mode Polarization".

• The time-dependent gravitational variation on the l-h-s of Eq.(1k) modifies the energy of the photons as they climb in and out of the potential well associated with large scale structures on its way to the detector. This is called ISW (Integrated Sachs-Wolfe) effect, which is seen mainly in the lowest multipoles in the power spectrum.

• At the spot where the CMB originated, both the time dependent potential and optical depth can be neglected in Eq.(1k), then + = constant is conserved or . This is known as SW (Sachs-Wolfe) Plateau, which reflects the initial condition when the power spectrum is flat, i.e., P(k) = Askns-1, with ns ~ 1. In such circumstance, both temperature and gravitational variations are proportional to (the density fluctuation). The origin of CMB sound wave can be derived by applying the equation of continuity and Euler's equation for compressible fluid (see "Navier-Stokes Equations"):

#### Figure 01a21 CMB Formation [view large image]

The acoustic oscillations (or Doppler peaks) determine 6 cosmic parameters : k, ,
cd, b, f, and the Hubble constant H0. See a list of the 10 parameters, also "Baryon Acoustic Oscillations (BAO)" and "Generation of CMBR".

• Gravitational red shift on the frequency of the radiation is another feature beyond the consideration of Eq.(1k). It is in the form: = 0R(t)[1+(t)] which is non-zero even in the absence of gravitational fluctuation.

Putting together Eq.(1k) and other considerations, a theoretical power spectrum is simulated. It only requests the input of some parameters to fit it to the observational profile. The task again involves maximing the likelihood function as discussed previously. This time it is formulated as:

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.

The animated CMB Analyzer below shows essentially the same process to arrive at a set of matching parameters.

(click image to access video); also check out "Max's Cosmic Cinema"

### Generation of the CMBR Power Spectrum

• The structure in the universe is seeded by random quantum fluctuations in the very beginning of the Big Bang. A period of rapid expansion, called inflation, caused these quantum fluctuations to be stretched into cosmic scales.
• The gravitational attraction in the density enhanced regions and radiation repulsion acted together to produce the incoherent acoustic oscillations (noise). Compressing a gas heats it up; letting it expand cools it down - this is the origin of the temperature variation. As shown in Figure 02, if the gravity and sonic motion (the alternate compression and rarefication) work together then the photons and baryons are compressed in the trough producing the first peak with large temperature fluctuation. However, if they counteract each others, a smaller second peak will be created.
• #### Figure 02 Acoustic Oscillations

• The size of these oscillations occurred on all scales (wavelength). Mathematically, the size of the variation at location x, i.e., (x) can be expressed by the Fourier series:

(x) = k{Gk cos(kx)} ---------- (1)

where the sum is over all values of k = 2/, is the wavelength. The coefficient Gk can be calculated from the inverse relation:
Gk = {(x) cos(kx)}, where the sum is over all x. For a given value of k, its harmonics are 2k, 3k, ...; k is called the fundamental mode. Thus, the CMBR power spectrum depends on the phase of the k-waves, i.e., cos(kx) as well.

• Theory of Inflation predicts that there should be as many hot spots as cold spots, i.e., its distribution curve is Gaussian (Figure 03a). Recent (2008) analysis of the WMAP data suggests that it may not be the case. The skewing, known as non-Gaussianity, shows up as a tiny effect with distortion in temperature distribution of the order 1 in 100000. More observations are needed to confirm such finding, which would falsify the theory of inflation. However, other researches indicated that the non-Gaussianity is caused by a large cold spot. The distribution remains Gaussian after removing this abnormal data.
• #### Figure 03a Gaussian Distribution

• As the universe expands and cools, the average energy of a photon falls until eventually hydrogen atoms are able to form. This is the epoch of recombination (Figure 03b) when the photons are released and stream off unimpeded as CMBR today. The acoustic oscillations stop at recombination (no more radiation pressure to produce the expansion). There is a special mode k1 for which the fluid just had enough time to compress once before frozen in at recombination (thus producing the maximum variation). The corresponding wavelength is called the sound horizon: k1 = /(2 sound horizon). Modes caught at oscillations with such wavelength become the peaks in the CMBR power spectrum and form a harmonic series based on k1.

### Observational Data

Since the observational data are obtained from a two-dimensional spherical surface in terms of angular coordinates, the temperature variation in the power spectrum plot is often expressed in terms of either the angle (as shown in the WMAP map) or its Fourier Transform counterpart (angular frequency or multipole as shown in Figure 01a15). Mathematically, the trigonometric function cos(kx) (the phase shift) in Eq.(1) is repalced by the Spherical Harmonics Ym(,), where = 0 denotes the monopole, = 1 the dipole, = 2 the quadrupole, ..., and m can be any integer between - and . The coefficient Gk is replaced by alm . Each alm constitutes a multipole mode. In effect, the x in Eq.(1) is replaced by the angular coordinates and its Fourier transform k is related to the multi-pole moment . Thus in terms of spherical harmonics, the temperature variation can be expressed as:

Similar to the definitions in Eqs.(1a, 1b), the correlation function is now denoted as C(). After integrating over :

#### 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()".

N.B. While it is not known exactly how the WMAP and Planck teams evaluate the observational data, the so-called "Hubble Tension" could be caused by the treatment of the lower terms as it is most affected by the difference in the transformation from a discrete sum to continuous integration (see Eq.(2b) above and an animated graph to illustrate the Hubble constant dependence on the lower multipole of the CMB spectrum); and as illustrated in Figure 06 for = 0, ( + 1)C/2 = 0, but the original form is (2 + 1)C/4 = C/4.

See CMB polarization in "Theories of Cosmic Inflation and B-mode Polarization".

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 largest-area, continent- and ocean-size undulations on the temperature map. Higher multipoles are like successively smaller-area 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 (2.726oK) 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 02-05 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.

### Photon Fluid Approximation

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 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 wave-length can re-appear first into the radiation era then to the matter era.

#### Figure 07 Hubble Horizon and Sound Wave

The shape of the observed power spectrum is determined by a number of factors:

1. The minima in Figure 05 between the peaks always reach down to zero. They are lifted upward by Doppler shift, which makes an out-of-phase contribution filling in the zeros (see Figure 06).
2. The varying height of the peaks in Figure 06 is due to the presence of attractive gravity, which causes more compression and less stretching, hence the odd peaks (#1, 3, ...) are higher (more compression) while the even peaks (#0, 2, ...) are lower (less stretching).
3. The first non-zero peak in Figure 06 corresponds to a flat space geometry. For a universe with positive curvature, it would shift to smaller k (to the left of the diagram); while the shift is to larger k for negative curvature.
4. For those waves (with higher k or shorter wavelength) emerging into the radiation era, they encounter a world of diluting density and gravity due to the cosmic expansion. The net effect is to cause the peaks to decrease with increasing k. The power spectrum eventually trails off at very high value of k.
5. For those waves (with lower k or longer wavelength) re-entering into the matter era when the increase in density is almost exactly balanced by the cosmic expansion. As a result, density, sound amplitude, and gravitational potential remain fairly constant through the matter era for small values of k.
6. Absence of the 0th peak at k = 0 is related to the fact that the sound wave re-enters into the matter era after recombination with no more radiation pressure, so there is no oscillation.
The topic on photon fluid approximation and the related diagrams as well as Figures 05, 07 are adopted from "The Zen in Modern Cosmology" by C. S. Lam; published in June 2008. The insert in Figure 07 is from the cover of his book. It is a Chinese painting imitating the curve in the diagram. Beside providing a lot of insights on the subject of power spectrum, the book contains in-depth presentation on the physics of modern cosmology notwithstanding the mentioning of Zen in the title.

### Theoretical Models

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 H0, 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 defects1).

#### Figure 08 CMB Power Spectrum Models

Theoretical power spectrum has become the modern computational tool for cosmology. There are essentially four components in its framework:

1. Friedmann-Robertson-Walker (FRW) Universe - It is used as the base for cosmic expansion. The effect of open, closed, or flat space are taken into consideration via the corresponding solutions in FRW.
2. Fluid equations - It is believed that the structure of the present universe has evolved from very small initial perturbations, which have grown due to gravity. The universe consists of several different particle species (e.g. photons, neutrinos, baryons and cold dark matter), which interact with each other and have different equations of state. Hence it is necessary to consider the coupled evolution of individual particle species in multicomponent sound wave.
3. Boltzmann equation - The equation that governs the temperature fluctuations is derived from the Boltzmann equation. The collision term describes the interaction of the photon with the electrons. The initial power spectrum is usually assumed to be in the form:
kns-1, where k denotes the momentum of the photon, and ns = 1 for flat space (ns is called the scalar tilt). Subsequently, there are altogether three processes to determine the finally shape of the CMBR power spectrum. The SW effect is the major contribution with the temperature fluctuation T/T ~ -, where is gravitational potential. Then there is the ISW (Integrated Sachs-Wolfe) effect, which modifies the energy of the photons as they climb in and out of the potential well associated with large scale structures. The ISW effect is seen mainly in the lowest multipoles in the power spectrum. The last one is the Doppler peaks. It is caused by electron movements in the plasma, because some of the electrons are moving towards the observer and some move away when they last scatter radiation. The temperature fluctuation is given by the formula: T/T ~ v/c with an angular size around 1o - 2o.
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

Some comments on the parameters in the animation:

• - This is the optical depth. It is used to measure the average distance a photon travelled before its original path is altered by hitting something, e.g., an electron. The CMBR would experiences a certain optical depth, if the Universe was reioninized long ago by quasars or early stars. This effect smears out small-scale features in the CMBR power spectrum, suppressing all accoustic peaks by a constant factor exp(-), while leaving the power spectrum on the large scale structures unaffected.
• k - The space curvature is measured by this parameter from a negative number (open universe) to zero (flat universe), and a positive number (closed universe).
• - This is the cosmological constant contribution to the cosmic acceleration discovered lately. It is sometimes referred to as vacuum energy density.
• cd or cd - This is the undetected cold matter density in term of the critical density c, i.e., cd = cd/c
• b or b - This is the observable luminous matter (baryons) density in term of the critical density c, i.e., b = b/c
• f - This is the undetected hot matter density such as neutrinos.
• ns - It describes the initial clumping of matter in space on different scales. This clumping in turn produced the galaxies and clusters of galaxies we see today. If these perturbations were the same on all physical scales, leading to the formation of similar structures on all scales, the power spectrum is said to have no tilt (ns = 1). The subscript "s" stands for scalar, which means the CMBR is un-polarized.
• nt - This is the tensor tilt for polarized CMBR.
• h - This is the Hubble's constant in unit of 100 km/sec-Mpc.

Table 01 below summarizes the parameters for the best fit theoretical curve to the WMAP data:

#### Table 01 Cosmological Parameters from the WMAP data.

Before the fifth upgrade on April 2009, the Hubble Space Telescope has rendered another valuable service by providing a more accurate value for the Hubble constant. The new value is estimated from the Type1a supernovae in six galaxies including NGC3021 and by the Cepheid variable stars data from NGC4258. The latest value of H0 = 74.2 km/sec-Mpc 3.6 corresponds to an age of the universe T = 1/H0 = 13.5x109 years.

### Density Power Spectrum

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 02-08. 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 two-point correlation function, where is a reference-density. For example, the observable universe with radius r ~ 13.8 Glys and total number of galaxies ~ 2x1012 within is used to define ~ 0.1 galaxy/Mly3.

#### Figure 10a 2-Point Correlation

The power spectrum of Figure 10 reveals some information about the cosmic structure:

• The fluctuations become weaker with larger scales. Weak fluctuations mean that the galaxy distribution is very close to homogeneous, exactly as proposed by Einstein's cosmological principle, which underlies all cosmological models.
• The plot does not follow a straight line. The deviation confirms that the dynamics of the universe have changed with time. From other astronomical observations, it is concluded that the universe is dominated by matter and dark energy. However, when the universe was less than 75,000 years old, photons were the dominant component. In such circumstance, gravity did not cause fluctuations to grow with time the same way they do today. That, in turn, caused the power spectrum to behave differently on the largest scales ( > 1,200 million light years).
• The exact scale of this deviation provides a measure of the total density of matter in the universe, and the result -
~ 2.5 x 10-30 gm/cm3 - in agreement with the value from other measurements.
• The plot strongly suggests that the dark matter is all of the cold variety (CDM). Hot dark matter (HDM) would smooth out the fluctuations in the galaxy distribution on smaller scales, i.e., the curve would dip to zero in contrary to observation.
As shown in Eq.(1a), the CMB power spectrum is derived from the variance. The density power spectrum is defined differently by the Fourier transform of the correlation of the density fluctuations at two spatial points :

Following is a brief description of its history in various cosmic eras (see Figures 11, 12, and 13).

• Inflation - In this era, quantum effect generated spatial density fluctuation between two points. It can be Fourier transformed as waves oscillating with wave length =2/k. The magnitude of these k-space components is described by the density power spectrum P(k). See "Quantum Fluctuations and Cosmic Structures" for more detail.

• Radiation - The (re-defined) power spectrum P(k)/k = Ak(n-1) grew linearly with k, i.e., P(k)/k = Ak which is known as H-Z (or primordial) spectrum (see Figure 13). However, the Hubble horizon dH is smaller than 10 Mpc in this era, the k-waves corresponding to k < 1/dH (or > dH) is frozen outside the horizon. Some of the k-waves would re-enter into the observable universe later as it expands (see lower right illustration in Figure 01a21). This is the era of sound wave generation via the alternative gravity attraction and repulsion by radiation pressure.

• Recombination - This is the era of CMB and Baryonic Acoustic Oscillations (BAO) generation. All the cosmic features behind the "last scattering shell" are supposed to be un-observable as the space earlier is opaque.

• Matter - This era occupies most of the cosmic time up until today. Figures 12 and 13 are the modern versions of the power spectrum which has been evolved to different shape from the original (see Figure 14). Nevertheless, it is possible to determine the constant A (called normalization) from the

#### Figure 13 Density Power Spectrum, Components

CMB data, giving a value .

The table below summarizes roughly the history of density power spectrum by defining :

Hubble Horizon = dh = c/(dR/dt) = Rc/H ----- (3a),
Power Spectrum = P(k)/k = Ak(n-1) ----- (3b)
The numerical values are derived from Figure 12. The index n is chosen by the combined effects of higher correlation, the limit posed by the Hubble horizon, and the evolution of large cosmic structures.

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 15 Power Spectrum Model [view large image]

(k) = P(k)/k = -ak2 + bk + c
where a, b, c are constants to be determined by observational data.

See "Evolution of the Power Spectrum and Self-Similarity in the Expanding One-dimensional Universe".

Figure 15 shows the general form of the formula. The constants can be adjusted such that the maximum shifts gradually to the left side from the earlier version (see Figure 14). In other words, they can be considered as functions of the time. Here's some special cases from

Using the data from the current epoch of (k) in Figure 12 :
P(0) ~ 0,    dP/dk ~ 4x106 for k ~ 0,    and kmax ~ 2x10-2, we obtain :
c ~ 0,    b ~ 4x106,    a ~ 108, which yield the derived parameters :
Pmax ~ 4x104,    and k0 ~ 4x10-2.

This simulated curve fits the observed data very well on the small "k" part of (k) up to the maximum, but fails to portray the long tail toward the higher value of "k". Nevertheless, it is instructive to examine the constants in Eq.(4) by comparing it to the Newtonian equation of motion for projectile in constant gravitational acceleration g :
d2y/dt2 = -g, the solution of which is y = y0 + v0t - (g/2)t2 ----- (5).
where y0 is the initial height, and v0 the initial velocity.

Thus, we can write down the "equation of motion" for (k) as d2(k)/dk2 = -2a, the solution of which is given by Eq.(4), where
"c" is zero as the contribution by the initial k-waves which is almost nil since the Hubble horizon is very small.
"b" should be related to the re-entering of k-waves with longer and longer wavelengths as the Hubble horizon expands.
"a" would be the diffusion damping to weaken the density correlation of large structures. The formula as described by Eq.(4) fails to take the effect into account properly. The remedy is to insert additional term e(k) into the equation :
(k) = P(k)/k = -ak2 + 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 17 Simulated Density Fluctuation

The correctional term e(k) = 0.95x108x(k - 0.02)2 has to be very precise. It would not work for just a little deviation from the value of 0.95x108. It has to be turned on only after the era of recombination.

The power spectrum (k) provides only half of the information to determine the strength of density fluctuation (see Eq.(3)). The other half is from the Spherical Bessel Function sin(kr)/kr (see Figure 01a2). The amplitude of this function decreases gradually with increasing value of kr leading to the fluctuation weakening in large scales as portrayed by the simulation in Figure 17 (in log-log plot). It is in good agreement with the measurement in Figure 10.

1 Cosmic strings are thought to be long, tube-like objects of high-energy material left over from the Big Bang. They are the most interesting type of topological defects because some cosmologists have suggested such material as an alternative source of the density irregularities, visible in CMBR. The WMAP measurements have shown that the actual form of the irregularities is inconsistent with those predicted by the string-based theories.