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Power Spectrum


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]


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

Correlation Function 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 01a1 Correlation Function

Bessel Function, Order Zero

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.

Correlation Function

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
Correlation Examples 2 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.

Figure 01a5 Power Spectrum
= Ak [view large image]


CMB Data Reduction (2019 Edition)

                CMB discovery via 1% of the TV static, circa 1964.


Generation of the CMBR Power Spectrum

Acoustic Oscillations
  • 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.

  • Gausssian Distribution
  • 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.
  • Figure 03b Recombination


    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 :
    Legendre Polymonials

    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".

    Multipole 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.

    Figure 04 Multipoles
    [view large image]


    Photon Fluid Approximation

    Simplified Power Spectrum Observed Power Spectrum 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]

    Hubble Horizon 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

    power spectrum 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.
    power spectrum animation 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

    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

    Cosmic Structure 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]

    2-Point Correlation 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

    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 :

    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.

    Power Spectrum Evolution Power Spectrum Model 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 = -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 :
    Power Spectrum, Simlated Density Fluctuation, Simulated (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 16 Power Spectrum, Simulated [view large image]

    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.