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Power Spectrum


Two-Point Correlation Function
The CMBR Power Spectrum
Generation of the CMBR Power Spectrum
Observational Data
Photon Fluid Approximation
Theoretical Models

Two-Point Correlation Function

Correlation Function The two-point correlation function in statistics can be defined as the probability of relating a function f(x) to another function f(y) at the spatial vector r in excess of the random distribution (Figure 01a1). This statement is valid only when it is averaged over a large number of such configurations in the space. It is expressed as < f(x)f(y) > in mathematical notation and sometimes denoted as (x,y) or (r). The coefficient P(k) in the Fourier transform to k-space (k = 2/) of the two-point correlation function is called Power Spectrum. Following is a brief sketch of the mathematics involved. BTW, the formulas become simpler in cosmology, which assumes an isotopic (rotational invariance) and homogeneous (translational invariance) 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 Si Function

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

Figure 01a3 Si Function
[view large image]

where j 0(kr) = sin(kr)/kr is the Spherical Bessel Function of the First Kind Order Zero (Figure 01a2). Its integration from 0 to x is called the Si Function (Figure 01a3). It is equal to /2 as x .

In cosmological applications, the function f(x) is usually designated by the fluctuation (x) = [f(x) - f] / f, where f is the average value. Then the correlation (r) = < (x)(y) >. The fluctuation 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. Theoretically, it is often calculated by assuming a particular form of the Power Spectrum P(k) = Akn for the primordial fluctuation, where the fluctuations in k-space are un-correlated by definition of the Fourier Transform and A is a constant. The power index n = 1 is a popular choice, because only such form can 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 density power spectrum, correlation, and fluctuation are illustrated in Figure 01a4, Diagram a, b, and c respectively. Figure 01a5 shows the power spectrum explicitly and also indicates the various processes that can modify the spectrum.

According to Eq.(1c) for P(k) = Ak, the correlation (r) for primordial density fluctuations at two spatial points with separation r is reduced to the form (Figure 01a4,b) :
Correlation Examples 2 Power Spectrum

Figure 01a4 Correlation Examples [view large image]

Figure 01a5 Power Spectrum, Modification

The conventional definition of the variance is as an indicator for fluctuation, uncertainty, spread, ... of the measurements on certain variable x,
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.


The CMBR Power Spectrum

power spectrum Theoretical physicists use the power spectrum from the observational data to determine the cosmic parameters. Essentially, the power spectrum (now denoted by C) is measured by taking the temperature fluctuations (n) between two points on the 2-D celestial sphere with angular separation (,) = (2,2) - (1,1) = n2 - n1, e.g., (n2) = [(T2 - T0)/T0] and (n1) = [(T1 - T0)/T0], with the correlation (,) = < (n1)(n2) > (where T0 = 2.726oK is the average temperature). The power spectrum at that particular angular separation (,) is calculated by averaging the Fourier transform of (,) over the whole sky similar to Eqs.(1a, 1b). Figure 01a6 shows three views of the temperature variation at different angular scales ~ 2/. The wavy curve is a theoretical model of the power spectrum C based on several parameters such as the total cosmic density, the baryon density (luminous matter) and the Hubble's constant as explained in more details below. There are literally millions of such models. The task is to obtain one that is best fit to the observational data.

Figure 01a6 Power Spectrum
[view large image]

The shape of the power spectrum in Figure 01a6 can be separated into sections corresponding to different underlying physical processes (since the matter-radiation de-coupling) as summarized below:
  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 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.
Density Power Spectrum Sound Wave Spectrum Figure 01b shows the strength of density fluctuations for the CMBR and other astronomical objects of various size as explained in the topic of "Superclusters". The corresponding power spectrum is depicted in Figure 01c. Actually, this is not the kind of sound wave we hear on Earth. Its wavelength is very long in the order of 1 - 1000 Mpc, and its medium is not the air but hot plasma with a mixture of photons and other elementary particles.

Figure 01b Density Fluctuation

Figure 01c Power Spectrum


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.

  • 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 distance 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 [view large image]


    Observational Data

    Since the observational data are obtained in a two-dimensional spherical surface in terms of angular size, the temperature variation in the power spectrum plot is often expressed in terms of either angular size (as shown in the WMAP map) or its Fourier Transform counterpart (angular frequency or multipole as shown in Figure 01a6). Mathematically, the trigonometric function 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. Thus in terms of spherical harmonics, the angular 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 k (in logarithmic scale)" .
    See Figure 06 for a pictorial sample, also see Eq.(1f) for the definition of Power Spectrum in its generic form.

    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). See an animated graph to illustrate the Hubble constant dependence on the lower multipole of the CMB spectrum.

    Multipole Any map drawn on a sphere, whether it be the CMBR's temperature or the topography of the earth, can be broken down into multipoles. The lowest multipoles are the 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 01a6). Figure 04 shows the multipoles with = 0, 1, 2. The red color represents variation above the average (green); while the blue color denots less.

    Figure 04 Multipoles
    [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. 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 Power Spectrum Models [view large image]

      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. Linearized equations of general relativity - The small perturbations of the metric tensor g produce the temperature fluctuations on the photons sitting in the photon-baryon fluid. This is called the SW (Sachs-Wolfe) effect.
    3. 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 as multicomponent fluid.
    4. 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 terms 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 effect. 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 button on the toolbar (the ) to view a stationary graph.

    Figure 09 Power Spectrum Animation [view 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.

    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.