Power Spectrum

Theoretical physicists use the power spectrum plot from the observational data to determine the cosmic parameters. Essentially, the power spectrum is a plot of the amount of fluctuation against the angular (or linear) size. The fluctuation is the difference in the two measurements at the corresponding points. It can be the fluctuation of temperature or density or any other kind of measurable quantity. Figure 01 shows the fluctuation of temperature at different (inverse) angular scales. It is a theoretical model 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.

Figure 01 Power Spectrum [view large image]

• 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.
• The size of these oscillations occurred on all scales (wavelength). Mathematically, the spatial variation can be expressed in the form of Fourier series F(x):
  
  F(x) = _k{G_k cos(kx)} ---------- (1)
  
  where the sum is over all values of k = 2/L, L is the wavelength.
  The coefficient G_k can be calculated from the inverse relation: G_k = {F(x) cos(kx)}, where the sum is over all x. For a given value of k, its harmonics are 2k, 3k, 4k, ...; k is called the fundamental mode. Inflation predicts that the amplitude G_k of each of the modes is random. In addition, the distribution of probabilities follows the shape of a bell curve (known as Gaussian) of very nearly the same width for each mode. The resulting power spectrum is called flat because of its lack of distinguishing features. Significant deviations from flatness should occur only in those modes produced at either the end or the beginning of inflation.
• 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 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 CMRB power spectrum and form a harmonic series based on k₁.
• 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 plot) or its Fourier series counterpart (as shown in Figure 01 using the angular frequency or multipole). Mathematically, the trigonometric functions in Eq.(1) are repalced by the spherical harmonics Y_lm(,), where l=0 denotes the monopole, l=1 the dipole, l=2 the quadrupole, ..., and m can be any integer between -l and l. The coefficient G_k is replaced by C_l. Each C_l constitutes a multipole mode. Thus in terms of spherical harmonics, the angular variation can be expressed as:
  
  F(,) = _l{C_lmY_lm(,) } ---------- (2)
  
  

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 (l=0) is the monopole - the entire sphere pulses as one. This is the average temperature (2.726oK) of the CMBR. The next lowest mode (l=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 Cl=2 because the real information about cosmic fluctuations begins with the quadrupole (l = 2). Note that the peak variation occurs at about l = 200 corresponding to an angular size of about 1 degree. Figure 02

Figure 02 Multipoles [view large image]

shows the multipoles with l = 0, 1, 2. The red color represents variation above the average (green); while the blue color denots less.

• Since photons were scattered randomly in the primordial plasma, it seems that the CMBR would be unpolarized. But on the small scales where photons can travel with relatively few scatterings, the directional information is retained. Polarization on larger scales was caused by scattering of CMBR photons at the re-ionization epoch.

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 03 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 03 Power Spectrum Models [view large image]