Fluctuations in the CMB (3)

Fluctuations In the CMB (3) ...

The Dipole Component Again

The dipole component of the CMB that we mentioned earlier corresponds to the L = 1 multipole of the power spectrum. Recall that the dipole component represents only the grossest angular features of the CMB temperature fluctuation: is it larger in one preferred direction in the sky than in the opposite direction? In light of the above discussion, this is because the multipole order for the dipole component is low (L = 1), so it is sensitive only to the most general features of the anisotropy, not to its fine details.

The Power Spectrum

To analyze the anisotropies quantitatively and to use them to determine information about cosmology, the angular anisotropies are decomposed into what astronomers call a multipole power spectrum. This is a standard mathematical technique in which a pattern is separated into a sum of pieces that when added together give the entire pattern. In this case, the decomposition is with respect to a set of terms labled by an integer L that is called the multipole order.

The physical meaning of L is that the larger the value of L, the smaller the average angular size to which one is sensitive. So, for example, if one starts with the low-L multipole pieces and adds pieces corresponding to successively higher values of L, the pattern will initially show only the coarsest features of the whole pattern and will fill in finer and finer detail as each successive multipole is added.

The following figure shows the multipole power decomposition corresponding to the WMAP temperature fluctuation map and to two additional ground-based experiments (ACBAR, located near the South Pole, and CBI, in the Chilean Andes Mountains) that are particularly sensitive to higher multipoles of the CMB because they have high angular resolution. A quantity called the temperature fluctuation power that is derived from the data is plotted as a function of the multipole order L. (The Technically Speaking box below contains more detailed information about what is plotted in this figure.) The uncertainty in the extracted fluctuation power is indicated by error bars on the data points.

The green curve corresponds to a prediction of a particular cosmological model called ΛCDM that combines cold dark matter (CDM) with a cosmological constant (Λ) to describe the spectrum of temperature fluctuations. Such models will be described further elsewhere in this chapter when we discuss the formation of large-scale structure.

Cosmological Parameters

Of particular interest is the set of peaks beginning around L = 200 and continuing for higher L. These are called acoustic peaks, and they have a definite physical meaning in terms of the preceding discussion of acoustic oscillations. By requiring the theoretical curve to fit the data, particularly the acoustic peaks, one can determine a number of important cosmological parameters. In fact almost all parameters of significance in cosmology enter into the exact form of the power spectrum, so if the power spectrum can be measured on small enough angular scales (to high enough L, that is), many of those parameters can be determined from the CMB data alone. Cosmological parameters determined from fits to WMAP data may be found in this table.

Technically Speaking: Multipole Components
of the Power Spectrum
For those students with more advanced math backgrounds, we offer a few more details of the multipole decomposition of the CMB temperature fluctuations in this box. If you are not interested in those details, you may skip this box.
Multipole Expansion
The multipole power spectrum described in the preceding paragraphs and displayed in the figure below is derived from mathematical expansion of the CMB temperature fluctuations in terms of the functions mathematicians call spherical harmonics. Spherical harmonics, which are functions of two angles, θ and φ, are denoted by the symbol
Y_{L M} (θ, φ)
where the integer index L is the multipole order and the index M ranges in integer steps between +L and -L for each L. The fractional variation in CMB temperature ΔT(θ, φ) / T(θ, φ) at angles θ and φ on the celestial sphere is expanded as the sum
ΔT(θ, φ) / T (θ, φ) = Σ F_{L M} Y_{L M} (θ, φ)
where the symbol Σ means to sum over all values of L and M and the coefficients that are adjusted to give the observed value of ΔT(θ, φ) are denoted by F_{L M} .
The quantity plotted on the vertical axis of the temperature fluctuation power plot above is essentially the square of these coefficients averaged over all values of M for each L. The green band around the theoretical curve in the angular power spectrum plot above represents the uncertainty introduced by the average over M and is called the cosmic variance. (The average over M is an average over the angle φ. Because there is no preferred direction for the CMB once the dipole component is subtracted, the average over M varies randomly with the observer's position in the cosmos; hence the name "cosmic variance".)
Geometrical Meaning of the Multipole Index L
The general meaning of the multipole index L in this sum is that the corresponding term is sensitive to structure down to an angular range of about π/L measured in radians, which corresponds to about 180/L measured in degrees. The top axis of the power spectrum above labeled "Angular Scale" is a reflection of this angular sensitivity (note that for low values of the multipole order the angular scale is around 90 degrees but for the highest values plotted it is less than a degree).
Example: Curvature and the Location of the First Acoustic Peak
One piece of information that can be determined from the temperature fluctuation power spectrum is the flatness of the Universe's geometry. This comes from a detailed fit of the theoretical curve to all of the data, but can be read qualitatively directly from the graph. In a flat Universe, calculations indicate that the dominant angular scale at the last scattering surface for the CMB is about 1 degree. From the power plot above (see the top and bottom axes), an angular scale of 1 degree implies an L of about 200. Therefore, a flat Universe requires that the first acoustic peak should occur around this L, as is seen to be true in the plot.
If the geometry is not flat, the position and strength of the first acoustic peak would change. For example, if the geometry of the Universe were open instead of flat, the first acoustic peak would move to higher L. This can be inferred from the previously shown schematic effect of curvature on the anisotropy pattern. Since the lensing effect of open spacetime would compress the observed fluctuations to smaller apparent angular size than they actually are, this would shift the first acoustic peak to higher L (which corresponds to smaller angular resolution).