1.2 Models and Measurements
1.2.3 Angular Power Spectrum
Maps such as the one shown in Figure 1.4 are not only visually striking, but also contain a wealth of information about the various physical processes responsible for producing the CMB temperature fluctuations (including those described in in the previous section). A powerful method for extracting this information is to examine the statistical properties of the fluctuations by computing their two-point correlation function in angular space. Since we observe the CMB on the surface of a sphere, it is natural to expand its temperature field on the sky in a geometrically appropriate basis of spherical harmonicsY`m:
Θ(ˆn) = ∞ X `=1 ` X m=−` a`mY`m(ˆn) (1.51)
where ˆnis a given direction on the sky,`andmare the multipole moment and azimuthal de- gree of freedom of the spherical harmonics, respectively, and we have ignored the monopole term`= 0 (i.e. the average temperature). The complex coefficientsa`m, which specify the
amplitude and phase of each harmonic mode, are given by:
a`m = Z d3k 2π2 (−i) `Y `m(ˆk) Θ`(k) (1.52)
where Θ`are the Fourier-space multipole moments of the temperature fluctuations Θ. Since
these coefficients describe a Gaussian random field, the expectation value for any particular
a`m must vanish; all their statistical information is thus contained within the two-point
correlation function, which may be written as:
ha`ma∗`0m0i=δ``0δmm0C` =δ``0δmm0 1 2π2 Z dk k Θ`(k) R(k) 2 PR(k) ! (1.53)
where R and PR are the scalar curvature perturbations and their dimensionless power
spectrum, respectively, as defined in §1.1.2. The quantitiesC`, which are expanded inside
the brackets in the above equation, are known as the angular power spectrum - they encode all the underlying physics responsible for generating the CMB temperature fluctuations we
Figure 1.5: Angular power spectrum of the CMB temperature fluctuations as measured by the Planck satellite and presented in Planck Collaboration et al. 2016b [92]. The spectrum is plotted as the power per logarithmic intervalD`=`(`+1)C`/(2π). The red curve represents
the best-fit cosmological model for the data, with the corresponding residuals plotted in the bottom panel. Note the difference in the data at high and low multipoles, as indicated by the dotted line at`= 30. This is due to the use of different spectral estimation algorithms.
observe today. While a highly precise measurement of this spectrum over a broad range of angular scales12 would be a valuable asset to modern cosmology, there is a fundamental
statistical limitation on the uncertainty which increases significantly at large angular scales (small `). Consider the best estimator ˆC` for the true angular power spectrumC` given a
measured set of spherical harmonic expansion coefficients ˜a`m:
ˆ C`= 1 2`+ 1 ` X m=−` |a˜`m|2 (1.54)
where we were able to take the average over all values of m due to the statistical isotropy of the temperature fluctuations (and thus the spectrum). Since the number of available
12
azimuthal degrees of freedom decreases at lower values of`, we should expect the statistical uncertainty of the estimation to increase; this is indeed the case, with the variance of the estimator defined in Equation 1.54 scaling as:
h( ˆC`−C`)2i=
2 2`+ 1C
2
` (1.55)
This “cosmic variance”, as it is more commonly known, is a reflection of the fact that we can only measure a single realization of the universe from our fixed location here on Earth. The CMB temperature angular power spectrum - or TT spectrum - measured by the Planck satellite is shown in Figure 1.5; it contains a number of interesting features that are directly related to some of the physics discussed earlier in this chapter. The most prominent of these is a series of harmonic peaks that start oscillating near a multipole of`∼200, and then continue with decreasing amplitude down to smaller angular scales. These peaks are the result of the acoustic oscillations in the photon-baryon fluid (§1.1.3): the first peak represents modes that have undergone a single compression between the time they entered the sound horizon and the time of photon decoupling, while the second represents those that have undergone both a compression and a rarefaction in the same time interval. The pattern continues at higher values of`, but with an increasingly damped amplitude due to the scattering of photons between hot and cold regions of the plasma toward the end of the decoupling epoch. The scale of the peaks is determined by the angular size the sound horizon at decoupling, which depends on the expansion history of the universe and therefore its curvature and the energy densities of its constituents. The peak amplitudes, on the other hand, are specifically sensitive to the baryon density, since a higher baryon/photon ratio enhances compression and reduces rarefaction. Note that the power spectrum does not vanish between peaks - this is due to maxima of the Doppler effect (§1.2.2) at scales where the fluid velocity is greatest. At relatively large angular scales (` < 30), the spectrum is reasonably flat, forming what is known as the Sachs-Wolfe plateau. It consists of modes that never entered the sound horizon prior to decoupling, and whose amplitude is determined by a combination of the inflationary perturbation spectrum (§1.1.2) and the Sachs-Wolfe effect.
e
Figure 1.6: An electron surrounded by a quadrupole temperature anisotropy in its local radiation environment. Red and blue lines indicate polarization components from hotter and colder temperatures, respectively. During Thomson scattering, this pattern produces a net linear polarization which aligns with the cold axis of the quadrupole.