1.4 How to Detect Inflation
1.4.1 CMB Temperature Anisotropies
Figure 1.2: The anisotropies of the CMB as observed by Planck. Image courtesy of ESA and Planck Collaboration.
(a) The temperature angular power spectrum of the CMB. The red points and error bars represent measurements, while the green shaded area represents cosmic variance. The vertical axis is `(`+ 1)C`/2π. Figure is from the 2013 release of Planck science results [1].
(b) The temperature angular power spectrum of the CMB with combined data from WMAP9 [17], ACT [25], and SPT [82]. The solid line shows a best fit model to the data. The dashed line shows the CMB-only component of that model. Figure from [25].
Figure 1.3: Current best measurements of the CMB power spectrum from Planck, SPT, and ACT.
The CMB temperature anisotropies have become one of the most important tools of modern experimental cosmology. To analyze these small temperature anisotropies, we de- compose them into spherical harmonics on the sky:
∆T T =X
`m
a`mY`m(θ, φ), (1.46)
where a`m are expansion coefficients and θ and φ are spherical polar angles on the sky.
From here, we take a power spectrum:
C` =h|a`m|2i. (1.47)
We typically multiply this quantity by `(`+ 1) for plots of the CMB power spectrum (see Fig. 1.3).
The CMB temperature anisotropy power spectrum is extremely valuable for understand- ing the early universe. From it, we can construct a model of what the universe looked like just prior to recombination and place valuable constraints on many of the fundamental pa- rameters of the universe. In order to understand the power spectrum, it is helpful to think about how the temperature anisotropies formed.
Small, random quantum mechanical fluctuations in the density of the universe were blown up to cosmological scales by the process of inflation. These density perturbations source gravitational potential wells that are the seeds of large-scale structure in the universe.
Over time, the density fluctuations grow, through gravitational instability, to become the first stars, galaxies, and clusters. A competition between the overdensity of the fluid in the gravitational potential wells and the gravitational redshifting of the photons as they climb out of the potential wells determines the observed CMB temperature fluctuations.
The temperature fluctuation due to redshifting is larger than that from the overdensities and thus, the overdense regions actually correspond to cold spots on the sky. These density fluctuations also cause the photons of CMB to have small variations in temperature. Since the photons of the CMB have propagated freely through the universe since decoupling, affected by little except cosmic expansion and reionization (which rescatters approximately 10% of the photon of the CMB), we still see the anisotropies today.
One of the most remarkable features of the CMB power spectrum is the series of peaks
known as the “acoustic peaks.” These peaks come from the physics of the photon-baryon fluid in gravitational potential wells prior to decoupling. The photon-baryon fluid com- presses in gravitational wells. As the fluid compresses, the radiation pressure of the fluid increases and provides a restoring force. The interplay of these two forces results in acoustic oscillations of the fluid. The compression of the fluid in the gravitational well causes it to heat up. The rarefaction of the fluid in corresponding gravitational “hills” (underdense regions) causes the fluid to cool. When the photons are released at recombination, the acoustic oscillations will be “frozen in” and we see the oscillations of the fluid as changes in the temperature of the CMB photons.
As mentioned in §1.3, inflation causes these random fluctuations to occur at all scales.
However, wave modes that have reached either the crest or trough of their oscillations at the time of recombination will have enhanced temperature fluctuations. The largest of these will be the mode that had time to compress exactly once before recombination, but not enough time to rebound. This wavenumber of this mode will correspond to π divided by the amount of distance that sound could travel prior to recombination — the sound horizon. Harmonics of this mode will also have enhanced temperature fluctuations for the same reason. These are the modes that were frozen in at exactly one of the extrema of their oscillations. We see the spatial variations in the temperature as angular scales on the sky today. The enhanced modes become the acoustic peaks of the CMB spectrum. Thus, the first peak in the CMB power spectrum corresponds to the angular size of the sound horizon at recombination, and all the subsequent peaks are the result of the harmonics of that mode. We can use estimates of the density of matter just prior to recombination to calculate the speed of sound in the photon-baryon fluid, which allows to evaluate the size of the particle horizon at the time of recombination.
This first peak in the CMB temperature power spectrum occurs around ` ' 220, or an angular scale of approximately one degree, which is in excellent agreement with the prediction for a flat universe (k = 0, ρ = ρc). BOOMERanG, in particular, is noted for being the first experiment to map the first acoustic peak of the CMB power spectrum [28].
(Their result was quickly followed by one from the MAXIMA experiment [44].) The flatness of the universe, as shown by the location of the first peak, indicates that the total density of the universe is very near the critical density.
The odd-numbered peaks in the CMB power spectrum are associated with modes that
were at their maximum compression at the time of recombination. Similarly, the even- numbered peaks are associated with modes at their maximum rarefaction. The baryons in the photon-baryon fluid add inertial and gravitational mass to the system, so a very high baryon density will enhance the compression of the fluid, and therefore the heights of the odd numbered peaks. We see from the power spectra that the height of the second peak is suppressed relative to the first and third peaks, and from this we can constrain the baryon density. The relative amplitudes of the higher acoustic peaks also allow us to constrain the dark matter density.
The oscillations at very high multipoles (small angular scales) are damped out due to the random walk that CMB photons make at the time of recombination. Photons within the distance traveled by a random walk will thermalize, and the temperature anisotropies on those scales will be averaged away. The damping scale provides another check on the curvature of the universe, as well as the baryon density. A higher baryon density will decrease the mean free path of the photon’s random walk, thereby decreasing the damping scale and shifting the damping tail of the CMB power spectrum to higher multipoles.
Our ability to measure the power spectrum at very low multipoles is fundamentally limited by sample variance, by which we mean that there are only 2`+ 1 msamples at each multipole. This leads to an error of
∆C` = r 2
2`+ 1C`. (1.48)
This “cosmic variance” becomes the limiting error on the power spectrum at very low multipoles.
There are additional, practical limits on how well the power spectrum can be measured due to the realities of experimental science [47]. Most instruments will observe only some fraction of the sky. This will increase the errors by a factor offsky−1/2:
∆C`= s
2
(2`+ 1)fskyC`. (1.49)
The noise of the experiment will also increase the error on the power spectrum:
∆C`=
s 2 (2`+ 1)fsky
(C`+N`), (1.50)
where N` is the power spectrum of the noise projected onto the sky. The error due to the noise will dominate at small angular scales (large multipoles) since the signal-to-noise of the instrument will be poor on scales smaller than the size of the instrument beam.
Lastly, averaging over bins in multipole space, ∆`≈`, will add an additional factor of
∆`−1/2 to the error:
∆C`=
s 2
(2`+ 1)fsky∆`(C`+N`). (1.51) The measurement of the CMB temperature power spectrum was the start of a new era in precision cosmology. Although many of the parameters estimated from it were supportive of the theory of inflation, definitive proof would have to come from an even more subtle measurement.