Piper at the Gates of Dawn

Variations in the microwave radiation capture important infor­mation about conditions in the early universe. To describe these variations, astronomers like to think about the power spectrum of the variations, or how much of the variation is on a particular angular scale. In this formalism, l is the angular frequency of varia­tions. For example, l = 2 corresponds to two cycles across the sky or variations over 100 degrees, showing the dipole of the Local Group motion mentioned earlier. The 7-degree limit of COBE cor­responded to l = 30 and the much better 0.3-degree resolution of WMAP reaches almost to l = 1000. Think of l as the number of waves to go all around the sky in a circle and l2 as the number of “tiles” needed to cover the sky. Having more tiles means each one covers a smaller area of sky. The shape of the angular power spec-

Multipole moment l

10 100 500 1000

Piper at the Gates of Dawn

Angular size

Figure 12.3. A graph showing the amount of temperature variations on the verti­cal scale, versus the angular scale of those variations on the horizontal scale. The quantity l is like a harmonic so it represents how many waves can fit across the sky, so larger l means smaller feature on the microwave map of the sky. The peak in microwave power on scales of a degree dates from 400,000 years after the big bang (NASA/WMAP Science Team/S. Larson).


trum is compared to predictions from the big bang model (figure 12.3). It’s fair to think of l as characterizing the “harmonics” of variation of the radiation.31

The physics of the early universe is esoteric, but we can gain insight by the analogy with sound as long as we don’t stretch the analogy to the breaking point. Before 380,000 years, while radia­tion was still trapped by matter, the electrons and photons acted like a gas, with the photons ricocheting off the electrons like little bullets. As in any gas, density disturbances moved at the speed of sound as a wave, a series of compressions and rarefactions. Com­pression would heat the gas and rarefaction would cool it, so the sound waves would manifest as a shifting series of temperature fluctuations. When electrons combined with protons to become neutral atoms, the photons from slightly hotter and cooler regions traveled unimpeded through the universe. So the temperature vari­ations that we see now are a “frozen” record of fluctuations from that time.

If there was a “piper at the gates of dawn,” then who or what was the piper?32 Inflation is presumed to be the mechanism in the
very early universe that rendered space flat and smooth, so it also must have been the source of the initial tiny variations. Those variations from inflation were hugely expanded quantum fluctua­tions, with the special property that the strengths of disturbances on all different scales were about equal. The disturbances are all produced at once from the very moment of creation, so they make sound waves that are synchronized. The result is sound waves with a series of harmonics or overtones, like the sounds from a flute with holes at regular intervals. Other models for the origin of the disturbances tend to be more chaotic or random, so they predict sound waves like those from a flute with holes at irregular or ran­dom intervals. Inflation is the music of the spheres dreamed of by Pythagoras.

In the flute analogy, the fundamental tone is a wave with its largest amplitude at either end of the tube and its minimum am­plitude in the middle. The overtones are whole number fractions of the fundamental tone, so one half its wavelength, one third, one quarter, and so on. In the early universe, however, the waves are oscillating in time as well as space, so the waves originate in the first iota of time at inflation and they end at the time the universe becomes transparent about 380,000 years later. The fundamental tone is a wave that has maximum positive displacement (or equiv­alently, maximum temperature) at inflation and has oscillated to maximum negative displacement (or minimum temperature) at the time of transparency. The overtones oscillate two, three, four, or more times faster and so cause successively smaller regions of space to reach their maximum amplitude 380,000 years later.

Thus, we have all the ingredients to interpret the graph of amount of temperature variation versus angular frequency, l, mea­sured by WMAP. There’s one more subtlety. Inflation predicts that all the harmonics should have the same strength. But sound with very short waves dissipates, because it’s carried by the collisions be­tween particles and when the wavelength is less than the distance a particle travels before hitting another particle, the wave dissipates. In the air, this is just 10-5 cm. But in the “empty space” of the universe before recombination, photons travel 10,000 light-years before colliding. So the high harmonics are reduced or damped out. After a thousand-fold expansion, those scales are now 10 mil­lion light-years. Thus, we don’t expect to see significant structure in the local universe on scales much more than ten times that size, and the clustering of galaxies is indeed weak on scales that large, chalking up another success for the big bang model. The piper at the gates of dawn is playing a strong fundamental note, with faint echoes from the higher harmonics, and a steady descent into high frequency “hiss.”

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