Figure 43 shows the CMB power spectrum,[4] which tells us the
intensity (loudness) of the cosmic sound as a function of the wave number k. Instead of k it is perhaps more intuitive to think of it as frequency, or the pitch. These are very low frequencies, way below our audible range, as the corresponding wavelengths are astronomically large, in the range of Mpc to thousands of Mpc.
As mentioned before, we see the sound through the CMB rather than hearing it. They are indicated by the green points in Fig. 43. These are points of low frequencies, or small k.
Those with larger k are not measurable with the present WMAP satellite but some of them will be accessible by the
Figure 42: The colored curves are the aH/c values as a function of time t in the inflationary era (blue), the radiation era (orange), and the matter era (red). The two vertical bars indicate the time when the radiation era turns into the matter era (eq), and when decoupling occurs (dec). The wave with comoving wave number k is inside the Hubble horizon to the left of the blue curve and to the right of the orange-red curve. It is out of the Hubble horizon between the blue and the orange-red curves. Depending on the value of k, the sound wave can re-enter the horizon either in the radiation era or the matter era. It is important to note that this diagram just shows the qualitative behavior. Nothing is drawn to the proper scale.
Planck satellite to be launched soon. For now, we rely on galaxy distributions to obtain the data points for higher frequencies (points with other colors). As mentioned in the roadmap, after decoupling gravity amplifies the disturbances and pulls them together into matter clumps and eventually galaxies. Hence, the distribution of galaxy sizes reflects the distribution of sound wavelengths, and the latter can be inferred from the former.
The tone we hear (see) in Fig. 43 is not the original tone generated in the inflationary era. Ordinary matter, dark matter,
Figure 43: The power spectrum plotted as a function of k. For our purposes, we can take h = 0.72, and think of the power spectrum as the square of the observed amplitude. The green data are the CMB points, the others are obtained from galaxy surveys. The solid and dashed curves are theoretical curves computed from CMB and galaxy formation theories using two different sets of parameters. Courtesy of Max Tegmark/SDSS Collaboration.
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expansion, as well as gravity act like acoustic materials in a room to modify the sound. Some frequencies are transparent to these materials, whereas others get partially absorbed. Since the universe has some 400,000 years to do it before we see the photons come to us at decoupling time, the frequency distribution of the sound that we see is no longer the same as the primordial distribution during the inflationary era.
However, recall from Fig. 42 that everything is frozen outside the Hubble horizon, when the wave number k is buried under the mountain. Thus, the acoustic materials can affect the sound pattern only to the right of the mountain. Moreover, I will explain later that the acoustic material is transparent to the low frequency sounds to the left of the peak in Fig. 43. Hence, that part of the distribution is still the primordial distribution in the inflationary era.
The original tone generated from quantum fluctuation in the inflationary era can be calculated. The wave number (frequency) distribution of the intensity turns out to be (k n/M V)( / )/
P
4 3− 2 ε1 2,
evaluated at the inflationary-era horizon exit point k = aH/c, namely, at the point when the dashed line in Fig. 42 begins to enter the mountain from the left.
I will explain the symbols in the next paragraph. A comparison of the observed data in Fig. 43 to the left of the peak with this formula yields the ‘COBE normalization’ (V/ε)1/4= 6.6 × 1016 GeV,
an important piece of information about the energy density in the inflationary era. It is truly amazing that such information about the very early universe 13.7 billion years ago can still be obtained.
In the formula above, V is related to the energy density ρ during inflation[5] by V = ρ(ħc)3, so that V1/4 has the unit of
energy. In short, V is just ρ expressed in the unit (GeV)4. H is the
Hubble parameter in the inflationary era, which is equal to the exponent α = (8πρG/3c2)1/2 of inflation discussed in Chap. 15.
Both V and H are approximately constant throughout the inflationary period. MP= (ħc/G)1/2= 2.18 × 10−8 kg = 1.22 × 1019
GeV/c2 is the Planck mass (see note 16[1] ). The parameter ε is
one of two slow-roll parameters characterizing how slowly the false vacuum decays. The other one is called η (not to be confused with either the conformal time η or the nucleon-to-photon number ratio η). These slow-roll parameters must be small in order to allow the universe to stay at the constant density long enough to inflate to a minimum multiple factor of 1025. The number n is
called the tilt. It determines the k distribution, and is related to the slow roll parameter by n = 1 − 6ε + 2η. Its best-fitted value from the 3-year WMAP data is 0.958 ± 0.016, close to the ‘scale invariant’ value of n = 1, and is consistent with the smallness of
the slow-roll parameters. Note that Fig. 43 is a log–log plot. The straight line below the peak at about 0.01 h/Mpc represents a power dependence on k, agreeing with the spectrum predicted by inflation and allowing the tilt to be extracted from it.
If ε can be measured independently, then from the COBE normalization (V/ε)1/4= 6.6 × 1016 GeV discussed above, V1/4
can be computed to give us the density of the universe in the inflationary era. Unless ε is terribly small, V1/4 is expected to be in
the 1015 to 1016 GeV range.[6]
In principle there is a way to measure ε (see the discussion on polarization later), but in practice this measurement is very difficult. If ε is somehow known, then tilt n determines the other slow-roll parameter η.
I mentioned in the last chapter that the dark energy density is much smaller than the energy density in the inflationary era. Let us see how much smaller. The dark energy density, being 73% of the critical density, is 4 × 10−6 GeV/(cm)3. In contrast,
the energy density in the inflationary era is ρ = V/(ħc)3. Taking
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To summarize, what we have learned by comparing the small
k part of Fig. 43 with the outcome of the quantum fluctuation
calculation in the inflationary era are: (i) the wave number (frequency) distribution of the sound intensity obeys a power law in k, with a tilt parameter equal to n = 0.958 ± 0.016; and (ii) the energy density V during inflation is constrained by the COBE normalization (V/ε)1/4= 6.6 × 1016 GeV, where the slow-
roll parameter ε is so far unknown but must be small. Otherwise inflation will not last long enough to produce an amplification factor of at least 1025 times for the size of the universe.