2 Perturbations from Inflation

We start from Einstein’s equation for the evolution of the Universe

2 8 − 1 Λ 2 2 ˙R (t) = 3πG ρ∘R (t) + 3-R − kc (1 )
where R describes the size of the Universe, G is the gravitational constant, ρ ∘ is the present density of the Universe, k is a measure of the curvature of space and Λ is the Cosmological constant which can be thought of as the zero energy of a vacuum. If the Cosmological term dominates (as the scalar field is expected to at very high temperatures), then the other two terms become negligible and it is possible to solve and find
[ 1 ] R (t) ∝ exp (Λ ∕3)2t . (2 )
Therefore, the inflationary theory describes an exponential expansion of space in the very early Universe. Amplification of initial quantum irregularities then results in a spectrum of long wavelength perturbations on scales initially bigger than the horizon size. Central to the theory of inflation is the potential V(ϕ ), which describes the self-interaction of the scalar inflaton field ϕ. Due to the unknown nature of this potential, and the unknown parameters involved in the theory, inflationary theory is bad (at the moment) at predicting the overall amplitude of the matter fluctuations at recombination. However, there is a reasonable agreement that the form of the fluctuation spectrum coming out of inflation should be given by
|δk|2 ∝ kn, (3 )
where k is the comoving wavenumber and n is the ‘tilt’ of the primordial spectrum. The latter is predicted to lie close to 1 (the case n = 1 being the Harrison–Zel’dovich, or ‘scale-invariant’ spectrum).

An overdensity in the early Universe does not collapse under the effect of self-gravity until it enters its own particle horizon when every point within it is in causal contact with every other point. The perturbation will continue to collapse until it reaches the Jean’s length, at which time radiation pressure will oppose gravity and set up acoustic oscillations. Since overdensities of the same size will pass the horizon size at the same time, they will be oscillating in phase. These acoustic oscillations occur in both the matter field and the photon field and so will induce ‘Doppler peaks’ in the photon spectrum.

The level of the Doppler peaks in the power spectrum depend on the number of acoustic oscillations that have taken place since entering the horizon. For overdensities that have undergone half an oscillation, there will be a large Doppler peak (corresponding to an angular size of ∼ 1°). Other peaks occur at harmonics of this. As the amplitude and position of the primary and secondary peaks are intrinsically determined by the number of electron scatterers and by the geometry of the Universe, they can be used as a test of the density parameter of baryons and dark matter, as well as other cosmological constants.

Prior to the last scattering surface, the photons and matter interact on scales smaller than the horizon size. Through diffusion, the photons will travel from high density regions to low density regions ‘dragging’ the electrons with them via Compton interaction. The electrons are coupled to the protons through Coulomb interactions, and so the matter will move from high density regions to low density regions. This diffusion has the effect of damping out the fluctuations and is more marked as the size of the fluctuation decreases. Therefore, we expect the Doppler peaks to vanish at very small angular scales. This effect is known as Silk damping [83].

Another possible diffusion process is free streaming. It occurs when collisionless particles (e.g. neutrinos) move from high density to low density regions. If these particles have a small mass, then free streaming causes a damping of the fluctuations. The exact scale this occurs on depends on the mass and velocity of the particles involved. Slow moving particles will have little effect on the spectrum of fluctuations as Silk damping already wipes out the fluctuations on these scales, but fast moving, heavy particles (e.g. a neutrino with 30 eV mass), can wipe out fluctuations on larger scales corresponding to 20 Mpc today [28].

Putting this all together, we see that on large angular scales (> 2°) we expect the CMB power spectrum to reflect the initially near scale-invariant spectrum coming out of inflation; on intermediate angular scales we expect to see a series of peaks, and on smaller angular scales (< 10 arcmin) we expect to see a sharp decline in amplitude. These expectations are borne out in the actual calculated form of the CMB power spectrum in what is currently the ‘standard model’ for cosmology, namely inflation together with cold dark matter (CDM). The spectrum for this, assuming Ω = 1 and standard values for other parameters, is shown in Figure 2View Image.

View Image

Figure 2: Power spectrum for standard CDM. Parameters assumed are Ω = 1, n = 1, H0 = 50 km s–1 Mpc–1 and a baryon fraction of Ωb = 0.04.

The quantities plotted are ℓ(ℓ + 1 )C ℓ, versus ℓ where Cℓ is defined via

2 ΔT-- ∑ Cℓ = ⟨|aℓm| ⟩, T (𝜃,ϕ) = almY ℓm (𝜃,ϕ), (4 ) ℓ,m
and the Yℓm are standard spherical harmonics. The reason for plotting ℓ(ℓ + 1)C ℓ is that it approximately equals the power per unit logarithmic interval in ℓ. Increasing ℓ corresponds to decreasing angular scale 𝜃, with a rough relationship between the two of 𝜃 ≈ 2∕ℓ radians. In terms of the diameter of corresponding proto-objects imprinted in the CMB, a rich cluster of galaxies corresponds to a scale of about 8 arcmin, while the angular scale corresponding to the largest scale of clustering we know about in the Universe today corresponds to 1/2 to 1 degree. The first large peak in the power spectrum, at ℓ’s near 200, and therefore angular scales near 1°, is known as the ‘Doppler’, or ‘Sakharov’, or ‘acoustic’ peak.

As stated above, the inflationary CMB power spectrum plotted in Figure 2View Image is that predicted by assuming the standard values of the cosmological parameters for a CDM model of the Universe. In order for an experimental measurement of the angular power spectrum to be able to place constraints on these parameters, we must consider how the shape of the predicted power spectrum varies in response to changes in these parameters. In general, the detailed changes due to varying several parameters at once can be quite complicated. However, if we restrict our attention to the parameters Ω, H0 and Ωb, the fractional baryon density, then the situation becomes simpler.

Perhaps most straightforward is the information contained in the position of the first Doppler peak, and of the smaller secondary peaks, since this is determined almost exclusively by the value of the total Ω, and varies as ℓpeak ∝ Ω −1∕2. (This behaviour is determined as mentioned above by the linear size of the causal horizon at recombination, and the usual formula for angular diameter distance.) This means that if we were able to determine the position (in a left/right sense) of this peak, and we were confident in the underlying model assumptions, then we could read off the value of the total density of the Universe. (In the case where the cosmological constant was non-zero, we would effectively be reading off the combination Ωmatter + Ω Λ.) This would be a determination of Ω free of all the usual problems encountered in local determinations using velocity fields etc.

Similar remarks apply to the Hubble constant. The height of the Doppler peak is controlled by a combination of H 0 and the density of the Universe in baryons, Ω b. We have a constraint on the combination 2 ΩbH 0 from nucleosynthesis, and thus using this constraint and the peak height we can determine H0 within a band compatible with both nucleosynthesis and the CMB. Alternatively, if we have the power spectrum available to good accuracy covering the secondary peaks as well, then it is possible to read off the values of Ωtot, Ωb and H0 independently, without having to bring in the nucleosynthesis information. The overall point here is that the power spectrum of the CMB contains a wealth of physical information, and that once we have it to good accuracy and have become confident that an underlying model (such as inflation and CDM) is correct, then we can use the spectrum to obtain the values of parameters in the model, potentially to high accuracy. This will be discussed further below, both in the context of the current CMB data, and in the context of what we can expect in the future.

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