4.2 Origin of Density Perturbations

Having decided the form of the density perturbations that is required, we now turn to the actual mechanism by which these are produced. The most natural choice, in the context of inflation, comes from the quantum fluctuations in the scalar field (t, x) driving the inflation. The computation of classical perturbations generated by a quantum field is a difficult and technically involved issue. Several questions of principle are still unresolved in this calculation (see e.g. 17, 94). Since this review is primarily intended for the astronomer, we limit discussion to the physical idea rather than to the technical aspects of the calculation.

During inflation, the universe was assumed to be, on the average, in a FRW state with small inhomogeneities. This implies that the source - which is a classical scalar field (t, x) - can be split as 0 (t) + f (t, x), where 0 (t) denotes the average, homogeneous part and f (t, x) represents the spatially dependent fluctuating part. Since the energy density due to a scalar field is c2 1/2 2, we obtain

33.

(where (t) c2 = 1/2 0 (t)2 and we have assumed f << 0). The Fourier transform will now give

34.

where we have put

35.

Since the average energy density during inflation is dominated by the constant term V0, we have the density contrast

36.

It might now appear that all we have to do is to compute the quantities 0 (t) and Qk (t) from the equation of motion for the scalar field. For 0 (t) we can use the mean evolution of the scalar field during the slow roll-over phase and determine 0 (t) from the classical solution. The fluctuating field f (t, x) is supposed to be some classical object mimicking the quantum fluctuations. Such a quantity is conceptually difficult to visualize and justify. What is usually done is to choose some convenient quantum mechanical measure for fluctuations and define Qk in terms of this quantity.

In quantum theory, the field (t, x) and its Fourier coefficients k(t) will become operators related by

37.

The quantum state of the field can be specified by giving the quantum state k (qk , t) of each of the modes k. (One can think of qk as coordinates of a particle and k (qk , t) as the wavefunction describing this particle.)

The fluctuations in qk can be characterized by the dispersion

38.

in this quantum state. (The mean value of the scalar field operator <(t, x)> = 0 (t) is homogeneous; therefore, we have set <k> to zero in the above expression. Note that we are interested only in the k 0 modes.) Expressing k in terms of (t, x), it is easy to see that

39.

In other words, the power spectrum of fluctuations k2 is related to the Fourier transform of the two-point-correlation function of the scalar field. Since k2 (t) appears to be a good measure of quantum fluctuations, we may attempt to define Qk (t) as

40.

This is equivalent to defining the fluctuating classical field f (t, x) to be

41.

42.

The procedure may be summarized as follows:

1. In quantum theory, the field (t, x) and its Fourier coefficient k(t) become operators. In any quantum state, the variables will have a mean value and fluctuations around this mean value.

2. Since the mean evolution of the scalar field is described by a homogeneous part 0 (t), we expect the mean values of k to vanish (for k 0); <|k(t)|> = 0. The fluctuations around these mean values, however, characterized by k2 (t) = <|k2|>, do not vanish.

3. We incorporate these quantum fluctuations in a semiclassical manner by taking the scalar field to be (t, x) = 0 (t) f (t, x), where f (t, x) is related to k (t) by (41).

4. The density perturbations are calculated by treating (t, x) as a classical object.

The expression derived above gives the value of (k, t) in the inflationary phase: ti < t < tf. To compare this with observations we need to know the value of (k, t) at t = tenter (k), that is, when the perturbations enter the Hubble radius. Fortunately, an approximate conservation law relates the value (k, tenter) with (k, texit), where texit (k) is the time at which the relevant perturbation ``leaves'' the Hubble radius in the inflationary epoch (8, 19, 46). This law can be stated as

43.

where W (t) is the ratio between pressure p (t) and the energy density (t) c2 of the background (mean) medium: W (t) = p (t) / (t) c2. In the inflationary phase with the scalar field,

44.

where we have used the fact 02 << V0. In the radiation-dominated phase (at t = tenter), 1 + W = 4/3. Therefore

45.

or using Eq. 42,

46.

This is the final result.

The problem now reduces to computing k (t) and 0 (t), which can be done once the potential V () is known. For a Coleman-Weinberg potential (see Section 3.2), detailed calculations give (see e.g. 17) the final result

47.

where we have taken the effective e-folding time N 50 and 0.1. We see that the density perturbations have the correct spectrum but too high an amplitude. To bring it down to the acceptable value of about 10-4, we need to take the dimensionless parameter to be about 10-13 ! This requires an extreme fine-tuning for a dimensionless parameter, especially since we have no other motivation for such a value.

This has been the most serious difficulty faced by all realistic inflationary models: They produce too large an inhomogeneity. The qualitative reason for this result can be found from Eq. 46. To obtain slow roll-over and sufficient inflation, we need to keep 0 small, and this tends to increase the value of . We could have saved the situation if it were possible to keep k arbitrarily small; unfortunately, the inflationary phase induces a fluctuation of about (H / 2) on any quantum field due to field theoretical reasons (see e.g. 18, 72). This lower bound prevents us from getting sensible values for unless we fine-tune the dimensionless parameters of V () (for a general discussion, see 92). Several solutions have been suggested in the literature to overcome this difficulty but none of them appears compelling (see e.g. 42, 54, 62, 86, 90, 94, 108).