This concludes the proof of what we stated above, namely that there is only one independent scalar degrees of freedom, corresponding to the free field v which propagates in the effective conformally flat space-time (5.9.28) with scale factor λ(η) = θ (η). Indeed the relevant point is that the fields u and v are not independent being related by a first order differential equation in η that can always be integrated yielding u in terms of v. Hence the effective field u can be quantized and the modes of both δϕ and Ψ can be uniquely expressed in terms of the modes of u.

5.9.3 Quantization of the Scalar Degree of Freedom

As a next step we can proceed to the canonical quantization of the scalar degree of freedom we have singled out in the previous sections. Following standard procedures we turn the classical field u(η, x) into an operator-valued distribution u(η,ˆ x) and we introduce the expansion of the latter into Fourier modes:

Inserting (5.9.55) into (5.9.54) we find that for each momentum vector k the corresponding wave function uk(η) satisfies the following equation:

u + κ2 − θ uk = 0

k θ

(5.9.56)

κ2 = k · k

We consider the case where the mass term θθ takes the slow-rolling approximate form derived in (5.9.41) and we obtain the following equation:

where the parameter μ allows to interpolate between various notable cases. If μ = 0 we are actually discussing propagation in Minkowski space. For μ = 1 we retrieve the propagation equation in de Sitter space (see (5.9.34)). For all the intermediate values 0 < μ < 1 we describe propagation in the background of a slow-rolling universe and the almost constant small parameter μ is given in (5.9.41).

Equation (5.9.57) is actually the Bessel equation in slightly modified variables and its solutions can be constructed by means of Bessel functions for all values of μ. Introducing the index

where πˆu(η, y) = ∂ηu(η,ˆ x), if the wave-function (5.9.60) is properly normalized in such a way that:

There are many choices of the wave function (coefficients c±) consistent with the normalization condition (5.9.67). Every such choice is associated with a different decomposition into creation and annihilation modes and therefore with a different vacuum |0! which, as usual, is defined by the condition:

A proper normalization of the wave function is provided by the observation that for late times η → ∞, or, equivalently, for very short wave-lengths κ → ∞, we approach an effective Minkowski scenario, where the effects of space-time curvature are negligible. Hence we can just choose the normalization of the wave-function which corresponds to the association with the creation operator of a standard outgoing wave in the late time regime, namely:

Having so done we are in a position to calculate the two-point correlation function of the field u(η,ˆ x) or better of the gravitational potential Φ(η,7 x), which is related to u(η,ˆ x) by (5.9.50).

Setting:

where, in the last line, we have used the definitions:

The function PΦ (κ) is named the power spectrum and it is the main target of all calculations since, supposedly, it is an experimentally accessible datum through the observation of anisotropies in the cosmic background radiation.