Go to previous page Go up Go to next page

7.3 Correlation functions for scalar metric perturbations

The two-point correlation function for the scalar metric perturbations induced by the inflaton fluctuations is thus given by
′ 2 ′ ⟨Φk (η)Φk′(η)⟩s = (2π) δ(⃗k + ⃗k ) (174 ) ∫ η ∫ η′ × dη1 dη2Gk (η,η1)Gk ′(η′,η2)⟨(ξ0i)k(η1)(ξ0i)k′(η2)⟩s. (175 )
Here the two-point correlation function for the stochastic source, which is connected to the stress-energy tensor fluctuations through the noise kernel, is given by,
1 ⟨(ξ0i)k(η1)(ξ0i)− k(η2)⟩s = 2⟨{(ˆt0i)k(η1),(ˆt0i)−k(η2)}⟩φϕ (176 ) 1 = -kikiφ′(η1)φ′(η2)G (k1)(η1,η2), (177 ) 2
where G (1k)(η1,η2) = ⟨{ ˆϕk(η1), ˆϕ−k(η2)}⟩ is the kth-mode Hadamard function for a free minimally coupled scalar field in the appropriate vacuum state on the Friedmann–Robertson–Walker background.

In practice, to make the explicit computation of the Hadamard function, we will assume that the field state is in the Euclidean vacuum and the background spacetime is de Sitter. Furthermore, we will compute the Hadamard function for a massless field, and will make a perturbative expansion in terms of the dimensionless parameter m ∕mP. Thus we consider

G¯(k1)(η1,η2) = ⟨0|{ˆyk(η1), ˆy−k(η2)}|0⟩ = 2ℛ (uk(η1)u∗k(η2)),


† ∗ ˆyk(η) = a(η)ϕˆk (η) = ˆakuk(η) + ˆa−ku− k(η ),

and where

uk = (2k)−1∕2eikη (1 − i∕η)

are the positive frequency k-modes for a massless minimally-coupled scalar field on a de Sitter background, which define the Euclidean vacuum state ˆak|0⟩ = 0 [34Jump To The Next Citation Point].

The assumption of a massless field for the computation of the Hadamard function is made because massless modes in de Sitter are much simpler to deal with than massive modes. We can see that this is nonetheless a reasonable approximation as follows: For a given mode, the m = 0 approximation is reasonable when its wavelength λ is shorter than the Compton wavelength, λc = 1∕m. In our case we have a very small mass m and the horizon size H −1, where H is the Hubble constant H = ˙a∕a (here a(t) with t the physical time dt = adη), satisfies H − 1 < λc. Thus, for modes inside the horizon, λ < λc and m = 0 are a good approximation. Outside the horizon, massive modes decay in amplitude as 2 ∼ exp(− m t∕3H ) whereas massless modes remain constant. Thus, when modes leave the horizon, the approximation will eventually break down. However, we only need to ensure that the approximation is still valid after 60 e-folds, i.e., H Δt ∼ 60 (Δt being the time between horizon exit and the end of inflation). But this is the case provided 3H2 ∕m2 > 60, since the decay factor ∼ exp[− (m2 ∕3H2 )H Δt ] will not be too different from unity for those modes that left the horizon during the last sixty e-folds of inflation. This condition is indeed satisfied given that m ≪ H in most slow-roll inflationary models [233, 284] and in particular for the model considered here, in which m ∼ 10− 6m P.

We note that the background geometry is not exactly that of de Sitter spacetime, for which a(η) = − (H η)−1 with − ∞ < η < 0. One can expand in terms of the “slow-roll” parameters and assume that, to first order, φ˙(t) ≃ m2P (m∕mP ), where t is the physical time. The correlation function for the metric perturbation (175View Equation) can then be easily computed; see [315, 316Jump To The Next Citation Point] for details. The final result, however, is very weakly dependent on the initial conditions as one may understand from the fact that the accelerated expansion of the quasi-de Sitter spacetime during inflation erases the information about the initial conditions. Thus, one may take the initial time to be η0 = − ∞ and obtain to lowest order in m ∕mP the expression

( )2 ⟨Φ (η )Φ ′(η ′)⟩ ≃ 8π2 -m-- k−3(2π )3δ(⃗k + ⃗k ′)cos k(η − η′). (178 ) k k s mP
From this result two main conclusions can be derived. First, the prediction of an almost scale-invariant Harrison-Zel’dovich spectrum for large scales, i.e., small values of k. Second, since the correlation function is on the order of (m ∕mP )2, a severe bound to the mass m is imposed by the gravitational fluctuations derived from the small values of the cosmic microwave background (CMB) anisotropies detected by COBE. This bound is on the order of − 6 (m ∕mP ) ∼ 10 [270, 334].

We now comment on some differences with [53, 60, 262, 263], which used a self-interacting scalar field or a scalar field interacting nonlinearly with other fields. In these works an important relaxation of the ratio m ∕mP was found. The long wavelength modes of the inflaton field were regarded as an open system in an environment made out of the shorter wavelength modes. Then, Langevin type equations were used to compute the correlations of the long wavelength modes driven by the fluctuations of the shorter wavelength modes. In order to get a significant relaxation on the above ratio, however, one had to assume that the correlations of the free long-wavelength modes, which correspond to the dispersion of the system’s initial state, were very small. Otherwise they dominate by several orders of magnitude those fluctuations that come from the noise of the environment. This would require a great amount of fine-tuning for the initial quantum state of each mode [316Jump To The Next Citation Point].

We should remark that in the linear model discussed here there is no environment for the inflaton fluctuations. When one linearizes with respect to both the scalar metric perturbations and the inflaton perturbations, the system cannot be regarded as a true open quantum system. The reason is that Fourier modes decouple and the dynamical constraints due to diffeomorphism invariance link the metric perturbations of scalar type with the perturbations of the inflaton field so that only one true dynamical degree of freedom is left for each Fourier mode. Nevertheless, the inflaton fluctuations are responsible for the noise that induces the metric perturbations.

  Go to previous page Go up Go to next page