Correlation functions for scalar metric perturbations

7.3 Correlation functions for scalar metric perturbations

The the two-point correlation function for the scalar metric perturbations induced by the inflaton fluctuations is thus given by

Here 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 = -⟨{(ˆt0i)k(η1),(ˆt0i)−k(η2)}⟩φϕ 2 1- ′ ′ (1) = 2kikiφ (η1)φ (η2)G k (η1,η2), (151 )

where $G(1)(η ,η ) = ⟨{ϕˆ (η ), ˆϕ (η )}⟩ k 1 2 k 1 −k 2$ is the $k$ -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)),

with

† ∗ 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, $ˆak |0⟩ = 0$ [25 ].

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, however, a reasonable approximation as follows. For a given mode the $m = 0$ approximation is reasonable when its wavelength $λ$ is shorter that 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 that $H −1 < λc$ . Thus, for modes inside the horizon, $λ < λ c$ and $m = 0$ is a reasonable approximation. Outside the horizon massive modes decay in amplitude as $2 ∼ exp(− m t∕H )$ , 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., $Ht ∼ 60$ , but this is the case since $60 m2 < H2$ given that $m ∼ 10− 6mP$ , and $m ≪ H$ as in most inflationary models [190 , 229 ].

The background geometry is not exactly that of de Sitter spacetime, for which $−1 a(η) = − (H η)$ 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 (150 ) can then be easily computed; see [253 , 254 ] 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 de quasi-de Sitter spacetime during inflation erases the information about the initial conditions. Thus one may take the initial time to be $η = − ∞ 0$ , and obtain to lowest order in $m ∕m P$ the expression

From this result two main conclusions are derived. First, the prediction of an almost Harrison–Zel’dovich scale-invariant spectrum for large scales, i.e., small values of $k$ . Second, since the correlation function is of 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 of the order of $− 6 (m ∕mP ) ∼ 10$ [265 , 218 ].

We should now comment on some differences with those works in [46 , 213 , 212 , 39 ] which used a self-interacting scalar field or a scalar field interacting nonlinearly with other fields. In those 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 initial state, had to be 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 [254 ]. We should remark that in the model discussed here there is no environment for the inflaton fluctuations. The inflaton fluctuations, however, are responsible for the noise that induces the metric perturbations.