Inflationary Perturbations: the Cosmological Schwinger Effect

Abstract

This pedagogical review aims at presenting the fundamental aspects of the theory of inflationary cosmological perturbations of quantum-mechanical origin. The analogy with the well-known Schwinger effect is discussed in detail and a systematic comparison of the two physical phenomena is carried out. In particular, it is demonstrated that the two underlying formalisms differ only up to an irrelevant canonical transformation. Hence, the basic physical mechanisms at play are similar in both cases and can be reduced to the quantization of a parametric oscillator leading to particle creation due to the interaction with a classical source: pair production in vacuum is therefore equivalent to the appearance of a growing mode for the cosmological fluctuations. The only difference lies in the nature of the source: an electric field in the case of the Schwinger effect and the gravitational field in the case of inflationary perturbations. Although, in the laboratory, it is notoriously difficult to produce an electric field such that pairs extracted from the vacuum can be detected, the gravitational field in the early universe can be strong enough to lead to observable effects that ultimately reveal themselves as temperature fluctuations in the Cosmic Microwave Background. Finally, the question of how quantum cosmological perturbations can be considered as classical is discussed at the end of the article.

Figures (6)

• Figure 1: Evolution of the quantity $\vert Q/\omega ^2\vert$ with time $\tau$ in the case of the Schwinger effect. In the limit $\tau/\sqrt{\Upsilon} \rightarrow \pm \infty$, $\vert Q/\omega ^2\vert$ vanishes and the notion of adiabatic vacuum is available.
• Figure 2: Evolution of the quantity $\vert Q/\omega ^2\vert$ with the quantity $k\eta$ for a typical model of inflation according to Eq. (\ref{['wkbinftest']}) (we have neglected the corrections proportional to the slow-roll parameters). In the limit $k\eta \rightarrow -\infty$, which corresponds to a wavelength much smaller than the Hubble radius, $\vert Q/\omega ^2\vert$ vanishes and the notion of an adiabatic vacuum is available.
• Figure 3: $68\%$ and $95\%$ confidence intervals of the two-dimensional marginalized posteriors in the slow-roll parameters plane, obtained at leading order in the slow-roll expansion MR. The shading is the mean likelihood and the left plot is derived under an uniform prior on $\epsilon _1$ while the right panel corresponds to an uniform prior on $\log \epsilon _1$.
• Figure 4: Wigner function (\ref{['wignercoh']}) for the coherent state $\vert \alpha \rangle$ at different times. The (arbitrary) values $q_0=1$, $p_0=1$ and $k=2$ have been used for this figure. This implies $\alpha _0=\sqrt{2}{\rm e}^{i\pi/4}$ and, see Eqs. (\ref{['harmoclass']}), $p_{\rm cl}=2\sqrt{2}\sin \left(\pi/4-t\right)$ and $q_{\rm cl}=\sqrt{2}\cos\left(\pi/4-t\right)$. The upper left panel represents the Wigner function (\ref{['wignercoh']}) at time $t=0$ while the upper right, lower right and lower left panels correspond to $W(p,q,t)$ at time $t=\pi/2$, $t=\pi$ and $t=3\pi/2$ respectively. The wave packet follows the periodic (ellipsoidal) classical trajectory in phase space and its shape remains unchanged during the motion.
• Figure 5: Wigner function of cosmological perturbations obtained from Eq. (\ref{['wignerfinal']}) (for a one-dimensional system). The squeezing parameter $r$ is chosen to be $r=0.1$, $r=0.5$, $r=1$ and $r=2$ for the left upper, right upper, left lower and right lower panels respectively (it is not the same time ordering as in Fig. \ref{['wignercoherent']} because, in the present case, the motion is not periodic). The other squeezing parameters are taken to be $\phi =\pi/6$ and $\theta =0$. As can be noticed in this figure, the Wigner function remains positive. Since the squeezing parameter increases with time, the different panels correspond in fact to the Wigner function at different times. At initial time, the quantum state is the vacuum and, therefore, the Wigner function is that of a coherent state, compare the left upper panel with Fig. \ref{['wignercoherent']}. Then, the Wigner function develops the "Dirac function behavior" discussed in the text that clearly appears on this plot.