Parametric amplification of metric fluctuations through a bouncing phase

Abstract

We clarify the properties of the behavior of classical cosmological perturbations when the Universe experiences a bounce. This is done in the simplest possible case for which gravity is described by general relativity and the matter content has a single component, namely a scalar field in a closed geometry. We show in particular that the spectrum of scalar perturbations can be affected by the bounce in a way that may depend on the wave number, even in the large scale limit. This may have important implications for string motivated models of the early Universe.

Figures (10)

• Figure 1: Scale factors as functions of the conformal time $\eta$ corresponding to the de Sitter-like solution [Eq. (\ref{['dS']}), full line] and its various levels of approximations stemming from Eq. (\ref{['aseries']}), namely up to quadratic (dashed), quartic (dotted), sixth (dot-dashed) and eighth power (dot-dot-dashed). The last two approximations, although clearly better from the point of view of the scale factor, do not lead to any new qualitative information as far as the evolution of the perturbations is concerned.
• Figure 2: Behavior of the scalar field and its coordinate time derivative as functions of the conformal time $\eta$ (varying between $-\pi/2$ and $\pi/2$ for the overall evolution of the Universe) for the de Sitter-like solution with $\eta_0=1.01$.
• Figure 3: The shape (\ref{['potphi']}) of the potential for the scalar field $\varphi$ (in units of the Planck mass $\kappa^{-1/2}=m_{_\mathrm{Pl}}/\sqrt{8\pi}$) for different values of the bounce characteristic conformal time $\eta_0$. The full lines are respectively for $\eta_0=1.001$ (above) and $\eta_0=1.01$ (below), the dashed line corresponds to $\eta_0=1.1$, and the dotted line is for $\eta_0=1.5$. In the strict de Sitter limit $\eta_0\to 1$, the potential goes to the constant value $V(\varphi)=3/(\kappa a_0^2)$, which explains why the $\eta_0=1.001$ seems almost constant as it oscillates with a very small amplitude around its central value [($3-1/\eta_0^2$) in these units].
• Figure 4: Absolute value of the effective potential $V_u(\eta)$ for the perturbation variable $u(\eta)$ for the de Sitter-like case (full line on both panels), for which it is constant and for the various approximation levels (from quadratic to eighth power of the scale factor). The left panel shows the potential as obtained by using the quadratic (dotted line) and quartic (dashed) expansions of the scale factor only, whereas the right panel presents the situation when quartic (dashed), sixth (dotted) and eighth (dot-dashed) terms are used. It is clear that the quadratic approximation is qualitatively wrong and cannot be used to describe a de Sitter bounce. The value $\eta_0=1.01$ has been used to derive these plots.
• Figure 5: Absolute value of the potential $V_u(\eta)$ as a function of rescaled conformal time $\eta/\eta_0$ for $\eta_0=1.01$ as derived using either the assumption that the scale factor behaves as a square root, i.e. $a=a_0\sqrt{1+\left( \eta/\eta_0 \right)^2 }$, (full line) or Eq. (\ref{['aseries']}) up to quadratic (dotted line) and quartic order with $\delta=0$ and $\xi=-2/5$ (dashed line). The quartic approximation is extremely close to the exact solution, exemplifying its accuracy, while the quadratic approximation appears to be at best qualitatively correct.