Bouncing models with a cosmological constant

Bouncing models have been proposed by many authors as a completion of, or even as an alternative to, inflation for the description of the very early and dense Universe. However, most bouncing models contain a contracting phase from a very large and rarefied state, where dark energy might have had an important role as it has today in accelerating our large Universe. In that case, its presence can modify the initial conditions and evolution of cosmological perturbations, changing the known results already obtained in the literature concerning their amplitude and spectrum. In this paper, we assume the simplest and most appealing candidate for dark energy, the cosmological constant, and evaluate its influence on the evolution of cosmological perturbations during the contracting phase of a bouncing model, which also contains a scalar field with a potential allowing background solutions with pressure and energy density satisfying $p=wϵ$, $w$ being a constant. An initial adiabatic vacuum state can be set at the end of domination by the cosmological constant, and an almost scale-invariant spectrum of perturbations is obtained for $w\approx 0$, which is the usual result for bouncing models. However, the presence of the cosmological constant induces oscillations and a running towards a tiny red-tilted spectrum for long-wavelength perturbations.

Figure 1

Behavior of the potential $V(t)$ given by Eq. (18) for three different values of $w$.

Figure 2

Behavior of the quantum bouncing potential given by Eq. (28) for three different values of $w$.

Figure 3

Numerical results of Eq. (44) for $w={10}^{-3}$, $1/8$ and $1/4$. Each curve shows $u_k(y)$ as a function of $y$ for different values of $k$, as indicated in the graphs.

Figure 4

Numerical results for the behavior of $|A_2(k)|$ for $w={10}^{-3}$, $1/8$ and $1/4$ evaluated at $y=-{10}^{-15}$. The solid lines show the numerical results and the dotted ones show, for comparison, a curve proportional to $k^{3(\omega -1)/2(3\omega +1)}$.

Figure 5

Numerical results for $n_S(k)$ evaluated at $y=-{10}^{-15}$, obtained using $w={10}^{-3}$. The solid line indicates the result obtained using ${\Omega }_{\Lambda }=0.7$, the dashed line for ${\Omega }_{\Lambda }={10}^{-3}$ and the dotted line for ${\Omega }_{\Lambda }={10}^{-6}$. Note that the oscillations become smaller for smaller ${\Omega }_{\Lambda }$, showing that they are due to the presence of the cosmological constant. This result is in agreement with Eq. (62).