Non-Gaussianity excess problem in classical bouncing cosmologies

The simplest possible classical model leading to a cosmological bounce is examined in the light of the non-Gaussianities it can generate. Concentrating solely on the transition between contraction and expansion, and assuming initially purely Gaussian perturbations at the end of the contracting phase, we find that the bounce acts as a source such that the resulting value for the post-bounce $f_{\mathrm{NL}}$ may largely exceed all current limits, to the point of potentially casting doubts on the validity of the perturbative expansion. We conjecture that if one can assume that the non-Gaussianity production depends only on the bouncing behavior of the scale factor and not on the specifics of the model examined, then many realistic models in which a nonsingular classical bounce takes place could exhibit a generic non-Gaussianity excess problem that would need to be addressed for each case.

Figure 1

Prototypical potential $V_u(\eta )$, and wave number squared (see [14, 15, 18] for explicit examples). The bounce itself occurs between ${\eta }_{-}$ and ${\eta }_{+}$. At the level of the two-point statistics, small scale perturbations (e.g. those of wave number $k_4$) remain unaffected, while long wavelength perturbations ($k_1$, $k_2$ or $k_3$) can be spectrally modified in different ways. For illustrative purposes, the time evolution of two modes, $u_{k_2}$ and $u_{k_3}$, is also shown. As shown in this paper, the bounce produces large non-Gaussianities for any $\{k_1,k_2,k_3\}$ configuration. The first peak before ${\eta }_{-}$ might represent an initial source for primordial perturbation enhancement, as e.g. a matter or ekpyrotic contraction, while the second peak, after ${\eta }_{+}$, could be understood as an inflationary stage subsequent to the bounce. Although during both these phases, further non-Gaussianity could be produced, we restrict attention here to the seemingly more harmless period between ${\eta }_{-}$ and ${\eta }_{+}$, i.e. the bounce itself.

Figure 2

Shape functions derived from Eq. (10) showing the relative contributions of the various possible non-Gaussian configurations. Top figures: $\log (|\mathcal{B}|)$ (left) and $\mathcal{C}$ (right) obtained assuming $K_1(k_i)K_{1,2}(k_j)\simeq K_3(k_1,k_2,k_3)$. Lower figures: almost scale-invariant “frozen” state approximation ($P_{{\mathrm{\Psi }}^{\prime }{\mathrm{\Psi }}^{\prime }}\ll P_{\mathrm{\Psi }{\mathrm{\Psi }}^{\prime }}\ll P_{\mathrm{\Psi }\mathrm{\Psi }}\propto k^{n_{\mathrm{s}}-4}$ with $n_{\mathrm{s}}=0.9603$); the figures are the shape functions obtained by combining Eqs. (20) to (24). In the figures on the left, $f_{\mathrm{NL}}\propto {\mathrm{{\rm Y}}}^{-1}$. In the figures on the right, $f_{\mathrm{NL}}\propto (k_1^2/\mathrm{{\rm Y}})$. In all four figures, $x_2=k_2/k_1$ and $x_3=k_3/k_1$. The differences in the amplitude as a function of the configuration $\{k_1,k_2,k_3\}$ highlight the dependence of the shape function on the details of $\mathit{P}$.