Primordial perturbations in a non singular bouncing universe model

TL;DR

This paper studies a nonsingular bouncing cosmology within general relativity by coupling a radiation fluid to a temporarily dominant negative-energy free scalar field. Using gauge-invariant perturbation theory, the authors derive coupled equations for the Bardeen potential and scalar-field perturbations, analyze their evolution across a symmetric bounce, and perform a careful matching of solutions. They find that, with vacuum initial conditions, the long-wavelength scalar spectrum has a spectral index of $n_S = -1$, indicating incompatibility with observations, although the bounce clarifies how perturbations are pumped and how standard matching conditions fail in this context. The work suggests that achieving a scale-invariant spectrum would require connecting the bounce to a slowly contracting phase, highlighting key constraints and guiding principles for constructing realistic, singularity-free bouncing cosmologies.

Abstract

We construct a simple non singular cosmological model in which the currently observed expansion phase was preceded by a contraction. This is achieved, in the framework of pure general relativity, by means of a radiation fluid and a free scalar field having negative energy. We calculate the power spectrum of the scalar perturbations that are produced in such a bouncing model and find that, under the assumption of initial vacuum state for the quantum field associated with the hydrodynamical perturbation, this leads to a spectral index n=-1. The matching conditions applying to this bouncing model are derived and shown to be different from those in the case of a sharp transition. We find that if our bounce transition can be smoothly connected to a slowly contracting phase, then the resulting power spectrum will be scale invariant.

Figures (8)

• Figure 1: Diagrams leading to instabilities in the theory (\ref{['action']}). $(a)$: the dynamical instability whereby the energy contained in the scalar field can be used to produce semi-classical perturbations, later to be identified with primordial fluctuations. $(b)$: Vacuum instability. As this process in non zero, the vacuum can spontaneously decay into a pair of negative energy scalar particles and a positive energy graviton.
• Figure 2: Potentials for the parametric oscillator equations giving the dynamics of the Bardeen potential and the scalar field perturbations. The full line shows the potential for the variable associated with $u_k$ [see Eq. (\ref{['uk']})], the dashed line the potential for the scalar field $w_k$ [Eq. (\ref{['wk']})], and the dotted line, showing the value of $(\eta_0 k)^2$, indicates visually the different regions where the different approximations hold. The points $x_1$ and $x_2$ are the matching points for these two fields.
• Figure 3: The bounce functions $f_i$ as functions of $x\equiv\eta/\eta_0$.
• Figure 4: Top panel: spectrum of scalar perturbations, i.e., $k^3 |{\Phi_k}\!\!\,\,|^2$ as function of the wavenumber $k$, normalized with $\eta_0$ as indicated. The long wavelength part of the spectrum, as expected, is well fitted by a power-law with spectral index $n_{_{\rm S}}=-1$. The full line is for vacuum initial condition for $\delta\varphi$, the dotted line is with $\delta\varphi=\delta\varphi'=0$ at the initial time ($y_{\rm ini}\equiv k\eta_{\rm ini}=-100$ in the numerical calculation), and the dashed curve represents the fully decoupled situation for which $\varphi'$ is assumed negligible all along. Bottom panel: Time evolution of the gravitational potential $|{\Phi_k}\!\!\,\,|^2$ for different wavelengths.
• Figure 5: Transfer function for the bounce model. Full line: ratio of the squared gravitational potential amplitude between horizon exit and re-entry. The dashed line shows the same multiplied by $\tilde{k}^6$ to emphasize the power law behavior obtained in Eq. (\ref{['trans']}).