Cosmological perturbations and the transition from contraction to expansion

TL;DR

This work analyzes scalar cosmological perturbations through a smooth contraction-to-expansion transition in a nearly flat FRW universe. By developing a general framework around two canonical variables, $u$ (related to the Bardeen potential $\\Psi$) and $v$ (related to the curvature perturbation $\\zeta$), the authors show that the post-transition spectral index critically depends on which variable remains regular, with $n=1-2q$ for regular $u$, and $n=3+2q$ (or $n=5-2q$ for $q>1/2$) for regular $v$, where $q$ characterizes the contraction rate. Through a general analysis and fast toy-model transitions, they demonstrate that both $u$ and $v$ cannot be regular across the transition for $-1/2<q$, explaining longstanding discrepancies in the literature and highlighting the instability of the scale-invariant prediction in many contracting scenarios. The results have significant implications for ekpyrotic and dilaton-driven string cosmology, and they underscore the need to specify transition dynamics (or invoke mechanisms like the curvaton) to obtain viable, observationally consistent spectra. Throughout, the study emphasizes the role of transition regularity and provides a rigorous theorem linking regularity to the underlying perturbation variables.

Abstract

We investigate both analytically and numerically the evolution of scalar perturbations generated in models which exhibit a smooth transition from a contracting to an expanding Friedmann universe. We find that the resulting spectral index in the late radiation dominated universe depends on which of the $Ψ$ or \$zeta$ variables passes regularly through the transition. The results can be parameterized through the exponent $q$ defining the rate of contraction of the universe. For $q \geq -1/2$ we find that there are no stable cases where both variables are regular during the transition. In particular, for $0<q\ll 1$, we find that the resulting spectral index is close to scale invariant if $Ψ$ is regular, whereas it has a steep blue behavior if $ζ$ is regular. We also show that as long as $q\leqslant 1$, perturbations arising from the Bardeen potential remain small during contraction in the sense that there exists a gauge in which all the metric and matter perturbation variables are small.

Figures (10)

• Figure 1: Here we illustrate the dominant spectral indices $n_u$ (solid) and $n_v$ (dashed) on super-Hubble scales which are obtained after a "stable", regular transition of the corresponding variable to the radiation era at sufficiently late time. This is also the resulting spectral index for the Bardeen potential on cosmologically relevant scales. The left panel shows the indices as a function of $\gamma_{-}$, the exponent of the pump field during the pre-big-bang phase. The right panel shows them as a function of $q$, the exponent of the contraction (expansion) law before the big bang. We have used $\gamma_{u_{-}}=-q$ and $\gamma_{v_{-}}=q$.
• Figure 2: This figure illustrates the evolution of various background quantities scaled in such a way that they are dimensionless (i.e., invariant under a change in $\eta_s$). As functions of $\eta/\eta_s$, they are (1) the scale factor $a(\eta)$, (2) the Hubble rate ${\cal H}\eta_s$ and (3) its first time derivative ${\cal H}'\eta_s^2$. On the bottom line, they are (4) the pre-factor of the comoving wave number $\Upsilon$, (5) the potential of the perturbation variable $u$, i.e., $V_u\eta_s^2=\left({\cal H}^2-{\cal H}'\right)\eta_s^2$ and (6) its rescaled square root $\sqrt{|V_u|/\Upsilon}~\eta_s$. The parameters used for these figures are $\epsilon = 10^{-2}$ and $q = 5 \cdot 10^{-2}$.
• Figure 3: Case 1, $\theta_1 \to \theta_1$, using $\tilde{\gamma}(\eta) = \iota(1-q)-1$. The $u$ potential (left) and spectrum, $P_u=|u|^2k^3$ (right) are shown for $q=5\cdot 10^{-2}$ and $\epsilon=10^{-2}$.
• Figure 4: Case 2, $\theta_1 \to \theta_2$, using $\tilde{\gamma}(\eta) =-\iota(2+q)+2$. The $u$ potential (left) and spectrum, $P_u=|u|^2k^3$ (right) are shown for $q=5\cdot 10^{-2}$ and $\epsilon=10^{-2}$.
• Figure 5: Case 3, $\theta_2 \to \theta_1$, using $\tilde{\gamma}(\eta) =\iota(2+q)-1$. The $u$ potential (left) and spectrum, $P_u=|u|^2k^3$ (right) are shown for $q=5\cdot 10^{-2}$ and $\epsilon=10^{-2}$.