Scale Invariance via a Phase of Slow Expansion

TL;DR

The paper extends the adiabatic ekpyrotic mechanism to an expanding background, showing that a rapidly evolving equation of state parameter $\epsilon$ during a finite-expansion transition yields a scale-invariant spectrum for the curvature perturbation $\zeta$ with the same two-point function as in inflation. Through analytic arguments and thorough numerical verification, it demonstrates that the expanding transition phase followed by a contracting ekpyrotic phase acts as a dynamical attractor, with a broader basin of attraction than the original contracting scenario. A linear perturbation analysis and phase-space study establish the robustness of the mechanism to a wide range of initial conditions and background evolutions, including starting from nonzero initial kinetic energy. The work highlights that, while the simplest models suffer from strong coupling and large non-Gaussianities, generalized potentials can preserve perturbativity over a finite range of scales, making this non-inflationary route to scale invariance a viable component of broader early-universe model-building, including cyclic or inflation-precursor scenarios.

Abstract

We consider a cosmological scenario in which a scale-invariant spectrum of curvature perturbations is generated by a rapidly-evolving equation of state on a slowly expanding background. This scenario generalizes the "adiabatic ekpyrotic" mechanism proposed recently in arXiv:0910.2230. Whereas the original proposal assumed a slowly contracting background, the present work shows that the mechanism works equally well on an expanding background. This greatly expands the realm of broader cosmological scenarios in which this mechanism can be embedded. We present a phase space analysis and show that both the expanding and contracting versions of the scenario are dynamical attractors, with the expanding branch having a broader basin of attraction. In both cases, a finite range of scale invariant modes can be generated within the regime of validity of perturbation theory.

Figures (5)

• Figure 1: Depiction of the "lifted exponential" potential, $V(\phi) = V_0 (1-e^{-c\phi/M_{\rm Pl}})$. At large field values the potential is nearly constant, and there is a steep waterfall around $\phi =0$.
• Figure 2: Numerical computation of the power spectrum $k^{3/2}\zeta_k$ vs. $k$ in both the expanding and contracting transition phase scenarios using an analytic expression for $z(\tau)$. This confirms that the spectrum is insensitive to whether the transition phase is contracting or expanding.
• Figure 3: Comparison of the power spectra for the exact numerical calculation and the integration using our analytic expression for $z(\tau)$. The curves are in excellent agreement.
• Figure 4: Phase portrait for the expanding (a) and contracting (b) cases, for $c=100$ and $V_0=10^{-4}$. The analytic solution is denoted by the black curve, with the transition phase taking place between the dotted lines. Colored dashed lines are particular numerical solutions to the system (\ref{['phidotphivector']}). Note that the arrows point in the direction of increasing time. This Figure confirms that the analytic solution is an attractor for a variety of initial conditions in both cases. However, at large positive $\phi$ where $V(\phi)\approx V_0$, the expanding solution is also an attractor, while the contracting solution is a repellor, due to the asymmetry between expanding vs. contracting de Sitter space.
• Figure 5: Detailed view of Fig. \ref{['globalview']}, zooming in on the transition phase. Both contracting and expanding transition solutions are attractors, but the expanding case has a slightly larger basin of attraction.