A Model of Graceful Exit in String Cosmology

TL;DR

The paper addresses the problem of achieving a graceful exit from a dilaton-driven inflationary phase to a decelerated FRW era in string cosmology. It proposes a two-stage mechanism: alpha-prime corrections create a high-curvature fixed point with a linearly growing dilaton, followed by NEC-violating quantum corrections that drive a branch change to a decelerated expansion. Using an effective four-dimensional string-frame action with alpha-prime and quantum corrections, the authors show NEC violation can complete the exit, though late-time evolution can be correction-dominated and unstable unless corrections are suppressed. They explore shut-off strategies via dilaton stabilization or radiation production and discuss the need for explicit string-theory inputs to establish physical viability, highlighting the rich, non-singular string-phase dynamics that precede standard cosmology.

Abstract

We construct, for the first time, a model of graceful exit transition from a dilaton-driven inflationary phase to a decelerated Friedman-Robertson-Walker era. Exploiting a demonstration that classical corrections can stabilize a high curvature string phase while the evolution is still in the weakly coupled regime, we show that if additional terms of the type that may result from quantum corrections to the string effective action exist, and induce violation of the null energy condition, then evolution towards a decelerated Friedman-Robertson-Walker phase is possible. We also observe that stabilizing the dilaton at a fixed value, either by capture in a potential minimum or by radiation production, may require that these quantum corrections are turned off, perhaps by non-perturbative effects or higher order contributions which overturn the null energy condition violation.

Figures (9)

• Figure 1: A solution to the classical equations with only $\alpha'$ corrections, ${\cal L}_q= {\cal L}_m=0$. (a) Evolution in the $(\dot \phi/3,H_S)$ plane. The four lines plotted are, in order of increasing slope, the $(+)$ branch vacuum ($\rho=0, \dot \phi=(3+3^{1/2}) H_S$), branch change ($\dot \phi=3 H_S$), Einstein frame "bounce" ($\dot \phi=2 H_S$) and $(-)$ branch vacuum ($\rho=0, \dot \phi=(3-3^{1/2}) H_S$). The cross is at the location of the fixed point. The remaining figures are various quantities plotted as a function of string time. (b) The Einstein frame Hubble parameter $H_E$. (c) The string frame source terms $e^\phi (\rho_S+p_S)$ and $e^\phi \rho_S$. (d) The Einstein frame source term $\rho_E+p_E$. (e) $\dot \phi-3 H_S$, a quantity indicating branch sign. (In section III this figure will be replaced by $e^\phi \rho_S$ from (c)), (f) $\phi$ evolution. Initial conditions at $t=0$, $\phi=-30, H_S=0.0148529, \dot \phi=0.0699602$. The following figures share these initial conditions, since quantum corrections are very small at this time. The $\phi$ evolution may be started at arbitrarily small values by evolving these initial conditions further backward towards the $(+)$ branch vacuum.
• Figure 2: $\hbox{\small $\frac{1}{2}$} {\cal L}_q^{\phi}=-(\nabla \phi)^4$, ${\cal L}_{m}=0$. See Fig. \ref{['f:gmv']} caption for details and initial conditions.
• Figure 3: ${\cal L}_q={\cal L}_q^\phi-\frac{1}{3}{\cal L}_q^{R^2}$, ${\cal L}_{m}=0$. Initial conditions at $t=0$, $\phi=-5$, $H_S=0.0926031$, $\dot \phi=0.383925$, $\dot H_S=0.018527$, $\ddot H_S=0.0260335$. See Fig. \ref{['f:gmv']} caption for explanation.
• Figure 4: $\hbox{\small $\frac{1}{2}$} {\cal L}_q^{R^2_{GB}}=e^{\phi} R^2_{GB}$, ${\cal L}_{m}=0$. See Fig. \ref{['f:gmv']} caption for details and initial conditions.
• Figure 5: ${\cal L}_q={\cal L}_q^{\phi}$, ${\cal L}_{m}=-0.02 \phi^2 e^{\phi}$. Quantities plotted and initial conditions are as in Fig. \ref{['f:gmv']}a. Also shown are the singularity curves, defined in the text. The right curve is for $\phi\rightarrow -\infty$ and the left curve is for the value of $\phi$ at which the solution hits the curve.