The graceful exit in pre-big bang string cosmology

TL;DR

This work addresses the graceful exit problem in pre-Big Bang string cosmology by incorporating the most general time-dependent classical corrections up to four derivatives and plausible string-loop corrections into the four-dimensional string effective action. It maps how the coefficients of the corrections and loop parameters must be chosen to achieve a smooth transition from the pre-Big Bang inflationary phase to the post-Big Bang FRW era while testing compatibility with entropic bounds. The results show that scale-factor duality-invariant truncations alone often fail to produce exits, whereas relaxing duality and including higher-derivative and loop terms—especially a two-loop term—yields viable graceful exits when supplemented by dilaton stabilization mechanisms. The findings indicate that non-singular cosmologies can respect Hubble entropy bounds in some parameter regions, but the precise role of unknown quantum corrections remains important for fully understanding the exit and its observational implications.

Abstract

We re-examine the graceful exit problem in the pre-big bang scenario of string cosmology, by considering the most general time-dependent classical correction to the Lagrangian with up to four derivatives. By including possible forms for quantum loop corrections we examine the allowed region of parameter space for the coupling constants which enable our solutions to link smoothly the two asymptotic low-energy branches of the pre-big bang scenario, and observe that these solutions can satisfy recently proposed entropic bounds on viable singularity free cosmologies.

Figures (5)

• Figure 1: On the left, we have represented in the $(c, d)$ plane the asymptotic state of the evolution when the classical corrections are included, with $a=-1$ and the remaining parameter $b$ is constrained by Eq. (\ref{['constraint']}). The red dots show the combinations of these parameters leading to a fixed point, with saturated Hubble parameter $H={\rm const}$ and a linearly growing dilaton. The black points indicate the set of coefficients forcing the evolution to head away from the $\dot{\bar{\phi}} \leq 0$ region. The black line represents the bound given in Brustein2. The figure on the right shows the fixed point distribution in the plane $(2\dot{\bar{\phi}}, H)$. The green line corresponds to a constant $c = 0$, whereas the red line stands for $d=16$. The cyan line shows the evolution for the SFD case with $c=4$ and $d=16$: the curve heads away from the $\dot{\bar{\phi}} \leq 0$ region, typical of the black dots area in the figure on the left.
• Figure 2: Hubble expansion in the S-frame as a function of the dilaton for the case $a=-1$, $b=c=0$ and $d=16$. The y-axis corresponds to $H$, and the x-axis to $2\dot{\phi}/3$. The initial conditions for the simulations have been set with respect to the lowest-order analytical solutions at $t_{S} = -1000$. The straight black lines describe the bounds quoted in Section II. The dotted magenta line shows the impact of the classical correction due to the finite size of the string. A $*$ denotes the fixed point. The contribution of the one-loop expansion is traced with a dashed cyan line ($A=4$). The dash-dotted blue line represents the incorporation of the two-loop correction without the Gauss-Bonnet combination ($B=-0.1$). Finally, the green plain line introduces radiation with $\Gamma_{\phi}=0.08$ and stabilises the dilaton.
• Figure 3: Hubble expansion in the S-frame as a function of the shifted dilaton $(2\dot{\bar{\phi}})$ for the SFD case, with $a=-1$, $b=-c=-16$ and $d=16$. Straight lines are identical to those in figure \ref{['fig1']}. The classical correction makes the curve turn the wrong way (dotted magenta). The inclusion of the one-loop correction (dashed) leads to a regime of instability: the evolution meets the curve corresponding to $\ddot{a} \rightarrow \infty$ (plain). Two cases are explicitly shown: $A=0.5$ in blue and $A=16$ in cyan, whereas the meeting points for other choices of the pre-factor $A$ are pictured with stars (*).
• Figure 4: Hubble expansion in the S-frame as a function of the shifted dilaton $(2\dot{\bar{\phi}})$ for the case $a=-1$, $b=-c=-4$ and $d=16$, with $A=+4$ (dashed cyan) and $B=-0.3$ (dash-dotted blue). The plain green line includes the effect of the particle creation, with $\Gamma_{\phi}=0.2$. The FRW and FP lines represent solutions for the fixed points given in Eq. (\ref{['fixed']}).
• Figure 5: $ln(S_{HB})_{t_{final}} - ln(S_{HB})_{t_{initial}}$ plotted as a function of $t_{final}$ for the setting $a=-1$, $b=-c=-4$ and $d=16$. The dotted line represents the classical correction, then the loop corrections with $A=+4$ (dashed line) and $B=-0.3$ (dash-dotted line). The plain green line also includes the effect of particle creation, with $\Gamma_{\phi}=0.2$.