Towards a non-singular pre-big bang cosmology

TL;DR

The paper analyzes the beta-function structure for spatially flat $(d+1)$-dimensional cosmologies in string theory, focusing on all-order $\alpha'$ corrections at tree level in $g_s$. It shows that constant-curvature solutions with a linear dilaton reduce to $(d+1)$ algebraic fixed points in the variables $(\dot\beta_i, \dot{\overline\phi})$, which can act as late-time attractors and regularize pre-big-bang singularities. A concrete first-order $\alpha'$ model with Gauss–Bonnet terms demonstrates explicit fixed points and a smooth flow from the dilaton phase to the string phase for isotropic backgrounds, with finite attraction basins for anisotropic initial data. The work clarifies the complementary roles of $\alpha'$-corrections and loops in completing the transition to standard cosmology, while noting the need for exact conformal realizations and loop/dilaton-potential effects for a fully realistic scenario.

Abstract

We discuss general features of the $β$-function equations for spatially flat, $(d+1)$-dimensional cosmological backgrounds at lowest order in the string-loop expansion, but to all orders in $α'$. In the special case of constant curvature and a linear dilaton these equations reduce to $(d+1)$ algebraic equations in $(d+1)$ unknowns, whose solutions can act as late-time regularizing attractors for the singular lowest-order pre-big bang solutions. We illustrate the phenomenon in a first order example, thus providing an explicit realization of the previously conjectured transition from the dilaton to the string phase in the weak coupling regime of string cosmology. The complementary role of $α'$ corrections and string loops for completing the transition to the standard cosmological scenario is also briefly discussed.

Figures (5)

• Figure 1: Curvature regularization of a pre-big bang background as a consequence of the first-order $\alpha^{\prime}$ corrections (in units $k\alpha^{\prime}=1$). The dashed curve shows the singular behaviour of the zeroth-order solution in $d=9$. The solid curves approach asymptotically the constant values of eq. (\ref{['48']}).
• Figure 2: Smooth evolution of the background from the perturbative vacuum, $\dot{\beta}=0=\dot\phi$, to the non-trivial fixed points with $\dot{\beta}$ and $\dot\phi$ constant reported in eq. (\ref{['48']}). The dashed curve corresponds to the singular zeroth-order solution.
• Figure 3: First-order $\alpha^{\prime}$ corrections (solid curves) to the branches of the singular zeroth-order solution (dashed curves).
• Figure 4: Evolution from the perturbative vacuum to the expanding fixed point (and its time-reversal) to first order in $\alpha^{\prime}$. The dashed lines represent the zeroth-order pre- and post-big bang solutions.
• Figure 5: Attraction basin of the fixed points, in the space of anisotropic pre-big bang initial conditions, for the case $d=2$, $n=1$, and for a particular fixed initial value of $\dot\beta$.