Nonsingular Bianchi type I cosmological solutions from 1-loop superstring effective action

TL;DR

The paper extends nonsingular cosmology from a 1-loop string effective action with a modulus-coupled Gauss-Bonnet term to anisotropic Bianchi type I spacetimes. By deriving the full anisotropic equations of motion and analyzing asymptotics, it shows that nonsingular flows exist near the isotropic limit, evolving from a Gauss-Bonnet-dominated past through a superinflationary phase to Friedmann- or Kasner-type futures, while revealing two main classes of singularities tied to the behavior of $\Delta$. The study also demonstrates that increasing anisotropy shrinks the nonsingular region, and it finds that energy conditions are violated in the GB-dominated past, with the long superinflationary stage failing to isotropize the geometry, thereby challenging the cosmic no-hair conjecture in this setting. These results provide a framework for understanding graceful exit mechanisms in string-inspired cosmologies and motivate future perturbation and stability analyses."

Abstract

Non-singular Bianchi type I solutions are found from the effective action with a superstring-motivated Gauss-Bonnet term. These anisotropic non-singular solutions evolve from the asymptotic Minkowski region, subsequently super-inflate, and then smoothly continue either to Kasner-type (expanding in two directions and shrinking in one direction) or to Friedmann-type (expanding in all directions) solutions. We also found a new kind of singularity which arises from the fact that the anisotropic expansion rates are multiple-valued function of time. The initial singularity in the isotropic limit of this model belongs to this new kind of singularity. In our analysis the anisotropic solutions are likely to be singular when the super-inflation is steep.

Figures (8)

• Figure 1: The $H$-$\sigma$ phase diagram of isotropic solutions ($\lambda=1$, $H>0$, $\dot\sigma>0$). Time flows from left to right since $\dot\sigma>0$. The nonsingular solutions are plotted with a solid line, and singular solutions with a dotted line. The bold line is a critical solution marking the border of singular and nonsingular solutions. All solution flows in the $H>0$, $\sigma<0$ quarter-plane continue smoothly to the $H>0$, $\sigma>0$ quarter-plane. In the $H>0$, $\sigma>0$ quarter-plane, however, only the flows below the critical solution continue to the $\sigma<0$ region.
• Figure 2: (a) The average expansion rate in the anisotropic case. The equations are solved from $\sigma=-10$, where the initial anisotropy is fixed as $X=0.1$, $Y=0.2$. For large initial $H_{\hbox{\scriptsize avr}}$, there appear singularities with which the solution flows terminate suddenly, keeping $H_{\hbox{\scriptsize avr}}$ and $\sigma$ finite. (b) The behavior of $H_{\hbox{\scriptsize avr}}$, $\dot H_{\hbox{\scriptsize avr}}$, $\sigma$, $\dot\sigma$, and $\ddot\sigma$ in a singular solution appearing in (a). Initial conditions are the same as in (a) except the initial $H_{\hbox{\scriptsize avr}}$ is set to 0.04. We can see $\dot H_{\hbox{\scriptsize avr}}$ and $\ddot\sigma$ diverge, but $H_{\hbox{\scriptsize avr}}$, $\sigma$, and $\dot\sigma$ stay finite.
• Figure 3: Constraint and $\Delta$ on the $\sigma=-10$ section of the $X$-$Y$ plane. $H_{\hbox{\scriptsize avr}}$ is 0.001, 0.005, 0.01 from above. $\Delta>0$, $\Delta<0$, and excluded regions are indicated by white, light-shaded, and dark-shaded areas, respectively. In the dark gray region the Hamiltonian constraint (\ref{['eqn:b1eom1']}) is not satisfied, and $\Delta$ cannot be defined. Cosmological solutions inhabit the white and light gray regions, and those in each region are separated by singularities since $p$, $q$, $r$ become infinite when $\Delta=0$ [see Eqs. (\ref{['eqn:b1eom2']}), (\ref{['eqn:b1eom3']}), and (\ref{['eqn:b1eom4']})].
• Figure 4: Solutions through the $\sigma=-10$ cross section. $H_{\hbox{\scriptsize avr}}$ is 0.001, 0.005, 0.01 from the left. NS means nonsingular. NSa leads to a Friedmann-type solution (expanding in all directions) and NSb leads to a Kasner-type solution (expanding in two directions and shrinking in one direction) in the future asymptotic region. S1 means it leads to a singularity where $\Delta\rightarrow 0$. S1a is the solution whose behavior near such a singularity is $p\dot p>0$, $q\dot q>0$, $r\dot r>0$, while S1b behaves as $p\dot p<0$, $q\dot q<0$, $r\dot r<0$, near the singularity. S2 leads to a singularity where $\Delta\rightarrow -\infty$.
• Figure 5: Behavior of solutions appearing in Fig. 4. $t=0$ is the time when $\sigma=-10$. These are solutions through the points $(X,Y)=(0.1,0.2)$, $(0.2,0.4)$, $(0.4,0.6)$, and $(0.8,0.9)$, respectively, in the $H_{\hbox{\scriptsize avr}}=0.01$, $\sigma=-10$ plane.