On Cosmological Singularities in String Theory

TL;DR

The paper analyzes time evolution in a 3+1D string-theory background built on $\\mathbb{R}_t \\times \\mathbb{S}^3_k \\times \\mathcal{M}_k$, focusing on time-dependent current-current (non-abelian Thirring) deformations and radius deformations of the $\\mathbb{S}^3$. Using a spacetime EFT for the deformation fields $φ$ and $χ$ in the large $k$ limit, the authors connect worldsheet RG flows to cosmological dynamics, uncovering two regimes: small perturbations generically drive big-bang/big-crunch singularities, while radius perturbations push the sphere to infinite size at finite time without collapsing to zero. In the isotropic, $SO(4)$-preserving sector, the radius deformation yields no big-bang/crunch singularities; instead, the sphere either oscillates around a finite radius with damping or expands to infinity in finite time, depending on the sign of the dilaton's time derivative. Altogether, the work highlights how worldsheet RG structure and stringy effects may resolve cosmological singularities in this class of models and outlines future directions, including alternative internal CFTs, higher-harmonic deformations, and observable definitions in time-dependent spacetimes, guided by a complementary analysis of climbing exponential potentials and their critical exponents.

Abstract

We study the time evolution of a $3+1$ dimensional spacetime, where space is a large three-sphere, due to small perturbations of the background fields. We focus on two classes of deformations. One corresponds on the worldsheet to time-dependent non-abelian Thirring deformations. The other to perturbations of the radius of the three-sphere. In the former case, we find that small deformations generically lead to big-bang and big-crunch singularities, near which the spacetime becomes highly anisotropic. We argue that string theory likely resolves these singularities. In the latter case, general solutions have the property that the radius of the three-sphere goes to infinity at a finite time, but there are no solutions in which it collapses to zero. We also discuss the interplay of these spacetime properties with the corresponding worldsheet RG flows.

Figures (8)

• Figure 1: Potential $V(\tilde{\phi})$ (\ref{['LKVtphi']}).
• Figure 2: Solutions of (\ref{['tphieom']}) and (\ref{['ddotPhi']}) with $\tilde{\phi}(0)=0$ and $\dot{\tilde{\phi}}(0)=-0.1/\sqrt{k\alpha'}$. The value of $\dot{\Phi}(0)$ is determined from (\ref{['H0']}). The singularities occur at $(t_{\text{m}in},t_{\text{m}ax})=\sqrt{k\alpha'}(-5.31,3.15)$ for $\dot\Phi(0)<0$, and $\sqrt{k\alpha'}(-7.06,2.69)$ for $\dot\Phi(0)>0$.
• Figure 3: The dependence of the lifetime of the universe, $t_\text{max}-t_\text{min}$, on the initial "velocity" $\dot{\tilde{\phi}}(0)$ for (a) $\dot \Phi(0)>0$, (b) $\dot \Phi(0)<0$.
• Figure 4: Solutions of (\ref{['tphieom']}) and (\ref{['ddotPhi']}) with $\tilde{\phi}(0)=0$ and $\dot{\tilde{\phi}}(0)=-2/\sqrt{k\alpha'}$. The value of $\dot{\Phi}(0)$ is determined from (\ref{['H0']}) using the negative root.
• Figure 5: (a) Solutions of $\tilde{\phi}$ for different values of $\dot{\tilde{\phi}}(0)$ with the values of $\dot{\Phi}(0)$ determined from (\ref{['H0']}) using the negative roots, and (b) the maximal value of $\tilde{\phi}$ diverging to $+\infty$ as $\dot{\tilde{\phi}}(0)$ approaches $-1.83/\sqrt{k\alpha'}$.