Universal accelerating cosmologies from 10d supergravity

TL;DR

The article develops a universal, time-dependent 10d Type IIA framework for obtaining 4d FLRW cosmologies by compactifying on Calabi–Yau, Einstein, or Einstein–Kähler manifolds. It demonstrates that the 10d equations reduce to a 1d dynamical system with flux-encoded potentials, and, in certain cases, to a 4d two-scalar truncation sharing its solutions with the universal sector of the compactifications. Analytic solutions exist for single-flux cases, while two-flux scenarios are analyzed via autonomous dynamical systems, revealing fixed points, invariant surfaces, and acceleration regions that determine periods of expansion. The results show a rich spectrum of cosmologies, including transient, eternal, and cyclic acceleration (rollercoaster) and non-singular Milne-like futures, with open universes offering the most expansive acceleration behavior. These findings illuminate how flux content and curvature shape acceleration without invoking explicit 4d de Sitter vacua, suggesting new avenues for string-inspired cosmology and swampland considerations.

Abstract

We study 4d Friedmann-Lemaître-Robertson-Walker cosmologies obtained from time-dependent compactifications of Type IIA 10d supergravity on various classes of 6d manifolds (Calabi-Yau, Einstein, Einstein-Kähler). The cosmologies we present are universal in that they do not depend on the detailed features of the compactification manifold, but only on the properties which are common to all the manifolds belonging to that class. Once the equations of motion are rewritten as an appropriate dynamical system, the existence of solutions featuring a phase of accelerated expansion is made manifest. The fixed points of this dynamical system, as well as the trajectories on the boundary of the phase space, correspond to analytic solutions which we determine explicitly. Furthermore, some of the resulting cosmologies exhibit eternal or semi-eternal acceleration, whereas others allow for a parametric control on the number of e-foldings. At future infinity, one can achieve both large volume and weak string coupling. Moreover, we find several smooth accelerating cosmologies without Big Bang singularities: the universe is contracting in the cosmological past ($T<0$), expanding in the future ($T>0$), while in the vicinity of $T=0$ it becomes de Sitter in hyperbolic slicing. We also obtain several cosmologies featuring an infinite number of cycles of alternating periods of accelerated and decelerated expansions.

Figures (18)

• Figure 1: Trajectories in phase space corresponding to two-flux cosmological solutions with accelerated expansion. The corresponding dynamical systems are given in Sections \ref{['case2']}, \ref{['sec:34d']}, \ref{['case4']}. The pair of fluxes that are turned on in each case is indicated below each subfigure. The trajectories (blue lines) interpolate between a fixed point on the equator in the past infinity, corresponding to a cosmology with a power-law scale factor $S\sim T^{\frac{1}{3}}$, and another fixed point at future infinity. For cases \ref{['fig:test3']} and \ref{['fig:test4']} the second fixed point is also on the equator, and thus the solution asymptotes the same scaling cosmology in the past and future infinities. Cases \ref{['fig:test1']} and \ref{['fig:test2']} asymptote at future infinity a fixed point in the interior of the sphere which corresponds to a cosmology with a power-law scale factor $S \sim T$ and $S\sim T^{\frac{19}{25}}$ respectively. Cases \ref{['fig:test3']} and \ref{['fig:test4']} describe the same system, but employ different parametrizations which interchange the role of the two fluxes that are turned on. The transient accelerated expansion corresponds to the portion of the trajectory within the acceleration region depicted in green.
• Figure 2: Three trajectories lying on the invariant plane $\mathcal{P}$ of the dynamical system obtained from the compactification with $k,\lambda \neq 0$. The point $p_0$ corresponding to a Milne universe is depicted in green. The point $p_1$ (and its mirror in the southern hemisphere) is drawn in blue and corresponds to a Milne universe with angular defect. The equator fixed points coincide with a scaling cosmology with $S\sim T^{\frac{1}{3}}$ and are illustrated in purple. The blue trajectory features transient acceleration with tunable number of e-foldings, whereas the red one corresponds to semi-eternal acceleration. Depicted in purple is the unique (fine-tuned) trajectory corresponding to eternal acceleration. The solution becomes de Sitter in the vicinity of the origin, which is reached at finite proper time. The solution can be geodesically completed in the past beyond the point at the origin, by gluing together its mirror trajectory in the southern hemisphere.
• Figure 3: Number of e-foldings $N$ as a function of $l$, which parametrizes the distance $\Delta$ between the $x$-coordinates of $p"$ and $p_1$ for the $(\lambda, k)$ model, viz. $l= - \log \Delta + \text{const.}$
• Figure 4: Plot log--log of the scale factor $S$ (in blue) as a function of the cosmological time $T$. It corresponds to the $(\lambda,k)$-compactification (with $l=2$). The green dot corresponds to the beginning of inflation; the red one to its end. The red dashed line coincides with the Milne fixed point $S \propto T$ and asymptotes the curve at future infinity.
• Figure 5: Example of a spiraling trajectory around $p_1$ in the $(m,k)$-compactification.