Time-dependent flux backgrounds in type IIB supergravity

TL;DR

<3-5 sentence high-level summary>The authors construct analytic time-dependent backgrounds in type IIB supergravity with a time-evolving 4D spacetime and a Ricci-flat 6D internal space, allowed to couple to nontrivial fluxes and a dynamic axiodilaton τ. They solve the full 10D equations with a metric ansatz that isolates the 4D and 6D sectors, demonstrate that the 10D and corresponding 4D energy conditions hold, and show the 4D Einstein-frame Ricci scalar is negative in explicit realizations, suggesting potential early-universe applications. They further extend the Maldacena–Nuñez no-go analysis to include a spacetime-dependent internal scale and partially wrapped fluxes, arguing that a positive $R_E$ cannot be excluded by the traditional no-go arguments in this generalized setup. The work also outlines the moduli and constraints on β, discusses late-time behaviour and possible cosmological implications, and provides a catalog of solvable flux configurations for time-dependent 4D cosmologies.

Abstract

We analytically construct families of type IIB supergravity backgrounds in ten dimensions in which the four-dimensional metric is time dependent, while the six-dimensional internal space is an arbitrary compact Calabi-Yau manifold (with no restriction on holonomy) up to an overall time-dependent scale factor. Our solutions include cases with all fluxes (three-form and self-dual five-form) switched on, as well as cases with subsets of these fluxes, together with a time-dependent axiodilaton in most cases. These constructions require no local sources. We show that the associated energy-momentum tensors (both 10D and the resulting 4D effective) satisfy the null, weak, strong, and dominant energy conditions. In our explicit constructions, the Ricci scalar of the four-dimensional Einstein frame metric is negative; such backgrounds may find applications to anisotropic or FLRW cosmologies in the early universe. We also revisit the Maldacena--Nuñez no-go analysis, incorporating new elements that appear in our constructions, namely an overall noncompact spacetime-dependent scale factor multiplying the internal metric, and field strengths with components partially covering the noncompact directions. We argue that, with these generalizations, a four-dimensional Einstein-frame metric with positive Ricci scalar cannot be ruled out by such an analysis.

Figures (7)

• Figure 1: Profiles of the time-dependent scale factors and Ricci scalar associated with the 4D Einstein frame metric for the case with vanishing $G_3$, nontrivial self-dual $\tilde{F}_5$, and constant $\tau$. The quantities are plotted against time for the representative choice of parameters specified in the text, with $\beta=-t$ for the top row and $\beta=-\ln[t+1]$ for the bottom row. In the left panels, both axes are shown on a logarithmic scale. The initial values of the plotted quantities are all nonzero, as given in the text.
• Figure 2: Profiles of the time-dependent scale factors and Ricci scalar associated with the 4D Einstein frame metric, and the imaginary part of axiodilaton, for the case with vanishing $G_3$, nontrivial self-dual $\tilde{F}_5$, and constant $\textrm{Re}\tau$. The quantities are plotted against time for the representative choice of parameters specified in the text, with $\beta=-t$. In the left panels, both axes are shown on a logarithmic scale. The initial values of the plotted quantities are all nonzero, as given in the text.
• Figure 3: Profiles of the time-dependent scale factors and Ricci scalar associated with the 4D Einstein frame metric, and the real and imaginary parts of axiodilaton, for the case with vanishing $G_3$ and nontrivial self-dual $\tilde{F}_5$. The quantities are plotted against time for the representative choice of parameters specified in the text, with $\beta=-t$. $\textrm{Re}\tau=\textrm{Re}\tau_{+}$ is displayed here only. In the top left panel, both axes are shown on a logarithmic scale. The initial values of the plotted quantities are all nonzero, as given in the text. The two plots in top row do not depend on the sign of $c_5$.
• Figure 4: Profiles of the time-dependent scale factors and Ricci scalar associated with the 4D Einstein frame metric, and the imaginary part of axiodilaton, for the case with vanishing $\tilde{F}_5$, nontrivial $G_3$ flux, and constant $\textrm{Re}\tau$. The quantities are plotted against time for the representative choice of parameters specified in the text, with $\beta=-t$. In the left panels, both axes are shown on a logarithmic scale. The initial values of the plotted quantities are all nonzero, as given in the text.
• Figure 5: Profiles of the time-dependent scale factors and Ricci scalar associated with the 4D Einstein frame metric, and the real and imaginary parts of axiodilaton, for the case with vanishing $\tilde{F}_5$ and nontrivial $G_3$. The quantities are plotted against time for the representative choice of parameters specified in the text, with $\beta=-t$. In the top left panel, both axes are shown on a logarithmic scale. The initial values of the plotted quantities are all nonzero, as given in the text.