In selecting a geometry for the compact extra dimensions, for instance, they chose a space that resembles a torus. “It’s a simple shape,” Bento said. A doughnut is an example of a 2D torus; it is considered “flat” because it can be made by rolling a flat sheet into a tube and then fastening the ends. Bento and Montero picked shapes of this general type, called 6D Riemann-flat manifolds, to house the extra dimensions in their model. Using this 6D space for the compactification gave them the physical properties they sought.

In comparison, the Silverstein team selected a much more complicated geometry to work with: negatively curved hyperbolic manifolds. That made their calculations dramatically harder.

Shortly after Bento and Montero published their paper, Gianguido Dall’Agata and Fabio Zwirner of the University of Padua published their own paper, in which they used a similar setup — also involving Riemann-flat manifolds — to compute the strength of the Casimir effect and show how it can be used to produce dark energy. “We use different techniques that are complementary,” Zwirner said.

Bento and Montero took things further than the Padua team, at least in terms of carrying out a full-fledged string compactification. But, Montero said, “it was nice that these two approaches agreed, because that provided a good check on the general idea.”

A Dose of Reality

The work of Bento and Montero comes with some substantial caveats, as the authors acknowledge.

First, their de Sitter solution is unstable; its dark energy, though positive, will diminish over time. A changeable, dynamical dark energy of this sort, Andriot pointed out, “is much easier to get from string theory” than a dark energy that remains fixed — a notion Einstein introduced in 1917 as the “cosmological constant.”

“Unstable,” in this case, has a specific meaning to physicists. It indicates that the period of stability, or constancy, of dark energy shouldn’t last much longer than a Hubble time — the estimated age of the universe, or about 14 billion years.

Until recently, most observations have been consistent with a universe containing a constant amount of dark energy. But recent results suggest that dark energy may be changing. In April 2024, the Dark Energy Spectroscopic Instrument presented tentative evidence that dark energy is weakening, and the finding was bolstered a year later. “If those results are here to stay, they are really hinting that the cosmological constant is not a constant,” Montero said.

In their pursuit of a de Sitter solution, Bento and Montero simplified their task by starting from M-theory (sometimes called “the mother of all string theories”). Whereas most versions of string theory require our universe to have six extra dimensions, M-theory requires it to have seven. Despite the larger number of dimensions, M-theory has fewer ingredients than string theory, so starting with M-theory made Bento and Montero’s calculations markedly easier. But subtracting the six extra dimensions curled into their manifold from the 11 total dimensions of M-theory left the theorists with a universe in 5D — one “D” too many.

The issue of landing on a 5D solution in a 4D universe is no small matter, and Bento and Montero consider resolving it a top priority. “If we cannot find the four-dimensional solution,” Bento said, “our work cannot be the final answer.”

“I hope it works, and they manage to get it [to work] in four dimensions,” Andriot said. However, he cautioned, given the myriad challenges string theorists have faced over the past few decades, he wouldn’t be surprised if the de Sitter problem threw at least a few more obstacles in their path.