In comparison, the Silverstein team chose a much more complex geometry to work with: negatively curved hyperbolic manifolds. This made his calculations dramatically more difficult.
Shortly after Bento and Monteiro published their paper, Gianguido Dall’Agata and Fabio Zwirner of the University of Padua published their paper, in which they used a similar setup – also involving Riemann-flat manifolds – to calculate the strength of the Casimir effect and show how it could be used to produce dark energy. “We use different technologies that are complementary,” Zwirner said.
Bento and Monteiro took things further than the Padua team, at least in terms of completing a full string compactification. But, Monteiro said, “It was good that these two viewpoints agreed, because it provided a good check on the common idea.”
a dose of reality
As the authors acknowledge, Bento and Monteiro’s work comes with some important caveats.
First, his de Sitter solution is unstable; Its latent energy, although positive, will diminish over time. This type of variable, dynamic dark energy, Andriot pointed out, is “much easier to obtain from string theory” than dark energy that remains constant—a notion that Einstein introduced in 1917 as the “cosmological constant.”
In this case, “unstable” has a specific meaning for physicists. This indicates that the stability, or persistence, of dark energy should not last longer than the 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 is changing. In April 2024, the Dark Energy Spectroscopic Instrument produced tentative evidence that dark energy is weakening, and this finding was confirmed a year later. “If those results hold up here, they’re really indicating that the cosmological constant is not constant,” Monteiro said.
In their search for the de Sitter solution, Bento and Monteiro simplified their task by starting with M-theory (sometimes called “the mother of all string theories”). While most versions of string theory require six extra dimensions in our universe, M-theory requires seven extra dimensions. Despite the larger number of dimensions, M-theory has fewer materials than string theory, so starting with M-theory made Bento and Monteiro’s calculations much easier. But subtracting six additional dimensions from M-theory’s 11 total dimensions into their manifolds left 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 Monteiro consider solving it a top priority. “If we can’t find a four-pronged solution, our work can’t be the final answer,” Bento said.
“I hope it works, and they succeed in getting it [to work] In four dimensions,” Andriot said. However, he cautioned that given the myriad challenges string theorists have faced over the past few decades, he would not be surprised if the de Sitter problem put at least a few more obstacles in their way.
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