In selecting a geometry for the compact extra dimensions, for instance, they chose a space that resembles a torus. \u201cIt\u2019s a simple shape,\u201d Bento said. A doughnut is an example of a 2D torus; it is considered \u201cflat\u201d 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.<\/p>\n
In comparison, the Silverstein team selected a much more complicated geometry to work with: negatively curved hyperbolic manifolds. That made their calculations dramatically harder.<\/p>\n
Shortly after Bento and Montero published their paper, Gianguido Dall\u2019Agata<\/a> and Fabio Zwirner<\/a> of the University of Padua published their own paper<\/a>, in which they used a similar setup \u2014 also involving Riemann-flat manifolds \u2014 to compute the strength of the Casimir effect and show how it can be used to produce dark energy. \u201cWe use different techniques that are complementary,\u201d Zwirner said.<\/p>\n 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, \u201cit was nice that these two approaches agreed, because that provided a good check on the general idea.\u201d<\/p>\n A Dose of Reality<\/p>\n The work of Bento and Montero comes with some substantial caveats, as the authors acknowledge.<\/p>\n 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, \u201cis much easier to get from string theory\u201d than a dark energy that remains fixed \u2014 a notion Einstein introduced in 1917 as the \u201ccosmological constant.\u201d<\/p>\n \u201cUnstable,\u201d in this case, has a specific meaning to physicists. It indicates that the period of stability, or constancy, of dark energy shouldn\u2019t last much longer than a Hubble time \u2014 the estimated age of the universe, or about 14 billion years.<\/p>\n 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<\/a>, and the finding was bolstered a year later<\/a>. \u201cIf those results are here to stay, they are really hinting that the cosmological constant is not a constant,\u201d Montero said.<\/p>\n