Sean smiles for the camera.
Sean smiles for the camera.
This guest post was written by Sean McBride, who is a postdoc at the University of British Columbia in Vancouver. He received his Ph.D. from the University of California, Santa Barbara in 2024 advised by Xi Dong. His research concerns the intersection of quantum gravity, quantum information, and string theory. When he’s not doing physics, you can most likely find Sean getting chased around by sea lions or playing with octopi while SCUBA diving.
Title: Candidate de Sitter Vacua
Authors: Liam McAllister, Jakob Moritz, Richard Nally, and Andreas Schachner
Authors’ Institutions: Cornell University, CERN, LMU Munich
Status: CERN report, posted to arXiv [open access]
String theory is, by far, our best attempt at unifying quantum mechanics and general relativity, two theories of the real world we hold as gospel. It has been useful theoretically in understanding strongly coupled quantum systems, efforts to resolve the black hole information paradox, and has generated a flurry of research topics in pure mathematics.
Connecting string theory to experiments has proven to be a Herculean venture. String theory is most naturally formulated in ten spacetime dimensions, as opposed to the four (three spatial, one time) dimensions that we inhabit. The extra six dimensions must be “compactified”: curled up on scales much smaller than we can observe. The geometry of these curled up dimensions determines what kind of particles a particular string compactification will contain; a given universe can have anywhere from zero particle species to several million. The number of compactifications possibly consistent with the Standard Model of particle physics is astronomical (pun intended), with estimates up to 10272,000; this is the much-maligned “landscape” of string theory.
The Standard Model is not the only lamppost to guide tests of string theory. In 1998, observations of Type 1a supernovae taught us that not only is the Universe expanding, but that expansion is accelerating, represented by a small but positive cosmological constant Λ. This revolutionary discovery solidified the ΛCDM model of cosmology, which accurately describes the timeline of our Universe. An idealized model of a universe with Λ > 0 is de Sitter space, a constantly accelerating spacetime whose energy content is entirely dark energy, with no matter nor radiation. De Sitter space is an excellent approximation to any cosmological era where effects caused by dark energy are more important than those caused by matter and light; two such eras are the inflationary epoch, which took place from around 10-36 to 10-32 seconds after the Big Bang, and the distant future, where the universe has expanded so much that matter is diluted and dark energy is all that remains. When Λ < 0, we instead have anti-de Sitter space, a universe with no physical relevance as it would have a decelerating expansion.
In string theory, all fundamental constants aren’t put in by hand; rather, they’re determined dynamically as minima of a potential function which depends on the geometry of the compactified dimensions. These minima are vacua; they are the quantum state with the lowest possible energy. Negative cosmological constants are well-suited to this approach, as the minima of their potentials always lie below zero; a universe with Λ < 0 will stay that way in perpetuity, or at least for an absurdly long time.
A plot of four different possible scalar potentials against volume. From lowest to highest: a dashed black line labeled SUSY AdS, a green line labeled Non-SUSY AdS, a pink line labeled de Sitter, and a blue line labeled Runaway. All except for the runaway potential decrease sharply at small volumes, rise again slightly, and then asymptotically trend towards 0 at large volumes. The runaway potential does not dip. Only the AdS potentials dip below 0.
Figure 1: Some of the string theoretic potentials that can give rise to a positive or negative cosmological constant. The cosmological constant of these solutions lies at the minimum of the potentials. The local minimum of the de Sitter potential (pink line) has Λ > 0 but will tunnel out to an asymptotic region with Λ = 0. Image Credit:McAllister et. al. 2024 (Fig 1).
On the other hand, Λ > 0 must arise only as a local minimum of a potential (see Figure 1); eventually, quantum effects will force tunneling to a different vacuum with zero or negative cosmological constant. This is the main challenge behind constructing de Sitter vacua in string theory; they are never stable, only metastable.
Furthermore, stable string theory vacua are constructed with an additional symmetry called supersymmetry. Supersymmetry imposes certain relations between bosons (e.g. photons) and fermions (e.g. electrons), but more importantly it ensures that many potentially harmful corrections are forbidden, ensuring a stable vacuum. We don’t observe supersymmetry in particle colliders, so it must be broken at low energies, which again complicates stability issues for string theoretic constructions.
How to Build a De Sitter Universe
Given all of these constraints, can we find just one string theory vacuum with Λ > 0? The most popular attempt to date is due to Kachru, Kallosh, Linde, and Trivedi from 2003, who proposed a straightforward algorithm to construct de Sitter universes. One starts with a supersymmetric, anti-de Sitter vacuum with an exponentially small, but still negative, cosmological constant (see the dashed line in Figure 1). One then adds exotic stringy objects called anti-D3-branes, whose role is twofold. They “uplift” the potential, raising the minimum just high enough that it’s still a minimum while keeping the valley deep enough to avoid tunneling out, at least on timescales of the age of the Universe. The uplifting also breaks supersymmetry, opening the doors for a set of fundamental particles akin to the Standard Model. This is a delicate balancing act; corrections to the potential function from a myriad of stringy effects (wild D-branes, curvature corrections, etc.) can cause these vacua to destabilize and drive the potential minimum back below zero, so in each case (and there are a lot of cases!) one must argue why a correction is small or vanishes by some symmetry. The claim in today’s paper is that almost all corrections in their string theory solutions are under analytic control; those that are not can be argued to be small or to only spoil a subset of the solutions found by searching through the landscape.
The potentials constructed in today's paper; a SUSY AdS potential and a de Sitter potential. The de Sitter potential is shifted above the AdS potential and towards slightly larger volumes, but is very similar in shape.
Figure 2: The cosmological constant potential in a candidate de Sitter vacuum (pink line) constructed in today’s paper by uplifting a supersymmetric anti-de Sitter vacuum (dashed line). Image Credit:McAllister et. al. 2024 (Fig 1).
Landscape or Swampland?
This is not to say the solutions constructed by McAllister et. al. are phenomenologically valid. The matter they contain is nothing like the matter, dark or otherwise, we observe in our Universe. There is no Big Bang nor any of the other cosmological eras we observe. Moreover, there’s been a growing concern in the string theory community that solutions of the type considered in today’s paper are simply forbidden. Rather than being part of the vast landscape of string theory, de Sitter vacua may be in the swampland, a bog home to theories which look physically reasonable from our human, low energy perspective but are ill-behaved when extended to stringy regimes. This has motivated some astronomers and string theorists to explore quintessence models of dark energy, in which the cosmological constant isn’t a constant at all, but is slowly changing with time (the “runaway” scenario in Figure 1). These candidate vacua, if shown to be stable, would demonstrate that de Sitter space belongs in the landscape instead of the swampland, and thus would be a significant step towards connecting string theory with realistic models of cosmology.
Astrobite edited by: Katherine Lee
Featured image credit: Polytope24/Wikimedia Commons