Flux repulsion may make a tiny C.C. natural

A "Darwinian" proposal to solve the cosmological constant problem

Since the cosmological observations in the late 1990s, most of us took for granted that the Universe is filled with dark energy (currently believed to represent 68% of the energy density \(\rho=T_{00}\)) whose character may be refined as the ordinary positive cosmological constant (the "C.C.") i.e. \[

p=-\rho, \quad T_{\mu\nu}=\rho g_{\mu\nu}

\] However, the energy density \(\rho\) seems to be extremely tiny in the apparently natural units of quantum gravity, \[

\rho\approx 10^{-123}\,m_{\rm Planck}^4

\] which is the worst known prediction of (dimensional analysis in) physics.



This problem, the cosmological constant problem, doesn't have any convincing explanation except for the "possible" explanation involving the multiverse. There are many possible values of \(\rho\) in different vacua of the theory of everything (we mean string/M-theory). Most of them don't admit any life but due to the large number, there are some vacua for which \(\rho\) is tiny and those are "more important" because intelligent life may emerge in them. We are supposed to live in one of them.




This multiverse scenario avoids the "straight contradiction" but one could argue that the right way to estimate the probability that the C.C. is tiny still leads to an intolerably small value so we haven't explained anything. Isn't there a better way to argue that the C.C. has to be tiny?




Adam R. Brown, Alex Dahlen, and Ali Masoumi propose a new "mixed" approach to the question in the new preprint

Compactifying de Sitter Naturally Selects a Small Cosmological Constant

They remind the readers of the Bousso-Polchinski-like (including KKLT) constructions in which the vacuum energy density is distributed "uniformly" so very tiny values of the C.C. are as unlikely as any other fine-tuned values.

However, they argue that some flux repulsion terms may heavily distort the distribution so that the vacua with \(|\rho|\ll 1\) in the Planck units are far more frequent.

They claim to have an example of anti de Sitter spaces with a tiny negative cosmological constant that get accumulated near \(\rho=0-\epsilon\) as well as (the phenomenologically relevant) numerous de Sitter vacua with \(\rho=0+\epsilon\). In both cases, the distribution is "expo-exponential" (my preferred word for any function similar to \(\exp[\exp (x)]\): they and others call it "doubly exponential").

These lower-dimensional vacua are said to have a rather naive geometry\[

dS_{D-Nq} \times (S^q)^N.

\] Here, \(D\) is the total spacetime dimension and the overall spacetime is being compactified \(N\) times on \(q\)-dimensional spheres, exploiting a \(q\)-form field strength. The compactification on a sphere is metaphorically identified with "having offspring" and the language of natural selection is applied there. The reason is that the "parent" vacua with a small C.C. are producing many more daughters and sons – the number of offspring scales like a negative power of the parent's C.C. So the "small C.C. vacua" are more viable in the Darwinian sense.



For the expo-exponential distribution to appear, we need \(N\geq 2\), i.e. at least two \(q\)-spheres, and the rank of the differential form i.e. the dimension of each sphere has to be \(q\geq 2\) for \(N\geq 3\) and \(q\geq 3\) for \(N=2\). If \(N,q\) are smaller than that, the divergence in the distribution isn't fast enough.

It means that at least six extra dimensions (the same number as we envision in compactified \(D=10\) superstring theory) are needed but they argue that the ordinary superstring vacua are probably not ready for their construction (three two-spheres or two three-spheres in the role of six extra dimensions sound too revolutionary in their simplicity and it would be shocking if such vacua had been overlooked – but I must recheck this expectation) so they propose that a realization of their scenario could occur in \(D\gt 10\) "supercritical" string theory, something I don't really like much, partly because of worries that such theories are non-perturbatively ill-defined, partly because they threaten us by a strictly infinite, "unbounded" landscape (the forefather spacetime dimension may be arbitrarily high).

Incidentally, when the number of vacua is strictly infinite (and it arguably is if we allow supercritical string theory), the number of vacua in any interval of the C.C. is infinite as well and the "relative proportion" of the different C.C. values (the probability distribution) depends on the way how we number the vacua so the accumulation could very well be just an unphysical artifact of a numbering scheme (these warnings are actually discussed in Brian Greene's popular book, The Hidden Reality).

Some of the annoying features of their approach have already been mentioned but there's one more: much like they predict a tiny C.C., they also predict a huge compactification radius. The distribution for the radius is similarly expo-exponentially peaked near the Hubble scale (which is obviously an unacceptable value). They may still get a "generic prediction of a small C.C." by demanding a much shorter compactification radius which they find exciting but it just means to trade one hierarchy problem for another, with some extra complications that their construction brings.

So I am mostly skeptical. But the point that some overlooked dynamical features could make the C.C. in a class of vacua to be "much more likely to be near zero" than naively expected is a point that I have shared for years. I am less certain about the validity of their particular method to achieve the accumulation of the tiny-C.C. vacua but "revolutionary enough model builders" should spend at least 10 minutes or an hour with thoughts whether such an unusually simple scenario is really impossible as they almost certainly believe while they are reading this sentence.