- Spacetime Dimensionality from de Sitter Entropy
By Arshad Momen and Rakibur Rahman
arXiv: 1106.4548 [hep-th]
We argue that the spontaneous creation of de Sitter universes favors three spatial dimensions. The conclusion relies on the causal-patch description of de Sitter space, where fiducial observers experience local thermal equilibrium up to a stretched horizon, on the holographic principle, and on some assumptions about the nature of gravity and the constituents of Hawking/Unruh radiation.
What they've done is to calculate the entropy and energy of the Unruh radiation in a causal patch of any one observer in a de Sitter spacetime with d spatial dimensions. Holding the energy fixed and making certain assumptions about the degrees of freedom of the particles in the radiation, the entropy has a local maxium at d= 2.97 spacelike dimensions, a minimum around 7 and goes to infinity for large d. Since the authors restrict themselves to d less or equal to 10, this seems to say for a given amount of energy the entropy is maximal for 3 spacelike dimensions. Assuming that the universe is created by quantum tunneling, the probability for creation is larger the larger the entropy, thus it would be likely then that we live in a space with 3 dimensions.
To calculate the entropy one needs a cutoff the value of which is fixed by matching it to the entropy associated with the de Sitter horizon, so that's where the holographic principle becomes important.
Not only is it crucial that they add an upper bound on the number of dimensions by some other argument, their counting also depends on the number of particles and the dimensions they can propagate into. They are assuming only massless particles contribute, and these are photons and gravitons. Massive particles even with small masses, the authors write, are "unacceptable" because then the cutoff could be sensitive to the Hubble parameter. By considering only photons and gravitons as massless particles they are assuming the standard model. So even in the best case one could say they have a correlation between the number of dimensions and the particle content. Also, in braneworld models the total number of spatial dimensions isn't necessarily the one determining degrees of freedom at low energy; a possibility the authors explicitly say they're not considering.
Thus, as much as I'd like to see a good answer to the question, I'm not very convinced by this one either.