This post in “Science After Sunclipse” reminded me of one of the most beautiful ideas to come out of the continuing attempts to combine String Theory and Cosmology. The idea of Robert Brandenberger and Cumrun Vafa, dubbed more recently as “String Gas Cosmology”, is a wonderfully creative attempt to explain why our world has three spatial directions. There is no other theory on the market where the dimensionality of space could be determined dynamically, or at least come out as a result of a calculation, rather than being put in as an input, so in String Theory this is a natural question to ask. The idea for an answer, provided by String Gas Cosmology, could be stated naturally and simply, which is what I try doing in this post. As usual, one has to remember that the devil is in the details, and those at the moment provide a real challenge for the idea.
One of the beautiful aspects of this approach is even asking the question: why four dimensions? famously, the dimension of spacetime in string theory is larger than the observed one, in the simplest scenarios 10 or 11 spacetime dimensions, depending on details. We of course only see 3 spatial directions to move in, and can only make sense out of one time direction. Isn’t that a clean Popperian falsification of String Theory?
As you may suspect, the answer to this question is no. There is nothing in the theory which requires all dimensions to be macroscopic, visible to the naked eye, part of our daily experience. Imagine an ant moving on a garden hose, that ant could move in two independent directions. In one of the directions the hose looks infinitely long (at least for an ant), and the other direction closes onto itself, having finite length. The ingenuity of the Brandenberger-Vafa approach is to realize that the natural question is not “how come most of the spatial dimensions in String Theory are so small?”, it is quite the opposite. String Theory has a single length scale, set by the tension of the fundamental strings. In generic situations, any other length scale should be more or less of that order of magnitude. The better question then is, how did some of the dimensions of space came to be so large?
The scenario of string gas cosmology goes as follows: imagine we start in the most natural initial state. In this state we have small dimensions in any direction in space, let’s imagine they are all circular, like in the example of the garden hose. We assume also that all the degrees of freedom of the theory are excited in some thermal equilibrium. Of course, the question of which initial conditions are natural for Cosmology has long and subtle history, but this certainly sounds plausibly generic. The idea is then to concentrate on “winding” strings wrapping those small circular directions (imagine a rubber band wrapping the small direction of the garden hose), which would be initially present. The claim is that those winding strings provide a tension that prevent the corresponding direction from expanding. As long as they are there, the direction stays small. Conversely, to allow for a particular direction to expand, reaching the macroscopic (cosmological, in fact) size we observe today, we need to somehow get rid of the strings wrapping it.
How do we get rid of those strings? this requires some interaction between them. If two winding strings meet each other, they can re-combine ends to create two non-winding strings. The strings that do not wind can then shrink and disappear. The upshot is that the basic interaction between two closed strings can make both of them disappear, allowing then some spatial dimensions to grow. The issue is now transformed to figuring out how many dimensions we can get to expand this way, by eliminating the strings wrapping them.
And here is the punchline: for two strings to have an appreciable change of interacting, they need to intersect each other. The chance of two strings intersecting each other depends very strongly on the number of dimensions. In one spatial dimension any two wrapped strings actually overlap with each other. In two spatial dimensions any two strings (wrapping independent directions) necessarily intersect each other, at each moment in time. Three spatial directions is the magic number: this allows for finite probability of intersection. Any larger number of directions leads to vanishing probability of string intersection. Then, the story goes, the winding strings would not have a chance of interacting with each other. They just stick around, preventing those extra dimensions from growing. The expansion stops at three spatial dimensions.
Isn’t this beautiful? if you are a theorist, the next step is to go and check all kinds of details. Do the winding strings really prevent expansion? is the probability for interaction really finite in three spatial dimensions (after all, the fundamental strings are both relativistic and quantum mechanical, maybe those classical pictures miss the point). How about experimental signature? for example statistics of fluctuations of the cosmic microwave background provides lots of constraints for any model of early universe cosmology. The upshot is, beautiful as it is, so far no cigar (to use the expression favored by my former graduate adviser). Some details don’t seem to work at the moment, which is the fate of the overwhelming majority of beautiful ideas. No other way to go but to keep trying…