Implication of NEC on compact extra dimension scenarios

Yesterday Paul Steinhardt went to our department to give a talk on his recent paper Dark Energy, Inflation, and Extra Dimensions . Well this all-inclusive title probably don’t tell you much details.

He is a very good speaker. I could understand the ideas pretty well. This is not so typical for high energy seminar. This results of their paper is simply impressive and stunning.

Let me summarize the main results. Of course, I can’t guarantee that I interpret 100% correctly, and you are strongly recommended to read the original paper. In their paper, they assume pretty standard assumptions, and then derive that to get the accelerating expansion that we observe now in the 4D world, the null energy conditions (NEC) are violated. NEC in simple terms in cosmology means . Violation of NEC is generally associated with many nasty consequences like superluminal propagation, instabilities, or violation of unitarity.

It is important to note what assumptions are made. First, GR is assumed to be the theory of gravity in both the 4D and extra dimensional manifold. Second, the 4D world is spatially flat. The extra dimensions are bounded, so the proof applies only to compact extra dimension models. But a lot models in string theory and extra dimensional model fall into this category. The metric assumed is Ricci-flat or conformally Ricci-flat. These assumptions are either observational fact or typical assumptions made in various kind of models. I don’t think they are controversial.

He then do some A-averaging, which is essentially average over all the spacetime point with some weighting factor, of the the quantities like . The kind of A-averaging is new, but it seems to me natural. The equations they derive are essentially the conservation equation. Then it becomes inevitable that to get accelerating expansion in CDM cosmology, NEC has to be violated at at least one point. NEC is even much much more strongly violated during inflation.

If nobody could find out any flaws in their proof, this paper is expected to have big repercussions as compact extra dimensions essentially to string theory, and a lot of models are built on this idea. Nonetheless, Steinhardt does support the idea of extra dimensions, and he urge people do models much more carefully. This may require some new way of thinking.