# How can contracting dimensions lead to cosmological inflation?

Using the Kasner metric, given by

$$ds^2 = -dt^2 + \sum_{j=1}^D t^{2p_j}(dx^j)$$

it is possible to not only describe the cosmological expansion of some space directions (the ones with positive Kasner exponents $p_j$, but this metric allows for some dimensions to contract too, those have negative $p_j$. The two Kasner conditions

$$\sum_{j=1}^{D-1} p_j = 1$$

and

$$\sum_{j=1}^{D-1} (p_j)^2 = 1$$

say that there have to be contracting and expanding dimensions at the same time, as the $p_j$ can not all have the same sign.

In a comment I have read, that in models with for example 3 expanding and $n>1$ contracting dimennsions, the contracting dimensions drive the inflation in the other directions by leading their expansion to accelerate without a cosmological constant. This is interesting and about this I'd like to learn some more.

So can somebody a bit more explicitely explain how such inflation models work? For example what exactly would the vacuum energy from a physics point of view be in this case? Up to now I only heard about inflation models where the vacuum energy density is the potential energy of some inflaton field(s) in a little bit more detail.

• Which dimensions have negative $p_j$? Knowing a little bit about the global topology might help with the vacuum energy question. – Chay Paterson Jun 12 '13 at 21:22
• @ChayPaterson there are for example 3 expanding spatial dimensions and $n>1$ contracting dimensions. – Dilaton Jun 12 '13 at 21:26
• Sure. Do we know if the contracting dimensions are compact or noncompact? – Chay Paterson Jun 12 '13 at 21:28
• @ChayPaterson about that there is unfortunately no information, I rather thought they should be compact ? – Dilaton Jun 12 '13 at 21:32
• Oh, ok. Well, my thinking was that the cosmological constant has a different meaning depending on if you are looking at just the 3+1 dimensional slice of the spacetime or the full D+1 dimensional spacetime: on the 3+1 dimensional slice, it might still be interpretable as the potential energy of a scalar field, but I think that will depend on whether or not you can separate out the dynamics of the compact dimensions. (Disclaimer: I'm not an expert) – Chay Paterson Jun 12 '13 at 21:45