The metric expansion can for example be described by the <PERSON>-Walker metric given by
$$ \mbox ds^2 = -\mbox dt^2 + a^2(t)\mbox d\Sigma^2 $$
where $a$ is the scaling factor and $\mbox d\Sigma^2$ is spatial part of the metric. If the scaling factor obeys the equation
$$ \dot{a} = a H $$
where dot means time derivative and H is the <PERSON> constant, the universe not just linearily expanding but the expansion is accelerated (or exponantial) which is called inflation.
From the observations we have up to now (?) it is known that the universe is rather flat, homogeneous, and isotropic. The isotropic means that the universe looks the same in all direction, which means in addition that all spacial directions have to be treated equally.
In string theory, there are small extra dimensions and it is therefore natural in string cosmology to correspondingly relax the condition of isotropy and treat the (compactified) extra dimensions different from the large macroscopic dimensions.
One possibility to implement this anisotropy due to not all spatial dimensions behaving the same is by assuming that the universe can be described by the <PERSON> metric
$$ \mbox ds^2 = -\mbox dt^2 + \sum_{j=1}^D t^{2p_j}(\mbox dx^j) $$
By this metric, it is possible to not only describe the cosmological expansion of some spatial directions (the ones with positive <PERSON> exponents $p_j$, but this metric allows for some dimensions to contract too, those have negative $p_j$. Carrying over to what is asked in the question, the expanding dimensions could for example correspond to the dimensions with positive <PERSON> exponents and the compactified extra dimensions could be described by a negative <PERSON> exponent. The <PERSON> exponents must satisfy the <PERSON> conditions
$$ \sum_{j=1}^{D-1} p_j = 1 $$
and
$$ \sum_{j=1}^{D-1} (p_j)^2 = 1 $$
which can be shown to boil down to the fact that not all exponents can have the same sign and therefore there have to be contracting and expanding directions present at the same time.
Also in string theory, the extent of the size of the (extra) dimensions is described by dilaton fields ;-) $\phi$ and they are related to the <PERSON> exponents in Models making use of the <PERSON> metric for example by
$$ \phi = -\left(1- \sum\limits_{j=1}^D p_j \right)\ln t $$
This introduces some kind of duality between expanding and contracting dimensions, such that something like a cyclic pre big bang cosmology can be described in this model in the following way:
1. The universe starts from a large, flat, cold state
2. It contracts until it reaches the small self dual radius where it is higly curved and hot (this is the big bang)
3. 764
Due to the mentioned duality the universe is in this state
equivalent to an expanding universe, which corresponds to the
universe we live in.
In such cosmological models, the contracting dimensions can cause the expanding dimensions to accelerate, but I did not fully understand this inflation mechanism. I am not sure if maybe the potential energy needed to keep the contracting dimensions small could take to role of an appropriate inflaton potential or something along these lines.
Other string cosmilogy models make use of brane worlds which are based on <PERSON>-Sundrum models
which describe our universe as a 3-brane connected by an extra dimension to an analog 3-brane called the hidden sector. In their "stationary form" where the branes dont move, these models give among other things an approach to solve the hieararchy problem in particular when warped dimensions are allowed for as for example nicely explained here. In cosmological applications, the branes are allowd to move along the extra dimension, such that cyclic ekpyrotic models are obtained where the universe always existed and the big bang is not the origin of time:
1. First, the two branes are located at the boundaries of the extra dimension (s). This state corresponds to an empty cold universe
2. The branes are attracted towards each other and collide in a big bang, matter and radiation are created from the collision energy
3. The branes move away from each other and the universe cools down
This cycle of events can take place repeatedly. 764