Anisotropic ’hairs’ in string cosmology

In most models, inflation is driven by the potential energy of a scalar field. This has similar effects as adding vacuum energy or, equivalently, a positive cosmological constant. For these models Wald [1] proved a cosmic no hair theorem, which implies that an initial shear is inflated away [2]. A couple of years ago, a new mechanism of inflation has been proposed, where inflation is due to the kinetic term of the dilaton which is always present in the low energy effective action from a string theory. These string cosmologies are symmetric under the duality transformation, t → −t and a → 1/a. Here t is cosmic time and a is the scale factor. An expanding solution at negative times is inflating [3]. Here, we want to study whether a ’no hair’ theorem is also valid for these pre-big-bang inflationary solutions. We investigate especially spatially homogeneous, but anisotropic cosmologies in the pre-big-bang scenario. The aim is to determine the evolution of a possible primordial shear. The pre-big-bang universe starts off at t → −∞ in a nearly Minkowski spacetime. But the hypothesis of low curvature and low coupling in the pre-bigbang scenario does not say anything about the possibility of starting with an anisotropic spacetime. Recently, the initial conditions of the pre-big-bang scenario have been criticized not to be natural [4]. Buonanno et al. [5] have addressed this issue and have shown that the pre-big-bang inflationary phase in the string frame is equivalent to gravitational collapse in the Einstein frame. They therefore have concluded that the initial conditions for pre-big-bang inflation are as natural as those for gravitational collapse. However, the pre-bigbang bubble picture suggested in this work [5] seems to favor anisotropic Kasner solutions. It is thus important to investigate whether such initial anisotropies are inflated away during the subsequent inflationary evolution. Below we show that this is not the case. We find that the behaviour of shear in pre-big-bang cosmology is quite different from its behavior in ordinary potential inflation and primordial shear is not inflated away. We first briefly recall the expressions for the Ricci tensor of spatially homogeneous models in the orthonormal frame. (Latin indices run from 0..3 and Greek indices from 1..3.) The metric (gab) of spatially homogeneous models can be written as