Scale Factor Duality and the Energy Condition Inequalities

We demonstrate, by a simple analysis, that cosmological line elements related by scale factor duality also exhibit a duality with respect to the conservation/violation of the Weak Energy Condition (WEC) by the matter that acts as the source in the one-loop β function equations for the metric coupling written explicitly in the form of the Einstein equations. Furthermore, a study of specific pairs of line elements (obtained via O(d,d) transformations) hints at a possible generalisation of the above duality w.r.t. WEC for the case of O(d, d) related spacetimes. Consequences and extensions thereof are also pointed out. Electronic Address : [email protected] 1 Scale factor duality as a symmetry of the string equations of motion in a curved background as well as low energy effective string actions was first discovered by Veneziano [1] in 1991. Shortly thereafter, it was extended to a more general symmetry of the effective action known as O(d, d) [2] where d denotes the number of coordinates of which the metric and matter fields are independent. Subsequent generalisations to background fields depending on more than one coordinate have been carried out in [4]. Such noncompact symmetries of string theory [5] have been exploited to a great extent to construct inequivalent string vacua and has yielded interesting background geometries representing black holes [6] and black p–branes [4], [7] as well as cosmologies [8], [9]. Stringy cosmologies (for reviews, see [8], [9] ) have certain characteristic features apart from the symmetry of scale factor duality. Firstly, one does not need to bring in an ad–hoc scalar field to get the necessary inflationary phase. The dilaton field which arises as one of the massless excitations of the string world sheet serves the purpose. Moreover, there exists the notion of a phase termed as pre–big–bang (t < 0) during which we have a rapidly inflating universe with a scale factor generically obeying a pole law ( a(t) = (−t) ; (β < 0 ). This new phase is expected to end at t = 0 where a FRW evolution (a(t) = t ; 0 < β < 1) takes over and we finally end up with our universe today. Unfortunately, the transition from the pre–big–bang epoch to the FRW phase is plagued by the presence of a singularity, which, if absent would have solved the problem of singularities in GR, at least, in a cosmological setting. The generic presence of a singularity in the transition epoch which spoils a smooth crossover has been termed as the graceful exit problem in string cosmology [10]. Recently, Rey [11] has claimed that by introducing quantum back reaction it is possible to avoid the graceful exit problem at least within the limits of a two dimensional model like that of CGHS [12]. Whether an extension of this to four dimensions is possible or not is still an open question. On the other hand, a quantum cosmology approach to graceful exit has been advocated in [13]. The low energy effective theory that emerges out of string theory is much like Einstein gravity. The equations are derivable from an action which resembles that of a Brans–Dicke 2 theory with the ω parameter set to −1. Specifically, one has : Seff = ∫