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.