Scale factor duality and the energy condition inequalities

Abstract

We demonstrate, by a simple analysis, that cosmological line elements related by scale factor duality also exhibit a duality with respect to the conserva- 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 ; (β < 0 ). This new phase is expected to end at t = 0 where a FRW evolution (a(t) = t β ; 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 :This is the bosonic sector of the genus-zero, low energy action for closed superstrings in the limit when inverse string tension (or α ′ ) goes to zero. Here, H µνλ is the third-rank antisymmetric tensor field, φ the dilaton field and V contains contributions from the dilaton potential and the cosmological constant. The β function equations which are obtained by imposing quantum conformal invariance in the worldsheet sigma model can be derived from this action by performing appropriate variations.Obviously, there are two frames in which the metrics look very different-the string (Brans-Dicke) and the Einstein frame. These frames are related to each other by a conformal transformation. We shall exclusively work in the string frame.The Einstein-like equations can be written in the form G µν = e 2φ T µν where T µν = T φ µν + T M µν (M denotes matter fields other than the dilaton). These equations are exactly those for Brans-Dicke theory with the parameter ω = −1. One might argue that it is not proper to rewrite the β function equations in an explicit Einstein form because at the level of the worldsheet sigma model g µν , φ or B µν -all have the same status-they ...

Naked singularities in low-energy, effective string theory

“…In particular, a duality is clearly seen for the ANEC along radial null geodesics. These facts substantiate the claim made earlier for black holes [18] and cosmologies [19] that there is a duality in the conservation/violation of the energy conditions for stringy spacetime geometries.…”

“…This fact was noted for the Einstein-scalar system in the Einstein frame by Roberts and there do exist recent verifications of this in the context of scalar field collapse through the numerical work of Choptuik and others [16] which indicate the existence of point-mass black holes (essentially naked singularities!). Finally, we demonstrate a reflection of string dualities in the conservation/violation of the local and global null energy condition-a fact which was noticed earlier for black holes and cosmologies in earlier papers by this author [18], [19].…”

Solving the graceful exit problem in superstring cosmology

“…Further, it can be shown that the Weak Energy Condition (WEC) is violated in this solution (Bose & Kar 1997). In fact this agrees with the recent result of Brustein & Madden (1997) and Kar (1996).…”

Numerical study of inhomogeneous pre-big-bang inflationary cosmology

“…The peculiarity of the PBB inflation is that even classically the weak energy condition [5] is violated during that epoch when the equation are written in the form of the Einstein equations [6], while the strong energy condition is violated but the weak energy condition is respected classically in the case of the usual (potential energy dominated) inflation. This suggests that the behaviour of inhomogeneities in the PBB inflation is quite different from that in the usual inflation.…”