The Tachyon in a Linear Expanding Universe

We investigate the tachyon coupling in a static Robertson–Walker like metric background. For a tachyon and dilaton field which are only time dependent one can rewrite this model as a SU(2) Wess–Zumino–Witten model and a scalar Feigin– Fuchs theory. In this case the restriction to a real exponential tachyon field fixes the level k of the Wess–Zumino–Witten model. For a spatially dependent tachyon the world radius and the dilaton are quantized in terms of k and the tachyon by two integers, i.e. one has a discrete set of fields. The spatial part of the tachyon is given by Chebyshev polynomials of the second kind. An investigation of the tachyon mass shows that the tachyon is massless for k = 1. Strings in cosmological background were discussed a lot in the last years. Mainly, there are two different approaches: 1) in terms of (gauged) Wess–Zumino–Witten (WZW) models yielding an exact 2d conformal field theory [1]; 2) via the σ model (or effective action) approach in which one gets results in the α′ expansion [2, 3]. In this paper we want to make some general remarks about the description of strings in a Robertson Walker universe especially for ǫ = +1 (for ǫ = 0 a general solution was found by Mueller [4]). The crucial point in this description is the fact that a Robertson–Walker (RW) space time is conformally flat and therefore it is always possible to find a coordinate system in which: Gμν = W (r) r2 ημν , where the scale factor W (r) describes the original time dependence of the RW metric. One possibility to handle models like this is given by the world sheet σ–model in which one looks for special solutions for the vanishing of the Weyl anomaly. After some general remarks about the σ–model approach we discuss the description of strings in RW space time. As one example we investigate a linear expanding universe in four space time dimensions. As it is known this model corresponds to a combined SU(2) WZW and Feigin–Fuchs theory [5, 6]. Our special interest in this model is to find out what the tachyon field looks like which reproduces in the flat limit the known results from the David–Distler–Kawai (DDK) model [7] (in general supplemented by spatial background charges and by the interpretation of the time as Liouville field [8]). Similar to this model the demand for a real exponential tachyon yields a restriction: if the tachyon is only time dependent the level of the WZW theory must be equal to one. Finally we discuss a spatial dependence and the mass of the tachyon field and find a discrete set of fields where the spatial part is given by the Chebyshev polynomials of the second kind. ∗e-mail: [email protected]