A new non-perturbative time-dependent string configuration

A time-dependent bosonic string configuration is discussed, in graviton and dilaton backgrounds, leading to Weyl-symmetry beta-functions which are homogeneous in X, to any order in α ′ . As a consequence, a string reparametrization can always be implemented, such that beta functions can be cancelled, to any order in α ′ . This non-perturbative conformal invariance is valid for any target space dimension, and leads to a power law expanding Universe, for which the power vanishes if a specific relation between the dimension and dilaton amplitude holds. Finally, D = 4 is the minimum dimension (in the case of a spherical world sheet) for which this configuration is consistent with a Wick rotation in a Minkowski target space. The usual approach in String Cosmology consists in cancelling one-loop (or two-loop) beta functions corresponding to Weyl invariance, and solving the corresponding differential equations in order to find the relevant backgrounds. An alternative approach is discussed here [1], where, instead of cancelling perturbative orders of the beta functions, we look for a configuration of the bosonic string leading to beta functions which are homogeneous functions of X, to all orders in α ′ . This property will be enough to recover Weyl invariance, as the two-loop and higher orders in α ′ of the beta functions are not fixed in a unique way, but depend on the renormalization scheme which is used when computing the trace of the energy-momentum tensor of the string [2]. Changing renormalization scheme corresponds to reparametrizing the string couplings, which in our case will not affect the time-dependence of the configuration, but will only rescale the metric by a constant and add another constant to the dilaton. In order to find a hint on what configuration can lead to homogeneous beta functions, we first present an exact functional method, dealing with the sigma model provided by the string action, where the evolution of the quantum theory with the amplitude of α ′ is derived. We will see that, if the target space metric is conformally flat and proportional to the second derivative of the dilaton (in the string frame), the latter is independent of the amplitude of quantum fluctuations. Homogeneous beta functions will then be obtained for a specific case, where the graviton is a power law of X. We consider a bosonic string on a spherical world sheet, with bare action