be a distinguishing feature of this setup [141, 958]. To obtain this result, several criteria for the background cosmology need to be satisfied. First of all, the background equations must indeed admit a quasi-static (loitering) solution. Next, our three large spatial dimensions must be compact. It is under this condition that [959, 960] the heat capacity C$_{V}$ as a function of radius r scales as r $^{2}$. Thirdly, thermal equilibrium must be present over a scale larger than 1mm during the stage of the early uni- verse when the fluctuations are generated. Since the scale of thermal equilibrium is bounded from above by the Hubble radius, it follows that in order to have thermal equilibrium on the required scale, the background cosmology should have a quasi-static phase. Finally, the dilaton velocity needs to be negligible during the time interval when fluctuations are generated [142, 965] $^{113}$. In string theory, higher-derivative corrections to the Einstein-Hilbert action appear al- ready classically (i.e at the tree level), but we do not preclude theories where such corrections (or strings themselves) appear at the loop level or even non perturbatively. From string field theory [967–973] (either light-cone or covariant) the form of the higher-derivative modi- fications can be seen to be Gaussian, i.e. there are e ✷ factors appearing in all vertices (i.e. ( e $^{✷}$φ ) $^{3}$). These modifications can be moved to kinetic terms by field redefinitions ( φ → e − $^{✷}$φ ). The non perturbative gravity actions that we consider here will be inspired by such stringy kinetic terms [974] $^{114}$. It was realized that if we wish to have both a ghost free and an asymptotically free theory of gravity $^{115}$, one has little choice but to look into gravity actions that are non-polynomial 3. Example of a non-singular bouncing cosmology 113 It is not easy to satisfy all of the conditions required for the mechanism proposed in [141, 958] to work. In the context of a dilaton gravity background, the dynamics of the dilaton is important. If the dilaton has not obtained a large mass and a fixed VEV at a high scale, then it will be rolling towards weak coupling at early times. This will lead to [943, 966] a phase in which the string frame metric is static, and thus the string frame Hubble radius will tend to infinity, i.e. H ≈ 0. 114 There are various discussions on singular bouncing cosmology in 4 d in the context of ‘ek-pyrotic’ and cyclic universe [139, 140, 975, 976]. These models are interesting in their own right. However, constructing and stabilizing a background with a singular bouncing cosmology is a non-trivial task, see for some related discussions [977–982]. 115 While perturbative unitarity requires the theory to be ghost free, in order to be able to address the singularity problem in General Relativity, it may be desirable to make gravity weak at short distances, 238