of those large dimensions. The initial paper, [942], has inspired a broad literature. One important generalization has been including branes in the gas of strings [945–948], and other space time topologies have also been considered [949, 950]. In particular, [951] showed that interesting cosmological dynamics happen when the expanding dimensions are still near the string scale. Importantly, several tests have been made of the BV mechanism for determining the number of macroscopic dimensionality of space, see [952–956]. These tests are all based on the fact that the winding strings will be unable to annihilate efficiently if their interaction rate, Γ, drops below the Hubble parameter for the expanding dimensions. Based on simple arguments from the low energy equations of motion and string thermodynamics it was demonstrated that the interaction rates of strings are negligible, so the common assumption of thermal equilibrium cannot be applicable [956, 957]. 2. Seed perturbations from a string gas Recently, a new structure formation scenario has been put forward in [141, 958] . It was shown that string thermodynamic fluctuations in a quasi-static primordial Hagedorn phase in 4 d , during which the temperature hovers near its limiting value, namely the Hagedorn temperature, T$_{H}$ [933], can lead to a scale-invariant spectrum of metric fluctuations. The crucial point of the mechanism is to note that provided three large spatial dimensions are compact, the heat capacity C$_{V}$ of a gas of strings in thermodynamical equilibrium scales are r 2 with the radius of the box [959–964], then the heat capacity determines the root mean square mass fluctuations, i.e. 〈 ( δM ) $^{2}$〉 = T $^{2}$C$_{V}$ . With the help of Poisson equation, ∇ $^{2}$Φ = 4 πGδρ , and the definition of power spectrum, P $_{Φ}$( k ) ≡ k $^{3}$| Φ( k ) | $^{2}$, one obtains [141, 958]: P $_{Φ}$( k ) = 16 π $^{2}$G2 $_{k}$− $^{1}$| δρ ( k ) | 2 = 16 π $^{2}$G$^{2}$k$^{2}$( δM ) $^{2}$( r ( k )) , ≈ 1920 π 2 $_{c}$− $^{1}$G$^{2}$T 4 $_{H}$( kr ) 2 T T$_{H}$ (1 − T/T$_{H}$ ) , (491) where c is the velocity of light, G is the Newton’s constant and T$_{H}$ is the Hagedorn tem- perature. The mean squared mass fluctuation | δM | $^{2}$( r ) in a sphere of radius r ( k ) = k − 1 is given by | δρ ( k ) | 2 = k $^{3}$| δM | $^{2}$( r ( k )). The tilt in the spectrum is scale invariant and the fine tuning in temperature has to be Δ T/T$_{H}$ ∼ 10 − 30 for M$_{S}$ ∼ 10 − $^{10}$M$_{P}$ . The tensor mode also requires similar level of fine tuning but with slight tendency towards a blue tilt, which could 237