Primeval Density Fluctuations

Another fundamental consequence of the quantum gravitational theory concerns the question of density fluctuations at the recombination epoch. In the standard approach to the problem of structure formation, no structure can be formed from a strictly constant matter density. The standard model of formation therefore assumes that the present structures originate from the gravitational growth of early fluctuations that existed at the recombination epoch. These structures have actually been observed through their temperature signature in the Cosmic Microwave Background Radiation field, though at a low level of δΤ/Τ ≈ 2 x 10^-5.

However, in the standard approach these fluctuations should themselves be understood in terms of still earlier fluctuations. But the strong isotropization that prevails during the prerecombination phase poses a difficult problem. The standard solution to this problem consists of making the postulate that the Universe would have known an inflation phase (i.e. exponential expansion) of unknown origin, which would have multiplied quantum fluctuations by an enormous factor.

We can suggest another solution to this problem in the scale-relativity framework, without any need of an inflationary phase. Indeed, as recalled in the Missing Mass Section, the fractality of space-time involves a transformation the geodesics equations in a Schrödinger-like equation. As a direct consequence, there is a tendency to form structures at any epoch: these structures are described by probability density distributions given by the square of the modulus of the probability amplitudes which are solutions of this gravitational Schrödinger equation. The classical approach, because of its deterministic and differentiable nature, predicts structures at a given epoch which are the result of an evolution from previous existing structures: these structures of the past are taken as initial conditions (in position and velocity). The new quantum-like approach is organized in a fundamentally different way. The loss of determinism of individual trajectories is compensated by a determinism of structures. At each epoch, stationary or steady-state solutions can be found in correspondence with the shape of the potential and the limiting and matching conditions. These structures do also evolve (as given by the time-dependent Poisson-Schrödinger system), in correspondence with the evolution of the environment.

Therefore one expects the occurence of quantum fluctuations (structures) at the decoupling epoch, but according to the macroscopic quantum theory based on
D = GM/2α_gc which applies at this epoch, instead of the microscopic quantum theory for which D = h/2m. No inflation is needed in the new framework to obtain a scale invariant, quantumlike, spectrum at z ≈ 1000.
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Quoted from: Nottale L., 2003, Chaos Solitons and Fractals, 16, 539
"Scale-relativistic cosmology"