  We present a new perspective on the symmetries that govern the formation of
large-scale structures across the Universe, particularly focusing on the
transition from the seeds of galaxy clusters to the seeds of galaxies
themselves. We address two main features of cosmological fluid dynamics
pertaining to both the linear and non-linear regimes. The linear dynamics of
cosmological perturbations within the Hubble horizon is characterized by the
Jeans length, which separates stable configurations from unstable fluctuations
due to the gravitational effect on sufficiently large (and therefore, massive
enough) overdensities. On the other hand, the non-linear dynamics of the
cosmological fluid is associated with a turbulent behavior once the Reynolds
numbers reach a sufficiently high level. This turbulent regime leads to energy
dissipation across smaller and smaller scales, resulting in a fractal
distribution of eddies throughout physical space. The proposed scenario
suggests that the spatial scale of eddy formation is associated with the Jeans
length of various levels of fragmentation from an original large-scale
structure. By focusing on the fragmentation of galaxy cluster seeds versus
galaxy seeds, we arrived at a phenomenological law that links the ratio of the
two structure densities to the number of galaxies in each cluster and to the
Hausdorff number of the Universe matter distribution. Finally, we introduced a
primordial magnetic field and studied its influence on the Jeans length
dynamics. The resulting anisotropic behavior of the density contrast led us to
infer that the main features of the turbulence could be reduced to a 2D Euler
equation. Numerical simulations showed that the two lowest wavenumbers
contained the major energy contribution of the spectrum.
