The original ideas of new theory – “Non-Inflationary Cosmology” (NIC) – and its consequences are manifested: 1. Global hierarchy of Bose-Einstein statistics’ (BES) at earliest Universe guarantees the formation of Bose-Einstein condensate in Matter Era (ME). 2. New phenomenon – Cosmological Small-Bang – has been disclosed as a consequence of further phase-transition process from BES to FDS in ME. 3. Time-evolution of Planck constant may be an alternative mechanism for the explanation of cosmological redshift. 4. Higgs boson’s cosmological mission with this hypothesis disclose fundamental new cosmic scales for ME, which already is perceives by modern physics. A broad scenario is revealing to unify into general model the mechanisms of generation of galaxies and their components (black holes, massive stars and supernovas). An original road map for theoretical investigations and astrophysical observations is suggested, aiming at explanation of morphological varieties of galaxies and their rotation’s profiles. The space-time-energy new fundamental measures in ME describe the evolution of Universe in ME, especially the hierarchic structure of galaxies’ groups and clusters, even illustrate the possible essence of giant voids. In suspense of “storage of frozen micro-pieces of BEC” within the galactic cosmic rays and relativistic jets, the experimental advantage of Higgs bosons assure a far-reaching program for the possible comparison of NIC results with experimental data of two major Russian projects: “OLIMPIA” and “Synthesis of Super Heavy Elements”.