Observations show that galaxies have magnetic fields with a component that is coherent over a large fraction of the galaxy with field strengths of order microGauss. These fields are supposed to be the result of amplification of an initial weak seed magnetic field of unknown nature. Understanding the origin and the evolution of these fields is one of the challenging questions of modern astrophysics. We have performed an interdisciplinary research focused on the following fundamental questions:  (1) What are the current and future observational constraints on large-scale correlated magnetic fields in the Universe? (2) How and when was the primordial seed magnetic field generated, and what does its strength need to be? (3) How does a seed field evolve during the evolution of the Universe, including phase transitions and the formation of cosmic structure? (4) To what extent can cosmological data, such as cosmic microwave background and large-scale structure measurements, test models of the magnetic field evolution?

Our collaboration (led by PI Tina Kahniashvili) has significantly improved the current understanding of the origin and evolution of a cosmic magnetic field and observational limits on, and predicted signatures of such a field. We have performed high-resolution numerical simulations of the hydrodynamical and magnetohydrodynamical (MHD) turbulence that helped to gain insight into the cosmic turbulence development.  Results of this analysis, in turn, can be applied to not only to cosmological scales, but to different astrophysical media (molecular clouds or even protoplanetary disks) or to laboratory plasma. The research project is not limited to theoretical astrophysics; it includes important parts related to high-energy physics, data analysis and interpretation, as well as numerical simulations of MHD processes and large-scale structure formation. The results of our research program will have important implications for many areas, including the early Universe physics, high-energy astrophysics, MHD modeling, and large-scale structure formation. Perhaps most importantly, it might convincingly establish that observations demand a cosmological magnetic field that cannot be generated by a mechanism operating within the confines of the Standard Model of particle physics.  Our major achievements are:

• We have studied the inflationary and cosmological phase transition magnetogenesis, and information we can get about the inflationary or phase transition physics through cosmological magnetic field measurements (or their upper bounds).
• After being generated the subsequent dynamics of the magnetic field are governed by decaying hydro-magnetic turbulence. We have studied the development of turbulence and have obtained several universal properties, including: decay laws, establishment of the Batchelor spectrum at large wavenumbers, and dissipation properties at small scales, Fig. 1-3. 
• We were first to recover the slow (compared to the case of the "causal" magnetic fields) rate of the inverse cascade (i.e. "absence" of the energy transfer from small to large scales) for the inflation generated helical magnetic field, Fig. 4.
• We were first to find an inverse transfer in non-helical MHD turbulence (originated from causal magnetic fields), with the spectral energy transfer rate about half as strong as with helicity. In both cases the magnetic field gains energy at large scales from velocity field interactions with smaller-scale magnetic fields, Fig. 5. 
• Primordial magnetic fields can be considered as seeds for observed magnetic fields in galaxies and clusters. The magnetic field strength bounds obtained in our analysis are consistent with the upper and lower limits of extragalactic magnetic fields, Fig. 6. 
• We have studied the gravitational wave signal from primordial turbu...

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