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We propose that the recent observations reported by the different PulsarTiming Array (PTA) collaborations (i.e.~IPTA, EPTA, PPTA, and NANOGrav) of acommon process over several pulsars could correspond to a stochasticgravitational wave background (SGWB) produced by turbulent sources in the earlyuniverse, in particular due to the magnetohydrodynamic (MHD) turbulence inducedby primordial magnetic fields. I discuss recent results of numericalsimulations of MHD turbulence and present an analytical template of the SGWBvalidated by the simulations. We use this template to constrain the magneticfield parameters using the results reported by the PTA collaborations. Finally,we compare the constraints on the primordial magnetic fields obtained from PTAwith those from blazar signals observed by Fermi Large Area Telescope (LAT),from ultra high-energy cosmic rays, and from the cosmic microwave background.We show that a non-helical primordial magnetic field produced at the scale ofthe quantum chromodynamics phase transition is compatible with such constraintsand it could additionally provide with a magnetic field at recombination thatwould help to alleviate the Hubble tension.

Gravitational waves from MHD turbulence at the QCD phase transition as a source for Pulsar Timing Arrays