Primordial magnetic field as a common solution of nanohertz gravitational waves and the Hubble tension

The origin of interstellar and intergalactic magnetic fields remains largely unknown. One possibility is that they are related to the primordial magnetic fields (PMFs) produced by, for instance, the phase transitions of the early Universe. In this paper, we show that the PMF-induced turbulence generated at around the QCD phase transition epoch--the characteristic magnetic field strength $B_{\rm ch}^* \sim \mathcal{O}(1)~\rm{\mu G}$ and coherent length scale $\ell_{\rm ch}^* \sim \mathcal{O}(1)~\rm{pc}$--can naturally accommodate nanohertz gravitational waves reported by pulsar timing array (PTA) collaborations. Moreover, the evolution of the PMFs to the recombination era with the form of $B_{\rm ch}\sim \ell_{\rm ch}^{-\alpha}$ can induce baryon density inhomogeneities, alter the recombination history, and alleviate the tension of the Hubble parameter $H_0$ and the matter clumpiness parameter $S_8$ between early- and late-time measurements for $0.88\leq \alpha \leq 1.17$ (approximate 95\% credible region based on three PTA likelihoods). The further evolved PMFs may account for the $\sim {\cal O}(10^{-16})$ Gauss extragalactic magnetic field inferred with GRB 221009A.