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The evidence of the nano-Hertz stochastic gravitational wave (GW) backgroundis reported by multiple pulsar timing array collaborations. While a prominentcandidate of the origin is astrophysical from supermassive black hole binaries,alternative models involving GWs induced by primordial curvature perturbationscan explain the inferred GW spectrum. Serendipitously, the nano-Hertz rangecoincides with the Hubble scale during the cosmological quantum chromodynamics(QCD) phase transition. The influence of the QCD phase transition can modifythe spectrum of induced GWs within the nano-Hertz frequency range,necessitating careful analysis. We estimate GWs induced by power-law powerspectra of primordial curvature perturbations taking account of the QCD phasetransition. Then we translate the implication of the NANOGrav data into theconstraint on the power spectrum of the primordial curvature perturbation,which suggests one would underestimate the amplitude by about $25\%$ and thespectral index by up to $10\%$ if neglecting the QCD effect.

Translating nano-Hertz gravitational wave background into primordial perturbations taking account of the cosmological QCD phase transition