Limits on primordial magnetic fields from direct detection experiments of gravitational wave background

Primordial magnetic fields (PMFs) can source gravitational wave background (GWB). In this paper, we investigate the possible constraints on small-scale PMF considering the ongoing and forthcoming direct detection observations of GWB. In contrast to the conventional cosmological probes, e.g., cosmic microwave background anisotropies, which are useful to investigate large-scale PMFs ($>1 {\rm Mpc}$), the direct detection experiments of GWB can explore small-scale PMFs whose scales correspond to the observed frequencies of GWB. We show that future ground-based or space-based interferometric gravitational wave detectors give a strong constraint of about $10^{2} {\rm nG}$ on much smaller scales of about $k\approx 10^{12} {\rm Mpc}^{-1}$. We also demonstrate that pulsar timing arrays have a potential to strongly constrain PMFs. The current limits on GWB from pulsar timing arrays can put the tight constraint on the amplitude of the PMFs of about $30 {\rm nG}$ whose coherent length is of about $k\approx 10^{6} {\rm Mpc}^{-1}$. The future experiments for the direct detection of GWB by the Square Kilometre Array could give much tighter constraints on the amplitude of PMFs about $5 {\rm nG}$ on $k\approx 10^{6} {\rm Mpc}^{-1}$, on which scales, it is difficult to reach by using the cosmological observations.