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.
