  We study the anisotropies of gravitational waves induced by weak
hypermagnetic fields which are randomly distributed and oriented during the
electroweak phase transition in the early universe. The theory setup of this
study is the standard model plus a real singlet scalar field, which can produce
the needed strongly first order electroweak phase transition. Then we
investigate how the hypermagnetic fields can convert to magnetic fields and we
compute the departure of energy difference between the symmetric phase and the
broken phase when the magnetic fields are turned on. It is found that the
presence of the hypermagnetic fields can increase the Euclidean action, thus
can decrease nucleation temperature, which can lead to a supercool plasma. We
point out that the hypermagnetic field can enhance the gravitational wave
production from a first order electroweak phase transition and the
inhomogeneity of primordial hypermagnetic field can lead to anisotropies of
gravitational waves. By examining three well-motivated distribution of
hypermagnetic fields, we calculate the corresponding angular power spectra of
stochastic gravitational wave background and find they can be significantly
larger than the contributions of the Sachs-Wolfe effects and integrated
Sachs-Wolfe effects. Our results show that the anisotropies of gravitational
wave could provide a novel probe to the primordial hypermagnetic field in the
electroweak phase transition epoch.
