Anisotropic gravitational waves induced by hypermagnetic fields during the electroweak phase transition epoch

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 the 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, and 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 the primordial hypermagnetic field can lead to anisotropies of gravitational waves. By examining three well-motivated distributions of hypermagnetic fields, we calculate the corresponding angular power spectra of stochastic gravitational wave backgrounds and find that 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 waves could provide a novel probe to the primordial hypermagnetic field in the electroweak phase transition epoch.