Anisotropy of phase transition gravitational wave and its implication for primordial seeds of the Universe

We quantitatively study how the primordial density fluctuations are imprinted on the anisotropy of the phase transition gravitational wave (PTGW). Generated long before recombination and free from Silk damping, the anisotropic PTGW might reveal the density perturbation seeded from inflation or alternatives. We find new behaviors of the PTGW anisotropy power spectrum. The PTGW anisotropy is stronger than the anisotropy of the cosmic microwave background temperature at all scales, and the high-$\ell$ multiples are enhanced about 1 order due to the early integrated Sachs-Wolfe effect. Furthermore, differences in primordial power spectra at small scales manifest themselves more significantly on the angular power spectrum of PTGW anisotropy compared to that of the cosmic microwave background. These properties might provide a novel clue to understanding the primordial density perturbation of our early Universe and thereby complete our understanding of inflation theory. Taking nanohertz PTGW from dark matter models as a typical example, we obtain amplitudes of PTGW anisotropy which are about 4 or 3 orders weaker than the isotropic PTGW energy spectra.