Higgs inflation and cosmological electroweak phase transition with N scalars in the post-Higgs era

We study inflation and cosmological electroweak phase transitions utilizing the Standard model augmented by $N$ scalars respecting a global $O(N)$ symmetry. We observe that the representation of the global symmetry is restricted by the inflationary observables and the condition of a strongly first order electroweak phase transition. Theoretical constraints including the stability, perturbativity and unitarity are used to bound the model parameter space. The Electroweak precision observables and Higgs precisions limit the representation of the symmetry. We evaluate the possibility to simultaneously address the inflation and the dark matter after considering the experimental constraints from the future leptonic colliders. When the $O (N)$ symmetry respected by the N-scalar is spontaneously broken to the $O (N-1)$ symmetry, both the one-step and two-step SFOEWPT can occur within the inflation viable parameter regions, which will be tested by the future CEPC, ILC and FCC-ee. The relation between the number of Goldstones and the SFOEWPT condition depends on phase transition patterns. The situation of Goldstone faking neutrinos and contributing to the dark radiation are investigated.