Multistep phase transition and gravitational wave from general <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi mathvariant="double-struck">Z</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math> scalar extension

Multistep phase transition provides a paradigm in which a broken symmetry during phase transition can be restored, enriching the phenomena of both dark matter and baryon asymmetry. We study the dynamics of the multistep phase transition in the standard model extension with additional isospin <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>N</a:mi></a:mrow></a:math>-plet scalar field <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi mathvariant="normal">Φ</c:mi><c:mn>2</c:mn></c:msub></c:math> under a discrete <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:msub><f:mi mathvariant="double-struck">Z</f:mi><f:mn>2</f:mn></f:msub></f:math> symmetry. We find that the multistep phase transition could be triggered if there is a moderately large coupling between the Higgs boson and the <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msub><i:mi mathvariant="normal">Φ</i:mi><i:mn>2</i:mn></i:msub></i:math> and this coupling is required to be larger as the mass of the <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:msub><l:mi mathvariant="normal">Φ</l:mi><l:mn>2</l:mn></l:msub></l:math> and/or isospin increases. The first-order phase transition at the first (second) step can be realized by the thermal loop (tree-level barrier) effects. Thus, it is more likely that a detectable spectrum of gravitational waves can be produced at the second step of the phase transition.