Domain Wall Constraints on Two Higgs Doublet Models with $Z_2$ Symmetry

The Two Higgs Doublet Model (2HDM) with spontaneously broken $Z_2$ symmetry predicts a production of domain walls at the electroweak scale. We derive cosmological constraints on model parameters for both Type-I and Type-II 2HDMs from the requirement that domain walls do not dominate the Universe by the present day. For Type-I 2HDMs, we deduce the lower bound on the key parameter $\tan\beta > 10^5$ for a wide range of Higgs-boson masses $\sim$ 100 GeV or greater close to the Standard Model alignment limit. In addition, we perform numerical simulations of the 2HDM with an approximate as well as an exact $Z_2$ symmetry but biased initial conditions. In both cases, we find that domain wall networks are unstable and, hence, do not survive at late times. The domain walls experience an exponential suppression of scaling in these models which can help ameliorate the stringent constraints found in the case of an exact discrete symmetry. For a 2HDM with softly-broken $Z_2$ symmetry, we relate the size of this exponential suppression to the soft-breaking bilinear parameter $m_{12}$ allowing limits to be placed on this parameter of order $\mu$eV, such that domain wall domination can be avoided. In particular, for Type-II 2HDMs, we obtain a corresponding lower limit on the CP-odd phase $\theta$ generated by QCD instantons, $\theta \ \stackrel{>}{{}_\sim}\ 10^{-11}/(\sin\beta \cos\beta)$, which is in some tension with the upper limit of $\theta \ \stackrel{<}{{}_\sim}\ 10^{-11}$--$10^{-10}$, as derived from the non-observation of a non-zero neutron electric dipole moment. For a $Z_2$-symmetric 2HDM with biased initial conditions, we are able to relate the size of the exponential suppression to a biasing parameter $\varepsilon$ so as to avoid domain wall domination.