Thermally Generated Gauge Singlet Scalars as Self-Interacting Dark Matter

We show that a gauge singlet scalar $S$, with a coupling to the Higgs doublet of the form ${\lambda }_S{S}^{†}{\mathrm{SH}}^{†}H$ and with the $S$ mass entirely generated by the Higgs expectation value, has a thermally generated relic density ${\Omega }_S\approx 0.3$ if $m_S\approx (2.9–10.5)({\Omega }_S/0.3\mathrm{}{)}^{1/5}(h/0.7\mathrm{}{)}^{2/5}\mathrm{MeV}$. Remarkably, this is very similar to the range $[m_S=(6.6–15.4\left){\eta }^{2/3}\mathrm{MeV}\right]$ required in order for the self-interaction $(\eta /4\mathrm{})(S^{†}S{)}^2$ to account for self-interacting dark matter when $\eta$ is not much smaller than 1. The corresponding coupling is ${\lambda }_S\approx (2.7\times {10}^{-10\mathrm{}}–3.6\times {10}^{-9\mathrm{}})({\Omega }_S/0.3\mathrm{}{)}^{2/5}(h/0.7\mathrm{}{)}^{4/5}$, implying that such scalars are very weakly coupled to the standard model sector.