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^{\dagger}S H^{\dagger}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)^{1/5}(h/0.7)^{2/5} MeV for Higgs boson masses in the range 115 GeV to 1 TeV. Remarkably, this is very similar to the range (m_{S} = (6.6-15.4)\eta^{2/3} MeV) required in order for the self-interaction (\eta/4)(S^{\dagger}S)^{2} to account for self-interacting dark matter when \eta is about 1. The corresponding coupling is lambda_{S} \approx (2.7 \times 10^{-10} - 3.6 \times 10^{-9})(Omega_{S}/0.3)^{2/5}(h/0.7)^{4/5}, implying that such scalars are very weakly coupled to the Standard Model sector. More generally, for the case where the S mass is at least partially due to a bare mass term, if m_{S} \approx 10 \eta^{2/3} MeV, corresponding to self-interacting dark matter, then in order not to overpopulate the Universe with thermally generated S scalars we require that lambda_{S} < 10^{-(9-10)}\eta^{-1/3}, making such scalars difficult to detect directly.