bar \chi \chi \bar N N$ contact operator, but with an additional $1/E_r$ suppression in the cross section. This gives a similar phenomenology as a light mediator being exchanged at tree-level with derivative coupling.

Note that the relative importance of these two scattering processes is highly model dependent. For example, if $n_\phi = 2$ the dominant scalar-DM coupling could be $\bar q q \phi_1 \phi_2^*/\Lambda_{12}$. In that case, the $2\to2$ operator above is $\propto y_\chi^{\phi_1} y_\chi^{\phi_2}$ and can be suppressed without reducing the $2\to3$ rate by taking $y_\chi^{\phi_2} \gg y_\chi^{\phi_1}$. The scattering behavior of both the $2\rightarrow3$ and $2\rightarrow2$ regimes necessitates a re-interpretation of all DM direct detection bounds. We will do this below.

Indirect Constraints {#s.indirectconstraints}
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Direct detection experiments probe the ratio $y_\chi/\Lambda$ and $y_\chi^2/\Lambda$ for $2\to3$ and $2\to2$ scattering respectively. However, indirect constraints on dmDM from cosmology, stellar astrophysics and collider experiments are sensitive to the Yukawa coupling and $\Lambda$ separately. In [@dmDM] we conduct an extensive study of these bounds, including the first systematic exploration of constraints on the $\bar q q \phi \phi^*/\Lambda$ operator with light scalars $\phi$. Since these constraints (in particular, Eqns. \[e.NScoolingbound\] and \[e.thermalrelic\] below) provide important context for our results on direct detection, we summarize the two most important results here. For details we refer the reader to [@dmDM].

The scalar mediator(s) of dmDM are most stringently constrained from stellar astrophysics and cosmology:

-   Avoiding overclosure requires $m_\phi \lesssim \ev$ [@Kolb:1990vq], so we take the heaviest stable $\phi$ to be essentially massless, making it a very subdominant dark matter component. This also satisfies structure formation, computed for light sterile neutrinos in [@Wyman:2013lza]. Measurements by the Planck satellite [@Ade:2013zuv] re