  We consider scenarios where the inflaton field decays dominantly to a hidden
dark matter (DM) sector. By studying the typical behavior of the Standard Model
(SM) Higgs field during inflation, we derive a relation between the primordial
tensor-to-scalar ratio $r$ and amplitude of the residual DM isocurvature
perturbations $\beta$ which is typically generated if the DM is thermally
decoupled from the SM sector. We consider different expansion histories and
find that if the Universe was radiation- or matter-dominated after inflation, a
future discovery of primordial DM isocurvature will rule out all simple
scenarios of this type because generating observable $\beta$ from the Higgs is
not possible without violating the bounds on $r$. Seen another way, the Higgs
field is generically not a threat to models where both the inflaton and DM
reside in a decoupled sector. However, this is not necessarily the case for an
early kination-dominated epoch, as then the Higgs can source sizeable $\beta$.
We also discuss why the Higgs cannot source the observed curvature perturbation
at large scales in any of the above cases but how the field can still be the
dominant source of curvature perturbations at small scales.
