  We propose a new interacting dark sector model, Stepped Partially Acoustic
Dark Matter (SPartAcous), that can simultaneously address the two most
important tensions in current cosmological data, the $H_0$ and $S_8$ problems.
As in the Partially Acoustic Dark Matter (PAcDM) scenario, this model features
a subcomponent of dark matter that interacts with dark radiation at high
temperatures, suppressing the growth of structure at small scales and thereby
addressing the $S_8$ problem. However, in the SPartAcous model, the dark
radiation includes a component with a light mass that becomes non-relativistic
close to the time of matter-radiation equality. As this light component
annihilates away, the remaining dark radiation heats up and its interactions
with dark matter decouple. The heating up of the dark sector results in a
step-like increase in the relative energy density in dark radiation,
significantly reducing the $H_0$ tension, while the decoupling of dark matter
and dark radiation ensures that the power spectrum at larger scales is
identical to $\Lambda$CDM.
