  We consider a recently proposed model in which dark matter interacts with a
thermal background of dark radiation. Dark radiation consists of relativistic
degrees of freedom which allow larger values of the expansion rate of the
universe today to be consistent with CMB data ($H_0$-problem). Scattering
between dark matter and radiation suppresses the matter power spectrum at small
scales and can explain the apparent discrepancies between $\Lambda$CDM
predictions of the matter power spectrum and direct measurements of Large Scale
Structure LSS ($\sigma_8$-problem). We go beyond previous work in two ways: 1.
we enlarge the parameter space of our previous model and allow for an arbitrary
fraction of the dark matter to be interacting and 2. we update the data sets
used in our fits, most importantly we include LSS data with full $k$-dependence
to explore the sensitivity of current data to the shape of the matter power
spectrum. We find that LSS data prefer models with overall suppressed matter
clustering due to dark matter - dark radiation interactions over $\Lambda$CDM
at 3-4 $\sigma$. However recent weak lensing measurements of the power spectrum
are not yet precise enough to clearly distinguish two limits of the model with
different predicted shapes for the linear matter power spectrum. In two
Appendices we give a derivation of the coupled dark matter and dark radiation
perturbation equations from the Boltzmann equation in order to clarify a
confusion in the recent literature, and we derive analytic approximations to
the solutions of the perturbation equations in the two physically interesting
limits of all dark matter weakly interacting or a small fraction of dark matter
strongly interacting.
