Decaying dark matter and the tension in $\sigma_8$
  We consider decaying dark matter (DDM) as a resolution to the possible
tension between cosmic microwave background (CMB) and weak lensing (WL) based
determinations of the amplitude of matter fluctuations, $\sigma_8$. We perform
N-body simulations in a model where dark matter decays into dark radiation and
develop an accurate fitting formula for the non-linear matter power spectrum,
which enables us to test the DDM model by the combined measurements of CMB, WL
and the baryon acoustic oscillation (BAO). We employ a Markov chain Monte Carlo
analysis to examine the overlap of posterior distributions of the cosmological
parameters, comparing CMB alone with WL+BAO. We find an overlap that is
significantly larger in the DDM model than in the standard CDM model. This may
be hinting at DDM, although current data is not constraining enough to
unambiguously favour a non-zero dark matter decay rate $\Gamma$. From the
combined CMB+WL data, we obtain a lower bound $\Gamma^{-1}\ge 97$ Gyr at 95 %
C.L, which is less tight than the constraint from CMB alone.
