  A striking signal of dark matter beyond the standard model is the existence
of cores in the centre of galaxy clusters. Recent simulations predict that a
Brightest Cluster Galaxy (BCG) inside a cored galaxy cluster will exhibit
residual wobbling due to previous major mergers, long after the relaxation of
the overall cluster. This phenomenon is absent with standard cold dark matter
where a cuspy density profile keeps a BCG tightly bound at the centre. We test
this hypothesis using cosmological simulations and deep observations of 10
galaxy clusters acting as strong gravitational lenses. Modelling the BCG wobble
as a simple harmonic oscillator, we measure the wobble amplitude, $A_w$, in the
BAHAMAS suite of cosmological hydrodynamical simulations, finding an upper
limit for the CDM paradigm of $A_w < 2$kpc at the 95% confidence limit. We
carry out the same test on the data finding a non-zero amplitude of
$A_w=11.82^{+7.3}_{-3.0}$kpc, with the observations dis-favouring $A_w = 0$ at
the 3$\sigma$ confidence level. This detection of BCG wobbling is evidence for
a dark matter core at the heart of galaxy clusters. It also shows that strong
lensing models of clusters cannot assume that the BCG is exactly coincident
with the large scale halo. While our small sample of galaxy clusters already
indicates a non-zero $A_w$, with larger surveys, e.g. Euclid, we will be able
to not only to confirm the effect but also to use it to determine whether or
not the wobbling finds its origin in new fundamental physics or astrophysical
process.
