peaked the density profile is in the centre and $\rm{r_s}$ is a specific radius depending on the extent of the cluster (here we define $\rm{r_s=r_{500}/C}$). $\rm{\rho_c}$ is the critical density of the universe at the cluster redshift. In the standard $\rm{\Lambda CDM}$ cosmology assumed in this study (see Sect. \[sec:introduction\]) the critical density is given as

$$\rm{\rho_c=\frac{3H^{2}(z)}{8\pi G}}$$

$$\rm{H^{2}(z)=H_{0}^{2}~(0.27~(1+z)^3+0.73)}$$

with gravitational constant $\rm{G\approx4.3\times10^{-3}~pc~M_{\odot}^{-1}~(km/s)^2}$.

Simulated spectra {#sec:simspec}
=================

We used simulated data to test our spectral stacking technique for detecting an additional line at 3.55keV. For the simulation we created fake spectra using `XSPEC`. Simulated spectra were obtained for the best fit model of every individual cluster with ten times the original exposure to get better statistics. We added a Gaussian line with the expected flux for each cluster assuming a mixing angle of $\rm{\sim7 \times 10^{-11}}$ and that all dark matter is made of the 7.1keV sterile neutrinos as in previous detections [see @2014ApJ...789...13B; @2014PhRvL.113y1301B]. Exactly the same merging and modelling procedure was performed on the simulations and the real data. Fig. \[fig:fakelinemerge\] shows the resulting stacked spectra of the simulation (compare to Fig. \[fig:linemerge\] for the real data). The additional line is recovered between 95 and 99.7 per cent 