First, let me explain why one would anyone dirty their hands to study X-ray spectra. In the most popular scenario the dark matter particle is a WIMP -- a particle in the GeV-TeV mass ballpark that has weak-strength interactions with the ordinary matter. This scenario may predict signals in gamma rays, high-energy anti-protons, electrons etc, and these are being searched high and low by several Earth-based and satellite experiments. But and in principle the mass of the dark matter particle could be anywhere between 10^-30 and 10^50 GeV, and there are many other models of dark matter on the market. One serious alternative to WIMPs is a keV-mass sterile neutrino. In general, neutrinos are dark matter: they are stable, electrically neutral, and are produced in the early universe. However we know that the 3 neutrinos from the Standard Model constitute only a small fraction of dark matter, as otherwise they would affect the large-scale structure of the universe in a way that is inconsistent with observations. The story is different if the 3 "active" neutrinos have partners from beyond the Standard Model that do not interact with W and Z bosons -- the so-called "sterile" neutrinos. In fact, the simplest UV-complete models that generate masses for the active neutrinos require introducing at least 2 sterile neutrinos, so there are good reasons to believe that these guys exist. A sterile neutrino is a good dark matter candidate if its mass is larger than ~keV (because of the constraints from the large-scale structure) and if its lifetime is longer than the age of the universe.
How can we see if this is the right model? Dark matter that has no interactions with the visible matter seems hopeless. Fortunately, sterile neutrino dark matter is expected to decay and produce a smoking-gun signal in the form of a monochromatic photon line. This is because, in order to be produced in the early universe, the sterile neutrino should mix slightly with the active ones. In that case, oscillations of the active neutrinos into sterile ones in the primordial plasma can populate the number density of sterile neutrinos, and by this mechanism it is possible to explain the observed relic density of dark matter. But the same mixing will make the sterile neutrino decay, as shown in the diagrams here. If the sterile neutrino is light enough and/or the mixing is small enough then its lifetime can be much longer than the age of the universe, and then it remains a viable dark matter candidate. The tree-level decay into 3 ordinary neutrinos is undetectable, but the 2-body loop decay into a photon and and a neutrino results in production of photons with the energy E=mDM/2. Such a monochromatic photon line can potentially be observed. In fact, in the simplest models sterile neutrino dark matter heavier than ~50 keV would produce a too large photon flux and is excluded. Thus the favored mass range for dark matter is between 1 and 50 keV. Then the photon line is predicted to fall into the X-ray domain that can be studied using X-ray satellites like XMM-Newton, Chandra, or Suzaku.
Until last week these searches were only providing lower limits on the lifetime of sterile neutrino dark matter. This paper claims they may have hit the jackpot. The paper use the XMM-Newton data to analyze the stacked X-ray spectra of many galaxy clusters where dark matter is lurking. After subtracting the background they see is this:
Although the natural reaction here is a resounding "are you kidding me", the claim is that the excess near 3.56 keV (red data points) over the background model is very significant, at 4-5 astrophysical sigma. It is difficult to assign this excess to any known emission lines from usual atomic transitions. If the excess is interpreted as a signal of new physics, one compelling (though not unique) explanation is in terms of sterile neutrino dark matter. In that case, the measured energy and intensity of the line correspond to the the neutrino mass 7.1 keV and the mixing angle of order 5*10^-5, see the red star in the plot. This is allowed by other constraints and, by twiddling with the lepton asymmetry in the neutrino sector, consistent with the observed dark matter relic density.
Clearly, a lot could possibly go wrong with this kind of analysis. For one thing, the suspected dark matter line doesn't stand alone in the spectrum. The background mentioned above consists not only of continuous X-ray emission but also of monochromatic lines from known atomic transitions. Indeed, the 2-10 keV range where the search was performed is pooped with emission lines: the authors fit 28 separate lines to the observed spectrum before finding the unexpected residue at 3.56 keV. The results depend on whether these other emission lines are modeled properly. Moreover, the known Ar XVII dielectronic recombination line happens to be nearby at 3.62 keV. The significance of the signal decreases when the flux from that line is allowed to be larger than predicted by models. So this analysis needs to be confirmed by other groups and by more data before we can safely get excited.
Decay diagrams borrowed from this review. For more up-to-date limits on sterile neutrino DM see this paper, or this plot. Update: another independent analysis of XMM-Newton data observes the anomalous 3.5 keV line in the Andromeda and the Perseus cluster.