Production of Sterile Neutrino Dark Matter and the 3.5 keV line

The recent observation of an X-ray line at an energy of 3.5 keV mainly from galaxy clusters has initiated a discussion about whether we may have seen a possible dark matter signal. If confirmed, this signal could stem from a decaying sterile neutrino of a mass of 7.1 keV. Such a particle could make up all the dark matter, but it is not clear how it was produced in the early Universe. In this letter we show that it is possible to discriminate between different production mechanisms with present-day astronomical data. The most stringent constraint comes from the Lyman-{\alpha} forest and seems to disfavor all but one of the main production mechanisms proposed in the literature, which is the production via decay of heavy scalar singlets. Pinning down the production mechanism will help to decide whether the X-ray signal indeed comprises an indirect detection of dark matter.


INTRODUCTION
A dream of all particle physicists, cosmologists, and astrophysicists is to discover the true nature of Dark Matter (DM), which makes up more than 80% of the matter in the Universe [1]. Motivated e.g. by supersymmetry, the generic candidate is a Weakly Interacting Massive Particle (WIMP), i.e. a particle which interacts as weakly as neutrinos and which has a mass larger than those of all the particles we know. However, the many recent attempts to directly detect such a particle [2] or to produce it at colliders [3], as well as the hunts for its annihilation products [4], have so far not found a clear indication. In this situation, the detection of an X-ray line in several galaxy clusters and in the Andromeda galaxy [5,7] has attracted the attention of the community. This line, if stemming from DM decay, would be a smoking gun signal for a very different type of DM particle: an extremely weakly interacting ("sterile") neutrino with a mass that is smaller than that of a WIMP by about seven orders of magnitude. Such a DM particle must be produced in a non-standard way in the early Universe and, putting all data together and taking the X-ray signal seriously, we can for the first time identify in this paper the most promising candidate out of all production mechanisms * amerle@mpp.mpg.de † aurel@physik.uzh.ch proposed. The case of sterile neutrino Dark Matter has become much stronger by the tentative signal, so that we must put all proposals for production mechanisms to the test to ultimately decide how good this strong competitor to WIMPs really is.
Standard Model (SM) particles or WIMPs are produced by thermal freeze-out [8]: at high temperatures, all particles have been in thermal equilibrium, but production and annihilation cease at some point and a final abundance of each stable species remains. Sterile neutrinos with keV-masses cannot be produced in this way because their interactions are too weak. However, even very feebly interacting particles can be gradually produced in the early Universe [9]. For sterile neutrinos this can be achieved by their tiny admixtures θ to active neutrinos, the so-called Dodelson-Widrow (DW) mechanism [10], but this is by now known to produce a too hot spectrum [11,12], i.e., too fast DM particles. However, active-sterile neutrino transitions could be resonantly enhanced if the background medium carries a net lepton number. This production proposed by Shi and Fuller [13] seems in a better shape when confronted with data [14], and it has been recently advocated to be able to produce DM in agreement with the line signal [15].
What is the status of the 3.5 keV line? Ref. [5] has reported evidences (> 3σ) in samples by XMM-Newton of the Perseus cluster (central region) [6], of nearby bright clusters, and of a set of 69 other clusters, leading to a combined significance above 4σ. This claim was independently supported by [7], where additional signals are reported in the outer regions of the Perseus cluster and in the center of the Andromeda galaxy. In contrast, Ref. [16] criticizes that, if the line was real, Chandra should also see a line signal from the center of the Milky way -which it does not. However the approach taken in [16] is in turn criticized in Refs. [17] and [18], respectively using the arguments that the galactic center is plagued by hard Xrays, which introduce uncertainties, and that the abundances of the chemical elements in the galaxy should not be taken as completely free parameters. Ref. [17] further explains that in general the outskirts of galaxy clusters should be looked at, which are cleaner, but no signal is found. This observation, however, is in fact consistent with [7] where no signal is found in the outer regions of Andromeda. The authors of [19] re-analyzed the data used in [7] and found no signal, however, Ref. [20] criticized the suitability of the background model assumed, and Ref. [21] further identifies a possible error in the use of the atomic data involved. Finally, the authors of Ref. [22] have found no signal in stacked XMM-Newton data from dwarf galaxies, which should provide a clean signal, and thus the claim made in Ref. [5] is disfavored by about 4.6σ. Nevertheless, the observation from Andromeda does not seem challenged. Obviously the situation is not clear at the moment and more data is required. On the other hand, the technical development of satellites proceeds slower than one would like, so that we cannot expect to see a very bright signal where nothing had been seen before. Ultimately, we should take the tentative 3.5 keV line as a motivation to scrutinize both the signal and its implications -the latter we will do here.
With the signal taken seriously, to find out whether DM decay causes it, further production mechanisms should be tested. If the sterile neutrino was charged beyond the SM gauge group [see 23, for a review of several such settings], freeze-out may be revived if a significant amount of entropy is produced to dilute the abundance [24], but this is strongly constrained by Big Bang Nucleosynthesis [25]. However, there is another production mechanism which is in a better shape, using a scalar that decays into sterile neutrinos: S → ν s ν s . This scalar could be the inflaton [26] or a general singlet S that is thermally produced in the early Universe by either freeze-out [27] or freeze-in [28]. Ultimately the production mechanism has an impact on the DM velocity profile and thus on structure formation.
In this paper we present the most important results of an extensive study to be available soon [29]. We show that, contrary to common believe, sterile neutrino production by scalar decays seems to be in better agreement with data than Shi-Fuller production, in particular when looking at the Lyman-α (Ly-α) bound. Knowing which mechanisms fit the data will be of uttermost importance when aiming at identifying whether DM decay could be behind the X-ray signal.

DARK MATTER PRODUCTION FROM DECAYS OF SCALAR SINGLETS
Just as the fermions in the SM obtain their masses by the so-called "Yukawa" couplings to the Higgs scalar field H, sterile neutrinos ν s could couple to a singlet scalar field S like y 2 Sν c s ν s + h.c. If S settles at its vacuum expectation value v S = S , this leads to a sterile neutrino mass m s = yv S similarly to the ordinary Higgs mechanism. However, the scalar field S is also allowed by all symmetries to couple to a SM Higgs field via a "portal" H † HS 2 . This portal coupling could produce sizable amounts of S particles (i.e., the physical components of S) which will decay with strength y into two sterile neutrinos. This mechanism can lead to efficient DM production.
Depending on how the scalar particle is produced in the first place, the sterile neutrinos may have some "memory" of the scalar production, i.e., their momentum distribution function will be influenced by the distribution of the parent scalar. However, if the scalar is very nonrelativistic at the time of the decay -which is typically a fair approximation since highly relativistic particles, just like atmospheric muons, do not decay very efficientlythen any memory of the original spectrum is lost. The only information that cannot be eliminated is that of the initial scalar abundance: since every scalar decay produces exactly two sterile neutrinos, the number of the latter inside a comoving volume will always be two times larger than that of the scalar particles.
The momentum distribution of a decay produced sterile neutrino with adjacent DW production is [30], where α DW = T DW /T ∼ 0.716 and α SD = T SD /T ∼ 0.156 (T being the photon-temperature). The normalization factors β SD and β DW depend on the details of the production mechanism and are fixed by the required DM abundance. This can be determined with the help of the empirical formula from Ref. [31], which gives an estimate for the fraction of DM produced non-resonantly via the active-sterile mixing θ. It becomes clear from Eq. (2) that the exact form of the momentum distribution depends effectively on two parameters, namely the mass m s of the sterile neutrino and the active-sterile mixing Θ ≡ sin 2 (2θ). Both parameters can be unambiguously determined from the energy spectrum and the flux of the observed X-ray line and are reported to be m s = (7.14 ± 0.07) keV and Θ = 6.8 +2.2 −1.7 · 10 −11 , respectively [5,32]. The resulting momentum distributions are plotted in Fig. 1.  [15]) and nonresonant production (green dotted). The black and the blue lines use parameters corresponding to the claimed signal [5]. The green dotted line assumes a larger mixing angle to allow for the right DM abundance. The red envelope around the black line corresponds to the 3σ C.L. from Ref. [5].
The black line corresponds to scalar decay production, cf. Eq. (1), while the thickness of the line illustrates mixing with different neutrino flavors. The surrounding red band designates the 3σ confidence level on the the flux measurement. The distribution exhibits two distinct maxima, one at very cold momenta coming from scalar decay production and a much smaller one at larger momenta associated with the subdominant active-sterile mixing. The blue dashed line in Fig. 1 shows the momentum distribution resulting from resonant production, calculated in [15]. It assumes a lepton number of L = 4.6 · 10 −4 and has a characteristic spike at low momenta due to the resonance in the active-sterile mixing. The green dotted line in Fig. 1 illustrates the standard non-resonant production based on the DW mechanism [10] as sole source of DM. The X-ray line measurement trivially excludes this mechanism, since non-resonantly produced sterile neutrinos would require a considerably larger mixing angle to make up for all of the DM in the Universe (and they would be too hot in any case). We nevertheless plot the non-resonant case as a reference, however, with a mixing angle Θ = 4 · 10 −10 required to obtain the correct DM abundance.

COSMOLOGICAL PERTURBATIONS
DM particles with a mass in the keV-range are usually categorized as warm DM (WDM) candidates because they generate an important amount of free streaming, which suppresses perturbations at dwarf galaxy scales. The free-streaming length (λ fs ) does however not only depend on the particle mass but also on the average particle momentum, i.e. λ fs ∼ q /m s . Since the average momentum of scalar-decay produced sterile neutrinos is comparatively small, the effect of free-streaming is expected to be reduced in comparison to production via active-sterile mixing. It is therefore important to properly calculate the free-streaming effect in order to see how strongly scalar decay sterile neutrinos suppress the collapse of dwarf galaxies and whether they act more like warm or cold DM [33].
We use the numerical Boltzmann solver CLASS [34] to compute the matter perturbations for the DM scenarios introduced above. The suppression of small structures with respect to pure CDM is shown in Fig. 2, where we plot the ratio of the transfer functions (i.e., the square-root of the linear power-spectrum T /T CDM = P/P CDM ). The black line with red envelope corresponds to the scalar decay momentum distribution, in agreement with the measured X-ray line and 3σ errors [5]. The blue dashed (green dotted) lines represent (non-)resonant production -the former is obtained from Ref. [15]. The gray shaded region illustrates the bound on the free-streaming from Ly-α data [35,36].
The transfer functions plotted in Fig. 2 all stem from sterile neutrinos with the exact same mass. The fact that they exhibit very different suppression scales illustrates the strong effect the momentum distribution, and thus the production mechanism, has on particle freestreaming. DM candidates in the keV mass range can either act as cold, warm, or hot DM, depending on their average momentum and their distribution. For this reason it is crucial to know the details of particle production to constrain sterile neutrino DM. Fig. 2 clearly illustrates the power of the Ly-α method to discriminates different DM scenarios. The nonresonant DW production mechanism of sterile neutrinos can be ruled out at high significance [37]. More surprising is the fact that the resonant production mechanism seems to be in tension with the Ly-α data, too, while the scalar decay production mechanism is perfectly consistent. However, the tension is at the 2.5σ level and hence not strong enough to exclude the resonant scenario as the driving production mechanism. Moreover, there could be sources of error in both data and theory that have to be clarified before drawing any final conclusions. Nevertheless it seems that, if the line signal was taken seriously, the most heavily advocated production mechanism for sterile neutrino DM would not be in a very good shape.
Potential systematics are the following: (i) Theory -The resonance of the active-sterile mixing is expected to be most efficient during the quark-hadron transition in the early Universe. This means that the momentum distribution of resonant production is sensitive to the exact evolution of the temperature in the plasma during that regime, which is not known in detail.
(ii) Lyman-α forest -The accuracy of the Ly-α analysis is somewhat controversial, because it requires a detailed modeling of the inter-galactic medium (IGM). The physics of the IGM may depend on hydrodynamical feedback effects which are poorly constrained and subject of intense current research [38]. (iii) X-ray observations -While the position of the X-ray line (and thus the mass) can be determined with high accuracy, the flux (and thus the mixing angle) is much more uncertain. However, recent constraints from Ref. [39] clearly disfavor mixing angles above Θ = 5 · 10 −11 . Taking these more stringent limits into account increases the tension between resonant production and Ly-α data even further (cf. Fig. 2 in Ref. [15]).
Despite these potential systematics, it is encouraging that present-day astronomical data can start to discriminate between different production mechanism of sterile neutrino DM. Future weak lensing surveys, such as EUCLID, are expected to provide robust constraints and yield an independent check of the Lyman-α results [40], which would form a solid basis to discriminate the known mechanisms.

HALO FORMATION
Understanding the formation of DM haloes -the building blocks of structure formation and main components of every galaxy -is crucial to distinguish different DM scenarios with astronomical data. Unfortunately, the smallest and most relevant scales are dominated by complex nonlinear physics of both gravitational and hydrodynamical origin. The modeling of the hydrodynamical effects is particularly cumbersome because it depends on various feedback mechanisms that are poorly understood and tend to suppress luminous sources, mimicking the expected suppression by WDM.
It has been known for a long time that dwarf galaxy number counts and internal kinematics are in conflict with predictions from N -body simulations in the standard ΛCDM scenario [41], i.e., the cosmological standard model including Dark Energy and cold DM. Suggestions to solve these small scale problems are numerous and go from invoking a more realistic treatment of baryonic physics [42] to postulating alternative DM scenarios [43].
Studying small scale structure formation of sterile neutrino DM in detail would therefore be very desirable. This would however imply running extensive numerical simulations and lies beyond the scope of this work. It is nevertheless possible to gain some insight into nonlinear clustering by applying an extended Press-Schechter (EPS) approach [44], which approximates structure formation by combining linear growth with idealized ellipsoidal collapse. Standard EPS models are designed for ΛCDM cosmologies and it turns out that they completely fail in the presence of suppressed power spectra. We therefore use a modified EPS approach constructed to cope with arbitrarily shaped power spectra and tested for warm and mixed DM cosmologies [45]. In this approach the halo mass function (i.e. the number density of haloes per logarithmic mass bin) can be written as where Θ H (x) is the Heaviside step-function and f (ν) = A 2ν/π(1 + ν −p )e −ν/2 is the 'first-crossing distribution' with ν = (1.686/σ) 2 , A = 0.322, and p = 0.3. A detailed description of the procedure can be found in Ref. [45]. In Fig. 3 we plot the halo mass function based on the transfer functions from Fig. 2, where the black line (with red band) corresponds to scalar decay production, the blue dashed line to resonant production, and the green dotted line to the standard non-resonant production. For comparison, the CDM halo mass function is given by the dashed black line and the relevant dwarf galaxy scales are highlighted in gray.
This figure shows that the halo mass function of scalar decay produced DM is in general nearly indistinguishable from CDM at dwarf galaxy scales. The halo abundance starts to be significantly suppressed below a mass of 10 7 M . Since haloes of this mass range are not able to form stars (their gravitational potentials are not deep enough to allow efficient cooling of the gas), the scalar decay scenario is expected to behave very similarly to CDM on astronomically relevant scales.
The situation is very different for the case of resonant production, where haloes below 10 9 M are strongly suppressed. While this scenario is in conflict with Ly-α data, it is expected to alleviate some of the small scale problems of CDM structure formation [46]. In the non-resonant scenario, finally, the halo mass function is suppressed in the entire dwarf galaxy range. The suppression is so strong that this scenario is not only ruled out by the Lyα forest but also because it predicts far fewer Milky-Way satellites than observed [47].

CONCLUSIONS
In this paper we have examined the assumption that the recently observed X-ray line in galaxy clusters stems from decays of sterile neutrino DM. In order to check the validity of such a scenario, it is crucial to understand the details of sterile neutrino production in the early Universe. We compared the most prominent production mechanisms and showed that they exhibit considerable differences in the growth of perturbations, which can be distinguished even with present-day astronomical observations. The Ly-α signal from high-redshift quasars gives the most stringent constraints and seems to disfavor all but one of the most named sterile neutrino production mechanisms, namely the production via scalar decay, while (non-)resonant oscillations of active into sterile neutrinos seem to be in tension with the Ly-α measurement. Indeed, production via the decay of heavy scalar singlets seems in perfect agreement with the data and could remain as the only valid production mechanism if the 3.5 keV line observation is solidified.
Additionally, we showed that scalar decay produced sterile neutrinos have an unusually cold momentum distribution. As a consequence, they are indistinguishable from cold DM at the relevant scales of dwarf galaxies, and do not contribute towards a solution of the highly contested small scale problems of ΛCDM.