Tight bonds between sterile neutrinos and dark matter

Despite the astonishing success of standard $\Lambda$CDM cosmology, there is mounting evidence for a tension with observations at small and intermediate scales. We introduce a simple model where both cold dark matter (DM) and sterile neutrinos are charged under a new $U(1)_X$ gauge interaction. The resulting DM self-interactions resolve the tension with the observed abundances and internal density structures of dwarf galaxies. At the ame time, the sterile neutrinos can account for both the small hot DM component favored by cosmological observations and the neutrino anomalies found in short-baseline experiments.


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This requires an effective number of additional neutrino species ∆N eff and an effective HDM mass of [37] ∆N eff | cmb = 0.61 ± 0.30 , m eff hdm = (0.41 ± 0.13) eV . (1.1) Here, we extend the standard model of particle physics (SM) by a spontaneously broken U(1) X gauge theory and introduce a sterile neutrino that is not charged under SM interactions. We demonstrate that all aforementioned problems of standard cosmology are resolved by coupling both CDM and sterile neutrinos, the latter automatically being promoted to the desired HDM component, to a U(1) X gauge boson of O (MeV) mass. This can be achieved for parameters that resolve the anomalies [40][41][42][43][44][45][46] reported in short-baseline neutrino oscillation experiments, in particular a sterile neutrino mass of ∼ 1 eV.
This article is organized as follows. We first set up our model and give a brief overview of the cosmological evolution of the non-standard degrees of freedom introduced here, as sketched in figure 1. We continue by showing that the properties of the thermally produced CDM sector can provide a solution to all ΛCDM problems at small scales. Next, we demonstrate that our sterile neutrino HDM component can simultaneously satisfy the cosmologically favored values stated above and describe the anomalies in short-baseline oscillation experiments. We conclude with a discussion and an outlook.

Model setup
We consider the extension of the SM gauge group, G SM = SU(3) c × SU(2) L × U(1) Y , by an Abelian gauge symmetry U(1) X with corresponding gauge boson V . We introduce a Dirac fermion χ at the TeV scale, which will form the CDM, and two right-handed neutrinos ν R 1,2 . Those new particles are neutral under G SM but carry U(1) X charges, while the SM particles are neutral under U(1) X . Anomaly cancellation requires the ν R 1,2 to carry charges of opposite sign with equal absolute value; for concreteness, we take the charges of (χ, ν R 1 , ν R 2 ) to be (1, X ν R , −X ν R ).
We further assume that the U(1) X is spontaneously broken at the MeV scale by the vacuum expectation value (VEV) v Θ of a complex Higgs field Θ, which is a representation (1, 0, 2X ν R ) under SU(2) L × U(1) Y × U(1) X , while the Higgs field φ responsible for the electroweak symmetry breaking is a (2, 1/2, 0). Another complex scalar ξ, with charges (1, 0, X ν R ) and VEV v ξ < v Θ , is introduced to enable active-sterile neutrino mixing.
After symmetry breaking the low-energy, effective Lagrangian of our theory reads Here, L SM denotes SM terms and L R contains in addition to kinetic terms and Majorana mass terms for the SM neutrinos. The active-sterile neutrino mixing arises from a dimension-5 operator with M LR ∼ v φ v ξ /Λ, suppressed by a scale Λ defined by the UV completion of the theory. The mass eigenstates (ν 1 , ν 2 , ν 3 , N 1 , N 2 ) are mixtures of the flavor eigenstates ν e , ν µ , ν τ , ν c R 1 , ν c R 2 . With the short-baseline anomalies in mind, Yukawa couplings between ν R 1,2 and Θ are chosen such that m N 2 ∼ M 2 ∼ MeV m N 1 ∼ M 1 ∼ eV. We note that even a very small mixing between ν R 1 and ν R 2 , with M RR /M 2 10 −6 , allows the cosmologically fast decay of ν R 2 .
3) where g X denotes the U(1) X gauge coupling. To ensure the stability of χ we might impose a discrete Z 2 symmetry under which only χ is assigned a negative parity. The symmetries also allow a kinetic mixing term L kin. mix. = − 2 F x µν F µν , where F xµν (F µν ) denotes the U(1) X (electromagnetic) field strength tensor. We assume 1 to satisfy the severe existing constraints on this parameter [47,48].
We refer to ref. [49] for a general discussion of the Higgs sector for Θ and φ as contained in L Higgs , adopting that m V and the mass of the new light Higgs boson h x are of the same order of magnitude in the relevant cases. The "Higgs portal" term in where we have assumed a negligible mixing between h x and the SM-like Higgs h in the last step, connects the SM and the new U(1) X sector. For simplicity, we assume the couplings of the additional portal terms |φ| 2 |ξ| 2 and |Θ| 2 |ξ| 2 to be negligibly small.

Thermalization via the Higgs portal and decoupling of the dark sector
In figure 1 we provide a schematic overview of the cosmology arising from our model represented by eq. (2.1). Before electroweak symmetry breaking, the 4-scalar interaction κ|Θ| 2 |φ| 2 keeps the U(1) X sector in thermal equilibrium with the SM bath if the thermalization rate Γ th = n hx σ th v rel ∼ 10 −3 κ 2 T is larger than the expansion rate H ∼ 10 T 2 /M pl . In those -3 -

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expressions, n hx denotes the number density of h x and σ th v rel the thermally averaged annihilation cross section of h x pairs. If we, e.g., require thermal equilibrium at temperatures below 10 TeV, i.e. above the CDM mass m χ , we obtain a lower bound on the Higgs portal coupling of κ 10 −6 . After electroweak symmetry breaking the relevant process becomes h with f corresponding to the heaviest relativistic SM fermion, so the decoupling temperature becomes T dpl For details on thermalization via the Higgs portal we refer to [50][51][52], where thorough calculations of h x abundances for m hx ∼ TeV were performed (while in our case h x decouples relativistically).
The particles in the dark sector are tightly coupled to each other due to the U(1) X interaction, and more weakly to the SM via the Higgs portal. Once the latter ceases to be effective, the whole U(1) X sector therefore decouples from the SM bath and entropy is conserved separately in the two sectors. Whenever a particle in equilibrium becomes non-relativistic it thus heats its bath, thereby increasing the temperature by a factor g before * ,ν/N 1 /g after * ,ν/N 1 counts the effective degrees of freedom (d.o.f.) determining the entropy density of the sector in thermal equilibrium with the species i. The non-standard contribution to the radiation density is then given by The maximal possible value of this quantity at the onset of big bang nucleosynthesis (BBN), at T ∼ 1 MeV, is then obtained if all new particles but the light sterile neutrino, N 1 , have become non-relativistic by then. This results in well within bounds from BBN [53][54][55] for T dpl x 1 GeV.

Self-interacting CDM
At high temperatures, the DM particles are kept in chemical equilibrium via χχ ↔ V V (for unit sterile neutrino charges, X ν R ∼ 1, also the annihilation into ν R ν R , h x h x and ξ * ξ via a virtual V becomes important). For TeV-scale DM the number density freezes out at sufficiently early times (T fo χ ∼ m χ /25) to still have T V = T . Assuming for simplicity X ν R 1, the CDM relic density then becomes up to O (1) corrections due to the Sommerfeld effect [58], which we fully take into account [33]. This fixes g X for a given m χ throughout this work. Kinetic decoupling [59] of χ happens much later and is determined by the elastic scattering rate for χN 1 ↔ χN 1 . For a thermal distribution of sterile neutrinos, the decoupling temperature is given by [33] T kd In the yellow area, the CDM self-interaction is strong enough to flatten density cusps in the inner parts of (dwarf) galaxies [30] and likely also solves the too big to fail problem (as explicitly demonstrated in N -body simulations for parameter values corresponding to the crosses [31]). The dark area is excluded by astrophysics [29,30,56,57]. The blue band addresses the missing satellites problem [33], with a normalization that -according to eq. (4.2) -is proportional to kd . Here, we show for reference the case of X ν R = 0.2 and (T N1 /T ) 4 kd = 0.46.
which translates into a cutoff in the power spectrum of matter density perturbations at M cut ∼ 1.7 × 10 8 T kd χ /keV −3 M . We note that the light mass eigenstates ν i also acquire a U(1) X charge from their ν c R component; this will further lower T kd χ if sin θ (T N 1 /T ν ) 3/2 . After structure formation, the U(1) X -induced Yukawa potential produces galaxy cores that match the observed velocity profiles of massive MW satellites, solving cusp vs. core [28,30] and too big to fail [31], while avoiding constraints on DM self-interactions on larger scales [30]. At the same time, the late kinetic decoupling addresses the missing satellites by suppressing the matter power spectrum at dwarf galaxy scales [33] (see also refs. [24,[60][61][62]). In figure 2, we show the desired parameter space for m V and m χ (based on ref. [33], but using an improved parameterization [63] of the Yukawa scattering cross section [28,[64][65][66][67]). The blue band, in particular, shows the range of masses that allow a solution of the missing satellites problem. Note that its normalization depends on the choice of X ν R and (T N 1 /T ) kd , whereas the m χ -dependence is uniquely determined by eq. (4.2) and the form of g X (m χ ) that corresponds to the correct thermal CDM relic density, cf. eq. (4.1). The dark area is excluded by the requirements to not disrupt galactic satellites and to avoid a gravothermal catastrophe [29,30,56,57].

The HDM component
We will now address the question whether the N 1 population in our model can account for the cosmologically preferred HDM component [36][37][38][39]. In the absence of any significant  additional N 1 production mechanism, see the discussion further down, we simply have By choosing the right decoupling temperature in eq. (3.2), which in our model corresponds to adjusting κ, we can then reproduce eq. (1.1). This is demonstrated in figure 3 where we show the allowed region of ∆N eff | cmb and m eff hdm [37] in terms of g * ,ν T dpl x and m N 1 .
The thermal production of the CDM component as treated here requires T dpl x T fo χ ∼ m χ /25. On the other hand, g * ,ν T dpl x cannot exceed the full number of SM d.o.f. even for very early U(1) X decoupling. Taken together, this points to 0.2 eV m N 1 1.2 eV.

Neutrino anomalies
Oscillation experiments observing neutrinos from accelerators [40,41], reactors [42,43] (but see [70]), and radioactive sources [44][45][46] reported anomalies that may indicate the existence of sterile neutrinos with a mass squared difference ∆m 2 ∼ 1 eV 2 to the SM neutrinos. In figure 3, we show the 1σ and 2σ ranges for ∆m 2 from [68,69] for orientation, assuming -6 -JCAP07(2014)042 m 2 N 1 = ∆m 2 . These ranges were obtained from a global fit of oscillation data assuming the existence of a single sterile neutrino (note that it is being debated to what extent the data can be consistently explained by oscillations alone and whether a second sterile neutrino is necessary to achieve a satisfactory fit [71], which would not be possible to accommodate in our setup). From figure 3 we find that the regions allowed by the HDM signal and neutrino oscillations indeed overlap. While this happens only at the 2σ level, we note that the corresponding range of ∆N eff is the same as independently suggested by m χ 1 TeV (as favored from figure 2).

Discussion
Before standard neutrino decoupling at T ∼ 1 MeV, the effective mixing angle θ m between active and sterile neutrinos is strongly suppressed due to the matter potential generated by the U(1) X couplings of the sterile neutrinos [72] (see also [73]). As the Universe cools, the effective mixing angle eventually reaches its vacuum value θ. This may give rise to an additional production of sterile neutrinos due to their U(1) X interaction. The largest effect on the scenario sketched above would result if the neutrinos completely re-thermalized, creating a thermal N -ν bath.
In that case, conservation of entropy density allows us to determine the temperature T N ν of the newly established neutrino bath as where N SM eff 3.046. Rather than eqs. (5.1), (5.2), we thus obtain Rewriting this as m N 1 = 2 √ 2 m eff hdm ∆N eff | cmb + N SM eff , we immediately see that in the re-thermalization case a sterile neutrino can still consistently explain the HDM signal -but only if its mass is considerably smaller than required by the neutrino anomalies.
Turning to potential constraints on our scenario, BBN limits are easily satisfied as already stressed earlier. CDM also decouples kinetically too early to imprint observable dark acoustic oscillation (DAO) features in the CMB [74]. Final state V radiation in the decay of SM particles [75], finally, does not constrain our scenario because V does not couple to left-handed neutrinos. An interesting aspect of our HDM component is that it does not necessarily manifest itself as perfectly free-streaming particles in the CMB or during structure formation, which in principle can be probed [76]; by comparing the elastic scattering rate with the Hubble expansion, we rather expect complete decoupling only at , where the last factor must be of order unity (see figure 2).
The dominant decay channel of our sterile neutrino is N 1 → ννν. Even though this is strongly enhanced compared to the analogous common decay via a virtual Z [77], we find the resulting lifetime for the best-fit neutrino mixings [68] to be which greatly exceeds the age of the Universe t 0 . We note that the decay N 1 → νγ is even more suppressed due to the necessarily small value of .

Conclusions
In this article we have considered a mixed DM model as favored by recent cosmological observations, which adds a small HDM component to the dominant CDM, the former consisting of an eV-scale sterile neutrino and the latter of a TeV-scale Dirac fermion. We have studied the cosmological consequences of equipping both these particles with charges under a new spontaneously broken U(1) X gauge theory, under which all SM particles are singlets. Thermalizing the U(1) X sector in the early universe via the so-called Higgs portal allows the thermal production of the CDM. The sterile neutrinos would also be thermally produced and elegantly form the HDM component, essentially because the U(1) X sector decouples much earlier than SM neutrinos. Remarkably, this is possible for a set of parameters that equip the CDM particles with a U(1) X mediated self-interaction that is of the right form and magnitude to provide a simultaneous solution to the small-scale problems of ΛCDM cosmology [33]. Finally, overproduction via mixings is likely prevented by the large thermal potential that the sterile neutrinos create by their U(1) X interactions [72,73]; in this case one can even address the neutrino oscillation anomalies within the same framework. In other words, a sterile neutrino as preferred by neutrino oscillation anomalies would not only be reconciled with cosmology but promoted to the desired HDM component.