Sterile neutrinos with eV masses in cosmology – how disfavoured exactly?
Abstract
We study cosmological models that contain sterile neutrinos with eVrange masses as suggested by reactor and shortbaseline oscillation data. We confront these models with both precision cosmological data (probing the CMB decoupling epoch) and lightelement abundances (probing the BBN epoch). In the minimal CDM model, such sterile neutrinos are strongly disfavoured by current data because they contribute too much hot dark matter. However, if the cosmological framework is extended to include also additional relativistic degrees of freedom beyond the three standard neutrinos and the putative sterile neutrinos, then the hot dark matter constraint on the sterile states is considerably relaxed. A further improvement is achieved by allowing a dark energy equation of state parameter . While BBN strongly disfavours extra radiation beyond the assumed eVmass sterile neutrino, this constraint can be circumvented by a small degeneracy. Any model containing eVmass sterile neutrinos implies also strong modifications of other cosmological parameters. Notably, the inferred cold dark matter density can shift up by 20–75% relative to the standard CDM value.
MPP2011100
TTK1129
a]Jan Hamann, a]Steen Hannestad, b]Georg G. Raffelt c]and Yvonne Y. Y. Wong
[a]Department of Physics and Astronomy
University of Aarhus, DK8000 Aarhus C, Denmark
[b]MaxPlanckInstitut fR̈u Physik (WernerHeisenbergInstitut)
Föhringer Ring 6, D80805 München, Germany
[c]Institut für Theoretische Teilchenphysik und Kosmologie
RWTH Aachen, D52056 Aachen, Germany
hamann@phys.au.dk \emailAddsth@phys.au.dk \emailAddraffelt@mppmu.mpg.de \emailAddyvonne.wong@physik.rwthaachen.de
1 Introduction
There is mounting evidence from reactor and shortbaseline neutrino oscillation experiments suggesting the existence of one or two sterile neutrinos with mass splittings relative to the active flavours in the neighbourhood of and fairly large mixing parameters (see, e.g., [1] for a review and global interpretation, and [2, 3, 4] for other recent analyses). In the early universe, flavour oscillations would bring these sterile states into thermal equilibrium prior to neutrino decoupling at MeV, thereby increasing the relativistic energy density (see, e.g., [5] for a review). This increase, commonly parameterised in terms of an additional contribution to the relativistic neutrino degrees of freedom , significantly modifies bigbang nucleosynthesis (BBN) of light elements, the cosmic microwave background (CMB) anisotropies, and the formation of largescale structures (LSS). By the same token, these eVmass sterile neutrinos would later play the role of a nonnegligible hotdark matter component.
The cosmological verdict on eVmass sterile neutrinos is somewhat mixed. Current CMB and LSS observations show a consistent preference for additional relativistic degrees of freedom beyond the standard model expectation of , with low to moderate statistical significance [6, 7, 8, 9, 10, 11]. On the other hand, if this putative radiation excess is interpreted in terms of sterile neutrinos, the usual hot dark matter limits constrain the sterile masses to the subeV regime. For example, in a 3+1 scenario consisting of three essentially massless active flavours and one fully thermalised sterile flavour, the sterile mass is constrained to eV [7]. Thus, cosmology appears to disfavour the existence of one or two sterile states that have mass and mixing parameters favoured by a global analysis of the laboratory data.
However, both the tentative evidence for extra radiation and these hotdark matter bounds are based on the simplest CDM framework; extended models may provide different answers. For example, if the dark energy equation of state parameter and curvature are used as fit parameters, the presence of eVmass sterile neutrinos favours a value much smaller than [12]. Here, we take the view that it may be more natural to look for extensions to CDM within the neutrino sector alone, for example in terms of additional radiation. (However, we note in passing that in certain dynamical dark energy models such as the mass varying neutrino scenario, a strongly modified neutrino sector is responsible for driving the dark energy evolution. In this sense, changing may not be an entirely unnatural solution to the massive sterile neutrino problem.) Multiple sterile righthanded states may exist and some of them may even be lighter that the eV range. Moreover, the existence of sterile neutrinos also provides a means for the generation of large neutrino chemical potentials [13], which may be exploited to circumvent standard BBN bounds on the number of relativistic degrees of freedom.
The premise of our work is that sterile neutrino states with eVrange masses are real and that they are fully thermalised prior to neutrino decoupling. In addition, we allow for radiation beyond that due to the three standard flavours (assumed to be effectively massless) and beyond the eVmass sterile neutrinos. We consider both data from precision cosmology and the constraints on imposed by BBN.
More specifically, in section 2 we investigate the impact of eVmass sterile neutrinos on the inference of cosmological parameters from CMB and LSS observations. In section 3 we evaluate the constraints on additional relativistic species during the BBN epoch, using measurements of the primordial abundances of several light elements. In section 4 we summarise and discuss our results.
2 CMB and LSS
2.1 Cosmological data
The CMB anisotropies and the LSS distribution are sensitive to both additional relativistic degrees of freedom and the mass scale of light, freestreaming particles. To explore the impact of eVmass sterile neutrinos on precision cosmology, we use CMB anisotropy data and their accompanying likelihood routines from the WMAP 7year data release [14], as well as the ACBAR [15], BICEP [16], and QuAD [17] experiments. In addition, we use the halo power spectrum extracted from the SDSSDR7 luminous red galaxy sample [18], and type Ia supernova (SN) data from the Union2 compilation [19]. Finally, we impose a constraint on the Hubble parameter based on the Hubble Space Telescope observations [20]. We refrain from using observational data pertaining to very small length scales, such as estimates of smallscale density fluctuation amplitudes from the Lyman forest and cluster abundance. While these measurements are in principle an extremely powerful tool for constraining neutrino masses, they are also currently dominated by systematic uncertainties. We have tested explicitly that adding data from the Atacama Cosmology Telescope (ACT) [21] does not alter our results; The main role of ACT is to break parameter degeneracies in CMBonly analyses. These degeneracies are however already broken by the addition of LSS data.
We construct allowed regions in cosmological parameter space using standard Bayesian inference techniques. All posterior probability density functions are sampled using the Markov Chain Monte Carlo (MCMC) code CosmoMC [22].
Parameter  Symbol  Prior 

Baryon density  
Cold dark matter density  
Hubble parameter  
Amplitude of scalar spectrum @  
Scalar spectral index  
Optical depth to reionisation  
Number of extra massless neutrino degrees of freedom  
Dark energy equation of state parameter 
2.2 Cosmological models
We consider three basic cosmological frameworks in which we embed light sterile neutrinos and analyse their consequences.

The CDM class of models is defined by a flat spatial geometry and six free parameters, . See table 1 for their definitions and prior ranges. Within this framework, we consider four possibilities in the neutrino sector: (a) 3 massless neutrinos (standard CDM), (b) 3 massless+1 sterile (0 eV), (c) 3 massless+1 sterile (1 eV), and (d) 3 massless+1 sterile (2 eV).

In the second class of models, which we dub CDM+, we include additional relativistic degrees of freedom beyond the 3+1 standard and sterile neutrinos. We consider three scenarios: (a) massless+1 sterile (0 eV), (b) massless+1 sterile (1 eV), and (c) massless+1 sterile (2 eV). Such models are especially sensitive to BBN constraints to be discussed in section 3.

The CDM+ framework is a variant of CDM+, in which we extend the dark energy sector to include the possibility that .
2.3 Goodnessoffit
We consider first the goodnessoffit, as quantified by the bestfit effective , of the cosmological models described in the previous section. Here the effective is defined as , where is the maximum likelihood of the data given the model. Table 2 summarises the bestfit for these models and, where appropriate, the preferred values of and , using the data sets described in section 2.1. Some comments are in order.

Within the CDM framework, a scenario with 3 massless neutrinos plus one fully thermalised massless sterile species offers a slightly better fit than standard CDM (). However, once the sterile states have mass, the quality of the fit deteriorates. The 1 eV scenario is already marginally worse than standard CDM by . The 2 eV model may be deemed unacceptable ().

The situation improves when we allow for additional radiation (CDM+). For example, if the sterile neutrino has a mass of 1 eV, the bestfit is comparable to that of standard CDM, albeit at the cost of admitting additional massless degrees of freedom. For a 2 eV sterile mass, we find and . This last result can be compared with the CDM+ model of reference [12], for which assuming a lighter 1.33 eV sterile neutrino. Therefore, introducing extra radiation appears to be somewhat superior to modifying the dark energy sector at resolving the sterile mass conundrum.

Even more improvement is available if, in addition, we allow the dark energy equation of state parameter to differ from (CDM+). In this class of models, we see that a scenario with one species of 1 eV sterile neutrinos in fact provides a better fit to the data than does standard CDM, with , at the expense of two additional free parameters.
Framework  Neutrino sector  
CDM  3 massless  –  –  
3 massless + 1 sterile (0 eV)  –  –  
3 massless + 1 sterile (1 eV)  –  –  
3 massless + 1 sterile (2 eV)  –  –  
CDM+  3+ massless + 1 sterile (0 eV)  –  
3+ massless + 1 sterile (1 eV)  –  
3+ massless + 1 sterile (2 eV)  –  
CDM+  3+ massless + 1 sterile (0 eV)  
3+ massless + 1 sterile (1 eV)  
3+ massless + 1 sterile (2 eV) 
2.4 Effects on other cosmological parameters
We have seen that precision cosmological observations can reasonably accommodate one fully thermalised species of massive sterile neutrinos if we allow also for additional massless degrees of freedom and/or a nonstandard dark energy equation of state. An interesting consequence is that the preferred values of other free parameters of the model also shift accordingly.
The most notable example in this regard is the cold dark matter density . Figure 1 illustrates the shift in as a function of the sterile neutrino mass within the CDM+ framework. Figure 2 is similar, but for the CDM+ models. See also table 2 for the bestfit values and credible regions. Clearly, the larger the sterile neutrino mass, the larger the preferred value of . In the case of a 2 eV sterile neutrino, the upward shift in can be as large as 75% in the CDM+ model, relative to the standard CDM inferred value. This shift in the cold dark matter density can have importance consequences for, e.g., the SUSY dark matter parameter space.
Another affected parameter is the scalar spectral index , whose preferred region widens in the presence of additional light species, as was also seen in our previous analysis [7].
3 Big bang nucleosynthesis
While not sensitive to the masses of neutrinos, BBN has long been used to probe the radiation content of the Universe at temperatures of order 1 MeV [23, 24, 25]. In this section, we explore the implications of the latest primordial element abundance measurements on the sterile neutrino scenario. We consider a general BBN model with three free parameters: the baryon density , an extra effective sterile neutrino species on top of the usual three fully thermalised standard neutrinos, and, eventually we also allow for the presence of a neutrino chemical potential . Table 3 summarises the prior ranges for these parameters.
Parameter  Symbol  Prior 

Baryon density  
Number of sterile neutrinos  
Neutrino chemical potential 
3.1 Analysis
We infer constraints on the BBN parameters with a modified version of the Markov Chain Monte Carlo (MCMC) sampler CosmoMC [22].^{1}^{1}1The numerical module for implementing the BBN likelihood in CosmoMC is available from the corresponding author (Jan Hamann) upon request. The PArthENoPE [26] code is employed to precalculate the primordial element abundances on a grid of points in space. For each set of parameter values the abundances are then obtained from the grid by 3dimensional cubic spline interpolation. We illustrate the dependence of the light element abundances on the cosmological parameters on selected slices through parameter space in figure 3.
3.2 Uncertainties of input data
The output of PArthENoPE is subject to theoretical uncertainties from two principal sources: the nuclear reaction rates and the free neutron lifetime . The former induce a % error for the D prediction, negligible error for He, and % for Li [27]. We fold these uncertainties into our definitions of the corresponding likelihood functions in section 3.3.
The main source of uncertainty for the predicted He abundance is the free neutron lifetime. The Particle Data Group recommends a value of s [28]. This, however, appears to be in strong tension with recent measurements performed by Serebrov et al. [29], who found s, and Pichlmair et al. [30], who measured s. Although a reanalysis of neutron lifetime data shows that previous estimates may have been biased by s [31], we will nonetheless consider the two extreme values and in the following, in order to illustrate the possible impact of this uncertainty on our inference. Going from to shifts the prediction by , whereas and change by which is negligible compared with the effects of other uncertainties (nuclear rates and astrophysical measurements).
3.3 Primordial element abundance data and likelihood functions
Deuterium
The primordial deuterium abundance can be inferred from measurements of the absorption of quasar light in highredshift lowmetallicity hydrogen clouds. Pettini et al. [32] derive a value of from the observation of the spectra of seven quasars. Taking into account the theoretical uncertainty of 1.6%, the corresponding likelihood function is
(1) 
Helium4
The mass fraction of He is determined by measuring helium and hydrogen emission lines in lowmetallicity HII regions of dwarf galaxies. Its primordial value can then be estimated by treating as a function of metallicity and extrapolating to . A linear regression of data from seven highquality objects with a careful treatment of systematic uncertainties yields , with the additional restriction to theoretically meaningful positive slopes [33].^{2}^{2}2It has been argued that a linear regression may not necessarily be realistic in view of certain models of early He production by Pop III stars [34], as for instance suggested in references [35, 36]. However, even in these cases, the difference between the primordial and the observed values of will not exceed , a possibility that is well covered by the lower limit on in equation (2). We therefore model the likelihood function as
(2) 
Lithium7
Lithium abundances have been determined from the spectra of metalpoor dwarf stars. At low metallicities, the lithium abundance appears to be independent of metallicity [37]. However, the measurements are subject to significant systematic uncertainties about the stars’ temperatures. Additionally, the measured plateau value may not be representative of the primordial abundance: Li can be generated by cosmic rays, or depleted through Population III stars or diffusion processes within the dwarf stars [38].
The Particle Data Group estimates an average of , which is significantly lower than the standard BBN expectation for realistic baryon densities – the wellknown lithium problem [39]. Approximating the 8% theoretical uncertainty with an absolute error of and adding the errors in quadrature, we arrive at the following likelihood function for lithium:
(3) 
which will be considered only where appropriate.
CMB+LSS prior on the baryon density
In addition to primordial abundance measurements, we also consider a prior on the baryon density from CMB+LSS data as an independent constraint. Since the bounds on are in principle modeldependent, using results from a fit to the standard CDM model as a prior for extended models would be technically incorrect. Instead we use the baryon density inferred within the CDM+ model, , to define our prior
(4) 
In practice, however, present CMB+LSS data are sufficiently sensitive to break any potential degeneracy between and , so that the constraints on show no appreciable variation with respect to the inclusion or otherwise of the parameter .
3.4 BBN with sterile neutrinos
We first consider a BBN scenario with effective sterile neutrino species. The additional radiation energy density contributed by the sterile neutrinos increases the expansion rate at BBN, leading to a higher freezeout temperature . As a consequence, the equilibrium neutrontoproton ratio , where is the neutronproton mass difference, is larger at neutron freezeout. The resulting is larger because almost all neutrons end up in He. Similarly, a larger neutron lifetime also increases the expected .
We show the constraints on the parameter space from our analysis in figure 4. The top two panels illustrate the wellknown fact that the deuterium abundance constrains mainly the baryon density, whereas constraints on are mostly driven by the helium data. However, once we impose the CMB+LSS prior on , even the deuterium data alone significantly constrain . This constraint is of interest in view of the helium data already being systematicslimited.
For a combined fit of the deuterium and the helium data, the bestfit value of is 0.86, with a 95%credible upper limit of (or 1.24 if the CMB+LSS prior on is included). Using the larger neutron lifetime , we obtain slightly lower values: a bestfit of 0.73 and a 95% upper limit of . The onedimensional marginalised posterior probability densities for are shown in figure 5. As already emphasised in reference [34], there is no strong indication for from BBN alone – lies well within the 90%credible interval in all cases – owing to the relatively weak lower limit on . While one fully thermalised sterile neutrino species is slightly favoured over , two fully thermalised sterile neutrino species are clearly incompatible with the data in this scenario.
3.5 Degenerate BBN with sterile neutrinos
If we imagine for a moment that there were two sterile neutrino species today, how could this be reconciled with the results of the previous section? One possibility would be incomplete thermalisation, such that the effective is smaller than 2; this scenario could be confirmed if future CMB+LSS data should find . Alternatively, the sterile neutrinos could be the decay products of a heavy particle species between BBN and decoupling. In this case, could be smaller than 2 at BBN, but equal to or larger than 2 at decoupling.
A third possibility is the degenerate BBN scenario [40], in which all standard neutrinos share a common nonzero chemical potential [41, 42, 43]. The presence of a chemical potential affects BBN in two ways. Firstly, it contributes an additional term to the effective radiation density,
(5) 
Secondly, the degeneracy in the electron neutrinos modifies the equilibrium neutrontoproton ratio,
(6) 
For , it is the latter effect that is most relevant for the resulting element abundances: a positive will reduce the number of available neutrons with respect to the case and thus suppress . This suppression can be used to circumvent the upper bound on found in the previous section.
Figure 6 shows the preferred region in the plane in the degenerate BBN scenario. In the 3dimensional parameter space , the deuterium and the helium data can only constrain two directions, leaving an unconstrained one which admits arbitrarily high values of . Only by adding the CMB+LSS prior on are we able to obtain any constraint at all: we find a bestfit of and a 95%credible upper limit of . Conversely, if we assume , a positive nonzero chemical potential is required: (95%credible interval), with a bestfit value of .
As can be seen from figure 3, larger values of and both lead to a lower prediction for the primordial Li abundance. Taking for instance , and , one obtains (compared to for ). While this is still standard deviations larger than the measured value, the discrepancy is not quite as serious as the 4.8 standard deviations one finds in standard BBN.
3.6 Combining BBN with CMB+LSS data
If one assumes that neither nor changed between the BBN era and the time of photon decoupling, one could perform a combined analysis of the CMB+LSS and BBN data in analogy to the previous literature [44, 45].
With the stringent upper bound on the radiation density from the standard BBN setting, it is clear that the scenarios with one eVmass sterile neutrino plus extra massless degrees of freedom considered in section 2 will fit the combined data rather badly: for a 1 eV (2 eV) sterile neutrino, the CDM+ model yields , relative to CDM. This, in turn suggests the necessity for further modifications of the cosmological model: either by dropping the assumption of (or ) being constant, or, as seen above, by admitting a nonzero neutrino chemical potential.
Aside of the massive sterile neutrino scenario, the combination of CMB+LSS and BBN data can of course also be used to constrain the commonly considered case of massless degrees of freedom. We find at 95% credibility, with the standard model expectation of outside the 99.5%credible region, similar to the limits reported in the recent work of Hou et al. [9].
4 Conclusions
In this work we have investigated the effects of eVmass sterile neutrinos, as suggested by global interpretations of neutrino oscillation data, on cosmology. Such sterile neutrinos can thermalise prior to neutrino decoupling, thus contributing to the relativistic energy density in the early universe. However, while the combination of CMB+LSS and BBN data does appear to prefer extra relativistic degrees of freedom at 99.5% credibility within the CDM framework, fully thermalised massive sterile neutrinos in the 1–2 eV mass range necessarily violates the hot dark matter limit on the maximum neutrino mass. In terms of the goodnessoffit, adding one massless sterile neutrino species improves the CMB+LSS fit by relative to standard CDM, whereas endowing this sterile state with a mass of 1 eV worsens the fit by relative to the same benchmark. Such a scenario is then excluded or strongly disfavoured.
Nonetheless, while it appears difficult to accommodate eVmass sterile neutrinos within the CDM framework, extending the framework with modifications in the neutrino sector improves to some extent the consistency of sterile neutrinos with precision cosmological data. The simplest such modification is to admit even more additional (effectively massless) relativistic degrees of freedom, not necessarily fully thermalised. Such a scenario improves somewhat the bad effect of the 1 eV sterile neutrino mass at the expense of introducing an additional 1.5 massless species, but the of the fit is still worse than standard CDM by units. Allowing in addition for a dark energy equation of state further improves the fit: for a 1 eV sterile neutrino, a model with and additional massless species in fact fits the data marginally better than standard CDM ().
Importantly, any model containing eVmass sterile neutrinos will induce an upward shift in the cold dark matter density inferred from precision cosmological data. This shift can have important consequence for, e.g., the SUSY dark matter parameter space.
However, as is well known, increasing the radiation content in the early universe can be problematic for BBN. We find that while standard BBN prefers roughly an extra relativistic degree of freedom, which we interpret here as a thermalised 1–2 eV sterile neutrino species, additional fully thermalised massless species are strongly disfavoured. Nevertheless, it is possible to circumvent these BBN constraints with the introduction of a chemical potential, which itself could have been created by activesterile oscillations in the early universe. Thus, the state of affairs can be summarised as follows. We need additional radiation to reduce the bad effect of hot dark matter on precision cosmology, and a small neutrino chemical potential to undo the bad effect of too much radiation on BBN. In principle, both of these ingredients can originate from the neutrino sector alone.
In summary, it is not trivial to accommodate a strongly mixed eVmass sterile neutrino in cosmology. Additional ingredients are required, such as additional radiation, a neutrino chemical potential, or a nontrivial parameter. In all cases, significant changes in the inferred values of other a priori unrelated cosmological parameters are also incurred, e.g., an increase in the cold dark matter density. Thus, should the experimental indications for eVmass sterile neutrinos become stronger, one must consider a fairly complex modification of the standard CDM cosmology. On the observational side, the upcoming precision measurement of by Planck [46, 47] remains one of the most promising windows to physics beyond the standard model.
Acknowledgements
We acknowledge computing resources from the Danish Center for Scientific Computing (DCSC). JH acknowledges support from a Feodor Lynenfellowship of the Alexander von Humboldt Foundation. GR acknowledges partial support by the Deutsche Forschungsgemeinschaft under grants No. TR 27 and EXC 153.
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