Statistical tests of sterile neutrinos using cosmology and short-baseline data
Johannes Bergström, M. C. Gonzalez-Garcia, V. Niro, J. Salvado
PPrepared for submission to JHEP
Preprint number: YITP-SB-14-21, ICCUB-14-054, UB-ECM-PF-14-80
Statistical tests of sterile neutrinos using cosmologyand short-baseline data
Johannes Bergstr¨om a M. C. Gonzalez-Garcia bc V. Niro a J. Salvado d a Departament d’Estructura i Constituents de la Mat`eria and Institut de Ciencies del Cosmos,Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain b Instituci´o Catalana de Recerca i Estudis Avan¸cats (ICREA),Departament d’Estructura i Constituents de la Mat`eria and Institut de Ciencies del Cosmos,Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain c C.N. Yang Institute for Theoretical Physics, State University of New York at Stony Brook,Stony Brook, NY 11794-3840, USA d Wisconsin IceCube Particle Astrophysics Center (WIPAC) and Department of Physics,University of Wisconsin, Madison, WI 53706, USA
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
In this paper we revisit the question of the information which cosmologyprovides on the scenarios with sterile neutrinos invoked to describe the SBL anomalies usingBayesian statistical tests. We perform an analysis of the cosmological data in ΛCDM+ r + ν s cosmologies for different cosmological data combinations, and obtain the marginalizedcosmological likelihood in terms of the two relevant parameters, the sterile neutrino mass m s and its contribution to the energy density of the early Universe ∆ N eff . We then presentan analysis to quantify at which level a model with one sterile neutrino is (dis)favoured withrespect to a model with only three active neutrinos, using results from both short-baselineexperiments and cosmology. We study the dependence of the results on the cosmologicaldata considered, in particular on the inclusion of the recent BICEP2 results and the SZcluster data from the Planck mission. We find that only when the cluster data is includedthe model with one extra sterile neutrino can become more favoured that the model withonly the three active ones provided the sterile neutrino contribution to radiation densityis suppressed with respect to the fully thermalized scenario. We have also quantified thelevel of (in)compatibility between the sterile neutrino masses implied by the cosmologicaland SBL results. Keywords:
Cosmology of Theories beyond the SM, Neutrino Physics a r X i v : . [ h e p - ph ] J u l ontents It is now an established fact that neutrinos are massive and leptonic flavors are not sym-metries of Nature [1, 2]. In the last decade this picture has become fully establishedthanks to the upcoming of a set of precise experiments. In particular, the results obtainedwith solar and atmospheric neutrinos have been confirmed in experiments using terrestrialbeams [3]. The minimum joint description of all these data requires mixing among allthe three known neutrinos ( ν e , ν µ , ν τ ), which can be expressed as quantum superpositionof three massive states ν i ( i = 1 , ,
3) with masses m i leading to the observed oscillationsignals with ∆ m = (7 . +0 . − . ) × − eV and | ∆ m | = (2 . ± . × − eV andnon-zero values of the three mixing angles [4].In addition to these well-established results, there remains a set of anomalies in neu-trino data at relatively short-baselines (SBL) (see Ref. [5] for a review) including theLSND [6] and MiniBooNE [7, 8] observed ν µ → ν e transitions, and the ¯ ν e disappearanceat reactor [9–11] and Gallium [12–14] experiments. If interpreted in terms of oscillations,each of these anomalies points out towards a ∆ m ∼ O (eV ) [15–23] and consequentlycannot be explained within the context of the 3 ν mixing described above. They requireinstead the addition of one or more neutrino states which must be sterile , i.e. elusive toStandard Model interactions, to account for the constraint of the invisible Z width whichlimits the number of light weak-interacting neutrinos to be 2 . ± .
008 [24].– 1 –everal combined analyses have been performed to globally account for these anoma-lies in addition to all other oscillation results in the context of models with one or twoadditional sterile neutrinos [20–23] with somehow different conclusions in what respects tothe possibility of a successful joint description of all the data. Generically these global fitsreveal a tension between disappearance and appearance results. But while Refs. [21, 22]seem to find a possible compromise solution for 3+1 mass schemes, the analysis in Ref. [23]concludes a significantly lower level of compatibility. In particular, Ref. [21] concludes thata joint solution is found with0 . ≤ ∆ m ≤ .
19 eV at 3 σ (1.1) | U e | ∼ . , | U µ | ∼ . . Alternative information on the presence of light sterile neutrinos is provided by Cos-mology as they contribute as extra radiation to the energy density of the early Universewhich can be expressed as ρ R = (cid:34) (cid:18) (cid:19) / N eff (cid:35) ρ γ , (1.2)where ρ γ is the photon energy density and the value of N eff in the Standard Model (SM)is equal to N SMeff = 3 .
046 [25]. The presence of extra radiation is then usually quantified interms of the parameter ∆ N eff ≡ N eff − N SMeff .Sterile neutrinos contribute to ρ R ( i.e. to ∆ N eff ) in a quantity which, in the absenceof other forms of new physics, is a function of their mass and their mixing with the activeneutrinos which determines to what degree they are in thermal equilibrium with those. Inparticular, in 3 + N s scenarios with the values hinted by the SBL results (of the order ofthose in Eq. (1.2)) the sterile neutrinos are fully thermalized (FT) and each one contributesto ρ R as much as an active one (see for example [26, 27]), so∆ N N s ,F T eff = N s . (1.3)Analyses of cosmological data have hinted for the presence of extra radiation, beyondthe standard three active neutrinos since several years (see for example Ref. [28] and ref-erences therein) and several authors have invoked the presence of eV-scale sterile neutrinoas a plausible source of the extra radiation [29–33] . In the last four years the statisticalsignificance of this extra radiation (or its upper bound) as well as the overall constrainton the neutrino mass scale has been changing as data from the Planck satellite [38], theAtacama Cosmology Telescope (ACT) [39], the South Pole Telescope (SPT) [40, 41], andmost recently of the BICEP2 [42] experiment, has become available [43–50].In particular, the recent measurement of B-mode signals from the BICEP2 [42] col-laboration excludes a zero scalar-to-tensor ratio at 7 σ and report a value of r = 0 . +0 . − . .This high value of r is compatible with previous results from the Planck data [38] if a Alternative scenarios without sterile neutrinos have been proposed as well (see for example Refs. [34–37]and references therein) – 2 –unning of the scalar spectral index dn s /d ln k is considered well beyond the characteristicvalue of 10 − of slow-roll inflation models. Alternatively the tension might be eased by thepresence of sterile neutrinos [44–48] without invoking such a large running of the scalarspectral index.Since cosmological data have the potential to test regions of parameter space of thesterile neutrino scenarios invoked to account for the SBL anomalies, the question of to whatdegree they support them has received an increasing attention in the literature [33, 51–54].The generic conclusion is that, in order to accommodate the cosmological observationswithin the 3 + N s scenarios motivated by SBL results, some new form of physics is requiredto suppress the contribution of the sterile neutrinos to the radiation component of theenergy density at the CMB epoch with respect to the FT expectation Eq. (1.3). Amongothers extended scenarios with a time varying dark energy component [31], entropy pro-duction after neutrino decoupling [55], very low reheating temperature [56], large leptonasymmetry [26, 57, 58], and non-standard neutrino interactions [59–61], have been con-sidered. All these mechanisms have the effect of diluting the sterile neutrino abundance orsuppressing the production in the early Universe.In this paper we revisit the question of the information which cosmology provides on thesterile scenarios introduced to explain the SBL anomalies using precise Bayesian statisticaltests which we briefly describe in Sec. 2. Section 3 contains the results of our cosmologicalanalysis in ΛCDM+ r + ν s cosmologies for three representative sets of cosmological data.With these results at hand we answer the question of how much cosmological data favour ordisfavour the scenario with sterile neutrino masses invoked by SBL anomalies, with respectto a model without sterile neutrinos? in Sec. 4, and we do so in terms of the departurefrom the fully thermalized expectation, Eq. (1.3). We also discuss the consistency ofsterile parameter constrains implied by cosmology and SBL. In Sec. 5 we summarize ourconclusions. Bayesian inference is a rigorous framework for inferring which, out of set of models orhypotheses H i , are favoured by a data set D . Bayes’ theorem is used to calculate theprobabilities of each of the hypotheses after considering the data, the posterior probabilities ,Pr( H i | D ) = Pr( D | H i ) Pr( H i )Pr( D ) . (2.1)Here Pr( D | H i ) is the probability of the data, assuming the model H i to be true, whilePr( H i ) is the prior probability of H i , which is how plausible H i is before considering thedata. Considering especially the case of a discrete set of models, one can compare two ofthem by calculating the ratio of posterior probabilities, the posterior odds , as O ij = Pr( H i | D )Pr( H j | D ) = Pr( D | H i )Pr( D | H j ) Pr( H i )Pr( H j ) = B ij Pr( H i )Pr( H j ) . (2.2)– 3 – log(odds) | odds Pr( H | D ) Interpretation < . (cid:46) (cid:46) .
75 Inconclusive1 . (cid:39) (cid:39) .
75 Weak evidence2 . (cid:39)
12 : 1 (cid:39) .
92 Moderate evidence5 . (cid:39)
150 : 1 (cid:39) .
993 Strong evidence
Table 1 . Jeffrey’s scale often used for the interpretation of model odds. The posterior modelprobabilities for the preferred model H are calculated by assuming only two competing hypotheses. In words, the posterior odds is given by the prior odds
Pr( H i ) / Pr( H j ) multiplied by the Bayes factor B ij = Pr( D | H i ) / Pr( D | H j ), which quantifies how much better H i describesthat data than H j . The prior odds quantifies how much more plausible one model is thanthe other a priory, i.e. , without considering the data. If there is no reason to favour oneof the models over the other, the prior odds equals unity, in which case the posterior oddsequals the Bayes factor.For a model H , containing the continuous free parameters Θ , Pr( D | H ) also called evidence of the model is given byPr( D | H ) = (cid:90) Pr( D , Θ | H )d N Θ = (cid:90) Pr( D | Θ , H ) Pr( Θ | H )d N Θ = (cid:90) L ( Θ ) π ( Θ )d N Θ . (2.3)Here, the likelihood function Pr( D | Θ , H ) is the probability (density) of the data as afunction of the assumed free parameters, which we often denote by L ( Θ ) for simplicity. Thequantity Pr( Θ | H ) is the correctly normalized prior probability (density) of the parametersand is often denoted by π ( Θ ). The assignment of priors is often far from trivial, but animportant part of a Bayesian analysis.From Eq. (2.3), we note that the evidence is the average of the likelihood over theprior, and hence this method automatically implements a form of Occam’s razor , sinceusually a theory with a smaller parameter space will have a larger evidence than a morecomplicated one, unless the latter can fit the data substantially better.Bayes factors, or rather posterior odds, are usually interpreted or “translated” intoordinary language using the so-called
Jeffreys scale , given in Tab. 1, where “log” is thenatural logarithm. This has been used in applications such as Refs. [62–64] (and Refs. [65,66] in neutrino physics), although slightly more aggressive scales have been used previously[67, 68].
In Bayesian statistics if one assumes a particular parametrized model to be correct, thecomplete inference of the parameters of that model is given by the posterior distribution– 4 –hrough Bayes’ theoremPr( Θ | D , H ) = Pr( D | Θ , H ) Pr( Θ | H )Pr( D | H ) = L ( Θ ) π ( Θ )Pr( D | H ) . (2.4)We see that the evidence here appears as a normalization constant in the denominator.Since the evidence does not depend on the values of the parameters Θ , it is usually igno-red in parameter estimation problems and the parameter inference is obtained using theunnormalized posterior.For the case in which we have a only a subset of parameters of interest, λ , so Θ = ( λ, η ),where η denotes the nuisance parameters , P ( λ, η ) = Pr( λ, η | D , H ) ∝ L ( λ, η ) π ( λ, η ) , (2.5)and inference on λ is obtained by marginalizing over the nuisance parameters in the usualway P ( λ ) = (cid:90) P ( λ, η )d η , (2.6)with no need to ever consider a likelihood L ( λ ) depending only on the parameter of interest.This is typically unproblematic in the case where the data is sufficiently informative toeliminate all practical prior dependence. Often, however, this is not the case, and therecan be large dependence on the prior chosen. In this case one can consider as a partialstep the marginal likelihood function L ( λ ) = (cid:90) L ( λ, η ) π ( η )d η , (2.7)such that P ( λ ) ∝ L ( λ ) π ( λ ) . This likelihood function then encodes the information on λ contained in the data(under H and after taking into account the uncertainty on the nuisance parameters), with-out needing to specify a prior π ( λ ). Note that, since the marginal likelihood is not aprobability density, it is not normalized to unity, and is not sufficient to perform the fullinference. Also, it is generally different from the profile likelihood, so regions definedby − L ( λ ) / L max ) < C will not, in general, be the same as those using the profilelikelihood, although arguments justifying defining regions in this way for profile likelihoodstypically also apply to marginal likelihoods. As shown in Ref. [69], the profile and marginallikelihoods are indeed similar for the cosmological data and models considered there. Inany case, the marginal likelihood can still be useful in scientific reporting as a rough guideto what information the data contains, for example, by considering regions defined by − L ( λ ) / L max ) < C [70].Furthermore, if two data sets do not share any common nuisance parameters, thetwo marginal likelihoods can simply be multiplied to obtain the total marginal likelihood.Notice, however, that the marginal likelihood still depends on the priors on the nuisanceparameters. If the nuisance parameters are well-constrained this dependence will be small,but in cosmology this is necessarily not always the case.– 5 – Cosmological analysis
In our cosmological analysis, we use data on Cosmic Microwave Background, large scalestructure (LSS) baryon acoustic oscillations (BAO) measurements, Hubble constant H ,and galaxy cluster counts. In particular, we define the following data combinations: • CMB: It includes the current Planck data [38] of the temperature anisotropy upto l = 2479, the high multipole values (highL), coming from ACT [39] and SPTdata [40, 41], that covers respectively the 500 < l < < l < • BAO: It includes the Data Release 11 (DR11) sample of the recent measurements bythe BOSS collaboration [72]. The DR11 sample is the largest region of the Universeever surveyed, covering roughly 8500 square degrees, with a redshift range 0 . 7. The measure of the sound horizon at the drag epoch has been evaluatedat redshift z =0.32 and at z =0.57, finding values in agreement with previous BAOmeasurements . • BICEP2: the 9 channels of the CMB BB polarization spectrum recently released bythe BICEP2 experiment [42]. • HST: the data from the Hubble Space Telescope [77] on H , obtained through thedistance measurements of the Cepheids: H = (73 . ± . 4) km s − Mpc − . (3.1) • PlaSZ: The counts of rich cluster of galaxies from the sample of Planck thermalSunyaev-Zel’Dovich catalogue [78]. It constrains the combination σ (Ω m / . . =0 . ± . In order to test the dependence of our results on the inclusion of the recent BICEP2data and on the tension with local HST and cluster PlaSZ results we perform the analysiswith three different combinations of the data sets above that we label: • DATA SET 1: CMB+BAO, where, as described above, CMB=Planck+WP+highL+lensingdata, and BAO=DR11. We have verified that the inclusion of the previous determinations of the BAO from the Sloan DigitalSky Survey (SDSS) Data Release 7 (DR7) [73, 74], the 6dF Galaxy Survey (6dFGS) [75], and WiggleZmeasurements Ref. [76] does not affect our results and therefore for the sake of simplicity we have notincluded them in the analysis. For simplicity we do not include the cosmic shear data of weak lensing from the Canada-French-Hawaii Telescope Lensing Survey (CFHTLenS) which constraints σ (Ω m / . . = 0 . ± . – 6 – DATA SET 2: CMB+BAO+BICEP2. We add to the previous data set, the resultsfrom BICEP2. • DATA SET 3: CMB+BAO+BICEP2+HST+PlaSZ. We add to the previous data setthe results from HST and Planck SZ counts of galaxy clusters. We consider in our analysis a ΛCDM cosmology extended with a free scalar-to-tensor ratio,and three active plus one sterile neutrino species with a hierarchical neutrino spectra ofthe 3+1 type which we denote as ΛCDM+ r + ν s . In this case the three active neutrinoshave masses m i =1 , (cid:46) (cid:112) | ∆ m | while the forth sterile neutrino has a mass m ≡ m s .As mentioned in the introduction, in the absence of other form of new physics thecontribution of the sterile neutrino to the energy density is completely determined by itsmass and its mixing with the active neutrinos. However in extended scenarios this maynot be the case. So generically we will consider that in the 3+1 scenario, irrespective of m s and mixings, the sterile neutrino contributes to ρ R as∆ N eff ≡ F NT ∆ N ,F Teff = F NT (3.2)where F NT is an arbitrary quantity which quantifies the departure from the fully-thermalizedactive-sterile neutrino scenario and which we will consider to be independent of T in therelevant range of T in the analysis. So in what follows we will label as ∆ N eff ≡ F NT thisparameter.The effect of m s is included in the analysis via the effective parameter m eff = (94 . s h , (3.3)with being h the reduced Hubble constant and Ω s ≡ ρ s /ρ c , with ρ s the sterile neutrinoenergy density and ρ c the current critical density. This effective mass is not equal to thephysical mass m s in general, and their relation depends on the assumed phase-space dis-tribution of the sterile neutrinos. For thermally distributed sterile neutrinos characterizedby a temperature T s (in general different from the temperature of the active neutrinos T ν ) m eff = (∆ N eff ) / m s , (3.4)while if they are produced by non-resonant oscillations (the so-called Dodelson-Widrowscenario) (DW) [82] the resulting phase-space distribution of the sterile neutrinos is equalto that of the active neutrinos up a constant factor. In this case m eff = ∆ N eff m s . (3.5)Altogether our analysis contain nine free parameters { ω b , ω c , Θ s , τ, log[10 A s ] , n s , r, m eff , N eff } , (3.6)where ω b ≡ Ω b h , ω c ≡ Ω c h being the physical baryon and cold dark matter energydensities, Θ s the ratio between the sound horizon and the angular diameter distance at– 7 –arameter PriorΩ b h . → . c h . → . s . → τ . → . A s ] 2 . → n s . → . r → m eff → N eff . → Table 2 . Uniform priors for the cosmological parameters considered in the analysis. The activeneutrinos have been fixed to one massive with a mass of 0.06 and two massless. In addition followingRef. [38] an upper constraint on m s defined in Eq. (3.4) of 10 eV is imposed which roughly definesthe region where the sterile neutrinos are distinct from cold or warm dark matter. decoupling, τ is the reionization optical depth, A s the amplitude of primordial spectrum, n s the scalar spectral index, and r the scalar-to-tensor ratio. To generate the marginalizedlikelihoods we use the CosmoMC package [83], implemented with the Boltzmann CAMBcode [84].The parameters in (3.6) are assigned uniform priors with limits as given in Tab. 2. Sincewe are interested in the constraints on sterile neutrino parameters, we follow Sec. 2.2, andaim to evaluate the marginal likelihoods of ( m eff , N eff ), which will then be used for thetests presented in Sec. 4. Thus in this analysis we consider { ω b , ω c , Θ s , τ, log[10 A s ] , n s , r } as the cosmological nuisance parameters . Note that we have employed uniform priors on m eff and N eff in the numerical analysis but that the marginal likelihoods to be presentedbelow do not depend on the priors, since these are “factorized out”. Furthermore, since thenuisance parameters are rather well-constrained the precise vales of the limits in Tab. 2 donot not affect any results, and also employing different shapes on these parameters wouldhave a small impact. The exception is r which is not so well-constrained (especially forcertain data sets) and for which the physical lower limit is important. The results of our analysis for the three data set combinations described in Sec. 3.1 areshown in Figs. 1 and 2.In the the upper left panel of Fig. 1 and in the left panels of Fig. 2 we plot the contoursof the marginal likelihood L ( N eff , m eff ) normalized to the value of L (∆ N eff = 0) (whichdoes not depend on m eff ) . The red contour delimits the regions for which m s in Eq. (3.4)exceeds 10 eV for which the sterile states becomes indistinguishable from cold or warm dark We do not include the lensing amplitude A L as a free nuisance parameter, even when adding the localmeasurements on σ and Ω m , for which a significant deviation from the standard value of unity is preferred.We have checked that adding A L only slightly shifts the preferred region for m eff to higher values. For likelihoods more than eight log-units away from the maximum value, we extrapolate using a con- – 8 –atter [38] and hence for values of the parameters below the curve, the marginal likelihoodis not evaluated. Black dashed contours are those of − L / L max ) < . , . , . σ , 2 σ , and 3 σ levels in two dimensions.In the upper right panel of Fig. 1 and the right panels of Fig. 2 we show the contoursof the marginal likelihood of ( N eff , m s ) for the thermally distributed sterile neutrinos whilein the lower left panel of Fig. 1 we show the corresponding marginal likelihood in the DWscenario. We see that for both DW and thermal ν s scenarios, m s becomes increasinglylarge for decreasing ∆ N eff , hence the distinctive appearance of large “flat” regions andweak constraints on m s for small ∆ N eff . Also as seen in Fig. 1 the results for the DWand the thermal ν s scenarios are qualitatively very similar. Only, since for fixed m eff and∆ N eff , m DW s = m TH s ∆ N − / , for ∆ N eff ≤ N eff increases.The impact of BICEP2 data can be seen by comparing the upper panels in Fig. 1 andFig. 2. The addition of BICEP2 data gives a preference for large values of the tensor-to-scalar ratio r and, due to its correlation with ∆ N eff , and therefore leads to the the shift toslightly larger values of ∆ N eff observed in upper panels in Fig. 2. Since r is now betterconstrained, also the preferred region becomes slightly smaller.The effects of adding HST+PlaSZ in the analysis are displayed in the lower panels inFig. 2 where we see a shift to larger values of both ∆ N eff and m eff . This is so becausethe constraints on σ of the PlaSZ measurement, which are in tension with the otherexperiments within this model, can be somewhat alleviated by an increase in m eff , whilethe inclusion of HST yields an increase ∆ N eff .All these results are in qualitative agreement with those in the analyses in Refs. [44–50]. However we notice that by showing the marginal likelihood we are explicitly notassigning any priors to the sterile parameters. This is an advantage in the absence of aphysical motivation for them, especially m s , since the data at hand is expected to leave alarge prior dependence. For example, if one used a prior on m s which is uniform in log m s instead of in m s , as in general the likelihood is non-negligible for a vanishing sterile mass,the derived Bayesian constraints on m eff and ∆ N eff would be very different. In principleone could embark on an extensive prior sensitivity analysis, but in this work we will insteadfocus on analyses for which the results have little or no prior dependence.Finally, in order to better illustrate how the constraints depend on the sterile masswe plot in Fig. 3, the slices of the marginal likelihood as a function of m s for fixed ∆ N eff .Also shown in the figure is the marginal likelihood for the SBL analysis in the 3+1 scenario(marginalized with respect to the lighter neutrino masses and all mixings) as given in Fig.1in Ref. [52]. In this section we perform the statistical tests on the 3+1 scenarios invoked to explainthe SBL anomalies using the results of the cosmological analysis presented in the previous stant value. It could be made more accurate by using non-constant functions such as polynomials but noqualitative change is expected. – 9 – MB+BAO Thermal ν s Dodelson-Widrow scenario Figure 1 . Marginal likelihood of ( m eff , ∆ N eff ) (upper right panel) and of ( m s , ∆ N eff ) for thermallydistributed ν s (upper left panel) and for the for DW scenario (lower panel) for the CMB+BAOcosmological data set ( SET 1 ). Black dashed contours are those of − L / L max ) < C , whichwould correspond to nominal 1,2,3 sigma levels. The red line denotes the region for which m s =10 eV for thermal ν s . section. In doing so we are going to assume that the contribution of the sterile neutrinoto the cosmological observables is independent of the active-sterile neutrino mixing (seediscussion around Eq. (3.2)). Under this assumption our tests will require the marginalizedlikelihood of the sterile neutrino mass m s obtained from the analysis of the SBL anomaliesin the 3+1 scenario (marginalized with respect to all other oscillation parameters in thescenario), and the marginalized likelihoods of ( m s , ∆ N eff ) from the cosmological analysispreviously presented.Concerning the SBL likelihood, we consider two different functions that can be inter-preted as the two limiting cases. In the first, we consider the precise SBL likelihood asgiven in Fig. 1 in Ref. [52] which we reproduce in Fig. 3 (in this case below roughly sevenlog-units from the maximum value, we set the SBL likelihood to a constant value). We– 10 – MB+BAO+BICEP2 Thermal ν s CMB+BAO+BICEP2+HST+PlaSZ Figure 2 . Same two upper panels in Fig. 1, but for CMB+BAO+BICEP2 cosmological data ( SET2 , upper panels), and CMB+BAO+BICEP2+HST+PlaSZ cosmological data ( SET 3 , lower panels). label this case in the following as “full SBL likelihood”. In the second case, we approximatethe SBL likelihood as a top-hat shaped likelihood which is constant and non-zero between0.86 eV and 1.57 eV and zero otherwise (which we illustrate by the arrow in Fig. 3). Thisis what we label in the following as “box SBL likelihood”. The first question we want to address is whether the current data shows evidence of theexistence of sterile neutrinos, and how strong this evidence is. Generically in Bayesiananalyses this takes the form of model comparison between a model without sterile neutri-nos and a model with sterile neutrinos using some posterior odds, Eq. (2.2). There areseveral ways to go about answering this question. In here we are interested in testing whatcosmology has to say on this comparison for the sterile models invoked to explain the SBLanomalies. This is, in this case the first model H is defined as a model with no sterileneutrino, which implies a cosmological model with ∆ N eff = 0. And the other model, H – 11 –MB+BAO CMB+BAO+BICEP2 m s /eV l og ( L ( ∆ N e ff , m s ) / L ( ∆ N e ff = )) SBL m s /eV l og ( L ( ∆ N e ff , m s ) / L ( ∆ N e ff = )) SBL CMB+BAO+BICEP2+HST+PlaSZ m s /eV l og ( L ( ∆ N e ff , m s ) / L ( ∆ N e ff = )) SBL Figure 3 . Marginal likelihoods as function of m s for fixed ∆ N eff (∆ N eff = 0 . , . , . , 1) andfor thermal ν s (solid lines) and for the DW scenario (dashed lines). We show the results for thethree cosmological data sets used as labeled in the figure. In all panels we also include the marginallikelihood for the SBL analysis in the 3+1 scenario (marginalized with respect to the lighter neutrinomasses and all mixings) as given in Fig.1 in Ref. [52]. We denote by the red arrow the width andheight of the box used to define “box SBL likelihood” (see text for details). is taken to include one sterile neutrino of mass m s as required to accommodate the SBLanomalies and which contributes to relativistic energy density in the early Universe as somefix ∆ N eff . – 12 –n this case we can define a posterior odds as: O = Pr( D c | D SBL , H )Pr( D c | D SBL , H ) Pr( H | D SBL )Pr( H | D SBL ) == Pr( D c | D SBL , H )Pr( D c | D SBL , H ) Pr( D SBL | H )Pr( D SBL | H ) Pr( H )Pr( H ) (4.1) ≡ B upd10 B SBL10 Pr( H )Pr( H ) , with B SBL10 ≡ Pr( D SBL | H )Pr( D SBL | H ) (4.2)and B upd10 ≡ Pr( D c | D SBL , H )Pr( D c | D SBL , H ) = Pr( D c | D SBL , H )Pr( D c | H ) = Pr( D c , D SBL | H )Pr( D c | H ) Pr( D SBL | H ) , (4.3)where in the last line we have used that SBL is not sensitive to any parameters effectingcosmology once we assume that there are no sterile neutrinos and hence D SBL and D c areindependent under H . The quantity B upd10 quantifies how much better the prediction ofcosmological data assuming sterile neutrino and SBL data is than the prediction assumingno sterile neutrinos.Now, the model H is inherently more complex than H since it contains additional pa-rameters (sterile mass and mixings), and this is as usual compensated for in a Bayesian ana-lysis. In the first row of Eq. (4.1) this is contained in the last factor of Pr( H | D SBL ) / Pr( H | D SBL ) = B SBL10 Pr( H )Pr( H ) . Now, the best-fit of the SBL data is significantly better if you have a sterileneutrino, but because of the added complexity it might not be totally unreasonable to havePr( H | D SBL ) / Pr( H | D SBL ) = O (1).In any case, it is the first factor in Eq. (4.1) the factor by which the cosmological datahave updated the SBL-only posterior odds to the final SBL+cosmology odds and whichwe will be using in our quantification. Furthermore, under the additional assumptionthat ∆ N eff is unconstrained by SBL data, Pr( D SBL | H ) does not depend on ∆ N eff , andhence B upd10 is in fact simply proportional to the combined marginal likelihood of ∆ N eff ,normalized such that B upd10 = 1 for ∆ N eff = 0.Before discussing the results, let us mention that here, as in any Bayesian analysis,the results are in principle always prior dependent, and we should consider how large thisdependence is in practice. First, as discussed in Sec. 3, the marginal likelihood depends onthe priors on the cosmological nuisance parameters, but this dependence is expected to besmall (except possibly for r ). More significantly, there is the dependence of the total Bayesfactor and odds on the prior on m s , even when it is well constrained by the combined dataset. However the value of B upd10 does not strongly depend on the the shape of the prior norits upper limit. This is so because the well-constraining SBL data is used to update theprior to a posterior (which is rather insensitive to the prior) which is then used to analyzethe cosmological data. We use a uniform prior between 0 and 10 eV as the nominal upper– 13 –imit, since, as described in Ref. [38], this roughly defines the region where (for the CMB)the particles are distinct from cold or warm dark matter.The results are shown in the left panel of Fig. 4. We see that for the CMB+BAOand CMB+BAO+BICEP2 data sets, B upd10 decreases quite steadily with ∆ N eff . Thesecosmological data hence disfavour models with sterile neutrinos required to explain theSBL anomalies over the model without sterile neutrinos independently of how much thecontribution of the sterile neutrino to the energy density is suppressed with respect to thefully thermalized expectation. We also see that the addition of the BICEP2 data has asmall impact on these conclusions.For the CMB+BAO+BICEP2+HST+PlaSZ data set, there is, on the contrary, a sig-nificant peak for intermediate values of ∆ N eff . The ΛCDM+ r model is significantly dis-favoured by this combination of cosmological data, while increasing ∆ N eff increases thecosmological likelihood for SBL-compatible masses. However, further increasing ∆ N eff ,cosmology requires a too small mass, so the B upd10 decrease again. Notice also that infact the ΛCDM+ r is so disfavored that it is in the region where the (too large) constantextrapolation is used. Hence, the exact B upd10 is expected to be even larger. In addition to comparing the models with and without sterile neutrinos we now addressthe question of the consistency of the parameter constraints from the different data setswithin the 3+1 model. A Bayesian test was formulated in [85], in which a model whereboth data sets are fitted by the same physical parameters is compared with a model inwhich each data set uses their own parameters. However, as is often the case in modelcomparison, the result can depend crucially on the prior on the parameters, in our case m s in particular. Since we cannot motivate using a specific shape or limits on the prior,we instead use the corresponding χ -test (these were also compared in [85]). Although notrigorous as the Bayesian test, it has the clear advantage that it is prior-independent.In particular, if we want to test how inconsistent the constraints on the sterile massfrom the different data sets are once a certain ∆ N eff is assumed, i.e. , without consideringhow favoured or disfavoured that ∆ N eff is by the cosmological data, we should evaluate∆ χ (∆ N eff ) = ˆ χ (∆ N eff ) − ˆ χ − ˆ χ (∆ N eff ) , , (4.4)where the hat denotes the value at the best fit i.e. , optimized over m s ( χ does notdepend on ∆ N eff ).The results of this test are shown on the right-hand side of Fig. 4. As expected, theresults for CMB+BAO and CMB+BAO+BICEP2 are quite similar and show an steadyincrease of the inconsistency with ∆ N eff . Also comparing the DW and thermal scenarios,we see that in general to obtain the same ∆ χ larger values of ∆ N eff are required in theDW scenario. This is so because, as explained before, in the DW scenario the preferredmasses are shifted to larger values by an amount which increases as ∆ N eff < m s , and over this wide range the cosmological likelihoodtypically varies significantly. The combined χ can then easily be reduced by finding thebest fit within this range.For the CMB+BAO+BICEP2+HST+PlaSZ combination, one observes in each curvea peak for small value of ∆ N eff , although it does not reach the level of 2 . σ . These peaksare due to the fact that (cf. Fig. 2) cosmology prefers large physical masses, too large to fitthe SBL data. As ∆ N eff increases, the mass constraints become compatible, but as ∆ N eff continues to decrease, cosmology requires the masses to be smaller than those which canfit the SBL data and the inconsistency increases.Again, we stress that this consistency test is in principle not affected by how favouredor disfavoured the considered value of ∆ N eff is, but instead consider that value to be “true”,and then test the compatibility of the constraints on the mass. For example, comparing theleft and right panels in the last raw of Fig. 4 we see that for the CMB+BAO+BICEP2+HST+PlaSZdata set, even though from the right panel we read that the mass ranges required for cos-mology and SBL become highly incompatible for ∆ N eff close to one, from the left hand sidewe see that these large values of ∆ N eff are not particularly disfavoured compared to small∆ N eff , for which the mass constraints are compatible. So what we see is that large ∆ N eff is disfavoured because the sterile mass required by SBL and cosmology are incompatibleand small ∆ N eff is disfavoured because it is so in the cosmological (and consequently inthe combined) analysis. So small and large ∆ N eff have comparable Bayes factors, but thisis only because they are both disfavoured. In this paper we have revisited the question of the information which cosmology provideson the scenarios with O(eV) mass sterile neutrinos invoked to explain the SBL anoma-lies (Eq. (1.2)) using Bayesian statistical tests and study how the results depend on theinclusion of the recently CMB polarization results of BICEP2 and on the inclusion of lo-cal measurements which show some tension with the Planck and LSS-BAO results whenanalyzed in the framework of the ΛCDM scenario.In order to do so we have first performed an analysis of three characteristic sets ofcosmological data in ΛCDM+ r + ν s cosmologies as described in Sec. 3. The result of ouranalysis is presented in Figs. 1 and 2 in the form of marginalized cosmological likelihoodsin terms of the two relevant parameters, the sterile neutrino mass m s and its contributionto the energy density of the early Universe ∆ N eff . The results clearly indicate that as longas the HST and SZ cluster data from Planck are not included, cosmological data favoursthe sterile neutrino mass m s clearly well below eV unless its contribution to the energydensity is suppressed with respect to the expected from a fully thermalized sterile neutrino.The inclusion of the BICEP2 data does not substantially affect this conclusion. Converselyincluding these HST and SZ cluster data higher sterile masses become favoured.With these results, we have performed in Sec. 4 two statistical test on their (in)compatibilitywith the corresponding likelihood derived from the analysis of the SBL results as given inRef. [52]. In the first test we have asked ourselves whether cosmology favours or disfavours– 15 – vidence Scenario SET 1 SET 2 SET 3/SBL Likel 3+1 Disfavoured 3+1 Disfavoured 3+1 FavouredTH/FULL [0 . , . 40] [0 . , . 40] [0 . , . ⊕ ≥ . . , . 44] [0 . , . 45] [0 . , . ⊕ [0 . , . . , . 51] [0 . , . 52] [0 . , . ⊕ ≥ . . , . 55] [0 . , . 58] [0 . , . ⊕ [0 . , . . , . ≥ . 40 [0 . , . . , . 66] [0 . , . ⊕ [0 . , . 68] [0 . , . . , . ≥ . 52 [0 . , . . , . 72] [0 . , . 83] [0 . , . ≥ . 86 – –Strong TH/BOX ≥ . 66 [0 . , . ⊕ ≥ . 68 –DW/FULL ≥ . 86 – –DW/BOX ≥ . ≥ . 83 – Table 3 . Ranges of ∆ N eff for which we find the evidence against or in favour of the 3+1 modelcompared to the model with only the 3 active neutrinos to be weak, moderate or strong. the 3+1 sterile models which explain the SBL results over a model without sterile neutri-nos. In order to do so we have constructed the Bayes factor defined in Eq. (4.3) which givesthe factor by which the cosmological data updates the SBL-only posterior odds of the 3+1vs 3+0 model to the final SBL+cosmology odds and we have studied its behaviour as afunction of ∆ N eff . The results of this test, shown in Fig. 4 implies that as long as the HSTand SZ cluster data from Plank is not included, the cosmological analysis disfavour the3+1 model with respect to 3+0. The inclusion of these cosmological data however favoursthe 3+1 model for an intermediate range of ∆ N eff . We summarize in Tab. 3 the rangesof ∆ N eff for which we find the evidence against or in favour of the 3+1 model comparedto the model with only the 3 active neutrinos to be weak, moderate or strong from theseanalyses.The second test performed deals with the (in)compatibility of the sterile mass con-straints as required to describe SBL and cosmology. For this we have evaluated the ∆ χ defined in Eq. (4.4) which we plot in the right panels in Fig. 4. Altogether we read thatthis test yields inconsistency on the m s required by cosmology and SBL larger than 3 σ for∆ N eff ≥ . 45 (0 . 54) [0 . 76] ([0 . , ∆ N eff ≥ . 39 (0 . 50) [0 . 65] ([0 . , (5.1)∆ N eff ≥ . 56 (0 . 67) [0 . 83] ([0 . , for thermal (WP) ν s scenario for the full [box] SBL likelihood.In summary, we find that the analysis of cosmological results from temperature andpolarization data on the CMB as well as from the BAO measurements from LSS data dis-favours the 3+1 sterile models introduced to explain the SBL anomalies over the scenariowithout sterile neutrinos, and also that their allowed/required ranges of m s are incompati-ble. This is so even if new physics is involved so that the contribution of the sterile neutrinoto the energy density of the Universe (and therefore to the cosmological observables) is sup-– 16 –ressed with respect to that of the fully thermalized case resulting from its mixing with theactive neutrinos. When the local measurement of the H by the Hubble Space Telescope,and the cluster SZ cluster data from the Planck mission is included, compatibility can befound between cosmological and SBL data, but still requires a substantial suppression ofthe ν s contribution to ρ R . Acknowledgments This work is supported by USA-NSF grants PHY-09-69739 and PHY-1316617, by CURGeneralitat de Catalunya grant 2009SGR502 by MICINN FPA2010-20807 and consolider-ingenio 2010 program grants CUP (CSD-2008-00037) and CPAN, and by EU grant FP7ITN INVISIBLES (Marie Curie Actions PITN-GA-2011-289442). J.S. acknowledges sup-port from the Wisconsin IceCube Particle Astrophysics Center (WIPAC) and U. S. De-partment of Energy under the contract DE-FG-02-95ER40896.– 17 –MB+BAO ∆ N eff l og ( B ) weakweakmoderatemoderatestrongstrong Full SBL likelihoodBox SBL likelihood ∆ N eff p ∆ χ Full SBL likelihoodBox SBL likelihood CMB+BAO+BICEP2 ∆ N eff l og ( B ) weakweakmoderatemoderatestrongstrong Full SBL likelihoodBox SBL likelihood ∆ N eff p ∆ χ Full SBL likelihoodBox SBL likelihood CMB+BAO+BICEP2+HST+PlaSZ ∆ N eff l og ( B ) weakweakmoderatemoderatestrongstrong Full SBL likelihoodBox SBL likelihood ∆ N eff p ∆ χ Full SBL likelihoodBox SBL likelihood Figure 4 . Left panels:Logarithm of the Bayes factor B upd10 as a function of ∆ N eff . Right panels:Consistency of mass constraints. 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