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Physics Letters B 581 (2004) 218–223

www.elsevier.com/locate/physlet

WMAPping out neutrino masses

Aaron Piercea, Hitoshi Murayamab,c

a Theoretical Physics Group, Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309, USAb Department of Physics, University of California, Berkeley, CA 94720, USA

c Theory Group, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

Received 3 November 2003; accepted 27 November 2003

Editor: H. Georgi

Abstract

Recent data from the Wilkinson microwave anisotropy probe (WMAP) place important bounds on the neutrino secprecise determination of the baryon number in the universe puts a strong constraint on the number of relativistic specBig Bang nucleosynthesis. WMAP data, when combined with the 2dF galaxy redshift survey (2dFGRS), also directly cthe absolute mass scale of neutrinos. These results impinge upon a neutrino oscillation interpretation of the resultliquid scintillator neutrino detector (LSND). We also note that the Heidelberg–Moscow evidence for neutrinoless doudecay is only consistent with the WMAP+ 2dFGRS data for the largest values of the nuclear matrix element. 2004 Published by Elsevier B.V.

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1. Introduction

Evidence for neutrino oscillation has steadmounted over the last few years, culminating in a pture that presents a compelling argument for finite ntrino masses. The observation of a zenith-angle dedent deficit ofνµ from cosmic ray showers at supeKamiokande [1], provided strong evidence for osciltions in atmospheric neutrinos. Recent results on sneutrinos at the sudbury neutrino observatory (SN[2] and reactor neutrinos at the KamLAND expement [3], have shed light on the solar neutrino prlem. These experiments have provided strong evidethat the solar neutrino problem is solved by osci

E-mail address: apierce@slac.stanford.edu (A. Pierce).

0370-2693/$ – see front matter 2004 Published by Elsevier B.V.doi:10.1016/j.physletb.2003.11.075

tions corresponding to the large mixing angle sotion [4]. Although clear oscillation data now existatmospheric, reactor, and solar neutrino experimeit remains to determine the significance of the resfrom the liquid scintillator neutrino detector (LSND[5,6], which claimed evidence for conversion ofνµ toνe with am2

ν of order 1 eV2.While these extraordinary advances in experim

tal neutrino physics were occurring, a concurrent rolution in experimental cosmology took place. Usered in by the Boomerang, MAXIMA, and DASI mesurements of the acoustic peaks in the cosmiccrowave background (CMBR) [7], an era has begwherein it is possible to make measurements of cmological parameters with previously unimaginaprecision. Most recently, the striking data [8] from tWilkinson microwave anisotropy probe (WMAP) ha

A. Pierce, H. Murayama / Physics Letters B 581 (2004) 218–223 219

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vastly improved our knowledge of several fundamtal cosmological parameters [9]. Because cosmolwould be significantly affected by the presence of ligspecies with masses of order 1 eV, the new WMdata strongly constrain neutrino masses in this ranWe will show this brings cosmology into some conflwith the LSND result in two ways.

First, WMAP determines the baryon to photontio very precisely. This removes an important sourceuncertainty in the prediction of Big Bang nucleosythesis (BBN) for the primordial abundance of4He.This allows for a strong limit to be placed on the nuber of relativistic species present at BBN, disfavorthe LSND result. Secondly, WMAP, when combinwith data from the 2 degree field galactic redshift svey (2dFGRS) [10], CBI [11], and ACBAR [12], iable to place stringent limits on the amount that ntrinos contribute to the critical density of the univerThis second constraint results in an upper mass-lon neutrinos that contradicts the LSND result in all bone “island” of parameter space not ruled out by otexperiments. The second constraint also impingethe recent evidence for neutrinoless double beta defrom the Heidelberg–Moscow experiment [13].

2. The LSND result

The LSND experiment used decays of stopped amuons at the LAMPF facility (Los Alamos) to loofor the appearance of anti-electron-neutrinos. Treported the oscillation probabilityP(νµ → νe) =(0.264±0.067±0.045)%,representing a 3.3σ signal.

If the result at the LSND experiment were a truedication of oscillations, it would have profound impcations for our understanding of neutrinos. Solar aatmospheric neutrinos have already determinedneutrino mass-squared differences to bem2

solar ∼10−4 eV2 and m2

atm ∼ 10−3 eV2. However, tak-ing into account the Bugey exclusion region [14], tLSND experiment points to a mass difference (Fig. 1) m2

LSND > 10−1 eV2. The presence of thicompletely disparate mass difference necessitateintroduction of a fourth neutrino.1 Because LEP ha

1 This statement assumes CPT. If neutrinos and anti-neutrhave different mass spectra, it may still be possible to accommo

Fig. 1. The LSND Allowed region, with Bugey and Karmen [1exclusion regions. The constraints from the global fit [16] as was the limit of [17] from the combination of WMAP and 2dFGRdata are also shown. There are two lines, corresponding to3+ 1 (normal) and 1+ 3 (inverted) spectra. The contours from tglobal fit would, of course, continue on to lower values ofm2, butRef. [16] did not show this region.

Fig. 2. Sample neutrino spectra in the light of LSND. Differepermutations are also possible.

determined the number of active neutrino speciebe three, this fourth neutrino must be sterile, havextraordinarily feeble couplings to the other particof the standard model.

The introduction of this fourth neutrino speciresults in principle in two characteristic typesspectra, 2+ 2 and 3+ 1. Two sample spectra of thetypes are shown in Fig. 2.

LSND together with solar, reactor, and atmospheric neutrinowithin three generations alone [18]. However, see also [19].

220 A. Pierce, H. Murayama / Physics Letters B 581 (2004) 218–223

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However, recent results from SNO [2] and supKamiokande [20], have indicated that the oscillatioresponsible for the atmospheric and solar neutanomalies involve transitions primarily between ative neutrinos. This means that it is difficult to put tsterile part of the neutrino in either the solar ormospheric pair in the 2+ 2 spectrum. A recent quantitative analysis [19] found this 2+ 2 spectrum to becompletely ruled out, while a 3+ 1 spectrum was allowed at the 99% confidence level [16,19]. The tesion for the 3+ 1 spectrum is in large part due to thlack of a signal in short-baseline disappearanceperiments such as CDHSW [21] and Bugey. Addadditional sterile neutrinos can only marginally improve this agreement [22]. In the next two sections,show how this allowed window is further constrainby cosmological considerations.

3. Big Bang nucleosynthesis

By measuring the primordial abundance of4He,one can place bounds on extra relativistic degrof freedom at the time of Big Bang nucleosynthe(BBN) [23]. These bounds are usually quotedterms of a number of effective allowed neutrispecies,Neff

ν . Additional degrees of freedom tendincrease the expansion rate of the universe, whcauses neutrons to freeze out at an earlier timea higher abundance. This abundance translatesmore primordial4He for a given baryon to photoratio, η. Therefore, knowledge of primordial4Heabundance along with a separate determinationηplaces a bound onNeff

ν . On the other hand, forfixed Neff

ν , a higherη results in a higher abundancfor primordial 4He; so, incomplete knowledge ofηdegrades the constraint onNeff

ν .In the era before precise CMBR measureme

BBN data alone were utilized to set the bound. Msurements of primordial deuterium or lithium weused for the separate determination ofη. An aggres-sive analysis by [24] cited a limit ofNeff

ν < 3.4 at 2σ ,and consequently found that LSND data were strondisfavored by BBN [25]. However, the data for primodial light element abundances were somewhat mdled, with some measurements of lithium and dterium preferring substantially lower values ofη thanothers. Due to the presence of these data, a cons

-

tive boundNν < 4 was often taken [26]. In fact, usinlithium data alone, [27], found that evenNeff

ν = 4.9was acceptable at the 95% confidence level.

However, after precise measurements of the CMthe situation has changed. The WMAP experimentdetermined [9]Ωbh

2 = 0.224± 0.001, corresponding to anη = 6.5+0.4

−0.3 × 10−10. For the central valueabove, the expected4He abundance,Yp , is roughlyYp = 0.249+ 0.013(Neff

ν − 3). The status of primordial helium measurements remains controversial. Ohelium measurement quotes a valueYp = 0.244±0.002 [29], while another quotesYp = 0.235± 0.002[28]. To deal with the discrepancy in these measuments, the particle data group (PDG) assigns an ational systematic error, takingYp = 0.238± 0.002±0.005 [30]. To be completely conservative, we wtake the higher helium abundance, and assign to iadditional systematic error of the PDG, namely,takeYp = 0.244± 0.002± 0.005. Using the formulae of [31] for 4He in terms ofNeff

ν and η, we findNeff

ν < 3.4 at the 95% (two-sided) confidence levleaving no room for the extra neutrino of LSND. Usinthe only slightly less conservative approach of adoing the PDG central value and error, we findNeff

ν < 3.0at the 95% (two-sided) confidence level.

Of course, additional systematic errors in thelium abundance measurements may be found. Thethat 3 neutrinos is barely consistent at the 95% cfidence level might cause some suspicion that thare unknown systematics at work. However, toNeff

ν = 4 at the 95% level would require inflating therrors on the PDG central value dramatically, toYp =0.238± 0.011.

It is possible that an asymmetry in the leptons coeffectively prevent the oscillation into sterile neutnos [32]. We find that a large pre-existing asymmtry of L(e) ∼ 10−2 would be sufficient to suppress thproduction of sterile neutrinos below the BBN costraint. Here,Le, represents the total asymmetry fby electron neutrinos,Le = 2Lνe + Lνµ + Lντ , withLνi = (Nνi −Nνj )/Nγ . SmallerLe (as low as∼ 10−5)can suppress sterile neutrino production, but osctions tend to erase asymmetries of this size. Whilepton asymmetry of 10−2 size does not bias the rafor the processes such asn + e+ → p + νe signifi-cantly enough to affect BBN, an asymmetry as laas 10−1 would. CMBR constraints also cannot excluthe possibility of a lepton asymmetry of 10−2, so this

A. Pierce, H. Murayama / Physics Letters B 581 (2004) 218–223 221

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possibility cannot be excluded. It does not appearneutrino oscillations themselves can create this asmetry [33]. One has to assume that the asymmetryisted before the BBN, possibly generated by a mecnism similar to that in [34].

4. Weighing neutrinos with large scale structure

WMAP has provided an additional constraintLSND. As noted, for example, in [35], galactic surveprovide a powerful tool to constrain the massesneutrinos. Neutrinos decouple at temperatures wabove those at which structure forms. They then frstream until they become non-relativistic. This tento smooth out structure on the smallest scales.scales within the horizon when the neutrinos wererelativistic, the power spectrum of density fluctuatiois suppressed as [35]:

(1)Pm

Pm

≈ −8Ων

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.

The 2dFGRS experiment used this fact to place a lon the sum of neutrino masses:

∑mν < 1.8 eV [10].

Recent data from WMAP greatly improve thmeasurement. A key contribution is the fact thWMAP and 2dFGRS overlap in the wavenumbprobed. This allows a normalization of the 2dFGRpower spectrum from the WMAP data. The WMAsatellite also precisely determinesΩm. Since depletionof power at small scales is sensitive to the ratioΩν/Ωm, a more accurate determination ofΩm leadsto a better bound on the neutrino mass. The ultimresult from combining data from 2dFGRS, ACBACBI, and WMAP isΩνh

2 < 0.0076 (95% confidenclevel) [9]. The bound onΩνh

2 places the boundmassesmν < 0.23 eV (3 degenerate neutrinos, 95confidence level). Note that using the WMAP da[17] finds a more conservative bound of

(2)mν < 0.33 eV (3 degenerate neutrinos, 95% CL).

The primary difference in the bounds is that [1allows the bias factor to float in the analysis.

In the case where there are four neutrinos,bound on neutrino masses is somewhat relaxednoted by [36], the bound on

∑mν is anti-correlated

with the value of the Hubble constant. On the oth

hand, limits onNν are correlated with Hubble constant. Playing these two effects against one anothelows the weakening of bounds onmν for Nν = 4. Forthe 3+ 1 spectrum shown in Fig. 2, again allowing tbias parameter to float, [17] finds a bound of

(3)mν < 1.4 eV (3+ 1 neutrinos, 95% CL).

This bound was derived assuming three degenerattive neutrino species. In the case where the neutrare not all degenerate, in principle one might expthe bound to by slightly modified, as the scale whfree-streaming stops would be shifted. In practihowever, this has only a very small quantitativefect [37], so we negelect it in our discussion. Also,above mass limit was placed assuming that the heneutrino has standard model couplings. These cplings determine when the neutrino decouples frthermal equilibrium. If the neutrino decoupled sufciently early, it might have been substantially dilutrelative to the active neutrinos. Consequently, it cocontribute a relatively small amount to the critical desity today. However, we do not expect this to becase for an LSND neutrino. While one must be caful to take into account plasma effects, [38], that migkeep sterile neutrinos out of equilibrium at high teperatures, these become negligible in time for LSneutrinos to thermalize before decoupling. Ref. [3found that an additional sterile neutrino in a 3+ 1scheme was nearly completely thermalized overentire favored LSND mixing region. Since the netrino ultimately decouples at temperatures of or10 MeV, abundance of these neutrinos will not beluted by the entropy produced at the QCD phase tsition. This assures us that the limit of Eq. (3) is aplicable for the heavy LSND neutrino as well.

Fitting the LSND result within a two neutrinoscillation picture requires (see Fig. 1) a neutrino mgreater than the square-root of smallest allowedm2.This givesmν 0.45 eV. Comparing this with thebound on the neutrino mass in the 3+ 1 scheme,Eq. (3), one sees that the minimum LSND resis significantly squeezed by the large scale strucmeasurement alone. Taking into account a full 3+ 1neutrino oscillation analysis, fully incorporating dafrom CDHSW and Bugey, we are forced into tsmall angle portion of the LSND allowed region. Thmeans higher masses. At the 99% confidence leveallowed region contains four islands, correspond

222 A. Pierce, H. Murayama / Physics Letters B 581 (2004) 218–223

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(4)mν 0.9 eV, 1.4 eV, 2.2 eV, 3.5 eV.

All but the first of these conflict with Eq. (3), thougthe second is marginal. If, unlike the analysis[17], one were to take a prior for the bias factthe conflict would become stronger. So the LSNexperiment is strongly constrained by large scstructure measurements alone.

If instead of the 3+ 1 spectrum, we had chosen tinverted 1+ 3 spectrum, the conflict would have besharper. In the inverted case, the bound coming flarge scale structure is stronger, (see Fig. 1), andLSND islands are easily excluded.

It is interesting to note that the WMAP experimealso detected a relatively early reionization perizreionize ∼ 20. This implies an early generationstars responsible for the energy of reionization durthis period. Early star formation disfavors warm damatter, consistent with the above statementsneutrinos make up a small fraction of the criticdensity.

5. Neutrinoless double beta decay

The limit on the neutrino mass from the combintion of WMAP and 2dFGRS data is also interestingthe context of the neutrinoless double beta decay.Heidelberg–Moscow experiment claimed a signalneutrinoless double beta decay [13], which woulddicate that neutrinos have Majorana masses. Thevant neutrino mass for the signal is the so-called eftive neutrino mass〈mν〉ee = |∑i mνiU

2ei |. The nuclear

matrix elements in [13] lead to the preferred ran〈mν〉ee = (0.11–0.56) eV, while the reanalysis in [39gives 0.4–1.3 eV using a different set of nuclear mtrix elements. This result does not require the prence of an additional (sterile) neutrino species, soBBN limits need not apply. However, this high valuof 〈mν〉ee together with solar, reactor, and atmospheneutrino data on mass splittings, require the three ntrinos to be nearly degenerate. In this case, the tdegenerate neutrino bound of Eq. (2) is appropriand the WMAP+ 2dFGRS data would therefore rquiremνi < 0.33 eV, ormνi < 0.23 eV, if the prior istaken on the bias factor. This large scale structure l

excludes the deduced range of the effective neutmass in [39] completely. However, using the largvalues of the nuclear matrix element in [13], a wdow is still allowed. Also, a recent review of the eidence for neutrinoless double beta decay assigsomewhat larger error for the matrix element, andlargest allowed values of the matrix element cocorrespond to an effective neutrino mass as sma0.05 eV [40]. So, the WMAP+ 2dFGRS constrainthe claimed evidence for the neutrinoless double bdecay, but this statement is dependent on what issumed about the nuclear matrix elements. Moreothe WMAP+ 2dFGRS result has nothing to say abothe Heidelberg–Moscow result if the neutrinoless dble beta decay arises from a source other than Mrana neutrinos, such as supersymmetric modelsR-parity violation [41].

6. Conclusions

Recent precise cosmological measurements hgiven strong indications against the presence ofadditional sterile neutrino in the range that wouexplain the LSND result. Bounds from BBN disfavthe presence of any additional neutrinos that dodecouple before the QCD phase transition. Large sstructure disfavors the presence of neutrinos with min the eV range.

It seems difficult to reconcile LSND with the comological data. We have already discussed the pobility of having a large pre-existing lepton asymmeof L ∼ 10−2. Another possibility is to have CPT volation. In this case, the BBN constraint disappebecause no new light species are introduced. Indition, the large scale structure constraint is amerated, as only an anti-neutrino would need to be hebut not its CPT neutrino partner. However, KamLANdata, when taken in concert with data from supKamiokande may disfavor this possibility [19]. Thneutrino mixing result of LSND will be tested directat the MiniBoone Experiment at Fermilab [42].

We also note that the cosmological data doprefer the neutrinoless double beta decay in the mrange claimed by Heidelberg–Moscow experimeunless the nuclear matrix element is very large.

A. Pierce, H. Murayama / Physics Letters B 581 (2004) 218–223 223

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Acknowledgements

A. Pierce thanks L. Dixon and M. Peskin for useand stimulating discussions that led to the ideathis work, as well as for their insightful commenon the draft. Both of us thank Kev Abazajian aSteen Hansen for useful discussions. As revisionthis Letter were being completed, the revision versof [43] appeared, which came to similar conclusioregarding the viability of a scenario with a large leptasymmetry. The work of A. Pierce was supportedthe US Department of Energy under Contract DAC03-76SF00515. The work of H. Murayama wsupported in part by the DOE Contract DE-AC076SF00098 and in part by the NSF grant PH0098840.

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