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Sterile neutrino hot, warm, and cold dark matter

Kevork Abazajian,∗ George M. Fuller,† and Mitesh Patel‡

Department of Physics, University of California, San Diego, La Jolla, California 92093-0319

(Dated: January 30, 2001)

We calculate the incoherent resonant and non-resonant scattering production of sterile neu-trinos in the early universe. We find ranges of sterile neutrino masses, vacuum mixing angles,and initial lepton numbers which allow these species to constitute viable hot, warm, and colddark matter (HDM, WDM, CDM) candidates which meet observational constraints. The con-straints considered here include energy loss in core collapse supernovae, energy density limitsat big bang nucleosynthesis, and those stemming from sterile neutrino decay: limits from ob-served cosmic microwave background anisotropies, diffuse extragalactic background radiation,and 6Li/D overproduction. Our calculations explicitly include matter effects, both effective mix-ing angle suppression and enhancement (MSW resonance), as well as quantum damping. Wefor the first time properly include all finite temperature effects, dilution resulting from the an-nihilation or disappearance of relativistic degrees of freedom, and the scattering-rate-enhancingeffects of particle-antiparticle pairs (muons, tauons, quarks) at high temperature in the earlyuniverse.

PACS numbers: 95.35.+d,14.60.Pq,14.60.St,98.65.-r

I. INTRODUCTION

In this paper we calculate the non-equilibrium resonantand non-resonant production of sterile neutrinos in theearly universe and describe cosmological and astrophysi-cal constraints on this dark matter candidate. Dependingon their masses and energy distributions, the sterile neu-trinos so produced can be either cold, warm, or hot darkmatter (CDM, WDM, HDM, respectively) and may helpsolve some contemporary problems in cosmic structureformation. We can define sterile neutrinos generically asspin-1/2, SU(2)-singlet particles which interact with thestandard SU(2)-doublet (“active”) neutrinos νe, νµ, andντ , solely via ordinary mass terms. Singlet neutrinos withmasses ∼ 1012 GeV arise naturally, for example, in “see-saw” models of neutrino mass in grand unified theories(GUTs) [1]. Recent solar, atmospheric, and acceleratorneutrino oscillation experiments [2, 3, 4, 5, 6]), however,imply the existence of four light neutrino species withmasses <∼ 10 eV, only three of which can be active, onaccount of limits on the invisible decay width of the Z0

boson [7]. The remaining neutrino must be sterile. Theexistence of multiple generations of quarks and leptonsin the standard model (SM) of particle physics, as well asmany independently-motivated extensions of the SM, im-ply that there are additional, more massive sterile neutri-nos with couplings to active neutrinos currently beyond

∗Electronic address: kabazajian@ucsd.edu†Electronic address: gfuller@ucsd.edu‡Electronic address: mitesh@physics.ucsd.edu

direct experimental reach. We describe herein the cos-mological implications of these heavier sterile neutrinos.

Some early constraints on massive sterile neutrino pro-duction in the early universe were derived in Refs. [8, 9].The first analytical estimates of the relic sterile neu-trino abundance from scattering-induced conversion ofactive neutrinos were made by Dodelson and Widrow[10]. They assumed a negligible primordial lepton num-ber, or asymmetry, so the neutrinos are produced non-resonantly. They found that sterile neutrinos could be aWDM candidate with interesting consequences for galac-tic and large scale structure. A process that can createsterile neutrino dark matter with a unique energy dis-tribution was proposed by Shi and Fuller [11]. In thatwork a non-vanishing initial primordial lepton numbergives rise to mass-level crossings which enhance sterileneutrino production, yielding dark matter with an en-ergy spectrum which is grossly non-thermal and skewedtoward low energies. In particular, the average energy issignificantly less than that of an active neutrino, and theneutrinos suppress structure formation on small scalesand behave like CDM on large scales (i.e., they behaveas WDM or “cool DM”).

At no previous time has there been a greater need fora more comprehensive study of the cosmological and as-trophysical consequences of active-sterile neutrino mix-ing. The continuing influx of data from neutrino experi-ments and new observations of galaxy cores and clustersdemand a more detailed understanding of the physics ofsterile neutrino dark matter production and more sophis-ticated calculations of their relic abundances.

We examine these experimental and observational is-sues in Secs. II and III, respectively. In Sec. IV we discuss

2

the present limits on primordial lepton asymmetries andpossible dynamical origins of these asymmetries. We at-tempt to unify the perspectives of previous work, so wetake the initial lepton asymmetry of the universe to be afree parameter within the rather generous bounds set byexperiment. After reviewing the physics of neutrino os-cillations and matter-affected neutrino transformation inSec. V, we compute the consequences of pre-existing lep-ton asymmetries for sterile neutrino dark matter scenar-ios in Secs. VI and VII. In Sec. VI, we give the Boltzmannequations governing the non-equilibrium conversion ofactive neutrinos into sterile neutrinos and solve them inthe limit of non-resonant conversion, a limit which ob-tains when the lepton asymmetry is sufficiently small, oforder the baryon asymmetry. In Sec. VII, we considerlarger asymmetries and examine in detail the physics ofresonant neutrino transformation in the early universe.For the non-resonant and several resonant cases we dis-play contours of constant relic density of sterile neutri-nos in the plane of neutrino mixing parameters. Theseresults confirm and extend previous work [10, 11]. Forboth sets of calculations we also take into account finite-temperature and finite-density effects for all three activeneutrino species; the dilution of sterile neutrino densitiesdue to heating of the photons and active neutrinos fromthe annihilation of particle-antiparticle pairs before, dur-ing, and after the quark-hadron (QCD) transition, andwe allow for the enhanced active neutrino scattering ratesdue to the increased number of scatterers in equilibriumat high temperatures. In Sec. VIII we calculate the colli-sionless damping scales relevant for structure formation,scales which classify the regions of dark matter param-eter space as HDM, WDM, or CDM regions. We con-sider in Sec. IX the limits on sterile neutrino dark matterfrom the diffuse extragalactic background radiation (DE-BRA), cosmic microwave background (CMB), big bangnucleosynthesis (BBN), and 6Li and D photoproduction.Finally, in Sec. X we examine the implications of active-sterile neutrino mixing in core-collapse (Type Ib/c, II)supernovae. Conclusions are given in Sec. XI. Through-out the paper we use natural units with h = c = kB = 1.

II. NEUTRINO ANOMALIES

Recent experiments have provided data indicating ev-idence for new neutrino physics. The most significantrecent evidence is the Super-Kamiokande Collaboration’sstatistically convincing result [2], verifying previous mea-surements [3], of a suppression of the atmospheric νµ/νµ

flux. The most persuasive Super-Kamiokande evidencefor neutrino oscillations is the measured zenith angle de-pendence of the νµ/νµ flux, an observation fit most sim-ply by maximal νµ ντ mixing in vacuum, with vacuum

mass-squared difference δm2 ∼ 3 × 10−3 eV2.On another front, the ground-breaking observations by

the Homestake Collaboration found a solar neutrino fluxfar below that predicted on the basis of sophisticated

solar models [4]. The solar neutrino problem has an in-teresting possible solution through matter-enhanced res-onant conversion via the Mikheyev-Smirnov-Wolfenstein(MSW) mechanism [12] or through vacuum or “quasi-vacuum” oscillations [13]. Depending on whether the so-lar solution involves two-, three-, or four-neutrino mixing,the parameter space of neutrino mass-squared differenceand vacuum mixing angle are constrained differently [5].

A third indication for neutrino oscillations comes fromthe Los Alamos Liquid Scintillator Neutrino Detector(LSND) Collaboration’s observations of excess νe andνe events in beams of νµ and νµ, respectively. Thesehave been interpreted as evidence for neutrino oscilla-tions in the νµ → νe and νµ → νe channels [6]. TheKarlsruhe Rutherford Medium Energy Neutrino (KAR-MEN) experiment probes the same channels but does notsee evidence for neutrino oscillations [14]. A joint analy-sis of the LSND and KARMEN data has found that thereare regions of neutrino mixing parameter space consistentwith both experiments’ results [15].

Efforts have been made to embed the above neutrinooscillation solutions within a three-neutrino framework[16, 17, 18, 19]. Leaving aside maximal vacuum mixingof all three neutrino species [16], these analyses gener-ally require the atmospheric neutrino oscillation lengthscale to be the one associated with the short-base-lineLSND experiment [17, 18], or the solar and atmosphericsolutions must be built on the same mass difference scale[19].

However, the zenith-angle dependence of the Super-Kamiokande measurement requires the atmospheric neu-trino oscillation length to be much larger than the cor-responding LSND scale [2, 6]. This disfavors the firstthree-neutrino scheme. Additionally, the three solarneutrino experimental modes presently available sug-gest an energy dependence in the νe survival probabil-ity which is likely inconsistent with the second three-neutrino scheme. Taking these two length scales, andthe results of global flux measurement fits for the solarneutrino oscillation interpretation [5], the three differ-ent oscillation length and energy scales require three dis-parate mass differences, which cannot be accommodatedin a three-neutrino framework. The CERN e+e− colliderLEP measurement of the Z0 width indicates the numberof active neutrinos with masses < mZ/2 is 3.00±0.06 [7],so the results are prima facie evidence for a light sterileneutrino species.

A number of neutrino mass models can provide themasses and mixings needed to accommodate all of theneutrino oscillation data [20, 21, 22, 23, 24, 25]. Forexample, in some string theories, higher-dimensional op-erators, suppressed by the powers of the ratio of someintermediate mass scale and the string scale, can givethe light and comparable Dirac and Majorana massesnecessary for appreciable active-sterile neutrino mixing[20]. In theories with light composite fermions, severalof the fermions may mix with standard model neutri-nos, giving light active and sterile neutrinos [21]. In the

3

minimal supersymmetric standard model (MSSM) withexplicit R-parity violation, a neutralino can provide therequired mixing for the atmospheric and solar neutrinoproblems [22, 23]. This model can also supply a sterileor “weaker-than-weakly” interacting particle that has asmall mixing with one or more active neutrino flavors.A low-energy extension of the Standard Model with anSO(3) gauge group acting as a “shadow sector” may re-sult in a neutral heavy lepton [24]. A model with severalsterile neutrino dark matter “particles” arises in brane-world scenarios. It invokes bulk singlet fermions couplingwith active neutrinos on our brane [25]. However, manymodels of bulk neutrinos as sterile neutrino dark mat-ter must be rather finely tuned and must avoid severalcosmological constraints [26].

Although these mass models have been proposed to ac-count for the existing neutrino anomalies, many of themalready contain or can be easily be extended to containadditional singlet states which also mix with active neu-trinos. As long as the new particles are sufficiently mas-sive and have sufficiently small mixings with active neu-trinos, they evade terrestrial constraints, and it is in-teresting and useful to speculate on their cosmologicaland astrophysical consequences. As we show later in thispaper, even apparently negligible active-sterile neutrinomixing is sufficient to induce the production of sterileneutrino dark matter in the early universe.

A specific model’s viability in producing a sterile neu-trino dark matter candidate depends on whether themass and mixing properties of the candidate(s) with theactive neutrinos lie within the range that produces an ap-propriate amount of dark matter, and whether the can-didate is stable over the lifetime of the universe, doesnot engender conflicts with observationally-inferred pri-mordial light element abundances, does not violate CMBbounds, and does not contribute excessively to the DE-BRA in photons [27]. Potential constraints on these sce-narios also may arise from deleterious effects associatedwith the neutrino physics of core-collapse (Type Ib/c, II)supernovae (see, e.g., Refs. [28, 29, 30] and Sec. X). Con-straints on massive sterile neutrinos (10 MeV ≤ ms ≤100 MeV) from the SN 1987A signal and BBN were con-sidered previously in Ref. [31].

The existence of massive sterile singlet neutrinos thatmix with νe has been probed in precision measurementsof the energy spectrum of positrons in the pion decayπ+ → e+ νe. The best current limits are from Brittonet al., [32], which constrain the mixing matrix element|Uex|2 < 10−7 for sterile neutrino masses 50 MeV < ms <130 MeV. These limits may be significantly improvedin future precision experiments [33]. Less stringent con-straints from peak and kink searches exist for smaller ms.In addition, searches for decay of massive νs have yieldedconstraints for |Uµx|2, |Uτx|2 [34], as well as |Uex|2 [7].

III. STRUCTURE FORMATION

Conflicts may have appeared between standard colddark matter theory and simulations and observations oflarge and small scale cosmological structure. Simulationspredict about 500 small halos (i.e., dwarf galaxies) withmass greater than 108 M⊙ around a galaxy like the MilkyWay, but only 11 candidates are observed near the MilkyWay [35], and only 30 in the local group out to ∼1.5Mpc [36]. In other words, simulations of the standardΛ-CDM model may predict more dwarf galaxies than areseen (but see Ref. [37]). Also, even if the dwarf halosare dark, their overabundance may hinder galactic diskformation [38].

Another potential problem with CDM simulations isthe appearance of singularities or “cusps” of high den-sity in the cores of halos. The observations of the in-nermost profiles of galaxy clusters are ambiguous [39],but rotation curves of the central regions of dark matter-dominated galaxies consistently imply low inner densi-ties [40]. Recent N -body calculations of the nonlinearclustering of WDM models have found that enhancedcollisionless damping can lower halo concentrations, in-crease core radii, and produce far fewer low mass satel-lites [41]. Also, the observed phase space density in dwarfspheroidal galaxies may suggest a primordial velocity dis-persion like that of WDM [42].

A promising new constraint on the nature of dark mat-ter may come from study of the Lyα forest in the spec-trum of high-redshift quasars [43, 44]. The structure ofthe Lyα forest at high redshifts has been used to con-strain the contribution of HDM and, therefore, the massof the active neutrinos [45]. The constraints presented inRef. [45] and the Gerstein-Zeldovich–Cowsik-McLellandbound [46] only pertain to fully populated active neu-trino seas with a thermal energy spectrum. For ex-ample, if an active neutrino has a mass mνα

∼ 1 keV(α = µ, τ) and the reheating temperature of inflation islow (TRH ∼ 1 MeV), then the corresponding active neu-trino sea will not be fully populated [47, 48] and can beWDM [49]. However, observations of the power spectrumof the Lyα forest cannot only constrain HDM scenarios,but also may be able to test the viability of WDM scenar-ios. Since the absolute normalization of the power spec-trum is uncertain, the relative presence of power betweenscales near ∼1 Mpc and near ∼100 kpc may constrain theWDM scenarios that have a relatively large contrast inpower between these scales [50]. In addition, the largenumber of high-redshift quasars found by the Sloan Dig-ital Sky Survey (SDSS) [51] can add considerably to theknowledge of the Lyα power spectrum, particularly atthe largest scales. The combination of the galactic powerspectrum from SDSS and the Lyα power spectrum willproduce even stronger constraints on the behavior of darkmatter on small scales.

4

IV. PRIMORDIAL LEPTON ASYMMETRY

The lepton number or asymmetry of a neutrino flavorα is defined to be

Lα ≡ nνα− nνα

nγ, (4.1)

where nναis the proper number density of neutrino

species να, and nγ = 2ζ(3)T 3/π2 ≈ 0.243T 3 is the propernumber density of photons at temperature T . The leptonnumber(s) of the universe is (are) not well constrained byobservation. The best limits come from the energy den-sity present during BBN and the epoch of decoupling ofthe CMB [52, 53, 54, 55]. In fact, the best current boundson the lepton numbers come from the observational limitson the 4He abundance, radiation density present at theCMB decoupling, and structure formation considerations[52, 54, 55]:

−4.1 × 10−2 ≤ Lνe≤ 0.79, (4.2)

|Lνµ,ντ| ≤ 6.0. (4.3)

The bound on positive Lνeis weaker than that for neg-

ative Lνesince it is possible to combine the effects of

neutron-to-proton ratio (n/p) reduction of positive Lνe

with a large Lνµor Lντ

, which increases the expan-sion rate and thus the n/p ratio entering BBN. Thiscancellation could provide a neutrino “degenerate” BBNthat could replicate not only the primordial 4He, butalso the D/H and 7Li abundances predicted by stan-dard BBN. See Refs. [52, 54] for a further discussion. Asstated, the limits (4.2), (4.3) depend on an assumptionof a roughly Fermi-Dirac, low-chemical-potential energyspectrum for each neutrino species. They do not ad-dress neutrino mass and do not take account of potentialbounds stemming from closure (i.e., age of the universe)considerations. The above limits do not change signifi-cantly when recent estimates of observationally-inferredprimordial abundances are employed [54]. In any case,it is clear that neutrino number density asymmetries lessthan about 10% of the photon number are easily allowed.However, we will see that lepton asymmetries at thislevel, or even several orders of magnitude smaller, couldhave a significant and constrainable effect if there aremassive sterile neutrinos (see Sec. VII).

An issue which arises whenever the lepton number(s)Lνα

differ from the baryon number B (or η = nb/nγ ≈2.79× 10−8Ωbh

2 ∼ 10−10, where nb is the proper baryonnumber density and Ωb is the baryon rest mass clo-sure fraction and h is the Hubble parameter in unitsof 100 km s−1 Mpc−1) concerns the process of baryoge-nesis. For example, electroweak baryogenesis predictsthe equality of these numbers B − L = 0. Note thatthere also exist simple processes that violate B − L andcreate lepton and/or baryon number either through theAffleck-Dine mechanism [56] or through non-equilibriumdecays of a heavy Majorana particle [57]. In Ref. [58],

several natural scenarios of producing a large lepton num-ber and the observed small baryon number were investi-gated. Furthermore, lepton number could arise sponta-neously through matter-enhanced active-sterile neutrinotransformation at energy scales below that of the baryo-genesis epoch [59, 60]. Finally, it should be recognizedthat through cancellation of lepton numbers we couldhave B −L = 0 (where L = Lνe

+ Lνµ+ Lντ

), while stillhaving significant lepton-driven weak potentials [see Eq.(5.12)] in the early universe.

V. MATTER-AFFECTED NEUTRINO

TRANSFORMATION

A. General framework

Neutrino mixing phenomena arise from the non-coincidence of energy-propagation eigenstate and theweak (interaction) eigenstate bases. Eigenstates of neu-trino interaction (flavor) include the active neutrinos (νe,νµ, ντ ) which are created and destroyed in the standardmodel weak interactions, as well as sterile neutrinos (e.g.,νs, ν′

s, ν′′s , ...) which do not participate in weak inter-

actions. Eigenstates of neutrino propagation are statesof definite mass and energy (or momentum) and evolveindependently of each other between weak interactionvertices. If the bases spanned by these sets of eigen-states happen to coincide, then active and sterile neu-trinos propagate independently between interactions. Ingeneral, however, the bases need not coincide, since thesymmetries of the standard model and its many proposedextensions do not require the unitary transformation be-tween the bases to be the identity transformation. Asa result, neutrinos can oscillate, or transform in flavor,between interactions.

This physics applies for any number of neutrino fla-vors, including the four (three active plus one sterile)which can accommodate the neutrino experiments. Inthe environments we consider, active-active mixing issuppressed due to the very similar matter effects for allactive species. In general, the 4-neutrino evolution mayhave multiple active species mixing with the sterile neu-trino, or in multiple-sterile scenarios, there may be mix-ing between the sterile neutrinos. In our analysis here,we consider the simplifying limit of two neutrino (active-sterile) mixing to explore the basic physics of sterile neu-trino dark matter production.

In the case of two-neutrino mixing, the unitary trans-formation between the bases can be written as

|να〉 = cos θ|ν1〉 + sin θ|ν2〉|νs〉 = − sin θ|ν1〉 + cos θ|ν2〉

where |να〉 and |νs〉 are active (α = e, µ, τ) and ster-ile neutrino flavor eigenstates, respectively, and |ν1〉 and|ν2〉 are neutrino mass (energy) eigenstates with masseigenvalues m1 and m2, respectively. The vacuum mix-ing angle θ parametrizes the magnitude of the mixing

5

(and, as we shall see, the effective coupling of the sterileneutrino in vacuum). We choose all of the neutrino fla-vor and mass eigenstates to be eigenstates of momentumwith eigenvalue p. Then a mass-energy eigenstate |νi〉(i = 1, 2) develops in time and space with the phase

ei~pi·~x = ei(p·x−Eit) = ei(px−√

m2i +p2t) ≈ e−ixm2

i /2p,(5.1)

where Ei =√

m2i + p2 is the energy of the eigenstate;

p ≡ |p| is the magnitude of the proper momentum of thespecies. If the neutrino mass eigenstates are relativisticso that Ei ≫ mi, we have Ei ≈ p + m2

i /2p and x ≈ t,yielding the last approximation in Eq. (5.1). (In this

last approximation we suppress the part of the evolutionoperator proportional to the trace, as this simply givesan overall common phase to the states.) In our studyof sterile neutrino production, the sterile neutrinos arealways relativistic during the epochs in which they wereproduced.

The difference of the squares of the vacuum neutrinomass eigenvalues is, for example, δm2 = m2

2 − m21. We

can follow the evolution of a coherently propagating neu-trino state |Ψν〉 in either the mass-energy or flavor ba-sis. In the flavor basis, a Schrodinger-like equation de-scribes how the flavor amplitudes, aα(x) = 〈να|Ψν(x)〉with α = e, µ, τ and as(x) = 〈νs|Ψν(x)〉, develop withtime-space coordinate x:

id

dx

(

aα

as

)

=

(

p +m2

1 + m22

4p+

V (x, p)

2

)

I +1

2

(

V (x, p) − ∆(p) cos 2θ ∆(p) sin 2θ∆(p) sin 2θ ∆(p) cos 2θ − V (x, p)

)(

aα

as

)

, (5.2)

where the first term is proportional to the identity and∆(p) ≡ δm2/2p. In the context of the early universe it ismost convenient to take x = t, the Friedman-Lemaitre-Robertson-Walker coordinate time (age of the universe),while in supernovae we take x to be position. The weakpotential V (x, p) represents the effects of neutrino neu-tral current and charged current forward scattering onparticles in the plasma carrying weak charge. In theearly universe we have V (T, p), but in supernovae, Valso depends on position x. We suppress the x and Tdependence of V in the rest of the paper. For a reviewof these issues and neutrino astrophysics, see, e.g., Refs.[61, 62, 63, 64].

Neutrino mixing can be modified by the presence ofa finite temperature background and any asymmetry inlepton number. The oscillation length is

lm =

∆2(p) sin2 2θ +[

∆(p) cos 2θ − V D − V T (p)]2−1/2

.

(5.3)

The effective matter-mixing angle is

sin2 2θm =∆2(p) sin2 2θ

∆2(p) sin2 2θ + [∆(p) cos 2θ − V D − V T (p)]2 .

(5.4)

Matter effects have been separated into finite density andfinite temperature potentials, V D and V T (p).

The finite density potential V D arises from asymme-

tries in weakly interacting particles (i.e, nonzero leptonnumbers), not from non-zero total densities alone. Ingeneral, the finite density potential is [64, 65]

V D =

√2GF

[

2(nνe− nνe

) + (nνµ− nνµ

) + (nντ− nντ

) + (ne− − ne+) − nn/2]

for νe νs,√2GF

[

(nνe− nνe

) + 2(nνµ− nνµ

) + (nντ− nντ

) − nn/2]

for νµ νs,√2GF

[

(nνe− nνe

) + (nνµ− nνµ

) + 2(nντ− nντ

) − nn/2]

for ντ νs.

(5.5)

The thermal potential V T arises from finite tempera-ture effects and neutrino forward scattering on the seas

of thermally created particles [64]:

V T (p) = − 8√

2GFpν

3m2Z

(〈Eνα〉nνα

+ 〈Eνα〉nνα

)

6

− 8√

2GFpν

3m2W

(〈Eα〉nα + 〈Eα〉nα) , (5.6)

where nα (nα) is the proper number density of leptons(anti-leptons) of flavor α and 〈Eα〉 (〈Eα〉) is the aver-age energy of the lepton (antilepton), and nνα

(nνα) and

〈Eνα〉 (〈Eνα

〉) are the proper number density and aver-age energy of the neutrinos (antineutrinos) of flavor α.The second term in Eq. (5.6) must be included wheneverthe lepton of the same flavor as the active neutrino inquestion is populated.

In this paper, we will assume that the initial neutrinodistribution functions are close to Fermi-Dirac black bod-ies, which for occupation of differential interval dp havethe form

dnνα≈ nνα

T 3F2 (ηνα)

p2dp

eE(p)/T−ηνα + 1

≈(

nγ

4T 3ζ (3)

)

p2dp

eE(p)/T + 1, (5.7)

where E(p) = (p2 + m2)1/2, and E(p) ≈ p in the rela-tivistic kinematics limit. In this expression ηνα

= µνα/T

is the degeneracy parameter (chemical potential dividedby temperature) for neutrino species να and F2(η) ≡∫∞

0x2dx/(ex−η + 1) is the relativistic Fermi integral of

order 2 [F2(0) = (3/2)ζ(3)]. The distribution functionfor a neutrino species α is

fα(p, t) = 1/(eE(p)/T−ηνα + 1). (5.8)

The last approximation in Eq. (5.7) follows if we takethe neutrino degeneracy parameter to be zero. This isfrequently a good approximation for almost all of therange of lepton numbers which are interesting for ourpurposes. However, it may not be valid over the broaderrange of allowed lepton numbers given in Eqs. (4.2) and(4.3). For small lepton numbers ηνα

≈ 1.46Lνα; in fact,

whenever να has relativistic kinematics, we can write

Lνα≈ 1

4ζ(3)

π2

3ηνα

+1

3η3

να

. (5.9)

This equation can easily be inverted to find ηνα(Lνα

).In the early universe, for temperature ranges where

the number of degrees of freedom is constant, the time-temperature relation is a simple power law and the quan-tity ǫ ≡ p/T is a comoving invariant. The differentialnumber density for small lepton number in this case isthen

dnνα≈(

nγ

4ζ (3)

)

ǫ2dǫ

eǫ + 1. (5.10)

B. Quantitative formulation for the early universe

If one makes the assumption that the only net leptonnumber in the universe is that required for electric charge

neutrality (i.e., half of the baryon number when thereare equal numbers of protons and neutrons), then the fi-nite density potential usually remains negligible. In somecases where δm2 < 0, an initially small asymmetry likethis can be amplified by matter-enhanced active-sterileconversion [59, 60].

The finite density potential in the early universe couldbe dominated by asymmetries in the lepton number, andso is often referred to as the “lepton potential.” It takesthe form

V D =2√

2ζ(3)

π2GFT 3

(

Lα ± η

4

)

, (5.11)

where we take “+” for α = e and “−” for α = µ, τ . Herewe define the net driving lepton number Lα in terms ofthe lepton numbers in each active neutrino species as

Lα ≡ 2Lνα+∑

β 6=α

Lνβ(5.12)

with the final sum over the active neutrino flavors otherthan να. Note that in Eq. (5.11), the baryon-to-photonratio is η, not to be confused with neutrino degeneracyparameter.

The total weak potential, V (p, T ) = V D(T )+V T (p, T ),experienced by an active neutrino να is approximately

V (p, T ) ≈ (40.2 eV)

( L10−2

)(

T

GeV

)3

− Bp

(

T

GeV

)4

.

(5.13)

As noted above, the thermal potential V T must take intoaccount the presence of populated leptons of the sameflavor. For νe this is required at all temperatures wherethe neutrinos are coupled; for νµ, the thermal muon termshould be included at temperatures T >∼ 20 MeV, wherethe µ is populated; and for ντ , the thermal term shouldbe included at T >∼ 180 MeV. Therefore, the coefficientB takes on the values:

B ≈

10.79 eV α = e

3.02 eV α = µ, τ(5.14)

for T <∼ 20 MeV;

B ≈

10.79 eV α = e, µ

3.02 eV α = τ(5.15)

for 20 MeV <∼ T <∼ 180 MeV;

B ≈ 10.79 eV α = e, µ, τ (5.16)

for T >∼ 180 MeV.The trend of the weak potential experienced by an

active neutrino species να is clear. When the quantityL ± η/4 is sufficiently large and positive (negative forνα) the potential will rise with increasing temperature,reach a maximum, and then turn over and eventually

7

0.4

0.3

0.2

0.1

0.0

(m2)eff (MeV2)

302520151050

T (100 MeV)

να

νs3

νs2

νs1

FIG. 1: An example of the temperature evolution of the ac-tive and sterile neutrino effective mass-squared, m2

eff . Theactive neutrino m2

eff is dominated by a positive finite densitypotential at lower temperatures and turns over when the ther-mal potential dominates. Resonance occurs at level crossings,where the active and sterile m2

eff tracks intersect.

become negative at a high temperature where thermalterms dominate. The potential as a function of tempera-ture for some representative parameters is shown as thesolid line in Fig. 1. With this behavior it is obvious thatneutrino mass level crossings in the temperature regimenot dominated by the thermal terms are possible onlyif the vacuum mass eigenvalues most closely associatedwith some sterile neutrinos are larger than those mostclosely associated with να.

VI. NON-EQUILIBRIUM PRODUCTION: THE

STERILE NEUTRINO BOLTZMANN

EQUATION

The Boltzmann equation gives the evolution of thephase-space density distribution function f(p, t) for aparticle species. For sterile neutrinos in the early uni-verse, it can be written as [66, 67]

∂

∂tfs(p, t) − H p

∂

∂pfs(p, t) =

∑

i

∫

Γi(p′α, p)fα(p′α, t) [1 − fs(p, t)] d3p′α

−∫

Γi(p′α, p)fs(p, t) [1 − fα(p′α, t)] d3p′α, (6.1)

where the rate of scattering production of νs correspond-ing to a particular channel i [see Eq. (6.8)] is Γi. Wemake two approximations regarding the scattering ker-nels Γi(p

′, p): (1) they are isotropic; and (2) they areconservative.

The first of these approximations is completely justi-fied given a homogeneous and isotropic universe. Thesecond—that an active neutrino scatters into a sterilestate of the same energy—is made purely to ease thecomputational complexity of following the evolving sys-tem of neutrinos. We note, however, that Ref. [68] es-timates the effects of relaxing this approximation in thecontext of a semi-analytic calculation and find that itmakes little qualitative or quantitative difference in theresults.

Certainly, however, a better treatment could be ex-tended to the scattering kernel. This could, for example,change the conditions required for coherence. For exam-ple, we note that the effects described in Ref. [69] may beimportant, especially regarding where coherence breaksdown. We show, however, in Sec. VII that coherent pro-duction of sterile neutrinos at MSW resonances is notimportant in our favored parameter regime.

In our discussion, we take fi(p, t) to be the momentum-and time-dependent distribution function of active (α =e, µ, τ) and sterile (s) neutrinos. The distribution func-tions fα are given by Eq. (5.8). The second left hand sideterm arises from the redshifting of the distribution.

However, Eq. (6.1) is semi-classical in its evolution ofthe neutrino distributions. To exactly follow the fullquantum development, including such quantum effects asdamping and off-diagonal contributions to the neutrinoHamiltonian in matter [70], one must follow the timeevolution of the density matrix [71, 72, 73, 74]. One canapproximate the effects of quantum damping through adamped conversion rate Γ(να → νs; p, t). Fermi blockingeffects are taken to be negligible, leaving the Boltzmannequation as

∂

∂tfs(p, t) − H p

∂

∂pfs(p, t)

≈ Γ(να → νs; p, t) [fα(p, t) − fs(p, t)] . (6.2)

The reverse term is important for the L 6= 0 case, wherefα and fs can become comparable for certain regions ofthe neutrino momentum distributions.

The conversion rate to sterile neutrinos is just theproduct of half of the total interaction rate, Γα, of theneutrinos with the plasma and the probability that anactive neutrino has transformed to a sterile:

Γ(να → νs; p, t) ≈ Γα

2〈Pm(να → νs; p, t)〉. (6.3)

This probability Pm depends on the amplitude of thematter mixing angle and the quantum damping rateD(p) = Γα(p)/2 [D(p) = Γα(p)/2] for neutrinos [antineu-trinos] [75, 76, 77, 78],

8

〈Pm(να → νs; p, t)〉 ≈ 1

2

∆(p)2 sin2 2θ

∆(p)2 sin2 2θ + D2(p) + [∆(p) cos 2θ − V D − V T (p)]2(6.4)

〈Pm(να → νs; p, t)〉 ≈ 1

2

∆(p)2 sin2 2θ

∆(p)2 sin2 2θ + D2(p) + [∆(p) cos 2θ + V D − V T (p)]2(6.5)

The full Boltzmann equation then is

∂

∂tfs(p, t) − H p

∂

∂pfs(p, t) ≈ Γα(p)

2〈Pm(να → νs, tin + τ)〉τ [fα(p, t) − fs(p, t)]

≈ Γα(p)

2sin2 2θm

[

1 +

(

Γα(p)lm2

)2]−1

[fα(p, t) − fs(p, t)] (6.6)

≈ 1

4

Γα(p)∆2(p) sin2 2θ

∆2(p) sin2 2θ + D2(p) + [∆(p) cos 2θ − V L − V T (p)]2 [fα(p, t) − fs(p, t)] ,

where 1+[Γα(p)lm/2]2−1 is the damping factor. Thereare analogous equations for Eqs. (6.1)-(6.3) and Eq. (6.6)for antineutrinos.

Previous calculations have approached the solution ofthis equation for the case of negligible lepton number(L ≈ 0) analytically and have been restricted to the shortepoch just prior to BBN where the Hubble expansion rateH (evolution of the scale factor) and time-temperaturerelations are simple power laws (the temperature is pro-portional to the inverse scale factor) [10, 68]. Such ap-proximations allow the reduction of the left-hand side ofEq. (6.6) to a single term. With the approximations thatlepton number L is always negligible, that the thermalterm V T is not modified by population of leptons, thatthe interaction rate is not enhanced due to populationof scatterers, that quantum damping is never important,and that the reverse rates (νs → να) are always negli-gible, then the right hand side can also be considerablysimplified, and the solution for the sterile neutrino darkmatter abundance is reduced to a simple integral. It isobvious, however, that many if not all of these approxi-mations are eventually invalid over at least some of theparameter range of interest for sterile neutrino dark mat-ter.

For example, in order to probe cases where sterile neu-trino dark matter production may lie above the QCDtransition (T >∼ 100 MeV), and in order to improve theaccuracy of the predicted dark matter contribution, wemust extend our calculation to epochs where the scalefactor-temperature and time-temperature relations arenot simple power laws (see the Appendix for details).The interactions contributing to Γα(p) which produce(and remove) sterile neutrinos are

να + νβ να + νβ

να + l± να + l± (6.7)

να + q να + q

να + να → l+ + l−.

Here, the να represent either neutrinos or anti-neutrinos,as appropriate, and q and l are any populated quark andcharged lepton flavors. The total interaction rate at tem-peratures 1 MeV <∼ T <∼ 20 MeV due to interactions ofthe neutrinos among themselves and the e± pairs is

Γα(p) ≈

1.27 G2FpT 4, α = e,

0.92 G2FpT 4, α = µ, τ.

(6.8)

At higher temperatures, other leptons and quarks arepopulated and contribute to the neutrino interactionrate. In our calculations, we have included the enhance-ment of the interaction rate due to the presence of thesenew particles in the plasma at high temperatures. In par-ticular, a significant increase to the scattering rate resultsat temperatures above the QCD scale. Interestingly, theresults of the production of sterile neutrino dark mattertherefore depend on the temperature of the QCD transi-tion, where the quark-antiquark pairs annihilate and areincorporated into color singlets.

The high scattering rate characteristic of the environ-ment of the early universe not only serves to populate thesterile neutrino sea, but can also suppress the productionof sterile neutrinos when the matter oscillation lengthis large compared to the mean free path of the neutri-nos. When the oscillation length is much larger than themean free path, the probability that an active neutrinohas transformed into a sterile state becomes very small.It can be shown that such scattering can force a quan-tum system to not evolve from the initial state [75, 79].Essentially, each scattering event resets the phase of thedeveloping neutrino state |Ψνα

〉. This is the so-calledquantum Zeno effect.

Consider the right hand side collision term in Eq. (6.2).The sterile neutrinos are initially not in thermal equi-librium, and therefore the reverse processes are initiallyunimportant. They will, however, become more impor-tant as the sterile neutrino sea is populated. Becausethe sterile neutrinos are never in equilibrium, the usual

9

simplifying principle of steady-state equilibrium [67, 80]cannot be made. Therefore, to calculate the productionof sterile species, we must start with the Boltzmann equa-tion in its “unintegrated” form Eq. (6.6).

We have directly solved the Boltzmann equation nu-merically for the L ≈ 0 case (e.g., L ≈ 10−10), and foundthe amount of sterile neutrino dark matter produced fora broad range of sterile neutrino mass and mixing angleswith each of the three active neutrino flavors. In treatingthe redshift of the sterile distribution, we have greatly ac-celerated the numerical calculation by following the red-shift of the sterile neutrino distribution function fs byredshifting momenta as p ∝ t−1/2, which is always truein a radiation dominated universe (a necessary conditionfor BBN and the CMB). This numerical method allowsus to correctly follow the redshifting of the sterile neu-trino without the second term of the Liouville operatorin the Boltzmann equation (6.6), while active neutrinosmay be reheated via annihilation of other species.

The contribution to the closure fraction of the universe,meeting all constraints (see Secs. IX and X) is shown inFig. 2 for νe νs and Fig. 3 for ντ νs. A general fitto our results for nonresonant production is

Ωνsh2 ≈ 0.3

(

sin2 2θ

10−10

)

( ms

100 keV

)2

. (6.9)

The maximum rate of sterile neutrino production occursat temperature [10, 81]

Tmax ≈ 133 MeV( ms

1 keV

)1/3

. (6.10)

The closure fraction contribution for νµ νs is nearlyidentical to ντ νs. The particular flavor(s) of the ac-tive neutrino with which the sterile neutrino mixes doesdetermine to some extent the ultimate closure fraction.However, this flavor dependence is negligible for larger νs

masses since, from Eq. (6.10), for sterile neutrinos withms

>∼ 2 keV, Tmax>∼ 180 MeV. At these temperatures,

e±, µ± and even τ± (due to the high entropy of theuniverse) are populated significantly, so that the ther-mal potentials are identical for all flavors. Therefore, themass fractions produced for all flavors with sterile neu-trino masses ms

>∼ 2 keV are very similar.The similarity of the results of production for νe, νµ,

and ντ mixing with a sterile neutrino is at odds withthe calculation of Ref. [68]. There are several differencesbetween our treatment and that of Ref. [68] that canaccount for the disparity in results. First, the primarycause for discrepancy for the ντ νs neutrino mixingcase is likely to lie in the fact that τ leptons are signifi-cantly populated at temperatures one-tenth of their mass(T >∼ 177 MeV) due to the high-entropy of the universe.This modifies the thermal term in Eq. (5.6) in an im-portant way, not included in Ref. [68]. In our work, wefollow the number and energy density of tau leptons ex-plicitly in our numerical evolution. One can see from Eq.(6.10) that for masses of sterile neutrinos greater than

FIG. 2: Regions of Ωνsh2 produced by resonant and nonres-onant νe ↔ νs neutrino conversions for selected net leptonnumber L, after applying all constraints (see Secs. IX andX). Regions of parameter space disfavored by supernova corecollapse considerations are shown with vertical stripes.

about 2 keV, production occurs at a temperature wherethe τ lepton is significantly populated. Second, the pop-ulation of massive species of scatterers (µ and τ leptonsand u, d, s, c, b quarks) which enhance the scattering rateis not included in Ref. [68], but are included in our cal-culations. Third, effects of re-heating on the dilutionof sterile neutrino dark matter are treated only approxi-mately in Ref. [68]. We explicitly include reheating anddilution in our calculation through our treatment of thetime-temperature evolution of active and sterile neutri-nos, as described in the Appendix.

VII. INITIAL LEPTON NUMBER AND

RESONANT PRODUCTION

In this section, we examine the resonant production ofsterile neutrino dark matter for a range of initial leptonnumbers. We will extend the treatment of this issue givenin Shi and Fuller [11] considering the effects of particleannihilation and reheating and enhanced scattering rateson coherent and incoherent evolution through MSW reso-nances. This allows us to consider resonant να νs con-version at high temperature epochs in the early universe,up to the electroweak transition regime at T ≈ 100 GeV.

Resonance, or mass level crossings as seen in Fig. 1, arecharacterized by maximal effective matter mixing angles(θm)res = π/4, where the να → νs conversion rate is,

10

FIG. 3: Same as Fig. 2, but for ντ ↔ νs.

consequently, enhanced. The resonance condition is

∆(p) cos 2θ − V L − V T (p) = 0

∆(p) cos 2θ − (40.2 eV)

L± η/4

10−2

(

T

GeV

)3

+ Bǫ

(

T

GeV

)5

= 0

( ms

1 keV

)2

cos 2θ ≈ 8.03 ǫ

L ± η/4

10−2

(

T

100 MeV

)4

+ 2ǫ2(

B

keV

)(

T

100 MeV

)6

. (7.1)

Choose one of the horizontal dotted lines laying be-low the peak of the solid line in Fig. 1. This indicatesthe “mass track” for a sterile neutrino species. Thisis, of course, simply the vacuum mass-squared value(m2

eff)s = m2s, and is flat and independent of tempera-

ture. The effective mass-squared track for an active neu-trino να could be as shown in Fig. 1. Mass level crossings(resonances) in the να νs system can occur when m2

s

lies below the peak value of (m2eff)να

. This peak will oc-

cur at temperature

TPEAK ≈(

2

3

)1/2

T0

≈(

4√

2ζ(3)

3π2

)1/2

ǫ−1/2L ± η/41/2

×[

GF (GeV)5

B

]1/2

≈ (2.98 GeV)ǫ−1/2

( L10−2

)1/2

, (7.2)

where T0 is the high temperature at which the effectivemass-squared track crosses zero, and where in the last nu-merical expression we have assumed that T > 180 MeV.

Note that the peak value of effective mass-squared, orequivalently, the largest value of δm2 cos 2θ for which alevel crossing can occur is,

m2eff (TPEAK) =

(

δm2 cos 2θ)

m2eff(PEAK)

(7.3)

≈(

4√

2ζ(3)

3π2

)3L ± η/43

ǫ

[

G3F (GeV)10

B2

]

.

For example, we can show that the largest νs mass whichcan have a resonance with a νe is roughly

(ms)PEAK ≈ 406 keV

ǫ1/2

( L10−2

)3/2

. (7.4)

As the universe expands, an active neutrino speciesνα will encounter two resonances with a sterile neutrinospecies, as long as this sterile species has m2

s less thanthe peak value of m2

eff in Eq. (7.3). At some epochs,both resonances may be present in a given active neu-trino spectrum since the resonance condition, Eq. (7.1),has more than one zero. This behavior is shown for aparticular case in Fig. 4.

What is the effect of resonance on sterile neutrino pro-duction in the early universe? The answer to this ques-tion depends on the rate of incoherent production at theresonance, whether the neutrinos are coherent as theypass through resonance and, if they are coherent, the de-gree of adiabaticity characterizing the evolution of theneutrino flavor amplitudes through the resonance. Thereare two processes by which resonance affects neutrinoconversion in the early universe: (1) enhanced incoherentconversion of those neutrinos at the resonance and (2) co-herent MSW transformation of neutrinos passing throughthe resonance. The rate of incoherent production can becalculated numerically through the semi-classical Boltz-mann evolution, Eq. (6.6).

The statistical formulation of both coherent and in-coherent production can be described by the time evo-lution of the density matrix for the two neutrino states[70, 71, 72, 73, 74, 75]. We show below that the massesand mixing angles of interest for sterile neutrino dark

11

0.05 0.1 0.5 1 5 10Ε = pT

1.´10-12

1.´10-10

1.´10-8

1.´10-6

0.0001

0.01

1si

n22Θ

m

FIG. 4: The effective matter mixing sin2 2θm is shown for acase with two resonances (solid line). For this case, ms =1keV, sin2 2θ = 10−9, L = 6 × 10−4, and T = 130 MeV. Thedashed line is the active neutrino energy distribution.

matter considered here give coherence across the rele-vant resonance width, but usually imply that the neu-trino amplitude evolution through resonance is nonadia-batic. (Adiabatic evolution at lower mixing angle couldoccur at epochs where the entropy is being transferredfrom annihilating particles and where, as a consequence,the temperature and density do not change rapidly or areconstant with time, e.g., the QCD transition [82].)

Though the expansion rate of the universe scales asT 2, the active neutrino scattering rate scales as G2

FT 5.For neutrino flavor evolution through resonances, the co-herence condition will be met only when the inverse ofthe scattering rate (the active neutrino mean free path)is much larger than the resonance width or

T ≪(

4π3

5

)1/6g1/6

m1/3pl G

2/3F tan1/3 2θ

≈ 140 MeV(g/100)

1/6

(

sin2 2θ/10−10)1/6

. (7.5)

(Here, mpl ≈ 1.22 × 1022 MeV is the Planck mass.)

Therefore, for the mixing angles relevant here (sin2 2θ <∼10−10), at temperatures below 140 MeV, the resonancecould in principle drive coherent να νs transformation.The efficiency of MSW conversion relies on the adiabatic-ity of the evolution through the resonance region.

The width of the resonance is the product of thelocal density scale height of weak charges and tan 2θ.In turn, the biggest share of the density scale heightis determined by the expansion rate of the universe,H ≈ (8π3/90)1/2g1/2T 2/mpl, where the statistical weightin relativistic particles, g, has contributions from bothbosons gb and fermions gf :

g =∑

i

(gb)i + 7/8∑

i

(gf )i. (7.6)

In fact, once neutrino flavor transformation begins, theinherent nonlinearity of this process can have a sizable,

even dominant effect on the density scale height. Ignor-ing this, the resonance width expressed in time is δt ≈(2/3)t(tan 2θ) = (1/3)H−1 tan 2θ, where t ≈ (1/2)H−1

is the age of the universe at an epoch with temperatureT in radiation dominated conditions. [The particle hori-zon is H−1, so that, absent large scale neutrino flavortransformation, the resonance width is a constant frac-tion (1/3) tan 2θ of the horizon scale.]

The effective matter mixing angle for the oscillationchannel να νs at an epoch with temperature T is

sin2 2θm =

1 +

[

1 − 2ǫTV/(

δm2 cos 2θ)]2

tan2 2θ

−1

. (7.7)

At resonance, sin2 2θm = 1, so that one resonance widthoff resonance this effective mixing will have fallen tosin2 2θm = 1/2. Clearly, the change in effective weakpotential over this interval is δV ≈ δm2 sin 2θ/(2ǫTres),where Tres is the resonance temperature for neutrinospectral parameter ǫ. From this it can be seen that(δV/V )res = tan 2θ and it follows that the resonancewidth is

δt =δt

δVδV ≈

∣

∣

∣

∣

∣

1

V· dV

dt

∣

∣

∣

∣

∣

−1

res

tan 2θ. (7.8)

The neutrino oscillation length at resonance is lresm =(4πǫTres)/(δm2 sin 2θ) = 2π/δV , and the adiabaticity pa-rameter is proportional to the ratio δt/lresm , and is definedas

γ ≡ 2πδt

lresm

≈ (δV )2

∣

∣

∣

∣

∣

dǫ

dV

/

dǫ

dt

∣

∣

∣

∣

∣

res

. (7.9)

When there are many oscillation lengths within the reso-nance width (i.e., when γ ≫ 1) neutrino flavor evolutionwill be adiabatic and we can then expect efficient flavorconversion at resonance. [Given the small vacuum mix-ing angles relevant for this work and consequent smallwidths, the Landau-Zener jump probability [83] with theform PLZ ≈ exp (−πγ/2) gives an adequate gauge of theprobability of να → νs conversion at resonance, 1−PLZ.]

The physical interpretation of Eq. (7.9) is straightfor-ward. The adiabaticity parameter is proportional to theenergy width of the resonance divided by the “sweeprate,” dǫ/dt, of the resonance energy through the neu-trino distribution function. The resonance sweep rateis determined mostly by the expansion rate of the uni-verse (an inverse gravitational time scale), but the rateof change of lepton number L as να → νs proceeds canbecome important, even paramount as lepton number isused up and L → 0.

The resonance energy width scaled by temperature isδǫ ≈ δV

∣

∣dǫ/dV∣

∣

res≈ ǫres tan 2θ, where ǫres is the reso-

nant value of the neutrino spectral parameter at Tres. Ifwe confine our discussion to resonances on the low tem-perature side of TPEAK, where the thermal terms in the

12

potential can be neglected, then

ǫres ≈ δm2 cos 2θ(

4√

2ζ(3)/π2)

GFT 4L(7.10)

≈ 0.1245

(

δm2 cos 2θ

1 keV2

)(

10−2

L

)(

100 MeV

T

)4

.

As the universe expands and cools with time, and fora given δm2, the resonance will sweep through the να

energy distribution function from low to high neutrinospectral parameter ǫ. In this same limit of resonancebelow TPEAK, the sweep rate is

dǫ

dt≈ 4ǫH

(

1 − L4HL

)

, (7.11)

where L is the time rate of change of the lepton num-ber resulting from neutrino flavor conversion. Since theexpansion rate scales as H ∼ T 2, the prospects for adia-baticity are better at lower temperatures and later epochsin the early universe, all other parameters being thesame.

From Eqs. (7.9), (7.10), and (7.11), we can estimatethat at resonance the degree of adiabaticity is

γ ≈ δm2

2ǫTres

sin2 2θ

cos 2θ

(

4T

T+

LL

)−1

(7.12)

≈ 3√

5ζ(3)3/4

217/8π3

(

δm2)1/4

mpl G3/4F L3/4

g1/2ǫ1/4∣

∣

∣1 − L/4HL

∣

∣

∣

sin2 2θ

cos7/4 2θ

≈( ms

1 keV

)1/2(

10.75

g

)1/2( L10−2

)3/41

∣

∣

∣1 − L/4HL

∣

∣

∣

×(

1

ǫres

)1/4sin2 2θ

7.5 × 10−10

, (7.13)

where in the second equality we assume the standardradiation-dominated conditions and expansion rate, andwhere in the final equality we have employed the approx-imation δm2 ≈ m2

s, valid when ms ≫ mνα, and where

we have assumed that the vacuum mixing angle is small.Here we see that for the mixing angles allowed by ourconstraints, sin2 2θ < 10−9(3×10−10) for νµ, ντ (νe) mix-ing with sterile neutrinos, and masses ms

>∼ 1 keV, theresonance is not adiabatic.

We conclude that the main effect of resonance is en-hancement of scattering-induced incoherent conversion ofneutrinos with energies in the resonant region. Therefore,the formulation of the semi-classical Boltzmann Equation(6.6) is appropriate for calculating the total productionof sterile neutrinos in the early universe.

The results of our numerical calculations can be seenin Fig. 2 for νe νs and in Fig. 3 for ντ νs forthe cases where initially L = 0.001, 0.01, 0.1. The calcu-lation includes both nonresonant scattering productionand matter-enhanced (resonant) production. Examples

FIG. 5: The sterile neutrino distribution for four cases of res-onant and non-resonant νe ↔ νs, as described in the text.The dotted line is a normalized active neutrino spectrum.The thick-solid, dashed, dot-dashed, and thin-solid lines cor-respond to cases (1)–(4), respectively. The inset shows a mag-nified view of the low momenta range of the distributions.

of the resulting sterile neutrino energy spectra are shownin Fig. 5. Resonantly produced sterile neutrinos tend tohave energy spectra appreciably populated only at thelow ǫ end. This results from the resonant energy start-ing at the lowest momenta and moving through highermomenta neutrinos [see Eq. (7.10)] as the universe coolsand lepton number is depleted through conversion into asterile neutrino population.

Figure 5 shows the resulting spectrum for four samplecases of sterile neutrino dark matter production:

(1) ms = 0.8 keV, sin2 2θ = 10−6, Linit = 0.01, resultingin Ωνs

h2 = 0.25 and 〈p/T 〉 = 2.9;

(2) ms = 1 keV, sin2 2θ = 10−7, Linit = 0.01, resultingin Ωνs

h2 = 0.13 and 〈p/T 〉 = 1.8;

(3) ms = 1 keV, sin2 2θ = 10−8, Linit = 0.01, resultingin Ωνs

h2 = 0.10 and 〈p/T 〉 = 2.0;

(4) ms = 10 keV, sin2 2θ = 10−8, Linit = 0.001, resultingin Ωνs

h2 = 0.57 and 〈p/T 〉 = 2.3.

A particularly interesting case is (4), where the resonancepasses through the distribution during the QCD transi-tion, where the disappearance of degrees of freedom heatsthe photon and neutrino plasma, forcing the universe tocool more slowly. For this period, the resonance movesmuch more slowly through the spectrum and is conse-quently more efficient in να → νs conversion throughthat region of the neutrino energy spectra. This pro-duces a “spike” in the sterile neutrino distribution (seeRef. [82]).

13

VIII. COLLISIONLESS DAMPING SCALE:

HOT, WARM OR COLD NEUTRINOS

Given the sterile neutrino energy spectrum and num-ber density, the transfer function for dark matter modelscan be calculated. Here, we instead give a rough guidebased on the free streaming length at matter-radiationequality, λFS . Structures smaller than λFS are damped.This is because at early epochs where sterile or activeneutrino species are relativistic, they can freely flow outof the regions of sizes smaller than λFS (very roughlythe horizon size at the epoch where the neutrinos revertto nonrelativistic kinematics). Previous numerical workhas shown that the free streaming scale is approximately[67, 84]

λFS ≈ 40 Mpc

(

30 eV

mν

)( 〈p/T 〉3.15

)

. (8.1)

The mass contained within the free streaming length isthen

MFS ≈ 2.6 × 1011 M⊙

(

Ωmh2)

(

1 keV

mν

)3( 〈p/T 〉3.15

)3

,

(8.2)

where Ωm is the contribution of all “matter” to closure.These values can give a guide as to the collisionless damp-ing scale of sterile neutrino dark matter. In Fig. 6, weshow contours of MFS for the ντ νs sterile neutrinoproduction channel with L = 0.01. In Fig. 6, we assumethat the universe is critically closed (Ωm = 1). Therefore,the contours are consistent with sterile neutrinos beingthe major constituent of dark matter near the dark grayregions where Ωνs

h2 = 0.1 − 0.5.In Figs. 2 and 3, we label the parameter regions corre-

sponding to MFS > 1014M⊙, 105M⊙ < MFS < 1014M⊙,and MFS < 105M⊙ as HDM, WDM, and CDM, respec-tively. These definitions are somewhat arbitrary and hereserve only as guides. Note that sterile neutrino mass byitself does not completely determine the free streaming(or collisionless damping) scale, since the inherent νs en-ergy spectrum at production also helps to determine theνs kinematics at a given epoch. This latter effect is es-pecially pronounced for nonthermal energy spectra, andis responsible, e.g., for the non-flat collisionless dampingmass scale lines in Fig. 6.

IX. COSMOLOGICAL CONSTRAINTS ON

STERILE NEUTRINOS

FIG. 6: The mass within the sterile neutrino free streaminglength at matter-radiation equality (MF S) in the ντ ↔ νs,L = 0.01 case. Regions of parameter space disfavored bysupernova core collapse considerations are shown with verticalstripes.

A. Diffuse extragalactic background radiation

The decay rate of a massive sterile neutrino with vac-uum mixing angle θ into lighter active neutrinos is [85]

Γνs≈ sin2 2θ G2

F

(

m5s

768π3

)

(9.1)

≈ 8.7 × 10−31 s−1

(

sin2 2θ

10−10

)

( ms

1 keV

)5

.

We have also included in our calculations the contri-butions to Γνs

from visible and hadronic decays esti-mated from the partial decay widths of the Z0 boson [7].The rate of the corresponding radiative decay branch issmaller by a factor of 27α/8π [86]:

Γνsγ ≈ sin2 2θ α G2F

(

9m5s

2048π4

)

(9.2)

≈ 6.8 × 10−33 s−1

(

sin2 2θ

10−10

)

( ms

1 keV

)5

.

That these sterile neutrinos can decay stems from thefact that they are not truly “sterile,” but have an effectiveinteraction strength ∼ sin2

m 2θ G2F = sin2 2θ G2

F, wherethe last equality is valid in vacuum (i.e., at late epochsin the universe). Obviously, if a particle is going to be adark matter candidate, it must have a lifetime τ at leastof order the age of the universe (ttoday

>∼ 10 Gyr). Re-

gions of ms vs sin2 2θ parameter space where τ < ttoday

and not otherwise constrained are so labeled in Figs. 7and 8. These parameters, corresponding to generally

14

high sterile neutrino masses, guarantee that these par-ticles decay away into lighter particles long before theyclump in gravitational potential wells.

The radiative decay channel for sterile neutrinos canprovide a constraint or a possible detection mode forsome regions of the ms − sin2 2θ parameter space. Inparticular, radiative decays of sterile neutrinos occuringbetween CMB decoupling and today could produce anappreciable flux of photons with energies comparable toms. In fact, there are firm upper limits on the flux froma diffuse photon component. For example, the total dif-ferential energy flux per unit solid angle for a DEBRAcomponent is [27, 67, 87]

dF/dΩ <∼ (1 MeV/E) cm−2sr−1s−1. (9.3)

As technology has progressed this limit on the true dif-fuse background has come down as distinct x-ray sourcesare resolved and their fluxes are removed from the count[88, 89]. At present the Chandra X-ray Observatorythreatens to resolve a considerable fraction of the ob-served x-ray background (in the 0.5 − 8 keV band) intopoint and extended sources, primarily active galaxies,QSO’s, and clusters.

Clearly, sterile neutrinos with radiative decay ratesgreater than an inverse Hubble time (H−1 ≈ 3.09 ×1017 h−1 s ≈ 9.78 h−1 Gyr) will tend to contribute back-ground photons before the sterile neutrinos fall into po-tential wells and form structure. These photons will thenproduce a DEBRA contribution which must not exceedthe overall limit in Eq. (9.3). Parameter regions violatingthis bound are labeled DEBRA in Figs. 7 and 8.

What about sterile neutrinos with much smaller radia-tive widths? These steriles could be decaying at morerecent epochs, even today. However, if these are thedark matter (CDM or WDM), then they are not dif-fuse, but are strongly clustered. In Figs. 7 and 8 theregions with widely spaced vertical lines correspond tosterile neutrino mass and mixing properties that wouldgive a DEBRA component in excess of the limit (9.3),if these steriles were distributed diffusely. Since most ofthe decay photons will be produced when the dark mat-ter is in structure rather than diffuse, this region doesnot as yet constitute a constraint. Rather, it serves todefine parameters that could give interesting x-ray fluxesin the gravitational potential wells of clusters of galaxiesor other structures, depending on redshift, the cosmolog-ical parameters, and transfer functions and collisionlessdamping scales of the sterile neutrinos. Improved obser-vations and models could lead to these regions turninginto true constraints and may result in more stringentconstraints or even lead to detection [90].

As an example, consider a large cluster of galaxies withdark matter mass M ≈ 1015 M⊙. The sterile neutrinodecay luminosity in photons is

L ≈ 7 × 1036 erg s−1

(

sin2 2θ

10−10

)

( ms

1 keV

)5

. (9.4)

The observed x-ray luminosity for such clusters in the1−10 keV band is Lcluster ∼ 1045 erg s−1 [91]. Clearly, forsterile neutrinos in our dark matter parameter space withmasses of 20− 30 keV and sin2 2θ ≈ 10−11, the predictedluminosities may be comparable to those observed. Theenergy spectra may be different, however, and this couldbe an avenue for constraint.

B. Cosmic microwave background

Another constraint stems from the increase in energydensity in relativistic particles due to massive sterileneutrino decay prior to cosmic microwave background(CMB) decoupling [92]. The BOOMERanG/MAXIMAobservations [93] currently limit the effective number ofneutrinos (Nν) at decoupling to Nν(CMB) < 13 at the95% confidence level [55]. Measurements to higher mul-tipole moments by the MAP and Planck probes will beable to further limit the relativistic energy present atdecoupling. MAP may constrain Nν(CMB) < 3.9, andthe Planck mission could reach a limit Nν(CMB) < 3.05[94]. The increase in energy density due to sterile neu-trino decays was found by calculating the energy pro-duced by decays between active neutrino decoupling(T ∼ 1 MeV) and photon decoupling (T ∼ 0.26 eV). TheBOOMERanG/MAXIMA constraint is shown in Figs. 7and 8 and the potentially stringent future “constraints”by MAP and Planck are also shown.

C. Big bang nucleosynthesis

The energy density in the sterile neutrino sea at weakfreeze-out just prior to primordial nucleosynthesis (T ≈0.7 MeV) must not be too large. The energy density con-tribution of a sterile neutrino is often described as thefraction of the energy density in a fully populated (ther-mal) single neutrino-antineutrino species plus the energydensity in the active neutrinos, Nν . Depending on one’sparticular adoption of observationally inferred primordialabundances, one can arrive at limits between Nν < 3.2and Nν < 4 [95, 96, 97]. We have calculated the energydensity contributed by the sterile neutrino at BBN, andits limits are shown in Figs. 7 and 8.

D. 6Li and D photoproduction

Another constraint arises from photoproduction ofdeuterium (D) and 6Li stemming from the decay of mas-sive sterile neutrinos after BBN [98]. Energetic cascadesdissociate 4He into excessive amounts of D, which isbounded observationally [99]. Also, energetic 3H and3He produced in the cascades can synthesize 6Li through3H(3He) +4 He →6 Li + n(p), overproducing 6Li, whichis also bounded observationally [100].

15

These constraints lie in regions of parameter spacewhich are almost identical to those bounded by CMBconsiderations. The 6Li limit can be seen to extend tothe small region just to the left of the CMB constraintsin Figs. 7 and 8 (closely hatched area, hatching orientedfrom the lower right to the upper left). It should benoted that the 6Li/D photoproduction constraints arefrom current observation, while the potential constraintsfrom MAP and Planck of the energy density present atCMB decoupling are a future possibility that can corrob-orate the photoproduction constraints.

FIG. 7: Contours of closure fraction are displayed as thicklines labelled with Ωνsh2 for negligible lepton asymmetry(L ≈ 10−10) for the three neutrino flavors mixing with a mas-sive sterile neutrino. Shown are constraints from the energydensity present at CMB decoupling, with the dark gray regioncorresponding to BOOMERanG/MAXIMA’s present limits.The medium gray region corresponds to MAP’s predictedfuture constraint, and the light gray region is the Planckmission’s potential constraint region. The constraints fromDEBRA, and BBN are so labelled. The region of potentialconstraints from supernovae (SN) is also shown as closelyspaced vertical lines. The region of widely spaced verticallines is where radiative decays today may give detectable X-ray signatures. Regions filled with horizontal lines are whereΩνsh2 > 0.5 and are inconsistent with the observed age ofthe universe. Sterile neutrinos with parameters in the regionlabelled by “τ < ttoday” decay without constrainable effectsand make no contribution to the present matter density.

FIG. 8: Contours of closure fraction are displayed as thicklines labelled with Ωνsh2 for L = 0.001, 0.01, 0.1. Constraintsare shown as in Fig. 7.

X. ACTIVE-STERILE NEUTRINO

TRANSFORMATION IN CORE-COLLAPSE

SUPERNOVAE

In the previous sections, we have calculated the rangesof sterile neutrino masses and mixing angles (and pri-mordial lepton asymmetries) for which sterile neutrinoscan account for some or all of the dark matter. In thissection we describe some of the implications for core-collapse (Type Ib/c, II) supernovae of active-sterile neu-trino transformation with parameters in these regions.Neutrinos play a dominant, pervasive role in core-collapsesupernovae, so the stakes are high whenever we introducenon-standard neutrino physics like neutrino mixing.

In broad brush the effects are simple: too much neu-trino conversion in a supernova results in too much en-ergy loss to sterile neutrinos, manifestly in conflict withobservation (the mixing angles considered here are suffi-ciently small that the sterile neutrinos are not trapped inthe core). Requiring the hemorrhaging in sterile neutri-nos not to be too great then places bounds on neutrinomixing parameters.

In fine detail, however, the procedure is not so sim-ple, since the coupled supernova neutrino transport-transformation problem is highly nonlinear and the dy-namics very difficult to treat analytically or numerically.

16

Confounding the matter is our relatively poor under-standing of the physics of the core-collapse phenomenonitself.

Nevertheless, we have boldly attempted (as have sev-eral workers before us [28, 29, 31, 68, 114]) to describesome of the salient effects of neutrino transformation in asupernova and have extracted some rather conservativelimits on mixing parameters, conservative, at least, forarguing the case for sterile neutrino dark matter. Our“limits” are by no means final, and much future workremains to be done in order ascertain the viability ofsterile neutrino dark matter. In what follows, we give abrief biography of a core-collapse supernova, describe therelevant neutrino transport and transformation physics,and indicate how we obtained the supernova “constraint”regions in Figs. 2, 3, 6, 7, and 8.

Core-collapse supernovae are the death throes of mas-sive stars. A star of mass >∼ 10M⊙ dies when its ≈ 1.4M⊙

iron core undergoes gravitational collapse to a neutronstar, an event releasing ≈ 99% of the ≈ 1053 erg gravi-tational binding energy of the neutron star in all activespecies of neutrinos (see Refs. [101, 102] for an introduc-tion). In the standard picture this event lasts ≈ 10−15 sand can be divided roughly into three phases: the infall,shock reheating, and r-process epochs.

During the infall epoch, the iron core collapses on anear gravitational time scale of ∼ 1 s to a dense proto-neutron star. As the core density rises, the forward reac-tion in

e− + “p” νe + “n” (10.1)

neutronizes the core until the neutrino trapping densityρ ∼ 1011 − 1012 g cm−3 is reached, a condition which al-lows the forward and reverse reactions in Eq. (10.1) toachieve β-equilibrium. This equilibrium is establishedwith relativistically degenerate electrons (µe ≈ 25 −220 MeV) and electron neutrinos (µνe

≈ 10 − 170 MeV).The quotes in Eq. (10.1) refer to free and bound

baryons. Most of the baryons are bound in large nuclei,since the infall-collapse phase is characterized by tem-peratures T ≈ 1 − 3 MeV and a low entropy-per-baryons ≈ 1.5 (in units where Boltzmann’s constant is unity).

The infall epoch ends when the core density reachesand exceeds the saturation density of nuclear matter.The inner core bounces, yielding an outward-propagatingshock wave at its boundary with the outer core. Theshock loses much of its energy dissociating the nucleiin the outer core and mantle of the pre-supernova starand eventually stalls some 500 km from the center of thecore, far short of generating a supernova explosion. Thepost-bounce core is a hot, dense proto-neutron star con-sisting of free baryons, electrons and positrons, and ac-tive neutrino-antineutrino pairs (and perhaps also muon-antimuon pairs and strange quark matter), all in ther-mal and chemical equilibrium at a temperature T ≈30 − 70 MeV and entropy per baryon s ≈ 10.

During the shock reheating epoch, at times post corebounce tpb ≈ 0.06− 1 s, neutrinos diffuse out of the core

and deposit energy behind the stalled shock, driving thesupernova explosion [103].

In the r-process epoch neutrinos continue to diffuseout of the star at times tpb ≈ 1 − 15 s. The neutrinosdrive mass loss from the proto-neutron star, possibly set-ting the physical conditions for r-process (heavy element)nucleosynthesis [104, 105].

The star’s nuclear composition becomes increasinglyneutron rich as the electron neutrinos depart the starand the reactions in Eq. (10.1) proceed with continuous,local re-establishment of β equilibrium. For example, atcore bounce (tpb = 0), the electron fraction (net numberof electrons per baryon) in the center of the core is Ye ≈0.35; a cold neutron star, the end point of evolution inthis case, has Ye

<∼ 0.01.Since neutrinos dominate the energetics of core-

collapse (Type Ib/c, II) supernovae, appreciable active-sterile neutrino transformation can completely alter thestandard picture of stellar collapse. Although this picturehas been refined by observations of neutrinos from SN1987A [106], large-scale numerical simulations [103, 107],and semi-analytical work [103, 108, 109, 110], it is notclear that this is the only way for supernovae to evolve.Nevertheless, by requiring that not too much energybe “lost” too quickly to singlet neutrinos in the proto-neutron star, and hence avoiding a conflict with stan-dard supernova theory and observations, we can delimitregions in the ms − sin2 2θ plane which may adverselyaffect its evolution. In fact, many supernova constraintson new physics rely on limiting the energy loss to exoticparticles in a proto-neutron star [111].

Sterile neutrinos can be produced in supernovae eithercoherently through mass level-crossings or incoherentlyvia scattering-induced wave function collapse. Whetherthe former or latter process dominates depends on thehierarchy of length scales relevant for neutrino trans-port and transformation: the local neutrino mean freepath λ, local neutrino oscillation length in matter lm,and resonance width δr [the supernova analogue of Eq.(7.8)]. Neutrino flavor eigenstate evolution is coherentand transformation is uninterrupted through resonanceif λ ≫ δr. Neutrino flavor transformation is also adia-batic if δr ≫ lresm , where lresm ≥ lm is the oscillation lengthat resonance. This is in complete analogy to the evolu-tion through resonances discussed above in Sec. VII forthe early universe. If instead neutrino conversion is inco-herent and transformation is interrupted often by colli-sions, then the conversion will be suppressed by quantumdamping if lm > λ.

It is easy to show that incoherent conversion domi-nates sterile neutrino production in supernova cores. Ina proto-neutron star of radius R = 10 − 50 km the den-sity varies relatively slowly with distance from the centerof the star, so the scale height of the weak potential isH = |d lnV/dr|−1 ≈ 10 − 100 km. Then the resonancewidth is δr = H tan 2θ ≈ (106−107 cm) tan 2θ. The meanfree path of a typical neutrino is about λ ≈ 10 cm (actu-ally λ ∼ 10 cm to ∼ 10 m, but this range makes little dif-

17

ference for our conclusions). Therefore, neutrino evolu-tion is coherent if sin2 2θ <∼ 10−12−10−10, and the maxi-mal resonance width giving coherent evolution is δrmax ≈10 cm. Now the oscillation length at resonance is lresm =4πE/δm2 sin 2θ ≈ 10−4 cm (10 keV/ms)

2/ sin 2θ, so co-herent evolution gives lresm

>∼ 10 − 100 cm(10 keV/ms)2.

In these conditions, however, the evolution is not adia-batic, since δrmax

<∼ lresm . As a result, mass level crossingscontribute only subdominantly to sterile neutrino pro-duction in supernovae. Conversion to sterile neutrinosless massive than 10 keV is even less efficient, and moremassive neutrinos will not have resonances.

Incoherent neutrino production is weakly dampedat (or away from) a resonance when sin2 2θ >∼10−10(10 keV/ms)

4. The local oscillation length in mat-ter away from resonance is typically much smaller thanthe oscillation length at resonance, so for sterile neutrinomasses ms

>∼ 10 eV weak damping generally obtains offresonance even if the condition above is not met. Fur-thermore, since resonances last only a fraction <∼ 10−8 ofthe neutrino diffusion time scale, as derived below, mostneutrino conversion in a core-collapse supernova occursin the weak damping regime.

The time rate of energy “loss” E to sterile neutri-nos (and sterile antineutrinos) per unit mass in a proto-neutron star is proportional to the active neutrino energy,scattering cross section on weak targets, and the averageconversion probability [28, 29, 111]:

E ≈ 1

mN

∫

dΦναE σνα

(E)1

2〈Pm(να → νs; p, t)〉

+1

mN

∫

dΦναE σνα

(E)1

2〈Pm(να → νs; p, t)〉,

(10.2)

where, for the sake of simplicity, we have suppressedPauli-blocking effects. In Eq. (10.2), the cross sectionsfor neutrino scattering on free baryons are σνα

(E) ≈σνα

(E) ≈ 1.66G2F E2, and the differential neutrino and

antineutrino fluxes are

dΦνα= c dnνα

≈ d3p

(2π)31

eE/Tνα−ηνα + 1

≈ 1

2π2

E2dE

eE/Tνα−ηνα + 1(10.3)

dΦνα= c dnνα

≈ d3p

(2π)31

eE/Tνα−ηνα + 1

≈ 1

2π2

E2dE

eE/Tνα−ηνα + 1(10.4)

for relativistic neutrinos, where ηνα= µνα

/Tναis the

denegeracy parameter of να as above. A simple butsomewhat crude criterion for avoiding conflict with su-pernova theory and observations of SN 1987A is E <∼1019 erg s−1 g−1 [111], a limit on the sterile neutrino emis-sivity equivalent to a loss of ∼ 10 MeV per baryon persecond.

The weak potentials [cf. Eqs. (5.5) and (5.6)] whichdrive neutrino flavor transformation take a slightly dif-ferent form in supernovae than in the early universe [64].For νe νs transformation the potential stemming fromfinite density effects is [from Eq. (5.5)]

V D =GF ρ√2mN

(

3Ye − 1 + 4Yνe+ 2Yνµ

+ 2Yντ

)

, (10.5)

where mN ≈ 931.5 MeV is an atomic mass unit, ni is thenumber density of species i, and Yi ≡ (ni − ni)/nB isthe net fraction of species i per baryon (nB = nn + np).In part, the form of Eq. (10.5) follows from local chargeneutrality and the assumption that muons, anti-muons,and strange quark matter are negligibly populated. Thefinite density potential for νµ νs transformation is[from Eq. (5.5)]

V D =GF ρ√2mN

(

Ye − 1 + 2Yνe+ 4Yνµ

+ 2Yντ

)

. (10.6)

The potential for ντ νs transformation follows fromEq. (10.6) upon switching the labels νµ and ντ . Thefinite temperature potential V T is negligible in super-novae [64], so the total potentials for neutrino and an-tineutrino transformation are V = V D + V T ≈ V D andV = −V D + V T ≈ −V D ≈ −V , respectively.

The average oscillation probabilities in Eq. (10.2) de-pend on the weak potentials and neutrino mixing param-eters, from Eqs. (6.4), (6.5):

〈Pm(να → νs; p.t)〉 ≈1

2

∆(E)2 sin2 2θ

∆(E)2 sin2 2θ + D2 + [∆(E) cos 2θ − V D]2(10.7)

〈Pm(να → νs; p, t)〉 ≈1

2

∆(E)2 sin2 2θ

∆(E)2 sin2 2θ + D2 + [∆(E) cos 2θ + V D]2, (10.8)

where ∆(p) = δm2/2p ≈ m2s/2E ≈ ∆(E) for rela-

tivistic neutrinos. As before the quantum damping rateD = Γνα

/2 =∫

dΦνασνα

(E)/2 for neutrinos is one-halfthe neutrino scattering rate; an analogous expression ap-plies for the antineutrino damping rate D. The coherenteffects described in Ref. [69] may modify these conversionrates.

The conditions for resonance ±V D = ∆(p) cos 2θ ≈m2

s cos 2θ/2E, the potentials in Eqs. (10.5), (10.6), andthe average oscillation probabilities in Eqs. (10.7), (10.8)imply a negative feedback between active (anti)neutrinotransformation and the potentials. If quantum dampingeffects are comparable for neutrinos and antineutrinos orare relatively unimportant, as argued above for most ofthe parameter range of interest, neutrino conversion is en-hanced relative to antineutrino conversion when V D > 0.The preferential conversion of neutrinos decreases the po-tential, since the finite density part of the potential de-pends on the relative excess (or deficit) of particles overantiparticles (the net contribution of particle-antiparticle

18

pairs is zero). As the potential decreases, the neutrinoand antineutrino conversion rates approach the commonrate they would have at vanishing potential. Since neu-trinos and antineutrinos of all flavors are produced inroughly equal numbers and their individual number den-sities are at least comparable to the electron density, thedynamical feedback mechanism may drive the potentialV D locally to zero throughout the proto-neutron star, aslong as the time scale for this process is less than thelocal neutrino diffusion time scale td ∼ λ(R/λ)2/c ∼ 3 s;a similar sequence will ensue if V D < 0. If this happens,the equal neutrino and antineutrino conversion rates en-sure that the potential will remain zero, with any localdeviation continuously smoothed out, up to the higherorder effects of neutrino diffusion.

In this sense zero potential is a fixed point of thedynamics of neutrino transport and conversion in core-collapse supernovae and the time scale for achieving zeropotential is a dynamical equilibration time scale. The sit-uation is more complicated if sterile neutrinos mix withmultiple active neutrinos or if damping is important. Forexample, an excess of neutrinos over antineutrinos im-plies damping is asymmetric, and this can retard equili-bration. These issues will be examined in more detail in aseparate work [112]. We note in passing that this behav-ior can also occur when neutrino scattering is rare andmass level crossings dominate sterile neutrino produc-tion [113]; some of this physics has also been discussedin connection with active-active neutrino transformationin proto-neutron stars [114].

We can estimate the equilibration time scale τV fordriving a potential V D

0 to zero by computing the timeover which the disparity in the rates of neutrino andantineutrino conversion will induce a potential δV D =−V D

0 which cancels the original potential V D0 . Now Eqs.

(10.5), (10.6) imply δV D = 2√

2GF (δnνα− δnνα

), whereδnνα

and δnναare the numbers of neutrinos and an-

tineutrinos converted per unit volume, respectively. Thenumber densities of (anti)neutrinos converted are pro-portional to the rates of (anti)neutrino conversion perbaryon, the baryon density, and the time over which theconversion takes place. If we assume τV is small com-pared to the time scales for the change of the density andtemperature in the star (a “static” approximation), wehave δnνα

= −nBτV

∫

dΦνασνµ

(E)12 〈Pm(νµ → νs; p, t)〉

and similarly for δnνα. Solving for τV and using nB =

ρ/mN gives the time scale for achieving dynamic equilib-rium in the να νs, να νs system:

τV =V D

0 mN

2√

2GF ρ

(

∫

dΦνασνα

(E)1

2〈Pm(να → νs; p, t)〉

−∫

dΦνασνα

(E)1

2〈Pm(να → νs; p, t)〉

)−1

. (10.9)

It is instructive to evalute the equilibration time scaleτV for the νµ νs, νµ νs system in two limits: (i) faraway form a resonance, for which |V D

0 | ≫ ∆(E) cos 2θ;

and (ii) very near or at a resonance, where |V D0 | ≈

∆(E) cos 2θ ≈ m2s/2E for θ ≪ 1. In the off-resonance

case, inserting Eqs. (10.3)-(10.4) and Eqs. (10.7)-(10.8)in Eq. (10.9) and expanding to lowest order yields

τoff−resV ≈ 96

1.66√

2

(

V D0

)4mN

G3F ρT 2m6

s sin2 2θ

≈ 1.5 × 10−8 s

sin2 2θ

(

1014 g cm−3

ρ

)

×(

50 MeV

T

)2(10 keV

ms

)6(V D

0

1 eV

)4

,

(10.10)

where the factor of 1.66 comes from the neutrino scat-tering cross section given earlier. We have derived Eq.(10.10) by assuming a locally constant density and tem-perature, consistent with our “static” approximation.We have also ignored the build up of a non-zero muonneutrino chemical potential µνµ

and accompanying Pauliblocking in neutrino scattering and pair production asequilibration proceeds, a simplification giving dΦνµ

≈dΦνµ

. Using the same simplifications, we can find thetime scale in the on-resonance case:

τon−resV ≈ 8π2

(1.66)45√

2ζ(5)

|V D0 |mN

G3F ρT 5

≈ 16ζ(3)

(1.66)7π2√

2ζ(5)

m2smN

G3F ρT 6

≈ 2 × 10−9 s

(

50 MeV

T

)5(1014 g cm−3

ρ

)∣

∣

∣

∣

V D0

1 eV

∣

∣

∣

∣

≈ 6.6 × 10−10 s

(

50 MeV

T

)6(1014 g cm−3

ρ

)

×( ms

10 keV

)2

. (10.11)

The second and fourth expressions of Eq. (10.11) followfrom the resonance condition |V D

0 | ≈ m2s/2E and taking

the neutrino energy E to be the average neutrino energy〈E〉 =

∫

dΦνµE/∫

dΦνµ≈ 7π4Tνµ

/180ζ(3) ≈ 3.15Tνµin

the proto-neutron star.The on- and off-resonance equilibration time scales

both vary inversely with the ambient density and tem-perature. Hotter conditions enhance the neutrino con-version rate per scatterer, and denser conditions en-hance the number density of scatterers. In either caseit takes less time to drive a pre-existing potential tozero. On the other hand, only the off-resonance timescale τoff−res

V depends on the vacuum mixing angle; theinverse dependence is natural, since the emissivity in ster-ile (anti)neutrinos is proportional to sin2 2θ. The on-resonance time scale τon−res

V is independent of sin2 2θ,because the relevant mixing angle at a resonance is π/4.If V D

0 > 0 neutrino conversion dominates antineutrinoconversion at resonance and vice versa if V D

0 < 0, so thistime scale is extremely short compared to the neutrino

19

diffusion time scale or indeed any time scale typically as-sociated with a core-collapse supernova. Individual reso-nances are fleeting and result in negligible energy loss tosterile neutrinos. As a result, we may safely assume thatmost neutrino transformation takes place off resonance,with an equilibration time scale given by Eq. (10.10).

We can delimit ranges of the neutrino mixing param-eters ms and sin2 2θ which may adversely affect core-collapse supernovae by evaluating the sterile neutrinoemissivity in Eq. (10.2) in the following manner.

In the case of νe νs transformation, the poten-tial V D is initially positive in the post-bounce supernovacore. If the neutrino mixing parameters happen to giveτoff−resV > td, core neutronization proceeds roughly on a

diffusion time scale, although the conversion of electronneutrinos accelerates the shift of β equilibrium towardneutron richness. Neutronization decreases the electronfraction Ye, and neutrino conversion decreases the elec-tron neutrino fraction Yνe

. As a consequence, the poten-tial decreases and eventually reaches a value such thatτoff−resV < td.Once this occurs, neutrino conversion quickly resets

the potential to zero and maintains this value as neu-tronization continues. If instead the neutrino mixing pa-rameters happen to give τoff−res

V < td at core bounce,conversion immediately resets and maintains the poten-tial at zero.

In the case of νµ νs or ντ νs transformation,the potential V D is initially negative in the post-bouncecore. Here conversion has no direct effect on the rate ofneutronization (notwithstanding feedback on the nuclearequation of state), which increases the magnitude of thepotential as the electron fraction falls to a value ∼ 0.01.Unless τoff−res

V < td from the outset, neutronization dic-tates the evolution of the potential.

The emissivity E depends on the local, instantaneousvalue of the potential, so the total energy lost to sterileneutrinos depends on the time history of the spatiallyvarying potential and, in particular, whether neutroniza-tion or neutrino conversion locally dominates its evolu-tion. An accurate estimate of the spatially integratedemissivity and, hence, of the regions in parameter spacewhich may alter the standard picture of stellar collapseclearly requires a detailed investigation of the dynamicsof these systems, as well as a better understanding of thephysics of core-collapse supernovae.

From the viewpoint of the viability of sterile neutri-nos as dark matter, however, we may conservatively esti-mate the emissivity by assuming the system achieves zeropotential relatively quickly, so that E is evaluated withV D = 0. Evaluating Eq. (10.2) in this limit and imposingthe condition E <∼ 1019 erg s−1 g−1 gives the disfavored re-

gions sin2 2θ >∼ 3×10−10 for νe νs transformation and

sin2 2θ >∼ 10−9 for νµ νs or ντ νs transformation forsterile neutrino masses ms

>∼ 10 eV. For smaller masses,the limits are significantly weaker, owing to the onset ofquantum damping. These limits are shown in Figs. 2, 3,6, 7, and 8. The emissivity falls with increasing potential

away from a resonance, so more detailed future work onthis complex problem may find actual constraints whichare weaker than the very conservative limits given here.

Sterile neutrino dark matter parameters which lie nearthe edges of the disfavored regions in Figs. 2, 3, 6, 7,and 8 could give interesting signals in current and fu-ture supernova neutrino detectors (see, e.g., Ref. [115]).These detectors possibly could discern unique signaturesfor sterile neutrinos. Such signatures would bolster thecase for sterile neutrino dark matter.

To summarize this section, we have delimited conserva-tively the regions in the parameter space of active-sterileneutrino mixing that are disfavored by energy-loss con-siderations in core-collapse supernovae. We have foundthat the coupled problem of neutrino transport and fla-vor transformation in hot and dense nuclear matter is aformidable one. It involves following the local evolutionof the weak potential which drives flavor transformation,including the feedback from diffusion and the conversionitself. We have described this physics roughly by esti-mating the competing time scales for lepton number dif-fusion and for cancellation of the potential. Dependingon the mixing parameters and the spacetime evolution ofthe potential, the potential may well be reset to zero inmuch of the proto-neutron star, so the effects of neutrinopropagation in matter need not suppress sterile neutrino(or antineutrino) production. As a result, core-collapsesupernovae can be significantly more sensitive to active-sterile neutrino mixing than they were found to be inprevious studies [28, 31, 68]. These studies used a spa-tially and temporally constant value of the potential, re-sulting in limits on mixing angles similar to ours but forsignificantly larger sterile neutrino masses >∼ 10 keV; forsmaller masses their limits on mixing angles are weakerbecause the putative matter effects suppress conversion[28, 31, 68].

Of course, we have deliberately chosen to be conser-vative in applying supernova limits, since our objectivein this work is to assess the viability of sterile neutrinodark matter. The true limits, quite interesting in theirown right and obtained from a self-consistent and propertreatment of the full, multi-dimensional Boltzmann evo-lution of the neutrino seas coupled with the nuclear equa-tion of state, may well lie somewhere in between the val-ues determined in this and previous studies. We leaveattempts at such investigations for a future work [112].

XI. CONCLUSIONS

We have estimated the resonant and non-resonant scat-tering prodution of sterile neutrinos in the early universe.The basis for the production of these sterile species is apresumed mixing with active neutrinos in vacuum. Ofcourse, such mixing renders these species not truly “ster-ile.” As a result, the “sterile” neutrinos can decay andthis, together with their overall contribution to energydensity, constitutes the basis for several stringent cos-

20

mological and astrophysical constraints which we havediscussed in detail.

Additionally, these sterile species may produce signif-icant effects in core-collapse supernovae. Some of theseeffects, such as massive core energy loss, could be thebasis for true constraints. However, it must be kept inmind that (1) the supernova explosion energy is onlysome ∼ 1% of the total energy resident in the activeneutrino seas and that (2) we do not yet understand indetail how supernovae explode (nor do we have a suf-ficiently detailed observed core-collapse neutrino signalto place stringent constraints). Better theoretical un-derstanding of supernova physics, perhaps coupled withthe neutrino signature of a Galactic core-collapse event,may allow the indicated regions on Figs. 7 and 8 to be-come true hard and fast constraints instead of simply“disfavored” parameter regions. In fact, deeper insightinto the time evolution of the potentials governing sterileneutrino production in the core may allow extension ofthe constrained parameter region to even smaller valuesof vacuum mixing angle.

Nevertheless, it is clear from our work that sterile “neu-trino” species with ranges of masses and vacuum cou-plings could be produced in quantities sufficient to ex-plain all of the non-baryonic dark matter, while evadingall present day laboratory and astrophysical constraints.Within the allowed ranges in mass and mixing parame-ters which give these dark matter solutions are regionswhere the sterile neutrino masses and/or energy spectracombine to produce collisionless damping scales corre-sponding to warm or even cold dark matter, subsumingthe interesting behavior range for large scale structure.

It is then a disturbing possibility that the dark mat-ter might not be “weakly interacting massive particles”(WIMPs), but rather “nearly non-interacting massiveparticles” (NNIMPs) which are likely not detectable inordinary dark matter detection experiments. This pos-sibility begs two questions: (1) how could we hope toconstrain or definitively detect this dark matter candi-date; and (2) what are sterile neutrinos?

The answer to the first question is more straightfor-ward than the resolution of the second. As outlinedabove, better understanding of the neutrino and equationof state physics of core-collapse supernovae could help usextend constraints. Improved observations and models ofx-ray emission from clusters of galaxies and other objectscould provide stringent constraints or, conceivably, evendetections of sterile neutrino photon decay.

As discussed above, future observations bearing on theclustering of dwarf galaxy or Lyman-α cloud halos at highredshift may provide evidence for or against WDM asopposed to CDM. Direct evidence for WDM would con-stitute a point in favor of sterile neutrinos, though not adefinitive one by any means. Likewise, and insidiously,our allowed parameter space for dark matter accommo-dates standard CDM behavior, even for lepton numbersequal to the baryon number.

Direct detection of WIMPs in the laboratory or detec-

tion of gamma rays associated with WIMP annihilationin galactic centers [116] obviously rule out sterile neu-trino dark matter, given the small vacuum mixing anglessuggested by our work. It is worth considering whetherβ-decay electron energy spectrum or pion decay experi-ments could be pushed in sensitivity to the point wheremassive sterile neutrinos in some of the allowed regionsof Figs. 7 and 8 could be constrained. This would re-quire an increase in sensitivity to ms sin2 2θ of at leastsome six orders of magnitude and this is clearly untenablewith current technology [33]. Finally, although a numberof extensions of the standard model motivate the exis-tence of multiple sterile neutrinos, it must be pointedout there is no independent physics suggestion for thesterile neutrino mass (∼ 1 keV to ∼ 10 MeV) and mix-ing (10−17 ≤ sin2 2θ ≤ 3× 10−10) parameters which giveviable dark matter candidates in our calculations.

As discussed above, sterile neutrino degrees of freedomwith ultra-large masses (e.g., of order the standard modelunification scale, or even the top quark mass) are in somesense “natural,” at least in the context of a see-saw ex-planation for the low masses of active neutrinos.

There is now a reasonable chance that new neutrinoexperiments scheduled to come to fruition in the nextfew years [e.g., the Sudbury Neutrino Observatory (SNO)[117], KamLAND [118], ORLaND [119], K2K [120], MI-NOS [121], ICARUS [122], and Mini-BooNE [123]] willallow us to deconvolve the neutrino mass and mixingspectrum. This would be an achievement heavy with im-plications for many aspects of physics and astrophysics.Will the requirement for sterile neutrinos remain?

It is now true that the interpretation of the current so-lar, atmospheric, and accelerator (LSND) data in termsof neutrino flavor mixing physics demands the introduc-tion of a sterile neutrino with a rest mass comparable tothat of some of the active neutrinos, i.e., light. Neces-sarily, then, this sterile neutrino is not the dark mattercandidate we speculate on in this work. However, theunambiguous establishment of the existence of a lightsterile neutrino would expose our ignorance of physics inthe neutrino sector in a stark and dramatic way. Onthis score, the Mini-BooNE experiment [123] and theSNO neutral-current experiment [117] are the most cru-cial ones for sterile neutrino dark matter. A confirmationof the LSND result invites speculation on the existenceof more massive sterile neutrino states.

ACKNOWLEDGMENTS

We thank A.B. Balantekin, J. Behr, S. Burles,D.O. Caldwell, J. Frieman, E. Gawiser, K. Griest,W. Haxton, C. Hogan, E.W. Kolb, R.N. Mohapatra,S. Reddy, R. Rothschild, B. Sadoulet, D. Tytler, andJ. Wadsley for useful discussions. M.P. and K.A. wouldlike to acknowledge partial support from NASA GSRP.This work was supported in part by NSF Grant PHY-9800980 at UCSD. G.M.F. would like to thank the In-

21

stitue for Nuclear Theory, the Physics Department, andthe Astronomy Department of the University of Wash-ington for hospitality.

APPENDIX: THE GENERAL

TIME-TEMPERATURE RELATION

In this appendix, we review the calculation of the tem-perature evolution in the early universe through periodsof varying statistical weight in relativistic particles, g,Eq. (7.6). The time derivative of the temperature can bewritten as [124]

dT

dt=

dr

dt

/

dr

dT(1)

where r ≡ ln(R3) and R is the scale factor. The expan-sion rate is determined by the Friedmann equation

H =dR

dt

1

R=

1

mpl

√

8π

3ρtot ≈ 0.207 s−1 g1/2 T 2, (2)

where ρtot = (π2/30)gT 4 is the total energy density, andT is in MeV in the last approximation of Eq. (2). There-fore, the first half of the time-temperature relation, Eq.(1), is straightforward: dr/dt = 3H . One can get the sec-ond half through the conservation of comoving energy,

d

dt

(

ρR3)

+ pd

dt

(

R3)

= 0 (3)

This can be rewritten into the desired form

dr

dT=

dρtot

dT(ρtot + ptot)

−1. (4)

With Eqs. (2) and (4) one has the general temperatureevolution.

In our case the sterile neutrinos can contribute sig-nificantly to the energy density and pressure. Approxi-mating all species other than the sterile neutrino to berelativistic, we have p∗ ≈ 1/3ρ∗, where ρ∗ and p∗ are theenergy and pressure in all particles other that the sterileneutrinos/antineutrinos. Therefore,

dT

dr=

(

4

3ρ∗ + ρs + ps

)(

dρ∗dT

+dρs

dT

)−1

. (5)

The rate of change of the standard energy density isstraightforward:

dρ∗dT

=4π2

30gT 3 +

π2

30T 4 dg

dT. (6)

The temperature derivative of the sterile neutrino energydensity is

dρs

dT=

2

π2

∫

(fνs(p) + fνs

(p))[

(p2 + m2s)/T 2

]1/2p2dp

− m2s

2π2

∫

[fνs(p) + fνs

(p)][

(p2 + m2s)/T 2

]−1/2p2dp.

This, together with the energy density and pressure inthe sterile neutrinos and antineutrinos calculated fromtheir time-dependent distribution functions, allows oneto readily arrive at a consistent time-temperature evolu-tion.

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