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Cosmic Neutrinos from Unstable Relic Particles

arXiv:hep-ph/9209236v1 15 Sep 1992UCLA/91/TEP/31 (revised)August 1992Cosmic Neutrinos from Unstable Relic ParticlesPaolo GondoloDepartment of Radiation Sciences, UppsalaUniversity, P.O. Box 535, 75121 Uppsala, SwedenGraciela GelminiDepartment of Physics, University of California,Los Angeles, CA 90024-1547, USASubir SarkarDepartment of Physics, University of Oxford,1 Keble Road, Oxford OX1 3NP, UKABSTRACTWe derive constraints on the relic abundance of a generic particle of mass∼1 −1014 TeV which decays into neutrinos at cosmological epochs, using datafrom the Fr´ejus and IMB nucleon decay detectors and the Fly’s Eye air showerarray.

The lifetime of such unstable particles which may constitute the dark mattertoday is bounded to be greater than ∼1014 −1018 yr, depending on the mass.For lifetimes shorter than the age of the universe, neutrino energy losses due toscattering and the expansion redshift become important and set limits to the abilityof neutrino observatories to probe the early universe.

1. IntroductionUpper limits on the flux of high energy cosmic neutrinos obtained from nucleondecay experiments and cosmic ray observatories constrain the relic cosmologicalabundance of heavy unstable particles which decay into neutrinos.Given theenergy spectrum of the decay neutrinos and the decay branching ratio, upperbounds can be obtained on the primordial abundance of the particle as a functionof its lifetime.

Earlier attempts to set such bounds [1,2] were made before anyexperimental data were available. We present here the bounds imposed by the non-observation of extraterrestrial high energy neutrinos in the Fr´ejus and IMB nucleondecay detectors and the Fly’s Eye air shower array.

We improve on previous workby taking into account the experimental energy thresholds, the neutrino opacity ofthe early universe, neutrino absorption in the Earth and the appropriate neutrinointeraction cross sections at high energies.In section 2, we discuss the cosmological absorption of high energy neutrinos,and in section 3 calculate the spectrum of neutrinos generated by heavy particledecay. A discussion of the expected signals is given in section 4 and the constraintsprovided by present observations are presented in section 5.

Our conclusions followin section 6.2. Cosmological neutrino absorptionHigh energy neutrinos can be absorbed in interactions with the relic ther-mal neutrino background and with nucleons in the early universe.

The dominantprocesses are the annihilation of a high energy neutrino (or antineutrino) with abackground antineutrino (or neutrino) and its inelastic scattering offa nucleon.We obtain below an analytical formula for the absorption redshift za(Ee) at whichthe neutrino opacity of the universe sν is unity for a neutrino emitted with energyEe; less than a fraction 1/e of the neutrinos emitted at redshifts larger than za(Ee)propagate to the present epoch.2

Consider a neutrino emitted with energy Ee at time te corresponding to redshiftze. The cosmological neutrino opacity sν(te, Ee) is the mean number of scatteringsundergone by the neutrino in which it could have been absorbed, given bysν(te, Ee) =t0Ztedtτν(t, Eν) ,(2.1)where t0 ∼0.65×1010 yr (Ω0h2)−1/2 is the present age of the universe and τν(t, Eν)is the mean free time between collisions at time t (and redshift z) for a neutrinoof energy Eν = Ee(1 + z)/(1 + ze).

Here Ω0 is the present mass density of theuniverse in units of the critical density ρc ≃1.9 × 10−29h2 g cm−3, where h is theHubble constant in units of 100 km s−1 Mpc−1 (0.4 <∼h <∼1). Taking into accountthe two absorption processes mentioned above:1τν(t, Eν) = 1τν¯ν+1τνN,(2.2)where the first term refers to neutrino-antineutrino annihilation and the second toneutrino-nucleon scattering.To obtain τν¯ν, we must average over the thermal energy distribution of therelic background (anti)neutrinos.Consider a decay neutrino and a backgroundantineutrino; the same formulae apply to a decay antineutrino and a backgroundneutrino.

Indicating by θν¯ν the angle between the two colliding particles in thecosmic frame, we have1τν¯ν=(1 −cos θν¯ν)σν¯νn¯ν ,(2.3)where (1 −cos θν¯ν) is the ν¯ν relative velocity (in units of c), σν¯ν is the total ν¯νannihilation cross section and n¯ν is the background antineutrino number density3

at temperature T¯ν, given byn¯ν = 34ζ(3)π2 T 3¯ν . (2.4)The angular brackets indicate an average over the antineutrino energy distribution,f¯ν(E¯ν) =12π2E2¯νeE¯ν/T¯ν + 1 .

(2.5)We consider only annihilations into charged fermion pairs, ν¯ν →f ¯f. In thiscase, σν¯ν = Pf σν¯ν→f ¯f, with the sum running over quarks and charged leptons.

Itis a good approximation for our purposes to consider massless fermions to estimatethe annihilation cross section and simply add a new fermion channel wheneverEνT¯ν > m2f. In this caseσν¯ν = G2Fs4πhNNCeffPZ(s) + NCCeffAW(s)i,(2.6)with s = 2EνE¯ν(1−cos θν¯ν).

The first term in the square brackets is due to neutralcurrents. The Z boson pole factor is defined asPZ(s) =M4Z(s −M2Z)2 + M2ZΓ 2Z.

(2.7)The second term takes into account the charge current contribution to the processesνe¯νe →e+e−, νµ¯νµ →µ+µ−, ντ ¯ντ →τ+τ−:AW(s) =M6W2s3 sin2 θW1 −M2Ws + M2W+ (1 −2a) sM2W+ as2M4W+ 2(a −1) ln1 +sM2W,(2.8)witha =12 −sin2 θWM2Z(s −M2Z)(s −M2Z)2 + Γ 2ZM2Z. (2.9)The coefficients Neffare the effective numbers of annihilation channels.

For neutral4

currents, this is calculated asNNCeff=XfθEνT¯ν −m2f 23nf1 −8t3fqf sin2 θW + 8q2f sin4 θW,(2.10)while for charged currents, the coefficientNCCeff= θEνT¯ν −m2l 163 sin2 θW(2.11)is non-zero only if the charged lepton l is in the same family as the annihilatingneutrino. Above, nf is the number of colours (1 for leptons, 3 for quarks), andt3f and qf are the third component of the weak isospin and the electric charge ofthe fermion in units of the positron charge respectively.

We take the electroweakmixing angle to be given by sin2 θW = 0.23. Inserting eq.

(2.6) into eq. (2.3), weobtain1τν¯ν= G2F4π(1 −cos θν¯ν)hNNCeffsPZ(s) + NCCeffsAW(s)i n¯ν ,(2.12)where s = 2EνE¯ν(1 −cos θν¯ν) is understood.The thermal average has a peak at EνT¯ν ≃M2Z/4, corresponding to the Z pole.All neutrinos emitted with energy Ee >∼M2Z/4T¯ν = 1.26 × 1013 TeV/(1 + ze) areabsorbed.

For Ee <∼M2W/4T¯ν, the factors PZ(s) and AW(s) in eq. (2.6) can be setequal to unity and τν¯ν is easily evaluated as1τν¯ν= σ0NeffEνρ¯ν = ρ¯ν0σ0Neff(1 + z)51 + zeEe .

(2.13)Here, Neff= NNCeff+ NCCeffis the effective total number of annihilation channels(which varies between 0.33 and 10.1), σ0 is defined asσ0 ≡23πG2F = 1.12 × 10−32 cm2 TeV−2,(2.14)andρ¯ν0 = 5.96 × 10−14 TeV cm−3(2.15)is the present antineutrino energy density (per species), taking the present photon5

temperature to be 2.74 K [3]. Thus, the annihilation mean free time isτν¯ν = 1.58 × 1027 yr N−1eff(1 + z)−5(1 + ze) EeTeV−1.

(2.16)Next we consider neutrino-nucleon scattering. The thermal motion of the non-relativistic nucleons can be neglected, and we have1τνN= nNσνN = nN0σνNEν(1 + z)41 + zeEe ,(2.17)where nN is the nucleon number density at redshift z and σνN is the neutrinonucleon scattering cross section at neutrino energy Eν.

The present nucleon meandensity is in the range nN0 ∼(0.25 −1.5) × 10−7 cm−3 according to Big Bangnucleosynthesis calculations; this reflects the observational uncertainty in primodial4He mass fraction, which is taken to be ∼0.21 −0.24 [4].For Eν <∼1 TeV, the ratio σνN/Eν is constant and equal to 0.67 × 10−35 cm2TeV−1 for neutrinos and to 0.34 × 10−35 cm2 TeV−1 for antineutrinos [5-6]. Theneutrino scattering mean free time at these energies isτνN ≃1024 yr (1 + z)−4(1 + ze) EeTeV−1.

(2.18)Comparing with the annihilation mean free time (2.16), we see that inelastic scat-tering upon nucleons dominates only at redshifts 1 + z <∼103N−1eff. However, nowτν ≃τνN >∼1015 yr ≫t0, i.e.

the universe has already become transparent to neu-trinos. At higher neutrino energies, σνN/Eν decreases and neutrino-nucleon scat-tering is even less important, becoming negligible at all redshifts for Eν >∼106 TeV.Thus inclusion of νN scattering affects za(Ee) only marginally.

For simplicity ofpresentation, we do not therefore write it explicitly in the formulae below, althoughwe have included it in the numerical calculations.6

The last ingredient necessary to compute the neutrino opacity is the relation-ship between the age of the universe and the redshift:t =(t0(1 + z)−3/2,for z < zeq,t0(1 + zeq)1/2(1 + z)−2,for z > zeq,(2.19)where 1+zeq = 2.25×104Ω0h2 is the redshift at which the energy density of matter(with present density parameter Ω0) begins to dominate over that of radiation.The absorption redshift obtained from integration of eq. (2.1), using eqs.

(2.13),(2.17) and (2.19), is shown as the diagonal full line in the Ee −te (or Ee −ze) planein figure 1 taking Ω0h2 = 1. The dot-dashed line separates the two regions whereannihilation and scattering absorption dominate.

The region of interest extendsfrom the present epoch (z = 0), through the epoch of matter-radiation equality(z ≃2 × 104), up to the epoch of light neutrino decoupling (z ≃1010), which areall indicated. The location of the Z boson pole is also shown as a diagonal dashedline.Approximate expressions for the absorption redshift za(Ee) can be obtained for1 ≪ze < zeq and ze ≫zeq.

In these cases, the result of the integration simplifiestosν =(3.5 × 10−17(Ω0h2)−1/2(1 + ze)5/2 (Ee/TeV) ,for 1 ≪ze < zeq ,0.81 × 10−14(1 + ze)2 (Ee/TeV) ,for ze ≫zeq . (2.20)The absorption redshift za(Ee) is then obtained by setting sν = 1:1 + za(Ee) =(3.8 × 106(Ω0h2)1/5 (Ee/TeV)−2/5,Ee >∼5.2 × 105 TeV(Ω0h2)−2,1.1 × 107 (Ee/TeV)−1/2,Ee <∼5.2 × 105 TeV(Ω0h2)−2.

(2.21)7

3. Neutrino spectrumWe now determine the present energy spectrum of neutrinos originating fromthe decay of an unstable heavy particle x with decay lifetime τx.

The number ofneutrinos of type νi (νi = νe, ¯νe, νµ, ¯νµ, . .

.) produced at time t, per unit comovingvolume and unit time, isγe(t) = BνiYx(t)τx= BνiYxpτxexp−tτx= BνiYx0τxexp−t −t0τx,(3.1)where Bνi is the number of neutrinos of type νi produced per decaying x particle,Yx(t) ≡nx(t)/nγ(t) is the x particle number density in ratio to the thermal photondensity nγ(t) (= 412.7(1 + z)3 cm−3), Yxp is its primordial value⋆and Yx0 is itsvalue today.The number of neutrinos absorbed in the same volume, γa(t), is proportionalto the comoving density of decay-generated neutrinos Yνi(t) = nνi(t)/nγ(t) and isgiven byγa(t) =Yνi(t)τνi(t, Ee) ,(3.2)where τνi(t, Ee) is the neutrino absorption mean free time (see section 2).The evolution of the comoving neutrino density Yνi(t) is governed bydYνi(t)dt= γe(t) −γa(t) == BνiYxpτxexp−tτx−Yνi(t)τνi(t, Ee) .

(3.3)with the following solution at the present epoch t0:Yνi0 = BνiYxpt0Z0exp−teτx−sνi(te, Ee) dteτx. (3.4)Now differentiating with respect to the present neutrino energy Eνi0 = Ee(1 +⋆This is conveniently measured at the earliest epoch following which the comoving photonnumber is conserved, say at T ∼0.01me, corresponding to t ∼10−3 yr; this is negligiblecompared to all other time-scales relevant here.8

ze)−1, one obtains the present neutrino fluxEνi0dφνi0dEνi0= φγ0BνiYxp κ teτxexp−teτx−sνi(te, Ee)θ(Ee −Eνi0),(3.5)wherete = t0Eνi0Eeκ,κ =(2,for Eνi0 < Ee(1 + zeq)−1,32,for Eνi0 > Ee(1 + zeq)−1,(3.6)and φγ0 = nγ0/4π = 0.98×1012 cm−2 s−1 sr−1 is the present background photon fluxper unit solid angle. The decay neutrino flux is shown in figure 2 for τx = 10−5t0and 3Ee = 105 TeV, 107 TeV, 109 TeV.

Notice that the present neutrino energyEν is redshifted from Ee.The dotted lines indicate what the flux would havebeen without cosmological neutrino absorption.These three curves are simpletranslations of each other, since the differential flux (3.5) with sνi = 0 dependsonly on the ratio Eνi0/Ee.Approximating the effect of the cosmological neutrino absorption with e−sνi ≃θ(te −ta), where ta < t0 corresponds to the absorption redshift za(Ee) at whichthe neutrino opacity is unity, eq. (3.5) can be easily integrated to obtain the totalneutrino flux today,φνi0 ≃φγ0BνiYxp(e−ta/τx −e−t0/τx).

(3.7)For τx ≪t0 −ta, this reduces toφνi0 ≃φγ0BνiYxpe−ta/τx,(3.8)while for τx ≫t0 −ta, it becomesφνi0 ≃φγ0BνiYx0t0 −taτx. (3.9)The neutrino flux is exponentially suppressed for τx ≪ta and reaches a maximumof φνi0 ≃BνiYxp × 1012 cm−2 s−1 sr−1 for τx ≃t0 −ta.

This flux is potentially9

enormous compared with present bounds on the diffuse extragalactic high energyneutrino flux (∼10−6 cm−2 s−1 sr−1 for Eν >∼1 TeV and ∼10−16 cm−2 s−1 sr−1for Eν >∼107 TeV) which may be inferred from data obtained with underground de-tectors and cosmic ray observatories (cf. section 4).

Hence very restrictive boundsmay be obtained on the abundance of the hypothetical decaying particle as demon-strated below.4. Expected signalsAt present, the best means to detect a diffuse background of high energy neutri-nos is through the production of an energetic charged lepton in the collision of sucha neutrino with a nucleon.

We consider three possible types of signal according towhere the interaction occurs. An event is called contained when the interactionoccurs inside an underground detector, such as Fr´ejus, IMB and Kamiokande.

Aflux of through-going muons is registered when interactions occur in the materialsurrounding the detector (rock in the underground experiments and water in theforthcoming DUMAND, GRANDE and Lake Baikal experiments). Finally, if theinteraction occurs in the atmosphere an extensive air shower (EAS) is generated,which can be detected by cosmic ray observatories such as Fly’s Eye and CASA.We consider the following experimental constraints on the total neutrino flux:(1) the rate of contained events in the Fr´ejus detector, with electron and/ormuon energies greater than 3 GeV, does not exceed 17.7 kton−1 yr−1 [12];⋆(2) the rate of contained events with energies between 100 MeV and 2.5 GeVin the IMB-3 detector is limited by 111.5 kton−1 yr−1 [13].†⋆We consider the 11 electron and 14 muon charged current events over 3 GeV observed in1.56 kton yr (see fig.

3 of ref. [12]), and apply the quoted identification efficiencies of 85%and 95% respectively.† From the total number of 422 contained events in 3.4 kton yr, we exclude the 43 eventsbelow 100 MeV (see fig.

2 of ref. [13]) where the track reconstruction and identificationefficiencies are low.

For comparison the IMB-1 detector recorded 401 contained events in3.77 kton yr [14].10

In fact, the observed contained events are well accounted for by the expectedneutrino flux from cosmic ray interactions in the atmosphere [15], within the un-certainty of ∼25% in these computations. Hence the bound on contained eventsof non-atmospheric origin can, in principle, be improved by up to a factor of ∼10and the limits to be derived strengthened proportionally.We also consider the following constraints on any extraterrestrial neutrino flux:(3) the flux of upward-going muons (from directions with zenith angle largerthan 98◦) with energy greater than 2 GeV registered by the IMB-1 detector is lessthan 2.65 × 10−13 cm−2 s−1 at the 90% confidence level, after subtraction of theexpected atmospheric component [16];‡(4) the Fly’s Eye array has set upper limits on the rate of neutrino-inducedEAS’s of 10−45 s−1 sr−1, 3.8×10−46 s−1 sr−1, 10−46 s−1 sr−1 and 3.8×10−47 s−1 sr−1,all at the 90% confidence level, for neutrino energies higher than 105 TeV, 106 TeV,107 TeV and 108 TeV respectively [18].For the isotropic neutrino flux (3.5), the rate of contained events per unitdetector mass, Rc, the upward-going muon flux, φµ, and the rate of EAS’s per unitsolid angle, J, can all be written in the formS =XiZdEνidφνidEνiPi(Eνi) Ωi(Eνi) ,(4.1)where the signal S is Rc, φµ or J, and the sum is over neutrino types (νi = νe,¯νe, νµ, ¯νµ, .

. .

). The effective aperture Ωi(Eνi), which takes account of neutrinoabsorption by the Earth (if any), and the transfer functions Pi(Eνi) depend on theexperimental data set considered.

(We have dropped the subscript 0 referring tothe present neutrino energy. )‡ The recent Kamiokande upper limit of 4×10−14 muons cm−2 s−1 sr−1 for zenith angles largerthan 150◦[17] is slightly less stringent than the IMB limit we consider, which correspondsto 3.7 × 10−14 muons cm−2 s−1 sr−1.11

For a simplified model of the Earth with uniform density ρ⊕= 5.5 g cm−3 andradius R⊕= 6.37 × 108 cm, the effective aperture is (neglecting the depth of theunderground detector relative to R⊕),Ωi(Eνi) =ZdΩexp−2R⊕ki(Eνi)| cos ϑ|θ(−cos ϑ),(4.2)where the integral extends over the geometrical aperture of the detector, ϑ is thezenith angle and ki(Eνi) is the neutrino absorption coefficient in the Earth, givenbyki(Eνi) = ρ⊕mNσνiN(Eνi),(4.3)with mN the nucleon mass and σνiN(Eνi) the total neutrino-nucleon cross section.For the neutrino energies under consideration, σνiN(Eνi) includes only the chargedcurrent cross section σCCνiN(Eνi), because the energy and momentum fractions trans-ferred to the nucleon in a neutral current process are negligible at these energies.The charged current cross section σCCνiN(Eνi) is well-known for Eνi <∼10 TeV:σCCνiN = 0.67 × 10−35 cm2 EνiTeV,(4.4)for a neutrino andσCC¯νiN = 0.34 × 10−35 cm2 EνiTeV,(4.5)for an antineutrino [5]. At higher energies the charged current cross section becomesmore and more uncertain — by as much as a factor of 10 at Eνi ≃109 TeV —because of the poor knowledge of nucleon structure functions at small arguments[11].

For this reason, we have not attempted a precise calculation of σCCνiN(Eνi)from a set of theoretical structure functions. We have used the differential chargedcurrent cross section up to Eνi = 107 TeV given in ref.

[8]. At still higher energies,we have matched the asymptotic form of the cross section in ref.

[7] to the resultsof ref. [8].12

In figure 3 we show the effect of absorption in the Earth by plotting the effectiveaperture (4.2) integrated below the horizon:Ωbelowi(Eνi) =2πσ⊕σνiN(Eνi)1 −exp−σνiN(Eνi)σ⊕,(4.6)with σ⊕= mN/2R⊕ρ⊕= 2.4×10−34 cm2. This effective aperture differs very littlefrom the one obtained in ref.

[19] using a more elaborate model of the Earth. Aswe see from the figure, the Earth severely attenuates the flux of neutrinos of energyexceeding ∼105 TeV, becoming nearly opaque at ∼1010 TeV.Note that for σ >∼10−33 cm2, i.e.

at Eνi >∼103 TeV, the effective aperture frombelow the horizon is inversely proportional to the neutrino-nucleon scattering crosssection,Ωbelowi(Eνi) ≃2πσ⊕σνiN(Eνi) . (4.7)The resonant reaction ¯νee−→W−→“anything” severely depletes the ¯νeflux from below the horizon at energies around E¯νe ≃7 × 103 TeV [20].

Howeversince we do not assume any predominant neutrino type in the decay neutrino flux,the ¯νe flux from below accounts for only one eighth of the total rate of containedevents. It is therefore a reasonable approximation, for our purposes, to neglect thisresonance.We present now the transfer functions Pi(Eνi) for the experimental data setsunder consideration.

For contained events we haveP conti(Eνi) = θ(Eνi −Eth)NnuclMmin(Ecut,Eνi)ZEthdElidσCCνiNdEli,(4.8)where Eth is the experimental energy threshold for the lepton energy Eli, Ecutis an experimental cutoff(2.5 GeV for IMB and infinite for Fr´ejus), M is thedetector mass and Nnucl = 6.02 × 1032 (M/kton) is the number of nucleons in13

the detector. If the charged current cross section is written in units of 10−38 cm2,σCCνiN(Eνi) = σi,38 × 10−38 cm2, then the transfer function for contained events isP conti(Eνi) ≃6.0 × 10−6 cm2 kton−1 θ(Eνi −Eth)min(Ecut,Eνi)ZEthdElidσi,38dEli.

(4.9)Both Fr´ejus and IMB data sets include neutrinos coming from all solid angles andtheir effective aperture computed from eq. (4.2) varies from 4π to 2π as the energyis increased; the neutrino flux is reduced at most by a factor of 2 at the highestenergies.In IMB, and other water-ˇCerenkov detectors, there is also the possibility thatthe hadronic fragments produced in the neutrino-nucleus collision give a detectableamount of ˇCerenkov light.

Their contribution P conti,hadr(Eνi) should then be addedto eq. (4.9).

In the appendix, we present an estimate of the contribution fromsuch ‘hadronic blasts’ and show that this is important only for very energetic neu-trinos, Eνi>∼107 TeV, where, however, the signal from EAS’s gives more stringentconstraints.The product P conti(Eνi) Ωconti(Eνi) thus obtained for the Fr´ejus and IMB con-tained events is shown in figure 4(a) for neutrinos (solid lines) and antineutri-nos (dotted lines) as function of the neutrino (or antineutrino) energy Eνi. Theunits are chosen such that the vertical axis directly gives the number of eventsper kiloton-year corresponding to a unit neutrino flux of 1 cm−2 s−1 sr−1.ForEνi >∼106 TeV we calculate P conti(Eνi) Ωconti(Eνi) ≃3.8 × 10−5 cm2 sr kton−1 σi,38for the Fr´ejus detector.

The IMB curve (curve 1) above 107 TeV is due to ‘hadronicblasts’ as discussed in the appendix.The transfer function for the flux of up-going muons is (see ref. [21])P µi (Eνi) = θ(Eνi −Eth)∞ZEthdE′µEνiZE′µdEµ∞Z0dX g (X, E′µ, Eµ) dσνiNdEµ,(4.10)for νi = νµ, ¯νµ, and P µi (Eνi) = 0 for the other neutrino types.Here Eµ and14

E′µ are the muon energies at production and at the detector respectively, andX = l ρrock/mN is the column density of rock, i.e. the number of nucleons perunit area encountered by a muon travelling a length l in rock.

Notice that P µi isadimensional. The probability that a muon with initial energy Eµ has an energybetween E′µ and E′µ + dE′µ after traversing an amount x of rock is denoted byg (X, E′µ, Eµ) dE′µ.

We assume a uniform rock density of ρrock = 2.6 g cm−3 in theregion surrounding the detector. Following ref.

[6], we make the approximationthat the final muon energy E′µ coincides with its mean value (with no dispersion):E′µ = (Eµ + ǫ)e−γX −ǫ,(4.11)with ǫ ≃0.51 TeV and γ−1 = 1.54 × 1029 cm−2. The integral over E′µ in eq.

(4.10)can then be performed, and we obtainP µi (Eνi) = θ(Eνi −Eth)EνiZEthdEµXth(Eµ)dσνiNdEµ,(4.12)for νi = νµ, ¯νµ. HereXth(Eµ) = γ−1 ln 1 + Eµ/ǫ1 + Eth/ǫ(4.13)is the column density traversed by muons produced with energy Eµ which reachthe detector with threshold energy Eth.

A plot of P µi (Eνi) times Ωµi (Eνi), obtainedby integrating eq. (4.2) over zenith angles larger than 98◦, is shown in figure 4(b).The decrease of P µΩµ for Eνi >∼107 TeV is due to absorption by the Earth.The final signal we consider is the rate of EAS’s per unit solid angle.Itstransfer function isP EASi(Eνi) =1ΩEASi(Eνi)σνiN(Eνi)θ(Eνi −Eth),(4.14)where the index i stands for νe and ντ, which can generate electromagnetic orhadronic showers in the atmosphere.

Muons from charged current νµ-nucleon in-teractions do not trigger electromagnetic showers, since their radiation length for15

bremstrahlung in air (105 g/cm2) is much larger than the atmosphere thickness(1030 g/cm2). The product P EASi(Eνi) ΩEASi(Eνi) is shown in figure 4(c) for thefour experimental energy thresholds of the Fly’s Eye detector [18].We are now in a position to compare the expected signals to the experimentallimits.5.

Present constraintsWe assume that the same numbers of neutrinos and antineutrinos of each type,Bνe = B¯νe = Bνµ = B¯νµ = . .

. ≡Bν, are produced in x decays and that theirproduction energy is always Ee = 13mx.

Inserting the decay-generated neutrinoflux, eq. (3.5), into eq.

(4.1) and using the appropriate transfer functions andeffective apertures described in section 3, we obtain the expected signals in termsof the decay lifetime τx and the quantity BνmxYxp = BνmxYx0et0/τx which isproportional to the primordial energy density of the decaying particles. Noticethat for τx ≫t0, this quantity is ∼25.5 eV (Bν Ωx0h2), where Ωx0 is the present xmass density in units of the critical cosmological density.We present the results in figure 5.

The shaded regions are excluded by thepresent experimental data. The solid lines refer to the limit on upward-going muonsfrom IMB, the dotted and short-dashed lines to the Fr´ejus and IMB containedevents, respectively, and the long-dashed lines to the Fly’s Eye EAS’s.Theseare essentially bounds on the relic energy density of the decaying particle, takingBν = 1; for Bν < 1, these lines are to be proportionally shifted upward.

As notedearlier, the experimental limits on contained events can, in principle, be improvedby a factor of ∼10 if the signal due to atmospheric neutrinos is accounted for; thecorresponding bounds should then be scaled downwards by the same factor. Theshort-dashed–dotted line corresponds to a present mass density Ω0h2 = 1 either inx particles (for τx >∼t0) or in its decay products (for τx <∼t0); the region Ω0h2 > 1is excluded by the observational lower limits to the age and present expansion rateof the universe.

For comparison we also show as a long-dashed–dotted line (in the16

lower left quadrant) the upper bound on mxYxp for unstable particles decayinginto electromagnetically interacting particles. This is obtained by requiring thatthe abundance of the primordially synthesised light elements D, 3He, 4He and7Li not be excessively altered from their observationally inferred values by theelectromagnetic cascades initiated by the decay products [22].

In fact this boundalso applies to unstable particles decaying into neutrinos, since the decay neutrinoscan initiate similar electromagnetic cascades through the process ν¯ν →e+e−,where the target (anti)neutrinos belong to the thermal background. (This hasalso been considered in ref.

[23]; however these authors do not calculate cascadegeneration correctly and obtain an overly restrictive bound. )Figures 5(a-d) correspond to mx = 1, 105, 106 and 1010 TeV respectively.As the x mass increases, the bounds at τx <∼t0 first shift to the left as the decayneutrinos become more energetic and the signals go further above the experimentalthresholds.

Then they proceed to move to the right because the neutrino absorptionredshift decreases with increasing neutrino energy, hence the neutrinos producedby relatively short-lived particles do not survive until the present. The mass at theturning point is given approximately by solving mx ≃3Eth[1 + za(13mx)] with thehelp of eqs.

(2.21); its value is 1 × 103 TeV and 1 × 102 TeV for contained events inFr´ejus and IMB respectively, 8 × 102 TeV for the IMB upward-going muons, and1 × 108 TeV, 1 × 109 TeV, 5 × 109 TeV and 3 × 1010 TeV for the four Fly’s Eyethresholds.When τx >∼t0, the best bounds on the relic abundance of the decaying particlescome from the IMB limit on upward-going muons at mx <∼5 × 105 TeV and theFly’s Eye limits on EASs at mx >∼5 × 105 TeV. We can invert the argument andconsider the interesting case Ωx0h2 ≃1, i.e.

when the x particles are assumed toconstitute the dark matter today.⋆A corresponding lower bound on its lifetime⋆In this case the actual spatial distribution of the relic particles should be taken into account,e.g. their likely concentration in the halo of our Galaxy.

This would yield even stricterconstraints. Preliminary work has been reported in ref.

[24] and a more detailed study is inprogress.17

versus its mass can then be inferred and is plotted in figure 6. For mx <∼30 TeV,this bound gets stronger with increasing mx as the mean energy of the decaygenerated neutrinos rises over the IMB detection threshold.

For mx >∼30 TeV, thelower bound on τx is inversely proportional to mx (cf. eq.

(3.9)), apart from thejumps at ≃105 TeV, ≃106 TeV, ≃107 TeV and ≃108 TeV corresponding to theFly’s Eye energy thresholds. No bound exists for mx >∼5 × 1014 TeV, since theuniverse is opaque to such high energy neutrinos at the present epoch.6.

ConclusionsWe have considered constraints on the lifetime, the abundance and the mass ofunstable relic particles decaying into neutrinos at cosmological epochs, taking intoaccount that both the early universe and the Earth are opaque to very high energyneutrinos.We have evaluated the signals expected from a diffuse backgroundof decay-generated neutrinos in underground nucleon decay experiments and atcosmic ray observatories. Comparing these to the present limits on the flux ofnon-atmospheric neutrinos, we find severe bounds on the relic abundance of suchheavy particles; in particular, such particles must be very long-lived indeed in orderto constitute the dark matter today.

These bounds are of relevance to massivemetastable particles such as technicolour baryons and ‘cryptons’ (bound states inthe hidden sector of superstring-inspired models) as discussed elsewhere [22].7. AcknowledgementsWe are grateful to the referee for suggesting that we consider the effects ofhadronic ‘blasts’ in water-ˇCerenkov detectors.This work was supported in part by the Department of Energy under thecontract DE-AT03-88ER-40384 Mod A006-Task C.Note Added18

Recently, we became aware of ref. [25], where the detection of neutrinos fromcosmic relic particles is also studied, in particular the effects due to absorption inthe early Universe.

However this work assumes that the decaying particles werethermally produced (with a calculable abundance) in the early Universe, whereaswe have presented results in a general form, applicable to any relic particle. This,in fact, is essential in order to consider particles with masses over a few hundredTeV.

We also believe that we have addressed experimental issues in more detail.APPENDIXHere we present an estimate of the contribution of ‘hadronic blasts’ to the IMBtransfer function for contained events. We calculate the visible energy equivalentto the ˇCerenkov light output from such a ‘blast’ and then compare it to the visibleenergy observable in the IMB detector.Let W be the energy transferred to the nucleus in the neutrino interaction.

Thisis also presumably the energy available to the hadronic shower generated by thenuclear fragments. The visible energy Evis is defined to be the energy of a fictitiousinitial electron generating an electromagnetic shower with the same ˇCerenkov lightoutput [13].

The physics of electromagnetic showers [26] then relates Evis to the‘detectable’ track length Xd,Evis = EcXd(W)X0F(z),(A.1)where Ec is the critical energy separating the domains where ionization and radi-ation energy losses dominate and is approximately given byEc ≃800 MeVZ + 1.2 = 71 MeV. (A.2)Here X0(= 36.08 g/cm2 [27]) is the electron radiation length in water and F(z) is19

approximatelyF(z) ≃ez 1 + z lnz1.526,(A.3)withz ≃4.58ZAEdEc≃Ed28 MeV(A.4)(the numerical values refer to water).In the IMB detector, Ed = 1.52 me +30 MeV + 140 MeV = 170 MeV for an electron [14], hence eq. (A.1) readsEvis = 4.8 × 10−4 MeV Xd(W)(A.5)with Xd(W) in g/cm2.It remains now to estimate Xd(W), the path length of charged particles in thehadronic shower with energy above the detection threshold.

The mean number ofcharged particles nch(W) as a function of the available energy W has been studiedin deep inelastic scattering and incorporated into the Lund Monte Carlo program[28]. By fitting fig.

8 of this reference we obtain:nch(W) = 1.67 + 0.211 exp3.06 ln1/2 WGeV. (A.6)We assume now that all charged particles in the first generation of the shower areabove the ˇCerenkov threshold.

This is a good approximation at the high energiesthat will turn out to be relevant for hadronic blasts. With this assumption, Xd(W)is the product of nch(W) and the mean path length of a charged particle.

A lowerbound to the latter is one nuclear interaction length, Xnucl = 84.9 g/cm2 in water.This leads to a lower bound for Xd(W):Xd(W) > Xmin(W) ≃nch(W)Xnucl. (A.7)An upper bound is obtained by multiplying Xmin(W) by the typical length of a20

hadronic shower in units of interaction lengths [29]:Xd(W) < Xmax(W) ≃nch(W)Xnucl5.45 + 0.89 ln WGeV. (A.8)A lower and an upper bound to the visible energy Evis(W) can then be obtainedfrom eq.

(A.5).This range then has to be compared with the IMB energy threshold for π±detection, Eπ = 1.52 mπ + 30 MeV + 140 MeV = 382 MeV, and with the highestenergy analyzed, Evis = 2500 MeV [13]. The result of such a comparison is that forW<∼102 TeV there is probably not enough ˇCerenkov light for the ‘blast’ to be seen,while for W>∼106 TeV the detector is probably overloaded (more than 900 PMTsfired [13]) or otherwise not sufficiently efficient to detect the blast.

However for102 TeV<∼W<∼106 TeV, such blasts, if they do occur, should already be present inthe sample of ref. [13], probably as multiple-ring events.However, the neutrino energy Eν required to have W>∼102 TeV is Eν>∼107 TeV.This comes from the kinematic relation W 2 ≃2mpEν(1−x)y, with mp the nucleonmass and x and y the usual deep inelastic scattering variables, together with theconsideration that at high energies the W or Z propagator restricts the importantvalues of x and y to x ≃0 and y ≃1.

So ‘hadronic blasts’ in ˇCerenkov detectorsturn out to be important only for very high energy neutrinos, where (at least for thepurposes of this paper) there already are much better bounds on cosmic neutrinofluxes from EAS arrays.For the sake of completeness, we show in fig. 4a (curve 1 at high energies) thecontribution of hadronic blasts in IMB, obtained by numerical integration of:P conthadr(Eν) = NnuclM2mpEνZ0d Q2W 2maxZW 2mind W 21(2mpEν)2y d σCCd x d y + d σNCd x d y.

(A.9)Here Q2 ≃2mpEνxy, Wmin = 102TeV, Wmax = 106TeV and the contributions fromcharged and neutral currents have been summed. The EHLQ structure functions21

have been used together with a McKay-Ralston asymptotic form at low x (as inref. [10]).22

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FIGURE CAPTIONSFig. 1.

The absorption redshift za (line 1) for cosmic neutrinos as a function of theneutrino energy at emission Ee taking Ω0h2 = 1. The other lines indicate:(2) the boundary between the regions where absorption due to annihilationand scattering dominate; (3) the present epoch; (4) the Z boson pole; (5) theepoch of matter-radiation equality; (6) the epoch of light neutrino decoupling.Fig.

2. The present energy spectrum of decay generated neutrinos for τx = 10−5t0and 3Ee = 105 TeV (line 1), 107 TeV (line 2) and 109 TeV (line 3).

The fulllines show the effects of cosmological neutrino absorption.Fig. 3.

The effective detector aperture, integrated below the horizon, for neutrinos(solid line) and antineutrinos (dotted line), demonstrating the opacity of theEarth at high energies.Fig. 4.

The product of the transfer function Pi(Eνi) and of the effective apertureΩi(Eνi) for the three experimental data sets we consider: (a) contained eventsin IMB (curve 1) and Fr´ejus (curve 2) (b) IMB upward-going muons and (c)Fly’s Eye EAS’s (four thresholds). The dotted lines corresponds to antineu-trinos.Fig.

5. The bound on the primordial energy density of the decaying particle mul-tiplied by the branching ratio into neutrinos, BνmxYxp, as a function of itslifetime τx, for various choices of its mass mx.

The shaded regions are ex-cluded by the present experimental data. The various lines refer to: IMBupward-going muons (solid lines), Fr´ejus and IMB contained events (dottedand short-dashed lines respectively), Fly’s Eye EAS’s (long-dashed lines).Also indicated are the upper bound on the total energy density Ω0h2 = 1(short-dashed–dotted line) and the upper bound inferred from considerationsof primordial light element abundances (long-dashed–dotted line).Fig.

6. The lower bound on the x particle lifetime versus its mass for a presentdensity Ωx0h2 = 1 in the relic particles, assuming unit branching ratio into25

neutrinos26

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