neutrino signatures from the first stars

PHYSICAL REVIEW D 72, 103007 (2005)

Neutrino signatures from the first stars

Frederic Daigne,1 Keith A. Olive,2 Pearl Sandick,3 and Elisabeth Vangioni11Institut d’Astrophysique de Paris, UMR 7095, CNRS, Universite Pierre et Marie Curie-Paris VI,

98 bis bd Arago, F-75014, Paris, France2William I. Fine Theoretical Physics Institute, School of Physics and Astronomy, University of Minnesota,

Minneapolis, Minnesota 55455 USA3Department of Physics, School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455 USA

(Received 20 September 2005; published 28 November 2005)

1550-7998=20

Evidence from the Wilkinson Microwave Anisotropy Probe (WMAP) polarization data indicates thatthe Universe may have been reionized at very high redshift. It is often suggested that the ionizing UV fluxoriginates from an early population of massive or very massive stars. Depending on their mass, such starscan explode either as type II supernovae or pair-instability supernovae, or may entirely collapse into ablack hole. The resulting neutrino emission can be quite different in each case. We consider here the relicneutrino background produced by an early burst of Population III stars coupled with a normal mode of starformation at lower redshift. The computation is performed in the framework of hierarchical structureformation and is based on cosmic star formation histories constrained to reproduce the observed starformation rate at redshift z & 6, the observed chemical abundances in damped Lyman alpha absorbers andin the intergalactic medium, and to allow for an early reionization of the Universe at z� 10� 20. We findthat although the high redshift burst of Population III stars does lead to an appreciable flux of neutrinos atrelatively low energy (E� � 1 MeV), the observable neutrino flux is dominated by the normal mode ofstar formation. We also find that predicted fluxes are at the present level of the SuperK limit. As aconsequence, the supernova relic neutrino background has a direct impact on models of chemicalevolution and/or supernova dynamics.

DOI: 10.1103/PhysRevD.72.103007 PACS numbers: 98.62.Sb, 97.60.Bw

I. INTRODUCTION

One of the most surprising results contained in the firstyear data obtained by the Wilkinson MicrowaveAnisotropy Probe (WMAP) [1] was large optical depthimplying that the Universe became reionized at high red-shift in the range, 11< z< 30 at 95% CL. This is muchhigher than previously believed, and it has been proposedthat a generation of very massive stars is necessarily re-sponsible [2]. Even a brief period of massive star formationat high redshift would have notable consequences forchemical evolution, in particular, the metal enrichment ofthe interstellar medium (ISM) and intergalactic medium(IGM) [3–5]. When combined with the observed cosmicstar formation rate (SFR) at z & 6 [6], a coherent picture ofthe star formation history of the Universe unfolds: the firststars (Population III) described by a top-heavy initial massfunction (IMF) were formed in primordial, metal-freestructures with masses of order 107M�. Once the metal-licity achieved a certain critical level (of order 10�4 timesthe solar metallicity [7]), the massive mode of star forma-tion yielded to a more normal distribution of stellar massesat a rate over an order of magnitude larger than the currentstar formation rate. The cosmic SFR is observed to havepeaked at redshift z � 3.

Among the consequences of this newly emerging viewof star formation is the predicted enhancement in the rateof core collapse supernovae. In addition to the sharp spikeof supernovae at very high redshift due to the explosions ofstars responsible for the early epoch of reionization, the

05=72(10)=103007(11)$23.00 103007

enhanced SFR of the normal mode of star formation atredshifts z & 6 leads to a supernova rate which is approxi-mately a factor of 30 times the current rate, and a factor of5 times the observed rate at z� 0:7 [8].

Another consequence of an enhanced SFR and SN rate isthe resultant neutrino background spectrum produced bythe accumulated core collapse supernovae. Early estimatesof the (anti-e)-neutrino flux based on simple models ofgalactic chemical evolution were in the range of1–10 cm�2s�1 MeV�1 [9], as were later estimates [10–12] based on peak SFRs of order 0:1M�yr�1Mpc�3 corre-sponding to supernova rates of order 10�3 yr�1Mpc�3. In[13], the first attempt at obtaining the neutrino flux fromvery massive Population III stars found fluxes at levelsexceeding 10 cm�2s�1 MeV�1, though the peak occursat lower energy due to the redshifted spectrum. Currentestimates of the SFR peak at values 3 times higher at aredshift z� 3 [6] with correspondingly higher SN rates.More startling is the possibility that SN rates at highredshift (z� 15) may be as large as 6�10�2 yr�1Mpc�3.Here, we incorporate fully developed chemical evolutionmodels which trace the history of pregalactic structures aswell as the IGM and are based on a �CDM cosmology andinclude a Press-Schechter model [14] of hierarchical struc-ture formation. We adopt the approach and chemical evo-lution models of Daigne et al. [4,5] and consider severalbimodal IMFs, each with a normal component of starformation as well as a massive component describingPopulation III stars. Given an IMF and a respective SFR,

-1 © 2005 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.72.103007

DAIGNE, OLIVE, SANDICK, AND VANGIONI PHYSICAL REVIEW D 72, 103007 (2005)

one can calculate the chemical history of cosmic structuresand the IGM, the reionization efficiency, and as we do here,the expected supernova relic neutrino (SRN) flux.

In Sec. II we describe our method for calculating theSRN flux, and in section III we discuss the star formationmodels considered. In Sec. IV we present our results for theSRN flux in a variety of models describing Population IIIstars. Opportunities for SRN detection are examined inSec. V, and the effects of neutrino oscillations are discussedin Sec. VI. Our conclusions are given in Sec. VII.

II. SUPERNOVA RELIC NEUTRINOBACKGROUND

In all of the models we will consider, star formationbegins at high redshift, dominated initially by massive starswhich may explode as core collapse or pair-instabilitysupernovae and provide for the reionization of the IGM.Each explosion, regardless of type, releases most of thestar’s gravitational energy in the form of neutrinos with aspecified energy spectrum and flux. Given a chemicalevolutionary model, or more specifically, given a rate ofstar formation along with the IMF, the integrated contribu-tion of SN to the neutrino background can be computed.

The expected differential flux of neutrinos at Earth withenergy E can be expressed as

dF�dE

�Z zi

0dz�1� z

��dtdz��Z Mmax

Mmin

dm��m �t

� ��mN��mdP�dE0

; (1)

where N��m is the total number of neutrinos of a givenspecies, �, emitted in the core collapse of a star of massm,� is the normalized IMF, is the SFR per unit comovingvolume, ��m is the lifetime of a star of mass m, anddP�=dE

0 represents the neutrino spectra in the comovingvolume at energy E0 � E�1� z. The integration limitsMmin and Mmax are the minimum and maximum massesin each model for which supernovae occur, and zi is theinitial redshift over which star formation occurs. It isassumed that star formation continues to the present (z �0).

When a star undergoes core collapse, the mass of theremnant is determined by the mass of the progenitor. Weassume that all stars of mass m * 8 M� will die as super-novae. For stars of mass 8 M� <m< 30 M�, the remnantafter core collapse will be a neutron star of m � 1:5 M�.More massive stars fall into two categories; black holes andpair-instability supernovae. Pair-instability supernovae arethought to occur for stars with 140 M� & m & 260 M�, inwhich case the explosion leaves no remnant. All other starscollapse to form black holes. There is of course consider-able uncertainty in the precise mass ranges correspondingto the final fate of a given star. The ranges we employshould only be viewed as being schematic [15]. Stars with

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30 M� <m< 100 M� become black holes with mass ap-proximately that of the star’s helium core before collapse[16]. We take the mass of the helium core to be

MHe �13

24 �m� 20 M� (2)

for a star with main sequence massm [15]. We assume thatstars with m> 260 M� collapse entirely to black holes.

The energy emitted in each core collapse, Ecc corre-sponds to the change in gravitational energy, 99% of whichis emitted as neutrinos [17]. In the cases where collapseresults in a neutron star, Ecc � 5� 1053ergs. For stars thatcollapse to black holes, Ecc is proportional to the mass ofthe black hole. Pair-instability supernovae experience amuch more powerful explosion than core collapse super-novae, however few neutrinos are emitted and with verylow energies such that they would not be observed [18]. Weaddress this issue further in Sec. III. It has been noted that ifthe collapse to a black hole proceeds without rotation, theneutrino luminosity will be diminished by �2 orders ofmagnitude [19], so this is really an upper limit to the fluxassuming rotation. Although several studies find a distincthierarchy in the partitioning of neutrino luminosity amongthe species during the different luminosity phases of corecollapse, equipartition of the total energy emitted by thestar is generally accepted [9,11,12]. For a comparison ofluminosity hierarchies found in recent simulations, seeKeil et al. [20].

In black hole formation, the neutrino luminosity isnearly constant for the first few seconds until the eventhorizon overtakes the neutrinospheres. Once the neutrino-spheres are inside the event horizon, the luminosity con-sists of neutrinos with lower average energies escapingfrom the outer layers of the star [19,20]. We assume thateach electron neutrino carries an average energy hE�ei �13:3 MeV, which is a reasonable approximation for theaverage neutrino energy over the two luminosity phases ofcore collapse to a black hole. For supernovae which do notcollapse to black holes, this energy is consistent withrecent simulation data [20]. The charged current reactionsthat prevent neutrinos from emerging from the star are�en! pe� and ��ep! ne�. The different trapping reac-tions result in different neutrinosphere radii, and thereforedifferent average energies for �e and ��e. We assumehE ��ei � 15:3 MeV which is the average energy over thetwo luminosity phases as above, following [13]. The otherspecies, denoted �x, undergo only neutral current interac-tions. The mechanism that governs their average tempera-ture at emission is more complicated, but the generallyaccepted hierarchy is hE�ei< hE ��ei< hE�xi. We havetaken hE�xi � 20 MeV. The total number of �� emittedby a star during core collapse is given by

N�� EcchE��i

: (3)

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NEUTRINO SIGNATURES FROM THE FIRST STARS PHYSICAL REVIEW D 72, 103007 (2005)

We discuss the sensitivity of our results to these choices insection V below.

The neutrino spectra can be described by a normalizedFermi-Dirac distribution,

dP�dE0

�2

3�3T3�

E02

eE0=T� � 1

; (4)

where T� � 180�3hE��i=7�4 is the effective neutrino tem-perature taken to be independent of the mass of the star. Weassume a flat �CDM cosmology with��dtdz

�� 9:78h�1Gyr

�1� z�� m�1� z

3p ; (5)

where �� 0:73, �m � 0:27, and h � 0:71 [1].

III. STAR FORMATION MODELS

The cosmic star formation histories we consider herehave been adopted from detailed chemical evolution mod-els [4,5]. These models are bimodal and are described by abirthrate function of the form

B�m; t � �1�m 1�t ��2�m 2�t; (6)

where �1�2 is the IMF of the normal (massive) componentof star formation, and 1�2 is the respective star formationrate. The normal component contains stars with massesbetween 0.1 and 100 M� and is primarily constrained byobservations at low redshift (z & 6). The massive compo-nent operates at high redshift and is required by the evi-dence for the reionization of the Universe at z� 17. Bothcomponents can contribute to the chemical enrichment ofgalaxy forming structures and the IGM. We consider threedifferent models of the massive mode as described below.

Given an IMF and SFR, it is straightforward to computethe rate of core collapse supernovae,

SNR �Z msup

max�8M�;mmin�tdm��m �t� ��m; (7)

where mmin�t is the minimum mass with lifetime less thant. Then for each model, the SRN flux is calculated usingEq. (1). In the two following subsections, we describe thetwo evolution models we have used, and their correspond-ing SFR and IMF.

A. Stage 1: Simplified models

We begin by describing a simplified set of hierarchicalmodels [4]. We assume star formation to occur betweenzi � 20 and zf � 0. The IMF for each mode is similar to aSalpeter mass function,

�i�m / m��1�x (8)

but has a slightly steeper slope of x � 1:7. Each IMF isnormalized independently by

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Z msup

minf

dmm�i�m � 1; (9)

differing only in the specific mass range of each model.The mass in cosmic structures will be denoted Mstruct�t,

which includes both the mass in stars and the mass in gas ofthe ISM. The mass in the IGM is MIGM and the total massMtot � MIGM �Mstruct is of course constant. The mass instructures evolves as

dMstruct

dt� ab�t � o�t; (10)

where ab�t is the cosmic baryon accretion rate due to thestructure formation process and o�t is the outflow fromstructures. In our simplified set of models, which wedenote as stage 1, we will ignore the effects of outflowand set o�t � 0. We will also assume structure formationto be exponentially decreasing from time t � 0,

ab�t �a�sMtote

�t=�s ; (11)

where a � 0:1 is the fraction of the total mass which iseventually accreted by structures and �s is the timescale ofthe accretion process. Here we consider both �s �0:01 Gyr and �s � 0:2 Gyr [4]. We also assume that starformation begins when the baryon fraction in structures is1%. This is our initial condition at z � 20 and correspondsto an estimate of the minimum baryon fraction wheresufficient dissipation occurs to allow star formation [21].In our more complex models we retain the condition on theonset of star formation, but determine its redshift using adetailed model of hierarchical structure formation (see nextsubsection).

The evolution of the gas mass in the structures is givenby :

dMISM

dt� � �t � e�t � ab�t � o�t: (12)

The stellar mass in the structures is simply Mstruct �MISM.Each term in Eq. (12) accounts for part of the gas budget ofthe ISM. The first term corresponds to the loss of gasthrough star formation while the second term correspondsto the ejected gas when the star dies. Here, we will use theinstantaneous recycling approximation (IRA) to evaluatethe rate at which gas is returned to the ISM, e�t

e�t �Z msup

0:9M�

dm��m �t�m�mr � R �t: (13)

where mr is the mass of the remnant, and R is the IRAreturn mass fraction.

The normal mode of star formation in [4] is referred to asModel 0 and provides a standard star formation history,with stellar masses in the range 0:1 M� � m � 100 M�.The SFR for Model 0 is proportional to the gas massfraction in cosmic structures, � � MISM=Mstruct,

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TABLE I. The model parameters for the normal mode of starformation (Model 2.0). Column 1 indicates the input value of theminimum mass for star forming structures. Column 2 is derivedfrom column 1, having assumed that fb � 1% when star for-mation begins. In columns 3, 4, and 5, parameter values for theefficiency of outflow and the SFR are given. The slope of theIMF is x � 1:3 for all models.

Normal modeMmin zinit � �1 �1

(M�) (Gyr�1) (Gyr)

106 18.2 2� 10�3 0.2 2.8107 16.0 3� 10�3 0.2 2.8108 13.7 5� 10�3 0.2 2.8109 11.3 10�2 0.2 3.01011 6.57 1:5� 10�2 0.5 2.2


1 � �1��t; (14)

where �1 � Mstruct�t=�1 with �1 � 5 Gyr which is a typi-cal timescale for star formation in the galactic disk. Thismodel alone is inadequate for high redshift reionization.

We consider three different models, labeled Models 1,2a, and 2b to describe the massive mode. They are distin-guished by their respective stellar mass ranges. In Model 1,the IMF is defined for stars with masses, 40 M� � m �100 M�. All of these stars die in core collapse supernovaeleaving a black hole remnant. They all contribute to thechemical enrichment of the ISM (and IGM when o�t � 0).This period of star formation is brief and is described by aSFR of the form

2 � �2e�t=�2 (15)

where �2 � f2Mstruct�t=�2 with a characteristic timescale�2 � 50 Myr. The constant f2 � 4:5%.

Model 2a is described by very massive stars whichbecome pair-instability supernovae. The IMF is definedfor 140 M� � m � 260 M� and the SFR for this modelis the same as that for Model 1, but must be reduced by afactor of 8 due to constraints on metal abundances in theISM. We consider the best case scenario for observation,where the energy emitted in neutrinos is the same as thatfor ordinary core collapse supernovae, but the averageneutrino energy is hE ��ei � 1:2 MeV [18].

The most massive stars are considered in Model 2b andfall in the range 270 M� � m � 500 M�, with the SFR asin Model 1. These stars entirely collapse into black holesand do not contribute to the chemical enrichment of eitherthe ISM or IGM. Collectively we will refer to stage 1models as 1.0, 1.1, 1.2a, and 1.2b. Unless otherwise noted,Models 1.1, 1.2a, and 1.2b (as well 2.1, 2.2a, and 2.2bdefined below) will correspond to the full bimodal model,i.e., they include the normal mode of star formation as wellas the particular massive mode.

B. Stage 2: Hierarchical models

We also consider a set of models with a more sophisti-cated treatment of the hierarchical growth of structurewhich we will call here stage 2. Complete details of thismodel can be found in [5]. Rather than assuming ananalytic form for the formation of structure as inEq. (11), we use the Press-Schechter formalism and take

ab�t � �b

�3H2

0

8�G

��dtdz

��1��dfb;struct

dz

�� 1:2h3 M�=yr=Mpc3

��b

0:044

��1� z

�� m�1� z

3q ��dfb;struct

dz

��: (16)

where fb is the baryon fraction in structures

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fb;struct�z �

R1Mmin

dMMfPS�M; zR10 dMMfPS�M; z

: (17)

In Eq. (17), fPS�M; z is the distribution function of halosand is computed using the method described in [14], andMmin is the minimum halo mass in which stars are formed.As before, we require an initial baryon fraction fb � 1%,which, for example, for Mmin � 107 M� corresponds to aninitial redshift for star formation, z � 16.

In [5], it was found that better fit to the global starformation rate and supernova rates is obtained when anormal mode SFR of an exponential form

1 � �1e�t=�1 (18)

is used, which corresponds to a SFR dominated by ellip-tical galaxies. Best fits for �1 and �1 are given in Table I. Inthese models the outflow is nonzero and the details forcomputing the outflow are given in [4,5]. The overallefficiency of outflow is parameterized by � whose valueis also given in Table I. The instantaneous recycling ap-proximation is no longer used here, and ejection ratesdepend on stellar lifetimes, ��m. This amounts to replac-ing �t with �t� ��m in the integral in Eq. (13).

The SFRs for the massive modes in stage 2 are deter-mined by the metallicity in the ISM,

2 � �2e�Z=Zcrit (19)

where Zcrit � 10�4 Z� is the critical metallicity at whichPopulation III star formation ends [7]. The IMF of bothmodes in this case has a slope x1�2 � 1:3. The massivemode SFR parameters are given in Table II for all modelsconsidered. The mass ranges and neutrino average energiesin stage 2 are the same as those in stage 1, respectively.

When a massive mode is added to the normal modedescribed by Model 2.0, the outflow efficiency must beadjusted so as to avoid the overproduction of metals in theIGM. However, there is a degeneracy in the massive modeparameters � and �2. In models labeled 2.1 and 2.2a, the

-4

0 2 4 6 8 10

0.5

1

1.5

2

0

Eν (MeV)

(c

m-2

s-1

MeV

-1)

dF dE

FIG. 1 (color online). Electron antineutrino fluxes from Model1.0 with different baryon accretion rates. We have chosen twomodels with ab � 0 with different baryon fractions,Mstruct=Mtot � :1 (dashed) and Mstruct=Mtot � :01 (dotted). Wehave also chosen two models with structure growth using �s �0:01 Gyr (thick), and �s � 0:2 Gyr (solid). In the latter cases,a � 0:1.

0 0.5 1 1.5 2 2.5 30

20

40

60

80

E (MeV)

(c

m-2

s-1

MeV

-1)

dF dE

2b

2a

1

TABLE II. Parameter values for the massive starburst Models2.1, 2.2a and 2.2b. Column 1 indicates the model number andcolumn 2 the input value of the minimum mass for star formingstructures. In columns 3 and 4, we show the adopted outflowefficiency and massive mode SFR.

Massive modeModel Mmin � �2

(M�) (Gyr�1)

2.1 107 2� 10�3 602.1e 107 6� 10�5 3402.2a 107 1� 10�3 92.2ae 107 8� 10�5 402.2b 107 3� 10�3 100


massive mode contributes roughly 50% of the IGM met-allicity at a redshift z � 2:5. By increasing �2 and decreas-ing �, this contribution can be increased to 90% and at thesame time increases the ionization capacity of the model.These cases are labeled 2.1e and 2.2ae.1

IV. RESULTS

A. Neutrino background

Stage 1 and Stage 2 models are discussed individually inSecs. IVA 1 and IVA 2. Then comparisons are made inSec. IVA 3 with other results. We will for the most partconsider only ��e, as they are the most easily detected atwater Cerenkov detectors. We will return to �e when weaddress thermonuclear neutrinos and detection.

1. Stage 1

We begin by examining the neutrino production fromsimplified models, labeled here as stage 1. As describedabove, these models employ the IRA and as such can to alarge extent be treated (semi)-analytically. In this class ofmodels, outflows are ignored and hierarchical growth issimply modeled by Eqs. (10) and (11). All of thePopulation III models we consider are bimodal (seeEq. (6)) and combine a normal stellar distribution (model0) along with a massive mode. As a reference point, we firstcompute the expected neutrino flux from the normal modealone. This is shown in Fig. 1, where we show the flux of ��eonly. The peak flux is about 2 cm�2s�1 MeV�1 and occursat E� � 2 MeV. While this flux is small compared to solarneutrino fluxes (for example, the flux of 8B neutrinos fromthe sun is of order 106 cm�2s�1 MeV�1), it is large com-pared to the nonlocalized atmospheric neutrino flux whichis less than 10�2 cm�2s�1 MeV�1 (see e.g. [10]).

Figure 1 shows the resultant neutrino flux for twochoices of the baryon fraction in structures, Mstruct=Mtot �:1 (large dashes) and for comparison, Mstruct=Mtot � :01

1Since stars associated with Model 2b do not contribute toelement enrichment, there is no Model 2be.

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(small dashes) when the growth of structures is neglected(i.e. ab � 0). As expected, the neutrino flux is in directproportion to the baryon fraction. We also show in Fig. 1,the neutrino flux when ab � 0, and the mass of the struc-ture grows. We show results for two different growthconstants, �s � 0:01 and 0.2 Gyr. Since the final baryonfraction is 10% in each case, we see that the integratedneutrino flux is quite similar.

The individual fluxes for the massive modes of Models1.1, 1.2a, and 1.2b are displayed in Fig. 2. For each of themassive modes, the duration of star formation is very brief,as the SFR is characterized by a time constant �2 �0:05 Gyr. The neutrino fluxes for Models 1.1 and 1.2b,shown by solid and dotted curves, respectively, are signifi-cantly larger than those found for a normal population ofstars. In these cases, the fluxes are approximately 30 and95 cm�2s�1 MeV�1, though the peak of the spectrum

ν

FIG. 2 (color online). Fluxes from the massive modes ofModels 1.1 (solid), 1.2a (dashed), and 1.2b (dotted).

-5

0 1 2 3 4 50

20

40

60

80

Eν (MeV)

(c

m-2

s-1

MeV

-1)

dF dE

0 0.05 0.1 0.15 0.2 0.25 0.30

5

10

15

20 2b

2a

1

2b

2a

1

FIG. 3 (color online). Total fluxes from Models 1.1 (solid),1.2a (dashed), and 1.2b (dotted). In all three cases, ab � 0 and�s � 0:01 Gyr. The insert shows the low energy peak due toPISN.

FIG. 4 (color online). The SFR for Model 2.1 (solid), 2.2a(dashed), and 2.2b (dotted) as a function of redshift. Each modelis indicated by the mass range associated with the massive mode.In all three cases, Mmin � 107 M�. Data are taken from [6].

0 2 4 6 8 100

2.5

5

7.5

10

12.5

15

17.5

Eν (MeV)

(c

m-2

s-1

MeV

-1)

dF dE

FIG. 5 (color online). Fluxes from Model 2.0 for five choicesof Mmin � 106 (solid), 107 (thick), 108 (dotted), 109 (dashed)and 1011 (dot-dashed ) M�.


occurs at lower energy, E� ’ 0:6 MeV. We note that theSRN flux from Model 1.2b is very similar to the flux due tothe population of rotating 300 M� stars with a SFR peakedat z � 17 in Iocco et al. [13]. The peak height obtained inour calculations is larger due primarily to the fact that theintegrated fraction of baryonic matter in population IIIstars is about 4 times greater in our model.

The flux from Model 1.2a is much smaller than thefluxes from Models 1.1 and 1.2b, and peaks at a lowerenergy. This is a consequence of less energy being releasedin neutrinos by pair-instability supernovae than by super-novae that collapse to form black holes, and that theaverage energy of each neutrino is limited by siliconburning and photodisintegration [18].

Comparison of Figs. 1 and 2 shows that the normalmode, though smaller in its peak flux, dominates the fullbimodal spectrum at energies E * 2:5 MeV. Recall thatthe massive mode is very localized at high redshift. As aresult, neutrinos produced by the massive mode have en-ergies which are redshifted from their initial value. Incontrast, the normal mode of star formation is peaked atz ’ 2� 3, which produces a broader spectrum today.

Finally, Fig. 3 shows the total flux for the bimodal IMFsin the case that the massive component is either Model 1.1,1.2a, or 1.2b. At low energies, the flux is dominated by themassive component. The insert shows the low energy peakdue to PISN.

2. Stage 2

We now use the full numerical results of the modelsdescribed in [5]. The overall form of these models issimilar to those in stage 1, i.e. they are described by abimodal birthrate function of the form in Eq. (6). The massranges for models 0, 1, 2a, and 2b are unchanged, thoughthe slope of the IMF in each case is now 1.3. As describedabove, these models include both outflow and the hierarch-

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ical growth of structure and we no longer employ the IRAin any of our calculations. The SFR is shown in Fig. 4 forModels 2.1, 2.2a, and 2.2b as indicated by their respectivemass ranges. In Models 2.1 and 2.2a, the massive modecontributes roughly equally with the normal mode to theIGM metallicity at z� 3. Since Model 2.2b produces noheavy elements, the SFR is chosen to be maximal toenhance its ionization potential. In each case, star forma-tion begins at z� 16 when the baryon fraction in a struc-ture of total mass 107 M� is fb � 1%. The massive burstends when the ISM metallicity has reached a critical valuetaken to be 10�4 Z�. The duration of the burst in Model2.2b is somewhat more prolonged as the metallicity isproduced solely by the normal component.

Model 2.0 fluxes are plotted in Fig. 5. Results are shownfor several choices of minimum halo masses, Mmin. Theneutrino fluxes found here are roughly a factor of 10 timeslarger than that found in the semianalytical model dis-cussed in the previous subsection. This is partly due tothe flatter IMF chosen here which greatly enhances thenumbers of massive stars and hence the supernova rate andneutrino flux. As we will see below these fluxes are large

-6

0 1 2 3 4 5

25

50

75

100

125

150

Eν (MeV)

(c

m-2

s-1

MeV

-1)

dF dE

0 0.05 0.1 0.15 0.20

2

4

6

8

10

12

2b

2a

1

2b

2a1

FIG. 8 (color online). Total fluxes for Model 2.1 (solid), 2.2a(dashed), and 2.2b (dotted).

0 1 2 3 4 50

20

40

60

80

Eν (MeV)

(c

m-2

s-1

MeV

-1)

dF dE

1

1e

FIG. 6 (color online). Fluxes from the massive modes ofModel 2.1 and 2.1e.


enough to be probed by current detectors. As Mmin isincreased, star formation occurs at later redshift and as aresult, the peak of the neutrino flux is shifted slightly tohigher energy.

The massive modes of Model 2.1 and 2.1e fluxes areplotted in Fig. 6 for the specific choice of Mmin � 107 M�which is the preferred case in [5]. As seen in Fig. 4,massive stars associated with Population III turn on at aredshift of approximately 16, but the duration of the burstis relatively brief. As a result, the peak of the flux distri-bution is at relatively low energy. More importantly, be-cause of the brevity of the burst, the entire neutrinospectrum is redshifted down, in contrast to the Model 2.0spectrum which extends to higher energy due to starsproduced at lower redshifts. As expected, the more extrememodel, 2.1e, has a peak flux which is about 5 times thatfound for Model 2.1. This is directly related to the in-creased SFR in Model 2.1e as characterized by the increasein �2.

Similarly, we show in Fig. 7 the resulting flux from verymassive Population III stars corresponding to Models 2.2a,2.2ae and 2.2b. As before, the fluxes from Models 2.2a arerelatively small and peak at very low energy as seen in the

0 1 2 3 4 5

20

40

60

80

100

120

140

160

Eν (MeV)

(c

m-2

s-1

MeV

-1)

dF dE

2b

0 0.05 0.1 0.15 0.20

10

20

30

40

50

2a

2ae

FIG. 7 (color online). Fluxes from the massive modes ofModel 2.2a (dashed), 2.2ae (dashed), and 2.2b (dotted).

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insert to the figure. In Figs. 8 (9), we show the total fluxesin Models 2.1, 2.2a, and 2.2b (2.1e and 2.2ae) with Mmin �107 M�. As one expects, the low energy spectrum is domi-nated by neutrinos produced in the massive mode, whereasthe spectrum at higher energies (E� * 3 MeV), is indis-tinguishable between the models and dominated by thenormal mode.

3. Comparisons to other models

The differences in the fluxes between stages 1 and 2models can be attributed entirely to differing IMF slopes,the baryon fraction, and treatment of the SFR. In order tocompare our calculations to previous ones, it is necessaryto single out individual models and discuss the differences.

Our Model 0 can be compared to models described in[9–12]. In [9], the calculated neutrino flux peaks at ’8 cm�2s�1MeV�1 with neutrino energies of �3 MeV. Inthe models considered, star formation only occurs for z �5 and the average neutrino energy at emission was assumedto be a step function in which energies are generally a fewMeV larger than our adopted value of hE ��ei � 15:3 MeV.Since the flux changes as

0 1 2 3 4 50

20

40

60

80

100

Eν (MeV)

(c

m-2

s-1

MeV

-1)

dF dE

0 0.05 0.1 0.15 0.20

5

10

15

20

25

30

2ae

1e2ae

1e

FIG. 9 (color online). Total fluxes for Model 2.1e (solid) and2.2ae (dashed).

-7

80

100

120

-1 M

eV-1

)


dFdE/

1

hE�i41

ex=hE�i � 1; (20)

where x depends on z, this both reduces the peak height andshifts the peak to larger neutrino energy at detection. Inaddition, they include only supernovae that collapse toneutron stars, with 8 M� � m � 50 M�. In the absenceof any component of stars which collapse to black holes,the flux should be smaller than that obtained by our calcu-lations. Ando, Sato and Totani [10] use a crude supernovamodel, but provide a more careful treatment of oscillations.As will be shown in section VI, our results are qualitativelysimilar despite the fact that our integrated flux differs fromtheirs by an order of magnitude.

The more modern approach taken in [10,11] is based onthe cosmic star formation rate. However, our Model 2.0spectra have significantly higher peaks than those in[10,11]. First, improved determinations of the SFR athigh redshift are higher by a factor of about 2. A moresignificant difference between our results can be traced toour inclusion of stars that collapse to black holes. Thechange in gravitational energy is much greater when theremnant is a black hole than when it is a neutron starleading to a larger value for Ecc, and therefore the neutrinoluminosity is also much greater. For comparison, if weconsider only stars that collapse to neutron stars, our fluxis reduced by a factor of �5.

In [12], the SFR is parametrized as a broken power law,flat for z > 1 and fit to the observed cosmic star formationrate. As in [9,10], star formation begins at z � 5. The effectof neglecting star formation at higher redshift is inconse-quential for the higher energy tail of the flux, but bothdiminishes the lower energy flux and shifts the peak for-ward. Both of these effects are consequences of the factthat neutrinos from the earliest supernova events will arriveat Earth with the lowest energies as a result of redshift.

The flux from a massive mode of star formation wascomputed in [13]. Population III stars were assumed to berotating 300 M� stars with a SFR approximated by a deltafunction peaked at z � 17. We observe the same peaklocation and spectral shape in our Models 1 and 2b. Ourfluxes peak at larger values than theirs for two mainreasons. First, our integrated fraction of baryonic matterin Pop III stars is larger. They use a fixed value of 10�3,whereas our value is derived for each model; 2:2� 10�3

TABLE III. These are the thermonuclear reactions consideredhere. The endpoint is indicated in MeV, and the percentage refersto the amount of the total thermonuclear neutrino flux due toeach reaction mechanism.

Reaction Endpoint %

p� p! e� � �e 0.42 1013N! 13C� e� � �e 1.20 4515O! 15N� e� � �e 1.73 45

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for Model 2.1 and 7:3� 10�3 for Model 2.2b. Second, inModel 2.2b the stars collapse entirely to black holes, whilein their model the energy emitted in neutrinos correspondsroughly to the gravitational energy released when the starscollapse to black holes with masses equivalent to the massof the helium core before collapse. As a result, a 300 M�star in our model emits�2:7 times the number of neutrinosas the same star in their model.

B. Thermonuclear neutrinos

In this section we consider briefly the neutrinos emittedby stars during the hydrogen-burning phase. The CNOcycle is the dominant neutrino production mechanism,responsible for 90% of the thermonuclear neutrino flux.The remaining 10% is due to the pp chains. To simplify thecalculations, we consider only the reactions given inTable III below.

The total number of neutrinos emitted by a star of massm during hydrogen burning has been estimated to be [13]

Ntherm�nucl � 0:2mmN

; (21)

where mN is the mass of a nucleon, and the energy spec-trum at emission is taken from Ref. [22]. The total flux ofthermonuclear electron neutrinos in Model 2.1 (includingthe normal mode) is shown in Fig. 10. As one can see, thetotal flux is quite large and significant up to E�e � 1 MeV.The flux from the massive component is cut off at E�e �0:1 MeV, which is far below the neutrino energy thresholdat gallium experiments such as SAGE and Gallex-GNO(Ethreshold � 0:233 MeV [22]). The normal mode thermo-nuclear flux is smooth and broad due to the larger SFR atlow redshift. However, the flux is overshadowed by severalorders of magnitude by solar neutrinos from the pp chainsand the CNO cycle. Since experiments sensitive to thisenergy range, like Borexino and LENS, do not have thecapability to resolve any directional information aboutincoming neutrinos, the only current possibility for distin-guishing SRN’s from solar neutrinos is by spectral shape.

0 0.2 0.4 0.6 0.8 1 1.2 1.40

20

40

60

Eν (MeV)

(c

m-2

sdF dE

FIG. 10 (color online). Total flux of thermonuclear neutrinosfrom Model 2.1.

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0 5 10 15 20 25 300

20

40

60

80

100

120

Eν (MeV)

Flu

x (

cm-2

s-1

)

FIG. 11 (color online). Detectable fluxes from Model 2.1 with(dashed) and without (solid) oscillations as a function of neutrinoenergy threshold.

TABLE IV. Predicted fluxes in cm�2s�1 in the models consid-ered here. Results are given for electron antineutrinos withenergies E ��e > 19:3 MeV for SK and for electron neutrinoswith 22:5 MeV<E�e < 32:5 MeV for SNO.

Model SK Flux SNO Flux

1.0 0.40 0.111.1 0.40 0.111.2a 0.40 0.111.2b 0.40 0.11

2.0 1.8 0.472.1 1.8 0.472.1osc 3.2 1.42.1e 1.9 0.492.2a 1.9 0.482.2ae 1.9 0.492.2b 1.8 0.472.2bosc 3.2 1.4


V. DETECTION

The detection of SRN’s is inhibited mainly by difficul-ties excluding background events which include solar neu-trinos, atmospheric neutrinos and antineutrinos, andantineutrinos from nuclear reactors. Cosmic ray muonsalso produce events that mimic the expected signal.

The solar neutrino flux at Earth is larger than the ex-pected flux from SRN’s by several orders of magnitude forE� & 19 MeV [11]. However, neutrinos, rather than anti-neutrinos, are produced in the thermonuclear reactions inthe sun and have a smaller cross section for detection byabout 2 orders of magnitude. This, with the directionalinformation from recoil electrons in the detector, allowsthis background to be excluded at SK, KamLAND, andSNO. But decays of spalled nuclei from cosmic ray muonsconstitute an unavoidable background in this range.

Atmospheric electron antineutrinos are cause for con-cern above �8 MeV, but the flux becomes larger than theexpected flux of SRN’s only for E� * 35 MeV [12]. Anatmospheric muon neutrino can interact with a nucleus toform a muon, which will be invisible in Cerenkov detectorsif its kinetic energy is below the Cerenkov radiation thresh-old of 53 MeV. This background is significant for19 MeV<E� < 35 MeV, but it can be described by theMichel spectrum and was subtracted off to obtain thecurrent upper limit for the flux of SRN’s at SK of1:2 cm�1s�1 for E� > 19:3 MeV [23].

It has been pointed out recently [24] that if the back-ground analysis from SK is coupled with the sensitivity toelectron neutrinos at SNO it will be possible to reduce theupper limit on the flux of electron neutrinos. SNO shouldbe sensitive to a flux of 6 cm�2s�1 in the range22:5 MeV<E�e < 32:5 MeV, which is an improvementon the Mont Blanc limit by 3 orders of magnitude.

In Fig. 11, we show the observable flux

F�Ethresh �Z 1Ethresh

dFdE

dE (22)

as a function of detector threshold energy. While the fluxesare quite appreciable at low threshold energies, they in factremain relatively high at larger energies due to the largeSFR associated with Model 0. Indeed, in all of our stage 2models, our predicted flux above 19.3 MeV already ex-ceeds the current bound of 1:2 cm�1s�1 from SuperK [23].The detailed flux predictions are given in Table IV, wherewe show the detectable flux for the viable energy windowsat SK and SNO. Although SRN’s will likely not be seen atSNO given these flux levels, in many of our models the SKbound is saturated by the expected flux, indicating thatSRN’s may be observed in the near future. This is inagreement with previous arguments made in [12] basedon simplified evolution models as well as arguments basedon SN1987A [25].

Despite the large fluxes displayed in Table IV relative tothe SK limit [23], one can not conclude that the stage 2

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models considered have already been excluded by experi-ment. There are of course many uncertainties built into ourchemical evolution models as well as uncertainties in theadopted neutrino physics. For example, one of the differ-ences between our stage 1 and stage 2 models, is our choiceof the IMF. In stage 1 models, the slope of the IMF wasfixed at 1.7 whereas in stage 2 models, it is fixed at 1.3. Theimpact of this difference lies in the strong suppression ofmassive stars, i.e. the precursors of neutrino producingsupernovae. As one can see from the table, the steeperIMF sufficiently suppresses the neutrino flux to satisfy theSK bound.

Another key uncertainty is the choice of the mass rangefor black hole formation in the normal mode. Black holeformation contributes significantly to the neutrino fluxfrom the normal mode despite the IMF favoring lowermass stars. As a result, our conclusions are sensitive toboth the number of stars collapsing to black holes (via the

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0 5 10 15 200

5

10

15

20

Eν (MeV)

(c

m-2

s-1

MeV

-1)

dF dE

FIG. 13 (color online). Total fluxes of electron antineutrinosfrom Model 2.1 with (dashed) and without (solid) oscillations.


normal mode IMF) and the number of neutrinos emittedduring collapse. If black hole formation begins to occur atmasses larger than 30 M�, our neutrino fluxes would belower. For example, if the mass range for black holes in thenormal mode were 25�40 � 100 M�, our SK flux wouldbecome 1:9�1:6 cm�2s�1 as opposed to the value of1:8 cm�2s�1 shown in Table IV when the mass range is30–100 M�.

Curiously, changes in the maximum mass of popIII starshas very little effect on our results. Indeed, in Models 1 and2b, up to finite lifetime effects, the neutrino flux in Eq. (1)does not depend on Mmax. This is because N�� / m and theintegration over m is identical to the normalization of �.For Model 2a, the effect of reducing the upper limit of themass range is actually to increase the peak height of theflux spectrum. As an extreme case, if Model 2a is chosen toinclude only stars with masses 140 M� <m< 150 M�,the peak height is increased by almost 30% due primarilyto changes in the IMF normalization.

One can also see from Table IV that the dominantcontribution to the integrated flux above the SK thresholdcomes from normal mode stars. Even the more extrememodel 2.1e only contributes 0:1 cm�2s�1 to the flux above19.3 MeV. However, these fluxes are very sensitive to ourassumed average neutrino energy. Recall our adopted valuefor E ��e is 15.3 MeV. In Fig. 12, we show the sensitivity ofthe flux above 19.3 MeV (F�19:3) to the average neutrinoenergy. In order to satisfy the SK limit of 1:2 cm�2s�1, wewould have to lower hE ��ei to 13.3 MeV. This is fullyconsistent with the range of neutrino energies in supernovamodels [20].

Ill-understood backgrounds and large detector energythresholds make detection of SRN’s difficult. Because theSRN flux peaks at or below 1 MeV, there is little hope thatthe peak will be probed with existing experiments.However, being able to exclude backgrounds at lowerenergies would greatly increase the probability for obser-vation at SK or a similar experiment.

8 10 12 14 16 18 200

1

2

3

< Eν (MeV) >

F(1

9.3

MeV

) (

cm-2

s-1

)

FIG. 12 (color online). The integrated flux above 19.3 MeV inModel 2.1 as a function of the average neutrino energy.

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VI. EFFECTS OF OSCILLATIONS

Because of our assumption of equipartition of energyamong the neutrino species, oscillation will only affect theflux if different species emerge from the explosion withdifferent average energies. Indeed most supernova modelcalculations do show a hierarchy of neutrino energies, andbecause the energies of E �� and E �� are generally higherthan E ��e , the effect of oscillations will in general increasethe observable flux. The effects of oscillations on theneutrino background were considered previously in [26].

Figure 13 shows the total flux of electron antineutrinosfrom Model 2.1 both with and without oscillations, wherewe use the neutrino average energies discussed in Sec. IIand maximal mixing has been assumed. Although the totalnumber of electron antineutrinos arriving at Earth issmaller due to oscillations by �16%, the flux of neutrinoswith energies greater than �9:5 MeV is larger. Note thatthe solid curve here is identical to that in Fig. 8. Here weclearly see the individual peaks due to the massive mode at& 1 MeV and normal mode at �3 MeV.

For E� > 19:3 MeV, oscillation effects would increasethe observable flux at SK by as much as 78%. This effect isseen in Fig. 11 where one sees that the integrated flux withoscillations (dashed curve) exceeds the flux when oscilla-tions are ignored for threshold energies greater than about6 MeV. The effect of oscillations on the SK observable fluxis seen in Table IV for the models labeled 2.1osc and2.2bosc. Reconciling these fluxes with the SK limit wouldrequire a further drop in the average neutrino energy or atightening of the assumed neutrino energy hierarchy.Similarly, the flux of electron neutrinos potentially observ-able at SNO is increased by almost 200% in the energywindow 22:5 MeV<E< 32:5 MeV. Although our fluxcalculated with oscillations is still less than 1=4 that whichis necessary to approach the projected SNO sensitivity[24], with a better understanding of backgrounds the pros-pects for detection in the near future are encouraging.

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VII. CONCLUSIONS

We have considered several scenarios for star formationwhich reproduce the observed chemical abundances andSFR for z � 6 and reionize the universe at high redshift.Each model of star formation here consists of a normalmode coupled to a Population III mode of massive starformation at high redshift. We examined the SRN flux fromthe core collapse supernova explosion as well as from thethermonuclear burning stage.

Because the massive mode of star formation is so briefand takes place at high redshift, the corresponding electronantineutrino fluxes peak at E� & 1 MeV. Thus despite thelarge fluxes produced by the massive mode, these lowenergy neutrinos will be difficult to detect. In contrast,the normal mode of star formation, which dominates theflux at observable energies, is peaked at a somewhat higherenergy and has a broad spectrum due to the production ofstars at lower redshift. The neutrinos produced duringthermonuclear burning are emitted with much lower ener-gies, on average, but also exhibit the behavior of a sharppeak due to the massive mode added to a broader spectrumfrom the normal mode. Thermonuclear neutrinos are un-likely to be observed at experiments which cannot at leastpartially resolve the direction of the incoming particle

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because they are produced in the electron flavor state andthe spectrum lies several orders of magnitude below that ofsolar neutrinos. Our calculated fluxes of SRN’s from corecollapse, however, saturate the SK bound of 1:2 cm�2s�1

for E� > 19:3 MeV in all stage 2 models. Although thereare uncertainties in the neutrino physics, such as the aver-age energies at emission, the prospects for observation inthe near future are good. We also examined the effect ofoscillations by calculating the flux with maximal mixing.With the accepted neutrino average energy hierarchy,hE�ei< hE ��ei< hE�xi, any oscillation will harden thehigh energy tail of both the �e and the ��e spectra.

Further refinement of the neutrino physics and measure-ment of the SFR out to higher redshift would allow for amore definite flux prediction. With decreased detectorthresholds and increased background rejection, observa-tion of the SRN flux will soon be possible.

ACKNOWLEDGMENTS

The work of K. A. O., F. D., and E. V. was supported bythe Project ‘‘INSU - CNRS/USA’’, and the work ofK. A. O. and P. S. was also supported partly by DOEGrant No. DE–FG02–94ER–40823.

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neutrino signatures from the first stars

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