# Probing Neutrino Dark Energy with Extremely High-Energy

## Transcript Of Probing Neutrino Dark Energy with Extremely High-Energy

DE06F7623

DEUTSCHES ELEKTRONEN-SYNCHROTRON

in der HELMHOLTZ-GEMEINSCHAFT

DESY 06-088 astro-ph/0606316 June 2006

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\X^*y

Probing Neutrino Dark Energy with Extremely High-Energy Cosmic Neutrinos

A. Ringwald, L. Schrempp

Deutsches Elektronen-Synchrotron DESY, Hamburg

ISSN 0418-9833

NOTKESTRASSE 85 - 22607 HAMBURG

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arXiv:astro-ph/0606316 v1 13 Jun 2006

DESY-06-088

Probing Neutrino Dark Energy with Extremely High-Energy Cosmic Neutrinos

Andreas Ringwald and Lily Schrempp

Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg, Germany E-mail: [email protected], [email protected]

Abstract. Recently, a new non-Standard Model neutrino interaction mediated by a light scalar

ﬁeld was proposed, which renders the big-bang relic neutrinos of the cosmic neutrino background a natural dark energy candidate, the so-called Neutrino Dark Energy. As a further consequence of this interaction, the neutrino masses become functions of the neutrino energy densities and are thus promoted to dynamical, time/redshift dependent quantities. Such a possible neutrino mass variation introduces a redshift dependence into the resonance energies associated with the annihilation of extremely high-energy cosmic neutrinos on relic anti-neutrinos and vice versa into Z-bosons. In general, this annihilation process is expected to lead to sizeable absorption dips in the spectra to be observed on earth by neutrino observatories operating in the relevant energy region above 1013 GeV. In our analysis, we contrast the characteristic absorption features produced by constant and varying neutrino masses, including all thermal background eﬀects caused by the relic neutrino motion. We ﬁrstly consider neutrinos from astrophysical sources and secondly neutrinos originating from the decomposition of topological defects using the appropriate fragmentation functions. On the one hand, independent of the nature of neutrino masses, our results illustrate the discovery potential for the cosmic neutrino background by means of relic neutrino absorption spectroscopy. On the other hand, they allow to estimate the prospects for testing its possible interpretation as source of Neutrino Dark Energy within the next decade by the neutrino observatories ANITA and LOFAR.

Probing Neutrino Dark Energy with Extremely High-Energy Cosmic Neutrinos

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1. Introduction

According to Big Bang Cosmology, in an expanding universe the freeze-out of a particle species occurs, when its interaction strength is too small to keep it in thermal equilibrium. Neutrinos, being the particles with the weakest known interactions, therefore are assumed to have already decoupled when the universe was just ≈ 1 s old, thereby guaranteeing a substantial relic neutrino abundance today with average number density nν0,i = nν¯0,i = 56 cm−3 per neutrino species i = 1, 2, 3. Yet, in turn, the weakness of the neutrino interactions so far has spoilt all attempts to probe this 1.95 K cosmic neutrino background (CνB), which is the analog of the 2.73 K cosmic microwave background (CMB) of photons, in a laboratory setting [1–4]. However, other cosmological measurements, such as the light element abundance, large scale structure (LSS) and the CMB anisotropies are sensitive to the presence of the CνB and therefore have provided us at least with indirect evidence for its existence (see e.g. Ref. [5] for a review).

Independently, Type Ia Supernova (SNIa) results (e.g. [6]), supported by CMB [7] and LSS data (e.g. Refs. [8, 9]), strongly suggest the existence of an exotic, smooth energy component with negative pressure, known as dark energy, which drives the apparent accelerated expansion of our universe. Recently, Fardon, Nelson and Weiner [10] have shown that the relic neutrinos, which constitute the CνB, are promoted to a natural dark energy candidate if they interact through a new non-Standard-Model scalar force – an idea which has great appeal. Neutrinos are the only Standard Model (SM) fermions without right-handed partners. Provided lepton number is violated, the active (left-handed) neutrinos are generally assumed to mix with a dark righthanded neutrino via the well-known seesaw mechanism [11–14], thus opening a window to the dark sector. Therefore, it would not seem to be surprising if neutrinos, whose interactions and properties we know comparably little about, were sensitive to further forces mediated by dark particles. Moreover, the scale relevant for neutrino mass squared diﬀerences as determined from neutrino oscillation experiments, δmν2 ∼ (10−2 eV)2, is of the order of the tiny scale associated with the dark energy, (2 × 10−3 eV)4.

As a consequence of the new interaction in such a scenario, an intricate interplay links the dynamics of the relic neutrinos and the mediator of the dark force, a light scalar ﬁeld called the acceleron. On the one hand, the neutrino masses mνi are generated by the vacuum expectation value A of the acceleron, mνi(A). Correspondingly, the A dependence of the masses mνi(A) is transmitted to the neutrino energy densities ρνi(mνi(A)) since these are functions of mνi(A). On the other hand, as a direct consequence, the neutrino energy densities ρνi(mνi, A) can stabilize the acceleron by contributing to its eﬀective potential Veﬀ (A, ρνi), which represents the total energy density of the coupled system. Moreover, cosmic expansion manifests itself in the dilution of the neutrino energy densities ρνi(z) ∼ (1 + z)3. Therefore, it crucially aﬀects the eﬀective acceleron potential Veﬀ (A, ρνi(mνi, z)) by introducing a dependence on cosmic time, here parameterized in terms of the cosmic redshift z. For a homogeneous

Probing Neutrino Dark Energy with Extremely High-Energy Cosmic Neutrinos

3

conﬁguration, the equilibrium value of the acceleron instantaneously minimizes its

eﬀective potential Veﬀ (A, ρνi(mνi, z)) and therefore also A(z) varies on cosmological time

scales. Finally, in turn, since the neutrino masses mνi(A) are sensitive to changes in A,

they are promoted to dynamical quantities depending on z, mνi(z), that is depending

on cosmic time. To summarize, the variation of the neutrino masses represents a clear

signature of the so-called Neutrino Dark Energy scenario.

In a subsequent work Fardon, Nelson and Weiner [15] presented a supersymmetric

Mass Varying Neutrino (MaVaN) model in which the origin of dark energy was

attributed to the lightest neutrino ν1 and the size of the dark energy could be expressed

in terms of neutrino mass parameters. By naturalness arguments the authors concluded

that the lightest neutrino still has to be relativistic today, thereby preventing potential

instabilities [16–18] which could occur in highly non-relativistic theories of Neutrino

Dark Energy.

The rich phenomenology of the MaVaN scenario has been explored by many authors.

The cosmological eﬀects of varying neutrino masses have been studied in Refs. [19, 20]

and were elaborated in the context of gamma ray bursts [21]. Apart from the time

variation, the conjectured new scalar forces between neutrinos as well as the additional

possibility of small acceleron couplings to matter lead to an environment dependence of

the neutrino masses governed by the local neutrino and matter density [10, 22, 23]. The

consequences for neutrino oscillations in general were exploited in Refs. [22, 24] and in

particular in the sun [25–27], in reactor experiments [26, 28] as well as in long-baseline

experiments [29].

In light of the possible realization of Neutrino Dark Energy in nature, a (more)

direct detection of the CνB should be thoroughly explored with special attention turned

to possible new physics beyond the SM. By this means, a time evolution of neutrino

masses could be revealed which would serve as a test of Neutrino Dark Energy. In

addition, the general importance of a (more) direct evidence for the existence of the

CνB lies in a conﬁrmation of standard cosmology back to the freeze-out of the weak

interactions and therefore thirteen orders of magnitudes before the time when photons

where imprinted on the last scattering surface.

An appealing opportunity to catch a glimpse of the CνB as it is today emerges

from the possible existence of extremely high-energy cosmic neutrinos (EHECν’s). Such

EHECν’s can annihilate with relic anti-neutrinos (and vice versa) into Z bosons, if their

energies coincide with the respective resonance energies E0re,is of the corresponding process νν¯ → Z [30–37]. These energies,

Eres = MZ2 = 4.2 × 1012 eV GeV

(1)

0,i 2mν0,i

mνi

in the rest system of the relic neutrinos, are entirely determined by the Z boson mass

MZ as well as the respective neutrino masses mνi. An exceptional loss of transparency of

the CνB for cosmic neutrinos results from the fact that the corresponding annihilation

cross-section on resonance is enhanced by several orders of magnitude with respect to

non-resonant scattering. As a consequence, the diﬀuse EHECν ﬂux arriving at earth is

Probing Neutrino Dark Energy with Extremely High-Energy Cosmic Neutrinos

4

expected to exhibit absorption dips whose locations in the spectrum are determined by the respective resonance energies of the annihilation processes. Provided that the dips can be resolved on earth, they could produce the most direct evidence for the existence of the CνB so far. Furthermore, as indicated by the resonance energies in Eq. (1), the absorption features depend on the magnitude of the neutrino masses and could therefore reﬂect their possible variation with time. Moreover, they are sensitive to the ﬂavor composition of the neutrino mass eigenstates as well as to various cosmological parameters. Accordingly, the possibility opens up to perform relic neutrino absorption spectroscopy as an independent means to probe neutrino physics and cosmology.

The existence of EHECν’s is theoretically well motivated and is substantiated by numerous works on possible EHECν sources of astrophysical nature (bottom-up) (see e.g. [38] for a review) or so-called top-down sources (see e.g. Ref. [39] for a review). In the latter case, EHECν’s with energies well above 1011 GeV are assumed to be produced in the decomposition of topological defects (TD’s) which originate from symmetry breaking phase transitions in the very early universe.

Depending on the underlying EHECν sources the EHECν ﬂuxes could be close to the current observational bounds set by existing EHECν observatories such as AMANDA [40] (see also Ref. [41]), ANITA-lite [42], BAIKAL [55], FORTE [43], GLUE [44] and RICE [45] which cover an energy range of 107 GeV < E0 < 1017 GeV (cf. Fig. 1). Promisingly, the sensitivity in this energy range will be improved by orders of magnitude (cf. Fig. 1) by larger EHECν detectors such as ANITA, EUSO [46], IceCube [47], LOFAR [48], OWL [49], SalSA [51] and WRST [48] which are planned to start operating within the next decade (cf. Fig. 1). Accordingly, the prospects of conﬁrming the existence of the CνB by tracking its interaction with EHECν’s have substantially improved since the original proposal in 1982 [30]. Moreover, in the likely case of appreciable event samples the valuable information encoded in the absorption features of the EHECν spectra could be revealed within the next decade (cf. Fig. 1), rendering the theoretical exploration of relic neutrino absorption spectroscopy a timely enterprise.

Note that the scenario introduced above has also attracted attention for another reason than the possible detection of the CνB– namely for the controversial possibility of solving the so-called GZK-puzzle to be discussed brieﬂy in the following. Beyond the Greisen-Zatsepin-Kuzmin (GZK) energy, EGZK = 4 × 1010 GeV, ultra-high energy nucleons rapidly lose energy due to the eﬀective interaction with CMB photons (predominantly through resonant photo-pion production) [56, 57]. In the so-called Zburst scenario, the observed mysterious cosmic rays above EGZK were associated with the secondary cosmic ray particles produced in the decays of Z bosons. The latter were assumed to originate from the neutrino annihilation process outlined above [58–63].

However, recently, ANITA-lite [42] appears to have entirely excluded the Z-burst explanation for the GZK-puzzle at a level required to account for the observed ﬂuxes of the highest energy cosmic rays. We would like to stress, that this only means that the GZK-puzzle stays unsolved (if there is any) . Moreover, this neither restricts the possible

Probing Neutrino Dark Energy with Extremely High-Energy Cosmic Neutrinos

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Figure 1. Current status and next decade prospects for EHECν physics, expressed in terms of diﬀuse neutrino ﬂuxes per ﬂavor, F = Fνα + Fν¯α , α = e, µ, τ . The upper limits from AMANDA [40], see also Ref. [41], ANITA-lite [42], FORTE [43], GLUE [44], and RICE [45] are plotted. Also shown are projected sensitivities of ANITA [42], EUSO [46], IceCube [47], LOFAR [48], OWL [49], the Pierre Auger Observatory in νe, νµ modes and in ντ mode (bottom swath) [50], SalSA [51], and WSRT [48], corresponding to one event per energy decade and indicated duration. Also shown are predictions from astrophysical Cosmic Ray (CR) sources [52], from inelastic interactions of CR’s with the cosmic microwave background (CMB) photons (cosmogenic neutrinos) [52, 53], and from topological defects [54].

success of producing evidence for the CνB by means of detecting absorption dips in the EHECν spectra nor does it spoil the possibility of gaining valuable information from performing relic neutrino absorption spectroscopy.

The goal of this paper is to carefully work out the characteristic diﬀerences in the EHECν absorption features which result from treating the neutrino masses as time varying dynamical quantities in comparison to constants. In our analysis, we incorporate the full thermal background eﬀects on the absorption process whose impact grows for smaller neutrino masses [36, 37]. This means, that in general relic neutrinos cannot be assumed to be at rest. Instead, they have to be treated as moving targets with a momentum distribution, if their mean momenta turn out to be of the order of the relic neutrino masses.

We illustrate our results for the diﬀuse neutrino ﬂuxes to be observed at earth ﬁrstly by considering astrophysical EHECν sources. Secondly, we calculate the neutrino spectrum (both for varying and constant neutrino masses) expected from the

Probing Neutrino Dark Energy with Extremely High-Energy Cosmic Neutrinos

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decomposition of a topological defect using the appropriate fragmentation functions and including the full thermal background eﬀects. By this means, for a given EHECν injection spectrum, we present all technical tools to interpret EHECν absorption dips as soon as they are observed at earth. Thereby the possibility opens up to test the CνB and its interpretation as source of Neutrino Dark Energy.

The paper is organized as follows. In Sec. 1 we discuss the MaVaN scenario and focus on a determination of the time dependence of the neutrino masses. Sec. 2 reviews the treatment of the absorption of an EHECν by the CνB in terms of the damping factor comprising the thermal background eﬀects. Furthermore, in order to make contact to treatments in the literature [30, 31, 33, 35–37], we include in the discussion common approximations [33,35,36] which neglect part or all of the dependence of the damping on the relic neutrino momenta. Moreover, we extend the complete analysis to incorporate a possible variation of the neutrino masses with time. In Sec. 3 we present and compare our results for the survival probabilities of EHECν’s with varying and constant masses which encode the physical information on all possible annihilation processes on their way from their source to us, again taking into account the thermal motion of the relic neutrinos. In order to gain more physical insight, in addition, we disentangle the characteristic features of the absorption dips caused by the mass variation by switching oﬀ all thermal background eﬀects. Sec. 4 illustrates the discovery potential of neutrino observatories for the CνB and gives an outlook for the testability of the MaVaN scenario. Therefore, both for astrophysical sources and for a topological defect scenario, we calculate the expected observable EHECν ﬂux arriving at earth which results from folding the survival probabilities with the corresponding EHECν source emissivity distribution. In the latter case, we perform the full state-of-the-art calculation with the help of fragmentation functions and by the inclusion of all thermal background eﬀects. In Sec. 5 we summarize our results and conclude.

2. Mass Varying Neutrinos (MaVaNs)

In Ref. [10] a new non-Standard Model interaction between neutrinos and a light ‘dark’ scalar ﬁeld, the so-called acceleron, was introduced. In essence, it serves as possible origin of the apparent accelerated expansion of the universe and promotes the CνB to a natural dark energy candidate. Furthermore, as a very interesting and intriguing secondary eﬀect, it causes a time evolution of neutrino masses.

A follow up publication [15] takes care of a possible stability problem of the model [16–18] and furnishes a viable model of the whole scenario.

Largely following Refs. [10,15], in this section we discuss the details of the complex interplay between the acceleron and the neutrinos that arises from a Yukawa coupling between them. Thereby, we will mainly focus on the determination of the resulting time variation of neutrino masses to be implemented later on in our analysis on relic neutrino absorption. For the latter it will turn out that the results are largely independent of the details of the model, since only a few generic features of the setting enter the

Probing Neutrino Dark Energy with Extremely High-Energy Cosmic Neutrinos

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investigation.

The new acceleron-neutrino interaction has a twofold eﬀect. On the one hand, as a

direct consequence, the neutrino masses mνi are generated from the vacuum expectation value (VEV) A of the acceleron and become functions of A, mνi(A), i = 1, 2, 3. On the other hand, the dependence of mνi on A turns the neutrino energy densities ρνi into implicit functions of A, since the energy densities ρνi (mνi(A)) depend on the masses mνi(A), i = 1, 2, 3. In this way, the energy density contained in a homogeneous background of neutrinos can stabilize the acceleron by contributing to its eﬀective potential Veﬀ(A). In other words, the dependence of the free energy on the value of A gets a contribution from the rest energy in neutrinos in addition to the pure scalar potential V0(A). The total energy density of the system parameterized by the eﬀective acceleron potential Veﬀ(A) takes the following form,

3

Veﬀ (A) = ρνi(mνi(A), z) + V0(A).

(2)

i=1

This is to be contrasted with the situation in empty space: if V0(A) is a ‘run-away

potential’, the acceleron does not possess a stable vacuum state but rolls to its state of

lowest energy given by the minimum of its pure potential V0(A).

Taking now the expansion of the universe into account, the dilution of the neutrino

energy densities ρνi(z) ∼ (1 + z)3 introduces a time dependence (here parameterized in

terms of the cosmic redshift z) into the eﬀective acceleron potential Veﬀ. Consequently,

in the adiabatic limit ‡, the equilibrium value A of the acceleron has to vary with time

in order to instantaneously minimize its eﬀective potential Veﬀ(A)§. Finally, as the

neutrino masses mνi(A) are directly aﬀected by changes in the A condensate, they are

promoted to dynamical quantities mνi(z) depending on cosmic time.

Note that Eq. (2) takes the neutrino energy density ρνi to be spatially constant.

To justify this assumption, the A condensate is not allowed to vary signiﬁcantly on

distances of the order of the inter-neutrino spacing r of the relic neutrinos, with currently

1/r ≃ 3361/3cm−1, where we have assumed a neutrino and anti-neutrino number density

of nν0,i = nν¯0,i ≃ 56 cm−3 per species i = 1, 2, 3. Consequently, remembering that the

range of the force mediated by a scalar ﬁeld is equal to its inverse mass, one arrives at

an upper bound on the A mass mA given by mA < 1/r ∼ O(10−4 eV) at the present

time.

Let us now determine the time evolution of the physical neutrino masses mνi(z).

Since the neutrino masses arise from the instantaneous equilibrium value A, we have

to analyze the minimum of the total energy density Veﬀ (A).

Assuming

∂mνi (A) ∂A

to be

‡ Under the assumption that the curvature scale of the potential is much larger than the expansion rate, ∂2Veﬀ (A)/∂A2 = m2A ≫ H2, the adiabatic solution to the equations of motion applies. In this case for |A| < MP l ≃ 3 × 1018 GeV the eﬀects of the kinetic energy terms can be safely ignored [10]. § Since therefore in the presence of the relic neutrinos the acceleron possesses a stable (time dependent)

vacuum state, in the literature both the acceleron and its VEV are referred to as A.

Probing Neutrino Dark Energy with Extremely High-Energy Cosmic Neutrinos

8

non-vanishing, one arrives at,

∂Veﬀ (A) = 3 ∂ρνi (mνi, z)

∂mνi(A) + ∂V0(A) = 0,

(3)

∂A i=1 ∂mνi mνi =mνi (A) ∂A ∂A

where [64]

∞

ρνi(mνi , z) = Tπν420 (1 + z)4 dy y2ey +y21+ x2i and (4)

0

xi

= mνi ,

(5)

Tν0 (1 + z)

with Tν0 = 1.69 × 10−4 eV denoting the neutrino temperature today. Note that the condition for the minimal energy density leads to a dependence of the neutrino masses

on the neutrino energy densities which evolve with z on cosmological time scales.

The smallness of the active neutrino masses mνi can be explained by letting them only indirectly feel the acceleron via the seesaw mechanism [11–14]. Therefore, following

Refs. [10,15], we introduce three ‘right-handed’ or ‘sterile’ neutrinos Ni with no Standard Model charges, whose masses MNi are constructed to vary with A due to a direct Yukawa interaction. In the seesaw mechanism the active neutrino masses mνi are functions of the sterile neutrino masses MNi(A). Consequently, the A dependence of the MNi(A) is transmitted to the active neutrino masses mνi(A) and causes them to change accordingly. Let us consider the interaction [10, 15]:

L ⊃ mDij Niνlj + κijNiNjA + h.c. + V0(A).

(6)

where i, j = 1, 2, 3 are the family-number indices and νli correspond to the left-handed active neutrinos. Furthermore, κA corresponds to the A dependent mass matrix of the sterile neutrinos and mD is the Dirac type matrix (originating from electroweak symmetry breaking). Assuming the eigenvalues of κA to be much larger than the eigenvalues of mD one can integrate out the sterile neutrinos Ni, arriving at the following eﬀective low energy Lagrangian [10, 15],

L

⊃ Mij(A) νliνlj + h.c. + V0(A), where

(7)

Mij (A) = (mTDκ−1mD)ij

(8)

A

represents the mass matrix of the active neutrinos.

In order to solve Eq. (3) for mνi(z) and to do MaVaN phenomenology the

fundamental scalar potential V0(A) has to be speciﬁed in an appropriate way. Namely,

the coupled neutrino acceleron ﬂuid has to act as a form of dark energy which is stable

against growth of inhomogeneities [16] and, as suggested by observations, must redshift

with an equation of state ω ∼ −1 today.

An appealing possibility arises in the framework of so-called hybrid models [65].

Those models were introduced to explain accelerated expansion in the context of

inﬂation. In essence, two light scalar ﬁelds interact in such a way that one of them

DEUTSCHES ELEKTRONEN-SYNCHROTRON

in der HELMHOLTZ-GEMEINSCHAFT

DESY 06-088 astro-ph/0606316 June 2006

(£*^

\X^*y

Probing Neutrino Dark Energy with Extremely High-Energy Cosmic Neutrinos

A. Ringwald, L. Schrempp

Deutsches Elektronen-Synchrotron DESY, Hamburg

ISSN 0418-9833

NOTKESTRASSE 85 - 22607 HAMBURG

DESY behält sich alle Rechte für den Fall der Schutzrechtserteilung und für die wirtschaftliche Verwertung der in diesem Bericht enthaltenen Informationen vor.

DESY reserves all rights for commercial use of information included in this report, especially in case of filing application for or grant of patents.

To be sure that your reports and preprints are promptly included in the HEP literature database

send them to (if possible by air mail):

DESY

Zentralbibliothek Notkestraße 85 22607 Hamburg Germany

DESY

Bibliothek Platanenallee 6 15738 Zeuthen Germany

arXiv:astro-ph/0606316 v1 13 Jun 2006

DESY-06-088

Probing Neutrino Dark Energy with Extremely High-Energy Cosmic Neutrinos

Andreas Ringwald and Lily Schrempp

Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg, Germany E-mail: [email protected], [email protected]

Abstract. Recently, a new non-Standard Model neutrino interaction mediated by a light scalar

ﬁeld was proposed, which renders the big-bang relic neutrinos of the cosmic neutrino background a natural dark energy candidate, the so-called Neutrino Dark Energy. As a further consequence of this interaction, the neutrino masses become functions of the neutrino energy densities and are thus promoted to dynamical, time/redshift dependent quantities. Such a possible neutrino mass variation introduces a redshift dependence into the resonance energies associated with the annihilation of extremely high-energy cosmic neutrinos on relic anti-neutrinos and vice versa into Z-bosons. In general, this annihilation process is expected to lead to sizeable absorption dips in the spectra to be observed on earth by neutrino observatories operating in the relevant energy region above 1013 GeV. In our analysis, we contrast the characteristic absorption features produced by constant and varying neutrino masses, including all thermal background eﬀects caused by the relic neutrino motion. We ﬁrstly consider neutrinos from astrophysical sources and secondly neutrinos originating from the decomposition of topological defects using the appropriate fragmentation functions. On the one hand, independent of the nature of neutrino masses, our results illustrate the discovery potential for the cosmic neutrino background by means of relic neutrino absorption spectroscopy. On the other hand, they allow to estimate the prospects for testing its possible interpretation as source of Neutrino Dark Energy within the next decade by the neutrino observatories ANITA and LOFAR.

Probing Neutrino Dark Energy with Extremely High-Energy Cosmic Neutrinos

2

1. Introduction

According to Big Bang Cosmology, in an expanding universe the freeze-out of a particle species occurs, when its interaction strength is too small to keep it in thermal equilibrium. Neutrinos, being the particles with the weakest known interactions, therefore are assumed to have already decoupled when the universe was just ≈ 1 s old, thereby guaranteeing a substantial relic neutrino abundance today with average number density nν0,i = nν¯0,i = 56 cm−3 per neutrino species i = 1, 2, 3. Yet, in turn, the weakness of the neutrino interactions so far has spoilt all attempts to probe this 1.95 K cosmic neutrino background (CνB), which is the analog of the 2.73 K cosmic microwave background (CMB) of photons, in a laboratory setting [1–4]. However, other cosmological measurements, such as the light element abundance, large scale structure (LSS) and the CMB anisotropies are sensitive to the presence of the CνB and therefore have provided us at least with indirect evidence for its existence (see e.g. Ref. [5] for a review).

Independently, Type Ia Supernova (SNIa) results (e.g. [6]), supported by CMB [7] and LSS data (e.g. Refs. [8, 9]), strongly suggest the existence of an exotic, smooth energy component with negative pressure, known as dark energy, which drives the apparent accelerated expansion of our universe. Recently, Fardon, Nelson and Weiner [10] have shown that the relic neutrinos, which constitute the CνB, are promoted to a natural dark energy candidate if they interact through a new non-Standard-Model scalar force – an idea which has great appeal. Neutrinos are the only Standard Model (SM) fermions without right-handed partners. Provided lepton number is violated, the active (left-handed) neutrinos are generally assumed to mix with a dark righthanded neutrino via the well-known seesaw mechanism [11–14], thus opening a window to the dark sector. Therefore, it would not seem to be surprising if neutrinos, whose interactions and properties we know comparably little about, were sensitive to further forces mediated by dark particles. Moreover, the scale relevant for neutrino mass squared diﬀerences as determined from neutrino oscillation experiments, δmν2 ∼ (10−2 eV)2, is of the order of the tiny scale associated with the dark energy, (2 × 10−3 eV)4.

As a consequence of the new interaction in such a scenario, an intricate interplay links the dynamics of the relic neutrinos and the mediator of the dark force, a light scalar ﬁeld called the acceleron. On the one hand, the neutrino masses mνi are generated by the vacuum expectation value A of the acceleron, mνi(A). Correspondingly, the A dependence of the masses mνi(A) is transmitted to the neutrino energy densities ρνi(mνi(A)) since these are functions of mνi(A). On the other hand, as a direct consequence, the neutrino energy densities ρνi(mνi, A) can stabilize the acceleron by contributing to its eﬀective potential Veﬀ (A, ρνi), which represents the total energy density of the coupled system. Moreover, cosmic expansion manifests itself in the dilution of the neutrino energy densities ρνi(z) ∼ (1 + z)3. Therefore, it crucially aﬀects the eﬀective acceleron potential Veﬀ (A, ρνi(mνi, z)) by introducing a dependence on cosmic time, here parameterized in terms of the cosmic redshift z. For a homogeneous

Probing Neutrino Dark Energy with Extremely High-Energy Cosmic Neutrinos

3

conﬁguration, the equilibrium value of the acceleron instantaneously minimizes its

eﬀective potential Veﬀ (A, ρνi(mνi, z)) and therefore also A(z) varies on cosmological time

scales. Finally, in turn, since the neutrino masses mνi(A) are sensitive to changes in A,

they are promoted to dynamical quantities depending on z, mνi(z), that is depending

on cosmic time. To summarize, the variation of the neutrino masses represents a clear

signature of the so-called Neutrino Dark Energy scenario.

In a subsequent work Fardon, Nelson and Weiner [15] presented a supersymmetric

Mass Varying Neutrino (MaVaN) model in which the origin of dark energy was

attributed to the lightest neutrino ν1 and the size of the dark energy could be expressed

in terms of neutrino mass parameters. By naturalness arguments the authors concluded

that the lightest neutrino still has to be relativistic today, thereby preventing potential

instabilities [16–18] which could occur in highly non-relativistic theories of Neutrino

Dark Energy.

The rich phenomenology of the MaVaN scenario has been explored by many authors.

The cosmological eﬀects of varying neutrino masses have been studied in Refs. [19, 20]

and were elaborated in the context of gamma ray bursts [21]. Apart from the time

variation, the conjectured new scalar forces between neutrinos as well as the additional

possibility of small acceleron couplings to matter lead to an environment dependence of

the neutrino masses governed by the local neutrino and matter density [10, 22, 23]. The

consequences for neutrino oscillations in general were exploited in Refs. [22, 24] and in

particular in the sun [25–27], in reactor experiments [26, 28] as well as in long-baseline

experiments [29].

In light of the possible realization of Neutrino Dark Energy in nature, a (more)

direct detection of the CνB should be thoroughly explored with special attention turned

to possible new physics beyond the SM. By this means, a time evolution of neutrino

masses could be revealed which would serve as a test of Neutrino Dark Energy. In

addition, the general importance of a (more) direct evidence for the existence of the

CνB lies in a conﬁrmation of standard cosmology back to the freeze-out of the weak

interactions and therefore thirteen orders of magnitudes before the time when photons

where imprinted on the last scattering surface.

An appealing opportunity to catch a glimpse of the CνB as it is today emerges

from the possible existence of extremely high-energy cosmic neutrinos (EHECν’s). Such

EHECν’s can annihilate with relic anti-neutrinos (and vice versa) into Z bosons, if their

energies coincide with the respective resonance energies E0re,is of the corresponding process νν¯ → Z [30–37]. These energies,

Eres = MZ2 = 4.2 × 1012 eV GeV

(1)

0,i 2mν0,i

mνi

in the rest system of the relic neutrinos, are entirely determined by the Z boson mass

MZ as well as the respective neutrino masses mνi. An exceptional loss of transparency of

the CνB for cosmic neutrinos results from the fact that the corresponding annihilation

cross-section on resonance is enhanced by several orders of magnitude with respect to

non-resonant scattering. As a consequence, the diﬀuse EHECν ﬂux arriving at earth is

Probing Neutrino Dark Energy with Extremely High-Energy Cosmic Neutrinos

4

expected to exhibit absorption dips whose locations in the spectrum are determined by the respective resonance energies of the annihilation processes. Provided that the dips can be resolved on earth, they could produce the most direct evidence for the existence of the CνB so far. Furthermore, as indicated by the resonance energies in Eq. (1), the absorption features depend on the magnitude of the neutrino masses and could therefore reﬂect their possible variation with time. Moreover, they are sensitive to the ﬂavor composition of the neutrino mass eigenstates as well as to various cosmological parameters. Accordingly, the possibility opens up to perform relic neutrino absorption spectroscopy as an independent means to probe neutrino physics and cosmology.

The existence of EHECν’s is theoretically well motivated and is substantiated by numerous works on possible EHECν sources of astrophysical nature (bottom-up) (see e.g. [38] for a review) or so-called top-down sources (see e.g. Ref. [39] for a review). In the latter case, EHECν’s with energies well above 1011 GeV are assumed to be produced in the decomposition of topological defects (TD’s) which originate from symmetry breaking phase transitions in the very early universe.

Depending on the underlying EHECν sources the EHECν ﬂuxes could be close to the current observational bounds set by existing EHECν observatories such as AMANDA [40] (see also Ref. [41]), ANITA-lite [42], BAIKAL [55], FORTE [43], GLUE [44] and RICE [45] which cover an energy range of 107 GeV < E0 < 1017 GeV (cf. Fig. 1). Promisingly, the sensitivity in this energy range will be improved by orders of magnitude (cf. Fig. 1) by larger EHECν detectors such as ANITA, EUSO [46], IceCube [47], LOFAR [48], OWL [49], SalSA [51] and WRST [48] which are planned to start operating within the next decade (cf. Fig. 1). Accordingly, the prospects of conﬁrming the existence of the CνB by tracking its interaction with EHECν’s have substantially improved since the original proposal in 1982 [30]. Moreover, in the likely case of appreciable event samples the valuable information encoded in the absorption features of the EHECν spectra could be revealed within the next decade (cf. Fig. 1), rendering the theoretical exploration of relic neutrino absorption spectroscopy a timely enterprise.

Note that the scenario introduced above has also attracted attention for another reason than the possible detection of the CνB– namely for the controversial possibility of solving the so-called GZK-puzzle to be discussed brieﬂy in the following. Beyond the Greisen-Zatsepin-Kuzmin (GZK) energy, EGZK = 4 × 1010 GeV, ultra-high energy nucleons rapidly lose energy due to the eﬀective interaction with CMB photons (predominantly through resonant photo-pion production) [56, 57]. In the so-called Zburst scenario, the observed mysterious cosmic rays above EGZK were associated with the secondary cosmic ray particles produced in the decays of Z bosons. The latter were assumed to originate from the neutrino annihilation process outlined above [58–63].

However, recently, ANITA-lite [42] appears to have entirely excluded the Z-burst explanation for the GZK-puzzle at a level required to account for the observed ﬂuxes of the highest energy cosmic rays. We would like to stress, that this only means that the GZK-puzzle stays unsolved (if there is any) . Moreover, this neither restricts the possible

Probing Neutrino Dark Energy with Extremely High-Energy Cosmic Neutrinos

5

Figure 1. Current status and next decade prospects for EHECν physics, expressed in terms of diﬀuse neutrino ﬂuxes per ﬂavor, F = Fνα + Fν¯α , α = e, µ, τ . The upper limits from AMANDA [40], see also Ref. [41], ANITA-lite [42], FORTE [43], GLUE [44], and RICE [45] are plotted. Also shown are projected sensitivities of ANITA [42], EUSO [46], IceCube [47], LOFAR [48], OWL [49], the Pierre Auger Observatory in νe, νµ modes and in ντ mode (bottom swath) [50], SalSA [51], and WSRT [48], corresponding to one event per energy decade and indicated duration. Also shown are predictions from astrophysical Cosmic Ray (CR) sources [52], from inelastic interactions of CR’s with the cosmic microwave background (CMB) photons (cosmogenic neutrinos) [52, 53], and from topological defects [54].

success of producing evidence for the CνB by means of detecting absorption dips in the EHECν spectra nor does it spoil the possibility of gaining valuable information from performing relic neutrino absorption spectroscopy.

The goal of this paper is to carefully work out the characteristic diﬀerences in the EHECν absorption features which result from treating the neutrino masses as time varying dynamical quantities in comparison to constants. In our analysis, we incorporate the full thermal background eﬀects on the absorption process whose impact grows for smaller neutrino masses [36, 37]. This means, that in general relic neutrinos cannot be assumed to be at rest. Instead, they have to be treated as moving targets with a momentum distribution, if their mean momenta turn out to be of the order of the relic neutrino masses.

We illustrate our results for the diﬀuse neutrino ﬂuxes to be observed at earth ﬁrstly by considering astrophysical EHECν sources. Secondly, we calculate the neutrino spectrum (both for varying and constant neutrino masses) expected from the

Probing Neutrino Dark Energy with Extremely High-Energy Cosmic Neutrinos

6

decomposition of a topological defect using the appropriate fragmentation functions and including the full thermal background eﬀects. By this means, for a given EHECν injection spectrum, we present all technical tools to interpret EHECν absorption dips as soon as they are observed at earth. Thereby the possibility opens up to test the CνB and its interpretation as source of Neutrino Dark Energy.

The paper is organized as follows. In Sec. 1 we discuss the MaVaN scenario and focus on a determination of the time dependence of the neutrino masses. Sec. 2 reviews the treatment of the absorption of an EHECν by the CνB in terms of the damping factor comprising the thermal background eﬀects. Furthermore, in order to make contact to treatments in the literature [30, 31, 33, 35–37], we include in the discussion common approximations [33,35,36] which neglect part or all of the dependence of the damping on the relic neutrino momenta. Moreover, we extend the complete analysis to incorporate a possible variation of the neutrino masses with time. In Sec. 3 we present and compare our results for the survival probabilities of EHECν’s with varying and constant masses which encode the physical information on all possible annihilation processes on their way from their source to us, again taking into account the thermal motion of the relic neutrinos. In order to gain more physical insight, in addition, we disentangle the characteristic features of the absorption dips caused by the mass variation by switching oﬀ all thermal background eﬀects. Sec. 4 illustrates the discovery potential of neutrino observatories for the CνB and gives an outlook for the testability of the MaVaN scenario. Therefore, both for astrophysical sources and for a topological defect scenario, we calculate the expected observable EHECν ﬂux arriving at earth which results from folding the survival probabilities with the corresponding EHECν source emissivity distribution. In the latter case, we perform the full state-of-the-art calculation with the help of fragmentation functions and by the inclusion of all thermal background eﬀects. In Sec. 5 we summarize our results and conclude.

2. Mass Varying Neutrinos (MaVaNs)

In Ref. [10] a new non-Standard Model interaction between neutrinos and a light ‘dark’ scalar ﬁeld, the so-called acceleron, was introduced. In essence, it serves as possible origin of the apparent accelerated expansion of the universe and promotes the CνB to a natural dark energy candidate. Furthermore, as a very interesting and intriguing secondary eﬀect, it causes a time evolution of neutrino masses.

A follow up publication [15] takes care of a possible stability problem of the model [16–18] and furnishes a viable model of the whole scenario.

Largely following Refs. [10,15], in this section we discuss the details of the complex interplay between the acceleron and the neutrinos that arises from a Yukawa coupling between them. Thereby, we will mainly focus on the determination of the resulting time variation of neutrino masses to be implemented later on in our analysis on relic neutrino absorption. For the latter it will turn out that the results are largely independent of the details of the model, since only a few generic features of the setting enter the

Probing Neutrino Dark Energy with Extremely High-Energy Cosmic Neutrinos

7

investigation.

The new acceleron-neutrino interaction has a twofold eﬀect. On the one hand, as a

direct consequence, the neutrino masses mνi are generated from the vacuum expectation value (VEV) A of the acceleron and become functions of A, mνi(A), i = 1, 2, 3. On the other hand, the dependence of mνi on A turns the neutrino energy densities ρνi into implicit functions of A, since the energy densities ρνi (mνi(A)) depend on the masses mνi(A), i = 1, 2, 3. In this way, the energy density contained in a homogeneous background of neutrinos can stabilize the acceleron by contributing to its eﬀective potential Veﬀ(A). In other words, the dependence of the free energy on the value of A gets a contribution from the rest energy in neutrinos in addition to the pure scalar potential V0(A). The total energy density of the system parameterized by the eﬀective acceleron potential Veﬀ(A) takes the following form,

3

Veﬀ (A) = ρνi(mνi(A), z) + V0(A).

(2)

i=1

This is to be contrasted with the situation in empty space: if V0(A) is a ‘run-away

potential’, the acceleron does not possess a stable vacuum state but rolls to its state of

lowest energy given by the minimum of its pure potential V0(A).

Taking now the expansion of the universe into account, the dilution of the neutrino

energy densities ρνi(z) ∼ (1 + z)3 introduces a time dependence (here parameterized in

terms of the cosmic redshift z) into the eﬀective acceleron potential Veﬀ. Consequently,

in the adiabatic limit ‡, the equilibrium value A of the acceleron has to vary with time

in order to instantaneously minimize its eﬀective potential Veﬀ(A)§. Finally, as the

neutrino masses mνi(A) are directly aﬀected by changes in the A condensate, they are

promoted to dynamical quantities mνi(z) depending on cosmic time.

Note that Eq. (2) takes the neutrino energy density ρνi to be spatially constant.

To justify this assumption, the A condensate is not allowed to vary signiﬁcantly on

distances of the order of the inter-neutrino spacing r of the relic neutrinos, with currently

1/r ≃ 3361/3cm−1, where we have assumed a neutrino and anti-neutrino number density

of nν0,i = nν¯0,i ≃ 56 cm−3 per species i = 1, 2, 3. Consequently, remembering that the

range of the force mediated by a scalar ﬁeld is equal to its inverse mass, one arrives at

an upper bound on the A mass mA given by mA < 1/r ∼ O(10−4 eV) at the present

time.

Let us now determine the time evolution of the physical neutrino masses mνi(z).

Since the neutrino masses arise from the instantaneous equilibrium value A, we have

to analyze the minimum of the total energy density Veﬀ (A).

Assuming

∂mνi (A) ∂A

to be

‡ Under the assumption that the curvature scale of the potential is much larger than the expansion rate, ∂2Veﬀ (A)/∂A2 = m2A ≫ H2, the adiabatic solution to the equations of motion applies. In this case for |A| < MP l ≃ 3 × 1018 GeV the eﬀects of the kinetic energy terms can be safely ignored [10]. § Since therefore in the presence of the relic neutrinos the acceleron possesses a stable (time dependent)

vacuum state, in the literature both the acceleron and its VEV are referred to as A.

Probing Neutrino Dark Energy with Extremely High-Energy Cosmic Neutrinos

8

non-vanishing, one arrives at,

∂Veﬀ (A) = 3 ∂ρνi (mνi, z)

∂mνi(A) + ∂V0(A) = 0,

(3)

∂A i=1 ∂mνi mνi =mνi (A) ∂A ∂A

where [64]

∞

ρνi(mνi , z) = Tπν420 (1 + z)4 dy y2ey +y21+ x2i and (4)

0

xi

= mνi ,

(5)

Tν0 (1 + z)

with Tν0 = 1.69 × 10−4 eV denoting the neutrino temperature today. Note that the condition for the minimal energy density leads to a dependence of the neutrino masses

on the neutrino energy densities which evolve with z on cosmological time scales.

The smallness of the active neutrino masses mνi can be explained by letting them only indirectly feel the acceleron via the seesaw mechanism [11–14]. Therefore, following

Refs. [10,15], we introduce three ‘right-handed’ or ‘sterile’ neutrinos Ni with no Standard Model charges, whose masses MNi are constructed to vary with A due to a direct Yukawa interaction. In the seesaw mechanism the active neutrino masses mνi are functions of the sterile neutrino masses MNi(A). Consequently, the A dependence of the MNi(A) is transmitted to the active neutrino masses mνi(A) and causes them to change accordingly. Let us consider the interaction [10, 15]:

L ⊃ mDij Niνlj + κijNiNjA + h.c. + V0(A).

(6)

where i, j = 1, 2, 3 are the family-number indices and νli correspond to the left-handed active neutrinos. Furthermore, κA corresponds to the A dependent mass matrix of the sterile neutrinos and mD is the Dirac type matrix (originating from electroweak symmetry breaking). Assuming the eigenvalues of κA to be much larger than the eigenvalues of mD one can integrate out the sterile neutrinos Ni, arriving at the following eﬀective low energy Lagrangian [10, 15],

L

⊃ Mij(A) νliνlj + h.c. + V0(A), where

(7)

Mij (A) = (mTDκ−1mD)ij

(8)

A

represents the mass matrix of the active neutrinos.

In order to solve Eq. (3) for mνi(z) and to do MaVaN phenomenology the

fundamental scalar potential V0(A) has to be speciﬁed in an appropriate way. Namely,

the coupled neutrino acceleron ﬂuid has to act as a form of dark energy which is stable

against growth of inhomogeneities [16] and, as suggested by observations, must redshift

with an equation of state ω ∼ −1 today.

An appealing possibility arises in the framework of so-called hybrid models [65].

Those models were introduced to explain accelerated expansion in the context of

inﬂation. In essence, two light scalar ﬁelds interact in such a way that one of them