MPI-PhT/2001-52, SISSA 9/2002/EP
Cosmological bounds on neutrino degeneracy improved by ?avor oscillations
arXiv:hep-ph/0201287v2 8 Apr 2002
A.D. Dolgov a,1, S.H. Hansen b , S. Pastor c, S.T. Petcov d,2, G.G. Ra?elt c, D.V. Semikoz c,3
section of Ferrara, Via del Paradiso 12, 44100 Ferrara, Italy University of Oxford, Keble road, OX1 3RH, Oxford, UK f¨ ur Physik, F¨ ohringer Ring 6, 80805 M¨ unchen, Germany
c Max-Planck-Institut d Scuola
Internazionale Superiore di Studi Avanzati and INFN section of Trieste, Via Beirut 2–4, 34014 Trieste, Italy
Abstract We study three-?avor neutrino oscillations in the early universe in the presence of neutrino chemical potentials. We take into account all e?ects from the background medium, i.e. collisional damping, the refractive e?ects from charged leptons, and in particular neutrino self-interactions that synchronize the neutrino oscillations. We ?nd that e?ective ?avor equilibrium between all active neutrino species is established well before the big-bang nucleosynthesis (BBN) epoch if the neutrino oscillation parameters are in the range indicated by the atmospheric neutrino data and by the large mixing angle (LMA) MSW solution of the solar neutrino problem. For the other solutions of the solar neutrino problem, partial ?avor equilibrium may be achieved if the angle θ13 is close to the experimental limit tan2 θ13 < ? 0.065. In the LMA case, the BBN limit on the νe degeneracy parameter, |ξν | < 0 . 07, now ap? plies to all ?avors. Therefore, a putative extra cosmic radiation contribution from degenerate neutrinos is limited to such low values that it is neither observable in the large-scale structure of the universe nor in the anisotropies of the cosmic microwave background radiation. Existing limits and possible future measurements, for example in KATRIN, of the absolute neutrino mass scale will provide unambiguous information on the cosmic neutrino mass density, essentially free of the uncertainty of the neutrino chemical potentials.
Key words: Physics of the Early Universe; Neutrino Physics
Preprint submitted to Elsevier Preprint
1 February 2008
The cosmic matter and radiation inventory is known with ever increasing precision, but many important questions remain open. The cosmic neutrino background is a generic feature of the standard hot big bang model, and its presence is indirectly established by the accurate agreement between the calculated and observed primordial light-element abundances. However, the exact neutrino number density is not known as it depends on the unknown chemical potentials for the three ?avors. (In addition there could be a population of sterile neutrinos, a hypothesis that we will not discuss here.) The standard assumption is that the asymmetry between neutrinos and anti-neutrinos is of ?10 . This would be order the baryon asymmetry η ≡ (nB ? nB ? )/nγ ? 6 × 10 the case, for example, if B ? L = 0 where B and L are the cosmic baryon and lepton asymmetries, respectively. While B ? L = 0 is motivated by scenarios where the baryon asymmetry is obtained via leptogenesis , there are models for producing large L and small B [2–5]. In order to quantify a putative neutrino asymmetry we assume that well before thermal neutrino decoupling a given ?avor is characterized by a Fermi-Dirac distribution with a chemical potential ?ν , fν (p, T ) = [exp (Ep /T ? ξν ) + 1]?1 , where ξν ≡ ?ν /T is the degeneracy parameter and Ep ? p since we may neglect small neutrino mass e?ects on the distribution function. For anti-neutrinos the distribution function is given by ξν ? = ?ξν . A neutrino chemical potential modi?es the outcome of primordial nucleosynthesis in two di?erent ways . The ?rst e?ect appears only in the electron sector because electron neutrinos participate in the beta processes which determine the primordial neutron-to-proton ratio so that n/p ∝ exp(?ξe ). 4 Therefore, a positive ξe decreases Yp , the primordial 4 He mass fraction, while a negative ξe increases it, leading to an allowed range ? 0.01 < ξe < 0.07 , (1)
compatible with ξe = 0 (see Refs. [7–10] and Sec. 5). A second e?ect is an increase of the neutrino energy density for any non-zero ξ which in turn increases the expansion rate of the universe, thus enhancing Yp . This applies to
Also at: ITEP, Bol. Cheremushkinskaya 25, Moscow 117259, Russia. Also at: Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 1784 So?a, Bulgaria. 3 Also at: Institute for Nuclear Research of the Academy of Sciences of Russia, 60th October Anniversary Prospect 7a, Moscow 117312, Russia. 4 We use the notation ξ ≡ ξ e νe etc. to avoid double subscripts. We never discuss charged-lepton chemical potentials so that there should be no confusion.
all ?avors so that the e?ect of chemical potentials in the ν? or ντ sector can be compensated by a positive ξe . Altogether the big-bang nucleosynthesis (BBN) limits on the neutrino chemical potentials are thus not very restrictive. Another consequence of the extra radiation density in degenerate neutrinos is that it postpones the epoch of matter-radiation equality. In the cosmic microwave background radiation (CMBR) it boosts the amplitude of the ?rst acoustic peak of the angular power spectrum and shifts all peaks to smaller scales. Moreover, the power spectrum of density ?uctuations on small scales is suppressed [11,12], leading to observable e?ects in the cosmic large-scale structure (LSS). A recent analysis of the combined e?ect of a non-zero neutrino asymmetry on BBN and CMBR/LSS yields the allowed regions  ? 0.01 < ξe < 0.22, |ξ?,τ | < 2.6, (2)
in agreement with similar bounds in [14,15]. These limits allow for a very signi?cant radiation contribution of degenerate neutrinos, leading many authors to discuss the implications of a large neutrino asymmetry in di?erent physical situations. These include the explanation of the former discrepancy between the BBN and CMBR results on the baryon asymmetry  or the origin of the cosmic rays with energies in excess of the Greisen-Zatsepin-Kuzmin cuto? . In addition, if the present relic neutrino background is strongly degenerate, it would enhance the contribution of massive neutrinos to the total energy density [18,19] and a?ect the ?avor oscillations of the high-energy neutrinos  which are thought to be produced in the astrophysical accelerators of high-energy cosmic rays. The limits in Eq. (2) ignore neutrino ?avor oscillations, an assumption which is no longer justi?ed in view of the experimental signatures for neutrino oscillations by solar and atmospheric neutrinos. For zero initial neutrino chemical potentials, the ?avor neutrinos have the same spectra so that oscillations produce no e?ect. This is true up to a small spectral distortion caused by the heating of neutrinos from e+ e? annihilations, an e?ect which is di?erent for electron and muon/tau neutrinos and which causes a small relative change in the ?nal production of 4 He of order 10?3 . This relative change is slightly enhanced by neutrino ?avor oscillations [22,23]. In the presence of neutrino asymmetries, ?avor oscillations equalize the neutrino chemical potentials if there is enough time for this relaxation process to be e?ective . If ?avor equilibrium is reached before BBN, then the restrictive limits on ξe in Eq. (1) will apply to all ?avors, in turn implying that the cosmic neutrino radiation density is close to its standard value. As a consequence, it is no longer necessary to use the neutrino radiation density as a ?t parameter for CMBR/LSS analyses, unless one considers exotic models with decaying massive particles. 3
The e?ects of ?avor oscillations on possible neutrino degeneracies have been considered in , where it was concluded that ?avor equilibrium was achieved before the BBN epoch if the solar neutrino problem was explained by the largemixing angle (LMA) solution. The LMA solution is favored by the current solar neutrino data. Thus, it was concluded that in the LMA case a large cosmic neutrino degeneracy was no longer allowed. We revisit this problem because the ?avor evolution of the neutrino ensemble is more subtle than previously envisaged if medium e?ects are systematically included. Contrary to the treatment of Ref. , the refractive e?ect of charged leptons can not be ignored, and actually is one of the dominant e?ects. While the background neutrinos produce an even larger refractive term, its e?ect is to synchronize the neutrino oscillations [25,26] which remain sensitive to the charged-lepton contribution. Still, equilibrium is essentially, but not completely, achieved in the LMA case so that our ?nal conclusion is qualitatively similar to that of Ref. . One counter-intuitive subtlety is that the neutrino self-potential actually can suppress oscillations in a situation where the excess of neutrinos in one ?avor is exactly matched by an excess of anti-neutrinos in another ?avor. In this case the synchronized oscillation frequency is zero so that oscillations begin only once the cosmic expansion has diluted the self-term. Therefore, one can construct cases where ?avor equilibrium is not achieved before BBN even in the LMA case. However, this is only possible for specially chosen initial conditions where |ξν | is equal for all ?avors, but the absolute signs may be di?erent. For the purpose of deriving limits on |ξν |, however, this case is equivalent to the one where equilibrium is achieved, the only important point being that |ξν | is approximately equal for all ?avors at the BBN epoch. Another subtlety appears in a full three-?avor analysis. We show that achieving equilibrium in the LMA case does not depend on the value of the mixing angle θ13 , to which strict limits from reactor experiments apply. Moreover, for the non-LMA solutions of the solar neutrino problem, partial ?avor equilibrium may be reached if the angle θ13 is small but non-zero. In Sec. 2 we set up the formalism to study primordial ?avor oscillations. We then turn in Sec. 3 to the primordial ?avor evolution of a simpli?ed system where νe mixes maximally with one other ?avor. This two-?avor case will illustrate many of the important subtleties of our problem. Then we turn in Sec. 4 to realistic three-?avor situations which involve yet further complications. In Sec. 5 we ?nally derive new limits on the degeneracy parameters and summarize our ?ndings.
Neutrino ?avor oscillations in the early universe
In order to study neutrino oscillations in the early universe we characterize the neutrino ensemble in the usual way by generalized occupation numbers, i.e. by 3×3 density matrices for neutrinos and anti-neutrinos as described in [27,28]. The form of the density matrices for a mode with momentum p is ρee ? ρ(p, t) = ? ρ?e ρτ e
ρe? ρ?? ρτ ?
ρeτ ? ρ?τ ? , ρτ τ
ρ ?ee ? ?e? ρ ?(p, t) = ? ρ ρ ?eτ
ρ ??e ρ ??? ρ ??τ
ρ ?τ e ? ρ ?τ ? ? ρ ?τ τ
where overbarred quantities refer to anti-neutrinos. The diagonal elements of the density matrices correspond to the usual occupation numbers of the di?erent ?avors. The de?nition of the density matrix for anti-neutrinos is “transposed” relative to that for neutrinos, allowing one to write the equations of motion in a compact form . The equations of motion for the density matrices relevant for our situation of interest are [27,29] √ M2 , ρp + 2GF 2p √ M2 i?t ρ ?p = ? ,ρ ?p + 2GF 2p 8p E+ρ?ρ ? , ρp + C [ρp ] , 3m2 W 8p ? ,ρ ?p + C [? ρp ] , + 2 E+ρ?ρ 3mW
i?t ρp = +
where we use the notation ρp = ρ(p, t) and [·, ·] is the usual commutator. Further, M 2 is the mass-squared matrix in the ?avor basis; in the mass basis 2 2 it would be diag(m2 1 , m2 , m3 ). The diagonal matrix E represents the energy densities of charged leptons. For example, Eee is the energy density of electrons plus that of positrons. The density matrix ρ is the integrated neutrino density matrix so that, for example, ρee is the νe number density while ρ ?ee is the ν ?e number density. The term proportional to (ρ ? ρ ?) is non-linear in the neutrino density matrix and represents self-interactions. Finally, C [·] is the collision term which is proportional to G2 F . We have neglected a refractive term proportional to the neutrino energy density which is much smaller than the (ρ ? ρ ?) term in our present situation where the neutrino asymmetries are assumed to be large. On the other hand, we have neglected the usual term which is proportional to the charged-lepton asymmetries. This asymmetric term is always negligible; at early times (high temperatures) it is negligible compared to the E term, while at temperatures near n/p freeze out (T ? 1 MeV) it is negligible compared to the vacuum term M 2 /2p if the mass-squared differences coincide with those characterizing the atmospheric neutrino and the solar neutrino LMA oscillations. 5
In the expanding universe we need to substitute ?t → ?t ? Hp?p with H the cosmic expansion parameter. We have taken this into account rewriting the equations in terms of comoving variables, as listed in the Appendix. We solve these equations numerically, calculating the evolution of ρ and ρ ? on a grid of neutrino momenta.
3.1 Equations of motion for the polarization vectors In order to develop a ?rst understanding of ?avor oscillation in the early universe it will be useful to study ?rst a two-?avor situation involving νe and ν? . The usual relation between ?avor and mass eigenstates is νe = cos θ ν1 + sin θ ν2 , ν? = ? sin θ ν1 + cos θ ν2 . The 2×2 density matrices are ρ(p, t) = ρ ?(p, t) = 1 ρee ρe? = [P0 (p, t) + σ · P(p, t)] , ρ?e ρ?? 2 1 ρ ?ee ρ ??e P 0 (p, t) + σ · P(p, t) . = ρ ?e? ρ ??? 2
Here, σi are the Pauli matrices while P(p, t) and P(p, t) are the usual polarization vectors for the neutrino and anti-neutrino modes p, respectively. We normalize ρ and ρ ? to a Fermi-Dirac distribution with zero chemical potential, f (p) = [exp (p/T ) + 1]?1 , so that for instance fνa (p, t) = ρaa (p, t)f (p). For the polarization vectors, the equations of motion are given by the usual spin-precession formula √ ?m2 8 2 GF p ?t Pp = + Eee ? z × Pp B? 2p 3m2 W √ + 2 GF (P ? P) × Pp + C [Pp ] , √ ?m2 8 2 GF p ? × Pp ?t Pp = ? Eee z B? 2p 3m2 W √ + 2 GF (P ? P) × Pp + C [Pp ] , (7)
2 where ?m2 = m2 2 ? m1 and B = (sin 2θ, 0, ? cos 2θ ) with θ the vacuum mixing angle. Further, ? z is a unit vector in the z -direction, Eee the electron-positron
energy density, and P and P are the integrated polarization vectors. We will see that a detailed treatment of the collision terms is not crucial. Therefore, we approximate them with a simple damping prescription of the form C [Pp ] = ?Dp PT p (8)
where the transverse part PT p consists of the x-y -projection of Pp . We use the damping functions as, for instance, in [28,30], but with slightly modi?ed coe?cients. We neglect a small dependence on ξ , i.e. we use the same damping functions for neutrinos and antineutrinos. We have checked that our results are ˙ as given ˙0 and P insensitive to the inclusion of the repopulation function, i.e. P 0 for instance in [28,30]. Moreover, we did not include the neutrino heating from e+ e? annihilations, since it is small even in the presence of degeneracies .
3.2 Evolution of simple ?avor asymmetry As a ?rst case we consider a simple situation where initially electron neutrinos do not have a chemical potential (ξe = 0) while muon neutrinos are asymmetric (ξ? = ?0.05). We use a rather small ν? asymmetry so that the anticipated equilibrium state ξe = ξ? will be close to the BBN limit on ξe in Eq. (2). The quantitative evolution with a large ξ? is the same. Moreover, we use maximal mixing and ?m2 = 4.5 × 10?5 eV2 as suggested by the LMA solution of the solar neutrino problem. In order to illustrate the relative importance of the various contributions to the equation of motion we ?rst calculate the primordial evolution of the integrated polarization vectors P and P for pure vacuum oscillations, including only the ?m2 /2p terms in Eq. (7), without any medium e?ects whatsoever. Our initial conditions ξe = 0 and ξ? = ?0.05 imply that there are more νe than ν? , i.e. P points initially in the +? z direction, while there are more ν ?? than ν ?e , i.e. P ? direction. In the upper panel of Fig. 1 we show the points initially in the ?z evolution of Pz as a function of cosmic temperature; the evolution of P z is similar, except with a negative initial value. The oscillations begin when the expansion rate has become slow enough at about T ? 30 MeV. The oscillation frequencies are di?erent for di?erent modes, leading to quick decoherence and thus to an incoherent equal mix of both ?avors. Evidently ?avor equilibrium is reached long before n/p freeze-out at T ? 1 MeV. As a next step we include the neutrino self-potential proportional to P ? P. The evolution is shown as a dotted line in the upper panel of Fig. 1. It is now dominated by the e?ect of synchronized oscillations, i.e. the self-potential 7
forces all neutrino modes to follow the same oscillation . Because P and P point in opposite directions, the common or synchronized oscillation frequency ωsynch is very much smaller than a typical ?m2 /2p. Explicitly we have  ωsynch = 1 |I| ?m2 ? d3 p ?m2 I · ( P + f ( p ) P ) ≡ p p (2π )3 2p 2pe? (9)
where I ≡ P ? P and ? I is a unit vector in the direction of I. Numerically our initial conditions imply pe? ? 132 T . Next we add to the vacuum term the e?ective potential caused by the e± background [see Eq. (4)], without including the self-term. The e± potential always points in the z -direction, thereby suppressing ?avor oscillations in the usual way. This e?ect is shown in the lower panel of Fig. 1 (solid line). P stays frozen at its initial value up to T ? 3 MeV where the medium potential becomes unimportant compared to the vacuum term. At these temperatures collisions are also about to become unimportant. Therefore it is not surprising that including collisions (dotted line) does not dramatically change the evolution of Pz . However, this impression is misleading as can be seen from Fig. 2 where we show the evolution of Px for the same cases as in the lower panel of Fig. 1. Without collisions (solid line) the evolution of P is a simple turning from the z - to the x-direction. The e± potential adiabatically disappears, leading to the usual MSW-type evolution. In this case the ?nal result corresponds to equal numbers of both ?avors, yet a coherent ?avor superposition. With collisional damping (dotted line) the x-component is damped, leaving little ?avor coherence of the ?nal state. Therefore, with or without collisional damping we reach “?avor equilibrium” in the sense of equal densities of both ?avors, but only collisions ensure the damping of the otherwise large transverse part of the ?nal P. Finally we study how oscillations evolve in the presence of all e?ects: vacuum, e± background, neutrino self-potential and collisions. The results are shown in Fig. 3 for di?erent strengths of the self-potential. In one run it was switched o? (“No Self”), it was taken at full strength (“Self”), or it was suppressed by a factor 104 or 105 . When the self-interaction term is close to its real value (solid line), again all modes oscillate in synch so that we see a combination of the synchronized oscillations and the MSW-like e?ect of the background medium. However, collisions ensure that no large transverse component survives—see the corresponding evolution of Px and Py in Fig. 4. Almost perfect ?avor equilibrium is achieved with or without self-interactions. This conclusion is not to be taken for granted. When the self-interaction term is arti?cially adjusted to be of comparable strength to the other terms (vacuum 8
or e± background), the evolution shows complicated features (Fig. 3). After some initial conversion, Pz remains constant for a long time, only to reach equilibrium at a much later time. This behavior is not easy to understand as it depends on all the ingredients of our calculation, i.e. the background potential, the neutrino self-potential and damping by collisions. We believe that the long phase where the polarization vector essentially stands still is caused by the synchronized oscillation frequency becoming very small. This is strictly the case in a situation where the chemical potentials for two ?avors are equal but opposite—see Sec. 3.3 below. We believe that in the present case a combination of decoherence of some of the modes and collisions drives the system into a state where the synchronized oscillations frequency vanishes so that oscillations stop until the cosmic expansion has diluted the neutrino self-term enough to eliminate this e?ect.
3.3 Equal but opposite asymmetries
Our discussion thus far suggests that for the parameters of interest it is generic to achieve ?avor equilibrium before the BBN epoch, even though the loss of coherence may not always be complete. However, this conclusion depends on the simple initial conditions that we have used thus far. The di?erent neutrino ?avors may show large asymmetries, yet the total cosmic lepton asymmetry could still be small if initially ξe = ?ξ? , which corresponds to conservation of the lepton number Le ?L? . These initial conditions imply that P = ?P so that the self-potential term (P ? P) is large while for each mode (Pp + Pp ) = 0 . This implies, in turn, that the synchronized oscillation frequency ωsynch = 0. Therefore, in this case the synchronization e?ect caused by the self-potential prevents ?avor oscillations entirely, at least until the self-term becomes weak, long after the BBN epoch. While ?avor equilibrium is not achieved in this case, from the start we have |ξe | = |ξ? | so that the BBN limits on |ξe | would apply to both ?avors.
4.1 Oscillation parameters
We now turn to the more interesting case of three-?avor neutrino oscillations. The neutrino ?avor eigenstates νe , ν? , and ντ are related to the mass eigen9
states via the mixing matrix
? ? ? ? ? ?s12 c23 ? ?
s12 s23 ? c12 c23 s13 ?c12 s23 ? s12 c23 s13 c23 c13
. ? c12 s23 s13 c12 c23 ? s12 s23 s13 s23 c13 ? ?
Here cij = cos θij and sij = sin θij for ij = 12, 23, or 13, and we have assumed CP conservation. The set of oscillation parameters is now ?ve-dimensional (see for instance ),
2 2 2 ?m2 sun ≡ ?m21 = m2 ? m1 2 2 2 ?m2 atm ≡ ?m32 = m3 ? m2
θsun θatm θ13
≡ θ12 ≡ θ23
We do not perform a global analysis of all possible values of these parameters, but ?x them to be in the regions that solve the atmospheric and solar neutrino ?3 problems [32,33]. In particular we take ?m2 eV2 and maximal atm = 3 × 10 mixing for θatm from the former, while from the solar analyses we consider the 2 ?5 ?6 ?7 ?11 following values for ?m2 sun in eV : 4.5 × 10 , 7 × 10 , 1 × 10 , 8 × 10 for the Large Mixing Angle (LMA), Small Mixing Angle (SMA), LOW and Vacuum regions, respectively. For the angle θsun we take the approximation of maximal mixing for all cases except SMA where we use θsun = 1.5? . 4.2 Simple three ?avor case We begin with the simpli?ed situation where initially only the muon neutrinos are asymmetric (ξ? = ?0.1). We perform a full three-?avor calculation, but for now set θ13 = 0. The evolution of the neutrino asymmetries is shown in Figs. 5 and 6 for the LMA and LOW cases, respectively, both with and without the neutrino self-interactions. For this choice of oscillation parameters the three-?avor oscillations e?ectively separate as two two-?avor problems for the atmospheric and solar parameters, respectively. The oscillations caused by the largest ?m2 are e?ective at T ? 20 MeV, as soon as the ?± background disappears completely. The presence of the self-term causes only a slight delay in the equilibration of ξ? and ξτ . The oscillations due to ?m2 sun and θsun are e?ective only when the vacuum term 10
overcomes the e± potential. In the LMA case, the conversions takes place above T ? 1 MeV, leading to nearly complete ?avor equilibrium before the onset of BBN. For the LOW parameters the synchronized oscillations just start at that epoch. The presence of the neutrino self-potential does not signi?cantly change the picture in the LMA case while for the LOW case one clearly observes the phenomenon of synchronized oscillations. For the SMA and Vacuum regions primordial oscillations involving νe are not e?ective before BBN if θ13 = 0. 4.3 Non-zero 13 mixing The angle θ13 is restricted to the approximate region tan2 θ13 < ? 0.065 (see for instance ) from a combined analysis of solar, atmospheric and reactor (CHOOZ) data. However, a small but non-zero θ13 does modify the oscillation behavior. This e?ect is shown in Fig. 7 for the LMA case and di?erent values of θ13 . Even small values of θ13 lead to conversion to the electronic ?avor at larger temperatures and enhance ?avor equilibration if θsun is in the LMA region. The e?ect of a non-zero θ13 is more important if the solar neutrino problem is solved by oscillations with parameters in a region other than LMA. This can be seen in Fig. 8, where we have calculated the evolution in the limit ?m2 sun ? 0 and θsun ? 0, thus corresponding approximately to the SMA, LOW and Vacuum regions. For values of tan2 θ13 close to the limit discussed at the beginning of this section, neutrino oscillations lead to almost complete ?avor equilibration, while for smaller θ13 angles the conversion is only partial. 4.4 Can the self-term prevent ?avor equilibrium? In the two-?avor case of Sec. 3.3 we saw that for equal but opposite asymmetries of the two ?avor distributions the synchronized oscillation frequency was strictly zero, suppressing oscillations entirely before the BBN epoch. In the three ?avor case the situation is more complicated so that again we need to raise the question if there is a special con?guration of initial neutrino distributions which suppresses ?avor oscillations by the neutrino self-potential. In analogy to Sec. 3.3 we ?rst consider situations where the asymmetries of two ?avors are equal but opposite, while the third asymmetry is strictly zero. In this case one expects that oscillations between the asymmetric ?avors are suppressed by the self-potential while oscillations into the third ?avor are unimpeded. We show the numerical results for two speci?c cases. In Fig. 9 we take ξ? = ?ξτ = ?0.1 and ξe = 0. The “atmospheric” oscillations between ν? and ντ are indeed blocked in the presence of self-interactions while the “solar 11
oscillations”, here the LMA case and tan2 θ13 = 0, proceed at the appropriate temperature. In Fig. 10 we take ξτ = 0 and ξe = ?ξ? = 0.1. Both the atmospheric and solar oscillations take place and equilibrate the ?avors essentially as in the previous cases with a simple initial condition. The question remains if one could devise special initial conditions such that |ξτ | and |ξ? | are large while |ξe | is small, and that this system avoids ?avor oscillations by reducing the synchronized oscillation frequency to a very small value. In this case our new limits would not apply. We do not believe that such a case can be constructed. To argue in favor of this claim we ?rst note that the most general initial condition consists of thermal equilibrium distributions which are diagonal in ?avor space, i.e. in the weak-interaction basis the initial 3×3 density matrices for neutrinos and anti-neutrinos are diagonal and characterized by Fermi-Dirac distributions. All o?-diagonal elements would be quickly damped by reactions which involve muons and electrons and thus “measure” the ?avor content of the participating neutrinos. Therefore, the most natural initial condition for our problem is indeed characterized by ξe , ξ? and ξτ . The general equations of motion Eq. (4) reveal that the quantity which develops in a synchronized fashion is ρ ? ρ ?, in agreement with the discussion of 2×2 synchronized oscillations in Ref.  where the picture of polarization vectors and the associated “internal magnetic ?elds” of the system was used. Therefore, the synchronized equation of motion is obtained by subtracting the two lines of Eq. (4) and integrating over all modes, d3 p M 2 , (ρp + ρ ?p ) (2π )3 2p √ 8p d3 p ? 2GF E, (ρp + ρ ?p ) (2π )3 3m2 W √ ?), (ρ ? ρ ?)] , + 2GF [(ρ ? ρ
i?t (ρ ? ρ ?) =
where we have dropped the collision term. The last term, of course, is identically zero. The second term vanishes also as long as the density matrices are diagonal in the ?avor basis since the matrix E is diagonal in that basis. Only the ?rst term, involving the mass matrix, represents a “force” such as to move ρ ? ρ ? away from being diagonal in the ?avor basis, i.e. which causes synchronized ?avor oscillations. This force is identically zero if the matrix d3 p ρp + ρ ?p 3 (2π ) 2p (13)
is proportional to the unit matrix. As the matrices ρp and ρ ?p are initially 12
diagonal and given by Fermi-Dirac distributions, the integral can be solved explicitly, leading to an expression proportional to
? ? ?
1 1 1
3 ? ? ?+ 2 ? π
? ? ?
If all ξ are initially equal, then the neutrinos are already in ?avor equilibrium and indeed oscillations trivially do not operate. The only nontrivial con?guration where synchronized oscillations proceed with a vanishing frequency is when one of the chemical potentials is opposite in sign to the others. We have checked numerically that indeed no ?avor conversion arises in this case as long as the self-potential is large, i.e. as long as we are in the synchronized regime. In the two-?avor case this corresponds to the situation discussed in Sec. 3.3. As in the two-?avor case we conclude that it is possible to avoid ?avor equilibrium by specially chosen initial conditions, but these conditions require |ξe | = |ξ? | = |ξτ |. Therefore, the strict BBN limits on |ξe | apply to all ?avors. 5 New limits on neutrino degeneracy
We conclude that in the LMA region the neutrino ?avors essentially equilibrate long before n/p freeze out, even when θ13 is vanishingly small. For the other cases the outcome depends on the magnitude of θ13 . In the LMA case it is thus justi?ed to derive new limits on the cosmic neutrino degeneracy parameters under the assumption that all three neutrino ?avors are characterized by a single degeneracy parameter, independently of the primordial initial conditions. We do not derive the corresponding limits for the other solar neutrino solutions, since they would strongly depend on the value of a non-zero θ13 . However, if that angle is close to the experimental limit, the bounds that we describe would approximately apply. We ?rst note that the energy density in one species of neutrinos and antineutrinos with degeneracy parameter ξ is ρν ν ?
2 30 4 7π ? = Tν 1+ 120 7
15 + 7
It is clear that the BBN limit will imply ξ ? 1 for all ?avors so that the modi?ed energy density and the resulting change of the primordial helium abun13
dance Yp will be negligibly small. If there are additional relativistic species, such as sterile neutrinos or majorons, then Eq. (2) will simply apply to all the active neutrinos |ξ | < 0.22 . (16)
Therefore, the only remaining BBN e?ect is the shift of the beta equilibrium by ξe . We recall that Yp is essentially given by n/p at the weak-interaction freezeout, and that n/p ∝ exp(?ξe ) ? 1 ? ξe where the latter expansion applies for |ξe | ? 1. Therefore, ?Yp ? ?Yp (1 ? Yp /2)ξe ? ?0.21 ξe . Modi?cations of Yp by new physics are frequently expressed in terms of the equivalent number of neutrino ?avors ?Nν which would cause the same modi?cation due to the changed expansion rate at BBN. If the radiation density at BBN is expressed in terms of Nν , the helium yield can be expressed by the empirical formula ?Yp = 0.012 ?Nν . Therefore, the e?ect of a small ξe on the helium abundance is equivalent to ?Nν ? ?18ξe . A conservative standard limits holds that BBN implies |?Nν | < 1 which thus translates into |ξe | < ? 0.057. A more detailed recent analysis reveals that the measured primordial helium abundance implies a 95% CL range Nν = 2.5 ± 0.8 or ?Nν = ?0.5 ± 0.8 [8,13]. We conclude that the BBN-favored range for the electron neutrino degeneracy parameter is at 95% CL ξe = 0.03 ± 0.04 . (17)
If all degeneracy parameters are the same, then this range applies to all ?avors. It should be noted that the actual limit we obtain on the neutrino degeneracy depends on the adopted BBN analysis. For instance ?Nν could be as high as 1.2 when the primordial abundance of lithium is used instead of that of deuterium . At any rate, a limit of |ξe | < ? 0.1 seems rather conservative and does not modify our conclusions. Using |ξ | < 0.07 as a limit on the one degeneracy parameter for all ?avors, the extra radiation density is limited by (?ρν ν ? )/ρν ν ? < 3 × 0.0021 = 0.0064, i.e. ?Nν < 0.0064. If the same radiation density were to be produced by the asymmetry of one single species, this would correspond to |ξ | < 0.12. For comparison with the future satellite experiments MAP and PLANCK that will measure the CMBR anisotropies, it was calculated that they optimistically will be sensitive to a single-species ξ above 0.5 and 0.25, respectively . However, with proper consideration of the degeneracy with the matter density, ωM , and the spectral index, n, a more realistic sensitivity is ξ ≈ 2.4 and 0.73, respectively . Turning this around we conclude that our new limits are so restrictive that the CMBR is certain to remain una?ected by neutrino 14
degeneracy e?ects so that |ξ | can be safely neglected as a ?t parameter in future analyses. If our new limits apply the number density of relic neutrinos is very close to its standard value. Therefore, existing limits and possible future measurements of the absolute neutrino mass scale, for example in the forthcoming tritium decay experiment KATRIN , will provide unambiguous information on the cosmic mass density in neutrinos, free of the uncertainty of neutrino chemical potentials. Note added: Very recently the authors of refs. [40,41] have also analyzed the equilibration of neutrino asymmetry from ?avor oscillations, providing further analytical insight and con?rming our conclusions.
We thank Alexei Smirnov, Gary Steigman, Karsten Jedamzik and Gianpiero Mangano for useful discussions and comments. In Munich, this work was partly supported by the Deutsche Forschungsgemeinschaft under grant No. SFB 375 and the ESF network Neutrino Astrophysics. S.H. Hansen and S. Pastor are supported by Marie Curie fellowships of the European Commission under contracts HPMFCT-2000-00607 and HPMFCT-2000-00445.
In this appendix we list in detail the evolution equations for the neutrino and anti-neutrino density matrices in the two-?avor case as in Eq. (5), while the generalization to the three-?avor case that we consider in Sec. 4 is straightforward. In our treatment of primordial neutrino oscillations we use the following dimensionless expansion rate and momenta x ≡ mR , y ≡ pR , (A.1)
where R is the universe scale factor and m an arbitrary mass scale that we choose to be 1 MeV. The neutrino and anti-neutrino density matrices are ρ(x, y ) = 1 1 [P0 (x, y ) + σ · P(x, y )] = 2 2 15 P0 + Pz Px + iPy Px ? iPy P0 ? Pz (A.2)
ρ(x, y ) =
1 1 P 0 (x, y ) + σ · P(x, y ) = 2 2
P0 + Pz P x + iP y
P x ? iP y P0 ? Pz
with the derivatives Hx dρ dρ (x, y ) = (t, p) , dx dt Hx dρ dρ (x, y ) = (t, p) . dx dt (A.4)
The initial conditions are for large temperatures as follows. For density matrices normalized to feq (y ) = (ey + 1)?1 and initial degeneracies ξα = ?ξα ? and ξβ = ?ξβ ? (for ?avor neutrinos να and νβ ) feq (y, ξα) + feq (y, ξβ ) feq (y ) feq (y, ξα) ? feq (y, ξβ ) P z (y ) = feq (y ) feq (y, ?ξα ) + feq (y, ?ξβ ) P 0 (y ) = feq (y ) feq (y, ?ξα ) ? feq (y, ?ξβ ) P z (y ) = feq (y ) P 0 (y ) = where feq (y, ξ ) = 1/[exp(y ? ξ ) + 1]. Finally, P x (y ) = P y (y ) = P x (y ) = P y (y ) = 0 . The components of the neutrino polarization vectors evolve as dP0 (x, y ) = Rα (x, y ) + Rβ (x, y ) , dx dPx D Vl ?m2 Py (x, y ) ? (x, y ) = cos 2θ ? Px (x, y ) , dx 2pHx Hx Hx D Vl ?m2 dPy Px (x, y ) ? (x, y ) = ? cos 2θ ? Py (x, y ) dx 2pHx Hx Hx ?m2 ? sin 2θ Pz (x, y ) , 2pHx ?m2 dPz (x, y ) = sin 2θ Py (x, y ) + Rα (x, y ) ? Rβ (x, y ) , dx 2pHx dP 0 (x, y ) = Rα (x, y ) + Rβ (x, y ) , dx Vl dP x ?m2 D (x, y ) = ? cos 2θ ? P y (x, y ) ? P x (x, y ) , dx 2pHx Hx Hx 16 (A.6)
Vl dP y ?m2 D (x, y ) = cos 2θ ? P x (x, y ) ? P y (x, y ) dx 2pHx Hx Hx ?m2 sin 2θ P z (x, y ) , + 2pHx ?m2 dP z (x, y ) = ? sin 2θ P y (x, y ) + Rα (x, y ) ? Rβ (x, y ) . dx 2pHx where D feq (y, ξα) 1 ? (P0 (x, y ) + Pz (x, y )) Hx feq (y ) 2 D feq (y, ξβ ) 1 Rβ (x, y ) = 2 ? (P0 (x, y ) ? Pz (x, y )) , Hx feq (y ) 2
Rα (x, y ) = 2
with Rα,β given by the same expressions with ξ → ?ξ . However, our numerical results do not change signi?cantly if the total neutrino and anti-neutrino number densities are taken to be constant (dP0 /dx = dP 0 /dx = 0). The di?erent terms in Eqs. (A.7)are given as follows. The vacuum oscillation terms are proportional to 1010 MP ?m2 = 2pHx 2 8π/3 ?m2 eV2 1 x2 √ ρ ? y (A.9)
where MP ≡ 1.221, ρ ? = (x/m)4 ρtot , and ρtot is the total energy density of the universe. The l+ l? background with l = ?, e the charged leptons (for ντ -ν? and νe -ν?,τ oscillations, respectively) is described by √ Vl 8 2 105 MP GF 1 y (A.10) ?l? ) =? (? ρl+ + ρ √ 4 Hx ρ ? x 3 8π/3 m2 W where GF ≡ 1.1664 and mW ≡ 80.42. Collisions are described by damping terms where, according to our calculations, D ? 2 × (4 sin4 θW ? 2 sin2 θW + 2)F0 , D ? 2 × (2 sin4 θW + 6)F0 ,
for ντ -ν? and νe -ν?,τ oscillations, respectively, where all collision terms are proportional to MP G2 y F0 F ζ (3) ymed 1 = √ Hx ρ ? x4 3 π 3 8π/3 17 (A.12)
and ζ (3) ? 1.20206 and ymed ? 3.15137. Finally one has to add the contribution from the neutrino-antineutrino background [27–29] Vasym Vsym J(x) ? J(x) ? U(x) + U(x) × P(x, y ) , Hx Hx Vsym Vasym J(x) ? J(x) + U(x) + U(x) × P(x, y ) , Hx Hx for dP/dx and dP/dx, respectively. Here we have de?ned J(x) = 1 2π 2 du u2 P(x, u) , U(x) = 1 2π 2 du u3 P(x, u) , (A.14)
and the same for J and U. Moreover, Vasym = Hx √ 2 1011 MP GF 8π/3 1 ρ ? x2 (A.15)
Vsym 8 2 105 MP GF 1 y = . cos2 θW √ 2 Hx ρ ? x4 3 8π/3 mW
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0.004 0.002 Pz 0 -0.002 -0.004 0.004 Pz 0.002 0 -0.002
Vacuum + e± potential Vacuum + e potential + Collisions
Pure vacuum Vacuum + Self
10 T (MeV)
Fig. 1. Evolution of the z -component of the integrated polarization vector P for di?erent situations as explained in the text.
0.004 0.002 Px 0 -0.002 -0.004 10
Vacuum + e± potential Vacuum + e± potential + Collisions
1 T (MeV)
Fig. 2. Evolution of the x-component of the integrated polarization vector P for the same cases as in the lower panel of Fig. 1.
0.004 Pz 0.002 0 -0.002 10 T (MeV)
Fig. 3. Evolution of the z -component of the integrated polarization vector P in the presence of the vacuum term, the e+ e? potential, collisional damping, and varying strength of the neutrino self-interaction.
No Self Self / 105 Self / 104 Self
0.004 0.002 Pi 0 -0.002 10 T (MeV)
Fig. 4. Evolution of the x and y components of the integrated polarization vector P in the presence of all e?ects and full-strength neutrino self-interactions, corresponding to the solid line in Fig. 3.
-0.04 ξν -0.06 -0.08
No Self Self 10 T (MeV) 1
Fig. 5. Evolution of the neutrino degeneracy parameters for the LMA case, θ13 = 0, and the initial values ξe = ξτ = 0 and ξ? = ?0.1.
0 νe -0.02 ντ
-0.04 ξν -0.06 -0.08
No Self Self 10 T (MeV) 1
Fig. 6. Evolution of the neutrino degeneracy parameters for the LOW case, θ13 = 0, and the initial values ξe = ξτ = 0 and ξ? = ?0.1.
-0.02 ντ -0.04 ξν -0.06 ν? -0.08 0.065 0.2 0 10 T (MeV) 1 -0.1
Fig. 7. Evolution of the neutrino degeneracies for di?erent values of tan2 θ13 in the LMA case for the initial conditions ξe = ξτ = 0 and ξ? = ?0.1.
-0.02 ντ -0.04 ξν -0.06 ν? -0.08 θsun=0 0.065 0.020 0.005 10 T (MeV) 1 -0.1
Fig. 8. Evolution of the neutrino degeneracies for di?erent values of tan2 θ13 in the limit ?m2 sun ? 0 and θsun ? 0 for the initial conditions ξe = ξτ = 0 and ξ? = ?0.1.
-0.05 ν? 10 T (MeV) No Self Self 1
Fig. 9. Evolution of the neutrino degeneracies if initially ξe = 0 and ξτ = ?ξ? = 0.1, with or without neutrino self-interactions.
-0.05 ν? 10 T (MeV) No Self Self 1
Fig. 10. Evolution of the neutrino degeneracies for the LMA case with θ13 = 0 with initial conditions ξτ = 0 and ξe = ?ξ? = 0.1, with or without neutrino self-interactions.
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