Constraining the cosmic radiation density due to lepton number with Big Bang Nucleosynthesis
PPrepared for submission to JCAP
IFIC/10-38
Constraining the cosmic radiationdensity due to lepton number with BigBang Nucleosynthesis
Gianpiero Mangano, a Gennaro Miele, a,b
Sergio Pastor, c OfeliaPisanti, a,b and Srdjan Sarikas a,b a Istituto Nazionale di Fisica Nucleare - Sezione di NapoliComplesso Universitario di Monte S. Angelo, I-80126 Napoli, Italy b Dipartimento di Scienze Fisiche, Universit`a di Napoli
Federico II
Complesso Universitario di Monte S. Angelo, I-80126 Napoli, Italy c Instituto de F´ısica Corpuscular (CSIC-Universitat de Val`encia),Ed. Institutos de Investigaci´on, Apdo. correos 22085, E-46071 Valencia, SpainE-mail: [email protected], [email protected], pastor@ific.uv.es,[email protected], [email protected]
Abstract.
The cosmic energy density in the form of radiation before and during BigBang Nucleosynthesis (BBN) is typically parameterized in terms of the effective numberof neutrinos N eff . This quantity, in case of no extra degrees of freedom, depends upon thechemical potential and the temperature characterizing the three active neutrino distributions,as well as by their possible non-thermal features. In the present analysis we determine theupper bounds that BBN places on N eff from primordial neutrino–antineutrino asymmetries,with a careful treatment of the dynamics of neutrino oscillations. We consider quite a widerange for the total lepton number in the neutrino sector, η ν = η ν e + η ν µ + η ν τ and theinitial electron neutrino asymmetry η in ν e , solving the corresponding kinetic equations whichrule the dynamics of neutrino (antineutrino) distributions in phase space due to collisions,pair processes and flavor oscillations. New bounds on both the total lepton number in theneutrino sector and the ν e − ¯ ν e asymmetry at the onset of BBN are obtained fully exploitingthe time evolution of neutrino distributions, as well as the most recent determinations ofprimordial H/H density ratio and He mass fraction. Note that taking the baryon fractionas measured by WMAP, the H/H abundance plays a relevant role in constraining the allowedregions in the η ν − η in ν e plane. These bounds fix the maximum contribution of neutrinos withprimordial asymmetries to N eff as a function of the mixing parameter θ , and point out theupper bound N eff (cid:46) .
4. Comparing these results with the forthcoming measurement of N eff by the Planck satellite will likely provide insight on the nature of the radiation content ofthe universe. Keywords:
Neutrinos, physics of the early universe, primordial asymmetries
ArXiv ePrint: a r X i v : . [ a s t r o - ph . C O ] M a r Introduction
The dynamics of neutrino oscillations in the early universe has been extensively studied inthe scientific literature and is expected to have produced an efficient mixing of flavor neutrinodistributions at a temperature T γ ∼ T γ ∼ m triggers efficient ν µ − ν τ mixing, aswell as ν x − ν e ( x = µ, τ ) conversions if the angle θ is not vanishing and sufficiently large.Finally, at T γ (cid:46) m set on and are large enough to achievestrong flavor conversions before BBN [1–4].These results have a major impact on the possible values for the cosmological leptonasymmetry stored in each neutrino flavor, which in analogy with the baryon-antibaryonasymmetry parameter η b = ( n b − n ¯ b ) /n γ , can be parameterized by the number density ratios η ν α = n ν α − n ¯ ν α n γ , α = e, µ, τ . (1.1)Based on the equilibration of lepton and baryon asymmetries by sphalerons in the very earlyuniverse, such a neutrino asymmetry should be of the same order of the cosmological baryonnumber η b = 273 .
93 Ω b h − , which is restricted to be a few times 10 − by present obser-vations, such as 7-year data from the WMAP satellite and other cosmological measurements[5]. Nevertheless, a cosmological neutrino asymmetry orders of magnitude larger than thisvalue is still an open possibility, with implications on fundamental physics in the early uni-verse, such as their potential relation with the cosmological magnetic fields at large scales [6].In particular, a non-zero lepton asymmetry leads to an enhanced contribution of neutrinosto the energy density in the form of radiation ρ r , which, after the e + e − annihilation phase, isusually parameterized as ρ r /ρ γ = 1 + 7 / / / N eff . The parameter N eff is the “effectivenumber of neutrinos” whose standard value is 3 in the limit of instantaneous neutrino de-coupling. Interestingly, recent data on the anisotropies of the cosmic microwave backgroundfrom WMAP [5] and the primordial He abundance [7–9] seem to favor a value of N eff > µ ν α , and each flavorneutrino asymmetry can be expressed in terms of the corresponding degeneracy parameter ξ ν α ≡ µ ν α /T ν α as η ν α = 112 ζ (3) (cid:18) T ν α T γ (cid:19) (cid:0) π ξ ν α + ξ ν α (cid:1) . (1.2)In this case, if flavor oscillations enforce the condition that all ξ ν α are almost the sameduring BBN, the stringent bound on the electron neutrino degeneracy, which directly entersthe neutron/proton chemical equilibrium [10], applies to all flavors. The common valueof the neutrino degeneracies is restricted to the range − . ≤ ξ ν α ≤ . α = e, µ, τ [11] (see also [12–19] for other analyses). This in turn also implies that, if neutrinos indeed,reach perfect kinetic and chemical equilibrium before they decouple , any large excess in cosmicradiation density, if observed, must be ascribed to extra relativistic degrees of freedom sincethe additional contribution to radiation density due to non vanishing ξ ν α is very small.Assuming that the neutrino distributions are given by their equilibrium form, the BBN bound– 2 –n the neutrino degeneracies leads to the following upper limit on the excess contribution to N eff [2], ∆ N eff = (cid:88) α = e,µ,τ (cid:34) (cid:18) ξ ν α π (cid:19) + 157 (cid:18) ξ ν α π (cid:19) (cid:35) (cid:46) . , (1.3)which is tiny, even compared with the value of N eff = 3 .
046 found solving the neutrino kineticequations in absence of asymmetries [20].Neutrino kinetic and chemical equilibrium is maintained in the early universe by purelyleptonic weak processes such as neutrino-neutrino interactions, ν - e ± scatterings and pairprocesses, ν ¯ ν ↔ e + e − , whose rates become of the order of the Hubble parameter at T γ ∼ − m take place when neutrinos are still fastly scattering off the surroundingmedium, so that the changes in their distribution due to oscillations are efficiently readjustedinto an equilibrium Fermi-Dirac function. Instead, flavor conversions due to ∆ m and θ occur around neutrino decoupling. This implies, at least in principle, that if neutrinossucceed in achieving comparable asymmetries in all flavors before BBN, their distributionsmight acquire distortions with respect to equilibrium values due to inefficient interactions.This can be easily understood by a simple example. Suppose that at temperatureshigher than 2 − η in ν e = − η in ν x (cid:54) = 0 andwe artificially switch-off scattering and pair processes. Due to solar-scale oscillations, in thecase of maximal mixing asymmetries in each flavor will eventually vanish, but the neutrinodistributions will not correspond to equilibrium, since averaging two equilibrium distributionswith a different chemical potential does not correspond to a Fermi-Dirac function. Onlyscatterings and pair-processes can turn it into an equilibrium distribution with, in this case,zero chemical potential.Though this example is quite extreme and unrealistic, nevertheless, it tells us that theinterplay of neutrino freeze–out and ∆ m oscillation phases might deserve a more carefulscrutiny, as first discussed in [21]. In fact, depending on the initial flavor neutrino asymme-tries and the value of θ , the final neutrino distributions at the onset of BBN might shownon-thermal distortions which change the neutron-proton chemical equilibrium due to thedirect role played by electron (anti)neutrinos. Moreover, this corresponds to an asymmetry-depending parameter N eff > ν e and ν µ,τ chemical potentials. It was shown that with fine-tuned initialasymmetries the BBN bound could be respected and at the same time an excess radiationdensity could survive, corresponding to values of ∆ N eff of order unity or larger.In the present work we extend the analysis of [21] in two ways. First, we considera wider range of values of the initial neutrino asymmetries and solve their evolution withthe corresponding kinetic equations, including both collisions and oscillations. Moreover,the obtained shape of the neutrino distributions is then plugged into the BBN dynamicsallowing, by the comparison between the theoretical results and the experimental data onprimordial abundances of deuterium and He, to find more accurate bounds on the totallepton asymmetry stored in the neutrino sector, as well as the way it was distributed at someearly stage in the ν e and ν x flavors. In fact, for all initial values of neutrino asymmetries which– 3 –re compatible with BBN bounds, the distortions in the neutrino distribution are typicallyquite small, see Section 2, so that it is accurate enough for our purposes to parameterize themin terms of a Fermi-Dirac function with two time-dependent parameters, which correspondto the first two moments of the actual distribution: an effective chemical potential ξ ν α andan effective temperature T ν α , or equivalently, the asymmetry in each flavor and the energydensity.The paper is organized as follows. After setting in Section 2 the formalism of kineticequations which rule the evolution of neutrino distributions written in the standard densitymatrix formalism and showing an example of their dynamics, we then study the BBN con-straints on neutrino asymmetries in Section 3. In particular, we discuss the experimentaldata which are used in our analysis for H/H and He mass fraction Y p , as well as the waywe have modified the public BBN numerical code PArthENoPE [22–24] to track neutrino evo-lution. Finally, we report the bounds on initial (at T γ ∼
10 MeV) neutrino asymmetries ortheir final values after flavor oscillation phase. In Section 4 we give our concluding remarks.
Our first aim is to calculate the evolution of the three active neutrino distributions in theepoch of the universe right before BBN, when these particles were interacting among them-selves and with electrons and positrons. The corresponding weak collision rate decreases veryfast with the expansion until neutrinos decouple at T γ ∼ × (cid:37) p for each neutrinomomentum p , where the diagonal elements are the usual flavor distribution functions (oc-cupation numbers) and the off-diagonal ones encode phase information and vanish for zeromixing.Oscillations in flavor space of the three active neutrinos are driven by two mass-squareddifferences and three mixing angles bounded by the experimental observations in the followingranges: the ”solar” ∆ m = 7 . +0 . − . × − eV , the ”atmospheric” | ∆ m | = 2 . +0 . − . × − eV and correspondingly, the large mixing angles sin θ = 0 . +0 . − . and sin θ =0 . +0 . − . (2 σ ranges from [28]). On the other hand, the third angle is quite small, sin θ ≤ .
053 (3 σ ) [28], even compatible with a vanishing value, though a mild evidence for sin θ > (cid:37) p are the same as those considered in reference[21], i d(cid:37) p dt = [ Ω p , (cid:37) p ] + C [ (cid:37) p , ¯ (cid:37) p ] , (2.1)and similar for the antineutrino matrices ¯ (cid:37) p . The first term on the r.h.s. describes flavoroscillations, Ω p = M p + √ G F (cid:18) − p m E + (cid:37) − ¯ (cid:37) (cid:19) , (2.2)where p = | p | and M is the neutrino mass matrix (opposite sign for antineutrinos). Mattereffects are included via the term proportional to the Fermi constant G F , where E is the 3 × (cid:37) − ¯ (cid:37) , where (cid:37) = (cid:82) (cid:37) p d p / (2 π ) and similar for antineutrinos. For the relevant values of neutrino asymmetries this matterterm dominates and leads to synchronized oscillations [2–4]. The last term in eq. (2.1)corresponds to the effect of neutrino collisions, i.e. interactions with exchange of momenta.Here we follow the same considerations of ref. [21], where the reader can find more detailson the approximations made and related references. In short, the collision terms for the off-diagonal components of (cid:37) p in the weak-interaction basis are momentum-dependent dampingfactors, while collisions and pair processes for the diagonal (cid:37) p elements are implementedwithout approximations solving numerically the collision integrals as in [20]. These lastterms are crucial for modifying the neutrino distributions to achieve equilibrium with e ± and, indirectly, with photons.We have solved numerically the EOMs for the matrices in flavor space of neutrinosand antineutrinos with non-zero initial asymmetries. The expansion of the universe is takeninto account using comoving variables as in [2], where it was shown that flavor oscillationsbetween muon and tau neutrinos take place at T γ >
10 MeV, when interactions are veryeffective. For any initial values of the muon or tau neutrino asymmetries, the combinedeffect of oscillations and collisions is able to equilibrate the two flavors and hence leads to η in ν µ = η in ν τ ≡ η in ν x . Therefore, we start our numerical calculations for each case at T = 10 MeVand initial parameters η in ν e and the total asymmetry η ν = η in ν e + 2 η in ν x . Note that hereafter by η ν we denote the initial value of the total asymmetry, which is kept constant until the onsetof e + − e − annihilations, when it is diluted by the increase of the photon density (as in thecase of the baryon asymmetry η b ).All neutrino mixing parameters, except for θ , are taken as the best-fit values in [28].Modifying these parameters within their allowed regions does not affect our results. Instead,the value of θ plays an important role in the evolution of the neutrino asymmetries [21].We thus consider either θ = 0 or sin θ = 0 .
04, a value close to the upper bound fromneutrino experiments [28–30].Let us describe the evolution of the flavor asymmetries with a specific example. Asshown in [21], large initial values of flavor primordial asymmetries could satisfy the BBNbound only if η in ν e and η in ν x have opposite signs and the total asymmetry is not very differentfrom zero. A case with η ν = 0 was the main example presented in [21]. Here we choose insteada benchmark case with non-zero negative total asymmetry, η ν = − .
41, and η in ν e = 0 .
82. Theevolution of the flavor asymmetries is found from the numerical solution of the EOMs and isshown in Figure 1. Other choices of the initial asymmetries will lead to different final values,but the overall behavior of the evolution is similar to the case shown here.One can see in Figure 1 the effect of flavor oscillations on the evolution of neutrinoasymmetries with the universe temperature and the dependence on the value of θ . If thismixing angle is close to the present experimental upper bound, flavor oscillations are effectivearound T γ ∼ The neutrino distributions evolve keeping an equilibrium form, andthe total asymmetry is almost equally distributed among the three flavors. However, thefinal value of the electron neutrino asymmetry is too different from zero and this case is notallowed by BBN, as we will see in Section 3. Instead, for θ = 0 flavor oscillations beginonly at T γ (cid:46) Here we consider only the case of normal hierarchy, ∆ m >
0. If we choose an inverted hierarchy theresults are very similar, except that equilibrium among the flavor asymmetries is reached slightly earlier. – 5 – N eu t r i no a sy mm e t r i e s T ! (MeV) " e " µ, $ " tot / 3 % =0sin % =0.04 Figure 1 . Evolution of the flavor neutrino asymmetries when η ν = − .
41, and η in ν e = 0 .
82. The solidcurves correspond to vanishing θ (outer black lines) and sin θ = 0 .
04 (inner red lines). The totalneutrino asymmetry is constant and equal to three times the value shown (blue dotted line). spectra in equilibrium. There is no full equipartition of the total η ν among the three flavors,although collisions lead to final values of the flavor asymmetries closer to η ν / θ (cid:46) − , the outcome is very close to the case of vanishing θ .In Figure 2 we show the final energy spectra of relic electron neutrinos and antineutrinosin arbitrary units for the same case of Figure 1 with vanishing θ . The upper (lower) solid linestands for the spectra of electron neutrinos (antineutrinos) calculated numerically, while thecorresponding dotted lines are described by a Fermi/Dirac distribution just characterized bythe same effective value of the electron neutrino degeneracy parameter (as used in all analysesbefore [21]). Both cases lead to the same value of the electron neutrino asymmetry but thereal calculation shows that an excess of radiation in neutrinos remains.In Figure 3 we show the evolution of the ratio of neutrino to photon energy densities, ρ ν /ρ γ , properly normalized so that it corresponds to N eff at early and late times as in [21].The fast drop of ρ ν /ρ γ at T ∼ . e + e − annihilations.The case without asymmetries (dotted line) ends at late times at N eff = 3 .
046 instead of 3because of residual neutrino heating [20]. We also show (solid lines) the evolution for ourmain example, where initially N eff = 4 .
16 for our choice of neutrino asymmetries. One cansee that as soon as oscillations become effective reducing the flavor asymmetries, the excessof entropy is transferred from neutrinos to the electromagnetic plasma, cooling the formerand heating the latter, but this process is only very effective for large values of θ . While thefinal N eff is 3 . θ = 0 .
04, for negligible θ a significant deviation from equilibriumsurvives and leads to a final enhanced value of N eff = 3 . N eu t r i no s pe c t r u m ( a . u . ) Comoving momentum y ! effective " ! e real Figure 2 . The final energy spectra of relic electron neutrinos and antineutrinos in arbitrary unitsfor the same case of Figure 1 with vanishing θ . Upper (lower) solid line stands for electron neutrino(antineutrino) calculated numerically (label ”real”). Upper (lower) dotted line stands for electron neu-trino (antineutrino) described by a Fermi/Dirac distribution just characterized by the same effectivevalue of the electron neutrino degeneracy parameter. As well known, BBN depends upon neutrino distribution functions in two ways. First of all,electron neutrinos and antineutrinos enter directly in the charge current weak processes whichrule the neutron/proton chemical equilibrium. A change in the effective temperature of thedistribution can shift the neutron/proton ratio freeze out temperature and thus modifies theprimordial He abundance. Similarly, a non-zero ν e − ¯ ν e asymmetry also changes chemicalequilibrium towards a larger or smaller neutron fraction for negative or positive values of ξ ν e ,respectively. Furthermore, lepton asymmetries in all flavors translate into a positive extracontribution to the neutrino energy density, speeding up the expansion rate given by theHubble parameter.Differently than in the approximated standard treatments, where both neutrino asym-metries and the extra contribution to N eff due to ξ ν α , see eq. (1.3), are considered as constantparameters, in the present analysis we exactly follow the evolution of the neutrino distribu-tion versus the photon temperature T γ , which is our evolution parameter. To this end wehave changed the public numerical code
PArthENoPE [22, 24] as follows. For any given initialvalues (at T γ = 10 MeV) for the total neutrino asymmetry η ν = (cid:80) α η ν α , unchanged by fla-vor oscillations, and electron neutrino asymmetry η in ν e we obtain, as described in the previous For another example of BBN calculations with arbitrarily-specified, time-dependent neutrino and antineu-trino distribution functions, see Ref. [31]. – 7 – .03.02.01.0 0.1 1 10 6.05.04.03.53.0 / ! " / ! / ( / ) / ! " / ! T (MeV) $ =0sin $ =0.04 Figure 3 . Evolution of the neutrino energy density for the same case as in Figure 1. The verticalaxis is marked with N eff , left before e + e − annihilation, right afterwards. The solid curves correspondto vanishing θ (upper black line) and sin θ = 0 .
04 (lower red line). The case without asymmetriesis shown for comparison (blue dotted line). section, the time dependent neutrino distributions. The latter are then fitted in terms ofFermi-Dirac functions with the two evolving parameters T ν α ( T γ ) and ξ ν α ( T γ ). An exampleof their evolution is shown in Figure 4 for the same choice of initial asymmetries as in thecase described in Section 2. Weak rates are then averaged over the corresponding electron(anti)neutrino distribution. The Hubble parameter is also modified to account for the actualevolution of total neutrino energy density.The final abundances of both the ratio H/H and the He mass fraction, Y p , are nu-merically computed as a function of the input parameters η ν and η in ν e and compared with thecorresponding experimental determinations. The baryon density parameter has been set tothe value determined by the 7-year WMAP result, Ω b h = 0 . ± . To get confidence intervals for η ν and η in ν e , one can construct the likelihood function L ( η ν , η in ν e ) ∝ exp (cid:0) − χ ( η ν , η in ν e ) / (cid:1) , (3.1)with χ ( η ν , η in ν e ) = (cid:88) ij [ X i ( η ν , η in ν e ) − X obsi ] W ij ( η ν , η in ν e )[ X j ( η ν , η in ν e ) − X obsj ] . (3.2)The proportionality constant can be obtained by requiring normalization to unity, and W ij ( η ν , η in ν e ) denotes the inverse covariance matrix [23], W ij ( η ν , η in ν e ) = [ σ ij + σ i,exp δ ij + σ ij,other ] − , (3.3) The allowed region of Ω b in extended cosmological models with free N eff does not differ significantly. – 8 – E ff e c t i v e T ! -1-0.75-0.5-0.2500.250.50.751 0.31310 E ff e c t i v e " ! T (MeV) $ =0sin $ =0.04 Figure 4 . Evolution of the effective comoving temperatures and degeneracy parameters of electron(solid lines) and muon or tau (dashed lines) neutrinos for the same case as in Figure 1. Both thecase of vanishing θ (thick black lines) and sin θ = 0 .
04 (thin red lines) are shown. The effectivetemperature for the case without asymmetries is shown in the upper panel for comparison (blue dottedline). where σ ij and σ i,exp represent the nuclear rate uncertainties and experimental uncertainties ofnuclide abundance X i , respectively [23], while by σ ij,other we denote the propagated squarederror matrix due to all other input parameter uncertainties ( τ n , G N , Ω b h , . . . ). In our casewe consider in eq. (3.2) as X i the quantities H/H and Y p only.Let us now briefly discuss the set of data we have used in our study. The H/H numberdensity is obtained by averaging seven determinations obtained in different Quasar Absorp-tion Systems, as in [11] H / H = (2 . ± . × − , (3.4)where the quadratic error has been enlarged by the value of the reduced χ to account forthe dispersion of measurements for this dataset, χ / .
6, see [11].For the He mass fraction we consider two different determinations. One is the resultof the data collection analysis performed in [11], Y p = 0 . ± . . (3.5)– 9 – .1 3.2 3.3 3.4 3.5 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Η Ν Η Ν e in (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Η Ν Η Ν e in Figure 5 . The 95% C.L. contours from our BBN analysis in the η ν − η in ν e plane for θ = 0 (left)and sin θ = 0 .
04 (right). The two contours correspond to the different choices for the primordial He abundances of eqs. (3.5) (blue) and (3.8) (purple). The (red) dot-dashed line is the set of valuesof η ν and η in ν e which, due to flavor oscillations, evolve towards a vanishing final value of electronneutrino asymmetry η fin ν e . We also report as dashed lines the iso-contours for different values of N eff ,the effective number of neutrinos after e + e − annihilation stage. More recently, new studies of metal poor H II regions have appeared in the literature [7–9].While these groups both agree on a larger central value with respect to the result of eq.(3.5), which incidentally seems to pin down a value for N eff > σ , a different estimate ofpossible systematic effects which dominate the total uncertainty budget is quoted in [7] and[8], with [8] quoting a larger error, of the order of 4%. Y p = 0 . ± . . ) ± . . ) [7] , (3.6) Y p = 0 . ± . . (3.7)Finally, in a recent paper [9], Markov Chain Monte Carlo method was exploited to determinethe He abundance, and the uncertainties derived from observations of metal poor nebulaefinding Y p = 0 . ± . . (3.8)In the following, we will use the two results of eq.s (3.5) and (3.8). While their uncertaintiesare the same, they differ for the central value, actually the smaller and higher of all resultsreported above, a fact which will produce two different bounds on the electron neutrinoasymmetry.The 95% C.L. contours for the total asymmetry η ν and the initial value of the electronneutrino parameter η in ν e are shown in Figure 5 for the adopted determinations of H and Heand for two different choices of θ . In both cases the contours are close to and aligned alongthe red dot-dashed line which represents the set of initial values for the asymmetries whicheventually evolve toward a vanishing final electron neutrino asymmetry, η fin ν e (cid:39)
0, which ispreferred by He data. We recall that He is strongly changed if neutron/proton chemical– 10 – (cid:45) (cid:45) (cid:45) (cid:45) Η Ν Η Ν e in Figure 6 . Bounds in the η in ν e - η ν plane for each nuclear yield. Areas between the lines correspondto 95% C.L. regions singled out by the He mass fraction (solid lines) of eq. (3.5), and Deuterium(dashed lines) as in eq. (3.4). (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Ξ Ν x in Ξ Ν e in (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Ξ Ν x in Ξ Ν e in Figure 7 . Same results as in Figure 5 in the plane of initial flavor degeneracy parameters ξ in ν x and ξ in ν e . equilibrium is shifted by a large value of ν e − ¯ ν e asymmetry around the freezing of weak rates( T γ ∼ . θ , oscillations efficiently mix all neutrino flavors and at BBN η ν α ∼ η ν /
3, so the bound on η ν is quite stringent, − . (cid:46) η ν (cid:46) .
1, if we adopt the value ofeq. (3.5) for Y p . Instead, for the choice of eq. (3.8) we find − . (cid:46) η ν (cid:46)
0, i.e. a larger valuefor Y p singles out slightly negative values for η ν (and η fin ν e ), since the theoretical predictionfor Y p grows in this case as the neutron-proton chemical equilibrium shift towards a larger– 11 – .1 3.2 3.3 3.4 3.5 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Η Ν x fin (cid:45) Η Ν e fin Η Ν e fin (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Η Ν x fin (cid:45) Η Ν e fin Η Ν e fin Figure 8 . Same results as in Figure 5 in the plane of η fin ν e and the difference η fin ν x − η fin ν x , where thesuperscript indicates that the asymmetries are evaluated at the onset of BBN, T γ ∼ η in ν α . θ = 0 sin θ = 0 . Y p = 0 . ± . − . < η ν < . − . < η ν < . Y p = 0 . ± . − . < η ν < . − . < η ν < . Table 1 . BBN bounds on the initial total neutrino asymmetry at 95% C.L. neutron fraction at freeze-out . On the other hand, for a vanishing θ the contours for η ν and η in ν e show a clear anticorrelation, and even values of order unity for both parameters are stillcompatible with BBN. The allowed regions of the total neutrino asymmetry are summarizedin Table 1.We stress that for any value of θ , the data on primordial deuterium, eq. (3.4), is crucialfor closing the allowed region that the He bound fixes along the η fin ν e (cid:39) H is less sensitive than He to neutrino asymmetries and effective temperature which enterthe Universe expansion rate, see e.g. [11], yet including it in the analysis breaks the degener-acy between η in ν e and η ν which is present when only He is used. This can be read from Figure6 where the 95% C.L. in the η in ν e - η ν plane are shown for He and H separately, for the case θ = 0. The solid lines bound the region of the plane compatible with the He measurementas in eq. (3.5), whereas the dashed contours correspond to Deuterium observation, see eq.(3.4). The different shape of these two regions is due to the different dependence of nuclideabundances on η in ν e and η ν , thus their combination breaks the degeneracy and leads to a closecontour as shown in Figure 5.It is also interesting to report our results in terms of other variables, as in Figures 7and 8. In the first case, the BBN contours are shown in the plane of initial flavor degeneracyparameters ξ in ν e and ξ in ν x while, in Figure 8, we consider a new pair of variables: the electronneutrino asymmetry at the onset of BBN η fin ν e , and the difference η fin ν x − η fin ν e , which in the– 12 –tandard analysis is usually assumed to be vanishing. One can see from this figure that,while the (95% C.L.) bound on η fin ν e is independent of the value of θ − . ≤ η fin ν e ≤ . , for Y p = 0 . ± . , (3.9) − . ≤ η fin ν e ≤ . , for Y p = 0 . ± . , (3.10)the difference between the final ν e and ν x asymmetries strongly depends upon this yet un-known mixing angle, as expected. In fact, for large θ we recover the standard result, η fin ν x ∼ η fin ν e , due to efficient mixing by oscillations and collisions, while for θ = 0 the twoasymmetries can be different. The following ranges are in good agreement with BBN dataat 95 % C.L., independently of the adopted value for Y p − . ≤ η fin ν x − η fin ν e ≤ . , sin θ = 0 , (3.11) − . ≤ η fin ν x − η fin ν e ≤ . , sin θ = 0 . . (3.12)We conclude that, in particular for θ = 0 oscillations lead to quite different neutrinoasymmetries in e and µ/τ flavors, still being in good agreement with BBN, differently thanwhat was previously assumed in the literature, see [1–4], and [11–19] for BBN analyses.In Figures 5-8 we also plot iso-contours for the value of the effective number of neutrinos, N eff , evaluated after e + e − annihilations. For large θ BBN data bound N eff to be very closeto the standard value 3.046, since all asymmetries should be very small in this case and flavoroscillations modify the neutrino distributions while neutrinos are still strongly coupled to theelectromagnetic bath. Therefore, we do not expect non-thermal features in the neutrinospectra in this case, since scatterings and pair processes allow for an efficient transfer of anyentropy excess. On the other hand, for vanishing θ , larger values of N eff are still compatiblewith BBN data, up to values of the order of 3.4 at 95% C.L. The dependence of the largestachievable value of N eff on the value of the mixing angle θ , obtained by spanning in theasymmetry parameter plane the region compatible with BBN, is reported in Figure 9.It is worth noticing that the final values of N eff , in particular for large final asymme-tries of ν x , are also slightly larger than the N eff that one would obtain using the equilibriumexpression of eq. (1.2). For example, if we take η fin ν e = 0 and η fin ν x = 0 .
3, a point on the bound-ary of the BBN contours (see Figure 8) and compute the corresponding effective chemicalpotentials, using eq. (1.3) one gets N eff = 3 .
2, while the actual value is larger, N eff = 3 . N eff coming fromrelativistic degrees of freedom other than standard active neutrinos. Their effect is knownto produce looser bounds on neutrino asymmetries, as they speed up expansion and thus,can compensate the effect of a positive ν e − ¯ ν e asymmetry. We have explicitly checked that,for some choices of primordial asymmetries, the addition of extra radiation does not modifythe evolution of flavor neutrino asymmetries. Of course, in such a case the contribution tothe energy density of the additional relativistic degrees of freedom adds up to the survivingexcess to N eff arising from neutrino asymmetries. In this paper we have calculated the evolution of neutrinos in the early universe with initialflavor asymmetries, taking into account the combined effect of collisions and oscillations. Our– 13 – -3 -4 m a x N e ff sin ! Figure 9 . Largest values of N eff from primordial neutrino asymmetries compatible with BBN, as afunction of θ . numerical results for the neutrino momentum distributions, which can develop non-thermalfeatures, were used to find the primordial production of light elements, employing a modifiedversion of the PArthENoPE
BBN code. Comparing with the recent data on primordial H and He abundances, we have found the allowed ranges for both the total asymmetry and theinitial asymmetry in electronic flavor. These BBN limits mostly depend on the value of θ ,the only unknown mixing angle of neutrinos. As can be seen from Figure 9, for sin θ (cid:46) − this implies an effective number of neutrinos bound to N eff (cid:46) .
4, independently of whichof the experimental He mass fraction of eq.s (3.5) and (3.8) is adopted, whereas for largervalues of θ the effective number of neutrinos is closely bound to the standard value 3.046obtained for vanishing asymmetries.In the near future it will be possible to improve our BBN constraints on the leptonnumber of the universe. On one hand, thanks to the better sensitivity of new neutrinoexperiments, either long-baseline or reactor, we expect to have a very stringent bound on θ or eventually its measurement (see e.g. [32]). Such results would lead to a more restrictiveBBN analysis on primordial asymmetries. On the other hand, in the next couple of yearsdata on the anisotropies of the cosmic microwave background from the Planck satellite [33]will largely reduce the allowed range of N eff , since the forecast sensitivity is of the order of0 . σ [34–36]. Indeed, suppose that Planck data confirm a value of N eff >
3. An excess ofradiation up to 0 . − . θ measurement. A vanishing θ would allowsuch possibility, but a measured θ in the next generation of experiments would imply thepresence of extra degrees of freedom others than active neutrinos. In case of a much larger– 14 –esult by Planck, namely N eff >
4, it would be impossible to explain such a result in terms ofprimordial neutrino asymmetries only, and alternative cosmological scenarios with additionalrelativistic species, such as sterile neutrinos (see for instance [37]) would be strongly favored.
Acknowledgments
We would like to thank Georg Raffelt for his useful comments on an earlier version of thispaper. G. Mangano, G. Miele, O. Pisanti and S. Sarikas acknowledge support by the
IstitutoNazionale di Fisica Nucleare
I.S. FA51 and the PRIN 2010 “Fisica Astroparticellare: Neutrinied Universo Primordiale” of the Italian
Ministero dell’Istruzione, Universit`a e Ricerca . S.Pastor was supported by the Spanish grants FPA2008-00319 and Multidark CSD2009-00064(MICINN) and PROMETEO/2009/091 (Generalitat Valenciana), and by the EC contractUNILHC PITN-GA-2009-237920. This research was also supported by a Spanish-ItalianMICINN-INFN agreement, refs. FPA2008-03573-E and ACI2009-1051.
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