Lepton asymmetry, neutrino spectral distortions, and big bang nucleosynthesis
LLA-UR-16-29176
Lepton asymmetry, neutrino spectral distortions, and big bang nucleosynthesis
E. Grohs , George M. Fuller , C. T. Kishimoto , , and Mark W. Paris Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA Department of Physics, University of California, San Diego, La Jolla, California 92093, USA Department of Physics and Biophysics, University of San Diego, San Diego, California 92110, USA and Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA (Dated: September 12, 2018)We calculate Boltzmann neutrino energy transport with self-consistently coupled nuclear reac-tions through the weak-decoupling-nucleosynthesis epoch in an early universe with significant lep-ton numbers. We find that the presence of lepton asymmetry enhances processes which give riseto nonthermal neutrino spectral distortions. Our results reveal how asymmetries in energy and en-tropy density uniquely evolve for different transport processes and neutrino flavors. The enhanceddistortions in the neutrino spectra alter the expected big bang nucleosynthesis light element abun-dance yields relative to those in the standard Fermi-Dirac neutrino distribution cases. These yields,sensitive to the shapes of the neutrino energy spectra, are also sensitive to the phasing of the growthof distortions and entropy flow with time/scale factor. We analyze these issues and speculate onnew sensitivity limits of deuterium and helium to lepton number.
PACS numbers: 98.80.-k,95.85.Ry,14.60.Lm,26.35.+c,98.70.Vc
I. INTRODUCTION
In this paper we use the burst neutrino-transportcode [1] to calculate the baseline effects of out-of-equilibrium neutrino scattering on nucleosynthesis in anearly universe with a nonzero lepton number, i.e. anasymmetry in the numbers of neutrinos and antineutri-nos. Our baseline includes: a strong, electromagnetic,and weak nuclear reaction network; modifications to theequation of state for the primeval plasma; and a Boltz-mann neutrino energy transport network. We do not in-clude neutrino flavor oscillations in this work. Our intentis to provide a coupled Boltzmann transport and nuclearreaction calculation to which future oscillation calcula-tions can be compared. In fact, the outstanding issuesin achieving ultimate precision in big bang nucleosyn-thesis (BBN) simulations will revolve around oscillationsand plasma physics effects. These issues exist in boththe zero and nonzero lepton-number cases, but are moreacute in the presence of an asymmetry.We self-consistently follow the evolution of the neu-trino phase-space occupation numbers through the weak-decoupling-nucleosynthesis epoch. There are many stud-ies of the effects of lepton numbers on light element,BBN abundance yields. Early work [2, 3] briefly ex-plored the changes in the helium-4 ( He) abundance inthe presence of large neutrino degeneracies. Later workconsidered how lepton numbers could influence the Heyield [4, 5] through neutrino oscillations. In addition,other works employed lepton numbers to constrain thecosmic microwave background (CMB) radiation energydensity [6, 7] or the sum of the light neutrino masses[8]. Refs. [9, 10] simultaneously investigated BBN abun-dances and CMB quantities using lepton numbers. Themost recent work has used the primordial abundancesto constrain lepton numbers which have been invokedto produce sterile neutrinos through matter-enhanced Mikheyev-Smirnow-Wolfenstein (MSW) resonances [11–13]. Currently, our best constraints on these lepton num-bers come from comparing the observationally-inferredprimordial abundances of either He or deuterium (D)with the predicted yields of He and D calculated in thesemodels.Previous BBN calculations with neutrino asymmetryhave made the assumption that the neutrino energydistribution functions have thermal, Fermi-Dirac (FD)shaped forms. In fact, we know that neutrino scatter-ing with electrons, positrons and other neutrinos andelectron-positron annihilation produce nonthermal dis-tortions in these energy distributions, with concomitanteffects on BBN abundance yields [1]. Though the nucle-osynthesis changes induced with self-consistent transportare small, they nevertheless may be important in the con-text of high precision cosmology. Anticipated Stage-IVCMB measurements [14, 15] of primordial helium and therelativistic energy density fraction at photon decoupling,coupled with the expected high precision deuterium mea-surements made with future 30-meter class telescopes[16–20] will provide new probes of the relic neutrino his-tory.In the standard cosmology with zero lepton numbers,neutrino oscillations act to interchange the populations ofelectron neutrinos and antineutrinos ( ν e , ν e ) with thoseof muon and tau species ( ν µ , ν µ , ν τ , ν τ ) [21]. Oncewe posit that there are asymmetries in the numbers ofneutrinos and antineutrinos in one or more neutrino fla-vors, then neutrino oscillations will largely determine thetime and temperature evolution of the neutrino energyand flavor spectra [22–30]. In this paper we ignore neu-trino oscillations and provide a baseline study of the re-lationship between neutrino spectral distortions arisingfrom the lengthy ( ∼
10 Hubble times) neutrino decou-pling process and primordial nucleosynthesis. This is anextension of the comprehensive study of this physics in a r X i v : . [ a s t r o - ph . C O ] M a r the zero lepton-number case with the burst code [1],and in other works [31–39]. We will introduce alterna-tive descriptions of the neutrino asymmetry to study theindividual processes occurring during weak decoupling.Our studies in this paper, together with the methods inother works, will be important in precision calculationsfor gauging the effects of flavor oscillations in the earlyuniverse.As we develop below, a key conclusion of a comparisonof neutrino-transport effects with and without neutrinoasymmetries is nonlinear enhancements of spectral dis-tortion effects on BBN in the former case. This suggeststhat phenomena like collective oscillations may have in-teresting BBN effects in full quantum kinetic treatmentsof neutrino flavor evolution through the weak decouplingepoch.The outline of this paper is as follows. Section II givesthe background analytical treatment of neutrino asym-metry, focusing on the equations germane to the earlyuniverse. Sec. III presents the rationale in picking theneutrino-occupation-number binning scheme and othercomputational parameters. We use the same binningscheme throughout this paper as we investigate how theoccupation numbers diverge from FD equilibrium, start-ing in Sec. IV. In Sec. V, we present a new way of charac-terizing degenerate neutrinos in the early universe. Sec.VI details the changes to the primordial abundances fromthe out-of-equilibrium spectra. We give our conclusionsin Sec. VII. Throughout this paper we use natural units, (cid:126) = c = k B = 1, and assume neutrinos are massless atthe temperature scales of interest. II. ANALYTICAL TREATMENT
To characterize the lepton asymmetry residing in theneutrino seas in the early universe, we use the followingexpression in terms of neutrino, ν , antineutrino, ν , andphoton, γ , number densities to define the lepton numberfor a given neutrino flavor L i ≡ n ν i − n ν i n γ , (1)where i = e, µ, τ . The photons are assumed to be in aPlanck distribution at plasma temperature T , with num-ber density n γ = 2 ζ (3) π T , (2)where ζ (3) ≈ . n ν i = T π (cid:90) ∞ d(cid:15) (cid:15) f ν i ( (cid:15) ) . (3)Here, T cm is the comoving temperature parameter andscales inversely with scale factor aT cm ( a ) = T cm ,i (cid:16) a i a (cid:17) , (4) where the i subscripts reflect a choice of an initial epochto begin the scaling. In this paper, we will choose T cm ,i such that T cm is coincident with the plasma temperaturewhen T = 10 MeV. For T > T cm ,i , the plasma temper-ature and comoving temperature parameter are nearlyequal as the neutrinos are in thermal equilibrium withthe photon/electron/positron plasma. T and T cm divergefrom one another once electrons and positrons begin an-nihilating into photon and neutrino/antineutrino pairsbelow a temperature scale of 1 MeV. The dummy vari-able (cid:15) in Eq. (3) is the comoving energy and related to E ν , the neutrino energy, by (cid:15) = E ν /T cm . The sets of f ν i are the phase-space occupation numbers (also referredto as occupation probabilities) for species ν i indexed by (cid:15) . In equilibrium the occupation numbers for neutrinosbehave as FD f (eq) ( (cid:15) ; ξ ) = 1 e (cid:15) − ξ + 1 , (5)where ξ is the neutrino degeneracy parameter related tothe chemical potential as ξ = µ/T cm . Unlike the leptonnumber for flavor i in Eq. (1), the corresponding degener-acy parameter ξ i is a comoving invariant. If we considerthe equilibrium occupation numbers in the expression fornumber density, Eq. (3), we find n (eq) ν = T π (cid:90) ∞ d(cid:15) (cid:15) e (cid:15) − ξ + 1 = T π F ( ξ ) , (6)where F ( ξ ) is the relativistic Fermi integral given by thegeneral expression F k ( ξ ) = (cid:90) ∞ dx x k e x − ξ + 1 . (7)We can define the following normalized number distribu-tion F ( (cid:15) ; ξ ) d(cid:15) ≡ dn (cid:82) dn = 1 F ( ξ ) (cid:15) d(cid:15)e (cid:15) − ξ + 1 . (8)Figure 1 shows F plotted against (cid:15) for three differentvalues of ξ .The expressions for the number densities in Eqs. (2)and (3) have different temperature/energy scales. As thetemperature decreases, electrons and positrons will an-nihilate to produce photons primarily, thereby changing T with respect to T cm . As a result, Eq. (1) decreasesfrom the addition of extra photons. To alleviate thiscomplication, we calculate the lepton number at a highenough temperature such that the neutrinos are in ther-mal and chemical equilibrium with the plasma. We cantake T cm = T at high enough temperature and write Eq.(1) as L (cid:63)i = 14 ζ (3) (cid:90) ∞ d(cid:15) (cid:15) [ f ν i ( (cid:15) ) − f ν i ( (cid:15) )] , (9)where we call L (cid:63)i the comoving lepton number. Eq. (9)simplifies further if we use the FD expression in Eq. (5) (cid:15) . . . . . . . F ( (cid:15) ; ξ ) ξ = 0 ξ = 3 . ξ = − . FIG. 1. Normalized number density plotted against (cid:15) forthree choices of degeneracy parameter: nondegenerate ( ξ = 0,solid blue), degenerate with an excess ( ξ = 3 .
0, dashed red),and degenerate with a deficit ( ξ = − .
0, dash-dotted green). and recognize that in chemical equilibrium the degener-acy parameters for neutrinos are equal in magnitude andopposite in sign to those of antineutrinos L (cid:63)i = π ζ (3) (cid:34) ξ i π + (cid:18) ξ i π (cid:19) (cid:35) , (10)where ξ i is the degeneracy parameter for neutrinos of fla-vor i . Eq. (10) provides an algebraic expression for relat-ing the lepton number to the degeneracy parameter withno explicit dependence on temperature. We will give ourresults in terms of comoving lepton number and use Eq.(10) to calculate the degeneracy parameter for input intothe computations. In this paper, we will only considerscenarios where all three neutrino flavors have identicalcomoving lepton numbers. Unless otherwise stated, wewill drop the i subscript and replace it with the neutrinosymbol, i.e. L (cid:63)ν , to refer to all three flavors.Degeneracy in the neutrino sector increases the totalenergy density in radiation. The parameter N eff is de-fined in terms of the plasma temperature and the radia-tion energy density ρ rad = (cid:34) (cid:18) (cid:19) / N eff (cid:35) π T . (11)Eq. (11) can be used at any epoch, even one in whichthere exists seas of electrons and positrons, e.g. Eq. (31)in Ref. [1]. We will consider ρ rad and T at the epoch T = 1 keV, after the relic seas of positrons and electronsannihilate. Assuming equilibrium spectra for all neutrinospecies, the deviation of N eff , ∆ N eff , from exactly 3 wouldbe∆ N eff ≡ N eff − (cid:88) i (cid:34) (cid:18) ξ i π (cid:19) + 157 (cid:18) ξ i π (cid:19) (cid:35) , (12) where the summation assumes the possibility of differentneutrino degeneracy parameters for each flavor [34, 40].We begin by presenting the case of instantaneous neu-trino decoupling with pure equilibrium FD distributions.Table I shows the deviations in energy densities for neu-trinos and antineutrinos with respect to nondegenerateFD equilibrium, the asymptotic ratio of T cm to T , andthe change to N eff , for various comoving lepton numbers.In this paper, we will colloquially refer to the asymptoteof any quantity as the “freeze-out” value. For the val-ues of L (cid:63)ν in Table I, a decade decrease in L (cid:63)ν producescomparable decreases in δρ ν and | δρ ν | . L (cid:63)ν is related tothe energy densities through the degeneracy parameterderived from Eq. (10), which is approximately linear in ξ for small L (cid:63)ν . The change in N eff is quadratic in ξ which is discernible for L (cid:63)ν = 10 − and L (cid:63)ν = 10 − atthe level of precision presented in Table I. The freeze-outvalue of T cm /T is not identically (4 / / , the canon-ical value deduced from covariant entropy conservation[41, 42]. Although the neutrino-transport processes areinactive for Table I and therefore the covariant entropyis conserved, finite-temperature quantum electrodynamic(QED) effects act to perturb T cm /T away from the canon-ical value [43, 44]. III. NUMERICAL APPROACH
For this work, changes to the quantities of interest willbe as small as a few parts in 10 . To ensure our resultsare not obfuscated by lack of numerical precision, we needan error floor smaller than the numerical significance ofa given result. In burst , we bin the neutrino spectra inlinear intervals in (cid:15) -space. The binning scheme has twoconstraints: the maximum value of (cid:15) to set the range; andthe number of bins over that range. We denote the twoquantities as (cid:15) max and N bins , respectively, and examinehow they influence the errors in our procedure.The mathematical expressions for the neutrino spectrahave no finite upper limit in (cid:15) . We need to ensure (cid:15) max islarge enough to encompass the probability in the tails ofthe curves in Fig. 1. As an example, consider the normal-ized number density in Eq. (8). We would numericallyevaluate the normalization condition as1 (cid:39) (cid:90) (cid:15) max d(cid:15) F ( (cid:15) ; ξ ) . (13)For large (cid:15) , F ∼ (cid:15) e − (cid:15) + ξ , and so we exclude a contri-bution to the above integral on the order of (cid:15) e − (cid:15) max if we take ξ = 0. If we are using double precision arith-metic, the contribution becomes numerically insignificantfor F ( (cid:15) max ; 0) < − , which corresponds to (cid:15) max (cid:39) (cid:15) max would seem like the natural value totake without loss of a numerically significant contribu-tion to the integral in Eq. (13). However, if we fix thenumber of abscissa in the partition used when integratingEq. (13) (i.e. fixing N bins in the binned neutrino spectra),we lose precision in the evaluation of the contribution to L (cid:63)ν δρ ν δρ ν T cm /T N eff − . − . . . − . × − − . × − . . − . × − − . × − . . − . × − − . × − . . . . ν and and ν energydensities compared to a FD energy distribution with zero degeneracy parameter at freeze-out. Column 4 shows the ratio ofcomoving temperature parameter to plasma temperature also at freeze-out. For comparison, (4 / / = 0 . N eff . N eff does not converge to precisely 3 . the integral from each bin as we increase (cid:15) max . Clearly,there is a trade off between (cid:15) max and N bins .Figure 2 examines the (cid:15) max versus N bins parameterspace by looking at the calculation of the equilibriumcomoving lepton number, in a scenario where ξ (cid:54) = 0. Wetake L (cid:63)ν to be exactly 0 . ξ . Next, we calculate neutrino and antineu-trino spectra with the equal and opposite degeneracy pa-rameters. We proceed to integrate Eq. (9) with the twospectra for different pairs of ( N bins , (cid:15) max ) values. The in-tegration is carried out using Boole’s rule, a fifth-orderintegration method for linearly spaced abscissas. Figure2 shows the filled contours of log values for the error in L (cid:63)ν δL (cid:63)ν = (cid:12)(cid:12)(cid:12)(cid:12) L (cid:63)ν [ (cid:15) max , N bins ] − . . (cid:12)(cid:12)(cid:12)(cid:12) , (14)for a given pair ( N bins , (cid:15) max ). We immediately see theloss of precision in the upper-left corner of the parameterspace, corresponding to small N bins and large (cid:15) max . Fur-thermore, for (cid:15) max (cid:46)
40, the error value flat lines withincreasing N bins , implying that the error is a result of atoo small choice for (cid:15) max . The black curve superimposedon the heat map gives the value of (cid:15) max with the lowesterror as a function of N bins . It monotonically increases for100 < N bins (cid:46) (cid:15) max ∼ N bins adds no more numerical significance.The computation time required to run burst scalesas N . In this paper, we attempt to be as comprehen-sive as possible when exploring neutrino energy transportwith nonzero lepton numbers. Therefore, we will choose100 bins for the sake of expediency. Fig. 2 guides us inpicking (cid:15) max = 25, and dictates a floor of ∼ − forour best possible precision. It would appear that thechoice ( N bins , (cid:15) max ) (cid:39) (300 ,
40) would give the absolutebest precision for calculating L (cid:63)ν . This is valid if usingthe linearly spaced abscissas as a binning scheme. Wehighlight both the precision and timing needs for a com-prehensive numerical study on binning schemes. Such astudy would be germane for the more general problemwhich includes neutrino oscillations and disparate lepton
200 400 600 800 1000 N bins (cid:15) m a x − . − . − . − . − . − . − . − . − . − . − . l og δ L ? ν FIG. 2. (cid:15) max versus N bins for contours of constant errorin L (cid:63)ν . The exact value of L (cid:63)ν is 0 .
1. The black line on thecontour space gives the value of (cid:15) max with the smallest erroras a function of N bins . numbers in the three active species [29, 30, 39].For more details on the numerics of burst , we referthe reader to Ref. [1]. We have essentially preserved thecomputational parameters except for a quantity relatedto the determination of nonzero scattering rates. Ref.[1] used ε (net / FRS) = 30, and in this work, we use ε (net / FRS) = 3.
IV. NEUTRINO SPECTRA
In this section we give a detailed accounting of how theneutrino energy spectra evolve through weak decouplingin the presence of zero and nonzero lepton numbers. Inthe first subsection we integrate the complete transportnetwork, including all the neutrino scattering processesin Table I of Ref. [1], from a comoving temperature pa-rameter T cm = 10 MeV down to T cm = 15 keV. In thesecond subsection we investigate how the different inter-actions between neutrinos and charged leptons affect thespectra.We compare our results to that of FD equilibrium. Forthe neutrino occupation numbers, we use the followingnotation to characterize the deviations from FD equilib-rium δf ξ ( (cid:15) ) = f ( (cid:15) ) − f (eq) ( (cid:15) ; ξ ) f (eq) ( (cid:15) ; ξ ) . (15)Here, f (eq) ( (cid:15) ; ξ ) is the FD equilibrium occupation num-ber for degeneracy parameter ξ given in Eq. (5). When itis obvious, we will drop the argument (cid:15) , i.e. δf ξ ( (cid:15) ) → δf ξ .As an example, δf gives the relative difference of the oc-cupation number from the nondegenerate, zero chemicalpotential FD equilibrium value.We also examine the absolute changes for the numberand energy distributions∆ (cid:18) dnd(cid:15) (cid:19) ξ ≡ T π (cid:15) [ f ( (cid:15) ) − f (eq) ( (cid:15) ; ξ )] (number) , (16)∆ (cid:18) dρd(cid:15) (cid:19) ξ ≡ T π (cid:15) [ f ( (cid:15) ) − f (eq) ( (cid:15) ; ξ )] (energy) . (17)When using the absolute change expressions, we normal-ize with respect to an equilibrium number or energy den-sity in order to compare to dimensionless expressions.For the energy density, we use the appropriate degener-acy factor ρ ξ ≡ T π (cid:90) ∞ d(cid:15) (cid:15) f (eq) ( (cid:15) ; ξ ) . (18)For the number density, we will exclusively use zero forthe degeneracy factor n ≡ T π (cid:90) ∞ d(cid:15) (cid:15) f (eq) ( (cid:15) ; 0)= 34 ζ (3) π T . (19)The out-of-equilibrium evolution of the neutrino occupa-tion numbers driven by scattering and annihilation pro-cesses with charged leptons does not proceed in a unitaryfashion. Consequently, the total comoving neutrino num-ber density increases. The increase in number results inan increase in energy density, and so we use ρ ξ to nor-malize the absolute changes in differential energy densitydistribution to compare with the initial distribution athigh temperature. However, the difference in numberdensity between neutrinos and antineutrinos, character-ized by the comoving lepton number in Eq. (9), does notchange with kinematic neutrino transport. In practice, burst follows the evolution of neutrino and antineutrinooccupation numbers separately, precipitating the possi-bility of numerical error. We will use the same normal-ization for neutrino and antineutrino differential number density distributions to study the relative error in L (cid:63)i .We will take the normalization quantity to be that of thenondegenerate number density in Eq. (19). A. All processes
Table II shows how neutrino transport alters neutrinoenergy densities, N eff , the ratio of comoving tempera-ture parameter to plasma temperature, and entropy perbaryon in the plasma, s pl . These quantities are computedfor a range of L (cid:63)i values and refer to the results at the endof the transport calculation, T cm ∼ δρ i = T π (cid:90) ∞ d(cid:15) (cid:15) f i ( (cid:15) ) − ρ ρ , ρ = 78 π T . (20)Columns 2 - 5 of Table II show the relative changes in en-ergy density at T cm = 1 keV, once the neutrino spectrahave converged to their out-of-equilibrium shapes. Wesee a monotonic decrease in δρ ν for the neutrinos, anda monotonic increase in δρ ν for the antineutrinos withdecreasing L (cid:63)ν . Column 6 gives the ratio of T cm /T at theend of the simulation. T cm /T increases with decreasinglepton number. However, the decrease is less than onepart in 10 between L (cid:63)ν = 0 . L (cid:63)ν = 0. The largerlepton number implies a larger total energy density whichincreases the Hubble expansion rate. The faster expan-sion implies a smaller time window for the entropy flowout of the plasma and into the neutrino seas. As a re-sult, the evolution of the plasma temperature is such thatlarger lepton numbers will maintain T at higher values,and the ratio T cm /T at freeze-out will decrease, albeit byan amount which is numerically insignificant. With thechanges in energy densities and temperature ratios, wecan calculate N eff N eff = (cid:20) T cm /T (4 / / (cid:21)
12 [(2+ δρ ν e + δρ ν e )+2(2+ δρ ν µ + δρ ν µ )] . (21)The coefficient in front of the second parenthetical ex-pression, equal in value to 2, results from the approxima-tion in taking the µ and τ flavors to behave identically.The approximation employed here is valid as there areno µ and τ charged leptons in the plasma and L i is thesame in all flavors. Both Refs. [36, 39] calculate weakdecoupling with a network featuring neutrino flavor os-cillations, which are absent in our calculation in TableII. However, Ref. [39] states that oscillations have no af-fect on the value of N eff at the level of precision whichthey use. The difference in our value of N eff versus thestandard calculation of Ref. [36] is most likely due to adifferent implementation of the finite-temperature QEDeffects detailed in Refs. [43, 44]. Ref. [36] uses the per-turbative approach outlined in Ref. [45] compared to ournonperturbative approach. We leave a detailed studyof the finite-temperature-QED-effect numerics to futurework.The final column of Table II shows the change in theentropy per baryon in the plasma. The relative changesin entropy for varying lepton numbers are large enoughto see a difference at the level of precision Table II uses,unlike T cm /T . With the faster expansion, neutrinos haveless time to interact with the plasma, yielding a smallerentropy flow.An increase in lepton number implies a larger energydensity for the neutrinos over the antineutrinos. Fig. 3shows four neutrino spectra after the conclusion of weakdecoupling in a scenario where L (cid:63)ν = 0 .
1. Plotted against (cid:15) is the relative difference in the neutrino occupationnumber with respect to a nondegenerate spectrum. Asseen in the first data row of Table II, δρ ν e obtains thelargest difference from equilibrium. The thick red linein Fig. 3 shows the final out-of-equilibrium spectrum for ν e . The ν e spectrum has the largest deviation from equi-librium, congruent with Table II. The black dashed linesshow equilibrium spectra for nondegenerate (flat, hori-zontal line) and degenerate cases. As (cid:15) increases, theneutrino curves diverge from the positive ξ ν spectrum inmuch the same manner as the antineutrino curves divergefrom the negative ξ ν . The primary difference in the out-of-equilibrium spectra is due to the initial condition thatthe neutrinos have larger occupation numbers over theantineutrinos for a positive lepton number.We would like to compare the out-of-equilibrium spec-tra to their respective equilibrium spectra. Such a com-parison allows us to examine how the initial asymmetrypropagates through the Boltzmann network. Fig. 4 showsthe T cm evolution of δf ξ for ν e (thick solid lines) and ν e (thin solid lines) for a scenario where L (cid:63)ν = 0 .
1. We onlyshow the relative differences for three unique values of (cid:15) ,namely (cid:15) = 3 , ,
7. The ν µ and ν µ spectra follow similarshapes, but are suppressed relative to the electron flavors.For comparison, we also plot the out-of-equilibrium spec-trum for ν e in the case of no initial asymmetry, i.e. L (cid:63)ν identically zero. It is unnecessary to show the spectrumfor ν e when L (cid:63)ν = 0 because it is exceedingly near the ν e spectrum (see Fig. [3] of Ref. [1]). For the degeneratespectra, the ν e show a larger divergence from equilibriumthan the ν e at these three specific (cid:15) values. This is con-sistent with Ref. [34] (see Figs. 8 and 9 therein) and isthe case for all (cid:15) after the neutrino spectra have frozenout. Fig. 5 shows the final freeze-out values of the rel-ative changes in the neutrino occupation numbers as afunction of (cid:15) . Fig. 5 is similar to Fig. 3 except for the useof the general δf ξ instead of δf . We have also includedthe transport-induced out-of-equilibrium spectra for ν e and ν µ in the nondegenerate scenario. For a given fla-vor, the relative changes in the nondegenerate spectrumare nearly averages of those in the ν and ν spectra. Wealso note that for (cid:15) (cid:46)
2, all of the relative differences are (cid:15) = E ν /T cm − × δ f solid : Non Eq . dashed : FD Eq .ξ ν = 0 . ξ = 0 ξ ¯ ν = − . ν e ν e ν µ ν µ FIG. 3. Relative differences in neutrino/antineutrino occupa-tion numbers plotted against (cid:15) at T cm = 1 keV. The relativedifferences are with respect to FD with zero degeneracy pa-rameter. The solid lines show the evolution for a scenariowhere L (cid:63)ν = 0 . ν e and ν e curves are colored red, and ν µ and ν µ curves are colored green. Neutrinos have thick line widthsand antineutrinos have thin line widths. Plotted for com-parison are black dashed curves representing the equilibriumrelative differences. The top dashed curve corresponds to a ν spectrum with ξ ν = 0 . ν spectrum with ξ ν = 0, and the bottom dashedcurve corresponds to a ν spectrum with ξ ν = − . negative, although this is obscured in Fig. 5 due to theclustering of lines. For small (cid:15) , the antineutrino occupa-tion numbers are larger than those of the neutrino, i.e.the δf ( ν ) ξ are not as negative.The weak interaction cross sections scale as σ ∼ G F E , where G F is the Fermi constant ( G F ≈ . × − MeV − ) and E is the total lepton energy. Wewould expect a larger difference from equilibrium for in-creasing (cid:15) . Except for the range 0 < (cid:15) (cid:46)
1, Figures 4 and5 clearly show an increase. The change in the energydistribution does not follow from a scaling relation. Fig.6 shows the normalized absolute difference in the energydistribution plotted against (cid:15) at the conclusion of weakdecoupling. The nomenclature for the six lines in Fig. 6is identical to that of Fig. 5. The energy distributionsall show a maximum at (cid:15) ∼
5. Similar to Fig. 5, thenondegenerate curves of Fig. 6 appear to be averages ofthe ν and ν curves in the degenerate scenario.In the positive lepton-number scenarios, the ν al-ways have larger occupation numbers than the ν ,when compared against the equilibrium degenerate spec-trum/distribution. This is not surprising as the occupa-tion numbers for antineutrinos are suppressed, implyingless blocking. When compared against its equilibriumdistribution, the ν have larger rates, leading to a largerdistortion. In Fig. 7, we compare the out-of-equilibriumnumber density distributions with those of the nondegen- L (cid:63)ν δρ ν e δρ ν e δρ ν µ δρ ν µ T cm /T N eff × ( s ( i )pl /s ( f )pl − − . − . . − . . . . − . × − − . × − . × − − . × − . . . − . × − . × − . × − . × − . . . − . × − . × − . × − . × − . . . . × − . × − . × − . × − . . . ν e , ν e , ν µ and ν µ energy densitiescompared to a FD energy distribution with zero degeneracy parameter. Comparisons are given for T cm ∼ / / = 0 . N eff as calculated by Eq. (21). Column8 gives the fractional change in the entropy per baryon in the plasma, s pl . − T cm (MeV)0 . . . . . . × δ f ξ (cid:15) = 7 (cid:15) = 5 (cid:15) = 3 ν e ( L ?ν = 0 . ν e ( L ?ν = 0) ν e ( L ?ν = 0 . FIG. 4. Relative differences in electron neutrino/antineutrinooccupation numbers plotted against T cm . The relative dif-ferences are with respect to FD with the same degeneracyparameters as Fig. 3. The solid lines show the evolution fora scenario where L (cid:63)ν = 0 .
1. The ν e (thin red curves) has alarger relative change than the ν e (thick red curves). Plottedfor comparison is the relative difference for ν e in a L (cid:63)ν = 0scenario (blue dash-dot curves). The relative differences areplotted for three values of (cid:15) , from bottom to top: (cid:15) = 3 , , erate case solely. In other words, the normalizing factor n is the same for each of the six curves in Fig. 7. Wehave adopted this nomenclature for the comparison ofnumber density distributions to study the change in thecomoving lepton number. None of the weak decouplingprocesses modify the lepton number in our model. Thetotal change in number density for ν should be identicalto the total change in number density for ν . Fig. 7 showsthis indirectly. We can see a difference; the ν curves areskewed to higher (cid:15) and have a larger maximum than the ν . The negative change in the distributions for the range0 ≤ (cid:15) (cid:46) ν become positive forsmaller (cid:15) than those of ν , implying there are more ν than ν for (cid:15) (cid:46)
2. Overall, when integrating the curves in Fig. (cid:15) = E ν /T cm × δ f ξ ν e ( L ?ν = 0 . ν e ( L ?ν = 0) ν e ( L ?ν = 0 . ν µ ( L ?ν = 0 . ν µ ( L ?ν = 0) ν µ ( L ?ν = 0 . ν µ ( L ?ν = 0 . ν µ ( L ?ν = 0) ν µ ( L ?ν = 0 . FIG. 5. Relative differences in neutrino/antineutrino oc-cupation numbers plotted against (cid:15) at T cm = 1 keV. Therelative differences are with respect to FD with the same cor-responding degeneracy parameters as Fig. 3. The solid linesare for a scenario where L (cid:63)ν = 0 .
1. Plotted for comparison isthe relative difference for ν e (blue dash-dot curve) and for ν µ (blue dotted curve) in a L (cid:63)ν = 0 scenario.
7, the total changes in number density for ν should bethe same as for ν . We have calculated this quantity andexpressed it as a relative change in the L (cid:63)i , taken to beexactly 0 . δL (cid:63)i ≡ ζ (3) (cid:90) ∞ d(cid:15) (cid:15) [ f ν i ( (cid:15) ) − f ν i ( (cid:15) )] − . . . (22)Eq. (22) gives the relative error in our calculation. Weconserve the comoving lepton number for both electronand muon flavor at approximately 7 × − . Also plot-ted in Fig. 7 are the absolute changes for ν e and ν µ inthe nondegenerate scenario. We do not directly comparethe lepton-number relative errors as the quantity is notdefined for the symmetric case. We do note that the non-degenerate curves are close to the average of the ν and ν (cid:15) = E ν /T cm − . . . . . . × ∆ (cid:16) d ρ d (cid:15) (cid:17) ξ / ρ ξ FIG. 6. Absolute change in the neutrino/antineutrino energydistributions plotted against (cid:15) at T cm = 1 keV. The changesare with respect to the same degeneracy parameters as thosein Fig. 5. Furthermore, the line colors and styles correspondto the same species and scenarios as Fig. 5. (cid:15) = E ν /T cm − . . . . . . . . × ∆ (cid:0) d n d (cid:15) (cid:1) ξ / n δL ?ν e ’ . × − δL ?ν µ ’ . × − FIG. 7. Absolute change in the neutrino/antineutrino num-ber distributions plotted against (cid:15) at T cm = 1 . δL (cid:63)i for i = e, µ ) show the relative error accumulated over the courseof a simulation. distributions, similar to that of Figs. 5 and 6.In Figs. 3 through 7, we have only presented the L (cid:63)ν = 0 . ν plotted against (cid:15) for other val-ues of L (cid:63)ν . The behavior of each curve is in line with thoseof Fig. 5. Not plotted are the curves for ν . They also be-have in a similar manner, where δf ν becomes larger than δf for increasing (cid:15) . The result is that with transport, (cid:15) = E ν /T cm − × δ f ξ ν e ν µ L ?ν = 0 L ?ν = 10 − L ?ν = 5 × − L ?ν = 10 − FIG. 8. Relative differences in neutrino occupation num-bers plotted against (cid:15) at T cm = 1 keV for various values of L (cid:63)ν . The relative differences are with respect to FD withthe corresponding degeneracy parameter, namely ξ = 0 . L (cid:63)ν = 10 − ), ξ = 0 . L (cid:63)ν = 5 × − ), ξ = 0 . L (cid:63)ν = 10 − ), and ξ = 0 ( L (cid:63)ν = 0). Shown are two sets ofcurves: the set with the larger relative differences correspondto the ν e spectral distortions, and the set with the smallerdifferences are ν µ . Within each set, the increase in L (cid:63)ν leadsto a decrease in δf ξ . For the antineutrinos, the relative dif-ferences behave in the opposite manner: increase in L (cid:63)ν leadsto an increase in δf ξ . L (cid:63)ν acts to increase the asymmetries in the occupationnumbers, which manifest in differences in the absolutechanges of the differential energy density. B. Individual processes
Figs. 5 and 6 demonstrate that the initial asymmetryin the neutrino energy density is maintained and evenamplified by scattering processes. We can dissect therelative contribution of various scattering processes tothis amplification.Figs. 9 and 10 show the absolute changes in the numberdensity distribution versus (cid:15) when we include only certaintransport processes. Fig. 9 contains three annihilationprocesses, schematically shown as: ν i + ν i ↔ e − + e + , i = e, µ, τ. (23)In this scenario, we have included only the annihila-tion channel into electron/positron pairs when comput-ing transport. The changes are with respect to the samedegeneracy parameters as those in Fig. 5. The line colorsin Fig. 9 correspond to the same species as Fig. 5. Be-cause of the close proximity of the neutrino and antineu-trino curves, we depart from the previous nomenclatureof emphasizing the ν curves with a thicker line width soas not to obscure the ν curves. For this plot, the absolute (cid:15) = E ν /T cm . . . . . . . . × ∆ (cid:0) d n d (cid:15) (cid:1) ξ / n δL ?ν e = 3 . × − δL ?ν µ = 5 . × − ν e ν e ν µ ν µ ν e ν e ν µ ν µ FIG. 9. Absolute change in the neutrino/antineutrino num-ber distributions plotted against (cid:15) at T cm = 1 . δL (cid:63)i for i = e, µ ) show the relativeerror accumulated over the course of a simulation. differences are normalized with respect to the equilibriumnumber density at temperature T cm with degeneracy pa-rameter ξ = 0. For a given neutrino species, the totalchange in number density should be equal to the changein number density for the corresponding antineutrino.Fig. 10 shows the effect of including 12 elastic scatter-ing processes: ν i + e − ↔ ν i + e − , (24) ν i + e + ↔ ν i + e + , (25)and the opposite- CP reactions, for neutrino flavors i = e, µ, τ . In this scenario, we have included only the elas-tic scattering channel with electrons/positrons (while ne-glecting the neutrino-antineutrino only channels) whencomputing transport. The changes are with respect tothe same degeneracy parameters as those in Fig. 5. Fur-thermore, the line colors and styles in Fig. 10 correspondto the same species and scenarios as Fig. 5. For this plot,the absolute differences are normalized with respect tothe equilibrium number density at temperature T cm withdegeneracy parameter ξ = 0. In an identical manner tothe processes in Fig. 9, the total change in ν number den-sity should be equal to the change in number density forthe corresponding ν in Fig. 10.The elastic scattering processes of Eqs. (24) and (25)(and the opposite- CP reactions) preserve the total num-ber of neutrinos and antineutrinos. The plasma ofcharged leptons acts to upscatter low energy neutrinosand antineutrinos to higher energies, precipitating an en-tropy flow. Fig. 10 vividly shows a deficit of neutrinosin the range 0 < (cid:15) (cid:46)
4, and the corresponding excess for (cid:15) (cid:38)
4. The deficit is more pronounced in Fig. 10 but alsoappeared in Figs. 5, 6, and 7 when computing the entireneutrino-transport network. The annihilation processes, (cid:15) = E ν /T cm − . − . − . − . . . . × ∆ (cid:0) d n d (cid:15) (cid:1) ξ / n δL ?ν e = 3 . × − δL ?ν µ = 5 . × − ν e ( L ?ν e = 0 . ν e ( L ?ν e = 0) ν e ( L ?ν e = 0 . ν µ ( L ?ν µ = 0 . ν µ ( L ?ν µ = 0) ν µ ( L ?ν µ = 0 . ν µ ( L ?ν µ = 0 . ν µ ( L ?ν µ = 0) ν µ ( L ?ν µ = 0 . FIG. 10. Absolute change in the neutrino/antineutrino num-ber distributions plotted against (cid:15) at T cm = 1 . δL (cid:63)i for i = e, µ ) show the relativeerror accumulated over the course of a simulation. shown in Fig. 9, do not preserve the total numbers of neu-trinos and antineutrinos and can fill the phase space va-cated by the upscattered neutrinos. The complete trans-port network, which includes annihilation, elastic scat-tering on charged leptons, and elastic scattering amongonly neutrinos/antineutrinos, is able to redistribute theadded energy by filling the occupation numbers for lowerepsilon. V. INTEGRATED ASYMMETRY MEASURES
In our presentation to this point, we have used the co-moving lepton number to describe the asymmetry in theearly universe. L (cid:63)i does not evolve with temperature inour model, except for errors in precision encountered byour code. Therefore, we introduce two integrated quanti-ties to examine how the initial asymmetry propagates tolater times. The quantities provide new means to analyzethe out-of-equilibrium spectra.The first integrated quantity we define is the leptonenergy density asymmetry R i ≡ ρ ν i − ρ ν i π T . (26)where i is the flavor index. Like the comoving leptonnumber in Eq. (9), we divide Eq. (26) by T so that R i is comoving and dimensionless. This will allow usto follow the evolution of R i to later times. At large T cm , all flavors have identical equilibrium FD spectra andlepton numbers/degeneracy parameters. For degeneracy0 − T cm (MeV)0 . . . . . . . . . . × δ R δR e (All) δR µ (All) δR e (Annih . ) δR µ (Annih . ) δR e (Scatt . ) δR µ (Scatt . ) δR e (All) δR µ (All) δR e (Annih . ) δR µ (Annih . ) δR e (Scatt . ) δR µ (Scatt . ) FIG. 11. The relative changes in R plotted against T cm for ν e and ν µ with different processes included in the trans-port calculation. Red lines correspond to ν e and green linescorrespond to ν µ . The process scheme is all processes (solidcurves), annihilation only (dash-dot curves), or elastic scat-tering only (dotted curves). parameter ξ , we calculate the equilibrium value of RR (eq) = sgn( ξ ) (cid:20)
78 + 154 (cid:18) ξπ (cid:19) + 158 (cid:18) ξπ (cid:19) − π e −| ξ | Φ( − e −| ξ | , , (cid:21) , (27)where sgn( x ) is the sign function with real-number argu-ment x , and Φ( z, s, v ) is the Lerch function (see Sec. 9.55of Ref. [46])Φ( z, s, v ) ≡ ∞ (cid:88) n =0 z n ( n + v ) s ; | z | < v (cid:54) = 0 , − , − , .... (28)Figure 11 shows the relative changes in R i from the R (eq) baseline ( δR i ), plotted against T cm for differentcombinations of transport processes. Solid lines (All)are for the complete calculation, whereas dash-dot curvesonly include the annihilation channels (Annih.) of the re-action shown in (23), and dotted curves only include theelastic scattering channels (Scatt.) of the reactions shownin (24), (25), and the opposite- CP reactions. Red linescorrespond to δR e and green lines to δR µ . δR i increasesfor all six combinations of flavor and transport process,until an eventual freeze-out. Indirectly, Figs. 7, 9, and10 all show that the neutrinos have larger changes in theenergy density distributions, increasing the asymmetry. Because of the charged-current process, δR e experiencesa greater enhancement. What is important to note isthat the total δR i , for either flavor, is not an incoherentsum of the two transport processes taken individually.There are two reasons for this.First, there are other transport processes in the fullcalculation. Neutrinos scattering on other neutrinos andantineutrinos will redistribute energy density. Second,the transport processes with the charged leptons are de-pendent on one another. Positron-electron annihilationinto neutrino-antineutrino pairs populates the lower en-ergy levels. Those particles upscatter on charged leptonsthrough elastic scattering. Positron-electron annihilationis then suppressed by the Pauli blocking of the upscat-tered particles. Both reasons change the evolution of thetotal R i , but do so in a flavor-dependent manner. For δR µ , the incoherent sum of annihilation and elastic scat-tering is smaller than that of the total asymmetry. For δR e , the total asymmetry is dominated by the contribu-tion from elastic scattering.In analogy with the lepton energy density asymmetry,we define the lepton entropy asymmetry asΣ i ≡ S ν i − S ν i π T (29)where S j is the entropic density for particle j , given by S j = − T π (cid:90) ∞ d(cid:15) (cid:15) [ f j ln f j + (1 − f j ) ln(1 − f j )] , (30)and we have suppressed the arguments of f j ( (cid:15) ; ξ ) forbrevity in notation. Under the equilibrium assumptions,we findΣ (eq) = R (eq) − π ξ (cid:104) ζ (3) | L (cid:63)ν | + e −| ξ | Φ( − e −| ξ | , , (cid:105) . (31)Fig. 12 shows the evolution of the relative change inΣ i away from Σ (eq) when divided into processes. Thenomenclature for the line styles and colors is identicalto that in Fig. 11. The evolution of the lepton entropyasymmetry shows more features than that of the leptonenergy density asymmetry.To understand the dynamics of Σ i in Fig. 12, we beginby considering how the entropy depends on perturbationsto the occupation numbers. We write the occupationnumbers as differences from FD equilibrium f j ( (cid:15) ; ξ ) = f (eq) j ( (cid:15) ; ξ ) + ∆ f j ( (cid:15) ; ξ ) . (32)We can calculate the change in the entropy producedby the out-of-equilibrium occupation numbers by sub-stituting Eq. (32) into Eq. (30). After dropping the (cid:15) argument, ξ argument, and species index for notationalbrevity, we find for small ∆ f − T cm (MeV) − . − . − . . . . . . × δ Σ FIG. 12. The relative changes in Σ plotted against T cm . Linecolors and styles correspond to the same transport processesand neutrino flavors in Fig. 11. S = − T π (cid:90) ∞ d(cid:15) (cid:15) [( f (eq) + ∆ f ) ln( f (eq) + ∆ f ) + (1 − f (eq) − ∆ f ) ln(1 − f (eq) − ∆ f )] (33) (cid:39) − T π (cid:90) ∞ d(cid:15) (cid:15) (cid:20) f (eq) ln f (eq) + (1 − f (eq) ) ln(1 − f (eq) ) + ∆ f ln f (eq) − f (eq) (cid:21) (34)= S (eq) − T π (cid:90) ∞ d(cid:15) (cid:15) ∆ f [ ξ − (cid:15) ] (35)= S (eq) − ξ ∆ n + ∆ ρ/T cm , (36)where ∆ n and ∆ ρ are the changes in number and energydensity, respectively, from equilibrium. The expressionfor the lepton entropy asymmetry isΣ i = Σ (eq) + 454 π T (cid:20) − ξ (∆ n ν i + ∆ n ν i ) + ∆ ρ ν i − ∆ ρ ν i T cm (cid:21) . (37)Lepton number is conserved in our scenarios, implying∆ n ν i = ∆ n ν i . As a result, we can write the lepton en-tropy asymmetry asΣ i = Σ (eq) + 454 π T (cid:20) − ξ ∆ n ν i + ∆ ρ ν i − ∆ ρ ν i T cm (cid:21) . (38)Eq. (38) shows how the lepton entropy asymmetrychanges for small perturbations to the occupation num-bers. Two trends are evident from this equation. First,adding particles (∆ n ν i >
0) decreases the asymme-try. Second, increasing the asymmetry in energy density(∆ ρ ν i − ∆ ρ ν i > (cid:15) space for all flavors (see Fig. 9). Therefore, the correspondingchanges in the energy density will also be the same, andthere will be no contribution to the change in Σ fromthe energy density terms. The dash-dot curves in Fig.12 shows the relative change in Σ for a run with onlythe annihilation channels active. Both the e and µ fla-vors show a suppression in Σ with decreasing T cm . Fig-ure 10 shows that for elastic scattering of neutrinos andcharged leptons, the neutrino and antineutrino numberdensity distributions are not coincident. Overall, eachneutrino species has zero net change in number density,as elastic scattering can only redistribute the number.Therefore, there will be no contribution to the changein Σ from the number density term. As there are moreneutrinos over antineutrinos for L (cid:63)ν >
0, elastic scatter-ing enhances the neutrino spectra over the antineutrinospectra. The result is a net positive change in the energydensity differences. Fig. 12 shows an increase in the rel-ative change in Σ for the elastic-scattering-only runs forboth flavors. When we add the elastic-scattering and an-nihilation channels together, along with the other trans-port processes which do not involve charged leptons, we2 − T cm (MeV) − . − . . . . . . × δ Σ δ Σ e δ Σ µ FIG. 13. Same as Fig. 12 except zoomed in on the solid curves. see that the two processes essentially cancel, leaving onlya modest change in Σ i as shown by the solid lines in Fig.12.The interesting thing to note in Fig. 12 is the asym-metry between flavors. Fig. 13 is a zoomed-in version ofthe solid lines in Fig. 12. We see that δ Σ µ is monotoni-cally increasing for decreasing T cm . The incoherent sumof the relative changes from the annihilation and elastic-scattering processes in Fig. 12 nearly gives the relativechange in Σ µ that we obtain when all transport processesare active. The same cannot be said for δ Σ e . The sumof the two transport processes is not incoherent, the evo-lution of δ Σ e is not monotonic, and the final freeze-outvalue of δ Σ e is of opposite sign from δ Σ µ . Although theelastic scattering would appear to produce a larger en-hancement of δ Σ e over the suppression of annihilation,the two processes do not have equal weight. We observethis by looking at the maxima in the number density dis-tributions in Figs. 9 and 10. The ratio of maxima in Fig.9 for annihilation is (cid:18) dn ν e d(cid:15) (cid:19)(cid:30) (cid:18) dn ν µ d(cid:15) (cid:19) (cid:39) . . ) (39)The ratio of maxima in Fig. 10 for elastic scattering is (cid:18) dn ν e d(cid:15) (cid:19)(cid:30) (cid:18) dn ν µ d(cid:15) (cid:19) (cid:39) . . ) (40)This shows that annihilation is more dominant in theelectron neutrino/antineutrino sector than it is in the µ sector. In Figs. 9 and 10, we have only showed the finaldistributions at freeze-out. Electron-positron annihila-tion into neutrinos is not always so dominant, as evi-denced by the positive values of Σ e for T cm (cid:38)
400 keV.The analysis of the lepton entropy asymmetry focusedon the transport processes which involve the charged lep-tons. The other scattering processes redistribute occupa-tion number and therefore change Σ i . However, we have verified that the contributions from the transport pro-cesses which involve only neutrinos or antineutrinos donot alter Σ i enough to explain the full evolution shownin Fig. 13. The transport processes which involve thecharged leptons play the dominant roles.We have considered the evolution of the integratedasymmetry measures for L (cid:63)ν = 0 . R i and Σ i at freeze-out for vari-ous values of L (cid:63)ν . Note that the positive relative changesfor L (cid:63)ν < L (cid:63)ν are beneath the error floor. VI. ABUNDANCES
Our calculations show potentially significant changesin lepton-asymmetric BBN abundance yields with neu-trino transport relative to those without. With the in-clusion of transport we find that the general trends ofthe yields of He and D with increasing or decreasinglepton number are preserved: positive L (cid:63)ν decreasing theyields of both, while negative lepton numbers increaseboth. In broad brush, Boltzmann transport makes lit-tle difference for helium, but gives a ≥ . reduction in the offset from the FD, zero lepton-number case withtransport. This change in the reduction is comparable touncertainties in BBN calculations arising from nuclearcross sections and from plasma physics and QED issues.For all BBN calculations, the baryon to photon ratio isfixed to be n b /n γ = 6 . × − (equivalent to thebaryon density ω b = 0 . . L (cid:63)ν . Columnswith the label “Boltz.” are the calculations in the fullBoltzmann neutrino-transport calculation. Relative dif-ferences are with respect to the appropriate abundancein the zero-degeneracy Boltz. calculation. The relativechanges in the abundances for the two different calcula-tions are quite close: δY P differs by 2 - 3 parts in 10 ; and δ D / H differs by 3 - 4 parts in 10 . Both differences areconsistent across L (cid:63)ν . We caution against any interpreta-tion that links the two calculations together, as the NoTrans. calculations ignore important physics related tonon-FD spectra, entropy flow, and the Hubble expansionrate.We have examined the detailed evolution of the spec-tra and integrated asymmetry measures in the Boltz.calculations. The electron neutrinos and antineutrinosbehave differently compared to muon and tau flavoredneutrinos. This behavior will have ramifications for theneutron-to-proton ratio and nucleosynthesis. To facili-tate the analysis of the effects of neutrino transport on3 L (cid:63)ν × δR e × δR µ × δ Σ e × δ Σ µ − .
20 8 . − . . − .
26 8 . − . . − .
27 8 . − . . − − .
20 8 . − . . − − .
26 8 . − . . − − .
24 8 . − . . L (cid:63)ν . BBN, we will introduce a model which uses additional ra-diation energy density. We will try to determine whetherthis simplistic “dark radiation” model [48, 49] – whichincludes radiation energy density distinct from photonsand active neutrinos, but does not include transport –can mock up the effects of the extra energy density whicharise from neutrino scattering and the associated spectraldistortions. We will compare this dark-radiation modelto the full neutrino-transport case. For ease in notationwhen comparing the two scenarios, we will abbreviate thedark-radiation model as “DR” and the full Boltzmannneutrino-transport calculation as Boltz.In the DR model, we introduce extra radiation en-ergy density, ρ dr , described at early times by the dark-radiation parameter δ dr ρ dr = 78 π δ dr T . (41)The FD Eq. calculation in Table IV used δ dr = 0. Wemandate that the dark radiation be composed of rela-tivistic particles which are not active neutrinos. We havechosen the specific form of Eq. (41) for conformity with N eff , namely ∆ N eff ≈ δ dr . The relation is not a strictequality due to the presence of finite-temperature-QEDcorrections to the electron rest mass [1, 43, 44, 50, 51].The DR model differs from the Boltz. calculation in mul-tiple respects. First, the DR model fixes the neutrinospectra to be in degenerate FD equilibrium. Second, neu-trino transport induces an entropy flow from the plasmainto the neutrino seas, absent in the DR model. Third,the entropy flow changes the phasing of the plasma tem-perature with the comoving temperature parameter ascompared to the case of instantaneous neutrino decou-pling in the DR model. The phasing is dependent on theHubble expansion rate and the flow of entropy. Althoughthe expansion rates are identical in the two scenarios, theentropy flows are not.For all calculations, we will fix δ dr = 0 . N eff between the DR modeland Boltz. calculation for the single case L (cid:63)ν = 0 .
1. Thechange in N eff depends on the Hubble expansion rate,which depends on the initial degeneracy. Therefore, ourchoice of δ dr will not ensure equal values of N eff betweenthe two scenarios for L (cid:63)ν (cid:54) = 0 .
1. Although our DR modelis not consistent across all L (cid:63)ν , the changes in N eff aresmall for the range of L (cid:63)ν we explored. Figure 14 shows the relative changes in abundancesversus the comoving lepton number for both calculations.Our baselines for comparison are the abundances in thenondegenerate case, L (cid:63)ν = 0, from the Boltz. calculation.As a result of the choice of baseline, the relative changesin abundances for the DR model will not converge to zeroas L (cid:63)ν →
0. We use a mass fraction to describe the heliumabundance, Y P , and relative abundances with respect tohydrogen to describe deuterium (D), helium-3 ( He), andlithium-7 ( Li). The solid lines in Fig. 14 show the rela-tive changes in the DR model. Positive relative changesin the abundances correspond to negative comoving lep-ton numbers, and negative changes to positive L (cid:63)ν . Wealso show individual points using the Boltz. calculationat three decades of L (cid:63)ν , namely log | L (cid:63)ν | = − , − , − L (cid:63)ν >
0, and circles to L (cid:63)ν < L (cid:63)ν . Anonzero comoving lepton number changes the occupationnumbers in the neutron-proton interconversion rates, andalso changes the Hubble expansion rate. The neutron-to-proton ratio ( n/p ) is sensitive to both quantities [52, 53],and Y P is the abundance most sensitive to n/p . In Fig.14, we see that Y P has the largest change from the non-degenerate baseline, while He has the least sensitivityto L (cid:63)ν . Deuterium and Li have a more intricate rela-tionship with L (cid:63)ν . As we increase L (cid:63)ν from large negativevalues towards zero, we see that the relative change forD is larger than that for Li until L (cid:63)ν ∼ − × − . Atthis point, Li appears to be more sensitive to L (cid:63)ν . Thetrend continues for L (cid:63)ν >
0, as the relative change in Liis more negative than that of D. The asymmetry between L (cid:63)ν > L (cid:63)ν < Liis present in Y P and He also. With the exception of Li,all abundances are more sensitive to negative L (cid:63)ν . Alltrends occur in both the DR model and Boltz. calcula-tion. These trends are similar but have minor differencesthan those discussed in Ref. [54].Table V gives the relative changes of Y P and D/H forvarious values of L (cid:63)ν in both scenarios. Columns withthe label “(DR)” are relative changes calculated withthe dark-radiation model and columns with the label“(Boltz.)” are relative changes in the full Boltzmannneutrino-transport calculation. The Boltz. columns inTable V are identical to the Boltz. columns in Table IV.For all four abundance columns, the relative changes arewith respect to the abundance calculated with the fullBoltzmann-transport network with degeneracy parame-4 L (cid:63)ν δY P (FD Eq.) δY P (Boltz.) δ (D / H) (FD Eq.) δ (D / H) (Boltz.)10 − − . − . − . × − − . × − − − . × − − . × − − . × − − . × − − − . × − − . × − − . × − − . × − . × − − . × − − . × − − . × − − . × − . × − − . × − − . × − − . × − − . × − − − . . . × − . × − − − . × − . × − . × − . × − − − . × − . × − − . × − . × − − . × − . × − . × − . × − . × − − . × − . × − . × − . × − . × − TABLE IV. Relative changes in primordial abundances of He and D in two calculations of BBN with nonzero comoving leptonnumbers L (cid:63)ν . FD Eq. signifies the calculation without transport. Boltz. signifies the full Boltzmann neutrino-transport networkcalculation. The abundances are given as relative changes from the zero degeneracy, full Boltzmann calculation. Column 1 isthe comoving lepton number. Column 2 gives the relative change of Y P at freeze-out in the no-transport model. Column 3gives the relative change of Y P at freeze-out in the Boltz. calculation. The relative changes for D/H are given in columns 4 and5. The four rows where | L (cid:63)ν | is not a power of 10 are projected sensitivity limits for 1% changes in the primordial abundances. − − − − | L ?ν |− . − . − . − . . . . . . R e l a t i v ec h a n g e L ?ν < L ?ν > Y P D / H Li / H He / H Y P D / H Li / H He / H FIG. 14. Relative changes in the primordial abundances plot-ted against the absolute value of the comoving lepton num-ber. Positive changes in abundances correspond to nega-tive comoving lepton numbers, and negative changes corre-spond to positive comoving lepton numbers. The solid linesuse the dark-radiation model described in the text. Indi-vidual points using the full Boltzmann-transport calculationare plotted for three decades of L (cid:63)ν . Squares correspond to L (cid:63)ν >
0, and circles for L (cid:63)ν <
0. The baryon density is fixedto be ω b = 0 . n b /n γ = 6 . × − ) for all calculations in both scenarios[47]. . The mean neutron lifetime is taken to be 885 . ter set to zero, consistent with the lines and points in Fig.14. For the Boltz. columns, the relative changes in Y P tend to be twice as large as those in D/H. Each decadechange in L (cid:63)ν induces close to a decade change in both rel-ative abundances. We have included calculations for setsof lepton numbers which aim for ±
1% changes in both Y P and D/H in the Boltz. calculation. For the DR model, the relative changes for He and deuterium are in line withthe Boltz. calculation for L (cid:63)ν = +0 .
1. Transport enhancesthe ν e occupation numbers over the ν e if L (cid:63)ν = +0 .
1. Theextra probability in the ν e spectrum enhances the rate of ν e + n → p + e − . As a result, the helium abundancedecreases further in the Boltz. calculation, which is ev-ident in Table V. Conversely, for L (cid:63)ν = − .
1, transportwill enhance the ν e over the ν e and we would expect anincrease in the He abundance. This is not the case inTable V - the DR model has a larger δY P than the Boltz.calculation.The error in the above logic resides in the treatmentof the rate which changes protons to neutrons, namely ν e + p → n + e + . This reaction has a threshold of Q ≡ δm np + m e (cid:39) . δm np and m e arethe neutron-to-proton mass difference and electron restmass, respectively. If we define the appropriate (cid:15) -valuefor Q to be q ≡ Q/T cm , we can see where and how thethreshold plays a role in (cid:15) and T cm space. Fig. 7 showsthe freeze-out distortion to the differential number den-sity distributions for L (cid:63)ν = +0 .
1. The ν e and ν e spectrawould be switched if we had plotted L (cid:63)ν = − .
1, i.e. a“mirror” of Fig. 7. At the start of the calculation at T cm = 10 MeV, the distortions are identically zero. Thecalculation proceeds and the peaks in ∆( dn/d(cid:15) ) for ν e and ν e grow. The locations of the peaks do change withdecreasing T cm , but we have verified that the shift inposition is small compared to peak location of (cid:15) ∼ ν e over ν e would increase the rate ν e + p → n + e + , butit is only the number density with (cid:15) -value larger than q which is able to increase the rate, thereby decreasing theneutron abundance. At T cm = 1 MeV, q (cid:39) ν e + p → n + e + .At T cm = 500 keV, q (cid:39) L (cid:63)ν δY P (DR) δY P (Boltz.) δ (D / H) (DR) δ (D / H) (Boltz.)10 − − . − . − . × − − . × − − − . × − − . × − − . × − − . × − − . × − − . × − . × − − . × − . × − − . × − − . × − − . × − − . × − . × − − . × − − . × − − . × − − . × − − − . . . × − . × − − − . × − . × − . × − . × − − − . × − . × − . × − . × − − . × − . × − . × − . × − . × − − . × − . × − . × − . × − . × − TABLE V. Relative changes in primordial abundances of He and D in two calculations of neutrino transport with nonzerocomoving lepton numbers L (cid:63)ν . DR signifies the dark-radiation model of neutrino transport and Boltz. signifies the full Boltzmannneutrino-transport network calculation. The columns are the same as Table IV with the replacement of “FD Eq.” by DR. T cm where the He abundance begins to depart fromnuclear statistical equilibrium [55]. Although the abun-dance is ∼
15 orders of magnitude smaller than its freeze-out value, the integration of the nuclear reaction networkis sensitive to the initial conditions, and already half ofthe peak width in the mirror of Fig. 7 is unavailable toenhance the rate and modify the neutron-to-proton ratio.The formation of He nuclei is typically ascribed to theepoch T cm = 100 keV, where q (cid:39)
20 and well larger thanthe range where the distortions in the mirror of Fig. 7could affect the rate for ν e + p → n + e + . Meanwhile,neutrino transport is inducing an increased populationon the high-energy tail of the ν e spectrum, which wouldincrease the neutron to proton rate ν e + n → p + e − .This reaction has no threshold, and so the entire peak inthe mirror of Fig. 7, integrated over the full range of T cm ,would increase the rate. Incidentally, e + + n → p + ν e hasno threshold and this process is also important in setting n/p . However, in this case, the spectral distortion effectswe described above would tend to hinder this process byproducing extra ν e blocking. The ν e in e + + n → p + ν e has a minimum energy of Q , and so the expected sup-pression of this rate from additional ν e number densitysuffers from the same sequence of events as mentionedabove.To summarize, transport-induced ν e and ν e spectraldistortions develop over such a long time span that thethreshold-limited ν e number density cannot overcome the ν e number density when calculating the neutron-protoninterconversion rates in the L (cid:63)ν = − . δY P for the Boltz. calculation comparedto the DR model.The DR model is tuned to have the same total energydensity as produced in the full Boltzmann calculationwhen L (cid:63)ν = ± .
1. If | L (cid:63)ν | (cid:54) = 0 .
1, the radiation energydensity, and by extension N eff , is slightly different. Theabundances are sensitive to the change in N eff , and asa result we see significant differences between the twomodels in Table V. An especially egregious example isthe L (cid:63)ν = 10 − scenario, where the relative changes in He are 2 orders of magnitude different and have different signs. We conclude that mocking up the effect of neutrinotransport in this model with dark radiation fails for smalllepton numbers. However, if we had tuned the DR modelfor N eff to agree when L (cid:63)ν = 10 − , we would have hadbetter agreement for smaller L (cid:63)ν . We note that for allcases with | L (cid:63)ν | ≤ × − , the changes in the abundancesare below current and projected error tolerances [14]. VII. CONCLUSION
We have done the first nonzero neutrino chemical po-tential calculations of weak decoupling and BBN withfull Boltzmann neutrino transport simultaneously cou-pled with all relevant strong, weak, and electromagneticnuclear reactions. We have performed these calculationswith a modified version of the burst code. This codeand the physics it incorporates is described in detail inRef. [1]. By design, our calculations here do not includeneutrino flavor oscillations. Our intent was to provide baseline calculations for comparison to future neutrinoflavor quantum kinetic treatments (see Refs. [56, 57] inthe early universe, and Refs. [58–60] in core-collapse su-pernova cores, for a discussion on the quantum kineticequations in their respective environments). One ob-jective of this baseline Boltzmann study was to iden-tify how a significant lepton number would affect out-of-equilibrium neutrino scattering and the concomitantneutrino scattering-induced flow of entropy out of thephoton-electron-positron plasma and into the decouplingneutrino component. A related objective was to assesswhether (and how) the scattering-induced neutrino spec-tral distortions develop differently in the case of a sig-nificant neutrino asymmetry. The third objective was touse a new description to connect the two previously men-tioned phenomena: macroscopic thermodynamics of en-tropy flow, and microscopic spectral distortions. Finally,the last objective was to assess the impact of these neu-trino spectral distortions and the accompanying changesin entropy flow and temperature/scale factor phasing onBBN light element abundance yields. A key finding of6our full Boltzmann neutrino-transport treatment is thatthe presence of a lepton-number asymmetry enhances the processes which give rise to distortions from equilib-rium, FD-shaped neutrino and antineutrino energy spec-tra. Our transport calculations show a positive feed-back between out-of-equilibrium neutrino scattering andany initial distortion from a zero chemical potential FDdistribution (see the elastic scattering of neutrinos withcharged leptons in Fig. 10). An initial distortion, forexample, stemming from a nonzero chemical potential, isamplified by neutrino scattering, at least for higher valuesof the comoving neutrino energy parameter (cid:15) = E ν /T cm .Of course, overall lepton asymmetry is preserved by thenonlepton number violating scattering processes we treathere.In broad brush, as the Universe expands entropy istransferred from the electron-positron component intophotons, with neutrinos receiving only a small portionof this entropy largess. The magnitude of this small en-tropy increase to the decoupling neutrinos is governedlargely by the out-of-equilibrium scattering of neutrinosand antineutrinos on the electrons and positrons, whichare generally “hotter” than the neutrinos. The neutrinoscattering cross sections scale like σ ∼ (cid:15) , and thereforehigher energy neutrinos are able to extract entropy fromthe photon-electron-positron component more effectivelythan neutrinos with lower energy. The result is that a“bump” or occupation excess (see Fig. 7) on the higherenergy end of the neutrino energy distribution functiongrows with time. Our transport calculations have allowedus to track both entropy flow between the neutrinos andthe plasma and the simultaneous development of neu-trino spectral distortions, all for a range of initial leptonasymmetries. For the larger values of lepton asymmetryconsidered here we found that the entropy transferred toneutrinos is decreased by a few tenths of a percent overthe zero lepton-number case (see Table II).The enhanced neutrino spectral distortions and en-tropy transfer revealed by our full Boltzmann-transportcalculations might be expected to translate into cor-responding nuclear abundance changes emerging fromBBN. Our full coupling between neutrino scattering andthe weak interaction sector and the nuclear reaction net-work is uniquely adapted to treat this physics. Indeed,for the zero neutrino chemical potential cases, the fullBoltzmann-neutrino transport resulted in a deuteriumBBN yield ∼ He abundance yield is sensitiveat the one percent level to an initial, comoving leptonnumber of L (cid:63)ν ≈ × − , while the deuterium abundanceyield is similarly sensitive to L (cid:63)ν ≈ . × − . This is sig-nificant because the next generation CMB experiments,e.g. proposed Stage-4 CMB observations [14], target pre-cisions for independent primordial helium abundance de-terminations at roughly the two percent level. Likewise,the next generation of large optical telescopes, for exam-ple 30-meter class telescopes [61–64], are touted as pro-viding a comparable level of precision in determining theprimordial deuterium abundance from quasar absorptionlines in high redshift damped Lyman-alpha systems. Ourcalculations show that we would need ∼ . He and D are sensitive to n/p , which itself is sensitiveto the ν e and ν e occupation numbers. Table IV showedthat the FD Eq. treatment of BBN closely matches theBoltz. calculation of Y P . Transport induces a relativechange in D/H nearly an order or magnitude larger thanthat of Y P . This finding is consistent with findings inthe zero-degeneracy case [1]. Tables II and V show thatthe primordial abundances are more sensitive to neutrinodegeneracy than N eff . Moreover, He is twice as sensi-tive to the degeneracy than D. CMB Stage-IV experi-ments [14, 65] and 30-meter-class telescopes will probe Y P , D/H, and N eff at the 1% level. If future observationswere to find little change in N eff from the standard pre-diction, but changes in the abundances matching the pat-terns in Table V, then this scenario would be consistentwith a degeneracy in the neutrino sector. However, theBoltz. calculations in Table V do not include the physicsof neutrino oscillations. In the presence of nonzero lep-ton numbers, oscillations may alter the scaling relationsof Table V and will necessitate a full quantum kineticequation treatment [66, 67].This brings us to the question of our selection of initiallepton asymmetries. We have chosen to examine valuesof these at and below usually accepted limits, and wehave examined only situations where the asymmetries are7the same across all flavors. The trends our Boltzmann-transport calculations reveal will likely hold for leptonasymmetries outside of the ranges considered here. How-ever, differences in lepton numbers between different fla-vors will drive medium-enhanced/affected neutrino fla-vor transformation which could lead to different conclu-sions in the neutrino sector. Comparing future quan-tum kinetic calculations which include both coherent andscattering-induced flavor transformation with our strictBoltzmann treatment might reveal BBN and N eff signa-tures of neutrino flavor conversion, although these maybe at levels well below what future observations and ex-periments can probe.Nevertheless, many beyond-standard-model physicsconsiderations invoke quite small initial lepton numbers[68–71]. Various models of sterile neutrinos in the earlyuniverse, including dark matter models, rely on leptonnumber-driven medium enhancements [72–74] or beyond-standard-model physics to create relic sterile-neutrinodensities (see Refs. [75, 76] and references therein for areview of sterile neutrino dark matter). Sterile neutrinosare an intriguing dark matter candidate [77], and couldconceivably be congruent with particle [78] and cosmo-logical bounds [79, 80]. For resonantly produced sterileneutrino dark matter, the models invoke lepton asymme-tries in the 10 − to 10 − range to match the relic dark-matter abundance, providing a motivation for our choiceof values for L (cid:63)ν .In fact, many models for baryon and lepton-number generation in the early universe [81–83] , e.g. the neutrinominimal standard model ( ν MSM) [84, 85] , can producelepton numbers in the ranges chosen for the the presentstudy. It will be interesting to see if future quantumkinetic calculations with neutrino flavor transformationwill yield deviations from the baseline calculations pre-sented here. Any such deviations would point to eithera different distribution of lepton numbers over neutrinoflavor than that considered here, or differences in the de-velopment of scattering-induced spectral distortions andattendant BBN abundance alterations over the standardscenario.
ACKNOWLEDGMENTS
We thank Fred Adams, J. Richard Bond, LaurenGilbert, Luke Johns, Joel Meyers, Matthew Wilson, andNicole Vassh for useful conversations. We acknowledgethe Integrated Computing Network at Los Alamos Na-tional Laboratory for supercomputer time. This researchused resources of the National Energy Research Scien-tific Computing Center, a DOE Office of Science UserFacility supported by the Office of Science of the U.S.Department of Energy under Contract No. DE-AC02-05CH11231. This work was supported in part by NSFGrant PHY-1307372 at UC San Diego, and LDRD fund-ing at Los Alamos National Laboratory. We thank theanonymous referee for their useful comments. [1] E. Grohs, G. M. Fuller, C. T. Kishimoto, M. W. Paris,and A. Vlasenko, “Neutrino energy transport in weakdecoupling and big bang nucleosynthesis,” Phys. Rev. D , 083522 (2016), arXiv:1512.02205.[2] Robert V. Wagoner, William A. Fowler, and Fred Hoyle,“On the Synthesis of elements at very high tempera-tures,” Astrophys.J. , 3–49 (1967).[3] D. N. Schramm and R. V. Wagoner, “Element productionin the early universe.” Annual Review of Nuclear andParticle Science , 37–74 (1977).[4] X. Shi, “Chaotic amplification of neutrino chemical po-tentials by neutrino oscillations in big bang nucleosyn-thesis,” Phys. Rev. D , 2753–2760 (1996), astro-ph/9602135.[5] D. P. Kirilova and M. V. Chizhov, “Neutrino degener-acy effect on neutrino oscillations and primordial he-lium yield,” Nuclear Physics B , 447–463 (1998), hep-ph/9806441.[6] S. H. Hansen, G. Mangano, A. Melchiorri, G. Miele, andO. Pisanti, “Constraining neutrino physics with big bangnucleosynthesis and cosmic microwave background radia-tion,” Phys. Rev. D , 023511 (2001), astro-ph/0105385.[7] V. Simha and G. Steigman, “Constraining the universallepton asymmetry,” J. Cosmology Astropart. Phys. ,011 (2008), arXiv:0806.0179 [hep-ph].[8] M. Shiraishi, K. Ichikawa, K. Ichiki, N. Sugiyama,and M. Yamaguchi, “Constraints on neutrino massesfrom WMAP5 and BBN in the lepton asymmetric uni- verse,” J. Cosmology Astropart. Phys. , 005 (2009),arXiv:0904.4396 [astro-ph.CO].[9] J. P. Kneller, R. J. Scherrer, G. Steigman, and T. P.Walker, “How does the cosmic microwave backgroundplus big bang nucleosynthesis constrain new physics?”Phys. Rev. D , 123506 (2001), astro-ph/0101386.[10] G. Mangano, G. Miele, S. Pastor, O. Pisanti, andS. Sarikas, “Constraining the cosmic radiation den-sity due to lepton number with Big Bang Nucleosyn-thesis,” J. Cosmology Astropart. Phys. , 035 (2011),arXiv:1011.0916.[11] K. Abazajian, N. F. Bell, G. M. Fuller, and Y. Y. Y.Wong, “Cosmological lepton asymmetry, primordial nu-cleosynthesis and sterile neutrinos,” Phys. Rev. D ,063004 (2005), astro-ph/0410175.[12] C. J. Smith, G. M. Fuller, C. T. Kishimoto, and K. N.Abazajian, “Light element signatures of sterile neutri-nos and cosmological lepton numbers,” Phys. Rev. D ,085008 (2006), astro-ph/0608377.[13] Y.-Z. Chu and M. Cirelli, “Sterile neutrinos, leptonasymmetries, primordial elements: How much of each?”Phys. Rev. D , 085015 (2006), astro-ph/0608206.[14] J. E. Carlstrom and et al., “CMB-S4 Science Book, FirstEdition,” ArXiv e-prints (2016), arXiv:1610.02743.[15] J. Richard Bond, George M. Fuller, E. Grohs, Joel Mey-ers, and Matthew Wilson, (2017), in preparation.[16] D. Kirkman, D. Tytler, N. Suzuki, J. M. O’Meara,and D. Lubin, “The Cosmological Baryon Density from the Deuterium-to-Hydrogen Ratio in QSO AbsorptionSystems: D/H toward Q1243+3047,” ApJS , 1–28(2003), astro-ph/0302006.[17] M. Pettini and R. Cooke, “A new, precise measurement ofthe primordial abundance of deuterium,” MNRAS ,2477–2486 (2012).[18] Ryan J. Cooke, Max Pettini, Regina A. Jorgenson,Michael T. Murphy, and Charles C. Steidel, “Precisionmeasures of the primordial abundance of deuterium,”The Astrophysical Journal , 31 (2014).[19] R. Cooke and M. Pettini, “The primordial abundance ofdeuterium: ionization correction,” MNRAS , 1512–1521 (2016), arXiv:1510.03867.[20] R. J. Cooke, M. Pettini, K. M. Nollett, and R. Jorgen-son, “The Primordial Deuterium Abundance of the MostMetal-poor Damped Lyman- α System,” ApJ , 148(2016), arXiv:1607.03900.[21] V. A. Kosteleck´y and S. Samuel, “Neutrino oscillationsin the early Universe with nonequilibrium neutrino dis-tributions,” Phys. Rev. D , 3184–3201 (1995), hep-ph/9507427.[22] M. J. Savage, R. A. Malaney, and G. M. Fuller, “Neu-trino oscillations and the leptonic charge of the universe,”ApJ , 1–11 (1991).[23] B. H. J. McKellar and M. J. Thomson, “Oscillating neu-trinos in the early Universe,” Phys. Rev. D , 2710–2728(1994).[24] A. Casas, W. Y. Cheng, and G. Gelmini, “Generation oflarge lepton asymmetries,” Nuclear Physics B , 297–308 (1999), hep-ph/9709289.[25] Y. Y. Wong, “Analytical treatment of neutrino asymme-try equilibration from flavor oscillations in the early uni-verse,” Phys. Rev. D , 025015 (2002), hep-ph/0203180.[26] K. N. Abazajian, J. F. Beacom, and N. F. Bell, “Strin-gent constraints on cosmological neutrino-antineutrinoasymmetries from synchronized flavor transformation,”Phys. Rev. D , 013008 (2002), astro-ph/0203442.[27] A. D. Dolgov, S. H. Hansen, S. Pastor, S. T. Petcov,G. G. Raffelt, and D. V. Semikoz, “Cosmological boundson neutrino degeneracy improved by flavor oscillations,”Nucl. Phys. B , 363–382 (2002), hep-ph/0201287.[28] J. Gava and C. Volpe, “CP violation effects on the neu-trino degeneracy parameters in the Early Universe,” Nu-clear Physics B , 50–60 (2010), arXiv:1002.0981 [hep-ph].[29] L. Johns, M. Mina, V. Cirigliano, M. W. Paris, andG. M. Fuller, “Neutrino flavor transformation in thelepton-asymmetric universe,” Phys. Rev. D , 083505(2016), arXiv:1608.01336 [hep-ph].[30] G. Barenboim, W. H. Kinney, and W.-I. Park, “Fla-vor versus mass eigenstates in neutrino asymmetries:implications for cosmology,” ArXiv e-prints (2016),arXiv:1609.03200.[31] A. D. Dolgov and M. Fukugita, “Nonequilibrium effectof the neutrino distribution on primordial helium syn-thesis,” Phys. Rev. D , 5378–5382 (1992).[32] A. D. Dolgov, S. H. Hansen, and D. V. Semikoz, “Non-equilibrium corrections to the spectra of massless neutri-nos in the early universe,” Nuclear Physics B , 426–444 (1997), hep-ph/9703315.[33] A. D. Dolgov, S. H. Hansen, and D. V. Semikoz, “Non-equilibrium corrections to the spectra of massless neutri-nos in the early universe,” Nuclear Physics B , 269–274 (1999), hep-ph/9805467. [34] S. Esposito, G. Miele, S. Pastor, M. Peloso, andO. Pisanti, “Non equilibrium spectra of degenerate relicneutrinos,” Nuclear Physics B , 539–561 (2000),astro-ph/0005573.[35] P. D. Serpico and G. G. Raffelt, “Lepton asymmetry andprimordial nucleosynthesis in the era of precision cosmol-ogy,” Phys. Rev. D , 127301 (2005), astro-ph/0506162.[36] G. Mangano, G. Miele, S. Pastor, T. Pinto, O. Pisanti,and P. D. Serpico, “Relic neutrino decoupling includ-ing flavour oscillations,” Nuclear Physics B , 221–234(2005), hep-ph/0506164.[37] M. S. Smith, B. D. Bruner, R. L. Kozub, L. F. Roberts,D. Tytler, G. M. Fuller, E. Lingerfelt, W. R. Hix, andC. D. Nesaraja, “Big Bang Nucleosynthesis: Impact ofNuclear Physics Uncertainties on Baryonic Matter Den-sity Constraints,” in Origin of Matter and Evolution ofGalaxies , American Institute of Physics Conference Se-ries, Vol. 1016, edited by T. Suda, T. Nozawa, A. Ohnishi,K. Kato, M. Y. Fujimoto, T. Kajino, and S. Kubono(2008) pp. 403–405.[38] N. Saviano, A. Mirizzi, O. Pisanti, P. D. Serpico,G. Mangano, and G. Miele, “Multimomentum and mul-tiflavor active-sterile neutrino oscillations in the earlyuniverse: Role of neutrino asymmetries and effectson nucleosynthesis,” Phys. Rev. D , 073006 (2013),arXiv:1302.1200 [astro-ph.CO].[39] P. F. de Salas and S. Pastor, “Relic neutrino decou-pling with flavour oscillations revisited,” J. CosmologyAstropart. Phys. , 051 (2016), arXiv:1606.06986 [hep-ph].[40] M. Shimon, N. J. Miller, C. T. Kishimoto, C. J. Smith,G. M. Fuller, and B. G. Keating, “Using Big Bang Nucle-osynthesis to extend CMB probes of neutrino physics,”J. Cosmology Astropart. Phys. , 037 (2010).[41] E. W. Kolb and M. S. Turner, The Early Universe. (Addison-Wesley Publishing Co., 1990).[42] S. Weinberg,
Cosmology, by Steven Weinberg. ISBN 978-0-19-852682-7. Published by Oxford University Press,Oxford, UK, 2008. (Oxford University Press, 2008).[43] A. F. Heckler, “Astrophysical applications of quan-tum corrections to the equation of state of a plasma,”Phys. Rev. D , 611–617 (1994).[44] N. Fornengo, C. W. Kim, and J. Song, “Finite tem-perature effects on the neutrino decoupling in the earlyUniverse,” Phys. Rev. D , 5123–5134 (1997), hep-ph/9702324.[45] G. Mangano, G. Miele, S. Pastor, and M. Peloso, “Aprecision calculation of the effective number of cosmo-logical neutrinos,” Physics Letters B , 8–16 (2002),astro-ph/0111408.[46] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, se-ries, and products , seventh ed. (Elsevier/Academic Press,Amsterdam, 2007) pp. xlviii+1171, translated from theRussian, Translation edited and with a preface by AlanJeffrey and Daniel Zwillinger, With one CD-ROM (Win-dows, Macintosh and UNIX).[47] Planck Collaboration, P. A. R. Ade, and et al., “Planck2013 results. XVI. Cosmological parameters,” A&A ,A16 (2014), arXiv:1303.5076.[48] E. Grohs, G. M. Fuller, C. T. Kishimoto, and M. W.Paris, “Probing neutrino physics with a self-consistenttreatment of the weak decoupling, nucleosynthesis, andphoton decoupling epochs,” J. Cosmology Astropart.Phys. , 017 (2015), arXiv:1502.02718. [49] S. Mukohyama, “Brane-world solutions, standard cos-mology, and dark radiation,” Physics Letters B , 241–245 (2000), hep-th/9911165.[50] J.-L. Cambier, J. R. Primack, and M. Sher, “Finite tem-perature radiative corrections to neutron decay and re-lated processes,” Nucl. Phys. B , 372–388 (1982).[51] R. E. Lopez and M. S. Turner, “Precision prediction forthe big-bang abundance of primordial He,” Phys. Rev. D , 103502 (1999), astro-ph/9807279.[52] C. J. Smith, G. M. Fuller, and M. S. Smith, “Bigbang nucleosynthesis with independent neutrino distri-bution functions,” Phys. Rev. D , 105001 (2009),arXiv:0812.1253.[53] E. Grohs and G. M. Fuller, “The surprising influence oflate charged current weak interactions on Big Bang Nu-cleosynthesis,” Nuclear Physics B , 955–973 (2016),arXiv:1607.02797.[54] J. P. Kneller and G. Steigman, “BBN for pedestrians,”New Journal of Physics , 117 (2004), astro-ph/0406320.[55] M. S. Smith, L. H. Kawano, and R. A. Malaney, “Ex-perimental, computational, and observational analysis ofprimordial nucleosynthesis,” ApJS , 219–247 (1993).[56] R. Barbieri and A. Dolgov, “Neutrino oscillations in theearly universe,” Nuclear Physics B , 743–753 (1991).[57] C. Volpe, D. V¨a¨an¨anen, and C. Espinoza, “Extendedevolution equations for neutrino propagation in astro-physical and cosmological environments,” Phys. Rev. D , 113010 (2013), arXiv:1302.2374 [hep-ph].[58] G. Raffelt and G. Sigl, “Neutrino flavor conversion ina supernova core,” Astroparticle Physics , 165–183(1993), astro-ph/9209005.[59] A. B. Balantekin and Y. Pehlivan, “Neutrino neutrinointeractions and flavour mixing in dense matter,” Journalof Physics G Nuclear Physics , 47–65 (2007), astro-ph/0607527.[60] C. Volpe, “Neutrino quantum kinetic equations,” In-ternational Journal of Modern Physics E , 1541009(2015), arXiv:1506.06222 [astro-ph.SR].[61] David Silva, Paul Hickson, Charles Steidel, and MichaelBolte, TMT Detailed Science Case: 2007 , 1945 (2015),arXiv:1505.01195 [astro-ph.IM].[63] P. McCarthy and R. A. Bernstein, “Giant Magellan Tele-scope: Status and Opportunities for Scientific Synergy,”in Thirty Meter Telescope Science Forum (2014) p. 61.[64] I. Hook (ed.),
The science case for the European Ex-tremely Large Telescope : the next step in mankind’squest for the Universe. (Cambridge, UK: OPTICON andGarching bei Muenchen, Germany: European SouthernObservatory (ESO), 2005).[65] K. N. Abazajian and et al., “Neutrino physics from thecosmic microwave background and large scale structure,”Astroparticle Physics , 66–80 (2015).[66] A. Vlasenko, G. M. Fuller, and V. Cirigliano, “Neu-trino quantum kinetics,” Phys. Rev. D , 105004 (2014),arXiv:1309.2628 [hep-ph].[67] D. N. Blaschke and V. Cirigliano, “Neutrino quantumkinetic equations: The collision term,” Phys. Rev. D , 033009 (2016), arXiv:1605.09383 [hep-ph].[68] J. A. Harvey and M. S. Turner, “Cosmological baryonand lepton number in the presence of electroweakfermion-number violation,” Phys. Rev. D , 3344–3349(1990).[69] M. Kawasaki, F. Takahashi, and M. Yamaguchi, “Largelepton asymmetry from Q-balls,” Phys. Rev. D ,043516 (2002), hep-ph/0205101.[70] F. Bezrukov, H. Hettmansperger, and M. Lindner,“keV sterile neutrino dark matter in gauge extensionsof the standard model,” Phys. Rev. D , 085032 (2010),arXiv:0912.4415 [hep-ph].[71] A. Merle, V. Niro, and D. Schmidt, “New productionmechanism for keV sterile neutrino Dark Matter by de-cays of frozen-in scalars,” J. Cosmology Astropart. Phys. , 028 (2014), arXiv:1306.3996 [hep-ph].[72] X. Shi and G. M. Fuller, “New Dark Matter Candidate:Nonthermal Sterile Neutrinos,” Physical Review Letters , 2832–2835 (1999), astro-ph/9810076.[73] K. Abazajian, G. M. Fuller, and M. Patel, “Sterile neu-trino hot, warm, and cold dark matter,” Phys. Rev. D , 023501 (2001), astro-ph/0101524.[74] C. T. Kishimoto, G. M. Fuller, and C. J. Smith, “Co-herent Active-Sterile Neutrino Flavor Transformation inthe Early Universe,” Phys. Rev. Lett. , 141301 (2006),astro-ph/0607403.[75] A. de Gouvea and et al., “Neutrinos,” ArXiv e-prints(2013), arXiv:1310.4340 [hep-ex].[76] R. Adhikari and et al., “A White Paper on keV Ster-ile Neutrino Dark Matter,” ArXiv e-prints (2016),arXiv:1602.04816 [hep-ph].[77] S. Dodelson and L. M. Widrow, “Sterile neutrinos as darkmatter,” Physical Review Letters , 17–20 (1994), hep-ph/9303287.[78] A. Kusenko, F. Takahashi, and T. T. Yanagida, “Darkmatter from split seesaw,” Physics Letters B , 144–148 (2010), arXiv:1006.1731 [hep-ph].[79] M. Yamaguchi, “Generation of cosmological large leptonasymmetry from a rolling scalar field,” Phys. Rev. D ,063507 (2003), hep-ph/0211163.[80] K. N. Abazajian, “Resonantly Produced 7 ˆA keV Ster-ile Neutrino Dark Matter Models and the Propertiesof Milky Way Satellites,” Phys. Rev. Lett. , 161303(2014), arXiv:1403.0954.[81] J. March-Russell, A. Riotto, and H. Murayama, “Thesmall observed baryon asymmetry from a large leptonasymmetry,” Journal of High Energy Physics , 015(1999), hep-ph/9908396.[82] P.-H. Gu, “Large lepton asymmetry for small baryonasymmetry and warm dark matter,” Phys. Rev. D ,093009 (2010), arXiv:1005.1632 [hep-ph].[83] L. Canetti, M. Drewes, and M. Shaposhnikov, “Ster-ile Neutrinos as the Origin of Dark and BaryonicMatter,” Physical Review Letters , 061801 (2013),arXiv:1204.3902 [hep-ph].[84] T. Asaka, S. Blanchet, and M. Shaposhnikov, “The ν MSM, dark matter and neutrino masses [rapid commu-nication],” Physics Letters B , 151–156 (2005), hep-ph/0503065.[85] M. Shaposhnikov and I. Tkachev, “The ν MSM, infla-tion, and dark matter,” Physics Letters B639