Aggregate Lepton Number Bounds Reconsidered

Lepton number is a quantum number defined to equal the total number of leptons (electrons, muons, tau leptons, electron neutrinos, muon neutrinos, and tau neutrinos) minus the total number of anti-leptons (positrons, anti-muons, anti-taus, electron antineutrinos, muon antineutrinos, tau antineutrinos). If there are more leptons than anti-leptons (making their ratio greater than one), then lepton number is a positive integer. If there are fewer leptons than anti-leptons (making their ratio less than one), then lepton number is a negative integer.

In the Standard Model of Particle Physics, lepton number is conserved in all interactions except one that only occurs in high energy circumstances (above 10 TeV), is predicted to be rare even then, and has never actually been observed. Neutrinoless double beta decay, if it was observed, would violate lepton number. These temperatures haven't been found in Nature in the universe since shortly after the Big Bang.

We don't have a direct measurement of the ratio of neutrinos to anti-neutrinos in the universe, and I have a post at Physics Stack Exchange asking about the experimental bounds on this ratio.

An anti-neutrino excess would favor an effective number of neutrino types (Neff) which is higher than the 3.042 predicted in the absence of a neutrino-antineutrino asymmetry, and higher proportions of primordial helium than predicted in vanilla Big Bang Nucleosynthesis, which are both present in the CMB data. But, the data are also consistent with no asymmetry between neutrinos and antineutrinos.

The expected primordial proportion of Helium-4 is about 25% and observation doesn't grossly differ from that value although it isn't a perfect fit. Per Wikipedia citing data through 2008:

Using this value, are the BBN predictions for the abundances of light elements in agreement with the observations? 

The present measurement of helium-4 indicates good agreement, and yet better agreement for helium-3. But for lithium-7, there is a significant discrepancy between BBN and WMAP/Planck, and the abundance derived from Population II stars. The discrepancy is a factor of 2.4―4.3 below the theoretically predicted value and is considered a problem for the original models,^[13] that have resulted in revised calculations of the standard BBN based on new nuclear data, and to various reevaluation proposals for primordial proton-proton nuclear reactions, especially the abundances of ⁷Be + n → ⁷Li + p, versus ⁷Be + ²H → ⁸Be + p.^[14]

One paper cited is this one.

The predicted value of Neff with the amount of neutrino asymmetry that is a best fit with observed helium-4 levels (as of 2012) would predict a Neff of about 3.146, when the observed value is 3.04 ± 0.18. This would involve a neutrino chemical potential of -0.2 which would imply a considerable excess of antineutrinos over neutrinos. 

There would be more anti-particles than particles in the universe by far, aggregate lepton number would be strongly negative and this negative amount would dwarf the positive baryon number of the universe. Even dark matter of keV mass or greater that carried lepton or baryon number would not overcome this asymmetry. For that matter, even a very modest 1% asymmetry in favor of antineutrinos would produce this result. This is because:

The number of baryons in the universe is about 4×1079, and the number of neutrinos in the universe is about 1.2×1089.

We know that the ratio of baryon antimatter to baryon matter (and the ratio of charged antileptons to charged leptons) is on the order of 10−11.

And, we know that to considerable precision there are 2 neutrons for every 14 protons in the universe (this is a confirmed prediction of Big Bang Nucleosynthesis), and that the number of charged leptons is almost identical to the number of protons in the universe.

The number of mesons and baryons other than neutrons and protons is negligible at any given time in nature since they are so short lived and generated only at high energies. (Also, the baryon number of a meson is zero.)

With an average neutrino mass of about 30 meV (which cosmology data combined with neutrino oscillation data suggests is order of magnitude correct), the share of neutrino mass of all Standard Model matter mass is about one part per 10,000. The ratio of nucleon mass to average neutrino mass is about 3*10^13 and the ratio of neutrinos to baryons is about 3*10^9. If neutrinos are disproportionately in the lightest neutrino mass eigenvalue, the fraction of all ordinary mass that comes from neutrinos is much lower.

The most recent papers I've located on the topic is from 2012. It's abstract states:

Recent observations of the cosmic microwave background (CMB) at smallest angular scales and updated abundances of primordial elements, indicate an increase of the energy density and the helium-4 abundance with respect to standard big bang nucleosynthesis with three neutrino flavour. This calls for a reanalysis of the observational bounds on neutrino chemical potentials, which encode the number asymmetry between cosmic neutrinos and anti-neutrinos and thus measures the lepton asymmetry of the Universe. We compare recent data with a big bang nucleosynthesis code, assuming neutrino flavour equilibration via neutrino oscillations before the onset of big bang nucleosynthesis. We find a slight preference for negative neutrino chemical potentials, which would imply an excess of anti-neutrinos and thus a negative lepton number of the Universe. This lepton asymmetry could exceed the baryon asymmetry by orders of magnitude.

Dominik J. Schwarz, Maik Stuke, "Does the CMB prefer a leptonic Universe?" (Submitted on 28 Nov 2012 (v1), last revised 11 Mar 2013 (this version, v3)). Published version here.

The body text explains:

In this work, we re-investigate the possibility of non-standard big bang nucleosynthesis, based on SPT results [3, 4], the final WMAP analysis [5], and the recent reinterpretation of the helium-4 and deuterium abundance [6, 7, 8, 9]. We use stellar observations and CMB data to constrain the influence of a possible neutrino or lepton asymmetry. To do so, we compare and combine different results for the abundance of primordial light elements with theoretical expectations including neutrino chemical potentials. We assume that neutrinos are Dirac fermions and that they are relativistic before and at the epoch of photon decoupling, i.e. mνi < 0.1 eV, i = 1, 2, 3. . . .

The difference in the energy densities is the observed extra radiation energy density, commonly expressed as additional neutrino flavour in the effective number of neutrinos 

∆Neff = (Nν − 3) +X f 30 7 ξf π 2 + 15 7 ξf π 4 , (1) 

with Nν = 3 for the three neutrino flavour f = e, µ, τ , and corresponding neutrino chemical potentials ξf = µνf /Tν at neutrino temperature Tν. Note that the standard model predicts Neff = 3.042, a small excess above 3 due to corrections from electronpositron annihilation (not included in (1), but taken into account in our numerical calculations below). . . . 

We will concentrate here only on neutrino asymmetry induced chemical potentials. Assuming relativistic neutrinos and a lepton asymmetry much bigger than the baryon asymmetry |l| ≫ b, but still |l| ≪ 1, one can link the neutrino chemical potentials to the lepton asymmetry l [13], 

ξf = µνf Tν = 1 2 l s T 3 , (2) 

where s denotes the entropy density. A large lepton asymmetry leads also to a second effect during BBN, due to interactions of electron neutrinos with ordinary matter. While all three neutrino flavour chemical potentials affect the Hubble rate independently of their sign, the electron neutrino chemical potential influences the beta-equilibrium e + p ↔ n + νe directly. It shifts the proton-to-neutron ratio, depending on the sign of µνe , and so modifies the primordial abundances of light elements. . . .

Recent CMB data, combined with priors obtained from BAO data and measurements of H0, point to a high helium fraction compared to standard BBN and observed in HII regions. At the same time there might be some extra radiation degrees of freedom, expressed in Neff. Introducing a single additional variable to the standard model of cosmology, a non-vanishing neutrino chemical potential induced by a large lepton asymmetry, leads naturally to higher primordial helium without affecting the abundance of primordial light elements too much. Allowing for the helium fraction Yp and Neff to be free parameters in the analysis of CMB data, gives, for the combination WMAP9+ACT11+SPT11+BAO+H0, Yp = 0.278+0.034 −0.032 and Neff = 3.55+0.49 −0.48 [5]. Also CMB alone (SPT12+WMAP7) points to a higher Yp = 0.314 ± 0.033 and Neff = 2.60 ± 0.67 [4]. . . .  [A] negative chemical potential would allow for a BBN best-fit model much closer to the maximum of the CMB posterior distribution of Yp and Neff. The chemical potential seems to do better than just adding extra degrees of freedom. 

As was shown in [4] the analysis of the CMB data combined with BAO and H0 shows a preference for an extension of the ΛCDM model with Neff and massive neutrinos. The CMB alone favours the one parameter extension of including a running of the spectral index of primordial density perturbations or a two-parameter extension with Yp and Neff as free parameters. The introduction of a neutrino chemical potential has the advantage that it can improve the fit to both data sets with only a single additional parameter. 

Here we suggested that recent CMB data could provide a first hint towards a lepton asymmetry of the Universe, much larger than the baryon asymmetry of the Universe. Today this lepton asymmetry would hide in the neutrino background. This scenario would have interesting implications for the early Universe, especially at the epochs of the cosmic quark-hadron transition [13] and WIMP decoupling [14]. The largest allowed (2σ) neutrino chemical potential |ξf | = 0.45 leads to ∆Neff = 0.266, fully consistent with CMB observations. 

The helium fraction reported by CMB observations results in a negative neutrino chemical potential ξf ∼ −0.2 and ∆Neff ∼ 0.1. In that case we would live in a Universe ruled by anti-neutrinos. From our analysis we conclude that the present abundance of light elements and CMB data are not able to rule out ξf = 0, the standard scenario of BBN. However, upcoming CMB data releases and improved measurements of primordial abundances will allow us to test the idea of a leptonic Universe. 

Unfortunately, the paper doesn't directly convert the neutrino chemical potential into a neutrino-antineutrino asymmetry ratio.

Since then, constraints on ∆Neff  have tightened considerably. "[A]s of 2015, the constraint with Planck data and other data sets was [Neff is equal to] 3.04 ± 0.18." Neff equal to 3.046 in a case with the three Standard Model neutrinos and neutrinos with masses of 10 eV or more do not count in the calculation.

But, this still doesn't rule out the kinds of neutrino-antineutrino asymmetries suggested.

There is also outdated information available to constrain this ratio, as I note at Physics Stack Exchange:

Another 2002 paper puts an upper bound on electron neutrino asymmetry at 3% of the number of electron neutrinos, and a bound on muon and tau neutrino asymmetry at 50% of the combined number of such neutrinos, but I'm not clear that this reflects current research or that its methods are sound.

This predates the latest CMB (cosmic background radiation) data by more than a decade, however, and also does not reflect the latest data on neutrino oscillations.