Cosmological lepton asymmetry with a nonzero mixing angle \theta_13

The thing to know: this is a competent, honest joint CMB+helium analysis of neutrino asymmetries using oscillation-corrected spectra, run for two values of θ13, plus a COrE forecast. The genuinely new content is the joint fit and the forecast table, not the θ13 dependence itself — that was already in the BBN-only work of Refs. [34,35], and the authors say so plainly rather than overselling it. That kind of candor is worth noting; they also decline to quote 95% intervals where the posterior doesn't vanish at the prior edge, and they flag that their ALL dataset gives a bimodal posterior and a spurious nonzero m1. Those are the marks of people who looked at their own chains honestly. The central mechanism — large θ13 equilibrates the flavors, so the helium bound on the electron flavor becomes a bound on the total — is sound and visible in Fig. 3. I don't think the stress-test's worry about the asymmetric interval (−0.071, +0.054) is well founded; Yp(ξe, ΔNeff) is nonlinear and ΔNeff is quadratic in ξ, so asymmetry is expected, not a table artifact. But the paper should say that in one sentence instead of leaving it unexplained. The real soft spot is the one the stress test names: the two-stage interpolation (spectra grid → ξe, ΔNeff → Yp) is validated only by the sentence 'we have checked that the agreement... is at the percent level.' One percent of Yp = 0.250 is 0.0025, essentially the 0.003 prior width that generates the whole bound. I'd read that as a smooth interpolation error that largely cancels in the derivative rather than a bias of the same size as the effect, so I don't think it breaks the result — but a residual plot and the grid resolution would settle it, and it sits directly under the headline number. Second, minor: the abstract's factor 6.6 is the θ13 = 0 case, which the paper itself calls disfavored; the favored case improves by 1.6. Reword. This is for people working on BBN, Neff, and extra radiation, and for CMB forecasters. It deserves a serious referee — the fix is a figure and a sentence, not a rewrite.

The central claim — that for sin²θ13 = 0.04 the helium bound on η_νe becomes a bound on total η_ν of [−0.071, 0.054] — is produced by a chain of two interpolations described in Sec. III: (i) precomputed neutrino spectra on a grid in (η_νe^in, η_ν) taken from Refs. [26,34], and (ii) a table mapping those to (ξ_e, ΔN_eff), which is then fed to a Yp(ξ_e, ΔN_eff, Ω_b h²) relation. The only validation offered is the sentence 'we have checked that the agreement between the interpolation scheme and the full BBN analysis is at the percent level.' No figure, no residual plot, and no grid resolution is given. This matters quantitatively: the adopted prior is Yp = 0.250 ± 0.003, i.e. 1.2% relative. A 'percent level' interpolation error in Yp is therefore ~0.0025, essentially the same size as the entire prior width that generates the bound. The unvalidated numerical error is thus of the same order as the effect being measured. The reader flagged that the comparison is never shown; the sharper point is that the claimed accuracy is not obviously sufficient for the quoted precision. This is not an acknowledged limitation in the text — the authors assert agreement rather than concede uncertainty — so it remains unresolved. Note also that the θ13 = 0.04 interval is asymmetric (−0.071, +0.054), a feature a coarse table could imprint; the paper does not explain its origin.

The paper does a competent, transparent job on a well-posed question: given the measured θ13, does flavor equilibration before BBN convert the helium bound on the electron-flavor asymmetry into a bound on the total asymmetry? The mechanism is sound and visible in Fig. 3, and the authors are candid that the Yp prior dominates, that a larger helium uncertainty (Ref. [55]) would weaken the bounds, that deuterium is omitted, and that the ALL dataset produces a bimodal posterior. Those acknowledgments are credit, not defects, and I do not re-raise them as findings. The reader's named weakest assumption — Yp prior dominance — is exactly the authors' own stated position, so it is not a new concern. Where the reader's rationale does land, and where I would sharpen it, is the interpolation scheme. The reader says the percent-level comparison is never shown; the quantitative point is that 'percent level' in Yp is ~0.0025, comparable to the 0.003 prior width, so the unvalidated numerical error is the same size as the constraint. That is a genuine, unresolved gap sitting directly under the headline number, and it is not conceded anywhere in the text. A secondary, smaller observation: Table II reports ΔN_eff ≤ 0.03 for sin²θ13 = 0.04 (N_eff ≤ 3.076), while the Conclusions say lepton asymmetries 'cannot increase N_eff significantly above 3.4'; the 3.4 figure appears to refer to the θ13 = 0 case (ΔN_eff ≤ 0.43). This is a loose internal statement rather than a load-bearing error, but it is worth tightening. Neither point overturns the central claim; both are addressable with one residual figure and one reworded sentence. I therefore keep the reader's CONDITIONAL verdict rather than moving it: the concern reinforces the existing conditional status and does not independently justify rejection. The COrE forecast is explicitly labeled optimistic on systematics, which is the right framing, and the factor-6.6 headline being attached to the disfavored θ13 = 0 case is a presentation issue the reader already noted.

The paper addresses a well-posed question with the standard machinery of the field and reports a result that is physically interpretable: a measured mixing angle sharpens cosmological lepton-asymmetry bounds because oscillations redistribute the asymmetry among flavors before BBN. The negative corollary — that with the Daya Bay/RENO value of θ13, lepton asymmetry cannot push Neff much above 3.4, so a large Neff would require new physics such as sterile neutrinos — is a useful and falsifiable statement. The analysis is transparent about its own weak point: the authors state explicitly that the Yp prior dominates, discuss alternative helium determinations ([52–56]), and decline to quote 95% intervals where the posterior does not vanish at the prior edge. The COrE forecast is a genuine forward-looking projection with tabulated experimental specifications (Table III) and fiducial parameters (Table IV). The main strengths are the consistent treatment of oscillation-damped spectra in a CMB pipeline and the honest separation of BBN-driven from CMB-driven constraints.

The paper asks how large a relic neutrino lepton asymmetry the universe can still be hiding, and argues that the answer depends sharply on the mixing angle $theta_{13}$, the last neutrino oscillation parameter to be measured. Before Big Bang Nucleosynthesis, flavor oscillations redistribute any initial asymmetries among electron, muon and tau neutrinos; if $theta_{13}$ is large the equilibration is efficient and the three flavors end up nearly equal, so the tight helium-4 limit on the electron-neutrino asymmetry becomes a limit on the total. The authors fold neutrino spectra computed with oscillations into a modified Boltzmann code and run Markov chains against WMAP data plus a Gaussian prior on the primordial helium mass fraction, then repeat the exercise with a mock future CMB experiment. The payoff is a factor of roughly ten between the $theta_{13}=0$ and $sin^2theta_{13}=0.04$ bounds on the total asymmetry, and a demonstration that present data leave the neutrino mass limit essentially untouched by the asymmetry. It matters because it says the recently measured $theta_{13}$ has already done cosmological work: it closes off the possibility that a large lepton asymmetry is propping up the effective number of neutrinos.

What carries the argument. The load-bearing object is the map from initial flavor asymmetries $(eta_nu_e^{in}, eta_nu)$ to the final neutrino spectra after oscillations, computed by solving the equations of motion for $3times 3$ density matrices with vacuum, matter and collision terms down to the BBN epoch. Those spectra are tabulated on a grid and interpolated during the Monte Carlo, yielding an effective degeneracy parameter $xi_e$ and an enhancement $Delta N_{rm eff}$ for each parameter pair; $Delta N_{rm eff}$ enters the expansion rate and $xi_e$ enters the helium-4 yield, which is then compared to the Gaussian prior on $Y_p$. The second piece of machinery is a modified CAMB interfaced with CosmoMC, so that the same tabulated spectra feed the CMB and matter power spectra. The comparison between the two $theta_{13}$ values is the experiment: the same pipeline, the same data, only the mixing angle changed.

Core claim. For $theta_{13}=0$ the 95% credible interval on the total asymmetry is $-0.64 leq eta_nu leq 0.72$; for $sin^2theta_{13}=0.04$ it shrinks to $-0.071 leq eta_nu leq 0.054$. The mechanism is equilibration: with large mixing the final asymmetries satisfy $eta_nu_e^{fin} approx eta_nu_x^{fin} approx eta_nu/3$, so the helium prior that forces the electron-neutrino asymmetry near zero now forces the total near zero, whereas with zero mixing the total can hide in the muon and tau flavors. The same effect shows up in the effective number of relativistic species, where the 95% upper bound on $Delta N_{rm eff}$ drops from $0.43$ to $0.03$. The authors emphasise that the constraint is not a CMB measurement: with WMAP data alone the asymmetry posteriors run to the edge of the prior, and it is the $Y_p = 0.250 pm 0.003$ prior of Eq. (4) that does the work. The neutrino mass limit, $m_1 leq 0.38$ eV, is unchanged between the two mixing cases, which the authors read as evidence that mass and asymmetry are essentially uncorrelated in the data. The COrE forecast then asks whether a future CMB experiment can beat the helium measurement, and answers that it can: $eta_nu$ would be pinned to $[-0.12, 0.09]$ for $theta_{13}=0$ and $[-0.048, 0.030]$ for $sin^2theta_{13}=0.04$, improvements of about $6.6$ and $1.6$ respectively.

The premise it rests on. The whole constraint rests on treating the primordial helium-4 mass fraction as a Gaussian measurement of $Y_p = 0.250 pm 0.003$, since the cosmic microwave background alone leaves the total asymmetry essentially unconstrained and the quoted intervals are that prior passed through the oscillation model.

Any excess in the effective number of neutrinos above the standard value cannot be attributed to lepton asymmetry if $theta_{13}$ is as large as Daya Bay and RENO indicate; the authors state that asymmetries cannot push $N_{rm eff}$ significantly above 3.4, so a persistent excess would require new physics such as sterile neutrinos.

Present limits on the total asymmetry are a restatement of the helium-4 abundance measurement rather than an independent cosmological probe, so improving them requires either better CMB sensitivity to $N_{rm eff}$ or a sharper primordial helium determination.

A CMB experiment with COrE-like sensitivity to lensing would constrain the total asymmetry to the few-percent level and the lightest neutrino mass to below about $0.05$ eV, entering the range of the oscillation mass splittings.

Because the CMB is insensitive to the sign of the asymmetries, helium and other BBN measurements remain necessary to determine whether the asymmetry is positive or negative.

Cosmological limits on the sum of neutrino masses are robust against the presence of lepton asymmetries, so mass bounds quoted in standard analyses need no correction.

If the primordial helium determination is revised to the larger-uncertainty values the authors mention, the $theta_{13}=0.04$ bound would widen by roughly a factor of three, which would put it close to the COrE forecast and weaken the claim that current data are already informative about the total asymmetry.

The same equilibration logic predicts that for inverted hierarchy, or for any $sin^2theta_{13} gtrsim 0.02$, the bound saturates: the authors note the cosmological effects are similar across that range, so a future precise $theta_{13}$ measurement would not further sharpen the limit, only confirm that it applies.

A joint analysis that used deuterium abundances alongside helium could close the contours in the $(eta_nu, eta_nu_e^{in})$ plane that the present work leaves open at $theta_{13}=0$, since deuterium is sensitive to the total radiation density rather than only to the electron flavor.

Because the tabulated spectra are interpolated from a finite grid, an independent check with a full non-equilibrium BBN code at several grid nodes would test whether the percent-level agreement claimed here holds where the posteriors are steepest.

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