The strongest bounds on active-sterile neutrino mixing after Planck data

This is the Planck-era version of the authors' own 2012 early-universe sterile neutrino calculation, and the honest description is: same machinery, new data, one genuinely new scan. The new part is real — they let two active-sterile angles be nonzero at once (abstract:s5), which their earlier work did not, and they run both active and sterile mass orderings. The oscillation-plus-collision framework is standard and correctly set up, and the resonance bookkeeping in Sec. II B and Fig. 1 is careful. The conclusion that standard cosmology with negligible lepton asymmetry is in tension with the short-baseline hints is right, and it is the conclusion most of us already expected after Planck 2013 [31]. That is worth having in one place with contours, and it is why I would cite it rather than redo it. On the reader's two methodological worries, I think one is misweighted and the other is real but secondary. The momentum-averaged ansatz (Sec. II B) does not carry the headline exclusion: at Delta m²_41 ~ 1 eV² all the splittings are positive, there are no resonances (the paper says so explicitly for the 5e-2 eV² case), and production is non-resonant collisional. The bound there needs less than ~42% thermalization, and no plausible error in a single-momentum closure turns saturated thermalization into 42%. Where the ansatz does bite is the small-Delta m² resonant contours, including the solar-upturn box, and there the error is not estimated for the contours actually shown. The paper leans on [46] and [41] for that, which are published, so it is supported — but a reader cannot tell how far those specific curves would move. The second worry, that Neff < 3.80 and m_eff,s < 0.42 eV are applied as independent cuts, is backwards: the rectangle contains the joint ellipse, so the exclusion is conservative, not overstated. The genuine load-bearing assumption is zero (or baryon-sized) lepton asymmetry. They flag it in the abstract and conclusions, but the L ~ 10⁻² escape hatch is quoted from prior work rather than computed for these contours, and that is the number a skeptic would want. Minor: the title's 'strongest bounds' overreaches given the asymmetry caveat, the 4-sigma statement in Sec. IV is asserted without a shown derivation, and there is a garbled sentence in Sec. III panel a ('for smaller values of sin² theta14 >~ 10⁻²'). No code or data tables, collision terms cited not specified. Send it out; it is a useful reference paper, not a surprising one.

The central claim is that the short-baseline region (Delta m^2_41 ~ 1 eV^2, sin^2 theta_14 ~ 10^-1, sin^2 theta_24 ~ 10^-2) is excluded by Omega_nu h^2 < 4.5e-3. At these masses all Delta m^2_4i > 0, so there are no MSW resonances (the paper says exactly this for Delta m^2_41 = 5e-2 eV^2 in Sec. II B), and production is non-resonant collisional. The bound requires (rho_ss + rho_bar_ss)/2 < 0.42, i.e. less than 42% thermalization. The paper's own statement in Sec. III, panel a, is that for sin^2 theta_24 > 10^-2, Delta m^2_41 > 10^-1 eV^2 is excluded 'independently on the value of sin^2 theta_14, since sterile neutrinos would be always produced with thermal abundances.' Saturated thermalization is robust: the averaged-momentum approximation can misestimate the approach to equilibrium by a factor, but it cannot turn full thermalization into 42% thermalization. So the SBL exclusion does not depend on the approximation. The approximation is load-bearing only for the small-Delta m^2 resonant contours, notably the solar-upturn region at Delta m^2_41 ~ 10^-5 eV^2, where resonance position and adiabaticity are momentum-dependent; that is a secondary claim, and the paper cites [46] for the approximation and [41] for the multi-momentum comparison, so it is supported by cited published work. The reader's second concern, that Neff < 3.80 and m_eff,s < 0.42 eV are applied as independent cuts, is conservative: the rectangle defined by the two cuts contains the true joint ellipse, so the paper's exclusion is weaker than the true joint exclusion, not stronger. The genuine load-bearing condition is the stated zero-asymmetry assumption, which the paper acknowledges and which the abstract flags as the escape route.

The reader identified a real methodological caveat but overweights it relative to the central claim. The paper's headline result, that the SBL region is excluded, lives at large Delta m^2_41 where the sterile is fully thermalized and the averaged-momentum approximation is least consequential; the result is a saturated upper bound, not a delicate rate calculation. The approximation matters for the small-Delta m^2 resonant contours (the solar-upturn exclusion), which is a secondary claim, and the paper cites [46] for the approximation and [41] for the multi-momentum comparison, so per the citation rule this is supported by published work rather than an unsupported internal assumption. The correlation concern is conservative: independent cuts on Neff and m_eff,s allow a rectangle that contains the true joint ellipse, so the paper's exclusion is weaker than the true joint exclusion. The genuine condition on the central claim is the stated zero-asymmetry assumption, which the paper acknowledges and which the abstract names as the escape route. The CONDITIONAL verdict is therefore appropriate, but for the stated-assumption reason rather than the approximation reason; my read does not change the reader's verdict. The concrete test above would confirm the saturation at the SBL point and bound the approximation's effect on the solar-upturn contour, settling whether the caveat lands anywhere that matters.

The paper is timely and useful. It applies the first Planck data release to a question the community was actively debating, and it does so with a systematic scan over four mass orderings rather than a single benchmark point. The treatment of the resonance pattern across the orderings (Fig. 1 and the accompanying discussion in Sec. II B) is clear and physically informative, and the exclusion contours are presented in a form that laboratory groups can use directly. The qualitative conclusion — that standard cosmology with negligible lepton asymmetry leaves little room for the eV-scale sterile neutrino invoked by short-baseline anomalies — has held up and is worth stating in the literature. The claim of a first treatment with two non-vanishing active-sterile angles is a genuine step beyond single-angle scans, though it should be qualified against earlier multi-flavour analyses as noted below.

The paper asks whether the light sterile neutrino invoked to explain short-baseline oscillation anomalies can survive the cosmological data. A fourth neutrino that mixes with the three active ones would have been resonantly and collisionally produced in the early universe, and its presence would leave two fingerprints: extra radiation at recombination, counted by $N_eff$, and extra mass density today, $N_eff$'s companion $m_{eff,s} = 94.1$ eV $h^2$ times the relic density. Planck's 2013 numbers, $N_eff = 3.30^{+0.27}_{-0.27}$ at 68% confidence and $N_eff < 3.80$ with $m_{eff,s} < 0.42$ eV at 95% confidence, are precise enough to bite. The authors evolve a (3+1) mixing scheme through the early universe for both active and sterile mass orderings, allowing two active-sterile mixing angles to be non-zero at once for the first time, and translate the Planck constraints into exclusion contours in the $N_eff$ plane. The conclusion they draw is a tension: the region the short-baseline experiments favour is excluded under standard cosmology, and relieving it requires extra physics such as a large primordial neutrino-antineutrino asymmetry.

What carries the argument. The engine is the momentum-averaged neutrino density matrix: a $4 N_eff 4$ matrix $N_eff$ obeying $i d N_eff/d t = [N_eff, N_eff] + C[N_eff]$, with the Hamiltonian carrying the vacuum mass matrix, the finite-temperature matter potential from $e^+e^-$ and neutrino energy densities, and a term proportional to the neutrino asymmetry, plus a collision term of order $G_F^2$. The oscillation term generates MSW-like resonances, which occur only when the relevant $N_eff$ is negative, so the resonance pattern is read off the mass ordering. The output of the evolution is the sterile density-matrix element $N_eff$, which converts to the observables through $N_eff = N_eff$ and $N_eff$; the Planck 95% confidence limits on those two numbers then become exclusion contours in the $N_eff$ plane. The approximation that makes an extensive scan feasible is replacing the full momentum distribution by its value at the mean thermal momentum, $N_eff$.

Core claim. The central claim is that the Planck 2013 determination of the relativistic energy density and of the neutrino mass sum excludes a large fraction of the (3+1) sterile neutrino parameter space, and that these cosmological bounds are more stringent than the corresponding laboratory limits. The authors scan $10^{-5} < N_eff$ for sterile-normal and sterile-inverted orderings, and for active normal and inverted hierarchies, plotting 95% confidence exclusion contours in $N_eff$ from the extra-radiation bound and from the energy-density bound. The shape of those contours is set by where Mikheyev-Smirnov-Wolfenstein-like resonances occur: production is enhanced whenever one of the active-sterile mass splittings is negative and the matter potential crosses it, so the number of resonances, and hence the strength of the bound, depends on the ordering pattern. Turning on a second mixing angle, $N_eff$ or $N_eff$, closes regions that would remain open if only one angle were non-zero. The authors conclude that sterile neutrinos with mass of order 1 eV, the range the short-baseline anomalies point to, would be excluded at more than $4$ $\sigma$, and that the solar-upturn region around $N_eff$ and $N_eff$ is excluded once $N_eff$.

The premise it rests on. The load-bearing premise is that the neutrino ensemble can be described by a single density matrix evaluated at the mean thermal momentum, $N_eff$, rather than by a full momentum-dependent distribution; the paper adopts this to make a broad parameter scan tractable but does not quantify how far the exclusion contours would shift if the full distribution were retained.

If the bounds hold, the $N_eff$ sterile neutrino region favoured by LSND, MiniBooNE, the reactor anomaly and the gallium anomaly is excluded under standard cosmology with negligible lepton asymmetry, so at least one of the two sets of evidence must give way.

The exclusion contours depend on how many active-sterile mass splittings are negative, so the ordering pattern of the four mass eigenstates becomes an observationally relevant question rather than a labelling convention.

Because the bound from the energy density today, $N_eff$, covers masses up to the eV scale where the radiation bound weakens, the two Planck constraints together close the parameter space from below and above.

The authors note that adding the primordial deuterium abundance to the Planck analysis would tighten the limit to $N_eff$ at 95% confidence, so improved quasar absorption measurements would translate directly into stronger mixing bounds.

Rescuing the laboratory hints requires modifying the standard cosmological model, for instance with a lepton asymmetry $N_eff$, which suppresses sterile production in the early universe.

The qualitative conclusion is more robust than the quantitative contours: a full momentum-dependent treatment would move the resonance positions and the collisional repopulation rate, but it would have to move them by a large factor to reopen the eV-mass window.

The same machinery, run in reverse, makes the sterile neutrino a probe of the primordial lepton asymmetry: if a future laboratory experiment confirms an eV-scale sterile state, the required $N_eff$ becomes a measurable cosmological parameter rather than a free escape hatch.

Independent constraints from big bang nucleosynthesis on the expansion rate during the first minutes probe the same production physics at a different temperature, so a joint CMB-plus-deuterium analysis is a sharper test than either alone.

The ordering dependence of the resonance count suggests a cross-check: scenarios with three resonances should show a step-like feature in the exclusion contour at the mass splitting where the third resonance switches on, which a finer scan could resolve.

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