Astronomy & Celestial Mechanics
the sky, computed
Cosmology

A bound below the floor is not a mass

Oscillation experiments prove that neutrinos weigh at least 0.059 eV between them. Cosmology's latest upper bound on their total is 0.064 eV, and a bound computed differently would sit below the floor. The usual reading is that cosmology has nearly weighed the neutrino. The better one is that the fit's likelihood peaks at a negative mass: the data see more clustering than any real neutrino allows, and the bound is a symptom of that tension rather than a measurement of anything.

Assumes Density parameters and Solar neutrinos.

15 min read 6 figures Measuring the unreachableSmall perturbations, large consequences

Neutrinos have mass. The proof came from the oscillations: a neutrino made as one flavour arrives as another after travelling a distance that depends on its energy, and the survival probability of the Sun’s electron neutrinos depends on energy in exactly the way oscillation between massive states requires. Oscillations measure differences between the squares of the masses, not the masses themselves, and two such differences are now known to a few per cent. They are enough to put a floor under the sum of the three masses, whatever the lightest one weighs.

Cosmology measures the sum from the other side. Neutrinos stream out of growing clumps and take their gravity with them, so a universe with heavier neutrinos has less small-scale structure, and the measurements of structure put an upper limit on the sum. Over the last decade that limit has fallen steadily — 0.12 electronvolts from the microwave background alone in 2018, 0.072 when the baryon acoustic scale from a large spectroscopic survey was added in 2024, 0.064 in 2025 — and the floor is 0.059.

The obvious headline is that cosmology is about to weigh the neutrino. This essay argues for a less comfortable reading, the one the essay on the neutrino’s hole in the structure ended on: the bound is falling not because the measurements are closing in on a mass but because they prefer a universe with more structure than massless neutrinos would leave. A bound that crosses the floor is the visible end of a tension, and the tension is in the clustering, not in the particles.

The floor

The three neutrino masses are not known individually. Their squared differences are: about 7.4×10−57.4 \times 10^{-5} eV² between the two closest, from solar and reactor neutrinos, and about 2.5×10−32.5 \times 10^{-3} eV² between those and the third, from atmospheric and accelerator neutrinos. Which way round the third sits — heavier than the pair or lighter — is not known, and the two possibilities are the normal and inverted orderings.

The sum of three masses nobody has weighed. The sum of the three neutrino masses against the mass of the lightest, for the two orderings oscillation experiments allow, from the measured splittings (squared-mass differences of 7.42 × 10⁻⁵ and 2.51 × 10⁻³ eV²). However light the lightest, the sum cannot fall below 0.059 eV in the normal ordering or 0.101 eV in the inverted one. The horizontal lines are upper bounds on the sum from cosmology at 95 per cent: 0.12 eV (microwave background, 2018), 0.072 eV (with baryon acoustic scale, 2024), 0.064 eV (with baryon acoustic scale, 2025). The last excludes the inverted ordering entirely and leaves the normal ordering only a sliver — or none, at a slightly higher confidence.
Fig. 1 The sum of the three neutrino masses against the mass of the lightest, for both orderings. However light the lightest, the sum cannot fall below 0.059 electronvolts in the normal ordering or 0.101 in the inverted. The horizontal lines are cosmology’s 95 per cent upper bounds on the sum, from 2018, 2024 and 2025. The last excludes the inverted ordering outright and leaves the normal ordering a margin of five thousandths of an electronvolt.

Read that way, the figure is an advertisement: the inverted ordering excluded by cosmology alone, and the lightest neutrino nearly pinned to zero. Laboratory experiments that weigh the neutrino directly, from the shape of the electron spectrum near the end point of tritium decay, bound it at a few tenths of an electronvolt — far above either line — and experiments trying to determine the ordering from oscillations have not yet settled it. If cosmology were right, it would have done both jobs first.

How a bound is made

The way the bound is computed matters as much as the data. An analysis fits the cosmological parameters, including the neutrino mass sum, to the measurements, and the fit produces a likelihood: how well each value of the sum describes the data. Because a mass cannot be negative, the analysis imposes a prior that the sum is zero or more, and quotes as the upper limit the value below which 95 per cent of the remaining probability lies.

If the likelihood peaks at a positive mass, the bound tracks the mass. If it peaks at zero, the bound measures the width of the likelihood. If, extended past zero, it would peak at a negative value, then the part above zero is only its tail, and the bound depends on how far into the tail the zero sits.

A bound that moves with the peak, not the mass. The 95 per cent upper limit on the sum of the neutrino masses that a Gaussian likelihood gives when the physical prior, sum ≥ 0, is imposed, against where the likelihood peaks — allowed, in the fit, to sit at negative values. For a likelihood of width 0.05 or 0.03 eV peaking at zero the bound is 0.098 or 0.059 eV. As the peak moves below zero the bound keeps falling, although no mass is being measured: with a width of 0.05 eV it crosses the normal-ordering floor of 0.059 eV when the peak reaches −0.08 eV. A bound below the floor says the data prefer less suppression of structure than any allowed mass gives — that they want, in the fit's terms, a negative mass.
Fig. 2 The 95 per cent upper limit, with the prior that the sum is zero or more, against where an underlying Gaussian likelihood would peak if allowed to go negative, for two widths. A likelihood of width 0.05 eV peaking at zero gives a bound of 0.098 eV. As the peak moves into negative values the bound keeps falling, although no mass is measured: it crosses the normal-ordering floor when the peak reaches about −0.08 eV. A tighter likelihood crosses it with its peak at zero.

So a bound below the floor does not need a light neutrino. It needs a likelihood that peaks below zero, or is narrow and peaks at zero. The narrower the measurements, the more readily the second happens, and the 2025 bound is close to the point where a likelihood peaking exactly at zero would produce it. That alone would be a perfectly honest result: cosmology sees no neutrino mass, at a precision where it should be starting to see one.

It helps to see the honest version of that case drawn.

One likelihood, read with three priors. A Gaussian likelihood in the effective neutrino mass sum, peaking at 0 eV with a width of 0.03 eV — of the shape that combinations of microwave-background and baryon-acoustic data have been reported to give. Read with no prior it says the data see no neutrino mass at all. Read with the physical prior, sum ≥ 0 (shaded right of zero), its 95 per cent upper limit is 0.059 eV, at the normal-ordering floor. Read with the oscillation prior, sum ≥ 0.059 eV, its limit is 0.091 eV — but only 2.5 per cent of the likelihood lies above the floor at all, which is the more informative number: the data and the oscillation experiments disagree at about 2.0 standard deviations.
Fig. 3 A likelihood that peaks exactly at zero with a width of 0.03 eV — cosmology seeing no neutrino mass at all, at a precision a little better than today’s. With the prior that the sum is zero or more, its 95 per cent bound lands on the floor. Nothing about it is in tension with the laboratory; the oscillation floor sits two standard deviations out on its wing. The same headline bound can come from this likelihood or from one peaking well below zero, and only the full distribution says which.

The effective mass that goes negative

To find out where the likelihood actually peaks, analyses since 2024 have done what the figure does: replaced the physical mass sum with an effective parameter that enters the equations the same way but is allowed to be negative. A positive effective mass suppresses structure as a real mass would. A negative one enhances it, by the same formula run backwards.

A fraction that has to be positive, extended through zero. The suppression of small-scale matter power, about eight times the neutrino share of the matter, against the sum of the neutrino masses, the share being the sum divided by 93.14 eV and by the physical matter density. The minimum sum allowed by oscillations, 0.059 eV, is a neutrino share of the matter of 0.45 per cent and a suppression of 3.5 per cent. The relation is linear and passes through zero, so a fit can extend it to negative sums without anything in the formula objecting: the left half of the line is where a data set lands when it measures more clustering than any physical neutrino allows. The shaded region is physically forbidden; the effective-mass analyses draw it anyway, to see where the data would go.
Fig. 4 The change in small-scale matter power against the neutrino mass sum: about eight times the neutrino share of the matter, which is the sum divided by 93.14 eV and by the physical matter density. The relation is linear and passes through zero. The lightest sum oscillations allow suppresses the power by three and a half per cent. Extended into the shaded region, the same line gives an enhancement — the region where a data set lands if it measures more structure than massless neutrinos would leave.

The answer, in several analyses of the 2024 and 2025 data, has been a likelihood peaking below zero, typically by a few hundredths to a tenth of an electronvolt, with a width of a few hundredths. The precise numbers depend on which microwave-background likelihood and which galaxy surveys are combined, and they move as the data improve; what has been robust is the sign.

One likelihood, read with three priors. A Gaussian likelihood in the effective neutrino mass sum, peaking at −0.08 eV with a width of 0.05 eV — of the shape that combinations of microwave-background and baryon-acoustic data have been reported to give. Read with no prior it says the data prefer a negative sum. Read with the physical prior, sum ≥ 0 (shaded right of zero), its 95 per cent upper limit is 0.059 eV, at the normal-ordering floor. Read with the oscillation prior, sum ≥ 0.059 eV, its limit is 0.102 eV — but only 0.3 per cent of the likelihood lies above the floor at all, which is the more informative number: the data and the oscillation experiments disagree at about 2.8 standard deviations.
Fig. 5 A likelihood of that shape — peaking at −0.08 eV with a width of 0.05 — read three ways. With no prior it prefers a negative sum. With the physical prior, sum ≥ 0, it gives a 95 per cent bound of 0.059 eV, right at the floor. With the oscillation prior, sum ≥ 0.059, it gives a bound of 0.10 — but only 0.3 per cent of the likelihood lies above the floor, a disagreement between cosmology and the laboratory of nearly three standard deviations.

The third reading is the informative one, and it is the one the headline number hides. A bound of 0.064 eV sounds like a measurement approaching the floor from above. A likelihood with almost nothing above the floor is a measurement that disagrees with the floor.

What a negative mass means

There is no particle with negative mass in the fit. The parameter is a stand-in for whatever the data want, and what they want is a universe in which structure has grown, or appears to have grown, more than the standard model with the minimum neutrino mass predicts.

What a neutrino mass does to the clustering of matter. The change in the matter power spectrum relative to massless neutrinos, against wavenumber, for neutrino mass sums of −0.1, 0.06, 0.12 eV, with the cold matter held fixed. Massive neutrinos stream out of clumps smaller than their free-streaming length and take their gravity with them, suppressing the small-scale power by about eight times their share of the matter: 6.0 per cent for −0.1 eV, −3.6 per cent for 0.06 eV, −7.2 per cent for 0.12 eV. The transition, drawn as a smooth step, sits at the free-streaming scale. A negative "mass" is not a particle: it is the same formula run backwards, an enhancement of small-scale clustering, which is what a fit finds when the measured clustering or lensing is stronger than massless neutrinos give.
Fig. 6 The change in the matter power spectrum relative to massless neutrinos, against wavenumber, for mass sums of −0.1, 0.06 and 0.12 eV. A real mass suppresses the power below its free-streaming scale by a few per cent; a negative effective mass of the size the fits prefer enhances it by six. The data that drive the preference do not measure this spectrum directly, but they measure things it controls — the lensing of the microwave background by the structure between, and the expansion history that sets the distances.

The neutrino mass enters the data in two ways, and both are implicated. The first is clustering: massive neutrinos reduce the gravitational lensing of the microwave background by the intervening structure, and the microwave background’s lensing signal is, if anything, slightly stronger than the standard model predicts — the same excess that made the background alone prefer a closed universe. The second is expansion: massive neutrinos behave as radiation early and matter late, changing the matter density today relative to what the background measured at recombination. The baryon acoustic scale measures the matter density through distances, and the survey’s measurement prefers a slightly lower matter density than the background’s best fit. Adding neutrino mass raises the matter density; the data push the other way; the fit, given the freedom, goes negative.

How much there is to weigh

The density the neutrinos contribute follows from how many there are, and that is fixed by the hot early universe. Neutrinos fell out of equilibrium with the rest of the plasma about a second after the beginning, when the temperature was a few million electronvolts, and have been cooling with the expansion ever since without interacting. Their temperature today is 1.95 kelvin — a little below the microwave background’s 2.73, because the background was heated when electrons and positrons annihilated after the neutrinos had already decoupled — and their number is fixed at about 336 per cubic centimetre, all three species together.

Multiply that number by the masses and the neutrino density follows. The constant that appears in the figures, 93.14 electronvolts, is the sum of the masses that would make the neutrinos’ density parameter, times the square of the dimensionless Hubble constant, equal to one. At the floor of 0.059 eV the neutrinos are about 0.14 per cent of the critical density — about half the stars’ share, and under half a per cent of the matter.

That small number is still measurable because of where it acts. The neutrinos’ thermal velocities today, for masses of a few hundredths of an electronvolt, are thousands of kilometres a second, enough to stream out of anything smaller than a large galaxy cluster. Their gravity is missing exactly where structure is growing fastest, and the deficit accumulates over the billions of years since they slowed enough to count as matter.

What else could produce it

A negative effective mass is a symptom with several possible causes, and the analyses that found it have listed them.

A statistical fluctuation. A preference at two to three standard deviations is the size that comes and goes as data accumulate. It would be nothing more than that if the next data release moved the peak back towards zero.

A systematic in the microwave background. The lensing excess in the background’s maps has been reduced, though not eliminated, by improved treatments of the foregrounds and the likelihood. A residual of a few per cent in the smoothing of the peaks moves the effective neutrino mass by about the amount the fits find.

Something wrong with the dark energy. The same survey that pushed the bound down also reported a preference for dark energy whose density changes with time. A dark energy that is not a constant changes the expansion history at late times, and when it is allowed, the neutrino likelihood relaxes towards positive masses. The two anomalies are correlated in the fits: one model of late-time expansion absorbs both.

Something that enhances clustering. A long-range force between dark-matter particles, a dark-matter component that clusters differently, a modification of gravity — any of these could produce more structure than the standard model allows, which a fit with only a neutrino mass to vary would report as a negative mass.

None of these is established. What the list shows is that the neutrino mass has become a sensitive test of the standard cosmological model as a whole: a small parameter, with a hard lower limit set by physics elsewhere, whose fitted value cannot fall below that limit without something in the model being wrong.

How the bound came down so fast

The microwave background alone constrains the neutrino mass weakly, because neutrinos of a tenth of an electronvolt were still relativistic when the background was released and left little imprint on its primary fluctuations. What it measures well is the matter density at recombination and the lensing of its own light by later structure. The constraint on the mass comes from comparing that early matter density with how much structure and expansion there is later — and that requires a second, late-time measurement.

The baryon acoustic scale supplies it. The same ruler measured ten billion years apart gives distances that, with the background’s calibration of the ruler, measure the expansion history at late times and so the late-time matter density. The spectroscopic survey whose results arrived in 2024 and 2025 measured that scale with millions of galaxies and quasars at a precision of better than a per cent at several redshifts, and the neutrino bound fell by nearly a factor of two in two years.

The speed of that improvement is part of why the preference for a negative effective mass appeared when it did. With the earlier, looser data, a likelihood peaking a little below zero and one peaking at zero were indistinguishable once the prior was imposed. Only when the width fell to a few hundredths of an electronvolt did the position of the peak start to matter to the bound — and to the comparison with the floor.

What the laboratory will add

The laboratory side is moving too, and it will put the cosmological result in a harder position whichever way it goes.

A large reactor-neutrino experiment in China, which began taking data in 2025, is designed to determine the mass ordering from the fine structure of the oscillation pattern within about six years. If it finds the inverted ordering, the floor rises to 0.101 eV and the cosmological bound, at 0.064, is in outright conflict with it — a conflict no fluctuation of the present size could explain, and one that would force a change somewhere in the cosmological model. If it finds the normal ordering, the floor stays at 0.059 and the comparison with cosmology rests on how the structure measurements evolve.

Direct mass measurements from tritium decay bound the effective electron-neutrino mass below about 0.45 eV — for three nearly equal masses, a sum some twenty times the cosmological bound — and their next generation aims at a factor of a few better. They are unlikely to reach the floor, but they measure the mass with no cosmological assumptions at all, and a direct detection above the cosmological bound would, like an inverted ordering, put the cosmological model rather than the neutrino under question.

Why the floor is a powerful test

The floor is what makes this more than another tension. Most cosmological parameters have no external value to compare with; the Hubble constant has a disputed local measurement, the matter density has nothing. The neutrino mass sum has a laboratory floor known to a few per cent, which no amount of cosmological reasoning can move.

So the standard model plus the floor makes a prediction: that the structure measurements will show the suppression of at least 0.059 eV of neutrinos — a three and a half per cent deficit in small-scale power, a corresponding reduction in lensing — when they become precise enough. The next surveys will reach a precision of about 0.02 electronvolts on the sum, three standard deviations on the floor. If they then see the suppression, the neutrino has been weighed and the present preference was a fluctuation. If they see none, or an enhancement, the standard model needs a new ingredient, and the neutrino will have found it.

The budget it belongs to

In the budget whose familiar part is five per cent, neutrinos are the smallest entry — a few tenths of a per cent of the matter at the floor, less than the stars. They matter because they are the only entry that changes category, radiation-like in the early universe and matter-like now, and so the only one whose density is tied to physics measured in a laboratory. The ordinary matter was found by a millisecond after twenty years of searching; the neutrinos, which outnumber the ordinary matter’s particles by a factor of a billion, have so far been found only by their absence.

Where the model stops

A Gaussian likelihood. The figures model the likelihood in the effective mass as a Gaussian. The published ones are not quite, and their tails, which set the 95 per cent bound, depend on the details of the data combination.

Illustrative numbers. The peak at −0.08 eV and width of 0.05 eV are of the size reported, not taken from any one analysis; different combinations of data give peaks from a few hundredths to more than a tenth below zero, with widths of a few hundredths.

A one-number suppression. The matter power suppression is drawn as eight times the neutrino share of the matter below a smooth step at the free-streaming scale. The real suppression grows slowly with wavenumber, depends on redshift, and is computed for each mass ordering separately, because three neutrinos of different masses free-stream on different scales.

Bounds as reported. The three bounds drawn are the headline 95 per cent limits of the analyses named, each depending on its own choice of data and priors.

The surprising connection

The neutrino bound is a case of the same statistical trap that made the tallest peak in a noise periodogram look like a signal, in reverse. There, a search reports the maximum of many noisy quantities and finds something where nothing is. Here, a fit reports a bound with a hard edge in its prior and finds a tighter constraint than the data justify, because the edge cuts away the part of the likelihood that was saying something. In both cases the number at the top of the paper is a correct calculation of the wrong question, and the right question — what the full distribution says — gives a different answer.

Still open: whether the clustering is really stronger

The preference for a negative effective mass is now present in several independent combinations of data, and its significance has grown with each release, but it has not passed the level at which a statistical fluctuation can be dismissed. The surveys reporting their next data in the coming years will measure the clustering of galaxies, the lensing of the microwave background and the baryon acoustic scale with enough precision to see the suppression of the minimum neutrino mass at three standard deviations, if it is there. Whether they see it — confirming the standard model and weighing the neutrino at last — or see more structure than any neutrino allows, breaking the model at its most constrained point, is the question this small, floor-bounded parameter has turned into.

Where this goes next

A mass measured by the structure it prevents. The fractional suppression of the small-scale matter power spectrum against the sum of the neutrino masses. The relic density follows from the thermal history with no astrophysics in it — the neutrinos’ share of the critical density, times h², is the mass sum divided by 93.14 eV exactly — and the suppression is about eight times the neutrinos' share of the matter, because a particle moving at a large fraction of the speed of light streams out of an overdensity while it is still growing and takes its own gravity with it. The two mass orderings put a floor under the sum at 58 meV and 100 meV; the cosmological upper bound is near 0.12 eV, which is 7.2 per cent of suppression. The floor and the ceiling are within a factor of two of each other, so this is a measurement about to happen or a model about to break, and the quantity it returns is a sum rather than any individual mass. A mass measured by what it stopped from forming Neutrinos were relativistic in the early universe and are not now, so they are the one entry in the cosmic budget that changes category. What cosmology measures is not their density but the hole they leave — they stream out of a growing clump and take their gravity with them, and the missing structure bounds a particle mass more tightly than any laboratory has. Every curvature has a Hubble constant that leaves the peaks where they are. The Hubble constant a universe needs in order to put the acoustic peaks of the microwave background at the measured angle, against its spatial curvature Ωₖ, with the physical densities of baryons, matter and radiation held at their fitted values so that the sound horizon is unchanged. Along this line the angle θ* — the sound horizon over the distance to last scattering — is exactly constant and the peak positions do not move. The flat member has H₀ = 67.4 km/s/Mpc; a closed universe with Ωₖ = −0.044 needs 53.5, with a matter density of 0.50; an open one with Ωₖ = +0.010 reaches the locally measured 73. The peaks alone therefore measure a combination of curvature and expansion rate, not either one, and the statement that the universe is flat to a few parts in a thousand needs a second distance measured somewhere else. A flatness that needs a second distance The universe is said to be flat to two parts in a thousand, and the figure is usually credited to the microwave background. On its own the background measures one angle, and one angle fixes one distance — so every curvature has a Hubble constant that reproduces the peaks exactly. Flatness is where that line crosses a second measurement, and it is only as good as the crossing. A survival probability with a step in the middle of the spectrum. The probability that an electron neutrino made in the core of the Sun is still an electron neutrino at the Earth, against its energy, from the resonant conversion it undergoes in the Sun's own electrons: mixing angle sin²θ₁₂ = 0.307, splitting Δm² = 7.4·10⁻⁵ eV², and an electron density at the point of production of 90 moles per cubic centimetre. At low energy the matter potential is small beside the vacuum term and the probability is the vacuum average, 0.54 at 0.3 MeV (the flat limit is 0.55). At high energy the matter term dominates, the neutrino is born almost entirely in the heavier mass state, and the probability falls to 0.32 at 10 MeV, close to sin²θ₁₂. The two terms are equal at 2.1 MeV, which falls in the gap between the ⁷Be line and the bulk of the ⁸B spectrum. The points are the measured survival probabilities for each source separately — pp, ⁷Be, pep and ⁸B from Borexino, and ⁸B from SNO — and every one of them lies within about two standard deviations of the curve: low-energy neutrinos survive about half the time, high-energy ones about a third. A survival that depends on the energy A low-energy neutrino from the Sun reaches the Earth still an electron neutrino about half the time, and a high-energy one about a third of the time. The difference is made by the Sun's own electrons, and the energy at which it switches over — near two million electronvolts — sits in the gap between the neutrinos that never meet a resonance and the ones that all do.

About the same objects

Not linked from either essay — found by the objects both name.

The objects this essay names

Each one links to every other essay that touches it.

Baryon acoustic oscillationsDensity parameterFree-streamingMatter power spectrumNeutrino massNeutrino oscillationPrior