The sound speed a neutron star needs

The heat that arrives before it could found that the ordinary equation for heat flow lets a disturbance spread faster than light, and that the relativistic repair gives heat a finite speed. The bath that pushes back and the body that has no temperature when it moves followed what thermodynamics does under a change of frame, and the column that is hotter at the bottom what it does in a gravitational field. The heat that turns into matter heated a gas until its thermal energy made new particles.

That leaves a quantity every fluid has and relativity constrains: the speed at which a disturbance in pressure travels through it, the speed of sound. Nothing is allowed to be rigid found that no material can transmit a push faster than light, so the sound speed is capped at $c$. There is a second, lower ceiling that appears naturally in relativistic matter, and for decades it was guessed to be the real one. This essay follows where that second ceiling comes from, why it was trusted, and why the heaviest neutron stars appear to break it.

The ceiling a gas approaches #

The speed of sound in a fluid is set by how much its pressure rises when it is compressed, relative to its inertia. For a relativistic fluid the inertia is the energy density, so the square of the sound speed is $dp/d\varepsilon$, the slope of pressure against energy density along a compression that adds no heat.

The figure computes it for the simplest relativistic matter, an ideal gas of particles of mass $m$ with the relativistic distribution of speeds. Cold, the particles move slowly, the energy density is almost all rest mass, and the sound speed squared is $5kT/3mc^2$: the familiar sound speed of a monatomic gas, a few hundred metres a second in air. As the gas is heated the particles become relativistic, their kinetic energy dominates their rest mass, and the sound speed rises — to 0.312 of $c^2$ when $kT$ equals the rest energy — approaching one-third from below. It never gets there and never passes it.

The one-third has a simple origin. A gas of massless particles, such as the photons of the gas that nobody counted, has pressure equal to exactly one-third of its energy density, whatever its temperature, because each particle’s momentum is its energy divided by $c$ and pressure counts momentum crossing a surface averaged over three directions. Then $dp/d\varepsilon = 1/3$ exactly, and the sound speed is $c/\sqrt{3}$, 58 per cent of the speed of light. Any rest mass makes the pressure a smaller fraction of the energy, because rest mass contributes energy but no momentum, so a gas of massive particles is always softer.

A conjecture about all matter #

The one-third is not special to ideal gases. Any theory with no intrinsic scale of mass or length — a conformal theory, whose physics looks the same at every magnification — has pressure equal to one-third of its energy density, for the same reason, and therefore a sound speed of exactly $c/\sqrt{3}$. The theory of the strong force, quantum chromodynamics, is not conformal: its interactions set a scale, and its particles have masses. But at very high energies its interactions weaken, the quarks and gluons behave more and more like free massless particles, and the theory approaches conformal behaviour. Calculations of hot quark–gluon matter, and of cold quark matter at enormous densities, find the sound speed squared approaching one-third from below, just as the ideal gas does.

It was therefore natural to conjecture that one-third is a universal ceiling: that no matter, however dense, has a sound speed above $c/\sqrt{3}$. The conjecture had support from an unexpected direction — a family of strongly interacting theories studied through their correspondence with gravity in higher dimensions, all of which obeyed the bound — and it had no known counterexample in any system where the calculation could be done reliably. It was a guess, but a well-motivated one, and it made a sharp prediction about the one place in the universe where cold matter is compressed beyond the density of an atomic nucleus.

Holding up a neutron star #

A neutron star is held up against its own gravity by the pressure of its matter, and the heaviest star any kind of matter can support depends on how stiff that matter is. The pressure that weighs down what it holds up found why there is a maximum at all: in general relativity pressure itself gravitates, so squeezing harder to hold up more mass eventually adds more weight than support. The stiffer the matter — the faster its pressure rises with density, the higher its sound speed — the heavier the maximum.

The question can be put to a model. Below about twice the density of an atomic nucleus, calculations from the theory of nuclear forces are believed to be controlled, and they give a pressure that rises steeply but from a small value. Above that density nothing is known reliably. The model used here takes a steep nuclear-like pressure up to 300 MeV per cubic femtometre, about twice nuclear density, and above that assumes the sound speed is constant, at a value that can be varied. It is a crude model, but it asks the right question: given what is trusted below, how stiff must matter be above to hold up what is observed?

The figure plots the sequence of stars each choice produces, from light stars with large radii to the heaviest, marked, beyond which the stars are unstable. With the sound speed held at the conformal value, one-third, the heaviest star is 1.92 solar masses. With one-half it is 2.32, and with the speed of light itself, the stiffest matter causality allows, 2.97.

The band across the figure is a measurement. The pulsar J0740+6620 orbits a white dwarf nearly edge-on, and as its pulses pass the white dwarf they are delayed by the white dwarf’s gravity — the delay of the light that goes past a mass — by an amount that fixes both stars’ masses. The pulsar weighs 2.08 solar masses, give or take 0.07. In this model, the conformal curve does not reach it.

How stiff is stiff enough #

Sweeping the assumed sound speed shows the threshold. The heaviest star grows steadily with the stiffness of the core, from under one and a half solar masses for soft matter to nearly three at the causal limit, and it crosses the pulsar’s mass at a sound speed squared of 0.394 — above one-third. A star of 2.08 solar masses cannot be built from this matter if the sound speed never exceeds $c/\sqrt{3}$.

This is the argument Paulo Bedaque and Andrew Steiner made in 2015 with more careful models, and it has been repeated since with every refinement: the heaviest known neutron stars require the sound speed to exceed the conformal value somewhere inside them, unless the nuclear physics below twice nuclear density is wrong in a specific direction. The conclusion is that dense nuclear matter is stiffer than any free gas — that at some density between the nucleus and the free-quark regime, the forces between its constituents make it push back harder under compression than massless particles would.

What makes matter stiffer than a gas #

A sound speed above $c/\sqrt{3}$ is not exotic in principle. It needs only forces that make compression cost more energy than it would for free particles, and the simplest such force gives the stiffest matter causality permits.

Yakov Zel’dovich pointed this out in 1961. Suppose the particles repel one another through the exchange of a massive vector particle — the way nucleons repel at short range through the omega meson. Then the energy of a dense collection includes, besides the particles’ rest mass and motion, an interaction energy proportional to the square of their number density: every particle repels every other within range, and the number of pairs goes as the square of the density. The pressure is the rate at which that energy rises under compression, and a term that grows as the square of the density gives a pressure equal to its own energy. At high enough density that term dominates both, pressure approaches energy density, and the sound speed squared approaches one — the speed of light. Zel’dovich’s point was that this is the maximum: a vector field cannot push harder, and the result respects causality exactly at its limit.

So there is a clear physical picture behind a stiff neutron-star core. At the density of an atomic nucleus the nucleons are close enough that their short-range repulsion is felt; at twice that density it is felt strongly; and a repulsion that grows faster than the kinetic pressure of free particles pushes the sound speed above the free-gas ceiling. The conformal ceiling was a statement about theories without forces of that kind. Real nuclear matter has them, and whether they persist, weaken or give way to deconfined quarks as the density rises further is what the peak in the sound speed is about.

Bounded from above as well #

The masses of neutron stars bound the stiffness from below; a single event in 2017 bounds it from above. When two neutron stars merged in the event called GW170817, the merger remnant — a hot, rapidly spinning star of about 2.7 solar masses — did not survive long as a neutron star: the electromagnetic signals that followed indicate it collapsed to a black hole within about a second. A remnant that heavy can survive briefly only because its rotation and heat support it, and one that collapsed quickly cannot have been supported by very stiff cold matter. Analyses of that event suggest the maximum mass of a cold, non-rotating neutron star is not much above 2.3 solar masses. In the model here that corresponds to a core sound speed squared of about one-half.

The heaviest pulsars push the stiffness up; the merger remnant’s collapse pushes it down. Between them the sound speed in the cores of the heaviest neutron stars is bracketed, with model-dependent edges, between a little above the conformal one-third and about one-half of $c^2$. The window is narrow enough to be a measurement rather than a range of opinions.

What the verdict depends on #

The qualification — unless the nuclear physics is wrong — deserves its own figure, because the argument is exactly as strong as the assumption behind it.

Suppose the conformal limit took over earlier, at a lower density. Then the figure shows the heaviest star rising, to 2.55 solar masses if the handover were at nuclear density itself. That looks paradoxical and is not: at nuclear density the real nuclear pressure is small and its sound speed low, so capping the sound speed at one-third there is not a restriction but a stiffening — it replaces soft matter with stiffer matter. The bound bites only if nuclear physics is trusted up to a density where its own sound speed is still well below one-third, so that the only way to reach two solar masses is to rise steeply above the handover. In this model the pulsar goes out of reach once the handover is later than about 1.6 times nuclear density.

So the statement that neutron stars break the conformal bound is really a statement that nuclear theory is reliable to about twice nuclear density. Calculations based on the effective theory of nuclear forces claim exactly that, with estimated uncertainties, and measurements of neutron-star radii — from X-ray timing of hot spots on pulsars and from the tidal deformation of neutron stars in merging binaries seen in gravitational waves — are consistent with them. The conclusion is well supported, but it is an inference with a stated premise, not a theorem.

The pressure can stay behind the sound speed #

The sound speed is the slope of pressure against energy density; the ratio of pressure to energy density is the average slope from zero. The two are different quantities, and the second has its own conjecture attached.

Because the pressure starts small at low density, the ratio $p/\varepsilon$ climbs only slowly towards the slope, and a core whose sound speed squared exceeds one-third can still have $p/\varepsilon$ below one-third at its centre. The figure shows this for the model with $c_s^2 = 0.4$, just stiff enough to hold up the pulsar: the ratio reaches 0.308 at the centre of its heaviest star, below one-third. Stiffer cores push it past: 0.374 for one-half, 0.666 for the causal limit.

The ratio has a meaning of its own. One-third minus $p/\varepsilon$ is proportional to the trace of the energy–momentum tensor, the quantity that vanishes exactly in a conformal theory, and in 2022 it was proposed that this trace, rather than the sound speed, might never change sign — that $p/\varepsilon$ stays below one-third in all matter, even when $dp/d\varepsilon$ does not. The model says the heaviest observed neutron stars can be built either way. Which is true depends on how quickly the stiffening sets in, and on whether the sound speed, having risen above one-third, falls back towards it at higher density, as the free-quark limit requires. A sound speed that rises through $c/\sqrt{3}$ and must eventually return to it has a peak somewhere, and the location and height of that peak are now among the main targets of neutron-star measurements.

Where the model stops #

The model’s equation of state is a device, not a description: a single steep power law for nuclear matter and a constant sound speed above it. Real analyses use nuclear calculations with uncertainty bands, piecewise or smoothly varying sound speeds, and statistical combinations of every measured mass and radius, and they find the same qualitative answer with quantitative differences: the threshold sound speed ranges from a little above one-third to around a half, depending on where the nuclear calculation is trusted to. The model ignores rotation, which raises the maximum mass by up to twenty per cent for the fastest spinning stars; the pulsar J0740+6620 spins slowly enough for the correction to be small. And the maximum masses depend on the matching density: the pressure that weighs down what it holds up found a causal ceiling of about 3.2 solar masses with a different matching, against this model’s 2.97.

What the pictures cannot show #

The figures show a sound speed that is a single number above a sharp density, when any real equation of state varies smoothly. They show the maximum-mass stars of each model and cannot show whether nature makes such stars: the heaviest neutron stars known may be well below the maximum their matter allows, and the maximum is inferred from the masses observed and the radii measured, not seen directly. And they say nothing about what the matter is — nucleons, hyperons, quarks, or some mixture — which is the question the stiffness is evidence about. A high sound speed is what dense matter does; why it does it, and what it is made of there, is a separate and still open question.

Still open: where the sound speed peaks, and what the peak is #

If the sound speed exceeds $c/\sqrt{3}$ at the densities inside heavy neutron stars and returns to it at the far higher densities where quarks are free, it must peak in between, and several proposals explain the peak as the signature of a change in the nature of matter — nuclei dissolving into a phase in which quarks are no longer confined to individual nucleons but have not yet become free. Measurements of neutron-star radii to a few per cent, gravitational waves from the tidal deformation of stars in mergers, and the maximum mass inferred from the heaviest pulsars and from the remnants of mergers are narrowing where the peak can be. Whether it is a smooth crossover or a sharp transition, and at what density, is unknown.

The habit worth carrying away is to ask what a proposed universal limit is a limit of. One-third of $c^2$ is the sound speed of anything without a mass scale, and a free gas approaches it from below — but a neutron star of two solar masses cannot be held up by matter that respects it, provided nuclear physics is trusted to twice nuclear density, so the densest matter in the universe must be stiffer than any free gas. Causality allows much more; the conformal value was a guess about the forces, and the stars have answered it.