Astronomy & Celestial Mechanics
the sky, computed
Cosmology

Two skies where the paradox comes out right

Olbers argued that every sight line should end on a source and the sky should blaze. In neutrinos and in gravitational waves it does — the walls are at one second and at no time at all — and both of those skies have now been detected.

Assumes Olbers's paradox and Microwave background.

17 min read 7 figures Light is the only messengerScales that defeat intuition

Olbers’ argument compares two lengths: how far a sight line runs before it ends on something, and how far anything has had time to come. Neither is a property of the universe alone. Both depend on what is doing the travelling, and the argument has only ever been run for one messenger.

Run it for the other two and the answer inverts. In neutrinos and in gravitational waves the sky is saturated — every sight line ends on a surface, or on a source — which is precisely what Olbers predicted and what the optical sky refuses to do. Both of those skies have been detected, one of them in 2023.

Three messengers, and the three walls they end on. The redshift of the last opaque surface, for the three things that cross the universe, on one logarithmic axis spanning thirty-one decades. Olbers' argument compares how far a sight line runs before it ends on something against how far anything has had time to come — and for starlight the first is 1.7·10¹⁸ Mpc against a horizon of 1.4·10⁴ Mpc, which is why the optical sky is dark. That comparison is not what decides the other two. A relic neutrino's mean free path against ordinary matter works out at 7.7·10³⁷ Mpc — 4·10¹⁹ times starlight's, so on the mean-free-path argument alone the neutrino sky should be darker still. It is not, because the quantity that terminates a sight line is the WALL, and the walls are at z = 1,090, z ≈ 6×10⁹ and nowhere at all. Every neutrino sight line ends on a surface from one second after the beginning; every gravitational-wave sight line runs to the beginning, or to a binary. Both of those skies are saturated — which is what Olbers' argument predicted, and what the optical sky refuses to do. The three are also wildly unequal in brightness: the microwave sky is 996 nW m⁻² sr⁻¹, and the gravitational-wave background at Ω = 10⁻⁹ is 0.018 — a saturated sky five decades fainter than a dark one.
Fig. 1 The redshift of the last opaque surface, for the three things that cross the universe, on one logarithmic axis spanning thirty-one decades. The mean free path is not what decides it: a relic neutrino travels 7.7×10377.7\times10^{37} Mpc against ordinary matter, twenty decades further than a photon’s path to a stellar surface, so on the mean-free-path argument alone the neutrino sky ought to be darker still. What terminates a sight line is a wall — the epoch before which the universe was opaque to that messenger — and the walls are at z=1,090z = 1{,}090, at z≈6×109z \approx 6\times10^{9}, and nowhere at all.

The quantity that turned out not to matter

The argument that resolved the paradox computed a mean free path to a stellar surface and compared it to a horizon. The ratio was thirteen orders of magnitude, and that ratio was the answer.

It is worth noticing how contingent that was. The calculation assumed the sources are stars, which are opaque, compact and rare — three properties that together make λ=1/nσ\lambda = 1/n\sigma enormous. Had the universe been filled with an absorbing medium rather than with discrete sources, the same argument would have produced a much shorter path and a bright sky.

And the universe is filled with such a medium, before recombination. Ionised hydrogen scatters photons by Thomson scattering, and the mean free path in the pre-recombination plasma was short enough that a photon’s path was a random walk rather than a line. The optical sky’s sight lines all terminate, and they terminate on that.

Two reasons the sky is dark, and only one of them is the expansion. The surface brightness of the sky accumulated out to a given distance, in units of the surface brightness of a star, both axes logarithmic. Every shell of thickness dr contributes the same amount, because the number of stars in it grows as r² and each one's flux falls as r⁻², so the straight rising line is the paradox: in an infinite static universe the sky reaches stellar surface brightness at the mean free path to a stellar surface, λ = 1/nσ, which for 10⁹ stars per cubic megaparsec of radius 0.6 R☉ is 1.74·10¹⁸ Mpc. The two suppressions are then computed separately, and they are wildly unequal. The finite age cuts the integral off at the particle horizon, 1.41·10⁴ Mpc, which is a factor of 1.23·10¹⁴ short of λ. The expansion then dims what is left by a further factor of 6.03, computed as the mean of (1+z)⁻² over the comoving distance to the horizon. The horizon does 14 orders of magnitude and the redshift does less than one. The common answer that the sky is dark because the universe is expanding is therefore very nearly the wrong answer: the sky is dark because the universe is young, and the expansion is a small correction on top.
Fig. 2 The accumulation the paradox is built from, for one messenger. The dashed line rises without limit in an infinite static universe until it saturates at a stellar surface; the horizon cuts it fourteen orders of magnitude short. What this drawing does not contain, because it was never about it, is the wall: the integral is cut by the time available rather than by an opaque surface, and the two happen to be at almost the same place for light because recombination was recent. For every other messenger they are not.

So the optical sky has a wall at z=1,090z = 1{,}090, and every sight line does end on an opaque surface after all. That is not a re-derivation of the paradox; it is the thermodynamic restatement of it, and the reason the sky is nonetheless dark is that the surface has cooled by a factor of 1,0901{,}090 since it was laid down. A surface brightness falls as the fourth power of a temperature, so the cooling costs a factor of 1.4×10121.4\times10^{12} — which is very nearly the whole of the deficit the star-counting argument produced by a different route.

A wall one second old

Neutrinos decoupled from the rest of the universe when it was about one second old and its temperature was around one million electronvolts.

The mechanism is the same as for photons and the arithmetic is the opposite way round. The weak interaction rate falls as the fifth power of temperature while the expansion rate falls only as the square, so there is a temperature below which a neutrino’s next scattering will not happen before the universe has doubled in size again. That temperature is about 1 MeV, and everything after it is free streaming.

So the neutrino sky’s wall sits at a redshift of about six billion, against the photon sky’s one thousand. The surface it ends on is the universe as it was one second after the beginning, and the sky in that messenger is saturated in exactly the sense Olbers meant: every direction terminates on an opaque surface, and the whole sky is that surface.

Three messengers, and the three walls they end on. The redshift of the last opaque surface, for the three things that cross the universe, on one logarithmic axis spanning thirty-one decades. Olbers' argument compares how far a sight line runs before it ends on something against how far anything has had time to come — and for starlight the first is 1.7·10¹⁸ Mpc against a horizon of 1.4·10⁴ Mpc, which is why the optical sky is dark. That comparison is not what decides the other two. A relic neutrino's mean free path against ordinary matter works out at 7.7·10³⁷ Mpc — 4·10¹⁹ times starlight's, so on the mean-free-path argument alone the neutrino sky should be darker still. It is not, because the quantity that terminates a sight line is the WALL, and the walls are at z = 1,090, z ≈ 6×10⁹ and nowhere at all. Every neutrino sight line ends on a surface from one second after the beginning; every gravitational-wave sight line runs to the beginning, or to a binary. Both of those skies are saturated — which is what Olbers' argument predicted, and what the optical sky refuses to do. The three are also wildly unequal in brightness: the microwave sky is 996 nW m⁻² sr⁻¹, and the gravitational-wave background at Ω = 10⁻⁹ is 0.018 — a saturated sky five decades fainter than a dark one.
Fig. 3 The same three walls, with what each sky is worth beside it. The microwave background carries about a thousand nanowatts per square metre per steradian; the neutrino background is drawn at its own temperature of 1.95 K and its 336 particles per cubic centimetre, slightly fewer than the microwave background’s own number density; the gravitational-wave background at Ω=10−9\Omega = 10^{-9} carries 0.018. The neutrino sky is nearly as bright as the microwave one and has never been directly detected, which is the strangest entry in the table: it is inferred from the expansion rate it produces, and from the effect its gravity had on the growth of structure, and from nothing that has ever been counted.

The reason it has not been detected is the same one that makes the mean free path so long. A relic neutrino today carries about 1.7×10−41.7\times10^{-4} eV of energy — the weak cross-section scales as the square of that — and the resulting interaction probability with any target anybody can build is beyond anything in reach. The most advanced proposed experiment aims to detect the capture of relic neutrinos on tritium, and would collect a handful of events a year from a hundred grams of it.

So there is a sky, three hundred and thirty-six particles per cubic centimetre, in every direction, older than anything else there is, and it is the one thing in this essay that has never been seen.

And a messenger with no wall at all

Gravitational waves are worse, or better, depending on the direction of the argument.

The universe has never been opaque to them. The graviton’s coupling is gravitational, its interaction rate falls off faster than the expansion rate at every temperature below the Planck scale, and no epoch after 10−4310^{-43} seconds has been dense enough to scatter one. There is no wall.

That means every gravitational-wave sight line runs all the way to the beginning — and Olbers’ argument, run without a cutoff, gives exactly what it always gave: a saturated sky. The only thing that stops the integral diverging is that the sources are finite in number and finite in strength.

Three messengers, and the three walls they end on. The redshift of the last opaque surface, for the three things that cross the universe, on one logarithmic axis spanning thirty-one decades. Olbers' argument compares how far a sight line runs before it ends on something against how far anything has had time to come — and for starlight the first is 1.7·10¹⁸ Mpc against a horizon of 1.4·10⁴ Mpc, which is why the optical sky is dark. That comparison is not what decides the other two. A relic neutrino's mean free path against ordinary matter works out at 7.7·10³⁷ Mpc — 4·10¹⁹ times starlight's, so on the mean-free-path argument alone the neutrino sky should be darker still. It is not, because the quantity that terminates a sight line is the WALL, and the walls are at z = 1,090, z ≈ 6×10⁹ and nowhere at all. Every neutrino sight line ends on a surface from one second after the beginning; every gravitational-wave sight line runs to the beginning, or to a binary. Both of those skies are saturated — which is what Olbers' argument predicted, and what the optical sky refuses to do. The three are also wildly unequal in brightness: the microwave sky is 996 nW m⁻² sr⁻¹, and the gravitational-wave background at Ω = 3·10⁻⁹ is 0.055 — a saturated sky five decades fainter than a dark one.
Fig. 4 The same three messengers with the gravitational-wave background drawn at three times the value the pulsar timing arrays report. The wall column is unchanged, because the absence of a wall is not a number that can be adjusted; only the brightness moves. A saturated sky can be faint, and this one is — five decades below the microwave background, and three below the accumulated light of every star ever formed — because the sources are rare and the coupling is weak, not because anything is cutting the integral short.

The sky that has been detected is not the primordial one. It is the background made by every supermassive black hole binary in every merged galaxy, all of them radiating at nanohertz frequencies, none of them individually resolvable. The pulsar timing arrays found it in 2023 by watching the arrival times of millisecond pulsars wander in a pattern correlated across the sky in the particular way a gravitational-wave background produces and nothing else does.

And the reason no individual source can be picked out of it is the confusion limit, which is Olbers’ argument stated as an instrumental problem. When the sources are numerous enough that more than one contributes to every resolution element, the sum is a background rather than a list. The optical sky escapes confusion because the horizon removes almost all the sources; the nanohertz sky does not, because nothing removes any of them.

What “detected” means for each of the three

The three skies are detected in three ways that have nothing in common, and the differences are more instructive than the similarities.

The microwave background is imaged. There is a map, at arcminute resolution, over the whole sky, and its statistics are the tightest constraint on the contents of the universe there is. It is the only one of the three that behaves like an astronomical observation in the ordinary sense.

The neutrino background is weighed, three times over, and never counted. Nothing has ever registered a relic neutrino; what has been measured is the energy density it must carry for the helium abundance to come out right, for the acoustic peaks to sit where they do, and for structure to have grown as it has.

The gravitational-wave background is correlated. No instrument responds to it directly; what is measured is that the arrival times of a few dozen millisecond pulsars, scattered around the sky, wander together in a pattern whose dependence on the angle between each pair of pulsars is a curve that only a quadrupolar wave background produces. The signal is in the relationship between the noise of one clock and the noise of another.

All three are detections of a saturated sky, and only one of them looks like an observation. That is a fair summary of what the paradox becomes once it is taken seriously: it predicts a sky in every messenger, and the predicted skies are found with instruments that would not have been recognised as telescopes.

The three answers side by side

Setting the messengers out together makes the structure of the paradox clearer than running it for one ever did.

For light, the wall is recent and has cooled a great deal, so the sky is faint. For neutrinos, the wall is ancient and has also cooled a great deal, so the sky is faint in energy and dense in number — the one place where counting particles and measuring energy give qualitatively different impressions of the same sky. For gravitational waves there is no wall, the sky is made of sources rather than of a surface, and it is faint because gravity is weak.

Every shell contributes the same. Concentric shells of equal thickness around an observer, with stars scattered uniformly in volume. The number of stars in a shell grows as its radius squared and the flux from each falls as the radius squared, so the two cancel exactly and every shell delivers the same total light — the drawn counts are 6, 15, 31, 55, 88, in proportion to r³ − r₀³, and the drawn sizes fall as 1/r. That cancellation is the paradox, and it is why no amount of dust helps: dust absorbs the light and then re-radiates it, and in a universe old enough for the sum to converge it would come to the same temperature as the stars. The sum diverges linearly with radius, so something has to stop it — and the only two candidates are that the shells eventually overlap, or that there are no shells beyond a certain distance because there has not been time for their light to arrive.
Fig. 5 The shell construction that every version of the argument runs, whatever the messenger. Equal thickness, equal contribution: the number of sources in a shell grows as the square of its radius and each one’s flux falls as the square, so the sum diverges linearly in the number of shells available. Nothing in that cancellation refers to light. It holds for neutrinos from every star that ever burned, for gravitational waves from every binary that ever merged, and for anything else emitted by a population spread uniformly through space — which is why the answer depends entirely on what stops the sum, and not at all on what is doing the emitting.

The cancellation being universal is what makes the three answers comparable. Each messenger inherits the same divergent sum and is rescued from it by a different mechanism, and the mechanism is the whole of the content.

Three messengers, and the three walls they end on. The redshift of the last opaque surface, for the three things that cross the universe, on one logarithmic axis spanning thirty-one decades. Olbers' argument compares how far a sight line runs before it ends on something against how far anything has had time to come — and for starlight the first is 1.7·10¹⁸ Mpc against a horizon of 1.4·10⁴ Mpc, which is why the optical sky is dark. That comparison is not what decides the other two. A relic neutrino's mean free path against ordinary matter works out at 3.9·10³⁸ Mpc — 2·10²⁰ times starlight's, so on the mean-free-path argument alone the neutrino sky should be darker still. It is not, because the quantity that terminates a sight line is the WALL, and the walls are at z = 1,090, z ≈ 6×10⁹ and nowhere at all. Every neutrino sight line ends on a surface from one second after the beginning; every gravitational-wave sight line runs to the beginning, or to a binary. Both of those skies are saturated — which is what Olbers' argument predicted, and what the optical sky refuses to do. The three are also wildly unequal in brightness: the microwave sky is 996 nW m⁻² sr⁻¹, and the gravitational-wave background at Ω = 10⁻⁹ is 0.018 — a saturated sky five decades fainter than a dark one.
Fig. 6 The neutrino mean free path recomputed against a fifth of the baryon density — the density of a cosmic void rather than the mean. It moves to 3.9×10383.9\times10^{38} Mpc, and the row it is drawn on does not move at all, because the row is indexed by the wall. That insensitivity is the figure’s argument in its sharpest form: the quantity Olbers’ argument turns on for light is, for the other two messengers, one that can be changed by an order of magnitude without changing anything.

The unifying statement is that a sky is saturated when the integral over sources reaches its own ceiling, and that there are two different ways to fail. Light fails because the integral is cut short by a horizon. Gravitational waves do not fail at all — the sky is saturated, and it is simply dim.

Neutrinos fail in neither way and are saturated for a third reason, which is worth separating: their sky is a wall rather than a sum over sources, so the integral never ran. The three messengers exhaust the possibilities.

How a wall is actually built

“Opaque before, transparent after” is a simplification, and the size of the simplification differs between the two walls in a way worth stating.

A messenger decouples when its interaction rate falls below the expansion rate, and that is a crossing rather than an event. The transition takes of order one expansion time, so a wall is a layer rather than a surface. For photons the layer is thick enough to matter: the last-scattering surface has a finite depth that blurs the smallest features in the microwave background, and the blurring is measured rather than assumed. What the background is a picture of is therefore a shell some tens of megaparsecs thick, not a sphere.

For neutrinos the layer is thicker still in redshift and far thinner in consequence, because nothing has ever been measured that would resolve it. The decoupling ran from about 3 MeV for muon and tau neutrinos down to about 1.5 MeV for electron neutrinos, which do not stop interacting at the same moment — the electron neutrino has a charged-current channel the others lack at those energies, and it keeps it slightly longer. So there is not one neutrino wall but three, separated by a factor of two in temperature.

That detail is not a refinement; it is one of the inputs to the primordial abundances. The electron neutrino’s extra scattering is what sets the neutron-to-proton ratio at the moment weak freeze-out ends, and that ratio determines the helium abundance of the entire universe. The thickness of a wall nobody can see is measured by counting helium in a dwarf galaxy.

A sky detected entirely by its gravity

The neutrino background is the only entry in the table that has never been observed and is nevertheless known to a few per cent. It is worth being clear about how.

Its energy density enters the expansion rate during nucleosynthesis, and the expansion rate sets how long the neutrons have to decay before they are locked into helium. Add a fourth neutrino species and the universe expands faster, fewer neutrons decay, and more helium comes out. The abundance is measured in metal-poor galaxies, and the number of species it allows is three.

It enters again, and independently, in the microwave background. A relativistic background that free-streams rather than oscillating with the plasma shifts the phase and damps the amplitude of the acoustic peaks, by an amount that depends on how much of the energy density it carries. The answer from that measurement is also three, to about a tenth of a species.

And it enters a third time, at low redshift, through mass. Massive neutrinos stream out of small overdensities before they can collapse, so they suppress the growth of structure below a scale their own speed sets — which is currently the tightest bound on the sum of the neutrino masses, and it is an astronomical measurement of a particle-physics quantity no accelerator has reached.

Three independent measurements of a sky nobody has detected, each of them of its gravitational effect rather than of the neutrinos themselves. Olbers’ argument says the sky is there; every one of these says how much of it.

The surprising equality

There is one coincidence in the table that is not a coincidence, and it is worth the arithmetic.

The photon and neutrino backgrounds have almost the same number density — 411 and 336 per cubic centimetre — and nearly the same temperature, 2.725 K and 1.95 K. Two completely separate decouplings, six billion in redshift apart, and the results land within a factor of two of each other.

The reason is that they were in equilibrium with each other when the neutrinos left, and nothing much happened afterwards except one event: electron–positron annihilation, at a temperature of about half an MeV, dumped its entropy into the photons and not into the neutrinos, which had already stopped listening. The ratio of the two temperatures is fixed by that alone, and it is (4/11)1/3=0.714(4/11)^{1/3} = 0.714 exactly.

So the near-equality is a conservation law with one correction, and the correction is a measurable number. It is also one of the few places where a prediction about a sky nobody has detected has been confirmed to a per cent — not by counting the neutrinos, but by measuring their gravitational effect on the expansion rate during nucleosynthesis and on the microwave background’s acoustic peaks, both of which depend on how much energy density the neutrinos carry.

Two of the three skies have no picture at all

The neutrino mean free path is computed for scattering off nucleons at rest. A relic neutrino is non-relativistic today if it has mass, which changes the kinematics and the cross-section, and the masses are not known. The number is right to within orders of magnitude, which is all the figure uses it for, and it is not a calculation anybody would publish.

The gravitational-wave background drawn is the astrophysical one, not the primordial one. The wall column says there is no wall, which is a statement about a primordial background that has not been detected; the brightness column is the nanohertz confusion background from black hole binaries, which has. Those are two different skies sharing a row, and the figure does not separate them.

And no drawing here shows what a saturated sky looks like, because two of the three are not images. A neutrino sky has no angular structure anybody has resolved, and the nanohertz gravitational-wave sky is measured as a correlation between pulsar pairs rather than as a map. Both are known to exist and neither has a picture.

An argument that outgrew its subject

The paradox was an argument about stars, made by people who thought the universe was static and infinite, and it has been retired and revived several times since.

What the three-messenger version shows is that the argument was never really about stars. It is about the relation between a source density, a cross-section and an available time, and its answer changes whenever any of those changes — which means it generates a different question for every new way of looking.

That is why the same reasoning recurs whenever a new messenger arrives. The first thing asked of a detection channel is what its sky looks like when integrated: whether it is a list of sources or a background, and if a background, what sets its level. For high-energy neutrinos of astrophysical origin the answer is a diffuse flux that is already at the confusion limit at the highest energies. For the ultra-high-energy cosmic rays it is a horizon of about 50 Mpc, set by pion production on the microwave background, which is the same calculation the gamma-ray horizon uses with a different projectile.

Every one of those is Olbers’ question, asked of a sky he could not have known existed. The answer has been different every time and the argument has never needed changing.

Three messengers, and the three walls they end on. The redshift of the last opaque surface, for the three things that cross the universe, on one logarithmic axis spanning thirty-one decades. Olbers' argument compares how far a sight line runs before it ends on something against how far anything has had time to come — and for starlight the first is 1.7·10¹⁸ Mpc against a horizon of 1.4·10⁴ Mpc, which is why the optical sky is dark. That comparison is not what decides the other two. A relic neutrino's mean free path against ordinary matter works out at 7.7·10³⁷ Mpc — 4·10¹⁹ times starlight's, so on the mean-free-path argument alone the neutrino sky should be darker still. It is not, because the quantity that terminates a sight line is the WALL, and the walls are at z = 1,090, z ≈ 6×10⁹ and nowhere at all. Every neutrino sight line ends on a surface from one second after the beginning; every gravitational-wave sight line runs to the beginning, or to a binary. Both of those skies are saturated — which is what Olbers' argument predicted, and what the optical sky refuses to do. The three are also wildly unequal in brightness: the microwave sky is 996 nW m⁻² sr⁻¹, and the gravitational-wave background at Ω = 10⁻¹⁴ is 0.000 — a saturated sky five decades fainter than a dark one.
Fig. 7 The same three rows with the gravitational-wave background drawn at 10−1410^{-14} — the order a primordial background from inflation is usually estimated at, five decades below the astrophysical one the timing arrays measured. The wall column is again unchanged. A primordial background has no wall in front of it and is nonetheless the faintest thing in the table by a wide margin, which is the cleanest demonstration that the two columns are independent: saturation says there is a sky in every direction, and says nothing whatever about whether anything can be built to detect it.

Still open: whether the nanohertz sky is entirely astrophysical

The background the pulsar timing arrays have measured is consistent with what a population of supermassive black hole binaries should produce, and it is also consistent with several cosmological sources — a phase transition in the early universe, or cosmic strings — that would put a genuinely primordial component underneath the astrophysical one.

Telling them apart needs the spectrum rather than the amplitude, since the binaries predict a particular power law and the alternatives do not. A population of circular binaries driven purely by gravitational radiation gives a characteristic strain falling as f−2/3f^{-2/3}, and the measured spectrum is consistent with that and not yet precise enough to exclude alternatives. That measurement is a decade of further timing away, and it is the closest anything has come to testing what a sky with no wall in front of it actually contains.

A second question sits underneath it and is nearer to the paradox’s own subject. If the nanohertz background is astrophysical, then it is a confusion limit — a sum over sources that individually could be resolved with enough sensitivity, exactly as a deep optical image resolves what a shallow one records as a background. The depth at which a background breaks into sources is a measurable quantity, and for the nanohertz sky nobody knows what it is. The first individually resolved binary would fix it, and would turn a background into a catalogue with one detection.

From here: down in energy, not out in messenger

The natural continuation is downward in energy rather than outward in messenger. Every argument so far has been about a sky integrated over all directions; none has asked what the fluctuations in that sky are worth. A background made of discrete sources is not smooth, and the angular power spectrum of its unresolved part carries the clustering of the sources that made it — which is a measurement of where the emitters are, taken from a sky in which none of them is visible.

That is a live technique for the optical background and for the infrared one, and it has been proposed for the neutrino sky and for the gravitational-wave sky. It also settles a question left open above: whether a saturated sky is a wall or a sum. A wall has no clustering in it and a sum does.

Where this goes next

Every shell contributes the same. Concentric shells of equal thickness around an observer, with stars scattered uniformly in volume. The number of stars in a shell grows as its radius squared and the flux from each falls as the radius squared, so the two cancel exactly and every shell delivers the same total light — the drawn counts are 6, 15, 31, 55, 88, in proportion to r³ − r₀³, and the drawn sizes fall as 1/r. That cancellation is the paradox, and it is why no amount of dust helps: dust absorbs the light and then re-radiates it, and in a universe old enough for the sum to converge it would come to the same temperature as the stars. The sum diverges linearly with radius, so something has to stop it — and the only two candidates are that the shells eventually overlap, or that there are no shells beyond a certain distance because there has not been time for their light to arrive. Why the sky is dark In an infinite universe of stars every sight line ends on a stellar surface, so the whole sky should be as bright as the Sun's disc. It is not, and the usual answer — that the expansion redshifts the light away — accounts for a factor of six out of a hundred trillion. A blackbody at 2.7255 kelvin, filling the sky. The Planck function at 2.7255 K in the units the measurement is reported in, with the peak marked where Wien's law in frequency puts it: x = hν/kT = 2.8214, so ν = 160.2 GHz and the intensity there is 384 MJy per steradian. The points are drawn at the twenty-one frequencies across the FIRAS band, displaced from the curve by a Gaussian of 50 parts per million of the peak, which is the root-mean-square deviation the instrument actually reported. At the scale of this plot that displacement is a fifth of a pixel and the points sit on the line — which is the entire finding. Nothing else in astronomy is a blackbody to a part in twenty thousand: a stellar spectrum is a blackbody with absorption lines cut into it and a continuum that is the wrong shape at both ends. A thermal spectrum this exact requires that the radiation was once in equilibrium with matter, which requires that the universe was once opaque, which requires that it was once hot and dense. The most perfect blackbody ever measured The whole sky glows at 2.7255 kelvin, and its spectrum matches the Planck curve to fifty parts in a million. Nothing else in astronomy is thermal to a part in twenty thousand, and a spectrum that exact is not a description of the radiation — it is a constraint on the history of everything that could have disturbed it. The sky's darkness, measured as an opacity. Optical depth to electron–positron pair production against gamma-ray energy, for sources at redshifts 0.03, 0.1, 0.3, 1, both axes logarithmic. A gamma ray of energy E is absorbed most readily by background photons near twice the square of the electron rest energy divided by E: at 1 TeV that is 2.37 µm and at 100 GeV it is 0.24 µm, so the energy axis is a wavelength axis for the background light, running backwards — and the background it is evaluated against is the same two-component spectrum the star formation history produced, not a flat number. Above the marked τ = 1 the universe is opaque. The depth is computed in the delta-function approximation, the cross-section replaced by 0.2 of the Thomson value over a bandwidth of order the energy, with the background's comoving density evolving as (1+z)^1.2. That is a factor-of-two calculation and the shape is what it gets right: the horizon closes from z = 0.59 at 100 GeV to z = 0.16 at 1 TeV. The measurement runs the other way. A blazar's spectrum is observed, the absorbed part is the difference between it and the spectrum the source is believed to have emitted, and that difference gives the background — in the near infrared, where no direct measurement can subtract the zodiacal light well enough to compete. A background weighed by what it stops The faintest light in the universe cannot be photographed from inside the Solar System, because the zodiacal foreground is a hundred times brighter. It can be weighed instead, by the bite it takes out of a blazar at a trillion electronvolts.

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.

Confusion limitCosmic neutrino backgroundGravitational-wave backgroundLast scatteringMean free pathOlbers's paradoxOptical depthParticle horizonSurface brightnessThomson scattering