Astronomy & Celestial Mechanics
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Cosmology

A drift in the tilt that is the tilt squared

The primordial spectrum's tilt has been measured to half a per cent. Whether the tilt itself changes with scale has not, and slow-roll inflation says why it will be hard: the running is fixed by the tilt, at about six ten-thousandths, ten times below the present error. The one measurement that reaches far enough along the spectrum to feel it — the μ-distortion of the microwave background's colour — would need a precision no planned instrument has.

Assumes Inflation.

18 min read 7 figures Measuring the unreachableScales that defeat intuition

The microwave background measures the primordial fluctuations as a power law. Their amplitude changes with wavelength by a fixed power, the spectral tilt, and the tilt is 0.965 — a little less than one, so the largest scales carry slightly more power than the smallest. That number is one of the firmest results in cosmology. It has been stable through every analysis since its first detection, it is now known to about half a per cent, and it is what remains of the predictions of the mechanism proposed for two coincidences after the tensor ratio has been bounded and the non-Gaussianity found to be absent.

A power law is an assumption, though, not a measurement. The tilt is fitted as a single number because the data have not yet asked for more, and the obvious next term is a change of the tilt with scale — the running, dns/dln⁡kdn_s/d\ln k. The Planck satellite’s analysis measured it at −0.0045±0.0067-0.0045 \pm 0.0067: consistent with zero, and with an error one and a half times as large as its central value.

What slow-roll inflation predicts for it is not a free parameter. It is the tilt’s own departure from one, squared and divided by a number of order one, and it comes out at about six ten-thousandths — ten times smaller than the present error. That prediction has two consequences, and this essay is about both. It makes the running almost impossible to measure. And it means that whatever running is measured, if one ever is, carries a verdict on slow roll rather than a parameter for it.

A second number the first one already sets

The tilt knows the slope of the potential and not its height: in slow roll it is ns−1=−6ϵ+2ηn_s - 1 = -6\epsilon + 2\eta, where ε measures the potential’s slope and η its curvature, both evaluated where the field sat when the observed scales left the horizon. For almost every potential still on the table, that combination reduces to a count. The tilt depends on N, the number of e-folds of expansion that remained between that moment and the end of inflation, as

1−ns=βN,1 - n_s = \frac{\beta}{N},

where β is a constant of order one belonging to the potential’s shape. Plateaus have β = 2, which is why their predicted tilt at fifty-five e-folds is 0.964; the power-law potentials ϕp\phi^p have β = (p + 2)/2.

The running follows from that relation with no further physics. A larger wavenumber leaves the horizon later, when fewer e-folds remain, so dln⁡k=−dNd\ln k = -dN and

dnsdln⁡k=−dnsdN=−βN2=−(1−ns)2β.\frac{dn_s}{d\ln k} = -\frac{dn_s}{dN} = -\frac{\beta}{N^2} = -\frac{(1-n_s)^2}{\beta}.

The running is the square of the tilt’s departure from one, divided by the shape’s constant. At the measured tilt, (1−ns)2(1-n_s)^2 is 1.2×10−31.2\times10^{-3}, and the running of every slow-roll model sits between about three and nine ten-thousandths, negative.

A running the measurement cannot yet tell from zero. The running of the spectral tilt, dnₛ/d ln k, against the tilt itself. The ellipses are the microwave-background measurement, nₛ = 0.9649 ± 0.0042 and a running of −0.0045 ± 0.0067, at one and two standard deviations. The three curves are what slow-roll inflation predicts once the tilt is fixed: a running of −(1 − nₛ)²/β, with β = 2 for the plateaus and every α-attractor (and for φ²), 4/3 for the steepest power law still drawn and 4 for a cubic hilltop. At the measured tilt they give −0.0009 to −0.0003, all within a band thinner than a line at this scale, 11 times smaller than the measurement's error.
Fig. 1 The running of the tilt against the tilt. Ellipses: the microwave-background measurement at one and two standard deviations. Curves: slow roll’s prediction once the tilt is fixed, −(1−ns)2/β-(1-n_s)^2/\beta, for β = 4/3, 2 and 4. All three lie within a band too thin to resolve on this scale, eleven times below the measurement’s error.

The ellipses are the measurement and the three curves are the theory, and the curves are drawn at the same scale as the ellipses. They are almost indistinguishable from the zero line. A prediction this small compared with its error bar is unusual in cosmology, where most parameters are measured to several standard deviations away from zero; it means the running is, for now, the one quantity where the theory has said something exact and the data have not yet reached it.

A factor of three across every shape

The relation −(1−ns)2/β-(1-n_s)^2/\beta turns the question of which potential nature chose into a question about β, and β barely varies.

Every model's running, once its tilt is fixed. The running of the tilt that slow-roll inflation predicts at nₛ = 0.9649, in units of 10⁻⁴, for eight potentials. Each has 1 − nₛ = β/N, so its running is −(1 − nₛ)²/β: one number, set by the tilt, divided by a constant of order one belonging to the shape. The plateaus and every member of the α-attractor family share β = 2 and run at −6.2 × 10⁻⁴ (the exact solution at α = 1 and 55 e-folds gives −6.5 at nₛ = 0.9639); the power laws and hilltops spread from −9.2 to −3.1. The open bars are potentials already excluded by their tensor ratio. The whole range is a factor of three wide and sits between three and nine ten-thousandths, against a present error of 67 × 10⁻⁴.
Fig. 2 The running predicted at nsn_s = 0.9649 for eight potentials, in units of 10−410^{-4}. Plateaus and every α-attractor run at −6.2; power laws and hilltops spread from −9.2 to −3.1. Open bars are potentials already excluded by their tensor ratio.

The plateau potentials, Starobinsky’s model among them, have β = 2. So does the whole family of α-attractors — the T-model potentials whose tensor ratio slides with their parameter while their tilt stays put, which a null result for gravitational waves would leave standing. Solved exactly in slow roll at fifty-five e-folds, the family’s running divided by (1−ns)2(1-n_s)^2 comes out at −0.500 whether α is a thousandth or a thousand. The quadratic potential, which is what the family becomes when α is large, also has β = 2, so the two ends of the family agree and so does everything between.

The other shapes spread the prediction. The linear potential and the two-thirds power, which arise in some string constructions, have β of 3/2 and 4/3 and run faster, at eight and nine ten-thousandths. Hilltop potentials, where the field rolls off a maximum, have β = 2(p − 1)/(p − 2) and run slower, at three to five. The quartic potential, at four, is excluded already by its tensor ratio; so is the quadratic. The full range of surviving slow-roll models covers a factor of three and sits between three and nine ten-thousandths. Measuring the running well enough to tell those shapes apart would need a precision of about a ten-thousandth, sixty times better than the present one.

The running counts e-folds

There is a second way to write the same result, and it is the surprising one. Since 1−ns=β/N1-n_s = \beta/N,

dnsdln⁡k=−1−nsN.\frac{dn_s}{d\ln k} = -\frac{1-n_s}{N}.

The shape’s constant has cancelled. For every potential in the class, the running is the tilt’s departure from one divided by the number of e-folds, and nothing else. At the measured tilt, a running of −6.4×10−4-6.4\times10^{-4} is fifty-five e-folds, and −5.4×10−4-5.4\times10^{-4} is sixty-five.

That makes the running a measurement of N independent of the potential, and N is a quantity set by the epoch nobody saw: the reheating that converted the inflaton’s energy into hot matter. A slow, matter-like reheating removes e-folds from the count and a fast or stiff one adds them, and the tilt alone cannot separate that from the shape, because a given tilt is a given ratio β/N. The running separates them. In principle a measured tilt and a measured running together fix both β and N — the shape of the potential and the history after it — with no reference to the tensor ratio at all.

In practice the difference between fifty-five and sixty-five e-folds is a difference of one ten-thousandth in the running, and that is the measurement nobody can make. The relation is still worth stating, because it says what kind of number the running is: not a fourth parameter of the potential but a clock, reading the same interval that reheating fills.

A tilt that changes across its own measurement

The microwave background covers a range of wavenumbers from about 10−410^{-4} to 0.2 Mpc−10.2\ \mathrm{Mpc^{-1}} — the largest scales visible on the sky down to the scales where the damping of the acoustic peaks removes the signal. That is a factor of two thousand, or 7.6 e-folds of k. Over that range, a slow-roll running changes the local tilt by its own value times 7.6.

A tilt that drifts across the scales it is measured over. The local tilt of the primordial spectrum, n(k) = nₛ + α ln(k/k₀), from 10⁻⁴ to 10⁴ per megaparsec, for no running, the slow-roll running and the measured running with its one-standard-deviation range, all through nₛ = 0.9649 at k₀ = 0.05 Mpc⁻¹. Across the microwave background's range, about 7.6 e-folds of k, the slow-roll tilt changes by −0.0047 — comparable to the ±0.0042 error on the tilt itself, which is why the tilt is always quoted at a pivot. At the scales of the μ-distortion the slow-roll tilt is 0.959 and the measured central running's 0.920.
Fig. 3 The local tilt n(k)=ns+αln⁡(k/k0)n(k) = n_s + \alpha \ln(k/k_0) from 10−410^{-4} to 10410^{4} Mpc⁻¹ for no running, slow roll, the measured central value and its one-standard-deviation range. Across the microwave background’s range the slow-roll tilt changes by −0.0047.

The result is a change of five thousandths, about the size of the error on the tilt itself. That is why the tilt is always quoted at a pivot — conventionally 0.05 per megaparsec, near the middle of the range in logarithm — and why the pivot is chosen where the tilt and the running are least correlated in the fit. A tilt quoted at a different pivot is a slightly different number. Under slow roll, the tilt measured from the largest angular scales of the sky should be about three thousandths bluer than the tilt measured from the damping tail, and the microwave background measures those two regions with different statistical power and different systematic errors, so neither is yet decisive.

The lines through the measured range all cross at the pivot, and from there they fan out. At the pivot the data fix the tilt; on either side of it the running decides where the line goes. The figure shows the problem in one picture. Within the shaded window every line is close to every other. The disagreements are outside it, at scales the microwave background cannot see.

Ten e-folds beyond the microwave sky

The primordial spectrum extends far beyond the scales the microwave background measures. The waves that became galaxies are at a few tenths per megaparsec; the waves that shape the hydrogen forest in quasar spectra reach a few per megaparsec; and below that, the fluctuations that would have made dwarf galaxies and the first star-forming haloes were erased by the time they could be observed, by the damping of the photon–baryon fluid and the free streaming of the dark matter. Very little now records the spectrum at a hundred per megaparsec, and nothing directly.

Eight decades of scale beyond the ones the tilt was measured on. The primordial power spectrum, Δ²(k) = Aₛ (k/k₀)^(nₛ−1+½α ln(k/k₀)) with Aₛ = 2.1 × 10⁻⁹ at k₀ = 0.05 Mpc⁻¹ and nₛ = 0.9649, from 10⁻⁴ to 10⁴ per megaparsec, for no running, the slow-roll running and the measured running with its one-standard-deviation range. The shaded strips are the scales each probe reaches: the microwave background up to about 0.2 Mpc⁻¹, galaxy clustering and the hydrogen forest to a few, and the μ-distortion of the background's spectrum from about 30 to 10⁴. Over the microwave background's own range every curve is the same to within its errors. At 1,000 Mpc⁻¹, ten e-folds beyond the pivot, the measured central value has removed 20% of the power and the slow-roll running 3%.
Fig. 4 The primordial power spectrum from 10−410^{-4} to 10410^{4} Mpc⁻¹ for the same five runnings. Shaded: the microwave background, galaxies and the hydrogen forest, and the μ-distortion window. At 1,000 Mpc⁻¹ the measured central running has removed 20% of the power and the slow-roll running 3%.

Extrapolated eight decades beyond the pivot, the different runnings diverge. The slow-roll running takes 3% of the power away at a thousand per megaparsec, compared with a pure power law. The central measured value takes a fifth. The lower edge of the present error removes two fifths of it, and the upper edge adds a tenth.

Those differences matter for more than inflation. The amount of small-scale power controls how many of the smallest dark-matter haloes formed and how early, and so the census of the faintest dwarf galaxies. A large positive running would also produce too many primordial black holes at very small scales, and the absence of their effects bounds the running from above, though only loosely. Some analyses that combined the hydrogen forest with the microwave background have preferred a negative running of about a hundredth, at two to three standard deviations; the forest’s own modelling — the temperature of the intergalactic gas, the pressure smoothing, the ultraviolet background — is uncertain enough that this is not regarded as a detection. The forest reaches about three e-folds beyond the pivot. What lies further out is recorded in only one place.

Sound waves that heated the spectrum

Before recombination, the photons and the baryons moved as one fluid, and the primordial fluctuations set up acoustic waves in it. On small scales those waves were damped: photons diffused out of the compressions into the rarefactions, and the same damping that blurs the smallest structure on the microwave sky erased every wave shorter than the distance a photon could random-walk in the time available. The energy in the waves did not disappear. It went into the photons, as heat.

Heat added to the photon bath distorts its spectrum unless something can redistribute it. Very early, at redshifts above about two million, double Compton scattering and bremsstrahlung could create and destroy photons fast enough to restore a perfect blackbody at a slightly higher temperature, and the added energy left no trace. Between about two million and fifty thousand, Compton scattering could still redistribute energy among the photons but could no longer change their number; the spectrum relaxed to a Bose–Einstein distribution with a small chemical potential, μ. After that, the spectrum could not even relax, and added energy left a different, y-type distortion.

Where the μ-distortion collects its power. The contribution of each decade of wavenumber to the μ-distortion, Δ²(k) W(k) per decade, in units of 10⁻⁹, for nₛ = 0.9649 with no running, the slow-roll running and the measured central running. W(k) is the window through which acoustic waves damped between redshifts of about two million and fifty thousand turn into a chemical potential in the background's spectrum: waves larger than about 30 Mpc⁻¹ were not yet damped by the time the photons could no longer thermalise, and waves smaller than about 10⁴ Mpc⁻¹ were damped while the spectrum could still relax back to a blackbody. The weight peaks near 76 Mpc⁻¹, 7 e-folds from the pivot, and 79% of it comes from 50 to 5,000 Mpc⁻¹. The area under each curve is μ: 1.81·10⁻⁸, 1.76·10⁻⁸ and 1.52·10⁻⁸.
Fig. 5 The contribution of each decade of wavenumber to the μ-distortion, for no running, slow roll and the measured central value. The weight peaks near 76 Mpc⁻¹ and 79% comes from 50 to 5,000 Mpc⁻¹. The area under each curve is μ: 1.81, 1.76 and 1.52 × 10⁻⁸.

The waves damped during the μ era were the ones short enough to be erased by then and long enough not to have been erased earlier — wavenumbers from about fifty to a few thousand per megaparsec. So the μ-distortion is an integral over the primordial power at those scales, through a window that can be calculated precisely from the physics of the damping and the thermalisation. Seven to eleven e-folds beyond the pivot, it weighs the spectrum where nothing else reaches. For a pure power law at the measured tilt and amplitude, the calculation gives μ of about 1.8×10−81.8\times10^{-8}. That is five thousand times below the bound the FIRAS instrument set in the 1990s, when it found the most perfect blackbody ever measured.

What a spectrometer would weigh

Because μ is a weighted integral of the power at fixed scales, it responds to the running directly: a negative running lowers the power there and lowers μ; a positive one raises it.

A distortion that weighs the running ten e-folds out. The μ-distortion of the microwave background's spectrum predicted from the primordial power alone, in units of 10⁻⁸, against the running of the tilt, for tilts of 0.957, 0.9649, 0.973. With no running and nₛ = 0.9649 it is 1.81·10⁻⁸; the slow-roll running lowers it by 2.4%, the measured central value by 16%. Near zero running μ changes by about 7.1·10⁻¹⁰ for each thousandth of running, so a measurement of μ to 10⁻⁹ would bound the running to about ±0.0014 from scales a thousand to a hundred thousand times smaller than the microwave background's — while the slow-roll prediction would still sit inside its error. The vertical band is the present measurement of the running, −0.0045 ± 0.0067; the FIRAS bound, |μ| below 9 × 10⁻⁵, lies 2300 times above the top of the figure.
Fig. 6 The μ-distortion predicted from the primordial power, in units of 10⁻⁸, against the running, for three tilts. The slow-roll running lowers μ by 2.4% and the measured central running by 16%. Shaded: the present measurement of the running. Near zero, μ changes by 7 × 10⁻¹⁰ per thousandth of running.

Over the range the microwave background allows, μ varies from 1.21.2 to 2.0×10−82.0\times10^{-8} — a factor of almost two. The tilt matters too, but much less, because μ integrates over the same lever arm that the tilt enters linearly and the running quadratically. Near zero running, each thousandth of running moves μ by about 7×10−107\times10^{-10}, and the slow-roll prediction moves it by about 4×10−104\times10^{-10} — two and a half per cent of the signal.

A μ-distortion measured to a precision of 10−910^{-9} would therefore bound the running to about ±0.0014, from scales a thousand to a hundred thousand times smaller than those the microwave background’s anisotropies use. That would be comparable with the best projections for the next generation of anisotropy and galaxy surveys, and independent of them in every systematic respect: different scales, different physics, a different instrument measuring a spectrum rather than a map. It would test whether the power law the anisotropies measure keeps its shape over a lever arm three times as long.

A precision the plans do not reach

The difficulty is the precision.

How well μ must be measured to say anything about the running. The one-standard-deviation bound on the running of the tilt that a measurement of the μ-distortion would give, against the precision of that measurement, with the tilt and amplitude fixed by the microwave background. The bound is the precision divided by μ's sensitivity, 7.1·10⁻¹⁰ per thousandth of running. To match the present microwave measurement (±0.0067) needs μ to about 4.8·10⁻⁹; to match a projected ±0.002, 1.4·10⁻⁹; to reach the slow-roll running itself, 4.4·10⁻¹⁰. Proposed spectrometers aim at a precision near 10⁻⁸, so the distortion is not a measurement of slow roll's running; it is a test of whether the spectrum keeps its measured shape over a lever arm ten e-folds long.
Fig. 7 The bound on the running that a μ measurement would give, against its precision. Matching the present microwave measurement needs μ to about 5 × 10⁻⁹; a projected ±0.002 needs 1.4 × 10⁻⁹; the slow-roll running itself needs 4 × 10⁻¹⁰.

The spectrometers that have been proposed since FIRAS — designed to measure the background’s spectrum with a Fourier-transform instrument against an onboard blackbody, from space — aim at a precision on μ of about 10−810^{-8}. That is enough to detect the predicted distortion at one or two standard deviations, which would itself be the first detection of a distortion caused by the primordial fluctuations, and it is not enough to say anything about the running beyond what the anisotropies already say. Matching the present anisotropy constraint needs μ to about 5×10−95\times10^{-9}; matching the next generation’s projected one needs 1.4×10−91.4\times10^{-9}; reaching the slow-roll running itself needs 4×10−104\times10^{-10}, twenty-five times better than the proposals.

The obstacle is not the instrument’s noise alone. The sky in front of the background at these frequencies is dominated by the Galaxy’s dust and synchrotron emission, by the cosmic infrared background and by the integrated y-distortion from hot gas in clusters, each of which has its own spectrum that must be separated from μ’s. The y-distortion from the hot gas that casts shadows on the background is a hundred times larger than the μ signal. Separating a signal of 10−910^{-9} from foregrounds a million times larger is a problem in spectral modelling before it is a problem in sensitivity.

A verdict rather than a parameter

All of this means the running’s slow-roll value is out of reach for the foreseeable future, through the anisotropies and through the spectrum alike. That is not the same as saying the running is uninteresting. It makes it a peculiar kind of test.

Slow roll predicts a running between three and nine ten-thousandths, negative. Any measurement whose error is larger than that will find zero if slow roll is right. A measurement that finds a running significantly different from zero — a hundredth, say, as the central values of some analyses have suggested — would not be a measurement of β or of N. It would be a statement that the simplest class of inflationary models is wrong, at least over the scales measured: that the potential has a feature, a step or a bump, where the field’s roll changed abruptly; that more than one field was rolling; or that the fluctuations were generated by something other than the inflaton.

There is a parallel in the test that can only fail one way: the single-field inflaton fixes the local non-Gaussianity at (5/12)(1−ns)(5/12)(1-n_s), also far below any measurement’s reach, so that any detection would remove the whole class. The running does the same with a different quantity, and its consistency condition is with the tilt squared rather than the tilt. Both say that the measurable signatures of the simplest models are the ones they predict to be unmeasurable, and that the value of improving the measurements is the chance of finding something they do not allow.

What the calculation assumes

The power is a running power law over sixteen e-folds. The figures extrapolate ln⁡Δ2\ln\Delta^2 as a quadratic in ln⁡k\ln k from the pivot to 10410^4 per megaparsec. A real slow-roll spectrum has a third derivative too, of order (1−ns)3(1-n_s)^3; it is negligible here. A model with a feature does not follow a quadratic at all, which is the case the μ-distortion would be most valuable for and the figures do not draw.

The window is an approximation. The μ window function used is a fit to the full thermalisation calculation, good to about ten per cent in its shape. The predicted μ for a given spectrum is uncertain at about that level, but the differences between runnings, which are what bound the running, are much less sensitive to it.

Other sources of μ are left out. The cooling of baryons as the universe expanded draws energy from the photons and produces a small negative distortion, about −3×10−9-3\times10^{-9}, which is known and would be subtracted; the decay or annihilation of any unstable particle in the μ era would add a positive one, with a size nobody knows. A measured μ constrains the primordial power only once those other contributions are bounded, and in practice a measured μ would bound a combination of them.

The tilt and the amplitude are held fixed. In reality they are measured with errors, and a joint fit would propagate them. The amplitude is known to about one and a half per cent, limited by the optical depth it arrives multiplied with, and that alone corresponds to a few times 10−1010^{-10} in μ, the size of the slow-roll running’s effect.

Still open: whether the spectrum bends where nobody has looked

The tilt is measured over 7.6 e-folds of scale and every analysis treats it as constant over them, because the data cannot yet ask otherwise. Slow roll says that this is very nearly right and quantifies the correction exactly: a running of −(1−ns)/N-(1-n_s)/N, about six ten-thousandths, set by the number of e-folds since the observed scales left the horizon.

The anisotropies of the next decade will reach a precision of a few thousandths, ten e-folds of lever arm will become available only through the background’s spectrum, and neither reaches six ten-thousandths. What they can do is look for a running that should not exist. If the spectrum at a hundred or a thousand per megaparsec turns out to carry a fifth less power than the power law, or a fifth more, the inflaton was not a single field rolling slowly down a smooth potential for the whole of its last sixty e-folds — and the first place that will show is a distortion of the background’s colour by a few parts in a hundred million.

Where this goes next

m²φ²: where the observed scales left, and where inflation ends. The m²φ² potential, drawn as a shape with its height divided out, against the field in reduced Planck masses. The field rolls downhill towards zero and inflation ends where the slow-roll parameter ε reaches one, at φ = 1.41. The shaded band is the stretch of field the scales now seen on the sky left the Hubble radius from: 60 e-folds before the end at φ = 15.56 and 50 before it at φ = 14.21. Nothing about the sky depends on the rest of the curve. Across that band the two numbers the tilt is made of are ε = 9.01e-3 and η = 0.0090, which give a spectral index of 0.9640 and a tensor-to-scalar ratio of 0.1441 at 55 e-folds. The height is not in either: it is fixed separately by the amplitude of the fluctuations, which puts the potential at (2.0 × 10¹⁶ GeV)⁴ there — and multiplying the whole curve by any constant leaves the band, the tilt and the ratio exactly where they are, because every slow-roll quantity is a ratio of the potential to its own derivatives. The field travels 13.49 Planck masses from the middle of the band to the end. The tilt knows the slope and not the height The measured spectral index, 0.965, is quoted as the strongest evidence for inflation, and it is a statement about two dimensionless numbers — how steeply the potential fell and how sharply that slope was changing, over the few e-folds the sky can see. The height of the potential is not in it at all, which is why potentials that look nothing alike reproduce it. A family that runs the full height of the plane. The spectral tilt and the tensor-to-scalar ratio (logarithmic) predicted by inflationary potentials, against the measured tilt (0.9649 ± 0.0042, shaded, two standard deviations) and the upper bound on the ratio (0.036, solid line). The two curves are the T-model α-attractor, V ∝ tanh²(φ/√6α), for α from ten thousandths to a thousand, at 50 and 60 e-folds before the end of inflation: at large α the model is a quadratic potential and sits above the bound; as α falls the ratio drops without limit while the tilt settles at 1 − 2/N. The dots on each curve are α = 1, close to Starobinsky's model, at a ratio of 0.0046 and 0.0032. The short segments are power-law potentials, all excluded or nearly so. The dashed line at 0.001 is a target of the next generation of polarisation experiments: a null result there would exclude α = 1 and leave every smaller α standing. A null that removes a model and not its family The plateau potentials that survive every measurement of inflation predict a tensor-to-scalar ratio a few thousandths high, and the next polarisation experiments are designed to reach it. A null result there would exclude Starobinsky's model. It would not exclude the family it belongs to: the α-attractors stretch one shape by one parameter, their tilt stays where the measurement put it, and their ratio slides down as α does, without a floor. 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.

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.

E-foldsInflationMicrowave backgroundReheatingScalar spectral indexSilk dampingSlow-rollSpectral distortion