A standing wave frozen at one instant

The spectrum of the microwave background is a blackbody to fifty parts in a million and carries almost no information beyond one temperature. The information is in the part that essay set aside: the temperature is not quite the same in every direction, and the pattern of the variation is a measurement of the geometry and contents of the universe.

The variation is small. Once the dipole from the Solar System’s own motion is removed, what is left is a mottling of hot and cold patches at the level of one part in a hundred thousand — about 30 microkelvin against 2.7255 kelvin. What makes it useful is that the mottling has a characteristic size.

Where a preferred scale comes from #

Before recombination the universe contains a fluid, and calling it that is precise rather than loose. Photons and free electrons scatter off each other constantly, electrons and protons are bound by electrostatics, so photons, electrons and baryons move as one substance with a single velocity field. It has enormous pressure, supplied by the photons, and it has mass, supplied mostly by the baryons. A medium with pressure and mass supports sound.

The sound speed is $c_{\rm s} = c/\sqrt{3(1+R)}$ where $R = 3\rho_{\rm b}/4\rho_\gamma$ measures how much the baryons weigh the fluid down. Early on the baryons are negligible and the sound speed is $0.577c$ — more than half the speed of light, because the fluid is mostly radiation and radiation is very stiff.

Now put a small overdensity in the dark matter somewhere. Dark matter does not feel the pressure and simply sits there, deepening; the photon–baryon fluid falls into the potential well it makes, overshoots, is pushed back out by its own pressure, and oscillates. Each initial overdensity therefore launches a spherical pressure wave that runs outward at $c_{\rm s}$.

The waves are not launched at random times. They all start together, at the end of inflation, and they all stop together, when recombination removes the free electrons and the photons cease to push. So at the moment the pattern is frozen, every wave has been running for the same length of time, and there is one distance that appears everywhere: the sound horizon.

That is the whole origin of the preferred scale. A region exactly one sound horizon across has had just enough time to complete a single compression, so it is at maximum density when the light leaves — hot. A region half that size has compressed and rebounded to maximum rarefaction — also an extremum, also visible. A region of some intermediate size is caught mid-swing and shows nothing. The spectrum of patch sizes therefore has a fundamental and a series of harmonics, which is what the peaks are.

The first peak is a protractor #

The angle actually measured is $\theta_* = 0.0104110 \pm 0.0000031$ radians, which is a precision of three parts in ten thousand and makes it the best-measured quantity in cosmology. What it constrains is the ratio $r_{\rm s}/D_M(z_*)$, where $D_M$ is the transverse comoving distance — the one of the four that converts a proper size into an angle, and the only one in which the curvature of space appears at all.

The result is $\Omega_k = 0.0007 \pm 0.0019$: the universe is spatially flat to a fifth of a per cent. That is a measurement, not an assumption, and the figure exists to make the point that a curved universe would have looked visibly different.

It comes with a caveat that the plot cannot show. The peak position constrains a combination of the curvature and the expansion history, so a curved universe with a compensating change in the matter density and the expansion rate can reproduce the same angle — the “geometric degeneracy”. Breaking it needs a second observable, and the second observable is the same acoustic scale measured in galaxy surveys at low redshift, where the geometry of the intervening space contributes almost nothing. The quoted flatness is a joint result, and it is the clearest example in the subject of two measurements of one length at two epochs doing what neither can do alone.

The same combination is what makes the peak position central to the disagreement about the expansion rate. The angle is measured to three parts in ten thousand and is not in dispute; what is in dispute is the length of the ruler, because that comes from the pre-recombination physics, and a seven per cent shorter sound horizon would move the inferred expansion rate by the whole of the discrepancy.

The odd peaks are taller, and that weighs the baryons #

This is the step that turns a map of temperature into a measurement of how much ordinary matter the universe contains. The answer, $\Omega_{\rm b}h^2 = 0.02237 \pm 0.00015$, is a 0.7 per cent determination of the density of protons and neutrons — from the relative heights of two bumps in a power spectrum. It agrees with the completely independent value from the light-element abundances to better than a per cent, and the two have nothing in common: one is a fluid oscillation at four hundred thousand years, the other a reaction network in the first three minutes.

The third peak does a different job. Its height relative to the second is sensitive to the total matter density, because the depth of the potential wells at the time depends on how much dark matter had already collected in them. That is where $\Omega_{\rm c}h^2$ comes from, and it is the reason the microwave background constrains dark matter — the same substance the rotation curves find, measured by a route with no galaxies in it at all.

Why the peaks die away #

Past $\ell \approx 1000$ the peaks shrink, and two separate effects are doing it. The second is Silk damping, and it is the more important of the two. Photons do not scatter instantaneously; they random-walk through the plasma with a mean free path that grows as recombination proceeds, and over the age of the universe at that time they diffuse a certain distance. Any temperature variation smaller than that diffusion length is smeared out by the photons themselves carrying heat from hot regions to cold ones. The diffusion scale at last scattering is about 10 comoving megaparsecs, which corresponds to $\ell \approx 1400$, and the exponential fall of the spectrum past there is its signature.

The damping tail is not a loss. Its slope depends on how quickly recombination happened, which depends on the baryon density and on the number of relativistic species present, so the region beyond the third peak is where the microwave background constrains the effective number of neutrinos. Measuring it was the main scientific case for the ground-based experiments that followed Planck.

Where the fluctuations came from #

Everything above takes the initial overdensities as given. They have to come from somewhere, and the sound waves cannot make them — a wave needs something to launch it. The reason this matters for the peaks is a question of phase. A wave that is launched at a random time contributes at a random point in its cycle, and averaging over many such waves washes out the peaks entirely. The peaks exist only because every wave of a given wavelength started at the same phase — at rest, with an overdensity already in place. That is what “adiabatic, growing-mode, super-horizon initial conditions” means, and it is a substantive prediction of inflation that could easily have failed. The peaks are sharp, so the initial conditions were coherent, and coherence over scales larger than the horizon at the time is not something a causal process inside the horizon can produce.

The spectrum of initial amplitudes is measured too, and it is nearly but not exactly scale-invariant: $n_{\rm s} = 0.9649 \pm 0.0042$. Exactly 1 would mean every scale started with equal power; the measured tilt is a four per cent departure, and it is significant at fifteen sigma.

The plateau at the left of the plot #

The peaks occupy the middle of the spectrum. To their left, below about $\ell = 30$, the curve flattens into a plateau, and that region is measuring something different from everything the peaks measure.

Scales larger than the sound horizon at last scattering were never inside the horizon while the fluid could oscillate, so no wave ever ran across them. They carry the primordial fluctuations unprocessed — the potential field as inflation left it, imprinted on the temperature by a single effect.

A photon climbing out of a gravitational potential well is redshifted; a photon emitted from a region where the clock ran slow appears to come from a different epoch. The two combine, and the result is that a temperature fluctuation of one third of the potential’s depth is imprinted directly. That is the Sachs–Wolfe effect, and it is the reason the plateau exists and is flat: a scale-invariant spectrum of potentials produces a flat plateau in the quantity conventionally plotted.

Its measurement is limited in a way nothing else on the plot is. There are only $2\ell+1$ independent modes at each multipole, so at $\ell = 2$ there are five numbers in the entire universe and no instrument will ever measure them better. That is cosmic variance, it is a hard floor, and it is why the leftmost points on any published spectrum carry error bars an order of magnitude larger than the ones in the middle.

The plateau is also where the largest-scale anomalies live — a quadrupole lower than expected, an alignment between the lowest multipoles, a hemispherical asymmetry in power. Each is a departure of a couple of standard deviations, each has been checked in two independent experiments, and none is significant enough to be a detection given how many such tests could have been performed. A region measured with five numbers cannot settle an argument, and the honest position is that the anomalies will remain exactly as unresolved as they are.

What is measured, and what is not #

The observable is a map of temperature over the sky, in several frequency bands, from which the Galaxy has to be subtracted. That subtraction is the dominant systematic below $\ell = 30$ and is why the largest angular scales have the largest error bars — there is only one sky, so the sample variance at low $\ell$ is irreducible, and the foregrounds are worst exactly there.

From the cleaned map the power spectrum is computed, and the power spectrum is not the map: it discards all the phase information and keeps only the variance per scale. That is the right thing to do if the fluctuations are Gaussian, because then the power spectrum contains everything. Whether they are Gaussian is a separate measurement, and so far they are, to the precision available.

From the power spectrum, six parameters are fitted. Every cosmological number quoted from Planck — the age, the densities, $H_0$, $\sigma_8$ — is an output of that fit and not a reading. The peak positions and heights are the data; the parameters are the interpretation.

What the picture cannot show #

The drawn curve is a representation of a measurement, not a derivation. Producing the spectrum from the parameters requires integrating the coupled photon, baryon, neutrino and dark-matter perturbations through recombination, which is a Boltzmann code’s work. The figure separates the two halves deliberately: the curve interpolates published extrema, and the computed content is the tick marks. Reading the interpolation between two peaks as a prediction would be reading a fabrication.

The spectrum is an average over the whole sky and cannot show where anything is. Two maps with identical power spectra can look entirely different, and one of the standing questions about the real sky — whether the largest-scale features have anomalous alignments — is invisible to this plot by construction.

And the plot has no error bars. At low $\ell$ the uncertainty is dominated by there being only $2\ell+1$ independent samples of each mode on one sky; at high $\ell$ by the instrument. The middle range is measured to a fraction of a per cent, and the impression a clean curve gives is accurate there and misleading at both ends.

Two of the parameters the spectrum measures are worth pushing well outside their fitted values, because the shape of the response is what makes each of them separable from the others.

Why six parameters, and what a seventh would do #

The model fitted to the spectrum has six free numbers, and the count is worth examining because it is the whole basis for calling the fit a success.

Two are densities: the physical density of baryons and of cold dark matter. One is the angular scale of the sound horizon, which stands in for the expansion history. Two describe the primordial fluctuations: an amplitude and a tilt. And one is the optical depth to reionisation, which suppresses the small-scale power by a constant factor and is degenerate with the amplitude until polarisation breaks it.

Six numbers, and the spectrum they are fitted to contains something like two thousand independent multipoles, each measured to a precision that is not trivial. So the fit is enormously over-determined, and the residuals are a test rather than a leftover.

That is the point people usually miss about the model’s success. A six-parameter fit to two thousand points that comes out with an acceptable chi-squared is a strong statement; the same fit with twenty parameters would not be. Every proposed extension — curvature, a running spectral index, a varying dark-energy equation of state, extra relativistic species, a neutrino mass — is a seventh parameter, and each is measured to be consistent with zero or with its standard value.

The consequence is a peculiar situation. The model is not favoured because it explains anything; it is favoured because nothing has been found that requires more than it. A theory that survives by not being contradicted is in a weaker position than one that predicts something new, and the field’s discomfort about that is why so much effort goes into the observations that could add a seventh number — the damping tail’s neutrino count, the polarisation’s tensor amplitude, and the growth of structure at low redshift.

The generalisation #

The move here is one of the oldest in physics: a resonance in a bounded medium turns a continuous spectrum into a discrete one, and the discrete positions measure the boundary.

An organ pipe of a given length produces a fundamental and its harmonics, and hearing the fundamental measures the pipe. Here the “pipe” is the time available before the photons let go, the harmonics are the peaks, and the measurement is the pipe’s length in comoving megaparsecs. The unusual feature is only that the bound is temporal rather than spatial — the wave stops not because it hits a wall but because the medium ceases to exist.

The same reasoning appears in this collection wherever a periodic system is used as a clock or a ruler. A pulsating star’s period is set by the sound travel time across it, which is why a longer period means a bigger and therefore brighter star. A resonance between two orbits clears a gap at a computable radius, and the radius measures the perturber. Spiral arms exist only between the Lindblad resonances, and the boundaries follow from the rotation curve.

What makes the cosmological case unusually clean is that the pipe’s length is set by physics that involves only the densities, and the densities are measured by the same spectrum. The ruler calibrates itself.

And the spectrum truncated at the multipole where the first generation of experiments stopped, which is where most of the parameters were first pinned down.

One more reading spans a wider range of the thickness the damping tail measures.

Where the ladder goes next #

The peaks give the angle. Turning the angle into a distance needs the ruler’s length, which needs the baryon density — and the baryon density has an entirely independent measurement, three minutes after the beginning, which is the next essay.

Later rungs on this anchor: polarisation, which is a second map from the same photons and settles several degeneracies the temperature alone cannot; lensing of the microwave background by the intervening structure, which measures the matter distribution at $z \approx 2$; the integrated Sachs–Wolfe effect at the largest scales, which is a direct probe of dark energy; the damping tail and the effective number of neutrinos; and the search for a $B$-mode polarisation signal from primordial gravitational waves, which would fix the energy scale of inflation and has so far produced only upper limits.