← Outreach

Why Matter Lowers the First CMB Peak

The peaks in the microwave background are sound waves, caught in the act of ringing. How tall the first one stands is not set by how much gravity pulls — it is set by how fast gravity was fading while the wave was ringing. That is why adding dark matter makes the first peak smaller, which is the opposite of what most people guess.

One oscillator, driven by a fading potential

Before the universe became transparent, photons and baryons moved as a single fluid. Gravity pulled that fluid into the dark matter’s potential wells; photon pressure pushed it back out. Each Fourier mode of the density therefore behaves as a harmonic oscillator, and it starts oscillating when its wavelength first fits inside the horizon.

The part that decides the peak heights is what the gravitational potential does while the mode rings. Radiation cannot cluster: its pressure resists collapse, so during radiation domination the potential a mode has just fallen into decays away beneath it. That decay is in step with the oscillation, and it pumps it — a resonant kick that can nearly double the amplitude. Once matter dominates, the potential stops decaying, the pumping stops, and a mode entering afterwards rings with no help at all.

So the question for any mode is simply: had matter taken over yet when I entered the horizon? And the answer is set by the matter density, because that is what fixes when matter–radiation equality happened.

What the three curves are. The gravitational potential is a landscape — a value at every point in space, hills and valleys. What panel 1 plots is not that landscape but its ingredients. Any landscape can be written as a sum of pure ripples of different wavelength, and each curve is the depth of one ripple: a single sinusoidal corrugation of the potential, 336, 154 and 101 Mpc from crest to crest. The strip below shows what that means: the same landscape early and at last scattering, with the short corrugations largely ironed out of it by the time the photons leave. Panel 5, further down, draws them against a time slider.

The reason this is worth doing — and the reason cosmologists always do it — is that while the perturbations are small the equations governing them are linear, so the ripples do not interact. Each one rises and falls entirely on its own, ignorant of the others, and the universe becomes a set of independent oscillators that can be followed one at a time. Panel 1 follows three of them.

They start at the same depth, because adiabatic initial conditions give every wavelength the same amplitude. What separates them is the horizon: a ripple can do nothing until the horizon has grown to its own scale. Written as \(k\eta = 1\), the usual convention, that puts the short ripple at \(\eta \approx 16\ \mathrm{Mpc}\), the middle at 25 and the long at 53. In the computed curves the visible bend comes a little after each — 24, 39 and 56 Mpc at a 5% departure — because the response is not instantaneous and because the crossing scale is a convention rather than a measurement. The ordering is what matters. The short ripple, entering earliest and deepest into the radiation era, decays the longest and ends with 18% of its depth; the long one keeps 60%. Everything on this page follows from that spread.

What the Weyl potential is. Two numbers are needed to describe a perturbed universe, not one. Writing the geometry as \(ds^2 = a^2(\eta)\,[-(1+2\Psi)\,d\eta^2 + (1-2\Phi)\,dx^2]\), \(\Psi\) is the Newtonian potential — the one a slowly moving particle falls into — and \(\Phi\) is the perturbation to spatial curvature. Light, being relativistic, feels both, and is deflected by their average \((\Phi+\Psi)/2\), which is the Weyl potential and is what CAMB reports. With no anisotropic stress \(\Phi = \Psi\) and the distinction collapses; free-streaming neutrinos spoil that in the radiation era, where they carry a fraction \(f_\nu = 0.409\) of the radiation and split the two by \(2f_\nu/5\), about 16%. Strictly the oscillator is driven by \(\Psi\) and by the rate at which \(\Phi\) decays, not by their average; the Weyl potential runs some 8% above \(\Psi\) while neutrinos matter and converges on it afterwards, which is close enough that nothing in the argument changes.

Reading the time axis. The first two panels run on conformal time, \(\eta = \int dt/a\). It is a time coordinate, but with \(c = 1\) it carries units of length — and that length is the comoving distance light has travelled since the Big Bang. So \(\eta\) is the comoving particle horizon: how far you can see, not how big things are. The size of the universe is the scale factor \(a\).

That makes \(\eta\) the natural ruler for these two panels. A mode begins to oscillate when \(k\eta \approx 1\) — the moment its wavelength first fits inside the horizon — and the sound horizon that sets the peak spacing is an integral along this very axis. The two quantities do rise together, so moving right is always later and larger, but not in step: \(a \propto \eta\) while radiation dominates and \(a \propto \eta^2\) once matter does. The redshift scale across the top of panel 1 is that conversion, and its ticks halve their spacing as you pass equality for exactly this reason.

What the four panels are saying

Drag the matter slider and watch panel 1 first. The vertical line is matter–radiation equality. More matter moves it left, to earlier times, and the potential curves flatten out sooner — there is simply less decay left for a mode to catch. At the Planck value the longest mode shown loses about 40% of its potential before last scattering; push the matter up and that falls to about a third.

Panel 2 shares panel 1's time axis and is the consequence. The modes still oscillate at the same rate — the sound speed has not changed — but they ring less hard, because they were pumped less. Nothing about the timing of the oscillation changes, which is why the peaks will not move sideways.

Panel 3 is the step that turns an oscillation into a spectrum. Last scattering happens everywhere at once, so every mode is frozen at whatever phase it had reached. Modes caught at maximum compression or maximum rarefaction have large amplitude; modes caught mid-swing have none. Plotted against wavenumber, that is a series of peaks — and they sit at \(k r_\mathrm{s} \approx \pi, 2\pi, 3\pi\), one for each half-oscillation that fitted into the time available.

Panel 4 is what a telescope sees, after that pattern is projected onto the sky at a distance of 13.9 billion parsecs. Because \(\theta_*\) is being held fixed here, the peaks stay put in \(\ell\) and only their heights move — which is the whole point.

The two kinds of matter pull in opposite directions

This is where intuition usually goes astray, and the second slider is there to show why. Baryons and cold dark matter are both matter, but they act on the oscillation by completely different routes.

The two rows hold different things fixed, deliberately, because that is what the two sliders do. Moving the baryon slider keeps \(\Omega_\mathrm{m}h^2\) constant and so trades dark matter for baryons — which is the cleanest way to see the seesaw, since the total gravitating mass does not change and only its composition does. Moving the matter slider keeps \(\Omega_\mathrm{b}h^2\) at the BBN value and lets the dark matter carry the whole change.

Baryons work through the depth of the well. They are heavy, and they add inertia to the photon–baryon fluid, so gravity drags it further in before pressure can turn it around. That displaces the zero point of the oscillation: compressions get deeper, rarefactions get shallower. Compressions are the odd peaks and rarefactions the even ones, so the first peak rises while the second falls. The ratio of the two is what measures the baryon density — and it agrees with the deuterium abundance left over from big bang nucleosynthesis, which is an entirely independent argument.

Dark matter works through the timing. It does not interact with the photons at all; it only decides when the universe stopped being radiation-dominated. More of it means an earlier equality, less potential decay, and less of the resonant pumping — so the whole envelope of the peaks comes down together, and the odd–even ratio hardly moves.

That difference is what lets a single photograph of the sky weigh the two separately. If both kinds of matter did the same thing to the peaks, the CMB could only report their sum.

A box with no walls

A harmonic series invites an obvious analogy: a particle in a box, or an organ pipe. Standing waves, a fundamental and its overtones, \(k_n = n\pi/L\). The analogy is worth taking seriously, because one half of it is exactly right and the other half is exactly backwards, and the backwards half is the more interesting one.

What maps. The box length is the sound horizon: \(L \to r_\mathrm{s}\), and the peaks of panel 3 really do sit on a ladder with rungs \(n\pi/r_\mathrm{s}\). Measured against the computed spectrum, consecutive peaks are spaced by 1.017, 0.991 and 1.014 in units of \(\pi/r_\mathrm{s}\) — the ladder is right to about one and a half percent. The troughs are real nodes too: in panel 3 they fall to \(2\times10^{-4}\) of the peak value, a line spectrum, just as a box would give. (The filled-in troughs of the observed \(C_\ell\) come from the Doppler term and from projection, neither of which panel 3 includes.)

What is backwards. There are no walls, and nothing is quantised in space. Every wavenumber exists, continuously; none is forbidden. The ladder comes from a condition in time: adiabatic initial conditions start every mode at rest with the same phase, and last scattering samples every mode at the same instant. The ones caught at full amplitude are those that completed a whole number of half-oscillations in the time available. It is selection by phase, not by eigenvalue — closer to a rank of pendulums of graded length, all released together and all photographed together, than to anything with a boundary.

So the box tells you where the peaks are, through \(r_\mathrm{s}\). It cannot tell you how tall they are: a box's harmonics may carry whatever amplitudes you like, while the envelope here is set by the driving and the damping of the first four panels. That division is the cleanest way to hold the whole picture — the box sets the positions, the matter density sets the heights — and it is why \(\theta_*\) is pinned throughout this page: the box length is held fixed so that only the heights move.

Panel 5 is the box, run forward. The three waves at the top are the same three wavenumbers as panels 1 and 2, now drawn along a line in space rather than against time; each swells, collapses and turns inside out at its own rate, and the heavy curve is their sum. Below, every mode in the range is sampled at that same instant. Early on there is no comb at all — almost nothing has entered the horizon and had time to ring. As the sound horizon grows the teeth appear one by one and sweep toward smaller \(k\), because the rungs are \(n\pi/r_\mathrm{s}\) and \(r_\mathrm{s}\) is getting longer. The box is lengthening. At last scattering it stops, and what is left is panel 3.

The faint vertical lines are the ideal ladder \(n\pi/r_\mathrm{s}\). The teeth sit slightly to the left of them, by a constant offset of about \(0.13\pi\): the first peak lands at 0.85 of \(n\pi/r_\mathrm{s}\), the second at 0.94, the third at 0.96, the fourth at 0.97 — exactly the behaviour of a fixed shift in phase. That offset is the acoustic phase shift from radiation driving and from neutrino free-streaming, and it is the clearest single sign that the box analogy is a description and not a derivation. A box would land on the rungs.

Panel 6 is the same physics seen the other way round, and it is the picture worth carrying away. Instead of decomposing into modes, put down a single point overdensity in the early universe and watch what happens to it. The dark matter, which feels no pressure, stays where it is and grows. The photon–baryon fluid cannot: pressure drives it outward as a spherical sound wave, a shell expanding at a speed close to \(c/\sqrt{3}\). When the photons decouple the pressure vanishes and the shell stops, stranded at about the sound horizon. The photons stream away and fade; the baryons are left in a shell, and the dark matter at the centre and the baryons in the shell then pull on one another until both carry both features.

The shell and the dashed line do not quite coincide, and the discrepancy is real rather than a drawing error. At last scattering the shell peaks a little inside \(r_\mathrm{s}\), near 133 Mpc against 144, because it is a broad feature and the sound horizon marks the wave front rather than the crest. It then keeps creeping outward and settles at about 149 Mpc — outside \(r_\mathrm{s}\) — because the baryons are not released when the photons are. Photon decoupling, which sets \(r_*\) and the CMB peaks, happens slightly before the baryons stop being dragged along, which sets \(r_\mathrm{drag}\) and the galaxy ruler. The two scales differ by a few Mpc, and keeping them apart matters when the CMB and a galaxy survey are compared.

What survives is a small excess probability of finding matter at a particular separation — about 150 Mpc, a fixed comoving length frozen into the universe. In the CMB it shows up as the peak spacing. In galaxy surveys it is the baryon acoustic oscillation, the standard ruler. The slider makes the connection concrete: push the matter density up and the sound horizon shrinks, the shell in panel 6 freezes at a smaller radius, and the ladder in panel 5 spreads out. They are one measurement, made twice.

How this was computed

Every curve on this page is CAMB output, not a model of CAMB. Panels 1 and 2 are the actual Weyl potential and photon overdensity from the Boltzmann hierarchy, panel 3 the same perturbation sampled at last scattering across wavenumber, and panel 4 the computed temperature spectrum. Forty-five models were run across a grid in the two densities and the page interpolates between them, which is why the sliders step in small increments rather than moving continuously.

Panels 5 and 6 come from a second pass, nine models along the matter axis at the Planck baryon density, carrying the photon overdensity across wavenumber at thirty-six times from \(\eta = 5\ \mathrm{Mpc}\) to the present. Panel 6 is a spherical Bessel transform of those same perturbations, after Eisenstein, Seo & White (2007): a point overdensity at early times is decomposed into modes, each evolved by CAMB, and transformed back, with the initial condition normalised to a unit point source and a 4 Mpc Gaussian applied to keep the central cusp finite. The time grid carries exact points at equality and at last scattering, so panel 5 at last scattering is the same curve as panel 3.

\(\theta_*\) is held fixed as the densities change, so \(H_0\) moves instead — from about 80 at the low end of the matter slider to about 58 at the high end. That is the right thing to hold fixed here: \(\theta_*\) is what the CMB measures best, and fixing it keeps the peaks from sliding sideways so that the change in height is unambiguous. The optical depth and the primordial amplitude and tilt are held at their Planck 2018 values throughout.

Further afield: the CMB power spectrum with all six parameters on sliders and the Planck data to fit; from sky map to power spectrum for how the curve is extracted from a photograph of the sky; \(\Lambda\)CDM and the six parameters.