Experiment: The Cosmic Microwave Background
Tests Phenomenon 48.12 and Equation (48.24). Assuming Equations (38.38) and (68.5).
Discovery and precision spectroscopy of the relic radiation: the blackbody spectrum, the anisotropies, and the parameters they fix in the cosmology of Evidence-Based Cosmology. The cosmic microwave background is the oldest light there is and the best-measured blackbody there is: found accidentally by Penzias and Wilson [Penzias:1965] and identified at once as the relic of a hot early universe [Dicke:1965], it now carries, in its spectrum [Mather:1994] [Fixsen:1996], its anisotropies [Smoot:1992] [Bennett:2013] [Aghanim:2020] and its polarization [Kovac:2002], the tightest constraints in all of cosmology.
This chapter is the evidence file for Evidence-Based Cosmology: the hot-big-bang prediction under test, the radiometers and spectrophotometers that tested it, and the numbers they returned. The blackbody physics being tested is that of Black-Body Radiation and Planck's Hypothesis; the boost kinematics that reads the dipole is that of Lorentz Transformations; and the general-relativistic perturbation theory that turns the anisotropy spectra into cosmological parameters belongs to Evidence-Based Cosmology and is quoted here, never silently rederived. What this chapter owns is the measuring: apparatus, procedure, the numbers with their uncertainties, and what they do and do not establish.
Historical context and the prediction under test
The hot-big-bang prediction
The prediction is a by-product of nucleosynthesis. In 1948 Alpher, Bethe and Gamow proposed that the chemical elements were assembled by neutron capture in a hot, dense early universe [Alpher:1948a]; the calculation works only if the baryons are outnumbered by thermal photons by a large, fixed factor, because it is the radiation that sets the temperature–time relation through which the reactions freeze out (Evidence-Based Cosmology). Later the same year Alpher and Herman followed that radiation forward: expansion cools a thermal photon gas as \(T\propto1/a\) while preserving its Planck form — the statement proved in Evidence-Based Cosmology — so the fireball must survive to the present as a cold blackbody filling all of space, at a temperature they estimated as about \(5\,\mathrm{K}\) [Alpher:1948b].
The prediction is threefold, and each part is separately testable. The relic must have a Planck spectrum, because it was thermalized when the universe was dense enough to reach equilibrium and free expansion cannot un-thermalize it; any measured distortion is a record of energy released since. It must be isotropic to high accuracy, because it fills the universe rather than emanating from anything in it. And it must carry small anisotropies, because the structure that now exists had to grow from seed inhomogeneities that were present at decoupling and must have left their imprint on the light released then. Spectrum, isotropy and anisotropy are the three sections of the observations below, in that order.
Nobody searched for it. The prediction lay dormant for seventeen years, unknown to the radio astronomers who by the late 1950s had the instruments to test it. In 1964 Dicke's group at Princeton rederived the hot early phase independently and set about the measurement deliberately: Roll and Wilkinson were building a purpose-designed radiometer when word arrived that a pair of radio astronomers fifty kilometres away, calibrating a communications antenna, had already found an excess they could not remove. The two papers were published back to back — the measurement [Penzias:1965] and its interpretation [Dicke:1965] — and the interpretation paper's senior author was the same Dicke whose switched radiometer, invented in 1946, is the instrument class behind nearly every measurement in this chapter [Dicke:1946].
Precursors
The radiation had in fact been detected before it was predicted. In 1941 McKellar analysed the interstellar absorption lines of the CN radical seen against hot stars, among them \(\zeta\) Ophiuchi. The first rotationally excited level of CN lies one photon of wavelength \(2.64\,\mathrm{mm}\) above the ground state, and the ratio of the absorption from the two levels measures how the molecules are distributed between them; McKellar translated the measured ratio into a “rotational temperature” of interstellar space of about \(2.3\,\mathrm{K}\) [McKellar:1941]. The molecules are thermometers immersed in the millimetre-wave radiation field, and they were reporting the cosmic microwave background — but with no prediction to connect it to, the number stood in the literature as a curiosity, and only after 1965 could it be read for what it is.
The CN temperature is a canonical case of the general lesson: a measurement with no theoretical frame attached does not function as a discovery, however accurate it is. McKellar's \(2.3\,\mathrm{K}\) sat unexplained for twenty-four years — within thirty per cent of the modern value of Phenomenon 51.3 — while the complementary failure ran in parallel: from 1948 to 1965 the Alpher–Herman prediction [Alpher:1948b] was a theory without a signal, unknown to the observers who could have tested it. The discovery of 1965 happened at the moment the two finally met [Penzias:1965] [Dicke:1965], and neither half would have sufficed alone.
Apparatus
Radiometers and the Holmdel horn
Every temperature in this chapter is an antenna temperature: the temperature of a matched resistive load that would deliver the same power to the receiver, a linear measure of received power in the Rayleigh–Jeans regime of Black-Body Radiation and Planck's Hypothesis. The obstacle to measuring a few kelvin of sky is that the receiver itself contributes hundreds or thousands of kelvin of noise, and its gain drifts. Dicke's solution of 1946 defines the instrument class: switch the receiver input many times per second between the sky and a stable reference load, and detect synchronously with the switch, so that receiver noise and gain drift, common to both half-cycles, cancel in the difference [Dicke:1946]. The residual fluctuation of an ideal radiometer of system temperature \(T_{\mathrm{sys}}\), predetection bandwidth \(\Delta\nu\) and integration time \(\tau\) is
within a factor of order unity fixed by the switching scheme [Dicke:1946]. The equation is the budget of the whole field: a receiver of \(T_{\mathrm{sys}}=100\,\mathrm{K}\) and \(\Delta\nu=100\,\mathrm{MHz}\) resolves — computed here from Equation (51.1) — about \(10\,\mathrm{mK}\) in one second, so the millikelvin dipole of Section 51.4.3 needs minutes, and the microkelvin anisotropies of Section 51.4.4 need years of integration and thousands of detectors.
The discovery instrument was not built for cosmology. The Holmdel antenna at Crawford Hill is a \(6\,\mathrm{m}\) horn-reflector erected for the Echo and Telstar satellite links, chosen for exactly the property a background measurement needs: the horn geometry shields the aperture from the ground, so the response in the back and side lobes is far below that of a paraboloid. Penzias and Wilson equipped it with a ruby travelling-wave maser preamplifier at \(4.08\,\mathrm{GHz}\) — a wavelength of \(7.35\,\mathrm{cm}\), computed here — and, crucially, with a reference termination immersed in liquid helium, giving the switched radiometer an absolute cold standard against which the sky could be measured rather than merely compared [Penzias:1965]. A year of systematic elimination followed: the atmosphere was measured by tipping the beam and fitting the secant law in zenith angle, the ohmic loss of the antenna throat was bounded, the response away from the beam was checked against the Sun and the Galaxy, the seams were taped, and the pair of pigeons roosting in the throat was evicted and their deposits removed. The excess survived everything, in every direction, in every season [Penzias:1965].
Satellites: COBE, WMAP, Planck
The atmosphere emits and absorbs strongly across the millimetre band, so the two precision programmes — the spectrum and the anisotropy — both culminated in space.
COBE (launched 1989) carried both instruments of record. FIRAS is a polarizing Michelson interferometer whose two inputs are the sky and an internal reference blackbody: the interferogram measures the difference between the two spectra, and the reference is servoed until that difference nearly vanishes, so the instrument spends its dynamic range on the deviation rather than on the signal. An external calibrator — a re-entrant absorbing cone that can swing over the horn and fill the beam entirely — provides the absolute standard; the accuracy of Section 51.4.2 is the accuracy of that cone. The spectral band of the monopole analysis runs from about \(60\,\mathrm{GHz}\) to \(630\,\mathrm{GHz}\) with a \(7^\circ\) beam [Mather:1990] [Mather:1994] [Fixsen:1996]. The companion DMR consists of differential microwave radiometers at \(31.5\,\mathrm{GHz}\), \(53\,\mathrm{GHz}\) and \(90\,\mathrm{GHz}\), each comparing two \(7^\circ\) beams \(60^\circ\) apart on the sky — a Dicke radiometer whose reference load is another patch of sky [Smoot:1992].
WMAP (2001–2010) took the differential principle to the second Lagrange point, where Sun, Earth and Moon sit together in a small solid angle behind the shields. Pseudo-correlation differencing assemblies at five frequencies from \(23\,\mathrm{GHz}\) to \(94\,\mathrm{GHz}\) measured temperature differences between beams separated by about \(141^\circ\), with angular resolution reaching about \(0.2^\circ\) in the highest band; nine years of data give the full-sky maps and power spectra of [Bennett:2013]. Planck (2009–2013), from the same orbit, spans \(30\,\mathrm{GHz}\) to \(857\,\mathrm{GHz}\) with two instrument families: the LFI radiometers at 30, 44 and \(70\,\mathrm{GHz}\), and the HFI bolometers from \(100\,\mathrm{GHz}\) upward, cooled to \(0.1\,\mathrm{K}\) and reaching about \(0.08^\circ\) resolution — enough to map the temperature anisotropy over the whole acoustic range [Aghanim:2020].
Balloons and ground stations
Between the discovery and the satellites, and alongside them since, the sub-orbital tier has repeatedly been first. The U-2 aircraft radiometers of 1976–77, flying above most of the atmosphere, found the dipole [Smoot:1977]. BOOMERanG, a bolometric telescope carried around Antarctica for about ten days at an altitude near \(38\,\mathrm{km}\) by a long-duration balloon, mapped a patch of sky at frequencies from \(90\,\mathrm{GHz}\) to \(400\,\mathrm{GHz}\) with sub-degree resolution and resolved the first acoustic peak [deBernardis:2000]. DASI, a thirteen-element interferometer at \(26\text{–}36\,\mathrm{GHz}\) sited at the South Pole — where the precipitable water vapour is among the lowest on Earth — made the first detection of polarization [Kovac:2002]. The modern ground tier is the arcminute programme: the \(10\,\mathrm{m}\) South Pole Telescope and the \(6\,\mathrm{m}\) Atacama Cosmology Telescope, with kilopixel arrays of transition-edge bolometers, extend the spectra into the damping tail and supply the secondary-anisotropy science of Section 51.4.7 [Hanson:2013] [Aiola:2020]; and the BICEP/Keck small-aperture polarimeters — refractors of tens of centimetres, built for the degree scale where a primordial B mode would peak — carry the search of Section 51.4.6 [Ade:2014] [Ade:2021].
Procedure
Four protocols cover every result reported below.
Absolute spectrophotometry. The sky is nulled against a calibrated blackbody and the residual spectrum is measured interferometrically; periodically the external calibrator replaces the sky entirely, transferring the laboratory standard to the celestial measurement. The result is a spectrum in absolute units at each frequency, from which one temperature and the distortion parameters of Section 51.4.2 are fitted [Mather:1994] [Fixsen:1996].
Differential mapping. Anisotropy instruments never measure the monopole: they measure temperature differences between sky directions, following Equation (51.1), and scan so that every pair of pixels is linked by many differences along many paths. The map is then the least-squares solution of the linear model relating the time-ordered differences to the pixel temperatures — an overdetermined system whose redundancy is also the systematics check, since a miscalibrated channel shows up as inconsistency between scan paths. The estimation machinery is that of Probability and Statistics [Smoot:1992] [Bennett:2013] [Aghanim:2020].
Component separation. The microwave sky is not only the background: the Galaxy radiates synchrotron and free–free emission, both falling steeply with frequency, and thermal dust emission, rising with frequency. The background alone has the achromatic signature of a temperature perturbation of a blackbody — the same \(\Delta T\) at every frequency when expressed as thermodynamic temperature. Multi-frequency coverage is therefore not a luxury but the separation variable itself; the Galactic plane is masked, and the components are fitted and removed pixel by pixel [Bennett:2013] [Aghanim:2020].
Harmonic analysis and likelihood. A full-sky anisotropy map is expanded in the spherical harmonics of Ordinary Differential Equations and Sturm–Liouville Theory,
and the cosmological information is carried by the angular power spectrum \(C_{\ell}\) (and its polarized counterparts \(C_{\ell}^{TE}\), \(C_{\ell}^{EE}\), \(C_{\ell}^{BB}\)), because statistical isotropy makes the individual \(a_{\ell m}\) zero-mean random variables with a common variance at each \(\ell\). That same statement fixes the floor of the whole enterprise. The unbiased estimator averages the \(2\ell+1\) independent squared moduli available at multipole \(\ell\), and for Gaussian fluctuations (Probability and Statistics) the relative scatter of such an average is
the cosmic variance: we observe one sky, and no instrument, however perfect, can beat the sample size. At \(\ell=2\) the floor is 63 per cent — computed here from Equation (51.3) — which is why the quadrupole is permanently uncertain while the acoustic peaks, with thousands of modes each, are measured to parts in a thousand. Parameters are then inferred by likelihood fits of model spectra to the measured \(C_{\ell}\), per Probability and Statistics [Bennett:2013] [Aghanim:2020].
Observations and data
Throughout this section, a number with an uncertainty is quoted from the cited primary paper; where a number is instead computed here from quoted inputs, the text says so.
Discovery
Penzias and Wilson measured the total antenna temperature at the zenith at \(4.08\,\mathrm{GHz}\) and subtracted everything they could account for. The budget is the experiment, and it is short:
The Holmdel antenna-temperature budget at \(4.08\,\mathrm{GHz}\), as reported by Penzias and Wilson [Penzias:1965]. All entries are quoted from the paper. The remainder on the last line, isotropic, unpolarized and stable through the seasons, is the cosmic microwave background.
| Contribution | Antenna temperature |
|---|---|
| Total zenith measurement | \(6.7\,\mathrm{K}\) |
| Atmospheric absorption | \(2.3 \pm 0.3\,\mathrm{K}\) |
| Ohmic losses of the antenna | \(0.8 \pm 0.4\,\mathrm{K}\) |
| Back-lobe ground pickup | $<0.1\,\mathrm{K}$ |
| Unaccounted excess | \(3.5 \pm 1.0\,\mathrm{K}\) |
A horn-reflector antenna operating at \(4.08\,\mathrm{GHz}\), after every identifiable contribution to the system noise had been measured and subtracted, retained an excess antenna temperature of about \(3.5\,\mathrm{K}\), uncertain by about \(1.0\,\mathrm{K}\). The excess was isotropic within the sensitivity of the instrument, unpolarized and unchanged with the season, so it could be neither a terrestrial source, nor the Galaxy, nor any known population of discrete extragalactic sources [Penzias:1965]. The companion paper published alongside identified it as relic radiation from a hot, dense early phase of the universe [Dicke:1965]. Rests on Equation (68.5), Phenomenon 68.5 and Definition 48.2.
Derivation. Derives Phenomenon 51.2. Two steps connect the reported number to a physical statement about the universe. The first is that an antenna temperature is a measure of received power — the temperature of a matched resistive load that would deliver the same power to the receiver — so equating it with the thermodynamic temperature of a blackbody filling the beam presupposes the Rayleigh–Jeans limit of the Planck law of Black-Body Radiation and Planck's Hypothesis, in which the spectral radiance is linear in \(T\). That limit requires \(h\nu\ll k_{\mathrm{B}}T\), and at the Holmdel frequency
so the departure from linearity is a few per cent, well inside the quoted uncertainty. The identification is therefore legitimate at this frequency — and would not have been at, say, \(100\,\mathrm{GHz}\), which is why the discovery had to wait for centimetre-wave receivers of very low noise.
The second step is that isotropy together with the absence of polarization already excludes the alternatives. A Galactic origin would concentrate the signal toward the plane; an unresolved population of extragalactic radio sources would carry their synchrotron spectrum, falling steeply with frequency rather than lying flat in \(T\); and any terrestrial, atmospheric or solar contribution would vary with elevation and with the season. All three were tested and excluded, and what remains must fill space uniformly — which is a cosmological statement, not an astronomical one.
∎The blackbody spectrum
One measurement at one frequency establishes an excess; only the spectrum can establish a relic. The prediction under test — the thermal form preserved by expansion, Evidence-Based Cosmology — was settled by FIRAS in stages, each tightening the last:
The spectrum campaigns. Every value is quoted from the cited paper. The deviation column bounds the departure of the measured spectrum from a Planck curve, as a fraction of the peak brightness.
| Measurement | Year | Temperature $T_{0}$ | Deviation |
|---|---|---|---|
| FIRAS, first nine minutes [Mather:1990] | 1990 | \(2.735 \pm 0.060\,\mathrm{K}\) | — |
| FIRAS, calibrated [Mather:1994] | 1994 | \(2.726 \pm 0.010\,\mathrm{K}\) | $<3\times10^{-4}$ |
| FIRAS, full data set [Fixsen:1996] | 1996 | \(2.728 \pm 0.004\,\mathrm{K}\) | $<5\times10^{-5}$ (rms) |
| Combined, recalibrated [Fixsen:2009] | 2009 | \(2.72548 \pm 0.00057\,\mathrm{K}\) | — |
The full-data-set analysis also bounds the two canonical distortion parameters: a chemical potential \(\abs{\mu}<9\times10^{-5}\) and a Compton parameter \(\abs{y}<1.5\times10^{-5}\), both at 95 per cent confidence [Fixsen:1996]. Each is a null with content, because each would be the fossil of energy released into the radiation: a \(\mu\) distortion records injection early enough for scattering to redistribute photons but too late to create them, a \(y\) distortion records injection later still, so the limits constrain the entire energy-release history of the universe since thermalization ceased (Evidence-Based Cosmology). The 2009 value combines the FIRAS data, recalibrated against later precision dipole measurements, with the other absolute measurements, and it is the temperature used throughout this book [Fixsen:2009].
The spectrum of the background radiation is a Planck spectrum. An absolute spectrophotometer flown above the atmosphere, nulling the sky against an on-board blackbody calibrator, found no deviation from the Planck form larger than about fifty parts per million of the peak brightness anywhere in the band it covered [Mather:1994] [Fixsen:1996]. The temperature is \(T_{0}=2.72548\,\mathrm{K}\), known to \(0.00057\,\mathrm{K}\) [Fixsen:2009]. No source constructed in a laboratory has ever been brought this close to thermal equilibrium, and the fit has no free parameter but the one temperature. Rests on Equation (68.5), Phenomenon 68.5 and Definition 48.2.
Derivation. Derives Phenomenon 51.3. Three numbers characterize the measured spectrum, and each follows from \(T_{0}\) and the Planck law of Black-Body Radiation and Planck's Hypothesis alone. The wavelength of maximum spectral radiance per unit wavelength is given by the Wien displacement law,
so the radiation peaks in the microwave band — which is why a radio receiver found it, and why measuring the spectrum required going above the atmospheric water lines. Per unit frequency the maximum instead lies at \(\nu_{\max}=2.821\,k_{\mathrm{B}}T_{0}/h \approx160\,\mathrm{GHz}\); the two maxima disagree because a spectral density is defined with respect to a chosen variable, and neither is “the” peak of the radiation.
The energy density follows from the Stefan–Boltzmann law,
with \(a=7.566\times10^{-16}\,\mathrm{J\,m^{-3}\,K^{-4}}\) the radiation constant, and dividing by the mean photon energy \(\approx2.70\,k_{\mathrm{B}}T_{0}\) gives a photon number density of about \(4.1\times10^{8}\) per cubic metre. That is roughly \(10^{9}\) photons for every baryon in the universe, and that ratio is the single free parameter of the nucleosynthesis calculation of Evidence-Based Cosmology — so the spectrum measured here and the light-element abundances measured there are two determinations of one number.
∎The dipole
The first anisotropy found is not primordial. Radiometers flown on a U-2 aircraft in 1976–77 detected a smooth cosine pattern across the sky [Smoot:1977], since measured to four significant figures by Planck [Akrami:2020]:
The dipole anisotropy. The inferred speed on the last line is computed here from the quoted Planck amplitude and the monopole temperature, via the boost formula Equation (51.4).
| Quantity | Value | Status |
|---|---|---|
| Amplitude, U-2 discovery [Smoot:1977] | \(3.5 \pm 0.6\,\mathrm{mK}\) | quoted |
| Amplitude, Planck [Akrami:2020] | \(3.3621 \pm 0.0010\,\mathrm{mK}\) | quoted |
| Direction, Galactic [Akrami:2020] | $(l,b)\approx(264.02^{\circ},48.25^{\circ})$ | quoted |
| Inferred solar-system speed | $\approx370\,\mathrm{km}/\mathrm{s}$ | computed here |
The background is not exactly isotropic. Superposed on the monopole is a pure dipole of amplitude about \(3.36\,\mathrm{mK}\), first detected from a U-2 aircraft [Smoot:1977] and now measured to four significant figures [Akrami:2020]. Its interpretation is kinematic: the solar system moves at about \(370\,\mathrm{km}/\mathrm{s}\) with respect to the frame in which the radiation is isotropic. This is the only large anisotropy of the microwave sky that is not primordial, and it must be removed before the intrinsic fluctuations of Section 51.4.4 can be seen at all — they are two orders of magnitude smaller. Rests on Theorem 38.4, Proposition 38.25 and Phenomenon 68.5.
Derivation. Derives Phenomenon 51.4. Let the radiation be an isotropic blackbody at temperature \(T_{0}\) in some frame, and let an observer move through it with speed \(v=\beta c\). Radiation arriving from the direction making an angle \(\theta\) with the motion is Doppler shifted by the factor \(\left[\gamma\left(1-\beta\cos\theta\right)\right]^{-1}\) of Lorentz Transformations. Every frequency in the spectrum is multiplied by that same factor, and the Planck spectrum of Black-Body Radiation and Planck's Hypothesis depends on frequency only through the combination \(h\nu/\left(k_{\mathrm{B}}T\right)\); the boosted spectrum is therefore again Planckian in every direction, with a direction-dependent temperature
To first order in \(\beta\) this is exactly a dipole of amplitude \(\Delta T=\beta T_{0}\), with no quadrupole or higher multipole at all — so the observed pattern being a clean dipole is itself evidence for the kinematic reading rather than an assumption of it. Inverting,
that is about \(370\,\mathrm{km}/\mathrm{s}\). The second-order terms of Equation (51.4) generate a kinematic quadrupole smaller than the dipole by a further factor \(\beta\approx10^{-3}\), which is comparable to the primordial signal and is accounted for explicitly in modern analyses. The same boost also aberrates the arrival directions by the aberration formula Equation (38.38) of Lorentz Transformations, displacing the whole anisotropy pattern by an angle of order \(\beta\approx1.2\times10^{-3}\) radians — about four minutes of arc, computed here — toward the apex.
∎The frame in which the dipole vanishes is physically distinguished: it is the rest frame of the radiation, and to high accuracy the rest frame of the average matter distribution. That this frame exists is a cosmological statement — the universe has a preferred state of motion at each point, defined by its contents — and its significance for the cosmological principle is discussed in Evidence-Based Cosmology. It contradicts nothing in Lorentz Transformations: the laws are Lorentz-invariant; the matter distribution is not.
Primary anisotropies
Underneath the dipole the sky is smooth to a few parts in \(10^{5}\), and finding the intrinsic structure took twenty-seven years of steadily improving null results. The COBE DMR maps of 1992 finally resolved it [Smoot:1992]:
The DMR discovery numbers [Smoot:1992]. The relative amplitude on the last line is computed here from the quoted rms and the monopole temperature.
| Quantity | Value | Status |
|---|---|---|
| Sky rms, \(10^\circ\) smoothing | \(30 \pm 5\,\mu\mathrm{K}\) | quoted |
| Quadrupole amplitude | \(13 \pm 4\,\mu\mathrm{K}\) | quoted |
| Relative amplitude $\Delta T/T_{0}$ | $\approx1.1\times10^{-5}$ | computed here |
Once the dipole of Phenomenon 51.4 is subtracted and the Galactic plane masked, the background retains intrinsic temperature fluctuations of relative amplitude \(\Delta T/T\sim10^{-5}\). They were first detected on angular scales of about ten degrees by the differential radiometers of COBE [Smoot:1992], twenty-seven years after the discovery of the monopole and after a long series of non-detections that had begun to constrain the models; they are now mapped over the whole sky to arcminute resolution [Bennett:2013] [Aghanim:2020]. On the largest angular scales the fluctuations track the gravitational potential on the last-scattering surface, the relation worked out by Sachs and Wolfe long before there was a measurement to apply it to [Sachs:1967]. These are the seed inhomogeneities from which all later structure grew, and their smallness is what makes linear perturbation theory sufficient for the epoch. Rests on Equation (42.7), Phenomenon 42.5 and Equation (51.2).
Derivation. Derives Phenomenon 51.5. Two things are derived here: what the reported numbers mean, and why a temperature fluctuation measures a gravitational potential. The statistical frame is Equation (51.2): a map smoothed to an angular scale \(\theta\) retains the multipoles up to \(\ell\sim\pi/\theta\), so the \(10^\circ\) smoothing of Table 51.4 reads the sky at \(\ell\lesssim20\) — the largest scales, causally disconnected at decoupling. Dividing the quoted rms by the monopole,
computed here from quoted inputs.
Why this measures gravity: a photon that last scatters at the bottom of a potential well must climb out, and arrives redshifted by the gravitational redshift of Equation (42.7), written for a general weak potential as \(\Delta\nu/\nu=-\Delta\Phi/c^{2}\). Since a uniform fractional shift of every frequency maps a Planck spectrum to a Planck spectrum — the same argument as in the dipole derivation above — the observed effect is a temperature perturbation \(\Delta T/T=\Delta\nu/\nu\): an overdense region, with \(\Phi<0\), is seen cold. The naive answer \(\Delta T/T=\abs{\Phi}/c^{2}\) overcounts, however, because the same perturbation also alters the plasma density, and hence the temperature, at the point of emission, and alters the rate of the clocks there. The general-relativistic accounting of all three pieces — carried out first by Sachs and Wolfe [Sachs:1967], and belonging in this book to Evidence-Based Cosmology — leaves, for adiabatic perturbations on scales larger than the acoustic scale of Section 51.4.5, the net result
one third of the naive redshift. Read through Equation (51.5), the DMR amplitude says \(\Phi/c^{2}\approx3\times10^{-5}\) on the last-scattering surface, computed here: the metric of the universe at decoupling departed from homogeneity by a few parts in \(10^{5}\). That smallness is the licence for every linear calculation in Evidence-Based Cosmology, and the potentials themselves are the seeds whose growth into the present galaxy distribution is followed there.
∎The acoustic peaks
Resolution below a degree turned the anisotropy from an amplitude into a spectroscopy. The peak structure was predicted three decades before it was resolved [Peebles:1970] [Sunyaev:1970], with photon diffusion cutting the spectrum off at small scales [Silk:1968]; BOOMERanG resolved the first peak in 2000 [deBernardis:2000], and the satellite spectra now trace the harmonics across the whole acoustic range [Bennett:2013] [Aghanim:2020].
The acoustic scale. Quoted values are from the cited papers; computed values follow from the quoted ones via Equation (51.7).
| Quantity | Value | Status |
|---|---|---|
| First-peak position, BOOMERanG [deBernardis:2000] | $\ell_{1}=197\pm6$ | quoted |
| Acoustic angular scale $100\theta_{*}$ [Aghanim:2020] | $1.0411\pm0.0003$ | quoted |
| Comoving sound horizon $r_{*}$ [Aghanim:2020] | \(144.4 \pm 0.3\,\mathrm{Mpc}\) | quoted |
| Comoving distance to last scattering | $\approx13.9\,\mathrm{Gpc}$ | computed here |
| Acoustic multipole $\ell_{A}=\pi/\theta_{*}$ | $\approx302$ | computed here |
| Spatial curvature $\Omega_{K}$ [Aghanim:2020] | $0.001\pm0.002$ | quoted |
The angular power spectrum of the temperature anisotropy is not featureless: it carries a series of harmonic peaks, the first and highest near multipole \(\ell\approx220\), with the spectrum cut off at high \(\ell\) by photon diffusion out of the shortest wavelengths [deBernardis:2000] [Aghanim:2020][Silk:1968]. These are the standing acoustic oscillations of the photon-baryon plasma before decoupling, predicted three decades before they were resolved [Peebles:1970] [Sunyaev:1970]. Their positions and relative heights measure geometry and contents at once: the angular scale of the first peak fixes the spatial curvature, which is consistent with zero with \(\abs{\Omega_{K}}\) bounded below about \(0.005\); the alternation of odd and even peak heights weighs the baryons; and the envelope constrains the total matter density [Aghanim:2020]. The same sound horizon, seen at low redshift in the galaxy distribution rather than projected on the last-scattering surface, is the acoustic ruler of Evidence-Based Cosmology. Rests on Theorem 48.3, Theorem 48.5 and Phenomenon 65.3.
Derivation. Derives Phenomenon 51.6. What is derivable at this chapter's level is the ruler and its projection; the dynamics of the oscillations themselves — the driven photon-baryon oscillator, the phase of each harmonic, the alternation of the peak heights and the damping envelope — is the business of Evidence-Based Cosmology, resting on [Peebles:1970] [Sunyaev:1970] [Silk:1968].
The ruler. Before decoupling, Thomson scattering (Radiation and Scattering of Electromagnetic Waves) locks photons and baryons into a single fluid whose pressure is the radiation pressure \(p=u/3\) of Black-Body Radiation and Planck's Hypothesis. For radiation alone the sound speed is \(c_{s}=c\sqrt{\pp p/\pp u}=c/\sqrt{3} \approx1.73\times 10^{5}\,\mathrm{km}/\mathrm{s}\), computed here; baryons add inertia without adding pressure and slow the wave, so
where \(r_{s}\), the comoving distance sound travels before the photons are released at \(t_{*}\), is the farthest any pressure wave ever got: the sound horizon. Evaluating the integral needs the full expansion and recombination history and is carried out in Evidence-Based Cosmology; the fitted value is \(r_{*}=144.4 \pm 0.3\,\mathrm{Mpc}\) [Aghanim:2020].
The projection. A comoving length \(r_{*}\) on the last-scattering surface, seen across the comoving distance \(D_{M}\) in a spatially flat geometry, subtends
the second relation because a spherical harmonic of degree \(\ell\) oscillates with angular period \(2\pi/\ell\) along a great circle (Ordinary Differential Equations and Sturm–Liouville Theory), so structure of angular size \(\theta\) concentrates its power near \(\ell\approx\pi/\theta\). The measured acoustic scale is \(100\theta_{*}=1.0411\pm0.0003\) [Aghanim:2020]; combining the two quoted numbers gives, computed here,
The first peak sits not at \(\ell_{A}\) but below it, at \(\ell_{1}\approx220\approx0.73\,\ell_{A}\): the oscillator is driven by the decaying gravitational potentials, which advances the phase of every harmonic by a calculable shift (Evidence-Based Cosmology). The BOOMERanG value \(\ell_{1}=197\pm6\) [deBernardis:2000] and the satellite value near \(220\) [Bennett:2013] [Aghanim:2020] bracket the same physics at very different calibrations and sky coverage.
The curvature. Equation (51.7) as written is the flat-space rule that angular size is length over distance. In a positively curved space light rays converge, and the same ruler at the same distance subtends a larger angle, pushing the peaks to lower \(\ell\); negative curvature does the reverse. With \(r_{*}\) fixed by the pre-decoupling plasma physics and the distance fixed by the measured expansion history, the observed \(\theta_{*}\) therefore pins the curvature; the fitted result is \(\Omega_{K}=0.001\pm0.002\) [Aghanim:2020]. Space, on the largest measurable scales, is Euclidean to the accuracy of the measurement — one of the sharpest geometric statements in physics, and an input assumption nowhere.
∎Polarization
The background is polarized, and its polarization was found where the atmosphere is driest. The DASI interferometer at the South Pole detected linear polarization at the microkelvin level in 2002, at a significance of \(4.9\sigma\) [Kovac:2002]; the correlation between temperature and polarization was then traced across the acoustic range by WMAP and Planck [Bennett:2013] [Aghanim:2020]; B modes generated by gravitational lensing were first detected with the South Pole Telescope [Hanson:2013]; and the degree-scale search for a primordial B mode produced the episode recorded in Remark 51.8, with the current bound \(r<0.036\) at 95 per cent confidence [Ade:2021].
The background is linearly polarized at the level of a few microkelvin — about a tenth of the temperature anisotropy — as first detected by an interferometer at the South Pole [Kovac:2002]. The polarization field decomposes into a curl-free E mode and a divergence-free B mode. The observed E mode is correlated with the temperature field in the sense and with the phase relation that the acoustic picture of Phenomenon 51.6 requires, which is a sharp consistency test of that picture rather than a new parameter [Bennett:2013] [Aghanim:2020]; and B modes generated by the gravitational lensing of E modes along the line of sight have been detected [Hanson:2013]. No primordial B mode has been detected. A degree-scale signal reported in 2014 [Ade:2014] was shown by joint analysis with multi-frequency data to be polarized emission from Galactic dust [Ade:2015], and the tensor-to-scalar ratio is now bounded by \(r<0.036\) [Ade:2021]. Rests on Phenomena 51.6 and 65.3.
Derivation. Derives Phenomenon 51.7. The polarization mechanism is Thomson scattering, and its geometry is derived here in full; the amplitude of the signal and the phase relation of \(C_{\ell}^{TE}\) to the temperature spectrum need the perturbation theory of Evidence-Based Cosmology and are quoted.
By Phenomenon 65.3, light Thomson-scattered through \(90^\circ\) emerges completely linearly polarized, its electric vector normal to the scattering plane: the driven electron re-radiates as a dipole, and a dipole does not radiate along its own axis (Radiation and Scattering of Electromagnetic Waves). Sit on an electron at last scattering and let the line of sight to the observer run along \(\hat{\vect{x}}\). Incident light arriving from \(\hat{\vect{z}}\) and scattered to the observer emerges polarized along \(\hat{\vect{y}}\); incident light from \(\hat{\vect{y}}\) emerges polarized along \(\hat{\vect{z}}\). If the incident intensity is the same from the two directions, the two scattered polarizations superpose to none: an electron in an isotropic bath scatters unpolarized light. A dipole anisotropy of the incident field cancels too: the light from \(+\hat{\vect{z}}\) and from \(-\hat{\vect{z}}\) is scattered with the same polarization axis, so only the sum \(I(\hat{\vect{z}})+I(-\hat{\vect{z}})\) enters, and a dipole leaves every such antipodal sum at its isotropic value — likewise for \(\pm\hat{\vect{y}}\). The lowest moment of the incident radiation that produces any net polarization is the difference between the two transverse axes, \(I(\hat{\vect{z}})-I(\hat{\vect{y}})\): a quadrupole. The polarized sky is therefore a map of the quadrupole anisotropy seen by each electron at the moment it last scattered.
That is also why the signal is small. Before decoupling, scattering is so frequent that the radiation each electron sees is isotropized and the quadrupole is erased; after decoupling there are no scatterers left. Polarization is generated only in the brief interval in between, as free streaming builds a local quadrupole while some scattering remains, and the outcome is a signal about an order of magnitude below the temperature anisotropy — a few microkelvin, exactly the level DASI detected [Kovac:2002]. The quantitative treatment belongs to Evidence-Based Cosmology.
The observed field is described by the Stokes parameters \(Q\) and \(U\) on the sphere — a spin-2 quantity, since a headless polarization axis returns to itself under rotation by \(180^\circ\) — and it decomposes into a curl-free E pattern and a divergence-free B pattern, the tensor analogue of splitting a vector field into gradient and curl parts. At linear order, scalar density perturbations generate E only; gravitational waves generate E and B alike. The decomposition machinery and that theorem are Evidence-Based Cosmology's; what matters for this chapter is the experimental logic it dictates, namely that the E mode and the TE correlation had to appear with the measured acoustic phases (they did [Kovac:2002] [Bennett:2013] [Aghanim:2020]), and that any primordial B mode is a direct gravitational-wave observable, which is why the degree scale is watched by dedicated instruments and why the bound \(r<0.036\) [Ade:2021] is a statement about the early universe rather than about photons.
∎In March 2014 the BICEP2 collaboration reported a degree-scale B mode well above the lensing expectation [Ade:2014], read at the time as primordial. The instrument observed at a single frequency, \(150\,\mathrm{GHz}\), so the measurement itself could not discriminate the achromatic background from Galactic foregrounds; the discrimination came from joining the BICEP2/Keck maps with the multi-frequency Planck dust maps, and the joint analysis attributed the signal to polarized thermal emission from Galactic dust, leaving no significant primordial component [Ade:2015]. The episode is kept in this book deliberately: it is the cleanest modern instance of the component-separation logic of Section 51.3 acting as the error bar — a single-frequency detection at the sky's faintest signal level is not yet a measurement of the sky.
Secondary anisotropies
Between the last-scattering surface and the receiver the radiation crosses the entire later universe, and three imprints of the crossing are measured. Each is a signal in its own right; none is primordial, and all must be modelled in the fits of Section 51.5.
Inverse-Compton scattering in clusters. Photons traversing the hot ionized gas of a galaxy cluster are Compton up-scattered by the thermal electrons, depleting the low-frequency side of the spectrum and enriching the high side — the Sunyaev–Zel'dovich effect [Sunyaev:1972]. In the Rayleigh–Jeans region the brightness change is
quoted from [Sunyaev:1972], with \(\sigma_{\mathrm{T}}\) the Thomson cross-section of Equation (65.5) and the integral running through the cluster. For a rich cluster \(y\sim10^{-4}\), giving a decrement of about \(0.5\,\mathrm{mK}\) — computed here from Equation (51.8) — which is why clusters appear as cold spots at low frequency; the distortion changes sign at about \(217\,\mathrm{GHz}\) (the null of the \(y\) spectral function, computed here as \(3.83\,k_{\mathrm{B}}T_{0}/h\)), a frequency signature no intrinsic fluctuation shares. Because \(y\) does not dilute with distance, the effect finds clusters at any redshift, and the arcminute surveys of ACT and SPT now use it as a routine cluster-finding channel [Aiola:2020].
Gravitational lensing. The anisotropy pattern is viewed through the deflecting mass of the whole line of sight, which remaps the sky by a few arcminutes coherently over degree scales: the acoustic peaks are slightly smoothed, characteristic higher-order correlations appear from which the projected mass map is reconstructed, and E-mode polarization is partly converted to the B modes detected in [Hanson:2013]. The lensing reconstruction is an integral part of the modern likelihood [Aghanim:2020], and it measures the growth of structure — the dark-sector census of The Dark Sector: Evidence Without Explanation rests partly on it.
Late-time scattering and evolving potentials. When the first stars reionized the universe, a fraction of the photons scattered again; the fitted optical depth is \(\tau=0.054\pm0.007\) [Aghanim:2020], so a fraction \(1-\ee^{-2\tau}\approx10\) per cent of the anisotropy power — computed here — is suppressed, and the re-scattering generates the large-angle E-mode “reionization bump” at \(\ell\lesssim10\) from which \(\tau\) is actually measured [Aghanim:2020]. Photons also gain or lose energy crossing potentials that evolve while being crossed — the integrated Sachs–Wolfe effect, the late-time counterpart of Equation (51.5) already identified in the original analysis [Sachs:1967] — which adds power at the lowest multipoles once dark energy begins to flatten the potentials (Evidence-Based Cosmology).
Interpretation
The measurements above are jointly fitted by a spatially flat six-parameter model — \(\Lambda\)CDM, the standard cosmology of Evidence-Based Cosmology — and the fit is both spectacularly economical and honestly incomplete.
The six-parameter fit and its principal derived quantities, from the final Planck analysis (temperature, polarization and lensing combined) [Aghanim:2020]. All values quoted.
| Parameter | Symbol | Value |
|---|---|---|
| Baryon density | $\Omega_{b}h^{2}$ | $0.02237\pm0.00015$ |
| Cold-dark-matter density | $\Omega_{c}h^{2}$ | $0.1200\pm0.0012$ |
| Acoustic scale | $100\theta_{\mathrm{MC}}$ | $1.04092\pm0.00031$ |
| Reionization optical depth | $\tau$ | $0.054\pm0.007$ |
| Scalar amplitude | $\ln(10^{10}A_{s})$ | $3.044\pm0.014$ |
| Scalar spectral index | $n_{s}$ | $0.9649\pm0.0042$ |
| Hubble constant | $H_{0}$ | \(67.4 \pm 0.5\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}\) |
| Matter fraction | $\Omega_{m}$ | $0.315\pm0.007$ |
| Age of the universe | $t_{0}$ | \(13.80 \pm 0.02\,\mathrm{Gyr}\) |
Two consistency checks can be computed here from the table alone. First, the age: for a flat universe with the quoted \(H_{0}\) and \(\Omega_{m}\), the expansion history of Evidence-Based Cosmology integrates in closed form to
in agreement with the quoted value. Second, the photon-to-baryon ratio: the quoted \(\Omega_{b}h^{2}\) corresponds to a present baryon number density of about \(0.25\,/\mathrm{m}^{3}\), and the measured \(T_{0}\) of Phenomenon 51.3 to a photon density of \(4.1\times10^{8}\) per cubic metre, so \(\eta=n_{b}/n_{\gamma}\approx6.1\times10^{-10}\) — the number the nucleosynthesis of Evidence-Based Cosmology requires independently from the light-element abundances. Two entirely different epochs of the universe, read with two entirely different instruments, return one number.
What the CMB establishes is the hot big bang itself and the skeleton of the model. The Planck spectrum of Phenomenon 51.3 establishes a hot, dense, thermalized early phase — no non-thermal explanation survives the FIRAS distortion limits. The isotropy at \(10^{-5}\) establishes the large-scale homogeneity assumed by the cosmological principle. The acoustic scale establishes spatial flatness at the few-per-mille level. The peak morphology establishes that the seeds were adiabatic and coherent, with a spectrum close to, but measurably different from, scale-invariant: \(n_{s}=0.9649\pm0.0042\) sits eight of its standard deviations below unity — computed here from the quoted values — which any account of the seeds must reproduce. And the fit requires the dark sector: \(\Omega_{c}h^{2}\) is five times \(\Omega_{b}h^{2}\), and the peaks cannot be fitted by baryons alone. What the radiation establishes is the gravitational existence of that sector, not its identity — the census and the failed identifications are The Dark Sector: Evidence Without Explanation's subject.
What the CMB does not establish must be said with equal care. There is no detection of a primordial B mode: the bound \(r<0.036\) [Ade:2021] constrains, but does not confirm, the inflationary origin of the seeds, and the mechanism that produced them remains unobserved. And the model that fits the CMB internally is in tension externally: the same \(\Lambda\)CDM parameters predict \(H_{0}=67.4 \pm 0.5\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}\), while the local distance ladder measures \(73.0 \pm 1.0\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}\) [Riess:2022] — a discrepancy of about \(5\sigma\), computed here from the two quoted uncertainties, unresolved at the time of writing and discussed further in Evidence-Based Cosmology. This book records it as an open disagreement between two mature measurements, not as a defect of either.
Modern repetitions and precision
Sixty years separate the entries of the table below, and every line was, in its year, the best measurement in the field. The monopole temperature improved from an uncertainty of \(1.0\,\mathrm{K}\) in 1965 to \(0.57\,\mathrm{mK}\) in 2009 — a factor of nearly two thousand, computed here; the anisotropy resolution improved from the \(7^\circ\) beams of DMR to the arcminute beams of the ground programme.
The precision lineage, instrument by instrument. Resolutions are the approximate beam scales of the instruments as described in Section 51.2.
| Years | Instrument | Contribution |
|---|---|---|
| 1965 | Holmdel horn [Penzias:1965] | discovery; $T$ to \(1.0\,\mathrm{K}\) |
| 1976–77 | U-2 radiometers [Smoot:1977] | the dipole |
| 1989–96 | COBE FIRAS [Mather:1994] [Fixsen:1996] | Planck spectrum to $5\times10^{-5}$ of peak |
| 1992 | COBE DMR [Smoot:1992] | anisotropy discovery at \(7^\circ\) |
| 2000 | BOOMERanG [deBernardis:2000] | first acoustic peak; flatness |
| 2002 | DASI [Kovac:2002] | first polarization detection |
| 2001–10 | WMAP [Bennett:2013] | full-sky spectra; precision $\Lambda$CDM |
| 2009–13 | Planck [Aghanim:2020] | acoustic range mapped to its cosmic-variance floor |
| 2007– | ACT, SPT [Aiola:2020] [Hanson:2013] | damping tail, lensing, SZ clusters |
| 2010– | BICEP/Keck [Ade:2015] [Ade:2021] | degree-scale B modes; $r<0.036$ |
Two closing observations belong to the record. First, the FIRAS spectral limits of 1996 [Fixsen:1996] are still unsurpassed after three decades: no flown instrument has improved on \(\abs{\mu}<9\times10^{-5}\) or \(\abs{y}<1.5\times10^{-5}\), and the proposed successor spectrometers are designed precisely against those numbers. A measurement can saturate its era's technology and stand for a generation. Second, the frontier has moved wholly to polarization: the temperature sky is measured to its cosmic-variance floor of Equation (51.3) over the acoustic range, so further temperature integration cannot help, and the open quantities — \(r\), the reionization history, the neutrino-mass imprint on lensing — are all polarization and secondary-anisotropy science [Ade:2021] [Aiola:2020] [Aghanim:2020].
Primary references
The discovery and interpretation pair [Penzias:1965][Dicke:1965]; the prediction [Alpher:1948b] and the precursor measurement [McKellar:1941]; the spectrum [Mather:1990] [Mather:1994] [Fixsen:1996] [Fixsen:2009]; the dipole [Smoot:1977] [Akrami:2020]; the anisotropies [Smoot:1992] [deBernardis:2000] [Bennett:2013] [Aghanim:2020]; the polarization [Kovac:2002] [Hanson:2013] [Ade:2014] [Ade:2015] [Ade:2021]; the theoretical predictions under test [Sachs:1967] [Silk:1968] [Peebles:1970] [Sunyaev:1970] [Sunyaev:1972].