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The Cosmic Background Radiation

We summarise the current status of cosmic microwave background spectrum and anisotropy measurements, and their theoretical interpretation. This is the update of the mini-review for the 1997 web-version of the Review of Particle Properties.


Introduction
The observed cosmic microwave background (CMB) radiation provides strong evidence for the hot big bang. The success of primordial nucleosynthesis calculations (see Sec. 16, "Big-bang nucleosynthesis") requires a cosmic background radiation (CBR) characterized by a temperature kT ∼ 1 MeV at a redshift of z ≃ 10 9 . In their pioneering work, Gamow, Alpher, and Herman [1] realized this and predicted the existence of a faint residual relic, primordial radiation, with a present temperature of a few degrees. The observed CMB is interpreted as the current manifestation of the hypothesized CBR.
The CMB was serendipitously discovered by Penzias and Wilson [2] in 1965. Its spectrum is well characterized by a 2.73 ± 0.01 K black-body (Planckian) spectrum over more than three decades in frequency (see Figs. 20.1 and 20.2). A non-interacting Planckian distribution of temperature T i at redshift z i transforms with the universal expansion to another Planckian distribution at redshift zr with temperature Tr/(1 + zr) = T i /(1 + z i ). Hence thermal equilibrium, once established (e.g. at the nucleosynthesis epoch), is preserved by the expansion, in spite of the fact that photons decoupled from matter at early times. Because there are about 10 9 photons per nucleon, the transition from the ionized primordial plasma to neutral atoms at z ∼ 1000 does not significantly alter the CBR spectrum [3].

The CMB frequency spectrum
The remarkable precision with which the CMB spectrum is fitted by a Planckian distribution provides limits on possible energy releases in the early Universe, at roughly the fractional level of 10 −4 of the CBR energy, for redshifts ∼ < 10 7 (corresponding to epochs ∼ > 1 year). The following three important classes of spectral distortions (see Fig. 20.3) generally correspond to energy releases at different epochs. The distortion results from the CBR photon interactions with a hot electron gas at temperature Te.
Compton scattering (γe → γ ′ e ′ ) of the CBR photons by a hot electron gas creates spectral distortions by transfering energy from the electrons to the photons. Compton scattering cannot achieve thermal equilibrium for y < 1, where (20.1) is the integral of the number of interactions, σ T ne(z) c dt, times the mean-fractional photon-energy change per collision [4]. For Te ≫ Tγ y is also proportional to the integral of the electron pressure nekTe along the line of sight. For standard thermal histories y < 1 for epochs later than z ≃ 10 5 .  The resulting CMB distortion is a temperature decrement ∆T RJ = −2y Tγ (20.2) in the Rayleigh-Jeans (hν/kT ≪ 1) portion of the spectrum, and a rapid rise in temperature in the Wien (hν/kT ≫ 1) region, i.e. photons are shifted from low to high frequencies. The magnitude of the distortion is related to the total energy transfer [4] ∆E by A prime candidate for producing a Comptonized spectrum is a hot intergalactic medium. A hot (Te > 10 5 K) medium in clusters of galaxies can and does produce a partially Comptonized spectrum as seen through the cluster, known as the Sunyaev-Zel'dovich effect. Based upon X-ray data, the predicted large angular scale total combined effect of the hot intracluster medium should produce y ∼ < 10 −6 [5].

20.2.2.
Bose-Einstein or chemical potential distortion: Early energy release (z ∼ 10 5 -10 7 ). After many Compton scatterings (y > 1), the photons and electrons will reach statistical (not thermodynamic) equilibrium, because Compton scattering conserves photon number. This equilibrium is described by the Bose-Einstein distribution with non-zero chemical potential: where x ≡ hν/kT and µ 0 ≃ 1.4 ∆E/E CBR , with µ 0 being the dimensionless chemical potential that is required.
The collisions of electrons with nuclei in the plasma produce free-free (thermal bremsstrahlung) radiation: eZ → eZγ. Freefree emission thermalizes the spectrum to the plasma temperature at long wavelengths. Including this effect, the chemical potential becomes frequency-dependent, (20.5) where x b is the transition frequency at which Compton scattering of photons to higher frequencies is balanced by free-free creation of new photons. The resulting spectrum has a sharp drop in brightness temperature at centimeter wavelengths [6]. The minimum wavelength is determined by Ω B . The equilibrium Bose-Einstein distribution results from the oldest non-equilibrium processes (10 5 < z < 10 7 ), such as the decay of relic particles or primordial inhomogeneities. Note that free-free emission (thermal bremsstrahlung) and radiative-Compton scattering effectively erase any distortions [7] to a Planckian spectrum for epochs earlier than z ∼ 10 7 .

20.2.3.
Free-free distortion: Very late energy release (z ≪ 10 3 ). Free-free emission can create rather than erase spectral distortion in the late universe, for recent reionization (z < 10 3 ) and from a warm intergalactic medium. The distortion arises because of the lack of Comptonization at recent epochs. The effect on the present-day CMB spectrum is described by where Tγ is the undistorted photon temperature, x is the dimensionless frequency, and Y ff /x 2 is the optical depth to free-free emission:

7)
Here h is Planck's constant, ne is the electron density and g is the Gaunt factor [8].

Spectrum summary:
The CMB spectrum is consistent with a blackbody spectrum over more than three decades of frequency around the peak. A least-squares fit to all CMB measurements yields: The limits here [9] correspond to limits [11][12][13] on energetic processes ∆E/E CBR < 2×10 −4 occurring between redshifts 10 3 and 5 × 10 6 (see Fig. 20.4). The best-fit temperature from the COBE FIRAS experiment is Tγ = 2.728 ± 0.002 K [11].

Deviations from isotropy
Penzias and Wilson reported that the CMB was isotropic and unpolarized to the 10% level. Current observations show that the CMB is unpolarized at the 10 −5 level but has a dipole anisotropy at the 10 −3 level, with smaller-scale anisotropies at the 10 −5 level. Standard theories predict anisotropies in linear polarization well below currently achievable levels, but temperature anisotropies of roughly the amplitude now being detected.
It is customary to express the CMB temperature anisotropies on the sky in a spherical harmonic expansion, 8) and to discuss the various multipole amplitudes. The power at a given angular scale is roughly ℓ m |a ℓm | 2 /4π, with ℓ ∼ 1/θ.

The dipole:
The largest anisotropy is in the ℓ = 1 (dipole) first spherical harmonic, with amplitude at the level of ∆T /T = 1.23 × 10 −3 . The dipole is interpreted as the result of the Doppler shift caused by the solar system motion relative to the nearly isotropic blackbody field. The motion of the observer (receiver) with velocity β = v/c relative to an isotropic Planckian radiation field of temperature T 0 produces a Doppler-shifted temperature The implied velocity [14] for the solar-system barycenter is β = 0.001236 ± 0.000002 (68% CL) or v = 371 ± 0.5 km s −1 , assuming a value T 0 = 2.728 ± 0.002 K, towards (α, δ) = (11.20 h ± 0.01 h , −7.22 • ± 0.08 • ), or (ℓ, b) = (264.31 • ± 0.17 • , 48.05 • ± 0.10 • ). Such a solar-system velocity implies a velocity for the Galaxy and the Local Group of galaxies relative to the CMB. The derived ve- where most of the error comes from uncertainty in the velocity of the solar system relative to the Local Group.
The Doppler effect of this velocity and of the velocity of the Earth around the Sun, as well as any velocity of the receiver relative to the Earth, is normally removed for the purposes of CMB anisotropy study. The resulting high degree of CMB isotropy is the strongest evidence for the validity of the Robertson-Walker metric. The rms quadrupole anisotropy amplitude is defined through Q 2 rms /T 2 γ = m |a 2m | 2 /4π. The current estimate of its value is 4 µK ≤ Qrms ≤ 28 µK for a 95% confidence interval [16]. The uncertainty here includes both statistical errors and systematic errors, which are dominated by the effects of galactic emission modelling. This level of quadrupole anisotropy allows one to set general limits on anisotropic expansion, shear, and vorticity; all such dimensionless quantities are constrained to be less than about 10 −5 . For specific homogeneous cosmologies, fits to the whole anisotropy pattern allow stringent limits to be placed on, for example, the global rotation at the level of about 10 −7 of the expansion rate [17].

Smaller angular scales:
The COBE-discovered [18] higher-order (ℓ > 2) anisotropy is interpreted as being the result of perturbations in the energy density of the early Universe, manifesting themselves at the epoch of the CMB's last scattering. Hence the detection of these anisotropies has provided evidence for the existence of primordial density perturbations which grew through gravitational instability to form all the structure we observe today. In the standard scenario the last scattering takes place at a redshift of approximately 1100, at which epoch the large number of photons was no longer able to keep the hydrogen sufficiently ionized. The optical thickness of the cosmic photosphere is roughly ∆z ∼ 100 or about 5 arcminutes, so that features smaller than this size are damped.
Anisotropies are observed on angular scales larger than this damping scale (see Fig. 20.5 and 20.6), and are consistent with those expected from an initially scale-invariant power spectrum (flat = independent of scale) of potential and thus metric fluctuations. It is believed that the large scale structure in the Universe developed through the process of gravitational instability, where small primordial perturbations in energy density were amplified by gravity over the course of time. The initial spectrum of density perturbations can evolve significantly in the epoch z > 1100 for causally connected regions (angles ∼ < 1 • Ω 1/2 tot ). The primary mode of evolution is through adiabatic (acoustic) oscillations, leading to a series of peaks that encode information about the perturbations and geometry of the Universe, as well as information on Ω 0 , Ω B , Ω Λ (cosmological constant), and H 0 [18]. The location of the first acoustic peak is predicted to be at ℓ ∼ 220 Ω −1/2 tot or θ ∼ 0.3 • Ω 1/2 tot and its amplitude increases is a calculable function of the parameters.
Theoretical models generally predict a power spectrum in spherical harmonic amplitudes, since the models lead to primordial fluctuations and thus a ℓm that are Gaussian random fields, and hence the power spectrum in ℓ is sufficient to characterize the results. The power at each ℓ is (2ℓ + 1)C ℓ /(4π), where C ℓ ≡ |a ℓm | 2 and a statistically isotropic sky means that all ms are equivalent. For an idealized full-sky observation, the variance of each measured C ℓ is [2/(2ℓ + 1)]C 2 ℓ . This sampling variance (known as cosmic variance) comes about because each C ℓ is chi-squared distributed with (2ℓ + 1) degrees of freedom for our observable volume of the Universe [19]. Thomson scattering of the anisotropic radiation field also generates linear polarization at the roughly 5% level [20]. Although difficult to detect, the polarization signal should act as a strong confirmation of the general paradigm. Fig. 20.7 shows the theoretically predicted anisotropy power spectrum for a sample of models, plotted as ℓ(ℓ + 1)C ℓ versus ℓ which is the power per logarithmic interval in ℓ or, equivalently, the two-dimensional power spectrum. If the initial power spectrum of perturbations is the result of quantum mechanical fluctuations produced and amplified during inflation, then the shape of the anisotropy spectrum is coupled to the ratio of contributions from density (scalar) and gravity wave (tensor) perturbations [21]. If the energy scale of inflation at the appropriate epoch is at the level of ≃ 10 16 GeV, then detection of the effect of gravitons is possible, as well as partial reconstruction of the inflaton potential. If the energy scale is ∼ < 10 14 GeV, then density fluctuations dominate and less constraint is possible. Figure 20.7: Examples of theoretically predicted ℓ(ℓ + 1)C ℓ or CMB anisotropy power spectra [22]. The plot indicates that precise measurements of the CMB anisotropy power spectrum could distinguish between models which are currently favored from galaxy clustering and other considerations. The textures model is from Ref. [23].
Fits to data over smaller angular scales are often quoted as the expected value of the quadrupole Q for some specific theory, e.g. a model with power-law initial conditions (primordial density perturbation power spectrum P (k) ∝ k n ). The full 4-year COBE DMR data give Q = 15.3 +3.7 −2.8 µK, after projecting out the slope dependence, while the best-fit slope is n = 1.2 ± 0.3, and for a pure n = 1 (scale-invariant potential perturbation) spectrum Q (n = 1) = 18 ± 1.6 µK [15,24]. The conventional notation is such that Q 2 /T 2 γ = 5C 2 /4π, and an alternative convention is to plot the "band-power" ℓ(2ℓ + 1)C ℓ /4π). The fluctuations measured by other experiments can also be quoted in terms of Q flat , the equivalent value of the quadrupole for a flat (n = 1) spectrum, as presented in Fig. 20.5.
It now seems clear that there is more power at sub-degree scales than at COBE scales (see Fig. 20.5), which provides some modeldependent information on cosmological parameters [18,25], for example Ω B . In terms of such parameters, fits to the COBE data alone yield Ω 0 > 0.34 at 95% CL [26] and Ω tot < 1.5 also at 95% CL [27], for inflationary models. Only somewhat weak conclusions can be drawn based on the current smaller angular scale data (see Fig. 20.5). A sample preliminary fit [28] finds Ω 0 h 1/2 ≃ 0.55±0.10 (68% CL).
However, new data are being acquired at an increasing rate, with a large number of improved ground-and balloon-based experiments being developed. It appears that we are not far from being able to distinguish crudely between currently favored models, and to begin a more precise determination of cosmological parameters. A vigorous suborbital and interferometric program could map out the CMB anisotropy power spectrum to about 10% accuracy and determine several parameters at the 10 to 20% level in the next few years.
There are also now two approved satellite missions: the NASA Microwave Anisotropy Probe (MAP), scheduled for launch in 2000; snd the ESA Planck Surveyor, expected for around 2004. The improved sensitivity, freedom from earth-based systematics and all-sky coverage allow a simultaneous determination of many of the cosmological parameters to unprecedented precision: for example, Ω 0 and n to about 1%, Ω B and H 0 at the level of a few per cent [29].
Furthermore, detailed measurement of the polarization signal provides more precise information on the physical parameters. In particular it allows a clear distinction of any gravity wave contribution, which is crucial to probing the ∼ 10 16 GeV energy range. The fulfillment of this promise may await an even more sensitive generation of satellites.