# Cosmic Microwave Background Radiation

Prediction and Discovery

The universe is filled with blackbody radiation whose temperature now is $T_0 = 2.725 \pm 0.002 {\rm ~K}$, so the frequency of the peak brightness is $\nu_{\rm max} \approx 160 {\rm ~GHz}$.  This cosmic microwave background (CMB) is a relict of the "big bang" creation of the universe and reveals precise values for a host of cosmological parameters.  Before the CMB was first observed, George Gamow and his students calculated the hydrogen and helium abundances produced in the presence of blackbody radiation when the universe was very hot (Alpher, R., Bethe, H., & Gamow, G. 1948, Phys. Rev. 73, 803), and Alpher, R. A., & Herman, R. 1948, Nature, 162, 774 contains the first estimate $T_0 \approx 5 {\rm ~K}$ for the present temperature of the universe.

The energy density of the early universe was dominated by hot blackbody radiation. By $t \approx 1 {\rm ~s}$ the temperature fell to $T \approx 10^{10} {\rm ~K}$ and neutrons decayed into protons and electrons or combined with protons to make deuterium nuclei, most of which combined to make helium nuclei during the first three minutes. The chemical composition of the early universe is a function of the photon/baryon ratio.  Baryonic matter now accounts for about 4.6% of the energy density of the universe.   Image credit

Although $5 {\rm K}$ is a very strong signal by radio astronomy standards, the CMB is not easy to detect because it is very nearly isotropic.  Unlike a compact radio source, the CMB cannot be detected by a differential measurement on two nearby regions of sky; it is difficult to distinguish from other noise sources in the receiver and the radio telescope.  It was accidentally discovered by Arno Penzias and Robert Wilson (Penzias, A. A., & Wilson, R. W. 1965, ApJ, 142, L419) at Bell Labs.  They reported careful measurements indicating an isotropic "excess antenna temperature" produced by their horn antenna at 4080 MHz.  Subsequent measurements, especially by the COBE (COsmic Background Explorer) satellite, have demonstrated that this "excess noise" matches a blackbody spectrum within 50 parts per million well into the far infared.  The largest (but still very small) anisotropies are (1) a dipole anistropy caused by the Earth's motion relative to the CMB and (2) contamination by sources in the galactic plane.

The observed CMB spectrum is a very close match to a $T = 2.725$ K blackbody. Image credit

The CMB is nearly isotropic. This COBE all-sky image in galactic coordinates spans only the temperature range $2.725 \pm 0.004$ K.  The dipole anisotropy is caused by the Earth's 600 km s$^{-1}$ motion, and some foreground emission in the galactic plane is visible. Image credit

Evolution

How does the blackbody CMB evolve as the universe expands?  The universe is spatially homogenous and isotropic, so the properties of the CMB can be calculated by considering what happens to blackbody radiation in a small cube whose side length $a(t)$ slowly grows with time $t$ since the big bang.  "Small" here means only much smaller than the Hubble distance $c / H_0 \approx 3.00 \times 10^5 {\rm ~km~s}^{-1} / 73 {\rm ~km~s}^{-1} {\rm ~Mpc}^{-1} \approx 4.1 \times 10^3 {\rm ~Mpc}$ so that relativistic effects can be ignored.  Although our imaginary cube has no walls, CMB radiation escaping from the cube is balanced by CMB radiation entering the cube from similar adjacent cubes.  Consequently the energy density and spectrum of CMB radiation within the cube is exactly the same as that in an adiabatically expanding cubical cavity having opaque walls.  The radiation remains in equilibrium during the slow expansion, so it remains blackbody radiation, but its temperature changes.  Blackbody radiation in a slowly expanding cube can be described by classical thermodynamics.  The first law of thermodynamics requires energy conservation as the total heat $Q$, internal energy $U$, and work $W$ done by the radiation in the cube change as the volume $V = a^3$ of the cube grows by a small amount $dV$: $$dQ = dU + dW = dU + p dV~,$$ where $p$ is the pressure exerted by the radiation.

Radiation exerts pressure because photons of frequency $\nu$ have momentum $h \nu / c$ as well as energy $h \nu$.  Consider a small flat mirror of unit area on the inside surface of the cavity.  According to Kirchoff's law, the mirror emits no radiation, so only reflections can cause radiation pressure.  Each cavity photon approaching the mirror at an angle $\theta$ from the normal to the mirror will be reflected at the same angle and transfer $2 h \nu \cos\theta$ of momentum to the mirror.  The radiation pressure equals the rate of momentum transfer per unit area.  Blackbody radiation is isotropic so the radiation pressure $p_\nu$ produced by photons of frequency $\nu$ is $$p_\nu = {2 \over c} \int B_\nu \cos^2 \theta d \Omega ,$$ where the second $\cos \theta$ factor comes from the projected area of the mirror, as seen by the incoming photon.  The total radiation pressure is $$p = \int_0^\infty p_\nu d \nu = {2 \over c} \int_0^\infty B_\nu d\nu \int_0^{2 \pi} d \phi \int_0^{\pi/2} \cos^2 \theta \sin \theta d\theta = {4 \pi B \over c} \int_0^1 x^2 dx$$ Substituting the radiation energy density $u = 4 \pi B / c$ and integrating relates the radiation pressure and energy density of isotropic radiation: $$p = {u \over 3}~.$$

Reflection of a photon by a mirror on the cavity wall.

The second law of thermodynamics relates the change in entropy $S$ to the change in heat $Q$ and the temperature $T$: $$dS = {dQ \over T}~.$$ Using the first law to eliminate $dQ$ and converting from pressure $p$ to energy density $u$ gives $$dS = {dU \over T} + p {dV \over T} = {d (uV) \over T} + {u \over 3} {dV \over T} = {u dV \over T} + {V du \over T} + {u dV \over 3 T}$$ $$dS = {4 u \over 3 T}dV + {V \over T} {du \over dT} dT$$ Setting $dS = 0$ for an adiabatic expansion and substituting $u = 4 \sigma T^4 / c$ yields $${dV \over V} = - 3{d T \over T}~.$$ Inserting the volume $V = a^3$ for a cube of side length $a$  gives $dT / T = - da /a$ so we find that the CMB temperature $T(t)$ is inversely proportional to the scale size of the universe: $$T(t) \propto a(t)^{-1}~.$$ The total number of photons in the expanding cube is unchanged, but the wavelength $\lambda$ of each photon grows linearly with the scale size $a(t)$.  Astronomers frequently use the term redshift defined by $$z \equiv {\lambda_{\rm o} - \lambda_{\rm e} \over \lambda_{\rm e}}= {\lambda_{\rm o} \over \lambda_{\rm e}} -1 ~,$$ where $\lambda_{\rm e}$ is wavelength emitted by a source at redshift $z$ and $\lambda_{\rm o}$ is the observed wavelength at $z = 0$.  The redshift $z$ and scale size $a$ are related by $$(1 + z) = a^{-1}~.$$ Thus the temperature of the CMB at any redshift $z$ is $$T = T_0 (1 + z)~,$$ where $T_0 \approx 2.725 {\rm ~K}$.

Prior to the time corresponding to the redshift $z_\star = 1091 \pm 1$ the temperature was $T > 3000 {\rm ~K}$, high enough to ionize the hydrogen atoms filling the universe and make the universe opaque to the CMB.  This "wall" beyond which we cannot see is called the surface of last scattering.  At the so-called recombination era when the age of the universe was $t_\star = 379 \pm 5 \times 10^3 {\rm ~yr}$, almost all of the free protons and electrons combined to form neutral hydrogen atoms and the universe became transparent to the CMB.  The CMB photons that we see today were last scattered at that time and have been traveling in straight lines ever since, so an image of the CMB shows its temperature distribution at $z_\star \approx 1091$.

A brief history of the universe. Image credit

Instrinsic Anisotropy

The time $t_\star$ can be calculated because the physics of the homogeneous and isotropic early universe was so simple.  During that time, small inhomogeneities in the density of dark matter were amplified by gravity.  Baryons tended to fall into the gravitational potential wells, and the CMB was coupled to this flow by Thomson scattering off the free electrons prior to $t_\star$.  Radiation pressure resisted this infall, setting up baryon acoustic oscillations analogous the sound vibrations in air, but with a sound speed $c_{\rm s}\approx c / \sqrt 3$. The strongest oscillations in the CMB brightess at recombination have wavelengths $$\lambda_\star \approx 2 c_{\rm s} t_\star \approx {2 c t_\star \over \sqrt 3} \approx {2 * 379 \times 10^3 \over \sqrt 3} {\rm ~ly} \approx 438 \times 10^3 {\rm ~ly} ~,$$ which is about $134 {\rm ~kpc}$.  After recombination, the photons decoupled and this "standard ruler" grew by a factor $1 + z_\star \approx 1092$ to its present size of $146 {\rm ~Mpc}$.  The power spectrum of CMB fluctuations observed by WMAP reveal that the angular size of this ruler is $\theta_\star = 0.010388 \pm 0.000027 {\rm ~rad}$, so the angular-size distance to the surface of last scattering is $D_{\rm A} = \lambda_\star / \theta_\star \approx 14.1 \pm 0.16 {\rm ~Gpc}$.  The largest extragalactic distance is actually the best measured!  If we live in a flat $\Lambda$CDM universe, its present age is $t = 13.75 \pm 0.13 {\rm ~Gyr}$.

The WMAP 7-year total-intensity image of the CMB after subtraction of the dipole anisotropy and the radio-source foreground. The intensity range is only $\pm 200\,\mu$K centered on the mean brightness $T_0 = 2.725$ K. Image credit