BIG BANG

The starting point for the Big Bang theory is Einstein's theory of general relativity in combination with the cosmological principle. General relativity is a metric theory of gravity, now verified to high precision via observations of the binary pulsar. The cosmological principle has also been verified to exquisite precision, as far as this can be achieved.

The cosmological principle asserts that the universe is statistically isotropic and homogeneous for any observer. The cosmic microwave background has demonstrated isotropy to a level of order 1 part in 100,000. There is no preferred direction such as might be associated with the geometrical center of the universe. Homogeneity has been demonstrated by galaxy redshift surveys that provide three-dimensional maps of the universe, given the strong empirical correlation discovered by Edwin Hubble in 1929 between redshift and distance. As one probes deeper and deeper into the universe, to distances as great as several gigaparsecs (1 parsec = 3.2 light-years), the density of galaxies is found to be uniform. Humankind definitely does not inhabit a fractal universe of vanishingly low density in the mean, as some have argued. One can set a limit on any large-scale nonuniformities of approximately 10 percent; otherwise, excessive perturbations would be induced in the Hubble diagram of galaxy redshift versus distance.

Friedmann-Lemaître Cosmology

The cosmological principle applied to the Einstein gravitational field equations led to a remarkable simplification. In 1917 Einstein found a static cosmological model that could only be prevented from collapsing under the relentless tug of gravity by invoking a repulsion force. This force was enshrined as the cosmological constant, a term that has no counterpart and no effect in Newtonian gravity, but is important only on cosmological scales. In modern parlance, one identifies the cosmological constant with the energy of the vacuum, and its introduction leads to the Einstein static universe.

In fact, Einstein had overlooked the only true cosmological solution to the field equations that satisfied the cosmological principle and that did not require the introduction of a cosmological constant. His mistake was soon rectified by Alexander Friedmann in 1924 and independently by Georges Lemaître in 1927, who discovered the expanding universe cosmological models.

The expanding universe was much later dubbed the Big Bang for the simple reason that it expanded from a pointlike singularity of infinite density. It was realized at the outset that this singularity was a mathematical artifact indicative of missing physics that was only supplied half a century later. Space itself was uniform, unbounded, and expanding. There was no center and no edge to space.

The Big Bang theory indeed predicted the expansion of the universe, a result that many scientists in the early decades of the twentieth century, including Hubble himself, found too radical to accept. The equations that describe the evolution of the universe come from Einstein's theory of general relativity. To describe expansion, one introduces the scale factor a (_t). Physical distance is d = ra (t ), where r is coordinate or comoving distance, just a fixed number conventionally evaluated with respect to a mass-scale at present. The Einstein equations are G_?v = 8?GT_?v + ?g_?v. Here, g_?? is the metric of the universe, which is incorporated into the Einstein tensor G_?? to describe gravity, and T_?? is the energy-momentum tensor that describes the matter and radiation content of the universe that acts as the source of gravity. Another important source of gravity that corresponds to the density of the vacuum is the cosmological constant ?.

Under the cosmological principle, equivalent to spherical symmetry about every point, the Friedmann-Lemaître equation of cosmology is obtained. This can be cast in the form of a cosmic energy equation: The first term on the left describes the kinetic energy of a shell of matter, and the second term is its gravitational potential energy. There are three distinct solutions in the absence of the cosmological constant term. These are conveniently described by the curvature of space that in the Newtonian limit corresponds to the total energy of an arbitrarily placed expanding shell of matter. The shell may have either zero total energy, in which case space is flat; negative energy, in which case space is positively curved like a spheroidal surface; or positive energy, which results in space being negatively curved like the surface of a hyperboloid. The constant -k represents the total energy of the shell per unit mass and may be -1, 0, or -1, corresponding to a universe of negative, zero, or positive energy. The flat and negatively curved spaces are infinite, and only the positively curved space is finite.

One can discriminate between the three solutions of the Friedmann-Lemaître equation by introducing the critical density. This is the density of the flat or Einstein–de Sitter universe and is equal to Here, H₀ is Hubble's constant. Because of the matter content, a universe without a cosmological constant is decelerating. At early times, the three spaces are indistinguishable. Only at late times do they deviate from one another, the negative-energy spatially closed model decelerating more strongly than the other models before reaching its maximum extent and then recollapsing to a future singularity.

To further examine the deceleration, one may apply conservation of mass-energy that leads to and an equation can now be derived for the deceleration of the universe:

From the deceleration equation, one learns that the universe actually accelerates if a negative energy condition is satisfied, ? + 3p ? 0. Indeed, the cosmological constant satisfies ? + 3p =0, and the solution is (t ) ? exp In this solution, the density is constant, equal to ?^? ? ?/H² where ?_? is the vacuum density corresponding to the cosmological constant relative to that of the k = 0 universe with density and H₀ is Hubble's constant, at present.

The Friedmann-Lemaître equation today reduces to ?^tot ? ?^mat + ?^rad + ?^? - 1 = k /a²H² where the mass density of matter ?_mat ?a^-3 and radiation ?_rada^-4. Three possibilities for the pressure content of the universe are p = p_m « ?c², applicable since the epoch of matter-radiation decoupling; p = ?_radc²/3, applicable prior to matterradiation decoupling; and p =w ?, a generalization of the cosmological constant to an arbitrary equation of state. In this latter case, ? ? a^{-3(1 + w )}, with w = -1 corresponding to the case of the cosmological constant. The universe is radiation-dominated prior to a /a₀ = ?_rad/?_mat or at redshift larger than 1 + z^eq = a⁰/a (t^eq) = 3,000(?^mat/0.3).

The Distance Scale

Lemaître formulated one of the greatest predictions of modern physics, that the universe should be expanding, into a relation that expressed the proportionality between the recession velocity of a distant galaxy and its distance. In 1929 Hubble verified the redshift-distance relation, which became known as Hubble's law, ? = H₀d , where ? is recession velocity and d is luminosity distance. The latter is measured by identifying a class of luminous variable stars, Cepheids, that were used to establish the size of the Milky Way galaxy and, more recently, the distances to its nearest neighbor galaxies. Hubble used the brightest stars in more distant galaxies as his basic distance indicators. He explored a region that extends to the Virgo cluster of galaxies. With hindsight, one knows that Hubble's distance indicators were erroneous, since he could not distinguish HII regions from stars. It is also known that the region between the Earth and the Virgo cluster, where Hubble's galaxies were located, is dominated by random motions. The uniformity of the universe only becomes manifest beyond Virgo. Nevertheless, Hubble in 1929 announced his discovery of the redshift-distance law. The redshift was produced by the Doppler effect and resulted in a systematic displacement toward longer wavelengths for a receding galaxy. Blueshifts would be indicative of approach; only a few of the nearest galaxies have blue-shifted spectra.

The prevalence of galaxy redshifts had been discovered in the first decades of the twentieth century by Vesto Slipher. The fainter the galaxy, on the average, the larger its redshift. However, the observers who tried to understand the relation between distance and redshifts paid too much attention to the theoretical cosmologists, who only knew about the possibility of redshifts in the de Sitter universe. The de Sitter cosmological model was a strange beast. It was an empty universe in which the distance depended exponentially on redshift. The nearby galaxies in this model displayed a quadratic dependence of distance on redshift. To his credit, Hubble did not care a great deal about theory. He reevaluated distances more precisely than his predecessors had done and inferred the linear relationship that is known as Hubble's law. To his dying day, Hubble could not accept that the universe was expanding, despite the prediction of Lemaître and Friedmann before him. Within a year of Hubble's announcement, most of the cosmological community seized on Hubble's law to infer that space was expanding.

It is difficult in retrospect to understand how Hubble inferred a linear law, given the enormous uncertainties in galaxy distances and the fact that Hubble only initially sampled such a small volume of the universe. Hubble's constant is measured in units of velocity per unit distance, in effect an inverse time. Hubble inferred a value of 600 km/s/Mpc. The modern value of H₀ is smaller by an order of magnitude, amounting to 70 km/s/Mpc with an uncertainty of about 15 percent. The inferred timescale 1/H₀ is a measure of the age of the universe if no deceleration (or acceleration) has occurred. The age inferred from Hubble's measurement was approximately 1.5 billion years and far less than the known age of the Earth. Hence, many astronomers were at first reluctant to accept the expanding universe interpretation.

What changed? First, the cosmologists were very ingenious. Under the influence of Lemaître, the cosmological constant, first introduced by Einstein to make the universe static, was reintroduced. Eddington and Lemaître advocated a universe that began from a static phase that would last as long as necessary before beginning to expand. Lemaître showed that galaxies could form in such a universe. A variant was an expanding universe that underwent an extended coasting phase as a consequence of the effect of the cosmological constant, with expansion eventually taking over. Such approaches greatly extended the age of the universe.

Most significantly, however, the observers revised the distance scale. This came about in part by recognition of Hubble's significant error in confusing the brightest stars with giant HII regions. Alan Sandage from 1960 onward was primarily responsible for developing a new distance calibrator that made use of the brightest galaxies in clusters as standard candles. This enabled him to probe the universe to great distances and to reduce Hubble's constant to 200 km/s/Mpc. A major breakthrough occurred when Walter Baade recognized that there are two types of Cepheid variable stars. The confusion between the two types only dissipated when Baade succeeded in identifying populations I and II in the Andromeda galaxy and realized that there were two types of Cepheids which differed appreciably in luminosity. He was able to double the distance scale. The remaining improvements happened more slowly. For nearly 40 years, cosmologists debated Hubble's constant within the range of 50 to 100 km/s/Mpc. Resolution came when the Hubble Space Telescope was able to resolve Cepheids in several galaxies outside our Local Group, in which supernovae were also found. The supernovae are of a type associated with the merger of a pair of white dwarfs that explode catastrophically once the Chandrasekhar mass limit on a white dwarf is exceeded. These SNeIa are found in old stellar populations in both spiral and elliptical galaxies and are luminous enough to be detectable at the edge of the observable universe and also to be reliable distance indicators. Type Ia supernovae seem to have identical luminosities, amounting to the light from a billion suns at maximum light and fading away after a year. The light curve is interpreted as resulting from the radioactive decay of 0.6 solar mass of Ni⁵⁶ produced in core collapse and provides the energy source for an ideal standard candle.

The Age of the Universe

Type Ia supernovae have been detected out to a look-back time of half the present age of the universe, from when light was redshifted by a factor of 2 in wavelength. The distance measurements are precise enough (to 15%) that acceleration of the universe has now been confirmed. Deviations from Hubble's linear law are found for the most distant supernovae. The measured age of the universe, inferred from Hubble's constant and the measured acceleration, is 15 billion years.

There are two completely independent measures of the age of the universe. Radioactive dating via thorium and uranium isotope measurements is applied to the abundances in old halo stars. Both thorium²³²with a half-life of 14 Gyr and uranium²³⁸ with a half-life of 4.5 Gyr have been detected in two halo stars, measured with the world's largest telescopes. Nuclear astrophysics theory provides an estimate of the initial abundances relative to iron. The observed ratio provides an estimate of the age of the universe since the supernova synthesized these elements and ejected them in the debris that eventually was incorporated into the molecular clouds from which stars such as the Sun formed.

Another age determination comes from application of the theory of stellar evolution to globular clusters. Globular star clusters are systems of millions of stars that predate our galaxy. One knows they are old because the abundances of metals as measured in stellar spectra are low compared to those in the Sun. Hence, the globular clusters must have formed long before the Sun. As stars radiate energy by thermonuclear burning of hydrogen into helium, they evolve in luminosity, becoming brighter as the fossil fuel is gradually exhausted and the central temperature rises. Heavier elements are burnt, first helium, then carbon, to provide the central temperature and pressure. Once the nuclear fuel supply of hydrogen, helium, and carbon is exhausted, the star soon runs out of fuel.

If the star initially weighed less than 8 solar masses, its final fate as the core heats up is that its envelope swells. The star becomes a luminous supergiant. The outer shell is expelled to become visible as a planetary nebula. The ejecta slowly, after some 10⁴ years, fade away, and a white dwarf is all that remains. If the star initially weighs more than 8 solar masses, its central pressure builds up to a level that the star core implodes via neutron capture and neutrino emission. A neutron star forms in the core, and the release of binding energy drives a supernova explosion of Type II.

In a globular cluster, the stars formed coevally, and so one has a snapshot of stars of different masses that have reached differing evolutionary points. One can thereby infer the age of the globular cluster from comparison with models of stellar evolution, the best estimate being 13 billion years. To this must be added about 1 billion years for the delay between the Big Bang and the formation of the globular cluster, to give an age for the universe of some 14 billion years. Remarkably, this agrees well with the independently determined ages from cosmology and uranium or thorium decays.

Cosmic Acceleration and Dark Energy

What causes the acceleration? Cosmologists have gone full circle, ending up with a value of the cosmological constant about 30 percent smaller than Einstein originally introduced for the static universe. One can interpret , the cosmological constant, as a constant energy density of the vacuum that has only recently begun to dominate the mass density of the universe. One does not observe any such energy directly. Hence, it is often referred to as dark energy. The matter density decreases as the universe expands. When the universe was about one-quarter of its present size, at redshift 4, the dark energy first became comparable to the matter density. One consequence is that the universe switched from deceleration under the influence of the gravitational attraction of matter to acceleration under the influence of the gravitational repulsion of the dark energy. The universe began to accelerate.

Dark energy produces acceleration because it has a large negative pressure, indeed p = -?c², where p is the dark energy pressure, and ? is the dark energy density. In a normal gas, pressure is positive, and Einstein's theory of relativity predicts that its contribution to gravity is attractive. Ordinary gas pressure acts as a source of gravity.

The ultimate fate of black hole formation by a collapsing massive star cannot be avoided by the action of gas pressure; in fact, it is enhanced. In the expanding universe, positive pressure produces deceleration, as does matter. As the universe expands, ordinary pressure does less and less work and produces less and less heat energy. However, negative pressure has the opposite effect. An elastic string when expanded gains energy. More energy means that the pressure of an elastic string is negative. In the expanding universe, negative pressure accordingly acts in the opposite way to positive pressure: more and more work is done as the universe expands. This is what drives acceleration. Negative pressure acts like antigravity: it is repulsive.

Dark energy accounts for two-thirds of the mass-energy density of the universe. There is no explanation for dark energy; it can simply be regarded as a contribution to the energy of the vacuum. Dark energy is completely uniform and does not cluster under the effect of gravity as does ordinary matter. It is only detectable via its effect on the acceleration of the expansion of the universe. In terms of fundamental units, the energy density associated with the cosmological constant is remarkably small, amounting to 10^-121, where the Planck mass m_pl is 1.2 × 10¹⁹ GeV. In conventional units, where the cosmological constant is an inverse square length, its magnitude is naturally the inverse Hubble length squared, or 10^-56cm^-2 or 10^-121.

Dark Matter

Dark matter, in contrast, is detectable. And it amounts to about a third of the total mass-energy density of the universe. The cosmic mass budget is best expressed with respect to the critical density for a universe that is spatially flat, the Einstein–de Sitter model, namely This can be expressed as 3 × 10¹¹h M_? Mpc^-3. The luminosity density of the universe is measured to be 2 × 10⁸h L_? Mpc^-3. The mass-to-light ratio for closure is therefore 1,500h M_?/L_?. This is a clear prediction for closure of the universe.

What is actually measured is far less. Galaxy clusters gave the first indication of the prevalence of dark matter on large scales as early as 1933. The first reliable values, however, came from galaxy rotation curves, which provided proof of dark matter dominance in ordinary galaxies and, in particular, in the Milky Way galaxy. The rotation curves for large spiral galaxies are generally flat at large distances, indicating that far from the Keplerian expectation, if mass traces light, the mass in fact increases with increasing galactocentric radius, M (<r ) ? r . Typical values of the mass-to-light ratio are 100h M_?/L_?, whereas within the half-light radius, one finds a value of approximately 10h M_?/L_?, the actual value depending slightly on the type of galaxy. Galaxy rotation curves are measured at low resolution via radio techniques using the 21 cm of atomic H and at high resolution in the optical band by H? emission lines. Consistent results are obtained, and dark matter is found to be ubiquitous on scales of up to 100 kpc.

In galaxy clusters, great progress has been made since the early determinations that used the virial theorem applied to the optically measured dispersion of radial velocities of cluster galaxies. Two independent techniques confirm the dynamical measurements. One utilizes X-ray measurements of the hot intracluster gas that is assumed to be in hydrostatic equilibrium, and another makes use of the gravitational lensing by the cluster of remote background galaxies and the consequent image distortions. All three methods consistently yield a value of 300h M_?/L_?. The scale probed is 1 Mpc.

On larger scales, there are no equilibrium gravitationally bound structures that can be reliably probed. One method utilizes the infall motion of galaxies into the Virgo Supercluster. This probes the dark matter density on scales of up to 20 Mpc. Another probe of the dark matter density on even larger scales, up to 100 Mpc, makes use of the variance in the counts of galaxies obtained in large-scale galaxy redshift surveys. The clustering of the galaxy distribution on large scales is measured by fluctuations in the galaxy counts averaged over randomly placed spheres. The matter on large enough scales must be correlated with the light. The fluctuations inferred in the matter density provide a gravitational source that induces perturbations in the Hubble flow, observable as random motions of galaxies and of galaxy clusters. The observed Hubble flow dispersion requires a value of the mass-to-light ratio that is equivalent to ?_m ? 0.3, in agreement with the mass-to-light ratio inferred for galaxy clusters. Were the universe at critical density, much larger Hubble flow distortions and galaxy peculiar velocities and cluster streaming motions would be observed amounting to 1,000 or more km/s. The observed random motions of galaxies amount to approximately 300 km/s. This method probes the dark matter out to 100 Mpc.

Similar conclusions are reached from studies of the peculiar velocity pattern of galaxies in the Virgo Supercluster, from large-area weak lensing of high redshift galaxies, and from the redshift evolution of the number density of clusters. The rich cluster abundance above a given mass is observed to only increase slowly as the universe expands. The theory of cluster formation predicts a rapid increase of the massive cluster abundance in a critical density universe, due to the growth of density fluctuations driven by gravitational instability, and this effect is systematically suppressed if the density of the universe is below the critical value.

Cosmic Blackbody Radiation

Only about 10 percent of the dark matter in the universe is baryonic. Nucleosynthesis of the light elements was predicted by George Gamow and his collaborators in the 1940s. This necessitated a hot origin to the universe and led, in turn, to the prediction of the cosmic radiation background by Ralph Alpher, George Gamow, and Robert Herman in 1950. The blackbody, and hence microwave nature, of the cosmological background radiation was first appreciated by Andrei Doroshkevich and Igor Novikov in 1964, and by Robert Dicke and his collaborators in1965. The search by the latter group was overtaken by the contemporaneous discovery by Arno Penzias and Robert Wilson of the Cosmic Microwave Background (CMB) in 1965.

In 1990 the COBE satellite confirmed the blackbody nature of the CMB to remarkable precision. No deviations to the blackbody spectrum are found to within a fraction of a percent. This provides eloquent testimony to a hot origin for the universe when matter and radiation were in thermal equilibrium. The blackbody temperature is measured to be 2.728 K, with an uncertainty of only 0.004 K. Gamow had already laid down the key ingredient of a hot universe. The present-day universe is cold and dominated by matter. But, the observed radiation density, while today only amounting to ?_rad = 10^-5 in mass density, would have dominated in the past as a consequence of the redshifting of the photon energy during expansion.

Baryon Density

Modern determinations of the abundances of He⁴, He³, D, and Li⁷ are found to be consistent with a Big Bang origin, and, now that the CMB blackbody temperature is measured, they provide an accurate accounting of the primordial baryon abundance. One finds that ?_baryonh² = 0.02, with an uncertainty of only 10 percent. Independent confirmation of the baryon fraction in the universe comes from studies of the intergalactic medium at two distinct epochs. At high redshift one sees intergalactic neutral hydrogen in absorption in the spectra of quasars. The gas exists in vast numbers of clouds and filaments, and one needs to apply the ionizing photon flux, measured directly via the quasar emission spectra, to infer the total amount of intergalactic gas. At low red-shift, the hot intracluster gas in galaxy clusters is measured via its X-ray emission flux to be approximately 10 percent of the total cluster mass. Since clusters are considered to be sufficiently massive to have preserved their original baryon content, one can also deduce the baryon content of the nearby universe. Both methods agree. There is a problem, however. One can only account for about half of the predicted baryon fraction today in known sources such as stars and diffuse intergalactic gas. There is also a dark baryonic matter problem.

Thermal History

The discovery of the CMB led to some remarkable insights into the beginning of the universe. The Big Bang was once a fireball. Only after redshift ?_m/?_rad, about 3 × 10⁴, did the universe become matter-dominated. Only in a universe dominated by matter could the density fluctuations be gravitationally unstable and grow in strength. Moreover, Stephen Hawking and Roger Penrose derived a theorem which proved that as a consequence of the dense past of the universe (inferred in order for the CMB to have thermalized and been isotropized by the photons scattering off free electrons, then under classical general relativity) the universe must inevitably have undergone a past singularity.

One could now begin to reconstruct the thermal history of the universe. Quantum gravity supplants general relativity on the Planck scale at an epoch of 10^-43 s or at a temperature of 10¹⁹ GeV. This is where unification of the four fundamental forces— electromagnetic, weak nuclear, strong nuclear, and gravity—occurred. There is as yet no preferred theory for this regime, although higher-dimensional theories of quantum gravity include models in which Planck scale physics is manifest at TeV energy scales. As the universe expanded and cooled below the Planck scale, the ensuing evolution can be sketched as follows.

Above 10¹⁶ GeV, the electromagnetic, weak and strong nuclear forces were indistinguishable and of equal strength. This was the grand unification (GUT) era. As the temperature dropped below 10¹⁶ GeV, the symmetry of grand unification was spontaneously broken. The resulting change in phase of the matter content of the universe involved the transient appearance of a scalar energy field that was responsible for the inflation of the universe. The universe expanded exponentially as long as this so called inflaton field was the dominant source of energy density. The universe is then 10^-36 s old. The inflaton is similar to the cosmological constant, except that its energy density was larger by about 120 factors of 10. The potential energy of the inflaton field dominates the kinetic energy, and this provides the constant energy density that drives the universe to inflate. The potential energy drops (by design), and inflation ends by about 10^-35 s. The enormous kinetic energy thermalizes, or turns into heat, and one is now again in the conventional hot Big Bang phase, initially dominated by radiation and relativistic particles.

At 100 GeV, the electroweak forces decouple, and the fundamental force strengths subsequently resemble those observed today, with the nuclear forces being strong and short-range compared to the feebler electromagnetic force and the vastly weaker gravitational interaction. At this epoch, the change in phase of the universe helps generate a small asymmetry in the baryon number, the number of particles minus antiparticles. The baryon number is expressed in dimensionless form as (N - N? )/(N + N?) and is only 10^-9. However, as the temperature drops further, all the strongly interacting particles annihilate into radiation. The radiation redshifts to become the CMB. There are 10⁹ CMB photons per proton in the universe. The relic particles freeze out of thermal equilibrium once the temperature drops below a fraction of the particle mass. Very few pp? pairs survive, because of the strong interactions that annihilate almost all the pairs. The baryon excess, however, means that the baryons, which have no anti-baryon counterparts, do survive to become the present matter content of the universe. The observed universe consists almost exclusively of matter: the anti-matter content is less than a hundredth of a percent, otherwise, one would see gamma rays from matter-antimatter annihilations.

If stable weakly interacting particles were present, these would freeze out in substantial numbers, regardless of whether there was any primordial asymmetry. The lightest supersymmetric particle or neutralino is such a possible stable relic. Its abundance is determined by its annihilation cross section, so that ?_x ? 10^-38cm²/<??>. For typical values of the weak cross section, the neutralino is a viable candidate for the nonbaryonic dark matter in the universe. Typical predicted mass scales are of order 0.1 to 1 TeV, the preferred supersymmetry energy scale.

The universe is now a soup of quarks, gluons, electron-positron pairs, neutrinos, and photons. At about 200 MeV, another phase transition occurs when the quarks and gluons form hadrons. The universe now contains protons and neutrons in thermal equilibrium with n_n/n_p ? e^-?m/kT, or about 0.1, where ?m is the mass difference between proton and neutron. Once the temperature drops below 1 MeV, the neutron-producing reactions stop, and neutrons freeze out. At 0.5 MeV, e⁺e^- pairs annihilate and neutrinos freeze out. The stage is now set for nucleosynthesis of the light elements that commences at0.1 Mev or 10⁹ K, when deuterium nuclei can first form. Subsequent reactions produce He³, He⁴, D, Li⁷, all of which are generated in abundances that are measurable today in primordial environments. Lack of stable nuclei at masses 5 and 8 means that nucleosynthesis peters out after He⁴ is synthesized. The predicted primordial He⁴ simply incorporates all the neutrinos: Y = 2n_n(n_p - n_n)^-1 ? 0.25 by mass.

One expects to find primordial helium in such unprocessed environments as the intergalactic medium, the outermost parts of galaxies, metal-poor galaxies, and even with suitable extrapolation, meteorites, and the atmosphere of Jupiter. All abundances are consistent with a universal baryon fraction ?_bh² = 0.02 to within 10 percent. The universe remained dense and hot enough for the thermal equilibrium of matter and radiation to be maintained until an epoch of about 1 month. This was when the cosmic blackbody radiation was effectively generated. Any spectral distortions would probe the physics of the universe at this epoch.

The temperature continued to drop. The hydrogen is ionized and the radiation scatters frequently. There are 10⁹ photons for every baryon, and these suffice to keep the hydrogen fully ionized until the temperature drops below 0.2 eV. At this point, there are too few photons with energy above the hydrogen ionization threshold of 13.6 eV to keep the hydrogen fully ionized. The protons and electrons combine to form hydrogen atoms. Unlike free electrons, these are very poor scatterers of electromagnetic radiation. Scattering of the photons abruptly stops. The universe is now transparent to the CMB, from a redshift of 1,000 or 300,000 years after the Big Bang, to the present.

CMB Fluctuations

Measurement of the 2.728 Kelvin blackbody radiation spectrum confirms the thermal history of the universe back to an epoch of a few days after the Big Bang. Detection of temperature fluctuations at a level of ?T /T ? 10^-5 has revealed the irregularities at the epoch of last scattering that trace the density fluctuations from which large-scale structure evolved. The primary temperature anisotropies that emerge from last scattering are measured at angular scales that range from the dipole (180 degrees), associated with motion relative to the CMB frame, to a few minutes of arc, which are induced at the moment of last scattering.

The density fluctuations prior to last scattering are like sound waves in a medium with a sound velocity approaching that appropriate to that of a relativistic plasma, c/?3 After last scattering, the radiation thermally decouples, and the sound speed drops to that of a gas at a few thousand degrees Kelvin. This means that the density fluctuations, which previously were pressure-driven sound waves, now respond only to gravity, the pressure being completely unimportant at least for fluctuations that contain the masses of even the smallest galaxies. Indeed, the minimum size for gravity to dominate, and thus for the first self-gravitating gas clouds to form, is about a million solar masses. As time proceeds, the clouds build up in mass by clustering together under the action of gravity to form a galaxy and eventually cluster mass clouds. The galaxy mass clouds are able to cool and fragment into stars. One ends up with galaxies and clusters of galaxies, the latter containing large amounts of gas that is too hot to have cooled.

The sound waves leave a remarkable imprint on the CMB. Inflation, or some equivalent theory, generates these waves that just begin to undergo their first compression peak when they enter the horizon. The wavelength simply spans the distance traveled by light since the Big Bang. Such waves that are cresting at last scattering for the first time have the largest amplitude. They produce a peak in the CMB fluctuations at an angular scale corresponding to the horizon scale at last scattering, about 1 degree. Shorter waves that are cresting for the second time at last scattering are amplified less and leave a smaller angular scale peak. Waves undergoing their first rarefaction also leave a peak on an intermediate scale, since rarefactions are measured in quadrature as fluctuations that are either negative or positive, the density field being random. There are a series of peaks predicted to be of decreasing strength until one reaches wavelengths that are so inefficient at scattering the radiation that there are no further fluctuations. It is then said the fluctuations are damped out, and this occurs at a physical scale corresponding to the thickness of the last scattering epoch, the distance a primordial sound wave could travel over the time the universe undergoes the transition from ionized to neutral. This amounts to approximately 30,000 years, so the smallest surviving primary fluctuations are on a scale of about one-tenth of a degree.

A series of peaks have been measured in the CMB temperature fluctuations. The first, second, and third peaks have been detected. The angular position of the peaks is sensitive to the curvature of the universe. If one lived, for example, in an open universe with hyperbolic geometry, the peaks are shifted to smaller angular scales, the universe acting like a giant concave lens. This effect is not observed: the universe is found to be flat to within an accuracy of 10 percent, in terms of the critical energy density, ?_m + ?_? ? 1.

The detection of the acoustic peaks is another independent confirmation of the dominance of nonbaryonic dark matter in the universe; the peaks are produced by baryons, scattering by electrons. From their strength, a value ?_b ? 0.04 is independently inferred. ?_m ? 0.3 is required in order to have enough fluctuation growth in the early universe to make the fluctuations as small as they are observed. From the locations of the peaks, the equation of state is also measured, and one infers from both large-scale structure and cosmic microwave background observations that w is less than approximately -0.5, not far from the value corresponding to the cosmological constant. Hence, an independent confirmation of holds: for the universe to be flat, ?_? × 0.7. This constitutes the concordance model of the Big Bang.

See also:Astrophysics; Big Bang Nucleosynthesis; Cosmology; Hubble Constant; Inflation

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Joseph I. Silk