Big bang theory

Studying the Universe

Measurement techniques

Historical background

The spiral nebulae

Implications of Hubble’s law

Other developments

The evolution of the Universe

Future work

Resources

The big bang theory is the conceptual model that scientists use to describe the origin and subsequent evolution of the Universe. It states that the universe began as a tiny, violent explosion about 14 billion years ago. That event produced all of the matter and energy in the Universe, including its hydrogen and helium. Some of these light atoms were forged in the cores of stars, over billions of years, into atoms of the heavier elements that exist today, including the atoms of which we ourselves are made. One consequence of the big bang is that today the Universe, which is of finite size and contains a finite amount of matter, is expanding; in fact, the occurrence of the big bang was originally deduced from the fact of the universe’s expansion. In recent years astronomers have made many observations that verify predictions of the big bang theory.

Studying the Universe

People have always wondered about the origin of the Universe. Questions about how and when Earth and the heavens formed have been pondered by tribal peoples, philosophers, religious thinkers, and scientists. The modern, scientific study of the origin and structure of the universe is known as cosmology.

For many centuries, cosmological thought was limited mostly to speculation. For example, it was not obvious to the ancients that the sun and the stars are objects of the same sort. The sun is a star much like the other stars and is only brighter because it is closer, but this fact took centuries to determine because it is difficult to measure the distance to most of the objects seen in the night sky. Early astronomers of the scientific era, although they knew that the stars are also suns, assumed that all stars have the same intrinsic brightness and thus that only their distance from Earth determines their apparent brightness. This was incorrect; in fact, enormous variations in brightness among individual stars exist. Examination of binary stars (paired stars that orbit each other) demonstrated these differences. (About half of all stars in binary or other multiple systems.) When binary star systems in which the two stars did not have the same brightness were observed, it became clear that the amount of light received from any given star is dependent on more than just its distance. Until the elementary task of measuring the position of astronomical objects could be pursued systematically, larger questions about the structure and history of the universe as a whole could not even begin to be answered.

Measurement techniques

All measurements of the stars must be made in the neighborhood of the Earth, since any distance achievable by a spacecraft in less than many years of travel still qualifies as “in the neighborhood of the Earth.” The nearest star other than the sun is more than four light-years away, and most objects seen from Earth, even with the naked eye, are much farther off. (A light-year is the distance that light travels in one year: 5.88 trillion miles [9.46 trillion kilometers], or about 60,000 times the distance from the Earth to the sun.)

There are two fairly direct ways to determine the distance to the nearest stars. The first is to measure their parallax or apparent change in position during the year. As the Earth circles the sun, stars are seen from a shifting vantage point. The furthest objects do not appear to move because the Earth’s change in position is too small to affect our view, but the nearest stars seem to move back and forth slightly during the course of a year. Parallax can be seen by holding up a finger a few inches before your eyes and closing first one eye, then the other, thus repeatedly shifting your point of view by the distance between your eyes: the finger seems to jump back and forth dramatically, while objects across the room move much less. The shift in the finger’s apparent position is its parallax. By measuring the parallax of a nearby star over a six-month period (during which the Earth moves from one side of its orbit to the other), and knowing the radius of the Earth’s orbit, it is a matter of straightforward trigonometry to determine the distance to that star.

Another technique to determine stellar distances is to measure the proper motion of a star. This is the apparent motion of a star with respect to other stars caused by the star’s actual motion through the sky. (All stars are moving, including the sun.) Although the motion of distant stars is too small to detect, closer stars can be seen changing position with respect to more distant stars over the years.

Such techniques are only applicable to a few of the nearest stars, however, and disclose nothing about the large-scale structure of the universe. More sophisticated methods had to be developed for this task, requiring different astronomical observations. One such method depends on the examination of a star’s (or other celestial object’s) spectrum, that is, the intensity of its radiation (including, but not limited to, its visible light) at various wavelengths. If the light from a star is divided into its component wavelengths using a prism, a continuous spread of wavelengths punctuated by a number of dark lines can be seen. (The visible part of our sun’s spectrum is the rainbow.) These absorption lines are caused by elements in the star’s outer atmosphere that absorb light at specific wavelengths. Each dark line in a star’s spectrum corresponds to a specific element; the absorption lines in a star’s spectrum thus give a catalogue of the substances in its outer layers. Furthermore, these lines can reveal how fast the star is approaching the Earth or receding from it using the Doppler effect, a fundamental property of all traveling waves (including light waves). The absorption lines in the spectra of objects moving away from the Earth are shifted to longer wavelengths, while absorption lines in the spectra of objects moving toward the Earth are shifted to shorter wavelengths. A shift to longer wavelengths is called a redshift because red light appears near the long-wavelength end of the visible spectrum, while a shift to shorter wavelengths is called a blueshift. Measurement of the Doppler shifting of spectral lines has made it possible to map the large-scale structure of the cosmos, and it is this structure that the theory of the big bang and the theory of general relativity explain.

Historical background

In 1905, Danish astronomer Ejnar Hertzsprung (1873–1967) compared the width of various stars’ absorption lines to the absolute luminosity, or brightness of the stars as determined from proper motion measurements. Hertzsprung found that that wider lines correspond to larger and brighter stars. This provided a way to determine the absolute brightness of a star from its spectrum. Knowing its absolute brightness, he could then determine its distance from Earth. This method applied to stars at any distance, as opposed to the parallax and absolute-motion methods, which applied only to stars quite near the sun, but was limited in accuracy.

In 1908, American astronomer Henrietta Swan Leavitt (1868—1921) discovered that Cepheid variables, a type of star in which the brightness changes in a regular manner, showing a well-defined relationship between period (time required for brightness to wax and wane through a full cycle) and absolute luminosity. Brighter Cepheid variable stars have longer periods while dimmer ones have shorter periods. Leavitt calculated a simple relationship between brightness and period. This discovery had a profound effect on stellar distance measurements. Now, any time a Cepheid could be found—in, say, a distant galaxy— the distance to it could be determined accurately.

The spiral nebulae

In the early twentieth century, there was a debate among astronomers over the nature of the spiral nebulae, the diffuse spiral-shaped structures visible (through telescopes) in most parts of the sky. Some believed these were nearby objects that were part of our Milky Way galaxy, while others thought that they were much further away, and in fact were “island universes,” or separate galaxies. If the distances to these objects could be measured, then the debate could be settled, and important knowledge gained about the structure of the Universe.

In 1914, American astronomer Vesto M. Slipher (1875–1969) presented figures for the velocities of 14 spiral nebulae, obtained by measuring the Doppler shifts of their spectral lines as described above. Slipher found that most of the nebulae were, surprisingly, moving away from Earth. If the nebulae’s motions were random, just as many of them would be expected to move toward Earth as were moving away. Why should the sun just happen to be at the center of an organized pattern of motion? Another puzzling finding was the large velocities at which these objects were receding. The nearby Andromeda nebula, for example, was speeding toward earth at 180 miles per second (300 km per second). Many astronomers interpreted this to mean that the nebulae must be outside our galaxy.

In 1923, American astronomer Edwin Hubble (1889–1953), using the 60-in and 100-in (152-cm and 254-cm) telescopes at Mount Wilson Observatory, succeeded in identifying Cepheid variables in the outer regions of two nebulae, M31 and M33. By measuring the periods of these Cepheids and using the formula developed by Leavitt several years earlier, he calculated that they are about 930,000 light years distant. From these distances and the observed sizes of the nebulae, their actual sizes could be calculated. These turned out to be similar to that of Earth’s galaxy, strongly supporting the idea that the nebulae are galaxies in their own right.

In 1929, Hubble plotted data on a number of galaxies on a graph. He plotted the distance to the galaxy along the horizontal axis and the velocity of the galaxy’s recession along the vertical axis. From this limited data, it was clear that a simple, linear relationship between the two quantities existed; on average, the velocity of a galaxy’s recession was proportional to the distance to the galaxy. (The Andromeda galaxy is a member of our local galactic group, which moves more or less and as a unit: other members of the Local Group do obey this general red-shift rule.) The constant of proportionality between distance and velocity, now called the Hubble’s constant (H ), was given by the slope of the line. From those data, Hubble’s constant was estimated to be 310 miles per second per megaparsec (500 km/s/Mpc)—that is, a galaxy 1 megaparsec (one million parsecs, where one parsec = 3.26 light-years) away would be moving away from our galaxy at 310 mi/s (500 km/s), while a galaxy ten times further away would be moving ten times as fast. Modern values for H are much smaller than Hubble’s estimate of 500 km/s/Mpc. The relationship between speed and distance described by Hubble’s constant is termed Hubble’s law.

Implications of Hubble’s law

Galaxies are moving away from Earth in all directions not because Earth is at the center of the universe but because all galaxies are receding from each other, that is, every galaxy is getting further away from every other galaxy. There is a simple way to visualize this effect. Imagine a partially blown up balloon, on the surface of which a number of spots (representing galaxies) are drawn. As the balloon inflates, each spot gets farther away from its neighbors; furthermore, the distance from any one spot to any other on the same side of the balloon (as measured on the surface of the balloon) grows faster the farther away from each other the two spots become. This can be seen by imagining three spots in a row (A, B, and C), with the balloon’s surface expanding evenly between them. The distance from A to B is growing at x inches per second, and so is the distance from B to C; the distance from A to C must therefore be growing at 2 x in/s (x + x = 2 x ). Thus no matter where an observer is located on the balloon, they will observe that all the other spots on the balloon’s surface obey Hubble’s law: the farther away they are, the faster they recede. There is no preferred direction, no preferred position, and no “center” of the balloon’s surface. The behavior of galaxies in our spatially three-dimensional universe is analogous to that of the spots on the two-dimensional surface of the balloon.

There is an important implication to this model. If all galaxies are moving away from each other with a velocity proportional to their separation, then at all earlier times the galaxies were closer together. If one goes back far enough, there must have been a time when all were at the same position—that is, there must have been a beginning to the universe. In fact, if the expansion has been constant for all time, the age of the universe is simply the inverse of Hubble’s constant. The situation is not quite so simple, as scientists expect Hubble’s constant to change over time (the gravitational attraction between the galaxies has a tendency to slow the rate of expansion, while recent observations show that the expansion of the universe is actually accelerating under the influence of a still-mysterious force that acts opposite to gravity).

Hubble’s original measurements gave an age of the universe of about two billion years. This immediately caused dispute, since it was known from measurements of radioactive decay that the age of the solar system is more than twice this value. How could the solar system have been formed before the universe itself? It is now known that Hubble’s original measurements were in error. Current measurements put Hubble’s constant in the range of 50–100, giving an age of 10–20 billion years. Geological and astronomical data from many sources have recently converged on a value for the age of the universe of about 13.7 billion years.

Other developments

When Einstein developed his general theory of relativity he added a term, the cosmological constant, to his equations in order to permit a static universe which was neither expanding nor contracting. He later came to regret this, calling it the worst mistake he ever made; however, the recent discovery that the expansion of the universe is actually accelerating has brought Einstein’s cosmological constant back into favor. Nevertheless, Russian mathematician Alexander Friedmann (1888–1925) and Belgian astronomer Georges LeMaître (1894–1966) found solutions (in 1922 and 1927, respectively) to Einstein’s equations that permitted an expanding universe. After Hubble’s 1929 discovery, there was a great deal of interest in these models, which could be used to explain the observations.

A big bang seemed an obvious implication of the new data—if everything was expanding, it must all once have been scrunched together—but steady-state models that avoided the embarrassment of a universe with a definite beginning had adherents for decades after Hubble’s measurements were made. One of the more promising models, constant creation, postulated that new hydrogen atoms form constantly and spontaneously throughout space, out of nothing, providing the material for new galaxies as the older galaxies move apart. On this theory, the universe has always been expanding, had no beginning, has always looked as it does now, and will always look as it does now. This theory predicted that nearby galaxies will look similar to those far away, but it was found that distant galaxies are in fact different from nearby ones, which agrees with the big bang’s claim that the universe is not in a steady state. It was one of the originators of this steady-state theory, British astronomer Fred Hoyle (1915–2001), who coined the term big bang, now used to describe the expanding-universe model based on Hubble’s observations. Hoyle chose the expression to ridicule the theory, but the name stuck.

The evolution of the Universe

The current picture of the big bang can be described briefly as follows. Because current formulations of the laws of physics break down very close to the big bang itself, the account will start one second after the event occurs. At this time, the temperature was 10,000,000,000K. This was too hot for atoms to exist, so their elementary particle constituents (electrons, protons, and neutrons) existed separately, along with photons (particles of light), and various exotic particles. Over the next 100 seconds, the temperature dropped by a factor of 10, enough to allow the nuclei of light elements such as deuterium (an isotope of hydrogen) and helium to form. As further cooling took place, these nuclei combined with electrons to form atoms.

At this point it should be stressed that the expansion of the Universe means that space itself is expanding. This differs fundamentally from an ordinary explosion, in which matter expands into a surrounding volume of space. The expansion of space itself can be compared to the increase of the surface area of the inflating balloon mentioned earlier; as the balloon expands, its surface area grows, but not by expanding into any larger, surrounding surface, as a circular ripple expands across the surface of a pond. Similarly, our universe is not expanding into any larger, surrounding volume of space.

The expansion of space has important cosmological implications. One is that as space expands, the average temperature of the Universe drops with it. This cooling has an important effect on the cosmic background radiation.

Early in the history of the universe, when its density was extremely high, particles and radiation were in equilibrium, meaning that there was a very uniform temperature distribution. Such a distribution gives rise to radiation with a particular spectrum, a blackbody spectrum, which has a well-defined peak wavelength. Radiation of this type currently pervades all space in the form of microwave radiation, the afterglow of the big bang. Due to expansion of the Universe, the peak of this radiation’s spectrum—its temperature—has by now been shifted to below 3K (–454°F [–270°C]), or three degrees above absolute zero, despite its initial high temperature. The cosmic background radiation was first detected by U.S. astrophysicist Arno Penzias (1933–) and U.S. radio astronomer Robert Wilson (1936–) in 1965. Measurements from the COBE spacecraft have shown that the spectrum is nearly perfect blackbody radiation at 2.73K (–454.5°F [–270.27°C]), as predicted by the big bang theory.

As described above, only the lightest elements were created in the big bang itself. As the Universe expanded, lumpiness developed, and regions of more-dense and less-dense gas formed. Gravity eventually caused the high-density areas to coalesce into stars and galaxies, which became luminous due to nuclear reactions in their cores. These reactions, which still power the stars today, take hydrogen and helium and create some of the heavier elements. Once its light-element nuclear fuels are exhausted, a star may explode in a supernova, creating still heavier elements in the process. It is these heavier elements from which the solar system, Earth, and humans are made. Every atom in the human body (of every element other than hydrogen; there is little or no helium in the body) was created in the core of an exploding star billions of years ago.

What will be the ultimate fate of the Universe? Will it continue to expand forever, or will it eventually contract in a “big crunch?” To understand this question, the analogy of a projectile being launched from the surface of Earth can be used. If a projectile is launched with enough velocity, it will escape the Earth’s gravity and travel on forever. If it is too slow, however, gravity will pull it back to the ground. This same effect is at work in the universe today. If there is enough mass in the universe, the force of gravity acting between all matter will eventually cause the expansion to slow, stop, and reverse, and the universe will become smaller and smaller until it ultimately collapses.

This does not seem likely. Astronomers have made estimates of the mass in the Universe based on the luminous objects they see, and calculated a total mass much less than that required to close the universe, that is, to keep it from expanding forever. From other measurements, they know that there is a large amount of unseen mass in the Universe as well, called dark matter. The amount of this dark matter is not known precisely, but greatly exceeds that of all the stars in the Universe. The nature of dark matter is being intensely researched and debated by astronomers. It should not be confused with “dark energy,” a separate phenomenon.

In 1998, astronomers studying a certain group of supernovas discovered that the older objects were receding at a speed about the same as the younger objects. According to the theory of a closed universe, the expansion of the universe should slow down as it ages, and older supernovas should be receding more rapidly than the younger supernovas. The fact that observations have shown the opposite has led many scientists to believe that the Universe is, in fact, open. Other theorists hold that the universe is flat—that is, that it will neither collapse nor expand forever, but will maintain a gravitational balance between the two and remain in a coasting expansion.

Starting in the late 1990s, various observations have proved—to astronomers’ astonishment—that the expansion of the universe is actually accelerating. It thus appears that the fate of the universe will be to expand without end. Eventually, all sources of energy will exhaust themselves, and after many trillions of years even the protons and neutrons of which ordinary matter is constructed will break down. If this vision is correct, the universe will end up as a diffuse, eternally expanding gas of subatomic particles at a uniform temperature.

Future work

Although the big bang model has done a good job of explaining what is seen of the universe and has been confirmed by numerous astronomical observations, there are still many unanswered questions. Until a few years ago, there was still disagreement about the exact value of the Hubble constant by approximately a

KEY TERMS

Cepheid variable star— A class of young stars that cyclically brighten and dim. From the period of its brightness variation, the absolute brightness of a Cepheid variable can be determined. Cepheid variables in distant galaxies give a measure of the absolute distance to those galaxies.

Hubble constant— The constant of proportionality in Hubble’s law which relates the recessional velocity and distance of remote objects in the universe whose motion is determined by the general expansion of the universe.

Light-year— The distance that light travels in one year, equal to 5.87 million miles 9.46 million km.

Parsec— 3.26 light-years.

factor of two. NASA’s WMAP (Wilson Microwave Anisotropy Probe) satellite began making observations in 2003 to resolve the question; as of 2006, a value for the Hubble constant of 70 km/s/Mpc (with an uncertainty of +2.4 to –3.2 km/s/Mpc) had been agreed upon by most astronomers. WMAP data has also revealed that the universe consists of only 4% ordinary matter—known particles of whatever kind. Twenty-two percent of the Universe is dark matter, the nature of which is unknown but which is known to gravitate like other matter, and 75% of the Universe consists of “dark energy,” the nature of which is also unknown but which serves to accelerate the Universe’s expansion. Thus, until breakthrough discoveries in fundamental physics reveal the nature of the dark matter and dark energy, we remain ignorant of the nature of 96% of the Universe.

Another open question is how galaxies actually formed from what was very close to a uniform, homogeneous medium in the early universe. From the uniformity of the microwave background radiation, it is known that this uniformity was better than one part in a thousand. Just by looking at the sky, however, a great deal of structure in the Universe is seen up to very large-size scales of clusters of galaxies and beyond. There must have been some type of clumping which occurred to start the process (with gravity helping the process along), but what started it? Physicists are seeking the answer in quantum mechanics, the science of very small events. The answer resides there because at the moment of the big bang, the universe was subatomic in size. Random fluctuations at the quantum level, which today appear only at the subatomic level, were then sufficient—for an extremely brief interval of time—to cause ripples throughout the entire Universe. These ripples persist today as large-scale clumpings of the galaxies, which have been observed by astronomers and found to agree with the big bang theory.

See also Blackbody radiation; Elements, formation of; Redshift.

Resources

BOOKS

Fleisher, Paul. The Big Bang. Breckenridge, CO: Twenty-First Century Books, 2005.

Lemonick, Michael D. The Echo of the Big Bang. Princeton, NJ: Princeton University Press, 2005.

Singh, Simon. Big Bank: The Origin of the Universe. New York: Harper Perennial, 2005.

PERIODICALS

Beers, Timothy C. “The First Generations of Stars.” Science. 309 (2005): 390-391.

Weinberg, David H. “Mapping the Large-Scale Structure of the Universe.” Science. 309 (2005): 564-565.

Wilczek, Frank. “Particle Physics: Did the Big Bang Boil?.” Nature. 443 (2006): 637-638.

Larry Gilman

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_μv = 8πGT_μv + Λg_μv. Here, g_μν is the metric of the universe, which is incorporated into the Einstein tensor G_μν to describe gravity, and T_μν 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 Λ.

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, ρ + 3p ≤ 0. Indeed, the cosmological constant satisfies ρ + 3p =0, and the solution is (t ) α exp In this solution, the density is constant, equal to Ω^Λ ≡ Λ/H² where Ω_Λ 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 Ω^tot ≡ Ω^mat + Ω^rad + Ω^Λ - 1 = k /a²H² where the mass density of matter ρ_mat αa^-3 and radiation ρ_rada^-4. Three possibilities for the pressure content of the universe are p = p_m « ρc², applicable since the epoch of matter-radiation decoupling; p = ρ_radc²/3, applicable prior to matterradiation decoupling; and p =w ρ, a generalization of the cosmological constant to an arbitrary equation of state. In this latter case, ρ α a^{-3(1 + w )}, with w = -1 corresponding to the case of the cosmological constant. The universe is radiation-dominated prior to a /a₀ = Ω_rad/ρ_mat or at redshift larger than 1 + z^eq = a⁰/a (t^eq) = 3,000(Ω^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, ν = H₀d , where ν 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 = -ρc², where p is the dark energy pressure, and ρ 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_θ Mpc^-3. The luminosity density of the universe is measured to be 2 × 10⁸h L_θ Mpc^-3. The mass-to-light ratio for closure is therefore 1,500h M_θ/L_θ. 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 ) α r . Typical values of the mass-to-light ratio are 100h M_θ/L_θ, whereas within the half-light radius, one finds a value of approximately 10h M_θ/L_θ, 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 Hα 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_θ/L_θ. 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 Ω_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 Ω_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 Ω_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 Ω_m/Ω_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 - N̄ )/(N + N̄) 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 pp̄ 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 Ω_x ≈ 10^-38cm²/<σν>. 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^-Δm/kT, or about 0.1, where δm 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 Ω_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 δT /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, Ω_m + Ω_Λ ≈ 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 Ω_b ≈ 0.04 is independently inferred. Ω_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, Ω_λ × 0.7. This constitutes the concordance model of the Big Bang.

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

Bibliography

Hu, Wayne. "The Physics of Microwave Background Anisotropies." <http://background.uchicago.edu/>.

NASA. "Cosmology: The Study of the Universe." <http://map.gsfc.nasa.gov/m_uni.html>.

"The N-Body Site." <http://star-www.dur.ac.uk/~moore/>.

"Ned Wright's Cosmology Tutorial." <http://www.astro.ucla.edu/~wright/cosmo_01.htm>.

Raine, D., and Thomas, E. An Intro to the Science of Cosmology (IoP, Philadelphia, 2001).

Scott, Douglas. "The Cosmic Microwave Background." <http://www.astro.ubc.ca/people/scott/cmb.html>.

Silk, J. The Big Bang (W. H. Freeman, New York, 2001).

"What Is Theoretical Cosmology?" <http://astron.berkeley.edu/~jcohn/tcosmo.html>.

Joseph I. Silk