2. MACROCOSMOS: FROM ANAXIMANDER TO EINSTEIN


  2.1 Greek and Roman World

    Anaximander of Miletus (c.610 - 545 B.C.) is said to have made a map of the inhabited world, and to have invented a cosmology that could explain the physical state of the Earth and its inhabitants. The infinite universe was said to be the source of an infinity of worlds, of which ours was but one, that separated off and gathered its parts together by their rotary motion. Masses of fire and air were supportedly sent outward and became the stars. The Earth was some sort of floating circular disc, and the Sun and the Moon were ring-shaped bodies, surrounded by air. The Sun acted on water to produce animate beings, and people were descended from fish.

    Anaximenes (c.545 B.C.) elaborates on Anaximander's ideas, and argues that air is the primeval infinite substance, from which bodies are produced by condensation and rarification, he produces logical arguments based on every day experience. Again he introduces rotary motions as the key to understanding how the heavenly bodies may be formed out of air and water.

    Pythagoras of Samos (c.560 - c.480 B.C.) took the cosmic ideas of Anaximander and Anaximenes one stage further, saying that the universe was produced by the heaven inhaling the infinite so as to form groups of numbers. The Pythagoreans proposed a geometrical model of the universe, involving a central fire around which the celestial bodies move in circles. The central fire was not the Sun, although the Earth was certainly of the character of a planet to it. To account for lunar eclipses, the Pythagoreans postulated a counter Earth.

    The discovery that the Earth is a sphere was traditionally assigned to Parmenides of Elea (c.515-c.450 B.C.), who was also said to have discovered that the Moon is illuminated by the Sun. A generation later, Empedocles (c.484 - c.424 B.C.) and Anaxagoras (c.500 B.C.) seem to have given a correct qualitative account of the reason for solar eclipses, namely the obscuration of the Sun's face by the intervening Moon.

    The discovery of the sphericity of the Earth, and of the advantages of describing the heavens as spherical, captured the imagination of the Greeks of the time of Plato (c.428 - c.347 B.C.) and Aristotle (384 - 322 B.C.), in the 4th century, and of one man particular: Eudoxus of Cnidus (c. 400 - c.355 B.C.), who produced a very remarkable planetary theory based entirely on spherical motions.

2.2 Aristotle's Universe

    At the time of the Greek philosopher and naturalist Aristotle (384-322 B.C.), the Earth and the universe were seen as constructed out of five basic elements: earth, water, air, fire, and ether. The natural place of the motionless Earth was at the centre of that universe. The stars in the heavens were made up of an indestructible substance called ether (aether) and were considered as eternal and unchanging. Aristotle's cosmology was the first "steady state" universe. The other basic elements - water, air, fire - were earthly elements. The celestial bodies, including the Sun, the planets, and the stars, were considered to be attached to rigid, crystalline spheres, which were supposed to revolve in perfect circles about the Earth. The three innermost spheres, closest to the Earth, contained water, air, and fire, respectively. The Egyptian-Greek astronomer Ptolemy (c.100-170 A.D.) left the Earth to be located at the centre of the universe but ascribed to the Sun and the planets a new place within the cosmological views of his time. The Sun and the planets would revolve in small circles whose centers revolve in large circles about the Earth ("epicycles"). The element of perfection and beauty ascribed to the divine heavens remained the circle out of which the orbits of heavenly bodies were composed. Basically, Ptolemaic ideas were devised to accomodate astronomical observations of planetary motions. His "Almagest" described the motions of the heavens on this geocentric basis, employing wheels within wheels (epicycles) to obtain a plausible match with observations. Ptolemy's geocentric system dominated western thought on astronomy (including cosmology) until the time of Copernicus, fourteen centuries later (Figure 3).

 


 

Click on figure for large version! (will open in new window)

Figure 3. Dante Alighieri's (1265-1321) scheme of the universe in illustration from "Paradies" in "The Divine Comedy" extends Aristotelian cosmology in a modern way. Dante traverses the material world from the icy core of Earth, the abode of Lucifer, to the Mount of Purgatory. He continues through the nine heavenly spheres, each sphere larger and more rapidly turning than the last, until he reaches the Primum Mobile, the ninth and largest sphere and the boundary of space. His goal was to see the Empyrean, the abode of God.

 


 

    Eratosthenes (c.270 - c.190 B.C.) was the first to accurately measure the radius of the Earth in 196 B.C. by determining the minimum angle between the Sun's direction and the vertical at Alexandria on the day of the summer solstice. He knew that a zero angle occurs approximately when the Sun was at its highest point at the city of Syene (now Aswan), and he knew the base of the triangle, i.e., the distance from Alexandria to Syene.

2.3 Copernicus's Universe

    Copernicus (1473-1543), a Polish astronomer, placed the Sun at the center of the solar system with the Earth orbiting around the Sun, thus proposing a heliocentric cosmology (De revolutionibus orbium coelestium, 1543). This entirely new basis of cosmological consideration did not fit the observations much better than the Aristotelian-Ptolemaic system but was justified by the "divine" principle of simplicity in comparison to the rather complicated construction using epicycles as employed in Ptolemy's cosmology. However, like Aristotle and Ptolemy, Copernicus retained the conventional idea that the planets moved in perfectly circular orbits and continued to believe that the stars were fixed and unchanging. Kepler (1571-1630), a German astronomer and physicist, adopted the Copernician system but introduced the concept of planets with elliptical orbits. He still believed that the Sun was the centre of the universe. The cosmologies of Aristotle and Ptolemy had nevertheless been abandoned.

2.4 Newton's Universe

    The Italian philosopher Bruno (1548-1600) laid the groundwork for Newtonian cosmology by emphasizing that the universe is infinite and stars are scattered outward through infinite space (see, however, Figure 4).

 


 

Click on figure for large version! (will open in new window)

Figure 4. The first diagram to illustrate the proposal that the universe is infinite. From the edition by Thomas Digges of his father's A Prognostication everlastinge..., published in 1576 in London, eight years before its publication by Giordano Bruno to whom the idea is often credited (By permission of The Royal Society).

 


 

    Bruno even went so far as to say that stars are Suns, perhaps with orbiting planets and life on them (De l'infinito, universo e mondi, 1584). This far-reaching statement signaled a transfer of attention away from the planets in the solar system to the stars in the Milky Way. The laws describing the behavior of planets are the same laws which describe the behavior of the stars. That the physical laws are of universal nature, and can be applied on Earth as well as in the heavens, was discussed on a philosophical basis by the French philosopher and mathematician Descartes (1596-1650). He compared the universe with a giant clock, obeying mechanical laws which later on had a major influence on Newton's (1643-1727) thinking. It was his belief that the universe was infinite and that the primary qualities of the universe were mathematical in nature.

    A major step in observational techniques was achieved through the invention of the reflecting telescope between, say, 1545 and 1559 by the Britain Leonhard Digges (c.1520-1559). But it was not until 1609 that the Italian astronomer and physicist Galileo (1564-1642) realized observationally that the Milky Way is actually a collection of individual stars. Galileo also observed mountains on the Moon and discovered four satellites around Jupiter. He had taken the first step toward deducing the structure of the Milky Way. The discovery of heavenly bodies that evidently did not circle the Earth, and his support for the Copernician heliocentric cosmology were described in his work Dialogo sopra i due massimi sistemi del mondo, Tolemaico e Copernicano (1632).

    Cassini (1625-1712), an Italian-French astronomer, used the telescope to make accurate measurements of the "dimensions of the universe", particularly determining the distance of the Earth from the planet Mars. A rigorous mathematical foundation of Descartes' notion of the universe as a giant mechanical clock was provided by Newton's theory of gravity and his laws of motion (Philosophiae naturalis principia mathematica, 1687). From these he explained Galileo's results on falling bodies, Kepler's three laws of planetary motion and the motion of the Moon, Earth and tides. Newton clearly realized that gravity is the dominant force for understanding the structure of the universe; however, he argued that the universe must be static in a famous letter he sent to the theologian Richard Bentley (1662-1742) in 1692. Mainly for religious reasons, at this time, constancy and stability were associated with the perfection of God and change with friction and decay. Philosophical and religious ideas served as the scaffolding upon which scientific systems of thought developed. It was Bentley who derived for the first time, based on Newton's gravitational theory, what is still considered to be one of the fundamental constants of nature, the gravitational constant.

    In Kant's (1724-1804) cosmology the gravitational attraction of stars for each other was exactly balanced by the orbital motions of the stars and he argued that forces can act at a distance without the necessity for a transmitting medium. In 1755 Kant proposed the nebular hypothesis for the formation of the solar system. In 1788 Laplace (1749-1827) attempted a mathematical proof of the stability of the solar system (Système du monde, 1796). Stability and order of the universe were considered as eternal principles in the heavens and on Earth. The cosmology of Copernicus, as refined by Kepler, is now believed to be essentially correct.

2.5 Einstein's Universe

    In 1915 Einstein (1879-1955) put forth his general relativity theory (a new theory of gravity); he applied this theory to cosmology in 1917. The theory describes gravity as a distortion of the geometry of space and time. Unlike Newton's theory of gravity, general relativity was consistent with special relativity, which Einstein had introduced in 1905. Cosmology, based on general relativity, broadened the problem into one of finding a model of the space-time structure of the universe. Einstein's original solutions of his gravitational field equations left the universe in a stable state of static equilibrium and he provided physical conditions required to maintain such static (time-independent) equilibrium (the "cosmological constant"). Only in 1922 Friedmann (1888-1925) succeeded in finding solutions of Einstein's field equations that evolved in time describing an expanding (or contracting, if one cares to reverse the sense of time) universe.

    Persuasive observational evidence that the universe is indeed expanding and changing in time was found by Hubble (1889-1953) in 1929 while employing the technique known as Doppler shift for measuring the red shift of colors in the spectrum of nebulae. "Modern cosmology" may be said to have began with Einstein (1917) and Friedmann (1922, 1924), based on observations of cosmological relevance made by Slipher (1875-1969) in 1914 and Hubble in 1929. Slipher discovered the redshift of nebulae, later on identified by Hubble to be entire galaxies similar to the Milky Way; however, until 1929 their cosmological significance remained obscure. In 1929 Hubble had counted a great number of galaxies (to determine their distribution throughout the observable universe), and plotted the galaxie's redshifts against magnitudes for the brightest E-type cluster galaxies (Hubble diagram). Hubble found evidence that the outward speed of a galaxy is directly proportional to its distance away from the observer (Hubble law). This observational fact was exactly what would be expected if the universe is expanding, as discussed in a paper of Lemaître (1894-1966) in 1927. Hubble's diagram reveals a linear increase of the magnitude of galaxies with increasing redshift. In 1956, Hoyle and Sandage developed the q0 criterion which could be used to distinguish one cosmological model from others. Lemaître's model and Hoyle's steady state' model were ruled out by predicting q0 = - 1, whereas Hubble's linearity gave q0 = + 1. Cosmology focused on the search for two numbers: H0, the rate of expansion of the universe at the position of the Milky Way; and q0, the deceleration parameter, characterizing the change of the rate of universal expansion. The value of q0is believed to lie in the range between 0 and 0.5; the value of H0 is thought to be uncertain by a factor of about 2 (H0 = 100 h km s-1 Mpc-1, where 0.5 smaller equal h smaller equal 1). Theoretical ideas on which Big Bang cosmology is based were contained in the publications of Friedmann and Lemaître by 1930 (for an in-depth review see North, 1965).

 

  2.6 Big Bang

    Astrophysical arguments were introduced into the cosmological models with Gamow's (1946) prediction that helium (and possibly heavier elements) were generated at an early stage of the evolution of the universe. At this time it was believed that the relative abundance of cosmic nuclei represents truly cosmic abundances. Based on Gamow's arguments the cosmological criterion of the origin of chemical elements in the primeval fireball was substantiated by Alpher and Herman (1949, 1950) and Alpher et al. (1953). However, after Burbidge et al. (1957) it became evident that the bulk of the chemical elements beyond helium where not synthesized at the early stages of the expansion of the universe but in the stars. Only the synthesis of helium (Hoyle and Tayler, 1964), which is not produced in sufficient quantities in stars, and deuterium (Peebles, 1966), which is destroyed during galactic evolution, continued to need Gamow's primordial nucleosynthesis to arrive at reasonable abundances as observed in the universe. Later on Wagoner et al. (1967) were able to show that in addition to D, 3He, and 4He, the only other cosmologically significant element was 7Li. In 1965 the microwave background radiation of 3 degrees Kelvin was discovered by Penzias and Wilson (1965), as predicted by Alpher and Herman in 1949 based on Gamow's considerations of a hot and dense origin of the universe.

    Today the Big Bang cosmological model, the temperature of the microwave background radiation, the abundances of D, 3He, 4He, 7Li, and the astrophysically observed average density of galactic material are starting points for developments in cosmology.

2.7 Beyond the Big Bang

    In the Big Bang cosmology, the universe has been expanding throughout its history. Mathematical calculations would suggest that the temperature and the density were infinite at the instant of the Big Bang. The universe is supposed to originate in the Big Bang, but the mathematical and physical structure of the model does not permit matter to originate. Additionally, the Big Bang model leaves unanswered several important questions regarding

  1. the number of protons and neutrons in the universe, relative to the number of photons,

  2. the large-scale homogeneity of the observed universe,

  3. the actual density of the universe which is close to the critical density (that average density of the universe which divides Big Bang models that will keep on expanding for ever from those that will ultimately recontract), and

  4. the origin of density perturbations from which small-scale (stars and systems of them) and large-scale (galaxies and systems of them) inhomogeneities have been developed (Kolb and Turner, 1990).

    A new approach to explaining some of the questions left over by Big Bang cosmology began with Guth's (1981) paper inventing an "inflationary" period in the evolution of the universe. The phase transition associated with the break-up of the unified force in the Grand Unification epoch could leave the universe in a state of false vacuum, in which the vacuum has a very high energy density. This vacuum energy density acts like a cosmic repulsion and the universe embarks on an exponential expansion which "inflates" the universe by a factor of 1030 in a brief instant of time. This inflationary period ends when the vacuum energy density transforms into matter and radiation and the expansion of the universe continues. Besides offering explanation of the four questions above, it makes one concrete prediction: The present universal expansion should exhibit flatness to the extent of Omega0 = 1 ± epsilon with epsilon < 10-6, if inflation was sufficient to extend to a distance greater equal cH0-1 today. Observations of the luminous matter content of the universe reveal only Omega0 < 0.002, and the quantity of dark or non-baryonic material necessary to explain the flat rotation curves of spiral galaxies and the virial equilibrium of large groups and clusters of galaxies requires at most Omega0 round 0.2. The only way that Omega0 = 1 from inflationary cosmological models can be reconciled with observation is by the existence of non-baryonic weakly interacting particles. The unknown nature of dark matter led to considerable activities in fundamental particle physics to speculate about new basic constituents of matter. Additionally, the inflationary universe picture has stimulated far-reaching speculation about the ultimate origin of the universe.

    Cosmology based on general relativity has thus moved close to fundamental particle physics based on quantum field theory. A new challenge emerged for physical theory and observation subsumed under the term "quantum gravity". Interaction between mathematics and physics, as exemplified by the role of Riemannian geometry in general relativity and functional analysis in quantum mechanics, became lively again. Cosmology is currently based on the two fundamental theories in twentieth-century physics, general relativity and quantum field theory. General relativity describes the gravitational force on an astronomical scale, while quantum field theory describes the weak, electromagnetic, and strong interaction of fundamental particles. A formal quantization of general relativity leads to unphysical infinities; encompassing gravity in quantum field theories leads to a new connection of theoretical physics to modern mathematics, called string theory. However, there is still no quantum cosmology. Particularly, an inflationary era may have taken place in the early universe, but there is no proof that it did so (Manin, 1981; Schmid, 1992).