Large-scale structure and expansion of the universe

Hubble inferred a uniformity in the spatial distribution of galaxies through number counts in deep photographic surveys of selected areas of the sky. This inference applies only to scales larger than several times 108 light-years. On smaller scales, galaxies tend to bunch together in clusters and superclusters, and Hubble deliberately avoided the more conspicuous examples in order not to bias his results. This clustering did excite debate among both observers and theorists in the earliest discussions of cosmology, particularly over the largest dimensions where there are still appreciable departures from homogeneity and over the ultimate cause of the departures. In the 1950s and early 1960s, however, attention tended to focus on homogeneous cosmological models because of the competing ideas of the big bang and steady state scenarios. Only after the discovery of the cosmic microwave background--which, together with the successes of primordial nucleosynthesis, signaled a clear victory for the hot big bang picture--did the issue of departures from homogeneity in the universe again attract widespread interest.

From a more pragmatic point of view, clusters and groups of galaxies are important to cosmological studies because they are useful in establishing the extragalactic distance scale. A fundamental problem that recurs over and over again in astronomy is the determination of the distance to an object. Individual stars in star clusters and associations provide an indispensable tool in gauging distances within the Galaxy. The brightest stars--in particular the brightest variable stars among the so-called Cepheid class--allow the distance ladder to be extended to the nearest galaxies; but at distances much larger than 107 light-years individual stars become too difficult to resolve, at least from the ground, and astronomers have traditionally resorted to other methods.

CLUSTERING OF GALAXIES

Clusters of galaxies fall into two morphological categories: regular and irregular. The regular clusters show marked spherical symmetry and have a rich membership. Typically, they contain thousands of galaxies, with a high concentration toward the centre of the cluster. Rich clusters, such as the Coma cluster, are deficient in spiral galaxies and are dominated by ellipticals and S0s. The irregular clusters have less well-defined shapes, and they usually have fewer members, ranging from fairly rich systems such as the Hercules cluster to poor groups that may have only a few members. Galaxies of all types can be found in irregular clusters: spirals and irregulars, as well as ellipticals and S0s. Most galaxies are to be found not in rich clusters but in loose groups. The Galaxy belongs to one such loose group--the Local Group. 

The Local Group.
The Local Group contains seven reasonably prominent galaxies and perhaps another two dozen less conspicuous members. The dominant pair in the group is the Milky Way and Andromeda, both giant spirals of Hubble type Sb and luminosity class II. The distance to the Andromeda system was first measured by Hubble, but his estimate was too low by a factor of two because astronomers at that time did not recognize the distinction between variable stars belonging to Population II (like those studied by Shapley) and Population I (those studied by Hubble). Another spiral in the Local Group--M33, Hubble type Sc and luminosity class III--is notable, but the rest are intermediate to dwarf systems, either irregulars or ellipticals. Most of the mass of the Local Group is associated with the Milky Way and Andromeda, and with a few exceptions the smaller systems tend to congregate about one or the other of these galaxies. The size of the Local Group is therefore larger only by about 50 percent than the 2 {times}106 light-years separating the Milky Way system and the Andromeda galaxy, and the centre of mass lies roughly halfway between these two giants.

The Andromeda galaxy is one of the few galaxies in the universe that actually has a velocity of approach with respect to the centre of the Galaxy. If this approach results from the reversal by the mutual gravitational attraction of a former recession, then the total mass of the Local Group probably amounts to a few times 1012 solar masses. This is greater than the mass inferred for the optically visible parts of the galaxies and is another manifestation of the dark matter problem. 

Neighbouring groups and clusters.

Beyond the fringes of the Local Group lie many similar small groups. The best studied of these is the M81 group, whose dominant galaxy is the spiral galaxy M81. Much like the Andromeda and Milky Way systems, M81 is of Hubble type Sb and luminosity class II. The distance to M81, as well as to the outlying galaxy NGC 2403, can be determined from various stellar calibrators to be at a distance of 107 light-years. It is not known whether NGC 2403 and its companion NGC 2366 are truly bound to M81 or whether they are an independent pair seen by chance to lie near the M81 group. If they are bound to M81, then a measurement of their velocity along the line of sight relative to that of M81 yields, by an argument similar to that used for the Andromeda and Milky Way galaxies, an estimate of the gravitating mass of M81. This estimate equals 2 {times} 1012 solar masses and exceeds by an order of magnitude what is deduced from measurements of the rotation curve of M81 inside its optically visible disk.

The M81 group also has a few normal galaxies with classifications similar to those of galaxies in the Local Group, and it was noticed by some astronomers that the linear sizes of the largest H II regions (which are illuminated by many OB stars) in these galaxies had about the same intrinsic sizes as their counterparts in the Local Group. This led Allan Sandage and the German chemist and physicist Gustav Tammann to the (controversial) technique of using the sizes of H II regions as a distance indicator, because a measurement of their angular sizes, coupled with knowledge of their linear sizes, allows an inference of distance.

This method can be used, for example, to obtain the distance to the M101 group, whose dominant galaxy M101 is a supergiant spiral--the closest system of Hubble type Sc and luminosity class I. Since Sc I galaxies are the most luminous spiral galaxies, with very large H II regions strung out along their spiral arms, determining the distance to M101 is a crucial step in obtaining the absolute sizes of the giant H II regions of these important systems. The sizes of the H II regions in the companion galaxies of M101 compared with the calibrated values for nearby galaxies of the same class yield a distance to the M101 group of approximately 2 {times}107 light-years.

Having calibrated the sizes of the giant H II regions in M101, Sandage and Tammann could then obtain the distances to 50 field Sc I galaxies. Once this had been done, it became possible to measure the absolute brightnesses of Sc I galaxies, and it was ascertained that all such systems have nearly the same luminosity. Since Sc I galaxies like M101 or M51 can be recognized on purely morphological grounds (well-organized spiral structure with massive arms dominated by giant H II regions), they can now be used as "standard candles" to help measure the distances to irregular clusters that contain such galaxies (e.g., the Virgo cluster containing the Sc I galaxy M100).

The Virgo cluster is the closest large cluster and is located at a distance of about 5 {times}107 light-years in the direction of the constellation Virgo. About 200 bright galaxies reside in the Virgo cluster, scattered in various subclusters whose largest concentration (near the famous system M87) is about 5 {times}106 light-years in diameter. Of the galaxies in the Virgo cluster, 68 percent are spirals, 19 percent are ellipticals, and the rest are irregulars or unclassified. Although spirals are more numerous, the four brightest galaxies are giant ellipticals, among them M87. Calibration of the absolute brightnesses of these giant ellipticals allows a leap to the distant regular clusters.

The nearest rich cluster containing thousands of systems, the Coma cluster, lies about seven times farther than the Virgo cluster in the direction of the constellation Coma Berenices. The main body of the Coma cluster has a diameter of about 2.5 {times}107 light-years, but enhancements above the background can be traced out to a supercluster of a diameter of about 2 {times} 108 light-years. Ellipticals or S0s constitute 85 percent of the bright galaxies in the Coma cluster; the two brightest ellipticals in Coma are located near the centre of the system and are individually more than 10 times as luminous as the Andromeda galaxy. These galaxies have a swarm of smaller companions orbiting them and may have grown to their bloated sizes by a process of "galactic cannibalism" like that hypothesized to explain the supergiant elliptical cD systems (see above).

The spatial distribution of galaxies in rich clusters such as the Coma cluster closely resembles what one would expect theoretically for a bound set of bodies moving in the collective gravitational field of the system. Yet, if one measures the dispersion of random velocities of the Coma galaxies about the mean, one finds that it amounts to almost 900 km/sec. For a galaxy possessing this random velocity along a typical line of sight to be gravitationally bound within the known dimensions of the cluster requires Coma to have a total mass of about 5 {times}1015 solar masses. The total luminosity of the Coma cluster is measured to be about 3 {times}1013 solar luminosities; therefore, the mass-to-light ratio in solar units required to explain Coma as a bound system exceeds by an order of magnitude what can be reasonably ascribed to the known stellar populations. A similar situation exists for every rich cluster that has been examined in detail. This dark matter problem for rich clusters was known to the Swiss astronomer Fritz Zwicky as early as 1933. The discovery of X-ray-emitting gas in rich clusters has alleviated the dynamic problem by a factor of about two, but a substantial discrepancy remains.

Superclusters.

In 1932 Harlow Shapley and Adelaide Ames introduced a catalog that showed the distributions of galaxies brighter than 13th magnitude to be quite different north and south of the plane of the Galaxy. Their study was the first to indicate that the universe might contain substantial regions that departed from the assumption of homogeneity and isotropy. The most prominent feature in the maps they produced in 1938 was the Virgo cluster, though already apparent at that time were elongated appendages that stretched on both sides of Virgo to a total length exceeding 5 {times}107 light-years. This configuration is the kernel of what came to be known later--through the work of Erik Holmberg, Gérard de Vaucouleurs, and George O. Abell--as the Local Supercluster, a flattened collection of about 100 groups and clusters of galaxies including the Local Group. The Local Supercluster is centred approximately on the Virgo cluster and has a total extent of roughly 2 {times}108 light-years. Its precise boundaries, however, are difficult to define inasmuch as the local enhancement in numbers of galaxies above the cosmological average in all likelihood just blends smoothly into the background.

Also apparent in the Shapley-Ames maps were three independent concentrations of galaxies, separate superclusters viewed from a distance. Astronomers now believe superclusters fill perhaps 10 percent of the volume of the universe. Most galaxies, groups, and clusters belong to superclusters, the space between superclusters being relatively empty. The dimensions of superclusters range up to a few times 108 light-years. For larger scales the distribution of galaxies is essentially homogeneous and isotropic--that is, there is no evidence for the clustering of superclusters. This fact can be understood by recognizing that the time it takes a randomly moving galaxy to traverse the long axis of a supercluster is typically comparable to the age of the universe. Thus, if the universe started out homogeneous and isotropic on small scales, there simply has not been enough time for it to become inhomogeneous on scales much larger than superclusters. This interpretation is consistent with the observation that superclusters themselves look dynamically unrelaxed--that is, they lack the regular equilibrium shapes and central concentrations that typify systems well mixed by several crossings. 

Statistics of clustering.

The description of galaxy clustering given above is qualitative and therefore open to a charge of faulty subjective reasoning. To remove human biases it is possible to take a statistical approach, a path pioneered by the American statisticians Jerzy Neyman and Elizabeth L. Scott and extended by H. Totsuji and T. Kihara in Japan and by P.J.E. Peebles and his coworkers in the United States. Their line of attack begins by considering the correlation of the angular positions of galaxies in the northern sky surveyed by C.D. Shane and C.A. Wirtanen of Lick Observatory, Mount Hamilton, Calif. If the intrinsic distribution in the direction along the line of sight is assumed to be similar to that across it, then it is possible to derive from the analysis the two-point correlation function that expresses the joint probability for finding two galaxies in certain positions separated by a distance r. Of special interest is the enhancement in the probability above a random distribution of locations well represented, up to scales of about 5 {times} 107 light-years, as a simple power law, (r/r0)-1.8, with r0 equal to about 2 {times} 107 light-years. Beyond 5 {times} 107 light-years, the enhancement drops more quickly with distance than r-1.8, but the exact way it does this is somewhat controversial.

To summarize, then, when one knows a galaxy to be present, there is a considerable statistical enhancement in the likelihood that other galaxies will be near it for distances of 5 {times} 107 light-years or less, whereas at much larger distances the probability drops off to the expectation for a purely random distribution in space. This result provides a quantitative expression for the phenomenon of galaxy clustering. A similar power-law representation seems to hold for the correlation of galaxy clusters; this provides empirical evidence for the phenomenon of superclustering.

 

 
In addition to angular positions, it is possible to derive empirical information about the large-scale distribution of galaxies in the direction along the line of sight by examining the redshifts of galaxies under the assumption that a larger redshift implies a greater distance in accordance with Hubble's law. A number of groups have carried out such a program, some in fairly restricted areas of the sky and others over larger regions but to shallower depths. A primary finding of such surveys is the existence of huge holes and voids, regions of space measuring hundreds of millions of light-years across where galaxies seem notably deficient or even totally absent. The presence of holes and voids forms, in some sense, a natural complement to the idea of superclusters, but the surprising result is the degree of the density contrast between the large-scale regions where galaxies are found and those where they are not (Figure 3).

  

GRAVITATIONAL THEORIES OF CLUSTERING

The fact that gravitation affects all masses may explain why the astronomical universe, although not uniform, contains structure. This natural idea, which is the basis of much of the modern theoretical work on the problem, had already occurred to Newton in 1692. Newton wrote to the noted English scholar and clergyman Richard Bentley:

 

It seems to me, that if the matter of our Sun & Planets & all ye matter in the Universe was eavenly scattered throughout all the heavens, & every particle had an innate gravity towards all the rest & the whole space throughout wch [sic] this matter was scattered was but finite: the matter on ye outside of this space would by its gravity tend towards all ye matter on the inside & by consequence fall down to ye middle of the whole space & there compose one great spherical mass. But if the matter was eavenly diffused through an infinite space, it would never convene into one mass but some of it convene into one mass & some into another so as to make an infinite number of great masses scattered at great distances from one to another throughout all yt infinite space. And thus might ye Sun and Fixt stars be formed supposing the matter were of a lucid nature.

Modes of gravitational instability.

It was the English physicist and mathematician Sir James Jeans who in 1902 first provided a quantitative criterion for the picture of gravitational instability speculated on by Newton. Jeans considered the idealized initial state of a homogeneous static gas of infinite extent and uniform temperature and asked under what conditions the compressed portions of a small sinusoidal fluctuation would continue to contract gravitationally and become denser and denser (eventually to form galaxies and stars presumably) rather than re-expand because of the increased internal pressure. He found that for gravitational instability to occur the wavelength of the density fluctuation had to exceed a certain critical value, now called the Jeans length, which is proportional to the square root of the ratio of temperature to density.

Two new considerations enter to modify the picture in a universe that begins with a hot big bang: the expansion of the background and the coexistence with matter of a thermal radiation field. The expansion of the background causes the dense portions of unstable small fluctuations to grow much more slowly, at least at first, than the static Jeans theory--as a power of time rather than as an exponential. The thermal radiation field causes greater complications.

First, the existence of a component in the universe other than ordinary matter, radiation, means that one has to specify--particularly in the early stages of the expansion when the energy density of radiation dominates that of matter--whether the radiation field fluctuates together with matter or whether it maintains a uniform level inside which matter fluctuates. Density fluctuations of the first type are called adiabatic perturbations, and those of the second type isothermal (isocurvature) perturbations (because the temperature of the radiation field remains uniform in space and the matter temperature locally equals that of the radiation when they are well coupled).

In the early universe when the radiation temperature was high and matter existed as a highly ionized plasma, neither adiabatic nor isothermal fluctuations could grow, because the intense radiation field resisted compression and, through its strong coupling to ionized matter, prevented the latter also from contracting relative to the overall expansion of the universe. Indeed, the tendency for the excess radiation in the compressed regions of adiabatic fluctuations to try to diffuse out of such regions implies that such fluctuations tend to decay. Therefore, given an arbitrary initial spectrum of adiabatic fluctuations, only those with a large enough scale can survive the decay for the age of the universe up to that point.

Decoupling between ordinary matter and radiation occurs when the temperature drops low enough for free (hydrogen) ions and electrons to recombine. When electrons become attached to atoms, they have a much smaller cross section for interaction with photons than when they were free. This occurs for reasonable cosmological models at a temperature of about 4,000 K. At this time, by coincidence (but perhaps ultimately one of great physical significance), the energy density also begins to drop below the rest-energy density of matter, and the universe turns from being radiation-dominated to being matter-dominated. Past the decoupling epoch, the density fluctuations of the type previously labeled isothermal can grow if they satisfy the original Jeans criterion, whereas those previously labeled adiabatic can grow only if they have survived the prior epoch of damping. Calculations indicate that the smallest unstable fragment of the former type has a mass comparable to that of a globular cluster, while that of the latter type has a mass comparable to that of a giant galaxy or of a large cluster of galaxies, depending on various assumptions.

Among these assumptions is the choice of the form of the dark matter or hidden mass. If the hidden mass is not ordinary matter but instead is contained in exotic forms of elementary particles whose properties have yet to be deciphered, then one needs to specify if and when this hidden mass decouples from the thermal radiation field. Two extremes are often considered: "warm" dark matter and "cold" dark matter. Warm dark matter is typified by such hypothetical particles as neutrinos that have small but nonzero rest mass, which decouple relatively early from the radiation field. Particles of this sort stream freely (nearly at the speed of light in the early universe) and erase initial fluctuations of all scale smaller than a critical coherence length (analogous to but larger than the critical scale introduced by photons for adiabatic fluctuations), above which self-gravity can finally cause growth (when the neutrinos are moving much less rapidly). Cold dark matter is typified by particles that interact only weakly with radiation and ordinary matter and that have sufficient rest mass so as always to possess random thermal motions much less than the speed of light at any stage relevant to the problem of galaxy formation. Density fluctuations of such particles can grow in a fashion similar to that described for isothermal fluctuations of ordinary matter after decoupling; therefore, on the scale of galaxies and larger groups, cold dark matter possesses no coherence length. In either picture, warm or cold, the dark component of the universe supposedly forms a lumpy background into whose concentrations ordinary matter falls eventually to produce galaxies and stars. 

Top-down and bottom-up theories.

The scenarios described in the previous subsection turn out, in the extremes, to lead to two different pictures for the origin of large-scale structure in the universe, which can be given the labels "top-down" and "bottom-up." In top-down theories the regions with the largest scale sizes, comparable to superclusters and clusters, collapse first, yielding flat gaseous "pancakes" of ordinary matter (a description coined by the primary proponent of this theory, the physicist Yakov B. Zeldovich of Russia) from which galaxies condense. In bottom-up theories the regions with the smallest scale sizes, comparable to galaxies or smaller, form first, giving rise to freely moving entities that subsequently aggregate gravitationally (perhaps by a hierarchal process) to produce clusters and superclusters of galaxies. Adiabatic fluctuations of ordinary matter tend to yield a top-down picture, and isothermal fluctuations a bottom-up picture. When hidden mass is added to the calculations, warm dark matter tends to give a top-down picture, and cold dark matter a bottom-up picture.

To make comparisons with observational data, the spectrum (dependence of amplitudes with size scale) of the initial fluctuations are needed as input to numerical simulations on a computer to follow the subsequent growth of structure. The shape of the spectrum is specified by heuristic arguments given first by Zeldovich and the American cosmologist Edward R. Harrison, and the results were later rederived from a first principles calculation of a quantum origin of the universe involving cosmic inflation (see below). Workers must use, however, measurements of the anisotropy of the cosmic microwave background to obtain (or set limits on) the absolute starting amplitudes. When this is done and models are computed, it is found that top-down theories tend to give a better but still imperfect account of the observed spatial distributions (flattened superclusters and large holes and voids) and streaming motions of galaxies. Unfortunately, cluster formation and galaxy formation take place at a redshift z less than 1, too recently relative to the present epoch to be compatible with the observational data. The measurements of the anisotropies of the cosmic microwave background severely limit the amount of power that can exist in the starting adiabatic perturbations, and so the growth to observed structures takes too long to complete. Moreover, neutrinos with their large coherence length probably cannot explain the hidden mass that is inferred to reside in the dark halos of individual galaxies.

Bottom-up theories that include cold dark matter can yield objects with the proper masses (i.e., dark halos), density profiles, and angular momenta to account for the observed galaxies, but they fail to explain the largest-scale structures (on the order of a few times 108 light-years) seen in the clustering data. A possible escape from this difficulty lies in the suggestion that the distribution of galaxies (made mostly of ordinary matter) may not trace the distribution of mass (made mostly of cold dark matter). This scheme, called biased galaxy formation, may have a physical basis if it can be argued that galaxies form only from fluctuations that exceed a certain threshold level. Local upward fluctuations in density on a small scale have a better chance to exceed the threshold if they happen to lie in a large region that has somewhat higher than average densities. This bias then produces galaxies with positions that correlate on a large scale better than the underlying distribution of dark matter whose gravitational clustering has no such threshold effect. Unfortunately, counter simulations show that no amount of biasing can reproduce both the large-scale spatial structure and the magnitude of the observed large-scale streaming motions.

On the problem of the formation of galaxies and large-scale structure by purely gravitational means, therefore, cosmologists face the following dilemma. The universe in the large appears to require aspects of both top-down and bottom-up theories. Perhaps this implies that the hidden mass consists of roughly equal mixtures of warm dark matter and cold dark matter, but adopting such a solution seems rather artificial without additional supporting evidence.

UNORTHODOX THEORIES OF CLUSTERING AND GALAXY FORMATION

Given the somewhat unsatisfactory state of affairs with gravitational theories for the origin of large-scale structure in the universe, some cosmologists have abandoned the orthodox approach altogether and have sought alternative mechanisms. One of the first to be considered was primordial turbulence. This idea enjoys little current favour for a variety of reasons, the most severe being the following. Because it tends to decay over time, turbulence of a magnitude sufficient to cause galaxy formation after decoupling would have had to be much larger during earlier epochs. This seems both unlikely and unnatural. Too delicate a balance is required for primordial turbulence to produce galaxies rather than, say, black holes.

Another suggestion is that energetic galactic explosions due to the formation of a first wave of massive stars may have compressed large shells of intergalactic gas that subsequently became the sites for further galaxy formation and more explosions. Such a picture is attractive because it predicts large holes and voids with galaxies at the interfaces, but it does not avoid the criticism that a "seed" galaxy needs to be formed at the centre of each shell by some other process. If such a process exists, why should it not be the dominant mechanism?

Finally, there is a suggestion that galaxy and cluster formation might take place by accretion around "cosmic strings." Cosmic strings, long strands or loops of mass-energy, are a consequence of some theories of elementary particle physics. They are envisaged to arise from phase transitions in the very early universe in a fashion analogous to the way faults can occur in a crystal that suffers dislocations because of imperfect growth from, say, a liquid medium. The dynamic properties of cosmic strings are imperfectly understood, but arguments exist that suggest they may give a clustering hierarchy similar to that observed for galaxies. Unfortunately, the same particle physics that produces cosmic strings also produces magnetic monopoles (isolated magnetic charges), whose possible abundance in the universe can be constrained by observations and experiments to lie below very low limits. Particle physicists like to explain the absence of magnetic monopoles in the Cosmos by invoking for the very early universe the mechanism of inflation (see below). The same mechanism would also inflate away cosmic strings.

In summary, it can be seen that mechanisms alternative to the growth of small initial fluctuations by self-gravitation all have their own difficulties. Most astronomers hope some dramatic new observation or new idea may yet save the gravitational instability approach, whose strongest appeal has always been the intuitive notion that the force that dominates the astronomical universe, gravity, will automatically promote the growth of irregularities. But, until a complete demonstration is provided, the lack of a simple convincing picture of how galaxies form and cluster will remain one of the prime failings of the otherwise spectacularly successful hot big bang theory. 

THE EXTRAGALACTIC DISTANCE SCALE AND HUBBLE'S CONSTANT

It was noted earlier that the galaxies in the Virgo cluster had an average recession velocity v (as measured by their redshift) of roughly 1,000 km/sec with respect to the Local Group. If the distance r to the Virgo cluster is 5 {times} 107 light-years and if the Virgo galaxies can be assumed to be far enough away to partake in the general Hubble flow, then the application of the Hubble law, v = H0r, yields Hubble's constant as H0 = 20 km/sec per million light-years. The reciprocal of Hubble's constant is called the Hubble time; for the value given above, H0-1 = 1.5 {times}1010 years.

The most naive interpretation for the Hubble time is that of a free expansion of the universe, wherein a Hubble time ago the distant galaxies started receding from one another (in particular, from the Milky Way system), reaching a distance r = vH0-1 in time H0-1 if they fly away at speed v, the fastest receding galaxies getting the farthest away. Rearranging terms yields the Hubble law v = H0r. The interpretation is naive in two respects: (1) it ignores the role of gravitation in slowing down the expansion, so that Hubble's "constant" does not always have the value it does at the present epoch; and (2) it overlooks the part played by gravitation in regulating the global structure of space-time, so that the interpretation of the "velocity" v and "distance" r is modified when distances or redshifts approach values such that v given by the above formula becomes comparable to or exceeds the speed of light. Nevertheless, as will be seen in the discussion of relativistic cosmologies below, the Hubble time does provide a useful rough estimate for the age of the universe. 

The exact value of Hubble's constant is an issue of great controversy among astronomers. Modern estimates for H0 range from 15 to 30 km/sec per million light-years. The source of the discrepancy lies partly in the interpretation of the amount of distortion superimposed atop a pure Hubble flow by the gravitational effects of the Local Supercluster in which the Local Group and the Virgo cluster are embedded and partly in the different calibrators used or emphasized by different workers for the distances to various extragalactic objects.

To avoid the first complication, the interpretation of the velocity field in the Local Supercluster, it is possible to examine the redshift-distance relation implied by Sandage's and Tammann's study of 50 Sc I galaxies. There is little controversy that these distant galaxies do empirically satisfy the idealized linear relationship of the Hubble law. The faintest galaxies in the sample have recession velocities of 9,000 km/sec, and, if they lie at the calibrated distance of 600 million light-years, then H0 = 15 km/sec per million light-years, the same value as Sandage and Tammann derived from their study of the Virgo cluster. Unfortunately, many workers do not accept the determination of Sandage and Tammann of the distances to the nearest Sc I galaxies (in particular, M101). They regard as suspect the technique using the sizes of H II regions as a distance indicator. These astronomers advocate using the relationship found to exist between the luminosity L of a spiral galaxy and the velocity V of its (flat) rotation curve, L proportional to V4, as a basis for measuring extragalactic distances, and they obtain values for H0 that lie on the high end of the range cited above.

As discussed earlier, the classical means of obtaining the distance to the Virgo cluster (a crucial accomplishment) relies on a bootstrap operation to pull the observer up the extragalactic distance ladder one step at a time. The problem with the method is that errors at one level propagate to the next. For this reason, some astronomers prefer using supernova explosions, which can be seen at great distances, to get from the Local Group to the Virgo cluster in one jump. Two basic methods have been developed, one using supernovas of type Ia and the other employing supernovas of type II.

Type Ia supernovas are believed to arise in interacting binaries from the thermonuclear explosion of a carbon-oxygen white dwarf pushed beyond the Chandrasekhar limit by mass transfer from a neighbouring companion star. In the process a fixed amount of radioactive nickel-56 is believed to be produced, whose subsequent decay into cobalt-56 and then to stable iron-56 is thought to power the entire light curve in these events. As a consequence of the uniformity of the underlying processes, type Ia supernovas serve, in principle, as excellent "standard candles" to obtain extragalactic distances. In practice, the uniformity of the underlying conditions has been questioned as being controversial.

Type II supernovas arise when evolved massive stars undergo core collapse, a partial rebound, and an expulsion of the (hydrogen-rich) envelope. Except for a scale factor, the shape of the subsequent light curve allows astronomers to infer a changing size for the rapidly expanding atmosphere. The scale can be obtained by measuring the Doppler shift (yielding the velocity, or time-rate of change of the radius, in kilometres per second) of the same layers of gas. Once the absolute size has been fixed, the absolute brightness can be deduced. 

From the deduced absolute brightness and the measured apparent brightness, the distance to the supernova can then be obtained. In principle, the method could be applied to supernovas of all types; in practice, good knowledge of the opacities is needed to correct for the difference in depth observed in the spectral lines (for the Doppler-shift measurements) and in the continuum light (for the light-curve measurements). Such knowledge is reliable only when the composition of the atmospheric layers is rich in hydrogen.

The supernova techniques tend to yield values of H toward the low end of the range 15 to 30 km/sec per million light-years. For the sake of definiteness, this article adopts the value H0 = 20 km/sec per million light-years, but it should be noted that uncertainties of the magnitude discussed still remain. The corollary of this warning is that the distances quoted for extragalactic objects also are uncertain by the same factor.

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