StudyWeb Award
Introduction to Cosmology

Milestones in the Birth of Modern Cosmology

1912 - Henrietta Levitt (American) published her work on Cepheid variables. These stars brighten and dim in a regular fashion. For example, the star Delta Cephei (in the constellation of Cepheus) varies in magnitude (see below) between m = 3.9 and 5.1 over period of 5.37 days. (John Goodricke discovered that Delta Cephei was a variable star on October 20, 1781, one month after the first Cepheid variable, Eta Aquilae, was discovered by Goodricke's friend Edward Pigott.) Levitt discovered that the average luminosity of these stars is related to their periods. Therefore, if we measure a Cepheid variable's period we can infer its absolute luminosity; that is, we can infer the total amount of energy the star emits per second.

Given the absolute luminosity L of a star and a measurement of the star's apparent flux f we can infer its distance by using the inverse square law

(1a) f = L/4pr2
 after making corrections for light absorption between the star and the earth. The absolute luminosity is the amount of energy emitted per second. The flux is the energy received per second per unit area. (The units are joules per second per square meter. One joule per second is equal to one watt. A 100-watt light bulb emits 100 joules of energy per second in the form of light and heat.) Astronomers, however, like to use a different (if somewhat strange) unit: the magnitudem. Magnitude and flux are related as follows: (1b) f = a 10-0.4m Notice that the bigger the magnitude the smaller the flux. Bigger magnitudes correspond to fainter objects! The faintest star that can be seen with the unaided eye is about magnitude 6. The sun has an apparent magnitude of -27, while the brightest stars have apparent magnitudes in the range 0 to -1.  The magnitude scale is devised so that a difference of 5 magnitudes corresponds to a difference of 100 in flux. For example, a star with m = 6 is 100 times fainter than one with m = 1.
    It is useful to have a measure of the true brightness of a stellar object. Astronomers have therefore defined a star's absolute magnitude M as the magnitude of a star if it were placed at a distance of 10 parsecs (32.6 light years) from earth.If the sun were placed at a distance of 10 pc from us it would be a faint star of magnitude of +4.8. On the other hand if we placed some of the brightest objects in the universe at 10 pc from earth then such is their energy output that they would outshine the midday sun!
    (Unfortunately, astronomers use lots of different kinds of magnitude. But there is one that has a direct relation to flux: the bolometric magnitude, for which the constant a = 2.54 x 10-8 watts per square meter. Given the bolometric magnitude of an object we can calculate the flux received from the object in watts per square meter.)

Cepheid variables have proven to be of immense value to observational cosmologists because these giant stars can be seen over immense distances and so provide a way to measure such distances accurately.

1915 - Albert Einstein published his theory of General Relativity. This theory describes the motion of objects through curved spacetimes. Einstein proposed that gravity is not a force but merely a manifestation of free (natural) motion in curved spacetimes. The geometry of spacetime is determined by the energy in the universe, that is, by the energy associated with mass, radiation and pressure.

Einstein proposed an hypothesis about the universe that has come to be known as the Cosmological Principle: the universe is isotropic (it looks the same in all directions), and its energy is uniformly distributed in space. If this principle is true it can only be so on a very large scale (hundreds of millions of light years) because on smaller scales the universe is decidedly non-uniform. The galaxies form clusters and these clusters form superclusters with huge voids between them. On the scale of superclusters the universe appears to have a honeycomb structure. On larger scales, however, it does seem that the cosmological principle holds true, at least approximately.  This principle is accepted by most cosmologists.

In accordance with the cosmological prejudice of his time, Einstein favored initially a static universe. But because his original equations predicted non-static universes he modified the equations to force them to predict what was then believed to be true, namely that the universe did not change. Later he would call this modification the biggest mistake of his life, for it caused him to fail to make the most profound prediction of twentieth century science: that the universe is dynamic, indeed expanding, and may have a beginning. The honor of discovering the universal expansion would belong to Edwin Hubble (1929).

1917 - Willem de Sitter (Dutch) discovered a static solution to Einstein's equation describing a universe in which light from distant objects becomes redder as the distance increases.

1922 - Alexander Friedmann (Russian) abandoned Einstein's static universe model and found solutions to Einstein's original equations that described an expanding universe filled with matter. It described a universe that expanded from a point, a finite time ago. Thus was born Big Bang cosmology.

1927 - Georges Abbe Lamaitre (Belgian) re-discovered the solutions, previously found by Friedmann. He too can be regarded as the founder of Big Bang cosmology. Georges Lemaitre was an interesting fellow. Not only was he a talented cosmologist but he was also a Roman Catholic priest, having been ordained in 1923! What a wonderful irony: a priest who was the founder of one of the cornerstones of modern scientific thought, which thought challenges the basis of the priest's  religious views. Lemaitre seems to have been silent about the degree to which he saw, or did not see, conflict between his religious and scientific views. My own speculation is that he probably saw no conflict, but instead took his scientific discoveries as evidence of the immense power and imagination of a supreme creator.

1928 - Howard Robertson (American) transformed de Sitter's solution into one describing an expanding universe. Unfortunately, his new solution described a universe devoid of matter! Robertson noted a connection between distance and velocity in this model universe. Alas for Robertson, his note was overshadowed by Hubble's spectacular announcement the following year.

1929 - Edwin Hubble formulated his recession law:  a linear relationship between the redshift of distant galaxies and their distances. Hubble assumed the redshifts to be due to the motion of the galaxies away from us. He found that larger redshifts, and by assumption larger velocities, were correlated with greater distances. He had discovered that the universe was expanding! It must be said, however, that others had suggested the idea before him, though without much evidence to support the suggestion. Hubble's breakthrough was to supply the observational evidence in a form that was convincingly clear. But he was surely influenced by previous work on model universes, especially the expanding models.

1932 - Having abandoned the static universe models, Einstein and de Sitter developed an expanding universe model in which the spatial geometry was flat, that is, the spatial geometry obeyed the laws of Euclid. The Einstein-de Sitter model of the universe is generally accepted as offering a reasonable description of the evolution of the universe at the present epoch.

Hubble's Law

Hubble's law states that the velocity of recession v, that is, the velocity with which a distant galaxy is receding from us is equal to the proper distance d, between the galaxy and us, times a constant:

Recall, the proper distance is the spatial distance between two points when the time at each point is the same. The constant, Ho, is called Hubble's constant. Its value is observed to be about 65 km/s per million parsecs (65 km/s/Mpc).
Example: A galaxy at a distance of 1 million parsecs (3.26 million light years) would have a recession velocity of 65 km/s; a galaxy at twice the distance would recede twice as fast, i.e., at 130 km/s and so on. Hubble inferred the recession velocities from the observed redshifts. If we retrace the paths of all the galaxies, and assume that their recession velocities have remained constant in time, we deduce that at a time equal to years ago the universe must have been crushed into a single point, from which it subsequently emerged in an explosive beginning that has been called the Big Bang. The time T is called the Hubble Time and is measured to lie between 10 to 20 billion years.

Of course, in reality gravity has been slowing down the universal expansion. So the galaxies are traveling slower today than they were in the past. We conclude therefore that the age of the universe must actually be less than the Hubble Time. In fact, in the model of Lamaitre the age of the universe is = (2/3)T.

Redshift

Atoms of the same element, for example hydrogen, emit light that consists of a definite set of colors. Each color corresponds to a different wavelength of light. Because an element is associated with a unique signature, that is, a unique set of wavelengths, it possible to recognize the type of atoms in distant galaxies by measuring the wavelengths of the light they emit and matching the measured wavelength values with the known element signatures.

When Hubble measured the wavelengths of the light from distant galaxies he found that the wavelengths were longer than those measured from stationary atoms on earth. Since red light has a longer wavelength than, for example, yellow light any wavelength that is longer than its usual value is said to be redshifted, that is, shifted towards the red end of the spectrum. The redshift z is defined to be

(3) z = (lo - le)/le, where lo is the observed wavelength and le is the emitted wavelength. The larger the value of z the bigger the redshift. The redshift can be caused by the Doppler effect: a source that moves away causes the light waves to be stretched out; a source that moves towards us causes the light waves to be squeezed (that is, blueshifted). The faster the motion the bigger the shift. For speeds v much smaller than that of light the speed is related to the redshift by the approximate formula v = cz, where c is the speed of light. (A more accurate formula is needed for speeds near the speed of light.) Using this formula we can write Hubble's law in terms of the redshift z, which is the quantity that can be measured directly by analyzing the light from galaxies: (4) z = d Ho/c
 
 

Universal Expansion

Even before Hubble formulated his law, the idea of an expanding universe had been considered by several people, chief amongst them Friedmann, Lemaitre and Robertson. They based their models of the universe on Einstein's Cosmological Principle hypothesis and on Einstein's equations of general relativity. The solutions to these equations are formulas for calculating the spacetime distance (the interval) between any two nearby events. Recall that an event is a given place at a given time; an event can therefore be labeled by four numbers, for example
(t, r, q, f). By studying how the interval changed in time we can gain an understanding of the evolution of the universe. (See expanding balloon.)

The Robertson-Walker Metric

Robertson and Walker showed thatthe Cosmological Principle requires the spacetime distance between any two nearby points to have the form

 (5) ds2 = (cdt)2 - a2(t)[dr2 + f(r,k)2(dq2+ sin2q df2)]. This formula is called the Robertson-Walker metric. The function f(r,k) describes the spatial geometry of the universe.

The case k = 0, f(r,k) = r, corresponds to the model developed by Einstein and de Sitter. It describes a universe whose spatial geometry is flat; that is, space is infinite in this model and obeys the geometrical laws of  Euclid.

The case k > 0, f(r,k) = sinÖkr/Ökr, describes a universe with a curved geometry (rather like that of a sphere, except this is curvature in 3 dimensions). Like a sphere (which is a 2-dimensional space) this 3-dimensional space is finite but has no boundary, that is closed. This means that if you set off from Earth in one direction and moved in as straight a line as possible (that is, you moved along a geodesic) eventually you would return back to Earth, without ever having turned back and without ever having reached a boundary!

The case k < 0, f(r,k) = sinhÖ|k|r/Ö|k|r, describes an infinite space, that is, open, with a negatively curved geometry. (See spatial geometry.)

From now on, we shall consider only the case k = 0; that is, the Einstein-de Sitter model.

The quantity a(t) that multiplies the spatial distance dl = dr2 + r2(dq2+ sin2q df2) is called the scale factor of the universe. It describes how the spatial part of the universe expands or contracts. In an expanding universe a(t) increases with time. This implies that the distance dl between any two nearby points increases by the factor a(t) as the universe evolves. Different models of the universe correspond to different formulas for the scale factor a(t).

Notice, that according to the Robertson-Walker metric, the coordinates (r, q, f) expand with the universe! They are called comoving coordinates. They can be pictured as a grid of lines, spread across space, that stretches with the expansion. The galaxies are assumed to be fixed with respect to this expanding grid. Therefore, "r" does not measure the real (that is, proper) radial distance between galaxies, merely their radial distance with respect to the grid.

The Scale Factor a(t)

We shall now show how the Robertson-Walker metric can explain Hubble's law.

Consider the figure below. Suppose that the Milky Way galaxy is situated at the point O with (arbitrarily chosen) coordinate position r = 0, in a spherical polar coordinate system. Now consider a light ray moving along a fixed direction in space to a nearby point P, a radial distance dr away. This is the distance to the point at some specific (but arbitrarily chosen) time. We have to specify the time at which we measure distances because distances are always changing due to the universal expansion. Let's assume that dr is the distance between O and P at the time t when the light leaves the Milky Way galaxy.  (Note, on the scale of the universe nearby could still be millions of light years!)
 
 

Because the direction is fixed the angles do not change; therefore, dq = df = 0. In this case the Robertson-Walker metric simplifies to

(6) ds2 = (cdt)2- a2(t)dr2 In a previous lecture we noted that light rays travel along null geodesics in spacetime, so we must set the interval ds = 0 for a light ray. This leads to (7) cdt = a(t)dr What this means is that in a short time interval dt (again short could still be millions of years) light can travel a distance equal cdt, where c is the speed of light. But by the time the light reaches the nearby point P that point will have moved an extra radial distance.  The distance between the nearby point P and the Milky Way galaxy O will have been stretched, by the universal expansion, to the value a(t)dr.

Now imagine two light rays emitted from the Milky Way at times t and t0. Suppose the first ray reaches the nearby point a short time later dt, while the second one reaches the point after a time interval dt0. In general, dt is not equal to dt0 because the universe will have expanded by different amounts at times t and t0. But we can relate these two time intervals using equation (7):

(8a) cdt   = a(t)dr
(8b) cdt0 = a(t0)dr The distance dr is the same in the above expressions because, by assumption, this is the distance between the Milky Way and the nearby point at some specific time, namely the time t. We can combine equations (8a) and (8b) to give (9) dt0/dt = a(t0)/a(t) This is an important relationship for it tells us how the relative change in travel time of light between nearby points is related to the universal scale factor at different times in the universe's history. In particular, when we interpret dt0 and dt as the oscillation periods of light waves we can use equation (9) to derive Hubble's law. We first note that the wavelengths of light at the two times t and t0 are related to the periods of oscillation of the light waves according to (10a) l   = cdt
(10b) l0 = cdt0 Therefore, when we put equation (10a) and (10b) into equation (9) we get (11) l0/l = a(t0)/a(t) If we take l0to be the wavelength observed on earth at the present epoch t0 and l to be the emitted wavelength at an earlier time t we arrive at the relationship (12) z = [a(t0)-a(t)]/a(t) between the redshift z and the scale factor a(t). The interval Dt = t0 - t is the time between emission of the light and its reception on Earth. It is called the look-back time. By multiplying it by the speed of light c we get the distance d=cDt that light had to travel to get from the galaxy to the Earth. This is the usual distance we quote for the distance to stellar objects. But notice that it is not the distance between the galaxy and Earth, because while the light was traveling towards the Earth the universe has been expanding. Now from equation (12) we can write (13) z = (d/cDt) [a(t0)-a(t)]/a(t). When we compare the above formula with Hubble's law equation (4) we can obtain that law if we identify a new parameter H(t) as follows: This parameter, called the Hubble parameter, is directly related to the universal scale factor, a(t).  The Hubble constant is the value of the Hubble parameter as we let the time t approach our epoch t0, that is, Ho = H(t->t0).

If we can measure the Hubble parameter accurately for different emission times t, we can deduce the form of the scale factor a(t) for the universe and thus test directly different predictions for the form of the scale factor. This is the most important task, at the moment, in observational cosmology. The form of the scale factor will tell us whether the universe will expand forever or whether at some distant future time the expansion will be reversed and lead, eventually, to a re-collapse of the entire cosmos.
 

Successes of, and Problems with, Big Bang Cosmology

Successes

There are three key successes of Big Bang cosmology: 1) derivation of the Hubble law, 2) prediction of the microwave background radiation, and its characteristics and 3) prediction of the abundances of the lightest elements: hydrogen, deuterium, helium and lithium.

Hubble Law: We saw above that the expansion models could explain the Hubble law. That is an important  success of these models. But that success is, arguably, insufficient reason to accept these models of the universe as good approximations to the truth. The other two succeses, however, are much more compelling.

Microwave Background: The expansion of the universe implies that in the distant past the universe was considerably smaller than it is at the present epoch. When matter is compressed it gets hotter, and radiates electromagnetic energy. Therefore, we expect that in its infancy the universe must have been fantastically hot and filled with radiation. In the beginning there was indeed light; lots of it!

As the universe expanded the different wavelengths of radiation were stretched out, that is redshifted, by the expansion.  Calculations using reasonable models for the scale parameter a(t), together with the measured value of Hubble's constant, predict that the primordial radiation should have been stretched to wavelengths of about 5 mm, at the present epoch. (This corresponds to a temperature of 2.7 Kelvin.) These wavelengths, which are about ten thousand times longer than the wavelength of light, are in the microwave part of the electromagnetic spectrum. The universe should be filled with microwave radiation, the afterglow of its fiery beginning, with a spectral shape indicative of radiation in thermal equilibrium. Radiation in thermal equilibrium follows a blackbody spectrum.

This microwave background radiation was indeed discovered, by accident, in 1965 by Arno Penzias and Robert Wilson, who worked at Bell Labs. The existence of  this radiation, with the black body spectral characteristics predicted by Big Bang cosmology, is considered by most cosmologists to be strong evidence in support of this cosmology.

Abundance of Elements: The universe is observed to consist largely of hydrogen and helium in the proportion 76% to 24% (by weight) with trace amounts of everything else. How do these proportions come about? According to the Big Bang cosmology these elements were created during the first 3 minutes of the universe's history by a process called nucleosynthesis. This is the creation of heavier nuclei from lighter ones. At these early times the temperature of the universe would have been about one billion degrees Kelvin, which is more than 60 times hotter than the center of the sun. Detailed calculations of the nuclear reactions taking place at that time predict a hydrogen to helium abundance in excellent agreement with observation. This constitutes another impressive confirmation of this theory.
 

Problems

The Big Bang cosmology has three major problems that are known as 1) the horizon problem, 2) the smoothness problem and 3) the flatness problem.

The Horizon Problem: The microwave background radiation is observed to be isotropric to a very high precision. Isotropic in the present context means that the radiation looks the same in all directions. For example, the radiation coming down onto the Earth's north pole has the same temperature (2.7 Kelvin) as that incident upon the Earth's south pole. We believe that the only way to get such isotropy is to suppose that the radiation from every part of the universe was able to interact with the radiation from every other part until the radiation achieved  a uniform temperature throughout the universe. But the problem is that it does not seem possible that the radiation coming from opposite sides of the universe could have interacted with each other. Why? Because there has not been enough time since the beginning of the universe for radiation to have traversed the gulf of space between one side of the universe and the other. The standard Big Bang cosmology offers no explanation for the observed isotropy.

The Smoothness Problem: We observe galaxies, clusters of galaxies, and clusters of clusters of galaxies. These structures must have arisen from tiny variations in the density of energy in the early universe. Where the densities were greatest is, presumably, where gravity caused matter to collapse into the structures we see today. The problem is that to explain these structures we must assume that the universe was created in an incredibly smooth non-chaotic manner. This seems extremely unlikely. The presumed extreme smoothness is not explained.

The Flatness Problem: Big Bang cosmology defines a critical density rc given by

rc = 3 H2/8pG where H is Hubble's constant and G is Newton's gravitational constant. When we put in the numbers the critical density comes out to about 3 hydrogen atoms per cubic meter. Not much! If the actual density of universe is called r we can define a parameter W = r/rc that measures how close the actual density is to the critical density. If W > 1 the universe will re-collapse, eventually. This is called the Big Crunch. If W < 1 the universe will expand forever. If W = 1 the universe will expand forever with a spatial geometry that is flat. That is, the universe will behave according to the Einstein-de Sitter model. Current measurements suggest that 0.1 < W < 2. The problem with this is that to achieve a value of W in that range at the present epoch requires a value of W that differs from unity by less than one part in a trillion, when the universe was no more than one second old! The Big Bang cosmology cannot explain why W in the past is so close, but not exactly equal, to one.

These three problems deal with the initial conditions of the universe's history. One solution is simply to assert that the universe just started with these highly unusual conditions. But most cosmologists do not find this satisfactory and want to find a deeper explanation. They want to know if the observable universe is inevitable or if is it merely an extraordinarily wonderful accident.
 

How Far is Far?

The Cartwheel Galaxy, shown on the right, lies at a distance of 500 million light years. What does this mean? This means that the light reaching us now, from that galaxy, has taken 500 million years to reach us. In other words the light left the Cartwheel Galaxy 500 million years ago, long before the Jurrasic period and long before what we now call the North American continent collided with what would become Africa. It also means that the light has traveled a distance of cDt, where c is the speed of light and the interval Dt is 500 million years. As noted above this interval Dt is called the look-back time.

At a given universal time t in the evolution of the universe the spacetime interval between any two points is called the proper distance between the points. In other words the proper distance between any two points is the distance between them when the time at each point is the same. The time at each point of spacetime is called the proper time; it is the time given by a clock attached to the point. The importance of the notions of proper time and proper distance is that all observers in the universe will agree on the values of these quantities. By thinking of spacetime as a continuous stack of 3-dimensional hypersurfaces ordered by proper time we can discuss, in a meaningful way, the evolution of the universe.

(The reason we can assign a universal proper time is because of the uniformity of the universe on very large scales and the fact that the galaxies are moving through space relatively slowly. The observed uniformity of the microwave background provides a natural universal frame of reference. Relative to this universal frame of reference the speeds of galaxies are low compared with that of light. Therefore, to a very good approximation they share the same proper time. This is fortunate. If the galaxies were instead moving at near light speed, relative to this universal frame of reference, each would have its own proper time and the concept "the age of the universe" would no longer be very meaningful, as the citizen's of every galaxy would assign to the universe a different age!)

The distance cDt is not the proper distance between the two galaxies because it depends on two times: when the light was emitted and when it's received. It is just the distance traveled by the light. To compute the proper distance we must find a formula for the spacetime interval between the two galaxies when each galaxy has the same time. We'll now outline how that can be done.

By definition, the proper distance between two nearby points is the interval between the two points when we set the time difference between them to zero, i.e., dt = 0, in equation (6). We find

(14a) ds = a(t)dr as the proper distance, at universal time t. If now we sum equation (14a) over a finite coordinate range r = 0 to r = r1, for a fixed time t we get (14b) r(t) = a(t) r0 as the proper distance between two points, at time t. The proper distance r(t) is just the sum of ds. The distance r0 is the sum of dr; it is just the difference in the grid coordinates between the two galaxies.

In big bang models the scale factor a(t) is zero at t = 0; that is, the universe starts from a point. For times greater than t = 0 the scale factor increases; that's what we mean by an expanding universe. At some universal time t0 the scale factor will be equal to unity. The time t0 at which this happens is not a fundamental quantity; it is a matter of convention. Usually, we define the scale of a(t) so that it is equal to 1 at the present epoch, denoted by t0.  With this choice we find, from equation (14b), r(t0) = r0. By our convention we have made r0 the proper distance between the two galaxies, at the present epoch.

Now consider equation (7), which defines the path of light rays. We can rewrite equation (7) as

(14c) dr = cdt/a(t). We would like to compute the proper distance r(t) between the Milky Way and the Cartwheel Galaxy at any universal time t, and in particular at the time the light left the Cartwheel 500 million years ago when, presumably, the galaxy was closer to us. Assuming that the current epoch corresponds to a time t0 = 15 billion years since the big bang the light we are now receiving left the Cartwheel at a time t1 = 15 billion - 0.5 billion = 14.5 billion years after the big bang. Since we want to compute r(t1), that is, the proper distance between the Cartwheel and us, when the universe was t1 = 14.5 billion years old we should use our general formula for the proper distance r(t), given in equation (14b). But for this formula to be useful we need to compute the proper distance r0 in terms of the universal times t1 and t0. That's easy; we just sum equation (14c) between the times t1 and t0. That is, we must do (15) r0 = òcdt/a(t) between t = t1 and t = t0. This may be a bit puzzling; r0 is the proper distance between the Milky Way and the Cartwheel at time  t0. Why should r0 have anything to do with the time t1? The paradox is resolved by noting that light travels on null intervals along which the proper time interval is zero. So from the point of view of the light ray the times t = t0 and t = t1 appear simultaneous! Therefore, the light ray can be used to measure proper distances.

To actually do the calculation we need a formula for a(t). For the Einstein-de Sitter universe it turns out that

(16) a(t) = (t/t0)2/3 Notice that when t = t0 we have a(t0) = 1, by construction. When the calculation is done we obtain finally (17) r0 = 3c t0[1- (t1/t0)1/3] as the proper distance, at the present epoch, between the two galaxies.

When we put in the numbers t0 = 15, t1=14.5 and c=1 we get r0 = 0.506 billion light years. With the times given in billions of years and the units for the speed of light chosen so that it has the value c = 1, the proper distance will come out in billions of light years.  So right now the Cartwheel Galaxy is at a proper distance of 506 million light years. But owing to the universal expansion the proper distance in the past must have been less than it is today. How can we calculate this?

Equation (14b) supplies the answer: the proper distance r(t) between the galaxies at any universal time t. Thus we arrive at

(18) r(t) = r0(t/t0)2/3 for the proper distance at any time t. When we set t = t1, the time at which the light, now being received, left the Cartwheel, we obtain a proper distance of r(14.5) = 0.506 (14.5/15)2/3
           = 0.494 billion light years, that is, 494 million light years. We therefore conclude that since the end of the Cambrian Age on earth to the present time the expansion of the universe has pushed the Cartwheel Galaxy away from us a proper distance of about 12 million light years!

To complete this discussion we shall derive a formula for the speed with which proper distances increase in an Einstein-de Sitter universe. That's easy (if you know a bit of calculus): just differentiate equation (18) with respect to the time t to obtain

(19) v(t) = (2/3) r0/(t1/3 t02/3) = (2/3)r(t)/t. The function v(t) gives the speed, as a fraction of that of light, with which proper distances increase at any given universal time t. For example, at the present epoch t0 = 15 (billion years) the proper distance between the Milky Way and the Cartwheel Galaxy is increasing at v(15) = (2/3) r(15)/15 = (2/3) 0.506/15 = 0.022 times the speed of light, that is, at the rate of 2.2% times the speed of light.

It is interesting to compare equation (19) with Hubble's law v = d Ho. We can write equation (19) as v(t0) = r(t0) [(2/3)/t0] at the present epoch t = t0. This leads to the identification t0 = (2/3)/Ho. By definition, the present time t0 is the age of the universe, which we see is equal to two-thirds times the inverse of the present value of the Hubble constant, a result first found by the Belgian priest Georges Lemaitre.

As we wind back the universal clock equation (19) predicts that the speed increases, while equation (18) shows that all proper distances decrease. Just for fun let's work out the epoch at which the Cartwheel Galaxy was receding from us at the speed of light. This we can do by simply setting v(t) = 1 in equation (19) and then solving for the time t. We find that this happened at about 170,000 years after the big bang. From equation (18) we calculate that the spatial location of the energy that would condense to form the Cartwheel was at a proper distance of just over 250,000 light years. The temperature of universe at that time was several thousand degrees Kelvin and the universe was still awash in a sea of intense light.

Equation (19) says even more. It predicts that for times earlier than 170,000 years after the big bang the Cartwheel-to-be was moving away at greater than the speed of light! And if we wind the clock all the way back to the earliest times the expansion velocity becomes arbitrarily large. But doesn't this violate the universal speed limit? No! Because here it is space that is expanding. Matter may not travel through space at relative speeds greater than that of light. But, according to these calculations, which are based on Einstein's theory of general relativity, space can expand as fast as it likes!

Thus we conclude that in the beginning the bang was truly BIG!

Interesting links:

Ned Wright's Cosmology Tutorial
The Cosmological Constant

Last updated October 6, 1999, Harrison B. Prosper