Given the absolute luminosity, L, of a star, that is, the amount of energy emitted per second by the star, and a measurement of the star's flux f we can infer its distance by using the inverse square law
(1a) f = L/4pr2
Instead of flux f, astronomers like to use a different (if somewhat strange) unit: the magnitude m. Magnitude and flux are related as follows:
The magnitude scale has been made more precise: the scale is now defined so that a difference of exactly 5 magnitudes corresponds to a difference of exactly 100 in flux. For example, a star with m = 6 emits 100 times less energy than one with m = 1. Using the modern definition of magnitude gives an apparent magnitude for the sun of -27, while the brightest stars have apparent magnitudes in the range 0 to -1.
So far we have considered apparent magnitudes, that is, the magnitudes of stellar objects as they appear to us on earth. The difficulty with apparent magnitude is that a star may appear brighter than another either because it is intrinsically brighter or because it is closer or both. Therefore, apparent magnitudes tell us nothing about the true relative luminosities of stellar objects. Astronomers have therefore defined a star's absolute magnitude M as the apparent magnitude of a star if it were placed at a distance of 10 parsecs (32.6 light years) from earth. (One parsec is the distance at which one astronomical unit subtends an angle of 1 second of arc, that is, 1/3600th of a degree.) If the sun were placed at a distance of 10 pc from us it would be a faint star of apparent magnitude +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 they would outshine the midday sun!
Distance Modulus - The difference m - M is the astronomer's preferred measure of distance, called the distance modulus. We can determine how it is related to the distance r by equating equations (1a) and (1b) and writing two equations: one for r = rMpc mega-parsecs (Mpc) and the other for 10-5 (Mpc), that is, 10 pc
L/4prMpc2 = a 10-0.4m,
L/4p(10-5)2 = a 10-0.4M.
Now divide one equation by the other, take logs to base 10 of both sides and re-arrange to get
(1c) m - M = 5 log10(rMpc) - 25.
Thus given a formula written in terms or the distance r, in
mega-parsecs, we can easily convert it to the distance modulus using equation
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. In this theory Einstein describes gravity not a force but rather as a manifestation of free (natural) motion in curved space-times. The geometry of space-time is determined by the energy in the universe, that is, by the energy associated with mass, radiation, pressure and the vacuum.
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 presumably 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, initially Einstein favored a static universe. However, his general relativistic equations predicted dynamic universes. Einstein therefore modified his equations (by adding an extra term) so that the equations would describe 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 what would have been one of the most profound predictions 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 Lamaître (Belgian) re-discovered the solutions, previously found by Friedmann. He too can be regarded as the founder of Big Bang cosmology. Georges Lemaître 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. Lemaître 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 - Following up on the work of others, notably the red-shift measurements by Slipher, Edwin Hubble formulated his recession law: a linear relationship between the red-shift of distant galaxies and their distances. Hubble assumed the red-shifts to be due to the motion of the galaxies away from us. He found that larger red-shifts, and by assumption larger velocities, were correlated with greater distances. He had discovered that the universe is 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 somewhat clearer.
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.
1940s - George Gamow (a Russian ex-student of Friedmann) and later Ralph Alpher and Robert Herman of Johns Hopkins University refined Lemaître's idea of a primeval atom. Alpher and Herman reasoned that far back in the past particles of matter would be constantly colliding with each other. These collisions would have generated a tremendous amount of heat that would create photons of very short wavelengths. The temperature of these primordial photons would be billions of degrees.
But as the universe expands all length scales are stretched by the expansion including the wavelengths of the primordial photons. Recall, that the longer the wavelength the lower a photon's energy. Therefore, as the universe aged, and expanded, the photon energies would be progressively lowered and the ensemble of photons would grow ever colder. Alpher and Herman predicted that the universe should now be bathed in a feeble radiation whose temperature would be just a few degrees above absolute zero. This radiation would be literally the afterglow of the earlier extremely hot dense phase of the universe. Alas for Alpher and Herman their ideas were more or less forgotten.
1965 - At Bell Labs in New Jersey, Arno Penzias and Robert Wilson were preparing a radio telescope to observe the Milky Way. They noted a persistent background noise wherever they pointed their telescope. They tried very hard to get rid of it, but couldn't. It finally dawned on them that this was not mere noise. In fact, they had discovered, by accident, photon radiation coming from outer space that was not associated with any known astronomical object. This radiation, which is in the microwave part of the electromagnetic spectrum, is now called the cosmic microwave background (CMB).
At the same time Bob Dicke and Jim Peebles (at Princeton), working on a suggestion by George Gamow that the universe might have been hot and dense in the past, were just getting ready to look for the afterglow radiation from the early universe when they were scooped by Penzias and Wilson. Sadly for Dicke and Peebles it was Penzias and Wilson who got the 1978 Nobel Prize for Physics for their accidental discovery of the microwave background! Such is life.
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 d, between the galaxy and us, times a constant:
Of course, in reality gravity has been slowing down the universal expansion. If so we would expect the galaxies to be 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 Lamaître the age of the universe is = (2/3)T.
It is tempting to think of the universe as a microscopic grain that subsequently exploded into a void. This is misleading. Firstly, according to general relativity, there was, and is, no void into which the primordial universe exploded. Space and time came into existence at the big bang. Secondly, the universe could well be infinite in extent, in which case it would have been infinite in extent at the big bang! Consequently, each point would be one from which matter and energy expanded. In that sense, the universe would have begun with infinitely many big bangs.
In 1914 Slipher measured the red-shifts of many nebulae. Hubble was able to identify Cepheid variables in many of them and was thus able to measure their distances using the Cepheid luminosity-period relationship discovered by Henrietta Leavitt. The inferred distances were so huge that it became absolutely clear that these nebulae were actually "island universes", or galaxies as we call them today. Hubble found that the wavelengths of the light from distant galaxies 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 red-shifted, that is, shifted towards the red end of the spectrum. The red-shift z is defined as
Ordinarily, a red-shift is caused by the Doppler effect: if the distance between us and the source increases then the light waves will be stretched out. If the separation is decreasing the light waves will be squeezed (that is, blue-shifted). The faster the relative motion the bigger the shift. For speeds v much smaller than that of light the speed is related to the red-shift by the approximate formula
(4) cz =
We said that the red-shift is normally the result of a Doppler effect, that is, the change in wavelength of an emitted wave due to the relative motion between the light source and the observer. However, as we shall see below, the red-shift of the galaxies (sometimes called the cosmological red-shift) is not the result of a Doppler effect. Rather it is a consequence of the stretching of wavelengths by the expansion of the universe.
The Robertson-Walker Metric
Robertson and Walker showed that the Cosmological Principle requires the space-time distance between any two nearby points to have the form
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, that is closed, but has no boundary. 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, an open space, with a negatively curved geometry. (Go here for an interesting philosophical discussion.)
From now on, we shall consider only the case k = 0; that is, models with flat spatial geometry. There is mounting evidence that the real universe has k = 0.
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).
Comoving coordinates - 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, the symbol r does not measure the real (that is, proper) radial distance between galaxies, merely their radial distance with respect to the grid. Of course, at any given time t the coordinate (that is, grid) distance r will correspond to some proper distance.
Cosmic Time - The time t in the metric formula requires some explanation. It is not at all obvious that we can assign a universal proper time throughout the universe, given what we have learnt from relativity theory. But it turns out that we can! The reason 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 cosmic 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 galaxy's proper time could differ by an arbitrarily large amount from that of other galaxies and the concept "the age of the universe" would no longer be very meaningful as the citizen's of each galaxy would assign to the universe a different age!
Happily we live in a rather more sedate universe. Therefore, we can think of space-time as a continuous stack of 3-dimensional spatial hyper-surfaces ordered by a proper time in terms of which we can discuss, in a meaningful way, the evolution of the universe.
In a way, this is a return to Newton's idea of an absolute time. However, unlike Newton's absolute time, the time t is not fundamental in the sense that it is merely an (approximate) consequence of the special condition of this universe.
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!)
NOTE: In the figure R(t) is the same thing as a(t)!
Because the direction is fixed the angles do not change; therefore, dq = df = 0. In this case the Robertson-Walker metric simplifies to
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, after traveling for a time interval dt, 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, from dr to the value a(t)dr.
(7) cdt = a(t)dr.
Let us now consider proper distances along the direction from O to P. By definition, the proper distance is the spatial distance between any two events that are simultaneous; that is, events for which the time difference between them dt = 0. From equation (6) this implies that the proper distance is just ds = a(t)dr. We can integrate this equation and write r0 = òdr as the coordinate (that is, grid) distance between two points, not necessarily near each other. Then r(t) = òds = a(t) r0 is the proper distance at time t between the two points.
The Robertson-Walker metric implies that all distances are scaled by a(t). In particular, this is true of the wavelength of light. If l = a(t)r0 is the wavelength of light at time t and l0 = a(t0) r0 is the wavelength of light at time t0, we deduce that
(10) z + 1= 1/a(t).
We see from equation (9) that the cosmological red-shift is not, in fact, a Doppler effect: it is not due to the recession speed! Rather it is due to the fact that when the light was emitted it had the wavelength l, but when it is later observed its wavelength will have been stretched to l0 because of the expansion that has occurred between the time of emission and the time of observation.
The interval Dt = t0 - t is the proper time between the emission of the light at time t and its reception on Earth at time t0. 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! Is d the distance that occurs in Hubble's law? No! To see this we begin with the scaling law for proper distance
(11) r(t) = a(t)r(t0),
where r(t0) is the proper distance between two points at our epoch t0. Now differentiate this expression with respect to cosmic time t. This gives
dr(t)/dt = v(t) = [da(t)/dt] r(t0)
since, by definition, the rate of change of distance is speed v(t). We can multiply the right hand side by a(t)/a(t), and use equation (11), to get
v(t) = (a/a) [da/dt] r(t0) = [(da/dt)/a] r(t),
which shows that the proper distance r(t), at time t, is proportional to the separation speed v(t), where H(t) = (da/dt)/a is called the Hubble parameter. The Hubble constant is the value of the Hubble parameter at our epoch t0, that is, H0 = H(t0). We see that Hubble's law is a consequence of the scaling law for proper distance, that follows from the Robertson-Walker metric. Moreover, we see that the distance in the Hubble law is the proper distance between any two points and that the speed is really the what one might call the proper speed.
(12) [(da/dt)/a]2 = (8pG/3)r + l/3 - k/a2.
The above differential equation is called Friedmann's equation. It follows from Einstein's equation and the Robertson-Walker metric. Once you have specified how the energy density r depends on the scale factor a(t) Friedmann's equation predicts how the scale factor should change with cosmic time t. Do not be confused by the parameter l in this equation; it is not a wavelength! It is the constant (called the cosmological constant) introduced by Einstein in his (ultimately unsuccessful) attempt to construct static models of the universe from his general relativity equations of 1915. The quantity k is related to the curvature of space. There is good evidence that the geometry of space is flat, that is, k = 0, and that's what we shall assume. Moreover, recent evidence suggests that l is a positive number.
If we can measure the Hubble parameter H(t) accurately for different cosmic times t, we can deduce the form of the scale factor a(t) for the universe and thus test directly different predictions for its functional form. This is the most important task at present in observational cosmology. The form of the scale factor tells us whether the universe will expand forever or whether in the distant future the expansion will be reversed and lead, eventually, to a collapse of the entire cosmos in a big crunch.
This equation can be solved for particularly simple cosmologies.
A dust filled universe - In the Einstein-de Sitter model, which describes a universe filled with pressure-less dust only, we have
r(t) = r(t0)/a3(t),
that is, the energy density decreases like the inverse of the cube of the scale factor. This is easy to understand: if we increase the side of a box by the factor a(t) the volume of the box will increase by the factor a3(t). Therefore, since the energy within the box is (assumed to be) constant the energy density will be reduced by the same factor of a3(t). The quantity r(t0) is the energy density at the present epoch. With this choice for the energy density one obtains
a(t) = (t/t0)2/3
for the scale factor by solving the Friedmann equation.
A photon filled universe - In a universe filled with photons only, the energy density would be reduced by another factor of a(t) over and above the factor a3(t) arising from the dilution due to the volume increase. This is because in addition to the stretching of the box, due to the expansion, the wavelength of every photon within it is also stretched (that is, red-shifted) thereby reducing its energy by the same factor of a(t). Therefore, for a universe filled with photons only (actually, any mass-less object) we have instead
r(t) = r(t0)/a4(t).
The solution of the Friedmann equation gives
a(t) = (t/t0)1/2
for the scale factor.
A dust filled universe with a cosmological constant
An exact solution of Friedmann's equation can also be found for a universe filled with pressure-less dust and a cosmological constant. The cosmological constant can be thought of as the energy of the vacuum, that is, the energy that is present even in the absence of ordinary matter and energy.
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 successes, 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 noted earlier, as the universe expanded the different wavelengths of radiation were stretched out, that is red-shifted, by the expansion. The temperature T of the universe falls like
T(t) = T(t0)/a(t),
where T(t0) is the temperature at the present epoch. 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, which corresponds to a temperature T(t0) = 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 cosmic 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 its favor.
Abundance of Elements: The universe is observed to consist largely of hydrogen and helium in the proportion 76% to 24% (by mass) 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.
The Horizon Problem: The microwave background radiation is observed to be isotropic 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 ensemble of photons 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
where H is Hubble's constant and G is Newton's gravitational constant. We can derive this formula from simple physical reasoning. Consider any large sphere of radius R in an expanding universe filled uniformly with matter and energy of density r. Because of the universal expansion the surface of this sphere expands at the speed
rc = 3H2/8pG,
v = HR.
If the speed v is greater than the escape speed vesc at the surface of the sphere the sphere will expand forever. If the speed v is less than the escape speed at the surface of the sphere the latter will eventually stop expanding and collapse. The escape speed
vesc = (2GM/R)1/2,
is determined by the mass and energy M contained within the sphere. For a flat geometry, M is given by
M = (4pR3/3)r.
If the total mass and energy M is just right, the expansion speed v will be exactly equal to the escape speed. This is precisely the condition that defines the critical density. That is,
v = vesc
HR = (2GM/R)1/2
HR = (2G(4pR3/3)r/R)1/2
H2R2 = (8pG/3)rR2
r = 3H2/8pG.
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
The problem with this value of W is this: to get an 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! 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
and is thus an extraordinarily wonderful accident. But most cosmologists do not find this satisfactory and want to find a
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 Jurassic 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.
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 space-time interval between the two galaxies when each galaxy has the same time. We'll now outline how that can be done.
Just to recap, 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
In big bang models the scale factor a(t) is zero at t = 0. 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 r0 is 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
To actually do the calculation we need a formula for a(t). For the Einstein-de Sitter universe it turns out that
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
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
It is interesting to compare equation (19) with Hubble's law v = H0d. 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)/H0. 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 Lemaître.
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!
and Cosmology of Non-Euclidean Geometry
Ned Wright's Cosmology Tutorial
The Cosmological Constant
The Big Bang, The Creation and Evolution of the Universe, by Joseph Silk (W. H. Freeman and Company, New York, 1980).
An Introduction to mathematical cosmology, J. N. Islam (Cambridge University Press, 1992).
Principles of Physical Cosmology, P. J. E. Peebles (Princeton University Press, Princeton, 1993).