10.1 Cornetti for Breakfast
Giuseppe was a simple market gardener, as was his father before him, and his grandfather before him, and so on. Giuseppe was not sure how long his family had been growing fruit and vegetables for the local market at Castellina in Chianti, Italy, but it had been for a very long time. Without doubt, this gave him some rights, or so he believed. He smiled to himself at the frustrated long line of vehicles behind his tiny three-wheeled Ape as, laden with vegetables, he made his way into town each Saturday along the narrow Tuscan roads.
Horticulture had always been Giuseppe’s life. Even at school, he had interpreted every lesson to see how it might be used to improve the vegetable gardens that one day would be his to run. This interest in advancing the family business had all started, strangely enough, when his class teacher explained some basic concepts of geometry, such as circles and spheres.
“It is a remarkable thing”, she used to tell the class, “that the circumference of a circle is proportional to its radius, while its surface is proportional to the square of the radius. That means that if you double the radius, the circumference only doubles, but the surface becomes four times as big.” She paused to let the children take in the implications of her remarks, then reached into a drawer and held up an apple that she had brought in for a snack. “But in a sphere,” she continued, “the volume becomes eight times larger.”
This lesson was not lost on the young Giuseppe, and at the market the next Saturday, he peered at a tray of strawberries, some as large as a tomato, and at tomatoes the size of a cherry, and an idea began to form. “If I can produce larger and larger fruits and vegetables, it will be much quicker to peel them, since the volume grows much faster than the surface to be peeled. And it is not only a question of time, but also of economy, since by peeling less, there will be less wastage.”
So in his spare time, Giuseppe began to carry out experiments in a little-used greenhouse at the rear of the farm. He was unsuccessful however, and eventually gave up. It was not until years later, when he finally inherited the farm, and was walking its perimeter, that he saw the old greenhouse and recalled his investigations. One morning, while he was spreading a thick layer of strawberry jam on his breakfast cornetto, Giuseppe let his mind drift back to those earlier experiments. He quaffed his macchiato in one gulp and reached for another two cornetti, and that was the precise moment that the solution came to him. He couldn’t wait to try out this new technique to control plant growth hormones.
This time Giuseppe was successful. He refused to divulge his secret to other growers, and slowly his products became sought after, and soon he could hardly keep up with demand. All his fruit and vegetables were grown to fixed weights, so that they could be sold without the need to weigh them, and with uniformly high quality. In addition, they were 100% “organic”, and not in the least genetically modified.
One morning a good friend of his, the cook of a local Greek restaurant, came to buy two 5 kg eggplants for preparing his renowned moussakas. He knew that their weight was reliable. Nevertheless, he insisted on their being weighed, so he could express his wonder, and repeat his favourite joke: “Next you’ll have to grow them cubic to better suit the shape of my baking pans And also grow them some legs, so they can run down the hill to my restaurant, saving my arthritic knees.” Giuseppe laughed at the old gag, and put the eggplants on his scales. To his great surprise, he found that they weighed 5.1 kg each, 100 g above their nominal weight.
Later, back home again, Giuseppe was still disturbed by the incident, and he decided to weigh his 5 kg fruits and vegetables. All of them were now 5.2 kg. He was very confused, but cheered up a little when he realised that at least now his profits would increase. Something, however, was definitely not right.
The next morning, as soon as he got out of bed, Giuseppe felt slightly uncomfortable. He seemed to require more of an effort to move. His stomach rumbled, but that was usual before breakfast. On an impulse, he walked into the bathroom and stepped onto the scales. Che Diavolo! He had gained 2 kg since yesterday morning.
He was really puzzled now, as he entered the kitchen and began to prepare his morning coffee, and warm up his cornetti. He switched on the radio to his usual news station, and caught the end of an item on a certain Mr Newton, and how experts believed his constant might have suddenly begun to change.
“Stupido,” he proclaimed aloud. “If constants aren’t constants, then I am not Giuseppe, and these eggplants are pineapples.” Annoyed, he changed the station, but there was more nonsense here as well. They also were talking about changing weights, but this time the culprit was Mr Higgs and some bosons he’d found. “What the hell are bosons, I wonder?” he muttered.
Giuseppe returned to the bathroom and stepped onto the scales again. His weight had already increased by one more kilo. It seemed that a force greater than anything he could muster was determined to make him gain weight. Well, so be it. At least there were compensations. “When there’s nothing to be done, it’s better not to resist” his father had once advised him. He returned to the breakfast table, and placed two more large cornetti on his plate. His saliva was already flowing as he covered them in jam and cream, and took the first massive bite of his breakfast. Whatever these bosons were, they clearly wanted him heavier, and who was he to argue?
***
As we shall see later in the next two Chapters, whether constants remain constant is an important question in the study of cosmology, and one that it is very difficult to answer.
10.2 The Very Beginning
In Chap. 2, we explored the possible origins of human intelligence and the evolution of logic. As these aptitudes evolved, early humans applied their new skills to make sense of the world in which they lived. Their curiosity extended to the search for an explanation of the origin of the universe, a study now known as cosmogony. They also pondered about their own origin, as well as making hypotheses for what might lay ahead. Thus were born a variety of religions in the different centres of human population.
Ancient mythology is a fascinating subject in its own right. However, what interests us here is the first appearance of ideas that are still part of modern cosmology. The concept of time was personified in pre-Socratic philosophy in the figure of Chronos, or Father Time, wielding his harvesting scythe, and wreaking havoc on all about. Was this a forerunner of the arrow of time and the Second Law of Thermodynamics (see Chap. 2), which states that the world moves from a state of order to disorder? The Second Law can perhaps also be recognised in Greek poet Hesiod’s five ages of the world, going from the best (golden) to the worst (iron), and in the four epochs of the great Indian texts, Vedas and Puranas.
In both the Chinese and the Hindu traditions, the evolution of the universe is conceived as a cyclic process, with the Hindu cosmology even quantifying the duration of each cycle in terms of billions of years. At the end of each cycle the universe is destroyed by fire, and then after an interlude, it is created again. As we shall see, this cyclic nature appears also in some modern conjectures about the cosmos, extending Einstein’s General Theory of Relativity beyond the initial (Big Bang) and possibly final singularity (Big Crunch).
Interesting as these ancient insights are, we cannot attach too much significance to them. There are a limited number of options for the evolution of the universe: static, cyclic, expanding and shrinking, so it is not surprising that primordial mythologies have suggested some of them. The important thing is that humans, so early in their history, pursued an explanation of the nature of their world.
The birth of what we regard as modern cosmogony had to await accurate observations of the heavens. These suggested that the earth was a sphere, and that the skies revolved around it diurnally. The use of the plural here is necessary because the sun and the moon revolved on a different layer (sky) from the stars. In addition, the observations also revealed a small number (five) of wandering stars, or planets.
The movements of the planets were not easy to understand, since sometimes they even went in the wrong direction with respect to the other stars. At the beginning of the second century CE, Claudius Ptolemy, an astronomer of Alexandria, elaborated a geocentric description of the universe, in which an articulated machinery explained all of the apparent anomalies in the motions of the celestial bodies.
Example of the Ptolemaic model to describe a planetary orbit, showing the use of a deferent and epicycle
This explanation of the motion of stars, sun, moon and planets held until the Sixteenth Century, when Copernicus revived an earlier, and forgotten, heliocentric model by Aristarchus. The troubles accompanying the Reformation, and conflict between different political powers (including the Roman Church), made the Copernican revolution particularly dramatic, as we know from the history of Galileo, who was forced to recant his belief in the Copernican model.
From the Seventeenth century, modern science became pervasive in all areas, including astronomy and cosmology. Telescopes allowed much more powerful observations of the sky. Newton’s theory of universal gravitation shed new light on celestial mechanics, and it was realised that planetary orbits were elliptical, not circular, removing the need for epicycles. At the end of the Nineteenth century, the description and interpretation of the universe in scientific terms began.
Zeiss Klein Planetarium 2P Projector. Image courtesy of Eryn Blaireová (https://commons.wikimedia.org/wiki/File:Zeiss_Klein_Planetarium_2P.jpg under Creative Commons Attribution Share Alike Licence)
10.3 Why is the Sky Dark at Night?
Let us begin our explorations of the universe by asking the simple question above, which on the face of it appears naïve to the point of childishness. Certainly the reaction of most people when first encountering this question is to reply somewhat scornfully: “because the sun is on the other side of the earth”. The retort is usually accompanied by a few derogatory asides. However, what on the surface appears an inane query contains within it some deeply puzzling aspects when we rigorously apply the tool of logic, which we have discussed in Chap. 3. So much so, that the question has been given the name Olbers’ Paradox after the German astronomer Heinrich Wilhelm Olbers (1758–1840), who was one of the first to raise it.1
To see the origin of the dilemma, we must first familiarise ourselves with the world view prevailing in the nineteenth century. The common representation of the Universe among astronomers (and scientists in general) at that time was of an infinite, substantially homogeneous, expanse of stars similar to the sun, with no location within the universe being more privileged than any other. Every object within the universe had its own evolutionary path, but anything dying was always replaced by something else being born, so that on the average the Universe would always appear to be the same, lasting forever, or at least indefinitely.
The different positions of the earth in its orbit around the sun results in a parallax effect whereby a nearby star appears in a different position against the distant background of stars
A simple example of parallax can be observed by holding up an object at arm’s length, and observing it against a distant background, shutting first one eye and then the other. The position of the object against the background will be different for each eye because of the separation of the two eyes. This effect becomes smaller as the object is moved farther away. Astronomers had verified that most visible stars (but not all) displayed zero parallax during the year, from which they deduced that these stars must lie at enormous distances from the earth (and the sun).
In the prevailing view at this time, there was no logical reason for any privileged place, or centre, of the universe. Why should the presence of stars end somewhere, leaving an infinite emptiness beyond? The universe must have no border and no centre.
What was true for space should also hold for time. There was no special moment here either; the universe had to be unchanging in time. In this case, the argument was more delicate and sensitive. If not to reject, at least to mark a complete separation from religion, no origin could be envisaged by science. A few centuries earlier, views such as these had led to Giordano Bruno, an Italian Dominican friar, philosopher, mathematician, poet, and astrologer, being charged with heresy and burned at the stake in 1600 CE. (Best-selling author, Morris West, has written a novel and a blank-verse stage play on the life of Giordano Bruno [1].) Astronomers in the 19th Century rejoiced in their hard-won emancipation and freedom of speech.
It is this conjecture of an infinite and uniform distribution of stars in the universe that gives rise to Olbers’ Paradox. If you assume that there are infinitely many stars scattered everywhere and at all distances, all lines of sight from your eye terminate sooner or later on the surface of a star. Of course, the farther you go, the smaller the apparent size of the star. However, even if the apparent size is reduced to an infinitesimal point, the luminosity will always be that of the surface of a star. As a result, the sky should always be as bright as the surface of the sun.
When this argument is proposed, it is generally received with a shake of the head, and the comment: “but, you are assuming that the skies are perfectly empty and transparent. Aren’t there dust clouds in space that cut off the light from stars behind them?” This is certainly true. The central region of our galaxy, the Milky Way, happens to lie in the southern skies, and as a consequence, the Milky Way in the Southern Hemisphere is brighter than in the Northern. Dust clouds are very noticeable against this prevailing brightness, especially a large one, known as the Coalsack, adjoining the iconic constellation, Crux, or the Southern Cross.
The Milky Way and its associated dust clouds. This image was taken by the ESO Ultra High Definition Expedition team at the location of the ALMA antennas on the Chajnantor plateau, at 5000 m altitude in northern Chile. Image credit: ESO/B. Tafreshi (twanight.org) (Image courtesy of the European Southern Observatory/B, Tafreshi (twanight.org), from Creative Commons at https://www.eso.org/public/images/uhd_img_2528_cc/)
The distributed dust clouds, not the bright stars themselves, form the “constellations” of Southern Hemisphere cultures, e.g. the emu of various Australian Aboriginal peoples. The Coalsack forms the head and beak of an emu, the elongated dark strip above it is the neck, and the many dark clouds around the galactic centre form the body of the emu, which in this photograph is inverted.
These dust clouds clearly obscure the stars behind them, so the objection raised earlier would appear to have merit. However, it stumbles against the alleged eternity of the universe: if the present age of the cosmos is infinite, dust, radiation, stars, planets, everything, should have had time to reach thermal equilibrium. A dust grain absorbs radiation having the same temperature as the stars, warms up a bit, then re-emits the absorbed energy in the infrared (even far infrared) band of frequency (not visible to our eyes). A similar process occurs when a stone wall becomes warm from the afternoon sun, and continues to radiate warmth (aka infrared radiation) long after the sun has set. However, if the system is in thermal equilibrium, the dust will have had an eternity to reach the same temperature as the incoming radiation, and will emit at the same frequency as is incident upon it. Again, the sky should be as bright as the surface of a star.
Another possible objection is: “light takes time to travel from the source to our eyes, so the photons from the farthest stars simply have not yet arrived on earth”. This remark implicitly assumes an origin of time, i.e. a beginning. If the universe is eternal, there will always be light arriving, no matter how far away the sources are located. After all, the light has forever to make the trip.
So we see that far from being naïve, the question: “why is the sky dark at night?” challenged the prevailing 19th Century view of the universe as homogeneous, eternal and infinite. (Any glance at the night sky will reveal that it is not really homogeneous, being full of constellations of stars, nebulae and dust clouds. However, we are considering here a much bigger picture, where these local concentrations are averaged out.) If we trust our logic, at least one of these three assumptions must be wrong, for our logic leads us to conclusions that are contradicted by observation. Despite all of the above objections, nineteenth century scientists did not appear to be overly troubled, and in the early times of Einstein, the stationary, uniform and infinite universe was the common wisdom, shared by Einstein himself.
In the next two Sections, we will see which of the 19th Century assumptions were erroneous, but we will also learn that the sky is indeed bright at night. It is simply that the “light” arriving from the depths of space has been red-shifted to frequencies far too low to be visible to our eyes, and must be detected by sophisticated equipment designed for that purpose. This discovery was one of the major achievements of 20th Century astronomy.
10.4 The Expanding Universe—Theoretical Framework
It is now time to advance to the Twentieth and Twenty-first Centuries, which saw an incredible expansion of our knowledge and interpretation of the world about us. This was brought about by the development of powerful new instrumentation, and by the revolutionary ideas of Quantum Mechanics and the Special and General Theories of Relativity. We have discussed these theories in Chaps. 5, 6 and 7 of this book, and we shall now see their importance in the understanding of the universe.
The driving force in the Cosmos is gravity. This may seem a little strange, because gravity is by far the weakest of the four fundamental forces that operate in nature. So what is going on here? Why is a force that has only a strength of about one billion trillion trillionth that of the weak nuclear force so important, both on earth (in holding us on the ground) and in outer space? The answer lies in the range of the forces, and the enormously large masses of the objects involved (compared with the masses of fundamental particles). The two nuclear forces have extremely short ranges and as a consequence do not extend beyond the radius of the nucleus.
Both the gravitational and electromagnetic forces are long range. However, the electromagnetic force can be either attractive or repulsive. Positive and negative electrical charges both exist in nature, and the force between similar charges (i.e. either both positive or both negative) is repulsive, and the force between dissimilar charges (i.e. one positive and one negative) is attractive. As the number of positive and negative charges are on the average equal and evenly distributed, the attractive and repulsive forces tend to cancel each other out. The gravitational force has no repulsive component, and so the forces between objects reinforce each other. As a result, when bodies are the size of stars and planets, the gravitational attraction between them is powerful enough to keep the planets in orbits about their suns, and to influence the creation and motion of galaxies.
One of the major achievements of Sir Isaac Newton was his Theory of Gravity. As we saw in Chap. 7, Einstein in his General Theory of Relativity, proposed a geometric interpretation, in which the curvature of space–time produces the effects of gravity. However, Einstein’s theory is formulated with equations that are horrendously difficult to solve. Only a few special cases lend themselves to an analytical solution. One of these, which we encountered in Chap. 7, is the FLRW model of a universe filled uniformly with a dust cloud.
The aspect of the solution of this model relevant to our discussion here is that such a universe must either expand or contract: no steady state is allowed. The universe can be pictured to be analogous to the surface of an inflating (or deflating) balloon. The two dimensional surface of the balloon represents space (which in reality is actually three-dimensional), with time along the radial direction. The dust grains, although locally at rest on the surface, drift apart (or get closer to one another) as the expansion (or contraction) proceeds. The steady state solution, where the radius of the balloon does not change, is not allowed by the equations.
Although this result was recognized to be mathematically correct by Einstein, he did not like it because in an expanding universe, going back in time, the matter/energy density becomes infinite at a finite point in the past. When he first proposed his theory, he considered such an infinite density to be unphysical.
In order to achieve a steady-state universe, and avoid this expansion or contraction, Einstein introduced an additional arbitrary term into his equations, known as the cosmological constantcosmological constant, Λ. For a stationary solution of the equations to exist, this new constant had to be fine-tuned, i.e. it had to have a very precise value. The new term is perfectly compatible with General Relativity and mathematically consistent. However, any arbitrarily small perturbation of the density of matter in the universe, or of the value of the cosmological constant, produces an irreversible switch to an expanding or contracting universe. It is a little like trying to balance a pencil on its point: although theoretically possible to do, it is in practice impossible. Einstein, according to a later testimony by George Gamow, called the cosmological constant the biggest blunder of his life [2].
In a letter to Georges LeMaître in 1947, Einstein wrote [3]: “I found it distasteful having to accept that the equation for the gravitational field must be put together out of two logically independent terms which are connected by addition. It is difficult to give arguments justifying such feelings, as I feel them in terms of logical simplicity. But I cannot fight against feeling them with all my strength, and I am not in a position to believe that such a repugnant thing could be realised in nature.”
It may seem strange to a non-scientist to encounter concepts such as ugliness, and its antithesis, beauty, when dealing with works of science. However, as we have seen in Chap. 3, scientists are often attracted by beauty or repelled by ugliness in their theories and equations. We will examine further far-reaching implications of this tendency in Chap. 12. Einstein’s dissatisfaction with his introduction of the cosmological constant may well have resided with its arbitrariness. In selecting a value to achieve a desired result, he is “tuning” his equations. The tuning of models by adjusting arbitrary parameters is a common enough practice in some areas of science, but is not generally regarded as good scientific practice.
Einstein, may have—or may not have, as there is some controversy about what he actually said to Gamow—described the cosmological constant as his greatest blunder, but that hasn’t prevented his successors reintroducing it to account for dark energy, which we will come to in the next Chapter. Not everybody is repelled by ugliness. Einstein was very modest about his achievements and freely admitted that he had committed other blunders. In a letter to his friend and colleague, Max Born, Einstein wrote: “I too committed a monumental blunder some time ago (my experiment on the emission of light with positive rays), but one must not take it too seriously. Death alone can save one from making blunders.”
The last sentence is a free translation from the original German, which reads: “Gegen das Böcke-Schießen hilft nur der Tod.” The literal translation of this is: “Only death helps against the shooting of goats.” If Einstein can so freely admit that he has shot a few goats in his life, then the rest of us should take heart. We have all of us in our various careers no doubt shot a goat or two, but most of us have tried to conceal it. Shooting goats (or the commission of blunders) is a by-product of innovative thought, even with the greatest minds in the world.
10.5 The Expanding Universe—Observational Evidence
Huge advances in the design of telescopes and in the observational techniques during the first decades of the twentieth century brought tremendous progress in the study of the sky. First, the number of visible objects formerly known as nebulae steadily increased. A nebula is a diffuse, cloud-like object. A well-known example, which can be observed with a small telescope or powerful binoculars, lies in the constellation of Orion, where the second “star” in the handle of the sword is diffuse. Improved resolution of telescopes revealed that these nebulae are actually comprised of many stars.
Next, an ongoing ancient debate on the whereabouts of these nebulae came to an end with the realization that they are located much farther away from us than are our surrounding stars. In fact, we now know that stars are grouped into huge conglomerations, or galaxies, as we call them today, from the Greek name for the Milky Way. The Milky Way, visible in our sky, is simply the inner part of our galaxy, seen from within.
The NGC 6744 galaxy photographed by the Wide Field Imager on the MPG/ESO 2.2-m telescope at La Silla. Image courtesy of European Southern Observatory (https://www.eso.org/public/images/eso1118a/ (accessed 2020/06/15))
Three observers on earth will see different densities of stars, depending on which direction they are looking out from the Milky Way Galaxy, because of the disc-like shape of the galaxy and the location of the earth in an off-centre position
Ultra-deep field picture of the sky in the near infrared domain, taken by the Hubble space telescope. Apart from a few stars (displaying a diffraction pattern) of our own galaxy, what we see are all galaxies. Image courtesy of NASA, ESA and R. Thompson (Univ. Arizona) (https://spacetelescope.org/images/heic0406b/ (accessed 2020/06/15))
Constructed image of bright-line (emission) spectrum of hydrogen. Other lines exist outside the visible range. The units of wavelength are in nanometres (nm); 1 nm = 1 billionth of a metre. Image courtesy of Patrick Edwin Moran (2009) (https://commons.wikimedia.org/wiki/File:Bright-line_Spectrum-Hydrogen.svg under the Creative Commons Attribution-Share Alike 3.0 Unported license (accessed 2020/06/15))
As we have already mentioned, the existence of galaxies does not introduce much change to the uniform, homogeneous model of the universe known as the FLRW model. Simply, instead of considering a dust of stars, the grains of the “universal dust” become whole galaxies.
To establish how far away these galaxies are from us, we need a method of measuring astronomical distances beyond the parallax method we discussed earlier, which is only applicable to nearby stars. This is an interesting problem in itself, and we discuss it in more detail in Appendix 10.1. Suffice it to say here that certain stars have a known intrinsic brightness. By measuring the observed brightness of the star and comparing it with this intrinsic brightness, we can estimate how far away the star is from the earth.
However even this approach will not be accurate for the most distant objects, and a third approach is necessary, which utilises the light spectra emitted from the distant stars. Chemical elements, when heated to incandescence, emit light of precise frequencies (or colours) specific to that particular element (see Fig. 10.8). This is known as the spectral signature of that element. We have already encountered this phenomenon in Chap. 8, when discussing the yellow emissions from heated sodium atoms.
If we were to observe the sky in the scenario of a static universe we would expect to recognize the signature of hydrogen (and of other elements) in the spectra of the stars. However, the stars are not stationary, and a phenomenon known as the Doppler Effect comes into play. Standing on a railway station, we will have observed a drop in the pitch of sound from an approaching train as it passes us, and races off into the distance. The wave-fronts of the sound wave emitted by the approaching train are closer together when they reach us than the wave-fronts from when the train is receding. This effect, named after Austrian physicist, Christian Doppler, is also observed for light waves, as anybody issued with a speeding ticket by the operator of a laser speed trap can testify. Studying the spectrum of the received light from the stars, it should be possible to recognise the signature of hydrogen, but we would expect the light to appear redder or bluer than in a corresponding terrestrial laboratory experiment due to the motion of the stars. In a static universe, we would expect to find approximately equal numbers of stars approaching us as receding from us.
In the case of an expanding or contracting universe, we expect to observe a similar colour shift in the spectra of stars, analogous to the Doppler Effect, which we have just described. (It is not exactly the same effect, as will be explained later.) The magnitude of the frequency shift allows us to calculate the velocity of the source away (or towards) us. Carrying out these measurements with galaxies, Hubble found that in almost all cases, the light was red-shifted, which meant that practically all of the galaxies were receding away from us. He also noticed, in cases where the distance to the galaxy was known, that the red shift was proportional to the distance. The constant of proportionality is now known as Hubble's constant. (Note the implicit assumption here that the expansion is uniform.) From Hubble's constant and the observed red shift we can calculate the distance to the farthest astronomical objects. So how do we estimate Hubble's constant?
Fortunately, there is a region of overlap between where luminosity measurements can be used to estimate distance, and where red shifts of the closest galaxies can be measured. A comparison of the two sets of observations enables us to obtain an estimate of Hubble's constant. It sounds simple, but decades of work have been undertaken to refine the accepted value of this constant. These estimates have fluctuated quite considerably. The estimated age of the universe is directly related to the Hubble constant.2
Measurement of the red-shift of galaxies became a turning point in the history of Cosmology. Einstein abandoned his cosmological constant, and the FLRW solution of the field equations of General Relativity lost its status as a mathematical curiosity and became accepted fully-fledged into the domain of physics. The expansion of cosmic space became generally accepted.
Earlier we considered a sphere as an analogy to the four dimensional universe. We develop this analogy further by considering the sphere to be an inflatable balloon, with the stars (and galaxies) as specks on the balloon’s surface. As the balloon expands, the specks move further apart from each other. Note that the specks do not move across the balloon’s surface, but rather the surface expands between the specks. Similarly, in the four-dimensional universe, the galaxies do not move through space, but space expands between them, making them further apart. This is what we meant earlier when we said that the red shift of the expanding universe was analogous to a Doppler shift, but not exactly the same. An important difference is that whereas the Theory of Relativity (see Chap. 9) restricts the velocity of stars moving through space to be less than the velocity of light, the stellar motion resulting from the cosmic expansion of space itself faces no such limitation. As we shall see in a later Section, this difference has important implications in the early life of the universe.
An immediate consequence of the expansion of space was the finite age of the universe, the current estimate of which is 13.8 billion years. If we could go backwards in time in such an expanding universe, we would find the mass/energy of all galaxies becoming increasingly squeezed by the reduced space that was available to it at these earlier times. The density of matter/energy would steadily increase until at the origin of the universe we would presumably encounter an infinite density condition, which we might call a space–time singularity.
Before moving on, it is opportune to discuss in more detail what we mean by singularity. Singularities in mathematics arise when some variable takes an infinite value. For instance, consider the reciprocal 1/x of a number x. The smaller the number, the larger the reciprocal. The reciprocal of 0.1 is 10, of 0.01 is 100, of 0.0001 is 10,000, and so on. Try taking the reciprocal of 0 with your calculator and you will obtain an error message; this is the calculator’s way of saying the answer is infinity and handling infinities is outside its job description, thank you very much.
Singularities are common enough in mathematics, but physicists don’t like them. Remember Einstein introduced his cosmological constant to avoid the very singularity that we are discussing above. Sometimes singularities bound a region that we cannot enter. For instance, we have seen in Chap. 6 that when we increase our speed, the closer we get to the velocity of light, the greater our mass becomes. If we could ever travel at the velocity of light our mass would become infinite, which indicates that it is impossible for us ever to attain that speed. In other cases, as we get closer to a singularity, the laws of physics that we are applying break down, and we need to find new laws that are applicable in this particular domain.
Upon hearing that the universe has an origin, the first naïve question that comes to mind is “what happened before the origin?” Such a question does not make sense, and is contradictory. The word “before” implies time, but the singularity is the origin of time, as well as of space. There is no “before”. Of course, physicists did not (and still do not) feel at ease with the idea of an origin of time, since it invokes scenarios outside the domain of science. This point is discussed further in Part 3.
Although the red shift has now become a routine tool in Astronomy, not everybody accepted its interpretation in terms of the expansion of the universe, and several scientists proposed alternative explanations, since abandoned. In 1948 British scientist, Fred Hoyle, proposed a mechanism that would have guaranteed, in his opinion, a steady state for the universe. (Independently, similar ideas were also considered by Hermann Bondi and Thomas Gold.) Hoyle accepted that the cosmic red-shift was due to an expansion, but rejected the corollary of a hotter past and cooler future. In particular, he rejected the idea of an initial singularity, which he jeeringly nicknamed “big bang”, never suspecting that the term would soon become a hit, even among the supporters of the FLRW model. Hoyle maintained that a steady state universe could be compatible with the drifting apart of galaxies if, due to a “creation field”, new matter were spontaneously and continuously generated in the intergalactic space. The amount of newly created matter required to ensure the steady state condition was very little: approximately one hydrogen atom per cubic kilometre per year. This is so small as to be totally unobservable.
The steady state theory was thus based on the assumption of an unobservable phenomenon, and as such might be thought incompatible with what we call “physics”. (Theories in physics should produce some predictions that are capable of being verified or refuted experimentally, either directly, indirectly, now or in the future.) Wolfgang Pauli, one of the greats of 20th Century Modern Physics, disparaged untestable theories as “not even wrong". This, in his eyes, was a far worse characteristic than being wrong, for the experimental testing of wrong theories often leads to unexpected new breakthroughs. However the Steady State Theory does predict observable differences with the FLRW model. Newly discovered radio sources (quasars and radio galaxies) were associated in the big bang theory with the early stages of the universe; they were therefore expected to be found only at large distances.
From the Steady State Theory, these unusual new objects were expected to be uniformly distributed throughout the universe. This disagreement between the Steady State Theory and the FLRW model led Steven Weinberg to write in 1972: “In a sense, this disagreement is a credit to the model; alone among all cosmologies, the steady-state model makes such definite predictions that it can be disproved even with the limited observational evidence at our disposal [4].”
The steady state theory was finally swept away by new observational evidence emerging in 1964 and analysed and perfected afterwards: namely, the relic radiation or cosmic microwave background (CMB), which we discuss in the next Section. Hoyle’s reluctance to accept the demise of his theory shows that even a front rank astrophysicist like Fred Hoyle may not be immune from prejudices.3
10.6 Cosmic Microwave Background
Now that we have established that our universe began with a Big Bang about 13.8 billion years ago, and has been expanding ever since, it is time to look at some of the events that have occurred along that timeline. We will discuss the earliest moments of the universe in the next Section, but first let us begin our story with an important period, the so-called recombination era, about 380,000 years after the Big Bang, and in so doing resolve Olber’s Paradox. At that time, according to the FLRW model, matter was in the form of a rarefied plasma and the estimated temperature was around 3000 K. A plasma in a terrestrial experiment is created when matter is heated until the temperature is so high that atoms cannot exist, but are split by the thermal energy into their constituents. Before the recombination era, charged particles (mostly electrons and protons) could not bind into atoms because the thermal radiation they continuously emitted and absorbed had enough energy to prevent the formation of stable bonds. The cosmic medium at that time resembled a bright white fog.
Afterwards, as the temperature decreased below the threshold of 3000 K, the average energy per photon became insufficient to break the atomic bonds when they formed, so that hydrogen atoms started to appear. Also photons, not having enough energy to ionise hydrogen, (i.e. strip its electron away from the atom) could no longer be absorbed, but only scatter elastically and the universe started to become transparent.
The light emitted from the hot plasma of the recombination era had a distribution of frequencies, or a spectrum, that is characteristic of what is called “black body radiation”. It may seem somewhat odd to call a white-hot, incandescent mass a “black body”. The terminology arises because when cold, the object is a perfect absorber of incident radiation, i.e. it is black. This is an ideal situation, because most objects reflect some of the incident radiation, and are therefore not perfect absorbers. As we have seen in Chap. 5, the attempt to explain black body radiation resulted in the birth of Quantum Mechanics.
We expect light escaping from the recombination era and arriving at earth today to display the type of black body radiation spectrum discussed in Chap. 5. However, if the light is not absorbed, due to the expansion of space it will undergo a red shift similar to the red-shift of light from stars in distant galaxies, which we discussed in the previous Section. In this case, however, the distance is so great that the radiation wavelength is stretched by a factor of approximately 1100. What was initially visible light now appears as microwave radiation, and is known as the Cosmic Microwave Background (CMB). It displays a typical thermal radiation spectrum, in which the initial 3000 K radiation spectrum of the primordial hot plasma is red-shifted to one corresponding to a temperature TCMB of just 2.72548 K.
In 1964 Arno Penzias and Robert Wilson, while testing a microwave antenna at the Bell Telephone Laboratories, accidentally found a “noise” coming uniformly from every direction in the sky. That “noise” was quickly recognized as being the cosmic microwave background, a.k.a. the relic radiation, arriving to us after having survived more than thirteen billion years of travel through space–time. Penzias and Wilson received the Nobel Prize for their discovery in 1978. Their experimental observation sounded the death knell for Hoyle’s steady state universe.
Also, now at last Olber’s paradox is explained. The sky is bright at night. However, it appears black because our eyes sense only a very small portion of the electromagnetic spectrum, and the white light from the recombination era has been red-shifted into the microwave region where our eyes see nothing. For a microwave antenna, the sky is uniformly bright, not as the surface of a suitably red-shifted star, but as a red-shifted hot plasma transitioning to a neutral gas.
The uniformity of the CMB, both in intensity and temperature is amazing, with fluctuations of only 1 part in 100,000. This uniformity raises an interesting question. As we have seen, the first 380,000 years of the universe’s life are impenetrable to our observations. When we look back in time through our telescopes,4 our view is stopped, as if by a curtain, by the recombination taking place at this time, at what is known as the last scattering surface. However, distant portions of the sky, from which the radiation comes, were at that time separated by more than 380,000 light years. As no physical “messenger” can travel faster than light, no causal interconnection can have taken place between these far-flung regions since the Big Bang. Then how is it possible that these independent regions can have kept such an amazing synchrony, arriving at the same average temperature in the same time? We shall come back to this point in the next section.
Distribution of the equivalent temperature of the CMB across the sky. False colours have been used: red means warmer; blue means cooler. The measurement has been made by the NASA space-based microwave telescope WMAP. Image courtesy of NASA/WMAP Science Team (https://map.gsfc.nasa.gov/media/121238/ilc_9yr_moll4096.png (accessed 2020/06/15))
Despite their smallness, the anisotropies in the CMB carry a wealth of important information about the primordial universe. An immediate remark, looking at Fig. 10.9, is that the distribution of warm and cool areas is not really random: some irregular structures do seem to exist. First, there is a concentration of red and yellow blots on the right side of the image and an area of prevailing dark blue in the middle (and again on the right border). Returning to the initial three-dimensional sphere, the warmest and coolest areas turn out to be in opposite directions in the sky: this is the cosmic anisotropy of the CMB.
The explanation is simple. If the earth were stationary with regard to the space surrounding it, we might indeed expect the observed red shift to be the same in all directions. In our balloon analogy, in this case the speck representing the earth does not move across the surface of the balloon; the balloon expands carrying the earth with it. However, the real earth is not stationary, but moves around the sun, the sun moves within the galaxy, and the galaxy moves through the cosmos. This motion introduces a further shift of frequencies (red or blue, depending on the direction of the earth’s motion relative to the incoming radiation) superimposed on that of the CMB. This additional shift is a true Doppler frequency shift due to the earth’s motion through space, unlike the cosmic red shift due to the expanding universe. From the observed anisotropy in the CMB relic radiation, the “absolute” velocity of the earth (i.e. the velocity with respect to the background) can be estimated, and is found to be 371 km/s in the direction of the constellation Leo.
Once we know its origin, we may as well subtract this Doppler anisotropy, so that we are left only with the primordial anisotropies. The study of the latter is extremely important in many aspects, which we cannot discuss here. However, a quick qualitative remark is that the tiny density fluctuations present at the recombination era are the “seeds” of the future galaxies. In fact, any density excess, via a gravitational positive feedback, would tend to grow, attracting matter from the surrounding less dense regions and in the process give rise to large scale structure.
Also, the pattern of density fluctuations of the last scattering surface carries the imprint of what happened before that era. Nothing else can arrive to us from earlier times closer to the Big Bang. The last scattering surface acts as a bright impenetrable curtain that prevents us from plunging deeper towards the singularity. The history of the universe before the recombination may be inferred to some extent from the accepted laws of physics, but not observed. This is certainly true for the information carried by known physical interactions. Various non-standard theories predict the existence of other “messengers” from the very early times of the universe. Besides these, it has been conjectured that gravitational waves could also exist as a relic of the pre-recombination era. However at present no observational evidence exists for any of this.
10.7 The Universe, as It Was Known at the End of the 20th Century
Let us now assemble facts and theories to construct a consistent cosmology, as it was accepted at the end of the twentieth century. General relativity was the basic conceptual framework, and the expanding universe had been well established in the FLRW formalism. A consistent chain of physical events may be described using known physics (including quantum mechanics for nuclear and sub-nuclear forces) from very close to the initial Big Bang up to the present time. Not everything is clear and understood, but the fundamental pillars are well established.
The initial singularity of the Big Bang remains outside the domain of physics, since nobody knows what to do with an infinite energy density. Immediately following the Big Bang, we don’t even know what the laws of physics were. At these incredible energies, the four forces may well be unified into a single force. We expect gravity to play a dominant role because of the enormous densities, and consequent curvature of space–time, in this region, but so far it has stubbornly resisted all of our attempts to bring it into the framework of quantum physics.
Despite these difficulties, theorists have proposed a mechanism to explain the puzzling homogeneity of the Cosmic Microwave Background that we discussed in the previous Section. Allan Guth and Andrei Linde in the 1980s conjectured that in the period leading up to 10–32 s after the Big Bang, the universe underwent an exponential expansion. This might seem an inconceivably short time period, but the Big Bang was proposed as initiating at a singularity, which is, as we have seen, a point with zero volume in space–time.
Such was the expansion rate that the scale of spatial distances increased by a factor of at least 1026. To achieve such an enormous rate, the recession velocities of the components of the universe, would have been well above the speed of light. As discussed earlier, these superluminal velocities are not a contradiction of relativity, because it was space–time itself that was expanding. This period of expansion was called inflation by Guth and Linde.
The development of inflation theory is highly mathematical, and beyond the scope of this book. However, a few qualitative observations are possible. The period of exponential expansion smooths out any density fluctuations present at the beginning of inflation. As a consequence, the CMB is incredibly homogeneous, as we have already seen. However, tiny residual inhomogeneities (see Fig. 10.9) do remain. Once the inflationary expansion is over, these inhomogeneities are frozen into the fabric of the universe because the regions containing them can no longer interact with each other. Such interactions would require faster-than-light travel, which is forbidden by Relativity.
To explain this period of inflation, Guth and Linde proposed a new field, called the inflaton, to describe matter/energy. As the expansion proceeds, the energy density decreases until the inflaton field decomposes, and the more familiar fields (strong nuclear interaction, weak interaction, and electromagnetism) decouple from each other. The end result is a type of phase transition resulting in the creation of the particles (gluons, quarks, etc.) carrying the above interactions. A phase transition is a common phenomenon in physics, where an entity can exist in different forms (e.g. as H2O can exist as ice, water and water vapour), and under particular conditions can transform from one to the other.
After the phase transition, the cosmic expansion continues at a much slower rate, and we enter a domain where we expect more familiar physics to prevail. The quarks and gluons begin to form protons and neutrons (hadrons). As the temperature falls, these hadrons combine to form the nuclei of atoms, such as deuterium (one proton and one neutron) and helium (two protons and two neutrons). The nucleus of deuterium is known as the deuteron, and the nucleus of helium is the α-particle, familiar from the earliest studies of radioactivity. The proton itself is the nucleus of the hydrogen atom.
At this stage an opportunity arises to use existing nuclear theory to predict the relative abundances of these lightest elements, as well as the slightly heavier ones of lithium and beryllium, in the primordial universe. Although the issue is not yet completely settled, the calculated relative abundance of helium (25%) is in good agreement with observation.
The next significant stage in the evolution of the universe is reached when, after approximately 380,000 years, the temperature drops sufficiently to allow electrons and atomic nuclei to combine to form neutral atoms. Before this time, the free electrons interact with any photons to prevent their passage. Once the electrons have been attached to nuclei to form atoms, their electromagnetic field is largely countered by the oppositely charged nuclei, and their ability to impede the passage of photons is greatly reduced. The universe becomes transparent, and visible through large telescopes on present-day earth, looking back through time and space to this past era.
With the short range nuclear forces contained within nuclei, and the electromagnetic force reduced in potency by the close proximity of oppositely charged particles, gravity becomes the driving force in the universe. The tiny inhomogeneities that survived inflation begin to grow under its influence. Slightly dense regions become denser as they attract more particles into their neighbourhood, with a resultant increase in temperature and pressure. The temperature rises in these regions until the nuclear fusion of hydrogen into helium begins, and the first generation stars are born. These are powered by the “burning” (nuclear fusion) of hydrogen.
From Fig. 10.9 we can infer that the scale of the inhomogeneities is much larger than the dimensions of stars. The stars themselves are subject to gravitational attraction, and group themselves slowly into clusters, or “galaxies”. Even the galaxies may become clustered. The time scale over which significant changes now manifest themselves is measured in millions (or billions) of years.
Stars pass through a life cycle of their own, converting their hydrogen into deuterium and helium, and depending on their mass, may commence burning their helium, converting it into heavier elements, up to iron and nickel. During this phase of nuclear fusion inside a star, a balance is achieved between the outwards radiation pressure from the fusion process, and the gravitational collapse of the outermost layers of matter. Following the exhaustion of the nuclear fuel in heavier stars, the radiation pressure falls until it can no longer oppose the gravitational forces. The core collapses, and there is a resultant huge release of energy. During this period of core collapse, nuclei heavier than iron are synthesised. A supernova, a star of immense brightness, appears in the sky for several days, or weeks. The outer layers of the star, containing traces of the heavier elements, are blown away into space, and the core collapses further into either a neutron star or a black hole. (Neutron stars and black holes have been discussed in Chap. 7.)
From the beginnings that we have outlined above, the universe has evolved into the amazing variety of objects, including planets and life-forms, that abound today. It is humbling to realise that the atoms, other than the very lightest, from which our bodies are constructed, were forged in the furnace of a star. This is the domain of astrophysics and astronomy.
What we have presented above is the standard picture of the universe as it was two decades ago. As often happens in physics, when physicists believe they are consolidating their theories and obtaining a glimmer of understanding, new discoveries occur that bring everything into question once more. In the next Chapter we will discuss some of these new observations and ideas, and the grey areas and open puzzles that require further investigation.