11.1 Hyperspace and Wormholes
Binh had been very apprehensive when he emigrated from Vietnam to Sydney with his young family. It was okay for him. He was a research fellow in astrophysics at a prestigious university, but his son had to adjust to a new school, and learn English before he could even begin his courses. However, Binh needn’t have worried. Danh was a bright boy and had soon made lots of friends. He’d also begun to pick up some of the local bad habits, including an over-fondness for video games. Binh was pleased when Danh’s English became good enough for him to read books. He’d bought him a few on the history of his old country, but Danh preferred adventure novels. He burst into Binh’s study carrying one.
At the unexpected entry, his father looked up from his desk. “What are you reading now?” he asked. Danh handed him the book. It was The Little Star Gatherer. “Any good?”.
“Good? It’s wicked.”
“Wicked, eh?” Binh smiled at the strange expression.
“There’s this boy called Rajveer. He’s got a starship so fast he can go to other solar systems and visit their planets. And every planet is different, with all sorts of animals and plants. And aliens.”
“Who all speak English, no doubt?”.
“Is it really possible to visit other planets, ba?”.
“You mean planets in other solar systems?” Danh nodded. “They’re called exoplanets. Well, let’s see if we can work it out, shall we?” Danh settled into the spare chair. He’d been through these didactic sessions with his father before, but he loved him too much to tell him how boring they were. “The nearest star system to us is Alpha Centauri, and yes, you’re in luck. It does have an exoplanet. But it’s four-and-a-half light years away. So even if you could travel at ten percent of the speed of light – which is unthinkable – it would still take you 45 years to get there.”
Danh pointed at the novel in Binh’s hand. “In there, Rajveer travels through hyperspace.”
“Does he now?”.
“What’s hyperspace, ba? Is it real?”.
“Now that’s a very good question.” Danh waited for an answer. It was the problem that he’d been pondering over all morning, and was the reason he’d come into his father’s study. Binh picked up a sheet of paper. Danh’s heart sank further. “Let’s imagine we are two-dimensional animals living in a two dimensional universe, like this sheet of paper.”
“Does it have anything to do with wormholes?” The boy’s interjection stopped Binh short, just as he was about to launch into a full explanation, and demonstrate with a folded paper how a two dimensional universe could be curved in a third dimension.
“What do you know about wormholes?” Binh was most impressed. He thumbed through the pages of The Little Star Gatherer to see what other pearls it might contain.
“A bit. But are they real?”.
“Maybe. Maybe not.”
“Ba! That’s no answer!”.
“No, but it’s the only one I can give you.” In truth Binh, who prided himself on his teaching skills, was a little annoyed that his son had interrupted his discourse before he had even got going. “If you want to know more, then you’ll just have to study physics when you get to high school, and then go on to university. And stop wasting so much time on video games.”
“Same old, same old,” muttered Danh under his breath. He did like video games, but he also liked science, and his grades were good. But his father was away again on his favourite hobby horse.
This time, however, the boy had a prepared response. He reached across for The Little Star Gatherer, and rose from his chair. At the door, he turned back towards his father, and called: “Hey, Ba, did I tell you our sports teacher thinks I should try out for the football team? He said I show a lot of promise. Never know. Might even get good enough to be a pro one day.” He closed the door quickly to hide his grin, and strode smirking down the passage to his own room.
***
Danh’s question is a good one. Wormholes, or tunnels connecting different events in space–time, are valid solutions of Einstein’s field equations. They do however raise the problem of time travel, where travellers might go back into the past and kill their parents, thereby threatening their own existence. Perhaps fortunately, there is no experimental evidence for the existence of wormholes.
11.2 The First 380,000 Years
In the last Chapter we have been diligent in our attempt not to stray too far from established theories in a search for an explanation of what is arguably the oldest and most puzzling question ever to have confronted humanity: how did the universe in which we live begin? This question is basic for all religions: physics cannot simply ignore it, and take refuge behind the defence that a direct observation of the Big Bang is impossible. However, when venturing to explore this region, caution is necessary, controversy unavoidable, and frankness laudable.
Although many people are working diligently in this field, it is openly admitted that the physics immediately following the Big Bang is completely unknown. Unfortunately, that period contains the original central singularity at the origin of the universe. It is where the two main cathedrals of modern physics, Quantum Mechanics and General Relativity—one probabilistic and the other deterministic in nature—are mutually inconsistent. We have discussed this conflict already in Chap. 9. Physicists have tended to skate around this issue, applying Quantum Mechanics when dealing with small particles, and General Relativity when gravity is the dominating force. However, at the Big Bang, and shortly thereafter, the disagreement must be confronted head on. There is no place to hide.
As an aside, let us remark that similar issues arise wherever singularities appear. Black holes are examples of singularities predicted by physical theory, and the physics close to the central singularity in a black hole is not understood for the same reason that we discussed in the preceding paragraph. In this region of intense gravitational attraction, the theories of Quantum Mechanics and General Relativity are in conflict, and the black hole is surrounded by an event horizon, from within which no light or other particles can emerge. Below the horizon, a cosmic censorship principle is acting: we cannot see in there, so it is regarded by many as beyond the purview of physics. Yet physicists have accepted the existence of black holes. For example, observations tell us that a massive object, compatible with a black hole, has been found at the centre of our galaxy. It has been named Sagittarius A*, and has a mass equal to four million solar masses. It is now believed that black holes are located at the centres of most galaxies.
In the last Chapter, we discussed the inflation model, and saw that it was necessary to introduce an entirely new field, the inflaton field, to explain the extraordinary homogeneity of the Cosmic Microwave Background. This uniformity could not be explained by conventional theories because, even over 13 billion years ago, the most distant parts of the universe were too far apart to be accessed, one from the other, at velocities less than the velocity of light.
Here, we are assuming that the velocity of light at this time was the same as it is today. This is in accord with our belief that the physical constants do not change, which we discussed in Chap. 3. Suppose, however, that our belief is wrong and that the speed of light in these early times was much larger than it is today. A Variable Speed of Light (VSL) model of early cosmology, as an alternative to the inflation model, was proposed by John Moffat in 1992 [1]. This work was largely ignored, until in 1998 another physicist, João Magueijo, published a similar idea in a more prestigious journal. An unpleasant controversy erupted over which author had priority, with the media largely ignoring Moffat’s work. Eventually the two physicists were reconciled, and have since published further papers jointly. Light speeds up to 60 times the current value of c have been suggested. A VSL model would have implications in special and General Relativity, Maxwell’s equations of electromagnetism and many other areas of physics.
Currently VSL models of early cosmology are outside the mainstream of physics. However, the is also not without its critics. Paul Steinhardt, one of the original contributors to inflation theory, is now concerned that the model requires fine tuning to explain the universe, as we know it. By varying parameters in the model, any conceivable universe can be explained. Steinhardt goes further and adds that inflation would produce not just one universe, but a multiverse containing all possible universes with all possible properties. There is no way to verify, or falsify, such a theory, which Wolfgang Pauli might well say is “not even wrong” (see Chap. 10). Steinhardt’s (and Pauli’s) rejection of non-testable theories is not a philosophical viewpoint shared by all physicists. Many take a different view, i.e. that a theory can simply provide an explanation that relates observations to underlying physical principles.
Steinhardt and collaborators have proposed that instead of inflation, the universe is cyclic in nature, and our universe has arisen from the remains of an earlier collapsed universe. This approach is not without its own difficulties, running afoul of the Second Law of Thermodynamics, which requires that entropy (or disorder) increase inexorably. A baby universe is in a low-entropy ordered state, and an old universe is in a much higher entropy state. A recycled universe would begin its life in an unacceptably high entropy state. Roger Penrose and others have made suggestions on how to overcome this problem by modifying some of the laws of physics.
In this section, we have probed into some of the controversy that underlies the physics of the early universe. We conclude by noting that the inflation theory, as we described it in Chap. 10, remains the most popular of the contending theories among astronomers and cosmologists. It is perhaps also worth giving ourselves the wry reminder that the most popular cosmological theory of the 1950s was the Steady State Theory, long since defunct. In the long run, popularity in science counts for very little.
11.3 The Strange Case of the Missing Mass
In the 1930s, from studies of the orbital speed of galaxies in clusters, and stars within galaxies, including the Milky Way, it was discovered that the velocities of objects gravitationally bound to each other were higher than expected [2, 3]. The observed velocities were explained by assuming that more mass existed in the galaxy, or the galaxy cluster, than was actually observed. We have noted already in Chap. 3 that interstellar dust is a feature of our galaxy. Matter is also present in the remnants of dead stars, planets and asteroids, none of which are visible from their radiation. However the sum total of all such obscure matter is estimated to contain less mass than the visible stars. In order to explain the anomalous orbital velocities, it would appear that there is a missing mass in the universe, and that it must be more than five times the visible mass.
Today, the term dark matter has been coined for this missing mass. If dark matter is assumed, the anomalous motion can be explained within the framework of General Relativity. The actual nature of the dark matter, however, remains a mystery. Many conjectures have been made about the components of dark matter (e.g. neutrinos and other exotic particles), but none have been satisfactorily substantiated.
Alternative explanations of the anomalous motion of stars and clusters involve modifications of the theory of General Relativity. One such theory is MOND, or Modified Newtonian Dynamics, created in 1983 by Israeli physicist Mordehai Milgrom [4]. In this theory, Newton’s law departs from the well-known inverse square law when the accelerations involved are very small, such as in the outer reaches of galaxies. The modification to Newton’s theory is too small to be observed on earth or within the solar system, where the accelerations are much larger. MOND has been successful in explaining a number of galactic phenomena, but fails to predict the behaviour of galactic clusters. It has attracted a small number of adherents, whereas the majority of cosmologists are committed to the dark matter solution of the observed anomalies.
Recent evidence of dark matter has been found by Seth Epps and Michael Hudson of the University of Waterloo [5]. Current theory predicts that galaxies are immersed in a halo of dark matter, and that galaxies that are relatively close to each other are connected by filaments of dark matter. Light passing in the vicinity of any matter is bent by the curvature induced in space–time by the presence of the mass. The effect is known as “gravitational lensing” (see Chap. 7). Epps and Hudson averaged the results of gravitational lensing from 23,000 pairs of neighbouring galaxies, and compared these with similar averages from pairs of galaxies that were in the same region of the sky, but actually well separated in distance.
Gravitational lensing between pairs of galaxies. The upper image shows a bridge between neighbouring galaxies which is absent in the results from well-separated galaxies (lower image). Image courtesy OUP (Image source: Seth D. Epps, Hudson, Michael J. The weak-lensing masses of filaments between luminous red galaxies (Mon. Not. R. Astron. Soc. 468 (3): 2605–2613, 2017, Fig. 3)
Further investigations are needed to confirm the results of Epps and Hudson: meanwhile the search for the elusive dark matter continues.
11.4 The Foamy Distribution of Galaxies
As we have seen earlier in our discussion of Olber’s Paradox, the prevailing view of the universe in the nineteenth century was a uniform distribution of stars throughout space. Subsequently it became clear that stars are grouped in galaxies, and it was presumed that these galaxies were homogeneously distributed. However, we now know that the galaxies are distributed in clusters and these clusters in superclusters. Looking at the broadest picture we still expect the superclusters to be spread homogeneously. In this belief, we are again applying Occam’s Razor, for a homogeneous distribution is one without structure, which involves fewer assumptions than one with structure.
Distribution of galaxies in a slice of sky, centred on the earth. The picture is taken from the Sloan Digital Sky Survey. Image courtesy of M. Blanton and the Sloan Digital Sky Survey (https://classic.sdss.org/includes/sideimages/sdss_pie2.jpg (accessed 2020/6/18))
We see from Fig. 11.2, in which every little dot is a galaxy, that the observed distribution of galaxies is anything but homogeneous. Close inspection reveals a spongy or foamy structure. The galaxies are located on walls and filaments that border large voids where nothing is visible. The underlying cause of this structure is unknown. It may be related to events that arose in the pre-recombination era, or to the prevalence of dark matter. Speculations abound, but we really don’t know.
11.5 The Accelerated Expansion
In Chap. 10, we discussed the understanding of the cosmos at the stage that it had reached towards the close of the 20th Century. However, at the end of 1998 and at the beginning of 1999, the complacency that was becoming common among cosmologists was shattered when two independent groups of astrophysicists, guided respectively by Adam Riess and Saul Perlmutter, published the results of extended surveys of type Ia supernovas located in other galaxies.
The measurement of distance in astronomy by the “standard candle” approach is described in Appendix 10.1. Type Ia supernovas are suitable standard candles; they all have the same intrinsic brightness, which can be estimated by measuring their apparent brightness at known distances. Further measurements of the apparent brightness of these supernovas in more distant galaxies can then provide an estimate of the distance to these galaxies.
A second independent measurement of this distance can be obtained by measuring the red shift from one of the spectral lines visible in the light emitted by a supernova, and applying the Hubble constant to calculate the distance to the supernova, and thus to the galaxy that contains it. Unfortunately, as is sometimes the case in physics when two independent approaches are available to measure the same quantity, the results from the two sets of measurement do not agree with each other. It is found that for type Ia supernovas at distances larger than approximately 1 billion light years, the supernovas appear to be fainter than they should be. Since both types of data come from the same sources, the explanation cannot be in terms of different distances.
A solution to this puzzle has already been hinted at in Sect. 10.5: when calculating the distance to a supernova by measuring the red shift and using the Hubble constant, the assumption is made implicitly that the Hubble constant is a constant. Its very name is an indication of our confidence in this assumption, which is an application of Occam’s Razor. The data from the two sets of measurements suggest that the expansion rate of the universe at the time of emission of the light (measured by the red shift) was less than it is today. The higher expansion rate in the universe today means that the path length travelled by the light to reach us (on which the apparent luminosity of the supernova observed through our telescopes depends) is longer than one would expect from the initial expansion rate. In other words: the expansion of the universe appears to be accelerating.
So how do these new results accord with Einstein’s Theory of General Relativity, which is the theory underlying most of our understanding of cosmology? Not very well at all, as it turns out. In fact, depending on the actual value of the average energy/matter density in the universe, General Relativity admits three different solutions, but in all cases the gravitational attraction between galaxies slows down the initial expansion.
Three different representations of the universe corresponding to a closed, positively curved universe; an open, negatively curved universe; and an open, flat universe. Image public domain, Courtesy: NASA/WMAP Science Team (https://commons.wikimedia.org/w/index.php?curid=647033 (accessed 2020/6/19))
The second solution occurs when the energy/matter density has a value of approximately five hydrogen masses per cubic meter. This value is critical, and even a small deviation from it will not lead to this solution of Einstein’s equations. In this solution, the expansion slows down and stops asymptotically at an infinite cosmic time. This would result in a flat and open universe.
In the third solution, which occurs for smaller densities, the expansion rate also slows down, but tends to an asymptotic value different from zero, so that the expansion never ceases. This would result in an open, but not flat universe. As we can see, under no circumstances do the equations of General Relativity yield a solution where there is an acceleration of the expansion rate. The conclusion is clear: either General Relativity is wrong, or we are missing something in our understanding of the universe.
In the preceding paragraphs we have used the terms “open” and “closed” to describe the universe, and “flat” and “curved” to describe space–time. What we mean by these can be illustrated by Fig. 11.3. The upper figure corresponds to a closed universe with curved space–time. The parameter Ω0 describes the energy/mass density in arbitrary units. In this space, Euclidean geometry does not apply, and the angles in a triangle sum to more than 180°. The space is said to have positive curvature, and corresponds to the first solution in the preceding paragraph. The middle figure corresponds to an open universe with negative curvature of space–time. Here the geometry is non-Euclidean and the angles of a triangle sum to less than 180°. This corresponds to the third solution of the General Relativity equations. The lower figure is an open universe with flat space–time. It is the second solution from the preceding paragraph. The geometry is Euclidean and the angles in a triangle sum to 180°.
The role of adjudicator of which of these solutions represents physical reality rests with the observational astronomer. There are several approaches that have been applied to this problem. One involves a study of the inhomogeneities in the Cosmic Microwave Background, which are visible in Fig. 10.9. These cosmic microwave temperature fluctuations are believed to have been imprinted shortly after the Big Bang, and describe fluctuations in the density of matter in the early universe. They have been frozen in time as the universe transitioned from a plasma to a transparent gas of atoms during the Recombination Era, 380,000 years after the Big Bang.
An analysis of the Cosmic Microwave Background, similar to the frequency analysis that is routinely performed to determine the harmonic structure of ordinary sound waves in air, reveals the fluctuations present at the time of the Recombination Era. (See Appendix 11.1 for more details of this procedure.) The measurements of the CMB shown in Fig. 10.9 were undertaken by the Wilkinson Microwave Anisotropy Probe (WMAP), launched by NASA in June, 2001, and operating until 2010. The Planck space observatory, operated by the European Space Agency from 2009 to 2013, improved on the observations made with WMAP. The results obtained from the Planck observatory show that the universe is topologically flat at large scales to within 0.5%. This is an impressive but somewhat strange result that warrants further explanation.
11.6 Dark Energy and the Concordance Model of the Universe
To achieve such a flat universe requires a very precise and critical matter/energy density. This fine tuning is disturbing because it is surprising that any physical quantity would assume a critical value, such that the smallest change will induce the universe either to collapse, or to expand forever at a finite expansion rate. However, throughout the universe there are a substantial number of quantities with values which, if modified even slightly, would change the nature of the universe such that life as we know it would not be possible.
An example is the strength of the strong nuclear interaction. A slight increase (2%) in the strength of this force would have enabled all the hydrogen in the early universe to bind into diprotons, rather than deuterium. (A diproton is a nucleus comprised of two protons, whereas deuterium is a nucleus formed from a proton and a neutron.) This would have drastically altered the physics of stars, where the elements essential to life are forged. One might say that unless these quantities had been adjusted such that human life is possible, we would not be here to observe and measure them. As we have seen earlier in the present Chapter, this concept is known as The Anthropic Principle.
Putting these philosophical speculations aside for the moment, the problem remains that visible matter accounts for only an estimated 5% of the required energy/matter density. Even including the dark matter required to explain the dynamical behaviour of star clusters, galaxies and galaxy clusters, which is estimated to form 24% of the energy density, the total “explained” density is only 29%. That leaves 71% of the energy/matter density in the universe unaccounted for. To add to the puzzle, this missing component must be something that does not gravitate, otherwise the accelerated expansion of the universe observed by astronomers would not be possible. Indeed, the missing component must exert a form of “negative pressure” to induce this acceleration. Since it produces physical effects, without being “matter”, this mysterious ingredient has been named “dark energy”.
We see therefore that from the end of the 20th Century, the picture of the universe accepted among astronomers has evolved, and converged towards what is now called the Cosmic Concordance Model (a.k.a. the Cosmic Standard Model). It is also known as the ΛCDM model. Here Λ (lambda) is the cosmological constant that Einstein dubbed as the biggest blunder of his life (see Chap. 10). Cosmologists have arbitrarily reintroduced that constant into the equations of General Relativity, and adjusted its value to account for the accelerated expansion of the universe. The mathematics is thus satisfied; however physics is left with the burden of finding a suitable interpretation for Λ.
As we have seen, because of the observed accelerating expansion of the universe, dark energy must function as a type of anti-gravity. We may think of it as being a fluid permeating the universe, but its density always stays the same. Normally we would expect that in an expanding space any fluid progressively dilutes, but this is not the case for dark energy, which dilutes much more slowly than the universe expands. Reintroducing the cosmological constant of Einstein to GR provides the simplest formalism to describe dark energy, since it is constant in both space and time.
The introduction of an ad hoc arbitrary constant into a physical theory is an indicator that one’s physical understanding is far from complete. Ideally the value of the constant should be derivable from other principles. However, attempts to derive the value of Λ from quantum field theories have been spectacularly unsuccessful, with the measured value of Λ being only 10–120 of the theoretically predicted value. This discrepancy of 120 orders of magnitude has been described as “the worst theoretical prediction in the history of physics” [6]. In summary, we do not know what dark energy is, but it seems to work, and provides a good fit to many cosmological observations. Of course, there is no guarantee that sometime in the future it will not be replaced by an altogether different theory. Such is the nature of science.
Let us turn now to the remaining CDM part of the ΛCDM model. The letters CDM stand for Cold Dark Matter. We have seen in an earlier Section of this Chapter that dark matter was introduced to explain the observed effects of gravity in large-scale clusters of matter. The various hypotheses proposed by theorists about the nature of dark matter may be grouped into three categories: cold, warm and hot dark matter. The three categories are related to the free-streaming length1 of the particles (or to their speed): cold means slow, i.e. non-relativistic; warm means weakly relativistic and hot means relativistic. None of the three types is expected to interact strongly with ordinary matter, but each has a different influence on the size of the structures (galaxies, galaxy clusters, and superclusters) that are observed in the universe. Of the three types, Cold Dark Matter provides the best description of the distribution and size of observed structures, and as a consequence it has been included in the Concordance Model.
Pictorial view of the evolution of the universe. The bell-shaped surface (which should be three-dimensional) is the border of the visible universe; cosmic time is measured along the surface starting from the initial singularity. Image: public domain, courtesy of NASA WMAP Mission (https://commons.wikimedia.org/wiki/File:CMB_Timeline300_no_WMAP.jpg (accessed 2020/6/19))
In Fig. 11.4, the three dimensional surface is depicted as a two dimensional expanding envelope. This is an example of the usual compromise forced on mapmakers when representing a three dimensional surface (e.g. the earth) on a plane sheet of paper. The third dimension of Fig. 11.4 is time. The universe comes into existence on the left with the sudden appearance of the Big Bang singularity. Immediately, the limitations of such a depiction become apparent when one asks what occurs in the region shown to the left of the singularity. The question cannot be answered. This region of the figure has no meaning because time (as well as space) is created at the instant of the singularity.
Following the Big Bang, and close to it, events are dominated by the interaction of strong gravity with quantum fluctuations. As we have indicated earlier, the laws of physics in this region are not well understood. Then follows an inflationary exponential expansion, and in the minutest fraction of a second, the universe explodes in scale from the microscopic to the cosmic.
With the burgeoning expansion, the temperature of the content of the universe decreases, and the synthesis of nuclear particles begins. The nuclear plasma continues to cool and the expansion rate of space slows down until the Recombination Era is reached at a cosmic age of 380,000 years, and neutral atoms are formed.
Following the Recombination Era, there ensues a “dark age”, during which atoms start to coalesce around positive fluctuations of the matter density, thereby increasing the local temperature and pressure. There are no visible light sources during this period, whence the name: Dark Age. The darkness is the reason our telescopes can probe through to the Recombination Era. Finally the temperature and pressure of the agglomerates increase sufficiently to ignite nuclear fusion, and the first stars are born. Once again light is emitted in the universe. Gravitational attraction groups the stars into galaxies, clusters, superclusters, and filaments, etc.
Up to this moment of time, the matter/energy density has been sufficient to allow gravity to slow the rate of expansion. However, the expansion progressively dilutes the matter density, thereby reducing this braking effect of gravity. Dark energy, which as we have noted above, does not become diluted with expansion, begins to dominate the process and convert the expansion from a decelerated to an accelerated stage. This is the phase in which we are now living. In Fig. 11.4, transverse sections of the bell are shown as circles, but in the real world they are spheres, and each of them corresponds to the frontier of the visible universe.
The Concordance Model is currently the most popular model of the universe among cosmologists. Its strength is in its relative simplicity and internal consistency. However, the fact that in this model 95% of the universe is “dark” (or unknown) presents a serious challenge to its credibility, and it is not without its doubters, as we shall see in the next Section.
11.7 Alternative Cosmological Models
The conceptual difficulties with Dark Energy that we have outlined in the last Section have spawned a vast literature describing attempts to find more intuitive or better founded solutions to the problem. It is not within the scope of this book to cover these endeavours in detail. However, some indication of what is involved is appropriate.
Many theories begin by introducing a field, or fluid, whose properties are tailored in an ad hoc fashion to reproduce the observed properties of the universe. The names used for these quantities are sometimes whimsical, such as “ghost energy” or “quintessence”. The latter is a reference to Aristotle’s “fifth essence”, an ether (aka aether) that was supposed to pervade the skies.
A common approach in physics is to draw an analogy between one area of physics and another, in the hope that the first will throw light on the second. For instance, in nuclear physics it is of use in some instances to compare the nucleus with a drop of liquid, and assign to it a “density” and “surface tension”. In cosmology, dark energy is sometimes treated as arising from the deformation of a four dimensional elastic continuum. This model is the Strained State Cosmology. For an elastic material in three dimensional space, the strain energy is related to the resistance of the material against an imposed deformation. The accumulation of strain energy hinders the deformation and forces the material to return to the unstrained state. In a four dimensional analogue, if we regard space–time as not just a mathematical artifice, but an entity with physical properties of its own, a strain energy density can be associated with the presence of curvature, and drives an accelerated expansion towards an asymptotic unstrained and completely flat state. It is the analogue of a twisted piece of elastic material returning to its flat state. This theory succeeds in explaining the observed accelerated expansion and also the exponential expansion close to the origin.
Another class of theories involves modifications to Einstein’s General Relativity, or even its replacement. However, at this moment, the ΛCDM model remains the conceptual framework that is used by cosmologists as the basis for interpreting observations, even though it contains ingredients that are counter-intuitive and difficult to comprehend.
So now we have arrived at the end of our journey through space–time, and find that in spite of the spectacular progress achieved, we should adopt a humble attitude with respect to our accumulated knowledge. When we look up at a clear night sky and behold the vast expanse of stars stretching across the heavens, we surely feel the same sense of awe and insignificance as our early ancestors. In the intervening millennia, we have learned much, but there is still much to learn. Not all our questions have been satisfactorily answered, and the future will surely show some of our current answers to be wrong. Science, by its nature, is evolutionary, and Homo sapiens is a creature driven by curiosity. Today’s physicists and cosmologists have left an abundance of puzzles to occupy the fresh minds and curiosity of succeeding generations.