the march of time - time and causality

Space and time expand with the universe because they are the universe. (Shallis 1983, 98)

If the universe expands, there must be a succession of events, and this succession may serve as a criterion for the direction of time. Both Kant and Leibniz establish a link between the anisotropy of time and the causal succession of events, a topic,

which will be investigated in this section (cf. van Fraassen 1970, Chap. VI; Whitrow 1980, Sect. 7.2). At least Leibniz situates this link on the global scale.

 

 

2.5.1   Immanuel  Kant (1724–1804)

 

In Saint Augustine and the British Empiricists, we have already come across the basic idea behind an idealist view of time. Saint Augustine, as shown, endorses a physical notion of time but declares individual minds its metric when he asks himself how the passage of time is to be grasped. Such a metric of time lacks regularity and invariance, since the perception of temporal intervals and durations is subject to the fluctuations of individual mental states. Saint Augustine therefore became the pioneer of internal time. The British Empiricists also accept a notion of physical time, as a succession of material events, but their major emphasis is on mental time. By contrast Kant proposes an objective idealist view of time but he too has to admit the importance of physical time. He holds that in the absence of human beings who perceive objects of the external world, there is no time. ‘Time is a purely subjective condition of our (human) intuition (which is always sensible, that is, so far as we are affected by objects)’. But Kant does not make individual minds the metrics of time. He holds that time and space are ‘pure forms of intuition’ (Kant 1787, B51/A35). He means that time and space belong to the a priori conditions of the human mind, amongst others, under which humans per- ceive the external world. All humans possess this form of intuition. Therefore, insofar as external objects appear to us and enter our experience, time ‘is neces- sarily objective’ (Kant 1787, B51/A35). It is objective, because all objects of the external world appear to the human mind in the same temporal order. But

it is no longer objective, if we abstract from the sensibility of our intuition, that is, from that mode of representation, which is peculiar to us, and speak of things in general. (Kant 1787, B51/A35)

Kant does not claim that the external, unperceived world is a timeless block universe. He simply states that the order of the ‘noumenal world’—the world of objects in itself—is unknowable to us. But in the ‘phenomenal world’ of our perception humans experience all perceivable objects as temporally ordered. Time and space may be a priori forms of intuition, still this inner and outer sense, respectively, must operate on the appearance of external objects. A similar situ- ation arose for Locke and Hume. Time may be merely a ‘succession of ideas’ but the ideas are formed through mental operations on impressions and these impressions follow a sequential order. Although Locke accepts that external things appear in some regular, temporal order, which constitute time, he places more importance on the mental idea of duration (passage of time). Hume simply reduces time to the succession  of perceptions.

Although these thinkers belittle the importance of physical time, they never- theless introduce an implicit distinction between physical and human time. Kant 

argues that time is a ‘subjective condition of human intuition’; he nevertheless makes a concession to physical time. In his chapters on the Analogies of Expe- rience (Kant 1787, B233–B256), he argues that a temporal order is produced through the application of the category of relation (subdivided into substance, causality, reciprocity) to perception. Kant’s achievement is to have shown how objective knowledge can be obtained from the perception of external objects. In an effort to obtain an objective notion of time, the Principle of Succession in Time, in accordance with the Law of Causality is of particular importance, since it is a rule, by which humans connect cause and effect. It states: ‘All alternations take place in conformity with the law of the connection of cause and effect’ (Kant 1787, B232; cf. Gardner 1999, Chap. 6, esp. pp. 174–175; Brittan 1978, Chap. 7). All changes occur according to the law of causality. It is this law of causality by which succession in time is determined (Kant 1787, B246).

Kant explains its application in the following terms:

• All appearances of succession in time are only changes in the determinations of substances, which themselves abide (A189/B233). The substances are thus the invariants, whose properties undergo change. For change to be regular, as Heraclitus already suggested, there must be some invariance through change.
• All changes occur through the law of causality, that of the connection of cause and effect (Kant 1787, A189/B233). The law of causality is a rule which states that the conditions, under which an event invariably and necessarily follows is to be  found  in  what  precedes  the  event  (Kant  1787,  B246).  Hence  the  event

incorporates the antecedent conditions (cause), which determine the consequent conditions (effect). The law of causality has a temporal dimension, since for an effect to be caused, the antecedent conditions must be prior to the consequent conditions. This situation even obtains when cause and effect occur simulta- neously, as in Kant’s example of the ball, which presses a dent in a cushion.

 

If I view as a cause a ball which impresses a hollow as it lies on a stuffed cushion, the cause is simultaneous with the effect. But I still distinguish the two through the time- relation of their dynamical connection. For if I lay the ball on the cushion, a hollow follows upon the previous flat smooth shape; but if (for any reason) there previously exists a hollow in the cushion, a leaden ball does not follow up it. (Kant 1787, B248–B249)

It is this law of causality, according to which we determine perceptions according to the succession in time, which must make a reference to external events, in order to avoid a vicious circle. The events, which succeed each other in a causal manner, do so irrespective of human awareness. The human mind, however, links them as ‘cause’ and  ‘effect’.

 

 

 

 

 

 

• Finally, objects can appear to humans as existing simultaneously, only insofar as they interact with each other. Again this perception presupposes an interaction between physical objects, before they can appear as simultaneous to the human mind. 

• Hence if time is to be an objective feature of the human mind, the rule of causality, by which succession in time is established, must be objective.2 That is,  in order to avoid a vicious circle the succession of objects must be located in the external world. Kant illustrates this contrast between subjective and objective succession through a judicious example: he compares the perception of a house to the perception of a moving ship. The order, in which the perceptions appear to the observer, is freely chosen in the first case (the house) but necessary in the second case (the ship). In the case of the house

  my perceptions could begin with the apprehension of the roof and end with the basement, or could begin from below and end above; and I could similarly apprehend the manifold of the empirical intuition either from right to left or from left to right. (Kant 1787, B238)

  But the case of the ship, moving downstream, is quite different:

  My perception of its lower position follows upon the perception of its position higher up in the stream, and it is impossible that in the apprehension of this appearance the ship should first be perceived lower down in the stream and afterwards higher up. The order in which the perceptions succeed one another in apprehension is in this instance determined, and to this order apprehension is bound down.’ (Kant 1787, B237)

  In the latter case  then,

  (t)he objective succession will therefore consist in that order of the manifold of appearance according to which, in conformity with a rule, the apprehension of that which happens follows upon the apprehension of that which precedes. (Kant 1787, B238)

  This rule is the rule of causality, according to which reason carries ‘the time- order over into the appearances and their existence’ (Kant 1787, B245). Kant never explicitly states that events follow each other according to a ‘before–after’ rela- tionship and thus constitute physical time. Kant is preoccupied with the necessity and universality of knowledge, which cannot be obtained from sensual experience alone. Kant locates the necessity and the objectivity of succession in apprehension in the rule of causality. Still, Kant is not interested in succession as a ‘merely subjective play of my fancy’ (Kant 1787, B247), but in its objectivity. Hence Kant implicitly acknowledges the need for physical time. Objective change can only be perceived and experienced if the change takes place objectively in the external world. However, his idealistic tendencies prevent him from calling this ‘objective succession’ simply (physical) time. But there is no escape. An objective idealist view of time must presuppose a succession of events in the physical world in order to secure the regularity and invariance of succession, which Kant requires for his theory of how appearances are turned into objective knowledge. ‘The function of causality is to order objectively the sequence of perceptions given by the sensi- tivity according to a rule’ (Kant 1787, B238–B239).

  Although the Kantian view was to find support in the Special theory of rela- tivity, it does not imply that time is a human illusion. As we shall see Kant 

  perceives time as an objective feature of the structure of the human mind, which produces the same succession of events in every human observer.

  According to Kant, the outer world causes only the matter of sensation, but our own mental apparatus orders this matter in space and time, and supplies the concepts by means of which we understand experience.  (Russell 2000/1946,  680)

  When Kant talks about time as a ‘subjective condition of our (human) intuition’, whilst claiming that there is no time in the absence of human observers, he is really talking about the structure of human time (Locke’s duration). There is indeed no human time—no calendars, no dating systems—in the absence of humans. Kant’s emphasis on human time is the result of his Copernican turn (Kant 1787, BXVII), according to which our experience of given objects, in so far as they are knowable, must conform to our concepts. As Kant is a transcendental idealist and empirical realist, he does not really deny the existence of an objective succession of events. This impression is confirmed by his evolutionary cosmology and by a ‘causal theory of time, which he embraces in his second analogy of experience’.3

   

   

  2.5.2  Gottfried  W.  Leibniz (1646–1716)

   

  The rule of causality guarantees the objective order of events. It was in fact G. W. Leibniz who grasped ‘the importance of the subject of order to the theory of time and space’ (cf. van Fraassen 1970, 35; Jammer 2007). As discussed in Chap. 3, Leibniz is famous for his ‘relational theory of space and time’, which he contrasted with Newton’s view of ‘absolute time’. What does Leibniz mean by ‘order’? Leibniz was a believer in the mechanical worldview, which adopted, amongst other principles, the axiom of the universal causal concatenation of all events.

  My earlier state involves a reason for the existence of my later state. And since my prior state, by reason of the connection between all things, involves the prior state of other things as well, it also involves a reason for the later state of these other things and is thus prior to them. Therefore, whatever exists is either simultaneous with other existences or prior or posterior. (Leibniz 1715,  666)

  From this viewpoint of determinism, which was shared by many 17th century thinkers, and culminated in Laplace’s famous demon (Weinert 2004, Sect. 5.1), Leibniz derives a causal order of events in the world, and hence a causal order of time.

• Time is the order of non-contemporaneous things. It is thus the universal order of change in which we ignore the specific kind of changes that have occurred. (Leibniz 1715, 202; italics in original; cf. van Fraassen 1970, 35–44)

  Thus Leibniz, too, embraces a causal theory of time. But it is a cause-effect relationship of events in the world, rather than an a priori rule. Leibniz postulates a universal chain of causality amongst events, thus ignoring ‘the specific kinds of changes that have occurred’. With his insistence that time is the order of the succession of events Leibniz proposes a topology of time. In his essay, written after 1714, he claims that this order is a causal order, not of particular events, but of a universal kind. The principle of universal causality is also a feature of clas- sical physics. But in the framework of classical physics, this metaphysical notion of causality is substantiated by the use of deterministic laws, which govern the material events. Newton’s classical mechanics operates with a notion of absolute of time, which Leibniz opposes with his relational view of time. Yet, despite these opposing views, there is the task of specifying the topological notion of the order of succession within the framework of classical physics. Leibniz himself hints at a possible solution when he points out that this order has a magnitude:

  (…) there is that which precedes and that which follows, there is distance and interval. (…). Thus although time and space consist in relations, they have their quantity none the less. (Leibniz 1715–1716, Fifth Paper, Sect. 54, 234–235)

  Thus if the spatial and temporal relations have quantity or magnitude, they are subject to regularities. For our discussion of the Newtonian and Leibnizian notions of time (Chap. 3), it is of great importance to note that, in accordance with his mechanical worldview, Leibniz holds that nature is subject to laws of nature:

  The natural forces of bodies are all subject to mechanical laws. (Leibniz 1715–1716, Fifth Paper, Sect. 124, 237–238)

  For the regularity, which is associated with ‘mechanical’ laws, gives rise to a metric of time. As will be discussed, Newton makes his notion of absolute time a prerequisite for the formulation of his laws of motion. In the present context, however, it suffices to remember that Leibniz’s view is concerned with the topology of time (it is a ‘before–after’ order of successive events), rather than a metric  of time  (a quantification of this order).

   

   

• The Topology of Time

•  

  Any adequate inquiry into ‘time’ is necessarily partly scientific and partly philosophical. (Denbigh 1981, Preface)

  Kant, with his evolutionary cosmic hypothesis, assumes a linear notion of time (an arrow of time). But is a forward-moving arrow of time the only direction con- ceivable for time? In  his view  of  the  evolution  of  the cosmos from  an  origina

  chaos to the observed order of galactic structures through the operation of mechanical laws, Kant underwrites two assumptions: one concerns the metric of time—or some device by which the passage of time can be measured and quan- tified; the other concerns the topology of time—or some device by which the shape of time—whether it is linear, circular or cyclic—can be ascertained. As has just been shown, Kant infers from the operation of mechanical laws that the evolution of the cosmos has taken millions of years to reach its present state of order and will take many more millions of years to mature to perfection in other cosmic worlds. This metric is not precise, since Kant did not know that the Earth is approximately

  4.5 billion years old and that our universe originated approximately 13.7 billion years ago. Kant’s cosmic evolution also assumes a linear topology of time. According to Kant’s evolutionary hypothesis the cosmos seems to follow a uni- directional arrow of events from chaos to order. The topology of time concerns the geometric shape of time rather than its quantitative measure. It is possible to have a topology without a metric. Galileo discovered the topology of the moon—its mountains and valleys indicate the geometric shape of the surface of the moon— but he did not calculate the height of the mountains, or the depth of its valleys. Kant and Leibniz had a clear notion of the topology of time but a crude metric. But considerations of the topology of time, even without metric considerations, are important in modern cosmology, where the question of the arrow of time has gained new significance (Chap.  4).

  Considerations of the topology of time run through the history of human time reckoning. They may be based on pure speculation or on a theoretical conjecture before evidence favours one or the other topology. An example of speculation occurs in archaic societies, which entertain two notions of time. One is profane time, in which people spend most of their ordinary lives; it is devoid of meaning. People are able to leave profane time and enter mythical time through participation in rituals and ceremonies (Eliade 1954, 34–46). Through the imitation and repe- tition of archetypes, Man is projected to a mythical epoch, in which the archetypes were first revealed. The Platonic structure of this archaic ontology is noteworthy. Objects or actions only become real if they imitate or repeat an archetype. Objects only acquire meaning and reality if they participate in an archetype. As partici- pation in the archetype involves repetition, archaic and pre-modern societies often entertained a cyclic conception of time. They regarded time as a succession of recurring events. Time performs a cycle, as illustrated in the quote by Nemesius, Bishop of Emesa, in the 4th century A.D.:

  Socrates and Plato and each individual man will live again, with the same friends and fellow citizens. They will go through the same experiences. Every city and village will be restored, just as it was. And this restoration of the universe takes place not once, but over and over again – to all eternity without end. (…) For there will never be any new thing other than that which has been before, but everything is repeated down to the minutest detail. (Quoted in Whitrow 1989,  42–43)

• The notion of cyclic time, however, is incoherent from a philosophical point of view, for it presupposes a linear conception of time.4 If each state of the universe recurs infinitely many times, then two possibilities need to be distinguished:

• Each individual state recurs repeatedly but separated by a temporal interval, Dt; hence each state, S, is identical to any other state, S0, (indistinguishable in their properties), except for their location in time. But then these states are distin- guishable with respect to their temporal properties. A cyclic or recurring theory of time presupposes a linear progression of time: ‘identical states of the universe recur but at a progressively later stage’ (Newton-Smith 1980, 66; cf. Fig. 2.3a). This spiral view implies that the same moment cannot be visited twice in time.

• The spiral view has no spatial analogue since it is possible to visit the same location  an indefinite number of times (Fig.  2.3b).

• A truly cyclic theory of time must affirm that all states, S1, S2, S3 are identical to states S10, S20, S30, in all, including their temporal properties. But this stipulation is inconsistent: (a) If S occurs before S0, but otherwise both are identical in their properties, then linearity obtains; (b) if S = S0 then S and S0 are indistinguishable

• in all their properties, including their temporal occurrence; they are exactly the same states and no recurrence has occurred. Case (b) employs Leibniz’s Principle of the Identity of Indiscernibles, which states that if entities A and B have all properties in common, then they are identical: A and B are the same entity.

  Even though, then, cyclic time is inconsistent, its impossibility does not exclude other topologies of time, based on theoretical conjectures. If time does not possess a cyclic pattern two geometric possibilities remain, in the absence of any empirical evidence.

   

2.6.1  Linearity of Time

 

There is a temporal relation, often regarded as the most fundamental relation (Zwart 1976, 34–37), which is ideally suited for the topological structure of temporal

linearity. As Saint Augustine already observed, it is the ‘before–after’ relation between events. There is also a philosophical theory of time—the relational theory—which is suited to this linear topology (van Fraassen 1970, 61). For the relational theory, in Leibniz’s words, time is the order of succession of events. Time unfolds, as events happen one after the other, in a one-dimensional line. But for the relational view, at least according to its classic statement, time has a beginning.    It coincides with the creation of the universe and it will last for as long as change happens in the universe. But the order of events cannot be changed: if A happens at the same time as B, then A and B happen simultaneously for all observers in the universe; and if A happens before B, it is logically impossible for B to happen before

A. Whilst this order relation sounds commonsensical, it has been cast into doubt in the Special theory of relativity. If the relational theory is to survive the findings of modern physics, it will have to be reformulated in its language.

The standard topology adopts the order type of linearity, which is represented by a line of real numbers (see Newton-Smith 1980, 51; van Fraassen 1970, 58ff; Lucas 1973). Real numbers, which include both rational and irrational numbers, are needed to express the properties of this order type. The linear view of time ascribes to time the topological properties of linearity (time is one-dimensional), density (a new instant can be inserted between any pair of distinct instants), and either finitude or infinity. It is a mathematical model of linear topology, which may not correspond, in every aspect, to physical time. For instance, if according to the theory of quantum gravity there exists a shortest unit of time—Planck time—then physical time would not be dense in a strict sense of the word; equally if the universe started with a Big Bang, then the universe may expand forever but it would have a definite beginning. There are several cosmological models, which approximate this linear  topology.

 

 

• The standard cosmological model is the Big Bang model, which makes the universe spatially homogeneous and isotropic (Weinberg 1978; Penrose 2005, 718; Kragh 2007, Chaps. 4, 5; Carroll 2010). The Big Bang model assumes that the Universe had a definite beginning, and started from a singularity (infinite temperature, density) in a gigantic explosion,  approximately  13.7 billion  (1.37 9 1010) years ago. At the very beginning the universe was  very  hot  (10110 °C), so hot that neither atoms, nor atomic nuclei or molecules could form; only quarks and other fundamental particles were present. But the universe began to cool very rapidly so that after the first 3 min, the early universe was filled with elementary particles (electrons, neutrinos, protons, photons). Over the next millions of years, the universe began to expand and cool so that ordinary matter—the first stars (after 150 million years), and galaxies (after 800 million years)—could form out of the soup of elementary particles. It took 3 billion years for our Milky Way to take shape. It is generally assumed that the entropy of the early universe was low so that its expansion is in accordance with the Second law of thermodynamics (Penrose 2005, Sects. 27.7, 27.13). If the universe had a definite beginning, what will be its ultimate fate? Will it expand forever and end in a Heat Death or will it grind to a halt and recollapse to a Big Crunch?

  According to the Standard Model its evolution depends on the amount of matter in the universe. As a threshold, cosmologists specify a critical density, qc, which is the largest density, which the universe can have and still expand. Cosmologists then define a parameter, X, as a ratio of actual and critical mass density:

   

   

  o                           -         -

  q        actual    mass    density

   

  X ¼       ¼

  

   

  ð2:1Þ

   

  qc           critical - mass - density

   

   

  The value of X = \ 1, 1 or [1 and this value indicates the future fate of the universe (Fig. 2.5).

  The value X = 1 signifies the famous Heat Death, which was predicted by  many cosmologists of the 19th century—the total dissipation of energy, such that no energy differentials would be left to perform useful work and sustain life. Is there any evidence in support of the Big Bang theory?

•  

• ¼

  In 1929 the American astronomer Edwin Hubble (1889–1953) discovered Hubble’s law, which states a relation between distance (or redshift) and radial velocity. The farther a galaxy is away, the faster it moves away from the observer: v Hd (where v = radial velocity of the  galaxy,  d = distance  to galaxy, H = Hubble’s constant). Hubble established, on the basis of observa- tions of the redshift of specific absorption lines in the spectra of galaxies that galaxies were moving away from each other at a velocity proportional to their distance. The inference from Hubble’s law is that the galaxies must have been closer in the past; and ultimately that the universe must have had a beginning some 13.7 billion years ago, at which point it was in a low entropy state.

   

• In 1965 A. Penzias and R. Wilson stumbled across the cosmic background radiation, at a uniform temperature of 2.7 °C, which was later understood to be

  the afterglow of the hot, young universe, shortly after the Big Bang. The microwave background radiation is a relic of the early stages of the universe. Some 380,000 years after the Big Bang the universe would have cooled down to temperatures, which allowed the formation of atoms like hydrogen and helium. From the abundance of hydrogen in the universe today, it can be inferred that prior to the formation of stable atoms, the early universe must have been filled with an enormous amount of radiation. As the universe expanded this radiation redshifted and cooled to approximately 3 K and this is the microwave radiation, which Penzias and Wilson detected in 1965.

  But the Big Bang model provides no indication as to whether the universe will expand forever—leading to a Big Chill (or what the 19th century called the Heat Death)—or will eventually recontract, leading to a Big Crunch. As indicated in Fig. 2.4 an important question from the point of view of the asymmetry of time, to be discussed later, is whether  the  Big  Crunch  will  resemble  the  Big  Bang.  The question of expansion or contraction essentially depends on the amount of dark matter and dark energy in the universe, which cannot be directly observed, but whose gravitational pull could be strong enough to bring the expansion of the universe to a halt and start the contraction phase. Current data indicate that the universe is actually accelerating so that at present the expansion seems set to continue forever. A further question, which the standard Big Bang model cannot answer, is how the Big Bang itself was caused and how the special (low-entropy) conditions for space–time inherent in it could have arisen from a singularity (Hawking 1988,  43f; Freedman  1998, 97).

  The standard Big Bang model also faced a number of anomalies,5 as further

  work revealed. Three of these problems are the horizon problem, the flatness problem and the smoothness problem (cf. Guth 1997, 183; Atkins 2003, 258ff; Penrose 2005, Sect. 28.3; Kragh 2007, 226; Carroll 2010, Chap. 14). The horizon problem relates to the extraordinary uniformity of the observed universe. This uniformity is most clearly revealed in the cosmic background radiation, which is at a uniform temperature of 2.7 °C in all directions across the sky. Yet the universe is so vast that very distant parts of it cannot communicate with each other because of the finiteness of the speed of light. The cosmic background radiation was released at about 380,000 years after the Big Bang, by which time two opposite points in space were already so widely separated that they could not have communicated through light signals. The only way to solve this problem is to stipulate that the universe began in a state of great uniformity.

  The second problem relates to the puzzle that the universe is remarkably close to being spatially flat. The fact that the curvature of space is very close to zero, and hence Euclidean, is referred to as the flatness problem. In terms of X (the ratio of actual mass density to critical mass density of the universe) this means that X  % 1

  today, so that the universe is expected to expand forever but the rate of expansion will tend to zero with time. But this gives X a very special value, already present at the beginning of the universe, for which the standard Big Bang model offers no explanation. It also conflicts with recent observations, which suggest that the expansion of the universe proceeds at an accelerated pace.

  The third problem is the smoothness problem, which refers to the even distri- bution of matter in the universe and the uniformity of space–time geometry. Kant already took note of this uniformity with his nebular hypothesis. Just as Kant’s reliance on Newtonian physics cannot account for this large-scale structure of the universe, so the standard Big Bang model cannot explain the distribution of matter throughout the universe. The standard model implies that the universe exploded so quickly that there was no time for information to be exchanged between its parts (Guth 1997, 213).

• Cosmologists responded to these problems by developing so-called inflationary models, which can both account for the uniformities and the irregularities (lumpiness) in the distribution of matter-energy in the universe. The cosmic background radiation shows similar irregularities. As J. Gribbon sums up the situation  in cosmology:

• All these problems would be resolved if something gave the Universe a violent outward push (in effect, acting like antigravity) when it was still about a Planck length in size. Such a small region of space would be too tiny, initially, to contain irregularities, so it would start off homogeneous and isotropic. There would have been plenty of time for signals traveling at the speed of light to have criss-crossed the ridiculously small volume, so there is no horizon problem – both sides of the embryonic universe are ‘aware’ of each other. And spacetime itself gets flattened by the expansion, in much the same way that the wrinkly surface of a prune becomes a smooth, flat surface when the prune is placed in water and wells up. As in the standard Big Bang model, we can still think of the universe as like the skin of an expanding balloon, but now we have to think of it as an absolutely enormous balloon that was hugely inflated during the first split-second of its existence. (Gribbon 1996, 220, italics and bold type have been removed from this quote)

  As just indicated, in order to deal with these anomalies cosmologists developed so-called inflationary models, which correspond to different topologies of time. Although there are now a number of inflationary models in circulation, in terms of the topology of time it is convenient to focus on the initial inflationary Big Bang model, due to the American particle physicist Alan Guth. It later gave way to the Chaotic Inflationary model (due to the Russian cosmologist Andrei Linde, as well as the American cosmologists Paul Steinhardt and Andreas Albrecht), which is now the standard version of inflation. Although the first inflationary model was developed by A. Starobinsky in Moscow (1979), the term ‘inflation’ was coined by Guth (in 1981), who developed his own version of an inflationary model. Inflation models deal with the earliest phase of the Universe, inspired by particle physics, and assume a very rapid expansion from a very small region (i.e., the Planck length