Inflation and the Birth of the Universe (Part 1)

The Universe Before the Big Bang

Inflation and the Birth of the Universe

The standard cosmological model, which describes the current Universe in terms of its matter and radiation components, and covers a seemingly long period of time – more than 10 billion years, from epochs preceding the synthesis of nuclear elements until now – legitimately represents one of the greatest achievements of twentieth century physics.

As already pointed out in Chap. 2, this model is based upon some crucial assumptions. One such assumption is that the geometry of the Universe and its time evolution are determined by the equations of general relativity. Another is that the whole particle content of our Universe can be described on cosmological distance scales in terms of a perfect fluid with two main components: matter and radiation, both uniformly distributed over space.

In this case the equations of general relativity successfully de- scribe the expansion of this cosmic fluid which progressively cools down, according to the laws of classical thermodynamics, starting from an initial state characterized by arbitrarily high temperature and density. This expansion, slowed down by gravitational attraction, then “separates out” the various components of the cosmological fluid: heavy particles separate from the radiation, condense and form the matter structures (stars, galaxies) that we observe today, while radiation – which was initially the dominant component of the fluid – is diluted faster by the expansion, and tends to become subordinate to heavy particles and lumpy matter.

As outlined in Chap. 2, the standard cosmological model can rightly claim a number of important successes. First of all, this model explains important astronomical observations such as the redshift of galaxies described by the well-known Hubble law. In particular, it provides us with a consistent theoretical framework for computing the regression velocity of galaxies as a function of the distance, including also their possible deceleration or acceleration (depending on the equation of state of the dominant cosmological fluid), giving good agreement with astronomical data.

Moreover, the standard cosmological model can account for the primordial formation of light elements (the process of nucle- osynthesis), since it provides a sufficiently hot environment for the required nuclear reactions to take place among the components of the primordial gas of particles. In addition, this model explains the existence of a cosmic background of electromagnetic radiation in thermal equilibrium, in full agreement with the progressively more precise measurements carried out today, confirming the presence of cosmic radiation with a black-body spectrum and a current temperature of about 3 degrees kelvin.

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Despite the numerous successes, there are some kinematic issues within the standard model that remain unsolved, in addition to the already mentioned problem of the initial singularity.

These kinematical problems concern the high degree of isotropy, homogeneity, and flatness characterizing the current Universe, and the large entropy associated with its background radiation. Why is the spatial geometry the Universe today so flat (i.e., so similar to the geometry of a three-dimensional Euclidean space)? Why – bar- ring some irregularities due to localized matter clumps – is the background radiation so uniformly distributed over the whole observable space? And – since the standard cosmological evolution is adiabatic, i.e., entropy-conserving – what is the origin of the large entropy encoded in this radiation?

Leaving the last question aside for the moment (it will be considered at the end of this chapter and in the next), let us focus our discussion on the other points, starting with the question about the curvature. If we select a spatial portion of the current Universe, and we (indirectly) measure its curvature, we find that the maximum allowed value for such a curvature, according to the most recent observations,1 is a few per cent of the total space-time curvature (which within the Einstein theory is represented by the square of the reciprocal Hubble radius, H2/c2). At first glance, the fact that the two curvatures, if not of the same order, are at least of comparable magnitude, might seem a quite reasonable and accept- able result.

The problem arises because in an expanding Universe the curvature of its geometry varies with time. According to the standard model, in particular, the curvature is currently decreasing so that its value was higher in the past. However, tracing the solutions of the standard cosmological model back in time, one finds that the curvature of the three-dimensional spatial sections of the Universe (henceforth referred to as the spatial curvature, for short), deter- mined by the reciprocal of the spatial radius R, grows much more slowly than the space-time curvature, determined by the Hubble parameter H. Hence, even if we start from a present configuration in which the values of the two curvatures are comparable, going deeply backward in time we necessarily end up with a primordial initial configuration where the spatial curvature is much smaller than the space-time curvature (instead of being of the same order of magnitude).

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This is not by any means a natural initial condition. Indeed, let us visualize – even if improperly – the cosmological space-time as the two-dimensional surface of a sheet of paper. We would have a sheet that, instead of being laid down onto a plane, would be wrapped like a narrow cylinder with one dimension (time) highly curved, and the other one (space) almost flat. Thus, we may naturally expect some previous phenomena to have occurred, determining such a highly asymmetric initial configuration through some peculiar mechanism. Indeed, it would not be satisfactory to assume that our Universe was born in that form solely because this con- figuration is the only initial condition able to produce the current cosmological state. This amounts to abandoning any attempt at scientific explanation.

In order to explain why, sometime in the past, our Universe was in a geometric state characterized by a spatial curvature much smaller than the space-time curvature, the standard cosmological model needs modifications at early epochs. The introduction of such modifications has led to the formulation of the so-called inflationary models, first developed at the beginning of the 1980s, starting from ideas almost simultaneously proposed by astrophysicists like Alexei Starobinski and Andrei Linde in the Soviet Union, and Alan Guth, Paul Steinhardt, and Andreas Albrecht in the United States. The term “inflationary” assigned to these models arises be- cause, within the framework they propose, the spatial part of the Universe at some point “inflates”, expanding at a very fast rate.

This feature is an essential ingredient of such models in order to explain the relative decrease in the spatial curvature, and to achieve (at the end of inflation) a geometric state that can suitably represent the initial configuration for the subsequent standard evolution (continued until the present epoch).

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The problem that we have just outlined is also called the flat- ness problem. Another issue pertaining to the standard cosmological model, and closely linked to the previous one, is the so-called horizon problem, which can be formulated as follows. The current Universe appears to be homogeneous and isotropic over the whole scale of the horizon, which is determined by the Hubble radius c/H (see Chap. 2). Going backward in time the space-time curvature increases, i.e., the Hubble parameter H increases, so that the radius c/H of the Hubble horizon shrinks. A problem then arises because, according to the standard model, the spatial radius R of the Universe shrinks with time more slowly than the Hubble radius.

Indeed, at a given epoch in the past, the radius of the spatial portion of the Universe that we are presently observing was much bigger than the radius of the Hubble horizon at the same epoch (see Fig. 5.1). This implies that in the past, according to the standard model, different portions of the currently observable Universe were included in different horizons, and thus were unable to interact and communicate with one another. If this is the case, why are the physical properties of the present Universe (e.g., the temperature of the radiation background) the same everywhere, as if all portions of space in the past were in causal contact, exchanging signals and interacting as if they were included within the same horizon?

inflation-and-the-birth-of-the-universe-pic-1-300x208 Inflation and the Birth of the Universe (Part 1)
FIGURE 5.1 The vertical axis corresponds to the time direction, while the horizontal axes are associated with the spatial dimensions. At the present time t = t0, the whole region of the Universe that we are able to observe is contained within the present Hubble horizon (shaded region). During the past (i.e., at any given time t1 < t0), the radius R of the current observable Universe was bigger than the corresponding Hubble radius c/H. The dashed line shows the time evolution of the horizon radius c/H, while the solid curve represents the qualitative evolution of the spatial radius R

The flatness and horizon problems (as well as other similar problems, all pertaining to the standard model) are strictly linked to each other. Both can be solved by assuming that the primordial Universe, before entering the phase of standard evolution (described in Chap. 2), has undergone a phase (called inflation) during which its spatial radius R expanded in accelerated fashion – hence faster than the horizon radius c/H which, during the standard phase, increases linearly with time, as illustrated in Fig. 5.1.

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The most conventional inflationary scenario is implemented by introducing a phase of de Sitter-like evolution, during which the spatial geometry of the Universe is subject to exponential expansion, while the space-time curvature (and thus the horizon radius) remain constant. The result is illustrated in Fig. 5.2, where the standard evolution is preceded by a phase of de Sitter inflation. With this modification, the currently observable portion of Universe at the beginning of inflation (i.e., at the time t = t2) was all included within the same Hubble horizon. Therefore, all of its parts were initially in causal contact, being able to interact and give rise to a homogeneous and isotropic patch of space-time.

During the inflationary phase the spatial radius R expands faster than the Hubble radius c/H. We can say, using the common jargon, that the Universe goes “outside the horizon”, i.e., it be- comes larger than the horizon itself, while its degree of homogeneity and isotropy remain unaffected since – as already mentioned in Chap. 3 – outside the horizon all physical properties “freeze out”.

At the end of the inflationary phase (i.e., for t = t1) this trend is in- verted: R starts to increase more slowly than c/H, and the current observable Universe begins to “re-enter the horizon”. The re-entry is eventually completed at t = t0.

inflation-and-the-birth-of-the-universe-pic-2-300x188 Inflation and the Birth of the Universe (Part 1)
FIGURE 5.2 Qualitative sketch of the evolution of the spatial radius R (solid curves) and the horizon radius c/H (dashed curves) for a model of the Universe which includes a conventional de Sitter-type inflationary phase. The evolution from t1 to t0 describes the standard regime (exactly the same as in Fig. 5.1). The evolution from t2 to t1 describes instead the inflationary de Sitter regime, characterized by a constant Hubble parameter and an exponentially expanding spatial radius. The whole portion of the Universe that we are currently observing (the shaded region at t = t0) was included within the same Hubble horizon at the beginning of inflation (the shaded region at t = t2)

From Fig. 5.2 it is evident that a successful inflationary phase must last for a sufficiently long period of time. In fact, going back- ward in time, the time interval between t1 and t2 has to be sufficiently long for the whole currently observable Universe to have enough time to re-enter the horizon. The minimal required duration of the inflationary phase also depends on the size of the horizon (as can be seen from the figure), and thus on the curvature of the Universe at the onset of inflation. For instance, if inflation takes place close to the Planck scale (i.e., at the edge of the domain of validity of the standard model), then the required minimal duration is longer than in the case of inflation occurring at lower curvature scales, where H is smaller and the horizon radius is larger. How- ever, if the amount of inflation is measured in units of the Hubble time 1/H, then the same amount of inflation requires a shorter du- ration at higher curvatures, where H is larger and the inflationary process is faster.

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It is interesting to check that the type of evolution described in Fig. 5.2 can also solve the flatness problem. To this end it will be enough to recall that the space-time curvature radius varies in time as the Hubble radius c/H, while the three-dimensional curvature radius varies as the spatial radius R. During inflation the space-time curvature remains constant, while the spatial curvature decreases as fast as R increases. At the end of the inflationary phase the space-time geometry is thus much more curved than its three- dimensional spatial sections, and this gives us a simple explanation for the origin of the “strange” initial condition characterizing the standard cosmological evolution.

The simplest models of inflation are usually characterized by a period of accelerated evolution during which the horizon either remains constant or slowly grows with time (this second case corresponds to the so-called slow-roll inflationary models). During the pre-Big-Bang phase typical of string cosmology models, how- ever, the curvature increases, H increases (as we have seen in the previous chapters), and therefore the radius of the horizon tends to decrease. Even in this case, however, the spatial radius during the pre-Big-Bang phase undergoes an accelerated evolution which complies with the realization of an inflationary regime: the horizon exit of our portion of the Universe, in that case, is somehow even more rapid and efficient than in the conventional inflationary scenario (see Fig. 5.3). For this reason this type of inflation (first introduced in the 1980s by Deshdeep Sahdev, Eward Kolb, David Lindley, David Seckel, and others, quite independently from string theory and string cosmology models) is also called super-inflation.

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In string cosmology models, the whole pre-Big-Bang phase may thus be considered as an inflationary phase, although an unconventional one, able to solve the kinematic problems of the standard cosmological model. The crucial difference – clearly illustrated by Figs. 5.2 and 5.3 – with respect to models of de Sitter-like inflation is that the initial size of the horizon at the beginning of inflation (i.e., the horizon at t = t2) is much bigger for string cosmology models than for conventional (de Sitter-like) models. As the horizon radius is the reciprocal of the curvature, this feature of string cosmology models is a direct consequence of the fact that their initial curvature is very small with respect to the Planck scale. In- deed, in pre-Big-Bang models, the Universe starts evolving from an initial state quite close to the string perturbative vacuum (which is flat), unlike conventional models where inflation starts in a regime of very large curvature (and small Hubble radius).

inflation-and-the-birth-of-the-universe-pic-3-300x212 Inflation and the Birth of the Universe (Part 1)
FIGURE 5.3 Qualitative sketch of the evolution of the spatial radius R (solid curves) and of the horizon radius c/H (dashed curves) for a model of the Universe in which the phase of standard cosmological evolution (from t1 to t0, the same evolution as depicted in Figs. 5.1 and 5.2) is complemented by a phase of pre-Big-Bang inflation (from t2 to t1), characterized by shrinking horizon and growing curvature. The figure illustrates the behavior of the pre-Big-Bang kinematics in both the string geometry, associated with an expanding spatial radius R, and the Einstein geometry, associated with a contracting spatial radius RE. The two geometrical representations are physically equivalent, but in the Einstein geometry the initial horizon is larger, and inflation is longer (in general), than in the associated string geometry

The physical origin of such a difference is ultimately related to the dual symmetry present at the heart of string cosmology and to the fact that, in pre-Big-Bang models, inflation precedes (rather than follows) the Big Bang epoch. Note that, from a technical point of view, the prediction of a small value for the curvature at the on- set of inflation should be considered as an advantage over models of inflation at large curvatures. In fact, if the curvatures are small, then the forces coming into play are weak, and the initial evolution is governed by well known low-energy physics and can be described in terms of simple, lowest-order equations. In the conventional inflationary scenario, on the other hand, the initial conditions are imposed at very high curvature scales, even inside the Planck scale, quantum gravity regime, where conventional low-energy results cannot be safely applied in general. This, in addition to the singularity problem, may also lead to the so-called trans-Planckian problem affecting the evolution of cosmological perturbations, as recently pointed out by Robert Brandenberger and Jerome Martin.

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Inflation and the Birth of the Universe (Part 2)