General Relativity and Standard Cosmology (Part 2)

The Universe Before the Big Bang: General Relativity and Standard Cosmology

Exploiting these assumptions (which may appear rather an oversimplification, but which are supported by various direct and indirect observations), one can exactly solve the Einstein gravitational equations, thus determining the evolution of the cosmic geometry and its gravitational sources. One then finds that the energy density of non-relativistic matter is inversely proportional to the spatial volume, i.e., to the third power of the spatial radius R

For its part, the radiation energy density is inversely proportional to the fourth power of the radius. Thus, in an expanding Universe where R grows in time, the radiation density decreases in time faster than the matter density. Indeed, current cosmological observations indicate that the radiation energy density is today about ten thousand times smaller (i.e., a factor of 10^−4 smaller) than the matter density.

Going backward in time, however, the energy density of radiation tends to increase with respect to that of matter until, at the so-called equality time, the two energy densities are equal. For times before the equality time, the energy density of radiation takes over, and the Universe undergoes a phase transition where its rate of expansion also undergoes a change.

Hence, according to the standard model, the Universe is characterized by two main stages: an initial phase, where radiation is the dominant form of energy filling the Universe and controlling its evolution, followed by a phase (possibly extended in time until the present epoch) where matter is dominant. The expansion rate of the space-time geometry (more precisely, the speed measuring the rate of change of R in time) is higher during the radiation phase, despite the fact that the energy density, and hence the mutual gravitational attraction, tends to decrease as a function of time.

In other words the equations of general relativity provide us with solutions describing a decelerated expansion, for both phases. This means, strictly speaking, that in these solutions the first derivative of the spatial radius R(t) is positive, while the second derivative is negative. Furthermore, during both phases the spatial curvature decreases uniformly as the inverse square of the spatial radius; the global, space-time curvature – which is closely linked to the expansion rate – also decreases continuously as a result of the decelerated expansion of R, but at a faster rate than the spatial curvature.

Quite independently from their particular kinematic proper- ties, it should be stressed that the possible existence of cosmological solutions describing an expanding Universe is one of the greatest achievements of the standard model. In fact, one can explain the famous Hubble law (discovered at the beginning of the twentieth century), according to which the light we receive from the various galaxies populating the Universe is characterized by a redshift that is an increasing function of the distance at which the emitting galaxy is located.

In fact, let us consider a light ray currently received on the Earth, and emitted from a galaxy many light-years away. Light propagates with a large but finite speed, so that the light ray has spent many years traveling in space to cover the distance separating us from that galaxy, from which it was emitted long ago. On the other hand, if the Universe is expanding, its radius was smaller in the past than its radius today, the cosmic energy density was more compressed, and the cosmological gravitational field more intense (so that the space-time geometry was more curved than today).

Thus, as previously discussed, such a light ray was emitted with a redshift (with respect to the flat space) which is greater than the one affecting light today. This is the reason why the light received from a distant source is redshifted with respect to the same light emitted today, or to the light emitted by a source located at a smaller distance.

Clearly, the redshift depends on the rate of change of the radius of the Universe during the intergalactic journey of the light ray, and thus on how far away the observed galaxy is. The exact relationship between redshift and distance is quite complicated in general, but to first order it can be approximated by a linear relationship. In this case the shift computed according to the standard model turns out to be directly proportional to the distance of the source, just as suggested by the observations leading to the Hubble law, and the proportionality constant H between redshift and distance is called the Hubble constant (or better, the Hubble parameter). In the standard model the Hubble parameter is just determined by the speed at which the spatial radius of the Universe changes with time, i.e., by the first derivative of the spatial radius divided by the spatial radius itself, and is also proportional to the space-time curvature (in agreement with the fact that the reddening of light is a gravitational effect).

Within the physics of electromagnetic waves, on the other hand, the reddening of light is not a new phenomenon. It is well known, even in a non-relativistic context, that the frequency of a periodic signal emitted by a moving source is shifted towards the red or towards the blue, depending on whether the source is receding or approaching the observer, respectively, according to the well-known Doppler effect.

The cosmological redshift described by the Hubble law can also be interpreted as a Doppler effect, associated with the fact that galaxies are mutually receding as a result of the expansion of the Universe. According to this interpretation, the redshift z (given to a first approximation by the product of the constant H and the distance of the source, divided by the speed of light c) also coincides with the ratio between the recession velocity of a galaxy and the speed of light. The recession velocity increases with the distance, and when the distance approaches the characteristic scale c/H, it approaches the speed of light. This is the reason why the parameter c/H (proportional to the inverse of the Hubble constant, and called the Hubble radius, or Hubble horizon) defines a limiting distance scale within which exchange of signals and, consequently, causal interactions are possible. It should be noted for subsequent applications that, since the parameter H is proportional to the space-time curvature, the horizon radius c/H is inversely proportional to the curvature. Thus, in a time-dependent geometry, the size of the horizon becomes large when the curvature decreases, while it shrinks when the curvature increases. We will come back to this point in next post, when we discuss the properties of inflationary cosmological models.

Let us now reconsider the solutions of the Einstein equation resulting from the assumptions of the standard cosmological model. Using these solutions we can make predictions about the future evolution of the Universe. It turns out that such evolution is completely determined once we know the present value of the Hubble parameter, the energy density, and the equation of state, i.e., the relation between the effective pressure and energy density, of the dominant gravitational sources. According to the simplest version of the standard model, the Universe should currently be matter-dominated, i.e., filled with a dust-like fluid, whose average pressure is zero. In this case, depending on the value of its energy density, there are two possible classes of future evolution.

If the density is below some critical threshold (which de- pends on the current value of the Hubble parameter, the speed of light, and Newton’s gravitational constant), the three-dimensional space must then have a constant, negative curvature. (This space, which is said to have hyperbolic geometry, is difficult to visualize.

Roughly speaking, it is rather like the central cavity of a saddle.) In this case there is an indefinite and gradual slowing down of the expansion rate, until the space-time becomes completely empty and flat. On the other hand, if the density is greater than this critical value, then the three-dimensional space must have a constant positive curvature (like a sphere). The expansion will decelerate until it eventually stops, whereupon the Universe will subsequently start contracting until it collapses into a final singularity. The limiting case in which the density is exactly equal to the critical value corresponds to a zero spatial curvature, and once again to an endlessly decelerated expansion.

Current effort in astronomy and astrophysics thus focuses on the possibility of measuring, either directly or indirectly, the values of the Hubble parameter and the total energy density of the present Universe. The most recent measurements seem to indicate that the spatial curvature is almost zero, in agreement with the predictions of inflationary models, and that the total energy density is very close to the critical value. However, oddly enough, the present Universe seems to be in a state of accelerated expansion.

These observational results, which are receiving more support all the time, clash with the cosmological solutions of the standard model, which do not actually predict any acceleration. They may therefore suggest that the current Universe is not matter- dominated, i.e., it may not be filled with a fluid with zero pressure.

Instead, its energy density may be dominated by some exotic element dubbed quintessence (as a reminder of the mysterious fifth element of post-Aristotelian philosophy). In order to induce the observed acceleration, this substance should have a nonzero negative pressure.

The simplest example of a source that could reproduce quintessence effects is certainly the famous cosmological constant, a term representing the vacuum energy density and introduced into the equations of general relativity by Einstein himself (although he publicly withdrew it, saying that it was the biggest blunder of his life!). Today, it would appear that Einstein was right after all.

It should be stressed, however, that the value of the cosmological constant needed to reproduce the observed cosmic acceleration has never received a satisfactory explanation, not even in the context of modern quantum theories which unify all interactions.

Besides predicting the future, the standard model allows us to go backwards in time through the history of the Universe, thus reconstructing its past. Using current observations one can compute, for instance, the temperature at which the energy densities of matter and radiation were of the same order. Since the radiation is in thermal equilibrium (as happens in an oven when it reaches its designated temperature), its energy density is proportional to the fourth power of the temperature, according to the well-known Stefan law of classical thermodynamics. On the other hand, ac- cording to the equations of the standard cosmological model, the energy density of radiation must also be inversely proportional to the fourth power of the spatial radius of the Universe, as mentioned previously. It follows that the radiation temperature is inversely proportional to the spatial radius, and consequently, as the Universe expands (i.e., as its radius increases), the temperature de- creases and the radiation gets colder.

This decrease in temperature is a gravitational effect quite similar to the redshift of frequencies. In the past, at the time of equality between matter and radiation energy density, the Universe was much hotter than it is today. But how much hotter, precisely?

We know that the current radiation temperature is about three degrees kelvin above absolute zero (i.e., about −270 degrees centi- grade, or 10^−4 eV, where the symbol eV denotes electron-volts, a typical unit of energy and temperature used in nuclear physics).

We also know, as already stressed, that the current value of the ratio between the radiation energy density and the matter density is about 10^−4. According to the standard model this ratio varies with time in a manner inversely proportional to the spatial radius, and it is therefore directly proportional to the temperature (see the previous paragraphs). The equality epoch, occurring when the ratio of the energies was about 104 times bigger than its current value, thus corresponds to a radiation temperature 104 times higher than the current value, i.e., about one electron-volt (or ten thousand degrees kelvin). Similar arguments can also be applied to compute the temperature at earlier epochs. We find therefore that, going backward in time according to the standard model, the Universe becomes not only denser and more warped, but also hotter. Cosmic history can then be traced back in terms of three possible evolution parameters: time, temperature, and space-time curvature (or, equivalently, the inverse of the curvature, the Hubble radius c/H). Obviously, these parameters are not mutually independent: time is proportional to the Hubble radius, while the temperature is inversely proportional to the spatial radius R which, in turn, depends on time.

The main stages of the standard cosmological model as a function of the above-mentioned evolution parameters are shown in Fig. 2.1. In this figure one proceeds backward in time from the current value of the Hubble radius down to the Planck radius, i.e., from the current epoch where the Universe has a curvature radius of about 1028 cm, or 10 billion light-years (the current value of the Hubble radius), until the beginning of the quantum gravity epoch, when the curvature radius was about 10^−33 cm (corresponding to the so-called Planck length LP).

Let us follow, for instance, the scale corresponding to the temperature of the cosmic microwave radiation, and proceed backward in time, starting from its current value of about 10^−4 electron-volts.

When the temperature is about one hundredth of an electron-volt, we reach the epoch of galaxy formation; as the temperature reaches the value of about one electron-volt, we reach the epoch of matter– radiation equality, corresponding roughly to the phase in which nuclei and electrons tend to combine into atoms. When the temperature grows to about 1 million electron-volts (1 MeV), we reach the epoch in which the primordial synthesis of the elements (or nucleosynthesis) was taking place. Neglecting many other processes, for the sake of brevity, we then get to a temperature of about 10^16 GeV (1 GeV = 1 billion electron-volts), corresponding to the epoch where the electromagnetic and nuclear (weak and strong) interactions were probably unified into a single fundamental interaction. Finally, at a temperature of about 10^19 GeV (the Planck temperature), we reach the beginning of the quantum gravity epoch where general relativity, together with the standard cosmological model, can no longer be unambiguously applied.

FIGURE 2.1 Simplified sketch of the evolution of the Universe according to the standard cosmological model. The figure summarizes the main stages of cosmic evolution, starting from the present epoch character- ized by the observed value of the Hubble radius, back in time down to the quantum gravity epoch characterized by the Planck radius. The past history of the Universe is parametrized in terms of three equivalent cos- mological variables: the cosmic time (measured in seconds), the radiation temperature (measured in electron-volts), and the space-time curvature radius (measured in centimeters)

It should be stressed that, for graphical reasons, the different scales used in Fig. 2.1 do not respect the relative lengths of the various epochs. However, one can nevertheless ascertain from the figure that the maximum energy scale currently attainable in a laboratory supplied with the most powerful available accelerators (about 1 TeV, i.e., one thousand GeV) is still well below the typical energies coming into play in the primordial Universe.

Tracing the evolution predicted by the standard model back- ward in time even beyond the Planck radius, the Universe (as shown in Fig. 2.1) necessarily reaches a singular stage where the temperature becomes infinite, the curvature radius c/H is then zero, and its reciprocal (the curvature) therefore becomes infinite.

The occurrence of arbitrarily high values of temperature, density and curvature has suggested the name Big Bang for such a singular phase, which the standard model identifies with the beginning of the expansion of the Universe.

If we fix the origin of the temporal axis (i.e., the coordinate t = 0) to coincide with the moment at which the Big Bang takes place (as in Fig. 2.1), the current epoch then corresponds to a time coordinate of about 10 billion years (i.e., about 1018 seconds). This number, often called the age of the Universe, actually represents the time interval elapsed between the Big Bang and the present epoch, as is clearly shown in the figure. It also coincides with the age of the Universe, strictly speaking, only if the Universe did not exist before the Big Bang. Otherwise, it simply represents the duration of the current phase of the Universe, i.e., of the epoch described by the standard model.

The origin of the time axis is obviously arbitrary, and we could have chosen to set the coordinate t = 0 at any other point in the graph. However, what is not arbitrary is the fact that in the standard model the Universe, at some point of its past evolution, necessarily reaches a singular stage where the temperature and the curvature become infinite. Beyond this point the time coordinate cannot be further extended, since the presence of a singularity makes any physical model meaningless.

The following question therefore naturally arises at this point: Does the singularity really mark the origin, the birth of the Uni- verse and the beginning of space-time itself, or is it only a short- coming of the standard model which could be removed within a more detailed, realistic and complete cosmological framework?

This is why there are dots and question marks in Fig. 2.1, at the beginning of the various scales. The presence of a singularity is a first, important reason that may suggest the possibility of modifying the standard cosmological model near the initial time. This is not the only reason, however. As will be shown in next post, there are also other kinematical issues and shortcomings. All of them can be sorted out, at least in principle, provided that the primordial evolution is modified by assuming that the initial Universe was not dominated by radiation (which would dominate only later), and that the initial expansion, contrary to the prediction of the standard model, is not decelerated. Rather, the initial phase of the Universe should be characterized by an accelerated expansion (also said to be inflationary) which only afterwards decelerates, eventually reducing to the standard one.

String cosmology suggests that this inflationary phase pre- ceding the standard one could be identified with the pre-Big-Bang phase introduced in the previous chapter. In the following chapters we shall therefore analyze the physical properties of this phase in more detail, starting from the motivations that would lead us to introduce it into the framework of a cosmological scenario inspired by string theory.

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General Relativity and Standard Cosmology (Part 1)