The Cosmological Background of Gravitational Radiation (Part 1)

The Universe Before the Big Bang

The Cosmological Background of Gravitational Radiation (P1)

There exist a number of physical effects characterizing the various models of the primordial Universe, and allowing us to discriminate between the different scenarios. All these effects are linked more or less directly to the radiation production which characterizes the transition from the phase of accelerated, inflationary evolution to the decelerated phase typical of our current Universe.

During this transition, there is in fact a copious production of photons, gravitons, dilatons, and other kinds of particles – including the most exotic ones – which are peculiar to the various cosmological models. The number of particles produced as a function of their energy is called the spectral distribution or spectrum, and the fundamental feature of these spectra is the fact that they represent a sort of snapshot of the primordial Universe, taken at different angles (i.e., corresponding to different kinds of energy) and at different times. By combining the various snapshots (i.e., analyzing the information encoded into the various spectra) it is then possible to reconstruct, step by step, the past history of our Uni- verse (unless the snapshots are of too low quality, i.e., the spectra are so weak that they escape detection).

In this chapter we focus attention on the production of gravitons, i.e., those particles carrying the gravitational force and representing the quanta of gravitational radiation (just as photons represent the quanta of electromagnetic radiation). Let us start by summarizing the basic properties of this type of radiation.

The possible existence of gravitational waves (which are ab- sent in the context of Newton’s theory) is likely to be one of the most interesting consequences of any relativistic theory of gravitation (hence, in particular, of general relativity). The production and propagation of gravitational waves is a phenomenon conceptually very similar to the one occurring in the electromagnetic context where, according to Maxwell’s equations, the oscillations of electric and magnetic fields propagate from one point to another at the speed of light. Similarly, according to Einstein’s equations, oscillations in the geometry can propagate from one point to another at a speed which (in vacuum) coincides with the speed of light.

Gravitational waves thus transmit information about how the gravitational field (i.e., the curvature of the space-time geometry) varies with time. Since the gravitational field is generated by masses and by their corresponding energies and momenta, it is the change in the status of motion of the gravitational sources – i.e., their acceleration – which generates perturbations of the local geometry, eventually propagating as a wave, and being transmitted to the whole surrounding space-time.

We can say, therefore, that the gravitational waves are produced by accelerated motion of masses, just as electromagnetic waves are produced by the accelerated motion of electrical charges.

Furthermore, as in the electromagnetic case, there is no hypothetical medium (similar to the aether of pre-relativistic physics) which starts vibrating when a gravitational wave passes by. Gravitational waves can only be detected through the motion they induce in an appropriate system of test masses (just as electromagnetic waves are detected by the oscillations they induce in an ensemble of charges).

However, the analogy between gravitational and electromagnetic waves – apparently quite close – terminates here. Beyond the formal similarities mentioned above there are indeed various crucial differences, which will be stressed below.

A first important difference concerns the kind of acceleration required of a massive body, or a system of massive bodies, in order for them to emit gravitational radation. In contrast to the electromagnetic case, the distribution of masses and accelerations has to be sufficiently asymmetric. More precisely, the source of gravitational waves must have a non-zero quadrupole moment, which varies at a sufficiently fast rate (in particular, its third time derivative must be different from zero).

Under such conditions, for instance, a spherically symmetric cloud of gas, radially collapsing under the influence of the mutual attraction of the various molecules, does not emit any outward gravitational radiation. In fact, in spite of the fact that the single molecules are radially accelerated, the total quadrupole moment of the cloud turns out to be zero because of the spherical symmetry of the system.

Another crucial difference concerns the fact that it is impossible to block the passage of a gravitational wave (at least, using macroscopic shields made of ordinary materials). The reason is that the particles composing the shield start to vibrate under the influence of the impinging wave, in such a way that they exactly re-emit the wave absorbed by the shield. Hence, the gravitational waves keep propagating both within the shield and beyond it.

In the case of electromagnetic waves the situation is different, because of the existence in nature of charges of opposite sign. Thanks to its content of positive and negative charges it is possible for an appropriate shield to reflect an incident electromagnetic wave, by rendering null the oscillatory part of the field in a given region of space. However, gravitational masses of opposite sign seem to be absent from nature (at least, there has so far been no observational evidence for their existence). As a consequence, gravitational radiation cannot be shielded or reflected as simply as electromagnetic radiation.

A further difference concerns the tensorial character of gravitational waves, in contrast to electromagnetic waves, which are vectorial. This means that the intrinsic angular momentum carried by a gravitational wave is twice that carried by an electromagnetic wave of the same intensity.

In fact gravitons, i.e., the elementary particles associated with the quantum description of gravitational waves, have zero mass, like photons (indeed, they propagate in vacuum with the speed of light), but their intrinsic angular momentum is twice that of pho- tons. This property, which emerges at the level of the quantum theory of gravitation, might seem to be irrelevant at the classical level. However, it is a crucial property of the gravitational inter- action, since it is precisely through its tensorial character that the gravitational force between two masses of the same sign is attractive, rather than being repulsive as happens for the electromagnetic force between charges of the same sign.

We should add, as a final difference, that the intensity of the gravitational field is much weaker than the corresponding intensity associated with the electromagnetic field. Let us consider, for instance, the ratio between the static gravitational force and the electric Coulomb force mutually exerted between two protons, located at an arbitrary distance. This ratio is constant and is deter- mined by the ratio between the square of the proton mass in Planck units – i.e., about (10^−19)2 – and the so-called electromagnetic fine structure constant α, whose value is about 1/137. The result is a tiny number of order 10−36. The fact that the gravitational force is so weak, together with the quadrupole-like nature of the corresponding radiation, imply that the emission of gravitational waves is a negligible process if compared with analogous processes associated with electromagnetic or nuclear forces. This weakness also explains why gravitational waves have not been directly observed so far in any laboratory.

In order to understand just how weak the power (i.e., the energy per unit time) emitted in the form of gravitational waves actually is, we may consider an oscillating mass (e.g., a pendulum, or a massive object attached to a spring, displaced from its equilibrium position). Its acceleration varies harmonically with time, so that its quadrupole moment and the time derivatives of the quadrupole moment are different from zero. Hence, the oscillating mass continuously emits gravitational waves. The emitted power (according to the theory of general relativity) is proportional to the mass squared, to the fourth power of the oscillation length, and to the sixth power of the oscillation frequency. However, the proportionality constant is the Newton constant divided by the fifth power of the speed of light! This is a truly tiny number, which gives gravitational waves of intensity too small to be detected by current instruments (at least if the mass and the frequency that we are considering are those typical of an oscillator produced artificially in a realistic laboratory).

This example suggests that in order to have more intense gravitational waves we should consider oscillations (or accelerations) of very big masses. Hence, we are naturally led to think that a class of promising sources of gravitational waves could be represented, in particular, by astrophysical processes in which the whole mass of a star is being accelerated. Indeed, a number of theoretical studies have shown that both high-velocity, close-orbit binary stars and collapsing/exploding stars (like the famous supernova observed in 1987) should radiate an intense flux of gravitational waves into the surrounding space.

Unfortunately, those sources are quite far away, and the radiation reaching us is so weak that it has not yet been directly observed by any of the currently operating gravitational antennas. However, there has been indirect evidence for the existence of those gravitational waves. In the binary system studied by Russel Hulse and Joseph Taylor, and associated with a pulsar (i.e., an extremely small and compact collapsed star), the orbits are shrinking with time.

This happens because the system, emitting gravitational waves, loses energy, so that the stars tend to fall toward one another. Now, the corresponding decrease in the orbital radius which has been observed is in full agreement with the prediction of general relativity in the case of gravitational wave emission. Thanks to this discovery, the two astrophysicists were awarded the Nobel Prize in 1993.

Since the intensity of the gravitational radiation grows with the mass, the strongest source of gravitational waves we may ever envisage is certainly present at the cosmological level, and coincides with the Universe itself. Indeed, as already pointed out at the beginning of this chapter, the processes that take place during the primordial epochs – in particular, the more or less sudden deceleration driving the initial inflationary evolution to an evolution typical of the standard model – are associated with a copious pro- duction of gravitational waves which have filled the Universe, and which should still be present as relics of the “prehistoric” cosmological epochs.

For this kind of process, however, gravitational wave emission cannot be directly associated with the motion of accelerated masses – in fact, the primordial Universe may have been empty.

Rather, it is the whole space-time itself which accelerates, and pro- duces gravitational waves according to a mechanism called para- metric amplification of vacuum fluctuations. Given the relevance of such a mechanism, which is quite effective not only for gravitational waves but also for other types of radiation (as we shall see in the next chapter), it is worth describing the basic principles in detail.

As already stressed in Chap. 2, the space-time geometry is fully determined, at the classical and macroscopic level, by the mass and energy distribution of the gravitational sources. However, at the microscopic level, there is still a tiny uncertainty in the local form of the geometry due to quantum mechanics, according to which all types of fields (including the gravitational field, and hence the geometry) can fluctuate. That is to say, they may undergo small local oscillations which, for a sufficiently short time interval, may drive them away from the value classically assigned to the field at a given point.

Such extremely fast fluctuations in the geometry are different at different space-time points, and their average value is zero. They can be treated as tiny “virtual” gravitational waves which are not freely propagating, being continuously emitted and immediately reabsorbed, locally, by space-time itself. Thus, according to quantum mechanics, space-time behaves as a sea which, even though it may be quiet and appear flat if seen from afar, shows a huge number of tiny “ripples”, continuously changing in a stochastic fashion, when viewed more closely.

The quanta of gravitational waves, on the other hand, are the gravitons. These small quantum disturbances of the geometry can thus be seen as due to “virtual” gravitons which are continuously produced and then suddenly destroyed. To avoid violations of various conservation laws (for instance the conservation of momentum, which is always valid), these gravitons must be produced and destroyed in pairs. And here we arrive at the crucial point for the mechanism of parametric amplification.

If the geometry is static and without horizons (like the geometry of the flat and empty Minkowski space, for instance), the situation of the graviton pairs is stationary: pairs are formed and destroyed in a chaotic way, but on average the net result is null, i.e., the average number of gravitons is still zero.

However, if the geometry expands rapidly enough (as happens during the inflationary phase), it is possible for two gravitons, after being produced, to be “dragged” away from one another (thanks to the background expansion) so rapidly that they are no longer able to come back together and annihilate each other. A large number of gravitons become somehow uncoupled, and the net result is a co- pious production of gravitons (i.e., of gravitational waves) directly from the space-time itself.

This mechanism of gravitational wave production, which does not require the presence of accelerated sources, is based upon fundamental principles of quantum mechanics, and it applies not only to the gravitational field but to all types of fields (for instance, pho- tons may be produced from the fluctuations of the electromagnetic field, and so on).

Furthermore, the mechanism described here is related to the effect that produces quantum radiation from a black hole,1 i.e., the radiation produced by a mass that has collapsed into a region of space so small that the gravitational force is strong enough to hold even light.

In fact, as explained by Stephen Hawking – the theoretical physicist who discovered that effect – even black hole radiation can be seen as due to the quantum fluctuations of the geometry, creating virtual pairs of particles in the region close to the horizon (i.e., close to the boundary of the region of space where light is trapped). Indeed, when one of the two particles is absorbed by the black hole, the other particle loses its “twin partner”, which it would normally rejoin and annihilate, and can therefore move away from the point of its quantum creation.

The resulting effect is a flux of radiation flowing out to infinity, which seems to emerge just from the black hole horizon. The process of particle production occurring in the context of inflationary cosmology can be described in a similar way. The main difference is that, in the cosmological case, the two virtual particles are separated not by the black hole horizon but by the Hubble horizon associated with the phase of accelerated expansion.

A key feature of the resulting radiation is its spectrum, describing the radiation intensity as a function of its energy. As already pointed out at the beginning of this chapter, this quantity provides direct information about the state of the Universe at the epoch during which the radiation was produced. The spectral distribution can also be given in terms of the frequency rather than the energy, since the energy of a particle is proportional to the frequency of the associated (quantum mechanical) wave, with a proportionality constant given by the well-known Planck constant.

There are various ways of determining the radiation intensity as a function of the frequency (i.e., the spectrum). One of them is to compute the number of particles produced within each frequency interval, and then multiply this number by the particle energy. An- other method, more intuitive and more suitable for illustration in the context of this book, is to represent the emission of gravitational radiation as a result of the amplification of the microscopic fluctuations that spontaneously emerge everywhere, as a consequence of quantum effects. The intensity of this amplification as a function of the frequency immediately provides us with the desired spectrum.

Let us focus on the production of gravitational waves, i.e., on the quantum fluctuations of the geometry. Such fluctuations can always be decomposed into waves oscillating at different frequencies. However, given their quantum origin, the fluctuations satisfy a crucial normalization condition: the initial amplitude of these waves is proportional to their frequency, and hence inversely proportional to their wavelength λ.

In an expanding Universe all frequencies decrease (see Chap. 2), and the amplitude of those oscillations therefore decreases with time, while λ increases. On the other hand, the radius of the Hubble horizon remains constant during a phase of standard inflationary evolution (it can also increase, but more slowly than λ); alternatively, it can decrease, as happens in some string cosmology models (see Chap. 5). In any case, even if the oscillations of the geometry initially have wavelengths much shorter than the Hubble radius c/H, it is inevitable that the two length scales will eventually be- come equal.

From that time it is no longer legitimate to talk about oscillations, given that λ is greater than the horizon. The oscillation of this wave turns out to be invisible to all physical effects and to all causally connected observers. Hence, the amplitude of the wave remains “frozen” (as happens to strings in the scenario described at the end of Chap. 3) at the value it had when λ = c/H. On the other hand, the amplitude of the oscillations is inversely proportional to λ, so the final amplitude is proportional to the value of H (i.e., the curvature) at the time of freezing.

The waves “de-freeze” after the end of inflation, when the standard phase begins and H starts decreasing. Then oscillations take place once again, and their amplitude starts to decrease again.

However, the freezing has prevented the amplitude from decreasing for a certain time interval, thus producing an effective amplification of the wave. The intensity of the final wave, which deter- mines the amplification in the various frequency bands (and thus the spectrum), depends upon the value of the wave amplitude at the freezing time, which in turn is determined by the value of H during the accelerated phase.

It must be noted, at this point, that waves with different frequencies will be frozen at different times. In general, the higher the initial frequency, the smaller the wavelength, hence the longer the time required for λ to increase enough to eventually satisfy the condition λ = c/H. Therefore, the final amplitude is the same for all waves only if H remains constant during the whole inflationary phase. If H increases with time, on the other hand, high-frequency waves will freeze much later, and will have a greater amplification than low-frequency waves; the opposite occurs if H decreases with time.

This conclusion, obtained for the amplification of the geo- metric fluctuations (associated with the process of graviton pro- duction), tends to be valid also for other types of fields. In general, we may summarize the previous discussion by saying that the frequency behavior of the spectrum tends to follow the time behavior of the curvature scale during the inflationary phase.

Therefore, within conventional inflationary models (where the curvature is either constant or decreasing with time, see Chap. 5) the particles produced in the process of amplification of the quantum fluctuations will have a spectrum which is either flat or decreasing with their frequency (or energy). In the context of string cosmology models, where the inflationary phase preceding the big bang is characterized by an increasing curvature scale, the resulting particles will have a spectrum which tends to increase with frequency, as pointed out by Massimo Giovannini and the present author.

It should be stressed that this crucial difference between string cosmology and the standard inflationary scenario brings advantages as well as disadvantages. The advantages (of phenomenological type) are quite evident: the more effective production at high energies yields to the formation of a cosmic background of relic particles which is more intense in the high-frequency regime, i.e., just where direct observation is easier, in principle. Moreover, thanks to this more copious production, all the matter and radiation currently present in our Universe may be the direct outcome of the transition between the pre-Big-Bang and post-Big-Bang epochs, i.e., the direct result of the decay of the initial state (the string perturbative vacuum), as anticipated in the previous chapter.

Let us consider, in particular, the radiation which determines the geometry of the Universe just at the beginning of the standard cosmological evolution (i.e., soon after the big bang). In the standard cosmological model this radiation is introduced ad hoc; its presence is indeed one of the underlying hypotheses of that model.

Within the conventional inflationary scenario the radiation is produced at the end of inflation as a consequence of quite complicated processes (phase transitions, inflaton decay, resonant oscillations, and so on), converting the potential energy of the dominant inflaton field into radiation. Within string cosmology, on the other hand, it is the energy of the initial perturbative vacuum that could transform itself into radiation; in that case, as discussed in the previous chapter, all the entropy we are currently observing could be obtained from quantum fluctuations of the initial state, amplified by the accelerated evolution of the space-time geometry.

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The Cosmological Background of Gravitational Radiation (Part 2)

The Cosmological Background of Gravitational Radiation (Part 3)