The Cosmological Background of Gravitational Radiation (Part 3)

The Universe Before the Big Bang

The Cosmological Background of Gravitational Radiation (P3)

Amongst all those alternative backgrounds, a very intense one could be associated with the gravitational radiation produced by what are known as topological defects, i.e., geometric configurations characterized by particular symmetry properties: spherical symmetry for monopoles, cylindrical symmetry for cosmic strings, planar symmetry for membranes, and so on. Such objects could have formed during the phase transition characterizing the breaking of the grand-unification symmetry, i.e., the violent process that made the transition from a single type of force to the various components corresponding to all the fundamental forces that we now observe in nature. These objects may have survived this transition, and their vibrations could have produced a cosmic background of gravitational waves. In the cosmic string case – studied in particular by Alexander Vilenkin, Bruce Allen, Richard Battye, Robert Caldwell, and Paul Shellard – the resulting background is characterized by a spectrum which is flat at high frequencies, but much more intense than the de Sitter spectrum, as shown in the figure.

We should mention that if the cosmic string network (generating the graviton background) has been produced in models of brane– antibrane inflation (see Chap. 10), then the peak intensity of the spectrum could be comparable to that predicted by pre-Big-Bang models. This possibility has been discussed recently by Edmund Copeland, Robert Myers, and Joseph Polchinski.

A globally flat graviton spectrum may also be produced in the context of the pre-Big-Bang scenario if the metric fluctuations, besides their primordial, direct amplification due to inflation, are indirectly amplified even during the post-Big-Bang era thanks to the presence of a cosmic background of particles called axions. These particles are also typical of string theory, and may be important for explaining the CMB anisotropy in a string cosmology context, as discussed in the next chapter. The spectrum produced by this “secondary” amplification has been computed by Filippo Vernizzi, Alessandro Melchiorri, and Ruth Durrer. It is flat at high frequencies, but less intense than the one produced by topological defects (see the curve labeled “axion seeds” in the figure).

Another possible background could be produced during the phase transition, typical of the so-called extended inflation models, signaling the end of the phase of accelerated evolution. Within these models, the end of inflation is characterized by the formation of “bubbles” in the space-time geometry (similar to the bubbles produced by shaking a bottle that contains a sparkling drink).

These bubbles can collide and emit gravitational waves. This possible source of cosmic gravitational radiation was suggested by Michael Turner and Frank Wilczeck at the beginning of the 1990s.

In this case, the intensity of the resulting background strictly de- pends upon the production temperature. In particular, the spectrum shown in the figure refers to a final temperature of about 10^9 GeV, i.e., more than a hundred billion billion degrees kelvin.

A graviton background could also emerge from the oscillation of the so-called inflaton, the field that ignites and maintains inflation within conventional models. At the end of inflation this fields enters an oscillating phase, and if the oscillations become resonant it is possible to produce a huge amount of radiation of any kind, including a gravitational component, as discussed by Bruce Bassett, Sergei Khlebnikov, and Igor Tkachev. This type of process is also called pre-heating of the Universe, referring to the fact the resulting radiation is not yet in thermal equilibrium, and that only at later stages will it be possible to define a cosmic temperature.

A further possibility is that, even within the standard inflationary scenario, metric fluctuations are amplified with a spectrum that is increasing at high frequencies (but not at low frequencies). This may happen for a class of models where the scalar field does not “freeze”, i.e., does not lose all its dynamical proper- ties at the end of inflation, but rather remains active, and can even significantly influence the expansion of the Universe by playing the role of the quintessential field (or dark energy) which seems to dominate current large-scale dynamics (see Chap. 9). We have shown in the figure (with the curve labeled “quintessence”) the possible spectrum obtained in the context of four-dimensional models studied by James Peebles, Alexander Vilenkin, and Massimo Giovannini, and (with the curve labeled “branes”) the possible spectrum obtained in the context of higher-dimensional models of braneworld inflation (see Chap. 10), studied by Varum Sahni, Mohammad Sami, and Tarun Souradeep.

Among the growing spectra shown in Fig. 6.2, we should also include the so called black body (or thermal) spectrum, with an effective temperature of about one degree kelvin, which could currently characterize a graviton background, possibly produced during the quantum gravity regime, when geometry and radiation were in thermal equilibrium. Within standard models, however, this background should have been significantly diluted by the sub- sequent inflationary expansion. Its current temperature should be much smaller than one degree kelvin, and its intensity should be reduced in consequence. Within the pre-Big-Bang scenario, how- ever, a graviton black body spectrum like the one illustrated in the figure is not forbidden. On the other hand, it would not correspond to an epoch of thermal equilibrium, but to such a fast transition between the pre-Big-Bang and the post-Big-Bang phases that the curvature, once it reached the maximum at the string scale, would immediately start decreasing.

As is clearly shown in Fig. 6.2, all those possible additional backgrounds are generally stronger than the one obtained by amplifying the vacuum fluctuations in the context of the standard inflationary scenario. However, they are not stronger than the graviton background obtained in the context of the pre-Big-Bang scenario and all models characterized by equivalent kinematics.

In any case, given the plethora of possible spectra present in the frequency band shown in Fig. 6.2, one question comes to mind: Are any of these primordial backgrounds currently observable?

The straight answer is no. Current detector sensitivity is too low for this purpose. However, the current experimental limit (see below) is rather close to the theoretical value ΩG = 10^−6 which marks the boundary of the allowed region, i.e., the maximal expected background intensity. In addition, there are promising prospects for the not too distant future. In order to dis- cuss this possibility, it is probably worth inserting a premise, albeit somewhat qualitative, providing a short description of the gravitational wave detectors and their fundamental operational principles.

The simplest detector we can think of consists of two masses mutually linked by a spring (at least two masses are needed, in or- der to make manifest the motion of one body relative to another).

When a gravitational wave passes by, the two masses start to vibrate incoherently, i.e., their displacements from the equilibrium position are not simultaneously the same. In fact, due to its quadrupole nature, the wave induces tidal forces in the two-mass system, so that the masses begin to move rhythmically to and fro, oscillating at the frequency of the incident wave. This effect induces oscillating tensions in the connecting spring. If these tensions can be amplified enough to be detected, we may be able to pinpoint the passage of the wave and measure its energy.

In practice, realistic gravitational detectors are not formed by two point-like masses, but by extended macroscopic bodies. The passage of the wave warps the space-time surrounding the body which works as a detector, so that the various constituent particles tend to follow the locally produced curvature. However, since different particles are located at different positions, each particle tends to follow different space-time trajectories. As a result, relative accelerations (also called geodesic deviations) are induced between the various points of the body, producing stresses which make the detector vibrate at the frequency of the incident wave.

Such a vibration tends to be damped by the friction present inside the body. However, a good detector is characterized by a characteristic frequency – its resonant frequency – at which the response to the incident wave is hugely amplified, and damping is ineffective.

With current technology the detectors (also called gravitational antennas), are mainly of two types: resonant bars and interferometers. Resonant bars are cylindrical metal objects (made, e.g., from aluminum), responding to passing gravitational waves with a vibration characterized by a typical resonant frequency of the order of one kilohertz (i.e., 10^3 Hz). The mechanical oscillations of the bar, induced by the gravitational wave, are transformed into electronic signals which are then efficiently amplified.

In order to observe the tiny vibrations of gravitational origin, it is mandatory to remove any other possible source of vibration, and in particular intrinsic, thermal oscillations. To this end, the bar is enclosed in an airtight container where a very high vacuum is created, and where the bar is cooled down to temperatures even smaller than one degree kelvin (this is the reason why such detectors are also called cryogenic detectors). We may well say that these detectors represent the coldest spot in the Universe, given that even deep intergalactic space turns out to be warmer! (As al- ready observed, the black body radiation filling the whole Universe has a temperature of 2.7 degrees kelvin.) By making use of all possible expedients, current technology would be able to detect oscillations with effective amplitude smaller than 10^−16 cm, a length scale a thousand times smaller than the radius of an atomic nucleus. Nevertheless, no gravitational signal has yet been observed with absolute certainty by a resonant bar detector.

The first bar detector was studied and built by Joseph Weber at the University of Maryland in the 1960s. Today there are various bars operating at different locations around the globe, and the most powerful ones are in Italy: NAUTILUS, at the INFN (National Institute of Nuclear Physics) Frascati labs, and AURIGA at INFN Legnaro labs. These two detectors are the result of the evolution and refinement of a previous detector: EXPLORER, built and used by research groups of two universities in Rome (Roma 1 “La Sapienza” and Roma 2 “Tor Vergata”), but located at CERN labs in Geneva. EXPLORER is older and functions less well than the new versions, but it is still working. Other resonant antennas are ALLEGRO, at Louisiana State University, and NIOBE, at the University of Western Australia.

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FIGURE 6.3 The AURIGA resonant bar, located at the Legnaro INFN labs, is enclosed within a large multi-layer container. In this photo the container is open, to give a full view of the cylinder located at the center.
The role of the container is to provide the bar with thermal and seismic isolation. (Picture courtesy of the AURIGA collaboration)

To get an idea of the main features of these detectors let us recall that they are cylindrical aluminum objects, with a typical weight of 2300 kg, and typically three meters long. EXPLORER is cooled to a temperature of two degrees kelvin using liquid supefluid helium, while NAUTILUS and AURIGA may reach 0.1 and 0.2 degrees kelvin, respectively. Figure 6.3 shows a picture of the resonant bar AURIGA and associated experimental apparatus.

The natural future evolution of bar detectors is represented by the spherical (or polyhedral) detectors, or resonant spheres (which are at present mainly in the design phase, however). The research activity for this type of detector began with the TIGA project at Lousiana University (the acronym stands for Truncated Icosahedron Gravitational Antenna), and continued with MiniGRAIL at Leiden University. Similarly to the bars, these spheres are made of metal and oscillate when a wave passes by. They can be filled or hollow, and in principle have various advantages over bars.

The first advantage is that one can determine the direction from which the wave originates, without comparison with another detector. Another advantage relies on the possibility of discriminating tensorial waves (associated with the propagation of gravitons) from scalar waves (associated with dilatons). This last property turns out to be quite interesting within string cosmology, which may also predict (as described in the following chapter) the possible formation of a cosmic background of dilatons, i.e., of scalar waves of gravitational intensity. The use of spherical antennas could there- fore be appropriate, in particular, for hunting a possible dilatonic component of the background radiation.

Finally, hollow resonant detectors are expected to reach a sensitivity about two orders of magnitude better than can presently be reached by solid bar detectors. Particularly promising is the so-called DUAL detector, an Italian INFN project for a wide-band gravitational antenna consisting of a massive solid cylinder suspended inside a larger hollow one.
The other class of currently operating gravitational antennas are the interferometric detectors, where the masses that vibrate due to the passage of the wave are two large mirrors located at the endpoints of the arms of the interferometer. These mirrors reflect the light of a laser beam and the reflected light, properly combined, forms interference patterns that change when the passage of the wave induces oscillations of the mirrors.

The interferometer arms (along which the laser beam is traveling) are oriented at approximately 90 degrees to one another and are made of long metal pipes with a diameter of about one meter. Inside these pipes there is a vacuum, created to get rid of air molecules that would otherwise disturb the beam. However, unlike the bar detectors, the antennas are not cooled. Another important difference is that their maximum sensitivity is attained in a lower frequency band than the one for the bars (i.e., 10–100 Hz rather than one kilohertz).

There are currently three large interferometers already in operation. One of them is VIRGO, located in Italy (at Cascina, near Pisa), while the other two are part of the American project LIGO, located in Washington state (in the north-west of the United States) and in Lousiana (essentially at the opposite side, i.e., south-east).

They are separated by 3030 kilometers. There are also smaller interferometric detectors (with smaller sensitivity): GEO600, located in Germany near Hannover, and TAMA300, located in Japan (the numbers 600 and 300 refer to the length of their arms, measured in meters). Actually, the sensitivity of these instruments increases with the interferometer arm-length. The LIGO arms are both 4 km long, while the VIRGO arms are 3 km long. Figure 6.4 shows an aerial view of the VIRGO interferometer.

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FIGURE 6.4 The VIRGO interferometer located at Cascina, in the countryside near Pisa. The picture – taken from a plane – shows an aerial view of the current interferometric configuration. The two arms (3 km long, oriented at 90 degrees to one another) contain the vacuum pipes within which the laser beam is transmitted. (Picture courtesy of the VIRGO collaboration)

The main limiting factor for increasing the sensitivity of these antennas beyond a certain limit, at low frequencies, is the presence of seismic vibrations on our planet, representing a background noise which cannot be completely erased. To get round this problem, the joint project LISA between the European (ESA) and American (NASA) space agencies is currently under way. The idea of the project is to send an interferometer into space, following an orbit around the Sun.

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FIGURE 6.5 The LISA interferometer comprises three unmanned space- craft, separated by a distance of five million kilometers, and orbiting around the Sun. Each of the spacecraft emits and receives a laser beam from the other two. (Picture courtesy of the LISA collaboration)

The LISA interferometer comprises three unmanned space- craft located at the vertices of an equilateral triangle with side five million kilometers (see Fig. 6.5). Each spacecraft sends a laser beam to the others and receives one from the others. The fact that the arm length is so large, and that there are no seismic effects (there are no earthquakes in space!) should make it possible to reach extremely high sensitivities at low frequencies (i.e., around one thousandth of a hertz). Other space interferometer projects are DECIGO, proposed by a Japanese collaboration and operating in the frequency band from 0.1 to one hertz, and BBO, a constellation of four space interferometers (operating in the same frequency range as DECIGO), which is currently being investigated by NASA.

Finally, it should be noted that the overall frequency band covered by interferometers and resonant mass detectors (both in operation and projected) ranges approximately from the millihertz to the kilohertz. At higher frequencies this type of detector is useless, and an alternative possibility is to build gravitational antennas operating over the kilohertz to megahertz range using resonant electromagnetic cavities, as suggested in the 1970s by Francesco Pegoraro, Emilio Picasso, and Luigi Radicati at the University of Pisa. Work is in progress on the possibility of using two coupled microwave cavities, but the sensitivity presently attain- able seems to be low compared with what is required to detect a cosmic graviton background. Other possibilities (currently under study) for the realization of very-high-frequency gravitational antennas are based on the use of electromagnetic waveguides, as recently suggested by Mike Cruise, or exploit the so-called Gertsenshtein effect – photon–graviton conversion in an external magnetic field – as proposed by Robert Baker, Zhengyun Fang, Fangyu Li, and Gary Stepenson, to detect waves with frequencies in the gigahertz range.

After this short review of the various types of gravitational detector currently in operation or still in the study phase, let us discuss the possibility of detecting the cosmological spectra shown in Fig. 6.2. To this end, it must be noted that a stochastically distributed ensemble of gravitational waves (as expected from cosmologically generated backgrounds) induces a stochastic signal in the detectors which is indistinguishable from other background noise that may already be present in the detector itself. Thus, for a reliable observation free from possible ambiguities of signal interpretation, at least two detectors are needed, in order to make a comparison and a cross-correlation of the registered data. A single detector can at most determine an upper limit for the energy density associated with a stochastic graviton background.

At present, the best level of sensitivity to a stochastic back- ground of cosmic gravitons has been reached through the cross- correlated analysis of the data of the two LIGO interferometers.3 It has been found that, for gravitational waves with frequencies in the range 51–150 Hz, there is no detectable signal associated with an energy density larger than about six hundred thousandths of the critical value. The cosmological graviton background must therefore satisfy the condition ΩG < 6×10^−5 at frequencies around one hundred hertz. This is not a surprising result since, looking at Fig. 6.2, we immediately realize that the value ΩG = 6 × 10^−5 is above the boundary of the allowed region. To have any chance of detecting a signal, the two cross-correlated detectors should be sensitive to at least the limiting value ΩG = 10^−6, which marks the boundary of the allowed region. This limiting sensitivity is expected to be reached by LIGO next year, clearly going beyond the nucleosynthesis bound at ΩG = 10^−5.

In order to penetrate well inside the allowed region, we must nevertheless wait for the realization of projects still under study, e.g., spherical resonant antennas, or hollow “dual” detectors, whose cross-correlation should allow us to push sensitivities to an energy density of about one ten millionth of the critical value.

Another possibility is provided by the second, advanced generation of interferometers (e.g., Advanced LIGO), whose cross-correlation should eventually reach sensitivity to an energy density of about one ten billionth of the critical value. A third possibility is pro- vided by space interferometers like LISA, which should achieve sensitivity to an energy density of about of one hundred billionth of the critical value, in a frequency band ranging between 3 and 10 millihertz.

For all these cases, the expected sensitivity is well within the allowed region. hence, future detectors could in principle detect a background of gravitons produced during a cosmological epoch preceding the Big Bang. A possible signal, extrapolated at the maximum (endpoint) frequency of the spectrum, could give us information about the peak value, thus providing the first experimental indication concerning the value of the fundamental string mass (about which there are, so far, only theoretical conjectures). But even the absence of signals in a given frequency band would pro- vide information, since it would tell us either that the spectrum peaks at higher frequencies or that it is completely absent, thus imposing significant constraints on the parameter space of string cosmology models (and other models) of inflation.

The important conclusion of this chapter is that the back- ground of relic gravitons, having such different intensities and properties for different models, provides a unique observational tool to test different cosmological scenarios. In particular, the background produced in the context of pre-Big-Bang models tends to be very intense in the high-frequency range, while it is so diminished in the low-frequency range that its contribution to the large-scale anisotropy of the CMB radiation (see next chapter) turns out to be completely negligible. On the other hand, the background produced within the standard inflationary scenario may contribute significantly to the CMB anisotropy, while at high frequencies – barring some peculiar mechanism of secondary graviton production, like those shown in Fig. 6.2 – is so low as to be barely detectable, even accepting the most optimistic predictions for the sensitivities of future detectors. Hence, a combined non-observation of gravitational wave contributions to the CMB anisotropy, together with a direct detection of relic gravitons at high frequency (in the allowed region of Fig. 6.2), could be interpreted as a strong signal in favor of the self-dual pre-Big-Bang scenario.

Needless to say, the direct observation of a cosmic background of gravitational waves would represent an event of importance comparable with the discovery of the cosmic background of electromagnetic radiation, first observed by Arno Penzias and Robert Wilson, who were awarded the Nobel Prize for this achievement.

We may even argue that the discovery of the gravitational radiation would be more important. Indeed, while the photons represents the relics associated with the Big Bang explosion, these gravitons would represent much older relics which – according to string models – could even bring direct information about an epoch prior to the Big Bang.

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The Cosmological Background of Gravitational Radiation (Part 1)

The Cosmological Background of Gravitational Radiation (Part 2)