The Cosmological Background of Gravitational Radiation (Part 2)

The Universe Before the Big Bang

The Cosmological Background of Gravitational Radiation (P2)

A too copious production of particles may present some disadvantages, however. The resulting radiation may not satisfy present experimental constraints, and may even be inconsistent with the cosmological model itself. In fact, the energy of the resulting particles could be large enough to change the cosmological evolution, forbidding the onset of the standard regime.

Fortunately, the second issue concerning the consistency of the model may be intrinsically avoided within a cosmological framework based upon string theory. Indeed, the minimum fundamental length scale Ls defines a mass Ms which, using the system of natural units already introduced in Chap. 5, can be simply writ- ten as the reciprocal of the string length, i.e., Ms = 1/Ls. Since the reciprocal of a minimum represents a maximum, this mass (multiplied by the square of the speed of light) determines a maximum energy corresponding to about 10^18 GeV, i.e., an energy one billion billion times bigger than the energy associated with the proton rest mass.

This energy is about ten times smaller than the energy corresponding to the Planck mass MP, whose value (of order 10^19 GeV) somehow represents the maximum energy which can be stored within the classical space-time geometry according to quantum mechanics. Beyond this energy scale the notion of space-time itself becomes uncertain. Within string cosmology, the maximum energy density of the produced particles is naturally determined by Ms. Hence, it cannot exceed the geometrical energy density – deter- mined by MP – and cannot have catastrophic consequences for the classical evolution of the space-time geometry on a cosmological scale.

Concerning the first issue, namely the compatibility of the resulting particles with the existing experimental constraints, the situation is more complicated, and it is worth addressing it in a detailed way. We shall focus the present discussion on a possible cosmic background of relic gravitational waves (or relic gravitons), which is the main subject of this chapter. No background of this type has yet been directly detected, but there are observations which already set constraints on its possible intensity.

For a clear discussion of such observations it is convenient to introduce the quantity ΩG, which represents the energy density of the cosmic graviton background measured in units of critical energy density. The critical density is the total energy density associated with a model of the Universe whose three-dimensional spatial sections have a vanishing curvature. Since the currently observed spatial curvature is quite small, the critical energy density represents a realistic estimate of the full energy density of the current Universe.

There are three main kinds of observations which set direct or indirect constraints on the possible value of ΩG. In all cases, the constraints emerge from the fact that a background of cosmic gravitons can be represented classically as an isotropic sea of gravitational waves distributed over all possible wavelengths, traveling and intersecting each other in all directions in a chaotic/stochastic fashion. These gravitational waves are perturbations of the space- time geometry, propagating at the speed of light, and universally coupled (even if very weakly) to all kinds of energy. Their presence thus produces a sort of cosmic disturbance, or background noise, on the homogeneous and isotropic large-scale geometry of the present Universe.

The most famous constraint on ΩG probably comes from the high degree of isotropy of the cosmic background of electromagnetic radiation – the CMB “black body” radiation – which currently fills the Universe. The existence of a gravitational-wave background, distorting the geometry, would also induce a distortion of the isotropy of the CMB radiation. The anisotropy measured in 1992 by the COBE (COsmic Background Explorer) satellite implies that, if there were gravitational waves in the cosmic background with a wavelength of the same order as the current value of the Hubble radius (i.e., of the radius of the observable Universe today), then their corresponding energy density could not exceed one ten billionth of the critical value. In other words, for those gravitational waves, ΩG < 10^−10. (We recall that the current value of the Hubble radius is about 10^28 cm, and corresponds to the distance that could be covered in about 10 billion years, traveling at the speed of light.) Another limit comes from the observation of the regularity in the beat of pulsating stars, the so-called pulsars. A background of gravitational waves, distorting the space-time geometry, would also produce a distortion of the observed pulsar timing. The absence of such an effect – confirmed in particular by Victoria Kaspi, Joseph Taylor, and M.F. Ryba in 1994 – implies that ΩG < 10^−8, i.e., that the energy density must be less than a hundred millionth of the critical value, for gravitational waves whose wavelength is about 108 cm (i.e., about one light-year).

Finally, an indirect (but important) limit comes from nucleosynthesis, i.e., from those primordial processes leading to the formation of atomic nuclei and hence to the synthesis of matter in its current form. These processes, which took place more than ten billion years ago, could not have happened undisturbed in the presence of too many gravitons in the background. Detailed computations give us a limit on the current energy density of such gravitons: it must be less than one hundred thousandth of the critical value, i.e., ΩG < 10^−5, for any wavelength. This limit, as well as the two previous limits, can be applied without distinction to any graviton background of primordial cosmological origin, quite irrespectively of the particular production mechanism.

Let us now recall that the gravitons obtained by amplifying the vacuum fluctuations within the conventional inflationary scenario would represent gravitational radiation coming from a constant or decreasing curvature phase, and would therefore be characterized by an energy spectrum ΩG which is constant or decreasing with frequency, respectively. Within the pre-Big-Bang scenario of string cosmology, on the other hand, the same gravitons would represent gravitational radiation coming from a growing-curvature phase, and would have a spectral energy density that increases with frequency.

Given the above-mentioned observational constraints, it follows that the graviton spectrum of highest allowed amplitude for the various inflationary models can be represented as in Fig. 6.1, where we have shown ΩG as a function of the frequency ω (expressed in hertz).

FIGURE 6.1 Possible spectra for a cosmic background of gravitons produced through the amplification of the vacuum fluctuations in the con- text of different inflationary scenarios. The plots show the logarithm of the energy density as a function of the logarithm of the frequency. The figure also shows the observational upper limits due to the CMB anisotropy, the pulsars, and nucleosynthesis. The spectrum labeled “de Sitter”, and associated with a phase of constant-curvature inflation, marks the boundary between the decreasing spectra (standard inflation, bottom left part of the figure) and the growing spectra (pre-Big-Bang inflation, top right part of the figure). All spectra have been plotted at the maximum intensity compatible both with present experimental constraints and with theoretical predictions for string theoretical parameters

The standard – flat or decreasing – spectra have been well known since the 1970s, thanks to the pioneering computations performed by the theoretical astrophysicists Leonid Grishchuk and Alexei Starobinski. Those spectra are shown in the bottom left part of Fig. 6.1, and it is evident that their maximum energy density is mainly constrained by the measured anisotropy of the CMB radiation. We note that for all these spectra, even in the limiting case of de Sitter inflation associated with a phase of constant curvature, there is a small jump at low frequencies, approximately located around the frequency scale 10^−16 Hz, which is the proper frequency of a wave that begins to oscillate again just at the epoch of matter– radiation equality. This small jump is due to a further production of gravitons which takes place after inflation, and which is associated with the transition between the radiation-dominated phase and the matter-dominated phase (see, e.g., Chap. 2). As for the present discussion, that jump is not important. What is crucial is the fact that the de Sitter spectrum represents the maximum energy density distribution for a background of cosmological gravitons produced from the amplification of the vacuum fluctuations within conventional inflationary models.

The spectra obtained in the context of self-dual string cosmology models, and associated with a phase of pre-Big-Bang inflation, are shown in the top right part of the figure. Since they are increasing with frequency, they are not influenced by the low-frequency CMB constraint. In the low-frequency regime these spectra increase very fast, as the third power of the frequency, while at higher frequencies this increase flattens out somewhat, eventually reaching a peak value which essentially depends upon the ratio between the maximum string energy scale Ms and the Planck energy scale MP (as shown in studies of Ram Brustein, Massimo Giovannini, Gabriele Veneziano, and the present author). Beyond that maxi- mum value, the energy density of the background rapidly tends to zero (as shown in the figure).

The peak intensity of ΩG for the pre-Big-Bang spectra is obtained from the fact that, at the end of inflation, the ratio between the graviton energy density ΩG and the total radiation energy density Ωrad is determined by the maximum curvature scale allowed by string theory, H = Ms, measured in Planck units and squared:

(Ms/MP)2. The value of the string mass (or its reciprocal, the string length Ls = 1/Ms) is still uncertain. (So far, there have been no direct measurements of these quantities.) However, we do expect the value of Ms to be not greater than one tenth of the Planck mass, if string theory has to represent a unified model of all fundamental interactions (see the discussion in Chap. 4).

On the other hand, the total radiation energy density is currently about one ten thousandth of the critical value: Ωrad = 10^−4.

Given that the ratio between the graviton energy density and the total radiation energy density has not changed with time, it follows that the maximum energy density currently allowed for a graviton background produced by a phase of pre-Big-Bang inflation is about one billionth of the critical value: ΩG = 10^−6 (see Fig. 6.1). We observe that, despite the uncertainty in the value of the string-to-Planck mass ratio, the peak value of the pre-Big-Bang spectra is automatically compatible with the upper limit set by nucleosynthesis.

Once the peak value has been fixed, the spectrum of pre-Big- Bang gravitons shown in Fig. 6.1 is still characterized by two arbitrary parameters: the slope and the width of the high-frequency portion of the spectrum. In fact, the figure shows three possible examples of spectra, characterized by different slopes and widths.

Such unknown parameters encode our lack of knowledge concerning the final stage of the pre-Big-Bang phase, where the curvature stops increasing and joins the subsequent phase of standard, curvature-decreasing evolution.

Actually, during those final stages, the curvature reaches values so high (close to the maximum allowed value) that it is mandatory to include purely “string-like” and quantum gravitational effects in the model. The low-energy approximation to the equations of motion is no longer legitimate: full, exact string- theoretical equations have to be implemented. This brings in technical difficulties that have so far prevented us from making detailed predictions, given the present status of knowledge about string theory. However, as this knowledge increases, it should be possible to develop a more complete model and to compute the high- frequency part of the spectrum explicitly, removing this kind of uncertainty.

It should be stressed at this point that the pre-Big-Bang spectra illustrated in Fig. 6.1 are those predicted by a particular class of “minimal models”, where the height and the frequency position of the peak are both determined by the ratio Ms/MP. Those models are characterized by three main cosmological phases. In the initial phase the curvature increases from the vacuum and the Universe “inflates” using the kinetic energy of the dilaton as a “pump” field. Then, during a second, intermediate phase – the so-called string phase – the Universe continues to expand in an accelerated fashion, while the curvature remains fixed around the maximum value allowed by string theory. Finally, the Universe enters the phase of standard evolution and decreasing curvature, where the resulting radiation fills the Universe and dominates over all other kinds of gravitational source (including the dilaton). Within this class of minimal models the peak of the spectrum is always fixed at a well-defined point in the plane of Fig. 6.1 (modulo a small uncertainty depending on our present ignorance about the exact value of Ms).

The frequency value (but not the height) of the peak could vary in the context of “non-minimal” models of pre-Big-Bang spectra where, after the string phase, the dilaton keeps growing even when the curvature is already decreasing, and where the resulting radiation becomes dominant only much later. In such models the peak of the spectrum, instead of being located at values of about one hundred billion hertz (as in Fig. 6.1), is set at smaller frequencies.

This shift toward smaller frequencies would be experimentally desirable, since detection is more favorable at smaller frequencies.

Indeed, as we shall see later, the peak sensitivity of currently operating gravitational wave detectors lies in a frequency range between ten and one thousand hertz. However, the non-minimal models appear to be less natural than the minimal ones. Barring possible “surprises”, it seems unlikely that nature would have chosen this solution.

It is worth recalling that the height of the peak may also differ from (and, in particular, be lower than) the one shown in Fig. 6.1 – even in the context of minimal pre-Big-Bang models – if the gravitons, after being produced in the transition between the inflationary and the standard phase, were diluted by a so-called re- heating phase, i.e., by a second, small Big Bang during which the Universe was heated up again by the generation of additional radiation and associated entropy. If we imagine the radiation as water, the Universe as a glass, and the gravitons as sugar, it is as if some- one were to add more water to a glass already containing sugared water, and then to mix everything together: eventually the water turns out to be less sweet, i.e., the fraction of sugar has dropped.

Similarly, the fraction of gravitons with respect to the full radiation content of the Universe would drop, and the present peak value of ΩG would be lower.

This possibility does not seem to be very natural either, in a cosmological framework based upon string theory, where all relevant physical phenomena should take place at earlier epochs, near the string scale, rather than at lower energy scales. In any case, this effect would not dramatically influence the intensity of the graviton background. For instance, even if 99% of the radiation entropy that we are currently observing were due to any of those reheating processes subsequent to the end of pre-Big-Bang inflation, it can be shown that the maximum graviton energy density would drop from one millionth to one hundred millionth of the critical density, i.e., from ΩG = 10^−6 to ΩG = 10^−8. Such a value is still well above the maximum high-frequency value allowed for a phase of constant- curvature inflation (corresponding to the curve labeled “de Sitter” in Fig. 6.1), ΩG = 10^−14, i.e., one hundred thousand billionth of the critical density.

Finally, it should be noted that a growing spectral distribution is a rather typical – almost “universal” – prediction of string cosmology models alternative to standard inflation. However, a high- level intensity of the graviton background like that obtained in the context of the (minimal or non-minimal) pre-Big-Bang scenario is not a property of all string cosmology models. An important example of a low-intensity spectrum is given by the so-called ekpyrotic models, suggesting a string cosmology scenario alternative to the standard inflationary one, but different from the self-dual pre-Big- Bang scenario. The ekpyrotic scenario, which will be illustrated in Chap. 10, is indeed characterized by a growing graviton spectrum, but the associated peak intensity can reach at most the standard inflationary value ΩG = 10^−14, at a maximum frequency of about 108 Hz.

Given the various uncertainties characterizing the current theoretical models, and the lack of precise predictions pertaining to the exact form of the graviton spectrum that we would expect from a phase preceding the Big Bang, it seems appropriate to determine the so-called allowed region in the plane of Fig. 6.1, i.e., the region that undoubtedly includes the spectrum of background gravitons.

This region corresponds to the portion of the plane “swept out” by the spectrum, varying all its parameters within the maximum allowed ranges (just as in the windscreen of a car, the region of glass swept by the wiper represents the allowed region within which the wiper itself has to be found).

FIGURE 6.2 Allowed region for the cosmic background of relic gravitons produced through the amplification of vacuum fluctuations in the context of the pre-Big-Bang scenario (upper trapezium) and the standard inflationary scenario (lower trapezium). The figure also shows (with dashed lines) the graviton spectrum of the ekpyrotic scenario, as well as other possible types of spectra produced through mechanisms other than the amplification of vacuum fluctuations. Since these spectra are localized at high frequencies, they are not constrained by the experimental bounds on the CMB anisotropy

In Fig. 6.2 we have compared the allowed region for a graviton background arising from the amplification of vacuum fluctuations within the self-dual pre-Big-Bang scenario (the region enclosed within the upper trapezium), with the allowed region of the standard inflation scenario (the region enclosed within the lower trapezium). It is easy to see that the first region is wider than the second by about eight orders of magnitude, since it allows a maximum spectral density of 10^−6, while in the de Sitter case the maximum, in the same frequency band, is only 10^−14. The figure focuses on the high frequency range, which is the phenomenologically relevant region for currently operating gravitational wave detectors; but the two allowed regions can be extended without modification down to the millihertz range, so as to include the sensitivity band of other detectors, such as the space interferometers currently under study (see below).

It is instructive to observe that the large enhancement of the string cosmology region with respect to standard inflation can also be explained by recalling that the peak intensity depends upon the maximum curvature scale associated with a given model of inflation. In fact, within pre-Big-Bang models the spectrum increases with frequency, and the peak energy is fixed by the maximum attainable curvature. The latter can be expressed in dimensionless units by taking the square of the ratio between the string mass and the Planck mass. Given that the maximum of Ms is about one tenth of MP (as previously discussed), we then obtain the maximum dimensionless ratio (Ms/MP) = 10^−2.

Within conventional inflationary models, however, the spectrum decreases with frequency, and the peak energy is constrained by the low-frequency bounds imposed by observations of the CMB anisotropy. According to these observations there is a maximum value of the anisotropy that can be induced in the CMB temperature by the relic gravitons. The ratio of the possible temperature variation over distance scales of the order of the Hubble radius and the mean temperature of the radiation has to be (approximately) at most of order 10^−5. This anisotropy ratio, on the other hand, corresponds to the amplitude of gravitational waves with wave- length of the order of the Hubble radius, which in turn corresponds to the space-time curvature H/MP (measured in Planck units) at the end of the inflationary phase. The observed CMB anisotropy thus sets a constraint on the curvature of conventional inflationary models, which cannot exceed a maximum value H such that (H/MP)2 = 10^−10.

Hence, going from pre-Big-Bang cosmology to the conventional one, we find that the maximum curvature changes from one hundredth to one ten billionth in Planck units, thus explaining the eight orders of magnitude of difference between the intensity of the spectra. It may be noted that, in conventional inflationary models, the scale associated with the maximum curvature is related to the grand unification mass scale MGUT (rather than to the string mass Ms). The energy associated with this mass defines the energy scale at which the fundamental forces active at the microscopic level (i.e., nuclear, electromagnetic, and weak) are expected to be unified into a single force.

For the sake of completeness, Fig. 6.2 also shows the graviton spectrum produced in the context of the ekpyrotic scenario, which is approximately superimposed on the last part of the standard inflationary spectrum. Finally, we have plotted (with dashed lines) the spectra associated with other gravitational backgrounds, possibly obtained even within standard inflation, but produced by different mechanisms from the inflationary amplification of vacuum fluctuations. All these spectra have a negligible low- frequency tail, and so are not constrained by CMB and pulsar observations. Let us briefly illustrate these new types of possible spectra.

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The Cosmological Background of Gravitational Radiation (Part 1)

The Cosmological Background of Gravitational Radiation (Part 3)