Relic Gravitons

The following sections are included:

• Preface
• Contents

The first few sentences of a scientific monograph customarily define its limits but such a definition deserves here more than a swift attention since the relic gravitons have not been directly detected and their spectra are still largely hypothetical. In spite of their uncertain origin, the wavelengths of cosmic gravitons cannot exceed the current value of the Hubble radius (i.e. about the inverse of the present value of the Hubble rate) because the distance between two successive crests (or two successive troughs) of the same wave must be resolved (at least in principle) otherwise any measurement of the wavelength (or of the frequency) is unfeasible. This naive remark is in fact quite wise and it implies that the observable frequencies of the relic gravitons cannot be smaller than 𝒪 (10⁻¹⁸) Hz. In terms of the prefixes of the international system of units (heavily used throughout this monograph), the smallest frequencies of the relic gravitons fall in the aHz range (1 aHz = 10⁻¹⁸ Hz) and, according to the current understanding, the low-frequency gravitons as well as the ones of much larger frequencies were produced in the early Universe (possibly during inflation) so that we strongly suspect that the spectrum of cosmic gravitons spans roughly 30 orders of magnitude from the aHz to the GHz. While at low frequencies (i.e. large wavelengths) the best electromagnetic detector of relic gravitons is the cosmic microwave background itself, at higher frequencies (i.e. smaller wavelengths) a number instruments have been proposed through the years but only the wide-band interferometers are now operating in the so called audio band (i.e. between few Hz and 10⁴ Hz). Throughout this monograph cosmogravitons is a synonym of relic gravitons and this wording also specifies the range and the limitations of our considerations: we shall not be dealing with the astrophysical gravitational waves that are, these days, one of the new avenues in astronomy but that have little to do with the early evolutionary stages of our Universe…

Even if the evolution of gravitational waves in flat space–time has been qualitatively anticipated in the previous chapter, a more accurate treatment of the problem is a prerequisite for the subsequent developments. While in the first part of the chapter the linearization of general relativity around Minkowski space–time is specifically analyzed, in the second part the conventional discussion is complemented by a non-covariant approach where the supplementary polarizations of the gravitational waves are explicitly identified…

The following sections are included:

• Gravitational and Electromagnetic Polarizations
  • Scalar spherical harmonics
  • Ladder operators
  • Vector spherical harmonics
  • Tensor spherical harmonics
• Polarizations of the Electromagnetic Waves
  • Linear and circular polarizations
  • Stokes parameters
• Polarizations of the Gravitational Waves
  • Linear polarizations
  • Circular polarizations
  • The sum over the polarizations
  • The polarization matrix
  • Stokes parameters
• Spin-Weight of a Function and Polarizations
  • Some examples
  • Spin-weighted spherical harmonics
  • Generalities on the E-mode and B-mode
  • Further considerations
• References

As established in the two previous chapters, in general relativity the gravitational waves are associated with a symmetric, solenoidal and traceless rank-two tensor in three (spatial) dimensions. When the gravity theory does not coincide with general relativity, up to four supplementary polarizations may arise in flat space–time and this happens, for instance, for the non-Einsteinian theories of gravity (e.g. scalar–vector, scalar–tensor, scalar–vector–tensor). If the Minkowski space–time is a solution of the field equations in the absence of matter sources, it is often plausible to ignore the fluctuations of the energy–momentum tensor but such a simplifying assumption is only justified far from the matter sources that are ultimately responsible for the curvature of the underlying space–time. The inhomogeneities of the matter sources in curved backgrounds entail the presence of supplementary polarizations not only in non-Einsteinian theories (as we saw in Chapter 2) but also in general relativity. The approach pioneered by Lifshitz [1], Grishchuk [2] and others considers the propagation of the tensor modes within a covariant description where the tensor indices are all kept four-dimensional but a complementary strategy, particularly useful in cosmological backgrounds, is to decompose the fluctuations of the metric in a non-covariant language by explicitly separating, form the very beginning, the spatial from the temporal components of the perturbed metric [3]. As in the case of non-Einsteinian theories of gravity in flat space–time (see Chapter 2), in four-dimensional curved backgrounds the 10 independent components of the perturbed metric can be reduced to six either by selecting a specific coordinate system or by defining an appropriate set of gauge-invariant fluctuations, as suggested in the context of the Bardeen approach [3]…

In flat space–time the evolution of the tensor modes follows either from the equations or motion perturbed to first-order or from the second-order fluctuations action. In curved backgrounds the same strategy discussed in the Minkowski case (see Chapter 2) leads to the effective action of the relic gravitons and also suggests a sound definition of their energy–momentum pseudo-tensor that is explicitly discussed in Chapter 7. The present and the following three chapters also address a series of related topics that are relevant for the forthcoming phenomenological applications…

Gravitational wave astronomy does not demand a quantum treatment since it successfully assumes a classical description of the tensor modes of the geometry and while this picture is not challenged by any direct observation, the purpose of this chapter is to motivate the quantum mechanical analysis which is mandatory in the case of the relic gravitons. The Universe evolves classically for most of its history but its inhomogeneities may have a quantum mechanical origin and this perspective is justified in all the situations where the classical inhomogeneities either vanish or, for some reason, are strongly suppressed. Most of the topics discussed hereunder involve the conditions under which the quantum treatment of the large-scale inhomogeneities is essential…

Since there exist different prescriptions for the energy–momentum (pseudo)-tensor of the relic gravitons, the purpose of this chapter is to compare them and to introduce the observables that are customarily employed for the analysis of concrete physical situations. From the practical viewpoint the problem is to find an appropriate definition of the energy density of the relic gravitons because the different prescriptions introduced through the years are not always equivalent and, in some cases, even contradictory. We could naively think that the energy–momentum tensor of the relic gravitons should always be uniquely defined but this is not the case for a number of reasons which are explored in this chapter…

In the previous chapter, we established that all the relevant observables effectively depend on the mutual relation between the frequency and the expansion rate. Therefore, even barring for other potential ambiguities, the present value of the spectral energy density is sensitive to the whole expansion history. The estimate of the spectral energy density of the relic gravitons is now conducted when the concordance paradigm is complemented, at early times, by a stage of inflationary expansion and since this potential signal encompasses different ranges of frequency the analysis is divided in three spectral regions that are discussed in the present and in the two following chapters. In the first part of this chapter, the motivations of inflation are reviewed by extending the preliminary analysis of Chapter 6 where the main focus was the distinction between classical and quantum fluctuations in cosmological background. The goal of the present and of the following two chapters is the derivation of the signal already illustrated at the bottom of Fig. 1.5 where the spectral energy density of the concordance paradigm has been compared with the black-body spectra of the photons and of the gravitons. The focus of Chapter 9 is on the wavelengths reentering the Hubble radius when the background is dominated by radiation and in this region of frequencies (between 100 aHz and 100 MHz) the spectral energy density is quasi-flat but it is also suppressed by various classes of physical effects like the free-streaming of the neutrinos or the late-time dominance of dark energy. Finally, in Chapter 10, we analyze the low-frequency part of spectrum (i.e. aHz < ν < 100 aHz) corresponding to the wavelengths that reenter either during the dust phase or when dark energy is already dominant; at the end of chapter, the partial results of each band are collected and the spectral energy density is illustrated in the whole range between few aHz and the MHz region…

The spectral energy density associated with the wavelengths crossing the effective horizon during the radiation epoch is specifically investigated in this chapter. Some of the relevant wavelengths of the spectrum are still larger than the Hubble radius after the end of inflation but they progressively reenter as the radiation plasma evolves. While, as expected, the long wavelengths have the same power spectrum they had during inflation, the short wavelengths exhibit strong Sakharov oscillations which practically disappear in the spectral energy density. As soon as the wavelengths reenter the Hubble radius their evolution is affected by various sources of damping that eventually suppress the high-frequency plateau of the spectral energy density. Indeed, for the short and intermediate frequency ranges examined here the evolution of the effective number of relativistic species and the neutrino free-streaming are the most prominent effects even if the former is less relevant than the latter. In the final part of this chapter the problem of the quantum mechanical normalization of the large-scale inhomogeneities is reexamined by specifically considering the possibility already discussed in Chapter 6 and illustrated in Fig. 6.9. The results illustrated here show, a posteriori, that different initial vacua only introduce a second-order correction on the leading-order expression of the scalar and tensor power spectra.

The wavelengths examined in the previous chapters reentered the effective horizon right after the end of inflation or during the radiation dominated stage of expansion. In this chapter (see Fig. 10.1), we analyze the longest wavelengths of the spectrum, namely the ones crossing the comoving Hubble radius during the matter-dominated stage of expansion or even when dark energy is already dominant. The physical situation discussed in this chapter is concisely summarized by Fig. 10.1 illustrating the wavelengths labeled by 2 and 3 in Fig. 8.4…

When the evolution of the comoving Hubble radius is not accurately known the spectral energy density and the other observables can only be evaluated with approximate methods and the considerations presented in this chapter are a natural extension of the results obtained in the conventional lore. The Wentzel–Kramers–Brioullin (WKB) approach is particularly handy for the analysis of the graviton spectra when the early completion of the concordance paradigm does not coincide with the conventional inflationary models. A relevant area of applications also involves the decelerated stages whose rate are either faster or slower than radiation. The results of this chapter are useful whenever an accurate knowledge of the transition regimes between the different stages of the dynamics is lacking. After some specific examples both in the unpolarized and in the polarized case, we discuss a different class of approximation methods based on a modification of the time parametrization. While the notations followed hereunder are standard and coincide with the ones previously established, in the first part of the chapter we present a general discussion that can be applied, with the necessary modifications, also to the analysis of the scalar modes. This choice is practical also in view of the analysis presented in Chapter 12 where the scalar modes of the geometry will be shown to produce an effective anisotropic stress which may eventually affect the tensor modes and produce secondary spectra of relic gravitons. In the first three sections of the chapter we present a number of general results whereas the final two sections discuss some applications with particular attention to the comparison between the exact and the WKB results. In the first part a key step involves the analysis of the turning points that may be either regular or singular. In the two complementary situations the spectral energy density must be estimated is different terms.

The primary spectra of the relic gravitons are directly produced by the variation of the background geometry. There are however various classes of secondary spectra. For instance, in the concordance paradigm the curvature inhomogeneities induce an effective anisotropic stress that indirectly affects the spectral energy density of the tensor modes. In this chapter, we discuss the simplest class of secondary spectra appearing in the standard scenario. This problem is already plagued with a number of gauge ambiguities that are explicitly addressed and its interest is mainly formal since the typical corrections to the primary spectra are typically 𝒪(10⁻⁸) and can therefore be neglected for all practical purposes…

For the present purposes, it is useful to consider the microwave background and its anisotropies as the largest electromagnetic detector of relic gravitons. The CMB is particularly sensitive in the aHz domain where the signal of the ΛCDM scenario dominates and, for this reason, it is appropriate to start the analysis of the phenomenological implications in the lowest frequency domain. After this step the intermediate and high-frequency ranges are examined, respectively, in Chapters 14 and 15. At higher frequencies the effects associated with unconventional scenarios are more important and may even represent the leading contribution in comparison with the signal of the concordance paradigm…

While the low-frequency gravitons primarily affect the temperature and the polarization anisotropies of the CMB, a number of different phenomena occur in the intermediate range that extends for about 12 orders of magnitude from the pHz range to a fraction of the Hz. In this region, the spectra are bounded by the millisecond pulsar’s pulses and are also constrained by the success big-bang nucleosynthesis (BBN). While the limits discussed here are crucial for various unconventional scenarios, they are however immaterial for the concordance paradigm where the expected signal remains always much smaller than the potential constraints. The variety of the phenomena and their phenomenological relevance explains why the first section is now devoted to a more thorough discussion of the general logic of the chapter.

Since well above the Hz the potential signals coming from the concordance paradigm remain very small and gradually disappear, the detection of relic gravitons for frequencies larger than the kHz would suggest that either the post-inflationary expansion history is not exactly the one of the concordance paradigm or that the early evolution does not coincide with the conventional inflationary hypothesis. Both conclusions are relevant for a wide range of suggestions examined in this chapter. Furthermore, there are little doubts that a careful analysis of the relic graviton spectra is the only way of pushing our potential observations in a range that is otherwise unreachable. The high-frequency domain ranges, approximately, from few Hz to 100 GHz and it encompasses not only the operating window of wide-band interferometers but also the regions potentially accessible to the electromechanical detectors. This broad domain can be partitioned into a high-frequency region (between the Hz and the MHz) eventually supplemented by an ultra-high-frequency range (between the MHz and the THz)…

The concordance paradigm suggests that the large-scale inhomogeneities should have a quantum mechanical origin and, from a purely a semiclassical viewpoint, this means that the travelling waves required by the normalization of the initial quantum state of the field turn into standing waves. In a more quantum mechanical perspective (explored in Chapter 6) the relic gravitons are produced in pairs with opposite momenta and are ultimately described by two-mode squeezed states. These multiparticle states are similar to the coherent states but exhibit a much larger degree of second-order coherence that even exceeds the one typical of chaotic mixtures. This property ultimately distinguishes the relic gravitons from other late-time sources: while astrophysical signals just lead to a collection of classical random fields, relic gravitons interfere as quantum states characterized by a large number of particles in each mode of the field. The degrees of first-order coherence (i.e. the tensor power spectra) cannot be distinguished, in practice, from tensor random fields and this aspect has been indirectly pointed out at the end of Chapter 7…

Throughout this book I discussed the relic gravitons and their potential interplay with the electromagnetic spectrum. As I already said at the beginning of this script, for me the most interesting of all sources has always been the primordial (or cosmological) stochastic background of gravitational waves since its detection promises important news about the early expansion history of Universe and these informations would be completely unaccessible otherwise. In the observed Universe we have at least two powerful electromagnetic detectors of relic gravitons namely the CMB itself and the pulsar timing arrays. These cosmological detectors complement the ground-based interferometers that operate in a much higher frequency range and while the astrophysical and astronomical interest of each frequency band cannot be disputed, the most intriguing physical aspect of the relic graviton backgrounds is their wide spectrum that covers, in principle, three decades of frequencies. It will be essential, in the years to come, to undertake a synergic approach encompassing the low-frequency region (where we might test the inflationary paradigm) and ultra-high-frequency domains (associated with various potential completions of the concordance scenario). In the future we might also investigate more directly the quantum mechanical nature of the relic gravitons and this is probably the most fundamental aspect of the problem. The primordial backgrounds of relic gravitons are also indirectly sensitive to the new physics beyond the standard lore of fundamental interactions and, on a more practical ground, the detection of diffuse backgrounds of gravitational radiation might represent a unique chance to improve the current bounds on the supplementary polarizations of gravitational waves as they arise in non-Einsteinian theories of gravity. All in all the considerations developed here seem to suggest that the relic gravitons really represent one of the possible bridges connecting the microworld of the standard model of fundamental interactions with the macroworld of gravity and cosmology…

The following sections are included:

• Appendix A The Basics of Riemannian Geometry
  • The Curves in Euclidean Space
  • The Surfaces in Euclidean Space
  • The Curvature of a Surface in Euclidean Space
  • The Transformations of the Metric
  • The Curvature Tensor
  • Covariant Derivatives
  • Geodesic Deviation and Levi-Civita Displacement
  • Lie Differentiation and Its Properties
• Appendix B The Essentials of General Relativity
  • The Metric and the Christoffel Symbols
  • The Riemann, Weyl and Ricci Tensors
  • The Dual Riemann and Weyl Tensors
  • Einstein–Hilbert Action
  • Conformal Mappings
  • Scalar–Tensor Theories
  • Palatini Gravity
• Appendix C Cosmological Evolution
  • The Friedmann–Robertson–Walker Metric
  • Remarks on Relativistic Hydrodynamics
    • Inviscid fluids
    • The role of the four-velocity
    • Covariant decompositions
    • Viscous fluids
    • Multiple fluids: Examples
  • Introduction to the Friedmann–Lemaître Equations
    • Cosmic time parametrization
    • Conformal time parametrization
  • Distances in Cosmology
    • The proper coordinate distance
    • The distance measure
    • The angular diameter distance
    • The luminosity distance
    • The particle and the event horizons
  • Matter Content of the ΛCDM Scenario
• Appendix D Second-order Fluctuations
  • Second-order Fluctuations of the Connections
  • Second-order Fluctuations of the Ricci Tensor
  • Specific Coordinate Systems
• Appendix E Arnowitt–Deser–Misner Decomposition
  • The Metric
  • The Christoffel Symbols
  • The Ricci Tensor
  • The Einstein Equations
• Appendix F Units and Frequently Used Symbols

The following section is included:

• Index