Journal Pre-proof Primordial backgrounds of relic gravitons Massimo Giovannini PII: S0146-6410(20)30021-1 DOI: https://doi.org/10.1016/j.ppnp.2020.103774 Reference: JPPNP 103774 To appear in: Progress in Particle and Nuclear Physics Please cite this article as: M. Giovannini, Primordial backgrounds of relic gravitons, Progress in Particle and Nuclear Physics (2020), doi: https://doi.org/10.1016/j.ppnp.2020.103774. This is a PDF file of an article that has undergone enhancements after acceptance, such asthe addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of thearticle. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. ' 2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Journal Pre-proof Primordial backgrounds of relic gravitons Massimo Giovannini∗ Department of Physics, CERN, 1211 Geneva 23, Switzerland INFN, Section of Milan-Bicocca, 20126 Milan, Italy Abstract The diffuse backgrounds of relic gravitons with frequencies ranging between the aHz band and the GHz region encode the ultimate information on the primeval evolution of the plasma and on the underlying theory of gravity well before the electroweak epoch. While the temperature and polarization anisotropies of the microwave background radiation probe the low-frequency tail of the graviton spectra, during the next score year the pulsar timing arrays and the wide-band interferometers (both terrestrial and hopefully space-borne) will explore a much larger frequency window encompassing the nHz domain and the audio band. The salient theoretical aspects of the relic gravitons are reviewed in a cross-disciplinary perspective touching upon various unsettled questions of particle physics, cosmology and astrophysics. CERN-TH-2019-166 Journal Pre-proof ∗Electronic address: [email protected] 1 Journal Pre-proof Contents 1 The cosmic spectrum of relic gravitons 4 1.1 Typical frequencies of the relic gravitons . 4 1.1.1 Low-frequencies . 5 1.1.2 Intermediate frequencies . 6 1.1.3 High-frequencies . 7 1.2 The concordance paradigm . 8 1.3 Cosmic photons versus cosmic gravitons . 9 1.4 Relic gravitons and large-scale inhomogeneities . 13 1.4.1 Quantum origin of cosmological inhomogeneities . 13 1.4.2 Weyl invariance and relic gravitons . 13 1.4.3 Inflation, concordance paradigm and beyond . 14 1.5 Notations, units and summary . 14 2 The tensor modes of the geometry 17 2.1 The tensor modes in flat space-time . 17 2.2 The tensor modes in curved space-times . 18 2.3 The tensor modes in cosmological space-times . 20 2.4 Relativistic cosmological fluctuations . 21 2.5 Gauge-invariant normal modes . 23 2.6 Supplementary polarizations? . 24 2.7 Tensor modes in Fourier space . 26 3 Effective action and energy density of the relic gravitons 28 3.1 Effective action for the relic gravitons . 29 3.1.1 Non-covariant derivation . 29 3.1.2 Covariant derivation . 31 3.1.3 Effective Hamiltonians . 32 3.1.4 Frame-invariance of the effective action . 32 3.2 Polarized backgrounds of relic gravitons and effective theories . 34 3.2.1 Spinor action for polarized backgrounds of relic gravitons . 35 3.3 Classical and quantum fluctuations in cosmology . 36 3.3.1 The quantum theory of parametric amplification . 37 3.3.2 Continuous mode squeezed states . 39 3.3.3 Two-point functions and power spectra . 41 3.4 Effective energy density of the relic gravitons . 42 3.4.1 The rate of variation of the background . 43 3.4.2 Super-adiabatic amplification . 45 3.4.3 Effective energy-momentum pseudo-tensor relic gravitons . 45 3.4.4 Drawbacks of other proposals for the energy density of the relic gravitons . 48 3.5 Relations among the observables . 50 4 Relic gravitons from the early universe 53 4.1 Scales exiting duringJournal inflation . Pre-proof . 54 4.1.1 Consistency conditions . 56 4.1.2 The maximal comoving frequency and the number of e-folds . 58 4.2 Sakharov oscillations and standing waves . 59 4.2.1 Standing waves in a radiation-dominated plasma . 59 2 Journal Pre-proof 4.2.2 Standing waves in a dust-dominated plasma . 60 4.3 Wavelengths reentering when the radiation dominates . 61 4.3.1 The power spectra in a plasma dominated by radiation . 61 4.3.2 The effect of the evolution of the relativistic species . 63 4.3.3 The effect of neutrino free streaming . 64 4.4 Wavelengths reentering when the dust dominates . 65 4.4.1 The power spectra in a plasma dominated by dust . 66 4.4.2 The concept of transfer function . 67 4.4.3 Approximate normalization of the spectrum . 69 4.5 Wavelengths reentering when the dark energy dominates . 70 4.5.1 Wavelengths that are currently inside the Hubble radius . 71 4.5.2 Wavelengths that are currently outside the Hubble radius . 72 4.6 The graviton spectra in the concordance paradigm . 73 5 The low-frequency band 74 5.1 Temperature and polarization anisotropies . 74 5.2 The effects of the relic gravitons . 76 5.3 The scalar and tensor Einstein-Boltzmann hierarchies . 77 5.4 The evolution of the temperature and of the polarization anisotropies . 80 5.5 Angular power spectra for the E-mode and B-mode polarizations . 84 5.6 Low-frequency modifications of the tensor power spectra . 85 5.7 Direct and indirect bounds on the tensor to scalar ratio . 88 5.8 Plateau-like potentials and a theory for the scalar spectral index . 89 6 The intermediate frequency band 91 6.1 Early deviations from the concordance scenario . 91 6.2 WKB approach and the blue spectral slopes . 94 6.3 Sufficient conditions for blue and violet spectra . 97 6.4 Early evolution of the refractive index of the relic gravitons . 100 6.5 Big-bang nucleosynthesis . 101 6.6 The big-bang nucleosynthesis bound . 103 6.7 The bounds from the pulsar timing array . 104 6.8 Potential signals from cosmic strings . 105 6.9 Potential signals from the electroweak epoch . 107 7 The high-frequency band 110 7.1 High-frequency gravitons and post-inflationary thermal history . 110 7.2 Spikes and scaling violations at high frequencies . 113 7.3 Secondary spectra of high-frequency gravitons from reheating . 114 7.4 Hanbury-Brown Twiss effect and the quantumness of relic gravitons . 116 8 Stochastic backgrounds of relic gravitons: observations 120 8.1 The early history of the detection of gravitational waves . 120 8.1.1 Resonant mass detectors . 120 8.1.2 Interferometric detection of gravitational waves . 121 8.2 Wide-band interferometersJournal in space . Pre-proof . 123 8.3 Bounds on relic gravitons from operating ground-based detectors . 126 8.4 Electro-mechanical detectors and high-frequency gravitons . 128 9 Concluding perspectives 129 3 Journal Pre-proof 1 The cosmic spectrum of relic gravitons The LIGO/Virgo collaboration recently published the three-detector observations of gravitational waves from black-hole coalescence [1], the evidence of gravitational waves from neutron star inspiral [2], and the observation of a 50-solar-mass binary black hole coalescence for a redshift z = 0.2 [3]. It is fair to say that the recent discoveries of wide-band interferometers [1, 2, 3] are fostering an unprecedented development of gravitational wave astronomy. The goal of this article is instead to scrutinize the theory of the relic gravitons, i.e. gravitons produced during the earliest stages of the evolution of the cosmic plasma. The energies and the frequencies of these massless particles are suppressed by the cosmic expansion so that their corresponding wavelengths are redshifted and may even be comparable with the diameter of the current Hubble patch. In many respects the cosmic gravitons bear analogy to the diffuse electromagnetic backgrounds with the notable difference that the microwave photons observed today last scattered free electrons and protons at a typical redshift zdec 1090, while the relic gravitons come from a stage when the Hubble rate was nearly Planckian. Even' if specific attention will be paid to the hoped sensitivities of the forthcoming detectors, the main emphasis of this article will be on the theoretical aspects and on the uncertainties of the potential signals. Table 1: Typical frequencies, energies and wavelengths of the electromagnetic spectrum. band frequency [Hz] energy [eV] wavelength [m] 21 6 13 γ-rays ZHz = 10 Hz 4 MeV = 4 10 eV 3 10− m 18 × 3 × 10 soft x-rays EHz = 10 Hz 4 keV = 4 10 eV 3 10− m 15 × × 7 ultraviolet PHz = 10 Hz 4 eV 3 10− m 12 3 × 4 infrared THz = 10 Hz 4 meV = 4 10− eV 3 10− m 9 × 6 × microwaves GHz = 10 Hz 4µeV = 4 10− eV 0.3 m 6 × 9 radio waves (HF) MHz = 10 Hz 4neV = 4 10− eV 300 m 3 × 12 5 radio waves (LF) kHz = 10 Hz 4 peV = 4 10− eV 3 10 m × 15 × 8 low frequency waves Hz 4 feV = 4 10− eV 3 10 m × × 1.1 Typical frequencies of the relic gravitons In a preliminary perspective it is useful to compare the gravitational and the electromagnetic spectra even if the origins of the two emissions are very different. The typical frequencies, wavelengths and energies of the electromagnetic spectra are illustrated in Tab.