- Open Access
Stochastic backgrounds of relic gravitons: a theoretical appraisal
© Giovannini et al 2010
- Received: 8 July 2009
- Accepted: 19 February 2010
- Published: 19 February 2010
Stochastic backgrounds or relic gravitons, if ever detected, will constitute a prima facie evidence of physical processes taking place during the earliest stages of the evolution of the plasma. The essentials of the stochastic backgrounds of relic gravitons are hereby introduced and reviewed. The pivotal observables customarily employed to infer the properties of the relic gravitons are discussed both in the framework of the ΛCDM paradigm as well as in neighboring contexts. The complementarity between experiments measuring the polarization of the Cosmic Microwave Background (such as, for instance, WMAP, Capmap, Quad, Cbi, just to mention a few) and wide band interferometers (e.g. Virgo, Ligo, Geo, Tama) is emphasized. While the analysis of the microwave sky strongly constrains the low-frequency tail of the relic graviton spectrum, wide-band detectors are sensitive to much higher frequencies where the spectral energy density depends chiefly upon the (poorly known) rate of post-inflationary expansion.
PACS codes: 04.30.-w, 14.70.Kv, 04.80.Nn, 98.80.-k
- Stokes Parameter
- Associate Legendre Function
- Tensor Mode
- Spectral Energy Density
- Curvature Perturbation
1.1 The frequencies and wavelengths of relic gravitons
Terrestrial and satellite observations, scrutinizing the properties of the electromagnetic spectrum, are unable to test directly the evolution of the background geometry prior to photon decoupling. The redshift probed by Cosmic Microwave Background (CMB in what follows) observations is of the order of zdec ≃ 1087 and it roughly corresponds to the peak of the visibility function, i.e. when most of the CMB photons last scattered free electrons (and protons). After decoupling the ionization fraction drops; the photons follow null geodesics whose slight inhomogeneities can be directly connected with the fluctuations of the spatial curvature present before matter-radiation decoupling. It is appropriate to mention that the visibility function has also a second (smaller) peak which arises because the Universe is reionized at late times. The reionization peak affects the overall amplitude of the CMB anisotropies and polarization. It also affects the peak structure of the linear polarization. The present data suggest that the typical redshift for reionization is zreion ~11 and the corresponding optical depth is 0.087. The optical depth at reionization actually constitutes one of the parameters of the concordance model (see below Eq. (2.7)).
The temperature of CMB photons is, today, of the order of 2.725 K. The same temperature at photon decoupling must have been of the order of about 2962 K, i.e. 0.25 eV (natural units ħ = c = kB = 1 will be adopted; in this system, K = 8.617 × 10-5eV). The CMB temperature increases linearly with the redshift: this fact may be tested empirically by observing at high redshifts clouds of chemical compounds (like CN) whose excited levels may be populated thanks to the higher value of the CMB temperature [1, 2].
The initial conditions for the processes leading to formation of CMB anisotropies are set well before matter radiation equality and right after neutrino decoupling (taking place for temperatures of the order of the MeV) whose associated redshift is around 1010. The present knowledge of particle interactions up to energy scales of the order of 200 GeV certainly provides important (but still indirect) clues on the composition of the plasma.
where g ρ denotes the weighted number of relativistic degrees of freedom at the onset of the radiation dominated evolution and 106.75 corresponds to the value of the standard model of particle interactions. Fermions and bosons contribute with different factors to g ρ . By assuming that all the species of the standard model are in local thermodynamic equilibrium (for instance for temperature higher than the top quark mass), g ρ will be given by g ρ = 28 + (7/8)90 = 106.75 where 28 and 90 count, respectively, the bosonic and the fermionic contributions.
In Eq. (1.1) it has been also assumed that H ≃10-5 MP as implied by the CMB observations in the conventional case of single-field inflationary models. In the latter case, absent other paradigms for the generation of (adiabatic) curvature perturbations, the condition H ≃ 10-5 MP is required for reproducing correctly the amplitude of the temperature and polarization anisotropies of the CMB. For more accurate estimates the quasi-de Sitter nature of the inflationary expansion must be taken into account. In the ΛCDM paradigm, the basic mechanism responsible for the production of relic gravitons is the parametric excitation of the (tensor) modes of the background geometry and it is controlled by the rate of variation of space-time curvature.
In the present article the ΛCDM paradigm will always be assumed as a starting point for any supplementary considerations. The reasons for this choice are also practical since the experimental results must always be stated and presented in terms of a given reference model. Having said this, most of the considerations presented here can also be translated (with the appropriate computational effort) to different models.
Given a specific scenario for the evolution of the Universe (like the ΛCDM model), the relic graviton spectra can be computed. The amplitude of the relic graviton spectrum over different frequencies depends upon the specific evolution of the Hubble rate. The theoretical error on the amplitude increases with the frequency: it is more uncertain (even within a specified scenario) at high frequencies rather than at small frequencies.
The experimental data, at the moment, do not allow either to rule in or to rule out the presence of a primordial spectrum of relic gravitons compatible with the ΛCDM scenario. The typical frequency probed by CMB experiments is of the order of νp = kp/(2π) ≃ 10-18 Hz = 1 aHz where kp is the pivot frequency at which the tensor power spectra are assigned. We are here enforcing the usual terminology stemming from the prefixes of the International System of units: aHz (for atto Hz i.e. 10-18 Hz), fHz (for femto Hz, i.e. 10-15 Hz) and so on. CMB experiments will presumably set stronger bounds on the putative presence of a tensor background for frequencies (aHz). This bound will be significant also for higher frequencies only if the whole post-inflationary thermal history is assumed to be known and specified.
The typical frequency window of wide-band interferometers (such as Ligo and Virgo) is located between few Hz and 10 kHz, i.e. roughly speaking, 20 orders of magnitude larger than the frequency probed by CMB experiments. The typical frequency window of LISA (Laser Interferometric Space Antenna) extends down to 10-100 μHz. While Virgo and Ligo are now operating, the schedule of LISA is still under discussion. The frequency range of wide-band interferometers will be conventionally denoted by νLV. To compute the relic graviton spectrum over the latter range of frequencies, the evolution of our Universe should be known over a broad range of redshifts. We do have some plausible guesses on the evolution of the plasma from the epoch of neutrino decoupling down to the epoch of photon decoupling. The latter range of redshifts corresponds to an interval of comoving frequencies going from νp ≃ 10-18 Hz up to νbbn ≃ 0.01 nHz ~10-11Hz (at most).
• the range νp <ν <νbbn will be generically referred to as the low-frequency domain; in this range the spectrum of relic gravitons basically follows from the minimal ΛCDM paradigm;
• the range νbbn <ν <νmax will be generically referred to as the high-frequency domain; in this range the spectrum of relic gravitons is more uncertain.
The high-frequency branch of the relic graviton spectrum, overlapping with the frequency window of wide-band detectors (see shaded box in the left plot of Fig. 1), is rather sensitive to the thermodynamic history of the plasma after inflation as well as, for instance, to the specific features of the underlying gravity theory at small scales. This is why we said that the theoretical error in the calculation of the relevant observables increases, so to speak, with the frequency.
In Fig. 1 (plot at the right) the electromagnetic spectrum is reported in its salient features. It seems instructive to draw a simple minded parallel between the electromagnetic spectrum and the spectrum of relic gravitons. Consider first the spectrum of relic gravitons (see Fig. 1, plot at the left): between 10-18 Hz (corresponding to νp) and 10 kHz (corresponding to νLV) there are, roughly, 22 decades in frequency. A similar frequency gap (see Fig. 1, plot at the right), if applied to the well known electromagnetic spectrum, would drive us from low-frequency radio waves up to x-rays or γ-rays. As the physics explored by radio waves is very different from the physics probed by γ rays, it can be argued that the informations carried by low and high frequency gravitons originate from two very different physical regimes of the theory. Low frequency gravitons are sensitive to the large scale features of the given cosmological model and of the underlying theory of gravity. High frequency gravitons are sensitive to the small scale features of a given cosmological model and of the underlying theory of gravity.
The interplay between long wavelength gravitons and CMB experiments will be specifically discussed in the subsection 1.2. The main message will be that, according to current CMB experiments, long wavelength gravitons have not been observed yet. The latter occurrence imposes a very important constraint on the low frequency branch of the relic graviton spectrum of the ΛCDM scenario whose salient predictions will be introduced in subsection 1.3. According to the minimal ΛCDM paradigm a very peculiar conclusion seems to pop up: the CMB constraints on the low-frequency tail of the graviton spectrum jeopardize the possibility of any detectable signal for frequencies comparable with the window explored by wide band interferometers (see subsection 1.4). The natural question arising at this point is rather simple: is it possible to have a quasi-flat low-frequency branch of the relic graviton spectrum and a sharply increasing spectral energy density at high-frequencies? This kind of signal is typical of a class of completions of the ΛCDM paradigm which have been recently dubbed TΛCDM (for tensor-ΛCDM). The main predictions of these models will be introduced in subsection 1.5. We shall conclude this introductory section with a discussion of two relevant constraints which should be applied to relic graviton backgrounds in general, i.e. the millisecond pulsar and the big-bang nucleosynthesis constraint (see subsection 1.6).
1.2 Long wavelength gravitons and CMB experiments
where kp is the pivot wave-number and is the amplitude of the power spectrum of curvature perturbations computed at kp; ns and nT are, respectively, the scalar and the tensor spectral indices. The value of kp is conventional and it corresponds to an effective harmonic ℓeff ≃ 30. The perturbations of the spatial curvature, conventionally denoted by ℛ are customarily employed to characterize the scalar fluctuations of the geometry since ℛ is approximately constant (in time) across the radiation-matter transition. As it is clear from Eqs. (1.2) and (1.3) there is a difference in the way the scalar and the tensor spectral indices are assigned: while the scale-invariant limit corresponds to ns → 1 for the curvature perturbations, the scale invariant limit for the long wavelength gravitons corresponds to nT → 0.
so, as anticipated, νp is of the order of the aHz. The amplitude at the pivot scale is controlled exactly by rT. In the first release of the WMAP data the scalar and tensor pivot scales were chosen to be different and, in particular, kp = 0.05 Mpc-1 for the scalar modes. In the subsequent releases of data the two pivot scales have been taken to coincide.
The change in determination of the parameters of the tensor background for three different choices of cosmological data sets.
Same as in Tab. 1 but assuming no running in the (scalar) spectral index (i.e. αS = 0).
The inferred values of the scalar spectral index (i.e. ns), of the dark energy and dark matter fractions (i.e., respectively, ΩΛ and ΩM0), and of the typical wavenumber of equality keq are reported in the remaining columns. While different analyses can be performed, it is clear, by looking at Tabs. 1 and 2, that the typical upper bounds on rT(kp) range between, say, 0.2 and 0.4. More stringent limits can be obtained by adding supplementary assumptions.
where ϵ measures the rate of decrease of the Hubble parameter during the inflationary epoch. The overdot will denote throughout the paper a derivation with respect to the cosmic time coordinate t while the prime will denote a derivation with respect to the conformal time coordinate τ.
where αT now measures the running of the tensor spectral index.
As already mentioned, among the CMB experiments a central role is played by WMAP [3–7] (see also [8–10] for first year data release and [11, 12] for the third year data release. In connection with [3–7], the WMAP 5-year data have been also combined with observations of the Acbar satellite [20–23] (the Arcminute Cosmology Bolometer Array Receiver (ACBAR) operates in three frequencies, i.e. 150, 219 and 274 GHz). The TT, TE and, partially EE angular power spectra have been measured by the WMAP experiment. Other (i.e. non space-borne) experiments are now measuring polarization observables, in particular there are
as well as various other experiments at different stages of development. Other planned experiments have, as specific target, the polarization of the CMB. In particular it is worth quoting here the recent projects Clover , Brain , Quiet , Spider  and EBEX  just to mention a few. In the near future the Planck explorer satellite  might be able to set more direct limits on rT by measuring (hopefully) the BB angular power spectra.
Following the custom the TT correlations will simply denote the angular power spectra of the temperature autocorrelations. The TE and the EE power spectra denote, respectively, the cross power spectrum between temperature and polarization and the polarization autocorrelations.
1.3 The relic graviton spectrum in the ΛCDM model
Having defined the frequency range of the spectrum of relic gravitons, it is now appropriate to illustrate the possible signal which is expected within the ΛCDM scenario.
Since ρcrit depends upon (i.e. the present value of the Hubble rate), it is practical to plot directly (ν, τ0) at the present (conformal) time τ0. The proper definition of ΩGW(ν, τ0) in terms of the energy-momentum pseudo-tensor in curved space-time is postponed to section 5. The salient features of the relic graviton spectra arising in the context of the ΛCDM scenario can be appreciated by looking carefully at Fig. 2.
Both in Eqs. (1.8) and (1.10) ΩM0 and ΩR0 denote, respectively, the present critical fraction of matter and radiation with typical values drawn from the best fit to the WMAP 5-yr data alone and within the ΛCDM paradigm. In Eq. (1.10) g ρ denotes the effective number of relativistic degrees of freedom entering the total energy density of the plasma. While νeq is still close to the aHz, νbbn is rather in the nHz range. In Fig. 2 (plot at the left) the spectral index nT is frequency independent; in the plot at the right, always in Fig. 2, the spectral index does depend on the wavenumber. These two possibilities correspond, respectively, to αT = 0 and αT ≠ 0 in Eq. (1.6). In the regime ν <νeq a numerical calculation of the transfer function is mandatory for a correct evaluation of the spectral slope. In the approximation of a sudden transition between the radiation and matter-dominated regimes the spectral energy density goes, approximately, as . The spectra illustrated have been computed within the approach developed in [46, 47] and include also other two effects which can suppress the amplitude of the quasi-flat plateau, i.e., respectively, the late dominance of the cosmological constant and the progressive reduction in the number of relativistic species. The latter two effects can be estimated analytically (see the final part of section 6) and they are, however, numerically less relevant than neutrino free streaming.
Apart from the modification induced by the neutrino free-streaming the slope of the spectral energy density for ν > νeq is quasi flat and it is determined by the wavelengths which reentered the Hubble radius during the radiation-dominated stage of expansion. The suggestion that relic gravitons can be produced in isotropic Friedmann-Robertson-Walker models is due to Ref.  (see also ) and was formulated before the inflationary paradigm. After the formulation of the inflationary scenario the focus has been to compute reliably the low frequency branch of the relic graviton spectrum. In [50–52] the low-frequency branch of the spectrum has been computed with slightly different analytic approaches but always assuming an exact de Sitter stage of expansion prior to the radiation-dominated phase. The analytical calculation (whose details will be described in section 6) shows that in the range νp <ν <νeq, the spectral energy density of the relic gravitons (see Eq. (1.7)) should approximately go as ΩGW(ν, τ0) ≃ ν-2. Within the same approximation, for ν > νeq the spectral energy density is exactly flat (i.e. ΩGW(ν, τ0) ≃ ν0). This result, obtainable by means of analytic calculations (see also [53–56]), is a bit crude in the light of more recent developments. To assess the accurately spectral energy density it is necessary to take into account that the infrared branch is gradually passing from a quasi-flat slope (for ν > νeq) to the slope ν-2 which is the one computed within the sudden approximation [53–56]. It is useful to quote some of the previous reviews which covered, in a more dedicated perspective, the subject of the stochastic backgrounds of relic gravitons. The review article by Thorne  does not deal solely with relic graviton backgrounds while the reviews of Refs. [58–60] are more topical.
where, as in Eqs. (1.2) and (1.3), denotes the amplitude of the power spectrum of curvature perturbations evaluated at the pivot wavenumber νp. It is worth noticing that between νbbn and νmax there are approximately 20 orders of magnitude in frequency. In the ΛCDM scenario the spectrum has, in this range, always the same slope (i.e. nT is frequency-independent in Eq. (1.2)).
Some details of the calculations leading to the spectral energy densities illustrated in Fig. 2 can be found in sections 5 and 6. Without dwelling on the details it is however clear, as anticipated, that the constraints on the long wavelength gravitons make it difficult (if not impossible) to have a detectable spectral energy density at the scale of wide-band interferometers. The latter statement, valid in the minimal ΛCDM scenario, will be sharpened in the following subsection.
1.4 Short wavelength gravitons and wide-band interferometers
In the ΛCDM scenario the spectral energy density of the relic gravitons has its larger amplitude in the low-frequency branch. As the frequency increases the spectral energy density diminishes so that it is plausible to expect a rather small amplitude over the frequencies corresponding to wide-band interferometers (see, for instance, Fig. 2 for ν ≃ νLV = 100 Hz).
Wide-band interferometers operate in a window ranging from few Hz up to 10 kHz (see also Fig. 1). The available interferometers are Ligo , Virgo , Tama  and Geo . In loose terms these instruments are Michelson interferometers with two important differences: the mirrors are suspended and Fabry-Pérot cavities are used to increase the optical path of the photons. It would be too pretentious to describe in detail, in the present script, also the experimental apparatus and we therefore suggest Ref.  where the basics of wide-band interferometers are introduced in a self-contined perspective.
The sensitivity of a given pair of wide-band detectors to a stochastic background of relic gravitons depends upon the relative orientation of the instruments. The wideness of the band (important for the correlation among different instruments) is not as large as 10 kHz but typically narrower and, in an optimistic perspective, it could range up to 100 Hz. The putative frequency of wide-band detectors will therefore be indicated as νLV, i.e. in loose terms, the Ligo/Virgo frequency. There are daring projects of wide-band detectors in space like the Lisa , the Bbo  and the Decigo  projects. The common feature of these three projects is that they are all space-borne missions and that they are all sensitive to frequencies smaller than the mHz (1 mHz = 10-3 Hz). While wide-band interferometers are now operating and might even reach their advanced sensitivities during the incoming decade, the wished sensitivities of space-borne interferometers are still on the edge of the achievable technologies.
In Fig. 3, the common logarithm of the spectral energy density is illustrated as a function of the common logarithm of rT.
The variable β is used in Eq. (1.12) just because this is the notation endorsed by the Ligo collaboration and there is no reason to change it. At the same time, in the present review, β will be used also with different meanings. In section 6, β quantifies the theoretical error on the maximal frequency of the relic graviton spectrum(see e.g. Eq. (6.48) and discussion therein). In section 7, β parametrizes a portion of the azimuthal structure of the Stokes parameters. Since none of these variables appear in the same context, potential clashes of conventions are avoided.
The parametrization of Eq. (1.12) fits very well with Fig. 3 where the pivot frequency νLV = 100 Hz coincides with the pivot frequency appearing in the parametrization (1.12). For the scale-invariant case (i.e. β = -3 in eq. (1.12)) the Ligo collaboration sets a 90% upper limit of 1.20 × 10-4 on the amplitude appearing in Eq. (1.12), i.e. ΩGW,-3. Using different sets of data (see [69, 71]) the Ligo collaboration manages to improve the bound even by a factor 2 getting down to 6.5 × 10-5. Thus Fig. 3 together with the upper limit of Eq. (1.12) shows that the current Ligo sensitivity is still too small to detect the relic graviton background arising within the ΛCDM paradigm.
1.5 Beyond the ΛCDM paradigm and high-frequency gravitons
where T denotes the observation time and SNR is the signal to noise ratio. Equation (1.13) is in close agreement with the sensitivity of the advanced Ligo apparatus  to an exactly scale-invariant spectral energy density [77–81]. Equation (1.13) together with the plots of Fig. 3 suggest that the relic graviton background predicted by the ΛCDM paradigm is not directly observable by wide-band interferometers in their advanced version.
CMB observations probe the aHz region of the spectral energy density of Fig. 2. Wide-band interferometers probe a frequency range between few Hz and 10 kHz. In both ranges, the signal of the ΛCDM scenario might be too small to be directly detectable.
the theoretical error in the estimate of the spectral energy density increases with the frequency;
departures from the standard post-inflationary thermal history can be directly imprinted in the primordial spectrum of the relic gravitons;
in the incoming decade the observations of wide-band interferometers could be analyzed in conjunction with more standard data sets (i.e. CMB data supplemented by large-scale structure data and by the observations of type Ia supernovae) to constrain the spectral energy density of the relic gravitons both at small and at high frequencies.
The presence of post-inflationary phases stiffer than radiation is, after all, rather natural and this was the original spirit of . We do not know which was the rate of the post-inflationary expansion and since guesses cannot substitute experiments it would be productive to use the TΛCDM paradigm as reference model for a unified analysis of the low-frequency data stemming from CMB and of the high-frequency data provided by wide-band interferometers. Already in  (see also [83, 84]) a rather special candidate for a post-inflationary phase stiffer than radiation was the case when the sound speed equals the speed of light, i.e. the case when the energy density of the sources driving the geometry is dominated by the kinetic term of a (minimally coupled) scalar field. This particular case was also prompted by various classes of quintessence models. A specific example of this dynamics was provided in .
A more detailed account of the techniques leading to Fig. 4 will be swiftly presented in section 6 and can be found in [46, 47]. Without going through the details it is however important to stress that the calculations should be accurate enough not only in the high-frequency region but also in the low-frequency part of the spectrum. Indeed, as stressed above, one of the purposes of the TΛCDM scenario is to convey the idea that low-frequency and high-frequency measurements of the relic graviton background can be analyzed in a single theoretical framework.
1.6 The millisecond pulsar bound and the nucleosynthesis constraint
where νpulsar roughly corresponds to the inverse of the observation time during which the pulsars timing has been monitored. The spectral energy densities illustrated in Figs. 2 and 4 satisfy the pulsar timing bound.
The most constraining bound for the high-frequency branch of the relic graviton spectrum is represented by big-bang nucleosynthesis. Gravitons, being relativistic, can potentially increase the expansion rate at the BBN epoch. The increase in the expansion rate will affect, in particular, the synthesis of 4He. To avoid the overproduction of 4He the expansion rate the number of relativistic species must be bounded from above.
where νbbn and νmax are given, respectively, by Eqs. (6.61) and (8.4). Thus the constraint of Eq. (1.18) arises from the simple consideration that new massless particles could eventually increase the expansion rate at the epoch of BBN. The extra-relativistic species do not have to be, however, fermionic  and therefore the bounds on ΔN ν can be translated into bounds on the energy density of the relic gravitons.
The spectral energy densities illustrated in Figs. 2 and 4 are both compatible with the big-bang nucleosynthesis bound. Thus the big-bang nucleosyntheis constraint does not forbid a potentially detectable signal in the high-frequency branch of the relic graviton spectrum. Potential deviations of the thermal history of the plasma must anyway occur before big-bang nucleosynthesis.
2.1 Basic notations
where Ωb0, Ωc0, ΩΛ denote, respectively, the (present) critical fractions of baryons, CDM particles and dark energy; h0 fixes the present value of the Hubble rate; ns, as already mentioned in section 1, is the spectral index of curvature perturbations and ϵ is the reionization optical depth.
At the beginning of the previous section we started by stressing analogies and differences between relic gravitons and relic photons. The most important one is that both gravitons and photons carry two polarizations. This observation is important for a quantitative understanding of the present endevours aimed at measuring the E-mode and the B-mode polarization of the CMB. In the present section the description of the polarization of the gravitons will be developed by stressing, when possible, the analogy with polarization observables of the electromagnetic field.
2.2 Linear and circular tensor polarizations
The transformation properties of the circular polarization under a rotation in the plane orthogonal to the direction of propagation are closely analog to the transformation properties, under the same rotation, of the polarization of the electromagnetic field. This analogy will now be exploited to introduce the E-mode and B-mode polarization.
Before proceeding with the discussion it is appropriate to recall a very basic aspect of rotations which can have, however, some confusing impact of the polarization analysis especially in the case of the tensor modes. Consider, for simplicity, a coordinate system characterized by two basis vectors, i.e. cos ϑ and sin ϑ. If we now perform a clockwise (i.e. right-handed) rotation of the axes and , the rotated basis will be given as in Eqs. (2.22) and (2.23) by replacing and . Some authors, for different reasons, instead of rotating the coordinate system prefer to rotate the polarization vector. If angles are in the right-handed sense for the rotation of the axes, they are in the left-handed sense for the rotation of the vectors.
2.3 Polarization of the CMB radiation field
Equations (2.24)-(2.25) and (2.37) express the fact that the polarization of the graviton and of the radiation field do change for a rotation on the plane orthogonal to the direction of propagation of the radiation (either gravitational or electromagnetic). It is possible to construct polarization observables which are invariant for rotations on the plane orthogonal to the direction of propagation of the radiation: because of their properties under parity transformations they are called E-and B-modes.
2.4 E- and B-modes
Overall, the existence of linear polarization allows for 6 different power spectra.
In the minimal version of the ΛCDM paradigm the adiabatic fluctuations of the scalar curvature lead to a polarization which is characterized exactly by the condition a2, ℓm= a-2, ℓm, i.e. = 0. This observation implies that, in the ΛCDM scenario, the non-vanishing angular power spectra are given by the TT, EE and TE correlations. In the TΛCDM scenario the TT, EE and TE angular power spectra are supplemented by a specific prediction for the B-mode autocorrelation (see section 7).
2.5 Spin-2 spherical harmonics
Spherical harmonics of higher spin appear in matrix elements calculations in nuclear physics (see e.g. the classic treatise of Blatt and Weisskopf , and, in a similar perspective the book of Edmonds ). The comprehensive treatments of Biedenharn and Louk  and of Varshalovich et al.  can also be usefully consulted.
The spin-s harmonics have been introduced, in their present form, by Newman and Penrose  and their group theoretical interpretation has been discussed in . The spin-s spherical harmonics have been applied to the discussion of CMB polarization induced by relic gravitons in a number of papers [98–100]. They are rather crucial in the formulation of the so-called total angular momentum approach. Discussions of the spin-weighted spherical harmonics in a cosmological context can also be found in [101, 102]. The spin weighted spherical harmonics will now be introduced by following the spirit of Ref.  which has been also used, with different conventions, in . In subsection 2.6 the (equivalent) approach of [99, 100] will be more specifically outlined.
where Pℓ(μ) are the Legendre polynomials and (μ) the associated Legendre functions. It is appropriate to mention here that the factor (-1) m (i.e. Condon-Shortley phase) can either be included in the normalization factor or (as it has been done) in the definition of the associated Legendre functions appearing in Eq. (2.55). When using the recurrence relations of the associated Legendre functions the Condon-Shortley phase introduces a sign difference every time m is odd. The conventions expressed by Eqs. (2.54) and (2.55) will be followed throughout the present discussion and, in particular, in section 7 where the correlation functions of the E-modes and of the B-modes will be specifically computed with different techniques.
where, as already mentioned, . In Eqs. (2.63) and (2.64) there appear only ordinary (i.e. spin-weight 0) spherical harmonics. This occurrence suggests a complementary approach to the problem: instead of expanding Δ± ( , τ) in terms of spin-2 spherical harmonics, fluctuations of spin-weight 0 can be directly constructed (in real space) from Δ± ( , τ) by repeated application of the ladder operators defined in Eqs. (2.52) and (2.53).
As discussed at the end of subsection 2.1 the sign of φ can be flipped. This possibility is not related to a parity transformation and it has to do with the way two-dimensional rotations are introduced. This aspect will also be relevant in section 7 for explicit derivations.
The transformation (2.71) implies that the two basis vectors defined in Eq. (2.70) transform as and , i.e. while does not change flips its sign under space inversion. It follows that space-inversion does not flip the sign of ΔE( ) but it does flip the sign of ΔB( ), i.e. under the transformation (2.71), ΔE( ) → ΔE( ) while ΔB( ) → -ΔB( ).
where L± and Lz obey the well known commutation relations [L±, L z ] = ∓L± and [L+, L-] = 2L z .
Looking at Eq. (2.79) it is tempting draw a parallel between the (orbital) ladder operators and the ladder operators raising (or lowering) the spin weight of a given function (see Eqs. (2.52) and (2.53). This problem has been discussed and solved in . It is possible to formulate the parallel in terms of a putative O(4) group. Half of the generators will be connected with the orbital angular momentum operators, while the other half will allow to increase (or decrease) the spin weight of a given function. The two sets of generators commute. The operators are not directly, though, the ladder operators stemming from the second set of generators. This has to do with the fact that in Eq. (2.51) the third Euler angle (i.e. γ) has been fixed to zero. The are ladder operators defined within a putative O(4) group in the case γ ≠ 0. When γ → 0 the dependence upon γ drops and we are left with Eqs. (2.52) and (2.53).
2.6 Polarization on the 2-sphere
satisfying P ab = P ba , and g ab P ab = 0, where is a unit vector in the direction (ϑ, φ). The sign of the off-diagonal entries in Eq. (2.81) is opposite with respect to the one obtained in Eq. (2.34). This is just because we want to match with the conventions adopted, for instance, in [100–102]. To avoid possible confusions, furthermore, the Latin indices a, b, c, d, .... run over the two-dimensional space.
is the Levi-Civita symbol on the 2-sphere. Notice that Nℓ differs from defined in Eqs. (2.46) (see also Eqs. (2.63) and (2.64)) by a factor . This difference will be ultimately relevant to relate and .
The two approaches to the spin weighted spherical harmonics described in the present section are equivalent and can be used interchangeably depending upon the specific problem.
3.1 Second-order fluctuations of the Einstein-Hilbert action
3.2 Lagrangian densities
All the three Lagrangian densities of Eqs. (3.14), (3.16) and (3.17) lead to the same Euler-Lagrange equations.
3.3 Hamiltonian densities
as it can be explicitly verified by using Eqs. (3.19), (3.20) and (3.25) into Eq. (3.26).
3.4 Evolution equations in different regimes
This form of the evolution equation for the tensor modes is the one required to compute the effects related to the finite value of the anisotropic stress.
There are analogies between the quantum state of relic gravitons and the quantum treatment of visible light. Quantum effects are not crucial to treat first-order interference of the radiation field (i.e. Young interferometry) . First-order interference in quantum optics correspond to the calculation of the two-point function of the relic gravitons. Quantum effects arise, in optics, from second-order interference, i.e. when computing (and measuring) the interference between the intensities of the radiation field. Second-order interference effects are associated with the possibilities of counting photons and have been pioneered by Hanbury-Brown and Twiss in the early fifties [104, 105]. Hanbury-Brown-Twiss interferometry is based on photon counting statistics.
Having said that we are not even close (experimentally) to study graviton counting statistics (as we do it with the photons), second order interference effects would allow, in principle, to assess the coherence properties of relic graviton backgrounds. The quantum state of the relic gravitons can be described in terms of a generalized coherent state usually called squeezed state. Squeezed states can be described in terms of quadrature operators where one of the modes of the radiation field is always broadened by the time evolution, while the other one is squeezed.
4.1 Heisenberg description
Since, by construction, the Hamiltonians of Eqs. (3.19) and (3.20) are related by canonical transformations, the mode functions of Eqs. (4.11) and (4.12) will have both to obey Eq. (4.13). In different terms, the commutation relations between field operators should be preserved by the time evolution and this is equivalent to the Wronskian normalization condition of Eq. (4.13).
4.2 Generalized coherent states of relic gravitons
There is a slight difference in the normalizations adopted between Eqs. (4.9)-(4.10) and Eqs. (4.25)-(4.26). This difference is due to the fact that, in Eqs. (4.9)-(4.10) the mode functions f k are normalized, asymptotically, in such a way that f k → 1/ . In Eqs. (4.15)-(4.16) the factors and have been included in the definition creation and annihilation operators.
which has exactly the same physical content of Eqs. (4.32) and (4.33). When the Universe expands, g k (τ) decreases and that the solution associated with A2(k) becomes progressively subleading. However, this observation does not imply that g k (τ) disappears since the evolution must be unitary. This feature of squeezed quantum state suggests the possibility of associating an effective entropy to the process of graviton production[106–110].
where the creation and destruction operators are the ones computed in τ0, i.e. by definition of Schrödinger description. The state |0⟩ is annihilated both by and by . These two-modes appear simultaneously since gravitons are produced from the vacuum whose total momentum vanish. The x and have been dubbed, in the literature, as superfluctuant operators (see, e. g., [106–108]).
the colons denote normal ordering and denotes the operator corresponding to the intensity of the radiation field. The normal ordering is related to the fact that, in the optical domain, most measurements of the electromagnetic field are based on the absorption of photons via the photoelectric effect. Needless to say that there is no analog of photoelectric detection for (single) relic gravitons. In this sense the following considerations should be regarded as a conditional predictions based on the analogy between squeezed states of photons and squeezed states of gravitons.
where ⟨ ⟩ = sinh2 r k is the multiplicity. The coherent state leads to a radiation field with Poissonian statistics. Thermal states (as well as squeezed states) have a statistics which is, according to the quantum optical terminology, superpoissonian. The latter statement is often dubbed by saying that if g(2)(0) > 1 photons are bunched while, in the opposite case (i.e. g(2)(0) < 1) the photons are said to be anti-bunched. The quantum optical language is much more effective for a mathematical description of the semi-classical limit than the usual considerations related to the limit ħ → 0. Squeezed states are genuine quantum states with many particles. They are, in some sense, like coherent states with the crucial difference that their statistics is super-Poissonian. The possibility of scrutinizing the statistical properties of many-gravitons systems would rely on our ability of resolving single gravitons which is not even close to the present technological capabilities.
In the literature relic graviton backgrounds are characterized in terms of different quantities and, in particular, the most common ones are:
It is understood that all the mentioned quantities can be expressed either in terms of the wave-number or in terms of the frequency since k = 2πν.
The three listed variables can be related in different regimes. For instance the power spectrum has a simple relation to the spectral energy density when the relevant wavelengths are inside the Hubble radius. In section 6 it will be argued that, for numerical applications, the transfer function for the spectral energy density is more practical to compute than the transfer function for the power spectrum or for the spectral amplitude itself. The power spectrum is actually a strongly oscillating function of the conformal time coordinate for wavelengths shorter than the Hubble radius (i.e. kτ > 1); in the same limit the spectral energy density is asymptotically constant.
5.1 The tensor power spectrum and the spectral amplitude
where (k) denotes the tensor power spectrum and where the factor 2 in front of the averages arises as a consequence of the appearing in Eq. (5.11). In Eqs. (5.11), (5.12) and (5.13) the conformal time coordinate is absent. In Eq. (5.10) the conformal time appears explicitly. Indeed, Eqs. (5.11), (5.12) and (5.13) tacitly assume that h⊕( , τ) = e⊕( )Te(k, τ) and that h⊗( , τ) = e⊗( )Te(k, τ). This factorization is related to the concept of transfer function for the amplitude which will be discussed in section 6. The decomposition of Eqs. (5.11), (5.12) and (5.13) is useful when all the polarization have to be treated simultaneously typically in problems involving long wavelength gravitons (see Eqs. (7.85)-(7.86) and discussion therein). Furthermore the decomposition of Eqs. (5.11), (5.12) and (5.13) allows to factorize the dependence upon the initial spectrum which is useful for numerical applications.
5.2 Energy-momentum tensors for the relic gravitons
By comparing Eqs. (5.38)-(5.39) with Eqs. (5.22)-(5.23) we can remark that the first term appearing in Eq. (5.38) is absent from Eq. (5.22). Moreover, also and seems to be superfficially different. As it will be shown in a moment the equivalence of the two approaches is clear as soon as the relevant wavelengths are larger than the Hubble radius at a given time.
5.3 The energy density of the relic gravitons
In the limit kτ > 1 we will have ℋ2 ≪ k2. Thus Eqs. (5.50) and (5.51) coincide (up to corrections (ℋ2/k2)). In this limit it is also possible to express ΩGW(k, τ) solely in terms of the power spectrum.
As long as the relevant wavelengths are shorter than the Hubble radius at a given time, different prescriptions for assigning the energy-momentum pseudo-tensor lead to the same result (see also the discussion in section 6). In the opposite limit different choices may exhibit quantitative differences. The limit of short wavelengths in comparison with the Hubble radius is the relevant one when discussing wide-band interferometers. Conversely, the initial conditions for the CMB anisotropies are set when the relevant wavelengths are larger than the Hubble radius before equality.
6.1 Inflationary power spectra
Within the present notations, as already established in Eq. (3.12), .
Equation (6.8) implies, recalling Eq. (6.5), that rT = -8nT. Since there is a direct relation of the tensor spectral index to rT, the number of the parameters can be reduced from two to one. In Tabs. 1 and 2 the values of rT have been reported as they can be estimated in few different analyses of the cosmological data sets.
6.2 Transfer functions for inflationary power spectra
In Eq. (6.10), (τ) denotes the approximate form of the mode function (holding during the matter-dominated phase); F k (τ) denotes, instead, the solution obtained by fully numerical methods. The averages appearing in Eq. (6.10) refer to the average over the oscillations: as the wavelengths are inside the Hubble radius, the solutions are all oscillating. The numerical average over the phases introduces some arbitrariness which can be cured by computing directly the transfer function for the spectral energy density.
where the typical value selected for is given by the sum of the photon component (i.e. = 2.47 × 10-5) and of the neutrino component (i.e. = 1.68 × 10-5): the neutrinos, consistently with the ΛCDM paradigm, are taken to be massless and their (present) kinetic temperature is just a factor (4/11)1/3 smaller than the (present) photon temperature.
Within the standard approach, Eq. (6.16) is customarily connected to the spectral energy density of the relic gravitons. In [46, 47] it has been observed that it is simpler and more accurate to compute directly the transfer function for the spectral energy density. In the following subsection this procedure will be illustrated in two different cases.
6.3 Transfer function for the spectral energy
To avoid unnecessary complications, the initial condition of the integrations illustrated in Figs. 5 and 6 have been set as (xi) = xi, i.e. the initial spectrum has been rescaled. The transfer function, by definition, must always depend only on the dynamics of the transition and not upon the features (e.g slope, amplitude) of the initial power spectrum.
Equation (6.28) permits the accurate evaluation of the spectral energy density of relic gravitons, for instance, in the minimal version of the ΛCDM paradigm.
Yet another relevant physical situation for the present considerations is the one where the back-ground geometry, after inflation, transits from a stiff epoch to the ordinary radiation-dominated epoch. In the primeval plasma, stiff phases can arise for various reasons. Zeldovich  (see also ) suggested this possibility in connection with the entropy problem. In [82–84, 74] it has been suggested that the stiff phase could take place after the inflationary phase with the main purpose of identifying a potential source of high-frequency gravitons. This possibility was also prompted by a possible post-inflationary dominance of a quintessence field.
to a radiation-dominated phase where cst = 1/41. Note that, according to Eqs. (6.29) and (6.30), = wt iff the (total) barotropic index is constant in time. In the limiting case wt = 1 = and the speed of sound coincides with the speed of light. As argued in , barotropic indices wt >1 would not be compatible with causality (see, however, ). The presence of a suitable stiff phase has been also discussed recently as an effective way of suppressing entropic fluctuations  which are observationally constrained by the WMAP 5-yr data.
where ks = . The value of ks can be computed in an explicit model but it can also be left as a free parameter. Taking into account that the energy density of the inflaton will be exactly , the value of ks (as well as the duration of the stiff phase) will be determined, grossly speaking, by Hi/ . In the context of quintessential inflation  (see also [83, 84]) ρRi ≃ .
6.4 Analytic results for the mixing coefficients
The analytic results for the mixing coefficients are rather useful to obtain the final expression of the various transfer functions. Indeed, defining as k* the typical wavenumber of the transition (e.g. k* = keq in the case of the radiation-mattter transition), the slope of the transfer function of the spectral energy density can analytically obtained in the limit κ ≫ 1. This observation helps when we have, for instance, to fit the numerical data points with an analytical expression which will however reproduce the data not only for κ > 1 (as Figs. 5 and 7 clearly show).
The logarithms arising in Eqs. (6.43) and (6.44) explain why, in Eq. (6.32), the transfer function of the spectral energy density contains logarithms. In spite of the fact that semi-analytical estimates can pin down the slope of the transfer functions in different intervals, they are insufficient for a faithful account of more realistic situations where the slow-roll corrections are relevant and when other dissipative effects (such as neutrino fee streaming) are taken into account.
6.5 Exponential damping of the mixing coefficients
Equations (6.45) and (6.46) are derived by assuming that, right after inflation, the radiation-dominated phase takes over. Furthermore, recalling the slow-roll dynamics, and V ∝ . In Eqs. (6.45) and (6.46) denotes, as already established, the amplitude of the curvature power spectrum evaluated at the pivot scale.
where the typical values of the slow-roll parameter have been derived by taking into account that, in the absence of running of the tensor spectral index, rT = 16ϵ; since, according to the WMAP 5-yr data alone, rT < 0.43, ϵ ≤ 0.01.
For τ → -∞ (i.e. τ ≪ -τi), a(τ) ≃ -ai/τ and the quasi de-Sitter dynamics is recovered. In the opposite limit (i. e. τ ≫ +τi), a(τ) ≃ ai τ and the radiation dominance is recovered. In Fig. 6 (plot at the left) the exponential damping of the mixing coefficients is numerically illustrated. The curve at the top (full line) illustrates the case κ = 1. The cases κ = 2 and κ = 3 are barely distinguishable at the bottom of the plot. Notice, always in the right plot, the rather narrow range of times which are reported in a linear scale. In the plot at the right the asymptotic values of the mixing coefficients are reported for different values of κ = k/kmax. By fitting the numerical data with with an equation of the form given in Eq. (6.48), the value of β = 6.33. Different examples can be presented on the same line of the one discussed in Fig. 6. While it is pretty clear that the decay is indeed exponential, the value of β may well vary. This can be summarized, for instance, in a rescaling of kmax, i.e. by positing, for instance that kmax → /β. Thus, the dynamics of the transition can slightly shift the numerical value of the upper cut-off by a numerical factor which depends upon the width of the transition regime.
6.6 Nearly scale-invariant spectra
where, we recall, H0 = 3.24078 × 10-18 h0 Hz.
By comparing Eqs. (6.50)-(6.51) to Eqs. (6.56)-(6.57), the amplitude for ν ≫ νeq differs, roughly, by a factor 2. This coincidence is not surprising since Eqs. (6.50)-(6.51) have been obtained by averaging over the oscillations (i.e. by replacing cosine squared with 1/2) and by imposing that |g k | = k|f k |. These manipulations are certainly less accurate than the procedure used to derive the transfer function for the spectral energy density.
At the present time gρ0 = 3.36 and gs0 = 3.90. In general terms the effect parametrized by Eq. (6.67) will cause a frequency-dependent suppression, i.e. a further modulation of the spectral energy density ΩGW(ν, τ0). The maximal suppression one can expect can be obtained by inserting into Eq. (6.67) the largest gs and g ρ . So, in the case of the minimal standard model this would imply that the suppression (on ΩGW(ν, τ0)) will be of the order of 0.38. In popular supersymmetric extensions of the minimal standard models g ρ and g s can be as high as, approximately, 230. This will bring down the figure given above to 0.29.
All the three effects estimated in the last part of the present section (i.e. free streaming, dark energy, evolution of relativistic degrees of freedom) have common features. Both in the case of the neutrinos and in the case of the evolution of the relativistic degrees of freedom the potential impact of the effect could be more pronounced. For instance, suppose that, in the early Universe, the particle model has many more degrees of freedom and many more particles which can free stream, at some epoch. At the same time we can say that all the aforementioned effects decrease rather than increasing the spectral energy density. Taken singularly, each of the effects will decrease ΩGW by less than one order of magnitude. The net result of the combined effects will then be, roughly, a suppression of ΩGW(ν, τ0) which is of the order of 3 × 10-2 (for 10-16 Hz <ν < 10-11 Hz) and of the order of 4 × 10-2 for ν > 10-11 Hz. These figures are comparable with the possible inaccuracies stemming from the calculation of the transfer function and, therefore, this is a further motivation, to use the transfer function of the spectral energy density. Finally the late time effects reduce a quantity which is already pretty small, i.e., as computed, (ν, τ0) ≃ 10-15 for ν ≫ νeq.
In the minimal realization of the ΛCDM scenario the scalar fluctuations of the geometry induce an E-mode polarization which has been observed and which is now subjected to closer scrutiny [3–7] The tensor modes of the geometry not only induce an E-mode polarization but also a B-mode polarization. The detected angular power spectra due to the presence of a putative (adiabatic) curvature perturbation are the temperature autocorrelation (TT angular power spectrum) the E-mode autocorrelation (EE angular power spectrum) and their cross correlation (i.e. the TE angular power spectrum). The various angular power spectra of the temperature and polarization observables have been already defined in section 2 (see, in particular, Eqs. (2.48)-(2.49) and discussions therein). Long wavelength gravitons contribute not only to the TT, EE and TE angular power spectra but also to the B-mode autocorrelations, i.e. the BB angular power spectra. The effect of long wavelength gravitons on the temperature and polarization observables can be studied by deriving the evolution equations of the brightness perturbations which are related, in loose terms, to the fluctuations of the Stokes parameters. The tensor nature of the fluctuation defined in Eq. (2.8) plays, in this respect, a decisive role. In particular the following two points should be borne in mind:
in the case of the scalar modes of the geometry the heat transfer equations have an azimuthal symmetry;
in the case of the tensor modes the fluctuations of the brightness do depend, both, upon μ = cos ϑ as well as upon φ ; this is ultimately, the rationale for the existence of a B-mode polarization.
where is the differential optical depth. In the differential optical depth enters not only the cross section but also the electron concentration and the ionization fraction xe. The notation for the differential optical depth varies: some authors prefer κ' some other . Given the notations used for the conformal time coordinate we will stick to the choice made in Eq. (7.1).
The right left side of Eq. (7.1) constitutes the collisionless term while the right hand side is the collisional contribution. At the right-hand side of Eq. (7.3) ℳ(Ω, Ω') is, in general, a matrix whose dimensionality depends upon the specific problem. As it will be shown ℳ(Ω, Ω') can be easily computed from Eq. (7.3). In similar terms f(Ω) should be understood as a column matrix whose components are the various Stokes parameters.
7.1 Collisionless Boltzmann equation for the tensor modes
Equations (7.17) and (7.18) are not symmetric for φ → -φ: while Eq. (7.17) is left unchanged, Eq. (7.18) acquires a minus sign. Conversely, the evolution equations of the scalar modes of the geometry are symmetric for φ → -φ.
7.2 Azimuthal structure of the collisional contribution
Equations (7.37) and (7.38) is just a useful warm-up in view of the realistic situation where:
the components of f(ϑ, φ) are not 2 but 3, i.e. ℐ x (ϑ, φ) and ℐ y (ϑ, φ) are supplemented by U (ϑ, φ);
all the 3 components of f(ϑ, φ) do depend upon φ ; in the analog of Eqs. (7.37) and (7.38) on top of the integration over μ' an integration over φ' will appear.
As in Eqs. (7.33)-(7.36), the second equality in each of Eqs. (7.45)-(7.49) follows immediately from Eqs. (7.27) and (7.28). A final remark on the symmetry properties of the various entries of ℳ(Ω, Ω') is in order:
ℳ11, ℳ12, ℳ21, ℳ22 and ℳ33 are all symmetric under the simultaneous transformation φ → -φ and φ' → -φ';
for the same transformation, the remaining entries flip their respective sign.
7.3 Different parametrizations of the full equation
Now the essential steps of the derivation are the following:
Eqs. (7.61)-(7.63) must be inserted, respectively, at left hand side of Eqs. (7.54)-(7.56);
Eqs. (7.70)-(7.72) must be inserted, respectively, at the right hand side of Eqs. (7.54)-(7.56).
As in the previous sections the and denote the time derivatives of the polarizations with respect to the conformal time coordinate τ. This notation has been avoided in the previous equations of the present section since it could have been confused with the angular variables describing the polarizations of the outgoing photons. From now on this possible clash of notations does not arise.
Concerning the result obtained in Eqs. (7.77) and (7.78) few comments are in order:
in Eqs. (7.77)-(7.78) denotes indifferently either or : the derivation reported in the case of h⊕ can be repeated in the case of h⊗ bearing in mind the differences in the angular structure (i.e. Eqs. (7.64)-(7.66) should be used instead of Eqs. (7.61)-(7.63));
by changing φ → -φ and φ' → -φ', Eqs. (7.61)-(7.63) and Eqs. (7.64)-(7.66) will be different because if the various sines appearing the various expressions; therefore the angular structure of the Stokes parameters will change but Eqs. (7.77) and (7.78) will keep their form.