(ed.). Gravitational waves (IOP, 2001)(422s).

The relic graviton background 309Table 16.3. Slope of the graviton spectrum.Scale factor Bessel index Spectrumde Sitter, constant curvature α =−1 3− 2ν = 0 flatPower-inflation, decreasing curvature α0 increasingIt is important to stress that the amplitude of the spectrum is controlledby g 1 = H 1 /M p , i.e. by the curvature scale in Planck units at the time of thetransition t 1 (a fundamental parameter of the given inflationary model). The slopeof the spectrum, 3−2ν, is instead controlled by the kinematics of the background.In fact, it depends on the Bessel index ν which, in its turn, depends on α, the powerof the scale factor (equation (16.89). The behaviour in frequency of the gravitonspectrum, in particular, tends to follow the behaviour of the curvature scale duringthe epoch of accelerated evolution, see table 16.3.The standard inflationary scenario is thus characterized by a flat ordecreasing graviton spectrum; in string cosmology, instead, we must expect agrowing spectrum. This has important phenomenological implications, that willbe discussed in the following section.16.6 The relic graviton backgroundAs discussed in the previous section, one of the most firm predictions of allinflationary models is the amplification of the traceless-transverse part of thequantum fluctuations of the metric tensor, and the formation of a primordial,stochastic background of relic gravitational waves, distributed over a quite largerange of frequencies (see [62] for a discussion of the stochastic properties of sucha background, and [63] for a possible detection of the associated ‘squeezing’[64]).In a string cosmology context, the expected graviton background has beenalready discussed in a number of detailed review papers [59,65,66]. Here we willsummarize the main properties of the background predicted in the context of thepre-big bang scenario.For a phenomelogical discussion of the spetrum, it is convenient to considerthe plane { G ,ω}. In this plane there are three main phenomenologicalconstraints:• A first constraint comes from the large scale isotropy of the CMB radiation.The degree of anisotropy measured by COBE imposes a bound on the energydensity of the graviton background at the scale of the present Hubble radius[67], G (ω 0 ) º 10 −10 , ω 0 ∼ 10 −18 Hz. (16.100)