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

Inflation 227the lower bound to the extension of the spectrum. To find the upper boundone must note that in our model the transition between the different regimestakes place instantaneously; in a more realistic situation such transitions havea typical duration time t, and consequently there is a cut-off physical frequencyof order 1/t. In our case the typical time of the transitions is of the order ofthe Hubble parameters at the transition points (H (η 1 ), H (η 2 )). This means thatfor k > k 2 = R2 ′ /R 2, i.e. for f ≥ 10 −16 Hz, equation (13.25) is not the correctexpression for N k ; above this frequency the radiation-matter transition does notaffect the spectrum, but the inflation-radiation one is still important. Hence,the number of gravitons created is given by the corresponding equation, with δreplaced by β( )HN k =|β k | 2 2= dsR(η 1 ) 2 22k 2 . (13.26)In turn, equation (13.26) has a high frequency cut-off given by k 1 = R1 ′ /R 1 =−1/η 1 ; η 1 , and consequently the corresponding value of the physical cut-offfrequency f 1 , is an almost free parameter of the model and depends on thereheating temperature. A typical value for f 1 is the one reported in [46] ( f 1 ∼10 10 Hz), but other (also much lower) values are possible, depending on themodel; in general one has f 1 > 1 kHz. Beyond f 1 there is no amplificationmechanism at all and the spectrum goes rapidly to zero.Once obtained the result for N k , it is straightforward to obtain its contributionto ρ gw , and consequently to gwdρ gw ( f ) = 2 · 2π f · N k ( f ) · 4π f 2 d f,where the factor of two is due to polarization, and f = k/2π R 0 is the physicalfrequency. The final result for gw in terms of the physical frequency is then⎧ ( )3H3 2 ( ) 2 ( ) 6ds R1 R1 18⎪⎨ 16π R 2 R 0 f 2 , k 0 < 2π f < k 2 gw ( f ) =3π H0 2M2 H 4 ( ) 4Pl ds R1, k 2 < 2π f < k 1⎪⎩ 4 R 00, 2π f > k 1(13.27)where M Pl is the Planck mass and H 0 is the present value of the Hubble parameter.Because of the flatness of gw ( f ) the strongest constraint turns out to be theone related to the observed anisotropy in CMBR. This constrains the spectrum asfollows( ) 2h 2 0 gw ≤ 7 × 10 −11 H0for f ∈ (10 −19 , 10 −16 ) Hz, (13.28)fwhich in turn impliesH ds ≤ 10 39 Hz ≃ 5 × 10 14 Gev ≃ 5 × 10 −5 M Pl .