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

170 Detection of scalar gravitational waveswith(z) ≡ j 2(q n0 z)q n0 R(z) ≡ j 2(q n2 z)q n2 R+ D y 2 (q n0 z)n0q n0 R− 3K n2j 2 (k n2 z)k n2 R(11.98)+ D y 2 (q n2 z)n2q n2 R − 3 y 2 (k n2 z)˜D n2k n2 R . (11.99)The absorption cross section, defined as the ratio of the absorbed energyto the incoming flux, can be calculated thanks to an optical theorem, as proved,for example, by Weinberg [26]. According to that theorem, the absorption crosssection for a signal of frequency ω close to ω N , say, the frequency of the detectormode excited by the incoming GW, is given by the expressionσ(ω) = 10πηc2ω 2 Ɣ 2 /4(ω − ω N ) 2 + Ɣ 2 /4(11.100)where Ɣ is the linewidth of the mode—which can be arbitrarily small, as assumedin the previous section—and η is the dimensionless ratioη = Ɣ gravƔ= 1 ƔP GWE osc(11.101)where P GW is the energy re-emitted by the detector in the form of GWs as aconsequence of it being set to oscillate by the incoming signal. In the followingwe will only consider the case P GW = P scalar−tensor with [6, 9]P scalar−tensor =2Gω 6 [5c 5 |Q kk (ω)| 2 + 1 ](2ω BD + 3)3 Q∗ ij (ω)Q ij(ω)(11.102)where Q ij (ω) is the quadrupole moment of the hollow sphere:∫Q ij (ω) = x i x j ϱ(x,ω)d 3 x (11.103)and ω BD is Brans–Dicke’s parameter.Antenna11.5 Scalar–tensor cross sectionsExplicit calculation shows that P scalar−tensor is made up of two contributions:P scalar−tensor = P 00 + P 20 (11.104)where P 00 is the scalar, or monopole contribution to the emitted power, while P 20comes from the central quadrupole mode which, as discussed in [6] and [9], isexcited together with monopole in JBD theory. One must, however, recall that