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

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The overlap reduction function 193Table 12.1. The normalization factor F for three different geometries of the detectors:interferometer (ITF), cylindrical bar (BAR) and sphere (SPH). A ⋆ denotes entries that canbe obtained from the symmetry of the table.SPHITF BAR m = 0 m =±1 m =±2ITF 2/5 ⋆ ⋆ ⋆ ⋆BAR 2/5 8/15 ⋆ ⋆ ⋆m = 0 0 2 √ 3/15 2/5 ⋆ ⋆SPH m =±1 0 0 0 2/5 ⋆m =±2 2/5 2/5 0 0 2/5where[ ρ0]ρ 1 (η) = 1[ 2η2 ][ ]−4η 2 j0ρFη 2 −4η 2 16η −20 j 1 (η), (12.45)2 η 2 −10η 35 j 2with j k (η) the standard spherical Bessel functions:j 0 (η) = sin ηη , j 1(η) = j 0(η) − cos η, j 2 (η) = 3 j 1(η)− j 0 (η).ηηIn the following, this formalism will be applied to treat the case in whichthe first detector is an interferometer and the other is, in turn, an interferometer, acylindrical bar and a sphere.12.3.1 Two interferometersFollowing equation (12.40), the detector tensor of the interferometer located at agiven point ⃗r k can be written asD k = 1 4 {(cos 2ξ k − cos 2ζ k )e + (ˆr k ) − (sin 2ξ k − sin 2ζ k )e × (ˆr k )} (k = 1, 2)(12.46)where ξ k and ζ k are the orientations of the kth interferometer arms measuredanticlockwise from the true north. The value of these angles, together withthe location of the central (corner) station on the Earth’s surface, for differentinterferometers are reported in table 12.2. By inserting this expression inequation (12.44) it is possible to compute the overlap reduction function for ageneric pair of interferometers. The results of this calculation in the case in whichone of the interferometers is VIRGO are shown in figure 12.1.Some important conclusions can be drawn from the inspection of this figure.First of all, owing to the oscillating behaviour with η ∝ fdof the spherical Bessel
194 Generalities on the stochastic GW backgroundTable 12.2. Site and orientation of Earth-based interferometric gravitational-wavedetectors. The latitude is measured in degrees north from the equator, and the longitudein degrees east of Greenwich, England. In the last column is reported the distance withrespect to VIRGO.Corner locationDistanceProject Latitude N (deg) Longitude E (deg) ξ (deg) ζ (deg) (km)VIRGO 43.63 10.50 71.5 341.5 —LIGO-LA 30.56 −90.77 108.0 198.0 7905.9LIGO-WA 46.45 −119.41 36.8 126.8 8158.2GEO-600 52.25 9.82 25.94 291.61 958.2TAMA-300 35.68 139.54 90.0 180.0 8864.1Figure 12.1. The overlap reduction function in the case of correlation between VIRGOand the other major interferometers.
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GRAVITATIONAL WAVES
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c IOP Publishing Ltd 2001All righ
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viContents4 Astrophysics of gravita
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viiiContents11.3 Gravitational wave
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xContents16.5 Cosmological perturba
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PrefaceGravitational waves today re
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2 Gravitational waves, theory and e
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10 Gravitational waves, theory and
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SynopsisGravitational waves and the
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Chapter 2Elements of gravitational
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Using the TT gauge to understand gr
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Interaction of gravitational waves
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Analysis of beam detectors 21This d
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Exercises for chapter 2 232. Show t
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Gravitational-wave detectors 25indi
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Gravitational-wave observables 27th
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The physics of interferometers 29
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The physics of interferometers 31Dr
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The physics of interferometers 33Fi
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The physics of resonant mass detect
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The physics of resonant mass detect
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A detector in space 39LISA has been
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Gravitational and electromagnetic w
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Chapter 4Astrophysics of gravitatio
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Sources detectable from ground and
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Variational principles and the ener
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Variational principles and the ener
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Practical applications of the Isaac
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Exercises for chapter 5 57(f)This s
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Expansion for the far field of a sl
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Application of the TT gauge to the
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6.4.1 Mass-quadrupole radiationEner
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Chapter 7Source calculationsNow tha
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The r-modes 73As we mentioned in ch
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The r-modes 75instability. In an id
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The r-modes 77Table 7.1. Gravitatio
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The r-modes 79evolution turns out t
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ReferencesWe have divided the refer
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References 83[19] van der Klis M 19
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Solutions to exercises 85For this p
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Solutions to exercises 87The gauge
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Chapter 8Resonant detectors for gra
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Sensitivity and bandwidth of resona
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8.2 Sensitivity for various GW sign
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Sensitivity for various GW signals
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Recent results obtained with the re
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Discussion and conclusions 101Figur
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Chapter 9The Earth-based large inte
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Introduction 105Figure 9.1. The sim
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h(sqrt(1/Hz))10 −610 −810 −10
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18.5 cmThe SA suspension and requir
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A few words about the Low Frequency
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Conclusion 113Figure 9.7. The R and
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Chapter 10LISA: A proposed joint ES
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Description of the LISA mission 117
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Expected gravitational-wave results
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Page 153 and 154: Expected gravitational-wave results
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Page 165 and 166: Introduction 153detect GWs with a s
Page 167 and 168: Testing theories of gravity 155•
Page 169 and 170: 0Testing theories of gravity 157the
Page 171 and 172: Taking the scalar product we findGr
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Page 181 and 182: The hollow sphere 169and introduced
Page 183 and 184: Scalar-tensor cross sections 171mon
Page 185 and 186: Scalar-tensor cross sections 173Tab
Page 187 and 188: Scalar-tensor cross sections 175Tab
Page 189 and 190: References 177Frossati G and Coccia
Page 192 and 193: Chapter 12Generalities on the stoch
Page 194 and 195: Introduction 183contribution to the
Page 196 and 197: Definitions 185The value of H 0 is
Page 198 and 199: Definitions 18712.2.2 The character
Page 200 and 201: Definitions 189or, dividing by the
Page 202 and 203: The overlap reduction function 191O
Page 206 and 207: The overlap reduction function 195F
Page 208 and 209: Achievable sensitivities to the SGW
Page 218 and 219: Observational bounds 207Table 12.10
Page 220 and 221: equation gives immediately(ρgwρ
Page 222 and 223: Chapter 13Sources of SGWBHere we re
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Page 228 and 229: Topological defects 217(i)A loop ra
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Page 234 and 235: Inflation 223Hubble radius and when
Page 236 and 237: Inflation 225as a first approximati
Page 238 and 239: Inflation 227the lower bound to the
Page 240 and 241: String cosmology 22913.3 String cos
Page 242 and 243: String cosmology 231HPre-big bangPo
Page 244 and 245: String cosmology 233Figure 13.6. h
Page 246 and 247: First-order phase transitions 23580
Page 248 and 249: Astrophysical sources 237which is t
Page 250 and 251: References 239of [71]). However, fo
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PART 4THEORETICAL DEVELOPMENTSHerma
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Chapter 15Gyroscopes and gravitatio
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270 Gyroscopes and gravitational wa
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Chapter 16Elementary introduction t
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282 Elementary introduction to pre-
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340 Post-Newtonian computation of b
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PART 5NUMERICAL RELATIVITYEdward Se
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362 Numerical relativityspite of mo
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364 Numerical relativityterms, of m
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366 Numerical relativityHere we hav
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368 Numerical relativityinformation
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370 Numerical relativityThe Palma,
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372 Numerical relativityHyperbolic
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374 Numerical relativityEinstein di
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376 Numerical relativityare the abi
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378 Numerical relativityeach cell i
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380 Numerical relativityFigure 18.1
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382 Numerical relativity1D code [45
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384 Numerical relativityfunctions,
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386 Numerical relativityto a very a
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388 Numerical relativity18.5 Comput
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390 Numerical relativitynumerics an
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392 Numerical relativityinitial dis
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396 Numerical relativityof these re
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400 Numerical relativityof Einstein
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402 Numerical relativity18.9.5.2 Co
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404 Numerical relativity[23] Ashby
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406 Numerical relativity[101] Shapi
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408 Numerical relativity(Singapore:
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410 IndexEhlers Lagrangian, 254Eins
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412 Indexscalarspherical harmonics,
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