Gravitational Waves: Generation and SourcesAlessandra BuonannoDepartment of Physics, University of MarylandAlessandra Buonanno March 27, 2007Lecture content• Generation problem• ApplicationsBinariesPulsarsSupernovaeStochastic background1Alessandra Buonanno March 27, 2007ReferencesLandau & Lifsh itz: Field Theory, Chap. 11, 13B. Schutz: A first course in general relativity, Chap. 8, 9S. Weinberg: Gravitation and Cosmology, Chap. 7, 10C. Misner, K.S. Thorne & A. Wheeler: Gravitation, Ch ap. 8S. Carroll, Spacetime and Geometry: An Introduction to GR, Chap. 7Course by K .S. Th orne available on the web: Lec tures 4, 5 & 6M. Maggiore: Gravitational waves: Theory and Experiments (2007)2Alessandra Buonanno March 27, 2007Relativistic units:G = 1 = c ⇒ Mass, space and time have same units1 sec ∼ 3 × 1010cm1M⊙∼ 5 × 10−6sec3Alessandra Buonanno March 27, 2007Multipolar decomposition of waves in linear gravity• Multipole expansion in terms of mass moments (IL) a nd mass-curr entmoments (JL) of the sourceh ∼can′t oscillatez}|{Gc2I0r+can′t oscillatez}|{Gc3˙I1r+mass quadrupolez}|{Gc4¨I2r+ ······ +Gc4˙J1r|{z}can′t oscillate+Gc5¨J2r|{z}current quadrupole+ ···• Typical strength: h ∼Gc4M L2P21r∼G(Ekin/c2)c2rIf Ekin/c2∼ 1M⊙, depending on r ⇒ h ∼ 10−23–10−174Alessandra Buonanno March 27, 2007Quadrupolar wave g eneration in linearized theoryryR R 0xx −y∂ρ∂ρ¯hµν= −16πGc4Tµν∂ν¯hµν= 0 ⇒like retarded potentials in E M:¯hµν(x)=4Gc4RTµν(y, t −Rc)d3y|x−y|R = |x − y| =pr2+ R20− 2r ·R0= R0q1 −2n·yR0+r2R20expanding in y/R0⇒ R ≃ R01 −n·yR0= R0− n · y¯hµν≃4Gc41R0RTµν(y, t −R0c+n·yc)d3y5Alessandra Buonanno March 27, 2007Quadrupolar wave g eneration in linearized theory [continued]ryR R 0xx −y•n·yccan be neglected if source mass distributiondoesn’t vary much du rin g t his time.If T typical time of variation of source⇒n·yc∼ac≪ T ∼λGWc⇒ λGW≫ a• Since T ∼av⇒ac≪av⇒vc≪ 1slow- moti on approximationFirst term in a multipolar expansion ( radiative zone λGW≪ R0):¯hµν≃4Gc41R0RTµν(y, t −R0c)d3y6Alessandra Buonanno March 27, 2007Quadrupolar wave g eneration in linearized theory [continued]For systems with significant self-gravity:¯hµν≃4Gc41R0R(Tµν+ τµν)(y, t −R0c)d3ytµν= Tµν+ τµνHere, we d isregard τµν, impose ∂νTµν= 0 and show thatRTijdV can be express ed only in terms of T007Alessandra Buonanno March 27, 2007Derivation of quadru pole formulaEq. (1):∂T0i∂xi−∂T00∂x0= 0Eq. (2):∂Tji∂xi−∂Tj0∂x0= 0multiplying Eq. (2) by xkand integrating on all spaceRxk∂Tji∂xidV =Rxk∂Tj0∂x0dV =∂∂x0RxkTj0dVintegrating by parts the LHS and assuming that the source decays sufficiently fast at ∞−RTjiδkidV =∂∂x0RxkTj0dVsymmetrizing ⇒RTkjdV = −12∂∂x0R(xkTj0+ xjTk0)dV8Alessandra Buonanno March 27, 2007Derivation of quadru pole formula [con tinued]Eq. (1):∂T0i∂xi−∂T00∂x0= 0Eq. (2):∂Tji∂xi−∂Tj0∂x0= 0multiplying Eq. (1) by xkxjand integrating on all space∂∂x0RT00xkxjdV =R∂T0i∂xixkxjdVintegrating by parts the R HS and assuming that the source decays sufficiently fast at ∞∂∂x0RT00xkxjdV = −R(xkTj0+ xjTk0)dVcombining ⇒RTkjdV =12c2∂2∂t2RT00xkxjdV9Alessandra Buonanno March 27, 2007Derivation of quadru pole formula [con tinued]T00= µ c2⇒¯hij=2Gc41R0∂2∂t2Rµ xkxjdVOther components of¯hµνare no n-radiative fields:¯h00=4Gc21R0ZµdV|{z }M¯h0k=4Gc31R0∂∂tZµ xkdV| {z }PIn TT gauge:hTTij=2Gc41R0PkiPlj¨Qklwith Qkl=Rd3xρxkxl−13x2δklPik= δik− nink10Alessandra Buonanno March 27, 2007¯h0kcomponentEq. (1):∂T0i∂xi−∂T00∂x0= 0Eq. (2):∂Tji∂xi−∂Tj0∂x0= 0multiplying Eq. (1) by xkand integrating on all space∂∂x0RT00xkdV =R∂T0i∂xixkdVintegrating by parts the R HS and assuming that the source decays sufficiently fast at ∞RT0kdV = −∂∂x0RT00xkdV11Alessandra Buonanno March 27, 2007Total power radiated in GWsPower radiated per unit solid angle in the direction n:dPdΩ= R20niτi0with τi0=c432πG∂0¯hβα∂i¯hαβ•dPdΩ=G8π c5(...Qijǫij)2for a given polarizationǫkk= 0 ǫklnk= 0 ǫklǫkl= 0•LGW≡ P =G5 c5(...Qij)2averaging over polarizations12Alessandra Buonanno March 27, 2007Useful relationsninj=13δijninjnknl=15(δijδkl+ δikδjl+ δilδjk)2 ǫijǫkl= 214{ninjnknl+ ninjδkl+ nknlδij−(ninkδjl+ njnkδil+ ninlδjk+ njnlδik)−δijδkl+ δikδjl+ δjkδil}13Alessandra Buonanno March 27, 2007Comparison between GW and EM luminosityLGW=G5c5(...I2)2I2∼ ǫ M R2R → typical source’s dimension, M → source’s mass, ǫ → deviation from sphericity...I2∼ ω3ǫ M R2with ω ∼ 1/P ⇒ LGW∼Gc5ǫ2ω6M2R4LGW∼c5Gǫ2G M ωc36R c2G M4⇒c5G= 3.6 × 1059erg/sec (huge!)• For a steel rod of M = 490 tons, R = 20 m and ω ∼ 28 rad/sec:GMω/c3∼ 10−32, Rc2/GM ∼ 1025→ LGW∼ 10−27erg/sec ∼ 10−60LEMsun!• As Weber noticed in 1972, if we introduce RS= 2GM/c2and ω = (v/c) (c/R)LGW=c5Gǫ2vc6RSR⇒|{z}v∼c, R∼RSLGW∼ ǫ2c5G∼ 1026LEMsun!14Alessandra Buonanno March 27, 2007GWs on the Eart h: comparison with other kind o f radiationSupernova at 20 kpc:• From GWs: ∼ 400ergcm2secfGW1kHz2h10−212during few msecs• From neutrino: ∼ 105ergcm2secduring 10 secs• From optical radiation: ∼ 10−4ergcm2secduring one week15Alessandra Buonanno March 27, 2007GW frequ ency spectru m extends over many decadesCMBPulsar timingLISAbars/LIGO/VIRGO/...10−1610 10 Hz−210−8h 16Alessandra Buonanno March 27, 2007Detecting GWs from comparable-mass BHs with LIGOMBH= 5–20M⊙or larger mass es if IMBH exists20 406080 100 120 140160180 200Binary total mass0369Signal-to-Noise RatioEqual massesat 100 Mpc10 100 1000f (Hz)10-2410-2210-2010-18Sh1/2 (Hz-1/2)Hanford 2kmHanford 4kmLivingston 4kmLIGO design (4km)(15+15) MsunfISCO= 4400/(m/M⊙) Hz17Alessandra Buonanno March 27, 2007Gravitational waves from compact binaries• Mass-quadrupole approximation: hij∼Grc4¨IijIij= µ (XiXj− R2δij)h ∝M5/3ω2/3rcos2Φfor quasi-circular orbits: ω2∼˙Φ2=GMR3Chirp: The signal continuously changesits frequency and the power emittedat any frequency is very small!chirptimehh ∼M5/3f2/3rfor f ∼ 100 Hz, M = 20M⊙r at 20 Mpc⇒ h ∼ 10−2118Alessandra Buonanno March 27, 2007Typical features of coalescing black-hole binaries050100150200250300t/M-0.050.000.05ψ4(from Pretorius 06)•Inspiral: q