Gravitational Waves: Generation and Sources. Alessandra Buonanno Department of Physics, University of Maryland

Transcription

1 Gravitational Waves: Generation and Sources Alessandra Buonanno Department of Physics, University of Maryland

2 Lecture content Generation problem Applications Binaries Pulsars Supernovae Stochastic background 1

3 References Landau & Lifshitz: Field Theory, Chap. 11, 13 B. Schutz: A first course in general relativity, Chap. 8, 9 S. Weinberg: Gravitation and Cosmology, Chap. 7, 10 C. Misner, K.S. Thorne & A. Wheeler: Gravitation, Chap. 8 S. Carroll, Spacetime and Geometry: An Introduction to GR, Chap. 7 Course by K.S. Thorne available on the web: Lectures 4, 5 & 6 M. Maggiore: Gravitational waves: Theory and Experiments (2007) 2

4 Relativistic units: G = 1 = c Mass, space and time have same units 1sec cm 1M sec 3

5 Multipolar decomposition of waves in linear gravity Multipole expansion in terms of mass moments (I L ) and mass-current moments (J L ) of the source h can toscillate {}}{ G I 0 c 2 r + G J 1 c 4 r + }{{} can toscillate + can toscillate {}}{ G I 1 c 3 r G J2 } c 5 {{ r} + current quadrupole mass quadrupole {}}{ G Ï 2 c 4 r + + Typical strength: h G M c 4 L2 1 P 2 r G(E kin/c 2 ) c 2 r If E kin /c 2 1M, depending on r h

6 Quadrupolar wave generation in linearized theory ρ ρ hµν = 16πG c 4 T µν ν hµν = 0 like retarded potentials in EM: r y R 0 x h µν (x) = 4G c 4 Tµν (y,t R c ) d3 y x y R x y R = x y = r 2 + R0 2 2r R 0 = R 0 1 2n y R 0 + r2 R0 2 ( ) expanding in y/r 0 R R 0 1 n y R 0 = R 0 n y h µν 4G c 4 1 R 0 Tµν (y,t R 0 c + n y c )d3 y 5

7 n y c Quadrupolar wave generation in linearized theory [continued] can be neglected if source mass distribution doesn t vary much during this time. r y R 0 x If T typical time of variation of source R x y n y c a c T λ GW c λ GW a Since T a v a c a v v c 1 slow-motion approximation First term in a multipolar expansion ( radiative zone λ GW R 0 ): h µν 4G c 4 1 R 0 Tµν (y,t R 0 c )d 3 y 6

8 Quadrupolar wave generation in linearized theory [continued] For systems with significant self-gravity: h µν 4G c 4 1 R 0 (Tµν + τ µν )(y,t R 0 c )d 3 y t µν = T µν + τ µν Here, we disregard τ µν, impose ν T µν = 0 and show that Tij dv can be expressed only in terms of T 00 7

9 Eq. (1): Derivation of quadrupole formula T 0i x i T 00 x 0 = 0 Eq. (2): T ji x i T j0 x 0 = 0 multiplying Eq. (2) by x k and integrating on all space x k T ji x i dv = x k T j0 x 0 dv = x 0 x k T j0 dv integrating by parts the LHS and assuming that the source decays sufficiently fast at T ji δ k i dv = x 0 x k T j0 dv symmetrizing T kj dv = 1 2 x 0 (xk T j0 + x j T k0 )dv 8

10 Eq. (1): Derivation of quadrupole formula [continued] T 0i x i T 00 x 0 = 0 Eq. (2): T ji x i T j0 x 0 = 0 multiplying Eq. (1) by x k x j and integrating on all space x 0 T00 x k x j dv = T 0i x i x k x j dv integrating by parts the RHS and assuming that the source decays sufficiently fast at x 0 T00 x k x j dv = (x k T j0 + x j T k0 )dv combining T kj dv = 1 2c 2 2 t 2 T00 x k x j dv 9

11 Derivation of quadrupole formula [continued] T 00 = µc 2 h ij = 2G c 4 1 R 0 2 t 2 µxk x j dv Other components of h µν are non-radiative fields: h 00 = 4G 1 c 2 R 0 µdv }{{} M In TT gauge: h TT ij h 0k = 4G 1 c 3 R 0 µx k dv } t {{} P with Q kl = d 3 xρ ( x k x l 1 3 x2 δ kl ) = 2G c 4 1 R 0 P k i Pl j Q kl P ik = δ ik n i n k 10

12 h 0k component Eq. (1): Eq. (2): T 0i x i T 00 x 0 = 0 T ji x i T j0 x 0 = 0 multiplying Eq. (1) by x k and integrating on all space x 0 T00 x k dv = T 0i x i x k dv integrating by parts the RHS and assuming that the source decays sufficiently fast at T0k dv = x 0 T00 x k dv 11

13 Total power radiated in GWs Power radiated per unit solid angle in the direction n: dp dω = R2 0 n i τ i0 with τ i0 = c4 32πG 0 h β α i hα β dp dω = G 8π c 5 (... Q ij ǫ ij ) 2 for a given polarization ǫ kk = 0 ǫ kl n k = 0 ǫ kl ǫ kl = 0 L GW P = G 5c 5 (... Q ij ) 2 averaging over polarizations 12

14 Useful relations n i n j = 1 3 δ ij n i n j n k n l = 1 5 (δ ij δ kl + δ ik δ jl + δ il δ jk ) 2 ǫ ij ǫ kl = {n i n j n k n l + n i n j δ kl + n k n l δ ij (n i n k δ jl + n j n k δ il + n i n l δ jk + n j n l δ ik ) δ ij δ kl + δ ik δ jl + δ jk δ il } 13

15 Comparison between GW and EM luminosity L GW = G 5c 5 (... I 2) 2 I 2 ǫm R 2 R typical source s dimension, M source s mass, ǫ deviation from sphericity... I 2 ω 3 ǫ M R 2 with ω 1/P L GW G c 5 ǫ 2 ω 6 M 2 R 4 L GW ( ) c5 G ǫ2 G M ω 6 ( c 3 R c 2 G M ) 4 c 5 G = erg/sec (huge!) For a steel rod of M = 490 tons, R = 20 m and ω 28 rad/sec: GMω/c , Rc 2 /GM L GW erg/sec L EM sun! As Weber noticed in 1972, if we introduce R S = 2GM/c 2 and ω = (v/c) (c/r) L GW = ( ) ( ) c5 G ǫ2 v 6 RS c R }{{} v c, R R S L GW ǫ 2 5 c G 1026 L EM sun! 14

16 GWs on the Earth: comparison with other kind of radiation Supernova at 20 kpc: From GWs: 400 erg cm 2 sec ( ) 2 fgw ( h ) 2 1kHz 10 during few msecs 21 From neutrino: 10 5 erg cm 2 sec during 10 secs From optical radiation: 10 4 erg cm 2 sec during one week 15

17 GW frequency spectrum extends over many decades h LISA Pulsar timing CMB bars/ligo/virgo/ Hz 16

18 Detecting GWs from comparable-mass BHs with LIGO M BH = 5 20M or larger masses if IMBH exists Signal-to-Noise Ratio 6 3 Equal masses at 100 Mpc S h 1/2 (Hz -1/2 ) (15+15) Msun Hanford 2km Hanford 4km Livingston 4km LIGO design (4km) Binary total mass f (Hz) f ISCO = 4400/(m/M ) Hz 17

19 Gravitational waves from compact binaries Mass-quadrupole approximation: h ij G rc 4 Ïij I ij = µ (X i X j R 2 δ ij ) h M5/3 ω 2/3 r cos 2Φ for quasi-circular orbits: ω 2 Φ 2 = GM R 3 h Chirp: The signal continuously changes chirp its frequency and the power emitted at any frequency is very small! time h M5/3 f 2/3 r for f 100 Hz, M = 20M r at 20 Mpc h

20 Typical features of coalescing black-hole binaries Inspiral: quasi-circular orbits Throughout the inspiral T RR T orb natural adiabatic parameter For compact bodies v2 GM c 2 c 2 r Chirping if T obs > ω/ ω ( ) ω v = O 5 ω 2 c 5 PN approximation: slow motion and weak field Inspiral: spin-precessing orbits T RR T prec T orb ; ω GW = {ω prec, 2ω} ψ Inspiral: eccentric orbits T RR T peri prec T orb ; ω GW = {N ω, } (from Pretorius 06) Last cycles-plunge-merger-ringdown t/m Numerical relativity; close-limit approx.; PN resummation techniques 19

21 Waveforms including spin effects Maximal spins M = 10M + 10M S 1 L Newt S h 0.00 h time time 20

22 Inspiral signals are chirps GW signal: chirp [duration seconds to years] (f GW 10 4 Hz 1kHz) NS/NS, NS/BH and BH/BH MACHO binaries (m < 1M ) [MACHOs in galaxy halos < 3 5%] GW frequency at end-of-inspiral (ISCO): f GW 4400 M M Hz ψ LIGO/VIRGO/... : M = 1 50M (from Pretorius 06) LISA: M = M t/m 21

23 Radial potential in black-hole spacetime 1.0 radial potential 0.9 Innermost Stable Circular Orbit (ISCO) radial separation 22

24 GW templates through 2PN order for binaries moving along circular orbits h(f) = A f 7/6 e iψ(f) ψ(f) = 2πft c φ c + 3 ( 128 (πmf) 5/3 1 5ˆα2 η 2/5 (πmf) 2/3 128 π 2 D M 336ω BD 3 λ 2 (πmf)2/3 g (1 + z) «9 η η 2/5 (πmf) 2/3 16π η 3/5 (πmf)+4β η 3/5 (πmf) η «ff 72 η2 η 4/5 (πmf) 4/3 10σ η 4/5 (πmf) 4/3 β = 1 12 " # 2X χ i 113 m2 i M η i=1 bl bs i, σ = η 48 χ 1χ 2 27 b S 1 bs b L bs 1 b L b S2 23

25 Inspiral: number of GW cycles predicted by PN theory M = ( )M f in = 40 Hz; f fin = 147 Hz χ = S /m 2 Number of cycles Newtonian: 1PN: 1.5PN Spin-orbit: +11.7χ χ 2 2PN +3.3 Spin-spin: 1.7χ 1 χ 2 2.5PN χ χ 2 3PN: PN:

26 Binary coalescence time E = 1 2 µv2 Gµ M r = Gµ M 2r r = Gµ M 2E ṙ = dr de de dt = 64 5 Gµ M 2 r 3 integrating r(t) = ( r GµM2 τ coal ) 1/4 If r(t f ) r 0 τ coal = 5 r Gµ M 2 Examples: LIGO/VIRGO/GEO/TAMA source: M = ( )M at r km, f GW 40Hz, T sec τ coal 1 sec LISA source: M = ( )M at r km, f GW Hz, T 0 11 hours τ coal 1 year 25

27 NR simulations for equal-mass binaries: quasi-circular evolution y i /m initial pre-merger AH shapes final pre-merger AH shapes initial enveloping AH shape late time AH shape est. late time co-rotating light ring x i /m [similar results from other NR goups] ṙ / (ω r) -16/5 (r/m) -5/2-16/5 (mω) 5/ (t - t CAH )/m [AB, Cook & Pretorius 06] 26

28 Equal-mass binary: one dominant frequency ω c from the coordinate separation ω c ω wave ω wave from the wave Decoupling between orbital frequency frequency and GW mω (t-t peak )/m [AB, Cook & Pretorius 06] 27

29 How well the Newtonian quadrupole formula works 0.01 numerical waveform numerical orbit, quadrupole formula wave (t-t peak )/m [AB, Cook & Pretorius 06] 28

30 Comparison NR and PN-adiabatic model The initial frequency ω NR /m (e.g., for a ( )M, f GW 70 Hz) [AB, Cook & Pretorius 06] 0.25 numerical relativity PN inspiral numerical relativity PN inspiral mω 0.15 ISCO PN wave ISCO Schw (t - t CAH )/m (t - t CAH )/m 29

31 Comparison NR and PN-adiabatic model: 16 cycles [Baker et al. 06 (NASA)] rh (M) time (M)

32 Comparison NR and effective-one-body model M end = 0.97 m and a end /M end = 0.78 [AB, Cook & Pretorius 06] Fundamental mode and two overtones included numerical relativity EOB inspiral-merger EOB ring-down numerical relativity EOB inspiral-merger EOB ring-down mω 0.15 wave (t - t CAH )/m (t - t CAH )/m 31

33 Detectability for ground-based detectors [AB, Cook & Pretorius 06;Baker et al. 06] h ~ (f) mω Numerical relativity f -7/6 f -2/3 SNR at 100 Mpc Numerical relativity PN inspiral EOB inspiral-merger-ring-down f (Hz) m (M sun ) Change of slope: f 7/6 f 2/3 increase of SNR for m > 40M 32

34 Gravitational waves from pulsars Body rotating rigidly around the x 3 principal axis with frequency ω s x 1, x 2, x 3 coordinate system fixed to the body x 1 = x 1 cos ω s t + x 2 sin ω s t x 3 x 2 = x 1 sin ω s t x 2 cos ω s t x 1 x 2 Q ij = R ρ x i x j d 3 x and I ij = R ρ(r 2 δ ij x i x j ) d 3 x Q 11 = Q 22 = 1 2 (I 1 I 2 ) cos 2ω s t Q 12 = 1 2 (I 1 I 2 ) sin 2ω s t Q 33 = I 3, Q 13 = Q 23 = 0 h 2ω2 s r (I 1 I 2 ) cos 2ω s t ǫ (I 1 I 2 )/I 3 33

35 Gravitational waves from spinning neutron stars: pulsars GW signal: (quasi) periodic (f GW 10 Hz 1 khz) Pulsars: non-zero ellipticity (or oblateness) h GW ( ) ( ) ( 26 ǫ I 3 10 kpc g cm 2 r ) ( ) 2 fgw 1 khz ǫ = I 1 I 2 I 3 ellipticity The crust contributes only 10% of total moment of inertia ǫ C is low Magnetic fields could induce stresses and generate ǫ M 0 Expected ellipticity rather low 10 7, unless exotic EOS are used search for known spinning neutron stars: Vela, Crab,... all sky search 34

36 (screensaver) Partecipate in LIGO pulsar data analysis by signing up! (B Allen, Univ. of Winsconsin, Milwakee) 35

37 Gravitational waves from stellar collapse GW signal: bursts [ few mseconds] or (quasi) periodic (f GW 1 khz 10kHz) Supernovae: Non-axisymmetric core collapse Material in the stellar core may form a rapidly rotating bar-like structure Collapse material may fragment into clumps which orbit as the collapse proceeds Pulsation modes of new-born NS; ring-down of new-born BH Dynamics of star very complicated GW amplitude and frequency estimated using mass- and current-quadrupole moments Numerical simulations Correlations with neutrino flux and/or EM counterparts Event rates in our galaxy and its companions < 30 yrs 36

38 Gravitational-wave strain from non-axisymmetric collapse h GW η eff ( 1 msec τ τ duration of emission ) 1/2 ( efficiency η eff = E M c ) 1/2 ( ) ( ) M 10 kpc 1 khz M r f GW 37

39 Summary of sources with first-generation ground-based detectors S h 1/2 (Hz -1/2 ) VIRGO NS/NS at 20 Mpc Crab (upper limit) GEO TAMA bars LIGO known pulsar at 10kpc ε=10-5 ε= f (Hz) Upper bound for NS-NS (BH-BH) coalescence with LIGO: 1/3yr (1/yr) 38

40 Advanced LIGO/VIRGO Higher laser power lower photon-fluctuation noise Heavier test masses 40 Kg lower radiation-pressure noise Better optics to reduce thermal noise Better suspensions and seismic isolation systems Signal-recycling cavity: reshaping noise curves 39

41 Summary of sources for second-generation ground-based detectors Sensitivity improved by a factor 10 event rates by Vela (upper limit) Crab (upper limit) S h 1/2 (Hz -1/2 ) BH/BH at 100Mpc NS/NS at 300Mpc Advanced LIGO (narrowband) Advanced LIGO (wideband) Low-mass X-ray binaries pulsar ε =10-6 at 10kpc f (Hz) Advanced Bars SCOX-1 Upper bound for NS-NS binary with Advanced LIGO: a few/month 40

42 GWs in curved space-time S = 1 16π G N d 4 x g R + S matter Isotropic and spatially homogenous FLRW background ds 2 = g µν dx µ dx ν = dt 2 + a 2 (t) d x 2 = a 2 (η) ( dη 2 + d x 2 ) Metric perturbations (δg µν = h µν ): h 2a k (η) + a h k (η) + k2 h k (η) = 0 Introducing the canonical field ψ k (η) = a h k (η) : ψ k + [ k 2 U(η) ] ψ k = 0 U(η) = a a 41

43 Semiclassical point of view Introducing the canonical field ψ k (η) = a h k (η) : ψ k + [ k 2 U(η) ] ψ k = 0 U(η) = a a desitter-like inflationary era: a = 1/(η H ds ) ˆ U(η) 1/η 2, (a H ds ) 1/η If k 2 U(η) ˆkη 1, k/a HdS, λ phys H 1 ds the mode is inside the Hubble radius ψ k e ±ik η h k 1 a e±ik η If k 2 U(η) : ˆkη 1, k/a HdS, λ phys H 1 ψ k a [ ] A k + B dη k a 2 (η) ds the mode is outside the Hubble radius h k A k + B k dη a 2 (η) 42

44 Amplification of quantum-vacuum fluctuations: semiclassical point of view h k 1 a e±ik η U( η ) h k A k + B k dη a 2 (η) η 43

45 Example: Stochastic GW background from slow-roll inflation Too low to be detected by first generations GW interferometers [Smith, Kamionkowsky and Cooray 05] 44