Gravitational Waves. Kostas Kokkotas. Department of Physics Aristotle University of Thessaloniki

Transcription

1 Gravitational Waves Kostas Kokkotas Department of Physics Aristotle University of Thessaloniki

2 ALLEGRO AURIGA EXPLORER NAUTILUS NIOBE 8/6-9/7/4 ISAPP-4

3 Grav. Waves: an international dream GEO6 (British-German) Hannover, Germany TAMA (Japan) Mitaka LIGO (USA) Hanford, WA & Livingston, LA AIGO (Australia), Wallingup Plain, Perth VIRGO (French-Italian) Cascina, Italy 8/6-9/7/4 ISAPP-4 3

4 About the lectures Theory of Gravitational Waves Gravitational Wave Detectors Signal Analysis Sources of Gravitational Waves 8/6-9/7/4 ISAPP-4 4

5 Gravitational Waves Why Gravitational Waves? Fundamental aspect of General Relativity Originate in the most violent events in the Universe Major challenge to present technology Why we have not seen them yet? They carry enormous amount of energy but They couple very weakly to detectors. How we will detect them? Resonant Detectors (Bars & Spheres) Interferometric Detectors on Earth Interferometers in Space 8/6-9/7/4 ISAPP-4 5

6 Gravitational vs E-M Waves EM waves are radiated by individual particles, GWs are due to nonspherical bulk motion of matter. I.e. the information carried by EM waves is stochastic in nature, while the GWs provide insights into coherent mass currents. The EM will have been scattered many times. In contrast, GWs interact weakly with matter and arrive at the Earth in pristine condition. Therefore, GWs can be used to probe regions of space that are opaque to EM waves. Stiil, the weak interaction with matter also makes the GWs fiendishly hard to detect. Standard astronomy is based on deep imaging of small fields of view, while gravitational-wave detectors cover virtually the entire sky. EM radiation has a wavelength smaller than the size of the emitter, while the wavelength of a GW is usually larger than the size of the source. Therefore, we cannot use GW data to create an image of the source. GW observations are more like audio than visual. Morale: GWs carry information which would be difficult to get by other means. 8/6-9/7/4 ISAPP-4 6

7 Uncertainties and Benefits Uncertainties The strength of the sources (may be orders of magnitude) The rate of occurrence of the various events The existence of the sources Benefits Information about the Universe that we are unlikely ever to obtain in any other way Experimental tests of fundamental laws of physics which cannot be tested in any other way The first detection of GWs will directly verify their existence By comparing the arrival times of EM and GW bursts we can measure their speed with a fractional accuracy ~1-11 From their polarization properties of the GWs we can verify GR prediction that the waves are transverse and traceless From the waveforms we can directly identify the existence of blackholes. 8/6-9/7/4 ISAPP-4 7

8 Information carried by GWs Frequency f 1 Hz ρ 1 gr/cm f Hz ρ 1gr/cm 4 3 Rate of frequency change Damping f dyn 1/ GM ~ ~( Gρ) 3 R 1/ Polarization Inclination of the symmetry plane of the source Test of general relativity Amplitude Information about the strength and the distance of the source (h~1/r). Phase Especially useful for detection of binary systems. 8/6-9/7/4 ISAPP-4 8

9 GW Frequency Bands High-Frequency: 1 Hz - 1 khz (Earth Detectors) Low-Frequency: Hz (Space Detectors) Very-Low-Frequency: Hz (Pulsar Timing) Extremely-Low-Frequency: Hz (COBE, WMAP, Planck) 8/6-9/7/4 ISAPP-4 9

10 Gravitation & Spacetime Curvature d x dt Newton U = 4 = U πgρ Einstein 1 8π G R g R+Λ g = T c d x µν f( g ) ds µν µν µν 4 µν µ ν ds = gµνdx dx Matter dictates the degree of spacetime deformation. Spacetime curvature dictates the motion of matter. GWs fundamental part of Einstein s theory 8/6-9/7/4 ISAPP-4 1

11 Linearized Gravity Assume a small perturbation on the background metric: The perturbed Einstein s equations are: g = g + h, h 1 µν µν µν µν h + g h h + R h R h = kt ; µ µν ; µ µν µ αβ ; µ αβ ; νµ µ ( α ; β ) µανβ µ ( α β ) αβ Far from the source (weak field limit) And by choosing a gauge: ~ h 1 h µν ; µ = α µν hµν nµν hα Simple wave equation: + = t h µν λ h µν µν kt λ 8/6-9/7/4 ISAPP-4 11

12 Transverse-Traceless (TT)-gauge Plane wave solution TT-gauge (wave propagating in the z-direction) A µν = h h µν µν + ε + + ε Riemann tensor Geodesic deviation and the tidal force ~ a A µν k = h ε µν = µν + R d A µν e ik a x µ µ k k = µ 1 µν 1 ε 1 = 1 TT h t TT jk jk ξ dt 1 TT k TT j 1 jk j R k j ξ = ξ f m R ξ k k j j h t 8/6-9/7/4 ISAPP-4 1

13 Gravitational Waves ΙΙ h µν = + h+ ε µν cos[ ω( t z)] 1 x = h+ cos [ ω( t z) ] x 1 y = h+ cos [ ω( t z) ] y in other words = h Polarization πft-k z 8/6-9/7/4 ISAPP-4 13

14 GW Polarizations 8/6-9/7/4 ISAPP-4 14

15 Stress-Energy carried by GWs GWs exert forces and do work, they must carry energy and momentum The energy-momentum tensor in an arbitrary gauge in the TT-gauge: it is divergence free For waves propagating in the z-direction for a SN exploding in Virgo cluster the energy flux on Earth The corresponding EM energy flux is: 1 1 3π ( GW ) t h ; h αβ ; h; h; h αβ ; h ; h αβ µν = αβ µ ν µ ν β αµ ν ; βhαν ; µ t 1 t = h h jk 3π ( GW ) t ν µν = ( GW ) jk TT TT µν ; µ ; ν ; c t t t h h c c 16π G ( GW ) ( GW ) ( GW ) = z = zz = ( GW ) π c f h z f h + + h 3 1 ergs = 4 G 1kHz 1 cm sec ~1 erg cm sec /6-9/7/4 ISAPP-4 15

16 Wave-Propagation Effects GWs affected by the large scale structure of the spacetime exactly as the EM waves The magnitude of h TT jk falls of as 1/r The polarization, like that of light in vacuum, is parallel transported radially from source to earth The time dependence of the waveform is unchanged by propagation except for a frequency-independent redshift received We expect Absorption, scattering and dispersion Scattering by the background curvature and tails Gravitational focusing Diffraction Parametric amplification Non-linear coupling of the GWs (frequency doubling) Generation of background curvature by the waves f f emitted 1 = 1 + z 8/6-9/7/4 ISAPP-4 16

17 The emission of grav. radiation If the energy-momentum tensor is varying with time, GWs will be emitted The retarded solution for the µν µν + h = kt (matter) linear field equation t µν µν T ( t x x', x') 3 h = d x' x x' For a point in the radiation zone in the slow-motion µν µν 3 jk h T ( t r, x') d x'~ Q ( t r) TT approximation r r t Where Q kl is the quardupole moment tensor Power emitted in GWs (, ) 1 3 kl k k l kl 3 Q ρ t x x x r δ d x de 1 G L = = Q Q GW 5 ij ij dt 5 c ij 8/6-9/7/4 ISAPP-4 17

18 Angular and Linear momentum emission Angular momentum emission dj dt GW i = ε Q Q 5 jkl ijk jl lk Linear momentum emission dp dt GW i 16 = Q jk Qjki + εijk Q jl Plk jk jkl Q ijk P ij : mass octupole moment : current quadrupole moment 8/6-9/7/4 ISAPP-4 18

19 Linearized GR vs Maxwell Potentials h ( x) ( x), ( ) Sources Lorentz gauge Wave equation Solution Solution (asymp) Radiated Power Einstein Maxw ell T αβ αβ ( Φ Ax) ( ρelec t, J ) αβ Φ h ; a = + A = t h ij = -8 πtij A = -4πJ ij ij 3 [ T ] ret h 3 [ J ] ret = d x ' A= ' x x ' d x x x ' ij Q p ij ret ret h = A = r r de 1 ij de = Q ij Q = p dt 5 dt 3 8/6-9/7/4 ISAPP-4 19

20 Quadrupole nature of GW Radiated power (by analogy with EM) Electric (l=1) d L = e a = d em Lgrav = m a = m mv = dt No mass dipole radiation in gravitational physics Magnetic (l=1) No magnetic dipole term ( ) µ ( r j ) ( r mv ) L µ = i i = = m m i i 8/6-9/7/4 ISAPP-4

21 Back of the envelope calculations! Characteristic time-scale for a mass element to move from one side of the system to another is: The quadrupole moment is approximately: Luminosity The amplitude of GWs at a distance r (R~R Schw ~1Km and r~1mpc~3x1 19 km): Radiation damping L GW MR Mυ Ens M ~ ~ ~ ~ 3 T T T R 8/6-9/7/4 ISAPP-4 1 Q ij T 3 R R R ~ ~ = ( / ) 1/ υ M R M 5 M G M 6 ~ ~ 5 5 υ 1/ 5/ 5 G c RSch υ ~ c R c R G R c 5 c 59 5 = erg / s =.3 1 M c / s G Q 1 MR 1 M 19 h ~ ~ ~ ~... ~ 1 r r T r R τ react 5/ Ekin R υ R = ~ T ~ T LGW M c RSchw 3 6

22 What we should remember Length variation = h Amplitude h jk r Q jk Power emitted L = de = 1 G Q Q GW 5 ij ij dt 5 c ij 8/6-9/7/4 ISAPP-4

23 Vibrating Quadrupole The position of the two masses The quadrupole moment of the system is The radiated gravitational field is The emitted power And the damping rate of the oscillator is x=± [ x + ξ sin( ωt)], x ξ kl ξ Q ( t r) 1+ sin ω( t r) Q x Q mx kl = mx 4mx kl ξ ω h = sin[ ω( t r)] Q 3 x r kl kl de G 16 G 6 = QQ ( 5 kl kl = mx 5 ξ ) ω dt 45c 15 c γ rad 1 de 16 G = = 5 mx E dt 15 c ω 4 8/6-9/7/4 ISAPP-4 3

24 Two-body collision The radiated power The energy radiated during the plunge from z= to z=-r If R=R Schw (M=1M & m=1m ) E =.19 m mc M de 8 G (3 ) 5 m z z z z dt = 15 c G E = m ( GM) 7/ 5 15 R c 5/ m Etrue = mc M erg Most radiation during R->R phase 4 t R/ υ R/ c 3 km/ c 1 s 8/6-9/7/4 ISAPP-4 4

25 Rotating Quadrupole (a binary system) THE BEST SOURCE FOR GWs Radiated power Energy loss leads to shrinking of their orbital separation Period changes with rate...and the system will coalesce after The total energy loss is Typical amplitude of GWs de 3 3 G G M µ = µ a ω = dt 5 c 5 c a 3 da 64 G µ M = 5 3 dt 5 c a P 96 G µ M = 5 4 P 5 c a inspiral /6-9/7/4 ISAPP-4 5 T = c G a µ M G 1 1 Erad = µ M a a /3 /3 µ 15 M f Mpc h 5 1.8M.7M 1Hz r

26 First verification of GWs PSR TODAY RESULTS! Nobel 1993 Hulse & Taylor 8/6-9/7/4 ISAPP-4 6

27 Discovery of a new binary pulsar Burgay et al Nature 3 Fastest known binary pulsar J Burgay et al (December 3) discovered a new pulsar in a binary J that is expected to open a new area of astrophysics/astronomy Strongly relativistic (period.5 Hrs), mildly eccentric (.88), highly inclined (i > 87 deg) Faster than PSR , J is the most relativistic neutron star binary Greatest periastron advance: dω/dt 16.8 degrees per year (thought to be fully general relativistic) very large compared to relativistic part of Mercury s perihelion advance of 4 per century 8/6-9/7/4 ISAPP-4 7

28 Binary systems (examples) PSR M =M ~1.4M, P=7h 45m 7s, r=5kpc, 1 h ~1, f~1 Hz, T ~3x1 yr earth theo insp dp /dt=-7. 1 s/yr dp /dt=-(6.9±6)x1 s/ yr M =M ~1.4M, f =1Hz, f T 1 final insp 1 final -1-1 obs The LIGO/VIRGO binary (1-1Hz) =1Hz, ~15min, after ~15 cycles (inspiral/merging 3Mpc) M =5M, M ~5M, f =1Hz, f =1Hz, (inspiral/merging 4 Mpc) -5 - The LISA binary (1-1 Hz) M =M ~1 M, f =1 Hz, f final M =M ~1 M, f =1 Hz, f final M=M~1M, f=1 Hz, f final f BH =.1Hz, (inspiral/merging at r~3gpc) =.1Hz, (inspiral/merging at r~3gpc) =1Hz, (inspiral at r~3gpc) Smaller Stars/BHs plunging into super-massive ones 1M ~ 1kHz M 8/6-9/7/4 ISAPP-4 8

29 An interesting observation The observed frequency change will be: The corresponding amplitude will be : h f M ~ 11/3 5/3 ~ f Mchirp = µ M 5/3 /3 chirp M 5/3 f /3 chirp r f = 3 f r Since both frequency and its rate of change are measurable quantities, we can immediately compute the chirp mass. The third relation provides us with a direct estimate of the distance of the source Post-Newtonian relations can provide the individual masses 8/6-9/7/4 ISAPP-4 9

30 A Quadrupole Detector Tidal force is the driving force of the oscillator Plane wave Displacement & Tidal force Equation of motion Solution Cross section ξ = + Weber s detector: M=141kg, L=1.5m, d=66cm, ω =166Hz, Q=x1 5. ξ iωt ω Lh e / ω ω + iω/ τ x h = h ε e µν µν i( ωt kz) + iωt iωt Lh+ e fx mlh+ ω e ξ + 1 ξ / τ + ω ξ = ω Lh+ e iωt ξmax = ωτ L h+ = Q L h/ 3π G σ = ω 3 Q M L 15 c 19 σ 3 1 cm Weber 8/6-9/7/4 ISAPP-4 3

31 Quadrupole Detector Limitations Very small cross section~3x1-19 cm. Sensitive to periodic GWs tuned in the right frequency of the detector Sensitive to bursts only if the pulse has a substantial component at the resonant frequency The width of the resonance is: h min Problems ν γ π ~ / ~1 Hz Thermal noise limits our ability to detect the energy of GWs. The excitation energy has to be greater than the thermal fluctuations E kt 1 15kT ω LQ M ~1 BURSTS Periodic signals which match the resonant frequency of the detector are extremely rare. A great number of events produces short pulses which spread radiation over a wide range of frequencies. The minimum detectable amplitude is 4π 1 / = 1 h min 1 ω L 9 55 r erg cm erg 1 M c!!! 3 kt eff πm ~1 16 The total energy of a pulse from the Galactic center (r=1kpc) which will provide an amplitude of h~1-16 or energy flux ~1 9 erg/cm. 8/6-9/7/4 ISAPP-4 31

32 Modern Bar Detectors 8/6-9/7/4 ISAPP-4 3

33 Laser Interferometers The output of the detector is L = F+ h+ () t + Fh () t = h() t L Technology allows measurements L~1-16 cm. For signals with h~ we need arm lengths L~1-1km. Change in the arm length by L corresponds to a phase change ϕ = 4πb L λ 9 ~1 rad The number of photons reaching the photo-detector is proportional to laser-beam s intensity [~sin ( φ/)] N N ϕ = out input sin ( / ) OPTIMAL CONFIGURATION Long arm length L Large number of reflections b Large number of photons (but be aware of radiation pressure) Operate at interface minimum cos(πb L/λ)=1. 8/6-9/7/4 ISAPP-4 33

34 International Network interferometers LIGO GEO Virgo TAMA ACIGA Simultaneously detect signal (within msec) detection confidence locate the sources decompose polarization of gravitational waves 8/6-9/7/4 ISAPP-4 34

35 Sensitivity of interferometers Gravity gradient and quantum uncertainty of mirror positions are the ultimate limiting factors Seismic noise Advanced Interferometers Thermal noise Initial Interferometers Photon shot noise 8/6-9/7/4 ISAPP-4 35

36 Current Sensitivity of GEO and LIGO Very good progress over the past years and we are fast approaching the designed sensitivity goals TAMA reached the designed sensitivity (3) LIGO (and GEO6) have reached within a factor of ~1 of their design sensitivity Currently focussing on: upper limit studies testing data analysis pipelines detector characterization VIRGO is following very fast 8/6-9/7/4 ISAPP-4 36

37 Future Gravitational Wave Antennas Advanced LIGO 6-8; planning under way 1-15 times more sensitive than initial LIGO High Frequency GEO 8? Neutron star physics, BH quasi-normal modes EGO: European Gravitational Wave Observatory 1? Cosmology 8/6-9/7/4 ISAPP-4 37

38 LISA the space interferometer LISA is low frequency detector. With arm lengths 5,, km targets at.1mhz.1hz. Some sources are very well known (close binary systems in our galaxy). Some other sources are extremely strong (SM-BH binaries) LISA's sensitivity is roughly the same as that of LIGO, but at 1 5 times lower frequency. Since the gravitational waves energy flux scales as F~f h, it has 1 orders better energy sensitivity than LIGO. 8/6-9/7/4 ISAPP-4 38

39 LISA Highlights Mapping the location of all binary compact objects in the galaxy (up to 1yr detection threshold). Detecting massive BH mergers anywhere in the universe Testing theories of MBH creation Compact objects around massive black holes (highly unequal mass systems) allows probing of GR near the event horizon easier to model and understand theoretically (perturbative approach) as the compact object spirals in, it s gravity radiation gives us a map of spacetime near the MBH horizon 8/6-9/7/4 ISAPP-4 39

40 Frequency range of astrophysics sources Audio band Space Terrestrial Gravitational Waves over ~8 orders of magnitude 8/6-9/7/4 ISAPP-4 4