The weakest link. Michael Agathos Student Seminar 08 Universiteit Utrecht

Transcription

1 The weakest link Michael Agathos 08 Universiteit Utrecht 1

2 Introduction Candidate Processes for Production GW Detection and Upcoming Experiments

3 Radiative Solutions are predicted by General Relativity (tiny ripples of spacetime) Because of extremely weak coupling, observation of such waves requires highly sophisticated technology The only available laboratory is the Universe itself. We obviously have no control over the conditions of the experiments. Sources can be of Astrophysical (massive bodies) or Cosmological (early Universe) nature 3

4 Why are we so interested in Gravitational Radiation? Detection of Gravitational waves will not only be undisputed support for GR but will also open a new window to the cosmos Study of Astrophysical systems that comprise sources of GW s Probe for a snapshot of the early Universe and validate or exclude cosmological models Challenge to push current technology to its extreme limits 4

5 Why are we so interested in Gravitational Radiation? Detection of Gravitational waves will not only be undisputed support for GR but will also open a new window to the cosmos Study of Astrophysical systems that comprise sources of GW s Probe for a snapshot of the early Universe and validate or exclude cosmological models Challenge to push current technology to its extreme limits 5

6 GR provides us with a set of dynamical equations for the geometry of spacetime. R g R GT 1 8 These equations are highly nonlinear wrt the metric and cannot be solved analytically in the generic case. One has to resort to a linearized version of the above equations in order to come up with radiative solutions for small field perturbations. Interactions? 6

7 Take metric perturbations around Minkowski spacetime g h h 1 and evaluate all quantities up to 1 st order in the perturbation The Christoffel connection reads : ( h )[ h h [ And the Ricci tensor : R v h 1 v h h h ] ( h [ h h h h h h ] v v [ h h h h ] R (1) v ] ) 7

8 The 10 Einstein equations are not functionally independent due to the 4 Bianchi identities G 0 the metric is underdetermined General covariance imposes diffeomorphism invariance on the theory (via coordinate xfms). This is the gauge freedom of GR : x x x ( x) So from the 6 remaining d.o.f. one can choose to fix the gauge by imposing certain properties on the metric and get rid of 4 more non-physical d.o.f. Remember EM gauge invariance 8

9 Now the perturbation will transform as : h h and h h trace reverse : h h h 1 We can always transform to a gauge that is convenient, here the Lorenz gauge : h 0 So the Einstein equation can be recast in the simple form h 16 G T N 9

10 Vanishing source term h 0 0 This allows us to bring the metric to the transverse traceless form : 0i h 0 h 0 and for a single plane wave Generic way of finding TT : h 0 h h h 0 TT hij h h P n n ij ij i j h P P P P h TT 1 ij ik jl ij kl kl ˆn 10

11 Minkowski is boring for cosmology. g g h The Christoffel connection reads : ( g h )[ g h g h g h ] 1 v v g [ g g g ] h [ g g g ] 1 1 v v g 1 [ vh h h ] ( h ) 1 h g g [ h vh h ] and we can define : 1 g [ h vh h ] h g g 11

12 And the Riemann tensor : R R R where we used the Riemann normal coord. General covariance implies: R The Ricci tensor : R h h h R ( ) And Einstein eqn : G R g R 1 in the gauge h 0 and h 0 transversality tracelessness G h R h 1 0 1

13 Residual symmetry 0 TT gauge In a spacetime with a nonvanishing source term one also gets spurious l-l and l-t contributions Φ, Θ, Ξ i They are not objective and are not detectable by any conceivable experiment. They are merely sinuosities in the co-ordinate system, and the only speed of propagation relevant to them is the speed of thought. A. S. Eddington TT part is gauge invariant observable and obeys a wave-like equation TT h 16 ij ij ij ij, lmtlm 13

14 Consider a spacetime with a source : h 8 GNT Introducing the Green s function : x L flat G x x x x (4) ( ) ( ) G( x x ) x x t x x t h G xg( x x) T x) 3 8 N d ( x For r = x >> L remember quadrupole formula ( x x r... ) r t 3 i j hij ( x) d xt 00( t r, x) x x 14

15 Linearized Einstein eqns no self-interactions Metric perturbations can be idealized as plane waves : 3 d P P ikx h ( ) 8 k ij GN ( ) ( ) 3/ ij kh e k P, ( ) As a free field theory of a relativistic spin- particle 3 d 1 * ( ) 8 [ ˆP P ikx ( ) ( ) ˆ P P ikx P h k ( ) ( ) ] ( ˆ ij GN b k h e b k h e ) 3/ k k ij P, ( ) k b ˆ ˆ, b ( k k ) k k b ˆ, bˆ bˆ, bˆ 0 k k k k 15

16 In Cosmology we are looking for the Stochastic background and Ω gw Astro vs. Cosmo Thermal history of the Universe Graviton decoupling Redshift and cooling of background radiation CMBR correspondence 16

17 Ignorance Blindness Suspicion Myopia 17

18 Ignorance Blindness Suspicion Myopia 18

19 Consider the case of CMB Photons decouple at z=1089 and T=3000K Equilibrium gives a black body spectrum Photons carry energy Photons cool down with expansion de 1 4Vf df hf / kt 3 hf e 1 c d em 3 de 8 h f df V c e 3 hf / kt 1 19

20 Recall neutrino decoupling and temperature relation with SM relativistic d.o.f. MPlc Gravitons decouple below Planck temperature T Pl 10 k when (at least!) : g ( ) * S Tdec And of course the physical wavelength is redshifted f k f a 0 * Assuming adiabatic evolution since emission : g T a g T a f a a * S * * S 0 So * 0 f * 1/ GeV 10 f g T * S * 3 K and 1/ Tg T K

21 Gravitational radiation carries energy t 00 1 h 1 h h ( x) h ( x) 16G 16G Express graviton density spectrum by a useful quantity 1 d gw ( f ) frequency dependence (spectrum, peak, amplitude?) cr dln gw f, cr 3H0 8 G This includes a measuring uncertainty of H 0. Better use h0 gw N N ij ij gw 1

22 Stochastic isotropic background : h ( t) df d e ) ij Spectral density : ˆ ˆ ) ift P (, ( ˆ hp f ij P, * * 1 h (, ˆ) (, ˆ ( ) ( ˆ, ˆ P f hq f ) f f ) PQSh( f ) 8 gw 4 3 f f S h f 3H0 ( ) ( )

23 In order to have a background of gravitational radiation from the early Universe, which is strong enough for us to detect in the present time, we have to look for extremely violent phenomena that occur throughout the Universe. Modern cosmological theories provide us with such scenarios. Some of them are: Fluctuation amplification during inflation Phase transitions Topological defects (cosmic strings) Brane world scenarios etc 3

24 Quantum fluctuations are amplified in an inflating Universe [Grishchuck-Starobinsky] Linearized eqns of TT perturbations give: 1 hij ( x) ( g ) g ) hij ( x) 0 g In FLRW : g diag[ 1, a ( t), a ( t), a ( t)] g a 3 () t a 3 a ( t) 1 ( ) ( ) ( ) 0 3 t a t h ij x t h ij x a a a( t) a ( t) 3 at ( ) 1 ikx t h ( ) 0 k e a( t) a ( t) 4

25 In conformal time : d dt at () i j gdx dx a ( ) d a ( ) ijdx dx And the box equation becomes a hk hk k hk 0 k( ) ahk( ) a ah ah k( ) ) k U( k 0 where we used the wave zone expansions g a 4 () t k k k ah ah ah k k k k 5

26 End up with an effective potential which gives a time varying harmonic oscillator a Simple case : de Sitter inflation a 1 H U( ) I Here we have to distinguish between sub-hubble and super-hubble modes depending on value of k. h h k 1 1 e k a A B ik d a 1 I 1 k k k I,, a H k a H k a 6

27 Recall : 1 3 G h ( t, x) h ( t, x) gw ij ij N V h a a h ah a H a a a a a 1 4 a d x d k d k ikx * ik x gw (, ) (, ) 4 V /3 /3 ij k e ij k e 3 GNa V 1 3 * (, ) (, ) 4 V d k ij k ij k 3 G av ij ij ij ij ij ij 4 N 7

28 After approximating for sub-hubble modes with the help of a few assumptions we work out the Green s function solution : Matching this solution with the one without a source 1 T a a g ( g aa V ) 3 TT ˆ d p Tij ( k ) ij, lm( k) p ( ) ( ) 3/ l pma p a k p ( ) TT ij(, k) k U( ) ij (, k) 16 GNTi j ( k ) 16 G TT ij (, k ) dsin k( ) a( ) Tij (, k) k i ij (, k ) Aij ( k )sin k( f ) Bij ( k )cos k( f ) will give us the amplitudes that will propagate to today 8

29 3 f f 4 G Nk TT TT TT Sk ( f ) ij d(, k) d k cos( Uk( ) a( ) T ij ( ij, k( ), k 1) 6 Gd NT sin( ij ( kk) ) a( ) Tij (, ) V k i, j i i After approximating for sub-hubble modes with the help of a few assumptions we work out dg the w Green s Sk ( f ) ( ) function solution : gw k 4 16 G dln k a f TT (, k ) dsin k( ) a( ) T (, k) Matching this solution with the one without a source 1 T a a g ( g aa V ) T k k p p p k p 3 TT ˆ d p ij ( ) ij, lm( ) ( ) ( ) 3/ l ma a ( ) ij k i ij (, k ) Aij ( k )sin k( f ) Bij ( k )cos k( f ) will give us the amplitudes that will propagate to today ij 9

30 In Standard Inflation spectrum is flat or descending and COBE bound sets it too weak to be detected ~10-13 even for LISA Bound is at Ultra low frequencies, so a theory with ascending spectrum could be detectable There are such theories : Pre-big-bang scenario (super-inflation) [Veneziano-1991] (1 3)/ a ( ) spectral slope n T =3 Bouncing Universe includes a ( ) o a slow contraction phase o a superinflation phase ae o Radiation dominating era and n T ~ +ε or n T ~ 3 Quintessential Inflation [Peebles-Vilenkin 1999] o ( ) non standard equation of state a E 1/ a 1/ o and n T ~ 1 30

31 Phase transitions in the early Universe can produce GW s. More specifically EW Primordial Universe is initially in a false vacuum state in high temperatures As T drops below the Higgs mass scale, the broken true vacuum shows but is still hidden by an energy barrier Tunneling takes action 31

32 Bubbles of true vacuum are nucleated and expanding t The latent heat left over by the transition becomes kinetic energy transferred to the bubble walls (and also reheats the plasma) [Kosowsky, Kamionkowski, Turner] peak Isotropy is spontaneously lost f 0 e H / Hz E fv.. 4 at* GW s are produced on collision surface or even by turbulent areas in plasma 1 T dte e d drr T N ˆ R ˆ ˆ, ) it ikxn ikx ij ( k (, ) 0 S e n 0 ij r t n1 3

33 Strongly 1 st order transition is needed SM predicts smooth crossover (m H >100GeV) MSSM needs light stop for 1 st order NMSSM can provide strong 1 st order (also gives baryon asymmetry 33

34 Strongly 1 st order transition is needed SM predicts smooth crossover (m H >100GeV) MSSM needs light stop for 1 st order NMSSM can provide strong 1 st order (also gives baryon asymmetry 34

35 Topological defects are produced after a phase transition via the Kibble mechanism Cosmic strings are stringy formations of enormous mass densities that extend throughout the universe Quantity of interest Loops can be created Pulsate relativistically and decay emitting GW s dm dl 35

36 Direct and indirect experimental evidence restrict amplitude of GW s Currently operating GW experiments : LIGO (x) (Luisiana-Washington, USA) VIRGO (Pisa, Italy) GEO 600 (Hannover, Germany) Important indirect bounds also given by BBN COBE and the Sachs-Wolfe effect msec Pulsars 36

37 37

38 Principle of detection: No geodesic deviation in 1 st order, only stretching of coordinates (physical distance travelled by a photon) different types of detection (none of which work yet) Resonant bar detectors [J. Weber 60s] MiniGRAIL spherical cryogenic antenna (Leiden) Laser interferometers 10 to 10 4 Hz LIGO VIRGO GEO 600 TAMA 300 dl 10 L 1 38

39 LIGO 3,000km LIGO VIRGO 39

40 Consider detector as an input/output system S ( t) s ( t) n ( t) The signal could be smaller than the noise. / / Correlate S T T 1 dt dt S 1( t) S( t) Q ( t t ) i i i T/ T/ Optimal filtering [Michelson-1987] Observing for a period of time T : S S, S s, s n, n s, s h ( f ) 1 n, n n ( f ) 1 ft ft n ( f) min ft 40

41 Laser Interferometer Space Antenna Joint project planned by NASA and ESA 41

42 3 spacecrafts equipped with same instrumentation 6 armlength L 510 km 5 Low frequency band 10 Hz f 1Hz with peak sensitivity at mhz ( 10 1 ) Possible because of no ground noise 0 o wrt earth s position, with a 60 o tilt Proof mass! (cool) 4

43 We overviewed the basic mechanisms of the early Universe that are expected (if they ever happened) to produce a stochastic gravitational wave spectrum. Considering the predicted numerical restrictions some mechanisms are not strong enough to be detected even by close future experiments. If some signal shows up it will provide a snapshot of the early Universe and will feed more arguing around cosmological models But detecting a tiny signal and extracting a stochastic background from it is still a great challenge Hopefully LISA will give some useful data 43

44

45 45