Gravitational Waves from Coalescing Binaries and the post-newtonian Theory

Transcription

1 Gravitational Waves from Coalescing Binaries and the post-newtonian Theory Riccardo Sturani Instituto de Física Teórica UNESP/ICTP-SAIFR São Paulo (Brazil) Ubu - Anchieta, April 16 th 2015 Riccardo Sturani (ICTP-SAIFR São Paulo) GWs & PN theory Ubu-Anchieta 1 / 27

2 Outline 1 Introduction to GW: sources and observations 2 GW detectors and sensitivities 3 Fundamental physics with GWs Riccardo Sturani (ICTP-SAIFR São Paulo) GWs & PN theory Ubu-Anchieta 2 / 27

3 Outline 1 Introduction to GW: sources and observations 2 GW detectors and sensitivities 3 Fundamental physics with GWs Riccardo Sturani (ICTP-SAIFR São Paulo) GWs & PN theory Ubu-Anchieta 3 / 27

4 Wave generation: localized sources Einstein formula relates h ij to the source quadrupole moment Q ij Q ij = d 3 xρ (x i x j 13 ) δ ijx 2 v 2 G N M/r h ij = 2G N D d 2 Q ij dt 2 2G Nµv 2 cos(2φ(t)) D de dt = D2 16πG N ( r f = 1kHz 14Km ( ) f 1/3 ( m v = 0.3 1kHz ) 3/2 ( M m ) 1/2 m ) 1/3 < 1 6 dω ḣ2 + + ḣ2 = 32ν2 5G N v 10 ν µ/m = m 1 m 2 (m 1 + m 2 ) 2 Riccardo Sturani (ICTP-SAIFR São Paulo) GWs & PN theory Ubu-Anchieta 4 / 27

5 Binary coalescence: a tale made of three stories Inspiral phase post-newtonian approximation: v/c Merger: fully non-perturbative Ring-down: Perturbed Kerr Black Hole a.u time(sec) Riccardo Sturani (ICTP-SAIFR São Paulo) GWs & PN theory Ubu-Anchieta 5 / 27

6 Data analysis technique: Matched filtering An experimental apparatus output: time series O(t) = h(t) + n(t) h(t) = D ij h ij (t) Noise is conveniently characterized by its spectral function ñ(f)ñ (f ) = δ(f f )S n (f) [Hz 1 ] Matched filter enhances the sensitivity 1 T T 0 T O(t)h(t) dt = 1 h 2 (t) dt + 1 T 0 T h 2 τ0 0 + T n 0h 0 T 0 n(t)h(t) dt Riccardo Sturani (ICTP-SAIFR São Paulo) GWs & PN theory Ubu-Anchieta 6 / 27

7 Hunting for tiny signals Detector s output is flooded with noise: Strain Noise Signal time(sec) Noise + GW signal from 2+12 M system at 50 Mpc distance Riccardo Sturani (ICTP-SAIFR São Paulo) GWs & PN theory Ubu-Anchieta 7 / 27

8 Matched filtering Matched filtering enhances sensitivity: O(f)h (f) O(t) MF (t) e 2πift df S n (f) a.u MF Signal a.u MF Signal time(sec) time(sec) MF with h h MF with h but requires good model of the signal h or a complete bank of h s Riccardo Sturani (ICTP-SAIFR São Paulo) GWs & PN theory Ubu-Anchieta 8 / 27

9 Outline 1 Introduction to GW: sources and observations 2 GW detectors and sensitivities 3 Fundamental physics with GWs Riccardo Sturani (ICTP-SAIFR São Paulo) GWs & PN theory Ubu-Anchieta 9 / 27

10 Detector locations All are now being upgraded to their Advanced LIGOs (Virgo) due to to start data taking in Sept 2015 ( spring 2016) for design sensitivity Old coincident runs terminated in October 2010 Riccardo Sturani (ICTP-SAIFR São Paulo) GWs & PN theory Ubu-Anchieta 10 / 27

11 Advanced detectors ] -1/2 Spectrum[Hz KAGRA Advanced AdvVirgo NS-NS BH-BH15M BH-BH100M Initial f GW [Hz] Sensitivity vs. 200Mpc w. optimal orientation 4 Riccardo Sturani (ICTP-SAIFR São Paulo) GWs & PN theory Ubu-Anchieta 11 / 27

12 Distance reach for compact binary coalescence Distance[Mpc] Advanced Initial TotalMass[M ] Riccardo Sturani (ICTP-SAIFR São Paulo) GWs & PN theory Ubu-Anchieta 12 / 27

13 Advanced LIGO/Virgo goals Make first direct detection of GWs from neutron stars and/or black holes Probe intermediate mass black hole range: M 100M Measure rate of binary coalescences Measure pulsar parameters Possible probe of neutron star interior/nuclear matter at high density Verify association between short GRB s and GW s Combine EM and GW detection Make strong tests of GR Use coalescing binaries as standard sirens for cosmology Riccardo Sturani (ICTP-SAIFR São Paulo) GWs & PN theory Ubu-Anchieta 13 / 27

14 Outline 1 Introduction to GW: sources and observations 2 GW detectors and sensitivities 3 Fundamental physics with GWs Riccardo Sturani (ICTP-SAIFR São Paulo) GWs & PN theory Ubu-Anchieta 14 / 27

15 GW detection Inspiral h = A cos(φ(t)) Virial relation: Ȧ A φ P (v) de dt v (G N Mπf GW ) 1/3 ν = m 1 m 2 (m 1 + m 2 ) 2 E(v) = 1 ( 2 νmv2 1 + #(ν)v 2 + #(ν)v ) PN corrections = 32 5G N v 10 ( 1 + #(ν)v 2 + #(ν)v ) E(v)(P(v)) known up to 3(3.5)PN 1 2π φ(t ) = 1 T v(t ) ω(v)de/dv ω(t)dt = dv 2π P (v) (1 + #(ν)v #(ν)v ) dv v 6 Riccardo Sturani (ICTP-SAIFR São Paulo) GWs & PN theory Ubu-Anchieta 15 / 27

16 The EFT point of view on the 2-body problem Different scales in EFT: Very short distance r s = G N M negligible up to 5PN (effacement principle) Short distance: potential gravitons k µ (v/r, 1/r) with r r s /v 2 Long distance: GW s k µ (v/r, v/r) coupled to point particles with moments W. Goldberger and I. Rothstein, PRD 06 Riccardo Sturani (ICTP-SAIFR São Paulo) GWs & PN theory Ubu-Anchieta 16 / 27

17 EFT for compact binary systems Fundamental Fundamental gravitational fields Fundamental coupling to particle world line exp [is eff (x a, h GW )] = Dh(x) exp [is EH(h + h GW ) + is pp(h + h GW, x a)] S pp = m S EH = 1 32πG N dτ [ d 4 x ( ih) 2 ( th) 2 + ] G N h( h) 2 + G N h 2 ( h) 2... Riccardo Sturani (ICTP-SAIFR São Paulo) GWs & PN theory Ubu-Anchieta 17 / 27

18 EFT for compact binary systems Fundamental Fundamental gravitational fields Fundamental coupling to particle world line exp [is eff (x a, h GW )] = Dh(x) exp [is EH(h + h GW ) + is pp(h + h GW, x a)] S pp = m S EH = 1 32πG N ( ) dtd 3 x δ( x x a(t)) h 00/2 + v ih 0i + v i v j h ij/2 + h [ d 4 x ( ih) 2 ( th) 2 + ] G N h( h) 2 + G N h 2 ( h) 2... Riccardo Sturani (ICTP-SAIFR São Paulo) GWs & PN theory Ubu-Anchieta 17 / 27

19 EFT for compact binary systems Fundamental Fundamental gravitational fields Fundamental coupling to particle world line exp [is eff (x a, h GW )] = Dh(x) exp [is EH(h + h GW ) + is pp(h + h GW, x a)] S pp = m S EH = dtd 3 x J µν(x)h µν (x) 1 d 4 x 32πG N [ ( ih) 2 ( th) 2 + G N h( h) 2 + G N h 2 ( h) 2... ] Riccardo Sturani (ICTP-SAIFR São Paulo) GWs & PN theory Ubu-Anchieta 17 / 27

20 EFT for compact binary systems Fundamental Fundamental gravitational fields Effective Generic v-dependent potential terms Fundamental coupling to Instantaneous couplings between particle world line world-line particles exp [is eff (x a, h GW )] = Dh(x) exp [is EH(h + h GW ) + is pp(h + h GW, x a)] S pp = m dtd 3 x J µν(x)h µν (x) 1 S EH = d 4 x 32πG N [ S eff = d 4 1 x m 1 2 v2 1 + GN m2 2r [ ( ih) 2 ( th) 2 + G N h( h) 2 + G N h 2 ( h) v4 1 + G2 N m 2 2r 2 ( GN m 1 2r ) ] + v1 2 + v 1rv 2r PN: Jaranowski, Schäfer, Damour; Blanchet, Faye; Itoh Futamase; Foffa & RS 4PN: Jaranowski, Schäfer, Damour ] Riccardo Sturani (ICTP-SAIFR São Paulo) GWs & PN theory Ubu-Anchieta 17 / 27

21 How to compute effective potential? Iteratively solve Einstein equations: dtv 1 2(t, r) dt dt d 3 x d 3 x J(x)G(x x )J(x ) = 32πG N m 1m 2 dt dt d 4 k e ik(x1(t1) x2(t2)) (2π) 4 k 2 k0 2 G N dt dt δ(t t m 1m 2 ) x 1(t) x 2(t ) where k 0 has been neglected. Considering effects of v V d 3 k eik(x 1 x 2 ) e ik(x 1 x 2 ( ) ) = 1 + k2 0 k 2 k0 2 k 2 k e ik(x 1 x 2 ( ) ) = 1 2 t e ik(x k 2 k x 2 ) = 2 k 2 ( 1 ) k v1k v k 2 Riccardo Sturani (ICTP-SAIFR São Paulo) GWs & PN theory Ubu-Anchieta 18 / 27

22 Trading knowledge over the full trajectory with knowledge of all derivatives of the trajectory at equal time time + v + v +... Riccardo Sturani (ICTP-SAIFR São Paulo) GWs & PN theory Ubu-Anchieta 19 / 27

23 Newtonian potential: S eff dt G N m 1 m 2 r L At 1PN: S eff = dt G2 N m2 1 m 2 r 2 Lv 2 v V 1P N = Gm 1m 2 2r 1 2 v v 2 [ 1 G Nm r 2 (v2 1) 7 2 v 1v v 1ˆrv 2ˆr ] + Riccardo Sturani (ICTP-SAIFR São Paulo) GWs & PN theory Ubu-Anchieta 20 / 27

24 Graviton self-interaction vertices Conservative dynamics V β 3 G 2 N m 1m 2 (m 1 + m 2 ) r 2 Emission L pp h ij β 3 (νmx i ẍ j ) β 3 β 3 Example of tagging fundamental physics effects β 3 is not a viable modification of General Relativity Riccardo Sturani (ICTP-SAIFR São Paulo) GWs & PN theory Ubu-Anchieta 21 / 27

25 Bound on self-interaction triple vertex from binary pulsars At present the binary pulsars give best constraint on non-conservative effect from β 3 P β3 = P GR (1 + cβ 3 ) c 3.21 P obs Given that P 1 0.1% = β 3 = (4.0 ± 6.4) 10 4 GR Conservative effect of β 3 already constrained by Lunar Laser Ranging, 1PN β 3 = β P P N < RS et al. 09 Riccardo Sturani (ICTP-SAIFR São Paulo) GWs & PN theory Ubu-Anchieta 22 / 27

26 Testing GR φ(t) = v 5 7 ( n=0 φ n + φ (l) n Bayesian model selection between two hypothesis ) log(v) v n H GR H modgr : one or more φ i s are not as predicted by GR Odds ratio: In absence of noise O modgr GR O modgr GR P (H modgr d, I) P (H GR d, I) > < 0 favours modgr GR Li, RS et al. JPCS (2012), PRD (2012) Riccardo Sturani (ICTP-SAIFR São Paulo) GWs & PN theory Ubu-Anchieta 23 / 27

27 Parameter estimation bias Waveform match (fitting factor) F F = df h 1 (f)h 2(f) + h 1 (f)h 2 (f) S n (f) for δχ 3 injections varying η (maximized over other params) GR deviation do not prevent detection, but considerable bias! Riccardo Sturani (ICTP-SAIFR São Paulo) GWs & PN theory Ubu-Anchieta 24 / 27

28 Simulated signals in noise Constant shift in φ 3 φ 3 (1 + δχ 3 ) SNR s limited to 8-25 Single sources with noise: Odds ratio overlaps 15-sources catalogs can disentangle fundamental effects Riccardo Sturani (ICTP-SAIFR São Paulo) GWs & PN theory Ubu-Anchieta 25 / 27

29 New era is about to start ( 4 years) GW astronomy will open a new window onto the Universe, both astrophysically and cosmologically Observational test of strong gravity field dynamics not available so far: GW detection will give access to strong gravity phenomena Riccardo Sturani (ICTP-SAIFR São Paulo) GWs & PN theory Ubu-Anchieta 26 / 27

30 School on Gravitational Waves: From data to theory and back August Lecturers: A. Buonanno, S. Foffa, S. Klimenko, E. Ramirer-Ruiz, C. Will, RS, A. Vecchio Astrophysics workshop Astro-GR 2015 August Riccardo Sturani (ICTP-SAIFR São Paulo) GWs & PN theory Ubu-Anchieta 27 / 27