When one black hole is not like the other

Transcription

1 When one black hole is not like the other Cal Poly, San Luis Obispo Center for Computational Relativity and Gravitation Rochester Institute of Technology 13 December 2010

2 Current gravitational-wave searches focus on binaries with mass ratios near one from Teviet Creighton We have searched for gravitational waves from coalescing low mass compact binary systems with a total mass between 2 and 35 solar masses LIGO Science collaboration, 2009

3 Current gravitational-wave searches focus on binaries with mass ratios near one from Teviet Creighton We have searched for gravitational waves from coalescing low mass compact binary systems with a total mass between 2 and 35 solar masses LIGO Science collaboration, 2009

4 the merging of two similar black holes movie from Harald Pfeiffer (

5 the merging of two similar black holes movie from Harald Pfeiffer (

6 soon we can look for more extreme mass ratios M = M µ = 90M spin = 0.9M e 0 = 0.79 a 0 = 23.6M ι 0 = 43

7 soon we can look for more extreme mass ratios M = M µ = 90M spin = 0.9M e 0 = 0.79 a 0 = 23.6M ι 0 = 43 how is this calculation possible?

8 mass = M semi-major axis = 6 M spin = 0.9 M eccentricity = 0.5 Setting the scale for LISA µpc

9 mass = M semi-major axis = 6 M spin = 0.9 M eccentricity = 0.5 For LISA: 10 5 M/M 10 7 µpc

10 mass = M semi-major axis = 6 M spin = 0.9 M eccentricity = 0.5 For LISA: 10 5 M/M 10 7 M r 3/2 t orbit (8 minutes) 10 6 M 6M M r 11/4 M/µ t radiation (1 day) 10 6 M 6M 10 5 µpc 1/2

11 mass = 500M semi-major axis = 4.5M spin = 0.9 eccentricity = Setting the scale for LIGO

12 mass = 500M semi-major axis = 4.5M spin = 0.9 eccentricity = For LIGO: 10 M/M 10 3

13 mass = 500M semi-major axis = 4.5M spin = 0.9 eccentricity = For LIGO: 10 M/M 10 3 M r 3/2 t orbit (0.25 seconds) 500M 6M M r 11/4 M/µ t radiation (3 seconds) 500M 6M 500 1/2

14 Observing the center of the Milky Way (from La Silla, Chile) from Max Planck Institute for Extraterrestrial Physics Infrared/Submillimeter Astronomy - Galactic Center Research - European Southern Observatory

15 Observing the center of the Milky Way (from La Silla, Chile) VLT - Cerro Paranal, Chile from Max Planck Institute for Extraterrestrial Physics Infrared/Submillimeter Astronomy - Galactic Center Research - European Southern Observatory

16 VLT - Cerro Paranal, Chile from Max Planck Institute for Extraterrestrial Physics Infrared/Submillimeter Astronomy - Galactic Center Research - European Southern Observatory

17 From Reinhard Genzel Max Planck Institute for Extraterrestrial Physics

18 Why the new interest for ground based detection?

19 Why the new interest for ground based detection?

20 Why the new interest for ground based detection? 2010??

21 ( Image from )

22 suppose: m 1 M, M = total mass ( Image from )

23 Kepler s suppose: law: m 1 M, f orbit (2 khz) M M = total mass M r 6M 3/2 ( Image from )

24 Kepler s suppose: law: m 1 M, f orbit (2 khz) M M = total mass M r 6M twice orbital frequency at innermost stable circular orbit for non-spinning black holes, in a binary with total mass 35 times that of the sun 3/2 ( Image from ) same, but for 2 solar masses

25 What can LIGO see: mass & spin targets M / M sun f ISCO > 2 Hz 4 Hz 8 Hz 16 Hz a / M

26 What can LIGO see: mass & spin targets M / M sun f ISCO > 2 Hz 4 Hz 8 Hz 16 Hz a / M

27 What can LIGO see: advanced LIGO range circular-equatorial orbit µ = 1.4M a = 0.2M a = 0 a = 0.2M from Mandel, Brown, Gair, Miller, 2008

28 open questions

29 open questions What waveforms can we use?

30 open questions What waveforms can we use? Will the cool stuff survive?

31 open questions What waveforms can we use? I will argue for a way to assess waveform models. Will the cool stuff survive?

32 open questions What waveforms can we use? I will argue for a way to assess waveform models. Will the cool stuff survive? I end with what I think is the cool stuff.

33 appraising waveform models cost (e.g. CPU-hours/waveform) Barack & Cutler generic kludge accuracy radiative (Teukolsky) self force to first order self force to second order

34 appraising waveform models cost (e.g. CPU-hours/waveform) post-newtonian orbits + post-newtonian evolution + quadrupole formula Currently used in Mock LISA Data Challenges Barack & Cutler generic kludge accuracy radiative (Teukolsky) self force to first order self force to second order

35 appraising waveform models cost (e.g. CPU-hours/waveform) post-newtonian orbits + post-newtonian evolution + quadrupole formula Currently used in Mock LISA Data Challenges Barack & Cutler generic kludge accuracy radiative (Teukolsky) self force to first order self force to second order Kerr orbits + Teukolsky-fitted evolution + quadrupole formula

36 quantifying the accuracy-axis (1) Two waveform models: true (GR) and approximate (AP). (2) The error θ i = θbf i θtr i, in the estimate for parameter is approximately θ i θ i =[Γ 1 (θ bf )] ij ( j h AP (θ bf ) n) +[Γ 1 (θ bf )] ij ( j h AP (θ bf ) h GR (θ true ) h AP (θ true )), where h is the waveform, n is the noise, ( ) is a noise-weighted inner product, and Γ ij is the Fisher matrix h Γ ij = h θ i θ j.

37 quantifying the accuracy-axis (1) Two waveform models: true (GR) and approximate (AP). (2) The error θ i = θbf i θtr i, in the estimate for parameter is approximately θ i θ i =[Γ 1 (θ bf )] ij ( j h AP (θ bf ) n) +[Γ 1 (θ bf )] ij ( j h AP (θ bf ) h GR (θ true ) h AP (θ true )), where h is the waveform, n is the noise, ( ) is a noise-weighted inner product, and Γ ij is the Fisher matrix h Γ ij = h statistical θ i θ j. error

38 quantifying the accuracy-axis (1) Two waveform models: true (GR) and approximate (AP). (2) The error θ i = θbf i θtr i, in the estimate for parameter is approximately θ i θ i =[Γ 1 (θ bf )] ij ( j h AP (θ bf ) n) +[Γ 1 (θ bf )] ij ( j h AP (θ bf ) h GR (θ true ) h AP (θ true )), where h is the waveform, n is the noise, ( ) is a noise-weighted inner product, and Γ ij is the Fisher matrix h systematic Γ ij = h statistical θ i θ j. error error

39 Barack & Cutler results S/M e LSO (ln M) 2.6e 4 5.6e 4 5.3e 5 2.7e 4 9.2e 4 7.7e 5 2.8e 4 2.5e 4 1.5e 4 (S/M 2 ) 3.6e 5 7.9e 5 4.5e 5 1.3e 4 6.3e 4 5.1e 5 2.6e 4 3.7e 4 2.6e 4 (ln µ) 6.8e 5 1.5e 4 7.4e 5 6.8e 5 9.2e 5 1.0e 4 6.1e 5 9.1e 5 1.0e 3 (e 0 ) 6.3e 5 1.3e 4 2.9e 5 8.5e 5 2.8e 4 3.2e 5 1.2e 4 1.1e 4 1.6e 4 (cos λ) 6.0e 3 1.7e 2 1.3e 3 1.3e 3 5.8e 3 2.4e 4 6.5e 4 8.4e 4 4.7e 4 (Ω s ) 1.8e 3 1.7e 3 7.9e 4 2.0e 3 1.7e 3 7.6e 4 2.1e 3 1.1e 3 6.7e 4 (Ω K ) 5.6e 2 5.3e 2 4.7e 2 5.5e 2 5.1e 2 4.7e 2 5.6e 2 5.1e 2 4.8e 2 ( γ 0 ) 4.0e 1 6.3e 1 3.8e 1 1.0e+0 6.1e 1 3.9e 1 9.3e 1 3.4e 1 3.9e 1 (Φ 0 ) 2.6e 1 6.7e 1 2.2e 1 1.4e+0 7.5e 1 2.7e 1 1.5e+0 1.7e 1 3.3e 1 (α 0 ) 6.2e 1 5.8e 1 5.5e 1 6.3e 1 5.9e 1 5.6e 1 6.4e 1 5.9e 1 5.9e 1 [ln(µ/d)] 8.7e 2 3.8e 2 3.7e 2 3.8e 2 3.7e 2 3.7e 2 3.8e 2 7.0e 2 3.7e 2 (t 0 )ν 0 4.5e 2 1.1e 1 3.3e 2 2.3e 1 1.3e 1 4.4e 2 2.5e 1 3.2e TABLE III. Parameter extraction accuracy for inspiral of a 10M CO onto a 10 6 M MBH at SNR=30 (based on data collected during the last year of inspiral). Shown are results for various values of the MBH s spin magnitude S and the final eccentricity e LSO.The rest of the parameters are set as follows: t 0 = t LSO (1/2)yr (middle of integration), γ 0 =0,Φ 0 =0,θ S = π/4, φ S =0,λ = π/6, α 0 =0, θ K = π/8, φ K =0. and since typical values are (Γ 1 ) ln(µ/d),ln(µ/d) 10 4,(Γ 1 ) ln µ,ln µ 10 9,and(Γ 1 ) ln µ,ln(µ/d) 10 9,wefind that indeed (ln D) [ln(µ/d)].] Finally, as a check, it is easy to see that [ln(µ/d)] must be greater than SNR 1 =0.033 (since the signal amplitude is linear in µ/d), which is indeed satisfied in every column of our tables. C. Comparison with other results in the literature Our angular resolution results can be compared to results by Cutler and Vecchio [43] on LISA s angular resolution

40 Barack & Cutler results S/M e LSO (ln M) 2.6e 4 5.6e 4 5.3e 5 2.7e 4 9.2e 4 7.7e 5 2.8e 4 2.5e 4 1.5e 4 (S/M 2 ) 3.6e 5 7.9e 5 4.5e 5 1.3e 4 6.3e 4 5.1e 5 2.6e 4 3.7e 4 2.6e 4 (ln µ) 6.8e 5 1.5e 4 7.4e 5 6.8e 5 9.2e 5 1.0e 4 6.1e 5 9.1e 5 1.0e 3 (e 0 ) 6.3e 5 1.3e 4 2.9e 5 8.5e 5 2.8e 4 3.2e 5 1.2e 4 1.1e 4 1.6e 4 (cos λ) 6.0e 3 1.7e 2 1.3e 3 1.3e 3 5.8e 3 2.4e 4 6.5e 4 8.4e 4 4.7e 4 (Ω s ) 1.8e 3 1.7e 3 7.9e 4 2.0e 3 1.7e 3 7.6e 4 2.1e 3 1.1e 3 6.7e 4 (Ω K ) 5.6e 2 5.3e 2 4.7e 2 5.5e 2 5.1e 2 4.7e 2 5.6e 2 5.1e 2 4.8e 2 ( γ 0 ) 4.0e 1 6.3e 1 3.8e 1 1.0e+0 6.1e 1 3.9e 1 9.3e 1 3.4e 1 3.9e 1 (Φ 0 ) 2.6e 1 6.7e 1 2.2e 1 1.4e+0 7.5e 1 2.7e 1 1.5e+0 1.7e 1 3.3e 1 (α 0 ) 6.2e 1 5.8e 1 5.5e 1 6.3e 1 5.9e 1 5.6e 1 6.4e 1 5.9e 1 5.9e 1 [ln(µ/d)] 8.7e 2 3.8e 2 3.7e 2 3.8e 2 3.7e 2 3.7e 2 3.8e 2 7.0e 2 3.7e 2 (t 0 )ν 0 4.5e 2 1.1e 1 3.3e 2 2.3e 1 1.3e 1 4.4e 2 2.5e 1 3.2e TABLE III. Parameter extraction accuracy for inspiral of a 10M CO onto a 10 6 M MBH at SNR=30 (based on data collected during the last year of inspiral). Shown are results for various values of the MBH s spin magnitude S and the final eccentricity e LSO.The rest of the parameters are set as follows: t 0 = t LSO (1/2)yr (middle of integration), γ 0 =0,Φ 0 =0,θ S = π/4, φ S =0,λ = π/6, α 0 =0, θ K = π/8, φ K =0. and since typical values are (Γ 1 ) ln(µ/d),ln(µ/d) 10 4,(Γ 1 ) ln µ,ln µ 10 9,and(Γ 1 ) ln µ,ln(µ/d) 10 9,wefind that indeed (ln D) [ln(µ/d)].] Finally, as a check, it is easy to see that [ln(µ/d)] must be greater than SNR 1 =0.033 (since the signal amplitude is linear in µ/d), which is indeed satisfied in every column of our tables. Some things can be measured really well, the rest can be measured, somewhat well. C. Comparison with other results in the literature Our angular resolution results can be compared to results by Cutler and Vecchio [43] on LISA s angular resolution

41 generic kludge results (SNR ~ 140 to 180) Last year of inspiral, at distance of about 1 Gpc, M = 10 6 M, µ = 10M If we average over 4 systems: a/m =0.1, 0.5 e final =0.01, 0.1 & σ 10 2 log D r 0 /M θ 0 /rad φ 0 /rad (θ, φ) sky /rad (θ, φ) detector /rad σ 10 7 to 10 5 log M log µ a/m 2 p 0 e 0 ι 0 /rad

42 generic kludge results (SNR ~ 140 to 180) Last year of inspiral, at distance of about 1 Gpc, M = 10 6 M, µ = 10M If we average over 4 systems: a/m =0.1, 0.5 e final =0.01, 0.1 & σ 10 2 log D r 0 /M θ 0 /rad φ 0 /rad (θ, φ) sky /rad (θ, φ) detector /rad σ 10 7 to 10 5 log M log µ a/m 2 p 0 e 0 ι 0 /rad but these results are unstable!

43 a closer look at generic kludge waveforms See Gair-Glampedakis (2006) for fluxes and Babak et al. (2007) for waveforms.

44 a closer look at generic kludge waveforms See Gair-Glampedakis (2006) for fluxes and Babak et al. (2007) for waveforms. (1) Solve Teukolsky-fitted flux rules for E(t),L z (t),q(t).

45 a closer look at generic kludge waveforms See Gair-Glampedakis (2006) for fluxes and Babak et al. (2007) for waveforms. (1) Solve Teukolsky-fitted flux rules for E(t),L z (t),q(t). (2) Solve modified Kerr geodesic equations, for osculating world line dx i dt = V i [r, θ; E(t),L z (t),q(t)]

46 a closer look at generic kludge waveforms See Gair-Glampedakis (2006) for fluxes and Babak et al. (2007) for waveforms. (1) Solve Teukolsky-fitted flux rules for E(t),L z (t),q(t). (2) Solve modified Kerr geodesic equations, for osculating world line dx i dt = V i [r, θ; E(t),L z (t),q(t)] (3) Get waves from quadrupole formula (or octupole) h jk = 2µ D d 2 dt 2 x j x k.

47 the spectral representation

48 the spectral representation (1) Solve Teukolsky-fitted flux rules for E(t),L z (t),q(t).

49 the spectral representation (1) Solve Teukolsky-fitted flux rules for E(t),L z (t),q(t). (2) Get coefficients in a spectral representation of geodesics at a handful of points (10-100) along that trajectory, r = kn r kn e i(kω θ+nω r )t, etc.

50 the spectral representation (1) Solve Teukolsky-fitted flux rules for E(t),L z (t),q(t). (2) Get coefficients in a spectral representation of geodesics at a handful of points (10-100) along that trajectory, r = kn r kn e i(kω θ+nω r )t, etc. (3) From these, use a series-multiplication algorithm to find the coefficients in a series representation of the quadrupole waveform h + ih = mkn h mkn e i(mω φ+kω θ +nω r )t

51 the spectral representation (4) Interpolate to find the coefficients of the inspiral waveform h + ih = mkn h mkn (t)e iw mkn(t) where the phases are integrals of the interpolated frequencies W mkn (t) = t 0 dt [mω φ (t )+kω θ (t )+nω r (t )]

52 how does this help? (1) parameter derivatives are now done entirely in terms of slowly evolving quantities Θ (h + ih x )= mkn [ Θ h mkn (t) ih mkn Θ W mkn ] e iw mkn where Θ W mkn = t 0 dt Θ [mω φ (t )+kω θ (t )+nω r (t )] These should be very stable.

53 how does this help?

54 how does this help? (2) The parameter derivatives Θ W mkn are trivial for all parameters other than Θ = a

55 how does this help? (2) The parameter derivatives Θ W mkn are trivial for all parameters other than Θ = a (3) The other parameter derivatives can be helped along using what we know from Teukolsky snapshots h Teukolsky mkn = 2 eimφ Dω 2 mkn l=2 Z lmkn S lmkn (θ)

56 Kludge-spectra generic spectral kludge waveforms as tools for general exploration h + ih = mkn h mkn e i(mω φ+kω θ +nω r )t agree well with Teukolsky-spectra h Teukolsky mkn = 2 eimφ Dω 2 mkn l=2 Z lmkn S lmkn (θ)

57 kludge vs. Teukolsky spectra kludge Teukolsky 10 2 (D/M) h M

58 no need to search for dominant modes in kludged spectra quadrupole h mkn, spin=0.9, eccentricity = 0.79, inclination = 43 deg frequency multiplier (k) radial frequency multiplier (n)

59 now some of the cool stuff

60 An example EMRI: spectral evolution M = 10 6 M a = 0.9M µ = 10M t inspiral = 3 years Spectra using Drasco & Hughes, PRD (2006). Inspiral using Gair & Glampedakis, PRD (2006).

61 An example EMRI: spectral evolution M = 10 6 M a = 0.9M µ = 10M t inspiral = 3 years Spectral Evolution Spectra using Drasco & Hughes, PRD (2006). Inspiral using Gair & Glampedakis, PRD (2006).

62 An example EMRI: spectral evolution M = 10 6 M a = 0.9M µ = 10M t inspiral = 3 years Spectra using Drasco & Hughes, PRD (2006). Inspiral using Gair & Glampedakis, PRD (2006).

63 regular orbits

64 resonant orbits

65 resonant orbits x 10 3 orbital frequencies (Hz) f f r f t (years)

66 resonant orbits frequency ratios x x 154 x x x 330 x ω r /ω θ 338 x 381 x 510 8/11 5/7 7/10 9/13 2/3 7/11 5/8 8/13 3/5 7/ days before merger x ω r /ω φ 571 x 683 x

67 resonant orbits horizon a * = 0.98 a * = ISCO 2: r/m (degrees) horizon ISCO 2:1 4: r/m Chris Hirata (2010)

68 resonant orbits from Flanagan & Hinderer (2010)

69 osculating orbits for black holes & Osculating orbit: perturbed motion as unperturbed orbit with constants of motion replaced by timedependent quantities. [Classic celestial mechanics topic] Recently done for Schwarzschild orbits & planar forces (Pound and Poisson, 2008) We have extended the formalism to Kerr orbits and arbitrary forces. (Gair, Flanagan, Drasco, Babak, Hinderer)

70 test application: gas drag M = 10 7 M a 0 =0.1 pc M 1 /M 2 =3 M disk =0.2M simulation by Jorge Cuadra (Pontificia Universidad Católica de Chile)

71 eccentricity semi-major axis (a o ) from Cuadra et al (2008)

72 eccentricity semi major axis (a 0 ) our gas drag test problem

73 conclusion - back to the open questions What waveforms can we use? Will the cool stuff survive?