How well can gravitational waves pin down merging black holes?

Transcription

1 How well can gravitational waves pin down merging black holes? Using gravitational wave information to point our telescopes and find the merger event on the sky Scott A. Hughes, MIT

2 How do we measure GWs? Sources of dynamical gravity: Radiation communicates changes in the field. Seen as tidal lines of force propagating through spacetime at the speed of light. Two polarizations act on distant test masses: plus, h+ cross, h+

3 How do we measure GWs? Sources of dynamical gravity: Radiation communicates changes in the field. Seen as tidal lines of force propagating through spacetime at the speed of light. Two polarizations act on distant test masses: plus, h+ cross, h+ circular

4 Measurable impact of GWs Oscillation in spacetime geometry can leave a measurable impact on propagation of light: Behavior of light in spacetime with wave: ds 2 = c 2 dt 2 + [1 + h(t, x)] dx 2 = Solve for the speed of light in this coordinate system: dx dt = c 1+h(t, x)

5 Measurable impact of GWs Oscillation in spacetime geometry can leave a measurable impact on propagation of light: Scott A. Hughes, MIT Behavior of light in spacetime with wave: ds 2 = c 2 dt 2 + [1 + h(t, x)] dx 2 = Solve for the speed of light in this coordinate system: dx dt = Now imagine that we have mirrors which fall freely in this spacetime. Bounce light between mirrors, record time between bounces. c 1+h(t, x)

6 Measurable impact of GWs Oscillation in spacetime geometry can leave a measurable impact on propagation of light: Behavior of light in spacetime with wave: ds 2 = c 2 dt 2 + [1 + h(t, x)] dx 2 = T = Time interval between bounces: dx dx/dt 1 c [1+ 12 h(t, x) ] dx Gravitational wave enters as an oscillation in interval from bounce to bounce. Scott A. Hughes, MIT NRC BE PAC, 6 November 26

7 LISA: Implementing freely falling mirrors in space Laser interferometry between spacecraft. Lasers used as optical transponders to track relative motions. Sensitive in band ~1-4 Hz < f < 1 Hz: Optimal for targeting processes involving massive black holes.

8 Character of binary black hole GWs GWs drain energy and angular momentum from binary, black holes spiral together until they merge. Nature of waveforms in different epochs encodes the nature of binary measurement of waves measures binary characteristics. Scott A. Hughes, MIT

9 Wave model Inspiral: Slow evolution driven by GW loss of orbital energy and angular momentum. Rather well understood. Waveform described by 17 parameters in general. 2 masses 6 spin components 1 initial eccentricity 2 position angles 1 initial periapsis 2 orientation angles longitude 1 distance 1 initial semi-major axis 1 initial orbit anomaly

10 Wave model Inspiral: Slow evolution driven by GW loss of orbital energy and angular momentum. Rather well understood. Waveform described by 15 parameters in most cases. Scott A. Hughes, MIT 2 masses 6 spin components 2 position angles 2 orientation angles 1 distance 1 initial semi-major axis 1 initial orbit anomaly 1 initial eccentricity 1 initial periapsis longitude Typically assume rapid circularization

11 Waves in simulated LISA noise Graphic courtesy John Baker, Goddard Space Flight Center m1 = 2 x 1 5 Msun m2 = 2 x 1 5 Msun z = 5 Last few dozen cycles stand above noise with no sophisticated filtering Scott A. Hughes, MIT

12 Post-Newtonian description Description that builds in the fact that Newtonian gravity is an excellent description for very wide binaries: Perfect for inspiral. Example: Equations describing center of mass motion of each member of the binary. Leading term: Newtonian gravity. Scott A. Hughes, MIT

13 Post-Newtonian description Description that builds in the fact that Newtonian gravity is an excellent description for very wide binaries: Perfect for inspiral. Example: Equations describing center of mass motion of each member of the binary. Leading term: Newtonian gravity. Relativity corrections Scott A. Hughes, MIT

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15 More corrections...

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17 And some more. [Blanchet 26, Liv Rev Rel 9, 4, Eq. (168)]

18 Waves from motion Use these equations of motion and laws of wave emission to build waveform describing two massive black holes spiralling together. Smooth chirp : Both amplitude and frequency steadily grow with time.

19 Waves from motion Use these equations of motion and laws of wave emission to build waveform describing two massive black holes spiralling together. Smooth chirp : Both amplitude and frequency steadily grow with time.

20 Encoding position in GWs The GWs from post-newtonian have an amplitude set by on binary s sky position and orientation: h + = A(M,D L,t)[1 + (n L) 2 ] cos[φ(t)] h =2A(M,D L,t)(n L) sin[φ(t)] Vector n: Points to the source on sky. Vector L: Points along orbit inclination. Measuring waves measures a combination of angles which determine position and orientation: Position is degenerate with orientation.

21 This is form as measured by a Scott A. Hughes, MIT stationary detector! Actual measurements last roughly a year... motion of LISA spacecraft around the sun leads to n that is effectively time varying. Modulations imposed by antenna motion break degeneracy between position and orientation: Can get positions with (roughly) degree scale precision.

22 FYI: Also get (redshifted)masses exquisitely Phase function of GWs depends very strongly on the binary s chirp mass: M = (m 1m 2 ) 3/5 (m 1 + m 2 ) 1/5 Simulations show that chirp mass [times (1+z)] can be measured with precision of roughly 1-5 [(amplitude signal-to-noise ratio number of measured wave cycles) -1 ].

23 Gravitomagnetism Angular momentum creates gravitomagnetic fields... drives precession of binary s spins. Orbital motion contribution. Contribution from other body s spin Leads to new forces, modifying the orbital acceleration felt by each body.

24 Gravitomagnetism Angular momentum creates gravitomagnetic fields... drives precession of binary s spins. Angular momentum is globally conserved: J = L + S1 + S2 = constant The orbital plane precesses to compensate for precession of the individual spins.

25 Impact of precession Video by Peter Reinhardt, MIT

26 Waves from motion & precession Redo waves including additional motions from spin-induced precessions. Example: Two rapidly spinning black holes. Clear modulations imposed on frequency and amplitude of GW.

27 Waves from motion & precession Redo waves including additional motions from spin-induced precessions. Example: Two rapidly spinning black holes. Clear modulations imposed on frequency and amplitude of GW.

28 Greatly improves ability to find sources on sky with GWs Time varying binary inclination breaks degeneracy of position with orientation! Can vastly improve ability to pinpoint GW sources. Scott A. Hughes, MIT Example from Lang & Hughes, PRD 26: 1 6 Mo, 3 x 1 5 Mo at z = 1. Including precession improves angular resolution by factor of roughly an order of magnitude in area.

29 Time evolution How does source localization evolve? Ellipses show how well a binary is localized (28, 21, 14, 7, 4, 2, 1, ) days before merging. Final position pinned down to a few to a few 1s of arcminutes... but most of that accuracy comes in the last day. 1 5!5!1 1 5!5 (a)!1 1!1! !1 (d) (g)!2! !5!1 5!5 5 25!25 (b)!1 1 (e)!5 5 (h)!5! !25 (c)!5!5 5 5!5 1 5!5 (f)!5 5 (i)!1!1 1 From Lang & Hughes, ApJ 28

30 FYI: Details on mass accuracy Spin precession breaks degeneracies among mass parameters: Individual masses are measured, not just chirp mass. Mass accuracy Binaries at z = 1 m1 = 1 6 Mo m2 = 3 x 1 5 Mo Over a wide range of redshift, redshifted black hole masses are typically measured with better than 1% accuracy.

31 FYI: Details on mass accuracy Spin precession induced modulation also encodes value of black hole spins: Can determine black hole spins from inspiral. Spin accuracy Binaries at z = 1 m1 = 1 6 Mo m2 = 3 x 1 5 Mo Over a wide range of redshift, black hole spins are typically measured with.1-1% accuracy.

32 Distance measurement Form of the GW amplitude directly encodes luminosity distance to the source: h + = A(M,D L,t)[1 + (n L) 2 ] cos[φ(t)] h =2A(M,D L,t)(n L) sin[φ(t)]

33 Distance measurement Form of the GW amplitude directly encodes luminosity distance to the source: h + = A(M,D L,t)[1 + (n L) 2 ] cos[φ(t)] h =2A(M,D L,t)(n L) sin[φ(t)] A = [G(1 + z)m/c2 ] 5/3 [πf(t)/c] 2/3 D L Phase determines (1 + z)m and f(t)... only remaining amplitude parameter is distance. Scott A. Hughes, MIT

34 Distance measurement Form of the GW amplitude directly encodes luminosity distance to the source: h + = A(M,D L,t)[1 + (n L) 2 ] cos[φ(t)] h =2A(M,D L,t)(n L) sin[φ(t)] A = [G(1 + z)m/c2 ] 5/3 [πf(t)/c] 2/3 D L Phase determines (1 + z)m and f(t)... only remaining amplitude parameter is distance. Binary inspiral GWs are standard candles... standardized by general relativity. Schutz, Nature, 1986; Chernoff & Finn, ApJ, 1993 Scott A. Hughes, MIT

35 Distance measurement Form of the GW amplitude directly encodes luminosity distance to the source: h + = A(M,D L,t)[1 + (n L) 2 ] cos[φ(t)] h =2A(M,D L,t)(n L) sin[φ(t)] A = [G(1 + z)m/c2 ] 5/3 [πf(t)/c] 2/3 D L Detailed analysis: Distance typically measured with accuracy (a few)/(signal-to-noise) Nearby (z ~ 1): Distant (z ~ 5): δd/d.2 -.4% is typical δd/d 2-5% is typical

36 Get distance... not redshift In stark constrast to most astronomical measurements, GWs alone cannot provide z. Why? Masses & spins enter waveform as timescales: m τ m = Gm/c 3 a S/m τ s = S/mc 2 Timescales undergo cosmological redshift; inferred masses/spins likewise redshift. All parameters that we measure end up degenerate with redshift.

37 Assume the underlying cosmology, invert from measured distance to source redshift. Redshift inferred from GW data with roughly same precision as distance measurement: Option 1 Allows us to break redshift degeneracy, associate masses and spins with z at accuracy of several percent: Map m(z), S(z) for black hole mergers. z z max [ ] D D, (cosmology) Scott A. Hughes, MIT

38 Option 2 If merger host can be identified, then distance & redshift might both be measured with high accuracy. Intrinsically precise... absolutely calibrated (by general relativity). Holz & Hughes, ApJ 25, 629, 15

39 Option 2 If merger host can be identified, then distance & redshift might both be measured with high accuracy. Intrinsically precise... absolutely calibrated (by general relativity). Intrinsic precision not really relevant: Waves subject to weak lensing by (unknown) distribution of intervening matter. Holz & Hughes, ApJ 25, 629, 15

40 How well can GWs pin down merging black holes? Pretty well! 1 5 (a) 1 5 (b) 5 25 (c) * Decent spatial localization (arcminutes to tens of arcminutes in best cases) * Good localization in z (a few to a few tens of percent)!5!1 1 5!5!1 1!1! !1 (d) (g)!2!2 2!5!1 5!5 5 25!25!1 1 (e)!5 5 (h)!5!5 5!25!5!5 5 5!5 1 5!5 (f)!5 5 (i)!1!1 1 Also get high precision data on redshifted masses, and measurements of the spins.

41 How well can GWs pin down merging black holes? However (a) 1 5 (b) 5 25 (c) Most of the localization accuracy comes very late in the coalescence.!5!1 1 5!5!1 1 (d)!1!1 1!5!1 5!5!1 1 (e)!5 5!25!5!5 5 5!5 (f)!5 5 Does something go bang at the end? 2 1!1 (g)!2! !25 (h)!5! !5 (i)!1!1 1 Better understand of possible/likely counterparts critical here.