Dynamics of star clusters containing stellar mass black holes: 1. Introduction to Gravitational Waves July 25, 2017 Bonn Seoul National University
Outline What are the gravitational waves? Generation of gravitational waves Detection methods Astrophysical Sources Detected GW Sources IMPRS Blackboard Lecture, July 25, 2017
Principle of Equivalence Principle of Equivalence: inertial mass = gravitational mass F = m i a = m g g g = GM r 3 r The trajectory of a particle in gravity does not depend on the mass: geometrical nature In a freely falling frame, one cannot feel the gravity. However, the presence of the gravity will cause tidal force. Again, this is similar to the difference between flat and curved space. IMPRS Blackboard Lecture, July 25, 2017
Curved Spacetime General geometry of space-time can be characterized by metric ds 2 = g µ dx µ dx In the absence of gravity, flat spacetime ds 2 = dt 2 + dx 2 + dy 2 + dz 2 g µ = = 0 B @ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1 C A The presence of gravity causes deviation from flat spacetime: curved spacetime IMPRS Blackboard Lecture, July 25, 2017
Tidal gravitational forces Let the positions of the two nearby particles be x and x+χ, then the equations of motions are d 2 x i ij @ (x) = dt2 @x j d 2 (x i + i ) dt 2 = If χ is small, one can expand Φ @ (x + ) @x j @ (x) @x j + @ @x k @ (x) @x j ij @ (x + ) @x j Then the relative acceletation between two particles becomes d 2 i dt 2 = ij @ 2 @x j @x k x k +... k IMPRS Blackboard Lecture, July 25, 2017 Tidal acceleration tensor
In GR, similar expression can be derived "Geodesic deviation" equation D 2 D 2 = R µ dx d dx d µ Riemann curvature tensor In the weak field limit, v c, Φ c 2. The only remaining components are μ=ν =0. Therefore, i.e., d 2 i dt 2 = Ri 0j0 j R i 0j0 = @ 2 @x i @x j Ricci tensor can be defined by contraction of Riemann tensor: R µ = R µ IMPRS Blackboard Lecture, July 25, 2017
Einstein's Field Equation In Newtonian dynamics, the motion of a particle is governed by the gravitational potential which can be computed by the Poisson's equation: i.e., summation over the quantities that describe the geodesic deviation (=tidal acceleration tensor). The Ricci tensor is similarly defined with Φ, i.e., summation of all tidal components. In vacuum, R µ =0, similar to r 2 =0 IMPRS Blackboard Lecture, July 25, 2017 r 2 = ij @ 2 @x i @x j =4 G In the presence of energy (mass, etc.), Einstein's field equation becomes 1 R µ 2 Rg µ = 8 G c 4 T µ Energy-momentum tensor Curvature scalar R = R µ µ
Field equation in weak field limit flat part small perturbation In the weak field limit, g µ = µ + h µ, h µ 1 Einstein's field equation becomes h µ = where @ 2 @x @x µ h Perform coordinate transformation, @ 2 @x @x h µ + @2 @x µ @x h = 16 GS µ S µ T µ 1 2 µ T and = @2 @t 2 + r2 x µ! x 0µ = x µ + µ Then, h 0 µ = h µ @ µ @x @ @x µ IMPRS Blackboard Lecture, July 25, 2017
Choice of guage (i.e., coordinates) Define a trace-reversed perturbation h µ = h µ 1 2 h µ and impose the Lorentz guage condition @ h µ @x =0 then equation for h µ becomes h µ = 16 GT µ IMPRS Blackboard Lecture, July 25, 2017
Further Properties In vacuum (i.e., Tμν=0), h µ =0 i.e, equation for plane waves The solution can be written in the form h µ = Re{A µ exp(ik x )} One can choose a coordinate system by rotating so that A µ µ = 0 (traceless), A = 0 purely spatial 0i The wave is transverse in Lorentz gauge. Therefore the metric perturbation becomes very simple in transverse-traceless (TT) gauge. Also h µ = h µ in TT coordinates. IMPRS Blackboard Lecture, July 25, 2017
Effects of GWs in TT gauge In TT guage, GWs traveling along z-direction can be written with only two components, h + and h x : they are called plus and cross polarizations 0 h TT µ = h + (t z) B @ 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 C A + h (t z) B @ 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 In these coordinates, the line element becomes 1 C A ds 2 = dt 2 +(1+h + )dx 2 +(1 h + )dy 2 + dz 2 +2h dxdy The lengths in x- and y- directions for h + then oscillate in the following manner L x = L y = Z x2 x 1 p 1+h+ dx (1 + 1 2 h +)L x0 ; Z y2 y 1 p 1 h+ dy (1 1 2 h +)L y0 h+ hx IMPRS Blackboard Lecture, July 25, 2017
Generation of gravitational waves The wave equation h µ = 16 GT µ gives formal solution of Z Tµ (t x x h 0, x 0 ) µ (t, x) =4 x x 0 d 3 x 4 r Z T µ (t x x 0, x 0 )d 3 x One can show that h jk (t, x) = 2 r where Ijk = Z d 2 I jk (t r) dt 2 x j x k d 3 x Using the identities @T tt @t + @T kt @x k IMPRS Blackboard Lecture,July 25, 2017 =0, and @2 T tt @t 2 = @2 T kl @x k @x l
Order of magnitude estimates of GW amplitude Note: the moment of inertia tensor is not exactly the same as the quadrupole moment tensor, but in TT guage, it does not matter. I jk = Z x j x k d 3 x Quadrupole kinetic energy Ï jk (mass) (size)2 (transit time) 2 quadrupole Kinetic E. = Mc 2 In most cases, ε is small, but it could become ~0.1 h jk 10 22 0.1 M M Q jk = Z 100Mpc r x j x k 1 3 r2 jk d 3 x IMPRS Blackboard Lecture,July 25, 2017
How small is h~10-21 which was the detected amplitude of the GW150914? IMPRS Blackboard Lecture, July 25, 2017
Measurement of GWs: 1. Tidal forces Equation of geodesic deviation becomes GW tidal acceleration: d 2 x j dt 2 = R j0k0 x k = 1 2ḧjkx k Riemann tensor R j0k0 is a gravity gradient tensor in Newtonian limit R j0k0 = @ 2 @x j @x k =!2 GW h jk Tidal force can cause resonant motion of a metallic bar = bar detector Force field Gravity gradiometer can be used as a gravitational wave detector IMPRS Blackboard Lecture,July 25, 2017
Measurement of GWs: 2. Laser Interferometer Consider a simple Michelson interferometer with lx ly l. The phase difference of returning lights reflected by x and y ends = x y 2! 0 [l x l y + lh(t)] Toward the laser: E laser / e i x + e i y Toward the photo-detector: E PD / e i x e i y = e i y (e i 1) For small Δφ, Time varying part I PD / e i 1 2 2 4! 2 0(l x l y ) 2 +8! 2 0(l x l y )lh(t) IMPRS Blackboard Lecture,July 25, 2017
Astrophysical sources Compact Binary Coalescence (neutron stars and black holes) Strong signal, but rare Computable waveforms Continuous (~ single neutron stars) Weak, but could be abundant Deviation from axis-symmetry is not known Burst (supernovae or gamma-ray bursts) GW amplitudes are now well known, not frequent Stochastic I jk = Z Superposition of many random sources Could be useful to understand distant populations of GW sources or early universe IMPRS Blackboard Lecture,July 25, 2017 x j x k d 3 x Black hole binaries
Confirmation of GW with Binary Pulsar Orbit of binary neutron stars shrinks slowly Hulse & Taylor received Nobel Prize in 1993 for the discovery of a binary pulsar P=7.75 hrs a=1.6 x 10 6 km M1=1.4 Msun M2=1.35 Msun Time to merge=3x10 8 yrs
IMPRS Blackboard Lecture,July 25, 2017
IMPRS Blackboard Lecture,July 25, 2017
Waveform tells many things Neutron star binary high f Black hole binary low f Comparison of waveforms between NS and BH mergers: Note differences in frequencies and shape. IMPRS Blackboard Lecture,July 25, 2017
GW from a merging black hole IMPRS Blackboard Lecture,July 25, 2017
Principles of GW Detector
Gravitational Wave Detector 1990 2002-2010 Initial LIGO 2015.9 ~ Advanced LIGO (10times better sensitivity)
Simple Estimates of Sensitivity of Interferometers If the length resolution is λlaser, detectable strain is h l l = laser l = 10 6 m 10 3 m = 10 9 Optical path length can be significantly increased by adopting optical cavity, but should be smaller than GW wavelength (~1000 km for 300 Hz) h l l eff laser GW 10 6 m 10 6 m = 10 12 However, due to quantum nature of the photons, the length resolution coud be as small as N -1/2 photonsλ laser. Thus sentivity could reach h N 1/2 photons laser GW IMPRS Blackboard Lecture,July 25, 2017
Shot Noise Collect photons for a time of the order of the period of GW wave 1/f GW N photons = P laser hc/ laser P laser hc/ laser 1 f GW For 1W laser with λlaser=1 µm, fgw=300hz, Nphotons=10 16 h l l eff N 1/2 photons GW laser 10 8 10 6 m 10 6 m = 10 20 By adopting high power laser (20W for O1) and power recycling, we can reach astrophysical sensitivity of ~10-22. IMPRS Blackboard Lecture,July 25, 2017
Other Noises Radiation pressure noise Make mirrors heavier Suspension thermal noise/ mirror coating brownian noise Increase beam size, monolithic suspension structure Seismic noise Multi-stage suspension, underground Newtonian Noise So far difficult to avoid. Seismic and wind measurement and careful modeling LIGO-G1601201-v3
GW Events from O1/O2 GW150914 (FAR<6x10-7 yr -1 ) LVT151012 (Candidate, FAR~0.37 yr -1 ) GW151226 (FAR<6x10-7 yr -1 ) GW170104 (<5x10-5 yr -1 IMPRS Blackboard Lecture,July 25, 2017
What made these detections interesting? LIGO detected gravitational waves from merger of black hole binaries, instead of neutron star binaries 7 6 BH Mass Distribu3on Black hole mass range was quite large: GW150914 is composed of 36 and 29 times of the mass of the Sun Most of the known black holes are much less massive (~10 times of the sun) Number 5 4 3 2 1 0 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 Mass (Msun) LVT151012 GW sources X-ray Binarie IMPRS Blackboard Lecture,July 25, 2017
What these results tell us? (Abbott et al., 2016, ApJL, 828, L22; PHYSICAL REVIEW X 6, 041015 (2016) PRL 118, 221101 (2017)) Proof of the existence of the black holes Existence of stellar mass black holes in binaries Individual masses in wide range (7-35 Msun) Formation of ~60 Msun BH BH appears to be much more frequent than previously thought Current estimation of the rate 12-210 yr -1 Gpc -1 IMPRS Blackboard Lecture,July 25, 2017
Summary of the GW sources GW150914: First Unambiguous detection of stellar mass black holes and a BH binary Accurate measurement of black hole masses (within ~10%) Higher mass of stellar mass BH than previously thought: low metallicity environment? GW151226: Lower masses than GW150914, similar to the X-ray binary BH mass Lower mass progenitor or high metallicity environment? GW170104 High mass (~50 Msun) Spin may not be aligned Origin Isolated or dynamical? IMPRS Blackboard Lecture,July 25, 2017 31
References. Weinberg, "Gravitation and Cosmology: Principles and applications of the general theory of elativity", 1972, John Wiley & Sons. B. Hartle, "Gravity, an introduction to Einstein's General Relativity". 2003, Addison Wesley ecture notes by K. Thorne: https://www.lorentz.leidenuniv.nl/lorentzchair/thorne/thorne1.pdf http://www.pmaweb.caltech.edu/courses/ph136/yr2012/ (together with R. Blandford, Chapters 25 & 27) ecent LIGO papers Abbott et al. 2016, "Abbott et al. 2016, "Observation of Gravitational Waves from a Binary Black Hole Merger", 2016, PRL, 116, 061102 Abbott et al., 2016, "ASTROPHYSICAL IMPLICATIONS OF THE BINARY BLACK HOLE MERGER GW150914", ApJL, 828, L22 Abbott et al. 2016, "GW150914: Implications for the Stochastic Gravitational-Wave Background from Binary Black Holes", PRL, 115, 131102 Abbott et al. 2016, "GW150914: The Advanced LIGO Detectors in the Era of First Discoveries", 2016, PRL, 116, 131103 Abbott et al., 2017, GW170104: Observation of a 50-Solar-Mass Binary Black Hole Coalescence, PRL, 118, 221101 IMPRS Blackboard Lecture, July 25, 2017