Scott A. Hughes- Gravitational waves from inspiral into massive black holes

8/3/2019 Scott A. Hughes- Gravitational waves from inspiral into massive black holes

1/7

Gravitational waves from inspiral intomassive black holesScott A. Hughes

Theoretical Astrophysics, California Institute of Technology, Pasadena, C A 91125

Abstract. Space-based gravitational-wave interferometers such as LISA will be sensi-tive to the inspiral of stellar mass compact objects into black holes with masses in therange of roughly 105 solar masses to (a few) 107 solar masses. During the last year ofinspiral, the compact body spends several hundred thousand orbits spiraling from sev-eral Schwarzschild radii to the last stable orbit. The gravitational waves emitted fromthese orbits probe the strong-field region of the black hole spacetime and can makepossible high precision tests and measurements of the black hole's properties. Measur-ing such waves will require a good theoretical understanding of the waves' properties,which in turn requires a good understanding of strong-field radiation reaction and ofproperties of the black hole's astrophysical environment which could complicate wave-form generation. In these proceedings, I review estimates of the rate at which suchinspirals occur in the universe, and discuss what is being done and what must be donefur ther in order to calculate the inspiral waveform.

One of the most exciting sources that should be measured by the space-basedgravitational- wave interferometer LISA (the Laser Interferometer Space Antenna[1]) is the inspiral of a "small" (1 - 10M0) compact body into a massive (105~7 M0)black hole. Such massive black holes reside at the cores of galaxies; the smallercompact bodies will become bound to the hole and spiral into them after undergoinginteractions with stars and other objects in the environment of the galactic center.The measurement of gravitational waves from such inspirals will make possible veryhigh precision tests of general relativity, an d probe th e nature of the galactic core'senvironment.To set the stage fo r understandingw hy these inspirals ar e such interesting objects,consider the following estimates: th e orbital energy of a small body in an equatorial,prograde orbit of a Kerr black hole is

b_ l - 2 +

where v = \JM/r and q = a/M. (I use units in which G = c = I throughout.) Theorbital frequency of the small body isCP523, Gravitational Waves: Third Edoardo Amaldi Conference, edited by S. Meshkov 2000 American Institute of Physics 1-56396-944-0/007$ 17.00

75

Downloaded 02 Oct 2007 to 131.215.225.176. Redistribution subject to AIP license or copyright, see http://proceedings.aip.org/proceedings/cpcr.jsp
http://dx.doi.org/10.1063/1.1291842


2/7

M1/2r3/2

Radiation reaction carries orbital energy away from th e system, causing the orbitto shrink. Eventually it shrinks enough that th e body reaches the innermo st stablecircular orbit ( ISC O) . Orbits inside this radius are dynamically unstable; fur the rradiative evolution tends to push the body into the black hole.Post-Newtonian theory allows us estimate th e rate at which the system losesenergy as a power series in the quantity u = ( M f l ) 1 / 3 (which is roughly the orbitalspeed) and the black hole's spin a. Reference [2] gives the energy loss in such apost-Newtonian expansion:

d E 32 /M2 1 0 r, , x f , xl__ _ __ / r" \ 1U f ( \ , f ( \1 /o\7 , K I A/f / L/SchwA"; ~r /spmva? u)\ ' V/c u e o \ J . y j. /The prefactor 32/5(/ / /M) 2 ti 1 0 is the result one gets applying the quadrupole for-mula to a binary described with Newtonian gravity; the function /schw.(^) ^s a(rather high-order) correction appropriate for zero-spin black holes, an d / sp i n ( a> ^ )is a correction incorporating information about the hole's spin. (Note that thisformula is only appropriate fo r // < C M : it does not incorporate any finite massratio corrections.)

Equations (l)-(3) can be used to estimate the t ime it takes for a small body tospiral from radius r x to r2, and the number of gravitational-wave cycles it emits inthat time:

[ t2 1 + fr2 dt s fr2 dE/dr j I d t = -dr I ' . , d r ,Jt , 7n d r J ri d E / d t 'ft2 / ^ T2 ^dt ^ fr2ndE/dr j= I fwdt = I -dr= I ' T dr .y J g w y dr -K dE/dt

/ A \(4 )',K\(5 )^ '

(O n th e last line, I have assumed that th e bulk of the radiation comes out in thequadrupole ra = 2m od e , so that f = 2fi /2?r . ) Consider now the inspiral of a 5M0body into a rapidly spinning (a c M) 106 M0 black hole. The small body spiralsfrom a radius of 8M (in Boyer-Lindquist coordinates) to the ISCO in one year,emitting around 5 x 105 gravitational-wave cycles as it does so. The gravitational-waves that it emits lie in the frequency band 3 x 1 0~ 3 Hz < / < 3 x 1 0~ 2 Hz theband to which LISA is most sensitive. Indeed, careful analyses of the detectabilityof the signal by LISA [3] indicate that such an inspiral should be detected out to adistance of roughly 1 Gigaparsec with am pl i tude signal-to-noise ratio of around 1 0to 100 (depending on factors such as the mass of the small body, the mass of theblack hole, and the black hole's spin). The fact that such a large number of cyclesar e emitted indicates that details of the waveform can in principle be measuredto very high precision. Detection of extreme mass-ratio inspirals by LISA o f f e r sthe possibility of very high precis ion measurements of the characteristics of extremestrong-field regions of spacetime.

76



3/7

The high precision tests that LISA will be able to make should allow us to directlymap the characteristics of the massive body's spacetime metric and confirm thatthey in fact exhibit th e Kerr metric. Most likely, the way that this will be donewill be to measure th e mult ipole m oments of the massive body. Fintan Ryan [ 4 ]ha s shown that th e gravitational waves which ar e emi t ted as a small body spiralsinto a massive compact object contain a "map" of the massive body's spacetime.By measuring the gravitational waves and decoding th e map, one learns th e massand current multipole moments which characterize the massive body. All m ultipolemomen ts of Kerr black holes are parameterized by the holes' mass M an d spin a:

Mi + iSt = M (ia)1 . (6 )Fo r Kerr black holes, knowledge of the moments M 0 = M and Si = aM deter-mines al l higher moments. This is one way of stating the "no-hair" theorem: Themacroscopic properties of a black hole are entirely determined by its mass and spin.(I neglect the astrophysically uninteresting possibility of charged holes.) By mea-suring gravitational waves from extreme mass ratio inspiral and thereby mappingth e m a s s i ve body's spacetime, LISA will test th e no-hair theorem fo r black holes,determining whether the m a s s i ve body has multipole moments characteristic of theKerr metric, or whether the body is someth ing more exotic, such as a boson star.

The waves emitted by extreme mass ratio inspiral are thus likely to be directlymeasureable by LISA, and are likely to be extremely interesting. One might nextwonder whether they occur often enough to be interesting. This question hasbeen examined in some detail by Mart in Rees and Steinn Sigurdsson [5,6]. Theyconsider the scatter of stellar mass black holes in the central density cusp of galaxiesinto tightly bound orbits of the galaxies' central black hole. Occasionally, such ascattering event will put the stellar mass hole into an orbit which is so tightlybound that its future dynamics are driven by gravitational-wave emission, and itbecomes an interesting source fo r LISA. They find that the rate of such events islikely to lie in the range

1 event < < 1eventrs^/v/~o , ^ ** * \' )r s ^ / ~ o , ^ **year G pc 6 mo nth Gpc

Obviously, there are large uncertainties in this calculation. The low mass end of themassive black hole population (which is most relevant to LISA observations) is notas well constrained as the population of very massive black holes (M > 108'M0),and there are uncertainties in the rate at which stellar mass black holes are "fed"into the central hole to produce extreme mass ratio binaries. However, the lowerend of the rate estimate (7) is based on very conservative estimates. We may ratherrobustly estimate that th e rate measured by LISA will be several events per yearout to a Gigaparsec [7].

The waves that LISA will measure from these sources will come from orbitsthat are rather eccentric and inclined with respect to the black hole's equatoria lplane [8]. To best interpret the measured waves (and, indeed, in order to improve

77



4/7

the odds of seeing the waves at all in the detector's noisy data stream), it will benecessary to have some theoretical model ing of the orbit and the waves that it emitsas gravi tational radiation reduces the orbit's energy and angular m o m e n t u m . Weexpect that the radius, inclination angle, and e ccentricity of the orbit will change asradiative backreaction drives the system's evolution. So me meanst)f understandingthese changes, in detail, is needed in order to model the waves' evolution accurately.Before discussing work in radiation reaction, it is necessary to take a sanity check.Relativity theorists often work in a very idealized universe: their extreme mass ratiobinary is likely to consist of a big black hole, a small body, and gravitational waves.In the real astrophysical world, there will be complications to this pretty (buthighly idealized) picture. One should worry whether th e complications render therelativity theorist's modeling invalid.

Perhaps the most important such complication arises from interaction betweenth e small inspiraling body and material accreting onto the massive black hole.Recently, Narayan has analyzed this interaction and concluded that, in almost al lcases, accretion induced drag is unlikely to significantly influence extreme massratio inspiral [9]. This conclusion is based on the fact that in the majority ofcases, th e rate at which th e central black hole accretes gas from its environmentis rather low (several orders of m agnitude less than the Eddington rate [10] ) . Fo rthese "normal" galaxies, much evidence [9,10] suggests that the gas accretes viaan advection dominated accretion flow (ADAF). Narayan's calculation [9] showsthat th e timescale fo r A DA F drag to change th e orbit's characteristics (e.g. , th eorbital angular momentum) is many (9 to 16) orders of magnitude longer thanthe timescale for radiation reaction to change th e orbit's characteristics. Thus,the relativity theorist's idealized view of an extreme mass ratio binary is probablyquite accurate: radiation reaction is likely th e most important factor driving theevolution of extreme mass ratio binaries.When the mass ratio is extreme, one can analyze the spacetime of the binary usinga perturbative expansion: the spacetime metr ic can be written as a "background"from the central object (which I will assume from now on is a Kerr black hole) ,plus a per turba t ion due to the inspiraling body:

^=9ST(M,o)+MA*). (8)The evolution of the perturbation f t a / ? ( / 4 ) should then describe th e dynamical evo-lution of the system. To linear order in the mass ratio p,/M (which should beadequate for extreme mass ratios), this evolution can be described using perturba-tion techniques, such as the Teukolsky equation 1 [11].When the mass ratio is extreme, the effects of radiation reaction ar e gentle enoughthat the system's evolution is adiabatic: the radiation reaction timescale is muchlonger than th e orbital timescale. At any given moment, the trajectory of thesmall body is very nearly a geodesic, parameterized by the three constants of Kerr* ) Th e Teukolsky equation actually describes the evolution of a curvature quantity related to theperturbation.

78



5/7

orbital mot i on : th e energy E1, th e (z -component o f) angular m o m e n t u m Lz, and the"Car ter constant" Q . Gravitat ional-wave emission causes these th ree constants tochange on the radiation reaction timescale. In this adiabatic l imi t , the evolut ion ofth e system can be understood in terms of the evolut ion of the quantities (E , Lz, Q ) .It is well known that gravitational waves carry energy and angular m o m e n t u m .One might think that they carry "Carter constant" as well, and that therefore onemight be able to deduce th e effects of radiation reaction by measur ing th e flux ofradiation at infinity and down th e event horizon. By measur ing the amount of E ,Z / 2 , and Q carried in the flux one should be able to deduce how much E, L z, andQ are lost from th e orbi t . This would then allow one to figure out orbits of Kerrblack holes radiatively evolve.

This approach does not work. One can deduce the change in the orbit's E andLz by examining radiation fluxes, but one cannot so deduce the change in Q:

oZorbit ~ vZt radiated >SQ orbit 7^ "Eradiated - (9)

T he change 6Q turns out to depend explicitly on the local radiation reaction force,f t* = dp^ /dr , which th e small body experiences due to radiative backreaction (see[14]) . The properties of this force (and programs to calculate it) are describedelsewhere in this volume [12]. Here, it is sufficient to note that an unders tandingof th e radiation react ion force fo r gravitational radiation reaction lies some time inth e fu tu re , so that w e cannot yet evolve generic Kerr black hole orbits.There are special cases where evolution of the Carter constant is not such a nastyimpediment . One case is the evolution of equatorial orbits. Equatorial orbits haveQ 0; and, one can easily show that an orbi t which starts off equa tor ia l remainsequatorial. In this case, one need only evolve the energy and angular m o m e n t u m .The local radiation react ion force is not needed in this case. Another case is theevolution of circular , non-equatoria l orbits. (For non-zero spin, "circular" mea ns"constant Boyer-Lindquist coordinate radius".) Such orbits have recently beenshown to r e ma i n circular under adiabatic radiation reaction [13]. Thus, in anadiabatic evolut ion, a system which is initially circular and inclined wil l remaincircular and incl ined: th e system evolves through a sequence of orbi ts changingonly i ts radius and inclinat ion angle. By imposing "circular goes to circular" , onecan write down a simple rule relating the change of the Carter constant to the fluxof energy and an gu la r m om e n tum : Q = Q(E, Lz).Recently, I have examined the evolution of these circular orbits under adiabaticradiation react ion, using a f lux-measur ing formalism based on the Teukolsky equa-t ion. The formalism and results are presented at length in [14]. Some highlights ofthe results are presented in Figure 1 .

Consider the left panel of Figure 1 . The horizontal axis is orbital radius r; thevert ical axis is inclinat ion angle i. This figure shows th e direct ion, in (r, i) phasespace, in which radiation reaction tends to push the orbit. This par t icu la r analysis

79



6/7

150

, 100

50

0 -

0.5 R

1000 2000t/M

3000

0.5

-0.52 4 6

Orbital radius r/M 10002000 3000

t/MFIGURE 1. The left panel illustrates the effect of radiation reaction on circular, inclined orbits.It is for a black hole with a = 0.8M. The tail of each arrow represents a particular orbit withradius r an d inclination angle i. The arrow indicates the direction in which radiation reactiondrives the orbit; the size of the arrow indicates how quickly it is so driven. The right panel is asnapshot of the gravitational waveform emitted during an inspiral into a hole with a = 0.95M.At this m o m e nt , the small body is spiraling through r = 7M, L = 62.43.

is for a black hole with a = 0.8M. Notice that th e orbits are in the strong-field ofth e hole; th e dotted line indicates th e maximum incl ination angle which th e orbitca n have and remain stable. Orbits tilted beyond this line are dynam ically unstableto small per tu rba t ions and plunge into the hole. The tail of each arrow represents aparticular orbit. The direction of the arrow gives the direction in which that orbittends to evolve due to gravitational-wave emission; the arrow's length indicatesth e relat ive rate of evolution. In all cases, th e direction of the arrow is such thatthe inclinat ion angle increases: radiation reaction tends to make tilted orbits moreinclined. This is exactly what one would have predicted by extrapolating frompost-Newtonian theory [15]. The rate at which this inclination angle increases israther slow the aspect of the arrows in Figure 1 is nearly flat. Indeed, th evalue of di/dt in this strong-field regime is roughly 3 t imes sm aller than w h a t post-Newtonian theory predicts2 . A n interesting fea ture of Figure 1 is the very longarrow at r 7 A/, L ~ 120. This orbit lies extrem ely close to the marginal ly stableorbi t: it is at L = 119.194; the marginal ly stable orbit is at L = 119.670. Thisorbi t is barely dynamically stable, so a small push has drastic effects.The right panel of Figure 1 shows a portion of the gravitat ional waveform emit tedduring inspiral. The central black hole in this case ha s a 0.95M; it lies at2) Recent analyses ar e showing that in the extreme strong-field of rapidly rotating holes, the incli-nation angle changes rather more dramatically, and in the opposite direction: radiation reactiontends to decrease the inclination angle. This work is in progress [16].

80



7/7

luminosity distance D from th e detector. The waveform here is shown as the smal lbody passes through r = 7M , i 62.43 . Note th e low-frequency modula t ionof both polar izat ions. This is due to the f rame dragging induced precession ofth e orbi ta l plane Lense-Thirring precession. Note also th e many sharp, short-t imescale features present in the two polar izat ions. When the spin is high, m an yharmonics of the small body's fundamenta l orbi ta l frequencies contribute to thegravitational waveform. This leads to a rather complicated structure; the energyspectrum corresponding to these waveforms extends to rather high frequencies.Accurate measurement of such complicated waveforms will be quite a challenge.However, the payoff is likely to be immense.I thank K ip Thorne and Steinn Sigurdsson for help and advice in writ ing thista lk; I also thank Ramesh Narayan for allowing me to use a pre l iminary draft of Ref.[9]. I am indebted to m an y people who helped me construct my radiation reactioncode, including (but no t limited to ) Sam Finn, Daniel Kennefick, Yuri Levin, Am osOri, Sterl Phinney, and "The Capra Gang": Lior Burko, Patrick Brady, EannaFlanagan, Eric Poisson, and Alan Wiseman. This research was support by NSFGrants AST-9731698 an d AST-9618537, an d NA SA Grants NAG5-6840 and NAG5-7034.

REFERENCES1. K. Danzmann et a/., LISA Proposal for a Laser-Interferometric Gravitat ional

W a v e Detector in 5pace, Max-Planck-Institut fu r Quantenoptik, Report M P Q 2 3 3(1998) .2. Y. Mino , M. Sasaki , M. Shibata, H. Tagoshi, and T. Tanaka, Prog. Theor. Phys.Supplement No. 128 (1997) .3 . L. S. Finn and K. S. Thorne, in preparation.4. F. D. Ryan, Phys. Rev. D 56, 1845 ( 1997) .5. S. Sigurdsson and M. J. Rees, Mon. Not . R. Astron. Soc. 284, 318 (1997) .6. S. Sigurdsson, Class. Quantum Grav. 14 , 1425 (1997) .7. S. Sigurdsson, private communication.8. D. Hils and P. L. Bender, Astrophys. J. 447, L7 (1995) .9. R. Narayan, Astrophys. J. submitted; also astro-ph/9907328.10. R. Narayan, R . Mahadevan, and E. Quataert, in Theory of Black Hole AccretionDi s k s , edited by M. A. Abramowicz , G. Bjornsson, and J. E. Pringle (CambridgeUniversity Press, Cambridge, 1998) .

11. S. A. Teukolsky, Astrophys. J. 185, 635 (1973) .12 . Y. M ino , these proceedings; L. Burko, these proceedings.13 . D . Kennefick and A. Ori, Phys. Rev. D 53, 4319 (1996); F. D. Ryan, Phys. Rev. D53, 3064 (1996) ; Y. Mino , unpublished Ph.D. thesis, Kyoto University (1996) .14. S . A. Hughes, Phys. Rev. D, in press; also gr-qc/9910091.15. F. D. Ryan, Phys. Rev. D 52, R3159 (1995) .16. S. A. Hughes, in preparation.

81

Scott A. Hughes- Gravitational waves from inspiral into massive black holes

Documents

Transcript of Scott A. Hughes- Gravitational waves from inspiral into massive black holes