Superradiant sca.ering in astrophysical binary systems

Transcription

1 Superradiant sca.ering in astrophysical binary systems João G. Rosa University of Aveiro Phys. Le.. B749, 226 (2015) [arxiv: [gr- qc]] + work in progress VIII Black Holes Workshop, IST Lisbon, 22 December 2015

2 Black hole superradiance Sca.ering in the Kerr space- ]me can result in amplifica]on of par]cular wave modes: Ω ω <mω Amplifica]on of up to 4.4% for EM waves and 138% for GW! [Zeldovich (1971); Starobinsky & Churilov (1973); Teukolsky & Press ( )] Does this occur in astrophysical systems? 2

3 Black hole superradiance Angular velocity of a Kerr black hole: Ω = ac 10M r khz + a2 M ã 1+ 1 ã 2 where ã = ac 2 /GM and J = amc. Need sources of low- frequency radia]on dominated by superradiant modes close to a black hole 3

4 Black hole superradiance The sca.ering problem can be studied using the Newman-. Penrose scalars: [Newman & Penrose (1962)] ψ 0 = C µνρσ l µ m ν l ρ m σ, ψ 4 = C µνρσ n µ m ν n ρ m σ. l µ = n µ = m µ = r 2 + a 2, 1, 0, a, 1 r 2 2ρ 2 + a 2,, 0,a, 1 i ia sin θ, 0, 1, 2 ρ sin θ, [Kinnersley (1969)] = r 2 + a 2 2Mr ρ = r + ia cos θ 4

5 Black hole superradiance The NP scalars admit a mode decomposi]on of the form:. ρ 4 ψ 4 = e iωt+imφ S jm (θ)r jm (r) j,m,ω which yields an angular equa]on (spheroidal harmonics) and the radial Teukolsky equa]on: [Teukolsky & Press ( )]. r 2 R jm + 2(s + 1)(r M) r R jm K 2 2is(r M)K + +4isωr A jm R jm =0 K(r) =(r 2 + a 2 )ω ma 5

6 Sca.ering of plane gravita]onal waves We then obtain:. de jm out dt where: Z jm (ω) Thus, superradiant modes have: =[1+Z jm (ω)] dejm in dt (ω mω), ω > 0 ( ω + mω), ω < 0 Also: (ω > 0 m>0) (ω < 0 m<0) 1 Z jm (ω) 1.38 [Teukolsky & Press ( )] 6

7 Sca.ering of plane gravita]onal waves A source sufficiently far from the BH will produce plane waves. Assume a generic plane wave at spa]al infinity as boundary condi]on: h ij = 1 2 e iωt+ik r h + e + ij + h e ij +c.c. e + ij =. sin 2 φ 0 cos 2 θ 0 cos 2 φ 0 (1 + cos 2 θ 0 ) cos φ 0 sin φ 0 sin θ 0 cos θ 0 cos φ 0 (1 + cos 2 θ 0 ) cos φ 0 sin φ 0 cos 2 φ 0 cos 2 θ 0 sin 2 φ 0 sin θ 0 cos θ 0 sin φ 0 sin θ 0 cos θ 0 cos φ 0 sin θ 0 cos θ 0 sin φ 0 sin 2 θ 0 e ij = cos θ 0 sin(2φ 0 ) cos θ 0 cos(2φ 0 ) sin θ 0 sin φ 0 cos θ 0 cos(2φ 0 ) cos θ 0 sin(2φ 0 ) sinθ 0 cos φ 0 sin θ 0 sin φ 0 sin θ 0 cos φ 0 0 7

8 Sca.ering of plane gravita]onal waves ψ 4 Boundary condi]on determined by NP scalars at spa]al. infinity (flat space): (h + ih )ω 2 e iωt jm a (2) e ikr j (kr) 5 + e ikr b(2) j 2Y kr jm(ˆk) 2 Y jm (ˆr) + (ω ω,h +, h +, ) The incident energy flux for each angular mode is then: h + ih 2 2 Y jm (θ 0, φ 0 ) 2, ω > 0 de jm in dt = π2 8 h + + ih 2 2 Y jm (θ 0, φ 0 ) 2, ω < 0 8

9 Sca.ering of plane gravita]onal waves Π 2 Π 2Y2m(θ0, φ0) m 1 m 2 m 0 m 2 m Π 2 Π θ 0 9

10 Sca.ering of plane gravita]onal waves Conclusion: For amplifica]on of a plane wave we need: 1) polariza]on: h + ih = h + + ih 2) incident direc]on parametrically close to the BH spin axis 3) low frequency: ω < m Ω Can we obtain this in realis]c systems? 10

11 Quadrupole formula (TT gauge) h ij = 2G c 4 1 r r p Pi k P l j 1 2 P ijp kl Q jk t r r p c where P ij = δ ij n i n j Q ij = I ij I k k δ ij Note that for an oscilla]ng quadrupole at large distance: Q ij e iω(t r r p /c) e iωt+ik r e ik r p 11

12 In ellipsoid frame: Qîĵ = Example 1: rota]ng ellipsoid [Bonazzola & Gourgoulhon (1995)] Qẑẑ / Qẑẑ / Qẑẑ z ω p ẑ In BH frame: Q ij = 3 2 Q ẑẑω 2 p sin β cos βe iωpt T (1) ij +2sinβe 2iωpt T (2) ij + c.c. T (1) ij = 0 0 i i 1 0, T (2) ij = 1 ±i 0 ±i

13 Example 1: rota]ng ellipsoid For example, for the double- frequency mode we obtain: h + ih 2 Y 2, 2(θ p, φ p ) h + + ih 2 Y 2,±2 (θ p, φ p ) so that waves are indeed polarized and amplifica]on can occur when the ellipsoid is parametrically close to the BH spin axis. Applica]on: pulsars orbi4ng stellar black holes! ω p Ω few khz Advanced LIGO/VIRGO 13

14 Example 2: compact binary z ω For a Keplerian circular orbit with two equal mass stars: Q ij = 2ω 2 pmr 2 e 2iωt T (2) ij + c.c The emi.ed GW are polarized like in the previous example! 14

15 Example 2: compact binary Applica]on: compact binary orbi]ng a supermassive BH 10 9 M Ω 0.1 mhz M BH LISA P binary 17 MBH 10 9 M hrs Note: Hulse- Taylor binary pulsar has period of 7.75 hours. 15

16 GW luminosity modula]on aligned P GW P GW N anti aligned S Orbital periods ω =2ω p =2Ω, ã =0.999 L p = 10λ 16

17 Summary Superradiant amplifica]on requirements (plane waves): 1. Low frequency 2. Polariza]on 3. Incidence parametrically close to BH spin axis There are realis]c systems that meet these criteria: 1. Pulsar + stellar BH 2. Compact binary + SMBH system Poten]al observa]onal signatures like luminosity modula]on 17