Gravitational wave fluxes at second-order in the mass-ratio h 2 R = - PowerPoint PPT Presentation

Gravitational wave fluxes at second-order in the mass-ratio □ h 2 R = δ 2 G [ h 1 , h 1 ] − □ h 2 P Niels Warburton Royal Society - SFI University Research Fellow University College Dublin Collaborators: Jeremy Miller, Adam Pound, Barry Wardell Advances and Challenges in Computational Relativity (virtual)@ICERM - 16th Sep. 2020

Overview Motivation Structure of the calculation Comparison with PN Comparison with NR

Motivation: extreme mass-ratio inspirals • Binary with an extremely small 
 mass ratio ϵ = m 2 / m 1 ≪ 1 m 1 • Primary: massive black hole • Secondary: compact object such as a 
 m 2 stellar-mass black hole, neutron star • For LISA EMRIs: = 10 -4 - 10 -7 ϵ Image credit: A. Pound Key Features: • Millihertz gravitational-wave source • Over 100,000+ orbits in strong field • Visible for months to years in LISA band • No spin alignment expected • Considerable eccentricity • Rich waveform phenomenology • Very low instantaneous SNR in LISA

Motivation: intermediate mass-ratio inspirals IMRIs: ϵ ≃ 10 − 2 − 10 − 4 Periastron advance: Le Tiec+, arXiv:1106.3278 Waveform phase: van de Meent + Pfei ff er, arXiv:2006.12036 For quasi-circular inspirals into non-rotating black holes there is evidence that the perturbation theory results can be very e ff ective even at large mass ratios

Black Hole Perturbation Theory A key question in any perturbative expansion is: how high in the expansion do I need to go in order to capture the physics I am interested in? Φ = ϵ − 1 Φ − 1 + Φ 0 + 𝒫 ( ϵ ) Waveform phase: Post-Adiabatic order Adiabatic From the orbit averaged piece Two contributions: of first-order self-force ⟨ F α 1 ⟩ •Oscillatory pieces of the first order self-force ⟨ F α 1 ⟩ can be related to the •Second-order orbit averaged fluxes, thus avoiding a local self-force ⟨ F α 2 ⟩ calculation of the self-force Good enough for detection and Needed to extract all sources rough parameter estimation for Needed for precision tests of GR astrophysics of EMRIs of bright Potential application to IMRIs sources

Black Hole Perturbation Theory: field equations g αβ + ϵ h (1) αβ + ϵ 2 h (2) G αβ [¯ αβ ] = 8 π T αβ ϵ n Field equations from coe ffi cients: ϵ 0 : G αβ [¯ g ] = 0 ϵ 1 : G 1 αβ [ h 1 ] = 8 π T αβ ϵ 2 : G 1 αβ [ h 2 ] + G 2 αβ [ h 1 , h 1 ] = 0

Black Hole Perturbation Theory: field equations g αβ + ϵ h (1) αβ + ϵ 2 h (2) G αβ [¯ αβ ] = 8 π T αβ ϵ n Field equations from coe ffi cients: ϵ 0 : G αβ [¯ g ] = 0 ϵ 1 : G 1 αβ [ h 1 ] = 8 π T αβ ϵ 2 : G 1 αβ [ h 2 ] = − G 2 αβ [ h 1 , h 1 ]

Black Hole Perturbation Theory: field equations g αβ + ϵ h (1) αβ + ϵ 2 h (2) G αβ [¯ αβ ] = 8 π T αβ ϵ n Field equations from coe ffi cients: ϵ 0 : G αβ [¯ g ] = 0 ϵ 1 : αβ [ h 1 S + h 1 R ] = 8 π T αβ G 1 ϵ 2 : αβ [ h 2 S + h 2 R ] = − G 2 G 1 αβ [ h 1 , h 1 ]

Black Hole Perturbation Theory: field equations g αβ + ϵ h (1) αβ + ϵ 2 h (2) G αβ [¯ αβ ] = 8 π T αβ ϵ n Mino, Sasaki, Tanaka 1997 Field equations from coe ffi cients: Quinn and Wald 1997 ϵ 0 : MiSaTaQuWa equations G αβ [¯ g ] = 0 ϵ 1 : G 1 αβ [ h 1 R ] = 8 π T αβ − G 1 αβ [ h 1 S ] ϵ 2 : G 1 αβ [ h 2 R ] = − G 2 αβ [ h 1 , h 1 ] − G 1 αβ [ h 2 S ] Pound 2012 Equations of motion Gralla 2012 u β ∇ β u α = F α self [ ∇ h 1 R , ∇ h 2 R ] - Non-compact - Diverges at the particle

Frequency domain implementation We perform a two-timescale expansion by introducing a “slow time” ˜ . This allows for a frequency domain decomposition: t = ϵ t □ 0 R 2 R = 2 δ 2 G 0 − □ 0 R 2 P − □ 1 R 1 t h 1 = · r 0 ∂ r 0 h 1 ∂ ˜ Hereafter we focus on quasi-circular inspirals into a Schwarzschild black hole ∇ β ¯ Work in the Lorenz gauge and solve radial equations h αβ = 0 on hyperboloidal slices From the monopole mode we can calculate the second-order binding energy: Phys. Rev. Lett. 124 2, 021101, arXiv:1908.07419 R 2 R By evaluating at infinity we can compute the second- ℱ (2) order flux to infinity See Miller+Pound, arXiv:2006.11263 for TT details

Expansion in the symmetric mass ratio So far we have been expanding using the small mass-ratio ϵ = m 2 / m 1 Let’s also introduce the large mass-ratio and the q = m 1 / m 2 = 1/ ϵ symmetric mass-ratio: ν = m 1 m 2 q where M = m 1 + m 2 = M 2 (1 + q ) 2 x = ( M Ω ) 2/3 Also instead of parametrising the orbit by we will use r 0 Using these definitions we can rewrite ℱ ( r 0 , ϵ ) = ϵ 2 · E (1) ( r 0 ) + ϵ 3 · E (2) ( r 0 ) + O ( ϵ 4 ) the form ℱ ( x , ν ) = ν 2 · ν ( x ) + ν 3 · E (1) E (2) ν ( x ) + O ( ν 4 ) where · ν = · E (1) , · ν = · ν ( · E (1) , · E (2) , d · E (1) E (2) E (2) E (1) / dx )

Comparison with post-Newtonian theory For this talk, let’s look at the mode l = 3, m = 1 The (3,3) and (3,1) fluxes were derived to 3.5 PN order in Faye+ arXiv:1409.3546 31 = ( 315 ) x 6 + ( − 4 ν 2 945 ) x 7 + ( 315 ) x 15/2 + O ( x 8 ) 1260 − ν 3 ν 2 945 + ν 3 63+ 4 ν 4 630 − 2 πν 3 πν 2 ℱ PN O ( v 3 ) We want to compare agains the pieces of this = − x 6 315 + x 7 315 π x 15/2 − 1291 x 8 2 31185 + 13 ℱ (2) PN 420 π x 17/2 63 − 31 ( ) + O ( x 19/2 ) − 389 π 2 7945938000 − log 2 (1024) + 4 log 2 (2) 26 log( x ) 120960 + 52 γ 6615 − 117030737 + 52 log(2) + x 9 6615 7875 315 6615 O ( ν 3 ) This is all the known terms at for the (3,1) mode up to 3.5PN

Comparison with post-Newtonian theory 10 - 7 10 - 9 10 - 11 ( 2 ) � � 10 - 13 10 - 15 ( l,m )=( 3,1 ) 10 - 17 0.02 0.05 0.10 0.20 x =( M � ) 2 / 3 = − x 6 315 + x 7 315 π x 15/2 − 1291 x 8 2 31185 + 13 ℱ (2) PN 420 π x 17/2 63 − 31 ( ) + O ( x 19/2 ) − 389 π 2 7945938000 − log 2 (1024) + 4 log 2 (2) 26 log( x ) + 52 log(2) 120960 + 52 γ 6615 − 117030737 + x 9 6615 7875 315 6615

Comparison with post-Newtonian theory 10 - 7 10 - 9 10 - 11 ( 2 ) � � 10 - 13 10 - 15 ( l,m )=( 3,1 ) 10 - 17 0.02 0.05 0.10 0.20 x =( M � ) 2 / 3 = − x 6 315 + x 7 315 π x 15/2 − 1291 x 8 2 31185 + 13 ℱ (2) PN 420 π x 17/2 63 − 31 ( ) + O ( x 19/2 ) − 389 π 2 7945938000 − log 2 (1024) + 4 log 2 (2) 26 log( x ) + 52 log(2) 120960 + 52 γ 6615 − 117030737 + x 9 6615 7875 315 6615

Comparison with post-Newtonian theory 10 - 7 10 - 9 10 - 11 ( 2 ) � � 10 - 13 10 - 15 ( l,m )=( 3,1 ) 10 - 17 0.02 0.05 0.10 0.20 x =( M � ) 2 / 3 = − x 6 315 + x 7 315 π x 15/2 − 1291 x 8 2 31185 + 13 ℱ (2) PN 420 π x 17/2 63 − 31 ( ) + O ( x 19/2 ) − 389 π 2 7945938000 − log 2 (1024) + 4 log 2 (2) 26 log( x ) + 52 log(2) 120960 + 52 γ 6615 − 117030737 + x 9 6615 7875 315 6615

Comparison with post-Newtonian theory 10 - 7 10 - 9 10 - 11 ( 2 ) � � 10 - 13 10 - 15 ( l,m )=( 3,1 ) 10 - 17 0.02 0.05 0.10 0.20 x =( M � ) 2 / 3 = − x 6 315 + x 7 315 π x 15/2 − 1291 x 8 2 31185 + 13 ℱ (2) PN 420 π x 17/2 63 − 31 ( ) + O ( x 19/2 ) − 389 π 2 7945938000 − log 2 (1024) + 4 log 2 (2) 26 log( x ) + 52 log(2) 120960 + 52 γ 6615 − 117030737 + x 9 6615 7875 315 6615

Comparison with post-Newtonian theory 10 - 7 10 - 9 10 - 11 ( 2 ) � � 10 - 13 |- 0.15 x 19 / 2 | reference line 10 - 15 ( l,m )=( 3,1 ) 10 - 17 0.02 0.05 0.10 0.20 x =( M � ) 2 / 3 Can estimate unknown O ( ν 3 ) PN terms = − x 6 315 + x 7 315 π x 15/2 − 1291 x 8 2 31185 + 13 ℱ (2) PN 420 π x 17/2 63 − 31 ( ) + O ( x 19/2 ) − 389 π 2 7945938000 − log 2 (1024) + 4 log 2 (2) 26 log( x ) + 52 log(2) 120960 + 52 γ 6615 − 117030737 + x 9 6615 7875 315 6615

̂ Comparison with numerical relativity For this comparison it’s useful to consider the flux normalised by the leading PN coe ffi cient, e.g., for the (2,2) PN flux we have 22 = 32 ν 2 x 5 + 32 105 ν 2 (55 ν − 107) x 6 + 128 5 πν 2 x 13/2 + O ( x 7 ) ℱ PN 5 22 = ℱ PN = 1 + 1 21 (55 ν − 107) x + 4 π x 3/2 + O ( x 2 ) 22 ℱ PN ℱ 0 PN 22 To compute the NR flux we write the waveform as h lm ( t ) = A lm ( t ) e i Φ lm ( t ) 1 16 π | · x ( t ) = ( M · h lm ( t ) | 2 ℱ NR Φ ( t )/ m ) 2/3 lm ( t ) = From these two we can compute ℱ NR lm ( x )

Comparison with numerical relativity 0.94 innermost stable circular orbit 0.92 3.5PN 0.90 � / � 0 PN 0.88 0.86 0.84 q = 10 ( l,m )=( 2,2 ) 0.82 0.05 0.10 0.15 0.20 x =( M � ) 2 / 3 3.5PN series from Faye+ arXiv:1204.1043

Comparison with numerical relativity 0.94 innermost stable circular orbit 3.5PN 0.92 NR 0.90 � / � 0 PN 0.88 0.86 0.84 q = 10 ( l,m )=( 2,2 ) 0.82 0.05 0.10 0.15 0.20 x =( M � ) 2 / 3 3.5PN series from Faye+ arXiv:1204.1043 NR data from SXS:BBH:1132

Comparison with numerical relativity 0.94 innermost stable 3.5PN circular orbit 0.92 NR 1GSF � 0.90 � / � 0 PN 0.88 0.86 0.84 q = 10 ( l,m )=( 2,2 ) 0.82 0.05 0.10 0.15 0.20 x =( M � ) 2 / 3 3.5PN series from Faye+ arXiv:1204.1043 NR data from SXS:BBH:1132

Comparison with numerical relativity 0.94 3.5PN innermost stable circular orbit NR 0.92 1GSF � 2GSF � 0.90 � / � 0 PN 0.88 0.86 0.84 q = 10 ( l,m )=( 2,2 ) 0.82 0.05 0.10 0.15 0.20 x =( M � ) 2 / 3 3.5PN series from Faye+ arXiv:1204.1043 NR data from SXS:BBH:1132