Gravitational self force in extreme-mass-ratio binary inspirals - PowerPoint PPT Presentation

Gravitational self force in extreme-mass-ratio binary inspirals Leor Barack University of Southampton (UK) December 16, 2010 IHES seminar December 16, 2010 1 / 26

Theory Meets Data Analysis at Comparable and Extreme Mass Ratios Perimeter Institute, June 2010 Conference summary by Steve Detweiler [arXiv 1009.2726, 15 September 2010] . . . As a member of the Capra community, I am pleased to report that we are reaching the end of a long, difficult adolescence. In the self-force portion of the meeting, a few serious meaningful applications of the gravitational self-force were described that allow for detailed comparisons among each other as well as with corresponding post-Newtonian analyses. The gravitational self-force has arrived. . . . IHES seminar December 16, 2010 2 / 26

In this review: Motivation: EMRIs as sources for LISA Self force theory Implementation methods Conservative effects of the gravitational self force IHES seminar December 16, 2010 3 / 26

2-body problem in relativity IHES seminar December 16, 2010 4 / 26

EMRIs as probes of strong-field gravity EMRI parameter extraction accuracies with LISA (SNR=30) S / M 2 0 . 1 0 . 1 0 . 5 0 . 5 1 1 0 . 1 0 . 3 0 . 1 0 . 3 0 . 1 0 . 3 e LSO ∆M / M 2 . 6e − 4 5 . 6e − 4 2 . 7e − 4 9 . 2e − 4 2 . 8e − 4 2 . 5e − 4 ∆ ( S / M 2 ) 3 . 6e − 5 7 . 9e − 5 1 . 3e − 4 6 . 3e − 4 2 . 6e − 4 3 . 7e − 4 ∆m / m 6 . 8e − 5 1 . 5e − 4 6 . 8e − 5 9 . 2e − 5 6 . 1e − 5 9 . 1e − 5 ∆( e 0 ) 6 . 3 e − 5 1 . 3 e − 4 8 . 5 e − 5 2 . 8 e − 4 1 . 2 e − 4 1 . 1 e − 4 ∆(cos λ ) 6 . 0 e − 3 1 . 7 e − 2 1 . 3 e − 3 5 . 8 e − 3 6 . 5 e − 4 8 . 4 e − 4 ∆(Ω s ) 1 . 8 e − 3 1 . 7 e − 3 2 . 0 e − 3 1 . 7 e − 3 2 . 1 e − 3 1 . 1 e − 3 ∆(Ω K ) 5 . 6 e − 2 5 . 3 e − 2 5 . 5 e − 2 5 . 1 e − 2 5 . 6 e − 2 5 . 1 e − 2 ∆[ln( µ/ D )] 8 . 7 e − 2 3 . 8 e − 2 3 . 8 e − 2 3 . 7 e − 2 3 . 8 e − 2 7 . 0 e − 2 ∆( t 0 ) ν 0 4 . 5 e − 2 1 . 1 e − 1 2 . 3 e − 1 1 . 3 e − 1 2 . 5 e − 1 3 . 2 e − 2 [LB & Cutler (2004)] IHES seminar December 16, 2010 5 / 26

“Self force” description of the motion Equations of motion 1 mu β ∇ β u α = F α self ( ∝ m 2 ) 2 � ¯ µν + 2 R α β µ ν ¯ h ret h ret αβ = − 16 π T µν self (¯ 3 F α self = F α h ret αβ ) = ? IHES seminar December 16, 2010 6 / 26

“Self force” description of the motion Equations of motion 1 mu β ∇ β u α = F α self ( ∝ m 2 ) 2 � ¯ µν + 2 R α β µ ν ¯ h ret h ret αβ = − 16 π T µν self (¯ 3 F α self = F α h ret αβ ) = ? Challenges: regularization make sense of “point particle” in curved space self-interaction is not instantaneous in curved space (“tail” effect) self force (and orbit) are gauge dependent Lorenz-gauge condition dictates geodesic motion IHES seminar December 16, 2010 6 / 26

Regularization: Dirac’s method and its failure in curved space Decomposition of the EM vector potential for an electron in flat space: 1 α ) + 1 A ret 2( A ret α + A adv 2( A ret α − A adv = α ) α F α self = e ∇ αβ A R → ≡ A S ≡ A R β α α Symmetric / Singular Radiative / Regular FLAT IHES seminar December 16, 2010 7 / 26

Regularization: Dirac’s method and its failure in curved space Decomposition of the EM vector potential for an electron in flat space: 1 α ) + 1 A ret 2( A ret α + A adv 2( A ret α − A adv = α ) α F α self = e ∇ αβ A R → ≡ A S ≡ A R β α α Symmetric / Singular Radiative / Regular Difficulty: Local Radiative potential becomes non-causal in curved space! FLAT CURVED IHES seminar December 16, 2010 7 / 26

Regularization of the gravitational self-force Mino, Sasaki & Tanaka (1997): via Hadamard expansion + integration across in a thin worldtube Mino, Sasaki & Tanaka (1997), Poisson (2003), Pound (2010): via Matched Asymptotic Expansions Quinn & Wald (1997): via an axiomatic approach based on comparison to flat space Gralla& Wald (2008): by taking “far/near”-zone limits of a family of spacetimes Harte (2010): from generalized Killing fields IHES seminar December 16, 2010 8 / 26

The gravitational self-force F α x → z ( τ ) ∇ αµν h tail = lim self µν x → z ( τ ) ∇ αµν � � h ret µν − h dir = lim µν IHES seminar December 16, 2010 9 / 26

Detweiler–Whiting reformulation (2003) Dirac-like decomposition of h ret αβ for a mass particle in curved space: 1 αβ ) − H αβ + 1 h ret 2( h ret αβ + h adv 2( h ret αβ − h adv = αβ ) + H αβ αβ → F α self = m ∇ αβγ h R ≡ h S ≡ h R βγ αβ αβ Symmetric / Singular Radiative / Regular IHES seminar December 16, 2010 10 / 26

Detweiler–Whiting reformulation (2003) Dirac-like decomposition of h ret αβ for a mass particle in curved space: 1 αβ ) − H αβ + 1 h ret 2( h ret αβ + h adv 2( h ret αβ − h adv = αβ ) + H αβ αβ → F α self = m ∇ αβγ h R ≡ h S ≡ h R βγ αβ αβ Symmetric / Singular Radiative / Regular h R αβ is a vacuum solution of the Einstein equations. Interpretation: orbit is a geodesic of g αβ + h R αβ . IHES seminar December 16, 2010 10 / 26

Mode-sum method [LB & Ori (2000-2003)] Define F ret / S ≡ m ∇ h ret / S (as fields), then write = ( F ret − F S ) | p F self ∞ � � �� F ℓ ret − F ℓ = ( ℓ -mode contributions are finite) � S p ℓ =0 ∞ ∞ � � � � � � F ℓ F ℓ = ret ( p ) − AL − B − C / L − S ( p ) − AL − B − C / L ℓ =0 ℓ =0 ∞ � � � F ℓ = ret ( p ) − AL − B − C / L − D (where L = ℓ + 1 / 2) ℓ =0 IHES seminar December 16, 2010 11 / 26

Mode-sum method [LB & Ori (2000-2003)] Define F ret / S ≡ m ∇ h ret / S (as fields), then write = ( F ret − F S ) | p F self ∞ � � �� F ℓ ret − F ℓ = ( ℓ -mode contributions are finite) � S p ℓ =0 ∞ ∞ � � � � � � F ℓ F ℓ = ret ( p ) − AL − B − C / L − S ( p ) − AL − B − C / L ℓ =0 ℓ =0 ∞ � � � F ℓ = ret ( p ) − AL − B − C / L − D (where L = ℓ + 1 / 2) ℓ =0 Regularization Parameters A α , B α , C α , D α derived analytically for generic orbits in Kerr [LB & Ori (2003), LB (2009)]. IHES seminar December 16, 2010 11 / 26

Implementations so far (geodesic orbits, no evolution yet) year Schwarzschild Kerr 2000 static 2000 head-on 2001 static 2002 head-on 2003 circular 2007 eccentric 2007 static 2007 circular 2009 circular-equatorial 2009 eccentric 2010 eccentric-equatorial 2010 circular-inclined gravitational self force / scalar-field toy model IHES seminar December 16, 2010 12 / 26

The gauge problem Original regularization formulated in Lorenz gauge (div ¯ h = 0). ◮ Linearized Einstein equation takes a neat hyperbolic form ◮ Particle singularity is “isotropic” and Coulomb-like IHES seminar December 16, 2010 13 / 26

The gauge problem Original regularization formulated in Lorenz gauge (div ¯ h = 0). ◮ Linearized Einstein equation takes a neat hyperbolic form ◮ Particle singularity is “isotropic” and Coulomb-like Unfortunately Lorenz-gauge equations are not easily amenable to numerical treatment. Options: Work out the singular gauge transformations, or develop methods to integrate the Lorenz-gauge equations. IHES seminar December 16, 2010 13 / 26

Direct Lorenz-gauge implementation [LB & Lousto (2005)] Start with 10 coupled perturbation equations + 4 gauge conditions: � ∞ δ [ x µ − z µ ( τ )] � ¯ h αβ + 2 R µανβ ¯ h µν = − 16 π m √− g u α u β d τ −∞ Z α ≡ ∇ β ¯ h αβ = 0 Add “constraint damping” terms, − κ t ( α Z β ) IHES seminar December 16, 2010 14 / 26

Direct Lorenz-gauge implementation [LB & Lousto (2005)] Start with 10 coupled perturbation equations + 4 gauge conditions: � ∞ δ [ x µ − z µ ( τ )] � ¯ h αβ + 2 R µανβ ¯ h µν = − 16 π m √− g u α u β d τ −∞ Z α ≡ ∇ β ¯ h αβ = 0 Add “constraint damping” terms, − κ t ( α Z β ) Expand in tensor harmonics, 10 � � ¯ h ( i ) lm ( r , t ) Y ( i ) lm h αβ = αβ l , m i =1 Obtain 10 coupled scalar-like eqs for h ( i ) lm ( r , t ) IHES seminar December 16, 2010 14 / 26

Direct Lorenz-gauge implementation [LB & Lousto (2005)] Start with 10 coupled perturbation equations + 4 gauge conditions: � ∞ δ [ x µ − z µ ( τ )] � ¯ h αβ + 2 R µανβ ¯ h µν = − 16 π m √− g u α u β d τ −∞ Z α ≡ ∇ β ¯ h αβ = 0 Add “constraint damping” terms, − κ t ( α Z β ) Expand in tensor harmonics, 10 � � ¯ h ( i ) lm ( r , t ) Y ( i ) lm h αβ = αβ l , m i =1 Obtain 10 coupled scalar-like eqs for h ( i ) lm ( r , t ) Solve numerically using time-domain evolution in characteristic coordinates Use as input for the mode-sum formula IHES seminar December 16, 2010 14 / 26

Sample numerical results [LB & Sago (2010)] Gravitational self-force in Schwarzschild ( p , e ) = (10 M , 0 . 2) ( p , e ) = (10 M , 0 . 5) 0.0025 0.006 F t F t (p,e)=(10,0.2) (p,e)=(10,0.5) F r F r /10 /10 0.002 0.004 0.0015 0.002 0.001 (M/ µ ) 2 F α (M/ µ ) 2 F α 0 0.0005 0 -0.002 -0.0005 -0.004 -0.001 -400 -300 -200 -100 0 100 200 300 400 -400 -200 0 200 400 t [in unit of M solar ] t [in unit of M solar ] IHES seminar December 16, 2010 15 / 26