Self-force calculations: synergies and invariants · 2016-07-28 ·...
Transcript of Self-force calculations: synergies and invariants · 2016-07-28 ·...
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Self-force calculations:synergies and invariants
Sam R Dolan
University of Sheffield, UK.
Perturbation Methods in GR @ Fields Institute, 20th May 2015.
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Overview
1 Motivation
GW astronomy with Adv. LIGO
2 Synergies: a review from 2012
ISCO shift and periastron advanceCalibrating EOB
3 Tides
Asymptotic matched expansionsInvariants for circular orbitsOctupoles
4 Effective One-Body theory
Tidally-interacting neutron stars
5 Prospects
Eccentric orbits. Kerr.
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“Large Two Forms”
Art Gallery of Ontario
Fµνωµ ∧ ων
“Large Two-Forms”
Fields Institute
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Intro Review Tides Octupoles EOB Summary
“Large Two Forms”
Art Gallery of Ontario
Fµνωµ ∧ ων
“Large Two-Forms”
Fields Institute
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Intro Review Tides Octupoles EOB Summary
Motivation: the general 2-body problem in relativity
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Intro Review Tides Octupoles EOB Summary
Motivation: the general 2-body problem in relativity
Effective One-Body (EOB) model [Buonanno & Damour 1999].
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Motivation: gravitational-wave astronomy
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Intro Review Tides Octupoles EOB Summary
Self-Force: Two complementary viewpoints
accelerated motion on abackground spacetime
µ~ag = ~Fself
m
geodesic motion in a perturbedregular vacuum spacetimeg + hR
µ~ag+hR = 0
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Self-Force ⇔ motion in a regular perturbed spacetime
Detweiler-Whiting split (’03):
h = hR + hS
R for Radiative / Regular
S for Symmetric / Singular
Motion of non-spinning compact body is geodesic ingR = g + hR.
We can compute hR at first-order O(µ/M), up to gaugefreedom,
hµν → hµν + ξ(µ;ν)
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Intro Review Tides Octupoles EOB Summary
Self-Force ⇔ motion in a regular perturbed spacetime
Detweiler-Whiting split (’03):
h = hR + hS
R for Radiative / Regular
S for Symmetric / Singular
Motion of non-spinning compact body is geodesic ingR = g + hR.
We can compute hR at first-order O(µ/M), up to gaugefreedom,
hµν → hµν + ξ(µ;ν)
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Intro Review Tides Octupoles EOB Summary
Three (related) methods for GSF calculations
1 Worldline integral (MiSaTaQuWa equation, schematically):
F selfa = local terms+µ2uµuν
∫ τ−
−∞∇[αGµ]νµ′ν′(z(τ), z(τ ′)uµ
′uν′dτ ′
2 Mode sum regularization: hretab =
∑ilm h
(i)lmab Y
(i)lm (θ, φ)
F aself =∞∑`=0
[F `ret(p)−AL−B − C/L
]−D
where L = l + 1/2.
3 Effective source / puncture schemes hR = h− hS
F aself = −µ2
(gab + uaub
) (2hRbc;d − hRcd;b
)ucud.
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Intro Review Tides Octupoles EOB Summary
Three (related) methods for GSF calculations
1 Worldline integral (MiSaTaQuWa equation, schematically):
F selfa = local terms+µ2uµuν
∫ τ−
−∞∇[αGµ]νµ′ν′(z(τ), z(τ ′)uµ
′uν′dτ ′
2 Mode sum regularization: hretab =
∑ilm h
(i)lmab Y
(i)lm (θ, φ)
F aself =∞∑`=0
[F `ret(p)−AL−B − C/L
]−D
where L = l + 1/2.
3 Effective source / puncture schemes hR = h− hS
F aself = −µ2
(gab + uaub
) (2hRbc;d − hRcd;b
)ucud.
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Intro Review Tides Octupoles EOB Summary
Harte (2012): Mechanics of extended masses in GR
A compact body with mass µM and spin s Gµ2/cundergoes parallel transport in the regular perturbedspacetime gR = g + hR.
ub∇bua = 0,
ub∇bsa = 0.
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Example: spin precession on circular orbit
ψ ≡ Precession angle per orbit / 2π.
∆ψ at O(µ) is gauge invariant.
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Synergies: Recap from Jan 2012
What were we excited about three years ago?
1 Comparison of gauge-invariant results with Post-Newtoniantheory (PN) and Numerical Relativity (NR):
ISCO shift due to conservative part of GSF
Perihelion advance of eccentric orbits
Benefits of using ‘symmetric mass-ratio’
2 Calibration of Effective One-Body (EOB) theory with GSF
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Synergies: Comparisons (I) The redshift invariant
Circular geodesic motion on Schwarzschild at radiusr > 3M ,
E =r − 2M√r(r − 3M)
µ,dE
dt= −Ft/ut0
The dissipative components, Ft and Fr, corresponding toenergy and angular momentum loss, are gauge-invariant(*).The conservative component Fr is gauge-dependent.Detweiler identified two quantities which are gaugeinvariant under transforms that respect the helicalsymmetry of the circular orbit.
1 Orbital frequency Ω ⇔ radius R ≡ (M/Ω2)1/3
2 Redshift z = 1/ut
Both defined w.r.t Schw. t coordinate of background.z(R) is a gauge-invariant relation.Results of Regge-Wheeler and Lorenz gauge calculationscompared by Detweiler, and Sago & Barack (’08).
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Synergies: Comparisons (II) The ISCO shift
Innermost stable circular orbit (ISCO) where dE/dr = 0.
For geodesic motion,
risco = 6M, Ωisco =(
63/2M)−1
.
The conservative part of GSF shifts the ISCO by O(µ).
∆Ωisco is invariant under gauge transformations thatrespect the helical symmetry of the circular orbit.
GSF prediction:
∆Ωisco
Ωisco= 0.4870µ/M
Barack & Sago, PRL 102, 191101 (2009), arXiv:0902.0573.
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Synergies: Comparisons (II) The ISCO shift
GSF prediction must be modified for comparison with PN,because Lorenz gauge is not asymptotically-flat(htt ∼ O(r0)).
Apply simple monopolar gauge transformation to get:
∆Ωisco
Ωisco= 1.2512µ/M
A challenge: can a resummed Post-Newtonian expansionmatch this strong-field result?
Challenge taken up in M. Favata, PRD 83, 024027 (2011),arXiv:1008.4622.
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Intro Review Tides Octupoles EOB Summary
Synergies: Comparisons (II) The ISCO shift
Table 1 in M. Favata, PRD 83, 024027 (2011),arXiv:1008.4622.
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Synergies: Comparisons (II) The ISCO shift on Kerr
(M + µ)Ωisco = MΩ(0)isco
(1 +
µ
MCΩ + . . .
)Isoyama et al., PRL 113, 161101 (2014).
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Synergies: Comparisons (III) The periastron advance
GR ⇒ periastron advance δ ≈ 6πM[(1−e2)p]
(e.g. 43” per century for Mercury).
Conservative part of GSF ⇒ ∆δ ∼ O(µ)
∆δ < 0 for all eccentric orbits
∆δ is gauge-invariant (within restricted class of gauges)
Numerical results in Barack & Sago, PRD 83, 084023(2011), arXiv:1101.3331.
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Synergies: Comparisons (III) Periastron advance
Periastron advance was compared between NR, PN, EOBand GSF in comparable mass regime 1/8 ≤ µ/M ≤ 1.
Le Tiec et al. PRL 107, 141101 (2011) [arXiv:1106.3278]
Remarkably, the GSF prediction works well even incomparable mass regime if we replace µ/M with symmetricmass ratio:
µ/M → ν = µM/(µ+M)2
Plots on next slide show K = Ωφ/Ωr = 1 + δ/(2π).
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Synergies: Comparisons (III) Periastron advance
From Le Tiec, Mroue, Barack, Buonanno, Pfeiffer, Sago andTaracchini, PRL 107, 141101 (2011), arXiv:1106.3278.
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Intro Review Tides Octupoles EOB Summary
Synergies: Comparisons (III) Periastron advance
From Le Tiec, Mroue, Barack, Buonanno, Pfeiffer, Sago andTaracchini, PRL 107, 141101 (2011), arXiv:1106.3278.
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Synergies: Calibration of EOB theory
Damour and collaborators have fed GSF results into theEOB model.Idea: Compare precession of small-eccentricity orbits atfirst-order in µ
Ω2r
Ω2φ
= 1− 6x+( µM
)ρ(x) +O
((µ/M)2
)where
x ≡ [(M + µ)Ωφ]2/3 .
PN theory gives the (weak-field) expansion
ρPN (x) = ρ2x2+ρ3x
3+(ρc4+ρlog4 lnx)x4+(ρc5+ρlog
5 lnx)x5+O(x6)
ρ2, ρ3 are given by 3PN.logarithmic contributions at 4PN and 5PN (ρlog
4 and ρlog5 )
have been derived by Damourρc4 and ρc5 were unknown in PN.
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Intro Review Tides Octupoles EOB Summary
Synergies: Calibration of EOB theory
Using accurate GSF results, ρ2, ρ3, ρlog4 , ρlog
5 may betested, and the unknown parameters ρc4 and ρc5 may beconstrained:
ρc4 = 69+7−4, ρc5 = −4800+400
−1200, ρlog6 < 0.
Determination of ρ(x) in the range 0 ≤ x ≤ 1/6 gives firstinfo on strong-field behaviour of a combination of EOBfunctions a(u) and d(u) [where u = G(M + µ)/(c2rEOB)].
Advantage of GSF calibration: Both GSF and EOB splitnaturally into conservative and dissipative effects.
GSF data for ρ(x) may be fitted with simple 2-point Padeapproximation that also makes use of PN information.
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Intro Review Tides Octupoles EOB Summary
Synergies: Calibration of EOB theory
From Barack, Damour and Sago, Phys. Rev. D 82, 084036 (2010)
[arXiv:1008.0935].
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Invariants lead to synergies . . .
Key Questions:
1 Does hR still have unexplored latent physical content?Yes.
2 What are the physical gauge-invariant quantitiesassociated with a geodesic γ with tangent vector ua
on a regular vacuum black hole spacetime gRab?
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Intro Review Tides Octupoles EOB Summary
Invariants lead to synergies . . .
Key Questions:
1 Does hR still have unexplored latent physical content?Yes.
2 What are the physical gauge-invariant quantitiesassociated with a geodesic γ with tangent vector ua
on a regular vacuum black hole spacetime gRab?
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Asymptotic Matched Expansions
image from
Zlochower et al.,
arXiv:1504.00286.
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Outer solution: Expansion about a worldline
Detweiler ’05: In THZ coordinates t, xi
gab = ηab + 2Hab + 3Hab +O(r4/R4)
with quadrupole part
2Habdxadxb = Eijxixj(dt2 + δkldx
kdxl) +4
3εklmBmi xlxidtdxk
and octupole part 3Habdxadxb =
−1
3E(ijk)x
ixjxk(dt2 + δkldx
kdxl)
+1
2εkpqBqijx
pxixjdtdxk +
−20
21
[Eij0xixjxk −
2
5r2Eik0x
i
]dtdxk
+5
21
[εjpqBqk0xix
pxk − 1
5r2εpqiBjq0xp
]dxidxj
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Intro Review Tides Octupoles EOB Summary
Inner solution: Tidally-perturbed black hole
A tidally-perturbed BH can be written
gab = gSchwab + 2hab + 3hab +O(r4/R4)
with quadrupole part
2habdxadxb = −Eijxixj
[(1− 2µ/r)2dt2 + dr2 + (r2 − 2µ2)dΩ2
]+
4
3εkpqBqi x
pxi(1− 2µ/r)dtdxk
and octupole part. . .
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Matching
Match in buffer regime µ r M
Analyze O(µ) parts
Coordinate + gauge subtleties (see e.g. Poisson;Johnson-McDaniel; Pound)
Key claim: The ‘external multipole moments’E(ijk), Eij0, etc., are those computed in regular perturbed
spacetime g + hR.
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Tidal dynamics?
Consider compact body with linear and angularmomentum pa and sa and electric & magnetic quadrupolemoments Qab(E) and Qab(B)
Hartle & Thorne (’85) used a matched asymptoticexpansion to obtain
dpidτ
= −Biasa −1
2EiabQab(E)
dsidτ
= −εiabQa(E)cEcb − 4
3εiabQ
a(B)cB
cb
Role for external quadrupole and octupole tidal tensors
BH : quadrupole moments are small, Qab(E) ∼ µ3a2 ∼ µ5
Neutron stars : not necessarily so.
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Tidal effects
Introduce electric- and magnetic-type tidal tensors:
Eab = Racbducud
Bab = R∗acbducud
where ∗ is (left) Hodge dual.
Eab generates geodesic deviation:
D2Xa
dτ2= −EabXb
Bab generates differential precession . . .
∆Ωa = BabXb
. . . and Papapetrou-Pirani force on a gyroscope
Dpa
dτ= −Babsb
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Intro Review Tides Octupoles EOB Summary
Tidal effects
Introduce electric- and magnetic-type tidal tensors:
Eab = Racbducud
Bab = R∗acbducud
where ∗ is (left) Hodge dual.
Eab generates geodesic deviation:
D2Xa
dτ2= −EabXb
Bab generates differential precession . . .
∆Ωa = BabXb
. . . and Papapetrou-Pirani force on a gyroscope
Dpa
dτ= −Babsb
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Intro Review Tides Octupoles EOB Summary
Tidal effects
Introduce electric- and magnetic-type tidal tensors:
Eab = Racbducud
Bab = R∗acbducud
where ∗ is (left) Hodge dual.
Eab generates geodesic deviation:
D2Xa
dτ2= −EabXb
Bab generates differential precession . . .
∆Ωa = BabXb
. . . and Papapetrou-Pirani force on a gyroscope
Dpa
dτ= −Babsb
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Intro Review Tides Octupoles EOB Summary
Tidal effects
Electric- and magnetic-type tidal tensors are . . .
Transverse: Babub = 0 = Eabub
Symmetric: Eab = Eba, Bab = BbaTraceless: Baa = 0, Eaa = 0 (in vacuum)
⇒ 2× 5 = ten degrees of freedom, like Weyl tensor
Introduce an orthonormal triad eai on γ anddefine 3× 3 tidal matrices:
Eij = Eabeai ebj , Bij = Babeai ebj
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Intro Review Tides Octupoles EOB Summary
Tidal effects
Electric- and magnetic-type tidal tensors are . . .
Transverse: Babub = 0 = Eabub
Symmetric: Eab = Eba, Bab = BbaTraceless: Baa = 0, Eaa = 0 (in vacuum)
⇒ 2× 5 = ten degrees of freedom, like Weyl tensor
Introduce an orthonormal triad eai on γ anddefine 3× 3 tidal matrices:
Eij = Eabeai ebj , Bij = Babeai ebj
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Intro Review Tides Octupoles EOB Summary
Tidal effects
3× 3 symmetric matrices Eij & Bij have
3 real eigenvalues
3 orthogonal eigenvectors
Traceless condition:
λE1 + λE2 + λE3 = 0 = λB1 + λB2 + λB3
Two orthogonal eigenbases define three Euler angles
Seven ‘intrinsic’ degrees of freedom in eigenvalues/vectorsof tidal matrices (2 + 2 + 3).
Three remaining degrees of freedom depend on choice oftriad along γ.
Could use parallel transport to define a preferred triadon γ⇒ three more Euler angles.
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Intro Review Tides Octupoles EOB Summary
Tidal effects
3× 3 symmetric matrices Eij & Bij have
3 real eigenvalues
3 orthogonal eigenvectors
Traceless condition:
λE1 + λE2 + λE3 = 0 = λB1 + λB2 + λB3
Two orthogonal eigenbases define three Euler angles
Seven ‘intrinsic’ degrees of freedom in eigenvalues/vectorsof tidal matrices (2 + 2 + 3).
Three remaining degrees of freedom depend on choice oftriad along γ.
Could use parallel transport to define a preferred triadon γ⇒ three more Euler angles.
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Intro Review Tides Octupoles EOB Summary
Tidal effects
3× 3 symmetric matrices Eij & Bij have
3 real eigenvalues
3 orthogonal eigenvectors
Traceless condition:
λE1 + λE2 + λE3 = 0 = λB1 + λB2 + λB3
Two orthogonal eigenbases define three Euler angles
Seven ‘intrinsic’ degrees of freedom in eigenvalues/vectorsof tidal matrices (2 + 2 + 3).
Three remaining degrees of freedom depend on choice oftriad along γ.
Could use parallel transport to define a preferred triadon γ⇒ three more Euler angles.
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Intro Review Tides Octupoles EOB Summary
Tidal effects
Alternatively, use the electric eigenbasis as the triad.
3 electric + 6 magnetic components - 2 traces = 7
Define relative precession three-vector Ωi = εijkΩjk,
Ωij = gabeai
Debjdτ
May also examine octupolar tensors
Eabc = Radbe;cudue, Babc = R∗adbe;cu
due
and resolve these in the electric eigenbasis.
cf. tidal tendexes and vortexes by Zimmerman et al.cf. gravitoelectromagnetism.
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Intro Review Tides Octupoles EOB Summary
Tidal effects
Alternatively, use the electric eigenbasis as the triad.
3 electric + 6 magnetic components - 2 traces = 7
Define relative precession three-vector Ωi = εijkΩjk,
Ωij = gabeai
Debjdτ
May also examine octupolar tensors
Eabc = Radbe;cudue, Babc = R∗adbe;cu
due
and resolve these in the electric eigenbasis.
cf. tidal tendexes and vortexes by Zimmerman et al.cf. gravitoelectromagnetism.
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Intro Review Tides Octupoles EOB Summary
Circular orbits
Define gauge-invariant relationships at O(µ) via
∆χ(y) = limµ→0
χ(y)− χ(y)
µ
where χ is the test-particle (µ = 0) function on BH
background, and y =(GMΩ/c3
)2/3is frequency-radius
Zero derivatives: Detweiler’s redshift invariant ∆U(’08)
First derivatives: spin precession invariant ∆ψ (’14)
Second derivatives: Three independent eigenvalues oftidal tensors ∆λE1 ,∆λ
E2 ,∆λ
B and one angle ∆χ.
Third derivatives: Octupolar invariants ∆E(ijk)
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Intro Review Tides Octupoles EOB Summary
Octupoles!
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Intro Review Tides Octupoles EOB Summary
Octupoles for circular orbits
Project Rabcd;eubud onto electric-quadrupolar eigenbasis,
i.e.χi0j... = χabc...e
ai u
becj
Three types of terms:
Eij0, Ei[j;k], and E(ijk),
Bij0, Bi[j;k], and B(ijk),
First two types are derived from quadrupole & dipoleinvariants
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Intro Review Tides Octupoles EOB Summary
Octupoles for circular orbits
Eij0 : all components zero except
E130 = ω (E11 − E33) , B120 = −ω B23, B230 = ω B12.
Ei[jk] : all components zero except
E2[23] = E1[31] =1
2ωB23,
E3[31] = E2[12] = −1
2ωB12,
B1[12] = B3[23] =1
2ω (E11 − E33) ,
E(ijk) and B(ijk) :
7 new dof = 10 symmetrized components - 3 traces.
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Intro Review Tides Octupoles EOB Summary
Symmetries for circular orbits
Triad:
ea1 points in radial / ‘electric-stretch’ eigendirection
ea2 points out of the plane
ea3 = εabcdubec1e
d2
Equatorial symmetry: Electric (magnetic) componentswith odd (even) number of ‘2’ components are zero.
Reversal symmetry: Electric and magnetic componentswith an odd number of ‘3’ components on the BHbackground are zero (but not in general).
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Intro Review Tides Octupoles EOB Summary
Equatorial symmetry, general triad
Eij + iBij =
E11 iB12 E13
· E22 iB23
· · E33
Align triad with electric eigenbasis,
Eij + iBij =
E11 iB12 0· E22 iB23
· · E33
On black hole background,
Eij + iBij =
E11 iB12 0· E22 0· · E33
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Intro Review Tides Octupoles EOB Summary
Equatorial symmetry: Electric (magnetic) componentswith odd (even) number of ‘2’ components are zero.
Just six components are non-zero on background:
E(111), E(122), E(133),
B(211), B(222), B(233).
Four components are zero on background due to reversalsymmetry:
E(311), E(322), E(333), B(123)
Three trace conditions:
E(111) + E(122) + E(133) = 0
B(211) + B(222) + B(233) = 0
E(311) + E(322) + E(333) = 0
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Intro Review Tides Octupoles EOB Summary
A classification: conservative / dissipative
For circular orbits,
∇n Conservative Dissipative
n = 0 Un = 1 ψ F3
n = 2 E11, E22, B12 B23
n = 3 E(111), E(122), B(211), B(222) E(311), E(322), B(123)
The dissipative quantities:
may be computed from 12 (hret − hadv)
“do not require regularization”
are zero on the background
have an odd number of ‘3’ legs
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Intro Review Tides Octupoles EOB Summary
Results: PN series in y ≡ (GMΩ/c3)2/3
Redshift [Detweiler (2008)]:
∆U = −y − 2y2 − 5y
3+
(4132π2 − 121
3
)y4
+(−
1157
15− 128
5γ + 677
512π2 − 256
5log 2− 64
5log y
)y5
+ . . .
Spin precession [Dolan, Harte et al. (2014)]
∆ψ = y2 − 3y
3 − 152y4
+(− 6277
30− 16γ + 20471
1024π2 − 496
15log 2
)y5 − 8y
6+ . . .
Tidal invariants [Dolan, Nolan et al. (2014); Bini & Damour (2014)]
∆E11 = 2y3
+ 2y4 − 19
4y5
+(
2273− 593
256π2)y6
+(− 71779
4800+ 768
5γ − 719
256π2
+ 15365
log 2)y7
+ . . .
∆E22 = −y3 − 32y4 − 23
8y5
+(
12491024
π2 − 2593
48
)y6
+(− 362051
3200− 256
5γ + 1737
1024π2 − 512
5log 2
)y7
+ . . .
∆E33 = −y3 − 12y4
+ 618y5
+(
11231024
π2 − 1039
48
)y6
+(
12297119600
− 5125γ + 1139
1024π2 − 1024
5log 2
)y7
+ . . .
∆B12 = 2y7/2
+ 3y9/2
+ 594y11/2
+(
276124− 41
16π2)y13/2
+ . . .
∆B23 =
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Intro Review Tides Octupoles EOB Summary
Results: PN series in y ≡ (GMΩ/c3)2/3
Octupoles [Nolan, Kavanagh, Dolan, Warburton, Wardell & Ottewill, arXiv:]
∆E(111) = −8y4
+ 8y5
+ 30y6 − ( 1711
6− 4681
512π2)y
7+
+( 136099
400− 6255
1024π2 − 2048
5γ − 4096
5log 2− 1024
5log y
)y8
+ . . .
∆E(122) = 4y4 − 7
3y5 − 9y
6+ ( 1369
8− 9677
2048π2)y
7+
+( 121369
7200+ 265
192π2
+ 10245γ + 2048
5log 2 + 512
5log y
)y8
+ . . .
∆E(133) = 4y4 − 17
3y5 − 21y
6+ ( 2737
24− 9047
2048π2)y
7 −
−( 2571151
7200− 14525
3072π2 − 1024
5γ − 2048
5log 2− 512
5log y
)y8
+ . . .
∆E(113) = 1285y13/2 − 108
5y15/2
+ 5125πy
8 − 46978105
y17/2
+ 379445
πy9
+ . . .
∆E(223) = − 325y13/2 − 18
5y15/2 − 128
5πy
8+ 8276
105y17/2 − 15242
315πy
9+ . . .
∆E(333) = − 965y13/2
+ 1265y15/2 − 384
5πy
8+ 38702
105y17/2 − 3772
105πy
9+ . . .
∆B(123) = 643y7
+ 365y8
+ 2563πy
17/2 − 534715
y9
+ 219715
πy19/2
+ . . .
∆B(211) = −8y9/2
+ 163y11/2 − 20y
13/2+ (− 677
2+ 5101
512π2)y
15/2+ . . .
∆B(222) = 6y9/2 − 4y
11/2+ 83
4y13/2
+ ( 10694− 7809
1024π2)y
15/2+ . . .
∆B(233) = 2y9/2 − 4
3y11/2 − 3
4y13/2
+ ( 2854− 2393
1024π2)y
15/2+ . . .
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Intro Review Tides Octupoles EOB Summary
Results: PN series vs Numerical data (conservative)
Relative difference between the numerical data and successive truncations
of the relevant PN series for conservative components.
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Intro Review Tides Octupoles EOB Summary
Results: PN series vs Numerical data (dissipative)
Relative difference between the numerical data and successive truncations
of the relevant PN series for dissipative components.
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Intro Review Tides Octupoles EOB Summary
Results: Light-ring divergences
Divergence of the conservative octupolar invariants as the orbital radius
approaches the light-ring: z = 1 − 3M/r0.
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Intro Review Tides Octupoles EOB Summary
Effective One-Body Theory
The Newtonian two-body problem may be reformulatedin terms of a single effective potential Veff with reducedmass
µ =m1m2
M, M = m1 +m2,
where
Veff =P 2φ
2µR2+ V (R)
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Intro Review Tides Octupoles EOB Summary
Effective One-Body Theory
In EOB formalism there is a relativistic radial potential,
Weff =√A(R) (µ2 + (Pφ/R)2)
A(R) is a radial function where
A(R) ≈ 1 + 2V (R)/µ as R→∞ ,
A(R) ≈ 1− 2M/R as m1,m2 → 0
A(R) modified by
Finite mass-ratio ν ⇒ repulsion
Tidal polarizability κ(`)i : ⇒ attraction
Tidal effects modify the radial potential:
A(R; ν;κ(`)A ) = A0(R; ν) +AT (R;κ
(`)i )
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Intro Review Tides Octupoles EOB Summary
Fig 1 in “Modeling the dynamics of tidally-interacting binary neutron stars
up to merger”, Bernuzzi, Nagar, Dietrich & Damour, Phys. Rev. Lett.
114, 161103 (2015), arXiv:1412.4553.
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Intro Review Tides Octupoles EOB Summary
Effective One-Body Theory
Electric (+) tidal contribution (u = M/R)
A(+)T (u; ν) = −
∑`=2
[κ
(`)1 u2`+2A
(`+)1 + (1↔ 2)
]Tidal polarizability:
κ(`)i = 2k
(`)i
(mi/M
Ci
)2`+1 mj
mi
where k(`)i are the dimensionless Love numbers and Ci
are stars’ compactnesses
‘Resummed’ term
A(2+)i = 1 +
3u2
1− rlru+XiA
(2+)1SF1
(1− rlru)7/2+X2i A
(2+)2SF2
(1− rlru)4
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Intro Review Tides Octupoles EOB Summary
Effective One-Body Theory
Effective Field Theory approach ⇒ terms Ai are built fromirreducible invariants
Je2 ≡ TrE2, Jb2 ≡ TrB2, Je3 ≡ TrE3, . . .
viaAJi =
√F (u)U−1J
U is redshift and F (u) is related to A(R).
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Intro Review Tides Octupoles EOB Summary
EOB : Irreducible invariants
EOB + Effective Field Theory ⇒ model calibrated withscalar invariants
TrE2, TrB2, TrE3, E(abc)E(abc), etc.
May be obtained from our component invariants, e.g.,
K3+ = E(abc)E(abc)
= E2(111) + E2
(333) + 3(E2
(122) + E2(133) + E2
(311) + E2(322)
)−6E2
(130)
∆K3+
K3+= δK3+ + 2h00
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Intro Review Tides Octupoles EOB Summary
EOB : Irreducible invariants
δK3+ = − 83
+ 35845y + 11848
675y2
+ (− 358190340500
+ 46811536
π2)y
3+
( 6147944832430000
− 79093192160
π2 − 2048
15γ − 4096
15log 2− 1024
15log y
)y4
+(− 759123028241
1020600000+ 431520437
11059200π2
+ 10707041575
γ + 354064225
log 2− 14587
log 3 + 5353521575
log y)y5 − 219136
1575πy
11/2
+( 12569905047667
2187000000− 1903269674027
1769472000π2 − 42147341
6291456π4
+ 181080056212625
γ − 123628168212625
log 2 + 7395335
log 3 + 90540028212625
log y)y6
+ 118163398165375
πy13/2
+ y7(
52369829422440012073990186120000000
− 4176344893416403990904320000
π2
+ 3512069844616039797760
π4 − 4143716714678
245581875γ + 1753088
1575γ2
− 6124042466966245581875
log 2 + 70123521575
γ log 2 + 70123521575
log2
2− 21435048930800
log 3 + 976562514256
log 5− 2071858357339245581875
log y
+ 17530881575
γ log y + 35061761575
log 2 log y + 4382721575
log2y − 32768
15ζ(3)
)+ 169822838237
245581875πy
15/2
+y8(
123440508676629175685507910812832430400000000
− 205165828703043199754214400000
π2 − 4004468043930067
11596411699200π4
+ 6403209826927357335219259375
γ − 819289024165375
γ2
+ 18668500151420029335219259375
log 2− 4048635776165375
γ log 2− 4434375616165375
log2
2− 4137804755289196196000
log 3 + 22744849
γ log 3
+ 22744849
log 2 log 3 + 11372449
log2
3− 88378906251111968
log 5 + 6300230470447357670438518750
log y − 819289024165375
γ log y
− 2024317888165375
log 2 log y + 11372449
log 3 log y − 204822256165375
log2y + 10678144
1575ζ(3)
)+(− 1048639996225198903
58998589650000π − 3506176
4725π3
+ 375160832165375
πγ + 750321664165375
π log 2 + 187580416165375
π log y)y17/2
+O(y9). (1)
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Intro Review Tides Octupoles EOB Summary
EOB : Tidally-interacting binary neutron stars
Punchline of Bernuzzi et al. (2015) : Compare red and bluelines
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Intro Review Tides Octupoles EOB Summary
Summary
The self-force approach yields various physical invariantsat O(µ/M)
Invariants allow for comparisons (with PN & NR) andcalibration (e.g. EOB)
Tidally-calibrated EOB models needed for (e.g.) Adv.LIGO data analysis for NS binaries
Frontiers:
Schwarzschild → Kerr BHs
Eccentric and non-equatorial orbits
Second-order self-force at O(µ2/M2)