Self-force calculations: synergies and invariants · 2016-07-28 ·...

Intro Review Tides Octupoles EOB Summary

Self-force calculations:synergies and invariants

Sam R Dolan

University of Sheffield, UK.

Perturbation Methods in GR @ Fields Institute, 20th May 2015.


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.


“Large Two Forms”

Art Gallery of Ontario

Fµνωµ ∧ ων

“Large Two-Forms”

Fields Institute


“Large Two Forms”

Art Gallery of Ontario

Fµνωµ ∧ ων

“Large Two-Forms”

Fields Institute


Motivation: the general 2-body problem in relativity


Motivation: the general 2-body problem in relativity

Effective One-Body (EOB) model [Buonanno & Damour 1999].


Motivation: gravitational-wave astronomy


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


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)


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)


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.


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.


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.


Example: spin precession on circular orbit

ψ ≡ Precession angle per orbit / 2π.

∆ψ at O(µ) is gauge invariant.


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


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).


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.



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.



Table 1 in M. Favata, PRD 83, 024027 (2011),arXiv:1008.4622.


Synergies: Comparisons (II) The ISCO shift on Kerr

(M + µ)Ωisco = MΩ(0)isco

(1 +

µ

MCΩ + . . .

)Isoyama et al., PRL 113, 161101 (2014).


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.


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).



From Le Tiec, Mroue, Barack, Buonanno, Pfeiffer, Sago andTaracchini, PRL 107, 141101 (2011), arXiv:1106.3278.



From Le Tiec, Mroue, Barack, Buonanno, Pfeiffer, Sago andTaracchini, PRL 107, 141101 (2011), arXiv:1106.3278.


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.



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.



From Barack, Damour and Sago, Phys. Rev. D 82, 084036 (2010)

[arXiv:1008.0935].


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?


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?


Asymptotic Matched Expansions

image from

Zlochower et al.,

arXiv:1504.00286.


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


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. . .


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.


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.


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


Tidal effects


Eab = Racbducud

Bab = R∗acbducud



D2Xa

dτ2= −EabXb


∆Ωa = BabXb


Dpa

dτ= −Babsb


Tidal effects


Eab = Racbducud

Bab = R∗acbducud



D2Xa

dτ2= −EabXb


∆Ωa = BabXb


Dpa

dτ= −Babsb


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


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


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.


Tidal effects


3 real eigenvalues



λE1 + λE2 + λE3 = 0 = λB1 + λB2 + λB3






Tidal effects


3 real eigenvalues



λE1 + λE2 + λE3 = 0 = λB1 + λB2 + λB3






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.


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.


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)


Octupoles!


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


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.


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).


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


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


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


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 =


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+ . . .


Results: PN series vs Numerical data (conservative)

Relative difference between the numerical data and successive truncations

of the relevant PN series for conservative components.


Results: PN series vs Numerical data (dissipative)

Relative difference between the numerical data and successive truncations

of the relevant PN series for dissipative components.


Results: Light-ring divergences

Divergence of the conservative octupolar invariants as the orbital radius

approaches the light-ring: z = 1 − 3M/r0.


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)



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 )


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.



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



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).


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


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)


EOB : Tidally-interacting binary neutron stars

Punchline of Bernuzzi et al. (2015) : Compare red and bluelines


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)

Date post:	21-Feb-2020
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Self-force calculations: synergies and invariants · 2016-07-28 ·...

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