Jonathan Thornburg - capra.obspm.frcapra.obspm.fr/IMG/pdf/thornburg.pdfScalar self-force for highly eccentric orbits in Kerr spacetime Jonathan Thornburg in collaboration with Barry

Scalar self-force for highly eccentric orbits

in Kerr spacetime

Jonathan Thornburg

in collaboration with

Barry WardellDepartment of Astronomy and Department of Astronomy

Center for Spacetime Symmetries Cornell University

Indiana University Ithaca, New York, USA

Bloomington, Indiana, USA

Jonathan Thornburg / Capra 19 @ Meudon 2016-06-29 1 / 18

Goals• Kerr

• highly-eccentric orbits:

• eLISA EMRIs: likely have e up to ∼ 0.8


Goals• Kerr


• eLISA EMRIs: likely have e up to ∼ 0.8• eLISA IMRIs (likely rare, but maybe very strong sources if they exist):

may have e up to ∼ 0.998


Goals• Kerr




• [now] (bound, geodesic) equatorial orbits: [future] inclined (generic geodesic)

• compute self-force (all around fixed orbit)


Goals• Kerr






⇒ easy access to ∆geodesic per orbit, but we haven’t done this yet⇒ [easy access to emitted radiation at J +, but we haven’t done this yet]


Goals• Kerr







Limitations

• O(µ)

• scalar field; develop techniques for gravitational field (Lorenz-gauge)


Goals• Kerr







Limitations

• O(µ)

• scalar field; develop techniques for gravitational field (Lorenz-gauge)

⋆ may have a solution to Lorenz gauge instabilities [with Sam Dolan]⇒ if this works, then extension to gravitational field looks doable


Overall Plan of the Computation

Effective-Source (puncture-field) regularization




• 4th order effective source and puncture field

• scalar field for now• gravitational field ⇒ Lorenz-gauge instabilities [Dolan & Barack]






may have a solution to these [joint with Dolan]







m-mode (e imφ) decomposition, time-domain evolution

• exploit axisymmetry of Kerr background

• can handle (almost) any orbit, including high eccentricity

• separate 2+1-dimensional time-domain (numerical) evolution for each m

• presently equatorial orbits, but no serious obstacles to generic orbits












• worldtube scheme

• worldtube moves in (r , θ) to follow the particle around the orbit














• hyperboloidal slices (reach horizon and J +)[Zenginoglu, arXiv:1008.3809 = J. Comp. Phys. 230, 2286]














• hyperboloidal slices (reach horizon and J +)[Zenginoglu, arXiv:1008.3809 = J. Comp. Phys. 230, 2286]

• (fixed) mesh refinement; some (finer) grids follow the worldtube/particle


Effective source (puncture field) regularization

Assume a δ-function particle with scalar charge q.

The particle’s physical (retarded) scalar field ϕ satisfies �ϕ = δ(

x − xparticle(t))

.ϕ is singular at the particle.





x − xparticle(t))


If we knew the Detwiler-Whiting decomposition ϕ = ϕsingular + ϕregular explicitly,we could compute the self-force via Fa = q(∇aϕregular)

∣

∣

particle.





x − xparticle(t))



∣

∣

particle. But it’s very hard

to explicly compute the decomposition.

Instead we choose ϕpuncture ≈ ϕsingular so that ϕresidual := ϕ− ϕpuncture

is finite and “differentiable enough” at the particle.





x − xparticle(t))



∣

∣

particle. But it’s very hard

to explicly compute the decomposition.

Instead we choose ϕpuncture ≈ ϕsingular so that ϕresidual := ϕ− ϕpuncture

is finite and “differentiable enough” at the particle. It’s then easy to see that

�ϕresidual =

{

0 at the particle−�ϕpuncture elsewhere

}

:= Seffective

If we can solve this equation for ϕresidual, then we can compute the self-force(exactly!) via Fa = q(∇aϕresidual)

∣

∣

particle.


Puncture field and effective source

We choose ϕpuncture so that |ϕpuncture − ϕsingular| = O(

‖x − xparticle‖n)

where the puncture order n ≥ 2 is a parameter.

ϕresidual is then C n−2 at the particle, and Seffective is C0.







The choice of the puncture order n is a tradeoff:Higher n ⇒ ϕresidual is smoother at the particle (good),but ϕpuncture and Seffective are more complicated (expensive) to compute.

We choose n = 4 ⇒ ϕresidual is C2 at the particle.







The choice of the puncture order n is a tradeoff:Higher n ⇒ ϕresidual is smoother at the particle (good),but ϕpuncture and Seffective are more complicated (expensive) to compute.

We choose n = 4 ⇒ ϕresidual is C2 at the particle.

The actual computation of ϕpuncture and Seffective involves lengthly seriesexpansions in Mathematica, then machine-generated C code. SeeWardell, Vega, Thornburg, and Diener, PRD 85, 104044 (2012) = arXiv:1112.6355

for details.

Computing Seffective at a single event requires ∼ 12 × 106 arithmetic operations.


The worldtube

Problems:

• ϕpuncture and Seffective are only defined in a neighbourhood of the particle


The worldtube

Problems:


• far-field outgoing-radiation BCs apply to ϕ, not ϕresidual


The worldtube

Problems:



Solution:

introduce finite worldtube containing the particle worldline

• define “numerical field” ϕnumerical =

{

ϕresidual inside the worldtubeϕ outside the worldtube

• compute ϕnumerical by numerically solving

�ϕnumerical =

{

Seffective inside the worldtube0 outside the worldtube


The worldtube

Problems:



Solution:



{



�ϕnumerical =

{


• now Seffective is only needed inside the worldtube


The worldtube

Problems:



Solution:



{



�ϕnumerical =

{


• now Seffective is only needed inside the worldtube

• ϕnumerical has a ±ϕpuncture jump discontinuity across worldtube boundary⇒ finite difference operators that cross the worldtube boundary

must compensate for the jump discontinuity


m-mode decomposition

Instead of numerically solving �ϕnumerical =

{


in 3+1 dimensions, we Fourier-decompose into e imφ modes and solve for eachFourier mode in 2+1 dimensions via

�m ϕnumerical,m =

{

Seffective,m inside the worldtube0 outside the worldtube

numerically

solve this

for each m

in 2+1D

The self-force is given (exactly!) by Fa = q∞∑

m=0

(∇aϕnumerical,m)∣

∣

particle


m-mode decomposition

Instead of numerically solving �ϕnumerical =

{


in 3+1 dimensions, we Fourier-decompose into e imφ modes and solve for eachFourier mode in 2+1 dimensions via

�m ϕnumerical,m =

{

Seffective,m inside the worldtube0 outside the worldtube

numerically

solve this

for each m

in 2+1D

The self-force is given (exactly!) by Fa = q∞∑

m=0

(∇aϕnumerical,m)∣

∣

particle

Advantages (vs. direct solution in 3 + 1 dimensions):

• can use different numerical parameters for different m

⋆ (this is crucial for our hoped-for solution to theLorenz-gauge instabilities in the gravitational case)

• each individual m’s evolution is smaller ⇒ test/debug code on laptop

• get moderate parallelism “for free” (run different m’s evolutions in parallel)


Moving the worldtube

We actually do m-mode decomposition before introducing worldtube⇒ worldtube “lives” in (t, r , θ) space, not full spacetime

The worldtube must contain the particle in (r , θ).But for a non-circular orbit, the particle moves in (r , θ) during the orbit.





Small eccentricity: can use a worldtube big enough to contain the entire orbit






Large eccentricity:

• must move the worldtube in (r , θ) to follow the particle around the orbit






Large eccentricity:

• must move the worldtube in (r , θ) to follow the particle around the orbit

• recall that our numerically-evolved field is

ϕnumerical :=

{

ϕ− ϕpuncture inside the worldtubeϕ outside the worldtube

this means then if we move the worldtube, we must adjust the evolvedϕnumerical: add ±ϕpuncture at spatial points which change from being insidethe worldtube to being outside, or vice versa


Code Validation

Comparison withfrequency-domainresults kindly providedby Niels Warburton


Code Validation

Comparison withfrequency-domainresults kindly providedby Niels Warburton

Typical example:(a, p, e) = (0.9, 10M , 0.5)⇒ results agree to

∼ 10−5 relative error

We have also compareda variety of otherconfigurations, withfairly similar results

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

6 8 10 12 14 16 18 20

(r/M)3 ||Fa||+(r

/M)3 ||

Fa|

| + (

q2 /M)

r (M)

(r/M)3 ||Fa dissipative part||+(r/M)3 ||Fa conservative part||+(r/M)3 ||difference in Fa dissipative part||+(r/M)3 ||difference in Fa conservative part||+


e = 0.8 orbit

(a, p, e) = (0.6, 8M , 0.8)

t

0

100

200

300

4 6 8 10 15 20 30 40

r3 Ft (

10-3

q2 /M

)

r (M)

spin=0.6 p=8M e=0.8 ∆R*=M/64 Self-force Loop

outwardsinwards

r

-120

-100

-80

-60

-40

-20

0

20

40

4 6 8 10 15 20 30 40

r3 Fr (

10-3

q2 /M

)

r (M)


outwardsinwards

φ

-2500

-2000

-1500

-1000

-500

0

4 6 8 10 15 20 30 40

r3 Fφ

(10-3

q2 /M

)

r (M)


outwardsinwards


e = 0.8 orbit

(a, p, e) = (0.6, 8M , 0.8)

t

0

100

200

300

4 6 8 10 15 20 30 40

r3 Ft (

10-3

q2 /M

)

r (M)


outwardsinwards

r

-120

-100

-80

-60

-40

-20

0

20

40

4 6 8 10 15 20 30 40

r3 Fr (

10-3

q2 /M

)

r (M)


outwardsinwards

φ

-2500

-2000

-1500

-1000

-500

0

4 6 8 10 15 20 30 40

r3 Fφ

(10-3

q2 /M

)

r (M)


outwardsinwards

Notice the “bump” in Fr

near r = 15M moving outwards

Maybe a caustic crossing?


Wiggles!

Higher-eccentricity orbit:(a, p, e) = (0.99, 7M , 0.9)

t

-200

0

200

400

4 6 8 10 15 20 30 40 50 60 70

r3 Ft (

10-3

q2 /M

)

r (M)


outgoingingoing

r

-200

-100

0

100

4 6 8 10 15 20 30 40 50 60 70

r3 Fr (

10-3

q2 /M

)

r (M)


outgoingingoing

φ

-2000

-1000

0

1000

4 6 8 10 15 20 30 40 50 60 70

r3 Fφ

(10-3

q2 /M

)

r (M)


outgoingingoing


Wiggles!

Higher-eccentricity orbit:(a, p, e) = (0.99, 7M , 0.9)

t

-200

0

200

400

4 6 8 10 15 20 30 40 50 60 70

r3 Ft (

10-3

q2 /M

)

r (M)


outgoingingoing

r

-200

-100

0

100

4 6 8 10 15 20 30 40 50 60 70

r3 Fr (

10-3

q2 /M

)

r (M)


outgoingingoing

φ

-2000

-1000

0

1000

4 6 8 10 15 20 30 40 50 60 70

r3 Fφ

(10-3

q2 /M

)

r (M)


outgoingingoing

Notice:

• wiggles on outgoing leg of orbit

• wiggles not seen on ingoing leg


Wiggles as Kerr Quasinormal Modes: Mode Fit

Leor Barack suggested that wiggles might be quasinormal modes of the(background) Kerr spacetime, excited by the particle’s close flyby. Test this byfitting decay of wiggles to a damped sinusoid with corrections for motion of theobserver (particle):

rF [m=1]r

(u) = background(u) + A exp(

−u − u0

τ

)

sin(

φ0 + 2πu − u0

T

)

where u := t − r∗

-1

-0.5

0

0.5

1

1.5

2

2.5

3

0 50 100 150 200

wiggle period = 12.892

wiggle 1/e decay time = 25.142

104 F

r

u = t-r*

spin=0.99 p=7M e=0.9 rFr(u = t-r*) wiggles fit

data (orbit 5)fn(u)


Wiggles as Kerr Quasinormal Modes: Mode Frequencies

Now compare wiggle-fit complex frequency ω := 2π/T − i/τvs. known Kerr quasinormal mode frequencies computed by Emanuele Berti.


Wiggles as Kerr Quasinormal Modes: Mode Frequencies

Now compare wiggle-fit complex frequency ω := 2π/T − i/τvs. known Kerr quasinormal mode frequencies computed by Emanuele Berti.

⇒ Nice agreement with least-damped corotating QNM!

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.0 0.2 0.4 0.6 0.8 1.0

Im[ω

]

Re[ω]

spin=0.99 p=7M e=0.9 rFr(u = t-r*) wiggles decay vs Kerr QNMs

wiggles decay fitKerr QNMs (m=+1)Kerr QNMs (m=-1)


Wiggles as Kerr Quasinormal Modes: Varying BH Spin

Repeat wiggle-fit procedure for other Kerr spins (0.99, 0.95, 0.9, and 0.8)⇒ Nice agreement with least-damped corotating QNM for all BH spins!

a = 0.99

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.0 0.2 0.4 0.6 0.8 1.0

Im[ω

]

Re[ω]



a = 0.95

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.0 0.2 0.4 0.6 0.8 1.0

Im[ω

]

Re[ω]



a = 0.9

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.0 0.2 0.4 0.6 0.8 1.0

Im[ω

]

Re[ω]



a = 0.8

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.0 0.2 0.4 0.6 0.8 1.0

Im[ω

]

Re[ω]




More wiggles

-300

-200

-100

0

100

200

300

-2000

-1000

0

1000

2000e9 configuration(a,p,e) = (0.99,7M,0.9)

(r/M

)3 {F

t,Fr}

(10

-3 q

2 /M)

(r/M

)3 Fφ

(10-3

q2 /M

)

(r/M)3 Ft (left scale)(r/M)3 Fr (left scale)(r/M)3 Fφ (right scale)

-600

-400

-200

0

200

400

600

-2000

-1000

0

1000

2000

e95 configuration(a,p,e) = (0.99,7M,0.95)

(r/M

)3 {F

t,Fr}

(10

-3 q

2 /M)

(r/M

)3 Fφ

(10-3

q2 /M

)


-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

-100 0 100 200 300 400-15000

-10000

-5000

0

5000

10000

15000ze98 configuration(a,p,e) = (0.99,2.4M,0.98)

(r/M

)3 {F

t,Fr}

(10

-3 q

2 /M)

(r/M

)3 Fφ

(10-3

q2 /M

)

t (M)


We now see wiggles formany configurations with

• high Kerr spin

• highly-eccentricprograde orbitwith close-inperiastron


More wiggles

-300

-200

-100

0

100

200

300

-2000

-1000

0

1000

2000e9 configuration(a,p,e) = (0.99,7M,0.9)

(r/M

)3 {F

t,Fr}

(10

-3 q

2 /M)

(r/M

)3 Fφ

(10-3

q2 /M

)


-600

-400

-200

0

200

400

600

-2000

-1000

0

1000

2000

e95 configuration(a,p,e) = (0.99,7M,0.95)

(r/M

)3 {F

t,Fr}

(10

-3 q

2 /M)

(r/M

)3 Fφ

(10-3

q2 /M

)


-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

-100 0 100 200 300 400-15000

-10000

-5000

0

5000

10000

15000ze98 configuration(a,p,e) = (0.99,2.4M,0.98)

(r/M

)3 {F

t,Fr}

(10

-3 q

2 /M)

(r/M

)3 Fφ

(10-3

q2 /M

)

t (M)


We now see wiggles formany configurations with

• high Kerr spin

• highly-eccentricprograde orbitwith close-inperiastron

Note non-sinsoidalwiggle shapes⇒ multiple QNMs?

maybe caustic crossingsare also important?(Ottewill & Wardell)


Gravitation: Unstable Lorenz Gauge Modes

How to extend this work to the full gravitational field?

• Work in Lorenz gauge ⇒ metric perturbation is isotropic at the particle.

• The effective-source (puncture-field) regularization works fine for thegravitational case.


Gravitation: Unstable Lorenz Gauge Modes

How to extend this work to the full gravitational field?

• Work in Lorenz gauge ⇒ metric perturbation is isotropic at the particle.

• The effective-source (puncture-field) regularization works fine for thegravitational case.

• Dolan and Barack [PRD 87, 084066 (2013) = arXiv:1211.4586] found that them = 0 and m = 1 modes have Lorenz gauge instabilities. These are modeswhich are consistent with the Lorenz gauge condition at any finite time, butblow up as t → ∞. They were ableto stabilize the m = 0 mode using adynamically-driven generalized Lorenzgauge (analogous to the Z

4 evolutionsystem in full nonlinear numericalrelativity).

• They were not able to stabilize them = 1 mode. The instability islinear in time. 0

10

20

30

40

50

60

70

0 50 100 150 200

||sta

te v

ecto

r||

t

standard


(Maybe) Stabilizing the m = 1 Lorenz Gauge Mode

Basic idea (schematic): [Joint work with Sam Dolan]

• First define (choose) an inner product on state vectors u.

• Then compute the growing mode G(t, x i ) := tugrowing(xi ). Notice that this

is a homogeneous solution of the evolution equation. ugrowing depends onlyon the spacetime, not on the initial data or particle orbit.







• Write m = 1 gravitational perturbation evolution as dudt

= R(u)








= R(u)

• Replace evolution eqn with dudt

= R(u)

where R(u) := R(u) + λugrowing

• Choose (update) λ “every so often”such that R(u) ⊥ ugrowing

(⊥ w.r.t. our chosen inner product).








= R(u)


= R(u)




• This is equivalent to Gram-Schmidtorthogonalizing R w.r.t. ugrowing.








= R(u)


= R(u)




• This is equivalent to Gram-Schmidtorthogonalizing R w.r.t. ugrowing.

0

10

20

30

40

50

60

70

0 50 100 150 200-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

||sta

te v

ecto

r||

λ

t

standardorthogonalizedλ

This seems to work: evolutions are stable!

• Now trying it for sourced evolution. . .


Conclusions

Methods

• effective source/puncture field regularization works very well

• Zenginoglu’s hyperboloidal slices work very well


Conclusions

Methods



• moving worldtube allows highly-eccentric orbits• we have done up to e = 0.98• higher is possible but expensive with current code

• details → Thornburg & Wardell, arXiv:1607.?????


Conclusions

Methods





Wiggles

• if high Kerr spin, a close particle passage excites Kerr quasinormal modes;these show up as “wiggles” in the local self-force

• are caustic crossings also important?


Conclusions

Methods





Wiggles



• Maarten van de Meent has now found wiggles in Ψ4 flux at J +


Conclusions

Methods





Wiggles



• Maarten van de Meent has now found wiggles in Ψ4 flux at J +

Gravitation [Joint work with Sam Dolan]

⋆ May have found a way to stabilize Lorenz gauge modes⇒ can do gravitational self-force etc.


Documents

Jonathan Thornburg - capra.obspm.frcapra.obspm.fr/IMG/pdf/thornburg.pdfScalar self-force for highly eccentric orbits in Kerr spacetime Jonathan Thornburg in collaboration with Barry