QFT methods for gravitational wave astronomy fileJan Steinhoff (AEI) QFT methods for gravitational wave astronomy HU Berlin, June 1st, 2016 3 / 19 pics/binarywave2 binary black holes

QFT methods for gravitational wave astronomyapplication to spin effects and dynamic tides

Jan Steinhoff

Max-Planck-Institute for Gravitational Physics (Albert-Einstein-Institute), Potsdam-Golm, Germany

Fields and Strings Seminar, Humboldt University Berlin, June 1st, 2016

Jan Steinhoff (AEI) QFT methods for gravitational wave astronomy HU Berlin, June 1st, 2016 1 / 19

Outline

1 IntroductionSpin and tidal effectsUpcoming ObservatoriesCommon view on analytic description of binariesEffective field theory for compact objects in gravity

2 Spin effectsTwo Facts on Spin in RelativityPoint Particle Action in General RelativityPost-Newtonian ApproximationSpin and GravitomagnetismResults for post-Newtonian approximation with spin (conservative)

3 Dynamical tidesNeutron starsNeutron Star Equations of StateDynamical tidesConvenient concept: response functionRelativistic effects on dynamic tides


Spin and tidal effects


pics/binarywave2

binary black holes



pics/binarywave2

binary black holes

GW150914 a.k.a. The Event

peak strain 10−21 at ∼ 1 Gly⇒ strain 10−7 at 1 AU

3 M radiated in a fraction of a second⇒ power > all stars in the visible universe



pics/binarywave2

binary black holes

pics/ligo



pics/binarywave2

binary black holes

pics/ligo

LIGO, www.ligo.org

black holes→ large spinstrong precession→ tests of gravitycompute spin effects!



pics/binarywave2

binary black holes

pics/ligo

LIGO, www.ligo.org


pics/neutronstar

www.astroscu.unam.mx/neutrones/



pics/binarywave2

binary black holes

pics/ligo

LIGO, www.ligo.org


pics/crust

crustneutron star model



pics/binarywave2

binary black holes

pics/ligo

LIGO, www.ligo.org


pics/crust


tidal forces↔ oscillation modes⇒ resonances & dynamic tides!



pics/binarywave2

binary black holes

pics/ligo

LIGO, www.ligo.org


pics/crust



pics/grb3

pics/swift

Swift/BAT



pics/binarywave2

binary black holes

pics/ligo

LIGO, www.ligo.org


pics/crust



pics/grb3

gamma ray burstsgravitationalwaves

→ mode spectrum

pics/swift

Swift/BAT



pics/binarywave2

binary black holes

pics/ligo

LIGO, www.ligo.org


pics/crust



pics/grb3


→ mode spectrum

pics/swift

Swift/BAT



pics/binarywave2

binary black holes

pics/ligo

LIGO, www.ligo.org


pics/crust



pics/grb3


→ mode spectrum

pics/spectral

pics/swift

Swift/BAT



pics/ligo

LIGO, www.ligo.org



pics/grb3


→ mode spectrum

pics/spectral

pics/swift

Swift/BAT

pics/binarywave2

pics/crust

−→ spin effects

−→ dynamical tides

Upcoming and Planned Observatories


Gravitational wave detectors:

pics/et




pics/et

Einstein Telescope

LIGO like detectors:

Virgo (Italy)LIGO-IndiaKagra (Japan)




pics/et

Einstein Telescope

pics/lisa

eLISA space mission





Radio astronomy:

pics/ska

Square Kilometre Array (SKA)


pics/et

Einstein Telescope

pics/lisa

eLISA space mission



Common view on analytic description of binariesmatching of zone, see, e.g., Ireland, etal, arXiv:1512.05650

various zones:inner zone (IZ)

around compact objectsnear zone (NZ)

for the orbitfar zone (FZ)

for the waves

in between: buffer zones (BZ)for the matching

Problematic: different gaugesin different zones


pics/zones

from Ireland, etal, arXiv:1512.05650





for the waves




pics/zones






for the waves




pics/zones


Effective field theory for compact objects in gravityGoldberger, Rothstein, PRD 73 (2006) 104029; Goldberger, arXiv:hep-ph/0701129

zones → scales

separation of scales:scale µobject size rs

orbital size rvelocity v→ frequency ∼ v

r

effective action replaces buffer zonesin traditional approach


pics/GRtower

from Goldberger, arXiv:hep-ph/0701129

Effective field theory for compact objects in gravityGoldberger, Rothstein, PRD 73 (2006) 104029; Goldberger, arXiv:hep-ph/0701129

zones → scales

separation of scales:scale µobject size rs

orbital size rvelocity v→ frequency ∼ v

r

effective action replaces buffer zonesin traditional approach


pics/GRtower

from Goldberger, arXiv:hep-ph/0701129

Outline





Two Facts on Spin in Relativity

1. Minimal Extension

R

V

ring of radius R and mass Mspin: S = R M Vmaximal velocity: V ≤ c⇒ minimal extension:

R =S

MV≥ S

Mc

2. Center-of-mass

fast & heavy

slow & light

∆zv

spin

now moving with velocity vrelativistic mass changes inhom.frame-dependent center-of-massneed spin supplementary condition:

e.g., Sµνpν = 0


Two Facts on Spin in Relativity

1. Minimal Extension

R

V

ring of radius R and mass Mspin: S = R M Vmaximal velocity: V ≤ c⇒ minimal extension:

R =S

MV≥ S

Mc

2. Center-of-mass

fast & heavy

slow & light

∆zv

spin

now moving with velocity vrelativistic mass changes inhom.frame-dependent center-of-massneed spin supplementary condition:

e.g., Sµνpν = 0


Point Particle Action in General RelativityWestpfahl (1969); Bailey, Israel (1975); Porto (2006); Levi & Steinhoff (2014)

SPP =

∫dσ[−m

√gµνuµuν + ...

]Legendre transformation in uµ (from Nambu-Goto to Polyakov action)

SPP =

∫dσ[pµ

Dzµ

dσ− λ

2H+

12

SµνΛAµDΛAν

dσ− pµSµν

pρpρDpνdσ− χµCµ

]

constraint, interactions inM: H := pµpµ +M2 = 0 mass-shell constraint

ΛAµ : frame field on the worldline, “body-fixed” frame

Action is invariant under a “spin gauge symmetry”:spin gauge constraint: Cµ := Sµν(pν + pΛ0

µ) ∼ generator of symmetry

action invariant under a boost of ΛAµ plus transformation of Sµν

gauge related choice of center → spin supplementary condition



SPP =

∫dσ[−m



SPP =

∫dσ[pµ

Dzµ

dσ− λ

2H+

12

SµνΛAµDΛAν

dσ− pµSµν


]









SPP =

∫dσ[−m



SPP =

∫dσ[pµ

Dzµ

dσ− λ

2H+

12

SµνΛAµDΛAν

dσ− pµSµν


]








Post-Newtonian Approximation

Application: post-Newtonian approximationfor bound orbitsone expansion parameter, εPN ∼ v2

c2 ∼ GMc2r 1 (weak field & slow motion)

propagation “almost” instantaneoustime dependence of metric gµν treated perturbatively

→ Kaluza-Klein decomposition useful [Kol, Smolkin, arXiv:0712.4116]

ds2 = gµνdxµdxν ≡ e2φ(dt − Aidx i )2 − e−2φγijdx idx j

Leads to gravitomagnetic analogy:φ : gravito-electric fieldAi : gravito-magnetic fieldγij : spin-2 field


























Spin and Gravitomagnetism


Interaction with gravito-magnetic field Ai ≈ −gi0:

12

SµνΛAµDΛAν

dσ

12

Sij∂iAj

S1

Ai AjS2

LS1S2 =12

Ski1 〈∂k Ai ∂Àj〉

12

S`j2

=12

Ski1

12

S`j2 δij (−16πG)∂

∂xk1 ∂x`2

∫dk

(2π)3ei~k(~x1−~x2)

~k2

= −GSki1 Sì2

∂

∂xk1 ∂x`2

(1

r12

)

Here: r12 = |~x1 − ~x2|Ignoring factors like δ(t1 − t2)

Feynman rules see e.g. [arXiv:1501.04956]

Status: NNLO (many more diagrams, loops)




12

SµνΛAµDΛAν

dσ

12

Sij∂iAj

S1

Ai AjS2

LS1S2 =12


12

S`j2

=12

Ski1

12


∂xk1 ∂x`2

∫dk

(2π)3ei~k(~x1−~x2)

~k2

= −GSki1 Sì2

∂

∂xk1 ∂x`2

(1

r12

)







12

SµνΛAµDΛAν

dσ

12

Sij∂iAj

S1

Ai AjS2

LS1S2 =12


12

S`j2

=12

Ski1

12


∂xk1 ∂x`2

∫dk

(2π)3ei~k(~x1−~x2)

~k2

= −GSki1 Sì2

∂

∂xk1 ∂x`2

(1

r12

)







12

SµνΛAµDΛAν

dσ

12

Sij∂iAj

S1

Ai AjS2

LS1S2 =12


12

S`j2

=12

Ski1

12


∂xk1 ∂x`2

∫dk

(2π)3ei~k(~x1−~x2)

~k2

= −GSki1 Sì2

∂

∂xk1 ∂x`2

(1

r12

)







12

SµνΛAµDΛAν

dσ

12

Sij∂iAj

S1

Ai AjS2

LS1S2 =12


12

S`j2

=12

Ski1

12


∂xk1 ∂x`2

∫dk

(2π)3ei~k(~x1−~x2)

~k2

= −GSki1 Sì2

∂

∂xk1 ∂x`2

(1

r12

)







12

SµνΛAµDΛAν

dσ

12

Sij∂iAj

S1

Ai AjS2

LS1S2 =12


12

S`j2

=12

Ski1

12


∂xk1 ∂x`2

∫dk

(2π)3ei~k(~x1−~x2)

~k2

= −GSki1 Sì2

∂

∂xk1 ∂x`2

(1

r12

)




Results for post-Newtonian approximation with spinconservative part of the motion of the binary

post-Newtonian (PN) approximation: expansion around 1c → 0 (Newton)

order c0 c−1 c−2 c−3 c−4 c−5 c−6 c−7 c−8

N 1PN 2PN 3PN 4PN

non spin " " " " "

spin-orbit " " "

S21 " " "

S1S2 " " "

Spin3 "(!)

Spin4 "(!)...

. . .

" known (!) partial " derived last yearWork by many people (“just” for the spin sector): Barker, Blanchet, Bohe, Buonanno, O’Connell,Damour, D’Eath, Faye, Hartle, Hartung, Hergt, Jaranowski, Marsat, Levi, Ohashi, Owen, Perrodin,Poisson, Porter, Porto, Rothstein, Schafer, Steinhoff, Tagoshi, Thorne, Tulczyjew, Vaidya


Outline





A neutron star model

pics/neutronstar

Neutron star picture by D. Pagewww.astroscu.unam.mx/neutrones/

”Lab“ for various areas in physics

magnetic field, plasmacrust (solid state)superfluiditysuperconductivityunknown matter in corecondensate of quarks, hyperons,kaons, pions, . . . ?

accumulation of dark matter ?

Related objects:pulsarsmagnetarsquark stars/strange stars


http://www.astroscu.unam.mx/neutrones/NS-Picture/NStar/NStar_l.gif


pics/neutronstar









pics/neutronstar








Neutron Star Equations of State

pics/NSMassRad

mpifr-bonn.mpg.de


Neutron Star Equations of State

pics/NSMassRad

mpifr-bonn.mpg.de


However, a measurement of the tidal defor-mations through gravitational waves can be agood alternative to radius measurements.

Dynamical tides of neutron starsJS etal, in preparation Hinderer etal, PRL 116 (2016) 181101 (Research Highlight in Nature)

Neutron stars can oscillate→ effective harmonic oscillator action

Ltide ∼DAµν

dσDAµνdσ

− ω20AµνAµν − I0

2AµνEµν +

K4

EµνEµν + ...

(electric) tidal field Eµν = Rαµβνuαuβ

Amplitude Aµν must be SO(3) irreducuble, i.e, symmetric-tracefree:

A[µν] = 0 = Aµµ, Aµνuν = 0

Here: quadrupolar amplitude, strongly couples to gravityMore oscillators in the action→ more modes→ spectrumIn Newtonian limit: easy to compute constants ω0, I0, K , . . .

How to get these constants in relativistic case? → matching

Other important references: Flanagan, Hinderer, PRD 77 (2008) 021502Hinderer, ApJ 677 (2008) 1216 Chakrabarti, Delsate, JS, arXiv:1304.2228Alexander, MNRAS 227 (1987) 843 Bini, Damour, Faye, PRD 85 (2012) 124034




Ltide ∼DAµν

dσDAµνdσ


2AµνEµν +

K4

EµνEµν + ...










Ltide ∼DAµν

dσDAµνdσ


2AµνEµν +

K4

EµνEµν + ...










Ltide ∼DAµν

dσDAµνdσ


2AµνEµν +

K4

EµνEµν + ...








Convenient concept: response functionChakrabarti, Delsate, Steinhoff, PRD 88 (2013) 084038 and arXiv:1304.2228

quadrupolar sector: ` = 2

linear responseF−→

external quadrupolar field −→ deformation −→ quadrupolar responseφ ∼ r `

∑2F1 φ ∼ r−`−1∑

2F1

0.00 0.05 0.10 0.15 0.20 0.25

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

frequency: Ω R 2 Π

resp

on

se:

F

R5

quadrupolarresponse fit:

F ≈∑

n

I2n

ω2n − ω2

ωn: mode frequencyIn: coupling constantR: radius

poles⇒ resonances!

pics/resonance

Tacoma Bridge


http://dx.doi.org/10.1103/PhysRevD.88.084038





∑2F1 φ ∼ r−`−1∑

2F1

0.00 0.05 0.10 0.15 0.20 0.25

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6


resp

on

se:

F

R5


F ≈∑

n

I2n

ω2n − ω2



pics/resonance

Tacoma Bridge







∑2F1 φ ∼ r−`−1∑

2F1

0.00 0.05 0.10 0.15 0.20 0.25

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6


resp

on

se:

F

R5


F ≈∑

n

I2n

ω2n − ω2



pics/resonance

Tacoma Bridge



Relativistic effects on dynamic tides

All that was said hold also in Newtonian gravity . . .

What is new in the relativistic case?

values of constants ω0, I0, K , . . .is differentNewtonian: K = 0relativistic K 6= 0redshift effectframe dragging effect∼ Zeeman effect

















frame of the neutron star is draggedin the direction of the orbital motion

Summary

black holes→ large spins→ strong gravitomagnetic effects→ interesting/new tests of gravity

NS structure largely unknown/mysterious→ tidal effects in gravitational waves and electromagnetic

counterparts can enlighten nuclear interactions

GW astronomy exciting new and interdisciplinary field

important to use common language→ effective field theorybut need to go a bit beyond standard techniquese.g., in-in formalism, see [Galley PRL 110, 174301 (2013)]“good” level of abstraction:

DeWitt, The Global Approach to Quantum Field Theory


QFT methods for gravitational wave astronomy fileJan Steinhoff (AEI) QFT methods for gravitational wave astronomy HU Berlin, June 1st, 2016 3 / 19 pics/binarywave2 binary black holes

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