2PN radiative dynamics of compact binary systems in the EFT approach - - PowerPoint PPT Presentation

Natália Tenório Maia

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2PN radiative dynamics of compact binary systems in the EFT approach

University of Pittsburgh Department of Physics and Astronomy Pittsburgh Particle Physics Astrophysics and Cosmology Center

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OUTLINE

• Motivation
• Brief review of the EFT formalism
• Radiative sector at 2PN order
• Final remarks

Adam Leibovich Ira Rothstein Zixin Yang {

{

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• B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration)
• Phys. Rev. Lett. 116, 061102

EFT (analytics) Numerics BHPT MOTIVATION

This is the signal of the first GW detected which came from a binary system of BHs. But now you ask me: how do we get an accurate theoretical template to compare with the signal detected and see if that is really a gravitational wave? Well, we don’t know the exact solution for the Einstein’s equations for this system. So what we do is to tackle the problem with different approaches for the different stages

• f the evolution of this system.

In the inspiral stage the relative velocity is low in comparison to the speed of light such that we can use approximations to find solutions for Einstein’s equations. But if we want the analytical descriptions to be valid up to late stages of the binary, the the evolution of a binary system has three main stages: inspiral merger ringdown

Hierarchy among scales during the inspiral stage:

rs ⌧ r ⌧ λ.

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MOTIVATION

rs : size of the compact bodies r : orbital radius 𝛍 : GW wavelength v : expansion parameter

rs r

𝛍

In the early inspiral stage, the relative velocity is low. inspiral merger ringdown

The EFT that I am going to talk about was to built to describe a binary system of compact bodies, which is the main source of GW that can be detected.

Non-Relativistic General Relativity

THE FORMALISM

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S = SEH + Spp

Spp → −m Z d¯ τ − m 2mP l Z d¯ τhµν ˙ xµ ˙ xν − m 8m2

P l

Z d¯ τ(hµν ˙ xµ ˙ xν)2 + ...

SEH → Z d4x  (∂h)2 + h(∂h)2 mP l + h2(∂h)2 m2

P l

+ ...

• =

=

( )

• 1+

+ + ...

Starting point: In the weak field limit, gµν = ηµν + hµν

mP l , and low velocity limit, we have

+ + ... +

∂αhµν ∼ v r hµν

∂0Hµν ∼ v r Hµν

Split fields:

hµν = hµν + Hµν

radiation mode potential mode ∂iHµν ∼ 1 r Hµν

The general scenario of NRGR:

eiSeff [x] = Z D¯ heiSNRGR[¯

h,x] =

Z D¯ hDHeiS[¯

h,H,x].

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hHµν (k, x0) Hαβ (k0, x0

0)i = i (2π)3 Pµναβδ (x0 x0 0) δ3 (k + k0) 1

k2

Pαβµν ≡ 1 2 (ηαβηµν − ηαµηβν − ηανηβµ)

∆F αβµν (x − y) = Pαβµν Z d4pie−ip0(t−t0)eip·x p2

0 − p2 + i✏

hµν (x) ∼ v r

Spp(H00) = − m 2mP l Z dtH00 (x) →

m mP l ∼ √ Lv

Propagator for the potential modes: Propagator for the radiation modes:

Hµν (x) ∼ √v r

v0

∼ √ Lv0

Power counting

Impose linearized harmonic gauge:

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v1

v0

∼ Lv0 ∼ Lv2

Leading order equation of motion: Next-to-leading order equation of motion:

Examples

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⤷ First effect due to the emission of GW

Accelerations without spin:

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RADIATION SECTOR

Scalar Field

Effective theory: Full theory:

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Formula for arbitrary STF tensors:

L = i1...il

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IL = Z d3x

∞

X

p=0

(2l + 1)!! (2p)!! (2l + 2p + 1)!! ⇢✓ 1 + 8p (l + p + 1) (l + 1) (l + 2) ◆ h ∂2p

0 T 00 |x|2p xLi ST F

+ ✓ 1 + 4p (l + 1) (l + 2) ◆ h ∂2p

0 T kk |x|2p xLi ST F −

✓ 4 l + 1 ◆ ✓ 1 + 2p l + 2 ◆ h ∂2p+1 T 0m |x|2p xmLi

ST F

+ ✓ 2 (l + 1) (l + 2) ◆ h ∂2p+2 T mn |x|2p xmnLi

ST F

• JL =

Z d3x

∞

X

p=0

(2l + 1)!! (2p)!! (2l + 2p + 1)!! ⇢✓ 1 + 2p l + 2 ◆ h ✏klmn@2p

0 T 0m |x|2p xnL−1i ST F

− ✓ 1 l + 2 ◆ h ✏klmr@2p+1 T mn |x|2p xnrL−1i

ST F

• RADIATION REACTION

Linearized Gravity

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RADIATION REACTION

Energy flux Waveform Radiation reaction

R [r±] = −1 5Iij

− (t) Iij(5) +

− 16 45Jij

− (t) Jij(5) +

− 1 189Iijk

− (t) Iijk(7) +

+ ...

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2PN MASS QUADRUPOLE MOMENT

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Partial Fourier transform of the pseudo tensor: For the limit , we have In this way, we read off the pseudotensor by matching to

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T00(2PN)

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Tij(1PN) Tij(0PN) T0i(1PN) T00(1PN)

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HIGHER PN CORRECTIONS TO THE PSEUDOTENSOR

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CONSISTENCY TESTS

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2PN MASS QUADRUPOLE MOMENT

Adding the reduced contributions But we still need the 2PN acceleration in the linearized harmonic gauge… and extracting the contributions of the components of the pseudo tensor, we obtain: We can now compute the power loss:

Will and Wiseman, Phys.Rev.D54:4813-4848; Gopakumar and Iyer, Phys.Rev.D56:7708-7731

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Diagrams that contribute to the Lagrangian at 2PN:

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2PN Lagrangian and EOM in the linearized harmonic gauge The Euler-Lagrangian equation gives us

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COMPARISON

Difference when comparing to Epstein-Wagoner or Blanchet-Damour-Iyer formalisms:

Considering the coordinate transformation

we obtain, up to 2PN,

Iij

EF T (¯

r) = ¯ Iij

aEF T (¯ r) = ¯ a

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For a circular orbit, the following relation holds Next step: obtain the next-to-next-to leading order radiation reaction (ongoing computation)

we need an accurate expression for the acceleration of the two-body system

a = a0PN + a1PN + a1.5PN + a2PN + a2.5PN + a3PN + a3.5PN + a4PN+ a4.5PN + . . .

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∆Φ = Z ωdt = Z ωf

ωi

ω ˙ ω dω

Orbital rate in terms of the frequency is obtained by: LIGO’s signal detection:

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FINAL REMARKS

Higher order effects are necessary for realistic/accurate descriptions NRGR: systematic way to obtain analytical results We provided an independent derivation of the 2PN correction to the mass quadrupole moment, to the acceleration and to the power loss in the EFT approach. At 2PN order, EFT results in the linearized harmonic gauge agrees with

• ther formalisms in the harmonic gauge once a well-defined coord. transf. is applied;

Dynamical Renormalization Group techniques: spin evolution (Zixin’s talk on Friday); NNLO radiation reaction Higher order spin EOM