2PN radiative dynamics of compact binary systems in the EFT approach Natália Tenório Maia University of Pittsburgh Department of Physics and Astronomy Pittsburgh Particle Physics Astrophysics and Cosmology Center 1
OUTLINE • Motivation • Brief review of the EFT formalism { { Adam Leibovich • Radiative sector at 2PN order Ira Rothstein Zixin Yang • Final remarks 2
MOTIVATION EFT (analytics) BHPT Numerics This is the signal of the first GW detected which came from a binary the evolution of a binary system has system of BHs. three main stages: inspiral But now you ask me: how do we get merger an accurate theoretical template to ringdown 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 di ff erent approaches for the di ff erent stages of 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 3 B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration) Phys. Rev. Lett. 116 , 061102
MOTIVATION 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. In the early inspiral stage, the relative velocity is low. r s r 𝛍 r s : size of the compact bodies Hierarchy among scales r : orbital radius during the inspiral stage: 𝛍 : GW wavelength r s ⌧ r ⌧ λ . v : expansion parameter 4
THE FORMALISM Non-Relativistic General Relativity Starting point: S = S EH + S pp In the weak field limit, g µ ν = η µ ν + h µ ν m P l , and low velocity limit, we have ( ∂ h ) 2 + h ( ∂ h ) 2 + h 2 ( ∂ h ) 2 � Z -1 + + ... ( ) d 4 x + + ... S EH → = m 2 m P l P l Z m Z m Z x ν ) 2 + ... x µ ˙ x µ ˙ τ h µ ν ˙ τ ( h µ ν ˙ x ν − + ... d ¯ d ¯ d ¯ S pp → − m + + τ − 8 m 2 = 2 m P l P l 5
∂ α h µ ν ∼ v Split fields: r h µ ν h µ ν = h µ ν + H µ ν ∂ 0 H µ ν ∼ v r H µ ν ∂ i H µ ν ∼ 1 radiation mode potential mode r H µ ν The general scenario of NRGR: Z Z e iS eff [ x ] = he iS NRGR [ ¯ h D He iS [ ¯ h,x ] = h,H,x ] . D ¯ D ¯ 6
Power counting Impose linearized harmonic gauge: Propagator for the potential modes: 0 ) δ 3 ( k + k 0 ) 1 0 ) i = � i (2 π ) 3 P µ ναβ δ ( x 0 � x 0 h H µ ν ( k , x 0 ) H αβ ( k 0 , x 0 k 2 Propagator for the radiation modes: P αβ µ ν ≡ 1 2 ( η αβ η µ ν − η α µ η βν − η αν η β µ ) d 4 pie − ip 0 ( t − t 0 ) e i p · x Z ∆ F αβ µ ν ( x − y ) = P αβ µ ν 0 − p 2 + i ✏ p 2 √ v m h µ ν ( x ) ∼ v √ Lv H µ ν ( x ) ∼ ∼ m P l r r v 0 m Z √ S pp ( H 00 ) = − dtH 00 ( x ) Lv 0 → ∼ 2 m P l v 7 xxxxxxx
Examples Leading order equation of motion: ∼ Lv 0 Next-to-leading order equation of motion: v 0 xxxxxxx v 0 v v ∼ Lv 2 xxxxxxx v 1 xxxxxxx v 1 v v xxxxxxx 8
Accelerations without spin: ⤷ First e ff ect due to the emission of GW 9
RADIATION SECTOR Scalar Field Effective theory: Full theory: 10
Formula for arbitrary STF tensors: L = i 1 ...i l 11
RADIATION REACTION Linearized Gravity ∞ ⇢✓ ◆ h Z (2 l + 1)!! 1 + 8 p ( l + p + 1) I L = 0 T 00 | x | 2 p x L i X ∂ 2 p d 3 x (2 p )!! (2 l + 2 p + 1)!! ( l + 1) ( l + 2) ST F p =0 ✓ 4 p ◆ h ✓ 4 ◆ ✓ 2 p ◆ h 0 T kk | x | 2 p x L i T 0 m | x | 2 p x mL i ∂ 2 p ∂ 2 p +1 + 1 + 1 + ST F − 0 ( l + 1) ( l + 2) l + 1 l + 2 ST F ✓ 2 ◆ h � T mn | x | 2 p x mnL i ∂ 2 p +2 + 0 ( l + 1) ( l + 2) ST F ∞ (2 l + 1)!! ⇢✓ 2 p ◆ h Z J L = 0 T 0 m | x | 2 p x nL − 1 i X ✏ k l mn @ 2 p d 3 x 1 + (2 p )!! (2 l + 2 p + 1)!! l + 2 ST F p =0 ✓ ◆ h � 1 T mn | x | 2 p x nrL − 1 i ✏ k l mr @ 2 p +1 − 0 l + 2 ST F 12
RADIATION REACTION Energy flux Waveform Radiation reaction R [ r ± ] = − 1 − 16 1 − ( t ) I ij (5) − ( t ) J ij (5) − ( t ) I ijk (7) 5 I ij 45 J ij 189 I ijk + ... + + − + 13
2PN MASS QUADRUPOLE MOMENT 14
Partial Fourier transform of the pseudo tensor: For the limit , we have In this way, we read off the pseudotensor by matching to 15
T 00(2PN) 16
T 0i(1PN) T ij(0PN) T ij(1PN) T 00(1PN) 17
HIGHER PN CORRECTIONS TO THE PSEUDOTENSOR 18
CONSISTENCY TESTS 19
2PN MASS QUADRUPOLE MOMENT Adding the reduced contributions and extracting the contributions of the components of the pseudo tensor, we obtain: We can now compute the power loss: But we still need the 2PN acceleration in the linearized harmonic gauge… Will and Wiseman, Phys.Rev.D54:4813-4848; Gopakumar and Iyer, Phys.Rev.D56:7708-7731 20
Diagrams that contribute to the Lagrangian at 2PN: 21
2PN Lagrangian and EOM in the linearized harmonic gauge The Euler-Lagrangian equation gives us 22
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Next step: obtain the next-to-next-to leading order radiation reaction (ongoing computation) a = a 0PN + a 1PN + a 1 . 5PN + a 2PN + a 2 . 5PN + a 3PN + a 3 . 5PN + a 4PN + a 4 . 5PN + . . . we need an accurate expression for the acceleration of the two-body system For a circular orbit, the following relation holds 24
<latexit sha1_base64="OWSr4myE7YRCg+ZxnkDSA6YrRkU=">ACS3icbZBLS8NAEMc39V1fVY9eFovgqSQq6EUQ9eCxgm2FpobNZtIubh7sToQS8v28ePHml/DiQREPbtsovgYW/vObndmdv59KodG2H63K1PTM7Nz8QnVxaXltba23tZJpji0eCITdeUzDVLE0EKBEq5SBSzyJXT8m9NRvXMLSoskvsRhCr2I9WMRCs7QIK/mu2cgkVG3ORD0iLoiRuomEfQZDbAE1/mEeHlYFN5XIoqCuqFivCRF7gYJfiYFDcpBXq1uN+x0L/CKUWdlNH0ag9mEM8iJFLpnXsVPs5Uyh4BKqptpSBm/YX3oGhmzCHQvH3tR0G1DAhomyhyzy5h+78hZpPUw8s3NiOFA/6N4H+1bobhYS8XcZohxHzyUJhJigkdGUsDoYCjHBrBuBLmr5QPmLEHjf1VY4Lze+W/or3bcPYauxf79eOT0o5skm2yA5xyAE5JuekSVqEkzvyRF7Iq3VvPVtv1vkasUqezbIj6jMfADKibSW</latexit> Orbital rate in terms of the frequency is obtained by: LIGO’s signal detection: Z ω f Z ω ∆Φ = ω dt = ω d ω ˙ ω i 25
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 other 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 26
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