Dynamical Renormalization Group Method Including Spin-Orbit - PowerPoint PPT Presentation

Dynamical Renormalization Group Method Including Spin-Orbit precession Zixin Yang Work with Adam K. Leibovich [arXiv:1908.05688], Based on the work of Galley & Rothestein [arXiv:1609.08268] University of Pittsburgh - PITT PACC QCD meets Gravity 2019 December 13, 2019 1 / 19

Motivation ◮ The binary inspiral orbit motion can be described by the post-Newtonian equations of motion. ◮ Adiabatic solutions average over the orbit, causing ambiguity and detail loss. ◮ Numerical approaches provide great precision but are time-consuming. ◮ Need analytic solutions without any averaging process to efficiently generate binary evolution covering full parameter space. 2 / 19

Binary Orbital Motion The binary acceleration in post-Newtonian Expansion: a = a N + a 1PN + a 2PN + ... + a SO + a SS + ... + a RR2.5PN + a RR3.5PN + ... + ..., a N = − M where r 2 ˆ n , a SO = 1 � �� ˆ � �� � � �� � 6ˆ 2 S + ∆ Σ 7 S + 3∆ Σ n n × v v × · − r 3 � � ��� + 3 ˙ ˆ 3 S + ∆ Σ r n × , � 136 M � 3 M a RR2.5PN = M 2 ν r − 8 M 2 ν + 72 v 2 � + v 2 � 15 r 4 ˙ r v , 5 r 3 r r where S = S 1 + S 2 and ∆ Σ = ( m 1 − m 2 )( S 2 /m 2 − S 1 /m 1 ). 3 / 19

Spin Precession Equations The equations that describe the time evolution of the spin vectors: S a = 1 � � 2 + 3 m b � �� ˙ L N × S a r 3 2 m a � − S b × S a + 3(ˆ n · S b )ˆ n × S a where { a, b } are the binary labels { 1 , 2 } , and L N = νM ( r × v ) is the Newtonian orbital angular momentum. 4 / 19

Dynamical Renormalization Group (Chen, Goldenfeld, Oono, 1995) ◮ Physically-motivated approach to ordinary differential equation problems involving multiple scales, boundary layers with technically difficult asymptotic matching, and WKB analysis. Conservative background quasi-circular orbit perturbed by radiation reaction. The perturbations that grow secularly with time can be reummed using the Dynamical Renormalization Group method. The resummed results are closed-form real-time solutions without any orbital or precession averaging. 5 / 19

The Procedure for DRG method 1. Determine the perturbation around a background solution given by the “bare paramters”. 2. Substitute the full solutions into EoMs and solve for the perturbations relative to an initial time t 0 . 3. Write the bare parameters as the renormalized parameters plus “counter-terms”, and introduce an arbitrary renormalization scale τ through t − t 0 = ( t − τ ) + ( τ − t 0 ) . 4. Use the counter-terms to cancel the secular terms that grow as ( τ − t 0 ) grows. 5. Find the Renormalization Group Equations of the renormalized parameters and solve the RGEs to obtain the resummed solutions. 6 / 19

The Moving Frame and the Euler Angle The fixed { ˆ x , ˆ y , ˆ z } frame: z z — the direction of the total ˆ y l l angular momentum J /J . n λ n , ˆ λ , ˆ ι The moving { ˆ l } frame: Φ n — the direction of the binary ˆ x l separation r = x 1 − x 2 . α y ˆ l — the direction of the orbital angular momentum L N . x α, ι are the spherical inclination and azimuthal angles of ˆ l . Φ is the z × ˆ ˆ l angle between ˆ n and ˆ x l = l | . z × ˆ | ˆ 7 / 19

DRG Step 1: Define the perturbations The equations of motion and the spin precession equations with 1.5PN spin-orbit + 2.5PN radiation reaction effects, written in the moving frame: 9 Degrees of freedom: � t S a = S a λ ˆ l ˆ n + S a λ + S a r ( t ) , ω ( t ) , φ ( t ) = ω ( t ) , n ˆ l . 8 / 19

DRG Step 1: Define the perturbations Write the solutions in the form of perturbations around background solutions, for example: r ( t ) = R B + δr ( t ) + δr S ( t ) , ω ( t ) = Ω B + δω ( t ) + δω S ( t ) ◮ Background solutions R B and Ω B are called the bare parameters only dependent on the initial time t 0 . ◮ δr ( t ) and δω ( t ) are the non-spinning 2.5PN perturbations with only radiation reaction present. ◮ δr S ( t ) and δω S ( t ) are the 4PN perturbations as the results of the interaction between spin-orbit and radiation reaction effects. 9 / 19

DRG Step 2: Perturbative solutions Expanding the EoMs up to the linear order of the unknown perturbation Solve for the perturbative terms, for instance: ◮ The constant coefficients for the general solutions that only depend on initial conditions are written as bare parameters. ◮ The solutions contains secular terms that grow as ( t − t 0 ), which are the “divergences” to be renormalized. 10 / 19

DRG Step 3: Parameters redefinition & subtraction scale Write the bare parameters as the renormalized parameters plus counter-terms: Isolating counter-terms removing divergences at different PN orders. Introduce the renormalization scale τ through replacing all ( t − t 0 ) by ( t − τ ) + ( τ − t 0 ) . 11 / 19

DRG Step 4: Remove the divergences Using the counter-terms to absorb all the secular terms, we can fix the value of counter-terms, for instance: After this step, the perturbative solutions are explicitly independent of t 0 . Choose the arbitrary renormalization scale τ = t to minimize the secular terms The solutions become functions of renormalized parameters. 12 / 19

DRG Step 5: RG equations The bare parameters are independent of the arbitrary scale τ , as in d R B / d τ = 0 , thus we can obtain the running of the renormalized R R ( τ ) though the derivative of counter-term d δ R / d τ . The RG flow of the renormalized parameters are described by The resummed solutions to the equations of motion are given by combining the perturbative solutions and the results of the RG equations . 13 / 19

Final result 14 / 19

Numerical Comparison: orbital radius r ( t ) m 1 / m 2 = 4 m 1 / m 2 = 1 20 20 15 15 Orbital Radius [ M ] Orbital Radius [ M ] 10 10 Resummed Resummed 5 5 Adiabatic Adiabatic Numerical Numerical 0 0 10 1 1 Resummed vs Numerical Resummed vs Numerical 1 Adiabatic vs Numerical Adiabatic vs Numerical 10 - 1 10 - 1 Fractional error Fractional error 10 - 2 10 - 2 10 - 3 10 - 3 10 - 4 10 - 4 10 - 5 10 - 5 0 5000 10000 15000 0 5000 10000 15000 20000 25000 t [ M ] t [ M ] 15 / 19

Numerical Comparison: orbital phase φ ( t ) 250 400 Resummed Resummed Adiabatic Adiabatic 200 Numerical Numerical 300 Orbital phase [ rad ] Orbital phase [ rad ] 150 200 100 100 50 0 0 10 - 1 1 Resummed vs Numerical Resummed vs Numerical 10 - 1 Adiabatic vs Numerical Adiabatic vs Numerical 10 - 2 10 - 2 Fractional error Fractional error 10 - 3 10 - 4 10 - 3 10 - 5 10 - 6 10 - 4 0 5000 10000 15000 0 5000 10000 15000 20000 25000 t [ M ] t [ M ] 16 / 19

Numerical Comparison: spin vector � S ( t ) 1.0 0.06 Resummed Resummed m 1 / m 2 = 1 m 1 / m 2 = 4 Numerical - Resummed Numerical - Resummed 0.04 0.5 Spin n - component [ M 2 ] Spin n - component [ M 2 ] 0.02 0.00 0.0 - 0.02 - 0.5 - 0.04 - 0.06 - 1.0 0.04 1.0 0.02 0.5 Spin λ - component [ M 2 ] Spin λ - component [ M 2 ] 0.00 0.0 - 0.02 - 0.5 - 0.04 - 1.0 - 0.06 1 1 10 - 1 10 - 1 10 - 2 Spin Vectors Angle [ rad ] Spin Vectors Angle [ rad ] 10 - 2 10 - 3 10 - 3 10 - 4 10 - 4 10 - 5 10 - 5 10 - 6 10 - 6 Angle betwen resummed and numerical spin vectors Angle betwen resummed and numerical spin vectors 0 5000 10000 15000 0 5000 10000 15000 20000 25000 t [ M ] t [ M ] 17 / 19

Binary motion and Spin Precession animation տ Equal-mass system with anti-aligned spins ← Unequal mass system with mis-aligned spins 18 / 19

Conclusion and Outlooks ◮ We obtain the analytic solutions to the spinnning binary dynamics at leading spin-orbit order with 2.5PN radiation reaction using the DRG method . ◮ We are adding the other non-spinning PN order terms in the equations of motion for a more complete solution to the dynamics. a = a N + a 1PN + a SO cons + a 2PN + a RR + a RR1PN + a SO RR + a RR2PN ⇐ (Natalia’s talk) ◮ More things to do: Large eccentricity and spin-spin order. 19 / 19