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 = aN + a1PN + a2PN + ... + aSO + aSS + ... + aRR2.5PN + aRR3.5PN + ... + ...,

where aN = − M r2 ˆ n, aSO = 1 r3

• 6ˆ

n ˆ n × v

• ·
• 2S + ∆Σ
• −
• v ×
• 7S + 3∆Σ
• + 3 ˙

r

• ˆ

n ×

• 3S + ∆Σ
• ,

aRR2.5PN = M2ν 15r4 ˙ r 136M r + 72v2 r − 8M2ν 5r3 3M r + v2 v,

where S = S1 + S2 and ∆Σ = (m1 − m2)(S2/m2 − S1/m1).

3 / 19

Spin Precession Equations

The equations that describe the time evolution of the spin vectors: ˙ Sa = 1 r3

• LN × Sa

2 + 3 2 mb ma

• − Sb × Sa + 3(ˆ

n · Sb)ˆ n × Sa

• where {a, b} are the binary labels {1, 2}, and LN = 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 t0.

• 3. Write the bare parameters as the renormalized parameters

plus “counter-terms”, and introduce an arbitrary renormalization scale τ through t − t0 = (t − τ) + (τ − t0).

• 4. Use the counter-terms to cancel the secular terms that

grow as (τ − t0) 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

x y z n λ ι α Φ xl yl l

The fixed {ˆ x,ˆ y,ˆ z} frame: ˆ z — the direction of the total angular momentum J/J. The moving {ˆ n,ˆ λ,ˆ l} frame: ˆ n — the direction of the binary separation r = x1 − x2. ˆ l — the direction of the orbital angular momentum LN.

α, ι are the spherical inclination and azimuthal angles of ˆ

• l. Φ is the

angle between ˆ n and ˆ xl =

ˆ z׈ l |ˆ z׈ l|.

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: r(t), ω(t), φ(t) = t ω(t), Sa = Sa

n ˆ

n + Sa

λˆ

λ + Sa

l 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) = RB + δr(t) + δrS(t), ω(t) = ΩB + δω(t) + δωS(t)

◮ Background solutions RB and ΩB are called the bare parameters

• nly dependent on the initial time t0.

◮ δr(t) and δω(t) are the non-spinning 2.5PN perturbations with

• nly radiation reaction present.

◮ δrS(t) and δωS(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 − t0), 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 − t0) by (t − τ) + (τ − t0).

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 t0. 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

dRB/dτ = 0,

thus we can obtain the running of the renormalized RR(τ) though the derivative of counter-term dδR/dτ. 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)

Resummed Adiabatic Numerical 5 10 15 20 Orbital Radius [M]

m1/m2=1

Resummed vs Numerical Adiabatic vs Numerical 5000 10000 15000 10-5 10-4 10-3 10-2 10-1 1 t [M] Fractional error Resummed Adiabatic Numerical 5 10 15 20 Orbital Radius [M]

m1/m2=4

Resummed vs Numerical Adiabatic vs Numerical 5000 10000 15000 20000 25000 10-5 10-4 10-3 10-2 10-1 1 101 t [M] Fractional error

15 / 19

Numerical Comparison: orbital phase φ(t)

Resummed Adiabatic Numerical 50 100 150 200 250 Orbital phase [rad] Resummed vs Numerical Adiabatic vs Numerical 5000 10000 15000 10-6 10-5 10-4 10-3 10-2 10-1 1 t [M] Fractional error Resummed Adiabatic Numerical 100 200 300 400 Orbital phase [rad] Resummed vs Numerical Adiabatic vs Numerical 5000 10000 15000 20000 25000 10-4 10-3 10-2 10-1 t [M] Fractional error

16 / 19

Numerical Comparison: spin vector S(t)

Resummed Numerical - Resummed

• 0.06
• 0.04
• 0.02

0.00 0.02 0.04 0.06 Spin n-component [M 2]

m1/m2=1

• 0.06
• 0.04
• 0.02

0.00 0.02 0.04 Spin λ-component [M 2] Angle betwen resummed and numerical spin vectors 5000 10000 15000 10-6 10-5 10-4 10-3 10-2 10-1 1 t [M] Spin Vectors Angle [rad] Resummed Numerical - Resummed

• 1.0
• 0.5

0.0 0.5 1.0 Spin n-component [M 2]

m1/m2=4

• 1.0
• 0.5

0.0 0.5 1.0 Spin λ-component [M 2] Angle betwen resummed and numerical spin vectors 5000 10000 15000 20000 25000 10-6 10-5 10-4 10-3 10-2 10-1 1 t [M] Spin Vectors Angle [rad]

17 / 19

Binary motion and Spin Precession animation

18 / 19

տ Equal-mass system with anti-aligned spins ← Unequal mass system with mis-aligned spins

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 =aN + a1PN + aSO

cons + a2PN

+ aRR + aRR1PN + aSO

RR + aRR2PN

⇐ (Natalia’s talk)

◮ More things to do: Large eccentricity and spin-spin order.

19 / 19