Progress Towards Multiscale EMRI Approximation: Zones and Scales J. Moxon 1 E. Flanagan 1 T. Hinderer 3 A. Pound 2 1 Cornell University Department of Physics 2 University of Southampton Mathematical Sciences 3 University of Maryland, College Park Maryland Center for Fundamental Physics Capra 2017 Post-adiabatic two-timescale Cornell University
The multiscale expansion ◮ ‘Multiscale’ - a combination of approximations ◮ Used to describe the use of two-timescale approximation where valid, combined with other methods: ◮ Near the small companion ◮ Far from the inspiral ◮ Near the SMBH horizon ◮ Our (ambitious) goals ◮ An algorithm built on existing SF tools for ensuring long scale ( t ∼ M/ǫ ) fidelity of: ◮ Post-adiabatic waveform ◮ Dynamical invariants of the inspiral for NR and PN comparison to second order Post-adiabatic two-timescale Cornell University
What we want from multiscale ◮ Post-adiabatic Waveform - important for parameter estimation for EMRIs, and possibly detection ◮ Phase accuracy throughout waveform ◮ Slowly varying memory effects Mass Ratio ◮ Dynamical invariants - highly useful for comparisons and confirmations EMRIs Self with NR and PN computations Force ◮ Redshift z [Detweiler] IMRIs Post ◮ Surface gravity [Zimmerman] Newtonian Numerical Relativity ◮ Precession of Perihelion [Le Tiec] ◮ Many of these are more Separation demanding for a multiscale scheme than waveforms Post-adiabatic two-timescale Cornell University
Multiscale requirements ◮ Waveforms Adiabatic Post-adiabatic Required Order Second Order Dissipative First Order Dissipative of Self-Force + First Order Conservative O ( ǫ 2 ) O ( ǫ ) Errors in Amplitude O (1) O ( ǫ ) Errors in Phase O ( ǫ ) O ( ǫ 2 ) Required oscillatory metric order O (1) O ( ǫ ) Required quasistatic metric order ◮ Dynamical Invariants (example: surface gravity) First order Second order Required Order Second Order Dissipative First Order Dissipative of Self-Force + First Order Conservative O ( ǫ 2 ) O ( ǫ ) Required oscillatory metric order O ( ǫ 2 ) Required quasistatic metric order O ( ǫ ) ◮ Requires quasistatic matching from distant regions Post-adiabatic two-timescale Cornell University
Zones and scales ◮ Interaction zone: − M/ǫ ≪ r ∗ ≪ M/ǫ ◮ Near small companion: distance from small Near Horizon companion ¯ r ≪ M ◮ Far zone: Near Small Companion r ∗ ≫ M ◮ Near-Horizon : Interaction Zone r ∗ ≪ − M matching Far Zone Post-adiabatic two-timescale Cornell University
Two-Timescale in interaction zone : − M/ǫ ≪ r ∗ ≪ M/ǫ ◮ Two-Timescale approximation promotes time dependence to multiple (temporarily) independent variables t → { ˜ t, q A } dq A t = µ ˜ dt = Ω(˜ M t ≡ ǫt t, ǫ ) ◮ Action angle variables q A coordinates on compact directions of the symplectic manifold ◮ Periodic behavior depends on q A , secular depends on ˜ t ◮ Worldline can be expressed using action angle variables and geodesic parameters P M ≡ { E, L z , Q } : dP M = ǫG (1) M ( P (0) M (˜ t ) , q A ) + O ( ǫ 2 ) dt dq A dt =Ω A ( P (0) M (˜ t )) + ǫg (1) A ( P (0) M (˜ t ) , q A ) Post-adiabatic two-timescale Cornell University
Improved long time fidelity ◮ Metric ansatz ( g (0) αβ taken to be Schwarzschild) g αβ = g (0) x i )+ ǫh (1) x i )+ ǫ 2 h (1) αβ (˜ t, q A , ¯ αβ (˜ t, q A , ¯ x i )+ O ( ǫ 3 ) αβ (¯ ◮ Worldline ansatz: z µ ( t ) = z (0) (˜ t, q A ) + ǫz (1) (˜ t, q A ) + O ( ǫ 2 ) ◮ Assume no resonances in the domain of interest ◮ Precision of approximation preserved: dephasing time is the entire inspiral ∼ M 2 /µ , rather than the standard result for black hole perturbation theory - geometric mean ∼ √ µM ◮ Our method applies the Two-Timescale approximation to metric perturbations to preserve field precision for the full inspiral Post-adiabatic two-timescale Cornell University
Breakdown of Two-Timescale at long distances ◮ Two-Timescale approximation assumes radiation timescale longer than all other scales of the system ◮ At each order, we solve the wave equation σ ρ h σρ = S, � q A h µν + R µ ν for some { ˜ t, q A , x i } -dependent source ◮ At long scales, inverting � q A is solving for perturbations assuming an eternal source ◮ Leading second-order source scales as ∼ Ω 2 /r 2 ◮ For second order static Green’s function, these contributions give divergent retarded field solution if integration domain r ′ ∈ [ a, ∞ ) ◮ similar problems arise at r ∗ → −∞ Post-adiabatic two-timescale Cornell University
Small companion puncture ◮ Two-timescale ansatz breaks down near small companion ◮ Use either Self-Consistent evaluated at each fixed ˜ t , or an extended Self-Consistent Near Small Companion ◮ Known puncture metric, derived by [Pound] ◮ Independent of matching conditions, dependent only on small companion structure ◮ Non-exact worldline z µ = z (0) µ + ǫz (1) µ + . . . requires a re-expansion from Self-Consistent ◮ Self-acceleration - direct re-expansion, up to slow time derivatives ◮ Puncture dipole correction - O ( µ ) displacement in worldline position ◮ Residual field derived in puncture region via relaxed EFE E µν [ h (2) R αβ ] = − E µν [ h (2) P αβ ] + S µν [ h (1) αβ , h (1) αβ ] + δT µν Post-adiabatic two-timescale Cornell University
Breakdown of Self-Consistent at long times ◮ Self-consistent formalism deals well with the slow evolution of the worldline by expanding the metric as a functional of the full worldline Near Small Companion g µν = g (0) µν [ x µ ] + ǫh (1) µν [ x µ ; z µ ] + ǫ 2 h (2) [ x µ ; z µ ] + O ( ǫ 3 ) ◮ Equations of motion are the Relaxed EFE and Lorenz gauge condition ◮ Slow evolution of background spacetime is incorrectly controlled ◮ Mass and spin evolution enter at the order of energy flux ∼ O ( ǫ 2 ) ◮ Entirely fixed by Lorenz gauge on initial data surface - no evolution during inspiral ◮ Linearly growing mass and spin at second order invalidates the result at a radiation-reaction time ◮ Direct two-timescale extension does not solve these problems, but a more involved incorporation can recover long-time fidelity Post-adiabatic two-timescale Cornell University
Zones and scales of approximation methods ◮ Interaction zone: | r ∗ | ≪ M/ǫ Two-Timescale expansion, worldline Two-Time ◮ Post-adiabatic evolution requires matching to adjacent regions ◮ Near small object : ¯ r ≪ M Puncture, Self-Consistent [Pound] Near Horizon ◮ Far zone: r ∗ ≫ M Puncture Geometric optics, with some Post-Minkowski techniques -[Extending Pound 2015] Two Timescale ◮ Near-Horizon : r ∗ ≪ − M matching Black hole perturbation theory Geometric Optics Post-adiabatic two-timescale Cornell University
Zones and scales of approximation methods ◮ Interaction zone: | r ∗ | ≪ M/ǫ Two-Timescale expansion, worldline Two-Time ◮ Post-adiabatic evolution requires matching to adjacent regions ◮ Near small object : ¯ r ≪ M Puncture, Self-Consistent [Pound] Near Horizon ◮ Far zone: r ∗ ≫ M Puncture Geometric optics, with some Post-Minkowski techniques -[Extending Pound 2015] Two Timescale ◮ Near-Horizon : r ∗ ≪ − M matching Black hole perturbation theory Geometric Optics Post-adiabatic two-timescale Cornell University
Geometric optics for the far zone x i ∼ ǫx i , on scale with slow inspiral ◮ Spatial scales vary with ˜ ◮ Construct ansatz with single fast variation parameterized by scalar function Θ( x ν ) /ǫ � � x ν , Θ � g µν ( x ν , ε ) = ε − 2 x ν ] + ε 2 j µν η µν + εh µν [˜ ˜ ε � x ν , Θ � � + ε 3 k µν + O ( ε 4 ) ˜ ε ◮ The rescaling of the coordinates grants an additional order to the weak waves, as they depend on 1 /r = ε/ ˜ r ◮ Define wave vector associated with the fast periodic dependence k µ = ∇ µ Θ ◮ Up to gauge, the leading dynamical equation enforces null wave vector k µ k ν η µν = 0 Post-adiabatic two-timescale Cornell University
First order - direct wave solutions ◮ Define tetrad { k, l, e A } such that l µ l µ = k µ k µ = 0 k µ e Aµ = l µ e Aµ = 0 k µ l µ = − 1 e α A e β B η αβ = δ AB ◮ Leading wave equation implies δ AB ∂ 2 Θ j AB = ∂ 2 Θ j ll = 0 ∂ 2 Θ j lA =0 ◮ Lorenz gauge not imposed, but compatible with the results after EFE is calculated ◮ Compatible with [Blanchet and Damour] Post-Minkowski leading order in 1 /r outgoing waves Post-adiabatic two-timescale Cornell University
Second order - propagation along null cones ◮ Null cone propagation at leading order gives simple 1 / ˜ r radiation dependence 1 r ∂ Θ j AB + ∂ ˜ r ∂ Θ j AB = 0 ˜ ◮ Subleading Lorenz gauge condition informs otherwise unconstrained parts (for instance, the ℓ = 0 , 1 parts not expressible as TT waves) ∇ µ j µν + k µ ∂ Θ k µν = 0 ◮ Remaining components fix the now nontrivial non-TT components of k µν : − 1 Θ k AB = − G (1 , 1) 2 δ AB ∂ 2 [ j ] kk − 1 Θ k lA = − G (1 , 1) 2 ∂ 2 kA [ j ] ∂ 2 Θ k ll =0 Post-adiabatic two-timescale Cornell University
Recommend
More recommend
