Incorporating eccentricity into GW templates

Transcription

1 Incorporating eccentricity into GW templates Manuel Tessmer & Achamveedu Gopakumar Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universität, Jena

2 The plan for this talk: Part I: 1. Compact binaries in eccentric orbits as LISA sources 2. Construction of accurate & efficient templates for above sources 3. Conclusions, time-& frequency domain waveforms & an application of Part I Journal-ref.: Mon.Not.Roy.Astron.Soc. 374 (27) Part II - WORK IN PROGRESS 1. How much residual eccentricity GEO, LIGO & VIRGO can tolerate? 2. Observations & provisional results Videoseminar, 25. June, M.Tessmer p.1/17

3 Astrophysical motivations I Compact binaries in eccentric orbits as LISA sources Chaurasia & Bailes 25: highly eccentric orbits as a natural consequence of an asymmetric kick imparted to neutron stars at birth Gusev et al. 22: assuming stationary distribution in the galaxy, LISA will see several NS-NS, NS-BH and BH-BH binaries in eccentric orbits. Benacquista 22, employing Monte Carlo simulations to model galactic globular clusters, observed that LISA may see several BH binaries in eccentric orbits Videoseminar, 25. June, M.Tessmer p.2/17

4 Astrophysical motivations II For LISA, stellar mass compact binaries [f Orb 1 3 Hz], can be modeled to move in slowly precessing ellipses. Effects of RR may be neglected. Reason: frequency sensitivity of LISA f LISA Hz [1 year observation] f k 1 7 Hz [due to the advance of periastron] f RR 1 9 Hz Videoseminar, 25. June, M.Tessmer p.3/17

5 Astrophysical motivations III It is desirable to have accurate & efficient eccentric GW templates GW templates consist of h Q & h + Q. We provided an accurate & efficient way to compute h Q & h + Q with fully 1PN accurate orbital motion. We also obtained the associated power spectrum. Videoseminar, 25. June, M.Tessmer p.4/17

6 h + (t) & h (t) I h + Q & h Q, (T. Damour, A. Gopakumar, B. R. Iyer (24) j «h Q (r, φ, ṙ, φ) = 2 G m η C G m c 4 R + r r 2 φ2 ṙ 2 sin2φ ff 2r ṙ φ cos2φ, h + Q (r, φ, ṙ, φ) =... where η = µ M = m 1 m 2 (m 1 +m 2 ) 2, C = cosi, (r, φ, ṙ, φ) are the dynamical variables. Videoseminar, 25. June, M.Tessmer p.5/17

7 h + (t) & h (t) II QK parameterization & geometrical meanings [T. Damour, N. Deruelle, (1985)] r = a r (1 e r cos u) φ φ = (1 + k) ν " 1 «1 # + eφ 2 u ν = 2 arctan tan 1 e φ 2 u eccentric anomaly, ν true anomaly, k advance of the periastron, a r, e r, e φ, n and e t are orbital elements a u 11 O R F v Videoseminar, 25. June, M.Tessmer p.6/17

8 h + (t) & h (t): III To obtain highly accurate h + (t) & h (t), we need to solve the 1PN accurate Kepler-equation l = n (t t ) = u e t sinu. We employed Mikkola s solution, detailed in the next section. [S. Mikkola (1987)] Videoseminar, 25. June, M.Tessmer p.7/17

9 MIKKOLA S solution: I Seppo Mikkola provided one of the most accurate and efficient numerical ways of solving the classical Kepler equation. A numerical solution to the KE usually employs Newton s method which requires an initial guess u depending on l and e t. A number of iterations will be required to obtain an approximate solution that has some desired accuracy. Videoseminar, 25. June, M.Tessmer p.8/17

10 MIKKOLA S solution: II 4 steps to solve KE 1. Replace variable u by s = sin(u/3) and write 3 arcsin s e t (3s 4s 3 ) = l 2. Truncating to the third order of Taylor expansion: 3 (1 e t )s + (4e t )s3 = l 3. Solution of this cubic equation & add correction term: ds =.78s 5 /(1 + e t ) solution provides accuracy of 1 3, a higher one is desired: 4. Take solution as an initial guess u for 4 th -order Newton s method: u = u + u 4 Videoseminar, 25. June, M.Tessmer p.9/17

11 MIKKOLA S solution: III Mikkola s method is a relatively simple and robust procedure to compute u(l) with a relative error that is not greater than It requires a solution of a cubic polynomial and only a one time evaluation of few trigonometric functions. It applies to all pairs (e t, l) with e 1. Videoseminar, 25. June, M.Tessmer p.1/17

12 What we have done: h +, (r, φ, ṙ, φ) Quasi-Keplerian parametrization h +, (r(u, n, e t ), φ(u, n, e t ), ṙ(u, n, e t ), φ(u, n, e t )) Mikkola s solution h +, (l, n, e t ) & the associated power spectrum Videoseminar, 25. June, M.Tessmer p.11/17

13 Results: I Time evolution & power spectrum toh + Q : 2e-5 2e-5 e t =.1 e t = e t =.1 e t =.4 1e-5 1e e-5.4-1e-5-2e H + Q -2e e-5-5e e 5e-5 t =.7 e t = e t =.7 e t = e e mean anomaly l frequency in units of f r Videoseminar, 25. June, M.Tessmer p.12/17

14 Results: II The ratio of the total power present in H Q & H + Q vs. orbital inclination: e t =.1 e t = e t =.7 e t = orbital inclination i i Videoseminar, 25. June, M.Tessmer p.13/17

15 Ongoing investigations Inclusion of RR Inclusion of finite-size effects 1. White dwarf binaries (noncompact objects) single most abundant & guaranteed source for LISA 2. Expected to be in highly eccentric orbits due to dynamics in GC 3. Monopole-quadrupole-interactions: for certain noncompact binaries, frequency shift in the observed GW spectrum that is expected to dominate the 1PN periastron advance 4. Internal structure visible for higher eccentricities LISA data analysis aspects Videoseminar, 25. June, M.Tessmer p.14/17

16 Part II: Inclusion of RR ICB (NS NS; NS BH, BH BH) in inspiralling eccentric orbits: plausible sources for ground - based detectors. GW reduce angular momentum & energy e & quasi-circularity appropriate Martel & Poisson(2) 1. Bank of circular templates with parameters M c & t c used to maximize the ambiguity function A( (s h) θ) =, def. FF := max θ A( θ) (s s)(h h) 2. Note: eccentric waveforms can be detected with circular templates up to e.2 for a FF >.97. Videoseminar, 25. June, M.Tessmer p.15/17

17 Part II: Inclusion of RR ICB (NS NS; NS BH, BH BH) in inspiralling eccentric orbits: plausible sources for ground - based detectors. GW reduce angular momentum & energy e & quasi-circularity appropriate Martel & Poisson(2) 1. Bank of circular templates with parameters M c & t c used to maximize the ambiguity function A( θ) (s h) =, def. FF := max θ A( θ) (s s)(h h) 2. Note: eccentric waveforms can be detected with circular templates up to e.2 for a FF >.97. Can this result be trusted? Videoseminar, 25. June, M.Tessmer p.15/17

18 Part II: Inclusion of RR ICB (NS NS; NS BH, BH BH) in inspiralling eccentric orbits: plausible sources for ground - based detectors. GW reduce angular momentum & energy e & quasi-circularity appropriate Martel & Poisson(2) 1. Bank of circular templates with parameters M c & t c used to maximize the ambiguity function A( θ) (s h) =, def. FF := max θ A( θ) (s s)(h h) 2. Note: eccentric waveforms can be detected with circular templates up to e.2 for a FF >.97. Can this result be trusted? MAYBE NOT! 3. Because Martel & Poisson used N + 2.5PN accurate orbital motion, while secular nonreactive effects like the PN accurate advance of periastron are neglected Videoseminar, 25. June, M.Tessmer p.15/17

19 Part II: inclusion of RR time evolution to h + Q : m 1 = 1.4M ; m 2 = 1.M ; e =.1; n 8Hz.2 "H_(1)_14_1.txt".2 "H_(1)_14_1.txt" (c) Full 2.5PN (d) N+RR Videoseminar, 25. June, M.Tessmer p.16/17

20 Provisional results We have reproduced the analysis by Martel & Poisson 2 We also developed an improved Mikkola s method for treating PN accurate KE Currently, we are computing FF with h Q (t) whose orbital motion is fully 2.5PN accurate Early resultes give low FFs compared to those obtained by Martel & Poisson Videoseminar, 25. June, M.Tessmer p.17/17