Binary black hole coalescence in semianalytic puncture evolution

Achamveedu Gopakumar and Gerhard Schäfer

Phys. Rev. D 77, 104023 – Published 19 May 2008

Abstract

Binary black hole coalescence is treated semianalytically by a novel approach. Our prescription employs the conservative Skeleton Hamiltonian that describes orbiting Brill-Lindquist wormholes (termed punctures in numerical relativity) within a waveless truncation to the Einstein field equations [G. Faye, P. Jaranowski, and G. Schäfer, Phys. Rev. D 69, 124029 (2004)]. We incorporate, in a transparent Hamiltonian way and in Burke-Thorne gauge structure, the effects of gravitational radiation reaction into the above Skeleton dynamics with the help of 3.5PN accurate angular momentum flux for compact binaries in quasicircular orbits to obtain a semianalytic puncture evolution to model merging black hole binaries. With the help of the TaylorT4 approximant at 3.5PN order, we perform a first-order comparison between gravitational-wave phase evolutions in numerical relativity and our approach for equal-mass binary black holes. This comparison reveals that a modified Skeletonian reactive dynamics that employs flexible parameters will be required to prevent the dephasing between our scheme and numerical relativity, similar to what is pursued in the effective one-body approach. A rough estimate for the gravitational waveform associated with the binary black hole coalescence in our approach is also provided.

Received 28 December 2007

DOI:https://doi.org/10.1103/PhysRevD.77.104023

Authors & Affiliations

Achamveedu Gopakumar and Gerhard Schäfer

Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany

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Vol. 77, Iss. 10 — 15 May 2008

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Images

Figure 1
Plots depicting various facets of SAPE, defined by Eqs. (30), for a $η = 0.25$ BBH having initial orbital separation $r = 10$ [recall that we have dropped the hat symbol appearing in the dimensionless variables defined by Eq. (21)]. In these plots, we terminate the SAPE when $ω$ reaches its maximum value, $ω_{mx} \sim 0.0896$ . The panel (i) provides a parametric plot of $x = r \cos ϕ$ versus $y = r \sin ϕ$ , indicating that the SAPE occurs along quasicircular orbits. The temporal evolutions for the dimensionless binding energy, $E \equiv H_{Sk}$ , and the orbital angular momentum $j$ are plotted in the panels (ii) and (iii). The plot for $p_{r} (t)$ suggests that even during the dynamical plunge, i.e. in the neighborhood of $ω_{mx}$ , the orbital motion is not that different from a quasicircular inspiral.Reuse & Permissions
Figure 2
Plots providing $ω (t)$ , $ω (r (t))$ , $r (t)$ , and $\dot{r} (t)^{2}$ for BBH evolution under SAPE [the binary configuration is the same as in Fig. 1 and all quantities are dimensionless]. The rapid changes in $ω$ , $r$ , and ${\dot{r}}^{2}$ occurring during the dynamical plunge is clearly visible and we terminate the SAPE when ! reaches its maximum value $ω_{mx}$ . Though ${\dot{r}}^{2} = (d r / d t)^{2}$ remains small near $ω_{mx}$ , its magnitude changes roughly fivefold during the plunge under SAPE. The black hole binary evolution that employs Eq. (31) for the initial value of $p_{r}$ is displayed with a thick line, while BBH evolution having $p_{r} = 0$ at the initial instant is displayed with dashed lines. The spurious eccentricity in SAPE is suppressed by using at $t = 0$ the expression for $p_{r}$ , given by Eq. (31), and this is also reported in Ref. 19.Reuse & Permissions
Figure 3
Plots displaying reactive evolutions of few interesting dynamical quantities and relations in the SAPE [the binary configuration is the same as in Fig. 1 and all quantities are dimensionless]. The top panel plots $v_{ω}^{2} = ω^{2 / 3}$ , $v_{G}^{2} = {\dot{r}}^{2} + r^{2} ω^{2}$ . The strong $r$ dependence of $v_{G}^{2}$ makes it less attractive to characterize the orbital velocity compared to $v_{ω}$ . In the middle panel, we plot the “Kepler combination” (KC) $ω^{2} r^{3}$ against $r$ and a gradual decrease is clearly visible throughout the inspiral. The quasicircularity (QC) condition ( $d E - \bar{ω} d j$ ) is displayed in the bottom panel and its smallness justifies the use of $v_{ω}$ in the 3.5 PN accurate expression for $d j / d t$ .Reuse & Permissions
Figure 4
Differences in the orbital phase evolutions associated with the SAPE and TaylorT4 approximant at 3.5PN order along with the associated $x \equiv ω^{2 / 3}$ evolutions [recall that we have dropped the hat symbol appearing in the dimensionless variables defined by Eq. (21)]. From the top to bottom panels, the orbital frequency ranges are [0.025–0.05], [0.025–0.045], and [0.022–0.045], respectively. The fractional differences in $ϕ$ are $\sim 7 %$ , 5.8%, and $\sim 6.1 %$ in the above three cases. A sharp increase in $ϕ_{Sk} (t) - ϕ_{T 4} (t)$ as we approach the plunge is also observed.Reuse & Permissions
Figure 5
Plots showing temporal evolution of scaled $h_{\times, +}$ , $ℜ (ψ_{4})$ , and $ℑ (ψ_{4})$ for equal-mass binary black hole coalescence in SAPE. We scale out $G m / c^{2} r^{'}$ from Eqs. (37, 42) and let $i = π / 3$ . The matching to the QNMs is based on Ref. 7 (see Sec. 4 and recall that $t$ is identical to $\hat{t}$ , defined by Eq. (21)).Reuse & Permissions

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