Large-Eccentricity Asymptotics and Fast Analytic Approximation for Fourier modes of Post-Newtonian Eccentric Waveforms

Author(s)

Liu, Xiaolin, Cao, Zhoujian

Abstract

In this work, we developed analytic asymptotic methods for computing the Fourier modes of gravitational waves from post-Newtonian binary systems in the quasi-Keplerian parametrization in the high eccentricity regime. We have also derived the large-eccentricity asymptotic expansion of the eccentricity enhancement function appearing in the tail contributions to the radiation. Furthermore, based on these results, we constructed an endpoint-constrained analytic approximation that significantly accelerate the computation of the Fourier modes at large eccentricity.The overall error of this analytic approximation is controlled within $10^{-3}$, and it remains valid for Fourier modes with $p\le200$. This approach provides an analytic building blocks for modeling frequency-domain gravitational wave from highly eccentric binaries.

Figures

Caption

Comparison of the numerical results (labeled as `Numeric') with the two approximation schemes—the $\Delta_e$ asymptotic expansion (labeled as `AE') and the uniform asymptotic expansion (labeled as `UAE')—for several representative integrals in the range $e>0.5$. The selected examples are $\feJ{(p, a=(1,3), 0 )}$, $\feK{(p, a=(1,3), 0)}$, and $\feJ{(p, a=(0,2), b=(1,2,3))}$. The upper panels correspond to the case $p=2$, while the lower panels show the case $p=10$.

Caption

Comparison of the numerical results (labeled as `Numeric') with the two approximation schemes—the $\Delta_e$ asymptotic expansion (labeled as `AE') and the uniform asymptotic expansion (labeled as `UAE')—for several representative integrals in the range $e>0.5$. The selected examples are $\feJ{(p, a=(1,3), 0 )}$, $\feK{(p, a=(1,3), 0)}$, and $\feJ{(p, a=(0,2), b=(1,2,3))}$. The upper panels correspond to the case $p=2$, while the lower panels show the case $p=10$.

Caption

Comparison of the numerical results (labeled as `Numeric') with the analytic approximated form of the PN-elliptic integrals (\ref{eq_approximated_regI}) (labeled as `Approx'). The selected examples are $\feJ{(p, a=(1,3), 0 )}$, $\feK{(p, a=(1,3), 0)}$, and $\feJ{(p, a=(0,2), b=(1,2,3))}$. The upper panels correspond to the case $p=10$, while the lower panels show the case $p=100$.

Caption

Comparison of the numerical results (labeled as `Numeric') with the analytic approximated form of the PN-elliptic integrals (\ref{eq_approximated_regI}) (labeled as `Approx'). The selected examples are $\feJ{(p, a=(1,3), 0 )}$, $\feK{(p, a=(1,3), 0)}$, and $\feJ{(p, a=(0,2), b=(1,2,3))}$. The upper panels correspond to the case $p=10$, while the lower panels show the case $p=100$.

Caption

Left panel: the error between the analytic approximation of the regularized integrals and the numerical results as a function of $p$. Different types of integrals are distinguished by different colors. Right panel: the comparison of the worse fitted case $\feJ{(p,1,3)}$, which corresponds to the most top line in the left panel, with the numeric result.

Caption

Left panel: the error between the analytic approximation of the regularized integrals and the numerical results as a function of $p$. Different types of integrals are distinguished by different colors. Right panel: the comparison of the worse fitted case $\feJ{(p,1,3)}$, which corresponds to the most top line in the left panel, with the numeric result.

Caption

Comparison of the 3PN leading-mode waveform $h_{22}$. In this example the binary has mass ratio $\nu=0.22$, initial frequency $v_0=0.17$, and the evolution ends at $v=1/\sqrt{6}$. The initial eccentricity is $e_0=0.7$. The gray line corresponds to the numeric results, obtained by summing $h_{22}\propto v^2 \eexp{-\ii 2(l-\lambda)} \sum_p \hat{H}_{2(-2)p}\eexp{\ii p l}$ until the required accuracy is reached. The blue and red lines correspond to the results obtained by computing each mode $\hat{H}_{2(-2)p}$ using our approximate expressions (\ref{eq_approximated_regI}), and then summing over $p$ up to $p=10$ (labeled by `Approx $p_{\max}=10$') and $p=30$ (labeled by `Approx $p_{\max}=30$'), respectively. The green line denotes the results of post-circular expansion truncated at $\order{e^{10}}$ (labeled by `PC'). The bottom panel shows the evolution of orbital eccentricity.

Caption

Comparison between the numerical results for the regularized EEFs (black cross markers) $(\varphi,\tilde\varphi,\beta,\tilde\beta,\gamma,\tilde\gamma,\chi,\tilde\chi)$ and the asymptotic expansion (red solid line) in the limit $\Delta_e\to1$.

Caption

Comparison between the numerical results for the EEFs (black cross markers) $(\alpha,\tilde\alpha,\theta,\tilde\theta)$ and the asymptotic expansion (red solid line) in the limit $\Delta_e\to1$.