Effective source for second-order self-force calculations: quasicircular orbits in Schwarzschild spacetime

Abstract

Recent years have seen the first production of "post-adiabatic" gravitational-waveform models based on second-order gravitational self-force theory. These models rely on calculations of an effective source in the perturbative second-order Einstein equation. Here, for the first time, we detail the calculation of the effective source in a Schwarzschild background, which underlies the second-order self-force results in [Phys. Rev. Lett. 127, 151102 (2021); ibid. 128, 231101 (2022); ibid. 130, 241402 (2023)]. The source is designed for use in the multiscale form of the Lorenz-gauge Einstein equation, decomposed in tensor spherical harmonics, or in the analogous second-order Teukolsky equation. It involves, among other things, contributions from (i) quadratic coupling of first-order field modes, (ii) the slow evolution of first-order fields, (iii) quadratic products of a first-order puncture field, and (iv) the second-order puncture field. We validate each of these pieces through numerical and analytical tests.

Figures (15)

• Figure 1: Visualisation of the transformation between rotated and unrotated coordinates, with blue arrows depicting rotations and solid orange arrows indicating the particle's velocity. The particle is rotated from $\phi=\phi_{p}$ to $\phi=0$, moved from the equator to the north pole and then has its velocity vector aligned with the curve $\beta=0$.
• Figure 2: Large-$\ell$ behaviour of the modes of a singular field with odd powers of $\rho$ for $|m'|=|s|$ (top) and $|m'|=|s|+2$ (bottom). Some cases, denoted by a $\pm$, have a directionally dependent value as $\Delta r \to 0$. Light coloured regions correspond to pieces required for the checks in Sec. \ref{['sec:validation']} while dark coloured regions correspond to extra pieces included in the expressions we provide online PuncturesRepository. Yellow coloured regions correspond to pieces included in the first order puncture that is used to compute the second order Ricci tensor (see Sec. \ref{['sec:d2R']}). As $|m'|$ increases from $|s|$ the columns shift leftwards. The $|m'|=1, 3, 4$ cases are not shown but are similarly included for the 'singular times regular' punctures, while the 'singular times singular' puncture includes the $|m'|=1$ case.
• Figure 3: Structure of the modes of a singular field with even powers of $\rho$ for $|m'|=|s|$. In this case we get additional logarithmic (in $\lambda$, $\Delta r$ and $\ell$) contributions compared to the odd-power case. Cases with a dash indicate that there is no large-$\ell$ contribution, although in some case there is a contribution for a specific $\ell$.
• Figure 4: (Left panel) The second-order Ricci tensor $\delta^2R_{122}[h^1_{\ell_{\rm max}},h^1_{\ell_{\rm max}}]$ for a variety of $\ell_{\rm max}$ values for a particle at $r_0 = 7.6M$. Curves of increasing $\ell_{\rm max}$ are stacked from bottom to top. (Right panel) Convergence of $\left|\delta^2R_{122}[h^1_{\ell_{\rm max}=50},h^1_{\ell_{\rm max}=50}] - \delta^2R_{122}[h^1_{\ell_{\rm max}},h^1_{\ell_{\rm max}}]\right|$ with $\ell_{\rm max}$ for a particle at $r_0 = 7.6M$. Here curves of increasing $\ell_{\rm max}$ are stacked from top to bottom. Note the convergence is very rapid far from the worldline, but it becomes unacceptably slow (while remaining formally exponential) close to the worldline.
• Figure 5: (Left panel) Convergence of $\left|\delta^2R_{122}[h^{\mathcal{P}1},h^{\mathcal{R}1}_{\ell_{\rm max}}] - \delta^2R_{122}[h^{\mathcal{P}1},h^{\mathcal{R}1}_{\ell_{\rm max=40}}]\right|$ with $\ell_{\rm max}$ for a particle at $r_0 = 7.6M$. (Right panel) Convergence of $\left|\delta^2R_{122}[h^{\mathcal{R}1}_{\ell_{\rm max}},h^{\mathcal{P}1}] - \delta^2R_{122}[h^{\mathcal{R}1}_{\ell_{\rm max=40}},h^{\mathcal{P}1}]\right|$ with $\ell_{\rm max}$ for a particle at $r_0 = 7.6M$. In both cases, curves of increasing $\ell_{\rm max}$ are stacked from top to bottom. Note the contrast to Fig. \ref{['fig:retret']}: even though the convergence is now polynomial rather than exponential, in practice this polynomial convergence produces a much more accurate result close to the worldline.