Merger breakdown

inspiral: analytic (many cycles!)

• Goal: Compute effective two-body Hamiltonian (+ radiative losses) $$\begin{aligned} H\sim G(1+v^2+v^4+v^6+\dots&)\\ + G^2(1+v^2+v^4+\dots&)\\ + G^3(1+v^2+\dots&) + \dots\\ \end{aligned}$$
• Various directions:
  • Conservative potential
  • Spinning objects
  • Radiation
  • Radiation reaction (tail effects)
  • Inner structure & tidal deformations (BH vs NS)
  • ...

• Post-Newtonian: expansion in $v^2\sim\frac{Gm}{r}\ll 1$
  • [Droste, Lorentz 1917; ...; Einstein, Infeld, Hoffmann 1938]
  • [Blanchet, Damour, Iyer, Faye, Bernard, Bohe’, Buonanno, Marsat; Jaranowski, Schäfer, Steinhoff; Will, Wiseman; Flanagan, Hinderer, Vines; Goldberger, Porto, Rothstein; Kol, Levi, Smolkin; Cristofoli, Mastrolia, Foffa, Sturani, Torres-Bobadilla; Blümlein, Maier, Marquard, Schäfer;...]

• Post-Minkowskian: expansion in $G$
  • "Classical GR" methods [Bertotti; Westpfahl, Goller; Bel, Damour, Deruelle, Ibanez, Martin; Ledvinka, Schäfer, Bicak; ...]
  • Using perturbation theory: Scattering amplitudes
  • First 3PM result [Bern, Cheung, Roiban, Shen, Solon, Zeng 2018/2019]
• Small mass-ratio/gravitational self-force: $m_2/m_1\ll 1$
  • [Regge, Wheeler 56; Zerilli 70; Teukolsky 72]
  • [Fujita, Poisson, Sasaki, Shibata, Khanna, Hughes, Bernuzzi, Harms, Nagar, Deitweiler, Whiting, Mino, Quinn, Wald, Tanaka, Barack, Ori, Pound, van de Meent, ...]

merger: numerical relativity (high velocities!)

• Simulation from the slide before!
• Breakthrough 2005 [Pretorius '05; Campanelli et al. '06, Baker et al. '06]

ringdown: blackhole perturbation theory

• Get the full waveform with from the Effective-One-Body (EOB) approach
• [Buonanno, Damour 1999]

Bound-to-boundary (B2B) relations

PM-EFT for a worldline action coupled to GR

• We are only interested in classical physics
• Perturbative expansion in $G$: can use particle physics/amplitudes toolbox
• Today: only interested in conservative effects

Full theory

$$\begin{align} S_{\rm EH} &= -2\Mp^2 \int \dd^4x \sqrt{-g} \, R[g]\\ S_{\rm pp} &= - \sum_a m_a \int \dd\sigma_a \sqrt{g_{\mu\nu}(x^\alpha_{a}(\sigma)) v_{a}^\mu(\sigma_a) v_{a}^\nu (\sigma_a)} + \dots\\ &\rightarrow -\sum_a \frac{m_a}{2} \int \dd\tau_a\, g_{\mu\nu}(x_{a}(\tau_a)) v_{a}^\mu(\tau_a) v_{a}^\nu (\tau_a)+\dots \end{align}$$

Extensions to finite-size effects and spinning bodies are simple to implement in this formalism.

EFT action

$$e^{i S_{\rm eff}[x_a] } = \int \cD h_{\mu\nu} \, e^{i S_{\rm EH}[h] + i S_{\rm GF}[h] + i S_{\rm pp}[x_a,h]}$$

We optimized the EH-Lagrangian by cleverly chosing gauge-fixing terms and adding total derivatives:

• 2-point Lagrangian: 2 terms
• 3-point Lagrangian: 6 terms
• 4-point Lagrangian: 18 terms
• 5-point Lagrangian: 36 terms

These numbers could even be further reduced by field redefinitions, but we don't want higher point WL-couplings.

PM deflection

In a Post-Minkowskian expansion: $$S_{\rm eff} = \sum_{n=0}^\infty \int \dd\tau_1\,G^n \cL_n [x_1(\tau_1),x_2(\tau_2)]$$ with $$\cL_0 = - \frac{m_1}{2} \eta_{\mu\nu} v_1^\mu(\tau_1) v_1^\nu(\tau_1)$$ E.o.m. from variation of the action $$-\eta^{\mu\nu}\frac{\dd}{\dd\tau_1} \left(\frac{\partial \cL_0}{\partial v^\nu_1}\right) = m_1 \frac{\dd v_1^\mu}{\dd\tau_1} = -\eta^{\mu\nu}\left(\sum_{n=1}^{\infty} \frac{\partial \cL_n}{\partial x^\nu_1(\tau_1)} - \frac{\dd}{\dd\tau_1} \left(\frac{\partial \cL_n}{\partial v^\nu_1}\right)\right)$$ allows us to compute the trajectories order by order: $$x^\mu_a(\tau_1) = b^\mu_a + u^\mu_a \tau_a + \sum_n G^n \delta^{(n)} x^\mu_a (\tau_a)$$ with $b=b_1-b_2$ the impact parameter and $u_a$ the incoming velocty at infinity, fulfilling $$u_1\cdot u_2 = \gamma\,, \quad u_a\cdot b = 0\,.$$

Scattering angle.

First we compute the deflection using above trajectories: $$\Delta p^\mu_1= m_1\Delta v^\mu_1 = - \eta^{\mu\nu} \sum_n \int_{-\infty}^{+\infty} \dd\tau_1 {\partial \cL_n \over\partial x^\nu_1}\,,$$ Physical scattering angle is then simply $$2\sin\left(\frac{\chi}{2}\right) = \frac{|\Delta \bp_{1{\rm cm}} |}{p_\infty}= \frac{\sqrt{-\Delta p_1^2}}{p_\infty}$$

Integration

Generic structure: $$\int \dd^Dq\frac{\delta(q\cdot u_1)\delta(q\cdot u_2)e^{i b\cdot q}}{(q^2)^m} \underbrace{\int \dd^D\ell_1\cdots\dd^D\ell_L\frac{\delta(\ell_1\cdot u_{a_1})\cdots\delta(\ell_L\cdot u_{a_L})}{(\ell_1\cdot u_{b_1}\pm i0)^{i_1}\cdots(\ell_L\cdot u_{b_L}\pm i0)^{i_L}(\textrm{sq. props})}}_{\textrm{Cut Feynman integrals with linear and square propagators}}$$

• Integration by part identities (IBPs): reduction to masters [Chetyrkin, Tkachov '81,...]
• Differential equations in $x$: $\gamma= (x^2+ 1)/(2x)$ [Parra-Martinez, Ruf, Zeng '20]
• Canonical basis: solve order by order in $\epsilon$ [Henn 13' 14']
• Static boundary conditions: potential region sufficient for conservative part $\rightarrow$ already known from PN-EFT

Final Fourier transform is known to all orders: $$\Delta p_a^\mu = \int \dd^Dq (A \overbrace{q^\mu}^{-i \frac{\partial}{\partial b_\mu}} + \underbrace{B u_1^\mu + C u_2^\mu}_{\substack{\textrm{bootstrap from}\\ \textrm{on-shell condition}\\ (\Delta p_a+p_a)^2=p_a^2}}) \frac{\delta(q\cdot u_1)\delta(q\cdot u_2)e^{i b\cdot q}}{(q^2)^{m+n}}$$

Results.

$$\frac{\chi}{2} = \sum_{n=1} \chi^{(n)}_b \left(\frac{GM}{b}\right)^n$$ Up to 3PM (agreeing with [Bern, Cheung, Roiban, Shen, Solon, Zeng 2018/2019]): $$\begin{align} \frac{\chi^{(1)}_b}{\Gamma} &= \frac{2\gamma^2-1}{\gamma^2-1}\\ \frac{\chi^{(2)}_b}{\Gamma} &= \frac{3\pi}{8} \frac{5\gamma^2-1}{\gamma^2-1}\\ \frac{\chi^{(3)}_b}{\Gamma} &= \frac{1}{(\gamma^2-1)^{3/2}}\Bigg[-\frac{4\nu}{3} \gamma\sqrt{\gamma^2-1}(14\gamma^2+25)\frac{(64\gamma^6-120\gamma^4+60\gamma^2-5)(1+2\nu(\gamma-1))}{3(\gamma^2-1)^{3/2}}\\ &\quad- 8\nu (4\gamma^4-12\gamma^2-3)\Arcsinh\sqrt{\frac{\gamma-1}{2}}\,\Bigg] \end{align}$$ with $\gamma = u_1\cdot u_2$, $\Gamma=\sqrt{1+2\nu(\gamma-1)}$, $\nu=m_1m_2/M^2$.

Boundary-To-Bound (B2B) dictionary

Conservative motion described by a Hamiltonian: $$H(\bp,r) = E \quad \Longrightarrow \quad \bp(r,E)$$ Scattering angle: $$\chi(b,E) +\pi = 2b \int_{r_{\rm min}}^\infty \frac{\dd r}{r\sqrt{r^2\bar\bp^2(r,E)-b^2}}$$ with $r_\textrm{min}$ the positive real root of $p_r$: $p_r^2(r,E)=\bp^2(r,E)-J^2/r^2$.
Periastron advance: $$\Delta \Phi + 2\pi = 2J \int_{r_-}^{r_+} \frac{\dd r}{r\sqrt{r^2\bp^2(r,E)-J^2}}$$ Don't forget: $J=p_\infty b$, $\bar\bp = \bp/p_\infty$.

Interlude: Impetus formula

In [1910.03008, w/ R. Porto] we found a formula relating the classical IR-finite part of the 2-to-2 scattering amplitude to the momentum along the trajectory: with $${\widetilde{\cM}}(r,E) \equiv \frac{1}{2E}\int \frac{\dd^3\bq}{(2\pi)^3}\, {\cal M}(\bq,\bp^2=p_\infty^2(E)) e^{-i\bq\cdot \br}$$ or in PM expanded form: $\widetilde{\cal M}_n(E) = P_n(E)$ with $$\widetilde{\cal M}(r,E) = \sum_{n=1}^\infty \widetilde{\cal M}_n(E) \left(\frac{G}{r}\right)^n,\quad\bp^2(r,E) = p_\infty^2(E) +\sum_{n=1}^\infty P_n(E) \left(\frac{G}{r}\right)^n$$

Firsov's formula

Let us invert $$\chi(b,E) = -\pi + 2b \int_{r_{\rm min}}^\infty \frac{\dd r}{r\sqrt{r^2\bar\bp^2(r,E)-b^2}} = \sum_{n=1} \chi^{(n)}_b(E) \left(\frac{GM}{b}\right)^n$$ [Firsov '53]: dependence on $r_\textrm{min}$ drops out These integrals are easy to perform in a PM-expanded form and one finds: $$\chi_b^{(n)} = \frac{\sqrt{\pi}}{2} \Gamma\left(\frac{n+1}{2}\right)\sum_{\sigma\in\mathcal{P}(n)}\frac{1}{\Gamma\left(1+\frac{n}{2} -\Sigma^\ell\right)}\prod_{\ell} \frac{f_{\sigma_{\ell}}^{\sigma^{\ell}}}{\sigma^{\ell}!}$$ The inversion thereof also exists.

Orbital Elements

$$r_{\textrm{min}} = \tilde r_- \overset{\textrm{Firsov}}{=} b \exp\left[ -\frac{1}{\pi} \int_{b}^\infty \frac{\chi(\tilde b,E)\dd\tilde b}{\sqrt{\tilde b^2-b^2}}\right] =b \prod_{n=1}^\infty e^{-\frac{(GM)^n\chi_b^{(n)}(E)\Gamma\left(\frac{n}{2}\right)}{b^n\sqrt{\pi}\Gamma\left(\frac{n+1}{2}\right)}}$$ Do an analytic continuation in $J=p_\infty b$ with $b\in i\mathbb{R}$ ($p_\infty^2\leq0$): $$\begin{align*} r_-(J) &= r_{\textrm{min}}(b)\\ r_+(J) &= r_-(-J) = r_{\textrm{min}}(-b) \end{align*}$$ These are the two real positive roots of $p_r$: $p_r^2(r,E)=\bp^2(r,E)-J^2/r^2$ with $J=p_\infty b$

$\textcolor{desyOrange}{\tiny r_\textrm{min}(-J)}$ $\textcolor{desyOrange}{\tiny r_\textrm{min}(J)}$ $$\begin{align} &=2J \int_{r_\textrm{min}(J)}^{\infty} \frac{\dd r}{r\sqrt{r^2\bp^2(r,E)-J^2}}\\ &\quad-2J \int_{r_\textrm{min}(-J)}^{\infty} \frac{\dd r}{r\sqrt{r^2\bp^2(r,E)-J^2}}\\ &=\chi(J,E)+\chi(-J,E)+2\pi \end{align}$$

Observables for the bound state

We already saw that the periastron advance can be extracted from the angle/amplitude. In PM language: $$\Delta\Phi = \pi \frac{\widetilde\cM_2}{\mu^2M^2 j^2} +\frac{3\pi}{4}\frac{1}{M^4 \mu^4 j^4}\big(\widetilde\cM_2^2+2\widetilde\cM_1\widetilde\cM_3+2p_\infty^2\widetilde\cM_4\big)+\cdots$$

The radial action

$$\frac{\Delta \Phi+2\pi}{2\pi} = - \frac{\partial \cS_r(J,\cE)}{\partial J}= -\frac{\partial}{\partial J} \frac{1}{\pi} \int_{r_-}^{r_+} \sqrt{\bp^2(r,{\cal E})-J^2/r^2}$$ Simply integrate in $J$ and find the boundary term: $$i_r = \frac{{\cal S}_r}{GM\mu} = {\rm sg}(\hat p_\infty)\chi^{(1)}_j(\cE) - j \left(1 + \frac{2}{\pi} \sum_{n=1} \frac{\chi^{(2n)}_j({\cE})}{(1-2n)j^{2n}}\right)$$ with $\chi_j^{(n)} = \left(\frac{p_\infty}{M\nu}\right)^n \chi_b^{(n)}$. Or in amplitudes language: $$i_r = -j + \frac{{\widetilde \cM}_1}{2 \sqrt{ -\hat p^2_\infty}M \mu^2} + \frac{{\widetilde \cM}_2}{2 jM^2\mu^2} + \frac{{\widetilde \cM}_2^2 +2 {\widetilde \cM}_1{\widetilde \cM}_3+2 p_\infty^2 {\widetilde \cM}_4}{8j^3M^4\mu^4}+\cdots$$

Observables: examples

• Periastron advance: $\frac{\Delta\Phi}{2\pi} = -\frac{\partial}{\partial j} \cS_r(j,\cE)-1$
• Periastron-to-periastron period: $\frac{T_p}{2\pi} = \frac{1}{\mu}\frac{\partial}{\partial \cE} \cS_r(j,\cE)$
• Radial frequency: $\Omega_r (j,\cE) = \frac{2\pi}{T_p}$
• Periastron frequency: $\Omega_p (j,\cE) = \frac{\Delta\Phi}{T_p}$
• Azimuthal frequency: $\Omega_\phi\equiv\Omega_r+\Omega_p=\frac{2\pi}{T_p}\left(1+\frac{\Delta\Phi}{2\pi}\right)$
• Averaged redshift: $\langle z_a\rangle = 1+\frac{\partial \mu}{\partial m_a} \cE -\Omega_r \frac{\partial}{\partial m_a}\cS_r(j,\cE,m_a)$ [LeTiec 15]

Classical potential

We can also extract the PM-expanded conservative two-body potential. We iteratively solve $$H(r,\bp^2) = E = \sqrt{\bp^2+m_1^2}+\sqrt{\bp^2+m_2^2} + V(r,\bp^2)$$ using $$E = E_1 + E_2 = \sqrt{p_\infty^2+m_1^2}+\sqrt{p_\infty^2+m_2^2}$$ and $$p_\infty^2(E) = \bp^2(r,E) - \sum_i P_i(E) \left(\frac{G}{r}\right)^i\,,$$ for the PM coefficient of the potential $$V(r,\bp^2) = \sum_{i=0}^{\infty} \frac{c_i(\bp^2)}{i!} \left(\frac{G}{r}\right)^i\,.$$

---

We find for the leading coefficient: $$c_0(\bp^2) = E(\bp^2) = E_1(\bp^2)+E_2(\bp^2) \equiv \sqrt{\bp^2+m_1^2} + \sqrt{\bp^2+m_2^2}$$ And a complicated (but purely combinatorial) expression for higher orders: $$c_i({\bp}^2) = \sum_{k=1}^{k=i} \frac{\sqrt{\pi}}{2\Gamma\left(\frac{3}{2}-k\right)} \frac{E_1({\bp}^2)^{2k-1}+E_2({\bp}^2)^{2k-1}}{(E_1({\bp}^2)E_2({\bp}^2))^{2k-1}}B_{i,k}\left({\cal G}_1({\bp}^2),\dots, {\cal G}_{i-k+1}({\bp}^2)\right)\,,$$ with Bell polynomials $B_{i,k}$ and $${\cal G}_m({\bp}^2) = -\sum_{s=0}^m\sum_{\ell=0}^s\frac{m!}{s!}P_{m-s}^{(\ell)}(c_0({\bp}^2))B_{s,\ell}\left(c_1({\bp}^2),\dots,c_{s-\ell+1}({\bp}^2)\right)$$

PM-EFT: Tidal effects for extended objects

$$S_{\rm pp} \mathrel{+}= \sum_a \!\int \!\dd\tau_a\!\left(c^{(a)}_{E^2} E_{\mu\nu} E^{\mu\nu} +c^{(a)}_{B^2} B_{\mu\nu} B^{\mu\nu} - c^{(a)}_{{\tilde E}^2} E_{\mu\nu\alpha} E^{\mu\nu\alpha} - c^{(a)}_{{\tilde B}^2} B_{\mu\nu\alpha} B^{\mu\nu\alpha}+\cdots \right)$$ with operators $$\begin{align} E_{\alpha\beta} &= R_{\mu\alpha\nu\beta} v^\mu v^\nu\,, & B_{\alpha\beta} &= R^\star_{\mu\alpha\nu\beta} v^\mu v^\nu, \\ E_{\alpha\beta\gamma} &= \nabla^\perp_{\{\alpha} R_{\beta\rho\gamma \} \nu} v^\rho v^\nu\,, & B_{\alpha\beta\gamma} &= \nabla^\perp_{\{\alpha} R^\star_{\beta\rho\gamma \} \nu} v^\rho v^\nu\end{align}$$ Use same setup as before and compute:

• Reproduced quadrupole corrections at NLO PM [Cheung, Solon 20]
• Novel octupole corrections at NLO PM
• Analytic continuation works as before
• Binding energy for circular orbits at $\cO(G^3)$

PM-EFT: Spinning extended objects

$$S_{\rm pp} \mathrel{+}= -\frac{1}{2}\sum_a \int \dd\tau_a S_{ab}(\tau_a)\omega_\mu^{ab}(\tau_a)\dot{x}_a^\mu(\tau_a)\,,$$ with spin connection $\omega_\mu^{ab}$.

• We can use the same setup for the PM expansion
• Need to simultaneously solve e.o.m. for the spin tensor: $$\frac{\dd S^{ab}}{\dd\tau} = \{S^{ab},\cL_\textrm{int}\}$$
• Work in progress with Z. Liu, R. Porto, Z. Yang.
• The amplitude at 2PM for spin-orbit effects has been worked out by [Bern, Luna, Roiban, Shen, Zeng 20]

Analytic continuation for aligned spins.

Idea: extend our map to the aligned spins for binary BH problem.
Motion is still in a plane!

• Explicit checks for known PN and PM results work neatly!
• Relies on the invariance of the (canonical) radial momentum $p_r$ under $J\rightarrow -J$, which is true for a quasi-isotropic gauge (given to us automatically by the amplitudes construction).
• We propose a version of the impetus formula to also hold for the aligned spin case.

Conclusions

• We developed an efficient PM EFT framework, allows us to include tidal effects and spin.
• Mapping (conservative & non-spinning) scattering data to orbital observables does not require a Hamiltonian. Analytic continuation does the job.
• Analytic continuation also works the special case of aligned spins.
• The (classical) amplitude, the scattering angle, and the momentum along the trajectory contain the same information.
• We have these relations in exact form, useful for e.g. numerical computation, as well as in PM expanded form to all orders (purely combinatorial).

Outlook

• Non-perturbative map invites for numerical studies: Self-force limit?
• Analytic continuation for arbitrary spin? Radiation? Radiation-reaction? Tail?
• PM-EFT with spin: soon!
• Conservative scattering angle at 4PM: We have all the technology.
• Input from self-force computation (numerical & analytical).
• We can resum certain quantities. More work required. (Ask me about it if you are interested!)

This research is supported by the ERC-CoG “Precision Gravity: From the LHC to LISA” provided by the European Research Council (ERC) under the European Union’s H2020 research and innovation programme (grant No. 817791), by the DFG under Germany’s Excellence Strategy ‘Quantum Universe’ (No. 390833306), and by the Knut and Alice Wallenberg Foundation (grant KAW 2018.0441).

Resummation

Let us truncate our theory at given order $n$, i.e. $\cM_m=f_m=0$ for $m \geq n$.
We can try to resum contributions to all orders in $G$, e.g. for the scattering angle:

• Resummation of $\Delta\phi$ works similar to $\chi$.
• We can resum parts of $\cS_r$.
• We can resum $f_{1,2}$ contributions for $r_\textrm{min}$ and $r_\pm$. ("closed form" for real positive roots of arbitrary order polynomial?)
• Difficult for $f_{1,2,3}$. Anyone can do it?