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$$ \definecolor{desyOrange}{RGB}{242,142,0} \DeclareMathOperator{\Arcsinh}{arcsinh} \DeclareMathOperator{\Arctan}{arctan} \DeclareMathOperator{\arcosh}{arcosh} \newcommand\dd{{\mathrm d}} \newcommand{\bb}{{\mathbf b}} \newcommand{\bk}{{\mathbf k}} \newcommand{\bl}{{\mathbf l}} \newcommand{\bm}{{\mathbf m}} \newcommand{\bn}{{\mathbf n}} \newcommand{\bp}{{\mathbf p}} \newcommand{\bq}{{\mathbf q}} \newcommand{\br}{{\mathbf r}} \newcommand{\bw}{{\mathbf w}} \newcommand{\bx}{{\mathbf x}} \newcommand{\by}{{\mathbf y}} \newcommand{\bz}{{\mathbf z}} \newcommand\cO{\mathcal{O}} \newcommand{\cD}{\mathcal{D}} \newcommand\cA{\mathcal{A}} \newcommand\cM{\mathcal{M}} \newcommand\cN{\mathcal{N}} \newcommand\cE{\mathcal{E}} \newcommand\cS{\mathcal{S}} \newcommand\cP{\mathcal{P}} \newcommand\cF{\mathcal{F}} \newcommand\cL{\mathcal{L}} \newcommand\Mp{M_{\rm Pl}} $$

A Post-Minkowskian EFT for the Gravitational Two-Body Problem

Work with Rafael Porto & Zhengwen Liu in
[1910.03008] [1911.09130]
[2006.01184] [2007.04977]

scattering orbit

Gregor Kälin

erc desy eu
AEI 27.01.2021

Motivation: Merger breakdown


ringdown: blackhole perturbation theory

merger: numerical relativity (high velocities!)

inspiral: analytic (many cycles!)

PN vs PM.

State of the art for the conservative potential (non-spinning) Buonanno@Amplitudes2018

0PM $$~~1~~$$ $$v^2$$ $$v^4$$ $$v^6$$ $$v^8$$ $$v^{10}$$ $$v^{12}$$ $$v^{14}$$ $$v^{16}$$
1PM $$1/r$$ $$v^2/r$$ $$v^4/r$$ $$v^6/r$$ $$v^8/r$$ $$v^{10}/r$$ $$v^{12}/r$$ $$v^{14}/r$$
2PM $$1/r^2$$ $$v^2/r^2$$ $$v^4/r^2$$ $$v^6/r^2$$ $$v^8/r^2$$ $$v^{10}/r^2$$ $$v^{12}/r^2$$
3PM $$1/r^3$$ $$v^2/r^3$$ $$v^4/r^3$$ $$v^6/r^3$$ $$v^8/r^3$$ $$v^{10}/r^3$$
4PM $$1/r^4$$ $$v^2/r^4$$ $$v^4/r^4$$ $$v^6/r^4$$ $$v^8/r^4$$
5PM $$1/r^5$$ $$v^2/r^5$$ $$v^4/r^5$$ $$v^6/r^5$$

Bound-to-boundary (B2B) relations


PM-EFT for a worldline action coupled to GR

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

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

We optimized the EH-Lagrangian by cleverly chosing gauge-fixing terms and adding total derivatives: 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 $$2\sin\left(\frac{\chi}{2}\right) = \frac{|\Delta \bp_{1{\rm cm}} |}{p_\infty}= \frac{\sqrt{-\Delta p_1^2}}{p_\infty}$$



Generic structure: $$ \int \dd^Dq\frac{\delta(q\cdot u_1)\delta(q\cdot u_2)e^{i b\cdot q}}{(q^2)^m} \underbrace{\int \dd^D\ell_1\cdots\dd^D\ell_L\frac{\delta(\ell_1\cdot u_{a_1})\cdots\delta(\ell_L\cdot u_{a_L})}{(\ell_1\cdot u_{b_1}\pm i0)^{i_1}\cdots(\ell_L\cdot u_{b_L}\pm i0)^{i_L}(\textrm{sq. props})}}_{\textrm{Cut Feynman integrals with linear and square propagators}} $$ Final Fourier transform is known to all orders: $$\Delta p_a^\mu = \int \dd^Dq (A \overbrace{q^\mu}^{-i \frac{\partial}{\partial b_\mu}} + \underbrace{B u_1^\mu + C u_2^\mu}_{\substack{\textrm{bootstrap from}\\ \textrm{on-shell condition}\\ (\Delta p_a+p_a)^2=p_a^2}}) \frac{\delta(q\cdot u_1)\delta(q\cdot u_2)e^{i b\cdot q}}{(q^2)^{m+n}}$$


$$\frac{\chi}{2} = \sum_{n=1} \chi^{(n)}_b \left(\frac{GM}{b}\right)^n $$ 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 \(\gamma = u_1\cdot u_2\), \(\Gamma=\sqrt{1+2\nu(\gamma-1)}\), \(\nu=m_1m_2/M^2\).

News @4PM

Update from last week: potential contributions @4PM presented by [Bern, Parra-Martinez, Roiban, Ruf, Shen, Solon, Zeng '21]

Where are we at 4PM?

[work with C. Dlapa, Z. Liu, G. Mogull, R. Porto ]

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:
\(\mathbf{p}^2(r,E) = p_\infty^2(E) + \widetilde{\cM}(r,E)\)
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
$$\bar\bp^2(r,E) = \exp\left[ \frac{2}{\pi} \int_{r|\bar{\bp}(r,E)|}^\infty \frac{\chi(\tilde b,E)\dd\tilde b}{\sqrt{\tilde b^2-r^2\bar\bp^2(r,E)}}\right] = 1 + \sum_{n=1}^\infty f_n(E) \left(\frac{GM}{r}\right)^n$$
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

scattering vs. orbit
$$r_{\textrm{min}} = \tilde r_- \overset{\textrm{Firsov}}{=} b \exp\left[ -\frac{1}{\pi} \int_{b}^\infty \frac{\chi(\tilde b,E)\dd\tilde b}{\sqrt{\tilde b^2-b^2}}\right] =b \prod_{n=1}^\infty e^{-\frac{(GM)^n\chi_b^{(n)}(E)\Gamma\left(\frac{n}{2}\right)}{b^n\sqrt{\pi}\Gamma\left(\frac{n+1}{2}\right)}}$$ 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 just saw that the periastron advance can be extracted from the angle: $$\Delta\Phi(J,E) = \chi(J,E)+\chi(-J,E)$$ It constains almost all information to compute the full radial action!

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

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:

EFT tidal graphs

PM-EFT: Spinning extended objects

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

Analytic continuation for aligned spins.

Idea: extend our map to the aligned spins for binary BH problem.
Motion is still in a plane!
$$\frac{\chi(J,\cE)+\chi(-J,\cE)}{2\pi} = \frac{\Delta\Phi(J,\cE)}{2\pi}$$

where \(J\) is now the total the total angular momentum, i.e. orbital angular momentum + spins.


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:

$$\begin{align*} \frac{\chi[f_1]}{2} &= \Arctan(y/2)\\ \frac{\chi[f_{1,2}]+\pi}{2} &= \frac{1}{\sqrt{1-{\cF}_2 y^2}}\left(\frac{\pi}{2} + \Arctan\left(\frac{y}{2\sqrt{1-{\cF}_2 y^2}}\right)\right) \end{align*}$$

with \(y \equiv G M f_1/b\) and \(\cF_2 \equiv f_2/f_1^2\)




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).

Circular orbit

Solve \(r_+(J) = r_-(J)\) $$\Leftrightarrow -2 \sum_{n=0}^\infty \left(\frac{1}{\sqrt{\pi}}\left(\frac{GM}{b}\right)^{2n+1}\frac{\Gamma\left(\frac{2n+1}{2}\right)}{\Gamma(n+1)}\chi_b^{(2n+1)}\right) = i \pi + 2\pi i \mathbb{N}$$ to find \(j(\cE) \) and compute the radial frequency: $$GM\Omega_{\rm circ} = \left(\frac{\dd\, j({\cal E})}{\dd\, {\cal E}}\right)^{-1}$$ Need to resum for a truncated theory \(f_i=0\) for \(i\geq n\)!
We can invert to write the binding energy \(\epsilon\equiv -2 \cE\): $$\begin{aligned} \epsilon =&x \left[1 - \frac{x}{12}(9+\nu) - \frac{x^2}{8}\left(27 -19\nu + \frac{\nu^2}{3}\right)\right.+\frac{x^3}{32}\left(\frac{535}{6}-\frac{5585\nu}{6}+135\nu^2-\frac{35\nu^3}{162}\right) \\ &+ \left.\frac{x^4}{384}\left(-10171+\frac{559993}{15}\nu-\frac{34027\nu^2}{3}+\frac{11354\nu^3}{9}+\frac{77\nu^4}{81}\right) + {\cal O}(x^5)\right] \end{aligned}$$ using the standard PN parameter \(x \equiv (GM \Omega_{\rm circ})^{2/3}\).