Challenges in the Post-Minkowskian Description of the Gravitational Two-Body Problem

Work with Christoph Dlapa, Zhengwen Liu,
Jakob Neef & Rafael Porto

[2207.00580]
[2210.05541]
[2304.01275]

Gregor Kälin

I would like to mention astrophysics; in this field, the strange properties of the pulsars and quasars, and perhaps also the gravitational waves, can be considered as a challenge. - Werner Heissenberg

Overview

Challenge 1: Integrand construction

• Efficiently and systematically computable.
• Minimal
  • Free of redundancies.
  • Only information that we want.
• Optimal representation.

A worldline EFT framework

[GK, Porto 2006.01184]

• Equivalent to solving classical equations of motion.
• Perturbative expansion in \(G\): particle physics/amplitudes toolbox
• EFT methodology: Full action \(\rightarrow\) effective action \(\rightarrow\) deflection/fluxes/waveform/...
• Complete: allows inclusion of radiation, finite size, spin, \(n\)-body

Full theory

Model the compact bodies by worldlines \(x_a^\mu(\tau)\) coupled to GR. $$\begin{align} S_{\rm EH} &= -2\Mp^2 \int \dd^4x \sqrt{-g} \, R[g]\\ S_{\rm pp} &= -\sum_a \frac{m_a}{2} \int \dd\tau_a\, g_{\mu\nu}(x_{a}(\tau_a)) \dot{x}_{a}^\mu(\tau_a) \dot{x}_{a}^\nu (\tau_a)+\dots\\ \end{align}$$ We can add more terms to describe tidal and spin effects in an EFT style.

Effective two-body action

First step: Integrate out the gravitational field (classical saddlepoint). $$g_{\mu\nu}=\eta_{\mu\nu} + \kappa h_{\mu\nu}$$ $$e^{i S_{\rm eff}[x_a] } = \int \cD h_{\mu\nu} \, e^{i S_{\rm EH}[h] + i S_{\rm GF}[h] + i S_{\rm TD}[h] + i S_{\rm pp}[x_a,h]}$$

Don't use Feynman rules!
- An amplitudist

Optimize the EH-Lagrangian by means of gauge-fixing terms and total derivatives:

Radiation reaction

[GK, Neef, Porto 2207.00580]

• For causal motion: in-in formalism \(\longrightarrow\) doubling of fields
  [Jordan 1986][Calzetta, Hu 1987][Galley, Tiglio 0903.1122] $$\begin{align} \cS[h_1,h_2] = \cS_\textrm{EH}[h_1] - \cS_\textrm{EH}[h_2] -\sum_{A=1}^2 \frac{\kappa m_A}{2}\int\dd\tau_A &\left[h_{1,\mu\nu}(x_{1,A}(\tau_A))\dot{x}_{1,A}^\mu(\tau_A)\dot{x}_{1,A}^\nu(\tau_A)\right.\\ &\quad\left.-h_{2,\mu\nu}(x_{2,A}(\tau_A))\dot{x}_{2,A}^\mu(\tau_A)\dot{x}_{2,A}^\nu(\tau_A)\right] \end{align} $$
• Rotate to Keldysh variables (advanced & retarded propagators) $$\begin{align} h_{\mu\nu}^- &= \frac{1}{2}(h_{1\mu\nu}+h_{2\mu\nu}) & h_{\mu\nu}^+ &= h_{1\mu\nu} - h_{2\mu\nu}\\ x_{a,+}^\mu &= \frac{1}{2}(x_{a,1}^\mu+x_{a,2}^\mu) & x_{a,-}^\mu &= x_{a,1}^\mu - x_{a,2}^\mu \end{align}$$
• Seems impractical, but it's not!

• Simple Feynman rules for variation of the action (in the physical limit)
  • Feynman\(\rightarrow\)retarded/advanced propagator; graviton vertices unchanged
  • Source:
    

    $$ = -\frac{i m}{2\Mp}\int\dd\tau \, e^{i\, k\cdot x} \dot{x}^\mu \dot{x}^\nu$$
  • Unique sink (hit by variation):
    

    $$=-\frac{i m}{2\Mp} \,e^{i\,k\cdot x} \left[ i\, k^\alpha \dot{x}^\mu \dot{x}^\nu-i\,k\cdot \dot{x}\, \eta^{\mu\alpha}\dot{x}^\nu-\eta^{\mu\alpha}\ddot{x}^\nu-i\,k\cdot\dot{x}\, \eta^{\nu\alpha}\dot{x}^\mu-\eta^{\nu\alpha}\ddot{x}^\mu\right]$$
• Variation of the eff. action
  $$\left.\frac{\delta\cS_\textrm{eff}[x_+,x_-]}{\delta x_{1,-}^\alpha}\right|_{\substack{x_-\to 0\\x_+\to x}}^\textrm{1PM}=$$

  

  $$\left.\frac{\delta\cS_\textrm{eff}[x_+,x_-]}{\delta x_{1,-}^\alpha}\right|_{\substack{x_-\to 0\\x_+\to x}}^\textrm{2PM}=$$

  
  $$\left.\frac{\delta\cS_\textrm{eff}[x_+,x_-]}{\delta x_{1,-}^\alpha}\right|_{\substack{x_-\to 0\\x_+\to x}}^\textrm{3PM}=$$

  

Computing observables

Second step: Solve equation of motions $$\left.\frac{\delta \cS_{\textrm{eff}}[x_+,x_-]}{\delta x_{b,-}^\mu(\tau)}\right|_{\substack{x_{a,-}\rightarrow 0\\x_{a,+}\rightarrow x_a}} = 0 \quad\Longleftrightarrow\quad m_b \ddot{x}_b^\mu(\tau) = \left.-\eta^{\mu\nu}\frac{\delta \cS_{\textrm{eff,int}}[x_+,x_-]}{\delta x_{b,-}^\nu(\tau)}\right|_{\substack{x_{a,-}\rightarrow 0\\x_{a,+}\rightarrow x_a}}$$ perturbatively, expanding around straight lines $$ x_a^\mu(\tau) = b_a + u_a \tau + \sum_{n=1}^\infty G^n \delta^{(n)}x_a^\mu(\tau) $$ Using above trajectories we can e.g. compute the impulse $$\Delta p^\mu_1= m_1 \int_{-\infty}^{+\infty}\dd\tau \ddot{x}_1^\mu = - \eta^{\mu\nu}\int_{-\infty}^{+\infty} \dd\tau \left.\frac{\delta \cS_{\textrm{eff,int}}[x_-,x_+]}{\delta x_{1,-}^\nu(\tau)}\right|_{\substack{x_{a,-}\rightarrow 0\\x_{a,+}\rightarrow x_a}}\,,$$ or the change of the mechanical angular momentum $$\Delta J^{\mu\nu}_1 = m_1 \int_{-\infty}^{+\infty} \dd\tau ( x_1^\mu \ddot{x}_1^\nu- \ddot{x}_1^\mu x_1^\nu)\,.$$

Challenge 1: Integrand construction

• Efficiently and systematically computable. Yes.
• Minimal
  • Free of redundancies. Iterations, on-shellness,...
  • Only information that we want. Yes.
• Optimal representation. Probably no.

Challenge 2: (Feynman) Integrals

• Analytic representation.
• vs. fast numerical evaluation.
• Minimal function space.
• Bootstrap.

Integration

• \(\tau\)-integrations are trivial to resolve: lead to delta functions and linear propagators.

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

• Single scale integrals in \(\gamma = u_1\cdot u_2\)
• We can use all modern integration methods from the amplitudes community
• One delta function per loop \(\rightarrow\) half of the linear propagators are cut
• Outer Fourier transform is easy.

Integration procedure

• Tensor reduction (standard PaVe) \(\rightarrow\) scalar integrals
• Integration-By-Part (IBP) reduction \(\rightarrow\) master integrals

  Technical challenges:

  • Huge system of equations with many redundancies.
  • Lack of easily applicable measure for good choice of masters.
  • Symmetries of retarded integrals.

• Method of differential equations: bring into \(\epsilon\)-form [Kotikov, Remiddi, Henn]
  • See [2304.01275] and newest version of INITIAL [Dlapa, Henn, Wagner '22]

  Technical challenge: Everything more complicated than polylogs.

• Method of regions to compute the boundary constants [Beneke, Smirnov]

Method of regions @3PM

$$I_1=\int_{\ell_1,\ell_2}\frac{\delta(\ell_1\cdot u_1)\delta(\ell_2\cdot u_2)}{\ell_1^2\ell_2^2(\ell_1+\ell_2-q)^2}$$

---

Shift \(k\equiv \ell_1+\ell_2-q\) and \(\ell\equiv\ell_2\) $$\begin{align} &\mathrm{potential:} & \ell&\sim (v_\infty,1)|{\bq}|\,, & k&\sim(v_\infty,1)|{\bq}|\,,\\ &\mathrm{radiation:} & \ell&\sim (v_\infty,1)|{\bq}|\,, & k&\sim(v_\infty,v_\infty)|{\bq}|\,. \end{align}$$ Choose a frame \(u_1=(1,0,0,0)\,, u_2=(\gamma, 0,0,v_\infty)\), effectively: $$\begin{align} &\mathrm{potential:} \quad (\bell\to \bell, \bk\to \bk)\,,\\ &\mathrm{radiation:} \quad (\bell\to \bell, \bk\to v_\infty \tilde\bk)\,,\quad \tilde\bk\sim \bq\,,\label{res3pm} \end{align}$$ Leading to $$\begin{align} I_{1}^\mathrm{pot} &= -\int_{\bell,\bk} \frac{1}{[(\bk-\bell+\bq)^2]\, [\bell^2]\, [\bk^2]} + \cO(v_\infty^2)\,, \\ I_{1}^\mathrm{rad} &= - \int_{\bell} \frac{1}{ [(\bell - \bq)^2]\,[\bell^2] }\, \int_{\tilde \bk} \frac{v_\infty^{d-2} }{ [\tilde\bk^2 - (\ell^z)^2] } +\cO(v_\infty^{d})\,, \end{align}$$

---

Feynman vs. causal $$\begin{align} I_{1,\rm Fey}^\mathrm{rad} &= - \int_{\bell} {1 \over [(\bell - \bq)^2]\,[\bell^2] }\, \int_{\bk} {v_\infty^{d-2} \over [\bk^2 - (\ell^z)^2 - i0] } +\cO(v_\infty^{d})\,, \\[0.35 em] I_{1,\rm ret}^\mathrm{rad} &= - \int_{\bell} {1 \over [(\bell - \bq)^2]\,[\bell^2] }\, \int_{\bk} {v_\infty^{d-2} \over [\bk^2 - (\ell^z + i0)^2] } +\cO(v_\infty^{d})\,. \end{align}$$

Method of regions @4PM

[Dlapa et al. 2304.01275]

$$I_2= \int_{\ell_1,\ell_2,\ell_3} \frac{ \delta (\ell_1\!\cdot\! u_1)\, \delta (\ell_2\!\cdot\! u_1)\, \delta(\ell_3\!\cdot\! u_2)} {\ell_1^2\,\ell_3^2\,(\ell_2 {-}q)^2\, (\ell_3 {-} q)^2\, (\ell_1 {-} \ell_2)^2\, (\ell_2 {-} \ell_3)^2\, (\ell_3{-}\ell_1)^2}$$ Relabel \(k_1=\ell_3-\ell_1\), \(k_2=\ell_2-\ell_3\), \(\ell = \ell_3\). Regions: $$\begin{align} &\mathrm{pot:} & k_1&\sim(v_\infty,1)|\bq|\,, & k_2&\sim(v_\infty,1)|\bq|\,, & \ell&\sim(v_\infty,1)|\bq|\,,\\ &\mathrm{1rad}^{(1)}: & k_1&\sim(v_\infty,v_\infty)|\bq|\,, & k_2&\sim(v_\infty,1)|\bq|\,, & \ell&\sim(v_\infty,1)|\bq|\,,\\ &\mathrm{1rad}^{(2)}: & k_1&\sim(v_\infty,1)|\bq|\,, & k_2&\sim(v_\infty,v_\infty)|\bq|\,, & \ell&\sim(v_\infty,1)|\bq|\,,\\ &\mathrm{2rad:} & k_1&\sim(v_\infty,v_\infty)|\bq|\,, & k_2&\sim(v_\infty,v_\infty)|\bq|\,, & \ell&\sim(v_\infty,1)|\bq|\,, \end{align}$$

---

Expand in these regions $$\begin{align} I_{2}^\textrm{pot} &= \int_{\bell,\bk_1,\bk_2} {1 \over [(\bell-\bk_1)^2]\, [\bell^2]\, [(\bk_2+\bell-\bq)^2]\, [(\bell-\bq)^2]\, [(\bk_1+\bk_2)^2]\, [\bk_2^2]\, [\bk_1^2]} +\cO(v_\infty^2)\,, \\ %% I_{2}^{\textrm{1rad}(1)} &= \int_{\bell,\bk_2} {1 \over [\bell^2]\, [\bell^2]\, [(\bk_2+\bell-\bq)^2]\, [(\bell-\bq)^2]\, [\bk_2^2]^2} \int_{\bk_1} {v_\infty^{d-2} \over \bk_1^2 - (\ell^z)^2} +\cO(v_\infty^{d})\,, \\ %% I_{2}^{\textrm{1rad}(2)} &= \int_{\bell,\bk_1} {1 \over [(\bell-\bk)^2]\, [\bell^2]\, [(\bell-\bq)^2]\, [(\bell-\bq)^2]\, [\bk_1^2]^2} \int_{\bk_2} {v_\infty^{d-2} \over \bk_2^2 - (\ell^z)^2} +\cO(v_\infty^{d})\,, \\ %% I_{2}^\textrm{2rad} &= \int_\bell {1 \over [\bell^2] \,[\bell^2]\, [(\bell-\bq)^2]^2} \int_{\bk_1,\bk_2} {v_\infty^{2d-6} \over [(\bk_1+\bk_2)^2]\, [\bk_2^2 - (\ell^z)^2]\, [\bk_1^2 - (\ell^z)^2]} +\cO(v_\infty^{2d-4})\,. \end{align}$$

• Dress with Feynman (conservative) or ret/adv (RR) \(i0\).
• All regions added up lead to a finite result.
• Perform inner integral first: Outer integral becomes lower-loop potential integral.

• rad2 corresponds to PN tail-type integrals

  

Results up to 4PM

[Dlapa, GK, Liu, Neef, Porto 2210.05541]

Challenge 2: (Feynman) Integrals

• Analytic representation. New challenge at every loop order!
• vs. fast numerical evaluation. Worth thinking about.
• Minimal function space. Not very relevant (until now).
• Bootstrap. Be creative

Challenge 3: Bound orbits and resummations.

• Completing the boundary-to-bound dictionary.
• Performing and understanding resummations.
• Apply amplitudes methods directly to bound problem.

Boundary-To-Bound (B2B)

[1911.09130, w/ Porto] [1910.03008, w/ Porto] Use scattering data, e.g. the scattering angle $$\chi(b,E) +\pi = 2b \int_{r_{\rm min}}^\infty \frac{\dd r}{r\sqrt{r^2\bar\bp^2(r,E)-b^2}}$$ to obain bound observables, e.g. the periastron advance: $$\begin{align} \Delta \Phi + 2\pi = 2J \int_{r_-=r_{\rm min}(J)}^{r_+=r_{\rm min}(-J)} \frac{\dd r}{r\sqrt{r^2\bp^2(r,E)-J^2}} &=2J \int_{r_\textrm{min}(J)}^{\infty} \frac{\dd r}{r\sqrt{r^2\bp^2(r,E)-J^2}}\\ &\quad-2J \int_{r_\textrm{min}(-J)}^{\infty} \frac{\dd r}{r\sqrt{r^2\bp^2(r,E)-J^2}}\\ &=\chi(J,E)+\chi(-J,E)+2\pi \end{align}$$ Don't forget: \(J=p_\infty b\), \(\bar\bp = \bp/p_\infty\).

Local vs. non-local for circular orbits

[2106.08276] Non-local contributions to unbound radial action $$\cI_{r\textrm{(nloc)}}^{\textrm{4PM}}=-\frac{E}{2\pi M^2\nu}\left(\int \frac{\dd\omega}{2\pi}\frac{\dd E}{\dd\omega}\log(4\omega^2e^{2\gamma_E})+\int \dd t\frac{\dd E}{\dd t}\log(r^2(t))\right)$$ Not all terms go through above B2B dictionary.

Challenge:
Remove non-local terms from unbound quantities & evaluate them instead on bound orbits.

Radiation

[2112.03976, w/ Cho, Porto]

In a similar way we can analytically continue radiative observables, e.g. the energy and angular momentum loss $$\begin{align} \Delta E_\textrm{ell}(J,\cE) &= \Delta E_\textrm{hyp}(J,\cE)-\Delta E_\textrm{hyp}(-J,\cE)\\ \Delta J_\textrm{ell}(J,\cE) &= \Delta J_\textrm{hyp}(J,\cE)+\Delta J_\textrm{hyp}(-J,\cE)\\ \end{align}$$

Firsov's formula

[w/ Porto 1910.03008]
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.

Firsov resummation

Challenge 3: Bound orbits and resummations.

• Completing the boundary-to-bound dictionary. Nonlocal. Spin. Waveform dictionary?
• Performing and understanding resummations. Why so good? What about bound?
• Apply amplitudes methods directly to bound problem. Challenge for all of you!

Conclusions

• PMEFT: systematic and efficient framework to study the gravitational 2-body dynamics.
• Modern integration techniques work for PM integrals with retarded propagators.
• Complete answer, including radiation-reaction, for the impulse/deflection up to 4PM.
• Resummed results agree well with numerical relativity simulations.
• Non-local B2B remains an important challenge to fulfill our promises to GW community.
• Need improved tools for higher orders.

Time to face these challenges!

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).

High Energy

High-energy + massless limit is divergent. Keeping \(s\equiv \gamma m_1 m_2\) fixed: $$\begin{align} c_{1b}^{(4)\mathrm{tot}} &\sim \frac{s^3}{m_1} + \cO(m_i)\\ \frac{\Delta E^{\mathrm{4PM}}_{\textrm{hyp}}}{\sqrt{s}} &\sim \frac{G^4 s^2}{b^4} \log\left(\frac{s}{m_1 m_2}\right)+\cO(m_i) \end{align}$$

PM to PN: The Magic of Firsov

Let's consider a simple example $${\Delta \Phi}(j,\cE)= \sum_{n=1} \Delta \Phi_j^{(2n)}(\cE)/j^{2n}$$ with \(j=J/(G M \mu)\). B2B states: $$\Delta \Phi_j^{(2n)}(\cE)= 4\, \chi^{(2n)}_j(\cE)$$ Let's assume we don't know anything about \(\chi^{(4)}_j\). Does this mean we don't know anything about \(\Delta \Phi_j^{(4)}\)? There's lot of lower PN information that our PM results up to \(\chi_j^{(3)}\) should contain. Where are they?

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.

Back to our problem

$$\frac{1}{4}\Delta \Phi_j^{(4)}= \chi^{(4)}_j = \left(\frac{p_\infty}{\mu}\right)^4\chi_b^{(4)}=\left(\frac{p_\infty}{\mu}\right)^4\frac{3\pi}{16}(2f_1f_3+f_2^2+2f_4)$$ and in turn $$\begin{align} f_1&=2\chi_b^{(1)}\\ f_2&=\frac{4}{\pi}\chi_b^{(2)}\\ f_3&=\frac{1}{3}\left(\chi_b^{(1)}\right)^3+\frac{4}{\pi}\chi_b^{(1)}\chi_b^{(2)}+\chi_b^{(3)} \end{align}$$ Prefectly reproduces 1 and 2PN information at order \(j^{-4}\).

Numerical integration using machine learning

[Jinno, GK, Liu, Rubira 2209.01091]

• Why numerical integration?
• Cross-checks are important and incredibly useful
• Might be the only available method at higher loops
• Hybrid: Analytical bootstrap
• Need fast algorithms for high precision!
• Our idea: teach a neural network to do the Monte-Carlo integration making use of the normalizing flows technology.

• Importance Sampling: Pick points for Monte-Carlo evaluation such that regions of large integrand \(|f|\) gain more weight.
• I.e. take a distribution \(x(G)\), \(\dd G = g(x) \dd x\) $$I = \int_\Omega \dd x f(x) = \int_{\tilde\Omega} \dd G \frac{f(x(G))}{g(x(G))}$$ that minimizes the variance $$\sigma^2=\frac{1}{N-1}\left[\frac{1}{N}\sum_i \left(\frac{f(G_i)}{g(G_i)}\right)^2-\left(\frac{1}{N}\sum_i \frac{f(G_i)}{g(G_i)}\right)^2\right]$$
• Hence, optimally \(g(x) = f(x)/I\), but it should also be invertible and fast to evaluate.

• Traditional Monte-Carlo algorithms assume \(g(x_1,\dots,x_n) = g(x_1) \dots g(x_n)\) and then use an \(N\)-dimensional histogram for each dimension.
• Often not true at all and leads to a bad sampling.
• Better: Machine learning in the form of a neural network setup
$$J =\left|\begin{pmatrix} 1 & 0 \\ \frac{\partial C}{\partial m}\frac{\partial m}{\partial x_A} & \frac{\partial C}{\partial x_B}\end{pmatrix}\right| = \left| \frac{\partial C}{\partial x_B} \right|$$

• Promising results for (potential) boundary integrals up to 4-loops