Gravitational wave forms for extreme mass ratio collisions from supersymmetric gauge theories
Francesco Fucito, Josè Francisco Morales, Rodolfo Russo
Abstract
We study the wave form emitted by a particle moving along an arbitrary (in general open) geodesic of the Schwarzschild geometry. The mathematical problem can be phrased in terms of quantities in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi mathvariant="script">N</a:mi> <a:mo>=</a:mo> <a:mn>2</a:mn> </a:math> supersymmetric gauge theories that can be calculated by using localization and the Alday-Gaiotto-Tachikawa correspondence. In particular through this mapping, the post-Newtonian expansion of the wave form is expressed as a double instanton sum with rational coefficients that resums all tail contributions into Gamma functions and exponentials. The formulas we obtain are valid for generic values of the orbital quantum numbers <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:mo>ℓ</d:mo> </d:math> and <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:mi>m</f:mi> </f:math> . For <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:mo>ℓ</h:mo> <h:mo>=</h:mo> <h:mn>2</h:mn> </h:math> , 3 we check explicitly that our results agree with the small mass ratio limit of the wave forms derived in the multipole post-Minkowskian and the amplitudes approaches. We show how the so-called tails and tails-of-tails contributions to the wave form arise in our approach. Finally, we derive a universal formula for the soft limit of the wave form that resums all logarithmic terms of the form <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"> <j:msup> <j:mi>ω</j:mi> <j:mrow> <j:mi>n</j:mi> <j:mo>−</j:mo> <j:mn>1</j:mn> </j:mrow> </j:msup> <j:mo stretchy="false">(</j:mo> <j:mi>log</j:mi> <j:mi>ω</j:mi> <j:msup> <j:mo stretchy="false">)</j:mo> <j:mi>n</j:mi> </j:msup> </j:math> .