Binary systems as sources of gravitational waves Theoretical research done at Nikhef/Vu Gideon Koekoek March 5th 2008

Overview of presentation Introduction: the binary system The Martel-Poisson method Methods used: current research Conclusions & outlook

Introduction General Relativity: any system with a quadrupole moment produces GWs Our prime target: the binary system! A sure source of GW’s: Hulse & Taylor Their GWs are about to be measured (LIGO, VIRGO experiments) Test-bed for GR and extensions

Introduction How to calculate gravitational waves? General scheme: Curvature of spacetime Source of the GWs General scheme: Take background metric gBμν add a disturbance hμν and plug into the Einstein Equations Differential equations that couple the gravitational wave hμν to the energy momentum Tμν

Introduction This is all well-known… In the case of a Minkowski background, this is an easy exercise. All constants The wave equation! Waves move with speed of light Waves have two (!) polarisations Are transversal This is all well-known…

M Introduction m How about the binary system? Variables! Approximation: one of the stars is very much heavier than the other (EMRI). The metric of the system is then that of the heavy star: Schwarzschild metric. Variables!

Introduction The challenge: Difficult! How to solve this system? Also, the source Tμν will be difficult, as the stars orbit each other in a non-trivial way. (Epicycle method; see last year’s presentation) The challenge: Non-constant background metric Non-trivial energy-momentum tensor 10 coupled, partial differential equations in 10 variables Difficult! How to solve this system?

Introduction Our goal: Expert groups are working on this by numerical methods: ..but this is a very demanding computation! Our goal: Try to find the gravitational waves for the binary system in an analytical way. We do this by using a formalism developed recently by Martel & Poisson, and work our way from there. (Frans Pretorius, 2006)

The Martel-Poisson method

The Martel-Poisson method Following a program started by John Wheeler (1957), Martel & Poisson devised a formalism (2004) The Martel-Poisson method Linearize the Einstein Equations in hμν Decompose the waves into tensorial spherical harmonics.. Plug into Einstein Equations and find the EOMs for the coefficients.. Do some very clever gauging to eliminate all unphysical degrees of freedom.. Think very hard..

The Martel-Poisson method ..in the end: only two linear, uncoupled Klein-Gordon equations remain. Great simplification! 10 coupled, partial differential equations in 10 variables The two Ψ(t,r) ’s (roughly) correspond to the two polarisations of the GWs. From these, we can directly calculate the GWs and the emitted energy!

The Martel-Poisson method A closer look at these two differential equations In which: Da is the generally covariant derivative, determined by the Schwarzschild metric VZM(r) and VRW(r) are ‘potentials’, fully determined by the Schwarzschild metric SZM(t,r) and SRW(t,r) are sources, fully determined by the motion of the small star around the bigger one These two equations can now be solved!

Methods used: current research

Methods used: current research How to solve these two equations? As always in such matters, there are two options: Numerical integration: write a clever C++ program that takes the initial conditions and extrapolates from there Approximation techniques: throw out some terms, do integral transforms, and try to find an approximate analytical solution We are doing them both!

Methods used: current research Analytical scheme: 4M > 6M Includes some approximations and is based on Laplace transforms. Solutions found are integrals in which the orbit of the smaller star can be freely specified, i.e. solutions work for general orbits status: implementing initial conditions; results expected shortly. Work in progress; results expected soon Numerical scheme: Spacetime is divided up into grid cells, worldline of the smaller star is plotted C++ code integrates the EOMs over the worldline Status: a first version of the code is available, and is ondergoing testing.

Conclusions & outlook

Conclusions & outlook Immediate future (i.e. working on this right now) Solve Martel & Poisson’s two EOMs both numerically and via an analytical scheme; compare the results. The greater plan: Blend these two formalisms, so the EMRI-condition can be relaxed..even for alternative gravities! Martel & Poisson’s method only works when one of the masses is small, because otherwise the curvature deviates from Schwarzschild We know a formalism to calculate deviations of the metric

Conclusions & outlook Work in progress..but results expected soon! Problem: the gravitational waves in a binary system are very challenging to calculate EMRI approximation: by assuming one of the stars to be much heavier than the other, the background metric can be taken as Schwarzschild Martel & Poisson method: enables us to find the GWs in a Schwarzschild spacetime for any source, by solving two scalar EOMs Status: Solving the two EOMs, by an analytical scheme using Laplace Transforms; making progress but no results yet. A numerical code is being developed: results will be compared. Work in progress..but results expected soon!

Any questions? gkoekoek@nikhef.nl