Production of ready-to-use gravitational wave templates for compact binary systems with the inclusion of spin and quadrupole effects and allowing eccentric and open orbits.

The sensitivity of the new generation ground-based interferometric
gravitational wave (GW) detectors has increased significantly. The associated developments certainly increase the chances
of the first direct GW detection. While the first detection will be of great importance on its own right, presumably it will also
teach us important lessons about gravitational interaction itself.

Currently the most widely accepted theory of gravity is general relativity, which is a metric theory of gravity. This implies, in particular, that the distribution and motion of matter fields and the spacetime metric are in an intimate relation. While the matter fields determine the spacetime geometry via Einstein's equations, the geometry is also involved in determining the evolution of matter fields via the pertinent field equations.

In general relativity the truly dynamical sources can only be treated by numerical and perturbative approaches. Whenever the gravitational field is weak and the motions are slow, the post-Newtonian approximation is applicable for the description of the motion of the sources. To describe the emitted gravitational radiation far from the source, it has to be combined with the post-Minkowskian multipolar expansion method. In this formalism, the evolution and the emitted waveform of black holes and/or neutron stars binaries can be described with sufficient precision.

The successful candidate will be expected to join to our ongoing research program aiming to determine the behaviour of astrophysically realistic binary systems and to provide ready-to use gravitational waveform templates. This is done such that the various physical parameters, such as spins, quadrupole moments and the eccentricity of the orbits; and radiation reaction effects are taken into account.

The most promising sources of gravitational radiation to be detected are compact binary systems composed of neutron stars and/or black holes in the last stage of their inspiral until the final plunge. To discuss the evolution of these systems to high precision the post-Newtonian (PN) approximation is a convenient method when the gravitational field is weak and the motion of the sources are slow. For the detailed description of the emitted gravitational radiation the PN approximation is matched with the post-Minkowskian multipolar expansion method in the near outer zone of the source system.

In the main stream investigations of the post-Newtonian description of binary systems the principal aim was to achieve the highest possible PN; currently 3.5 PN order ; level since then the frequency of the associated GW signals is supposed to be high enough to be relevant for the ground-based interferometric GW detectors. This required the use of a great number of simplifying assumptions concerning the underlying binary system. For instance in most of the cases the investigations were restricted to the case of circular orbits and the spin-orbit, spin-spin and quadrupole-monopole interactions were completely neglected. During the last decade more and more investigations take into account these additional astrophysically important effects and the eccentricity of the orbits [1],[2]. There are important preliminary results concerning the full description of the motion of the binary [3], [4], nevertheless the equations of motion either are not solved or some of the above mentioned contributions are still missing.

More than a decade ago an intensive study of the PN framework has been started under the leadership of Zoltán Perjés in our institute, at the KFKI Research Institute for Particle and Nuclear Physics. Our principal aim was to include all the astrophysically important effects in describing the motion of the binary and also to determine the created GW signals. Because these effects were missing from the former studies in our investigations we had to reproduce all the succeeding PN levels of corrections starting with the lowest order above the Newtonian. In the associated process we have given a new, detailed parametrization independent method in describing the motion of the binary system including the usual 1 PN corrections and spin-orbit contributions manifest themselves through spin precession, orbital precession and orbital frequency ; and to determine the and polarization states of the emitted waveform [5],[6],[7] up to 1.5 PN order while keeping all the terms relevant for compact binaries.

Our results contain the explicit description of the motion including angular evolution and the analytic form of the polarization states with the use of the generalized true anomaly parametrization [8]. Following the steps given in the general method we have given time-dependent templates in the circular orbit limit. The associated results are currently written up for publication. These results justify the significance of the eccentricity of the orbits and the effects of the spin-orbit interaction. For instance, the importance of spin effects on the waveform can be seen at 1.5 PN order in the equal or nearly equal mass case which is supposed to be highly relevant for neutron star binaries. Since the non-spinning contributions are all multiplied by the difference of the masses of the objects, the spin-orbit interaction dominates the waveform at 1.5 PN order, in these particular cases. As opposed to this our first results demonstrate that in the general case the spin effects highly changes the characteristic properties of the waveforms even in the circular orbit special case from 1.5 PN order, although at lower orders these effects are definitely less significant. It is an important byproduct of the fact that the eccentricity of the orbits is also taken into account that higher harmonics of the waveforms appear in the analytic expressions. This effect enhances the probability of the detection of binary sources during the inspiralling era. The inclusion of the eccentricity is also supported by the fact that there has to be a significant fraction of the binaries, where the circularization of the orbits do not come to an end due to gravitational radiation in advance to the coalescence.

Based on our present results the following three main research programs manifest themselves as the ones which can also be considered as natural continuation of our former investigations:

- The most straightforward is to complete our research at 1.5 PN order. This work includes the detailed investigation of the waveforms, the evaluation of ready-to-use GW templates, the determination of the effects of the different dynamical and geometrical parameters on the waveforms. An additional, more ambitious goal is to provide, based on our theoretical model, a method which can be applied in data analyzing processes to determine the astrophysical parameters (masses, spins, etc.) of the sources by making use of the detected GW signals. Our analytic results could also be useful in providing analytic initial conditions for numerical simulations describing coalescence.
- The next reasonable step to do is the investigation of spin-spin and quadrupole-monopole interactions at 2 PN order. The first one is mainly important in the case of BH-BH and rapidly rotating BH-NS, NS-NS sources, and the second one is relevant in the case of slowly rotating NS-NS binaries. Perhaps the importance of these effects is underlined by the fact that the inclusion of spin-spin interaction is not compatible with the usual assumptions guaranteeing the circularization of the orbits [9].

Along this line of investigations at first the spin-precession is to be discussed, and following this the decoupled angular equations can be given for the evolution of the system. With the help of these results the method for the evaluation of the polarization states will be worked out.

The integration of the yielded equations requires a suitable generalization of the applied parametrization to include all the 2 PN corrections. Following this we will be able to integrate the equations of motion and provide a detailed description of the motion. With the use of the general method the explicit waveform contributions are also to be given.

At this point the investigations of the 1.5 PN order results are to be repeated at the new level, and more accurate waveform templates will immediately be produced.

In the next two years the possible endpoint of this line can be the investigation of the radiational reaction at 2.5 PN order. Since all the constants of the motion changes due to radiation, and since these constants play important role in our evaluation methods, technically it will be very challenging to incorporate this effect in our description. Nevertheless, it is important to be done, since it is crucial in describing the typical increase of the frequency of the; signal of the inspiralling binaries.

- The investigation of binaries following parabolic or hyperbolic orbits is also very promising since they yield unique, non-periodic and presumably strong enough GW signals. As former investigations show [10], the event rate is higher than one per year at the frequency range of the Virgo detector. Although these binaries appear to be relevant for the operating GW observatories, very few sufficiently detailed investigations of these types of systems have been done. The theoretical description of these sources is in a preliminary state, i.e., the parametrization of the orbit is available up to 1 PN order [11], and the description of the waveform is only at the Newtonian level.

We plan to generalize our methods to investigate these sources. In doing so first we have to work out a suitable form of the parametrization of the orbit, and repeat the integration processes. Since the PN framework we have developed is already suitable to take into account various effects relevant for open orbits up to 1.5 PN order our results should extend without experiencing technical difficulties to the case of these systems.

Finally, we would like to emphasize that once the detection of GW signals will become a daily practice---this is expected to happen once Advanced Virgo starts to operate---the most important goal will be to determine the physical properties of the sources of the observed GW signals. The above outlined research program promises that such a determination of the astrophysical parameters will be possible in case of the investigated sources.

**Bibliography**:

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1 J. Majár and M. Vasúth, Phys. Rev. D74, 124007 (2006).

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1 J. Majár and M. Vasúth, Phys. Rev. D77, 104005 (2008).

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Such as group composition and experience, equipment:

Our host institute, the KFKI Research Institute for Particle and Nuclear Physics of the Hungarian Academy of Sciences, shortly RMKI, is carrying out scientific research and related R&D activities in international collaborations. The main mission of RMKI is to conduct experimental and theoretical studies in particle and nuclear physics, gravitation and other field theoretical disciplines, plasma physics, space science and biophysics at an internationally recognized level. The main Hungarian centre for research in gravity, especially for general relativity, is also hosted by RMKI since the beginning of 70s. This theory group provided the core for the Hungarian VESF group, the MTA-KFKI-RMKI Group which joined to VESF in 2006 and which also involves researchers from the Eötvös University and from the University of Szeged. This year a new research group of RMKI has been formed which is dominated by experimental physicists aiming to participate in the Virgo-EGO scientific collaboration. In the last meeting of the Virgo Scientific Committee the application of the RMKI group for the membership was approved. One of our main contributions for the Virgo -EGO collaboration is to take part in the all sky search for GWs emitted by coalescing binaries formed by neutron stars and black holes with the inclusion of non-negligible spin effects (Coalescing Binaries, CB, working group). With our present proposal we are aiming to open a VESF fellowship for theoretical research on GW detection. With the inclusion of a talented new post-doctoral fellow, whose main task will be to produce ready-to-use GW templates relevant for binary systems, we would like to strengthen the performance of the RMKI group. As for the guarantees ensuring the success of the VESF program is concerned, there are two members of the RMKI-Virgo group, István Rácz and Mátyás Vasuth who are actively working in this field and are going to be able to provide scientific support for the newly applied post-doctoral fellow.

In addition we would like to mention that the RMKI group is supported by advanced computing resources of our host institute. For instance, the largest cluster of HunGrid VO, the official Hungarian Virtual Organization of EGEE, is hosted by the computer centre of RMKI and administrated by the members of the RMKI group. This computational capacity will be suitable to produce the software injected fake GW signals and, if it will be needed, the new template banks for the Virgo scientific collaboration as it was indicated in details in the research plan.

The research program is naturally structured by the principal aims addressed in the detailed research plan. An outline of this plan and the associated schedule can be given as follows:

1st year:

- Since we believe that our current results which are worked out up to 1.5 PN order will be found to be of sufficient interest for the data analyzing processes, our primary aim is to produce software injected fake GW signals as soon as it is possible. Assuming that the outcome of these investigations will be convincing enough in the succeeding period, we are going to work out the template banks fitting to the requirements of the CBC group of the Virgo collaboration. This part should be completed in 6 months after the start of the VESF program.

- Following this, as an immediate application of our former results, during the second half of the first year we are going to extend the production of software injected fake GW signals relevant for binaries following parabolic or hyperbolic orbits yielding burst type signals. If the current template banks are lacking the sensitivity to these types of GW signals we are ready to provide the corresponding, possibly the first, template banks up to 1.5 PN order for the burst group.

- In parallel to these activities we shall also start to investigate the extendibility of our results up to 2 PN order.

2nd year:

- As a more theoretical starting at the beginning of the second year emphasis will be given to the work out of the full description of the dynamics of the selected binary systems. In parallel we also determine the produced gravitational waveforms up to 2 PN order.

- Based on these results we are going to upgrade the template banks produced during the first year of this research plan for both bounded and unbounded binary systems.

- Finally, if it will be possible we would like to make some progress in the investigation of radiation reaction effects at 2.5 PN order.

-- GergelyDebreczeni - 19 Jan 2009

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