CBwaves: A C++ code producing physically adequate precise waveforms for spinning and eccentric binaries

RMKI Virgo group (Hungary, Budapest)

Motivations

• provide an effective tool for the parameter estimation group
• if possible increase the sensitivity of the CBC searching pipelines
• apply it in modelling some of the burst type events

• The waveforms are calculated in time domain
• They also can be determined in frequency domain by using the implemented FFT.
• The spin weighted (s=-2) spherical harmonics expansion of the radiative field is also given.

In determining the motion of the bodies and the waveforms yielded the setup proposed by Kidder http://xxx.lanl.gov/abs/gr-qc/9506022 --- which is known to be accurate upto 2.5 PN order had been applied. A short summary of the implemented methods and expressions and the specifications of the parameters can be found in the cbwaves-desc.pdf file located in the doc directory coming together with the software. The equations of motion are integrated numerically. The applied method is known to be 4th order accurate. The radiation field is determined in the time domain by evaluating the analitic waveforms relevant for the yielded motion of the sources.

The examples/cbwgen.pl perl executable is an example script which demonstrates how to generate .ini files and .des files for submission to condor clusters.

The generated job description files can be submitted to condor clusters by launchin the command:

• condor_submit <name of .des file>

Download

tgz	cbwaves-1.0.0-4.tgz
rpm (src)	cbwaves-1.0.0-4.src.rpm
rpm (x86_64)	cbwaves-1.0.0-4.x86_64.rpm
rpm (i386)	cbwaves-1.0.0-4.i386.rpm

The simulation

Questions, problems

1. Since it is not possible to describe these templates with closed analytical formula, it is difficult to determine the optimal placing and the number of templates necessary for a given minimal match. The only method is a brute-force trial-and-error. Does anybody have a better idea ?
2. The generation of the templates takes quite a lot of time. What about pre-generating a template bank with high enough tmeplate density, so it can just be downloadad and downsampled according to the actual PSD of the data if necessary. It would safe quite a lot of time !
3. When calculating the overlap how we should deal with tc and phic ? Does it have a meaning in this case ? (I guess not).
4. Is there any widely accepted method how to deal with seven parameter (m1,m2, ecc, s1,s2, phi1, phi2) template bank ? Or how to reduce the number of dimension in parameter space ?
5. Should we try to figure out the optimal spacing or simple equal steps in parameter space in enough ?

The simulation

The input parameters for the simulations are the

1. the relative distance of the two mass
2. the relative velocity
3. m1, m2
4. s1, s2
5. phi1, phi2, the angle of the spins respect to the orbital angular momentum

The initial relative speed is determined as if the two mass were orbiting on a Keplerian orbit around each other, having epsilon eccentricity.The simulation starts in one of the turning point of the radial motion. The initial orbital frequency is set to 19 Hz (as such, the emitted gravitational wave frequency is 38 Hz) due to the 40 Hz low frequency cut-off of the data analysis pipelines.

'''Due to the direct integration of motion, the simulation of any spin, angle and eccentricity configuration is possible with this methd.'''

Results

We have started the investigation step by step from basic non spinning, circular templates and examined the effect of various configurations on the produced waveform.

Circular, non-spinning waveforms

We started with the construction of a non-spinning circular waveform template bank to serve as a reference for the forthcoming studies. An examples is shown for a circular non spinning orbit and for the emitted waveform:

The slow rise in the amplitude and frequency is observable, results of earlier studies are nicely reproduced.

Eccentric waveforms, frequency modulation

Due to the effect of radiation the motion tends to circularize, however no in all circumstances. The actual eccentricity of the system can always be calculated from the following equation epsi = (rmin-rmax)/(rmin+rmax). Where rmin and rmax is the minimum and maximum distance between the two mass, i.e. the distance in the turning points. This can easily be determined during simulation, and we obtain the eccentricity and its evolution during the inspiral. An example of this is shown in the picture below:

Due to non negligible eccentricity the waveforms suffer a frequency modulation, demonstrated in the figure below. The orbit of an eccentric binaries and their produced waveform (extract only, not showing the full evolution):

Effect of spin, amplitude modulation

In a case of spin the emitted gravitational wave suffers an amplitude modulation. This is demonstrated in the pictures. On the left picture only one of the star has spin of 0.7 and is aligned with the orbital angular momentum. On the right picture the both having spin (s1=0.7, phi1=0 ; s2=0.4, phi2=pi/3).

The high modulation is clearly visible in both cases.

Generic spinning, eccentric waveforms

Having armed with all the machinery developed so far we are ready to produce some completely general spinning, eccentric templates best describing physical reality.

The nice simultaneous effect of amplitude and frequency is very well visible! The fitting of an analytic formula to this waveform is really difficult. It is interesting to see, how the eccentricity is changing during such an inspiral:

Ther has remaind quite a non negligible eccentricity even at the very end of the inspiral phase.

Template bank generation

The method

We generate the templates in the following way:

1. Generate the template and downsampling it to 4096 Hz
2. Since the longest template is only around 14 sec, we allocate a 65536 long memory area.
3. The template is then copied into this memory area shifting it to to end, so state as Swarzschild ISCO is always at the very end of the memory area. The rest of the area is filled with zero.
4. Using the usual overlap function the overlap is calculated between two neighbouring template. (No scan on tc or phic)

As an example:

The idea of an offline template bank

Since the computation of the templates is a quite time consuming process, we plan to setup an offline template bank. The corressponding fcuntion in lalapps_inspiral would just download the pre-generated templates from this bank.

Spinning and eccentric inspiral waveforms - RMKI Virgo group (Hungary, Budapest)

The target

The method

We use the results of Kidder http://xxx.lanl.gov/abs/gr-qc/9506022 to describe the relative acceleration of the binary and the radiation field far from the source. His expressions are valid to 2.5 PN order (radiation reaction involved, higher order effects can be added later).

For more details see: http://www.kfki.hu/~vasuth/CBwaves.pdf

The equations of motion are integrated numerically with the 4th order Runge–Kutta method, and then inserted into the general expression of the radiation field. As a result we generate time domain inspiral waveforms and stop the calculations at the Schwarzschild ISCO, 6M.

The simulation

The input parameters for the simulations are the

1. the relative distance of the two mass
2. the relative velocity
3. m1, m2
4. s1, s2
5. phi1, phi2, the angle of the spins respect to the orbital angular momentum

The initial relative speed is determined as if the two mass were orbiting on a Keplerian orbit around each other, having epsilon eccentricity.The simulation starts in one of the turning point of the radial motion. The initial orbital frequency is set to 19 Hz (as such, the emitted gravitational wave frequency is 38 Hz) due to the 40 Hz low frequency cut-off of the data analysis pipelines.

'''Due to the direct integration of motion, the simulation of any spin, angle and eccentricity configuration is possible with this methd.'''

Results

Circular, non-spinning waveforms

The slow rise in the amplitude and frequency is observable, results of earlier studies are nicely reproduced.

Eccentric waveforms, frequency modulation

Due to non negligible eccentricity the waveforms suffer a frequency modulation, demonstrated in the figure below. The orbit of an eccentric binaries and their produced waveform (extract only, not showing the full evolution):

Effect of spin, amplitude modulation

In a case of spin the emitted gravitational wave suffers an amplitude modulation. This is demonstrated in the pictures. On the left picture only one of the star has spin of 0.7 and is aligned with the orbital angular momentum. On the right picture the both having spin (s1=0.7, phi1=0 ; s2=0.4, phi2=pi/3).

The high modulation is clearly visible in both cases.

Generic spinning, eccentric waveforms

Having armed with all the machinery developed so far we are ready to produce some completely general spinning, eccentric templates best describing physical reality.

The nice simultaneous effect of amplitude and frequency is very well visible! The fitting of an analytic formula to this waveform is really difficult. It is interesting to see, how the eccentricity is changing during such an inspiral:

Ther has remaind quite a non negligible eccentricity even at the very end of the inspiral phase.

Template bank generation

The method

1. Generate the template and downsampling it to 4096 Hz
2. Since the longest template is only around 14 sec, we allocate a 65536 long memory area.
3. The template is then copied into this memory area shifting it to to end, so state as Swarzschild ISCO is always at the very end of the memory area. The rest of the area is filled with zero.
4. Using the usual overlap function the overlap is calculated between two neighbouring template. (No scan on tc or phic)

As an example:

The idea of an offline template bank

Questions, problems

1. Since it is not possible to describe these templates with closed analytical formula, it is difficult to determine the optimal placing and the number of templates necessary for a given minimal match. The only method is a brute-force trial-and-error. Does anybody have a better idea ?
2. The generation of the templates takes quite a lot of time. What about pre-generating a template bank with high enough tmeplate density, so it can just be downloadad and downsampled according to the actual PSD of the data if necessary. It would safe quite a lot of time !
3. When calculating the overlap how we should deal with tc and phic ? Does it have a meaning in this case ? (I guess not).
4. Is there any widely accepted method how to deal with seven parameter (m1,m2, ecc, s1,s2, phi1, phi2) template bank ? Or how to reduce the number of dimension in parameter space ?
5. Should we try to figure out the optimal spacing or simple equal steps in parameter space in enough ?