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< < | ## CBwaves: A C++ code producing physically adequate precise waveforms for spinning and eccentric binaries## RMKI Virgo group (Hungary, Budapest) | |||||||

> > | ##
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> > | ## RMKI Virgo group (Hungary, Budapest) | |||||||

## MotivationsThe cbwaves software calculates the gravitational waves emitted by generic configuration compact binary systems but it is also capable to follow the time evolution of open systems. Our principal aim was to construct highly accurate templates describing waveforms generated by inspiral spinning and eccentric binaries within the PN ramework. We have already carried out some preliminary investigations measuring the effectivity of the template banks, applied currently by the CBC working groups, in recognizing them. | ||||||||

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## The methodThe cbwaves software calculates the gravitational waves emitted by generic binary neutron stars (BNSs) or binary black holes (BBHs) --- with arbitrary orientation of the spins and with arbitrary of value the eccentricity --- by direct integration of the equation of motion of the bodies. | ||||||||

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< < | - 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.
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> > | - The waveforms are calculated in time domain (BUT)
- The waveforms can also be determined in frequency domain by using implemented FFT.
- The spin weighted (s=-2) spherical harmonics of the waveforms are also be provided.
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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. | ||||||||

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> > | ## The most important input parameters- r ------- # the initial (minimal) separation of the two bodies [m]
- m1,m2 ------- # the mass of the two bodies [m]
- ε ------- # the initial eccentricity
- s1,s2 ------- # the spin of the objects is s_A, where A = 1,2 and s_A = sqrt(s_Ax^2+s_Ay^2+s_Az^2). [For a black hole 0. < s_A < 1, for most neutron star models 0 < s_A < 0.7]
- δ1,δ2 ------- # the orientation of the spin vectors SA with respect to the orbital angular momentum L
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< < | 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. | |||||||

> > | ||||||||

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< < | The generated job description files can be submitted to condor clusters by launchin the command: - condor_submit <name of .des file>
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> > | The simulation starts at the turning point of the radial motion determined by the minimal distance of the bodies. 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 usual analysis pipelines but its value is at will. | |||||||

## Download
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< < |
## The simulation
## The simulationThe input parameters for the simulations are the - the relative distance of the two mass
- the relative velocity
- m1, m2
- s1, s2
- 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 | |||||||

> > |
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< < | 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. | |||||||

> > | A sample .ini file can be found in the etc directory of the package. - The examples/cbwgen.pl perl executable is an example script which demonstrates how to generate .ini files
- The .des files for submission to condor clusters.
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< < | ## Circular, non-spinning waveformsWe 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. | |||||||

> > | The generated job description files can be submitted to condor clusters by launchin the command: - condor_submit <name of .des file>
To contact the maintainers or send patches please us the following email address: <cbwaves@rmki.kfki.hu> :::::: Please feel free to use it but do not forget to refer to "cbwaves" :-).
##
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< < | ## Eccentric waveforms, frequency modulation | |||||||

> > | Due to the effect of radiation the motion tends to circularize, however not as fast as generally believed. The temporary value of the eccentricity can always be determined by the relation - ε = (rmin-rmax)/(rmin+rmax),
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< < | 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: | |||||||

> > | where rmin and rmax is the minimum and maximum distance between the two mass, i.e. the distance in the turning points. | |||||||

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< < | 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): | |||||||

> > | Due to non negligible eccentricity the waveforms suffer a frequency modulation. [On the figures below only a short section of the evolution is indicated.] | |||||||

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< < | ||||||||

## Effect of spin, amplitude modulation | ||||||||

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< < | 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). | |||||||

> > | In a case of the inclusion of spins the emitted gravitational wave acquires a considerable large amplitude modulation. On the left panel only one of the bodie possesses spin (s1=0.7) which is aligned with the orbital angular momentum. On the right panel the both of the bodies possess spins (s1=0.7, phi1=0 ; s2=0.4, phi2=pi/3). The modulation is clearly visible in both cases. | |||||||

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< < | The high modulation is clearly visible in both cases.
## Generic spinning, eccentric waveforms | |||||||

> > | ## Generic waveforms for spinning and eccentric binaries | |||||||

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

> > | As indicated above the cbwaves software is capable to determine the evolutions and the yielded waveforms of completely general spinning, eccentric binaries. The simultaneous effect of amplitude and frequency modulation is transparent! Fitting analytic formulas to all the possible waveforms seems not to be feasible. | |||||||

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< < | 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 methodWe generate the templates in the following way: - Generate the template and downsampling it to 4096 Hz
- Since the longest template is only around 14 sec, we allocate a 65536 long memory area.
- 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.
- 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 | |||||||

> > | According to our investigations circularisation happens but a tiny eccentricity is retained by binaries with initial eccentricity ε=0.4-0.7. | |||||||

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< < | 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. | |||||||

> > | ||||||||

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< < | ## Questions, problems- 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 ?
- 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 !
- When calculating the overlap how we should deal with tc and phic ? Does it have a meaning in this case ? (I guess not).
- 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 ?
- Should we try to figure out the optimal spacing or simple equal steps in parameter space in enough ?
## The simulation | |||||||

> > | For those who are interested in the effect of this tiny eccentricity on the SNR it might be informative to look at the figure below indicating the loss of SNR where on the horizontal axis the value of the retained part of the initial eccentricity ε=0.4 is indicated at 40Hz frequency cut. [Here the overlap of our purely circular and eccentric templetes were determined.] | |||||||

Changed: | ||||||||

< < | The input parameters for the simulations are the - the relative distance of the two mass
- the relative velocity
- m1, m2
- s1, s2
- phi1, phi2, the angle of the spins respect to the orbital angular momentum
| |||||||

> > |
## 2) Burst pipelines | |||||||

Changed: | ||||||||

< < | 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.'''
## ResultsWe 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 waveformsWe 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 modulationDue 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 modulationIn 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 waveformsHaving armed with all the machinery developed so far we are ready to produce some completely general spinning, eccentric templates best describing physical reality.
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> > | An indication that one can use our cbwaves software to investigate burst type events generated by an open binary. | |||||||

Changed: | ||||||||

< < | 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: | |||||||

> > |
## Orbital evolution of an eccentric binary and the associated waveform | |||||||

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< < | ||||||||

> > | m1=24, m2=8, ε0=0.8, D=2.5 10^23 m | |||||||

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

> > | ||||||||

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< < | ## Template bank generation## The method | |||||||

> > |
## Orbital evolution of an eccentric double spinning binary and the associated waveform | |||||||

Changed: | ||||||||

< < | We generate the templates in the following way: - Generate the template and downsampling it to 4096 Hz
- Since the longest template is only around 14 sec, we allocate a 65536 long memory area.
- 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.
- Using the usual overlap function the overlap is calculated between two neighbouring template. (No scan on tc or phic)
| |||||||

> > | m1=24, m2=8, ε0=0.8, s1=s2=1, δ1=45°, δ2=135° , D=2.5 10^23 m | |||||||

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< < | As an example: | |||||||

> > | ||||||||

Changed: | ||||||||

< < | ## The idea of an offline template bank | |||||||

> > |
## The Foruier spectrum of the early part of the evolution of an eccentric binary | |||||||

Changed: | ||||||||

< < | 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. | |||||||

> > |
## The time dependence of the eccentricity and the satisfaction of the energy conservation | |||||||

Changed: | ||||||||

< < | ## Questions, problems- 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 ?
- 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 !
- When calculating the overlap how we should deal with tc and phic ? Does it have a meaning in this case ? (I guess not).
- 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 ?
- Should we try to figure out the optimal spacing or simple equal steps in parameter space in enough ?
| |||||||

> > |

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