---

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

---

RMKI Virgo group (Hungary, Budapest)

Motivations

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

In short terms our aims are to:

• 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 method

The 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 value of the eccentricity --- by direct integration of the equation of motion of the bodies. The radiation field is determined simultaneously in the time domain by evaluating the analytic waveforms relevant for the yielded motion of the sources.

• Hence 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.

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 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 sA, where A = 1,2 and sA^2 = sAx^2+sAy^2+sAz^2. [For a black hole 0. < sA < 1, for most neutron star models 0 < sA < 0.7. The spin vectors are given as: SA=sA?mA^2.]
• θ1,θ2 --- the orientation of the spin vectors SA with respect to the orbital angular momentum L

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

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

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.

The generated job description files can be submitted to condor clusters by launching 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" :-).

  ---

---

Some of the characteristic results

---

1) CBC pipelines

We have started with the investigation of non-spinning, circular binaries and examined latter the changes of the motion and the yielded waveforms while eccentricity and spins were turned on in a step by step approach.

Circular, non-spinning motions and waveforms

We constructed a non-spinning circular waveform template bank to serve as a reference for the forthcoming studies. Samples for a circular non spinning orbit and for the emitted waveform with slow rise in the amplitude and frequency:

---

Eccentric motions and 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),

where rmin and rmax denote the minimum and maximum distances between the two masses, i.e. the distances at the turning points.

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

Effect of spin, amplitude modulation

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 bodies 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, θ1=0; s2=0.4, θ2=π/3). The modulation is clearly visible in both cases.

Generic waveforms for spinning and eccentric binaries

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.

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

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 templates were determined.]

2) Burst pipelines

An indication that one can use our cbwaves software to modell single burst type events emitted by an open binary.

Orbital evolution of an eccentric binary and the associated waveform

m1=24, m2=8, ε=0.8, D=2.5 10^23 m

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

m1=24, m2=8, ε=0.8, s1=s2=1, θ1=45°, θ2=135° , D=2.5 10^23 m

The Foruier spectrum of the early part of the evolution of an eccentric binary

The time dependence of the eccentricity and the satisfaction of the energy conservation