Comparison of CBwaves, Columbia, NR and hybrid waveforms

RMKI Virgo group (Budapest, Hungary)

PN contributions

Both CBwaves and the code of the Columbia group developed by Janna Levin use the PN equations to describe the evolution and the emitted waveforms of binary systems. The main differences in the applied approximation can be summarized as:

• Equations of motion
  - PN terms: PN, 2PN, 3PN are identical.
  - Spin terms: SO (denoted as PN-SO by Janna), SS (PN-SS), RRSO (3.5PN-SO), RRSS (3.5PN-SS) are identical.
  In addition to the above spin terms the latest version of CBwaves conatins the 1PN correction to the SO term, called PNSO, which gives non negligible contribution.
  - Radiation reaction: the two codes use different gauges, BT (CBwaves) and DD (Columbia) at 2.5PN and 3.5PN.
• Spin precession
  The usual SO and SS terms are included in both CBwaves and the code of the Columbia group. In addition to these contributions CBwaves contains the 1PN correction to the SO term in spin precession.
• Waveform
  Both codes include the usual Q, P0.5Q, P1.Q, P1.5Q, PQSO, P1.5QSO and PQSS terms. The new term in CBwaves is the 2PN contribution P2.Q at 4.5PN. A sample plot with the relative order of all these contributions can be found here.

NR waveforms

A comparsion with the provided (thanks to Valentin) NR waveform is shown here. Initial values: m1=m2=r0=10 Msun, the binary is circular, D=500 Mpc.

Here we plot the output of CBwaves (left column) and the code of the Columbia group (right column) together with the NR data.

Issues:

• One has to check the "hybridisation" region of NR waveforms. We should exactly know how they were hybridized.

The properties of the two different RR gauges BT and DD can be easily analyzed with CBwaves. To give a more detailed explanation of the above results we show additional plots generated by CBwaves.

The waveform and the evolution of the orbtial separation in the BT and DD gauges

The evolution of the total energy and the post-Newtonian parameter v^2

Hybrid waveforms

Ninja2 waveforms are hibrydized and available for comparsion. The plot below shows the /BAM_D11spp50_96.T4.hyb.n2.gwf waveform compare with CBwaves as an example. Parameters are:

• m1 = m2 = 2.48 Msun
• ecc = 0.0061
• D = 11 M
• s1z = s2z = 0.5

Comparsion is not straightforward because:

• Initial conditions for the NR simulations are not completely known
• Have to split hybrid waveforms so get only the NR part
• There could be (is) a phase shift between the waveforms that has to be checked / modified

BT and DD gauges

The radiation reaction terms at 2.5PN and 3.5PN depends on 2 and 6 parameters, see e.g. IyerWill95. These arbitrary parameters represent the residual gauge freedom in the relative coordinate x=x1-x2 at 2.5PN and 3.5PN leaving the formulas for energy and angular momentum flux invariant. CBwaves and the code of the Columbia group use different gauges BT and DD, but CBwaves can also analyize the DD gauge thanks to the built-in and configurable gauge transformation. With the appropriate coordinate transformation x'=x+δx (applied at 2.5PN order in the figure below for a circular binary with m1=m2=r0=10 Msun) we can change from one system to the other.

Conclusions

• The inclusion of all the PN contributions is important
• The use of the BT gauge provides results similar to NR data
• With the appropriate change of the coordinates the results in the BT and DD gauges can be transformed to each other
• Anomalies are excluded if one takes into account the domain of validity of PN and the requirements for energy conservation