Gravitational Waveforms


Wave Strain

We analyze the gravitational waveforms from our numerical relativity simulations and compare them with the simulation done at LIGO. If the length of the interferometer arms is $L$, the strain $\frac{\delta L(t)}{L}$ on the arms due to the gravitational waveform is given by $$h \equiv \frac{\delta L(t)}{L}=F_{+}(\theta,\phi,\psi)h_{+}(t;\theta_{GW},\phi_{GW})+ F_{\times}(\theta,\phi,\psi)h_{\times}(t;\theta_{GW},\phi_{GW}),$$ where $F_{+}$ and $F_{\times}$ are antenna pattern functions determined by the orientation of the LIGO detectors with respect to the binary (see Fig. 1): $$F_{+}(\theta,\phi,\psi)=\frac{1}{2}(1+\cos^{2}\theta)\cos2\phi\cos2\psi -\cos\theta\sin2\phi\sin2\psi,$$ $$F_{\times}(\theta,\phi,\psi)=\frac{1}{2}(1+\cos^2\theta)\cos2\phi\sin2\psi -\cos\theta\sin2\phi\cos2\psi.$$

Fig. 1: The relative 
	orientation of the sky and detector frames (right panel) and the effect 
	of a rotation by the angle $\psi$ in the sky frame (left panel).

Fig. 1: The relative orientation of the sky and detector frames (left panel) and the effect of a rotation by the angle $\psi$ in the sky frame (right panel). (Reference: http://relativity.livingreviews.org/Articles/lrr-2009-2/download/lrr-2009-2Color.pdf)


Here, $\theta_{GW}$ and $\phi_{GW}$ are the spherical coordinates of the observer from the binary's frame, taking the angular momentum of the binary to point along the z-axis. These values are tabulated below.

Viewing Angle with respect to Orbital Plane
Angle Value LIGO Ref
$\theta_{GW}$ $\pi/6$ https://dcc.ligo.org/public/0122/P1500218/01
2/GW150914_parameter_estimation_v13.pdf
$\phi_{GW}$ 0 Arbitrary

To obtain $\theta$, $\phi$, and $\psi$ for calculating $F_{+}$ and $F_{\times}$, we need the find the orientation of the binary with respect to the observers. LIGO found that on September 14 2015, 09:50:45.39 UTC, the source position was consistent with: $$ \begin{align} \text{RA} &= 8\text{h} \\ \text{DEC} &= -70^{\circ} \end{align} $$

LIGO Ref: https://dcc.ligo.org/public/0122/P1500218/012/GW150914_parameter_estimation_v13.pdf

The location sites of the observatories are given and listed below.

Detector Orientation
Detector Latitude Longitude X-Arm Azimuth
LIGO Hanford 46°27'19"N 119°24'28"W N 36°W
LIGO Livingston 30°33'46"N 90°46'27"W W 18°S

LIGO Ref: http://www.ligo.org/scientists/GW100916/GW100916-geometry.html


Using these values, we then calculate $\theta$ and $\phi$, and setting $\psi=0$.

Source Orientation with respect to Detectors
Detector   Angle     Value  
(rad)
Ref
LIGO Hanford
$\theta$
$\phi$
$\psi$
2.364
3.018
0
Liu code
Liu code
Arbitrary
LIGO Livingston
$\theta$
$\phi$
$\psi$
1.952
1.607
0
Liu code
Liu code
Arbitrary

Liu code can be found here


From this, we get $$ F_{+} = 0.7339 \\ F_{\times} = 0.1754 $$ Our simulation calculates $h_{+}^{NR}$ and $h_{\times}^{NR}$ at an extraction radius of $r_{ext}=4924.5M_{\odot}=7272\text{km}$. The luminosity distance between the source and LIGO has been observed to be $D_{L} = 410 \text{Mpc} = 1.265\times 10^{22} \text{km}$. Thus, we must scale our data to obtain the measured strain by LIGO: $$ h_{+} = \frac{r_{ext}}{D_{L}}h_{+}^{NR} = 5.7483\times 10^{-19} h_{+}^{NR}\\ h_{\times} = \frac{r_{ext}}{D_{L}}h_{\times}^{NR} = 5.7483\times 10^{-19} h_{\times}^{NR}\\ $$ The final equation for measuring the strain from our numerical simulation is then $$h \equiv \frac{\delta L(t)}{L} = 5.7483\times 10^{-19}\left(0.7339\, h_{+}^{NR}(t;\pi/6, 0) +0.1754 \, h_{\times}^{NR}(t;\pi/6, 0)\right) $$