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LALInspiral 5.0.3.1-5e288d3
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LALInspiral 5.0.3.1-5e288d3
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This is a module which generates inspiral waveforms.
The type of waveform generated is determined by specifying the value of the enums Approximant and LALPNOrder All Approximants generate the restricted post-Newtonian (PN) waveform given by (we use units in which \(c=G=1\)):
\begin{equation} h(t) = A v^2 \cos(\phi(t)), \end{equation}
where the coefficient \(A\) is set equal to 1 for non-spinning binaries and to the appropriate modulation function for spinning binaries (see below), \(v\) is the PN expansion parameter, and \(\phi(t)\) is the appropriately defined phasing formula which is either an explicit or an implicit function of time. In what follows we summarize the basic formulas used in generating the waveforms. As we shall see there are are a number of ways in which these waveforms can be generated. In the inspiral package these are controlled by two enums: The first enum called LALPNOrder controls the PN order to which the various quantities are defined and the second enum called Approximant controls the approximant used in generating the waveforms.
Theoretical calculations have given us PN expansions (i.e. a series in terms of ascending powers of \(v\)) of an energy function \(E(x=v^2)\) and a gravitational wave (GW) luminosity function \(\mathcal{F}(v):\)
\begin{equation} E(x) = E_N \sum_n E_k x^k,\ \ \ \ {\cal F}(v) = {\cal F}_N \sum_j {\cal F}_j v^j. \end{equation}
One can use kinematical equations \(dt = (dt/dE)(dE/dv) dv,\) and \(d\phi/dt = 2\pi F\) and the energy balance equation relating the luminosity in gravitational waves to the rate of change of binding energy \({\cal F} = - dE/dt\) , to obtain a phasing formula [15] :
\begin{eqnarray} \label{InspiralWavePhasingFormula} t(v) & = & t_\textrm{ref} + m \int_v^{v_\textrm{ref}} \, \frac{E'(v)}{{\cal F}(v)} \, dv, \\ \phi (v) & = & \phi_\textrm{ref} + 2 \int_v^{v_\textrm{ref}} v^3 \, \frac{E'(v)}{{\cal F}(v)} \, dv, \end{eqnarray}
where \(E'(v)=dE/dv,\) \(v=(\pi m F)^{1/3}\) is an invariantly defined velocity, \(F\) is the instantaneous GW frequency, and \(m\) is the total mass of the binary. There are basically three ways of solving the problem:
Leave \(E^{\prime}(v)/\mathcal{F}(v)\) as it is and integrate the equations numerically. Using standard PN expansions for the energy and flux functions one generates the Approximant called TaylorT1. If instead one uses the P-approximant for the energy and flux functions [15] ,[12] then one generates the approximant called PadeT1. In reality, it is computationally cheaper to use two ordinary differential equations instead of the integrals. These are:
\begin{equation} \label{eq_ode2} \frac{dv}{dt} = - \frac{\mathcal{F}(v)}{m E^{\prime}(v)},\ \ \ \ \frac{d \phi(t)}{dt} = \frac{2v^{3}}{m}. \end{equation}
These are implemented in module LALInspiralWave1.c.
Re-expand \(E^{\prime}(v)/\mathcal{F}(v)\) in a Taylor expansion in which case the integrals can be solved analytically to obtain a parametric representation of the phasing formula in terms of polynomial expressions in the auxiliary variable \(v\)
\begin{eqnarray} \label{eq_InspiralWavePhase2} \phi(v)&=& \phi_\textrm{ref} + \phi^v_N (v)\sum_{k=0}^{n} {\phi}^v_k v^k,\\ t(v)&=& t_\textrm{ref} +t_N(v) \sum_{k=0}^{n} {t}_k v^k, \end{eqnarray}
This corresponds to TaylorT2 in the enum Approximant [15] . These are implemented in module LALInspiralWave2.c
\begin{equation} \label{eq_InspiralWavePhase3} \phi(t)=\phi_\textrm{ref}+\phi_N^t \sum_{k=0}^{n} {\phi}^t_k\theta^k, \ \ \ F(t)= F_N \sum_{k=0}^{n} {F}_k \theta^k, \end{equation}
where \(\theta=[\eta (t_\textrm{ref}-t)/(5m)]^{-1/8}\), \(F \equiv d \phi/ 2 \pi dt =v^3/(\pi m)\) is the instantaneous GW frequency and \(\eta=m_1 m_2/m^2\) is the symmetric mass ratio. This corresponds to theApproximant TaylorT3 [5] , [6] , [15] . These are implemented in module LALInspiralWave3.c The expansion coefficients in PN expansions of the various physical quantities are summarized in this table and this table.
Consider a GW signal of the form,
\begin{equation} h(t)=2a(t)\cos\phi(t)= a(t) \left [ e^{-i \phi(t)} + e^{i \phi(t)} \right ], \end{equation}
where \(\phi(t)\) is the phasing formula, either specified as an explicit function of time or given implicitly by a set of differential equations [15] . The quantity \(2\pi F(t) = {d\phi(t)}/{dt}\) defines the instantaneous GW frequency \(F(t)\), and is assumed to be continuously increasing. (We assume \(F(t)>0\).) Now the Fourier transform \(\tilde h(f)\) of \(h(t)\) is defined as
\begin{equation} \tilde{h}(f) \equiv \int_{-\infty}^{\infty} dt e^{2\pi ift} h(t) = \int_{-\infty}^{\infty}\,dt\, a(t) \left[ e^{2\pi i f t - \phi(t)} + e^{2\pi ift +\phi(t)}\right ]. \end{equation}
The above transform can be computed in the stationary phase approximation (SPA). For positive frequencies only the first term on the right contributes and yields the following usual SPA:
\begin{equation} \label{eq_inspiralspa1} \tilde{h}^\textrm{uspa}(f)= \frac {a(t_f)} {\sqrt {\dot{F}(t_f)}} e^{ i\left[ \psi_f(t_f) -\pi/4\right]},\ \ \psi_f(t) \equiv 2 \pi f t -\phi(t), \end{equation}
and \(t_f\) is the saddle point defined by solving for \(t\), \( d \psi_f(t)/d t = 0\), i.e. the time \(t_f\) when the GW frequency \(F(t)\) becomes equal to the Fourier variable \(f\). In the adiabatic approximation where the value of \(t_f\) is given by the following integral:
\begin{equation} \label{eq_InspiralTimeAndPhaseFuncs} t_f = t_\textrm{ref} + m \int_{v_f}^{v_\textrm{ref}} \frac{E'(v)}{{\cal F}(v)} dv, \phi (v) = \phi_\textrm{ref} + 2 \int_v^{v_\textrm{ref}} dv v^3 \, \frac{E'(v)}{{\cal F}(v)}, \end{equation}
where \(v_\textrm{ref}\) is a fiducial reference point that sets the origin of time, \(v_f \equiv (\pi m f)^{1/3},\) \(E'(v)\equiv dE/dv\) is the derivative of the binding energy of the system and \({\cal F}(v)\) is the gravitational wave flux. Using \(t_f\) and \(\phi(t_f)\) in the above equation and using it in the expression for \(\psi_f(t)\) we find
\begin{equation} \label{eq_InspiralFourierPhase} \psi_f(t_f) = 2 \pi f t_\textrm{ref} - \phi_\textrm{ref} + 2\int_{v_f}^{v_\textrm{ref}} (v_f^3 - v^3) \frac{E'(v)}{{\cal {\cal F}}(v)} dv . \end{equation}
This is the general form of the stationary phase approximation which can be applied to all time-domain signals, including the P-approximant and effective one-body waveforms. In some cases the Fourier domain phasing can be worked out explicitly, which we now give:
Using PN expansions of energy and flux but re-expanding the ratio \(E'(v)/{\cal F}(v)\) in Eq. \eqref{eq_InspiralFourierPhase} one can solve the integral explicitly. This leads to the following explicit, Taylor-like, Fourier domain phasing formula:
\begin{equation} \label{eq_InspiralFourierPhase_f2} \psi_f(t_f) = 2 \pi f t_\textrm{ref} - \phi_\textrm{ref} + \psi_N \sum_{k=0}^5 {\psi}_k (\pi m f)^{(k-5)/3} \end{equation}
where the coefficients \({\psi}_k\) up to 2.5 post-Newtonian approximation are given by:
\[\psi_N = \frac{3}{128\eta},\ \ \ \psi_0 = 1,\ \ \ \psi_1 = 0,\ \ \ \psi_2 = \frac{5}{9} \left ( \frac{743}{84} + 11\eta\right ),\ \ \ \psi_3 = -16\pi,\]
\[\psi_4 = \frac{5}{72}\left(\frac{3058673}{7056} + \frac{5429}{7}\eta + 617\eta^2\right),\]
\[\psi_5 = \frac{5}{3} \left ( \frac{7729}{252} + \eta \right ) \pi + \frac{8}{3} \left ( \frac{38645}{672} + \frac{15}{8} \eta \right ) \ln \left ( \frac{v}{v_\textrm{ref}} \right )\pi.\]
Eq. \eqref{eq_InspiralFourierPhase_f2} is (one of) the standardly used frequency-domain phasing formulas. This is what is implemented in LALInspiralStationaryPhaseApprox2() and corresponds to TaylorF2 in the enum Approximant.
Alternatively, substituting (without doing any re-expansion or re-summation) for the energy and flux functions their PN expansions or the P-approximants of energy and flux functions and solving the integral in Eq. \eqref{eq_InspiralFourierPhase} numerically one obtains the T-approximant SPA or P-approximant SPA, respectively. However, just as in the time-domain, the frequency-domain phasing is most efficiently computed by a pair of coupled, non-linear, ODE's:
\begin{equation} \label{eq_frequencyDomainODE} \frac{d\psi}{df} - 2\pi t = 0, \ \ \ \ \frac{dt}{df} + \frac{\pi m^2}{3v^2} \frac{E'(f)}{{\cal F}(f)} = 0, \end{equation}
rather than by numerically computing the integral in Eq. \eqref{eq_InspiralFourierPhase}. However, the current implementation in LALInspiralStationaryPhaseApproximation1 solves the integral and corresponds to TaylorF1 in the enum Approximant.
The derivative of the frequency that occurs in the amplitude of the Fourier transform, namely \(1/\sqrt{\dot{F}(t)},\) in Eq. \eqref{eq_inspiralspa1}, is computed using
\begin{equation} \dot{F(t)} = \frac{dF}{dt} = \frac{dF}{dv}\frac{dv}{dE}\frac{dE}{dt} = \frac{3v^2}{\pi m}\left [\frac{{-\cal F}(v)}{E'(v)} \right ], \end{equation}
where we have used the fact that the gravitational wave flux is related to the binding energy \(E\) via the energy balance equation \({\cal F} = -dE/dt\) and that \(F=v^3/(\pi m).\) At the Newtonian order \(E=-\eta m v^2/2,\) and \({\cal F} = 32\eta^2 v^{10}/5,\) giving \(\dot{F}(t(v)) = 96\eta v^{11}/(5\pi m^2).\) Taking \(2a(t(v)) = v^2\) (i.e., \(h(t) = v^2 \cos (\phi(t)),\) this gives, the total amplitude of the Fourier transform to be
\[\frac{a(t(v))}{\sqrt{\dot{F}(t(v))}} = \sqrt{\frac{5\pi m^2}{384\eta}} v_f^{-7/2}.\]
This is the amplitude used in most of literature. However, including the full PN expansion in \(\dot{F}(t),\) gives a better agreement between the time-domain and Fourier domains signals and this code therefore uses the full PN expansion [15] .
The Fourier transform of a chirp waveform in the restricted post-Newtonian approximation in the stationary phase approximation is given, for positive frequencies \(f,\) by [cf. Eq. \eqref{eq_InspiralFourierPhase_f2}]
\begin{equation} \tilde h(f) = h_0 f^{-7/6} \exp \left [ \sum_k \psi_k f^{(k-5)/3} \right ], \end{equation}
where \(h_0\) is a constant for a given system and \(psi_k\) are parameters that depend on the two masses of the binary. Since the time-domain waveform is terminated at when the instantaneous GW frequency reaches a certain value \(F_\textrm{cut}\) (which is either the last stable orbit or the light-ring defined by the model) and since the contribution to a Fourier component comes mainly from times when the GW instantaneous frequency reaches that value, it is customery to terminate the Fourier transform at the same frequency, namely \(f_\textrm{cut} = F_\textrm{cut}.\) In otherwords, the Fourier transform is taken to be
\begin{equation} \tilde h(f) = h_0 f^{-7/6} \theta(f-f_\textrm{cut}) \exp \left [ \sum_k \psi_k f^{(k-5)/3} \right ], \end{equation}
where \(\theta(x<0)=0\) and \(\theta(x\ge 0) =1.\) We have seen that there are different post-Newtonian models such as the standard post-Newtonian, P-approximants, effective one-body (see Sec. Effective one-body approach), and their overlaps with one another is not as good as we would like them to be. The main reason for this is that matched filtering is sensitive to the phasing of the waves. It is not clear which model best describes the true GW signal from a compact binary inspiral although some, like the EOB, are more robust in their predictions than others. Thus, Buonanno, Chen and Vallisneri proposed [9] ,[10] a phenomenological model as a detection template family (DTF) based on the above expression for the Fourier transform. Indeed, they proposed to use a DTF that depends on four parameters \((\psi_0,\, \psi_3,\, f_\textrm{cut},\, \alpha)\) for non-spinning sources [9] and on six parameters \((\psi_0,\, \psi_3,\, f_\textrm{cut},\, \alpha_1,\, \alpha_2,\, \beta)\) for spinning sources [10] . We have implemented both the non-spinning and spinning waveforms in LALBCVWaveform and LALBCVSpinWaveform, respectively.
The proposed waveform has structure similar to the one above:
\begin{equation} \label{eq_BCV_NonSpinning} \tilde h(f) = h_0 f^{-7/6} \left (1 - \alpha f^{2/3} \right) \theta(f-f_\textrm{cut}) \exp \left [ \psi_0 f^{-5/3} + \psi_3 f^{-2/3} \right ], \end{equation}
where the motivation to include an amplitude correction term \(\alpha\) is based on the fact that the first post-Newtonian correction to the amplitude would induce a term like this. Note carefully that the phasing does not include the full post-Newtonian expansion but only the Newtonian and 1.5 post-Newtonian terms. It turns out this four-parameter family of waveforms has good overlap with the two-parameter family of templates corresponding to different post-Newtonian models and their improvments.
In the generic case of spinning black hole binaries there are a total of 17 parameters characterizing the waveform amplitude and shape (see Sec. Spinning Modulated Chirps). However, the phasing of the waves is determined, in general, by about 10 parameters. Apostolatos et al. [2] studied the time-evolution based on which Apostolatos found [3] that far fewer parameters can be used to capture the full structure of the spinning black hole binary waveforms. BCV suggested [10] that one could use a DTF of waveforms that resembles the non-spinning case. Their proposal has the following structure:
\begin{equation} \label{eq_BCV_Spinning} \tilde h(f) = h_0 f^{-7/6} \left [1 + \alpha_1 \cos ( \beta f^{-2/3} ) + \alpha_2 \sin ( \beta f^{-2/3} ) \right] \theta(f-f_\textrm{cut}) \exp \left [ \psi_0 f^{-5/3} + \psi_3 f^{-2/3} \right ], \end{equation}
where \(\alpha_1,\) \(\alpha_2\) and \(\beta\) are the parameters designed to capture the spin-induced modulation of the waveform.
The entry EOB in the enum Approximant corresponds to the effective one-body (EOB) approach of Buonanno and Damour [7] ,[8] ,[13] ,[16] . The EOB formalism allows one to evolve the dynamics of two black holes beyond the last stable orbit into the plunge phase thereby increasing the number of wave cycles, and the signal-to-noise ratio, that can be extracted. Here a set of four ordinary differential equations (ODEs) are solved by making an anzatz for the tangential radiation reaction force.
In practical terms, the time-domain waveform is obtained as the following function of the reduced time \(\hat{t}=t/m\):
\begin{equation} \label{eq_4_1} \quad h(\hat{t}) = {\cal C} \, v_{\omega}^2 (\hat{t}) \cos (\phi (\hat{t}))\,, \quad v_{\omega} \equiv \left( \frac{d \varphi}{d \hat{t}} \right)^{\frac{1}{3}}\,, \quad \phi \equiv 2\varphi \,. \end{equation}
(The amplitude \(\cal C\) is chosen to be equal to 1 in our codes.) The four ODEs correspond to the evolution of the radial and angular coordinates and the corresponding momenta:
\begin{eqnarray} \label{eq_3_28} &&\frac{dr}{d \hat{t}} = \frac{\partial \widehat{H}}{\partial p_r} (r,p_r,p_\varphi)\,, \\ \label{eq_3_29} && \frac{d \varphi}{d \hat{t}} = \widehat{\omega} \equiv \frac{\partial \widehat{H}}{\partial p_\varphi} (r,p_r,p_\varphi)\,, \\ \label{eq_3_30} && \frac{d p_r}{d \hat{t}} + \frac{\partial \widehat{H}}{\partial r} (r,p_r,p_\varphi)=0\,, \\ \label{eq_3_31} && \frac{d p_\varphi}{d \hat{t}} = \widehat{\cal F}_\varphi(\widehat{\omega} (r,p_r,p_{\varphi}))\,. \end{eqnarray}
The reduced Hamiltonian \(\widehat{H}\) (of the one-body problem) is given, at the 2PN approximation, by
\begin{equation} \label{eq_3_32} \widehat{H}(r,p_r,p_\varphi) = \frac{1}{\eta}\,\sqrt{1 + 2\eta\,\left [ \sqrt{A(r)\,\left (1 + \frac{p_r^2}{B(r)} + \frac{p_\varphi^2}{r^2} \right )} -1 \right ]}\,,\\ \end{equation}
where
\begin{equation} \label{eq_3_34} A(r) \equiv 1 - \frac{2}{r} + \frac{2\eta}{r^3} \,, \quad \quad B(r) \equiv \frac{1}{A(r)}\,\left (1 - \frac{6\eta}{r^2} \right )\,. \end{equation}
The damping force \(\cal{F}_{\varphi}\) is approximated by
\begin{equation} \label{eq_damp} \widehat{{\cal F}}_{\varphi} =-\frac{1}{\eta v_\omega ^3}{\cal F}_{P_n}(v_\omega)\,, \end{equation}
where \( {\cal F}_{P_n} (v_{\omega}) = \frac{32}{5}\,\eta^2\,v_\omega^{10}\, {\widehat{\cal F}}_{P_n} (v_{\omega})\) is the P-approximant to the flux function.
The initial data \((r_0, p_{r}^0, p_{\varphi}^0)\) are found using
\begin{equation} r_0^3 \left [ \frac {1 + 2 \eta (\sqrt{z(r_0)} -1 )}{1- 3\eta/r_0^2} \right ]- \omega_0^{-2} = 0,\ \ p^0_\varphi = \left [\frac {r_0^2 - 3 \eta}{r_0^3 - 3 r_0^2 + 5 \eta} \right ]^{1/2}r_0,\ \ p^0_r = \frac {{\cal F}_\varphi(\omega)}{C(r_0,p^0_\varphi) (dp^0_\varphi/dr_0)}\ \ \end{equation}
where \(z(r)\) and \(C(r,p_\varphi)\) are given by
\begin{equation} z(r) = \frac{r^3 A^2(r)}{r^3-3r^2+5 \eta},\ \ C(r,p_\varphi) = \frac{1}{\eta \widehat{H} (r,0,p_\varphi) \sqrt{z(r)}} \frac{A^2(r)}{(1-6\eta/r^2)}. \end{equation}
The plunge waveform is terminated when the radial coordinate attains the value at the light ring \(r_\textrm{lr}\) given by the solution to the equation,
\begin{equation} \label{eq_LightRing} r_\textrm{lr}^3 - 3 r_\textrm{lr}^2 + 5 \eta = 0. \end{equation}
Waveforms from spinning black hole binaries at 2PN order can be generated using the choice SpinTaylorT3 in the enum Approximant. Current implementation closely follows Ref. [2] . The orbital plane of a binary consisting of rapidly spinning compact objects precesses causing the polarization of the wave received at an antenna to continually change. This change depends on the source location on the sky and it might therefore be possible to resolve the direction to a source. It is therefore essential to specify the coordinate system employed in the description of the waveform. As in Ref. [2] the coordinate system is adapted to the detector with the x-y plane in the plane of the interferometer with the axes along the two arms.
In the restricted post-Newtonian approximation the evolution of a binary system comprising of two bodies of masses \(m_1\) and \(m_2,\) spins \({\mathbf S}_1=(s_1,\, \theta_1, \, \varphi_1)\) and \({\mathbf S}_2=(s_2,\, \theta_2, \, \varphi_2),\) orbital angular momentum \({\mathbf L}=(L,\, \theta_0, \, \varphi_0),\) is governed by a set of differential equations given by:
\begin{eqnarray} \label{eqn_precession1} \dot{\mathbf {L}} = \left [ \left ( \frac{4m_1+3m_2}{2m_1m^3} - \frac{3\, {\mathbf S}_2 \cdot {\mathbf L}}{2\, L^2m^3} \right ) {\mathbf S}_1 + \left ( \frac{4m_2+3m_1}{2m_2m^3} - \frac{3}{2}\frac{ {\mathbf S}_1 \cdot {\mathbf L}}{L^2m^3} \right ) {\mathbf S}_2 \right ] \times {\mathbf L} v^6 -\frac{32\eta^2 m}{5L} {\mathbf L} v^7, \end{eqnarray}
\begin{equation} \label{eqn_precession2} \dot{ \mathbf S}_1 = \left [ \left ( \frac{4m_1+3m_2}{2m_1m^3} - \frac{3\, {\mathbf S}_2 \cdot {\mathbf L} }{2\, L^2m^3} \right ) {\mathbf L} \times {\mathbf S}_1 + \frac{ {\mathbf S}_2 \times {\mathbf S}_1 }{2m^3} \right ] v^6, \end{equation}
\begin{equation} \label{eqn_precession3} \dot{ \mathbf S}_2 = \left [ \left ( \frac{4m_2+3m_1}{2m_2m^3} - \frac{3\, {\mathbf S}_1 \cdot {\mathbf L} }{2\, L^2m^3} \right ) {\mathbf L} \times {\mathbf S}_2 + \frac{ {\mathbf S}_1 \times {\mathbf S}_2 }{2m^3} \right ] v^6. \end{equation}
where as before \(v=(\pi m f)^{1/3},\) \(f\) is the gravitational wave frequency, \(m = m_1+m_2\) is the total mass, and \(\eta = m_1m_2/m^2\) is the (symmetric) mass ratio. An overdot denotes the time-derivative. In the evolution of the orbital angular momentum we have included the lowest order dissipative term [the second term containing \(v^7\) in Eq. \eqref{eqn_precession1}] while keeping the non-dissipative modulation effects caused by spin-orbit and spin-spin couplings [the first term within square brackets containing \(v^6\) in Eq. \eqref{eqn_precession1}]. The spins evolve non-dissipatively but their orientations change due to spin-orbit and spin-spin couplings. Though the non-dissipative terms are not responsible for gravitational wave emission, and therefore do not shrink the orbit, they cause to precess the orbit.
In the absence of spins the antenna observes the same polarization at all times; the amplitude and frequency of the signal both increase monotonically, giving rise to a chirping signal. Precession of the orbit and spins cause modulations in the amplitude and phase of the signal and smear the signal's energy spectrum over a wide band. For this reason we shall call the spin modulated (sm) chirp, a smirch (an anagram of sm and chir).
The strain \(h(t)\) produced by a smirch at the antenna is given by
\begin{eqnarray} \label{eqn_waveform} h(t) & = & -A(t)\cos[\phi(t)+\varphi(t)], \end{eqnarray}
where \(A(t)\) is the precession-modulated amplitude of the signal, \(\phi(t)\) is its post-Newtonian carrier phase that increases monotonically and \(\varphi(t)\) is the polarization phase caused by the changing polarization of the wave relative to the antenna. (We have neglected the Thomas precession of the orbit which induces additional, but small, corrections in the phase.) For a source with position vector \({\mathbf N} = (D,\, \theta_S,\, \varphi_S)\) the amplitude is given by,
\begin{equation} \label{eqn_amplitude} A(t) = \frac{2\eta m v^2}{D} \left[ \left( 1 + (\hat {\mathbf L}\cdot \hat {\mathbf N})^2\right )^2 F_{+}^2(\theta_S,\varphi_S,\psi) + 4 \left ( \hat {\mathbf L}\cdot \hat {\mathbf N} \right )^2 F_\times^2(\theta_S,\varphi_S,\psi) \right]^{1/2}. \end{equation}
Here \(\hat{\mathbf L} = {\mathbf L}/L,\) \(\hat{\mathbf N} = {\mathbf N}/D\) ( \(D\) is the distance to the source) and \(\psi(t)\) is the precession-modulated polarization angle given by
\begin{equation} \label{eqn_psi} \tan \psi(t) = \frac{\hat{\mathbf L}(t)\cdot\hat{\mathbf z} - (\hat{\mathbf L}(t)\cdot\hat{\mathbf N}) (\hat{\mathbf z}\cdot\hat{\mathbf N})} {\hat{\mathbf N}\cdot(\hat{\mathbf L}(t)\times\hat{\mathbf z})}. \end{equation}
Also, \(F_+\) and \(F_\times\) are the antenna beam pattern functions are given by
\begin{equation} F_+(\theta_S,\varphi_S,\psi) = \frac{1}{2}\left(1+\cos^2\theta_S\right)\cos{2\phi_S}\cos{2\psi} - \cos\theta_S\sin{2\phi_S}\sin{2\psi}, \end{equation}
\begin{equation} F_{\times}(\theta_S,\phi_S,\psi) = \frac{1}{2}\left(1+\cos^2\theta_S\right)\cos{2\phi_S}\sin{2\psi} + \cos\theta_S\sin{2\phi_S}\cos{2\psi}. \end{equation}
Next, the polarization phase \(\varphi(t)\) is
\begin{equation} \label{eqn_varphi} \tan \varphi(t) = \frac{ 2 \hat {\mathbf L}(t) \cdot \hat {\mathbf N}\, F_\times(\theta_S,\varphi_S,\psi)} {\left [ 1 + \left ({\mathbf L}(t)\cdot {\mathbf N} \right )^2 \right ] F_{+}(\theta_S,\varphi_S,\psi)}. \end{equation}
And finally, for the carrier phase we use the post-Newtonian expression, but without the spin-orbit and spin-spin couplings. These spin couplings modify the carrier phase by amounts much smaller than the post-Newtonian effects. To second post-Newtonian order the carrier phase is given by
\begin{equation} \label{eqn_phi} \phi(t) = \frac{-2}{\eta\theta^5}\left[1+\left(\frac{3715}{8064}+\frac {55}{96}\eta\right)\theta^2-\frac{3\pi}{4}\theta^3 +\left(\frac{9275495}{14450688}+\frac{284875}{258048}\eta+\frac{1855}{2048}\eta^2\right)\theta^4\right]. \end{equation}
where \(\theta=[\eta (t_C-t)/(5m)]^{-1/8},\) \(t_C\) being the time at which the two stars merge together and the gravitational wave frequency formally diverges.
Modules | |
| Header LALInspiral.h | |
| Header file for the template generation codes. | |
1.9.4