\documentclass[12pt]{article} \usepackage{amsmath} \begin{document} \title{Parameter space metric for combined diurnal and orbital motion} \author{Ian Jones, Ben Owen, and David Whitbeck} \date{April 2005 version + typo fixes} \maketitle This is an old document describing how the equations in the \texttt{PtoleMetric} function in LAL were derived. It is a fast approximate parameter space metric for continuous-wave searches for isolated sources. It was used in some early Einstein@Home S5 searches, and a version of this document appeared as part of David Whitbeck's 2006 Ph.D. thesis. I've finally uploaded it to the DCC, where it now has number LIGO-T0900500. ---Ben \section{General formalism} We want to generate a parameter space metric to search for continuous nearly sinusoidal signals. We assume templates of the form \begin{equation} u = \exp i (2\pi f t_c + \phi) \equiv \exp i\psi. \end{equation} The canonical time $t_c$ is a function of detector time and any other parameters important in the problem, e.g. sky location $(\alpha, \delta)$ or spin-down parameters $\{f_k\}$: \[ t_c = t_c(\alpha, \delta, f_k, \dots). \] We will collect the template parameters together as a vector \begin{equation} \theta^\mu = (\phi, f, \alpha, \delta, f_k, \dots) \end{equation} and define an inner product \begin{equation} = \frac{1}{2T} \int_0^T \!\! dt \, \, u^*(t) v(t) + u(t) v^*(t). \end{equation} Our detection statistic will be: \begin{equation} D(\theta, \Delta \theta) = ^2. \end{equation} The metric is then given by \begin{equation} \label{eq:general_metric} g_{\mu \nu} = -\frac{1}{2} \left. \frac{\partial^2 D} {\partial \Delta \theta^\mu \partial \Delta \theta^\nu}\right|_{\Delta \theta = 0} \end{equation} which, for our particular detection statistic and waveform, can be shown to reduce to: \begin{equation} g_{\mu \nu} = \frac{1}{T} \int^T_0 \left.\frac{\partial \psi}{\partial\Delta \theta^\mu}\right|_{\Delta \theta=0} \left.\frac{\partial \psi}{\partial\Delta \theta^\nu}\right|_{\Delta \theta=0} \, dt. \end{equation} The quantity $\psi$ is given by \[ \psi = 2 \pi (f+\Delta f) (t_c + \Delta t_c) + \phi + \Delta \phi , \] where \[ t_c + \Delta t_c = t_c(\theta + \Delta \theta). \] The partial derivatives are as follows: \begin{eqnarray} \left.\frac{\partial \psi}{\partial\Delta \phi}\right|_{\Delta \theta=0} & = & 1 \label{eq:spiffy} \\ \left.\frac{\partial \psi}{\partial\Delta f}\right|_{\Delta \theta=0} & = & 2 \pi t_c \\ \left.\frac{\partial \psi}{\partial\Delta \alpha}\right|_{\Delta \theta=0} & = & 2 \pi f \frac{\partial t_c}{\partial \alpha} \\ \left.\frac{\partial \psi}{\partial\Delta \delta}\right|_{\Delta \theta=0} & = & 2 \pi f \frac{\partial t_c}{\partial \delta} \\ \left.\frac{\partial \psi}{\partial\Delta f_k}\right|_{\Delta \theta=0} & = & 2 \pi f \frac{\partial t_c}{\partial f_k} \end{eqnarray} \section{Doppler modulation only} Let's include both the Earth's spin and orbital motions, and compute the metric for \[ \theta^\mu = (\phi, f, \alpha, \delta). \] The canonical timing function for an epicyclic (Ptolemaic) detector motion is: \[ t_c = t + \cos\delta \cos\alpha (R_o \cos\phi_{o} + R_s \cos\lambda \cos\phi_{s}) \] \[ \hspace{1cm} + \cos\delta \sin\alpha (R_o \cos\iota \sin\phi_{o} + R_s \cos\lambda \sin\phi_{s}) \] \begin{equation} \hspace{0cm} + \sin\delta (R_o \sin\iota \sin\phi_{o} + R_s \sin\lambda). \end{equation} Here $R_o$ is an astronomical unit, $R_s$ is the Earth's (equatorial) radius, $\lambda$ is the latitude of the detector, $\iota$ is the angle between the Earth's spin and orbital angular momenta, $\phi_o$ is the orbital phase (measured from vernal equinox), and $\phi_s$ is the rotation phase (measured from detector midnight). It is then straightforward (but tedious) to evaluate the partial derivatives above and insert them into (\ref{eq:general_metric}) to evaluate the metric. We will make use of the small quantities \[ \frac{v_{s}}{c} \sim \frac{R_s \omega_s}{c} \] \[ \frac{v_{o}}{c} \sim \frac{R_o \omega_o}{c} \] to evaluate each metric component only to leading non-zero order. The corresponding accuracy is then summarised in table \ref{table:Doppler_accuracy}. The fractional errors of order $v/c$ propagate during the projections to lower dimensional metrics, but the fractional errors in any projected component will never be worse than $v/c$. \begin{table} \caption{Accuracy of computed metric components} \label{table:Doppler_accuracy} \begin{tabular}{ll} Component & Terms missing \emph{beyond} leading order\\ \hline $g_{\phi,\phi}$ & Exact \\ $g_{\phi, f}$ & $v/c$ missing \\ $g_{\phi,\alpha}$ & Exact \\ $g_{\phi,\delta}$ & Exact \\ $g_{f, f}$ & $v/c$ and $(v/c)^2$ missing\\ $g_{f, \alpha}$ & $v/c$ missing\\ $g_{f, \delta}$ & $v/c$ missing\\ $g_{\alpha,\alpha}$ & Exact \\ $g_{\alpha, \delta}$ & Exact \\ $g_{\delta, \delta}$ & Exact \end{tabular} \end{table} \subsection{Useful definitions} These simply make the metric expression more manageable. For any quantity $Q$: \begin{equation} \Delta Q \equiv Q_{\rm final} - Q_{\rm initial}. \end{equation} In particular \begin{equation} \Delta \phi_o \equiv \phi_{o, \rm final} - \phi_{o, \rm initial} = \omega_o T \end{equation} \begin{equation} \Delta \phi_s \equiv \phi_{s, \rm final} - \phi_{s, \rm initial} = \omega_s T, \end{equation} where $T$ is the integration time, and for trig expressions \begin{equation} \Delta \sin \phi \equiv \sin \phi_{\rm final} - \sin \phi_{\rm initial} \end{equation} \begin{equation} A_1 = R_o \frac{\Delta \sin \phi_o}{\Delta \phi_o} + R_{\rm s} \cos \lambda \frac{\Delta \sin \phi_s}{\Delta \phi_s} \end{equation} \begin{equation} A_2 = R_o \cos i \frac{\Delta \cos \phi_o}{\Delta \phi_o} + R_s \cos \lambda \frac{\Delta \cos \phi_s}{\Delta \phi_s} \end{equation} \begin{equation} A_3 = -R_o \sin i \frac{\Delta \cos \phi_o}{\Delta \phi_o} + R_s \sin \lambda \end{equation} \begin{equation} A_4 = R_o \left[ \frac{\sin \phi_{o, \rm final}}{\Delta \phi_o} + \frac{\Delta \cos \phi_o}{(\Delta \phi_o)^2} \right] \end{equation} \begin{equation} A_5 = R_s \left[ \frac{\sin \phi_{s, \rm final}}{\Delta \phi_s} + \frac{\Delta \cos \phi_s}{(\Delta \phi_s)^2} \right] \end{equation} \begin{equation} A_6 = R_o \left[ -\frac{\cos \phi_{o, \rm final}}{\Delta \phi_o} + \frac{\Delta \sin \phi_o}{(\Delta \phi_o)^2} \right] \end{equation} \begin{equation} A_7 = R_s \left[ -\frac{\cos \phi_{s, \rm final}}{\Delta \phi_s} + \frac{\Delta \sin \phi_s}{(\Delta \phi_s)^2} \right] \end{equation} \begin{equation} A_8 = R_o^2 \left( 1 + \frac{\Delta \sin 2 \phi_o}{2\Delta \phi_o} \right) \end{equation} \begin{equation} A_9 = R_o R_s \left[ \frac{\Delta \sin (\phi_o-\phi_s)}{\Delta \phi_o-\Delta \phi_s} + \frac{\Delta \sin (\phi_o+\phi_s)}{\Delta \phi_o+\Delta \phi_s} \right] \end{equation} \begin{equation} A_{10} = R_s^2 \left( 1 + \frac{\Delta \sin 2 \phi_s}{2\Delta \phi_s} \right) \end{equation} \begin{equation} A_{11} = R_o^2 \frac{\Delta \cos 2 \phi_o}{2\Delta \phi_o} \end{equation} \begin{equation} A_{12} = R_o R_s \left[ -\frac{\Delta \cos (\phi_o-\phi_s)}{\Delta \phi_o-\Delta \phi_s} + \frac{\Delta \cos (\phi_o+\phi_s)}{\Delta \phi_o+\Delta \phi_s} \right] \end{equation} \begin{equation} A_{13} = R_o R_s \left[ \frac{\Delta \cos (\phi_o-\phi_s)}{\Delta \phi_o-\Delta \phi_s} + \frac{\Delta \cos (\phi_o+\phi_s)}{\Delta \phi_o+\Delta \phi_s} \right] \end{equation} \begin{equation} A_{14} = R_s^2 \frac{\Delta \cos 2 \phi_s}{2\Delta \phi_s} \end{equation} \begin{equation} A_{15} = R_o^2 \left( 1 - \frac{\Delta \sin 2 \phi_o}{2\Delta \phi_o} \right) \end{equation} \begin{equation} A_{16} = R_o R_s \left[ \frac{\Delta \sin (\phi_o-\phi_s)}{\Delta \phi_o-\Delta \phi_s} - \frac{\Delta \sin (\phi_o+\phi_s)}{\Delta \phi_o+\Delta \phi_s} \right] \end{equation} \begin{equation} A_{17} = R_s^2 \left( 1 - \frac{\Delta \sin 2 \phi_s}{2\Delta \phi_s} \right) \end{equation} \begin{equation} A_{18} = R_0 R_s \frac{\Delta \sin\phi_o}{\Delta \phi_o} \end{equation} \begin{equation} A_{19} = R_s^2 \frac{\Delta \sin\phi_s}{\Delta \phi_s} \end{equation} \begin{equation} A_{20} = R_0 R_s \frac{\Delta \cos\phi_o}{\Delta \phi_o} \end{equation} \begin{equation} A_{21} = R_s^2 \frac{\Delta \cos\phi_s}{\Delta \phi_s} \end{equation} \begin{equation} B_{1} = A_4 + A_5 \cos\lambda \end{equation} \begin{equation} B_{2} = A_6 \cos i + A_7 \cos\lambda \end{equation} \begin{equation} B_{3} = A_6 \sin i + \frac{R_s \sin\lambda}{2} \end{equation} \begin{equation} B_{4} = A_8 + 2 A_9 \cos\lambda + A_{10} \cos^2\lambda \end{equation} \begin{equation} B_{5} = A_{11} \cos i + A_{12} \cos\lambda + A_{13} \cos i \cos\lambda + A_{14} \cos^2\lambda \end{equation} \begin{equation} B_{6} = A_{15} \cos^2 i + 2 A_{16} \cos i \cos\lambda + A_{17} \cos^2\lambda \end{equation} \begin{equation} B_{7} = -A_{11} \sin i + 2 A_{18} \sin\lambda - A_{13} \sin i \cos\lambda + A_{19} \sin2\lambda \end{equation} \begin{equation} B_{8} = A_{15} \sin i \cos i - 2 A_{20} \cos i \sin\lambda + A_{16} \sin i \cos\lambda - A_{21} \sin2\lambda \end{equation} \begin{equation} B_{9} = A_{15} \sin^2 i - 4 A_{20} \sin i \sin\lambda + 2 R_s^2 \sin^2\lambda \end{equation} \subsection{The metric components} The metric components can then be written rather compactly as follows. \begin{equation} g_{\phi \phi} = 1 \end{equation} \begin{equation} g_{\phi f} = \pi T \end{equation} \begin{equation} g_{\phi \alpha} = - 2 \pi f \cos \delta [ A_1 \sin \alpha + A_2 \cos \alpha ] \end{equation} \begin{equation} g_{\phi \delta} = 2 \pi f [ -A_1 \sin \delta \cos \alpha + A_2 \sin \delta \sin \alpha + A_3 \cos \delta ] \end{equation} \begin{equation} g_{f f} = \frac{(2 \pi T)^2}{3} \end{equation} \begin{equation} g_{f \alpha} = (2\pi)^2 f \cos \delta T [ - B_1 \sin \alpha + B_2 \cos \alpha ] \end{equation} \begin{equation} g_{f \delta} = (2\pi)^2 f T [ - B_1 \sin \delta \cos \alpha - B_2 \sin \delta \sin \alpha + B_3 \cos \delta ] \end{equation} \begin{equation} g_{\alpha \alpha} = 2 (\pi f \cos \delta)^2 [ B_4 \sin^2\alpha + B_5 \sin2\alpha + B_6 \cos^2\alpha ] \end{equation} \[ g_{\alpha \delta} = 2 (\pi f)^2 \cos \delta [ B_4 \sin\alpha \cos\alpha \sin\delta - B_5 \sin^2\alpha \sin\delta \] \[ \hspace{2.0cm} - B_7 \sin\alpha \cos\delta + B_5 \cos^2\alpha \sin\delta \] \begin{equation} \hspace{2.8cm} - B_6 \sin\alpha \cos\alpha \sin\delta + B_8 \cos\alpha \cos\delta ] \end{equation} \[ g_{\delta \delta} = 2 (\pi f)^2 [ B_4 \cos^2\alpha \sin^2\delta + B_6 \sin^2\alpha \sin^2\delta \] \[ \hspace{2.0cm} + B_9 \cos^2\delta - B_5 \sin2\alpha \sin^2\delta \] \begin{equation} \hspace{2.8cm} - B_8 \sin\alpha \sin2\delta - B_7 \cos\alpha \sin2\delta ] \end{equation} \subsection{Refinement for short duration observations} The function \texttt{PtoleMetric} has difficulty in calculating the metric for short duration observations. It seems that this is caused by finite accuracy errors in subtracting nearly equal quantities involving the orbital phase. To give the computer a helping hand, the following replacements are useful, where a Taylor series expansion in the orbital phase change, $\Delta \phi_o$ is used. This allows the leading order terms to be cancelled by hand. For short duration observations the quantities $A_4$ and $A_6$ are form of subtractions involving terms of order $1/\Delta \phi_o$ to give a result of order unity. These can be recast in the form \[ A_4 = R_o \left[\frac{S_1}{\Delta \phi_o} \cos \phi_{o, \rm final} + S_2 \sin \phi_{o, \rm final} \right] \] \[ A_6 = R_o \left[\frac{S_1}{\Delta \phi_o} \sin \phi_{o, \rm final}- S_2 \cos \phi_{o, \rm final} \right] \] where \[ S_1 = \frac{\Delta \phi_o}{2!} - \frac{\Delta \phi_o^3}{4!} + \dots \] \[ S_2 = \frac{\Delta \phi_o}{3!} - \frac{\Delta \phi_o^3}{5!} + \dots \] The trigonometric subtractions can be recast as: \[ \frac{\Delta \sin \phi_o}{\Delta \phi_o} = S_1 \sin \phi_{o, \rm final} + \frac{\sin{\Delta \phi_o}}{\Delta \phi_o} \cos \phi_{o, \rm final} \] which appears in $A_1$ and $A_{18}$. \[ \frac{\Delta \cos \phi_o}{\Delta \phi_o} = S_1 \cos \phi_{o, \rm final} - \frac{\sin{\Delta \phi_o}}{\Delta \phi_o} \sin \phi_{o, \rm final} \] which appears in $A_2$, $A_3$ and $A_{20}$. Similarly \[ \frac{\Delta \sin 2\phi_o}{2\Delta \phi_o} = S_3 \sin 2\phi_{o, \rm final} + \frac{\sin{2\Delta \phi_o}}{2\Delta \phi_o} \cos 2\phi_{o, \rm final} \] which appears in $A_8$ and $A_{15}$, and $S_3$ is the same as $S_2$ with the replacement $\phi_o \rightarrow 2\phi_o$: \[ S_3 = \frac{2\Delta \phi_o}{3!} - \frac{(2\Delta \phi_o)^3}{5!} + \dots \] Finally \[ \frac{\Delta 2\cos \phi}{2\Delta \phi} = S_3 \cos 2\phi_{o, \rm final} - \frac{\sin{2\Delta \phi_o}}{2\Delta \phi_o} \sin 2\phi_{o, \rm final} \] which appears in $A_{11}$. \section{Spin-down only} Now include spin-down only, i.e. compute the metric for \[ \theta^\mu = (\phi, f, f_k). \] and \begin{equation} \label{eq:tcspindown} t_c = t + \sum_{k=1} \frac{f_k}{k+1} (t-t_0)^{k+1}. \end{equation} The relevant partial derivatives are now: \begin{eqnarray} \frac{\partial \phi}{\partial \phi_0} & = & 1 \\ \frac{\partial \phi}{\partial f} & = & 2 \pi t_c \\ \frac{\partial \phi}{\partial f_k} & = & 2 \pi f \frac{\partial t_c}{\partial f_k} \end{eqnarray} We will evaluate each metric component to leading non-zero order, making use of the smallness of $\{f_k\}$. We obtain: \begin{eqnarray} g_{\phi \phi} & = & 1 \\ g_{\phi f} & = & \pi T \\ g_{\phi f_k} & = & \frac{2 \pi f T^{k+1}}{(k+1)(k+2)} \\ g_{f f} & = & \frac{(2\pi T)^2}{3} \\ g_{f f_k} & = & \frac{(2\pi)^2 T^{k+2}f}{(k+1)(k+3)} \\ g_{f_j f_k} & = & \frac{(2\pi f T)^2 T^{j+k}}{(j+1)(k+1)(j+k+3)} \end{eqnarray} The corresponding accuracies are given in table \ref{table:spin-down_accuracy}. \begin{table} \caption{Accuracy of computed metric components for spin-down only.} \label{table:spin-down_accuracy} \begin{tabular}{ll} Component & Terms missing \emph{beyond} leading order\\ \hline $g_{\phi,\phi}$ & Exact \\ $g_{\phi, f}$ & $f_k$ missing \\ $g_{\phi,f_k}$ & Exact \\ $g_{f, f}$ & $f_k$ and $f_k^2$ missing\\ $g_{f, f_k}$ & $f_k$ missing\\ $g_{f_j, f_k}$ & Exact \end{tabular} \end{table} \section{Doppler modulation \emph{and} spin-down} The canonical time can now be approximated as \[ t_c = t + \Delta t_{\rm Doppler} + \Delta t_{\rm spin-down} \] where the two $\Delta$ terms on the right hand side are the previously computed delays due to Doppler effects and spin-down. Note that this neglects a Doppler---spin-down term which would account for the fact that the intrinsic frequency of the source would change in the time it takes the gravitational waves to propagate across the Earth's orbit. It can be shown that this is always a negligible effect. If we continue to evaluate each metric component to leading order only, we can readily convince ourselves that very little new work need be done. The components break down into four classes: \begin{enumerate} \item Those independent of Doppler modulation and spin-down (and therefore identical to those computed in both earlier calculations): \begin{quotation} $g_{\phi \phi}, g_{\phi f}, g_{f f}$ \end{quotation} \item Those identical to the Doppler modulation only case: \begin{quotation} $g_{\phi \alpha}, g_{\phi \delta}, g_{f \alpha}, g_{f \delta}, g_{\alpha \alpha}, g_{\alpha \delta}, g_{\delta \delta}$ \end{quotation} \item Those identical to spin-down only case: \begin{quotation} $g_{\phi f_k}, g_{f f_k}, g_{f_j f_k}$ \end{quotation} \item New Doppler modulation---spin-down components: \begin{quotation} $g_{\alpha f_k}, g_{\delta f_k}$ \end{quotation} \end{enumerate} The new components are as follows: \[ g_{f_k \alpha} = \frac{(2\pi f)^2}{(k+1) T} \{-\cos\delta \sin\alpha [R_o I_{k,c}^{o} + R_s \cos\lambda I_{k,c}^{s}] \] \begin{equation} \hspace{2cm} + \cos\delta \cos\alpha [R_o \cos\iota I_{k,s}^{o} + R_s \cos\lambda I_{k,s}^{s}]\} \end{equation} \[ g_{f_k \delta} = \frac{(2\pi f)^2}{(k+1) T} \{-\sin\delta \cos\alpha [R_o I_{k,c}^{o} + R_s \cos\lambda I_{k,c}^{s}] \] \[ \hspace{2cm} - \sin\delta \sin\alpha [R_o \cos\iota I_{k,s}^{o} + R_s \cos\lambda I_{k,s}^{s}] \] \begin{equation} \hspace{2cm} + \cos\delta [R_o \sin\iota I_{k,s}^{o} + R_s \sin\lambda \frac{T^{k+2}}{k+2}]\} \end{equation} where \[ I_{k,c}^{o} = \int^T_0 t^{k+1} \cos \phi_{o} \, dt \] \[ I_{k,s}^{o} = \int^T_0 t^{k+1} \sin \phi_{o} \, dt \] and similarly for the spin quantities. These are best evaluated iteratively using the following set of equations: \begin{eqnarray} I_{k+1, c} & = & \frac{T^{k+1}}{\omega} \sin\phi_{\rm final} - \frac{k+1}{\omega^2} \left\{-T^k \cos\phi_{\rm final} + k I_{k-1, c}\right\} \\ I_{-1,c} & = & \int^T_0 \cos\phi \, dt = \frac{\Delta \sin\phi}{\omega} \\ I_{0,c} & = & \int^T_0 t \cos\phi \, dt = T \frac{\sin\phi_{\rm final}}{\omega} + \frac{\Delta \cos\phi}{\omega^2} \\ I_{k+1, s} & = & -\frac{T^k}{\omega} \cos\phi_{\rm final} + \frac{(k+1)}{\omega^2} \left\{T^k \sin\phi_{\rm final} - k I_{k-1, s} \right\} \\ I_{-1,s} & = & \int^T_0 \sin\phi \, dt = -\frac{\Delta \cos\phi}{\omega} \\ I_{0,s} & = & \int^T_0 t \sin\phi \, dt = -\frac{T}{\omega}\cos\phi_{\rm final} + \frac{\Delta \sin\phi}{\omega^2} \end{eqnarray} \section{Projecting out the Phase} The phase offset of the waveform $\phi$ is a nuisance parameter, and is projected out so that the template spacing will only be determined by the metric on the intrinsic parameter subspace of the full parameter space. The projected metric $\gamma$ is calculated by the standard formula \begin{equation} \gamma_{\mu\nu} = g_{\mu\nu}-\frac{g_{\mu\phi}g_{\nu\phi}}{g_{\phi\phi}} \end{equation} Analytically that means that since \begin{equation} g_{\mu\nu} = \langle \partial_\mu\psi\partial_\nu\psi \rangle \end{equation} Then we have \begin{equation} \gamma_{\mu\nu} = \langle \partial_\mu\psi\partial_\nu\psi \rangle - \langle \partial_\mu\psi \rangle \langle \partial_\nu\psi \rangle \end{equation} by virtue of (\ref{eq:spiffy}). These projections are numerically computed in PtoleMetric.c. \newpage \begin{table} \caption{Definitions of angles, lengths, etc...} \label{table:definitions} \begin{tabular}{ll} Symbol & definition\\ \hline $u$ & Signal template \\ $f$ & A fixed frequency \\ $t_c$ & Canonical time \\ $t$ & Detector time \\ $\phi$ & Phase of signal at $t_c=0$ \\ $\psi$ & Signal phase $=2\pi f t_c + \phi$ \\ $\alpha, \delta$ & Source RA and dec \\ $\{f_k\}$ & Spin-down parameters (see Eq. (\ref{eq:tcspindown}) \\ $\theta^\mu$ & Vector of signal parameters \\ $D$ & Detection statistic \\ $T$ & Duration of observation \\ $R_s$ & Radius of Earth \\ $R_o$ & Radius of Earth's orbit \\ $\phi_o, \phi_s$ & Earth's orbital and spin phases \\ $\lambda$ & Polar angle giving detector latitude ($\lambda=0$ at N pole) \\ $\iota$ & Misalignment of Earth's spin and orbital vectors (approx $23^\circ$). \\ $\omega_o, \omega_s$ & Angular velocities of Earth's orbit and spin motions \\ $\Delta \phi_o, \Delta \phi_s$ & Change in orbital, spin phases over $T$. \\ \end{tabular} \end{table} \end{document}