1 Introduction
2 Physical and Historical Motivations of TDI
$$\begin{array}{c}\Delta C\left(t\right)=C(t-2{L}_{1})-C(t-2{L}_{2}).\end{array}$$ | (1) |
$$\begin{array}{c}\left|\stackrel{~}{\Delta C}\right(f\left)\right|\simeq \left|\stackrel{~}{C}\right(f\left)\right|4\pi f\left|\right({L}_{1}-{L}_{2}\left)\right|.\end{array}$$ | (2) |
$$\begin{array}{ccc}{y}_{1}\left(t\right)& =& C(t-2{L}_{1})-C\left(t\right)+{h}_{1}\left(t\right)+{n}_{1}\left(t\right),\end{array}$$ | (3) |
$$\begin{array}{ccc}{y}_{2}\left(t\right)& =& C(t-2{L}_{2})-C\left(t\right)+{h}_{2}\left(t\right)+{n}_{2}\left(t\right).\end{array}$$ | (4) |
$$\begin{array}{c}{\stackrel{~}{y}}_{1}\left(f\right)=\stackrel{~}{C}\left(f\right)\left[{e}^{4\pi if{L}_{1}}-1\right]+{\stackrel{~}{h}}_{1}\left(f\right)+{\stackrel{~}{n}}_{1}\left(f\right).\end{array}$$ | (5) |
$$\begin{array}{c}{\stackrel{~}{y}}_{1}^{T}\equiv {\int}_{-T}^{+T}{y}_{1}\left(t\right){e}^{2\pi ift}dt={\int}_{-\infty}^{+\infty}{y}_{1}\left(t\right)H\left(t\right){e}^{2\pi ift}dt,\end{array}$$ | (6) |
$$\begin{array}{c}{y}_{1}\left(t\right)-{y}_{2}\left(t\right)=C(t-2{L}_{1})-C(t-2{L}_{2})+{h}_{1}\left(t\right)-{h}_{2}\left(t\right)+{n}_{1}\left(t\right)-{n}_{2}\left(t\right).\end{array}$$ | (7) |
$$\begin{array}{ccc}{y}_{1}(t-2{L}_{2})-{y}_{2}(t-2{L}_{1})& =& C(t-2{L}_{1})-C(t-2{L}_{2})+{h}_{1}(t-2{L}_{2})-{h}_{2}(t-2{L}_{1})\end{array}$$ |
$$\begin{array}{ccc}& & +{n}_{1}(t-2{L}_{2})-{n}_{2}(t-2{L}_{1}).\end{array}$$ | (8) |
$$\begin{array}{c}X\equiv \left[{y}_{1}\right(t)-{y}_{2}(t\left)\right]-\left[{y}_{1}\right(t-2{L}_{2})-{y}_{2}(t-2{L}_{1}\left)\right].\end{array}$$ | (9) |
3 Time-Delay Interferometry
$$\begin{array}{c}{\mathcal{D}}_{i}x\left(t\right)=x(t-{L}_{i}),\end{array}$$ | (10) |
$$\begin{array}{ccc}{s}_{1}& =& {s}_{1}^{\text{gw}}+{s}_{1}^{\text{optical path}}+{\mathcal{D}}_{3}{p}_{2}^{\prime}-{p}_{1}+{\nu}_{0}\left[-2{\hat{n}}_{3}\cdot {\stackrel{\u20d7}{\delta}}_{1}+{\hat{n}}_{3}\cdot {\stackrel{\u20d7}{\Delta}}_{1}+{\hat{n}}_{3}\cdot {\mathcal{D}}_{3}{\stackrel{\u20d7}{\Delta}}_{2}^{\prime}\right],\end{array}$$ | (11) |
$$\begin{array}{ccc}{\tau}_{1}& =& {p}_{1}^{\prime}-{p}_{1}-2{\nu}_{0}{\hat{n}}_{2}\cdot \left({\stackrel{\u20d7}{\delta}}_{1}^{\prime}-{\stackrel{\u20d7}{\Delta}}_{1}^{\prime}\right)+{\mu}_{1}.\end{array}$$ | (12) |
$$\begin{array}{ccc}{s}_{1}^{\prime}& =& {s}_{1}^{{}^{\prime}\text{gw}}+{s}_{1}^{{}^{\prime}\text{optical path}}+{\mathcal{D}}_{2}{p}_{3}-{p}_{1}^{\prime}+{\nu}_{0}\left[2{\hat{n}}_{2}\cdot {\stackrel{\u20d7}{\delta}}_{1}^{\prime}-{\hat{n}}_{2}\cdot {\stackrel{\u20d7}{\Delta}}_{1}^{\prime}-{\hat{n}}_{2}\cdot {\mathcal{D}}_{2}{\stackrel{\u20d7}{\Delta}}_{3}\right],\end{array}$$ | (13) |
$$\begin{array}{ccc}{\tau}_{1}^{\prime}& =& {p}_{1}-{p}_{1}^{\prime}+2{\nu}_{0}{\hat{n}}_{3}\cdot \left({\stackrel{\u20d7}{\delta}}_{1}-{\stackrel{\u20d7}{\Delta}}_{1}\right)+{\mu}_{1}.\end{array}$$ | (14) |
4 Algebraic Approach to Cancelling Laser and Optical Bench Noises
^{ $\text{1}$ } A module is an Abelian group over a ring as contrasted with a vector space which is an Abelian group over a field. The scalars form a ring and just like in a vector space, scalar multiplication is defined. However, in a ring the multiplicative inverses do not exist in general for the elements, which makes all the difference!
4.1 Cancellation of laser phase noise
$$\begin{array}{c}\begin{array}{ccc}{s}_{1}& =& {\mathcal{D}}_{3}{p}_{2}-{p}_{1},\\ {s}_{1}^{\prime}& =& {\mathcal{D}}_{2}{p}_{3}-{p}_{1}\end{array}\end{array}$$ | (15) |
$$\begin{array}{c}{\sum}_{i=1}^{3}{q}_{i}{s}_{i}+{q}_{i}^{\prime}{s}_{i}^{\prime}=0.\end{array}$$ | (16) |
$$\begin{array}{c}\mathbf{s}={\mathbf{D}}^{T}\cdot \mathbf{p},{\mathbf{s}}^{\prime}=\mathbf{D}\cdot \mathbf{p},\end{array}$$ | (17) |
$$\begin{array}{c}\mathbf{D}=\left(\begin{array}{ccc}-1& 0& {\mathcal{D}}_{2}\\ {\mathcal{D}}_{3}& -1& 0\\ 0& {\mathcal{D}}_{1}& -1\end{array}\right).\end{array}$$ | (18) |
$$\begin{array}{c}{\mathbf{q}}^{T}\cdot \mathbf{s}+{{\mathbf{q}}^{\prime}}^{T}\cdot {\mathbf{s}}^{\prime}=({\mathbf{q}}^{T}\cdot {\mathbf{D}}^{T}+{{\mathbf{q}}^{\prime}}^{T}\cdot \mathbf{D})\cdot \mathbf{p}=0,\end{array}$$ | (19) |
$$\begin{array}{c}{\mathbf{q}}^{T}\cdot {\mathbf{D}}^{T}+{\mathbf{q}}^{\prime}\cdot \mathbf{D}=0.\end{array}$$ | (20) |
4.2 Cancellation of laser phase noise in the unequal-arm interferometer
$$\begin{array}{c}{\phi}_{1}\left(t\right)=\phi (t-2{L}_{1})-\phi \left(t\right)\equiv ({\mathcal{D}}_{1}^{2}-1)\phi \left(t\right).\end{array}$$ | (21) |
$$\begin{array}{c}{\phi}_{2}\left(t\right)=\phi (t-2{L}_{2})-\phi \left(t\right)\equiv ({\mathcal{D}}_{2}^{2}-1)\phi \left(t\right).\end{array}$$ | (22) |
$$\begin{array}{c}X\left(t\right)=\left[{\phi}_{2}\right(t-2{L}_{1})-{\phi}_{2}(t\left)\right]-\left[{\phi}_{1}\right(t-2{L}_{2})-{\phi}_{1}(t\left)\right],\end{array}$$ | (23) |
$$\begin{array}{ccc}X\left(t\right)& =& ({\mathcal{D}}_{1}^{2}-1){\phi}_{2}\left(t\right)-({\mathcal{D}}_{2}^{2}-1){\phi}_{1}\left(t\right)\end{array}$$ |
$$\begin{array}{ccc}& =& \left[\right({\mathcal{D}}_{1}^{2}-1\left)\right({\mathcal{D}}_{2}^{2}-1)-({\mathcal{D}}_{2}^{2}-1\left)\right({\mathcal{D}}_{1}^{2}-1\left)\right]\phi \left(t\right)\end{array}$$ |
$$\begin{array}{ccc}& =& 0.\end{array}$$ | (24) |
4.3 The module of syzygies
$$\begin{array}{c}\begin{array}{ccc}{q}_{1}+{q}_{1}^{\prime}-{\mathcal{D}}_{3}{q}_{2}^{\prime}-{\mathcal{D}}_{2}{q}_{3}& =& 0,\\ {q}_{2}+{q}_{2}^{\prime}-{\mathcal{D}}_{1}{q}_{3}^{\prime}-{\mathcal{D}}_{3}{q}_{1}& =& 0,\\ {q}_{3}+{q}_{3}^{\prime}-{\mathcal{D}}_{2}{q}_{1}^{\prime}-{\mathcal{D}}_{1}{q}_{2}& =& 0.\end{array}\end{array}$$ | (25) |
$$\begin{array}{c}\begin{array}{ccc}{q}_{1}& =& -{q}_{1}^{\prime}+{\mathcal{D}}_{3}{q}_{2}^{\prime}+{\mathcal{D}}_{2}{q}_{3},\\ {q}_{2}& =& -{q}_{2}^{\prime}+{\mathcal{D}}_{1}{q}_{3}^{\prime}+{\mathcal{D}}_{3}{q}_{1}\\ & =& -{\mathcal{D}}_{3}{q}_{1}^{\prime}-(1-{\mathcal{D}}_{3}^{2}){q}_{2}^{\prime}+{\mathcal{D}}_{1}{q}_{3}^{\prime}+{\mathcal{D}}_{2}{\mathcal{D}}_{3}{q}_{3},\end{array}\end{array}$$ | (26) |
$$\begin{array}{c}(1-{\mathcal{D}}_{1}{\mathcal{D}}_{2}{\mathcal{D}}_{3}){q}_{3}+({\mathcal{D}}_{1}{\mathcal{D}}_{3}-{\mathcal{D}}_{2}){q}_{1}^{\prime}+{\mathcal{D}}_{1}(1-{\mathcal{D}}_{3}^{2}){q}_{2}^{\prime}+(1-{\mathcal{D}}_{1}^{2}){q}_{3}^{\prime}=0.\end{array}$$ | (27) |
4.4 Gröbner basis
$$\begin{array}{c}{u}_{1}=1-{\mathcal{D}}_{1}{\mathcal{D}}_{2}{\mathcal{D}}_{3},{u}_{2}={\mathcal{D}}_{1}{\mathcal{D}}_{3}-{\mathcal{D}}_{2},{u}_{3}={\mathcal{D}}_{1}(1-{\mathcal{D}}_{3}^{2}),{u}_{4}=1-{\mathcal{D}}_{1}^{2}.\end{array}$$ | (28) |
$$\begin{array}{c}\mathcal{G}=\{{\mathcal{D}}_{3}^{2}-1,{\mathcal{D}}_{2}^{2}-1,{\mathcal{D}}_{1}-{\mathcal{D}}_{2}{\mathcal{D}}_{3}\}.\end{array}$$ | (29) |
4.5 Generating set for the module of syzygies
$$\begin{array}{c}\begin{array}{ccc}{X}^{\left(1\right)}& =& \left({\mathcal{D}}_{2}-{\mathcal{D}}_{1}{\mathcal{D}}_{3},0,1-{\mathcal{D}}_{3}^{2},0,{\mathcal{D}}_{2}{\mathcal{D}}_{3}-{\mathcal{D}}_{1},{\mathcal{D}}_{3}^{2}-1\right),\\ {X}^{\left(2\right)}& =& \left(-{\mathcal{D}}_{1},-{\mathcal{D}}_{2},-{\mathcal{D}}_{3},{\mathcal{D}}_{1},{\mathcal{D}}_{2},{\mathcal{D}}_{3}\right),\\ {X}^{\left(3\right)}& =& \left(-1,-{\mathcal{D}}_{3},-{\mathcal{D}}_{1}{\mathcal{D}}_{3},1,{\mathcal{D}}_{1}{\mathcal{D}}_{2},{\mathcal{D}}_{2}\right),\\ {X}^{\left(4\right)}& =& \left(-{\mathcal{D}}_{1}{\mathcal{D}}_{2},-1,-{\mathcal{D}}_{1},{\mathcal{D}}_{3},1,{\mathcal{D}}_{2}{\mathcal{D}}_{3}\right).\end{array}\end{array}$$ | (30) |
$$\begin{array}{c}\begin{array}{ccc}{G}^{\left(1\right)}& =& \left(-{\mathcal{D}}_{1},-{\mathcal{D}}_{2},-{\mathcal{D}}_{3},{\mathcal{D}}_{1},{\mathcal{D}}_{2},{\mathcal{D}}_{3}\right),\\ {G}^{\left(2\right)}& =& \left({\mathcal{D}}_{2}-{\mathcal{D}}_{1}{\mathcal{D}}_{3},0,1-{\mathcal{D}}_{3}^{2},0,{\mathcal{D}}_{2}{\mathcal{D}}_{3}-{\mathcal{D}}_{1},{\mathcal{D}}_{3}^{2}-1\right),\\ {G}^{\left(3\right)}& =& \left(-{\mathcal{D}}_{1}{\mathcal{D}}_{2},-1,-{\mathcal{D}}_{1},{\mathcal{D}}_{3},1,{\mathcal{D}}_{2}{\mathcal{D}}_{3}\right),\\ {G}^{\left(4\right)}& =& \left(-1,-{\mathcal{D}}_{3},-{\mathcal{D}}_{1}{\mathcal{D}}_{3},1,{\mathcal{D}}_{1}{\mathcal{D}}_{2},{\mathcal{D}}_{2}\right),\\ {G}^{\left(5\right)}& =& \left({\mathcal{D}}_{3}(1-{\mathcal{D}}_{1}^{2}),{\mathcal{D}}_{3}^{2}-1,0,0,1-{\mathcal{D}}_{1}^{2},{\mathcal{D}}_{1}({\mathcal{D}}_{3}^{2}-1)\right).\end{array}\end{array}$$ | (31) |
4.6 Canceling optical bench motion noise
$$\begin{array}{ccc}{s}_{1}& =& {\mathcal{D}}_{3}\left[{p}_{2}^{\prime}+{\nu}_{0}{\hat{\mathbf{n}}}_{3}\cdot {\stackrel{\u20d7}{\Delta}}_{2}^{\prime}\right]-\left[{p}_{1}-{\nu}_{0}{\hat{\mathbf{n}}}_{3}\cdot {\stackrel{\u20d7}{\Delta}}_{1}\right],\end{array}$$ | (32) |
$$\begin{array}{ccc}{s}_{1}^{\prime}& =& {\mathcal{D}}_{2}\left[{p}_{3}-{\nu}_{0}{\hat{\mathbf{n}}}_{2}\cdot {\stackrel{\u20d7}{\Delta}}_{3}\right]-\left[{p}_{1}^{\prime}+{\nu}_{0}{\hat{\mathbf{n}}}_{2}\cdot {\stackrel{\u20d7}{\Delta}}_{1}^{\prime}\right],\end{array}$$ | (33) |
$$\begin{array}{ccc}{\tau}_{1}& =& {p}_{1}^{\prime}-{p}_{1}+2{\nu}_{0}{\hat{\mathbf{n}}}_{2}\cdot {\stackrel{\u20d7}{\Delta}}_{1}^{\prime}+{\mu}_{1},\end{array}$$ | (34) |
$$\begin{array}{ccc}{\tau}_{1}^{\prime}& =& {p}_{1}-{p}_{1}^{\prime}-2{\nu}_{0}{\hat{\mathbf{n}}}_{3}\cdot {\stackrel{\u20d7}{\Delta}}_{1}+{\mu}_{1}.\end{array}$$ | (35) |
$$\begin{array}{c}{z}_{1}\equiv \frac{1}{2}({\tau}_{1}-{\tau}_{1}^{\prime})={\phi}_{1}^{\prime}-{\phi}_{1},\end{array}$$ | (36) |
$$\begin{array}{c}\begin{array}{ccc}{\phi}_{1}^{\prime}& \equiv & {p}_{1}^{\prime}+{\nu}_{0}{\hat{\mathbf{n}}}_{2}\cdot {\stackrel{\u20d7}{\Delta}}_{1}^{\prime},\\ {\phi}_{1}& \equiv & {p}_{1}-{\nu}_{0}{\hat{\mathbf{n}}}_{3}\cdot {\stackrel{\u20d7}{\Delta}}_{1},\end{array}\end{array}$$ | (37) |
$$\begin{array}{c}\begin{array}{ccc}{s}_{1}& =& {\mathcal{D}}_{3}{\phi}_{2}^{\prime}-{\phi}_{1},\\ {s}_{1}^{\prime}& =& {\mathcal{D}}_{2}{\phi}_{3}-{\phi}_{1}^{\prime}.\end{array}\end{array}$$ | (38) |
$$\begin{array}{ccc}{\eta}_{1}& \equiv & {s}_{1}-{\mathcal{D}}_{3}{z}_{2}={\mathcal{D}}_{3}{\phi}_{2}-{\phi}_{1},{\eta}_{{1}^{\prime}}\equiv {s}_{{1}^{\prime}}+{z}_{1}={\mathcal{D}}_{2}{\phi}_{3}-{\phi}_{1},\end{array}$$ | (39) |
$$\begin{array}{ccc}{\eta}_{2}& \equiv & {s}_{2}-{\mathcal{D}}_{1}{z}_{3}={\mathcal{D}}_{1}{\phi}_{3}-{\phi}_{2},{\eta}_{{2}^{\prime}}\equiv {s}_{{2}^{\prime}}+{z}_{2}={\mathcal{D}}_{3}{\phi}_{1}-{\phi}_{2},\end{array}$$ | (40) |
$$\begin{array}{ccc}{\eta}_{3}& \equiv & {s}_{3}-{\mathcal{D}}_{2}{z}_{1}={\mathcal{D}}_{2}{\phi}_{1}-{\phi}_{3},{\eta}_{{3}^{\prime}}\equiv {s}_{{3}^{\prime}}+{z}_{3}={\mathcal{D}}_{1}{\phi}_{2}-{\phi}_{3},\end{array}$$ | (41) |
4.7 Physical interpretation of the TDI combinations
$$\begin{array}{c}\alpha =[{\eta}_{{1}^{\prime}}+{\mathcal{D}}_{2}{\eta}_{{3}^{\prime}}+{\mathcal{D}}_{1}{\mathcal{D}}_{{2}^{\prime}}{\eta}_{{2}^{\prime}}]-[{\eta}_{1}+{\mathcal{D}}_{3}{\eta}_{2}+{\mathcal{D}}_{1}{\mathcal{D}}_{3}{\eta}_{2}],\end{array}$$ | (42) |
$$\begin{array}{c}\zeta -{\mathcal{D}}_{1}{\mathcal{D}}_{2}{\mathcal{D}}_{3}\zeta ={\mathcal{D}}_{1}\alpha -{\mathcal{D}}_{2}{\mathcal{D}}_{3}\alpha +{\mathcal{D}}_{2}\alpha -{\mathcal{D}}_{3}{\mathcal{D}}_{1}\beta +{\mathcal{D}}_{3}\gamma -{\mathcal{D}}_{1}{\mathcal{D}}_{2}\gamma .\end{array}$$ | (43) |
$$\begin{array}{c}\begin{array}{ccc}{\mathcal{D}}_{1}X& =& {\mathcal{D}}_{2}{\mathcal{D}}_{3}\alpha -{\mathcal{D}}_{2}\beta -{D}_{3}\gamma +\zeta ,\\ P& =& \zeta -{\mathcal{D}}_{1}\alpha ,\\ E& =& {\mathcal{D}}_{1}-{\mathcal{D}}_{1}\zeta ,\\ U& =& {\mathcal{D}}_{1}\gamma -\beta ,\end{array}\end{array}$$ | (44) |
$$\begin{array}{c}X=\left[({\eta}_{1}^{\prime}+{\mathcal{D}}_{{2}^{\prime}}{\eta}_{3})+{\mathcal{D}}_{{2}^{\prime}}{\mathcal{D}}_{2}({\eta}_{1}+{\mathcal{D}}_{3}{\eta}_{2}^{\prime})\right]-\left[({\eta}_{1}+{\mathcal{D}}_{3}{\eta}_{2}^{\prime})+{\mathcal{D}}_{3}{\mathcal{D}}_{{3}^{\prime}}({\eta}_{1}^{\prime}+{\mathcal{D}}_{{2}^{\prime}}{\eta}_{3})\right],\end{array}$$ | (45) |
5 Time-Delay Interferometry with Moving Spacecraft
$$\begin{array}{c}{\mathcal{D}}_{i}\Psi ={\Psi}_{,i}\equiv \Psi (t-{L}_{i}(t\left)\right).\end{array}$$ | (46) |
$$\begin{array}{ccc}{\mathcal{D}}_{j}{\mathcal{D}}_{i}\Psi ={\Psi}_{;ij}& \equiv & \Psi (t-{L}_{j}(t)-{L}_{i}(t-{L}_{j}\left(t\right)\left)\right)\end{array}$$ |
$$\begin{array}{ccc}& \simeq & \Psi (t-{L}_{j}(t)-{L}_{i}(t)+{\dot{L}}_{i}(t\left){L}_{j}\right)\end{array}$$ |
$$\begin{array}{ccc}& \simeq & {\Psi}_{,ij}+{\dot{\Psi}}_{,ij}{\dot{L}}_{i}{L}_{j}.\end{array}$$ | (47) |
$$\begin{array}{ccc}{\mathcal{D}}_{k}{\mathcal{D}}_{j}{\mathcal{D}}_{i}\Psi ={\Psi}_{;ijk}& =& \Psi (t-{L}_{k}(t)-{L}_{j}(t-{L}_{k}\left(t\right))-{L}_{i}(t-{L}_{k}\left(t\right)-{L}_{j}(t-{L}_{k}(t\left)\right)\left)\right)\end{array}$$ |
$$\begin{array}{ccc}& \simeq & {\Psi}_{,ijk}+{\dot{\Psi}}_{,ijk}\left[{\dot{L}}_{i}({L}_{j}+{L}_{k})+{\dot{L}}_{j}{L}_{k}\right],\end{array}$$ | (48) |
$$\begin{array}{c}{D}_{i}^{-1}\Psi \left(t\right)\equiv \Psi (t+{L}_{i}(t+{L}_{i}\left)\right).\end{array}$$ | (49) |
5.1 The unequal-arm Michelson
$$\begin{array}{ccc}{\eta}_{1}+{\eta}_{{2}^{\prime},3}& =& \left({\mathcal{D}}_{3}{\mathcal{D}}_{{3}^{\prime}}-I\right){\phi}_{1},\end{array}$$ | (50) |
$$\begin{array}{ccc}{\eta}_{{1}^{\prime}}+{\eta}_{3,{2}^{\prime}}& =& \left({\mathcal{D}}_{{2}^{\prime}}{\mathcal{D}}_{2}-I\right){\phi}_{1},\end{array}$$ | (51) |
$$\begin{array}{c}X=\left[{\mathcal{D}}_{{2}^{\prime}}{\mathcal{D}}_{2}-I\right]({\eta}_{1}+{\eta}_{{2}^{\prime},3})-\left[\left({\mathcal{D}}_{3}{\mathcal{D}}_{{3}^{\prime}}-I\right)\right]({\eta}_{{1}^{\prime}}+{\eta}_{3,{2}^{\prime}}).\end{array}$$ | (52) |
$$\begin{array}{ccc}\left[({\eta}_{{1}^{\prime}}+{\eta}_{3;{2}^{\prime}})+({\eta}_{1}+{\eta}_{2;3}{)}_{;{22}^{\prime}}\right]& =& \left[{D}_{{2}^{\prime}}{D}_{2}{D}_{3}{D}_{{3}^{\prime}}-I\right]{\phi}_{1},\end{array}$$ | (53) |
$$\begin{array}{ccc}\left[({\eta}_{1}+{\eta}_{{2}^{\prime};3})+({\eta}_{{1}^{\prime}}+{\eta}_{3;{2}^{\prime}}{)}_{;{3}^{\prime}3}\right]& =& \left[{D}_{3}{D}_{{3}^{\prime}}{D}_{{2}^{\prime}}{D}_{2}-I\right]{\phi}_{1}.\end{array}$$ | (54) |
$$\begin{array}{ccc}{X}_{1}& =& \left[{D}_{2}{D}_{{2}^{\prime}}{D}_{{3}^{\prime}}{D}_{3}-I\right]\left[({\eta}_{21}+{\eta}_{12;{3}^{\prime}})+({\eta}_{31}+{\eta}_{13;2}{)}_{;{33}^{\prime}}\right]\end{array}$$ |
$$\begin{array}{ccc}& & -\left[{D}_{{3}^{\prime}}{D}_{3}{D}_{2}{D}_{{2}^{\prime}}-I\right]\left[({\eta}_{31}+{\eta}_{13;2})+({\eta}_{21}+{\eta}_{12;{3}^{\prime}}{)}_{;{2}^{\prime}2}\right].\end{array}$$ | (55) |
5.2 The Sagnac combinations
$$\begin{array}{c}\alpha =[{\eta}_{{1}^{\prime}}+{\mathcal{D}}_{{2}^{\prime}}{\eta}_{{3}^{\prime}}+{\mathcal{D}}_{{1}^{\prime}}{\mathcal{D}}_{{2}^{\prime}}{\eta}_{{2}^{\prime}}]-[{\eta}_{1}+{\mathcal{D}}_{3}{\eta}_{2}+{\mathcal{D}}_{1}{\mathcal{D}}_{3}{\eta}_{2}],\end{array}$$ | (56) |
$$\begin{array}{ccc}[{\eta}_{{1}^{\prime}}+{\mathcal{D}}_{{2}^{\prime}}{\eta}_{{3}^{\prime}}+{\mathcal{D}}_{{1}^{\prime}}{\mathcal{D}}_{{2}^{\prime}}{\eta}_{{2}^{\prime}}]& =& [{\mathcal{D}}_{{2}^{\prime}}{\mathcal{D}}_{{1}^{\prime}}{\mathcal{D}}_{{3}^{\prime}}-I]{\phi}_{1}\end{array}$$ | (57) |
$$\begin{array}{ccc}[{\eta}_{1}+{\mathcal{D}}_{3}{\eta}_{2}+{\mathcal{D}}_{1}{\mathcal{D}}_{3}{\eta}_{2}]& =& [{\mathcal{D}}_{3}{\mathcal{D}}_{1}{\mathcal{D}}_{2}-I]{\phi}_{1}.\end{array}$$ | (58) |
$$\begin{array}{c}{\alpha}_{1}=[{\mathcal{D}}_{3}{\mathcal{D}}_{1}{\mathcal{D}}_{2}-I][{\eta}_{{1}^{\prime}}+{\mathcal{D}}_{{2}^{\prime}}{\eta}_{{3}^{\prime}}+{\mathcal{D}}_{{1}^{\prime}}{\mathcal{D}}_{{2}^{\prime}}{\eta}_{{2}^{\prime}}]-[{\mathcal{D}}_{{2}^{\prime}}{\mathcal{D}}_{{1}^{\prime}}{\mathcal{D}}_{{3}^{\prime}}-I][{\eta}_{1}+{\mathcal{D}}_{3}{\eta}_{2}+{\mathcal{D}}_{1}{\mathcal{D}}_{3}{\eta}_{2}].\end{array}$$ | (59) |
$$\begin{array}{c}{\dot{\phi}}_{1,{1231}^{\prime}{2}^{\prime}{3}^{\prime}}\left[\left({\dot{L}}_{1}+{\dot{L}}_{2}+{\dot{L}}_{3}\right)\left({L}_{1}^{{}^{\prime}}+{L}_{2}^{{}^{\prime}}+{L}_{3}^{{}^{\prime}}\right)-\left({\dot{L}}_{1}^{{}^{\prime}}+{\dot{L}}_{2}^{{}^{\prime}}+{\dot{L}}_{3}^{{}^{\prime}}\right)\left({L}_{1}+{L}_{2}+{L}_{3}\right)\right].\end{array}$$ | (60) |
$$\begin{array}{ccc}{\eta}_{3,3}-{\eta}_{{3}^{\prime},3}+{\eta}_{1,{1}^{\prime}}& =& \left[{D}_{3}{D}_{2}-{D}_{{1}^{\prime}}\right]{\phi}_{1},\end{array}$$ | (61) |
$$\begin{array}{ccc}{\eta}_{{1}^{\prime},1}-{\eta}_{2,{2}^{\prime}}+{\eta}_{{2}^{\prime},{2}^{\prime}}& =& \left[{D}_{{3}^{\prime}}{D}_{{2}^{\prime}}-{D}_{1}\right]{\phi}_{1},\end{array}$$ | (62) |
$$\begin{array}{c}{\zeta}_{1}=\left[{D}_{{3}^{\prime}}{D}_{{2}^{\prime}}-{D}_{1}\right]\left({\eta}_{31,{1}^{\prime}}-{\eta}_{32,2}+{\eta}_{12,2}\right)-\left[{D}_{2}{D}_{3}-{D}_{{1}^{\prime}}\right]\left({\eta}_{13,{3}^{\prime}}-{\eta}_{23,{3}^{\prime}}+{\eta}_{21,1}\right).\end{array}$$ | (63) |
6 Optimal LISA Sensitivity
$$\begin{array}{c}{\alpha}_{1}\left(t\right)\simeq \alpha \left(t\right)-\alpha (t-{L}_{1}-{L}_{2}-{L}_{3}),\end{array}$$ | (64) |
$$\begin{array}{c}\eta \left(f\right)\equiv {a}_{1}(f,\stackrel{\u20d7}{\lambda})\stackrel{~}{\alpha}\left(f\right)+{a}_{2}(f,\stackrel{\u20d7}{\lambda})\stackrel{~}{\beta}\left(f\right)+{a}_{3}(f,\stackrel{\u20d7}{\lambda})\stackrel{~}{\gamma}\left(f\right)+{a}_{4}(f,\stackrel{\u20d7}{\lambda})\stackrel{~}{\zeta}\left(f\right),\end{array}$$ | (65) |
$$\begin{array}{c}{\text{SNR}}_{\eta}^{2}={\int}_{{f}_{l}}^{{f}_{u}}\frac{{\left|{a}_{1}{\stackrel{~}{\alpha}}_{s}+{a}_{2}{\stackrel{~}{\beta}}_{s}+{a}_{3}{\stackrel{~}{\gamma}}_{s}+{a}_{4}{\stackrel{~}{\zeta}}_{s}\right|}^{2}}{\langle {\left|{a}_{1}{\stackrel{~}{\alpha}}_{n}+{a}_{2}{\stackrel{~}{\beta}}_{n}+{a}_{3}{\stackrel{~}{\gamma}}_{n}+{a}_{4}{\stackrel{~}{\zeta}}_{n}\right|}^{2}\rangle}df.\end{array}$$ | (66) |
$$\begin{array}{c}{\text{SNR}}_{\eta}^{2}={\int}_{{f}_{l}}^{{f}_{u}}\frac{{\left|{a}_{1}{\stackrel{~}{\alpha}}_{s}+{a}_{2}{\stackrel{~}{\beta}}_{s}+{a}_{3}{\stackrel{~}{\gamma}}_{s}\right|}^{2}}{\langle {\left|{a}_{1}{\stackrel{~}{\alpha}}_{n}+{a}_{2}{\stackrel{~}{\beta}}_{n}+{a}_{3}{\stackrel{~}{\gamma}}_{n}\right|}^{2}\rangle}df.\end{array}$$ | (67) |
$$\begin{array}{c}(\mathbf{C}{)}_{rt}\equiv \langle {\mathbf{x}}_{r}^{\left(n\right)}{\mathbf{x}}_{t}^{\left(n\right)*}\rangle .\end{array}$$ | (68) |
$$\begin{array}{c}{\text{SNR}}_{\eta}^{2}={\int}_{{f}_{l}}^{{f}_{u}}\frac{{\mathbf{a}}_{i}{\mathbf{A}}_{ij}{\mathbf{a}}_{j}^{*}}{{\mathbf{a}}_{r}{\mathbf{C}}_{rt}{\mathbf{a}}_{t}^{*}}df,\end{array}$$ | (69) |
$$\begin{array}{c}\frac{\partial}{\partial {\mathbf{a}}_{k}}\left[\frac{{\mathbf{a}}_{i}{\mathbf{A}}_{ij}{\mathbf{a}}_{j}^{*}}{{\mathbf{a}}_{r}{\mathbf{C}}_{rt}{\mathbf{a}}_{t}^{*}}\right]=0,k=1,2,3.\end{array}$$ | (70) |
$$\begin{array}{c}\left({\mathbf{C}}^{-1}{)}_{ir}\right(\mathbf{A}{)}_{rj}({\mathbf{a}}^{*}{)}_{j}=\left[\frac{{\mathbf{a}}_{p}{\mathbf{A}}_{pq}{\mathbf{a}}_{q}^{*}}{{\mathbf{a}}_{l}{\mathbf{C}}_{lm}{\mathbf{a}}_{m}^{*}}\right]({\mathbf{a}}^{*}{)}_{i},i=1,2,3,\end{array}$$ | (71) |
$$\begin{array}{c}{\text{SNR}}_{\eta \text{opt}}^{2}={\int}_{{f}_{l}}^{{f}_{u}}{\mathbf{x}}_{i}^{(s)*}({\mathbf{C}}^{-1}{)}_{ij}{\mathbf{x}}_{j}^{(s)}df.\end{array}$$ | (72) |
6.1 General application
$$\begin{array}{c}\mathbf{C}=\left(\begin{array}{ccccc}{S}_{\alpha}& & {S}_{\alpha \beta}& & {S}_{\alpha \beta}\\ {S}_{\alpha \beta}& & {S}_{\alpha}& & {S}_{\alpha \beta}\\ {S}_{\alpha \beta}& & {S}_{\alpha \beta}& & {S}_{\alpha}\end{array}\right).\end{array}$$ | (73) |
$$\begin{array}{c}{\mu}_{1}={\mu}_{2}={S}_{\alpha}-{S}_{\alpha \beta},{\mu}_{3}={S}_{\alpha}+2{S}_{\alpha \beta}.\end{array}$$ | (74) |
$$\begin{array}{c}{\mathbf{v}}_{1}=\frac{1}{\sqrt{2}}(-1,0,1){\mathbf{v}}_{2}=\frac{1}{\sqrt{6}}(1,-2,1){\mathbf{v}}_{3}=\frac{1}{\sqrt{3}}(1,1,1).\end{array}$$ | (75) |
$$\begin{array}{c}A=\frac{1}{\sqrt{2}}\left(\stackrel{~}{\gamma}-\stackrel{~}{\alpha}\right)E=\frac{1}{\sqrt{6}}\left(\stackrel{~}{\alpha}-2\stackrel{~}{\beta}+\stackrel{~}{\gamma}\right)T=\frac{1}{\sqrt{3}}\left(\stackrel{~}{\alpha}+\stackrel{~}{\beta}+\stackrel{~}{\gamma}\right),\end{array}$$ | (76) |
$$\begin{array}{ccc}{S}_{A}\left(f\right)={S}_{E}\left(f\right)& =& 16{sin}^{2}\left(\pi fL\right)[3+2cos(2\pi fL)+cos(4\pi fL\left)\right]{S}_{y}^{\text{proof mass}}\left(f\right)\end{array}$$ |
$$\begin{array}{ccc}& +& 8{sin}^{2}\left(\pi fL\right)[2+cos(2\pi fL\left)\right]{S}_{y}^{\text{optical path}}\left(f\right),\end{array}$$ | (77) |
$$\begin{array}{ccc}{S}_{T}\left(f\right)& =& 2[1+2cos(2\pi fL){]}^{2}\left[4{sin}^{2}\left(\pi fL\right){S}_{y}^{\text{proof mass}}+{S}_{y}^{\text{optical path}}\left(f\right)\right].\end{array}$$ | (78) |
6.2 Optimization of SNR for binaries with known direction but with unknown orientation of the orbital plane
$$\begin{array}{c}{\text{SNR}}_{\text{network}}^{2}={\text{SNR}}_{+}^{2}+{\text{SNR}}_{\times}^{2}.\end{array}$$ | (79) |
$$\begin{array}{c}{\kappa}_{a}\left(f\right)=\frac{{\text{SNR}}_{a}\left(f\right)}{{\text{SNR}}_{max[X,Y,Z]}\left(f\right)},\end{array}$$ | (80) |
7 Concluding Remarks
8 Acknowledgement
A Generators of the Module of Syzygies
$$\begin{array}{c}(1-xyz){q}_{3}+(xz-y){q}_{1}^{\prime}+x(1-{z}^{2}){q}_{2}^{\prime}+(1-{x}^{2}){q}_{3}^{\prime}=0,\end{array}$$ | (81) |
$$\begin{array}{c}\mathcal{G}=\{{g}_{1}={z}^{2}-1,{g}_{2}={y}^{2}-1,{g}_{3}=x-yz\}.\end{array}$$ | (82) |
$$\begin{array}{c}{f}_{i}={d}_{ij}{g}_{j},{g}_{j}={c}_{ji}{f}_{i},\end{array}$$ | (83) |
$$\begin{array}{c}d=\left(\begin{array}{ccc}-1& -{z}^{2}& -yz\\ y& 0& z\\ -x& 0& 0\\ -1& -{z}^{2}& -(x+yz)\end{array}\right),c=\left(\begin{array}{cccc}0& 0& -x& {z}^{2}-1\\ -1& -y& 0& 0\\ 0& z& 1& 0\end{array}\right).\end{array}$$ | (84) |
$$\begin{array}{c}\begin{array}{ccc}{a}_{1}& =& \left({z}^{2}-1,0,x-yz,1-{z}^{2}\right),\\ {a}_{2}& =& \left(0,z\left(1-{z}^{2}\right),xy-z,y\left(1-{z}^{2}\right)\right),\\ {a}_{3}& =& \left(0,0,1-{x}^{2},x\left({z}^{2}-1\right)\right),\\ {a}_{4}& =& \left(-{z}^{2},xz,yz,{z}^{2}\right).\end{array}\end{array}$$ | (85) |
$$\begin{array}{c}{p}_{12}={y}^{2}{g}_{1}-{z}^{2}{g}_{2}={z}^{2}-{y}^{2}.\end{array}$$ | (86) |
$$\begin{array}{c}\begin{array}{ccc}{b}_{1}^{*}& =& \left({z}^{2}-1,y\left({z}^{2}-1\right),x\left(1-{y}^{2}\right),\left({y}^{2}-1\right)\left({z}^{2}-1\right)\right),\\ {b}_{2}^{*}& =& \left(0,z\left(1-{z}^{2}\right),1-{z}^{2}-x\left(x-yz\right),\left(x-yz\right)\left({z}^{2}-1\right)\right),\\ {b}_{3}^{*}& =& \left(-x+yz,z-xy,1-{y}^{2},0\right).\end{array}\end{array}$$ | (87) |
B Conversion between Generating Sets
$$\begin{array}{c}\begin{array}{ccc}\alpha & =& (-1,-z,-xz,1,xy,y),\\ \beta & =& (-xy,-1,-x,z,1,yz),\\ \gamma & =& (-y,-yz,-1,xz,x,1),\\ \zeta & =& (-x,-y,-z,x,y,z).\end{array}\end{array}$$ | (88) |
$$\begin{array}{c}\begin{array}{ccc}{a}_{1}& =& \gamma -z\zeta ,\\ {a}_{2}& =& \alpha -z\beta ,\\ {a}_{3}& =& -z\alpha +\beta -x\gamma +xz\zeta ,\\ {a}_{4}& =& z\zeta ,\\ {b}_{1}^{*}& =& -y\alpha +yz\beta +\gamma -z\zeta ,\\ {b}_{2}^{*}& =& (1-{z}^{2})\beta -x\gamma +xz\zeta ,\\ {b}_{3}^{*}& =& \beta -y\zeta .\end{array}\end{array}$$ | (89) |
$$\begin{array}{c}\begin{array}{ccc}\alpha & =& {X}^{\left(3\right)},\\ \beta & =& {X}^{\left(4\right)},\\ \gamma & =& -{X}^{\left(1\right)}+z{X}^{\left(2\right)},\\ \zeta & =& {X}^{\left(2\right)}.\end{array}\end{array}$$ | (90) |
Note: The reference version of this article is published by Living Reviews in Relativity