Optimal LISA Sensitivity

6 Optimal LISA Sensitivity

All the above interferometric combinations have been shown to individually have rather different sensitivities [15 *], as a consequence of their different responses to gravitational radiation and system noises. Since LISA has the capability of simultaneously observing a gravitational-wave signal with many different interferometric combinations (all having different antenna patterns and noises), we should no longer regard LISA as a single detector system but rather as an array of gravitational-wave detectors working in coincidence. This suggests that the LISA sensitivity could be improved by optimally combining elements of the TDI space.

Before proceeding with this idea, however, let us consider again the so-called “second-generation” TDI Sagnac observables: $(α ,α ,α ) 1 2 3$ . The expressions of the gravitational-wave signal and the secondary noise sources entering into $α1$ will in general be different from those entering into $α$ , the corresponding Sagnac observable derived under the assumption of a stationary LISA array [2 *, 15 *]. However, the other remaining, secondary noises in LISA are so much smaller, and the rotation and systematic velocities in LISA are so intrinsically small, that index permutation may still be done for them [58 *]. It is therefore easy to derive the following relationship between the signal and secondary noises in $α1$ , and those entering into the stationary TDI combination $α$ [45 , 58 ],

where $Li$ , $i = 1,2,3$ , are the unequal-arm lengths of the stationary LISA array. Equation (81 *) implies that any data analysis procedure and algorithm that will be implemented for the second-generation TDI combinations can actually be derived by considering the corresponding “first-generation” TDI combinations. For this reason, from now on we will focus our attention on the gravitational-wave responses of the first-generation TDI observables $(α, β,γ,ζ )$ .

As a consequence of these considerations, we can still regard $(α, β,γ,ζ)$ as the generators of the TDI space, and write the most general expression for an element of the TDI space, $η(f)$ , as a linear combination of the Fourier transforms of the four generators $(^α,β^, ^γ, ^ζ)$ ,

where the ${ai(f,⃗λ)}4 i=1$ are arbitrary complex functions of the Fourier frequency $f$ , and of a vector $⃗λ$ containing parameters characterizing the gravitational-wave signal (source location in the sky, waveform parameters, etc.) and the noises affecting the four responses (noise levels, their correlations, etc.). For a given choice of the four functions ${ai}4i=1$ , $η$ gives an element of the functional space of interferometric combinations generated by $(α,β, γ,ζ)$ . Our goal is therefore to identify, for a given gravitational-wave signal, the four functions ${ai}4 i=1$ that maximize the signal-to-noise ratio $SNR2 η$ of the combination $η$ ,

| |2 ∫ fu ||a1 ^αs + a2 ^βs + a3^γs + a4^ζs|| SNR2 η = ⟨-|--------------------------|-⟩-df. (83 ) fl | ^ ^|2 |a1 ^αn + a2 βn + a3^γn + a4ζn|

In Eq. (83 *) the subscripts s and n refer to the signal and the noise parts of $(^α, ^β,^γ, ^ζ)$ , respectively, the angle brackets represent noise ensemble averages, and the interval of integration $(fl,fu)$ corresponds to the frequency band accessible by LISA.

Before proceeding with the maximization of the $SNR2 η$ we may notice from Eq. (44 *) that the Fourier transform of the totally symmetric Sagnac combination, $^ζ$ , multiplied by the transfer function $1 − e2πif(L1+L2+L3)$ can be written as a linear combination of the Fourier transforms of the remaining three generators $(^α,β^, ^γ)$ . Since the signal-to-noise ratio of $η$ and $(1 − e2πif(L1+L2+L3))η$ are equal, we may conclude that the optimization of the signal-to-noise ratio of $η$ can be performed only on the three observables $α,β, γ$ . This implies the following redefined expression for $SNR2 η$ :

|| ||2 ∫ fu |a1 ^αs + a2β^s + a3 ^γs| SNR2η = ⟨|--------------------|2⟩--df. (84 ) fl ||a1 ^αn + a2 ^βn + a3 ^γn||

The $2 SNR η$ can be regarded as a functional over the space of the three complex functions $3 {ai} i=1$ , and the particular set of complex functions that extremize it can of course be derived by solving the associated set of Euler–Lagrange equations.

In order to make the derivation of the optimal SNR easier, let us first denote by $x(s)$ and $x (n)$ the two vectors of the signals $^ (α^s, βs,^γs)$ and the noises $^ (^αn,βn,^γn )$ , respectively. Let us also define $a$ to be the vector of the three functions $3 {ai}i=1$ , and denote with $C$ the Hermitian, non-singular, correlation matrix of the vector random process $xn$ ,

If we finally define $(A )ij$ to be the components of the Hermitian matrix $x (si)x(js)∗$ , we can rewrite $2 SNR η$ in the following form,

where we have adopted the usual convention of summation over repeated indices. Since the noise correlation matrix $C$ is non-singular, and the integrand is positive definite or null, the stationary values of the signal-to-noise ratio will be attained at the stationary values of the integrand, which are given by solving the following set of equations (and their complex conjugated expressions):

After taking the partial derivatives, Eq. (87 *) can be rewritten in the following form,

which tells us that the stationary values of the signal-to-noise ratio of $η$ are equal to the eigenvalues of the the matrix $C −1 ⋅ A$ . The result in Eq. (87 *) is well known in the theory of quadratic forms, and it is called Rayleigh’s principle [36 , 42 ].

In order now to identify the eigenvalues of the matrix $− 1 C ⋅ A$ , we first notice that the $3 × 3$ matrix $A$ has rank 1. This implies that the matrix $C −1 ⋅ A$ has also rank 1, as it is easy to verify. Therefore two of its three eigenvalues are equal to zero, while the remaining non-zero eigenvalue represents the solution we are looking for.

The analytic expression of the third eigenvalue can be obtained by using the property that the trace of the $3 × 3$ matrix $−1 C ⋅ A$ is equal to the sum of its three eigenvalues, and in our case to the eigenvalue we are looking for. From these considerations we derive the following expression for the optimized signal-to-noise ratio $SNR2 ηopt$ :

We can summarize the results derived in this section, which are given by Eqs. (84 *) and (89 *), in the following way:

Among all possible interferometric combinations LISA will be able to synthesize with its four generators $α$ , $β$ , $γ$ , $ζ$ , the particular combination giving maximum signal-to-noise ratio can be obtained by using only three of them, namely $(α, β,γ)$ .
The expression of the optimal signal-to-noise ratio given by Eq. (89 *) implies that LISA should be regarded as a network of three interferometer detectors of gravitational radiation (of responses $(α,β,γ )$ ) working in coincidence [20 , 40 *].

6.1 General application

As an application of Eq. (89 *), here we calculate the sensitivity that LISA can reach when observing sinusoidal signals uniformly distributed on the celestial sphere and of random polarization. In order to calculate the optimal signal-to-noise ratio we will also need to use a specific expression for the noise correlation matrix $C$ . As a simplification, we will assume the LISA arm lengths to be equal to their nominal value $L = 16.67 s$ , the optical-path noises to be equal and uncorrelated to each other, and finally the noises due to the proof-mass noises to be also equal, uncorrelated to each other and to the optical-path noises. Under these assumptions the correlation matrix becomes real, its three diagonal elements are equal, and all the off-diagonal terms are equal to each other, as it is easy to verify by direct calculation [15 *]. The noise correlation matrix $C$ is therefore uniquely identified by two real functions $Sα$ and $Sαβ$ in the following way:

( ) S α S αβ Sαβ C = ( S αβ S α Sαβ ) . (90 ) S αβ S αβ Sα

The expression of the optimal signal-to-noise ratio assumes a rather simple form if we diagonalize this correlation matrix by properly “choosing a new basis”. There exists an orthogonal transformation of the generators $(^α, ^β,^γ )$ , which will transform the optimal signal-to-noise ratio into the sum of the signal-to-noise ratios of the “transformed” three interferometric combinations. The expressions of the three eigenvalues ${μi}3i=1$ (which are real) of the noise correlation matrix $C$ can easily be found by using the algebraic manipulator Mathematica, and they are equal to

Note that two of the three real eigenvalues, ( $μ1$ , $μ2$ ), are equal. This implies that the eigenvector associated to $μ3$ is orthogonal to the two-dimensional space generated by the eigenvalue $μ1$ , while any chosen pair of eigenvectors corresponding to $μ 1$ will not necessarily be orthogonal. This inconvenience can be avoided by choosing an arbitrary set of vectors in this two-dimensional space, and by ortho-normalizing them. After some simple algebra, we have derived the following three ortho-normalized eigenvectors:

Equation (92 *) implies the following three linear combinations of the generators $^ (^α, β,^γ)$ :

where $A$ , $E$ , and $T$ are italicized to indicate that these are “orthogonal modes”. Although the expressions for the modes $A$ and $E$ depend on our particular choice for the two eigenvectors ( $v1,v2$ ), it is clear from our earlier considerations that the value of the optimal signal-to-noise ratio is unaffected by such a choice. From Eq. (93 *) it is also easy to verify that the noise correlation matrix of these three combinations is diagonal, and that its non-zero elements are indeed equal to the eigenvalues given in Eq. (91 *).

In order to calculate the sensitivity corresponding to the expression of the optimal signal-to-noise ratio, we have proceeded similarly to what was done in [2 *, 15 *], and described in more detail in [56 *]. We assume an equal-arm LISA ( $L = 16.67 s$ ), and take the one-sided spectra of proof mass and aggregate optical-path-noises (on a single link), expressed as fractional frequency fluctuation spectra, to be $S pyroofmass= 2.5 × 10 −48[f∕1 Hz ]−2 Hz− 1$ and $Syopticalpath = 1.8 × 10−37[f ∕1 Hz]2 Hz− 1$ , respectively (see [15 *, 5 ]). We also assume that aggregate optical path noise has the same transfer function as shot noise.

The optimum SNR is the square root of the sum of the squares of the SNRs of the three “orthogonal modes” $(A, E, T)$ . To compare with previous sensitivity curves of a single LISA Michelson interferometer, we construct the SNRs as a function of Fourier frequency for sinusoidal waves from sources uniformly distributed on the celestial sphere. To produce the SNR of each of the $(A, E, T )$ modes we need the gravitational-wave response and the noise response as a function of Fourier frequency. We build up the gravitational-wave responses of the three modes $(A,E, T )$ from the gravitational-wave responses of $(α, β,γ)$ . For 7000 Fourier frequencies in the $∼ 10−4 Hz$ to $∼ 1 Hz$ LISA band, we produce the Fourier transforms of the gravitational-wave response of $(α,β,γ )$ from the formulas in [2 *, 56 ]. The averaging over source directions (uniformly distributed on the celestial sphere) and polarization states (uniformly distributed on the Poincaré sphere) is performed via a Monte Carlo method. From the Fourier transforms of the $(α, β,γ )$ responses at each frequency, we construct the Fourier transforms of $(A, E,T )$ . We then square and average to compute the mean-squared responses of $(A, E, T)$ at that frequency from $4 10$ realizations of (source position, polarization state) pairs.

We adopt the following terminology: We refer to a single element of the module as a data combination, while a function of the elements of the module, such as taking the maximum over several data combinations in the module or squaring and adding data combinations belonging to the module, is called as an observable. The important point to note is that the laser frequency noise is also suppressed for the observable although it may not be an element of the module.

The noise spectra of $(A, E,T )$ are determined from the raw spectra of proof-mass and optical-path noises, and the transfer functions of these noises to $(A,E, T )$ . Using the transfer functions given in [15 *], the resulting spectra are equal to

S (f) = S (f) = 16 sin2(πf L) [3 + 2 cos(2 πfL ) + cos(4πf L )]Sproof mass(f ) A E y +8 sin2(πfL )[2 + cos(2πfL )]Syopticalpath(f), (94 ) 2[ 2 proofmass opticalpath ] ST (f) = 2[1 + 2cos(2πf L)] 4 sin (πf L) Sy + Sy (f) . (95 )

Let the amplitude of the sinusoidal gravitational wave be $h$ . The SNR for, e.g., $A$ , $SNRA$ , at each frequency $f$ is equal to $h$ times the ratio of the root-mean-squared gravitational-wave response at that frequency divided by $∘ -------- SA(f) B$ , where $B$ is the bandwidth conventionally taken to be equal to 1 cycle per year. Finally, if we take the reciprocal of $SNRA ∕h$ and multiply it by 5 to get the conventional $SNR = 5$ sensitivity criterion, we obtain the sensitivity curve for this combination, which can then be compared against the corresponding sensitivity curve for the equal-arm Michelson interferometer.

In Figure 7 * we show the sensitivity curve for the LISA equal-arm Michelson response ( $SNR = 5$ ) as a function of the Fourier frequency, and the sensitivity curve from the optimum weighting of the data described above: $∘ ------------------------ 5h ∕ SNR2A + SNR2E + SNR2T$ . The SNRs were computed for a bandwidth of 1 cycle/year. Note that at frequencies where the LISA Michelson combination has best sensitivity, the improvement in signal-to-noise ratio provided by the optimal observable is slightly larger than $√ -- 2$ .

Figure 7: The LISA Michelson sensitivity curve (SNR = 5) and the sensitivity curve for the optimal combination of the data, both as a function of Fourier frequency. The integration time is equal to one year, and LISA is assumed to have a nominal armlength $L$ = 16.67 s.

Figure 8: The optimal SNR divided by the SNR of a single Michelson interferometer, as a function of the Fourier frequency $f$ . The sensitivity gain in the low-frequency band is equal to $√ -- 2$ , while it can get larger than 2 at selected frequencies in the high-frequency region of the accessible band. The integration time has been assumed to be one year, and the proof mass and optical path noise spectra are the nominal ones. See the main body of the paper for a quantitative discussion of this point.

Figure 9: The SNRs of the three combinations $(A, E, T)$ and their sum as a function of the Fourier frequency $f$ . The SNRs of $A$ and $E$ are equal over the entire frequency band. The SNR of $T$ is significantly smaller than the other two in the low part of the frequency band, while is comparable to (and at times larger than) the SNR of the other two in the high-frequency region. See text for a complete discussion.

In Figure 8 * we plot the ratio between the optimal SNR and the SNR of a single Michelson interferometer. In the long-wavelength limit, the SNR improvement is $√ -- 2$ . For Fourier frequencies greater than or about equal to $1∕L$ , the SNR improvement is larger and varies with the frequency, showing an average value of about $√3--$ . In particular, for bands of frequencies centered on integer multiples of $1∕L$ , $SNRT$ contributes strongly and the aggregate SNR in these bands can be greater than 2.

In order to better understand the contribution from the three different combinations to the optimal combination of the three generators, in Figure 9 * we plot the signal-to-noise ratios of $(A, E, T)$ as well as the optimal signal-to-noise ratio. For an assumed $h = 10− 23$ , the SNRs of the three modes are plotted versus frequency. For the equal-arm case computed here, the SNRs of $A$ and $E$ are equal across the band. In the long wavelength region of the band, modes $A$ and $E$ have SNRs much greater than mode $T$ , where its contribution to the total SNR is negligible. At higher frequencies, however, the $T$ combination has SNR greater than or comparable to the other modes and can dominate the SNR improvement at selected frequencies. Some of these results have also been obtained in [40 *].

6.2 Optimization of SNR for binaries with known direction but with unknown orientation of the orbital plane

Binaries are important sources for LISA and therefore the analysis of such sources is of major importance. One such class is of massive or super-massive binaries whose individual masses could range from $103M ⊙$ to $108M ⊙$ and which could be up to a few Gpc away. Another class of interest are known binaries within our own galaxy whose individual masses are of the order of a solar mass but are just at a distance of a few kpc or less. Here the focus will be on this latter class of binaries. It is assumed that the direction of the source is known, which is so for known binaries in our galaxy. However, even for such binaries, the inclination angle of the plane of the orbit of the binary is either poorly estimated or unknown. The optimization problem is now posed differently: The SNR is optimized after averaging over the polarizations of the binary signals, so the results obtained are optimal on the average, that is, the source is tracked with an observable which is optimal on the average [40 *]. For computing the average, a uniform distribution for the direction of the orbital angular momentum of the binary is assumed.

When the binary masses are of the order of a solar mass and the signal typically has a frequency of a few mHz, the GW frequency of the binary may be taken to be constant over the period of observation, which is typically taken to be of the order of an year. A complete calculation of the signal matrix and the optimization procedure of SNR is given in [39 *]. Here we briefly mention the main points and the final results.

A source fixed in the Solar System Barycentric reference frame in the direction $(𝜃B,ϕB )$ is considered. But as the LISA constellation moves along its heliocentric orbit, the apparent direction $(𝜃L,ϕL )$ of the source in the LISA reference frame $(xL,yL,zL)$ changes with time. The LISA reference frame $(xL,yL, zL)$ has been defined in [39 *] as follows: The origin lies at the center of the LISA triangle and the plane of LISA coincides with the $(xL,yL)$ plane with spacecraft 2 lying on the $xL$ axis. Figure (10 *) displays this apparent motion for a source lying in the ecliptic plane, that is with $𝜃B = 90∘$ and $ϕB = 0∘$ . The source in the LISA reference frame describes a figure of 8. Optimizing the SNR amounts to tracking the source with an optimal observable as the source apparently moves in the LISA reference frame.

Figure 10: Apparent position of the source in the sky as seen from LISA frame for $∘ ∘ (𝜃B = 90 ,ϕB = 0 )$ . The track of the source for a period of one year is shown on the unit sphere in the LISA reference frame.

Since an average has been taken over the orientation of the orbital plane of the binary or equivalently over the polarizations, the signal matrix $A$ is now of rank 2 instead of rank 1 as compared with the application in the previous Section 6.1. The mutually orthogonal data combinations $A$ , $E$ , $T$ are convenient in carrying out the computations because in this case as well, they simultaneously diagonalize the signal and the noise covariance matrix. The optimization problem now reduces to an eigenvalue problem with the eigenvalues being the squares of the SNRs. There are two eigen-vectors which are labeled as $⃗v+,×$ belonging to two non-zero eigenvalues. The two SNRs are labelled as $SNR+$ and $SNR ×$ , corresponding to the two orthogonal (thus statistically independent) eigenvectors $⃗v +,×$ . As was done in the previous Section 6.1 F the two SNRs can be squared and added to yield a network SNR, which is defined through the equation

The corresponding observable is called the network observable. The third eigenvalue is zero and the corresponding eigenvector orthogonal to $⃗v+$ and $⃗v×$ gives zero signal.

The eigenvectors and the SNRs are functions of the apparent source direction parameters $(𝜃L,ϕL)$ in the LISA reference frame, which in turn are functions of time. The eigenvectors optimally track the source as it moves in the LISA reference frame. Assuming an observation period of an year, the SNRs are integrated over this period of time. The sensitivities are computed according to the procedure described in the previous Section 6.1. The results of these findings are displayed in Figure 11 *.

It shows the sensitivity curves of the following observables:

The Michelson combination $X$ (faint solid curve).
The observable obtained by taking the maximum sensitivity among $X$ , $Y$ , and $Z$ for each direction, where $Y$ and $Z$ are the Michelson observables corresponding to the remaining two pairs of arms of LISA [2 ]. This maximum is denoted by $max [X, Y, Z]$ (dash-dotted curve) and is operationally given by switching the combinations $X$ , $Y$ , $Z$ so that the best sensitivity is achieved.
The eigen-combination $⃗v+$ which has the best sensitivity among all data combinations (dashed curve).
The network observable (solid curve).

It is observed that the sensitivity over the band-width of LISA increases as one goes from Observable 1 to 4. Also it is seen that the $max [X, Y, Z]$ does not do much better than $X$ . This is because for the source direction chosen $𝜃 = 90∘ B$ , $X$ is reasonably well oriented and switching to $Y$ and $Z$ combinations does not improve the sensitivity significantly. However, the network and $⃗v+$ observables show significant improvement in sensitivity over both $X$ and $max [X, Y,Z ]$ . This is the typical behavior and the sensitivity curves (except $X$ ) do not show much variations for other source directions and the plots are similar. Also it may be fair to compare the optimal sensitivities with $max [X,Y, Z ]$ rather than $X$ . This comparison of sensitivities is shown in Figure 12 *, where the network and the eigen-combinations $⃗v+,×$ are compared with $max [X, Y,Z ]$ .

Figure 11: Sensitivity curves for the observables: Michelson, $max [X, Y,Z ]$ , $⃗v +$ , and network for the source direction ( $∘ 𝜃B = 90$ , $∘ ϕB = 0$ ).

Figure 12: Ratios of the sensitivities of the observables network, $⃗v+,×$ with $max [X, Y,Z ]$ for the source direction $𝜃B = 90∘$ , $ϕB = 0∘$ .

Defining

where the subscript $a$ stands for network or $+$ , $×$ , and $SNRmax [X,Y,Z]$ is the SNR of the observable $max [X, Y,Z ]$ , the ratios of sensitivities are plotted over the LISA band-width. The improvement in sensitivity for the network observable is about 34% at low frequencies and rises to nearly 90% at about 20 mHz, while at the same time the $⃗v+$ combination shows improvement of 12% at low frequencies rising to over 50% at about 20 mHz.

	Abstract
1	Introduction
2	Physical and Historical Motivations of TDI
3	Time-Delay Interferometry
4	Algebraic Approach for Canceling Laser and Optical Bench Noises
	4.1	Cancellation of laser phase noise
	4.2	Cancellation of laser phase noise in the unequal-arm interferometer
	4.3	The module of syzygies
	4.4	Gröbner basis
	4.5	Generating set for the module of syzygies
	4.6	Canceling optical bench motion noise
	4.7	Physical interpretation of the TDI combinations
5	Time-Delay Interferometry with Moving Spacecraft
	5.1	The unequal-arm Michelson
	5.2	The Sagnac combinations
	5.3	Algebraic approach to second-generation TDI
	5.4	Solutions with one arm nonfunctional
6	Optimal LISA Sensitivity
	6.1	General application
	6.2	Optimization of SNR for binaries with known direction but with unknown orientation of the orbital plane
7	Experimental Aspects of TDI
	7.1	Time-delays accuracies
	7.2	Clocks synchronization
	7.3	Clocks timing jitter
	7.4	Sampling reconstruction algorithm
	7.5	Data digitization and bit-accuracy requirement
8	Concluding Remarks
	Acknowledgement
A	Generators of the Module of Syzygies
B	Conversion between Generating Sets
	References
	Footnotes
	Updates
	Figures