A statistical inference approach to time-delay interferometry for gravitational-wavedetection

Quentin Baghi,∗ James Ira Thorpe, Jacob Slutsky, and John BakerGoddard Space Flight Center

Mail Code 6638800 Greenbelt Rd, Greenbelt, Maryland 20771, USA

(Dated: October 15, 2020)

The future space-based gravitational wave observatory LISA will consist of a constellation ofthree spacecraft in a triangular constellation, connected by laser interferometers with 2.5 million-kilometer arms. Among other challenges, the success of the mission strongly depends on the qualityof the cancellation of laser frequency noise, whose power lies eight orders of magnitude above thegravitational signal. The standard technique to perform noise removal is time-delay interferometry(TDI). TDI constructs linear combinations of delayed phasemeter measurements tailored to cancellaser noise terms. Previous work has demonstrated the relationship between TDI and principalcomponent analysis (PCA). We build on this idea to develop an extension of TDI based on a modellikelihood that directly depends on the phasemeter measurements. Assuming stationary Gaussiannoise, we decompose the measurement covariance using PCA in the frequency domain. We obtain acomprehensive and compact framework that we call PCI for “principal component interferometry,”and show that it provides an optimal description of the LISA data analysis problem.

I. INTRODUCTION

Only five years since the historic first detection ofan astrophysical signal, the field of gravitational-waveastronomy continues to rapidly advance through up-grades to ground-based facilities [1], plans for ambitiousfollow-on terrestrial detectors [2], and a steady increasein sensitivity of pulsar timing arrays [3]. Adding tothis in the 2030s will be the space mission known asthe Laser Interferometer Space Antenna (LISA) [4, 5].LISA will open the source-rich frequency window be-tween 0.1 mHz and 1 Hz, enabling the detection of thou-sands of gravitational-wave emitting systems of variedorigin and at distances ranging from our galactic neigh-borhood (kpc) to cosmological redshift (z = 15 and be-yond). The types of sources will include from compactbinary stars, the capture of stellar-remnant black holes bymassive black holes, to mergers of (super)massive blackhole binaries. LISA will form a triangular constellationof 3 satellites with trailing Earth on heliocentric orbits.Each satellite will house inertial test-masses whose tra-jectories will be monitored through a network of interfer-ometric laser links, connecting the spacecrafts separatedfrom each other by 2.5 million kilometers. Incoming grav-itational waves (GW) will affect the space-time acrossthe constellation, introducing a characteristic shift in thelight travel time along the six one-way optical links.

Due to the large distance between the inertial refer-ences, LISA needs 9 interferometric measurements to op-erate optimally [6]. In this setup, the phasemeter mea-surements are sensitive to noise fluctuations of the laserfrequencies at a level of 10−13 Hz−1/2. The typical metricperturbations induced by the target GW sources being

∗ quentin.s.baghi@nasa.gov

about 10−21, the laser noise dominates them by eight or-ders of magnitude. Recovery of the GW signal is possiblebecause the correlations between the individual link sig-nals differ for GWs and laser noise. Laser frequency noiseis almost entirely canceled by a post-processing techniquecalled time-delay interferometry (TDI) [7], which con-structs linear combinations of delayed phasemeter mea-surements tailored to cancel the laser frequency noise upto ranging errors. Some TDI variables can be interpretedphysically by synthetically retracing the path of light raystraveling in a classical Michelson interferometer. Thefeasibility of TDI has been studied extensively over thepast decade (see, e.g., Ref. [8] for an overview), includ-ing the implementation of interpolation filters needed toproduce the laser-free data streams [9]. Further analyseshave recently been tackled, revealing expected TDI noiseartifacts, such as flexing-filtering [10] and clock jitterseffects [11].

Different generations of TDI variables achieving dif-ferent accuracy levels can be formed depending on as-sumptions about the spacecraft motion. TDI generations1.0, 1.5, and 2.0, respectively, assume a rigid and staticconstellation, a rigid and rotating configuration, and aflexing, rotating configuration where the armlengths arelinearly varying in time. Recent work also identifiednew TDI combinations by using the explicit dependenceof arm delays on the satellites’ velocities and accelera-tions [12].

In addition to its geometric interpretation, TDI can beviewed algebraically as the group of solutions of an equa-tion involving six-tuples polynomials in the delay oper-ators [13], which encodes the cancellation of noise. Be-yond laser noise cancellation, the analysis of LISA’s mea-surement can also be interpreted as an inference prob-lem. We observe the same physical effect (the perturba-tion of the metric due to incoming gravitational waves)through different sensors (interferometric arms) affected

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by fundamental errors (the laser noise and other stochas-tic noises). Everything works as if we recorded the samesound using different microphones. The goal of the anal-ysis is generally to estimate parameters of astrophysicalsources emitting GWs. Thus, we can infer them by di-rectly writing the likelihood to observe the phasemetermeasurements given a prescribed model. The model de-scribes how we expect the GW signal to appear in thevarious measurements and the relationships between thenoises present in each data stream. In Ref. [14], Val-lisneri et al. recently formalized this idea in the timedomain, by marginalizing the likelihood with respect tolaser noises. In this study, we develop a similar approachin the frequency domain, using an eigen-decompositionof the covariance.

Typically, the likelihood depends on the inverse cor-relation matrix of all measured variables. Applying itsinverse to the model residuals yields the weighed, un-correlated squared errors involved in any optimal pa-rameter estimation scheme. We refer to the process ofgenerating orthogonal variables as principal componentanalysis (PCA). In a pioneering work [15], Romano andWoan show that we can derive the TDI combinationsfrom the eigenvectors of the single-link covariance ma-trix through a demonstration based on a simplified time-domain analysis assuming white noise and short time se-ries. In this work, we generalize this study by building ananalytic formalism implementing PCA, based on a ma-trix formulation in the frequency domain readily usablefor gravitational-wave data analysis. We refer to this ap-proach as “Principal Component Interferometry” or PCIfor short.

PCI implements the generalized analog of the orthog-onal TDI channels A, E T [16]. While A, E, T’s originalconstruction relies on assumptions such as equal armsand uncorrelated acceleration noises, PCI yields the opti-mal variance in a unified, data-driven formalism. We firstdecompose the data on the eigenvector basis of the laser-noise covariance matrix. We then estimate the frequency-domain covariance matrix (or spectral matrix) of eigen-streams, assuming that the acceleration noise is station-ary. The resulting model likelihood allows us to fit fornoise parameters encoding all types of noise correlations,which we account for by construction in the inferenceprocess.

In Sect. II, we present the algebraic formalism used todevelop PCI. In Sect. III, we derive the principal compo-nents of the covariance in the simple case of rigid arm-lengths. In Sect. IV, we demonstrate PCI’s performancethrough simple numerical simulations, with an exampleof an application where we estimate conjointly the laserlight propagation delays (ranging), the noise covarianceelements, and the parameters of a compact galactic bi-nary source. We conclude and discuss the further gener-alization to more complex cases in Sect. V.

II. MODELING SINGLE-LINKMEASUREMENTS USING MATRIX

OPERATORS

A. Derivation of the likelihood

In this section, we present the conventions and the for-malism that we adopt throughout the study. Using con-ventions in Fig. 1 of Ref. [10], we consider a spacecraftlabeled i, and the signal si obtained by comparing thelaser light coming from the distant spacecraft i+1 to thelocal oscillator of optical bench i. The measurement siis where gravitational waves imprint their presence andis called the science interferometer signal. The scienceinterferometer measurements si(t) = (νi(t)− ν0) /ν0 areexpressed as the relative deviation of instantaneous fre-quency with respect to the carrier frequency ν0. For thetwo optical benches onboard spacecraft i, we have, ateach time t:

si = hi+2 +Di+2pi′+1 − pi + ni;

si′ = hi+1 +Di′+1pi+2 − pi′ + ni′ , (1)

where Dix(t) = x(t− c−1Li

)denotes the operator ap-

plying the light travel time delay c−1Li along arm i,hi is the integrated frequency shift along arm i due toincoming gravitational waves, pi is the frequency noisecontribution of the laser in optical bench i, and ni gath-ers all other noises affecting the science measurement onthat bench for all i ∈ 1, 2, 3. We adopt the conventionof cyclic indexing where pi actually means pi−3b(i−1)/3c,and simple indices refer to links and light travel timespointing clockwise, whereas prime indices denote coun-terclockwise directions. In the following, we assume thatthe lasers in the two optical benches are identical, so that

pi = pi′ ∀i ∈ 1, 2, 3 . (2)

This assumption allows us to simplify the analysis, butcan be adopted without loss of generality. Note that al-though the laser sources are different in reality, they willbe compared through reference interferometer measure-ments.

The measurements si in Eq. (1) will be sampled at acadence of fs = 2 Hz or more over a finite duration T .The resulting time series can be represented by a columnvector si of size N = fsT . In this discretized versionof the measurement, the delay operators Di acting onany variable x(t) can be represented by N × N matri-ces depending on the arm lengths Li. Using consistentnotation, all lines in Eq. (1) can be re-written as

y = h+Mp+ n, (3)

where y ∈ R6N is the column vector stacking thephasemeter measurements for all spacecrafts and all op-tical benches so that y(i) = si and y(i + 3) = si′∀i ∈ 1, 2, 3. Vectors h, p, and n respectively representthe GW signals, laser noises, and other noises. They havein general the same structure and dimensions as y. Note

3

that under the assumption outlined in Eq. (2), we keeponly the first half of p so that p ∈ R3N .

Matrix M encodes the mixing and delaying of lasernoises and can be written as a block matrix:

M =

−IN D3 0N0N −IN D1

D2 0N −IN−IN 0N D2′

D3′ −IN 0N0N D1′ −IN

. (4)

Now that the observation equation is written in ma-trix form, we can derive the corresponding likelihood.Gravitational-wave source parameters θ are usually ex-tracted using Bayesian inference, which estimates theirposterior distribution given the data:

p (θ|y) =p (y|θ) p (θ)

p (y), (5)

where p (y|θ) is the model likelihood, p (θ) is the priordistribution of the parameters, and p (y) is the evidence,acting as a normalization.

Assuming a zero-mean Gaussian distribution for allnoises, the likelihood follows from Eq. (3):

p (y|θ) =exp

− 1

2 (y − h)†Σ−1 (y − h)

√

(2π)6N |Σ|, (6)

where † denotes the Hermitian conjugate, and Σ is the6N × 6N covariance matrix of the observations y, whoseexpression derives from Eq. (3):

Σ = MΣpM† + Σn, (7)

where Σp and Σn are respectively the covariance ma-trices of the laser noises p and of the other noises n,assuming no intrinsic correlation between the two.

In the following, we make the convenient but realisticassumption that all noises are stationary (at least for arelatively short period of time). In that case, their co-variance matrices are Toeplitz, and for a sufficiently largeN , they are approximately diagonalizable in the discreteFourier basis W which form its eigenvectors and writes

W (k, n) = e2πjnkN with j =

√−1 being the complex num-

ber. We can therefore re-write the covariance matrix inEq. (7) in the Fourier domain as

Σ = MSpM† + Sn, (8)

where Sp and Sn are the covariance matrices of thediscrete Fourier-transformed data, that we call spectralmatrices. Due to stationarity, Sp (respectively Sn)includes 3 × 3 (respectively 6 × 6) blocks which areN × N diagonal matrices, whose diagonal elementsare given by the noise cross-spectra. Spectral matricesare Hermitian, with real positive diagonal blocks andcomplex conjugate off-diagonal blocks. If we restrict theanalysis to a specific set of Nf frequencies, then each

block has size Nf ×Nf . In the following, we look for theprincipal components of the covariance matrix.

III. PRINCIPAL COMPONENT ANALYSIS OFSINGLE-LINK MEASUREMENTS

A. Principle of PCA

PCA aims at finding a transformation of the obser-vations that converts them into uncorrelated variables,ordered according to their variance. The process is oftenused to reduce the dimension of the problem by discard-ing the highest variance components. Here, we aim atfinding a unitary transformation matrix V where the co-variance matrix can be diagonalized as

Σ = V ΛV †, (9)

where Λ is a diagonal matrix. Then the log-likelihoodcan conveniently be re-written as

log p (y|θ) = −1

2(y − h)

†V Λ−1V † (y − h)

−1

2log |Λ| . (10)

However, finding a full decomposition like Eq. (9) canbe tricky, unless we make a few key assumptions, whichwe do in the following.

B. PCI for equal noises

A way to find a decomposition of the form (9) is tofind the eigenvectors of the covariance matrix. To easetheir calculation, we make two assumptions.

First, we assume that all delays are constant in time,which implies that the delay operators are commutative,i.e. D1D2 = D2D1. In this case, the Fourier basis alsoprovides approximate eigenvectors for the delay opera-tors Di. Hence, in the Fourier basis, the delay operatorsare approximately diagonal, and in the limit of large N ,we can write their diagonal elements as:

Di|k,l = e−2πjfkLiδkl. (11)

Second, we assume for now that all non-laser noiseshave the same power spectral density (PSD). Thus, alldiagonal blocks Sn,i of the spectral matrix Sn are equal:Sn,i = λn∀i.

Based on these assumptions, from the calculation ofthe characteristic polynomial of matrix Σ we find thatthere are 6 eigenvalues per frequency bin, hence 6Nfeigenvalues for the full problem. Half of them (whichwe label Λn) are degenerate and equal to the non-lasernoise PSD values, so that Λn = diag (λn,λn,λn). Wecan write the 6Nf × 3Nf matrix gathering their associ-

ated eigenvectors Vn analytically as

4

Vn =

D†2

(D†1′D

†1 − I

)D†3′ − D†1D†2 D†2′D

†2 − I

D†1′ − D†2D†3 D†3′D†3 − I D†3

(D†2′D

†2 − I

)D†1′D

†1 − I D†1

(D†3′D

†3 − I

)D†2′ − D†1D†3

0 0 I − D†1D†2D†30 I − D†1D†2D†3 0

I − D†1D†2D†3 0 0

. (12)

Note that Vn is also a basis for the null space of thelaser-noise part of the covariance, so that we have ΣVn =SnVn.

The 3Nf other eigenvalues, that we label as Λp =diag (λp1,λp2,λp3), have more complicated expressionsbut are all proportional to the laser noise PSD Sp. We

denote by Vp the associated eigenvector matrix, which

has the same dimensions as Vn. We plot the laser-noisedominated eigenvalues λpi in gray as a function of fre-quency in Fig. 1, along with the degenerate laser-noisefree eigenvalues λn in blue. This figure confirms that theformer are much larger than the latter.

10 4 10 3 10 2 10 1

Frequency [Hz]

10 20

10 18

10 16

10 14

Eige

nvalu

es p1

p2

p3

n

FIG. 1. Laser-noise dominated eigenvalues λpi (gray) andlaser-noise free eigenvalues λn (blue) of the phasemeter co-variance matrix as a function of frequency.

As a result, we can partition the eigenvector matrixas V =

(Vp Vn

). Let us consider the data transforma-

tion e ≡ V †y. As they correspond to different eigenval-ues, the eigenvector matrices Vn and Vp are orthogonal.Therefore, the covariance of e is block diagonal:

Cov (e) =

(V †p ΣVp 0

0 V †nSnVn

)≡(Cp 00 Cn

), (13)

where we defined the 3Nf ×3Nf matrix Cn (respectively

Cp) as the covariance of the projected data en ≡ V †n y(respectively ep ≡ V †p y) onto the laser-noise free (respec-tively laser-noise dominated) basis.

Then it is possible to separate the laser noise-dominated eigenbasis Vp from the laser noise-free eigen-

basis Vn in the calculation of the likelihood:

log p (y|θ) = −1

2

(y − h

)†VpC

−1p V †p

(y − h

)−1

2

(y − h

)†VnC

−1n V †n

(y − h

)−1

2(log |Cp|+ log |Cn|) . (14)

Similarly to Romano and Woan’s example in Ref. [15],the laser noise variance being much larger than othernoises, the term in the second line in Eq. (14) is almostconstant as a function of θ compared to the term in thefirst line. Therefore, we can safely approximate the log-likelihood by

log p (y|θ) ≈ −1

2

(y − h

)†VnC

−1n V †n

(y − h

)−1

2log |Cn| . (15)

In the next section, we detail how we compute the inverseof Cn.

C. Orthogonalization with respect to non-lasernoise

In the previous section, we saw that the approximatelog-likelihood depends on the covariance C of the dataprojected onto the eigenbasis associated with null eigen-values of the laser-noise covariance. If the noise param-eters (PSD levels and light travel time delays) are as-sumed to be known and fixed in the inference scheme,this matrix, and its inverse, can be computed once forall. However, if we need to update the delays and thenoise model along the way, we must compute C−1n at ev-ery parameter update for all frequencies. In this work,we perform this computation by numerically diagonaliz-ing Cn with its eigenvectors Φ and eigenvalues ΛK , andthen computing C−1n = ΦΛ−1K Φ†. We use the NumPylibrary [17, 18], which includes efficient algorithms whenthe number of frequency bins is not large (< 1000). Itmay be more efficient for larger frequency bands to usean analytical formula for 3 × 3 Hermitian matrices asderived by Ref. [19].

5

This diagonalization is a generalization of the orthog-onalization process that leads to TDI variables A, E,T [8, 20]. Indeed, we can apply the formalism devel-oped in this section to any linear transformation thatcancels laser frequency noise. For example, TDI trans-formations can be encoded by some matrix T instead ofVn in Sect. III B. While channels A, E, and T are or-thogonal under specific conditions (including equal arm-lengths, non-rotating constellation, identical accelerationnoise levels, and uncorrelated noises), the rationale lead-ing to Eq. (15) does not rely on any of these assumptions.

D. PCI with no prior knowledge of noise PSDs

While the projection onto the null space of the lasernoise covariance matrix (i.e., the calculation of Vn) is in-dependent of the laser noise spectra, the orthogonaliza-tion that we outlined in Sect. III C relies on our knowl-edge of the other noises’ spectral matrix Sn. Althoughwe may have a physical model describing accelerationand OMS noises, we must expect deviations from thetheory when dealing with future LISA data. Therefore,it is necessary to have a formalism that also allows usto estimate Cn robustly. Estimating the full covariancematrix elements is not commonly done in gravitational-wave data analysis, but it can be performed in similar toPSD estimation methods, extending them to off-diagonalterms. This type of problem relates to spectral anal-ysis of co-stationary multivariate time series, for whichseveral approaches are available, such as estimating thecomponents of the generalized Cholesky decompositionof the spectral matrix or its inverse [21]. Regardless ofthe model we adopt, one has to ensure that the estimatedspectrum is a positive definite matrix and is continuousas a function of frequency. To benefit from fast con-ditional steps when it comes to posterior sampling, wechoose to model the covariance elements themselves withthe regression scheme proposed by Ref. [22], which takesadvantage of conjugate priors. To this end, let us con-sider a single frequency f , and the 3×3 covariance of thecorresponding elements:

Cn(f) ≡ Cov (e(f)) , (16)

where we labeled as e(fk) ≡(yk, yNf+k, y2Nf+k

)Tthe

vector of eigenstream elements associated with frequencybin fk. Thus, we assume that the covariance has the form

Cn(f) = Ψ +Bx(f)x†(f)B†, (17)

where Ψ is a constant 3 × 3 Hermitian matrix, x(f) isa q × 1 design matrix depending on frequency, and B isa 3 × q matrix of regression parameters. For example,x can have the form of a polynomial in frequency with

elements x =(1, f, . . . , fq−1

)T.

In this model, Ψ and B are unknown and must beestimated. For a sufficiently short frequency range, wecan even approximate the covariance by a constant term

across the band, as in Ref. [23]. Under this assumption,Eq. (17) reduces to Cn(f) = Ψ, and Ref. [22]’s sam-pling scheme amounts to using the conjugate prior forthe Gaussian distribution, i.e. the inverse-Wishart priorIW (Ψ0, ν0). We adopt this simplification in what fol-lows, where the conditional posterior of Ψ given the de-lays L and GW parameters θGW is also inverse-Wishart:

p (Φ|y,L,θGW) = IW(Ψ0 + Ψ, ν0 +Nf

), (18)

where Nf is the number of frequency bins and Ψ is the3× 3 sample covariance of the eigenstream residuals:

Ψ = V †n

(y − h

)(y − h

)†Vn. (19)

We implement this step in Python using statisti-cal packages from the Scipy library [24]. FollowingRef. [22]’s suggestions we set ν0 = d + 2, where d = 3is the dimension of Ψ0. We choose Ψ0 to be the medianof the frequency bins’ sample covariances after a first runobtaining a rough estimate of the eigenstreams en.

E. Frequency-domain implementation of delays

Up to now, we assumed that time series have a quasi-infinite length so that the asymptotic frequency-domainformulation of the delay operator in Eq. (11) is valid.In practice, we analyze relatively short measurementsfor which this approximation breaks. Applying Eq. (11)on Fourier-transformed data leads to large edge effects.To mitigate this behavior, we use a time-window thatsmoothly drops to zero at the time series’ edges. How-ever, such an operation usually requires transforming thedata back time domain, which is computationally expen-sive compared to the usual cost of one likelihood eval-uation. Therefore, we perform the equivalent computa-tion in Fourier space (i.e., a discrete convolution) usinga sparse approximation of the convolution kernel, simi-larly as in covariance approximation techniques [25]. Thedelay operation amounts to the following matrix multi-plication:

Dtap = ΩD, (20)

where D is the asymptotic delay operator as given byEq. (8) and Ω is the tapered convolution matrix whoseelements are given by

Ωk,p =

∑N−1n=0 w(n)e−2πjn

k−pN if |k − p| ≤ p0;

0 otherwise.(21)

We denoted by w(n) the time-domain window functionand p0 an integer threshold for the row-column difference,above which the matrix elements are zero.

6

IV. CASE STUDY

To demonstrate the developed approach’s perfor-mance, we consider the simple case where phasemeterdata only contain a single GW source buried into station-ary Gaussian noise. Unless otherwise stated, we assumearbitrary armlengths and noise PSDs. The parametersgoverning the estimation model are:

• Laser light travel time delaysL = (L1, L2, L3, L1′ , L2′ , L3′);

• GW source parameters that we restrict to intrinsic

ones θGW =(θ, φ, f0, f0

);

• Non-laser noise covariance parameters Ψ.

We use the likelihood function in Eq. (15) that we max-imize over extrinsic GW amplitudes. In this function,the frequency-domain waveform h depends both on θGW

and on L, while the laser noise covariance eigenvectorsVn depend on L only.

A. Simulation parametrization

a. Noise. We simulate one month of LISA observa-tions that yield single-link time series by implementingEq. (1) with a Python code. We generate noises at asampling cadence of 2 Hz. We first applied the delays us-ing time-domain Lagrange interpolation filters, with thesame parametrization as in LISANode [10], and checkedthat the PCI algorithm was successfully canceling lasernoise. However, we noted that the applied delays’ accu-racy was not enough to be unnoticed when recovering de-lays from data simulated over long periods (one month).In other words, the delay values optimally canceling lasernoise were slightly biased compared to injected delay val-ues. This mismatch is understandable, as fractional delayfilters have a frequency response that only approximatesthe ideal delay filter [26]. Therefore, we chose to simu-late the data used in this study directly in the frequencydomain, relying on noise stationarity, following Ref. [27].

We then filter the data using a Kaiser finite-responsefilter and downsample it to 0.2 Hz to generate the out-puts. We assume a rigid, rotating LISA constellation sothat the effective armlengths do not vary in time andthat the light travel time is sensitive to the direction ofpropagation due to the Sagnac effect.

The noise PSD model includes three components: laserfrequency noise, test-mass (TM) acceleration noise, andoptical metrology system (OMS) noise. We assume thatthe noises affecting two different optical benches are un-correlated, so that matrices Sp and Sn are block diago-nal. Matrix Sn has 6 diagonal blocks Sni of the form:

Sni|k,l = αi (STM(fk) + SOMS(fk)) δkl, (22)

where αi is a positive coefficient depending on opti-cal bench i. Expressions for noise PSDs STM(f) and

SOMS(f) are given in Appendix A. Thus, noise spectrahave the same shape for every optical benches, up to acoefficient accounting for possible noise level discrepan-cies.b. Gravitational-wave signal. We assume that the

gravitational signal comes from the loudest verifica-tion compact galactic binary known to date, called HMCnc [28, 29]. We simulate single-link gravitational-wavesignals sampled at 0.2 Hz in the time domain using thesame code as in Ref. [30] that we adapted by remov-ing the TDI transfer function. We also relaxed the low-frequency approximation, using a Fourier series decom-position similar to Cornish and Littenberg’s implemen-tation in Ref. [31]. For all parameters, we use uniformpriors around the true parameter values. The source’scharacteristics, along with prior boundaries, are summa-rized in Table I.

TABLE I. Values of the source parameters used in the simu-lations, with their uniform prior boundaries.

Parameter Value Prior range

Frequency [mHz] 6.22 [6.12, 6.32]Frequency derivative [mHz/s] 3.47 ×10−9

[10−9, 10−8

]Ecliptic latitude [rad] -0.0821 [0, π]Ecliptic longitude [rad] 2.102 [−π, π]Amplitude [strain] 6.4 × 10−23 MarginalizedInclination [rad] 0.6632 MarginalizedInitial phase [rad] 5.78 MarginalizedPolarization [rad] 3.97 Marginalized

B. Projection on null space

The first simulation we consider includes single-linkmeasurements where all noises are generated from thesame PSD, as given in Appendix A. It also contains onesingle GW source, as described in Sect. IV A. For thesake of description, here we assume that the light traveltime delays Li/c are known. We compute the null-space

eigenvector matrix Vn analytically using Eq. (12) and

the other eigenvector matrix Vp numerically. Thanks tothese matrices, we apply the PCI transformation to ob-tain eigenstreams that we orthogonalize as described inSect. III C. We label as e⊥i these orthogonal streams.

We plot the periodogram of the PCI transformationsin Fig. 2, which shows the laser-noise dominated eigen-streams (gray) along with the null space eigenstreams(blue), lying 8 orders of magnitude below. We alsoplot the GW signal transformed in the null space eigen-streams, which emerges from the noise. We show thatthis noise is indeed limited by the acceleration and OMSerrors by plotting the theoretical PSD (light blue) com-puted from the diagonal elements of the covariance ma-trix Cov (e⊥) = Φ†V †nSnVnΦ. These plots demonstratethe ability of the frequency-domain algorithm to separatethe two orthogonal spaces correctly.

7

10 23

10 20

10 17

10 14

|e1|

[Hz

1/2 ]

Laser noise-dominatedLaser noise-free

Theor. PSD GW signal

10 23

10 20

10 17

10 14

|e2|

[Hz

1/2 ]

10 4 10 3 10 2

Frequency [Hz]10 23

10 20

10 17

10 14

|e3|

[Hz

1/2 ]

FIG. 2. PCI eigenstreams expressed in relative frequency de-viation for the laser-noise free subspace (blue) and the laser-noise dominated subspace (gray) for a month-long simulation.The theoretical PSD function of laser-noise free eigenstreamsare plotted in light blue. The red vertical lines denote thegravitational signal from the verification galactic binary HMCnc that shows up in all channels as a quasi-monochromaticsignal.

C. Sensitivity analysis

In this section we investigate the theoretical perfor-mance of the PCI approach in two cases: i) all non-lasernoise levels are the same, i.e., αi = 1∀i and ii) non-lasernoise levels are different depending on optical benches,with α = (4, 0.25, 16, 0.1, 0.4, 1). In case ii), some op-tical bench noises have larger amplitudes than the base-line, while others have smaller amplitudes. Overall, themean noise level is larger than in case i) by a factor 3.6.In Fig. 3, we plot in blue the generalized sensitivity toan ultra-compact galactic binary source observed duringone month. The source we consider has the same param-eters as HM Cnc, except its frequency, which we allowto vary. Here, for any frequency f , “sensitivity” refers tothe signal-to-noise ratio (SNR) of a source of frequencyf measured in the frequency bin f . This calculationtakes into account the fact that the covariance matrixis non-diagonal in general, as described in Appendix B.For comparison, we plot in red and orange generalizedsensitivities of unoptimized TDI combinations A, E, and

T. They are obtained from combining TDI Michelson X,Y, and Z strictly as derived in [16], relying on the as-sumption of equal noises and equal arm lengths. Thus,we compute associated sensitivities assuming that A, E,T’s covariance matrix is perfectly diagonal; hence we callthem “unoptimized”.

10 4 10 3 10 2 10 1

Frequency [Hz]

10 2

10 1

100

101

SNR

Equal noises PCIEqual noises TDIUnequal noises PCIUnequal noises TDI

FIG. 3. Sensitivity of laser-noise free channels as a functionof frequency obtained with PCI eigenstreams (blue or gray)and unoptimized TDI channels A, E, T (red or orange). Thesolid lines represent the case of equal noise; dashed lines cor-respond to unequal noise levels, computed for a source likeHM Cnc. PCI and TDI sensitivities are almost identical inthe case of equal noises, while unoptimized TDI undergoes alarger SNR loss due to mis-orthogonalization.

PCI and unoptimized TDI yield almost the same sen-sitivity when noises are equal, as shown by the superim-position of the blue and red, solid curves. This similarityconfirms that the A, E, T formulation is nearly optimalin this configuration because the assumptions made intheir derivation are almost met, except for the equal armlengths hypothesis, which plays a minor part in the or-thogonalization process.

As expected, regardless of the method used, the figureshows that the unequal-noise case (dashed lines) yieldsa degraded SNR compared to the case where all noisesare equal (solid lines). However, the SNR loss is smallerwith the PCI method because it includes the frequency-dependent orthogonalization by design, providing a gen-eral extension to classic TDI. This comparison illustratesthe importance of off-diagonal terms in the covarianceCn, which are not equal in the case of heterogeneousnoise levels. Not taking this discrepancy into accountmay result in sub-optimal performance.

D. Parameter inference scheme

We further assess the performance of the developedmethod by using it to recover injected parameters fromthe numerical simulations described in Sect. IV A. Esti-

8

mated parameters include light travel time delays, sourceparameters, and noise covariance. We restrict the infer-ence to a portion of the frequency band between 6.13and 6.31 mHz around the binary’s frequency. We sam-ple the posterior distribution of delays and GW param-eters through parallel-tempered Markov Chain MonteCarlo (PTMCMC) sampling, using the ptemcee al-gorithm [32], a parallel-tempered version of the affine-invariant ensemble sampler emcee [33].

We modify the sampling algorithm to include noisecovariance parameters in the inference, using a two-stepBlocked Gibbs sampling scheme where noise parametersare sampled conditionally to delays and GW parameters:

Step 1: L,θGW ∼ p (L,θGW|y,Φ,B) ;

Step 2: Φ ∼ p (Φ|y,L,θGW) . (23)

While step 1 is still based on PTMCMC, Step 2 usesdirect sampling as described in Sect. III D. We describesampling results in the next section.

E. Inference results

We present the results of the inference of delays, GWparameters, and covariance parameters applied to thesynthetic data described in Sect. IV A with the samplingscheme presented in Sect. IV D.

First, we use two data sets: one corresponding to theequal noise case i) described in Sect. IV C, the other forthe unequal noise case ii). We run 40 chains in parallelwith 10 different temperatures, and we retain 4 × 105

samples after chains have reached convergence.a. Delays. We plot in Fig. 4 the delays posteri-

ors marginalized over other parameters in the case ofequal (dashed lines) and unequal (solid lines) accelerationnoises, using the ChainConsumer package [34]. We ex-press delays in equivalent inter-spacecraft distances. Inthe case of equal noise levels, posteriors obtained fromPCI and unoptimized TDI are almost equal to each other,confirming the result found in Fig. 3.

Delays distributions are broader in the case of unequalnoises because, in this example, the overall noise poweris larger than in the case of equal noises. However, formost delays, posteriors have a larger variance with classicTDI than with the PCI analysis, because PCI accountsfor the change of off-diagonal covariance terms, maintain-ing orthogonalization. This result is consistent with themore significant SNR loss shown in Fig. (3) when usingunoptimized TDI combinations. Note that the fact thatthe centers of L1’s posteriors (upper-left panel) are lyingcloser to the true value with unequal noises than withequal noises is fortuitous: we would obtain a differentoutcome with another noise realization.

We remark that the uncertainty in estimating equiv-alent arm lengths is of order 40 m, which is enough tocancel laser noise in this particular case. Should we wishto, we could obtain a better precision in using the entirefrequency data instead of restricting it to a narrow band.

−60−3

0 0 30 60

δL1 [m]

PCI equal noises

TDI equal noises

PCI unequal noises

TDI unequal noises

−100 −5

0 0 50

δL2 [m]−1

00 −50 0 50

δL3 [m]

−100 −5

0 0 50 100

δL1′ [m]−4

0 0 40 80

δL2′ [m]−7

5−5

0−2

5 0 25

δL3′ [m]

FIG. 4. Posterior distribution of the 6 light-travel time delaysexpressed in equivalent arm lengths. Solid lines correspondto the case of equal acceleration noises, whereas dashed linescorrespond to unequal noises. Posteriors obtained with PCIare in blue or gray, and posteriors obtained with unoptimizedTDI are in red or orange. Thin vertical black dashed linesrepresent true values, and shaded areas under the curves coverthe 1σ region.

b. GW parameters. Then, in Fig. 5, we examine theposterior of the GW source’s frequency and frequencyderivative, marginalized over all other parameters. Herewe focus on PCI results only, comparing equal (solid bluelines) and unequal (gray dashed lines) noises. The fig-ure shows that this couple of parameters is accuratelyrecovered by the PCI analysis, even when performed si-multaneously with the estimation of laser light delays.As expected, we observe a difference between equal andunequal noise cases because of SNR’s drop. The firstmode’s width increases compared to the equal-noise case,and the secondary mode significantly broadens. However,the adaptive orthogonalization built in the PCI processminimizes the impact of the noise heterogeneity.

The GW source sky location posteriors that we plotin Fig. 6 exhibit the same behavior as for the frequencyparameters. The PCI analysis of the equal-noise dataaccurately spots the source’s right location in the skyrepresented by the yellow star, as shown by the blue pos-terior. In the case of unequal noises, the loss in SNRbrings two secondary modes in the distribution, with themost westwards carrying more posterior weight. Hence,there is an ambiguity in the sky location appearing inthe case of unequal noises in spite of the optimal noiseweighting, which could be resolved by extending the ob-servation duration.

c. Covariance. We collect the covariance parametersamples computed in the Gibbs steps in Fig. 7. We usethem to compute the chains of covariance values. We

9

6.21

90

6.21

95

6.22

00

6.22

05

6.22

10

f0 [mHz]

3.00

3.25

3.50

3.75

4.00

f 0[µ

Hz2

]

Equal noises

Unequal noises

FIG. 5. Joint posterior distribution of GW source’s frequencyf0 and frequency derivative f0, obtained with PCI in the caseof equal (solid blue) and unequal (dashed gray) noises appliedto a one-month long simulation of phasemeter measurements.Contours correspond to 1σ and 2σ regions.

-60.0°-60.0°

-30.0°-30.0°

0.0°

30.0°

60.0°

-68° -58° -48°8°

18°

28°Equal noises Unequal noises

FIG. 6. Sky localization posterior distribution of the GWsource in celestial coordinates, obtained with PCI in the caseof equal (solid blue) and unequal (dashed gray) noises. Theorange star represents the true location.

plot the estimated posterior of the diagonal terms in lightgreen in Fig. 7 as a 3σ interval around the mean, and wecompare it to the true value represented by the solid line.The real value is located within the 3σ interval, demon-strating the covariance estimate’s accuracy. We obtainsimilar-looking plots for off-diagonal covariance elements(real and imaginary parts), confirming the accurate char-acterization of non-laser noise and the proper orthogonal-ization of the eigenstreams.

We also plot the periodograms of the three eisgen-streams using the delays’ actual values (solid blue line),along with the 3σ interval of the posterior samples. Theposterior closely encompasses the target value, showingthat the ranging estimates’ variability is acceptable.

Finally, we plot the 3σ interval (red shaded area) of

the GW waveform samples against the true value of thesignal (solid red curve), showing the right consistency be-tween the two. As a result, Fig. 7 provides a summary ofthe multi-parameter inference enabled by the PCI frame-work.

0.5

1.0

1.5

|e 1| [

Hz1/

2 ]

1e 20Eigenstreams Noise PSD GW signal

0.5

1.0

1.5

|e 2| [

Hz1/

2 ]

1e 20

6.20 6.21 6.22 6.23 6.24 6.25Frequency [mHz]

0.5

1.0

1.5|e 3

| [Hz

1/2 ]

1e 20

FIG. 7. True value (solid lines) and estimated 3σ posteriorinterval (light shaded areas) of the eigenstream periodograms(blue), the GW signal (red) and the noise PSDs (diagonalcovariance entries) expressed in relative frequency deviation.Posterior intervals are computed using 1000 MCMC samples.The data shown here corresponds to the case of unequal non-laser noises.

V. DISCUSSION AND PROSPECTS

We revisited space-based GW data modeling by writ-ing the model likelihood directly as a function ofphasemeter measurements. Based on Romano andWoan’s idea in Ref. [15], we restricted the likelihood toits lowest variance terms using principal component anal-ysis. This work’s main contribution is to formalize thePCA approach in the frequency domain, using an asymp-totic formulation of the delay operators with sparse ma-trices. This formalization provides a framework that isreadily applicable to parameter inference, yielding opti-mal precision. We show that it allows us to handle theTDI transfer function of both signal and noise in a singlecompact, matrix-based formalism.

10

With a simplified example of simulated LISA data, weshow that the method allows us to consistently and simul-taneously fit for inter-spacecraft phase delays, an ultra-compact galactic binary source’s parameters, and noisecovariance parameters. We show that a numerical anddata-driven diagonalization of the covariance yields anoptimal sensitivity to gravitational waves and minimalsource parameter uncertainty.

This work lays the foundation for a more robust anal-ysis of LISA data. First, it provides a way to derivesensitivities from instrumental noises, tracking all corre-lations straightforwardly. Second, it generalizes the con-cept of orthogonal TDI variables to arbitrary armlengthsand noise correlations. However, to apply to in-orbitdata, the approach must be extended to time-varyinginter-spacecraft distances. Future work will focus on for-malizing this time-dependence in the frequency domainto maintain computational efficiency. Furthermore, wecan extend the developed framework to a complete set ofLISA measurements by dropping the assumption of iden-tical spacecraft’s laser noise and include reference inter-ferometer measurements. Preliminary work has alreadydemonstrated the successful decomposition into large andlow variance components in this configuration, that wewill present in follow-up studies.

Finally, our work highlights the importance of L0 dataincluding raw phasemeter measurements in LISA’s sci-ence analysis. The availability of such data will ensurethat alternative processing methods complementary tothe standard TDI pipeline are possible. This diversityof approaches is an essential tool for cross-checking andvalidating the data in an off-line interferometry step thatis crucial for the precise characterization of astrophysicalsources.

Appendix A: Expressions of noise PSD model

The noise PSD models used in this study are given infractional frequency per Hertz (Hz−1) as

Slaser(f) =

(a0ν0

)2

;

STM(f) =

(aTM

2πfc

)2(

1 +

(f1f

)2)(

1 +

(f

f2

)4)

;

SOMS(f) = a2OMS

(2πf

c

)2(

1 +

(f3f

)4). (A1)

We indicate the values of the noise model parameters inTable II.

Appendix B: Computation of sensitivity tomonochromatic binaries

In this section, we derive the expression for the gener-alized SNR plotted in Fig. 3, which extends the classic

TABLE II. Values of the noise parameters used in the simu-lations.

Noise type Parameter Value

Lasera0 28.2 Hz.Hz−1/2

ν0 281759 GHz

Test-massaTM 3 fms−2Hz−1/2

f1 0.4 mHzf2 8 mHz

OMSaOMS 15 pmHz−1/2

f3 2 mHz

SNR calculation to the case where the noise covarianceused for the estimation is different than the correct one.

Let us consider some linear transformation of thesingle-link measurements, encoded by a 6× 3 matrix Wthat can represent any TDI or PCI transformation inthe frequency domain. Matrix W is defined for a givenfrequency bin f . Applying W to the phasemeter mea-surements y(f) yields a collection of 3 channels that wedenote by e(f).

We define the generalized SNR ρW (f0) of a monochro-matic GW of frequency f0 obtained with the data trans-formation W as the ratio between the absolute value ofits gravitational-wave strain amplitude h0 and the stan-dard deviation σ0 of its maximum likelihood estimate:

ρW (f0) ≡ |h0|σ(f0)

. (B1)

We assume that the GW signal appears in the single-link measurements y as

y(f) = A(f)h0, (B2)

where A is a 6 × 1 design matrix. From Eq. (3), thistransformation maps the GW amplitude as

e(f) = W (f)†A(f)h0 + ε(f), (B3)

where ε(f) is the noise contribution to the data streamvector e at frequency f . The effective 3 × 1 transferfunction matrix is thus H(f) ≡W (f)†A(f).

Let us assume that we analyse the data with the fol-lowing likelihood model:

p (e(f)|h0) =exp

− 1

2 (e−Hh0)†C−1 (e−Hh0)

√

(2π)6N |C|,

(B4)where C is some model of the covariance of e, and wedropped the frequency dependence for clarity. Assumingthat C is fixed, the maximum likelihood estimator er-ror on h0 from data at frequency f is then given by theinverse Fisher matrix, which is defined as

σ20 = E

[(∂ log p

∂h0

)2]−1

. (B5)

11

Inserting Eq. (B4) into Eq. (B5) yields

σ20 =

H†C−1ΣeC−1H

(H†C−1H)2 , (B6)

where Σe is the true variance of data streams e, and isgiven by

Σe = W †ΣW , (B7)

where Σ is the covariance of y. We can remark 2 prop-erties:

• If W is an orthogonal transformation with respectto the noise, then Σe is diagonal by construction.

• IfC = Σe (i.e., the model covariance is equal to the

true one), then ρW (f0) = |h0|√H†Σ−1e H which

is exactly equivalent to the standard SNR formulaused in the gravitational-wave literature.

ACKNOWLEDGMENTS

We would like to thank Tyson Littenberg and JessicaPage for enlightening conversations. We would also liketo acknowledge interesting feedback from Martin Staaband Shane Larson. This work is supported by an appoint-ment to the NASA Postdoctoral Program at the GoddardSpace Flight Center, administered by Universities SpaceResearch Association under contract with NASA.

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