# Stochastic gravitational wave background reconstruction
for a non-equilateral and unequal-noise LISA constellation

Olaf Hartwig<EMAIL_ADDRESS>SYRTE, Observatoire de Paris, Université
PSL, CNRS, Sorbonne Université, LNE, 61 avenue de l’Observatoire 75014 Paris,
France  Marc Lilley<EMAIL_ADDRESS>SYRTE, Observatoire de Paris,
Université PSL, CNRS, Sorbonne Université, LNE, 61 avenue de l’Observatoire
75014 Paris, France  Martina Muratore<EMAIL_ADDRESS>Max-Planck-
Institut für Gravitationsphysik (Albert-Einstein-Institut),
Am Mühlenberg 1, 14476 Potsdam, Germany  Università degli studi di Trento, via
Calepina, 14 - I-38122 Trento, Italy  Mauro Pieroni<EMAIL_ADDRESS>Department of Theoretical Phyics, CERN, 1211 Geneva 23, Switzerland

###### Abstract

We explore the impact of choosing different sets of Time-Delay Interferometry
(TDI) variables for detecting and reconstructing Stochastic Gravitational Wave
Background (SGWB) signals and estimating the instrumental noise in LISA. Most
works in the literature build their data analysis pipelines relying on a
particular set of TDI channels, the so-called AET variables, which are
orthogonal under idealized conditions. By relaxing the assumption of a
perfectly equilateral LISA configuration, we investigate to which degree these
channels remain orthogonal and compare them to other TDI channels. We show
that different sets of TDI variables are more robust under perturbations of
the perfect equilateral configuration, better preserving their orthogonality
and, thus, leading to a more accurate estimate of the instrumental noise.
Moreover, we investigate the impact of considering the noise levels associated
with each instrumental noise source to be independent of one another,
generalizing the analysis from two to twelve noise parameters. We find that,
in this scenario, the assumption of orthogonality is broken for all the TDI
variables, leading to a misestimation of measurement error for some of the
noise parameters. Remarkably, we find that for a flat power-law signal, the
reconstruction of the signal parameters is nearly unaffected in these various
configurations.

††preprint: CERN-TH-2023-050

###### Contents

  1. I Introduction
  2. II Measurement characterization
    1. II.1 Data model
      1. II.1.1 Single link signal response
      2. II.1.2 Single link noise spectra
    2. II.2 TDI variables
      1. II.2.1 Base variable definition
      2. II.2.2 Signal and noise projection on the TDI variables
  3. III Relaxing the equilateral assumption: spectral analysis
    1. III.1 Equal noise levels
    2. III.2 Unequal noise levels
  4. IV SGWB and noise parameter reconstruction for AET and AE$\zeta$
  5. V Conclusions
  6. A Useful relationships amongst TDI variables
    1. A.1 Unequal arms
    2. A.2 Relationship between orthogonal channels in the equal arms limit
  7. B Noise and signal analytic approximations
    1. B.1 Noise PSDs and CSDs for an equal arms configuration
    2. B.2 Noise PSDs and CSDs for an unequal arm-lengths configuration
    3. B.3 Low-frequency signal response for an equal arm-length configuration
    4. B.4 Low-frequency signal response for an unequal arm-length configuration
  8. C Data analysis technical details
  9. D Noise parameter reconstruction for AET and AE$\zeta$
    1. D.1 Toy model
    2. D.2 Equal noises
    3. D.3 Unequal noises

##  I Introduction

The Laser Interferometer Space Antella (LISA) Amaro-Seoane et al. (2017) is a
space mission led by the European Space Agency (ESA), also involving NASA,
which is planned to be launched in the mid-2030s. LISA will consist of a
constellation of three satellites separated by nearly 2.5 million kilometers,
operating as a gravitational wave (GW) observatory in the milliHertz (mHz)
range. LISA is expected to detect tens of thousands of resolvable sources
during the expected 4-year mission duration, including Stellar Origin Binary
Black Holes (SOBBHs), Compact Galactic Binaries (CGBs) comprised mostly of
Double White Dwarfs (DWDs), Super Massive Black Holes (SMBHs) and Extreme Mass
Ratio Inspirals (EMRIs). For a review of detection prospects for all these
sources see, e.g., Amaro-Seoane et al. (2023) and references therein. In
addition to the resolvable sources enumerated above, a much larger number of
weak and unresolvable sources will superimpose incoherently, leading to the
generation of a Stochastic GW Background (SGWB). There are (at least) two
guaranteed components contributing to the astrophysical SGWB in the LISA
band111Depending on the detection rate, the SGWB due to EMRIs signal might, or
might not, be detectable with LISA Amaro-Seoane et al. (2023).: at frequencies
lower than few mHz, the dominant contribution will come from CGBs Evans et al.
(1987); Bender and Hils (1997), while at higher frequencies another
contribution is expected to originate from SOBBH mergers Abbott et al. (2019).
Beyond astrophysical components, LISA could also be sensitive to cosmological
SGWBs generated by violent processes taking place in the very early Universe.
Detecting these signals would open a new window on energy scales beyond the
reach of all the other probes used in particle physics/cosmology. Among the
possible sources of cosmological SGWBs for LISA, let us mention inflation
Bartolo et al. (2016), cosmological Phase Transitions (PTs) Caprini et al.
(2020), Cosmic String (CS) networks Auclair et al. (2020), and second-order
scalar induced tensor perturbations, typically associated with Primordial
Black Holes production LISA Cosmology Working Group (To appear). For reviews
see, e.g., Caprini and Figueroa (2018); Auclair et al. (2022). One of the main
challenges for SGWB detection is that these signals appear in the detector
data stream as an additional noise source that has to be distinguished from
the instrumental one. For this reason, SGWB detection and characterization
requires dedicated methods Allen and Romano (1999); Romano and Cornish (2017)
that are quite different from the ones commonly employed for resolvable
sources.

For what concerns instrumental noise sources, the main contribution in LISA
will come from laser frequency noise LISA Consortium (2020). This critical
noise source needs to be suppressed by several orders of magnitude to allow
any GW detections with LISA. This noise suppression will be achieved using an
on-ground data processing technique called Time-Delay Interferometry (TDI)
Tinto and Dhurandhar (2021). The results of the TDI algorithm are synthesized
data streams representing several laser-noise-free virtual interferometers. As
shown in Armstrong et al. (1999); Prince et al. (2002); Shaddock (2004);
Shaddock et al. (2003); Tinto et al. (2004); Vallisneri (2005); Muratore et
al. (2020); Muratore et al. (2022a), it is possible to form several TDI
channels which can have different sensitivities to GW signals and instrumental
noise. The most commonly used TDI channels for LISA data analysis are the
three Michelson like-variables, typically dubbed X, Y, and Z, often re-
combined into the (quasi-)orthogonal channels, typically dubbed A, E, and T
Prince et al. (2002). Note that there are several motivations to advocate the
search for orthogonal channels. For example, working in a diagonal TDI basis
is convenient for data analysis algorithms. Indeed, since a diagonal matrix is
trivial to invert, this avoids problems related to numerical inversion, thus
improving the numerical stability and possibly significantly speeding up
likelihood evaluations.

The AET basis is exactly orthogonal if LISA is a perfect equilateral triangle
and if all secondary noise sources, i.e., noises that dominate the LISA data
stream after laser noise suppression through TDI, such as Test Mass (TM)
acceleration and Optical Metrology System (OMS) noise, are perfectly equal in
all three spacecraft. In fact, for realistic motion of the satellites, the
arm-lengths will be unequal (and time-varying) at the percent level Martens
and Joffre (2021), and LISA will not be a perfectly equilateral triangle.
Moreover, the noises appearing in the different spacecraft will not be equal,
breaking another assumption underlying the idealized derivation of the A, E,
and T variables. Therefore the standard expressions for these channels will
not be perfectly orthogonal when computed using actual LISA data222While, in
principle, it is still possible to find other sets of TDI variables that form
an orthogonal basis, this computation would have to be performed on the fly on
the real data, vastly increasing the computational costs of the data analysis
pipelines..

In this work, we first quantify the impact that a LISA configuration with
unequal fixed armlengths and different noise levels among the satellites would
have on the orthogonality of different sets of TDI variables by an explicit
calculation of the noise and signal power spectral density (PSD) and cross-
spectral density (CSD). For simplicity, we focus on the so-called first-
generation TDI variables Armstrong et al. (1999), which only fully suppress
laser noise for a static LISA constellation, i.e., with arm-lengths that do
not evolve in time. We expect the main conclusions of this work to remain
valid under our working assumptions for the second-generation variables
Hartwig and Muratore (2022), which achieve laser noise suppression even for
time-varying arm-lengths. Besides the most commonly used GW-sensitive
Michelson variables XYZ and the corresponding quasi-orthogonal AET channels,
we consider the Sagnac variables $\alpha\beta\gamma$ and their corresponding
set of quasi-orthogonal channels, which we denote
$\mathcal{A}\mathcal{E}\mathcal{T}$. In addition, we consider the fully-
symmetric Sagnac variable $\zeta$, which shares with T and $\mathcal{T}$ the
property that it is quasi-insensitive to GWs and can be used instead of these
channels to form a quasi-orthogonal set with A, E or $\mathcal{A}$,
$\mathcal{E}$.

We then focus specifically on AET and AE$\zeta$ and use simulated data
together with Markov Chain Monte Carlo (MCMC) Parameter Estimation (PE) as
well as the Fisher Information Matrix (FIM) formalism to further study the
sensitivity of both sets of TDI variables to a power law SGWB and to
instrumental noise as defined in Amaro-Seoane et al. (2017), when cross-
correlations are included or neglected. While both sets of variables are
expected to be orthogonal sets and perfectly equivalent in the idealized
situation of equal LISA arms and equal noise amplitudes, differences arise
when the assumptions underlying the construction of the orthogonal channels
are broken.

The paper is organized as follows. In section II, we describe the data model
employed and derive the signal response of LISA to a SGWB for unequal LISA
arms. We also show how the two dominant noise sources left after TDI (TM and
OMS) appear in the LISA measurements. We then introduce the TDI formalism and
derive a general formula for the noise and signal PSD and CSD for all the TDI
channels considered in this work. In section III, we discuss the noise and
signal PSDs and CSDs when the noise levels are assumed to be the same for all
TM and OMS components, or when each TM and OMS component is assumed to be
different. In either case, we compare the signal and noise correlations for
equal and unequal LISA arms. In section IV, as mentioned above, for a specific
set of noise parameters and for a power-law SGWB signal, we produce simulated
LISA data and perform PE using MCMC to compare the performance of two sets of
TDI variables, namely AET and $AE\zeta$, when the cross-correlations in the
TDI matrix are neglected. Using FIM, we also study the impact of including or
neglecting those cross-correlations on the PE results. We do so for both equal
and unequal noise levels and report on the results obtained for the SGWB in
the main text, while those for the noise parameters can be found in an
appendix. We then conclude in section V.

Our paper includes four appendices. Appendix A discusses the relationships
among the different sets of TDI variables considered. In appendix B we provide
useful analytic approximations for the signal and noise spectra. In appendix
C, we give an overview of the data analysis method employed in section IV.
Finally, appendix D contains a detailed analysis of the noise reconstruction
which serves to complement the SGWB signal parameter reconstruction provided
in section IV.

##  II Measurement characterization

In this section, we describe the signal and noise components of the LISA data
stream. After writing down a general model for the data as a superposition of
signal and noise in section II.1, we derive the instrument response for an
isotropic SGWB signal and the propagation of the different noise components in
a single LISA link in section II.1.1 and section II.1.2, respectively. In
section II.2 we introduce the TDI variables that we will use in this paper.
After providing the definitions of the base variables in section II.2.1, we
introduce, in section II.2.2, a general method to compute the TDI PSDs and
CSDs using the single link spectra computed in section II.1.1 and section
II.1.2.

###  II.1 Data model

Let us start by assuming that all transient signals and glitches in the noise
have been subtracted from the data stream, which is necessary to assume the
noise to be stationary and Gaussian. For the moment, let us also restrict to
the case of a single detector. Under these assumptions, the time domain data
$d(t)$, can be expressed as a combination of the GW signal $s(t)$ plus the
instrumental noise $n(t)$ as:

$d(t)=s(t)+n(t)\;.$ (1)

While in reality, data will be sampled at a finite rate, in the following, we
assume them to be continuous functions in the interval $[-T/2,T/2]$, with $T$
the observation time333This choice would only impact the high-frequency part
of the data, where the frequency gets close to the Nyquist frequency. Since
for LISA, the sampling rate is expected to be $\gtrsim$ 2Hz, and since in the
analysis we only consider data up to $.5Hz$, this assumption should not impact
the main results of this work.. Assuming that signal and noise are
uncorrelated, these quantities can be discussed separately. We start by
discussing the signal properties, whose Fourier transform reads:

$\tilde{s}(f)=\int_{-T/2}^{T/2}\textrm{e}^{2\pi ift}\;s(t)\;\textrm{d}t\;.$
(2)

Since $s(t)$ is real, $\tilde{s}$ obeys $\tilde{s}(f)=\tilde{s}^{*}(-f)$.
Assuming stationarity, the expectation value of the signal’s Fourier modes
reads:

$\langle\tilde{s}(f)\tilde{s}^{*}(f^{\prime})\rangle=\frac{1}{2}\delta(f-f^{\prime})S^{\mathrm{GW}}(f)\;,$
(3)

where $S^{\mathrm{GW}}(f)$ is a real and positive function with
$S^{\mathrm{GW}}(f)=S^{\mathrm{GW}}(-f)$, which for a homogeneous and
isotropic power spectrum444In general, the spectrum
$P_{h}^{\lambda\lambda^{\prime}}$, with $\lambda$, $\lambda^{\prime}$ running
over the two GW polarizations, defines a $2\times 2$ matrix. The four entries
of this matrix are typically expressed in terms of the Stokes parameters
$I,V,Q,U$. In the L/R basis (see eq. 10), $I$ and $V$, with $I$ the intensity
and $V$ the circular polarization, only contribute to the diagonal while $Q$
and $U$ appear in the off-diagonal terms only. Homogeneity and isotropy
correspond to vanishing $Q$ and $U$ Stokes parameters, implying that
$P_{h}^{AB}$ is diagonal with $P_{h}^{RR}=I+V$ and $P_{h}^{LL}=I-V$. Finally,
for a non-chiral background, $V=0$, which implies
$P_{h}^{RR}=P_{h}^{LL}=P_{h}$. $P^{\lambda}_{h}$, with $\lambda$ denoting the
two GW polarizations, see eq. 10 for the definition of the polarization
tensors, can be expressed as:

$S^{\mathrm{GW}}(f)=\sum_{\lambda}\mathcal{R}_{\lambda}(f)P^{\lambda}_{h}(f)\;,$
(4)

where $\mathcal{R}_{\lambda}$, is the sky-averaged LISA response function.
Note that since the LISA spacecraft lie in a plane, LISA cannot distinguish
between chiralities without making use of the motion of the constellation Seto
and Taruya (2007, 2008); Smith and Caldwell (2017); Domcke et al. (2020).
Here, we assume the signal to be non-chiral i.e., $P^{L}_{h}=P^{R}_{h}$, so
that eq. 4 reduces to $S^{\mathrm{GW}}(f)=2\mathcal{R}(f)P_{h}(f)$. Absorbing
the factor 2 into $\mathcal{R}$ and assuming the signal to be parity even
(i.e., $P^{\lambda}(\vec{k})=P(k)$) leads to

$S^{\mathrm{GW}}(f)=\mathcal{R}(f)\,P_{h}(f)\;.$ (5)

We conclude our discussion of the signal by recalling the expression of the
signal power spectrum in units of the energy density parameter:

$\Omega_{GW}h^{2}\equiv\frac{4\pi^{2}}{3(H_{0}/h)^{2}}f^{3}P_{h}(f)\;,$ (6)

where $H_{0}\approx 3.24\times 10^{-18}h_{0}$ Hz is the Hubble constant today
and $h_{0}=0.6766\pm 0.0042$ is its dimensionless value Aghanim et al. (2020).

The expectation value of the noise’s Fourier modes reads:

$\langle\tilde{n}(f)\tilde{n}^{*}(f^{\prime})\rangle=\frac{1}{2}\delta(f-f^{\prime})S^{\mathrm{N}}(f)\;,$
(7)

where we have introduced the noise power spectrum $S^{\mathrm{N}}(f)$, which
satisfies the same properties as $S^{\mathrm{GW}}(f)$. Note that in the
following sections, we consider the case of several data streams. In this
generalized and more realistic scenario, the response function and noise
spectra will be replaced by positive-definite Hermitian matrices.

####  II.1.1 Single link signal response

To compute the GW response, we start by expressing a GW signal
$h_{ab}(\vec{x},t)$ as a superposition of plane waves Maggiore (2000); Smith
and Caldwell (2017); Romano and Cornish (2017) (note that $c=1$):

$h_{ab}(\vec{x},t)=\int_{-\infty}^{\infty}\differential
f\int\differential\Omega_{\hat{k}}\;\textrm{e}^{2\pi
if(t-\hat{k}\cdot\vec{x})}\sum_{A}\tilde{h}_{A}(f,\hat{k})e^{A}_{ab}(\hat{k})\;,$
(8)

with $f$ the GW frequency, $\hat{k}$ the outward vector in the direction of
the incoming GW, $d\Omega_{\hat{k}}$ the infinitesimal solid angle and
$e^{A}_{ab}(\hat{k})$ the polarization tensors. Following the convention of
Domcke et al. (2020); Bartolo et al. (2018); Flauger et al. (2021), given the
normalized wave-vector $\hat{k}$, we can introduce the two vectors:

$\hat{u}(\hat{k})\equiv\frac{\hat{k}\times\hat{\rm
e}_{z}}{|\hat{k}\times\hat{\rm
e}_{z}|}\;,\qquad\qquad\hat{v}(\hat{k})\equiv\hat{k}\times\hat{u}\;,$ (9)

where $\times$ denotes the external product and $\hat{\rm e}_{z}$ is the
z-component vector of an arbitrarily oriented reference system. The $+/\times$
and $L/R$ polarization tensors are defined in terms of $\hat{u}(\hat{k})$ and
$\hat{v}(\hat{k})$ as:

$e^{+}_{ab}(\hat{k})\equiv\hat{u}_{a}\hat{u}_{b}-\hat{v}_{a}\hat{v}_{b}\;,\qquad
e^{\times}_{ab}(\hat{k})\equiv\hat{u}_{a}\hat{v}_{b}+\hat{v}_{a}\hat{u}_{b}\;,\qquad
e^{L/R}_{ab}(\hat{k})\equiv e^{+}_{ab}(\hat{k})\mp
ie^{\times}_{ab}(\hat{k})\,,$ (10)

expressed in the L/R or $+/\times$ bases. We proceed by assuming that the LISA
constellation is static and in a flat background spacetime. The time delay
induced by GWs on a photon leaving at time $t-L_{ij}$ ($L_{ij}$ being the
distance $|\vec{x}_{i}-\vec{x}_{j}|$) from $\vec{x}_{j}$ and reaching
$\vec{x}_{i}$ at time $t$, can be expressed, at lowest order in $h_{ab}$, as:

$\Delta
t_{ij}(t)\simeq\int_{0}^{L_{ij}}\frac{\hat{l}^{a}_{ij}\;\hat{l}^{b}_{ij}}{2}h_{ab}(t(s),\vec{x}(s))\;\differential
s\;,$ (11)

where $\hat{l}_{ij}=(\vec{x}_{j}-\vec{x}_{i})/|\vec{x}_{j}-\vec{x}_{i}|$ is a
unit vector pointing from $i$ to $j$, and $t(s)\equiv t-L_{ij}+s$,
$\vec{x}(s)=\vec{x}_{j}-s\,\hat{l}_{ij}$ are respectively the time and
position along the photon path expressed in terms of the affine parameter $s$.
By inserting eq. 8 into eq. 11, and by considering the fractional frequency
shift rather than the time delay induced by the GWs, we obtain:

$\eta_{ij}^{\mathrm{GW}}(t)=\frac{\differential}{\differential t}\Delta
t_{ij}(t)=i\int_{-\infty}^{\infty}\differential
f\,\frac{f}{f_{ij}}\textrm{e}^{2\pi
if(t-L_{ij})}\int\differential\Omega_{\hat{k}}\left[\textrm{e}^{-2\pi
if\hat{k}\cdot\vec{x}_{i}}\sum_{A}\xi^{A}_{ij}(f,\hat{k})\tilde{h}_{A}(f,\hat{k})]\right]\;,$
(12)

where we have introduced the characteristic frequencies $f_{ij}\equiv(2\pi
L_{ij})^{-1}$ and the functions $\xi^{A}_{ij}\left(f,\hat{k}\right)$ defined
as:

$\xi^{A}_{ij}\left(f,\hat{k}\right)=e^{-2\pi
if\hat{k}\cdot\vec{L}_{ij}}\mathcal{M}_{ij}(f,\hat{k})\;\mathcal{G}^{A}(\hat{k},\hat{l}_{ij})\;,$
(13)

with:

$\mathcal{M}_{ij}(f,\hat{k})\equiv\mathrm{e}^{\pi
ifL_{ij}(1+\hat{k}\cdot\hat{l}_{ij})}\;\mathrm{sinc}\left(\pi
fL_{ij}(1+\hat{k}\cdot\hat{l}_{ij})\right)\quad\mathrm{and}\quad\mathcal{G}^{A}(\hat{k},\hat{l}_{ij})\equiv\frac{\hat{l}^{a}_{ij}\hat{l}^{b}_{ij}}{2}e^{A}_{ab}(\hat{k})\;.$
(14)

Using eq. 3, the CSD for $\tilde{\eta}_{ij}^{\mathrm{GW}}(f)$, i.e. the
Fourier transform of eq. 12, then reads555This equation assumes $Tf\gg 1$, so
that the finite-time delta functions, arising from the Fourier transform, can
be replaced with real delta functions.:

$S^{\eta,\mathrm{GW}}_{ij,mn}(f)\equiv\sum_{A}\mathcal{R}^{A}_{ij,mn}\,P_{h}^{AA}(f)=\frac{f^{2}}{f_{ij}f_{mn}}e^{-2\pi
if(L_{ij}-L_{mn})}\sum_{A}P_{h}^{AA}(f)\;\Upsilon_{ij,mn}^{A}(f)\;,$ (15)

with:

$\Upsilon_{ij,mn}^{A}(f)=\int\frac{\differential\Omega_{\hat{k}}}{4\pi}\;\textrm{e}^{-2\pi
if\hat{k}\cdot(\vec{x}_{i}-\vec{x}_{m})}\;\xi^{A}_{ij}(f,\hat{k})\,\xi^{A}_{mn}(f,\hat{k})^{*}\;,$
(16)

where we have used the statistical properties of $\langle
h_{A}(f,\hat{k})\,h_{B}^{*}(f^{\prime},\hat{k}^{\prime})\rangle$ for a
homogeneous, isotropic and non-chiral SGWB, i.e.:

$\langle\tilde{h}_{A}(f,\hat{k})\,\tilde{h}_{B}^{*}(f^{\prime},\hat{k}^{\prime})\rangle=\delta(f-f^{\prime})\delta(\hat{k}-\hat{k}^{\prime})\delta_{AB}\frac{P_{h}^{AB}(f)}{16\pi}\;\qquad\qquad\langle\tilde{h}_{A}(f,\hat{k})\,\tilde{h}_{B}(f^{\prime},\hat{k}^{\prime})\rangle=0\;,$
(17)

with $P_{h}^{AB}(f)$ the one-sided PSD. The integral appearing in eq. 15 can
generally be computed numerically and in the low-frequency approximation, it
can be computed analytically (see appendix B, in particular section B.3).

####  II.1.2 Single link noise spectra

In this section we discuss the two main unsuppressed secondary noises limiting
the performance of LISA. We consider TM acceleration noise, and OMS noise
Amaro-Seoane et al. (2017). These noise sources enter the single link
measurement as666For a more detailed computation considering the “split
interferometry” scheme see Hartwig (2021). Note that the TM and OMS noise
contributions have different correlation properties, which will ultimately
cause the two components to have different transfer functions in the various
TDI channels.:

$\eta_{ij}^{\mathrm{N}}(t)=n^{\text{OMS}}_{ij}(t)+D_{ij}n_{ji}^{\text{TM}}(t)+n_{ij}^{\text{TM}}(t)\;,$
(18)

where $D_{ij}$ is the delay operator, which in the static LISA arm
approximation used in this work acts on any time-dependent function $x(t)$ as
$D_{ij}x(t)=x(t-L_{ij})$. We formally define the single links noise CSDs as:

$\langle\tilde{\eta}_{ij}^{\mathrm{N}}(f)\,\tilde{\eta}_{lm}^{\mathrm{N}*}(f^{\prime})\rangle=\frac{1}{2}\,S_{ij,lm}^{\eta,\mathrm{N}}(f)\,\delta(f-f^{\prime})\;,$
(19)

and assume the individual noise terms to be stationary, zero mean, and
uncorrelated, such that cross-terms between noises vanish, with the only non-
zero terms given by:

$\displaystyle\langle\tilde{n}^{\text{OMS}}_{ij}(f)\,\tilde{n}^{\text{OMS*}}_{ij}(f^{\prime})\rangle$
$\displaystyle=\frac{1}{2}\,S^{\text{OMS}}_{ij}(f)\,\delta(f-f^{\prime})\;,$
(20a)
$\displaystyle\langle\tilde{n}^{\text{TM}}_{ij}(f)\,\tilde{n}^{\text{TM*}}_{ij}(f^{\prime})\rangle$
$\displaystyle=\frac{1}{2}\,S^{\text{TM}}_{ij}(f)\,\delta(f-f^{\prime})\;.$
(20b)

Here, $S^{\text{OMS}}_{ij}(f)$ and $S^{\text{TM}}_{ij}(f)$ are the PSDs of the
individual OMS and TM acceleration noise terms. Considering the definition of
the noise CSD and the single link measurement, given eq. 18 and eq. 20, we can
then directly compute the non-zero entries of the noise CSD of the single
links as:

$\displaystyle S_{ij,ij}^{\eta,\mathrm{N}}(f)$
$\displaystyle=S^{\text{OMS}}_{ij}(f)+S^{\text{TM}}_{ij}(f)+S^{\text{TM}}_{ji}(f)\;,$
(21a) $\displaystyle S_{ij,ji}^{\eta,\mathrm{N}}(f)$ $\displaystyle=e^{2\pi
ifL_{ji}}S^{\text{TM}}_{ij}(f)+e^{-2\pi ifL_{ij}}S^{\text{TM}}_{ji}(f)\;.$
(21b)

In the following, we will further assume all noises of the same type to have
the same spectral shape given by Amaro-Seoane et al. (2017):

$\displaystyle S^{\text{TM}}_{ij}(f)$ $\displaystyle=A_{ij}^{2}\times
10^{-30}\;\times\quantity(1+\quantity(\frac{$0.4\text{\,}\mathrm{mHz}$}{f})^{2})\quantity(1+\quantity(\frac{f}{$8\text{\,}\mathrm{mHz}$})^{4})\times\quantity(\frac{1}{2\pi
fc})^{2}\;\times(\textrm{m}^{2}/\textrm{s}^{3})\;,$ (22a) $\displaystyle
S^{\text{OMS}}_{ij}(f)$ $\displaystyle=P_{ij}^{2}\times
10^{-24}\;\times\quantity(1+\quantity(\frac{$2\text{\times}{10}^{-3}\text{\,}\mathrm{Hz}$}{f})^{4})\times\quantity(\frac{2\pi
f}{c})^{2}\;\times(\textrm{m}^{2}/\textrm{Hz})\;,$ (22b)

such that each noise depends only on a single constant amplitude parameter
$A_{ij}$ for TM and $P_{ij}$ for OMS. These parameters are dimensionless so
that the overall noise PSDs are given in units of fractional frequency
deviations.
In this work, we consider two scenarios:

  1. 1.

The noise amplitudes of TM on the one hand and OMS on the other are equal,
i.e., $A_{ij}=A$ and $P_{ij}=P$, with central values respectively given by
$A=3$ and $P=15$ Babak et al. (2021).

  2. 2.

The noise parameters $A_{ij}$ and $P_{ij}$ have random values lying within a
standard deviation of $20\%$ around the central values777Randomly drawing the
noise amplitudes with a standard deviation of $20\%$ causes the largest and
smallest noise terms to be approximately within a factor 2. This is roughly in
line with what was observed in LISA PathFinder (LPF), where the observed noise
levels were within a factor of a few of their anticipated values Castelli
(2020). Note that while the LPF noise measured in flight did not agree with
the predicted noise level nor with its shape (especially at frequencies below
$10^{-3}$ Hz), here, we make the rather strong assumption that the noise
shapes are perfectly known.. The exact values of the $A_{ij}$ and $P_{ij}$
(with $\left\\{ij\right\\}\in\mathcal{I}=\\{12,23,31,21,32,13\\}$) are given
in section III.2.

For reference, a plot of the noise levels for the case $A_{ij}=A=3$ and
$P_{ij}=P=15$, is shown in fig. 1.

Figure 1: TM and OMS PSDs as a function of frequency in units of $1/$Hz.

###  II.2 TDI variables

The TDI variables that we will consider throughout this paper are based on the
standard X, Y, and Z Michelson combinations, as well as the $\alpha$, $\beta$,
and $\gamma$ Armstrong et al. (1999) Sagnac variables. We will also discuss
the orthogonal channels A, E, and T and $\mathcal{A}$, $\mathcal{E}$ and
$\mathcal{T}$ build out of these sets of base variables, respectively, as well
as the fully symmetric Sagnac (null channel) $\zeta$ Hogan and Bender (2001);
Muratore et al. (2022a); Tinto and Dhurandhar (2021). For simplicity, we
consider the first-generation versions of all these variables, which fully
suppress laser noise only for a constellation with three unequal but constant
arms888Laser noise suppression with more realistic orbits requires additional
“virtual loops” in the photon paths that make up the TDI combinations.
Following Vallisneri (2005); Muratore et al. (2020); Muratore et al. (2022a);
Hartwig and Muratore (2022), there exist several second-generation versions of
each of the first-generation TDI variables. The “standard” choices that are
found in the literature can be approximated as:  $\displaystyle{\rm X}_{2}$
$\displaystyle=(1-D_{31}^{2}D_{12}^{2}){\rm X}\;,$ (23a)
$\displaystyle\alpha_{2}$ $\displaystyle=(1-D_{12}D_{23}D_{31})\alpha\;,$
(23b) $\displaystyle\zeta_{2}$ $\displaystyle=(D_{31}-D_{12}D_{23})\zeta\;.$
(23c) Note that $\zeta_{2}$ here refers to the variable recently found in
Muratore et al. (2020), not the traditional variable given in Tinto et al.
(2004), which is less effective at suppressing laser noise than other second-
generation TDI variables. These additional pre-factors should not
significantly impact the sensitivity of the TDI variables taken individually,
as they apply equally to both noise and signal. However, we remark that the
factors applied to ${\rm X}_{2}$, ${\rm Y}_{2}$ and ${\rm Z}_{2}$ are
different, such that the full CSD matrix of the second-generation Michelson
variables cannot be trivially derived from the first-generation versions. It
is different for the Sagnac variables, where all three permutations are
modified by the same overall factor., and compute their signal and noise CSDs
in terms of the single-link CSDs given in the previous sections.

####  II.2.1 Base variable definition

Let us start by defining the first-generation version of the XYZ Michelson
variables. The X variable is defined as:

${\rm
X}=(1-D_{13}D_{31})(\eta_{12}+D_{12}\eta_{21})+(D_{12}D_{21}-1)(\eta_{13}+D_{13}\eta_{31})\;,$
(24a) while Y and Z are cyclic permutations of X. The Sagnac variable $\alpha$
is defined as:
$\alpha=\eta_{12}+D_{12}\eta_{23}+D_{12}D_{23}\eta_{31}-(\eta_{13}+D_{13}\eta_{32}+D_{13}D_{32}\eta_{21})\;,$
(24b) with again $\beta$, $\gamma$ defined as cyclic permutations of $\alpha$.
Finally, the fully symmetric $\zeta$ variable is defined as:
$\zeta=D_{12}(\eta_{31}-\eta_{32})+D_{23}(\eta_{12}-\eta_{13})+D_{31}(\eta_{23}-\eta_{21})\;.$
(24c)

Following Prince et al. (2002), we introduce the so-called quasi-orthogonal
TDI channels, which are usually given as:

$\mathcal{A}=\frac{\gamma-\alpha}{\sqrt{2}}\;,\qquad\mathcal{E}=\frac{\alpha-2\beta+\gamma}{\sqrt{6}}\;,\qquad\mathcal{T}=\frac{\alpha+\beta+\gamma}{\sqrt{3}}\;,$
(25)

for the Sagnac variables. An analogous procedure can be carried out for the
Michelson variables, giving:

${\rm A}=\frac{{\rm Z}-{\rm X}}{\sqrt{2}}\;,\qquad{\rm E}=\frac{{\rm X}-2{\rm
Y}+{\rm Z}}{\sqrt{6}}\;,\qquad{\rm T}=\frac{{\rm X}+{\rm Y}+{\rm
Z}}{\sqrt{3}}\;.$ (26)

These channels are designed to be orthogonal for both signal and noise, at
least in the idealized case of equal arm and equal and symmetric noise levels
in the three base variables. Having orthogonal channels drastically simplifies
the computation of the inverse of the covariance matrix appearing in the
likelihood, which makes these channels attractive for the practical
application of LISA data analysis. We will discuss in section III the extent
to which the orthogonality of these channels survives if one relaxes some of
the assumptions used to derive them. Note finally that the different TDI
variables discussed here are not independent, see appendix A for more details.

####  II.2.2 Signal and noise projection on the TDI variables

In order to compute the noise PSDs and the response to a SGWB, we first need
to evaluate the Fourier transform of any TDI variable $V$, for which we use
the compact vector notation:

$\tilde{V}(f)=\sum_{ij\in\mathcal{I}}\,c^{V}_{ij}(f)\;\tilde{\eta}_{ij}(f)\;,$
(27)

where, as in section II.1.2, $\mathcal{I}=\\{12,23,31,21,32,13\\}$ denotes the
pairs of indices that define the six inter-satellite links and where the
coefficients $c^{V}_{ij}$ map the single-link measurements onto the TDI
variable $V$. Since we work in the assumption of constant delays, we can
directly read off the coefficients $c^{V}_{ij}$ from eq. 24 by replacing each
delay in the time domain, $D_{ij}$, with the corresponding frequency domain
expression $e^{-i2\pi fL_{ij}}$. Assuming $\tilde{U}$ and $\tilde{V}$ to be
any two TDI variables, which, similarly to eq. 19, obey:

$\langle\tilde{U}(f)\,\tilde{V}^{*}(f^{\prime})\rangle=\frac{1}{2}\,S^{UV}(f)\,\delta(f-f^{\prime})\;,$
(28)

we can substitute eq. 27 into this expression to get:

$\displaystyle\langle\tilde{U}(f)\tilde{V}^{*}(f^{\prime})\rangle$
$\displaystyle=\sum_{ij,mn\in\mathcal{I}}c^{U}_{ij}(f)\,c^{V*}_{mn}(f^{\prime})\langle\tilde{\eta}_{ij}(f)\,\tilde{\eta}_{mn}^{*}(f^{\prime})\rangle\;,$
(29a)
$\displaystyle=\frac{1}{2}\underbrace{\sum_{ij,mn\in\mathcal{I}}\underbrace{c^{U}_{ij}(f)\,c^{V*}_{mn}(f)}_{C^{UV}_{ij,mn}(f)}\,S^{\eta}_{ij,lm}(f)}_{S^{UV}(f)}\,\delta(f-f^{\prime})\;.$
(29b)

For any particular $(\tilde{U}\,,\tilde{V})$ combination, the coefficients
$c^{U}_{ij}(f)$ and $c^{V}_{ij}(f)$ can be combined to form a $6\times 6$
matrix999A compact notation one could use in order to consider several, say
$n$, TDI variables would require for the coefficients to be arranged into
$n\times 6$ matrices, and the $C^{UV}$ to be replaced by a rank 4 tensor
($n\times n\times 6\times 6$), which maps the $6\times 6$ single-link
correlations onto the $n\times n$ TDI variable correlations. $C^{UV}$. We
remark that this procedure is similar to what has very recently (and
independently from our work) been proposed in Baghi et al. (2023). Note that
the coefficient matrices $C^{UV}$ depend only on the choice of TDI variables,
while the actual noise or signal correlations are encoded in the previously
computed single-link correlation matrix $S^{\eta}$. In the following sections,
we will use:

$S^{UV,\mathrm{N}}=C^{UV}S^{\eta,\mathrm{N}},\quad
S^{UV,\mathrm{GW}}=C^{UV}S^{\eta,\mathrm{GW}}\;,$ (30)

to identify the noise and signal TDI covariance, respectively. Explicit
expressions for the CSDs are given in appendix B.

##  III Relaxing the equilateral assumption: spectral analysis

In this section, we present the signal response, the noise spectra, and the GW
sensitivities for all the TDI variables considered in this work. We consider
both the case of an equilateral and that of a non-equilateral LISA
configuration. We perform a detailed correlation analysis to test the
robustness of the orthogonalization procedure for the different TDI bases. The
section is divided into two parts: the first part concerns the case of equal
noise levels while the second part treats the case of unequal noise levels.
All plots presented in this section will show the equal-arm model with solid
lines while the unequal-arm results are plotted using dashed lines. Analytic
expressions for the noise CSDs in the equal and (in the low-frequency limit)
unequal arms case are provided in appendix B. The full expressions for the
noise spectra are provided as supplementary material. The signal response
shown in the plots was evaluated numerically (analytic expressions for the
signal exist only in the low-frequency approximation, which are also provided
in appendix B).

Figure 2: Signal response (first row), noise spectra (second row), and strain
sensitivities (last row) defined as in eq. 30, for the XYZ Michelson variables
(first column) and for their orthogonal AET combinations as well as for the
$\zeta$ variable (second column) considering equal (solid lines) and un-equal
(dashed-line) arm-lengths.

###  III.1 Equal noise levels

Let us focus on the case where all the noises of the same type are
characterized by a single amplitude parameter, i.e., $A_{ij}\equiv A$,
$P_{ij}\equiv P$ so that $S_{ij}^{\mathrm{OMS}}\equiv S^{\mathrm{OMS}}$ and
$S_{ij}^{\mathrm{TM}}\equiv S^{\mathrm{TM}}$ for all links. Figure 2 shows the
three quantities that best describe the self-correlations of the XYZ Michelson
variables, the AET orthogonal variables, and as well as the $\zeta$ channel.
In the top row, we plot the quadratic signal response $\mathcal{R}$, as
defined in section II.1, in units of squared fractional frequency deviation.
We observe that while the equal and unequal arms models agree with one another
for the signal-dominated channels A and E, they disagree at low frequencies
for the null channels T and $\zeta$, especially for T. In the second row, we
show the full noise spectra, $S^{UU,N}$, in units of squared fractional
frequency deviation per $\mathrm{Hz}$. We notice that the equal and unequal
arms models agree for all channels with the exception of T. Finally, the third
row shows the strain sensitivity101010Note that the strain sensitivity is a
function of the noise model, the instrument response, and the TDI variable,
but is independent of the GW signal. computed as the ratio between the noise
PSD $S^{UU,N}$ and the signal response $\mathcal{R}$. We observe that the
strain sensitivities of X, Y, Z, A, and E are well approximated by the equal
arm model, while there are sizeable differences for the two null channels. T,
in particular, is no longer a null channel below $1\text{\,}\mathrm{mHz}$, and
becomes as sensitive to the signal as A and E (as also reported in Adams and
Cornish (2010) and more recently in Muratore et al. (2022a)). Notice that
$\zeta$ also becomes more sensitive to GWs, but remains significantly less
sensitive than A and E.

Figure 3: Signal response (first row), noise spectra (second row) and strain
sensitivities (last row) defined as in eq. 30, for the Sagnac
$\alpha,\beta,\gamma$ variables (first column) and for their orthogonal
combinations $\mathcal{A}$, $\mathcal{E}$, and $\mathcal{T}$ as well as for
the $\zeta$ variable (second column).

In fig. 3 we show the same quantities as in fig. 2, but for the Sagnac
variables, the orthogonal variables $\mathcal{A}$, $\mathcal{E}$ and
$\mathcal{T}$, and again the channel $\zeta$. Once again, the base variables
$\alpha$, $\beta$, $\gamma$ as well as the sensitive channels $\mathcal{A}$,
$\mathcal{E}$ are well approximated by the equal arms model. The null channel
$\mathcal{T}$ behaves very similarly to $\zeta$, meaning that its noise
spectrum is also well described using an equal arms model. It remains a valid
null channel at low frequencies and slightly outperforms $\zeta$ for unequal
arms. Let us also note that the noise spectra of the base variables $\alpha$,
$\beta$, $\gamma$, and $\zeta$ have no zeros within the LISA band. While this
property is not fully retained by $\mathcal{A}$ and $\mathcal{E}$, they do
have fewer zeros than A and E (compare fig. 2).

Figure 4: Noise correlations considering equal noise levels for the TM
acceleration and OMS noises and equal (solid) vs. unequal arms (dashed lines).
The first column shows XYZ (first row), AET (second row), and AE$\zeta$. The
second column shows $\alpha\beta\gamma$ (first row), $\mathcal{AET}$ (second
row), and $\mathcal{AE}\zeta$.

In fig. 4 we show the square root of the noise coherence, defined by
$|S^{UV,\mathrm{N}}|/\sqrt{S^{UU,\mathrm{N}}\,S^{VV,\mathrm{N}}}$, for the
three Michelson and Sagnac variables, for their orthogonal combinations, using
the signal-insensitive variables T, $\mathcal{T}$ and $\zeta$. In the first
row, we show the noise coherence for the base variables X, Y, and Z and
$\alpha$, $\beta$, and $\gamma$. Both sets of three channels are strongly
correlated, and the Sagnac channels are almost 100% correlated at low
frequencies. In the second and third rows, we show the results obtained for
the orthogonal channels. AE$\zeta$, $\mathcal{A}\mathcal{E}\mathcal{T}$ and
$\mathcal{A}\mathcal{E}\zeta$ remain strongly uncorrelated (except close to
the zeros). $\mathcal{A}\mathcal{E}\mathcal{T}$ is slightly less correlated
than $\mathcal{A}\mathcal{E}\zeta$ at low frequencies while it is more
strongly correlated in the oscillatory region at high frequencies. AE$\zeta$
and $\mathcal{A}\mathcal{E}\zeta$ are very similar. T, however, becomes
strongly correlated to A and E across a broad range of frequencies. Note that
there are no solid lines in the second and third rows. Indeed, by
construction, the orthogonal variables are fully uncorrelated when assuming
equal arms and equal noise levels.

In fig. 5 we show the square root of the signal response coherence
$|\mathcal{R}^{UV}|/\sqrt{\mathcal{R}^{UU}\mathcal{R}^{VV}}$ for all TDI
variables considered. In the first row, we give the square root of the signal
coherence for the base XYZ variables and for $\alpha\beta\gamma$. Both sets of
TDI variables are strongly correlated. The equal and unequal arms models agree
very well at low frequencies but show some differences near the peaks around
the zeros at high frequencies. Note that while the noise spectra of $\alpha$,
$\beta$, and $\gamma$ do not have any zeros in the LISA band, the CSDs do have
zeros, but not as densely spaced as the ones for the Michelson variables. The
second row shows the square root of the signal response coherence for the
orthogonal AET and $\mathcal{A}\mathcal{E}\mathcal{T}$ variables. Note that A
and E, and similarly $\mathcal{A}$ and $\mathcal{E}$, are not as correlated as
the other pairs of TDI variables in the orthogonal sets. This is also true
when T and $\mathcal{T}$ are replaced with $\zeta$, as shown in the third row,
such that all null channels show strong low-frequency correlations with the
other variables. We note that, at the same time, the T variable becomes
significantly more sensitive to GWs than $\mathcal{T}$ or $\zeta$, as shown in
in fig. 2 and fig. 3. This implies that the same loss in signal orthogonality
for T, $\mathcal{T}$ and $\zeta$ more significantly impacts the overall
orthogonality of AET than that of the other sets.

Figure 5: Signal response correlations for the Michelson variables XYZ (first
row, first column), the Sagnac variable $\alpha\beta\gamma$ (first row, second
column), their respective orthogonal channels AET (second row, first column),
$\mathcal{A}\mathcal{E}\mathcal{T}$ (second row, second column) and
combinations including the null channels AE$\zeta$ (last row, first column),
$\mathcal{A}\mathcal{E}\zeta$ (last row, second column). Equal arms are shown
with solid lines and unequal arms with dashed lines.

###  III.2 Unequal noise levels

In this section, we consider the more general and realistic case in which all
the individual TM and OMS noise levels, $A_{ij}$ and $P_{ij}$ (with
$ij\in\\{12,23,31,21,32,13\\}$) are unequal. The values of $A_{ij}$ and
$P_{ij}$ are generated as explained in section II.1.2 and, in the specific
example we discuss here, they take the following values:

$A_{ij}=\\{3.61,3.02,2.87,3.43,2.65,3.45\\}\;,\qquad
P_{ij}=\\{14.00,16.93,9.43,21.55,17.04,20.83\\}$ (31)

As before, the equal arms results are shown using solid lines while the
unequal arms results are shown with dashed lines. Note that we do not show
plots for the overall noise PSDs of the variables, as they remain
qualitatively unchanged with respect to the equal noise case ones. In
addition, given that only the instrumental noise levels are changed, the
conclusions reached for the signal response and sensitivity discussed in
section III.1 remain valid.

Figure 6: Noise correlations considering six unequal levels for the TM and OMS
noises and equal (solid) vs. unequal arms (dashed lines). The first column
shows XYZ (first row), AET (second row), and AE$\zeta$. The second column
shows $\alpha\beta\gamma$ (first row), $\mathcal{AET}$ (second row), and
$\mathcal{AE}\zeta$.

The first row of fig. 6 shows the square root of the noise coherence for X, Y,
and Z and for $\alpha$, $\beta$, and $\gamma$. Both sets of variables behave
similarly to the equal noise case, with the exception that some of the zeros
are smoothed out and that the Michelson variables have slightly different
levels of coherence at low frequencies. The results for the orthogonal
channels are shown in the second row of the figure. Both AET and
$\mathcal{A}\mathcal{E}\mathcal{T}$ now show levels of coherence that reach
$\sim 10\%$ for AE at all frequencies, and several percent for AT and ET even
for equal arms. These levels are typically higher than those induced by the
inequality of the arm-lengths discussed in the previous section, that is to
say, the inequality in the levels of the noise has a stronger impact than the
inequality of the LISA arms for most pairs of variables. The results obtained
for unequal noise levels are approximately identical for all variables (with
the exception of the peaks near the zeros), whether the LISA arms are equal or
not, the only exception being T, which shows stronger correlations to A and E
when the LISA arms are unequal.

Finally, the third row of fig. 6 shows the correlations obtained when
replacing T and $\mathcal{T}$ with $\zeta$. In both cases, the overall
behaviour is very similar to that of $\mathcal{A}\mathcal{E}\mathcal{T}$, and
seems to be dominated by the correlations due to unequal noise levels at
almost all frequencies, such that the equal arm and unequal arm results are
superimposed. Differences again appear near the peaks close to the zeros, for
which $\mathcal{A}\mathcal{E}\zeta$ seems to slightly outperform both
AE$\zeta$ and $\mathcal{A}\mathcal{E}\mathcal{T}$.

##  IV SGWB and noise parameter reconstruction for AET and AE$\zeta$

As shown in the previous section, relaxing the assumption that all arms are of
equal length significantly breaks the orthogonality of the AET channels while
that of AE$\zeta$, $\mathcal{A}\mathcal{E}\mathcal{T}$, and
$\mathcal{A}\mathcal{E}\zeta$ is preserved to a large degree. Moreover,
relaxing the assumption of equal secondary noises on each spacecraft also
breaks the orthogonality of all TDI variable sets. In this section, we
restrict ourselves to the unequal arms case and study how, in practice,
neglecting the off-diagonal terms impacts the uncertainty in SGWB
reconstruction in the equal and unequal noise cases. For each of the two
cases, we perform Fisher forecasts, validated with Markov Chain Monte Carlo
(MCMC) runs111111The reader is referred to appendix C for the technical
details of the MCMC data analysis. to test the goodness of the Gaussian
approximation. We restrict the following analysis to AET and AE$\zeta$. The
MCMC runs are performed considering only the diagonal terms of the AET and
AE$\zeta$ TDI matrices121212The main reason to restrict our analysis to this
simplified scenario is that the present version of the data compression
techniques described in appendix C are not suitable for application to
problems including off-diagonal terms in the TDI matrix. Including those terms
would thus require considering the full (and uncompressed) data set, which
would significantly increase the computational cost of the MCMC runs.. We
proceed by comparing the results obtained with pure Fisher analyses,
including, or neglecting, the off-diagonal terms in the TDI matrices, in order
to assess their impact on the precision with which the determination of signal
parameters can be made. A similar analysis for the noise parameters is
presented in appendix D. We conclude by comparing the MCMC posteriors for the
signal parameters obtained with the different levels of complexity for the
LISA configuration we have discussed throughout this work.

For all the analyses discussed in this section, we assume the LISA frequency
range to be $3\times 10^{-5}\leq f\leq 5\times 10^{-1}$ Hz and the signal to
be described by a simple power-law template:

$\Omega_{\rm GW}h^{2}(f)=10^{\alpha}\,\left(\frac{f}{f_{*}}\right)^{n_{T}}\;,$
(32)

where the pivot frequency $f_{*}$ is chosen to be the geometric mean of the
minimal and maximal frequency, i.e., $f_{*}\simeq 3.873\times 10^{-3}$Hz. In
all analyses presented in this section (and similarly in appendix D), we
consider a signal with zero slope, i.e., $n_{\rm T}=0$, and $\alpha=-11.5$,
corresponding to a Signal-to-Noise Ratio (SNR) $\simeq 271$ (see eq. 51).
Finally, The single link TDI transfer function is computed as outlined in
section II.1.1 and the noise is modeled as indicated in section II.1.2. Again,
note that for brevity, we only focus on the AET and AE$\zeta$ TDI variables.

Figure 7: Signal (bottom right) and noise (AA in top left, EE in top right,
$\zeta\zeta$ in bottom left) reconstruction using the AE$\zeta$ TDI basis
assuming unequal arm-lengths with equal noise levels.

In the following (and also in appendix D), we denote quantities recentered on
zero and normalized by the fiducial values with an overbar, and we denote
quantities shifted with respect to the chain means and normalized using the
fiducial value parameter values with a tilde. Thus, while the latter provide
information on the posterior widths but also on the best-fit parameter values,
the former are best suited to compare posterior widths and shapes.

Figure 8: Signal (bottom right) and noise (AA in top left, EE in top right,
$\zeta\zeta$ in bottom left) reconstruction using the AE$\zeta$ TDI basis
assuming unequal arm-lengths with unequal noise levels.

Let us first present, in fig. 7, the reconstruction of the signal and noise
spectra (in $\Omega$ units) for unequal arms but equal noise levels using the
diagonal of the AE$\zeta$ matrix. Both the signal parameters and the two noise
parameters, and thus the signal and noise levels plotted in fig. 7, are
compatible at two sigmas with the injected signal and noise (as shown by the
green contours in fig. 10(a) for the signal and as shown also in fig. 12(b)).
Note that the noise reconstruction for the different TDI channels is so
accurate that the error bands are not visible in fig. 7. On the other hand,
while the reconstructed signal is still compatible with the injection, the
error band for the signal, shown in the bottom right panel, is sufficiently
large to be clearly visible. Although we do not include a plot of the
reconstructed signal and noise spectra obtained for AET, the posteriors for
the signal parameters are shown in fig. 10(a), while those for the noise are
shown in fig. 12(b).

Let us conclude our discussion of the unequal arms but equal noise case by
comparing the results of the MCMC runs with those of the FIM analysis, see
fig. 9(a). This comparison focuses on the posterior widths and the
degeneracies between parameters, such that all posteriors are recentered on
zero. This figure illustrates that there is nearly perfect agreement between
the results obtained with the FIM analysis performed using the diagonal of the
TDI covariance matrix and the MCMC results, and also between AE$\zeta$ and
AET. It also shows a comparison with the FIM results that include the CSDs in
the AET and AE$\zeta$ analyses and demonstrates that the reconstruction of the
signal parameters would only be marginally affected by the inclusion of these
terms.

Let us now discuss the case in which the TM and OMS noise levels are unequal,
with amplitudes $A_{ij}$ and $P_{ij}$ given in eq. 31. The signal and noise
reconstructions in the AE$\zeta$ basis are shown in fig. 8. As for the equal
noise case considered above, a detailed discussion of the noise parameter
reconstruction is presented in section D.3. Let us simply mention here that
all the noise parameters are compatible with the injection parameters at the
one or two-sigma level, see fig. 14(a) and fig. 14(b). As far as the signal
reconstruction is concerned, we can see from the bottom-right panel of fig. 8
and from the blue contours in fig. 10(a), that the reconstructed values are
within the two sigma region from the injected values. Once again, as can be
seen in fig. 9(b), we find excellent agreement between the FIM analysis and
the MCMC results for both AET and AE$\zeta$, and find that the precision with
which the signal parameters can be recovered is similar for both choices of
TDI variables. Finally, fig. 9(b) also demonstrates that the results for the
signal parameters remain the same whether one includes or excludes the off-
diagonal terms of the TDI covariance matrix.

To sum up, using fig. 9(a) and fig. 9(b), we can conclude that, for the simple
power-law model considered in this work, the precision with which the
reconstruction of the SGWB parameters can be achieved is not sensitive to the
inclusion of the off-diagonal terms in the TDI covariance matrix, and that AET
and AE$\zeta$ give comparable results.

Let us end this section by further commenting on the contour plots shown in
fig. 10. The figure provides a comparison of the MCMC results obtained for
three different scenarios: 1) equal arms and equal noises, 2) unequal arms and
equal noises, and lastly 3) unequal arms and unequal noises. Figure 10(a)
demonstrates that the reconstructed signal is compatible with the injection
parameters for all cases. Figure 10(b) demonstrates that the posterior widths
remain largely unchanged across all scenarios. These plots demonstrate that
the reconstruction of the signal parameters is only marginally sensitive to
the complexity of the underlying LISA scenario. This is one of the main
results of the present work.

(a) Equal noise case.

(b) Unequal noise case.

Figure 9: GW parameter posteriors in the case of unequal arms and equal or
unequal noises. The results of the MCMC runs, the FIM analyses using the
diagonal or the entire TDI covariance matrix are shown to superimpose for AET
and $AE\zeta$. Note that the results are centered on zero and rescaled by the
values of the fiducial parameters.

(a) SGWB parameters vs. fiducial.

(b) SGWB parameters recentered on zero.

Figure 10: Comparison of MCMC SGWB parameter reconstruction for AET and
AE$\zeta$, for all the configurations considered in this work, with UA/EA
standing for un/equal arms, and UN/EN standing for un/equal noise amplitudes.
Note that for this particular figure, the posteriors aren’t normalized by the
fiducial.

##  V Conclusions

In this work, see in particular section III, we first studied the impact of a
non-equilateral but stationary configuration for the LISA constellation and
the effect induced by considering independent noise levels for each test mass
(TM) and optical metrology system (OMS) on the orthogonality of the most well-
known TDI bases, namely the Michelson XYZ, AET, and AE$\zeta$, and the Sagnac
$\alpha\beta\gamma$, $\mathcal{A}\mathcal{E}\mathcal{T}$ and
$\mathcal{A}\mathcal{E}\zeta$. While the two noises propagate differently in
the TDI variables, leading to different cross-correlations in the TDI bases,
TM noise contributions dominate the low frequencies for signal-sensitive
variables. On the other hand, for signal-orthogonal variables, OMS noise
dominates the whole frequency range. For the case of equal noise levels, we
confirmed the result, already known in the literature Adams and Cornish
(2010), that the null channel T, built from the Michelson XYZ variables, loses
its signal and TM noise orthogonality if the constellation is not perfectly
equilateral, becoming similar to the signal-sensitive variables.
Interestingly, the loss of orthogonality between the signal-sensitive A and E
channels and the null channel, T, is worse at low frequencies.

We also showed, for the first time, that other null channels, e.g., the
$\mathcal{T}$ channel, built from the $\alpha\beta\gamma$ Sagnac TDI variables
Prince et al. (2002), or the variable $\zeta$ Armstrong et al. (1999), prove
to be more robust under perturbations of the equilateral configuration.
Moreover, these alternative TDI bases achieve the same level of laser noise
suppression when extended to second generation, requiring fewer loops around
the LISA satellites in their definitions Muratore et al. (2020). Accordingly,
this would generate fewer zeros in the response functions, reducing the
frequency window lost in the TDI definition process Hartwig and Muratore
(2022). For both unequal arms and unequal noises, we demonstrated that all TDI
bases exhibit sizeable cross-correlations in the noise and signal. For the T
channel, the arm-length mismatch still produces the largest effect at low
frequencies, while for the other channels, the impact of unequal noises is
most significant across the whole band.

We then studied the signal and noise parameter reconstruction in section IV
and appendix D respectively, using a Fisher Information Matrix (FIM) approach
and Markov Chain Monte Carlo (MCMC) runs for two out of the total of six TDI
bases, namely AET and AE$\zeta$. While the FIM analysis is a convenient tool
that provides information on the posterior widths, the MCMC is computationally
costly but also determines the best-fit parameters by confronting the signal
and noise models with the LISA data. In our MCMC runs, for the reasons
explained in section IV, we ignored the cross-correlations among various TDI
variables. On the other hand, we used FIM analysis to compare results obtained
when the cross-correlations were either included or excluded.

For the equal noise case, the large cross-correlations among the AET
(quasi-)orthogonal TDI variables induced by the non-equilateral configuration
of the LISA spacecraft result in a mild under-estimation of the TM
acceleration amplitude parameter $A$ and a slight over-estimation of the OMS
noise parameter $P$ in the signal and noise parameter reconstruction performed
using MCMC and the FIM analysis. This can be understood using the toy model of
section D.1. Note that, as mentioned already, the frequency range over which
the analysis is performed can play an important role, as shown by comparing
the two contours plots of fig. 12. These findings do not apply to the
$\mathcal{AE}\zeta$ TDI basis, which, rather than being sensitive to the arm-
length mismatch, is only affected by the inequality in the noise amplitudes.
Both in the equal and unequal noise cases, the OMS noise parameters are
typically better determined than the TM noise parameters. This is mainly due
to the fact that, while the information on the TM noises mostly comes from the
low-frequency part of the frequency band, information on OMS noises comes from
the high-frequency spectrum (for the null channels, OMS noise dominates the
entire frequency range), which as larger weight in the likelihood since it
contains more data points. While the above discussion pertains to the MCMC
runs, which were performed using only the diagonal of the TDI matrices, we
remind the reader that, as shown in the FIM results of fig. 16, including
cross-correlations has a significant impact on the uncertainty with which
these noise parameters can be obtained.

Let us stress that even if some of the noise parameters are rather loosely
constrained, the overall signal and noise shapes can be recovered with
sufficient accuracy for all the scenarios tested in this work. While some
previous analyses, e.g., Adams and Cornish (2010, 2014); Wang et al. (2022),
have already considered different levels for the TM and OMS noises, we also
studied and quantified their impact on signal parameter reconstruction for
different sets of TDI variables, going beyond the usual XYZ and AET bases. It
is particularly noteworthy that the signal parameter reconstruction does not
vary significantly among the different configurations considered, see fig. 10,
and that it is not sensitive to the inclusion of the off-diagonal terms of the
TDI matrix, see fig. 9(a) and fig. 9(b). For all these reasons, the results
presented in this work represent an important contribution to the community’s
understanding of cosmological SGWB data analysis for LISA.

We conclude by commenting on some of the assumptions of our work and on the
possibility of relaxing them in future analyses. First, let us stress that a
major limitation of our work resides in the fact that we assume perfect
knowledge of the functional form of the instrumental noise, while it is
possible that unknown sources of noise will be present in the real data.
Future studies are needed in order to test the impact of these effects on
noise and signal reconstruction. For existing works setting the path in this
direction, see, e.g., Karnesis et al. (2020); Muratore et al. (2022b); Baghi
et al. (2023).

Let us point out that the present analysis does not include any of the
possible time dependencies that might be present in the data. Beyond
transients, the arm-lengths are expected to change over time, and the noise
levels, or even the signal (e.g., the SGWB due to CGBs is expected to feature
an annual modulation Adams and Cornish (2014)), might have some modulations
over time, making the configuration non-static. Similarly, while we have
restricted our study to isotropic SGWBs, the signal might have a non-trivial
angular structure, whose reconstruction should be one of the targets to be
included in an SGWB data analysis pipeline. Existing works in this direction
range from more theoretical studies Bartolo et al. (2022) to numerical
techniques involving the decomposition of the LISA response function in pixel
space Contaldi et al. (2020) / in spherical harmonics Banagiri et al. (2021).

Finally, in this work, we have considered a simplified scenario where a single
SGWB of cosmological origin, described by a simple power-law mode with zero
tilt, is present in the LISA band. In reality, as already mentioned in this
work, at least two SGWB of astrophysical origin will be present, implying SGWB
measurement will require component separation techniques to disentangle the
different components contributing to the observed signal. See Georgousi et al.
(2022); Pozzoli et al. (2023) for studies estimating some of the astrophysical
SGWBs for LISA, using an iterative source subtraction technique Karnesis et
al. (2021), and see, e.g., Pieroni and Barausse (2020); Flauger et al. (2021);
Boileau et al. (2021, 2022, 2023) for studies attempting a simultaneous
detection of astrophysical and cosmological SGWBs. Furthermore, given that
several cosmological mechanisms can generate spectra with a more elaborate
frequency structure, one should test the robustness of the results obtained in
this work when the assumption of a power-law signal is relaxed. Future studies
aiming to be more realistic would have to include a combination of some of,
and possibly all, these effects.

###### Acknowledgements.

The authors thank Aurelien Hees and Peter Wolf, as well as Quentin Baghi,
Jean-Baptiste Bayle, Valerie Domcke, Nikolaos Karnesis, Jonathan Gair, and
Angelo Ricciardone for fruitful discussions. O. H’s work was supported by the
Programme National GRAM of CNRS/INSU with INP and IN2P3 co-funded by CNES.
O.H. and M.L. gratefully acknowledge support from the Centre National d’Études
Spatiales (CNES). M. M. gratefully acknowledges support by the Deutsches
Zentrum fur Luft- und Raumfahrt (DLR) with funding from the Bundesministerium
fur Wirtschaft und Technologie (Project Ref. No. 50 OQ 2301 based on work done
under Project Ref. No. 50 OQ 1801). M. P. ’s work was partially funded by STFC
grant ST/T000791/1 and by the European Union’s Horizon 2020 Research Council
grant 724659 Massive-Cosmo ERC-2016-COG. M. P. thanks Imperial College London
for hosting him during the early stages of this work.

##  Appendix A Useful relationships amongst TDI variables

In this Appendix, we provide some useful relationships between the different
TDI channels introduced in section II.2. In particular, in section A.1, we
show how the Sagnac variables are related to the Michelson variables in the
unequal arms case, such that we would expect any independent set of three TDI
channels to contain almost the same information. Then, in section A.2, we give
explicit relationships between the different sets of (quasi-)orthogonal
channels. These expressions are given assuming equal LISA arms, where they can
be formulated concisely as properties of the TDI variables themselves,
regardless of the noise or signal correlations in the actual data. We also
provide simplified low-frequency expansion of these relationships, which
remain valid in the more general unequal-arm case for all covariance matrices
not involving the Michelson T channel.

###  A.1 Unequal arms

It is known that for a constellation with three constant, but unequal arms,
one can exactly reproduce any TDI variable as a linear combination of four
generators, with the set $\\{\alpha,\beta,\gamma,\zeta\\}$ as one possible
basis Dhurandhar et al. (2002). Furthermore, these four generators are
themselves not fully independent, but can be related by Armstrong et al.
(1999):

$(1-D_{23}D_{31}D_{12})\zeta=(D_{23}-D_{31}D_{12})\alpha+(D_{31}-D_{23}D_{12})\beta+(D_{12}-D_{23}D_{31})\gamma\,.$
(33)

This means that we can derive time-delay relationships between different
variables using just three variables as a basis. For example, the following
relationships can be used to express $\alpha$ and $\zeta$ in terms of X, Y,
and Z:

$\displaystyle\begin{split}(D_{23}^{2}-1)(D_{31}^{2}-1)(D_{12}^{2}-1)\alpha&=(1+D_{23}D_{31}D_{12})(D_{23}^{2}-1)X\\\
&\quad+(D_{23}D_{31}+D_{12})(D_{31}^{2}-1)Y\\\
&\quad+(D_{31}+D_{23}D_{12})(D_{12}^{2}-1)Z\end{split}$ (34a)
$\displaystyle\begin{split}(D_{23}^{2}-1)(D_{31}^{2}-1)(D_{12}^{2}-1)\zeta&=(D_{23}+D_{31}D_{12})(1-D_{23}^{2})X\\\
&\quad+(D_{31}+D_{23}D_{12})(1-D_{31}^{2})Y\\\
&\quad+(D_{12}+D_{23}D_{31})(1-D_{12}^{2})Z\;.\end{split}$ (34b)

If we exclude Fourier frequencies at which some of the delay operator
combinations in the previous equations lead to an exact cancellation of the
signal131313For instance, the Fourier transform of terms of the type
$(1-D_{ij}^{N})$ is $(1-e^{-2\pi fNd})$, which is exactly zero if $fd$ is an
integer. At such “singular” frequencies, the response of one variable can be
exactly zero (while that of another is not). In this case, the two variables
are no longer equivalent. In reality, we expect these zeros to be smoothed out
to some extent due to numerical limitations and other noise sources, such that
the discrepancies between two related variables will be extended to a small
frequency band around each zero Vallisneri (2005); Hartwig and Muratore
(2022)., we expect all sets of three independent TDI variables to contain
exactly the same information. This is verified by the fact that we get
identical results in all cases when using the full TDI covariance matrices.

However, as we discuss in this paper, some TDI variables, such as the
traditionally used Michelson T channel, prove to be particularly sensitive to
deviations from the equal-arm assumption. This means they potentially require
more elaborate models to achieve the same scientific output as more robust
channels, such as $\mathcal{T}$ or $\zeta$. Furthermore, the exact
cancellation of the TDI channels at some Fourier frequencies leads to zeros in
the corresponding covariance matrix, making it non-invertible. Since parameter
estimation typically requires inverting the noise covariance matrix this
favors variables with fewer zeros, for which fewer Fourier frequencies need to
be excluded from the analysis. $\mathcal{A}\mathcal{E}\zeta$ is the optimal
set considered in this work from this viewpoint, with the first singularity of
the covariance matrix appearing at $f=1/L\approx$0.12\text{\,}\mathrm{Hz}$$.

###  A.2 Relationship between orthogonal channels in the equal arms limit

In the limit of equal LISA arms, we can derive simple relationships between
the TDI coefficient matrices $C^{UV}$, as defined in eq. 29, for the different
sets of quasi-orthogonal channels considered in this manuscript. We can
further perform a low-frequency expansion of these relationships, the results
of which are given alongside the full expressions after the $\simeq$ signs:

$\displaystyle C^{\rm{AA}/\rm{EE}/\rm{AE}}=4\cos^{2}(\pi
fL)C^{\mathcal{A}\mathcal{A}/\mathcal{E}\mathcal{E}/\mathcal{A}\mathcal{E}}\qquad$
$\displaystyle\simeq$
$\displaystyle\,4C^{\mathcal{A}\mathcal{A}/\mathcal{E}\mathcal{E}/\mathcal{A}\mathcal{E}}\,,$
(35a) $\displaystyle C^{\rm TT}=\frac{16}{3}\sin^{2}\left(\pi
fL\right)\sin^{2}\left(2\pi fL\right)C^{\zeta\zeta}\qquad$
$\displaystyle\simeq$
$\displaystyle\,\frac{64}{3}L^{4}\pi^{4}f^{4}C^{\zeta\zeta}\,,$ (35b)
$\displaystyle C^{\mathcal{T}\mathcal{T}}=\frac{\left[1+2\cos(2\pi
fL)\right]^{2}}{3}\;C^{\zeta\zeta}\qquad$ $\displaystyle\simeq$
$\displaystyle\,3C^{\zeta\zeta}\,,$ (35c) $\displaystyle
C^{\rm{AT}/\rm{ET}}=-\frac{4\sin^{2}(2\pi fL)}{1+2\cos(2\pi
fL)}C^{\mathcal{A}\mathcal{T}/\mathcal{E}\mathcal{T}}\qquad$
$\displaystyle\simeq$
$\displaystyle\,-\frac{16L^{2}\pi^{2}f^{2}}{3}C^{\mathcal{A}\mathcal{T}/\mathcal{E}\mathcal{T}}\,,$
(35d) $\displaystyle C^{\rm{AT}/\rm{ET}}=\frac{e^{4i\pi fL}-2i\sin(2\pi
fL)-1}{\sqrt{3}}C^{{\rm A}\zeta/{\rm E}\zeta}\qquad$ $\displaystyle\simeq$
$\displaystyle\,-\frac{8L^{2}\pi^{2}f^{2}}{\sqrt{3}}C^{{\rm A}\zeta/{\rm
E}\zeta}\,,$ (35e) $\displaystyle
C^{\mathcal{A}\mathcal{T}/\mathcal{E}\mathcal{T}}=\frac{1+2\cos(2\pi
fL)}{\sqrt{3}}C^{\mathcal{A}\zeta/\mathcal{E}\zeta}\qquad$
$\displaystyle\simeq$
$\displaystyle\,\sqrt{3}C^{\mathcal{A}\zeta/\mathcal{E}\zeta}\,.$ (35f)

The notation $C^{UV/WL}$ indicates that the same expression is valid for
$C^{UV}$ and $C^{WL}$ independently, but does not imply any relationship
between $C^{UV}$ and $C^{WL}$.

Let us first note that the coefficient matrices of the different sets of
quasi-orthogonal channels can be related by overall frequency-dependent
scaling factors. This implies that the CSD and PSD terms in the matrix
$S^{UV}$ (see eq. 29) will inherit the same relationships, irrespective of the
single-link correlation matrix $S^{\eta}$. Since this applies equally to
either noise or signal in the data, this further implies that the different
sets all have almost exactly the same signal-to-noise ratio and are therefore
almost equivalent for the purpose of data analysis, at least in the equal-arm
approximation. The only caveat of this statement, as mentioned already in
section A.1, is that the frequency-dependent factors can be vanishing at
singular frequencies. Therefore, the “simpler” variables with fewer zeros,
such as $\mathcal{A}$, $\mathcal{E}$ or $\zeta$, in principle contain slightly
more information than the more “complex” ones, like A, E, T, and
$\mathcal{T}$.

Going to the low-frequency expansion simply gives a constant scaling factor
for most cases. The T channel, however, shows a significantly stronger low-
frequency suppression when compared to the otherwise equivalent $\mathcal{T}$
or $\zeta$ channels. We remark that T is also the only channel for which these
low-frequency expansions do not remain valid approximations for the unequal
arm scenario. This might explain why we find it to be more susceptible to
deviations from the equal-arm assumptions at low frequencies.

##  Appendix B Noise and signal analytic approximations

In this appendix, we provide the analytic expressions of the noise PSDs and
CSDs for the different configurations considered in this work. In appendix B,
we consider identical TM and OMS noise terms on all spacecraft for both the
Michelson and Sagnac variables, as well as for their orthogonal channels, for
the case of equal LISA arms. In section B.2, we provide the corresponding
expressions in the case of unequal but constant arms. Given that the full
expressions are not particularly enlightening, we shall write down only their
low-frequency (i.e., $f\ll c/(2\pi L)$) expansions. The full expressions are
made available as supplementary material. It is possible to compute the signal
response over the entire LISA band by performing a numerical integration of
eq. 16. In order to obtain an analytic expression, one instead has to expand
this expression at low frequencies. We do so for equal and unequal arms in
section B.3 and section B.4, respectively.

All the expressions involving unequal arms which we consider below are
computed by assuming that LISA can undergo two main distortion modes, namely
$\delta_{c}$ and $\delta_{d}$ Muratore et al. (2022a). These distortions leave
the average arm-length unchanged and can therefore be used to characterize
arm-length mismatches across the three LISA arms. This approach helps in
identifying the effects of each mode on the PSDs and CSDs of the TDI
variables.

###  B.1 Noise PSDs and CSDs for an equal arms configuration

Let us write down the expressions for the noise CSDs and PSDs of all channels
assuming equal arms. We have

$\displaystyle S^{\rm XX,N}=S^{\rm YY,N}=S^{\rm ZZ,N}$
$\displaystyle=16\sin^{2}(2\pi fL)\left\\{\left[3+\cos(4\pi
fL)\right]S^{\text{TM}}+S^{\text{OMS}}\right\\}\;,$ (36a) $\displaystyle
S^{\rm XY,N}=S^{\rm XZ,N}=S^{\rm YZ,N}$ $\displaystyle=-4\sin(2\pi
fL)\sin(4\pi fL)\left(S^{\text{OMS}}+4S^{\text{TM}}\right)\;,$ (36b)

for the Michelson variables, and

$\displaystyle S^{\alpha\alpha,\rm
N}=S^{\beta\beta,\mathrm{N}}=S^{\gamma\gamma,\mathrm{N}}$
$\displaystyle=6S^{\text{OMS}}+4\left[3-2\cos(2\pi fL)-\cos(6\pi
fL)\right]S^{\text{TM}}\;,$ (37a) $\displaystyle
S^{\alpha\beta,\mathrm{N}}=S^{\alpha\gamma,\mathrm{N}}=S^{\beta\gamma,\mathrm{N}}$
$\displaystyle=2\left[2\cos(2\pi fL)+\cos(4\pi
fL)\right]S^{\text{OMS}}-4\left[1-\cos(2\pi fL)\right]S^{\text{TM}}\;,$ (37b)

for the Sagnac variables.

For the orthogonal channels A, E, we find:

$\displaystyle S^{\rm AA,N}=S^{\rm EE,N}$ $\displaystyle=8\sin^{2}(2\pi
fL)\left\\{S^{\text{OMS}}\left[\cos(2\pi fL)+2\right]+2\left[3+2\cos(2\pi
fL)+\cos(4\pi fL)\right]S^{\text{TM}}\right\\},$ (38a) $\displaystyle S^{\rm
AE,N}=S^{\rm AT,N}=S^{\rm ET,N}$ $\displaystyle=0\;.$ (38b)

As expected, the CSDs between the orthogonal channels in the case of equal
arm-lengths are zero, since A, E, and T are defined to be orthogonal. Finally,
the expression for $\zeta$ reads:

$S^{\zeta\zeta,\mathrm{N}}=6\left\\{S^{\text{OMS}}+2\left[1-\cos(2\pi
fL)\right]S^{\text{TM}}\right\\}\;.$ (39)

The expressions for any other set of orthogonal variables follow from eq. 35.

###  B.2 Noise PSDs and CSDs for an unequal arm-lengths configuration

While it is possible to obtain exact expressions for the noise CSDs and PSDs
in the unequal arm case, as mentioned before, the resulting expressions are
rather large and cumbersome, and thus not particularly enlightening. For this
reason, we shall only write down simpler expressions, by working in the low-
frequency limit. Furthermore, as discussed in Muratore et al. (2020), one can
express the arm-lengths $L_{ij}$ in terms of the breathing modes of the LISA
triangle, $\delta_{c}$ and $\delta_{d}$ Nussbaum (1968) as:

$\displaystyle L_{23}(t)$
$\displaystyle=L\left[1+\frac{1}{2}\left(\sqrt{3}\,\delta_{c}-\delta_{d}\right)\right]\;,$
(40a) $\displaystyle L_{31}(t)$ $\displaystyle=L\left(1+\delta_{d}\right)\;,$
(40b) $\displaystyle L_{12}(t)$
$\displaystyle=L\left[1-\frac{1}{2}\left(\sqrt{3}\,\delta_{c}+\delta_{d}\right)\right]\;.$
(40c)

and further expand the noise CSDs and PSDs in powers of $\delta_{c}$ and
$\delta_{d}$. Indeed, while
$L=\frac{L_{12}+L_{23}+L_{31}}{3}\approx$8.3\text{\,}\mathrm{s}$$ is the
average arm-length, the small parameters $\delta_{c}$ and $\delta_{d}$ are
typically of the order $1\text{\,}\mathrm{ms}$ to $10\text{\,}\mathrm{ms}$ for
realistic orbits. Evidently, the case $\delta_{c}=\delta_{d}=0$ corresponds to
the equal LISA arms scenario.

Aside from $S^{\rm TT,N}$, none of the PSDs are modified by the arm-length
mismatch at leading order in $\delta_{c,d}$. In particular, we
have141414Expanding the TDI coefficient matrix $C^{TT}$ computed according to
eq. 29 to leading order in frequency shows that it _only_ contains terms that
are second order in $\delta_{c}$ and $\delta_{d}$. This implies that T’s
leading order dependence on the arm-length mismatch is a generic effect, and
we should expect $S^{\rm TT,N}$ to exhibit low-frequency deviations for any
kind of noise correlations we model in $S^{\eta}$.:

$S^{\rm TT,N}\simeq 12(\delta_{c}^{2}+\delta_{d}^{2})(2\pi
f)^{2}L^{2}(S^{\text{OMS}}+4S^{\text{TM}})\;.$ (41)

Furthermore, none of the CSDs between quasi-orthogonal channels remain exactly
zero if the arm-lengths are unequal:

$\displaystyle S^{\rm AE,N}$ $\displaystyle=-12\delta_{c}\delta_{d}L^{2}(2\pi
f)^{2}(S^{\text{OMS}}+4S^{\text{TM}})=4S^{\mathcal{A}\mathcal{E},\rm N}\;,$
(42a) $\displaystyle S^{\rm AT,N}$
$\displaystyle=12\sqrt{2}\delta_{c}L^{2}(2\pi
f)^{2}(S^{\text{OMS}}+4S^{\text{TM}}),$ (42b) $\displaystyle S^{\rm A\zeta,N}$
$\displaystyle=\sqrt{6}\delta_{c}L^{2}(2\pi
f)^{2}(S^{\text{OMS}}+12S^{\text{TM}})\;,$ (42c) $\displaystyle S^{\rm ET,N}$
$\displaystyle=12\sqrt{2}\delta_{d}L^{2}(2\pi
f)^{2}(S^{\text{OMS}}+4S^{\text{TM}}),$ (42d) $\displaystyle S^{\rm E\zeta,N}$
$\displaystyle=\sqrt{6}\delta_{d}L^{2}(2\pi
f)^{2}(S^{\text{OMS}}+12S^{\text{TM}})\;,$ (42e) $\displaystyle
S^{\mathcal{A}\mathcal{T},\rm N}$
$\displaystyle=-6\sqrt{2}\delta_{c}L^{2}(2\pi f)^{2}S^{\text{TM}},$ (42f)
$\displaystyle S^{\mathcal{A}\zeta,\rm N}$
$\displaystyle=2\sqrt{6}\delta_{c}L^{2}(2\pi
f)^{2}(S^{\text{OMS}}+3S^{\text{TM}})\;,$ (42g) $\displaystyle
S^{\mathcal{E}\mathcal{T},\rm N}$
$\displaystyle=-6\sqrt{2}\delta_{d}L^{2}(2\pi f)^{2}S^{\text{TM}},$ (42h)
$\displaystyle S^{\mathcal{E}\zeta,\rm N}$
$\displaystyle=2\sqrt{6}\delta_{d}L^{2}(2\pi
f)^{2}(S^{\text{OMS}}+3S^{\text{TM}})\;.$ (42i)

We note that $S^{\rm AT}$ and $S^{\rm ET}$ are proportional to $\delta_{c}$
and $\delta_{d}$, and are thus 2 orders of magnitude larger than $S^{\rm TT}$,
assuming $\delta_{c}\approx 0.01$. This highlights the importance of the CSDs
in the low-frequency regime when using A, E, and T with unequal arms. This is
shown explicitely in section D.2 in a comparison of $AET$ and $AE\zeta$ using
FIM and MCMC in the low frequency range. $S^{\rm AE}$, on the other hand, is
proportional to $\delta_{c}\delta_{d}$, and is therefore 4 orders of magnitude
smaller than $S^{\rm AA}$ or $S^{\rm EE}$. $S^{A\zeta}$ and $S^{E\zeta}$ are
again proportional to $\delta_{c}$ and $\delta_{d}$, such that they are also
suppressed by two orders of magnitude with respect to the diagonal terms
$S^{\rm AA}$, $S^{\rm EE}$ and $S^{\zeta\zeta}$. We can therefore conclude
that A, E, and $\zeta$ remain almost orthogonal even in the unequal arms case.
This is consistent with the discussion of section III and with the results of
section D.2.

###  B.3 Low-frequency signal response for an equal arm-length configuration

One can obtain an analytic expression for eq. 16 by first expanding its kernel
for low frequencies. Doing so, one obtains in turn an analytic expression for
the $S^{\eta,\mathrm{GW}}_{ij,mn}(f)$ CSD given by eq. 15. This expression can
then be used in eq. 29 with the TDI coefficients, and the resulting expression
expanded in powers of frequency. By doing so, one obtains the following
expressions in the equal arm approximation:

$\displaystyle S^{\rm XX,GW}=S^{\rm YY,GW}=S^{\rm ZZ,GW}$
$\displaystyle=\frac{384}{5}(fL\pi)^{4}\;,$ (43a) $\displaystyle S^{\rm
XY,GW}=S^{\rm XZ,GW}=S^{\rm YZ,GW}$ $\displaystyle=-\frac{1}{2}S^{\rm
XX,GW}\;,$ (43b)

for the Michelson variables, and:

$\displaystyle S^{\alpha\alpha,\rm GW}=S^{\beta\beta,\rm
GW}=S^{\gamma\gamma,\rm GW}$ $\displaystyle=\frac{96}{5}(fL\pi)^{4}\;,$ (44a)
$\displaystyle S^{\alpha\beta,\rm GW}=S^{\alpha\gamma,\rm
GW}=S^{\beta\gamma,\rm GW}$ $\displaystyle=-\frac{1}{2}S^{\alpha\alpha,GW}\;,$
(44b)

for the Sagnac variables. Finally, we can compute equivalent expressions for
AET, AE$\zeta$, $\mathcal{A}\mathcal{E}\mathcal{T}$ and
$\mathcal{A}\mathcal{E}\zeta$:

$\displaystyle S^{\rm AA,GW}$ $\displaystyle=S^{\rm EE,GW}=\frac{3}{2}S^{\rm
XX,GW}\;,\qquad S^{\mathcal{A}\mathcal{A},\rm
GW}=S^{\mathcal{E}\mathcal{E},\rm GW}=\frac{3}{2}S^{\alpha\alpha,\rm GW}\;,$
(45a) $\displaystyle S^{\rm TT,\rm GW}$
$\displaystyle=\frac{256}{63}(fL\pi)^{10},\qquad S^{\mathcal{T}\mathcal{T},\rm
GW}=\frac{4}{7}(fL\pi)^{6}\;,\qquad S^{\zeta\zeta,\rm
GW}=\frac{4}{21}(fL\pi)^{6}\;,$ (45b)

where all cross-terms are vanishing.

###  B.4 Low-frequency signal response for an unequal arm-length configuration

In order to obtain reasonably short expressions in the case of unequal arms,
we follow the same procedure as in section B.3 but also expand the resulting
expressions in powers of the breathing modes $\delta_{c}$ and $\delta_{d}$
introduced in eq. 40. Similarly to the results obtained in section B.2, we
again find that most channels are unaffected by the leading order in
$\delta_{c}$ and $\delta_{d}$. Contrary to section B.2, we now find that all
three null channels T, $\mathcal{T}$, and $\zeta$ are modified with respect to
the equal arm case. Concretely, we get

$S^{\rm
TT,GW}=16S^{\mathcal{T}\mathcal{T},GW}=\frac{16}{3}S^{\zeta\zeta,GW}=\frac{288}{5}f^{4}\pi^{4}L^{4}\left(\delta_{c}^{2}+\delta_{d}^{2}\right)\;,$
(46)

such that each of the three channels now shows the same low-frequency
behaviour as the GW-sensitive channels. However, comparison with eq. 45b shows
that for $\mathcal{T}$ and $\zeta$, this deviation corresponds to a change in
slope by $f^{2}$ with respect to the equal arm scenario, while for T, the
slope changes by a factor $f^{6}$. This, together with the findings in section
B.2, explains the qualitatively very different behaviour between the T channel
and the other null channels observed in section III. Furthermore, we observe
that we now have $S^{\mathcal{T}\mathcal{T},GW}=\frac{1}{3}S^{\zeta\zeta,GW}$,
whereas we had $S^{\mathcal{T}\mathcal{T},GW}=3S^{\zeta\zeta,GW}$ in the equal
arm case. Since the noise curves of these two channels are, at leading order,
unaffected by the arm-length mismatch, which explains why we found
$\mathcal{T}$ to be a slightly better null channel than $\zeta$ in section
III.

Finally, we note that while the CSDs of the base Michelson XYZ and Sagnac
$\alpha\beta\gamma$ variables are again unaffected at leading order, all
orthogonal channels exhibit non-vanishing cross-terms, given by:

$\displaystyle S^{\rm AE,GW}$ $\displaystyle=-\frac{\sqrt{2}}{2}\,S^{\rm
AT,GW}=-\frac{576}{5}f^{4}\pi^{4}L^{4}\delta_{c}\;,$ (47a) $\displaystyle
S^{\rm ET,GW}$
$\displaystyle=\frac{288}{5}\sqrt{2}f^{4}\pi^{4}L^{4}\delta_{d}\;,$ (47b)
$\displaystyle S^{\rm{A}\zeta,\rm GW}$
$\displaystyle=\frac{72}{5}\sqrt{6}f^{4}\pi^{4}L^{4}\delta_{c}\;,$ (47c)
$\displaystyle S^{\rm{E}\zeta,\rm GW}$
$\displaystyle=\frac{72}{5}\sqrt{6}f^{4}\pi^{4}L^{4}\delta_{d}\;,$ (47d)
$\displaystyle S^{\mathcal{A}\zeta,GW}$
$\displaystyle=\frac{\sqrt{6}}{4}S^{\mathcal{A}\mathcal{E},GW}=-\frac{36}{5}\sqrt{6}f^{4}\pi^{4}L^{4}\delta_{c}\;,$
(47e) $\displaystyle S^{\mathcal{A}\mathcal{T},GW}$
$\displaystyle=-\frac{36}{5}\sqrt{2}f^{4}\pi^{4}L^{4}\delta_{c}\;,$ (47f)
$\displaystyle S^{\mathcal{E}\zeta,GW}$
$\displaystyle=\sqrt{3}S^{\mathcal{E}\mathcal{T},GW}=-\frac{36}{5}\sqrt{6}f^{4}\pi^{4}L^{4}\delta_{d}\;.$
(47g)

##  Appendix C Data analysis technical details

As discussed in section II a data stream of LISA (and in general of any GW
experiment) can be modeled as a superposition of some signal and detector
noise. Let us assume that the data $d_{i}(t)$, with the index $i$ labeling the
TDI channel, are provided in the time domain, and for simplicity let us assume
$d_{i}(t)$ to be stationary. This corresponds to assuming rigid arms during
the full mission duration. This assumption is clearly not going to be valid
for LISA, but for the sake of our discussion, which rather focuses on the
impact of unequal arm-lengths and noise levels on the orthogonality, as well
as on the constraining power, of the different TDI variables, we restrict
ourselves to this simplified scenario. Given the total observation time of the
detector $T_{d}$, which in this paper is always assumed to be $4$ years with
$100\%$ efficiency, we can divide it into a given number, say $N_{d}$, of
segments, of duration $T_{d}/N_{d}$ each. We can thus define the data
$\tilde{d}^{s}_{i}(f_{\textmd{k}})$, in the frequency domain, where the index
$s$ runs over all segments, and the index k runs over frequencies in the
detector range. The frequency resolution for any given segment is directly
given by $\Delta f=N_{d}/T_{d}$. In the following, we will assume the time
duration of each segment to be $\sim$ 11.5 days corresponding to $\Delta f\sim
10^{-6}$ Hz. We also assume different frequencies to be uncorrelated. As
discussed in section II, both signal and noise are also assumed to be
Gaussian-distributed with vanishing mean and variance given by their
respective power spectral densities. Under all these assumptions, we can
generate $N_{d}$ statistical realizations of the signal and of all the noise
components. Following the procedure described in Caprini et al. (2019);
Flauger et al. (2021), we define a new set of (averaged) data
$\bar{D}^{\textmd{k}}_{IJ}\equiv\tilde{d}^{i}_{J}(f_{\textmd{k}})\tilde{d}^{i}_{I}(f_{\textmd{k}})/N_{d}$,
which we down-sample using a coarse-graining procedure. By applying these
techniques, we obtain a new data set $D^{k}_{IJ}$, where $k$ runs now over a
sparser set of frequencies $f^{k}_{IJ}$, with weights $w^{k}_{IJ}$
corresponding to the number of points we average over in the coarse-graining
procedure. The down-sampled data set will have statistical properties similar
to the ones of the $\bar{D}^{\textmd{k}}_{IJ}$ while being easier to handle
numerically Caprini et al. (2019); Flauger et al. (2021).

It would seem natural to describe the data using a Gaussian likelihood of the
form:

$\ln\mathcal{L}_{\rm
G}(\vec{\theta}|D^{k}_{IJ})=-\frac{N_{d}}{2}\sum_{k}\sum_{I,J}w^{k}_{IJ}\left[1-D^{k}_{IJ}/D^{\rm
Th}_{IJ}(f_{k},\vec{\theta})\right]^{2}\;,$ (48)

where $\vec{\theta}=\\{\vec{\theta}_{s},\vec{\theta}_{n}\\}$ is the vector of
parameters (with $\vec{\theta}_{s}$, $\vec{\theta}_{n}$ being the signal and
noise parameters, respectively), and $D^{\rm
Th}_{IJ}(f_{k},\vec{\theta})=\Omega_{IJ,\ {\rm
GW}}(f_{k},\vec{\theta})+\Omega_{IJ,\,n}(f_{k},\vec{\theta})$ is the
theoretical model for the data (with $\Omega_{IJ,\,{\rm
GW}}(f_{k},\vec{\theta})$ and $\Omega_{IJ,\ n}(f_{k},\vec{\theta})$, being the
signal and noise model, respectively). However, it is known Bond et al.
(2000); Sievers et al. (2003); Verde et al. (2003); Hamimeche and Lewis (2008)
that this compressed likelihood does not account for the mild non-Gaussianity
of the full likelihood, giving systematically biased results. This bias might
be corrected by introducing a log-normal likelihood:

$\ln\mathcal{L}_{\rm
LN}(\vec{\theta}|D^{k}_{IJ})=-\frac{N_{d}}{2}\sum_{k}\sum_{I,J}w^{k}_{IJ}\ln^{2}\left[D^{\rm
Th}_{IJ}(f_{k},\vec{\theta})/D^{k}_{IJ}\right]\;,$ (49)

and considering the final likelihood to be:

$\ln\mathcal{L}(\vec{\theta})=\frac{1}{3}\ln\mathcal{L}_{\rm
G}(\vec{\theta}|D^{k}_{IJ})+\frac{2}{3}\ln\mathcal{L}_{\rm
LN}(\vec{\theta}|D^{k}_{IJ})\;.$ (50)

Consistently with the discussion in section II.1.2, we include a prior for all
the noise parameters in our analysis. These priors are chosen to be Gaussian,
centered around the face values, i.e., $A_{ij}=3$, $P_{ij}=15$,
$\forall\;\\{ij\\}$ combinations, and with a 20$\%$ width. Finally, to sample
the parameter space we use the emcee sampler Foreman-Mackey et al. (2013), and
results are visualized in contour plots using Chainconsumer Hinton (2016).

For completeness, we report the expression of the SNR Romano and Cornish
(2017):

$\textrm{SNR}\equiv\sqrt{T_{d}\;\sum_{i}\int\textrm{d}f\left(\frac{S_{i}^{\rm
GW}}{S_{i}^{\rm N}}\right)^{2}}\;,$ (51)

where the index $i$ runs over all the TDI channels, which for simplicity, are
assumed to be orthogonal.

##  Appendix D Noise parameter reconstruction for AET and AE$\zeta$

In this section, we present the analysis of the noise parameter
reconstruction, which is complementary to the discussion of section IV. For
this purpose, we start, in section D.1, by considering a simple toy model,
which provides useful insight for the interpretation of our results. We then
proceed, in section D.2 and section D.3, with the discussion of the noise
parameter reconstruction for the equal and unequal noise level cases,
respectively.

###  D.1 Toy model

Let us consider two time series, $D_{1}=N_{1}+N_{2}$ and $D_{2}=N_{1}$, where
$N_{1}$ and $N_{2}$ are uncorrelated noises with PSDs $a$ and $b(f/f_{\star})$
respectively, i.e., while the PSD of $N_{1}$ is a constant, that of $N_{2}$
has a frequency dependence. The covariance $C_{ij}$, with
$\\{i,j\\}\in\\{1,2\\}$, of the two data sets is given by:

$C_{ij}\equiv\mathrm{Cov}(D_{i},D_{j})=\left(\begin{array}[]{cc}a+b\frac{f}{f_{\star}}&a\\\
a&a\end{array}\right)\;.$ (52)

Assuming the data to be Gaussian, the log-likelihood reads:

$-\log\mathcal{L}(D_{i}|a,b)\propto\sum_{f}\left\\{\ln\left[\det(C_{ij})\right]+D_{i}C^{-1}_{ij}D_{j}^{*}\right\\}\;,$
(53)

and it’s easy to show that the Fisher matrix $F_{ij}$ can be computed as:

$F_{ij}\equiv-\left.\frac{\partial^{2}\log\mathcal{L}}{\partial\theta_{i}\partial\theta_{j}}\right|_{\vec{\theta}=\vec{\theta}_{0}}=\sum_{f}\Tr\left[C^{-1}\frac{\partial
C}{\partial\theta_{i}}C^{-1}\frac{\partial C}{\partial\theta_{j}}\right]\;,$
(54)

where $\vec{\theta}_{0}$ are the best-fit parameters, which for eq. 53 are
given by $C_{ij}=D_{i}D^{*}_{j}$. To simplify the analysis, in the following,
we replace the sum over finite frequencies with a continuous integral over the
frequency range (going from some $f_{\mathrm{min}}$ to $f_{\mathrm{max}}$)
multiplied by the total observation time $T$151515Assuming the integration
time is sufficiently long to have a reasonably fine sampling in $f$, the
results in the two cases will coincide.. From the equations above, it is easy
to compute the covariance matrix (which is given by the inverse of the Fisher
matrix) of the parameters $a$ and $b$. It reads:

$\mathrm{Cov}(a,b)=\frac{1}{T(f_{\mathrm{max}}-f_{\mathrm{min}})}\left(\begin{array}[]{cc}a^{2}&0\\\
0&b^{2}\end{array}\right)\;.$ (55)

Since $D_{2}$ gives an independent readout of $N_{1}$, and $N_{2}$ can be
estimated from $D_{1}-D_{2}$, we can get independent estimates of the two
parameters $a$ and $b$, whose variances scale with $a^{2}$ and $b^{2}$,
respectively. To have better insight into the impact of the off-diagonal terms
of eq. 52 on eq. 55, we repeat this calculation in the case in which these
terms are neglected. Before entering into details, let us first note that if
we consider a frequency range defined by $f_{\mathrm{min}}=0$ and
$f_{\mathrm{max}}\rightarrow\infty$, the information diverges, the variance
goes to zero, and we recover the above result. Let us proceed by considering
$f_{\mathrm{max}}\gg f_{\mathrm{min}}$, with $bf_{\mathrm{max}}/f_{\star}\ll
a$. In this case, the signal $D_{2}$ is subdominant over the entire frequency
range and eq. 55 evaluates to:

$\mathrm{Cov}(a,b)\simeq\frac{1}{Tf_{\mathrm{max}}}\left(\begin{array}[]{cc}0.8\,a^{2}&-1.2\,a^{2}\frac{f_{\star}}{f_{\mathrm{max}}}\\\
-1.2\,a^{2}\frac{f_{\star}}{f_{\mathrm{max}}}&4.8\,a^{2}\frac{f_{\star}^{2}}{f_{\mathrm{max}}^{2}}\end{array}\right)\;,$
(56)

implying that the variance of $a$ is underestimated compared to the true value
(which is equal to $a^{2}$), while that of $b$ depends on $a$ and is large
compared to the true value (which is $b^{2}$). The second case we consider is
$f_{\mathrm{max}}\gg f_{\mathrm{min}}$, with $bf_{\mathrm{max}}/f_{\star}=2a$.
Then, the two signals are of similar amplitude, and eq. 55 reads:

$\mathrm{Cov}(a,b)\simeq\frac{1}{Tf_{\mathrm{max}}}\left(\begin{array}[]{cc}0.88\,a^{2}&-0.8\,ab\\\
-0.8\,ab&5\,b^{2}\end{array}\right)\;.$ (57)

In this case, the covariance of $a$ continues to be underestimated by a factor
of order $4/5$, while that of $b$ is overestimated by a factor $\sim 5$.
Finally, we consider $f_{\mathrm{max}}\gg f_{\mathrm{min}}$, with
$bf_{\mathrm{max}}/f_{\star}\gg a$, so that $D_{2}$ is much larger than
$D_{1}$ and eq. 55 reads:

$\mathrm{Cov}(a,b)\simeq\frac{1}{Tf_{\mathrm{max}}}\left(\begin{array}[]{cc}a^{2}&\frac{a^{2}f_{\star}}{f_{\mathrm{max}}}\\\
\frac{a^{2}f_{\star}}{f_{\mathrm{max}}}&b^{2}\end{array}\right)\;.$ (58)

In this case, the frequency band is sufficiently broad such that either signal
is clearly visible in at least part of it, and it is possible to recover the
true covariance to a good approximation despite our having neglected the off-
diagonal terms. Notice that in all three cases, neglecting off-diagonal terms
in eq. 52, introduces degeneracies in the estimates of $a$ and $b$.

The findings of this toy model are consistent with the rationale that the off-
diagonal elements add information that helps break degeneracies between model
parameters, and that when they are neglected, the dominant signal, if present
in multiple data streams, will have artificially more statistical weight and
will thus be recovered with artificially small error bars, while the
subdominant signal will be difficult to identify with much accuracy.

###  D.2 Equal noises

Figure 11:  Left: TM, OMS, and GW PSDs in the case of unequal arms but equal
noise spectra. Right: TM, OMS, and GW CSDs divided by the square root of the
product of the PSDs for AA and TT or $\zeta\zeta$ (i.e., the square root of
the component-wise coherence), in the case of unequal arms but equal noises.

In order to better understand the constraints on the noise parameters, we
start by presenting in fig. 11 the PSDs of the TM, OMS, and GW signal for the
AA, TT, and $\zeta\zeta$ variables in the left-hand plot, and the CSDs of TM,
OMS and GW divided by the square root of the product of the PSDs for AA and TT
or $\zeta\zeta$ in the right-hand plot (i.e., the component-wise square-root
of the coherence). For example, “AT TM” is computed as
$|S^{\mathrm{AT},\mathrm{TM}}|/\sqrt{(S^{\mathrm{AA},\mathrm{N}}+S^{\mathrm{AA},\mathrm{GW}})(S^{\mathrm{TT},\mathrm{N}}+S^{\mathrm{TT},\mathrm{GW}})}$.
Restricting the focus to the low-frequency range ($f<10^{-3}$ Hz), and to the
variable T, we find that OMS is suppressed with respect to TM, reaching levels
almost as small as GW. The TM cross-correlations in AT are at the 10% level,
while those for OMS and GW are several orders of magnitude smaller. Turning
our attention to $\zeta$, we find that in the low-frequency range, contrary to
T, the TM and OMS PSDs remains large compared to GW. Moreover, the cross-
correlations for AE$\zeta$ are significantly smaller than those of AET. It is
also worth noting that the hierarchy between the TM, OMS, and GW PSDs remains
the same throughout the entire frequency range ($3\times 10^{-5}$ to 0.5 Hz)
for $\zeta\zeta$, whereas the TM and OMS spectra intersect at $10^{-3}$ Hz for
TT. The TM and OMS cross-correlations intersect at $f\simeq 5\times 10^{-3}$
Hz for both AT and A$\zeta$. Finally, at frequencies above $10^{-2}$ Hz, TM,
OMS, and GW have comparable amplitudes in AET and AE$\zeta$. What differs is
mainly the oscillatory features introduced by the TDI transfer functions.
While the GW cross-correlations are strongly suppressed at large frequencies,
the TM and OMS ones are much larger and of order 1% and 10% respectively.

(a) Low frequency range.

(b) Full frequency range.

Figure 12: AET diag. MCMC vs. AE$\zeta$ diag. MCMC stochastic background and
noise parameter reconstruction for unequal arms and equal noises.

Let us proceed by discussing the parameter reconstruction in the case of equal
noise levels, first restricting the analysis to the low-frequency ($f\leq
10^{-3}$Hz) range. The results of the MCMC runs for this configuration are
presented in fig. 12(a). As shown in fig. 11, at low frequencies, while the
$AE\zeta$ basis is nearly orthogonal, correlations are relevant for AET.
Indeed, for AET, neglecting off-diagonal terms significantly impacts the
posterior for the OMS amplitude (see the marginalized posterior for parameter
$P$ in fig. 12(a)) which becomes largely underconstrained. This is due to the
fact that OMS is strongly suppressed relative to TM in AET, such that much
information could be gained from including the relatively large CSDs shown in
the right-hand plot of fig. 11. The situation is in fact similar to that of
$N_{2}$, and its associated parameter $b$, discussed in the toy model of
section D.1. On the other hand, the posteriors for all other parameters are
slightly narrower than when considering the full AET matrix. This is once
again consistent with what was found in section D.1, for $N_{1}$ and
associated parameter $a$. It is also worth noting that $\alpha$ and $n_{T}$
are strongly degenerate in both AET and AE$\zeta$. This happens in the low-
frequency range, since the pivot frequency
($f_{\star}=$3.873\text{\times}{10}^{-3}\text{\,}\mathrm{Hz}$$) is located
outside this range such that a change in $n_{T}$ induces a strong change in
the GW signal’s amplitude within the frequency range. In this sense, including
the pivot frequency in the frequency range considered, or widening the
frequency range would break this degeneracy. This is indeed what happens once
we consider the full frequency range, as we discuss below.

We conclude this section by including the higher frequency range (up to
$5\times 10^{-1}$Hz, consistent with the analysis in section IV), which we
expect will include significant information to the GW and noise parameters for
both AET and AE$\zeta$. Indeed, the results shown in fig. 12(b) show a great
improvement in the determination of most parameters. In particular, the OMS
noise parameter $P$ gains a factor of order 500 in the width of the
marginalized posterior for AET, and a factor of order 50 for AE$\zeta$. This
is consistent with the findings obtained in the example of section D.1, where
it was found that by extending the frequency range, one could capture
additional information on sub-dominant components in the signal. Finally,
comparing with fig. 12(b), we notice that including higher frequencies breaks
the degeneracy between $\alpha$ and $n_{T}$ parameters. Indeed, as shown in
fig. 11, the GW PSD grows at frequencies above $10^{-2}$ Hz, while the CSD
drops significantly. While not explicitly shown in fig. 12, all these results
are consistent with the corresponding Fisher matrix analyses.

###  D.3 Unequal noises

Figure 13:  Left: TM, OMS, and GW PSDs in the case of unequal arms and unequal
noise spectra. Right: TM, OMS, and GW CSDs divided by the square root of the
product of the PSDs for AA and TT or $\zeta\zeta$ (i.e., the square root of
the component-wise coherence), in the case of unequal arms and unequal noise
spectra.

In this section, we discuss the situation of most interest, the case of
unequal arms and unequal noise levels. Comparing the left-hand plot of fig. 13
with the one of fig. 11, one finds that the TM, OMS, and GW PSDs in AA, TT,
and $\zeta\zeta$, are practically unchanged with respect to the equal noise
levels case. Turning attention to the right-hand plot of fig. 13, one can make
the following observations. Firstly, the OMS noise cross-correlation in
A$\zeta$ is much larger than its counterpart in the equal noise case, over the
entire frequency band. Indeed, this quantity is of the same order as the one
for AT for frequencies $\gtrsim 5\times 10^{-4}$Hz and is larger for lower
frequencies. Secondly, the A$\zeta$ TM cross-correlations are also enhanced,
by close to an order of magnitude in the band between $10^{-5}$ and $10^{-1}$
Hz. Thirdly, as expected, the GW cross-correlations are not affected by the
introduction of independent noise levels and, contrarily to the equal noise
level case, they stay smaller than the TM and OMS ones over the entire range.
Finally, while the cross-correlations reach above the 10% for both noises and
for both TDI basis, the GW signal off-diagonal terms remain below 10% for both
noises and are significantly smaller for A$\zeta$. Drawing from these
observations, one anticipates that the largest impact of neglecting off-
diagonal terms will be seen in the OMS noise.

(a) Test mass (TM) acceleration noise

(b) Optical metrology system (OMS) noise

Figure 14: AET diag. MCMC vs. AE$\zeta$ diag. MCMC for the TM and OMS
parameter posteriors obtained from the MCMC runs in the case of unequal arms,
unequal noises, and over the full frequency range.

(a) AET diag. Fisher vs. MCMC diag. TM on the first line and OMS on the second
line.

(b) AE$\zeta$ diag. Fisher vs. MCMC diag. TM on the first line and OMS on the
second line.

Figure 15: TM and OMS one-dimensional parameter posteriors obtained from
Fisher forecasts and MCMC runs in the case of unequal arms and unequal noise
levels.

Let us now focus on the AET and AE$\zeta$ MCMC results. Figure 14(a) and fig.
14(b) show the results of the AET and AE$\zeta$ runs for the TM and OMS
parameters, respectively. As shown in the right-hand plot of fig. 13, the TM
cross-correlations are significantly larger in AET than in AE$\zeta$ while the
OMS cross-correlations are non-negligible and of similar amplitude in both
AE$\zeta$ and AET. Furthermore, as shown in the left-hand plot of that figure,
the TM and OMS power spectral densities are significantly larger in $\zeta$
compared to T over a significant fraction of the frequency band. One can
therefore conclude that similar to what was found in the toy model of section
D.1, ignoring cross-correlations will induce differences in the TM and OMS
posterior widths for AET than in AE$\zeta$, and in particular, may lead to
variance underestimation for AET. This is true for 4 out of the 6 $A_{ij}$’s
and 3 out of the 6 $P_{ij}$’s. Let us recall that a Gaussian prior centered
around the face values $A=3$, $P=15$, and with a standard deviation equal to
the 20$\%$ of the central value was used for all the TM and OMS noise
parameters. With this in mind, we note that, while all the $P_{ij}$ have
posteriors that are considerably narrower than the priors, some of the
$A_{ij}$ constraints are actually comparable in size, implying that the data
fails to provide significant information on those parameters. Finally, we note
that there exist degeneracies between the $A_{ij}$ and $A_{ji}$, and $P_{ij}$
and $P_{ji}$ pairs. This is expected because whether AET or AE$\zeta$, all
measurements that involve the quantity $\eta_{ij}$ also involve its time-
delayed counterpart $\eta_{ji}$. For what concerns the comparison between the
AET and AE$\zeta$ results obtained via MCMC sampling and the FIM analysis
assuming a diagonal TDI correlation matrix, while we do not present the full
corner plots, we show in fig. 15(a) and fig. 15(b) the 1D posteriors for AET
and AE$\zeta$, respectively. As is clear from these plots, the constraints
obtained with the approaches agree.

(a) TM acceleration noise.

(b) OMS noise.

Figure 16: TM and OMS posterior widths obtained from FIM analysis in the case
of unequal arms and unequal noise levels when including or neglecting the off-
diagonal terms in the TDI correlation matrix.

We conclude this appendix with a comparison of the Fisher analyses performed
with the diagonal and full TDI covariance matrices. Since AET and AE$\zeta$
show slightly different, but qualitatively similar behaviours, we only show
the corner plot for the latter, see fig. 13. Given that the AE$\zeta$ TM noise
CSDs are relatively small, the constraints on the $A_{ij}$ parameters
constraints are not expected to change significantly in going from a diagonal
to a full correlation matrix analysis. This is verified in the 1D marginalized
posteriors of fig. 16(a). The introduction of correlations in the analysis
does introduce correlations among parameters, with the result that the surface
area of 2D contours shrinks when considering the full matrix. Let us now turn
our attention to OMS. Given the CSDs of fig. 13 which for AE$\zeta$ (and also
for AET), exhibit correlations that are greater than $10\%$ for a broad set of
frequencies, we expect that including or excluding correlations of the TDI
variable will significantly modify the constraints on the $P_{ij}$’s. Indeed,
as illustrated in the toy model of section D.1, ignoring cross-correlations
means that the amplitude parameters of all subdominant noise components are
largely unconstrained. This is confirmed by the contour plots of fig. 16(b).

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