# A Stochastic Gravitational Wave Background in LISA from Unresolved White
Dwarf Binaries in the Large Magellanic Cloud

Steven Rieck,1,2 Alexander W. Criswell,1⋆22footnotemark: 2 Valeriya Korol,3,4
Michael A. Keim,5 Malachy Bloom,6 Vuk Mandic1
1School of Physics and Astronomy, University of Minnesota, Minneapolis, MN
55455, USA
2Department of Physics, University of Cincinnati, Cincinnati, OH 45221, USA
3Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Straße 1, 85741
Garching, Germany
4Institute for Gravitational Wave Astronomy & School of Physics and Astronomy,
University of Birmingham, Birmingham, B15 2TT, UK
5Department of Astronomy, Yale University, PO Box 208101, New Haven, CT
06520-8101, USA
6Department of Physics and Astronomy, Carleton College, Northfield, MN 55057,
USA
E-mail<EMAIL_ADDRESS>(SR<EMAIL_ADDRESS>(AWC)S. Rieck
and A.W. Criswell contributed equally to this work and are co-first authors.

(Accepted 2024 May 10. Received 2024 May 10; in original form 2023 September
2)

###### Abstract

The Laser Interferometer Space Antenna (LISA) is expected to detect a wide
variety of gravitational wave sources in the mHz band. Some of these signals
will elude individual detection, instead contributing as confusion noise to
one of several stochastic gravitational-wave backgrounds (SGWBs) – notably
including the ‘Galactic foreground’, a loud signal resulting from the
superposition of millions of unresolved double white dwarf binaries (DWDs) in
the Milky Way. It is possible that similar, weaker SGWBs will be detectable
from other DWD populations in the local Universe, including the Large
Magellanic Cloud (LMC). We use the Bayesian LISA Inference Package
($\tt{BLIP}$) to investigate the possibility of an anisotropic SGWB generated
by unresolved DWDs in the LMC. To do so, we compute the LMC SGWB from a
realistic DWD population generated via binary population synthesis, simulate
four years of time-domain data with BLIP comprised of stochastic contributions
from the LMC SGWB and the LISA detector noise, and analyze this data with
BLIP’s spherical harmonic anisotropic SGWB search. We also consider the case
of spectral separation from the Galactic foreground. We present the results of
these analyses and show, for the first time, that the unresolved DWDs in the
LMC will comprise a significant SGWB for LISA.

###### keywords:

gravitational waves – white dwarfs – Magellanic Clouds

††pubyear: 2024††pagerange: A Stochastic Gravitational Wave Background in LISA
from Unresolved White Dwarf Binaries in the Large Magellanic Cloud–8

##  1 Introduction

The launch of the Laser Interferometer Space Antenna (LISA) (Amaro-Seoane et
al., 2017) in 2035 will revolutionize gravitational wave (GW) astronomy. A
space-based gravitational observatory, LISA will detect GWs in the millihertz
frequency band, a range inaccessible to both pulsar timing arrays such as the
International Pulsar Timing Array (IPTA; Manchester, 2013) and current ground-
based detector networks such as the Laser Interferometer Gravitational-wave
Observatory (LIGO), the Virgo detector, and the Kamioka Gravitational Wave
Detector (KAGRA) (Aasi et al., 2015; Acernese et al., 2015; Adhikari et al.,
2020).

LISA is expected to detect a wide variety of astrophysical GW sources,
including millions of double white dwarfs (DWDs) in the Milky Way (MW) and the
nearby Universe (for a review see Amaro-Seoane et al., 2022). Some of these
sources will be individually resolvable, whereas others will contribute to
several potential stochastic gravitational wave backgrounds (SGWBs; e.g.,
Bonetti & Sesana, 2020; Babak et al., 2023; Pozzoli et al., 2023). SGWBs arise
from confusion noise formed by the overlap of many unresolved astrophysical or
cosmological sources; evidence for such a signal in the nanohertz band has
recently been detected (Agazie et al., 2023). While many Galactic DWDs will be
individually resolvable by LISA — some serving as verification binaries for
the instrument (e.g. Stroeer & Vecchio, 2006; Savalle et al., 2022; Finch et
al., 2023) — a far greater number of Galactic DWDs will contribute to a
stochastic GW signal distributed mostly along the Galactic plane, comprised of
the superposition of millions of individually unresolvable DWDs (Edlund et
al., 2005). Characterization of this anisotropic MW foreground (so-called due
to its prominence above the LISA detector noise) will be necessary in order to
subtract it from the LISA data and identify other signals. Additionally, the
foreground is of scientific interest in its own right as a means of studying
MW structure and star formation history (SFH) (e.g. Benacquista & Holley-
Bockelmann, 2006; Breivik et al., 2020; Georgousi et al., 2023).

The MW is not the only host of DWDs detectable with LISA. Recent simulations
show that nearby dwarf galaxies including the Large Magellanic Cloud (LMC),
Small Magellanic Cloud, and Sagittarius Dwarf contain DWDs that will appear as
individually resolvable LISA sources (Roebber et al., 2020a). The number of
resolvable DWDs depends on a dwarf galaxy’s mass, distance, and SFH (Korol et
al., 2020). However, as in the MW, the majority of DWDs in dwarf galaxies will
not generate resolvable signals; for instance in the LMC population of
${\mathcal{O}}(10^{6})$ DWDs only ${\mathcal{O}}(10^{2})$ will be individually
detectable (Korol et al., 2020). It is possible that, as in the MW, these
unresolved DWDs will contribute to an anisotropic SGWB detectable with LISA.
To our knowledge, the detectability of SGWBs from nearby dwarf galaxies with
LISA anisotropic SGWB searches has not been investigated prior to this work.

Due to its high mass and relative proximity, the LMC is an ideal first
candidate to evaluate the possibility of a SGWB from DWDs outside of the MW.
Recent work has considered the detectability of individual DWDs in the LMC by
constructing model populations based on hydrodynamic simulations and
electromagnetic observations of its SFH and stellar density (Keim et al.,
2022). Keim et al. found that while the LMC likely has only tens or hundreds
of detectable DWDs with LISA signal-to-noise ratio (SNR) $>7$, it contains
approximately two million DWDs in the LISA frequency band111The LMC is
expected to contain approximately 61 million DWDs in total, but a vast
majority are non-interacting with large orbital separations and are negligible
sources of GWs. A frequency cut-off of $10^{-4}\;\mathrm{Hz}$ reduces this
number to two million in the LISA frequency band (Keim et al. (2022), by
correspondence with the author).. This quantity is significantly less than the
DWDs in the MW; the LMC stellar mass (e.g. Marel et al., 2002) is roughly an
order of magnitude less than the mass of the MW ($2.7\times 10^{9}$ M⊙ vs
several $10^{10}$ M⊙). At $\sim 50\;\mathrm{kpc}$ from Earth, the LMC signal
is also reduced by distance, as GW amplitudes scale as the inverse of the
distance. Given these considerations, we may expect the LMC SGWB to have
approximately $1\%$ to $2\%$ the strength of the MW signal. On the other hand,
while the MW DWDs are distributed across a large fraction of the sky, the LMC
DWDs are focused in 77 square degrees, making the LMC a good target for an
anisotropic SGWB search.

In this work, we simulate and recover the SGWB signal in LISA from a model LMC
population using the Bayesian LISA Inference Package (BLIP) (Banagiri et al.,
2021). In §2 we describe the model population used to simulate the LMC signal
and our code for simulation and recovery. Results are presented in §3 and the
conclusions of this study alongside possible future extensions are discussed
in §4.

##  2 Methods

To investigate the stochastic signal from the LMC we use $\tt{BLIP}$,
described at length in Banagiri et al. (2021). BLIP is a Python package
designed for the end-to-end simulation and Bayesian analysis of stochastic GW
signals with LISA. In this study, we use the BLIP spherical harmonic
anisotropic stochastic search first presented in Banagiri et al. (2021), which
is explained in brief in §2.1. BLIP can simulate a wide variety of anisotropic
stochastic GW signals; we make use of its capability to simulate a SGWB from a
realistic simulated population of the unresolved DWDs in the LMC. This
population is further described in §2.2, while the simulated and recovered
models in BLIP are described in §2.3 and §2.4 respectively.

###  2.1 Anisotropic SGWBs in BLIP

Simulation and recovery of anisotropic SGWBs in BLIP is performed in the
spherical harmonic basis. Several studies – considering ground-based, space-
based, and pulsar timing-based analyses – have used versions of the spherical
harmonic basis for expanding the sky distribution of GW power in e.g.
(Cornish, 2001a; Ungarelli & Vecchio, 2001; Kudoh & Taruya, 2005; Taruya &
Kudoh, 2005; Taruya, 2006; Thrane et al., 2009; Mingarelli et al., 2013;
Taylor & Gair, 2013; Renzini & Contaldi, 2018). However, constraining the
spherical harmonic distribution to be real and non-negative everywhere is a
non-trivial problem that can hamper the accurate characterization of highly
anisotropic sources such as the Galactic foreground — or, indeed, the LMC.
This is especially true for Bayesian analyses.

Banagiri et al. (2021) developed an explicitly Bayesian version of the
spherical harmonic SGWB analysis for LISA wherein this problem was solved by
fitting the square root of the GW power. Specifically, the spatial
distribution of the SGWB on the sky (for purposes of both simulation and
inference) is represented by the spherical harmonic coefficients $b_{\ell m}$.
The $b_{\ell m}$s describe the spherical harmonic expansion of the square root
of the GW power on the sky $\mathcal{S}(\mathbf{n})$. The $b_{\ell m}$s are
related to the usual spherical harmonic coefficients and functions of the GW
power on the sky, $a_{\ell m}$ and $Y_{\ell m}$, respectively, via

$\begin{split}\mathcal{S}(\mathbf{n})=\sqrt{\mathcal{P}(\mathbf{n})}&=\left[\sum_{\ell=0}^{\ell_{\rm
max}^{a}}\sum_{m=-\ell}^{\ell}a_{\ell m}Y_{\ell m}(\mathbf{n})\right]^{1/2}\\\
&=\sum_{\ell=0}^{\ell_{\rm max}^{b}}\sum_{m=-\ell}^{\ell}b_{\ell m}Y_{\ell
m}(\mathbf{n}),\end{split}$ (1)

where $\ell_{\mathrm{max}}^{b}=\ell_{\mathrm{max}}^{a}/2$ (Banagiri et al.,
2021). The $a_{\ell m}$ and $b_{\ell m}$ terms are directly related to each
other via simple linear transformations involving Clebsch-Gordan coefficients
(Banagiri et al., 2021). Characterizing the GW anisotropy in this way
mathematically ensures that the GW power in every proposed sample is real and
non-negative across the entire sky (see Banagiri et al. (2021) for details).

In practice, the BLIP anisotropic search infers and produces posterior
distributions for each $b_{\ell m}$ coefficient (alongside spectral
parameters; see §2.4) up to some
$\ell_{\mathrm{max}}^{b}=\ell_{\mathrm{max}}^{a}/2$.222As the usual spherical
harmonic $\ell_{\mathrm{max}}$ referred to in the literature is
$\ell_{\mathrm{max}}^{a}$, we will quote this truncation $\ell_{\mathrm{max}}$
in terms of $\ell_{\mathrm{max}}^{a}$ throughout this work. Using a higher
$\ell_{\mathrm{max}}^{a}$ for the anisotropic search increases the angular
resolution of the search, but also increases the number of parameters that one
must infer as
$N_{\mathrm{par,sph}}\propto\ell_{\mathrm{max}}^{a}(\ell_{\mathrm{max}}^{a}+1)/2$.
Additionally, as the BLIP anisotropic search considers the LISA detector
response to each spherical harmonic, the computational resources required to
analyze data at large $\ell_{\mathrm{max}}^{a}$ can become limiting.

This latter point is also a limitation for simulation of anisotropic SGWBs
with $\tt{BLIP}$, as the SGWB spatial distribution is simulated in the
spherical harmonic basis. Accordingly, simulations of anisotropic signals in
BLIP similarly employ a truncation $\ell_{\mathrm{max}}^{a}$. This, of course,
results in highly-localized signals (like the LMC) spreading out over an area
much larger than their true spatial extent on the sky. However, a study of
BLIP’s angular resolution (Bloom et al., in-prep) has shown that the value of
$\ell_{\mathrm{max}}^{a}$ used in the SGWB simulation does not impact the
final spatial recovery so long as
$\ell_{\mathrm{max,simulation}}^{a}\geq\ell_{\mathrm{max,recovery}}^{a}$.
(Simply put, our analysis is insensitive to variations on smaller scales than
it parameterizes, as one would expect intuitively.) Development work is
ongoing to improve BLIP’s performance for both simulation and analysis at high
$\ell_{\mathrm{max}}^{a}$ ($\gtrsim 8$), but these computational limitations
remain relevant at present.

###  2.2 Simulated LMC DWD Population

To date, no DWD has been observed in the LMC. Even within the MW most of the
known LISA-detectable DWDs are found within a few kpc (e.g. Kupfer et al.,
2024); this is mainly due to the faint nature of white dwarf stars.
Nonetheless, this highlights an opportunity for LISA to reveal the DWD
population inaccessible to electromagnetic observatories as far as the LMC. To
model the LMC DWDs we employ a mock catalogue compiled by Keim et al. (2022).
It is based on a fiducial DWD population synthesis model computed with the
SeBa code (Portegies Zwart & Verbunt, 1996; Toonen et al., 2012), which has
been calibrated on the observed DWDs (albeit in the Solar neighborhood) and,
therefore, is in good agreement with the observed DWD space density and mass-
ratio distribution (Toonen et al., 2012, 2017).

Synthetic DWDs are distributed across the sky and assigned formation times and
ages based on the Magellanic Clouds Photometric Survey and the observed,
spatially resolved 2D SFH from Harris & Zaritsky (2009, for a visual
representation see their fig. 4). We refer to Keim et al. (2022, see their
‘Model 1’) for further details.

For the assumed LMC total stellar mass of $2.7\times 10^{9}$ M⊙ (van der Marel
et al., 2002), the adopted model yields $\sim 2\times 10^{6}$ DWDs in the LISA
frequency band. For this model, only about $\sim$500 DWD are individually
resolved with SNR $>7$, assuming the mission lifetime of 4 years with 100%
duty cycle. The detectable binaries have frequencies $>$1.7 mHz (or
equivalently binary orbital periods of $<$20 min) due to LISA’s selection
effects. The total number of LISA sources in the LMC represents about $8\%$ of
the MW DWD population. As detailed in Keim et al. (2022), the difference
between the two populations is twofold. Firstly, the number of LISA sources
(and stars in general) scales linearly with the total mass of the host galaxy.
The lower mass of the LMC thus decreases individual DWD detections. Secondly,
unlike the MW, the LMC is an active site of star formation, and so a
significant fraction of DWD in the adopted model have formed only
$\sim\mathcal{O}(10^{2})$ Myr ago. This active star formation increases
detections of individual DWDs in the LMC.

###  2.3 Simulated LISA Data

The simulation of stochastic GW signals from DWD population-synthesis
catalogues is a novel BLIP feature demonstrated for the first time in this
work. For each catalogue binary, we compute the (assumed monochromatic) strain
amplitude from its binary masses and orbital frequency, following the
conventions in Wagg et al. (2022). We use the catalogue sky position and
distance (as seen in the Solar System Barycentre frame) to bin the population
in both frequency and sky direction. Binning on the sky is performed on a
Healpix (Gorski et al., 2005) map, with user-specified skymap pixel
resolution, quantified by the Healpix $\tt{nside}$. In this work we use an
nside of 8 to generate our simulated signal. At our chosen skymap resolution,
the area of each pixel equals approximately 53 square degrees. The angular
size of the LMC is approximately 77 square degrees (Roebber et al., 2020b).
Thus, in our initial simulated skymap the entire LMC is contained within only
a few pixels. Simulating the LMC with a higher nside would incur significantly
higher computational cost for little-to-no ultimate effect due to limitations
on the sky resolution of our analysis (see §2.1).

To compute the associated SGWB spectrum of the DWD population, we assume all
DWD systems with individual SNR $>7$ are individually resolvable and can be
subtracted from the data (Keim et al., 2022). We use LEGWORK (Wagg et al.,
2022) to calculate the SNR of every DWD considering the instrumental noise and
Milky Way foreground given in Robson et al. (2019), and remove from the
population those DWDs with SNR $>7$. Disentangling the resolved and unresolved
DWDs — let alone the entire cacophony of LISA sources — is beyond the scope of
this work, requiring a global, simultaneous solution (e.g., Littenberg &
Cornish 2023). We assume all other GW sources are perfectly characterized and
subtracted from the data, and we first simulate a signal that includes only
the unresolved LMC DWDs. In a second analysis, we also include a simple model
of the MW foreground (see §2.3.1). Our simulation of the LMC DWDs is identical
in each analysis. The monochromatic strains of the remaining unresolved
binaries are then binned in frequency at a frequency resolution determined by
the LISA nominal mission duration of 4 years, i.e. $\Delta
f=1/T_{\mathrm{obs}}\simeq 8\times 10^{-9}~{}\mathrm{Hz}$. We consider a
frequency range of $f\in[10^{-4},10^{-2}]$ Hz, as this will be the most-
sensitive band of the LISA detector.

After the population skymap and spectrum are computed, BLIP simulates a time
series of the corresponding stochastic signal. It does so by computing the
spherical harmonic representation of the population skymap up to some
$\ell_{\mathrm{max}}^{a}$ (we consider a simulation $\ell_{\mathrm{max}}^{a}$
of 4 due to computational limitations; see §2.1). BLIP convolves both this
spherical harmonic expansion and the population spectrum with the time-varying
LISA response across frequency and all considered spherical harmonic modes
(see Banagiri et al. 2021 for details). Note that this process explicitly
models the orbits of the LISA constellation and as such naturally accounts for
the time-varying amplitude of highly anisotropic SGWBs like that of the LMC
and the MW. The simulated population skymap as represented in the spherical
harmonic basis can be found in Fig. 1(a).

The resulting GW time series is added to Gaussian detector noise with the
spectral form given in the LISA proposal (Amaro-Seoane et al., 2017),
reproduced below in Eqs. (2) and (3), with $N_{p}=9\times 10^{-42}$ and
$N_{a}=3.6\times 10^{-49}$ Hz-4 for the position and acceleration noise
contributions, respectively:

$S_{p}(f)=N_{p}\left[1+\left(\frac{2mHz}{f}\right)^{4}\right]Hz^{-1},$ (2)
$S_{a}(f)=\left[1+\left(\frac{0.4mHz}{f}\right)^{2}\right]\left[1+\left(\frac{f}{8mHz}\right)^{4}\right]\times\frac{N_{a}}{\left(2\pi
f\right)^{4}}Hz^{-1}.$ (3)

Throughout this study we simulate and model LISA data using the X — Y — Z
time-delay interferometry (TDI) channels (see Tinto & Dhurandhar (2014) for a
review of TDI in LISA). For further details on the BLIP data simulation
procedure, see Banagiri et al. (2021). The simulated spectrum, as it appears
in the detector, along with the simulated detector noise, is included in Fig.
3.

(a)

(b)

Figure 1: (a) The simulated sky distribution of
$\Omega_{\mathrm{GW}}(1\mathrm{mHz})$ for the LMC SGWB generated by our model
DWD population. (b) The marginalized posterior sky distribution of
$\Omega_{\mathrm{GW}}(1\mathrm{mHz})$ inferred by our analysis of the LMC in
isolation. Both skymaps are in the spherical harmonic basis at
$\ell_{\mathrm{max}}^{a}=4$ and display distribution of the dimensionless GW
energy density $\Omega_{\mathrm{GW}}$ evaluated at 1 mHz. These skymaps do not
include LISA instrumental noise. The black star marks the position of the LMC.
The recovered sky distribution is consistent with both the simulated sky
distribution and the position of the LMC.

####  2.3.1 Simple Milky Way Foreground

We also include a simple analytic (i.e., non-population) simulation of the MW
foreground. Its spatial distribution follows the simple bulge + disc model
described in Breivik et al. (2020). We use the "thin" model (see Breivik et
al. (2020) for details), with radial scale height $r_{\mathrm{h}}=2.9$ kpc and
vertical scale height $z_{\mathrm{h}}=0.3$ kpc. This simulated Galaxy is then
used to create a skymap in the Solar System Barycentre frame as described in §
6 of Banagiri et al. (2021). As throughout the rest of this work, we represent
this spatial distribution in the spherical harmonic basis. For the MW
foreground spectrum, we use a tanh-truncated power law similar to that of
(e.g.,) Robson et al. (2019), such that

$\Omega(f)=\Omega_{\mathrm{ref}}\left(\frac{f}{f_{\mathrm{ref}}}\right)^{\alpha}\left(1+\tanh\left(\frac{f_{\mathrm{cut}}-f}{f_{\mathrm{scale}}}\right)\right),$
(4)

where for this simulation $\Omega_{\mathrm{ref}}=2\times 10^{-5}$,
$f_{\mathrm{ref}}=25$ Hz, $f_{\mathrm{cut}}=2$ mHz, and
$f_{\mathrm{scale}}=0.4$ mHz. Although this more simplistic analytic function
does not account for iterative subtraction of resolved MW DWDs, it remains a
sufficient approximation for our purposes given the large uncertainties in the
overall amplitude and shape of the MW foreground signal. This skymap and
spectrum are then used to compute the GW time-series contribution of the MW
foreground in the same manner as described above for the LMC.

###  2.4 Model Recovery in BLIP

After generating the simulated data, BLIP performs Bayesian parameter
estimation via nested sampling with dynesty (Speagle, 2020). This process is
described in brief below; reference Banagiri et al. (2021) for a more detailed
treatment. The BLIP anisotropic search simultaneously models the LISA detector
noise, the SGWB spectral distribution, and the SGWB spatial distribution,
inferring posterior distributions for each of the parameters described below.

LISA’s instrumental noise is modelled in terms of the position and
acceleration noise amplitudes $N_{p}$ and $N_{a}$, with the spectral form
given by Eqs. (2) and (3). We characterize the LMC SGWB spectrum using a
power-law spectral model of the form

$\Omega(f)=\Omega_{\mathrm{ref}}\left(\frac{f}{f_{\mathrm{ref}}}\right)^{\alpha},$
(5)

where $\Omega_{\mathrm{ref}}=\Omega(f_{\mathrm{ref}}=25\ \mathrm{Hz})$ is the
power-law amplitude at the reference frequency $f_{\rm ref}$ and $\alpha$ is
the power-law spectral index (slope). The value of $f_{\rm ref}$ is an
arbitrary choice. BLIP recovers both $\Omega_{\mathrm{ref}}$ and $\alpha$ as
free parameters. The majority of the LMC SGWB spectrum can be approximated as
a power law, although this model will be unable to capture the high-frequency
turnover in the spectrum; as this work focuses on establishing the LMC SGWB as
a significant signal in LISA, more complex spectral models are left to future
work (see §4).

As discussed in §2.1, the spatial distribution of the LMC SGWB on the sky is
inferred in the spherical harmonic basis. Our final spatial posteriors are
given in terms of the $b_{\ell m}$s, from which it is straightforward to
compute the corresponding $a_{\ell m}$s and SGWB power skymap. We choose an
analysis $\ell_{\mathrm{max}}^{a}$ of 4, in keeping with our choice for the
simulated LMC spatial distribution.

The Fourier-domain likelihood used in BLIP’s nested sampling is a complex
multivariate Gaussian (Adams & Cornish, 2010) whose covariance is a function
of the parameters in the previous four equations:
$\mathcal{L}(\tilde{d}|N_{p},N_{a},\Omega_{\mathrm{ref}},\alpha,\\{b_{\ell,m}\\})$.
The likelihood is given by Eq. $32$ from Banagiri et al. (2021):

$\begin{split}&\mathcal{L}(\tilde{d}|N_{p},N_{a},\Omega_{\mathrm{ref}},\alpha,\\{b_{\ell,m}\\})=\\\
&\prod_{f,t}\frac{1}{2\pi
T_{\mathrm{seg}}|C_{IJ}(f,t)|}\times\mathrm{exp}\left(-\frac{2\tilde{d}^{*}_{f,t}C_{IJ}(f,t)^{-1}\tilde{d}_{f,t}}{T_{\mathrm{seg}}}\right)\end{split}$
(6)

where $T_{\mathrm{seg}}$ is the length of each time segment, $C_{IJ}(f,t)$ is
the channel covariance matrix, and $\tilde{d}_{f,t}$ is the array of data in
the Fourier domain for the three LISA channels measured in the time segment
labelled by $t$ and at frequency $f$. For explicit definitions of these terms
see discussion in Banagiri et al. (2021) and original derivations in Cornish
(2001b); Cornish & Larson (2001).

####  2.4.1 Joint Model with the Milky Way Foreground

We also consider a joint model that simultaneously infers the LMC SGWB
alongside the MW foreground. This is a simplified, prototype demonstration of
the full, flexible spectral separation infrastructure developed for BLIP
(Criswell et al., in prep.). Accordingly, we restrict ourselves to a simple MW
model: we assume the MW spatial distribution is well-measured a priori from
the resolved Galactic DWDs, and fix its skymap to the analytic distribution
described in §2.3.1. We then use the spectral model of Eq. (4), fixing
$f_{\mathrm{scale}}=0.4$ mHz, and inferring the set of free parameters
$\vec{\theta}_{\mathrm{MW}}=\\{\Omega_{\mathrm{ref,MW}},\alpha_{\mathrm{MW}},f_{\mathrm{cut}}\\}$.
The joint likelihood is then
$\mathcal{L}(\tilde{d}|\vec{\theta}_{\mathrm{n}};\vec{\theta}_{\mathrm{LMC}};\vec{\theta}_{\mathrm{MW}})$,
where $\vec{\theta}_{\mathrm{n}}=\\{N_{p},N_{a}\\}$ describe the noise and
$\vec{\theta}_{\mathrm{LMC}}=\\{\Omega_{\mathrm{ref}},\alpha,\\{b_{\ell,m}\\}\\}$
describe the LMC as discussed above. We leave a full discussion of BLIP’s
approach to spectral separation to Criswell et al. (in prep.). We stress that
this simple model is a first pass at resolving the LMC SGWB in the presence of
the MW foreground. A detailed treatment of spectral separation between the LMC
and MW signals is sufficiently involved so as to warrant its own dedicated
study.333See §4 for further discussion as to what such a study could entail.
As such, more complicated models are outside the scope of this initial work,
which primarily seeks to establish the LMC SGWB as a significant stochastic
contribution in LISA.

##  3 Results

We include results from two simulations. In the first, we simulate the LMC
SGWB generated from the population described in §2.2 with LISA instrumental
noise. In §3.2 we present the results of the recovery process described in
§2.4. In §3.3 we present a recovery of the LMC in the presence of a simple
realization of the MW foreground, as described in §2.4.1.

###  3.1 LMC SGWB Spectrum

The population-derived power spectrum of the LMC SGWB is shown in Fig. 2.
Notably, the amplitude of the LMC signal is comparable to — and even exceeds —
that of the expected SGWB from extragalactic stellar-origin binary black holes
(SOBBHs), shown here using the observationally-driven estimate of Babak et al.
(2023). The LMC signal will therefore comprise a significant SGWB for LISA,
and will be important to consider in efforts to characterise the SOBBH SGWB
and other underlying SGWBs. This result is the first demonstration of the LMC
SGWB as a relevant signal for LISA.

Figure 2: The simulated, population-derived LMC SGWB PSD. The LISA
instrumental noise spectrum and the Babak et al. (2023) interquartile
prediction for the LISA SOBBH SGWB are shown for reference. Both SGWB PSDs are
shown convolved with the LISA response. Note that the LMC SGWB amplitude
exceeds that of the SOBBH signal in the relevant frequency band.

###  3.2 Recovery of the LMC SGWB in Isolation

We present here an analysis of the LMC SGWB in isolation (i.e., assuming the
MW foreground has been subtracted) using an integration time of $1.26\times
10^{8}$ seconds, approximately the planned LISA mission duration of four
years, and considering a frequency band of $f\in[10^{-4},10^{-2}]$ Hz. We
simulate and recover the LMC SGWB in the spherical harmonic basis, use a power
law to model the SGWB spectrum, and model the LISA detector noise according to
the spectral form given in Eqs. (2) and (3). The corresponding marginalized
posterior skymap computed from the inferred $b_{\ell m}$s is shown in Fig.
1(b), and the marginalized posterior detector-convolved power spectral density
(PSD) is shown in Fig. 3 (alongside the PSDs of the simulated detector noise
and of the SGWB due to the LMC DWD population). Posterior samples for all
parameters are shown in Fig. 6.

Figure 3: The simulated and inferred power spectral density of the LMC SGWB
and the LISA detector noise. For the inferred spectra, the solid lines and
shaded regions are the median and 95% credible intervals, respectively, of the
marginalized posterior spectral fit. As can be seen in Fig. 6, the noise
spectrum is recovered extremely precisely; as a result the 95% credible
intervals are difficult to see by eye. Note that the power-law spectral fit
has highest fidelity to the simulated LMC spectrum over the sensitive band of
1-4 mHz.

As seen in Fig. 1(b), the inferred distribution of power on the sky is
consistent with both the true position of the LMC and the simulated LMC SGWB
skymap (Fig. 1(a)). While more precise localization of the LMC SGWB could in
principle be achieved with higher $\ell_{\mathrm{max}}^{a}$ or a targeted
directional search that takes advantage of the known position of the LMC, we
leave these avenues of exploration to future work.

The inferred power-law spectrum of the LMC SGWB is shown in Fig. 3, alongside
the simulated and inferred noise spectra and the simulated population-derived
spectrum of the LMC SGWB. The inferred amplitude and slope of the power-law
model used in this study are most impacted by the shape of the LMC spectrum at
frequencies where its SNR is largest — namely 1-4 mHz, where the simulated LMC
spectrum is closest to the LISA noise curve. At frequencies outside this
range, the power-law model does not adequately describe the complexity of the
simulated LMC SGWB spectrum, and hence it overestimates the contribution from
the LMC signal at these frequencies. We leave treatment of more complex or
nonparametric spectral models to future work, although we note that the
overall low SNR of the LMC may make constraining highly-complex models
difficult (unless the dimensionality of the inference problem is otherwise
reduced by, for example, a targeted directional search). The noise spectra is
recovered extremely well, due to the fact that we recover it using the exact
functional form that we initially simulate. Ultimately the noise spectral
shape will not be precisely known, which will introduce additional error.

Finally, we perform model comparison via Bayes factor and consider two cases:
our power-law spherical harmonic model including LISA noise and the LMC SGWB,
and a noise-only model. Using the same four-year dataset including the LMC
SGWB described in §2.3, we repeat our analysis using a model that only
accounts for the LISA detector noise in terms of $N_{p}$ and $N_{a}$ as given
in Eqs. (2) and (3) (neglecting the presence of any kind of underlying SGWB).
Computing the Bayesian evidences of each model ($\mathcal{Z}_{1}$ for the
noise + SGWB model; $\mathcal{Z}_{2}$ for the noise-only model) is trivial due
to our use of nested sampling via dynesty, which produces the Bayesian
evidence as its primary product (Speagle, 2020). We compute the log Bayes
factor to be

$\log K=\log\mathcal{Z}_{1}-\log\mathcal{Z}_{2}=310\pm 3,$

constituting decisive evidence444For reference, a log Bayes factor of $1$ is
substantial to strong evidence, and any log Bayes factor $>2$ is typically
considered decisive evidence in favor of one model over another (Kass &
Raftery, 1995). in favor of our SGWB plus noise model over the noise-only
model. We conclude that — in the absence of the MW foreground signal and for
the case of stationary, Gaussian noise with a fixed, equilateral LISA
constellation — we are able to detect and characterize the LMC SGWB signal.
Relaxing any of these assumptions will reduce LISA’s sensitivity to SGWBs
(see, e.g., Hartwig et al., 2023; Muratore et al., 2024) and, accordingly,
impact the ability of the LISA to detect and characterize the LMC SGWB. While
fully accounting for these factors is beyond the scope of this work, we
present a simplified treatment of a search for the LMC SGWB in the presence of
the MW foreground in the following section.

###  3.3 Recovery of the LMC SGWB with the MW Foreground

We now turn to the case of the LMC SGWB in the presence of the MW foreground.
We additionally include in our simulated data a simple MW foreground as
described in §2.3.1; the simulation procedure for the LISA instrumental noise
and LMC SGWB is otherwise unchanged. This new dataset is then analyzed with
the joint inference model described in §2.4.1; all other quantities of
interest (integration time, frequency range, etc.) are identical to the
procedure described in §3.2 for the LMC in isolation.

We find that, despite the presence of the MW foreground, we are again able to
detect and characterize the simulated LMC SGWB. The recovered spectral
distribution of the LMC SGWB in the presence of the MW foreground is shown in
Fig. 4, alongside those of the noise and the MW foreground. As before, we
display the simulated and inferred spectra for each of our model components.
Our recovered model successfully describes the LISA instrumental noise, MW
foreground, and LMC SGWB simultaneously. Posterior samples for all spectral
parameters are shown in Fig. 7. The presence of the MW does affect the
recovered LMC SGWB, reducing the recovery quality below $\sim 3$ mHz causing
the power law to even more dramatically overestimate the LMC SGWB. Above $\sim
3$ mHz, the recovered power law follows closely above the simulated LMC SGWB
spectrum. It is again clear that the majority of information is being gleaned
from the region around $\sim 3$ mHz where the LMC SNR would be highest; more
refined spectral models may be able to leverage this fact in future.

Figure 4: The simulated and inferred PSDs of the LMC SGWB, the MW foreground,
and the LISA detector noise. For the inferred spectra, the solid lines and
shaded regions are the median and 95% credible intervals, respectively, of the
marginalized posterior spectral fit. The precise recovery of both the LISA
noise and MW foreground renders their respective medians and 95% credible
intervals nearly indistinguishable. The simple power law model for the LMC
signal again results in an overestimation of power at low frequencies. The
signal is most accurately recovered above 3 mHz where the contribution from
the MW foreground is minimal. Figure 5: The marginalized posterior sky
distribution of $\Omega_{\mathrm{GW}}(1\mathrm{mHz})$ inferred by our analysis
for the LMC SGWB in the presence of the MW foreground. The simulated LMC in
this simulation is identical to Fig. 1(a). We represent the signal in the
spherical harmonic basis at $\ell_{\mathrm{max}}^{a}=4$. This skymap does not
include LISA instrumental noise or the MW foreground, though both are present
in the simulation. The black star marks the true position of the LMC. The
recovered sky distribution is consistent with both the simulated signal and
the true position of the LMC.

The LMC SGWB spatial recovery in the presence of the MW foreground can be seen
in Fig. 5. It is important to note that this figure only displays the inferred
distribution of power on the sky (i.e. the spherical harmonic spatial model
for the LMC SGWB), and does not include the contribution from the MW (which is
assumed known and therefore not inferred; see §2.4.1). The associated
posterior samples are shown in Fig. 8. As would be expected, the quality of
the spatial recovery is degraded somewhat in the presence of the MW (and with
a more statistically-complex signal model). While the extent of the inferred
LMC spatial distribution is similar to the simulated skymap and the true
position of the LMC is included in our recovered spatial distribution, it does
experience some bias, shifting slightly off of the true position of the LMC.

Finally, we again perform a second analysis of the same simulated LISA data
(including LISA instrumental noise, the MW foreground, and the LMC SGWB) with
a model which accounts for LISA instrumental noise and the MW foreground, but
neglects the presence of the LMC. We compute the log Bayes factor
$(\mathrm{log}K)$ for this case using the Bayesian evidences of each model
($\mathcal{Z}_{1}$ for the LMC-included model; $\mathcal{Z}_{2}$ for the LMC-
absent model):

$\log K=\log\mathcal{Z}_{1}-\log\mathcal{Z}_{2}=92\pm 4,$

While this Bayes factor is reduced compared to that for the LMC in isolation —
as expected, the MW foreground makes the LMC SGWB more difficult to recover —
it still constitutes extremely decisive evidence in favor of the model that
includes the LMC SGWB.

##  4 Discussion and Conclusions

In this work, we evaluate for the first time the existence and prospects for
LISA of an anisotropic SGWB arising from the unresolved DWDs in the LMC. We
use a population catalog generated using realistic stellar synthesis codes to
create a model of the LMC, which we then use to simulate its DWD-generated
SGWB with $\tt{BLIP}$. We use BLIP’s spherical harmonic, Bayesian search for
anisotropic SGWBs to demonstrate a proof-of-concept recovery of the LMC SGWB
both in isolation and in the presence of the MW foreground.

We find that the simulated SGWB from the unresolved DWDs in the LMC can be
recovered in the presence of LISA instrumental noise using BLIP with four
years of integration time and a power-law spherical harmonic signal model.
Model comparison between the noise + SGWB power-law spherical harmonic model
and a noise-only model yields decisive evidence in favor of the presence of
the LMC SGWB signal. The recovered position of the LMC on the sky is
consistent with its true location, and the LMC SGWB spectrum can be well
modeled as a simple power law over the sensitive frequency band (roughly 1-4
mHz).

Additionally, we find that we are able to simultaneously recover the LMC SGWB
and a rudimentary model of the MW foreground. While the presence of the MW has
a noticeable, adverse effect on the recovery of the LMC SGWB, the recovered
spatial distribution remains consistent with the true position of the LMC, and
our power-law spectral model only slightly overestimates the LMC spectrum
above $~{}3$ mHz. As in the LMC-only case, model comparison via Bayes factor
yields decisive evidence in favor of the presence of the LMC SGWB signal.
While a detailed treatment of spectral separation between realistic,
population-derived realizations of the MW and LMC signals is required to make
a strong statement of detectability — and remains a subject of future work —
this result is nonetheless extremely promising for the prospects of LISA to
detect and characterize the LMC SGWB.

While the power-law spectral model employed here is accurate to the simulated
LMC spectrum where the LISA noise curve is lowest and the MW foreground has
dropped off, outside these areas, it does not capture the full spectral shape
of the LMC SGWB. Further characterization of the LMC SGWB with more complex
spectral and/or spatial models is one promising avenue of future work. One
could, for example, leverage the known location of the LMC to infer only its
spectral distribution while holding its spatial distribution fixed, thereby
reducing model complexity along one axis and allowing for (e.g.) a truncated
or broken power-law spectral model to better capture the cutoff in the LMC
SGWB spectrum. Such a model could also be informed by our theoretical
knowledge of the LMC SGWB, either by setting astrophysically-motivated priors
on its parameters, or fixing those parameters that see little variation across
different population-synthesis realizations of the LMC. Conversely, ongoing
efforts to incorporate non-parametric spectral models into BLIP could enable
more accurate characterization of the LMC spectrum, at the cost of increased
difficulty of spectral separation from the MW foreground. With more precise
spectral models, it may be possible to characterize the LMC SGWB well enough
to gain information about the distribution of DWDs in the LMC and learn about
its structure, mass, and/or SFH. Methods have been proposed to study the MW in
this way using the unresolved Galactic DWDs (e.g. Breivik et al., 2020), so it
is possible that similar techniques could be used to study the LMC. In
particular, it may be possible to achieve a measurement of the LMC mass via a
similar approach to the one described in Korol et al. (2021), which used the
resolvable binaries in the LMC. Additionally, the analysis presented in this
work is generalizable to simulation and recovery of the (albeit weaker) SGWBs
from the Small Magellanic Cloud and other dwarf galaxy satellites of the Milky
Way.

Finally, the development of refined approaches to concurrent characterization
of the LMC SGWB and the MW foreground will be vital moving forward. The
spectral overlap between these signals is significant; neglecting to properly
account for the LMC SGWB could lead to spectral biases for analyses of the MW
foreground. Despite their close proximity in terms of LISA’s angular
resolution, the spatial distributions of the MW and LMC are distinct on the
sky and — as demonstrated in this work — can be used to aid in spectral
separation between these signals. In particular, the spatial distribution of
the LMC on the sky is well known from electromagnetic observations; our
anisotropic search at high $\ell_{\mathrm{max}}^{a}$ and/or a targeted
directional search could leverage this fact. One could also incorporate
concurrent GW localization measurements of the resolved DWDs in the LMC,
improving prospects for resolving the LMC SGWB by jointly modelling the 3D
spatial distribution of the LMC population (as has been proposed for the MW
population (Adams et al., 2012)). Finally, a pixel-basis method to describe
the spatial distribution of a signal provides a promising alternative to a
spherical-harmonic basis approach, which by necessity describes the entire sky
rather than the region containing the LMC specifically. This method would be
well-suited to enabling realistic spectral separation of the stochastic
contributions from unresolved MW and LMC DWDs.

Proper, joint treatment of both the LMC SGWB and MW foreground will likely be
crucial for detecting and characterizing other, lower-amplitude SGWBs. The
SOBBH background (e.g., Babak et al., 2023) is likely of comparable or lower
amplitude in comparison to the LMC SGWB; see Fig. 2. Characterization of the
LMC SGWB is thus extremely relevant when considering the search for the SOBBH
SGWB, as well as other, underlying backgrounds — including those of
cosmological origin.

## Acknowledgements

This work is supported by the NASA grant 90NSSC19K0318, and utilized computing
resources provided by the Minnesota Supercomputing Institute at the University
of Minnesota. Packages used for this work include numpy (Harris et al., 2020),
scipy (Virtanen et al., 2020), ChainConsumer (Hinton, 2016), and Matplotlib
(Hunter, 2007). The authors would like to thank Sharan Banagiri, Joe Romano,
and Jessica Lawrence for their work on BLIP and many helpful conversations, as
well as the anonymous reviewer for their thorough and insightful comments and
suggestions.

## Data Availability

All data and code used in this study is publicly available. The simulated LMC
population data along with all generated SGWB data and resulting posterior
distributions are available at https://zenodo.org/records/10783952. The BLIP
package is open source and is available at https://github.com/sharanbngr/blip.

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##  Appendix A Additional Figures

Corner plots of the sampled posterior distributions for each of the analyses
discussed are found on this and the following pages: Fig. 6 for the LMC in
isolation with LISA instrumental noise, and Fig. 7 (Fig. 8) for the spectral
(spatial) parameters of the analysis with the LMC + MW + LISA instrumental
noise.

Figure 6: Corner plot for the analysis in §3.2 of the LMC SGWB in isolation,
showing the 1- and 2-dimensional marginalized posterior samples for each of
our model parameters. These are (moving left to right): the LISA position and
acceleration noise amplitudes ($\mathrm{log}_{10}(N_{\mathrm{p}})$ and
$\mathrm{log}_{10}(N_{\mathrm{p}})$, respectively); the SGWB power-law model
slope ($\alpha$) and log amplitude ($\log_{10}(\Omega_{\mathrm{ref}})$); and
the magnitude and phase of the $b_{\ell m}$ spherical harmonic coefficients up
to $\ell_{\mathrm{max}}^{b}=2$ ($\ell_{\mathrm{max}}^{a}=4$). The true values
of the noise parameters are marked with green dashed lines. The remaining
parameters do not have defined true values, as our simulated signal is
generated from a DWD population. Contours shown are 1- and $2\sigma$. A
careful eye will note a slight bias in the recovery of the position noise
contribution, $N_{\mathrm{p}}$. This is a result of our power-law spectral
model being an imperfect fit for the population-derived, non-power-law
spectrum of the LMC SGWB; repeating this study without the inclusion of the
LMC signal results in unbiased noise recoveries. Potential future approaches
to fitting the LMC signal with higher fidelity are discussed in §4. Figure 7:
Spectral parameters corner plot for the analysis in §3.3 of the LMC SGWB
alongside a simple simulation of the MW foreground, showing the 1- and
2-dimensional marginalized posterior samples for all spectral model
parameters. Spatial parameter samples are shown in Fig. 8. Included parameters
are (moving left to right): the LISA position and acceleration noise
amplitudes ($\mathrm{log}_{10}(N_{p})$ and $\mathrm{log}_{10}(N_{a})$,
respectively); the LMC SGWB power-law model slope ($\alpha_{\mathrm{LMC}}$)
and log amplitude ($\log_{10}(\Omega_{\mathrm{ref,LMC}})$); and the MW
foreground truncated power-law model slope ($\alpha_{\mathrm{MW}}$), log
amplitude ($\log_{10}(\Omega_{\mathrm{ref,MW}})$), and log cutoff frequency
($\log_{10}(f_{\mathrm{cut,MW}})$). True values are marked with green dashed
lines. As before, the LMC model parameters do not have defined true values.
Contours shown are 1- and $2\sigma$. Figure 8: Spatial parameters corner plot
for the analysis in §3.3 of the LMC SGWB alongside a simple simulation of the
MW foreground, showing the 1- and 2-dimensional marginalized posterior samples
for the LMC spatial model parameters (the MW spatial model is fixed; see
§3.3). Parameters shown are the magnitude and phase of the $b_{\ell m}$
spherical harmonic coefficients up to $\ell_{\mathrm{max}}^{b}=2$
($\ell_{\mathrm{max}}^{a}=4$). Contours shown are 1- and $2\sigma$.