KCL-PH-TH/2021-71

# Impact of Schumann resonances on the Einstein Telescope
and projections for the magnetic coupling function

Kamiel Janssens  Department of Physics, Universiteit Antwerpen, Antwerpen,
Belgium  Artemis, Université Côte d’Azur, Observatoire Côte d’Azur, Nice,
France
Katarina Martinovic  Theoretical Particle Physics and Cosmology Group, Physics
Department,
King’s College London, University of London, Strand, London WC2R 2LS, UK
Nelson Christensen  Artemis, Université Côte d’Azur, Observatoire Côte d’Azur,
Nice, France
Patrick M. Meyers  School of Physics, University of Melbourne, Parkville, VIC
3010, Australia  OzGrav, University of Melbourne, Parkville, VIC 3010,
Australia  Mairi Sakellariadou  Theoretical Particle Physics and Cosmology
Group, Physics Department,
King’s College London, University of London, Strand, London WC2R 2LS, UK

###### Abstract

Correlated magnetic noise in the form of Schumann resonances could introduce
limitations to the gravitational-wave background searches of future Earth-
based gravitational-wave detectors. We consider recorded magnetic activity at
a candidate site for the Einstein Telescope, and forecast the necessary
measures to ensure that magnetic contamination will not pose a threat to the
science goals of this third-generation detector. In addition to global
magnetic effects, we study local magnetic noise and the impact it might have
on co-located interferometers. We express our results as upper limits on the
coupling function of magnetic fields to the interferometer arms, implying that
any larger values of magnetic coupling into the strain channel would lead to a
reduction in the detectors’ sensitivity. For gravitational-wave background
searches below $\sim 30$ Hz it will be necessary for the Einstein Telescope
magnetic isolation coupling to be two to four orders of magnitude better than
that measured in the current Advanced LIGO and Virgo detectors.

##  I Introduction

When searching for an isotropic gravitational-wave background (GWB)
Christensen (2018) one typically uses cross-correlation techniques between
spatially separated interferometers to detect a correlated signal that is
below the local noise of either individual instrument. However, globally
correlated noise sources remain present in the cross-correlated data and can
therefore affect the analysis. An example of such a source is the Schumann
resonances Schumann (1952a, b). Schumann resonances are extremely low
frequency ($<$ 50Hz) electromagnetic excitations in the cavity formed by the
Earth’s surface and the ionosphere, driven by lightning strikes across the
globe. Given their global character, the Schumann resonances are correlated
over distances of several thousands of kilometers and longer.

Schumann resonances have magnetic strengths of the order of 1 pT at the
fundamental mode, which has an observed frequency of 7.8 Hz. They are expected
to couple magnetically to gravitational-wave (GW) interferometers via, e.g.
the mirror suspension systems, electric cables and the electronics, thereby
inducing a correlated signal of terrestrial origin Thrane  _et al._ (2013,
2014); Fiori  _et al._ (2020); Nguyen  _et al._ (2021).

The impact of Schumann resonances on the ongoing searches for a GWB with the
current generation of Earth-based interferometers (LIGO Aasi  _et al._ (2015),
Virgo Acernese  _et al._ (2015) and KAGRA Aso  _et al._ (2013)) has been
thoroughly investigated Thrane  _et al._ (2013, 2014); Coughlin  _et al._
(2016); Himemoto and Taruya (2017); Coughlin  _et al._ (2018); Himemoto and
Taruya (2019); Meyers  _et al._ (2020). The latest results in a search for an
isotropic GWB showed that Schumann resonances are below the detector
sensitivity; however as the sensitivity increases they could limit the search
Abbott  _et al._ (2021a). More importantly, one must study their effect on
searches with future Earth-based interferometers which aim to have
approximately one order of magnitude improvement in sensitivity compared to
current instruments.

In this study we focus on the European proposal for a third-generation (3G) GW
interferometer: the Einstein Telescope (ET) Punturo  _et al._ (2010). The
current proposal consists of three interferometers with an opening angle
$\pi$/3 forming an equilateral triangle. The no-longer operational Sos Enattos
mine in Sardinia, Italy, is one of the possible locations to host the future
ET interferometer, another being the Euregio Rhein-Maas at the intersection of
the Belgian, Dutch and German borders Amann  _et al._ (2020). Please note that
this paper does not contain a site comparison and typical magnetic spectra
will be used to make statements on the impact of magnetic fields on ET,
regardless of its exact location.

The research and development, and design phase of ET is ongoing. With this
consideration we investigate the impact of fundamental magnetic noise sources,
such as Schumann resonances, on the ET interferometers. We construct the
maximal allowed magnetic coupling function so that the fundamental magnetic
sources are not limiting the sensitivity of ET, either as an instrument itself
or in its use for a search for an isotropic GWB.

In addition to low-frequency Schumann resonances, there are high-frequency
correlated magnetic fields from individual lightning strikes Kowalska-
Leszczynska  _et al._ (2017), mainly situated in the frequency range 100 Hz -
1 kHz Ball  _et al._ (2020). We will investigate the impact of these on ET’s
sensitivity, as well as consider possible limitations due to infrastructural
noise.

The paper is organised as follows. In Sec. II we discuss the key ingredients
of the formalism of the search for an isotropic GWB and the way correlated
magnetic noise can enter in the analysis. This formalism can be rewritten to
allow us to calculate the maximal allowed magnetic coupling function to
prevent magnetic contamination. Sec. III focuses on the different magnetic
data sets that will be used to make projections about the magnetic coupling
function for ET. In Sec. IV the magnetic coupling functions are constructed
and we will discuss the impact of our results and their interpretation. A
conclusion is presented in Sec. V.

##  II Gravitational-Wave background

The key figure of merit when searching for an isotropic GWB is $\Omega_{\rm
GW}(f)$ Christensen (2018, 1992); Allen and Romano (1999); Romano and Cornish
(2017a):

$\Omega_{\rm GW}(f)=\frac{1}{\rho_{\rm c}}\frac{\text{d}\rho_{\rm
GW}}{\text{d}\ln f}~{},$ (1)

where we have defined the energy density, $\text{d}\rho_{\rm GW}$, contained
in a logarithmic frequency interval, $\text{d}\ln f$, divided by the critical
energy density $\rho_{\rm c}=3H_{0}^{2}c^{2}/(8\pi G)$ for a flat Universe.
$H_{0}$ is the Hubble constant, $c$ is the speed of light and $G$ is Newton’s
constant. We use the 15-year Planck value of 67.9 km s-1 Mpc-1 for $H_{0}$ Ade
_et al._ (2016).

To perform a search for a GWB, one typically uses a cross-correlation
statistic that is an unbiased estimator for $\Omega_{\rm GW}(f)$111This is
statement holds when one assumes the GWB is isotropic, Gaussian, stationary
and unpolarized. Allen and Romano (1999); Romano and Cornish (2017a):

$\hat{C}_{IJ}(f)=\frac{2}{T_{\textrm{obs}}}\frac{{\rm{Re}}[\tilde{s}^{*}_{I}(f)\tilde{s}_{J}(f)]}{\gamma_{IJ}(f)S_{0}(f)}~{},$
(2)

for interferometers $I$ and $J$, where $\tilde{s}_{I}(f)$ is the Fourier
transform of the time domain strain data $s_{I}(t)$ measured by interferometer
$I$, and $\gamma_{IJ}$ the normalized overlap reduction function (ORF) which
encodes the baseline’s geometry Christensen (1992); Romano and Cornish
(2017b). We approximate ET as three interferometers and ignore the details of
the xylophone configuration Hild  _et al._ (2011); this does not affect our
results.

In what follows, we assume the three co-located, 10km long arm interferometers
in triangle configuration of ET, namely ${\rm ET}_{1},{\rm ET}_{2},{\rm
ET}_{3}$, as having identical sensitivity. Furthermore, we neglect the
difference in $\gamma_{IJ}$ between the baseline pairs $IJ={\rm ET}_{1}{\rm
ET}_{2},{\rm ET}_{1}{\rm ET}_{3},{\rm ET}_{2}{\rm ET}_{3}$, where we will be
using $\gamma_{{\rm ET}_{1}{\rm ET}_{2}}$ from now on. For frequencies under 1
kHz the relative differences are Andersson  _et al._ (2009):

$\frac{|\gamma_{{\rm ET}_{1}{\rm ET}_{2}}-\gamma_{{\rm ET}_{1}{\rm
ET}_{3}}|}{\gamma_{{\rm ET}_{1}{\rm ET}_{2}}}<5\times 10^{-7},$
$\frac{|\gamma_{{\rm ET}_{1}{\rm ET}_{2}}-\gamma_{{\rm ET}_{2}{\rm
ET}_{3}}|}{\gamma_{{\rm ET}_{1}{\rm ET}_{2}}}<2\times 10^{-7},$

justifying our choice to neglect the difference between baseline pairs.

The normalisation factor $S_{0}(f)$ is given by
$S_{0}(f)=(9H_{0}^{2})/(40\pi^{2}f^{3})$ and $T_{\textrm{obs}}$ is the total
observation time of the data-collecting period 222The form of $S_{0}(f)$ for
ET differs from that one of e.g. LIGO by a factor of 3/4. This is due to the
difference in opening angle between interferometers’ arms ($\pi$/2 for LIGO
and $\pi$/3 for ET) that leads to different normalization factors in the ORF
of LIGO and ET baseline pairs Romano and Cornish (2017b).. In the absence of
correlated noise this cross-correlation statistic is an unbiased estimator of
$\Omega_{\rm GW}(f)$.

Equivalent to the cross-correlation statistic, one can construct a magnetic
cross-correlation statistic Thrane  _et al._ (2013, 2014)

$\displaystyle\hat{C}_{{\rm mag},{\rm ET}_{1}{\rm ET}_{2}}(f)$
$\displaystyle=|\scalebox{1.5}{$\kappa$}_{\rm ET}(f)|^{2}M_{{\rm ET}_{1}{\rm
ET}_{2}},$ (3) $\displaystyle\mbox{where~{}~{}}M_{{\rm ET}_{1}{\rm ET}_{2}}$
$\displaystyle=\frac{2}{T_{\textrm{obs}}}\frac{|\tilde{m}^{*}_{{\rm
ET}_{1}}(f)\tilde{m}_{{\rm ET}_{2}}(f)|}{\gamma_{{\rm ET}_{1}{\rm
ET}_{2}}(f)S_{0}(f)}~{},$

and $\scalebox{1.5}{$\kappa$}_{\rm ET}(f)$ describes the coupling from
magnetic fields to interferometer ${\rm ET}_{1}$, where we have used
$\scalebox{1.5}{$\kappa$}_{\rm ET}(f)=\scalebox{1.5}{$\kappa$}_{{\rm
ET}_{1}}(f)=\scalebox{1.5}{$\kappa$}_{{\rm ET}_{2}}(f)$. We denote by
$\tilde{m}_{{\rm ET}_{1}}(f)$ the Fourier transform of the time domain data
$m_{{\rm ET}_{1}}(t)$ measured by a magnetometer at site ${\rm ET}_{1}$, and
by $T_{\textrm{obs}}$ the duration of the segments used when Fourier
transforming the magnetic data333Note the different meaning of
$T_{\textrm{obs}}$ in Eq. (2) and Eq. (3), since in Eq. (2) the full data set
is analysed in one segment, whereas in Eq. (3) we divide the data in multiple
segments after which they are combined.. To construct a conservative magnetic
cross-correlation statistic we take the modulus of $\tilde{m}^{*}_{{\rm
ET}_{1}}(f)\tilde{m}_{{\rm ET}_{2}}(f)$ rather than taking only the real part
into account Abbott  _et al._ (2019, 2021a).

When analysing data in a search for an isotropic GWB using interferometers $I$
and $J$, one typically constructs the magnetic cross-correlation statistic
$\hat{C}_{{\rm mag},IJ}(f)$ to investigate if the observed magnetic fields
might result in correlated noise in the analysis Abbott  _et al._ (2019,
2021a). The magnetic coupling functions $\scalebox{1.5}{$\kappa$}_{I,J}(f)$
are measured by injecting magnetic fields with known amplitude and frequency
and observing their impact on the output in the GW data $s_{I,J}(t)$ Fiori
_et al._ (2020); Nguyen  _et al._ (2021).

However, in this study we will consider a different approach: given a desired
sensitivity and a magnetic spectrum $|\tilde{m}^{*}_{{\rm
ET}_{1}}(f)\tilde{m}_{{\rm ET}_{2}}(f)|$, we estimate the maximal allowed
magnetic coupling function $\scalebox{1.5}{$\kappa$}_{\rm ET}(f)$ such that ET
will not be limited by this magnetic noise.

The sensitivity for a GWB search is different to the instantaneous sensitivity
of the ET interferometer, referred to as the one-sided amplitude spectral
density (ASD) $P_{\rm ET}(f)=P_{{\rm ET}_{1}}(f)=P_{{\rm ET}_{2}}(f)=P_{{\rm
ET}_{3}}(f)$. In the case of an isotropic GWB search, the sensitivity is given
by the standard deviation on the cross-correlation statistic defined in Eq. 2.
In the small signal-to-noise ratio (SNR) limit, this is given by Christensen
(1992); Allen and Romano (1999); Romano and Cornish (2017a)

$\sigma_{{\rm ET}_{1}{\rm ET}_{2}}(f)\approx\sqrt{\frac{1}{2T_{\rm obs}\Delta
f}\frac{P_{\rm ET}^{2}(f)}{\gamma_{{\rm ET}_{1}{\rm
ET}_{2}}^{2}(f)S_{0}^{2}(f)}}~{},$ (4)

where $\Delta f$ is the frequency resolution, and $\sigma_{{\rm ET}_{1}{\rm
ET}_{2}}(f)$ defines our uncertainty and therefore our sensitivity in a single
frequency bin used in the analysis. However, when searching for an isotropic
GWB, one typically expects a broadband signal and often assumes a power-law
GWB model. Hence, a useful measure of GWB sensitivity is also the power-law
integrated (PI) curve. The PI curve, $\Omega^{\rm PI}_{{\rm ET}_{1}{\rm
ET}_{2}}(f)$ is constructed using $\sigma_{{\rm ET}_{1}{\rm ET}_{2}}(f)$ so
that its tangent at any frequency represents the sensitivity at which one
could detect a power-law $\Omega_{\rm GW}(f)$ with an SNR of 1 for the ${\rm
ET}_{1}{\rm ET}_{2}$ baseline Thrane and Romano (2013). Therefore, it serves
as a good figure of merit to identify broadband noise sources that could limit
the sensitivity of a GWB search.

To ensure that magnetic noise does not obstruct the isotropic search for a
GWB, we construct an upper limit on the magnetic coupling function that we
label “GWB,” in the following way. We use Eq. (3) and take the upper limit for
the magnetic cross-correlation $\hat{C}_{{\rm mag},{\rm ET}_{1}{\rm
ET}_{2}}(f)$ to be the 1$\sigma$-PI sensitivity curve – $\Omega^{\rm PI}_{{\rm
ET}_{1}{\rm ET}_{2}}$ – after one year of taking data:

$\scalebox{1.5}{$\kappa$}_{\rm ET}^{\rm GWB}(f)\equiv\sqrt{\frac{\Omega^{\rm
PI}_{{\rm ET}_{1}{\rm ET}_{2}}}{M_{{\rm ET}_{1}{\rm ET}_{2}}}}~{}.$ (5)

In addition, we explore a complimentary method for computing upper limits of
the magnetic coupling function $\scalebox{1.5}{$\kappa$}_{\rm ET}(f)$. To
investigate the impact of magnetic noise sources on the ASD of an individual
interferometer, we construct what we refer to as the “ASD” upper limit:

$\scalebox{1.5}{$\kappa$}_{\rm ET}^{\rm ASD}(f)\equiv k~{}\frac{P_{\rm
ET}(f)}{P_{\rm mag}(f)}~{},$ (6)

where $k$ is set to be $1/10$ to require any single technical noise
contribution to be a factor of 10 lower than ET’s ASD, $P_{\rm ET}(f)$.
$P_{\rm mag}(f)$ is the one-sided ASD of the magnetometers witnessing the
local noise of ET, assuming the magnetic noise to be the same for the three
individual ET interferometers ${\rm ET}_{1}$, $\rm ET_{2}$ and $\rm ET_{3}$.
If magnetic fields couple significantly at this level, they will limit the
expected sensitivity of the interferometer. The “ASD” upper limit should be
investigated to prevent magnetic noise from drastically impacting all science
goals of the GW interferometer.

To make full use of ET’s capabilities when searching for an isotropic GWB, one
should use $\scalebox{1.5}{$\kappa$}_{\rm ET}^{\rm GWB}(f)$. On the contrary,
to find the level at which magnetic fields might directly impact the
instantaneous sensitivity achieved by ET, then $\scalebox{1.5}{$\kappa$}_{\rm
ET}^{\rm ASD}(f)$ is the relevant measure.

##  III Data

In this analysis we use observed, rather than simulated, magnetic data.
However, since the location and exact positioning of ET is unknown we will
estimate $|\tilde{m}^{*}_{{\rm ET}_{1}}(f)\tilde{m}_{{\rm ET}_{2}}(f)|$ using
a variety of observed magnetic spectra that we describe below.

Schumann resonances have a relatively similar amplitude regardless where on
Earth they are measured, with variations of a factor of 2-3 in some cases
Coughlin  _et al._ (2018). The amplitude of the fundamental mode ($\sim$ 8 Hz)
rarely exceeds 1 pT444See Fig. 1 of Coughlin  _et al._ (2018) for a comparison
of Schumann resonance measurements across the globe.. Moreover, daily and
seasonal variations of the resonances’ amplitude occur.

As a part of site characterisation of the Sos Enattos mine for ET Naticchioni
_et al._ (2020); Romero , magnetic field measurements were taken from 5Hz to
100Hz in the exceptionally quiet environment inside the unused mine. We use 48
days of data taken from Nov 14 2019 to Dec 31 2019, using a single-axis
Metronix MFS-06e magnetometer positioned inside the Sos Enattos mine about 200
m below ground level. The magnetometer is sampled at 250 Hz. Given these on-
site measurements are available for Sos Enattos but not for the Euregio Rhein-
Maas ET-candidate site, we will use the magnetic spectrum observed at Sos
Enattos as a reference estimate for the magnetic spectrum at the future ET
site: $\tilde{m}_{{\rm ET}}(f)=2\tilde{m}_{\rm SosEnattos}(f)$. The factor 2
allows for higher magnetic fields at the final site and to be more
conservative.

Further investigations into magnetic noise at Euregio Rhein-Maas could help
compare the two currently proposed sites, but we do not make any statement
about which site will be used as the actual location of the ET detector. Site-
specific amplifying or reducing ambient magnetic fields are typically a second
order effect that can be ignored. However, as shown by measurements at KAGRA,
local amplification of ambient magnetic fields may not always be negligible
Atsuta  _et al._ (2016); Akutsu  _et al._ (2018).

We show in Fig. 1 percentiles of the magnetic ASD as measured at Sos Enattos.
We use the shape of the 10% magnetic percentile curve, since it captures the
peaks of the Schumann spectrum very well. We then scale the amplitude to match
the amplitude of the 95%-percentile curve at 7.8 Hz, the first Schumann mode.
This gives a good prediction for the Schumann resonance spectrum at Sos
Enattos.

Figure 1: Sos Enattos magnetic ASD constructed using 48 days of data from Nov
14 2019 to Dec 31 2019. A one-directional magnetometer was employed to collect
the data in the mine approximately 200 m below ground level. The line at 50Hz
is coming from the power mains.

This low-frequency range is where most of the sensitivity is for the ongoing
isotropic GWB searches, with the most recent LIGO-Virgo observing run
containing 99% sensitivity to a flat $\Omega_{\rm GW}$ spectrum below 100 Hz
Abbott  _et al._ (2021a). The sensitivity at high frequencies is suppressed
since the ORF for the LIGO and Virgo baseline pairs drops significantly with
frequency and approaches zero at a few hundreds of Hz. The same, however, may
not be true for the ET detectors, whose geometry leads to no suppression at
high frequency, and the anticipated ORF remains approximately constant up to 1
kHz, e.g. $\gamma_{{\rm ET}_{1}{\rm ET}_{2}}$(1 kHz)/$\gamma_{{\rm ET}_{1}{\rm
ET}_{2}}$(1 Hz) = 99.48%. We therefore look to place upper limits on the
magnetic coupling function at high frequencies.

We estimate the correlated magnetic noise spectrum above 100Hz by computing
the cross spectral density (CSD) between magnetometers at LIGO Hanford and
LIGO Livingston. The separation of several thousands of kilometers between the
two sites ensures that the estimated spectrum only contains fundamental,
global effects rather than local contributions. We consider the Hanford-
Livingston baseline as the most conservative choice for magnetic CSD since
these are the closest interferometers and the impact of individual lightning
strikes at frequencies above $\sim$ 100 Hz drastically attenuates with
increasing distance. Hanford and Livingston both have two low-noise
magnetometers on site, positioned at low-noise locations orientated along the
interferometers arms. We make use of an “omni-directional” magnetic CSD, where
we take into account all possible cross-correlation combinations between the
magnetometer pairs at both interferometer sites,

$\displaystyle{\rm CSD}_{\rm HL}=$
$\displaystyle\left[\right.|\tilde{m}^{*}_{H_{x}}(f)\tilde{m}_{L_{x}}(f)|^{2}+|\tilde{m}^{*}_{H_{x}}(f)\tilde{m}_{L_{y}}(f)|^{2}$
(7)
$\displaystyle+|\tilde{m}^{*}_{H_{y}}(f)\tilde{m}_{L_{x}}(f)|^{2}+|\tilde{m}^{*}_{H_{y}}(f)\tilde{m}_{L_{y}}(f)|^{2}\left.\right]^{1/2}~{}.$

Here $H_{x},H_{y}$ ($L_{x},L_{y}$) represent the two orthogonal magnetometers
at Hanford (Livingston) pointing along the interferometer’s $x$\- and
$y$-arms. We use data recorded from Apr 2 2019 to Mar 27 2020, which
approximately matches the LIGO-Virgo third observing run, named O3.

We assume ${\rm CSD}_{\rm HL}$ is a realistic estimation for the magnetic
noise at ET, where we include an additional factor of 2 reflecting our
uncertainty on the magnetic spectrum 555A factor of 2 at the level of
individual magnetic fields $\tilde{m}_{{\rm ET}}(f)$, corresponds to a factor
of $2^{2}$ at the level of $|\tilde{m}^{*}_{{\rm ET}_{1}}(f)\tilde{m}_{{\rm
ET}_{2}}(f)|$.: $\frac{2}{T_{obs}}|\tilde{m}^{*}_{{\rm
ET}_{1}}(f)\tilde{m}_{{\rm ET}_{2}}(f)|=4{\rm CSD}_{\rm HL}$.

We also investigate the impact of local magnetic noise. One should study local
magnetic noise carefully as this could be a correlated noise source as well
because each local corner station for the triangular ET set-up will likely
house mirrors for two of the three interferometers (separated by $\sim 300$ m
and in different vacuum tubes). To model local magnetic fields of a GW
interferometer, we consider the local magnetic noise measured in the central
building (CEB) at Virgo site Fiori  _et al._ (2020). We use the 90% magnetic
percentile of data collected between Feb 10 2020 and Feb 16 2020 - during the
second half of the third observing run, O3b. This spectrum is quite similar to
the one observed in the Virgo CEB during O2 Cirone  _et al._ (2019).

We assume the local noise at Virgo is a realistic estimation for the local
magnetic noise at ET, where we include an additional factor of 2 reflecting
our uncertainty on the magnetic spectrum: $\tilde{m}_{{\rm
ET}_{1}}(f)=2\tilde{m}_{\rm V_{CEB}}(f)$.

##  IV Results

We construct two complementary measures—$\scalebox{1.5}{$\kappa$}_{\rm
ET}^{\rm GWB}(f)$ and $\scalebox{1.5}{$\kappa$}_{\rm ET}^{\rm ASD}(f)$—of the
magnetic coupling function. In our analysis, we use two preliminary design
studies, called $\rm ET_{\rm Single}$ and $\rm ET_{\rm Xylophone}$666In
previous literature $\rm ET_{\rm Single}$ and $\rm ET_{\rm Xylophone}$ have
also been referred to as respectively ET-B, ET-D. Hild  _et al._ (2009, 2011).
In the remainder of this paper we will use the notation ET-S and ET-X as
abbreviations to refer to $\rm ET_{\rm Single}$, respectively $\rm ET_{\rm
Xylophone}$. In Fig. 2 we show the sensitivity curves for both of these design
options in the left panel. In the right panel we show the PI curves for each
of these design options for 1 year of integration time. The ET-X configuration
is an order of magnitude more sensitive at low frequencies as compared to
ET-S. Fig. 2, shows ET-X could be sensitive to a GWB at the $\Omega_{\rm
GW}(f)\sim 10^{-12}$ range, with an SNR=1 after 1 year of observation. This
seems to be consistent with earlier investigations stating a GWB with strength
$\Omega_{\rm GW}(f)=2\cdot 10^{-12}$ would be detected with an SNR=5 after 1.3
years of observation timeSharma and Harms (2020).

The upper limits presented here assume the reduction of the magnetic coupling
is the only pursued method to prevent magnetic fields from coupling
significantly to the interferometer. Methods such, as the use of Wiener
filters Coughlin  _et al._ (2018), could also be used to reduce the effects of
correlated magnetic noise. However, the best strategy for ET will be to design
the magnetic isolation to be as good as possible.

Figure 2: The ET configurations – ET-S, ET-X Hild  _et al._ (2009, 2011) – and
their anticipated sensitivity curves (left panel) as well as their power-law
integrated curves after a year-long observation (right panel).

In Fig. 3 we show limits on the coupling imposed by our target sensitivity
measures. The $\scalebox{1.5}{$\kappa$}^{\textrm{GWB}}_{\rm ET-X}(f)$ limits
are indicated by the solid blue curve, while
$\scalebox{1.5}{$\kappa$}^{\textrm{ASD}}_{\rm ET-X}(f)$ limits are denoted by
the dash-dotted yellow curve. The average coupling measurements made at Virgo
Fiori  _et al._ (2020); Cirone  _et al._ (2018), LIGO Hanford and Livingston
Davis  _et al._ (2021); Nguyen (2021); Nguyen  _et al._ (2019); Merfeld  _et
al._ (2019) during the O3 run are indicated by the green circles, smaller
orange circles, and magenta stars respectively. Please note that the magnetic
coupling measurements only start from 11Hz, 9Hz and 7Hz for respectively the
Virgo, LIGO Hanford and LIGO Livingston interferometers. In Fig. 4 and Fig. 5
we show high and low-frequency coupling limits,
$\scalebox{1.5}{$\kappa$}^{\textrm{GWB}}_{\textrm{ET}}(f)$, respectively for
each of the design options (solid blue for ET-S, yellow dashed for $\rm
ET-X$). The average coupling measurements at Virgo, Hanford, and Livingston
are the same as in Fig. 3.

We notice that the upper limit on the magnetic coupling at certain frequencies
is allowed to be greater than the magnetic coupling at current
observatories777Above 100 Hz the weekly measurements of the magnetic coupling
are often upper limits rather than an actual measurements. However during some
periods of “extreme” magnetic coupling such high values are actually measured
as well Fiori  _et al._ (2020)., for example $\gtrsim 30$ Hz in the
$\scalebox{1.5}{$\kappa$}^{\textrm{ASD}}_{\rm ET-X}(f)$ curve in Fig. 3. This
means that one can be less concerned about Schumann resonance magnetic noise
coupling into ET as compared to LIGO/Virgo detectors. The reason for this
result can be seen by considering the difference between the magnetic coupling
measured in units of [$\rm{T^{-1}}$] 888The unit of magnetic coupling function
[$\rm{T^{-1}}$] is often referred to as [strain $\rm{T^{-1}}$]. and in units
of [$\rm{m~{}T^{-1}}$]. The latter takes into account the arm-length of the
interferometer, $\textit{L}_{\rm arm}$,

$\scalebox{1.5}{$\kappa$}[\rm{T^{-1}}]=\frac{\scalebox{1.5}{$\kappa$}\,[\rm{m~{}T^{-1}}]}{\textit{L}_{\rm
arm}[m]}~{}.$

Since ET is planned to have a 10 km arm-length, instead of 4 km (LIGO) or 3 km
(Virgo), the test masses displacements due to magnetic effects measured in the
units [$\rm{m~{}T^{-1}}$] is allowed to be larger compared to existing
interferometers.

We emphasise that Fig. 3 and Fig. 4 use data measured at the Sos Enattos
candidate site for ET. However, given the similar amplitude of the Schumann
resonances around the globe, these results could be transferable to another
location and more specifically a second candidate site in the Euregio Rhein-
Maas. That is assuming there is no extreme magnification of magnetic fields
due to local effects, as observed at KAGRA Atsuta  _et al._ (2016); Akutsu
_et al._ (2018). To account for such a possibility we introduced a factor of
1/2 in our estimates of the coupling upper limits.

We illustrate in Fig. 5 the effect of magnetic fields at ET above a 100 Hz. An
example of an analysis that could target a signal at these higher frequencies
is the study of a GWB from unresolved millisecond pulsars. This search is
complementary to the standard continuous-wave search of individual pulsars,
and can help constrain ellipticity of rotating neutron stars Talukder  _et
al._ (2014); Abbott  _et al._ (2020, 2021b). Current forecasts predict an
improvement of one to two orders of magnitude in the sensitivity to
ellipticity going from LIGO-Virgo to ET. However, correlated magnetic noise at
high frequencies could weaken the ellipticity constraints (see, Fig. 6 in
Maggiore  _et al._ (2020)), and should thus be treated carefully.

In our high-frequency analysis we have used the magnetic CSD measured between
LIGO Hanford and LIGO Livingston, two widely separated sites, to ensure we are
not dominated by local effects. However, since magnetic fields from the
sources contributing to this magnetic noise (e.g. individual lightning strikes
Ball  _et al._ (2020)) are attenuated over long distances, the fundamental
magnetic spectrum at ET could be stronger compared to our predictions.
Therefore one should be cautious interpreting the upper limits presented in
Fig. 5. For the magnetic coupling above 100Hz, we do not show the upper limits
calculated using the “ASD” formalism, Eq. (6), since these limits are less
stringent than the measured magnetic coupling at present day interferometers.

Figure 3: “ASD” and “GWB” magnetic coupling function upper limits for $\rm
ET-X$ design sensitivity. Also included are the average of the measurements of
the coupling functions at LIGO Hanford, LIGO Livingston and Virgo during the
O3 run for comparison. Figure 4: Variation in the “GWB” magnetic coupling
function upper limits for the different ET designs. Also included are the
average of the measurements of coupling functions at LIGO Hanford, LIGO
Livingston and Virgo during the O3 run for comparison. Figure 5: Upper limits
on “GWB” magnetic coupling function of ET-S and -X at high frequencies. Also
included is the average of the measurements of coupling functions at LIGO
Hanford, LIGO Livingston and Virgo during the O3 run for comparison.

Figure 6: Needed improvement factor as a function of frequency for the “ASD”
(left panels) and “GWB” (right panels) upper limits on the ET magnetic
coupling function. The low-frequency (top panels) magnetic coupling poses a
greater challenge for the operation of ET compared to the high-frequency
(bottom panels) magnetic coupling, while “GWB” upper limits on the magnetic
coupling are more constraining than the “ASD” ones. In all panels, the dash-
dotted blue line indicates the line where no improvement is necessary.

We summarise these results in Fig. 6, where we express the estimated “ASD” and
“GWB” upper limits as a factor of improvement needed in the ET coupling
function relative to Hanford, Livingston and Virgo coupling functions. We show
how this factor varies with frequency, as well as how it changes with the
choice of ET design sensitivity. For frequencies below $\sim 30$ Hz, the
magnetic coupling would need to drastically reduced, with factors of
improvement of the order $10^{2}-10^{4}$ needed for the isotropic GWB search,
see top right panel. High-frequency coupling, on the other hand, would only
require up to a factor of 10 reduction in magnetic coupling to run a
successful GWB search (bottom right panel). Meanwhile, from the “ASD” upper
limits we do not require any improvement in coupling function at frequencies
above 30 Hz, see two left panels, confirming that this is a less conservative
constraint on the magnetic coupling.

Note that above 30 Hz Advanced LIGO’s magnetic coupling is dominated by
induction of currents in cables Nguyen  _et al._ (2021). One mitigation
strategy that could be followed in this scenario is using, as much as
possible, a cabling network of optical fibers. The implementation of a large-
scale optical fiber network has been investigated and implemented at CERN
Shoaie  _et al._ (2018). An important factor for the reduced magnetic coupling
for Advanced LIGO compared to Advanced Virgo is that LIGO uses an
electrostatic test mass actuators whereas Virgo uses magnetic actuation Nguyen
_et al._ (2021); Cirone  _et al._ (2018). Further reducing the number of
magnets attached to the suspensions should reduce magnetic coupling.
Additional magnetic shielding can be a complementary method to reduce the
magnetic coupling Cirone  _et al._ (2019). Ultimately, if methods for magnetic
coupling reduction are insufficient one could consider the cancellation of
magnetic noise, similar to what is considered in the context of Newtonian
Noise Harms and Paik (2015); Badaracco and Harms (2019), albeit using
magnetometers instead of seismometers.

Finally, if one is unable to reduce the effect from local magnetic fields
originating from e.g. used infrastructure, local magnetic noise sources will
dominate the fundamental magnetic noise discussed above. This leads to the
most stringent coupling upper limits, reported in Fig. 7. To construct these
upper limits the magnetic noise as observed in the Virgo central building is
used. This represents a realistic magnetic environment in present-day
interferometers, however it may not be the most conservative.

In the case of Virgo this local noise does not pose a serious problem in the
GWB search since it is uncorrelated with local magnetic noise at far-away
Hanford and Livingston detectors. Between co-located ET interferometers,
however, the local noise could become correlated. This can lead to drastically
more stringent upper limits on the magnetic coupling. If one wants to fully
utilize the data for GWB searches, the magnetic coupling should be well below
the magnetic coupling measured at Hanford, Livingston and Virgo, see Fig. 7.
Below $\sim$ 30 Hz, the instantaneous detector sensitivity will be limited by
magnetic noise if the coupling is not reduced below the current day magnetic
coupling of Hanford, which is already significantly smaller compared to the
coupling measured at Livingston and Virgo.

Figure 7: “ASD” and “GWB” magnetic coupling function upper limits of all ET
design sensitivities in the case the local magnetic noise is the same level as
the CEB at Virgo during O3. Also included are the average of the measurements
of coupling functions at LIGO Hanford, LIGO Livingston and Virgo during the O3
run for comparison.

##  V Conclusion

ET is a powerful and promising instrument for detecting a GWB, with an
unprecedented low-frequency sensitivity compared to LIGO and Virgo Maggiore
_et al._ (2020). Methods have been proposed to separate an astrophysical GWB
(from compact binary mergers) from a cosmological one Regimbau  _et al._
(2017); Sachdev  _et al._ (2020); Martinovic  _et al._ (2021). ET could be
sensitive at the SNR=1 level to a cosmological GWB $\Omega_{\rm GW}(f)\sim
10^{-12}$, but this depends critically on the low-frequency performance of ET.
The ability to detect a GWB through correlation methods between multiple
detectors assumes the absence of correlated noise Christensen (2018), however
globally coherent magnetic fields have been identified as a limiting noise
source for the present GW detector network Thrane  _et al._ (2013, 2014);
Meyers  _et al._ (2020). As we have shown, this is also the case for ET. More
precisely, we have shown that the magnetic coupling functions for ET must be
better than those of LIGO and Virgo by a factor of $10^{2}-10^{4}$ for
frequencies below 30 Hz, in order to avoid correlated noise from Schumann
resonances affecting GWB searches.

Reducing the magnetic coupling to prevent a significant impact on the
interferometers, and also to ensure that local magnetic noise is as small as
possible, is the best strategy for ET. This could be achieved by reducing the
number of magnets attached to the suspensions Nguyen  _et al._ (2021),
additional shielding Cirone  _et al._ (2019), and using optical fibers as much
as possible for signal transmission Nguyen  _et al._ (2021). There could also
be a synergy with noise subtraction methods, such as Wiener filters Coughlin
_et al._ (2018). Such methods have been investigated to cancel Newtonian Noise
Harms and Paik (2015); Badaracco and Harms (2019). However they also could be
used to reduce the effects of correlated magnetic noise and loosen the
requirements on the magnetic coupling, as presented here.

Not reaching the reported upper limits on the magnetic coupling functions
could have a direct impact on the search for a GWB with ET. Note, however,
that reaching these upper limits is not necessarily a guarantee that there are
no effects by magnetic fields on the search for a GWB. The importance of
correlated magnetic noise coupling will need to be considered as ET is
designed and constructed.

###### Acknowledgements.

The authors acknowledge access to computational resources provided by the LIGO
Laboratory supported by National Science Foundation Grants PHY-0757058 and
PHY-0823459. Furthermore the authors acknowledge the Sos Enattos former mine,
hosting the instrumentation used for this study, the Istituto Nazionale di
Fisica Nucleare (INFN), the Istituto Nazionale di Geofisica e Vulcanologia
(INGV), the European Gravitational Observatory (EGO), the University of
Sassary and the Regione Autonoma Sardegna for the support to the site
characterization activities. Furthermore the authors would like to thank
Michael Coughlin, Irene Fiori and Jan Harms for useful comments and
discussions. This paper has been given LIGO DCC number P2100356, Virgo TDS
number VIR-1050A-21 and ET TDS number ET-0427A-21. K.J. is supported by FWO-
Vlaanderen via grant number 11C5720N. K.M. is supported by King’s College
London through a Postgraduate International Scholarship. M.S. is supported in
part by the Science and Technology Facility Council (STFC), United Kingdom,
under the research grant ST/P000258/1.

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