# Anomalies and open issues of the MICROSCOPE Space Test
of the Weak Equivalence Principle

Anna M. Nobili  Dept. of Physics “E. Fermi”, University of Pisa, Largo B.
Pontecorvo 3, 56127 Pisa, Italy  Alberto Anselmi  Thales Alenia Space Italia,
Strada Antica di Collegno 253, 10146 Torino, Italy (ret.)

###### Abstract

MICROSCOPE’s final results report no violation of the Weak Equivalence
Principle (Universality of Free Fall) for Pt and Ti test masses quantified by
an E$\ddot{\rm o}$tv$\ddot{\rm o}$s parameter $\eta\simeq 10^{-15}$, an
improvement by about two orders of magnitude over the best ground tests. The
measurement is limited by random noise with $1/\sqrt{\nu}$ frequency
dependence attributed to thermal noise from internal damping occurring in the
grounding wires. From information available and the physics of internal
damping we calculate the differential acceleration noise spectral density at
the signal frequency, and show it varies widely between experiment sessions.
Such large variations are inexplicable if translated into physical quantities
such as the quality factor. While calibrations interspersed with measurement
sessions may cause some such changes, they cannot explain jumps between
consecutive sessions without recalibration. A potential explanation is
conjectured related to a fluctuating zero depending on measurement
initialization errors. The experiment was severely affected by “glitches”
–anomalous acceleration spikes related to radiation from the Earth– injecting
significant power at the signal frequency and its harmonics. The procedure
used to deal with the glitches depends on introducing artificial data and
leaves spurious effects potentially mimicking a violation signal or canceling
a real one. An alternative procedure, relying only on real measured data, is
proposed, already used in ground tests of the Weak Equivalence Principle by
the Eöt-Wash group. Future experiments aiming to exploit the full potential of
space must resolve these issues, rely solely on measured data, and, more
generally, readdress the experiment design.

##  I Introduction

MICROSCOPE is the first experiment on the Weak Equivalence Principle (WEP)
performed in low Earth orbit. A potential violation of the WEP is quantified
by the E$\ddot{\rm o}$tv$\ddot{\rm o}$s ratio $\eta$, the fractional
differential acceleration between two test masses of different composition as
they fall in the gravitational field of a source body, the Earth in this case.

The instrument complement included two sensor units (SU), one with test masses
(TM) of different composition for the WEP test (SUEP), the other with equal
composition TM’s for control (SUREF). Each SU includes two test masses
configured as coaxial hollow cylinders, the common axis being the sensitive
axis. Each TM forms part of an independent accelerometer. A set of electrodes
is used for both measuring the position of the TM and applying the voltages
that maintain it “motionless” with respect to its cage. A thin gold wire
provides electric grounding and polarization of the electrostatically
levitated mass. TM motion is detected by capacitive sensing (displacements
induce capacitance variations). In orbit, the axis of symmetry of the SU’s is
in the orbit plane and a putative violation signal is an Earth pointing vector
whose size oscillates at the orbit frequency of $1.6818\times
10^{-4}\,\rm{Hz}$ plus or minus the satellite spin rate. In the valid
experiment sessions, spin rates opposite the orbit motion were used, resulting
in signal frequencies of $\nu_{{}_{EP_{V2}}}=0.92499\times 10^{-3}\,\rm{Hz}$
and $\nu_{{}_{EP_{V3}}}=3.11133\times 10^{-3}\,\rm{Hz}$, the synodic
frequencies relative to the Earth in V2 and V3 spin mode respectively.

The measurements were taken in sessions of 120 orbits (8.26 days) or less, the
upper limit being driven by periodic correction of clock drift. The experiment
was affected by anomalous acceleration peaks (named “glitches”) occurring
simultaneously in the four test masses. The issue arose in past geodesy
missions and was solved in GOCE, mostly thanks to a stiffer solution for the
multi-layer insulation. In MICROSCOPE glitches are found to produce large
accelerations at the same frequency as the signal and its harmonics. In order
to cope with glitches large portions of data were removed and replaced with
artificial data.

Early results published in 2017 MS-PRL2017  were based, for the SUEP sensor,
on the analysis of one measurement session (#218) lasting $8.26$ days and
reported a fractional difference of $\simeq 10^{-14}$ in the accelerations of
Pt and Ti test bodies. A complete analysis of 94 days in 19 sessions reports
no violation at the level of $\eta\simeq 10^{-15}$ MS-PRL2022 ;
CQGspecialissueMS2022 .

Establishing the nature and source of the noise limiting the measurement is
crucial for any experiment and a prerequisite for future improvements. The
early report MS-PRL2017  showed plots of the square root of the power spectral
density (PSD) of the differential acceleration measured in two sessions, and
the comment explicitly pointed out, in the frequency region of interest, a
$1/\sqrt{\nu}$ trend attributed to random thermal noise from internal damping
occurring in the gold wires. Two years later CQG2019 , an improved analysis of
the same two sessions was interpreted in the same way. The final report MS-
PRL2022  does not mention the limiting noise and no longer shows any PSD plot.
In the batch of companion papers CQGspecialissueMS2022  only a couple of plots
are provided TouboulCQG2022-9 , which appear to confirm the $1/\sqrt{\nu}$
trend. In one case LiorzouCQG2022-2  a potential minor contribution from
electrostatic patch noise is conjectured, with unknown frequency dependence.
In general CQGspecialissueMS2022 , wire damping is mentioned as the most
likely cause but the authors seem to refrain from drawing any firm conclusions
as to the nature of the noise. Lacking any contrary information, we assume the
wire damping hypothesis in our analysis.

Systematic effects that were a matter of concern MicroscopeLimitations  turned
out to be dominated by temperature variations; after calibration and a
posteriori correction of all sessions, residual systematic errors were much
smaller than random errors (SUEP) or close to them (SUREF)
RodriguesSystematicsCQG2022 ; TouboulCQG2022-9 .

The stiffness of the gold wires, whose damping gives rise to the dominant
thermal noise, exceeds expectations based on ground tests by two orders of
magnitude, in addition to showing unexplained features (Sec. II).

We first focus on the thermal noise and the physical parameters it depends
upon: the stiffness and the quality factor of the wires. They are properties
of the physical plant that must remain constant unless specific actions are
undertaken which alter the instrument setup. Alterations may occur during
calibrations, since they may offset the TM and affect the stiffness (Sec. II).
However, an anomalous behavior of thermal noise occurs also in some sequential
sessions not affected by calibrations, and even within the same session. We
conjecture an explanation in terms of initialization errors (Sec. III).

Glitches are very numerous, between 14000 and 40000 in 120 orbits. They
produce large effects at the frequency of the signal and its harmonics. In the
procedure used to deal with glitches, many data points are removed and
replaced with artificially reconstructed data. Such a procedure introduces new
potential sources of error and is not mandatory. The WEP ground tests by the
Eöt-Wash group Adel2008  provide a successful example of data analysis in the
presence of missing data carried out without introducing artificial data,
exploiting the fact that the frequency and phase of the target signal are
known. We suggest that such an analysis is the most appropriate and would
definitively settle the experiment result (Sec. IV).

Conclusions are drawn in (Sec. V).

##  II Thermal noise from internal damping with a hundredfold higher stiffness

Once systematics have been taken care of –and if readout noise is not an
issue– the measurement is ultimately limited by random thermal noise. It sets
the length of the integration time required for the signal to emerge. A
thermal noise $10$ times larger requires an integration time $100$ times
longer to detect the same signal NobiliWEPfocusissue .

In both sensors each TM constitutes an independent accelerometer. Unlike
common mode effects, random noise is not reduced by taking the difference of
the individual accelerations. Assuming that they are uncorrelated, the
differential acceleration noise is obtained by adding them in quadrature.

In the case of internal damping, each test cylinder of mass $m$ is subjected
to a thermal acceleration with a PSD whose square root is
SaulsonInternalDamping :

$S_{a_{th}}^{{}^{1/2}}(\nu_{{}_{EP}})=\sqrt{4k_{{}_{B}}T\cdot\frac{k}{m^{2}Q}}\cdot\frac{1}{\sqrt{2\pi\nu_{{}_{EP}}}}$
(1)

with $k_{{}_{B}}T$ ($k_{{}_{B}}$ the Boltzmann constant) the random disordered
energy of thermal equilibrium at temperature $T$, $k$ the stiffness and $Q$
the quality factor of the wire, $\nu_{{}_{EP}}$ the frequency of the target
WEP violation signal.

WEP tests with rotating torsion balances operate close to the thermal noise
limit due to dissipation in the suspension wire with quality factor $Q\simeq
6000$ AdelCQGfocusissue2012 . In PRD2014integrationtime2014  various sources
of thermal noise have been investigated for the GG proposed space test of the
WEP, finding that gas damping (frequency independent) would contribute more
than internal damping because the frequency of the signal is upconverted from
the orbital frequency to the much higher spin frequency ($1\,\rm Hz$) in order
to exploit the $1/\sqrt{\nu}$ dependence of internal damping noise
PRLthermalnoise .

Along the symmetry axis of MICROSCOPE test cylinders (the only one sensitive
to WEP violation) displacements of the TM from the “zero” position are
measured with the “area variation” principle rather than the “gap variation”
principle used for the other two linear axes ONERA1999 . In the former case
there should be no electrostatic stiffness, except for small boundary effects
depending on the gap and larger for larger gaps. Instead, with “gap variation”
the “zero” is a point of unstable equilibrium, the (negative) electrostatic
stiffness exceeding the (positive) stiffness of the wire by far. A stiffness
of $5\times 10^{-5}\,\rm{Nm^{-1}}$ was measured for the “area variation”
CAESAR accelerometer ONERA1999 .

For each TM the total stiffness was measured in orbit by applying a large
displacement signal and measuring the acceleration in response to it
ChhunCQG2022-5 . The acceleration over displacement ratio is $k_{tot}/m$ and
yields the total stiffness $k_{tot}>0$, showing that the system behaves like a
harmonic oscillator with natural frequency of oscillation
$\omega_{\circ}=\sqrt{k_{tot}/m}$ forced at $\omega<\omega_{\circ}$. The
stiffness measured is attributed entirely to the wire because it is found
unaffected by the electric voltages CQG2019 . Alas, it is two orders of
magnitude bigger than in ground measurements WT-RSI2000b .

An ad-hoc electrostatic torsion pendulum measured the stiffness of a wire by
applying a force perpendicular to it, the theoretical prediction being
$k_{\perp}=3\pi Er^{4}/\ell^{3}$ for a beam of radius $r$ and length $\ell$
with $E=7.85\times 10^{10}\,\rm{Nm^{-2}}$ the elastic Young modulus of gold.
The agreement was good and on that basis the expectation for the similar wire
of the space experiment ($3.5\,\mu\rm{m}$ radius, $2.5\,\rm{cm}$ length) was
$k_{\perp}=7\times 10^{-6}\,\rm{Nm^{-1}}$, while the measured value is $\simeq
10^{-3}\,\rm{Nm^{-1}}$. Furthermore, although the four wires are all made of
gold and all have the same geometry, their measured stiffnesses differ up to
$7$ times.

If the wire is attached parallel to the cylinder’s $X$ axis –as we guess– only
the radial stiffness (in any direction of the plane perpendicular it) obeys
the $k_{\perp}$ formula, contributing a positive stiffness which is negligible
if compared to the much larger electrostatic negative stiffness, as it is
confirmed by the measurements along the $Y,\,Z$ axes ChhunCQG2022-5 . Instead,
for the sensitive $X$ axis what matters is the stiffness under the effect of a
force along the wire. For a beam with section $\pi r^{2}$ the theoretical
stiffness under the effect of a longitudinal force is
$k_{\parallel}=\pi\,r^{2}E/\ell$, which in this case would be a huge factor
$1.7\times 10^{7}$ bigger than the expected $k_{\perp}$. This is not the case,
hinting that the wire is, to some extent, slack.

It is argued CQG2019  that the very large measured values may be due to a
contribution from the stiffness along the axis of the wire, which comes into
play if the wire is pushed or pulled along its axis. However, there is no
indication as to how large this contribution would be and why it can vary so
much. In WT-RSI2000b  a sketch shows the wire under test as if it were slack,
but the effect is not discussed. The fact is that the pendulum measures
$k_{\perp}$, while what is needed is $k_{\parallel}$. Its dependence on the
wire being more or less slack/taut is likely to make it vary between sessions
if a calibration occurs in between, because in the calibration the quadratic
acceleration term must be kept sufficiently small, which is achieved by
offsetting the TM to a new “zero”, hence affecting $k_{\parallel}$. Since
losses mostly occur at the two ends of the wire in correspondence of the glued
points of attachment, they are unlikely to undergo substantial changes after
launch, hence we expect the quality factor not to vary with calibrations.

Within the mission team, measurements ChhunCQG2022-5  have been questioned by
BergeCQG2022-10 . However, their model is incorrect: for a wire along the $X$
axis it gives zero stiffness in the $Y,Z$ directions, while it is
$k_{\perp}=3\pi Er^{4}/\ell^{3}$.

A higher stiffness by $\lambda\simeq 100$ yields a higher thermal acceleration
noise by $\sqrt{\lambda}$ (Eq. (1)), hence an integration time $\lambda$ times
longer to detect the same signal. The acceleration of the signal and
systematics are unchanged, but produce $\lambda$ times smaller displacements,
the displacement being the physical quantity measured by each capacitance
sensor. With the measured positive stiffness each electrostatically levitated
TM is a harmonic oscillator with natural frequency of $\simeq
10^{-2}\,\rm{Hz}$.

##  III Analysis of thermal noise and observed anomalies

For each sensor with TM’s $m_{1},m_{2}$, stiffnesses $k_{1},k_{2}$, and the
same $Q$ for both masses (if not, the lower dominates), the square root of the
PSD of the differential acceleration thermal noise is:

$S_{\Delta a_{th}}^{{}^{1/2}}(\nu)=f_{{}_{TkQ}}\mdot\frac{1}{\sqrt{2\pi\nu}}$
(2)

with:

$f_{{}_{TkQ}}=\sqrt{4k_{{}_{B}}T\mdot\left(\frac{k_{1}}{m_{1}^{2}}+\frac{k_{2}}{m_{2}^{2}}\right)\mdot\frac{1}{Q}}\
\ \ \ \ \ .$ (3)

For a mechanical suspension the value of $Q$ is known to be frequency
dependent (higher at higher frequencies). However, the frequencies of interest
in V2 or V3 spin mode are too close to expect any relevant difference in $Q$,
and we therefore take $f_{{}_{TkQ}}$ as frequency independent. Hence, Eq.(2)
represents, in a typical log-log plot (e.g. Fig. 2 of MS-PRL2017 ), a straight
line with negative $1/2$ slope whose position along the vertical axis is set
by the factor $f_{{}_{TkQ}}$ (in $\rm{ms^{-2}}$).

For a given value of $f_{{}_{TkQ}}$ the measured noise is fitted by the same
straight line. If the signal frequency increases, this noise decreases (as
$1/\sqrt{\nu}$), but once the oscillator has been set up and launched, $k$’s
and $Q$’s are fixed, and (except for a mild dependence on $T$) the value of
$f_{{}_{TkQ}}$ is fixed and so is the straight line (2). However, after a
calibration we cannot exclude a variation of the stiffness (Sec. II), hence of
$f_{{}_{TkQ}}$ and of the spectral density of differential accelerations. To
the contrary, in sequential sessions, or within the same session analyzed with
equally valid methods, we expect no such variations.

In order to verify that the measured thermal noise does not change whenever it
is expected not to, we need –for all measurement sessions– the spectral
density of differential accelerations at the frequency of the signal. Since
this quantity is not available in papers MS-PRL2022 ; CQGspecialissueMS2022
we derive it indirectly as follows.

At the frequency of the signal $S_{\Delta a_{th}}^{{}^{1/2}}(\nu_{{}_{EP}})$
given by (2) also obeys the equation PRD2014integrationtime2014 :

$S_{\Delta a_{th}}^{{}^{1/2}}(\nu_{{}_{EP}})=\sqrt{t_{int}}\mdot\delta\mdot
g_{{}_{drive}}$ (4)

where $g_{{}_{drive}}=7.9\,\rm{m\,s^{-2}}$ is the driving signal of WEP
violation at the satellite orbit, $t_{int}$ is the session duration and
$\delta\mdot g_{{}_{drive}}$ is a “nominal” violation signal with signal to
noise ratio SNR=1 that would not be detected because of thermal fluctuations
($\sigma$), which decrease with the session duration as more data are
available. For a given $\delta$, the lower $\sigma$, the lower the integration
time for a violation signal to emerge and be detected.

Tables 6, 7 in TouboulCQG2022-9  give $\delta$ (and its $\sigma$) for all
sessions based on two different methods of data analysis, named $M$ and $A$
(discussed below). In all sessions, except for SUREF sessions #294 and #380,
the value of $\delta$ is smaller than $2\sigma$ TouboulCQG2022-9 . With
$\delta$ and the duration of the session MissionScenarioCQG2022 , Eq. (4)
gives $S_{\Delta a_{th}}^{{}^{1/2}}(\nu_{{}_{EP}})$; this value, using (2) at
$\nu=\nu_{{}_{EP}}$, yields $f_{{}_{TkQ}}$, from which –given the equilibrium
temperature TouboulCQG2022-9 , the masses and the measured stiffnesses– the
quality factor $Q$ is inferred. An extensive series of ground measurements is
available for comparison and validation WT-RSI2000b . Since $k_{1}\,,k_{2}$
may vary because of calibrations we compute also $k/Q$ assuming the same
unknown $k$ for both TM’s. These quantities are listed in Table 1 for SUEP and
in Table 2 for SUREF, along with the percentage $G$ of artificial data that
have been introduced in each session after the elimination of glitches.

Table 1: SUEP sensor. Session number in V2/V3 spin mode; session $\delta$ and percentage of glitches $G$ from Tables 6,7 and 4,5 of TouboulCQG2022-9 ; calculated values of $S_{\Delta a_{th}}^{{}^{1/2}}(\nu_{{}_{EP}})$, $Q$, $k/Q$. $\downarrow$ indicates a jump between sequential sessions; * an $M$ vs $A$ discrepancy. | $\delta$ | $S_{\Delta a_{th}}^{{}^{1/2}}(\nu_{{}_{EP}})$ | $Q$ | $k/Q$ |  G
---|---|---|---|---|---
| $10^{-15}$ | $10^{-11}\rm ms^{-2}/\sqrt{Hz}$ |  | $10^{-3}\rm{N/m}$ |  %
Session |  $M$ |  $A$ |  $M$ |  $A$ |  $M$ |  $A$ |  $M$ |  $A$ |
$\downarrow$210 (V3) |  ${-30.1}$ |  ${-29.2}$ |  13 |  13 |  0.75 |  0.79 |  1.2 |  1.1 |  18
212 (V3) |  ${10.4}$ |  ${9.5}$ |  4.9 |  4.5 |  5.2 |  6.3 |  0.17 |  0.15 |  17
*218 (V3) |  $3.6$ |  $6.7$ |  2.4 |  4.5 |  22 |  6.3 |  0.042 |  0.14 |  15
234 (V3) |  $5.6$ |  $5.9$ |  3.3 |  3.4 |  12 |  11 |  0.078 |  0.086 |  18
$\downarrow$236 (V3) |  $2.7$ |  $2.6$ |  1.8 |  1.7 |  39 |  42 |  0.024 |  0.022 |  21
238 (V3) |  $6.1$ |  $5.8$ |  4.1 |  3.9 |  7.6 |  8.4 |  0.12 |  0.11 |  24
252 (V3) |  $-14.7$ |  $-14.9$ |  9.2 |  9.3 |  1.5 |  1.4 |  0.62 |  0.64 |  26
$\downarrow$254 (V3) |  $-14.2$ |  $-14.1$ |  9.5 |  9.4 |  1.4 |  1.4 |  0.65 |  0.64 |  27
256 (V3) |  $-4.7$ |  $-5.3$ |  3.1 |  3.5 |  13 |  10 |  0.071 |  0.091 |  28
326-1 (V3) |  $-10.1$ |  $-16.3$ |  4.8 |  7.7 |  5.5 |  2.1 |  0.16 |  0.43 |  12
326-2 (V3) |  $-11.1$ |  $-10.4$ |  3.9 |  3.7 |  8.1 |  9.2 |  0.11 |  0.099 |  7
358 (V3) |  $15.4$ |  $15.8$ |  9.0 |  9.2 |  1.6 |  1.5 |  0.59 |  0.62 |  14
402 (V2) |  $27.3$ |  $28.4$ |  7.1 |  7.3 |  8.5 |  7.9 |  0.11 |  0.12 |  35
*404 (V3) |  $6.3$ |  $4.7$ |  4.2 |  3.1 |  7.1 |  13 |  0.13 |  0.071 |  23
406 (V3) |  $6.0$ |  $5.9$ |  1.6 |  1.6 |  47 |  49 |  0.019 |  0.019 |  23
*438 (V2) |  $-12.5$ |  $-23.4$ |  4.3 |  8.1 |  23 |  6.5 |  0.040 |  0.14 |  21
*442 (V2) |  $-10.7$ |  $-1.5$ |  4.1 |  0.58 |  25 |  1273 |  0.037 |  0.00072 |  21
*748 (V2) |  $-17.5$ |  $-23.4$ |  5.2 |  7.0 |  15 |  8.7 |  0.059 |  0.11 |  25
750 (V3) |  $66.6$ |  $66.9$ |  11 |  12 |  0.97 |  0.95 |  0.94 |  0.97 |  19

Tables 1 and 2 show that, contrary to expectations, large jumps in $S_{\Delta
a_{th}}^{{}^{1/2}}(\nu_{{}_{EP}})$, $Q$ and $k/Q$ occur between sequential
sessions (sessions are numbered by even numbers) in three cases for SUEP and
in three cases for SUREF.

Values of $Q<1$ and close to critical damping observed in two SUEP sessions
are unrealistic.

Two methods of analysis, $M$-$ECM$ ($M$) and $ADAM$ ($A$), have been employed
to estimate the experiment parameters, including $\delta$ and $\sigma$. Both
methods use artificially reconstructed data to account for the missing data
resulting from the elimination of glitches.

Method $M$ estimates in the time domain the missing data and then computes the
least-squares estimate of the regression parameters, maximizing the likelihood
conditional on the observed data. An estimation of the PSD is also produced in
the process. Method $A$ performs the parameter estimation in the frequency
domain. As such it requires an uninterrupted, regularly spaced time series,
which is obtained by filling the gaps left by the removal of glitches with the
artificial data estimated by $M$.

The two methods are stated to be equivalent, leading to the same results, but
in a few cases the $\delta$’s calculated by $M$ and $A$ differ considerably
(by a factor $\simeq 2$ in sessions #218, #438 and by a factor 7 in #442),
implying that the same data stream leads to a different PSD according to one
or the other method. Replicating the analysis by two (partially) independent
methods is meant to enhance the confidence in the results, but the confidence
is undermined if the methods give different results and the difference is not
explained.

SUEP session #442 is an instructive extreme case. As reported in
TouboulCQG2022-9 , Table 7, it has: $(\delta\pm\sigma)_{M}=(-10.7\pm
19.0)\times 10^{-15}$, $(\delta\pm\sigma)_{A}=(-1.5\pm 19.1)\times 10^{-15}$.
That is, the noise is essentially the same but the values of $\delta$ differ
by a factor 7; $\delta$ representing a “nominal” violation with SNR=1. How is
it possible that, for the same measurement session, in the same experimental
conditions, the same data with the same level of noise can lead to almost one
order of magnitude difference in the evaluation of $\delta$?

The difference is particularly evident when the results are interpreted in
terms of the physics of the dominant thermal noise from internal damping
(widely different $Q$’s, up to an embarrassing level in session #442).

SUREF session #778-1 is the only one, among all 32 measurement sessions ($19$
SUEP $+$ $13$ SUREF), not affected by glitches, hence it has no artificial
data. In this case TouboulCQG2022-9 , Table 6, reports
$(\delta\pm\sigma)_{M}=(-8.1\pm 4.5)\times 10^{-15}$,
$(\delta\pm\sigma)_{A}=(-8.1\pm 4.7)\times 10^{-15}$. That is, the results are
exactly the same for $\delta$ and only slightly different for $\sigma$, as one
expects if the same data set is analyzed with two equally valid methods.

Sessions #442 and #778-1 show that the $M$ and $A$ methods are in agreement
when applied to a time series of real measured data (no missing points and no
reconstructed data), but give different results (in particular different
$\delta$’s) in the presence of artificial reconstructed data. Since the
artificial data in $A$ are those estimated by $M$, it is the way the two
methods manipulate them that makes the difference. In any case, no matter how
artificial data are generated and/or manipulated, they should not introduce
any physical information, i.e. their effect on the $\delta$ estimated in each
session ought to be null; otherwise, the more artificial data are used in a
session, the more “artificial” will be the value of $\delta$ obtained for that
session.

It is not clear whether the artificial data are included in the calculation of
the the $\sigma$’s. If they are, all $\sigma$’s are underestimated (the more
dummy data, the lower the noise), and this has an impact on the reported
global result (weighted by $\sigma^{-2}$) whose noise would be underestimated
too.

Table 2: Same as Table 1 for SUREF sensor. A $\bullet$ has been added to indicate sessions with $Q$ values much larger than the largest value of $118$ measured in ground tests WT-RSI2000b . | $\delta$ | $S_{\Delta a_{th}}^{{}^{1/2}}(\nu_{{}_{EP}})$ | $Q$ | $k/Q$ |  G
---|---|---|---|---|---
| $10^{-15}$ | $10^{-11}\rm ms^{-2}/\sqrt{Hz}$ |  | $10^{-3}\rm{N/m}$ |  %
Session |  $M$ |  $A$ |  $M$ |  $A$ |  $M$ |  $A$ |  $M$ |  $A$ |
$\downarrow$*$\bullet$120-1 (V2) |  $-3.1$ |  $-4.2$ |  0.89 |  1.2 |  268 |  146 |  0.042 |  0.077 |  4
120-2 (V2) |  $-16.8$ |  $-15.1$ |  8.2 |  7.4 |  3.1 |  3.9 |  0.36 |  0.29 |  15
$\downarrow$174 (V2) |  $7.8$ |  $8.0$ |  4.4 |  4.5 |  11 |  10 |  0.10 |  0.11 |  25
$\bullet$176 (V2) |  $1.7$ |  $1.8$ |  0.82 |  0.86 |  317 |  283 |  0.0036 |  0.0040 |  40
294 (V3) |  $-8.0$ |  $-7.7$ |  4.2 |  4.1 |  3.5 |  3.7 |  0.33 |  0.30 |  17
376-1 (V2) |  $-3.4$ |  $-4.1$ |  1.2 |  1.5 |  135 |  93 |  0.0083 |  0.012 |  14
376-2 (V2) |  $-5.7$ |  $-6.4$ |  1.8 |  2.1 |  62 |  49 |  0.018 |  0.023 |  11
380-1 (V3) |  $7.6$ |  $7.4$ |  3.1 |  3.1 |  6.4 |  6.7 |  0.18 |  0.17 |  7
380-2 (V3) |  $9.3$ |  $8.9$ |  3.3 |  3.2 |  5.7 |  6.3 |  0.20 |  0.18 |  5
452 (V2) |  $-4.3$ |  $-4.8$ |  1.5 |  1.7 |  101 |  81 |  0.011 |  0.014 |  20
454 (V2) |  $-3.1$ |  $-3.7$ |  1.4 |  1.7 |  111 |  78 |  0.010 |  0.014 |  22
$\downarrow$778-1 (V2) |  $-8.1$ |  $-8.1$ |  3.0 |  3.0 |  23 |  23 |  0.049 |  0.049 |  0
*$\bullet$ 778-2 (V2) |  $-2.3$ |  $-3.2$ |  0.59 |  0.83 |  599 |  309 |  0.0019 |  0.0036 |  6

In SUEP session #218, the values of $\delta$ differ by a factor of about 2
depending on the type of analysis. This session (120 orbits) was the basis of
the early results MS-PRL2017 , and was further elaborated two years later
CQG2019 , ending up with a different value of the spectral density at the
signal frequency. Even discarding the earlier results because systematics had
not yet been reduced and/or glitches had not yet been treated, it is apparent
that even the latest analysis cannot be considered conclusive. $Q$ values
differing by a factor 3.5, for the same oscillator in the same conditions, are
inexplicable.

The reference sensor SUREF behaves quite differently from SUEP. Thermal noise
is about a factor $\sqrt{2}$ smaller than for SUEP (assuming the same $Q$),
due to a factor $2$ larger $(k_{1}/m_{1}^{2}+k_{2}/m_{2}^{2})$ term, and
residual systematic errors are comparable to the stochastic errors
TouboulCQG2022-9 . Moreover, 4 out of 9 sessions have been split because of
sudden jumps in the mean value of the differential acceleration that required
the two data segments to be treated as distinct experiments. Unlike glitches
these jumps do not occur on all accelerometers, hence they cannot be
attributed to the spacecraft but are likely to originate in the accelerometers
themselves, and last much longer (tens of seconds). In SUEP only 1 session
(#326) out of 18 has been split. In SUREF, in addition to the three cases
mentioned above of anomalous jumps in $S_{\Delta
a_{th}}^{{}^{1/2}}(\nu_{{}_{EP}})$, $Q$ and $k/Q$ in sequential sessions, we
observe three sessions with large $M$ vs $A$ discrepancies (see Table 2).
However, the values of $Q$ are typically larger than in SUEP and in three
cases much larger than the largest value of $118$ reported in ground
measurements WT-RSI2000b .

Early results MS-PRL2017  were based on session #176 (62 orbits). The spectral
density $S_{\Delta a_{th}}^{{}^{1/2}}(\nu_{{}_{EP}})$ repored in MS-PRL2017
was confirmed in CQG2019  and attributed to $1/\sqrt{\nu}$ thermal noise,
yielding $\delta=3.75\times 10^{-15}$ and $Q=65$, consistent with ground
measurements. Instead, the final values of $\delta_{{M}}$ and $\delta_{{A}}$
are much smaller (Table 2), and the corresponding $Q$’s of 317 and 283 are far
too high to be realistic on the basis of ground tests. Note that session #176
has $40\%$ of reconstructed data.

An explanation of the anomalies observed in sequential sessions may be
conjectured as follows.

Accelerations are inferred (after calibration) from displacements measured
with very high precision relative to a position assumed as a zero force point
that the TM of each independent accelerometer is forced to maintain at all
times. This capacitance zero is affected by inevitable uncertainties; it may
change after calibration, and after initialization of a new session.

Within this design an elastic force much larger than expected was found to
dominate the motion of each levitated TM. The zero of the elastic force and
the capacitance zero that the TM is forced to, are unlikely to coincide, but
the problem has not been investigated. With a large stiffness, and a dominant
elastic force, the issue of an unstable zero is especially relevant because
the elastic force is linear with the displacement from its physical zero to
the capacitance zero (as in the case of the tidal force).

Aside from the gold wire, an unstable zero is an issue of its own in measuring
the effects of extremely small forces. In WEP tests it is related to the
problem of initial conditions (or release) errors that affect all experiments
which are not intrinsically null experiments, including those with laser
tracked satellites, celestial bodies and cold atoms Blaser2001 ; Comandi2003 ;
Nobili2008 ; Nobili2016 . Instead, in torsion-balance tests and the proposed
GG space experiment the physical observable is a true null PerBraginskyPLA2018
.

##  IV Glitches, gaps and reconstructed data

The experiment was plagued by anomalous short-duration ($<5\rm s$)
acceleration spikes originating in the spacecraft and occurring simultaneously
in the four test masses.

Since 2001 five space missions carrying ONERA’s ultra-sensitive accelerometers
have been launched –CHAMP, GRACE, GOCE, GRACE Follow-On (GFO) and MICROSCOPE–
with a total of seven satellites, and all but one have experienced such spikes
(named “clanks” in CHAMP and GOCE, “twangs” in GRACE, “glitches” in
MICROSCOPE). At the time of the design of GOCE, reports of such effects in
CHAMP caused alarm, and countermeasures were adopted at design and test level,
which were successful, and no spikes were seen. The succeeding missions
discovered the problem anew.

Although as far as we know a comprehensive analysis across all the missions
has not been carried out, there seems to be a consensus that the spikes are
triggered by energy input from the Earth causing micro-vibration, e.g. release
of stress energy in the spacecraft materials such as the multi-layer
insulation (MLI) GRACEglitchesTwangs1 ; GRACEglitchesTwangs2 . Alternative
hypotheses point to small electric discharges in the spacecraft surfaces, or
changes in the solar array currents. The GOCE countermeasures program was
based on the mechanical hypothesis, and in GFO a stiffer design of the nadir
insulating foil has been adopted, thus, too, assuming a mechanical origin of
the spikes.

Most important for the MICROSCOPE test of the WEP, the spacecraft is spinning,
hence the occurrence of glitches is not random but it correlates with the
synodic frequency of the spacecraft relative to the Earth (i.e. the frequency
of WEP violation), and its harmonics.

In Ref. MicroscopeGlitchesCQG2022-008  it is concluded that glitches must be
removed from the data because it is impossible to accurately model their
effects. In the time series of every session glitches are identified as all
outliers above $4.5\,\sigma$ from the moving average, some time is allowed
before and after each outlier to account for transient effects, and all data
points identified in this way are removed, up to 35% in SUEP and up to 40% in
SUREF. By comparison, in the Eöt-Wash experiment the amount of data removed
due to sporadic spikes in the ion pump current, or due to abrupt changes of
the turntable axis from the local vertical, was 7% of the total Adel2008 .

For MICROSCOPE, unlike Eöt-Wash, the choice has been made to reconstruct the
missing data.

An interesting comparison is reported in MicroscopeGlitchesCQG2022-008
between the FFT of the differential acceleration as obtained from the original
measured data, and the FFT after glitches were removed and the missing data
were reconstructed. This was done for SUREF session #380 lasting 120 orbits
(of which two segments of 46 and 34 orbits each have been used in the final
analysis) Fig. 1, copied from Fig. 10 in MicroscopeGlitchesCQG2022-008 , shows
the two plots, measured data in black, reconstructed data in red.

Figure 1: Same as Fig. 10 in MicroscopeGlitchesCQG2022-008 , where the caption
reads: “Spectra of SUREF’s $x$ axis differential acceleration for session 380,
before (black) and after (red) glitches masking and data reconstruction.” ($x$
is the sensitive axis of the sensor)

This session is in V3 mode, hence the synodic frequency relative to the Earth
is $\nu_{{}_{EP_{V3}}}=3.1\times 10^{-3}\,\rm{Hz}$, at which no WEP violation
is expected. At this frequency the black curve in Fig. 1 shows a differential
acceleration line of $6$ to $7\times 10^{-14}\,\rm{m\,s^{-2}}$. Lines at
$2\nu_{{}_{EP_{V3}}}$, $3\nu_{{}_{EP_{V3}}}$, $4\nu_{{}_{EP_{V3}}}$ and
$5\nu_{{}_{EP_{V3}}}$ are visible too, indicating that glitches produce
effects not only at the signal frequency but at its higher harmonics as well.

The largest effect occurs at $2\nu_{{}_{EP_{V3}}}$ and is dominated by the
main component of the Earth tide; it must appear both in the original and in
the reconstructed data, hence we guess that the black line is hidden behind
the red line. The tidal effect at twice the signal frequency, being
deterministic, is modelled to derive the offset between the TM’s, which is
then used to quantify and subtract the tidal component at the same frequency
and phase as the signal. Therefore the glitch effect at twice the signal
frequency must be sufficiently smaller than the tidal effect at the same
frequency. If the physical origin of the glitches is related to thermal energy
from the Earth, one expects a non-negligible glitch component with half the
synodic period, as the spacecraft goes from facing deep space to facing the
warm planet.

After glitches have been removed and gaps have been filled with reconstructed
data ($46\%$ of the total according to MicroscopeGlitchesCQG2022-008 ), the
red curve in Fig. 1 shows a reduction of the lines previously ascribed to
glitches. At twice the signal frequency the reduction cannot be quantified
because of the dominant tidal effect.

More importantly, the pattern of lines related to glitches shows that the FFT
based on the reconstructed data retains memory of the removed glitches. The
reconstructed data, like the gaps, must necessarily follow the glitches, which
have specific frequencies, hence the imprinting of glitches in the
reconstructed data. Such a spurious effect at the signal frequency would mimic
a violation signal or cancel a real one.

To demonstrate the correctness of the procedure, a fake violation signal was
injected in the data before preprocessing, and two reconstructed data sets
were built from the original set (one without and one with the fake signal)
from which two values of $\delta$ are recovered: if their difference yields
the fake signal this is taken as evidence that the reconstructed data provide
a reliable representation of the real measured data. However, the glitch
spikes are so large that their removal is not affected by the fake signal; the
resulting gaps follow the pattern of the frequencies at which glitches occur,
and so do the data generated to fill them, as shown in Fig. 1. Thus, the two
reconstructed data sets are the same except for the fake signal, which is
obviously recovered.

The use of a fake signal also shows a specific anomaly. In the only case
without glitches, SUREF session #778-1, when a fake signal was added with
$\delta_{fake}=3.4\times 10^{-14}$, it was recovered with the largest error of
all other SUREF and SUEP sessions to which the same fake signal was added
(TouboulCQG2022-9 , Table 8 and Sec. 5.6), somehow suggesting that real data
may in fact be more noisy than those containing artificial data. To make
things even more confusing, when the fake signal added to the same session is
10 times weaker, it is recovered correctly.

Filling the gaps is mandatory as long as the analysis is performed in the
frequency domain. However, a different approach is possible because the
frequency and phase of a differential acceleration due to WEP violation in the
field of a source body (the Earth in this case) are known; only its amplitude
and sign are unknown.

At any given time, in the reference frame rotating with the MICROSCOPE
spacecraft, the position of the Earth and the phase of the sensitive axis are
known, and the offset of the TM’s can be derived from the tidal effect. Hence,
in the time series of differential accelerations the violation signal is a
sinusoid with the synodic period of the spacecraft relative to the Earth (at
zero spin it would coincide with the orbital period) whose maximum size
(unknown) occurs twice per synodic period when the sensitive axis of the
sensor points towards or away from Earth (sign unknown). This demodulated
phase lock-in signal can be fitted to the time series of the differential
acceleration data –only the real measured data– in order to determine its
amplitude and sign, which therefore would not be affected by whatever gaps.

In addition, lock-in detection makes it possible to separate spurious effects
that have the same frequency of the signal, but not the same phase, as in the
case of the glitch-induced effect (provided the spurious effect is not much
bigger). Although both the glitch and the signal come from the Earth, it is
very unlikely that the sensitive axes point exactly to the center of mass of
the Earth when the glitch effect is maximum. Therefore, should a residual
spurious effect remain after the elimination of glitches, it would not be
confused with the signal.

A procedure of this kind has been successfully used in the Eöt-Wash ground
tests of the WEP Adel2008 ; AdelCQGfocusissue2012 . The same procedure,
applied to the MICROSCOPE data, would establish beyond question that the
result of the experiment is not affected by the gaps and the artificial data
generated to fill them.

##  V Conclusions

A space test of the weak equivalence principle in the field of the Earth has
enormous potential for a leap forward in precision, by building on a strong
driving signal and better instrument isolation in space than in any ground
laboratory, hence easier control of systematic effects from local
disturbances. A satellite test of the WEP has been over 40 years in the
making, a number of designs were proposed, one -MICROSCOPE- has made it to
flight, and has reported two orders of magnitude improvement over the best
laboratory experiments to date MS-PRL2022 .

This work points at anomalies in the experiment and in the data analysis that
need to be addressed for the experiment result to be consolidated, and to help
designing a better space test of the future.

The MICROSCOPE experiment appears to be limited by thermal noise from damping
in the grounding wires. Using the published information CQGspecialissueMS2022
we calculate the PSD for all sessions, and express it in terms of the dominant
thermal noise and the physical quantities it depends upon. Anomalous jumps are
observed in sequential sessions (and even within the same session); the
corresponding quality factors vary widely and are sometimes unrealistically
high or low. We suggest an unstable zero may be the cause of such
fluctuations, an issue that would require careful investigation even if the
grounding wire were to be replaced by an active discharger.

The experiment was plagued by a large number of “glitches”, anomalous releases
of energy, originating in the spacecraft, producing large differential
accelerations at the signal frequency and its harmonics. We find evidence that
the way they were treated (removed and replaced with artificial reconstructed
data) leaves extant effects at the critical frequencies that could mimic a
violation signal or cancel a real one. We suggest that an analysis as
successfully employed in the Eöt-Wash laboratory tests, would be tolerant of
the missing data without affecting the sensitivity of the experiment.

Any future WEP experiment in space must prove by its very design that it will
not suffer from glitches (GOCE showed the way), and must avoid introducing
artificial data potentially compromising the analysis. Over and above that,
future experiments aiming to exploit the full potential of space must
readdress the experiment design: null experiment, stable zero, minimal thermal
noise by fast spin MicroscopeLimitations ; PerBraginskyPLA2018 .

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