Current address: ] ††thanks: Deceased. Current address: ] ††thanks: Deceased.
Current address: ] Current address: ] Current address: ] Current address: ]
††thanks: Deceased. Current address: ] Current address: ]

LISA Pathfinder Collaboration††thanks: Corresponding authors:
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
# In-depth analysis of LISA Pathfinder performance results:
time evolution, noise projection, physical models, and implications for LISA

M Armano  [  H Audley  J Baird  P Binetruy  M Born  D Bortoluzzi  E Castelli
[  A Cavalleri  A Cesarini  V Chiavegato  A M Cruise  D Dal Bosco  K Danzmann
M De Deus Silva  I Diepholz  G Dixon  R Dolesi  L Ferraioli  V Ferroni  E D
Fitzsimons  M Freschi  L Gesa  D Giardini  F Gibert  [  R Giusteri  C Grimani
J Grzymisch  I Harrison  M S Hartig  G Heinzel  M Hewitson  [  D Hollington  D
Hoyland  M Hueller  H Inchauspé  [  O Jennrich  P Jetzer  B Johlander  N
Karnesis  B Kaune  N Korsakova  C J Killow  [  J A Lobo  J P López-Zaragoza  R
Maarschalkerweerd  D Mance  V Martín  L Martin-Polo  F Martin-Porqueras  J
Martino  P W McNamara  J Mendes  L Mendes  N Meshksar  M Nofrarias  S
Paczkowski  M Perreur-Lloyd  A Petiteau  E Plagnol  J Ramos-Castro  J Reiche
F Rivas  [  D I Robertson  G Russano  [  L Sala  J Slutsky  C F Sopuerta  T
Sumner  D Texier  J I Thorpe  D Vetrugno  S Vitale  G Wanner  H Ward  P Wass
W J Weber  L Wissel  A Wittchen  C Zanoni  P Zweifel

(May 8, 2024)

###### Abstract

We present an in-depth analysis of the LISA Pathfinder differential
acceleration performance over the entire course of the mission science
operations, spanning about 500 days. We show that the frequency-independent
Brownian noise level, detected in the acceleration amplitude spectral density
(ASD), for frequencies $f\gtrsim$1\text{\,}\mathrm{m}\mathrm{H}\mathrm{z}$$,
in all measurement runs, has been evolving consistently with the decaying
pressure due to the outgassing of a single gaseous species, with an activation
temperature of $(7.0\pm 0.2)\,\text{kK}$. In excess to the Brownian noise, the
acceleration ASD always shows a sub-mHz tail which is reasonably well fit,
between $f=$36\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$$ and
$1\text{\,}\mathrm{mHz}$, to $\widetilde{S}_{\Delta
g}^{1/2}(1\,\text{mHz}/f)$. A Bayesian estimate of $\widetilde{S}_{\Delta
g}^{1/2}$ on a partition of the entire set of measurements in 27 data
stretches, each 2.75 d long, gives $\widetilde{S}_{\Delta g}^{1/2}=(1.1\pm
0.3)\,$\mathrm{f}\mathrm{m}\,\mathrm{s}^{-2}\mathrm{/}\sqrt{\text{Hz}}$$, with
no particular time pattern over the course of the mission. The width of such a
posterior estimate contains, in excess of the purely statistical uncertainty,
a true physical fluctuation of $\widetilde{S}_{\Delta g}^{1/2}$ from run to
run, of about
$0.2\text{\,}\mathrm{f}\mathrm{m}\,\mathrm{s}^{-2}\mathrm{/}\sqrt{\text{Hz}}$,
with no confidently identified correlation with specific operating conditions.
Below $f=$36\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$$, at the lowest
considered frequency of $f=$18\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$$, the
ASD significantly deviates from the $1/f$ behavior, because of temperature
fluctuations that appear to modulate a quasi-static pressure gradient,
sustained by the asymmetries of the outgassing pattern. We also present the
results of a projection of acceleration noise on the potential sources for
which we had either a direct correlation measurement, or a quantitative
estimate from dedicated experiments. We show that these sources account for
about 40% of the noise power in the frequency range where the ASD shows the
$1/f$ behavior. We discuss the possible sources of the unaccounted-for
fraction, present a series of analyses that rule many of them out, and
identify the possible measures that may be taken to keep the remaining ones
under control in LISA. Finally, we illustrate in the appendix our data
processing methods, some of which may be useful beyond the current work.

###### pacs:

04.80.Nn, 07.05.Kf, 95.55.-n

##  I Introduction

The European Space Agency (ESA) launched and operated the LISA Pathfinder
(LPF) mission [1, 2] between December 2015 and July 2017. The scientific goal
of LPF was to demonstrate that parasitic forces on a test mass (TM), to be
used as a geodesic reference in the future LISA gravitational wave
observatory, may be suppressed below the required noise level [3].

To that aim, the mission carried a miniature version of one of the LISA
interferometric arms, that is, two kilogram-size free-orbiting TMs, separated
by a few tens of centimeters, and an interferometric readout measuring their
relative acceleration along the line joining their respective centers of mass
[4].

The main results of the mission have been reported in [1, 2], showing that the
mission had surpassed its goals. In the interest of brevity, those papers
focused on just two experimental acceleration measurement runs, and we
presented a rather synthetic discussion on the possible physical origin of the
observed residual acceleration noise.

Throughout the 16 months of the mission, however, we performed many more
acceleration measurements that allowed us to observe the time evolution of the
performance, and its sensitivity to the mission operating conditions.

In addition, during all those acceleration noise measurements, we have been
measuring a set of other physical parameters, like magnetic fields,
temperatures, parasitic torques acting on the TMs, and others. Those
measurements have allowed us to set limits on the role of some of the possible
physical sources of the observed acceleration noise.

In this paper, we present the evolution of the acceleration performance of the
mission, and present a quantitative analysis of the correlation of
acceleration data with those additional physical parameters. Based on that, we
present a more in-depth analysis of the possible physical origin of the
observed acceleration noise, and discuss the implication for the performance
of the LISA observatory.

The paper is organized as follows:
In Sec. II we summarise the key features of the experiment, the operating
conditions and the measurement runs.
In Sec. III we describe, for all runs, the measured acceleration power spectra
and their key features.
Sec. IV focuses on Brownian noise and its time evolution.
Sec. V describes the evolution of the noise in the excess of the Brownian
contribution, including some long-term drift of the acceleration. This section
concludes the part of the paper dedicated to experimental findings. Given the
extension of the observations and of the performed analyses, we have divided
the discussion part into major sections.
In Sec. VI we discuss the implication of the findings on the Brownian noise
and very low-frequency behavior for the gas environment of the test masses.
In Sec. VII and Sec. VIII we discuss several possible noise contributions for
which we have quantitative estimates.
In Sec. IX we discuss the possible sources for the part of the noise that we
could not account for in the previous sections, and identify possible measures
to keep it under control in LISA.
In Sec. X we finally give some short concluding remarks.
To make the main text more readable, we put significant information in the
appendix, which is somewhat extended. In particular, App. B contains a
detailed description of our spectral estimation methods, some aspects of which
we could not trace in the standard literature on data processing.

##  II Summary description of the experiment

###  II.1 The LISA Technology Package

The instrument flown on LPF, the LISA Technology Package (LTP), has been
described in detail in [4]. Here we summarize its essential features.

FIG. 1: Rendering of the LISA Technology Package. The rendering shows the two
test masses hosted inside their respective electrode housings (some of the
electrodes are not represented), and the vacuum chambers enclosing both test
masses and electrode housings. The picture also shows the high stability
optical bench hosting all interferometric readouts, and many other features of
the instrument, as the launch lock, UV-light based test mass neutralizer, etc.
that are not relevant here. [Credits: ESA/ATG medialab]

The LTP, depicted in Fig. 1, carried two cubic Au-Pt TMs each with a mass of
$M=$1.928\text{\,}\mathrm{k}\mathrm{g}$$ and size
$l=$46\text{\,}\mathrm{m}\mathrm{m}$$. During operations, these TMs had no
mechanical contact with their surroundings, and were nominally _free falling_
, each one at the center of a housing leaving 3 to
$4\text{\,}\mathrm{m}\mathrm{m}$ clearance gaps to the faces of the TM. Each
of these _electrode housings_ (EH) carried a series of electrodes facing all
faces of the respective TM. These electrodes were used for two purposes.
First, they were part of a capacitive sensor of the motion of the TM, relative
to its housing, for all degrees of freedom (DOF). Second, they were used to
apply electrostatic forces and torques to the TM, whenever needed.

Each EH, with its respective TM, was hosted inside a vacuum chamber, called
vacuum enclosure (VE), sealed by a dedicated vacuum valve. The valve connected
the VE to a second outer volume that was connected by a vent duct to the
outside of the spacecraft.

The VE was needed both to allow for vacuum preparation on ground and to shield
the TM from the spacecraft internal outgassing that had been predicted to be
too large to achieve the desired vacuum level around a non-shielded TM.
Further details about the vacuum preparation and handling are given a bit
later.

In what follows we call the Gravitational Reference Sensor (GRS) the system of
the TM, its EH and VE, and all related accessories.

The main sensor for the relative motion of the TMs was a heterodyne laser
interferometric system, called the Optical Metrology System (OMS) [5]. For the
purpose of this paper, it is important to recall that the interferometer
measured six different DOF: the relative displacement $\Delta x(t)$ between
the TMs along the sensitive $x$-axis, joining their respective centers of
mass; the relative displacement $x_{1}(t)$, along the $x$-axis, of one of the
TM, called TM1, relative to the interferometer optical bench and, as a
consequence, also relative to the spacecraft; the two angles of rotation
$\eta(t)$ and $\phi(t)$ for both TMs, around the $y$-axis and $z$-axis
respectively. These six TM DOF, and also the remaining six, were also measured
at all times by the capacitive sensors [6]. However, the interferometric
readout was approximately three orders of magnitude more sensitive than the
capacitive one for all DOF for which they were both available. All the data
series analyzed in this paper have a sampling rate of
$10\text{\,}\mathrm{Hz}$.

Besides the measurement of the TM motion, other physical quantities have been
measured, by dedicated instruments, throughout the mission. In particular, we
measured: the magnetic field vector at various locations, via a dedicated set
of magnetometers [7]; the temperature at various critical locations, via a
dedicated set of thermistors [8]; the cosmic ray flux with a radiation monitor
[9, 10].

Before closing this section, we want to specify some details of the vacuum
handling that are useful for some of the discussions in the following
sections.

Once the VE was evacuated on ground, the vacuum valve was closed and the VE
remained sealed for a few months, until the valve was opened again once in
orbit. Valve opening took place, within an hour for both VE, on February 3,
2016, i.e. $62.4\text{\,}\mathrm{d}$ after the launch. In the rest of the
paper, we call $t_{v}$ this time of valve opening, while we call $t_{0}$ the
time of the launch, December 3, 2015 04:04:00 UTC. The outer volume, with its
vent duct, was not evacuated on ground, and, having no seal, was exposed to
space immediately after $t_{0}$.

Besides the TM and EH, the VE contained various metal structural elements, two
piezoelectric motors, used to center the TM within the EH and release it into
free fall, and various thermistors and heaters. The VE also contained the
$\sim 40$ cables needed for all these appliances, which crossed the VE wall
through a set of vacuum feed-throughs. Finally, also included in the VE was a
$\sim$2\text{\,}\mathrm{k}\mathrm{g}$$ Tungsten balance mass used to suppress
the gravitational field at TM location [11]. The outer chamber contained a
high-output paraffin motor [12] used to activate the vacuum valve and the TM
launch lock, with all the necessary cables.

###  II.2  Dynamical controls and data series formation

LPF was a controlled dynamical system consisting of the spacecraft and the two
TMs. More specifically, the spacecraft was forced to follow one of the TMs
(TM1) along $x$ via an active control loop, using the spacecraft cold gas
micro-thrusters as actuators [13], known as drag-free control.

Each TM rotation along $\phi$ and $\eta$ was kept fixed relative to the
spacecraft by an active loop using electrostatic torques. These torques were
applied via the above-mentioned electrodes.

No electrostatic actuation force was applied along $x$ on TM1, while a control
loop (electrostatic suspension) kept the distance between the two TMs
nominally fixed, by applying a suitable electrostatic force along $x$ on the
other TM (TM2). All other degrees of freedom were also controlled, but the
details are not relevant here.

As the distance between the TMs was actively controlled, the external, out-of-
the-loop, differential disturbance force per unit mass acting on the TMs,
$\Delta g_{\text{ext}}$, could not be identified with the in-loop relative
acceleration measurement $\Delta\ddot{x}$ [2]. On the contrary,
$\Delta\ddot{x}$, which was estimated numerically [14], had to be corrected
for the known applied feedback forces per unit mass $g_{c}(t)$ [2].

In addition, acceleration data series were also corrected for the following
effects.

  * •

Measured inertial forces per unit mass due to spacecraft rotation $g_{i}(t)$,
which include the centrifugal and the Euler force [1]. These effects will not
be relevant for LISA [15].

  * •

The forces per unit mass generated by the motion of the TMs through static
force gradients in the spacecraft, as LISA data can also be corrected for
those. Such force acting on TM$i$ is well approximated by the linear model
$-\omega_{i}^{2}x_{i}$, as described in [1].

  * •

The interferometer spurious pick up $g_{\text{CT}}(t)$ of spacecraft motion
along DOF different from $x$, due to cross-talk [16, 17]. This also includes
the pick-up of the common mode motion of the TMs, described by a term
$\delta_{x_{1}}\ddot{x}_{1}$. Some of these effects will also be present in
LISA, and will be analogously corrected [15].

Thus the out-of-the-loop, differential force per unit mass data series used in
the following analyses, can be written as:

$\begin{split}\Delta g(t)=&\Delta\ddot{x}_{\text{OMS}}(t)+\omega_{2}^{2}\Delta
x_{\text{OMS}}(t)+(\omega_{2}^{2}-\omega_{1}^{2})x_{1,\text{OMS}}(t)\\\
&-g_{c}(t)-g_{i}(t)-g_{\text{CT}}(t).\end{split}$ (1)

Note that, in Eq. (1), we have attached the suffix OMS to all coordinates, to
indicate that these have been measured by the relevant interferometers, and
not by the capacitive sensors. For these we will use the GRS suffix. Note also
that $\omega_{1}^{2}$ and $\omega_{2}^{2}$ above, as well as $\delta_{x_{1}}$,
have been measured in dedicated calibration experiments [18]. In particular
$\omega_{2}^{2}\approx$-4.5\text{\times}{10}^{-7}\text{\,}\mathrm{s}^{-2}$$ is
negative, while the differential stiffness $(\omega_{2}^{2}-\omega_{1}^{2})$
is roughly 20 times smaller and thus neglected in this discussion here.

The feedback force $g_{c}(t)$ in Eq. (1), is the time series of the force that
the onboard computer has commanded to the electrostatic force system. It was
calibrated through an extensive series of experiments [18], consisting of the
application to the TMs of a series of forces oscillating at frequencies
$\gtrsim$1\text{\,}\mathrm{m}\mathrm{H}\mathrm{z}$$, and of the measurement of
the resulting acceleration, a quantity which is intrinsically calibrated in
terms of the laser wavelength and of the onboard clock calibration.

This calibration campaign led us to discover a systematic, non-linear, error
in the electronics, originating from an overlooked truncation of the digital
voltage commands [19], that resulted in an amplitude-dependent fluctuation of
the calibration. Therefore $g_{c}$ had to be recalculated considering such an
extra truncation. After this crucial correction, the calibration of $g_{c}$
was found to be stable at better than $<1\%$, independently of the amplitude
of the applied forces and torques, throughout the entire mission [18].

$\Delta g(t)$ in Eq. (1) is our best estimate for the external differential
force per unit mass $\Delta g_{\text{ext}}$. However the series is corrupted
by the noise $n_{\text{OMS}}(t)$ in the differential interferometer readout
$\Delta x_{\text{OMS}}$. Such disturbance enters into $\Delta g(t)$ in Eq.
(1), both through $\Delta\ddot{x}_{\text{OMS}}(t)$, and through
$\omega_{2}^{2}\Delta x_{\text{OMS}}(t)$. Thus, the residual noise in $\Delta
g$ can be evaluated as:

$\Delta g=\Delta
g_{\text{ext}}(t)+\ddot{n}_{\text{OMS}}(t)+\omega_{2}^{2}n_{\text{OMS}}(t).$
(2)

Finally, it is important to mention that occasional force transients were
observed in the data [20]. In ordinary runs these glitches occurred at an
average rate of $\simeq 1\,\text{d}^{-1}$. These glitches have been removed
from the data, as described in Ref. [20], before any noise analysis.

###  II.3 Data runs

The mission scientific operations lasted from March 1, 2016 to July 18, 2017.
During these more than 16 months, we performed many uninterrupted _noise runs_
during which the TMs and the satellite were in steady control conditions, with
no purposely applied stimulus of any nature.

We have performed many such noise runs, however we restrict the main analysis
to those with an overall duration of at least 2.5 days. Such a duration allows
an estimate of acceleration power spectral density (PSD), down to about
$\simeq$18\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$$ with reasonable accuracy
(see following sections).

We list in Table 1, for all these runs, the start and stop dates, the
duration, and the instrument operating temperatures. Temperature values are
the average of the 8 thermometers placed on the two GRS as described in [8].
We note that 11 of these runs were performed at an operating temperature
around 21-22 $\mathrm{\SIUnitSymbolCelsius}$, while run 10 and 11 were
operated at around 11 $\mathrm{\SIUnitSymbolCelsius}$.

TABLE 1: List of the considered noise runs. Run 10 is the February 2017 run published in [1]. N. | Start date | Duration | Mean time |  Temperature
---|---|---|---|---
|  | [d] | from launch [d] | [K]
1 |  2016-03-21 | 5.3 | 112 | $295.37\pm 0.04$
2 | 2016-04-04 | 9.3 | 127 | $295.30\pm 0.03$
3 | 2016-05-16 | 3.2 | 166 | $294.97\pm 0.06$
4 | 2016-06-19 | 4.8 | 202 | $294.93\pm 0.01$
5 | 2016-07-17 | 2.8 | 229 | $296.62\pm 0.05$
6 | 2016-07-24 | 5.3 | 237 | $296.50\pm 0.02$
7 | 2016-09-28 | 2.8 | 302 | $296.50\pm 0.02$
8 | 2016-11-16 | 9.9 | 354 | $296.86\pm 0.04$
9 | 2016-12-26 | 18.5 | 398 | $295.38\pm 0.04$
10 | 2017-02-14 | 13.3 | 446 | $284.72\pm 0.03$
11 | 2017-05-18 | 4.3 | 535 | $284.2\;\pm 0.1$
12 | 2017-05-29 | 6.8 | 547 | $295.78\pm 0.01$
13 | 2017-06-08 | 8.6 | 558 | $295.91\pm 0.03$

The detailed operating conditions have been slightly different for different
runs. We have listed in Table 2, in App. A, the few differences of settings
that may bear some relevance for the noise performance.

For runs 1 to 11, the total charge of both TMs was maintained within the
interval of $\pm\,3\times 10^{7}\,e$ [21], in order to keep the noise caused
by voltage fluctuations negligible [22] (see Sec. VIII.4). For those runs,
charge was measured just before and just after each run. For runs 12 and 13,
the charge was measured just once before run 12. We could however extrapolate
the missing final values from the measurement of cosmic ray flux [21, 23]. For
run 12, such extrapolated final value is still in the aforementioned interval,
while for run 13 it might have gone up to $4\times 10^{7}\,e$.

TM neutralization was performed using a non-contacting UV discharging system
[24] before the start of each run. As cosmic rays resulted in a steady
increase of the charge, for most of the runs the charge was brought to a
negative value within the $\pm\,3\times 10^{7}\,e$ interval at the start of
the run, to have it drifting through zero during the run.

Two major events must be mentioned. First ESA operated LPF, until April 6,
2017, on a Lissajous orbit near the L1 point of the Earth-Sun system. This
orbit was unstable, and mission control had to make station-keeping maneuvers
every few weeks to maintain it. On April 6, 2017 in preparation for the end of
the mission, mission control performed a final maneuver, called “de-orbiting
burn”, to place the spacecraft on a trajectory that would not risk
intercepting the Earth. The maneuver lasted a few days, and after that, and
until the end of the mission, no further station-keeping maneuver was
performed.

In addition, between April 30, 2017, and May 16, 2017, with the purpose of
further reducing noise beyond the level reached in February 2017 [1], we tried
to operate the instrument near $0\text{\,}\mathrm{\SIUnitSymbolCelsius}$, a
value well outside its nominal operating range. The instrument entered into a
rather unstable state, with a very high rate of glitches [20], and went back
to its ordinary behavior only when the temperature was raised back to a value
within its operating range.

During such $0\text{\,}\mathrm{\SIUnitSymbolCelsius}$ cooling, we were able to
perform a noise run lasting from 2017-05-03 to 2017-05-09 (for a total of
$5.8\text{\,}\mathrm{d}$), not listed in Table 1. In this run, despite the low
quality of the data, we were able to perform some noise measurements at
frequencies around 1 mHz, which we will mention later in the paper, Sec. IV.

Unfortunately, the temperature of this _cold run_ , has not been measured
directly, as the electronics of the GRS thermometers clipped at
$\simeq$7\text{\,}\mathrm{\SIUnitSymbolCelsius}$$. However we found that, upon
a proper calibration, the average of a group of thermometers just outside the
instrument bay was a good proxy of the average of the GRS thermometers, at all
temperatures above $\simeq$7\text{\,}\mathrm{\SIUnitSymbolCelsius}$$. This
proxy gives, for the cold-run,
$T\simeq$1.7\text{\,}\mathrm{\SIUnitSymbolCelsius}$\simeq$274.8\text{\,}\mathrm{K}$$
with an overall uncertainty not larger than $0.5\text{\,}\mathrm{K}$.

##  III Acceleration PSD

During all noise runs we have estimated the PSD of $\Delta g(t)$, $S_{\Delta
g}(f)$ (or, equivalently, its square root, the amplitude spectral density
(ASD)), as a function of frequency $f$, by using the periodogram method
explained in App. B.2. The method gives a Bayesian estimate for the posterior
distribution of $S_{\Delta g}(f_{i})$, over a given set of frequencies
${f_{i}}$. The frequencies within the set have been chosen such that the PSD
estimates at different frequencies have minimal statistical correlations (see
App. C and [1]). In addition, for practical reasons, we have adjusted the
selection such that the fourth frequency is 0.1 mHz, the lower bound of the
official LISA band.

The blue points in Fig. 2 illustrate the result of the procedure for run 10 of
Table 1111Note that, for the rest of the paper, with the ‘$n\sigma$ interval’
we indicate the equal-tailed, $p$-credible interval of any probability
distribution, with $p$ the probability for a normal random variable to fall in
the interval $\pm n\sigma$, with $\sigma$ its standard deviation. In
particular, for $n=1$, $p=0.683$, and for $n=2$, $p=0.955$. In addition,
unless otherwise specified, dots in plots represent the median of the
distribution, and error bars the $1\sigma$ confidence interval., which is the
February 2017 run that we published in [1].

FIG. 2: Blue points: ASD of $\Delta g(t)$, $S_{\Delta g}^{1/2}$ for run 10 of
Table 1 (February 2017) as a function of frequency $f$. The vertical error
bars delimit the 1$\sigma$ interval (see footnote on page 1 for the
definition). The horizontal bars indicate the effective width of the spectral
window contributing to each of the points. Red points: estimated excess over
Brownian noise. Green thin band: uncertainty band for the estimate of the
Brownian noise.

The figure shows that, as already noted in [1], the ASD, and the PSD, have
three different branches:

  1. 1.

a low frequency branch with an approximate $f^{-1}$ behavior ($f^{-2}$ for the
PSD);

  2. 2.

an approximately frequency-independent branch above about 1 mHz;

  3. 3.

a rising branch above about 10 mHz.

As explained in [2], the rising branch is due to the interferometer readout
noise $n_{\text{OMS}}$ and will be not discussed any further.

In [2] we have attributed the frequency independent branch, with PSD value
$S_{\text{Brown}}$, to Brownian noise due to gas damping. To separate it from
the other two branches, we use, for each run, the following procedure.

FIG. 3: Blue dots: PSD of Brownian noise $S_{\text{Brown}}$ as a function of
$(t-t_{v})$, the time since venting of the vacuum enclosure. Error bars
correspond to 1$\sigma$ intervals. For convenience, call-outs repeat the mean
temperatures $T$ of the runs from Table 1. Blue dot: $S_{\text{Brown}}$ for
the cold run ($T\sim$275\text{\,}\mathrm{K}$$). Data may be converted into the
mean of the pressures $P_{\text{H}_{2}\text{O}}$ in the two GRS (right
vertical scale) by using the conversion factor in [25], and by assuming that
the residual gas consists of water molecules. Red dots: PSD of Brownian noise
of ordinary runs, scaled as
$S_{\text{Scaled}}=S_{\text{Brown}}\,e^{-T_{a}\left(\frac{1}{T}-\frac{1}{T_{0}}\right)}$,
with $T_{0}=$293\text{\,}\mathrm{K}$$ and $T_{a}=$7.0\text{\,}\mathrm{kK}$$.
Dashed black line: $S_{\text{Scaled}}(t)\propto(t-t_{v})^{-0.80}$. See text
for details. Dotted gray line: best fit assuming adsorption quasi-equilibrium.
See Sec. VI for details.

  * •

We fit the experimental PSD data $\Pi_{\Delta g,k}(f_{i})$ of run $k$ to the
PSD model

$S_{\Delta g,k}(f_{i})=S_{\Delta g_{e},k}(f_{i})+S_{\text{Brown},k},$ (3)

that is, with a frequency independent term $S_{\text{Brown},k}$, plus the
excess $S_{\Delta g_{e},k}(f_{i})$ that depends on the frequency $f_{i}$.

  * •

To avoid numerical instabilities, we limit the analysis to
$f<$3\text{\,}\mathrm{m}\mathrm{H}\mathrm{z}$$, that is, to the lowest 10
frequencies of Table 6 in pag. 6. However, to better constrain
$S_{\text{Brown},k}$ we add one further data point consisting of the minimum
value of $\Pi_{\Delta g,k}(f_{i})$ for $i>10$, $\Pi_{\Delta g,k,\text{min}}$.
We fit this data point with the simpler model $S_{\Delta
g,k,\text{min}}=S_{\text{Brown},k}$, that is, with no excess over the Brownian
noise; this is a reasonable assumption, also supported by the results of the
following analyses. This additional term also prevents the fit to put
$S_{\text{Brown},k}$ to zero.

  * •

The fit is a Monte Carlo Markov Chain (MCMC) estimate of the posterior

$\begin{split}&p\left(S_{\Delta g_{e},k}(f_{i}),S_{\text{Brown},k}|\Pi_{\Delta
g,k}(f_{i}),\Pi_{\Delta g,k,\text{min}}\right)_{i\in[1,10]}\propto\\\
&\propto\prod_{i=1}^{10}p\left(\Pi_{\Delta g,k}(f_{i})|S_{\Delta
g_{e},k}(f_{i})+S_{\text{Brown},k}\right)\times\\\ &\times p\left(\Pi_{\Delta
g,k,\text{min}}|S_{\text{Brown},k}\right)\end{split}$ (4)

Here the function $p(\Pi|S)$, the probability distribution of the experimental
PSD, is that given in App. B.2.

  * •

We scan the domain of the _logarithm_ of all parameters $S_{\Delta
g_{e},k}(f_{i})$ and $S_{\text{Brown},k}$, and do not multiply the posterior
in Eq. (4) by any explicit prior. This is equivalent to using the Jeffreys
prior on all parameters [26]. However, again to avoid numerical instabilities,
we constrain all PSD parameters, both Brownian and excess, to be larger than
$1\,\text{am}^{2}\,\text{s}^{-4}/\text{Hz}$.

The results of such procedure are again shown in Fig. 2 for run 10 of Table 1,
and for all runs in Fig. 25 and 26 in App. D.

Note that Fig. 25 and 26 show that the procedure gives consistent results for
all runs, with the Brownian uncertainty band always very close to the average
of native data in the range $$2\text{\,}\mathrm{mHz}$\leq
f\leq$5\text{\,}\mathrm{mHz}$$.

##  IV Time evolution of Brownian noise

From the procedure described in the previous section, we obtain an estimate of
$S_{\text{Brown}}$ for all runs. Results are plotted in Fig. 3, as blue dots,
as a function of $t-t_{v}$, the time since venting to space of the VE.

As discussed in [2, 1], the PSD of Brownian noise due to gas damping is
expected to be proportional to the gas pressure $P$, with a coefficient of
proportionality that depends on the geometry of the TM environment. Ref. [25]
estimates such coefficient to be
$\kappa\simeq$1.7\text{\,}\mathrm{f}\mathrm{m}^{2}\,\mathrm{s}^{-4}\,\mathrm{Hz}^{-1}\mathrm{/}\mathrm{\SIUnitSymbolMicro}\mathrm{Pa}$$,
for a single LPF TM, and for a gas consisting of water molecules. Thus, if
indeed the frequency-independent branch represents Brownian noise,
$P_{\text{H}_{2}\text{O}}=S_{\text{Brown}}/(2\kappa)$, which can be read on
the right vertical axis of Fig. 3, gives a measurement of the mean of the
pressures of the two GRS. Note that, with the effective conductance from the
interior of the EH to the outer space estimated to be
$\simeq$19\text{\,}\mathrm{L}\mathrm{/}\mathrm{s}$$ for water, a pressure of
$1\text{\,}\mathrm{\SIUnitSymbolMicro Pa}$ corresponds to an outgassing rate
of
$1.9\text{\times}{10}^{-7}\text{\,}\mathrm{m}\mathrm{bar}\,\mathrm{L}\mathrm{/}\mathrm{s}$.

The figure also shows that, remarkably, the data can be scaled to follow a
straight line, by just multiplying them by a single “activation” factor
$e^{-T_{a}\left(\frac{1}{T}-\frac{1}{T_{0}}\right)}$, with $T_{a}$ a properly
chosen activation temperature, and $T_{0}$ an arbitrarily chosen, convenient
common temperature for the scaled data.

The scaled data may be fit, with comparable qualities, to a few different
power-law models, reminiscent of the standard pump-down behavior of vacuum
systems [27]. This is discussed in more detail in Sec. VI. The figure reports
the result for the case of a VE-dominated outgassing:

$S_{\text{Brown}}(t)=a\left(\frac{t_{v}}{t-t_{v}}\right)^{\gamma}$ (5)

where $a$ and $\gamma$ are two fitting parameters. As usual for this kind of
pump-down laws, Eq. (5) holds only after the initial rapid pressure decay
phase, and should not be extrapolated to $t\to t_{v}$ [28].

Actually a simultaneous fit leaving also $T_{a}$ as a free parameter, gives
$T_{a}=(7.0\pm 0.2)\leavevmode\nobreak\ \text{kK}$, $\gamma=(0.80\pm 0.02)$,
and finally $a=(27.0\pm 0.7)\leavevmode\nobreak\
$\mathrm{f}\mathrm{m}^{2}\,\mathrm{s}^{-4}\mathrm{/}\mathrm{Hz}$$.

Given the uncertainty on the temperature and on the quality of the PSD data,
we have left the cold run datum out of re-scaling and fitting. This is further
discussed in Sec. VI. Here we only note that the cold run datum is lower than
those for the lowest temperature ordinary runs. This shows that the TM
acceleration noise in the mHz bandwidth is truly Brownian down to a level well
below
$2\text{\,}\mathrm{f}\mathrm{m}\,\mathrm{s}^{-2}\mathrm{/}\sqrt{\text{Hz}}$
per TM, and not saturating to a significant level reflecting other important
noise sources in this band.

For the sake of the discussion in Sec. VI, we also report in Fig. 3 the best
fit for one common model for vacuum evolution under the hypothesis of quasi-
equilibrium between surface water readsorbtion and outgassing. The discussion
on Brownian and vacuum environment continues in the mentioned Sec. VI.

##  V  Stability and time evolution of excess noise

In this section we discuss the stability and evolution of $S_{\Delta g_{e}}$,
the excess noise over Brownian, over the duration of the mission. The details
of the variations of $S_{\Delta g_{e}}$ can be tracked in Fig. 25 and 26 in
App. D. Here we present a few summary analyses with the main aim of
quantifying noise stationarity.

We first discuss the compatibility of the observations at each frequency with
a model considering a single stationary, run-independent, noise process.

Having identified significant variations from run to run of the lowest
frequency datum, mostly in the initial phase of the mission, we separate the
discussion on the time evolution of the excess noise for
$f>$18\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$$ from that at
$f=$18\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$$, that we treat in Sec.V.6
together with long-term drifts.

###  V.1 Fit to a common stationary Gaussian excess for all runs

In Fig. 4 we report, at each frequency $f_{i}$, the inferred posterior
distribution for $S_{\Delta g_{e}}(f_{i})$ (blue data points) assuming that
the excess is a common number to all runs, $S_{\Delta
g_{e},k}(f_{i})=S_{\Delta g_{e}}(f_{i})$. We call this the _common-noise
model_. The inference is done by using a collective posterior consisting of
the product of the posteriors in Eq. (4), for all values of $k$, having
dropped the dependence of the excess on $k$.

FIG. 4: Summary of ASD of $\Delta g$ excess noise, over all runs of Table 1
and Figs. 25 and 26, as a function of the frequency. Blue data: median and
$\pm 1\sigma$ quantiles of the inferred posterior distribution of the excess
for the entire set of runs, common-noise model (see text for details). Red
data: median and $\pm 1\sigma$ quantiles of the inferred posterior
distribution, independent-runs model (see text for details). Red data have
been slightly shifted in frequency for the sake of readability, but they all
refer to the same frequency as that of blue data. Also reported are current
LISA requirements adapted from [29].

We have done a simple Akaike information criterion comparison [30] between the
common-noise model and the model consisting of the separate Bayesian fits to
the data of the 13 different runs discussed in Sec. III. The posterior for the
latter is just the product of the posteriors used for the separated fits, and
thus depends on 143 parameters: 10 excess noise coefficients for each of the
13 runs, plus 13 Brownian noise coefficients. The common excess noise model
depends instead on only 10 excess noise coefficients, and 13 Brownian noise
ones, for a total of 23 coefficients.
The Akaike test favors the common-noise model, with a relative likelihood
ratio of $\simeq 10^{-20}$. This indicates that the hypothesis of completely
independent excess noise values across the runs substantially over-fits the
data. However, as usual with Bayesian model comparison, this does not tell if
the best of the two fits is a good fit.

We have then done a posterior predictive check [31] on the common-noise model
described in the following. We first find the set of parameter values
$\hat{\theta}_{\text{best}}=\\{S_{\Delta g_{e}}(f_{1}),S_{\Delta
g_{e}}(f_{2}),\dots,S_{\text{Brown},1},S_{\text{Brown},2},\dots\\}$ that
maximizes the posterior likelihood.

Having $\hat{\theta}_{\text{best}}$, for any set of data, either observed or
simulated from the posterior, we can then calculate, as a measure of
discrepancy at any given frequency $f_{i}$, the log-likelihood of the data at
that frequency, conditional on $\hat{\theta}_{\text{best}}$

$\Lambda_{i}=\sum_{k=1}^{13}\log\,p\left(\Pi_{\Delta
g,k}(f_{i})|\hat{\theta}_{\text{best}}\right)$ (6)

Note that in an ordinary least square fit to independent Gaussian data, the
log-likelihood would be, modulo an irrelevant additive constant, just the sum
of the square residuals, a quantity commonly used to measure the discrepancy
between the data and the model. Thus the use of $\Lambda_{i}$ may be seen as
just an extension of the method of square residuals to Wishart-distributed
data.

We have then generated simulated data, by sampling the posterior for the
parameters $\\{S_{\Delta g_{e}}(f_{2}),S_{\Delta
g_{e}}(f_{3}),\dots,S_{\text{Brown},1},S_{\text{Brown},2},\dots\\}$, and by
generating then simulated periodograms $\Pi_{\Delta g,k}(f_{i})$ from the
proper Wishart distribution. For each simulated periodogram, we have
calculated $\Lambda_{i}$, with $1\leq i\leq 9$ obtaining then an expected
posterior predictive distribution for each of these parameters.

We have restricted the analysis to frequencies $f_{i}\leq f_{9}$, as, at
higher frequencies, the model breaks down anyway due to the dominance of
Brownian noise, particularly for the earlier runs.

We have then calculated the probability $p$ that $\Lambda_{i}$ is less than
value $\Lambda_{i,0}$ calculated for the real data, that is the cumulative
distribution function for $\Lambda_{i}$ evaluated in $\Lambda_{0,i}$. For
$i\in\\{2,7,8,9\\}$ we find that $p>0.2$, while for the other frequencies we
find $p\leq 0.002$. This shows that the common-noise model, despite being more
informative than the separated fits, is not predictive of the observations.

To make it predictive, the common-noise model posterior must be widened by a
(frequency-dependent) factor. To estimate how large this factor should be, we
have repeated the simulation above, this time by generating the simulated
periodogram for run $k$ at frequency $f_{i}$, by multiplying the sample for
$S_{\Delta g_{e}}^{1/2}(f_{i})$ from the original posterior, common to all
runs, by a random variable $\gamma_{k,i}$. For each frequency $f_{i}$, all
$\gamma_{k,i}$, were extracted from the same gamma distribution with unit mean
value and standard deviation $\sigma_{\gamma_{i}}$. This model in essence
allows $S_{\Delta g_{e}}^{1/2}(f_{i})$ to fluctuate from run to run by a mean
relative amount $\sigma_{\gamma_{i}}$. This fluctuation adds up to the natural
fluctuation of the periodogram for stationary Gaussian time series, in
generating the observed data. We call this model the _independent-runs_ model.

The independent-runs model becomes predictive of the observation at all
frequencies, $p\geq 0.2$, if $\sigma_{\gamma_{i}}=0$ for
$i\in\\{2,7,8,9,10\\}$, $\sigma_{\gamma_{i}}=0.4$ for $i=1$, and
$\sigma_{\gamma_{i}}=0.2$ for $i\in\\{3,4,5,6\\}$. We report the data for the
independent-runs posterior, in Fig. 4, red points.

Note that the largest variation is for $i=1$, that is
$f=$18\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$$. This is dominated by the
large initial decrease of the ASD in the initial part of the mission that can
be noticed by inspection in Fig. 25 on page 25. We will discuss this further
in Sec. V.6.

The common-noise model is more predictive for subsets of data including a
reduced number of runs. We find for instance that for the groups including
runs 1 to 4 of Table 1, and for that including run 11 to 13, the model is
reasonably predictive at all frequencies, while for the groups 5 to 8 and 9 to
10, the model is predictive at all frequencies but one. It must be said that
other groups of runs, with no temporal continuity, may give similar results,
and that this apparent improvement might just be a consequence of the reduced
discrimination power of the posterior test if applied to a smaller number of
data.

###  V.2  $1/f$ model, and evolution of the excess noise for
$f>$18\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$$

To study if the observed variations of excess noise level are correlated with
operational conditions we have found a useful ‘figure of merit’ that
summarizes the excess noise across the entire band $f>f_{1}$. We introduce it
in the following.

We take advantage of the fact that, as said, the PSD in this frequency band
scales approximately as $1/f^{2}$. Thus the coefficient $\widetilde{S}_{\Delta
g}$ of the $1/f^{2}$ term in a fit with the model

$S_{\Delta g}(f_{i})=\widetilde{S}_{\Delta
g}\left(\frac{$1\text{\,}\mathrm{mHz}$}{f_{i}}\right)^{2}+S_{\text{Brown}},$
(7)

gives a reasonable measure of the average power, smoothing any frequency-to-
frequency variation. Consistently with the discussion so far, we will focus on
the evolution of its square root $\widetilde{S}_{\Delta g}^{1/2}$ and will
call this the ‘$1/f$’ model222We stress that in this model it is the ASD that
depends on the frequency as $1/f$. This should not be confused with the
popular model for flicker noise, for example in electronics, where it is the
PSD that depends on the frequency as $1/f$..

To make the analysis consistent, and avoid mixing any effect of the different
duration of the various runs, we have partitioned the data for all runs in
non-overlapping stretches, all of the same duration of about 2.75 d. This is
the duration of the shortest of the runs (run 5 of Table 1). We have then
fitted the PSD data for the 27 resulting _partitions_ , to the model of Eq.
(7).

The fit was again a Bayesian MCMC analysis, supported by the posterior
predictive test based on the likelihood, in close analogy with that used above
to test the common-noise model. The only difference is that we now sum the
log-likelihood on all frequencies considered. We find that all fits have
$p\geq 0.1$ except for one for which $p=0.02$. Actually for 2/3 of the 27
short runs $p>0.5$. The results for $\widetilde{S}_{\Delta g}^{1/2}$ are shown
in Fig. 5.

FIG. 5: Time evolution of the $1/f$ tail amplitude of $S_{\Delta g}^{1/2}$,
expressed as the corresponding ASD at 1 mHz $\widetilde{S}_{\Delta
g_{e}}^{1/2}$, as a function of time since launch $(t-t_{0})$, over the entire
mission duration. Blue data: values for all the 2.75 d long, non-overlapping
stretches of data into which we partition the full set of the 13 run data
series. The gray vertical bands with dashed edges represent the time span of
the original runs to which the partition belongs. The picture also shows the
date of the initial venting (red line), that of the beginning of the science
operations (blue line), the epoch of the de-orbiting burn (green band), and
the date of the end of the science operations (magenta line). In cyan the
epochs during which the temperature was held at about
$11\text{\,}\mathrm{\SIUnitSymbolCelsius}$; runs 10 and 11 of Table 1 are both
included in this span. Finally, in blue the epoch of the
$0\text{\,}\mathrm{\SIUnitSymbolCelsius}$ cooling.

Notice that the LISA requirements for the ASD of $\Delta g$ in Fig. 4, include
a $\propto 1/f$ branch, though just down to the lower corner at 0.1 mHz. This
branch would correspond to $\widetilde{S}_{\Delta
g_{e}}^{1/2}=$1.7\text{\,}\mathrm{f}\mathrm{m}\,\mathrm{s}^{-2}\mathrm{/}\sqrt{\text{Hz}}$$,
significantly above all the points in Fig. 5. This is consistent with Fig. 4.

One feature that stands out in Fig. 5 is that data fluctuate significantly
more than the uncertainty on the single datum, even for the partitions of the
same original run. Actually, the fluctuation of data from partition of run 9,
span a similar range than that of the remaining data. This large run-to-run
fluctuation appears consistent with the poor fitting to the common-noise
model.

Just to confirm that these two facts are consistent, we have simulated the
data assuming a common value for $\widetilde{S}_{\Delta g_{e}}^{1/2}$ for all
27 partitions, and Gaussian, stationary noise. This common value is extracted
from the posterior for $\widetilde{S}_{\Delta g_{e}}^{1/2}$ of a cumulative
fit to all 27 partitions, with a common $1/f$ noise. The model is found, as
expected, as non-predictive as the common-noise model for the general case.

To get a useful quantitative statistics to compare the simulated data to the
actual ones, we form the sample of the 27 maximum likelihood values for
$\widetilde{S}_{\Delta g_{e}}^{1/2}$, in essence, for the real sample, the
dots of Fig. 5, and then calculate the sample standard deviation divided by
the sample mean $\rho\equiv\sigma_{\tilde{S}_{\Delta
g_{e}}}/\langle{\widetilde{S}_{\Delta g_{e}}}\rangle$. For the sample of the
actual data, we find $\rho\simeq 0.23$.

The simulation gives instead a mean value for $\rho$ of
$\langle\rho\rangle\simeq 0.11$ and projects a probability $<10^{-3}$ of
observing a value as large as $\rho=0.23$.

Coherently with findings for the common-noise model, to give such a large
value some reasonable probability, we had to widen the posterior, allowing for
a 0.2 relative fluctuation of $\widetilde{S}_{\Delta g_{e}}^{1/2}$ from
partition to partition.

This is consistent with the observation that, joining the posterior for all 27
partitions, we get that $\widetilde{S}_{\Delta g_{e}}^{1/2}=(1.1\pm
0.3)\,$\mathrm{f}\mathrm{m}\,\mathrm{s}^{-2}\mathrm{/}\sqrt{\text{Hz}}$$,
while the width of the posterior for each run fluctuates of about $\simeq
0.1\,$\mathrm{f}\mathrm{m}\,\mathrm{s}^{-2}\mathrm{/}\sqrt{\text{Hz}}$$,
implying a true fluctuation of about $\simeq
0.25\,$\mathrm{f}\mathrm{m}\,\mathrm{s}^{-2}\mathrm{/}\sqrt{\text{Hz}}$$ from
partition to partition, beyond the statistical uncertainty.

While the data of Fig. 5 do not display any clear systematic long term trend,
or correlation with the experimental configuration parameters of Table 2, they
show a comparatively large difference, $\simeq
0.45\,$\mathrm{f}\mathrm{m}\,\mathrm{s}^{-2}\mathrm{/}\sqrt{\text{Hz}}$$,
between the mean value of the 4 partitions of run 10 (438-451 days after
launch in Fig. 5), and that of the six last partitions of runs 11 to 13
(531-561 days after launch in Fig. 5).

Actually the difference is not much less, $\simeq
0.32\,$\mathrm{f}\mathrm{m}\,\mathrm{s}^{-2}\mathrm{/}\sqrt{\text{Hz}}$$, if,
instead of just taking the mean of the partitions of run 10, one also adds the
6 partitions belonging to run 9, as run 9 and 10 could be reasonably fit
together with the common-noise model, as could also runs 11 to 13. The
observation is particularly intriguing, as the
$0\text{\,}\mathrm{\SIUnitSymbolCelsius}$ cooling, with its rather dramatic
consequences [20], and the de-orbiting burn took place just between these two
epochs.

In the attempt to assess if the observed variation could still be due to a
random fluctuation, given the comparatively large spread of the values in Fig.
5, we have resorted to a classical permutation test. We have done random
permutations of the time coordinates of the data of Fig. 5, and counted the
number of times a step of any sign, larger than the observed one, could be
found between two adjoining sets of data, one 6-sample long, like that of run
11 to 13, and the other either 4- or 10-sample long.

We find that the fraction of permutations in which we find a step $\leq
0.45\,$\mathrm{f}\mathrm{m}\,\mathrm{s}^{-2}\mathrm{/}\sqrt{\text{Hz}}$$,
between the 6-sample and the 4-sample sets, is $p=0.09$, while for the 10- and
6-sample case, the chances of a step $\leq
0.32\,$\mathrm{f}\mathrm{m}\,\mathrm{s}^{-2}\mathrm{/}\sqrt{\text{Hz}}$$ is
$p=0.12$. These fractions are both too large to allow us to rule out the
hypothesis that the observed increase is due to a random fluctuation.

###  V.3  Gaussianity

Besides the lack of stationarity discussed so far, non-Gaussian distributed
data may also contribute to the discrepancy between a simple common-noise
model and the data. We have performed a coarse test on the Gaussianity of the
periodograms used to calculate all experimental PSDs.

We have first taken the real and the imaginary parts of all periodograms used
to produce the PSD at a given frequency and for a given run, and merged them
into a single sample. This to increase the statistics for those frequencies
and those runs for which the number of averaged periodograms was very small.
We have done this merging taking advantage of the fact that for Gaussian data
the real and the imaginary part are independent and have the same
distribution.

We have then standardized each sample, by subtracting its mean and by dividing
by its standard deviation. Limiting the analysis to the lowest 9 frequencies
we then have $9\times 13=117$ of these samples, as the number of runs is 13.
The number of periodograms in each sample may vary quite significantly,
ranging from 2, for the lowest frequency bin and the shortest runs, up to 512
for $f_{9}$ and run 9. We exclude cases with just one periodogram, as the
normalization procedure would be not significant.

As all elements of these 117 samples are normalized, if data are Gaussian,
they should all come from the same zero-mean, unit variance Gaussian
distribution. We have then merged them all in a single sample containing 12684
standardized periodograms. In Fig. 6 we report the histogram of this sample
and, for comparison, the probability density function of the zero-mean unit
variance Gaussian distribution.

FIG. 6: Green bars: histogram of the periodogram amplitudes standardized as
explained in the text. Solid black line: probability density function of the
zero-mean unit variance Gaussian distribution.

The agreement between the two is supported by an Anderson-Darling test for
Gaussianity, that gives $p>0.15$. We note that the test is rather crude, the
statistic being dominated by the longest runs and the higher frequencies.
However, the absence of any outlier significantly above $5\sigma$ excludes the
presence of significant long tails.

###  V.4  Associated angular acceleration noise

For the purpose of characterizing the properties of the excess acceleration
noise described above, we have investigated if there is any torque associated
with it. To this purpose, we have measured the torque acting on the test
masses around $z$ and $y$.

As the rotation of the test masses around all axes is controlled, and as the
angular acceleration of the spacecraft, which is common mode for both TM, is
rather large, in total analogy with the $\Delta g$ and the motion along $x$,
we form the differential, open-loop torque (per unit moment of inertia) acting
on the test mass:

$\begin{split}&\Delta\gamma_{\phi}=\ddot{\phi}_{2}-\ddot{\phi}_{1}-\frac{N_{z_{2}}-N_{z_{1}}}{I_{zz}}+\omega_{\phi_{2}}^{2}\phi_{2}-\omega_{\phi_{1}}^{2}\phi_{1}\\\
&\Delta\gamma_{\eta}=\ddot{\eta}_{2}-\ddot{\eta}_{1}-\frac{N_{y_{2}}-N_{y_{1}}}{I_{yy}}+\omega_{\eta_{2}}^{2}\eta_{2}-\omega_{\eta_{1}}^{2}\eta_{1},\end{split}$
(8)

Here ${\phi}_{i}$ and ${\eta}_{i}$ are the angular displacements of TM$i$
along $z$ and $y$ respectively, $N_{z_{i}}$, $N_{y_{i}}$ are the control
torques applied to TM$i$ along $z$ and $y$ respectively, $I_{zz}$ and $I_{yy}$
are the TM moments of inertia around $z$ and $y$ respectively, and finally the
$\omega$’s are the angular stiffnesses. All signals had to be properly
calibrated to ensure that the large, common mode angular acceleration of the
spacecraft was duly suppressed.

The PSDs $S_{\Delta\gamma_{\phi}}(f)$ and $S_{\Delta\gamma_{\eta}}(f)$ are
reported in Fig. 7 for run 10 of Table 1.

FIG. 7: PSD of $\Delta\gamma_{\phi}(f)$ and $\Delta\gamma_{\eta}(f)$, as a
function of frequency $f$, for run 10 of Table 1 (February 2017). The green
band represents the uncertainty range for the projection of the PSD of the
rotational Brownian noise around $\phi$, from the corresponding Brownian noise
for $\Delta g$ (see text for details). The reported error on the projected
value accounts just for the propagation of statistical errors. Additional
inaccuracy due to various sources may amount to some additional 10%.

The figure also shows the PSD of rotational Brownian noise along $z$,
projected from the measured value of the PSD of the Brownian noise in $\Delta
g$ for the same run. The projection is based on calculating the rotational
viscous damping coefficient from the linear one, as explained in [25].

Both $S_{\Delta\gamma_{\phi}}(f)$ and $S_{\Delta\gamma_{\eta}}(f)$ show a
minimum much larger than the projected Brownian noise. This minimum is set by
the cross-over between the torque noise and the readout noise due to the
angular interferometer, which is comparatively more noisy than the linear one.

To assess if the excess $\Delta g$ noise carries any torque, we have measured,
together with the PSDs, also the cross-spectral densities among
$\Delta\gamma_{\phi}$, $\Delta\gamma_{\eta}$, and $\Delta g$.

To get a simple parametrization of the associated torque we use the simple
model of a vector point-like force ${f}(t)$, applied to one of the test masses
at the point $(x_{0},y_{0},z_{0})$, relative to the test mass center, and with
component $(f_{x}(t),f_{y}(t),f_{z}(t))$. Such force would also apply a torque
$N(t)$ with $z$ and $y$ components given, respectively, by
$N_{z}(t)=x_{0}f_{y}(t)-y_{0}f_{x}(t)$ and
$N_{y}(t)=z_{0}f_{x}(t)-x_{0}f_{z}(t)$. For such a force,

$\begin{cases}&S_{\Delta\gamma_{\phi}}(f)=S_{\Delta\gamma_{\phi},0}(f)+y_{0}^{2}\left(\frac{M}{I_{zz}}\right)^{2}S_{\Delta
g}(f)\\\ &S_{\Delta\gamma_{\phi}\Delta
g}(f)=-y_{0}\left(\frac{M}{I_{zz}}\right)S_{\Delta g}(f)\\\ \end{cases}$ (9)

and similarly for y, with $-y_{0}\rightarrow z_{0}$. Here,
$S_{\Delta\gamma_{\phi},0}(f)$ ($S_{\Delta\gamma_{\eta},0}(f)$) is the
spectral density of any part of $\Delta\gamma_{\phi}(t)$
($\Delta\gamma_{\eta}(t)$) that is not correlated to $\Delta g(t)$.

The parametrization holds for an arbitrary system of forces, but while for a
real point-like force $\lvert y_{0}\rvert,\lvert z_{0}\rvert\leq
l=$46\text{\,}\mathrm{m}\mathrm{m}$$, for an arbitrary system both parameters
can take any value. One important example is that of the force due to a
voltage on only one of the electrodes facing the $x$-faces of the TM, for
which $|y_{0}|\simeq$11\text{\,}\mathrm{m}\mathrm{m}$$ while $z_{0}=0$.

We have estimated the Bayesian posterior for the parameters
$S_{\Delta\gamma_{\phi},0}(f)$, $S_{\Delta\gamma_{\eta},0}(f)$, $y_{0}$ and
$z_{0}$ with an MCMC calculation, assuming Gaussian data, a Jeffreys prior for
$S_{\Delta\gamma_{\phi},0}(f)$, $S_{\Delta\gamma_{\eta},0}(f)$, and a uniform
prior for $y_{0}$ and $z_{0}$. We performed this analysis including
frequencies from the second to the seventh frequency bins
($36\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$ – $0.60\text{\,}\mathrm{mHz}$),
as in some of the runs, some eighth-frequency data already include the
interferometer rising branch.

We have performed the calculation for both the individual runs, and also by
assuming that the same value of $y_{0}$ or $z_{0}$ may fit all runs. The
results are reported in Fig. 8.

FIG. 8: Top panel. Data points: effective lever arm $y_{0}$ of force noise
around the $z$-axis as a function of the epoch $t$ of the run. Error bars are
$1\sigma$ width of the posterior. Light gray band $y_{0}=(1.13\pm
0.15)\,\text{mm}$, the estimated value of the lever arm assuming a common
value for all runs. Bottom panel. Same as the top panel, but for the effective
lever arm $z_{0}$ of force noise around the $y$-axis. The light gray band is
for $z_{0}=(0.5\pm 0.8)\,\text{mm}$.

We note that for $y_{0}$ a systematic pattern appears, with the global fit
giving $y_{0}=(1.13\pm 0.15)\,\text{mm}$, excluding then
$y_{0}=$0\text{\,}\mathrm{m}\mathrm{m}$$ and
$y_{0}=$11\text{\,}\mathrm{m}\mathrm{m}$$ with very large significance. On the
contrary, for $z_{0}$ data scatter on both sides of zero and the global fit
gives $z_{0}=(0.5\pm 0.8)\,\text{mm}$, i.e. a lever arm not significantly
different from zero, though within an error which is significantly larger than
that for $y_{0}$.

Ref. [32] analyzes the force and torque noise due to gain fluctuations in the
LPF TM actuation systems (see also Sec. VIII.4). That study has found that the
$\phi$ angular acceleration noise excess at low frequencies is largely
explained by actuation gain fluctuations in the circuitry used to produce both
$x$ forces and $\phi$ torques. Additionally, that study observes a slight
correlation between $\Delta g$ and $\Delta\gamma_{\phi}$ arising from the gain
fluctuations in these shared $x$-$\phi$ actuators, a correlation that is
consistent with the $y_{0}$ armlength in the top panel of Fig. 8. The slightly
positive armlength comes from a slight asymmetry in the electrode voltage
noises, as better explained in Ref. [32].

We have compared the results of our analysis, in particular for run 10 of
Table 1, to those in [32], and have found full compatibility between them,
indicating that the detected torque noise correlated with $\Delta g$, is
entirely explained by the shared gain fluctuations. We will further discuss
the implications of these findings later in the paper, Sec. IX.

###  V.5  Summary note on the $1/f$ tail

In conclusion, the overall picture is that, for frequencies in the range
[$36\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$ – $0.60\text{\,}\mathrm{mHz}$]
the noise ASD for $\Delta g_{e}$ is:

  * •

basically compatible with the $1/f$ model within a single run of $\simeq
2.5\,\text{d}$ duration;

  * •

the ‘true’ amplitude of the $1/f$ branch may fluctuate on the average by
$\simeq\pm 20\%$ from one of these runs to another, this fluctuation being in
addition to that expected from Gaussian stationary noise;

  * •

the amplitude of such a non-stationary extra fluctuation does not seem to
follow any long-term pattern, neither decaying nor increasing over the course
of the mission;

  * •

there is no proven correlation between these extra fluctuations and any
identifiable operational condition, though we cannot exclude that the
$0\text{\,}\mathrm{\SIUnitSymbolCelsius}$ cooling, or the de-orbiting burst
may have had some effect on the noise after day 460 (runs 11-13);

  * •

there is a small correlated torque associated with such $1/f$ force noise. The
mm-size effective radius points to forces acting toward the center of the TM,
or to forces that are almost uniform over the TM faces, ruling out simple
possibilities like the force due to a noisy voltage on one of the
$x$-electrodes;

  * •

finally, it is worth noting that the approximate stability of the $1/f$ tail’s
amplitude persisted despite numerous changes to the operational environment
including station keeping maneuvers, planned experiments with the LTP and DRS
payloads, and unplanned spacecraft anomalies. By comparison, LISA operations
will be simpler with no orbital station keeping or planned experiments post
commissioning.

###  V.6 Evolution of the lowest frequency datum and of long-term drifts

In Sec. V, we noted that the first-frequency point
($f_{1}=$18\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$$) deviates from the $1/f$
ASD behavior, consistently showing a noise level above that predicted by the
$1/f$ fit model. To quantify the deviation from the model, we define
$\widetilde{S}_{\Delta g_{e}}(f_{1})$, i.e. the excess noise at frequency
$f_{1}$, above $\widetilde{S}_{\Delta g}$:

$\widetilde{S}_{\Delta
g_{e}}(f_{1})\equiv(f_{1}/$1\text{\,}\mathrm{mHz}$)^{2}S_{\Delta
g_{e}}(f_{1})-\widetilde{S}_{\Delta g}$ (10)

We have multiplied the data by the frequency factor
$(f_{1}/$1\text{\,}\mathrm{mHz}$)^{2}$ to make the comparison with the $1/f$
branch more immediate.

In Fig. 9 (grey points) we show the time evolution of $\widetilde{S}_{\Delta
g_{e}}(f_{1})$ at the lowest frequency333Note that some of the PSD values in
Fig. 9 take negative values, which would make the ASD imaginary. This is the
reason why we prefer to use PSD.. Despite the large errors, the picture shows
a significant initial decay pattern. In addition, all values of
$\widetilde{S}_{\Delta g_{e}}(f_{1})$ are significantly above zero, i.e. there
is a noise excess above the $1/f$ model at
$18\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$ at all times during the mission.

FIG. 9: Evolution of the excess noise at the lowest frequency
$f_{1}=$18\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$$ over the course of the
mission. Grey points: $\widetilde{S}_{\Delta
g_{e},1}\equiv\left(f_{1}/$1\text{\,}\mathrm{m}\mathrm{H}\mathrm{z}$\right)^{2}S_{\Delta
g_{e}}(f_{1})$, for the 13 runs of Table 1, as a function of the mean time of
the run since launch $(t-t_{0})$. Red points: residual after correcting for
the effect of the temperature, (see text at the end of the section for
details). Only data for runs with more than one periodogram at
$18\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$ could be corrected, hence in some
cases there is no red point (see. App. B.3.2).

Before discussing this behavior in further detail, and before introducing the
temperature correction shown by the red points in the figure, we have to
discuss the long-term drift, over scales of several hours and days, that
affects $\Delta g$ data for all runs. We will show that such drifts share many
of the features shown by $\widetilde{S}_{\Delta g,e}(f_{1})$.

This is not surprising as the relative width of the spectral window at $f_{1}$
is wider, by construction, than that for all other frequencies (See App. B.2).
Thus a significant spectral leakage from the frequency band $f\ll f_{1}$, i.e.
from the long-term drifts may be naturally expected.

An example of the drift that, in addition to quasi-stationary noise, affects
$\Delta g$ data for all runs is shown in Fig. 10.

FIG. 10: Long-term evolution of $\Delta g$ for run 7 of Table 1 (purple line),
and of various disturbances that may contribute to it. Green line:
gravitational signal from propellant tank depletion. Yellow line: mechanical
distortion calculated assuming the nominal value of effective stiffness
$\omega_{d}^{2}$. Orange line: temperature contribution from best fit to data.
Dashed cyan line: residual drift. See text for details.

The sources for such a drift may be many. There are two predictable ones:

  * •

motion of massive parts of the instrument relative to the TMs, due to any kind
of mechanical distortion, also produces a time-varying gravitational field and
then a time-varying $\Delta g_{\text{Dist}}$. One important mechanical
distortion is the rigid translation, along $x$, of each GRS relative to its
own TM due to expansion and contraction of the optical bench to GRS
separation. By GRS we mean here vacuum chamber, electrode housing, etc., but
not the TM itself.

Such distortion is rather likely, as the GRS has a rather complex interface,
on one hand to the satellite, via a set of tens of centimeters long ceramics
struts, and, on the other, to the optical metrology, through a sophisticated
metal-glass interface. This mounting method is rather ‘soft’ and prone to
strain, while the GRS itself is a much more rigid assembly.

Fortunately, the capacitive motion sensor gives a measurement of the relative
motion of the GRS relative to its own TM, $\Delta X\equiv\left(\Delta
x_{\text{OMS}}-\Delta x_{\text{GRS}}\right)$, and allows to predict $\Delta
g_{\text{Dist}}$, as $\Delta g_{\text{Dist}}=\omega_{d}^{2}\Delta X$ (see Fig.
10) with $\omega_{d}^{2}$ a stiffness factor which is known to within a
sufficient accuracy. This is again explained in App. E.

We note that this contribution to the overall drift is rather small, if not
negligible, for all runs, except that for runs 7, 11, and 12 of Table 1.

  * •

The use of propellant for the drag-free control slowly depletes the propellant
tanks [13]. The propellant in the tanks produces a gravitational field at the
TM’s location that results in a differential acceleration of the TMs with a
significant component $\Delta g_{\text{Tanks}}$ along $x$. Due to depletion,
this acceleration drifts in time (see Fig. 10). In App. E we show that, for a
run in which the propellant tank $i$ and the thruster branch $j$ are used,
$\Delta g_{\text{Tanks}}(t)=\kappa_{t,i}\kappa_{b,j}\Delta
g_{\text{Tank},0,i,j}(t)+c$, where $\Delta g_{\text{Tank},0,i,j}(t)$ is a
signal that can be entirely calculated, $\kappa_{t,i}$ and $\kappa_{b,j}$
(where $t$ stands for tanks and $b$ stands for branch) are two factors that
reflect calibration uncertainty, and that are both 1 for the nominal case; $c$
is a constant, depending on $i$ and $j$, with no real relevance for the
following discussion.

Once the two sources above have been subtracted from the $\Delta g$ data
series, the residual shows an evident correlation with the instrument
temperature $T$, defined in Sec. II.3, and its long-term variations (Fig. 10).
Once also such correlation has been suppressed, by linear least squares
fitting $T$ to $\Delta g$ and by subtracting the best fit, still, some
residual drift remains (Fig. 10).

We note from Fig. 10 that the raw drift (purple line) is of order
$0.5\text{\,}\mathrm{p}\mathrm{m}\,\mathrm{s}^{-2}\mathrm{/}\mathrm{d}$, while
the residual unmodelled drift (dashed blue line) is of order
$60\text{\,}\mathrm{f}\mathrm{m}\,\mathrm{s}^{-2}\mathrm{/}\mathrm{d}$.

To get some quantitative estimate of this residual drift, we have
preliminarily corrected the data for the two predictable effects, $\Delta
g_{\text{Tanks}}$ and $\Delta g_{\text{Dist}}$, and then fitted the corrected
data to a linear combination of temperature and time as described in App. E.
As explained therein, such fitting gives, for each run, a value for the mean
partial derivative of $\Delta g$ relative to temperature, $\partial\Delta
g/\partial T$, and one for that relative to time $\partial\Delta g/\partial
t$.

The above-mentioned preliminary correction requires some assumptions on the
value of $\omega_{d}^{2}$, and on those for $\kappa_{t,i}$ and $\kappa_{b,j}$.
The results for the nominal calibration $\kappa_{t,i}=\kappa_{b,j}=1$ and
$\omega_{d}^{2}=-3.32\times 10^{-7}\text{\,s}^{-2}$ are shown, as black
points, in Fig. 11 as a function of the time of the run.

FIG. 11: Upper panel: partial derivative of $\Delta g$ relative to temperature
$T$, $\partial\Delta g/\partial T$, as a function of the time of the run. Data
refer to the nominal configuration, but those for the optimized configuration
(see text for the definition) are numerically indistinguishable from them.
Vertical errors are derived as explained in App. E. Horizontal bars represent
the time span of the run. Lower panel: partial derivative of $\Delta g$
relative time $t$, $\partial\Delta g/\partial t$, as a function of the time of
the run. Black dots refer to the nominal configuration, while the red ones to
the optimized configuration. The meaning of the error bars is the same as that
for the upper panel.

The relative rapid decay of both quantities in the early phase of the mission
led us to investigate the existence of any possible correlation with the
similar decay of Brownian noise.

We have found a quite significant linear relation between $\partial\Delta
g/\partial T$ and $\partial S_{\text{Brown}}/\partial T$; we calculate the
latter, from Sec. IV, as $\partial S_{\text{Brown}}/\partial
T=(T_{a}/T^{2})S_{\text{Brown}}$ (see Fig. 12).

FIG. 12: $\partial\Delta g/\partial T$ vs $\partial S_{\text{Brown}}/\partial
T$ for the nominal calibration. Points are the data while the dashed line is
the linear, least square best fit $\partial\Delta g/\partial
T=\alpha_{T}\;\partial S_{\text{Brown}}/\partial T$, with $\alpha_{T}=(1.49\pm
0.07)\times 10^{18}\,\text{s/m}$.

A fit with the simple proportionality relation $\partial\Delta g/\partial
T=\alpha_{T}\partial S_{\text{Brown}}/\partial T$, gives $\alpha_{T}=(1.49\pm
0.07)\times 10^{18}\,\text{s/m}$ (see Fig. 12). The chi-squared test for the
goodness of such fit gives $p=0.8$.

Note however that the factor $(T_{a}/T^{2})$ changes at most by some 6% from
run to run, thus the correlation between $\partial\Delta g/\partial T$ and
$S_{\text{Brown}}$ itself is as strong as the other. Nevertheless, we will
continue to discuss the case for $\partial S_{\text{Brown}}/\partial T$ for
reasons that will be clear in the following.

A plot of $\partial\Delta g/\partial t$ as a function of $S_{\text{Brown}}$
shows a rather low level of linear correlation. The linear correlation is
instead rather significant between $\partial\Delta g/\partial t$ and $\partial
S_{\text{Brown}}/\partial t=(-\gamma/(t-t_{v}))S_{\text{Brown}}$ (following
from Eq. (5)). A plot is shown in Fig. 13 together with a best fit to the data
with $\partial\Delta g/\partial t=\alpha_{t}\partial S_{\text{Brown}}/\partial
t$ that gives $\alpha_{t}=(1.4\pm 0.2)\times 10^{18}\,\text{s/m}$.

FIG. 13:  $\partial\Delta g/\partial t$ vs $\partial S_{\text{Brown}}/\partial
t$. Black points are the data for the nominal calibration, and the black
dashed line is the linear, least square best fit to the black points with
$\partial\Delta g/\partial t=\alpha_{t}\partial S_{\text{Brown}}/\partial t$,
and $\alpha_{t}=(1.4\pm 0.2)\times 10^{18}\leavevmode\nobreak\ \text{s/m}$.
Red data and line have the same meaning, but for the optimized calibration. In
this case $\alpha_{t}=(1.44\pm 0.07)\times 10^{18}\leavevmode\nobreak\
\text{s/m}$.

The goodness of the fit to nominal data is rather poor, with negligible
$p$-value. As the gravitational drift is large, even a limited change to the
values of $\omega_{d}^{2}$, $\kappa_{t,i}$ and $\kappa_{b,j}$, used for its
subtraction, may change the value of the residual drift and then of
$\partial\Delta g/\partial t$. Thus the uncertainty on those coefficients
projects a large uncertainty on the true value of $\partial\Delta g/\partial
t$, uncertainty that is not taken into account while performing the goodness
of fit test.

To show that this is the case, we have searched if values of $\omega_{d}^{2}$,
$\kappa_{t,i}$ and $\kappa_{b,j}$, other than the nominal ones, but still
within the range of their uncertainties, may improve the quality of the fit.
Actually, we have searched for the values that just give the maximum $p$-value
in the goodness of fit test.

We find that the maximum is attained when $\kappa_{t,1}=1.00$,
$\kappa_{t,2}=1.05$, $\kappa_{t,3}=0.95$, $\kappa_{b,A}=1.00$,
$\kappa_{b,B}=0.92$, and $\omega_{d}^{2}=-3.31\times 10^{-7}\,\text{s}^{-2}$.
With this ‘optimized’ calibration, $\alpha_{t}=(1.44\pm 0.07)\times
10^{18}\,\text{s/m}$, and the $p$-value is $p=0.18$ (See Fig. 13). Remarkably,
moving to this optimized calibration, within errors, does not change the slope
of the line.

The most important observation though, is that
$\alpha_{T}=\alpha_{t}\equiv\alpha$ within the errors. This supports, with all
the limitations and caveats that come from the empirical approach we had to
use to process the data, that, for run $k$, the long-term evolution of $\Delta
g$ obeys:

$\Delta g_{k}(t)=\alpha S_{\text{Brown}}(t)+\Delta g_{0,k}=2\alpha\kappa
P_{\text{H}_{2}\text{O}}+\Delta g_{0,k}$ (11)

with $\Delta g_{0,k}$ a constant that depends on the run, and is affected by
many operational factors that may be different in different runs.

In Eq. (11) we have used the conversion from Brownian noise PSD to the mean
pressure around the TM that we have discussed in Sec. IV. It is interesting to
note that also $\Delta g$ may be converted into an equivalent difference of
pressure between the $x$-faces of one of the TM’s, as $\Delta
P_{\text{H}_{2}\text{O}}=M\Delta g/L^{2}$, with $M$ the mass of the TM and $L$
the length of one of its edges. Then $\Delta
P_{\text{H}_{2}\text{O}}=((2M\alpha\kappa)/L^{2})P_{\text{H}_{2}\text{O}}=(4.5\pm
0.2)\times 10^{-3}\,P_{\text{H}_{2}\text{O}}$. If the pressure difference was
similar and opposite on both TM’s, not unlikely given the mirror symmetry of
the instrument, then all figures should be divided by 2.

The temperature dependence of $\Delta g$ at these very low frequencies is
consistent with the transient behavior of the lowest frequency datum reported
in Fig. 9. Indeed a Bayesian decorrelation of temperature, following the
method of App. B.3.2, gives the red points in Fig. 9, which show a suppressed
initial transient, and a significantly reduced discrepancy from the $1/f$
tail.

Remarkably, the coefficient obtained from the decorrelation,
$\left(\partial\Delta g_{e}/\partial T\right)_{\text{noise}}$, is in
quantitative agreement with the values of $\partial\Delta g_{e}/\partial T$
reported in Fig. 12 (see Fig. 14). This confirms our modelization.

FIG. 14: Temperature coefficient $\left(\partial\Delta g_{e}/\partial
T\right)_{\text{noise}}$ from the Bayesian temperature decorrelation of the
lowest frequency datum, vs the temperature coefficient $\partial\Delta
g_{e}/\partial T$ of Fig. 12. The dashed line is the result of a linear least
square fit, which gives $\left(\partial\Delta g_{e}/\partial
T\right)_{\text{noise}}=\left(1.0\pm 0.1\right)\partial\Delta g_{e}/\partial
T+(-0.3\pm 0.2)\,\text{pm\,s}^{-2}\,\text{K}^{-1}$, with a reduced chi-square
$\chi^{2}\simeq 0.6$.

We will discuss in Sec. VI the implication of such findings for the nature of
the vacuum environment of the TMs.

##  VI Discussion I: Brownian noise, long-term drift, and the TM vacuum
environment

In this and in the following section we discuss the physical information we
can gather from the observation reported so far.

We start discussing the Brownian noise and the very low-frequency behavior,
both drift and ASD, as both shed light on the nature of the vacuum environment
surrounding the test mass.

We will discuss in some detail the possible physical sources of the $1/f$ tail
in a following section.

###  VI.1 Brownian noise

Power-law evolution of pressure over time is very often observed in vacuum
systems during initial pump down [27]. When the outgassing surfaces consist
predominantly of metals, like stainless steel, aluminum, or titanium, a
$\propto 1/t$ behavior is very often observed, consistent with models in which
water vapor is in quasi-equilibrium between thermal outgassing from the metal
walls, and re-adsorption onto them. To check if this model would be consistent
also with our observed $\propto 1/t^{0.8}$ behavior, we have attempted to fit
our observations to it. To this aim, we have integrated the differential
equation (see [28])

$\frac{1}{P}\frac{dP}{dt}=-\frac{\frac{1}{\tau}+\frac{P_{s}}{P}\frac{d\theta}{dT}\frac{dT}{dt}}{1+P_{s}\frac{d\theta}{dP}}$
(12)

that, within such model, describes the time evolution of pressure. Here

  * •

$\theta\left(P,T\right)$ is the fraction of occupied adsorption sites.

  * •

$P_{s}=N_{s}k_{B}T/V$, with $N_{s}$ the total number of available adsorption
sites, and $V$ the vacuum enclosure volume.

  * •

$\tau$ is the vacuum relaxation time of the vacuum enclosure, in the absence
of outgassing and adsorption, which is set by $V$ and by the pumping speed of
the vent duct and is of the order of 0.2 s.

Eq. (12) can only be solved in combination with a choice for the functional
form of $\theta\left(P,T\right)$. A versatile form that reproduces
observations in many cases, is the Temkin isotherm [33].

$\theta\left(P,T\right)=\frac{T}{T_{2}-T_{1}}\log\left(\frac{1+\frac{P}{P_{0}e^{-T_{2}/T}}}{1+\frac{P}{P_{0}e^{-T_{1}/T}}}\right)$
(13)

With $P_{0}$, $T_{1}$ and $T_{2}$ free parameters of the model. We have done a
least squares fit of the observed values of $S_{\text{Brown}}/(2\kappa)$ to
the prediction of Eq. (12), as a function of $P_{s}$, $\tau$, $P_{0}$,
$T_{1}$, and $T_{2}$. For each choice of the parameters we have integrated Eq.
(12), and calculated the chi-square of the deviation of the data from the
resulting solution. To integrate the equation, we have used the mean measured
GRS temperature, mentioned before, throughout the entire integration time
interval, except for the short duration epoch of the cold run, where we have
used the proxy made with the LTP bay thermometers, also mentioned before.

The best-fit line is the dotted grey line in Fig. 3. The reduced chi-square of
the minimum is rather poor $\simeq 2.0$, though this can be, at least in part,
due to a significant underestimation of the true uncertainties in the model.

A more serious limitation with such an interpretation lies in the best-fit
parameter values. Let us first mention that, provided $P_{0}\,e^{-T_{2}/T}\ll
P\ll P_{0}\,e^{-T_{1}/T}$, the chi-square is found to depend only on $P_{0}$
and on the combination $c=(T_{2}-T_{1})/(\tau P_{s})$. Such a simplification
can also be readily derived from an inspection of Eqs. 12 and 13. For these
two parameters, the fit gives a broad minimum for $\log P_{0}$,
$\log_{10}\left(P_{0}/$1\text{\,}\mathrm{k}\mathrm{P}\mathrm{a}$\right)=(1.4\pm
0.3)$, and a relatively narrow minimum for $c$, $c=(0.328\pm
0.004)\leavevmode\nobreak\
$\mathrm{K}\,\mathrm{d}^{-1}\,\mathrm{\SIUnitSymbolMicro}\mathrm{Pa}^{-1}$$.

The value for $P_{0}$,
$P_{0}\simeq$25\text{\,}\mathrm{kPa}$\simeq$200\text{\,}\mathrm{T}\mathrm{o}\mathrm{r}\mathrm{r}$$
is orders of magnitude smaller than the quoted, for in instance, in [33] of
about $6\times 10^{8}$ Torr, and within the theory discussed in that same
reference, the only way to get such a small number would be to assume an
unreasonably low density of adsorbing sites or an equally unreasonably high
molecular attempt frequency.

Furthermore, of the four parameters that enter in the definition of $c$,
$\tau$ can be independently evaluated from the estimated conductance of the
vacuum valve and the vent duct, and from the volume of the vacuum enclosure,
to be $\tau\simeq$0.3\text{\,}\mathrm{s}$$. A possible choice for the
temperature values is given in [33], and is
$T_{1}\simeq$5.3\text{\,}\mathrm{kK}$$ and
$T_{2}\simeq$11\text{\,}\mathrm{kK}$$. The value for $T_{1}$ is only
marginally fulfilling the condition on the pressure range. The closest value
that still keeps the chi-square at its minimum is
$T_{1}=$4.5\text{\,}\mathrm{kK}$$. With this choice for $\tau$, $T_{1}$, and
$T_{2}$, we get $P_{S}=$6.0\pm 0.1\text{\,}\mathrm{kPa}$$. In turn, from this
value for $P_{s}$, assuming, as in [33] a density of sites $\simeq 3\times
10^{15}\,\text{cm}^{-2}$, we get an estimate for the internal area of the
vacuum enclosure of $\simeq 180\,\text{m}^{2}$, a couple of orders of
magnitude larger than the physical area.

Thus to reconcile $P_{0}$ with the model one would need fewer absorption
sites, while for $P_{s}$ one would need more. We believe that the chance that
water re-adsorption has played a major role in our vacuum system is highly
unlikely.

An alternative to the model of water desorption from walls, known to show as
well power-law evolution over time, is diffusion-limited outgassing. In the
case of hydrogen outgassing from stainless steel, for instance, a $1/\sqrt{t}$
evolution is often observed, in agreement with Fick’s law prediction for one-
dimensional diffusion [28]. Approximate outgassing power-law evolution with an
exponent even larger than 0.5 is observed in the presence of polymers and is
attributed to water diffusion through them. In addition, experiments on
polymer samples show that the temperature dependence of the outgassing rate is
well described by a single activation exponential, as is the case of our
observation [28].

Given the abundance of polymers in cable bundles, motors, and connectors
within the GRS, and given in general the complex geometry of the TM
surroundings with abundant solid interfaces, it looks rather likely that
desorption from walls is playing a minor role, and that gas diffusion out of
some of the GRS components is dominating the pressure environment.

###  VI.2  Long-term drift, very low frequency noise and pressure gradient

The results of Sec. V.6 support the idea that there was a permanent difference
of pressure across one or both the TMs, whose amplitude scales with the
overall pressure as measured by the Brownian noise level. This pressure
gradient makes $\Delta g$ both sensitive to temperature, because of the
corresponding temperature sensitivity of pressure, and drifting, because of
the pressure drift due to venting to space.

We have simulated with Molflow+ [34] the effect of localized outgassing on the
pressure on the TM. Simulations consistently show that, due to the rather
complex geometry of the TM surroundings, which creates a rather asymmetric
molecular flow impedance pattern, it takes very little asymmetry of outgassing
to support a pressure gradient across the TM.

To give a scale of the phenomenon, a large fraction of any flow of molecules
out of the cavity formed by the outer wall of the EH and the Tungsten balance
mass (see Fig. 15 of Ref. [20]), penetrates inside the EH, through a hole in
the EH $x$-wall, symmetric to that for the laser beam. These molecules diffuse
in the gaps surrounding the TM and, in a time of the order of ms, eventually
leave the EH. In their flow within the EH, these molecules exert both a mean
pressure on the TM, and a pressure difference across the $x$-axis. We find
that this difference is approximately 30% of the overall mean pressure
contribution around the TM.

On the contrary, simulations show that outgassing from sources farther away
from the EH, like, for instance, from the inner wall of the VE, diffuses and
equalizes outside the EH and does not create any significant pressure
difference across the TM.

###  VI.3 In summary: the vacuum environment of the TMs

The observed tight link between the total residual pressure, as measured by
the Brownian noise level, and the decaying drift in $\Delta g$, with both time
and temperature, implies, as a minimum, the same desorption properties for the
gas setting the overall residual pressure and the pressure gradients.

The most simple explanation for this is that the dominant source of outgassing
in the GRS is diffusion out of sources close to the TM, that is, located
inside the EH or just outside its $x$-walls. These sources may include the
electrodes, the insulators and the structure of the EH, the Tungsten balance
mass, the cables placed close to the outer $x$-walls of the EH, and a few
other elements.

This recommends, in preparation for LISA, to conduct in-depth qualification
studies on the outgassing properties of these elements.

##  VII Discussion II: excess noise and possible observational artifacts

Here we begin discussing the possible sources of the observed $1/f$ tail. We
first consider the role of some possible observational artifacts, while a
detailed projection of the noise onto the possible modeled sources is
discussed in the next section.

###  VII.1  Role of interferometer noise.

The contribution of interferometer noise $n_{\text{OMS}}$, with ASD
$S_{n_{\text{OMS}}}^{1/2}$, to $S_{\Delta g}^{1/2}$ is $S_{\Delta
g,n}^{1/2}=S_{n_{\text{OMS}}}^{1/2}\left(4\pi^{2}f^{2}+\lvert\omega_{2}^{2}\rvert\right)$.

Thus, for $f\leq\lvert\omega_{2}\rvert/2\pi$, a branch of
$S_{n_{\text{OMS}}}^{1/2}$, raising rapidly enough upon decreasing frequency,
may have contributed to $S_{\Delta g,e}^{1/2}$.

In our ordinary noise measurements, there was no way to separate $S_{\Delta
g,n}^{1/2}$ from the contribution of true forces. We have however two
independent methods to put an upper limit to the former.

The first method is to use the interferometer data we collected during an
epoch in which the test masses were held fixed by the blocking mechanism [35].
This gave us the chance to measure the interferometer output noise in open
loop, that is, with no active control of the TM positions, which is the
quantity appearing in Eq. (2). The results for $S_{\Delta g,n}^{1/2}$ are
reported in Fig. 15 and compared to the lowest $S_{\Delta g_{e}}^{1/2}$ data,
that is those from run 10 of Table 1444The measurement epoch is the same as
that used by Ref. [35], though we have used a longer, and possibly noisier
data series, in order to be able to reach the lowest possible frequency..

FIG. 15: Role of interferometer noise in the overall ASD of $\Delta g$: Red
points: $S_{\Delta g_{e}}^{1/2}$ for run 10 (February 2017), as in Fig. 2.
Dark blue points: $S_{\Delta g,n}^{1/2}$ measured with blocked TMs. Light blue
points: ASD of the difference $\Delta g-\Delta g_{\text{GRS}}$ for run 10.

The second method exploits the independent capacitive measurement of the
relative motion of the TMs $\Delta X_{\text{GRS}}$. As the ordinary $\Delta g$
is calculated using $\Delta X_{\text{OMS}}$, we can calculate the analogous
$\Delta g_{\text{GRS}}$, using $\Delta X_{\text{GRS}}$. $\Delta
g_{\text{GRS}}$ is significantly more noisy than $\Delta g$ at all frequencies
above $80\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$, while the agreement of
$\Delta g$ and $\Delta g_{\text{GRS}}$ at lower frequencies, with their
independent readouts, indicates that we are observing the true force noise.

Neglecting any distortion, the difference between the two is just $\Delta
g-\Delta
g_{\text{GRS}}=\lvert\omega_{2}^{2}\rvert\left(n_{\text{OMS}}-n_{\text{GRS}}\right)+\ddot{n}_{\text{OMS}}-\ddot{n}_{\text{GRS}}$,
and the ASD of this difference is then $S_{\Delta g-\Delta
g_{\text{GRS}}}^{1/2}=\left(\lvert\omega_{2}^{2}\rvert+(2\pi
f)^{2}\right)\sqrt{S_{n_{\text{OMS}}}+S_{n_{\text{GRS}}}}$. This provides then
an upper limit for $\left(\lvert\omega_{2}^{2}\rvert+(2\pi
f)^{2}\right)S_{n_{\text{OMS}}}^{1/2}$. Note that $n_{\text{GRS}}$ is an all
encompassing figure and includes not just electronic noise but also the
spurious pickup of degrees of freedom other than $\Delta x$.

Figure 15 shows the values $S_{\Delta g-\Delta g_{\text{GRS}}}^{1/2}$ for run
10. It is worth to add that the values for $(\lvert\omega_{2}^{2}\rvert+(2\pi
f)^{2})S_{\Delta X_{\text{GRS}}}^{1/2}\simeq(\lvert\omega_{2}^{2}\rvert+(2\pi
f)^{2})S_{n_{\text{GRS}}}^{1/2}$, not shown in the figure, coincide almost
exactly with those for $S_{\Delta g-\Delta g_{\text{GRS}}}^{1/2}$, which shows
that the role of $n_{\text{OMS}}$ in these is negligible.

Both methods give an upper limit to the interferometer contribution, likely
rather pessimistic. Indeed within the blocked TM measurement the contrast was
poor and the interferometer performance at $f>10\,\text{mHz}$ was at least one
order of magnitude worse than that with the free TMs. In addition, the
interferometer output might still have included some residual relative motion
of the TMs, due to any mechanical distortion of the instrument. On the other
hand the data $S_{\Delta g-\Delta g_{\text{GRS}}}^{1/2}$ are clearly dominated
by the GRS noise, with the noise from the interferometer, and the TM motion,
not contributing more than 1% in power.

In conclusion we consider that measured with the TM fixed as the relevant
upper limit on interferometry noise contribution to $\Delta g$, a contribution
which is less than some 1-2% in power, and probably significantly less than
this figure.

###  VII.2  Excess noise as a flow of undetected glitches

As soon as transient events, known as glitches, were observed in the data [20]
and removed from them, the question arose if the excess noise might be due to
an undetected and non-removed fraction of glitches. Not only is the question a
legitimate one, but it is also made particularly relevant by the observation
that both the glitch properties and the excess noise appeared rather stable
throughout the mission, despite the changes in operation conditions [20].

We have addressed the question by performing extensive simulations. The detail
of such simulation work is reported in App. G.

The basic conclusion is that the excess noise might be due to a Poisson flow
of undetected glitches, but those glitches would belong to a distribution
quite well separated in properties from that of the detected ones. In
addition, the flow rate should be high enough that the resulting noise would
be stationary and Gaussian, bearing no detectable feature proving its Poisson
nature.

Thus if the $1/f$ tail is made of Poisson noise this is not related to the
observed glitches, and its Poisson nature does not show up in the data.

##  VIII Discussion III: projection of excess noise on modeled noise sources

In this section, we estimate the contribution of modeled sources of force
noise to the observed excess over the Brownian noise.

Our approach is to give a quantitative estimate of those contributions
whenever they appear to be significantly different from zero, while we try to
establish an upper limit whenever the resolution of our methods limits the
estimate.

We focus our analysis on the run performed in February 2017, run 10 of Table
1, which is the lowest noise one. We will also discuss, whenever relevant, the
case for the other runs.

We consider two categories of sources: the first includes the effect of
physical quantities that have been measured during noise runs, synchronously
with $\Delta g$; the second includes effects for which we have an estimate
from different experiments, performed at different times from those of the
noise runs.

For sources of the first category, we infer their contribution to the ASD of
$\Delta g$ via a ‘decorrelation’ method. For those in the second category, we
just give the best estimate of the contributed noise ASD.

###  VIII.1 Decorrelation of synchronous time series: framework

During noise runs we have measured, synchronously with $\Delta g$:

  1. a.

The gravitational force loss due to fuel depletion, $\Delta g_{\text{Tanks}}$
defined in Sec. V.6;

  2. b.

The relative motion of the two GRS, $\Delta X$, as defined in App. E;

  3. c.

Temperature $T$, defined in Sec. II.3;

  4. d.

The two temperature differences across the two electrode housings, in the
$x$-direction;

  5. e.

The three magnetic field components at four different locations;

  6. f.

A series of spurious low-frequency voltages that have unintentionally been
applied to the electrodes via the actuation circuitry non-linearity [19].

We treat these disturbances as small and Gaussian and consider their effect
only to first order, within the simple model:

$\begin{cases}\Delta g(t)&=\Delta
g_{0}(t)+\sum_{i=1}^{r}\int_{0}^{\infty}\alpha_{i}(t^{\prime})y_{i}(t-t^{\prime})\mathrm{d}t^{\prime}\\\
z_{i}(t)&=y_{i}(t)+n_{i}(t)\end{cases}$ (14)

where $z_{i}(t)$ is any of the measured time series above, consisting of the
“true” physical disturbance $y_{i}(t)$, superimposed to its readout noise
$n_{i}(t)$. $\Delta g_{0}$ is the _residual_ acceleration, not correlating to
$y_{i}$.

Given the nature of the disturbances considered, the susceptibilities
$\alpha_{i}(f)$, i.e. the Fourier transforms of the $\alpha_{i}(t)$, should
be, for most of them, real and constant, but we also consider the general case
to take into account the possibilities of delays and other more complex
correlations.

With the model in Eq. (14), the elements of the cross-spectral density (CSPD)
matrix $\bm{\Sigma}$ of $\Delta g$ and the $z_{i}$ are given by

$\begin{split}&\Sigma_{1,1}=S_{\Delta g}(f)=S_{\Delta
g_{0}}(f)+\sum_{i,j=1}^{r}\alpha_{i}(f)\alpha_{j}^{*}(f)S_{y_{i},y_{j}}(f)\\\
&\Sigma_{i+1,1}=S_{y_{i},\Delta
g}(f)=\sum_{j=1}^{r}\alpha_{j}^{*}(f)S_{y_{i},y_{j}}(f)\\\
&\Sigma_{i+1,j+1}=S_{z_{i},z_{j}}(f)=S_{y_{i},y_{j}}(f)+\delta_{i,j}S_{n_{i}}(f)\end{split}$
(15)

In Eq. (15), $S_{y_{i},y_{j}}(f)$ is the cross-spectral density between
$y_{i}$ and $y_{j}$, while $S_{n_{i}}$ is the PSD of $n_{i}$. In all cases,
except $\Delta X$ which we discuss separately, the readout noise $n_{i}$ is
assumed to be independent of any of the other time series.

We deal with two broad cases. In the first case, we have an independent
knowledge of the value of $\alpha_{i}(f)$, but we know or suspect that the
readout noise $S_{n_{i}}(f)$ is significant, even dominating
$S_{z_{i},z_{j}}(f)$. In the second case we have limited knowledge of
$\alpha_{i}(f)$, but we are confident that $S_{n_{i}}(f)$ is negligible for
the purpose of noise analysis. The general case of unknown susceptibilities
and unknown readout noise is overdetermined and cannot be treated.

###  VIII.2 Decorrelation of synchronous time series: results for noisy series

####  VIII.2.1  Temperature effects and LTP distortion

The instrument distortion $\Delta X(t)$ and temperature $T(t)$ give rather
significant contributions, $\Delta g_{\text{Dist}}(t)=\omega_{d}^{2}\Delta
X(t)$, and $\Delta g_{T}(t)=\partial\Delta g/\partial T\times T(t)$, to the
quasi-DC long-term evolution of $\Delta g$ (see Sec. V.6). In this section, we
address their possible roles in contributing to the in-band excess noise, for
$f\geq f_{2}$. As we found a significant correlation between these series
[36], we consider them together.
Unfortunately, these two series are affected by a significant readout noise.
Here briefly follows our knowledge of the properties of such noise.

_Instrument distortion_ — The experimental PSD of $\Delta X$ peaks at about
0.1 mHz, and slowly decays above that [36]. The experimental CPSD between
$\Delta X$ and $\Delta g$ parallels somewhat this behavior, its real part
becoming significantly different from zero and positive above about 0.3 mHz,
slowly decaying above that. In the frequency range in which the real part is
significant, the imaginary part is also significantly different from zero,
this time with a frequency-dependent sign. This behavior of the CPSD is hardly
compatible with the gravitational signal from mechanical distortion, which is
virtually instantaneous, and then free of imaginary CPSD, and supports instead
the existence of a dominating readout noise of electrical origin.
Actually, the linear instrument distortion $\Delta X$ is the combination of
four signals. Each of these signals consists of an independent differential
capacitance measurement [6]. We find that only two of these signals bear some
significant correlation with $\Delta g$, the correlation being in different
frequency ranges for the two signals.
Both these signals refer to TM2, to whom the largest actuation is applied, and
cross-correlation is likely due to electronic cross-talk between the actuation
command signals and the TM motion sensing ones. At low frequencies the $\Delta
g(t)$ time series is dominated by the actuation contribution $g_{c}(t)$ in Eq.
(1), which brings an indirect correlation between $\Delta g$ and TM motion
sensing within the GRS.

Analogously to the linear distortion $\Delta X$, the GRS allows to measure the
angular distortion
$\Delta\Phi\equiv\left(\Delta\phi_{\text{OMS}}-\Delta\phi_{\text{GRS}}\right)$,
which is built with different combinations of the same four capacitance
measurements. Once corrected with the analogous interferometer signal
$\Delta\phi_{\text{OMS}}$, this signal measures the much suppressed angular
distortion, while being insensitive to the linear one. Hence, it carries the
same readout noise that affects $\Delta X$, though in a different combination.
Including $\Delta\Phi$ in our model should lead to a better constraint of the
contribution of these sources to $\Delta g$.

_Temperature_ — The average temperature at the location of the test masses was
measured by averaging the readings of various thermistors located on the
external $x$-faces of the two electrode housings.
The time series of these thermistors were dominated, above about
$\sim$30\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$$, by excess electronic noise
[8]. This fact was made particularly evident by the loss of mutual coherence
among the time series of the different thermistors above that frequency, a
coherence that below $\sim$30\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$$ was
nearly complete.
The noise was caused by a subtle interaction between drift and a nonlinearity
of the analog-to-digital converter. For two of the eight available
thermistors, such noise was so large that we had to discard the corresponding
time series.

Regarding susceptibilities, we take those from the long-term behavior
discussed in Sec. V.6. In addition, we measured the dependence of $\Delta X$
on $T$ by using the “low distortion” runs, that is by excluding run 7, 11, and
12 (see Sec. V.6). We find consistent results both by measuring the slope
$\gamma=\partial\Delta X/\partial T$ in a linear fit over one-day-long data
stretches, and from the CPSD between the two data series at
$18\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$. The distributions which we use
for the susceptibilities are given in App. F. We recall here that
$\omega_{d}^{2}$, $\alpha_{T}(f)$, and $\gamma(f)$ are, respectively, the
susceptibilities of $(\Delta g,\Delta X)$, $(\Delta g,T)$, and $(\Delta X,T)$.

We address the problem with two different approaches, which are described in
App. F.
The first approach is “naive”, as it assumes that the time series are not
affected by any readout noise. It is represented as a dashed line in Fig. 16.
The bound is completely dominated by the instrument distortion term, the
effect of temperature alone being orders of magnitude smaller. However, it
overcomes the entire excess noise above 0.1 mHz, resulting in a non-physical
limit. Hence, this result carries no information.

FIG. 16: Upper bound ($1\sigma$) of the contribution of LTP distortion and of
pressure-mediated temperature effect to $S_{\Delta g_{e}}^{1/2}$. Red data:
ASD of the excess over Brownian $\Delta g_{e}$ from Sec. III. Dashed blue
line: $1\sigma$ upper quantile of the estimate assuming time series free of
readout noise, “naive” model. Solid blue line: $1\sigma$ upper quantile
considering the presence of readout noise, as described in the text.

The second approach, on the contrary, is more refined as it takes into account
that time series are affected by readout noises, possibly cross-correlating.
Given the experimental assumptions on the susceptibilities, we model the
contribution to $\Delta g$ of the time series $(\Delta X,T,\Delta\Phi)$,
imposing that the residual noise matrix is positive-definite, as a CPSD matrix
should always be. This way, we take into account that time series may be
affected by readout noise. Details in App. F. The $1\sigma$ upper bound is
reported in Fig. 16, solid line.

This upper limit is once again dominated by the effect of $\Delta X$, and
constrained by its correlation with $\Delta\Phi$; the role of temperature is
completely negligible. Despite being tighter than the “naive” one, this upper
limit is still set by the large correlation between $\Delta g$ and the readout
noise of $\Delta X$. Note that, as the temperature is irrelevant, the effect
would amount to some non-thermal distortion of the LTP, such as long-term
creep due to stress release. One would expect that, below the system
mechanical resonances, all greater than $10\text{\,}\mathrm{H}\mathrm{z}$, the
ASD of this kind of effects to be some, possibly rather steep power law of
frequency [37]. But even for a $\propto f^{-1/2}$ dependence, the value of the
limit at $f_{2}$, would set an upper limit at the highest frequencies at least
a factor 4 in power smaller than that indicated by the solid line in Fig. 16.
Thus this limit again probably significantly overestimates the effect.

Finally, note that this technique does not give an explicit lower limit, as
the hypothesis that time series are constituted by readout noise only is fully
compatible with the observations (solution $S_{X}=0$, $S_{T}=0$).

####  VIII.2.2  Tank depletion

The case for the gravitational signal from tank depletion is a bit easier than
the previous one. However, here too the PSD of the readout noise of the
propellant flow meter is unknown. The susceptibility is
$\alpha_{\text{Tanks}}=\kappa_{t1}\kappa_{bB}$ (see Sec. V.6) a real,
frequency-independent figure known to within a 10% uncertainty.

We estimate the contribution of this source of noise to the total excess noise
as $\alpha_{\text{Tanks}}S_{\Delta g,\Delta g_{\text{Tanks}}}$, as explained
in App. B.3.1, with $\Delta\tilde{g}_{\text{Tanks}}$ the true gravitational
noise, free of any readout noise contamination.

We take the posterior for $S_{\Delta g,\Delta g_{\text{Tanks}}}$ from the
proper Wishart distribution (see App. B.3.1), and that for
$\alpha_{\text{Tanks}}$ to be a normal distribution with mean and standard
deviation, respectively, 0.92 and 0.10 (see Sec. V.6). Results are shown in
Fig. 17.

FIG. 17: Contribution of gravitational noise due to tank depletion to the ASD
of $\Delta g_{e}$. Red data: ASD of the excess over Brownian $\Delta{g_{e}}$
from Sec. III. Blue data: $\pm 1\sigma$ posterior interval for the
gravitational noise contribution from CPSD and susceptibility posteriors.

Note that $\sum_{i,j=1}^{r}\alpha_{i}(f)\alpha_{j}^{*}(f)S_{y_{i},y_{j}}(f)$
is a positive real number, while there is no guarantee that a posterior sample
for $\sum_{i=1}^{r}\alpha_{i}(f)S_{y_{i},\Delta g}(f)$ is even real.

We have dealt with this problem in two ways. We have first checked that the
posterior distribution of the imaginary part of
$\sum_{i=1}^{r}\alpha_{i}(f)S_{y_{i},\Delta g}(f)$ was statistically
compatible with zero.

We have then taken the posterior just for the real part of
$\sum_{i=1}^{r}\alpha_{i}(f)S_{y_{i},\Delta g}(f)$, checking that, whenever
the distribution extended to negative values, the zero was within the $\pm
1\sigma$ credible interval.

Except for the lowest frequency, the $+1\sigma$ limit is well below 10% in
power. At the opposite end, except for one frequency, zero falls always within
the $\pm 1\sigma$ interval.

###  VIII.3 Decorrelation of synchronous time series: results for series with
negligible readout noise

For the time series we analyze here, we have limited prior knowledge about the
susceptibilities, except that they are real and frequency-independent. This
because delays between the time series and the forces they exert on the TMs
are negligible.

We do then a simultaneous Bayesian fit at all frequencies as explained in App.
B.3.2. This way we get a posterior for the residual $S_{\Delta g_{0}}(f)$, for
the susceptibilities $\alpha_{i}$, and for the joint contribution to
$S_{\Delta g}$,
$S_{c}(f)=\sum_{i,j=1}^{r}\alpha_{i}\alpha_{j}S_{y_{i},y_{j}}(f)$ (see Eq.
(15)).

$S_{c}(f)$ may be subject to a large positive bias in case of large $r$. For
that reason we calibrate the method with extensive simulations, as explained
again in App. B.3.2.

####  VIII.3.1 Spurious actuation due to in-band noise from digitized
electrostatic actuation

The capacitive actuation design employed digitally-synthesized audio-frequency
sinusoidal voltages [19]. As the force is quadratic in the voltage, this
applies a force at low frequency, proportional to the mean square value of the
sinusoid, while avoiding mixing down low-frequency voltage noise from the
final amplifiers.

The truncation error mentioned earlier affects the above mean square value,
and, if uncorrected, gives a wrong estimate of the applied force.

The nominal sinusoidal voltage signal has zero time average and then zero dc
value. The truncated signal acquires a non-zero average, that is a quasi-dc
voltages $\delta V_{i}(t)$ of order $\sim$\mathrm{\SIUnitSymbolMicro V}$$
amplitude that we were able to calculate from the commanded voltages with
negligible numerical noise.

These voltages are able to apply electrostatic forces onto the test masses by
coupling to its charge and to the parasitic voltages biasing the electrodes
[38]. To linear terms in the $\delta V_{i}(t)$’s this force is given by:

$\Delta g_{D}(t)=\sum_{i\in[1,24]}\alpha_{i}\delta V_{i}(t)$ (16)

where the sum is over the 24 electrodes that surround the two TMs, and where
the susceptibility $\alpha_{i}$ is a linear combination of charge and patch
potentials, and cannot be predicted except, possibly, in order of magnitude.

In reality, due to cosmic rays, the TM charge is subject to an approximately
linear variation in time, $q(t)\sim q_{0}+\dot{q}t$ [21], with
$\dot{q}\sim+23\leavevmode\nobreak\ $\mathrm{e}\mathrm{/}\mathrm{s}$$. On the
charge itself the relative effect may be large, as we have often operated by
putting a proper negative charge on both TMs at the beginning of a noise run,
to find an equal but opposite charge on them at the end of the run.

Hence, a complete analysis should in principle include 48 charge parameters.
To circumvent this problem, we have taken advantage of the fact that the
differential force contribution due to the coupling of the $\delta V_{i}(t)$
to the drifting charge is
$\left(\dot{q}_{2}t\Delta_{x,2}-\dot{q}_{1}t\Delta_{x,1}\right)$. Here
$\Delta_{x,i}$ is linear combination, with $\pm 1$ coefficients, of the
$\delta V_{i}(t)$ of the $x$-electrodes of GRS$i$, as defined in [38], and
$\dot{q}_{i}$ is the charging rate of TM$i$ [38].

We have then formed the two time series $t\Delta_{x,1}(t)$ and
$t\Delta_{x,2}(t)$ and added them to our analysis. This obviously breaks the
hypothesis of noise stationarity, but should still help highlight the
existence of significant correlation.

To keep $r$ not too large, we have done two separate analyses. In the first we
have included the 24 voltage series only. Note that with $r=24$ the analysis
could only be performed for $f\geq f_{4}$ ($0.1\text{\,}\mathrm{mHz}$) as
lower frequencies would not have a sufficient number of periodograms (see App.
B.3.2).

In the second we have considered the two series $t\Delta_{x,1}(t)$ and
$t\Delta_{x,2}(t)$, but we have also added the two series $\Delta_{x,1}(t)$
and $\Delta_{x,2}(t)$, to consider the complete effect of the charge, and to
reduce the bias due to direct correlation between $\Delta g$ and the $\delta
V_{i}(t)$ involved in the $\Delta_{x,i}$. This allows us to include just 4
signal time series, hence being able to analyze a wider set of frequencies. We
show in Fig. 18 the results of the second analysis.

FIG. 18:  Decorrelation of the coupling of spurious actuation to TM charging
for run 10, over the [$0.036\text{\,}\mathrm{m}\mathrm{H}\mathrm{z}$,
$0.77\text{\,}\mathrm{m}\mathrm{H}\mathrm{z}$] frequency band. Frequencies are
slightly shifted for clarity. Grey points: ASD of total excess noise over
Brownian, $S_{\Delta g_{e}}^{1/2}$, as in Fig. 2. Red points: Bayesian
posterior $\pm 1\sigma$ interval for the residual ASD after decorrelation of
the spurious actuation along the x axis. Blue points: estimated ASD
$S_{c}^{1/2}$ of the contribution of spurious actuation along the $x$-axis.

The results of Fig. 18 clearly show that this effect is compatible with zero
within the resolution of the measurement. The $1\sigma$ error, on the other
hand, is compatible with a contribution slightly less than $\simeq 1\%$ of
total power at $f\geq f_{5}$, and much less below that. A similar result is
obtained from the analysis of the 24 voltage series.

The posteriors of the susceptibilities are all compatible with zero, except,
perhaps, for one. More specifically we have analyzed, for all the 24
$\alpha_{i}$, the likelihood, that we call $\mathcal{L}_{0}$, assigned by the
posterior to the less likely of the two tails $\alpha_{i}<0$ and
$\alpha_{i}>0$, as a very low likelihood of one of the two tails would
indicate that $\alpha_{i}$ has a well-defined sign. All likelihoods are found
to be larger than 5%, except for one, the voltage applied to one of the
$x$-electrodes of TM1, which is $\mathcal{L}_{0}\simeq 1\%$. For this last
series the susceptibility is $\alpha=(10\pm 5)\text{ fm s}^{-2}/\text{mV}$. To
give an order of magnitude, if the coupling was due to a uniform stray voltage
on the said electrode, this should be $(0.07\pm 0.03)\,\text{V}$, a figure in
the range of observed patch potentials [38].

This slightly significant susceptibility is reflected, in the analysis with a
linearly drifting charge, in an equivalently significant susceptibility to
$\Delta_{x,1}(t)$, that contains the series above, while the susceptibilities
to $t\Delta_{x,1}(t)$ and $t\Delta_{x,2}(t)$ are, within their large
uncertainty, both compatible with zero and with the observed values of the
charging rates [22].

####  VIII.3.2 Magnetic Fields

Below $\sim$1\text{\,}\mathrm{m}\mathrm{H}\mathrm{z}$$, the noise part of our
magnetometer signals, as shown by their almost complete cross-correlation, and
by the absence of any measurable fluctuating gradient [7], was dominated by
the interplanetary magnetic field and had negligible readout noise. Because of
their negligible gradient, the interplanetary field fluctuations interact with
the TMs only as they induce, via the TM residual diamagnetism, a fluctuating
magnetic moment that couples to any static magnetic field gradient at the TM
location. The force due to this interaction is given by:

$\Delta g(t)=\frac{\chi
L^{3}}{M\mu_{0}}\left(\bm{\nabla}B_{x,\text{DC,2}}-\bm{\nabla}B_{x,\text{DC,1}}\right)\cdot\bm{B}(t),$
(17)

where $\bm{\nabla}B_{x,\text{DC},i}$ is the magnetic gradient of the static
$B_{x}$ component averaged over the volume of TMi, $\mu_{0}$ is the vacuum
magnetic permeability, and $\chi=(-3.37\pm 0.15)\times 10^{-5}$ the magnetic
susceptibility of the test masses [39, 40].

Note that the static magnetic gradient may be different from the value that
can be extrapolated, at the test mass location, from the differences of the
magnetometer readings, which typically is
$\simeq$0.5\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{T}\mathrm{/}\mathrm{m}$$.
Indeed a gradient up to
$10\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{T}\mathrm{/}\mathrm{m}$, is
expected to be created by a series of thermistors, containing ferromagnetic
materials, placed on the outer surface of the EH.

These thermistors are too far away from the magnetometers to give any
significant measurable signal. Thus we had no in-flight information on the
gradient that these thermistors were creating at the location of both TMs. A
dedicated, in-flight experiment using oscillating magnetic fields [40] was
able to give an estimate just for $\partial_{x}B_{x,\text{DC},1}$, that is,
just for one of the three required components of the static magnetic gradient
and at the location of just one of the TMs.

This lack of knowledge of the static magnetic gradient difference at the TM
locations left us with only a possible order of magnitude estimate of the
corresponding susceptibilities. Hence, we have performed our decorrelation
analysis with no prior assumptions on the susceptibilities. The results are
shown in Fig. 19.

FIG. 19:  Decorrelation of low frequency magnetic fields for run 10, over the
[$36\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$,
$0.77\text{\,}\mathrm{m}\mathrm{H}\mathrm{z}$] frequency band. Frequencies are
slightly shifted for clarity. Grey points: ASD of total excess noise over
Brownian, $S_{\Delta g_{e}}^{1/2}$, as in Fig. 2. Red points: Bayesian
posterior $\pm 1\sigma$ interval for the residual ASD after decorrelation of
the 3 magnetic field time series. Blue points: estimated ASD $S_{c}^{1/2}$ of
the contribution of magnetic fields to $\Delta g_{e}$.

At all frequencies, $S_{c}(f)$ is significant at $1\sigma$, while the lower
$2\sigma$ quantile of the posterior is negative.
Some significance is also supported by the susceptibilities. In particular,
while $\mathcal{L}_{0}\geq 0.16$ for the $x$ and $z$ components of the field,
$\mathcal{L}_{0}\simeq 0.001$ for the $y$ component. The corresponding
susceptibility is $\alpha_{y}=(-8\pm
2)\,$\mathrm{f}\mathrm{m}\,\mathrm{s}^{-2}\mathrm{/}\mathrm{\SIUnitSymbolMicro}\mathrm{T}$$,
that, if due to a gradient acting on just one of the two TMs, would correspond
to $\lvert\delta\partial_{y}B_{x,\text{DC}}\rvert=(6\pm
2)\,$\mathrm{\SIUnitSymbolMicro}\mathrm{T}\mathrm{/}\mathrm{m}$$, close to
what one would expect because of the magnetic thermistors [40]. The $\delta$
is meant to highlight that the measured quantity is the difference of magnetic
gradients at the two TM locations.

We note that our estimate of the magnetic force noise ASD, based on
decorrelation, is consistent with that reported in [39], which is based
instead on a calculation from measured or estimated values of all the involved
quantities. In particular, besides the in-flight measurements of the magnetic
field fluctuations, that we also use, Ref. [39] uses the aforementioned
measured value for $\partial_{x}B_{x,\text{DC},1}$, while the other components
of $\bm{\nabla}B_{x,\text{DC,}i}$ are estimated based on a model for the
statistical distribution of the magnetic dipoles associated with above
mentioned thermistors.

####  VIII.3.3 Thermal gradients

In addition to single thermistor readout, that we used to form the average
temperature signal, we also had two differential readouts, one for each GRS,
each reading a pair of the thermistors located on the opposite faces of the EH
of the corresponding GRS.

As the drift, which was the source of extra noise on the single readouts, is
largely common mode among the thermistors , these channels were basically
immune to the nonlinearity noise that plagued the average temperature, and
fall then in the category of negligible noise time series.

Thermal gradients are the source of various forces, dominating ones being
radiometer effect and asymmetric outgassing [41]. While for the radiometer
effect the susceptibility can be estimated to within some 30%,
$\alpha_{R}\simeq 2\times
10^{3}\,$\mathrm{f}\mathrm{m}\,\mathrm{s}^{-2}\mathrm{/}\mathrm{K}$$, for
asymmetric outgassing the uncertainty is much larger [41], not better than an
order of magnitude. The results of the decorrelation procedure are shown in
Fig. 20.

FIG. 20:  Decorrelation of thermal gradients for run 10, over the
[$36\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$,
$0.77\text{\,}\mathrm{m}\mathrm{H}\mathrm{z}$] frequency band. Frequencies are
slightly shifted for clarity. Grey points: ASD of total excess noise over
Brownian, $S_{\Delta g_{e}}^{1/2}$, as in Fig. 2. Red points: Bayesian
posterior $\pm 1\sigma$ interval for the residual ASD after decorrelation of
the 2 thermal gradients time series. Blue points: estimated ASD $S_{c}^{1/2}$
of the contribution of thermal gradients to $\Delta g_{e}$.

This contribution is clearly undetectable. Note that the susceptibilities,
$(1\pm 5)\times 10^{3}\text{ fm s}^{-2}/\text{K}$, and $(1\pm 7)\times
10^{3}\text{ fm s}^{-2}/\text{K}$, are zero within errors, errors that
comfortably include the value expected of the radiometric effect. This lack of
contribution is due substantially to the good stability of the thermal
gradient, with an ASD in the considered range of
$\simeq$40\text{\,}\mathrm{\SIUnitSymbolMicro
K}$/\sqrt{\text{Hz}}\sqrt{0.1\,\text{mHz}\,/\,f}$.

###  VIII.4 Modeled contributions from different experiments

As said, besides the disturbances discussed in the previous section, there is
another set of effects that we could estimate from dedicated measurements
separated from the noise measurements. The results of these experiments have
been published in dedicated publications [22, 32, 42]. We list these
disturbances in the following and use the results from those experiments.

  * •

_Actuation gain noise_. This is the largest known contribution to the LPF
noise in $\Delta g_{e}$. It consists of the effect of the gain fluctuations of
the amplifiers that apply the audio-frequency actuation voltages to the TMs.
During LPF operations we performed an extensive experimental campaign to model
this force. The details of the measurements, and of their results are the
subject of a dedicated paper [32]. In Fig. 21, blue points, we report from
[32] the $\pm 1\sigma$ credible interval of the posterior for its contribution
to $S_{\Delta g}$. Note that gain fluctuations also affect rotational
actuation [32], and induce a correlation between $\Delta g$ and
$\Delta\gamma_{\phi}$. The figures in Fig. 21 are in agreement with the
effective cross-talk arm discussed in Sec. V.4.

  * •

_In-band voltage noise_. Actuation voltage fluctuations _within the
measurement frequency band_ induce noisy forces on the test masses by coupling
to their dc counterparts. Results of the measurements on this effect performed
during LPF operations can also be found in [32], and reported again in Fig. 21
as orange points.

  * •

_Random charging_. The effect of noisy charging due to cosmic rays was
estimated, in [22], to be in quantitative agreement with an equivalent Poisson
flow of single elementary charges arriving at a rate $\lambda_{\text{eff}}\sim
1\times 10^{3}\text{ s}^{-1}$. This Poisson charge flow converts into a force
noise through the effective dc-voltage across the TM electrode capacitor
system $\Delta_{x}$, as described in Ref. [22]. Considering residual dc-
voltages – after compensation as in Table 5 – of
$|\Delta_{x,1}|\sim|\Delta_{x,2}|\sim$5\text{\,}\mathrm{m}\mathrm{V}$$, the
effect of random charging noise is shown as brown points in Fig. 21.

  * •

_Laser radiation pressure_ Fluctuation of the radiation pressure of the
measurement laser beam induces a differential force on the TMs. The
measurement beam reflects off TM1 and TM2, with nominal power
$P_{1}=$2.4\text{\,}\mathrm{m}\mathrm{W}$$ and
$P_{2}=$1.2\text{\,}\mathrm{m}\mathrm{W}$$ respectively. Relative fluctuations
$\delta P_{1}/P_{1}$ and $\delta P_{2}/P_{2}$ induce a force $\Delta
g\simeq$8\text{\,}\mathrm{p}\mathrm{m}\,\mathrm{s}^{-2}$\,(\delta
P_{1}/P_{1})+$4\text{\,}\mathrm{p}\mathrm{m}\,\mathrm{s}^{-2}$\,(\delta
P_{2}/P_{2})$. Unfortunately, on LPF there was no direct measurement of the
instantaneous total optical power reflected off the test masses. However, [42]
proposes a thorough analysis of the range of values of the contribution of
this phenomenon to $S_{\Delta g}$. The analysis gets two possible estimates
depending on the (unknown) sign of the correlation between different light
polarizations. In Fig. 21 (green points) we report the overall range spanned
by the $\pm 1\sigma$ uncertainty of both options.

FIG. 21:  Estimated contributions to acceleration noise, for run 10 and within
the [$36\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$,
$0.77\text{\,}\mathrm{m}\mathrm{H}\mathrm{z}$] frequency band, of: actuation
gain fluctuations (blue points, data adapted from [32]); in-band voltage
fluctuations (orange points, data adapted from [32]); cosmic ray charging
fluctuations (brown points, estimate taken from [22]); laser radiation
pressure fluctuations (green points, data adapted from [42]). For reference,
we also report the ASD of the excess noise over Brownian $S_{\Delta
g_{e}}^{1/2}$ as in Fig. 2, red points.

###  VIII.5 Summary of modeled contributions to excess noise

In Fig. 22 we report the ASD of the sum $S_{c,\text{tot}}$ of all
contributions to the excess over Brownian $S_{\Delta g_{e}}$ that we have
found to be, at least at some frequency in the
[$36\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$,
$0.77\text{\,}\mathrm{m}\mathrm{H}\mathrm{z}$] band, statistically different
from zero: tank depletion gravitational noise, magnetic fields, actuation gain
fluctuations (that dominates the sum), in-band voltage fluctuations, random
charging, and laser radiation pressure.

FIG. 22:  Total modeled contributions to $S_{\Delta g_{e}}^{1/2}$, for run 10
in the [$36\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$,
$0.77\text{\,}\mathrm{m}\mathrm{H}\mathrm{z}$] frequency band. Red points: ASD
$S_{\Delta g_{e}}^{1/2}$ of excess over Brownian. Dark blue points: modeled
contributions $S_{c,\text{tot}}^{1/2}$, as described in the text. Light brown
points: posterior for difference $\sqrt{S_{\Delta g_{e}}-S_{c,\text{tot}}}$.
The dashed line is the best fit to the difference with exponent $n=1.00$,
$\sqrt{S_{\Delta
g_{e}}-S_{c,\text{tot}}}=$0.67\text{\,}\mathrm{f}\mathrm{m}\,\mathrm{s}^{-2}\mathrm{/}\sqrt{\text{Hz}}$\,($1\text{\,}\mathrm{mHz}$/f)^{1.00}$.

We have built the posterior for $S_{c,\text{tot}}$, by adding, whenever
available, the samples of the $S_{c}$ posteriors of the various disturbances.
For random charging and laser radiation pressure, for which we only had an
error interval, we have assumed a Gaussian distribution with the $\pm 1\sigma$
interval coinciding with the said error one. Given the smallness of their
contributions, the results are largely independent of the specific choice of
such an equivalent posterior.

From the posterior for $S_{c,\text{tot}}$ and that for $S_{\Delta g_{e}}$, we
have built that for their difference, our best estimate of the residual noise
after subtraction of the effect of all modeled disturbances listed above. The
results are shown again in Fig. 22 as the ASD $\sqrt{S_{\Delta
g_{e}}-S_{c,\text{tot}}}$.

Such an ASD can be fit, in the range $36\mu\text{Hz}\leq f\leq
460\mu\text{Hz}$ to a power law $\widetilde{S}_{0}^{1/2}(1\,\text{mHz}/f)$
(see Fig. 22), with $\widetilde{S}_{0}^{1/2}=(0.67\pm
0.05)\,\text{fm\,s}^{-2}/\sqrt{\text{Hz}}$ and a reduced chi-square $\simeq
1$, which allows a direct comparison with the value of $\widetilde{S}_{\Delta
g_{e}}^{1/2}$ (see Sec. V.2) for run 10, of $\widetilde{S}_{\Delta
g_{e}}^{1/2}=(0.91\pm 0.04)\,\text{fm s}^{-2}/\sqrt{\text{Hz}}$. The ratio of
their square, $\widetilde{S}_{0}/\widetilde{S}_{\Delta g_{e}}=(0.54\pm 0.09)$
gives the fraction of noise power that remains unexplained after our noise
projection, the basic result of the procedure.

We have repeated our decorrelation procedure for the other runs, in order to
assess if some of the observed variability of $\widetilde{S}_{\Delta g_{e}}$
(see Fig. 5) might be due to corresponding variations of some of the
considered disturbances, and we could not find any evidence of that.

Among the disturbances in Sec. VIII.4, actuation gain fluctuations are the
dominating ones. Unfortunately, we cannot assess the variability of those
sources, as they were all determined in dedicated experiments. To justify the
observed 20% variability of $\widetilde{S}_{\Delta g_{e}}$ by a corresponding
variability of those, the latter should be of order 50%.

On the opposite end, if the entire variability is due to the 55% unjustified
fraction, this should fluctuate by some 40%.

Finally, we have not included, in the above analysis, the contributions that
we have found to be compatible with zero: LTP distortion, spurious actuation,
and thermal gradients. While the addition of the latest two would not change
the results in any appreciable way, the case of the distortion is different.
Actually, the sum of the $+1\sigma$ value of $S_{c,\text{tot}}$ in Fig. 22 and
the $1\sigma$ limit for $S_{c,\text{tot}}$ of Fig. 16, falls within $\pm
1\sigma$ credible interval of $S_{\Delta g_{e}}$ for $f\geq f_{5}$.

As said in Sec. VIII.2.1, however, this limit is likely to be significantly
overestimated. The argument discussed in Sec. VIII.2.1, for instance, would
reduce this contribution to no more than some 10% in power at the highest
frequencies.

##  IX  The unexplained excess: possible sources and implications for LISA

The previous section concludes that $(0.54\pm 0.09)$ of the measured excess
noise remains unexplained by the sources for which we had a quantitative
estimate. In this section we discuss the most likely sources of this
unexplained fraction, and the measures one can possibly take to ensure that
they do not compromise LISA performance.

###  IX.1  Patch potential fluctuations

Patch potentials [43] may cause force noise in many ways [38]: through the
interaction of their time fluctuations with any static potential applied to
the TM, including that due to the TM charge; by creating a quasi-static
potential difference that may interact with electronic noise and charge
fluctuations; through the interaction of their quasi-static part with their
own fluctuations.

The first two effects have already been discussed in Sect. VIII.4. An analysis
of the third effect, the self-interaction of the patch fields [44], also based
on the results of the experiments with charge bias [32] in LPF, has appeared
recently. The analysis indicates contamination as the most likely source of
patches, and shows that a model of diffusion-driven fluctuations of
contaminant density, that has been considered to explain self-heating in ion
traps [45], may indeed predict force noise with $\propto 1/f^{2}$ PSD, with an
amplitude in the range of the observed excess, for model parameter values that
are not unreasonable.

Though such scenario remains unproven, its possibility would suggest a few
precautions to be followed in the development of LISA. These have been
discussed in [44], and we report them here for convenience.

First, torsion pendulum experiments with LISA-like TM have achieved
sensitivities [38, 46] that may allow a direct detection of, or a significant
upper limit to the noise we are discussing here. This would require though to
reduce the gaps around the TM to around 1 mm or smaller. Needless to say, a
direct measurement would bring this potential source of noise under full
control.

In the absence of a direct measurement, a measure of precaution is to
investigate the nature and the extent of the adsorbate that may have been
present on the surface of LPF TM and EH during its operations. The objective
is that of keeping TM contamination in LISA close to, or better than that in
LPF.

This requires a campaign of surface characterisation on samples that have
undergone a similar preparation history to that of LPF test masses. A
systematic experimental study, with the Kelvin probe technique, of the quasi-
static distribution of patch potentials, would be an important part of such a
characterisation campaign.

It is reasonable to assume that if no new contaminants are introduced in LISA,
that had not been present in LPF, and if the amount of contamination can be
kept below that of LPF, the noise performance of LPF, that fulfills LISA
requirements, may confidently be reproduced.

###  IX.2 Actuation force calibration and additional voltage noise

An obvious source of excess noise could be an inaccuracy in the calibration of
the applied forces $g_{c}(t)$, which dominate the spectrum at sub-mHz
frequencies; similarly, unaccounted nonlinearities in the applied voltages
time series could lead to inaccuracies in the applied forces time series.
Experiments were carried out in flight to calibrate the actuation system,
against differences between commanded and applied forces/torques. In
particular [18, 32], sinusoidal “calibration tones” of amplitude
$100\text{\,}\mathrm{fN}$ and frequency $10\text{\,}\mathrm{mHz}$ were
injected out-of-loop, effectively inducing a controlled force on TM2. These
experiments yielded fluctuations in the calibration coefficient $<0.5\%$,
which could not account for the observed excess noise.

However, these experiments also led to the discovery of the truncation error,
already introduced in Sec. II.2, [19], which required a deterministic
correction on the actuation voltage amplitudes, of the order of
$10\text{\,}\mathrm{\SIUnitSymbolMicro V}$ RMS, out of a nominal bit
resolution of $153\text{\,}\mathrm{\SIUnitSymbolMicro V}$. This correction was
relevant to the success of the “calibration tone” experiments: the injection
of external sinusoidal forces resulted in an apparently time-varying actuation
gain factor, as well as a series of spurious lines at several harmonics of the
injection frequency. These lines could be effectively suppressed – within
statistical errors – only by taking into account such correction.

Driven by these results, we performed simulations to understand if additional
nonlinearities, smaller than the truncation error RMS value, could result in
increased noise levels in noise-only runs, but at the same time go unnoticed
in calibration tone experiments. We showed that, if the effective amplitudes
of applied voltages $V_{x}$ and $V_{\phi}$ (see [18, 32] for definitions) were
affected by additional small nonlinearities of the order of 1-2
$\mathrm{\SIUnitSymbolMicro V}$ RMS – i.e. of the order of 1/100th of the
nominal bit resolution –, neither the “calibration tone” experiment, nor any
other in-flight experiment could rule out the presence of such nonlinearities.

These nonlinearities would however have a non-negligible impact on noise-only
runs. With forces applied during February 2017, run 10 of Table 1, the
presence of a $1\text{\,}\mathrm{\SIUnitSymbolMicro V}$ RMS or a
$2\text{\,}\mathrm{\SIUnitSymbolMicro V}$ RMS nonlinearity would indeed have a
relevant impact on the total modeled noise as shown in Fig. 23. For each level
of RMS deviation ($1\text{\,}\mathrm{\SIUnitSymbolMicro
V}$-$2\text{\,}\mathrm{\SIUnitSymbolMicro V}$), we simulated 10 realizations
and showed their joint posterior distribution. In Fig. 23, the red points and