# Estimate of force noise from electrostatic patch potentials in LISA
Pathfinder

S Vitale<EMAIL_ADDRESS>Department of Physics, University of Trento,
and Trento Institute for Fundamental Physics and Applications, INFN, 38123
Povo, Trento, Italy  V Ferroni  Department of Physics, University of Trento,
and Trento Institute for Fundamental Physics and Applications, INFN, 38123
Povo, Trento, Italy  L Sala  Department of Physics, University of Trento, and
Trento Institute for Fundamental Physics and Applications, INFN, 38123 Povo,
Trento, Italy  W J Weber  Department of Physics, University of Trento, and
Trento Institute for Fundamental Physics and Applications, INFN, 38123 Povo,
Trento, Italy

(April 12, 2024)

###### Abstract

This paper discusses force noise in LISA and LISA Pathfinder arising from the
interaction of patch potentials on the test mass and surrounding electrode
housing surfaces with their own temporal fluctuations. We aim to estimate the
contribution of this phenomenon to the force noise detected in LISA Pathfinder
in excess of the background from Brownian motion. We introduce a model of that
approximates the interacting test mass and housing surfaces as concentric
spheres, treating patch potentials as isotropic stochastic Gaussian processes
on the surface of these spheres. We find that a scenario of patches due to
surface contamination, with diffusion driven density fluctuations, could
indeed produce force noise with the observed frequency $f^{-2}$ dependence.
However there is not enough experimental evidence, neither from LISA
Pathfinder itself, nor from other experiments, to predict the amplitude of
such a noise, which could range from completely negligible to explaining the
entire noise excess. We briefly discuss several measures to ensure that this
noise is sufficiently small in LISA .

##  1  Introduction

Electrostatic patch potentials are among the possible sources of force
disturbance in LISA and LISA Pathfinder (LPF) [1]. In particular they may be a
candidate source to explain part of the excess noise observed in LPF, above
the Brownian noise level [2], in the frequency band $0.02-1\text{ mHz}$.

Forces from the stray electrostatic fields produced by patch potentials, have
been a leading driver for both the design and operations of the system of the
test mass (TM) and surrounding electrode housing (EH), known collectively as
the gravitational reference sensor (GRS). This includes, for instance, the
adoption of relatively large (2.9-4 mm) TM/EH gaps [3], the avoidance of
applied DC voltages in the electrostatic actuation system [4], and the in-
flight measurement and compensation of the averaged stray field that couples
TM charge into force [5, 6]. Our quantitative modelling and experimental
understanding of the relevant stray potentials and the resulting forces is
quite good for stray potentials over relatively macroscopic scales; force
measurement upon controlled modulation of TM charge or electrode potentials
allows measurement of the relevant stray potentials averaged over,
respectively, the size of the TM face ($\approx 2000$ mm2) or that of an
electrode ($\approx 500$ mm2). As such, coupling of patch potentials, both
steady and noisy, to TM charge and actuation voltages is well modelled and
projected to the LPF and LISA performance [5, 6, 4, 7].

The effect of variations of patch fields on microscopic levels, smaller than
the electrodes or even gap dimensions, is much less weakly constrained by
experiment. Force noise from the “patch-to-patch” interactions on smaller
scales are independent of TM charge or averaged electrode potentials, and thus
both inaccessible in the modulation experiments mentioned above and always
present. Quantifying the force noise contribution from the interaction of the
quasi-static patch fields with their own temporal fluctuations requires
modelling, in addition to experimental probes, and this is the subject of this
current paper.

In the literature the problem has been analysed either within the model of two
parallel infinite conducting planes, covered by a layer of dipoles and
separated by a gap [8], or within one in which a set of random shaped plane
capacitors, formed by tiles of the surface of the cubic shaped LISA test-mass
(TM), face the equivalent tiles on the inner surface of the electrode housing
(EH) surrounding the TM [5].

The first of the two models above allows a quantitative parametrization of the
patch potential distribution, but lacks the essential topology of a finite TM
fully enclosed within a cavity. This is particularly relevant in the case of
LISA, where gaps between the TM and the EH may be as large as 10% of the TM
size, so that the infinite plane model may seriously lack in
representativeness.

The second model is limited in its assignment of equipotential domains and by
the inaccuracy of approximations in the relevant capacitance derivatives for
domains that are smaller than the gap separating test mass from electrode
housing.

In the attempt of overcoming these limitations, we investigate a model wherein
the TM and the EH are two concentric spheres separated by a vacuum gap. The
shape of the TM and EH is not the actual one, but the basic topology is, thus
the model may be considered as a topological improvement of the infinite
planes one.

The spherical geometry allows taking advantage of the expansion of the
electrostatic potentials into spherical harmonics, in analogy with the
potential Fourier expansion of Ref. [8]. The expansion in spherical harmonics
has been discussed in [9], though with an aim and an approach significantly
different from ours. We will thus work out the necessary equations from the
first principles.

Our spherical model also presents limitations in describing the electrostatic
interaction with a cubic test mass. For instance, patch fields on a spherical
TM produce no torque sensitivity to the total TM charge, while the same is not
true for a cubic TM, with a non-zero torque sensitivity which in fact has been
directly measured [5, 4, 10]. Similarly, the spherical geometry does not allow
simple implementation of electrodes with variable voltages, which would allow
simulation of force and torque experiments with modulated voltages. However,
we will show that the concentric sphere model does allow to draw useful,
quantitative conclusions on the patch-to-patch force noise. Future studies
with finite element modelling will be useful to further anchor these
conclusions to the realistic cubic geometry.

The paper is structured as follows: in Sect. 2 we describe the concentric
sphere model and its key parameters. In Sect. 3 we summarise the spherical
harmonic formalism for the calculation of the electrostatic potential within
our model. In Sect 4 we use these results to calculate the force on the inner
sphere, i.e. on the TM, as a function of the amplitude of the spherical
harmonic components. In Sect. 5 we parametrise the statistical properties of
such spherical harmonic amplitudes, and calculate the statistical properties
of the force noise. In Sect. 6 we apply the derived framework to the analysis
of a series of experiment related to patch potentials, performed on LPF by
means of a properly applied TM charge bias. In Sect. 7 we finally use our
model, and the information form LPF, and other relevant experiments reported
in the literature, to evaluate the force noise that may have been caused by
the patch potential fluctuations in LPF. Sect 8 gives a few concluding
remarks, especially related to the impact of our findings for the LISA
mission.

##  2  The concentric spheres model

In the model we discuss here, we treat the TM outer surface and the EH inner
surface, as two concentric, spherical conducting surfaces of radius $R_{in}$
and $R_{out}$ respectively (see Fig. 1 ). Both surfaces are assumed to be
covered by a continuous layer of dipoles, each dipole being radially oriented.
Finally, the inner surface is electrical floating while the outer is grounded.

When performing numerical calculation we adopt a inner sphere with the same
area as the cubic LISA test mass $R_{in}=L\sqrt{6/(4\pi)}\simeq 32\text{ mm}$,
with $L=46\text{ mm}$ the side-length of the LISA cubic TM. This way the inner
sphere has approximately the same surface area as the original TM. We will
also take $R_{out}=R_{in}+d=35.5\text{ mm}$, with $d=3.5\text{ mm}$ an
approximate average, over the three faces of the TM, of the gaps between TM
and EH in LISA.

Figure 1: Schematics of the test-mass - electrode housing system: two
concentric spherical surfaces of radius $R_{in}$ and $R_{out}=R_{in}+d$
respectively.

##  3  Spherical harmonic analysis of electrostatic potential

###  3.1  Potential of a spherical surface dipole distribution.

The expansion of the Green function for the Laplace equation, in _real_
spherical harmonics of spherical coordinates $r$, $\theta$, and $\phi$, is:

$\begin{split}&G\left(r_{1},\theta_{1},\phi_{1},r_{2},\theta_{2},\phi_{2}\right)=\frac{1}{\sqrt{r_{1}^{2}+r_{2}^{2}-2r_{2}r_{1}\left(\sin\left(\theta_{1}\right)\sin\left(\theta_{2}\right)\cos\left(\phi_{1}-\phi_{2}\right)+\cos\left(\theta_{1}\right)\cos\left(\theta_{2}\right)\right)}}=\\\
&=\sum_{l=0}^{\infty}\frac{4\pi}{2l+1}\sum_{m=-l}^{l}\frac{r_{{\scriptscriptstyle<}}^{l}}{r_{\scriptscriptstyle>}^{l+1}}Y^{-m}_{l}(\theta_{1},\phi_{1})Y^{m}_{l}(\theta_{2},\phi_{2})\end{split}$
(1)

where $r_{{\scriptscriptstyle<}}$ is the smallest between $r_{1}$ and $r_{2}$,
and $r_{{\scriptscriptstyle>}}$ the largest.

If a surface charge distribution $\sigma\left(\theta,\phi\right)$ covers the
surface of a sphere of radius $R$, the electrical potential
$V\left(r,\theta,\phi\right)$ at the point of spherical coordinates $r$,
$\theta$, and $\phi$ is :

$\begin{split}&V\left(r,\theta,\phi\right)=\\\
&=\frac{1}{\epsilon_{o}R}\sum_{l=0}^{\infty}\frac{1}{2l+1}\sum_{m=-l}^{l}\left(\frac{R}{r}\right)^{l+1}Y^{m}_{l}(\theta,\phi)c_{l,m}\text{
for }r>R\\\
&=\frac{1}{\epsilon_{o}R}\sum_{l=0}^{\infty}\frac{1}{2l+1}\sum_{m=-l}^{l}\left(\frac{r}{R}\right)^{l}Y^{m}_{l}(\theta,\phi)c_{lm}\text{
for }r<R\\\ \end{split}$ (2)

with

$c_{l,m}=R^{2}\int_{0}^{\pi}\int_{0}^{2\pi}Y^{m}_{l}(\theta^{\prime},\phi^{\prime})\sigma\left(\theta^{\prime},\phi^{\prime}\right)\sin{\theta^{\prime}}d\theta^{\prime}d\phi^{\prime},$
(3)

the spherical harmonic amplitudes of $R^{2}\sigma(\theta,\phi)$.

Spatially varying potential on the surface of conductors have been described
by a distribution of electrostatic dipoles normal to the surface, with surface
density $D$ [11]. For a conducting sphere, we can treat this as two surface
charge distributions of opposite polarity, $\sigma_{+}(\theta,\phi)$ and
$\sigma_{-}(\theta,\phi)$ on two spheres of radius $R+\delta/2$ and
$R-\delta/2$ respectively, and then take the limit for $\delta\to 0$. As the
total charge must be zero, we assume that
$\sigma_{+}(\theta,\phi)=\left(R^{2}/(R+\delta/2)^{2}\right)\sigma(\theta,\phi)$
and
$\sigma_{-}(\theta,\phi)=-\left(R^{2}/(R-\delta/2)^{2}\right)\sigma(\theta,\phi)$

Then the potential inside the inner sphere is

$\begin{split}&V\left(r,\theta,\phi\right)=\\\ &=\lim_{\delta\to
0}\frac{1}{\epsilon_{o}}\sum_{l=0}^{\infty}\frac{1}{2l+1}\sum_{m=-l}^{l}r^{l}Y^{-m}_{l}(\theta,\phi)c_{lm}\times\\\
&\times\left(\frac{1}{\left(R+\delta/2\right)^{l+1}}-\frac{1}{\left(R-\delta/2\right)^{l+1}}\right)=\\\
&=-\frac{1}{\epsilon_{o}R^{2}}\sum_{l=0}^{\infty}\frac{l+1}{2l+1}\sum_{m=-l}^{l}\left(\frac{r}{R}\right)^{l}Y^{m}_{l}(\theta,\phi)b_{l,m}\end{split}$
(4)

with the dipole _harmonic amplitudes_ $b_{l,m}$ defined by:

$\begin{split}&b_{l,m}=\\\ &\lim_{\delta\to
0}R^{2}\int_{0}^{\pi}\int_{0}^{2\pi}Y^{m}_{l}(\theta^{\prime},\phi^{\prime})\left(\sigma\left(\theta^{\prime},\phi^{\prime}\right)\times\delta\right)\sin(\theta^{\prime})d\theta^{\prime}d\phi^{\prime}\equiv\\\
&\equiv
R^{2}\int_{0}^{\pi}\int_{0}^{2\pi}Y^{m}_{l}(\theta^{\prime},\phi^{\prime})D\left(\theta^{\prime},\phi^{\prime}\right)\sin(\theta^{\prime})d\theta^{\prime}d\phi^{\prime}\end{split}$
(5)

which defines the surface dipole density as

$D\left(\theta,\phi\right)=\lim_{\delta\to
0}\left(\sigma\left(\theta,\phi\right)\times\delta\right)=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\frac{b_{l,m}}{R^{2}}Y_{l}^{m}(\theta,\phi)$
(6)

The formula could have been obtained by deriving the second line of Eq. 2
relative to $R$. A similar calculation for the potential outside the sphere
gives

$\begin{split}&V\left(r,\theta,\phi\right)=\\\
&=\frac{1}{\epsilon_{o}R^{2}}\sum_{l=0}^{\infty}\frac{l}{2l+1}\sum_{m=-l}^{l}\left(\frac{R}{r}\right)^{l+1}Y^{m}_{l}(\theta,\phi)b_{l,m}\end{split}$
(7)

Eqs. 4 and 7 show that the potential has a discontinuity at $r=R$ given by

$\Delta
V\left(r,\theta,\phi\right)=\frac{1}{\epsilon_{o}R^{2}}\sum_{l=0}^{\infty}\sum_{m=-l}^{l}Y^{m}_{l}(\theta,\phi)b_{l,m}=\frac{D\left(r,\theta,\phi\right)}{\epsilon_{o}}$
(8)

###  3.2  Images

If the sphere, on which the dipole distribution lies, is conducting, there
will be an induced surface charge that can be calculated by the method of
images. Table 1 summarises the key features of the images of a point charge,
and of a point radial dipole, in the presence of a conducting spherical
surface of radius $R$.

Table 1: Charge and dipole images on a grounded conducting sphere of radius
$R$ of a point charge and a point dipole with radial orientation. Relations
hold for both $R>r$ and $R<r$. For a neutral sphere, instead of a grounded
one, and $R<r$, the total charge of images is neutralised by an opposite point
charge in the center.
$\begin{array}[]{|c|c|c|c|c|c|}\hline\cr\vrule\lx@intercol\hfil\text{Source}\hfil\lx@intercol\vrule\lx@intercol&\vrule\lx@intercol\hfil\text{Image}\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\text{radial}&\text{charge}&\text{dipole}&\text{radial}&\text{charge}&\text{dipole}\\\
\hline\cr\text{position}&\text{}&\text{}&\text{position}&\text{}&\text{}\\\
\hline\cr r&q&-&R^{2}/r&-qR/r&-\\\ \hline\cr
r&-&p&R^{2}/r&pR/r^{2}&pR^{3}/r^{3}\\\ \hline\cr\end{array}$

Now assume that our radially aligned dipole, of dipole moment $p$, lies in
between our two concentric spheres at a position $R_{s}$, with $R_{in}\leq
R_{s}\leq R_{out}$. The dipole will produce images on both spheres, images
that will be in turn reflected by the spheres. We calculate with Mathematica
the sequence in Table 2

Table 2: Positions and intensities of images of a radial dipole of moment $p$
sitting at position $R_{s}$ between two conducting spheres of radius $R_{in}$
and $R_{out}$ respectively
$\begin{array}[]{|c|c|c|c|c|c|}\hline\cr\vrule\lx@intercol\hfil\text{Inner
images}\hfil\lx@intercol\vrule\lx@intercol&\vrule\lx@intercol\hfil\text{Outer
images}\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\text{Position}&\text{Dipole}&\text{Charge}&\text{Position}&\text{Dipole}&\text{Charge}\\\
\hline\cr\frac{R_{in}^{2}}{R_{s}}&\frac{pR_{in}^{3}}{R_{s}^{3}}&\frac{pR_{in}}{R_{s}^{2}}&\frac{R_{out}^{2}}{R_{s}}&\frac{pR_{out}^{3}}{R_{s}^{3}}&\frac{pR_{out}}{R_{s}^{2}}\\\
\frac{R_{in}^{2}R_{s}}{R_{out}^{2}}&\frac{pR_{in}^{3}}{R_{out}^{3}}&0&\frac{R_{out}^{2}R_{s}}{R_{in}^{2}}&\frac{pR_{out}^{3}}{R_{in}^{3}}&0\\\
\frac{R_{in}^{4}}{R_{out}^{2}R_{s}}&\frac{pR_{in}^{6}}{R_{out}^{3}R_{s}^{3}}&\frac{pR_{in}^{2}}{R_{out}R_{s}^{2}}&\frac{R_{out}^{4}}{R_{in}^{2}R_{s}}&\frac{pR_{out}^{6}}{R_{in}^{3}R_{s}^{3}}&\frac{pR_{out}^{2}}{R_{in}R_{s}^{2}}\\\
\frac{R_{in}^{4}R_{s}}{R_{out}^{4}}&\frac{pR_{in}^{6}}{R_{out}^{6}}&0&\frac{R_{out}^{4}R_{s}}{R_{in}^{4}}&\frac{pR_{out}^{6}}{R_{in}^{6}}&0\\\
\frac{R_{in}^{6}}{R_{out}^{4}R_{s}}&\frac{pR_{in}^{9}}{R_{out}^{6}R_{s}^{3}}&\frac{pR_{in}^{3}}{R_{out}^{2}R_{s}^{2}}&\frac{R_{out}^{6}}{R_{in}^{4}R_{s}}&\frac{pR_{out}^{9}}{R_{in}^{6}R_{s}^{3}}&\frac{pR_{out}^{3}}{R_{in}^{2}R_{s}^{2}}\\\
\frac{R_{in}^{6}R_{s}}{R_{out}^{6}}&\frac{pR_{in}^{9}}{R_{out}^{9}}&0&\frac{R_{out}^{6}R_{s}}{R_{in}^{6}}&\frac{pR_{out}^{9}}{R_{in}^{9}}&0\\\
\frac{R_{in}^{8}}{R_{out}^{6}R_{s}}&\frac{pR_{in}^{12}}{R_{out}^{9}R_{s}^{3}}&\frac{pR_{in}^{4}}{R_{out}^{3}R_{s}^{2}}&\frac{R_{out}^{8}}{R_{in}^{6}R_{s}}&\frac{pR_{out}^{12}}{R_{in}^{9}R_{s}^{3}}&\frac{pR_{out}^{4}}{R_{in}^{3}R_{s}^{2}}\\\
\frac{R_{in}^{8}R_{s}}{R_{out}^{8}}&\frac{pR_{in}^{12}}{R_{out}^{12}}&0&\frac{R_{out}^{8}R_{s}}{R_{in}^{8}}&\frac{pR_{out}^{12}}{R_{in}^{12}}&0\\\
\hline\cr\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\ \hline\cr\end{array}$

###  3.3 Potential due to images

Take now a surface distribution of radial dipoles on a sphere of radius
$R_{s}$, with surface density given by Eq. 6, with $R=R_{s}$ Each elementary
dipole of intensity
$p=\frac{b_{l,m}}{R_{s}^{2}}Y_{l}^{m}(\theta,\phi)R_{s}^{2}\sin{\theta}d\phi
d\theta$, at position $\theta$ and $\phi$ on the sphere will have its own
sequence of images with intensity and radial positions given in Table 2, and
the same angular position as the elementary dipole.

As a consequence the entire surface distribution will produce images
consisting of surface distribution of charge and dipoles, with the same
angular dependance of the original distribution, with amplitudes scaled from
the original by the same factor by which $p$ is scaled in Table 2, and finally
lying on a sphere with radius given again by Table 2.

Note that, in principle, one should neutralise the inner sphere by adding a
charge opposite to the total charge of the images. However, for $l>0$,
$\int_{0}^{\pi}\int_{0}^{2\pi}Y_{l}^{m}(\theta,\phi)\sin(\theta)d\theta
d\phi=0$, and carry no net charge.

As for the component with $l=0$, that is a spherically symmetric dipole
density, it requires no image, and its only effect is to change the potential
of the inner, electrically floating, conducting surface by a constant, without
inducing any surface charge on either of the spheres. Its presence is
irrelevant for force calculations so that, from now on, all sums will start
from $l=1$.

To the original distribution, and to all its images, one can apply the
formulas of Sect. 3.1, and calculate the resulting potential. The angular
dependance will be the same for all images, while the radial dependance will
produce a term proportional either to $r^{l}$ or to $r^{-(l+1)}$.

These can be grouped together for all images. We obtain, with the help of
Mathematica:

$\begin{split}&V_{I}(r,\theta,\phi)=\frac{1}{R_{in}^{2}\epsilon_{o}}\sum_{l=1}^{\infty}\frac{1}{2l+1}\sum_{m=-l}^{l}b_{l,m}Y_{l}^{m}(\theta,\phi)\left(\frac{R_{in}}{R_{s}}\right)^{l+2}\times\\\
&\times\left(\frac{\left((l+1)R_{in}^{2l+1}+lR_{s}^{2l+1}\right)}{R_{in}^{2l+1}-R_{out}^{2l+1}}\left(\frac{r}{R_{in}}\right)^{l}-\right.\\\
&\left.-\frac{\left((l+1)R_{out}^{2l+1}+lR_{s}^{2l+1}\right)}{R_{in}^{2l+1}-R_{out}^{2l+1}}\left(\frac{R_{in}}{r}\right)^{l+1}\right)\end{split}$
(9)

One can check that

$\begin{split}&\lim_{r\to R_{in}}V_{I}(r,\theta,\phi)=\\\
&=\frac{1}{R_{s}^{2}\epsilon_{o}}\sum_{l=1}^{\infty}\frac{1}{2l+1}\sum_{m=-l}^{l}b_{l,m}Y_{l}^{m}(\theta,\phi)(l+1)\left(\frac{R_{in}}{R_{s}}\right)^{l}\end{split}$
(10)

which is the opposite of the potential created by the original distribution at
$r=R_{in}$, while, equivalently,

$\begin{split}&\lim_{r\to R_{out}}V_{I}(r,\theta,\phi)=\\\
&=-\frac{1}{R_{s}^{2}\epsilon_{o}}\sum_{l=1}^{\infty}\frac{1}{2l+1}\sum_{m=-l}^{l}b_{l,m}Y_{l}^{m}(\theta,\phi)l\left(\frac{R_{s}}{R_{out}}\right)^{l+1}\end{split}$
(11)

again the opposite of the potential created by the original distribution at
$r=R_{out}$

###  3.4 Total potential for the actual configuration

We will now consider two surface distributions of dipoles, one with
$R_{s}=R_{in}$ and one with $R_{s}=R_{out}$, with coefficients $b_{l,m}^{in}$
and $b_{l,m}^{out}$ respectively, with aim of calculating the force they
produce on the inner sphere. To calculate the force we need two potentials:

  * •

The total potential generated by all sources on the surface of the inner
conducting sphere. This is needed to calculate the total electric field at the
conductor surface, and, from that, the resulting surface charge density.
Notice that the potential due to the dipoles on the inner sphere must be
calculated by taking $R_{s}>R_{in}$ and only at the end of any calculation
take the limit $R_{s}\to R_{in}$.

  * •

The potential due to sources on the outer spheres only. These include the
dipoles lying on the inner surface of the outer sphere, and the images located
at $R_{s}>R_{out}$. This is needed to calculate the field and its gradient
that exert forces on the dipoles and the charges lying on the surface of the
inner sphere.

The first term, the total potential at $r\simeq R_{in}$ can be derived from
Eqs. 4, 7, and 9 to be

$\begin{split}&V_{tot}(r,\theta,\phi)=\frac{1}{R_{in}^{2}\epsilon_{o}}\sum_{l=1}^{\infty}\sum_{m=-l}^{l}Y_{l}^{m}(\theta,\phi)\times\\\
&\times\left(\left(\frac{\alpha^{2l+1}}{1-\alpha^{2l+1}}+\frac{l+1}{2l+1}\right)b_{l,m}^{in}+\frac{\alpha^{l+2}}{1-\alpha^{2l+1}}b_{l,m}^{out}\right)\times\\\
&\times\left(\left(\frac{R_{in}}{r}\right)^{l+1}-\left(\frac{r}{R_{in}}\right)^{l}\right)\end{split}$
(12)

with $\alpha=R_{in}/R_{out}\simeq 0.90$, while the second is

$\begin{split}&V_{out}(r,\theta,\phi)=-\frac{1}{R_{\text{in}}^{2}\epsilon_{o}}\sum_{l=1}^{\infty}\sum_{m=-l}^{l}Y_{l}^{m}(\theta,\phi)\times\\\
&\left(\frac{\alpha^{2l+1}}{1-\alpha^{2l+1}}b_{l,m}^{in}+\frac{\alpha^{l+2}}{1-\alpha^{2l+1}}b_{l,m}^{out}\right)\left(\frac{r}{R_{in}}\right)^{l}\end{split}$
(13)

Before concluding this section it is useful to establish a connection between
the formulas above, and the potentials one would measure just outside the
dipole distribution lying on the surface of the inner sphere, or just inside
the surface of the outer sphere, the real ‘patch potentials’.

For the former, as the potential due to the dipoles on the outer sphere is
independently canceled by images at the inner sphere, one needs to evaluate
the contribution of the dipoles on the inner sphere only.

This can be obtained by adding the potential in Eq. LABEL:eq:limrin, evaluated
for $R_{s}\to R_{in}$ and $b_{l,m}=b_{l,m}^{in}$, to that in Eq. 7, evaluated
for $R\to R_{in}$, $r\to R_{in}$, and $b_{l,m}=b_{l,m}^{in}$. Such calculation
gives:

$\begin{split}&V_{K}(\theta,\phi)=\frac{1}{R_{in}^{2}\epsilon_{o}}\sum_{l=1}^{\infty}\sum_{m=-l}^{l}b_{l,m}^{in}Y_{l}^{m}(\theta,\phi)=\frac{D^{in}(\theta,\phi)}{\epsilon_{o}}\end{split}$
(14)

with $D_{in}(\theta,\phi)$ the surface density of dipoles on the inner sphere.
This result is consistent with the starting assumption that patch potentials
can be described as a surface distribution of dipoles.

Note that the result is independent of $R_{out}$, and thus that would be the
potential also in the case $R_{out}\to\infty$, that is in the case of an
isolated floating sphere covered by a dipole distribution. This may represent
the case of an isolated sample one would investigate with a Kelvin probe, to
detect the spatial variation of the potential, as done in Ref. [12]. This is
the reason for the suffix $K$ in Eq. 14

A similar calculation gives for the potential just inside the outer sphere:

$\begin{split}&V^{\prime}_{K}(\theta,\phi)=-\frac{1}{R_{out}^{2}\epsilon_{o}}\sum_{l=1}^{\infty}\sum_{m=-l}^{l}b_{l,m}^{out}Y_{l}^{m}(\theta,\phi)=-\frac{D^{out}(\theta,\phi)}{\epsilon_{o}}\end{split}$
(15)

with $D^{out}(\theta,\phi)$ the surface dipole density on the outer sphere.

One can check that the solution of Laplace equation for the potential in the
gap between the spheres, with the boundary conditions expressed by Eqs. 14
and15, matches that obtained by summing up the potential due to the images and
that due to the original dipole distributions, a useful test of consistency
for our calculations.

##  4  Field, gradient and force on the inner sphere

###  4.1 General formula for the force

To evaluate the component of the force on the inner sphere along some
direction, we calculate the effect of the field and field gradient due to
sources on the outer sphere, on the dipole and charge distributions lying on
the surface of the inner sphere.

The alternative approach, preferred in planar geometries, of calculating the
derivative of the electrostatic energy with respect to a displacement of the
inner sphere along the selected direction leads to cumbersome formulas, as
such a displacement would break the spherical symmetry.

As the system is statistically isotropic, we can pick any of the three
Cartesian component. In spherical coordinates calculations are significantly
easier for the z component that we calculate as:

$\begin{split}&F_{z,c}=R_{in}^{2}\int_{0}^{\pi}\int_{0}^{2\pi}\left(D^{in}(\theta,\phi)\partial_{r}E_{out,z}(\theta,\phi)+\right.\\\
&\left.+\sigma(\theta,\phi)E_{out,z}(\theta,\phi)\right)\sin(\theta)d\theta
d\phi\end{split}$ (16)

In Eq. 16 we have used:

  * •

the surface density of dipoles sitting on the inner sphere, the explicit
expansion of which we repeat here for convenience:

$D^{in}(\theta,\phi)=\sum_{l=1}^{\infty}\sum_{m=-l}^{l}\frac{b_{l,m}^{in}}{R_{s}^{2}}Y_{l}^{m}(\theta,\phi);$
(17)

  * •

the surface charge due to images on the inner sphere

$\sigma(\theta,\phi)=\epsilon_{o}E_{tot,r}(\theta,\phi)\equiv=-\epsilon_{o}\left(\frac{\partial
V_{tot}(r,\theta,\phi)}{\partial r}\right)_{r=R_{in}};$ (18)

  * •

the z component of the field created, at the surface of the inner sphere, by
the dipoles and images on the outer sphere:

$\begin{split}&E_{out,z}(\theta,\phi)=-\hat{z}\cdot\left(\vec{\nabla}V_{out}(r,\theta,\phi)\right)_{r=R_{in}}=\\\
&=-\left(\cos(\theta)\frac{\partial V_{out}(r,\theta,\phi)}{\partial
r}-\right.\\\ &\left.\sin(\theta)\frac{1}{r}\frac{\partial
V_{out}(r,\theta,\phi)}{\partial\theta}\right)_{r=R_{in}};\end{split}$ (19)

  * •

and, finally, the radial derivative of the above

$\begin{split}&\partial_{r}E_{out,z}(\theta,\phi)=\\\
&=-\left(\cos(\theta)\frac{\partial^{2}V_{out}(r,\theta,\phi)}{\partial
r^{2}}-\right.\\\ &\left.-\sin(\theta)\frac{\partial}{\partial
r}\left(\frac{1}{r}\frac{\partial
V_{out}(r,\theta,\phi)}{\partial\theta}\right)\right)_{r=R_{in}}.\end{split}$
(20)

###  4.2 Charge density from images

Differentiating the potential in Eq. 12 and following Eq. 18 and 19, we
obtain:

$\begin{split}&\sigma(\theta,\phi)=\frac{1}{R_{in}^{3}}\sum_{l=1}^{\infty}\sum_{m=-l}^{l}Y_{l}^{m}(\theta,\phi)\times\\\
&\times\frac{\left(l\alpha^{2l+1}+l+1\right)b_{l,m}^{\text{in}}+(2l+1)\alpha^{l+2}b_{l,m}^{\text{out}}}{1-\alpha^{2l+1}}\end{split}$
(21)

###  4.3 Field and its radial derivative

As for $E_{out,z}(\theta,\phi)$ one can check with Mathematica that

$\begin{split}&\cos(\theta)\frac{\partial
r^{l}Y_{l}^{m}(\theta,\phi)}{\partial r}-\frac{\sin(\theta)}{r}\frac{\partial
r^{l}Y_{l}^{m}(\theta,\phi)}{\partial\theta}=\\\
&=r^{l-1}\sqrt{\frac{2l+1}{2l-1}}\sqrt{l^{2}-m^{2}}Y_{l-1}^{m}(\theta,\phi)\end{split}$
(22)

From this it follows that

$\begin{split}&E_{out,z}(\theta,\phi)=\frac{1}{R_{in}^{3}\epsilon_{o}}\sum_{l=1}^{\infty}\sum_{m=-(l-1)}^{l-1}\left(\frac{\alpha^{2l+1}}{1-\alpha^{2l+1}}b_{l,m}^{in}+\right.\\\
&\left.+\frac{\alpha^{l+2}}{1-\alpha^{2l+1}}b_{l,m}^{out}\right)\sqrt{\frac{2l+1}{2l-1}}\sqrt{l^{2}-m^{2}}Y_{l-1}^{m}(\theta,\phi),\end{split}$
(23)

and that

$\begin{split}&\partial_{r}E_{out,z}(\theta,\phi)=\frac{1}{R_{in}^{4}\epsilon_{o}}\sum_{l=1}^{\infty}\sum_{m=-(l-1)}^{l-1}\left(\frac{\alpha^{2l+1}}{1-\alpha^{2l+1}}b_{l,m}^{in}+\right.\\\
&\left.+\frac{\alpha^{l+2}}{1-\alpha^{2l+1}}b_{l,m}^{out}\right)\sqrt{\frac{2l+1}{2l-1}}\sqrt{l^{2}-m^{2}}(l-1)Y_{l-1}^{m}(\theta,\phi)\end{split}$
(24)

Note that in the sum in Eq. 24, the term with $l=1$ is zero due to the
multiplication by $l-1$. Thus the first term contains $Y_{1}^{m}$ and would
exert no force on the $Y_{0}^{0}$ component of the dipole surface density of
the inner sphere. This confirms the legitimacy of ignoring the term with $l=0$
in all calculations.

###  4.4 Final formula for the force

One can now evaluate the integral in Eq 16. To do that we will first consider
that $Y_{l}^{m}=0$ for both $|m|>l$ and $l=-1$, so that in Eqs. 23 and 24 it
is better to make the substitution $l\to l+1$. Using then the orthonormality
of spherical harmonics one gets

$\begin{split}&F_{z}=\\\
&=\frac{1}{R_{\text{in}}^{4}\epsilon_{o}}\sum_{l=1}^{\infty}\sum_{m=-l}^{l}\frac{\sqrt{(2l+3)(2l+1)((l+1)^{2}-m^{2})}}{\left(1-\alpha^{2l+1}\right)\left(1-\alpha^{2l+3}\right)}\alpha^{3+l}\times\\\
&\times\left(\alpha^{l}b_{l,m}^{in}b_{l+1,m}^{in}+b_{l,m}^{in}b_{l+1,m}^{out}+\right.\\\
&\left.+\alpha^{2l+2}b_{l,m}^{out}b_{l+1,m}^{in}+\alpha^{l+2}b_{l,m}^{out}b_{l+1,m}^{out}\right)\end{split}$
(25)

To describe patch potential fluctuations, we will assume in the following that
the dipole densities are time-dependent and that the harmonic amplitudes are
then functions of time.

##  5  Statistical distribution of harmonic amplitudes, and autocorrelation of
the force

We need now to discuss the possible distributions of the harmonic amplitudes
$b_{l,m}^{in}$ or $b_{l,m}^{out}$ that would reproduce a corresponding
realistic distribution of patches. We will discuss first the static part of
those amplitudes.

###  5.1 Isotropic random fields on a sphere

The idea of distributions consisting of spherical harmonics with random
amplitudes is not new. One can find a vast literature on _isotropic gaussian
random fields on the sphere_ (see for instance [13]). These are random fields
defined over the surface of a unit sphere, for which, among other things, the
autocorrelation function at two points on the sphere, with coordinates
$\\{\theta_{1},\phi_{1}\\}$ and $\\{\theta_{2},\phi_{2}\\}$ respectively, is
just a function of the spherical angle
$\omega=\arccos\left(\cos(\theta_{1})\cos(\theta_{2})+\sin(\theta_{1})\sin(\theta_{2})\cos(\phi_{1}-\phi_{2})\right)$
between the two points.

As discussed in [13], if our surface dipole density is one of these processes,
the autocorrelation can be expanded as:

$\begin{split}&\langle
D\left(\theta_{1},\phi_{1}\right)D\left(\theta_{2},\phi_{2}\right)\rangle=\\\
&=\frac{1}{R^{4}}\sum_{l=0}^{\infty}\frac{2l+1}{4\pi}A_{l}\text{P}_{l}\left(\cos(\omega)\right)=\\\
&=\frac{1}{R^{4}}\sum_{l=0}^{\infty}\sum_{m=-l}^{l}A_{l}Y_{l}^{m}(\theta_{1},\phi_{1})Y_{l}^{m}(\theta_{2},\phi_{2}),\end{split}$
(26)

The second equality is based on the well known expansion of Legendre
polynomials of the cosine of the spherical angle, into complex spherical
harmonics of the two spherical positions, but it can be easily shown to hold
also for real spherical harmonics.

One can then check, by using Eq. 5, that $\langle
b_{l,m}b_{j,n}\rangle=A_{l}\delta_{l,j}\delta_{m,n}$, thus the $A_{l}$’s are
positive numbers that play the role of a power spectral density (PSD) as a
function of the ‘spherical frequency’ $l$.

Note that the division by $R^{4}$ on the right hand side of the first line,
and on the second line of Eq. 26, is necessary to obtain such a direct
relation between $\langle b_{l,m}^{2}\rangle$ and $A_{l}$.

If the dipole surface density is time varying but stationary, in the sense of
stochastic processes, then Eq. 26 becomes

$\begin{split}&\mathcal{R}_{D}(\omega,\Delta t)\equiv\langle
D\left(\theta_{1},\phi_{1},t\right)D\left(\theta_{2},\phi_{2},t+\Delta
t\right)\rangle=\\\
&=\frac{1}{R^{4}}\sum_{l=0}^{\infty}\frac{2l+1}{4\pi}A_{l}(\Delta
t)\text{P}_{l}\left(\cos(\omega)\right)\end{split}$ (27)

From Eq. 27 one can easily calculate that, as $P_{l}(1)=1$,

$\begin{split}&\langle D(\theta,\phi,t)D(\theta,\phi,t+\Delta
t)\rangle=\frac{1}{R^{4}}\sum_{l=0}^{\infty}\frac{2l+1}{4\pi}A_{l}(\Delta
t)\end{split}$ (28)

The cross coherence of the two time-domain, statistically identical stochastic
processes $D\left(\theta_{1},\phi_{1},t\right)$ and
$D\left(\theta_{2},\phi_{2},t\right)$ is then given by:

$\begin{split}&\frac{\langle D(\theta,\phi,t)D(\theta,\phi,t+\Delta
t)\rangle}{\sqrt{\langle D^{2}(\theta,\phi,t)\rangle\langle
D^{2}(\theta,\phi,t+\Delta t)\rangle}}=\\\
&=\frac{\sum_{l=0}^{\infty}\frac{2l+1}{4\pi}A_{l}(\Delta
t)\text{P}_{l}\left(\cos(\omega)\right)}{\sum_{l=0}^{\infty}\frac{2l+1}{4\pi}A_{l}(\Delta
t)}\end{split}$ (29)

Eq. 27 allows to fix the scale of the $A_{l}(0)$ by linking it to root mean
square fluctuation, at any given time, of patch potentials over the surface,
something you may measure with a Kelvin probe. This, provided that the
measurement is faster than the typical time scale on which $A(l)$ changes
significantly, or in general, if the time variations can be neglected.

Indeed, by using Eq. 14, one gets

$\begin{split}&V_{K,rms}^{2}\equiv\langle
V_{K}(\theta,\phi,t)^{2}\rangle=\frac{1}{\epsilon_{o}^{2}}\langle{D^{in}(\theta,\phi,t)}^{2}\rangle=\\\
&=\frac{1}{\epsilon_{o}^{2}R_{in}^{4}}\sum_{l=0}^{\infty}\frac{2l+1}{4\pi}A_{l}(0)\end{split}$
(30)

These relations between voltage fluctuations and the $A_{l}(0)$ suggest to
define $V_{l}^{2}(\Delta t)\equiv A_{l}(\Delta
t)/(\epsilon_{o}^{2}R_{in}^{4})$, a coefficient the use of which makes many of
the formulas more transparent.

###  5.2 Autocorrelation of force

We now calculate the autocorrelation of force on the inner sphere, assuming
that the both the inner and outer dipole distribution are stationary and
isotropic stochastic processes with $b_{0,0}^{in}(t)=b_{0,0}^{out}(t)=0$,
$\langle b_{l,m}^{in}(t)\rangle=\langle b_{l,m}^{out}(t)\rangle=0$, $\langle
b_{l,m}^{in}(t)b_{j,n}^{in}(t+\Delta t)\rangle=\alpha^{4}\langle
b_{l,m}^{out}(t)b_{j,n}^{out}(t+\Delta t)\rangle=A_{l}(\Delta
t)\delta_{l,j}\delta_{m,n}$, and finally $\langle
b_{l,m}^{in}(t)b_{j,n}^{out}(t+\Delta t)\rangle=0$. The correction by radius
ratio $\alpha=R_{in}/R_{out}$ is needed to give same dipole surface density to
both spheres.

With these prescriptions, as $\langle b_{l,m}^{in}\rangle=\langle
b_{l,m}^{out}\rangle=0$, as $\langle b_{l,m}^{in}b_{l+1,m}^{in}\rangle=0$, and
same for the outer dipoles, then $\langle F_{z}\rangle=0$. The force has
instead a non zero autocorrelation given by:

$\begin{split}&\mathcal{R}_{F}(\Delta t)\equiv\langle F_{z}(t)F_{z}(t+\Delta
t)\rangle=\\\
&\frac{1}{R_{\text{in}}^{8}\epsilon_{o}^{2}}\sum_{l=1}^{\infty}K(l)A_{l}(\Delta
t)A_{l+1}(\Delta t)=\\\
&=\epsilon_{o}^{2}\sum_{l=1}^{\infty}K(l)V_{l}^{2}(\Delta t)V_{l+1}^{2}(\Delta
t)\end{split}$ (31)

with

$\begin{split}&K(l)=(2l+1)(2l+3)\left(\sum_{m=-l}^{l}\left((l+1)^{2}-m^{2}\right)\right)\times\\\
&\times\left(\frac{\alpha^{l+3}}{(1-\alpha^{2l+1})(1-\alpha^{2l+3})}\right)^{2}\times\\\
&\times\left((\alpha^{l})^{2}+\alpha^{-4}\left((\alpha^{2l+2})^{2}+1\right)+\alpha^{-8}(\alpha^{l+2})^{2}\right)\end{split}$
(32)

which gives

$\begin{split}&K(l)=\frac{1}{3}(l+1)(4l(l+2)+3)^{2}\frac{\alpha^{2l+2}\left(1+\alpha^{2l}\right)\left(1+\alpha^{2l+4}\right)}{\left(1-\alpha^{2l+1}\right)^{2}\left(1-\alpha^{2l+3}\right)^{2}}\end{split}$
(33)

Before picking a spectral shape for $A_{l}$ and calculating $\langle
F_{z}(t)F_{z}(t+\Delta t)\rangle$, it is interesting to look at the dependence
on $l$ of the ‘transfer’ factor $K(l)$, which we show in Fig. 2, for the value
of the gap between the spheres, $d=3.5\text{ mm}$, we are using here, but
also, for the sake of comparison, for narrower and wider gaps.

Figure 2: The transfer factor $K(l)$ defined in Eq. 33, as a function of the
harmonic number $l$. Black continuous line: the case for the gap value
$d=3.5\text{ mm}$ we are using in all calculations. Dashed coloured lines: the
case for both wider and narrower gaps as shown in the legend

For all values of the gap, $K(l)$ reaches a maximum, and decays quite rapidly
after that. By defining $l_{100}$, the value of $l$ for which $K(l)$ attains
$\simeq 1\%$ of such a maximum, we find that, within, at worst, a 10%
accuracy, $l_{100}\simeq 2.4\pi R_{in}/d$, with $l_{100}=72$ in our case of
$d=3.5\text{ mm}$.

Thus when the ‘wavelength’ $\pi R_{in}/(l/2)$ of the $l$-th harmonic
component, becomes as large as $\simeq 1.2d$, the force effect is already
strongly suppressed. Crudely speaking, this is because smaller wavelengths
correspond to smaller patches, and the electric field lines due to patches
smaller than the gap close on the adjacent patches on the same sphere rather
than crossing the gap and closing somewhere on the other sphere. This
obviously suppresses the electrical interaction between the spheres.

Anyway this result shows that under realistic hypotheses, numerical
calculations can be safely truncated at $l$ a few times $l_{100}$

As a final note, the force is bilinear in the stochastic processes
$b_{l,m}^{in}(t)$ and $b_{l,m}^{out}(t)$. So, even if these are Gaussian, the
force in principle is not. However the force results from the incoherent sum
of many processes, and thus, unless just one or two dominate its fluctuation,
the central limit theorem should guarantee that it may be considered Gaussian
for the purposes of the present work.

###  5.3 The measurement of quasi-static components

The experiments on patch potentials we have mentioned above [6] [5] [12],
report values for a dc part of the patch potentials and/or of the force they
induce.

It is important to realise that instead, within the model we are discussing
here, as both $\langle V_{K}(t)\rangle=0$ and $\langle F_{z}(t)\rangle=0$,
there is strictly speaking no static part of both the potential and the force.

However a measurement of

$\overline{F}_{z}(t)=\frac{1}{T}\int_{t}^{t+T}F_{z}(t)dt$ (34)

though still with zero mean value, has autocorrelation

$\begin{split}&\langle\overline{F}_{z}(t)\overline{F}_{z}(t+\Delta
t)\rangle=\frac{1}{T^{2}}\int_{t}^{t+T}\int_{t+\Delta t}^{t+\Delta t+T}\langle
F_{z}(t)F_{z}(t^{\prime})\rangle dt\;dt^{\prime}=\\\
&=\frac{1}{T}\int_{-T}^{T}\mathcal{R}_{F}\left(\Delta t^{\prime}-\Delta
t\right)\left(1-\frac{\lvert\Delta t^{\prime}\rvert}{T}\right)d\Delta
t^{\prime}\end{split}$ (35)

.

Thus a single sample will be a random number from a distribution with a width
$D_{\overline{F}}=\sqrt{\langle\overline{F}_{z}(t)\overline{F}_{z}(t)\rangle}$,
varying over a time scale set by the evolution of $\mathcal{R}_{F}\left(\Delta
t\right)$ with $\Delta t$.

Such a picture is not in contrast with the experimental evidence that has
often witnessed very long term evolution of the results of nominally static
measurements [12] [6]. We will discuss this aspect again when examining the
details of the LPF experiment.

##  6 LPF experiments with charge bias, and the implications for $A_{l}(\Delta
t)$

Let us now discuss how the LPF measurements on patch potentials can be
imported in our model, with the aim of using their results to derive
$V_{l}^{2}(\Delta t)$.

###  6.1 Charge bias in the concentric sphere model and in LPF

In LPF patch potentials were investigated by applying a static charge bias $Q$
to the TM and by measuring both the resulting static force, and the resulting
increase of the force noise [6].

In our model, a charge $Q$ on the TM creates a uniform charge density on its
surface $\sigma_{Q}=Q/(4\pi
R_{in}^{2})=Y_{0}^{0}(\theta,\phi)2\sqrt{\pi}/(4\pi R_{in}^{2})$. This surface
density couples to $E_{z,out}$ in Eq. 23, to create a force:

$\begin{split}&F_{Q,z}(t)=Q\sqrt{\frac{3}{4\pi}}\frac{1}{R_{in}^{3}\epsilon_{o}}\frac{\alpha^{3}}{1-\alpha^{3}}\left(b_{1,0}^{in}(t)+b_{1,0}^{out}(t)\right)\end{split}$
(36)

To compare this formula with that used in the relevant LPF literature,
[5],[6], [4], it is useful to define the effective, rescaled potential
difference $\Delta
V_{K,1}(t)=\frac{b_{1,0}^{in}(t)+b_{1,0}^{out}(t)}{\epsilon_{o}R_{in}^{2}}$,
the autocorrelation of which is $\langle\Delta V_{K,1}(t)\Delta
V_{K,1}(t+\Delta t)\rangle=\frac{1+\alpha^{4}}{\alpha^{4}}V_{1}^{2}(\Delta
t)$. With such a definition:

$\begin{split}&F_{Q,z}(t)=\frac{Q\Delta
V_{K,1}(t)}{R_{in}}\sqrt{\frac{3}{4\pi}}\frac{\alpha^{3}}{1-\alpha^{3}}\end{split}$
(37)

In LPF, the coupling $dF_{Q,z}/dQ$ was measured by introducing a step change
in the TM charge, through the UV discharge system [CHARGE2017], and then
applying the appropriate compensation voltage to the four identical electrodes
facing, in pairs, the two opposite TM x-faces, to null the effect.

The resulting compensation force scales as
$dF_{Q,z}^{c}/dQ=-Q\Delta_{x}(1/C_{T})(\partial C_{X}/\partial x)$, where
$C_{T}$ is the total capacitance of the TM to ground, $C_{X}$ is that of one
of the electrodes relative to the TM, and where $\Delta_{x}$ is the sum of the
bias voltages applied to the electrode pair on one x-face, minus the sum of
the voltages applied to the opposite pair.

Equating this equation to Eq. 37 we get

$\begin{split}&\Delta
V_{K,1}(t)=\frac{R_{in}}{\sqrt{\frac{3}{4\pi}}\frac{\alpha^{3}}{1-\alpha^{3}}}\frac{1}{C_{T}}\frac{\partial
C_{X}}{\partial x}\Delta_{x}(t)\equiv\beta\Delta_{x}(t)\end{split}$ (38)

with $\beta\simeq 0.2$

By using Eq. 38, we may use the measurement of $\Delta_{x}$ from [6] and [4],
to infer the properties of $V_{1}^{2}(\Delta t)$.

###  6.2 LPF charge bias experiment: quasi static force, noise increase and
the estimate of $V_{1}^{2}(\Delta t)$

LPF performed both measurements of static values of $\Delta_{x}$, at several
points over $\simeq 15$ months of the mission [6], and a measurement of the
in-band noise in $\Delta_{x}$ [6, 4]. The quasi-static measurements of
$\Delta_{x}$ were performed with controlled steps in the TM charge, as
mentioned above, while the in-band noise was measured by detecting the force
noise increase with a highly charged TM ($\simeq 2\times 10^{8}e$, or $\simeq
1$ V TM bias). Let’s discuss first the results of the first of these two
measurements.

$\Delta_{x}$ on each of the step was measured by averaging over $T\simeq
10^{4}\text{ s}$. Measured values, over a roughly 15 month span, displayed
variations of several mV for each TM, around average values observed to be -20
mV and +1 mV for the two TM [6].

Similar long term variations of the average quasi-static potentials on
centimetre scale samples have been consistently observed in all experiments on
patch potentials [12][10][5]. They have also been observed to be sensitive to
the vacuum environment [12], something that is commonly taken as an indication
that patch potentials, in these kinds of systems, are dominated by the
deposition of volatile surface contaminants.

Before proceeding further we need to clarify one point. Within the model we
are discussing here, as $\langle V_{K}(t)\rangle=0$ there is strictly speaking
no static part in any of the random quantities we are dealing with here,
including $\Delta_{x}(t)$. However, or averaged measurement:

$\overline{\Delta}_{x}(t)=\frac{1}{T}\int_{t}^{t+T}\Delta_{x}(t^{\prime})dt^{\prime}$
(39)

though still with zero mean value, has autocorrelation

$\begin{split}&\langle\overline{F}_{z}(t)\overline{F}_{z}(t+\Delta
t)\rangle=\\\ &=\frac{1}{T^{2}}\int_{t}^{t+T}\int_{t+\Delta t}^{t+\Delta
t+T}\langle\Delta_{x}(t)\Delta_{x}(t^{\prime})\rangle dt\;dt^{\prime}=\\\
&=\frac{1}{T}\int_{-T}^{T}\mathcal{R}_{\Delta_{x}}\left(\Delta
t^{\prime}-\Delta t\right)\left(1-\frac{\lvert\Delta
t^{\prime}\rvert}{T}\right)d\Delta
t^{\prime}\simeq\mathcal{R}_{\Delta_{x}}\left(\Delta t\right)\end{split}$ (40)

where $\mathcal{R}_{\Delta_{x}}(\Delta t$ is the autocorrelation of
$\Delta_{x}(t)$, and where the last approximation hold for a
$\mathcal{R}_{\Delta_{x}}(\Delta t)$ decaying on a much longer time than $T$.

In this last scenario, of a very slowly decaying autocorrelation, a single
sample of $\overline{\Delta}_{x}(t)$ will be a random number from a
distribution with a width $\sqrt{\mathcal{R}_{\Delta_{x}}(0)}$, while the
relative fluctuation from one sample to another one, measured after a time
$\Delta t$, will be of the order $\sqrt{1-\mathcal{R}_{\Delta_{x}}(\Delta
t)/\mathcal{R}_{\Delta_{x}}(0)}$. Such a picture seems consistent with the
long term variations we were discussing above.

Coming back to LPF measurements, we have done a Bayesian fit of the LPF
results 111We use the data reported in [CHARGE2017], plus an additional
unpublished data point that had not been processed at the time of the
publication of that paper assuming that for both TM $\Delta_{x}(t)$ is a
stochastic process with autocorrelation given either by
$\langle\Delta_{x}(t)\Delta_{x}(t+\Delta
t)\rangle=\sigma_{\Delta_{x}}^{2}e^{-\lvert\Delta
t\rvert/\tau}\equiv\sigma_{\Delta_{x}}^{2}f_{1}(\Delta t,\tau)$ or by
$\langle\Delta_{x}(t)\Delta_{x}(t+\Delta
t)\rangle=\sigma_{\Delta_{x}}^{2}e^{-\lvert\Delta
t\rvert/\tau}\left(1+\lvert\Delta
t\rvert/\tau\right)\equiv\sigma_{\Delta_{x}}^{2}f_{2}(\Delta t,\tau)$.

The (single-sided) Fourier transforms of $f_{1}(\Delta t,\tau)$ and
$f_{2}(\Delta t,\tau)$ are given by $f_{1}(f,\tau)=4\tau/(1+(2\pi f\tau)^{2})$
and $f_{2}(f,\tau)=8\tau/(1+(2\pi f\tau)^{2})^{2}$ respectively. This means
that the corresponding PSD $\sigma_{\Delta_{x}}^{2}f_{1}(f,\tau)$ and
$\sigma_{\Delta_{x}}^{2}f_{2}(f,\tau)$, for $f\gg(1/\tau)$, are $\propto
1/f^{2}$ and $\propto 1/f^{4}$ respectively.

We obtain $\sigma_{\Delta_{x}}=9.1_{-2.0}^{+3.0}\text{ mV}$ and
$\tau=\left(1.8_{-0.7}^{+0.5}\right)\times 10^{8}\text{ s}$ for the first
model and $\sigma_{\Delta_{x}}=14._{-4.}^{+9.}\text{ mV}$ and
$\tau=\left(3.7_{-1.3}^{+2.0}\right)\times 10^{7}\text{ s}$ for the second.
Considering the long time constants the two models indeed produce noise with
$1/f^{2}$ and $1/f^{4}$ PSD in the LISA band above 0.1 mHz.

A posterior predictive check, based on the likelihood itself gives p$>0.45$
for both models.

Turning now to the in-band excess noise induced by charge bias, the
acceleration noise has been found, in LPF, to indeed increase upon a constant
charge bias.

The PSD of equivalent the noise in $\Delta_{x}$, $S_{\Delta_{x}}(f)$, scales
as $1/f$, with an ASD of $S_{\Delta_{x}}^{1/2}(0.1\text{ mHz})\simeq
100\;\mu\text{V}/\sqrt{\text{Hz}}$. The $1/f$ scaling is observed both for the
force and torque coupling to the TM charge. The measurements are consistent
with uncorrelated additive voltage noise on the 4 x-faces electrodes, such as
that due the actuation electronics, and are quantitatively compatible with the
ground tests of such electronics noise [4].

On the other hand, if the in-band fluctuations of the patch potentials were
responsible for a substantial part of the measured total $1/f^{2}$
acceleration noise excess measured in LPF, then there would be a $1/f^{2}$
patch potential contribution to the above in-band noise power. A Bayesian
estimate on how large such a component could be in the observed noise, gives a
$2-\sigma$ upper limit $S_{\Delta_{x}}^{1/2}(f)\leq
65\;\mu\text{V}/\sqrt{\text{Hz}}(0.1\text{ mHz}/f)$.

Note that the two models for the quasi static force measurements discussed
above, despite having similar posterior predictive power, have an important
difference in their ability to predict the fluctuations of $\Delta_{x}$ within
the LPF measurement band.

The purely exponential model projects a noise ASD for $\Delta_{x}$ at 0.1 mHz
of $S_{\Delta_{x}}^{1/2}(f=0.1\text{ mHz})=2.2_{-0.6}^{+0.9}\text{
mV}/\sqrt{\text{Hz}}$, a factor 10 larger than the upper limit above, while
the second, much more ‘red’ model, projects instead
$S_{\Delta_{x}}^{1/2}(f=0.1\text{
mHz})=0.5_{-0.2}^{+0.3}\;\mu\text{V}/\sqrt{\text{Hz}}$, a completely
negligible contribution to $S_{\Delta_{x}}(f)^{1/2}$.

Thus, all in all, the results of these observation may be translated into the
following model for the coefficient $V_{1}^{2}(\Delta t)$

$\begin{split}&V_{1}^{2}(\Delta
t)=\frac{\alpha^{4}}{1+\alpha^{4}}\langle\Delta V_{K,1}(t)\Delta
V_{K,1}(t+\Delta t)\rangle=\\\
&=\beta^{2}\frac{\alpha^{4}}{\alpha^{4}+1}\left\langle\Delta_{x}(t)\Delta_{x}(\text{$\Delta$t}+t)\right\rangle=\\\
&=V_{dc,1}^{2}f_{2}(\Delta t,\tau_{dc})+V_{n,1}^{2}(\Delta t)\end{split}$ (41)

with parameter values given in Table 3, and with $V_{n,1}^{2}(\Delta t)$ a
term representing the putative in-band noise fluctuations, with Fourier
transform $S_{V_{n,1}}(f)=S_{o}(0.1\text{ mHz}/f)^{2}$. From
$S_{\Delta_{x}}^{1/2}(f)\leq 65\;\mu\text{V}/\sqrt{\text{Hz}}(0.1\text{
mHz}/f)$, and
$S_{V_{n,1}}(f)=\beta^{2}\alpha^{4}/(\alpha^{4}+1)S_{\Delta_{x}}(f)$,
$S_{o}^{1/2}\leq 8\;\mu\text{V}/\sqrt{\text{Hz}}$

Table 3: Parameter values for Eq. 41
$\begin{array}[]{|c|c|}\hline\cr\text{Parameter}&\text{Value}\\\ \hline\cr
V_{dc,1}&1.8_{-0.6}^{+1.2}\text{ mV}\\\
\hline\cr\tau_{dc}&\left(3.7_{-1.3}^{+2.0}\right)\times 10^{7}\text{ s}\\\
\hline\cr\end{array}$

An obvious way of modelling $V_{n,1}^{2}(\Delta t)$ is $V_{n,1}^{2}(\Delta
t)=V_{n,1}^{2}f_{1}(\Delta t,\tau_{n})$. The parameters in this formula must
obey various constraints. First, to produce a $1/f^{2}$ tail from $(1+(2\pi
f\tau_{n})^{2})^{-2}$ in the PSD, within the full LPF band down to about
$\simeq 20\;\mu\text{Hz}$, $2\pi(0.01\text{ mHz })\tau_{n}\gtrsim 2$, that is
$\tau_{n}\gtrsim 1.5\times 10^{4}\text{ s}$.

In addition, if this noisy part must be negligible relative to the term
$V_{dc,1}^{2}f_{2}(\Delta t,\tau_{dc})$, then both $V_{n,1}\ll V_{dc,1}$ and
$\tau_{n}<\tau_{dc}$. Finally, to obey the constraint on the maximum PSD
compatible with observations, $4V_{n,1}^{2}/((2\pi(0.1\text{
mHz})^{2}\tau_{n})\leq(0.008\text{ mV}/\sqrt{\text{Hz}})^{2}$.

Thus:

$\begin{split}&V_{n,1}\leq\frac{S_{o}^{1/2}(2\pi\;0.1\text{
mHz})}{2}\sqrt{\tau_{dc}}\simeq\\\ &\simeq 3\times 10^{-6}\text{
mV}\times\sqrt{\text{Hz}}\times\sqrt{\tau_{dc}}\end{split}$ (42)

a number that turns out to be $\leq 0.003V_{dc,1}$ for the entire posterior
behind Table 3, thus fulfilling also $V_{n,1}\ll V_{dc,1}$

##  7 Projection of force noise in LPF in the absence of charge bias

We now use the results of the preceding section to estimate the force noise
due to patch potentials in LPF. As the experimental results above only give
information on $V_{1}^{2}(\Delta t)$, our estimate will require some
modelling, and will be conditional to the specific model adopted. We will
discuss a few possibilities, based on the available literature.

###  7.1 Some model assumptions

We will assume that, as for the case of $l=1$, the potential has both a quasi-
static component, and an independent, much smaller noise component,
responsible for the patch potential fluctuation within the LPF frequency band.

In addition we will assume that the slow drift behaviour observed in
$V_{1}^{2}(\Delta t)$, is shared by the entire quasi-static component, while
we leave open the frequency dependence of the single harmonic amplitudes of
the noise component. We will discuss further the validity of such an
assumption based on available evidence.

Under this assumption, the potential $V_{K}(\theta,\phi,t)$ has
autocorrelation

$\begin{split}&\mathcal{R}_{V_{K}}(\omega,\Delta t)=f_{2}(\Delta
t,\tau_{dc})\sum_{l=0}^{\infty}\frac{2l+1}{4\pi}V_{dc,l}^{2}P_{l}(\cos(\omega))+\\\
&+\sum_{l=0}^{\infty}\frac{2l+1}{4\pi}V_{n,l}^{2}(\Delta
t)P_{l}(\cos(\omega))\equiv\\\ &\equiv f_{2}(\Delta
t,\tau_{dc})\mathcal{R}_{dc}(\omega)+\mathcal{R}_{n}(\omega,\Delta
t)\end{split}$ (43)

where $V_{n,l}^{2}(\Delta t)\ll V_{dc,l}^{2}f_{2}(\Delta t,\tau_{dc})$.

Within this model, to linear terms in $\mathcal{R}_{n}(\omega,\Delta t)$, the
autocorrelation of the the noisy part of the force becomes

$\begin{split}&\mathcal{R}_{F}(\Delta t)=\epsilon_{o}^{2}f_{2}(\Delta
t,\tau_{dc})\sum_{l=1}^{\infty}K(l)\left(V_{dc,l}^{2}V_{n,l+1}^{2}(\Delta
t)+\right.\\\ &\left.+V_{dc,l+1}^{2}V_{n,l}^{2}(\Delta t)\right)\end{split}$
(44)

plus a term proportional to the sum of terms containing
$V_{dc,l}^{2}V_{dc,l+1}^{2}F_{2}^{2}(\Delta t,\tau_{dc}$. One can check that
under all hypotheses for the distribution of $V_{dc,l}^{2}$ discussed in the
following, the contribution to the force PSD of the Fourier transform of this
term, which is $\propto 1/f^{6}$, is completely negligible.

###  7.2 Models for $V_{dc,l}$

Let us now discuss possible models for $V_{dc,l}$ and then for
$\mathcal{R}_{dc}(\omega)$, the angular autocorrelation of the quasi-static
patch potentials.

Not all isotropic autocorrelation functions that are viable on a flat surface
are allowed on a sphere [15]. Indeed, some of the coefficients
$2\pi\int_{0}^{\pi}\mathcal{R}(\omega)P_{l}\left(\cos(\omega)\right)\sin(\omega)d\omega$,
of the expansion of a generic one, $\mathcal{R}(\omega)$, in Legendre
polynomials, may turn to be negative and cannot then be taken as the variance
of the relative random harmonic amplitude.

Fortunately the exponential
$\mathcal{R}(\omega)=\delta^{2}e^{-\omega/\omega_{o}}$, with $\delta^{2}$ the
variance and $\omega_{o}$ a positive angular scale factor, the simplest model
to describe correlation up to a given angular scale, is a viable one. For such
an autocorrelation, the coefficients of the expansion are independent of $l$
up to $\simeq 1/\omega_{o}$, and decay rapidly after that. If such a roll-off
is well above $l\simeq 75$, where the coefficient $K(l)$ begins to display a
significant attenuation then in practice $V_{dc,l}$ may be considered as a
constant in the calculations of the force autocorrelation.

To try to understand if a simple exponential model may describe the actual
distribution of patches, and to get an idea of the length scale of the related
exponentials, we now try a comparison with the observations available in the
literature.

We start noting that there is an analytic relation between the variance and
the coefficient for $l=1$. If $\mathcal{R}_{dc}(\omega)\simeq
V_{K,rms}^{2}e^{-\omega/\omega_{dc}}$, we calculate that

$V_{dc,1}^{2}=V_{K,rms}^{2}2\pi\frac{\left(1-e^{-\frac{\pi}{\omega_{dc}}}\right)\omega_{dc}^{2}}{1+4\omega_{dc}^{2}}$
(45)

Thus from our estimate of $V_{dc,1}$ we can get an estimate of $V_{K,rms}$ as
a function of the coherence length $\omega_{dc}$.

For $\omega_{dc}\ll 1/2$, and taking the most likely value for $V_{dc,1}$,
$V_{dc,1}=1.8\text{ mV}$, the formula predicts $V_{K,rms}\simeq 23\text{
mV}\times\left(1\text{ mm}/(R_{in}\omega_{dc})\right)$.

For the $\simeq 0.5\;\mu\text{m}$ kind of correlation length shown in the
measurement of Ref. [16], this would predict $V_{K,rms}\simeq 40\text{ V}$
with a peak to peak variation, over the kind of samples considered there, in
excess of 100 V. This is orders of magnitude larger than the observed
fluctuations, that span at most 250 mV.

Thus the small scale patches detected in [16] are not related to the potential
observed in LPF. In addition, if these have exponential autocorrelation, it is
unlikely that the correlation length may be in the micrometer range.

A more detailed comparison can be made to the results of Ref. [12], that
reports various scanning of 1 cm$\times$1 cm gold-coated samples. The scans
were performed at a much coarser resolution than those of Ref. [16], namely at
steps of 0.5 mm in both directions, and with a 3 mm tip, which should give a
spatial resolution of a fraction of $\pm 3\text{ mm}/2$ 222N. Robertson,
private communication.

In particular Ref. [12] reports, in _their_ Fig. 3 the contour plots for two
Kelvin probe scanning of two nominally equal Au-coated samples taken at a
pressure of $\simeq 10^{-7}\text{ mbar}$, not too different from that of LPF,
a few months apart. The plots are quite different, one (marked February 2004)
being dominated by some long range gradient-like pattern, the other (August
2004) having a more random appearance.

We have qualitatively reproduced both scans, with the results shown in Fig 3
and a brief description of the simulation input here below. We stress that the
simulation involves also the algorithm to produce the contour plot from the
data, the results of which may affect the appearance of the plot. Thus the
comparison is just merely qualitative.

Figure 3: Simulated patch potential distributions to be compared with the
Kelvin probe scans for the two Au-coated samples shown in Ref. [12]. Left:
simulated patch distribution with an exponential autocorrelation function
$\mathcal{R}_{dc}=V_{K,rms}^{2}e^{-\omega/\omega_{o}}$, with
$R_{in}\omega_{o}=10\text{ mm}$, and $V_{K,rms}$ adjusted to give
$V_{dc,1}=1.7mv$. A random number from a Gaussian distribution with
$\sigma_{V}=1\text{ mV}$ has been superimposed to each sample of the simulated
scan. This plot is intended to mimic the upper left panel of Fig.3 of Ref.
[12]. Right: contour plot for a matrix of random Gaussian numbers with
$\sigma_{V}=1.7\text{ mV}$. This plot is intended to mimic the middle left
panel of Fig.3 of Ref. [12]. Both plots show a projection of the domain of the
TM surface with $R_{in}\theta\in[R_{in}\pi/2-5\text{ mm},R_{in}\pi/2+5\text{
mm}]$ and $R_{in}\phi\in[-5\text{ mm},+5\text{ mm}]$.

Nevertheless, the August 2004 scan is reasonably well reproduced by a matrix
of $20\times$20 independent random numbers extracted by a Gaussian
distribution with $\sigma_{V}=1.7$ mV. Such would be the case both if the scan
is limited by the noise of the technique, or if the effective resolution is
better than the scanning step of 0.5 mm. In the second case, the scan would
result from the averaging of a distribution with a correlation length $\ll
0.5$mm, for which $V_{dc,1}\simeq 70\;\mu$V.

The February 2004 scan is reproduced instead with a distribution of patches
with a correlation length $R_{in}\omega_{o}=10$ mm, and $V_{dc,1}=1.7$ mV,
like in LPF, superimposed to a random field, like in the previous case, but
this time with $\sigma_{V}\simeq 1$ mV.

The implication of the coincidence in the value of $V_{dc,1}$ in this
simulation and that in LPF should not be overstated but still is an indication
of a similar range of phenomena. The lower value of $\sigma_{V}$, relative to
the February scan, may be an indication that only part of the distribution in
the latter might be due to instrumental uncertainty.

We want also to point out that Ref. [12] also observed a voltage bias of the
entire cm-sized sample in the hundreds of mV scale, in addition to those shown
in the pictures. This bias was observed to change randomly, again by hundred
mV, for the same sample, both from one installation into the vacuum chamber to
the other, but also in different days but within the same installation.

These fluctuations are not included in the distribution we used for the
simulations in Fig. 3 and, if due to patch potential and not to some other
instrumental artefact, must be included by hand.

As already mentioned, this may be a sign that samples get almost uniformly
coated by contaminants, with small relative density/composition variations
that account for the fluctuations of Fig. 3.

A series or recent studies with torsion pendulums also provide some valuable
information. A linear scanning of a gold plated torsion pendulum planar blob,
in steps of 0.125 mm, and with a probe 5 mm wide, showed mV variation on a
$\simeq 5\text{ mm}$ scan [18]. A crude periodogram analysis of the reported
data gives indeed a $1/\nu^{2}$ behaviour, with $\nu$ the spatial frequency.
This is what one would expect from an exponential autocorrelation, if the
coherence length is longer than $\simeq 5\text{ mm}/(2\pi)\simeq 0.8\text{
mm}$, the inverse of the spectral resolution.

In addition, a recent measurement of the force gradient between a gold-coated
plate suspended as a torsion pendulum element and a conducting membrane of cm
dimensions [19], points to a patch scale size $\simeq 0.5\text{ mm}$.

Finally, in discussing the available evidence on the distribution of
$V_{dc,l}$, it is worth mentioning the observations of the patch induced
torques on the gyroscopes of the Gravity Probe B (GP-B) mission [9]. Such
observations, that refer though to Nb-coated spheres, are compatible with
$V_{dc,l}\propto l^{-n}$ with $n>0$.

Our tentative conclusion from this overview is that $\mathcal{R}_{dc}(\omega)$
in systems likely dominated by contamination, may be compatible with an
exponential shape, but the coherence length must be significantly larger than
micrometers. The absolute scale of the phenomenon seems to put $V_{K,rms}$ in
the 1-10 mV range, probably depending on the nature and the extent of the
contaminants.

Thus, one can obtain a cautious upper limit for the force autocorrelation, by
assuming a constant value for $V_{dc,l}^{2}$, $V_{dc,l}^{2}=V_{dc,1}^{2}$,
instead of a decaying one because of the cut-off in the exponential or because
of some power law behaviour.

###  7.3 Models for $V_{n,l}^{2}(\Delta t)$

Let us now review the available evidence on $\mathcal{R}_{n}(\omega,\Delta t)$
or, more in general, on the possible in-band fluctuations of the higher
harmonics of the patch distribution.

Experiments with torsion pendulums [5, 10, 18], that suffer from the same
limitation of resulting from mm-to cm-wide surface average measurements as for
LPF, seem to give slightly conflicting information.

In particular Ref. [5], a measurement of the torsional analogue of
$\Delta_{x}$ on a LISA-like TM, did not detect any significant patch noise
down to 0.1 mHz, and could only put an upper limit for its ASD, an upper limit
completely superseded by that form LPF discussed above.

Ref. [10], using a plate shaped test-mass close (0.1-1 mm) to a metal plate,
did instead detect some noise, with a ‘surprisingly white’ (in the words of
the authors) ASD down to 0.1 mHz and $1/f$ tail in the ASD below that. This
noise seems to exceed the upper limit form LPF discussed above, however the
two experiments are different enough that a direct comparison might be
misleading.

Ref. [18] reports a noise ASD of the potential of their 5 mm probe of about
$0.5\text{ mV}/\sqrt{\text{Hz}}$, which seems definitely higher than the upper
limit in [5]. However the authors only give the ASD of the overall electrical
potential and do not discuss the role of other possible sources of electrical
noise.

An attempt of Ref. [12] to measure time fluctuations of the patch potentials
turned to be limited by the electronic noise of the Kelvin probe.

Some relevant information on electric field fluctuations nearby metal surfaces
is available from studies on anomalous heating of ions in ion traps [20].
Patch field fluctuations are considered there as one of the possible sources
of electric field fluctuations responsible for the heating.

Among all possible mechanisms that may make the patch potential to fluctuate,
diffusion limited fluctuation of the adsorbate density, which is expected to
exhibit a linear relation to electrostatic potential, is supported by some
evidence, mostly the fact that it predicts $1/f^{3/2}$ behaviour of the PSD of
the electric field observed in some experiments [20]. Though those experiments
refer to the MHz frequency range, there is no obvious reason why such a
spectrum should not extend down to low frequencies.

The model calculate the density autocorrelation on the surface $S$ of a planar
electrode as:

$\begin{split}&\langle\delta n(\vec{r}_{1},t+\Delta t)\delta
n(\vec{r_{2}},t)\rangle=\\\ &=\int_{S}G(\vec{r}_{1},t+\Delta
t,\vec{r}_{3},t)\langle\delta n(\vec{r}_{2},t)\delta n(\vec{r_{3}},t)\rangle
d\vec{r}_{3},\end{split}$ (46)

with

$G(\vec{r}_{1},t+\Delta
t,\vec{r}_{3},t)=e^{-\lvert\vec{r}_{1}-\vec{r}_{3}\rvert^{2}/(\tilde{D}\lvert\Delta
t\rvert)}/(\tilde{D}\lvert\Delta t\rvert)$ (47)

the Green function for diffusion on a plane, with diffusion coefficient
$\tilde{D}$.

The analogous Green function for diffusion on a sphere or radius $R_{in}$ can
be calculated to be [21, 22]

$\begin{split}&G(\theta_{1},\phi_{1},t+\Delta t,\theta_{3},\phi_{3},t)=\\\
&=\frac{1}{R_{in}^{2}}\sum_{l=0}^{\infty}\frac{2l+1}{4\pi}e^{-l(l+1)\lvert\Delta
t\rvert/\tau_{d}}P_{l}(\cos(\omega))\end{split}$ (48)

where, as usual, $\omega$ is the angular distance between the two points
$(\theta_{1},\phi_{1})$ and $(\theta_{3},\phi_{3})$, and where
$\tau_{d}=R_{in}^{2}/\tilde{D}$.

Taking $V_{K}(\theta_{1},\phi_{1},t)\propto\delta n(\theta_{1},\phi_{1},t)$,
then $\langle\delta n(\theta_{2},\phi_{2},t)\delta
n(\theta_{3},\phi_{3},t)\rangle\propto\mathcal{R}_{n}(\cos(\omega),0)$. By
using the expansion of Legendre polynomial into spherical harmonics, the
integral in Eq. 46, translated into our spherical geometry, can be performed
explicitly, giving:

$\begin{split}&\mathcal{R}_{n}(\omega,\Delta
t)=\sum_{l=0}^{\infty}\frac{2l+1}{4\pi}V_{n,l}^{2}(0)e^{-l(l+1)\lvert\Delta
t\rvert/\tau_{d}}P_{l}(\cos(\omega))\end{split}$ (49)

We note that the model gives a well defined time scale $\tau_{d}$,
proportional to the time for diffusion around the entire sphere. This is a
remarkable difference with the infinite plane model where the analogous time
scale becomes infinite.

Still on $\tau_{d}$, we also note that it takes two different values for the
inner and the outer sphere, as the radii are different. The variations is
about 20% which, for the purpose of the current work, is negligible compared
to the relative uncertainties on all time parameters like $\tau_{dc}$. In the
following calculation we will thus neglect the effect of this small
inaccuracy.

By using Eq. 49 into Eq. 44, Fourier transforming and taking the square root,
we can get the ASD of the predicted force noise as:

$\begin{split}&S_{F}^{1/2}(f)\leq\epsilon_{o}V_{dc,1}V_{n,1}\times\\\
&\times\left(\sum_{l=1}^{\infty}K(l)\left(\left(\frac{V_{dc,l+1}V_{n,l}}{V_{dc,1}V_{n,1}}\right)^{2}g(l,f,\tau_{dc},\tau_{d})\right.\right.+\\\
&\left.\left.+\left(\frac{V_{dc,l}V_{n,l+1}}{V_{dc,1}V_{n,1}}\right)^{2}g(l+1,f,\tau_{dc},\tau_{d})\right)\right)^{1/2}\end{split}$
(50)

where $V_{n,l}^{2}\equiv V_{n,l}^{2}(0)$, $g(l,f,\tau_{dc})$ is the Fourier
transform of $f_{2}(\Delta t,\tau_{dc})\times e^{-l(l+1)\lvert\Delta
t\rvert/\tau_{d}}$, and where, finally, the coefficients of $g(l,f,\tau_{dc})$
and $g(l+1,f,\tau_{dc})$ have been normalised to those for $l=1$ to make the
sum independent of any common scale factor.

To proceed further we need the dependency of $V_{n,l}^{2}$ on $l$, that is the
shape of $\mathcal{R}_{n}(\omega,0)$. Again the simple exponential decay seems
general enough for the purpose of the present discussion. In addition one
would expect that the density of diffusing adatoms should be proportional to
the density of adsorbate, so that
$\mathcal{R}_{n}(\omega,0)\propto\mathcal{R}_{dc}(\omega)$.

As said, with such an exponential distribution, the assumption that maximises
the sum and then the ASD, is that of a very short coherence length, that makes
$V_{n,l}/V_{n,1}=V_{dc,l}/V_{dc,1}=1$ for all values of $l$ well beyond the
cutoff of $K(l)$.

With this assumption in hand, we only need an estimate of $V_{n,1}^{2}$ to
calculate the ASD. Consider that the model in Eq. 49 predicts that
$V_{n,1}^{2}(\Delta t)=V_{n,1}^{2}e^{-3\lvert\Delta t\rvert/\tau_{d}}$ and
that $S_{V_{n,1}}(f)\simeq 4/((2\pi f)^{2}\tau_{d}/3)$.

Taking the maximum bound for such a spectral density, $\tau_{d}\simeq
3\tau_{dc}$ and $V_{n,1}^{2}=(\tau_{dc}S_{o}(2\pi 0.1\text{ mHz})^{2})/4$, the
ASD in Eq. 50 can be calculated numerically using values for $\tau_{dc}$ and
$V_{dc,1}$ extracted form the posterior behind Table 3.

We report in Fig. 4 the $+2\sigma$ upper bound to the force noise for such a
calculation. For comparison the figure also includes the the excess force
noise ASD measured during the February 2017 noise run of LPF [2].

Figure 4: Projected force noise ASD for a single test-mass from the diffusion
model Eq. 50, with $\tau_{d}=3\tau_{dc}$, $V_{n,1}$ from the upper bound for
the LPF measured noise, and using the posterior for $\tau_{dc}$ and $V_{dc,1}$
in Table 3. Red dashed line: the case for
$V_{dc,l}/V_{dc,1}=V_{n,l}/V_{n,1}=1$. Blue dashed line: the case for both
$V_{dc,l}/V_{dc,1}$ and $V_{n,l}/V_{n,1}$ calculated form an exponential
correlation with a coherence length of 1 mm. Green dashed line: the case for
both $V_{dc,l}/V_{dc,1}$ and $V_{n,l}/V_{n,1}$ calculated form an exponential
correlation with a coherence length of 10 mm. Black line: best fit to $1/f$
function, of the ASD of excess force noise, expressed as single test-mass
disturbance, measured during the February 2017 noise run of LPF [2]

The picture clearly shows that the model predicts indeed a noise ASD with
$1/f$ frequency dependence. However, the $+2\sigma$ upper limit predicted by
this very conservative limit exceeds the total observed excess noise by almost
a factor 2 in the limit of vanishing coherence length.

We note that this conservative upper limit would be reduced considerably, even
for small coherence length, if the observed in-band noise is indeed explained
just by electronics noise even at the lowest frequency.

What also significantly reduces the value of the predicted noise is dropping
the very conservative assumption of an infinitely short coherence length of
the geometrical patch distribution. To illustrate the effect we also report in
Fig. 4 the $+2\sigma$ upper limit calculated under the hypothesis that
$V_{n,l}\propto V_{dc,l}$ and that $\mathcal{R}_{dc}(\omega)\propto
e^{-\omega/\omega_{o}}$, for both $R_{in}\omega_{o}=10\text{ mm}$, and
$R_{in}\omega_{o}=1\text{ mm}$, distributions close to our simulations of Fig.
3. The picture shows that the noise decreases non-linearly with increasing
coherence length, becoming negligible compared to the LPF noise excess for
$R_{in}\omega_{o}\simeq 10\text{ mm}$.

The assumption $\tau_{d}=3\tau_{dc}$, may seem a bit unnatural, giving to the
noisy part of the potential a similar time evolution as the quasi-dc one.
However the posterior for the diffusion coefficient one gets from such an
hypothesis has a $\pm 1\sigma$ range of
$\tilde{D}=\left(1.4_{-0.7}^{+1.5}\right)\times 10^{-11}\text{
m}^{2}\text{s}^{-1}$. This is well within the range $10^{-15}-10^{-9}\text{
m}^{2}{s}^{-1}$ quoted by [20].

One should also mention that, in the case of very short coherence length
$V_{dc,l}/V_{dc,1}=V_{n,l}/V_{n,1}=1$, the effect of a shortened $\tau_{d}$ is
that of bending the $1/f$ line of the predicted ASD, into a plateau at the
lowest frequency. For instance, already $\tau_{d}=0.3\tau_{dc}$ would make the
bending visible at around 0.1 mHz, without affecting the value of the $1/f$
tail at higher frequencies.

On the contrary, the cutoff in $V_{dc,l}$ introduced by the long coherence
length, makes also the $1/f$ behaviour less sensitive to $\tau_{d}$, so that
even taking $\tau_{d}=0.03\tau_{dc}$, does not affect significantly the shape
of the ASD if $R_{in}\omega_{o}=10\text{ mm}$. Note that
$\tau_{d}=0.03\tau_{dc}$ would give $D=\left(1.4_{-0.7}^{+1.5}\right)\times
10^{-9}\text{ m}^{2}\text{s}^{-1}$, still within the range mentioned above.

##  8  Conclusions

The concentric sphere model has proven to be a useful tool to analyse the
effect of the fluctuation of patch potentials in the actually topology of a
finite size, three-dimensional test mass enclosed in a hollow housing.

Nevertheless, its application to the current available information on the
space distribution, and the time evolution of these potentials, from LPF and
other relevant experiments, does not allow to rule out that they may have
contributed to the force noise detected in LPF, in excess of the Brownian
plateau below $\simeq 1$ mHz.

We have actually sketched a physically reasonable scenario of adsorbate
patches with density fluctuations correlated over a length of millimetres,
accompanied by diffusion of adatoms on their surface, which may have caused a
noise with the right frequency dependence and the right order of magnitude, to
explain all or part of the unexplained excess.

Though such scenario remains unproven, its possibility would suggest a few
precautions to be followed in the development of LISA.

First, torsion pendulum experiments with LISA-like TM have achieved
sensitivities [5, 23] that may allow a direct detection of, or a significant
upper limit to the noise we are discussing here.

Such an experiment is under analysis, but would require reducing the gaps
around the TM to about or less than 1 mm, considering the scaling with gap in
Fig. 2, and would likely detect torque noise rather than force. Such a direct
measurement would supersede much of the analysis presented here and provide a
stronger experimental anchor to keep this potential noise source under
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 of 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 fulfils LISA
requirements, may be confidently be reproduced.

##  9 Aknowledgement

We thank Norna Robertson and Annika Lang for their patience and their very
valuable feedback to our questions. We thank Vittorio Chiavegato for providing
the posterior of the $1/f^{2}$ component of $\Delta_{x}$ noise.

This work has been made possible by the LISA Pathfinder mission, which is part
of the space-science program of the European Space Agency, and has been
supported by Istituto Nazionale di Fisica Nucleare (INFN) and Agenzia Spaziale
Italiana (ASI), Project No. 2017-29-H.1-2020 “Attività per la fase A della
missione LISA”.

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