Analyzing Power Law Extensions of Newtonian Gravity Using Differential Force Measurements

1. Introduction

The success of the Standard Model (SM) in describing matter and interactions cannot be over stated, but it is not a complete description: it does not explain dark matter and dark energy [1], it predicts CP violations in the strong force which have not yet been observed [1,2,3], and there is no quantized description of gravity [1]. This incompleteness has lead to many theories to fill the gaps of the SM.

One such approach is hypothesizing an interaction mediated by an as of yet undiscovered boson [4,5]. If the hypothetical boson is massive it leads to Yukawa-like interactions [6], but if it is massless the interaction will be parameterized with a power law [7]. This power law is typically written as a correction to Newtonian gravity, and for two point masses is expressed as

$$U=\frac{-G{m}_1{m}_2}{r}\left(1+{\Lambda }_n{\left(\frac{r_0}{r}\right)}^{n-1}\right)$$

(1)

where G is Newton’s gravitational constant, $m_1$ and $m_2$ are point masses separated by a distance r, ${\Lambda }_n$ is the strength of the correction relative to gravity for a particular power of n, and $r_0$ is a constant used to preserve the dimension of the interaction; in this work $r_0=1\times {10}^{-15}$ m [8].

Previous experiments to probe both power law and Yukawa-like deviations in Newtonian gravity [8,9,10]. One experiment to probe hypothetical Yukawa-like interactions was carried out in 2016 [11]. The Yukawa-like interaction is of the form

$$U=\frac{-G{m}_1{m}_2}{r}\left(1+\alpha e^{r/\lambda }\right)$$

(2)

was probed and placed the best limits on $\alpha$ for a range of $\lambda \in (30,8000)$ nm, where $\alpha$ is the strength of the correction and $\lambda$ is the Compton wavelength of a massive hypothetical boson [11]. The experiment consisted of a spherical test mass attached to micro-mechanical oscillator, which was brought within 200 nm of a source mass of Au-Si sector and the force was measured between the two masses. The set up was not designed to probe power law extensions of the SM and was expected to not be sensitive enough to improve those limits. However, a reanalysis of the force measurements from [11] and a full analysis with the data was never carried out in the context of a power law.

Power law models are not explicitly mentioned in the pursuit of experimental evidence of dark matter [12,13,14,15,16,17,18,19]. This work does a full analysis considering power law extensions of the SM on the force measurements in [11] as well as on new data. The limits obtained are not an improvement over the current best limits [8,10], so we discuss what would need to be done for the approach in [11] to improve limits on power law extensions to the SM.

2. Materials and Methods

The study published in 2016 [11] used a differential force measurement technique between a spherical test mass attached to a micro-mechanical oscillator and a source mass. The test mass was a sphere composed of 3 layers; a central sapphire core with a 149.3 ± 0.2 $\mu$m radius covered by 10 nm of Cr followed by 250 nm of Au (see Figure 1 for a cross sectional diagram of the test and source mass). The sphere was glued to a 500 × 500 $\mu$m micro-mechanical oscillator and the system had a quality factor Q ∼ 7200. The deflection of the test mass was measured through a change in capacitance between the oscillator’s plate and electrodes located below the plate.

The source mass was a layered structure of BK7 Schott glass followed by a 2.10 ± 0.02 $\mu$m layer of alternating sectors of Au and Si. Both Au and Si sectors shared a common layer of a 10 ± 1 nm Cr wetting layer on top of which was a 150 nm Au layer covering the sample. The shared top Au layer thickness was chosen to be larger than the effective penetration depth of the Casimir force. In this way the contribution due to the Casimir force is the same whether the test mass is located over a Si sector or a Au sector and leads to a Casimir-less measuring technique as described in [20]. The test mass was brought to within 200 nm of the source mass’s surface and at this separation the minimum detectable force is 12 fN/$\sqrt{\mathrm{Hz}}$. The source mass was rotated so that the sectors alternated under the test mass at the oscillator’s resonant frequency. Doing so makes the experiment select the first harmonic of the force commensurate with the period of the samples.

While the source masses used to set limits on the Yukawa-like interaction [11] had upwards of 300 Au-Si sectors, there were source masses that had two, 1 mm wide, sectors of alternating gold and silicon, as depicted in Figure 2. One sample had an inner radius of 2.5 mm and the other had an inner radius of 5 mm. The layered structure is the same as the 300 sector samples except the wetting layer is Ti. The data taken with these larger source masses were previously not analyzed.

To extract limits on ${\Lambda }_n$, the force due to the potential expressed in Eq. 1 was calculated by integrating over the experimental geometry. First, the interaction between a spherical test mass and an arbitrary point in the source mass was calculated analytically with a coordinate system centered at the sphere, shown as the unprimed coordinate system in Figure 3, using spherical coordinates ($\tilde{r},\theta ,\varphi$) where $\theta$ and $\varphi$ are the polar and azimuthal angle respectively. The potential energy between the sphere and the source mass is

$$\begin{array}{cc}\hfill & U=-2\pi G{\rho }_1{\rho }_2{\Lambda }_n{r}_0^{n-1}\int \int \int \left(\frac{1}{\psi (2-n)}\left[\frac{R{(\psi -R)}^{3-n}}{3-n}\right.\right.\hfill \\ & +\left.\left.\frac{{(\psi -R)}^{4-n}}{(4-n)(3-n)}+\frac{R{(\psi +R)}^{3-n}}{3-n}-\frac{{(\psi +R)}^{4-n}}{(4-n)(3-n)}\right]\right)d{V}_{sm}\hfill \end{array}$$

(3)

where R is the radius of the sphere, ${\rho }_1$ and ${\rho }_2$ are the densities of the sphere and the point respectively, $\psi$ is the distance from the center of the sphere to the arbitrary point in the source mass and $d{V}_{sm}$ is the differential volume of the source mass, see Appendix A for details. It was verified that Eq. 3 does not diverge for $n=$ 2, 3, and 4 by taking the limit of Eq. 3 as $n\to$ 2, 3, and 4 respectively, these limits can be seen in Appendix A. The integrals over the source mass were carried out using cylindrical coordinates ($\varrho$, ${\theta }^{\prime }$, $z^{\prime }$) in the primed coordinate system, see Figure 3, centered in the middle of the source mass. In the primed coordinate system

$$\psi =\sqrt{{\varrho }^2-2\varrho {\varrho }_{s}cos({\theta }^{\prime }-{\theta }_s)+{\varrho }_s^2+{(z^{\prime }-z_s)}^2}$$

(4)

where $\varrho$ is the radial variable integrated between the inner and outer radius of the sample, ${\theta }^{\prime }$ is the angular extent of the sample, ${\varrho }_s$ is the radial distance to the sphere, ${\theta }_s$ is the angular postion of the sphere, $z_s$ is the vertical position of the center of the sphere, and $z^{\prime }$ is the vertical coordinate integrated over the thickness of the sample.

The experiment is only sensitive to forces in the vertical direction, normally calculating the force in the z direction would be done by

$$F_z=-\frac{\partial }{\partial z^{\prime }}U$$

(5)

where $d{F}_z$ is the differential force in the vertical direction that needs to be integrated over the geometry of the source mass. However, since Eq. 5 needs to be integrated along $z^{\prime }$ doing the derivative explicitly can be avoided because the operations are the inverse of each other; meaning the integral of Eq. 5 over the source mass in z is simply the difference of Eq. 3 evaluated at the $z^{\prime }$ integration limits of the source mass, $z_1$ and $z_2$.

$$F_z^{\left(n\right)}=\int \int (U^{\left(n\right)}\left(z_1\right)-U^{\left(n\right)}\left(z_2\right))d{A}_{sm}$$

(6)

where $d{A}_{sm}$ is the area element of the source mass which remains to be integrated. A polar coordinate system centered at the center of the sample, the primed coordinate system in Figure 3, is used for the last two integrals. The area element is expressed as $d{A}_{sm}=\varrho d\varrho d{\theta }^{\prime }$ where ($\varrho$, ${\theta }^{\prime }$) are the polar coordinates in the primed system. The integrals over $\varrho$ and ${\theta }^{\prime }$ were done numerically with Python code using SciPy packages [21].

In order to get the correct interaction from the layered geometry, the numerical integrations were carried out three times, once for each of the the three different layers of the test mass. Each test mass layer was considered to be a solid sphere with corresponding radii of $R_{sap}=$ 149.30 $\mu$m, $R_{Cr}=$ 149.31 $\mu$m, and $R_{Au}=$ 149.56 $\mu$m for sapphire, Cr, and Au respectively. As an example for n = 3 the integrations for each layer are

$$\begin{array}{c}\hfill F_{Au}^{\left(3\right)}=2\pi G{r}_0^2\int \int \left(\frac{R_{Au}(\psi \left(z_2\right)-\psi \left(z_1\right))}{\psi \left(z_1\right)\psi \left(z_2\right)}+ln\left(\frac{(\psi \left(z_1\right)-R_{Au})(\psi \left(z_2\right)+R_{Au})}{(\psi \left(z_1\right)+R_{Au})(\psi \left(z_2\right)-R_{Au})}\right)\right)d{A}_{sm}\end{array}$$

(7)

$$\begin{array}{c}\hfill F_{Cr}^{\left(3\right)}=-2\pi G{r}_0^2\int \int \left(\frac{R_{Cr}(\psi \left(z_2\right)-\psi \left(z_1\right))}{\psi \left(z_1\right)\psi \left(z_2\right)}+ln\left(\frac{(\psi \left(z_1\right)-R_{Cr})(\psi \left(z_2\right)+R_{Cr})}{(\psi \left(z_1\right)+R_{Cr})(\psi \left(z_2\right)-R_{Cr})}\right)\right)d{A}_{sm}\end{array}$$

(8)

$$\begin{array}{c}\hfill F_{Sap}^{\left(3\right)}=-2\pi G{r}_0^2\int \int \left(\frac{R_{Sap}(\psi \left(z_2\right)-\psi \left(z_1\right))}{\psi \left(z_1\right)\psi \left(z_2\right)}+ln\left(\frac{(\psi \left(z_1\right)-R_{Sap})(\psi \left(z_2\right)+R_{Sap})}{(\psi \left(z_1\right)+R_{Sap})(\psi \left(z_2\right)-R_{Sap})}\right)\right)d{A}_{sm}\end{array}$$

(9)

Equations 7,8 and 9 are the integrals over the source mass geometry carried out after the limit of Eq. 3 was taken as $n\to$ 3, where $\psi \left(z_1\right)$ and $\psi \left(z_2\right)$ are Eq. 4 evaluated at $z_1$ and $z_2$ respectively. The integrals for the other powers were done in the same manner. Once the integrals for each test mass layer is carried out, the total force due to the layered structure of the source mass and test mass (Figure 1) is

$$\begin{array}{cc}\hfill F_z^{\left(3\right)}={\Lambda }_3({\rho }_{Au}-& {\rho }_{Si})(({\rho }_{Sap}-{\rho }_{Cr})F_{Sap}^{\left(3\right)}+({\rho }_{Cr}-{\rho }_{Au})F_{Cr}^{\left(3\right)}+{\rho }_{Au}{F}_{Au}^{\left(3\right)})\hfill \end{array}$$

(10)

where ${\rho }_{Au}$, ${\rho }_{Cr}$, ${\rho }_{sap}$, and ${\rho }_{Si}$ are the densities of Au, Cr, sapphire, and Si respectively, $F_z^{\left(3\right)}$ indicates the total force in the z-direction for the power $n=3$, and the factor out front, $({\rho }_{Au}-{\rho }_{Si})$, accounts for the difference when the sphere is over a Au or Si in the source mass. Further more, since the first two layers are a shared layer of Au and Cr, the contributions of these layers to the interaction gets subtracted out.

The total force was calculated with the test mass at different angular positions (${\theta }_s^{\prime }$), Figure 4. The amplitude of the first harmonic for a particular power, $A_1^{\left(n\right)}$, of the force commensurate with the period, $\Theta$, of the sectors was equated to the error bars of the force measurements.

$$A_1^{\left(n\right)}=\frac{2}{\Theta }{\int }_0^{\Theta }{F}_z^{\left(n\right)}cos\left(\frac{2\pi {\theta }_s}{\Theta }\right)d{\theta }_s$$

(11)

The error bar value is $f_{err}=0.13$ fN at a separation of 300 nm, as shown in Figure 5. Equating the calculated first harmonic of the force to the experimental error bar allows limits on ${\Lambda }_n$ to be extracted.

$${\Lambda }_n=\frac{f_{err}}{A_1^{\left(n\right)}}$$

(12)

The same method to determine limits was used for the powers of $n=$ 1 to 5. Table 1 shows the limits placed on ${\Lambda }_n$ from both the 300 sector source masses used in [11] and the 4 sector source mass with inner radius of 2.5 mm as depicted in Figure 2.

3. Results

The limits presented in Table 1 were calculated using results from an experiment that was not designed to probe power law extensions to the SM. If the source masses were designed with a power law in mind, however, the limits could be further improved. Figure 6 shows the potential limits for ${\Lambda }_5$ as a function of the sample’s radial extent ($\varrho$) for a thickness of 2 mm. The limits have a similar functional dependence with the other dimensions of the sample as the radius increases. The improvement is attributed to the larger volume of interaction improving the sensitivity of the measurement yielding better limits. However, the improvement gained in the limits by increasing the source mass’s volume quickly diminishes.

Changing the test mass to also have a larger volume of interaction could provide a small improvement on power law limits. Considering a system, like the one proposed in [22], where the test mass is cylindrical and carrying out a similar analysis as described above with a new test mass, the expected limits using a cylindrical test mass can be calculated. The details of the calculation can be found in Appendix B. The following is the expression of the force between a cylindrical test mass and an infinite slab with thickness $t=2$ mm along the vertical direction

$$\begin{array}{c}\hfill F_5=\frac{-{\pi }^{2}G{\rho }_1{\rho }_{2}L{\Lambda }_5{r}_0^4}{3}\left(\frac{A_2^2}{(R^2-A_2^2)\sqrt{A_2^2-R^2}}\right.-\left.\frac{A_1^2}{(R^2-A_1^2)\sqrt{A_1^2-R^2}}\right)\end{array}$$

(13)

where $A_1=d+R_{Au}$, $A_2=A_1+t$, and $L=500\mu$m the length of the cylinder. The expected limits for the powers n = $1-5$ using a cylindrical test mass are listed in Table 2 calculate with the separation $d=200$ nm. Only the power $n=5$ is expected to improve by about an order of magnitude using a cylinder.

4. Conclusions

While these types of mechanical measurements have a high precision even at separations ∼200 nm, this work shows that the size of the masses limit the sensitivity when probing power law interactions. The limit on ${\Lambda }_5$, reported here to be $1.0\times {10}^{43}$, is expected to improved about an order of magnitude over the best current limits [8] by designing the experiment for power law extensions to Newtonian gravity. Improving the limits of power law extensions to gravity is a challenging endeavor and new methods are needed to continue to make improvements on short range interaction experiments. Several experimental approaches are being developed to test gravity at short distances, seeking signs of deviations potentially due to dark matter effects [12,13,14,15,16,17,18,19]. The description of these new approaches does not explicitly state probing power law extensions to the SM. Researcher should evaluate whether these new techniques are equally suited to test power law models as well as Yukawa-like interactions, which could result in a larger selection of theories being ruled out.

Appendix B.1. n=1

$$\begin{array}{cc}\hfill & U=\frac{-2\pi G{\rho }_{sm}{\Lambda }_1}{(2-1)(3-1)}\left(z^{\prime 3-1}-{\left(z^{\prime }+t\right)}^{3-1}\right)\hfill \end{array}$$

(A17)

$$\begin{array}{cc}\hfill & \overrightarrow{g}=-2\pi G{\rho }_{sm}{\Lambda }_{1}t\widehat{z}\hfill \end{array}$$

(A18)

$$\begin{array}{cc}\hfill & F_z^{\left(1\right)}=\overrightarrow{g}{m}_{tm}=-2\pi G{\rho }_{sm}{\Lambda }_{1}t\widehat{z}\left({\rho }_{tm}\pi R^{2}L\right)\hfill \end{array}$$

(A19)

Appendix B.2. n=2

$$\begin{array}{cc}\hfill & dU=-G{\rho }_{sm}{\Lambda }_2{r}_0\int \int \int \frac{d{V}_{sm}}{r^2}\hfill \end{array}$$

(A20)

$$\begin{array}{cc}\hfill & =-G{\rho }_{sm}{\Lambda }_2{r}_0\int \int \int \frac{\rho d\rho d\theta dz}{{\rho }^2+{(z^{\prime }-z)}^2}\hfill \end{array}$$

(A21)

Here the limits of integration for $\rho$ are $\rho \in [0,{\rho }_{+}]$. Integrating with respect to $\theta$ and making the following substitution $s={\rho }^2+{(z^{\prime }-z)}^2$ gives

$$\begin{array}{cc}\hfill & dU=-\pi G{\rho }_{sm}{\Lambda }_2{r}_0{\int }_{{(z^{\prime }-z)}^2}^{{\rho }_{+}^2+{(z^{\prime }-z)}^2}\frac{ds}{s}\hfill \end{array}$$

(A22)

$$\begin{array}{cc}\hfill & U=-\pi G{\rho }_{sm}{\Lambda }_2{r}_0{\int }_0^{-t}ln\left(\frac{{\rho }_{+}^2+{(z^{\prime }-z)}^2}{{(z^{\prime }-z)}^2}\right)dz\hfill \end{array}$$

(A23)

$$\begin{array}{cc}\hfill & \overrightarrow{g}=-\frac{dU}{dz}\widehat{z}=\pi G{\rho }_{sm}{\Lambda }_2{r}_0\frac{d}{dz}{\int }_0^{-t}ln\left(\frac{{\rho }_{+}^2+{(z^{\prime }-z)}^2}{{(z^{\prime }-z)}^2}\right)dz\hfill \end{array}$$

(A24)

$$\begin{array}{cc}\hfill & =\pi G{\rho }_{sm}{\Lambda }_2{r}_0\left(ln\left(\frac{{\rho }_{+}^2+{(z^{\prime }+t)}^2}{{(z^{\prime }+t)}^2}\right)-ln\left(\frac{{\rho }_{+}^2+z^{\prime 2}}{z^{\prime 2}}\right)\right)\hfill \end{array}$$

(A25)

$$\begin{array}{cc}\hfill & =\pi G{\rho }_{sm}{\Lambda }_2{r}_0\left(ln\left(\frac{{\rho }_{+}^2+{(z^{\prime }+t)}^2}{{\rho }_{+}^2+z^{\prime 2}}\right)-ln\left(\frac{{(z^{\prime }+t)}^2}{z^{\prime 2}}\right)\right)\hfill \end{array}$$

(A26)

Since we are considering the source mass as an infinite slab we take ${\rho }_{+}\to +\infty$

$$\begin{array}{c}\hfill \underset{{\rho }_{+}\to +\infty }{lim}ln\left(\frac{{\rho }_{+}^2+{(z^{\prime }+t)}^2}{{\rho }_{+}^2+z^{\prime 2}}\right)=0\end{array}$$

(A27)

$$\begin{array}{c}\hfill \overrightarrow{g}=-\pi G{\rho }_{sm}{\Lambda }_2{r}_{0}ln\left(\frac{{(z^{\prime }+t)}^2}{z^{\prime 2}}\right)\end{array}$$

(A28)

$$\begin{array}{cc}\hfill & F_z^{\left(2\right)}=\int \int \int \overrightarrow{g}\left(z^{\prime }\right){\rho }_{tm}dxdyd{z}^{\prime }\hfill \end{array}$$

(A29)

$$\begin{array}{cc}\hfill & F_z^{\left(2\right)}=-\pi G{\rho }_{sm}{\rho }_{tm}{\Lambda }_2{r}_0\int \int \int ln\left(\frac{{(z^{\prime }+t)}^2}{z^{\prime 2}}\right)dxdyd{z}^{\prime }\hfill \end{array}$$

(A30)

The limits of integration over the cylindrical test mass are $x\in [-L/2,L/2]$, $y\in [0,\sqrt{R^2-r^2}]$, and $r\in [-R,R]$ with $r=d+R-z^{\prime }$ and $dr=-d{z}^{\prime }$

$$\begin{array}{cc}\hfill & F_z^{\left(2\right)}=-\pi G{\rho }_{sm}{\rho }_{tm}{\Lambda }_2{r}_{0}L{\int }_{-R}^R{\int }_0^{\sqrt{R^2-r^2}}ln\left(\frac{{(d+R+t-r)}^2}{{(d+R-r)}^2}\right)dy(-dr)\hfill \end{array}$$

(A31)

$$\begin{array}{cc}\hfill & =-\pi G{\rho }_{sm}{\rho }_{tm}{\Lambda }_2{r}_{0}L{\int }_{-R}^R\sqrt{R^2-r^2}ln\left(\frac{{(d+R+t-r)}^2}{{(d+R-r)}^2}\right)(-dr)\hfill \end{array}$$

(A32)

We could not find an analytical solution to the final integral, and it was solved numerically.

Appendix B.3. n=3

$$\begin{array}{c}\hfill dU=-G{\rho }_{sm}{\Lambda }_3{r}_0^2\int \int \int \frac{\rho d\rho d\theta dz}{{({\rho }^2+{(z^{\prime }-z)}^2)}^{3/2}}\end{array}$$

(A33)

Integrating $\theta$ and substituting $s^2={({\rho }^2+{(z^{\prime }-z)}^2)}^{1/2}$

$$\begin{array}{cc}\hfill & U=-2\pi G{\rho }_{sm}{\Lambda }_3{r}_0^2{\int }_0^{-t}{\int }_{z^{\prime }-z}^{\infty }\frac{dsdz}{s^2}\hfill \end{array}$$

(A34)

$$\begin{array}{cc}\hfill & =-2\pi G{\rho }_{sm}{\Lambda }_3{r}_0^2{\int }_0^{-t}\frac{dz}{2(z^{\prime }-z)}\hfill \end{array}$$

(A35)

$$\begin{array}{cc}\hfill & =\pi G{\rho }_{sm}{\Lambda }_3{r}_0^{2}ln\left(\frac{z^{\prime }+t}{z^{\prime }}\right)\hfill \end{array}$$

(A36)

$$\begin{array}{cc}\hfill & \overrightarrow{g}=-\pi G{\rho }_{sm}{\Lambda }_3{r}_0^2\left(\frac{1}{z^{\prime }+t}-\frac{1}{z^{\prime }}\right)\hfill \end{array}$$

(A37)

$$\begin{array}{c}\hfill F_z^{\left(3\right)}=-\pi G{\rho }_{sm}{\rho }_{tm}{\Lambda }_3{r}_0^2\int \int \int \left(\frac{1}{z^{\prime }+t}-\frac{1}{z^{\prime }}\right)dxdyd{z}^{\prime }\end{array}$$

(A38)

$$\begin{array}{cc}\hfill F_z^{\left(3\right)}=-\pi G{\rho }_{sm}{\rho }_{tm}{\Lambda }_3{r}_0^{2}L& {\int }_{-R}^R{\int }_0^{\sqrt{R^2-r^2}}\left(\frac{-dydr}{d+R+t-r}\right.\hfill \\ \hfill +& \left.\frac{dydr}{d+R-r}\right)\hfill \end{array}$$

(A39)

$$\begin{array}{cc}\hfill F_z^{\left(3\right)}=-\pi G{\rho }_{sm}{\rho }_{tm}{\Lambda }_3{r}_0^{2}L& {\int }_{-R}^R\left(\frac{-\sqrt{R^2-r^2}}{d+R+t-r}\right.+\hfill \\ & \left.\frac{\sqrt{R^2-r^2}}{d+R-r}\right)dr\hfill \end{array}$$

(A40)

$$\begin{array}{c}\hfill F_z^{\left(3\right)}=-\pi G{\rho }_{sm}{\rho }_{tm}{\Lambda }_3{r}_0^{2}L\left(\frac{-\pi {(d+R+t)}^2}{2{({(d+R+t)}^2-R^2)}^{3/2}}\right.\\ \hfill \left.+\frac{\pi {(d+R)}^2}{2{({(d+R)}^2-R^2)}^{3/2}}\right)\end{array}$$

(A41)

Appendix B.4. n=4

Starting with Eq. A15 for $n=4$

$$U=\frac{-2\pi G{\rho }_{sm}{\Lambda }_4{r}_0^3}{2}\left(\frac{1}{z^{\prime }}-\frac{1}{z^{\prime }+t}\right)$$

(A42)

$$\overrightarrow{g}=\pi G{\rho }_{sm}{\Lambda }_4{r}_0^3(-z^{\prime -2}+{(z^{\prime }+t)}^{-2})$$

(A43)

$$F_z^{\left(4\right)}=-\pi G{\rho }_{sm}{\Lambda }_4{r}_0^3\int \int \int (z^{\prime -2}-{(z^{\prime }+t)}^{-2})dxdyd{z}^{\prime }$$

(A44)

$$F_z^{\left(4\right)}=-\pi G{\rho }_{sm}{\Lambda }_4{r}_0^{3}L{\int }_0^{\sqrt{R^2-r^2}}{\int }_{-R}^R(z^{\prime -2}-{(z^{\prime }+t)}^{-2})dy(-dr)$$

(A45)

$$F_z^{\left(4\right)}=-\pi G{\rho }_{sm}{\rho }_{tm}{\Lambda }_4{r}_0^{3}L\left(\frac{A_2\pi }{\sqrt{A_2^2-R^2}}-\frac{A_1\pi }{\sqrt{A_1^2-R^2}}\right)$$

(A46)

$A_2=d+R+t$ and $A_1=d+R$

Appendix B.5. n=5

Starting with Eq. A15 for $n=5$

$$U=\frac{-\pi G{\rho }_{sm}{\Lambda }_5{r}_0^4}{3}\left(\frac{1}{z^{\prime 2}}-\frac{1}{{(z^{\prime }+t)}^2}\right)$$

(A47)

$$\overrightarrow{g}=\frac{-2\pi G{\rho }_{sm}{\Lambda }_5{r}_0^4}{3}\left(\frac{1}{z^{\prime 3}}-\frac{1}{{(z^{\prime }+t)}^3}\right)$$

(A48)

$$F_z^{\left(5\right)}=\frac{-2\pi G{\rho }_{sm}{\Lambda }_5{r}_0^4}{3}\int \int \int \left(\frac{1}{z^{\prime 3}}-\frac{1}{{(z^{\prime }+t)}^3}\right)dxdyd{z}^{\prime }$$

(A49)

$x\in [-L/2,L/2]$, $y\in [0,\sqrt{R^2-r^2}$, $r=d+R-z^{\prime }$,

$dr=-d{z}^{\prime }$

$$F_z^{\left(5\right)}=\frac{-\pi G{\rho }_{sm}{\Lambda }_5{r}_0^{4}L}{3}\left(\frac{A_1^2}{{(A_1^2-R^2)}^{3/2}}-\frac{A_2^2}{{(A_2^2-R^2)}^{3/2}}\right)$$

(A50)

$A_2=d+R+t$ and $A_1=d+R$