The Yukawa potential of a non-homogeneous sphere, with new limits on an ultralight boson

TL;DR

This work develops a comprehensive framework for the Yukawa potential of a non-homogeneous sphere, introducing the hyperbolic form factor Φ(x) = ⟨ sinh(mr)/(mr) ⟩ to encode finite-size and density-profile effects with x = mR. By solving the Poisson-like equation and exploiting multishell and continuum representations, the authors express Φ(x) and the effective density ȳρ(x), enabling accurate exterior potentials even for complex Earth density profiles. They apply MICROSCOPE’s dissymmetric Eötvös parameter δTi-Pt measurements to derive 95% CL limits on ultralight boson couplings to B, B−L, or L for both spin-1 and spin-0 mediators, with explicit final bounds and a detailed analysis of confidence-level effects. A simplified five-shell Earth model demonstrates how density stratification shifts the limits, and the results are presented with explicit formulae for Φ(x), its expansions, and the scaling factor G(x) that amplifies limits at larger mediator masses. Overall, the paper provides a general, scalable methodology for extracting finite-range-force constraints from satellite data in the presence of realistic, non-homogeneous sources, with practical implications for ultralight dark matter searches and beyond.

Abstract

Extremely weak long-range forces may lead to apparent violations of the Equivalence Principle. The final MICROSCOPE result, leading at 95 % c.l. to $|δ| < 4.5 \times 10^{-15}$ or $6.5 \times 10^{-15}$ for a positive or negative Eötvös parameter $δ$, requires taking into account the spin of the mediator, and the sign of $Δ(Q/A_r)_{\rm{Ti-Pt}}$ ($Q$ denoting the new charge involved). A coupling to $B-L$ or $B$ should verify $|g_{B-L}|<1.1 \times 10^{-25}$ or $|g_{B}| < 8 \times 10^{-25}$, for a spin-1 mediator of mass $m < 10^{-14}$ eV$/c^2$, with slightly different limits of $1.3 \times 10^{-25}$ or $\,6.6 \times 10^{-25}$ in the spin-0 case. The limits increase with $m$, in a way which depends on the density distribution within the Earth. This involves an hyperbolic form factor, expressed through a bilateral Laplace transform as $Φ(x=mR)= \langle\,\sinh mr/mr \,\rangle$, related by analytic continuation to the Earth form factor $Φ(ix)= \langle \,\sin mr/mr \,\rangle $. It may be expressed as $Φ(x) = \frac{3}{x^2}\, (\cosh x - \frac{\sinh x}{x}) \times\, \barρ(x)/ρ_0\,$, where $\barρ(x)$ is an effective density, decreasing from the average $ρ_0$ at $m=0$ down to the density at the periphery. We give general integral or multishell expressions of $Φ(x)$, evaluating it, and $\barρ(x)$, in a simplified 5-shell model. $Φ(x)$ may be expanded as $\, \sum \frac{x^{2n}}{(2n+1)!} \frac{\langle \,r^{2n}\,\rangle}{R^{2n}} \simeq 1 + .0827\ x^2 + .00271 \ x^4 + 4.78 \times 10^{-5}\,x^6 + 5.26\times 10^{-7}\, x^8 +\ ... \ $, absolutely convergent for all $x$ and potentially useful up to $x\approx 5$. The coupling limits increase at large $x$ like $mR \ e^{mz/2}/\sqrt{1+mr}$ ($z=r-R$ being the satellite altitude), getting multiplied by $\simeq 1.9,\ 34$, or $1.2\times 10^9$, for $m = 10^{-13},\ 10^{-12}$ or $10^{-11}$ eV$/c^2$, respectively.

Figures (4)

• Figure 1: The MICROSCOPE result $\,\delta_{\hbox{\scriptsize ,Ti-Pt}}=\,(\,-1.5\pm 2.3_{\text{\,stat}} \pm 1.5_{\text{\,syst}}\,)\times 10^{-15}$, represented proportionally to $\,e^{-(\delta - \delta_0)^2\!/\,2\sigma^2}$ (in blue), with $\delta_0 \simeq -1.5\times 10^{-15}$ and $\sigma\simeq 2.75 \times 10^{-15}$. It leads for an unconstrained $\delta$ to $\,-\,7\times 10^{-15}\!< \,\delta< 4\times 10^{-15}$ at the $2\sigma$ CL (versus $\,-\,5.5\times 10^{-15}\!< \,\delta< 5.5\times 10^{-15}$ for $\delta_0=0$, cf. dashed curve in orange). The negative $\delta_0$ leads to limits on $|\delta|$increased to about $4.5 \times 10^{-15}$ for $\delta>0$, and decreased to $6.5 \times 10^{-15}$ for $\delta < 0$, at the 95 % CL. 29.24 % of the area under the blue curve corresponds to $\delta >0$ (including 1.462 % for $\delta > 4.5\times 10^{-15}$), and 70.76 % for $\delta < 0$ (including 3.538 % for $\delta < -\,6.5 \times 10^{-15}$).
• Figure 2: For an interaction of range $\lambda$ the Earth generates the same outside Yukawa potential as an homogeneous sphere of density $\bar{\rho} (x)= \rho_0\,\Phi(x)/\phi(x)$, function of $x=mR=R/\lambda$. It decreases regularly from about $\rho_0\simeq 5.51$ g/cm$^3$ for $\lambda$ larger than the Earth radius, down to $\mathrel{ \hbox{$<$}{\hbox{$\sim$}}}$ 3.5 g/cm$^3$ for $\lambda \mathrel{ \hbox{$<$}{\hbox{$\sim$}}}$ 300 km. $\bar{\rho} (x)$, given by eqs. (\ref{['defrho']}-\ref{['limrho']}), is then representative of an average density around a depth $d= R-r \approx \lambda$, and is evaluated in Sec. \ref{['sec:5spheres']} in a simple 5-shell model. For $x\simeq 3$ i.e. $\lambda\simeq 2\,100$ km, $\bar{\rho}\simeq$ 4.92 g/cm$^3$ is close to the inner mantle density, while for $x\simeq 20$ i.e. $\lambda\simeq 320$ km, $\bar{\rho}\simeq$ 3.57 g/cm$^3$ is close to the outer mantle density (cf. Table \ref{['phiPhirho']}).
• Figure 3: The hyperbolic form factors $\phi(x)=3\,(x\cosh x-\sinh x)/x^3$ for an homogeneous sphere (in orange), and $\Phi(x)$, as approximated in a 5-shell model of the Earth (in blue), as functions of $x=mR=R/\lambda$. Their ratio $\Phi(x)/\phi(x) = \bar{\rho}(x)/\rho_0$ defines the effective density $\bar{\rho}(x)$ represented earlier in Fig. \ref{['rho']}.
• Figure 4: Upper limits on $|g_{B-L}|$ or $|g_L|$ (in blue), and $|g_B|$ (in orange), depending on the mediator mass $m$ (or range $\lambda = \hbar/mc$), and spin, at the 95 % CL. The limits for $\delta<0$ are larger than for $\delta >0$ by $\simeq \sqrt{6.5/4.5}\,\simeq 1.2\,$. For $\lambda \gg R$ they are about $1.1\times 10^{-25}$ for $|g_{B-L}|$ in the spin-1 case or $|g_L|$ in the spin-0 case (solid blue line); and $1.3\times 10^{-25}$ for $|g_{B-L}|$ in the spin-0 case or $|g_L|$ in the spin-1 case (dashed blue line). For $|g_B|$ they are about 6 times larger, at $7.7\times 10^{-25}$ for spin 1 (solid orange) and $6.4 \times 10^{-25}$ for spin-0 (dashed orange). The limits increase with $m$ proportionally to $e^{mr/2}\,[\,\phi(x)\,(1+mr)\, \bar{\rho}(x)/\rho_0\,]^{-1/2}$, behaving at large $x$ like $mR\ e^{mz/2}/\sqrt{1+mr}\,$. The limits at the 90 % CL are slightly lower, down to $10^{-25}$ in the spin-1 case for $B-L$.