A No-Go Theorem for the Mass-Radius Relation of Solitons
We prove a no-go theorem for the mass-radius relation of localized and stable field configurations, known as solitons. Defining the mass-radius index by $Γ\equiv \frac{\rm{d}\ln M}{\rm{d}\ln R}$, for real scalar field theories in $d$ spatial dimensions, we show that typical non-topological, non-relativistic, and spherically symmetric solitons cannot have $Γ$ in the range $[0, d]$. The forces considered originate from gradient energy, self-interaction, and gravitation, with the typicality assumption excluding the fine-tuned region of the parameter space where all three forces have comparable strength. Importantly, the theorem works for an arbitrary self-interaction that, in the relativistic theory, is allowed to be non-power-law in the field, be non-analytic around the classical vacuum (where the field amplitude vanishes), or to include derivative couplings. Additionally, the theorem makes no assumptions about the explicit form of the soliton's density profile or the behavior of $Γ$ as a function of $R$. We also argue that the same exclusion applies to compact objects formed from self-gravitating, non-relativistic, barotropic fluids with arbitrary equations of state. As a consequence for cosmology, it is worth noting that observations favor a core in the centers of dark matter halos with $Γ\simeq 1.7$, which (for $d=3$) lies approximately in the middle of the excluded range. Therefore, proposals such as ultra-light or fluid-like dark matter models are essentially ruled out as natural explanations for halo cores, provided other astrophysical effects are negligible.