Self-gravitating vector dark matter

We derive the nonrelativistic limit of a massive vector field. We show that the Cartesian spatial components of the vector behave as three identical, noninteracting scalar fields. We find classes of spherical, cylindrical, and planar self-gravitating vector solitons in the Newtonian limit. The gravitational properties of the lowest-energy vector solitons—the gravitational potential and density field—depend only on the net mass of the soliton and the vector particle mass. In particular, these self-gravitating, ground-state vector solitons are independent of the distribution of energy across the vector field components and are indistinguishable from their scalar-field counterparts. Fuzzy vector dark matter models can therefore give rise to halo cores with observational properties that are identical to the ones in scalar fuzzy dark matter models. We also provide novel hedgehog vector soliton solutions which cannot be observed in scalar-field theories. The gravitational binding of the lowest-energy hedgehog halo is about 3 times weaker than the ground-state vector soliton. Finally, we show that no spherically symmetric solitons exist with a divergence-free vector field.

Figure 1

The spherical hedgehog solution, ${\mathcal{A}}^j=({\mathcal{A}}^r,{\mathcal{A}}^{\theta },{\mathcal{A}}^{\phi })=(h(|\mathbf{x}|)e^{-im{E}_{h}t},0,0)$, given in terms of the rescaled quantities from Eq. (28). The solution is normalized so that $\int {\tilde{h}}^2(\tilde{r})4\pi {\tilde{r}}^{2}dr=1$.

Figure 2

The hedgehog cylindrical solution, ${\mathcal{A}}^j=({\mathcal{A}}^{\rho },{\mathcal{A}}^{\theta },{\mathcal{A}}^z)=(g(\rho )e^{-im{E}_{g}t},0,0)$, given in terms of the rescaled quantities from Eq. (28). The solution is normalized so that ${\tilde{g}}^{\prime }(\tilde{\rho }=0)=1$.

Figure 3

The ground-state Cartesian VDM soliton, ${\mathcal{A}}^j=({\mathcal{A}}^x,{\mathcal{A}}^y,{\mathcal{A}}^z)=f(|\mathbf{x}|)e^{-imEt}(w^x,w^y,w^z)$, for arbitrary $w^j$ given in terms of the rescaled quantities from Eq. (28). The $\tilde{f}(\tilde{r})$ and $\tilde{\mathrm{\Phi }}(\tilde{r})$ profiles are identical to those of the scalar FDM ground-state soliton solution; see, e.g., Ref. [17]. We use the same normalization as in Fig. 1.

Figure 4

The lowest-energy cylindrical solution, ${\mathcal{A}}^j=({\mathcal{A}}^x,{\mathcal{A}}^y,{\mathcal{A}}^z)=q(\rho )e^{-im{E}_{q}t}(w^x,w^y,0)$, given in terms of the rescaled quantities from Eq. (28). The $\tilde{q}(\tilde{\rho })$ and $\tilde{\mathrm{\Phi }}(\tilde{\rho })$ solutions are identical to the ones of the scalar FDM lowest-energy cylindrical solution; see, e.g., [60]. The solution is normalized so that it gives rise to the same asymptotic behavior of the gravitational potential, $\tilde{\mathrm{\Phi }}(\tilde{\rho }\to \infty )\to (\pi /2)\ln (\tilde{\rho })+\text{const}$, as that of the hedgehog string from Fig. 2.

Figure 5

The ground-state solution for the ansatz in Eq. (a21) with ${\tilde{M}}^{(x)}=0.690$, ${\tilde{M}}^{(y)}=0.248$. Note that ${\sum }_j{\tilde{M}}^{(j)}=1$. The $\tilde{\mathrm{\Phi }}(\tilde{r})$ and $\sqrt{{\sum }_j{\tilde{\psi }}^{j2}(\tilde{r})}$ are identical to the blue and orange curves in Fig. 3, respectively. Since the data in Fig. 3 are equivalent to, e.g., ${\tilde{M}}^{(x)}=1$, one concludes that the gravitational properties of the ground-state VDM halos, i.e., the gravitational potential, $\tilde{\mathrm{\Phi }}(\tilde{r})$, and energy density, ${\sum }_j{\tilde{\psi }}^{j2}(\tilde{r})$, are independent of the ${\tilde{M}}^{(j)}$ values.