Scalar stars and lumps with AdS or dS cores

We explore the possibility of embedding regular compact objects with an (anti-)de Sitter [(A)dS] core as solutions of Einstein’s gravity minimally coupled to a real scalar field. We consider, among others, solutions interpolating between an inner, potential-dominated core and an outer, kinetic-term-dominated region. Owing to their analogy with slow-roll inflation, we term them gravitational vacuum inflative stars, or gravistars for short. We systematically discuss approximate solutions of the theory describing either the core or the asymptotically flat region at spatial infinity. We extend nonexistence theorems for smooth interpolating solutions, previously proved for black holes, to compact objects without event horizons. This allows us to construct different classes of exact (either smooth or nonsmooth) singularity-free solutions of the theory. We first find a smooth solution interpolating between an AdS spacetime in the core and an asymptotically flat spacetime (a Schwarzschild solution with a subleading $1/r^2$ deformation). We proceed by constructing nonsmooth solutions describing gravistars. Finally, we derive a smooth scalar lump solution interpolating between ${\mathrm{AdS}}_4$ in the core and a Nariai spacetime at spatial infinity.

Figure 1

Metric function $U$ and potential $V$ for exact solutions with AdS core. (a) Metric solution as a function of $r/\ell$. (b) Potential as a function of $r/\ell$.

Figure 2

Plot of the effective potential (30) as a function of $r/\ell$.

Figure 3

Plot of the function $\mathcal{F}(r_0/y_0)$ (45) for different values of $\beta$. Since $r_0<y_0$, the plot stops at 1. We note that only for values of $\beta \le 0.1$, $c_2$, given by Eq. (44), is negative.

Figure 4

(a) Junction of the two metric functions $U$ for selected values of the parameters. (b) Junction of the two radial function for selected values of the parameters. For both figures, the inner solution is the dashed blue line, while the exterior one is given by the dashed orange line. The black solid line corresponds to the full jointed solution. The vertical line, instead, corresponds to the position $y_0$ of the shell. We set $\beta =0.1$, $r_0/y_0=0.1$.

Figure 5

Contour plot for surface energy density $\sigma$ and pressure $\mathcal{P}$ of the shell. (a) Contour plot for the surface energy density $\sigma (y_0)$ as a function of the parameters $y_0$ and $r_0$. (b) Contour plot for $\sigma (y_0)+\mathcal{P}(y_0)$ as a function of the parameters $y_0$ and $r_0$. For both figures, $\beta =0.1$, while the others are fixed to require continuity of the induced metric (see Sec. 6c). As can be seen, both the WEC and NEC are satisfied in these intervals of parameters.

Figure 6

Metric (blue solid line), scalar field (dashed orange line), and scalar potential (dotted green line) as functions of $r/\ell$ for our scalar lumps. We set $c_2={\ell }^{-2}$ and rescaled each curve by their maximum.