Dynamical formation of Proca stars and quasistationary solitonic objects

We perform fully nonlinear numerical simulations within the spherically symmetric Einstein-(complex)-Proca system. Starting with Proca field distributions that obey the Hamiltonian, momentum and Gaussian constraints, we show that the self-gravity of the system induces the formation of compact objects, which, for appropriate initial conditions, asymptotically approach stationary solitonlike solutions known as Proca stars. The excess energy of the system is dissipated by the mechanism of gravitational cooling in analogy to what occurs in the dynamical formation of scalar boson stars. We investigate the dependence of this process on the phase difference between the real and imaginary parts of the Proca field, as well as on their relative amplitudes. Within the timescales probed by our numerical simulations the process is qualitatively insensitive to either choice: the phase difference and the amplitude ratio are conserved during the evolution. Thus, whereas a truly stationary object is expected to be approached only in the particular case of equal amplitudes and opposite phases, quasistationary compact solitonic objects are, nevertheless, formed in the general case.

Figure 1

Evolution of the radial profile of $A_t$ (top panel) and $A_r$ (bottom panel) at intermediate to late times, for model 3. In the inset we focus near the node of $A_t$.

Figure 2

Early time evolution, $t\in [100,1000]$, of the radial profile of $A_t$ for model 3.

Figure 3

Radial profile of the energy density ${\rho }_e$ for model 3 at selected times in between times 6000 and 6900 and for $r\in [0,35]$. During the evolution we can see how the peak amplitude changes in an oscillatory manner, confined in the inner region defined by $r\simeq 30$. The red dashed line corresponds to the radial profile of ${\rho }_e$ for an isolated, stable Proca star with a similar mass.

Figure 4

Proca energy in spheres of different radii, for model 3. The magenta, red, green, blue and violet lines correspond to, respectively, $r^{*}=10$, 30, 50, 100, 200. The black line represents the total energy.

Figure 5

Spacetime diagram of the Proca energy density ${\rho }_e$ for model 3. This is a logscale plot and the darker areas represent zones where ${\rho }_e$ is higher.

Figure 6

Top panel: Time evolution of the real and imaginary parts of the scalar potential $\mathrm{\Phi }$ at $r=5$ for model 3. The inset shows a shorter time window $t\in [25,75]$, where the $\pi /2$ phase difference becomes clear. Bottom panel: Corresponding Fourier transform of the two time series of $\mathrm{\Phi }$ in the top panel. The peaks for the real and imaginary parts completely overlap, so that only the blue line appears. The units in the vertical axis are arbitrary.

Figure 7

Phase difference $\delta$ between the real and imaginary parts of the scalar potential $\mathrm{\Phi }$ calculated at radius $r=5$ for the models 1,2,3,4, corresponding, respectively, to an initial phased difference $\delta =0$, $\pi /3$, $\pi /2$, $\pi$.

Figure 8

Time evolution of the norm of the vector potential amplitude, $|a_r|$, at $r=5$ for an initial phase difference of $\pi /2$ (model 3) and $\pi /3$ (model 2).

Figure 9

Time evolution of the energy contribution from the real part of the Proca field (black line) and imaginary part (red line), for model 6. In the inset we multiply the imaginary part by a factor 4 to compare the two curves.

Figure 10

Spacetime diagram of the Proca energy density ${\rho }_e$ for the real oscillaton model 5. This is a logscale plot and the darker areas represent zones where ${\rho }_e$ is higher.