Boson stars in $f(T)$ extended theory of gravity

Spherically symmetric configurations of the noninteracting massive complex scalar field, representing nonrotating boson stars, are considered within the framework of the modified torsion based $f(T)$ gravity, with $f(T)=T+\alpha T^2/2$. We find that with sufficiently large negative value of $\alpha$ the mass of the boson stars can be made arbitrarily large. This is in contrast to general relativity where an upper bound, $M_{\max }\sim M_{\text{Planck}}^2/m$, to the mass of the boson stars built from the noninteracting scalar field exists and where the masses of boson stars in the astrophysical regime can be obtained only with the introduction of the scalar field self-interaction. With sufficiently large negative $\alpha$ we also find negative gravitational binding energy for all masses, which can be seen as an indication of the stability of such configurations. In its positive regime, $\alpha$ can not be made arbitrarily large as a phase transition in the stress-energy components of the $f(T)$-fluid develops. This phenomenon has already been reported to occur in polytropic stars constructed within the $f(T)$ gravity theory.

Figure 1

Gravitational mass $M$ (thick black dashed line, in units of $M_{\text{Planck}}^2{m}^{-1}$), particle number $N$ (thin black dashed line, in units of $M_{\text{Planck}}^2{m}^{-2}$), and energy of scalar field quantum $\omega$ (thick gray dashed line, in units of $m$) in boson stars with field mass $m$, no field self interaction, central field amplitude ${\sigma }_0=\sqrt{4\pi }\times 0.1$, and a range of values of $\tilde{\alpha }=\alpha m^2$.

Figure 2

Gravitational mass $M$ (thick black lines), particle number $N$ (thin black lines), and energy of scalar field quantum $\omega$ (thick gray lines) of boson stars with field mass $m$, no field self interaction, a range of values of central field amplitude ${\sigma }_0=\sqrt{4\pi }{\varphi }_0$, and $\tilde{\alpha }=0$ (solid lines) and $\tilde{\alpha }=-5$ (dashed lines).

Figure 3

Gravitational mass $M$ (thick lines) and particle number $N$ (thin lines) shown as functions of energy of scalar field quantum $\omega$ in boson stars with field mass $m$, no field self interaction, and $\tilde{\alpha }=0$ (solid lines) and $\tilde{\alpha }=-5$ (dashed lines).

Figure 4

Radial profiles of stress-energy tensor components for ${\sigma }_0=\sqrt{4\pi }\times 0.1$ and $\tilde{\alpha }=2.048$ (close-to-critical configuration). Upper plot: effective energy density (solid line), energy density due to boson fluid (long dashed line), and energy density due $f(T)$-fluid (short dashed line). Lower plot: effective radial and transverse pressures (black and gray solid lines, respectively), radial and transverse pressures due to boson fluid (black and gray long dashed lines), and radial and transverse pressures due to nonlinear terms in $f(T)$ (black and gray short dashed lines). All quantities are shown relative to value of effective energy density at $r=0$.