Spherical boson stars as black hole mimickers

We present spherically symmetric boson stars as black hole mimickers based on the power spectrum of a simple accretion disk model. The free parameters of the boson star are the mass of the boson and the fourth-order self-interaction coefficient in the scalar field potential. We show that even if the mass of the boson is the only free parameter, it is possible to find a configuration that mimics the power spectrum of the disk due to a black hole of the same mass. We also show that for each value of the self-interaction a single boson star configuration can mimic a black hole at very different astrophysical scales in terms of the mass of the object and the accretion rate. In order to show that it is possible to distinguish one of our mimickers from a black hole, we also study the deflection of light.

Figure 1

(Top panel) Sequences of equilibrium configurations for different values of $\Lambda$ are shown as a function of the central value of the scalar field ${\varphi }_0(0)$; each point in the curves corresponds to a solution of the eigenvalue problem and represents a boson star. The filled circles indicate the critical solution that separates the stable from the unstable branch. Those configurations to the left of the maxima represent stable configurations. The inverted triangles indicate the point at which the binding energy is zero. The mass of the configurations is given in units of the squared Planck mass over the mass of the boson, which implies a scale invariance of this plot under the change of $m$. (Bottom panel) We show the compactness of the equilibrium configurations. We define compactness as the ratio $2N_{99}/R_{99}$, where $R_{99}$ is the radius at which 99% of the total number of particles $N_{99}$ is contained. In this plot we also show the critical point with a filled circle and indicate that the critical configurations are the most compact ones among the stable configurations for each value of $\Lambda$. Those configurations to the right of the critical solutions are even more compact; however, they are unstable and are not considered here. In our results below we will also use the critical solutions for reference as the most compact stars.

Figure 2

The emission spectra for cases A (top panel) and B (bottom panel) due to the black hole and two different boson stars: the most compact of the stable branches for the $\Lambda =0$ case ($M=0.633M_{\mathrm{pl}}^2/m$; see Fig. 1) and an adequate configuration ($M=0.473M_{\mathrm{pl}}^2/m$) that presents a power spectrum that mimics that of the black hole. The most compact boson star is used only as a reference. The configuration with $M=0.473$ is used for the two different astrophysical situations. For case A, $m=1.2\times {10}^{-27}\mathrm{GeV}$, and for case B, $m=2.51\times {10}^{-22}\mathrm{GeV}$. The spectrum due to the black hole and the one due to the mimicker lie approximately on top of each other. For similar physical configurations the spectra are comparable to those obtained in 5, where the redshift was also taken into account.

Figure 3

The emission spectra for cases A (top panel) and B (bottom panel) due to the black hole and two different boson stars: the most compact of the stable branches for the $\Lambda =50$ case ($M=2.257M_{\mathrm{pl}}^2/m$; see Fig. 1) and an adequate configuration ($M=0.9898M_{\mathrm{pl}}^2/m$) that presents a power spectrum that mimics that of the black hole. As in the previous case, the same configuration with $M=0.9898$ is used for the two different astrophysical situations. For case A, $m=1.88\times {10}^{-27}\mathrm{GeV}$, and for case B, $m=5.25\times {10}^{-22}\mathrm{GeV}$, respectively. The spectrum due to the black hole and that due to the mimicker lie approximately on top of each other.

Figure 4

The mimicker configurations are indicated with stars for values of $\Lambda =0$, 10, 20, 30, 40, 50, 60, 70. It is expected that for every value of $\Lambda$, there is a mimicker configuration that lies near the ones indicated for the values shown here.

Figure 5

We show the mass of the boson in GeVs in terms of the mass of the black hole candidate in solar masses for mimicker configurations and various values of $\Lambda$. We extend the case $\Lambda =0$ in order to show that, for the axion mass case, the object would be very small.

Figure 6

We show a set of null geodesics for the black hole and its boson star mimicker. We choose the mimicker for $\Lambda =0$. We use units for the coordinates on the equatorial plane $x$ and $y$ scaled with $M$ (the mass of the BHC), and therefore this plot corresponds to both cases A and B for $\Lambda =0$, where the scales for cases A and B are obtained when the value of $M$ of the BHC is substituted.

Figure 7

We show the deflection angle of the null rays for the black hole and its boson star mimicker, corresponding again to case A and $\Lambda =0$ presented above. In this plot $r_p$ is the photon sphere radius of the black hole, $r_S$ is the Schwarzschild radius, and the label $r_c$ indicates the closest approach radius. We only show the deflection angle for $r>M$ because in spherical coordinates the geodesic equations become stiff; a detailed study of this region would need a patch with different coordinates which is not within the scope of this paper.