Spherical structures in conformal gravity and its scalar-tensor extension

We study spherically symmetric structures in conformal gravity and in a scalar-tensor extension and gain some more insight about these gravitational theories. In both cases we analyze solutions in two systems: perfect fluid solutions and boson stars of a self-interacting complex scalar field. In the purely tensorial (original) theory we find in a certain domain of parameter space finite mass solutions with a linear gravitational potential but without a Newtonian contribution. The scalar-tensor theory exhibits a very rich structure of solutions whose main properties are discussed. Among them, solutions with a finite radial extension, open solutions with a linear potential and logarithmic modifications and also a (scalar-tensor) gravitational soliton. This may also be viewed as a static self-gravitating boson star in purely tensorial conformal gravity.

Figure 1

Perfect fluid solutions with $\kappa =0$: (a) the profiles of three solutions; (b) plots of several characteristics of the solutions as a function of the parameter $A$.

Figure 2

Perfect fluid solutions with $\kappa =0.1$: (a) the profiles of three solutions; (b) plots of several characteristics of the solutions as a function of the parameter $A$. Note that unlike the $\kappa =0$ case, here $B^{\prime \prime }(0)$ does not vanish as $A\to 2/3$.

Figure 3

Boson stars with $\kappa =0.1$. (a) the profiles of a typical conformal boson star solution with $\lambda =1$; (b) plots of several characteristics of the solutions as a function of $\lambda$: The inertial mass $M_I$ and particle number $Q$ are plotted in units $8\pi$, while the gravitational mass $M_G$ is given in units $4\pi$.

Figure 4

First branch vacuum solutions for the case $\nu =0$: Two profiles for $B_2=4/3$ with $b=1$ and $b=3$ (analytic solution). Color online distinguishes between the curves. In the black and white version notice that $S(x)$ curves are the two decreasing ones, while those of $B(x)S^2(x)$ start on the $x$ axis.

Figure 5

The two branch structure of the $\nu =0$ solutions. The bullets indicate the corresponding analytic solutions. For the meaning of the various parameters, see the text.

Figure 6

Second branch vacuum solutions for the case $\nu =0$: Three profiles for $B_2=1/3$ with $b=2$, $b=1$, and $b=0.25$. (a) curves for $B(x)$ and $B^{\prime }(x)$; (b) curves for $S(x)$, $xS(x)$, and $S^2(x)B(x)$.

Figure 7

A typical solution with $\nu >0$ (online red) compared with a $\nu =0$ (online black) solution.

Figure 8

Open solutions of the vacuum scalar-tensor system with $\nu =0.2$ for three asymptotic values of the scalar field.

Figure 9

Several profiles of the regular solutions of the scalar-tensor system with $\nu =0$ for different central value of the scalar field: $S(0)=1$, $S(0)=4$, $S(0)=16$. Notice the oscillations for large $S(0)$.

Figure 10

Perfect fluid open solutions in the scalar-tensor theory with $\nu =1$. $\kappa =0.1$: (a) the profiles of two solutions; (b) plots of several characteristics of the solutions as a function of the parameter $A$.

Figure 11

Perfect fluid solutions in the scalar-tensor theory: dependence on the parameter $\kappa$. The other parameters are $\nu =0.2$, $A=1/3$.

Figure 12

Boson stars in the scalar-tensor theory corresponding to $\lambda =1$, $\nu =\mu =0$, $\kappa =0.1$. (a) mass, particle number $Q$ and the value $f(0)$ as a function of $S(0)$; (b) details of the profile for $S(0)=100$. The high value of $S(0)$ was chosen to get noticeable oscillations—see the text.

Figure 13

The value $B^{\prime \prime }(0)$ and the inertial mass of the field S for STBS (online red) and STR soliton (online black) as functions of $S(0)$. Notice the gaps in the STBS (red) curves where no solutions exist. This can be used to distinguish between the cases in a black and white plot.

Figure 14

Open boson star solutions in the scalar-tensor theory: (a) two profiles for $\nu =0.2$ and $\nu =1$ with $\lambda =1$, $\mu =0$, $\kappa =0.1$; (b) plots of several characteristics of the solutions as a function of the parameter $\lambda$ with $\nu =0.2$, $\mu =0$, $\kappa =0.1$.

Figure 15

Open boson star solutions in the scalar-tensor theory: dependence on the parameter $\kappa$. The other parameters are $\lambda =1$, $\nu =0.2$, $\mu =0$.