Newtonian analysis of multiscalar boson stars with large self-couplings

We study solutions for boson stars in the multiscalar field theory with global symmetry $[U(1){]}^N$. The properties of the boson stars are investigated by the Newtonian approximation with the large coupling limit. Our purpose is to study the models bringing about exotic mass distributions which explain flat rotation curves of galaxies. We propose plausible models in which coupling matrices are associated with various graphs in graph theory.

Figure 1

The behavior of the density profile of scalar bosons as the function of the rescaled distance $x$. The upper curve represents $S_2={\rho }_1+{\rho }_2$ and the lower curve appeared for $x<x_2=0.35$ represents ${\rho }_1$. The couplings are set as $C_{11}=C_{22}=1$, $C_{12}=0.9$. The kink in the curve is due to the approximation of the large coupling limit.

Figure 2

The behavior of the gravitational potential $\mathrm{\Phi }(x)$ as the function of the rescaled distance $x$. The couplings are set as $C_{11}=C_{22}=1$, $C_{12}=0.9$ and $x_2$ is chosen as $x_2=0.35$.

Figure 3

The fraction of the particle number of the boson 2, which lives inner region of the boson star as a function of $x_2$, when $C_{11}=C_{22}=1$, $C_{12}=0.9$.

Figure 4

The ratio of the Newtonian binding energy to the total mass of the boson star as a function of $x_2$, when $C_{11}=C_{22}=1$, $C_{12}=0.9$.

Figure 5

The ratio of the Newtonian binding energy to the total mass of the boson star as a function of the fraction of the particle number of the boson 2, when $C_{11}=C_{22}=1$, $C_{12}=0.9$.

Figure 6

The behavior of the total density of scalar bosons as the function of the rescaled distance $x$, for $x_2=0.3$, 0.35, 0.4, 0.45, 0.5, when $C_{11}=C_{22}=1$, $C_{12}=0.9$.

Figure 7

The galaxy rotation curves induced by the bi-scalar boson stars for $x_2=0.3$, 0.35, 0.4, 0.45, 0.5, when $C_{11}=C_{22}=1$, $C_{12}=0.9$. Note that overall scale can be arbitrarily chosen.

Figure 8

The behavior of the total density of scalar bosons as the function of the rescaled distance $x$, for $x_2=0.8$, 1.0, 1.2, 1.4, 1.6, when $C_{11}=C_{22}=1$, $C_{12}=0$.

Figure 9

The galaxy rotation curves induced by the biscalar boson stars for $x_2=0.8$, 1.0, 1.2, 1.4, 1.6, when $C_{11}=C_{22}=1$, $C_{12}=0$. Note that an overall scale can be arbitrarily chosen.

Figure 10

The behavior of the total density of ten-scalar bosons as the function of the rescaled distance $x$ when $C_{11}=\cdots =C_{10,10}=1$, $C_{ij}=0$ for $i\ne j$. We take $\{x_{10},x_9,x_8,x_7,x_6,x_5,x_4,x_3,x_2\}=\{0.43,0.48,0.53,0.58,0.63,0.68,0.73,0.78,0.83\}$ as indicated by gray vertical lines. The most right line indicates $x_1$, the surface of the boson star.

Figure 11

The galaxy rotation curves induced by the ten-scalar boson stars when $C_{11}=\cdots =C_{10,10}=1$, $C_{ij}=0$ for $i\ne j$. We take $\{x_{10},x_9,x_8,x_7,x_6,x_5,x_4,x_3,x_2\}=\{0.43,0.48,0.53,0.58,0.63,0.68,0.73,0.78,0.83\}$ as indicated by gray vertical lines. The most right line indicates $x_1$, the surface of the boson star. Note that overall scale can be arbitrarily chosen.

Figure 12

The behavior of the total density of five-scalar bosons as the function of the rescaled distance $x$ when $C_{11}=\cdots =C_{55}=1$, $C_{ij}=0$ for $i\ne j$. We take $\{x_5,x_4,x_3,x_2\}=\{0.85,0.9,0.95,1\}$ as indicated by gray vertical lines. The most right line indicates $x_1$, the surface of the boson star.

Figure 13

The galaxy rotation curves induced by the five-scalar boson stars when $C_{11}=\cdots =C_{55}=1$, $C_{ij}=0$ for $i\ne j$. We take $\{x_5,x_4,x_3,x_2\}=\{0.85,0.9,0.95,1\}$ as indicated by gray vertical lines. The most right line indicates $x_1$, the surface of the boson star. Note that overall scale can be arbitrarily chosen.

Figure 14

The behavior of the total density of five-scalar bosons as the function of the rescaled distance $x$ when $\mathcal{C}=\mathrm{diag}(5,4,3,2,1)$. We take $\{x_5,x_4,x_3,x_2\}=\{1.25,1.35,1.45,1.55\}$ as indicated by gray vertical lines. The most right line indicates $x_1$, the surface of the boson star.

Figure 15

The galaxy rotation curves induced by the five-scalar boson stars when $\mathcal{C}=\mathrm{diag}(5,4,3,2,1)$. We take $\{x_5,x_4,x_3,x_2\}=\{1.25,1.35,1.45,1.55\}$ as indicated by gray vertical lines. The most right line indicates $x_1$, the surface of the boson star. Note that overall scale can be arbitrarily chosen.

Figure 16

A complete graph $K_5$.

Figure 17

The behavior of the total density of five-scalar bosons as the function of the rescaled distance $x$ in the model associated with a complete graph $K_5$ and $\gamma =1$. We take $\{x_5,x_4,x_3,x_2\}=\{0.83,1.03,1.23,1.43\}$ as indicated by gray vertical lines. The most right line indicates $x_1$, the surface of the boson star.

Figure 18

The galaxy rotation curves induced by the five-scalar boson stars in the model associated with a complete graph $K_5$ and $\gamma =1$. We take $\{x_5,x_4,x_3,x_2\}=\{0.83,1.03,1.23,1.43\}$ as indicated by gray vertical lines. The most right line indicates $x_1$, the surface of the boson star. Note that overall scale can be arbitrarily chosen.

Figure 19

The contribution of each boson species in the model associated with $K_5$ and $\gamma =1$. We take $\{x_5,x_4,x_3,x_2\}=\{0.83,1.03,1.23,1.43\}$. The curves indicate $S_k(x)=\sum _{i=1}^k{\rho }_i(x)$, $k=5$ to 1, from the upper to the lower.

Figure 20

A cycle graph $C_{10}$.

Figure 21

A star graph with five vertices.

Figure 22

The behavior of the total density of five-scalar bosons as the function of the rescaled distance $x$ in the model associated with a complete graph $K_5$ and $\gamma =1$, in the presence of the external gravitational source (see text) with ${\omega }_e=\sqrt{5}{\omega }_5$ and ${\overline{\rho }}_{\mathrm{ext}}(0)/S(0)=0.5$. The gray curve in the figure shows ${\overline{\rho }}_{\mathrm{ext}}(x)$. We take $\{x_6,x_5,x_4,x_3,x_2\}=\{0.44,0.83,0.95,1.07,1.19\}$ as indicated by gray vertical lines. The most right line indicates $x_1$, the surface of the boson star.

Figure 23

The galaxy rotation curves induced by the five-scalar boson stars in the model associated with a complete graph $K_5$ and $\gamma =1$, in the presence of the external gravitational source (see text) with ${\omega }_e=\sqrt{5}{\omega }_5$ and ${\overline{\rho }}_{\mathrm{ext}}(0)/S(0)=0.5$. We take $\{x_6,x_5,x_4,x_3,x_2\}=\{0.44,0.83,0.95,1.07,1.19\}$ as indicated by gray vertical lines. The most right line indicates $x_1$, the surface of the boson star. Note that overall scale can be arbitrarily chosen.