Decay of boson stars with application to glueballs and other real scalars

One of the most interesting candidates for dark matter is massive real scalar particles. A well-motivated example is from a pure Yang-Mills hidden sector, which locks up into glueballs in the early universe. The lightest glueball states are scalar particles and can act as a form of bosonic dark matter. If self-interactions are repulsive, this can potentially lead to very massive boson stars, where the inward gravitational force is balanced by the repulsive self-interaction. This can also arise from elementary real scalars with a regular potential. In the literature it has been claimed that this allows for astrophysically significant boson stars with high compactness, which could undergo binary mergers and generate detectable gravitational waves. Here we show that previous analyses did not take into proper account $3\to 2$ and $4\to 2$ quantum mechanical annihilation processes in the core of the star, while other work misinterpreted the classical $3\to 1$ process. In this work, we compute the annihilation rates, finding that massive stars will rapidly decay from the $3\to 2$ or $4\to 2$ processes (while the $3\to 1$ process is typically small). Using the Einstein-Klein-Gordon equations, we also estimate the binding energy of these stars, showing that even the densest stars do not have quite enough binding energy to prevent annihilations. For such boson stars to live for the current age of the universe and to be consistent with bounds on dark matter scattering in galaxies, we find the following upper bound on their mass for $\mathcal{O}(1)$ self-interaction couplings: $M_{*}\lesssim {10}^{-18}{M}_{\mathrm{sun}}$ when $3\to 2$ processes are allowed and $M_{*}\lesssim {10}^{-11}{M}_{\mathrm{sun}}$ when only $4\to 2$ processes are allowed. We also estimate destabilization from parametric resonance which can considerably constrain the phase space further. Furthermore, such stars are required to have very small compactness to be long-lived.

Figure 1

Exact versus large $g$ approximation profile. Boson star profile $\mathrm{\Phi }(r)$ for the core value ${\mathrm{\Phi }}_0\approx 0.028M_{\mathrm{Pl}}$ and parameter $g=100$. The orange is the exact numerical solution, while the purple is from scaling the large $g$ approximation (this is similar in form to a plot in Ref. [34]).

Figure 2

Exact versus large $g$ approximation mass. Ratio of the mass of a boson star using the large $g$ approximation scheme to the exact mass using the full equations (albeit always in the single harmonic approximation) with ${\beta }_0=1.585$. Note that the ratio approaches 1 at large $g$.

Figure 3

Fixing coupling $\lambda$ to its maximum value. Decay rates ${\mathrm{\Gamma }}_d$ (in units ${\mathrm{Gyr}}^{-1}$) of boson stars as a function of the star’s mass $M_{*}$ (units $M_{\mathrm{sun}}$) for different choices of particle mass $m_{\varphi }$ (solid red lines) and compactness $C$ (dotted gray lines). Here we have imposed the coupling $\lambda =(2\pi {)}^2$, which is on the order of the maximum allowed by unitarity (appropriate for small gauge groups). The black dashed line is the present Hubble rate $H_0$. In the upper right orange region the stars would collapse to black holes. The lower right green shaded region is excluded by bullet cluster bounds. The yellow band is LISA, and the blue band is LIGO, which is only appreciable at the top right where the stars are compact. The light blue shaded region indicates where parametric resonance may occur according to a simple analysis. Upper panel: $3\varphi \to 2\varphi$ processes when there are odd powers of $\varphi$ in $V$. Lower panel: $4\varphi \to 2\varphi$ processes when there are only even powers of $\varphi$ in $V$.

Figure 4

Fixing scattering to core-cusp preferred value. Decay rates ${\mathrm{\Gamma }}_d$ (in units ${\mathrm{Gyr}}^{-1}$) of boson stars as a function of the star’s mass $M_{*}$ (units $M_{\mathrm{sun}}$) for different choices of coupling $\lambda =(4\pi /N{)}^2$ (solid blue lines). Here we have imposed the scattering cross section ${\sigma }_{2\to 2}/m_{\varphi }=1{\mathrm{cm}}^2/\mathrm{g}$ to possibly explain the cores of galaxies. The black dashed line is the present Hubble rate $H_0$. In the upper right orange region the stars would collapse to black holes. In the upper left green region there are no stars, since once would need couplings $\lambda \gg 1$ violating unitarity. The yellow band is LISA, and the blue band is LIGO, which is only appreciable at the top right where the stars are compact. The light blue shaded region indicates where parametric resonance may occur according to a simple analysis. Upper panel: $3\varphi \to 2\varphi$ processes when there are odd powers of $\varphi$ in $V$. Lower panel: $4\varphi \to 2\varphi$ processes when there are only even powers of $\varphi$ in $V$.

Figure 5

Fixing perturbative decay rate to Hubble and exploring $\{m_{\varphi },\lambda \}$ parameter space. Parameter space of solutions after imposing the decay rate is equal to the current Hubble rate ${\mathrm{\Gamma }}_d=H_0$. The solid brown lines are contours of fixed star mass $M_{*}$ and the dotted black lines are contours of fixed compactness $C$. The lower right green shaded region is excluded by bullet cluster bounds. The lower left shaded orange region would have compactness so high the stars would collapse to black holes. The yellow band is LISA, and the blue band is LIGO, which is only appreciable at the lower left where the stars are compact. The light blue shaded region indicates where parametric resonance may occur according to a simple analysis. Upper panel: $3\varphi \to 2\varphi$ processes when there are odd powers of $\varphi$ in $V$. Lower panel: $4\varphi \to 2\varphi$ processes when there are only even powers of $\varphi$ in $V$.

Figure 6

Fixing perturbative decay rate to Hubble and exploring $\{C,\lambda \}$ parameter space. Parameter space of solutions after imposing the decay rate is equal to the current Hubble rate ${\mathrm{\Gamma }}_d=H_0$. The solid brown lines are contours of fixed star mass $M_{*}$. The upper right green shaded region is excluded by bullet cluster bounds. In this figure we have not indicated the LISA or LIGO bands or the black hole regime since these are only important at high compactness, which is far off the top of the plot. The light blue shaded region indicates where parametric resonance may occur according to a simple analysis. Upper panel: $3\varphi \to 2\varphi$ processes when there are odd powers of $\varphi$ in $V$. Lower panel: $4\varphi \to 2\varphi$ processes when there are only even powers of $\varphi$ in $V$.

Figure 7

Boson star’s energy/mass (red line) and number (blue line) as a function of parameter ${\beta }_0$ within the large $g$ approximation. (The left solid curves are the solutions of most interest in this work, with the maximum value $M_{\max }\approx 0.03\sqrt{{\lambda }_{4\mathrm{eff}}}{M}_{\mathrm{Pl}}^3/m_{\varphi }^2$. The right dashed curves might suffer from other sorts of instabilities.)

Figure 8

Change in boson star’s energy with respect to the particle number. (The upper left solid curve shows the regular solutions of most interest in this work. The lower right dashed curve might suffer from other sorts of instabilities.)