Phase transitions between dilute and dense axion stars

We study the nature of phase transitions between dilute and dense axion stars interpreted as self-gravitating Bose-Einstein condensates. We develop a Newtonian model based on the Gross-Pitaevskii-Poisson equations for a complex scalar field with a self-interaction potential $V(|\psi {|}^2)$ involving an attractive $|\psi {|}^4$ term and a repulsive $|\psi {|}^6$ term. Using a Gaussian Ansatz for the wave function, we analytically obtain the mass-radius relation of dilute and dense axion stars for arbitrary values of the self-interaction parameter $\lambda \le 0$. We show the existence of a critical point $|\lambda {|}_c\sim (m/M_P{)}^2$, where $m$ is the axion mass and $M_P$ is the Planck mass, above which a first-order phase transition takes place. We qualitatively estimate general relativistic corrections on the mass-radius relation of axion stars. For weak self-interactions $|\lambda |<|\lambda {|}_c$, a system of self-gravitating axions forms a stable dilute axion star below a general relativistic maximum mass $M_{\max ,\mathrm{GR}}^{\text{dilute}}\sim M_P^2/m$ and collapses into a black hole above that mass. For strong self-interactions $|\lambda |>|\lambda {|}_c$, a system of self-gravitating axions forms a stable dilute axion star below a Newtonian maximum mass $M_{\max ,\mathrm{N}}^{\text{dilute}}=5.073M_P/\sqrt{|\lambda |}$ [Phys. Rev. D 84, 043531 (2011)], collapses into a dense axion star above that mass, and collapses into a black hole above a general relativistic maximum mass $M_{\max ,\mathrm{GR}}^{\text{dense}}\sim \sqrt{|\lambda |}{M}_P^3/m^2$. Dense axion stars explode below a Newtonian minimum mass $M_{\min ,\mathrm{N}}^{\text{dense}}=98.9m/\sqrt{|\lambda |}$ and form dilute axion stars of large size or disperse away. We determine the phase diagram of self-gravitating axions and show the existence of a triple point $(|\lambda {|}_{*},M_{*}/(M_P^2/m))$ separating dilute axion stars, dense axion stars, and black holes. We make numerical applications for QCD axions and ultralight axions. Our approximate analytical results are in good agreement with the exact numerical results of Braaten et al. [Phys. Rev. Lett. 117, 121801 (2016)] for Newtonian dense axion stars. They are also qualitatively similar to those obtained by Helfer et al. [J. Cosmol. Astropart. Phys. 03 (2017) 055] for general relativistic axion stars, but they differ quantitatively for weak self-interactions presumably due to the use of a different self-interaction potential $V(|\psi {|}^2)$. We point out analogies between the evolution of self-gravitating axions (bosons) at zero temperature evolving from dilute axion stars to dense axion stars and black holes and the evolution of compact degenerate (fermion) stars at zero temperature evolving from white dwarfs to neutron stars and black holes. We also discuss some analogies between the phase transitions of Newtonian axion stars at zero temperature and the phase transitions of Newtonian self-gravitating fermions at nonzero temperature. Finally, we suggest that a dense axionic nucleus may form at the center of dark matter halos through the collapse of a dilute axionic core (soliton) passing above the maximum mass $M_{\max ,\mathrm{N}}^{\text{dilute}}$. It would have a mass $1.11\times {10}^9(f/m)M_{\odot }$, a radius $0.949/(m{f}^{1/3})\mathrm{pc}$, a density $2.10\times {10}^{-8}(m^2{f}^2)\mathrm{g}/{\mathrm{m}}^3$, a pulsation period $8.24/(m{f}^{1/3})\mathrm{yr}$, and an energy $-5.59\times {10}^{62}(f/m)\mathrm{erg}$, where the axion mass $m$ is measured in units of ${10}^{-22}\mathrm{eV}/{\mathrm{c}}^2$ and the axion decay constant $f$ is measured in units of $10^{15}\mathrm{GeV}$. This dense axionic nucleus could be the remnant of a bosenova associated with the emission of a characteristic radiation [Phys. Rev. Lett. 118, 011301 (2017)].

Figure 1

The function $f(x)$ characterizing the equation of state of axions. For comparison, we have plotted as a dashed line the function obtained by keeping only the first two terms in the expansion of $f(x)$ for $x\to 0$ [see Eq. (42)]. It corresponds to the simplified equation of state of Sec. 3c studied in this paper.

Figure 2

Sketch of the phase diagram obtained by Helfer et al. [120] by numerically solving the KGE equations with the instantonic potential (28). The dashed line corresponds to the Newtonian maximum mass of dilute axion stars given by Eq. (66) obtained in Ref. [34]. It is valid for $f\ll M_P{c}^2$ (strong self-interactions). We see that it gives a fair agreement with the numerical results of Ref. [120] up to the triple point at $(f,M)\sim (0.3M_P{c}^2,2.4M_P^2/m)$.

Figure 3

Effective potential $V(R)$ as a function of the radius $R$ for $\delta ={10}^{-3}$ and $M=0.9<M_{\max }=1$. It presents a local minimum (M) corresponding to a metastable dilute axion star, a global minimum (S) corresponding to a fully stable dense axion star, and a local maximum (U) corresponding to an unstable axion star.

Figure 4

Mass-radius relation of QCD axion stars ($\delta =8.68\times {10}^{-17}$). The dashed line corresponds to the mass-radius relation of dilute axion stars with $\delta =0$ [34]. The dotted line corresponds to the TF approximation of dense axion stars (see Sec. 6).

Figure 5

Mass-density relation of QCD axion stars ($\delta =8.68\times {10}^{-17}$).

Figure 6

The minimum radius $R_{\min }(\delta )$ and the corresponding mass $M_{*}(\delta )$ as a function of $\delta$. We note, parenthetically, that these curves cross each other at $\delta =1$ (at that point, $R_{\min }=M_{*}=\sqrt{3}$).

Figure 7

The curve $M_{*}(R_{\min })$ (short dashed) and the curve $M_e(R_e)$ (long dashed) defined in Sec. 5h. For comparison, we have also plotted the mass-radius relation (105) of dilute axion stars with $\delta =0$ in full lines.

Figure 8

The radius $R_{*}(\delta )$ corresponding to the maximum mass $M_{\max }(\delta )$ and the radius $R_{*}^{\prime }(\delta )$ corresponding to the minimum mass $M_{\min }(\delta )$ as a function of $\delta$. They coincide at the critical point ${\delta }_c$ (see Sec. 5i). We have also plotted the minimum radius $R_{\min }(\delta )$ for comparison.

Figure 9

The maximum mass $M_{\max }(\delta )$ and the minimum mass $M_{\min }(\delta )$ as a function of $\delta$. They coincide at the critical point ${\delta }_c$ (see Sec. 5i). Since the functions $\delta (R_e)$ and $M_e(R_e)$ are maximum at the critical point (see Sec. 5i), the curve $M_e(\delta )$ presents a spike at $\delta ={\delta }_c$. We have also plotted the mass $M_{*}(\delta )$ corresponding to the minimum radius for comparison (we note, parenthetically, that $M_{*}=M_{\max }$ for $\delta =1.74\times {10}^{-3}$ and $M=1.005$).

Figure 10

Mass-radius relation for different values of the interaction parameter ($\delta ={10}^{-10},{10}^{-5},{\delta }_c,{10}^5,{10}^{10}$).

Figure 11

Total energy $E_{\mathrm{tot}}$ as a function of the radius $R$ for axion stars with $\delta ={10}^{-3}$ (for illustration, we have taken a relatively large value of $\delta$ to facilitate the reading of the figure). The dashed line corresponds to dilute axion stars with $\delta =0$.

Figure 12

Total energy $E_{\mathrm{tot}}$ as a function of the mass $M$ for axion stars with $\delta ={10}^{-3}$. The dashed line corresponds to dilute axion stars with $\delta =0$.

Figure 13

Eigenenergy $NE$ as a function of the radius $R$ for axion stars with $\delta ={10}^{-3}$. The dashed line corresponds to dilute axion stars with $\delta =0$.

Figure 14

Eigenenergy $NE$ as a function of the mass $M$ for axion stars with $\delta ={10}^{-3}$. The dashed line corresponds to dilute axion stars with $\delta =0$.

Figure 15

Complex pulsation as a function of the radius $R$ for axion stars with $\delta ={10}^{-3}$. The dashed line corresponds to dilute axion stars with $\delta =0$.

Figure 16

Same as Fig. 15 showing the branch of dense axion stars with high pulsations (${\omega }^2$ has been divided by 1000 for clarity).

Figure 17

Complex pulsation as a function of the mass $M$ for axion stars with $\delta ={10}^{-3}$. The dashed line corresponds to dilute axion stars with $\delta =0$.

Figure 18

Same as Fig. 17 showing the branch of dense axion stars with high pulsations (${\omega }^2$ has been divided by 1000 for clarity). The minimum pulsation ${\omega }_4$ corresponds to a large mass $M_4=202$ that is not represented on the figure.

Figure 19

The mass $M_{\mathrm{E}}(\delta )$ at which the total energy vanishes as a function of $\delta$. The upper branch $M_{\mathrm{E},\mathrm{II}}(\delta )$ corresponds to unstable axion stars, and the lower branch $M_{\mathrm{E},\mathrm{III}}(\delta )$ corresponds to stable dense axion stars. For comparison, we have also plotted the maximum and minimum masses. Finally, we have indicated the sign of the energy of the different equilibrium states in each domain. For example, $(++-)$ means that the dense stable axion star (left) has a positive energy, the unstable axion star (middle) has a positive energy, and the stable dilute axion (right) star has a negative energy. The equilibrium states in the triplet $(++-)$ are ordered according to the value of their radius (see Fig. 23 below).

Figure 20

Total energy at the point of maximum mass (lower branch) and at the point of minimum mass (upper branch) as a function of $\delta$.

Figure 21

Series of equilibria $E(M)$ for different values of the interaction parameter $\delta$. The dashed line corresponds to dilute axion stars with $\delta =0$.

Figure 22

Stability of the equilibrium states from the Poincaré theory of linear series of equilibria. We have performed the Maxwell construction to distinguish fully stable (S), metastable (M), and unstable (U) equilibrium states.

Figure 23

Effective potential $V(R)$ as a function of the radius for $\delta ={10}^{-3}$. For $M<M_{\min }=0.601$ (here, $M=0.5$), the dilute axion stars are fully stable, and the dense axion stars are nonexistent. For $M_{\min }=0.601<M<M_t=0.700$ (here, $M=0.65$), the dilute axion stars are fully stable, and the dense axion stars are metastable. For $M=M_t=0.700$, the dense axion stars have the same energy as the dilute axion stars. For $M_t=0.700<M<M_{\max }=1.00$ (here, $M=0.9$), the dense axion stars are fully stable, and the dilute axion stars are metastable. For $M>M_{\max }=1.00$ (here, $M=1.1$), the dense axion stars are fully stable, and the dilute axion stars are nonexistent.

Figure 24

Series of equilibria illustrating the collapse from a dilute axion star to a dense axion star above $M_{\max }$ (see Sec. 8b) or the explosion from a dense axion star to a dilute axion star below $M_{\min }$ (see Sec. 8c).

Figure 25

Phase diagram of Newtonian axion stars. The $H$ zone corresponds to a hysteretic zone where the actual phase depends on the history of the system. If the system is initially prepared in a dilute state, it will remain dilute until the maximum mass $M_{\max }$ at which it will collapse and become dense. Inversely, if the system is initially prepared in a dense state, it will remain dense until the minimum mass $M_{\min }$ at which it will explode and become dilute. We show in Appendix pp11 that the transition mass $M_t(\delta )$, which is by definition in the interval $[M_{\min }(\delta ),M_{\max }(\delta )]$, approaches the mass $M_{*}(\delta )$ corresponding to the minimum radius as $\delta \to 0$. We recall, however, that the transition mass $M_t(\delta )$ is usually not physically relevant because of the very long lifetime of metastable states (see the main text).

Figure 26

Mass-radius relation of Newtonian axion stars for different values of the interaction parameter ($\delta ={10}^{-10},{10}^{-5},{\delta }_c,{10}^5,{10}^{10}$). The intersection between the $M(R)$ curve and the black hole line $M=R{c}^2/2G$ ($M=0.0303R/\delta$ in dimensionless form) determines the limit of validity of the Newtonian treatment and the order of magnitude of the general relativistic maximum mass of axion stars. The mass-radius relation (solid line) and the black hole line (dashed line) corresponding to the same $\delta$ can be easily identified by using the fact that the black hole line and the TF curve (dotted line) intersect each other at $M\sim M_0$ (i.e., when the TF curve presents a change of slope).

Figure 27

Mass-radius relation of weakly self-interacting axion stars with $\delta >{\delta }_{*}$ (for illustration, we have taken $\delta ={10}^5$). The Newtonian relation is only valid for $M\ll M_{\max ,\mathrm{GR}}^{\text{dilute}}$ and $R\gg R_{*,\mathrm{GR}}^{\text{dilute}}$. Close to $M_{\max ,\mathrm{GR}}^{\text{dilute}}$, the real (relativistic) mass-radius relation $M(R)$ should form a spiral like in the case of noninteracting boson stars (see Appendix pp1-s5).

Figure 28

Mass-radius relation of strongly self-interacting axion stars with $\delta <{\delta }_{*}$ (for illustration, we have taken ${\delta }_{\mathrm{QCD}}=8.68\times {10}^{-17}$). The Newtonian relation is only valid for $M<M_{\max ,\mathrm{GR}}^{\text{dense}}$. Close to $M_{\max ,\mathrm{GR}}^{\text{dense}}$, the exact (general relativistic) mass-radius relation $M(R)$ may form a spiral or stop suddenly (see footnote 28).

Figure 29

Compactness $R/R_S=0.0303R/(M\delta )$ as a function of the mass $M$ for QCD axion stars ($\delta =8.68\times {10}^{-17}$). For $\delta \to 0$, we find that $(R/R_S)(M_{\max })\sim 0.0303/\delta$ and $(R/R_S)(M_{\min })\sim 0.0114/\delta$, and $(R/R_S{)}_{\mathrm{coll}}\sim 0.0551/{\delta }^{2/3}$. We note that $R/R_S$ is constant ($R/R_S=0.0152/\delta$) on the part of branch (II) corresponding to unstable axion stars where $M\sim 2R$ (nongravitational limit). The axion stars are Newtonian when $M\ll M_{\max ,\mathrm{GR}}^{\text{dense}}$ (i.e., $R/R_S\gg 1$) and general relativistic when $M\sim M_{\max ,\mathrm{GR}}^{\text{dense}}$.

Figure 30

Phase diagram of axion stars using dimensionless variables.

Figure 31

Phase diagram of axion stars in terms of the scattering length $|a_s|$.

Figure 32

Phase diagram of axion stars in terms of the axion decay constant $f$.

Figure 33

Sketch of the phase diagram of axion stars obtained by Helfer et al. [120]. It can be compared with the phase diagram of Fig. 32 obtained from our analytical study.

Figure 34

Mass-radius relation of QCD axion stars with dimensional variables obtained from the Gaussian Ansatz (recall that within the Gaussian Ansatz $R_{99}=2.38167R$). The Newtonian maximum mass and the corresponding radius of dilute axion stars are $M_{\max ,\mathrm{N}}^{\text{dilute}}=6.92\times {10}^{-14}{M}_{\odot }$ and $R_{*,\mathrm{N}}^{\text{dilute}}=71.5\mathrm{km}$. The radius of the dense axion star resulting from the collapse of a dilute axion star with the maximum mass is $R_{\mathrm{coll}}=0.576\mathrm{m}$. The general relativistic maximum mass and the corresponding radius of dense axion stars are $M_{\max ,\mathrm{GR}}^{\text{dense}}=9.60M_{\odot }$ and $R_{*,\mathrm{GR}}^{\text{dense}}=28.3\mathrm{km}$. Close to $M_{\max ,\mathrm{GR}}^{\text{dense}}$, the mass-radius relation $M(R)$ may form a spiral or stop suddenly. The Newtonian minimum mass and the corresponding radius of dense axion stars are $M_{\min ,\mathrm{N}}^{\text{dense}}=1.38\times {10}^{-20}{M}_{\odot }$ and $R_{*,\mathrm{N}}^{\text{dense}}=0.533\mathrm{cm}$. The radius of the dilute axion star resulting from the explosion of a dense axion star with the minimum mass is $R_{\exp }=7.20\times {10}^8\mathrm{km}$, suggesting that the star is dispersed away. From the exact results of Ref. [35], we get $M_{\max ,\mathrm{N}}^{\text{dilute}}=6.46\times {10}^{-14}{M}_{\odot }$ and $R_{*,99,\mathrm{N}}^{\text{dilute}}=227\mathrm{km}$. From the exact results of Ref. [111], we get $M_{\min ,\mathrm{N}}^{\text{dense}}=1.2\times {10}^{-20}{M}_{\odot }$, $R_{*,\mathrm{N}}^{\text{dense}}=1.81\mathrm{cm}$, and $R_{\exp }=2.17\times {10}^9\mathrm{km}$. The authors of Ref. [111] find no solution with $M_{\max }^{\text{dense}}=1.9M_{\odot }$ (see Appendix pp5-s2). From the exact result of Appendix pp5, we get $R_{\mathrm{coll}}=0.756\mathrm{m}$. We have also plotted the lines $M_{<}=2\times {10}^{-9}{M}_{\odot }$ and $M_{>}=15M_{\odot }$ between which MACHOs are excluded from observations [123, 124]. We see that axionic DM can be in the form of gases of axions, dilute axion stars, dense axion stars with $M<2\times {10}^{-9}{M}_{\odot }$, or dense axion stars (or axionic black holes) with $M>15M_{\odot }$.

Figure 35

Mass-radius relation of ULA clusters (this concerns more precisely their solitonic core) with dimensional variables obtained from the Gaussian Ansatz. The Newtonian maximum mass and the corresponding radius of dilute clusters are $M_{\max ,\mathrm{N}}^{\text{dilute}}=1.07\times {10}^8{M}_{\odot }$ and $R_{*,\mathrm{N}}^{\text{dilute}}=0.313\mathrm{kpc}$. The radius of the dense cluster resulting from the collapse of a dilute cluster with the maximum mass is $R_{\mathrm{coll}}=0.567\mathrm{pc}$. The general relativistic maximum mass and corresponding radius of dense clusters are $M_{\max ,\mathrm{GR}}^{\text{dense}}=1.29\times {10}^{15}{M}_{\odot }$ and $R_{*,\mathrm{GR}}^{\text{dense}}=0.124\mathrm{kpc}$. Close to $M_{\max ,\mathrm{GR}}^{\text{dense}}$, the mass-radius relation $M(R)$ may form a spiral or stop suddenly. However, the general relativistic maximum mass $M_{\max ,\mathrm{GR}}^{\text{dense}}=1.29\times {10}^{15}{M}_{\odot }$ is too large to describe the solitonic core of ULA clusters suggesting that ULA clusters never form black holes. The Newtonian minimum mass and corresponding radius of dense clusters are $M_{\min ,\mathrm{N}}^{\text{dense}}=7.19\times {10}^4{M}_{\odot }$ and $R_{*,\mathrm{N}}^{\text{dense}}=0.0789\mathrm{pc}$. The radius of the dilute cluster resulting from the explosion of a dense cluster with the minimum mass is $R_{\exp }=931\mathrm{kpc}$, suggesting that the cluster is dispersed away. From the exact results of Ref. [35], we get $M_{\max ,\mathrm{N}}^{\text{dilute}}={10}^8{M}_{\odot }$ and $R_{*,\mathrm{N}}^{\text{dilute}}=1\mathrm{kpc}$. From the exact results of Ref. [111], we get $M_{\min ,\mathrm{N}}^{\text{dense}}=6.28\times {10}^4{M}_{\odot }$, $R_{*,\mathrm{N}}^{\text{dense}}=0.269\mathrm{pc}$, and $R_{\exp }=2810\mathrm{kpc}$. The authors of Ref. [111] find no solution with $M_{\max }^{\text{dense}}=2.56\times {10}^{14}{M}_{\odot }$ (see Appendix pp5-s2). From the exact result of Appendix pp5, we find $R_{\mathrm{coll}}=0.745\mathrm{pc}$.

Figure 36

Mass-density relation of QCD axion stars with dimensional variables obtained from the Gaussian Ansatz (recall that within the Gaussian Ansatz ${\rho }_{99}=0.0740\rho$). The density of dilute axion stars at the Newtonian maximum mass is $\rho =8.98\times {10}^4\mathrm{g}/{\mathrm{m}}^3$ (the exact value obtained from the results of Ref. [35] is $\rho =2.62\times {10}^3\mathrm{g}/{\mathrm{m}}^3$). The constant density of dense axion stars in the $\mathrm{TF}+\text{nongravitational}$ regime, which includes the dense axion star resulting from the collapse of a dilute axion star with the maximum mass, is ${\rho }_{\text{dense}}=1.73\times {10}^{20}\mathrm{g}/{\mathrm{m}}^3$ (the exact value obtained from the results of Appendix pp5 is ${\rho }_{\text{dense}}=7.07\times {10}^{19}\mathrm{g}/{\mathrm{m}}^3$). The density of dense axion stars at the general relativistic maximum mass is $\rho =2.01\times {10}^{20}\mathrm{g}/{\mathrm{m}}^3$. The density of dense axion stars at the Newtonian minimum mass is $\rho =4.31\times {10}^{19}\mathrm{g}/{\mathrm{m}}^3$ (the exact value obtained from the results of Ref. [111] is $\rho =9.61\times {10}^{17}\mathrm{g}/{\mathrm{m}}^3$). The density of the dilute axion star resulting from the explosion of a dense axion star with the minimum mass is ${\rho }_{\exp }=1.75\times {10}^{-23}\mathrm{g}/{\mathrm{m}}^3$ (the exact value obtained from the results of Ref. [111] is $\rho =5.58\times {10}^{-25}\mathrm{g}/{\mathrm{m}}^3$), suggesting that the star is dispersed away.

Figure 37

Mass-density relation of ULAs clusters (this concerns more precisely their solitonic core) with dimensional variables obtained from the Gaussian Ansatz. The density of dilute axionic clusters at the Newtonian maximum mass is $\rho =5.63\times {10}^{-17}\mathrm{g}/{\mathrm{m}}^3$ (the exact value obtained from the results of Ref. [35] is $\rho =1.62\times {10}^{-18}\mathrm{g}/{\mathrm{m}}^3$). The constant density of dense axionic clusters in the $\mathrm{TF}+\text{nongravitational}$ regime, which includes the dense axionic cluster resulting from the collapse of a dilute axionic cluster with the maximum mass, is ${\rho }_{\text{dense}}=9.47\times {10}^{-9}\mathrm{g}/{\mathrm{m}}^3$ (the exact value obtained from the results of Appendix pp5 is ${\rho }_{\text{dense}}=3.88\times {10}^{-9}\mathrm{g}/{\mathrm{m}}^3$). The density of dense axionic clusters at the general relativistic maximum mass is $\rho =1.09\times {10}^{-8}\mathrm{g}/{\mathrm{m}}^3$. The density of dense axionic clusters at the Newtonian minimum mass is $\rho =2.37\times {10}^{-9}\mathrm{g}/{\mathrm{m}}^3$ (the exact value obtained from the results of Ref. [111] is $\rho =5.21\times {10}^{-11}\mathrm{g}/{\mathrm{m}}^3$). The density of the dilute axionic cluster resulting from the explosion of a dense axionic cluster with the minimum mass is ${\rho }_{\exp }=1.44\times {10}^{-30}\mathrm{g}/{\mathrm{m}}^3$ (the exact value obtained from the results of Ref. [111] is $\rho =4.57\times {10}^{-32}\mathrm{g}/{\mathrm{m}}^3$), suggesting that the cluster is dispersed away.

Figure 38

Collapse time of axion stars as a function of $\delta$ for $M=2$ and $R_0=1$. We note that it presents a small maximum at $\delta =6.16\times {10}^{-4}$ before decreasing. The dashed line corresponds to the result obtained in Ref. [73] by taking $\delta =0$. It provides an excellent approximation of the collapse time of QCD axions (${\delta }_{\mathrm{QCD}}=8.68\times {10}^{-17}$) and ULAs (${\delta }_{\mathrm{ULA}}=9.93\times {10}^{-10}$).