Dynamical boson stars
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Abstract
The idea of stable, localized bundles of energy has strong appeal as a model for particles. In the 1950s, John Wheeler envisioned such bundles as smooth configurations of electromagnetic energy that he called geons, but none were found. Instead, particlelike solutions were found in the late 1960s with the addition of a scalar field, and these were given the name boson stars. Since then, boson stars find use in a wide variety of models as sources of dark matter, as black hole mimickers, in simple models of binary systems, and as a tool in finding black holes in higher dimensions with only a single Killing vector. We discuss important varieties of boson stars, their dynamic properties, and some of their uses, concentrating on recent efforts.
Keywords
Numerical relativity Boson stars Solitons1 Introduction
Particlelike objects have a very long and broad history in science, arising long before Newton’s corpuscles of light, and spanning the range from fundamental to astronomical. In the mid1950s, John Wheeler sought to construct stable, particlelike solutions from only the smooth, classical fields of electromagnetism coupled to general relativity (Wheeler 1955; Power and Wheeler 1957). Such solutions would represent something of a “gravitational atom”, but the solutions Wheeler found, which he called geons, were unstable (but see the discussion of geons in AdS in Sect. 6.3). However, in the following decade, Kaup replaced electromagnetism with a complex scalar field (Kaup 1968), and found Klein–Gordon geons that, in all their guises, have become wellknown as today’s boson stars (see Sect. II of Schunck and Mielke 2003 for a discussion of the naming history of boson stars).
As compact, stationary configurations of scalar field bound by gravity, boson stars are called upon to fill a number of different roles. Most obviously, could such solutions actually represent astrophysical objects, either observed directly or indirectly through its gravity? Instead, if constructed larger than a galaxy, could a boson star serve as the dark matter halo that explains the flat rotation curve observed for most galaxies?
The equations describing boson stars are relatively simple, and so even if they do not exist in nature, they still serve as a simple and important model for compact objects, ranging from particles to stars and galaxies. In all these cases, boson stars represent a balance between the dispersive nature of the scalar field and the attraction of gravity holding it together.
This review is organized as follows. The rest of this section describes some general features about boson stars. The system of equations describing the evolution of the scalar field and gravity (i.e., the Einstein–Klein–Gordon equations) are presented in Sect. 2. These equations are restricted to the spherical symmetric case (with a harmonic ansatz for the complex scalar field and a simple massive potential) to obtain a bosonstar family of solutions. To accommodate all their possible uses, a large variety of bosonstar types have come into existence, many of which are described in more detail in Sect. 3. For example, one can vary the form of the scalar field potential to achieve a larger range of masses and compactnesses than with just a mass term in the potential. Certain types of potential admit solitonlike solutions even in the absence of gravity, leading to socalled Qstars. One can adopt Newtonian gravity instead of general relativity, or construct solutions from a real scalar field instead of a complex one. It is also possible to find solutions coupled to an electromagnetic field or a perfect fluid, leading respectively to charged boson stars and fermion–boson stars. Rotating boson stars are found to have an angular momentum which is not arbitrary, but instead quantized, and can even coexist with a Kerr black hole. Multistate boson stars with more than one complex scalar field are also considered. Recently, stars made of a massive vector field have been constructed which more closely match the original geon proposal because such a field has the same unit spin as Maxwell.
We discuss the dynamics of boson stars in Sect. 4. Arguably, the most important property of bosonstar dynamics concerns their stability. Approaches to analyzing their stability include linear perturbation analysis, catastrophe theory, and fully nonlinear, numerical evolutions. The latter option allows for the study of the final state of perturbed stars. Possible endstates include dispersion to infinity of the scalar field, migration from unstable to stable configurations, and collapse to a black hole. There is also the question of formation of boson stars. Full numerical evolutions in 3D allow for the merger of binary boson stars, which display a large range of different behaviors as well producing distinct gravitationalwave signatures.
Finally, we review the impact of boson stars in astronomy in Sect. 5 (as astrophysical objects, black hole mimickers, gravitationalwave sources, and sources of dark matter) and in mathematics in Sect. 6 (appearing in critical behavior, the Hoop conjecture, other dimensions and antide Sitter spacetimes, and gravitational analogs). We conclude with some remarks and future directions.
1.1 The nature of a boson star
There are, of course, many other soliton and solitonlike solutions in three dimensions finding a variety of ways to evade Derrick’s theorem. For example, the fieldtheory monopole of ’t Hooft and Polyakov is a localized solution of a properly gauged triplet scalar field. Such a solution is a topological soliton because the monopole possesses false vacuum energy which is topologically trapped. The monopole is one among a number of different topological defects that requires an infinite amount of energy to “unwind” the potential energy trapped within (see Vilenkin and Shellard 1994 for a general introduction to defects and the introduction of Ryder 1996 for a discussion of relevant classical field theory concepts).
Boson stars are then either a collection of stable fundamental bosonic particles bound by gravity, or else a collection of unstable particles that, with the gravitational binding, have an inverse process efficient enough to reach an equilibrium. They can thus be considered a Bose–Einstein condensate (BEC), although boson stars can also exist in an excited state as well.
Despite their connection to fundamental physics, one can also view boson stars in analogy with models of neutron stars. In particular, as we discuss in the following sections, both types of stars demonstrate somewhat similar mass versus radius curves for their solutions with a transition in stability at local maxima of the mass. There is also a correspondence between (massless) scalar fields and a stiff, perfect fluid (see Sect. 2.1 and Appendix A of Brady et al. 2002), but the correspondence does not mean that the two are equivalent (Faraoni 2012). More than just an analogy, boson stars can serve as a very useful model of a compact star, having certain advantages over a fluid neutron star model: (i) the equations governing its dynamics avoid developing discontinuities, in particular there is no sharp stellar surface, (ii) there is no concern about resolving turbulence, and (iii) one avoids uncertainties in the equation of state.
1.2 Other reviews
A number of other reviews of boson stars have appeared. Most recently, Schunck and Mielke (2003) concentrate on the possibility of detecting BS, extending their previous reviews (Mielke and Schunck 1999, 2002). In 1992, a number of reviews appeared: Jetzer (1992) concentrates on the astrophysical relevance of BS (in particular their relevance for explaining dark matter) while Liddle and Madsen (1992) focus on their formation. Other reviews include Straumann (1992); Lee and Pang (1992). Most recently, Mielke (2016) reviewed rotating boson stars, while Herdeiro and Radu (2015a) reviewed Kerr black holes with scalar hair.
2 Solving for boson stars
In this section, we present the equations governing bosonstar solutions, namely the Einstein equations for the geometry description and the Klein–Gordon equation to represent the (complex) scalar field. We refer to this coupled system as the Einstein–Klein–Gordon (EKG) equations.
The covariant equations describing boson stars are presented in Sect. 2.2, which is followed by choosing particular coordinates consistent with a \(3+1\) decomposition in Sect. 2.3. A form for the potential of the scalar field is then chosen and solutions are presented in Sect. 2.4.
2.1 Conventions
Throughout this review, Roman letters from the beginning of the alphabet \(a,b,c,\ldots \) denote spacetime indices ranging from 0 to 3, while letters near the middle \(i,j,k,\ldots \) range from 1 to 3, denoting spatial indices. Unless otherwise stated, we use units such that \(\hbar =c=1\) so that the Planck mass becomes \(M_{\mathrm {Planck}} = G^{1/2}\). We also use the signature convention \((,+,+,+)\) for the metric.
2.2 The Lagrangian, evolution equations and conserved quantities
2.3 The \(3+1\) decomposition of the spacetime

specify the choice of coordinates The spacetime is foliated by a family of spacelike hypersurfaces, which are crossed by a congruence of time lines that will determine our observers (i.e., coordinates). This congruence is described by the vector field \(t^a = \alpha n^a +\beta ^a\), where \(\alpha \) is the lapse function which measures the proper time of the observers, \(\beta ^a\) is the shift vector that measures the displacement of the observers between consecutive hypersurfaces and \(n^a\) is the timelike unit vector normal to the spacelike hypersurfaces.
 decompose every 4D object into its 3 + 1 components The choice of coordinates allows for the definition of a projection tensor \({\gamma ^a}_b \equiv \delta ^a_b + n^a\, n_b\). Any fourdimensional tensor can be decomposed into \(3+1\) pieces using the spatial projector to obtain the spatial components, or contracting with \(n^a\) for the time components. For instance, the line element can be written in a general form asThe stress–energy tensor can then be decomposed into its various components as$$\begin{aligned} ds^2 =  \alpha ^2\, dt^2 + \gamma _{ij} (dx^i + \beta ^i dt)\, (dx^j + \beta ^j dt). \end{aligned}$$(16)$$\begin{aligned} \tau \equiv T^{ab}\, n_a\, n_b, \quad S_i \equiv T_{ab}\, n^a\,{\gamma ^a}_i, \quad S_{ij} \equiv T_{ab}\, {\gamma ^a}_i\, {\gamma ^b}_j. \end{aligned}$$(17)
 write down the field equations in terms of the 3 + 1 components Within the framework outlined here, the induced (or equivalently, the spatial 3D) metric \(\gamma _{ij}\) and the scalar field \(\phi \) are as yet still unknown (remember that the lapse and the shift just describe our choice of coordinates). In the original \(3+1\) decomposition (ADM formulation Arnowitt et al. 1962) an additional geometrical tensor \(K_{ij} \equiv \left( 1/2 \right) \mathcal{L}_{\mathbf {n}} \gamma _{ij} = 1/\left( 2\alpha \right) \left( \partial _t\mathcal{L}_\beta \right) \gamma _{ij}\) is introduced to describe the change of the induced metric along the congruence of observers. Loosely speaking, one can view the determination of \(\gamma _{ij}\) and \(K_{ij}\) as akin to the specification of a position and velocity for projectile motion. In terms of the extrinsic curvature and its trace, \(\mathrm {trK} \equiv {K_i}^i\), the Einstein equations can be written as$$\begin{aligned}&{R_i}^i + \left( \mathrm {trK}\right) ^2  {K_i}^j\, {K_j}^i = 16\, \pi \, G\, \tau \, \end{aligned}$$(18)$$\begin{aligned}&\nabla _j\;\left( {K_i}^j  \mathrm {trK}\;{\delta _i}^j \right) = 8\, \pi \, G\, S_i \end{aligned}$$(19)In a similar fashion, one can introduce a quantity \(Q \equiv  \mathcal{L}_{\mathbf {n}} \phi \) for the Klein–Gordon equation which reduces it to an equation first order in time, second order in space$$\begin{aligned}&\left( \partial _t  \mathcal {L}_\beta \right) K_{ij} = \, \nabla _i \nabla _j \alpha + \alpha \nonumber \\&\quad \left( R_{ij} 2{K_i}^k {K_{jk}} + \mathrm {trK}\,K_{ij}  8\pi G \left[ S_{ij}{\frac{\gamma _{ij}}{2}}\left( \mathrm {trS}  \tau \right) \right] \right) \qquad \end{aligned}$$(20)$$\begin{aligned} \partial _t (\sqrt{\gamma }\, Q)  \partial _i (\beta ^i \sqrt{\gamma } Q) + \partial _i (\alpha \, \sqrt{\gamma }\, \gamma ^{ij}\, \partial _j \phi ) = \alpha \, \sqrt{\gamma }\, \frac{d V}{d \phi ^2} \phi . \end{aligned}$$(21)
 enforce any assumed symmetries Although the boson star is found by a harmonic ansatz for the time dependence, here we choose to retain the full timedependence. However, a considerably simplification is provided by assuming that the spacetime is spherically symmetric. Following Lai (2004), the most general metric in this case can be written in terms of spherical coordinates aswhere \(\alpha (t,r)\) is the lapse function, \(\beta (t,r)\) is the radial component of the shift vector and a(t, r), b(t, r) represent components of the spatial metric, with \(d\varOmega ^2\) the metric of a unit twosphere. With this metric, the extrinsic curvature only has two independent components \(K^i_j = {\mathrm {diag}} \left( {K^r}_r, {K^{\theta }}_{\theta }, {K^{\theta }}_{\theta } \right) \). The constraint equations, Eqs. (18) and (19), can now be written as$$\begin{aligned} ds^2 = \left(  \alpha ^2 + a^2\, \beta ^2 \right) dt^2 + 2\,a^2\,\beta \,dt\,dr + a^2\, dr^2 + r^2\, b^2\, d\varOmega ^2, \end{aligned}$$(22)$$\begin{aligned}&\frac{2}{a r b} \left\{ \partial _r \left[ \frac{\partial _r (r b)}{a} \right] + \frac{1}{r b} \left[ \partial _r \left( \frac{r b}{a} \partial _r \left( r b\right) \right)  a \right] \right\} + 4 {K^r}_r\, {K^{\theta }}_{\theta } + 2 {K^{\theta }}_{\theta }\, {K^{\theta }}_{\theta }\nonumber \\\end{aligned}$$(23)$$\begin{aligned}&\quad = \frac{8 \pi G}{a^2} \left[ \left \varPhi \right ^2 + \left \varPi \right ^2 + a^2 V\left( \phi ^2\right) \right] \end{aligned}$$(24)where we have defined the auxiliary scalarfield variables$$\begin{aligned}&\partial _r {K^{\theta }}_{\theta } + \frac{\partial _r\left( r b\right) }{r b} \left( {K^{\theta }}_{\theta }  {K^{r}}_{r}\right) = \frac{2 \pi G}{a} \left( \bar{\varPi } \varPhi + \varPi \bar{\varPhi } \right) , \end{aligned}$$(25)The evolution equations for the metric and extrinsic curvature components reduce to$$\begin{aligned} \varPhi \equiv \partial _r \phi , \quad \varPi \equiv \frac{a}{\alpha } \left( \partial _t \phi  \beta \partial _r \phi \right) . \end{aligned}$$(26)$$\begin{aligned}&\partial _t a = \partial _r (a \beta )  \alpha a {K^{r}}_{r} \end{aligned}$$(27)$$\begin{aligned}&\partial _t b = \frac{\beta }{r} \partial _r (r b)  \alpha b {K^{\theta }}_{\theta } \end{aligned}$$(28)Similarly, the reduction of the Klein–Gordon equation to first order in time and space leads to the following set of evolution equations$$\begin{aligned}&\partial _t {K^{r}}_{r}  \beta \partial _r {K^{r}}_{r} =  \frac{1}{a} \partial _r \left( \frac{\partial _r \alpha }{a} \right) \nonumber \\&\quad + \,\alpha \left\{  \frac{2}{a r b} \partial _r \left[ \frac{\partial _r (r b)}{a} \right] + \mathrm {trK}\, {K^r}_r  \frac{4\pi \,G}{a^2} \left[ 2 \varPhi ^2 + a^2 V(\phi ^2) \right] \right\} \nonumber \\&\partial _t {K^{\theta }}_{\theta }  \beta \partial _r {K^{\theta }}_{\theta } = \frac{\alpha }{(r b)^2}  \frac{1}{a (r b)^2} \partial _r \left[ \frac{\alpha r b}{a} \partial _r (r b) \right] \nonumber \\&+ \,\alpha \left[ \mathrm {trK}\, {K^{\theta }}_{\theta }  4\pi \,G V(\phi ^2) \right] . \end{aligned}$$(29)$$\begin{aligned} \partial _t \phi= & {} \beta \varPhi + \frac{\alpha }{a} \varPi \end{aligned}$$(30)$$\begin{aligned} \partial _t \varPhi= & {} \partial _r \left( \beta \varPhi + \frac{\alpha }{a} \varPi \right) \end{aligned}$$(31)This set of equations, Eqs. (23)–(32), describes general, timedependent, spherically symmetric solutions of a gravitationallycoupled complex scalar field. In the next section, we proceed to solve for the specific case of a boson star.$$\begin{aligned} \partial _t \varPi= & {} \frac{1}{(r b)^2} \partial _r \left[ (r b)^2 \left( \beta \varPi + \frac{\alpha }{a} \varPhi \right) \right] + 2 \left[ \alpha {K^{\theta }}_{\theta }  \beta \frac{\partial _r (r b)}{r b} \right] \varPi  \alpha a \frac{d V}{d \phi ^2} \phi .\nonumber \\ \end{aligned}$$(32)
2.4 Miniboson stars
As above, boson stars are spherically symmetric solutions of the Eqs. (38–40) with asymptotic behavior given by Eqs. (41–45). For a given value of the central amplitude of the scalar field \(\phi _0(r=0) =\phi _c\), there exist configurations with some effective radius and a given mass satisfying the previous conditions for a different set of n discrete eigenvalues \(\omega ^{(n)}\). As n increases, one obtains solutions with an increasing number of nodes in \(\phi _0\). The configuration without nodes is the ground state, while all those with any nodes are excited states. As the number of nodes increases, the distribution of the mass as a function of the radius becomes more homogeneous.
As the amplitude \(\phi _c\) increases, the stable configuration has a larger mass while its effective radius decreases. This trend indicates that the compactness of the boson star increases. However, at some point the mass instead decreases with increasing central amplitude. Similar to models of neutron stars (see Sect. 4 of Cook 2000), this turnaround implies a maximum allowed mass for a boson star in the ground state, which numerically was found to be \(M_{\max } = 0.633\,M^2_{\mathrm {Planck}}/m\). The existence of a maximum mass for boson stars is a relativistic effect, which is not present in the Newtonian limit, while the maximum of baryonic stars is an intrinsic property.
3 Varieties of boson stars
Quite a number of different flavours of boson stars are present in the literature. They can have charge, a fermionic component, or rotation. They can be constructed with various potentials for the scalar field. The form of gravity which holds them together can even be modified to, say, Newtonian gravity or even no gravity at all (Qballs). To a certain extent, such modifications are akin to varying the equation of state of a normal, fermionic star. Here we briefly review some of these variations, paying particular attention to recent work.
3.1 Selfinteraction potentials
Originally, boson stars were constructed with a freefield potential without any kind of selfinteraction, obtaining a maximum mass with a dependence \(M \approx M^2_{\mathrm {Planck}}/m\). This mass, for typical masses of bosonic particle candidates, is much smaller than the Chandrasekhar mass \(M_{\mathrm {Ch}} \approx M^3_{\mathrm {Planck}}/m^2\) obtained for fermionic stars, and so they were known as miniboson stars. In order to extend this limit and reach astrophysical masses comparable to the Chandrasekhar mass, the potential was generalized to include a selfinteraction term that provided an extra pressure against gravitational collapse. To preserve the global U(1) invariance, and hence to retain a conserved particle number, such a potential should be a function of \(\phi \).
Many subsequent papers further analyze the EKG solutions with polynomial, or even more general nonpolynomial, potentials. One work in particular (Schunck and Torres 2000) studied the properties of the galactic dark matter halos modeled with these boson stars. They found that a necessary condition to obtain stable, compact solutions with an exponential decrease of the scalar field, the series expansion of these potentials must contain the usual mass term \(m^2\phi ^2\).
More exotic ideas similarly try to include a pressure to increase the mass of BSs. Agnihotri et al. (2009) considers a form of repulsive selfinteraction mediated by vector mesons within the meanfield approximation. However, the authors leave the solution of the fully nonlinear system of the Klein–Gordon and Proca equations to future work. Barranco and Bernal (2011) models stars made from the condensation of axions, using the semirelativistic approach with two different potentials. Mathematically this approach involves averages such that the equations are equivalent to assuming the axion is constituted by a complex scalar field with harmonic time dependence.
It has been shown recently that very compact boson stars can also be found by using a Vshaped potential proportional to \(\phi \) (Hartmann et al. 2012). The same Vshaped potential with an additional quadratic massive term has been considered in Kumar et al. (2015)
Bhatt and Sreekanth (2009) considers a chemical potential to construct BSs, arguing that the effect of the chemical potential is to reduce the parameter space of stable solutions. Boson stars with a thermodynamically consistent equation of state, leading to an isotropic pressure, were considered in Chavanis and Harko (2012). The solutions, obtained by integrating the TOV equations, reached compactness smaller (but comparable) to neutron stars. The extension to boson stars with finite temperature was considered in Latifah et al. (2014).
Related work modifies the kinetic term of the action instead of the potential. Adam et al. (2010) studies the resulting BSs for a class of K field theories, finding solutions of two types: (i) compact balls possessing a naked singularity at their center and (ii) compact shells with a singular inner boundary which resemble black holes. Akhoury and Gauthier (2008) considers coherent states of a scalar field instead of a BS within kessence in the context of explaining dark matter. Dzhunushaliev et al. (2008) modifies the kinetic term with just a minus sign to convert the scalar field to a phantom field. Although, a regular real scalar field has no spherically symmetric, local static solutions, they find such solutions with a real phantom scalar field.
3.2 Newtonian boson stars
The possibility of including selfinteraction terms in the potential was considered in Guzmán and UreñaLópez (2006), studying also the gravitational cooling (i.e., the relaxation and virialization through the emission of scalar field bursts) of spherical perturbations. Nonspherical perturbations were further studied in Bernal and Guzmán (2006b), showing that the final state is a spherically symmetric configuration. Single Newtonian boson stars were studied in Guenther (1995), either when they are boosted with/without an external central potential. Rotating stars were first successfully constructed in Silveira and Sousa (1995) within the Newtonian approach, and, more recently, analytical approximate solutions for rotating boson stars in four and five dimensions has been achieved (Kan and Shiraishi 2016). Numerical evolutions of binary boson stars in Newtonian gravity are discussed in Sect. 4.2.
Recent work by Chavanis with Newtonian gravity solves the Gross–Pitaevskii equation, a variant of Eq. (58) which involves a pseudopotential for a Bose–Einstein condensate, to model either dark matter or compact alternatives to neutron stars (Chavanis 2011, 2012, 2015; Chavanis and Harko 2012; Chavanis and Matos 2017). However, see a rebuttal to some of this work (Mukherjee et al. 2015).
Much recent work considers boson stars from a quantum perspective as a Bose–Einstein condensate involving some number, P, of scalar fields. Michelangeli and Schlein (2012) studies the collapse of boson stars mathematically in the mean field limit in which \(P \rightarrow \infty \). Kiessling (2009) argues for the existence of bosonic atoms instead of stars. Bao and Dong (2011) uses numerical methods to study the mean field dynamics of BSs.
3.3 Charged boson stars
We look for a time independent metric by first assuming a harmonically varying scalar field as in Eq. (33). We work in spherical coordinates and assume spherical symmetry. With a proper gauge choice, the vector potential takes a particularly simple form with only a single, nontrivial component \(A_a=\left( A_0(r),0,0,0\right) \). This choice implies an everywhere vanishing magnetic field so that the electromagnetic field is purely electric. The boundary conditions for the vector potential are obtained by requiring that the electric field vanishes at the origin because of regularity, \(\partial _r A_0 (r=0) = 0\). Because the electromagnetic field depends only on derivatives of the potential, we can use this freedom to set \(A_0 (\infty ) = 0\) (Jetzer and van der Bij 1989).
With these conditions, it is possible to find numerical solutions in equilibrium as described in Jetzer and van der Bij (1989). It was shown that bound stable configurations exist only for values of the coupling constant less than or equal to a certain critical value, such that solutions are found for \({\tilde{e}}^2 \equiv e^2\,M^2_{\mathrm {Planck}}/(8\,\pi \,m^2)<1/2\). For \({\tilde{e}}^2 >1/2\) the repulsive Coulomb force is bigger than the gravitational attraction and no solutions were found, although it has been shown recently that, due to the binding energy, solutions with \({\tilde{e}}^2 =1/2\) and even slightly higher are also allowed (Pugliese et al. 2013). This bound on the BS charge in terms of its mass ensures that one cannot construct an overcharged BS, in analogy to the overcharged monopoles of Lue and Weinberg (2000). An overcharged monopole is one with more charge than mass and is, therefore, susceptible to gravitational collapse by accreting sufficient (neutral) mass. However, because its charge is higher than its mass, such collapse might lead to an extremal Reissner–Nordström BH, but BSs do not appear to allow for this possibility. Interestingly, Sakai and Tamaki (2012) finds that if one removes gravity, the obtained Qballs may have no limit on their charge.
Recent work with charged BSs includes the publication of Maple^{1} routines to study boson nebulae charge (Dariescu and Dariescu 2010; Murariu and Puscasu 2010; Murariu et al. 2008) and charged boson stars in the presence of a cosmological constant (Kumar et al. 2016). New regular solutions of charged scalar fields in a cavity are presented in Ponglertsakul et al. (2016), which are stable only when the radius of the mirror is sufficiently large.
Other work generalizes the Qballs and Qshells found with a certain potential, which leads to the signumGordon equation for the scalar field (Kleihaus et al. 2009, 2010). In particular, shell solutions can be found with a black hole in its interior, which has implications for black hole scalar hair (for a review of black hole uniqueness see Chruściel et al. 2012).
One can also consider Qballs coupled to an electromagnetic field, a regime appropriate for particle physics. Within such a context, Eto et al. (2011) studies the chiral magnetic effect arising from a Qball. Other work in Brihaye et al. (2009) studies charged, spinning Qballs.
Charged BSs in antide Sitter spacetimes have attracted some interest as noted at the end of Sect. 6.3.
3.4 Oscillatons
As mentioned earlier, it is not possible to find timeindependent, spacetime solutions for a real scalar field. However, there are nonsingular, timedependent nearequilibrium configurations of selfgravitating real scalar fields, which are known as oscillatons (Seidel and Suen 1991). These solutions are similar to boson stars, with the exception that the spacetime must also have a time dependence in order to avoid singularities.
Although the geometry is oscillatory in nature, these oscillatons behave similarly to BSs. In particular, they similarly transition from longlived solutions to a dynamically unstable branch separated at the maximum mass \(M_{\max } = 0.607\,M^2_{\mathrm {Planck}}/m\). Figure 6 displays the total mass curve, which shows the mass as a function of central value. Compact solutions can be found in the Newtonian framework when the weak field limit is performed appropriately, reducing to the socalled Newtonian oscillations (UreñaLópez et al. 2002). The dynamics produced by perturbations are also qualitatively similar, including gravitational cooling, migration to more dilute stars, and collapse to black holes (Alcubierre et al. 2003). More recently, these studies have been extended by considering the evolution in 3D of excited states (Balakrishna et al. 2008) and by including a quartic selfinteraction potential (ValdezAlvarado et al. 2011). In Kichenassamy (2008), a variational approach is used to construct oscillatons in a reduced system similar to that of the sineGordon breather solution. Such localized solutions have also been constructed in AdS (see Sect. 6.3), and numerical evolutions suggest that they are stable below some critical density (Fodor et al. 2015).
Closely related, are oscillons that exist in flatspace and that were first mentioned as “pulsons” in Bogolyubskiĭ and Makhan’kov (1977). And so just as a Qball can be thought of as a BS without gravity, an oscillon is an oscillaton in the absence of gravity. Extensive literature studies such solutions, many of which appear in Fodor et al. (2008). A series of papers establishes that oscillons similarly radiate on very long time scales (Fodor et al. 2008, 2009a, b, c). An interesting numerical approach to evolving oscillons adopts coordinates that blueshift and damp outgoing radiation of the massive scalar field (Honda 2000; Honda and Choptuik 2002). A detailed look at the long term dynamics of these solutions suggests the existence of a fractal boundary in parameter space between oscillatons that lead to expansion of a truevacuum bubble and those that disperse (Honda 2010). Dymnikova et al. (2000) examines the collision of two of these bubbles in the context of a first order phase transition. The reheating phase of inflationary cosmology generally feature oscillons which may produce observable gravitational waves (Antusch et al. 2017; Antusch and Orani 2016).
3.5 Rotating boson stars
Recently, their stability properties were found to be similar to nonrotating stars (Kleihaus et al. 2012). Rotating boson stars have been shown to develop a strong ergoregion instability when rapidly spinning on short characteristic timescales (i.e., 0.1 s–1 week for objects with mass \(M=110^{6}\,M_{\odot }\) and angular momentum \(J> 0.4\,G M^2\)), indicating that very compact objects with large rotation are probably black holes (Cardoso et al. 2008). The presence of light rings (i.e., a region of space where photons are forced to travel in closed orbits) around rotating boson stars is studied in Grandclément (2016), while geodesics on the spacetime of these solutions are studied in Grandclément et al. (2014).
A recent review by Mielke (2016) focuses on rotating boson stars. Further discussion concerning the numerical methods and limitations of some of these approaches can also be found in the Ph.D. thesis by Lai (2004).
3.6 Fermion–boson stars
The possibility of compact stellar objects made with a mixture of bosonic and fermionic matter was studied in Henriques et al. (1989, 1990a). In the simplest case, the bosonic component interacts with the fermionic component only via the gravitational field, although different couplings were suggested in Henriques et al. (1990a) and have been further explored in de Sousa et al. (1998), Pisano and Tomazelli (1996). Such a simple interaction is, at the very least, consistent with models of a bosonic dark matter coupling only gravitationally with visible matter, and the idea that such a bosonic component would become gravitationally bound within fermionic stars is arguably a natural expectation.
The existence of slowly rotating fermion–boson stars was shown in de Sousa et al. (2001), although no solutions were found in previous attempts (Kobayashi et al. 1994). Also see Dzhunushaliev et al. (2011) for unstable solutions consisting of a real scalar field coupled to a perfect fluid with a polytropic equation of state.
3.7 Multistate boson stars
It turns out that excited BSs, as dark matter halo candidates, provide for flatter, and hence more realistic, galactic rotation curves than ground state BSs. The problem is that they are generally unstable to decay to their ground state. Combining excited states with the ground states in what are called multistate BSs is one way around this.
Similar results were found in the Newtonian limit (UreñaLópez and Bernal 2010), however, with a slightly higher stability limit \(N^{(1)} \ge 1.13\, N^{(2)}\). This work stresses that combining several excited states makes it possible to obtain flatter rotation curves than only with ground state, producing better models for galactic dark matter halos (see also discussion of boson stars as an explanation of dark matter in Sect. 5.4).
Hawley and Choptuik (2003) considers two scalar fields describing boson stars that are phase shifted in time with respect to each other, studying the dynamics numerically. In particular, one can consider multiple scalar fields with an explicit interaction (beyond just gravity) between them, say \(V\left( \phi ^{(1)}\, \phi ^{(2)} \right) \). Brihaye et al. (2009), Brihaye and Hartmann (2009) construct such solutions, considering the individual particlelike configurations for each complex field as interacting with each other.
3.8 Proca stars
3.9 Kerr black holes with scalar hair and superradiance
Closely related to a BS, one can instead construct stable configurations of a complex scalar field around a rotating black hole (Hod 2012). Such solutions are akin to a BS with a black hole embedded at its center. As such, the scalar field serves as scalar hair (see a recent review about noscalarhair theorems by Herdeiro and Radu 2015b).
Instead of solving the full system of equations, a first approximation can be obtained by solving the linearized scalar field equations on a fixed spacetime (Herdeiro and Radu 2015a). Within such a linear approximation, one finds that nonrotating (Schwarzschild) BHs do not allow for bound states with strictly real \(\omega \) (Herdeiro and Radu 2014b). However, quasibound states can exist with \(\mathfrak {I}(\omega ) <0\) in which the scalar field decays, infalling into the BH.
For a Kerr black hole with angular momentum J, mass M, and horizon radius in the equatorial plane \(r_H\), one can identify the angular velocity of the horizon as \(\varOmega _H\equiv J/(2 M^2 r_H)\). For such rotating BHs, there is a critical frequency \(\omega _c \equiv m \varOmega _H\) separating disparate behavior. For \(\omega =\omega _c\), the frequency is strictly real allowing for regular bound states known as scalar clouds.
As \(\omega \) increases above \(\omega _c\), its imaginary part becomes negative, allowing again only for quasibound states with a timedecaying scalar field. In contrast, as \(\omega \) decreases below \(\omega _c\), its imaginary part becomes positive, indicating growth of the scalar field in Eq. (83). This growth of the massive field is called the superradiant instability (for a recent review of superradiance see Brito et al. 2015) and results in the extraction of energy, charge, and angular momentum from the black hole. For a rigorous treatment of this instability and a proof of boundedness see the work of Dafermos et al. (2014).
In Kühnel and Rampf (2014), an analog of a boson star (see Sect. 6.4 for physical analogs of BSs) is used to study superradiance. BSs have also been found as the zero radius limit of hairy black holes in AdS\({}_4\) (see Sect. 6.3 for BSs in AdS), and these hairy BHs are proposed as the end state of the superradiant instability (Dias and Masachs 2017).
These solutions persist when solving the fully nonlinear system in which the harmonic ansatz of Eq. (83) implies that the stress–energy tensor is independent of \(\{t,\phi \}\), and are generically known as Kerr BHs with scalar hair (Herdeiro and Radu 2014b). As reviewed by Herdeiro and Radu (2015a), solutions can be parametrized in such a way that connects pure Kerr BHs with pure BSs. In particular, defining \(q \equiv kN/J\) where N is the number of bosonic particles as in Eq. (15) and where k is the integer “quantum” number associated with the angular momentum as in Eq. (71), Kerr BHs are described by the vanishing of the scalar field, \(q=0\), and BSs are described by the vanishing of the horizon, \(q=1\). Fig. 11 shows the space of solutions interpolating between these two limits.
More recent work has extended these solutions. For example, a selfinteracting potential with a quartic term was considered in Herdeiro et al. (2015b, 2016b), producing a larger amplitude scalar field but not a more massive black hole than with the nonselfinteracting potential. Coupling the scalar field to the electromagnetic field allows for charged clouds (Delgado et al. 2016). Kerr black holes with Proca hair (see Sec. 3.8 for a description of Proca stars) were constructed in Herdeiro et al. (2016a). Superradiant instabilities are likely to be weaker for hairy black holes than for Kerr black holes with the same global charge (Herdeiro and Radu 2014a). A recent review on the physical properties of Kerr black holes with scalar hair can be found in Herdeiro and Radu (2015a), and prospects for testing whether BHs have hair is reviewed in Cardoso and Gualtieri (2016).
3.10 Alternative theories of gravity
Instead of modifying the scalar field potential, one can consider alternative theories of gravity. Constraints on such theories are already significant given the great success of general relativity (Will 2014), and more strict bounds might be set with present and future astrophysical observations (Berti et al. 2015). However, the fast advance of electromagnetic observations and the rise of gravitationalwave astronomy promise much more in this area, in particular in the context of compact objects that probe strongfield gravity.
An ambitious effort is begun in Pani et al. (2011), which studies a very general gravitational Lagrangian (“extended scalar–tensor theories”) with both fluid stars and boson stars. The goal is for observations of compact stars to constrain such theories of gravity. General theoretical bounds on the mass to radius ratio of stable compact objects (i.e., both neutron and boson stars) can be set for extended gravity theories, in particular for scalar tensor theories (Burikham et al. 2016).
It has been found that scalar tensor theories allow for spontaneous scalarization in which the scalar component of the gravity theory transitions to a nontrivial configuration analogously to ferromagnetism with neutron stars (Damour and EspositoFarèse 1996). Such scalarization is also found to occur in the context of bosonstar evolution (Alcubierre et al. 2010) and scalarized hairy black holes (Kleihaus et al. 2015).
One motivation for alternative theories is to explain the apparent existence of dark matter without resorting to some unknown dark matter component. Perhaps the most well known of these is MOND (modified Newtonian dynamics) in which gravity is modified only at large distances (Milgrom 1983, 2011) (for a review see Famaey and McGaugh 2012). A nonminimal coupling of the scalar field to the Ricci curvature scalar results in configurations that better resemble dark energy stars than ordinary boson stars (Horvat and Marunović 2013; Marunović 2015) . Boson stars are studied within TeVeS (Tensor–Vector–Scalar), a relativistic generalization of MOND (Contaldi et al. 2008). In particular, their evolutions of boson stars develop caustic singularities, and the authors propose modifications of the theory to avoid such problems. Bosons star solutions also exist in biscalar extensions of Horndeski gravity (Brihaye et al. 2016), in the framework of teleparallel gravity (Horvat et al. 2015), and within conformal gravity and its scalar–tensor extensions (Brihaye and Verbin 2009, 2010). Charged boson stars with torsioncoupled field have been considered in Horvat et al. (2015).
Recently there has been renewed interest in Einstein–Gauss–Bonet theory, which appears naturally in the low energy effective action of quantum gravity models. This theory only differs from General Relativity for dimensions \(D>4\), and so the easiest nontrivial case is to consider \(D=5\). Boson star have been found in (\(4+1\))dimensional Gauss–Bonnet gravity (Hartmann et al. 2013). Rotating configurations were constructed in Brihaye and Riedel (2014), and its classical instability and existence of ergoregions studied in Brihaye and Hartmann (2016). Rotating boson stars in odddimensional asymptotically antide Sitter spacetimes in Einstein–Gauss–Bonnet gravity are studied in Henderson et al. (2015). A nonminimal coupling between a complex scalar field and the Gauss–Bonnet term was studied in Baibhav and Maity (2017). Coupling Einstein gravity to a complex selfinteracting boson field as well as a phantom field allows for new type of configurations, namely boson stars harboring a wormhole at their core (Dzhunushaliev et al. 2014).
3.11 Gauged boson stars
In 1988, Bartnik and McKinnon published quite unexpected results showing the existence of particlelike solutions within SU(2) Yang–Mills coupled to gravity (Bartnik and McKinnon 1988). These solutions, although unstable, were unexpected because no particlelike solutions are found in either the Yang–Mills or gravity sectors in isolation. Recall also that no particlelike solutions were found with gravity coupled to electromagnetism in early efforts to find Wheeler’s geon (however, see Sect. 6.3 for discussion of Dias et al. (2012), which finds geons within AdS).
Bartnik and McKinnon generalize from the Abelian U(1) gauge group to the nonAbelian SU(2) group and thereby find these unexpected particlelike solutions. One can consider, as does Schunck and Mielke (2003) (see Sect. IIp), these globally regular solutions (and their generalizations to SU(n) for \(n>2\)) as gauged boson stars even though these contain no scalar field. One can instead explicitly include a scalar field doublet coupled to the Yang–Mills gauge field (Brihaye et al. 2005) as perhaps a more direct generalization of the (U(1)) charged boson stars discussed in Sect. 3.3.
Dzhunushaliev et al. (2007) studies BSs formed from a gauge condensate of an SU(3) gauge field, and Brihaye and Verbin (2010) extends the Bartnik–McKinnon solutions to conformal gravity with a Higgs field.
4 Dynamics of boson stars
In this section, the formation, stability and dynamical evolution of boson stars are discussed. One approach to the question of stability considers small perturbations around an equilibrium configuration, so that the system remains in the linearized regime. Growing modes indicate instability. However, a solution can be linearly stable and yet have a nonlinear instability. One example is Minkowski space, which, under small perturbations, relaxes back to flat, but, for sufficiently large perturbations, leads to blackhole formation, decidedly not Minkowski. To study nonlinear stability, other methods are needed. In particular, full numerical evolutions of the Einstein–Klein–Gordon (EKG) equations are quite useful for understanding the dynamics of boson stars.
4.1 Gravitational stability
A linear stability analysis consists of studying the time evolution of infinitesimal perturbations about an equilibrium configuration, usually with the additional constraint that the total number of particles must be conserved. In the case of spherically symmetric, fermionic stars described by a perfect fluid, it is possible to find an eigenvalue equation for the perturbations that determines the normal modes and frequencies of the radial oscillations (see, for example, Font et al. 2002). Stability theorems also allow for a direct characterization of the stability branches of the equilibrium solutions (Friedman et al. 1988; Cook et al. 1994). Analogously, one can write a similar eigenvalue equation for boson stars and show the validity of similar stability theorems. In addition to these methods, the stability of boson stars has also been studied using mainly two other, independent methods: by applying catastrophe theory and by solving numerically the time dependent Einstein–Klein–Gordon equations. Recently, a method utilizing information theory shows promise in analyzing the stability of equilibrium configurations. All these methods agree with the results obtained in the linear stability analysis.
4.1.1 Linear stability analysis
The stability of the star depends crucially on the sign of the smallest eigenvalue. Because of time reversal symmetry, only \(\sigma ^2\) enters the equations (Lee and Pang 1989), and we label the smallest eigenvalue \(\sigma _0^2\). If it is negative, the eigenmode grows exponentially with time and the star is unstable. On the other hand, for positive eigenvalues the configuration has no unstable modes and is therefore stable. The critical point at which the stability transitions from stable to unstable therefore occurs when the smallest eigenvalue vanishes, \(\sigma _0 = 0\).
Linear perturbation analysis provides a more detailed picture such as the growth rates and the eigenmodes of the perturbations. For instance, Macedo et al. (2013a) studies the free oscillation spectra of different types of boson stars via perturbation theory.
Gleiser and Watkins (1989) carries out such an analysis for perturbations that conserve mass and charge. They find the first three perturbative modes and their growth rates, and they identify at which precise values of \(\phi _c\) these modes become unstable. Starting from small values, they find that ground state BSs are stable up to the critical point of maximum mass. Further increases in the central value subsequently encounter additional unstable modes. This same type of analysis applied to excited state BSs showed that the same stability criterion applies for perturbations that conserve the total particle number (Jetzer 1989c). For more general perturbations that do not conserve particle number, excited states are generally unstable to decaying to the ground state.
A more involved analysis by Lee and Pang (1989) uses a Hamiltonian formalism to study BS stability. Considering first order perturbations that conserve mass and charge (\(\delta N = 0\)), their results agree with those of Gleiser and Watkins (1989); Jetzer (1989c). However, they extend their approach to consider more general perturbations which do not conserve the total number of particles (i.e., \(\delta N \ne 0\)). To do so, they must work with the second order quantities. They found complex eigenvalues for the excited states that indicate that excited state boson stars are unstable. More detail and discussion on the different stability analysis can be found in Jetzer (1992).
Catastrophe theory is part of the study of dynamical systems that began in the 1960s and studies large changes in systems resulting from small changes to certain important parameters (for a physicsoriented review see Stewart 1982). Its use in the context of boson stars is to evaluate stability, and to do so one constructs a series of solutions in terms of a limited and appropriate set of parameters. Under certain conditions, such a series generates a curve smooth everywhere except for certain points. Within a given smooth expanse between such singular points, the solutions share the same stability properties. In other words, bifurcations occur at the singular points so that solutions after the singularity gain an additional, unstable mode. Much of the recent work in this area confirms the previous conclusions from linear perturbation analysis (Tamaki and Sakai 2010, 2011a, b, c) and from earlier work with catastrophe theory (Kusmartsev et al. 1991). Another recent work using catastrophe theory finds that rotating stars share a similar stability picture as nonrotating solutions (Kleihaus et al. 2012). However, only fast spinning stars are subject to an ergoregion instability (Cardoso et al. 2008).
A recent and promising alternative method to determine the stability bounds of selfgravitating astrophysical objects, and in particular of boson stars, makes use of a new measure of shape complexity known as configurational entropy (Gleiser and Jiang 2015). Their results for the critical stability region agree with those of traditional perturbation methods with an accuracy of a few percent or better.
4.1.2 Nonlinear stability of single boson stars
The dynamical evolution of spherically symmetric perturbations of boson stars has also been studied by solving numerically the Einstein–Klein–Gordon equations (Sect. 2.3), or its Newtonian limit (Sect. 3.2), the Schrödinger–Poisson system. The first such work was Seidel and Suen (1990) in which the stability of the ground state was studied by considering finite perturbations, which may change the total mass and the particle number (i.e., \(\delta N \ne 0\) and \(\delta M \ne 0\)). The results corroborated the linear stability analysis in the sense that they found a stable and an unstable branch with a transition between them at a critical value, \(\phi _{\mathrm {crit}}\), of the central scalar field corresponding to the maximal BS mass \(M_{\max }=0.633\,M^2_{\mathrm {Planck}}/m\).
The perturbed unstable configurations will either collapse to a black hole or migrate to a stable configuration, depending on the nature of the initial perturbation. If the density of the star is increased, it will collapse to a black hole. On the other hand, if it is decreased, the star explodes, expanding quickly as it approaches the stable branch. Along with the expansion, energy in the form of scalar field is radiated away, leaving a very perturbed stable star, less massive than the original unstable one.
More recently, the stability of the ground state was revisited with 3D simulations using a Cartesian grid (Guzmán 2004). The Einstein equations were written in terms of the BSSN formulation (Shibata and Nakamura 1995; Baumgarte and Shapiro 1999), which is one of the most commonly used formulations in numerical relativity. Intrinsic numerical error from discretization served to perturb the ground state for both stable and unstable stars. It was found that unstable stars with negative binding energy would collapse and form a black hole, while ones with positive binding energy would suffer an excess of kinetic energy and disperse to infinity.
That these unstable stars would disperse, instead of simply expanding into some less compact stable solution, disagrees with the previous results of Seidel and Suen (1990), and was subsequently further analyzed in Guzmán (2009) in spherical symmetry with an explicit perturbation (i.e., a Gaussian shell of particles, which increases the mass of the star around 0.1%). The spherically symmetric results corroborated the previous 3D calculations, suggesting that the slightly perturbed configurations of the unstable branch have three possible endstates: (i) collapse to BH, (ii) migration to a less dense stable solution, or (iii) dispersal to infinity, dependent on the sign of the binding energy.
Closely related is the work of Lai and Choptuik (2007) studying BS critical behavior (discussed in Sect. 6.1). They tune perturbations of boson stars so that dynamically the solution approaches some particular unstable solution for some finite time. They then study evolutions that ultimately do not collapse to BH, socalled subcritical solutions, and find that they do not disperse to infinity, instead oscillating about some less compact, stable star. They show results with increasingly distant outer boundary that suggest that this behavior is not a finiteboundaryrelated effect (reproduced in Fig. 14). They use a different form of perturbation than Guzmán (2009), and, being only slightly subcritical, may be working in a regime with nonpositive binding energy. However, it is interesting to consider that if indeed there are three distinct endstates, then one might expect critical behavior in the transition among the different pairings. Nonspherical perturbations of boson stars have been studied numerically in Balakrishna et al. (2006) with a 3D code to analyze the emitted gravitational waves.
Much less is known about rotating BSs, which are more difficult to construct and to evolve because they are, at most, axisymmetric, not spherically symmetric. However, as mentioned in Sect. 3.5, they appear to have both stable and unstable branches (Kleihaus et al. 2012) and are subject to an ergoregion instability at high rotation rates (Cardoso et al. 2008). To our knowledge, no one has evolved rotating BS initial data. However, as discussed in the next section, simulations of mini BS binaries (Mundim 2010; Palenzuela et al. 2008) have found rotating boson stars as a result of merger.
4.2 Dynamics of binary boson stars
The dynamics of binary boson stars is sufficiently complicated that it generally requires numerical solutions. The necessary lack of symmetry and the resolution requirement dictated by the harmonic time dependence of the scalar field combine so that significant computational resources must be expended for such a study. However, boson stars serve as simple proxies for compact objects without the difficulties (shocks and surfaces) associated with perfect fluid stars, and, as such, binary BS systems have been studied in the twobody problem of general relativity. When sufficiently distant from each other, the precise structure of the star should be irrelevant as suggested by Damour’s “effacement theorem” (Damour 1987).
First attempts at binary bosonstar simulations assumed the Newtonian limit, since the SP system is simpler than the EKG one. Numerical evolutions of Newtonian binaries showed that in headon collisions with small velocities, the stars merge forming a perturbed star (Choi 1998). With larger velocities, they demonstrate solitonic behavior by passing through each other, producing an interference pattern during the interaction but roughly retaining their original shapes afterwards (Choi 2002). Choi (1998) simulated coalescing binaries, although the lack of resolution in these 3D simulations did not allow for strong conclusions.
The first simulations of boson stars with full general relativity were reported in Balakrishna (1999), where the gravitational waves were computed for a headon collision. The general behavior is similar to the one displayed for the Newtonian limit; the stars attract each other through gravitational interaction and then merge to produce a largely perturbed boson star. However, in this case the merger of the binary was promptly followed by collapse to a black hole, an outcome not possible when working within Newtonian gravity instead of general relativity. Unfortunately, very little detail was given on the dynamics.
Much more elucidating was work in axisymmetry (Lai 2004), in which headon collisions of identical boson stars were studied in the context of critical collapse (discussed in Sect. 6.1) with general relativity. Stars with identical masses of \(M = 0.47 \approx 0.75\,M_{\max }\) were chosen, and so it is not surprising that for small initial momenta the stars merged together to form an unstable single star (i.e., its mass was larger than the maximum allowed mass, \(M_{\max }\)). The unstable hypermassive star subsequently collapsed to a black hole. However, for large initial momentum the stars passed through each other, displaying a form of solitonic behavior since the individual identities were recovered after the interaction. The stars showed a particular interference pattern during the overlap, much like that displayed in Figs. 1 and 16.
Another study considered the very high speed, headon collision of BSs (Choptuik and Pretorius 2010). Beginning with two identical boson stars boosted with Lorentz factors ranging as high as 4, the stars generally demonstrate solitonic behavior upon collision, as shown in the insets of Fig. 25. This work is further discussed in Sect. 6.2.
The orbital case was later studied in Palenzuela et al. (2008). This case is much more involved both from the computational point of view (i.e., there is less symmetry in the problem) and from the theoretical point of view, since for the final object to settle into a stationary, rotating boson star it must satisfy the additional quantization condition for the angular momentum of Eq. (71).
One simulation consisted of an identical pair each with individual mass \(M=0.5\), with small orbital angular momentum such that \(J \le N\). In this case, the binary merges forming a rotating bar that oscillates for some time before ultimately splitting apart. This can be considered as a scattered interaction, which could not settle down to a stable boson star unless all the angular momentum was radiated.
Other simulations of orbiting, identical binaries have been performed within the conformally flat approximation instead of full GR, which neglects gravitational waves (Mundim 2010). Three different qualitative behaviors were found. For high angular momentum, the stars orbit for comparatively long times around each other. For intermediate values, the stars merged and formed a pulsating and rotating boson star. For low angular momentum, the merger produces a black hole. No evidence was found of the stars splitting apart after the merger.
Three dimensional simulations of solitonic core mergers colliding two or more boson stars in the Newtonian limit (Schrödinger–Poisson) are studied in the context of dark matter with different mass ratios, phases and orbital angular momentum (Schwabe et al. 2016). The final core mass does not depend strongly on the phase difference nor on the angular momentum. Cotner (2016) also studies collisions within the Schrödinger–Poisson system and discusses implications for dark matter. However, this work focuses on the headon case and includes effects of different mass ratios, relative phases, selfcouplings, and separation distances. Interestingly, analytic estimates are compared to the numerical simulations (Cotner 2016).
The dynamics of particularly compact boson stars are interesting to contrast with the dynamics of black holes because, at least in part, we now have observations of the gravitational waves from binary BH mergers (discussed more in Sect. 5.3). To this end, the study of the headon collision of solitonic boson stars (which can be quite compact Cardoso et al. 2016) found the dynamics to be qualitatively similar to those observed previously with miniboson stars (Palenzuela et al. 2007). However, the gravitational waves emitted displayed significant differences and, in some cases, closely resembled the signal from a binary black hole merger.
5 Boson stars in astronomy
Scalar fields are often employed by astronomers and cosmologists in their efforts to model the Universe. Most models of inflation adopt a scalar field as the inflaton field, the vacuum energy of which drives the exponential inflation of the Universe. Dark energy also motivates many scalar field models, such as kessence and phantom energy models. It is therefore not surprising that boson stars, as compact configurations of scalar field, are called upon to provide consequences similar to those observed.
5.1 As astrophysical stellar objects
We have already discussed a number of similarities between boson stars and models of neutron stars. Just as one can parameterize models of neutron stars by their central densities, one can consider a 1parameter family of boson stars according to the central magnitude of the scalar field. The mass is then a function of this parameter, and one finds the existence of a local maximum across which solutions transition from stable to unstable, just as is the case for neutron stars. Similarly, models of neutron stars can be constructed with different equations of state, whereas boson stars are constructed with differing scalar field potentials.
One difference of consequence concerns the stellar surface. Neutron stars of course have a surface at which the fluid density is discontinuous, as discussed for example in Gundlach and Leveque (2011), Gundlach and Please (2009). In contrast, the scalar field that constitutes the boson star is smooth everywhere and lacks a particular surface. In its place, one generally defines a radius that encompasses some percentage (e.g. 99%) of the stellar mass. Such a difference could have observational consequences when matter accretes onto either type of star.
It is still an open question whether some of the stars already observed and interpreted as neutron stars could instead be astrophysical boson stars. In a similar fashion, it is not known whether many, if not all, of the stars we observe already have a bosonic component that has settled into the gravitational well of the star (see Sect. 3.6 for a discussion of fermion–boson stars). The bosonic contribution may arise from exotic matter, which could appear at high densities inside the neutron star or from some sort of dark matter accretion (Güver et al. 2014). This possibility has gained popularity over the last years and there have been several attempts to constrain the properties of weakly interacting dark matter particles (WIMPs) by examining signatures related to their accretion and/or annihilation inside stars (for instance, see Kouvaris and Tinyakov 2010 and works cited in the introduction).
Recently, it was suggested that, due to the stronger gravitational field of neutron stars compared to other stars such as white dwarfs and main sequence stars, WIMPs will accrete more efficiently, leading to two different possibilities. If the dark matter is its own antiparticle, it will selfannihilate and heat the neutron star. This temperature increase could be observable in old stars, especially if they are close to the galactic center (Kouvaris and Tinyakov 2010; de Lavallaz and Fairbairn 2010). If WIMPs do not selfannihilate, they will settle in the center of the star forming a fermion–boson star (as discussed in Sect. 3.6). The accretion of dark matter would then increase the star’s compactness until the star collapses (de Lavallaz and Fairbairn 2010) (see discussion of BSs as a source of dark matter in Sect. 5.4). Núñez et al. (2011) follows such work by considering the result of a collision between a BH and a boson star. In particular, they consider the problem as a perturbation of a black hole via scalar accretion and analyze the resulting gravitationalwave output.
Because of the similarities between boson stars and neutron stars, one finds that boson stars are often used in place of the other. This is especially so within numerical work because boson stars are easier to evolve than neutron star models. One can, for example compare the gravitationalwave signature of a bosonstar merger with that of more conventional compact object binaries consisting of BHs and/or NSs. Differentiating BSs from other compact objects with gravitationalwave observations is discussed further in Sect. 5.3.
With the continued advancement in observation, both in the electromagnetic and gravitational spectra, perhaps soon we will have evidence for these questions. At the same time, further study of boson stars can help identify possible distinguishing observational effects in these bands. One example where knowledge is lacking is the interaction between boson stars with a magnetic field. Whereas a neutron star can source its own magnetic field and a neutral star can obtain an induced charge when moving with respect to a magnetic field, we are aware of no studies of the interaction of boson stars with a magnetic field.
5.2 Compact alternatives to black holes
As a localized scalar field configuration, a boson star can be constructed as a noninteracting compact object, as long as one does not include any explicit coupling to any electromagnetic or other fields. In that respect, it resembles a BH, although it lacks a horizon. Can observations of purported BHs be fully explained by massive boson stars? See Psaltis (2008) for a review of such observations.
Neutron stars also lack horizons, but, in contrast to a boson star, have a hard surface. A hard surface is important because one would expect accretion onto such a surface to have observable consequences. Can a boson star avoid such consequences? Yuan et al. (2004) consider the the viability of \(10\,M_{\odot }\) boson stars as BH candidates in Xray binaries. They find that accreting gas collects not at the surface (which the star lacks), but instead at the center, which ultimately should lead to Type I Xray bursts. Because these bursts are not observed, the case against boson stars as black hole mimickers is weakened (at least for BH candidates in Xray binaries).
Guzmán and RuedaBecerril (2009) considers a simplified model of accretion and searches for bosonstar configurations that would mimic an accreting black hole. Although they find matches, they argue that light deflection about a boson star will differ from the BH they mimic because of the lack of a photon sphere. Further work studies the scalar field tails about boson stars and compares them to those of BHs (LoraClavijo et al. 2010). If indeed a boson star collapses to a BH, then one could hope to observe the QNM of the massive scalar field, as described in Hod (2011). Differences between accretion structures surrounding boson stars and black holes are analyzed in Meliani et al. (2015), showing that the accretion tori around boson stars have different characteristics than in the vicinity of a black hole. Further studies on the subject include disk (Meliani et al. 2016) and supersonic winds (GraciaLinares and Guzman 2016) accreting onto boson stars.
Some of the strongest evidence for the existence of BHs is found at the center of most galaxies. Observational evidence strongly suggests supermassive objects (of the order of millions of solar mass) occupying a small region (of order an astronomical unit), which is easily explained by a supermassive BH (Boehle et al. 2012). While definitive evidence for a BH horizon from conventional electromagnetic telescopes is perhaps just on the “horizon” (Johannsen et al. 2016; Broderick et al. 2011), there are those who argue for the viability of supermassive boson stars at galactic centers (Torres et al. 2000). There could potentially be differences in the (electromagnetic) spectrum between a black hole and a boson star, but there is considerable freedom in adjusting the boson star potential to tweak the expected spectrum (Guzmán 2007). However, there are stringent constraints on BH alternatives to Sgr A* by the low luminosity in the near infrared (Broderick and Narayan 2006). In particular, the low luminosity implies a bound on the accretion rate assuming a hard surface radiating thermally and, therefore, the observational evidence favors a black hole because it lacks such a surface. In particular, although a BS lacks a surface, any material it accretes would accumulate and that material would have a surface that would radiate thermally.
At least two possible ways to test the nature of astrophysical black hole candidates are apparent; either with Xray observations or with millimeter very long baseline interferometry (VLBI).
The analysis of Xray reflection spectroscopy with data provided by the current Xray missions can only provide weak constraints on boson stars (Cao et al. 2016), Proca stars (Shen et al. 2017), and hairy Kerr BHs (Ni et al. 2016). The quasiperiodic oscillations (QPOs) observed in the Xray flux emitted by accreting compact objects also provide a powerful tool both to constrain deviations from Kerr and to search for exotic compact objects. Therefore, a future eXTP mission or LOFTlike mission could set very stringent constraints on black holes with bosonic hair and on (scalar or Proca) boson stars (Franchini et al. 2017).
VLBI, on the other hand, may be able to resolve Sgr A*, the closest supermassive black hole located at the center of our galaxy. The Event Horizon Telescope (EHT) promises to resolve angular scales of the order of the horizon scale, and so soon there will be accurate images of the closest surroundings of the supermassive compact object at the center of the Galaxy. These images will allow the study of socalled BH shadows, that is, the gravitational lensing and redshift effect due to the BH on the radiation from background sources.
It has also been shown in Cunha et al. (2015) that hairy Kerr BHs can exhibit very distinct shadows from those of their vacuum counterparts when the light source is sufficiently far away from the BH. These differences remain, albeit less dramatically, when the BH is surrounded by an emitting torus of matter (Vincent et al. 2016).
Other studies have also studied the difference in appearance of a BS with that of the presumed BH in the center of our galaxy. BinNun (2013) argues that, because BSs have an extended mass distribution that is transparent to electromagnetic radiation, the resulting strong gravitational lensing images of the S stars in the galactic center would yield much brighter images than a BH of similar mass. Horvat et al. (2013) studies BSs with a nonminimally coupled scalar field and makes a similar argument about bright images.
One can also consider differences between the motion of celestial bodies about BSs versus BHs. In particular, finding general geodesic motion of test particles in the space–time of boson stars generally requires numerical integration. Geodesics around noncompact boson star were studied in Diemer et al. (2013), finding additional bound orbits of massive test particles close to the center of the star that are not present in the Schwarzschild case and that could be used to make predictions about extrememassratio inspirals (EMRIs), such as the stars orbiting Sagittarius A*. One can also compute the mass parameters of compact objects from redshifts and blueshifts emitted by geodesic particles around them (Becerril et al. 2016). The motion of charged, massive test particles in the spacetime of charged boson stars was considered in Brihaye et al. (2014).
There are other possible BH mimickers, and a popular recent one is the gravastar (Mazur and Mottola 2001). Common among all these alternatives is the lack of an event horizon. Both gravastars and BSs undergo an ergoregion instability for high spin \(\hbox {J}/(\hbox {GM}^2) >0.4\) (Cardoso et al. 2008). As mentioned above for BSs, gravitational waves may similarly be able to distinguish gravastars from BHs (Pani et al. 2009). In order to reach the high compactnesses needed to mimic a BH, one can adopt specialized potentials (Cardoso et al. 2016), but an alternative is to embed the BS within a global monopole as studied in Reid and Choptuik (2016) and Marunović and Murković (2014).
5.3 As source of gravitational waves
The era of gravitationalwave (GW) astronomy began in 2015, precisely 100 years after Einstein’s development of GR. In particular, LIGO directly detected the gravitational waves from the inspiral, merger, and ringdown of a BH binary (Abbott 2016b). This observation has since been followed by a second BH binary and many more observations are expected from the facility (Abbott 2016a). The excitement about these first direct detections should also help ensure the completion of other gravitational wave observatories such as LISA (Armano 2017), KAGRA (Flaminio 2016), and next generation detectors (Abbott 2017).
Now that we have actual GW observations in hand, it behooves us to extract as much science as possible from this new window on the Universe. Much work has already appeared examining the implications of these initial detections (Yunes et al. 2016; Yagi and Stein 2016; Abbott 2016c). In this paper, of course, we are concerned with the implications for BSs: (i) could these extent observations actually represent the signal from a pair of boson stars instead of BHs? (ii) might we observe a signal from boson stars, and, if so, what templates will we need? or (iii) can we place tight bounds excluding the existence of boson stars?
A BS binary system is the most natural GW source. However, at early times, the precise structure of the stars is irrelevant and the signatures are largely the same whether the binary is composed of NSs, BHs, or BSs. However, during the late inspiral and merger, internal structure becomes important. In particular for boson stars, the relative phase determines the GW signature (Palenzuela et al. 2007, 2008; Cardoso et al. 2016).
Gravitational waves may be an ideal messenger for revealing dark matter (discussed in Sect. 5.4). If new dark sector particles can form exotic compact objects (ECOs) of astronomical size, then the first evidence for such objects—and their underlying microphysical description—may arise in gravitationalwave observations. The relationship between the macroscopic properties of ECOs, such as their GW signatures, with their microscopic properties, and hence new particles, was studied in Giudice et al. (2016). The GW efficiency of compact binaries generally is examined in Hanna et al. (2017).
Along the same lines, the tidal Love numbers for different ECOs, including different families of boson stars, are calculated in Cardoso et al. (2017). The tidal Love number, which encodes the deformability of a selfgravitating object within an external tidal field, depends significantly both on the object’s internal structure and on the dynamics of the gravitational field. Present and future gravitationalwave detectors can potentially measure this quantity in a binary inspiral of compact objects and impose constraints on boson stars. Direct numerical simulations in headon collision already have shown similarities in the gravitational waves emitted by black holes and boson stars in some cases (Cardoso et al. 2016). Fig. 23 compares the expected GW signal of a BH binary with various BS binaries.
5.4 As origin of dark matter
Studies of stellar orbits within various galaxies produce rotation curves, which indicate galactic mass within the radius of the particular orbit. The discovery that these curves remain flat at large radius suggests the existence of a large halo of massive, yet dark, matter that holds the galaxy together despite its large rotation (see Feng 2010 for a review). However, what precise form of matter could fulfill the observational constraints is still very much unclear. Scalar fields are an often used tool in the cosmologist’s toolkit, but one cannot have a regular, static configuration of scalar field to serve as the halo (Pena and Sudarsky 1997) (see Dias et al. 2011 as discussed in Sect. 6.3 for a discussion of rotating boson stars with embedded, rotating BH solutions). Instead, some form of boson star represents a possible candidate for providing the necessary dark mass.
Compact binaries are the primary target of LIGO, but instead of neutron stars or black holes, Soni and Zhang (2017) studies the expected signal from binaries consisting of SU(N) glueball objects, one of the simplest models of dark matter. More discussion of the merger of two BSs and the production of GW can be found in Sect. 4.2. At the lower frequencies targeted by LISA, if galaxies generally possess some extended, supermassive configuration, then the inspiral of small compact body into this field will result in both dynamical friction and dark matter accretion, in addition to radiationreaction (Macedo et al. 2013b). These dynamical effects may potentially be encoded on observable gravitational waves from the inspiral.
Boson stars can be matched onto the observational constraints for galactic dark matter halos (Lee 2010a; Sharma et al. 2008). However, multistate boson stars that superpose various bosonstar solutions (e.g., an unexcited solution with an excited solution) can perhaps find better fits to the constraints (UreñaLópez and Bernal 2010). Boson stars at the galactic scale may not exhibit general relativistic effects and can be effectively considered as Bose–Einstein condensates (BEC) with angular momentum (RindlerDaller and Shapiro 2012).
Representing dark matter as BSs also offers certain computational benefits, avoiding some of the costs of modeling the particles themselves with an Nbody scheme. For example, Davidson and Schwetz (2016) studies structure formation of an axion dark matter model with ground state solutions of the appropriate SchrödingerPoisson system along with quantum pressure term (see Eq. 58). Even if dark matter consists of clumps of weakly interacting massive particles (WIMPs) instead of BSs, Mendes and Yang (Mendes and Yang 2016) map clumps of such particles to perturbed boson stars and study their tidal deformability, bypassing the large computational cost of studying the dynamics of these WIMPs with an Nbody code. Tidal deformability of BSs was also studied recently in the context of testing strongfield general relativity (Cardoso et al. 2017).
Instead of galactic scale BSs, one could instead argue for the accumulation of bosonic field in neutron stars. Such solutions contain the “normal” fermionic matter as well as a bosonic component (discussed above in Sect. 3.6). However, the accumulation of additional mass in a neutron star, already the expected last stage before complete collapse to black hole, might conceivably lead to the star’s collapse. If indeed collapse can be expected, then the existence of old neutron stars would place constraints on such a form of dark matter (Yz et al. 2012; Jamison 2013; Bramante et al. 2013). In the face of such arguments, Kouvaris and Tinyakov (2013), Bell et al. (2013) instead argue that a broad range of realistic models survive such constraints. Most recently, Brito et al. (2015) argue with perturbation and numerical methods that old stars are in fact stable to the accretion of light bosons by an efficient gravitational cooling mechanism (see also the Ph.D. thesis by Brito 2016).
Another dark matter model arising from a scalar field is wave dark matter (Bray and Goetz 2014; Bray and Parry 2013; Goetz 2015a, b). In particular, they examine Tully–Fisher relationships predicted by this wave dark matter model (Bray and Goetz 2014; Goetz 2015b). High resolution simulations of a nonrelativstic Bose–Einstein condensate within this model reproduce the large scale structure of standard cold dark matter while differing inside galaxies (Schive et al. 2014).
Other studies solve the Gross–Pitaevskii equation for a Bose–Einstein condensate as a model of dark matter stars and study its stability properties (Li et al. 2012; Madarassy and Toth 2015; Marsh and Pop 2015).
The solitonic nature of boson stars (see Fig. 1) lends itself naturally to the wonderful observation of dark matter in the Bullet Cluster (Lee et al. 2008). Lee and Lim (2010b) attempts to determine a minimum galactic mass from such a match.
Interestingly, Barranco et al. (2011) foregoes boson stars and instead looks for quasistationary scalar solutions about a Schwarzschild black hole that could conceivably survive for cosmological times. Another approach is to use scalar fields for both the dark matter halo and the supermassive, central object. AmaroSeoane et al. (2010) looks for such a match, but finds no suitable solutions. Quite a number of more exotic models viably fit within current constraints, including those using Qballs (Doddato 2012).
Section 4.2 discusses the dynamics of boson stars including some references commenting on the implications of the dynamics for dark matter.
6 Boson stars in mathematical relativity
Although the experimental foundation for the existence of boson stars is completely lacking, on the theoretical and mathematical front, boson stars are well studied. Recent work includes a mathematical approach in terms of large and small data (Frank and Lenzmann 2009a), followed up by studying singularity formation (Lenzmann and Lewin 2011) and uniqueness (Frank and Lenzmann 2009b; Lenzmann 2009) for a certain boson star equation. In Cho et al. (2009), they study radial solutions of the semirelativistic Hartree type equations in terms of global wellposedness. Bičák et al. (2010) demonstrates stationarity of time periodic scalar field solutions.
Already discussed in Sect. 3.9 has been the no hair conjecture in the context of BSs holding a central BH within. Beyond just existence, however, boson stars are often employed mathematically to study dynamics. Here, we concentrate on a few of these topics that have attracted recent interest.
6.1 Black hole critical behavior
If one considers some initial distribution of energy and watches it evolve, generally one arrives at one of three states. If the energy is sufficiently weak in terms of its gravity, the energy might end up dispersing to larger and large distances. However, if the energy is instead quite large, then perhaps it will concentrate until a black hole is formed. Or, if the form of the energy supports it, some of the energy will condense into a stationary state.
In his seminal work (Choptuik 1993), Choptuik considers a real, massless scalar field and numerically evolves various initial configurations, finding either dispersion or blackhole formation. By parameterizing these initial configurations, say by the amplitude of an initial pulse p, and by tuning this parameter, he was able to study the threshold for blackhole formation at which he found fascinating blackhole critical behavior. In particular, his numerical work suggested that continued tuning could produce as small a black hole as one wished. This behavior is analogous to a phase transition in which the blackhole mass serves as an order parameter. Similar to phase transitions, one can categorize two types of transition that distinguish between whether the blackhole mass varies continuously (Type II) or discontinuously (Type I). For Choptuik’s work with a massless field, the transition is therefore of Type II because the blackhole mass varies from zero continuously to infinitesimal values.
Subsequent work has since established that this critical behavior can be considered as occurring in the neighborhood of a separatrix between the basins of attraction of the two end states. For \(p=p^*\), the system is precisely critical and remains on the (unstable) separatrix. Similarly other models find such threshold behavior occurring between a stationary state and blackhole formation. Critical behavior about stationary solutions necessarily involve blackhole formation “turningon” at finite mass, and is therefore categorized as Type I critical behavior.
The critical surface, therefore, appears as a codimension 1 surface, which evolutions increasingly approach as one tunes the parameter p. The distance from criticality \(pp^*\) serves as a measure of the extent to which a particular initial configuration has excited the unstable mode that drives solutions away from this surface. For Type II critical behavior, the mass of the resulting blackhole mass scales as a power law in this distance, whereas for Type I critical behavior, it is the survival time of the critical solution that scales as a power law. See Gundlach et al. (2007) for a recent review.
We have seen that boson stars represent stationary solutions of Einstein’s equations and, thus, one would correctly guess that they may occur within Type I blackhole critical behavior. To look for such behavior, Hawley and Choptuik (2000) begin their evolutions with bosonstar solutions and then perturb them both dynamically and gravitationally. They, therefore, included in their evolutionary system a distinct, free, massless, real scalar field which couples to the boson star purely through its gravity.
A very different type of critical behavior was also investigated by Lai (2004). By boosting identical boson stars toward each other and adjusting their initial momenta, he was able to tune to the threshold for blackhole formation. At the threshold, he found that the time till blackhole formation scaled consistent with Type I critical behavior and conjectured that the critical solution was itself an unstable boson star. This is one of the few fully nonlinear critical searches in less symmetry than spherical symmetry, and the first of Type I behavior in less symmetry. A related study colliding neutron stars instead of boson stars similarly finds Type I critical behavior (Jin and Suen 2007) and subsequently confirmed by Kellermann et al. (2010).
The gauged stars discussed in Sect. 3.11 also serve as critical solutions in spherical symmetry (Choptuik et al. 1996, 1999; Millward and Hirschmann 2003).
6.2 Hoop conjecture
They, therefore, numerically collide boson stars headon at relativistic energies to study blackhole formation from just such dynamical “squeezing”. Here, the nature of boson stars is largely irrelevant as they serve as simple bundles of energy that can be accelerated (see Fig. 25). However, unlike using boosted blackhole solutions, the choice of boson stars avoids any type of bias or predisposition to formation of a black hole. In addition, a number of previous studies of boson star headon collisions showed interesting interference effects at energies below the threshold for blackhole formation (Choi et al. 2009; Choi 2002; Lai 2004; Mundim 2010). Indeed, it has been proposed that such an interference pattern could be evidence for the bosonic nature of dark matter because of evidence that an ideal fluid fails to produce such a pattern (González and Guzmán 2011).
Choptuik and Pretorius (2010) find that indeed blackhole formation occurs at energies below that estimated by the Hoop Conjecture. This result is only a classical result consistent with the conjecture, but if it had not held, then there would have been no reason to expect a quantum theory to be consistent with it.
6.3 Other dimensions and antide Sitter spacetime
Much work has been invested recently in considering physics in other dimensions. Motivation comes from various ideas including string theory (more dimensions) such as the AdS/CFT correspondence and holography (one fewer dimensions) (Maldacena 1998; McGreevy 2010; Polchinski 2010). Another source of motivation comes from the fact that higher dimensional black holes can have different properties than those in three spatial dimensions (Emparan and Reall 2008). Perhaps BSs will similarly display novel properties in other dimensions.
In lower dimensional AdS (\(2+1\)) spacetimes, early work in 1998 studied exact solutions of boson stars (Sakamoto and Shiraishi 1998a; Degura et al. 2001; Sakamoto and Shiraishi 1998). Higher dimensional scenarios were apparently first considered qualitatively a few years later in the context of brane world models (Stojkovic 2003). This discussion was followed with a detailed analysis of the 3, 4, and 5 dimensional AdS solutions (Astefanesei and Radu 2003).
More recently, Fodor et al. (2010b) considers oscillatons in higher dimensions and measures the scalar mass loss rate for dimensions 3, 4, and 5. They extend this work considering inflationary spacetimes (Fodor et al. 2010a). Brihaye et al. (2014) and Herdeiro et al. (2015a) construct higher dimensional black hole solutions (Myers–Perry BHs) with scalar hair, and, in so doing, they find higher dimensional, rotating BS solutions.
The axisymmetric rotating BSs discussed in Sect. 3.5 satisfy a coupled set of nonlinear, elliptic PDEs in two dimensions. One might therefore suspect that adding other dimensions will only make things more difficult. As it turns out, however, moving to four spatial dimensions provides for another angular momentum, independent of the one along the zdirection (for example). Each of these angular momenta are associated with their own orthogonal plane of rotation. And so if one chooses solutions with equal magnitudes for each of these momenta, the solutions depend on only a single radial coordinate. This choice results in the remarkable simplification that one need only solve ODEs to find rotating solutions (Kunz et al. 2006).
The work of Dias et al. (2011) makes ingenious use of this 5D ansatz to construct rotating black holes with only a single Killing vector. They set the potential of Hartmann et al. (2010) to zero so that the scalar fields are massless and they add a (negative) cosmological constant to work in antide Sitter (AdS). Some of their solutions represent a black hole embedded inside a rotating BS. They find solutions for rotating black holes in 5D AdS that correspond to a bar mode for rotating neutron stars in 3D (see also Shibata and Yoshino 2010 for a numerical evolution of a black hole in higher dimensions which demonstrates such bar formation; see Emparan and Reall 2008 for a review of black holes in higher dimensions).
One might expect such a nonsymmetric black hole to settle into a more symmetric state via the emission of gravitational waves. However, AdS provides for an essentially reflecting boundary in which the black hole can be in equilibrium. The distortion of the higher dimensional black hole also has a correspondence with the discrete values of the angular momentum of the corresponding boson star. For higher values of the rotational quantum number, the black hole develops multiple “lobes” about its center. Very compact BSs constructed with this single Killing vector posses an ergoregion (Brihaye et al. 2015).
This construction can be extended to arbitrary odddimensional AdS spacetimes (Stotyn et al. 2012). Finding the solutions perturbatively, they explicitly show that these solutions approach (i) the boson star and (ii) the Myers–Perry blackhole solutions in AdS (Myers and Perry 1986) in different limits. Boson stars, along with neutron stars and black holes, in five dimensions are discussed in Brihaye and Delsate (2016), and see Emparan and Reall (2008) for a review of black holes in higher dimensions.
In AdS\({}_4\) this ansatz cannot be used, and the construction of spinning boson stars requires the solution of the appropriate multidimensional PDEs as is done in Radu and Subagyo (2012).
Interest in the dynamics of AdS spacetimes increased significantly with the work of Bizoń and Rostworowski (2011) who studied the collapse of a scalar field in spherically symmetric, global AdS\({}_4\). They argued that a nonzero initial amplitude for the scalar field would generically result in gravitational collapse to black hole via turbulent instability. In particular, fully nonlinear numerical evolutions of small amplitude configurations of scalar field generically resulted in a continued sharpening of the initial pulse as it reflected off the AdS boundary. This instability in the bulk is considered the mechanism that achieves thermal equilibration in the conformal theory on the boundary.
Many studies followed trying to answer the many questions arising from this work. Did this instability extend to any initial amplitude? Was the instability tied to the precise structure of AdS or instead simply to the fact that the spacetime was bounded?
One question in particular concerned the implications of this instability for localized solutions which might naturally be expected to extend their stability in asymptotically flat spacetimes. To that end, Buchel et al. (2013) studied boson stars in AdS, and found that indeed they are stable. In the course of understanding how the boson stars were stable, this work found a whole class of initial data that appear immune to the instability. Later work added to this class, namely breather solutions in AdS (Fodor et al. 2015). Linear perturbation analysis of spherically symmetric Proca stars in AdS suggests that these too will be stable (Duarte and Brito 2016b).
The same authors of (Dias et al. 2011) also report on the existence of geons in \(3+1\) AdS “which can be viewed as gravitational analogs of boson stars” (Dias et al. 2012) (recall that boson stars themselves arose from Wheeler’s desire to construct local electrovacuum solutions). These bundles of gravitational energy are stable to first order due to the confining boundary condition adopted with AdS. The instability of these geons, black holes, and boson stars were studied in Dias et al. (2011) in the context of the turbulent instability reported in Bizoń and Rostworowski (2011), but later these authors argued for their nonlinear stability (Dias et al. 2012).
Basu et al. (2010) also studies blackhole solutions in 5D AdS. They find solutions for black holes with scalar hair that resemble a boson star with a BH in its center. The stability of charged boson stars with a massive scalar field in fivedimensional AdS was studied in Brihaye et al. (2013). Also in AdS\({}_5\), Buchel studies boson stars in a type IIB supergravity approximation to string theory in which the U(1) symmetry of the complex field is gauged instead of global (Buchel 2015; Buchel and Buchel 2015). A range of solutions, including Qballs and shell solutions, for different values of the cosmological constant have similarly been constructed (Hartmann et al. 2013; Hartmann and Riedel 2012, 2013).
Basu et al. (2010) also studies black hole solutions in 5D AdS. They find solutions for black holes with scalar hair that resemble a boson star with a BH in its center.
Earlier work with BSs in lower dimensional AdS was reported in Astefanesei and Radu (2003).
Boson stars in AdS with charge are constructed in Hu et al. (2012), and they are also used as the background for a study of entanglement entropy (Nogueira 2013) (for a review of holographic entanglement entropy see Rangamani and Takayanagi 2017). Charged boson stars with spin in AdS have also been studied (Kichakova et al. 2014). See Gentle et al. (2012) for a review of charged scalar solitons in AdS.
6.4 Analog gravity and physical systems
The study of the correspondence between gravitating systems and analogous physical systems goes by the name of analog gravity (Barceló et al. 2011). One example of such an analog is the acoustic or dumb hole, analogous to a black hole, that requires information to flow in a particular direction. For such a system the analog of Hawking radiation is expected, and, remarkably, such radiation may have already been measured (Unruh 2014).
Analogs exist for BS as well. Recent work of Roger et al. (2016) finds an interesting optical analog of Newtonian BSs. So far this analog appears to be mostly associated with corresponding equations of motion as opposed to some deep physical correspondence that might reveal critical insight.
A more concrete analog is the formation of a Bose–Einstein condensation such as studied in Kühnel and Rampf (2014) in the context of superradiance (see Sect. 3.9). However, note that as mentioned in Sect. 1.1, ground state BSs can be considered as condensed states of bosons without invoking any analogy (Chavanis 2015; Chavanis and Matos 2017).
7 Final remarks
Boson stars have a long history as candidates for all manner of phenomena, from fundamental particle, to galactic dark matter. A huge variety of solutions have been found and their dynamics studied. Mathematically, BS are fascinating solitonlike solutions. Astrophysically, they represent possible explanations of black hole candidates and dark matter, with observations constraining BS properties.
Remarkably, in just the last 5 years since the first version of this review, two incredibly significant experimental results have appeared, and a third may soon be on its way. The Higgs particle has been found by the LHC, the first scalar particle. While its instability makes it less than promising as the fundamental constituent of boson stars, perhaps its discovery heralds a new period of scalar discoveries.
Far from the quantum particle regime of the LHC, LIGO has directly detected gravitational waves completely consistent with the merger of a binary blackhole system as predicted by general relativity. Not only does this put an end to the nagging questions about whether LIGO can really detect such extremely weak signals, but, as said often in the wake of these detections, it opens a new window into some of the most energetic events in the Universe. Although it is impossible to predict what new phenomena will be observed, one can hope that gravitational waves will further illuminate the nature of compact objects.
In the electromagnetic spectrum, the EHT just completed its 10 day observation, with an image expected by early next year (2018). The images from EHT are anxiously awaited because of their potential for demonstrating explicitly the presence of a horizon in Sgr A*. Of course, nature often surprises and so perhaps these images may instead lend credence to BSs.
With all of this experimental and observational data, physicists need to provide unambiguous tests and explicit predictions. Much work on that front is ongoing, trying to tease out observational differences from alternative models of gravity or alternatives to the standard compact objects (BHs and NSs) (Berti et al. 2015, 2016; Choptuik et al. 2015). Black holes were once exotic and disbelieved, but now BHs are the commonly accepted standard while BSs are proposed as just one of many exotic compact objects.
Perhaps future work on boson stars will be experimental, if fundamental scalar fields are observed, or if evidence arises indicating the boson stars uniquely fit galactic dark matter. But regardless of any experimental results found by these remarkable experiments, there will always be regimes unexplored by experiments where boson stars will find a natural home.
Footnotes
Notes
Acknowledgements
It is our pleasure to thank Juan Barranco, Francisco Guzmán, Carlos Herdeiro, and Luis Lehner for their helpful comments on the manuscript. We especially appreciate the careful and critical reading by Bruno Mundim. We also thank Gyula Fodor and Péter Forgács for their kind assistance with the section on oscillatons and oscillons. SLL also thanks the Perimeter Institute for their hospitality where the first version of this work was completed. This research was supported by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. This work was also supported by NSF Grants PHY1607291, PHY1308621, PHY0969827, and PHY0803624 to Long Island University. CP acknowledges support from the Spanish Ministry of Economy and Competitiveness Grants FPA201341042P and AYA201680289P (AEI/FEDER, UE), as well as from the Spanish Ministry of Education and Science by the Ramon y Cajal Grant.
References
 Aad G et al (2012) Observation of a new particle in the search for the standard model Higgs boson with the ATLAS detector at the LHC. Phys Lett B 716:1–29. https://doi.org/10.1016/j.physletb.2012.08.020. arXiv:1207.7214 ADSCrossRefGoogle Scholar
 Abbott BP et al (2016a) GW151226: observation of gravitational waves from a 22solarmass binary black hole coalescence. Phys Rev Lett 116:241103. https://doi.org/10.1103/PhysRevLett.116.241103. arXiv:1606.04855 ADSCrossRefGoogle Scholar
 Abbott BP et al (2016b) Observation of gravitational waves from a binary black hole merger. Phys Rev Lett 116:061102. https://doi.org/10.1103/PhysRevLett.116.061102. arXiv:1602.03837 ADSMathSciNetCrossRefGoogle Scholar
 Abbott BP et al (2016c) Tests of general relativity with GW150914. Phys Rev Lett 116:221101. https://doi.org/10.1103/PhysRevLett.116.221101. arXiv:1602.03841 ADSCrossRefGoogle Scholar
 Abbott BP et al (2017) Exploring the sensitivity of next generation gravitational wave detectors. Class Quantum Gravity 34:044001. https://doi.org/10.1088/13616382/aa51f4. arXiv:1607.08697 ADSCrossRefGoogle Scholar
 Adam C, Grandi N, Klimas P, SánchezGuillén J, Wereszczyński A (2010) Compact boson stars in k field theories. Gen Relativ Gravity 42:2663–2701. https://doi.org/10.1007/s1071401010064. arXiv:0908.0218 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Agnihotri P, SchaffnerBielich J, Mishustin IN (2009) Boson stars with repulsive selfinteractions. Phys Rev D 79:084033. https://doi.org/10.1103/PhysRevD.79.084033. arXiv:0812.2770 ADSCrossRefGoogle Scholar
 Akhoury R, Gauthier CS (2008) Galactic halos and black holes in noncanonical scalar field theories. ArXiv eprints arXiv:0804.3437
 Alcubierre M (2008) Introduction to \(3+1\) numerical relativity, International Series of Monographs on Physics, vol 140. Oxford University Press, OxfordzbMATHCrossRefGoogle Scholar
 Alcubierre M, Becerril R, Guzmán FS, Matos T, Núñez D, UreñaLópez LA (2003) Numerical studies of \(\phi ^{2}\)oscillatons. Class Quantum Gravity 20:2883–2903. https://doi.org/10.1088/02649381/20/13/332. arXiv:grqc/0301105 ADSzbMATHCrossRefGoogle Scholar
 Alcubierre M, Degollado JC, Núñez D, Ruiz M, Salgado M (2010) Dynamic transition to spontaneous scalarization in boson stars. Phys Rev D 81:124018. https://doi.org/10.1103/PhysRevD.81.124018. arXiv:1003.4767 ADSCrossRefGoogle Scholar
 Alic D (2009) Theoretical issues in numerical relativity simulations. PhD thesis, Universitat de les Illes Balears, Palma. http://hdl.handle.net/10803/9438
 AmaroSeoane P, Barranco J, Bernal A (2010) Constraining scalar fields with stellar kinematics and collisional dark matter. J Cosmol Astropart Phys 2010(11):002. https://doi.org/10.1088/14757516/2010/11/002. arXiv:1009.0019 CrossRefGoogle Scholar
 Antusch S, Orani S (2016) Impact of other scalar fields on oscillons after hilltop inflation. J Cosmol Astropart Phys 2016(03):026. https://doi.org/10.1088/14757516/2016/03/026. arXiv:1511.02336 MathSciNetCrossRefGoogle Scholar
 Antusch S, Cefala F, Orani S (2017) Gravitational waves from oscillons after inflation. Phys Rev Lett 118:011303. https://doi.org/10.1103/PhysRevLett.118.011303. arXiv:1607.01314 ADSCrossRefGoogle Scholar
 Armano M et al (2017) Chargeinduced forcenoise on freefalling test masses: results from LISA pathfinder. Phys Rev Lett 118:171101. https://doi.org/10.1103/PhysRevLett.118.171101. arXiv:1702.04633 ADSCrossRefGoogle Scholar
 Arnowitt R, Deser S, Misner CW (1962) The dynamics of general relativity. In: Witten L (ed) Gravitation: an introduction to current research. Wiley, New York, pp 227–265. https://doi.org/10.1007/s1071400806611. arXiv:grqc/0405109 Google Scholar
 Arodź H, Karkowski J, Świerczyński Z (2009) Spinning Qballs in the complex signumGordon model. Phys Rev D 80:067702. https://doi.org/10.1103/PhysRevD.80.067702. arXiv:0907.2801 ADSCrossRefGoogle Scholar
 Astefanesei D, Radu E (2003) Boson stars with negative cosmological constant. Nucl Phys B 665:594–622. https://doi.org/10.1016/S05503213(03)004826. arXiv:grqc/0309131 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Baibhav V, Maity D (2017) Boson stars in higherderivative gravity. Phys Rev D 95:024027. https://doi.org/10.1103/PhysRevD.95.024027. arXiv:1609.07225 ADSCrossRefGoogle Scholar
 Balakrishna J (1999) A numerical study of boson stars: Einstein equations with a matter source. PhD thesis, Washington University, St. Louis. arXiv:grqc/9906110
 Balakrishna J, Seidel E, Suen WM (1998) Dynamical evolution of boson stars. II. Excited states and selfinteracting fields. Phys Rev D 58:104004. https://doi.org/10.1103/PhysRevD.58.104004. arXiv:grqc/9712064 ADSCrossRefGoogle Scholar
 Balakrishna J, Bondarescu R, Daues G, Guzmán FS, Seidel E (2006) Evolution of 3d boson stars with waveform extraction. Class Quantum Gravity 23:2631–2652. https://doi.org/10.1088/02649381/23/7/024. arXiv:grqc/602078 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Balakrishna J, Bondarescu R, Daues G, Bondarescu M (2008) Numerical simulations of oscillating soliton stars: excited states in spherical symmetry and ground state evolutions in 3d. Phys Rev D 77:024028. https://doi.org/10.1103/PhysRevD.77.024028. arXiv:0710.4131 ADSCrossRefGoogle Scholar
 Bao W, Dong X (2011) Numerical methods for computing ground states and dynamics of nonlinear relativistic hartree equation for boson stars. J Comput Phys 230:5449–5469. https://doi.org/10.1016/j.jcp.2011.03.051 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Barceló C, Liberati S, Visser M (2011) Analogue gravity. Living Rev Relativ 14:lrr20113. https://doi.org/10.12942/lrr20113. http://www.livingreviews.org/lrr20113. arXiv:grqc/0505065
 Barranco J, Bernal A (2011a) Constraining scalar field properties with boson stars as black hole mimickers. In: UreñaLópez LA, MoralesTécotl HA, LinaresRomero R, SantosRodríguez E, EstradaJiménez S (eds) VIII workshop of the gravitation and mathematical physics division of the Mexican Physical Society, American Institute of Physics, Melville, NY, AIP conference proceedings, vol 1396, pp 171–175. https://doi.org/10.1063/1.3647542. arXiv:1108.1208
 Barranco J, Bernal A (2011b) Selfgravitating system made of axions. Phys Rev D 83:043525. https://doi.org/10.1103/PhysRevD.83.043525. arXiv:1001.1769 ADSCrossRefGoogle Scholar
 Barranco J, Bernal A, Degollado JC, DiezTejedor A, Megevand M, Alcubierre M, Núñez D, Sarbach O (2011) Are black holes a serious threat to scalar field dark matter models? Phys Rev D 84:083008. https://doi.org/10.1103/PhysRevD.84.083008. arXiv:1108.0931 ADSCrossRefGoogle Scholar
 Bartnik R, McKinnon J (1988) Particlelike solutions of the Einstein–Yang–Mills equations. Phys Rev Lett 61:141–144. https://doi.org/10.1103/PhysRevLett.61.141 ADSMathSciNetCrossRefGoogle Scholar
 Basu P, Bhattacharya J, Bhattacharyya S, Loganayagam R, Minwalla S, Umesh V (2010) Small hairy black holes in global AdS spacetime. J High Energy Phys 10:045. https://doi.org/10.1007/JHEP10(2010)045. arXiv:1003.3232 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Battye RA, Sutcliffe PM (2000) Qball dynamics. Nucl Phys B 590:329–363. https://doi.org/10.1016/S05503213(00)00506X. arXiv:hepth/0003252 ADSCrossRefGoogle Scholar
 Baumgarte TW, Shapiro SL (1999) Numerical integration of Einstein’s field equations. Phys Rev D 59:024007. https://doi.org/10.1103/PhysRevD.59.024007. arXiv:grqc/9810065 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Baumgarte TW, Shapiro SL (2010) Numerical relativity: solving Einstein’s equations on the computer. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
 Becerril R, ValdezAlvarado S, Nucamendi U (2016) Obtaining mass parameters of compact objects from redshifts and blueshifts emitted by geodesic particles around them. Phys Rev D 94:124024. https://doi.org/10.1103/PhysRevD.94.124024. arXiv:1610.01718 ADSCrossRefGoogle Scholar
 Bell NF, Melatos A, Petraki K (2013) Realistic neutron star constraints on bosonic asymmetric dark matter. Phys Rev D 87:123507. https://doi.org/10.1103/PhysRevD.87.123507. arXiv:1301.6811 ADSCrossRefGoogle Scholar
 Bernal A, Guzmán FS (2006a) Scalar field dark matter: headon interaction between two structures. Phys Rev D 74:103002. https://doi.org/10.1103/PhysRevD.74.103002. arXiv:astroph/0610682 ADSCrossRefGoogle Scholar
 Bernal A, Guzmán FS (2006b) Scalar field dark matter: nonspherical collapse and latetime behavior. Phys Rev D 74:063504. https://doi.org/10.1103/PhysRevD.74.063504. arXiv:astroph/0608523 ADSCrossRefGoogle Scholar
 Bernal A, Barranco J, Alic D, Palenzuela C (2010) Multistate boson stars. Phys Rev D 81:044031. https://doi.org/10.1103/PhysRevD.81.044031. arXiv:0908.2435 ADSCrossRefGoogle Scholar
 Berti E, Cardoso V (2006) Supermassive black holes or boson stars? Hair counting with gravitational wave detectors. Int J Mod Phys D 15:2209–2216. https://doi.org/10.1142/S0218271806009637. arXiv:grqc/0605101 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Berti E, Cardoso V, Crispino LCB, Gualtieri L, Herdeiro C, Sperhake U (2016) Numerical relativity and high energy physics: recent developments. Int J Mod Phys D 25:1641022. https://doi.org/10.1142/S0218271816410224, proceedings, 3rd Amazonian Symposium on Physics and 5th NRHEP Network Meeting is approaching: Celebrating 100 Years of General Relativity: Belem, Brazil. arXiv:1603:06146
 Berti E et al (2015) Testing general relativity with present and future astrophysical observations. Class Quantum Gravity 32:243001. https://doi.org/10.1088/02649381/32/24/2430011501.07274 ADSCrossRefGoogle Scholar
 Bezares M, Palenzuela C, Bona C (2017) Final fate of compact boson star mergers. Phys Rev D 95:124005. https://doi.org/10.1103/PhysRevD.95.1240051705.01071 ADSCrossRefGoogle Scholar
 Bhatt JR, Sreekanth V (2009) Boson stars: chemical potential and quark condensates. ArXiv eprints arXiv:0910.1972
 Bičák J, Scholtz M, Tod P (2010) On asymptotically flat solutions of Einstein’s equations periodic in time II. Spacetimes with scalarfield sources. Class Quantum Gravity 27:175011. https://doi.org/10.1088/02649381/27/17/175011. arXiv:1008.0248
 BinNun AY (2013) Method for detecting a boson star at Sgr A* through gravitational lensing. ArXiv eprints arXiv:1301.1396
 Bizoń P, Rostworowski A (2011) On weakly turbulent instability of antide Sitter space. Phys Rev Lett 107:031102. https://doi.org/10.1103/PhysRevLett.107.031102. arXiv:1104.3702 ADSCrossRefGoogle Scholar
 Boehle A, Ghez A, Schoedel R, Yelda S, Meyer L (2012) New orbital analysis of stars at the Galactic center using speckle holography. In: AAS 219th meeting, American Astronomical Society, Washington, DC, Bull. Am. Astron. Soc., vol 44Google Scholar
 Bogolyubskiĭ IL, Makhan’kov VG (1977) Dynamics of spherically symmetrical pulsons of large amplitude. JETP Lett 25:107–110ADSGoogle Scholar
 Bona C, PalenzuelaLuque C, BonaCasas C (2009) Elements of numerical relativity and relativistic hydrodynamics: from Einstein’s equations to astrophysical simulations. Lecture Notes in Physics, vol 783, 2nd edn. Springer, BerlinzbMATHCrossRefGoogle Scholar
 Brady PR, Choptuik MW, Gundlach C, Neilsen DW (2002) Blackhole threshold solutions in stiff fluid collapse. Class Quantum Gravity 19:6359–6376. https://doi.org/10.1088/02649381/19/24/306. arXiv:grqc/0207096 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Bramante J, Fukushima K, Kumar J (2013) Constraints on bosonic dark matter from observation of old neutron stars. Phys Rev D 87:055012. https://doi.org/10.1103/PhysRevD.87.055012. arXiv:1301.0036 ADSCrossRefGoogle Scholar
 Bray HL, Goetz AS (2014) Wave dark matter and the Tully–Fisher relation. ArXiv eprints arXiv:1409.7347
 Bray HL, Parry AR (2013) Modeling wave dark matter in dwarf spheroidal galaxies. ArXiv eprints arXiv:1301.0255
 Brihaye Y, Delsate T (2016) Boson stars, neutron stars and black holes in five dimensions. ArXiv eprints arXiv:1607.07488
 Brihaye Y, Hartmann B (2009) Angularly excited and interacting boson stars and \(q\) balls. Phys Rev D 79:064013. https://doi.org/10.1103/PhysRevD.79.064013. arXiv:0812.3968 ADSCrossRefGoogle Scholar
 Brihaye Y, Hartmann B (2016) Minimal boson stars in 5 dimensions: classical instability and existence of ergoregions. Class Quantum Gravity 33:065002. https://doi.org/10.1088/02649381/33/6/065002. arXiv:1509.04534
 Brihaye Y, Riedel J (2014) Rotating boson stars in fivedimensional Einstein–Gauss–Bonnet gravity. Phys Rev D 89:104060. https://doi.org/10.1103/PhysRevD.89.104060. arXiv:1310.7223 ADSCrossRefGoogle Scholar
 Brihaye Y, Verbin Y (2009) Spherical structures in conformal gravity and its scalar–tensor extension. Phys Rev D 80:124048. https://doi.org/10.1103/PhysRevD.80.124048. arXiv:0907.1951 ADSCrossRefGoogle Scholar
 Brihaye Y, Verbin Y (2010) Spherical nonAbelian solutions in conformal gravity. Phys Rev D 81:044041. https://doi.org/10.1103/PhysRevD.81.044041. arXiv:0910.0973 ADSMathSciNetCrossRefGoogle Scholar
 Brihaye Y, Hartmann B, Radu E (2005) Boson stars in SU(2) Yang–Millsscalar field theories. Phys Lett B 607:17–26. https://doi.org/10.1016/j.physletb.2004.12.020. arXiv:hepth/0411207 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Brihaye Y, Caebergs T, Delsate T (2009a) Chargedspinninggravitating Qballs. ArXiv eprints arXiv:0907.0913
 Brihaye Y, Caebergs T, Hartmann B, Minkov M (2009b) Symmetry breaking in (gravitating) scalar field models describing interacting boson stars and Qballs. Phys Rev D 80:064014. https://doi.org/10.1103/PhysRevD.80.064014. arXiv:0903.5419 ADSCrossRefGoogle Scholar
 Brihaye Y, Hartmann B, Tojiev S (2013) Stability of charged solitons and formation of boson stars in fivedimensional antide Sitter spacetime. Class Quantum Gravity 30:115009. https://doi.org/10.1088/02649381/30/11/115009. arXiv:1301.2452 ADSzbMATHCrossRefGoogle Scholar
 Brihaye Y, Diemer V, Hartmann B (2014a) Charged Qballs and boson stars and dynamics of charged test particles. Phys Rev D 89:084048. https://doi.org/10.1103/PhysRevD.89.084048. arXiv:1402.1055 ADSCrossRefGoogle Scholar
 Brihaye Y, Herdeiro C, Radu E (2014) Myers–Perry black holes with scalar hair and a mass gap. Phys Lett B 739:1–7. https://doi.org/10.1016/j.physletb.2014.10.019. arXiv:1408.5581 ADSzbMATHCrossRefGoogle Scholar
 Brihaye Y, Hartmann B, Riedel J (2015) Selfinteracting boson stars with a single Killing vector field in antide Sitter spacetime. Phys Rev D 92:044049. https://doi.org/10.1103/PhysRevD.92.044049. arXiv:1404.1874 ADSCrossRefGoogle Scholar
 Brihaye Y, Cisterna A, Erices C (2016) Boson stars in biscalar extensions of Horndeski gravity. Phys Rev D 93:124057. https://doi.org/10.1103/PhysRevD.93.124057. arXiv:1604.02121 ADSMathSciNetCrossRefGoogle Scholar
 Brito R (2016) Fundamental fields around compact objects: massive spin2 fields, superradiant instabilities and stars with dark matter cores. PhD thesis, Lisboa University. arXiv:1607.05146
 Brito R, Cardoso V, Okawa H (2015a) Accretion of dark matter by stars. Phys Rev Lett 115:111301. https://doi.org/10.1103/PhysRevLett.115.111301. arXiv:1508.04773 ADSCrossRefGoogle Scholar
 Brito R, Cardoso V, Pani P (2015b) Superradiance, Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/9783319190006. arXiv:1501.06570 Google Scholar
 Brito R, Cardoso V, Herdeiro CAR, Radu E (2016a) Proca stars: gravitating Bose–Einstein condensates of massive spin 1 particles. Phys Lett B 752:291–295. https://doi.org/10.1016/j.physletb.2015.11.051. arXiv:1508.05395 ADSCrossRefGoogle Scholar
 Brito R, Cardoso V, Macedo CFB, Okawa H, Palenzuela C (2016b) Interaction between bosonic dark matter and stars. Phys Rev D 93:044045. https://doi.org/10.1103/PhysRevD.93.044045. arXiv:1512.00466 ADSCrossRefGoogle Scholar
 Broderick AE, Narayan R (2006) On the nature of the compact dark mass at the Galactic center. Astrophys J Lett 638:L21–L24. https://doi.org/10.1086/500930. arXiv:astroph/0512211 ADSCrossRefGoogle Scholar
 Broderick AE, Loeb A, Reid MJ (2011) Localizing Sagittarius A* and M87 on microarcsecond scales with millimeter very long baseline interferometry. Astrophys J 735:57. https://doi.org/10.1088/0004637X/735/1/57. arXiv:1104.3146 ADSCrossRefGoogle Scholar
 Buchel A (2015) AdS boson stars in string theory. ArXiv eprints arXiv:1510.08415
 Buchel A, Buchel M (2015) On stability of nonthermal states in strongly coupled gauge theories. ArXiv eprints arXiv:1509.00774
 Buchel A, Liebling SL, Lehner L (2013) Boson stars in AdS spacetime. Phys Rev D 87:123006. https://doi.org/10.1103/PhysRevD.87.123006. arXiv:1304.4166 ADSCrossRefGoogle Scholar
 Burikham P, Harko T, Lake MJ (2016) Mass bounds for compact spherically symmetric objects in generalized gravity theories. Phys Rev D 94:064070. https://doi.org/10.1103/PhysRevD.94.064070. arXiv:1606.05515 ADSMathSciNetCrossRefGoogle Scholar
 Cao Z, CardenasAvendano A, Zhou M, Bambi C, Herdeiro CAR, Radu E (2016) Iron k\(\alpha \) line of boson stars. J Cosmol Astropart Phys 2016(10):003. https://doi.org/10.1088/14757516/2016/10/003. arXiv:1609.00901 CrossRefGoogle Scholar
 Cardoso V, Gualtieri L (2016) Testing the black hole ‘nohair’ hypothesis. Class Quantum Gravity 33:174001. https://doi.org/10.1088/02649381/33/17/174001. arXiv:1607.03133 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Cardoso V, Pani P, Cadoni M, Cavaglià M (2008) Ergoregion instability of ultracompact astrophysical objects. Phys Rev D 77:124044. https://doi.org/10.1103/PhysRevD.77.124044. arXiv:0709.0532 ADSCrossRefGoogle Scholar
 Cardoso V, Hopper S, Macedo CFB, Palenzuela C, Pani P (2016) Gravitationalwave signatures of exotic compact objects and of quantum corrections at the horizon scale. Phys Rev D 94:084031. https://doi.org/10.1103/PhysRevD.94.084031. arXiv:1608.08637 ADSCrossRefGoogle Scholar
 Cardoso V, Franzin E, Maselli A, Pani P, Raposo G (2017) Testing strongfield gravity with tidal Love numbers. Phys Rev D 95:084014. https://doi.org/10.1103/PhysRevD.95.084014. arXiv:1701.01116 ADSCrossRefGoogle Scholar
 Chatrchyan S et al (2012) Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC. Phys Lett B 716:30–61. https://doi.org/10.1016/j.physletb.2012.08.021. arXiv:1207.7235 ADSCrossRefGoogle Scholar
 Chavanis PH (2011) Massradius relation of Newtonian selfgravitating Bose–Einstein condensates with shortrange interactions. I. Analytical results. Phys Rev D 84:043531. https://doi.org/10.1103/PhysRevD.84.043531. arXiv:1103.2050 ADSCrossRefGoogle Scholar
 Chavanis PH (2012) Growth of perturbations in an expanding universe with Bose–Einstein condensate dark matter. Astron Astrophys 537:A127. https://doi.org/10.1051/00046361/201116905. arXiv:1103.2698 ADSCrossRefGoogle Scholar
 Chavanis PH (2015) Selfgravitating Bose–Einstein condensates. In: Calmet X (ed) Quantum aspects of black holes, Fundamental Theories of Physics, vol 178. Springer, Cham, pp 151–194. https://doi.org/10.1007/9783319108520_6 Google Scholar
 Chavanis PH, Harko T (2012) Bose–Einstein condensate general relativistic stars. Phys Rev D 86:064011. https://doi.org/10.1103/PhysRevD.86.064011. arXiv:1108.3986 ADSCrossRefGoogle Scholar
 Chavanis PH, Matos T (2017) Covariant theory of BoseEinstein condensates in curved spacetimes with electromagnetic interactions: the hydrodynamic approach. Eur Phys J Plus 132:30. https://doi.org/10.1140/epjp/i2017112924. arXiv:1606.07041 CrossRefGoogle Scholar
 Cho Y, Ozawa T, Sasaki H, Shim Y (2009) Remarks on the semirelativistic Hartree equations. Discrete Contin Dyn Syst A 23:1277–1294. https://doi.org/10.3934/dcds.2009.23.1277 MathSciNetzbMATHGoogle Scholar
 Chodosh O, ShlapentokhRothman Y (2015) Timeperiodic Einstein–Klein–Gordon bifurcations of Kerr. ArXiv eprints arXiv:1510.08025
 Choi D, Lai CW, Choptuik MW, Hirschmann EW, Liebling SL, Pretorius F (2009) Dynamics of axisymmetric (headon) boson star collisions. http://laplace.physics.ubc.ca/Group/Papers/choietalprd05/choietalprd05.pdf, unpublished
 Choi DI (1998) Numerical studies of nonlinear Schrödinger and Klein–Gordon systems: techniques and applications. PhD thesis, The University of Texas, Austin. http://laplace.physics.ubc.ca/Members/matt/Doc/Theses/
 Choi DI (2002) Collision of gravitationally bound Bose–Einstein condensates. Phys Rev A 66:063609. https://doi.org/10.1103/PhysRevA.66.063609 ADSCrossRefGoogle Scholar
 Choptuik MW (1993) Universality and scaling in gravitational collapse of a massless scalar field. Phys Rev Lett 70:9–12. https://doi.org/10.1103/PhysRevLett.70.9 ADSCrossRefGoogle Scholar
 Choptuik MW, Pretorius F (2010) Ultrarelativistic particle collisions. Phys Rev Lett 104:111101. https://doi.org/10.1103/PhysRevLett.104.111101. arXiv:0908.1780 ADSCrossRefGoogle Scholar
 Choptuik MW, Chmaj T, Bizoń P (1996) Critical behavior in gravitational collapse of a Yang–Mills field. Phys Rev Lett 77:424–427. https://doi.org/10.1103/PhysRevLett.77.424. arXiv:grqc/9603051 ADSCrossRefGoogle Scholar
 Choptuik MW, Hirschmann EW, Marsa RL (1999) New critical behavior in Einstein–Yang–Mills collapse. Phys Rev D 60:124011. https://doi.org/10.1103/PhysRevD.60.124011. arXiv:grqc/9903081 ADSMathSciNetCrossRefGoogle Scholar
 Choptuik MW, Lehner L, Pretorius F (2015) Probing strongfield gravity through numerical simulations. In: Ashtekar A, Berger BK, Isenberg J, MacCallum M (eds) General relativity and gravitation: a centennial perspective. Cambridge University Press, Cambridge, pp 361–411. https://doi.org/10.1017/CBO9781139583961.011. arXiv:1502.06853 CrossRefGoogle Scholar
 Chruściel PT, Costa JL, Heusler M (2012) Stationary black holes: uniqueness and beyond. Living Rev Relativ 15:lrr20127. https://doi.org/10.12942/lrr20127, http://www.livingreviews.org/lrr20127. arXiv:1205.6112
 Coleman SR (1985) Qballs. Nucl Phys B 262:263–283. https://doi.org/10.1016/05503213(85)90286X ADSMathSciNetCrossRefGoogle Scholar
 Colpi M, Shapiro SL, Wasserman I (1986) Boson stars: gravitational equilibria of selfinteracting scalar fields. Phys Rev Lett 57:2485–2488. https://doi.org/10.1103/PhysRevLett.57.2485 ADSMathSciNetCrossRefGoogle Scholar
 Contaldi CR, Wiseman T, Withers B (2008) TeVeS gets caught on caustics. Phys Rev D 78:044034. https://doi.org/10.1103/PhysRevD.78.044034. arXiv:0802.1215 ADSCrossRefGoogle Scholar
 Cook GB (2000) Initial data for numerical relativity. Living Rev Relativ 3:lrr20005. https://doi.org/10.12942/lrr20005, http://www.livingreviews.org/lrr20005, arXiv:grqc/0007085
 Cook GB, Shapiro SL, Teukolsky SA (1994) Rapidly rotating neutron stars in general relativity: realistic equations of state. Astrophys J 424:823–845. https://doi.org/10.1086/173934 ADSCrossRefGoogle Scholar
 Cotner E (2016) Collisional interactions between selfinteracting nonrelativistic boson stars: effective potential analysis and numerical simulations. Phys Rev D 94:063503. https://doi.org/10.1103/PhysRevD.94.063503. arXiv:1608.00547 ADSCrossRefGoogle Scholar
 Cunha PVP, Herdeiro CAR, Radu E, Runarsson HF (2015) Shadows of Kerr black holes with scalar hair. Phys Rev Lett 115:211102. https://doi.org/10.1103/PhysRevLett.115.211102. arXiv:1509.00021 ADSzbMATHCrossRefGoogle Scholar
 Dafermos M, Rodnianski I, ShlapentokhRothman Y (2014) A scattering theory for the wave equation on Kerr black hole exteriors. ArXiv eprints arXiv:1412.8379
 Damour T (1987) The problem of motion in Newtonian and Einsteinian gravity. In: Hawking SW, Israel W (eds) Three hundred years of gravitation. Cambridge University Press, Cambridge, pp 128–198Google Scholar
 Damour T, EspositoFarèse G (1996) Tensor–scalar gravity and binarypulsar experiments. Phys Rev D 54:1474–1491. https://doi.org/10.1103/PhysRevD.54.1474. arXiv:grqc/9602056 ADSCrossRefGoogle Scholar
 Danzmann Kea (2017) LISA: laser interferometer space antenna. a proposal in response to the ESA call for L3 mission concepts. In: Technical report, Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Potsdam. https://www.elisascience.org/files/publications/LISA_L3_20170120.pdf
 Dariescu C, Dariescu MA (2010) Boson nebulae charge. Chin Phys Lett 27:011101. https://doi.org/10.1088/0256307X/27/1/011101 CrossRefGoogle Scholar
 Davidson S, Schwetz T (2016) Rotating drops of axion dark matter. Phys Rev D 93:123509. https://doi.org/10.1103/PhysRevD.93.123509. arXiv:1603.04249 ADSCrossRefGoogle Scholar
 Degura Y, Sakamoto K, Shiraishi K (2001) Black holes with scalar hair in (\(2+1\))dimensions. Gravit Cosmol 7:153–158 arXiv:grqc/9805011 ADSMathSciNetzbMATHGoogle Scholar
 de Lavallaz A, Fairbairn M (2010) Neutron stars as dark matter probes. Phys Rev D 81:123521. https://doi.org/10.1103/PhysRevD.81.123521 ADSCrossRefGoogle Scholar
 Delgado JFM, Herdeiro CAR, Radu E, Runarsson H (2016) KerrNewman black holes with scalar hair. Phys Lett B 761:234–241. https://doi.org/10.1016/j.physletb.2016.08.032. arXiv:1608.00631 ADSzbMATHCrossRefGoogle Scholar
 de Sousa CMG, Tomazelli JL, Silveira V (1998) Model for stars of interacting bosons and fermions. Phys Rev D 58:123003. https://doi.org/10.1103/PhysRevD.58.123003. arXiv:grqc/9507043 ADSCrossRefGoogle Scholar
 de Sousa CMG, Silveira V, Fang LZ (2001) Slowly rotating boson–fermion star. Int J Mod Phys D 10:881–892. https://doi.org/10.1142/S0218271801001360. arXiv:grqc/0012020 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Derrick GH (1964) Comments on nonlinear wave equations as models for elementary particles. J Math Phys 5:1252–1254. https://doi.org/10.1063/1.1704233 ADSMathSciNetCrossRefGoogle Scholar
 Dias ÓJC, Masachs R (2017) Hairy black holes and the endpoint of AdS\(_4\) charged superradiance. J High Energy Phys 2017(02):128. https://doi.org/10.1007/JHEP02(2017)128. arXiv:1610.03496 MathSciNetCrossRefGoogle Scholar
 Dias ÓJC, Horowitz GT, Santos JE (2011) Black holes with only one Killing field. J High Energy Phys 2011(07):115. https://doi.org/10.1007/JHEP07(2011)115. arXiv:1105.4167 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Dias ÓJC, Horowitz GT, Marolf D, Santos JE (2012) On the nonlinear stability of asymptotically antide Sitter solutions. Class Quantum Gravity 29:235019. https://doi.org/10.1088/02649381/29/23/235019. arXiv:1208.5772 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Dias ÓJC, Horowitz GT, Santos JE (2012) Gravitational turbulent instability of antide Sitter space. Class Quantum Gravity 29:194002. https://doi.org/10.1088/02649381/29/19/194002. arXiv:1109.1825 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Dias ÓJC, Santos JE, Way B (2016) Numerical methods for finding stationary gravitational solutions. Class Quantum Gravity 33:133001. https://doi.org/10.1088/02649381/33/13/133001. arXiv:1510.02804 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Diemer V, Eilers K, Hartmann B, Schaffer I, Toma C (2013) Geodesic motion in the spacetime of a noncompact boson star. Phys Rev D 88:044025. https://doi.org/10.1103/PhysRevD.88.044025. arXiv:1304.5646 ADSCrossRefGoogle Scholar
 DiezTejedor A, GonzalezMorales AX (2013) Nogo theorem for static scalar field dark matter halos with no noether charges. Phys Rev D 88:067302. https://doi.org/10.1103/PhysRevD.88.067302. arXiv:1306.4400 ADSCrossRefGoogle Scholar
 Doddato F, McDonald J (2012) New Qball solutions in gaugemediation, Affleck–Dine baryogenesis and gravitino dark matter. J Cosmol Astropart Phys 2012(06):031. https://doi.org/10.1088/14757516/2012/06/031. arXiv:1111.2305 CrossRefGoogle Scholar
 Duarte M, Brito R (2016) Asymptotically antide Sitter Proca stars. Phys Rev D 94:064055. https://doi.org/10.1103/PhysRevD.94.064055. arXiv:1609.01735 ADSMathSciNetCrossRefGoogle Scholar
 Dymnikova I, Koziel L, Khlopov M, Rubin S (2000) Quasilumps from first order phase transitions. Grav Cosmol 6:311–318 arXiv:hepth/0010120 ADSzbMATHGoogle Scholar
 Dzhunushaliev V, Myrzakulov K, Myrzakulov R (2007) Boson stars from a gauge condensate. Mod Phys Lett A 22:273–281. https://doi.org/10.1142/S0217732307022669. arXiv:grqc/0604110 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Dzhunushaliev V, Folomeev V, Myrzakulov R, Singleton D (2008) Nonsingular solutions to Einstein–Klein–Gordon equations with a phantom scalar field. J High Energy Phys 2008(07):094. https://doi.org/10.1088/11266708/2008/07/094. arXiv:0805.3211 MathSciNetCrossRefGoogle Scholar
 Dzhunushaliev V, Folomeev V, Singleton D (2011) Chameleon stars. Phys Rev D 84:084025. https://doi.org/10.1103/PhysRevD.84.084025. arXiv:1106.1267 ADSCrossRefGoogle Scholar
 Dzhunushaliev V, Folomeev V, Hoffmann C, Kleihaus B, Kunz J (2014) Boson stars with nontrivial topology. Phys Rev D 90:124038. https://doi.org/10.1103/PhysRevD.90.124038. arXiv:1409.6978 ADSCrossRefGoogle Scholar
 Eby J, Kouvaris C, Nielsen NG, Wijewardhana LCR (2016) Boson stars from selfinteracting dark matter. J High Energy Phys 2016(02):028. https://doi.org/10.1007/JHEP02(2016)028. arXiv:1511.04474 CrossRefGoogle Scholar
 Eckart A, Hüttemann A, Kiefer C, Britzen S, Zajaček M, Lämmerzahl C, Stöckler M, ValenciaS M, Karas V, GarcíaMarín M (2017) The Milky Way’s supermassive black hole: how good a case is it? A challenge for astrophysics & philosophy of science. Found Phys 47:553–624. https://doi.org/10.1007/s1070101700792. arXiv:1703.09118
 Emparan R, Reall HS (2008) Black holes in higher dimensions. Living Rev Relativ 11:lrr20086. https://doi.org/10.12942/lrr20086, http://www.livingreviews.org/lrr20086. arXiv:0801.3471
 Eto M, Hashimoto K, Iida H, Miwa A (2011) Chiral magnetic effect from Qballs. Phys Rev D 83:125033. https://doi.org/10.1103/PhysRevD.83.125033. arXiv:1012.3264 ADSCrossRefGoogle Scholar
 Famaey B, McGaugh SS (2012) Modified Newtonian Dynamics (MOND): observational phenomenology and relativistic extensions. Living Rev Relativ 15:lrr201210. https://doi.org/10.12942/lrr201210, http://www.livingreviews.org/lrr201210. arXiv:1112.3960
 Fan Yz, Yang Rz, Chang J (2012) Constraining asymmetric bosonic noninteracting dark matter with neutron stars. ArXiv eprints arXiv:1204.2564
 Faraoni V (2012) Correspondence between a scalar field and an effective perfect fluid. Phys Rev D 85:024040. https://doi.org/10.1103/PhysRevD.85.024040. arXiv:1201.1448 ADSCrossRefGoogle Scholar
 Feng JL (2010) Dark matter candidates from particle physics and methods of detection. Ann Rev Astron Astrophys 48:495–545. https://doi.org/10.1146/annurevastro082708101659. arXiv:1003.0904 ADSCrossRefGoogle Scholar
 Flaminio R (2016) The cryogenic challenge: status of the KAGRA project. J Phys Conf Ser 716:012034. https://doi.org/10.1088/17426596/716/1/012034 CrossRefGoogle Scholar
 Fodor G, Forgács P, Horváth Z, Lukacs A (2008) Small amplitude quasibreathers and oscillons. Phys Rev D 78:025003. https://doi.org/10.1103/PhysRevD.78.025003. arXiv:0802.3525 ADSCrossRefGoogle Scholar
 Fodor G, Forgács P, Horváth Z, Mezei M (2009a) Computation of the radiation amplitude of oscillons. Phys Rev D 79:065002. https://doi.org/10.1103/PhysRevD.79.065002. arXiv:0812.1919 ADSCrossRefGoogle Scholar
 Fodor G, Forgács P, Horváth Z, Mezei M (2009b) Oscillons in dilaton–scalar theories. J High Energy Phys 2009(08):106. https://doi.org/10.1088/11266708/2009/08/106. arXiv:0906.4160
 Fodor G, Forgács P, Horváth Z, Mezei M (2009c) Radiation of scalar oscillons in 2 and 3 dimensions. Phys Lett B 674:319–324. https://doi.org/10.1016/j.physletb.2009.03.054. arXiv:0903.0953
 Fodor G, Forgács P, Mezei M (2010a) Boson stars and oscillatons in an inflationary universe. Phys Rev D 82:044043. https://doi.org/10.1103/PhysRevD.82.044043. arXiv:1007.0388 ADSCrossRefGoogle Scholar
 Fodor G, Forgács P, Mezei M (2010b) Mass loss and longevity of gravitationally bound oscillating scalar lumps (oscillatons) in Ddimensions. Phys Rev D 81:064029. https://doi.org/10.1103/PhysRevD.81.064029. arXiv:0912.5351 ADSCrossRefGoogle Scholar
 Fodor G, Forgács P, Grandclément P (2015) Selfgravitating scalar breathers with negative cosmological constant. Phys Rev D 92:025036. https://doi.org/10.1103/PhysRevD.92.025036. arXiv:1503.07746 ADSMathSciNetCrossRefGoogle Scholar
 Font JA (2008) Numerical hydrodynamics and magnetohydrodynamics in general relativity. Living Rev Relativ 11:lrr20087. https://doi.org/10.12942/lrr20087. http://www.livingreviews.org/lrr20087
 Font JA, Goodale T, Iyer S, Miller M, Rezzolla L, Seidel E, Stergioulas N, Suen WM, Tobias M (2002) Threedimensional numerical general relativistic hydrodynamics. II. Longterm dynamics of single relativistic stars. Phys Rev D 65:084024. https://doi.org/10.1103/PhysRevD.65.084024. arXiv:grqc/0110047 ADSMathSciNetCrossRefGoogle Scholar
 Franchini N, Pani P, Maselli A, Gualtieri L, Herdeiro CAR, Radu E, Ferrari V (2017) Constraining black holes with light boson hair and boson stars using epicyclic frequencies and quasiperiodic oscillations. Phys Rev D 95:124025. https://doi.org/10.1103/PhysRevD.95.124025. arXiv:1612.00038 ADSCrossRefGoogle Scholar
 Frank RL, Lenzmann E (2009a) On ground states for the \(l^2\)critical boson star equation. ArXiv eprints arXiv:0910.2721
 Frank RL, Lenzmann E (2009b) Uniqueness of ground states for the \(l^2\)critical boson star equation. ArXiv eprints arXiv:0905.3105
 Friedberg R, Lee TD, Pang Y (1987a) Minisoliton stars. Phys Rev D 35:3640–3657. https://doi.org/10.1103/PhysRevD.35.3640 ADSCrossRefGoogle Scholar
 Friedberg R, Lee TD, Pang Y (1987b) Scalar soliton stars and black holes. Phys Rev D 35:3658–3677. https://doi.org/10.1103/PhysRevD.35.3658 ADSCrossRefGoogle Scholar
 Friedman JL, Ipser JR, Sorkin RD (1988) Turningpoint method for axisymmetric stability of rotating relativistic stars. Astrophys J 325:722–724. https://doi.org/10.1086/166043 ADSCrossRefGoogle Scholar
 Gentle SA, Rangamani M, Withers B (2012) A soliton menagerie in AdS. J High Energy Phys 2012(05):106. https://doi.org/10.1007/JHEP05(2012)106. arXiv:1112.3979 CrossRefGoogle Scholar
 Giudice GF, McCullough M, Urbano A (2016) Hunting for dark particles with gravitational waves. J Cosmol Astropart Phys 2016(10):001. https://doi.org/10.1088/14757516/2016/10/001. arXiv:1605.01209 CrossRefGoogle Scholar
 Gleiser M (1988) Stability of boson stars. Phys Rev D 38:2376–2385. https://doi.org/10.1103/PhysRevD.38.2376 ADSCrossRefGoogle Scholar
 Gleiser M, Jiang N (2015) Stability bounds on compact astrophysical objects from informationentropic measure. Phys Rev D 92:044046. https://doi.org/10.1103/PhysRevD.92.044046. arXiv:1506.05722 ADSCrossRefGoogle Scholar
 Gleiser M, Watkins R (1989) Gravitational stability of scalar matter. Nucl Phys B 319:733–746. https://doi.org/10.1016/05503213(89)906275 ADSCrossRefGoogle Scholar
 Goetz AS (2015a) The Einstein–Klein–Gordon equations, wave dark matter, and the Tully–Fisher relation. PhD thesis, Duke University. arXiv:1507.02626
 Goetz AS (2015b) Tully–Fisher scalings and boundary conditions for wave dark matter. ArXiv eprints arXiv:1502.04976
 González JA, Guzmán FS (2011) Interference pattern in the collision of structures in the Bose–Einstein condensate dark matter model: comparison with fluids. Phys Rev D 83:103513. https://doi.org/10.1103/PhysRevD.83.103513. arXiv:1105.2066 ADSCrossRefGoogle Scholar
 Gourgoulhon E (2012) \(3+1\) formalism in general relativity: bases of numerical relativity. Lecture Notes in Physics, vol 846. Springer, Berlin. https://doi.org/10.1007/9783642245251. arXiv:grqc/0703035 zbMATHCrossRefGoogle Scholar
 GraciaLinares M, Guzman FS (2016) Accretion of supersonic winds on boson stars. Phys Rev D 94:064077. https://doi.org/10.1103/PhysRevD.94.064077. arXiv:1609.06398 ADSCrossRefGoogle Scholar
 Grandclément P (2016) Light rings and light points of boson stars. ArXiv eprints arXiv:1612.07507
 Grandclément P, Fodor G, Forgács P (2011) Numerical simulation of oscillatons: extracting the radiating tail. Phys Rev D 84:065037. https://doi.org/10.1103/PhysRevD.84.065037 ADSCrossRefGoogle Scholar
 Grandclément P, Somé C, Gourgoulhon E (2014) Models of rotating boson stars and geodesics around them: new type of orbits. Phys Rev D 90:024068. https://doi.org/10.1103/PhysRevD.90.0240681405.4837 ADSCrossRefGoogle Scholar
 Guenther RL (1995) A numerical study of the time dependent Schrödinger equation coupled with Newtonian gravity. PhD thesis, The University of Texas, Austin. http://laplace.physics.ubc.ca/Members/matt/Doc/Theses/
 Gundlach C, Leveque RJ (2011) Universality in the runup of shock waves to the surface of a star. J Fluid Mech 676:237–264. https://doi.org/10.1017/jfm.2011.42. arXiv:1008.2834 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Gundlach C, MartínGarcía JM (2007) Critical phenomena in gravitational collapse. Living Rev Relativ 10:lrr20075. https://doi.org/10.12942/lrr20075, http://www.livingreviews.org/lrr20075. arXiv:0711.4620
 Gundlach C, Please C (2009) Generic behaviour of nonlinear sound waves near the surface of a star: smooth solutions. Phys Rev D 79:067501. https://doi.org/10.1103/PhysRevD.79.067501. arXiv:0901.4928 ADSCrossRefGoogle Scholar
 Güver T, Emre Erkoca A, Hall Reno M, Sarcevic I (2014) On the capture of dark matter by neutron stars. J Cosmol Astropart Phys 2014(05):013. https://doi.org/10.1088/14757516/2014/05/013. arXiv:1201.2400 CrossRefGoogle Scholar
 Guzmán FS (2004) Evolving spherical boson stars on a 3D Cartesian grid. Phys Rev D 70:044033. https://doi.org/10.1103/PhysRevD.70.044033. arXiv:grqc/0407054 ADSCrossRefGoogle Scholar
 Guzmán FS (2007) Scalar fields: at the threshold of astrophysics. J Phys Conf Ser 91:012003. https://doi.org/10.1088/17426596/91/1/012003 CrossRefGoogle Scholar
 Guzmán FS (2009) The three dynamical fates of boson stars. Rev Mex Fis 55:321–326. http://www.scielo.org.mx/scielo.php?pid=S0035001X2009000400011&nrm=iso&script=sci_arttext
 Guzmán FS, RuedaBecerril JM (2009) Spherical boson stars as black hole mimickers. Phys Rev D 80:084023. https://doi.org/10.1103/PhysRevD.80.084023. arXiv:1009.1250 ADSCrossRefGoogle Scholar
 Guzmán FS, UreñaLópez LA (2006) Gravitational cooling of selfgravitating Bose condensates. Astrophys J 645:814–819. https://doi.org/10.1086/504508. arXiv:astroph/0603613 ADSCrossRefGoogle Scholar
 Hanna C, Johnson MC, Lehner L (2017) Estimating gravitational radiation from superemitting compact binary systems. Phys Rev D 124042. https://doi.org/10.1103/PhysRevD.95.124042. arXiv:1611.03506
 Harrison BK, Thorne KS, Wakano M, Wheeler JA (1965) Gravitation theory and gravitational collapse. University of Chicago Press, ChicagoGoogle Scholar
 Hartmann B, Riedel J (2012) Glueball condensates as holographic duals of supersymmetric Qballs and boson stars. Phys Rev D 86:104008. https://doi.org/10.1103/PhysRevD.86.104008. arXiv:1204.6239 ADSCrossRefGoogle Scholar
 Hartmann B, Riedel J (2013) Supersymmetric Qballs and boson stars in (\(\text{ d }+1\)) dimensions. Phys Rev D 87:044003. https://doi.org/10.1103/PhysRevD.87.044003. arXiv:1210.0096 ADSCrossRefGoogle Scholar
 Hartmann B, Kleihaus B, Kunz J, List M (2010) Rotating boson stars in five dimensions. Phys Rev D 82:084022. https://doi.org/10.1103/PhysRevD.82.084022. arXiv:1008.3137 ADSCrossRefGoogle Scholar
 Hartmann B, Kleihaus B, Kunz J, Schaffer I (2012) Compact boson stars. Phys Lett B 714:120–126. https://doi.org/10.1016/j.physletb.2012.06.067. arXiv:1205.0899 ADSCrossRefGoogle Scholar
 Hartmann B, Kleihaus B, Kunz J, Schaffer I (2013a) Compact (A)dS boson stars and shells. Phys Rev D 88:124033. https://doi.org/10.1103/PhysRevD.88.124033. arXiv:1310.3632 ADSCrossRefGoogle Scholar
 Hartmann B, Riedel J, Suciu R (2013) Gauss–Bonnet boson stars. Phys Lett B 726:906–912. https://doi.org/10.1016/j.physletb.2013.09.050. arXiv:1308.3391 ADSzbMATHCrossRefGoogle Scholar
 Hawley SH, Choptuik MW (2000) Boson stars driven to the brink of black hole formation. Phys Rev D 62:104024. https://doi.org/10.1103/PhysRevD.62.104024. arXiv:grqc/0007039 ADSCrossRefGoogle Scholar
 Hawley SH, Choptuik MW (2003) Numerical evidence for ‘multiscalar stars’. Phys Rev D 67:024010. https://doi.org/10.1103/PhysRevD.67.024010. arXiv:grqc/0208078 ADSCrossRefGoogle Scholar
 Henderson LJ, Mann RB, Stotyn S (2015) Gauss–Bonnet boson stars with a single Killing vector. Phys Rev D 91:024009. https://doi.org/10.1103/PhysRevD.91.024009. arXiv:1403.1865 ADSMathSciNetCrossRefGoogle Scholar
 Henriques AB, Liddle AR, Moorhouse RG (1989) Combined boson–fermion stars. Phys Lett B 233:99–106. https://doi.org/10.1016/03702693(89)906230 ADSCrossRefGoogle Scholar
 Henriques AB, Liddle AR, Moorhouse RG (1990) Combined boson–fermion stars: configurations and stability. Nucl Phys B 337:737–761. https://doi.org/10.1016/05503213(90)90514E ADSCrossRefGoogle Scholar
 Henriques AB, Liddle AR, Moorhouse RG (1990) Stability of boson–fermion stars. Phys Lett B 251:511–516. https://doi.org/10.1016/03702693(90)907899 ADSCrossRefGoogle Scholar
 Herdeiro C, Radu E (2014a) Ergosurfaces for Kerr black holes with scalar hair. Phys Rev D 89:124018. https://doi.org/10.1103/PhysRevD.89.124018. arXiv:1406.1225 ADSCrossRefGoogle Scholar
 Herdeiro C, Radu E (2015a) Construction and physical properties of Kerr black holes with scalar hair. Class Quantum Grav 32:144001. https://doi.org/10.1088/02649381/32/14/144001. arXiv:1501.04319 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Herdeiro C, Kunz J, Radu E, Subagyo B (2015) Myers–Perry black holes with scalar hair and a mass gap: unequal spins. Phys Lett B 748:30–36. https://doi.org/10.1016/j.physletb.2015.06.059. arXiv:1505.02407 ADSzbMATHCrossRefGoogle Scholar
 Herdeiro C, Radu E, Runarsson H (2016a) Kerr black holes with Proca hair. Class Quantum Grav 33:154001. https://doi.org/10.1088/02649381/33/15/154001. arXiv:1603.02687 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Herdeiro CAR, Radu E (2014b) Kerr black holes with scalar hair. Phys Rev Lett 112:221101. https://doi.org/10.1103/PhysRevLett.112.221101. arXiv:1403.2757 ADSCrossRefGoogle Scholar
 Herdeiro CAR, Radu E (2015b) Asymptotically flat black holes with scalar hair: a review. Int J Mod Phys D 24:1542014. https://doi.org/10.1142/S0218271815420146, proceedings, 7th Black Holes Workshop 2014: Aveiro, Portugal, December 18–19, 2014. arXiv:1504.08209
 Herdeiro CAR, Radu E, Rúnarsson H (2015b) Kerr black holes with selfinteracting scalar hair: hairier but not heavier. Phys Rev D 92:084059. https://doi.org/10.1103/PhysRevD.92.084059. arXiv:1509.02923 ADSCrossRefGoogle Scholar
 Herdeiro CAR, Radu E, Rúnarsson HF (2016b) Spinning boson stars and Kerr black holes with scalar hair: the effect of selfinteractions. Int J Mod Phys D 25:1641014. https://doi.org/10.1142/S0218271816410145, proceedings, 3rd Amazonian Symposium on Physics and 5th NRHEP Network Meeting is approaching: Celebrating 100 Years of General Relativity: Belem, Brazil. arXiv:1604.06202
 Hod S (2011) Quasinormal resonances of a massive scalar field in a nearextremal Kerr black hole spacetime. Phys Rev D 84:044046. https://doi.org/10.1103/PhysRevD.84.044046. arXiv:1109.4080 ADSCrossRefGoogle Scholar
 Hod S (2012) Stationary scalar clouds around rotating black holes. Phys Rev D 86:104026. https://doi.org/10.1103/PhysRevD.86.104026. [Erratum: Phys. Rev. D 86 (2012) 129902]. arXiv:1211.3202
 Honda EP (2000) Resonant dynamics within the nonlinear Klein–Gordon equation: much ado about oscillons. PhD thesis, The University of Texas, Austin. http://laplace.physics.ubc.ca/Members/matt/Doc/Theses/. arXiv:hepph/0009104
 Honda EP (2010) Fractal boundary basins in spherically symmetric \(\phi ^4\) theory. Phys Rev D 82:024038. https://doi.org/10.1103/PhysRevD.82.024038. arXiv:1006.2421 ADSCrossRefGoogle Scholar
 Honda EP, Choptuik MW (2002) Fine structure of oscillons in the spherically symmetric \(\phi ^4\) Klein–Gordon model. Phys Rev D 65:084037. https://doi.org/10.1103/PhysRevD.65.084037. arXiv:hepph/0110065 ADSMathSciNetCrossRefGoogle Scholar
 Horvat D, Marunović A (2013) Dark energylike stars from nonminimally coupled scalar field. Class Quantum Gravity 30:145006. https://doi.org/10.1088/02649381/30/14/145006. arXiv:1212.3781 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Horvat D, Ilijić S, Kirin A, Narančić Z (2013) Formation of photon spheres in boson stars with a nonminimally coupled field. Class Quantum Gravity 30:095014. https://doi.org/10.1088/02649381/30/9/095014. arXiv:1302.4369 ADSzbMATHCrossRefGoogle Scholar
 Horvat D, Ilijic S, Kirin A, Narancic Z (2015) Note on the charged boson stars with torsioncoupled field. Phys Rev D 92:024045. https://doi.org/10.1103/PhysRevD.92.024045. arXiv:1503.02480 ADSzbMATHCrossRefGoogle Scholar
 Horvat D, Ilijić S, Kirin A, Narančić Z (2015) Nonminimally coupled scalar field in teleparallel gravity: boson stars. Class Quantum Gravity 32:035023. https://doi.org/10.1088/02649381/32/3/035023. arXiv:1407.2067 ADSzbMATHCrossRefGoogle Scholar
 Hu S, Liu JT, Pando Zayas LA (2012) Charged boson stars in AdS and a zero temperature phase transition. ArXiv eprints arXiv:1209.2378
 Jamison AO (2013) Effects of gravitational confinement on bosonic asymmetric dark matter in stars. Phys Rev D 88:035004. https://doi.org/10.1103/PhysRevD.88.035004. arXiv:1304.3773 ADSCrossRefGoogle Scholar
 Jetzer P (1989a) Dynamical instability of bosonic stellar configurations. Nucl Phys B 316:411–428. https://doi.org/10.1016/05503213(89)900382 ADSCrossRefGoogle Scholar
 Jetzer P (1989b) Stability of charged boson stars. Phys Lett B 231:433–438. https://doi.org/10.1016/03702693(89)906898 ADSCrossRefGoogle Scholar
 Jetzer P (1989c) Stability of excited bosonic stellar configurations. Phys Lett B 222:447–452. https://doi.org/10.1016/03702693(89)903420 ADSCrossRefGoogle Scholar
 Jetzer P (1990) Stability of combined boson–fermion stars. Phys Lett B 243:36–40. https://doi.org/10.1016/03702693(90)909523 ADSCrossRefGoogle Scholar
 Jetzer P (1992) Boson stars. Phys Rep 220:163–227. https://doi.org/10.1016/03701573(92)90123H ADSCrossRefGoogle Scholar
 Jetzer P, van der Bij JJ (1989) Charged boson stars. Phys Lett B 227:341–346. https://doi.org/10.1016/03702693(89)909416 ADSCrossRefGoogle Scholar
 Jin KJ, Suen WM (2007) Critical phenomena in headon collisions of neutron stars. Phys Rev Lett 98:131101. https://doi.org/10.1103/PhysRevLett.98.131101. arXiv:grqc/0603094 ADSCrossRefGoogle Scholar
 Johannsen T, Wang C, Broderick AE, Doeleman SS, Fish VL, Loeb A, Psaltis D (2016) Testing general relativity with accretionflow imaging of Sgr A*. Phys Rev Lett 117:091101. https://doi.org/10.1103/PhysRevLett.117.091101. arXiv:1608.03593 ADSCrossRefGoogle Scholar
 Kan N, Shiraishi K (2016) Analytical approximation for Newtonian boson stars in four and five dimensions–a poor person’s approach to rotating boson stars. Phys Rev D 94:104042. https://doi.org/10.1103/PhysRevD.94.104042. arXiv:1605.02846 ADSCrossRefGoogle Scholar
 Kasuya S, Kawasaki M (2000) Qball formation through the Affleck–Dine mechanism. Phys Rev D 61:041301. https://doi.org/10.1103/PhysRevD.61.041301. arXiv:hepph/9909509 ADSCrossRefGoogle Scholar
 Kaup DJ (1968) Klein–Gordon geon. Phys Rev 172:1331–1342. https://doi.org/10.1103/PhysRev.172.1331 ADSCrossRefGoogle Scholar
 Kellermann T, Rezzolla L, Radice D (2010) Critical phenomena in neutron stars: II headon collisions. Class Quantum Gravity 27:235016. https://doi.org/10.1088/02649381/27/23/235016. arXiv:1007.2797 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Kesden M, Gair JR, Kamionkowski M (2005) Gravitationalwave signature of an inspiral into a supermassive horizonless object. Phys Rev D 71:044015. https://doi.org/10.1103/PhysRevD.71.044015. arXiv:astroph/0411478 ADSCrossRefGoogle Scholar
 Khachatryan V, Sirunyan AM, Tumasyan A, Adam W, Bergauer T, Dragicevic M, Erö J, Friedl M, Frühwirth R, Ghete VM et al (2015) Precise determination of the mass of the higgs boson and tests of compatibility of its couplings with the standard model predictions using proton collisions at 7 and 8. Eur Phys J C 75:212. https://doi.org/10.1140/epjc/s1005201533517. arXiv:1412.8662 ADSCrossRefGoogle Scholar
 Kichakova O, Kunz J, Radu E (2014) Spinning gauged boson stars in antide Sitter spacetime. Phys Lett B 728:328–335. https://doi.org/10.1016/j.physletb.2013.11.061. arXiv:1310.5434 ADSCrossRefGoogle Scholar
 Kichenassamy S (2008) Soliton stars in the breather limit. Class Quantum Gravity 25:245004. https://doi.org/10.1088/02649381/25/24/245004 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Kiessling MKH (2009) Monotonicity of quantum ground state energies: Bosonic atoms and stars. J Stat Phys 137:1063–1078. https://doi.org/10.1007/s1095500998439. arXiv:1001.4280 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Kleihaus B, Kunz J, List M (2005) Rotating boson stars and Qballs. Phys Rev D 72:064002. https://doi.org/10.1103/PhysRevD.72.064002. arXiv:grqc/0505143 ADSMathSciNetCrossRefGoogle Scholar
 Kleihaus B, Kunz J, List M, Schaffer I (2008) Rotating boson stars and Qballs. II. Negative parity and ergoregions. Phys Rev D 77:064025. https://doi.org/10.1103/PhysRevD.77.064025. arXiv:0712.3742 ADSMathSciNetCrossRefGoogle Scholar
 Kleihaus B, Kunz J, Lämmerzahl C, List M (2009) Charged boson stars and black holes. Phys Lett B 675:102–109. https://doi.org/10.1016/j.physletb.2009.03.066. arXiv:0902.4799 ADSMathSciNetCrossRefGoogle Scholar
 Kleihaus B, Kunz J, Lämmerzahl C, List M (2010) Boson shells harboring charged black holes. Phys Rev D 82:104050. https://doi.org/10.1103/PhysRevD.82.104050. arXiv:1007.1630 ADSCrossRefGoogle Scholar
 Kleihaus B, Kunz J, Schneider S (2012) Stable phases of boson stars. Phys Rev D 85:024045. https://doi.org/10.1103/PhysRevD.85.024045. arXiv:1109.5858 ADSCrossRefGoogle Scholar
 Kleihaus B, Kunz J, Yazadjiev S (2015) Scalarized hairy black holes. Phys Lett B 744:406–412. https://doi.org/10.1016/j.physletb.2015.04.014. arXiv:1503.01672 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Kobayashi Y, Kasai M, Futamase T (1994) Does a boson star rotate? Phys Rev D 50:7721–7724. https://doi.org/10.1103/PhysRevD.50.7721 ADSCrossRefGoogle Scholar
 Kouvaris C, Tinyakov P (2013) (not)Constraining heavy asymmetric bosonic dark matter. Phys Rev D 87:123537. https://doi.org/10.1103/PhysRevD.87.123537. arXiv:1212.4075 ADSCrossRefGoogle Scholar
 Kouvaris C, Tinyakov PG (2010) Can neutron stars constrain dark matter? Phys Rev D 82:063531. https://doi.org/10.1103/PhysRevD.82.063531 ADSCrossRefGoogle Scholar
 Kühnel F, Rampf C (2014) Astrophysical Bose–Einstein condensates and superradiance. Phys Rev D 90:103526. https://doi.org/10.1103/PhysRevD.90.103526. arXiv:1408.0790 ADSCrossRefGoogle Scholar
 Kumar S, Kulshreshtha U, Kulshreshtha DS (2015) Boson stars in a theory of complex scalar field coupled to gravity. Gen Relativ Gravit 47:76. https://doi.org/10.1007/s1071401519180. arXiv:1605.07015 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Kumar S, Kulshreshtha U, Kulshreshtha DS (2016) Charged compact boson stars and shells in the presence of a cosmological constant. Phys Rev D 94:125023. https://doi.org/10.1103/PhysRevD.94.125023 ADSCrossRefGoogle Scholar
 Kunz J, NavarroLerida F, Viebahn J (2006) Charged rotating black holes in odd dimensions. Phys Lett B 639:362–367. https://doi.org/10.1016/j.physletb.2006.06.066. arXiv:hepth/0605075 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Kusenko A, Steinhardt PJ (2001) \(q\)Ball candidates for selfinteracting dark matter. Phys Rev Lett 87:141301. https://doi.org/10.1103/PhysRevLett.87.141301. arXiv:astroph/0106008 ADSCrossRefGoogle Scholar
 Kusmartsev FV, Mielke EW, Schunck FE (1991) Gravitational stability of boson stars. Phys Rev D 43:3895–3901. https://doi.org/10.1103/PhysRevD.43.3895. arXiv:0810.0696 ADSMathSciNetCrossRefGoogle Scholar
 Lai CW (2004) A numerical study of boson stars. PhD thesis, The University of British Columbia, Vancouver. http://laplace.physics.ubc.ca/Members/matt/Doc/Theses/, arXiv:grqc/0410040
 Lai CW, Choptuik MW (2007) Final fate of subcritical evolutions of boson stars. ArXiv eprints arXiv:0709.0324
 Landea IS, García F (2016) Charged Proca stars. Phys Rev D 94:104006. https://doi.org/10.1103/PhysRevD.94.104006. arXiv:1608.00011 ADSCrossRefGoogle Scholar
 Landsberg GL (2006) Black holes at future colliders and beyond. J Phys G Nucl Part Phys 32:R337–R365. https://doi.org/10.1088/09543899/32/9/R02. arXiv:hepph/0607297 ADSCrossRefGoogle Scholar
 Latifah S, Sulaksono A, Mart T (2014) Bosons star at finite temperature. Phys Rev D 90:127501. https://doi.org/10.1103/PhysRevD.90.127501. arXiv:1412.1556 ADSCrossRefGoogle Scholar
 Lee JW (2010) Is dark matter a BEC or scalar field? J Kor Phys Soc 54:622 arXiv:0801.1442 Google Scholar
 Lee JW (2010) Minimum mass of galaxies from BEC or scalar field dark matter. J Cosmol Astropart Phys 2010(01):007. https://doi.org/10.1088/14757516/2010/01/007 ADSCrossRefGoogle Scholar
 Lee JW, Lim S, Choi D (2008) BEC dark matter can explain collisions of galaxy clusters. ArXiv eprints arXiv:0805.3827
 Lee TD (1987) Soliton stars and the critical masses of black holes. Phys Rev D 35:3637–3639. https://doi.org/10.1103/PhysRevD.35.3637 ADSCrossRefGoogle Scholar
 Lee TD, Pang Y (1989) Stability of miniboson stars. Nucl Phys B 315:477–516. https://doi.org/10.1016/05503213(89)903659 ADSCrossRefGoogle Scholar
 Lee TD, Pang Y (1992) Nontopological solitons. Phys Rep 221:251–350. https://doi.org/10.1016/03701573(92)900647 ADSMathSciNetCrossRefGoogle Scholar
 Lenzmann E (2009) Uniqueness of ground states for pseudorelativistic hartree equations. Anal PDE 2:1–27. https://doi.org/10.2140/apde.2009.2.1 MathSciNetzbMATHCrossRefGoogle Scholar
 Lenzmann E, Lewin M (2011) On singularity formation for the \(l^2\)critical boson star equation. Nonlinearity 24:3515–3540. https://doi.org/10.1088/09517715/24/12/009. arXiv:1103.3140 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Li XY, Harko T, Cheng KS (2012) Condensate dark matter stars. J Cosmol Astropart Phys 2012(06):001. https://doi.org/10.1088/14757516/2012/06/001. arXiv:1205.2932 CrossRefGoogle Scholar
 Liddle AR, Madsen MS (1992) The structure and formation of boson stars. Int J Mod Phys D 1:101–143. https://doi.org/10.1142/S0218271892000057 ADSzbMATHCrossRefGoogle Scholar
 LoraClavijo FD, CruzOsorio A, Guzmán FS (2010) Evolution of a massless test scalar field on boson star spacetimes. Phys Rev D 82:023005. https://doi.org/10.1103/PhysRevD.82.023005. arXiv:1007.1162 ADSCrossRefGoogle Scholar
 Lue A, Weinberg EJ (2000) Gravitational properties of monopole spacetimes near the black hole threshold. Phys Rev D 61:124003. https://doi.org/10.1103/PhysRevD.61.124003. arXiv:hepth/0001140 ADSMathSciNetCrossRefGoogle Scholar
 Lynn BW (1989) Qstars. Nucl Phys 321:465–480. https://doi.org/10.1016/05503213(89)903520 ADSCrossRefGoogle Scholar
 Macedo CFB, Pani P, Cardoso V, Crispino LCB (2013a) Astrophysical signatures of boson stars: quasinormal modes and inspiral resonances. Phys Rev D 88:064046. https://doi.org/10.1103/PhysRevD.88.064046. arXiv:1307.4812 ADSCrossRefGoogle Scholar
 Macedo CFB, Pani P, Cardoso V, Crispino LCB (2013b) Into the lair: gravitationalwave signatures of dark matter. Astrophys J 774:48. https://doi.org/10.1088/0004637X/774/1/48. arXiv:1302.2646 ADSCrossRefGoogle Scholar
 Macedo CFB, Cardoso V, Crispino LCB, Pani P (2016) Quasinormal modes of relativistic stars and interacting fields. Phys Rev D 93:064053. https://doi.org/10.1103/PhysRevD.93.064053. arXiv:1603.02095 ADSCrossRefGoogle Scholar
 Madarassy EJM, Toth VT (2015) Evolution and dynamical properties of Bose–Einstein condensate dark matter stars. Phys Rev D 91:044041. https://doi.org/10.1103/PhysRevD.91.044041. arXiv:1412.7152 ADSCrossRefGoogle Scholar
 Maldacena JM (1998) The large\(n\) limit of superconformal field theories and supergravity. Adv Theor Math Phys 2:231–252. https://doi.org/10.1023/A:1026654312961. arXiv:hepth/9711200 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Marsh DJE, Pop AR (2015) Axion dark matter, solitons and the cuspcore problem. Mon Not R Astron Soc 451:2479–2492. https://doi.org/10.1093/mnras/stv1050. arXiv:1502.03456 ADSCrossRefGoogle Scholar
 Marunović A (2015) Boson stars with nonminimal coupling. ArXiv eprints arXiv:1512.05718
 Marunović A, Murković M (2014) A novel black hole mimicker: a boson star and a global monopole nonminimally coupled to gravity. Class Quantum Gravity 31:045010. https://doi.org/10.1088/02649381/31/4/045010. arXiv:1308.6489 ADSzbMATHCrossRefGoogle Scholar
 Mazur PO, Mottola E (2001) Gravitational condensate stars: an alternative to black holes. ArXiv eprints arXiv:grqc/0109035
 McGreevy J (2010) Holographic duality with a view toward manybody physics. Adv High Energy Phys 2010:723105. https://doi.org/10.1155/2010/723105. arXiv:0909.0518 zbMATHCrossRefGoogle Scholar
 Meliani Z, Vincent FH, Grandclément P, Gourgoulhon E, MonceauBaroux R, Straub O (2015) Circular geodesics and thick tori around rotating boson stars. Class Quantum Gravity 32:235022. https://doi.org/10.1088/02649381/32/23/235022. arXiv:1510.04191 ADSzbMATHCrossRefGoogle Scholar
 Meliani Z, Grandclément P, Casse F, Vincent FH, Straub O, Dauvergne F (2016) GRAMRVAC code applications: accretion onto compact objects, boson stars versus black holes. Class Quantum Gravity 33:155010. https://doi.org/10.1088/02649381/33/15/155010 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Mendes RFP, Yang H (2016) Tidal deformability of dark matter clumps. ArXiv eprints arXiv:1606.03035
 Michelangeli A, Schlein B (2012) Dynamical collapse of boson stars. Commun Math Phys 311:645–687. https://doi.org/10.1007/s0022001113417. arXiv:1005.3135 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Mielke EW (2016) Rotating boson stars. Fundam Theor Phys 183:115–131. https://doi.org/10.1007/9783319312996_6 CrossRefGoogle Scholar
 Mielke EW, Scherzer R (1981) Geontype solutions of the nonlinear Heisenberg–Klein–Gordon equation. Phys Rev D 24:2111–2126. https://doi.org/10.1103/PhysRevD.24.2111 ADSCrossRefGoogle Scholar
 Mielke EW, Schunck FE (1999) Boson stars: early history and recent prospects, gravitation and relativistic field theories. In: Piran T, Ruffini R (eds) The Eighth Marcel Grossmann Meeting on recent developments in theoretical and experimental general relativity. World Scientific, Singapore, pp 1607–1626 arXiv:grqc/9801063 Google Scholar
 Mielke EW, Schunck FE (2002) Boson and axion stars. In: Gurzadyan VG, Jantzen RT, Ruffini R (eds) The Ninth Marcel Grossmann Meeting: on recent developments in theoretical and experimental general relativity, gravitation, and relativistic field theories. World Scientific, Singapore, pp 581–591CrossRefGoogle Scholar
 Milgrom M (1983) A modification of the Newtonian dynamics: implications for galaxies. Astrophys J 270:371–383. https://doi.org/10.1086/161131 ADSCrossRefGoogle Scholar
 Milgrom M (2011) MOND—particularly as modified inertia. Acta Phys Pol B 42:2175–2184. https://doi.org/10.5506/APhysPolB.42.21751111.1611 CrossRefGoogle Scholar
 Millward RS, Hirschmann EW (2003) Critical behavior of gravitating sphalerons. Phys Rev D 68:024017. https://doi.org/10.1103/PhysRevD.68.024017. arXiv:grqc/0212015 ADSCrossRefGoogle Scholar
 Mukherjee A, Shah S, Bose S (2015) Observational constraints on spinning, relativistic Bose–Einstein condensate stars. Phys Rev D 91:084051. https://doi.org/10.1103/PhysRevD.91.084051. arXiv:1409.6490 ADSCrossRefGoogle Scholar
 Mundim BC (2010) A numerical study of boson star binaries. PhD thesis, The University of British Columbia, Vancouver. http://laplace.physics.ubc.ca/Members/matt/Doc/Theses/. arXiv:1003.0239
 Murariu G, Puscasu G (2010) Solutions for Maxwellequations’ system in a static conformal spacetime. Rom J Phys 55:47–52MathSciNetzbMATHGoogle Scholar
 Murariu G, Dariescu C, Dariescu MA (2008) Maple routines for bosons on curved manifolds. Rom J Phys 53:99–108Google Scholar
 Myers RC, Perry MJ (1986) Black holes in higher dimensional spacetimes. Ann Phys (NY) 172:304–347. https://doi.org/10.1016/00034916(86)901867 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Ni Y, Zhou M, CardenasAvendano A, Bambi C, Herdeiro CAR, Radu E (2016) Iron k\(\alpha \) line of Kerr black holes with scalar hair. J Cosmol Astropart Phys 607(07):049. https://doi.org/10.1088/14757516/2016/07/049. arXiv:1606.04654 ADSCrossRefGoogle Scholar
 Nogueira F (2013) Extremal surfaces in asymptotically AdS charged boson stars backgrounds. Phys Rev D 87:106006. https://doi.org/10.1103/PhysRevD.87.106006. arXiv:1301.4316 ADSCrossRefGoogle Scholar
 Núñez D, Degollado JC, Moreno C (2011) Gravitational waves from scalar field accretion. Phys Rev D 84:024043. https://doi.org/10.1103/PhysRevD.84.024043. arXiv:1107.4316 ADSCrossRefGoogle Scholar
 Okawa H (2015) Nonlinear evolutions of bosonic clouds around black holes. Class Quantum Gravity 32:214003. https://doi.org/10.1088/02649381/32/21/214003 ADSzbMATHCrossRefGoogle Scholar
 Page DN (2004) Classical and quantum decay of oscillations: oscillating selfgravitating real scalar field solitons. Phys Rev D 70:023002. https://doi.org/10.1103/PhysRevD.70.023002. arXiv:grqc/0310006 ADSCrossRefGoogle Scholar
 Palenzuela C, Olabarrieta I, Lehner L, Liebling SL (2007) Headon collisions of boson stars. Phys Rev D 75:064005. https://doi.org/10.1103/PhysRevD.75.064005. arXiv:grqc/0612067 ADSCrossRefGoogle Scholar
 Palenzuela C, Lehner L, Liebling SL (2008) Orbital dynamics of binary boson star systems. Phys Rev D 77:044036. https://doi.org/10.1103/PhysRevD.77.044036. arXiv:0706.2435 ADSCrossRefGoogle Scholar
 Pani P, Berti E, Cardoso V, Chen Y, Norte R (2009) Gravitational wave signatures of the absence of an event horizon: nonradial oscillations of a thinshell gravastar. Phys Rev D 80:124047. https://doi.org/10.1103/PhysRevD.80.124047. arXiv:0909.0287 ADSCrossRefGoogle Scholar
 Pani P, Berti E, Cardoso V, Read J (2011) Compact stars in alternative theories of gravity: Einstein–Dilaton–Gauss–Bonnet gravity. Phys Rev D 84:104035. https://doi.org/10.1103/PhysRevD.84.104035. arXiv:1109.0928 ADSCrossRefGoogle Scholar
 Park SC (2012) Black holes and the LHC: a review. Prog Part Nucl Phys 67:617–650. https://doi.org/10.1016/j.ppnp.2012.03.004. arXiv:1203.4683 ADSCrossRefGoogle Scholar
 Pena I, Sudarsky D (1997) Do collapsed boson stars result in new types of black holes? Class Quantum Gravity 14:3131–3134. https://doi.org/10.1088/02649381/14/11/013 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Petryk RJW (2006) Maxwell–Klein–Gordon fields in black hole spacetimes. PhD thesis, The University of British Columbia, Vancouver. http://laplace.physics.ubc.ca/Members/matt/Doc/Theses/
 Pisano F, Tomazelli JL (1996) Stars of WIMPs. Mod Phys Lett A 11:647–651. https://doi.org/10.1142/S0217732396000667. arXiv:grqc/9509022 ADSCrossRefGoogle Scholar
 Polchinski J (2010) Introduction to gauge/gravity duality. In: Proceedings, theoretical advanced study institute in elementary particle physics (TASI 2010). String theory and its applications: from meV to the Planck scale: Boulder, Colorado, USA, June 1–25, 2010, pp 3–46. https://doi.org/10.1142/9789814350525_0001. arXiv:1010.6134
 Ponglertsakul S, Winstanley E, Dolan SR (2016) Stability of gravitating chargedscalar solitons in a cavity. Phys Rev D 94:024031. https://doi.org/10.1103/PhysRevD.94.024031. arXiv:1604.01132 ADSMathSciNetCrossRefGoogle Scholar
 Power EA, Wheeler JA (1957) Thermal geons. Rev Mod Phys 29:480–495. https://doi.org/10.1103/RevModPhys.29.480 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Psaltis D (2008) Probes and tests of strongfield gravity with observations in the electromagnetic spectrum. Living Rev Relativ 11:lrr20089. https://doi.org/10.12942/lrr20089, http://www.livingreviews.org/lrr20089, arXiv:0806.1531
 Pugliese D, Quevedo H, Rueda HJA, Ruffini R (2013) Charged boson stars. Phys Rev D 88:024053. https://doi.org/10.1103/PhysRevD.88.024053. arXiv:1305.4241 ADSCrossRefGoogle Scholar
 Radu E, Subagyo B (2012) Spinning scalar solitons in antide Sitter spacetime. Phys Lett B 717:450–457. https://doi.org/10.1016/j.physletb.2012.09.050. arXiv:1207.3715 ADSCrossRefGoogle Scholar
 Rangamani M, Takayanagi T (2017) Holographic entanglement entropy, Lecture Notes in Physics, vol 931. Springer, Cham. https://doi.org/10.1007/9783319525730. arXiv:1609.01287 zbMATHCrossRefGoogle Scholar
 Reid GD, Choptuik MW (2016) Nonminimally coupled topologicaldefect boson stars: static solutions. Phys Rev D 93:044022. https://doi.org/10.1103/PhysRevD.93.044022. arXiv:1512.02142 ADSCrossRefGoogle Scholar
 RindlerDaller T, Shapiro PR (2012) Angular momentum and vortex formation in Bose–Einsteincondensed cold dark matter haloes. Mon Not R Astron Soc 422:135–161. https://doi.org/10.1111/j.13652966.2012.20588.x. arXiv:1106.1256 ADSCrossRefGoogle Scholar
 Roger T, Maitland C, Wilson K, Westerberg N, Vocke D, Wright EM, Faccio D (2016) Optical analogues of the Newton–Schrödinger equation and boson star evolution. ArXiv eprints arXiv:1611.00924
 Rosen G (1966) Existence of particlelike solutions to nonlinear field theories. J Math Phys 7:2066–2070. https://doi.org/10.1063/1.1704890 ADSCrossRefGoogle Scholar
 Rousseau B (2003) Axisymmetric boson stars in the conformally flat approximation. Master’s thesis, The University of British Columbia, Vancouver. http://laplace.physics.ubc.ca/Members/matt/Doc/Theses/
 Ruffini R, Bonazzola S (1969) Systems of selfgravitating particles in general relativity and the concept of an equation of state. Phys Rev 187:1767–1783. https://doi.org/10.1103/PhysRev.187.1767 ADSCrossRefGoogle Scholar
 Ruiz M, Degollado JC, Alcubierre M, Núñez D, Salgado M (2012) Induced scalarization in boson stars and scalar gravitational radiation. Phys Rev D 86:104044. https://doi.org/10.1103/PhysRevD.86.104044. arXiv:1207.6142 ADSCrossRefGoogle Scholar
 Ryder LH (1996) Quantum field theory, 2nd edn. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
 Sakai N, Tamaki T (2012) What happens to Qballs if \(q\) is so large? Phys Rev D 85:104008. https://doi.org/10.1103/PhysRevD.85.104008. arXiv:1112.5559 ADSCrossRefGoogle Scholar
 Sakamoto K, Shiraishi K (1998) Boson stars with large selfinteraction in (\(2+1\))dimensions: an exact solution. J High Energy Phys 1998(07):015. https://doi.org/10.1088/11266708/1998/07/015. arXiv:grqc/9804067 zbMATHCrossRefGoogle Scholar
 Sakamoto K, Shiraishi K (1998b) Exact solutions for boson fermion stars in (\(2+1\))dimensions. Phys Rev D 58:124017. https://doi.org/10.1103/PhysRevD.58.124017. arXiv:grqc/9806040 ADSMathSciNetCrossRefGoogle Scholar
 SanchisGual N, Herdeiro C, Radu E, Degollado JC, Font JA (2017) Numerical evolutions of spherical Proca stars. Phys Rev D 95:104028. https://doi.org/10.1103/PhysRevD.95.104028. arXiv:1702.04532 ADSCrossRefGoogle Scholar
 Schive HY, Chiueh T, Broadhurst T (2014) Cosmic structure as the quantum interference of a coherent dark wave. Nat Phys 10:496–499. https://doi.org/10.1038/nphys2996. arXiv:1406.6586 CrossRefGoogle Scholar
 Schunck FE, Mielke EW (1996) Rotating boson stars, numerics, visualization. In: Hehl FW, Puntigam RA, Ruder H (eds) Relativity and scientific computing: computer algebra. Springer, Berlin, pp 138–151CrossRefGoogle Scholar
 Schunck FE, Mielke EW (2003) General relativistic boson stars. Class Quantum Gravity 20:R301–R356. https://doi.org/10.1088/02649381/20/20/201. arXiv:0801.0307 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Schunck FE, Torres DF (2000) Boson stars with generic selfinteractions. Int J Mod Phys D 9:601–618. https://doi.org/10.1142/S0218271800000608. arXiv:grqc/9911038 ADSGoogle Scholar
 Schwabe B, Niemeyer JC, Engels JF (2016) Simulations of solitonic core mergers in ultralight axion dark matter cosmologies. Phys Rev D 94:043513. https://doi.org/10.1103/PhysRevD.94.043513. arXiv:1606.05151
 Seidel E, Suen WM (1990) Dynamical evolution of boson stars: perturbing the ground state. Phys Rev D 42:384–403. https://doi.org/10.1103/PhysRevD.42.384 ADSCrossRefGoogle Scholar
 Seidel E, Suen WM (1991) Oscillating soliton stars. Phys Rev Lett 66:1659–1662. https://doi.org/10.1103/PhysRevLett.66.1659 ADSCrossRefGoogle Scholar
 Seidel E, Suen WM (1994) Formation of solitonic stars through gravitational cooling. Phys Rev Lett 72:2516–2519. https://doi.org/10.1103/PhysRevLett.72.2516. arXiv:grqc/9309015 ADSCrossRefGoogle Scholar
 Sharma R, Karmakar S, Mukherjee S (2008) Boson star and dark matter. ArXiv eprints arXiv:0812.3470
 Shen T, Zhou M, Bambi C, Herdeiro CAR, Radu E (2017) Iron K\(\alpha \) line of Proca stars. J Cosmol Astropart Phys 2017(08):014. https://doi.org/10.1088/14757516/2017/08/014. arXiv:1701.00192 CrossRefGoogle Scholar
 Shibata M, Nakamura T (1995) Evolution of threedimensional gravitational waves: harmonic slicing case. Phys Rev D 52:5428–5444. https://doi.org/10.1103/PhysRevD.52.5428 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Shibata M, Yoshino H (2010) Barmode instability of rapidly spinning black hole in higher dimensions: numerical simulation in general relativity. Phys Rev D 81:104035. https://doi.org/10.1103/PhysRevD.81.104035. arXiv:1004.4970 ADSCrossRefGoogle Scholar
 Silveira V, de Sousa CMG (1995) Boson star rotation: a Newtonian approximation. Phys Rev D 52:5724–5728. https://doi.org/10.1103/PhysRevD.52.5724. arXiv:astroph/9508034 ADSCrossRefGoogle Scholar
 Sirunyan AM et al (2017) Search for black holes in highmultiplicity final states in proton–proton collisions at sqrt(s) = 13 TeV. ArXiv eprints arXiv:1705.01403
 Smolić I (2015) Symmetry inheritance of scalar fields. Class Quantum Gravity 32:145010. https://doi.org/10.1088/02649381/32/14/145010. arXiv:1501.04967 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Soni A, Zhang Y (2017) Gravitational waves from SU(N) glueball dark matter. Phys Lett B 771:379–384. https://doi.org/10.1016/j.physletb.2017.05.077. arXiv:1610.06931 ADSCrossRefGoogle Scholar
 Stewart I (1982) Catastrophe theory in physics. Rep Prog Phys 45:185–221. https://doi.org/10.1088/00344885/45/2/002 ADSMathSciNetCrossRefGoogle Scholar
 Stojkovic D (2003) Nontopological solitons in brane world models. Phys Rev D 67:045012. https://doi.org/10.1103/PhysRevD.67.045012. arXiv:hepph/0111061 ADSMathSciNetCrossRefGoogle Scholar
 Stotyn S, Mann RB (2012) Another mass gap in the BTZ geometry? J Phys A 45:374025. https://doi.org/10.1088/17518113/45/37/374025. arXiv:1203.0214 MathSciNetzbMATHCrossRefGoogle Scholar
 Stotyn S, Park M, McGrath P, Mann RB (2012) Black holes and boson stars with one Killing field in arbitrary odd dimensions. Phys Rev D 85:044036. https://doi.org/10.1103/PhysRevD.85.044036. arXiv:1110.2223 ADSCrossRefGoogle Scholar
 Stotyn S, Chanona M, Mann RB (2014a) Numerical boson stars with a single Killing vector. II. The d \(=\) 3 case. Phys Rev D 89:044018. https://doi.org/10.1103/PhysRevD.89.044018. arXiv:1309.2911 ADSCrossRefGoogle Scholar
 Stotyn S, Leonard CD, Oltean M, Henderson LJ, Mann RB (2014b) Numerical boson stars with a single Killing vector I. The \(d\ge 5\) case. Phys Rev D 89:044017. https://doi.org/10.1103/PhysRevD.89.044017. arXiv:1307.8159 ADSCrossRefGoogle Scholar
 Straumann N (1984) General relativity and relativistic astrophysics. Springer, Berlin. https://doi.org/10.1007/9783642844393 CrossRefGoogle Scholar
 Straumann N (1992) Fermion and boson stars. In: Ehlers J, Schäfer G (eds) Relativistic gravity research with emphasis on experiments and observations. Lecture Notes in Physics, vol 410. Springer, BerlinGoogle Scholar
 Tamaki T, Sakai N (2010) Unified picture of Qballs and boson stars via catastrophe theory. Phys Rev D 81:124041. https://doi.org/10.1103/PhysRevD.81.124041. arXiv:1105.1498 ADSCrossRefGoogle Scholar
 Tamaki T, Sakai N (2011a) Gravitating Qballs in the Affleck–Dine mechanism. Phys Rev D 83:084046. https://doi.org/10.1103/PhysRevD.83.084046. arXiv:1105.3810 ADSCrossRefGoogle Scholar
 Tamaki T, Sakai N (2011b) How does gravity save or kill Qballs? Phys Rev D 83:044027. https://doi.org/10.1103/PhysRevD.83.044027. arXiv:1105.2932 ADSCrossRefGoogle Scholar
 Tamaki T, Sakai N (2011c) What are universal features of gravitating Qballs? Phys Rev D 84:044054. https://doi.org/10.1103/PhysRevD.84.044054. arXiv:1108.3902 ADSCrossRefGoogle Scholar
 Thorne KS (1972) Nonspherical gravitational collapse: a short review. In: Klauder JR (ed) Magic without magic: John Archibald Wheeler. A collection of essays in honor of his sixtieth birthday. W.H. Freeman, San Francisco, pp 231–258Google Scholar
 Torres DF, Capozziello S, Lambiase G (2000) Supermassive boson star at the Galactic center? Phys Rev D 62:104012. https://doi.org/10.1103/PhysRevD.62.104012. arXiv:astroph/0004064 ADSCrossRefGoogle Scholar
 Unruh WG (2014) Has Hawking radiation been measured? Found Phys 44:532–545. https://doi.org/10.1007/s1070101497780, proceedings, Horizons of Quantum Physics: Taipei, Taiwan, October 1418, 2012. ArXiv eprints arXiv:1401.6612
 UreñaLópez LA, Bernal A (2010) Bosonic gas as a galactic dark matter halo. Phys Rev D 82:123535. https://doi.org/10.1103/PhysRevD.82.123535. arXiv:1008.1231 ADSCrossRefGoogle Scholar
 UreñaLópez LA, Matos T, Becerril R (2002) Inside oscillatons. Class Quantum Gravity 19:6259–6277. https://doi.org/10.1088/02649381/19/23/320 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 ValdezAlvarado S, Becerril R, UreñaLópez LA (2011) \(\phi ^{4}\) oscillatons. ArXiv eprints arXiv:1107.3135
 ValdezAlvarado S, Palenzuela C, Alic D, UreñaLópez LA (2013) Dynamical evolution of fermion–boson stars. Phys Rev D 87:084040. https://doi.org/10.1103/PhysRevD.87.084040. arXiv:1210.2299 ADSCrossRefGoogle Scholar
 Vilenkin A, Shellard EPS (1994) Cosmic strings and other topological defects. Cambridge Monographs on Mathematical Physics. Cambridge University Press, CambridgezbMATHGoogle Scholar
 Vincent FH, Gourgoulhon E, Herdeiro C, Radu E (2016a) Astrophysical imaging of Kerr black holes with scalar hair. Phys Rev D 94:084045. https://doi.org/10.1103/PhysRevD.94.084045. arXiv:1606.04246 ADSCrossRefGoogle Scholar
 Vincent FH, Meliani Z, Grandclément P, Gourgoulhon E, Straub O (2016b) Imaging a boson star at the Galactic center. Class Quantum Gravity 33:105015. https://doi.org/10.1088/02649381/33/10/105015. arXiv:1510.04170 ADSCrossRefGoogle Scholar
 Wald RM (1984) General relativity. University of Chicago Press, ChicagozbMATHCrossRefGoogle Scholar
 Wheeler JA (1955) Geons. Phys Rev 97:511–536. https://doi.org/10.1103/PhysRev.97.511 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Will CM (2014) The confrontation between general relativity and experiment. Living Rev Relativ 17:lrr20144. https://doi.org/10.12942/lrr20144. http://www.livingreviews.org/lrr20144. arXiv:1403.7377
 Yagi K, Stein LC (2016) Black hole based tests of general relativity. Class Quantum Gravity 33:054001. https://doi.org/10.1088/02649381/33/5/054001. arXiv:1602.02413 ADSMathSciNetzbMATHCrossRefGoogle Scholar
 Yoshida S, Eriguchi Y (1997) Rotating boson stars in general relativity. Phys Rev D 56:762–771. https://doi.org/10.1103/PhysRevD.56.762 ADSMathSciNetCrossRefGoogle Scholar
 Yuan YF, Narayan R, Rees MJ (2004) Constraining alternate models of black holes: type I Xray bursts on accreting fermion–fermion and boson–fermion stars. Astrophys J 606:1112–1124. https://doi.org/10.1086/383185. arXiv:astroph/0401549 ADSCrossRefGoogle Scholar
 Yunes N, Yagi K, Pretorius F (2016) Theoretical physics implications of the binary blackhole mergers GW150914 and GW151226. Phys Rev D 94:084002. https://doi.org/10.1103/PhysRevD.94.084002. arXiv:1603.08955 ADSCrossRefGoogle Scholar
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