In this work we present some new results which we have obtained in a study of the phase diagram of charged compact boson stars in the theory involving massive complex scalar fields coupled to the U(1) gauge field and gravity in a conical potential in the presence of a cosmological constant Λ which we treat as a free parameter taking positive and negative values and thereby allowing us to study the theory in the de Sitter and Anti de Sitter spaces respectively. In our studies, we obtain four bifurcation points (possibility of more bifurcation points being not ruled out) in the de Sitter region. We present a detailed discussion of the various regions in our phase diagram with respect to four bifurcation points. Our theory is seen to have rich physics in a particular domain for positive values of Λ which is consistent with the accelerated expansion of the universe.Introduced long ago [1][2][3], boson stars represent localized self-gravitating solutions studied vary widely in the literature [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. Such theories are being considered in the presence of positive [14][15][16] as well as negative [17][18][19][20] values of the cosmological constant Λ. The theories with positive values of Λ (corresponding to the de Sitter (dS) space) are relevant from observational point of view as they describe a more realistic description of the compact stars in the universe since all the observations seem to indicate the existence of a positive cosmological constant. Such theories are also being used to model the dark energy of the universe. However, the theories with negative values of Λ (corresponding to the Anti de Sitter (AdS) space) are meaningful in the context of AdS/CFT correspondence [24][25][26].In fact, cosmological constant, the value of the energy density of the vacuum of space is the simplest form of dark energy and it provides a good fit to many cosmological observations. A positive vacuum energy density resulting from a positive cosmological constant (implying a negative) pressure gives an accelerated expansion of the universe consistent with the observations. Our theory is seen to have rich physics in a particular domain for positive values of Λ.have studied In a recent paper [15], we have studied the boson stars and boson shells in a theory of complex scalar field coupled to U (1) gauge field A µ and the gravity in the presence of a fixed positive cosmological constant Λ (i.e. in the de Sitter space). In the present work we study this theory of complex scalar field coupled to U (1) gauge field A µ and the gravity in the presence of a potential: V (|Φ|) := (m 2 |φ| 2 + λ|φ|) (with m and λ are constant parameters) and a cosmological constant Λ which we treat as a free parameter and which takes positive as well as negative values and thereby allowing us to study the theory in the dS as well as in the AdS space. We investigate * sanjeev.kumar.ka@gmail.com † ushakulsh@gmail.com, ushakuls@iastate.edu ‡ dskulsh@gmail.com, dayakuls@iastate.edu the properties...

We study boson shells and boson stars in a theory of a complex scalar field coupled to the gauge field and Einstein gravity with the potential . This could be considered either as a theory of a massive complex scalar field coupled to an electromagnetic field and gravity in a conical potential, or as a theory in the presence of a potential that is an overlap of a parabolic and conical potential. Our theory has a positive cosmological constant . Boson stars are found to come in two types, having either ball-like or shell-like charge density. We studied the properties of these solutions and also determined their domains of existence for some specific values of the parameters of the theory. Similar solutions have also been obtained by Kleihaus, Kunz, Laemmerzahl and List, in a V-shaped scalar potential.

Compact boson stars, whose scalar field vanishes identically in the exterior region, arise in a theory involving a massless complex scalar field with a conical potential, when coupled to gravity. Their charged compact generalizations, obtained in the presence of a U(1) gauge field, exhibit further interesting features. On the one hand, charged compact boson shells can arise, whose scalar field vanishes also in the central region, while on the other hand, the domain of existence of charged compact boson stars exhibits bifurcation points. First 2D phase diagrams have been studied before. Here we extend these earlier studies to a larger range of the variables and study additional phase diagrams. We then extend these studies to obtain 3D phase diagrams and present these with a detailed discussion of their various regions with respect to the bifurcation points and argue, that there is an infinite series of such bifurcation points. Thus the theory is seen to contain rich physics in a particular domain of the phase diagrams. We also discuss the dependence of the fields on the dimensionless radial coordinate for some representative points of the phase trajectories in the phase diagrams of the theory.

We study boson stars in a theory of complex scalar field coupled to Einstein gravity with the potential: V (|Φ|) := m 2 |Φ| 2 + 2λ |Φ| (where m 2 and λ are positive constant parameters). This could be considered either as a theory of massive complex scalar field coupled to gravity in a conical potential or as a theory in the presence of a potential which is an overlap of a parabolic and a conical potential. We study our theory with positive as well as negative values of the cosmological constant Λ . Boson stars are found to come in two types, having either ball-like or shell-like charge density. We have studied the properties of these solutions and have also determined their domains of existence for some specific values of the parameters of the theory. Similar solutions have also been obtained by Hartmann, Kleihaus, Kunz, and Schaffer, in a V-shaped scalar potential.Keywords Gravity Theories · Boson stars · Boson shells · Q-balls · Q-shells A study of boson shells and boson stars in scalar electrodynamics with a self-interacting complex scalar field Φ coupled to Einstein gravity is of a very wide interest in the gravity theories[1]- [26] . Hartmann, Kleihaus, Kunz, and Schaffer (HKKS) [2,3] have recently studied boson stars in a theory of complex scalar field coupled to Einstein gravity in a V-shaped scalar potential: V (ΦΦ * ) ≡ V (|Φ|) = λ c |Φ| (where λ c is a constant). They have found that the boson stars come in two types, having either ball-like or shell-like charge density. They have studied the properties of these solutions and have also determined their domains of existence.Actually, the boson stars represent localized self-gravitating solutions [6,7,8], that have been considered in many different contexts [9,10,11,12,13]. To obtain boson stars, typically a complex scalar field Φ is considered. The U(1) invariance of the theory then provides a conserved Noether current.The properties of the boson stars depend strongly on the self-interaction employed. In particular, as discussed by Lee and collaborators [14] and by Coleman [15], the existence of a flat space-time limit of these localized solutions, i.e., the existence of Q-balls, puts constraints on the types of self-interaction possible.Arodz and Liz showed, that besides Q-balls another type of localized solution could also arise in flat space, when a V-shaped self-interaction is employed in the presence of a gauge field [16,17,18]. In this type of solution, the energy density is no longer ball-like as in the case of Q-balls, but it is instead shell-like.