Boson Stars under Deconstruction
aa r X i v : . [ g r- q c ] F e b N. Kan 1
Boson Stars under Deconstruction
Nahomi Kan a ) and Kiyoshi Shiraishi b )( a ) Yamaguchi Junior College, Hofu-shi, Yamaguchi 747-1232, Japan ( b ) Yamaguchi University, Yamaguchi-shi, Yamaguchi 753-8512, Japan
Abstract
We study solutions for boson stars in multiscalar theory. We start with simple modelswith N scalar theories. Our purpose is to study the models in which the mass matrixof scalars and the scalar couplings are given by an extended method of dimensionaldeconstruction. The properties of the boson stars are investigated by the Newtonianapproximation with the large coupling limit. Boson Stars (BSs) have been studied in expectation of solving the rotation curve (RC) problem. Manyauthors have attempted to explain RC of galaxies by assuming the existence of the galactic scale BS. Inorder to fit the observable data, the mass density of BS needs to be widely distributed. This configurationcan be constructed by the models such as BSs with scalar particles in excited states or the rotating BSs,but these BSs are unstable. Whereas Newtonian BSs with all the particles in only one ground state isstable, it is difficult to illustrate the realistic RC. Alternative models to solve these problems have beenstudied by Matos and Ure˜na-L´opez [1], and recently by Bernal et al. [2]. They considered the multi-stateBS, i. e. scalar fields both in ground and in excited states, with no (quadratic) self-couplings.In the present work, We consider multi-kind scalar BS, not multi-states. We suggest several models. Thefirst model contains two scalar particles with self- and mutual-couplings. The second model is build upunder dimensional deconstruction (DD). This model contains N scalar particles interacting with oneselfand with adjacent scalars. DD has an aspect of latticized extra dimensions, and the latter model couldbe an alternative to a higher dimensional BS. In each model, we examine BS with large coupling limit.We also consider BSs under extended DD. In this model, DD is generalized to field theory based on a graph , and the interactions between scalar particles are restricted by supersymmetry (SUSY). We consider a BS model, in which two scalar particles ψ and ψ with self- and mutual-couplings g ij ,described by the Hamiltonian: H − µ ˜ N − µ ˜ N = ~ m |∇ ψ | + ~ m |∇ ψ | + (cid:0) m | ψ | + | ψ | (cid:1) φ + 18 πG ( ∇ φ ) − µ | ψ | − µ | ψ | + 14 ~ c (cid:18) g m | ψ | + 2 g m m | ψ | | ψ | + g m | ψ | (cid:19) , (1)where ˜ N i and µ i are the number density and the chemical potential of the i -th scalar, respectively,whereas φ is the gravitational potential. We also normalize the particle number to N i = R d r | ψ i | . Inthe large coupling limit [6], the equations of motion are as follows: ∇ φ = 4 πG ( m | ψ | + m | ψ | ) , (2) m φψ + 12 (cid:18) g m | ψ | + g m m | ψ | (cid:19) = µ ψ , (3) m φψ + 12 (cid:18) g m | ψ | + g m m | ψ | (cid:19) = µ ψ , (4) Email address: [email protected] Email address: [email protected] Interacting boson stars and Q-balls have been studied by Brihaye et al. [3–5].
BSs under DDwhere ~ = c = 1. In the core of the BS, where ψ = 0 , ψ = 0, the gravitational potential becomes φ + const. ∝ − sin( ωr ) r , (5)where r is the distance from the center of the BS, and ω = 8 πG m g − m m g + m g g g − g . (6)The outside of the BS, where ψ = 0 , ψ = 0, the gravitational potential is φ + const. ∝ − sin( ω r + δ ) r , (7)with ω = 8 πGm /g . The typical structure of BS are shown in Fig. 1 and the rotational curves inFig. 2. We can find that the gravitational potential is spread out by the existence of Ψ , and which leadsto an improvement of RC. If a single scalar field model is considered, which realized by Ψ = 0 in (1), the r Figure 1: The behavior of the scalar fieldsΨ , Ψ and the gravitational potential φ asthe function of the rescaled distance r . Thesolid line, the broken line and the dotted linerepresent Ψ , Ψ and φ , respectively. Thepotential is spread out by the existence ofΨ . r v Figure 2: The behavior of the rotational ve-locity v as the function of the rescaled dis-tance r . The multi-scalar configuration im-prove RC.range of the gravitational potential becomes narrow (Fig. 3), and the RC looks far from a satisfactoryexplanation of the observational data (Fig. 4). We propose three models of BSs under the DD scheme.
We consider self-interacting U (1) scalar field theory in DD. This model is described by the action: S B = Z d x √− g N X i =1 ( | ∂ µ φ i | − m | φ i | − f | φ i +1 − φ i | − ˜ λN | φ i | ) . (8)If f = 0, [ U (1)] N symmetry recovers. Ansatz for a static and spherical BS: φ i ( x ) = √ N φ ( r ) e − iωt + iθ i ,where θ i is a constant number, leads to the square of a scalar boson mass: m + f N N X i =1 | − e i ( θ i +1 − θ i ) | ≡ m b , (9). Kan 3 r Figure 3: The behavior of the scalar field andthe gravitational potential as the function ofthe rescaled distance r . The solid line repre-sents the scalar field, whereas the dotted linerepresents the gravitational potential. Com-pared to the two scalar field model (Fig. 1),the range of the potential becomes narrow. r v Figure 4: The behavior of the rotational ve-locity v as the function of the rescaled dis-tance r . The RC looks far from a satisfactoryexplanation of the observational data.and to the charge density: N X i =1 i ( φ ∗ i ∂ φ i − φ i ∂ φ ∗ i ) = 2 ωφ ( r ) , (10)where the charge are equally distributed on each site. The BS mass made of a single scalar with mass m b becomes M BS ∼ M pl m b for no coupling, whereas M BS ∼ p ˜ λ M pl m b for large coupling, where we use theconvention of Jetzer [7]. Stars with these maximum masses are stable. If θ = · · · = θ N , or N → ∞ ,Kaluza-Klein (KK) theory is recovered and BSs are made of the zero-mode field with a minimum mass( m b ) min = m . We extend DD to the model based on graph theory. In this model, a continuum limit is not necessary,and U (1) interactions at each site (vertex) of an arbitrary graph is still invariant. We also assume SUSYin order to restrict the other interactions [8], and then simplify the interaction terms. This model isdescribed by the action: S B = Z d x √− g N X i,j =1 (cid:26) | ∂ µ φ i | − m | φ i | − f φ ∗ i △ ij φ j − N Λ ij | φ i | | φ j | (cid:27) , (11)where λ = P i,j Λ ij , and △ is graph laplacian. Unfortunately, this model also describes a single U (1)charge in general. Thus the most probable BS in this model is made of a scalar field with the minimummass. Characteristic matrices associated with a graph are the graph laplacian △ and identity matrix. We expandthe scalar field by the eigenvector of the graph laplacian, such as ~φ = { φ , φ , · · · , φ N } = P Na =1 φ a ~x a ,where △ ~x a = λ a ~x a , and ~x a · ~x b = δ ab , as usual. In this notations, BS based on the graph with p verticesand q edges is described by the action: S B = Z d x √− g (cid:26) ∂ ~φ † · ∂ ~φ − m | ~φ | − f ~φ † △ ~φ − Λ p | ~φ | ) − Λ q | ~φ | ~φ † △ ~φ − Λ r ~φ † △ ~φ ) (cid:27) . (12) BSs under DDIf φ = · · · = φ N = 0 and λ = 0, a boson mass becomes m a = p m + f λ a and interaction terms are − Λ p (cid:18) | ψ | m + | ψ | m (cid:19) − Λ q λ | ψ | m (cid:18) | ψ | m + | ψ | m (cid:19) − Λ r λ | ψ | m , (13)where ψ a ≡ √ m a φ a e im a t . Self- and mutual-couplings g ij can be read from (13), such as g = Λ q / We have examined the Newtonian boson star with two U (1) charges in the large-coupling limit. Theunderstanding of rotation curves of galaxies are improved in this model. We have also examined BSsunder deconstruction. In the continuum limit, it is found that the possible BS is made of “zero mode”field. We have suggested two models of BSs based on graph theory.As future work, we will consider general relativistic BSs and graph-oriented models with many charges orarbitrary couplings. We will also investigate excited states under the condition of fixing the mass and sizeof BSs. Time dependent solutions or oscillations are also interesting. We wish to study these subjects,elsewhere. Acknowledgements
The authors would like to thank K. Kobayashi for useful comments, and also the organizers of JGRG19.
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