Oscillating soliton stars with network of domain walls
aa r X i v : . [ h e p - t h ] J un Oscillating soliton stars with network of domain walls
Stephen Owusu ∗ Departamento de F´ısica, Universidade Federal de Campina GrandeCaixa Postal 10071, 58429-900 Campina Grande, Para´ıba, Brazil
Francisco A. Brito † Departamento de F´ısica, Universidade Federal de Campina Grande Caixa Postal 10071,58429-900 Campina Grande, Para´ıba, Brazil andDepartamento de F´ısica, Universidade Federal da Para´ıba,Caixa Postal 5008, 58051-970 Jo˜ao Pessoa, Para´ıba, Brazil
In this work we study oscillating soliton stars (oscillatons) with network of domain walls. Weconsider a Lagrangian with three scalar fields coupled among themselves by a specific potential.We choose an appropriate potential to admit the formation of network of domain walls on theoscillatons. With small perturbations applied to this potential, we then compute the Einstein-Klein-Gordon (EKG) equations numerically and analyze the mass profile of this new object. Fromthe results we discuss how the stability of the oscillatons is affected by the network. At someconditions the network provides a ‘bouncing stability’ to the oscillatons.
PACS numbers: 11.27.+d, 11.30.Er, 97.10.Cv
I. INTRODUCTION
A new class of astronomical objects was proposed by Friedberg, Lee, and Pang (1987) and called these objectssoliton stars [1–3]. These stars are objects of interest because they exhibit some remarkable properties. Accordingto general relativity, if a star becomes massive enough, or captures enough matter greater than a critical value M c ,the star would undergo violent processes, either by expelling some of its mass and becoming a neutron star or awhite dwarf with M < M c , or by collapsing into a black hole. Soliton star can exist with a very large stable coldmass configuration without becoming a black hole. Ordinary cold star like a white dwarf or a neutron star cannothave mass greater than five solar masses (5 M ⊙ at zero angular momentum) [4, 5]. Soliton star is a type of stellarconfiguration that can have a very large mass ( > M ⊙ ), a very small volume and very high density. For example, amini-soliton star could have a radius 6 × cm , a mass of 10 kg and density 10 times that of a neutron star [6].In quantum field theory, when a discrete symmetry is spontaneously broken, for instance a Z symmetry, a domainwalls can form. For other Z N symmetries, with N > SU (3)group whose center is the discrete Z symmetry. At the same scale, primordial black holes can also appear [9–11].Thus, the network once formed may affect the stability of the soliton stars and can drive the rate of formation of suchblack holes.It has been long known that classical field theories admit non-topological soliton solution [12]. These include Q-balls [13], scalar soliton stars and boson stars [14]. They are configurations made up of complex scalar fields, throughnonlinear couplings of the scalar field to itself to other matter fields or to gravity. In the cases presented above, theconserved current is as a result of the global U(1) symmetry of the complex fields. But in the case of real scalar field,because of the absence of such symmetry there is no non-topological soliton solution. It can be shown that a massivereal scalar field satisfying the Klein-Gordon equation can form a self-gravitating solitonic object when coupled toEinstein gravity. This new class of objects is not static, but rather periodic in time. We call such objects oscillatingsoliton stars [15].In this work we will investigate the formation of networks with the oscillating soliton star and study how the networkaffect the stability of the star by examining the mass profile. We execute this idea starting with appropriate model,described by three real scalar fields, introduced in line with [16]. An adjustment in the form of small perturbations ∗ Electronic address: [email protected] † Electronic address: [email protected] applied to the potential will indicate where the network is formed with respect to the oscillating soliton star. Thismay give rise to three possibilities, that is having the network formed inside, or on the surface, or outside the star.We analyze the stability of this new type of self-gravitating object and its significant role it can play in cosmology,as a dark matter candidate or as a significant source of gravitational wave. Since primordial black holes can alsobe candidates to dark matter [17] these new oscillating soliton stars (oscillatons) with network of domain walls canoffer the possibility to uncover new scenarios. As source of gravitational waves some interesting new scenarios canalso happen. For instance, rotating sources of two such objects can suffer transition to two black holes before theycan be merged — a signature that may appear in the detection of gravitational waves [18] and electromagnetic wavesfrom the same event. As we shall show, the stability os such oscillatons is affected by slightly perturbing the networkfrom their surface. Then approximating sources of such objects tidal forces can deform their surface in relation to thenetwork destroying their stability driving them to collapse to black holes.This work will be organized as follows: In Sec. II we will introduce the basic aspects of the study of oscillatingsoliton star. Sec. III will contain the oscillating soliton star with network of domain walls. Finally Sec. IV will holdcomment, discussion and summary of the work.
II. OSCILLATING SOLITON STARS
In this section let us quickly review the non self-interacting single scalar field model. The most complete actiondescribing a self-gravitating real scalar field in curved space-time is S = Z d x √− g (cid:20) R κ + L m (cid:21) , (1)where κ = 8 πG , g = det g µν , R is the Ricci scalar, and L m is the Lagrangian for matter field. In this section, workingwith one scalar field we define the Lagrangian as L m = 12 [ g µν ∂ µ φ∂ ν φ − V ( φ )] , (2)where φ is the scalar field and V ( φ ) = m φ is the potential when we consider free-field case with m being the massof the field. Variation of this action with respect to the scalar field gives the Klein-Gordon equation, as (cid:3) φ = 12 ∂V∂φ (3)being (cid:3) ≡ √− g ∂ µ ( √− gg µν ∂ ν ) the D’Alembertian in curved space. On the other hand, the variation of the actionwith respect to the metric g µν leads to the Einstein field equations given by R µν − g µν R = κ T µν (4)where T µν = ∂ µ φ∂ ν φ − g µν L m is the energy-momentum tensor.Consider therefore a general time dependent, spherically symmetric geometry in a simple example of oscillatingsoliton stars, as a real massive Klein-Gordon scalar field, coupled to gravity as described in [15, 19, 20]. In the absenceof angular momentum we consider the soliton solution to be spherically symmetric. The spherically symmetric lineelement is written in the form ds = − N ( t, r ) dt + g ( t, r ) dr + r dθ + r sin ( θ ) dϕ , (5)with N ( t, r ) being the lapse function and g ( t, r ) the radial metric function. Let us start by taking the case analyzedby [15], and write the coupled Einstein-Klein-Gordon (EKG) equations as: the t − t component ( g ) ′ = − g (cid:18) g − r (cid:19) + 4 πGrg ˙ φg N + φ ′ + g m φ ! , (6) the r − r component ( N ) ′ = N ( g − r ) + 4 πGr ( N φ ′ − N g m φ + ˙ φ ) , (7) the t − r component ˙ g = 4 πGrg ˙ φφ ′ . (8)We follow equation (3) to write the Klein-Gordon equation as¨ φ − ˙ N ˙ φN = ( N ) ′ g φ ′ + N g (cid:20) φ ′′ − ( g ) ′ φ ′ g − g ˙ gN (cid:21) + 2 N φ ′ g r − m N φ, (9)where the over-dot represent ∂∂t and the prime denote ∂∂r . For numerical convenience we define the dimensionlessquantities r → r/m , t → t/m , Φ → φ √ κ where we note that the bosonic mass is the natural scale for time anddistance. Also in order to deal with the non-linearity present in the EKG equations, it is convenient to introduce newvariables where A ( r, t ) = g and C = [ g ( r, t ) /N ( r, t )] [19–22]. The EKG equations (6)-(9) now becomes A ′ = − A (cid:18) A − r (cid:19) + Ar h C ˙Φ + Φ ′ + A Φ i , (10) C ′ = 2 Cr (cid:2) A (Φ r − (cid:3) , (11) C ¨Φ + 12 ˙ C ˙Φ = Φ ′′ + Φ ′ (cid:18) r − C ′ C (cid:19) − A Φ , (12)˙ A = 2 rA ˙ΦΦ ′ . (13)These equations have no equilibrium solutions which are time independent (static metric component). All knownstatic solutions to the system either have singularities or are topologically nontrivial [23]. The nature of equations(10)-(12) suggests periodic expansion of the form A ( r, t ) = ∞ X j =0 A j ( r ) cos(2 jωt ); (14) C ( r, t ) = ∞ X j =0 C j ( r ) cos(2 jωt ); (15)Φ( r, t ) = ∞ X j =1 φ j − ( r ) cos[(2 j − ωt ] , (16)where ω is the fundamental frequency and will be absorbed in the redefinition of the time variable in the numericalcalculations. The actual solution is an infinite Fourier expansion of the above form that is convergent [15, 19]. A. Eigenvalue problem and the boundary conditions for equilibrium configurations
We put these expansion into equations (10)-(12), and set each Fourier coefficient to zero, to obtain a systemof coupled nonlinear ordinary first-order differential equations for A j ( r ) and C j ( r ), and second order differentialequations φ j − ( r ). Eq. (13) is used as an algebraic equation to determine A .The boundary conditions are given by the following requirements:i) Asymptotic flatness: this requires that as r → ∞ , A ( r = ∞ , t ) = 1 and φ ( r = ∞ , t ) = 0. The coefficient A ( ∞ ) = 1, A j ( ∞ ) = 0 for j = 0, φ j ( ∞ ) = 0 for all j , and C j ( ∞ ) = 0 for j = 0.ii) Non-singularities: at r = 0, the absence of a conical singularity implies A j ( r = 0) = 0. The requirement that themetric coefficients be finite at r = 0 implies φ ′ ( r = 0) = 0.The set of equations (10)-(12) becomes an eigenvalue problem. Thus, it is necessary to determine the initial values φ j − (0), C j (0), corresponding to a given central value φ (0). To proceed, we truncate the system of equations aftera certain maximum j = j max = 1, numerically solve the eigenvalue problem by using the Runge-Kutta fourth-ordermethod, and study the convergence of the series as function of j = j max .The soliton star total mass M . Asymptotically any soliton star (or boson) metric resembles Schwarzschild metric,which allows us to associate the metric coefficient g rr = (1 − M/r ) − , where M is the ADM mass defined for anasymptotically flat spacetime and g rr = g ( r, t ) = A ( r, t ). It can be calculated as M = lim r →∞ r (cid:20) − A ( r, t ) (cid:21) M P lanck m (17)where r is the outermost point of numerical domain. For a recent discussion on this issue and boson stars, see forexample [24, 25].A typical oscillaton solution is shown in Fig. 1. The radial metric function A ( r, t ), for the case of φ (0) = 0 . A j for the first few values of j . Though we are solving non-linear equations, the Fourier seriesconverges rapidly.In Fig. 2 the mass M of the star is plotted as a function of the central field φ (0). This mass curve is similarto those of white dwarfs, neutron stars, and boson stars, with a maximum mass given by M c ≈ . M planck /m at φ c (0) ≈ .
0. The branch to the left of the maximum is the stable branch traditionally called the S-branch. Stabilityhere means that the stars on this branch move to new lower mass configurations on the same branch under smallperturbations. To the right is the unstable branch called U-branch. U-branch are stars inherently unstable to smallperturbations and collapse to black hole. In this regime, even small increase in mass will induce the collapse of thestar into a black hole or lose of mass through scalar radiation will cause it to migrate to the S-branch.
FIG. 1: A typical solution to the truncated eigenvalue equations ( j max = 1) of the expansion of the metric A ( r, t ). The solidline shows A and the dotted line shows A . III. OSCILLATING SOLITON STARS WITH NETWORK OF DOMAIN WALLS
Let us explore the idea of having a network of domain walls living on an oscillating soliton star. In this presentwork we offer a model that contains combined ideas of the basic mechanism seen in section II and the ideas treated inRefs. [7, 16]. The model will ultimately lead to the scenario of oscillating soliton stars hosting a network of domainwalls (or defects). This will be done by employing a Lagrangian which is made up of three scalar fields that is coupledamong themselves by a potential. In line with [7], we will consider the Lagrangian to be L m = 12 ∂ µ φ∂ µ φ + 12 ∂ µ χ∂ µ χ + 12 ∂ µ σ∂ µ σ − V ( φ, χ, σ ) . (18)This model is made up of three real scalar fields that is coupled among themselves by the potential V ( φ, χ, σ ). Puttingthis potential into the action given by equation (1) and varying it with respect to the scalar fields ( σ, φ, χ ) lead to the FIG. 2: The total mass M of the oscillating soliton star (in units of M Planck /m ) is plotted as a function of central density φ (0)for stars in their first excited state φ (0). The circles represent actual configurations resulting from solutions to the eigenvalueequations. following Klein-Gordon equations 1 √− g ∂ µ ( √− gg µν ∂ ν φ ) − ∂V ( φ, χ, σ ) ∂φ = 0 , (19)1 √− g ∂ µ ( √− gg µν ∂ ν χ ) − ∂V ( φ, χ, σ ) ∂χ = 0 , (20)1 √− g ∂ µ ( √− gg µν ∂ ν σ ) − ∂V ( φ, χ, σ ) ∂σ = 0 . (21) A. The model at Minkowski space
Generally we are interested in the model that should be able to describe a spherical soliton star via the scalar field σ (host field), by breaking its Z symmetry under the shift σ → σ − σ , to entrap the other two fields ( φ, χ ) with a Z symmetry on its surface [7]. We obtain this by considering the potential of the form V = 12 µ σ ( σ − σ ) + λ ( φ + χ ) − λ φ ( φ − χ ) + [ λµ ( σ − σ ) − λ ]( φ + χ ) , (22)where σ = 0 and σ = σ are the true and false vacua corresponding to the standard soliton star.We use the equations of motion to see that in the false and true vacua σ ≈ σ and σ = 0, respectively, the fields( φ, χ ) turn out to be zero. For scalar fields φ , χ = 0 the theory (22) allows the field σ to form a soliton solution. Wenote that at Minkowski space such a solution can be found by using the following first order differential equation dσdR = µσ ( σ − σ ) = W σ , (23)where W σ = dWdσ . The function W = µ ( σ / − σ σ /
2) is well-known as the ‘superpotential’ that defines the potential V ( σ, ,
0) = (1 / W σ [26]. Integrating equation (23) gives the solution σ = σ (cid:20) − tanh µσ ( R − R )2 (cid:21) . (24)This solution indicates that at the surface of the star ( R ≃ R ) the sigma field goes to (1 / σ . This representapproximately [27] a spherical wall (the soliton star surface) with surface tension [7] t h ≃ | W ( σ ) − W (0) | = 16 µσ . (25)When σ ≃ (1 / σ the other two scalar fields ( φ, χ ) develops vacua that respects the Z symmetry, and describethree-junctions of domain walls which allow the formation of a network on the surface [7, 16]. The solution alsoindicates that outside ( R > R ) and inside ( R < R ) the star the sigma field goes to < (1 / σ and > (1 / σ respectively. B. Einstein-Klein-Gordon equations for three scalar fields
In this section, we will calculate the EKG equations for three scalar fields using the potential defined by equation(22) in curved space. The soliton star solutions now can be found only numerically, as we shall see shortly.The Einstein’s equation for three scalar fields coupled with each other aided by a potential are obtained from thevariation of the action given with respect to the metric tensor g µν that leads to R µν − g µν R = κ T ( φ,χ,σ ) µν , (26)where T ( φ,χ,σ ) µν is the energy-momentum tensor for the scalar fields.To proceed we will consider the following conditions. The constants are assumed λ = µ = 1 on the GeV scale[3, 6] and the sigma field is considered to be periodic in time, whereas the other two fields ( φ and χ ) are static. Thepotential then becomes V = 12 σ ( σ − σ ) + ( φ + χ ) − φ ( φ − χ ) + (cid:20) ( σ − σ ) − (cid:21) ( φ + χ ) . (27)The general energy-momentum tensor associated with it is T ( φ,χ,σ ) µν = ∂ µ φ∂ ν φ + ∂ µ χ∂ ν χ + ∂ µ σ∂ ν σ − g µν [ ∂ α φ∂ α φ + ∂ α χ∂ α χ + ∂ α σ∂ α σ + V ( φ, χ, σ )] . (28)Using the spherical symmetric line element given in equation (5), we follow Refs. [14, 15, 19, 20] and write Einstein’sequations as: the t − t component ( g ) ′ = − g (cid:18) g − r (cid:19) + 4 πGg r (cid:20) g N ˙ σ + σ ′ + φ ′ + χ ′ (cid:21) + 4 πGg r (cid:20) g (cid:18) σ ( σ − σ ) + ( φ + χ ) − φ ( φ − χ ) + [( σ − / σ ) − / φ + χ ) (cid:19)(cid:21) , (29) the r − r component ( N ) ′ = N (cid:18) g − r (cid:19) + 4 πGr h g ˙ σ + N ( σ ′ + φ ′ + χ ′ ) i + 4 πGr (cid:20) N g (cid:18) σ ( σ − σ ) + ( φ + χ ) − φ ( φ − χ ) + [( σ − / σ ) − / φ + χ ) (cid:19)(cid:21) , (30) the t − r component ˙ g = 8 πGrg ˙ σσ ′ . (31)In order to find the Klein-Gordon equations we use Eqs. (19)-(21) with the potential defined in equation (27). Theycan be written as ¨ σ − ˙( N ) ˙ σ N = ( N ) ′ σ ′ g + N g " σ ′′ − ( g ) ′ g σ ′ − ˙( g ) N ˙ σ + 2 N σrg − N V σ , (32) φ ′′ = ( g ) ′ g φ ′ − ( N ) ′ N φ ′ − φ ′ r + g V φ , (33) χ ′′ = ( g ) ′ g χ ′ − ( N ) ′ N χ ′ − χ ′ r + g V χ , (34)where V σ = ∂V ( σ,φ,χ ) ∂σ , V φ = ∂V ( σ,φ,χ ) ∂φ , and V χ = ∂V ( σ,φ,χ ) ∂χ .As done in Sec. II, we consider change of variables ( A ( r, t ) = g , C ( r, t ) = [ g ( r, t ) /N ( r, t )] ) in other to deal with thenon-linearity present in the Einstein’s equation. It is also convenient to perform suitable re-scaling of the parametersleading to dimensionless quantities. For this reason, we use the following dimensionless variables 4 πG = 1, r → r/m σ , C → Cm σ /ω and t → ωt . Here m σ is the mass of the σ field computed at the true vacuum i.e., m σ = V σσ | σ =0 = σ .The coupled Einstein-Klein-Gordon equations now take the form A ′ = A (cid:18) − Ar (cid:19) + Ar h C ˙ σ + σ ′ + φ ′ + χ ′ i + A r (cid:20) σ ( σ − σ ) + ( φ + χ ) − φ ( φ − χ ) + [( σ − / σ ) − / φ + χ ) (cid:21) , (35) C ′ = 2 Cr − CAr + 2
CAr (cid:20) σ ( σ − σ ) + ( φ + χ ) − φ ( φ − χ ) + [( σ − / σ ) − / φ + χ ) (cid:21) , (36) C ¨ σ = −
12 ˙ C ˙ σ + σ ′′ + σ ′ (cid:18) r − C ′ C (cid:19) − AV σ , (37) Cφ ′′ = C ′ φ ′ − φ ′ Cr + CAV φ , (38) Cχ ′′ = C ′ χ ′ − χ ′ Cr + CAV χ , (39)˙ A = rA ˙ σσ ′ . (40)The simplest solutions to equations (35)-(39) are periodic expansions of the form A ( r, t ) = J max X j =0 A j ( r ) cos(2 jωt ) , (41) C ( r, t ) = J max X j =0 C j ( r ) cos(2 jωt ) , (42) σ ( r, t ) = J max X j =1 σ j − ( r ) cos[(2 j − ωt ] , (43)where ω is the fundamental frequency and again will be absorbed in the time variable. J max is the value of j at whichthe series is truncated for numerical computation. C. Numerical analysis
We want to study the effect of network of domain walls with an oscillating soliton star, with the help of a Lagrangiancontaining three scalar fields ( σ, φ, χ ), where the sigma field ( σ ) serves as shell of the star (or the host field) and theremaining two scalar fields ( φ and χ ) are responsible for the formation of the network. In other to accomplish thistask it is necessary to consider two cases where we use two types of perturbations.
1. Case 1. Perturbations of the potential
In this case we study the formation of network of domain walls with oscillating soliton stars, by changing σ to workaround the parameter σ . This leads to small perturbations of the potential, resulting in the possible disturbance ofthe network around the surface of the star. This is done by considering σ ≈ σ + η , where η ≪ σ ( η is a perturbationfield). Depending on how we choose the relation between σ and σ based on equation (24) will tell us whether thenetwork is entrapped inside the star, or the network is exactly on its surface, or outside the star. Thus leading tothese three possibilities:a) The network forming inside the star ( σ ≈ σ + η )b) The network forming on the surface of the star ( σ ≈ (1 / σ + η )c) The network forming outside the star ( σ ≈ (1 / σ + η ).By imposing these conditions (a, b, and c) on equations (35)-(39), we find sets of EKG equations for each possibilityunder this kind of perturbations. We solve the EKG equations by invoking the same boundary conditions used insection (II A) only that in this case we are working with three scalar fields. The series is truncated after j max = 1 andthe Fourier coefficient are set to zero. The Figs. 3 demonstrate typical numerical results for the metric g rr = A ( r, t ). (a)Network inside the oscillaton (b)Network on the surface the oscillaton(c)Network outside the oscillaton FIG. 3: A typical solution to the truncated eigenvalue equations ( j max = 1) of the expansion of the metric quantity A ( r, t ) forthe three possibilities under this perturbations. These simulations are computed for the central densities σ (0) ≈ . φ (0) ≈ χ (0) ≈
0. The solid line shows A and the dotted line shows A . The convergence of the series is significant, regardless of the non-linearity of the EKG equations.Let us study the dynamics of the mass profiles of the oscillaton stars under this perturbation. In Fig. 4(a) themass increases monotonically until it reaches a value M c ≈ . M P lanck /m σ (critical mass) after which it start todecrease. The critical mass in this sense is the combined maximum mass (mass of the star and the network) that the (a)Oscillaton with network inside (b)Oscillaton with network on the surface (c)Oscillaton with network outside FIG. 4: The total mass M of the oscillating soliton star (in units of M Planck /m σ ) is plotted as a function of central density σ (0), for all three possibilities. The central densities for the other two fields are fixed at φ (0) ≈ χ (0) ≈
0. The circlesrepresent actual configurations resulting from solutions to the eigenvalue equations. star attains before it collapses into a black hole. Similar to the case without network, the branch to the left of themaximum point is the stable branch, and to the right is the unstable branch. Looking at the critical point σ c (0) ≈ . M L ) M L ≈ . M P lanck /m σ ,after which it starts to increase again rapidly. This presents a different behavior from the mass profile well-knownin other theories. This exhibits a behavior which we call “the bouncing stability”. We can explain this behavior asthe shifting of the critical mass before collapse further. Meaning the star have the ability to accrue more mass beforedecaying into a black hole, and this leads to the increment of the lifespan of the star. Indeed, we can associate thisfeature to the network of domain walls present on the surface of the star.Looking at the behavior shown in Fig. 4(c), it physically makes sense because in this simulation, the network isoutside the star, which is the same as saying there is no network on the surface, or inside the star, hence resulting inthis behavior.
2. Case 2. Perturbations of the shell of the star
The focus of this case is to study small perturbations applied to the ‘surface term’ ( σ − σ ) in the potential andto investigate the effect that these perturbations will have on the mass profile of the oscillaton. We perturb the shellby the introduction of a perturbation field η into the surface term in the potential and fix the parameter σ ≈ V = 12 σ ( σ − + ( φ + χ ) − φ ( φ − χ ) + [( σ − η ) −
94 ]( φ + χ ) . (44)Carefully evolving the η field around σ , produces small perturbations of shell around the network, by slightly shiftingthe shell to entrap the network inside star, or leaving the network outside the star. Here when the shell coincideexactly with the surface of the network, the surface term in the potential vanishes i.e., ( σ − η ) ≈
0; based on theknowledge that on the surface of the star σ ≈ σ . Thus we have to send the surface term in equation (44) to zero.This generates three possibilities:a) The network forming inside the star (when we consider η ≈ σ − η ) ≈ η ≈ ).In order to find solutions to the EKG equations for all the three possibilities under this case, we invoke the argumentspresented above (a, b and c) on the potential. These conditions serve as the source of the perturbations leading tothe said possibilities. Following the usual numerical routines, we extract and discuss the resulting graphs.We can see from Fig. 5, a typical radial metric function A ( r, t ), for the case of central density of σ = 0 . (a)Network of domain walls inside the oscillaton (b)Network of domain walls on the surface the oscillaton(c)Network of domain walls outside the oscillaton FIG. 5: A typical solution to the truncated eigenvalue equations ( j max = 1) of the expansion of the metric A ( r, t ) for the threepossibilities under this perturbations. These simulations are computed for the central densities σ (0) ≈ . φ (0) ≈
0, and χ (0) ≈ (a)Oscillaton with network of domainwalls inside (b)Oscillaton with network of domainwalls on the surface (c)Oscillaton with network of domainwalls outside FIG. 6: The total mass M of the oscillating soliton star (in units of M Planck /m σ ) is plotted as a function of central density σ (0), for all three possibilities. The central densities for the other two fields are fixed at φ (0) ≈ χ (0) ≈
0. The circlesrepresent actual configurations resulting from solutions to the eigenvalue equations.
The mass profile from the numerical results for different oscilatons as function of σ (0) are shown in Fig. 6. InFig. 6(a) we have the mass profile for the simulation when the network is sitting inside the oscillaton. This casereveals a different behavior as compared to that of first case. Here we see that the mass profile exhibit again “thebouncing stability” with a local maximum M L ≈ . M P lanck /m σ and a critical central density of σ c ≈ . M L ≈ . M P lanck /m σ . It indicates that, again the star is prevented from decaying into a black hole due to thepresence of network on its surface. We find that in both perturbations it is convenient to have the network on thesurface of the star if we want to ensure the stability of the star.The Fig. 6(c), shows a general behavior similar to the mass profile in the first case with a maximum mass M c ≃ . M P lanck /m σ at σ c (0) ≃ .
0. By comparison, this case present an oscillaton with a delayed decay.
IV. DISCUSSIONS AND CONCLUSIONS
In this work we presented a model that admits formation of network of domain walls living on oscillatons. This wasdone by introducing a Lagrangian that contains three scalar fields that is coupled among themselves by a suitablepotential. The potential was chosen to provide the standard spherical oscillating soliton star with a network of domainwalls living on [16].We used two types of perturbations to simulate the three possibilities; having the network of domain walls inside thestar, or the network forming on the surface of the star, or the network sitting outside the star. These three scenarioscan be thought of as three possible different experiences that an oscillaton can go through in the cosmological evolution.Sec. III C 1 presented the first case where we considered small perturbations to the whole potential of the system. Thiswas made possible by working the host field σ around σ with the help of a perturbation field η . The next sectionpresented the second case where we perturbed the surface term in the potential, by carefully evolving the η fieldaround σ . These simulations are challenging because of the non-linearity of the Einstein-Klein-Gordon equationsand also due to dynamical characteristics of oscillating soliton stars, with no equilibrium configurations having staticmetric components. In this work we truncated the system of equations after a maximum of j = j max = 1. Althoughwe have not proved analytically that the series represents an exact solution to the Einstein-Klein-Gordon equationsfor the three scalar field system, we have given strong evidence that the series indeed converges rapidly, in both cases(Figs. 3 and 5).Since the main objectives of this work is to investigate the effect of network of domain walls on the stability of thestar, we particularly invested much effort in the mass profiling of all configurations in both perturbations. From thiswe have demonstrated that the network of domain walls indeed affects the stability of the star. Figs. 4(b), 6(a), and6(b) present a different behavior from the mass profiles well-known in other theories (for solitons and boson stars).We can say that the present object of study has two S-branches, the first S-branch is for the part of the star wherethere is no network and the second S-branch can be related to the network of domain walls on the star. From thesimulations, we can say that it is energetically favorable for the network to be formed on the surface of the star,meaning, the formation of network on the surface of the star can be a cosmological mechanism that can ensure thestability of an oscillating soliton star. Also, we saw interesting results, when we simulated the possibility of havingthe network sitting outside the star — Figs. 4(c) and 6(c). We find out that their mass profiles are similar to that inusual theories. This physically makes sense, since network outside the star can be interpreted as no network on thestar.To our knowledge this is the first time oscillating soliton stars with network of domain walls are studied. Thisinvestigation has led to the discovery of new type of self-gravitating objects and their existence could have significantcosmological and astrophysical implications. They can interplay the role with primordial black holes new cosmologicalscenarios via transition induced by perturbation of the network. As source of gravitational waves they can providenew signatures that may be detected in the form of gravitational and electromagnetic waves in the same event. Whensuch objects are sufficiently close from each other, tidal forces can deform their surface in relation to the networkdestroying their stability driving them to collapse to black holes.Finally, noticed that in this work all the analysis were done without the consideration of temperature, i.e., wedid not consider the effect of temperature on the network. The model presented gives room for the inclusion oftemperature in the form of thermal corrections that will modify the Lagrangian. For instance, one might want tocompute the critical temperature to form network due to QCD phase transition. This issue and many others will beaddressed in the near future.2 Acknowledgments
We would like to thank J.R.L. Santos and C.A.S. Silva for discussions and CNPq and CAPES for partial financialsupport. [1] T.D. Lee.
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