Boson stars in general scalar-tensor gravitation: Equilibrium configurations

Diego F. Torres*Departamento de Fı´sica, Universidad Nacional de La Plata, C.C. 67, C.P. 1900, La Plata, Buenos Aires, Argentina

~Received 3 April 1997!

We study equilibrium configurations of boson stars in the framework of general scalar-tensor theories ofgravitation. We analyze several possible couplings, with acceptable weak field limits and, when known,nucleosynthesis bounds, in order to work in the cosmologically more realistic cases of these kinds of theories.We find that for general scalar-tensor gravitation the range of masses boson stars might have is comparablewith the general relativistic case. We also analyze the possible formation of boson stars along different eras ofcosmic evolution, allowing for the effective gravitational constant far out from the star to deviate from itscurrent value. In these cases, we find that the boson star masses are sensitive to these kinds of variations, withina typical few percent. We also study cases in which the coupling is implicitly defined, through the dependenceon the radial coordinate, allowing it to have significant variation in the radius of the structure.@S0556-2821~97!00218-X#

PACS number~s!: 04.50.1h, 04.40.Dg, 95.35.1d

I. INTRODUCTION

Boson stars, stellar structures first proposed by Ruffiniand Bonazzola@1#, are gravitationally bound macroscopicquantum states made up of scalar bosons. They differ fromneutron stars, their fermionic counterparts, in that their pres-sure support is derived from the uncertainty relation ratherthan Pauli’s exclusion principle. Although the seminal workof Ref. @1# was published in 1969, it was followed up only inthe past decade. In recents years, cosmology has been refur-bished with the introduction of several ideas concerning thecritical role that a scalar field may have had in the evolutionof the universe. This revived the possibility of constructingstellar objects made up of these scalars instead of conven-tional fermions.

Considering bosons as described by a noninteracting,massive, complex scalar field, Ruffini and Bonazzola solvedthe equations of motion given by the Einstein field equationsand the Klein-Gordon equation. The general setting theyused is the same as the one explained below. They found thatthe masses of such boson stars were of the order ofM.MPl

2 /m, where MPl is the Planck mass andm is theboson mass. This model served to open the possibility thatboson stars might indeed exist in nature, although theirmasses were small enough to discard them as a viable solu-tion to the dark matter problem. The order of magnitude ofthese boson stars coincides with simple computations. For aquantum state confined into a region of radiusR, and withunits given byh5c51, the boson momemtum isp51/R. Ifthe star is moderately relativisticp.m, thenR.1/m. If weequateR with the Scharszchild radius 2MPl

22M ~recall thatG5MPl

22! we getM.MPl2 /m. The later work of Colpi, Sha-

piro, and Wasserman@2# introduced a self-interaction termfor the scalar field. With this addendum, stellar equilibriumconfigurations had masses of orderMPl

3 /m2, which are of thesame order of the Chandrasekhar mass~.1 solar masses!.

The stability of these objects have also been analyzed withsimilar results to those of the neutronic case@3#. Taking thework of Ruffini and Bonazzola and Colpiet al. as startingpoints, several extensions have been proposed. Jetzer andvan der Bij @4# considered the inclusion of a U~1! gaugecharge and Jetzer@5# studied its stability properties. Non-minimal couplings for the scalars have also been analyzed in@6#. These and other related models are reviewed in@7#. Inaddition, provided that many of these objects have a primor-dial origin, while being formed in a gas of boson and fermi-ons, it is expected that boson-fermion stars might exist. Thiswas studied by Henriqueset al. @8# without interaction and,recently, by de Sousaet al. with current-current–type inter-action @9#. Finally, the possible understanding of galactichalo properties by means of boson stars models has also beenproposed@10#.

Boson stars solutions have been, however, scarcely ana-lyzed in the framework of alternative theories of gravitation.We are particularly interested in scalar-tensor gravity, inwhich the effective gravitational constant is a field variable@11#. Historically, most interest has been given to Brans-Dicke ~BD! gravity, in which the coupling functionv~f!, afree function these theories have, is constant. To ensure thatthe weak field limit of this theory agrees with current obser-vations,v has to be big enough@12#. But in general, whenvvaries, we need thatv→` and v23dv/df→0 whent→`, so that the weak field limit of scalar-tensor gravitationagrees with general relativity~GR! tested predictions. After-wards, it was realized that these general scalar-tensor theo-ries would admit significant deviations from GR of the past@11# and that they could be a useful tool in the understandingof early universe models. The interest in them was recentlyrekindled by inflationary scenarios@13# and fundamentaltheories that seek to incorporate gravity with other forces ofnature@14#. In general, almost all studies made on scalar-tensor gravitation focus in the cosmological models theylead. This is in order to put several constraints upon thecoupling function. Observational bounds, mainly comingfrom weak field tests@12# and nucleosynthesis@15–18# aremore restrictive if exact analytical solutions are known for*Electronic address: dtorres@venus.fisica.unlp.edu.ar

PHYSICAL REVIEW D 15 SEPTEMBER 1997VOLUME 56, NUMBER 6

560556-2821/97/56~6!/3478~7!/$10.00 3478 © 1997 The American Physical Society

the cosmological equations. A few years ago, Barrow@19#,Barrow and Mimoso@20#, and Mimoso and Wands@21# de-rived algebraic numerical methods that allow Friedmann-Robertson-Walker~FRW! solutions to be found in modelswith matter content in the form of a barotropic fluid for anykind of couplingv~f!. Some of these methods were recentlyextended to incorporate nonminimally coupled theories, evenin the cases in which the functions involved in the Lagrang-ian do not posses analytical inverses@22,23#. These exten-sions showed the possibility to classify the cosmological be-havior of scalar-tensor theories in equivalence sets, wherethe field itself is a class variable.

In an astrophysical setting, if a scalar-tensor theory de-scribes gravitation, the value of the effective gravitationalconstant far out from the star must not necessarily be theNewton constant, but the value given by the evolution of acosmological model at the time of formation of the stellarobject. As we shall see below, this may change the boundaryconditions of the problem. Within this gravitational frame-work, boson stars were analyzed only in the simplest case.Gunderson and Jensen@24# addressed the possible existenceof such objects in Brans-Dicke gravity, with and withoutself-interaction. They adopted a fixed boundary condition forthe field equal to 1—the dimensionless Newton constant—and found that, in general, for almost allv, the Brans-Dickemodel of boson stars gives a maximum mass smaller than thegeneral relativistic model in a few percent@24#. A similarwork addressed the existence of boson stars in a gravitationaltheory with a dilaton@25# and its results coincides with theprevious case.

The aim of this work is to present a comprehensive studyon the possible existence of boson stars solutions ingeneralscalar-tensor theories. In the case of their existence, we wantto analyze the values they give for masses of typical objectsand other dynamical variables of interest, such as the typicalbehavior of the scalar. We also want to see if modificationsin the boundary condition for the Brans-Dicke scalar—due tocosmological evolution—produce any noticeable deviationin the masses of equilibrium configurations. We also implic-itly define the coupling in order to allow substantive varia-tions of it in the radius of the structure. We then study equi-librium configurations in these schemes.

The rest of the paper is organized as follows. In Sec. II weintroduce the formalism for the boson star construction to-gether with the numerical recipes used. In Sec. III we choosecoupling functions and in Sec. IV the results for them arepresented. The last section deals with our conclusions.

II. FORMALISM

A. Gravitational theory and boson system

We first derive the equations that corresponds to the gen-eral scalar-tensor theory. The action for this kind of general-ized BD theory is

S51

16p E A2gdx4FfR2v~f!

fgmnf ,mf ,n116pŁmG ,

~1!

whereg5Det gmn , R is the scalar curvature,v is the cou-pling function, andLm represents the matter content of the

system. We take thisLm to be the Lagrangian density of acomplex, massive, self-interacting scalar field. This Lagrang-ian reads

Lm521

2gmnc ,m* c ,n2

1

2m2ucu22

1

4lucu4. ~2!

Varying the action with respect to the dynamical variablesgmn andf we obtain the field equations

Rmn21

2gmnR5

8p

fTmn1

v~f!

f S f ,mf ,n21

2gmnf ,af ,aD

11

f~f ,m;n2gmnhf!, ~3!

hf51

2v13 F8pT2dv

dff ,af ,aG , ~4!

where we have introducedTmn as the energy-momentum ten-sor for matter fields andT as its trace. This energy-momentum tensor is given by

Tmn51

2~c ,m* c ,n1c ,mc ,n* !

21

2gmnS gabc ,a* c ,b1m2ucu21

1

2lucu4D . ~5!

The covariant derivative of this tensor is null. That may beproved either from the field equations, recalling the Bianchiidentities, or by intuitive arguments because of the minimalcoupling between the fieldf and the matter fields. This im-plies

c ,m,m2m2c2lucu2c* 50. ~6!

We now introduce the background metric. That is the cor-responding to a spherically symmetric system, because of thesymmetry we impose upon the star. Then

ds252B~r !dt21A~r !dr21r 2dV2. ~7!

We also demand a spherically symmetric form for the fieldwhich describes the boson; i.e., we adopt

c~r ,t !5x~r !exp@2 iÃt#. ~8!

Semiclassically, we are able to think aboutx expanded increation and annihiliation operators andTmn as an expecta-tion value in a given configuration with a large number ofbosons@1#. Using the metric~7! and the equation definingthe form of the boson field~8! together with the energy-momemtum tensor~5! in the field equations~3!,~4! we getthe equations for the structure of the star. Before we explic-itly write them we are going to introduce a rescaled radialcoordinate by

x5mr. ~9!

From now on, a prime will denote a differentiation with re-spect to the variablex. We also define dimesionless quanti-ties by

56 3479BOSON STARS IN GENERAL SCALAR-TENSOR . . .

V5Ã

m, F5

f

MPl2

, s5A4px~r !, and L5l

4p S MPl

m D 2

,

~10!

whereMPl is the Planck mass. In order to consider the totalamount of mass of the star within a radiusx we change thefunction A in the metric to its Schwarzschild form

A~x!5S 122M ~x!

x D 21

. ~11!

Then, the total mass will be given byM (`) and will corre-spond to

M star5M ~`!

mMPl

2 , ~12!

for a given value ofm. With all these definitions, the equa-tions of structure reduce to

s91s8S B8

2B2

A8

2A1

2

xD1AF S V2

B21Ds2Ls3G50,

~13!

F91F8S B8

2B2

A8

2A1

2

xD22A

2v13 F S V2

B22Ds22

s82

A

2Ls4G11

2v13

dv

dFF8250, ~14!

B8

xB2

A

x2 S 121

AD5A

F F S V2

B21Ds21

s82

A2

L

2s4G

1v

2 S F8

F D 2

2A

F

2

2v13 F S V2

B22Ds2

2s82

A2Ls4G1S F9

F2

1

2

F8

F

A8

A D1

1

2v13

dv

dF

F82

F~15!

2BM8

x25

B

F F S V2

B11Ds21

s82

A1

L

2s4G1

v

2

B

A S F8

F D 2

1B

F

2

2v13 F S V2

B22Ds22

s82

A2Ls4G

2B

A~2v13!

dv

dF

F82

F2

1

2

F8

F

B8

A. ~16!

It is important to note that these equations reduce to theknown BD ones of reference@24# whenv is taken as a con-stant and to those of GR of Ref.@2# when F→F0 is con-stant.

B. Numerical procedure and boundary conditions

We shall carry out a numerical integration from the centerof the star outwards towards radial infinity. The boundarycondition for the system are the following. Concernings, we

require a finite mass, which impliess(`)50 and nonsingu-larity at the origin, i.e.,s(0) a finite constant ands8(0)50.We shall look for zero node solutions because, as remarkedin @2#, it is reasonable to regard them as the lowest energybound states. We shall demand asymptotic flatness, whichmeansB(`)51 andA(`)51; this last condition is ensuredby the equations themselves. Nonsingularity at the originalso requiresM (0)50. Finally,F(`) must take the value ofF in an appropiate cosmological model at the time of stellarformation. If not otherwise specified, it will be considered as1. Note that this boundary condition differs from the othersin being less restrictive, in fact, preliminarily, it impose onlyanacceptable boundary value prescription. We shall managethis boundary condition by stopping the integration at theasymptotic region, whereF5F(`), (61026), and the de-rivative F8(`) tends to zero. All these boundary conditionsgenerate an eigenvalue problem forV. In order to abtainaccurate results, this eigenvalue has to be specified with atleast seven significant figures in a typical case; this is withinthe capability of a double precision numerical method andwas also the case in the general relativistic cases@8#. Notethat due to the form of the equations, which are linear inB,we can integrate the system without impose the boundarycondition onB from the beginning. Instead, we ultimatelyrescaleB andV in order to satisfy flatness requirements.

The numerical method we shall use is a fourth-orderRunge-Kutta and is based in the recipes of@26#. Some of thesubroutines were modified in order to test the possibility ofsatisfying the boundary conditions at each step of the inte-gration ~see @1# for details on this! and some others werebuilt in order to search for the eigenvalue and satisfy thespecific boundary conditions. The program was tested in thelimiting cases of equations~13!–~16!, i.e., GR and BD, andagreement was found with reported results.

III. THE COUPLING FUNCTION

As there is noa priori prescription about the form or thevalue ofv, we are interested in ascertaining the general be-havior displayed by a wide range of scalar-tensor stars. Thefirst group of couplings we are going to analyze have theproperty of tending to infinity asf→f0 , wheref0 may betaken as the present valuef(t) or, equivalently, as the in-verse of the Newton constant. We shall take the forms thatBarrow and Parsons recently analyzed in a cosmological set-ting @27#. They are the following.

Theory 1. 2v1352B1u12f/f0u2a, with a.0 andB1.0 constants.

Theory 2. 2v1352B2 lnuf/f0u22d, with d.0 andB2.0 constants.

Theory 3. 2v1352B3u12(f/f0)bu21, with b.0 andB3.0 constants.

The behavior of these theories in a FRW metric was ana-lytically studied in@27# and weak field limit constraints uponthe parameters were provided there also. Note that, whileusing these couplings in our equations, the functions depen-dences shift from cosmic time to radial coordinate. Cosmo-logical solutions for this group allowsf approach tof0 frombelow, i.e.,fP(0,̀ ) or from above,fP(`,0). This impliesthat the boundary condition inF may be equal to, less

3480 56DIEGO F. TORRES

than, or bigger than 1. Theories 1–3 approach BD whenF→0 and to next theory~theory 4! whenF→`. Note alsothat the weak field constraints are, in fact, independent of theform of the cosmological solutions providedF→F0 , whent is big enough. It is this latter requirement which introducefurther restrictions upon the the parameter space, especiallyin the exponents, which vary as a function of the cosmic era@27#.

The second group of coupling functions will be repre-sented by the following.

Theory 4. 2v135v0fn, with n.0 andv0 constants. Italso has an analytical solution@20# and even we know nu-cleosynthesis bounds for it@18#. This group differs from thefirst in that, although growing with time, they only reach GRwhenf→`, that is, whent→`, f does not tend tof0 . Tonormalize we may setf(t5today)51.

Finally, the third group we shall analyze consists oflocalimplicitly defined functionsof the following form:

Theory 5.v5v(x)5v@x(f)#. The aim in doing this isto explicitly see how one can manage the behavior ofvwithin the radius of the star. Note that these are implicitdefinitions, being ultimately necessary to invertf(x) to getthe correct dependence of the coupling function. Iff(x) is amonotonous function, then, the existence of this inverse isanalytically ensured. It is worth recalling that the limitx→`is of crucial importance. Far out from the star we would wantto recover a scalar-tensor theory with a cosmo-

logical well-behaved evolution.

IV. RESULTS

A. Group I and II couplings

Although formally and conceptually different when con-sidering a cosmological setting, the theories described asgroup I and group II resulted in being similar concerningboson stars solutions. In addition, most of the similaritiesarise inside group I couplings, where simulations based onthese theories suggest that anyone can be mapped into theother for convenient choices of each particular set of theirfree parameters. Concerning theory 4, some differences haveto be remarked on, and so we do below, but its general be-havior does not differ very much from group I couplings.Taking this into account, we shall present in deep detail onlythe case of theory 1 and we shall make some comments onother special situations.

We shall first consider boson stars based on theory 1gravitation. We takeB155 anda52 and look for models inequilibrium for different values of the central density andstrengh of the self-interaction. What we found is sketched inFig. 1. Recall that the valuea52 is one extremum of theinterval which admits convergency to GR in a cosmologicalevolution with a perfect fluid model as the matter source@gP(0,4/3);p5(g21)r# @27#. We can note from there thatthe general form of the graph is preserved when comparedwith both the GR and BD cases. The boson stars massesincrease from the BD case withv56 presented by Gunder-son and Jensen. In fact, results for values ofa greater than 1are extremely similar to those of general relativity and theBD scalar is almost numerically constant along the bosonstar structure. This is something that is to be expected in thecase where solutions may exist, because of the rapidly ap-proaching scheme to GR that theory 1 develops whena isbig enough. Things change when considering values ofa

FIG. 1. Boson stars masses of theory 1 forB155 anda52 anddifferent values ofL ands(0). There are 34 models for each valueof L. Numerical values in this graph are very similar to the onesderived for general relativity boson stars masses. FIG. 2. Behavior ofs as a function of the radial coordinate for

two typical models of scalar-tensor boson stars; theory 1 withB155 anda50.5.

TABLE I. Boson stars masses for theory 1 withB155 anda50.5.

L s(0) B(0) F~0! M (`)

0 0.325 0.4231 1.0007 0.62710 0.225 0.4163 1.0010 0.919

100 0.100 0.3853 1.0011 2.248200 0.070 0.4256 1.0009 3.128

56 3481BOSON STARS IN GENERAL SCALAR-TENSOR . . .

smaller than 1. Table I presents computations for modelswith B155 anda50.5. Recall that this value ofa is thesmaller value that preserves the weak field limit in a cosmo-logical evolution and one of the extremum which guaranteesconvergency to GR in the case of the radiation era@27#. Notethat the equilibrium configuration in each case alwayschooses a bigger value ofF at the center of the star, whichimplies less gravitationally bounded objects than those ofGR. This was also the case of the BD models. The values ofthe masses are smaller than the GR ones but still greater thanthe ones which do not behave as required for a cosmologicalsetting, as for instance the BD case withv56. So, theory 1has viable solutions for boson star structures where massesare compatible with simplest cases. Concerning the behaviorof s as a function ofx, it has the same convexity propertiescommented on for the GR and BD models. Figure 2 showsits behavior for typical values of the parameter space, as doesFig. 3. for the behavior ofF. The dependences of the massesof the equilibrium structures upon the parameters of thegravitational theory was tested in further detail. It was foundthat for values ofa greater than 1, changes inB1 do notproduce noticeable changes in the mass. The opposite hap-pens for smaller values ofa. Table II represents these trends

in a more quantitative form forL5100 ands(0)50.100.Finally, we address the possible variation ofM (`) with adeviation in the boundary condition forF, the effectivegravitational constant far out from the star. This is aimed atgetting a first insight of possible boson stars formation alongdifferents eras of cosmic evolution. As theory 1 admits cos-mological solutions with values ofF greater or smaller than1 we consider both cases as possible boundary conditions.Table III showsM (`) and the correspondingF~0! value foreach choice of the boundary condition in three particularmodels. It is interesting to note that, in the first place,massesare sensitive to variations in the boundary condition of thescalar within a few percent as a typical order of magnitude,and second,the behavior of the models varies withL. If L isbig enough~greater than 10! a growing mass appears with agrowing boundary condition. Otherwise, the models show apeak in the masses within the range explored forF(`).

Concerning theory 4 it has to be noted that the parameterspace is not mainly constrained by weak field test@20#,which does not limit the values ofn—providedn.0—but itis by nucleosynthesis processes@18#. These bounds, whichresult in lower limits forn, are provided once the cosmologi-cal parametersV0 and the Hubble constantH0 are given. Acommon characteristic of this group is that masses of equi-librium configurations are smaller than the cases previouslystudied and much more smaller when compared with GR. Atypical example is theL5100 ands(0)50.100 model. Forv052 and n53, M (`)51.870. Note, however, that thisvalue ofn produce acceptable nucleosynthesis consequencesonly in a range ofV0h2,0.25. Another thing to note is thattheory 4 resulted in the ones which are more dependent onthe parameters shown for small values ofn andv0 . There,variations may reach a typical 10% in mass.

B. Group III couplings

The third and last group of couplings we shall analyzeconsists of implicitly defined functions of the form of theory

FIG. 3. Behavior ofF as a function of the radial coordinate fortwo typical models of scalar-tensor boson stars; theory 1 withB155 anda50.5.

TABLE II. Dependences of boson stars masses upon the param-eter space for theory 1 withL5100, s(0)50.100.

B1 a F(0) M (`)

2 0.5 1.0053 2.2455 0.5 1.0010 2.2488 0.5 1.0004 2.2492 1.0 1.0000 2.2508 1.5 1.0000 2.2508 2.0 1.0000 2.250

TABLE III. Boson stars masses as a function of the boundary condition forF. The first set shows themodel withL5100 ands(0)50.100, the second set showsL510 ands(0)50.225, while the third showsL50 ands(0)50.325. These models are for theory 1 withB155 anda50.5.

F~`! F~0! M (`) F~0! M (`) F~0! M (`)

0.90 0.9127 2.096 0.9128 0.875 0.9111 0.6100.95 0.9593 2.164 0.9593 0.893 0.9581 0.6161.00 1.0009 2.253 1.0010 0.920 1.0007 0.6271.05 1.0615 2.263 1.0617 0.916 1.0600 0.6181.10 1.1167 2.295 1.1170 0.921 1.1145 0.614

3482 56DIEGO F. TORRES

5. As an example we choose several forms of the couplings,results for which can be seen in Table IV, for the modelgiven by L50 and s(0)50.325. These functions areenough to get a feeling of at the idea is. Locally there is noprescription uponv, while far out from the star we wouldwant to recover a scalar-tensor theory cosmologically wellbehaved. Scalar-tensor theories which deviates more fromGR are those which can be compared with BD theories ofsmall v. The choice of the different functions is focused toencompass GR at the asymptotic region while admitting se-vere deviation inside the structure. For all cases, whenx→`, v→`, making these theories cosmologically accept-able. If not otherwise specified, all cases present monotonousF functions. The correct dependence of the coupling,v~F!may now be obtained from the inverse of the functionx(F).It is worth recalling that the dependence ofv with F changeswhenever the model changes. For instance, in passingthrough different values ofL, the functional form ofF(x)changes, as does its inverse. This implies that even withoutchanging v5v(x) we are changingv5v(F). For thecases studied in Table IV we found that the order of magni-tude of boson stars masses remains the same, although somecases with very small masses may arise—also in the cases ofLÞ0. These, in general, mild variations in the boson starproperties must be explained in terms of the complex struc-ture of the differential system. The terms proportional to thederivatives of the couplings are also proportional to the de-rivative of F, which in turn must be obtained from the solu-tion of the system.

V. CONCLUSIONS

In the last few years, the possibility of constructing com-plete cosmologies by encompassing exact analytical solu-

tions of general scalar-tensor gravitation, has raised an enor-mous interest in these kind of theories, which has to beadded to those developed by the applicability of them toinflationary scenarios. Once the cosmological setting is fixed,we have to analyze possible astrophysical consequences ofhaving, for instance, a different value for the gravitationalconstant or a different rate of expansion. As an example, weshould mention a recent work on the primordial formationand evaporation of black holes@28#.

In this work, we analyzed the possible existence of bosonstars solutions within the framework of general scalar-tensortheories. It then extends a previous paper by Gunderson andJensen@24# where solutions to the Brans-Dicke equationswere considered. We have shown the theoretical constructionof such systems in general cases of alternative gravity, allcontrasting with the gravity tests known up to date. Differentkinds of couplings with exact cosmological solutions andothers that allow a significant variation within the radius ofthe star were considered. We found that the order of magni-tude of the general relativistic boson star masses do not varywhen these more realistic cases of scalar-tensor gravity arethe basis of the gravitational theory. In general, and becausegeneral forms of couplings can be expanded in the form of aseries of group I and/or group II couplings, we may state thatboson stars might exist for any of these gravitational settings.We also found an interesting situation concerning the evolu-tion of boson stars masses as a function of the time of for-mation of the stellar object. It varies appreciably, typicallywithin a few percent, when cosmological time scales are con-sidered. Whether this fact may provide a useful basis forsearching for new observational consequences and/or boundsupon the coupling function is currently under study. Finallythe question of stability remains to be considered, for whichwe have no results, even in the Brans-Dicke case. We hopeto report on it in a forecoming work.

ACKNOWLEDGMENTS

It is the author’s pleasure to acknowledge I. Andruchow,S. Grigera, T. Grigera, and, especially, O. Benvenuto fortheir help with the numerical procedure and H. Vucetich foruseful comments and a critical reading of the manuscript.The author also acknowledges D. Krmpotik and R. Borzi forthe use of the computing facilities of Office 23 at UNLP andpartial support by CONICET.

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TABLE IV. Masses of boson stars for implicitly defined scalar-tensor theories. Results for the modelL50 ands(0)50.325 areshown. A small star point that theF function is not monotonous,typically in the innermost region. Boundary condition on theBDscalar was set equal to 1 although deviations provided by anasymptotic derivative ofF of order 1024 were accepted.

Theory 5:v5v(x) M (`) F~0!

0.1x 0.538 1.101110x 0.624 1.0075ln(x)* 0.577 1.0754exp(0.01x) 0.539 1.0760

56 3483BOSON STARS IN GENERAL SCALAR-TENSOR . . .

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