VOLUME 57, NUamER 20 PHYSICAL REVIEW LETTERS 17 NOvEMmR 1986

Boson Stars: Gravitational Equilibria of Self-Interacting Scalar Fields

Monica Colpi, t'l Stuart L. Shapiro, and Ira WassermanCenter for Radiophysics and Space Research, Cornell Uni Uersity, Ithaca, wYork I4853

(Received 13 August 1986)

Spherically symmetric gravitational equilibria of self-interacting scalar fields p with interaction poten-tial V(p) ,k

~

p)" are determined. Surprisingly, the resulting configuration may differ markedly fromthe noninteracting case even when A, &&1. Contrary to generally accepted astrophysical folklore, it isfound that the maximum masses of such boson stars may be comparable to the Chandrasekhar mass forfermions of mass mf . -X, '~4mb

PACS numbers: 04.20Jb, 11.10.-z, 95.30.Sf

Recent speculations in particle physics and cosmologyhave emphasized the potentially critical role played byevolving scalar fields in the development of the earlyUniverse, ' and raised the possibility that various exoticbosons comprise the cosmological missing mass. z3 Inview of this development, a detailed examination of thegravitational equilibria of massive fields in asymptotical-ly flat space-times "boson stars" seems appropriate.

Earlier work has considered equilibrium configura-tions of noninteracting boson fields. 4 Such objects aremacroscopic quantum states, that are only preventedfrom collapsing gravitationally by the Heisenberg uncer-tainty principle. This requiress that a state of charac-teristic size R has a typical boson momentum p 1/R(here and throughout we set h = 1 =c ). In a moderatelyrelativistic boson star one has p -rn, where m is the bo-son mass, so that R I/m. Since hydrostatic equilibri-um requires that the total mass M-R/G for relativisticbound states, we find M -(Gm) ' M)~k/m for mar-ginally relativistic boson stars. This mass is generallymuch smaller than the Chandrasekhar mass,-M)l, k/m, characteristic of marginally relativistic

fermion st rass.In this paper we consider arbitrarily relativistic equih-

bria of self interacting c-omplex scalar fields. We as-sume an interaction potential V(p) ,' A, ( p ( where A, isa dimensionless coupling constant. We demonstrate ex-plicitly that euen if A, ((1 the structure of the resultingboson stars may differ radically from the A, 0 limitTo see why, first notice that the characteristic energydensity inside a A, 0 configuration is p-M)hkm inthe relativistic case. Since the energy density of nonin-teracting bosons is p m ( p ~, we find that

( p ~ -Mp~, k inside relativistic X 0 boson stars. Next,consider the effect of increasing X, from zero. The im-portance of the interaction potential is measured by theratio V(p)/m ( p ~ 2 of interaction energy to kinetic ener-gy. At sufficiently small X,, this ratio is justA,M)t, k/m . Thus, self-interactions may only be ig-nored if

((m /M)tgngk (6 7 x 10 GeV )m

Surprisingly, for A))1 we find that M-A'I2M(~, ,k/m-X' Mch, so that the masses of relatiuistic bosonstars may be comparable to those of their ferrnion counterparts ifk- l.

The starting point for our calculation is the scalar-field Lagrangean

J'- - lg""e,'-O,. &m'I OI' l& I O I-',

which implies an energy-momentum tensor

T," --'g" (~~,,+~,~,'.)

(3)

'&."&g"-~~-e;.+m'I e I'+ '& I ~ I'j -(4)We explicitly ignore interactions of p with any otherfields. In particular, Eq. (3) assumes negligible couplingto gauge fields, which is a good approximation for gaugecoupling constant e ((m/Mp~, k. Calculations for largervalues of e are currently under way.

We consider spherically symmetric, time-independentsolutions of Einstein's field equations

6" 8trGT"

in Schwarzschild coordinates

ds2 B(r)dtz+A(r)dr2+r2d Q, .For such solutions to exist we require that

y(r, t) @(r)e

(5)

(6)

where @(r) is a real function. [We could equally wellconsider "antiboson stars" with p(r, t) @(r)e' '. ] Forconvenience we actually solve the (;) and (,') com-ponents of Eq. (5) coupled with the scalar wave equation

~-m'~-&I e I 'e-0, (8)

which may be derived from Eq. (5) using Eq. (4) and

Moreover, families of gravitational equilibria may beparametrized by the single dimensionless quantity

A XM)(ak /4%m

1986 The American Physical Society 2485

pHYSI &L REVIEW LETTERS yEMBER 1988657 NUMBER 20VQLUME

uations aret e i ties The resultmg equthe Bianchi identities. e u

A 4+ a')z+ +I cr+ a+A'+1A2x x2

(9A 4 (a')'I 08' 1 4

ABx x2 A(9b)

2+ 8'g + + 1 a Acr 0,(9c)

d/dx, a (4rrG ) 1/2Crimes denote x,/, dAi gi by q.(4 )' @/Mpi,g, Q co rri anIf we ~rite

(10)A (x) - [1 2AI(x)/x]te E . (9a) aswe may rewrite q.

2 (9a')+I a ~cr+At'(x) -x' +

) w ' ted for Eq. (9a) in our in-', was substitute . in our inE uation 9a we ', hE. (10 use

8o o1 ~ h dFollowing Ruffini an ona

h h). It seems reasonab e o')'/2~ T..s of the cr

conditionsth contributions

subject to the boundary(9) are solved s j

d ~( ) 1 (i.e., reg"0) -0, an it)() (0) a a ' fit ess atproblem for & a"

~,(r) have no no e'

Our renumerical r

diffent valu~ of

~ .

1 for severawith those o

shown ing..

ellent agreementAosuits for e en'c boson-i, i that characteristic

l' b

Ftrend momasses aas a function o i2 is the relation

m=0 22A' '~Rane) /rii.~max =

11 for A)&1 and is dederivedh lds asymptotica ywhich o s~ 3be seen from Fig.

below.erivation may e

lues of A:01 b tdiff t Iates the differentTh' fl ill

11 and large values os cr(x) drops smoo

h ra't""t" l'ngt .

''a

chara t rz d b ltels are cnentia ecout to radii r-A

mass in the large-g

A increases.g"

dimensionaltzationlternative nonFigure 3 suggests an a

I II

I I II II

I I I

00

II

.2 4II0 I I I I I I I I I

300200

0~r a. -(4m) e.&/2M as a function o o,. 1. Boson-star mass aFIG. 1. a

for A 0,1, 10, 30, 100,function of A.asses as a unboson-star mas nn

The so I. c1'd curve is the asymp

VoLUME 57, NUMBER 20 PHYSICAL REVIEW LETTERS 17 NovEnmER 1986

I I If

I I I I)

I I I I

00 10 20

00

I I I I I I I I I I I I I I

1 2

io (n'/B(o))FIG. 3. Scalar field o as a function of dimensionless radius

x for cz, 0.1 but A 0 and A 300. The dots are the solutionto Eqs. (12)-(15) with 02/8(0) from the exact cr, O. l,A 300 model, scaled to A 300.

of Eqs. (9a'), (9b), and (9c) accurate at large A:o, oA'/, x, xA ', and At, At/A'/ Ignor. ingterms O(A '), the scalar wave equation may be solvedalgebraically to yield

cr (0 /8 1)'

FIG. 4. Boson-star mass vs 02/8(Q) for the limiting equili-bria computed from Eqs. (12)-(l5).

Because of self-gravity, the ground state of the bosonfield is not a zero-energy state. Moreover, the scalarfield, at large A, only varies on a relatively large lengthscale A'/ m '&)m ', so that we can solve the scalarwave equation locally, ignoring derivatives, to get Eq.(12). This results in an effective equation of state, Eqs.(15a) and 15(b), for the boson star, or on elimination offI2/8 and restoration of dimensional quantities

which may be substituted into the field equations to give,to the same accuracy, p -poF(p/po). (16)

and

JKy 4' y Py (13) where pn m4/4X and+(pjpn) - 9 ((I+,' pipn) '~' I 1'.

1 11 8'~,x2

where prime denotes d/dx, and'/+)( '/

p -'~-&(~2/g 1)2

(14)

(15a)(15b)

Equations (12)-(15) become essentially exact forA . At large A these equations may be used to gen-erate approximate solutions. Figure 3 compares n(x)based on Eqs. (12)-(15) with the exact o (x) forA 300. As expected, the two calculations agree wellexcept at very large radii. Because A does not appearexplicitly in Eqs. (13)-(15), we can use these limitingequations to determine the rescaled mass At~ M/(A'~ M(~,qjrn) as a function of the single free parame-ter 0 /8(0). The results, shown in Fig. 4, imply thepeak value At *=0.22 used in Eq. (11).

Equation (11) is then equivalent to the statement thatM~,-M$~,,q/p) for a fluid star with an equation ofstate of the form of Eq. (16). It is straightforward toshow that the well-known theorems on stability of fluidstars may be applied to the A oo limiting scalar-fieldequilibria satisfying Eqs. (12)-(15). In particular, M denotes the boundary along the sequence between stableand unstable equilibria as A . It seems reasonableto suppose that the same stability criteria may be appliedfor all (finite) A, although we have not proven this state-ment in general.

Generally accepted astrophysical folklore maintainsthat boson stars, should they exist, must have negligiblysmall masses. %'bile detailed studies have corroboratedthis prejudice for bound states of noninteracting bosons,the situation for self-interacting scalar fields may be verydifferent. In this paper we have shown that for an in-teraction potential V(p) -,'k

~ p( much larger masseswill result provided that A=A/4x(Jm ))1, an inequality

VOLUME 57, NUMBER 20 PHYSICAL REVIEW LETTERS 17 NOVEMBER 1986

that may be satisfied even at A, &&1 for reasonablescalar-boson masses. When A)&1 we have found thatthe maximum boson-star mass is

M~gg 0.22A M)[spy /m-(0.10 Gev')~. ~'"/m',

which is comparable to the Chandrasekhar mass for fer-mions of mass m/1, 'l . It is conceivable that boson starswith masses approaching Eq. (18) could arise in thecourse of gravitational condensation of bosonic darkmatter in the early Universe. Equation (18) would im-ply masses possibly near, but below, the stellar massrange for scalar neutrinos with m 1 GeV, but wouldappear to require ridiculously large masses,M~,-10 X, ' Mo, for cosmologically relevant axions3(with m 10 eV) except for exceedingly tiny )I, .

It is a pleasure to acknowledge useful conversationswith J. Hartle, R. Ruffini, and M. Sasaki. This researchwas supported in part by the National Science Founda-tion under Grant No. AST 84-15162 to Cornell Univer-sity. One of us (I.W. ) acknowledges the hospitality ofthe Harvard-Smithsonian Center for Astrophysics,where part of this work was carried out; I. %. is also anAlfred P. Sloan Fellow. M. C. thanks the AstronomyDepartment at Cornell University for kind hospitalityduring the course of this research, and acknowledgesfinancial support from the Ministero della Pubblica Is-truzione, Italy.

' On leave from Dipartimento di Fisica, Universita degli

Studi di Milano, 20133 Milano, Italy.tA. Linde, Phys. Lett. 1088, 389 (1982); A. Albrecht and

P. J. Steinhardt, Phys. Rev. Lett. 4$, 1220 (1982); A. Al-brecht, P. J. Steinhardt, M. S. Turner, and F. Wilczek, Phys.Rev. Lett. 4$, 1437 (1982). See also M. S. Turner, inProceedings of the Cargese School on Fundamental Physicsand Cosmology, edited by J. Adouze and J. Tran Thanh Van(Editions Frontieres, Gif-Sur-Yvette, France, 1985), and refer-ences therein.

2J. S. Hagelin, G. L. Kane, and S. Raby, Nucl. Phys. 8241,638 (1984).

3J. Preskill, M. Wise, and F. Wilczek, Phys. Lett. 120$, 127(1983); L. Abbott and P. Sikivie, Phys. Lett. 1208, 133(1983); M. Dine and W. Fischler, Phys. Lett. 1208, 137(1983).

4R. Ruffini and S. Bonazzola, Phys. Rev. 1$7, 1767 (1969).The calculations performed in this reference pertain to com-plex scalar fields, contrary to explicit statements made by theauthors.

5See, e.g., E. Takasugi and M. Yoshimura, Z. Phys. C 26,241 (1984).

sEquation (8) and its complex conjugate may also be derivedby considering variations of S~ Jd'xc with respect to p andp, respectively.

7See B. K. Harrison, K. S. Thorne, M. Wakano, and J. A.Wheeler, Gravitation Theory and Gravitational Collapse(Univ. of Chicago Press, Chicago, 1965), Chaps. 3, 5, and 7,and Appendix B; S. Chandrasekhar, Phys. Rev. Lett. 12, 114,437 (1964). Just as for fluid stars, Eq. (14) may be derived byconsideration of first-order variations in the total mass, definedin Eq. (13), at fixed boson number, with use of Eqs. (12) and(15). Second-order variations yield the usual stability theoryfor the analogous fluid stars. Extension of these results to gen-eral A, for which Eqs. (12) and (15) do not hold, does not ap-pear to be straightforward.