Table Of ContentGravitational memory of boson stars
Diego F. Torres♣
Departamento de F´ısica, Universidad Nacional de La Plata, C.C. 67, 1900 La Plata, Argentina
Andrew R. Liddle♦ and Franz E. Schunck♠
Astronomy Centre, University of Sussex, Falmer, Brighton BN1 9QJ, United Kingdom
(February 7, 2008)
‘Gravitational memory’ refers to the possibility that, in cosmologies with a time-varying gravi-
tational ‘constant’, objects such as black holes may retain a memory of conditions at the time of
their birth. We consider this phenomenon in a different physical scenario, where the objects under
consideration are boson stars. We construct boson star solutions in scalar–tensor gravity theories,
and consider their dependence on the asymptotic value of the gravitational strength. We then
discuss several possible physical interpretations, including the concept of pure gravitational stellar
8 evolution.
9
9 PACS numbers: 04.40.Dg, 04.50.+h Sussex preprint SUSSEX-AST97/10-2, gr-qc/9710048
1
n
a
J I. INTRODUCTION is scalar–tensor theories [3], which have an action
5
√ g ω(φ)
1 A few years ago, Barrow introduced the concept of S = − dx4 φR ∂ φ∂µφ+16π . (1)
µ m
gravitational memory [1] by posing the following prob- Z 16π (cid:20) − φ L (cid:21)
2
lem: what happens to black holes during the subsequent Here g is the metric, R is the scalar curvature, φ is
v µν
evolution of the universe if the gravitational coupling G the Brans–Dicke field, is the Lagrangianof the mat-
8 m
4 evolves with time? He envisaged two possible scenarios. ter content of the systLem, and ω(φ) gives the strength
0 In the first, dubbed scenario A, the black hole evolves of the coupling between the scalar φ and the metric.
0 quasi-statically,inordertoadjustitssizewiththechang- The special case of a constant ω is the Jordan–Brans–
1 ing G. If true, this means that there are no static black Dicke (JBD) theory, and general relativity is regained
7 holes, even classically, during any period in which G in the limit of large ω. The gravitational constant G
9
changes. In the alternative possibility, scenario B, the of the Einstein–Hilbert action is replaced by a dynam-
/
c local value of G within the black hole is preserved,while ical field, φ−1. In the usual applications, it is either
q the asymptotic value evolves with a cosmological rate. spatially-constant but time-varying (cosmological solu-
-
r This would mean that the black hole keeps a memory of tions) or spatially-varying but time-independent (astro-
g the strength of gravity at the moment of its formation. physical solutions). Our situation will be unusual in in-
:
v Further analysis of the striking phenomena which arise cluding both types of variation.
i in both of these scenarios was made in Ref. [2]. Black Our choice of stellar object is motivated by two con-
X
holes are of particular interest in this context, because cerns. Firstly, it should be simple enough as to allow us
ar primordialblackholesmayhaveformedintheveryearly to isolate the effects causedby a varying G. Secondly, in
stagesoftheUniverse. Atthosetimes nodirectevidence order to show the changes due to the particular gravita-
of the strength of gravity is available; nucleosynthesis is tional theory, it should be fully relativistic, in the sense
the earliest epoch at which significant constraints apply. thatitsequilibriumconfigurationmustbeobtainedfrom
Gravitational memory may provide a unique probe of Einstein-like field equations. There is one type of stel-
these early epochs. However, in this paper our aim is lar object which meets these criteria, though so far it is
to consider the gravitational memory phenomenon in a known to exist only as a theoretical construct. It is the
completely stellar arena. The motivation is twofold. On boson star, the analogue of a neutron star formed when
oneside,sincetherearenosingularitiesoreventhorizons a large collection of bosonic particles (normally taken to
asintheblackholecase,therearemanycalculationalad- be scalars) becomes gravitationally bound. Such con-
vantageswhichmayprovidea more directroutetowards figurations were introduced by Kaup [4] and by Ruffini
sheddinglightonwhichofthescenariosstatedabovewill and Bonazzola[5] in the nineteen sixties, but currentin-
happen in practice. On the other, a comprehensive un- terest was sparked by Colpi et al. [6] who showed that,
derstandingoftheeffectsofavarying-Gcosmologyupon provided the scalar field has a self-interaction, the bo-
astrophysicalobjects is far from complete. sonstarmasses couldbe of the same orderofmagnitude
To make a proper study of a varying gravitational as, or even much greaterthan, the Chandrasekharmass.
strength, it is imperative to operate within a self- Thisledtothestudyofmanyproperties,whicharesum-
consistent framework rather than simply writing in a marizedin two reviews[7]. It seems possible for them to
variation ‘by hand’. The most useful such framework form through gravitational collapse of a scalar field [8],
1
thoughlittleisknownasyet. Theobservationalstatusof ψ(r,t)=χ(r)exp[ i̟t]. (6)
−
bosonstars wasanalyzedvery recently in Ref. [9], where
it was asked whether radiating baryonic matter moving To write the equations of structure of the star, we use a
around a boson star could be converted into an observa- rescaled radial coordinate, given by
tionalsignal. Unfortunately,anydirectdetectionlooksa
x=mr. (7)
long way off.
As far as we are aware, only three papers have stud-
From now on, a prime will denote a derivative with re-
ied boson stars in scalar–tensor theories. Gunderson
spect to the variable x. We also define dimensionless
and Jensen [10] studied the JBD scenario, concluding
quantities by
that typically the mass of equilibrium solutions would
be reduced by a few percent. This work was general- ̟ φ χ(r) λ m 2
ized by Torres [11], who looked at several coupling func- Ω= , Φ= , σ =√4π , Λ= Pl ,
m m2 m 4π m
tions ω(φ), chosen to be compatible with known weak- Pl Pl (cid:16) (cid:17)
field limit and nucleosynthesis constraints. Similar sce- (8)
narioswere further studied in Ref. [12]. In this paper we
shall study boson stars formed at different times of cos- where mPl ≡ G0−1/2 is the present Planck mass. Our
mic history and in addition consider the possible phys- observed gravitational coupling implies Φ = 1. In order
ical evolution of such objects. One might further hope to consider the total amount of mass of the star within
that results found for boson stars might be representa- a radius x we change the function A in the metric to its
tive of other stellar candidates, especially neutron stars Schwarzschildform,
but perhaps even indicative of what might happen with
long-lived hydrogen-burning stars such as our own Sun. 2M(x) −1
A(x)= 1 . (9)
(cid:18) − xΦ( )(cid:19)
∞
II. EQUILIBRIUM CONFIGURATIONS TheissueofthedefinitionofmassinJBDtheoryisquite
a subtle one [13]. The above gives the Schwarzschild
mass; there are arguments that the tensor mass is more
We begin with a review of the formalism, which can
appropriate [13], but we have verified numerically that
be found in more detail in Ref. [11]. The material from
forthe largecouplingswe use the difference is negligible.
which the boson star is made is a complex, massive,
Note that a factor Φ( ) appears in Eq. (9). This is
self-interacting scalar field ψ, which is unrelated to the ∞
crucial to obtain the correct value of the mass, which is
Brans–Dicke scalar already described. Its Lagrangianis
given by
1 1 1
Lm =−2gµν∂µψ∗∂νψ− 2m2|ψ|2− 4λ|ψ|4. (2) M =M( )Φ( )m2Pl , (10)
star
∞ ∞ m
The U(1)symmetry leadstoconservationofbosonnum-
ber. Varying the action with respect to gµν and φ we for a given value of m. The Φ( ) factor allows for the
∞
asymptotic gravitational coupling to be different from
obtain the field equations:
that presently observed.∗ With all these definitions, the
1 8π ω(φ) 1 non-trivial equations of structure are [11]
R g R= T + φ φ g φ,αφ
µν µν µν ,µ ,ν µν ,α
− 2 φ φ (cid:18) − 2 (cid:19)
B′ A′ 2
1 σ′′+σ′ +
+ (φ g 2φ) , (3) (cid:18)2B − 2A x(cid:19)
,µ;ν µν
φ −
Ω2
1 dω +A 1 σ Λσ3 =0, (11)
2φ= 8πT φ,αφ,α , (4) (cid:20)(cid:18)B − (cid:19) − (cid:21)
2ω+3(cid:20) − dφ (cid:21)
whereTµν istheenergy–momentumtensorforthematter B′ A′ 2 1 dω
fields (Eq. (5) of Ref. [11]) and T, its trace. Commas Φ′′+Φ′ + + Φ′2
(cid:18)2B − 2A x(cid:19) 2ω+3dΦ
and semicolonsare derivatives and covariantderivatives,
respectively. 2A Ω2 2 σ2 σ′2 Λσ4 =0, (12)
Atfirst,weseekstaticequilibriumsolutions. Themet- −2ω+3(cid:20)(cid:18)B − (cid:19) − A − (cid:21)
ric of a spherically-symmetric system can be written as
ds2 = B(r)dt2+A(r)dr2+r2dΩ2. (5)
−
∗Note that the Φ(∞) factor was mistakenly forgotten in
We also demand a spherically-symmetric form for the ψ Ref.[11]. Thisaffects TableIIIof thatwork,where M(∞)is
field. Crucially, the most general ansatz consistent with actually M(∞)/Φ(∞). The Φ(0) values and the conclusions
thestaticmetricpermitsatime-dependentψ oftheform extracted from it are unaffected.
2
B′ A 1 A Ω2 σ′2 Λ
1 = 1 σ2+ σ4
xB − x2 (cid:18) − A(cid:19) Φ (cid:20)(cid:18)B − (cid:19) A − 2 (cid:21)
ω Φ′ 2 Φ′′ 1Φ′A′ 1 dω Φ′2
+ + +
2 (cid:18)Φ(cid:19) (cid:18) Φ − 2 Φ A(cid:19) 2ω+3dΦ Φ
A 2 Ω2 σ′2
2 σ2 Λσ4 , (13)
−Φ2ω+3(cid:20)(cid:18)B − (cid:19) − A − (cid:21)
2BM′ B Ω2 σ′2 Λ ωB Φ′ 2
= +1 σ2+ + σ4 +
x2Φ( ) Φ (cid:20)(cid:18)B (cid:19) A 2 (cid:21) 2 A (cid:18)Φ(cid:19)
∞
B 2 Ω2 σ′2
+ 2 σ2 Λσ4
Φ2ω+3(cid:20)(cid:18)B − (cid:19) − A − (cid:21)
B dω Φ′2 1Φ′B′
. (14)
−A(2ω+3)dΦ Φ − 2 Φ A
To solve these equations numerically, we use a fourth-
order Runge–Kutta method, for which details may be
found in Ref. [11]. Here we shall study two kinds of
theories.
1. JBD theory with ω = 400. This is comparable to
current observational limits [14,15].
2. A scalar–tensor theory with a coupling function of
the form
2ω+3=2B 1 Φ−α,
1
| − |
where we choose α = 0.5 and B = 5. The cos-
1
mologicalsettingofthismodelhasbeenstudiedby
FIG.1. BosonstarmassesasafunctionofΦ(∞);squares
Barrow and Parsons [16].
are for the JBD theory and circles the scalar–tensor theory.
TheupperpanelshowsmodelswithΛ=100andσ(0)=0.06.
As explained in Ref. [11], the results for these couplings
The lower one has Λ=0 and σ(0)=0.1.
maybe thoughtofasthe generalbehaviorofanyscalar–
tensor theory, by suitable expanding a general coupling
in a Taylor or Laurent series. The positive exponent of of general relativity, namely a constant φ and a t1/2.
this theory must be obtained from power-law couplings, ∝
This changeswith the onsetofmatter domination,when
but the behavior of these was found to be similar to the
the attractor solution becomes [17,18]
pure Brans–Dicke ones.
In addition to the freedom to choose the fundamen- a(t) t(2−n)/3 ; G(t) t−n, (15)
tal parameters, there are two free boundary conditions. ∝ ∝
One is the value of the boson field at the centre of the where n = 2/(4+ 3ω), so G exhibits a slow decrease.
star, σ(0), which we call the central density. The sec- Assumingthatmatter–radiationequalitytookplacenear
ond is the asymptotic value of the Brans–Dicke field, the general relativity value, at z = 24000Ω h2, then,
eq 0
whichdeterminestheasymptoticstrengthofgravity. The for critical density and h = 0.5, the fractional change in
other boundary conditions are fixed by demanding non- G since equality is
singularity and finite mass [7,11]. This still leaves an
infinite discrete set of solutions with different ̟, cor- G(t )
0 =6000−1/(1+ω). (16)
responding to a different number of nodes in σ(x). We G(t )
eq
choosethenodelesssolution,whichistheonlystableone.
Higher node solutions are generated in Ref. [12]. For ω = 400, the ratio is 0.98, so G will have changed
value by about two percent since equality.
Informed by the likely range of variation, we compute
III. QUASI-STATIC EVOLUTION static configurations with different values of the asymp-
totic gravitational strength, denoted Φ( ) = 1/G. In
∞
We first assess the likely cosmological variation of φ, Fig. 1, we see that if Λ and the central density σ(0)
concentrating on JBD theory. During radiation domi- are kept fixed, then the mass changes with a different
nation, the attractor behavior is actually exactly that Φ( ). In the JBD case, the mass is a growing function
∞
3
which led Barrow to propose the gravitational memory
hypothesis. We describe the possible subsequent histo-
ries a boson star might have, after forming at a time t
f
with a mass M(t ).
f
A. The gravitational memory hypothesis
The star remains completely static, without change in
the values of its mass or central density. Such a situa-
tionwouldbe reminiscentsay ofa virializedgalaxyafter
gravitationalcollapse,whichhasbecomedecoupledfrom
thecosmologicalexpansionoftheUniversearounditand
in particular does not participate in further expansion.
In such a scenario the mass M(t ) is a function of the
FIG.2. Thisshowsthebosonmassasafunctionofparti- f
formation time, and hence of Φ( , t ), as well as the
clenumberinthetwomodels,withsquaresfortheJBDtheory ∞ f
central density; the star keeps memory of the value of G
andtrianglesforthescalar–tensor one. BothhaveΛ=0and
at formation. This situation clearly cannot be precisely
σ(0) = 0.1. The leftmost point in each case corresponds to
Φ(∞) = 0.9 and the rightmost to Φ(∞) = 1.1. As M(N) is correct since the asymptotic gravitational constant does
evolve, but it may well be that the effect on the star is
single valued,there can be no change of stability.
much slower even than the cosmological evolution of G,
so that in practice the star can be viewed as static.
of Φ( ), while in the more complicated second model In this scenario, there is the interesting feature that
∞
there is actually a small peak in the mass at our present starsofthe same massmaydiffer inotherphysicalprop-
gravitational strength. That peak is due to the special erties (their radius, for example) depending on the for-
form of the coupling function, which selects out our own mation time.
observed coupling strength.
Let us now suppose that evolution does drive the sys-
tem through a series of quasi-static states, and na¨ıvely B. Quasi-static evolution hypothesis
estimatethe sortsofenergiesinvolved. Fig.1showsthat
variationofΦ( )byafewpercentcanchangethemasses
The opposite regime has the adjustment time for the
∞
ofconfigurationsby a few percent. As ithappens, this is
starbeingmuchshorterthantheasymptoticevolutionof
very similar to, or even greater than, the fraction of the
G. If true, the star should quasi-statically evolve, either
mass–energywhichcanbe liberatedbynuclearreactions
changing its mass or its central density or both. In this
overthe lifetime ofastar,aprocesswhichfor moststars
scheme, when intervals of time are short enough com-
occurs on similar timescales to the cosmological ones of
pared with the scale of cosmological evolution, the star
the time-varying G.
can be taken as static and the solutions computed hold
Before analyzing this in further detail, we shall first
in this limit.
showthecomputationforthenumberdensityofthecon- An interesting subcase is pure gravitational evolution.
figurations, defined by
This assumes that the mass of the star is preserved all
alongthequasi-staticevolution,throughthecentralden-
m2 ∞ A
N =Ω σ2 x2dx, (17) sity evolving in the appropriate way. This leads to the
m2Pl Z0 rB remarkable conclusion that stars may be able to evolve
evenwithoutabsorptionoremissionofenergy,andwith-
We foundthat, forboth modelsunder consideration,the
out nuclear burning. Such quiescent evolution would be
number of bosons is a growing function of Φ( ). Fig. 2
∞ entirely gravitationalin its nature.
shows the bifurcation diagram, number density against
Purely gravitational evolution would doubtless have
mass. There is no cusp structure in the sense of catas-
significantconsequencesfor stellar evolution,and indeed
trophe theory, which implies that there is no change in
may well be quite strongly constrained. Yet within the
the stability criterionfor values ofG closeto the present
quasi-static evolution hypothesis, it is actually rather
one [19]. In fact, we found this result for a wider range
conservative, because the other possibility is that mass
of Φ( ) than shown.
∞ is lost during the evolution, if dissipative processes are
requiredto keep trackingthe evolutionarysequence. We
have seen that the variation in G can easily lead to
IV. SCENARIOS
changes in the mass of up to a few percent, which can
be ofthe sameorderasthe energyliberatedina conven-
Now we discuss possible interpretations of the com-
tional star by nuclear burning.
putational results described above, in the same context
4
The reduction in central density seems reasonable step would be numerical simulation including dynamical
when we recall the force balance in polytropic stars. evolution,startingfromastaticsolutionandvaryingthe
Since gravityis reducing instrengthas time goesby, the asymptotic gravitational coupling. This looks a promis-
equilibrium configurations can become more diffuse and ingavenuefordeterminingwhichoftheproposedscenar-
hence drop in central density. In an extreme situation, ios is the correct one, and that knowledge may allow a
one might wonder whether the continuation of this pro- test of general relativity not just at the present epoch,
cess can in fact lead to the complete destruction of the but in the distant past, through astrophysicalsystems.
star, though this may be prevented by the initial nega-
tive binding energy of a stable boson star, since boson
number is conserved. ACKNOWLEDGMENTS
Thequasi-staticevolutionhypothesishasanimportant
difference from the gravitational memory hypothesis; in D.F.T. was supported by a British Council Fellowship
it, stars of the same mass are identical in all their other and CONICET, A.R.L. by the RoyalSociety and F.E.S.
physical properties too. by an European Union Marie Curie TMR fellowship.
We thank John Barrow, Eckehard W. Mielke, Hisa-aki
Shinkai and Andrew Whinnett for useful discussions.
C. Feedback on the asymptotic gravitational
constant?
Finally, one can ask whether the formation of boson
stars can significantly influence the ‘asymptotic’ gravi- ♣ E-mail address: [email protected]
tational constant; when the stars form gravity becomes ♦ E-mail address: [email protected]
weaker in their interior as the φ value increases. Follow- ♠ E-mail address: [email protected]
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5
