Table Of ContentNumerical studies of Φ2-Oscillatons
Miguel Alcubierre1, Ricardo Becerril2, F. Siddhartha Guzma´n3,
Tonatiuh Matos4, Dar´ıo Nu´n˜ez1, and L. Arturo Uren˜a-L´opez5∗
1Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico, A.P. 70-543, 04510 M´exico, D. F., M´exico.
2Instituto de F´ısica y Matem´aticas, Universidad Michoacana,
Edif. C-3, Ciudad Universitaria 58040, Morelia, Michoac´an, M´exico.
3Max-Planck-Institut fu¨r Gravitationsphysik, Am Mu¨hlenberg 1, D-14476 Golm, Germany.
4Departamento deF´ısica,Centro deInvestigaci´on ydeEstudios Avanzados delIPN,A.P.14-740, 07000M´exicoD.F.,M´exico.
5Astronomy Centre, University of Sussex, Brighton BN1 9QJ, United Kingdom.
(Dated: January,2003)
WepresentanexhaustiveanalysisofthenumericalevolutionoftheEinstein-Klein-Gordonequa-
tions for the case of a real scalar field endowed with a quadratic self-interaction potential. The
self-gravitating equilibrium configurations are called oscillatons and are close relatives of boson
stars, their complex counterparts. Unlike boson stars, for which the oscillations of the two compo-
3
nents of the complex scalar field are such that the spacetime geometry remains static, oscillatons
0
giverise toageometry thatistime-dependentandoscillatory in nature. However,theycan still be
0
2 classified into stable (S-branch) and unstable (U-branch) cases. We have found that S-oscillatons
areindeedstableconfigurationsundersmall perturbationsandtypicallymigrate tootherS-profiles
n
when perturbed strongly. On theother hand,U-oscillatons are intrinsically unstable: theymigrate
a
totheS-branchiftheirmassis decreased andcollapse toblack holesiftheirmass isincreased even
J
bya small amount. TheS-oscillatons can also bemadeto collapse to black holes if enough mass is
4 addedtothem,butsuchcollapsecanbeefficientlypreventedbythegravitationalcoolingmechanism
2
in the case of diluted oscillatons.
1 PACSnumbers: 04.25.Dm,95.30.Sf,95.35.+d,98.62.Ai,98.80.-k
v
5
0 I. INTRODUCTION oftheevolutionoftheEKGequations,muchinthesame
1
way as it has been done for boson stars [2, 3, 4]. The
1
main aim is to investigate whether oscillatons are sta-
0
ble. For simplicity, we only consider here the spherically
3 In a seminal paper Seidel & Suen [1] found that there
0 symmetric case of a real scalar field Φ endowed with a
exist non-singular, time-dependent equilibrium configu-
/ quadraticscalarpotentialoftheformV(Φ)=(m2/2)Φ2.
c rations of self-gravitating real scalar fields. These os-
Other cases will be treated in future publications.
q cillating soliton stars are called oscillatons, and are so-
- This paper is organized as follows. In section II, we
r lutions of the Einstein-Klein-Gordon (EKG) system of present the necessary mathematical background to find
g equations for minimally coupled real scalar fields. The
equilibriumconfigurationsandtofurtherevolvetheEKG
:
v time-dependence of these solutions appears as a funda- equations. As we shall see, the equilibrium configura-
i mentalingredientthatallowssingularitiestobe avoided,
X tions may be classified in two different groups: the S
incontrasttostaticsolutionsoftheEKGequationswith
and U-branches. The numerical methods and tests are
r real scalar fields, where singularities frequently appear.
a presented in section III. In sections IV and V, we ana-
The caseof realscalarfields is quite different to the case
lyze the results obtained from the evolution of the EKG
of complex scalar fields, for which the EKG equations
equations for different oscillatons. We separately study
providethe so-calledbosonstars,whicharenon-singular
the behavior of S and U-oscillatons by adding to them
solitonicsolutionwithastaticgeometry(thecomponents
small and strong perturbations that change their total
ofthe complexfieldoscillateinpreciselysuchawaythat
mass. We conclude in section VI.
the stress-energy tensor is time independent). However,
the stability of oscillatons has not been studied in as
much detail as in the case of boson stars [2, 3, 4]. Such
II. MATHEMATICAL BACKGROUND
studiesarenecessarybecauseofthepossibleroleofoscil-
latons in astrophysics and cosmology, where real scalar
To begin with, we consider the spherically symmetric
fieldshavebeenproposedascandidatesforthedarkmat-
line element
ter in the Universe [5, 6, 7].
In this paper we want to complement previous stud- ds2 = α2dt2+a2dr2+r2 dθ2+sin2(θ)dϕ2 (1)
−
ies[1,3,8,9,10]onoscillatonswithanumericalanalysis (cid:0) (cid:1)
with α(r,t) the lapse function anda(r,t) the radialmet-
ric function. We choose the polar-areal slicing condition
(i.e. we force the line element to have the above form
∗Presentaddress: InstitutodeF´ısicadelaUniversidaddeGuana- at all times, so that the area of a sphere with r = R is
juato,A.P.E-143,C.P.37150, Leo´n,Guanajuato, M´exico. alwaysequalto4πR2); this choiceofgaugewillforcethe
2
lapse function α(r,t) to satisfy an ordinary differential Equations (7-12) form the complete set of differential
equation in r. Through out the work, we will be using equations to be solved numerically. The evolution equa-
units such that c = ~ = 1, and we express the gravita- tion for Π above is further transformed into the equiva-
tional constant in terms of he Planck mass: G=1/m2 . lent form:
Pl
The energy momentum tensor of a scalar field Φ
d x2αΨ
endowed with a quadratic self-interaction potential Π =3 aαΦ. (13)
V(Φ)=(m2/2)Φ2 is ,t dx3 (cid:18) a (cid:19)−
T =Φ Φ gµν Φ,αΦ +m2Φ2 . (2) Notice that the first term on the right hand side of this
µν ,µ ,ν − 2 ,α equation includes now a first derivative with respect to
(cid:2) (cid:3)
x3 (andnotathirdderivative). Thereasonfordoingthis
The non-vanishing components of T are
µν
transformation has to do with the numerical regulariza-
T0 = ρ = 1 α−2Φ˙2+a−2Φ′2+m2Φ2 , (3) tion near the origin of the 1/x2 factor in equation (11)
0 Φ
− 2h i above (see Ref. [11]).
T = =Φ˙Φ′ , (4)
01 Φ
P
1
T11 = pr = α−2Φ˙2+a−2Φ′2 m2Φ2 , (5) A. Eigenvalue problem for equilibrium
2h − i configurations
1
T22 = p⊥ = α−2Φ˙2 a−2Φ′2 m2Φ2 , (6)
2h − − i Inordertofindthe equilibriumconfigurationsofoscil-
and also T33 = T22. These different components are latons,eqs.(7-12)aresolvedusingFourierexpansionsfor
identified as the energy density ρΦ, the momentum den- boththemetricandthescalarfieldfunctions[1,8,9,10].
sity Φ, the radialpressure pr, and the angular pressure We brieflydescribe herethe procedureusedin[9]to find
P
p⊥. The parameter m is interpreted as the mass of the such equilibrium configurations.
scalar particles. Over-dots denote ∂/∂t and primes de- In order to deal with the non-linearities presentin the
note ∂/∂r. EKG equations, it is convenient to introduce the new
The evolution of the metric functions α and a can be variables A(t,r) = a2(t,r), C(t,r) = (a/α)2, for which
obtainedfromtheEinsteinequationsGµν =κ0Tµν,with eqs. (7-12) take the form
κ =8πG. In order to write appropriateevolutionequa-
0
tions, we now introduce the firstordervariables Ψ=Φ,r A′ = Ax CΦ˙2+Φ′2+AΦ2 + A(1 A), (14)
andΠ=aΦ,t/α. Also,wedefinethedimensionlessquan- 2 (cid:16) (cid:17) x −
tities r = x/m, t → t/m, Φ → Φ/√κ0, where we note C′ = 2C 1+A 1x2Φ2 1 , (15)
that the bosonic mass is the natural scale for time and x (cid:20) (cid:18)2 − (cid:19)(cid:21)
distance. ′
1 2 C
Using these new variables,the Hamiltonian constraint CΦ¨ = C˙Φ˙ +Φ′′+Φ′ AΦ, (16)
−2 (cid:18)x − 2C(cid:19)−
becomes
A˙ = xAΦ˙Φ′. (17)
a 1 a2 x
,x = − + Ψ2+Π2+a2Φ2 , (7)
a 2x 4 The lapse function is later obtained as α2(t,x) =
(cid:2) (cid:3)
and the polar-arealslicing condition takes the form A(t,x)/C(t,x). Notice that Eq. (17) is a consequence of
themomentumconstraint(the t,r partoftheEinstein
α a a2 1 { }
,x = ,x + − xa2Φ2. (8) equations.)
α a x − We shall consider the Fourier expansions
The evolution of the scalar field Φ is governed by the
jmax
Klein-Gordon (KG) equation, which appears as a conse-
Φ(t,x) = φ (x)cos(jωt),
quence of the conservation equations of the scalar field j
Xj=1
energy-momentum tensor (2) in the form
jmax
Tµν;ν =Φ,µ 2−m2 Φ=0, (9) A(t,x) = Aj(x)cos(jωt), (18)
where2=gαβ isthe(cid:0)d’Alam(cid:1)bertianoperator. The Xj=0
α β
KG equation i∇s eq∇uivalent to the following set of first jmax
C(t,x) = C (x)cos(jωt),
order differential equations j
Xj=0
α
Φ = Π, (10)
,t a where ω is called the fundamental frequency and jmax
1 x2αΨ is the mode at which the Fourier series are truncated.
Π,t = x2 (cid:18) a (cid:19) −aαΦ, (11) SolutionsareobtainedbyintroducingtheFourierexpan-
,x sions(18)ineqs.(14-17), andsettingeachFouriercoeffi-
αΠ cient to zero; that is, the EKG equations are reduced to
Ψ = . (12)
,t
(cid:18) a (cid:19) a set of coupled ordinary differential equations.
,x
3
The boundary conditions are determined by requir- is not a fixed value but instead oscillates in time. How-
ing non-singular and asymptotically flat solutions, for ever, as we shall see below (see Fig. 8), the amplitude of
which equations (14-17) become an eigenvalue problem. suchoscillationsisquitesmallandtheinitialvaluecanbe
Thus, it is only necessary to determine the initial val- takenasrepresentativeofeachoscillaton. Themaximum
ues φ (0), C (0) (the fundamental frequency is an out- mass M = 0.607m2 /m is reached for a central value
i i c Pl Φ
put value) corresponding to a given central central value φ (0) = 0.48, to which also corresponds a fundamental
1c
φ (0), to obtain different equilibrium configurations. frequency Ω = 0.864. The fundamental frequency is al-
1
ways such that Ω 1 for all oscillatons, with Ω = 1 for
≤
the trivial solution. In general, more massive oscillatons
0.3 φ
0.25 φφ53*1*10012 oscillate with a smaller fundamental frequency.
0.2
0.15
Φ
0.1
1 0.7
0.05
0.98
0.6
0 0.96
0.94 0.5
-0.05
0 5 10 15 20 25 0.92
x 0.4
0.2 Ω 0.9 MT
0.3
0.88
0.15
0.86 0.2
0.1
0.84
0.05 0.1
0.82 MΩT
g-1rr -0.005 0.80 0.1 0.2 0.3 φ0.4 0.5 0.6 0.7 0.8 0
1(0)
-0.1 0.7
-0.15
0.6
-0.2 g g*rr100-12
grr2*103 0.5
-0.25 rr4
0 5 10 15 20 25
x 0.4
T
M
FIG. 1: Non-zero Fourier coefficients of the scalar field 0.3
Φ and the radial metric coefficient g − 1 (see eq. (18))
rr
for a configuration with φ1(0) = 0.2828. The total mass 0.2
is M = 0.5726m2Pl/m and the fundamental frequency is
(ω/m) = 0.9128. The solution shown here was calculated 0.1
uptothe6thFouriermode(j =6ineqs.(18)). Thecon-
max
vergenceoftheFourierseriesismanifest(noticethere-scaling 0
0 20 40 60 80 100 120
of the higher Fourier modes). R (0)
max
A typical oscillaton solution is shown in Fig. 1. Even FIG. 2: (Top) Total masses MT (in units of m2Pl/mΦ) and
fundamentalfrequencies(Ω)ofdifferentoscillatons. Thecrit-
though we are solving non-linear equations, the Fourier
series converges rapidly. A particular feature of the so- ical (maximum) mass is Mc = 0.607m2Pl/mΦ for a configu-
lutions is that they are represented only by odd Fourier ration with a central value φ1c(0) = 0.48; its corresponding
frequencyis(ω/m)=0.864. (Bottom) Plot of thetotal mass
coefficients of the scalar field Φ and the even coefficients M vsR (0) (the latter in units of m−1), the radius at
T max
of the metric functions A and C. whichthemetriccoefficientg reachesitsmaximumvalueat
rr
In Fig. 2, we show the calculated total mass (MT), t=0.
the fundamentalfrequency (Ω ω/m)andthe radius at
≡
which the radial metric coefficient reaches its maximum
value at t = 0 (R (0)) for different configurations. In
max
thecaseofoscillatons,thepositionofthemaximumofa2
4
III. NUMERICS Π is even and Ψ is odd. Notice that we can integrate Φ
all the way to the boundary point x = ∆x/2, since
0
−
A. Numerical algorithm for the evolution of the its evolution equation does not require the evaluation of
system spatial derivatives.
At the outer boundary we also need to impose bound-
ary conditions. Notice again that we do not need to ap-
In order to integrate the Klein-Gordon equa-
plyaboundaryconditionforΦ,asitsevolutionequation
tions (10,11,12) we used a method of lines with second
canbe integratedallthewaytothe boundarypoint. For
order centered differences in space. For the time inte-
Π we assume that, for large enough x, it behaves as an
gration we use a method inspired in the Iterative Crank
outgoing wave pulse of the form:
Nicholson(ICN)schemewiththreeiterations(seeforex-
ample [12, 13]). The standardICN is usedto integrate a
Π=u(x t)/x, (26)
system of evolution equations of the form: −
∂u with u an arbitrary function. In differential form this
i
∂t =Si(uj,∂xkuj). (19) becomes
Given values of the variables u at time step t = n∆t, ∂ Π+∂ Π+Π/x=0, (27)
i x t
one updates the values of the variables to time step t =
(n+1)∆t in the following way: which, performing finite difference can be solved for the
unknown boundary value at the new time level. On the
otherhand,it isnotdifficult toconvinceoneselfthatthe
u(1) = un+(∆t/2)Sn , (20) function Ψ does not behave as an outgoing wave at the
i i i
u(k) = un+(∆t/2)S(k−1) k =2,...,N 1,(21) boundary. However, we are assuming that Π does and,
i i i − as a consequence, so does Φ. In particular, the outgoing
un+1 = un+∆tS(N−1) , (22) wave boundary condition applied to Φ can be seen to
i i i
imply that at the boundary:
where un =u (t=n∆t), un+1 =u (t=(n+1)∆t), and
i i i i
u(k) are intermediate values. Taking N =3 is enough to Ψ= Π Φ/x. (28)
i − −
obtainasecondorderaccurate,stablescheme[12,13](in
This equation can then be used to obtain boundary val-
fact, taking N = 2 is enough for second order accuracy,
ues for Ψ once those of Φ and Π are known.
but it is unstable).
Finally, we need to mention the boundary conditions
Forourpurposeswehavemodifiedtheabovealgorithm
used for the ordinary differential equations that have to
for the case N =3 in the following way:
be solvedtofindthe metricfunctions α anda. Fora, we
u(1) = un+(∆t/3)Sn , (23) use the fact that local flatness implies that a(x=0)=1
i i i
and ∂ a(x = 0) = 0. These two conditions imply that
u(i2) = uni +(∆t/2)Si(1) , (24) a(x0)x=a(x1)=1+ (∆x)3. Weusethesetwoboundary
O
un+1 = un+∆tS(2) , (25) valuesatthefirsttwogridpointsandintegratethesecond
i i i
order hamiltonian constraint outwards.
Thereasonforthismodificationisthattheabovescheme For the lapse function α we use the fact that, on a
is considerably less dissipative than standard three-step vacuum, our slicing condition implies that we are in
ICN. In fact, for linear equations one can show that the Schwarzschild coordinates, so we must have α = 1/a.
above scheme is third order in time. We have found our We then assume that our boundaries are sufficiently far
modified scheme ICN to be very stable and robust in away as to be always in a vacuum, and impose α = 1/a
practice. as an outer boundary condition. The slicing condition is
Once we have advanced the variables Φ, Ψ and Π one then integratedinwards. One could presumably improve
time step using the above algorithm, we substitute their on this by setting α(x = 0) = constant, ∂xα(x = 0) = 0
new values into equations (7) and (8). These are simple asboundaryconditionsontheorigin,integratingtheslic-
ordinary differential equations on the radial coordinate ingconditionoutwards,andthenre-scalingitsothatthe
thatwesolveusingastandardsecondorderRunge-Kutta lapse goes as 1+k/x far away (k is a constant; notice
scheme. that the slicing condition is scale invariant).
B. Boundary conditions C. Code tests
Ourset ofequationsis singularatx=0. Toavoidthe In order to illustrate how all these ingredients work
singularity,westaggertheoriginandtakeaspatialgridof together properly, we study now the accuracy of our nu-
the form x =(i 1/2)∆x. The fictitious point at x = merical methods in some particular cases. In Figure 3
i 0
−
∆x/2 is used to impose appropriate parity conditions: we show the convergence of a system initially consisting
−
5
of a Gaussian pulse of scalar field. The equation we use Themainresultspresentedinthis sectioncanbesum-
to calibrate our techniques is the (r,t) component of the marized as follows. i) S-oscillatons are stable against
Einstein’s equations (the momentum constraint): small perturbations. ii) Besides the fundamental oscil-
lations of the system, there is an overallvibration of the
1 oscillaton when slightly perturbed. The frequencies (f)
β :=a xαΦΠ=0, (29)
,t
− 2 ofthesevibrationsshouldbeidentifiedwiththeso-called
quasi-normalmodesofthesystem,andtheresultingplot
whichshouldbesatisfiedforanexactsolution[14]. What
fvsM is a particular feature of the Φ2-oscillatons. As
weshowintheplotistheL2normofthevalueofβacross
we shall see below, such quasi-normal modes are very
the grid as a function of time for three different resolu-
important in the study of evolved profiles.
tions. The initial Gaussian corresponds to an oscillaton
We take the scalar field profiles obtained by solving
with M = 0.575, so that it is in the stable branch and
eqs.(17-16) to be the initial data for the evolutionequa-
hasalongtime life; theboundaryislocatedatx=50so
tions. In this form, the initial conditions for the scalar
that the evolutions were carried out up to 100 crossing
field are (see eqs. (18))
times. Thefactthatthevalueofβ goesdownbyafactor
of 4 every time the resolution is doubled shows that the
jmax
code remains second order convergent.
Φ(t=0,x) = φ (x), (30)
j
Xj=1
6e-06 jmax
Fine Φ′(t=0,x) = φ′(x) (31)
j
5e-06 Medium
Coarse Xj=1
2
orm 4e-06 Φ˙(t=0,x) = 0. (32)
n
n
n i 3e-06 andthenthemetricfunctionsα,aarecalculatedthrough
o
ati theEinsteinequations. Inthismanner,wewillalsocheck
qu 2e-06 the consistency between the eigenvalue problem and the
e
(r,t) 1e-06 nthuemmereitcrailcefvuonlcuttiioonns,sitnhceefnuondaadmdietniotnaallfrienqfouremncaytioωn, e(ltick.e)
is taken initially.
0 AsatypicalexampleofaS-branchoscillaton,weshow
0 500 1500 2500 3500 4500 inFig.4theevolutionoftheinitialprofileshowninFig.1,
t characterized by the central value φ1(0)=0.2828. From
the figure we see that such an oscillaton is stable, main-
FIG.3: Themomentumconstraint(29)forthreedifferentres- taining the same oscillatory pattern from t 300 (see
≃
olutions: Fine (dr = 0.01), Medium (dr = 0.02) and Coarse Fig. 5) up to times t 20,000 (in the plot we are show-
≃
(dr=0.04). TheinitialconditionscorrespondstoaGaussian ing only a fraction of the run). That the evolution is
pulse of the form Φ(0,x) = 0.296e−x2/5.35, and total mass stable can also be seen from Fig. 5, in which we show
M =0.575. the accuracy of the numerical evolution by plotting the
momentumconstraint. InFig6weshowtheevolutionof
the total integrated mass. We can see a small apprecia-
ble adjustment of the original mass at around t = 300,
IV. EVOLUTION OF OSCILLATONS: THE indicating a small ejection of scalar field from the sys-
S-BRANCH AND THE QUASI-NORMAL MODES tem. The overall linear decay of the mass can be shown
to be consistent (using convergence tests) with a small
We shall call S-branch oscillatons those equilibrium amount of numerical dissipation still present in our nu-
configurations to the left hand side of the critical config- merical method, and is therefore not an intrinsic decay
uration in the plot Mvsφ (0) in Fig. 2 (or equivalently, of the oscillaton.
1
those configurations to the right hand side in the plot of Thesimulationsshowthattheevolvedprofileoscillates
MvsR ). Asafirststepinthestudyofthestabilityof periodically at two different time scales. In particular,
max
oscillatons, we will numerically evolve these equilibrium thereis ashort-periodoscillation,whichwemayidentify
configurations. withthefundamentaloscillation,andanoverallvibration
We should keep in mind that, compared to realistic withalongerperiod[1]. Followingtheliteratureofboson
oscillatons, the solutions presented in Fig. 2 are already stars [2], we shall refer to the later as the “quasi-normal
perturbed. These perturbations arise because of the in- modes”.
herentdiscretizationerrorofthenumericalsolutionsand To determine the oscillatory scales of the solution,
(mainly)ofthetruncatedFourierexpansions. Therefore, we calculated the power spectrum of the evolution of
the eigen-solutions of section IIA will be treated as al- (g ) in Fig. 4, and plotted it in Fig. 7. Notice that
rr max
ready slightly perturbed profiles[1]. the dominant frequencies are the quasi-normal one and
6
Slightly perturbed S-oscillaton: φ1(0)=0.2828 0
0.18
-0.0005
0.178
0
0.176 M
-0.001
(t)-1ax 00..117724 M-M) / 0 -0.0015
(g)rrm 0.17 00 x ( -0.002
1
0.168
-0.0025
0.166
0.164 -0.003
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
t t
1
t=4500 FIG.6: Totalintegratedmassasafunctionoftimeduringthe
t=4750
t=5000 evolution of the same oscillaton. We see that the oscillaton
0.01 ejects a very small amount of matter at around t=300m−1.
Theoveralllineardecreaseisduetothenumericaldissipation
0.0001 still present in the code.
x)
ρ(t, 1e-06
2x (twice) the fundamental angular frequency, which are,
1e-08 respectively, (f/m) = 4.52 10−2 and (ω/m) = 0.9119.
×
Highermodescorrespondingtoeven-multiplesofthefun-
1e-10 damental frequency are also present, but their power is
smaller. Observethatthefundamentalfrequencyisalit-
tlebitreadjustedtoo,butitsvalueisconsistentwiththe
1e-12
0 10 20 30 40 50 60 70 80 90 100 eigen-solution in Fig. 1.
x
We systematically evolve all other S-branch equilib-
rium configurations shown in Figs. 2, calculating their
FIG. 4: (Top) Maximum values of the radial metric function
g (t,x)=a2(t,x). The initial configuration is that of Fig. 1 corresponding power spectrum and then their quasi-
rr
with ∆x = 0.01,∆t/∆x = 0.5. The boundary is at x = 100 normal and fundamental frequencies. In all cases, the
andtheevolutionisshowntot=5,000m−1,some50crossing evolvedprofilesbehavedaccordinglywiththedescription
times. Infact,wehavefollowedtherunuptot=20,000m−1 givenabovefor the (representative)case φ (0)=0.2828.
1
and no change was observed. Thus, we can conclude that For instance, the values of the fundamental frequen-
theinitialconfigurationisstableagainstsmallperturbations. cies were just a little bit readjusted in all cases, but
(Bottom) The density profile ρΦ(t,x) times x2 is plotted for its values were always consistent with those shown in
late times.
Figs. 2. In general, the power spectrum of more dilute
oscillatons is dominated by the fundamental frequency,
6
while the quasi-normal frequency dominates for oscilla-
tons near the critical point. Also, the linear numerical
5
dissipationof mass was as in Fig. 5, suggesting also that
7 itshouldbe attributedto the numericalmethodandnot
x 104 to the intrinsic properties of oscillatons.
T) 0tr3 toTahnealryezseultthinegevpololuttfiovnsoMf binosFoing.st7ahrsa[s2,be4e],navnedrywuesewfuilll
κ
G - tr2 saeemaitxiims uumsefaunldfoar sohsacirlplatdoencslinaesnweealrl.thTehceriptilcoatlsmhoawsss.
(
This is a typical behavior indicating the transition from
1 stable to unstable configurations.
‘To further demonstrate that the S-oscillatonsare sta-
0 ble and that the frequencies shown in Fig. 7 are their
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
t intrinsic quasi-normalmodes, we show in Fig. 8 the evo-
lution of the total mass M and R compared to the
T max
FIG.5: L2normofthemomentumconstraintforthenumer- equilibrium configurations as shown in Fig. 2. It is clear
ical evolution of theoscillaton shown in Figs. 1,4 that the slightly perturbed S-oscillatons oscillate with
. very small amplitudes around the original equilibrium
7
6 Slightly perturbed S-oscillatons
0.65
4
φ1(0)=0.45
2 φ1(0)=0.2828
m 0.6
ctru 0 φ1(0)=0.25
e
er sp -2 MT 0.55
w -4
o
P
-6 0.5 φ1(0)=0.15
-8
-10 0.45
0.001 0.01 0.1 1 10 2 3 4 5 6 7 8 9 10 11
ω R
max
Quasinormal modes of S-oscillatons
5 FIG. 8: The evolution of the total mass MT and Rmax is
4.5 (((∆∆∆ttt ===111000000000))) CAB Migration sthootwn=fo5r,0d0iff0emre−n1t.S-Iotscciallnatobnes.cleTahrelyrusenesnwtehraetfotlhloeweevdoluup-
(∆t =4000) D
4 (∆t =4000) E tionsproceedjustbysmalloscillationsaroundtheinitialpro-
(∆t =1000) F files. Actually,sincetheprofileinFig.2usestheinitialvalue
)Φ 3.5 R (0), the oscillations are shifted to the left of the solid
m max
3 3 line. Hence, it is manifest that these oscillatons are not mi-
-10 2.5 grating nor decaying. For a comparison with truly migrating
x oscillatons, see Figs. 10,12,13.
f ( 2
1.5
1 1. Perturbed S-oscillatons
0.5
0.1 0.2 0.3 0.4 0.5 0.6 0.7 Ourinterestnowisto determinewhether S-branchos-
M
T cillatons are stable against strong perturbations. That
is, we want to know the conditions under which such os-
FIG. 7: (Top) The power spectrum of the evolution of the
metriccoefficientg forthecaseinFig.4. Theconfiguration cillatons will either collapse into a black hole, disperse
rr
vibrateswiththequasi-normalfrequency(f/m)=4.52×10−2 away or form another oscillaton. The latter is a rather
while oscillates with even-multiples of the fundamental an- interesting possibility since, as we shall show below, it
gular frequency (ω/m) = 0.9119. The latter coincides with implies the migration of oscillatons.
the result of the eigenvalue problem, see Fig. 1. (Bottom) We have included Gaussian-like perturbations in the
Thefrequenciesofthequasi-normalmodesobtainedfromthe
original equilibrium configurations, like that shown in
evolution of slightly perturbed S-branch oscillatons. We also
Fig. 9, which can be seen just as perturbations to the
show the migration of perturbed S-branch and U-branch os-
originalmassoftheoscillatons. (Weshallcalloriginalthe
cillatons, labelled A-F; the details of their evolution can be
profilesshownin Figs.2 and their correspondingparam-
found in sections IV 1,V. In theplot, ∆t represents thetime
eters). The main results are summarized as follows. iii)
intervals at which the evolved mass was measured for each
case. If an oscillaton is perturbed in such a way that its mass
is less than the critical mass M 0.606(m2 /m),[15]
c ≃ Pl
it migrates to another solution in the S-branch. iv) On
the other hand, if the initial mass is larger than M , the
c
oscillaton can either migrate to an S-oscillaton or col-
lapse to a black hole. The collapse to a black hole can
configurations. Inotherwords,theyarenotmigratingto
be prevented by the oscillaton ejecting the mass excess,
other oscillatons nor decaying. This can be compared to
the so-called gravitational cooling mechanism [3]. This
the migrating oscillatons shown in Figs. 10,12,13 below.
mechanism is highly effective for dilute oscillatons. In
both cases (iii,iv) above, the evolution of the perturbed
The results presented above give evidence for the profiles can be tracked according to their vibration fre-
points(i-ii) outlinedatthe beginningofthis section. We quencies, i.e., the quasi-normal modes. We have plotted
wouldliketostressherethattheimportantresultisthat sometypicalevolutionsofperturbedS-branchoscillatons
theredoexiststableoscillatons(atleastintheS-branch). in Fig. 10 which show the different behaviors described
As a side-effect, we have also proven the consistency be- above.
tween the eigenvalue problem and the numerical evolu- We start by discussing point (iii). In the first case, we
tion code. decreasedthe initial mass of a φ (0)=0.45-oscillatonto
1
8
φ
Strongly perturbed S-oscillaton: 1(0)=0.1 Strongly perturbed S-oscillatons
0.12 0.07 0.65
Original
Perturbed
0.06
0.1 0.6 φ1(0)=0.1, +40%
0.05
0.08
0.55
0.04
φ 0.06 0.03 g-1rr MT 0.5 φ1(0)=0.45, -20%
0.04
0.02
0.45
0.02
0.01
0 0 0.4
0 10 20 30 40 50 60 70 80 90 100 0 5 10 15 20 25
x R
max
Strongly perturbed S-oscillatons
FIG. 9: Examples of the scalar field and grr profiles of a 0.7
strongly perturbed S-oscillaton by means of a Gaussian per-
0.68
turbation. Shownisthecaseofaφ1(0)=0.1-oscillatonwhose
original mass was increased 40%. The evolution of thisoscil- 0.66 φ1(0)=0.1, +66%
laton is given in Figs. 7 (label B),11,10. See also text below 0.64
for details. 0.62 φ1(0)=0.2, +25%
T 0.6
M
0.58
an initial value M = 0.484 (20% less than its original
i 0.56
mass M = 0.605). We can observe that initially the os-
0.54
cillaton expands and then bounces back while it looses
mass, to finally settle down on the S-branch. The mi- 0.52
gration is manifest in the trajectory shown in Fig. 10, 0.5
that oscillates around the stable equilibrium configura- 0 5 10 15 20
R
tionsdiscussedintheprevioussection. Duringtheevolu- max
tion,theoscillatonmaintainsafixedvibrationfrequency.
FIG. 10: Evolutions of different strongly perturbed S-
Its path of migration is labelled A in Fig. 7, and shows
oscillatons,seetextfordetailsandcaseslabelledA-CinFig.7.
that the oscillaton will stop at a diluted one.
The percentage figures are the perturbations made in their
A second case corresponds to a φ1(0) = 0.1-oscillaton originalmasses. Ingeneral,aperturbedS-oscillatonmigrates
which was strongly perturbed as to have an initial mass back to the S-branch if its initial mass is less than the criti-
of Mi = 0.579 (40% over its original mass M = 0.414). cal value Mc =0.606. The migration proceeds by the loss of
Despite the strong perturbation, the oscillaton collapses mass, throughtheso-called gravitational cooling mechanism.
and looses enough mass to settle down onto another S- Thismechanismissoefficientinthecaseofdiluteoscillatons,
that it can prevent the formation of a black hole even if the
oscillaton. Bymeasuringthevibrationfrequency,wefind
initial mass is larger than the critical value. However, this
that the migrating oscillaton follows the path labelled B
coolingmechanismissometimesnotenoughfordenseoscilla-
in Fig. 7, which suggests that the oscillaton is migrating
tons. Theasterisksdenotethestartingpoints. Therunswere
to a φ1(0) = 0.15-like profile. This can also be seen in followed up to t=5,000m−1.
Fig. 11, in which the profile of the metric coefficient g
rr
rapidlyapproachesandoscillatesaroundthefinalconfig-
uration.
To illustrate point (iv), a φ (0) = 0.1-oscillaton was of the hole. However, we have observed that the same
1
perturbed to have an initial mass of M =0.687,a value oscillatondoesmigratetotheS-branchiftheinitialmass
i
largerthanthecriticalone. However,wecanobservethat is somewhat less perturbed, while still being larger than
theoscillatonrapidlylosesmassandmigratesbacktothe the critical one. This is also a generic phenomenon we
S-branch. Its migration path is labeled C in Fig. 7. This have observed by perturbing other S-oscillatons. This
is a typical example of the efficiency of the gravitational tellsusthatthegravitationalcoolingmechanismishighly
cooling mechanism in enough dilute oscillatons. efficient if the scalar field is diluted enough, like in the
cases shown in the original paper of Seidel & Suen [3].
The opposite case is shown by a perturbed φ (0) =
1
0.2-oscillaton, whose initial mass is M = 0.657. The Theresultspresentedsofarpointoutthat,apartfrom
i
gravitational cooling mechanism is not efficient enough being stable configurations, S-branch oscillatons are in-
in this case to prevent the formation of a black hole. deed the final states in the evolution of other perturbed
We see this by noticing that its trajectory stops at the S-oscillatons. These two properties (stability and final
value ofR correspondingto the Schwarzschildradius state-quality) are the imprint of S-oscillatons. To end
max
9
φ φ
Migration: 1(0)=0.1 --> 1(0)=0.15 Slightly perturbed U-oscillatons
0.12 0.45
t=40t=000 φ1φ(10(0)=)=00.5.85
t=4250 0.4
0.1 t=4500
t=4750
φ t=5000 0.35
0.08 1(0)=0.15
g-1rr 0.06 g)(t)-1rrmax 000.2..235
0.04 (
0.15
0.02
0.1
0 0.05
0 5 10 15 20 25 30 35 40 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
x t
Slightly perturbed U-oscillatons
FIG. 11: Evolved profiles of the metric coefficient grr for a 0.64
stronglyperturbedφ1(0)=0.1-oscillaton(seealsoFigs.9,10), Migration φ1(0)=0.6
which is migrating to a φ1(0) = 0.15-oscillaton. This is con- 0.62
sistent with the path (labeled B) in Fig. 7. The times and φ1(0)=0.65
distances are given in unitsof m−1. 0.6
φ1(0)=0.7
0.58 φ1(0)=0.8
T
M
this section, we would like to stress here the important
0.56
role of the quasi-normal modes to follow the migration
process of oscillatons. The examples presented so far 0.54
show that the vibrationfrequency of the systemremains
0.52
the same during its evolution, even while the system
looses mass.
0.5
2 4 6 8 10 12 14 16 18 20
R
max
V. EVOLUTION OF OSCILLATONS: THE
FIG.12: (Top)Evolutionofthemaximumvalueoftheradial
U-BRANCH
metriccoefficient g for twoslightly perturbedoscillatons in
rr
the U-branch, whose corresponding central initial values are
We shall call U-branch oscillatons those equilibrium φ1(0) = 0.55,0.8. Larger values of the latter mean more in-
configurations on the right (left) hand side of the crit- stability. (Bottom) The migration of different U-oscillatons
ical configuration in the plot Mvsφ1(0) (MvsRmax) is manifest in a graph of MvsRmax(t). By looking at the
plot fvsM in Fig. 7, we can see at which point the different
in Fig. 2. We also evolved these equilibrium configu-
oscillatons will stop at. For instance, the labels D,E corre-
rations, under the same idea that the eigen-solutions of
spond totheφ1(0)=0.7,0.8-oscillatons shown in this graph,
sectionIIAarealreadyslightlyperturbedconfigurations. respectively. The time is in units of m−1 and the asterisks
The main result of this section is that U-oscillatons are
denotethe starting points.
intrinsically unstable since, as we shall see below, they
decay and migrate to the S-branch under small pertur-
bations.
A. Perturbed U-oscillatons
Shown in Fig. 12 are some instances of slightly per-
turbed U-oscillatons, and the plot MvsR speaks by
max
itself when compared to Fig. 8. The U-branch oscilla- WehavealsoperturbedtheU-branchequilibriumcon-
tonsareintrinsicallyunstable;theymigratetoandsettle figurationsandstudied whether they migrate orcollapse
downontotheS-branch,evenundersmallperturbations. into black holes. The main results of this section are as
The larger the central value (φ (0) > 0.48), the more follows. i) If the initial mass somewhat larger than the
1
unstable they are. Their migration to the S-branch also original mass of the eigen-configuration, the oscillaton
confirmsthestabilityoftheS-oscillatonsandtheirimpor- collapses into a black hole. This result is independent of
tant role as the final states in the evolution of migrating thekindofperturbationmadeandthenitcanbecharac-
oscillatons. terized by the excess in mass only. ii) If the mass of the
As we have done before, the migration of oscillatons original configuration is decreased by the perturbation,
can be tracked by determining their quasi-normal fre- the oscillaton migrates to the S-branch.
quencies. For example, the slightly perturbed φ (0) = The φ (0) = 0.8-oscillaton is a typical example of U-
1 1
0.7,0.8-oscillatons shown in Fig. 12 are labeled D,E, re- oscillaton and its evolution under strong perturbations
spectively, in Fig. 7. is plotted in Fig. 13. If the initial mass of this oscilla-
10
Strongly perturbed U-oscillatons: φ1(0)=0.8 Strongly perturbed U-oscillaton: φ1(0)=0.8
0.64 80 1
t=0
t=24
0.62 +2% 60 t=48
t=72
0.6 0.5
40
0.58 -2%
20
MT 0.56 g-1rr 0 0 -g-1tt
0.54
-20
0.52 -0.5
0.5 -40
0.48 -60 -1
0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10
R x
max
FIG. 13: The paths of the evolution of a perturbed φ1(0) = FIG. 14: Plots of the metric coefficients gtt,grr for a per-
0.8-oscillatoninaplotMvsR . Thecollapseintotheblack turbedconfigurationwithanexcessmassof2%. Eventhough
max
holeisrepresentedbythehorizontallinegoingtotheleft,end- thevalueoftheinitialmassiswellbelowM ≃0.606, theos-
c
ingattheSchwarzschildradiusofthefinalblackhole;seealso cillaton collapses into a black hole.
Fig.14. Thesameoscillatonwasalsoperturbedbydecreasing
its original mass, in which case it migrates and settles down
φ φ
intotheS-branch,seetextandFig.15fordetails. Theaster- Migration: 1(0)=0.8 --> 1(0)=0.15
0.45
isksdenotethestartingpoints. Therunswerefollowed upto t=0
t=5,000m−1. 0.4 tt==12000000
t=3000
0.35 t=4000
t=4500
0.3 φ t=5000
ton is perturbed so that is 2% more massive that the 1(0)=0.15
original mass, the final stage of the collapse is a black 1 0.25
hole,eventhoughtheinitialmassislessthanthecritical g-rr 0.2
value. This can also be seen in Fig. 14, where we see
0.15
plots of the metric coefficients g and g . The coeffi-
tt rr
0.1
cientg showsthewell-know“collapseofthe lapse”(the
tt
lapse drops to zero) characteristic of black holes, while 0.05
the coefficient grr shows the “grid stretching” effect (te 0
0 5 10 15 20 25 30 35 40
radial metric grows exponentially), also typical of black
x
hole evolutions. Onthe other hand, if the initial mass of
the sameoscillatonis2%lessthanthe originalmass,the FIG. 15: Evolved profile of the metric coefficient g for a
rr
oscillaton rapidly migrates to the S-branch and settles φ1(0)=0.8-oscillaton migrating to a φ1(0)=0.15-oscillaton.
downintoastableoscillaton. ItsevolutionpathinFig.7 This is consistent with the graphs in Fig. 7 (label E) and in
(labelF),overlapswiththatofthemigratingS-oscillaton Fig. 13. The times and distances are given in unitsof m−1.
showninFig.9,11(labelBinFig.7). Thus,thefinalstate
ofthe evolutionseems to be a φ (0)=0.15-oscillaton,as
1
can also be seen in Fig. 15. This last fact reveals again B. The S-U transition point
the usefulness of the plot fvsM to determine the final
configuration of a migrating oscillaton. Now we take a closer look at the equilibrium configu-
We wouldliketosummarizeheretheresultspresented rations near the critical profile, the one being the most
so far. We have found that the imprint of U-branch os- massivein Figs. 2, which we shall callthe S-U transition
cillatons is that they are intrinsically unstable. That is, point.
evensmallperturbationsprovoketheirmigrationtoother To begin with, let us recall the basic characteristicsof
configurations, instead of the small oscillations around the S,U-branches. The basic imprint of S-branch oscil-
the original ones. This latter fact again confirms the latons is that they are stable and then oscillate around
stability and final state-status of S-oscillatons. Another the initial profiles when slightly perturbed, see Fig. 8.
particular characteristic of U-oscillatons is that the for- Onthe other hand,U-branchoscillatonsareintrinsically
mationofa black hole is verylikely if the originalprofile unstable and they migrate to the S-branch even when
is perturbed by adding massto it. This phenomenonoc- slightly perturbed, like those cases shown in Fig. 12.
curs even if the perturbed initial mass is less than the The evolution of the profiles near the S-U transition
critical one, so that the gravitationalcooling mechanism point when slightly perturbed are shown in Fig. 16. We
is not efficient at all in the U-branch. observethattheresultssummarizedaboveareagaincon-
