Geodesics around oscillatons made of exponential scalar field potential

2 MB·

by  Ali. Mahmoodzadeh

Geodesics around oscillatons made of
exponential scalar field potential
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Ali. Mahmoodzadeh∗, B. Malakolkalami†
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n
a Faculty of Science, University of Kurdistan, Sanandaj, P. O. Box :416, Iran,
J
Tel : [phone]

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Abstract
]
c
Oscillatons are spherically symmetric solutions to the Einstein-Klein-
q
- Gordon (EKG) equations for soliton stars made of real time-dependent
r scalar fields. These equations are non singular and satisfy flatness con-
g
ditions asymptotically with periodic time dependency. In this paper, we
[
investigate the geodesic motion of particles moving around an oscillaton

1 related to a time-dependentscalar field. Bound orbital is found for these
v particles under the condition of particular values of angular momentum
3 L and initial radial position r0. We discuss this topic for an exponential
1 scalar field potential which could be of the form of V(Φ) = V0eλ√k0Φ
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with a scalar field Φ(r,t) and investigate whether the radial coordinates
6
ofsuchparticlesoscillate intimeornotandtherebywecouldpredictthe
0
corresponding oscillating period as well as amplitude. It is necessary to
.
1 recall,ingeneralrelativity,ageodesicgeneralizesthenotionofa"straight
0
line" to curved space-time. Importantly, the world line of a particle free
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fromallexternal,non-gravitationalforces,isaparticulartypeofgeodesic.
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In other words, a freely moving or falling particle always moves along a
:
v geodesic. In general relativity,gravity can beregarded as not aforce but
i aconsequenceofacurvedspace-timegeometrywherethesourceofcurva-
X
tureisthestress–energytensor(representingmatter,forinstance). Thus,
r
a for example, thepath of aplanet orbiting around a star istheprojection
ofageodesicofthecurved4-Dspace-timegeometryaroundthestaronto
3-D space.
keywords Einstein-Klein-Gordon equations, metric functions, geodesics
[email protected]
∗
[email protected]
†
1
1. Introduction
As we know the most accepted cosmological models for understanding the
mechanism of structure formation are those that contain dark matter and dark
energy,withapossiblynegativeequationofstateasthemainconstituentsofthe
Universe. Both dark matter and dark energy can be described by a dynamical
scalarfield ΦthatrollsinitspotentialV(Φ)[1,2]. Ontheotherhand,nowadays
oscillatons are well known as astronomical objects with a spherically symmet-
ric and time-dependent metrics which are asymptotically flat and non singular
solutions to the Einstein-Klein-Gordon (EKG) equations and their properties
have been the subject of recent studies as a possibility for the role of dark
matter [3-6]. From astrophysical point of view oscillatons are gravitationally
bound objects with the real scalar field Φ endowed with a scalar field potential
V(Φ) which can be the subject of scalar field dark matter(SFDM) hypothesis
at the galactic scale [7-9]. First of all we know that the space-time around
an oscillaton varies with time depending on the form of the scalar field and
scalar field potential. But on the other side we know that there is not yet a
general agreement on the correct form that this scalar potential V(Φ) should
have, see for instance [1] and references there in. In some previous works, for
instance[10],hasbeensuggestedanexponential–likescalarfieldpotentialwhich
fitsverywelltheconstraintsfrombigbangnucleosynthesis,duetoits(socalled)
tracker solutions, and then fine tuning can be avoided. The most simple exam-
ple of a potential with both exponential behavior and a minimum is a cosh
potential[11]. Recently we propose another form of the scalar field potential,
described by V(Φ) = V0eλ√k0Φ,[12] where V0 and λ are free parameters which
should be fixed through the cosmological observation [13] or constraints which
areimposedthroughtheformulationoftheproblem[12]. Thusthegeodesicmo-
tion of real particles around an oscillaton varies depending on the form of the
potential. In previous works, the geodesics around an oscillaton made out of a
quadratic V(Φ)= 1m2Φ2 scalar field potential has been carriedout[14] and in
2 Φ
this paper we are interested in to do this for the new exponential form of the
potentialmentionedabove. Secondifweassumethatagalaxyisbasicallymade
out of scalar field dark matter described by an oscillaton, one would hope to
observe periodic variations of stars, path around the galaxy. These two topics
are the main motivations for studying what we are interested in. A summary
of this study is as follows:
In section two we review the mathematical background of oscillatons en-
dowedwithanexponentialscalarfieldpotentialV(Φ). Inthirdsectionwesolve
the geodesic equations numerically and investigate their properties. In the last
part we summarize the results and make some final comments for next investi-
gations.
2. Mathematical background of oscillatons
Oscillatonsaretimedependentsphericallysymmetricandasymptoticallyflat
2
solutions for the coupled Einstein-Klein-Gordon (EKG) equations. Numerical
solutions have been found for these equations by using the Fourier expansions.
But non-linearity of these equations has caused only few modes have been em-
ployed [13, 16]. The most general spherically symmetric metric is written as
ds2 =g dxαdxβ = eν µdt2+eν+µdr2+r2(dθ2+sin2θdϕ2), (1)
αβ −
−
where ν = ν(t,r) and µ = µ(t,r) are functions of time and radial position
(the units are so chosen in which~ = c = 1). The energy-momentum tensor
endowedforarealscalarfieldΦ(t,r)withascalarfieldpotentialV(Φ)isdefined
as [9]
T =Φ Φ 1g [Φ,γΦ +2V(Φ)]. (2)
αβ ,α ,β − 2 αβ ,γ
The non-vanishing components of T are
αβ
.
T0 =ρ = 1[e (ν µ)Φ2+e (ν+µ)Φ′2+2V(Φ)], (3. a)
− 0 Φ 2 − − −
. ′
T =p =ΦΦ , (3. b)
01 Φ
.
T1 =p = 1[e (ν µ)Φ2+e (ν+µ)Φ′2 2V(Φ)], (3. c)
1 r 2 − − − −
.
T2 =p =1[e (ν µ)Φ2 e (ν+µ)Φ′2 2V(Φ)], (3. d)
2 ⊥ 2 − − − − −
and we have also T3 = T2 . Over dots denote ∂ and primes denote
3 2 ∂t
∂ . The different components of the tensor for the scalar field mentioned
∂r
above are identified as the energy density, the momentum density, the radial
pressureandtheangularpressurerespectively. Einsteinequations,G =R
αβ αβ
−
1g R=k T areusedtoobtaindifferentialequationsforfunctions ν, µ, then
2 αβ 0 αβ
(ν+µ). =k rΦ.Φ′, (4. a)
0
.
ν′ = k0r(e2µΦ2+Φ′2) , (4. b)
2
µ′ = 1[1+eν+µ(k r2V(Φ)-1)] , (4. c)
r 0
where R and R are the Ricci tensor and Ricci scalar respectively and
αβ
k = 8πG = 8π ,where the gravitational constant, G , is the inverse of the
0 m2
pl
reduced Planck mass, m , squared. The conservation of the scalar field tensor
pl
reads as
Tαβ =Φ,α[(cid:3)Φ dV(Φ)]=0 , (5)
;β − dΦ
where(cid:3)=∂ ∂α =g ∂α∂β(α,β=0,1,2,3)isthe d,Alembertianoperator.
α αβ
Therefore we can obtain the Klein-Gordon (KG) equation for the scalar field
Φ(t,r),
3
Φ′′+Φ′(2 µ′) eν+µdV(Φ) =e2µ(Φ.. +µ.Φ.). (6)
r − − dΦ
If we choose Φ(r,t)=σ(r)φ(t) , then Eq. (6) reads as
φ{σ′′+σ′(2r −µ′)}−λ√k0V0eν0+µ0eλ√k0σ(r)φ(t) =e2µσ(φ..+µ.φ.). (7)
The following Fourier expansions
e f(x)cos(2θ) =I (f(x))+2 ∞ ( 1)nI (f(x))cos(2nθ) , (8)
± 0 n
P ±
n=1
where I (z) are the modified Bessel functions of the first kind, require that
n
Eq. (7) changes into
1 σ′′ +σ′(2 µ′) λ√k V eν+µ(I (λ√k σ)+2I (λ√k σ)) =
eσ2{µ(φ..+µ.φ.)r+−λ√k0−V0eν+µ[0I (0λ√k σ0)+2I 0(λ√k σ)1] 0 } (9)
φ σφ 0 0 1 0
It is necessaryto recallthat, we havetakeninto accountonly the firstterm
of the exponential field expansionfor simplicity to obtain Eq. (9). If we choose
√k Φ(r,t)=2σ(r)cos(ωt) , (10)
0
whereω isthe fundamentalfrequencyofthescalarfield,thenintegratingon
the Eq. (4.a) formally yields
′
ν+µ=(ν+µ) +rσσ cos(2ωt), (11)
0
where (ν +µ) is a function of r-coordinate only. The boundary conditions
0
which will be discussed latter require that
′
ν +µ =rσσ . (12)
1 1
It is useful to performthe followingvariable changesfornumericalpurposes
of the following form
x=mΦr , Ω = mωΦ , eν0 →Ωeν0, eµ0 →Ω−1eµ0. (13)
HencebyusingEqs. (4.a-6),(10,12)andsettingeachFouriercomponentto
zero the metric functions and the radial part of the scalar field are obtained as
ν′ =x[e2µ0σ2(I (2µ ) I (2µ ))+σ′2] , (14)
0 0 1 − 1 1
ν1′ =x[e2µ0σ2(2I1(2µ1)−I0(2µ1)−I2(2µ1))+σ′2] , (15)
µ′0 = x1{1+eν0+µ0[x2(cid:16)I0(xσσ′)I0(2λσ)+2I1(xσσ′)I2(2λσ)(cid:17)−I0(xσσ′)]},(16)
4
µ′1 = x2eν0+µ0[x2(cid:16)I0(xσσ′)I2(2λσ)+I1(xσσ′)I0(2λσ)+I2(Xσσ′)I(2λσ)(cid:17)−
′
I (xσσ )] , (17)
1
σ′′ =−σ′(x2 −µ′0− 12µ′1)+λeν0+µ0I0(2λσ)(cid:16)I0(xσσ′)+I1(xσσ′)(cid:17)−
e2µ0σ[I0(2µ1)(1 µ1)+I1(2µ1)+I2(2µ1)] . (18)
−
Now primes denote d . The two last equations are true if we put , k V =
dx 0 0
m2. With initial boundary values for ν (x = 0) , ν (x = 0) , µ (x = 0) ,
Φ 0 1 0
µ (x=0) , σ(x =0) and parameter λ, these equations are soluble numerically
1
and the answers are in the form of Interpolation Functions. Therefore the
metric functions and then the metric coefficients are completely determined.
Singularity points for the metric functions are also obtained through boundary
conditions. For example if we put ν (x = 0) = 0.11 , ν (x = 0) = 0.17 ,
0 1
−
µ (x = 0) = 0.11 , µ (x = 0) = 0.17 , σ(x = 0) = 0.3 , and λ = 0.1 then
0 1
− −
singularity happens at point x=1.62380. ( with increasing the values of λ and
σ, singularity points will happen at less and less values of x ). The important
pointthatshouldbementionedhereisthattheexponentialandquadraticscalar
potentialsdescribedbyV(Φ)=V eλ√k0Φ andV(Φ)= 1m2Φ2 respectivelywith
0 2 Φ
thesameinitialboundaryconditionscausedifferentmetricfunctionsandmetric
coefficients[4,12,16].
For next purposes the answers obtained from Eqs. (14-17) are fitted to six
order which are in compliance with equations (19-22). The results of these
fittings are shown in Fig. 1. From now on the metric functions are presented
by these fitted functions of the following form:
ν (x)= 0.106045 0.35163x+3.1483x2 9.5576x3+13.2842x4 8.45323x5+
0
− − − −
2.01531x6, (19)
ν (x)=0.171322 0.12740x+1.08631x2 3.76961x3+5.48473x4 3.68322x5+
1
− − −
0.93388x6, (20)
µ (x)=0.178029 1.72671x+12.168x2 33.2923x3+44.0517x4 27.2995x5+
0
− − −
6.44389x6, (21)
µ (x) = 0.169913 0.008487x+ 0.096403x2 0.242323x3 + 0.32989x4
1
− − −
0.202957x5+0.0397515x6. (22)
5
0.5 0.30
fittedfunction fittedfunction
0.4
(cid:1)0 0.25 (cid:1)1
0.3
n n 0.20
o o
ncti 0.2 ncti
u u 0.15
f f
d d
e e
fitt 0.1 fitt 0.10
0.0
0.05
-0.1
0.00
0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5
r r
fittedfunction -0.160
2.5
(cid:1)0
-0.165
2.0
function 1.5 function -0.170
d d -0.175
e e
fitt 1.0 fitt
-0.180 fittedfunction
0.5
-0.185 (cid:1)1
0.0
0.0 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
r r
Fig. 1. The metric functions and their corresponding fitted functions.
Thenextstepistospecifythegeodesicsaroundoscillatonsforrealparticles.
As we mentioned above, this work for a real particles around oscillatons de-
scribed by a quadratic scalarfield potential has been done in previous work[14]
and investigation of geodesics for real particles around oscillatons made of an
exponential scalar field potential discribed by[12], is the subject of this paper.
For this work we continue the process as follow:
If we redefine the metric coefficients as the following form
g = eν µ = B(x,t) , g =eν+µ =A(x,t), C(x,t)= A(x,t), (23)
tt − − − rr B(x,t)
then the Einstein equations can be rewritten as
∂A =k A∂Φ∂Φ , (24. a)
∂t 0 ∂t ∂x
∂A = k0Ax[C(∂Φ)2+(∂Φ)2+2AV(Φ)] , (24. b)
∂x 2 ∂t ∂x
∂C = 2C[1+A(k x2V(Φ) 1] , (24. c)
∂x x 0 −
6
C∂2Φ = 1∂C∂Φ + ∂2Φ + ∂Φ[2 1 ∂C] AdV(Φ) . (24. d)
∂t2 −2 ∂t ∂t ∂x2 ∂x x − 2C ∂x − dΦ
It is clear that by combining of Eq. (24. c) and (24. d) the non-linearity of
these equationscanbe minimized. By applyingA(x,t) , B(x,t) andC(x,t) the
non-vanishing components of T can be renovated as
αβ
T0 =ρ = 1[C(x,t)(∂Φ)2+ 1 (∂Φ)2+2V(Φ)] , (25. a)
− 0 Φ 2 A(x,t) ∂t A(x,t) ∂x
T0 =p =(∂Φ)(∂Φ) , (25. b)
1 Φ ∂t ∂x
T1 =p = 1[C(x,t)(∂Φ)2+ 1 (∂Φ)2 2V(Φ)] , (25. c)
1 r 2 A(x,t) ∂t A(x,t) ∂x −
T2 =p = 1[C(x,t)(∂Φ)2 1 (∂Φ)2 2V(Φ)] . (25. d)
2 ⊥ 2 A(x,t) ∂t −A(x,t) ∂x −
2.1. Boundary Conditions And Numerical Results
We can solve the (EKG) equations by considering the following Fourier ex-
pansions [13].
√k Φ(x,t)=2σ(x)cos(ωt) , (26. a)
0
nmax
A(x,t)= A (x)cos(nωt)=eν+µ =
n
P
n=0
nmax
eν0+µ0[I0(ν1+µ1)+2 In(ν1+µ1)cos(2nωt)] , (26. b)
P
n=1
nmax
C(x,t)= C (x)cos(nωt)=e2µ =
n
P
n=0
nmax
e2µ0[I0(2µ1)+2 In(2µ1)cos(2nωt)], (26. c)
P
n=1
where n is the Fourier mode at which the series are truncated. It is
max
remarkable that in Eq. (26 a) we have used the first term of the scalar field
nmax
, √k Φ(x,t) = σ(x)cos(nωt) , for simplicity, because the first term of the
0
P
n=1
approximation is sufficient to yield the main properties of oscillaton[16]. Non-
singularity and asymptotically flatness are two main characteristics of an oscil-
laton which determine the boundary conditions. Solutions of (EKG) equations
′
must be regular at x = 0, this means that σ (0) = 0 and A(x = 0) = 0. The
latter condition is equivalent to ν (0) + µ (0) = 0 , ν (0)+ µ (0) = 0 and
0 0 1 1
A (x = 0) = 0. The conditions σ( ) = 0 and A(x = ,t)= 1 are imposed
n>0
∞ ∞
by asymptotically flatness solutions as well as A ( )=1, A ( )=0, these
0 n>0
∞ ∞
conditionsareequivalenttoν ( )+µ ( )=0,ν ( )=0andµ ( )=0. But
0 0 1 1
∞ ∞ ∞ ∞
theconditionofν ( )+µ ( )=0yieldsC ( )=0andC ( )=1because
0 0 n>0 0
∞ ∞ ∞ ∞ 6
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this variable determines the fundamental frequency Ω =pC0(∞)=e−ν0(∞)as
anoutputvalue aftersolvingtheoscillatonequations. Thenextstepistochoose
σ(0)asthecentralvalueandC forfulfillingofboundaryconditions,Thereby
n 2
≥
we obtain a set of eigenvalues for each central value. It is quotable that Eqs.
(26. b) and (26. c) confine the metric coefficients to even nodes only.
3. Geodesic equations of motion
Spherically symmetric metric allows us to write the geodesics equations for
the metric (1). By using the geodesics equations which are in the form of
d2xκ = Γκ dxµdxν , (27)
dτ2 − µν dτ dτ
where Γk = 1gkη( g +g +g ) is affine connection, κ, η, µ, ν=0,
1, 2, 3 and xµνκ, xµ2 , xν−=(tµν,,ηr , θν,ηφ,µ), forηµa,νspecial case (θ = π) we have:
2
.t.= 1 ∂Bt.2 1 ∂Br.t. 1 ∂Ar.2, (28. a)
−2B ∂t − B ∂r −2B ∂t
.r.= 1 ∂Bt.2 1 ∂Ar.t. 1 ∂Ar.2+ rφ.2 , (28. b)
−2A ∂r −A ∂t −2A ∂r A
φ.. = 2r.φ. , (28. c )
−r
where dot is derivative with respect to proper time(τ).
Theseequationsaresolublenumericallybecauseitisnotpossibletowritethe
radialequation(28b)asusualas 1r.2+U(r)=Ewhere 1r.2isthekineticenergy,
2 2
U(r)aneffectivepotentialandE theparticlestotalenergy. TheconditionL=0
equivalent to φ=φ =0 means that a particle undergoes a direct straight line
0
free fall from its initial position (r ,φ ) . For L = 0 a particle will undergo
0 0
6
a variety of paths, on the other hand for L > 0 with increasing φ, particles
undergointhe counterclockwise directionbut forL<0 andwith increasingφ
, particles undergo in the clock wise direction. Needless to say that the motion
depends on the central value of the radial part of the scalar field , σ(x = 0) ,
the angular momentum L, the initial radial position r and initial radial speed
0
. .
r . In turning point where r = 0 , one of these two options happen: either a
0 0
particle goes away to infinity or oscillates with a certain period and amplitude
which could be found by solving the geodesic equations numerically. Fig. 2.
shows a path at which a particle goes away to infinity gradually under the
.
initial conditions r (τ = 0) = 1.63 , r (τ = 0)= 0 and two different values for
0 0
momentum ,L = 1.5 and 2.8 for the case σ(x = 0) = 0.3 (at which singularity
happensatpointx=1.62380)for arangeofpropertime [τ =0,40]. As wecan
see with increasing τ , the path of the particles approaches to a closed circles.
Changing the radial position to r (τ = 0) = 2.63 and momentum L = 1.5 and
0
L = 2.8 but this time the proper time τ = 100 causes particles undergo closed
orbital. See Fig. 3.
8
2 2
1 1
y 0 y 0
-1 -1
-2 -2
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
x x
Figs. 2. The orbitsforparticlesunder the boundaryconditions: σ(x=0)=
.
0.3, r (τ =0)=1.62381 , r (τ =0)=0 and L=1.5 for left plot and L=2.8
0 0
for right plot at which proper time varies from zero to 40.
2 2
1 1
y 0 y 0
-1 -1
-2 -2
-2 -1 0 1 2 -2 -1 0 1 2
x x
Fig. 3. The orbits for particles under the boundary conditions: σ(x=0)=
.
0.3, r (τ =0)=2.62381 , r (τ =0)=0 and L=1.5 for left plot and L=2.8
0 0
for right plot at which proper time varies from zero to 100.
Reiteration of the results mentioned above but this time for σ(x=0)=0.7
and the same λ= 0.1 show that singularity happens at point x=1.43696 for
− .
metric functions, therefore for initial values of r (τ =0)=1.44 , r (τ =0)=0
0 0
and two different values for momentum , L = 1.5 and L = 2.8 ,when proper
time varies from zero to 40, the geodesics for real particles are shown in Fig. 4
and Fig. 5.
9
1.5 1.5
1.0 1.0
0.5 0.5
y 0.0 y 0.0
-0.5 -0.5
-1.0 -1.0
-1.5 -1.5
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
. x x
Fig. 4. The orbits for particles under the boundary conditions: σ(x=0)=
.
0.7, r (τ =0)=1.43697 , r (τ =0)=0 and L=1.5 for left plot and L=2.8
0 0
for right plot at which proper time varies from zero to 40.
2 2
1 1
y 0 y 0
-1 -1
-2 -2
-2 -1 0 1 2 -2 -1 0 1 2
x x
Fig. 5. The orbits for particles under the boundary conditions: σ(x=0)=
.
0.7, r (τ =0)=2.43697 , r (τ =0)=0 and L=1.5 for left plot and L=2.8
0 0
for right plot at which proper time varies from zero to 40.
4. conclusions
In this study we presented the simplest approximation for solving a mini-
mal coupled Einstein-Klein-Gordon equations for a spherically symmetric os-
cillating soliton object called oscillaton endowed with an exponential scalar
field potential V(Φ) = V0eλk0Φ and an harmonic time-dependent scalar field
Φ(r,t)=2σ(r)cos(ωt). BytakingintoaccounttheFourierexpansionsofdiffer-
ential equations and with regard to the boundary conditions which require the
nonsingularityandasymptoticallyflatness,thesolutions(themetricfunctions)
are obtained numerically. It is more convenient to fit these solutions which are
in the form of Interpolation Functions to new functions, In present study this
workhasbeendonetoorderfour,althoughbetterresultsareobtainedbytaking
10