Galactic dark matter in the phantom field

Ming-Hsun Li and Kwei-Chou Yang

Department of Physics, Chung Yuan Christian University, Chung-Li 320, Taiwan(Received 26 September 2012; published 20 December 2012)

We investigate the possibility that the galactic dark matter exists in a scenario where the phantom field

is responsible for the dark energy. We obtain the statically and spherically approximate solution for this

kind of galaxy system with a supermassive black hole at its center. The solution of the metric functions is

satisfied with gtt ¼ �g�1rr . Constrained by the observation of the rotational stars moving in circular orbits

with nearly constant tangential speed in a spiral galaxy, the background of the phantom field which is

spatially inhomogeneous has an exponential potential. To avoid the well-known quantum instability of the

vacuum at high frequencies, the phantom field defined in an effective theory is valid only at low energies.

Under this assumption, we further investigate the following properties. The absorption cross section of

the low-energy S-wave excitations of the phantom field into the central black hole is shown to be the

horizontal area of the central black hole. Because the infalling phantom particles have a total negative

energy, the accretion of the phantom energy is related to the decrease of the black hole mass, which is

estimated to be much less than a solar mass in the lifetime of the Universe. Using a simple model with the

cold dark matter very weakly coupled to the ‘‘low-frequency’’ phantom particles that are generated from

the background, we show that these two densities can be quasistable in the galaxy.

DOI: 10.1103/PhysRevD.86.123015 PACS numbers: 95.36.+x, 04.70.Dy, 11.25.Db, 98.35.Gi

I. INTRODUCTION

The recent experimental data have shown that thecurrent Universe is undergoing a phase of acceleratedexpansion [1,2]. Considering the universe filled with abarotropic perfect fluid which corresponds to the darkenergy component, its equation of state w<�1=3 isrequired for cosmic acceleration, where w ¼ p=� with� and p being the density and pressure, respectively.Recently observations suggest that the equation ofstate lies in a narrow strip around w ¼ �1 [3,4], wherew ¼ �1 corresponds to a cosmological constant � andw<�1 is allowed. A specific form of the dark energycorresponding to the phantom field was proposed torealize the possibility of late-time acceleration withw<�1 [5,6], while the quitom model has w crossing�1 [7]. The peculiar property of the phantom darkenergy is the violation of the dominant energy condi-tion, so that the energy density and curvature may growto infinity in a finite time, which is referred to as a BigRip singularity [6,8].

Observations related to the cosmic microwave back-ground and the large-scale structure support that theUniverse is very close to spatially flat geometry [3,9]. Atthe present time our universe is dominated by dark energywith the fraction �72%. The most accessible componentof the Universe is baryonic matter which amounts to only4.6%. The main remaining part that is nonbaryonic andnonluminous is believed to be the so-called dark matterresponsible for �23%. The weakly interacting massiveparticles (WIMPs) are considered one of the main candi-dates for cold dark matter which is dustlike with equationof state w ’ 0. A dark matter halo, demonstrated by its

gravitational effect on a spiral galaxy’s rotation curve,dominates the mass of a galaxy and can stretch up to belarger than 50 kpc from the center of a galaxy.The presence of the interaction between the dark matter

and dark energy field may modify the distribution of darkmatter in a galaxy. A homogeneous scalar field with nega-tive kinetic energy has been used to model the phantomenergy (with w<�1) and then to study the cosmologicalevolution [10–12], where the scalar field varies only withtime but does not change spatially. However, for a galaxydue to the gravitational instabilities, the phantom field maynot be spatially homogeneous. Some works related to theinhomogeneous dark energy properties have been com-pleted [13–21]. In this paper, we are interested in the staticsolution of the Einstein equations that can describe thedark matter halo with the existence of the supermassiveblack hole at its center and the background of the spatiallyinhomogeneous phantom field. We adopt the standardassumption that the dark matter halo consists of theWIMPs with w ’ 0. We find that an approximate solutionof the metric exists for describing this galactic halo sce-nario when we take the metric functions to be satisfied withgtt ¼ �g�1

rr . In general, several static and spherically sym-metric exact solutions of Einstein equations are obtained interms of the parameter �, defined by T�

� ¼ Trrð1� �Þ, for

which some related works can be found in Refs. [22–25].We also obtain the approximate solution for the spatiallyinhomogeneous phantom field in a galaxy. Our resultindicates that the corresponding exponential potential ofthe phantom field is relevant to the stage of structureformation.We further study the stability of the space-time structure

for the galaxy. Since the phantom dark energy field can

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cause the quantum instability of the vacuum at highfrequencies, we therefore treat it as an effective theoryvalid at low energy, i.e., we add a cutoff in the momentumintegral [10]. The stringent limit for the cutoff is � &3 MeV, which was obtained from the diffuse gamma raybackground [26]. We compute the accretion rate of thephantom particles into the black hole. Since the infallingphantom particles have a total negative energy, the blackhole mass diminishes in the process. On the other hand,due to the conservation of the angular momentum ofthe individual WIMPs and their tiny interaction rate, thecapture cross section for dark matter particles by the super-massive black hole is sufficiently small. Therefore, thedark matter that we consider here is quite stable.

This paper is organized as follows. In Sec. II, we considerthe pressless massive dark matter (WIMP) directly or indi-rectly interacting with a phantom field which is associatedwith the acceleration of the Universe. We solve the Einsteinequations with the condition gtt ¼ �g�1

rr , and the approxi-mate solution is then obtained. As a byproduct, wewill showthat the exact solutions can be extended to some typical limitswhich are related to Schwarzschild, Reissner-Nordstrom andSchwarzschild—de Sitter/anti—de Sitter solutions, respec-tively. We calculate the distribution function of the phantomfield and its potential in the region of the galactic halo. InSec. III, we will examine if the spatially inhomogeneousphantom field can stably survive with the existence of asupermassive black hole at the center. Considering thespherically symmetric space with a supermassive blackhole at the center, we first semiclassically calculate theKlein-Gordon equation of a S-wave phantom particle thatis excited from the background field. We obtain the absorp-tion probability of the infalling phantom particle into thecentral black hole and further show that the absorptive crosssection is approximately the area of the horizon. We showthat the accretion rate of the phantom particle, which isaccompanied by the decreasing rate of the black hole mass,could be small enough, so that the space structure of a galaxyis stable comparedwith the life of theUniverse. InSec. IV,weconsider a toymodel of phantomparticles coupled tomassivedark matter and show that both the dark matter and lowfrequency phantom densities can be quasistable for a suffi-ciently small coupling constant. Finally, we give the sum-mary in Sec. V.

II. THE EXACT SOLUTION IN THE STATIC LIMIT

We consider the real phantom field minimally coupledto gravity

S¼Zd4x

ffiffiffiffiffiffiffi�gp �

R

2�2þ1

2g��@��@���Vð�ÞþLmþLI

�;

(1)

where �2 ¼ 8 �G is the reduced Planck mass, Vð�Þ is thephantom field potential, the Lagrangian term Lm accounts

for the massive dark matter in the galaxy, andLI describespossible interactions between the phantom field and thedark matter. Due to the small coupling between the phan-tom field and the dark matter, LI can be negligible in thepresent calculation. To investigate static, spherically sym-metric solutions, we employ the metric ds2 ¼ �e�dt2 þe�dr2 þ r2ðd�2 þ sin2�d2Þ, adding the ansatz � ¼ ��.This condition is satisfied with the exact solutions, like theSchwarzschild, Reissner-Nordstrom and de Sitter/anti—deSitter solutions. For the static situation, the Einstein equa-tions read

gttRtt � 1

2R ¼ e��

�1

r2� �0

r

�� 1

r2¼ �2Tt

t; (2)

grrRrr � 1

2R ¼ e��

�1

r2þ �0

r

�� 1

r2¼ �2Tr

r; (3)

g��R���1

2R¼e��

2

��00þ�02

2þ�0��0

r��0�0

2

�¼�2T�

�;

(4)

gR�1

2R¼e��

2

��00þ�02

2þ�0��0

r��0�0

2

�¼�2T

;

(5)

where the energy-momentum tensor corresponds to themassive dark matter in the background of the phantomfield,

Ttt ¼ �� ¼ ��ph � �DM ¼ 1

2e���02 � Vð�Þ � �DM;

(6)

Trr ¼ pr ¼ pr;ph ¼ � 1

2e���02 � Vð�Þ; (7)

T�� ¼ p� ¼ p�;ph ¼ 1

2e���02 � Vð�Þ; (8)

T ¼ p ¼ p;ph ¼ 1

2e���02 � Vð�Þ; (9)

and a prime denotes the differentiation with respect to r. Inthe spherical coordinate, the total energy-momentum ten-sor is denoted by diagð��; pr; p�; pÞ, the phantom field

background is diagð��ph; pr;ph; p�;ph; p;phÞ, and the cold

dark matter can be approximated as diagð��DM; 0; 0; 0Þ.It is interesting to note that we can find solutions that are

satisfied with the condition � ¼ ��, i.e., gtt ¼ �g�1rr , in

the limit Ttt ! Tr

r. Moreover, it will be shown in thefollowing that for the mass density of WIMPs to be �DM ¼e��02 > 0, the present case can be satisfied with thiscondition for which the existence of the correspondingsolution is due to the fact that the sign of the kinetic term

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of the phantom field is opposite compared to the ordinaryscalar field (quintessence field); for the quintessence darkenergy, there exists no such solution.

To find the solution satisfied with the condition � ¼ ��,we first set � ¼ lnð1�UÞ and substitute it into Eqs. (2)and (4) [or into (3) and (5)]. We then obtain

r2U00 þ 2�rU0 þ 2ð�� 1ÞU ¼ 0; (10)

where we have set T�� ¼ T

¼ Tttð1� �Þwith � being a

constant. The solution of this equation is

U ¼ rsr� r2ð1��Þ

r�; for � �

3

2; (11)

or

U ¼ 1

r

�rs � a ln

�r

jaj��

; for � ¼ 3

2; (12)

i.e., the corresponding metric is

ds2 ¼ ��1� rs

rþ r2ð1��Þ

r�

�dt2 þ

�1� rs

rþ r2ð1��Þ

r�

��1dr2

þ r2ðd�2 þ sin2�d2Þ; (13)

or

ds2 ¼ ��1� rs

rþ a

rln

�r

jaj��

dt2

þ�1� rs

rþ a

rln

�r

jaj���1

dr2

þ r2ðd�2 þ sin2�d2Þ; (14)

where rs, r�, and a � 0 are the integration constants.Before we continue, we discuss the obtained exact solu-

tions in some typical limits. In addition to the present caseof the galactic dark matter interacting with the phantomfield, the following ones can be satisfied with the relationTrr ¼ Tt

t. First, for � ¼ 1 with r� ! 1, it gives theSchwarzschild metric corresponding to T�

� ¼ 0, and rs isthe Schwarzschild radius. Second, for � ¼ 2, the solutionis the Reissner-Nordstrom metric, where r�1

� ¼ GQ2 andQis the charge of the black hole. Third, for � ¼ 0 resultingin Tt

t ¼ Trr ¼ T�

� ¼ T ¼ 3=ð8�Gr�Þ, it gives the

Schwarzschild-de Sitter/anti-de Sitter solutions which areequivalent to the replacement r� � �3=�, with� being thepositive/negative cosmology constants.

In this paper, we are interested in the metric that candescribe the motions of stars in the galaxy. For the severalobserved cases, the rotational stars with radius rhalo > r �rs in a spiral galaxy, where rhalo * 50 kpc denotes theradius of a typical halo in a galaxy, are moving in circularorbits with nearly constant tangential speed v whichroughly ranges from 10�4 to 10�3. For the region with r >rhalo, the dark matter may become very dilute. In the darkmatter dominant region, where the test particle stablymoves in constant rotational curve, the metric function�gtt was estimated in the form [27]

�gtt ffi�r

r0

�2v2

; (15)

where r0 is a constant and �gtt ¼ e� will be denoted as fin the following discussion. Note that the form of gtt ismodel independent but grr is not [27,28]. Because 2v2 �10�8–10�6 is a tiny magnitude, we can approximate themetric function in the form

f ffi�r

r0

�2v2

¼ elnð rr0Þ

2v2 ffi 1þ 2v2 lnr

r0: (16)

On the other hand, we focus on our solution with � � 1but � 1, for which the galaxy has a supermassive blackhole of the Schwarzschild radius rs � 10�7 pc at its centerand is predominated by the massive dark matter (WIMPs)in the phantom dark energy background. In the following,we will further exhibit that if we put a test particle in such agalaxy, it is possible to find a solution for which the testparticle moves in circular orbit with nearly constant tan-gential speed consistent with the result given in Eq. (16)(see the result given in Fig. 1). Compared with Eq. (16), wefind that in the same dark-matter dominant region, if themetric function in Eq. (13) is given by

f ¼ 1� rsrþ

�r

�r0

�� ffi 1þ e

lnð r�r0Þ�

ffi 1þ

�1þ � ln

r

�r0

�; (17)

where j lnð r�r0Þ�j< 1, jj, j�j � 1, 1� � ¼ �=2, r�1� ¼

=�r�0 , these two equations are approximately equal under

the following conditions: � ¼ 2v2 � 10�8–10�6 and ¼ 2v2 lnð�r0=r0Þ. In Fig. 1, we consider the extremecase of the dark-matter galaxy with the supermassive blackhole at its center and show the corresponding metric func-tion �gtt as a function of r. We find that the obtained

0 100 200 3005.

4.

3.

2.

1.

r kpc

106

lng t

t

FIG. 1 (color online). The metric function �gtt as a functionof r. The solid curve corresponds to �gtt ¼ 1� rs

r þ ð r�r0Þ� and

the dashed curve is for �gtt ¼ ð rr0Þ2v2, where we have adopted

the following parameters: rs ¼ 10�7 pc, r0 ¼ 6 Mpc, � ¼�0:155, v2 ¼ 10�7. For comparison, using the same r0, theupper and lower dot-dashed curves are for �gtt ¼ ð rr0Þ2v

2but

with v2 ¼ 10�7=4 and v2 ¼ 4� 10�7, respectively.

GALACTIC DARK MATTER IN THE PHANTOM FIELD PHYSICAL REVIEW D 86, 123015 (2012)

123015-3

metric function can be consistent with the result given inEq. (15) under the certain condition.

In the region with the nearly constant tangential speed ofthe stars, the energy density is given by � ’ v2m2

pl=4�r2,

where mpl ¼ G�1=2 is the Planck mass. Moreover, from

Eqs. (2), (3), (6), and (7), we have �DM ¼ e��02b in the

static limit, where �b is the classical phantom field back-ground. On the other hand, from Eqs. (6)–(8), and using� � 1, we find that � ¼ e��02

b =� ’ �DM ¼ e��02b , i.e.,

�ph ’ 0 and V ¼ e��02b =2þ �ð1� �Þ ’ e��02

b =2. Since

the metric function is e� ’ 1, combining these gives

�bðrÞ ��1 � vmplffiffiffiffiffiffiffi4�

p ln

�r

~r0

�; (18)

Vð�bÞ �v2m2

pl

8�~r20e�ffiffiffiffiffi16�

p ð�b��1Þvmpl ; (19)

during the distances rhalo > r � rs, where the integrationconstant ~r0 is roughly* 6 Mpc, the intergalactic distance.Here we choose the positive sign of �b. As for r * ~r0, thespace becomes flat, the phantom field �b ¼ �1 is spa-tially uniform, and its potential is responsible for thecurrent accelerated expansion for which we have 3H2

0 ’8�Vð�1Þ=m2

pl, where at the present epoch the Hubble

parameter H0 � 10�42 GeV. Consistently, we get

~r0 � v

H0

� 6 Mpc: (20)

III. THE ABSORPTION OF THE PHANTOM FIELDBY THE SUPERMASSIVE BLACK HOLE

The WIMPs (dark matter) are quite stable owing to theconservation of the angular momentum of the individualparticles and the tiny interaction among them, so that thecapture cross section for dark matter particles by the super-massive black hole is sufficiently small. Therefore, to studythe stability of the space-time structure for the galaxy, wecompute the accretion rate of the excited phantom particlesinto the black hole. If the nonrelativistic dark matter can belong lived enough compared with the age of the universe,in the following we will estimate the absorption probabil-ity, absorption cross section, and the accretion rate of thephantom excitation wave into the black hole. At low ener-gies, the dominant effect comes from the excited phantomparticle with lowest angular momentum. Therefore, weconsider its S state, which is the excitation from the back-ground, �� ¼ �ðt; rÞ ��bðrÞ. The phantom potential isassumed to be around the local maximum,

Vð�Þ � v2m2pl

8�~r20e�ffiffiffiffiffi16�

p ðFð�Þ��1Þvmpl ; (21)

where Fð�Þ ’ �b þ ðF��ð�bÞ=2Þ��2 (since the phantomparticle with the negative kinetic energy might evolve to the

maximum of the local potential). In other words, the localpotential is around the locally stable point, V�ð�bÞ ¼ 0.Here F�� � d2F=d�2 and V� � dV=d�. Taking intoaccount the backreaction, the wave for the phantom excita-tion is satisfied with the Klein-Gordon equation

�1

f@20��þ 1

r2@rðr2f@r��Þ¼ðm2

þ�h�2iÞ���m2eff��;

(22)

where

m2 ¼ Vð�bÞ

ffiffiffiffiffiffiffiffiffi16�

pvmpl

F��ð�bÞ

is the mass squared of the excited phantom field ��, � is thedimensionless coupling constant describing the interactionbetween the excited phantom particle and the dark matter inthe form of LI ¼ ð�=2Þ��2 �2, and h�2i stands for theaverage of the quantum fluctuation of the dark matter field�. We will show in the next section that when � is smallenough, the densities ��� and �� exhibit a stable oscillatorybehavior, so that one can take m2

eff ’ m2. In this paper, the

conclusion is also applicable for m2 < 0.

The excited phantom wave can be variable separated inthe form of ��ðt; rÞ ¼ <½ðrÞe�i!t. In the present study,we reasonably assume the tiny phantommass and thereforeneglect its spatial dependence. We redefine a new coordi-nate variable by the relation

d ¼ dr

r2f: (23)

Then, the differential equation of reads

½@2 þ ð!2 �m2efffÞr4 ¼ 0: (24)

Very close to the horizon r � rs, where f � 0 and we canhave the approximation ð!2 �m2

efffÞr4 ’ !2r4s , the

ingoing wave is then given by ðrÞ ’ 0e�i!r2s . In this

case, because the infalling phantom particles have a totalnegative energy, the phantom energy accretes, resultingin the decrease of the black hole mass [29]. Extending tothe region where r � !r2s and the rotational speed, v,

of the star is constant, corresponding to f � ðr=r0Þ2v2

[27], the solution can be approximated as

ðrÞ / 1� i!

�1�m2

eff

!2

�12r2s

¼ 1þ id!r2sr2v

2

0

r1þ2v2

1

1þ 2v2; (25)

where d ’ ½1�m2eff=!

21=2. On the other hand, using the

dimensionless variable � ¼ !r, we find that the differen-tial equation for can be rewritten by

@2�þ�@�f

fþ 2

�

�@�þ

�1

f2� m2

eff

!2f

� ¼ 0: (26)

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For the distances rhalo > r � rs, this equation is approxi-mated as

@2�þ�2v2 þ 2

�

�@�þ

�1� 2�m2

eff

!2

� ¼ 0: (27)

The corresponding solution is

ð�Þ ¼ ��ð12þv2ÞðAJ12þv2ðd�Þ þ BN1

2þv2ðd�ÞÞ: (28)

The overlap region between the two solutions, Eqs. (25)and (28), is !2r2s � !r � 1, where Eq. (28) reduces to

’�d

2

�12þv2 A

�ð32 þ v2Þ ��

2

d!2

�12þv2 �ð12 þ v2ÞB

�r1þv2 : (29)

Matching Eq. (29) onto Eq. (25) in the overlap region,we find

B

A¼ �i

�d!

2

�1þ2v2 d!r2s�

�ð32 þ v2Þ�ð12 þ v2Þr�2v0 ð1þ 2v2Þ

¼ �i� � �id2!2r2s : (30)

From the above result, we expect jBj � jAj for !rs � 1.For distances !r � 1, the solution in Eq. (28) asymptoti-cally behaves as

� 1

!rffiffiffiffiffiffiffiffiffi2�d

p ½ðA� iBÞei� þ ðAþ iBÞe�i�; (31)

where � ¼ d!r� �v2=2� �=2. Thus, we know that theabsorption probability of a spherical S wave is

� ¼ 1���������A� iB

Aþ iB

��������2ffi 4�

ð1þ �Þ2 � 4d2!2r2s : (32)

We then further calculate the absorption cross section for theingoing S wave. For an incident plane wave ðzÞ ¼ eikz,which can be expanded in the partial-wave amplitudes

eikr cos� ¼ X‘

ð2‘þ 1ÞP‘ðcos�Þ eikr � e�iðkr�‘�Þ

2ikr; (33)

the absorption cross section of the S-wave component is

�abs ¼ number of particles absorbed by the area of spherical surface per unit time

number of incident particles crossing unit area per unit time¼ j 1

2kr j24�r2�1

¼ 4�r2sd2!2

k2¼ 4�r2s ;

(34)

where k ¼ d!, r is the area of the spherical surface, and4�r2s is exactly the area of the Schwarzschild horizon.The result is consistent with the low-energy cross sectionfor massless minimally coupled scalars [30]. It should bestressed that the result is independent of r. Because thephantom field has a negative kinetic term, the phantomenergy flux onto the black hole is T0r ¼ ���;t��;r; whichhas the opposite sign compared with the ordinary matterfluid. For the ultralight jm2

j, the potential term can benegligible for r < rhalo, and we can approximately take theJacobson solution �� ¼ _��1½tþ rs lnð1� rs=rÞ near thehorizon [31], where ��1 is the excitation of the phantomfield in the absence of the black hole. The excited phantomparticles carry negative energies. As they fall into the blackhole, the black hole diminishes with the rate dMBH=dt ¼4�r2T0r ¼ �4�r2sð _��1Þ2, accompanied by the accretionof the phantom energy [29,32,33]. In Refs. [32,33], the fullnonlinear absorption of a phantom field by a black hole hasbeen taken into account. Alternatively, the above calculationcan be obtained using the Eddington-Finkelstein coordinatesfor which the phantom energy flux across the absorption area(horizon) is Tvv ¼ �ð _��1Þ2, where the advance time coor-dinate v ¼ tþ rþ rs ln

r�rsrs

. Thus the decrease of the blackhole mass with the same rate, dMBH=dt ¼ 4�r2sTvv ¼�4�r2sð _��1Þ2.

The boundary condition far away from the galaxy centeris based on the assumption that the phantom field is

spatially uniform. We adopt the simple potential modelVð�Þ ¼ m2�2 with m� 10�33 eV [34] to describe theacceleration phase at the present epoch. We can match thismodel with Eq. (19), when � is very close to �1�mpl,

If the kinetic energy of the phantom field tends to besubdominant compared with its potential energy, from the

equation of motion we can have the approximation _�1 ’V�=ð3HÞ ¼ mmpl=

ffiffiffiffiffiffiffi6�

p. In this model no Big Rip occurs

and it is satisfied with w ! �1 for t ! 1. It is expectedthat the perturbations ��1 of the phantom field are of order10�5�1 [10]. Therefore, if the kinetic energy of the excitedphantom states is also expected to be much less than or eventhe same order of magnitude as that of the background,

ð _��1Þ2 �Oð _�21Þ, the decrease rate of the black hole canbe estimated to be dMBH=dt & �10�21M� yr�1, where wehave used MBH � 106M�. Therefore, the decrease of theblack hole mass is much less than a solar mass in the lifetimeof the Universe.

IV. STABILITY

In this section, we consider the classical evolution of asimple system for which a nonrelativistic scalar dark mat-ter couples to the excitation of the phantom field in agalaxy. Note that to avoid the quantum instability of thevacuum at high frequencies, the phantom dark energy fielddefined in an effective theory is valid at low energy [10,26].

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The stringent limit for the momentum cutoff, which wasestimated from the diffuse gamma ray background, is lessthan 3 MeV [26]. As for the phantom field defined in aneffective theory at low energies, we will exhibit that thissystem can be quasistable for a weak coupling. It should benoted that even for low frequencies, the negative energy ofphantom particles may cause the system to become un-stable since the positive energy dark matter particles couldincrease the energy to any level as long as the phantomparticles decrease the same magnitude of the energy. Asimilar system with the phantom excitation coupled to themassless graviton had been considered in Ref. [10].

The relevant Lagrangian L is

L ¼ � 1

2g��@��@��� 1

2m2

��2 þ 1

2g��@���@���

þ 1

2m2

��2 þ 1

2��2��2; (35)

where � is the nonrelativisitic dark matter field and �� isthe excitation of the phantom field as in the previous study.The spatial variations of them are relatively small in thisconsideration because � is nonrelativisitc and the accre-tion rate of the phantom excitations is typically small. Thusthe energy densities are

��’1

2_�2þ1

2m2

��2; ���’�1

2_��2�1

2m2

��2; (36)

and the interaction term ���� ¼ �ð�=2Þ�2��2. Notethat here ��2 summarize only for low-energy modes.The equations of motion of these two kinds of particles are

��00 ’�½1� �� ���2 ��; ���00 ’��m2

m2�

þ �� ��2

����; (37)

where we have used the dimensionless variables

�� ¼ �=M;

��� ¼ ��=M;

�� ¼ �ðM=m�Þ2; ¼ m�t;

(38)

where �� ’ v2m2pl=4�r

2, and in the dark-matter dominant

region of the galaxy, we have adopted the approximation forthe metric function f ’ 1. The perturbations, �� and�, areexpected to be of orderM� 10�5mpl [10]. Note that in this

section a prime denotes the differentiation with respect to .We plot the evolutions of densities in Fig. 2, where we have

adopted the initial conditions, ��0 ¼ ���0 ¼ 0 and �� ¼��� ¼ 1 at ¼ 0. Although we have used m ¼

10�33 eV and m� ¼ 1011 eV as inputs, we note that evo-lution curves do not obviously change if the conditionm � m� is satisfied. We obtain that for 0 � �� � 0:37,

the densities oscillate with a stable behavior, where ��� �0. However, for �� < 0, �� and ���� grow quickly. Inshort, we could conclude that the stability for the galacticcold dark matter existing in the phantom field backgroundis possible.

V. SUMMARY

We have studied the possibility that the galactic darkmatter exists in a scenario where the phantom field, respon-sible for the dark energy, may not be spatially homogeneousin a galaxy. We have obtained the statically and sphericallyapproximate solution for this kind of galaxy system with asupermassive black hole at its center. The static exact solu-tion of the metric functions is satisfied with gtt ¼ �g�1

rr , ofwhich the relation is also consistent with the black holesolutions in the vacuum, electromagnetic, and cosmologicalconstant sources, corresponding to Schwarzschild, Reissner-Nordstrom, and Schwarzschild—de Sitter/antide Sittermetrics, respectively.Constrained by the observation that the rotational stars in a

spiral galaxy are moving in circular orbits with nearly con-stant tangential speed, we have obtained that the background

0.2

0 10 20 30 40 500.5

0

0.5

1.

1.5

2.

0.1

0 10 20 30 40 5060

40

20

0

20

40

60

FIG. 2 (color online). Evolution of the energy densities of a coupled pair of the dark matter denoted by the solid curve and phantomexcitation denoted by the dashed curve. The initial condition is ��0 ¼ ���0 ¼ 0 and �� ¼ ��� ¼ 1 starting at ¼ 0. The energydensities are displaced in units of m2

�M2=2 and satisfied with the constraint �� þ ��� þ ���� ¼ constant.

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of the phantom field which is spatially inhomogeneous has(i) the small density�ph ’ 0with negligible fluctuation if the

coupling between the excitation phantom field and darkmatter is small, and (ii) an exponential potential.

To avoid the well-known quantum instability of the vac-uum at high frequencies, the phantom field that we considerhere is an effective theory valid at low energies. Under thiscondition, we have computed the absorption cross section ofthe S-wave excitations, arising from the phantom back-ground, into the central black hole and shown that it is equalto 4�r2s , the horizontal area of the central black hole.Because the infalling phantom particles have a total negativeenergy, we estimate that the black hole mass thus diminishesat the quite small rate dMBH=dt & �10�21M� yr�1, so that

the decrease of the black hole mass is much less than a solarmass in the lifetime of the Universe. Furthermore, using asimple model with the cold dark matter very weakly coupledto the ‘‘low-frequency’’ phantom particles which are gen-erated from the background, we have shown that the darkmatter and phantom densities can be quasistable.

ACKNOWLEDGMENTS

We are grateful to Dr. Ho-Chin Tsai for useful discus-sions. This research was supported in part by the NationalCenter for Theoretical Sciences and the National ScienceCouncil of the Republic of China under Grant No. NSC99-2112-M-033-005-MY3.

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