2015-11-181 boson stars 陈次星王通. 2015-11-182 boson stars 陈次星王通 department of...

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Boson StarsBoson Stars

陈次星王通陈次星王通

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Boson StarsBoson Stars

陈次星王通陈次星王通 Department of Astronomy, USTCDepartment of Astronomy, USTC2011-6-22011-6-2

23/4/2023/4/20 33

AbstractAbstract

The reports of all kinds of boson stars and The reports of all kinds of boson stars and their interaction with environments are their interaction with environments are given . In particular, the BSs in given . In particular, the BSs in gravitation theories with torsion are put gravitation theories with torsion are put forward. I also give my plan of studying a forward. I also give my plan of studying a kind of quantum boson star. kind of quantum boson star.

23/4/2023/4/20 44

ContentsContents

1.1. IntroductionIntroduction

2.2. General relativistic boson starsGeneral relativistic boson stars

3.3. Boson stars in alternative theories of Boson stars in alternative theories of gravitygravity

4.4. How to detect boson starHow to detect boson star

5.5. Our idea: boson stars in gravitation Our idea: boson stars in gravitation theories with torsion; a kind of quantum theories with torsion; a kind of quantum boson star.boson star.

23/4/2023/4/20 55

1. Introduction1. Introduction

(1)Do fundamental scalar fields exist in nature?(1)Do fundamental scalar fields exist in nature?• Higgs particle (the possible discovery)Higgs particle (the possible discovery)• Charged pions (complex scalar fields)Charged pions (complex scalar fields)• Neutral (a complex KG field)Neutral (a complex KG field)• In string theories (possible existence)In string theories (possible existence)• As sources of dark matterAs sources of dark matter• The possible role in primordial phase transitionsThe possible role in primordial phase transitions• AxionAxion• InflatonInflaton

23/4/2023/4/20 1. Introduction1. Introduction 66

(2) Boson stars:(2) Boson stars:

• If the scalar fields exist in nature, it is If the scalar fields exist in nature, it is possible that they form gravitationally bound possible that they form gravitationally bound state which is called the boson star via a state which is called the boson star via a Jeans instability. Jeans instability.

• compact starscompact stars


(3)Why does one study the “scalar compact objects” seriously?(3)Why does one study the “scalar compact objects” seriously?• Scalar fields play an important role in theories of Scalar fields play an important role in theories of

fundamental forcesfundamental forces• Extensive use of scalar fields is made in order to model the Extensive use of scalar fields is made in order to model the

physics of the early Universe, from phase transitions to the physics of the early Universe, from phase transitions to the formation of large scale structureformation of large scale structure

• A potential dark matter candidate (nontopological solitons or A potential dark matter candidate (nontopological solitons or boson stars)boson stars)

• Ideal laboratories to study the role of gravity in the physics of Ideal laboratories to study the role of gravity in the physics of compact objects (need not equation of state)compact objects (need not equation of state)

• As a kind of source of gravitational wavesAs a kind of source of gravitational waves• (a cold boson star) As a self-gravitating Bose-Einstein (a cold boson star) As a self-gravitating Bose-Einstein

condensate on astrophysical scalecondensate on astrophysical scale


In order to classify boson stars, let’s see------In order to classify boson stars, let’s see------(4) General physical systems include:(4) General physical systems include:• Basic constituents: Basic constituents: bosonic field (scalar), fermionic field (spinor), bosonic field (scalar), fermionic field (spinor),

bosonic fluid , fermionic fluid (neutron, proton), classical particles, bosonic fluid , fermionic fluid (neutron, proton), classical particles, gauge field (photon, Yang-Mills fields), gravitational fields and so ongauge field (photon, Yang-Mills fields), gravitational fields and so on

• How to interact (couple): How to interact (couple): One describe the interaction by using One describe the interaction by using Lagrangian (action, equation of motion) constructed from some Lagrangian (action, equation of motion) constructed from some physical motivationphysical motivation

• Three conditions: Three conditions: initial, boundary, joint conditionsinitial, boundary, joint conditions• Physical situationPhysical situation• Results: Results: One solves the mathematical model and evaluate some One solves the mathematical model and evaluate some

interesting physical quantitiesinteresting physical quantities• Observables: Observables: One compares these physical quantities with One compares these physical quantities with

corresponding observables corresponding observables


So one gets------------So one gets------------(5) All kinds of boson stars (BS)(5) All kinds of boson stars (BS)• General relativistic BS: General relativistic BS: non-rotatingnon-rotating: : mini-BS (with a free massive mini-BS (with a free massive

scalar field), BS with a self-interacting massive scalar field, charged scalar field), BS with a self-interacting massive scalar field, charged BS, BS with Yang-Mills fields, non-topological soliton stars, oscillating BS, BS with Yang-Mills fields, non-topological soliton stars, oscillating BS, boson-boson BS, boson-fermion BS, Q-stars, charged q-stars, BS, boson-boson BS, boson-fermion BS, Q-stars, charged q-stars, dilaton star, axion star, BS with non-minimal energy-momentum tensor, dilaton star, axion star, BS with non-minimal energy-momentum tensor, charged D star, hot BS, the Universe (a large BS); charged D star, hot BS, the Universe (a large BS); rotatingrotating::

rotating BS, rotating charged BS, BS in a gravitation theory with rotating BS, rotating charged BS, BS in a gravitation theory with dilatondilaton; ; dynamical evolution (radial) & non-radial perturbeddynamical evolution (radial) & non-radial perturbed: : BS with BS with self-interacting massive scalar fieldsself-interacting massive scalar fields

• BS in alternative theories of gravity:BS in alternative theories of gravity: non-rotating non-rotating: : BS in Newtonian BS in Newtonian gravity, BS in general scalar-tensor gravitation, BS in massive dilatonic gravity, BS in general scalar-tensor gravitation, BS in massive dilatonic gravity, BS in Brans-Dicke theorygravity, BS in Brans-Dicke theory; ; dynamical evolution (radial):dynamical evolution (radial):BS in BS in BD theoryBD theory

• BS in gravitation theories with torsion: BS in gravitation theories with torsion: to be studiedto be studied

23/4/2023/4/20 1010

2. General relativistic BS2. General relativistic BS

------non-rotating------non-rotating: : (1). Mini-BS (1). Mini-BS （（ D. J. Kaup, Phys. D. J. Kaup, Phys.

Rev. 172 (1968) 1331; R. Rev. 172 (1968) 1331; R. Ruffini, S. Bonazzola, Phys. Ruffini, S. Bonazzola, Phys. Rev. 187 (1969) 1767Rev. 187 (1969) 1767))

• Constituents: ,Constituents: ,complex scalar complex scalar field field

• Action: (Action: (motivation: motivation: stardard,free massive scalar stardard,free massive scalar fieldfield))

• Boundary conditions: Boundary conditions: • Physical situation: Physical situation: where is the frequencywhere is the frequency

)sin( 22222)(2)(2 ddrdredteds rr

g

)16

( 24 MgG

Rgxds

0)(,0)(,0)0(,0)0( rr

tierPtr )(),(

23/4/2023/4/202.General relativistic BS (1)mini-2.General relativistic BS (1)mini-

BSBS 1111

• Results: Results: critical mass critical mass ( if an effective radius of ( if an effective radius of

) ) where where is the Planck mass;is the Planck mass; Particle number Particle number Mass,- (Mass,- ( 图）图） ; - radius; - radius

(( 图）图） F1• ObservablesObservables ：： to compare with to compare with

observables: observables: solar mass for 1 Gev boson,solar mass for 1 Gev boson, fm, therefore, it seems that fm, therefore, it seems that

the mass is too small to be a the mass is too small to be a dark matter candidate.dark matter candidate.

mMM plcrit /6366.0 2

2

247.247.2

c

GMRR sv

GcM pl /

3)/(~ mMN plcrit

)0(P

1910~ M

1~R

)(rP

23/4/2023/4/20 2. General relativistic BS2. General relativistic BS 1212

(2). BS with self-interacting massive (2). BS with self-interacting massive scalar fields (scalar fields (M. Colpi, S. L. M. Colpi, S. L. Shapiro, I. Wasserman,Shapiro, I. Wasserman,

Phys. Rev. Lett. 576 (1986) 2485Phys. Rev. Lett. 576 (1986) 2485))• Constituents: Constituents: , complex scalar , complex scalar

field field • Action: Action: (motivation: simple quartic (motivation: simple quartic

coupling)coupling)• Boundary conditions: Boundary conditions: • Physical situation: Physical situation: • Results: Results: relation(relation( 图图 t11),),F2

the fractional anisotropy as a the fractional anisotropy as a function of the radial coordinatefunction of the radial coordinate

• Observables: Observables: to compare with to compare with observables: solar observables: solar mass. For and ,mass. For and ,

solar mass of MACHO’ssolar mass of MACHO’s

g

422;;

4

416

mg

G

RgxdS

0)(,0)(,0)0(,0)0( rr

)()8()(,)( 02

1

0 rGrer ti

cM ~

rmx

22 )/(1.0 mcGevM crit 1 2/1~ cGevmp

1.0~critM


(3). Charged BS (BS with (3). Charged BS (BS with Yang-Mills fields)Yang-Mills fields)

• Constituents: , Constituents: , complex scalar field complex scalar field

gauge field gauge field • Action: Action: (motivation: (motivation: gauge field theory) gauge field theory) with with for charged BS; orfor charged BS; or

g

)(, a)3,2,1;2,1)((, kaAA k

FFmDDgGRgxdS

4

1

4)()(16/

4224

ieADAAF ,

kkaaaaaa FFmDDgGRgxdS

4

1)(

4)()(16/ 224

23/4/2023/4/202. General relativistic BS 2. General relativistic BS

(3).Charged BS(Y-M)(3).Charged BS(Y-M) 1414

With With

where is the structure where is the structure constants of the Lie constants of the Lie algebra of G (SU(2)) ,for algebra of G (SU(2)) ,for BS with Yang-Mills fields.BS with Yang-Mills fields.

The following is only for The following is only for charged BS------charged BS------

• Boundary conditions: Boundary conditions: • Physical situation:Physical situation: (One has only electric field (One has only electric field

and no magnetic one )and no magnetic one )

),(2

1 mlmllmk

kkk AAAAfAAF

)3,2,1;2,1(,)(2

kaAgi

D kbabk

aa

lmkf

.0)0(,0)(,0)(,0)0(

,0)()(,0)0(.,)0(

000'0

'00

'00

AA

const

).0,0,0,(,)(),( 00 AAertr ti

23/4/2023/4/202. General relativistic BS (3) 2. General relativistic BS (3)

Charged BS(Y-M)Charged BS(Y-M) 1515

• Results: Results: the mass the mass (( 图图 t1）） the particle numberthe particle number (( 图图 t2）） F3F3 the maximal mass has the maximal mass has

the following asymptotic the following asymptotic behaviorbehavior

for ;for ; for , when is close for , when is close

to the critical charge, to the critical charge, where where

)0(~ 0M

)0( RN ~

)0(

)/()~~(44.0 221

maxmMeeM plcrit

0

~ 8/

~)/()~~(226.0 23

max2

1

mMeeM plcrit

0~

e~

222222 8/,8/~ mMmMee plpl


(4). Non-topological soliton stars(4). Non-topological soliton stars• Constituents : Constituents : , complex , complex

scalar field scalar field real scalar fieldreal scalar field • Actions: Actions: (motivation: Theory has non-(motivation: Theory has non-

topological soliton solution in the topological soliton solution in the absent of the gravitational field;absent of the gravitational field;

If the potential is If the potential is renormalizable,renormalizable,

If renormalizability is no longer If renormalizability is no longer required, required,

g

),(

2

116/4

UggGRgxdS

2220

22 )(2

1)/1(

2

1),(

mmU

220

2 )/1(),( mU

23/4/2023/4/202. General relativistic BS (4) Non-2. General relativistic BS (4) Non-

topological soliton starstopological soliton stars 1717

• Boundary condition: Boundary condition: （（ a). In the interior of the a). In the interior of the

soliton star, is in the soliton star, is in the false vacuum and false vacuum and approximately ;approximately ;

Outside is Outside is essentially in the normal essentially in the normal vacuum state vacuum state

(b). If , (b). If , (interior region); (interior region); (exterior region). (exterior region). The metric is asymptotically The metric is asymptotically

flat.flat.

0

00 0)( r

0)( r

23/4/2023/4/202. General relativistic BS (4) Non-2. General relativistic BS (4) Non-

topological soliton starstopological soliton stars 1818

• Physical situation: Physical situation: • Results: Results: the critical the critical

mass (estimated)mass (estimated)

tiertrr )(),(),(

20

220

4 /~,/~ mMRmMM plplc


(5). BS with non-minimal energy-(5). BS with non-minimal energy-momentum tensormomentum tensor

• Constituents: Constituents: , complex , complex scalar field scalar field

• Action: Action: (motivation: a non-minimal (motivation: a non-minimal

coupling term can arise naturally coupling term can arise naturally in the effective Lagrangian in in the effective Lagrangian in the process of dimensional the process of dimensional reduction, when considering reduction, when considering Kaluza-Klein, supergravity or Kaluza-Klein, supergravity or superstring theories in high superstring theories in high dimensions.)dimensions.)

• Boundary conditions: Boundary conditions:

g

}])16/(1{[224

mgRGgxdS

0)()(,0)0(.,)0(,0)0( const

23/4/2023/4/202. General relativistic BS (5) BS 2. General relativistic BS (5) BS

(non-minimal)(non-minimal) 2020

• Physical situation: Physical situation: • Results: Results: Similar to the previous cases, Similar to the previous cases,

the mass of the star as a function of the mass of the star as a function of which is related to the central which is related to the central density, increases, reaches a maximum density, increases, reaches a maximum value then drop a little and oscillates to value then drop a little and oscillates to reach an asymptotic reach an asymptotic

value. For large ,value. For large , Similarly, Similarly, For any For any there will be a critical value for the center there will be a critical value for the center

density beyond which gravitational density beyond which gravitational collapse is unavoidable.collapse is unavoidable.

generalization:generalization: one can include a more one can include a more general potential as well as extend it to a general potential as well as extend it to a

charge scalar field. (motivation: standardcharge scalar field. (motivation: standard self-interaction, gauge field theory)self-interaction, gauge field theory)

tiertr )(),(

)0(

)(,/66.0).(,/73.0 22122

1

max mMmMM plpl

)(,/72.0).(,/88.0 2221222

1

max mMmMN plpl0


------rotating------rotating: :

(6). Rotating BS ((6). Rotating BS (F. E. Schunck, E. F. E. Schunck, E. W. Mielke, Phys. Lett. A 249 W. Mielke, Phys. Lett. A 249 (1998) 389-394; also see: S. (1998) 389-394; also see: S. Yoshida, Y. Eriguchi Phys. Rev. Yoshida, Y. Eriguchi Phys. Rev. D, 56 (1997) 762D, 56 (1997) 762))

• Constituents: , Constituents: , complex complex scalar fieldscalar field

• Action: Action: • Boundary conditions: Boundary conditions:

asymptotically flat spacetimeasymptotically flat spacetime• Physical situation: Physical situation:

)(),(),(2),( 222),(222 drdredrldtdrkdtrfds r

g

)()16/(2

;;4

UgGRgxdS 0

2,0|2,0|2,0|2,0|2,0|

lkfP

tiia eerPtr ),(),,,(

23/4/2023/4/202. General relativistic BS (6) 2. General relativistic BS (6)

Rotating BSRotating BS 2222

• Results: Results: LettingLetting one has one has

wherewhere • Observables: Observables: If sub-millisecond If sub-millisecond

pulsars would be detected, today’s pulsars would be detected, today’s “realistic” EOS for neutron stars had “realistic” EOS for neutron stars had to be subjected to a major revision. to be subjected to a major revision. Central cores built from strange Central cores built from strange matter or even fundamental bosons matter or even fundamental bosons would possibly be need for denser would possibly be need for denser stars during an even faster rotation. stars during an even faster rotation. The core of a pulsar resemble a The core of a pulsar resemble a rotating BS whose physical rotating BS whose physical quantities could be connected to quantities could be connected to observablesobservables

22)(,500 mUa

)/(675.36),/(07335.0),/(07331.0 2222 mMJmMNmMM plplpl

GM pl1:


------dynamical evolution (perturbed ------dynamical evolution (perturbed radially metric):radially metric):

(7). BS with a quartic self-(7). BS with a quartic self-interacting massive scalar fieldinteracting massive scalar field

• Constituents: , Constituents: , complex complex scalar field scalar field

• Action: Action:

(motivation: a standard quartic (motivation: a standard quartic self-interaction term)self-interaction term)

• Boundary conditions: Boundary conditions: Regularity Regularity conditions require that conditions require that

and haveand have

)sin(),(),( 222222222 ddrdrrtgdtrtNds

g

)4

1

2

1

2

1(

16

1 4244

mggxdRgxdG

S

,1)0( rg21,,, Ng

23/4/2023/4/202. General relativistic BS (7) BS 2. General relativistic BS (7) BS

(quartic self-interaction)(quartic self-interaction) 2424

..

vanishing first spatial derivatives at vanishing first spatial derivatives at where . The boundary where . The boundary

condition on the scalar field is an condition on the scalar field is an outing scalar wave conditionouting scalar wave condition

• Physical situation: Physical situation: • Results:Results: (a). Ground state (a). Ground state S a new S-branch S a new S-branch

configuration;configuration; U BH (+mass); a new U BH (+mass); a new

equilibrium of a lower mass (-mass). equilibrium of a lower mass (-mass). (b). Excited state [both U (different (b). Excited state [both U (different

instability time scales)]instability time scales)] If they cannot lose enough mass to If they cannot lose enough mass to

go to the ground, they become BH go to the ground, they become BH or totally disperse.or totally disperse.

(c). Its oscillation frequency (c). Its oscillation frequency (( 图）图） F4F4

0r G 4

),( tr

23/4/2023/4/20 2525

3. BS in alternative theories of 3. BS in alternative theories of gravitygravity

(8). Rotating BS in Newtonian gravity (8). Rotating BS in Newtonian gravity ((V. Silveira, C. M. G de Sousa, V. Silveira, C. M. G de Sousa, Phys. Rev. D 52 (1995) 5724Phys. Rev. D 52 (1995) 5724)(The )(The central density or the total mass of central density or the total mass of the star is not higher than a certain the star is not higher than a certain critical limit)critical limit)

If special relativistic effects are not If special relativistic effects are not important, the only relevant important, the only relevant component of is component of is

• Constituents: , Constituents: , complex scalar complex scalar field field

• Action: Action: (motivation: standard free massive (motivation: standard free massive

scalar field)scalar field)• Boundary conditions:Boundary conditions: The metric must satisfy the The metric must satisfy the

asymptotically flat. asymptotically flat.

g

24

16Mg

RgxdS

]sin,,1,1[

,1,,

222

2

rrdiag

hhgdxdxgds

h ),(200 rVh

0|),2,,,(),,,( rrtrt

23/4/2023/4/203. BS( alternative, gravity) (8) 3. BS( alternative, gravity) (8)

Rotating BS (Newtonian)Rotating BS (Newtonian) 2626

• Physical situation: Physical situation: • Results: Results: One presents the One presents the

numerical results for numerical results for (the ground state)(the ground state) and forand for (the excited states) (the excited states) where the states are identified where the states are identified

by the values ofby the values of (( 图图 Fig 1t3, Fig 2t4,, Fig 3t5) Fig 1t3, Fig 2t4,, Fig 3t5)

F5F5 Note: come fromNote: come from

1),(,),(),( rhheetrtr imti

0,0 ml

0,2;1,1;0,1 mlmlml

ml,

)(),ˆ(ˆ rVrR l

dddrrMNml

mllNGMN

rRGNrRrVNrV

MrNrPrVrVPrRr ll

lml

mll

sin2,)!(

)!)(12(ˆ

),ˆ(ˆ)2(ˆ)(),,ˆ(ˆˆ),(

,ˆˆ),()(),(),()(4

1),(

222

2122

0

23/4/2023/4/20 3. BS (alternative, gravity) 3. BS (alternative, gravity) 2727

(9). Charged scalar-tensor BS ((9). Charged scalar-tensor BS (A. W. A. W. Whinnett, D. F. Torres, Phys. Rev. Whinnett, D. F. Torres, Phys. Rev. D 60 (1999) 104050D 60 (1999) 104050))

• Constituents: Constituents: , a scalar , a scalar field ,a complex scalar field , field ,a complex scalar field , gauge field gauge field

• Action: Action: the matter Lagrangianthe matter Lagrangian wherewhere (motivation: It is the low energy limit of (motivation: It is the low energy limit of

string theory;string theory; and the scalar gravitational field arises and the scalar gravitational field arises

from dimensional reduction of from dimensional reduction of higher dimensional theory. Sever al higher dimensional theory. Sever al model of inflation are driven by the model of inflation are driven by the same scalar field of ST gravity. same scalar field of ST gravity.

is based on gauge field , quarticis based on gauge field , quartic

g

A

mLgRgxdS

16

)(

16

1,,

4

FFmDDgLm 16

1)(

4

1

2

1))((

2

1 22

ieADAAF ,

mL

23/4/2023/4/203. BS (alternative, gravity) (9) 3. BS (alternative, gravity) (9)

Charged ST BSCharged ST BS 2828

• Boundary conditions:Boundary conditions: (The solutions are regular (The solutions are regular

at the origin.)at the origin.)• Physical situation: Physical situation: (To prove this, one (To prove this, one

minimizes the total energy minimizes the total energy of the BS subject to the of the BS subject to the constraint that particle constraint that particle number be conserved) number be conserved)

(There are only electric (There are only electric charges and no magnetic charges and no magnetic ones) ones)

0)0()0()0(,1)0( ''' CA 0,

1~,

1,

1~,

11~

nnnn xCC

xxBB

xA

1,,,2

0 nxMM

AC

plpl

tiertr )(),(

)0,0,0),(( 0 rAA

23/4/2023/4/203. BS (alternative, gravity) (9) 3. BS (alternative, gravity) (9)

Charged ST BSCharged ST BS 2929

• Results: Results: denote the tensor, Newtonian, denote the tensor, Newtonian,

Keplerian mass respectively. Keplerian mass respectively. 图图 (a). BD theory(t6)F6(a). BD theory(t6)F6 (b). BD theory((t9)(b). BD theory((t9) (c). BD theory(t8)(c). BD theory(t8) (d). The power law ST (d). The power law ST

theory(t7)F7theory(t7)F7 (e). BD theory(t10)F8(e). BD theory(t10)F8The effect of introducing a quartic The effect of introducing a quartic

term (a). Increases the mass of term (a). Increases the mass of the BS but don’t the value of the BS but don’t the value of

(GR); (b). Not only increases the (GR); (b). Not only increases the mass but also slightly the mass but also slightly the charge limit of the strong field charge limit of the strong field solutions (ST theory).solutions (ST theory).

KNT MMM ,,

0~, NT MM

NNT

n

MMM

e

nnd

c

b

a

~

.,1,1).(

;4,3

43)(2).(

;0,1,1).(

;0,1,1).(

;0,1,500).(

maxq

23/4/2023/4/20 3. BS (alternative, gravity)3. BS (alternative, gravity) 3030

(10). Rotating charged BS (the slow (10). Rotating charged BS (the slow rotation) (rotation) (Y. Kobayashi, M. Y. Kobayashi, M. Kasai, T. Futamase, Phys. Rev. Kasai, T. Futamase, Phys. Rev. D 50 (1994) 7721D 50 (1994) 7721) )

where where • Constituents: , Constituents: , a complex a complex

scalar field , gauge fieldscalar field , gauge field • Action:Action: where where (motivation: gauge field theory, (motivation: gauge field theory,

simple quartic coupling)simple quartic coupling) • Boundary conditions:Boundary conditions: for largefor large , ,

])(sin[ 22222)(22)(22)(22 dtddredredteds

g A

),(),,(),,( rrr

]4

1

4)()(16/[

4224

FFmDDgGRgxdS

AAFiqAD ,

0)0()0(,0)0(,)0(

.,)0(.,)0(,0)0()0(''

0

''0

''

aa

constconstC

1)(,)( sl

sl rrarr

r

r

23/4/2023/4/203.BS (alternative, gravity) (10) 3.BS (alternative, gravity) (10)

Rotating charged BSRotating charged BS 3131

• Physical situation: Physical situation: • Results: Results:

The boson star made The boson star made of a complex scalar of a complex scalar field alone does not field alone does not rotate at least rotate at least perturbatively. i.e.perturbatively. i.e.

)0,,,(),(

)sin)(,0,0,0(),(),,(),(),(

),0,0,0),((),,(),()]()()[(

,)(),,,(),(,sin

1)(),(

),()(),(),()(),(),()(),(

,

,00

000210

01

ll

lll

lll

le

ll

le

lti

lll

tibb

l

ll

ll

ll

llll

l

PkPhPfrA

PrarArArArA

rCArAAAPerirr

ertrtrd

dPrr

PrBrPrArPrNr

01

23/4/2023/4/20 3. BS (alternative, gravity)3. BS (alternative, gravity) 3232

------dynamical evolution:------dynamical evolution:

(11). BS in Brans-Dicke theory ((11). BS in Brans-Dicke theory (J. J. Balakrishna, H. Shinkai, Phys. Balakrishna, H. Shinkai, Phys. Rev. D 58 (1998) 044016Rev. D 58 (1998) 044016) )

• Constituents: Constituents: the the gravitational (real and massless) gravitational (real and massless) scalar field (the BD field) scalar field (the BD field)

the bosonic matter (complex the bosonic matter (complex and massive) scalar fieldand massive) scalar field

• Action: Action: wherewhere

)sin(),(),( 222222222 ddrdrrtgdtrtNds

g~

)](~2

1[~]~)(

~[~

16

1 414

VggxdgRgxdS

500.)(,)(42

)( 22

constm

V BD

23/4/2023/4/203. BS (alternative,gravity) (11) BS 3. BS (alternative,gravity) (11) BS

in BD theoryin BD theory 3333

(motivation: a quartic self-action, the (motivation: a quartic self-action, the experimental test in the solar experimental test in the solar system system

the low energy limit of string theory. the low energy limit of string theory. The so-called extended inflation The so-called extended inflation models based on BD gravity explain models based on BD gravity explain the completion of the phase the completion of the phase transition in a more natural manner, transition in a more natural manner, which requires no fine-tuning)which requires no fine-tuning)

• Boundary condition: Boundary condition: Regularity Regularity dictates that the radial metic be dictates that the radial metic be equal to 1 at the origin. The boson equal to 1 at the origin. The boson field and the BD field both specified field and the BD field both specified at the origin. The boson field goes at the origin. The boson field goes to zero at and the BD field goes to zero at and the BD field goes to a constant. The derivatives of all to a constant. The derivatives of all the metrics vanish at the point .the metrics vanish at the point .

for the asymptotic region.for the asymptotic region.

500BD

)(1

),(c

rtF

rtr

23/4/2023/4/203.BS (alternative, gravity) (11)BS 3.BS (alternative, gravity) (11)BS


For the boson , an asymptotic For the boson , an asymptotic solution of the form to solution of the form to order is assumed.order is assumed.

Note:Note:

where is the effective gravitational where is the effective gravitational constant in the Einstein frame.constant in the Einstein frame.

• Physical situation: Physical situation: • Results: Results: One starts the evolutions One starts the evolutions

of an S-branch equilibrium state of an S-branch equilibrium state with a tiny perturbation. One finds with a tiny perturbation. One finds that the system begins oscillating that the system begins oscillating with a specific fundamental with a specific fundamental frequency, called a quasinormal frequency, called a quasinormal mode (QNM) frequency.mode (QNM) frequency.

forfor see fig.4 (see fig.4 ( 图）图） F9F9

ree tikr /

r/1)(21,

32)(

a

BD

eGa

G

),(44),,( rteGGrt ti

trgrr ~)0(, ,600,2.0 BDc

23/4/2023/4/203. BS (alternative, gravity) (11)BS 3. BS (alternative, gravity) (11)BS


(result) (result) under large perturbations,the under large perturbations,the maximum radial metric maximum radial metric

and the central BD field as a function and the central BD field as a function of time are shown in Fig. 6 and 7 of time are shown in Fig. 6 and 7 respectively for As in respectively for As in GR, we have also seen migrations GR, we have also seen migrations of the stars to the stable branch of the stars to the stable branch when one removes enough scalar when one removes enough scalar field from some region of the star:field from some region of the star:

Contrary to the above example, if we Contrary to the above example, if we add a small mass to U-branch add a small mass to U-branch stars , we can see the formation of stars , we can see the formation of a BH in its evolution; F10a BH in its evolution; F10

As in GR , exited states of BS in As in GR , exited states of BS in general are not stable. They form general are not stable. They form BH if they cannot lose enough to go BH if they cannot lose enough to go to the ground state: F11to the ground state: F11

g

.60,600BD

)10.(~/),9.(~)0(, figtMMfigtrg initrr

);11(~00 figrg

).12(,~ figtrgrr

23/4/2023/4/203.BS( alnative, gravity) (11)BS in 3.BS( alnative, gravity) (11)BS in

BD theoryBD theory 3636

(12). BS in Brans-Dicke theory(12). BS in Brans-Dicke theory.(D.F. .(D.F. Torres, Phys. Rev. D. vol 57 No 8 Torres, Phys. Rev. D. vol 57 No 8 (1998)4821; M.A. Gunderson, (1998)4821; M.A. Gunderson, L.G. Jensen, Phy. Rev. D Vol 48 L.G. Jensen, Phy. Rev. D Vol 48 No 12(1993)5628)No 12(1993)5628)

• ConstituentsConstituents ：，：， BD BD field ,a complex ,massive, field ,a complex ,massive, self-interacting scalar fieldself-interacting scalar field

• Action: Action: (motivation: a self-consistent (motivation: a self-consistent

framework for a study of a framework for a study of a varying gravitational strength. varying gravitational strength. The low energy limits of string The low energy limits of string theory, a quaritic self-interaction)theory, a quaritic self-interaction)

.)()( 22222 drdrrAdtrBds

g

422

4

1

2

1

2

1

,16)(

16

mgL

LRg

S

m

m

)tan( tconsBD ）（

23/4/2023/4/203. BS(alnative, gravity) (11)BS in 3. BS(alnative, gravity) (11)BS in


• Boundary conditionBoundary condition ：： non-singularity, finite mass non-singularity, finite mass

• Physical situation: Physical situation: • Results: Results: one studies two one studies two

kinds of theory (1) JBD kinds of theory (1) JBD theory with (2)A theory with (2)A scalar-tensor theory with a scalar-tensor theory with a coupling function of the coupling function of the form form

Boson star mass Boson star mass Fig1(Fig1( 图）图） F12 note: F12 note:

))(2

1)((0)0(,1)(,)0(

0)0(,0)0(,32

421

1

''

x

xMxAMB

G

c

）（

]exp[)(), tirtr （

400

5,5.0,1232 1 BB

)1.0)0(,006.0)0(,100)((~ andM

.4

,)(

4,,,2

2

m

m

m

r

mmmrx pl

plpl

23/4/2023/4/203.BS(alnative, gravity)(11).BS in 3.BS(alnative, gravity)(11).BS in


Fig2(there is no change in the stability Fig2(there is no change in the stability criterion for values of G close to the criterion for values of G close to the present one) Note:present one) Note:

In addition, one also gets two In addition, one also gets two hypothesis: a. gravitational memory hypothesis: a. gravitational memory (Stars of the same mass may differ (Stars of the same mass may differ in other physical properties (R) in other physical properties (R) depending on the formation time.) b. depending on the formation time.) b. quasistatic evolution (purely quasistatic evolution (purely gravitation evolution). gravitation evolution).

• Compare with observableCompare with observable since as in GR, these solution have since as in GR, these solution have

a mass on the order of a mass on the order of Chandrasekhar mass , one has a Chandrasekhar mass , one has a significant contribution to the significant contribution to the existence of dark matter in a existence of dark matter in a universe in which the force of universe in which the force of gravity is determined by BD field.gravity is determined by BD field.

NM ~

dxxB

AN

M

m

pl

2

0

22

2

23/4/2023/4/20 3939

4. How to detect boson stars4. How to detect boson stars

A large physical system: BS + its A large physical system: BS + its environmentenvironment

System 1: a complex scalar field BS System 1: a complex scalar field BS +baryonic matter (photons)+baryonic matter (photons)

• Constituent: , Constituent: , a complex scalar a complex scalar field , baryonic matter (photons)field , baryonic matter (photons)

• Action: Action: only gravitational action only gravitational action (motivation: simple).(motivation: simple).

• Boundary condition: Boundary condition: its exterior solution its exterior solution is asymptotically Schwarzschild, is asymptotically Schwarzschild, regularity at the origin.regularity at the origin.

• Physical situation: Physical situation: the spherically the spherically symmetric BS metric, baryonic matter symmetric BS metric, baryonic matter discdisc

• Results: Results: an accretion disc forms outside an accretion disc forms outside the BS, nearly beyond its effective the BS, nearly beyond its effective radius the BS look similar to an AGN, radius the BS look similar to an AGN, giving a non-singular solution where giving a non-singular solution where emission can occur even from the emission can occur even from the center.center.

g

23/4/2023/4/20 4. How to detect BS, system 14. How to detect BS, system 1 4040

(result) (result) a. Rotation curves: geodesics a. Rotation curves: geodesics of a collisionless circular orbit obey of a collisionless circular orbit obey

Rotation curves for the cases Rotation curves for the cases were calculated for a critical mass BSwere calculated for a critical mass BS

( ( Schunck F E and Liddle A R Schunck F E and Liddle A R Phys. Lett. B 404,25 (1997)).Phys. Lett. B 404,25 (1997)). Baryonic matter rotating with Baryonic matter rotating with maximal velocity of about c/3 maximal velocity of about c/3 possesses an impressive kinetic possesses an impressive kinetic energy of up to 6% of its rest mass. energy of up to 6% of its rest mass. If one supposes that each year a If one supposes that each year a mass of 1 solar mass transfersmass of 1 solar mass transfers

this amount of kinetic energy into this amount of kinetic energy into radiation, a BS would have a radiation, a BS would have a liminosity of liminosity of

erpk

r

rM

erpk

eeerv

r

r

2

2'

2

)(2

)1(2

1

2

1

300,10,0

144 .10 serg


（（ result) result) b. Gravitational red b. Gravitational red shift shift

where emitter and receiver where emitter and receiver are located at are located at

and ,respectively. and ,respectively. With increasing self-With increasing self-

interaction coupling interaction coupling constant also the constant also the maximal redshift grows. In maximal redshift grows. In general, observed redshift general, observed redshift values would consist of values would consist of combination of combination of cosmological and cosmological and gravitational redshift :gravitational redshift :

gz)()( int/1 RR

g eez ext

intR

extR

)1)(1()1( gc zzz

23/4/2023/4/20 4. How to detect BS4. How to detect BS 4242

System2: a transparent spherical System2: a transparent spherical symmetric min-BS+ symmetric min-BS+ photon( Gravitational lensing, photon( Gravitational lensing, Cerenkov radiation)Cerenkov radiation)

• Constituent: , Constituent: , a complexa complex scalar field, photonscalar field, photon• Action: Action: the complex scalar has the complex scalar has

no interaction except gravityno interaction except gravity (motivation: standard free (motivation: standard free

massive scalar field , simple)massive scalar field , simple)• Boundary condition: Boundary condition:

g

24

16Mg

G

RgxdS

0)(,0)(,0)0(,0)0( ' rr


• Physical situation: Physical situation: The BS The BS interior is empty of baryonic interior is empty of baryonic matter, so the deflected matter, so the deflected photons can travel freely photons can travel freely through the BS. through the BS.

• Result: Result: a. Gravitational lensing : a. Gravitational lensing : the deflection angle is then the deflection angle is then given bygiven by

where b is the impact parameter, where b is the impact parameter, denotes the closest distance denotes the closest distance between a light ray and the between a light ray and the center of the BS. center of the BS.

The lens equation for small The lens equation for small deflection anglesdeflection angles

tierPtr )(),(

dr

berr

ber

r022

2/

0 2)(ˆ

0r

ˆsin)sin(os

ls

D

D


(result) (result) ,are the distances ,are the distances form the lens to the source and form the lens to the source and from the observer to the source, from the observer to the source, respectively.respectively.

For large deflection angle, the lens For large deflection angle, the lens equationequation

b. Cerenkov radiation from BSs: b. Cerenkov radiation from BSs: the gravitational BS field can act the gravitational BS field can act as a medium with an refractive as a medium with an refractive index. index.

If ,If ,Cenenov radiation(CR) is allowed.Cenenov radiation(CR) is allowed. It was found the stable mini-BSs It was found the stable mini-BSs

and stable BSs can generate and stable BSs can generate CR, whereas unstable BSs canCR, whereas unstable BSs can..

osls DD ,

,

)],ˆtan([tan)sin(arcsincoscossin

where

D

D

D

D

os

ls

os

ls

n iiRk

kn20

02 1

1)(

0)(..0 02 kneiR i

i


System 3: BS and a compact System 3: BS and a compact object of a solar mass.object of a solar mass.

• Constituents: Constituents: , a , a compact object of a solar masscompact object of a solar mass

• Action: Action: only gravitational only gravitational interaction between the scalar interaction between the scalar field and the compact starfield and the compact star

• Boundary condition: Boundary condition: all kinds of all kinds of BSs have different boundary BSs have different boundary condition condition （见前）（见前）

• Physical situation: Physical situation: The compact The compact object is observed to be object is observed to be spiraling into central one with spiraling into central one with much larger mass (BS).much larger mass (BS).

• Interesting physical quantities Interesting physical quantities (result): (result): the gravitational wave.the gravitational wave.

,g


• Compare with observables: Compare with observables: From the emitted gravitational From the emitted gravitational

waves (observable), the values waves (observable), the values of the lowest few multipole of the lowest few multipole moments of the central object moments of the central object can be extracted, such as mass can be extracted, such as mass M, angular momentum J, mass M, angular momentum J, mass quadrupole moment , the quadrupole moment , the spin octopole moments spin octopole moments

For example,For example,

2M

3S

)int,(4/,24/

)(1/,1/

)(19/,24/,01.0/

323

22

323

22

323

22

2

eractionselfBSJMSJMM

BHJMSJMM

rotatingBSJMSJMMMJ


System 4: axionic BSs + white System 4: axionic BSs + white dwarf (or neutron stars) dwarf (or neutron stars)

• Constituent: Constituent: metric, axion field metric, axion field a ,oscillating electric fields, a ,oscillating electric fields, magnetic fields, matter of white magnetic fields, matter of white dwarfs or neutron starsdwarfs or neutron stars

• Action Action (Equations of motion) , In (Equations of motion) , In addition, the axion couples with addition, the axion couples with electromagnetic fields in the electromagnetic fields in the following way.following way.

with , is the with , is the decay constant of the axion.decay constant of the axion.

(motivation(motivation: : J.E.Kim , Phys. Rep, J.E.Kim , Phys. Rep, 150,1(1987) 150,1(1987)

• Boundary condition: Boundary condition: regularity of regularity of

the spacetime .the spacetime .

),(4

),(4

,2

2

2

)(

2222'

2222'

2'''

aamaGr

hh

aamaGr

hh

amahh

ra

ahha

rr

rt

rtrt

PQ

a f

BEacL

137/1 PQf

0)0( rhr


• Physical situation: Physical situation: • Results: Results: Axion stars are Axion stars are

possible sources for generating possible sources for generating energy in magnetized energy in magnetized conducting media. Denoting the conducting media. Denoting the conductivity of the media by conductivity of the media by and assuming ohm’s law, we and assuming ohm’s law, we find that an axion star with find that an axion star with radius R dissipate an energy W radius R dissipate an energy W per unit time. In addition, in per unit time. In addition, in actual collision process, the actual collision process, the axion stars must be axion stars must be deformed ,also, strongly torn by deformed ,also, strongly torn by the tidal force of a white dwarf the tidal force of a white dwarf even without direct collisions.even without direct collisions.

,1

),sin()1()1(

,

2222222

rt

rt

h

ddrdrhdthds

.

)1(10//102.2

4//

2

2

1212

223

2320

222320

222

G

B

M

M

scm

cserg

RaBcRaBcW

sunm

m

23/4/2023/4/20 4949

5. Our idea: Boson Stars in 5. Our idea: Boson Stars in gravitation theories with torsion; a gravitation theories with torsion; a

kind of quantum boson starkind of quantum boson star(1). Why does one introduce torsion.(1). Why does one introduce torsion.• GR is a good theory but nonrenormalizable, GR is a good theory but nonrenormalizable,

nonunitarynonunitary• Torsion theory have something to do with Torsion theory have something to do with

modern string theory, and avoid the gravitational modern string theory, and avoid the gravitational singularities.singularities.

• A propagating torsion model (Saa’s model) is A propagating torsion model (Saa’s model) is derived from the requirement of compatibility derived from the requirement of compatibility between minimal action and minimal coupling between minimal action and minimal coupling procedure in Riemann-Cartan (RC) spacetimeprocedure in Riemann-Cartan (RC) spacetime

23/4/2023/4/20 5. Our idea: torsion,quantum5. Our idea: torsion,quantum 5050

(2). RC manifolds: (2). RC manifolds: Torsion Torsion tensor tensor

The metric – compatible The metric – compatible connection can be written connection can be written as as

where is traceless where is traceless part andpart and

is the trace of the torsion is the trace of the torsion tensor,tensor,

The curvature tensor The curvature tensor

)(2

1

S

).(3

2~,

,,

SgSgKKAlso

SSSKwhereK

K~

S

SS

,

R

23/4/2023/4/205. our idea (torsion,quantum) (2) 5. our idea (torsion,quantum) (2)

RC manifoldRC manifold 5151

After some algebraic After some algebraic manipulations, one get manipulations, one get the following the following expression for the expression for the scalar of curvature R:scalar of curvature R:

where is the where is the Riemannian scalar of Riemannian scalar of curvature, calculated curvature, calculated from the Christoffel from the Christoffel symbols . In Saa’s symbols . In Saa’s model,model,

KKSSSDgRRggR~~

3

164,(),(

),(

gR

.),()( 42 xdgedvolxxS

23/4/2023/4/20 5.our idea (torsion,quantum)5.our idea (torsion,quantum) 5252

(3). All kinds of BSs in (3). All kinds of BSs in theories with torsion. theories with torsion.

• Constituents: Constituents: torsiontorsion scalar field , gauge scalar field , gauge

field , spinor fieldfield , spinor field fermionic fieldfermionic field classical particlesclassical particles • Action: Action: a. Vacuuma. Vacuum: :

g ),~

,( K

A ),,( Pu

ix

termsurfKKgRgxed

RgxedSS grav

.~~

3

16),(24

24

23/4/2023/4/20 5. our idea (torsion, quantum,BS)5. our idea (torsion, quantum,BS) 5353

(action)(action)

b. only scalar field: b. only scalar field:

c. only gauge field:c. only gauge field:

d. only spinor field:d. only spinor field:

2

224

22,

mggxedSwhereSSS scalscalgrav

.4

124max

FFxedSSSS gravgarv

.],[)((8

1

,,,,)()(,

)()()((2

,

,

0

24

a

a

F

Fgrav

xexwith

mxxi

gxedSwhere

SSS

23/4/2023/4/20 5. our idea (torsion,quantum BS)5. our idea (torsion,quantum BS) 5454

(action)(action) e. gauge field and e. gauge field and

classical particleclassical particle f. fermion field and f. fermion field and

classical particle classical particle g. gauge field and g. gauge field and

fermion fieldfermion field （陈次星，张家铝，（陈次星，张家铝，天文学报，天文学报， vol.45, vol.45,

No.2,2004 (141))No.2,2004 (141))

.)((

,

)),((

16

1,

4

1,

,,

2

2

42

dt

dx

eg

txxqJand

txxdt

ds

eg

mJAL

FFLRLwhere

LLLLxdgLeS

i

ii

ii

iipf

M

emM

G

pfM

emM

G

dxdxxgemc

xdgcec

xdgRek

cS

)(

)(1

22

42242

.1

16

11

)(1

2

42

42242

xdgAJc

FFec

xdgcec

xdgRek

cS

23/4/2023/4/20 5. our idea (torsion ,quantum,BS)5. our idea (torsion ,quantum,BS) 5555

• Boundary conditions Boundary conditions （见前）（见前） • Physical situationPhysical situation （见前）（见前）• Result: Result: We will study the physical quantities We will study the physical quantities

as a function of torsion (sensitive ?)as a function of torsion (sensitive ?)

23/4/2023/4/20 5.our idea (torsion, quantum BS)5.our idea (torsion, quantum BS) 5656

(4)Our plan of studying a kind of quantum BS(4)Our plan of studying a kind of quantum BS11 由于磁场穿过大多数天体物理系统，它的存由于磁场穿过大多数天体物理系统，它的存

在有许多宇宙学和天体物理意义。确实，磁在有许多宇宙学和天体物理意义。确实，磁场大大影响大尺度和单个星系的形成过程，场大大影响大尺度和单个星系的形成过程，因此必须考虑磁场，我们选择一种含磁的拉因此必须考虑磁场，我们选择一种含磁的拉氏量来研究相应的玻色星，它可以用来早期氏量来研究相应的玻色星，它可以用来早期宇宙特性。必须注意早期宇宙有量子效应。宇宙特性。必须注意早期宇宙有量子效应。根据拉氏量，引入根据拉氏量，引入 WignerWigner 函数，给出量子函数，给出量子BoltzmannBoltzmann 方程，并导出方程，并导出 EinsteinEinstein 方程，以方程，以及及

23/4/2023/4/20 5.our idea(torsion, quantum BS)5.our idea(torsion, quantum BS) 5757

其它基本方程，它们的导出方法也可适用其它拉氏量，其它基本方程，它们的导出方法也可适用其它拉氏量，如挠率引力理论和超引力理论。我们采用的拉氏量如挠率引力理论和超引力理论。我们采用的拉氏量为较为复杂的形式（参考为较为复杂的形式（参考 K. Dimopoulos, T. K. Dimopoulos, T. Prokopec, O. Tornkvist, A. C. Davis Physical Prokopec, O. Tornkvist, A. C. Davis Physical Review vol.65(2002) 063505Review vol.65(2002) 063505 ），），其系统有三种标量其系统有三种标量场，为简单，先讨论三种标量场单独存在的情况，场，为简单，先讨论三种标量场单独存在的情况，在此基础，最后讨论它们同时存在的情形；在此基础，最后讨论它们同时存在的情形；

22 考虑弱引力情形即牛顿情形下的基本方程，它们可考虑弱引力情形即牛顿情形下的基本方程，它们可按球谐函数展开；按球谐函数展开；

23/4/2023/4/20 5. our idea(torsion, quantum BS)5. our idea(torsion, quantum BS) 5858

33 考虑史瓦西和克尔背景的扰动，研究玻色星考虑史瓦西和克尔背景的扰动，研究玻色星的引力辐射；的引力辐射；

44 以上方程组可采用以上方程组可采用 3+13+1 方法求解；方法求解；55 研究相应玻色星特性，如质量与半径的关系，研究相应玻色星特性，如质量与半径的关系，

质量与星体中心处标量数值之间的关系质量与星体中心处标量数值之间的关系 ;;

66 研究经典极限下试验粒子在玻色星中的测地研究经典极限下试验粒子在玻色星中的测地运动运动 ;;

77 我们也可研究暗能量星及相应的吸积问题我们也可研究暗能量星及相应的吸积问题 ..

23/4/2023/4/20 5959

谢谢大家！谢谢大家！

2015-11-181 boson stars 陈次星 王通. 2015-11-182 boson stars 陈次星 王通 department of...

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2015-11-181 boson stars 陈次星王通. 2015-11-182 boson stars 陈次星王通 department of...