Boson Stars

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Boson Stars

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Title: Boson Stars

1
Boson Stars

??? ??

2
Boson Stars

??? ??
Department of Astronomy, USTC
2011-6-2

3
Abstract

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

4
Contents

Introduction
General relativistic boson stars
Boson stars in alternative theories of gravity
How to detect boson star
Our idea boson stars in gravitation theories
with torsion a kind of quantum boson star.

5
1. Introduction

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

(2) Boson stars
If the scalar fields exist in nature, it is
possible that they form gravitationally bound
state which is called the boson star via a Jeans
instability.
compact stars

(3)Why does one study the scalar compact
objects seriously?
Scalar fields play an important role in theories
of fundamental forces
Extensive use of scalar fields is made in order
to model the physics of the early Universe, from
phase transitions to the formation of large scale
structure
A potential dark matter candidate (nontopological
solitons or boson stars)
Ideal laboratories to study the role of gravity
in the physics of compact objects (need not
equation of state)
As a kind of source of gravitational waves
(a cold boson star) As a self-gravitating
Bose-Einstein condensate on astrophysical scale

In order to classify boson stars, lets see------
(4) General physical systems include
Basic constituents bosonic field (scalar),
fermionic field (spinor), bosonic fluid ,
fermionic fluid (neutron, proton), classical
particles, gauge field (photon, Yang-Mills
fields), gravitational fields and so on
How to interact (couple) One describe the
interaction by using Lagrangian (action, equation
of motion) constructed from some physical
motivation
Three conditions initial, boundary, joint
conditions
Physical situation
Results One solves the mathematical model and
evaluate some interesting physical quantities
Observables One compares these physical
quantities with corresponding observables

So one gets------------
(5) All kinds of boson stars (BS)
General relativistic BS non-rotating mini-BS
(with a free massive scalar field), BS with a
self-interacting massive scalar field, charged
BS, BS with Yang-Mills fields, non-topological
soliton stars, oscillating BS, boson-boson BS,
boson-fermion BS, Q-stars, charged q-stars,
dilaton star, axion star, BS with non-minimal
energy-momentum tensor, charged D star, hot BS,
the Universe (a large BS) rotating
rotating BS, rotating charged BS, BS in a
gravitation theory with dilaton dynamical
evolution (radial) non-radial perturbed BS
with self-interacting massive scalar fields
BS in alternative theories of gravity
non-rotating BS in Newtonian gravity, BS in
general scalar-tensor gravitation, BS in massive
dilatonic gravity, BS in Brans-Dicke theory
dynamical evolution (radial)BS in BD theory
BS in gravitation theories with torsion to be
studied

10
2. General relativistic BS

------non-rotating
(1). Mini-BS (D. J. Kaup, Phys. Rev. 172 (1968)
1331 R. Ruffini, S. Bonazzola, Phys. Rev. 187
(1969) 1767)
Constituents ,complex scalar field
Action (motivation stardard,free massive scalar
field)
Boundary conditions
Physical situation
where is the frequency

Results critical mass
( if an effective radius of
)
where
is the Planck mass
Particle number
Mass,- (?) - radius(?)F1
Observablesto compare with observables
solar mass for 1 Gev boson,
fm, therefore, it seems that the
mass is too small to be a dark matter candidate.

(2). BS with self-interacting massive scalar
fields (M. Colpi, S. L. Shapiro, I. Wasserman,
Phys. Rev. Lett. 576 (1986) 2485)
Constituents , complex scalar field
Action
(motivation simple quartic coupling)
Boundary conditions
Physical situation
Results relation(?t11),F2 the
fractional anisotropy as a function of the radial
coordinate
Observables to compare with observables
solar mass. For and
,
solar mass of MACHOs

(3). Charged BS (BS with Yang-Mills fields)
Constituents , complex scalar field
gauge field
Action (motivation
gauge field theory)
with
for charged BS or

With
where is the structure constants of the
Lie algebra of G (SU(2)) ,for BS with Yang-Mills
fields.
The following is only for charged BS------
Boundary conditions
Physical situation
(One has only electric field and no magnetic
one )

Results the mass
(?t1)
the particle number
(?t2) F3
the maximal mass has the following
asymptotic behavior
for
for , when is close to the
critical charge, where

(4). Non-topological soliton stars
Constituents , complex scalar field
real scalar field
Actions
(motivation Theory has non-topological
soliton solution in the absent of the
gravitational field
If the potential is renormalizable,
If renormalizability is no longer required,

Boundary condition
(a). In the interior of the soliton star,
is in the false vacuum and approximately
Outside is essentially in the
normal vacuum state
(b). If ,
(interior region)
(exterior region).
The metric is asymptotically flat.

Physical situation
Results the critical mass (estimated)

(5). BS with non-minimal energy-momentum tensor
Constituents , complex scalar field
Action
(motivation a non-minimal coupling term can
arise naturally in the effective Lagrangian in
the process of dimensional reduction, when
considering Kaluza-Klein, supergravity or
superstring theories in high dimensions.)
Boundary conditions

Physical situation
Results Similar to the previous cases, the mass
of the star as a function of which
is related to the central density, increases,
reaches a maximum value then drop a little and
oscillates to reach an asymptotic
value. For large ,
Similarly,
For any
there will be a critical value for the center
density beyond which gravitational collapse is
unavoidable.
generalization one can include a more
general potential as well as extend it to a
charge scalar field. (motivation standard
self-interaction, gauge field theory)

------rotating
(6). Rotating BS (F. E. Schunck, E. W. Mielke,
Phys. Lett. A 249 (1998) 389-394 also see S.
Yoshida, Y. Eriguchi Phys. Rev. D, 56 (1997) 762)
Constituents , complex scalar field
Action
Boundary conditions
asymptotically flat spacetime
Physical situation

Results Letting
one has
where
Observables If sub-millisecond pulsars would be
detected, todays realistic EOS for neutron
stars had to be subjected to a major revision.
Central cores built from strange matter or even
fundamental bosons would possibly be need for
denser stars during an even faster rotation. The
core of a pulsar resemble a rotating BS whose
physical quantities could be connected to
observables

------dynamical evolution (perturbed radially
metric)
(7). BS with a quartic self-interacting massive
scalar field
Constituents , complex scalar field
Action
(motivation a standard quartic
self-interaction term)
Boundary conditions Regularity conditions
require that
and have

24
.

vanishing first spatial derivatives at
where . The boundary condition
on the scalar field is an outing scalar wave
condition
Physical situation
Results
(a). Ground state
S a new S-branch configuration
U BH (mass) a new equilibrium of
a lower mass (-mass).
(b). Excited state both U (different
instability time scales)
If they cannot lose enough mass to go to
the ground, they become BH or totally disperse.
(c). Its oscillation frequency (?)F4

25
3. BS in alternative theories of gravity

(8). Rotating BS in Newtonian gravity (V.
Silveira, C. M. G de Sousa, Phys. Rev. D 52
(1995) 5724)(The central density or the total
mass of the star is not higher than a certain
critical limit)
If special relativistic effects are not
important, the only relevant component of
is
Constituents , complex scalar field
Action
(motivation standard free massive scalar
field)
Boundary conditions
The metric must satisfy the asymptotically flat.

Physical situation
Results One presents the numerical results for
(the ground state)
and for
(the excited states)
where the states are identified by the values
of
(?Fig 1t3, Fig 2t4,, Fig 3t5) F5
Note come from

(9). Charged scalar-tensor BS (A. W. Whinnett, D.
F. Torres, Phys. Rev. D 60 (1999) 104050)
Constituents , a scalar field ,a
complex scalar field , gauge field
Action
the matter Lagrangian
where
(motivation It is the low energy limit of
string theory
and the scalar gravitational field arises from
dimensional reduction of higher dimensional
theory. Sever al model of inflation are driven by
the same scalar field of ST gravity.
is based on gauge field , quartic

Boundary conditions
(The solutions are regular at the origin.)
Physical situation
(To prove this, one minimizes the total energy
of the BS subject to the constraint that particle
number be conserved)
(There are only electric charges and no
magnetic ones)

Results
denote the tensor, Newtonian, Keplerian mass
respectively. ?
(a). BD theory(t6)F6
(b). BD theory((t9)
(c). BD theory(t8)
(d). The power law ST theory(t7)F7
(e). BD theory(t10)F8
The effect of introducing a quartic term (a).
Increases the mass of the BS but dont the value
of
(GR) (b). Not only increases the mass but also
slightly the charge limit of the strong field
solutions (ST theory).

(10). Rotating charged BS (the slow rotation) (Y.
Kobayashi, M. Kasai, T. Futamase, Phys. Rev. D 50
(1994) 7721)
where
Constituents , a complex scalar field
, gauge field
Action
where
(motivation gauge field theory, simple
quartic coupling)
Boundary conditions
for large ,

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

------dynamical evolution
(11). BS in Brans-Dicke theory (J. Balakrishna,
H. Shinkai, Phys. Rev. D 58 (1998) 044016)
Constituents the gravitational (real
and massless) scalar field (the BD field)
the bosonic matter (complex and massive)
scalar field
Action
where

(motivation a quartic self-action, the
experimental test in the solar system
the low energy limit of string theory. The
so-called extended inflation models based on BD
gravity explain the completion of the phase
transition in a more natural manner, which
requires no fine-tuning)
Boundary condition Regularity dictates that the
radial metic be equal to 1 at the origin. The
boson field and the BD field both specified at
the origin. The boson field goes to zero at
and the BD field goes to a constant. The
derivatives of all the metrics vanish at the
point .
for the asymptotic region.

For the boson , an asymptotic solution of
the form to order is
assumed.
Note
where is the effective gravitational
constant in the Einstein frame.
Physical situation
Results One starts the evolutions of an S-branch
equilibrium state with a tiny perturbation. One
finds that the system begins oscillating with a
specific fundamental frequency, called a
quasinormal mode (QNM) frequency.
for
see fig.4 (?)F9

(result) under large perturbations,the maximum
radial metric
and the central BD field as a function of
time are shown in Fig. 6 and 7 respectively for
As in GR, we have also seen
migrations of the stars to the stable branch when
one removes enough scalar field from some region
of the star
Contrary to the above example, if we add a
small mass to U-branch stars , we can see the
formation of a BH in its evolution F10
As in GR , exited states of BS in general are
not stable. They form BH if they cannot lose
enough to go to the ground state F11

(12). BS in Brans-Dicke theory.(D.F. Torres,
Phys. Rev. D. vol 57 No 8 (1998)4821 M.A.
Gunderson, L.G. Jensen, Phy. Rev. D Vol 48 No
12(1993)5628)
Constituents ,BD field ,a complex
,massive, self-interacting scalar field
Action
(motivation a self-consistent framework for
a study of a varying gravitational strength. The
low energy limits of string theory, a quaritic
self-interaction)

Boundary condition non-singularity, finite mass
Physical situation
Results one studies two kinds of theory (1) JBD
theory with (2)A scalar-tensor
theory with a coupling function of the form
Boson star mass
Fig1(?)F12 note

Fig2(there is no change in the stability
criterion for values of G close to the present
one) Note
In addition, one also gets two hypothesis a.
gravitational memory (Stars of the same mass may
differ in other physical properties (R) depending
on the formation time.) b. quasistatic evolution
(purely gravitation evolution).
Compare with observable
since as in GR, these solution have a mass
on the order of Chandrasekhar mass , one has a
significant contribution to the existence of
dark matter in a universe in which the force of
gravity is determined by BD field.

39
4. How to detect boson stars

A large physical system BS its environment
System 1 a complex scalar field BS baryonic
matter (photons)
Constituent , a complex scalar field ,
baryonic matter (photons)
Action only gravitational action (motivation
simple).
Boundary condition its exterior solution is
asymptotically Schwarzschild, regularity at the
origin.
Physical situation the spherically symmetric BS
metric, baryonic matter disc
Results an accretion disc forms outside the BS,
nearly beyond its effective radius the BS look
similar to an AGN, giving a non-singular solution
where emission can occur even from the center.

(result) a. Rotation curves geodesics of a
collisionless circular orbit obey
Rotation curves for the cases
were calculated for a critical mass BS (
Schunck F E and Liddle A R Phys. Lett. B 404,25
(1997)). Baryonic matter rotating with maximal
velocity of about c/3 possesses an impressive
kinetic energy of up to 6 of its rest mass. If
one supposes that each year a mass of 1 solar
mass transfers
this amount of kinetic energy into
radiation, a BS would have a liminosity of

(result) b. Gravitational red shift
where emitter and receiver are located at
and ,respectively.
With increasing self-interaction coupling
constant also the maximal redshift
grows. In general, observed redshift values would
consist of combination of cosmological and
gravitational redshift

System2 a transparent spherical symmetric
min-BS photon( Gravitational lensing, Cerenkov
radiation)
Constituent , a complex
scalar field, photon
Action the complex scalar has no interaction
except gravity
(motivation standard free massive scalar
field , simple)
Boundary condition

Physical situation The BS interior is empty of
baryonic matter, so the deflected photons can
travel freely through the BS.
Result a. Gravitational lensing the deflection
angle is then given by
where b is the impact parameter, denotes
the closest distance between a light ray and the
center of the BS.
The lens equation for small deflection angles

(result) ,are the distances form
the lens to the source and from the observer to
the source, respectively.
For large deflection angle, the lens equation
b. Cerenkov radiation from BSs the gravitational
BS field can act as a medium with an refractive
index.
If ,
Cenenov radiation(CR) is allowed.
It was found the stable mini-BSs and stable BSs
can generate CR, whereas unstable BSs can.

System 3 BS and a compact object of a solar
mass.
Constituents , a compact object of
a solar mass
Action only gravitational interaction between
the scalar field and the compact star
Boundary condition all kinds of BSs have
different boundary condition (??)
Physical situation The compact object is
observed to be spiraling into central one with
much larger mass (BS).
Interesting physical quantities (result) the
gravitational wave.

Compare with observables
From the emitted gravitational waves
(observable), the values of the lowest few
multipole moments of the central object can be
extracted, such as mass M, angular momentum J,
mass quadrupole moment , the spin
octopole moments
For example,

System 4 axionic BSs white dwarf (or neutron
stars)
Constituent metric, axion field a ,oscillating
electric fields, magnetic fields, matter of white
dwarfs or neutron stars
Action (Equations of motion) , In addition, the
axion couples with electromagnetic fields in the
following way.
with , is the
decay constant of the axion.
(motivation J.E.Kim , Phys. Rep, 150,1(1987)
Boundary condition regularity of
the spacetime .

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

49
5. Our idea Boson Stars in gravitation theories
with torsion a kind of quantum boson star

(1). Why does one introduce torsion.
GR is a good theory but nonrenormalizable,
nonunitary
Torsion theory have something to do with modern
string theory, and avoid the gravitational
singularities.
A propagating torsion model (Saas model) is
derived from the requirement of compatibility
between minimal action and minimal coupling
procedure in Riemann-Cartan (RC) spacetime

(2). RC manifolds Torsion tensor
The metric compatible connection can be
written as
where is traceless part and
is the trace of the torsion tensor,
The curvature tensor

After some algebraic manipulations, one get the
following expression for the scalar of curvature
R
where is the Riemannian scalar of
curvature, calculated from the Christoffel
symbols . In Saas model,

(3). All kinds of BSs in theories with torsion.
Constituents torsion
scalar field , gauge field ,
spinor field
fermionic field
classical particles
Action
a. Vacuum

(action)
b. only scalar field
c. only gauge field
d. only spinor field

(action)
e. gauge field and classical particle
f. fermion field and classical particle
g. gauge field and fermion field
(???,???,
????,vol.45, No.2,2004 (141))

Boundary conditions (??)
Physical situation(??)
Result We will study the physical quantities as
a function of torsion (sensitive ?)

(4)Our plan of studying a kind of quantum BS
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