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Boson Stars

Boson Stars. 陈次星 王通. Boson Stars. 陈次星 王通 Department of Astronomy, USTC 2011-6-2. Abstract.

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Boson Stars

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  1. Boson Stars 陈次星 王通

  2. Boson Stars • 陈次星 王通 • 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

  6. (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 1. Introduction

  7. (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 1. Introduction

  8. In order to classify boson stars, let’s 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 1. Introduction

  9. 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 1. Introduction

  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

  11. Results: critical mass ( if an effective radius of ) where is the Planck mass; Particle number Mass,- (图）; - radius(图）F1 • Observables：to 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.General relativistic BS (1)mini-BS

  12. (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(图t11),F2 the fractional anisotropy as a function of the radial coordinate • Observables: to compare with observables: solar mass. For and , solar mass of MACHO’s 2. General relativistic BS

  13. (3). Charged BS (BS with Yang-Mills fields) • Constituents: , complex scalar field gauge field • Action: (motivation: gauge field theory) with for charged BS; or 2. General relativistic BS

  14. 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 ) 2. General relativistic BS (3).Charged BS(Y-M)

  15. Results: the mass (图t1） the particle number (图t2） F3 the maximal mass has the following asymptotic behavior for ; for , when is close to the critical charge, where 2. General relativistic BS (3) Charged BS(Y-M)

  16. (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, 2. General relativistic BS

  17. 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. 2. General relativistic BS (4) Non-topological soliton stars

  18. Physical situation: • Results: the critical mass (estimated) 2. General relativistic BS (4) Non-topological soliton stars

  19. (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: 2. General relativistic BS

  20. 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) 2. General relativistic BS (5) BS (non-minimal)

  21. ------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: 2. General relativistic BS

  22. Results: Letting one has where • Observables: If sub-millisecond pulsars would be detected, today’s “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 2. General relativistic BS (6) Rotating BS

  23. ------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 2. General relativistic BS

  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 (图）F4 2. General relativistic BS (7) BS (quartic self-interaction)

  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.

  26. 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 (图Fig 1t3, Fig 2t4,, Fig 3t5) F5 Note: come from 3. BS( alternative, gravity) (8) Rotating BS (Newtonian)

  27. (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 3. BS (alternative, gravity)

  28. 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) 3. BS (alternative, gravity) (9) Charged ST BS

  29. Results: denote the tensor, Newtonian, Keplerian mass respectively. 图 (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 don’t the value of (GR); (b). Not only increases the mass but also slightly the charge limit of the strong field solutions (ST theory). 3. BS (alternative, gravity) (9) Charged ST BS

  30. (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 , 3. BS (alternative, gravity)

  31. Physical situation: • Results: The boson star made of a complex scalar field alone does not rotate at least perturbatively. i.e. 3.BS (alternative, gravity) (10) Rotating charged BS

  32. ------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 3. BS (alternative, gravity)

  33. (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. 3. BS (alternative,gravity) (11) BS in BD theory

  34. 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 (图）F9 3.BS (alternative, gravity) (11)BS in BD theory

  35. (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 3. BS (alternative, gravity) (11)BS in BD theory

  36. (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) 3.BS( alnative, gravity) (11)BS in BD theory

  37. 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(图）F12 note: 3. BS(alnative, gravity) (11)BS in BD theory

  38. 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. 3.BS(alnative, gravity)(11).BS in BD theory

  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.

  40. (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 4. How to detect BS, system 1

  41. （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 : 4. How to detect BS, system 1

  42. 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: 4. How to detect BS

  43. 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 4. How to detect BS, system 2

  44. (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. 4. How to detect BS, system 2

  45. 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 （见前） • 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. 4. How to detect BS

  46. 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, 4. How to detect BS, system 3

  47. 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 . 4. How to detect BS

  48. Physical situation: • Results: Axion stars are possible sources for generating energy in magnetized conducting media. Denoting the conductivity of the media by and assuming ohm’s 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. 4. How to detect BS, system 4

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

  50. (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 5. Our idea: torsion,quantum

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