General Relativity and Gravitation, Vol. 33, No. 5, 2001
Are Axidilaton Stars Massive Compact Halo Objects? Eckehard W. Mielke* and Franz E. Schunck† Received November 6, 2000 Boson stars built from a very light Kalb–Ramond axion, the dilaton or other moduli fields of effective string models could provide a considerable fraction of the non-baryonic part of dark matter. Gravitational microlensing of ∼ 0.5 M6 MACHOs within the halo of galaxies may indirectly “weighing” the mass of the constituent scalar particle, resulting in ∼ 10 − 10 eV/ c2 . KEY WORDS: String model; dilaton star; conformal transformation
1. INTRODUCTION: BOSON STARS AS DARK MATTER?
Dark matter candidates fall into two broad classes: astrophysical size objects called MAssive Compact Halo Objects (MACHOs), almost certainly detected by gravitational microlensing [1], and the hypothesized Weakly Interacting Massive Particles (WIMPs). Actually, via Bose–Einstein condensation, these classes could possibly be interrelated, as we are going to propose here. The suggestion [2, 3] that MACHOs could be low-mass primordial black holes formed during the early QCD epoch in the inflationary scenario, falls short in no providing constrictions on the mass scales. For cosmological dark matter, bound states of gravitational waves, so-called ‘gravitational geons’ built from spin-2 bosons, were also considered [4]. Since the standard model of elementary particles as well as its superstring extensions involve several light scalar fields, there arises the alternative possibil*Departamento de F´ısica, Universidad Auto´ noma Metropolitana-Iztapalapa, Apartado Postal 55-534, C.P. 09340, Me´ xico, D.F., Mexico. E-mail:
[email protected] † Institut fu ¨ r Theoretische Physik, Universita¨ t zu Ko¨ ln, 50923 Ko¨ ln, Germany. E-mail:
[email protected] 805 0001-7701/ 01/ 0500-0805$19.50/ 0 2001 Plenum Publishing Corporation
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ity [46] that primordial boson stars (BSs) account for this non-baryonic part of dark matter [5]. Boson stars are descendants of the so-called geons of Wheeler [6, 7], except that they are built from scalar particles (spin-0) instead of electromagnetic fields, i.e. spin-1 bosons. If scalar fields exist in nature, such localized gravitational solitons kept together by their self-generated gravitational field in a stable [8, 9] configuration could be rather massive and therefore a viable alternative to neutron stars or even black holes. This is supported by the likely discovery of the Higgs boson of mass mH c 114.5 GeV/ c2 at the Large Electron Positron (LEP) collider at CERN [79]. Accordingly, at least the existence of mini-boson stars appears not only a mathematically established [78], but may also be an astrophysically realistic possibility. 2. AXIONS AND DILATONS FROM EFFECTIVE STRING MODELS
String theory [10, 11] is an attractive approach of unifying the standard model with gravity on the quantum level. Commonly, the effective theory in four dimensions makes the prediction [12] that the tensor field gmn of gravity is accompanied by several scalar fields. Besides the familiar dilaton J, another scalar field of the effective string Lagrangian is predicted [12], the ‘universal’ axion j , a pseudo-scalar potential for the Kalb–Ramond (KR) three form H :c eJ/ f J *dj . A further modulus field b arise through the spontaneous compactification from ten dimensions onto an isotropic six torus of radius eb . (Since the axion field is the ‘superpotential’ [13] for the dual of the antisymmetric KR field strength H, there have been attempts [14–18] to identify it with the axial torsion of post-Riemannian spacetimes [19].) In the string frame g˜ mn c e − J/ f J gmn the effective string Lagrangian reads L eff
f
c | g˜ | e−J/ f J
−
[
R˜ + g˜ mn ∂mJ∂n J 2k
冢
− 6∂m b∂n b −
1 2J / f J e ∂m j ∂n j 2
]
1 J/ f J ˜ e U(J, j ) , 2
冣 (1 )
where the axidilaton part exactly corresponds to Eq. (11) of Dereli et al. [20]. f A conformal f change [21, 22, 23] of the metric via gmn r g˜ mn c C gmn , with | g˜ | c C 2 | g | converts the curvature scalar density into R˜
f
Then for C the form
| g˜ | c
[
CR −
3 mn g (∂m C )(∂n C ) 2C
c exp(J/ f J ) and f J c 1/
f
] f| |
g + 3∂m (
f
| g | gmn ∂n C ).
(2 )
2k the Lagrangian (1) can be rewritten in
Are Axidilaton Stars Massive Compact Halo Objects?
L eff
f
c |g|
−
[
R 2k
− gmn 冢 12
]
1 2J / f J ˜ e U(J, j ) + 2
807
∂mJ∂n J + 6∂m b∂n b + 3
f
2k
f
∂m (
1 2J / f J e ∂m j ∂n j 2
| g | gmn ∂n J).
冣 (3)
In this Einstein frame conformally related to the string frame, the kinetic dilaton term changes sign and thus allows to formally combine [24] the axion and the dilaton into a single complex scalar field F :c j + i fJ e − J/ f J , the axidilaton. Then all physical quantities depend universally on the axidilaton as in the case for the conjectured S-duality, cf. [10, 11]. With this complex scalar field the Lagrangian reads L eff
f
c |g|
[
R 2k
− gmn 冢 12 e2J/ f J ∂m F∂n F * + 6∂m b∂n b冣 −
]
1 2J / f J ˜ e U(J, j ) , 2
(4) where the boundary term has now been suppressed. For constructing boson stars, we will restrict ourselves to the electric neutral U(1) sector of this effective string model, in order to utilize a Noether symmetry F r e − ic F for establishing the global stability [8, 9] of the star. Let us stress that we regard F as a scalar field and not just as a complex parameter of the Neveu–Schwarz/ Neveu–Schwarz action [25]. When both, axion and dilaton, fuse into F, a conserved particle number arises, as a result of a global U(1) symmetry. Additionally, a self-interaction emerges consisting in lowest order of a mass term responding to the residual U(1) symmetry; this process resembles the well-known symmetry breaking method. (There, a potential with even powers of Fis transformed into a potential where odd powers occur as well, cf. our proposal in Section 4). We expect an inverse symmetry breaking mechanism in our case. In this approximation, the induced mass term reads
˜ j ) −∼ m2j | F | 2 U(J, j ) :c e2J/ f J U(J,
c m2j (j 2 + f J2 mJ2 e− 2J/ f J ) ∼− U(j ) + 3
f J2 U(J), f j2
(5)
where the masses m and decay constants f of the bosonic particles are related via mJ f J ∼ − mj f j ∼ − mp f p ∼ − 1016 eV2 to those of the pion [28]. Let us compare this with phenomenological assumptions on topological non-trivial configurations in the axion sector interacting with instantons, where usually two parts are inferred:
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1. A dilaton potential [26, 29] of a form similar to the emergence of potential terms from gluino condensates
U(J) c
1 [2mJ2 f J2 exp(J / f J ) + m2j f j2 exp( − 2J / f J )] − mJ2 f J2 3
c
1 2 2 m f [2 exp(J / f J ) + exp( − 2J / f J ) − 3] 3 J J
⬵
mJ2 [J 2
− J3 / 3f J + J4 / 4f J2 − O(J5 )].
(6 )
(7 )
2. The effective axion potential [30]
U(j ) c 2m2j f j2 [1 − cos(j / f j )] 2 2 ⬵ mj [j
−j / 4
12f j2
(8 ) 6
+ O(j )]
(9 )
created by instantons [26, 27] satisfying F c ±i∗F exhibiting an enumerable set of equidistant vacua, where F :c DF is the gluon field strength. As indicated by the expansion, both these potentials exhibit a mass term in leading order. Conventionally, the mass of the dilaton J is supposed to be related [31] to ∼ 10 − 3 (mSUSY / TeV/ c2 ) eV/ c2 . the supersymmetry breaking scale mSUSY by mJ − However, such a rather large mass is not the only f possibility. In the case of ∼ broken S-duality with a decay constant f J − 1/ 2k close to the Planck scale, the potential (7) hints for non-perturbatively [29] generated tiny dilaton mass mJ ∼ − ( f j / f J )mj ∼ 10 − 6 mj . Imposing the bound Q J / Q j < 0.1 on the respective contribution to the critical density of the Universe, implies a fairly strong alignment [32] of the dilaton. Then the paradigm of chaotic inflation suggest that local patches of the early Universe with a small variance of J survive and will expand, while overclosed patches collapse. On the other hand, from the Hubble scale H eq ∼ 10 − 27 eV/ c2 of matterradiation equilibrium and the temperature T m ∼ 100 MeV of mass generation at the epoch of chiral symmetry restoration, one can derive [33] the condition mj > (T m / eV)2 H eq . This allows a very light axion mass mj c 7.4 × (107 GeV/ f j ) eV/ c2 > 10 − 11 eV/ c2 with a decay constant f j close to the inverse Planck time, thus a prime candidate for dark matter, cf. [34]. In the context of string cosmology, massless axions are able to seed the observed anisotropy of the Cosmic Microwave Background (CMB). The same holds for the Kalb–Ramond axion, provided it lies in an ultra-low mass window. This KR axion is not necessarily to be identified with the “invisibel” axion advocated for solving the strong CP problem of QCD and therefore can evade
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the existing cosmological lower bounds on its mass. In the recent pre-big bang model of Gasperini and Veneziano [35], there exist a branch of a less efficient relaxation of the axion mass given by mj > 10 − 11 eV for which the axion energy density remains well undercritical, i.e. Q j > 0.1. The controversies [36] on mass scales in string cosmology are yet not completely resolved. 3. MAXIMAL MASSES OF BOSON STARS
In the following we consider for simplicity arguments from linearized stability [37], which do not require an exact Noether symmetry. In a nut-shell, a BS is a stationary solution of a (non-linear) Klein–Gordon equation RF c − ∂U / ∂F c m2 F for a complex scalar field in its own gravitational field; cf. [38–40]. The self-generated spacetime curvature affects the resulting radial Schro¨ dinger equation [∂r∗2
− V eff (r) + q 2 − m2 ]P c 0
(10)
for the radial function P(r) :c Feiqt essentially via an effective gravitational potential V eff (r *) c en d U/ (d |F |2 ) + en l(l + 1)/ r 2 + (n ′ − l ′ )en − l / 2r, when written in terms of the tortoise coordinate r * :c ∫e(l − n)/ 2 dr. Then, it can be easily f realized that localized solutions fall off asymptotically as P(r) ∼ (1/ r) exp ( − m2 − q 2 r) in a Schwarzschild-type asymptotic background. As first shown by Kaup [41], cf. [42], no Schwarzschild type event horizon nor an initial singularity develop: Metric and curvature of a BS remain completely regular. Due to the conserved Noether current ∂m jm c 0, in the spherically symmetric case, we could show via catastrophe theory [8, 43] that these BSs have an absolutely stable branch within a definite range of masses M, effective radii Reff ∞ ∞ :c (1/ N ) ∫0 j 0 rdr, and conserved particle number N c ∫0 j 0 dr. For a mini-BS with U( | F | ) c m2 | F | 2 as sole self-interaction, having an effective radius Reff ∼ − (p/ 2)2 RS close to the last stable Kepler orbit of a black hole of Schwarzschild radius RS , the critical mass or the so-called Kaup limit, cf. Ref. [44, 45], is
f
∼ (2/ p)M Pl2 / m ≥ 0.633M Pl2 / m c K Kaup . M crit −
(11)
Here M Pl :c ¯hc/ G is the Planck mass and m the mass of a bosonic particle. As is typical for solitonic solutions, it becomes heavier for weaker couplingU( | F | ), i.e. for smaller constituent mass m in our case. In building star-sized BS, one needs particles of ultra-low mass m or a Higgs-type [46–48] repulsive self-interaction U( | F | ) c m2 | F | 2 (1 + L( | F | 2 )/ 8). In the latter case, it turns out that one can apply again the Kaup limit (11), but
Mielke and Schunck
810
f
for a rescaled mass m r mresc :c m/ 1 + L/ 8. Depending on the constituent mass and the coupling constant, the maximal mass of a BS can then reach the Chandrasekhar limit M Ch. :c M 3Pl / m2 or even extend the limiting mass of 3.23 M 6 for (rotating) neutron stars [49, 77]. Three surveys [44, 50, 51] summarize the present status and formation of non-rotating configurations, a more recent review including rotating BSs [52–55] can be found in [45, 46].
4. ARE MACHOS AXIDILATON OR MODULI STARS?
Moreover, BSs could be the solution for the MACHO problem, as we are going to analyze in some more detail: Stable axidilaton stars (ASs) [56–58] exist at central densities lower than the maximum mass (11), which depends inversely on the particle mass m. For a mass 10 − 10 eV/ c2 of the boson close to the lower bound of axions our analysis reveals the following critical values: M crit c 0.846 M 6 , mNcrit c 0.873 M 6 and r c c 9.1 × r nucl , where r nucl c 2.8 × 1017 kg/ m3 is the average density of nuclei. Since non-interacting bosons are very “soft”, ASs are extremely dense objects with a critical density higher than for neutron or strange stars [59]. The stable ASs have radii larger than the minimum at 20.5 km. We stress that the total mass of these relativistic ASs is just in the observed range of 0.3 to 0.8 M 6 for MACHOs [1]. This conclusion is not changed much, if we take the full Brans– Dicke type dilaton interaction for the bosons into account or consider a very light dilaton J being stabilized [26] through the axion, cf. [58]. Phenomenologically, one should rather turn this argument around: By identifying the MACHOs with known gravitational mass of about 0.5 M 6 with ASs, we are essentially “weighing”, via M Kaup / N crit ∼ − m, the constituent mass to be − 10 2 10 eV c . It is gratifying to note that such an ultra-low mass is perfectly mj ∼ − / compatible with the constraints on the mass range of the Kalb–Ramond axion seeding the large-scale CMB anisotropy, cf. the recent results of Gasperini and Veneziano [35, 60, 61] within low-energy string cosmology. ASs which will during their evolution accumulate a mass larger than M crit will start to oscillate [8] and may get rid of excess mass due to gravitational cooling [62, 63] or will collapse to a black hole (BH) in the upper range of the MACHO mass. These axion induced BHs [64, 65] do not carry scalar hair, which could serve to distinguish them from primordial black holes, cf. [66], but could have some remnant of P-violation or even “axion hair” [15], if j is interpreted as a superpotential for torsion. In the pre-big bang scenarios, the dilaton would produce a detectable gravitational wave background [67]. Therefore, if such string-inspired scalar fields would exist in Nature, axions could not only solve the non-baryonic dark matter problem [68], but their grav-
Are Axidilaton Stars Massive Compact Halo Objects?
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itationally confined mini-clusters, the axidilaton stars, would also represent the observed MACHOs in our Galaxy. According to a recent speculation of Iwazaki [69], the collision of ASs with neutron stars could also be responsible for gamma ray bursts without invoking an adhoc ‘beaming mechanism’ due to strong magnetic fields. This leads to the estimate E m ∼ 1044 / mj erg which, for our fit mj ∼ − 10 − 10 eV to the MACHO data, is again just in the range of the most energetic gamma ray burst, cf. also [70]. Instead of combining the dilaton with the axion,fone could also take recurse to the other moduli field. Define b :c f J exp(J / f J )/ 12 and F˜ :c f F exp(J / f J ) exp(ij ). Then, the Lagrangian (1) takes the following form
L eff
f
c |g|
[
R − gmn
˜ F˜ * − 冢 2 ∂ J∂ J + 2 ∂ F∂ 冣 2e 1
1
m
n
1
m
n
2J / f J
]
˜ , U(F)
(12)
which is, up to a missing factor eJ/ f J in front of the kinetic term of the complex scalar field, the Lagrangian for a BS in the Jordan–Brans–Dicke theory in the Einstein frame. In the latter case, several numerical investigations [71–73] have been recently carried out. f Another possibility is to consider the complex Ka¨ hler form field Fˆ :c J + i 12b akin to T-duality [74–76], without including the axion. The Lagrangian is then close to the situation after a symmetry breaking: one real massive scalar field plus a real massless one. We assume that an additional potential for b exists without a quadratic term so that the potential after symmetry breaking to the true ˆ c m2 | Fˆ | 2 + O( | Fˆ | 4 ) where m is the vacuum state for Fˆ takes the form U(F) mass of Fˆ being in the same order of magnitude as mJ . In contrast to the usual symmetry breaking process, we have here the transition between two minima; this ensures masses of the scalar fields before and after the breaking.
ACKNOWLEDGEMENTS We dedicate this work to Heinz Dehnen on the occasion of his 65th birthday and thank him for hospitality at the University of Konstanz. Moreover, we would like to thank John Barrow, Mariusz D¸abrowski, Rainer Dick, Marcelo Gleiser, Andrew Liddle, Alfredo Mac´ıas, Remo Ruffini, and Diego Torres for useful discussions, literature hints, and support. This work was partially supported by CONACyT, grant No. 28339-E, the grant P/ FOMES 98-35-15, and the joint German–Mexican project DLR-Conacyt E130-1148 and MXI 010/ 98 OTH. One of us (E.W.M.) thanks Noelia and Markus Ge´ rard Erik for encouragement.
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