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Axion Phantom Energy View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Digital.CSIC PHYSICAL REVIEW D 69, 063522 ͑2004͒ Axion phantom energy Pedro F. Gonza´lez-Dı´az Centro de Fı´sica ‘‘Miguel A. Catala´n,’’ Instituto de Matema´ticas y Fı´sica Fundamental, Consejo Superior de Investigaciones Cientı´ficas, Serrano 121, 28006 Madrid, Spain ͑Received 6 October 2003; published 31 March 2004͒ The existence of phantom energy in a universe which evolves to eventually show a big rip doomsday is a possibility which is not excluded by present observational constraints. In this paper it is argued that the field theory associated with a simple quintessence model is compatible with a field definition that is interpretable in terms of a rank-3 axionic tensor field, whenever we consider a perfect-fluid equation of state that corresponds to the phantom energy regime. Explicit expressions for the axionic field and its potential, both in terms of an imaginary scalar field, are derived, which show that these quantities both diverge at the big rip, and that the onset of phantom-energy dominance must take place just at present. DOI: 10.1103/PhysRevD.69.063522 PACS number͑s͒: 98.80.Cq, 14.80.Mz INTRODUCTION Chaplygin-like equations of state ͓16͔, it seems to be the simplest and most natural possibility stemming from phan- Cosmology has become the current common place where tom energy. On the other hand, the big rip cosmic scenario the greatest problems of physics are being concentrated or interestingly adds an extra qualitative feature to the known have most clearly manifested themselves. Along with the na- set of cosmological models as it introduces a curvature sin- ture of dark matter, the origin of supermassive black holes gularity, other than that at the big bang, at a finite, nonzero and several unexplained huge amounts of released astro- value of the cosmic time. However, in spite of the feature physical energy are only some illustrative examples. How- that the blowing up of the scale factor appears to be unavoid- ever, perhaps the biggest problem of all in physics is the able in models with equation of state pϭ␻␳ and ␻ϽϪ1, a so-called dark energy problem ͓1͔ in which it is known that complete account of the nature and properties of the field nearly 70% of the total energy of the Universe is in the form theory associated with such models has not been done yet. of an unobserved vacuum energy which is responsible for the This paper aims at investigating the characteristics of the present accelerating expansion of the Universe ͓2,3͔ and massless scalar field which allow emergence of a big rip whose nature remains a mystery ͓4͔. There has been a rather singularity in the simplest of such phantom-energy models, large influx of papers in recent years trying to shed some assuming a perfect-fluid equation of state. It will be shown light on the existence and kind of possible stuff that may that the stuff making up phantom energy can be interpreted make up dark energy ͓5͔. to be a vacuum sea of cosmic axions, which can be described Four main candidates to represent dark energy have been in terms of the kind of rank-3 tensor field strengths predicted hitherto suggested: A positive cosmological constant ͓6͔, the in supergravity and string theories. quintessence fields ͑which may ͓7͔ or may not ͓8͔ be This paper can be outlined as follows. In Sec. II we argue tracked͒, some generalizations of the Chaplygin gas ͓9͔, and in favor of the idea that superlight axions are the source of the so-called tachyon model of Padmanabhan et al. ͓10͔. cosmic phantom energy. A simple cosmological model ac- Even though such models could all be accommodated by counting for a big rip singularity in the case that the vacuum present observational constraints, these models pose new or is filled with phantom energy is discussed in Sec. III. Section traditional problems in such a way that none of them be- IV contains the solution of the phantom field theory within comes completely satisfactory. In particular, cosmological the simple cosmological model of Sec. III. We check that data from current observations do not exclude ͓11͔, but may both the scalar field and its potential also have a singularity suggest ͓12͔, values of the parameter ␻ in the perfect-fluid at the big rip. The results are summarized in Sec. V. equation of state pϭ␻␳ of the most popular quintessence Ϫ models which are smaller than 1. If this were the case, AXIONS AS THE SOURCE OF PHANTOM ENERGY then dark energy would become what is now named phantom energy ͓13͔, an entity violating the dominant energy condi- The definition of the massless scalar field ␾ which is tion and thereby allowing the natural occurrence of worm- assumed to make up dark energy is usually taken to be the holes and ringholes and even their corresponding time ma- conventional simplest one; that is, in terms of pressure p and chines in the universe ͓14͔. energy density ␳, Caldwell, Kamionkowski, and Weinberg have recently ͓ ͔ noticed 15 that in the framework of quintessence models 1 1 phantom energy may lead to a doomsday for the universe— ␳ϭ ␾˙ 2ϩV͑␾͒, pϭ ␾˙ 2ϪV͑␾͒, ͑1͒ which would take place at a big rip singularity—once clus- 2 2 ters, galaxies, stars, planets, and, ultimately, nucleons and leptons in it are all ripped apart. Even though such a big rip where V(␾) is the field potential. From the equation of state does not take place in some phantom-energy models that use pϭ␻␳ and Eq. ͑1͒ it immediately follows that 0556-2821/2004/69͑6͒/063522͑6͒/$22.5069 063522-1 ©2004 The American Physical Society PEDRO F. GONZA´ LEZ-DI´AZ PHYSICAL REVIEW D 69, 063522 ͑2004͒ ␾˙ 2 is equipped with a potential V(␾), then an extra term ␳ϭ . ͑2͒ Ϫ͐TdtNA3V(␾) should be added to the action. In that case, 1ϩ␻ 0 the axionic action can be obtained by simply rotating ␾ ! ⌽ Thus, if the weak energy condition ␳у0 is taken to be al- i in an action containing the above potential term, with- ways satisfied ͓17͔, the requirement pϩ␳Ͻ0, ␻ϽϪ1 from out any extra boundary term. phantom energy in this kind of model necessarily implies Such a property, which was first noticed in Euclideanized that the field ␾ ought to be purely imaginary; that is to say, if solutions such as wormholes and other instantons describing ͓ ͔ the vacuum energy density for a phantom vacuum as referred nucleation of baby universes 19 , is actually independent of to any timelike observer has to be positive, then the massless the metric signature and, therefore, applies also to our scalar field making up the phantom stuff should be purely Lorentzian cosmological context. Thus, one can interpret that imaginary. I will argue in what follows that, in the classical the stuff that makes up phantom energy can be regarded as a framework, such a massless, pure imaginary scalar field ac- rank-3 antisymmetric tensor axionic field. It is usually tually represents an axion describable as a rank-3 antisym- thought that axions can explain the absence of an electrical metric tensor field, considering later the associated cosmic dipole moment for the neutron and thereby solve the so- ͓ ͔ theory for such a field. In fact, the Lorentzian action that called strong CP problem 20 . The axions are chargeless couples such an axion field to gravity can generally be writ- and spinless particles with very tiny mass which interact with ten as ordinary matter only very weakly. Such particles are believed to have been abundantly produced in the big bang. It is worth R noticing that, whereas relic axions are an excellent candidate Sϭ ͵ d4xͱϪgͩ ϪA2ϩL ͪ , ͑3͒ for the dark matter in the universe ͓21͔, their vacuum quan- 16␲G m tum background could make cosmic phantom energy. 2 The coincidence and fine-tuning problems could be where Lm is the Lagrangian for observable matter and A ␮␯␣ thought to become exacerbated in the present scenario where ϵA␮␯␣A if we choose the axion to be given by a three- ␻ ϽϪ form AϭdB field strength so that dAϭ0, that is, as a rank-3 one sets constant and 1. However, in dark energy ͓ ͔ antisymmetric tensor field strength of a type arising in models such as the generalized Chaplygin gas 9 and ͓ ͔ supergravity-theory motivated quantum-gravity solutions tachyon models 10 , dark energy and dark matter are de- ͓18͔. The equations of motion derived from action ͑3͒ are scribed as separate limiting cases from an existing unique field. Partly inspired by these models, one could naively as- 1 1 sume the existence of a unique axion field which, when ex- Ϫ ϭ ␲ ͩ 2 Ϫ 2ϩ (m) ͪ ͑ ͒ R␮␯ g␮␯R 16 G 3A␮␯ g␮␯A T␮␯ , 4 cited, would make dark matter, and when at its vacuum 2 2 ground state would be the source of phantom energy. Coin- cidence time could then be interpreted as the time when both d*Aϭ0, ͑5͒ the vacuum and the excited states are approximately equally 2 ␣␤ populated. Of course, this would not solve the coincidence in which A␮␯ϭA␮␣␤A␯ , the asterisk denotes the Hodge and fine-tuning problems but provided some explanation of dual, and T(m) is the momentum-energy tensor for ordinary ␮␯ these problems and of the prediction of generating such observable matter. One can now check that any explicit so- strangely small amounts of homogeneously distributed ax- lution to these equations of motion subject to the usual ions.
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