Physics Letters B 624 (2005) 147–157 www.elsevier.com/locate/physletb Effective equation of state for dark energy: Mimicking quintessence and phantom energy through a variable Λ Joan Solà a,b, Hrvoje Štefanciˇ c´ a,1 a Departament d’Estructura i Constituents de la Matèria, Universitat de Barcelona, Av. Diagonal 647, 08028 Barcelona, Catalonia, Spain b CER for Astrophysics, Particle Physics and Cosmology 2, Spain Received 13 May 2005; received in revised form 27 July 2005; accepted 12 August 2005 Available online 24 August 2005 Editor: G.F. Giudice Abstract While there is mounting evidence in all fronts of experimental cosmology for a non-vanishing dark energy component in the Universe, we are still far away from understanding its ultimate nature. A fundamental cosmological constant, Λ,isthe most natural candidate, but many dynamical mechanisms to generate an effective Λ have been devised which postulate the existence of a peculiar scalar field (so-called quintessence, and generalizations thereof). These models are essentially ad hoc, but they lead to the attractive possibility of a time-evolving dark energy with a non-trivial equation of state (EOS). Most, if not all, future experimental studies on precision cosmology (e.g., the SNAP and PLANCK projects) address very carefully the determination of an EOS parametrized a la quintessence. Here we show that by fitting cosmological data to an EOS of that kind can also be interpreted as a hint of a fundamental, but time-evolving, cosmological term: Λ = Λ(t). We exemplify this possibility by studying the effective EOS associated to a renormalization group (RG) model for Λ. We find that the effective EOS can correspond to both normal quintessence and phantom dark energy, depending on the value of a single parameter of the RG model. We conclude that behind a non-trivial EOS of a purported quintessence or phantom scalar field there can actually be a running cosmological term Λ of a fundamental quantum field theory. 2005 Published by Elsevier B.V. 1. Introduction During the last few years we are witnessing how E-mail addresses: [email protected] (J. Solà), [email protected] cosmology is rapidly becoming an experimental (H. Štefanciˇ c).´ 1 On leave of absence from the Theoretical Physics Division, Rud- branch of physics. It is no longer a pure realm of philo- jer Boškovic´ Institute, Zagreb, Croatia. sophical speculation; theoretical models can be tested, 2 Associated with Institut de Ciències de l’Espai-CSIC. and new and more accurate data in the near future will 0370-2693/$ – see front matter 2005 Published by Elsevier B.V. doi:10.1016/j.physletb.2005.08.051 148 J. Solà, H. Štefanˇci´c / Physics Letters B 624 (2005) 147–157 restrict our conceptions of the Universe to within few the cosmological term. Actually, it was long ago that percent accuracy. Although the list of unsolved prob- it was considered the general possibility that the cos- lems in cosmology does not run short, there is a pre- mological term could evolve with time [8,9] or even to eminent one that seems to overshoot the strict domain be a dynamical scalar field variable [10,11], but only of cosmology and remains boldly defiant since its first in more recent times this idea took the popular form formulation by Zeldovich in 1967 [1]. We are referring of the quintessence proposal mentioned above [7,12]. to the famous cosmological constant (CC) problem In fact, so popular that all parametrizations of the DE [2,3]. Its ultimate solution desperately cries out for seem to presume it. help, hopefully to come from theoretical physics at its The reason why the quintessence idea can be use- deepest level. The CC problem is the problem of un- ful, in principle, is because if χ is a time-evolving field derstanding the theoretical meaning and the measured it may help to understand another aspect of the CC value of the cosmological term, Λ, in Einstein’s equa- problem which is also rather intriguing, the so-called tions. As it is well known, the quantum field theory “coincidence problem”, to wit: why the presently mea- (QFT) contributions prove to be exceedingly large as sured value of the CC/DE is so close to the matter compared to the measured Λ inferred from the accel- density? In other words, why the current cosmolog- erated expansion of the Universe [4], the anisotropies ical parameters ΩΛ and ΩM are of the same order? of the CMB [5] and the large scale structure [6]. Unfortunately, in spite of its virtues the quintessence In recent times the CC problem has become man- idea has a big theoretical drawback: the typical mass ifold and has been rephrased in a more general way, of the quintessence field should be of the order of the −33 namely one interprets the observed accelerated expan- Hubble parameter now: mχ ∼ H0 ∼ 10 eV, mean- sion of the Universe as caused by a generic entity ing a particle mass 30 orders of magnitude below the called the dark energy (DE) component, ρD,oftheto- very small mass scale associated to the measured value tal energy density ρ . Within this new conception the ≡ 1/4 ∼ −3 T of the cosmological constant: mΛ Λ0 10 eV. DE could be related to the existence of a dynamical One may wonder if by admitting the existence of an field that would generate an effective CC. Obviously ultralight field like χ (totally unrelated to the rest the very notion of CC in such broader context becomes of the particle physics world) is not just creating a degraded, the CC could just be inexistent or simply rel- problem far more worrisome than the CC problem it- egated to the status of one among many other possible self! In view of these facts, it is more than advisable candidates. For example, an alternate candidate to DE to seek for alternatives to quintessence which never- that has spurred an abundant literature goes under the theless should preserve the major virtue of that pro- name of quintessence [7], meaning some scalar field posal, such as the possibility to have a dynamical DE χ which generates a non-vanishing ρD from the sum that can help explaining why the CC is very small at of its potential and kinetic energy term at the present present (comparable to the matter density) and perhaps ={ ˙ 2 + } time: ρD (1/2)ξχ V(χ) t=t0 .Hereξ is a coef- much larger in the past. One possibility is to have a ficient whose sign can be of some significance, as we “true”, but variable, Λ parameter. This idea has been shall see. If the kinetic energy for χ is small enough, cherished many times in the literature, but only on it is clear that ρD looks as an effective cosmological purely phenomenological grounds [8,9,13].InRefs. 3 constant Λeff. The scalar field χ is in principle un- [14,15], however, a proposal was put forward aiming related to the Higgs boson or any other field of the at a model of variable Λ stemming from fundamental Standard Model (SM) of particle physics, including all physics: viz. the renormalization group (RG) methods of its known extensions (e.g., the supersymmetric gen- of QFT in curved space–time. The basic idea is that eralizations of the SM); in other words, the χ field is in QFT the CC should be treated as a running para- an entirely ad hoc construct just introduced to mimic meter, much in the same way as the electric charge in QED or the strong coupling constant in QCD.4 More 3 In our notation, Λ has dimensions of energy density. The CC 4 term λgµν in Einstein’s equations is related to our Λ by λ = 8πGΛ, See Refs. [15,16] for attempts to relate the running of Λ and of where G is Newton’s constant. the DE to neutrino physics. J. Solà, H. Štefanˇci´c / Physics Letters B 624 (2005) 147–157 149 recently this RG cosmological model has been shown with 4-vector velocity field U µ is given by to be testable in the next generation of precision ex- ˜ = + periments [17,18]. The general idea of a running CC Tµν Tµν gµνΛ has been further elaborated in [19–21], and its phe- = (Λ − p)gµν + (ρ + p)UµUν, (1) nomenological consequences have been explored in where T is the ordinary matter-radiation energy– great detail in [22] (see also the framework of [23]). µν momentum tensor, p is the proper isotropic pressure However in practice—meaning in all future experi- and ρ is the proper energy density of matter-radiation. mental projects for precision cosmology (like SNAP The basic cosmological equations with non-vanishing and PLANCK [24])—the general strategy to explore Λ are the Friedmann equation the properties of the DE is to assume that there is 2 an underlying equation of state (EOS), pχ = ωχ ρχ , a˙ 8πG k H 2 ≡ = (ρ + Λ) − , (2) that describes the field χ presumably responsible for a 3 a2 the accelerated expansion of the universe [25].Ifωχ together with the dynamical field equation for the scale lies in the interval −1 <ωχ < −1/3, the field χ is factor a standard quintessence field; if ωχ < −1, then χ is called a “phantom field” because this possibility is 4π a¨ =− G(ρ + 3p − 2Λ)a. (3) non-canonical in QFT (namely it enforces ξ<0 in its 3 kinetic energy term) and violates the weak energy con- LetusfirstassumethatG = G(t) and Λ = Λ(t) can dition. Still, it cannot be discarded at present because it be both arbitrary functions of the cosmic time.