Effective Equation of State for Dark Energy: Mimicking Quintessence and Phantom Energy Through a Variable Λ

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. Štefanˇcić / 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. Štefanˇcić / 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. This seems to be slightly preferred by the combined analy- is allowed by the cosmological principle embodied in sis of the supernovae and CMB data [26].5 the FLRW metric. Then one can check that the Bianchi At variance with the idea of a canonical or non- identities lead to the following first integral of the pre- canonical scalar field description of the DE, a funda- vious system of differential equations mental CC (whether strictly constant or a variable one) d can only have a “trivial” EOS: ωΛ =−1. Notwith- G(Λ + ρ) + 3GH (ρ + p) = 0. (4) standing, one may describe such a variable Λ within dt the scalar field parametrization of the DE and try to Equivalently, this also follows from Eq. (1) and µ ˜ uncover what is the effective EOS for the running CC ∇ Tµν = 0. When G is constant, the identity above term. A main result of this work is that a fundamental implies that Λ is also a constant, if and only if the running Λ can mimic the effective vacuum energy of a ordinary energy–momentum tensor is individually µ dynamical field χ both in the quintessence and phan- conserved (∇ Tµν = 0), i.e., ρ˙ + 3H(ρ + p) = 0. tom mode. At the same time our analysis will illustrate However, a first non-trivial situation appears when that an eventual determination of an EOS from exper- G = const but Λ = Λ(t). Then (4) boils down to iment should not necessarily be interpreted as a sign Λ˙ +˙ρ + 3H(ρ+ p) = 0. (5) that there is a dynamical field responsible for the DE component of the Universe. This scenario exemplifies that a time-variable Λ = Λ(t) cosmology may exist such that transfer of energy may occur from matter-radiation into vacuum energy, and vice versa. The solution of a generic cosmolog- 2. Running Λ versus quintessence ical model of this kind is contained in part in the coupled system of differential equations (2) and (5) Let us compare an scenario with a variable Λ together with the equation of state p = p(ρ) for mat- with one with a DE component represented by a ter and radiation. However, still another equation is quintessence field χ. In the first case the full energy– needed to completely solve this cosmological model momentum tensor of the cosmological perfect fluid in terms of the basic set of cosmological functions (H (t), ρ(t), p(t), Λ(t)). At this point one may either resort to any of the various phenomenological mod- 5 See, e.g., [27,28] for some recent literature on phantom DE. els available in the market [13] or use some new idea. 150 J. Solà, H. Štefanˇcić / Physics Letters B 624 (2005) 147–157

The particular case of a continually decaying Λ has contribution is absent because it corresponds to terms ∝ 4 been examined long ago [9]. In the absence of a funda- Mi that give an extremely fast evolution. These are mental calculation to specify how rapidly the vacuum to be banished if we should describe a successful phe- energy decays and how it couples to non-relativistic nomenology; actually from the renormalization group matter and radiation, these authors decided to make point of view they are excluded because, as noted some assumptions and examine the potential phenom- above, µ Mi for all known masses. In practice only enological consequences. Here we generalize this ap- the first term n = 1 is needed, with Mi of the order proach for a variable Λ = Λ(t) that can either increase of the highest mass available. We may assume that the or decrease with time, and show that this kind of cos- dominant masses Mi are all of order of a high mass mological scenario could emerge from QFT. To illus- scale M near the Planck mass MP . Let us define (as in trate the last possibility, we are going to make use of [17]) the ratio the renormalization group model of Refs. [14,17,18].6 σ M2 In a few words this model is based on an RG equation ν = . 2 (7) for Λ of the general form 12π MP ∞ Here σ =±1 depending on whether bosons or fermi- dΛ = A µ2n. (6) ons dominate in their loop contributions to (6). Then, d ln µ n n=1 to within very good approximation, the solution of the renormalization group equation (6) reads Here µ is the energy scale associated to the RG running. One can argue that µ can be identified with the 2 Λ(t) = C0 + C1H (t), (8) Hubble parameter µ = H at any given epoch [14,17, 18,20]. Since H evolves with the cosmic time, the cos- with mological term Λ inherits a time-dependence (which 3ν 2 2 3ν 2 one may transform for convenience into redshift de- C0 = Λ0 − M H ,C1 = M , (9) 8π P 0 8π P pendence) through its primary scale evolution with the = renormalization scale µ. Coefficients A are obtained where H(t) is given by (2).Fort t0 we just get n = after summing over the loop contributions of fields of Λ(t0) Λ0, the value of the CC at present. More- over, for t around t the variation of Λ is δΛ(t ) ∼ different masses Mi and spins σi . The general behav- 0 0 4−2n νH2M2 ∼ H 2M2. This is numerically in the ballpark ior is An ∼ M [14,19]. Therefore, for µ Mi , 0 P i of Λ for M M . As we see, this provides the fourth the series above is an expansion in powers of the small 0 P ∼ 2 equation Λ = Λ(t) needed to solve the cosmological quantities µ/Mi . Given that A1 Mi , the heavi- est fields give the dominant contribution. This feature model. It is well-behaved and it predicts a small evo- (“soft-decoupling”) represents a generalization of the lution of Λ around our time, which nevertheless may decoupling theorem in QFT [29]—see [14,19,22] for a have some measurable effects [17,22]. In the next sec- more detailed discussion. In fact, it is characteristic of tion we will translate these effects into the language of the Λ parameter because it is the only dimension-4 pa- the quintessence parametrization of the DE. rameter available in the SM, whereas quantum effects But before doing that, let us recall how the cosmo- on dimensionless couplings and masses just decouple logical picture becomes modified when one trades the in the standard way. Now, since µ = H ∼ 10−33 eV CC for a dynamical scalar field χ, with an EOS of the 0 = the condition µ M is amply met for all known par- general form pχ ωχ ρχ . Consider the present time i ticles, and the series on the r.h.s. of Eq. (6) converges where ρ ρM and p 0. Then Eqs. (2) and (3) be- extremely fast. Notice that only even powers of µ = H come are consistent with general covariance [17].Then = 0 8πG k H 2 = (ρ + ρ ) − , (10) 3 M χ a2 6 A more general RG cosmological model with both running G and and running Λ can also be constructed within QFT in curved space– time, see Ref. [20]. However, for simplicity hereafter we limit our- 4πG a¨ =− ρM + (1 + 3ωχ )ρχ a. (11) selves to the case G = const. 3 J. Solà, H. Štefanˇcić / Physics Letters B 624 (2005) 147–157 151

= From the last equation it is clear that for ρ → 0 ρχ (z) ρχ (0)ζ(z) M the expansion will accelerate if ω < −1/3. However, z χ + ω (z ) − = 1 χ for χ to mimic a positive CC one needs ωχ 1 where ζ(z) exp 3 dz . (14) − 1 + z (quintessence). If ωχ < 1 the Universe will still ac- 0 celerate, but the χ ﬁeld is non-canonical (phantom) If we plug this equation into (10) we may write the because it should have a small, and negative, kinetic Hubble expansion rate as a function of the redshift and term at present: the unknown (z-dependent) barotropic index ωχ = ωχ (z) as follows: 1 ˙ 2 − pχ 2 ξχ V(χ) ωχ ≡ = −1 2 = 2 ˜ 0 + 3 + ˜ 0 + 2 + ˜ 0 1 2 H (z) H0 ΩM (1 z) ΩK (1 z) Ωχ ζ(z) . ρχ ξχ˙ + V(χ) t=t 2 0 (15) if |ξ|˙χ2 V(χ)and ξ<0. (12) If one expands

Here we assumed a positive potential for χ,thesim- ωχ (z) = ω0 + ω1z +··· (16) = 2 2 plest possibility being V(χ) (1/2)mχ χ . The field then for small redshifts one can replace ζ(z) in (15) χ is usually thought of as a high energy field (unre- with lated to SM physics), i.e., χ MX where MX is some 3ω z 3(1+ω −ω ) high energy scale typically around MP . Neglecting the ζ(z) e 1 (1 + z) 0 1 , (17) contribution from the kinetic term at the present time, where one expects ω −1 and |ω |1 in order that such scalar field model would produce an effective 0 1 χ can mimic a slowly varying CC. In Eq. (15) we have cosmological constant of the order of the measured ˜ ˜ defined the cosmological parameters ΩM and ΩK in one, Λeff V(χ) |t=t Λ0, provided the mass of 0 − the usual way. The tilde indicates that they are presum- that (high-energy) field is m ∼ H ∼ 10 33 eV, χ 0 ably determined from a fit to experimental data assum- which looks rather contrived—to say the least. Even ing a true quintessence model. This notation will help if (by some unknown mechanism) χ would be related − to distinguish them from the cosmological parameters to the electroweak scale (say χ G 1/2 300 GeV, F associated to the aforementioned RG model (more on where GF is Fermi’s constant in electroweak theory) ˜ 0 −12 this in the next section). Finally, we have defined Ωχ the previous condition would imply mχ ∼ 10 eV. ρ ( ) ={( / )ξχ˙ 2 + V(χ)} = This mass scale is 21 orders of magnitude larger than in (15) as the value of χ 0 1 2 z 0 before, but still one billion times smaller than the tiny in units of the critical density at present. mass scale associated to the measured value of the 1/4 ∼ −3 cosmological constant: Λ0 10 eV. It is very dif- 3. Effective equation of state for Λ ficult to understand the mass mχ in particle physics, and this is of course a serious problem underlying the Let us now come back to the RG cosmological quintessence models. model. Solving the system (2), (5) and (8) one finds The corresponding full energy–momentum tensor [17] ρ = ρ(z; ν) and Λ = Λ(z; ν) as explicit functions ˜ χ replacing (1) in this case is Tµν = Tµν + Tµν , where of the redshift and depending on the single additional one assumes that the two components are of perfect parameter ν,Eq.(7). These functions can be substi- fluid form and are conserved separately. For the χ part, tuted back into Eq. (2) to obtain the expansion para- µ χ ∇ Tµν = 0 leads to meter as a function of the redshift: H 2(z; ν) ρ˙ + 3(1 + ω )Hρ = 0, (13) − χ χ χ (1 + z)3(1 ν) − 1 Ω0 = H 2 + Ω0 + K 0 1 M − − instead of (5). We can easily convert this into a red- 1 ν 1 3ν − shift equation using the correspondence between time (1 + z)3(1 ν) − 1 × (1 + z)2 − 1 − 2ν . derivatives and redshift derivatives: d/dt =−(1 + 1 − ν z)H d/dz. Then integrating (13) we have (18) 152 J. Solà, H. Štefanˇcić / Physics Letters B 624 (2005) 147–157

For ν = 0 we recover the standard form corresponding follows. We first note from this equation that to strictly constant Λ. Here the cosmological parame- 1 1 dζ ters are denoted without tilde because they need not ω (z) =−1 + (1 + z) . (21) eff 3 ζ dz to be the same ones as in (15). In fact, in Ref. [22] it has been shown how to fit the high-z supernovae data Next we compute the redshift derivative of (15) and using this RG model. The fit crucially depends on the arrive at luminosity distance function, which is determined by dζ d H 2 Ω˜ 0 = − Ω˜ 0 ( + z) − Ω˜ 0 ( + z)2. the explicit structure of (18), so that the fitting para- χ 2 2 K 1 3 M 1 0 0 0 dz dz H0 meters ΩM ,ΩΛ,ΩK can be different from those ob- (22) tained by substituting the alternate function (15) in the The pending derivative on the r.h.s. of this equation luminosity distance function. The potential differences can be computed from (18). Finally we insert the result between these parameters, for dζ/dz in (21). In doing this we keep non-vanishing parameter differences ( Ω = 0) in (19). The final Ω = Ω0 − Ω˜ 0 , Ω= Ω0 − Ω˜ 0 , M M M Λ Λ χ result is obtained after a straightforward calculation, = 0 − ˜ 0 but in the non-flat case (Ω0 , Ω˜ 0 = 0) the result is a ΩK ΩK ΩK (19) K K bit too cumbersome and will not be quoted here. Let can play a role in our discussion, but the main effect us quote here only the result for the flat-space case under consideration would be there even if these dif- 0 = ˜ 0 = (ΩK ΩK 0). This should be enough to illustrate ferences would exactly be zero. What we are really the basic facts, and moreover it is the most realistic searching for is an effective dark energy EOS situation in the light of the present data. One finds the following barotropic index function for the effective pD = ωeffρD (20) EOS of the running Λ model: associated to the running Λ model that gives rise to the ωeff(z)| Ω =0 expansion rate (18). This means the following. In prac- =−1 + (1 − ν) tice we would have experimental data, and we would Ω0 (1 + z)3(1−ν) − Ω˜ 0 (1 + z)3 usually fit it to a quintessence-like DE model in or- × M M . 0 [ + 3(1−ν) − ]− − [ ˜ 0 + 3 − ] der to determine its EOS. But suppose that the RG ΩM (1 z) 1 (1 ν) ΩM (1 z) 1 model described above should be the correct one and (23) that the experimental data would follow the Hubble If the parameter differences (19) vanish, this yields function (18) for some value of ν. In that case the data ωeff(z)| Ω=0 would actually adapt perfectly well to a fundamental =− + − running Λ. But of course it could be that we just ignore 1 (1 ν) Ω0 (1 + z)3[(1 + z)−3ν − 1] this fact, and insist in fitting the data to a quintessence- × M . − − 0 + 0 + 3[ + −3ν − + ] like model (15) with ωχ replaced by an effective ωeff. 1 ν ΩM ΩM (1 z) (1 z) 1 ν Then the natural questions that emerge are the follow- (24) ing: (i) what would be the effective barotropic index, In the next section we analyze some phenomenolog- ωeff, for the EOS of this model? (ii) would it appear as ical consequences and perform a detailed numerical a normal quintessence model (ωeff −1)?, (iii) could analysis of these formulae. it effectively behave as a phantom model (ωeff < −1) for some values of ν and/or in some range of redshift?; (iv) what is the impact on these questions if 4. Effective quintessence and phantom behavior we have non-vanishing parameter differences (19) in the two independent fits of the same data? To answer Before embarking on an exact numerical analysis these points we have to solve for the barotropic index of the formulae for ωeff = ωeff(z) found in the previ- function ωeff = ωeff(z) obtained after equating (15) ous section, we can identify some interesting features and (18). Since ωeff(z) appears in the integral at the from simple analytical methods. Let us concentrate on exponent of (14), the procedure can be simplified as Eq. (24). First of all, as it could be expected, for ν = 0 J. Solà, H. Štefanˇcić / Physics Letters B 624 (2005) 147–157 153

Fig. 1. (a) Numerical analysis of ωeff,Eq.(23), as a function of the redshift for ﬁxed ν = ν0 > 0, Eq. (7), and for various values of Ω in (19). 0 = 0 = 0 = The Universe is assumed to be spatially ﬂat (ΩK 0) with the standard parameter choice ΩM 0.3, ΩΛ 0.7; (b) Extended z range of the plot (a).

one retrieves the pure CC behavior ωeff =−1atall redshift indicates that a significant effective phan- redshifts. On the other hand, for non-vanishing ν and tom phase can actually be reached already for red- z →∞we get ωeff → 0 (for ν>0) and ωeff →−ν shifts of order 1 corresponding to our “recent” Uni- (for ν<0). And in the infinite future (z →−1) the verse. For example, for the standard flat-space choice =− 0 0 = EOS recovers again a pure CC behavior ωeff 1for (ΩM ,ΩΛ) (0.3, 0.7), and a typical value of ν as any ν<1. We have seen that ν is a naturally small pa- in (25), we get ωeff (−1.2, −1.5) for z = (1, 1.5), rameter. For example, if M = MP in (7) then ν = ν0, respectively. Even for ν>0 ten times smaller (ν = where 0.1ν0) we get a non-negligible phantom-like behavior − = 1 ωeff 1.1 near z 2. These results are approximate, ν0 ≡ 0.026. (25) but the exact numerical analysis of Eqs. (23)–(24) is 12π shown in Figs. 1–3 where we have also included the | | In general we expect ν ν0 because from the effec- possibility of having non-vanishing parameter differ- tive field theory point of view we should have M ences Ω in (19).Forν ν0 at z = 1, the differences MP . This is also suggested from the bounds on ν ob- between the exact result and the approximate one (26) tained from nucleosynthesis [17,22] and also from the are of order of a few percent, and for z = 1.5 there is a CMB, although in this latter case the preferred values difference of 10%; in this last case the more accurate for ν are smaller [30]. Therefore it is natural to ex- value reads ω (z = 1.5) =−1.67. eff pand the previous results for small ν 1. Again we If we consider now the impact of the parameter dif- take the simplest case (24) and we find, in linear ap- ferences (19), we see that in the case ν = ν0 the phan- proximation in ν (and for not very large values of the tom effect can either be more dramatic (if Ω < 0) redshift): or it can be smoothed out, and even disappear, for Ω0 small z when Ω > 0. In the last case the phantom ω (z) =− − ν M ( + z)3 ( + z). eff 1 3 0 1 ln 1 (26) behavior is nevertheless retrieved at larger redshifts, ΩΛ see Fig. 1(b). In the same figure we show the behav- This result is simple and interesting, and contains the ioroftheν>0 models for an extended redshift range basic qualitative features of our analysis. Of course up to z = 10. Of course this behavior cannot be de- it boils down to ωeff =−1forν = 0. But for ν>0 scribed with the approximate expression (26), only it shows that we can get an (effective) phantom-like with the full equation (23). At very large z one at- → behavior (ωeff < −1)! The cubic enhancement with tains very slowly the asymptotic limit ωeff 0(cf. 154 J. Solà, H. Štefanˇcić / Physics Letters B 624 (2005) 147–157

Fig. 2. As in Fig. 1,butforν =−ν0.

Fig. 3. As in Fig. 1, but assuming Ω = 0in(19): (a) for three values ν>0; (b) for three values ν<0.

Fig. 1(b)). But well before reaching this limit one can On the other hand, there is the class of models with appreciate a kind of divergent behavior, e.g., around ν<0, with an entirely different qualitative behav- z 2forthe Ω = 0 case. It is due to the denom- ior. Here we have normal quintessence (ωeff −1) inator of Eq. (23) which vanishes at that point. This for z>0 whenever Ω 0. This is obvious from can only happen for ν>0. Of course there is nothing Eq. (26). For example, if we fix ν =−ν0, then for odd going on here because the presumed fundamen- z = (1, 1.5) we find ωeff (−0.82, −0.62) respec- tal RG model is well-behaved for all values of z— tively using the exact formula. For ν ten times smaller cf. Eq. (18). It is only the effective EOS description (ν =−0.1ν0), we have ωeff (−0.98, −0.95) at the that displays this fake singularity, which is nothing respective redshift values. Moreover, from Fig. 2 but an artifact of the EOS parametrization of a true (which displays the exact numerical analysis of the Λ model. If we would discover a sort of anomaly like case ν<0) it is apparent that this model can eas- this when fitting the data we could suspect that there ily accommodate the possibility of a relatively recent is no fundamental dynamical field behind the EOS but EOS transition from a quintessence phase into a phan- something else, like e.g., the RG model under discus- tom phase. This would indeed happen for Ω < 0 sion. in Eq. (19). If, instead, Ω > 0, then at small z J. Solà, H. Štefanˇcić / Physics Letters B 624 (2005) 147–157 155 the index ωeff increases with redshift faster than for preceded by a long quintessence-like regime. This is Ω 0. However, in all cases with negative ν the exactly the kind of behavior that the effective EOS of effective barotropic index climbs fast with z up to pos- our RG model predicts for ν<0 and Ω < 0 (as can itive values before reaching the asymptotically small be seen in Fig. 2). value ωeff →−ν>0(cf.Fig. 2(b)). For example, for Let us recall that the RG model underlying the ef- ν =−ν0 one achieves ωeff +0.2 around z = 5. This fective EOS under consideration predicts a redshift positive behavior of ωeff effectively looks as additional evolution of the cosmological constant. An approxi- radiation, and it is sustained for a long redshift in- mate formula for the relative variation of Λ (valid for terval. Finally, in Fig. 3(a) and (b) we plot ωeff in small ν and not very high redshift z) reads [17] detail for various values of ν and both signs, but for = Λ(z; ν) − Λ Ω0 vanishing parameter differences Ω 0in(19).It δ ≡ 0 = ν M ( + z)3 − . Λ 0 1 1 (27) is patent that the effects (both normal quintessence Λ0 ΩΛ and phantom-like behavior) should be visible even for Again taking the flat-space case with Ω0 = 0.3, |ν| 0.1ν0, i.e., for ν of order of a few per mil. M 0 = = = It is interesting to compare the previous result for ΩΛ 0.7, and ν ν0, one obtains δΛ 16.3% for the effective EOS with usual expansions like (16), z = 1.5 (reachable by SNAP [24]). This effect is big (17). One could naively think that the parameter ω1 enough to be measurable in the next generation of high is the direct analog of ν for the RG model. In fact it precision cosmological experiments. At the end of the is, but only in part. Already from the approximate for- day we see that, either by direct measurement of the mula (26) it is patent that the first two terms in the evolution of the cosmological constant, or indirectly expansion (16) describe very poorly the redshift be- through the rich class of qualitatively different behav- havior of the RG model. This is because the coefficient iors of its effective EOS, it should be possible to get a ν is highly enhanced by the cubic powers of 1 + z, handle on the underlying RG cosmological model. Fi- 0 = whereas ω1 is just the coefficient of the linear term nally, let us clarify that in the non-flat case (ΩK 0) in z. It means that if one would enforce the data fit to we have checked that the numerical results are not sig- be of the linear form (16) the quality of the EOS could nificantly different from those presented here for the be rather bad—e.g., if the data would hypothetically flat Universe. A more complete numerical analysis of adapt perfectly well to the RG model under discus- these effective EOS models, including the possibil- sion. There are alternative parametrizations of the EOS ity of a running Newton’s constant, will be presented that may overcome some of these difficulties [25], elsewhere. but the example (26) shows that the effective EOS of variable Λ models can have a much stronger redshift dependence than usually assumed for scalar field 5. Conclusions models of the DE. This issue can be further illustrated using, e.g., the (model-independent) analysis of the We have illustrated the possibility that a “true” or SNe(Gold) + CMB data [4,5] performed in Ref. [31]. fundamental cosmological term Λ can mimic the be- In this analysis a polynomial fit to the expansion para- havior expected for quintessence-like representations meter and EOS of the DE is made as a function of z. of the dark energy. Specifically, we have shown that The results show that the fitted function ωeff = ωeff(z) a running cosmological constant based on the princi- in the redshift range 0 z 1.7 does uphold the pos- ples of quantum field theory—more concretely on the sibility of a slowly varying ωeff(z) which is monotoni- renormalization group (RG)—can achieve this goal. cally increasing with z from ωeff(0)<−1 (today) and This suggests that the usual description of the dark then reaching a long period ωeff(z) > −1 at higher red- energy in terms of a dynamical field should be cau- shifts, with a crossing of the CC threshold ωeff =−1 tiously interpreted more as a general parametrization at some intermediate redshift in that interval. In other rather than as a fundamental one. That is to say, the words, these model-independent fits of the data show fact that the cosmological precision data may turn out the effective dynamical evolution of the DE can be to be adjustable to an equation of state (EOS) of a dy- assimilated to a phantom-like behavior near our time namical field does not necessarily mean that in such 156 J. Solà, H. Štefanˇcić / Physics Letters B 624 (2005) 147–157 case we would have proven that there is such a field are quite sizable even for the relatively close redshift there. It could be an effective description of funda- range z = 1–2 and for values of the ν parameter of or- mental physics going on at higher energy scales, for der of a few per mil. This should be welcome because example near the Planck scale. This physics could be the next generation of supernovae experiments, such based on just the cosmological constant, Λ,asthe as SNAP, is going to scan intensively that particular ultimate explanation for the dark energy, except that redshift range. The net outcome of our analysis is that Λ should then be a running parameter, Λ = Λ(µ), an experimental determination, even with high preci- namely one that evolves with an energy scale µ char- sion, of a non-trivial EOS for the dark energy must be acteristic of the cosmological system. A picture of Λ interpreted with great care, whether it results into nor- like this is not essentially different from the quantum mal quintessence or into phantom energy. A running field theoretical running of, say, the electromagnetic cosmological constant, based on the standard princi- charge, e = e(µ), in QED. While in the latter case ples of quantum field theory, could still be responsible µ should√ be in the ballpark of the collider energy, for the observed dark energy of the universe. e.g., µ s in a e+e− interaction at LEP, in cosmology the scale µ should be suitably identified from some testable ansatz. In previous work [14] the appro- Acknowledgements priate running scale µ for the cosmological context was identified with H(t), because the expansion rate J.S. thanks A.A. Starobinsky for useful correspon- gives the typical energy of the cosmological gravi- dence. The work of J.S. has been supported in part by tons. Indeed H is of the order of the square root of MECYT and FEDER under project 2004-04582-C02- the 4-curvature scalar of the FLRW metric. From this 01, and also by the Dep. de Recerca de la Generalitat ansatz the primary renormalization group running of de Catalunya under contract CIRIT GC 2001SGR- the cosmological term with µ, i.e., Λ = Λ(µ), can 00065. The work of H. Štefanciˇ c´ is supported by the be easily converted into time-evolution, or alterna- Secretaria de Estado de Universidades e Investigación tively into redshift dependence Λ = Λ(z). And this of the Ministerio de Educación y Ciencia of Spain redshift dependence can then be matched to the gen- within the program Ayudas para la mobilidad de pro- eral quintessence-like behavior, leading to an effective fesores, investigadores, doctores y tecnólogos extran- EOS for the DE, pD = ωeffρD, where ωeff = ωeff(z) jeros en España. He also thanks the Dep. E.C.M. of is a non-trivial function of the redshift precisely deter- the Univ. of Barcelona for the hospitality. mined by the RG model. Remarkably enough it turns out that this effective EOS for Λ can be both of normal quintessence and of phantom type, depending on References the value and sign of a single parameter, ν,intheRG cosmological model. In this respect we should recall [1] Ya.B. Zeldovich, JETP Lett. 6 (1967) 883. that the present data suggest some tilt of the dark en- [2] S. Weinberg, Rev. Mod. Phys. 61 (1989) 1. [3] See, e.g., V. Sahni, A. Starobinsky, Int. J. Mod. Phys. A 9 ergy EOS into the phantom phase. Further remarkable (2000) 373; is that the effective EOS of our RG model follows, T. Padmanabhan, Phys. Rep. 380 (2003) 235; with striking resemblance, the qualitative behavior de- S. Nobbenhuis, gr-qc/0411093. rived in some model-independent fits to the most re- [4] A.G. Riess, et al., Astron. J. 116 (1998) 1009; S. Perlmutter, et al., Astrophys. J. 517 (1999) 565; cent data [31]. These fits suggest that ωeff > −1for R.A. Knop, et al., Astrophys. J. 598 (2003) 102; a long (quintessence-like) period in the past, and at A.G. Riess, Astrophys. J. 607 (2004) 665. the same time they suggest that the universe has just [5] D.N. Spergel, et al., Astrophys. J. Suppl. 148 (2003) 175; entered a phantom phase ωeff < −1 near our present. See also WMAP Collaboration, http://map.gsfc.nasa.gov/. Irrespective of the credit we may wish to give to this [6] M. Tegmark, et al., Phys. Rev. D 69 (2004) 103501. possibility at present, our analysis shows how to pos- [7] C. Wetterich, Nucl. Phys. B 302 (1988) 668; B. Ratra, P.J.E. Peebles, Phys. Rev. D 37 (1988) 3406; sibly account for anomalies of this sort without resort- For a review, see, e.g., P.J.E. Peebles, B. Ratra, Rev. Mod. ing to a true phantom scalar field. Finally, we have Phys. 75 (2003) 559, and the long list of references therein. shown that the effects resulting from the effective EOS [8] M. Reuter, C. Wetterich, Phys. Lett. B 188 (1987) 38. J. Solà, H. Štefanˇcić / Physics Letters B 624 (2005) 147–157 157

[9] K. Freese, F.C. Adams, J.A. Freeman, E. Mottola, Nucl. Phys. I.L. Shapiro, J. Solà, JHEP Proc. AHEP2003/013, astro- B 188 (1987) 797. ph/0401015. [10] A.D. Dolgov, in: G. Gibbons, S.W. Hawking, S.T. Tiklos [23] A. Bonanno, M. Reuter, Phys. Rev. D 65 (2002) 043508; (Eds.), The Very Early Universe, Cambridge Univ. Press, Cam- E. Bentivegna, A. Bonanno, M. Reuter, JCAP 01 (2004) 001. bridge, 1982; [24] See all the relevant information for SNAP in: http://snap. F. Wilczek, Phys. Rep. 104 (1984) 143. lbl.gov/, and for PLANCK in: http://www.rssd.esa.int/index. [11] R.D. Peccei, J. Solà, C. Wetterich, Phys. Lett. B 195 (1987) php?project=PLANCK. 183; [25] J.A. Frieman, D. Huterer, E.V. Linder, M.S. Turner, Phys. Rev. J. Solà, Phys. Lett. B 228 (1989) 317; D 67 (2003) 083505; J. Solà, Int. J. Mod. Phys. A 5 (1990) 4225. T. Padmanabhan, T.R. Choudhury, Mon. Not. R. Astron. [12] R.R. Caldwell, R. Dave, P.J. Steinhardt, Phys. Rev. Lett. 80 Soc. 344 (2003) 823; (1998) 1582; H.K. Jassal, J.S. Bagla, T. Padmanabhan, Mon. Not. R. Astron. I. Zlatev, L. Wang, P.J. Steinhardt, Phys. Rev. Lett. 82 (1999) Soc. Lett. 356 (2005) L11; 896. E.V. Linder, Phys. Rev. D 70 (2004) 023511; [13] See, e.g., R.G. Vishwakarma, Class. Quantum Grav. 18 (2001) S. Hannestad, E. Mortsell, JCAP 0409 (2004) 001. 1159; [26] A. Melchiorri, L. Mersini, C.J. Odman, M. Trodden, Phys. Rev. J.M. Overduin, F.I. Cooperstock, Phys. Rev. D 58 (1998) D 68 (2003) 043509; 043506, and the long list of references therein. T. Roy Choudhury, T. Padmanabhan, Astron. Astrophys. 429 [14] I.L. Shapiro, J. Solà, JHEP 0202 (2002) 006, hep-th/0012227. (2005) 807; [15] I.L. Shapiro, J. Solà, Phys. Lett. B 475 (2000) 236, hep- S. Hannestad, E. Mortsell, JCAP 0409 (2004) 001; th/9910462. A. Upadhye, M. Ishak, P.J. Steinhardt, astro-ph/0411803; [16] R.D. Peccei, Phys. Rev. D 71 (2005) 023527; R. Lazkoz, S. Nesseris, L. Perivolaropoulos, astro-ph/0503230; R. Fardon, A.E. Nelson, N. Weiner, Astropart. Phys. 10 (2004) S. Hannestad, astro-ph/0504017; 005. C. España-Bonet, P. Ruiz-Lapuente, hep-ph/0503210. [17] I.L. Shapiro, J. Solà, C. España-Bonet, P. Ruiz-Lapuente, Phys. [27] R.R. Caldwell, Phys. Lett. B 545 (2002) 23; Lett. B 574 (2003) 149, astro-ph/0303306. H. Štefanciˇ c,´ Phys. Lett. B 586 (2004) 5; [18] I.L. Shapiro, J. Solà, Nucl. Phys. B (Proc. Suppl.) 127 (2004) R.R. Caldwell, M. Kamionkowski, N.N. Weinberg, Phys. Rev. 71, hep-ph/0305279. Lett. 91 (2003) 071301; [19] A. Babic, B. Guberina, R. Horvat, H. Štefanciˇ c,´ Phys. Rev. S.M. Carroll, M. Hoffman, M. Trodden, Phys. Rev. D 68 D 65 (2002) 085002; (2003) 023509; B. Guberina, R. Horvat, H. Štefanciˇ c,´ Phys. Rev. D 67 (2003) S. Nojiri, S.D. Odintsov, Phys. Rev. D 70 (2004) 103522; 083001; E. Elizalde, S. Nojiri, S.D. Odintsov, P. Wang, hep-th/0502082; A. Babic, B. Guberina, R. Horvat, H. Štefanciˇ c,´ astro- V.K. Onemli, R.P. Woodard, Phys. Rev. D 70 (2004) 107301. ph/0407572. [28] H. Štefanciˇ c,´ Eur. Phys. J. C 36 (2004) 523. [20] I.L. Shapiro, J. Solà, H. Štefanciˇ c,´ JCAP 0501 (2005) 012, hep- [29] T. Appelquist, J. Carazzone, Phys. Rev. D 11 (1975) 2856. ph/0410095. [30] P. Wang, X.H. Meng, Class. Quantum Grav. 22 (2005) 283; [21] F. Bauer, gr-qc/0501078; R. Opher, A. Pelinson, Phys. Rev. D 70 (2004) 063529. B. Guberina, R. Horvat, H. Štefanciˇ c,´ JCAP 05 (2005) 01, [31] U. Alam, V. Sahni, A.A. Starobinsky, JCAP 0406 (2004) 008; astro-ph/0503495. U. Alam, V. Sahni, T.D. Saini, A.A. Starobinsky, Mon. Not. R. [22] C. España-Bonet, P. Ruiz-Lapuente, I.L. Shapiro, J. Solà, Astron. Soc. 354 (2004) 275. JCAP 0402 (2004) 006, hep-ph/0311171;