Relaxing the Cosmological Constraints on Unparticle Dark Component

Eur. Phys. J. C (2009) 62: 579–586 DOI 10.1140/epjc/s10052-009-1056-5

Relaxing the cosmological constraints on unparticle dark component

Hao Weia Department of Physics, Beijing Institute of Technology, Beijing 100081, China

Received: 7 February 2009 / Revised: 22 April 2009 / Published online: 3 June 2009 © Springer-Verlag / Società Italiana di Fisica 2009

Abstract Unparticle physics has been an active field since where P is the 4-momentum; Adu is the normalization fac- the seminal work of Georgi. Recently, many constraints on tor; du is the scaling dimension. When du = 1, (1) reduces to unparticles from various observations have been considered the familiar one of a massless particle. The scaling dimen- in the literature. In particular, the cosmological constraints sion du is constrained by the unitarity of the conformal alge- on the unparticle dark component put it in a serious situa- bra and the requirement to avoid the singular behavior [1, 4]. tion. In this work, we try to find a way out of this serious The theoretical bounds are 1 ≤ du ≤ 2 (for bosonic unparti- situation, by including the possible interaction between dark cles) or 3/2 ≤ du ≤ 5/2 (for fermionic unparticles) [4]. Typ- energy and the unparticle dark component. ically, values 1 ≤ du ≤ 2 have been extensively considered in the literature. PACS 98.80.Es · 14.80.-j · 95.36.+x · 98.80.-k The pressure and energy density of thermal unparticles have been derived in [23]. They are − T 2(du 1) C(d ) 1 Introduction p = g T 4 u , u s 2 (2) Λu 4π − Recently, the so-called unparticle physics has been an active T 2(du 1) C(d ) ρ = ( d + )g T 4 u , u 2 u 1 s 2 (3) field since the seminal work of Georgi [1, 2]. It was based Λu 4π on the hypothesis that there could be an exact scale invariant C = + + hidden sector resisted at a high energy scale. A prototype where (du) B(3/2,du)Γ (2du 2)ζ(2du 2), while B, model of such a sector is given by the Banks–Zaks theory Γ , ζ are the Beta, Gamma and Zeta functions, respec- which flows to an infrared fixed point at a lower energy scale tively. These are the results for the bosonic unparticles. The pressure and energy density of fermionic unparticles can Λu through dimensional transmutation [3]. Recently, there − + C [ − (2du 1)]C has been a flood of papers on the unparticle phenomenol- be obtained by replacing (du) by 1 2 (du). ogy. We refer to e.g. [3, 4] for some brief reviews. In fact, Therefore, the equation-of-state parameter (EoS) of both the soon after the seminal work of Georgi [1, 2], some authors bosonic and fermionic unparticles is given by [23] considered the constraints on unparticles from various ob- ≡ pu = 1 servations, such as collider experiments [5–7], the new long wu . (4) ρu 2du + 1 range force experiments [8–11], the solar and reactor neu- = = trinos data [12, 13], as well as the observations from astro- When du 1, we have wu 1/3, which is the same as radi- →∞ → physics and cosmology [11, 14–22, 47]. These observations ation. When du , we find wu 0, which approaches put fairly stringent constraints on the unparticle. the EoS of pressureless matter. In the intermediate case, the The unparticle does not have a definite mass and instead EoS of unparticles is different from the one of radiation or has a continuous spectral density as a consequence of scale cold dark matter and generically lies in between. Since the invariance [1, 2](seealsoe.g.[22, 23, 48]), unparticle interacts weakly with standard model particles, it is “dark” in this sense. Some authors regard the unparticle as − 2 = 0 2 2 du 2 a candidate of dark matter [24, 25](seealso[22, 23]). Not- ρ P Adu θ P θ P P , (1) ing that the EoS of unparticles (wu > 0) is different from the one of cold dark matter (wm = 0), we instead call it “dark a e-mail: [email protected] component” to avoid confusion. 580 Eur. Phys. J. C (2009) 62: 579–586

In [22], the cosmological constraints on the unparticle So, the total energy conservation equation ρ˙tot + 3Hρtot × dark component have been considered, by using the type Ia (1 + wtot) = 0 is preserved. H ≡˙a/a is the Hubble para- supernovae (SNIa), the shift parameter of the cosmic mi- meter; a = (1 + z)−1 is the scale factor of the universe (we crowave background (CMB), and the baryon acoustic os- set a0 = 1); the subscript “0” indicates the present value of cillation (BAO). The authors of [22] found that du > 60 the corresponding quantity; z is the redshift; a dot denotes at 95% confidence level (C.L.) for the ΛUDM model in the derivative with respect to cosmic time t. The interac- which the unparticle is the sole dark matter. As mentioned tion forms extensively considered in the literature (see for above, however, the theoretical bounds on unparticles are instance [26–34, 49] and references therein) are Q ∝ Hρm, ≤ ≤ ≤ ≤ ˙ 2 1 du 2 (for bosonic unparticles) or 3/2 du 5/2 (for Hρde, Hρtot, κρmφ (where κ ≡ 8πG), and so on. In this fermionic unparticles) [4]. The situation is serious. Even for work, for simplicity, we consider the interaction term the ΛUCDM model in which the unparticle dark component co-exists with cold dark matter, they found that the unparti- Q = 3αHρu, (8) cle dark component can at most make up a few percent of the total cosmic energy density if du < 10; so that it cannot where α is a constant. So (6) becomes ρ˙u + 3Hρu × + eff = eff ≡ − be a major component [22]. In fact, it is easy to understand (1 wu ) 0, where wu wu α. It is easy to find that these results. Since the unparticle dark component scales as −3(1+wu) −3 −3(1+weff) a whereas cold dark matter scales as a (here a is ρu = ρu0a u . (9) the scale factor of the universe), for wu > 0, the energy den- −3 sity of the unparticle dark component decreases faster than On the other hand, from (7), we have ρm = ρm0 a .For the one of cold dark matter. So it is not surprising that the convenience, we introduce the fractional energy density 2 energy density of the unparticle dark component cannot be Ωi ≡ (8πGρi)/(3H ), where i = de, u and m. comparable with cold dark matter at the present epoch. To In the following sections, similar to [22], we consider the make a considerable contribution to the total cosmic energy cosmological constraints on the unparticle dark component density, du should be very large to make wu 0 so that the which interacts with dark energy, by using the observations unparticle dark component could mimic cold dark matter. of SNIa, the shift parameter R of CMB, and the distance pa- In this work, we try to find a way out of the serious situa- rameter A from BAO. In Sect. 2, we present the cosmolog- tion mentioned above. Our physical motivation is very sim- ical data and the methodology used to constrain the mod- ple. If dark energy, the major component of the universe, can els. In Sects. 3 and 4, we consider the observational con- decay into unparticles, the energy density of the unparticle straints on the interacting ΛUCDM model and the interact- dark component should decrease more slowly. Since both ing XUCDM model, respectively. As expected, we find that dark energy and unparticle are unseen, the interaction be- the cosmological constraints on the unparticle dark compo- tween them is not prevented. If the dilution rates of unpar- nent can be significantly relaxed, thanks to the possible in- ticle dark component and cold dark matter are comparable, teraction between dark energy and unparticle dark compo- it is possible to have a considerable energy density of the nent. The tension between theoretical bounds and cosmolog- unparticle dark component at the present epoch, without re- ical constraints could be removed. Finally, some concluding quiring d to be very large. In fact, this is similar to the key u remarks are given in Sect. 5. point of the interacting dark energy models which are extensively considered in the literature to alleviate the cosmological coincidence problem (see e.g. [26–34, 49] and references therein). 2 Cosmological data Here, we consider a flat universe which contains dark energy, the unparticle dark component and pressureless mat- In this work, we perform a χ2 analysis to obtain the con- ter (including cold dark matter and baryons). We assume straints on the model parameters. The data points of the 307 that dark energy and unparticle dark component exchange Union SNIa compiled in [35] are given in terms of the dis- energy according to (see e.g. [26–34, 49] and references tance modulus μobs(zi). On the other hand, the theoretical therein) distance modulus is defined as

ρ˙ + 3Hρ (1 + w ) =−Q, (5) ≡ + de de de μth(zi) 5log10 DL(zi) μ0, (10) ρ˙ + 3Hρ (1 + w ) = Q, (6) u u u ≡ − where μ0 42.38 5log10 h and h is the Hubble constant whereas pressureless matter (including cold dark matter and H0 in units of 100 km/s/Mpc, whereas baryons) evolves independently, i.e., z dz˜ DL(z) = (1 + z) , (11) ρ˙m + 3Hρm = 0. (7) 0 E(z˜; p) Eur. Phys. J. C (2009) 62: 579–586 581 in which E ≡ H/H0; p denotes the model parameters. The index ns is taken to be 0.960, which has been updated from χ2 from the 307 Union SNIa are given by the WMAP5 data [41]. So the total χ2 is given by

[μ (z ) − μ (z )]2 χ2 =˜χ2 + χ2 + χ2 , (17) χ2(p) = obs i th i , μ CMB BAO μ 2 (12) σ (zi) i ˜ 2 2 = − 2 2 where χμ is given in (14), χCMB (R Robs) /σR and 2 = − 2 2 where σ is the corresponding 1σ error. The parameter μ0 is χBAO (A Aobs) /σA. The best-fit model parameters are a nuisance parameter but it is independent of the data points. determined by minimizing the total χ2. As is well known, 2 ≡ 2 − 2 ≤ One can perform an uniform marginalization over μ0.How- the 68% C.L. is determined by Δχ χ χmin 1.00, ever, there is an alternative way. Following [36–38], the min- 2.30, 3.53, 4.72 and 5.89 for n = 1, 2, 3, 4 and 5, respec- imization with respect to μ0 can be made by expanding the tively, where n is the number of free model parameters. Sim- 2 ilarly, the 95% C.L. is determined by Δχ 2 ≡ χ2 − χ2 ≤ χμ of (12) with respect to μ0 as min 4.00, 6.17, 8.02, 9.70 and 11.3forn = 1, 2, 3, 4 and 5, 2 = ˜ − ˜ + 2 ˜ respectively. χμ(p) A 2μ0B μ0C, (13) In the following sections, we consider the observational where constraints on the interacting ΛUCDM model and the interacting XUCDM model, respectively, by using the method- [μ (z ) − μ (z ; μ = 0, p)]2 A(˜ p) = obs i th i 0 , ology presented here. σ 2 (z ) i μobs i μ (z ) − μ (z ; μ = 0, p) B(˜ p) = obs i th i 0 , 3 Cosmological constraints 2 σ (zi) i μobs on the interacting ΛUCDM model 1 C˜ = . 2 In this section, we consider the interacting ΛUCDM model σ (zi) i μobs in which the role of dark energy is played by the decay- ing cosmological constant (vacuum energy), as in the well- Equation (13) has a minimum for μ = B/˜ C˜ at 0 known Λ(t)CDM model [50–53]. In this case, (5) becomes ˜ B(p)2 − + eff ˜ 2 = ˜ − ˙ =− =− 3(1 wu ) χμ(p) A(p) . (14) ρΛ 3αHρu 3αρu0Ha . (18) C˜ Noting that the case without interaction (α = 0) was con- 2 =˜2 Since χμ,min χμ,min obviously, we can instead minimize sidered in [22], there are two other different cases with χ˜ 2 which is independent of μ . The shift parameter R of eff = − eff =− μ 0 wu 1 and wu 1 for this differential equation. We CMB is defined by [39, 40] will study them one by one. z∗ dz˜ ≡ 1/2 3.1 The interacting ΛUCDM model with weff = −1 R Ωm0 , (15) u 0 E(z)˜ If weff = −1, the solution for (18)isgivenby where the redshift of recombination z∗ = 1090 which has u been updated in the Wilkinson Microwave Anisotropy Probe αρu0 − + eff ρ = a 3(1 wu ) + const., (19) 5-year (WMAP5) data [41]. The shift parameter R relates Λ + eff 1 wu the angular diameter distance to the last scattering surface, the comoving size of the sound horizon at z∗ and the angular where const. is an integration constant. Obviously, this solu- eff =− scale of the first acoustic peak in CMB power spectrum of tion diverges for wu 1 which will be considered in the temperature fluctuations [39, 40]. The value of R has been next subsection. Inserting (19) into the Friedmann equation, 2 = 2 + + = = updated to 1.710 ± 0.019 from the WMAP5 data [41]. On H κ (ρΛ ρu ρm)/3, and requiring E(z 0) 1, one the other hand, the distance parameter A of the measurement can determine the integration constant. Finally, we find that of the BAO peak in the distribution of SDSS luminous red α + eff galaxies [42–45] is given by E2 = Ω (1 + z)3 + 1 + Ω (1 + z)3(1 wu ) m0 1 + weff u0 u z 2/3 / − 1 b dz˜ α A ≡ Ω1 2E(z ) 1/3 , (16) + 1 − Ω − 1 + Ω . (20) m0 b ˜ m0 + eff u0 zb 0 E(z) 1 wu where zb = 0.35. In [46], the value of A has been determined There are four free model parameters, namely, Ωm0, Ωu0, −0.35 2 to be 0.469 (ns/0.98) ± 0.017. Here the scalar spectral du and α. By minimizing the corresponding total χ in (17), 582 Eur. Phys. J. C (2009) 62: 579–586

Fig. 1 (Color online) The 68% C.L. (black solid lines) and 95% C.L. (red dashed lines) contours in the Ωu0 − du plane and the α − du plane for eff = − the interacting ΛUCDM model with wu 1. The best ﬁt is indicated by the red solid point

we find the best-fit parameters Ωm0 = 0.280, Ωu0 = 0.005, = = 2 = du 1.722 and α 0.116, while χmin 311.924. In Fig. 1, we present the corresponding 68% and 95% C.L. contours in the Ωu0 −du plane and the α −du plane for the interacting eff = − ΛUCDM model with wu 1, while the other parameters are taken to be the corresponding best-fit values. Obviously, from the left panel of Fig. 1, we see that the unparticle dark component can at most make up a few percent of the total cosmic energy density, so that it cannot be a major component in this case.

eff =− 3.2 The interacting ΛUCDM model with wu 1

As mentioned above, the solution (19) diverges for eff =− wu 1. Therefore, one is required to consider this case eff =− separately. If wu 1, we find that Fig. 2 (Color online) The 68% C.L. (black solid line) and 95% C.L. 1 − = + = + (red dashed line) contours in the Ωu0 du plane for the interacting α 1 wu 1 . (21) eff =− 2du + 1 ΛUCDM model with wu 1. The best fit is indicated by the red solid point So, α is no longer a free model parameter. In this case, (9) becomes ρ = ρ . Now, the solution for (18) is given by u u0 In this case, there are three free model parameters, namely, ρ =−3αρ ln a + const., (22) Ωm0, Ωu0 and du. By minimizing the corresponding total Λ u0 2 χ in (17), we find the best-fit parameters Ωm0 = 0.270, = = 2 = where const. is an integration constant. Then, inserting (22) Ωu0 0.020 and du 0.273, while χmin 313.168. In into the Friedmann equation and requiring E(z = 0) = 1, Fig. 2, we present the corresponding 68% and 95% C.L. one can determine the integration constant. Finally, we find contours in the Ωu0 − du plane for the interacting ΛUCDM eff =− that model with wu 1, while the other parameters are taken to be the corresponding best-fit values. It is a pleasure to find 1 E2 = Ω (1 + z)3 + 3Ω 1 + ln(1 + z) that the unparticle dark component can have a considerable m0 u0 2d + 1 u contribution to the total cosmic energy density (Ωu0 > 0.1) + (1 − Ωm0). (23) in the 95% C.L. region while 1 ≤ du ≤ 2 (for bosonic unpar- Eur. Phys. J. C (2009) 62: 579–586 583 ticles) or 3/2 ≤ du ≤ 5/2 (for fermionic unparticles). The the XUCDM model without interaction, while the other pa- serious situation found in [22] could be relaxed in this case. rameters are taken to be the corresponding best-fit values. Obviously, from the left panel of Fig. 3, we see that the unparticle dark component can at most make up a few percent 4 Cosmological constraints of the total cosmic energy density, so that it cannot be a ma- on the (interacting) XUCDM model jor component in the case without interaction.

eff = In this section, we consider the (interacting) XUCDM model 4.2 The interacting XUCDM model with wu wx in which the EoS of dark energy wx is a constant. We can eff = recast (5)as If wu wx, the solution for (24) is given by dρx 3 3α − + eff αρu0 − + eff − + + ρ (1 + w ) =− ρ a 3(1 wu ). (24) ρ = a 3(1 wu ) + const. × a 3(1 wx), (26) x x u0 x eff − da a a wu wx In the following, we first consider the case without inter- where const. is an integration constant. Obviously, this solu- action (α = 0). Then, we consider two different cases with eff tion diverges for w = wx, which will be considered in the eff = eff = u wu wx and wu wx for the differential equation (24). next subsection. Inserting (26) into the Friedmann equation, 2 2 H = κ (ρx + ρu + ρm)/3, and requiring E(z = 0) = 1, one 4.1 The XUCDM model without interaction can determine the integration constant. Finally, we find that It is natural to see first what will happen if there is no inter- α + eff E2 = Ω (1 + z)3 + 1 + Ω (1 + z)3(1 wu ) action between dark energy and unparticle dark component. m0 eff − u0 wu wx In this case, α = 0. One can easily write down α + 1 − Ωm0 − 1 + Ωu0 2 3 3(1+wu) eff − E = Ωm0(1 + z) + Ωu0(1 + z) wu wx + 3(1+wx) 3(1 wx) + (1 − Ωm0 − Ωu0)(1 + z) . (25) × (1 + z) . (27)

There are four free model parameters, namely, Ωm0, Ωu0, du There are five free model parameters, namely, Ωm0, Ωu0, 2 and wx. By minimizing the corresponding total χ in (17), du, α and wx. By minimizing the corresponding total 2 we find the best-fit parameters Ωm0 = 0.283, Ωu0 = 0.0, χ in (17), we find the best-fit parameters Ωm0 = 0.277, = =− 2 = = = = =− du 0.540 and wx 1.006, while χmin 311.981. In Ωu0 0.214, du 13.478, α 0.589, and wx 1.666, 2 = Fig. 3, we present the corresponding 68% and 95% C.L. while χmin 311.077. In Fig. 4, we present the correspond- contours in the Ωu0 − du plane and the wx − du plane for ing 68% and 95% C.L. contours in the Ωu0 − du plane

Fig. 3 (Color online) The 68% C.L. (black solid lines) and 95% C.L. (red dashed lines) contours in the Ωu0 − du plane and the wx − du plane for the XUCDM model without interaction. The best ﬁt is indicated by the red solid point 584 Eur. Phys. J. C (2009) 62: 579–586

Fig. 4 (Color online) The 68% C.L. (black solid lines) and 95% C.L. (red dashed lines) contours in the Ωu0 − du plane and the α − du plane for eff = the interacting XUCDM model with wu wx. The best ﬁt is indicated by the red solid point

Fig. 5 (Color online) The 68% C.L. (black solid lines) and 95% C.L. (red dashed lines) contours in the Ωu0 − du plane and the wx − du plane for eff = the interacting XUCDM model with wu wx. The best ﬁt is indicated by the red solid point

− eff = and the α du plane for the interacting XUCDM model 4.3 The interacting XUCDM model with wu wx eff with w = wx, while the other parameters are taken to be u As mentioned above, the solution (26) diverges for the corresponding best-fit values. It is fortunate to find that eff = wu wx. Therefore, one is required to consider this case the unparticle dark component can have a sizable contribu- eff = separately. If wu wx, we find that tion to the total cosmic energy density (0.1 <Ωu0 < 0.2 = − = 1 − in the 68% C.L. region and 0.1 <Ωu0 < 0.25 in the 95% α wu wx wx. (28) 2du + 1 C.L. region) while 1 ≤ du ≤ 2 (for bosonic unparticles) or So, α is no longer a free model parameter. In this case, the 3/2 ≤ d ≤ 5/2 (for fermionic unparticles). The serious sit- u solution for (24) is given by uation found in [22] could be significantly relaxed in this −3(1+wx) case. ρx = (const. − 3αρu0 ln a)a , (29) Eur. Phys. J. C (2009) 62: 579–586 585 where const. is an integration constant. Inserting (29)into them is not prevented. If the dilution rates of unparticle dark the Friedmann equation and requiring E(z = 0) = 1, one component and cold dark matter are comparable, it is pos- can determine the integration constant. Finally, we find that sible to have a considerable energy density of the unparticle dark component at the present epoch, without requiring du E2 = Ω (1 + z)3 m0 to be very large. In fact, this is similar to the key point of the 1 interacting dark energy models which are extensively con- + (1 − Ωm0) + 3Ωu0 − wx ln(1 + z) 2du + 1 sidered in the literature to alleviate the cosmological coinci- + dence problem (see e.g. [26–34, 49] and references therein). × + 3(1 wx) (1 z) . (30) As shown above, the cosmological constraints on the un- There are four free model parameters, namely, Ω , Ω , d particle dark component can be relaxed in the interacting m0 u0 u eff =− w χ2 ΛUCDM model with wu 1, and both the interacting and x. By minimizing the corresponding total in (17), eff eff XUCDM models with w = wx and w = wx. However, we find the best-fit parameters Ωm0 = 0.270, Ωu0 = 0.0, u u = =− 2 = one might note that the unparticle dark component with cos- du 1.969 and wx 0.954, while χmin 313.138. In Fig. 5, we present the corresponding 68% and 95% C.L. mological constant is constrained by the observational data relatively more severe than in the case with dark energy. This contours in the Ωu0 − du plane and the wx − du plane for the eff = is also easy to understand. Notice that the EoS of dark en- interacting XUCDM model with wu wx, while the other parameters are taken to be the corresponding best-fit values. ergy wx is a free model parameter, whereas the EoS of the − It is a pleasure to find that the unparticle dark component can cosmological constant is a constant 1. Therefore, the num- have a considerable contribution to the total cosmic energy ber of free model parameters in the interacting XUCDM model is larger than the one in the interacting ΛUCDM density (Ωu0 > 0.1) in the 95% C.L. region while 1 ≤ du ≤ 2 model. As is well known, statistically, the constraints should (for bosonic unparticles) or 3/2 ≤ du ≤ 5/2 (for fermionic unparticles). The serious situation found in [22] could also be looser in this case. be relaxed in this case. Acknowledgements We thank the anonymous referee for quite use- ful comments and suggestions, which helped us to improve this work. We are grateful to Professors Rong-Gen Cai, Shuang Nan Zhang, 5 Concluding remarks Xuelei Chen, and Xinmin Zhang, for helpful discussions. We also thank Minzi Feng, as well as Yan Gong, Xin Zhang and Dao-Jun Liu, Unparticle physics has been an active field since the seminal for kind help and discussions. We would like to express our gratitude to IHEP (Beijing) for kind hospitality during the period of National Cos- work of Georgi [1, 2]. Recently, many constraints on un- mology Workshop. This work was supported by the Excellent Young particles from various observations were considered in the Scholars Research Fund of Beijing Institute of Technology. literature. In particular, the cosmological constraints on the unparticle dark component put it in a serious situation [22]. In the present work, we try to find a way out of this seri- References ous situation, by including the possible interaction between dark energy and the unparticle dark component. In fact, as 1. H. Georgi, Phys. Rev. Lett. 98, 221601 (2007). hep-ph/0703260 we have shown above, the cosmological constraints on un- 2. H. Georgi, Phys. Lett. B 650, 275 (2007). arXiv:0704.2457 particle dark component can be relaxed in the interacting 3. K. Cheung, W.Y. Keung, T.C. Yuan, AIP Conf. 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