arXiv:0710.5129v3 [hep-ph] 23 Jan 2009 n h rpriso h xadn nvre[] tde- factor since It scale temperature metric [1]. given universe a at expanding ρ density the determin- energy in the of role termines properties crucial a the plays ing carrier, energy of species ω eeeainsnetedclrto aaee spropor- with 3 is (CDM) + (1 parameter to deceleration tional the since deceleration rmuprilswoeESparameter EoS whose unparticles from invoked particular? often in the there particles besides is massive And matter interacting is universe? weakly dark the What to of [3]. alternative side’ matter matter ‘dark any dark this ordinary by of of nature carried instead the is energy pre- universe dark the our in and of in discovery majority budget recent the energy the that observations has to cosmological this due And cision urgent [2]. parameters en- even EoS relic become their various and determine densities to ergy cosmology important an modern is for It universe. task our of expansion the accelerate rcinta htnwhose photon than traction ω D n htnwith photon and CDM ncsooyadatohsc.I nepniguniverse, expanding an impacts In its astrophysics. of and some cosmology investigate on we Then matter. ordinary dubbed be hra natce:ANwFr fEeg est nthe in Density Chen Energy Shao-Long of Form New Universe A Unparticles: Thermal edopitrqet to requests offprint Send 1 2 3 wl eisre yteeditor) the by inserted be (will No. manuscript EPJ a Q h ai fpesr ( pressure of ratio the , eateto hsc n etrfrTertclSciences, Theoretical for Center and Physics of Department en etrfrCsooyadPril srpyis Nati Astrophysics, Particle and Cosmology for Center Leung eateto hsc,Nna nvriy ini 300071 Tianjin University, Nankai Physics, of Department vle ihteFidanRbrsnWle (FRW) Friedmann-Robertson-Walker the with evolves ncsooyteeuto fsae(o)parameter (EoS) state of equation the cosmology In nti okw eosrt oe ido e energy new of kind novel a demonstrate we work this In [email protected] < − 1 / ubtv ehius–1.5H ofra edter,alg theory, field Conformal 11.25.Hf – techniques turbative and ρ PACS. if natceeeg density energy unparticle sgvnb 1 by given is Abstract. date version: Revised / date Received: pc tutr.W n htteeuto fsaeparameter state of equation the that find We structure. space n omlgclconstant cosmological and 3 U unmatter ρ ( U T ω ρ γ ( 0 M U T ω tpeetpoo temperature photon present at ) ( D 1 ) 88.qPril-hoyadfil-hoymdl fteear the of models field-theory and Particle-theory 98.80.Cq rvdsasrne rvttoa at- gravitational stronger a provides 0 = T , tahg eopigtemperature decoupling high a at ) ρ 2 γ 0 R o xml,wiecl akmatter dark cold while example, For . ioGn He Xiao-Gang , sn htnupril neatosfrillustration. for interactions photon-unparticle using ) Unparticle ob itnuse rmCMand CDM from distinguished be to , as / : (2 ω R d U U p − oeeg est ( density energy to ) qa o1 to equal 3(1+ )poiiganwfr feeg noruies.I nexpan an In universe. our in energy of form new a providing 1) + ω U γ ω ihsaigdimension scaling with 1 = ) tas xstert of rate the fixes also It . ρ 1 U , / 2 ( ,qitsec with quintessence 3, / u-egHu Xue-Peng , T (2 vle rmtclydffrnl rmta o htn.Fo photons. for that from differently dramatically evolves ) Λ d ω U U with ) tmight It 1). + isbetween lies T γ 0 ω ρ ag nuht lyterl fdr atr ecalculate We matter. dark of role the play to enough large , Λ ,fra for ), = T 3 D n iLiao Yi and , d − U svr ml,i spsil ohv ag ei density relic large a have to possible is it small, very is 1 a euirtemlpoete u oisuiu phase unique its to due properties thermal peculiar has ainlTia nvriy Taipei University, Taiwan National nlTia nvriy Taipei University, Taiwan onal h density the clydffrn hnta o htn.For photons. for that than different ically svr ml,i spsil ohv ag ei density relic large a have to possible is it ρ small, very is dniy nta,isknmtc smil eemndby determined an mainly such dimension to is scaling apply its kinematics not its does instead, named mass identity; of scale, notion the The below unparticle. freedom of degree scale-invariant ainta eti iheeg hoywt nontriv- a scale with some theory at fixed-point energy infrared high ial certain that vation matter. dark of role the play to un u ob eeal ieetfo hto photons. of that from different which generally properties, be basic to thermodynamics their out out from turns work directly we work unparticles ana- this of an In as mind. photon massless in invoked the logue are with EoS invariance unparticle conformal the for or of processes, arguments physics naive particle the ordinary tempera- in zero occurring at as treated phenomenologi- ture are unparticle unparticles of either glut studies, a cal In [11]. to few [7,8] [9,10], a issues implications mention theoretical astrophysical and to tests cosmological [5,6] and precision effects on from physics [4], physics collider Georgi and unparticle a of of work been seminal has aspects the various There since (EFT). activities de- theory of well burst field be effective can in interaction the scribed and to relevant; feebly, however physically particles, be with interact must unparticle h eairo natceeeg density energy unparticle of behavior the U ( h ocp fupril 4 tm rmteobser- the from stems [4] unparticle of concept The T bacsrcue 48.jOhrparticles Other 14.80.-j – structures ebraic γ 0 tpeetpoo temperature photon present at ) a 3 ω U h ai fpesr oeeg density, energy to pressure of ratio the , ρ yUies 11.kOhrnonper- Other 11.15.Tk – Universe ly U ( T D tahg eopigtemperature decoupling high a at ) d U ne cl rnfrain.The transformations. scale under iguies,the universe, ding r d U Λ > U T ρ ,even 1, γ U 0 d a eeo a develop may ag enough large , ( U T T > sdramat- is ) D ,ee if even 1, T D 2 Shao-Long Chen et al.: Thermal Unparticles: A New Form of Energy Density in the Universe
The thermodynamics of a gas of bosonic particles with of B function, the apparent singularity at d = 1 can be mass µ is determined by the partition function: removed explicitly: U
4 2 d p 0 0 gsV (d ) ln Z(µ )= gsV 2π2p θ(p ) ln Z = C U , (7) 4 3 2(dU 1) 2 − (2π) β (βΛ ) − 4π Z U 2 2 p0β δ(p µ ) ln(1 e− ), (1) × − − with (d ) = B(3/2, d )Γ (2d + 2)ζ(2d +2). It is now C U U U U 1 straightforward to work out the quantities: where V, β = T − are the volume and inverse temperature in natural units respectively, and gs accounts for degrees of 2(dU 1) freedom like spin. The density of states in four-momentum 4 T − (d ) p = gsT C U , space is proportional to the δ function due to the disper- U Λ 4π2 U sion relation for particles. There is no such a constraint in 2(dU 1) 4 T − (d ) the case of unparticles, whose density of states is dictated ρ = (2d + 1)gsT C U . (8) U U Λ 4π2 by the scaling dimension d of the corresponding field to U be proportional to [4]: U Again the case of massless particles is recovered correctly 4 by setting d = 1 and (1) = 2π4/45. The above results d p 0 2 2 dU 2 U θ(p )θ(p )(p ) − . (2) imply the following EoSC parameter for unparticles: (2π)4 1 ω = (2d + 1)− . (9) Nevertheless, we can interpret it in terms of a continuous U U collection of particles with the help of a spectral function 2 2 2 dU 2 The results for fermionic unparticles can be obtained by ̺(µ ) θ(µ )(µ ) − [5]: (2dU +1) ∝ replacing (d ) by (1 2− ) (d ). U U 4 It is clearC that ω −is very differentC from that for pho- 0 2 2 d p 2 2 2πθ(p )δ(p µ ) ̺(µ )dµ (3) tons or CDM, and genericallyU lies in between for d > 1. − (2π)4 This is in contrast to the naive expectation based onU con- In this construction, compared to the case of particles of formal theory arguments and the massless photon ana- a definite mass, µ2 serves as a new quantum number to logue. This arises essentially from the fact that unparti- be summed over with the weight ̺(µ2). cles exist only below a finite energy scale Λ as reflected U To write down the partition function for unparticles, in the spectral function ̺(µ2) while a conventional confor- we have to normalize ̺ correctly. Since unparticles exist mal theory is not characterized by such a scale. If the limit only below the scale Λ , the spectrum must terminate Λ were naively taken, which means there would be U → ∞ there [12]. Beyond the scale,U unparticles can be resolved no unparticles in the infrared, ρ would vanish trivially. U and are no more the suitable degrees of freedom to cope This is indeed not the case interested in here. The factor 2(1 dU ) with. This also implies that we should require βΛ > 1 for Λ − in p , ρ acts as an effective parameter in the U U self-consistency. We thus find the normalized spectrum,U lowU temperature theory, and the presence of Λ reflects its connection to the underlying theory that producesU the 2 2(1 dU ) 2 2 dU 2 ̺(µ ) = (d 1)Λ − θ(µ )(µ ) − , (4) unparticle. This connection between low and high energy U − U theories is completely expected, as for instance, in ther- which has the correct limit δ(µ2) as d 1+. Note that modynamics of solids viewed from atomic physics. integrability at the lower end of µ2 requiresU → d 1. The The ensemble of unparticles thus provides a new form partition function for unparticles is U ≥ of energy density in our universe, which will have impor-
2 tant repercussions for cosmology. We now study their im- ΛU plications in our expanding universe by concentrating on ln Z = dµ2̺(µ2) ln Z(µ2) 0 their contribution to the energy density in the universe. Z 2 (βΛU ) The unparticle energy density at present is determined by gsV (d 1) dU 2 = U − dy y − its initial value at the decoupling temperature TD where 2 3 2(dU 1) −4π β (βΛ ) − 0 unparticles drop out of the thermal equilibrium with stan- U Z ∞ √x dard model (SM) particles, and its evolution thereafter dx √x y ln(1 e− ) (5) × − − which is closely related to the EoS parameter. Zy In an FRW expanding universe, the energy density For βΛ > 1, the above integrals factorize to good preci- ρ(T ) (or ρ(R)) of a species after decoupling from equi- sion dueU to the exponential: librium is given by
3(1+ω) gsV (d 1) 2B(3/2, d 1) R ln Z = U − U − D 2 3 2(dU 1) ρ(R)= ρ(RD) , (10) 4π β (βΛ ) − (2d + 1) R U U Γ (2d + 2)ζ(2d + 2), (6) U U × where RD is the scale factor of the expanding universe at where Γ, B, ζ are standard functions and integration by decoupling. From now on, we will interchange the nota- parts has been used for 2d +1 > 0. Using the definition tions ρ(T ) and ρ(R) freely. Since photon expansion follows U Shao-Long Chen et al.: Thermal Unparticles: A New Form of Energy Density in the Universe 3
A practical study with a global fitting should make a sur- vey of all such interactions and those induced by thermal 1000 effects. For the purpose of illustration here, we consider
L below the unparticle-photon interactions: BBN
T dU µν dU µν H ˜ ˜ Γ 100 = λΛ− F Fµν + λΛ− F Fµν , (13) r
L L U U U U 0 Γ T H
Γ ˜
r where F, F are respectively the electromagnetic field ten- 10 sor and its dual, and the coefficients λ, λ˜ can be expressed in terms of the standard ones in Ref. [14]. We will treat the two interactions one by one. 1 The above interactions can bring photons and unpar- 1.0 1.2 1.4 1.6 1.8 2.0 ticles into equilibrium. Taking the λ term as an example, dU the cross section for γγ is 0 →U Fig. 1. The double ratio rγ (Tγ )/rγ (TBBN) as a function of dU . 1 s dU 1 σ(s)= λ2 A , (14) 4 Λ2 s dU RD/R = Tγ/TD, we have U where 3(1+ωU ) Tγ 5/2 ρ (Tγ)= ρ (TD) , 16π Γ (d +1/2) U U TD AdU = U (15) (2π)2dU Γ (d 1)Γ (2d ) 4 U U Tγ − ργ (Tγ)= ργ (TD) , (11) is a normalization factor for the unparticle density of states TD suggested in Ref. [4], and the interaction rate is where ργ,Tγ are the quantities for photons. For d > 1, U 2dU the unparticle energy density decreases more slowly than ζ(3)A U 2T Γ n σ(s)c = d λ2T , (16) the photon’s as the universe cools down. ≃ γ 8π2 Λ If unparticle is always in thermal equilibrium with pho- U ton, its energy density drops faster than photon when tem- where nγ is the photon number density, and we have used perature goes down. However, after unparticle freezes out s = (2T )2. of equilibrium, the situation is different. The ratios of the This rate is compared with the Hubble parameter H = 1/2 2 energy densities, rγ (T ) = ρ (T )/ργ(T ), at two tempera- 1.66g T /mPl in the radiation dominated era to deter- U ∗ tures T1 and T2 are related by mine at what temperature unparticles decouple from pho-
2dU −2 tons [1]. Here g is the total number of degrees of freedom 2d +1 ∗ 19 T1 U at the decoupling temperature and mPl =1.22 10 GeV rγ (T2)= rγ (T1) . (12) is the Planck mass. When Γ
dU −dU Table 1. Upper bound (λ/ΛU )max (in units of GeV ) and (close to 2), the scale can still be as low as a few hundred min the corresponding TD (in units of GeV) for various values for GeV which may be reached at LHC and ILC colliders. dU . Appropriate g∗ has been used for the given energy with The standard BBN theory explains data well. It is SM particles and a scalar unparticle. therefore important to make sure that at TBBN unparti- dU 4/3 5/3 2 cles do not cause problems. A simple criterion is to require dU −14 −13 −11 (λ/ΛU )max 1.04 × 10 7.17 × 10 5.11 × 10 that at this temperature the unparticle energy density be min 6 3 TD 7.37 × 10 2.70 × 10 3.68 × 10 less than the photon’s. With this restriction, it is interest- ing to see whether one can still have large relic unmatter at present. We find this is indeed possible. Although there
Table 2. ΛU and rγ (T ) = ρU (T )/ργ (T ) as functions of are many cases shown in Table 2 where rγ (TBBN) is larger 0 0 2 −47 4 0 ΩU (Tγ ). We have used ρcr(Tγ ) = 8.0992h × 10 GeV and than one, circumstances with sizable Ω (Tγ ) but small +0.04 U taken the central value for h = 0.73−0.03 [13]. rγ (TBBN) also appear at large d . This can easily be un- derstood from eq. (12) and fromU Fig.1. For d > 1, a small dU = 4/3 0 U 0 rγ (TBBN) can result in a sizable Ω (Tγ ). A universe dom- ΩU (Tγ ) 1.0 0.161 0.01 U 6 8 inated by unparticle between the BBN era and the matter ΛU (GeV) − 7.37 × 10 4.78 × 10 min −1 −2 or dark energy dominated universe is possible. rγ (TD ) − 9.90 × 10 6.13 × 10 rγ (TBBN) − 6.22 × 10 3.85 In the above discussions, the interactions of unparti- 2dU +1 dU = 5/3 cles with photons lead to an interaction rate Γ T 0 ∼ ΩU (Tγ ) 1.0 0.20 0.01 which brings unparticles and SM particles into equilib- 4 4 5 ΛU (GeV) 1.46 × 10 4.87 × 10 4.61 × 10 rium at a high temperature, and they decouple at a lower min −1 −2 −3 rγ (TD ) 2.48 × 10 4.97 × 10 2.48 × 10 temperature if the unparticle dimension d is larger than −1 U rγ (TBBN) 2.35 × 10 4.73 2.36 × 10 1. There are many possible ways unparticles can interact dU = 2 with SM particles , but not all interactions will have the 0 ΩU (Tγ ) 1.0 0.2 0.01 same properties as far as thermal equilibrium is concerned. 2 2 3 ΛU (GeV) 4.31 × 10 9.65 × 10 4.32 × 10 For example, we find that all of the operators involving SM min −2 −3 −4 rγ (TD ) 4.57 × 10 9.11 × 10 4.55 × 10 fermions listed in Ref.[14] will result in an interaction rate −1 −2 2dU 1 rγ (TBBN) 3.06 6.14 × 10 3.05 × 10 proportional to T − . If unparticles and SM fermions are required to be in thermal equilibrium at a high tem- perature and then decouple at a lower one, d must be larger than 3/2. On the contrary, if d is lessU than 3/2, In order that the relic density of unparticles is not then unparticles and SM fermions willU not be in thermal too large, say as large as the critical energy density (ρcr) equilibrium at a high temperature in the first place, but which would over close the universe, for a given TD one will be at a lower temperature till the epoch of matter has to choose a big enough Λ besides the requirement U dominated universe. Since when in thermal equilibrium, Λ > TD. This provides a way to constrain the scale U the unparticle density dilutes faster than SM particles, its Λ directly. We illustrate our results in Table 2 for sev- relic density today will be negligibly small if the equilib- eralU representative values of the ratio of energy densities, 0 0 0 0 rium sets in before or just after BBN with the unparticle Ω (Tγ )= ρ (Tγ )/ρcr(Tγ ), at Tγ . In our analysis, we as- relic density not larger than photon density. This is an U Umin sume TD = TD , as shown in Table 1. By equating ρ (TD) U interesting scenario to study, which may lead to sensitive that is obtained via eq. (8) on the one hand with the one information about the unparticle scaling dimension. from backward evolution via eq. (11) on the other, we can There is much to be explored for the roles that thermal determine Λ for each given d . Also shown are the values U minU unparticles can play in our universe. It is important to of the ratio rγ = ρ /ργ at TD and TBBN. Note that we analyze available cosmological and astrophysical data for do not assume a valueU for the dimensionless coupling λ; min a global fit with unparticle energy density integrated. We instead, it is fixed by Λ and TD = TD via eq. (17). U leave this for a detailed future study. For d = 4/3, we find that it is not possible to sat- U To summarize, we have studied for the first time the urate the critical density, nor the dark matter density thermal properties of unparticles. Due to its peculiar phase ˜ ΩDM = 0.2 [13]. With the constraint that TD < Λ , the space structure we found that the EoS parameter ω is 0 ˜ U U largest Ω (Tγ ) is 0.16 which occurs at TD = Λ . This, given by 1/(2d + 1), providing a new form of energy in U U of course, still leaves enough room for unparticle to play our universe. InU an expanding universe, the behavior of a significant role as dark matter. For d = 5/3 and 2, unparticle energy density ρ (T ) is dramatically different U we see that the present unparticle relic density can easily than that for photons. ForUd > 1, even if its value at U saturate the critical density and dark matter density. In a high decoupling temperature TD is very small, it could 0 all cases, ρ (TD) is smaller (in most cases much smaller) evolve into a sizable relic density ρ (Tγ ) at present, large U U than ργ (TD). Requiring the present relic of unmatter to enough to play the role of dark matter. We have exempli- be less than these densities one obtains conservative lower fied this with photon-unparticle interactions, and found ˜ min bounds on Λ for given TD = TD . For small d , the that it is indeed feasible to obtain a large relic energy U U scale Λ˜ is constrained to be very large, making low en- density of unparticles with the most stringent constraints ergy searchU for unparticle effects difficult. But for large d saturated. U Shao-Long Chen et al.: Thermal Unparticles: A New Form of Energy Density in the Universe 5
Acknowledgments This was supported in part by the grants NSC95-2112-M-002-038-MY3,NCTS (NSC96-2119- M-002-001), NCET-06-0211, and NSFC-10775074.
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