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Thermal Unparticles: a New Form of Energy Density in the Universe

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Thermal Unparticles: a New Form of Energy Density in the Universe EPJ manuscript No. (will be inserted by the editor) Thermal Unparticles: A New Form of Energy Density in the Universe Shao-Long Chen1,2, Xiao-Gang He1,2, Xue-Peng Hu3, and Yi Liao3 a 1 Department of Physics and Center for Theoretical Sciences, National Taiwan University, Taipei 2 Leung Center for Cosmology and Particle Astrophysics, National Taiwan University, Taipei 3 Department of Physics, Nankai University, Tianjin 300071 Received: date / Revised version: date Abstract. Unparticle U with scaling dimension dU has peculiar thermal properties due to its unique phase space structure. We find that the equation of state parameter ωU , the ratio of pressure to energy density, is given by 1/(2dU + 1) providing a new form of energy in our universe. In an expanding universe, the unparticle energy density ρU (T ) evolves dramatically differently from that for photons. For dU > 1, even if ρU (TD) at a high decoupling temperature TD is very small, it is possible to have a large relic density 0 0 ρU (Tγ ) at present photon temperature Tγ , large enough to play the role of dark matter. We calculate TD 0 and ρU (Tγ ) using photon-unparticle interactions for illustration. PACS. 98.80.Cq Particle-theory and field-theory models of the early Universe – 11.15.Tk Other nonper- turbative techniques – 11.25.Hf Conformal field theory, algebraic structures – 14.80.-j Other particles In cosmology the equation of state (EoS) parameter the behavior of unparticle energy density ρ (T ) is dramat- ω, the ratio of pressure (p) to energy density (ρ), for a ically different than that for photons. ForUd > 1, even if U species of energy carrier, plays a crucial role in determin- the density ρ (TD) at a high decoupling temperature TD ing the properties of the expanding universe [1]. It de- is very small,U it is possible to have a large relic density 0 0 termines the energy density at a given temperature since ρ (Tγ ) at present photon temperature Tγ , large enough ρ evolves with the Friedmann-Robertson-Walker (FRW) toU play the role of dark matter. 3(1+ω) metric scale factor R as R− . It also fixes the rate of deceleration since the deceleration parameter is propor- tional to (1 + 3ω)ρ. For example, while cold dark matter (CDM) with ωM = 0 provides a stronger gravitational at- The concept of unparticle [4] stems from the obser- traction than photon whose ωγ = 1/3, quintessence with vation that certain high energy theory with a nontriv- arXiv:0710.5129v3 [hep-ph] 23 Jan 2009 ωQ < 1/3 and cosmological constant Λ with ωΛ = 1 ial infrared fixed-point at some scale Λ may develop a − − accelerate the expansion of our universe. It is an important scale-invariant degree of freedom belowU the scale, named task for modern cosmology to determine various relic en- unparticle. The notion of mass does not apply to such an ergy densities and their EoS parameters [2]. And this has identity; instead, its kinematics is mainly determined by become even urgent due to the recent discovery in pre- its scaling dimension d under scale transformations. The cision cosmological observations that the majority of the unparticle must interactU with particles, however feebly, to energy budget in our universe is carried by dark matter be physically relevant; and the interaction can be well de- and dark energy instead of ordinary matter [3]. What is scribed in effective field theory (EFT). There has been a the nature of this ‘dark side’ of the universe? And is there burst of activities since the seminal work of Georgi [4], on any alternative to dark matter besides the often invoked various aspects of unparticle physics from precision tests weakly interacting massive particles in particular? and collider physics effects [5,6] to theoretical issues [7,8] In this work we demonstrate a novel kind of new energy and cosmological and astrophysical implications [9,10], to from unparticles whose EoS parameter ω lies between mention a few [11]. In a glut of unparticle phenomenologi- CDM and photon with ω equal to 1/(2d U+ 1). It might cal studies, either unparticles are treated at zero tempera- be dubbed unmatter, to beU distinguishedU from CDM and ture as occurring in ordinary particle physics processes, or ordinary matter. Then we investigate some of its impacts the naive arguments of conformal invariance are invoked on cosmology and astrophysics. In an expanding universe, for the unparticle EoS with the massless photon as an ana- logue in mind. In this work we work out thermodynamics Send offprint requests to: of unparticles directly from their basic properties, which a [email protected] turns out to be generally different from that of photons. 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.
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