2-loop Quantum Yang-Mills Condensate as Dark Energy
a r X i v :0710.0077v 1 [a s t r o -p h ] 29 S e p 20072-loop Quantum Yang-Mills Condensate as Dark
Energy
T.Y.Xia and Y.Zhang[*]Astrophysics Center University of Science and Technology of China Hefei,Anhui,China Abstract In seeking a model solving the coincidence problem,the e?ec-tive Yang-Mills condensate (YMC)is an alternative candidate for dark energy.A study is made for the model up to the 2-loop order of quantum corrections.It is found that,like in the 1-loop model,for generic initial conditions during the radiation era,there is always a desired tracking solution,yielding the cur-rent status ?Λ?0.73and ?m ?0.27.As the time t →∞the dynamics is a stable attractor.Thus the model naturally solves the coincidence problem of dark energy.Moreover,if YMC de-cays into matter,its equation of state (EoS)crosses -1and takes w ～?1.1,as indicated by the recent observations.
PACS numbers:95.36.+x,98.80.Cq,04.40.Nr,04.62+v
Key words:dark energy,Yang-Mills ?eld,accelerating universe
e-mail:yzh@https://www.sodocs.net/doc/1e10615769.html,
1.Introduction
The energy content of the present Universe is such that?Λ?0.73and?m?0.27,as indicated by the observational results from SN Ia[1]and CMB anisotropies[2],and from studies of the large scale structure as well[3].The origin of the cosmic dark energy is now a big challenge to astronomy and physics.There have been several types of models proposed to interpret the dark energy.The simplest one is the cosmological constantΛ.Since the matter density evolves asρm∝a(t)?3andρΛ～constant throughout the history of the Universe,the initial value ofρΛ/ρm has to be chosen with great precision to achieveρm～0.37ρΛtoday. This is the coincidence problem[4].Moreover,ifΛwere interpreted to arise from the vacuum ?uctuations of quantum?elds[5],then there is a“?ne-tuning”di?culty,i.e.,why at present the vacuum energy of a scale10?3ev is so tiny compared to the typical scales in particle physics.Another model is the Steady State Universe[6],in which the accelerating expansion is driven by some C-?eld with negative energy,di?cult to accept as a physical?eld.The e?ective gravity[7]is a model outside the scope of General Relativity,which has a task to to explain all observed features of gravity.The Born-Infeld quantum condensate[8]is a model, which uses gauge?elds as a candidate for the dark energy.One interesting speculations is that the dark energy may result from some dynamic?eld with a tracker behavior,i.e.,the ?eld is subdominant during early stages of expansion,later becomes dominant as the dark energy.Among this scalar kind of models are the quintessence[9],k-essence[10],phantom [11],quintom[12],etc.For a dynamic?eld model to solve the coincidence problem,it is required not to spoil the standard Big Bang cosmology,i.e.,the Big Bang nucleosynthesis and the recombination must occur as usual,and the matter era must be long enough for structure formation.Moreover,the solution needs to be a stable tracker,insensitive to the initial conditions.But so far the scalar models have di?culty to ful?l these criteria[13].
In our previous work[14][15][16]a dynamic model is proposed,in which the e?ective YMC,up to1-loop quantum corrections,serves as the dark energy.For quite generic initial conditions the model always has the desired tracking behavior,naturally solves the coincidence problem,in the sense that it satis?es those criteria mentioned above.When YMC is coupled with matter,EoS of YMC crosses-1and takes w～?1.1,as indicated by the?ttings of recent preliminary observational data[17][18][19][20].One may ask,does the model still work for more quantum corrections?As will be seen in this paper,the YMC dark energy model including the2-loop quantum corrections has a tracking solution that solves the coincidence
2.The E?ective Lagrangian of2-Loop YM Condensate
The quantum e?ective YM condensate is described by the following Lagrangian up to
2-loops[21,22]
L eff=b
eκ2
|+ηln|ln|
F
3(4π)2
representing the1-loop part,
andη≡2b1
3(4π)4
representing the2-loop contribution,the parameterκis the renormalization scale with dimension of squared mass,and F≡?1
bτ?
2b1
τ2
+O(
1
2
F τ+2+η ln|τ+δ|+2
6
F τ?2+η ln|τ+δ|?2
.(5)
When one setsη=0in the above expressions,the1-loop model is recovered[15].For comparison,we plotρy,p y,and w in Fig.1for both the2-loop and the1-loop order,respectively, where the variable is y≡τ+1=ln|F/κ2|.We see thatρy,p y,and w in the2-loop order have similar shapes to,and higher magnitude than,those in the1-loop one,respectively.At high energies y→∞,ρy and p y are positive,and the EoS of YMC approaches to that of a radiation,w→1/3,as is expected for an e?ective theory.At low energies y<2.3,p y and w become negative,and at y
3.The Cosmic Evolution By the YM Condensate
The spacetime of the universe is described by a spatially?at Robertson-Walker metric
ds2=dt2?a2(t)δij dx i dx j,(6)
and is?lled with the dark energy,the matter,including both baryons and dark matter,and the radiation.In our model,the dark energy component is represented by the YMC.To be concordant with the isotropy of the Universe,one may take an SU(2)YM?eld with a highly symmetric con?guration A a0=0and A a i=φ(t)δai,whereφ(t)is a scalar function and a=1,2,3[29].The overall cosmic expansion is determined by the Friedmann equations
(˙a
3
(ρy+ρm+ρr),(7)
whereρm andρr are the energy density of the matter and radiation components respectively. The dynamical evolutions of the three components are
˙ρy+3
˙a
˙ρr+3
˙a
2
bκ2,and r≡ρr/1
dN =
?4y+4η(ln|y?1+δ|+1
h
[y+1+η(ln|y?1+δ|+2
y+2+η(ln|y?1+δ|+3
(y?1+δ)2
)
,(12) dx
h y+1+η(ln|y?1+δ|+2
dN
=?4r,(14)
whereγ≡Γ/H,H≡ x+r+e y[y+1+η(ln|y?1+δ|+2
ρri
≤2.28×10?2.(16) The solutions for y i=54and for y i=15(i.e.ρyi
around z～0.6it becomes dominant.Therefore,the matter era is long enough for formation of large scale structure.The fractional densities?y=0.73and?m=0.27are achieved at z=0(y=?0.97)as the current status of the Universe.Notice that this tracking behavior is always achieved for any initialρyi in Eq.(16)whose range stretches over～25order in magnitude.For a smaller initialρyi,say y i～15,ρy(t)only tracksρr(t)for a shorter period correspondingly,and then becomes the constant.Moreover,the dynamical equation for(y,x) at t→∞has a?xed point(y f,x f)=(?1.043,0.114),which has been examined to be stable, i.e.,the solution is an attractor.So,by the criteria mentioned in introduction,the coincidence problem is solved also in this2-loop model.Furthermore,the EoS of YMC crosses w=?1 around z?1.7for the initial y i=54,but for smaller y i the crossing occurs earlier.For any initial YMC,w??1.14at z=0is obtained,which qualitatively agrees with the?ttings of preliminary observational data[19][20][31][32].These behaviors are quite similar to those of the1-loop model[15],even though the expressions involved in the2-loop model are much lengthier.
Parallel to the above of a constantγ,the desired evolution is also realized in an extended range of parameters,γ?(0～0.6),andδ?(3～7).By calculation,we?nd that largerδandγyield a lower w at z=0.For instances,at a?xedγ=0.5,δ=7?w??1.18,more interestingly,at a?xedδ=3,γ=0.1?w??1.02,consistent with theΛ-model,indicated by WMAP[2],SNLS[17]and ESSENCE[18],and the crossing of?1occurs rather late at z～0.35,quite close to the?tting of Gold+HST sample[19].The desired evolution can be also realized forγbeing a generic function of the YMC F.We have examined a variety of couplings,sayγ=0.13e y/4(2+y)/(1+y)describing the YMC decaying into fermions and gauge bosons[33],andγ=1.18e y[15],etc..In the simple case of non-coupling,γ=0,the dynamic evolution is similar to the coupling cases,but w does not cross,only approaches to ?1.Thus,if future observations do con?rm EoS w0is needed in our model;otherwise,γ=0.
4.Summary
The dark energy model based on the2-loop quantum e?ective YMC,under generic initial conditions and various forms of coupling,has a stable tracking solution of dynamics,thus naturally solves the coincidence problem.Moreover,with coupling to the matter,EoS of YMC crosses?1and w??1.1at z=0.In this model the matter era ends rather late at z～0.6,
subdominant during the early stages,so the nuclear synthesis and the recombination occur as in the standard Big Bang cosmology.These conclusions are the same as for the1-loop model.With the Lagrangian being?xed by quantum corrections,the model depends on only two physical parameters:the couplingΓ?(0～0.6)H and the energy scaleκ1/2.As said earlier,since the“?ne tuning”problem is not solved by the model,so the energy scale needs to be viewed as a new scale of physics.
ACKNOWLEDGMENT:We thank the referee for helpful discussions.Y.Zhang’s work was supported by the CNSF No.10773009,SRFDP,and CAS.
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Figure1:ρy,p y,and w as functions of the variable y=ln(F/κ2).The2-loop model (solid)is compared with the1-loop one(dotted)[15].
Figure2:The evolution of energy densities in the2-loop model withγ=0.5andδ=3.0. Starting from z i=108the evolutions for two cases y i=15and54are given.For a whole wide range of initial YMCρyi=(10?27～10?2)ρri,there always exists a tracking solution,andρy(t)becomes dominant around z～0.6.Due to the coupling,ρm(t)will level o?in future.
Figure3:The evolution of fractional energy densities in the same model as Fig..2.At z?3454the equality of radiation-matter?m=?r occurred,and the values?y=0.73 and?m=0.27are arrived at z=0.
Figure4:The evolution of EoS in the same model as Fig.2and Fig.3.In early stage at z→∞,the YMC behaves like a radiation,w→1/https://www.sodocs.net/doc/1e10615769.html,ter on w reduces smoothly.Due to the decay of YMC,EoS crosses?1,and w??1.14at z=0.
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/2)
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