I propose a new mechanism to account for the observed tiny but finite dark
energy in terms of a non-Abelian Higgs theory, which has infinitely many
perturbative vacua characterized by a winding number, in the framework of
inflationary cosmology without introducing any tiny numbers.Comment: To be published in Proc. the XVIIIth IAP meeting "On the Nature of
  Dark Energy.

Yokoyama, Jun'ichi

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arXiv

I propose a new mechanism to account for the observed tiny but finite dark energy in terms of a non-Abelian Higgs theory, which has infinitely many perturbative vacua characterized by a winding number, in the framework of inflationary cosmology without introducing any tiny numbers

Yokoyama, J

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Degenerate vacua as the origin of the dark energyJun’ichi Yokoyama11Department of Earth and Space Science, Graduate School of Science,Osaka University, Toyonaka 560-0043, JapanAbstract. I propose a new mechanism to account for the observed tiny but nitedark energy in terms of a non-Abelian Higgs theory, which has innitely many per-turbative vacua characterized by a winding number, in the framework of inflationarycosmology without introducing any tiny numbers.One of the greatest mysteries of present-day cosmology is the origin of darkenergy or vacuum-like component of cosmic energy indicated by the analysis oftype Ia supernovae observations [1], whose magnitude is some 120 orders smallerthan its natural value, the Planckian density [2]. I propose a novel mechanismto account for the observed tiny but nite dark energy without introducing anysmall numbers in the framework of inflationary cosmology [3] making use of anon-Abelian Higgs theory which is a part of typical unied theories of elementaryinteractions.Inflation stretches preexistent inhomogeneities and yields an extremely largesmooth domain in which the current Hubble volume is contained. It also pro-vides a generation mechanism of density fluctuations which originate in thequantum nature of the inflation-driving scalar eld [4]. In calculating perturba-tions, we consider the eld to be in the vacuum state and calculate its zero-pointquantum fluctuations. This treatment should be an extremely good approxima-tion, if not exact, as long as we focus on perturbation within currently observableHubble volume. This is because, even if the scalar eld was in some excited stateinitially, subsequent inflation would exponentially dilute the preexistent quanta.Thus for all the practical purposes we can regard that the quantum state of thescalar eld is no dierent from the vacuum state in this context. That is to say,the vacuum state is dynamically selected as a result of cosmological inflation.Next let us consider a non-Abelian gauge theory with a gauge group Gwhich has innitely many perturbative vacuum states classied by the windingnumber n. Most gauge groups used in grand unied theories such as G =SU(N),SO(N), and Sp(N) with N  2 possess this property. For deniteness, however,we focus on the simplest theory with this property, G =SU(2), below. In thesetheories perturbative vacuum states with dierent winding numbers cannot betransformed from each other by a continuous gauge transformation [5] and thereis an energy barrier between them. Quantum transition, however, is possibleand described by an instanton solution which is an Euclidean solution joiningtwo adjacent perturbative vacua with a nite Euclidean action S0 = 8pi2/g2.Here g is the gauge coupling constant [6].As a result, the true vacuum state is expressed by a superposition of pertur-1bative vacua, jni, asjθi =1Xn=−1einθjni, (1)where θ is a real parameter [5, 7]. One can easily nd that this state is areal eigenstate of the Hamiltonian H by calculating the amplitude hθ0j e−HT jθiwith the dilute instanton approximation [7], namely, adding contributions ofm instantons and m anti-instantons keeping in mind that each (anti-)instantonchanges the winding number by (−1)1.hθ0j e−HT jθi =Xn,n′Xm,m¯1m!1m!(KV Te−S0m+m¯δm−m¯,n−n′einθ−in′θ′ (2)= exp(2KV Te−S0 cos θδ (θ − θ0) .Here KV Te−S0 represents contribution of a single instanton or anti-instantonwhere K is a positive constant and V T represents spacetime volume.The above equality (2) implies that each θ-vacuum jθi has a dierent energydensity than the perturbative vacuum jni by ρ = −2Ke−S0 cos θ. Apparentlythe θ = 0 vacuum has the lowest energy, but one cannot conclude that thisis the only vacuum state, because the other θ-vacua are also stable againstgauge-invariant perturbations [5, 7]. In fact, the factor K is divergent dueto the contribution of arbitrary large instantons in pure gauge theory [8]. Inorder to obtain a physical cuto scale, let us introduce an SU(2) doublet scalareld  with a potential V [] = λ(jj2 −M2/2)2/2 following ’t Hooft [9]. ForM 6= 0, instead of the instanton solution of pure gauge theory we should use aconstrained instanton solution which also connects adjacent perturbative vacuawith a nite Euclidean action [9]. Furthermore a gauge-invariant expression forthe winding number can be found in the presence of a Higgs eld [10]. Thenthe prefactor is given by [11] K = (8pi/g2)4M4/2 and the expectation value ofthe vacuum energy density in perturbative vacuum jni and that in θ-vacuumjθi are given byhnjHjni = ρd + ρv(θ = 0), (3)ρv(θ)  hθjHjθi = ρd(1− cos θ) + ρv0, (4)respectively. Here ρd  (8pi/g2)4M4e−8pi2g2 , and ρv(θ = 0)  ρv0 is the vacuumenergy density in the state jθ = 0i. Since dierent vacuum states have dierentvacuum energy density, they behave dierently in the presence of gravity. Henceany good solution to the conventional cosmological constant problem, namelywhy the vacuum energy density is vanishingly small compared with the Planckscale, should specify at which vacuum state the cosmological constant vanishes.Unfortunately, since there is no completely satisfactory solution to this long-standing problem yet, let us simply normalize the vacuum energy density tovanish in the CP-symmetric θ = 0 vacuum state and set ρv0 = 0 for the moment2just for deniteness1. As will be seen below, however, this choice is not essentialand our mechanism can work even if the vacuum energy density vanishes at someother vacuum except at jθ = pii.Now let us consider what quantum state is chosen by cosmological inflationin this non-Abelian-Higgs system. Let us assume the energy scale of the Higgseld, M , is much larger than the Hubble parameter during inflation, H , whichis shown to be the case later. As in the case of singlet scalar eld, sucientlylong inflation would homogenize the Higgs eld conguration over an exponen-tially large domain with jj = M/p2. This in turn means that the state withvanishing winding number jn = 0i is practically realized as long as we concen-trate on the scales within the current Hubble volume, because this is the onlystate with homogeneous scalar eld conguration among many possible vacuumstates, perturbative jni or real jθi. In terms of the energy eigenstates this stateis expanded asjn = 0i =Z 2pi0dθ2pijθi, (5)which consists of superposition of all possible real vacuum state jθi with equalweight.While we can calculate the expectation value of vacuum energy density inthis state as (3), it is more appropriate to discuss the probability distributionfunction (PDF) of vacuum energy because this state consists of superpositionof energy eigenstates with dierent energy density [14]. Using (5) we can easilycalculate the PDF of vacuum energy density, P (ρv), in this state asP (ρv) =Dδρv − ρv(θ)Eθ=Z 2pi0dθ2piδρv − ρd(1− cos θ)=(1pi(2ρvρd−ρ2v)1/2for 0  ρv  2ρd,0 otherwise.(6)It is sharply peaked at ρv = 0 corresponding to θ = 0 and ρv = 2ρd correspond-ing to θ = pi and the probability to nd ρv  2ρd is fairly large.Thus, if the perturbative vacuum jn = 0i is stable in the cosmological timescale, the PDF of the vacuum energy density is still given by (6) today in ouruniverse which experienced inflation. Hence it is by no means surprising thatwe observe a nite dark energy around 2ρd today.We can easily match the predicted value,  2ρd, with the observed one,10−120M4G, by appropriately choosing M and g, where MG is the reduced Planckscale. We also demand that the tunneling rate from jn = 0i, Γ  Ke− 16pi2g2 , issmall enough to guarantee that there is no transition within the current Hubblevolume, H−30 , in the cosmic age, ΓH−40< 1. Then we obtainpi/α + 2 lnα = 60 ln 10 + 2 ln (M/MG) , and M > αMG, (7)1We point out that the wormhole mechanism [12] is an example of proposed solutions tothe conventional cosmological constant problem which predicts the vacuum energy densityvanishes at θ = 0 vacuum [13].3where α  g2/(4pi) is the coupling strength at the energy scale M/p2. If theabove inequality is marginally satised, we nd α = 1/44.4 and M = 5 1016GeV, which is much larger than the upper bound on both the amplitude ofquantum fluctuation, H/(2pi), during inflation [15] and the reheat temperatureafter inflation [16]. If, on the other hand, we take M = MG so that the cutoscale of instanton is identical to the presumed eld-theory cuto, we nd α =1/47.We note that it is not mandatory to assume that the vacuum energy densityvanishes at the state jθ = 0i. One may take other vacuum state to normalizethe origin of the vacuum energy density. For example, if one assumes that ρvvanishes at jni or jθ = pi/2i, P (ρv) will be peaked at ρv = ρd instead, whichwe may interpret that we live in a universe with ρv  ρd. Then the values ofthe model parameters should only be slightly changed accordingly.In conclusion, we have reached a striking conclusion that the observed tinydark energy can be realized by the interplay of high energy unied gauge the-ory and inflationary cosmology. Furthermore by virtue of the exponential de-pendence of the cosmological constant on the action of the instanton, we canreproduce its observed value without introducing any tiny numbers.References[1] S. Perlmutter et al., Astrophys. J. 517, 565 (1999); B.P. Schmidt et al.,Astrophys. J. 507, 46 (1998); A.G. Riess et al., Astron. J. 116, 1009 (1998).[2] S. Weinberg, Rev. Mod. Phys. 61, 1 (1988).[3] For a review of inflation, see, e.g. A.D. Linde, Particle Physics and Infla-tionary Cosmology (Harwood, Chur, Switzerland, 1990).[4] S.W. Hawking, Phys. Lett. 115B, 295(1982); A.A. Starobinsky, Phys. Lett.117B, 175(1982); A.H. Guth and S-Y. Pi, Phys. Rev. Lett. 49, 1110(1982).[5] R. Jackiw and C. Rebbi, Phys. Rev. Lett. 37, 172 (1976).[6] A.M. Polyakov, Phys. Lett. 59B, 82 (1975); A.A. Belavin, A.M. Polyakov,A.S. Schvartz, and Yu.S. Tyupkin, Phys. Lett. 59B, 85 (1975).[7] C.G. Callan Jr, R.F. Dashen, and D.J. Gross, Phys. Lett. 63B, 334 (1976).[8] G. ’t Hooft, Phys. Rev. Lett. 37, 8 (1976); Phys. Rev. D14, 3432 (1976);D18, 2199 (1978)(E).[9] I. Aeck, Nucl. Phys. B 191, 429 (1981);M. Nielsen and N.K. Nielsen, Phys. Rev. D61, 105020 (2000).[10] E. D’Hoker and E. Farhi, Nucl. Phys. B248, 59(1984);M. Carena, Phys. Lett. B211, 177 (1988).4[11] J. Yokoyama, Phys. Rev. Lett. 88, 151302 (2002).[12] S.R. Coleman, Nucl. Phys. B 310, 643 (1988).[13] H.B. Nielsen and M. Ninomiya, Phys. Rev. Lett. 62, 1429 (1989).[14] J. Yokoyama, gr-qc/0205104.[15] V.A. Rubakov, M.V. Sazin, and A.V. Veryaskin, Phys. Lett. 115B,189(1982).[16] T. Moroi and M. Kawasaki, Prog. Theor. Phys., 93, 879 (1995).5

Degenerate vacua as the origin of the dark energy

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