ON INFLATION AND TORSION IN COSMOLOGY

Transcription

1 Vol. 36 (005) ACTA PHYSICA POLONICA B No 10 ON INFLATION AND TORSION IN COSMOLOGY Christin G. Böhmer The Erwin Schrödinger Interntionl Institute for Mthemticl Physics Boltzmnngsse 9, A-1090 Wien, Austri nd Institut für Theoretische Physik, Technische Universität Wien Wiedner Huptstrsse 8-10, A-1040 Wien, Austri boehmer@hep.itp.tuwien.c.t (Received April 14, 005) In recent letter by H. Dvoudisl, R. Kitno, T. Li nd H. Murym The new Miniml Stndrd Model (NMSM) ws constructed which incorportes new physics beyond the Miniml Stndrd Model (MSM) of prticle physics. The uthors follow the principle of miniml prticle content nd therefore dopt the viewpoint of prticle physicists. It is shown tht generlistion of the geometric structure of spcetime cn lso be used to explin physics beyond the MSM. It is explicitly shown tht for exmple infltion, i.e. n exponentilly expnding universe, cn esily be explined within the frmework of Einstein Crtn theory. PACS numbers: h, Jk 1. Introduction There re mny ides how physics beyond the Miniml Stndrd Model my be explined, however none of them so fr ws ble to give consistent output tht could explin consistently ll experimentl results of prticle physics nd cosmology. In contrst to these modern pproches the uthors of [] dopt conservtive prticle physicist s point of view nd include the miniml number of new degrees of freedom to formulte the NMSM tht cn explin Drk Energy, non-bryonic Drk Mtter etc. From geometricl point of view it my be preferble to llow more generl geometric structures rther thn incresing the number of required prticles. Therefore the guiding principle of this note my be clled the principle of miniml geometry content. The cosmologicl principle sttes tht the universe is sptilly homogeneous nd isotropic. Mthemticlly speking the four-dimensionl (4d) (841)

2 84 C.G. Böhmer spcetime (M, g) is folited by 3d spcelike hypersurfces of constnt time which re the orbits of Lie group G cting on M with isometry group SO(3). All fields re invrint under the ction of G. The cosmologicl principle implies L ξ mg µν = 0, nd L ξ mt λ µν = 0, (1) where ξ m re the six Killing vectors (lbelled by m) generting the spcetime isometries. g µν denotes the metric tensor nd T λ µν stnds for the torsion tensor, Greek indices lbel the holonomic components. By imposing the restrictions (1), the metric tensor is of Robertson Wlker type ( ) ds = dt (t) + 1 k (dx + dy + dz ) = η ije i e j, () 4 r where r = x + y + z nd where the 3-spce is sphericl for k = 1, flt for k = 0 nd hyperbolic for k = 1. The vielbein 1-forms in () red e t = dt, e x = (t) 1 k 4 rdx, ey = (t) 1 k 4 rdy, ez = (t) 1 k, (3) 4rdz where Ltin indices lbel the nholonomic components. When the restrictions (1) re imposed on the torsion tensor [8], the (nonvnishing) llowed components re where we closely follow the nottion of [5]. T xxt = T yyt = T zzt = h(t), (4) T xyz = T zxy = T yzx = f(t), (5). Einstein Crtn theory in cosmology In the following it is shown tht infltion cn be explined without introducing dditionl fields but considering spcetime with torsion. The simplest theory of this type is Einstein Crtn theory which is derived from the Einstein Hilbert ction by vrying the vielbein nd the spin-connection independently. Then the field equtions re [6] R i j 1 Rδi j + Λδ i j = 8π t i j, (6) T i jk δ i j T l lk δ i k T l jl = 8π s i jk, (7) where t i j is the cnonicl energy-momentum tensor nd s i jk is the tensor of spin.

3 On Infltion nd Torsion in Cosmology 843 By tking the cosmologicl principle into ccount the field equtions (6) of Einstein Crtn theory simplify to ( ( 3 h + ȧ ) ) 1 4 f Λ = 8πρ, (8) ( ( h + ȧ ) ) ( ( 1 4 f h + ȧ ) + ȧ ( h + ȧ ) ) +Λ = 8πP. (9) The torsion field equtions (7) become f = 8πs, s(t) = S xyz = S zxy = S yzx, (10) h = 8πq, q(t) = S xxt = S yyt = S zzt. (11) If no torsion source is present s = q = 0, the lgebric equtions of motion imply the vnishing of the torsion tensor f = h = 0. Without torsion, the field equtions (8) nd (9) reduce to the stndrd Friedmn equtions of cosmology. Let us hve closer look t the field equtions (8) (11) in cse of q = h = 0, i.e. only the skew-symmetric prt of the torsion tensor, cf. [7]. Then the field equtions simplify to ( (ȧ ) ) 3 Λ = 8πρ, (1) ( (ȧ ) ) 1 4 f ( (ȧ 1 4 f (ȧ ) ) + Λ = 8πP, (13) ) + f = 8πs, (14) which implies the following conservtion eqution ρ 3 + ȧ (ρ + P) + s ( f + ȧ ) f = 0. (15) With (14) the two remining independent field equtions cn be reformulted to give ( (ȧ ) ) 3 = 8πρ eff = 8πρ + Λ (8πs), (16) (ȧ ) ä k = 8πP eff = 8πP Λ 1 4 (8πs). (17)

4 844 C.G. Böhmer In (16) nd (17) the mtter dominted er of cosmology is defined by P = 0 nd ρ = ρ m where in ddition it is ssumed tht the torsion contribution is sufficiently smll, which is indeed very resonble s shll be seen. The rdition dominted er is defined by the eqution of stte P = ρ/3 nd ρ = ρ r, gin with sufficiently smll torsion contribution. For ske of simplicity we ssume the following setup for the torsion dominted er, in which the universe is exponentilly incresing: Assume tht torsion in (16) nd (17) is the leding contribution, such tht one my neglect the others. In the erly time of the universe the prticle density ws high nd therefore the probbility of hving some non-vnishing mcroscopic spin is the higher the denser the mtter distribution is. On the other hnd it is resonble tht the verged spin density is exponentilly decresing with time, s exp( t/τ), where τ is chrcteristic time scle. Putting this into (15) yields ṡ s = 1 τ = ȧ, (18) which simply implies tht the scle fctor is n exponentilly incresing function of time, exp(t/τ) if the torsion function is exponentilly decresing nd if the torsion contribution is the leding one. Hence physiclly intuitive ssumption on the behviour of torsion cn explin the infltion er of cosmology without introducing further prticles. Since the torsion is rpidly decresing, its contribution to (16) nd (17) will indeed be sufficiently smll fter the short period of infltion. This implies tht tody s cosmologicl mesurements possibly should detect some smll non-vnishing torsion contribution, (see e.g. [3]). This torsion remnnt could then be used to solve the sign problem of the cosmologicl constnt, s ws shown by the uthor in [1]. It is neither the uthor s im to criticise the motivtion nd derivtion of the NMSM nor to criticise the successful wy tht led to the MSM. We try to show tht other, eqully conservtive, pproches my lso work. It should be emphsised tht the considertion of torsion is nerly s old s generl reltivity itself (see e.g. [4] for historicl review). Thus the guiding principle of miniml geometry content might be s successful s the miniml prticle content principle. Only the experiment will decide which of these two principles is the one describing nture correctly. I wish to thnk Herbert Blsin nd Wolfgng Kummer for vluble comments. Moreover, I wish to thnk Dominik J. Schwrz for the useful discussion. The work ws supported by the Junior Reserch Fellowship of The Erwin Schrödinger Interntionl Institute for Mthemticl Physics.

5 On Infltion nd Torsion in Cosmology 845 REFERENCES [1] C.G. Böhmer, Clss. Qunt. Grv. 1, 1119 (004). [] H. Dvoudisl, R. Kitno, T. Li, H. Murym, Phys. Lett. B609, 117 (005). [3] L.C. Grci de Andrde, Int. J. Mod. Phys. D8, 75 (1999). [4] H.F.M. Goenner, Living Rev. Rel. 7, (004). [5] H.F.M. Goenner, F. Müller-Hoissen, Clss. Qunt. Grv. 1, 651 (1984). [6] F.W. Hehl, P. von der Heyde, G.D. Kerlick, J.M. Nester, Rev. Mod. Phys. 48, 393 (1976). [7] P. Minkowski, Phys. Lett. B173, 47 (1986). [8] M. Tsmprlis, Phys. Lett. A75, 7 (1979).