Domain Wall Start to Inflation with Contributions to Off Diagonal GR Stress Energy Tensor Terms

Transcription

1 Domin Wll Strt to Infltion with Contributions to Off Digonl GR Stress Energy Tensor Terms Andrew Beckwith Chongqing University Deprtment of Physics; e mil: beckwith@uh.edu Chongqing, PRC, 000 Abstrct We represent how n off digonl representtion of stress energy in GR s given by Dodelson cn be ffected by n xion style domin wll tretment of stress energy tensor in GR, s given by Kolb nd Turner.We rgue tht this is wy of presenting how domin wll physics my impct grviton production which in turn hs, through Dodelson nd his off digonl stress energy terms consequences s to non uniform evolution of spce time cosmology. We close with tretment of xions ( cndidte for DM) s impcting GW, nd through Dodelson hving consequences which we outline t the end of this document. The off digonl terms of the stress energy tensor T μν lluded to re llegedly for lrge scle spce-time evolution, but the trnsition for spce-time from the big bng to the Electrowek regime is mny thousnds of times lrger thn Plnckin spce-time, so we rgue tht the off digonl reltion so used still holds. 1

2 1. Introduction To begin with, we will review wht Kolb nd Turner suggested for domin wll tretment for initil stress energy tensor terms in the erly universe. This tretment mkes use of n xion type of contribution to GR spcetime evolution nd we write out domin wll tretment for Lgrngin s given by Kolb nd Turner [1], pge 1, leding to, in turn n eqution of motion for φ φ + = z λ φ ( φ λ ) 0 This in turn will led to n eqution for phi, in Eq.(1) which we write s contributing to Strese energy tensor contribution we give s given by Kolb nd Turner s, with z the xis of domin wll evolution, so tht the GR expression becomes [1] λ = σ cosh ( / Δ ) ( 1,1,1, 0) () μ Tν z dig This will constitute the pressure term of stress energy term. Tking the pressure s the negtive of density ccording to the initil infltion condition given s p = ρ [1]. We will then go to the Visser expression for grviton contributions to the Stress energy Tensor which we will write up s [] (1) T uv m GM r GM = exp + lpλ g r λ g r (3) In writing up Eq. (3) this wy we re treting it s the time component of the GR stress energy tensor, so s to mke it in mgnitude equl to the terms in Eq. () bove, reltionship which we will use to good effect in this document. Wht we will do is to ssume tht the mssive grvitons in Eq. (3) bove hve their genesis in the domin wll contributions given in Eq. () nd from there mke some order of mgnitude contributions to chnge in energy s given by Lee [3], below which will influence how energy is djusted nd ffected by entropy production, which for the ske of comprison will be conflted with erly universe grviton production. The term M refers to the

3 initil mss of the universe which is enormous. This provides, lso for tie in for grviton production t the onset of DM, which my led to the dynmics of DE s given by []. Energy, chnge in entropy nd the cretion of erly universe grvitons Lee [3] mkes sttement of chnge in entropy s given by the following expression =Δ = Δ = Δ mc E TU S S π c kb () If the mss m, for the initil mss is set to M, s consistent with Eq. (3) bove, nd ~ c is ccelertion (of the net universe) nd chnge in Δx 50 enthropy between the strt of infltion, essentilly no entropy, to Δ S ~10 bout the electrowek regime, we re essentilly buying Ng s [5] counting lgorithm pproch to entropy, which then sys it is necessry to hve enormous initil ccelertion for the onset of infltion, s well s the following reltionship between the terms s we relte them in proportionl rtios. Nmely let the chnge in x be conflted with z, in Eq. () bove, nd we then get the following proportionlity reltionships, nmely if z is the distnce domin wll trvels up to ner the electrowek regime, nd [1], [3] μ λ Tν σ = cosh ( z / Δ) ~ GM r GM Tuv = exp ~ m 0 + lpλ g z λ g z Δ E = TU Δ S = ΔS π c k B (5) Note tht in Eq. (5) we re ssuming following Kolb s convention tht the xion domin wll hs thickness given by [1] Δ ~ λ / σ (6) 3

4 Also, for the σ term for the minimum of the potentil term [1] λ V ( φ) = ( φ σ ) (7) Leds to representing σ ( m ) / λ ± with the plus nd minus terms representing wht to do with ssorted mxim nd minim vlues, ccording to the convention tht for when z goes to infinity, tht the plus term is picked, nd for when z goes to infinity tht the minus term is picked, leding to n energy term of the vcuum being proportionl to hving vcuum energy equl to mss, nd the lmd term s given by Kolb nd Turner [1] s: ρv = m /λ (8) These terms ply role in the subsequent off digonl stress energy terms we llude to next. 3. Looking t the formtion of off digonl terms for initil GR T μν To do this, we look t the reltion given to us by Dodelson, s to off digonl T μν which is given s, by pge 168 of Dodelson s [6]: 3 Ik δt H / k 3 Ik δt H / k = δt δt 0 0 I I I I 0 3 Ik δt H / k = δt Vcuum energy I I (9) This mens tht if H = is n erly universe Hubble prmeter, nd k I derivble from the tensor reltionship for GW given by h nd h where we hve when k k tht then we hve, for the h I = 1 I nd h k I obeying, if H = h α + h α + k hα = 0 (10) This is true for the Fourier trnsform of spce-time x, so tht by [6]

5 h h h + h + k h = 0 + = k (11) α α α α α hα hα Where the significnce of Eq.(11) is in tht the digonl vcuum energy given by δt Vcuum energy hs the opposite sign of the off digonl δt I terms. We will comment upon wht the sign difference tells us next. Conclusion: Exming wht Dodelson s cross term contribution to GR stress-energy sy bout wht Eq.(5) cn tell us bout inhomogenities in the erly universe. The opposite signs of the digonl to off digonl vcuum energy terms s given by Eq. (9) nd Eq. (11) with their scled dependence, llude to inhomogeneties being inevitble likely well before the CMBR decoupling of photons to shine forth 310 thousnd yers fter the strt of infltion. Tht Eq. (9) with Eq. (11) suggest tht there re opposite signs of terms contributing to the vrition of vcuum energy spce-time, nd tht this is tied into GW tensors, suggest tht turbulence is inevitble, nd not by product of just initil E nd B fields in qurk - gluon plsms, s of bout the Electrowek er, s implied by Duerrer [7]. More exctly, tht simple model of xion type domin wlls, s given by Eq. (1) nd Eq. () s suggested by Kolb nd Turner [1], with xions nother model of DM, suggest this is tied into GW, which my be n emerging dynmic contributing to DE, s suggested by both Beckwith [8], nd Alves, Mirnd. nd de Arujo [3]. The ide, tht if DM is tied into initil xion wll dependence, nd n initil DM spce-time is inevitbly shered into DE contribution by the off dignonl δt I terms, with I = 1,,3,, tht the off digonl terms re the beginning of GW genertion, with GW closely linked to DE style speed up of the universe lter on. Also to consider is wht δt I terms contribute to the development of Mgueijo style non Gussin contributions to the CMBR spectr [9]. Tht will be lter project by the Author. Note tht this ssumes n erly universe infinite quntum sttistics counting lgorith for grvitons, in lieu of Ng s [5], [10], [11] work which hs yet to be experimentlly verified. Acknowledgements This work is supported in prt by Ntionl Nture Science Foundtion of Chin grnt No

6 References 1. E. Kolb nd M. Turner, The Erly Universe, Westview Books, 199. M. Visser, 3. M. Alves, O. Mirnd nd J. de Arujo, Cn Mssive Grvitons be n Alterntive to Drk Energy?, rxiv: (July 009). Je-Weon Lee, On the Origin of Entropic Grvity nd Inerti, Foundtions of physics, Volume, Y.J. Ng, Entropy 10(), pp1-61 (008); Y.J. Ng nd H. vn Dm, FoundPhys30, pp (000) 6. S. Dodelson, Modern Cosmology, Acdemic Press, Elsevier, R. Durrer, Mssimilino Rinldi, Phys.Rev.D79:063507,009, 8. A. Beckwith, Identifying Kluz Klein Tretment of Grviton Permitting decelertion prmeter Q (Z) s n Alterntive to Stndrd DE, Journl of Cosmology, 011, Vol J.C.R. Mgueijo, Non Gussin CMBR Angulr power spectr, ArXIV stro-ph/ My 1995; Y.J. Ng nd H. vn Dm, Phys. Lett. B77, pp9 35 (000) 11. Y.J. Ng, Quntum Fom nd Drk Energy, in the conference Interntionl work shop on the Drk Side of the Universe, ergy.pdf 6