Why We Don t See Cosmic Strings

Transcription

1 International Journal of Theoretical and Mathematical Phsics, : 6- DOI:.9/j.ijtmp..7 Wh We Don t See Cosmic Strings Reinoud Jan Slagter Department of Phsics, Universit of Amsterdam and ASFYON, Bussum, EP, The Netherlands Abstract Cosmic strings can be formed in smmetr-breaking phase transition in the earl stage of the universe. The are topological defects, analogous to flux tubes in tpe-ii superconductors, or to vortex filaments in superfluid helium. Cosmic strings consist of trapped regions of false vacuum in U( gauge theories with spontaneous smmetr breaking. Densit perturbations that would be produced b these strings of GUT scale, G μ = η /M pl = -6, where G=/M pl is Newton's constant, M pl the Planck mass, μ the mass per unit length of the string and η the smmetr breaking scale, could have served as seeds for the formation of galaxies and clusters. However, recent observation of the cosmic microwave back-ground (CMB radiation disfavored this scenario. The WAMP-data prove that cosmic strings can't contribute more than an insignificant proportion of the primordial densit perturbation,g μ -6. The space time around a cosmic string is conical, with an angle deficit θ ~ μ. The should produce axiall smmetric gravitational lensing effect, not found b observations. Recentl, braneworld scenarios suggest the existence of fundamental strings, predicted b superstring theor. These super-massive cosmic strings, G μ ~, could be produced when the universe underwent phase transitions at energies much higher than the GUT scale. To overcome the conflict with observational bounds, we present the "classical" Nielsen-Olesen string solution on a warped five dimensional space time, where we solved the effective four dimensional equations from the fivedimensional equations together with the junction and boundar conditions. Where the mass per unit length in the bulk can be of order of the Planck scale, in the brane it will be warped down to unobservable GUT scale. It turns out that the induced four dimensional space timedoes notshow asmptotic conical behaviour. So there is no angle deficit and the space time seems to be un-phsical, at least under fairl weak assumptions on the stress-energ tensor and without a positive brane tension. The results are confirmed b numerical solutions of the field equations. ewords Cos mic Strings, Warped Space Times, BraneWorlds, Higher Dimensional Cosmolog. Introduction The standard model is extremel successful up to scales M EW ~ GeV. The fundamental scale of gravit is the Planck scale, M Pl ~ 9 GeV. It is the scale where quantum gravit will act. The discrepanc between these two scales is called the hierarch problem. Electro-weak interactions have been tested up to M EW, while gravit, on the other hand, has been tested to several millimetres, orders of magnitude above M Pl. Braneworld models could overcome the hierarch problem. The idea originates from string theor. One of the predictions of string theor is the existence of branes embedded in the full bulk space time. Gravitons can then propagate into the bulk, while other fields are confined to these branes. It also predicts that space time is -dimensional, with 6 of them are ver compact and small, not verifiable b an experiment. There are man models which attacked the hierarch problem. Essentiall there are * Corresponding author: reinoudjan@gmail.com (Reinoud Jan Slagter Published online at Copright Scientific & Academic Publishing. All Rights Reserved globall two categories: flat compact extra dimensions and warped extra dimensions. Recentl, there is growing interest in the second categor, i.e., the Randall-Sunum(RS warped -dimensional geometr ([],[]. We live in a + dimensional space time embedded in a -dimensional space time, with an extra dimension which can be ver large compared to the ones predicted in string theor. One estimates that the extra dimension can be as large as - cm, which is the under-bound of Newton's law in our world. The observed -dimensional Planck scale M pl = M is no longer the fundamental scale but an effective one, an important consequence of the extra dimensions, which is now M, the Planck scale in D. The weakness of gravit can be understand b the fact that it "spreads" into the extra dimension and onl a part is felt in the -dimensional brane. Spontaneousl broken gauge theories give rise to topological defects, such as cosmic strings, monopoles and domain walls. The simplest model is a U( gauge theor, involving a complex scalar field Φ interacting with the electromagnetic gauge field and a "sombrero"-potential V(Φ and described b the Lagrangian densit ([] µ µν L = D ΦD Φ * Fµν F V ( Φ µ, (

2 7 International Journal of Theoretical and Mathematical Phsics, : 6- withf μν the electromagnetic tensor and D μ the gauge-covariant derivative. Cosmic strings occur as topological defects, consisting of confined regions of false vacuum in gauge theories with spontaneous smmetr breaking. When the temperature in the earl universe falls down,the scalar field will develop a locus of trapped points of valse vacuum. This formation of vortices is equivalent to the Ginzburg-Landau theor of tpe II superconductivit. If local strings appeared in phase transitions in the earl universe,the could have served as seeds for the formation of galaxies. However, observations of the cosmic microwave background, would rule out this model. Cosmic strings possess a conicalspace time, with an angle deficit which could produce axiall smmetric gravitational lens effects, not found b observations ([]. M-theor, the improved version of superstring theor, allo ws, via braneworld scenarios, macroscopic fundamental strings that could pla a role ver similar to that of cosmic strings ([],[6],[7] The resulting super-massive cosmic strings are even more exotic, because the could develop singular behaviour at finite distance of the core of the string ([8],[9]. We will investigate on an axiall smmetric -dimensional warped space time the modifications of the behaviour of a gauge cosmic string in the Abelian Higgs model. The general relativistic cosmic string was first investigated b Garfinkle ([] and later extended b Der and Marleau ([]. In section we outline the revised U( cosmic string in D and present some numerical solutions. In section we de investigate the angle deficit and the changes with respect to the D model. F = Λ F κ F( X + c ( F = ΛF + κ F ( X + c F (6 X = F ( X with c some constant. For c = there exist exact solutions for F( and X(. It is remarkab le that these equations for F( and X( are separable. F (7. The Model Let us consider the warped dimensional space time Art (, ds = F( [ e ( dt + dz + Art (, ( + (, r t e dφ ] + d F( the warp factor and the extra dimension. For F= and = we recognize the D axiall smmetric space time. We will follow the notations of references[]-[]. If there is a bulk scalar field Φ (no coupling to a D gauge field A μ, as in the D case, which could stabilize the branes ([6], then we have for the D equations (from now on all the indices run from.. = Λ + ( G µν gµν κ Tµν µ Φ = µ ( with T μυ the energ momentum tensor of the bulk scalar field. When we write Φ= Xe iφ, then it turns out that the -dependent part is separable. We obtain for F( and X( the set equations Fi gure. The warpfactor F( in the case of an empt bulk for positive bulk cosmological constant (upper and negative one From now on we will consider the case of an empt bulk (c and the field equations on the brane time-independent. There are two solutions for F, plotted in figure c F( = + Dsinh( 6 Λ Λ 9c + D Λ ± cosh( 6 Λ, Λ c F D ( = + cos( 6 Λ Λ 9c D Λ ± sin( 6 Λ Λ (8

3 Reinoud Jan Slagter: Wh We Don t See Cosmic Strings 8 with D i some constants. From the Einstein equations one obtains a constraint equation A r r rr ( r A = c (9 Constraint sstems appear whenever a theor has gauge smmetr, in general relativit equivalent to coordinate change. Gauge invariance implies here general covariance under coordinate transformations. In terms of fields on a space time one sas that the theor has redundanc and the constraint equations are preserved in time ([7]. Let us now investigate the induced field equations on the brane. Following[] we obtain ( Fi gure.the -dimensional cosmic string ( ( µν = Λ gµν + κ Tµν + κ G eff S E µν µν ( whereλ eff =½(Λ +κ λ and λ the vacuum energ in the brane ( brane tension. ( T μν is the dimensional energ momentum tensor. The equations for the D scalar and gauge fields are the same as in the D case([]. The first correction term S μν is the quaatic term in the energ-momentum tensor arising from the extrinsic curvature terms in the projected Einstein tensor ( ( ( αβ ( + gµν [ Tαβ T T ] ( ( ( ( α Sµν = T Tµν Tµα T ν ( The second correction term E μν is given b γ δ ( α ( β Eµν = Cαγβδ n n gµ gν ( and is a part of the D Wel tensor and carries information of the gravitational field outside the brane and is constrained b the motion of the matter on the brane, i.e., the Codazzi equation. The induced metric on the brane is given b ( g μν = g μν - n μ n ν, with n μ the unit normal vector on the brane. E μν represents the D graviton effects, i.e., aluza-lein modes in the linearized theor ([],[]. In the static case of the space time (, one can evaluate the equations for and A and the gauge fields X and P. In order to compare the change in behaviour of these equations with respect to D counterpart equations, we take the same combinations of the components of the Einstein equations as in the D case. The result is 6 Θ r = [ κ(ρr σ + ρφ c + κ (ξr ξt + ξφ + 6 Λeff ] Θ r +Θ r = 6 [ κ( ρr + ρφ c + κ ( ξr + ξφ + Λeff ] A rr X XP e rr X = + X ( X + rpr rrp = rpr A + + αx P ( ( (6 where we used the same notations as in the D case, i.e., Θ = r A and Θ = r. Further, we write the energ mo mentum tensors as T μν =σk t k t +ρ z k z k z +ρ φ k φ k φ +ρ r k r k r, S μν =ξ t k t k t +ξ z k z k z +ξ φ k φ k φ +ξ r k r k r, with k i a set of orthonormal vectors on the D space time. We also used the constraint equation of (9. In the D case, the equations are, r Θ =κ (ρ r +ρ φ and r Θ =½κ (ρ r -σ+ρ φ, so we recognize the κ -terms in ( and (. In the special case of c = 8Λ eff, we see on the right hand side that the κ term has the same combinations of the components of the energ-mo mentum tensor S μν. On the left hand side we observe a difference due to the consequence of the E μν term entering the equations. We will use these equations to evaluate the angle deficit in our model. If we would take κ = λ κ /6 ([] we then have parameters for the model, i.e., β, e, η, λ and κ. The effective field equations are supplemented b the D equations and the conservationof stress-energ. We can compare the numerical solution of the D-brane equations with the "classical" Nielsen-Olesen solution. In figure we plotted two solutions with the same initial conditions and parameters used b Garfinkle([]. We used a standard routine using solel initial conditions. In order to find an "acceptable" solution, one has to fine tune the initial values ([]. We observe that e -A behaves differentl. In the D case there is onl a small deviation fro m Minkowski for large r, representing an angle deficit.

4 9 International Journal of Theoretical and Mathematical Phsics, : 6- Fi gure. T wo characteristic solutions of the braneinduces U( gauge string. We used a standard "shooting" routine. We observe the same behaviour of X and P, but a significant different behaviour of e -A. The dashed line represents the Minkowski space time

5 Reinoud Jan Slagter: Wh We Don t See Cosmic Strings Figure. Numerical solution of the metric component e -A for different values of the brane and bulk cosmological constants. The last one represents the most realistic situation, where λ > and Λ <. The behaviour don't change significantl with increasing η In figure we plotted e -A for different values of the parameters. We don t find a tpical cosmic string behaviour as in the D case.for negative bulk cosmological constant, it seems to be possible to find a behaviour of e -A which is comparable with the classical D behaviour ([]. However, this is not a tpical cosmic string situation..analsis of The Angle Deficit The angle deficit can be calculated for a class of static translational smmetric space times which are asmptoticall Minkowski minus a wedge. If we denote with l the length of an orbit of (/φ a in the brane, then the angle

6 International Journal of Theoretical and Mathematical Phsics, : 6- deficitis given b[] with l = π g dl ( θ = lim r π (7 ϕ dϕ a b ( ( dϕ = πe ab (8 Using boundar conditions at the axis, we obtain d A θ = π ( e dθ dθ Θ π [ ( ( ] = e e Θ Θ. (9 In order to evaluate the right hand side, we first calculate, as in the D case, the total derivative of the expression (κ ρ r using the conservation of stress energ, νµ νµ νµ k T = ( T k T k µ ν ν µ ν µ = ( ρ ρ + ( σ + ρ A = r r φ r φ r ( and the two field equations for A and. We find after a straightforward calculation d ( d = ( ΘΘ + Θ ρ r Θ κ ( Using the case where c =8Λ eff and boundar conditions at the axis, we obtain Θ Θ + Θ Θ = κ ρ r ( Further, we used that S μν ~ (T μν, so ( κ Sµν / λ Tµν ( κ Tµν λ ( Further we assumed ( T μν << λ ([]. As in the D case we have also lim σ r and σ > ρ r > ρ φ. From the field equations we then have that Θ and Θ approach constant values k and k as r. The solution of (then ields = ± Θ ( Θ where we denote with Θ and Θ the asmptotic values. So we have for r A A = r ( ± r The solutions for A and are then = kr + a ln( kr + a A = + a ± with a and a some constants. The space time becomes a ± = ( + [ + ] + + a + e ( kr + a dφ ds e k r a dt dz (6 (7 Let us compare our relation ( with the D result: Θ (Θ -/ Θ =κ ρ r. The solution Θ = A = results in a conical space time ([]. In our case this solution is no option. We have now two possibilities for the asmptotic space time: both non-asner-like and non-conical. Combined with the warp factor F(, the space time becomes or ds = F( [ e ( k r + a [ dt + dz ] + a.8 a e ( kr + a dφ ] + d ds = F( [ e ( k r + a [ dt + dz ] + a. a. + e ( kr + a dφ ] + d (8 (9 with F( in the empt bulk situation given b (8. These solutions are in general un-phsical. The behaviour depends on the sign of k /a. Under less restrictive conditions, for example, c 8Λ eff and with special choices of the parameters, the numerical solutions show some regular behaviour.now we can evaluate the angle deficit. We can make a linear combination of (,( and ( in order to isolate the term e -A σ ( as in the D case. We obtain: A e [ ( ΘΘ Θ + Θ 6 θ = κ µ π ( de ( Θ de ( Θ + ] 6 The first term is again the linear energ densit of the string and is of order η. The correction terms are in contrast with the D case, unbounded and will give chaotic results, as is seen in the numerical solutions of figure. Onl for positive brane tension and negative bulk tension (the D braneworld preferred values there seems to be a stable solution for e -A. However, this solution cannot be classified as a cosmic string.. Conclusions Cosmic strings produced in the Einstein-Higgs-gauge field model at GUT-scales give rise to a conical space time and could have important cosmological effects. Strings produced at much higher energ scales, i.e., G μ >> -6,show singular behaviour. The seem to be inconsistent with the observed CMB spectrum. One believes that cosmic super-strings with energ scale close to the Planck scale arising at the end of brane inflation in braneworld

7 Reinoud Jan Slagter: Wh We Don t See Cosmic Strings cosmological models, could have comparable properties with respect to the GUT cosmic strings. Warped braneworld models with large extra dimensions could overcome the conflict with observation bounds. Here we considered on a warped D space time the "classical" self-gravitating Nielsen-Olesen vortex. It seems possible that the absence of evidence of cosmic strings in observational data could be explained b our model, where the conical feature disappears. In the super-massive case of Laguna and Garfinkle ([9], i.e., where the linear mass per unit length G μ >> -6, a continuous transition occurs from a conical space time to a asner-tpe with a curvature singularit at finite distance of the core for increasing smmetr breaking energ scale. In our induced brane space time we find a different result. On the brane, there is no conical space time, measured far from the core of the string. The solutions don't change significantl for increasing smmetr breaking scale η. We find an exact expression for the warp factor, which will warp down the found asner-like solutions. The numerical solutions confirm our conclusion. [] L. Randall,R. Sunum, Phs. Rev. Lett. 8, 69, 999. [] H. B. Nielsen, P. Olesen, Nucl. Phs. B 6,, 97 [] A. Vilenkin, E. P. S. Shellard, Cosmic Strings and Other Topological Defects, Cambridge Monographs, Cambridge, 99 [] A. Vilenkin, arxiv: hep-th/8 [6] A. C. Davis, T. W. B. ibble, arxiv: hep-th/ [7] R. Gregor, arxiv: hep-th/99 [8] P. Laguna-Castillo, R. A. Matzner, Phs. Rev. D 6,66, 987 [9] P. Laguna, D. Garfinkle,Phs. Rev. D,, 989 [] D. Garfinkle, Phs. Rev. D,, 98 [] C. Der, F. Marleau,Phs. Rev. D, 88, 99 [] N. Arkani-Hamed, S. Dimopoulos,G. R. Dvali, Phs. Lett. B 9, 6, 998 [] T. Shiromizu,. Maeda. M. Sasaki, arxiv: gr-qc/9976 [] R. Maartens, arxiv: gr-qc/9 REFERENCES [] L. Randall, R. Sunum, Phs. Rev. Lett. 8, 7, 999. []. Aoanagi,. Maeda, arxiv: hep-th/69 [6] W. D. Goldberger, M. B. Wise, arxiv: hep-ph/9977 [7] M. Bojowald,Canonical Gravit and Applications, Cambridge Universit press, Cambridge,