Brane cosmology driven by the rolling tachyon

Transcription

1 Brane cosmolog driven b the rolling tachon Shinji Mukohama Department of Phsics, Harvard Universit Cambridge, MA, 2138, USA (April 11, 22) Brane cosmolog driven b the tachon rolling down to its ground state is investigated. We adopt an effective field theoretical description for the tachon and Randall-Sundrum tpe brane world scenario. After formulating basic equations, we show that the standard cosmolog with a usual scalar field can mimic the low energ behavior of the sstem near the tachon ground state. We also investigate qualitative behavior of the sstem beond the low energ regime for positive, negative and vanishing 4-dimensional effective cosmological constant Λ 4 = κ 4 5V (T ) 2 /12 Λ 5 /2, where κ 5 and Λ 5 are 5-dimensional gravitational coupling constant and (negative) cosmological constant, respectivel, and V (T ) is the (positive) tension of the brane in the tachon ground state. In particular, for Λ 4 < the tachon never settles down to its potential minimum and the universe eventuall hits a big-crunch singularit. I. INTRODUCTION Pioneered b Sen [1], the stud of non-bps objects such as non-bps branes, brane-antibrane configurations or spacelike branes [2] has been attracting phsical interests in string theor [3]. These objects are epected to be important for our understanding non-perturbative dualities beond the BPS level. Moreover, the ma pla roles in cosmolog. Sen showed that classical deca of unstable D-brane in string theories produces pressure-less gas with non-zero energ densit [4]. Gibbons took into account coupling to gravitational field b adding an Einstein-Hilbert term to the effective action of the tachon on a brane, and initiated a stud of tachon cosmolog, cosmolog driven b the tachon rolling down to its ground state [5]. Fairbairn and Ttgat considered possibilit of inflation driven b the rolling tachon [6]. Another subject in which branes pla important roles is the brane-world scenario. Actuall, in this scenario a brane in a higher dimensional bulk spacetime is supposed to be our universe itself. Randall and Sundrum [7] showed that, in a 5-dimensional AdS background, 4-dimensional Newton s law of gravit can be reproduced on the world-volume of a 3-brane, despite the eistence of the infinite fifth dimension. There are also cosmological solutions in this scenario, in which the standard cosmolog is restored at low energ, provided that a parameter representing the mass of a bulk black-hole is small enough [8 13]. Considering that the tachon is a degree of freedom on a brane, it is perhaps interesting to consider the brane-world scenario to take into account gravit. In this case, the Einstein-Hilbert term is introduced in the bulk action rather than in the brane action. Hence, purpose of this paper is to initiate the brane-world version of the tachon cosmolog. We adopt an effective field theoretical description for the tachon and Randall-Sundrum tpe brane world scenario. After formulating basic equations, we show that the standard cosmolog with a usual scalar field can mimic the low energ behavior of the sstem near the tachon ground state. An obvious consequence of this result is that if the tachon potential has a minimum at finite distance from the maimum and if it oscillates about the minimum then the tachon will behave like pressure-less gas. We also investigate qualitative behavior of the sstem beond the low energ regime for positive, negative and vanishing 4-dimensional effective cosmological constant. The rest of this paper is organized as follows. In Sec. II we briefl review the effective field theoretical description for the rolling tachon. In Sec. III we consider the Randall-Sundrum tpe brane cosmolog driven b the rolling tachon and investigate the low energ behavior of the sstem near the tachon ground state. In Sec. IV we investigate a simple double-well potential and a run-awa potential to see qualitative behavior of the sstem beond the vicinit of the tachon ground state. Sec. V is devoted to a summar of this paper. II. ROLLING TACHYON Let us consider an n-brane in D-dimensional spacetime. The imbedding of the world volume of the brane can be described b the parametric equation M = Z M (), (1) 1

2 where { M } is a coordinate sstem in the D-dimensional bulk and denotes a set of (n + 1) parameters { µ }.The parameters µ will pla a role of (n + 1)-dimensional coordinates on the world-volume of the n-brane. Throughout this paper we shall adopt the effective field theoretical description of non-bps branes proposed in refs. [14 16], which includes a tachon. The effective action for the brane with the tachon field T is S brane = d n+1 det q) V (T ), (2) where T (V ) is a tachon potential, q µν is the induced metric defined b q µν = q µν + µ T ν T, (3) Z M Z N q µν = g MN µ ν, (4) and g MN is the bulk metric. Note that the choice of the tachon potential V (T ) is ver important. Different potentials give different dnamics of the tachon and the brane universe. Hence, if we could, we would like to use potentials predicted b rigorous calculations based on boundar string field theor or conformal field theor for configurations of our interest. As we shall see, we would like to consider potentials with a positive value at the tachon ground state. This situation ma be epected for, for eample, deca of stacked ND-branes and N D-branes with N N. However, as far as the author knows, there is no such calculation in the literature so far. Hence, in this paper we shall take an alternative approach: we shall consider ver simple forms of the tachon potential. In Sec. IV we shall consider a simple double-well potential V (T )=(T 2 1) 2 + V and a simple run-awa potential V (T )=1/ cosh(t )+V to investigate qualitative behavior of the sstem. The surface stress energ tensor S µν is given b S µν 2 δs brane = q δq µν det q det q V (T )( q 1 ) µν. (5) The equation of motion for the tachon is 1 [ ] µ det qv (T )( q 1 ) µν ν T V (T )=. (6) det q For a homogeneous isotropic brane, we can assume the following form of the induced metric q µν and the tachon field T without loss of generalit. q µν d µ d ν = dt 2 + a(t) 2 Ω K ij di d j, T = T (t), (7) where Ω K ij isthemetricofthen-dimensional constant curvature space with the curvature constant K: Ω K ij di d j = dρ2 1 Kρ 2 + ρ2 dω 2 n 1. (8) Here, dω 2 n 1 isthemetricofthe(n 1)-dimensional unit sphere. Positive, zero and negative values of K correspond to S n, R n and H n, respectivel. With the above form of q µν and T, the equation of motion and the surface stress energ tensor are reduced to V T 1 T + nȧ 2 a V T + V =, (9) S ν µ = ρ p... p, (1) 2

3 where a dot denotes derivative with respect to t, and V ρ = 1 T, 2 p = V 1 T 2. (11) The equation of motion is formall equivalent to the conservation equation µ S µ ν =,or ρ + nȧ (ρ + p) =. (12) a III. BRANE COSMOLOGY Now we consider brane cosmolog driven b the rolling tachon. We consider Randall-Sundrum brane world scenario on a 3-brane (n = 3) in a 5-dimensional bulk spacetime (D = 5) [7]. We assume that the bulk is invariant under the Z 2 reflection along the brane and described b 5-dimensional Einstein pure gravit with a negative cosmological constant and that the brane motion is described b Israel s junction condition [17]. With these assumptions, the bulk geometr is AdS-Schwarzschild spacetime [18] and the evolution of the brane is governed b [19,2] (ȧ a ) 2 = κ ρ2 K a 2 + µ a 4 1 l 2, (13) where κ 5 is the 5-dimensional gravitational coupling constant, l = 6/ Λ 5 is the length scale of the bulk cosmological constant Λ 5,andµ is the mass parameter of the bulk black hole. This equation looks ver different from the standard Friedmann equation in the sense that the first term in the right hand side is proportional to ρ 2. Nonetheless, the standard cosmolog is restored at low energ if brane tension is properl introduced [8 13]. The term µ/a 4 is due to Wel tensor in the bulk [21] and can be understood as dark radiation [11]. Thus, our basic equations for brane cosmolog driven b the rolling tachon are the equation of motion (9) and the generalized Friedmann equation (13) with ρ given b (11). Let us analze behavior of the sstem near the tachon ground state T = T and show that the standard cosmolog with a usual scalar field can mimic the low energ behavior of the sstem. For this purpose, we assume that [V (T ) V (T )]/V (T ) lv (T )/V (T ) T 2 l T O(ɛ), (14) where ɛ is a dimensionless small parameter. The actual value of T can be either finite or infinite. With this assumption, the generalized Friedmann equation (13) and the tachon equation (9) are reduced to (ȧ a ) 2 = 8πG N 3 [ ] 1 2 φ 2 + V eff (φ) K a 2 + Λ µ a 4 + O(ɛ2 ), (15) φ +3ȧ a φ + V eff (φ) =O(ɛ2 ), (16) where φ = V (T )T, V eff (φ) =V (T ) V (T ), G N = κ 4 5 V (T )/48π and Λ 4 = κ 4 5 V (T ) 2 /12 3l 2. These equations are the same as the corresponding equations in the standard cosmolog driven b a usual scalar field φ with the potential V eff (φ), the cosmological constant Λ 4 and the dark radiation µ/a 4, up to corrections of order O(ɛ 2 ). Consistenc of the above reduced equations with the assumption (14) requires that Kl 2 a 2 l 2 Λ 4 µl 2 a 4 O(ɛ). (17) In particular, we need to impose a fine-tuning between the tachon vacuum energ V (T ) and the bulk cosmological constant Λ 5 so that the 4-dimensional effective cosmological constant Λ 4 is small compared to Λ 5. It is eas to introduce other matter fields on the brane, following refs. [8 13]. In this case, the generalized Friedmann equation (13) becomes (ȧ a ) 2 = 8πG N 3 [ ] 1 2 φ 2 + V eff (φ)+ρ matter K a 2 + Λ µ a 4 + O(ɛ2 ), (18) 3

4 where ρ matter is energ densit of other matter fields on the brane, and the equation of motion of the tachon is unchanged. Here, we have assumed that ρ matter /V (T )=O(ɛ). Therefore, the standard cosmolog can mimic the low-energ behavior of the brane cosmolog driven b the tachon. It is well-known in the standard cosmolog that a scalar field oscillating about a potential minimum behaves like pressure-less gas. Hence, if the tachon potential in the effective field theor has a minimum at finite distance from the maimum and if it oscillates about the minimum then the tachon will behave like pressure-less gas. Note that calculations in boundar string field theor suggest that the minimum of the tachon potential is a finite distance awa from the maimum [22,23]. It is probabl worth while mentioning Sen s result that deca of unstable D-branes in string theor produces pressure-less gas with non-zero energ densit but that the minima must be at infinit in the effective field theor description [4]. There is an interesting accidental coincidence (production of pressure-less gas) between Sen s result in conformal field theor and the above conclusion based on the effective field theor and the brane world. IV. SIMPLE EXAMPLES To see behavior of the sstem beond the low energ regime satisfing (14) and (17), we need to analze the generalized Friedmann equation (13) and the tachon equation (9) directl. In the following, for simplicit we consider a spatiall flat brane (K = ) in the pure AdS bulk (µ = ). B introducing dimensionless quantities τ = t/l, = T/l, = τ and z = τ a/a, our basic equations in this case are written as τ =, τ = (1 2 ) [ ] 3z + v (), v() (19) where v() =κ 2 5 lv (T )/6, v () = v(), and z is given b z 2 = v()2 1. (2) 1 2 There are the epanding branch (z >) and the contracting branch (z <). Hence, the projection to the -plane is two-fold. Hereafter, we shall consider the following three cases separatel: (i) v( ) > 1(Λ 4 > ), where is a potential minimum; (ii) v( ) < 1(Λ 4 < ); (iii) v( )=1(Λ 4 = ). In the case (i), epanding and contracting branches are disconnected. In the case (ii) the two branches are connected in the z-space through the intersection of the surfaces v() 2 =1 2 and z =. In the critical case (iii), two branches are just touching at a point (,, z) =(,, ) in the z-space. (See ref. [24] for discussion about the standard scalar field cosmolog with negative potentials.) Hereafter, unless otherwise stated, we shall consider the epanding branch. The behavior of the sstem can be easil understood b plotting the vector field ( τ, τ )inthe-plane. First, let us consider the critical case (iii). For a simple double-well potential v() =( 2 1) 2 + 1, figure 1 shows a plot near the minimum = 1. This plot shows that the tachon field oscillates about the minimum. Since the conservation equation (12) implies that epansion of the brane universe dilutes the energ densit ρ, the amplitude of the oscillation decas. In order to see a global picture of the vector field, it is convenient to introduce a normalized vector field (N τ, N τ ), where N =1/ ( τ ) 2 +( τ ) 2. Figure 2 shows the global picture of the normalized vector field. It is eas to see how the tachon decas into the minimum with oscillation. Hence, for the critical case (iii), we ma generall epect that the tachon rolls down to the minimum of the potential, that it oscillates about the minimum with decaing amplitude and that the universe approaches to Minkowski spacetime (H = z/l =). Net, let us consider the case (i). Figure 3 and figure 4 show plots for the potential v() =( 2 1) In this case, rolling down of the tachon is slower than the case (iii) due to the larger cosmic friction term 3z in (19). For the same reason, deca of the oscillation about the minimum in the case (i) is faster than the case (iii). In the case (i) the universe approaches to de Sitter spacetime (H = z/l > ). Finall, let us consider the case (ii). Figure 5 and figure 6 show the epanding branch and the contracting branch, respectivel, for the potential v() =( 2 1) These two branches are connected in the z-space through a throat. In each figure, the throat is the boundar between regions with and without arrows. Hence, even if the universe was initiall in the epanding branch, the sstem approaches to the throat and goes into the contracting branch. Eventuall, the universe hits a big-crunch singularit. Note that the tachon never settles down to the potential minimum in the case (ii). 4

5 The general qualitative behavior eplained here is consistent with the result of ref. [24] in which the standard scalar field cosmolog was analzed, provided that V in ref. [24] is replaced b the 4-dimensional effective cosmological constant Λ 4 = κ 4 5 V (T ) 2 /12 Λ 5 / FIG. 1. A plot of the vector field ( τ, τ) for the double-well potential v() =( 2 1) The diamond at (, ) =(1, ) represents an attractor corresponding to Minkowski spacetime with the tachon in its ground state FIG. 2. The global picture of the normalized vector field for the double-well potential v() =( 2 1) The diamond at (, ) =(1, ) represents an attractor corresponding to Minkowski spacetime with the tachon in its ground state. 5

6 FIG. 3. A plot of the vector field ( τ, τ ) for the double-well potential v() =( 2 1) The diamond at (, ) =(1, ) represents an attractor corresponding to de Sitter spacetime with the tachon in its ground state FIG. 4. The global picture of the normalized vector field for the double-well potential v() =( 2 1) The diamond at (, ) =(1, ) represents an attractor corresponding to de Sitter spacetime with the tachon in its ground state. 6

7 FIG. 5. The global picture of the normalized vector field for the double-well potential v() =( 2 1) in the epanding branch. The region without arrows is not allowed. The boundar between the allowed and disallowed regions corresponds to the throat through which the sstem evolves to the contracting branch shown in figure FIG. 6. The global picture of the normalized vector field for the double-well potential v() =( 2 1) in the contracting branch. The region without arrows is not allowed.the boundar between the allowed and disallowed regions corresponds to the throat through which the sstem evolves from the epanding branch shown in figure 5 to the contracting branch shown here. The above eamples of double-well potentials have minima at finite T. The finiteness is consistent with the result in boundar string field theor that the minimum of the tachon potential is a finite distance awa from the maimum [22,23]. However, this is not, at least apparentl, consistent with the result in conformal field theor that the tachon evolves to the minimum without oscillation [25,4]. Sen suggested that the minima must be at infinit in the effective field theor description [4]. Hence, it ma be relevant to consider not a double-well potential but a run-awa potential. As a simple eample, let us consider a ran-awa potential v() =1/ cosh() + 1. Figure 7 shows a plot of the vector ( τ, τ ). In order to see a global behavior of the sstem, figure 8 plots the normalized vector field (N τ, 7

8 N τ ), where N =1/ ( τ ) 2 +( τ ) 2. From these two figures, it is eas to see how rolling down of the tachon slows down. Hence, we ma epect inflation on the brane b the rolling tachon, depending on the form of the potential. Actuall, the effective potential V eff (φ) corresponding to the run-awa potential v() =1/ cosh() +1 is V eff (φ) =6κ 2 4 l 2 / cosh(κ 4 φ/ 6) ep( κ 4 φ/ 6) (κ 4 φ 1), where κ 2 4 = κ2 5 /l, and it is known that in the standard cosmolog the so called power-law inflation occurs for this potential V eff (φ) [26 28] FIG. 7. A plot of the vector field ( τ, τ) for the ran-awa potential v() =1/ cosh() + 1. The dotted line represents points satisfing τ = FIG. 8. The global picture of the normalized vector field for the ran-awa potential v() = 1/ cosh() + 1. The dotted line represents points satisfing τ =. 8

9 V. SUMMARY We have investigated brane cosmolog driven b the tachon rolling down to its ground state. We have adopted the effective action (2) for the tachon and Randall-Sundrum tpe brane world scenario. We have shown that the standard cosmolog with a usual scalar field can mimic the low energ behavior of the sstem near the tachon ground state. In particular, if the tachon potential in the effective field theor has a minimum at finite distance from the maimum and if it oscillates about the minimum then the tachon will behave like pressure-less gas. There is an interesting accidental coincidence (production of pressure-less gas) between Sen s result in conformal field theor [25,4] and the above conclusion based on the effective field theor and the brane world. We have also analzed qualitative behavior of the sstem beond the low energ regime for a spatiall flat brane in pure AdS bulk. As an eample we have considered a double-well potential. For the double-well potential, qualitative behavior of the sstem is classified b the sign of Λ 4 = κ 4 5 V (T ) 2 /12 Λ 5 /2: (i) Λ 4 > ; (ii) Λ 4 < ; (iii) Λ 4 =. In the case (i) the tachon rolls down the potential hill and starts oscillating about a minimum. The amplitude of the oscillation decas due to cosmic epansion and the universe approaches to de Sitter spacetime. The case (iii) is similar to the case (i), but the universe approaches to Minkowski spacetime. In the case (ii), even if the universe was initiall epanding, it starts contracting and eventuall hits a big-crunch singularit. In this case the tachon never settles down to the potential minimum. Finall, we considered a run-awa potential, for which the ground state is at infinit. For the run-awa potential with Λ 4 =, rolling down of the tachon slows down and power-law inflation can occur. ACKNOWLEDGMENTS The author would like to thank Lev Kofman, Shiraz Minwalla, Ricardo Schiappa, Andrew Strominger and Lisa Randall for useful discussions and/or comments. He would be grateful to Werner Israel for continuing encouragement. This work is supported b JSPS Postdoctoral Fellowship for Research Abroad. [1] A. Sen, JHEP 986, 7 (1998) [hep-th/983194]; A. Sen, JHEP 988, 1 (1998) [hep-th/98519]; A. Sen, JHEP 988, 12 (1998) [hep-th/98517]. [2] A. Strominger and M. Gutperle, Spacelike Branes, hep-th/2221. [3] For review, see A. Sen, Non-BPS states and Branes in String Theor, hep-th/99437; A. Lerda and R. Russo, Stable non-bps states in string theor: a pedagogical review, hep-th/9956; J. Schwarz, TASI Lectures on Non-BPS D-brane Sstems, hep-th/ [4] A. Sen, Tachon matter, hep-th/ [5] G. W. Gibbons, Cosmological Evolution of the Rolling Tachon, hep-th/248. [6] M. Fairbairn and M. H. G. Ttgat, Inflation from a Tachon Fluid?, hep-th/247. [7] L. Randall and R. Sundrum, Phs. Rev. Lett. 83, 469 (1999). [8] J. M. Cline, C. Grojean and G. Servant, Phs. Rev. Lett (1999) [hep-ph/996523]. [9] E. E. Flanagan, S. H. H. Te, I. Wasserman, Phs. Rev. D62, 4439 (2) [hep-ph/991498]. [1] P. Binétru, C. Deffaet, U. Ellwanger and D. Langlois, Phs. Lett. B477, 285 (2) [hep-th/991219]. [11] S. Mukohama, Phs. Lett. B473, 241 (2) [hep-th/ ]. [12] P. Kraus, JHEP 9912, 11 (1999) [hep-th/991149]. [13] D. Ida, JHEP 9, 14 (2) [gr-qc/99122]. [14] M. R. Garousi, Nucl. Phs. B584, 284 (2) [hep-th/3122]. [15] E. A. Bergshoeff, M. de Roo, T. C. de Wit, E. Eras and S. Panda, JHEP 5, 9 (2) [hep-th/3221]. [16] J. Kluson, Phs. Rev. D62, 1293 (2) [hep-th/416]. [17] W. Israel, Nuovo Cim. B44, 1 (1966); Erratum-ibid. B48, 463 (1967). [18] S. Mukohama, T. Shiromizu and K. Maeda, Phs. Rev. D62, 2428 (2), Erratum-ibid. D63, 2991 (21) [hepth/ ]. [19] P. Binetru, C. Deffaet and D. Langlois, Nucl. Phs. B565, 269 (2) [hep-th/99512]. [2] C. Csaki, M. Graesser, C. Kolda and J. Terning, Phs. Lett. B462, 34 (1999) [hep-ph/996513]. [21] T. Shiromizu, K. Maeda and M. Sasaki, Phs. Rev. D62, 2412 (2). [22] A. A. Gerasimov and S. L. Shatachvili, JHEP 1, 34 (2) [hep-th/913]. [23] D. Kutasov, M.Marino and G. W. Moore, JHEP 1, 45 (2) [hep-th/9148]. 9

10 [24] G. Felder, A. Frolov, L. Kofman and A. Linde, Cosmolog With Negative Potentials, hep-th/2217. [25] A. Sen, Rolling Tachon, hep-th/ [26] F. Lucchin and S. Matarrese, Phs. Rev. D32, 1316 (1985). [27] J. Halliwell, Phs. Lett. B185, 341 (1987). [28] Y. Kitada and K. Maeda, Class. Quantum Grav. 1, 73 (1993). 1