Rolling Solution for Tachyon Condensation in Open String Field Theory
aa r X i v : . [ h e p - t h ] A p r DAMTP-2008-23
Rolling Solution for Tachyon Condensationin Open String Field Theory
Liudmila Joukovskaya ∗ DAMTP, Centre for Mathematical Sciences, University of Cambridge,Wilberforce Road, Cambridge CB3 0WA, UK
Abstract
Open string field theory in the level truncation approximation is considered. It isshown that the energy conservation law determines the existence of rolling tachyonsolution. The coupling of the open string field theory action to a Friedmann-Robertson-Walker metric is considered and as a result the new time dependentrolling tachyon solution is presented and possible cosmological consequences arediscussed.
Consideration of fundamental theories such as M/String Theory in the cosmological con-text continues to attract attention in the literature.One of the interesting questions is the role of the tachyon in String Theory andCosmology. The great progress in our understanding about tachyon condensation wasmade in the past decade[2, 1] but a lot of interesting issues are still open. Among themost important ones is a better understanding of the dynamics in tachyon condensationprocess.Open String Field Theory (OSFT) [3] gives us a tachyon effective action[4, 5] which isderived from first principles and correctly describes tachyon physics and could representmost exhaustive framework for studying tachyon dynamics.The tachyon dynamics even in the case without gravity represents a nontrivial taskwhich was widely studied [6, 8, 9, 7, 10], while considering OSFT coupled to the gravity ina Friedmann-Robertson-Walker (FRW) type of metric makes this problem more nontrivialbecause of the d’Alambert operators in curved spaces that would appear in the action.This fact makes mathematical investigation of this problem much more complicated butwill benefit us with desired time-dependent solution interpolating between different vacua. ∗ E-mail: [email protected]
1n this work new numerical rolling solution for tachyon condensation will be presented.We will show that consideration of OSFT coupled to the gravity which represents morerealistic situation from cosmological point of view allows the existence of the rollingtachyon configuration which was forbidden by energy conservation law in the case withoutgravity.
The action of bosonic cubic string field theory has the form S = − g Z ( 12 Φ · Q B Φ + 13 Φ · (Φ ∗ Φ)) , (1)where g is the open string coupling constant, Q B is BRST operator, ∗ is noncommutativeproduct and Φ is the open string field containing component fields which correspond toall the states in string Fock space.Considering only tachyon field φ ( x ) at the level (0,0) the action (2) becomes S = 1 g Z d x (cid:20) α ′ φ ( x ) (cid:3) φ ( x ) + 12 φ ( x ) − K Φ ( x ) − Λ (cid:21) , (2)where α ′ is the Regge slope, K = √ , φ is a scalar field, Φ = e k (cid:3) g φ , k = α ′ ln K , (cid:3) = √− g ∂ µ √− gg µν ∂ ν and Λ = K − was added to the potential to set the local minimumof the potential to zero according Sen’s conjecture [11].The action (2) leads to equation of motion( α ′ (cid:3) + 1) e − k (cid:3) Φ = K Φ . (3)The Stress Tensor for our system is T αβ ( x ) = − g αβ (cid:18) φ − α ′ ∂ µ φ∂ µ φ − K Φ − Λ (cid:19) − α ′ ∂ α φ∂ β φ (4) − g αβ k Z dρ (cid:2) ( e kρ (cid:3) K Φ )( (cid:3) e − kρ (cid:3) Φ) + ( ∂ µ e kρ (cid:3) K Φ )( ∂ µ e − kρ (cid:3) Φ) (cid:3) +2 k Z dρ ( ∂ α e kρ (cid:3) K Φ )( ∂ β e − kρ (cid:3) Φ) . The energy is defined as E ( t ) = T and pressure as P ( t ) i = − T ii (no summation) andfor our system are E = E k + E p + Λ + E nl + E nl , P = E k − E p − Λ − E nl + E nl Note that here and below integration over ρ is understood as limit of the following regularization Z dρf ( ρ ) = lim ǫ → +0 lim ǫ → +0 Z − ǫ ǫ dρf ( ρ ) . E k = α ′ ∂φ ) , E p = − φ + K E nl = k Z dρ (cid:0) e kρ (cid:3) K Φ (cid:1) (cid:0) − (cid:3) e − kρ (cid:3) Φ (cid:1) , E nl = − k Z dρ (cid:0) ∂e kρ (cid:3) K Φ (cid:1) (cid:0) ∂e − kρ (cid:3) Φ (cid:1) . In this paper we will be interested in spatially homogeneous configurations for whichBeltrami-Laplace operator used above takes the form (cid:3) g = − ∂ . To avoid calculationof e − kρ∂ term which is much harder to compute then e kρ∂ ( k >
0) as computation ofthe former results in an ill-posed problem we will use the following representation fornonlocal energy terms E nl and E nl which are valid on the equation of motion for thescalar field E nl = k Z dρ (cid:16) ( − α ′ ∂ + 1) e (2 − ρ ) k∂ Φ (cid:17) (cid:16) ∂ e kρ∂ Φ (cid:17) , E nl = − k Z dρ (cid:16) ∂ ( − α ′ ∂ + 1) e (2 − ρ ) k∂ Φ (cid:17) (cid:16) ∂e kρ∂ Φ (cid:17) . Claim 1 The Energy E = α ′ ∂φ ) − φ + K + Λ + k Z dρ (( − α ′ ∂ + 1) e (2 − ρ ) k∂ Φ) ←→ ∂ ( ∂e kρ∂ Φ) , is conserved on the solutions of equation of motion (3)( − α ′ ∂ + 1) e k∂ Φ = K Φ where A ←→ ∂ B = A∂B − B∂A . Proof. dE ( t ) dt = α ′ ∂φ∂ φ − φ∂φ + Φ ∂ Φ + k Z dρ (( − α ′ ∂ + 1) e (2 − ρ ) k∂ Φ) ←→ ∂ ( ∂e kρ∂ Φ) . Using following identity[18] Z dρ ( e ρ∂ ϕ ) ←→ ∂ ( e (1 − ρ ) ∂ φ ) = ϕ ←→ e ∂ φ, equation of motion and definition of field Φ, we have dE ( t ) dt = α ′ ∂φ∂ φ − φ∂φ +Φ ∂ Φ+ ∂ Φ ←→ e k∂ ( α ′ ∂ − e k∂ Φ = ∂ Φ h Φ + ( α ′ ∂ − e k∂ Φ i = 0 . Similar theorem was proved in [6], but the variant we use is more useful for numerical calculations,because in order to define action of the exponential operator we need to do only one well definedintegration instead of the summation over infinite series expansions with which one always need to bevery careful about the convergence and related issues.
3t is straightforward to generalize this statement to arbitrary potential and only finitenumber of fields. Let us consider physical consequence of the energy conservation law.
Claim 2 There doesn’t exist continuous solution of equation (3) which satisfies theboundary conditions lim Φ( t ) = ( , t → ∞ ,K − , t → −∞ (5)or vice-versa (in terms t → − t ). Proof . Let us assume existence of such solution and calculate energy at the extremumpoints, we get E (Φ = 0) = Λ and E (Φ = K − ) = − K − + Λ, i.e. energy values at t → + ∞ and t → −∞ are different what contradicts the energy conservation theorem.As we can see energy conservation law plays crucial role in the existence of the timedependent solutions of equation of motion for the case of level truncation approxima-tion for OSFT. The above statement could probably be generalized to the case of fullOSFT action because for the action with cubic interaction solution interpolating betweenmaximum and minimum in the effective potential has to interpolate between vacua withdifferent energy. In this section we consider more realistic case when gravity is included S = 1 g Z d x √− g (cid:18) m p R + 12 φ (cid:3) g φ + 12 φ − K Φ − Λ (cid:19) , (6)here m p = g M pl and we will work in units where α ′ = 1. As a particular metric we willconsider a FRW one ds = − dt + a ( t )( dx + dx + dx ) , for which the Beltrami-Laplace operator for spatially-homogeneous configurations takesthe form (cid:3) g = − ∂ − H ( t ) ∂ = − D . Scalar field and Friedmann equations are( − D + 1) e k D Φ = K Φ , (7)3 H = 1 m p E , H + 2 ˙ H = − m p P . (8)Inclusion the gravity drastically changes the question of existence of the dynamical in-terpolation between maximum and minimum of the scalar field potential. This happensbecause then there are no restrictions from energy conservation law. The similar claim for the p-adic string model was proved in [6], which rules out the possibility thatthe tachyon may roll monotonically down from one extremum reaching the tachyon vacuum. F -0.04-0.020.020.040.060.08 V H F L Figure 1: The potentialAccording to the Friedmann and scalar field equations we can expect the scalar fieldrolling solution and Hubble function satisfying the following boundary conditionslim Φ( t ) = ( , t → ∞ ,K − , t → −∞ lim H ( t ) = ( (18 K ) − / , t → ∞ , , t → −∞ (9)or vice-versa (in terms of t → − t ). Note that from cosmological perspective we areinterested only in positive values for the Hubble function and thus we do not considernegative sign in front of the square root in (9).To analyze physical situation let us consider potential in which motion is expected.Naive extraction of potential from the model action (6) results in V (Φ) = − Φ + K Φ + Λ. The constant Λ represents the D-brane tension which according to Sen’sconjecture must be added to cancel the negative energy appearing due to the presenceof tachyon. We have obtained two type of solutions. The first one is an ordinary rollingsolution which starts from Φ = 0 and goes towards configuration Φ = K − which isassociated with the true vacuum. This solution can be interpreted as a description of theD-brane decay. The second one is a rolling solution which goes in the opposite direction,which appears in this model possibly because of the non-locality in the interaction. It isknown that nonlocal dynamics has many interesting properties which are not present inthe local case. In particular the “slop effect” [6, 8, 14] which is present in the obtainedsolutions (Fig. 2, 3) when the scalar field goes beyond the values from which the scalarfield configuration starts – situation which is not possible in the local models. Potentiallya similar effect can initiate non-symmetry in the potential in ekpyrotic [15] and cycliccosmology [16]. For numerical calculations we operate with scalar field equation of motion (7) and thedifference of equations (8)( − D + 1) e k D Φ = K Φ , ˙ H = − m p ( P + E ) . (10)5he outline of the numerical scheme is the following • For equations (10) we introduce lattice in t variable and then solve resulting systemof nonlinear equations using iterative relaxation solver using discrete L norm tocontrol error tolerance. • The nontrivial thing from computational point of view is efficient evaluation of e kρ D Φ for ρ ∈ [0 , ∂ ρ ϕ ( t, ρ ) = ∂ t ϕ ( t, ρ ) + 3 H ( t ) ∂ t ϕ ( t, ρ ) , (11) ϕ (0 , t ) = Φ( t ) , ϕ ( ρ, ±∞ ) = Φ( ±∞ ) . Once solution of this equation is constructed we have e kρ D Φ( t ) = ϕ ( ρ, t ). • To solve (11) we used second order Crank-Nicholson scheme which is based onapproximation e k ∆ ρ ˜ D ϕ = (cid:16) k ∆ ρ ˜ D (cid:17) (cid:16) − k ∆ ρ ˜ D (cid:17) − ϕ + o (∆ ρ k ˜ D k ) , where ˜ D is a D operator on the t -lattice (it thus has a finite norm) and ∆ ρ isa step size along ρ variable. Derivatives in t variable were approximated using 4thorder finite differences on uniform lattice (symmetric scheme). • In order to exclude possible artifacts of this specific numerical scheme we triedChebyshev-pseudospectral method which is known to have impressive exponentialconvergence [17]. This scheme is known to have very different properties [17] com-pared to finite difference scheme described above, but it produced the same resultsup to the approximation error which gives us confidence in the existence of therolling solution reported in this work.
Solutions of (7) and (8) are presented on Fig. 2 and 3. As we can see we obtained accel-erating rolling solutions for tachyon scalar field Φ. It is natural to address cosmologicalissues in the context of rolling tachyon solution [19]. Taking into account that acceleration It is important that the described below scheme is general and was also used for solving cosmologicalequations for the case of Cubic Fermionic Field Theory with the quartic interaction term in (6) insteadof cubic one as well as for p-adic string model at least for p = 2 ,
3. Obtained interpolating solutionsbetween corresponding maximum and minimum of the tachyon potential look very similar to one whichwill be presented in section 5 and will be presented in [18]. It looks that cosmology effaces differencebetween cubic and quartic interaction for the type of solutions indicated above.
30 -20 -10 10 20 30 t -0.2-0.10.10.20.30.40.5 j H t L -30 -20 -10 10 20 30 t H H t L -30 -20 -10 10 20 30 t a H t L Figure 2: Solutions of the scalar field (7) and Friedmann equation (8) Φ, H and a (leftto right) for m p = 1.of the Universe is one of the most fascinating processes of the modern cosmology manyauthors tried to explain among other possibilities with a possible explanation via scalarfield. It is interesting that cosmology gives us this solution owing to coupling our action(2) to a FRW metric and the consequent inclusion of a Hubble friction term which leadsto time-dependent rolling solution with exponentially decreasing oscillations around theminimum. Moreover because generally speaking string scale does not exactly coincidewith Plank mass we obtain some freedom in settling m p parameter for numerical calcula-tions which enters into Friedmann equations and as a result govern the value of Hubblefunction H ( t ). Thus decreasing the value of m p leads to more smooth profile for rollingsolution while increasing m p results in higher oscillations of the solution in comparisonto those presented on the Fig. 2 and 3, more details will be presented in [18]. During theprocess of completion of this work appeared [10] in which OSFT tachyon in the dilatonbackground was considered and time-like rolling tachyon solution were obtained. Becausedilaton appears from the same string sector as graviton including the dilaton into thetachyon action can qualitatively reproduce behavior of the tachyon in the curved spaces.Concluding this section we would like to summarize that we obtained time dependentaccelerating solution interpolating between unstable and the true vacua (see Fig. 2) whichcan be interpreted as being responsible for acceleration of the Universe during this rollingfrom unstable vacuum to the true vacuum, after which it disappears. Evolution of thescalar field in the opposite direction is also possible with the Hubble function in form ofincreasing kink when scale factor a ( t ) starts from the constant plateau and exponentiallygrows, which seems counter intuitive but can be related to late time acceleration. -30 -20 -10 10 20 30 t -0.10.10.20.30.40.5 j H t L -30 -20 -10 10 20 30 t H H t L -10 -5 5 10 t a H t L Figure 3: Solutions of the scalar field (7) and Friedmann equation (8) Φ, H and a (leftto right) for m p = 1. 7 Conclusion
The Witten’s cubic open bosonic string filed theory in the level truncation approximationwas considered. It was shown that the energy conservation law determines existenceof rolling tachyon solution. As a result it was explicitly shown that the non-existenceof the rolling solution in the Minkowski case is a necessary consequence of the energyconservation law of the system. The modification of conservation law in the presence ofthe gravity is discussed. The first rolling solution for tachyon condensation in this theoryis presented and possible cosmological consequences are discussed. Although only lowestexcitation in the full OSFT were taken into account there are solid reasons to supposethat the general picture for the tachyon condensation process will be the same in the caseof full OSFT.
Acknowledgements
The author would like to thank I. Aref’eva, R. Bradenberger, A.-C. Davis, J. Khoury,N. Nunes, F. Quevedo, D. Seery, D. Wesley and especially D. Mulryne and Ya. Volovichfor useful discussions. The author gratefully acknowledge the use of the UK NationalSupercomputer, COSMOS, funded by PPARC, HEFCE and Silicon Graphics. This workis supported by the Centre for Theoretical Cosmology, in Cambridge.
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