Moduli Evolution in the Presence of Matter Fields and Flux Compactification
aa r X i v : . [ h e p - t h ] S e p Preprint typeset in JHEP style - PAPER VERSION
Moduli Evolution in the Presence of Matter Fieldsand Flux Compactification
Carsten van de Bruck
Department of Applied MathematicsUniversity of SheffieldSheffield, S3 7RH. UK
Ki-Young Choi
Department of Physics and AstronomyUniversity of SheffieldSheffield, S3 7RH. UK [email protected]
Lisa M.H. Hall
Department of Applied MathematicsUniversity of SheffieldSheffield, S3 7RH. UK [email protected]
Abstract:
We provide a detailed analysis of the dynamics of moduli fields in the KKLTscenario coupled to a Polonyi field, which plays the role of a hidden matter sector field. Itwas previously shown that such matter fields can uplift AdS vacua to Minkowski or de Sittervacua. Additionally, we take a background fluid into account (which can be either matteror radiation), which aids moduli stabilisation. Our analysis shows that the presence of thematter field further aids stabilisation, due to a new scaling regime. We study the systemboth analytically and numerically.
Keywords: physics of the early universe, string theory and cosmology, cosmologicalapplications of theories with extra dimensions. ontents
1. Introduction 12. Background Equations 33. Properties of the Potential 54. Review of Single Field Dynamics 7 b ≪ b .
5. Two Real Fields with a Matter Background Fluid 10 b ≪ b . b . b ≪
6. Dynamics of Complex Fields 177. Radiation ( γ = ) 208. Conclusion 22A. Useful Formulae 24B. Tracking Solution 25C. Stability of the critical points 26
1. Introduction
Theories beyond the standard model, such as string theory, predict the existence of massless(or nearly massless) scalar fields. These so-called moduli fields may contain, for example,the information about the dynamics of extra spatial dimensions, or, as it is the case ofthe string theory dilaton, they determine the coupling strength of gauge fields. In addi-tion, these fields usually interact, with gravitational strength, to other particles, therebymediating a new (“fifth”) force. Since so far we have neither observed any significant– 1 –ime-dependence of gauge couplings nor detected new forces, the moduli fields have to bestabilised [1]. This should happen at some stage during the cosmological evolution, whichis one of the problems addressed in string cosmology. It was emphasized in [2] that sta-bilising the string theory dilaton in a cosmological framework is a rather difficult, becauseof the the small barrier height separating a local Minkowski (or de Sitter) vacuum andthe steepness of the potential. The situation can be improved if background fluids (eitherradiation or matter) or temperature dependent corrections are taken into account (for workin this direction see e.g. [3, 4, 5, 6]).In order to stabilise the moduli fields, a mechanism is required which gives them amass. Additionally, the resulting vacuum energy should be very small and non-negativeand the vacuum itself quasi-stable (i.e. with a life-time much longer than the age of theuniverse). Recently, compactification mechanisms with fluxes on internal manifolds havebeen extensively studied (see [7], a review can be found in [8]); a well-investigated scenariois the KKLT proposal [9]. Within this latter setup, moduli fields are stabilised and theresulting cosmological constant can be fine-tuned to be either zero or very small. To obtaina realistic scenario, a crucial ingredient in these mechanisms is a “lifting” procedure, whichraised a supersymmetric AdS-vacuum to become a Minkowski or even a (quasi) de-Sittervacuum. In the original KKLT proposal, an anti-D3 brane was added, explicitly breakingsupersymmetry. In an alternative proposal, D-terms were considered to up-lift the AdS-vacuum, which requires the existence of charged matter fields [10, 11, 12].Recently it was pointed out that interactions of the moduli fields with a hidden mattersector can also result in an effective uplifting to Minkowski or de Sitter vacua. The ideaof this procedure is to take matter fields into account (such fields will be present in anyrealistic theory). The resulting interaction terms in the scalar potential lead to spontaneoussupersymmetry breaking, with local minima which have a small and positive vacuum energydensity. This procedure is known as F-term uplifting (for recent work, see see e.g. [13, 14,15, 16, 17, 18, 19, 20] and references therein). An appealing feature is that this procedurecan result in a small gravitino mass, while maintaining a high potential barrier, resultingin long-lived de-Sitter vacua [14, 17, 18, 21].In this paper, we investigate the cosmological dynamics of the moduli fields and matterfields in the the context of F-term uplifing. As a toy model, we will couple the KKLT modelwith the Polonyi model [22], following [14, 20, 21, 23]. This model is simple enough to studythe dynamical consequences of the interactions between the moduli fields and matter fields,but we believe that our results can be generalised to more complex models. In particularwe are interested in whether the interaction will facilitate or impede the stabilisation of themoduli fields in the cosmological context. The paper is organized as follows: In Section 2we formalise the set-up and write down the essential equations. In Section 3, we discussthe properties of the potential. We review past results, in the context of our potential,in Section 4 and generalise to multi-field scenarios in Section 5 (two real fields). The fulldynamics for two complex fields is presented in Section 6. We briefly consider the effectsof a radiation background fluid in Section 7. We conclude in Section 8. We give furtheruseful formulae in the Appendices and derive the new scaling regime.– 2 – . Background Equations
We begin by stating the four-dimensional N = 1 SUGRA action, which is of the form S = − Z √− g (cid:18) κ P R + K i ¯ j ∂ µ Φ i ∂ µ ¯Φ ¯ j + V (cid:19) d x, (2.1)where K i ¯ j = ∂ K∂ Φ i ∂ ¯Φ ¯ j is the K¨ahler metric, Φ i are complex chiral superfields and V (Φ) isthe scalar potential. κ P is the 4-dimensional Newton constant, κ P = 8 πG N = 1 . (2.2)The effective scalar potential is given, with given K¨ahler metric K and superpotential W ,as V = e K (cid:16) K i ¯ j D i W D j W − W W (cid:17) , (2.3)where K i ¯ j is the inverse K¨ahler metric and D i W = ∂ i W + ∂K∂ Φ i W .The following equations of motion for the real and imaginary parts of superfieldsare [26] ¨ ϕ iR + 3 H ˙ ϕ iR + Γ ijk ( ˙ ϕ jR ˙ ϕ kR − ˙ ϕ jI ˙ ϕ kI ) + 12 K i ¯ j ∂ j R V = 0 , ¨ ϕ iI + 3 H ˙ ϕ iI + Γ ijk ( ˙ ϕ jI ˙ ϕ kR + ˙ ϕ jR ˙ ϕ kI ) + 12 K i ¯ j ∂ j I V = 0 , (2.4)where ϕ iR ( ϕ iI ) refers to the real (imaginary) part of the scalar fields and ∂ j R ( ∂ j I ) are usedto denote the derivative of the potential with respect to the real (imaginary) parts of thefields, respectively. The connections on the K¨ahler manifold are given byΓ nij = K n ¯ l ∂K j ¯ l ∂ Φ i . (2.5)There are two different sectors in the model considered here; one involves a modulusfield T , whereas the other sector contains a matter field C . Both fields will be taken ascomplex fields. The corresponding K¨ahler potential [21], which arises in type IIB andheterotic string theory, is K = − T + ¯ T ) + | C | ,W = W ( T ) + W ( C ) . (2.6)Following [21, 23], for example, we consider the combination of the KKLT [9] and thePolonyi model, for which the superpotential is given by W ( T ) = W + Ae − aT , W ( C ) = c + µ C, (2.7)so that the resulting scalar potential is given by– 3 – = e C ¯ C T + ¯ T ) (cid:16)(cid:12)(cid:12) − Aae − aT ( T + ¯ T ) − W + Ae − aT + c + µ C ) (cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) µ ¯ C + ( W + Ae − aT + c + µ C ) (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) W + Ae − aT + c + µ C (cid:12)(cid:12)(cid:12) ! . (2.8)The differential equations describing the dynamics of the field T are given by¨ T r + 3 H ˙ T r − T r ( ˙ T r − ˙ T i ) + 2 T r ∂ T r V =0 , ¨ T i + 3 H ˙ T i − T r ˙ T r ˙ T i + 2 T r ∂ T i V =0 , (2.9)where we write T = T r + iT i . Furthermore, we consider a background fluid with density ρ b and with equation of state γ − ≡ p b /ρ b . Energy conservation dictates that˙ ρ b + 3 Hγρ b = 0 . (2.10)We consider C to be a complex field, C = C r + iC i , so that the equations of motion canbe written as ¨ C r + 3 H ˙ C r + 12 ∂ C r V =0 , ¨ C i + 3 H ˙ C i + 12 ∂ C i V =0 . (2.11)The Friedmann equation reads3 H = 34 T r ( ˙ T r + ˙ T i ) + ( ˙ C r + ˙ C i ) + V + ρ b . (2.12)Since T r is not a canonical field, it is convenient to define a new field φφ ≡ r
32 ln T r . (2.13)Additionally, the equations for the fields are easier to solve if we define a new set of variablesas follows x ≡ ˙ φ √ H , y ≡ √ V √ H , z ≡ e − q φ ˙ T i H , (2.14)and p ≡ ˙ C r √ H , q ≡ ˙ C i √ H . (2.15)With these variables, the Friedmann equation becomes1 = x + y + z + p + q + Ω b , (2.16)– 4 –here Ω b = ρ b / H . The evolution of Hubble parameter H is H ′ = − H (cid:2) x + 2 z + 2 p + 2 q + γ (1 − x − y − z − p − q ) (cid:3) , (2.17)where the prime denotes differentiation with repect to e -fold number N , defined by N ≡ Z Hdt. (2.18)The equations of motion for the fields are now given by x ′ = − x + λ r y − z + 32 x (cid:2) x + 2 z + 2 p + 2 q + γ (1 − x − y − z − p − q ) (cid:3) ,y ′ = − λ r xy − η r yz − δ √ yp − θ √ yq + 32 y (cid:2) x + 2 z + 2 p + 2 q + γ (1 − x − y − z − p − q ) (cid:3) , (2.19) z ′ = − z + η r y + 2 xz + 32 z (cid:2) x + 2 z + 2 p + 2 q + γ (1 − x − y − z − p − q ) (cid:3) , and p ′ = − p + δ √ y + 32 p (cid:2) x + 2 z + 2 p + 2 q + γ (1 − x − y − z − p − q ) (cid:3) ,q ′ = − q + θ √ y + 32 q (cid:2) x + 2 z + 2 p + 2 q + γ (1 − x − y − z − p − q ) (cid:3) . (2.20)In these equations, we have defined λ and η to be λ ≡ − V ∂V∂φ , η ≡ − r e q φ V ∂V∂T i , (2.21) δ ≡ − V ∂V∂C r , θ ≡ − V ∂V∂C i . (2.22)For the remainder of this paper, we solve the above equations, in order to investigate thestabilisation of the moduli fields, T r and T i , in the presence of the matter fields, C r and C i , and the background fluid, ρ b .
3. Properties of the Potential
As a specific example in this paper, following [21], we take the potential parameters to be A = 1 , a = 12 , µ = 10 − , c = µ (cid:16) − √ (cid:17) . (3.1)Additionally, W is assumed to be real and is fine-tuned such that the potential is zero(or the cosmological constant Λ) at the local minimum. For the values above, the localminimum is found at the co-ordinates T (lm) r ≃ . φ (lm) ≃ . , T (lm) i = 0 , C (lm) r ≃ − . , C (lm) i = 0 . (3.2)– 5 – igure 1: Scalar potential of the KKLT + Polonyi model in units of 10 − κ − p . We plot thepotential in the T r - C r plane (left hand side) and T r - T i plane (right hand side), with the other fieldsset at their local minimum values shown in the text. Figure 2:
Scalar potential of the KKLT + Polonyi model in units of 10 − κ − p . We plot thepotential in the φ - T i plane (left hand side) and φ - C i plane (right hand side), with the other fieldsset at their local minimum values shown in the text. In Figs. 1 and 2 we show the shape of this potential with the parameters given above.Since there are four independent fields, we fix two fields at their local minimum values andplot the remaining two fields. For convenience, we plot the canonical field, φ , instead of T r . In Fig. 1 the local minimum of this model is seen, where φ is stabilised at φ ≈ . φ → ∞ , the potential vanishes. Along the C r direction, the potential increases as C r departs from the minimum. As can be seen, the minimum of C r depends directly on thevalue of φ . Around the local minimum, C r ∼ − .
71. For φ . −
5, the potential can beapproximated by V ∼ A C r e C r e √ φ (3.3)with T i = C i = 0, where the minimum of C r is ∼
0. Note that the C r direction is much– 6 – a) λ (b) δ Figure 3:
The contours of λ and δ (defined in Eqns. (2.21-2.22)) over the range of φ and C r considered in the text, for two real fields ( σ i = C i = 0). steeper than along the φ direction. On the left hand side of Fig. 2, we show the sinusoidalshape of T i , with periodicity of 2 π/a , with one minimum at zero. On the right hand side,the minimum of C i is seen to exist at zero and shows symmetry about C i → − C i becausewe assume T i = 0 in the figure. The values of λ and δ (defined in Eqns. (2.21-2.22)) overuseful ranges are plotted in Fig. 3.It should be noted that, as the T i and C i evolve away from their minima, the shapeof the potential changes from that shown in the figures. This will become important whenwe consider the evolution of the imaginary parts of the fields, C and T .
4. Review of Single Field Dynamics
We begin by reviewing known results [4, 25] and consider only real fields; we put theimaginary fields in the minimum point, T i = C i = 0, so that these fields do not evolve.This will enable isolation of the effects of each field (real and imaginary parts). Some ofthe results obtained will prove vital later.In this simplifed scenario, the potential becomes V = e C r − aT r T r h A { C r + 4 aT r (3 + aT r ) } + 3 e aT r ( cC r + µ + µ C r + ωC r ) + 6 Ae aT r { c ( C r + 2 aT r ) + µ C r + C r ( µ + 2 aµ T r ) + ωC r + 2 aωT r } i . (4.1)Further, to compare to single field dynamics, for negatively large φ , we set C r in its localminimum ( C r = 0) initially, such that it does not evolve until the final stages of theevolution. Even with only φ evolving, due to the background fluid and the potential, thedynamics are non trivial.In this section, we will closely follow [26] and will refer to an example trajectoryshown in Fig. 4. We assume that φ starts with zero kinetic energy from a point φ f (thisterminology will become useful later) and Ω b is close to unity initially (Ω b = 0 . a) (b) Figure 4:
Single Field Dynamics. Initial conditions are taken to be Ω b = 0 . C r = 0 and φ = −
15. The background fluid is matter, with γ = 1. Note that ˙ C r ( p ) is zero until the last fewe-folds. field starts to evolve, as long as λ is small enough, the potential energy quickly dominatesthe dynamics, seen in Region 1 of Fig. 4(b). As the potential becomes more steep ( λ increases sharply around φ ≈ − φ f (in Region 3), and the field only starts rollingagain once this fluid has sufficiently diluted.As the field recommences to roll, one of two things can happen; either the scalar fielddominates again (Region 1) or a scaling regime is found, in which the potential and kineticenergies track the background fluid (Region 4), which occurs for our trajectory. The choiceof regime is specified by a scaling condition:3 γ < λ , (4.2)where we restate that γ = 1 + p b /ρ b . For our choice of potential, this condition is alwayssatisfied when φ & − φ f < φ (lm) , the end of scaling occurs near the minimum and the fieldstabilises. If φ f > φ (lm) but smaller than the local maximum value, the field will stillstabilise. Conversely, if φ f is larger than the local maximum value, the field will roll toinfinity. In our case, the fields are stabilised.In the following, we state the analytical solutions for the single field scenario. Inparticular, two initial branches can be distinguished; Ω b ≪ b . x , y etc. (defined earlier) for analysis of the evolution.– 8 – .1 Ω b ≪ b ≪
1, following [24], the initial evolution is given by x ′ = λ r y , y ′ = − λ r xy (4.3)where we assume λ ′ ≈
0, for which the solution is x = p (1 − Ω b ) tanh λ r N ! , (4.4) y = p (1 − Ω b ) sech λ r N ! . (4.5)This solution is valid until λ ′ is no longer insignificant ( φ ≈ − x ≃ y ≃ . λ , no approximation can be found for this period. However, during this stage, due to theinsignificance of Ω b , x increases to almost unity, as the kinetic energy fully dominates theevolution. Due to the expansion of the universe, the kinetic energy is diluted and slowly thebackground fluid becomes significant (Ω b → x → φ f .When the potential is insignificant, the system during kination is greatly simplifiedand can be described by [4]: x ′ = − x + 32 x (cid:2) x + γ (cid:0) − x (cid:1)(cid:3) . (4.6)If x starts at a value x , the solution is x = (cid:18) − x x e − γ ) N (cid:19) − . (4.7)During this decay, the field moves a distance, ∆ φ , given by∆ φ = √ − γ ) ln (cid:18) x − x (cid:19) . (4.8)after which the field freezes at φ f . (Note that x is never exactly unity since Ω b = 0). Ω b . b .
1, the the kinetic energy of φ never fully dominates, in that x <
1, beforethe background fluid takes over. As seen in Fig. 4(a), the background fluid is never totallynegligible, which complicates the evolution. However, kination starts when λ starts toincrease ( φ ≈ − x ≃ y ≃ . y is negligible, φ moves a further distance givenby Eqn (4.8) before freezing at φ f . Although this regime cannot be solved analytically,one point can be made: in this case, the field freezes earlier (due to less kinetic energyoverall, x < φ f is smaller than that when Ω b (0) ≪ γ = 1 and Ω b = 0 .
93 (an example that will prove usefullater), stabilisation only occurs when φ f & −
17 and φ f .
1. We define these points as φ min and φ max . – 9 – . Two Real Fields with a Matter Background Fluid This section extends the previous single field scenario, by considering the evolution of anadditional scalar field. Specifically, we consider the evolution of C r , in which direction thepotential is much steeper than the φ -field previously considered. We will show that thisleads to a new mechanism for stabilisation, due to the presence of a new scaling regime.We will explain the dynamics using several representative trajectories for which C r = 0shown in Fig. 5:I φ (0) = − , C r (0) = 10 , Ω b (0) = 0 .
01 [solid line (blue)]II φ (0) = − , C r (0) = 3 , Ω b (0) = 0 .
01 [dotted line (red)]III φ (0) = − , C r (0) = 3 , Ω b (0) = 0 .
01 [dashed line (green)]IV φ (0) = − , C r (0) = 10 , Ω b (0) = 0 . γ = 1 (a background matter fluid) and consider two cases Ω b = 0 .
01 and Ω b = 0 . φ and C r against e -folding number, N , and thetrajectories in the φ - C r plane. The evolution of each energy component for trajectories Iand II are shown in Figs. 6 and 7. Trajectory III is identical to Trajectory II (althoughstabilisation does not occur) and is not shown. The evolution of the energy componentsfor trajectory IV is shown in Figs. 8. Note that Fig. 6(b) strongly resembles Fig. 2 of [26].As in the single field case, similar regimes can be identified: (1*) field domination, (2*)kination, (3*) freeze-out and (4*) scaling. The asterix (*) denotes two-field evolution, inorder to separate it from the single field case. When C r approaches zero, however, theseregimes are followed by a period of oscillation around C r = 0 (Region 5*), further followedby the regimes (1-4) of single field evolution (as in the previous section). After this, thefield is either stabilised or rolls over. It should be noted that, due to the steepness in the C r direction ( δ > λ ), the C r field arrives in its minimum quickly with the φ field stillrunning toward the minimum. Due to this, the second half of the dynamics follows singlefield evolution, as was discussed in the previous section.We will now derive some useful approximations for the first part of the evolution, when | C r | 6 = 0. Depending on the initial value of Ω b , there are two branches of evolution, A andB. These are shown as a flowchart in Fig. 9, which also shows the path of each exampletrajectory. Ω b ≪ b = 0 .
01. Initially, fora very brief time, the potential energy dominates but since the solution of field dominationis not stable (see Appendix C) this energy is quickly transfered to the kinetic energy ofboth fields, φ and C r . This change of energy can be described by x ′ = λ r y , p ′ = δ √ y , y ′ = − λ r xy − δ √ py, (5.1)– 10 – a) (b)(c) (d) Figure 5:
Various trajectories, each with differing initial conditions for φ , C r and Ω b . Only realfields are considered ( T i = C i = 0). In (a), for Ω b = 0 .
01, three trajectories are shown (I - solidblue, II - dotted red, III - dashed green). In (b), for Ω b = 0 .
9, only one trajectory is shown (IV- dotted magenta). The greyed region signifies initial conditions for trajectories which do not leadto stabilisation. The horizontal lines and regions i-iv are discussed in the text. In (c) and (d), theevolution of the fields are shown with respect to the e -folding number, N , for these four trajectories. for which the solutions are given by x = √ − Ω b q δ λ tanh " λ r r δ λ N ,p = √ − Ω b q λ δ tanh " λ r r δ λ N ,y = p (1 − Ω b ) sech " λ r r δ λ N . (5.2)This is the extended solution of Eqns. (4.4) and (4.5) for two fields. The maxima of thesesolutions, which can be considered to define the start of kination, are approximately given– 11 – a) (b) Figure 6:
Breakdown of energy for Trajectory I with initial conditions Ω b = 0 . φ = − C r = 10. We consider only the real parts of the fields ( T i = C i = 0). (a) (b) Figure 7:
Breakdown of energy for Trajectory II with initial conditions Ω b = 0 . φ = − C r = 3. We consider only the real parts of the fields ( T i = C i = 0). by x ≃ λ λ + δ (1 − Ω b ) ,p ≃ δ / λ + δ (1 − Ω b ) ,y ≃ . (5.3)Here we have neglected the evolution of Ω b , but the error induced should be small. We shalluse the notation that a subscripted bracket will denote the end of the regime of interest,here kination (Region 2*), but we omit the asterix for brevity.– 12 – a) (b) Figure 8:
Breakdown of energy for Trajectory IV with initial conditions Ω b = 0 . φ = − C r = 10. We consider only the real parts of the fields ( T i = C i = 0). InitialConditions A : Ω b ≪ B : Ω b . γ < λ + δ / b ≈ . Figure 9:
Flowchart of possibilities (Colors: Blue - Trajectory I, Red - Trajectory II, Green-Trajectory III, Magenta - Trajectory IV)
During kination, we may assume y ≃ x = x (1 − Ω b (2) ) + Ω b (2) e − γ )( N − N (2) ) ,p = p (1 − Ω b (2) ) + Ω b (2) e − γ )( N − N (2) ) , Ω b = Ω b (2) e − γ )( N − N (2) ) (1 − Ω b (2) ) + Ω b (2) e − γ )( N − N (2) ) . (5.4)– 13 –hen the potential energy negligible compared to the kinetic energy, from the Klein-Gordon equation ¨ φ + 3 H ˙ φ ≃ , (5.5)for which the solution is ˙ φ ∝ a − . Therefore the kinetic energy is proportional to a − . Thisis the same for the C r field.Alternatively, in this region, we can obtain p ′ x ′ = px = √ C ′ r φ ′ = constant , (5.6)where we have used the original definition of x and p for the third term. This gives a directrelation between x and pp = p (2) x (2) x, C r − C r (2) = p (2) √ x (2) ( φ − φ (2) ) , (5.7)where we have used the initial values of kination from Eqn. (5.3). The freeze-out point canbe calculated by integrating these solutions∆ φ = x (2) p − Ω b (2) √ − γ ) " sinh − s − Ω b (2) Ω b (2) ! − sinh − s − Ω b (2) Ω b (2) e − − γ ) N/ ! , ∆ C r = p (2) p − Ω b (2) √ − γ ) " sinh − s − Ω b (2) Ω b (2) ! − sinh − s − Ω b (2) Ω b (2) e − − γ ) N/ ! . (5.8)where ∆ φ = φ (3) − φ (2) and ∆ C r = C r (3) − C r (2) Note however, that when the initialbackground density is small enough, the transfer of potential energy to kinetic energy isso quick that we can consider starting at a point φ , C r with kinetic energies given by x (2) , p (2) . Ω b . b is initially large, then thepotential energy can never dominate and kination is not achieved. Instead, the equationof motion for the background fluid is given byΩ ′ b = − b (cid:2) ( γ − − Ω b ) + 2 y (cid:3) . (5.9)As potential energy is converted to kinetic energy, y becomes negligible in a few e-foldsand we quickly obtain the solutionΩ b = (cid:18) − Ω b Ω b e γ − N (cid:19) − . (5.10)Though the fields obtain some kinetic energy from the potential energy, it is suppressedcompared to the background density. Quickly, Ω b → φ (3) ≈ φ and C r (3) ≈ C r – 14 – .3 Scaling We calculate the full scaling solution and condition in Appendix B. We will use the mainresults in this section. The two-field scaling condition is given by3 γ < λ + δ / . (5.11)At the freeze-out point, if the condition for scaling is satisfied, then the fields track thebackground fluid. We define Γ ij = V V ij V i V j . (5.12)Making the assumption that Γ φφ ≃ Γ C r C r ≃ Γ φC r ≃ , (5.13)which is reasonable in the ranges φ . − | C r | & x c = r
32 ˜ γλ , p c = √ δ ˜ γλ , y c = 32 ˜ γλ (cid:18) − ˜ γ − δ λ ˜ γ (cid:19) , (5.14)where ˜ γ = γ (cid:18) δ λ (cid:19) − . (5.15)This is a valid solution only when Γ ij ≈
1. When C r = 1, however, δ changes form andΓ CrCr begins to decrease (and later increases again). This deviation from Γ ≈ b (as seen around N ∼
35 in Fig. 6). Scaling proceeds until C r goes toaround 0, where the scaling condition is violated.During scaling, the background fluid density takes the formΩ b = 1 − x c − p c − y c = 1 − γλ = 1 − γλ + δ / p c x c = δ √ λ = √ dC r dφ . (5.17)From Fig. 3, we may take λ approximately constant and from Eqn. (4.1) δ ≈ − C r + 1 C r ) (5.18)so that φ = φ (3) − λ C r C r (3) . (5.19)– 15 –t the end of the scaling regime, as found numerically, the fractional energy of a matterbackground fluid is Ω b ≃ . , (5.20)and the kinetic energy of φ is small. We find that this fractional background density isnumerically independent of the initial values of the dynamics and is determined only bythe existence of scaling solution (i.e. potential dependent). Additionally, from Eqn. (5.19),when the trajectory reaches C r = 0, we find φ (5) ≈ φ (3) − λ C r (3) , (5.21)where either [ φ (3) , C r (3) ] are calculated from Eqn. (5.8) (Branch A) or [ φ , C r ] (Branch B).For a matter background and Ω b ≈
1, trajectories always scale. When Ω b ≪
1, onlytrajectories with | C r | & If the trajectory undergoes scaling, then it is clear from Eqn. (5.14) that, for our potentialat least, p c > x c and there is more kinetic energy in C r than in φ . When the trajectoryreaches the C r minimum ( C r = 0), the trajectory oscillates but φ is almost constant; itacts as if it is almost frozen at a point φ f with Ω b ≈ . φ not fixed. Due to the relative gradients, λ and δ ,the kinetic energy is shared between the fields, but the expansion of the universe acts todamp the oscillations. Finally, the background fluid dominates and the fields stop rollingat a point φ f . At this point, we find numerically that Ω b ≈ . C r ≈ We shall characterise a stabilising trajectory by the initial conditions for φ and C r whichlead to final stabilisation. Samples of the stabilisation regions for two real fields can be seenin Fig. 5. Greyed out regions denote initial field values for trajectories which do not resultin stabilisation. We will see that the important factor in stabilisation is the point φ f , thevalue at which the φ - field freezes on the C r = 0 line, after either scaling or oscillation. Ω b . b = 0 . | C r | will follow similardynamics as Trajectory IV. After freezing at a point [ φ f , C r ( f ], the fields will quicklyreach a scaling regime. When C r ≈
0, scaling ends and the C r -field oscillates around the– 16 –valley” and φ is effectively fixed at φ f . Henceforth, the trajectory follows single fielddynamics, with initial conditions φ f and Ω b ≈ . − < φ f < b = 0 .
93, as is seen in Fig. 5(b).The slight asymmetry of the stabilisation region (seen for large φ ) is due to the asymmetryof the potential; the local minimum does not occur at C r = 0. Trajectories which startclose to the minimum are easier to stabilise.We have numerically simulated trajectories only within | C r | <
10. Note however, thatall trajectories with φ f within the stabilisation bounds at C r = 0 will lead to stabilisation.Therefore the allowed C r region extends to | C r | → ∞ , due to the scaling solution. Ω b ≪ b ≪ φ -fieldfreezing on the C r axis between − < φ f < C r = 0. Thus the field has enoughkinetic energy at the crossing point to oscillate around C r = 0 (seen in Fig. 5(b)). Asdescribed earlier, the trajectory settles into the minimum at C r = 0 at a point φ f . Aslong as φ f is within the range − < φ f <
1, the trajectory will lead to stabilisation.Therefore Trajectory II stabilises ( φ f ≈ −
12) whilst Trajectory III does not ( φ f ≈ − b , the kinetic energies are less damped and oscillate more widely. Sincethe field gains more kinetic energy initially (from potential energy) and is less damped, thefields travel further before settling at a point φ f . Therefore as Ω b decreases, the conicalbound moves left and the stabilisation area increases.The asymmetry seen between regions (iii) and (iv) is once more again to the asymmetryof the potential; trajectories starting in region (iii) are pushed towards the minimum (dueto the gradient) and stabilise, whereas trajectories from region (iv) are pushed away fromthe minimum.
6. Dynamics of Complex Fields
In this section, we include the dynamics of the imaginary parts of the fields, C i and T i , inorder to fully generalise our model. The existence of non zero imaginary parts changes theshape of the potential in the φ - C r plane during evolution. As an example we consider atrajectory with initial values: φ = − , C r = 0 . , T i = 0 , C i = 1 , Ω b = 0 . . (6.1)The evolution of the fields for this trajectory are shown in Fig. 10. It can be seen that,even though T i = 0, this field is given a “kick” since C i = 0 We show the evolution of each– 17 – igure 10: An example trajectory with evolving σ i and C i , when Ω b = 0 .
9. The initial conditionsare given in Eqn. (6.1). (a) (b)
Figure 11:
The evolution of each energy component for the trajectory given by the initial conditionsin Eqn. (6.1). (a) The breakdown of kinetic energies. (b) The total breakdown. Note that thekinetic energy of T i is negligible since T i = 0; it is not a generic result. energy component in Fig. 11, where the kinetic energy of T i is seen to be negligible. Thisis due to the fact that T i = 0 and is not generic.In Fig. 12 we show the contour plots of the potential in the φ − C r plane for the abovetrajectory at e-folding numbers N = 26 , , ,
30. In the top lefthand plot, where T i ≃ .
1– 18 – igure 12:
The potential at different efolding values ( N = 26 , , , T i and C i are non-zero), showing how the existence of the minimum (in the φ - C r plane) dependson the imaginary field values. The initial conditions are given in Eqn. (6.1). At N ≈
26, the localminimum has almost disappeared since C i is large, but reappears at later times when C i ≈
0. Theevolution of the potential works to nudge the fields into the local minimum. and C i ≃ .
7, the local minimum almost disappears. However, at larger N and smaller C i ,the potential changes shape and finally around C i ≃ b C i A C i ,T i =0 (%) A C i ,T i =0 (%)10 −
20 68 1810 65 240 39 910 −
20 71 2210 66 220 54 110 . Table 1:
Table showing the stabilisation measure, A ab , (percentage of the total area which leadsto stabilisation) as shown graphically in Figure 13. These numbers are indicative of trends only, asexplained in the text. In order to understand the full stabilisation region, a coarse 4-dimesional grid of tra-jectories was analysed numerically, in which each point represents the initial conditions for[ φ , T i , C r , C i ]. Three such grids were run, for Ω b = 10 − , − , .
9. The results are shownin Fig. 13 for the 3-D volume φ - C r - C i . We do not show the T i direction, since all valueslead to identical surface plots (this is due to the symmetry seen in Fig. 2).It is convenient to define a measure of goodness of stability. Firstly, to determinewhether the background fluid adds stabilisation, we define an overall normalised, 4-D vol-ume, V , which is given by the volume of stabilised trajectories as a percentage of the totalvolume. In addition, we may calculate the ratio of stabilised trajectories to the total tra-jectories on a given 2-D plane (again, as a percentage), A ab , where a and b denote the fixedvalues of the remaining fields. (For example A C i =10 ,T i =0 represents the stabilisation areaof the φ - C r plane when C i = 10 and T i = 0).The numerical results for the measure of goodness are given in Tables 1 and 2. Itshould be well noted that the numbers quoted are intended to show a trend and are onlyindicative. Numerically, we rely on a finite volume, with finite (coarse) spacing. We chosea grid size that was numerically feasible within timescales. The may preclude us fromresolving all the fine structure. We do note, however, that the fine structure arises fromthe asymmetry of the potential and the fine-tuning of the trajectories. This will be morerelevant when we consider a radiation background fluid.Our results indicate that a background matter fluid aids stabilisation, consistent withprevious work [4, 25, 26]. In addition, the presence of matter fields further aids stabilisation.Importantly, while a non-zero value of C r assists the stabilisation of φ , the imaginary part C i further increases the stabilsation regions.
7. Radiation ( γ = ) For completeness, we briefly consider the effects of a radiation background fluid, in place– 20 – igure 13:
Matter (left column) and Radiation (right column), from top to bottom: Ω b =10 − , − , .
9. A measure of stabilisation is the percentage of the total area which leads tostabilisation. Values are given in Table 1. of a matter fluid. All the equations in the preceding sections are valid, where γ = 4 / γ = 1.– 21 –atter RadiationΩ b V (%) V (%)10 −
42 1310 −
48 150 . Table 2:
Table showing the stabilisation measure, V , (percentage of the total 4-D volume whichleads to stabilisation). 2-D cross-sections are shown graphically in Figure 13. These numbers areindicative of trends only, as explained in the text.
2. Increasing Ω b again aids stabilisation.3. Due to the reduced cosmological friction acting on the fields, more fine-tuning isnecessary for trajectories to find the local minimum.4. We observe the same trend of stabilisation when C r and C i are non-zero as for thematter background fluid, although the effects are less pronounced.
8. Conclusion
In this paper, we have considered moduli stabilisation in the KKLT scenario coupled to aPolonyi matter field, a model which has been previously studied in the context of particlephenomenology. In this scenario, the AdS vacuum is uplifted by F-terms in supergravity.We have examined the evolution of two complex fields (one modulus and one matter field),taking also a background fluid into account, either matter or radiation. We find that thepresence of both the background fluid and the matter field enlarge the region leading tostabilisation of the moduli fields, due to a new scaling regime. Although a more detailedtreatment is necessary, we believe that our conclusions will hold for similar scenarios,where the new direction is steep. A matter background fluid aids stabilisation more thanradiation, consistent with previous studies.
Acknowledgments
The authors acknowlege the use of the package “SuperCosmology” to calculate the scalarpotential [27] and would like to thank STFC for financial support.
References [1] M. Dine and N. Seiberg, “Is The Superstring Weakly Coupled?,” Phys. Lett. B (1985)299.[2] R. Brustein and P. J. Steinhardt, “Challenges for superstring cosmology,”, Phys. Lett. B (1993) 196 [arXiv:hep-th/9212049].[3] N. Kaloper and K. A. Olive, “Dilatons in string cosmology,” Astropart. Phys. (1993) 185. – 22 –
4] T. Barreiro, B. de Carlos and E. J. Copeland, “Stabilizing the dilaton in superstringcosmology,” Phys. Rev. D (1998) 083513 hep-th/9805005.[5] T. Barreiro, B. de Carlos and N. J. Nunes, “Moduli evolution in heterotic scenarios,” Phys.Lett. B (2001) 136 [arXiv:hep-ph/0010102].[6] G. Huey, P. J. Steinhardt, B. A. Ovrut and D. Waldram, “A cosmological mechanism forstabilizing moduli,” Phys. Lett. B , 379 (2000) [arXiv:hep-th/0001112].[7] S. B. Giddings, S. Kachru and J. Polchinski, “Hierarchies from fluxes in stringcompactifications,” Phys. Rev. D (2002) 106006 [arXiv:hep-th/0105097].[8] M. R. Douglas and S. Kachru, “Flux compactification,” Rev. Mod. Phys. (2007) 733[arXiv:hep-th/0610102].[9] S. Kachru, R. Kallosh, A. Linde and S. P. Trivedi, “De Sitter vacua in string theory,” Phys.Rev. D (2003) 046005 hep-th/0301240.[10] C. P. Burgess, R. Kallosh and F. Quevedo, “de Sitter string vacua from supersymmetricD-terms,” JHEP (2003) 056 [arXiv:hep-th/0309187].[11] K. Choi, A. Falkowski, H. P. Nilles and M. Olechowski, “Soft supersymmetry breaking inKKLT flux compactification,” Nucl. Phys. B (2005) 113, [arXiv:hep-th/0503216].[12] A. Achucarro, B. de Carlos, J. A. Casas and L. Doplicher, “de Sitter vacua from upliftingD-terms in effective supergravities from realistic strings,” JHEP , 014 (2006)[arXiv:hep-th/0601190].[13] A. Saltman and E. Silverstein, “The scaling of the no-scale potential and de Sitter modelbuilding,” JHEP (2004) 066 [arXiv:hep-th/0402135].[14] O. Lebedev, H. P. Nilles and M. Ratz, “de Sitter vacua from matter superpotentials,” Phys.Lett. B (2006) 126 hep-th/0603047.[15] M. Gomez-Reino and C. A. Scrucca, “Locally stable non-supersymmetric Minkowski vacua insupergravity,” JHEP (2006) 015 [arXiv:hep-th/0602246].[16] F. Brummer, A. Hebecker and M. Trapletti, “SUSY breaking mediation by throat fields,”Nucl. Phys. B (2006) 186 [arXiv:hep-th/0605232].[17] R. Kallosh and A. Linde, “O’KKLT,” JHEP (2007) 002 [arXiv:hep-th/0611183].[18] E. Dudas, C. Papineau and S. Pokorski, “Moduli stabilization and uplifting with dynamicallygenerated F-terms,” JHEP (2007) 028 [arXiv:hep-th/0610297].[19] P. Brax, A. C. Davis, S. C. Davis, R. Jeannerot and M. Postma, “Warping and F-termuplifting,” arXiv:0707.4583 [hep-th].[20] H. Abe, T. Higaki and T. Kobayashi, “More about F-term uplifting,” arXiv:0707.2671[hep-th].[21] O. Lebedev, V. Lowen, Y. Mambrini, H. P. Nilles and M. Ratz, “Metastable vacua in fluxcompactifications and their phenomenology,” JHEP (2007) 063 hep-ph/0612035.[22] J. Polonyi, “Generalization Of The Massive Scalar Multiplet Coupling To The Supergravity,”Budapest preprint KFK-1977-93 (1977).[23] H. Abe, T. Higaki, T. Kobayashi and Y. Omura, “Moduli stabilization, F-term uplifting andsoft supersymmetry breaking terms,” Phys. Rev. D (2007) 025019 arXiv:hep-th/0611024. – 23 –
24] S. C. C. Ng, N. J. Nunes and F. Rosati, “Applications of scalar attractor solutions tocosmology,” Phys. Rev. D (2001) 083510 astro-ph/0107321.[25] R. Brustein, S. P. de Alwis and P. Martens, “Cosmological stabilization of moduli with steeppotentials,” Phys. Rev. D (2004) 126012. hep-th/0408160[26] T. Barreiro, B. de Carlos, E. Copeland and N. J. Nunes, “Moduli evolution in the presence offlux compactifications,” Phys. Rev. D (2005) 106004 hep-ph/0506045.[27] R. Kallosh and S. Prokushkin, “Supercosmology,” arXiv:hep-th/0403060. Appendix
A. Useful Formulae
From the definition of λ , δ , η and θ , one obtains λ ′ λ = √ xλ (cid:2) − Γ φφ (cid:3) + √ zη (cid:2) − Γ φT i (cid:3) + √ pδ (cid:2) − Γ φC r (cid:3) + √ qθ (cid:2) − Γ φC i (cid:3) ,δ ′ δ = √ xλ (cid:2) − Γ φ C r (cid:3) + √ zη (cid:2) − Γ C r T i (cid:3) + √ pδ (cid:2) − Γ C r C r (cid:3) + √ qθ (cid:2) − Γ C r C i (cid:3) ,η ′ η = 2 x + √ xλ (cid:2) − Γ φT i (cid:3) + √ zη (cid:2) − Γ T i T i (cid:3) + √ pδ (cid:2) − Γ T i C r (cid:3) + √ qθ (cid:2) − Γ T i C i (cid:3) ,θ ′ θ = √ xλ (cid:2) − Γ φC i (cid:3) + √ zη (cid:2) − Γ C i T i (cid:3) + √ pδ (cid:2) − Γ C r C i (cid:3) + √ qθ (cid:2) − Γ C i C i (cid:3) , where we define Γ ij ≡ V V ij V i V j . We take a change of variables ǫ φ ≡ λ , ǫ T i ≡ η , ǫ C r ≡ δ , ǫ C i ≡ θ , and we define X , Y , Z , P and Q to be x = ǫ φ X, y = ǫ φ Y, z = ǫ T i Z, p = ǫ C r P, q = ǫ C i Q. With these new variables we obtain: H ′ H = − (cid:2) ǫ φ X + 2 ǫ T i Z + 2 ǫ C r P + 2 ǫ C i Q + γ (cid:0) − ǫ φ X − ǫ φ Y − ǫ T i Z − ǫ C r P − ǫ C i Q (cid:1) (cid:3) – 24 –nd X ′ = λ ′ λ X − X + r Y − ǫ T i ǫ φ Z − H ′ H XY ′ = λ ′ λ Y − r XY − r ZY − √ P Y − √ QY − H ′ H YZ ′ = η ′ η Z − Z + r (cid:18) ǫ φ ǫ T i (cid:19) Y + 2 ǫ φ XZ − H ′ H ZP ′ = δ ′ δ P − P + √ (cid:18) ǫ φ ǫ C r (cid:19) Y − H ′ H PQ ′ = θ ′ θ Q − Q + √ (cid:18) ǫ φ ǫ C i (cid:19) Y − H ′ H Q
Finally, from above, we find ǫ ′ φ ǫ φ = − n √ X (cid:2) − Γ φφ (cid:3) + √ Z (cid:2) − Γ φT i (cid:3) + √ P (cid:2) − Γ φC r (cid:3) + √ Q (cid:2) − Γ φC i (cid:3)o (A.1) ǫ ′ C r ǫ C r = − n √ X (cid:2) − Γ φC r (cid:3) + √ Z (cid:2) − Γ C r T i (cid:3) + √ P (cid:2) − Γ C r C r (cid:3) + √ Q (cid:2) − Γ C r C i (cid:3)o (A.2) ǫ ′ T i ǫ T i = − n ǫ φ X + √ X (cid:2) − Γ φT i (cid:3) + √ Z (cid:2) − Γ T i T i (cid:3) + √ P (cid:2) − Γ T i C r (cid:3) + √ Q (cid:2) − Γ T i C i (cid:3)o (A.3) ǫ ′ C i = − ǫ C i n √ X (cid:2) − Γ φC i (cid:3) + √ Z (cid:2) − Γ C i T i (cid:3) + √ P (cid:2) − Γ C r C i (cid:3) + √ Q (cid:2) − Γ C i C i (cid:3)o (A.4) B. Tracking Solution
In the following, we generalise the scaling regime found in [24] for two real fields. Wetherefore consider T i = C i = 0 in the following. The simplified equations of motionconsidered (with Q = Z = 0) are X ′ = −√ φφ − X −√ φC r − XP − X + r Y (B.1)+ 32 X (cid:2) ǫ φ X + ǫ C r P ) + γ (1 − ǫ φ X − ǫ φ Y − ǫ C r P ) (cid:3) ,Y ′ = −√ φφ − XY −√ φC r − Y P − r XY − √ P Y (B.2)+ 32 Y (cid:2) ǫ φ X + ǫ C r P ) + γ (1 − ǫ φ X − ǫ φ Y − ǫ C r P ) (cid:3) ,P ′ = −√ φC r − XP −√ C r C r − P − P + √ ǫ φ ǫ C r Y (B.3)+ 32 P (cid:2) ǫ φ X + ǫ C r P ) + γ (1 − ǫ φ X − ǫ φ Y − ǫ C r P ) (cid:3) . – 25 –mall ǫ φ and ǫ C r or Γ ij ≈ ǫ i is almost constant, as seen in Equations (A.1) and(A.2). Therefore the “instant critical point” is obtained from solving X ′ = Y ′ = P ′ = 0.From Eqns. (B.1)-(B.3), we obtain Y = − X + √ X − √ XP, (B.4) Y ( P − ǫ φ √ ǫ Cr X ) = 2(Γ φφ − Γ φC r ) X P + √ φC r − Γ C r C r ) XP . (B.5)It is reasonable to assume (in the regions we consider in the paper)Γ φφ ≃ Γ C r C r ≃ Γ φC r ≃ , from which we find P ≃ ǫ φ √ ǫ C r X. Plugging this solution into Eqn. (B.4), both P = P ( X ) and Y = Y ( X ). Solving Eqn. (B.1)(using X ′ = 0) leads to x c = r
32 ˜ γλ , p c = √ δ ˜ γλ , y c = 32 ˜ γλ (cid:18) − ˜ γ − δ λ ˜ γ (cid:19) , (B.6)where ˜ γ = γ (cid:18) δ λ (cid:19) − . (B.7) C. Stability of the critical points
We expand about the critical points X = X c + u, Y = Y c + v, P = P c + w, which yield, to first order, the equations of motion u ′ v ′ w ′ = M uvw . Assuming Γ’s are 1 (this assumption is quite good for the region of approximately φ < − | C r | > m i , of M . We also assume 0 ≤ γ ≤ Fluid-dominated solution
For this solution, x c = p c = y c = 0. We find a saddle point for 0 < γ < m = 32 γ, m = m = −
32 (2 − γ ) . – 26 – inetic-dominated solutions Here, x c + p c = 1 and y c = 0. We find an unstable node for λx c + ( δ/ √ p c < √ λx c + ( δ/ √ p c > √ m = 0 , m = 3(2 − γ ) , m = r (cid:18) √ − λx c − δp c √ (cid:19) . Scalar field dominated solution
In this case, x c = λ √ , p c = δ √ , y c = 1 − (cid:0) λ + δ (cid:1) . There is a stable node for λ + δ / < γ and a saddle point for 3 γ < λ + δ / < m , = − δ + 2 λ ) , m = − γ + 12 ( δ + 2 λ ) . Scaling solution
For the scaling solution, x c , y c and p c are given in Eqns. (B.6) and (B.7). We find a stablenode for 3 γ < λ + δ / < γ / (9 γ −
2) and a stable spiral for λ > γ / (9 γ − m = −
32 (2 − γ ) ,m , = −
34 (2 − γ ) h ± s − γ ( λ + δ / − γ ))( λ + δ / − γ ) i ..