弦宇宙學的研究

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(1)國立台灣師範大學物理研究所碩士論文. 弦宇宙學的研究 Topics on String Cosmology. 學生：王建勛指導教授：林豐利博士. 中華民國九十八年十一月.

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(3) 致謝能夠完成碩士學歷，要感謝林豐利老師給予我指導，老師教我許多做研究的態度和方法，並且給予我的研究題目許多寶貴的建議和想法。積極的研究精神和自信心的培養是我在研究過程中最大的收穫。. 另外要感謝父母親給予我精神和物質上的支持，讓我有自由能做自己想做的事。尤其是母親在我修業中最艱難的時刻，給我鼓勵和建議，讓我能順利克服難關。. 在此要感謝研究室的同學在研究所兩年的時光互相打氣和扶持。謝謝曹國興、李欣翰、潘鳳鳴、劉智豪、錢宜新、張維中、昌文宗、林宜樺以及趙博文學長。他們的陪伴使我的研究生生涯更多采多姿. 最後，我必需感謝我鍾愛的師父 Osho，是他鼓勵我發覺自己的興趣，使我有勇氣轉讀物理。我感覺到他一直陪伴著我。願我能成為我自己。.

(4) To my parents.

(5) Preprint typeset in JHEP style - HYPER VERSION. Topics on String Cosmology. Chien-Hsun Wang [email protected] Department of Physics, National Taiwan Normal University, Taipei, 116, Taiwan, R.O.C.. Abstract: This thesis consists two parts. In part I we study the dynamics of vacuum bubble and junction condition with higher order correction and show that it is difficult to obtain the equation of motion in the fourth order theory of gravity. Then we study the five dimensional true vacuum bubble in Einstein-Gauss-Bonnet’s theory of gravity. We find a new trapped solution. In part II we review brane inflation model and D3 brane DBI inflation models. We propose a D4 DBI inflation model. We study its attractor behavior and calculate the spectral indices and tensor-to-scalar ratio..

(6) Contents 1. Part I Introduction. 3. 2. Bubble dynamics with Einstein’s theory of gravity. 6. 3. The junction condition with higher order corrections. 12. 4. 5D true vacuum bubble dynamics. 15. 4.1. 5D true vacuum bubble dynamics in Einstein’s theory of gravity. 4.2. 5D true vacuum bubble dynamics in Einstein-Gauss-Bonnet’s theory. 4.3. 15. of gravity. 17. Hamiltonian formalism. 21. 5. Summary. 21. 6. Part II Introduction. 24. 7. Brane Inflation. 26. 7.1. Brane Inflation in Warped Geometry. 8. D3 brane DBI inflation model 8.1 8.2. 27 28. DBI inflation in AdS5. 29. Attractor behavior of the DBI inflation model. 9. D4 DBI Inflation model. 31 33. 9.1. DBI Inflation in the Witten’s geometry. 33. 9.2. Attractor behavior of the D4 DBI inflation model. 36. 9.3. Phase diagram of attractor behavior. 37. 10. Power spectrum. 38. 11. Summary. 40. 12. Appendix A Israel junction condition. 41. –1–.

(7) 13. Appendix B derivation of higher derivative terms in junction condition. 42. –2–.

(8) 1. Part I Introduction There are two kinds of vacuum bubbles, false vacuum bubbles and true vacuum bubbles. The false vacuum bubble may nucleate in the early universe and one might ask whether we can create a universe in the laboratory out of a false vacuum bubble [1]. On the other hand, true vacuum bubbles are discussed in the case of false vacuum decay and tunneling in string landscape [2][3]. There are two motivations to study the dynamics of false vacuum bubbles. First, many true vacuum bubbles may nucleate simultaneously around a sphere and leave a false vacuum bubble trapped in a true vacuum. It is interesting to know the fate of the false vacuum bubble. Second, if our universe originated from an inflating false vacuum, given sufficient technology can we create a universe in a laboratory by forming a small region of high vacuum energy? Although the analysis of the dynamics of general false vacuum bubbles is quite intricate, we can assume spherical symmetry to simplify the problem and keep the essential feature. The classical solutions show that a false vacuum bubble can expand to infinity surrounded by black horizon and causally disconnected from the original universe. The result shows that a false vacuum bubble can subsequently produce many child universes. Applying the Penrose theorem, Farhi and Guth show that it is impossible to create such a false vacuum without an initial singularity [4]. The Penrose theorem states that in a connected spacetime, if a) there exists a noncompact Cauchy surface.b)Rµν k µ k ν ≥ 0 for all null k µ , and c) there exists an anti-trapped surface then there exists an initial singularity. If the laboratory exists in a space which is asymptotically flat and has a well-defined Cauchy development, then condition a) is satisfied. As a consequence of the Einstein equations Rµν = 8πGTµν , condition b)is Tµν k µ k ν ≥ 0. Because all standard matter obeys energy condition, we also assume that the machine would create a child universe also obeys the energy condition. Then condition b) is satisfied. Since the existence of an anti-trapped surface means the existence of the cosmological horizon, the inflationary universe created in a laboratory should also satisfy condition c). Then, a false vacuum bubble large enough to be an inflationary universe could not avoid initial singularity. To avoid this obstacle, many groups [4] [5] [6] [7] develop different ways to evade the theorem or consider quantum decay from a small bubble without an initial singularity to a large bubble which becomes an inflationary universe. One proposal we consider here is to modify Einstein’s theory of gravity with stringy correction. Our result shows that it is necessary to consider. –3–.

(9) a finite thickness of the domain wall, the surface of the bubble, instead of the thin wall approximation in Einstein’s theory gravity discussed above. The inflationary universe scenario proposes that our universe in the early stage expanded exponentially [8]. It indicates that the observed universe originates from a false vacuum which has a positive cosmological constant and can be regard as de Sitter space. Due to the instability of de Sitter space, a false vacuum may decay into a vacuum which has a smaller or no cosmological constant. Many vacuum bubbles would nucleate inside the false vacuum and expand rapidly. Because the false vacuum has a larger cosmological constant and expands faster than the vacuum bubbles, those vacuum bubbles will be separated from each other and causally become disconnected [1]. Although some bubbles may nucleate close enough and collide. String theory requires that the spacetime dimension is ten so the extra six dimension should be compactified. After compactification many massless scalar fields called moduli appear. Using fluxes and other ingredients in string theory one can stabilize all the moduli [9]. Different configuration of moduli space correspond to the different four dimensional vacua and laws of physics [10]. The possible vacua are viewed as the local minima of a complicated potential function called the string landscape [2]. There are many peaks and valleys in the string landscape. These local minima are separated by the barriers. If the potential of the minimum is positive, the valley is a de Sitter vacuum. The de Sitter vacuum is a false vacuum which will tunnel to another vacuum [11] [12] [13]. The possibility of tunneling to larger cosmological constant is rather small so we can concentrate on the case of tunneling to a smaller cosmological constant vacuum. When tunneling event happens, a vacuum bubbles nucleates. The bubble with smaller vacuum energy nucleates inside the de Sitter space. There is a de Sitter space with a smaller cosmological constant inside the bubble. The new universe is born from a neighboring valley of the landscape by tunneling. The bubble with smaller vacuum energy expands but the de Sitter region outside the bubble inflates fast enough that it is not consumed. Therefore the global universe is an eternally inflating region that is continually producing bubble universes. Our universe went through a series of tunneling events and new bubbles nucleate within the bubbles. The last tunneling led to the universe which settled into a valley of the landscape. For simplicity we assume that there are only two vacua. One has a positive cosmological constant Λ and the other has approximately vanishing cosmological constant.. –4–.

(10) When the false vacuum tunnels to a vacuum with approximately no cosmological constant which is the case of our universe, a true vacuum bubble forms. The dynamics of the wall of the bubble is determined by the junction condition [14] and the analysis states that if the size of the true vacuum bubble is large enough at the time when it nucleates, it would expands eternally. However this analysis is based on Einstein’s theory of gravity only, we wonder if by adding stringy correction whether the dynamics of the bubble would change. The string quantum correction would give the Einstein-Hilbert and some higher order terms. The next leading order terms are proportional to the R2 , Rµν Rµν and Rµνρσ Rµνρσ . However, arbitrary combinations of these terms would give the fourth order equation of motion which is too complicated to solve. A special combination is the Gauss-Bonnet term [15] and the equation of motion would remain second order. However, Gauss-Bonnet is a topological invariant in four dimension and it would not contribute to the equation of motion, but the GaussBonnet term becomes dynamical above four dimension. Therefore, we would consider a five dimensional landscape toy model and see the effect of the stringy correction. The topology of the bubble would be a three sphere. We will consider the five dimensional bubble universe in the Einstein’s theory of gravity. Then we will add the Gauss-Bonnet term and analyze the dynamics of the true vacuum bubble. In section 2 we review the bubble dynamics in the Einstein’s theory of gravity and show that the false vacuum bubble can produce a child universe, and also review the true vacuum bubble dynamics in string landscape. In section 3 we will try to formulate the equation of motion for the domain wall of the bubble and discuss the conflict between thin wall approximation and higher derivative gravity theory. In section 4 we will study the 5D bubble dynamics in Einstein’s theory of gravity and Einstein-Gauss-Bonnet’s theory of gravity respectively. In section 5 we will discuss a possible resolution for higher derivative correction and the result of the dynamics of the five dimensional true vacuum bubbles. In the appendix A we will review the derivation of the Israel junction condition by using Gaussian normal coordinate formalism because it is the easiest way to see the discontinuity of the metric. In the appendix B we calculate the higher order terms of the junction condition in R2 gravity and collect the more singular terms that are used in section 3.. –5–.

(11) 2. Bubble dynamics with Einstein’s theory of gravity We will briefly review the dynamics of a false vacuum bubble surrounded by a true vacuum [1], and idealize the system by assuming a spherical region with nonzero energy density inside the wall surrounded by an infinite region of true vacuum with a thin domain wall separating the two regions. It raises a paradox; if the false vacuum region is sufficiently large, then an observer deep within it would expect to see inflation. However, an observer watching the domain wall would note that the false vacuum region has negative pressure and is surrounded by the zero pressure true vacuum, so the pressure force is inward. The spherical symmetry implies that the metric region has the Schwarzschild form, so the gravitational field is not expected to oppose the force of the pressure gradient. Thus, the second observer would not expect to see inflation. The resolution of this paradox depends on the highly non-Euclidean geometry, allowing the false vacuum region to inflate without moving outward into the true vacuum region. The observer out-side of the domain cannot see the domain wall moving outward because the false vacuum bubble which inflates is hidden by the event horizon. The setup involves a spherical false vacuum region with energy per unit volume ρf separated from a true vacuum region by a thin wall with an energy per unit area σ. The motion of the thin wall is determined by the Isreal junction condition [14] which can be derived from Einstein’s equation. The true vacuum region solution is the Schwarzschild geometry. The Schwarzschild metric coordinate is given as 2GM 2GM −1 2 ds2 = −(1 − )dTS2 + (1 − ) dR + R2 dΩ2 R R where dΩ2 = dθ2 + sin2 θdφ2 , and M is the total energy of the bubble. Since these coordinates do not cover all Schwarzschild space and have a singularity at R = 2GM we can introduce Kruskal-Szekeres coordinates (VS , US )which cover the entire manifold and have no singularities. The metric can be written as ds2 = 32. (GM )3 R exp(− )(−dVS2 + dUS2 ) + R2 dΩ2 R 2GM. where R R − 1)exp( ) = (−VS2 + US2 ) 2GM 2GM Lines of constant TS are straight lines through the origin of the (VS , US ) plane, with (. TS = 4GM tanh−1 (VS /US )if |VS /US | < 1 TS = 4GM tanh−1 (US /VS )if |VS /US | > 1. –6–.

(12) Inside the bubble the geometry is de Sitter space. The static metric coordinate is ds2 = −(1 − χ2 R2 )dTD2 + (1 − χ2 R2 )−1 dR2 + R2 dΩ2 These coordinate have a coordinate singularity at R = χ and can be avoided in terms of Gibbons-Hawking coordinates (VD , UD ): ds2 = χ−2 (1 + χR)2 (−dVD2 + dUD2 ) + R2 dΩ2 where 1 − χR = (−VD2 + UD2 ) 1 + χR Lines of constant TD are straight through the origin of the (VS , US ) plane, with TD = χ−1 tanh−1 (VD /UD )if |VD /UD | < 1 TD = χ−1 tanh−1 (UD /VD )if |VD /UD | > 1 The motion of the thin wall separating these two regions is determined by junction condition [1] [14] i [Kji − δji K]+ − = κSj. (2.1). Kij = −1/2∂y hij is extrinsic curvature By taking trace we obtain K = aπGS and substitute back i i [Kji ]+ − = −8πG[Sj − 1/2δj S]. using thin wall approximation we can assume that Sij = −σhij . hij is the induced metric on the thin wall. Then the junction condition is reduced to i [Kji ]+ − = −4πσGδj. Due to spherical symmetry we can assume the metric on the thin wall ds2 = −dτ 2 + r2 (τ )dΩ2 Calculate the component of the extrinsic curvature Kθθ = 1/2∂y hθθ = 1/2∂y r2 = 1/2ξ µ ∂µ r2. –7–. (2.2).

(13) In the Schwarzschild side ξ µ = ((1 −. 2GM )∂τ r, ±(1 r. Kθθ (Schwarzschild) = ±r(1 −. −. 2GM r. + ∂τ r2 )1/2 , 0, 0) so. 2GM + (∂τ r)2 )1/2 r. Similarly, the de Sitter side Kθθ (deSitter) = ±r(1 − χr2 + (∂τ r)2 )1/2 Because the spherical symmetry Kφφ = sin2 θKθθ and Kτ τ is just the proper time derivative of Kθθ we have only one equation of motion βS − βD = −4πGσr 2GM βS = ±(1 − + (∂τ r)2 )1/2 r βD = ±(1 − χr2 + (∂τ r)2 )1/2 βS is positive(negative) if the polar angle tan−1 (VS /US ) is increasing(deceasing)along the trajectory.βD is positive(negative) if the polar angle tan−1 (VD /UD ) is increasing(deceasing)along the trajectory. We can arrange the equation into the form which describes a particle representing the position of the thin wall moving with a potential: (∂τ r)2 + V (r, M ) = −1 V (r, M ) = −(2GM )2/3 (χ2 + κ2 )1/3 [. (2.3). (1 − z 3 )2 1 + ] γ 2z4 z. κ = 4πGσ z3 =. χ2 + κ2 3 r 2GM. γ2 =. 4κ2 χ2 + κ2. There is critical mass defined by V (zm , Mcr ) = −1. For M < Mcr there are bounded solutions and bounce solutions. For M > Mcr there are monotonic solutions which are the most interesting. The potential is plotted in fig 1. The monotonic solutions in the Penrose diagram is shown in fig 2 and the shaded region is de Sitter space. To the right of the bubble wall (a heavy line with an arrow on it)the diagram represents. –8–.

(14) Figure 1: .. Figure 2: .. a region of Schwarzschild space and the false vacuum region is to the left, where the metric describes part of the de Sitter space. The evolution of the false vacuum bubble can be illustrated by the diagram in fig 3. The diagrams are drawn by suppressing one dimension of the hypersurface and embedding the resulting two dimensional surface into a three dimensional space so that the curvature can be displayed. The false vacuum region expands into a bulge which soon disconnects completely from the original space, forming an isolated child universe. A black hole singularity is left in the original spacetime, but it would soon evaporate. The discussion shows that if inflation occurred in an inhomogeneous universe where an inflating region. –9–.

(15) Figure 3: .. surrounded by a non-inflating region, many isolated child universes would have been ejected. True vacuum bubbles nucleate when false vacuum decay to a vacuum with vanishing cosmological constant Λ = 0 [2]. The thin wall of the bubble separates the two vacuum regions [3]. The metric inside the thin wall is Minkowski space and outside the thin wall is de Sitter space. We will study the dynamics of the bubble in Einstein’s gravity. If we use static coordinates on both sides, then the metric on each side has the form ds2 = −f (r)dt2 +. 1 dr2 + r2 dΩ2 f (r). with f (r) = 1 in the flat part and f (r) = 1 − r2 /R2 in the de Sitter region, where R is the de Sitter radius. We can add a mass M at the center of the flat region, r = 0, so that the metric inside the wall is Schwarzschild and the metric outside remains pure de Sitter space or adds a mass M at the center of the de Sitter region, so that the metric is Schwarzschild-de Sitter. The motion of the thin wall can be described as r(τ ) where τ denotes proper time along the domain wall trajectory. First we discuss adding a mass in the de Sitter region. for r < r(τ ) the metric is ds2 = −fin dt2 +. 1 2 dr + r2 dΩ2 fin. with fin = 1. – 10 –.

(16) The metric for r > r(τ ) is ds2 = −fout dt2 +. 1 fout. dr2 + r2 dΩ2. with fout = (1 −. r2 2GM − ) 2 R r. The motion of the thin wall position r(τ ), is determined by the Israel junction condition. p. fin (r) + r˙ 2 −. p. fout (r) + r˙ 2 = 4πGσr. where fin (r) is the metric the function inside the bubble, r˙ is the derivative of the radial coordinate with respect to proper time, and σ is the tension of the domain wall. The junction condition can be rearranged to look like an energy conservation equation for the domain wall motion, √ 4πσr2 1 + r˙ 2 − (. 1 + 8π 2 Gσ 2 )r3 = M 2GR2. Each term has a physical interpretation. The square root term is the usual kinetic term for the domain wall, and −8π 2 Gσ 2 r3 is the gravitational self-energy of a spherical wall. The bubble of flat space replaces a region of positive vacuum energy 1 3 with a region of zero vacuum energy, resulting in a change − 2GR in the energy. 2r. We square the above equation and the equation becomes r˙ 2 + V (r) = −1. (2.4). which has the form an energy conservation equation for a non-relativistic particle with potential. The energy is given as M and the potential is V (r) = −(. M +B 2 ) Ar2. 1 where A = 4πσ and B = ( 2π2 Gσ2 +8π The potential is shown in figure 2 Gσ 2 ).. 4. The classical evolution is that the bubble contracts from infinity, bouncing off the barrier and expands eternally. For M = 0 there are two solutions. One is the original bounce solution. The other is a static solution located at the minimum of potential at r = 0. It represents bubble of zero radius and is just a de Sitter universe. Quantum mechanically, the static solution is unstable and it can tunnel out through the barrier to the bounce solution. It signifies the instability of de Sitter and nucleation of bubbles from decay.. – 11 –.

(17) Figure 4: .. 3. The junction condition with higher order corrections In the previous section we showed that a false vacuum bubble can nucleate in the true vacuum region, but with an initial singularity. We want to ask what kind of physical process can generate this solution and how to avoid the initial singularity. The Penrose theorem, which assume a weak energy condition Tµν k µ k ν ≥ 0 states that there is no way to avoid the initial singularity in Einstein’s classical gravity because it indicates that the outward velocity required for monotonic solutions is so large that it can only emerge from an initial singularity. However, if we include higher order corrections, we can invalidate the theorem and the initial singularity might be avoided. We also discuss that once the true vacuum bubble nucleates, it would expand eternally. Is it also true in the presence of higher order corrections? The dynamics of the bubble is determined by the junction condition, so we need to derive the junction condition modified by higher order corrections. A Lagrangian including higher order corrections derived from high energy theory like string theory may have the form L = R − 2Λ + αR2 + βRµν Rµν + γRαβγδ Rαβγδ. (3.1). A special case for second order gravity is α = γ = 1, β = −4 that is Einstein-GaussBonnet gravity. However in four dimensions these terms are topological invariant which does not contribute to the equation of motion. We need to consider other examples to meet real world situations. For simplicity, we consider L(2) = R − 2Λ +. – 12 –.

(18) αR2 first and show that thin wall approximation is not reliable. The field equation √ from variation of gL is [15] 1 σµν = (1 + 2αR)Gµν + αR2 gµν − 2αDµ Dν R + 2αgµν ¤R2 + Λgµν 2 1 σij = Λhij + Gij + 2αRGij + α R2 hij + 2α[−Di Dj R + Kij ∂y R + hij (∂yy R + ¤R − K∂y R)] 2 In order to extract the discontinuity from equation of motion, we assume the metric is continuous at the wall, but it has a kink. Its first derivative has a step function discontinuity and its second derivative has a delta function term [16] + hij (y) = h− ij θ(−y) + hij θ(y). ∂h− ∂h+ ∂hij (y) ij (y) ij (y) = θ(−y) + θ(y) ∂y ∂y ∂y ∂ 2 h− ∂ 2 h+ ∂h− ∂h+ ∂ 2 hij (y) ij (y) ij (y) ij (y) ij (y) = θ(−y) + θ(y) + (− + )δ(y) 2 2 2 ∂ y ∂ y ∂ y ∂y ∂y. (3.2). In Einstein’s gravity, substitute (3.2) into Gij = κ2 Tij and match the delta function term, so that we can get the Israel junction condition. In order to get the junction condition for the R2 correction gravity, we need to work out following quantity ³ ´ 1 ¯ ij RGij = 2∂y K∂y (Kij − Khij ) + 2∂y K 2Kil Klj + ∂y Klj − KKij + hij (T rK 2 + K 2 ) + G 2 2 2 ¯ + ∂y (Kij − Khij )(−T rK − K + R) (3.3) ¯2 R2 = [2∂y K − T rK 2 − K 2 + R]. (3.4). ¯ ∂y RKij = [2∂yy K − ∂y T rK 2 − ∂y K 2 + ∂y (R)]K ij. (3.5). ¯ ∂yy R = 2∂yyy K − ∂yy T rK 2 − ∂yy K 2 + ∂yy (R). (3.6). ¯ K∂y R = K(2∂yy K − ∂y T rK 2 − ∂y K 2 + ∂y (R). (3.7). We consider that induced metric hij is continuous across the domain wall, but discontinuous across the wall. However terms like R2 would give δ( y)2 and ∂y R would. – 13 –.

(19) give ∂y δ(y) and ∂yy R would give ∂yy δ(y). Those more singular terms are not well defined. One may require higher continuity of the induced metric hij , but this would make the equation of the domain wall trivial. We cannot assume this condition in our case. We propose to include more singular source terms on the wall to see if the equation of motion can be solved consistently. The derivation of the left hand side in the equation of motion is left in the appendix. Introducing source terms T µν = S µν (xi )δ(y) + U µν (xi )δ(y)2 + V µν (xi )∂y δ(y) + W µν (xi )∂yy δ(y) Due to the conservation law∇ν T µν = 0 ∇ν T iν = (∇j S ij + 2Kji S jy + T rKS iy )δ(y) + S iy ∂y δ(y) +(∇j U ij + 2Kji U jy + T rKU iy )δ(y)2 + U iy 2δ(y)∂y δ(y) +(∇j V ij + 2Kji V jy + T rKV iy )∂y δ(y) + V iy ∂yy δ(y) +(∇j W ij + 2Kji W jy + T rKW iy )∂yy δ(y) + W iy ∂yyy δ(y) =0 We should have the following constraints. ∇j S ij + 2Kji S jy + T rKS iy = 0 ∇j U ij + 2Kji U jy + T rKU iy = 0 S iy + ∇j V ij + 2Kji V jy + T rKV iy = 0 U iy = 0 ∇j W ij + 2Kji W jy + T rKW iy + V iy = 0 W iy = 0 Matching the left hand side and right hand side of equation for the first and second derivatives of the delta function and the square of the delta function respectively. It gives four equations, where one being proportional to δ(y) is rather cumbersome, so we neglect it, but it will not affect our result − i ∂yy δ(y) −→ −2αhij+ (∂h+ ij − ∂hij )hij = Wij (x ). α +kl α + − − − − +mn (∂y h+ [h (∂y h+ δ(y)2 −→ h+kl (∂y h+ mn − ∂y hmn )]hij kl − ∂y hkl )(∂y hij − ∂y hij ) + kl − ∂y hkl )h 2 2 α α +ij + − − − 2 +mn − + [hkl+ (∂y h+ (∂y h+ [h (∂y h+ mn − ∂y hmn )(∂y hij + ∂y hij )] + ij − ∂y hij )] hij = Uij kl − ∂y hkl )h 4 2. – 14 –.

(20) α +kl + + − +kl +ij +kl [h ∂y h+ ∂y h− (h+kl ∂y h+ ∂y h− ij ∂y hkl + h ij ∂y + h ij ∂y hkl + h ij ∂y + hkl )hij 2 − +ij +ij 2 + +ij − 2(∂y h+ij ∂y h+ ∂y h− ∂y hij + h+ij ∂y h− (∂y h+ ij + ∂y h ij + h ij + ∂y h ij − ∂y hij ). ∂y δ(y) −→. − +ij +ij 2 + −ij + h+ij (∂y h+ ∂y h+ ∂y hij − ∂y h−ij ∂y h− ∂y h− ij − ∂y hij ) + ∂y h ij + h ij − h ij )hij + − − + − − +ij +kl + (h+ik h+jl (∂y h+ h (∂y h+ ij ∂y hkl − ∂y hij ∂y hkl ) + h ij ∂y hkl − ∂y hij ∂y hkl ))hij ] = Vij. We can see that these three equations are very different and cannot be solved simultaneously. It shows the breakdown of the thin wall approximation. Instead, we should consider finite thickness of the domain wall to avoid severe singular behavior.. 4. 5D true vacuum bubble dynamics In section 3, we show that the gravity theory with generic higher order terms is too difficult to solve. We would focus the gravity theory whose equation of motion remains second order. The known theory which remains second order is the EinsteinGauss-Bonnet’s theory of gravity [15]. However, Gauss-Bonnet term is a topological invariant in four dimension and it would not contribute to the equation of motion, but the Gauss-Bonnet term becomes dynamical above four dimension. Therefore, we would consider a five dimensional landscape toy model and see the effect of the stringy correction. The topology of the bubble would be a three sphere. We will consider the five dimensional bubble universe in the Einstein’s theory of gravity. Then we will add the Gauss-Bonnet term and analyze the dynamics of the true vacuum bubble by using phase diagram analysis. 4.1 5D true vacuum bubble dynamics in Einstein’s theory of gravity The Einstein Hilbert action with cosmological constant and bubble is given as Z √ 1 S = 2 d5 x −g[R − 2Λ] + Sbubble (4.1) 2κ Varying the action, we can obtain the Einstein equation Gab + Λgab = κ2 Tab. (4.2). We can also obtain the Israel junction condition [14] from the Einstein equation 2 [Kab − Khab ]+ − = −κ Sab. The induced metric on the bubble wall which is a three sphere ds2 = −dτ 2 + r2 (τ )dΩ23. – 15 –. (4.3).

(21) The bulk metric inside the bubble is ds2 = −fin (r)dt2 + fin (r)−1 dr2 + r2 dΩ23 Minkowski fin (r) = 1 The bulk metric outside the bubble ds2 = −fout (r)dt2 + fout (r)−1 dr2 + r2 dΩ23 Schwarzschild-de Sitter fout (r) = 1 −. Λ 2 µ r − 2 6 r. matter source of the bubble wall Sab = −σhab so τ τ component is the tension of the bubble wall Sτ τ = σ We can choose an embedding such that t = τ , then we substitute the above ingredients into the Israel junction condition and we will get the equation of motion for the bubble wall. Note that there is only one degree of freedom r(t) due to spherical symmetry [15]. We only need to calculate the tt component of the junction condition. −. p √ κ2 σ r = r˙ 2 + fout − r˙ 2 + 1 3. (4.4). Squaring again and rearranging, we can obtain an equation like energy conservation of a point particle with a potential 1 2 r˙ + V = 0 2. (4.5). where V =. 1 (Br2 + µ/r2 )2 − 2 8A2 r2. where A = κ2 σ/3 and B = A2 + Λ/6. The graph of the potential is shown in figure 8.. – 16 –.

(22) Figure 5: potential for true vacuum bubble in Einstein gravity.. There are three kinds of solutions of the equation 4.5. They are bounded, bounce and monotonic. Let us analyze these three solutions in phase diagram. We can regard the equation 4.5 as a Hamiltonian constraint H = 0. The momentum p = r˙ and the coordinate q = r, We can obtain the Hamilton’s equations of motion q˙ = p. p˙ +. ∂ ³ 1 (Bq 2 + µ/q 2 )2 ´ − =0 ∂q 2 8A2 q 2. (4.6). (4.7). The three different initial conditions correspond to the three solutions of the equation of motions. These three solutions are shown in figure 6. The bounded, the monotonic and the bounce solution correspond to the curve A, B and C respectively. These solutions are the same as the solutions of the four dimensional true vacuum bubble. 4.2 5D true vacuum bubble dynamics in Einstein-Gauss-Bonnet’s theory of gravity The action with the Gauss-Bonnet term Z √ 1 2 S = 2 d5 x −g[R − 2Λ + αRGB ] + Sbubble 2κ where 2 = R2 − 4Rµν Rµν + Rµνρσ Rµνρσ RGB. – 17 –. (4.8).

(23) Figure 6: phase diagram of three solutions. A: bounded solution. B: monotonic solution. C: bounce solution.. The equation of motion Gab + Λgab + 2αHab = κ2 Tab. (4.9). where Hab is the Lovelock tensor 1 Hab = RRab − 2Rac Rbc − 2Rcd Racbd + Racde Rbcde − gab (R2 − 4Rcd Rcd + Rcdes Rcdes ) 4 The generalized junction condition for Einstein-Gauss-Bonnet’s theory of gravity is given as [17] cd + 2 [Kab − Khab ]+ − + 2α[3Jab − Jhab + 2Pacdb K ]− = −κ Sab. where 1 Jab = (2KKac Kbc + Kcd K cd Kab − 2Kac K cd Kdb − K 2 Kab ) 3 and Pabcd = Rabcd + 2Rb[c gd]a − 2Ra[c gb]b + Rga[c gd]b The induced metric on the bubble wall which is a three sphere ds2 = −dτ 2 + r2 (τ )dΩ23 The bulk metric inside the bubble is ds2 = −fin (r)dt2 + fin (r)−1 dr2 + r2 dΩ23. – 18 –. (4.10).

(24) Minkowski fin (r) = 1 The bulk metric outside the bubble is ds2 = −fout (r)dt2 + fout (r)−1 dr2 + r2 dΩ23 Boulware-Deser-Schwarzschild-de Sitter [18] [19] r2 fout (r) = 1 + (1 − 4α. r 1+. 4αΛ 8αµ + 4 ) 3 r. matter source of the bubble wall Sab = −σhab so τ τ component is the tension of the bubble wall Sτ τ = σ We can choose an embedding such that t = τ , then we substitute the above ingredients into the generalized junction condition and we will get the equation of motion of the bubble wall. Note that there is only one degree of freedom r(t) due to spherical symmetry. We only need to calculate the τ τ component of the generalized junction condition [20]. h pr˙ 2 + f (r) ³ 2 τ κ Sτ = 3 − 2α 2 r. ³p. r˙ 2. ´3 + f (r) r3. p 2. − 6(1 + r˙ ). r˙ 2 + f (r) ´i+ (4.11) r3 −. Squaring twice and rearranging, we can obtain an equation. where k = κ2 σ ´2 ³ 4α 4α − (r˙ 2 + fout )[3 − 2 (−2r˙ 2 + fout − 3)]2 − k 2 r2 − (r˙ 2 + 1)[3 − 2 (−2r˙ 2 − 2)]2 r r 4α +4k 2 r2 (r˙ 2 + 1)[3 − 2 (−2r˙ 2 − 2)]2 = 0 r We can rearrange the equation into the form X(r)r˙ 2 + Y (r)r˙ 4 + Z(r)r˙ 6 + V (r) = 0. – 19 –. (4.12).

(25) Figure 7: potential of true vacuum bubble with Gauss-Bonnet correction.. where 768α2 k 2 − 8r2 αΛk 2 r2 12µ 96αΛµ 384α2 Λµ 48αk 2 µ 288αµ2 1152α2 µ2 + 2 − − − − − 6 r r2 r4 r2 r r8 r ³ 8 4αΛ 8αµ 12µ 32αΛµ 96αµ2 ´ + −2r2 Λ − r2 αΛ2 − 2 − 1+ − + 4 2 6 3 r r r 3 r. X(r) = 2r2 Λ − 8r2 αΛ − 32α2 Λ2 + 36r2 k 2 + 384αk 2 +. Y (r) = −16α2 Λ2 + 192αk 2 +. Z(r) =. 768α2 k 2 192α2 Λµ 576α2 µ2 − − r2 r4 r8 256α2 k 2 r2. r4 Λr4 4r4 Λ2 4 4 3 r4 k2 2 2 2 2 2 2 2 + 2Λr + − − 8r αΛ − 16α Λ − r αΛ + 36r k + 8α2 4α 3 27 2α 2 2 12µ 3µ 96αΛµ 256α k − 2r4 Λk 2 − 8r2 αΛk 2 − r4 k 4 + 2 + − 16Λµ − +192αk 2 + 2 r r 2α r2 2 2 2 2 2 2 2 192α Λµ 8 2 48αk µ 48µ 288αµ 576α µ 16αΛµ 32αµ3 2 − − αΛ µ − 12k µ − − − − − − r4 3 r2 r4 r6 r8 r4 r8 ³ r4 r4 Λ 2 4 2 8 2 2 r4 k 2 2 4 2 12µ 2µ 32αΛµ 2 + − 2r Λ − − r Λ − r αΛ − − r Λk − 2 − − 8Λµ − 2 8α 3α 3 3 2α 3 r αr r2 ´ 24µ2 96αµ2 4αΛ 8αµ −4k 2 µ − 4 − 1+ + 4 6 r r 3 r V (r) = −. The potential is shown in figure 7.. – 20 –.

(26) 4.3 Hamiltonian formalism Let r˙ be the momentum p and r be coordinate q. The Hamiltonian of this system is given as H = X(q)p2 + Y (q)p4 + Z(q)p6 + V (q). (4.13). Substitute into the Hamilton’s equations of motion ∂H ∂p ∂H −p˙ = ∂q q˙ =. We would obtain two first order differential equations. p˙ +. q˙ = 2X(q)p + 4Y (q)p3 + 6Z(q)p5. (4.14). ∂X 2 ∂Y 4 ∂Z 6 ∂ p + p + p + V (q) = 0 ∂q ∂q ∂q ∂q. (4.15). There are also bounded, bounce and monotonic solutions in the Einstein Gauss Bonnet’s theory of gravity. We will set the cosmological constant Λ = 1, mass µ = 1, α = 0.01 and tension of the bubble κ = 1 for illustrative purpose. The phase diagram is shown in figure 8. The bounded, the monotonic and the bounce solution correspond to the curve A, B and C respectively. If we choose to set the tension of the bubble a smaller value κ = 0.027, we will find a trapped solution which is shown in figure 9. It means that in the EinsteinGauss-Bonnet’s theory of gravity, the bubble can expand eternally only if the bubble has large enough tension and proper initial conditions.. 5. Summary The dynamics of vacuum bubbles are important for early cosmology. False vacuum bubbles may explain the origin of the inflationary universe. True vacuum bubbles nucleated by tunneling between false vacua are crucial to understand string landscape. Their evolution with the domain wall separating the two different vacuum regions is described by the junction condition. We are eager to see whether quantum effect influences the evolution of the bubble and this can be studied by adding higher order corrections to the theory of gravity. One kind of these corrections remaining. – 21 –.

(27) Figure 8: A: bounded solution. B: monotonic solution. C: bounce solution.. Figure 9: B: monotonic solution. C: bounce solution. D: trapped solution. The bounded solution is omitted.. second order gravity is the Gauss Bonnet term but it is a topological invariant in four dimensions. so it will not contribute to the equation of motion. Another simple choice is R2 however this theory became fourth order. Detail study shows that the junction condition generated by this correction contains highly singular terms, like the first and second derivatives of the delta function and the square of the delta function. The junction condition will contain the delta function and its derivatives. This is not applicable. One may impose a stronger junction condition on the induced metric to get a regular junction condition, but it will cause the bubble wall to be static.. – 22 –.

(28) We try to include higher order correction to the matter source of the thin wall and match the delta function, the first and second derivatives of the delta function and the square of the delta function terms respectively. Then, we decompose the junction condition into four equations and show that these four equations are not consistent and cannot be solved simultaneously. This result indicates that we cannot use the thin wall approximation and should replace the delta function with a fast varying function across the wall, but this formalism has not been derived explicitly in the literature. Moreover, beyond thin wall approximation, the junction condition with higher order corrections would contain over one hundred terms and it is almost impossible to solve analytically. One would need to develop a numerical method to solve this problem, but third order derivative and fourth order derivative terms will still make the analysis very difficult. Dynamics of vacuum bubbles seem to be tractable only in the second order gravity theory. We need to seek another theory which contains higher order corrections with the equation of motion still being the second order in four dimension. In order to see the effects of Gauss-Bonnet terms, we study the dynamics of the five dimensional true vacuum bubbles. In the Einstein’s theory of gravity, the bubble dynamics only depends on the initial conditions. If the kinetic energy of bubble is smaller than potential and the initial size of the bubble is small, the solution is a bounded solution. If the kinetic energy of bubble is smaller than potential and the initial size of the bubble is large enough, the solution is a bounce solution. If the kinetic energy of bubble is larger than potential, the solution is a monotonic solution. The monotonic solution means that when the bubble nucleates, it will expand eternally and approaches to move at the speed of light. In the EinsteinGauss-Bonnet’s theory of gravity, The above argument still holds. However, there is a trapped solution that the bubble can not expand eternally.. – 23 –.

(29) 6. Part II Introduction The Standard Big Bang model can explain the origin of the universe quite successfully [21] [22], but it also has some problems. Current astrological observations indicate that the universe is flat. However, the flatness problem states that it is very difficult to form a flat universe. The universe is homogeneous and CMB is uniform. However different regions in universe are separated by more than the horizon scale and cannot interact. Horizon problem means that the observed universe had become uniform in temperature at a very early time. If the universe achieved thermal equilibrium so early, there would be no galaxies or planets at all. Inflation can solve the problems of the Standard Big Bang. For the flatness problem, the space was driven to be flat during inflation. For the Horizon problem, The comoving Hubble length reduced and our present observable universe originated from a tiny region inside the Hubble radius during inflation. Inflation can explain the homogeneity and isotropy of the universe. It can also predict the density perturbation and the gravitational waves perturbation of the CMB. The density perturbations create anisotropies in the CMB. It is also the seeds of the formation and clustering of the galaxies. The fluctuation of the inflaton causes the fluctuation of the energy density of the universe. The inflationary perturbation predicts an almost scale invariant power spectrum with Gaussian distribution. The prediction of the inflation is consistent with current observations of the CMB. The WMAP 5 year shows that the spectral index ns = 0.960 ± 0.013 and tensor-to-scalar ratio of r < 0.22(0.95CL) [23]. Although the mechanism of inflation is successful, the source of inflation is still a open problem. Which field play the role of the inflation? Where does its potential come from? This question can be addressed in a fundamental theory, like string theory. String theory requires ten spacetime dimension so the extra 6 dimension should be compactified. In the brane world scenario, a pair of d branes is considered. The distance between the branes is described by the scalar field. We identify the scalar field as inflaton. The D3 anti-D3 barne system [24] is considered. The coulomb potential between the D3 brane and anti-D3 brane is C − µφ−4 , where C and µ are constants and φ is the inflaton field. The slow roll parameter η for the coulomb potential can not be small. The slow roll conditions can not be satisfied in this. – 24 –.

(30) model. The geometry of the internal space which admits a positive cosmological constant is constructed by KKLT [9]. The internal space is a Calabi-Yau geometry with a throat. The anti-D3 branes is placed at the tip of the throat. In this background, the warp factor modifies the Coulomb potential. In the KKLMMT model [25], they probe another D3 brane in the throat geometry. The slow roll condition can be satisfied because the warped factor of the throat geometry makes the coulomb potential flatter. However, conformal coupling generates a mass term to the inflaton potential. The potential of the inflaton is too steep to satisfy slow roll condition. Later, a new D-cceleration inflation model [26] is proposed. This kind of model is often called DBI inflation model. The kinetic term is of the Dirac-Born-Infield form, which contains an infinite sum of higher derivative kinetic terms and they take the full DBI action into account without expansion. The DBI inflation is that the D3 brane moves down in a warped throat relativistically. At the UV end the throat joins smoothly onto a Calabi-Yau space. The throat is cut off at the IR scale. Let the φ denotes the position of the brane then for φir < φ < φuv we can approximate the throat by an AdS5 of radius r. A crucial point about the action is that it imposes a speed limit on how quickly φ can change. For a homogeneous field, reality of the action imposes a speed limit. In order to neglect the quantum gravity effect, the radius of the AdS5 space should be large. Thus the speed limit would require the inflaton move slowly even though the potential is steep. The throat geometry used in D3 brane DBI inflation is AdS5 with a infinitely long throat so the IR cut off in the D3 brane DBI inflation is imposed by hand. we want to construct a DBI inflation model with a natural IR cut off. Another warped geometry used in hologrphic QCD is the Witten’s geometry [28] in the type IIA string theory. This geometry is the near-horizon geometry of the D4 brane and we compactify one spatial direction x4 . The radius of the compact spatial circle is obtained from the condition that the compact direction x4 shrinks away smoothly at r = rKK . The geometry has a natural IR cut off rKK . We probe a D4 brane in the throat. We treat the radial position in the thraot as the inflaton. The D4 brane moves toward the tip of the throat and drives the inflation. The warp factor of the witten’s geometry [28] is different from AdS5 . The observational quantity may be different. In the next section, we review the brane inflation model and the problem of the. – 25 –.

(31) model. Then in section 10 we discuss the D3 brane DBI inflation model. Inflation can be predictive if it has the attractor behavior. Therefore we use the phase portrait [27] [30] [31] to show that the DBI inflation model exhibits attractor behavior. In section 9 we consider the D4 brane DBI inflation model. We also show the attractor behavior of the model. In section 10 we calculate the spectra index and tensor-to scalar-ratio of these two models. We compare these with the observational data. In section 11 we discuss the difference between the D3 DBI inflation model and D4 DBI inflation model.. 7. Brane Inflation In this section we review the inflation and brane inflation model. The pressure of the matter source must be negative in order to satisfy the condition of inflation. Scalar field has this property and the field responsible for inflation is also called inflaton. The Lagrangian of scalar field is L = −g µν ∂µ φ∂ν φ − V (φ) = φ˙ 2 − ∇φ · ∇φ − V (φ). (7.1). We assume spatial homogeneousness and drop spatial derivative terms in Lagrangian. The energy density and pressure of the scalar field are 1 ρ = φ˙ 2 + V (φ) 2 1 ˙2 p = φ − V (φ) 2 V (φ) is the potential of the scalar field. Different models correspond to different choices for the potential. Substitute ρ and p into the Friedmann and continuity equations, and we can obtain the equation of motions H2 =. 1 1 ˙2 [ φ + V (φ)] 3Mpl2 2. dV φ¨ + 3H φ˙ = − dφ If the energy density and pressure obey φ˙ 2 < V (φ), the condition for inflation is satisfied. With a flat potential, then even if this condition is not obeyed initially, it would come to be satisfied quickly. Slow-roll approximation is the standard technique. – 26 –.

(32) to analyze inflation. This approximation drops the last term of the equation and the first term of the equation. H2 '. 1 V (φ) 3Mpl2. dV 3H φ˙ ' − dφ. For this approximation to be valid, it is necessary for two conditions to hold. These are ²(φ) ¿ 1. η(φ) ¿ 1. where the slow-roll parameters ² and η are defined by [21] [22] ²(φ) =. Mpl2 V 0 2 ( ) 2 V. η(φ) = Mpl2. V 00 V. The brane inflation model is considered in order to realize the inflation in string theory. Dvali et al. [24] proposed a brane anti-brane inflation model. The brane and antibrane are initially separated by a distance in the compact manifold. They experience Coulomb attraction due to gravity and RR fields. The position of the brane can be regarded as an inflaton field seen by a 4D observer and the potential takes the form V (r) = 2T3 (1 −. µ ) r4. 8 where µ = T33 /2π 3 M10,pl and T3 is D3 brane tension. However the slow roll condition. η(φ) = Mpl2 V 00 /V ¿ 1 cannot be satisfied due to the relation of the four dimensional Planck mass to the ten dimensional Planck scale and the six dimensional compact 8 6 6 space scale by MP2 l = M10,P l L . This implies that η ∼ (L/r) and slow-roll condition. can be satisfied only if r > L, which is impossible. 7.1 Brane Inflation in Warped Geometry If brane anti-brane inflation occurs in warped geometry, the slow roll condition may be satisfied [25]. A mobile D3 brane moves in the throat region of the compact manifold. The anti-D3 branes are at the tip of the throat r0 so the additional D3 brane will experience an attractive force and moves toward the tip. The throat region far from the tip can be approximately described as AdS5 . ds2 =. ´ R2 r2 ³ 2 i j −dt + δ dx dx + 2 dr2 ij R2 r. – 27 –. (7.2).

(33) The motion of the D3 brane is then described by DBI action with the potential between the brane and antibrane S = −T3. Z ³ √. r Z r4 R4 µν V (r) ´ −gd x( 4 ) 1 + 4 g ∂µ r∂ν r + + T3 C4 R r T3 4. (7.3). The potential between brane located r and antibrane located at r0 is given by V (r) = 2T3. 1 r04 r04 (1 − ) R4 N r4. (7.4). Assuming the slow motion of the D3 brane we can expand the action to the second order and the Chern Simone term can be ignored. The action is reduced to Z S=. ³ 1 1 r04 ´ r4 d4 x − T3 g µν ∂µ r∂ν r − 2T3 04 (1 − ) 2 R N r4. In terms of canonically normalized scalar φ =. √. T3 r, and including four dimensional. Einstein-Hilbert action Z √ ³ R 1 µν 4π 2 φ40 1 φ40 ´ − g ∂ φ∂ φ − (1 − ) S = d4 x −g µ ν 2MP2 l 2 N N φ4. (7.5). The slow roll parameter is η=−. 20 2 φ40 M N P l N φ6. The slow roll can be easily achieved because the warp factor of the throat geometry makes the potential flatter. However, due to the Conformal coupling Rφ2 , it will modify the potential and generate a mass term to the inflaton. 2 m2 ∼ Vds ∼ H 2 3 This term would lead η ∼ 2/3 and inflation would be short. Some fine-tuning is required to satisfy the slow roll condition.. 8. D3 brane DBI inflation model In this section we review the DBI inflation model and explain why it works. Then we study the attractor behavior of the model.. – 28 –.

(34) 8.1 DBI inflation in AdS5 Silverstein et.al [26] consider the relativistic motion of a D3 brane and take the full DBI action into account without expansion. The geometry of the throat is also AdS5 . Assuming spatial homogeneity and dropping spatial derivative. We consider a spatially flat universe and four dimensional metric. ds2 = −dt2 + a(t)2 dx2 the action with four dimensional gravity can be written s Z Z 4 √ √ φ4 MP2 l 1 φ λ S= dx4 −gR − d4 x −g[ 1 − 4 φ˙ 2 − + V (φ)] 2 gs λ φ λ. (8.1). ˙ This maximum speed limit This lagrangian imposes a speed limit restriction on φ. is required to make the action consistent. The terms in the square root should be positive 1−. λ ˙2 φ ≥0 φ4. it implies the speed limit φ4 φ˙ 2 ≤ λ. (8.2). Although the relativistic motion of the D3 brane seems to violate slow roll inflation, the speed limit will slow down the inflaton making inflation possible. In addition the speed limit is sensitive to the warp factor of the throat geometry. The equation of motion of DBI inflation is 6φ˙ 4φ3 λφ˙ 2 4φ3 λφ˙ 2 φ¨ − + + 3H(1 − 4 )φ˙ + (V 0 (φ) − )(1 − 4 )3/2 = 0 φ λ φ λ φ. (8.3). For a general potential V , the Friedmann equation reads 1 ρ gs Mpl2 1 a ¨ 2 + 2H 2 = − P a gs Mpl2 3H 2 =. where the energy density ρ and pressure P is [29] ρ(φ, X) = p. φ4 1 − 2λX/φ4. – 29 –. −. φ4 + V (φ) λ. (8.4) (8.5).

(35) φ4 p φ4 1 − 2Xλ/φ4 + − V (φ) λ λ where φ is the inflaton field and X = φ˙ 2 /2. The slow roll parameter for the DBI P (φ, X) = −. type inflationary model is given as [29] H˙ XP,X = H2 Mpl2 H 2. ²=−. η=. ²˙ ²H. It can be shown that the slow conditions ² ¿ 1 and η ¿ 1 can be satisfied if the inflaton field is large enough. In addition, the inflationary model can solve the cosmological problem of the Big Bang theory if the model can produce a large amount of inflation. The amount of inflation is characterized by the ratio of the scale factors at time t and at the end of inflation. The ratio is the number of e-folding N (t) defined by Z. a(tend ) N (t) = ln = ai (t). tend t. a˙ dt = a. Z. Z. tend. φend. Hdt = t. H φi. dφ φ˙. (8.6). The initial condition for the inflaton which can produce 60 e-folding is φi = 100. Inflation occurs when the brane starts to move far from the tip. When the inflaton is very near the fixed point which is the origin in this model, the velocity of inflaton decelerates very quickly. Although the speed limit on the inflaton force the speed of the inflaton slow near the tip of the throat, the speed of the inflaton will approach to zero very quickly. Therefore the amount of the inflation near the tip of the throat is negligible. The speed of sound cs is cs =. s 1−. λ ˙2 φ φ4. (8.7). Even though the inflaton rolls relativistically, the speed limit requires the positivity of the argument of the square root in the action. We can approximate the inflaton speed during inflation as cs ¿ 1 φ2 φ˙ ' ± √ λ The requirement ² ¿ 1 implies that the potential energy dominates during inflation. The Friedmann equation reduces to H2 =. 1 V (φ) 3gs Mpl2. – 30 –. (8.8).

(36) The DBI inflation model is different from the KKLMMT model. For the KKLMMT model, the potential is too steep to satisfy the slow-roll condition. However, for the DBI inflation model, the speed limit requires the inflaton change slowly. In order to neglect the quantum gravity effect, the AdS radius should be large enough so the speed of the inflaton is initially small. When the D3 brane moves down to the IR region, the speed limit makes the D3 brane move slower and slower. When the D3 brane is very near the tip of the throat, the speed of the inflaton approaches zero very quickly. The speed limit makes inflation work even though the potential is steep. 8.2 Attractor behavior of the DBI inflation model Inflation can be predictive only if the solution has the attractor behavior [27] [30] [31]. The difference between solutions of different initial condition will vanish quickly. The Hamilton-Jacobi formulation is an effective method for analyzing the inflationary attractor property. In this formulation, we can regard inflaton field φ as the time variable φ = φ(t) and require that the inflaton field φ does not change sigh during inflation. We also consider H = H(φ) so the inflaton field φ is monotonic and H will not have oscillatory behavior in φ. we can take the time derivative of the first Friedmann equation and use the equation of motion of the inflaton field φ, we can find 6HH 0 φ˙ = −. φ˙ 2. 3H q gs Mpl2. 1 − λφ˙ 2 /φ4. we can solve it for φ˙ −2H 0. φ˙ = q. 1/gs2 Mpl4 + 4λH 0 φ˙ 2 /φ4. Then, we can substitute the φ˙ into the first Friedmann equation and use the expression for ρ to get an expression for the potential V (φ) in terms of the Hubble parameter H(φ) V = 3(gs Mpl2 )H 2 −. gs Mpl2 φ4 q 2 2 φ4 1/gs Mpl + 4H 02 λ/φ4 + λ λ. Suppose that H(φ)0 is a solution of the above equation and consider a small perturbation about H(φ) = H0 (φ) + δH(φ). The linearized equation for δH becomes, q 0 0 2H0 δH = 3H0 δH 1/(gs2 Mpl4 ) + 4λH00 /φ4. – 31 –.

(37) Figure 10: Attractor curves of D3 DBI inflation.. with the general solution hZ. q. φ. δH(φ) = δH(φi )exp. 1/gs2 Mpl2 + 4λH00 /φ4. 3H0 φi. dφ i 2H00. where δH(φi ) is the value at some initial point φi . Since H00 and dφ have the opposite sign, if H0 is an inflationary solution, all linear perturbations are damped at least exponentially. Thus we conclude that the inflationary attractor property is satisfied for the inflaton field φ. To study the attractor behavior [27] [30] [31], rewrite the equation of motion as a set of two first order equations with two independent variables x ≡ φ and y ≡ φ˙ : y = x˙. y˙ −. (8.9). 6y 4x3 λy 2 4x3 λy 2 + + 3H(1 − 4 )y + (V 0 (x) − )(1 − 4 )3/2 = 0 x λ x λ x. (8.10). The Freidmann equation can be also re written as H2 =. 1 x4 q ( 3gs MP2 l λ 1 −. R4 2 y x4. −. x4 + V (x)) λ. (8.11). The numerical analysis shows the trajectories in the phase diagram (x, y) with gs = 1 × 10−8 and R4 = 100 and m2 = 10−3 . The result shows that DBI inflation indeed exhibits the attractor behavior which is independent of initial conditions.. – 32 –.

(38) 9. D4 DBI Inflation model In this section we propose a new DBI inflation model. The throat geometry is the Witten’s geometry [28] and we probe a D4 brane to drive inflation. Then we discuss the property of the model and study the attractor behavior of the model. 9.1 DBI Inflation in the Witten’s geometry Warped geometry provides a successful framework for building inflation models. If some branes in the compact manifold become very heavy, they will warp the compact manifold and form a throat geometry. We consider D4 branes to condensate in a 5D internal manifold and use this background for DBI inflation. The setup follows. D4 branes fill five dimensional noncompact spacetime and warp the internal manifold and compactify one spatial direction x4 . The metric is given by [28] r 3 R 3 dr2 ds2 = ( ) 2 (ηµν dxµ dxν + f (r)(dx4 )2 ) + ( ) 2 ( + r2 d2 Ω4 ) R r f (r). F4 =. (9.1). 2πNc r 3 r3 ²4 , eφ = gs ( ) 4 , f (r) = 1 − KK , R3 = πgs Nc ls3 Ω4 R r3. The metric describes a throat geometry with the x4 circle smoothly shrinking to a 1/2. tip at the IR endpoint r = rKK by requiring the period of x4 to be 4πR3/2 /3rKK ≡ 2π/MKK to avoid a conical singularity. The action of a D4 brane oriented along the compact space is given as Z Z p 1 5 −φ SD4 = − dx e −detgmn + C5 (2π)4 ls5 D4. (9.2). r r R 1 5 dxm dxn = ( )3/2 f (r)(dx4 )2 + [( )3/2 gµν + ( )3/2 gmn ∂µ r∂ν r]dxµ dxν R R r f (r) gµν dxµ dxν = −dt2 + a2 (t)dx2 where m, n = 0, ..., 4 and µ, ν = 0, ...3 and the five-form potential obtained from the Hodge dual F4 is C5 =. 3 1 Nc 3 r a(t) dx0 ∧ dx1 ∧ dx2 ∧ dx3 ∧ dx4 2 3 (2πls ) 4π R. – 33 –.

(39) Put all these together with the Einstein-Hilbert action and use R3 = πgs Nc ls3 . and we get r Z Z √ √ MP2 l 1 1 2π r3 R3 µν r3 4 4 dx −gR − d x −g[ f + g ∂ r∂ r − +V] µ ν 2 2(π)4 ls5 gs MKK R3 r3 R3 Transform the coordinate variable r into the field theory variable φ = r/ls2 and the IR cutoff is φKK = rKK /ls2 . The ’t Hooft coupling is λ = πgs ls Nc . For the zeroth order evolution, we can assume the inflaton φ is spatially homogeneous and g µν ∂µ φ∂ν φ = −φ˙ 2 . Then the action becomes s Z Z 3 √ √ MP2 l 1 φ λ µν φ3 1 2π 4 dx4 −gR − d x −g[ f + g ∂ φ∂ φ − + V (φ)] µ ν 2 2(π)4 ls gs MKK λ φ3 λ ˙ the The feature of this lagrange is that it automatically imposes a restriction on φ, speed limit f h3 ≥ φ˙ 2. (9.3). The speed of the inflaton field φ will become slow as the brane approaches the tip of the throat. The equation of motion for inflaton field is φ˙. d (q dt. f−. h−3 φ˙ 2. 3H φ˙ + 3h2 h0 f + 12 h3 f 0 − 32 h−1 h0 φ˙ 2 q + (V − h3 )0 = 0 f − h−3 φ˙ 2. )+. Where dot denotes. d dt. and prime denotes. d dφ. (9.4). and h3 = φ3 /λ. We can expand the. equation and obtain the equation of motion for U 3 3 3 ˙2 φ3 ¨ 3 φ3KK ˙ 2 ˙ − φKK − λ φ˙ 2 ) + ( 3φKK − 9 φ )(1 − φKK ) φ¨ − KK φ − φ + 3H φ(1 φ3 2 φ4 φ3 φ3 2λφ 2φ φ3 2 3 3 2 3 3φ φ 3φKK ˙ 2 3φ φKK λ ˙ 2 3/2 2 0 + (1 − KK ) − φ + (V − )(1 − − φ ) =0 λ φ3 2φ4 λ φ3 φ3. For a general potential V , the Friedmann equation reads Te3 ρ Mpl2 Te3 a ¨ 2 + 2H 2 = − 2 P a Mpl 3H 2 =. where Te3 =. 1 2π 1 2(π)4 ls gs MKK. and the energy density ρ and pressure P is [29] h3 f. ρ= p. f−. 2h−3 X. − h3 + V (φ). – 34 –. (9.5).

(40) P = −h3. p. f − 2h−3 X + h3 − V (φ). (9.6). φ˙ 2 . 2. where φ is the inflaton field and X =. The inflationary model can be successful if the model satisfy the slow roll condition. We can define a quantity which is the pressure 9.6 of the model The slow roll parameter for the DBI type inflationary model is given as XP,X H˙ = 2 2 2 H Mpl H. ²=−. η=. (9.7). ²˙ ²H. (9.8). It can be shown that the slow conditions ² ¿ 1 and η ¿ 1 can be satisfied if the inflaton field is large enough. In addition, the inflationary model can solve the cosmological problem of the Big Bang theory if the model can produce a large amount of inflation. The amount of inflation is characterized by the ratio of the scale factors at time t and at the end of inflation. The ratio is the number of e-folding N (t) defined by a(tend ) N (t) = ln = ai (t). Z. tend. Z. a˙ dt = a. t. Z. tend. φend. Hdt = t. H φi. dφ φ˙. (9.9). We set the tip of the throat at φKK = 0.1. Using the numerical calculation, we can show that the initial condition for the inflaton which can produce 60 e-folding is φi = 100. This means that inflation occurs when the brane starts to move far from the tip. During the inflation, the factor f is near 1 f =1−. φ3KK 0.13 ' 1 − '1 φ3 1003. The action of the D4 DBI inflationary model reduces to the form MP2 l 2. Z dx. √ 4. Z −gR − Te3. √. d4 x −g[. 3. s. φ λ. 1+. λ µν φ3 g ∂ φ∂ φ − +V] µ ν φ3 λ. (9.10). The speed of sound cs is s cs =. 1−. λ ˙2 φ φ3. – 35 –. (9.11).

(41) Even though the inflaton rolls relativistically, the speed limit requires the positivity of the argument of the square root in the action. We can approximate the inflaton speed during inflation as cs ¿ 1 3/2. φ φ˙ ' ± 1/2 λ. The requirement ² ¿ 1 implies that the potential energy dominates during inflation. The Friedmann equation reduces to H2 =. Te3 V (φ) 3Mpl2. This model is similar to the D3 DBI inflation model. The warp factor of the D4 DBI inflation model is cubic. 9.2 Attractor behavior of the D4 DBI inflation model Inflation can be predictive only if the solution has the attractor behavior. The difference between solutions of different initial condition will vanish quickly. The Hamilton-Jacobi formulation is an effective method for analyzing the inflationary attractor property. In this formulation, we can regard inflaton field φ as the time variable φ = φ(t) and require that the inflaton field φ does not change sigh during inflation. We also consider H = H(φ) so the inflaton field U is monotonic and H will not have oscillatory behavior in φ. we can take the time derivative of the first Friedmann equation and use the equation of motion of the inflaton field φ, we can find 6HH 0 φ˙ = −. φ˙ 2. Te3 3H q Mpl2. f − h−3 φ˙ 2. we can solve it for φ˙ √ −2H 0 f ˙ φ= r f3 T + 4H 0 h−3 M2 pl. Then, we can substitute the φ˙ into the first Friedmann equation and use the expression for ρ to get an expression for the potential V (φ) in terms of the Hubble parameter H(φ) V =3. Mpl2 Te3. H2 − h. v u uT f2 f t 34 + 4H 02 h−3 + h3 Mpl. Mpl2 p 3 Te3. – 36 –.

(42) Suppose that H(φ)0 is a solution of the above equation and consider a small perturbation about H(φ) = H0 (φ) + δH(φ). The linearized equation for δH becomes, s Te3 2H00 δH 0 = 3H0 δH + 4H00 h−3 Mpl2 with the general solution hZ. s. φ. δH(φ) = δH(φi )exp. 3H0 φi. i Te3 0 −3 dφ + 4H0 h Mpl2 2H00. where δH(φi ) is the value at some initial point φi . Since H00 and dφ have the opposite sign, if H0 is an inflationary solution, all linear perturbations are damped at least exponentially. Thus we conclude that the inflationary attractor property is satisfied for the inflaton field φ. 9.3 Phase diagram of attractor behavior We will study numerically the dynamics of the φ for the simplest potential that may lead to a power law acceleration 1 V = m2 φ2 2. (9.12). This potential is simple and the higher power terms in the effective potential will be small compared to the quadratic term without tuning as discussed in [26] [32]. We will fix the parameter λ = 100 in the following discussion. To study the attractor behavior [27], we can rewrite the equation of motion as ˙ In terms of x and y, the speed a set of two independent variables x ≡ φ and y ≡ φ. limit translate into x3 φ3KK (1 − ) ≥ y2 R3 x3. (9.13). The evolution equation of the inflaton field φ can be written as a set of two simultaneous equations from (9.4): y = x˙. y˙ −. 3 φ3KK 2 φ3KK λ 2 3φ3KK 9y 2 φ3KK φ3KK y ˙ − y + 3Hy(1 − − y ) + ( − )(1 − ) x3 2 x4 x3 x3 2λx 2x x3 3x2 φ3 2 3φ3KK 2 3x2 φ3KK λ 2 3/2 2 + (1 − KK ) − y + (m x − )(1 − − y ) =0 λ x3 2x4 λ x3 x3. – 37 –.

(43) Figure 11: Attractor curves of D4 DBI inflation model.. with the Friedmann equation r 3 e3 x3 T φ φ3 λ 2 x3 KK (1 − 3 )( 1 − KK − y + V − ) 3H 2 = 2 MP l λ x x3 x3 λ. (9.14). In the following we start the numerical analysis of the dynamical system. We will set Te3 = 108 and m2 = 10−3 and the tip of the throat φKK = 0.1. The diagram is shown in figure 11. We can see that there is a curve that attracts all he trajectories which correspond to the slow-roll curve. This confirms our conclusion in the HamiltonJacobi formulation analysis. For all allowed initial condition, φ tends to the slow-roll curve quickly after it begins to evolve. The dynamics of φ can be divided into three stages. In the first stage, y is damped to the slow roll curve quickly with x kept almost constant. In the second stage, x decreases while y kept almost constant. In the final stage, the attractor curve coincides with the limit curve that satisfies the speed limit, (x3 /R3 )(1 − φ3KK /x3 ) = y 2 . The inflaton field φ will behave in this way before it reaches the critical point. The result shows that the D4 DBI inflation also exhibits the attractor behavior.. 10. Power spectrum In order to compare the observation data, we calculate the power spectrum of the DBI inflation model. The power spectrum of scalar perturbations and tensor perturbations in these models are computed in [29]. We calculate the spectral index which represents the tilt of the spectrum and power in tensor mode in this section.. – 38 –.

(44) We set the coupling constant of D3 brane DBI inflation model gs = 1. We set the tension of the D4 brane DBI inflation model Te = 1. The scalar power spectrum and gravitation power spectrum are computed in [29]. Pkζ =. 1 ρ2 1 = W (φ)V (φ)2 4 2 36π Mpl cs (P + ρ) 36π 2 Mpl4. (10.1). 2ρ 2V (φ) = 4 3π 2 Mpl 3π 2 Mpl4. (10.2). Pkh =. where W (φ) is the warp factor of the throat geometry and V (φ) is the inflaton potential. The warp factor of the D3 brane DBI inflation is λ/φ4 . The spectral index can be computed. dlnPkζ ns − 1 = = dlnk. √. 3Mpl φ2 ³ 4 2V 0 ´ √ − + φ V Vλ. (10.3). For the potential V (φ) = m2 φ2 , the spectral indices vanishes. The spectral index is a second-order quantity in the D3 brane DBI inflation model. The tensor-to-scalar ratio is r=. Pkh Pkζ. =. 24φ2 λm2. (10.4). CMBR observation requires the initial value of inflaton slightly above Planck scale. COBE normalization gives us a mass of the inflaton in the range m ∼ 1013 → 1014 GeV [32]. Fitting the scalar power spectrum Pkζ = 2.4 × 10−9 , we can get the parameter λ ' 9 × 1012 . The tensor to scalar ratio is in the range 0.02 ≤ r ≤ 0.4. The value is consistent with WMAP 5 year result [23]. Next, we calculate the spectral index and tensor to scalar ratio of the D4 brane DBI inflation. The warp factor of the D4 brane DBI inflation is λ/φ3 . The scalar power spectrum is Pkζ =. 1 1 ρ2 λ = V (φ)2 4 4 3 2 2 36π Mpl cs (p + ρ) 36π Mpl φ. (10.5). 2ρ 2V (φ) = 2 4 4 2 3π Mpl 3π Mpl. (10.6). Pkh =. The spectral indices can also be computed √ dPkζ 3Mpl φ3/2 ³ 3 2V 0 ´ √ ns − 1 = = − + dlnk φ V Vλ. – 39 –. (10.7).

(45) For the potential V (φ) = m2 φ2 , the spectral indices is √ 3 Mpl dPkζ ns − 1 = =√ dlnk λφ m. (10.8). For the quadratic potential, the spectral index does not vanish. The spectral index for the DBI D4 brane inflation is first order. The tensor-to-scalar ratio is r=. Pkh Pkζ. =. 24φ λm2. (10.9). CMBR requires the initial value of inflaton slightly above Planck scale ∼ 1018 . COBE normalization gives us a mass of the inflaton in the range m ∼ 1013 → 1014 GeV [32]. Fitting the scalar power spectrum Pkζ = 2.4 × 10−9 , we can get the parameter λφ ' 9 × 1012 . The spectral indices is 1 ± 10−8 and the tensor to scalar ratio is the same as the one of the D3 model. The power spectrum of this model is almost scale invariant and it is consistent with the observation data [23].. 11. Summary We have reviewed the mechanism of brane inflation and how the DBI inflation works. The DBI type kinetic term imposes the speed limit on the speed of the inflaton. The speed limit is time dependent and depends on the warp factor of the throat geometry. The speed limit makes the inflation possible even though the potential is steep. We propose another DBI inflation by using the D4 brane. The D4 brane moves in the throat region of Witten’s geometry. There is a IR cutoff in the Witten’s geometry. The D4 brane starts to move far from the tip of the throat and approaches the IR cutoff. The speed limit also make the inflaton slow down. We show that the D4 DBI inflation also exhibits attractor behavior. The initial speed of the inflaton U˙ should obey the speed limit. There is a slow roll curve which attracts different initial value of the inflaton U . The speed of the inflaton approaches zero very quickly as the D4 brane is very near the tip of the throat. In order to produce amount of e-folding, the initial value of the inflaton must be large. Therefore the IR cutoff factor is negligible during inflation. To conclude, the mechanism of the D3 and D4 DBI brane inflation models is the same. There are two differences between the D3 model and D4 model. One is that there is a natural IR cut off in the D4 model, while the IR cutoff in the D3 model is imposed by hand. Another is that warp factor of these two model is different. The. – 40 –.

(46) warp factor of the D3 model is 4th power and the warp factor of the D4 model is 3rd power. The IR cutoff can be omitted during inflation so the only difference is the warp factor. To compare with the observational data, we calculate the spectral indices and tensor-to-scalar ratio. Because the warp factor is different, the scalar perturbations are not the same up to first order. The spectral indices of the D3 model is 1. The spectral indices of the D4 model is near 1. The tensor-to-scalar ratio of D4 model is the same as the D3 model. The power spectrum of the D4 DBI inflation model is almost scale invariant and consistent with the WMAP observation.. 12. Appendix A Israel junction condition Let metric be taken into 3+1 decomposition in the Gaussian Normal coordinate to simplify calculation. define three dimensional domain wall induced metric as hij and extrinsic curvature as Kij ds2 = dy 2 + hij dxi dxj. Kij = −. 1 ∂hij 2 ∂y. and calculate the component of Rienmann tensor ∂Kij + Klj Kil ∂y = ∇j Kik − ∇k Kij. Ryiyj = Ryijk. ¯ lijk + [Kij Klk − Kik Klj ] Rlijk = R ¯ lijk is three dimensional Riemann tensor constructed by hij .Then contracting Where R index to give component of Ricci tensor ∂Kij + T rK 2 ∂y Ryi = ∇i K − ∇j Kij. Ryy = hij. ¯ ij + ∂Kij + 2Kil Klj − KKij Rij = R ∂y. – 41 –.