A Brief Survey of Single Field, Multiple Field, and D-brane Inflation

Transcription

1 A Brief Survey of Single Field, Multiple Field, and D-brane Inflation Thomas Rudelius May 2012 Thesis Adviser Liam McAllister Department of Physics

2 Contents 1 Introduction 2 2 Perturbations Preliminary Definitions The Primordial Power Spectra Non-Gaussianity Multiple Field Inflation Multiple Field Perturbations Multiple Field Non-Gaussianity D-brane Inflation Evolution of the Scale Factor and Change of Variables Potential DBI Inflation Numerical Analysis and Results Conclusions 38 6 Acknowledgments 39 A Friedmann-Robertson-Walker Metric 39 B Geometry 39 1

3 Abstract In 1, we examine the basic principles of the theory of inflation and the cosmological problems it solves. In 2, we follow the treatment of D. Baumann in [8] to explore the quantum perturbations produced during inflation. In 3, we present an abridged version of B. van Tent s work in [2] on multiple field inflation and the perturbations produced during multiple field inflation. In 4, we briefly examine the underlying theory of one particular multiple field inflationary model known as D-brane inflation. We study the possible inflationary scenarios resulting from different initial conditions of a D3-brane confined to a warped throat region and find very little inflation in one particular scenario. 1 Introduction The theory of inflation provides one possible solution to two seemingly contradictory puzzles: The large-scale homogeneity of the universe. The small-scale inhomogeneities of the universe. Figure 1: The CMB (source: NASA/WMAP Science Team). Small anisotropies lie against a large homogeneous background. The image of the cosmic microwave background (CMB) is a visual representation of this fact. Figure 1 depicts an image from a seven-year WMAP study of the CMB, showing that the early universe was largely homogenous but with relatively small anisotropies [WMAP picture]. It seems that a prerequisite for this large-scale homogeneity is causal contact between all regions of space visible to us. If information were unable to travel between these regions, as is predicted by the standard model of the Big 2

4 Bang, there would be no reason to expect any homogeneity between them. For the Friedmann-Robert-Walkerson metric (c.f. Appendix A), [ ] dr ds 2 = dt 2 + a 2 2 (t) 1 kr 2 + r2 (dθ 2 + sin 2 θdφ 2 ) we may define the comoving horizon τ to be the maximum distance a light ray can travel between time 0 and time t according to τ t 0 dt a a(t ) = da 0 Ha 2 = a 0 ( ) 1 d ln a ah where H ȧ a is called the Hubble constant and (ah) 1 is known as the comoving Hubble radius. Hence, τ measures the maximum separation between causally connected regions of space. Under the standard Big Bang model, τ grows with time according to (1) (2) τ a 1 2 (1+3w) (3) for a universe dominated by a fluid of pressure p, density ρ, and equation of state w p ρ 0. This means that τ grows monotonically with time, so that regions out of causal contact now would always have also been out of causal contact at last-scattering. However, the high degree of homogeneity shown in Figure 1 strongly suggests communication between regions of space at the time of CMB decoupling, an apparent contradiction. The existence of a period of time in the early stages of the universe for which the comoving horizon size (ah) 1 is decreasing can solve this puzzle (referred to as the horizon problem ). If two regions are separated by a distance greater than τ, they never could have communicated. However, if they are separated by a distance larger than the comoving Hubble radius (ah) 1, then they cannot communicate presently. This means that if τ is much larger than (ah) 1, then the entire observable universe could indeed have been in causal contact at last scattering, as shown in Figure 2, even though it is not any longer. Furthermore, a decreasing comoving Hubble radius would also stretch out the universe and make it appear flat, explaining the fact that our universe is indistinguishable from flat space. 3

5 Figure 2: Solution to the horizon problem (adapted from [8]). The comoving Hubble radius was originally large to allow causal contact, shrunk during inflation, and grew after the end of inflation. Physically, this corresponds to accelerated expansion, d2 a > 0, and negative dt 2 pressure, p < ρ 3. Experimental evidence suggests that a period of inflation occured at around t = s and caused an increase of the scale factor, a, by at least 60 e-folds [8]. The second problem, that of small-scale anisotropies, is typically explained by considering a scalar field, φ, called the inflaton, sitting in a potential V (φ). When V (φ) dominates the kinetic energy term 1 φ 2 2, inflation occurs. As the kinetic term approaches the potential term, inflations ends. Mathematically, this can be represented by a correction to the Einstein-Hilbert action, yielding S = d 4 x g[ 1 2 R gµν µ φ ν φ V (φ)] = S EH + S φ (4) We begin our analysis of the inflaton field by considering the Friedmann equations, ȧ ρ a = (5) 3 ä a = 1 (ρ + 3p) (6) 6 where c 8πG 1. If the inflaton comes to rest in a false (local but not absolute) minimum of the potential, the density ρ becomes a nonzero constant. The first Friedmann equation then implies exponential growth of a, which would indeed yield inflation. However, observation tells us that the period of inflation in the early universe did in fact come to an end, with the universe transitioning to a radiation-dominated state. The only explanation for such a termination would be a quantum tunneling of the inflaton out of a 4

6 false minimum and into a true minimum. It turns out that such a transition could not actually have occured [21, 2]. Instead, it is common to consider the next best thing to a false minimum: a slope of V (φ) so gradual that the inflaton is allowed to remain at high V for a sufficiently long period before the kinetic term dominates the potential term, as in Figure 3. This is known as slow-roll inflation. Figure 3: A typical potential for slow-roll inflation. Inflation occurs while the inflaton travels along the flat part of the potential, then ends when it falls off the cliff. To study slow-roll inflation, we consider the equations for the energy density, ρ, and the pressure, p, as functions of φ, φ, and V (φ): ρ = 1 2 φ 2 + V (φ) (7) p = 1 2 φ 2 V (φ) (8) Furthermore, we can manipulate the Friedmann equations to get, Or, where w p ρ. This yields ρ ρ = 3ȧ (ρ + p) (9) a d ln ρ = 3(1 + w) d ln a a 3(1+w) Plugging this result into the first Friedmann equation, we arrive finally at { t 2/3(1+w) w 1 a(t) e Ht w = 1 (10) Thus, for p ρ, i.e. V (φ) 1 φ 2 2, we find that the scale factor is approximately exponential in time. So, when the potential term dominates the 5

7 kinetic, inflation occurs. It is necessary, therefore, to determine the shape of the inflaton potential, since the places in which the potential is relatively flat will correspond to periods of time in the slow-roll regime, and hence periods of inflation. To describe the shape of V (φ), we define two dimensionless parameters, ε V (φ) M pl 2 ( ) 2 Vφ (11) 2 V η V (φ) Mpl 2 V φφ (12) V with slow-roll inflation occuring when ɛ V, η V 1. For the remainder of this paper, we will take M pl = 1 unless otherwise noted. According to models of hybrid inflation, the inflaton eventually exits the slow-roll regime and begins to oscillate about the minimum of the potential. When its frequency of oscillation reaches a resonance, the inflaton couples to another scalar field χ via a potential such as V = 1 4λ (M 2 λχ 2 ) m2 φ g2 φ 2 χ 2 (13) and particle production takes place, allowing for a graceful termination of inflation and initiating a phase of reheating [2]. The inflaton φ may or may not decay completely, though an incomplete decay would necessitate some amount of fine tuning to avoid producing a cold, empty, lifeless universe [15]. 2 Perturbations Fluctuations are always present at quantum mechanical scales. During inflation, the wavelengths of quantum fluctuations in the early universe grow larger than the horizon, becoming classical in nature. Hence, a classical perturbation spectrum is produced from quantum effects, solving the problem of cosmic anisotropies [2]. 2.1 Preliminary Definitions As shown in Figure 1, the early universe was largely uniform, with only small inhomogeneities. Thus, it makes sense to treat these inhomogeneities as linear perturbations next to the homogenous background discussed previously. To this end, we define perturbations around the background φ(t) and ḡ µν (t) for the inflaton and metric, respectively [8], so that φ(t, x) = φ(t) + δφ(t, x) (14) g µν (t, x) = ḡ µν (t) + δg µν (t, x) (15) The metric perturbations can be decomposed into a scalar (S), vector (V), and tensor (T) component via ds 2 = (1 + 2Φ)dt 2 + 2aB i dx i dt + a 2 [(1 2Ψ)δ ij + E ij ]dx i dx j (16) 6

8 where the perturbations B i and E ij are decomposed into scalars, vectors, and tensors by and B i i B S i, i S i = 0 (17) E ij 2 ij E + 2 (i F j) + h ij, i F i = 0, h i i = i h ij = 0. (18) Φ can be identified with the Newtonian gravitational potential in the weak-field limit. The vector perturbations S i and F i are not created by inflation, hence they can be ignored. The tensor perturbations are gauge-invariant, but the scalar perturbations are not. We may, however, define three gauge-invariant scalar quantities, the curvature perturbation on uniform-density hypersurfaces the comoving curvature perturbation ξ Ψ + δρ, (19) Ḣ ρ R Ψ and the inflaton perturbation on spatially flat slices Q = δφ + H δq, (20) ρ + p φ H Ψ (21) where ρ and p are the energy density and pressure of the background, respectively, and δq is the scalar part of the 3-momentum density Ti 0 = i δq. During inflation, Ti 0 = φ i δφ, so that R = Ψ + Ḣ φ δφ (22) Of particular interest is the power spectrum of R, defined by R k R k = (2π) 3 δ(k + k )P R (k) (23) 2 R = k3 2π 2 P R(k) (24) where... gives the ensemble average of the fluctuations. We may further define the scalar spectral index and the running of the spectral index n s 1 d ln 2 R d ln k α s dn s d ln k (25) (26) 7

9 The power spectrum can be approximated by a power law, ( ) k ns(k ) αs(k ) ln(k/k ) R(k) = A s (k ) (27) k If R is Gaussian, then all of the statistical information is encoded in the the power spectrum. Higher-order correlation functions contain information about the non-gaussianity of R. In the case of single field slow-roll inflation, the departure from Gaussianity is small. However, for the multiple field inflation models to be discussed later, this is not always the case. Finally, we give analogous formulas for the power spectrum of the two polarization modes of h ij, h k h k = (2π) 3 δ(k + k )P h (k) (28) 2 t 2 2 R = k3 π 2 P h(k) (29) and n t d ln 2 t d ln k ( ) k nt(k ) 2 t (k) = A t (k ) k (30) (31) 2.2 The Primordial Power Spectra Fluctuations are produced at all length scales, though the cosmologically relevant ones begin inside the horizon (Hubble radius). As inflation occurs, the Hubble radius shrinks, so that these fluctuations exit the horizon. Once inflation ends and reheating begins, these fluctuations re-enter the horizon. The scalars R and ξ are conserved outside the horizon. Thus, to determine R at horizon re-entry, it suffices to calculate R at horizon exit. To do so, we begin with the action for single field slow-roll inflation, S = 1 d 4 x g[r ( φ) 2 2V (φ)] (32) 2 and work in the gauge for which δφ = 0, g ij = a 2 [(1 2R)δ ij + h ij ], δ i h ij = h)i i = 0 (33) In this gauge, all scalar degrees of freedom are parametrized by R. This action can be expanded to second order in R (c.f. Appendix B of [8]), giving S (2) = 1 d 4 x a φ H 2 [Ṙ2 a 2 ( i R) 2 ]. (34) From here, we define the Mukhanov variable v zr, with z 2 a 2 φ 2 8 H 2. (35)

10 And transition to conformal time τ t/a to get S (2) = 1 ] dτd 3 x [(v ) 2 + ( i v) 2 + z 2 z v2. (36) where indicates a derivative with respect to τ. Varying the action and taking the Fourier transform of v, d 3 k v(τ, x) = (2π) 3 v k(τ)e ik x, (37) we find the Mukhanov Equation, ( ) v k + k 2 z v k = 0 (38) z The Muhkanov Equation, thus, is the equation of motion for a harmonic oscillator with time-dependent frequency. The Muhkanov Equation must be solved numerically for a general inflationary model, though in some simple situations it is possible to derive approximate analytical solutions. Per the usual trick, we promote the field v to a quantum operator, ˆv, with ˆv = d 3 k [ (2π) 3 v k (τ)â k e ik x + vk (τ)â e ik x] k. (39) The creation and annihilation operators â k and â k satisfy the canonical commutation relations, [â k, â k ] = (2π) 3 δ(k k ) (40) provided the mode functions are normalized according to v k, v k v i(v k v k v k v k ) = 1. (41) We must choose a vacuum state, satisfying â k 0 = 0 (42) Typically, the vacuum state is chosen to be the Minkowski vacuum of an observer in the distant past, τ. With this choice, Eq. 38 becomes v k + k2 v k = 0, (43) which is simply the equation for a simple harmonic oscillator. vacuum to be the ground state of this oscillator, so that We set the lim v k = e ikτ (44) τ 2k Eqs. 41 and 44 give the boundary conditions for the mode functions on all scales. 9

11 To simplify matters further, we consider the limit for which H 0, also known as the de Sitter limit. Thus, we set z z = a a = 2 τ 2, (45) so that Eq. 38 gives v k + (k2 2 τ 2 )v k = 0, (46) An exact solution to Eq. 46 is v k = α e ikτ 2k (1 i kτ ) + β eikτ 2k (1 + i kτ ) (47) for free paramters α and β. However, imposing the boundary conditions, Eqs. 41 and 44, sets α = 1, β = 0, so that v k = e ikτ 2k (1 i kτ ) (48) From this, we can compute the power spectrum of the field ˆψ k a 1ˆv k, ˆψ k (τ) ˆψ k (τ) = (2π) 3 δ(k + k ) v k(τ) 2 Plugging in the result of Eq. 48, we find a 2. (49) ˆψ k (τ) ˆψ k (τ) = (2π) 3 δ(k + k ) H2 2k 3 (1 + k2 τ 2 ) (50) Outside the horizon, kτ 0, so that ˆψ k (τ) ˆψ k (τ) (2π) 3 δ(k + k ) H2 2k 3 (51) At the time of horizon crossing t, a(t )H(t ) = k. Evaluating Eq. 51 gives us the power spectrum for R = Ḣ ψ = Ḣ v φ φ a, R k R k = (2π) 3 δ(k + k ) H2 (t ) 2k 3 H 2 (t ) φ 2 (t ) (52) Plugging this into Eqs. 23 and 24 gives a final result of 2 R(k) = H2 (t ) (2π) 2 H 2 (t ) φ 2 (t ) (53) R remains approximately constant outside the horizon, so knowing the spectrum at the time of horizon crossing t tells us the spectrum of a particular fluctuation until inflation ends and it re-enters the horizon. 10

12 A very similar calculation applies to the case of tensor perturbations. Expanding the Einstein-Hilbert action to second-order in the tensor fluctuations yields S (2) = 1 dτd 3 xa 2 [(h 8 ij) 2 ( l h ij ) 2 ] (54) We can define the Fourier transform, d 3 k h ij = (2π) 3 s=+, ɛ s ij(k)h s k (τ)eik x (55) where the sum ranges over the two polarization modes, ɛ ii = k i ɛ ij = 0, and ɛ s ij (k)ɛs ij (k) = 2δ ss. We further define the canonically normalized field, v s k a 2 hs k (56) With this, the action of Eq. 54 becomes S (2) = ] 1 dτd 3 k [(v sk ) 2 (k 2 a 2 a )(vs k )2 s=+, (57) This action is essentially identical to the one in Eq. 36, except for the sum over polarizations. Each polarization, therefore, can be treated as renormalized, massless field in de Sitter space. Thus, we can simply apply the result in the case of scalar perturbations to the fields, yielding h s k = 2ψs k, ψs k = v k a (58) ( ) 2 t = 2 2 h (k) = 2 H(t ) 2 (59) π with t the time of horizon crossing. Putting together Eqs. 53 and 59, we find that the ratio of tensor to scalar perturbations is given by r 2 t 2 R = 16ε(t ) (60) where ε d ln H dn and N is the number of e-folds of inflation that have occurred. From Eq. 25, we have Now, n s 1 = d ln 2 s dn d ln 2 s dn ln H = 2d dn dn d ln k d ln ε dn (61) (62) 11

13 And, some minor manipulations give d ln φ 2 d ln H dn = 2ε, d ln ε = 2(ε η). (63) dn where η dn. Further, since we are concerned with the time of horizon crossing t, we have k = ah, or ln k = N + ln H, so that dn d ln k = ( ) d ln k 1 ( = 1 + d ln H ) ε (64) dn dn Putting this all together gives, to first order in ε and η And similarly, n s 1 = 2η(t ) 4ε(t ) (65) n t = 2ε(t ) (66) A perfectly scale-invariant theory would have n s = 1, n t = 0. Experimental constraints on n s, r, and α s are shown in Table 1 below [9]. Model Parameter 5-year WMAP WMAP + CMB WMAP + ACBAR08 Power-law n s ± ± ± Running n s ± ± ± α s ± ± ± Tensor n s ± ± ± r < 0.43 < 0.36 < 0.40 Tensor + Running n s ± ± ± r < 0.58 < 0.64 < 0.54 α s ± ± ± Table 1: Experimental constraints on the primordial power spectra. 2.3 Non-Gaussianity The power spectrum 2 R (k) is related to the Fourier transform of the two-point correlation function of R according to R k (t)r k (t) = (2π) 3 δ(k + k )P R (k), 2 R(k) = k3 2π 2 P R(k) (67) Non-Gaussian contributions to higher-order correlation functions would manifest themselves in the CMB, so they serve as an important tool for determining the interaction terms of the inflation Lagrangian. Of particular interest is the bispectrum, related to the the Fourier transform of the three-point function according to R k1 R k2 R k3 = (2π) 3 δ(k 1 + k 2 + k 3 )B R (k 1, k 2, k 3 ) (68) 12

14 in analogy to Eq. 67. B R is symmetric in its arguments and satisfies B R (λk 1, λk 2, λk 3 ) = λ 6 B R (k 1, k 2, k 3 ). (69) We may encode non-gaussian corrections to R in a parameter fnl loc according to R(x) = R g (x) f NL loc [ Rg (x) 2 R g (x) 2 ]. (70) This is known as local non-gaussianity. The corresponding bispectrum is given by B R (k 1, k 2, k 3 ) = 6 5 f loc NL [P R (k 1 )P R (k 1 ) + P R (k 2 )P R (k 3 ) + P R (k 3 )P R (k 1 )] For a scale-invariant spectrum, in which P R (k) = Ak 3, this becomes B R (k 1, k 2, k 3 ) = 6 [ ] 5 f NL loc A 2 1 (k 1 k 2 ) (k 2 k 3 ) (k 3 k 1 ) 3 (71) (72) We suppose without loss of generality that k 3 k 2 k 1. Then, the bispectrum reaches a supremum as k 3 0 and k 1 k 2. In this limit, we have lim B R (k 1, k 2, k 3 ) = 12 k 3 k 2 k 1 5 f NLP loc R (k 1 )P R (k 3 ) (73) The most recent constraints placed on fnl loc by study of the CMB are, at a 95% confidence level [14], 4 < fnl loc < 80. (74) Study of the large-scale structure of the universe, in particular luminous red galaxies, gives a 95% CI of [1] 29 < f loc NL < 70. (75) Thus, there is not yet strong evidence of departure from Gaussianity. The Planck satellite and CMBPol are projected to narrow down the range to a standard deviation of about 5. In the case of single field inflation, Creminelli and Zaldarriaga showed that [20] lim R k 1 R k2 R k3 = (2π) 3 δ(k 1 + k 2 + k 3 )(1 n s )P R (k 1 )P R (k 3 ) (76) k 3 0 For perfectly scale-invariant perturbations, n s = 1, so the right hand side vanishes. The presence of non-gaussianity in this limit would therefore eliminate the possibility of single field inflation. Regardless of the size of k 3, however, single field slow-roll inflation would always lead to negligible non-gaussianity. To explain any observed Gaussianity, either models with higher-order derivatives or models with multiple fields are often considered. In the following section, we discuss the latter of these options in depth. 13

15 3 Multiple Field Inflation Thus far, we have considered only the case of a single inflaton field, φ. However, there are several reasons to consider inflation scenarios with multiple fields. First of all, the use of multiple fields makes it is easier to construct models without unnatural or arbitrary parameter constraints. Secondly, most high energy theories (e.g. supergravity in string theory) involve multiple scalar fields, and it seems unlikely that only one of these fields would contribute to inflation. The fields involved in multiple field inflation must behave as scalars under spacetime transformations, as a nonzero global background vector or spinor field would violate Lorentz invariance [2]. We consider the case of scalar fields φ = (φ a ), which represent coordinates on a field manifold with metric G. Supersymmetry requires that the manifold must be a Kähler manifold, (i.e. a complex manifold with a real scalar function K(z, z) encoding the local geometric properties of the manifold). For the purposes of this discussion, we do not restrict ourselves to the case of a Kähler manifold, but instead examine the more general case of real scalar fields φ a, as any complex scalar field can be split into two real scalar fields. The Langrangian density for such a theory that is quadratic in the derivatives can be written L = ( g 1 ) 2 µ φ µ φ V (φ) (77) where g is the spacetime metric and the dot product is taken over the field metric G. This leads to an equation of motion of g µν (D µ δ λ ν Γ λ µν) λ φ G 1 T V = 0 (78) which simplifies, using the FRW spacetime metric, to the equation for the motion of the scalar fields φ, D t φ + 3H φ + G 1 T V = 0 (79) The system is considered to be in the slow-roll regime if D t φ 3H φ and 1 2 φ 2 V. Following the treatment of [2], we introduce a basis {e n } on the field manifold using the Gram-Schmidt orthogonalization process. We define e 1 to the be the unit vector pointing in the direction of the field velocity φ. Next, we define e 2 to be the unit vector that points along the component of acceleration D φ perpendicular to e 1. Setting φ (1) φ, φ (n) D (n 1) t φ, n 2 (80) we recursively define e n to point along the direction of the part of φ (n) that is orthogonal to the subspace spanned by {e 1,..., e n 1 }. We further define P n and 14

16 Pn to be the projection operators onto {e n } and the orthogonal complement of {e 1,..., e n }, respectively. We set P P 1 and P P1. In terms of these operators, we note that any vector v can be decomposed as v = v + v, and any matrix A can be decomposed as A = A + A + A + A, with v = P v, A = P AP, and so on. The slow-roll parameters ɛ and η generalize to the case of multiple field inflation as well. We may define ɛ = D th H 2, η(n) = D(n 1) t φ H n 1 φ (81) We note that and for convenience set e 1 = η (1) (82) η η (2), ξ η (3). (83) We can decompose η = η + η and ξ = ξ + ξ and set With these parameters, Eqs. 5 and 79 become H = 1 3 ξ n e n (84) V ( ɛ ) 1/2 (85) and φ + 2 G 1 2 ɛ T V = V ɛ η + 3 ɛe 1 (86) ɛ The system is in the slow-roll regime provided ɛ, ɛ η, ɛ η 1 (87) We note further that in the case of single field inflation, η = η and η = 0. η, hence, will turn out to be important in determining whether or not multiple field effects are important in regards to quantum perturbations. Eq. 86 is a system of coupled differential equations which may be nontrivial to solve. However, if ɛ 1, then we can neglect the right-hand side of Eq. 86 to first order in the slow-roll parameters, and to leading order we find ɛ = 1 V 2 2 V 2 (88) η ɛ = T r(m 2 ) V (89) 15

17 η = T r[(m 2 ) (M 2 ) ] where M 2 G 1 T V. For later reference, we define a matrix Z by V (90) Z mn (Z T ) mn 1 H e m D t e n (91) Z is antisymmetric, first order in the slow-roll parameters, and has vanishing entries except on the superdiagonal and subdiagonal. In matrix form, Z is given by 0 η 0 η 0 ξ 3 Z = 0. η ξ 3 0 η... (92) With this, we are in a position to examine the perturbations arising from multiple field models of inflation. 3.1 Multiple Field Perturbations As in the case of single field inflation, we may treat the quantum fluctuations as linear perturbations to the inflaton, so that φ(η, x) = φ(η) + δφ(η, x). (93) It will turn out that conformal time η t a is the most appropriate choice of time coordinate for this calculation. Similarly, for the metric tensor, we may write g µν (η, x) = ḡ µν (η) + δg µν (η, x). (94) These may be decomposed, analogously to Eq. 16, into scalar, vector, and tensor components via ds 2 = a 2 (1+2Φ)dη 2 +2a 2 ( i B S i )dηdx i +a 2 [(1 2Ψ)δ ij +2 ij E+2 (i F j) +h ij ]dx i dx j where (95) i S i = 0, i F i = 0, i h ij = h i i = 0 (96) Once again, to linear order, the scalar, vector, and tensor perturbations decouple and may be handled independently. We may use our gauge freedom to set B = E = F = 0 (97) We begin by examining the scalar perturbations. The linearized Einstein equation yields the following expression for the energy-momentum tensor: δt 0 0 = 1 a 2 (φ D η δφ φ 2 Φ + a 2 V δφ) (98) 16

18 δt 0 i = 1 a 2 i(φ δφ) (99) δt i j = 1 a 2 δi j(φ D η δφ φ 2 Φ + a 2 V δφ) (100) Furthermore, through calculation of the Christoffel symbols, we find δg 0 0 = 2 a 2 ( Ψ + 3H(Ψ + HΦ)) (101) δg 0 i = 2 a 2 i(ψ + HΦ)) (102) δg i j = 2 [Ψ a ] 2 (Φ Ψ) + H(Φ + 2Ψ ) + (2H + H 2 )(Φ + Ψ) δ ij ij (Φ Ψ) (103) where indicates a derivative with respect to η, is the Laplacian, and H a a. Since Tj i = 0 for i j, we see by taking Fourier transforms that k i k j (Φ Ψ) = 0 (104) and hence, Φ = Ψ in general. Furthermore, combining the diagonal components of the Eistein equation gives Φ + 6HΦ + 2(H + 2H 2 )Φ Φ = a 2 ( V δφ) (105) while the (0,i) components give Φ + HΦ = 1 2 φ δφ = 1 2 φ δφ (106) where we have made the decomposition δφ = δφ e 1 + δφ as discussed previously. From here, one can calculate the equation of motion for scalar perturbations (c.f. [2]), yielding a final result of ( ) Dη 2 + 2HD η + a 2 M 2 a 2 R( φ, φ) δφ = 4Φ φ 2a 2 ΦG 1 T V (107) where R is the curvature tensor. Using the background equation, Eq. 79, along with the constraint equation 106, we find ( ) 1 a 2 V δφ = 2(Φ + HΦ) φ (D ηφ ) e 1 + 2H + (D η φ ) δφ. (108) Inserting this into Eq. 105 yields Φ + 2 (H φ ) φ Φ + 2 (H H φ ) φ Φ Φ = (D η φ ) δφ. (109) Using the definitions of the slow-roll functions and basis vectors, this can be rewritten as, Φ 2H η Φ 2H 2 ( ɛ + η )Φ Φ = 2H 2 η η e 2 δφ. (110) 17

19 As in the case of single field inflation, we introduce Mukhanov variables, u a φ Φ = Φ, q a (δφ + ΦH ) φ (111) 2 ɛh Plugging the definition of q into Eq. 107 gives 1 [ Dη 2 H 2 (2 ɛ) + a 2 M 2 a 2 R( a φ, φ) ] q+2φ [ D η φ + 2Hφ + a 2 G 1 T V ] Φ [D 2 H ηφ 2H 2 (1 + ɛ)φ + a 2 M 2 φ a 2 R( φ, φ)φ ] φ [ ] Φ 2H η Φ 2H 2 ( ɛ + η )Φ Φ H ( ) 1 2 H D ηφ + (2 + ɛ + η )φ [Φ + H(1 + ɛ)φ] = 0 (112) The terms contained within the second and fourth pairs of brackets vanish by Eq. 79, reparametrized with respect to conformal time. Furthermore, differentiating Eq. 79 once more gives D 2 ηφ 2H 2 (1 + η)φ + a 2 (M 2 R( φ, φ))φ = 0 (113) which is precisely the term in the third pair of brackets, so that this term also vanishes. Eq. 106 can be rewritten in terms of q as Φ + H(1 + ɛ)φ = 1 2 φ q a (114) These results may be amalgamated to give a homogenous equation for the Fourier mode k of q D 2 ηq k + (k 2 + H 2 Ω)q k = 0 (115) where Ω 1 H 2 (M 2 R( φ, φ)) ( (2 ɛ)i 2 ɛ (3 + ɛ)p + e 1 η + η ). (116) by Hence, Before determining the equation of motion for u, we define the quantity θ θ ( θ = H 1 + ɛ + η ), θ H a φ = 1 a 2 ɛ (117) θ θ = H2 ( 2 ɛ + η + 2( η ) 2 ( η ) 2 ξ ) (118) Substituting the definitions of u and q into Eq. 110, we arrive at ( ) u k + k 2 θ θ u k = H η e 2 q k 18 (119)

20 Eqs. 115 and 119 show that distinct Fourier modes of u and q decouple, so that we may henceforth consider a generic mode k and eliminate the subscripts. Definining q n e n q, we may write Differentiating once yields u θ θ = 1 2 q 1 (120) u θ θ u = 1 2 ) (q 1 + θ θ q 1 (121) Using Eq. 119, this becomes k 2 u = H η q ) (q 1 + θ θ q 1 (122) We now quantize q, which will allow us to relate q to u via Eq In terms of the {e n } basis, our Lagrangian is L = 1 2 (q + HZq) T (q + HZq) 1 2 qt (k 2 + H 2 Ω)q (123) This gives rise to the Hamiltonian H = 1 2 (π HZq)T (π HZq) qt ( k 2 + H 2 (Ω + Z 2 ) ) q (124) where π L is the canonical momentum. Rather than cluttering our commutation relations with indices, we use arbitrary vectors α and β that are inde- q T pendent of q and π, and hence we may write the (time-independent) canonical commutation relations [α T ˆq, β T ˆq] = [α T ˆπ, β T ˆπ] = 0, [α T ˆq, β T ˆπ] = iα T β. (125) The Hamilton equations for this system are q = H π T = π HZq, π = H q T = (k2 + H 2 Ω)q HZq (126) We let Q and Π denote complex matrix solutions to the Hamilton equations such that q = Qa 0 + c.c., π = Πa 0 + c.c. (127) for any fixed complex vector a 0 and c.c. denotes the complex conjugate. Substituting into the Hamilton equations and combining gives a second order differential equation for Q, ( Q + 2HZQ + k 2 + H 2 [Ω + 1 ) H Z + (1 ɛ)z + Z 2 ] Q = 0 (128) 19

21 We can eliminate the first order time derivative by making a change of variables Q R Q, where R represents a rotation and We thus have Q + (k 2 + H 2 Ω) Q = 0, R + HZR = 0, Ω R 1 ΩR (129) Π = R Q (130) We may expand ˆq and ˆπ in terms of creation and annihilation operators to give ˆq = Qâ + Q â = R Qâ + R Q â, ˆπ = Πâ + Π â (131) The creation and annihilation operators obey the commutation relations [α T â, β T â] = [α T â, β T â ] = 0, [α T â, β T â ] = α T β (132) Comparing with the commutation relations in Eq. 125 gives Q Q T QQ T = Π Π T ΠΠ T = 0, Q Π T QΠ T = ii (133) We assume that the system is initially in the vacuum state 0 (with â 0 = 0) and free of particle production. This means that the Hamiltonian must be devoid of terms with â T â and (â ) T â, so that (Π HZQ) T (Π HZQ) + Q T ( k 2 + H 2 (Ω Z T Z) ) Q = 0. (134) This leads to the Hamiltonian operator, Ĥ = 1 2âT [ (Π HZQ) T (Π HZQ) + Q T ( k 2 + H 2 (Ω Z T Z) ) Q ] â + c.c. (135) The solutions to Eqs. 133 and 134 at the beginning of inflation may be parametrized by a unitary matrix U, provided k 2 is the dominant scale, giving Q i = 1 2k U, k Π i = i 2 U (136) Working in the Heisenberg picture, we denote the initial operators by â, â and those at a later time η by ˆb(η), ˆb (η). We my write ˆq(η) and ˆπ(η) either in terms of the â or ˆb operators according to Q(η)â + c.c. = ˆq(η) = 1 2k V ˆb (η) + c.c. (137) k Π(η)â + c.c. = ˆπ(η) = i V ˆb (η) + c.c. (138) 2 where V is a unitary matrix. These equations can be solved to give ˆb(η), ˆb (η) as functions of â, â, ˆb(η) = (V ) 1 i 2k ( ) (ikq(η) Π(η))â + (ikq (η) Π (η))â 20 (139)

22 ˆb (η) = V 1 i 2k ( ) (ikq(η) + Π(η))â + (ikq (η) + Π (η))â (140) We may calculate the expectation value of the operator ˆb ˆb in the vacuum state, finding a 0 (ˆb ) T (η)ˆb(η) 0 a = 1 2k (ikq(η) Π(η)) T (ikq(η) Π(η)). (141) This gives the number of particles created during the transistion that occurs when a perturbation exits the horizon. An order-of-magnitude calculation shows that this is a huge number, so that the system indeed changes from quantum-mechanical to classical in nature. We now solve the system of equations to first order in the slow-roll parameters. There are three limiting cases to consider: the sub-horizon region (H k), the superhorizon region (H k), and the transition region (H k). We examine each in turn. In the sub-horizon region, we may solve Eq. 129 explicitly to find Hence, Q(η) = 1 2k R(η)e ik(η η i U (142) Q (η)q T (η) = 1 2k I (143) Thus, the sub-horizon region has no impact on the correlators at the end of inflation. In the super-horizon region, we may use u to compute the correlator of the Newtonian potential Φ. Neglecting the k 2 in Eq. 119, the exact solution of u is given by η dη u k (η) = u P k + C k θ + D k θ η H θ 2 (η (144) ) where η dη η u P k = θ η H θ 2 dη Hθ η q 2k (145) η H We may rewrite u P using only the background quantities discussed previously. To first-order, Eq. 128 becomes Q + H( δ (1 ɛ)i + Z)Q = 0 (146) where We define Q SR by δ 1 3 ( ) Ω 2I + (1 ɛ) 2 (147) Q SR a H ɛh a ɛ QQ 1 H (148) 21

23 where Q H is defined to be the leading-order asymptotic expression for Q evaluated at η H and η H is given by H(η H ) = k. Eq. 146 yields [ ] Q SR + H δ + (2 ɛ + η + Z) Q SR = 0 (149) with Q SR (η H ) = I. This is solved by integration, [ η ( Q SR (η) = exp dη H δ (2 ɛ + η )I Z) ] (150) η H Q and u may be related, after some calculation, using Eq. first-order result for D k of Eq. 144, 121 to give a where ˆD k = 1 2 2k θ He T 1 E H â + c.c. (151) E H (1 ɛ H )I + (2 γ ln 2)δ H (152) with γ the Euler constant. Combining Eqs. 111, 144, 145, and 151 gives a first-order expression for the gravitational potential operator ˆΦ in the super-horizon region near the end of inflation, ˆΦ k (t) = H ( ) H 2k 3/2 ɛ a(t H, t)e T 1 + Ũ p T (t) E H â k + c.c. (153) H where the C k term has been neglected, as it decays to 0, and A(t H, t) H a η η H dη a 2 ɛ = 1 H a t t H dt a (154) Ũ T P (t) H a t t H dt a ɛu T P (t ), U T P (t) 2 ɛ H t dt H η a H t H ɛ a et 2 QQ 1 H (155) Plugging in Eq. 150 yields, to first-order, UP T = 2 t dt H η e T 2 exp t H [ t dt H t H ( δ (2 ɛ + η )I Z) ] (156) In the transition region, we define η by H 2 (η ) ɛ 3/2 k 2 to be the time at which the sub-horizon limit no longer holds. We take the initial conditions of Q(η ) = 1 2k I, Q (η ) = i k 2 I, R(η ) = I (157) 22

24 We further define z kη. With this, the solution of Eq. 129 for R in the transition region becomes ( z R(z) = z ) 1 1 ɛ H Z H (158) To first order, we have Ω = Ω H in the transition region, so that ( Ω = R 1 (z)ω H R(z) = Ω H R(z) = Ω H + 3[δ H, Z H ] ln z + 3 ) z H 4 ln ɛ H (159) Defining Ω 1 R H ΩR H, Eq. 129 can be rewritten with 2 z Q + Q ν2 H 1 4 z 2 Q = 0 (160) ν 2 H = 9 4 I + 3δ H (161) The solution of this equation is given in terms of a Hankel function, π Q(z) = zh (1) ν 4k H (z) (162) with ν H = 3 2 I + δ H. To leading order, this is Q = eiπδ H i 2k E H ( z z H ) I δh (163) Finally, we look at the vector and tensor perturbations from Eq. 95. Scalar fields cannot generate vector perturbations, so we only need consider tensor perturbations, as in the case of single field inflation. In regards to the tensor perturbations, the Einstein equation gives h ij + 2Hh ij h ij = 0 (164) h ij has only two independent components, as it is symmetric, transverse, and traceless. Thus, h ij can be decomposed as h ij (η, x) = s=+, 2 a ψ s(η, x)e s ij (165) where e s ij, s = +, are two constant, symmetric, transverse, traceless polarization tensors normalized according to e s ij eijr = δ sr. Taking the Fourier transform yields the equation of motion ψ s k + (k 2 a a ) ψ sk = 0 (166) 23

25 The Lagrangian associated with this equation of motion is L = 1 2 (ψ s k ) 2 1 ) (k 2 a (ψ sk ) 2 (167) 2 a With this, we may quantize the tensor perturbations according to ĥ ijk (η) = 2 ( ) a es ij ψ sk (η)â s k + c.c. s=+, (168) where the creation and annihilation operators satisfy the commutation relations [â sk, â r k ] = δ rsδ(k k ) (169) We henceforth drop the subscripts s and k, since distinct Fourier modes and polarizations decouple. As in the case of scalar perturbations, we impose the constraint that the Hamiltonian does not contain any particle creation or annihilation terms initially. Combining this with the canonical commutation relations gives These may be solved, yielding ψ ψ ψψ = i, (ψ ) 2 + k 2 ψ 2 = 0 (170) ψ i = 1 2k e iα, k ψ i = i 2 eiα (171) where α is a phase factor and is irrelevant to the correlator. Thus, without loss of generality, we may set α = 0. Now, a treatment similar to the case of scalar perturbations gives, to first-order for an interval around η H, πz eiπ ɛ ( ) H z 1 ɛh ψ(z) = 4k H(1) 3/2+ ɛ H (z) = i 2k [1 + ɛ H(1 γ ln 2)] (172) z H where again, z = kη and γ is the Euler constant. In the super-horizon region, we have z dz ψ(z) = Ca + Da z H a 2 (173) The first-order approximation for a near η = η H gives ) 1 ɛh (174) ( z a(z) = a H z H Using Eq. 172, we find an expression for C of Eq. 173, 1 C = [1 + ɛ H (1 γ ln 2)] (175) 2kaH The D term in Eq. 173 decays quickly and may be neglected. Hence we arrive at 2 ĥ ijk = k 3/2 H H [1 + ɛ H (1 γ ln 2)] e s ijâ s k + c.c. (176) s=+, In comparison with Eq. 153, Eq. 176 shows that the tensor perturbations do not contain the large 1/ ɛ H factor out front, so the tensor perturbations will generally be smaller than the scalar perturbations. 24

26 3.2 Multiple Field Non-Gaussianity Perturbations generally come from one of two sources: initial density fluctuations or stresses in matter [23]. The former lead to adiabatic perturbations, which are present in the case of single field inflation. The latter lead to isocurvature perturbations, which may be found in models of multiple field inflation. It has been shown that the CMB bispectrum is influenced not only by the adiabatic and isocurvature bispectrum but also by cross-correlation terms, which leave a distinct signature on the CMB [18]. Of particular importance for studying perturbations in a multiple field inflationary scenario is the curvature parameter η discussed in the first part of 3. Using the number of e-folds N as a time parameter, η = [( N H H )φ ] φ (177) where indicates a derivative with respect to N and the in the numerator indicates that we are taking the component perpendicular to the tangent vector φ. A brief calculation shows that this reduces to η = φ 2 φ 2 (φ φ )2 φ 2. (178) This measure of acceleration perpendicular to φ plays an important role in determinining the shape of the quantum perturbations. In particular, if η is consistently small throughout the evolution of a trajectory, then little bending of the trajectory will occur, and perpendicular fluctuations δφ will be relatively unimportant in determining the variation in the number of e-folds produced (c.f. Figure 4 (left)). However, if η is sizeable near the end of inflation, then perpendicular fluctuations occurring earlier will lead to significant variation in the length of the inflationary phase (c.f. Figure 4 (right)), which will imprint in the CMB as non-gaussianities. 25

27 Figure 4: The effect of δφ in a trajectory with negligible η (left) and significant η (right). δφ plays a significant roll in determining the length of the inflationary phase only if η is large. Non-Gaussianity in two-field models of inflation are explored in detail in [10]. We present here an abridged version of their calculation in the case for which the Lagrangian can be separated, that is, L = 1 2 µ φ µ φ µ χ µ χ + U(φ) + V (χ) (179) for inflaton fields φ and χ. Varying the action gives the equations of motion, φ + 3H φ + U φ = 0, χ + 3H χ + V χ = 0. (180) The background Friedmann equations then become, H 2 = 1 ( φ ) 2 χ2 + U + V (181) Ḣ = 1 2 ( φ2 + χ 2) (182) The slow-roll parameters ɛ and η can once again be generalized in a convenient way by defining and ɛ φ 1 2 ( ) 2 Uφ, ɛ χ 1 W 2 η φ U φφ W, V χχ W ( ) 2 Vχ (183) W (184) where W U + V is the total potential term in the Lagrangian so that ɛ = ɛ φ + ɛ χ. 26

28 Given a time t = t c after the time of Hubble crossing t, we define the number of e-folds of inflation elapsed in the usual way, In this case, we can rewrite N as N = N(t c, t ) = tc t tc t H dt. (185) U tc dφ U φ t V V χ dχ. (186) There is also an integral of motion C defined by dφ dχ C +. (187) U φ V χ The nonlinear parameter f NL (introduced in 2.3) can be rewritten in this scenario according to N,I N,J N,IJ 6 5 f NL = P I,J=φ,χ (1 + f) + 2P R ( (188) N,K) 2 2 K where P R is the power spectrum of the curvature perturbation R, P H(t )2, 4π 2 and f = f(k 1, k 2, k 3 ) 1 F 2. (189) i k3 i The first term in the expression for f NL has been found experimentally to be small. The second term, N,I N,J N,IJ 6 5 f (4) NL I,J=φ,χ ( K N 2,K) 2, (190) must be large if f NL is to be larger than unity. The following calculation gives an explicit expression for f (4) NL in terms of the slow-roll parameters. To begin, we note from Eqs. 186 and 187 that [( ) ( ) ( ] dn = U χc V Uφ φ V φc U χ c φ U φ dφ )c [( ( ) ( ] (191) + V Vχ ) χc V χ V φc U χ c χ U φ dχ )c and dφ c = dφc dc dχ c = dχc dc ( ) C φ dφ + C χ dχ ( ) (192) C φ dφ + C χ dχ 27

29 where C = 1, φ (U φ ) C = 1. (193) χ (V χ ) We enforce the condition that ρ = 0 everywhere along the hypersurface corresponding to t = t c, so that during slow-roll inflation, U(φ c ) + V (χ c ) = const. (194) or, equivalently, dφ c dc (U φ) c + dχ c dc (V χ) c = 0 (195) Eqs. 187 and 195 combine to give [ ( )] 1 dφ c dc = (U φ ) 1 c + 1 (U φ ) 2 c (V χ) 2 c [ ( )] 1 (196) dχ c dc = (V χ ) 1 c (U φ + 1 ) 2 c (V χ) 2 c And, plugging Eqs. 193 and 49 into Eq. 192 yields φ c φ χ c φ ( ) = Wc ɛ χ c ɛ φ 1/2 c W ɛc, ɛ φ ) 1/2, = Wc W ɛ φ c ɛc ( ɛ χ c ɛ φ φ c χ χ c χ ( ) = Wc ɛ χ c ɛ φ 1/2 c W ɛc ɛ χ = Wc W ɛ χ c ɛc ( ɛ φ c ɛ χ ) 1/2 (197) The derivatives with respect to the initial fields now read N = 1 U + Z c, φ 2ɛ φ W N = 1 V Z c (198) χ 2ɛ χ W where Z c (V c ɛ φ c U c ɛ χ c )/ɛ c. Differentiating Eq. 198 again gives expressions for the second derivatives of N, 2 N φ 2 = 1 ηφ U + Z c 2ɛ φ + W 1 W 2ɛ φ Z c φ (199) 2 N χ 2 2 N φ χ = = 1 ηχ V Z c 1 Z 2ɛ χ c (200) W W 2ɛ χ χ 1 Z c 1 Z = c (201) W 2ɛ φ χ W 2ɛ χ φ Eq. 197 can be used to show ɛ φ Z c = ɛ χ Z c = 2W A (202) φ χ 28

30 where A W c 2 ɛ φ c ɛ χ c W 2 ɛ c ) (1 ηss c ɛ c and η ss (ɛ χ η φ + ɛ φ η χ )/ɛ. Plugging this all into Eq. 190 gives (203) 6 u2 (1 ηφ 5 f (4) NL = 2 ɛ φ u) + v2 2ɛ φ ɛ χ (1 ηχ 2ɛ χ v) + ( u ɛ φ ( u2 ɛ φ v ɛ χ )2 A + v2 ɛ χ )2 (204) where u U + Z c W, v V Z c W. (205) Of the three terms in the sum of Eq. 204, the first two are O(ε) (where ε denotes both ɛ and η), while the the third is O(A). Hence, in the slow-roll regime where η and ɛ are small, only the third term is significant. 4 D-brane Inflation We have seen that it is possible to construct models of inflation using an inflaton field φ. Although these models solve the horizon problem and the problem of small-scale inhomogeneities, they do not quite settle the matter. The inflaton itself is, as far as we know, physically unobservable. Thus, the task ahead is to find a theoretical framework for inflation that will someday lead to experimentally verifiable results. D-brane inflation provides one possible framework based on concepts of string theory. In string theory, D-branes are objects on which the endpoints of open strings lie. Open strings (i.e. strings whose endpoints are disconnected) are endowed with one of two boundary conditions: Neumann or Dirichlet. In Neumann boundary conditions, the endpoints are free to move about. In Dirichlet boundary conditions, the endpoints are fixed throughout. These Dirichlet boundary conditions will arise if the string endpoints are attached to some physical object. These objects are called D-branes and are further characterized by their dimensionality [3]. A D2-brane, for instance, is a twodimensional surface on which the endpoints of strings are constrained to move. Of particular interest in the case of inflation are D3-branes, which can be thought of as point-like particles living in a 6+1 dimensional space. The three dimensions of the D3-brane correspond to the three observable dimensions of space, while the six spatial dimensions housing the D3-brane correspond to the six unobservable dimensions postulated by superstring theory. The action governing the evolution of a Dp-brane is given by [13] S p = T p d p+1 ξe Φ [ det(g ab + B ab + 2πα F ab )] 1/2. (206) 29

31 where G ab (ξ) = Xµ ξ a X ν ξ b G µν(x(ξ)), B ab (ξ) = Xµ ξ a X ν ξ b B µν(x(ξ)). (207) Here, T p is a constant called the D-brane tension, α is a constant called the Regge slope, ξ a, a = 0,..., p are coordinates on the brane, X(ξ) are fields on the brane, G µν is symmetric, B µν is anti-symmetric, and F ab corresponds to the gauge field A a (ξ). T p and α also satisfy the recursion relation, T p = T p 1 /2πα 1/2. (208) We may boost and rotate to set F ab to block-diagonal form and then use the relation dx 1 [1 + ( 1 X 2 ) 2 ] 1/2 = dx 1 [1 + (2πα F 12 ) 2 ] 1/2 (209) to eliminate the F ab term from the action while introducing a product of factors. The action in this form is referred to as the Born-Infeld action and is relevant for DBI inflation, which is discussed in 4.3. In one model of D-brane inflation, the inflaton φ is taken as the separation between a D3-brane and an anti-d3-brane in a 6-dimensional Calabi-Yau manifold. A weak Coulomb interaction between the two branes leads to a relatively flat potential, which gives rise to slow-roll inflation [11]. Curvature coupling induces a large inflaton mass, which would prevent significant inflation were it not counteracted by a relatively strong potential. The inflaton feels a potential from various sources, causing it to meander about the space before eventually meeting the anti-d3-brane, at which point annihilation occurs and inflation ends. When the brane and anti-brane annhilate, strings are produced. The universe transitions from an inflationary phase into a radiation-dominated phase. As the photons become red-shifted and lose energy, the universe transitions into a matter-dominated phase, consistent with cosmological observation [17]. In another possible scenario of D-brane inflation, studied here and in [22], the anti-d3-brane is absent from the tip of the conifold. In this model, inflation ends not with annihilation, but rather when the inflaton φ exits the slow-roll regime and begins to oscillate near the tip of the conifold. Friction removes energy from the inflaton, and its frequency eventually reaches a resonanant frequency, at which point particle creation occurs and the universe transitions into a radiation-dominated phase. Moduli stabilization creates corrections to the inflaton potential. In [5], it was found that for a certain fine-tuned range of compactification parameters, these corrections largely cancel, allowing for a relatively flat inflaton potential. In [17], explicit trajectories were computed far from the tip of the conifold (c.f. Appendix B). It was found that the probability of N e-folds of inflation follows a power law, P (N) N α, where α 3. In particular, the probability of observing 60 or more e-folds of inflation in optimal regions of the parameter 30

32 space was about 1 in 1000, though more rigorous experimental constraints limit the probability of observing a realistic inflationary scenario to 1 in 10, 000. In [22], trajectories were computed in the near-tip region using the exact geometry of the conifold, discussed in Appendix B. Results of these simulations are given in 4.4. Additional information on D-brane inflation can be found in [5, 7, 6, 17, 19]. 4.1 Evolution of the Scale Factor and Change of Variables The inflaton of D-Brane inflation is a six-dimensional object, φ = (φ 1,..., φ 6 ). The generalization of the kinetic energy and potential energy to 6-d is as we would expect, T = 1 2 g ij φ i φj, V V (φ 1,..., φ 6 ) (210) so that the pressure and density equations become ρ = T + V = 1 2 g ij φ i φj + V (φ 1,..., φ 6 ) (211) and the Friedmann equations remain p = T V = 1 2 g ij φ i φj V (φ 1,..., φ 6 ) (212) 3(ȧ a )2 = ρ = T + V (213) 6ä = (ρ + 3p) = 2V 4T (214) a where i, j {1, 2, 3, 4, 5, 6}. To find the evolution of the scale factor a(t), we must first solve the Euler-Lagrange equations for the evolution of φ in the potential V (φ) on the geometry of the conifold. The Lagrangian density is given by L = 1 2 g ij φ i φj V (φ 1,..., φ 6 ) So that the Lagrangian itself is L = a 3 (t)l = a 3 (t) [ ] 1 2 g ij φ i φj V (φ 1,..., φ 6 ) (215) The effect of the factor of a 3 (t) is a damping known as Hubble friction. From here, a(t) is determined by the Friedmann equations. For purposes of numerical calculation, it is useful to introduce a time parameter N such that dn = Hdt = ȧ dt (216) a Under this change of variables, a(n) = da dn, so that a(n) = en. Thus, N(t) is the number of e-folds of inflation that have occured by time t. For the 31

33 remainder of the paper, we will denote derivatives with respect to t by a dot ( ) and derivatives with respect to N with a prime ( ). By changing the integration variable from t to N, we have also made H, rather than a, the parameter of interest, since a is fixed by a(n) = e N. This means that the Friedmann equations must be reformulated. The first Friedmann equation gives, clearly, 3H 2 = ρ = T + V (217) We note that ä a = Ḣ + H2 = H H + H 2 so that the second Friedmann equation becomes 6(H H + H 2 ) = 2V 4T (218) Substituting for H 2 from the first Friedmann equation leads gives So, Finally, since 6(H H + T + V 3 ) = 2V 4T H = T/H (219) T = 1 2 g ij φ i φj = H g ij(φ ) i (φ ) j we define a new function, K, by K 1 2 g ij(φ ) i (φ ) j = T/H 2 (220) So that, in the final analysis, the Friedmann equations with respect to N become V H = (221) 3 K H = KH (222) These two equations, along with the six Euler-Lagrange equations that result from varying the Lagrangian, give eight ordinary differential equations for eight variables, φ 1,...φ 6, H, and H. These can be solved numerically once the shape of the potential and the initial conditions are specified. 32

34 4.2 Potential The potential can generally be divided into four parts [17], V (φ) = V 0 + V C (φ) + V R (φ) + V B (φ) (223) where V 0 represents sources of supersymmetry breaking, V C represents the Coulomb potential between the D3-brane and anti-d3-brane, V R represents the contributions from curvature, and V B is the remainder. We consider the case in which the anti-d3-brane is absent from the tip of the conifold, so that the Coulomb potential V C is absent. With this assumption, the leading order terms to the potential come from bulk effects. Here, the potential is given by V B = T 3 Φ, where Φ is harmonic to first order [22], 2 (0) Φ(1) = 0. (224) At second order Φ is sourced by the harmonic modes of G, 2 (0) Φ(2) = g s 96 Φ(0) + G(0) 2 (225) where g s is the string coupling constant. For the relevant mode, we have [6] ( ) Φ (0) + G(1) = i l fg l m Ω m j k (226) i j k where i, j, k,... = 1, 2, 3 run over z 1, z 2, z 3, the three independent complex coordinates on the conifold. In the near-tip Klebanov-Tseytlin throat region, the dominant mode has f KT (τ, Ψ) = cz a, a = 1, 2, 3, 4. (227) Farther from the tip, in the Klebanov-Strassler throat, we find that the solution f KS (τ, Φ) = 4 c a z a (228) is Harmonic, regular at the tip, and coincides with Eq. 227 in the neartip region. In general, all four of these modes may be turned on, so c a are expected to be arbitrary complex numbers. The solution to Eqs. 225 and 226 then becomes Φ = g s i f j 32 f. (229) This gives rise to, V B (τ, Ψ) = g ( ) ( ) s 32 T 3 g i j z i z i c i c 4 c j c 4. z 4 z 4 (230) Here g i j is the inverse Kähler metric on the deformed conifold, and i and j run over only 1,2,3 (not 4). The constants c i may be taken such that c 1 = c 2 = 0, 33 a=1