Following the density perturbations through a bounce with AdS/CFT Correspondence

A bounce universe model, known as the coupled-scalar-tachyon bounce (CSTB) universe, has been shown to solve the Horizon, Flatness and Homogeneity problems as well as the Big Bang Singularity problem. Furthermore a scale invariant spectrum of primordial density perturbations generated from the phase of pre-bounce contraction is shown to be stable against time evolution. In this work we study the detailed dynamics of the bounce and its imprints on the scale invariance of the spectrum. The dynamics of the gravitational interactions near the bounce point may be strongly coupled as the spatial curvature becomes big. There is no a prior reason to expect the spectral index of the primordial perturbations of matter density can be preserved. By encoding the bounce dynamics holographically onto the dynamics of dual Yang-Mills system while the latter is weakly coupled, via the AdS/CFT correspondence, we can safely evolve the spectrum of the cosmic perturbations with full control. In this way we can compare the post-bounce spectrum with the pre-bounce one: in the CSTB model we explicitly show that the spectrum of primordial density perturbations generated in the contraction phase preserves its stability as well as scale invariance throughout the bounce process.


Introduction
A universe with a bounce process (see for example [1,2] for two most updated reviews) is a possible solution of the cosmic singularity problem [3,4] in the standard cosmology within the inflation paradigm [5,6]. The bounce universe postulates that a phase of matter-dominated contraction precedes the big bang during which the scale factor of the universe reaches a non-zero minimal value. There have been many attempts to extend the standard cosmology beyond the Big Bang, the most notable first effort being the Pre-Big-Bang cosmology [7,8], and then the Ekpyrotic cosmology [9]. A breakthrough was due to the key observation by D. Wands [10] in which he pointed out a scale invariant spectrum of primordial density perturbations can be generated during a matter dominated contraction. Although the spectrum generated in his naive model was later proved to be unstable, it opens a new chapter in cosmological modeling of the early universe.
Building on many pioneering works to utilize AdS/CFT correspondence [11] in cosmological studies , in this paper, we use the correspondence to study how a spectrum generated during the contraction phase can evolve through the bounce in a particular bounce universe model. We are going to conduct our investigations on the coupled scalar tachyon bounce (CSTB) model [40] constructed earlier, which is based on the D-brane and anti-D-brane dynamics in Type IIB string theory.
The CSTB model has been shown to solve the singularity, horizon and flatness problem [41]; it can produce a scale invariant as well as stable spectrum of primordial density perturbations [42,43]. Furthermore predictions testable using dark matter direct detections have been extracted (for a wide class of bounce models) [44][45][46][47]. An out-of-thermal-equilibrium dynamics of matter production in the the bounce universe makes the bounce scenario very distinct [44] from the standard model of cosmology in which thermal equilibrium dynamics washes out early universe information. A short review of the key ideas can be found [45,48]. We would like to further corroborate our model by investigating the fluctuations across the bounce.
The fact that CSTB is a string-inspired model and the bounce point may be strongly gravitationally-coupled prompts us to use the AdS/CFT correspondence [11] to study the evolution of the primordial density fluctuations in a Type IIB string background. We take the bulk spacetime metric to be a time dependent AdS 5 × S 5 with its four dimensional part being a FLRW (Friedmann-Lematre-Robertson-Walker) spacetime. In [49] a recipe is provided to map the bulk dynamic, fro and back, to the boundary. In this work we improve on their recipe by finding a solution to the dilaton dynamic equation of motion with more realistic Type IIB fields configurations.
According to the AdS/CFT correspondence, which is a strong/weak duality, i.e. when the bulk fields are strongly coupled the boundary is described by a weakly coupled field theory, and vice versa, the bulk fields have dual-operators prescribed by the boundary theory. The dilaton field is related to the square of the gauge field strength, and the gauge coupling of the boundary theory is determined by the vev of the dilaton φ. Therefore the first step is to find a time dependent solution of dilaton equation which, in turn, determines the dynamics of gauge fields on the boundary. Consequently when the boundary gauge field theory becomes weakly coupled during the contraction, we can map the bulk fluctuations onto the boundary and observe its evolution through the bounce.
The bounce process in the bulk could be potentially violent or highly singular in nature -although this is not the case for the CSTB model which enjoys a string theoretical completion at high energy and has a minimum radius -the gauge fields on the boundary, however, evolves most smoothly.
After the bounce, we map the evolved fluctuations -using again the AdS/CFT dictionary -back to the bulk as the gravitational dynamics return to a weakly coupled state. The operation described above hence allows comparing the post-bounce spectrum with the pre-bounce spectrum and checking whether the scale invariance of the spectrum is respected by the bounce process.
The paper is organized as follows. In section 2 we present a time dependent dilaton solution with nonzero Ramond-Ramond charges in Type IIB string theory. We describe the cosmic background in which CSTB model can be constructed. In section 3, we use the results of the previous section to solve the equation of motion of the boundary gauge fields near bounce point, and match the solutions at different evolutionary phases; and finally check whether the spectrum is altered during the bounce. In section 4, we summarize our findings, discuss a potential caveat and remedies. We conclude with outlook on further studies with alternative solutions.

A time dependent dilaton solution to Type IIB supergravity
First of all we would like to find a solution of the dilaton in Type IIB supergravity with nonzero Ramond-Ramond potentials [50]. The CSTB cosmos is a string cosmological model that can be embedded into an exact string background with appropriate time dependence. The time dependence is necessary for cosmological studies. Altogether we need to generalize the AdS/CFT correspondence to incorporate time dependence in order to study how the spectrum of primordial density perturbations, generated before the bounce, is affected by the bounce dynamics.
The low energy effective theory of Type IIB string is given by [51]: where the field strengths are defined as The p-forms fields arise from the Ramond-Ramond sector and couple to D-branes of various dimensions; whereas φ is the dilaton field we are interested in. Note that there is an added constraint which should be imposed on the solution that the 5-form field strength: F 5 is self-dual, F 5 = * F 5 . The field equations derived from the action (2.1) should be consistent with, but do not imply, it.
The deformed AdS 5 × S 5 spacetime metric we will be working on is, where dΩ 2 5 being the metric of the unit S 5 and a(t) being the scale factor of the 4dimensional FLRW universe and L the AdS radius.
We need to make some sensible assumptions to solve this formidable array of equations. A common formula for the self-dual F 5 is [53]: as we would like r to be a constant. Note that we can assume that B 2 and C 2 live on the AdS 5 part and dC 4 lives on the S 5 part.
In the orthonormal basis, we can express them as: (2.10) where i = 1, 2, 3, {dy µ } are the orthonormal basis, i.e. dy µ = √ g µµ dx µ . To lessen the influence and difficulty caused by forms we assume that the coefficients f 1 · · ·, g 1 · · · are at most linear in y 0 and y 4 , then we can get the expression for the AdS 5 part of F 5 : (2.12) We will take these f i to be constant and g j to be linear in y 0 and y 4 , then the constant, r, mentioned above becomes: where h 1 = ∂g 1 ∂y 4 + ∂g 3 ∂y 0 , h 2 = ∂g 2 ∂y 0 and h 3 = ∂g 2 ∂y 4 . Since f 4 and g 4 won't appear in the equations of forms, we'll take them to be zero. Therefore Note here we consider the axion field C 0 to be linear in time, y 0 , and in, y 4 , the spatial direction transverse to our 4-dimensional universe inside the AdS 5 . So far what we do is to represent the forms by the coeffcients f i and h j . In addition, we solve for (2.4) which is the Euler-Lagrange equation of φ: Putting Equations (2.17) to (2.20) into (2.3), we get the equations of φ when µν = 00, ii, 44 (with the metric (2.2)): These are quadratic first-order partial differential equations of φ. Normally they are hard to solve, however, if we view them as linear equations ofφ 2 , φ 2 ,z and φ 2 ,i , life becomes much easier: We would like φ to be spatially homogeneous, i.e. φ 2 ,i = 0; we can take such an approximation of e 2φ that the right side of equation (2.25) equals to zero, then Substituting it into (2.24) we obtaiṅ The constant captures the effects of form fields C 2 , B 2 , C 0 on the dilaton, φ. In the next section we will see that it isφ that matters. Note that we should not solve (2.27) directly since it is actually a result of an approximation instead of an exact solution. If we want exact solutions to Equations (2.24) to (2.26) then the second partial derivatives of φ should satisfy a constraint equation, ∂φ ∂z = ∂φ,z ∂t .

The evolution of the gauge-field fluctuations
The boundary gauge theory is described by N = 4 SYM theory, we will follow the notations in [49]. The Yang-Mills coupling is determined by the dilaton by g 2 YM = e φ . The boundary theory is strongly coupled in the far past. As the universe contracts, the bulk gravity theory becomes more and more strongly coupled. Before we approach the bounce point, we map the fluctuations onto the boundary as it becomes weakly coupled at this point. We let the gauge field evolve well after the bounce ends and the bulk returns to a weakly coupled state.
After rescaling and gauge fixing, the equations of motion for the Fourier modes of the gauge fields A becomes [30]: Let us now zoom into the cosmic dynamics near the bounce point and consider the three phases of universe evolution in the CSTB model [40]: Smooth bounce : a = cosh(Ht), −t 1 ≤ t ≤ t 1 (3.4) Inflation : a = e Ht , t 1 < t < t f ; (3.5) where t 1 is the time when inflation starts and t f is when it ends. The bounce process is symmetric about t = 0. The mapping happens at deflation and inflation phases while the bulk becomes strongly coupled. We solve the equations of motion in each phases: where β ≡ √ −k 2 − M .

Smooth bounce:
Taking the first order of t we obtain . (3.9) 3. Inflation: In this case everything is same as deflation except the value of t. Therefore (3.10) We denote ±t 0 as the time of mapping and for the sake of convenience, we set the two modes of A k to have the same amplitudes after the first mapping, i.e.
We assume the arguments of both Airy functions to be small and that the (−P ) 1 3 t term dominates. Then we can asymptotically expand the Airy functions to first power in q ≡ −k 2 −Q−P t (−P ) 2 3 : Now we can match A k and its derivative at the end of deflation and at the beginning of inflation, which we denote as −t 1 and t 1 respectively. Matching A k yields: (3.14) where q 1 ≡ q(−t 1 ) and q 2 ≡ q(t 1 ).
Matching˙ A k yields: Solving Equations (3.14) to (3.17), we get We are interested in small wave number limit compared to time scales above, i.e. kt 1 1. In the typical inflationary process Ht f ∼ 10 2 and Ht 1 ∼ 10 −2 . Combining these two facts, we can assume In addition, Q ∼ M , we have . (3.21) Similar argument goes for q 2 as well. From (3.20) and (3.21) we can see that I 1 , I 2 and J 1 , J 2 are independent of k when kt 1 1. From (3.11) we know Putting these two into (3.18) and (3.19), we obtain, after the second matching, A k , has the form both G 1 and G 2 being independent of k. All in all we can conclude that after the bounce the spectral index is not altered.
The reconstruction of the bulk data from boundary data is elucidated in [49], we do not reproduce the arguments here. The punch line is that the k−dependence of the bulk fluctuations is completely determined by the k−dependence of the gauge field fluctuations, A k (t), which implies, in turn, that the evolution of the gauge fluctuations will preserve scale invariance across the bounce.

Conclusion and discussion
In this paper we used the AdS/CFT correspondence to show that, when kt 1 1 or in the long-wave limits of the dual gauge fields on the boundary, the spectral index of the dilaton fluctuations is not altered as the universe described by the CSTB model undergoes a contraction prior to an expansion. The first step was to find a timedependent solution of the dialton in a type IIB supergravity on a time-dependent AdS 5 × S 5 . We generalize the previous proposal of [49] in which a certain behavior of the dilaton φ was assumed. We then utilize the dilaton solution to solve for the dynamics of gauge fields living on the boundary of the AdS 5 . We study the gauge fields near the bounce point and matched their behavior at the transitional points in the different phases of cosmic evolution. The combined profile of gauge field evolution is smooth across the bounce point. The bounce process merely alters the amplitudes of the modes in the density perturbations, and it affects them in the same manner. Therefore it cannot alter the intrinsic scale dependence in the spectrum of matter perturbations generated during the phase of cosmic contraction prior to the bounce. Nevertheless as we can see from (3.18) and (3.19), as k becomes larger and larger, i.e. if we do not take long wavelength approximation, the dependence in k begins to show up in the spectrum, the implications of which are under investigation.
A clarifying remark is perhaps needed here to distinguish the two kinds of k-modes, and their time dependence, involved in the above discussion. The CST bounce universe undergoes a deflation, before the bounce point, accompanied by horizon crossings with modes with different k's crossing at different times. This makes each k-mode in the primordial density perturbations pick up an implicit time dependence: only after this implicit time dependence is carefully taken into account can the spectrum has no overall time dependence. This is the key to the stability analysis on the spectrum generated from the CSTB model [42,43]. But this commonly concerned k-dependence in the primordial spectrum is not what we have discussed so far in this work. The k-modes in (3.1) are the k-modes of the gauge fields living on the boundary of the AdS. They are involved in the mapping procedure and merely encode the bulk dynamics holographically at some particular points on the boundary. Therefore they cannot inject or remove any time dependence in the primordial spectrum. Once the dynamics are mapped onto the boundary, there is no more horizon crossing, the gauge fields evolve under their own equations of motion.
We have made several assumptions and approximations throughout the analysis. Different solutions of the dilaton could be obtained with different ansatz of the Ramond-Ramond field configurations. We have simply chosen the most manageable configuration yet retaining interesting physics. With higher orders of time dependence in the dilaton field we have to expand M 2 YM to the higher order in t when solving (3.1). A systematic study of the field configurations and the corresponding effects on the dilaton field is beyond the scope of this paper. These are nevertheless interesting effects together with higher α effects to the whole analysis, which we hope to address in a future publication.
Another line of research would be to properly set up and study the D-brane and anti-D-brane annihilation process for cosmological modeling. This is the basis for building early universe model from string theory. Going beyond effective field theory approach and beyond kinematic analysis or symmetry arguments can give a more realistic touch to string cosmology. What kind of string compactifications can give rise to a nonsingular universe matching up to the array of precision cosmological observations should be the ultimate question to answer for string cosmologists.