1. Motivation and Introduction

The standard big bang cosmology is hugely successful in describing the evolution of our universe from the time of nucleosynthesis onwards. However, a central assumption is that the universe started out in a hot big bang and in a special state: extrapolating back from current knowledge, we know that early on the universe must have been very flat, homogeneous, and isotropic, with in addition small density perturbations with a nearly scale-invariant spectrum and a nearly gaussian distribution. Hence, the “initial” state of the universe was far from random, and its specialness prompts us to try and explain it via a dynamical mechanism.

The most studied such mechanism is the model of inflation, which assumes that there was a phase of rapid, accelerated expansion preceding the hot big bang; for a comprehensive review see [1]. Such a phase can be modeled by having a scalar field (the “inflaton”) with a positive and suitably flat potential. Inflation has the property of flattening the universe, so that, if it lasts long enough, the flatness of the “initial” state can be explained. Moreover, inflation possesses the remarkable byproduct that it generates nearly scale-invariant spectra of scalar and tensor perturbations by amplifying quantum fluctuations. The predicted scalar perturbations are in excellent agreement with current observations, but the tensor perturbations have yet to be observed—their discovery would be a strong indication for the correctness of the inflationary picture. However, inflation also presents a number of conceptual problems: for example, even though the inflationary phase is supposed to erase all memory of initial conditions, this is not really the case. In order for inflation to start in a given patch of space, that patch must be reasonably smooth over several Planck lengths and the inflaton field must have a small initial velocity (the “patch” and “overshoot” problems, respectively; see e.g., [1, 2]). Also, it has been realized not long ago that inflation is geodesically incomplete towards the past, which means that the predictions of the theory depend on the specification of data on a space-like initial hypersurface [3]. In other words, inflation requires its own initial conditions. Hence, if inflation is correct, it will only form a part of the story. More worrying is the problem of unpredictability, which is associated with the quantum nature and the effectiveness of inflation. Inflation ends when the inflaton field oscillates around a minimum of its potential and “reheats” the universe by decaying into standard model particles. However, for generic initial conditions there will always be regions in which rare but large quantum fluctuations kick the inflaton field back up its potential and keep a fraction of the universe in the inflationary phase. In most of the concrete realizations of inflation, the region that keeps inflating expands so fast that it quickly dominates the overall volume of the universe. Hence, inflation never ends, and the global picture of this process of “eternal inflation” is that of an empty de Sitter universe punctured by an infinite number of small pockets where inflation has ended (at a random time) [4]. Because inflation ends at a random moment in these pocket universes, the pockets might have become sufficiently flattened or not, they might have acquired scale-invariant perturbations or not. Without a measure which would determine the relative likelihood of the various pockets, it becomes difficult to know exactly what eternal inflation predicts. These problems do not mean that the idea of inflation is wrong, but, if inflation continues to be supported by observations, they will have to be addressed. In the meantime, the seriousness of these open problems means that it is worthwhile considering alternative models for the early universe in parallel.

The present paper deals with one such model in particular, namely, the ekpyrotic model and its extension, the cyclic universe; for a comprehensive overview see [5]. In this model, the inflationary phase is replaced by the ekpyrotic phase, which is a slowly contracting phase preceding the big bang. The ekpyrotic phase can be modeled by having a scalar field with a negative and steep potential. As described in detail below, it also manages to flatten a given region of the universe and generates nearly scale-invariant scalar perturbations but no observable tensor fluctuations. At the linear level, the scalar fluctuations are virtually indistinguishable from the perturbations produced by inflation, but at higher orders the predictions differ. Since primordial gravitational waves might turn out to be rather elusive to measure over the coming years, the most promising way of distinguishing between alternative models of the early universe is therefore by studying these higher-order, nongaussian signatures.

There is a simply, intuitive argument for why the predictions regarding higher-order corrections to the linear perturbations should differ for models of inflation and ekpyrosis. For a scalar field fluctuation the semiclassical probability density is roughly given by where is the euclidean action [6]. Since inflation requires a very flat potential, the inflaton is an almost free field. For a free field, the action is quadratic in the field, and hence the probability distribution is simply a gaussian distribution. For an exact gaussian distribution the 3-point function vanishes, and hence for inflation, where the field is almost free, we would expect the 3-point function to be nonzero, but small. For ekpyrosis, on the other hand, the potential is steep, and hence the scalar field is necessarily significantly self-coupled. This has the consequence that ekpyrotic models generally predict significant levels of nongaussianity. In fact, some versions of ekpyrosis are already on the borderline of being ruled out by observations, while the best-motivated models predict values that are measurable by near-future experiments. Thus, the nongaussian predictions are crucial in assessing the viability of various cosmological models, and promise to significantly enhance our understanding of the physics of the early universe.

The plan of this paper is to start with a brief summary of the main ideas behind ekpyrotic and cyclic models of the universe. We will then discuss in some detail the generation of linear cosmological perturbations (a good understanding of the linear perturbations greatly facilitates an understanding of the higher-order ones), before turning to the main subject of the paper, namely, the nongaussian corrections to these linear perturbations. We will conclude with a discussion of the nongaussian predictions and in particular their observability and relation to current observational limits, as well as the consequences of a potential detection.

2. Ekpyrotic and Cyclic Cosmology

The ekpyrotic phase is the cornerstone of ekpyrotic and cyclic models of the universe: it is a conjectured, slowly contracting phase preceding the big bang, and it resolves the standard cosmological puzzles [7, 8]. The main feature of ekpyrosis is that during this phase the equation of state is very large (here and denote the average pressure and energy density of the universe). Let us briefly explore the most direct consequences of such an ultrastiff equation of state. Consider a Friedmann-Robertson-Walker (FRW) metric ( I will mostly use natural units and .) where denotes the scale factor of the universe and for an open, flat, or closed universe, respectively. If the universe is filled with a number of fluids interacting only via gravity and with energy densities and constant equations of state then the equations of continuity (where dots denote derivatives with respect to time ) imply that they will evolve according to The Einstein equations for this system contain a constraint equation, better known as the Friedmann equation, which involves the Hubble parameter : The ’s are constants giving the energy densities at scale factor of the various constituents of the universe: we consider the universe to be composed of nonrelativistic matter (subscript ), radiation (), and the energy density associated with anisotropies in the curvature of the universe (). In addition, we consider there to be ekpyrotic (scalar) matter, denoted by the subscript , and, as usual, there is a contribution due to the average curvature of space.

As the universe contracts, components whose energy density scales with a higher negative power of the scale factor will successively come to dominate, first matter, then radiation, then anisotropies, and eventually, since by assumption, the ekpyrotic matter. This means that the relative energy densities in curvature and anisotropies, for example, become smaller and smaller, the longer the ekpyrotic contracting phase lasts. In other words, if ekpyrosis lasts long enough, the flatness problem is solved. We will make this statement quantitative below. Strictly speaking, for the flatness problem to be solved, all we need is a matter component with In the next section, we will see that for realistic ekpyrotic models, typically In passing, we should also point out that there is no horizon problem in ekpyrotic and cyclic models, as there is plenty of time before the big bang for different parts of our currently observable universe to have been in causal contact with each other.

But what form of matter can have the large equation of state that we require? A simple way to model the ekpyrotic matter is to have a scalar field with a steep and negative potential A concrete example is provided by the negative exponential where and are constants—see Figure 1. In the context of string theory, such scalar fields appear very naturally, and the ekpyrotic potential can then correspond to an attractive force between branes—this picture will be briefly described below.

Figure 1: The potential during ekpyrosis is negative and steeply falling; it can be modeled by the exponential form

Given an explicit form of the potential, such as (6), we can solve for the evolution of the universe. In fact it is straightforward to generalize the treatment to having many scalars with potentials Then, in a flat FRW background and neglecting other matter components, the equations of motion become where with no summation implied. If all the fields have negative exponential potentials and if for all then the Einstein-scalar equations admit the scaling solution Thus, we have a very slowly contracting universe with (constant) equation of state We are using a coordinate system in which the big crunch occurs at in other words, the time coordinate is negative during the ekpyrotic phase. Here, the parameter corresponds to the fast-roll parameter and is typically of ; its definition is identical with that in inflation, where its value is typically of and where, correspondingly, it is called the slow-roll parameter.

Using this explicit solution, we can get an idea for how long the ekpyrotic phase has to last in order for the flatness problem to be solved. Quantitatively, the problem can be formulated as follows: dividing the Friedmann equation (5) by we can see that the fractional energy density stored in the average curvature of the universe is given by At the present time, observations imply that this quantity is smaller than in magnitude [9]. If we assume a radiation-dominated universe, which is a good approximation for this calculation, then , and hence, if we extrapolate back to the Planck time, the fractional energy density in curvature must have been smaller than an incredibly small number. However, from (8), we can see that during the ekpyrotic phase the scale factor remains almost constant, while the Hubble parameter Hence grows by a factor of as long as where the subscripts and refer to the beginning and the end of the ekpyrotic phase, respectively. As will be discussed in the next section, we need in order to obtain the observed amplitude of cosmological perturbations, so that we need This is the minimum time the ekpyrotic phase has to last in order to solve the flatness problem. Cosmologically speaking, this is a very short time, attesting to the effectiveness of the ekpyrotic phase.

Before discussing the cosmological perturbations produced during the ekpyrotic phase, it is useful to provide a quick overview of how the ekpyrotic phase might fit into a more complete cosmological model. The crucial ingredient in any such model is the proposed mechanism for how the ekpyrotic contracting phase (with ) and the subsequent radiation-dominated expanding phases (with ) should link up. The Einstein equations provide the relation All forms of matter that are currently known to exist obey the null energy condition which implies and which thus precludes a smooth transition between a contracting and an expanding universe. This leaves two possibilities for achieving such a transition: either the NEC is violated during the transition, or the transition is classically singular.

In new ekpyrotic models [10–12], a smooth reversal from contraction to expansion is achieved by adding a further matter component to the universe which can violate the NEC. The particular example that these models consider is the so-called ghost condensate, which corresponds to the gravitational equivalent of a Higgs phase [13]. It is not clear yet whether or not the ghost condensate can be obtained from a fundamental theory such as string theory [14] (in the more restricted framework of quantum field theory it seems impossible to construct a stable ghost condensate model [15]); however, it is interesting that string theory contains many objects (orientifolds, negative-tension branes) which do violate the NEC. Of course, simply adding such a component is not enough: it must become relevant as the universe contracts, and vanish again as the universe expands. The simplest way to achieve this is by assuming that the ghost condensate itself also plays the role of the ekpyrotic matter, and that after the transition to expansion, it decays into ordinary matter fields. This scenario requires the ghost condensate to possess both a particular form for its kinetic term and a particular potential; for details regarding possible realizations see [10, 12].

The cyclic model of the universe [16, 17] is based on the braneworld picture of the universe, in which space time is effectively 5-dimensional, but with one dimension not extending indefinitely but being a line segment; see Figure 2. The endpoints of this line segment (orbifold) are two -dimensional boundary branes. In the full string theory setup, there is in addition a 6-dimensional internal manifold at each point in the 5-dimensional space time, for a total of 11 dimensions [18]. This description of the universe stems from string theory and in particular the duality, known as Hořava-Witten theory [19], between 11-dimensional supergravity and the heterotic string theory. All matter and forces, except for gravity, are localized on the branes, while gravity can propagate in the whole space time. Our universe, as we see it, is identified with one of the boundary branes and, as long as the branes are far apart, can interact with the other brane only via gravity. The cyclic model assumes that there is an attractive force between the two branes, which causes the branes to approach each other. This force is modeled by a potential of the form shown in Figure 3. Note that the potential incorporates an ekpyrotic part. From the higher-dimensional point of view, the ekpyrotic phase has the rather nonintuitive property that it flattens the branes to a very high degree. Eventually the two branes collide and bounce off each other. It is this collision that, from the point of view of someone living on one of the branes, looks like the big bang. Classically, the collision is singular, as the orbifold dimension shrinks to zero size. The collision is slightly inelastic and produces matter and radiation on the branes, where the standard cosmological evolution now takes place. However, due to quantum fluctuations, the branes are slightly rippled and do not collide everywhere at exactly the same time. In some places, the branes collide slightly earlier, which means that the universe has a little bit more time to expand and cool. In other places, the collision takes place slightly later, and those regions remain a little hotter. This provides a heuristic picture of the way temperature fluctuations are naturally produced within the model. Shortly after the branes have separated, the distance between the boundary branes gets almost stabilized, but the branes start attracting each other again very slightly. This very slight attraction acts as quintessence and is identified with the dark energy observed in the universe. After a long time, and as the branes become closer again, they start attracting each other more strongly so that we get another ekpyrotic phase and eventually another brane collision with the creation of new matter. In this way, a cyclic model of the universe emerges. Before continuing, we should mention the main open issues related to the cyclic model: the first one concerns the potential, which at this point is simply conjectured. It will be important to see if a potential of the required shape can be derived from microphysics. And the second is the brane collision, which so far has been extensively studied at the classical and semiclassical level [20], but a full quantum treatment has remained elusive.

Figure 2: The braneworld picture of our universe. Think of a sandwich: the 5-dimensinonal bulk space time is bounded by two 4-dimensional boundary branes. There is no space “outside” of the sandwich, but the branes can be infinite in all directions perpendicular to the line segment (orbifold). In the M-theory embedding, there are 6 additional internal dimensions at each point of the sandwich.

Figure 3: The potential for the cyclic universe integrates the ekpyrotic part and a quintessence epoch but is irrelevant at the brane collision. A possible form for the potential is with , and tends to unity for and to zero for Reproduced with permission from [8].

In the discussion above, we have mostly focussed on models involving one effective scalar field. However, there are two good reasons to extend the analysis to two or more scalars: first, in embedding the ekpyrotic and cyclic models in M-theory, there are two universal scalars, namely, the radion mode (which determines the distance between the branes) and the volume modulus of the internal 6-dimensional manifold [21]. There can be many more scalar fields (such as the shape moduli of the internal space), but we always must consider these two universal scalars. And secondly, as we will see in the next section, it is much more natural to generate a nearly scale-invariant spectrum of curvature perturbations (in agreement with observations) in models with two scalars than in models with only one. However, multifield ekpyrotic models present some qualitatively new features, which we discuss briefly here.

The 4-dimensional effective action can be obtained as the low-energy limit of Hoava-Witten theory, where and are related by a field redefinition to the radion and the internal volume modulus [22]. We are assuming that during the ekpyrotic phase, both fields feel an ekpyrotic-type potential, for example, Then it is much more natural to discuss the dynamics in terms of the new variables and pointing transverse and perpendicular to the field velocity, respectively [23, 24]; they are defined, up to unimportant additive constants which we will fix below, via with It is also useful to define the angle of the trajectory in field space via [25] In terms of these new variables, the potential can be re-expressed as where for exact exponentials of the form (17), one has and However, in the absence of a microphysical derivation of the potential, we will simply take and express all results in terms of See also Figure 4 for an illustration of the potential. The ekpyrotic scaling solution becomes with the angle being constant. The solution corresponds to motion along a ridge in the potential, as is evident from the figure. Hence, in contrast to the single field case, the multifield ekpyrotic background evolution is unstable to small perturbations [26, 27]. This implies that the trajectory must be localized near the ridge with extreme precision at the beginning of the ekpyrotic phase, the condition being that the field should stray no more than a value of (at best) in Planck units from the ridge at the beginning of ekpyrosis [11]. Thus, at first sight, it looks as if the multifield ekpyrotic phase has not managed to solve the problem of initial conditions. However, there currently exist two approaches addressing this issue: the authors of [11] considered the existence of a “pre-ekpyrotic” phase during which the potential is curved upwards and during which the trajectory is localized. Meanwhile, in the context of the cyclic universe, there is a natural resolution of the issue of initial conditions, not involving any new ingredients of the model: indeed, the multifield cyclic universe selects those regions that happen to correspond to trajectories sufficiently close to the ridge, in the sense that these regions are vastly amplified over the course of one cycle due to the phases of radiation, matter, and dark energy domination (note that the ekpyrotic phase shrinks the universe by a negligible amount). At the same time, the regions corresponding to trajectories not sufficiently close to the ridge (this would include the vast majority of trajectories) do not undergo a full ekpyrotic phase, and after these regions undergo chaotic mixmaster behavior close to the big crunch, they simply collapse (presumably they will end up forming black holes) and stop both growing and cycling. In this way the global structure of the universe becomes of the phoenix type, in which vast habitable regions are interspersed with small collapsed ones. The important point is that the habitable regions, which are the only regions of interest to us here, automatically correspond to the regions that had the right “initial conditions” at the beginning of their preceding ekpyrotic phase. This is discussed in detail in [28]; see also the essay [29].

Figure 4: After a rotation in field space, the two-field ekpyrotic potential can be viewed as composed of an ekpyrotic direction () and a transverse tachyonic direction (). The ekpyrotic scaling solution corresponds to motion along the ridge of the potential. Perturbations along the direction of the trajectory are adiabatic/curvature perturbations while perturbations transverse to the trajectory are entropy/isocurvature perturbations.

3. Linear Perturbations

3.1. Single Scalar Field

In the last section, we have dealt with the classical evolution during the ekpyrotic phase. We will now add quantum fluctuations, and we will see that, just as in inflation, the quantum fluctuations get amplified into classical density perturbations. Hence, on top of resolving the standard cosmological puzzles, the ekpyrotic phase can also be the source of the primordial temperature fluctuations whose imprint is seen in maps of the cosmic microwave background, provided that the amplitude and spectrum of the fluctuations match observations.

From the study of inflationary models, we have developed the intuition that quantum fluctuations that get stretched to superhorizon scales turn into classical perturbations, roughly speaking because the fluctuations go out of causal contact with themselves, do not remember locally that they are in fact fluctuations, and end up as local, classical perturbations to the background evolution. In inflation, this effect occurs because the horizon is approximately constant in size while the wavelengths of the quantum modes get stretched exponentially with time (the scale factor of the universe grows exponentially). For ekpyrosis, the scaling solution (8) shows that the scale factor is almost constant, so that the mode wavelengths remain almost constant too. However, the horizon, which is proportional to shrinks rapidly as and hence the modes automatically become of superhorizon size. (Since tensor modes/gravitational waves depend on the evolution of the scale factor alone and since the scale factor shrinks imperceptively slowly during ekpyrosis, there are no substantial gravity waves produced during the ekpyrotic phase (the background space time is almost Minkowski) [30]. In fact, the dominant gravitational waves that are produced from ekpyrosis are those that arise from the backreaction of the scalar fluctuations onto the metric, at second order in perturbation theory [31] .) We will now discuss in some detail what amplitude and spectrum these modes obtain. We will first concentrate on the single field case, before discussing two fields.

Since the scale factor evolves very little during the ekpyrotic phase, one is tempted to simply turn gravity off as a first approximation and to consider the theory consisting only of a scalar field with a steep and negative potential [32]: Then, if we define scalar fluctuations via where denotes the background evolution, the equation of motion for the fluctuations is given by where We then expand the fluctuation field into Fourier modes where the s are the positive frequency mode functions (due to the assumed cosmological symmetries, they depend only on the magnitude ). We proceed to quantize the field by imposing the canonical commutation relations In the process, the ’s have been promoted to (annihilation) operators, and the vacuum state is defined by The mode functions obey the equation of motion which admits the two solutions However, as the modes should asymptote to the Minkowski space-free particle state (note that in that limit (26) reduces to the equation of a simple harmonic oscillator), and this fixes the solution to be Towards the end of the ekpyrotic phase, we have and then the solution can be well approximated by

The quantum fluctuations have a mean, that is, zero, However, the variance which is defined by , does not vanish. It is conventional to write the variance as where denotes a reference scale and is the spectral index.

A related concept in momentum space is the power spectrum defined by It is the Fourier transform of the 2-point correlation function, and we can equivalently define it as where isotropy dictates that only depends on We will find this definition useful later on.

In our case, we have that at late times so that the variance is given by The variance is independent of , and hence we obtain a scale-invariant spectrum for This looks very promising. However, we really must include gravity in our analysis and calculate the spectrum for the curvature perturbation , which is the quantity, that is, measured to have a nearly scale-invariant spectrum of perturbations.

Once we add gravity, it is easiest to perform the calculation in so-called -gauge, where the perturbations in the scalar field are gauged away and all perturbations are expressed via dilatations of the 3-metric where Then, using the background scaling solution (8), the action reduces to an action for which is given by [33] During ekpyrosis, is typically nearly constant. In fact, in the scaling solution used above, we have already made the approximation that is constant, and with this approximation, the equation of motion for resulting from the action above is particularly simple: in Fourier space it is given by If we use conformal time , defined via and the notation the above equation becomes After a further change of variables to and the equation turns into a Bessel equation with since . Hence the solutions are given by the Hankel functions , and with the boundary condition that we want as we obtain the solution (up to a phase) (Useful asymptotic expressions are when and when and for .) At late times, the variance becomes and hence we get a spectral index This spectrum is blue as there is more power on smaller scales, and it is in disagreement with observations [32–34]. Hence, the scale-invariant spectrum of the scalar perturbation in the no-gravity theory did not get transferred to the curvature perturbation A closer analysis reveals that these two perturbations correspond to two physically distinct modes, the former is a time-delay mode to the big crunch, and the latter a local dilatation in space. In a contracting universe, these two modes are distinct, and they do not mix. It is conceivable that they might mix at the big crunch/big bang transition [35, 36], in which case the scale-invariant contribution would be the dominant one on the large scales of interest, but this possibility is still insufficiently understood to make definite predictions. As we will show next, this is also unnecessary, as there is a very natural entropic mechanism which generates scale-invariant curvature perturbations before the big bang, as long as there is more than one scalar field present. (Recently, Khoury and Steinhardt have also pointed out that right at the onset of the single-field ekpyrotic phase, a range of scale-invariant modes can be produced [37]. However, contrary to the cases that we have discussed so far, this adiabatic mechanism requires the universe to already be contracting when the equation of state is still near If viable, this mechanism would produce an interesting nongaussian signal; but as it is currently not known how to incorporate this mechanism into a more complete cosmological model, we will not discuss this mechanism here. See also [38] for the challenges that this scenario must address.) But before continuing, it might be useful to add a few remarks concerning the validity of our approach: indeed, the reader might be worried about the validity of perturbation theory, since the background quantities, such as the Hubble rate, as well as the perturbations themselves blow up as However, as shown in [33], after switching to synchronous gauge, it is straightforward to see that the universe evolves to become closer and closer to the unperturbed background solution, and hence perturbation theory is valid. Also, even though the background quantities blow up as seen from the 4-dimensional viewpoint, in fact in the higher-dimensional colliding branes picture the ekpyrotic phase has the effect of flattening the branes and hence of rendering the curvatures small as the big crunch is approached [20].

3.2. Two Fields: The Entropic Mechanism

As discussed at the end of the last section, it is rather unnatural to consider only a single scalar field in the effective theory, since there are two universal scalars that are always present in a higher-dimensional context: the radion field, determining the distance between the two end-of-the-world branes, and the volume modulus of the internal manifold. But as soon as there is more than one scalar field present, one can have entropy, or isocurvature, perturbations, which are growing mode perturbations in a collapsing universe [39]. Entropy perturbations can source the curvature perturbation, and hence (provided the entropy perturbations acquire a nearly scale-invariant spectrum), nearly scale-invariant curvature perturbations can be generated just before the bounce [26, 40]. These then turn into growing mode perturbations in the ensuing expanding phase.

For the two scalar fields, we will again assume the potential (20), but with the slight generalization that we allow the fast-roll parameter to be slowly varying. There are two gauge-invariant scalar perturbation modes: the entropy perturbation corresponds to perturbations transverse to the background trajectory, see Figure 4, while the adiabatic, or curvature, perturbation is the gauge-invariant quantity expressing perturbations along the background trajectory; see [25, 41] for a detailed exposition. For a straight trajectory (), the linearized equation of motion for is where denotes the second derivative of the potential with respect to In conformal time, and for the rescaled variable we obtain To proceed, we must relate and to the fast-roll parameter and its derivative with respect to the number of e-folds of expansion , where By requiring to vary slowly, what is meant is that we will keep terms in but not higher-order terms such as Then, by differentiating twice, and using as well as one can derive the following expressions, valid to subleading order in : Using in addition that (41) finally reads In analogy with our discussion of the single-field case, this equation can be solved in terms of the Hankel functions, supplemented by the boundary condition of approaching the Minkowski vacuum state in the far past, to yield (up to a phase) At late times , and we obtain implying that at the end of the ekpyrotic phase, the entropy perturbation is given by Following the same steps as in the single field case above, it is straightforward to see that the spectral index of the entropy perturbation is now given by [26] The first term on the right-hand side is a gravitational contribution, which, being positive, tends to make the spectrum blue. The second term is a nongravitational contribution, which tends to make the spectrum red. A simple way to estimate the natural range of is to rewrite the above expression in terms of , the number of e-folds before the end of the ekpyrotic phase (where ): In this expression, measures the equation of state during the ekpyrotic phase, which decreases from a value much greater than unity to a value of order unity in the last e-folds. If we estimate [42], then the spectral tilt is Here we see that the sign of the tilt is sensitive to . For nearly exponential potentials (), the spectral tilt is , slightly blue, because the first term dominates. However, in the cyclic model the steepness of the potential must decrease in order for the ekpyrotic phase to come to an end, and parameterizes these cases. If , the spectral tilt is red. For example, for . These examples represent the range that can be achieved by the entropic mechanism, roughly These are in good agreement with the present observational bounds obtained by the WMAP satellite, which are at the level [9].

Now that we have shown how an approximately scale-invariant spectrum of entropy perturbations may be generated by scalar fields in a contracting universe, we will discuss how these perturbations may be converted to curvature perturbations. Since the entropy perturbations of interest are all of superhorizon scales, we can now restrict our study to large scales only, where spatial gradients can be neglected. On these scales, the evolution equation for the curvature perturbation is given by [25] Hence, as soon as the background trajectory bends (), the entropy perturbations become a source for the curvature perturbations.

There are at least two ways in which such a bending can occur: the first makes use of the instability of the two-field ekpyrotic potential; compare again Figure 4. If the background trajectory strays sufficiently far from the ridge of the potential, the trajectory will turn and fall off one of the steep sides of the potential [10, 23, 24]. The turning of the trajectory then immediately results in the conversion of entropy into curvature perturbations. Since this conversion occurs during the ekpyrotic phase, we will term this process ekpyrotic conversion. It is straightforward to estimate the amplitude of the resulting curvature perturbation (its spectrum will be identical to the spectrum of the entropy perturbations, as (51) is -independent): if we approximate the entropy perturbation as remaining constant during the conversion process and assume a total bending angle of order unity, then the resulting curvature perturbation after conversion will be given by We should mention straight away that the approximations just made will not be good enough in calculating the nongaussian corrections to the linear calculation, but for the present purposes, they will do. Since the fast-roll parameter , we find that

The second way in which a bending of the trajectory can occur is suggested by the embedding of the cyclic model in M-theory [26] and applies when the background field trajectory remains straight throughout the ekpyrotic phase. At the end of the ekpyrotic phase, the potential turns off, and the evolution becomes dominated by the kinetic energy of the two scalar fields. This kinetic phase corresponds to the final approach of the branes in the higher-dimensional picture. During this approach, there is a generic effect that occurs, but that cannot be seen in the 4-dimensional effective theory. The branes that are bound to collide with each other are of opposite tension. Now, it turns out that just before the collision, it always happens that at the location of the negative tension brane, the internal 6-dimensional manifold tries to shrink to zero size [21]. However, just about any type of matter present on the negative tension brane will smoothly cause the internal manifold to grow again [43]. This effect is due to the special properties of gravity on a negative tension object. When this effect is translated back into the effective theory that we have been using all along, the location in field space where the internal manifold reaches zero size is given by the line. This line thus constitutes a boundary to field space. And the presence of matter on the negative tension brane gives rise to an effective repulsive potential in the vicinity of the line. Hence, during the kinetic phase, the background trajectory automatically bends, just before the trajectory shoots off to where the brane collision/big bang occurs. What we have just discussed is a concrete example originating from string theory of how a bend in the trajectory can occur during the kinetic phase. However, more generally our results will apply whatever the microphysical cause of the bending and of the effective repulsive potential; see Figure 5 for an illustration of the general case that we have in mind. Again, it is quite straightforward to estimate the amplitude of the curvature perturbation resulting from this process of kinetic conversion. On large scales, the linearized equation of motion for the entropy perturbation is given by where, incidentally, we have the useful relation Then, during the kinetic phase and away from the repulsive potential, the Einstein equations immediately yield Thus, (54) simplifies to which implies that the entropy perturbation grows logarithmically during the potential-free kinetic phase. We can ignore this insignificant growth. However, the evolution of the entropy perturbation during the process of conversion turns out to be important. We can estimate it by assuming that the trajectory bends with a constant where denotes the duration of the conversion process. We can further assume that the repulsive potential depends only on Then and can all be related to evaluated during the conversion, and it is not difficult to show that this leads to where corresponds to the time halfway through the bending of the trajectory [44]. For the particular example where the cyclic model is embedded in M-theory, we have that and specializing to this example, we have Hence, (54) becomes (where we can neglect the term in ) and thus, during the conversion, the entropy perturbation evolves sinusoidally where denotes the time at which the trajectory starts to bend, and Now we can immediately evaluate the resulting linear curvature perturbation by integrating (51) to get Thus, the amplitude is very similar in magnitude to the value estimated above for the process of ekpyrotic conversion.

Figure 5: After the ekpyrotic phase, the trajectory in scalar field space enters the kinetic phase and bends - this bending is described by the existence of an effective repulsive potential (the potentials are indicated by their contour lines). A trajectory adjacent to the background evolution can be characterized by the entropy perturbation at the end of the ekpyrotic phase, leading to a corresponding off-set or equivalently at the time of bending.

We are now in a position to calculate the variance of the generated curvature perturbation, which, on account of (46), is given by Hence, the amplitude is in agreement with the current WMAP bounds of [9], as long as that is, the minimum of the potential has to be roughly at the grand unified scale for models using kinetic conversion [26]. This scale is also the natural scale of Hořava-Witten theory, and thus it is the scale where one would expect the potential to turn around. Note that for models using ekpyrotic conversion, this result implies that the bending must occur at a specific time, namely, when the potential reaches the grand unified scale. In the latter models, this may or may not also correspond to the bottom of the potential.

Finally, we should state an important assumption that we have been making implicitly up to now, namely, we assumed that the curvature perturbation passes through the big crunch/big bang transition essentially unchanged. The reason for doing so is that the perturbations we are considering are vastly larger than the horizon size around the time of the crunch, and hence, due to causality, it seems reasonable to assume that long-wavelength modes suffer no change—this viewpoint is discussed in much more detail in [33]. In new ekpyrotic models, in which the bounce is smooth and describable entirely within a 4-dimensional effective theory, this assumption certainly holds true. In the case of a classically singular bounce, this remains an assumption subject to possible revision in the future. (In this context, we can also mention the possibility that no conversion of entropy to curvature perturbations might occur before the big crunch, but that this conversion could happen during the phase shortly following the bang through modulated reheating [45]: if massive matter fields are produced copiously at the brane collision and dominate the energy density immediately after the bang and if, furthermore, these fields couple to ordinary matter via a function of , then their decay into ordinary matter will occur at slightly different times depending on the value of In this way, the ordinary matter perturbations would also inherit the entropic perturbation spectrum.)

4. Higher-Order Perturbations and Predictions for Nongaussianity

4.1. Definitions and Local Nongaussianity

Now that we have seen in detail how the ekpyrotic phase generates linear, nearly scale-invariant density perturbations via the entropic mechanism, we can inquire as to whether the higher-order corrections might lead to an observable signal. We will only calculate nongaussian corrections for perturbations generated via the entropic mechanism, because, as discussed in the previous section, this is the only robust and well-understood mechanism to date that generates ekpyrotic perturbations in agreement with observations. As we saw earlier, the linear perturbations are related to observations of the 2-point correlation function. Similarly, quadratic and cubic corrections to these perturbations are related to observations of the 3- and 4-point functions, respectively. For an exactly gaussian probability distribution, all -point functions for which is odd vanish while for even, the -point functions are related to the 2-point function. Thus, the simplest way in which we could detect a departure from exact gaussianity would be due to the presence of a nonvanishing 3-point function.

In momentum space, the 3-point function corresponds to a configuration of 3 momenta, which form a closed triangle due to momentum conservation. Hence, the 3-point function is specified not only by its magnitude on different scales but also by its magnitude for different shapes of the triangle. Or, turning this reasoning around, when we make predictions for nongaussianity, we must predict both the amplitude and the shape of the momentum space triangle that we would like to observe. Let us make all of this more precise now. Earlier, we defined the power spectrum as the Fourier transform of the 2-point function Similarly, the bispectrum, which is the Fourier transform of the 3-point function, is given by the trispectrum, the Fourier transform of the 4-point function, via and so on. The -functions result from momentum conservation, while and are shape functions (for a triangle and a quadrangle, resp.). In the last expression, the subscript indicates that we only need to consider the connected part of the 4-point function, that is, the part that is not captured by products of 2-point functions.

For nongaussianity of the so-called local form, it is useful to define (in real space) the following expansion of the curvature perturbation on uniform energy density surfaces, with being the linear, gaussian part of The factors of are a historical accident; they arose because this type of expansion was first defined for a different variable. In momentum space, is then given by as can be verified straightforwardly by combining (60), (61), and (63). Similarly, the momentum space 4-point function corresponding to nongaussianity of the local form can be expressed as where and parameterize the two relevant shape functions; see, for example, [46] for more details. For cosmological models in which the perturbations originate from the fluctuations of a single field (in our case the entropy field), is directly related to the square of explicitly Concentrating now on the bispectrum, we can see that, since for a scale-invariant spectrum we have This is the typical momentum dependence for local nongaussianity [47], which is also the relevant one for ekpyrotic models, as we will show shortly. The signal is the largest when one of the momenta is very small—this automatically requires the other two momenta to be almost equal, and hence the local form of nongaussianity corresponds to having the largest signal generated for squeezed triangles in momentum space.

It is instructive to calculate explicitly the tree-level 3-point function for the entropy perturbation generated during the ekpyrotic phase, that is, before the conversion to curvature perturbations. (Note that quantum corrections from loop diagrams will be suppressed by factors of .) Maldacena described in [48] how the expectation value for the 3-point function is given by where is the cubic interaction Hamiltonian. In Fourier space, this can be rewritten as [12, 49]