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Advances in Astronomy
Volume 2010 (2010), Article ID 638979, 43 pages
Primordial Non-Gaussianities from Inflation Models
Center for Theoretical Cosmology, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK
Received 15 February 2010; Accepted 9 June 2010
Academic Editor: Sarah Shandera
Copyright © 2010 Xingang Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This is a pedagogical review on primordial non-Gaussianities from inflation models. We introduce formalisms and techniques that are used to compute such quantities. We review different mechanisms which can generate observable large non-Gaussianities during inflation, and distinctive signatures they leave on the non-Gaussian profiles. They are potentially powerful probes to the dynamics of inflation. We also provide a nontechnical and qualitative summary of the main results and underlying physics.
An ambitious goal of modern cosmology is to understand the origin of our Universe all the way to its very beginning. To what extent this can be achieved largely depends on what type of observational data we are able to get. Thanks to many modern experiments, we are really making progress in this direction.
One of the representative experiments is the Wilkinson Microwave Anisotropy Probe (WMAP) satellite [1–6]. It measures the light coming from the last scattering surface about 13.7 billions years ago. This cosmic microwave background (CMB) is emitted at about 379,000 years after the Big Bang, when electrons and protons combine to form neutral hydrogen atoms and photons start to travel freely through the space. Our Universe was very young at that moment and the large scale fluctuations were still developing at linear level. So the CMB actually carries valuable information much earlier than itself, which can potentially tell us about the origin of the Big Bang.
There are two amazing facts about the CMB temperature map. On the one hand, it is extremely isotropic, despite the fact that the causally connected region at the time when CMB formed spans an angle of only about degree on the sky today. On the other hand, we do observe small fluctuations, with .
The inflationary scenario [7–9] naturally solves the above two puzzles. It was proposed nearly thirty years ago to address some of the basic problems of the Big Bang cosmology, namely, why the universe is so homogeneous and isotropic. In this scenario, our universe was once dominated by dark energy and had gone through an accelerated expansion phase, during which a Hubble size patch was stretched by more than 60 e-folds or so. Inhomogeneities and large curvature were stretched away by this inflationary epoch, making our current observable universe very homogenous and flat. In the mean while, the fields that were responsible for and participated in this inflationary phase did have quantum fluctuations. These fluctuations also got stretched and imprinted at superhorizon scales. Later they reentered the horizon and seeded the large scale structures today [10–14].
The inflationary scenario has several generic predictions on the properties of the density perturbations that seed the large scale structures. (i)They are primordial. Namely, they were laid down at superhorizon scales and entering the horizon after the Big Bang. (ii)They are approximately scale-invariant. This is because, during the 60 e-folds, each mode experiences the similar expansion when they are stretched across the horizon. (iii)They are approximately Gaussian. In simplest slow-roll inflation models, the inflaton is freely propagating in the inflationary background at the leading order. This is also found to be true in most of the other models and for different inflationary mechanisms. So the tiny primordial fluctuations can be treated as nearly Gaussian.
The CMB temperature anisotropy is the ideal data that we can use to test these predictions. The obvious first step is to analyze their two-point correlation functions, that is, the power spectrum. All the above predictions are verified to some extent . The presence of the baryon acoustic oscillations proves that the density perturbations are indeed present at the superhorizon scales and reentering the horizon as the horizon expands after the Big Bang. The spectral index, , is very close to one, therefore, the density perturbations are nearly scale invariant. Several generic types of non-Gaussianities are constrained to be less than one thousandth of the leading Gaussian component.
But is this enough?
Experimentally, the amplitude and the scale-dependence of the power spectrum consist of about numbers for WMAP. For the Planck satellite, this will be increased up to about . However, we have about one million pixels in the WMAP temperature map alone, and 60 millions for Planck. So the information that we got so far is highly compressed comparing to what the data could offer in principle. This high compression is only justified if the density perturbations are Gaussian within the ultimate sensitivities of our experiments, so all the properties of the map is determined by the two-point function. Otherwise, we are expecting a lot more information in the non-Gaussian components.
Theoretically, inflation still remains as a paradigm. We do not know what kind of fields are responsible for the inflation. We do not know their Lagrangian. We also would like to distinguish inflation from other alternatives. Being our very first data on quantum gravity, we would like to extract the maximum number of information from the CMB map to understand aspects of the quantum gravity. All these motivate us to go beyond the power spectrum.
To give an analogy, in particle physics, two-point correlation functions of fields describe freely propagating particles in Minkowski spacetime. More interesting objects are their higher-order correlations. Measuring these are the goals of particle colliders. Similarly, the power spectrum here describes the freely propagating particles in the inflationary background. To find out more about their interaction details and break the degeneracies among models, we need higher-order correlation functions, namely, non-Gaussianities. So the role non-Gaussianities play for the very early universe is similar to the role colliders play for particle physics.
With these motivations in mind, in this paper, we explore various mechanisms that can generate potentially observable primordial non-Gaussianities, and are consistent with the current results of power spectrum. We will not take the approach of reviewing models one by one. Rather, we divide them into different categories, such that models in each category share the same physical aspect which leaves a unique fingerprint on primordial non-Gaussianities. On the one hand, if any such non-Gaussianity is observed, we know what we have learned concretely in terms of fundamental physics. On the other hand, explicit forms of non-Gaussianities resulted from this exploration provide important clues on how they should be searched in data. Even if the primordial density perturbations were perfectly Gaussian, to test it, we would still go through these analyses until various well-motivated non-Gaussian forms are properly constrained.
1.1. Road Map
The following is the outline of the paper. For readers who would like to get a quick and qualitative understanding of the main results instead of technical details, we also provide a shortcut after the outline.
In Section 2, we review the essential features of the inflation model and density perturbations.
In Section 3, we review the first-principle in-in formalism and related techniques that will be used to calculate the correlation functions in time-dependent background.
In Section 4, we compute the scalar three-point function in the simplest slow-roll model. We list the essential assumptions that lead to the conclusion that the non-Gaussianities in this model is too small to be observed.
In Section 5, we review aspects of inflation model building, emphasizing various generic problems which suggest that the realistic model may not be the algebraically simplest. We also introduce some terminologies used to describe properties of non-Gaussianities.
Sections 6, 7, and 8 contain the main results of this paper. We study various mechanisms that can lead to large non-Gaussianities, and their distinctive predictions in terms of the non-Gaussian profile.
In Section 9, we give a qualitative summary of the main results in this paper. Before conclusion, we discuss several useful relations among different non-Gaussianities.
Here is a road map for readers who wish to have a non-technical explanation and understanding of our main results. After reading the short review on the inflation model and density perturbations in Section 2, one may read the first and the last paragraph of Section 4 to get an idea of the no-go statement, and then directly proceed to read Section 5. After that, one may jump to Section 9 where the main results are summarized in non-technical terms.
The subject of the primordial non-Gaussianities is a fast-growing one. There exists many nice reviews and books in this and closely related subjects. The introductions to inflation and density perturbations can be found in many textbooks [15–20] and reviews [21–25]. Inflationary model buildings in particle physics, supergravity, and string theory are reviewed in [26–32]. Comprehensive reviews on the developments of theories and observations of primordial non-Gaussianities up to mid 2004 can be found in [33, 34]. Recent comprehensive reviews on theoretical and observational developments on the bispectrum detection in CMB and large-scale structure has appeared in [35, 36]. A recent comprehensive review on non-Gaussianities from the second-order postinflationary evolution of CMB, which acts as contaminations of the primordial non-Gaussianities, has appeared in . A recent review on how primordial non-Gaussianities can be generated in alternatives to inflation can be found in .
2. Inflation and Density Perturbations
In this section, we give a quick review on basic elements of inflation and density perturbations. We consider the simplest slow-roll inflation. The action is The first term is the Einstein-Hilbert action. The second term describes a canonical scalar field coupled to gravity through the metric . This is the inflaton, which stays on the potential and creates the vacuum energy that drives the inflation. We first study the zero-mode background evolution of the spacetime and the inflaton. The background metric is where is the scale factor and is the comoving spatial coordinates. The equations of motion are The first equation determines the Hubble parameter , which is the expansion rate of the universe. The second equation is the continuity condition. The third equation describes the evolution of the inflaton. Only two of them are independent.
The requirement of having at least e-fold of inflation imposes some important conditions. By definition, to have this amount of inflation, the Hubble parameter cannot change much within a Hubble time . This gives the first condition We also require that the parameter does not change much within a Hubble time, In principle, can be close to but kept small. In such a case, grows exponentially with e-folds and the inflation period tends to be shorter. More importantly, such a case will not generate a scale-invariant spectrum, as we will see shortly, thus cannot be responsible for the CMB. The two conditions (6) and (7) are called the slow-roll conditions. Using the background equations of motion, we can see that the slow-roll conditions impose restrictions on the rolling velocity of the inflaton. The first condition (6) implies that So the energy driving the inflation on the right-hand side of (3) is dominated by the potential. Adding the second condition (7) further implies that So the first term in (5) is negligible and the evolution of the zero-mode inflaton is determined by the attractor solution Using (10), the slow-roll conditions can also be written in a form that restricts the shape of the potential, They are related to and by So the shape of the potential has to be rather flat relative to its height. We emphasize that, although in this example several definitions of the slow-roll conditions are all equivalent, the definitions (6) and (7) are more general. In other cases that we will encounter later in this paper, these two conditions are still necessary to ensure a prolonged inflation and generate a scale-invariant spectrum, but the others no longer have to be satisfied. For example, the shape of potential can be steeper, or the inflationary energy can be dominated by the kinetic energy.
Now let us study the perturbations. To keep things simple but main points illustrated, in this section, we will ignore the perturbations in the gravity sector and only perturb the inflaton, We also ignore terms suppressed by the slow-roll parameters, which we often denote collectively as . For example, the mass of the inflaton is , and will be ignored. The quadratic Lagrangian for the perturbation theory is simply and the equation of motion follows: where we have written it in the comoving momentum space, The solution to the differential equation (15), , is called the mode function. It is not difficult to check that To quantize the perturbations according to the canonical commutation relations between and its conjugate momentum , we decompose with the commutation relations One can check that the commutation relations (18) and (21) are equivalent because of (17), given that the constant on the right-hand side of (17) is specified to be . This gives the normalization condition for the mode function.
We now write down the mode function explicitly by solving (15): where we have used the conformal time defined as . The infinite past corresponds to and the infinite future . We also used the relation . This mode function is a superposition of two linearly independent solutions with the normalization condition followed from (17). Consider the limit in which the mode is well within the horizon, that is, its wavelength much shorter than the Hubble length , and consider a time period much shorter than a Hubble time. In these limits, the mode effectively feels the Minkowski spacetime, and the first component in (22) with the positive frequency asymptotes to the vacuum mode of the Minkowski spacetime as we can see from (23). We choose this component as our vacuum choice, and it is usually called the Bunch-Davies state. The annihilation operator annihilates the corresponding Bunch-Davies vacuum, .
The mode function has the following important properties. It is oscillatory within the horizon . As it gets stretched out of the horizon , the amplitude becomes a constant and frozen. Physically this means that, if we look at different comoving patches of the universe that have the superhorizon size, and ignore the shorter wavelength fluctuations, they all evolve classically but with different . This difference makes them arrive at , the location of the end of inflation, at different times. This space-dependent time difference leads to the space-dependent inflationary e-fold difference Again we ignore terms that are suppressed by the slow-roll parameters. This e-fold difference is the conserved quantity after the mode exits the horizon, and remains so until the mode reenters the horizon sometime after the Big Bang. It is the physical quantity that we can measure, for example, by measuring the temperature anisotropy in the CMB, . The information about the primordial inflation is then encoded in the statistical properties of this variable. So we would like to calculate the correlation functions of this quantity. Using (25), (19), (24), and (8), we get the following two-point function: where is defined as the power spectrum and in this case it is The spectrum index is defined to be where the relation is used. If , the spectrum is scale invariant. The current data from CMB tells us that . So as we have mentioned, this requires a small , which is also a value that tends to give more e-folds of inflation.
If this were the end of story, all the primordial density perturbations would be determined by this two-point function and they are Gaussian. The rest of the paper will be devoted to making the above procedure rigorous and to the calculations of higher-order non-Gaussian correlation functions in this and various other models.
3. In-In Formalism and Correlation Functions
In this section, we review the in-in formalism and the related techniques that are used to calculate the correlation functions in time-dependent background. The main procedure is summarized in the last subsection.
3.1. In-In Formalism
We are interested in the correlation functions of superhorizon primordial perturbations generated during inflation. So our goal is to calculate the expectation value of an operator , which is a product in terms of field perturbations and , at the end of inflation. The subscript labels different fields. In inflation models, these fields are, for example, the fluctuations of the scalars and metric and their conjugate momenta. In the Heisenberg picture, where is the end of inflation and is the vacuum state for this interacting theory at the far past .
We start by looking at how the time-dependence in is generated.
The Hamiltonian of the system is a functional of the fields and their conjugate momenta at a fixed time . On the left-hand side of (30), we have suppressed the variable and index which are integrated or summed over. The and satisfy the canonical commutation relations and their evolution is generated by following the equations of motion:
We consider a time-dependent background, and which are c-numbers and commute with everything, and the perturbations, and , The background evolution is determined by the classical equations of motion, The commutation relations (31) become those for the perturbations, We expand the Hamiltonian as where we use to denote terms of quadratic and higher-orders in perturbations.
Using (34), (35), and (36), the equations of motion (32) become So the evolution of the perturbations, and , is generated by . It is straightforward to verify that the solutions for (37) are where satisfies with the condition at an initial time being
To have a systematic scheme to do the perturbation theory, we split into two parts, The is the quadratic kinematic part, which in the perturbation theory will describe the leading evolution of fields. Fields whose evolution are generated by are called the interaction picture fields. We add a superscript “” to label such fields. They satisfy The solutions are where satisfies with
So the idea is to encode the leading kinematic evolution in terms of the interaction picture fields, and calculate the effects of the interaction through the series expansion in terms of powers of . To do this, we rewrite (29) as where Using (39), (44), and (41), we get with The solution to (48) and (49) can be written in the following way, where the operator means that, in each term in the Taylor series expansion of the exponential, the time variables have to be time-ordered. The operator will be used to mean the reversed time-ordering.
In summary, the expectation value (29) is Notice that in all the field perturbations are in the interaction picture.
The perturbation theory is also often done in terms of the Lagrangian formalism. In the following, we show that they are equivalent. In the above, we perform perturbations on the Hamiltonian, and define by perturbing , (here we use to denote the functional derivative) The Hamiltonian is defined by (36). So using the definition together with the classical equations of motions (34) and , the definition (36) for becomes If we perturb the Lagrangian directly, we keep the part of the Lagrangian that is quadratic and higher in perturbations and , The is defined directly as where in the second step (56) has been used. So these two definitions of are equivalent. The Hamiltonian is defined through , Again, using (56), we can see that the two definitions of are equivalent.
3.2. Mode Functions and Vacuum
The Hamiltonian in the above formalism is typically chosen to be the quadratic kinematic terms for field perturbations without mixing, So they describe free fields propagating in the time-dependent background. The , , and are some time-dependent background fields, and they are all positive. The solutions to the equations of motion (42) in momentum space, , are called the mode functions, where denotes the comoving momentum. They satisfy the Wronskian condition Note that we have specified the time-independent constant on the right-hand side of (60) to be for the same reason that we see in Section 2. Namely, we decompose as where the annihilation and creation operators satisfy the following relations, These commutation relations are equivalent to (35) because of (60), but the constant needs to be . This gives the normalization condition for the mode functions.
Being the solutions of the second-order differential equation, generally the mode function is a linear superposition of two independent solutions. So we need to specify the initial condition. For inflation models, as long as the field theory applies, one can always find an early time at which the physical momentum of the mode is much larger than the Hubble parameter and study a time interval much less than a Hubble time. Under these conditions, the equations of motion approach to those in the Minkowski limit, in which the mode function is a linear superposition of two independent plane waves, one with positive frequency and another negative. The ground state in the Minkowski spacetime is the positive one. The mode function which approaches this positive frequency state in the Minkowski limit is called the Bunch-Davies state. In physical coordinates, this limit is proportional to , (for ), where is the physical momentum. In terms of the conformal time and the comoving momentum coordinate which we often use, this limit is proportional to . We have seen an example in Section 2 and will see more similar examples later with different , and . The corresponding vacuum is the Bunch-Davies vacuum and annihilated by defined in (61), .
We also would like to write the vacuum of the interacting theory (51) in terms of the vacuum of the free theory defined above. Unlike the scattering theory where the vacuum of the free theory is generally different from the vacuum of the interaction theory, the process that we are studying here do not generate any nontrivial vacuum fluctuations through interactions. This is a direct consequence of the identity So we can replace in (51) with the Bunch-Davies vacuum that we have specified above.
The integrand in (50) is highly oscillatory in the infinite past due to the behavior of the mode function . Their contribution to the integral is averaged out. For the Bunch-Davies vacuum, this regulation can be achieved by introducing a small tilt to the integration contour or performing a Wick rotation . The imaginary component turns the oscillatory behavior into exponentially decay, making the integral well defined.
When evaluating (51), one encounters (anti)time-ordered integrals, of which the integrands are products of fields, such as and , or and , sandwiched between the vacua. In contrast to the Minkowski space, in the inflationary background, we do not have a simple analogous Feynman propagator which takes care of the time ordering. Therefore, we will just evaluate the integrands, but leave the complication of the time ordering to the final integration.
To evaluate the integrand, one can shift around the orders of fields in that product, following the rules of the commutation relations. A contraction is defined to be a nonzero commutator between the following components of two fields, , where and denote the first and second term on the right-hand side of (61), respectively. After normal ordering, namely moving annihilation operators to the right-most and creation operators to the left-most so that they give zeros hitting the vacuum, it is not difficult to convince oneself that the only terms left are those with all fields contracted. Feynman diagrams can be used to keep track of what kind of contractions are necessary.
In the following, we demonstrate this using a simple example. We consider a field and quantize it as usual, So a contraction between the two terms, on the left and on the right, is defined to be For example, we want to compute a contribution to the four-point function from a tree-diagram containing two three-point interactions of the following form: These two s come from expanding or in (51). The corresponding Feynman diagram is Figure 1:
In Figure 1, the two cubic vertices each represent the three-point interaction (66). Each line represents a contraction. The four outgoing legs connect to the four in . The following is a term from the perturbative series expansion of (51). We demonstrate in the following one set of contractions represented by the diagram in Figure 1,(67)
Note that all terms are contracted. The result can be further evaluated using (65). After integration over momenta indicated in (66), the final momentum conservation will always manifest itself as . There are other sets of contractions represented by the same diagram for the same term. Namely, there are three ways of picking two of the three s (s), so we have a symmetry factor 9; also, there are 24 permutations of the four s. We need to sum over all these possibilities. We also need to sum over all possible terms containing two s in the perturbative series, which are not listed here, with their corresponding time ordered integral structure.
3.4. Three Forms
Now we deal with the time ordered integrals in the series expansion. There are two ways to expand (51).
In the first form, we simply expand the exponential in (50). For example, for an even , the th order term is Each term in the above summation contains two factors of multiple integrals, one from and another from . Each multiple integral is time ordered or antitime-ordered, but there is no time ordering between the two. We call this representation the factorized form.
In the second form, we rearrange the factorized form so that all the time variables are time-ordered, and all the integrands are under a unique integral. The th order term in this form is  We call this representation the commutator form.
Each representation has its computational advantages and disadvantages.
The factorized form is most convenient to achieve the UV () convergence. As mentioned, after we tilt or rotate the integration contour into the positive imaginary plane for the left integral, and negative imaginary plane for the right integral, all the oscillatory behavior in the UV becomes well-behaved exponential decay. However, this form is not always convenient to deal with the IR () behavior. For cases where the correlation functions have some nontrivial evolution after modes exit the horizon, as typically happens for inflation models with multiple fields, the convergence in the IR is slow. Cancellation of spurious leading contributions from different terms in the sum (68) can be very implicit in this representation, and could easily lead to wrong leading order results in analytical estimation or numerical evaluation.
The commutator form is most convenient to get the correct leading order behavior in the IR. The mutual cancellation between different terms are made explicit in terms of the nested commutators, before the multiple integral is performed. However, such a regrouping of integrands makes the UV convergence very implicit. Recall that the contour deformation is made to damp the oscillatory behavior in the infinite past. In the commutator form, for any individual term in the integrand, we can still generically choose a unique convergence direction in terms of contour deformation. Although the directions are different for different terms, they can be separately chosen for each of them. But now the problem is, if these different terms have to be grouped as in the nested commutator so that the IR cancellation is explicit, the two directions get mixed. Hence, the explicit IR cancellation is incompatible with the explicit UV convergence in this case.
To take advantage of both forms, we introduce a cutoff and write the IR part of the in-in formalism in terms of the commutator form and the UV part in terms of the factorized form , This representation is called the mixed form. This form is particularly efficient in numerical computations when combined with the Wick-rotations in the UV.
We will not always encounter all these subtleties in every model, but there does exist such interesting examples, as we will see in Section 7.1.
To end this section, we summarize the procedure that we need to go through to calculate the correlation functions in the in-in formalism.
Starting with the Lagrangian , we perturb it around the homogenous solutions and of the classical equations of motion, Keep the part of the Lagrangian that is quadratic and higher in perturbations and denote it as . Define the conjugate momentum densities as . We can also equivalently expand the Hamiltonian by perturbing and .
Work out the Hamiltonian in terms of and , and separate them into the quadratic kinematic part , which describes the free fields in the time-dependent background, and the interaction part . Relabel s and s in the Hamiltonian density as the interaction picture fields, s and s. These latter variables satisfy the equations of motion followed from the . We quantize and in terms of the annihilation and creation operators as in (61) and (62). The mode functions are solutions of the equations of motion from , normalized according to the Wronskian conditions (60) and specified by an initial condition such as the Bunch-Davies vacuum. The correlation function for is given by where is a product in terms of and . If we want to work with and instead of and , we replace with using the relation .
Choose appropriate forms in Section 3.4 and series-expand the integrand in powers of to the desired orders. Perform contractions defined in Section 3.3 for each term in this expansion. Each term gives a nonzero contribution only when all fields are contracted. Draw Feynman diagrams that represent the correlation functions, and use them as a guidance to do contractions. Finally sum over all possible contractions and perform the time-ordered integrations.
4. A No-Go Theorem
Simplest inflation models generate negligible amount of non-Gaussianities that are well below our current experimental abilities [47, 48]. By simplest, we mean (i)single scalar field inflation (ii)with canonical kinetic term (iii)always slow-rolls (iv)in Bunch-Davies vacuum (v)in Einstein gravity.
This list is extracted based on Maldacena's computation of three-point functions in an explicit slow-roll model . We now review this proof. The notations here follow [49, 50] and will be consistently used later in this paper.
The Lagrangian for the single scalar field inflation with canonical kinetic term is the following: where is the inflaton field, is the canonical kinetic term and is the slow-roll potential. The first term is the Einstein gravity and is the reduced Planck mass. For convenience we will set the reduced Planck mass . The signature of the metric is .
The inflaton starts near the top of the potential and slowly rolls down. As we have reviewed in Section 2, to ensure that the inflation lasts for at least e-folds, the potential is required to be flat so that the slow-roll parameters (11) are both much less than one most of the time. The energy of the universe is dominated by the potential energy, and the inflaton follows the slow-roll attractor solution (10). Also as discussed in Section 2, we will use the following more general slow-roll parameters:
To study the perturbation theory, it is convenient to use the ADM formalism, in which the metric takes the form The action becomes where the index of can be lowered by the 3D metric and is the 3D Ricci scalar constructed from . The definitions of and are In the ADM formalism, the variables and are Lagrangian multipliers whose equations of motion are easy to solve. In single field inflation, we have only one physical scalar perturbation . We choose the uniform inflaton gauge (also called the comoving gauge) in which the scalar perturbation appears in the three dimensional metric in the following form: and the inflaton fluctuation vanishes. The is the homogeneous scale factor of the universe, so is a space-dependent rescaling factor. In this paper we do not consider the tensor perturbations.
We plug (75) and (78) into the action (76) and solve the constraint equations for the Lagrangian multipliers and . We then plug them back to the action and expand up to the cubic order in in order to calculate the three-point functions. To do this, in the ADM formalism, it is enough to solve and to the first-order in . This is because their third-order perturbations will multiply the zeroth order constraint equation which vanishes, and their second-order perturbations will multiply the first-order constraint equation which again vanishes. After some lengthy algebra, we obtain the following expansions: where Here is the inverse Laplacian and is the variation of the quadratic action with respect to the perturbation . We now can follow Section 3 and proceed to calculate the correlation functions. For simplicity, we will always neglect the superscript “” on various interaction picture fields.
We restrict to the case where the slow-roll parameters are always small and featureless. We first look at the quadratic action. In this case, we can analytically solve the equation of motion followed from (79) in terms of the Fourier mode of , and get the mode function where is the conformal time. The normalization is determined by the Wronskian condition (60). We have chosen the Bunch-Davies vacuum by imposing the condition that the mode function approaches the vacuum state of the Minkowski spacetime in the short wavelength limit , The dynamical behavior of that has been explained around (24) and (25) is made precise here. In particular, is exactly massless without dropping any suppressed terms. In addition, from (78), we can see that, for superhorizon modes, the only effect of is to provide a homogeneous spatial rescaling. And is the only scalar perturbation. So the fact that is frozen after horizon exit will not be changed by higher-order terms.
If we choose the spatially flat gauge, we make disappear and the scalar in this perturbation theory becomes the perturbation of . The relation between and in (25) (with corrections) is thus a gauge transformation through a space-dependent time shift.
We quantize the field as with the canonical commutation relation . We can easily compute the two-point function at the tree level, where Since remains constant after it exits the horizon, the and are both evaluated near the horizon exit.
We next look at the cubic action. For single field models, . Keeping in mind that is proportional to , one can see that the first line of (80) is proportional to . For the featureless potential, , where collectively denotes either or . So the second line of (80) is proportional to , and negligible. The third line can be absorbed by a field redefinition . The only term in that will contribute to the correlation function is written out explicitly in (83). All the others involve derivatives of so vanish outside the horizon. Thus this redefinition eventually introduces an extra term, According to (72), we expand the exponential to the first-order in to get the leading result
To estimate the order of magnitude of the bispectrum, we only need to keep track of the factors of and . For example, from the first term in (80), we have , where we used the conformal time and the prime denotes the derivative to . Using , , we see that this three-point vertex contributes . Together with the three external legs and the definition , we get Similar results can be obtained for the other two terms in the first line of (80). As we will define more carefully later, the size of the three-point function is conventionally characterized by the number , which is defined as . So the contribution from the first line of (80) is . The extra term due to the redefinition (90) contributes . This completes the order-of-magnitude estimate. To get the full non-Gaussian profile, we need to compute the integrals explicitly and get where where and the permutations stand for those among , , and .
The slow-roll parameters are of order , so for these models. Even if we start with Gaussian primordial perturbations, nonlinear effects in CMB evolution will generate , and a similar number for large scale structures due to the nonlinear gravitational evolution or the galaxy bias . It seems unlikely that we can disentangle all these contaminations and detect such small primordial non-Gaussianities in the near future.
5. Beyond the No-Go
5.1. Inflation Model Building
The following are two examples of slow-roll potentials in the simplest inflation models that we studied in Section 4: The first type (95) belongs to the small field inflation models. The slow-roll conditions (11) require the potential to be flat enough relative to its height, that is, the mass of the inflaton should satisfy . The second type (96) belongs to the large field inflation models. The potential also needs to be flat relative to its height, but here one achieves this by making the field range very large, typically . The other conditions that we listed in Section 4 should also be satisfied by these models. These are the classic examples, which exhibit algebraic simplicities and illustrate many essential features of inflation.
However, when it comes to the more realistic model building in a UV complete setup, such as in supergravity and string theory, situations get much more complicated. For example, it is natural that we encounter multiple light and heavy fields, and the potentials for them form a complex landscape. These multiple fields live in an internal space, whose structure can be very sophisticated. In string theory, some of them manifest themselves as extra dimensions and can have intricate geometry and warping. All these elements have to coexist with the inflationary background that introduces profound back-reactions.
Even with varieties of model building ingredients, it has been proven to be very subtle to construct an explicit and self-consistent inflation model. Indeed various problems have been noticed over the years in the course of the inflation model building. For example, consider the following problems.
(i) The -Problem for Slow-Roll Inflation 
As we have seen, in order to have slow-roll inflation [8, 9], the mass of the inflaton field has to be light enough, , to maintain a flat potential. However, in the inflationary background, the natural mass of a light particle is of order . This can be seen in many ways, and in some ideal situations they are equivalent to each other. For example, one way to see this is to consider the coupling between the Ricci scalar and the inflaton, ~. In the inflationary background . Unless we have good reasons to set the coefficient of this term to be much less than one, it will give inflaton a mass of order , spoiling the inflation. Another way to see this is to note that the effective potential in supergravity takes the form other terms. Here schematically is the Kahler potential and its dependence on is normalized as such to give the canonical kinetic term for . So the first term in the expansion of is of order and model independent. Therefore, either symmetry needs to be imposed or other tuning contributions introduced to solve this -problem.
(ii) The -Problem for DBI Inflation 
DBI inflation  is invented to generate inflation by a different mechanism. It makes use of the warped space in the internal field space [55, 56]. These warped space impose speed limits for the scalar field, so even if the potential is steep, the inflaton is not allowed to roll down the potential very quickly. A canonical example of warped space is where is the extra dimension (or internal space), is the warp factor, and is the length scale of the warped space. The position of a dimensional brane in the -coordinate is the inflaton. So the inflaton velocity is limited by the speed limit in the -direction, . In order to provide a speed limit that is small enough for inflation, the warp factor has to be small enough, . However one of the Einstein equations with the metric (97) takes the following form where the second term on the left-hand side is due to the back-reaction of the inflationary spacetime. It is easy to see that the naive should be modified for , precisely where the inflation is supposed to happen. Without contributions from other source terms, such a deformed geometry closes up too quickly and leads to an unacceptable probe-brane back-reaction if we demand the inflaton still follow the speed limit. Therefore, either symmetry, or tuning using other source terms from the right-hand side of (98), is necessary to solve this -problem. The -problem and -problem are closely related in an AdS/CFT setup.
(iii) The Field Range Bound [57, 58]
Large field inflation models require the field range to be much larger than . In supergravity and string theory, starting from a ten-dimensional theory with 10-dim Planck mass , the 4-dim Planck mass is obtained by integrating out the six extra dimensions, where we use and to denote the size and volume of the extra dimensions, respectively. The field range often appears as the distance in the extra dimensions, , with the factor being the proportional coefficient. Clearly, . If the field range manifests itself within a warped throat with a length scale , we still require , and so . Together with , we get We further note that the microscopic length scale has to be much larger than the 10-dim Planck length for the field theory to make sense. So , and the field range in these models is generically sub-Planckian. For example, for a warped throat with charge , , we have We have ignored a detailed numerical coefficient appearing on the right-hand side of (101), which is model dependent. For example, considering the volume to be the sum of the throat and a generic bulk volume, it is ; considering an extreme case where the throat does not attach to a bulk, it is . Notice that, due to the dependence of on the volume , increasing the volume only makes the bound tighter.
(iv) The Variation of Potential 
Even in cases where there is no fundamental restriction on the excursion of fields, one encounters problems constructing the large field inflationary potential. Large field potentials that arise from a fundamental theory take the following general from: where represents typical scales in the theory. For field theory descriptions to hold, such scales are much less than . For example, can be the higher dimensional Planck mass, string mass, or their warped scales. The s are dimensionless couplings of order . Unless some symmetries are present to forbid an infinite number of terms in (102), or a high degree of fine-tuning is assumed, the shape of potential (102) varies over a scale of order . This variation is too dramatic for the potential to be a successful large field slow-roll potential.
None of the arguments in the above list is meant to show that the specific type of inflation is impossible. In fact, these have been the driving forces for the ingenuity and creativity in the field of inflation model building. This list is used to demonstrate some typical examples of complexities in reality. Often times, solving one problem will be companied by other structures that make the model step beyond the simplest one. So we may want to keep an open mind that the algebraic simplicity may not mean the simplicity in Nature.
Following is a partial list of possibilities that allow us to go beyond the no-go theorem in Section 4. (i)Instead of single field inflation, we can consider quasisingle field or multifield inflation models (Sections 7 and 8).(ii)Instead of canonical kinetic terms, there are models where the higher derivative kinetic terms dominate the dynamics (Section 6.1).(iii)Instead of following the attractor solution such as the slow-roll precisely, features can be present in the potentials or internal space, that temporarily break the attractor solution, or cause small but persistent perturbations on the background evolution (Sections 6.2 and 6.3).(iv)Instead of staying in the Bunch-Davies vacuum, other excitations can exist due to, for example, boundary conditions or low scales of new physics (Section 6.4).(v)Although strong constraints, from experimental results and theoretical consistencies, exist on non-Einstein gravities, early universe may provide an opportunity for their appearance. We use this category to include a variety of possibilities, such as modified gravities, noncommutativity, nonlocality and models beyond field theories.
There are also strong motivations from data analyses for us to search and study different large non-Gaussianities. The signal-to-noise ratio in the CMB data is not large enough for us to detect primordial non-Gaussianities model-independently. A well-established method is to start with a theoretical non-Gaussian ansatz, and construct optimal estimators that compare theory and data by taking into accounts all momenta configurations. This then gives constraints on the parameters characterizing the theoretical ansatz. Therefore, the following two important possibilities exist. First, the primordial non-Gaussianities exist in data could be missed if we did not start with a right theoretical ansatz. Second, even if a non-Gaussian signal was detected with one ansatz, it does not mean that we have found the right one. So different well-motivated non-Gaussian templates are needed for clues on how corresponding data analyses should be formed. From a different perspective, even if the primordial density perturbations were Gaussian, we would still do the similar amount of work and reach the conclusion after various well-motivated non-Gaussian forms are properly constrained.
5.2. Shape and Running of Bispectra
In this paper, we will be mainly interested in the three-point correlation functions of the scalar primordial perturbation . They are also called the bispectra. In this subsection, we introduce some simple terminologies that we often encounter in studies of bispectra.
The three-point function is a function of three momenta, , , and , which form a triangle due to the translational invariance. Assuming also the rotational invariance, we are left with three variables, which are their amplitudes, , , and , satisfying the triangle inequalities. The information is encoded in a function that we define as where is the fiducial power spectrum, and we fix it to be a constant , where . We have chosen the above definition so that it can be uniformly applied to different types of bispectra that we will encounter in this paper. In the literature, different notations have been used. The differences are simple and harmless. For example, different functions such as or are sometimes defined. We choose since it is dimensionless and, for scale-invariant bispectra, it is invariant under a rescaling of all momenta. This quantity is the combination that is used to plot the profiles of bispectra in the literature any way, despite of different conventions. Also, the precise power spectrum instead of is often used in the definition (103). Here, we absorb the momentum dependence of in . This is because the three-point function is an independent statistic relative to the two-point. In cases where both the power spectrum and bispectrum have strong scale dependence, it is not convenient if they are defined in an entangled way.
Under different circumstances, different properties of are emphasized. The conventions involved may not always be precisely consistent with each other, since they are chosen to best describe the case at hand. Following are some typical examples.
The dependence of on , , and is usually split into two kinds.
One is called the shape of the bispectrum. This refers to the dependence of on the momenta ratio and , while fixing the overall momentum scale . Several special momentum configurations are shown in Figure 2.
Another is called the running of the bispectrum. This refers to the dependence of on the overall momentum scale , while fixing the ratio and .
For bispectra that are approximately scale invariant, the shape is a more important property [50, 60]. We will encounter such cases in Sections 6.1, 7.1, and 8.1. The amplitude, also called the size, of the bispectra is often denoted as by matching In this case, is approximately a constant but can also have a mild running, that is, a weak dependence on the overall momentum [61, 62]. An index is introduced to describe this scale dependence. The power spectrum also has a mild running, . In this paper, when we give explicit forms of in the approximately scale-invariant cases, for simplicity, we mostly ignore these mild scale dependence and concentrate on shapes. Shapes of bispectra have been given names according to the overall dependence of on momenta. For example, for the equilateral bispectrum, peaks at the equilateral triangle limit and vanishes as ~ in the squeezed triangle limit (). The local bispectrum peaks at the squeezed triangle limit in the form ~, such as the two shape components in (94). To visualize the shapes, we often draw 3D plots , where and vary from to and satisfy the triangle inequality .
There are also cases where the running becomes the most important property, while the shape is relatively less important [63, 64]. In such cases, the bispectra are mostly functions of . So defined in (104) has strong scale dependence. Instead, one can choose a constant to describe the overall running amplitude. We will encounter such cases in Sections 6.2 and 6.3. In these cases, the shape plot may look nontrivial but this is because it does not fix .
The above dissection will become less clean for cases where both properties become important.
One thing is clear. The , that is always used to quantify the level of non-Gaussianities, is only sensible with an extra label that specifies, at least qualitatively, the profile of the momentum dependence, such as shapes and runnings.
It is useful to quantify the correlations between different non-Gaussian profiles, because as we mentioned in data analyses an ansatz can pick up signals that are not completely orthogonal to it. In real data analyses this is performed in the CMB -space. To have a simple but qualitative analogue in the -space, we define the inner product of the two shapes as and normalize it to get the shape correlator [60, 65] Following , we choose the weight function to be so that the -scaling is close to the -scaling in the data analyses estimator. Later in this review, when we use this correlator to estimate the correlations between shapes, we take the ratio between the smallest and largest to be , close to that in WMAP. A more precise correlator should be computed in the -space in the same way that the estimator is constructed. We refer to  for more details.
In typical data analyses [66–70], the estimator involves a triple integral of the bispectrum over the three momenta . To have practical computational costs, it is necessary that this integral can be factorized into a multiplication of three integrals, each involves only an individual . This requires the bispectrum to be of the form , or a sum of such forms. Such a form is called the factorizable form or separable form. The factor may be tolerable since it can be written as . If the analytical result is too complicated, to make contact with experiments we will try to construct simple factorizable ansatz or template to capture the main features of the original one. New methods that are applicable to nonfactorizable bispectrum forms and are more model-independent are under active development .
6. Single Field Inflation
In this section, we relax several restrictions of the no-go theorem on single field inflation models and study how large non-Gaussianities can arise. We present the formalisms and compute the three-point functions. We emphasize how different physical processes during inflation are imprinted as distinctive signatures in non-Gaussianities. Obviously, any mechanism that works for single field inflation can be generalized to multifield inflation models.
6.1. Equilateral Shape: Higher Derivative Kinetic Terms
In this subsection, we study large non-Gaussianities generated by noncanonical kinetic terms in general single field inflation models, following .
Consider the following action for the general single field inflation : Comparing to (73), we have replaced the canonical form with an arbitrary function of and . This is the most general Lorentz-invariant Lagrangian as a function of and its first derivative. It is useful to define several quantities that characterize the differential properties of with respect to [49, 72]: where is called the sound speed and the subscript “” denotes the derivative with respect to . The third derivative is enough since we will only study the three-point function here.
It is a nontrivial question which forms of will give rise to inflation. The model-independent approach we take here is to list the conditions that an inflation model has to satisfy, no matter which mechanism is responsible for it. Namely, we generalize the slow-roll parameters in (74) to the following slow-variation parameters: and require them to be small most of the time during the inflation. The smallness of these parameters guarantees the Hubble constant , the parameter , and the sound speed to vary slowly in terms of the Hubble time. Similar to the arguments given in the case of slow-roll inflation in Section 2, these are necessary to ensure a prolonged inflation as well as an approximately scale-invariant power spectrum that we observed in the CMB.
Following the same procedure that is outlined in Section 4, we get the quadratic and cubic action for the scalar perturbation [47, 49, 50]. The quadratic part is If the slow-variation parameters are always small and featureless, we can analytically solve the equation of motion followed from (111) and get the following mode function: Notice the appearance of comparing to (85). The two-point function is with the power spectrum where the variables are evaluated at the horizon crossing of the corresponding -mode.
To calculate the bispectrum, we look at the cubic action. In the following, we list three terms that are most interesting for this subsection, The full terms can be found in in [50, equations ()–()].
The order of magnitude contribution from these three terms can be estimated similarly as we did in (92), but now we not only keep factors of and , but also factors of . Take the first term as an example, we write it in terms of the conformal time, Comparing (112) with (85), we see that there is an extra factor of companying . So we estimate and . Also, , but . Overall, the vertex (116) contributes Multiplying the three external legs , and using the definition and , we get The other two terms are similar. A detailed calculation reveals that where we have decomposed the shape of the three-point function into six parts. The first two come from the leading order terms that we listed in (115), In terms of their sizes are The next four terms come from the subleading terms that we did not list explicitly in (115) as well as the subleading contributions from the first two terms. Their orders of magnitude are The detailed profiles can be found in .
The full results we obtained can be used in different regimes.(i)If we look at the limit, or , the leading order results give two shape components, and . This result can also be obtained using a simple method of considering only the fluctuations in scalar field while neglecting those in gravity [73, 74]. Intuitively, this is because the higher derivative terms are responsible for the generation of large non-Gaussianities, and the gravity contribution is expected to be small as we saw in Section 4. Therefore, one expands using . The derivatives of with respect to are ignored because the inflation and scale invariance imposes an approximate shift symmetry on in terms of inflaton . We then get two terms in the cubic Lagrangian density, This gives two leading bispectra the same as (121) and (122). The approach that we present here gives a rigorous justification to such an method. The subleading order component may be observable as well. At this limit where the higher derivative terms of the inflaton field are dominant, the Lagrangian of the above effective field theory are generalized  to include, for example, the ghost inflation  whose Lagrangian cannot be written in a form of . Another two slightly different equilateral shapes arise. However, it is worth to mention that, generally in single field models and Einstein gravity, going beyond requires adding either terms that explicitly break the Lorentz symmetry, or terms with higher time derivatives on which cannot be eliminated by partial integration, such as . Different treatment of such terms and discussions on their effects can be found in [77–79].(ii)If we take the opposite, slow-roll limit, and , we recover the two shape components and that we got in Section 4, with unobservable size .(iii)We can also look at the intermediate parameter space. In slow-roll inflation models, one can also add higher derivative terms [49, 80]. But in order not to spoil the slow-roll mechanism, the effect of these terms can only be subdominant. This corresponds to and . Using the full results, we can see that the size of the non-Gaussianity is . Therefore, it is important to emphasize that, for the class of models we consider here, nonslow-roll inflationary mechanisms, such as the example that will be given below, are necessary to generate observable large non-Gaussianities.(iv)The other terms that we did not list in (115) (see ) and their canonical limit (80) are also useful. These terms are exact for arbitrary values of , , and , so the usage of the action is beyond the category of models that we focus on in this subsection. As we will see in Sections 6.2 and 6.3, it can be applied to the cases of sharp or periodic features where these parameters do not always remain small.
In the rest of this subsection, we focus on the first case.
In Figures 3 and 4, we draw the shapes of and . The two shapes are similar. They both peak at the equilateral limit, and behave as in the squeezed limit . We call these shapes the equilateral shapes. There are some small differences between and , for example, around the folded triangle limit . A factorizable shape ansatz for the equilateral shape that is often used in data analyses is the following : and is shown in Figure 5. As we can see, it represents the most important features of Figures 3 and 4.
The shape of is more complicated, but we expect they have the similar shapes as the equilateral one because their squeezed limits behave the same . The three other shapes , and are all close to the local shapes as their squeezed limit scale as for .
The scale dependence in , and will introduce mild running for the three-point function. We usually regard only the contributions from and as the running of the non-Gaussianity.
The underlying physics of the equilateral shape can be readily understood in terms of their generation mechanism. In single field inflation, the long wavelength mode that exits the horizon are frozen and can have little interaction with modes within the horizon. The large interaction only occurs among modes that are crossing the horizon at about the same time. These modes then have similar wavelengths. This is why the shape of the non-Gaussianity peaks at the equilateral limit in momentum space.
This physical origin also suggests the caveat that, as long as there are large interactions involving modes with similar wavelengths, an equilateral-like shape may arise. For example, such cases can happen in multifield models where there are particle creation [82, 83] (see however ).
(i) An Example: Dirac-Born-Infeld (DBI) Inflation
An explicit example of the above general results is the DBI inflation [54, 73, 85–92]. These inflation models describe a 3 + 1 dimensional brane moving in warped extra dimensions. The location of the brane is a scalar field in 4D effective field theory, and it is the inflaton. The warped extra dimensions provide a nontrivial internal field space for the inflaton. In terms of the 4D effective field theory, the action is The nontrivial part is the kinetic term involving the square-root. It can be understood as a generalization of the following two familiar situations. It is a higher dimensional generalization of the action of a relativistic point particle where the speed of light varies with . It is also a relativistic generalization of the usual canonical kinetic term in the nonrelativistic limit , Because the speed limit of the inflaton can vary in the internal space, if it can be made small enough near the top of potential where the inflaton is about to roll down, the warped space restricts the rolling velocity even if the potential is too steep for slow-roll inflation to happen. So the inflaton rolls ultra relativistically, but with very small velocity, and this generates the DBI inflation.
The physical consequence is now easy to obtain using the general results in this subsection. In our notation the Lagrangian is The sound speed is which is the inverse of the Lorentz boost factor , so . The component (123) vanishes identically, and we have a large bispectrum of shape with size (124).
DBI inflation is still driven by the potential energy. The general single field inflation models also include the k-inflation , where the inflation is driven by the inflaton kinetic energy. Model construction of single field k-inflation can be found in [93–95]. The bispectra for such models are computed in [50, 94].
Multifield generalization have been studied in [96–102], where this type of kinetic terms are generalized to multiple fields. The three-point functions involving these different fields have the same or similar shapes.
6.2. Sinusoidal Running: Sharp Feature
Although various slow-variation parameters in (110) have to be small most of the time during inflation, they can become temporarily large. Such cases can happen if there are sharp features in inflaton potentials or internal field space, so the behavior of inflatons temporarily deviates from the attractor solution, and then relaxes back within several Hubble time, or stay longer but with small deviation amplitudes. Motivations for such models include the following. It may be possible explanations for features in power spectrum [103–106], and if so the associated non-Gaussianity is a cross-check. And there are brane inflation models that are very sensitive to sharp features present in the potential or in the internal space .
As an example, we study a sharp feature in the slow-roll potential. The fact that a sharp feature in potential can enhance non-Gaussianities has long been anticipated and qualitative estimates have been made by different methods [108–110]. The precise method of analyzing the size, running and shape of such non-Gaussianities [63, 64] is made possible with the developments of the formalisms that we reviewed in Sections 3.1, 4, and 6.1. This will be the subject of this subsection.
We start by studying the behavior of the slow-roll parameters. We use a small step in potential as an example and will ignore numerical coefficients. We use to denote the relative height of the step, and the width of the step. In the attractor solution, the inflaton velocity is given by . As it falls down the step, the potential energy gets converted to the kinetic energy, so we have The amplitude of density perturbations is given by , so such a sharp feature causes glitches in the power spectrum. It will leave a dip with relative size since increases first, followed by oscillations caused by a nonattractor component of the mode function before it settles down again in the attractor solution. To fit the CMB data, cannot change much. As we can see, the sensitivity of the power spectrum to the step size is proportional to , and we need . Reducing the width of sharp feature increases the amplitude of the glitches, but this is only for a large over which the inflaton spends more than one e-fold to cross. Further reducing will not change the amplitude of the glitches since (133) is saturated, but the sharpness will determine how deep within the horizon the modes are affected.
So does not change much, . But it changes within a very short period, . So can be very large, It is also clear that the feature is associated with a characteristic physical scale and generates a scale-dependent power spectrum and higher-order correlation functions.
With these qualitative behavior in mind, we now study the three-point function. An important fact of the formalisms in Sections 4 and 6 is that the expansion is exact in terms of the slow-variation parameters. So it is valid even if these parameters are not always small, as long as the expansion in is perturbative.
In all terms in the cubic expansion (80), appears at most with one time derivative; the field redefinition gives a term that is proportional to at the end of the inflation; and the other terms are all suppressed by powers of , which remains small even in the presence of a sharp feature. So the most important term is in which the coupling is proportional to . The correlation function is dominated by Precise evaluation of this expression has to be done numerically. But it is not difficult to see the generic properties of bispectra associated with a sharp feature.
For long wavelength modes that already crossed the horizon at the time of the sharp feature, , the mode function is already frozen and the integration (136) gives vanishing contribution. For short wavelength modes that are still well within the horizon, the modes are not affected if their momenta are larger than the inverse of the time scale characterizing the sharpness of changes in slow-roll parameters. The modes most affected are those which are near the horizon crossing. These modes are all oscillatory, . As we have studied, is temporarily boosted, so it can be roughly approximated as several hat-functions that satisfy . Examples of such behavior are shown in Figure 6. If we simply approximate the hat-functions by several delta-functions, , the integration (136) will give something like where is the scale corresponding to the location of feature, is a phase and Comparing with the effect on the power spectrum, one can keep the size of glitches in the power spectrum small while make large, for example, by fixing and decreasing .
This ansatz describes the most important running behavior of this bispectrum. Notice that the oscillatory frequency in the -space is of order , which is the scale of the feature. A rescale in can be compensated by a rescale in all . Also notice that the oscillatory frequency, , along the <