# Large non-Gaussianity in multiple-field inflation

###### Abstract

We investigate non-Gaussianity in general multiple-field inflation using the formalism we developed in earlier papers. We use a perturbative expansion of the non-linear equations to calculate the three-point correlator of the curvature perturbation analytically. We derive a general expression that involves only a time integral over background and linear perturbation quantities. We work out this expression explicitly for the two-field slow-roll case, and find that non-Gaussianity can be orders of magnitude larger than in the single-field case. In particular, the bispectrum divided by the square of the power spectrum can easily be of –, depending on the model. Our result also shows the explicit momentum dependence of the bispectrum. This conclusion of large non-Gaussianity is confirmed in a semi-analytic slow-roll investigation of a simple quadratic two-field model.

## I Introduction

The assumption that inflationary fluctuations are Gaussian is a good starting point for the study of cosmological perturbations, but it is only true to linear order in perturbation theory. Since gravity is inherently non-linear, and most inflation models have (self-)interacting potentials, non-linearity must be present at some level in all inflation models. Hence the issue is not whether inflation is non-Gaussian, but how large the non-Gaussianity is. With increasingly precise CMB data becoming available in the near future from the WMAP and Planck satellites and other experiments, we might well hope to detect this non-Gaussianity. This would offer us another key observable to help constrain or confirm specific inflation models and the underlying high-energy theories from which they are derived. As a rough order of magnitude estimate, we note that non-Gaussianity will be detectable by Planck if the bispectrum (the Fourier transform of the three-point correlator) is of the order of the square of the power spectrum komatsu .

It follows that to compute the predicted amount of non-Gaussianity in specific inflation models we need to go beyond linear-order perturbation theory. In gp2 ; formalism we introduced a new formalism to deal with the non-linearity during inflation. We will not again summarise the other work dealing with this subject, references for which can be found in formalism or a recent review ngreview . Our formalism is distinguished by being based on a system of fully non-linear equations for long wavelengths, while stochastic sources take into account the continuous influx of short-wavelength fluctuations into the long-wavelength system as the inflationary comoving horizon shrinks. The variables used incorporate both scalar metric and matter perturbations self-consistently and they are invariant under changes of time slicing in the long-wavelength limit.

The advantages of our method are threefold: (i) it is physically intuitive and relatively simple to use for quantitative analytic and semi-analytic calculations; (ii) it is valid in a very general multiple-field inflation setting, which includes the possibility of a non-trivial field metric; and (iii) it is well-suited for direct numerical implementation. The first point was already demonstrated in sf , where we computed the non-Gaussianity in single-field slow-roll inflation, while the third point is the subject of a forthcoming paper RSvTnum . The present paper is dedicated to the exploration of the second point, as well as the first.

In sf we found, confirming
what was known in the literature beforehand (see e.g. maldacena ), that non-Gaussianity in the single-field case
is too small to be realistically observable, because it is
suppressed by slow-roll factors (actually the scalar spectral
index ). However, there have been long-standing claims in the
literature (see e.g. salopek ; uzan ) that specific
multiple-field models can, in principle, create significant
non-Gaussianity. Indeed, there has been growing recent interest
in models which can produce large primordial non-Gaussianity
large ng . A feature shared by most of these models though
is that this non-Gaussianity involves some mechanism operating
*after* inflation, usually (p)reheating or later domination
of a curvaton field. In this paper we investigate for the first
time general multiple-field inflation, not just specific models,
presenting a method by which to accurately calculate the resulting
three-point correlator *during* inflation. We find that it is
possible to get significant primordial non-Gaussianity without
invoking some post-inflationary mechanism even for the simplest
two-field quadratic potential.

The key mechanism for the production of this large non-Gaussianity is the superhorizon influence of isocurvature perturbations on the adiabatic mode. The former feed into the latter when the background follows a curved trajectory in field space. Note that the example studied in section V.4 illustrates that there is no need for the potential to be interacting. Our aim is to push forward towards a tractable non-Gaussian methodology for the new era of precision cosmology which confronts us.

This work is organised as follows. In section II we give the equations from formalism that are used as the starting point for the present investigations. In section III we then derive the general solution for the relevant quantities in multiple-field inflation, culminating in a general expression for the three-point correlator of the adiabatic component of the curvature perturbation, without any slow-roll approximations. This integral solution - equation (29) - is a very useful calculational tool because it gives the three-point correlator entirely in terms of background quantities and linear perturbation quantities at horizon crossing. In section IV we make a leading-order slow-roll approximation to work out the various contributions in the general solution more explicitly. Finally in section V we calculate the bispectrum in an analytic treatment of the two-field case with constant slow-roll parameters. We find that the result can be orders of magnitude larger than for single-field inflation. This result is confirmed with a semi-analytic slow-roll calculation of an explicit model with a quadratic potential in section V.4. Our method yields the full momentum dependence, not just an overall magnitude, and we find that there can be a difference of the order of a few between opposite extreme momentum limits. We conclude in section VI. Parts of this paper, in particular section V, are rather technical, so some readers might be interested in referring to mf2 first, which contains a simplified derivation of only the dominant non-Gaussian contributions in multiple-field inflation, along with a summarised discussion.

## Ii Multiple-field setup

Since in this paper we are explicitly working out the general non-linear multiple-field formalism of formalism , we refer the reader to that paper for derivations and more details of the initial equations. Here we just briefly describe the context and give the relevant equations and definitions to be used as starting point for further calculations.

We start from a completely general inflation model, with an
arbitrary number of scalar fields (where labels the
different fields) and a potential with arbitrary
interactions. We also allow for the possibility of a non-trivial
field manifold with field metric . We will consider only
scalar modes and make the long-wavelength approximation (i.e. consider only wavelengths larger than the Hubble radius ,
where second-order spatial gradients can be neglected compared to
time derivatives)^{1}^{1}1Formally this corresponds with taking
only the leading-order terms in the gradient expansion. We expect
higher-order terms to be subdominant on long wavelengths during
inflation, but this statement has only been rigorously verified at
the linear level. A calculation to higher order in spatial
gradients, or, even better, a full proof of convergence of the
expansion, would be desirable. See gradient for more
details on the validity of the gradient expansion beyond linear
theory.. The spacetime metric and matter Lagrangean
are given by

(1) |

with the local scale factor and the lapse function. The local expansion rate is defined as , where the dot denotes a derivative with respect to . The proper field velocity is , with length . We also define local slow-roll parameters as

(2) |

where and is a covariant derivative with respect to the field . For the first part of the paper we will not make a slow-roll approximation, and consider these definitions as just a short-hand notation. When we do make this approximation, from section IV, and are first order in slow roll, while is second order. Finally, we choose the gauge where

(3) |

In this gauge horizon exit of a mode, , occurs simultaneously for all spatial points and calculations are simpler.

We will make use of a preferred basis in field space, defined as follows. The first basis vector is the direction of the field velocity. Next, is defined as the direction of that part of the field acceleration that is perpendicular to . One continues this orthogonalisation process with higher derivatives until a complete basis is found. Explicit expressions can be found in formalism , here we only give . Now one can take components of vectors in this basis and we define, for example for (defined below in (6)) and :

(4) |

Note that, unlike for the index , there is no difference between upper and lower indices for the . By construction there are no other components of , so that one can write . We also define

(5) |

where is the covariant time derivative containing the connection of the field manifold. is antisymmetric and only non-zero just above and below the diagonal, and first order in slow roll. Its explicit form in terms of slow-roll parameters can be found in vantent ; here we only need that .

As discussed in gp2 ; formalism it is useful to work with the following combination of spatial gradients to describe the fully non-linear inhomogeneities:

(6) |

which is invariant under changes of time slicing, up to second-order spatial gradients gp ; formalism . Note that, when linearised, (the =1 component of ) is the spatial gradient of the well-known from the literature, the curvature perturbation. In formalism we derived a fully non-linear equation of motion for without any slow-roll approximations:

(7) |

where is the velocity corresponding with and

where and with the curvature tensor of the field manifold. Although for the first part of the paper we will not make a slow-roll approximation, we give here immediately the leading-order slow-roll approximation of (7), which we will be using in the second part, to show that things simplify considerably in that case:

(9) |

The stochastic source terms and are given by

(10) |

where c.c. denotes the complex conjugate. The perturbation quantity is the solution from linear theory for the multiple-field generalisation of the Sasaki-Mukhanov variable . It can be computed exactly numerically, or analytically within the slow-roll approximation vantent . The are a set of Gaussian complex random numbers satisfying

(11) |

The quantity is the Fourier transform of an appropriate smoothing window function which cuts off modes with wavelengths smaller than the Hubble radius; we choose it to be a Gaussian with smoothing length , where 3–5:

(12) |

Since and are smoothed long-wavelength variables, the appropriate initial conditions are that they should be zero at early times when all the modes are sub-horizon. Hence,

(13) |

A key aspect of the system (7) or (9) is that it is fully non-linear. All functions in the coefficients on the left-hand side of the equation, like , and in the sources on the right-hand side are inhomogeneous and depend on and via a basic set of three constraint equations:

(14) | |||||

(15) | |||||

(16) |

Using only these three constraints one can compute the spatial derivative of all relevant quantities, keeping in mind that . Note that in our gauge depends on only and does not get any non-linear contributions.

## Iii General analytic solution

In this section we investigate how to solve the system (7) analytically and give formal expressions for the solution. In the next sections we will investigate cases where we can determine the solution more explicitly. We start by rewriting the system (7) into a single vector equation:

(17) |

Here the indices label the components within this -dimensional space (with the number of fields). The matrix can be read off from (7) and has the following form: , and , where is the matrix between parentheses in the second equation of (7) and . All other entries of are zero.

Equation (17) is non-linear since the matrix and the the source term are inhomogeneous functions in space and depend on the through (14)–(16). It can be solved perturbatively as an infinite hierarchy of linear equations with known source terms at each order (see also formalism ). We expand the relevant quantities as

(18) |

Then the equation of motion for is

(19) |

Let us recapitulate the meaning of the various indices, to avoid confusion. The index labels the components of spatial vectors. The indices label components in field space. These indices will not occur in the rest of the paper, since they have been replaced by the indices that label components in field space within the special basis as defined in (4). Next, the indices label components within the -dimensional space consisting of both and as defined in (17). Finally there are the labels within parentheses that denote the order in the perturbative expansion defined above. Only with the and is there a difference between upper and lower indices.

We now show that the source term is known from the solutions for up to order . The equation for is linear by construction: depends only on the homogeneous background quantities, and the only x dependence in is in the , for the rest it depends on homogeneous background quantities. All of these are in the end functions of just , , and via their definitions. To go beyond linear order all these background quantities are perturbed as follows ( stands for any of the quantities to be perturbed, for example , , etc.):

(20) |

where we use (14)–(16) to compute and is some homogeneous (space-independent) vector that is the result of that calculation. Next, to compute one simply repeats this process with the vector , and continues in this way order by order (of course there is also a term at second order, etc.). Then it is easy to see that depends only on , and hence is a known quantity at each order.

The solution of equation (19) for can be written as

(21) |

with the Green’s function satisfying^{2}^{2}2To be
precise, the Green’s function is actually defined as the solution
of (22) with on the right-hand side instead
of zero. The solution is then a step function times what we call
the Green’s function. This step function has been taken into
account by changing the upper limit of the integral in
(21) from to .

(22) |

It is important to note that this Green’s function is homogeneous, a solution of the background equation. It has to be computed only once, and can then be used to calculate the solution at each order using the different source terms as in (21). We write explicitly for the first two orders:

(23) |

Comparison with (10) shows that is given by the following equations:

(24) |

The quantity is derived from as in (20), but in addition it also contains the perturbation of the basis vector inside the . In the same way we define .

Using the solution (21), valid at each order, and the relations (11) to compute the averages, it is now straightforward to write down the general expressions for the two-point and three-point correlators of the adiabatic () component of , which is the component of , or rather their Fourier transforms, the power spectrum and the bispectrum. Making use of the short-hand notation

(25) |

we find for the power spectrum:

(26) |

We emphasise here that the quantity defined by (25) is simply made up of the linear , mode functions. One needs to evaluate the Green’s function and to perform the integral (25) when the linear source terms , in are known only up to horizon crossing, . However, where analytic solutions are available for , or in a fully numerical approach, we can dispense with the integral (25) by using the linear solution on super-horizon scales.

To find the bispectrum the calculation is slightly longer. One first has to compute . As we noted in the single-field case in sf , is indeterminate. To remove this ambiguity and also require that perturbations have a zero average, we define . Expanding , the three-point correlator will be a combination of the different permutations of , and the bispectrum is its Fourier transform. The intermediate steps are given in more detail in the explicit calculation in section V; here we go directly to the end result for the bispectrum of the adiabatic component:

(27) |

with

(28) |

If is real, this simplifies to

(29) |

This integral expression is a key result of this paper. Using our methodology, the three-point correlator with full momentum dependence has been expressed as a single time integral over quantities determined by the background model and the linear perturbations, that is, respectively the matrix and the solution embedded in (24) and (25) (in both of which the background is also implicit). Of course, one also has to find the Green’s function from (22), but, like the equation for in formalism , this is a linear ordinary differential equation for which there is no serious impediment to finding a numerical solution, in cases where an analytic or semi-analytic solution is unknown. The integral solution (29), then, demonstrates that the calculation of the three-point correlator is straightforward and tractable. It is, in principle, similar to calculations of the power spectrum, where accurate estimates can be found from background quantities, for example, in the slow-roll approximation. Here, we only have to supplement this with the amplitudes of the linear perturbation mode functions and the closely related Green’s function . In section V.4, using this methodology, we provide some quantitative semi-analytic results for the bispectrum of a two-field inflation model with a quadratic potential.

Before closing this section a final comment is in order. A feature of (29) which may at first sight cast doubt on its utility for quantitative calculations is its apparent dependence on the ad hoc choice for the functional form of the window function . Closer inspection reveals that the second term of (29) (the term) does not depend on for scales sufficiently larger than the horizon. This is evident from the fact that (25) is simply the solution to linear theory smoothed on scales larger than the horizon. Any properly normalised window function with for scales sufficiently larger than the horizon will produce the same final answer. The term represents the non-linear evolution on superhorizon scales and, as we show below, it describes an integrated effect which can lead to large non-Gaussianity. In contrast, the term arises from perturbations around horizon crossing and may depend on . We find below that this term is localized around horizon crossing and that it does not give rise to observationally interesting effects. Section V.5 provides more discussion on these points. In mf2 we explicitly show that taking a step function instead of a Gaussian as window function does not change the leading-order integrated effects.

## Iv Slow-roll approximation

The perturbation quantity can be computed
exactly numerically, or analytically within the slow-roll
approximation where all slow-roll parameters are assumed to
be smaller than unity. The latter was done in vantent to
next-to-leading order in slow roll:^{3}^{3}3Compared with the
solution in vantent there is an extra factor of
. It has to be introduced to take into account the
difference between the classical Gaussian random numbers ,
which have , and the quantum creation/annihilation operators
, , which have . In sf we introduced this factor of
in the analogue of equation (10), which
leads of course to identical results.

(30) |

where the matrices and are defined by

(31) |

with Euler’s constant. Overall unitary factors that are physically irrelevant have been omitted. Using this expression the source terms are given by

(32) |

Now when computing as defined in (23), or any higher-order terms in the perturbative expansion, we would in principle have to make the background quantities in dependent on x and perturb them according to (20). However, from (30) we see that depends on x only beyond leading order in slow roll (to leading order it is just given by ). Hence in a leading-order slow-roll approximation the only non-linear parts in the source terms are the factors in front of the integrals in (32), plus the basis vector inside the .

Within the leading-order slow-roll approximation, we now look at the two-field case, to make things a bit more explicit. In that case the matrix in (17) is given by

(33) |

The quantity is first order in slow roll. Here we used (9) and the relation, valid to leading order in slow roll (see e.g. vantent ),

(34) |

Using the constraints (14)–(16) we can compute the spatial derivatives that are needed to calculate . Some of these were given in a general form in formalism ; to first order in slow roll for the coefficients and to second order for the coefficients we find in the two-field case,

(35) | |||||

introducing the two second-order slow-roll quantities and as short-hand notation:

(36) |

with . The reason for this specific definition of will become clear later on. Since we have only two fields, the notation is unambiguous. To compute we used that in the two-field case, because of the orthonormality of the basis vectors, . All slow-roll parameters in these expressions take their homogeneous background values. From this we find that the rank-3 matrix (defined below (24)) is

(37) |

In the same way we find that the matrices and , defined in (24), are given by