1 Introduction

Abstract

We consider theories which explain the flatness of the power spectrum of scalar perturbations in the Universe by conformal invariance, such as conformal rolling model and Galilean Genesis. We show that to the leading non-linear order, perturbations in all models from this class behave in one and the same way, at least if the energy density of the relevant fields is small compared to the total energy density (spectator approximation). We then turn to the intrinsic non-Gaussianities in these models (as opposed to non-Gaussianities that may be generated during subsequent evolution). The intrinsic bispectrum vanishes, so we perform the complete calculation of the trispectrum and compare it with the trispecta of local forms in various limits. The most peculiar feature of our trispectrum is a (fairly mild) singularity in the limit where two momenta are equal in absolute value and opposite in direction (folded limit). Generically, the intrinsic non-Gaussianity can be of detectable size.

INR-TH-2011-10

Non-Gaussianity of scalar perturbations

[0.3cm] generated by conformal mechanisms

M. Libanov, S. Mironov, V. Rubakov

Institute for Nuclear Research of the Russian Academy of Sciences,

60th October Anniversary Prospect, 7a, 117312 Moscow, Russia;

Physics Department, Moscow State University

Vorobjevy Gory, 119991, Moscow, Russia

1 Introduction

Scalar perturbations in the Universe may well originate from inflation [1]. In that case, approximate flatness of their power spectrum is due to the approximate de Sitter symmetry of inflating background. As pointed out in Ref. [2], a possible alternative to the de Sitter symmetry in this context is conformal invariance. Concrete models employing conformal invariance, instead of the de Sitter symmetry, for explaining the flat scalar spectrum include conformal rolling [3] and Galilean Genesis [4]. Despite substantially different motivations and dynamical features, the latter models end up in one and the same mechanism that generates scalar perturbations. From this prospective, the theory boils down to a model with two scalar fields and and the Lagrangian

(1)

where governs the dynamics of . Under scaling transformations, the field scales as ; it provides for a non-trivial background . The field scales as ; the perturbations of the field serve as predecessors of the adiabatic perturbations. One makes sure that at the time the perturbations of are generated, gravitational effects on the dynamics of the two fields are irrelevant (this is achieved in different ways in Ref. [3] and Ref. [4]) and assumes that the background field is spatially homogeneous. Then conformal invariance of the field equation for implies that

(2)

where is conformal time in the conformal rolling model and is cosmic time in the Galilean Genesis scenario. In such a background, the field behaves exactly in the same way as massless scalar field (minimally coupled to gravity) in the de Sitter space-time. The modes of its perturbations about a certain background value ,

start off in the WKB regime and freeze out when , where is (conformal) momentum. Assuming that the field is originally in its vacuum state, one obtains, at the linearized level, the flat power spectrum of the Gaussian field at late times1, when . The conformal stage ends up at some point, and the -perturbations are reprocessed into adiabatic ones at much later epoch via, e.g., curvaton [6] or modulated decay [7] mechanism.

A common feature of the conformal models of Refs. [3, 4] is the existence of the perturbations of the field about the background (2). The Lagrangian contains a small parameter, call it , then ( is quartic self-coupling in the conformal rolling model; we relate the parameter to the parameters of the galileon Lagrangian in the Galilean Genesis scenario in Section 2.1, see Eq. (9)). As discussed in Refs. [3, 4], the perturbations have rather peculiar properties. Nevertheless, as we review in Section 2.1, these perturbations are exactly the same (modulo overall amplitude) in the two, apparently very different models. Furthermore, in Section 2.2 we give a general argument showing that the properties of the linearized perturbations are common to a large class of models employing conformal invariance.

There is a subtlety here. Strictly speaking, our observation is valid in a theory without gravity or in the case when the energy density of the field is small compared to the total energy density in the Universe (spectator approximation). Though is is likely that our results are valid in much more general setting, the corresponding analysis is yet to be done. We proceed in this paper by neglecting gravity altogether.

So, it is of interest to study the effects of the perturbations on the field and, in the end, on the adiabatic perturbations. There are at least two of these effects, namely, the statistical anisotropy and non-Gaussianity. Of course, the non-Gaussianity may well be generated also at the time the -perturbations are converted into the adiabatic ones. In this respect the conformal models are nothing special as compared to inflationary theories equipped with the curvaton or modulated decay mechanism; for this reason we are going to disregard this conversion-related non-Gaussianity. What we are interested in is the intrinsic non-Gaussianity, which is due to the interaction of the field with perturbations .

The resulting phenomenology depends strongly on what happens to the field after the conformal stage (2) ends up. One option is that the field starts evolving again, and its evolution proceeds until it becomes superhorizon in the conventional sense. This option is fairly natural in the conformal rolling model of Ref. [3] and more contrived in the Galilean Genesis2 of Ref. [4]. Both statistical anisotropy and non-Gaussianity generated in this case are studied in Ref. [8].

Here we consider the opposite case, i.e., we assume that the field does not evolve after the end of the conformal stage. This option is particularly natural in the Galilean Genesis, but it is not contrived in the conformal rolling scenario either. In models from this sub-class, the adiabatic perturbations inherit the statistical properties of -perturbations that exist already at the late conformal stage, when . The statistical anisotropy in the field is induced by long-ranged perturbations . This effect has been studied in Ref. [9] with the result that the leading statistical anisotropy in the power spectrum of , and hence of adiabatic perturbation , has the quadrupole form,

where is isotropic (and nearly flat) power spectrum obtained at the linearized level, and are unit 3-vector and unit traceless 3-tensor of a general form, is the present value of the Hubble parameter, the parameter is of order 1 and is positive and logarithmically enhanced.

The main purpose of this paper is to study the intrinsic (as opposed to conversion-related) non-Gaussianity in this sub-class of models. In the absence of the cubic self-interaction of the field , the intrinsic bispectrum vanishes, so we have to consider the intrinsic trispectrum. Perhaps the most striking feature of the trispectrum is the singularity in the limit where two momenta are equal in absolute value and have opposite directions (folded limit, in nomenclature of Ref. [10]). The singular part of the connected four-point function has been calculated in our earlier paper [11] with the result

(3)

i.e., the trispectrum blows up as . This is in contrast to trispectra obtained in single-field inflationary models [10, 12, 13, 14, 15], and, indeed, there are general arguments [13] showing that in these models, the four-point function is finite in the limit . The singularity in the four-point function (3) is due to the enhancement of the perturbations at low momenta, see the discussion of the infrared properties of the conformal models in Ref. [9]. We will see in Section 3.3 that the most relevant features of the trispectrum are captured by its singular part.

Whether the intrinsic non-Gaussianity dominates over the conversion-related one, and whether the former is detectable depends on both the underlying conformal model and the mechanism that converts the -perturbations into the adiabatic ones. Generically, the linear order relationship between and is , where is the dilution factor. With our normalization (1), (2), the power spectrum of -perturbations is . The background value cannot be estimated in a model independent way. It can be as large as , so that the correct adiabatic amplitude is obtained at . This can be the case, e.g., in the Galilean Genesis model. In such a situation, the conversion-related non-Gaussianities induced by the curvation or modulated decay mechanism are fairly small (, are roughly of order 1 [16, 17, 18, 19]). On the other hand, the size of the intrinsic non-Gaussianity (see Eq. (36) for its definition) is governed by the amplitude of the perturbations , so that

see Eq. (37) for numerical coefficient. There is no general reason to expect that the parameter is particularly small, except for the mild requirement ensuring the self-consistency of the conformal scenario. This shows that the intrinsic non-Gaussianity may well be dominant and detectable (say, ), though it size cannot be predicted because of our ignorance of the value of the parameter . The situation is more subtle in the conformal rolling model; we consider this point in Section 3.3 with the result that dominant and detectable intrinsic non-Gaussianity is possible, but not generic.

This paper is organized as follows. To set the stage, we consider in Section 2.1 the linearized perturbations and in the Galilean Genesis scenario and compare the resulting expressions with those obtained in the conformal rolling model. This Section contains nothing new as compared to Refs. [3, 4]; the main point is to show that the perturbations and are identically the same in the two scenarios. In Section 2.2 we give a general argument showing that the properties of are uniquely determined by conformal invariance (modulo the overall constant amplitude). Hence, the peculiarities of the adiabatic perturbations that we study in this paper, as well as the results of Refs. [9, 11], are common to the whole class of conformal mechanisms. We turn to the non-Gaussianity in Section 3, where we perform the complete calculation of the intrinsic trispectrum. We confirm our earlier result (3) concerning the singular part of the trispectrum. We then consider various limits, following the nomenclature of Refs. [10, 15], and compare them with local models [20, 21]. We conclude in Section 4. Some details of our calculations are collected in Appendix.

2 Perturbations in conformal scenarios

2.1 Galilean Genesis vs. conformal rolling

The galileon model has been introduced in Ref. [22]. The rolling galileon serves as the field entering (1) and (2). In the Minkowski space-time, the Lagrangian of the simplest conformally-invariant version [4] of the model is (mostly negative signature)

(4)

where . The field equation in Minkowski space-time admits the homogeneous solution such that

(5)

where and

Hence, one defines

(6)

so that the background solution is given by (2).

The quadratic action for perturbations about this solution is

It is convenient to introduce the variable . It follows from (6) that is the perturbation of the field about the background . Its action reads

The field equation has a simple form,

(7)

Its properly normalized solution for given momentum is

(8)

where and are creation and annihilation operators obeying the standard commutational relation , and

(9)

The theory is weakly coupled in the relevant range of momenta provided that .

The mode (8) oscillates at early times, when , while at late times it behaves as follows,

This behavior can be interpreted as local time shift [3, 4]. Indeed, for time-shifted galileon solution (5) we have

Hence,

Thus, is independent of time at late times, and its power spectrum is red,

(10)

In the Galilean Genesis scenario, the field is introduced as an additional field precisely for the purpose of generating the scalar perturbations. By conformal invariance, its quadratic Lagrangian has the form of the second term in (1). In the background , its modes are

(11)

where and is another set of creation and annihilation operators. At early times, the field is in the WKB regime, while for the mode stays constant in time. The resulting late-time power spectrum is flat,

(12)

This result is valid at the linearized level. The lowest order interaction between and is described by the interaction Hamiltonian, whose density is (in the interaction picture)

(13)

It is this interaction that is responsible for the intrinsic non-Gaussianity which we study in Section 3.

In the above discussion we neglected gravity effects. This is legitimate in the Galilean Genesis scenario at the early stage of Genesis, when the energy density of the galileon field is small, while the energy density of other fields is assumed to vanish. The latter assumption is relaxed in the conformal rolling scenario whose main ingredient is a complex scalar field conformally coupled to gravity. Unlike in the galileon case, the scalar potential does not vanish; it is assumed to be negative and is quartic by conformal symmetry, . One makes use of the parametrization

(14)

Then the background field rolls down the potential according to (2), where is now conformal time [3]. The Lagrangian for coincides with the second term in (1), while the perturbations are again governed by Eq. (7). The properly normalized solutions for the field and perturbations coincide with (11) and (8), respectively, with being now the quartic self-coupling and . So, the dynamics of perturbations in the conformal rolling scenario is identical to that in the Galilean Genesis.

2.2 General argument

Let us see that the form of Eq. (7) that governs the perturbations at the linearized level is completely determined by conformal invariance. Namely, let us consider any weakly coupled theory, in which the classical field equation for is second order in derivatives, the action for is local, invariant under space-time translations and spatial rotations and invariant under scaling

and inversion

One designs a theory in such a way that it admits the runaway solution (one can always set the overall constant in equal to 1 by field redefinition). This solution is invariant under both scaling and inversion. So, the quadratic action for perturbations about this solution must also be invariant. Let us write for the quadratic action

(15)

where is second order differential operator, and is some constant whose choice is specified below. Since the background depends only on time, the operator does not contain spatial coordinates explicitly, but may contain time. Also, it is invariant under spatial rotations.

Invariance under scaling implies that modulo overall constant, the part of that involves derivatives is

(16)

where is independent of time. We can choose the constant in (15) in such a way that the term with two time derivatives enters with the coefficient . Scale invariance alone is insufficient3 for obtaining . However, the requirement of invariance under inversion uniquely specifies . The term without derivatives is almost uniquely determined from the requirement of invariance under scaling and inversion,

(17)

where is yet undetermined constant. To complete the argument we notice that the invariance of the original action under time translations implies that must be a solution to equation . This gives . Thus, the whole quadratic action for perturbations is uniquely determined by conformal invariance, modulo an overall constant factor, and the linearized equation for has one and the same form in the whole class of conformally-invariant models with the Lagrangians of the general form (1). To the leading non-linear order, the properties of the -perturbations are identical in these models, as they are governed by the interaction Hamiltonian (13).

3 Trispectrum

3.1 Generalities

In models with the Lagrangians of the form (1), the bispectrum of the field vanishes. To calculate the trispectrum we make use of the in-in formalism, cf. Ref. [23]. We use the shorthand notation

with understanding that we are interested in the formal limit

(18)

Then the four-point function reads

where subscript refers to interaction picture and the interaction Hamiltonian density is given by (13). Of course, we are going to calculate the connected part of the four-point function.

To this end we need the two-point functions of the linear fields and ,

By making use of (11) and (8) we find

(19)
(20)

Both of these pairing functions satisfy

We also need the (anti)-product of the field . We write

(21)

and

(22)

The connected four-point function is a sum of the three terms, one of which has the following form,

(23)

and the two others are

(24)

In the 3-dimensional momentum representation, these expressions reduce to the integrals over and . Our definition of the -point function in the momentum representation is

which corresponds to

With our convention, the power spectrum is related to the two-point function by

and at the linearized level is given by (12).

3.2 Leading singularity

Let us calculate the contribution to the trispectrum due to the term explicitly written in square brackets in (23). In the limit (18) it involves the following combination

(25)

where

(26)

and

(27)

with . So, we obtain for the contribution under study (in a certain sense this is the leading contribution, hence the notation)

where

and we made use of the fact that . To regularize integrals at infinity we insert into the integrand and take the limit in the end of the calculation. This is equivalent to the replacement which is the standard prescription for calculating vacuum expectation values in the interaction picture (see, e.g., Ref. [23]). At all integrals are converging. We observe that the integrals over and factor out and obtain

(28)

where

(29)

and

(30)

The expression (29) is singular in the folded limit , while vanishes in this limit. The crossing terms in (23), as well as the terms (24) are regular as (the latter property is established by direct calculation in Appendix). It is convenient to define the singular part of in the following way,

so that

Then the singular part of the four-point function is

(31)

In the limit we recover the result (3), which can be written in a symmetric form,

where

Note that the terms of order and , which one could off hand expect from (29), cancel out. Thus, the field has rather mild infrared behavior, even though it interacts with infrared-enhanced modes of (the power spectrum of is red, see (10)). The reason for this property is discussed in Ref. [9].

3.3 Shapes

The calculation of the contribution (24) is not so straightforward. We perform this calculation in Appendix, where we also present the complete result for the trispectrum. The definition of the trispectrum in our case is

where . The combinations and are not independent, since

It is convenient for the purpose of illustration to decompose the trispectrum into the part , which is singular as either or or , and the regular part ,

The singular part is obtained4 from (31),

(32)

while the rest can be read off from (42) and (43). We emphasize that there are no other singularities in : one can check that the logarithmic divergences appearing in intermediate formulas cancel out when one takes into account all contributions.

Following Ref. [10], we are going to compare our trispectrum to the trispectra of the local forms. The latter are obtained from the Ansatz in real space [20, 21]

(33)

where is the Gaussian field and and are constants. One has [10]

where the two local shapes are

(34)
(35)

Let us quantify the strength of the non-Gaussianity. The standard estimator is related to the four-point function in the regular tetrahedron limit, ,

(36)

Note that the standard definition involves in the right hand side. This is appropriate for the local Ansatz (33), and the sizes in the local models are [10]

On the other hand, the quantity we can directly calculate in our model is

where , see (12), and the size of the non-Gaussianity in in our model is obtained from (42) and (43),

(37)

The adiabatic perturbation is proportional to , namely,

(38)

where is the homogeneous background value of the scalar field, and is a dilution factor, which is independent of for both curvaton and modulated decay mechanism of conversion of the -perturbations into adiabatic ones. Therefore, in our model the size of the non-Gaussianity of the adiabatic perturbations is

As discussed in Section 1, there are no model-independent constraints on and , so the intrinsic non-Gaussianity we study in this paper may well dominate over conversion-related one and be detectable.

Let us point out, however, that in the conformal rolling scenario of Ref. [3], the adiabatic power spectrum is itself proportional to . The reason is that is the phase field. With our normalization, its background value is bounded from above, see Eq. (14),

and without fine tuning . Equation (38) then gives

so that

Therefore, the intrinsic non-Gaussianity can be sizeable only for small dilution factor . If the -perturbations are converted into adiabatic ones by the curvaton mechanism, the conversion-related non-Gaussianity is large at small ,  [16], so its contribution to the trispectrum is very roughly of the same order as that due to the intrinsic non-Gaussianity. The analysis of the detectability of the intrinsic non-Gaussianity in this case deserves further study. The same remark applies to the bulk of the modulated decay models, where as well [17]. There is an exception, however [18]: if the width of decaying particles is linear in or has the form (which is more plausible from particle physics prospective), then , are roughly of order 1 even for small , and hence large . We conclude that in the conformal rolling scenario, the domination and detectability of the intrinsic non-Gaussianity is possible, but not at all generic.

Let us now turn to the shapes. To compare them, we set in what follows

then the sizes of the trispectra in all models are similar.

Let us consider various limits of the shape function . We use the nomenclature of Refs. [10, 15]. The first three panels in Figs. 1, 2, 3 and 4 show the complete trispectrum , the contribution of the singular part and the regular part in our model, respectively. The fourth and fifth panels show the two local trispectra and . Note that vertical scales in the last two panels are different from each other and from vertical scales in the first three panels. The ranges of arguments in these figures are limited due to various inequalities obeyed by the momenta. In particular,