On the scalar consistency relation away from slow roll

On the scalar consistency relation away from slow roll

V. Sreenath,    Dhiraj Kumar Hazra    and L. Sriramkumar
today
Abstract

As is well known, the non-Gaussianity parameter , which is often used to characterize the amplitude of the scalar bi-spectrum, can be expressed completely in terms of the scalar spectral index in the squeezed limit, a relation that is referred to as the consistency condition. This relation, while it is largely discussed in the context of slow roll inflation, is actually expected to hold in any single field model of inflation, irrespective of the dynamics of the underlying model, provided inflation occurs on the attractor at late times. In this work, we explicitly examine the validity of the consistency relation, analytically as well as numerically, away from slow roll. Analytically, we first arrive at the relation in the simple case of power law inflation. We also consider the non-trivial example of the Starobinsky model involving a linear potential with a sudden change in its slope (which leads to a brief period of fast roll), and establish the condition completely analytically. We then numerically examine the validity of the consistency relation in three inflationary models that lead to the following features in the scalar power spectrum due to departures from slow roll: (i) a sharp cut off at large scales, (ii) a burst of oscillations over an intermediate range of scales, and (iii) small, but repeated, modulations extending over a wide range of scales. It is important to note that it is exactly such spectra that have been found to lead to an improved fit to the CMB data, when compared to the more standard power law primordial spectra, by the Planck team. We evaluate the scalar bi-spectrum for an arbitrary triangular configuration of the wavenumbers in these inflationary models and explicitly illustrate that, in the squeezed limit, the consistency condition is indeed satisfied even in situations consisting of strong deviations from slow roll. We conclude with a brief discussion of the results.


On the scalar consistency relation away from slow roll

  • Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India

  • Asia Pacific Center for Theoretical Physics, Pohang, Gyeongbuk 790-784, Korea

E-mail: sreenath@physics.iitm.ac.in, dhiraj@apctp.org, sriram@physics.iitm.ac.in

 

 

1 Introduction

Over the past two decades, cosmologists have dedicated a considerable amount of attention to hunting down credible models of inflation. The inflationary scenario, which is often invoked to resolve certain puzzles (such as the horizon problem) that plague the hot big bang model, is well known to provide an attractive mechanism for the origin of perturbations in the early universe [1, 2, 3]. In the modern viewpoint, it is the primordial perturbations generated during inflation that leave their signatures as anisotropies in the Cosmic Microwave Background (CMB) and later lead to the formation of the large scale structure. Ever since the discovery of the CMB anisotropies by COBE [4], there has been a constant endeavor to utilize cosmological observations to arrive at stronger and stronger constraints on models of inflation. While the CMB anisotropies have been measured with ever increasing precision by missions such as WMAP [5, 6, 7], Planck [8, 9, 10] and, very recently, by BICEP2 [11, 12], it would be fair to say that we still seem rather far from converging on a small class of well motivated and viable inflationary models (in this context, see Refs. [13, 14, 15]).

The difficulty in arriving at a limited set of credible models of inflation seems to lie in the simplicity and efficiency of the inflationary scenario. Inflation can be easily achieved with the aid of one or more scalar fields that are slowly rolling down a relatively flat potential. Due to this reason, a plethora of models of inflation have been proposed, which give rise to the required or so e-folds of accelerated expansion that is necessary to overcome the horizon problem. Moreover, there always seem to exist sufficient room to tweak the potential parameters in such a way so as to result in a nearly scale invariant power spectrum of the scalar perturbations that lead to a good fit to the CMB data. In such a situation, non-Gaussianities in general and the scalar bi-spectrum in particular have been expected to lift the degeneracy prevailing amongst the various inflationary models. For convenience, the extent of non-Gaussianity associated with the scalar bi-spectrum is often expressed in terms of the parameter commonly referred to as  [16], a quantity which is a dimensionless ratio of the scalar bi-spectrum to the power spectrum. The expectation regarding non-Gaussianities alluded to above has been largely corroborated by the strong limits that have been arrived at by the Planck mission on the value of the parameter [17]. These bounds suggest that the observed perturbations are consistent with a Gaussian primordial distribution. Also, the strong constraints imply that exotic models which lead to large levels of non-Gaussianities are ruled out by the data.

Despite the strong bounds that have been arrived at on the amplitude of the scalar bi-spectrum, there exist many models of inflation that remain consistent with the cosmological data at hand. The so-called scalar consistency relation is expected to play a powerful role in this regard, ruling out, for instance, many multi-field models of inflation, if it is confirmed observationally (for early discussion in this context, see, for instance, Refs. [18, 19]; for recent discussions, see Refs. [20]; for similar results that involve the higher order correlation functions, see, for example, Refs. [21]). According to the consistency condition, in the squeezed limit of the three-point functions wherein one of the wavenumbers associated with the perturbations is much smaller than the other two, the three-point functions can be completely expressed in terms of the two-point functions111It should be added here that, in a fashion similar to that of the purely scalar case, one can also arrive at consistency conditions for the other three-point functions which involve tensors [22, 23, 24, 25, 26, 27].. In the squeezed limit, for instance, the scalar non-Gaussianity parameter can be expressed completely in terms of the scalar spectral index as  [18, 19]. As we shall briefly outline later, the consistency conditions are expected to hold [27] whenever the amplitude of the perturbations freeze on super-Hubble scales, a behavior which is true in single field models where inflation occurs on the attractor at late times (see Refs. [1]; in this context, also see Refs. [28]). While the scalar consistency relation has been established in the slow roll scenario, we find that there has been only a limited effort in explicitly examining the relation in situations consisting of periods of fast roll [29, 30]. Moreover, it has been shown that there can be deviations from the consistency relation under certain conditions, particularly when the field is either evolving away from the attractor [31] or when the perturbations are in an excited state above the Bunch-Davies vacuum [32]. In this work, our aim is to verify the validity of the scalar consistency relation in inflationary models which exhibit non-trivial dynamics. By considering a few examples, we shall explicitly show, analytically and numerically, that the scalar consistency relation holds even in scenarios involving strong deviations from slow roll.

The remainder of this paper is organized as follows. In the next section, we shall quickly summarize a few essential points and results concerning the scalar power spectrum and the bi-spectrum. We shall also briefly revisit the proof of the scalar consistency relation in the squeezed limit. In the succeeding section, we shall explicitly verify the validity of the consistency condition analytically in the cases of power law inflation and the Starobinsky model which is described by a linear potential with a sudden change in its slope. We shall then evaluate the scalar bi-spectrum numerically for an arbitrary triangular configuration of the wavenumbers in three inflationary models that lead to features in the power spectrum, and examine the consistency condition in the squeezed limit. We conclude with a brief discussion on the results we obtain.

A few remarks on our conventions and notations seem essential at this stage of our discussion. We shall work with natural units wherein , and define the Planck mass to be . We shall adopt the signature of the metric to be . We shall assume the background to be the spatially flat, Friedmann-Lemaître-Robertson-Walker (FLRW) line element that is described by the scale factor and the Hubble parameter . As is convenient, we shall switch between various parametrizations of time, viz. the cosmic time , the conformal time  or e-folds denoted by . An overdot and an overprime shall represent differentiation with respect to the cosmic and the conformal time coordinates, respectively. We shall restrict our attention in this work to inflationary models involving the canonical scalar field. Note that, in such a case, the first and second slow roll parameters are defined as and .

2 The scalar bi-spectrum in the squeezed limit

In this section, we shall quickly summarize the essential definitions and governing expressions concerning the scalar power spectrum, the bi-spectrum and the corresponding non-Gaussianity parameter . We shall also sketch a simple proof of the consistency relation obeyed by the non-Gaussianity parameter in the squeezed limit of the scalar bi-spectrum.

2.1 The scalar power spectrum and bi-spectrum

Consider the following line-element which describes the spatially flat, FLRW spacetime, when the scalar perturbations, characterized by the curvature perturbation , have been taken into account:

(2.1)

Let denote the Fourier modes associated with the curvature perturbation at the linear order in the perturbations. In the case of inflation driven by the canonical scalar field that is of our interest here, the modes satisfy the differential equation [2, 3]

(2.2)

where . Upon quantization, the curvature perturbation can be decomposed in terms of the Fourier modes as

(2.3)

where and are the usual creation and annihilation operators that obey the standard commutation relations.

The scalar power spectrum is defined in terms of the two-point correlation function of the curvature perturbation as follows:

(2.4)

where denotes the Bunch-Davies vacuum annihilated by the operator  [33]. In terms of the modes , the scalar power spectrum is given by

(2.5)

The inflationary model governed by a given potential determines the behavior of the quantity . In order to arrive at the scalar power spectrum using the above expression, one first solves the differential equation (2.2) for the modes with the Bunch-Davies initial conditions [33], and then evaluates the amplitude of the modes at sufficiently late times when they are well outside the Hubble radius during inflation. The scalar spectral index is defined as

(2.6)

It should be stressed here that the scalar spectral index proves to be a constant only in simple situations such as power law and slow roll inflation. In general, when the power spectrum contains features, the quantity depends on the wavenumber .

The scalar bi-spectrum evaluated, say, in the vacuum state , is defined in terms of the three-point function of the curvature perturbation as follows [6]:

(2.7)

Note that the delta function on the right hand side imposes the triangularity condition, viz. that the three wavevectors , and have to form the edges of a triangle. For the sake of convenience, we shall set

(2.8)

The non-Gaussianity parameter  that is often used to characterize the extent of non-Gaussianity indicated by the bi-spectrum is introduced through the relation [16]

(2.9)

where denotes the Gaussian part of the curvature perturbation. Using the above relation and Wick’s theorem (which applies to Gaussian perturbations), one can arrive at the following expression for the dimensionless non-Gaussianity parameter in terms of the bi-spectrum and the scalar power spectrum :

(2.10)

The scalar bi-spectrum generated during inflation can be evaluated using the Maldacena formalism [18]. The approach basically makes use of the third order action governing the curvature perturbation and the standard rules of perturbative quantum field theory to arrive at the scalar three-point function [18, 34, 35]. It is found that, in the case of inflation driven by the canonical scalar field, the third order action consists of six terms and the scalar bi-spectrum receives a contribution from each of these ‘vertices’. In fact, there also occurs a seventh term which arises due to a field redefinition, a procedure which is necessary to reduce the action to a simpler form. One can show that the complete contribution to the scalar bi-spectrum in the perturbative vacuum can be written as [36, 37, 38]

(2.11)

where are the Fourier modes in terms of which we had decomposed the curvature perturbation at the linear order in the perturbations [cf. Eq. (2.3)]. In the above expression, the quantities with correspond to the six vertices in the interaction Hamiltonian (obtained from the third order action), and are described by the integrals

(2.12a)
(2.12b)
(2.12c)
(2.12d)
(2.12e)
(2.12f)

These integrals are to be evaluated from a sufficiently early time, say, , when the initial conditions are imposed on the modes until very late times, say, towards the end of inflation at . The additional, seventh term arises due to the field redefinition and its contribution to the bi-spectrum is given by

(2.13)

2.2 The consistency relation

The squeezed limit refers to the case wherein one of the wavenumbers of the triangular configuration vanishes, say, , leading to . Or, equivalently, one of the modes is assumed to possess a wavelength which is much larger than the other two. The long wavelength mode would be well outside the Hubble radius. In models of inflation driven by a single scalar field, the amplitude of the curvature perturbation freezes on super-Hubble scales, provided the inflaton evolves on the attractor at late times [28, 31]. As a result, the long wavelength mode simply acts as a background as far as the other two modes are concerned. If is the amplitude of the curvature perturbation associated with the long wavelength mode, then the unperturbed part of the original FLRW metric will be modified to

(2.14)

In other words, the effect of the long wavelength mode is to modify the scale factor locally, which is equivalent to a spatial transformation of the form , with the components of the matrix being given by . Under such a transformation, the modes of the curvature perturbation transform as . Further, we have and . Utilizing these relations, the scalar two-point function can be written as

(2.15)

where the suffix on the two-point function indicates that the correlator has been evaluated in the presence of a long wavelength perturbation. Upon using the above expression for the scalar power spectrum, we can write the scalar bi-spectrum in the squeezed limit as [19, 26, 27]

(2.16)

On making use of this expression for the scalar bi-spectrum in the squeezed limit and the definition of the scalar power spectrum, one can immediately arrive at the consistency relation for , viz. that  [18, 19, 20].

3 Analytically examining the validity of the condition away from slow roll

As was outlined in the previous section, the only requirement for the validity of the consistency relation is the existence of a unique clock during inflation. Hence, in principle, this relation should be valid for any single field model of inflation irrespective of the detailed dynamics, if the field is evolving on the attractor at late times. Therefore, it should be valid even away from slow roll. In this section, we shall analytically examine the validity of the consistency condition in scenarios consisting of deviations from slow roll. After establishing the relation first in the simple case of power law inflation, we shall consider the Starobinsky model which involves a brief period of fast roll.

3.1 The simple example of power law inflation

We shall first consider the case of power law inflation with no specific constraints on the power law index, so that the behavior of the scale factor can be far different from that of its behavior in slow roll inflation. In power law inflation, the scale factor can be written as

(3.1)

where and are constants, and . In such a background, the Fourier modes associated with the curvature perturbation that satisfy the Bunch-Davies initial conditions are found to be [37, 39]

(3.2)

where the first slow roll parameter is a constant given by . Note that denotes the Hankel function of the first kind [40], while the scale factor is given by Eq. (3.1). For real arguments, the Hankel functions of the first and the second kinds, viz. and , are complex conjugates of each other [40]. Moreover, as , the Hankel function has the following form

(3.3)

Upon using this behavior, one can show that the corresponding scalar power spectrum, evaluated at late times, i.e. as , is given by

(3.4)

where represents the Gamma function [40]. The scalar spectral index corresponding to such a power spectrum is evidently a constant and can be easily determined to be . If the consistency condition is true, it would then imply that the scalar non-Gaussianity parameter has the value in the squeezed limit.

Let us now evaluate the scalar bi-spectrum in the squeezed limit using the Maldacena formalism and illustrate that it indeed leads to the above consistency condition for . It should be clear that, in order to arrive at the complete scalar bi-spectrum, we first need to carry out the integrals (2.12) associated with the six vertices, calculate the corresponding contributions for , and lastly add the contribution [cf. Eq. (2.13)] that arises due to the field redefinition. However, since is a constant in power law inflation, the second slow roll parameter vanishes identically. As a result, the contribution corresponding to the fourth term that is determined by the integral (2.12d) as well as the seventh term prove to be zero. Moreover, in the squeezed limit of our interest, i.e. as , the amplitude of the mode freezes and hence its time derivative goes to zero. Therefore, terms that are either multiplied by the wavenumber corresponding to the long wavelength mode or explicitly involve the time derivative of the long wavelength mode do not contribute, as both vanish in the squeezed limit. Due to these reasons, one finds that it is only the first and the second terms, determined by the integrals (2.12a) and (2.12b), that contribute in power law inflation. After an integration by parts, we find that, in the squeezed limit, these two integrals can be combined to be expressed as

(3.5)

where we have set and . One can show that the derivative can be written as

(3.6)

Therefore, upon using this expression for the derivative , the behavior (3.3), the following asymptotic form of the Hankel function

(3.7)

and the integral [40]

(3.8)

we find that the bi-spectrum in the squeezed limit can be written as

(3.9)

This expression and the definition (2.10) for the scalar non-Gaussianity parameter then leads to , which is the result suggested by the consistency relation. We should add here that such a result has been arrived at earlier using a slightly different approach (see the third reference in Refs. [20]).

3.2 A non-trivial example involving the Starobinsky model

The second example that we shall consider is the Starobinsky model. In the Starobinsky model, the inflaton rolls down a linear potential which changes its slope suddenly at a particular value of the scalar field [41]. The governing potential is given by

(3.10)

where , , and are constants. An important aspect of the Starobinsky model is the assumption that it is the constant  which dominates the value of the potential around . Due to this reason, the scale factor always remains rather close to that of de Sitter. This in turn implies that the first slow roll parameter remains small throughout the domain of interest. However, the discontinuity in the slope of the potential at causes a transition to a brief period of fast roll before slow roll is restored at late times. One finds that the transition leads to large values for the second slow roll parameter and, importantly, the quantity grows to be even larger, in fact, behaving as a Dirac delta function at the transition. As we shall discuss, it is this behavior that leads to the most important contribution to the scalar bi-spectrum in the model [36, 42, 43].

Clearly, it would be convenient to divide the evolution of the background quantities and the perturbation variables into two phases, before and after the transition at . In what follows, we shall represent the various quantities corresponding to the epochs before and after the transition by a plus sign and a minus sign (in the super-script or sub-script, as is convenient), while the values of the quantities at the transition will be denoted by a zero. Let us quickly list out the behavior of the different quantities which we shall require to establish the consistency relation.

The first slow roll parameter before and after the transition is found to be [41, 36, 42, 43]

(3.11a)
(3.11b)

where , is the Hubble parameter determined by the relation , and denotes the conformal time when the transition takes place. The second slow roll parameter is given by

(3.12a)
(3.12b)

In fact, to determine the modes associated with the scalar perturbations and to evaluate the dominant contribution to the scalar bi-spectrum, we shall also require the behavior of the quantity . One can show that can be expressed as

(3.13)

where , and it should be stressed that this is an exact relation. It should be clear that the first term in the above expression involving will lead to a Dirac delta function due to the discontinuity in the first derivative of the potential in the case of the Starobinsky model. Hence, the dominant contribution to at the transition can be written as [42, 43]

(3.14)

where denotes the value of the scale factor when . Post transition, the dominant contribution to is found to be [36]

(3.15)

Due to the fact that the potential is linear and also since the first slow roll parameter remains small, the modes governing the curvature perturbation can be described by the conventional de Sitter modes to a good approximation before the transition. For the same reasons, one finds that the scalar modes can be described by the de Sitter modes soon after the transition as well. However, due to the transition, the modes after the transition are related by the Bogoliubov transformations to the modes before the transition. Therefore, the scalar mode and its time derivative before the transition can be written as [41, 36, 44, 42, 43]:

(3.16a)
(3.16b)

Whereas, the mode and its derivative after the transition can be expressed as follows:

(3.17a)
(3.17b)

with and denoting the Bogoliubov coefficients. Upon matching the above modes and their time derivatives at the transition, the Bogoliubov coefficients can be determined to be

(3.18a)
(3.18b)

where denotes the mode that leaves the Hubble radius at the transition. At late times, the scalar mode behaves as

(3.19)

where . Therefore, the scalar power spectrum, evaluated as , can be expressed as

(3.20)

where the quantities , and are given by

(3.21a)
(3.21b)
(3.21c)

Note that, because of the features in the power spectrum, the corresponding scalar spectral index depends on the wavenumber , and is found to be

(3.22)

where , and are given by

(3.23a)
(3.23b)
(3.23c)

If the consistency condition is indeed satisfied, then the non-Gaussianity parameter, as predicted by the relation, would prove to be

Let us now examine whether we do arrive at the same result upon using the Maldacena formalism to compute the scalar bi-spectrum. It is known that, when there exist deviations from slow roll, it is the fourth vertex that leads to the most dominant contribution to the bi-spectrum. In other words, we need to focus on the contribution  that is governed by the integral (2.12d). Notice that the integral involves the quantity . In the Starobinsky model, at the level of approximation we are working in, , with being a constant [cf. Eqs. (3.12a) and (3.11a)]. Hence, as well as the integral vanish during the initial slow roll phase, prior to the transition. However, as we discussed above, due to the discontinuity at , is described by a delta function at the transition [cf. Eq. (3.14)], whereas, post transition, it is given by Eq. (3.15). Since the mode and its derivative are continuous, the contribution due to the delta function at the transition can be easily evaluated using the modes and the corresponding derivative [cf. Eqs. (3.16)]. Since we are interested in the squeezed limit, the contribution at the transition can be written as

(3.25)

The corresponding contribution to the bi-spectrum can be easily evaluated using the late time behavior (3.19) of the mode . The contribution after the transition is governed by the integral