Sharp inflaton potentials and bi-spectra: Effects of smoothening the discontinuity
Sharp shapes in the inflaton potentials often lead to short departures from slow roll which, in turn, result in deviations from scale invariance in the scalar power spectrum. Typically, in such situations, the scalar power spectrum exhibits a burst of features associated with modes that leave the Hubble radius either immediately before or during the epoch of fast roll. Moreover, one also finds that the power spectrum turns scale invariant at smaller scales corresponding to modes that leave the Hubble radius at later stages, when slow roll has been restored. In other words, the imprints of brief departures from slow roll, arising out of sharp shapes in the inflaton potential, are usually of a finite width in the scalar power spectrum. Intuitively, one may imagine that the scalar bi-spectrum too may exhibit a similar behavior, i.e. a restoration of scale invariance at small scales, when slow roll has been reestablished. However, in the case of the Starobinsky model (viz. the model described by a linear inflaton potential with a sudden change in its slope) involving the canonical scalar field, it has been found that, a rather sharp, though short, departure from slow roll can leave a lasting and significant imprint on the bi-spectrum. The bi-spectrum in this case is found to grow linearly with the wavenumber at small scales, a behavior which is clearly unphysical. In this work, we study the effects of smoothening the discontinuity in the Starobinsky model on the scalar bi-spectrum. Focusing on the equilateral limit, we analytically show that, for smoother potentials, the bi-spectrum indeed turns scale invariant at suitably large wavenumbers. We also confirm the analytical results numerically using our newly developed code BINGO. We conclude with a few comments on certain related points.
a]Jérôme Martin, b]L. Sriramkumar c]and Dhiraj Kumar Hazra
Sharp inflaton potentials and bi-spectra
Sharp inflaton potentials and bi-spectra: Effects of smoothening the discontinuity
Institut d’Astrophysique de Paris, UMR7095-CNRS, Université Pierre et Marie Curie, 98bis boulevard Arago, 75014 Paris, France.
Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India.
Asia Pacific Center for Theoretical Physics, Pohang, Gyeongbuk 790-784, Korea.
Keywords: Cosmic Inflation, Cosmic Microwave Background, Non-Gaussianities
- 1 Inflationary models, discontinuities and the scalar bi-spectrum
- 2 Essential aspects of the Starobinsky model
- 3 The dominant contribution to the scalar bi-spectrum
- 4 Effects of smoothening the discontinuity: A simple analytical treatment
- 5 Smoothening the transition: A more general treatment
- 6 Comparison with the numerical results from BINGO
- 7 Discussion
1 Inflationary models, discontinuities and the scalar bi-spectrum
The inflationary scenario is a very efficient paradigm to resolve the puzzles of the standard cosmological model and to simultaneously describe the origin of perturbations in the early universe [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. Even the simplest of models lead to a sufficiently long duration of inflation that is required to overcome the horizon problem. Moreover, many of these models permit inflation of the slow roll type, which generates a nearly scale invariant primordial power spectrum that is remarkably consistent with the observations of the anisotropies in the Cosmic Microwave Background (CMB) and other cosmological data [22, 23, 24, 25, 26, 27, 28].
While attempting to identify the correct inflationary scenario, apart from the power spectrum, the non-Gaussianities and, in particular, the scalar bi-spectrum, also play a significant role. Indeed, the recent Planck data have shown that the non-Gaussianities are consistent with zero, with the three parameters that characterize the scalar bi-spectrum constrained to be: , and . These constraints imply that the correct model of inflation cannot deviate too much from the standard single field inflation of the slow roll type, involving the canonical kinetic term. In other words, inflation seems to be a non-trivial (i.e. , where denotes the scalar spectral index), but ‘non-exotic’ (viz. ) mechanism [29, 30, 31]. On the theoretical front, the most complete formalism to calculate the three-point correlation functions involving scalars and tensors in a given inflationary model is the approach due to Maldacena . In the Maldacena formalism, the three-point functions are evaluated using the standard rules of perturbative quantum field theory, based on the interaction Hamiltonian that depends cubically on the perturbations [32, 33, 34, 35, 36, 37, 38]. The resulting expressions for the three-point function of primary interest, viz. the scalar bi-spectrum, involves integrals over combinations of the background quantities such as the scale factor and the slow roll parameters as well as the modes describing the curvature perturbation (see, for instance, Refs. [39, 40]). Evaluating the bi-spectrum analytically for a generic inflationary model proves to be a non-trivial task. But, as in the case of the power spectrum, the bi-spectrum can be calculated analytically under the slow roll approximation [32, 33, 35, 41, 42, 43, 44].
As we have already mentioned, slow roll inflation driven by a single scalar field seems to be the most likely possibility to describe the early universe. Nevertheless, it is also interesting to consider other scenarios which lead to larger levels of non-Gaussianities. Such analyses can help us gain a better understanding of the constraints imposed by the Planck data on the parameters characterizing these class of models. Moreover, these exercises can actually allow us to assess the degree of fine-tuning implied by the CMB data from Planck and WMAP on the non-minimal alternatives (see, for example, Ref. [45, 46, 47]). Further, the recent claim of the detection of the imprints of the primordial tensor modes by BICEP2 and the indication of a relatively high tensor-to-scalar ratio [48, 49], if confirmed, implies that we cannot completely rule out non-trivial possibilities either. Studying non-standard scenarios is however not a simple task, since, in these situations, the calculation of non-Gaussianities can be highly non-trivial and one often has to rely on numerical calculations (for numerical analysis of specific models, see Refs. [50, 51, 52, 53, 54, 55, 56]; for a broader discussion on the procedures involved and applications to a few different classes of models, see Ref. ; in this context, also see Ref. ). However, occasionally, it is also possible to evaluate the bi-spectrum analytically in non-trivial situations such as scenarios involving departures from slow roll [59, 60, 61]. One such example that permits an analytic evaluation of the power spectrum and the complete bi-spectrum (at least, in the equilateral limit) even in the presence of fast roll, is the model originally due to Starobinsky . This model has recently attracted quite a lot of attention, but different physical conclusions with regards to the shape of the bi-spectrum have been reached. The present paper is aimed at considering the question again in order to clarify the situation.
As we shall soon outline, the Starobinsky model involves a canonical scalar field and is described by a linear potential with a sudden change in the slope at a given point. The sharp change in the slope leads to a brief period of fast roll sandwiched between two epochs of slow roll. Typically, in such situations, the power spectrum is expected to turn scale invariant when slow roll has been restored, and it is indeed what happens in the case of the Starobinsky model. The scalar power spectrum has a step like feature with a burst of oscillations connecting the two levels of the step (see, for instance, Fig. 3 of Ref. ). The flat regions of the step reflect the two epochs of slow roll, while the oscillations in between arise as a result of the period of fast roll.
Naively, one would have expected that the scalar bi-spectrum too would exhibit a similar behavior, viz. that it would turn scale invariant when slow roll has been restored. However, strikingly, when considered without adequate care, it is found that the bi-spectrum grows linearly with the wavenumbers at small scales (see, Refs. [62, 63]; in this context, also see Figs. 7 and 11 in Ref.  and the recent work Ref. ). From a theoretical point of view, evidently, it is imperative to firmly establish the predictions of the model and settle upon the correct behavior in a physically relevant and realistic situation. It is also worth noting here that, given the extent of accuracy of the measurements of non-Gaussianities by Planck, upon comparing with the data, the two behavior mentioned above would probably lead to very different constraints on the Starobinsky model. With these motivations in mind, in this work, as we have already pointed out, we intend to revisit the issue.
Clearly, the fact that the scale invariance of the bi-spectrum is not restored on large scales must be unphysical and, in this paper, we shall show that this arises due to the discontinuity in the second derivative of the potential. In fact, the point that the growing term is indeed unrealistic could have been easily guessed from the very beginning, since it exactly corresponds to a well-known and well-studied situation which was investigated long ago in the context of particle production by time-dependent, classical, gravitational fields . In what follows, we shall quickly recall the main results and conclusions arrived at in the earlier work, as the phenomenon closely resembles the behavior of the bi-spectrum encountered in the Starobinsky model.
The earlier work  considers a scalar field, say, , that is non-minimally coupled to gravity, and is evolving in a spatially, flat, Friedmann-Lemaître-Robertson-Walker (for convenience, simply FLRW, hereafter) metric. Such a scalar field is governed by the following equation of motion:
where denotes the scalar curvature, while an arbitrary constant. In a time-dependent background such as the FLRW universe, it is common knowledge that, upon quantization, pairs of particles associated with the scalar field will, in general, be produced, provided the coupling is not conformal, i.e. . Let denote the scale factor of the FRLW universe, with being the conformal time coordinate. Upon Fourier transforming the scalar field and redefining the Fourier modes, say, , as , one finds that the differential equation satisfied by can be written as
where an overprime represents differentiation with respect to the conformal time , while denotes the comoving wavenumber. The quantity is given by
The above differential equation for can also be cast as an integro-differential equation as follows:
At early stages of the expansion, the mode function can be expected to behave as , which essentially corresponds to choosing the field to be in the vacuum state initially. At late times, one has , where and are the standard Bogoliubov coefficients that relate the modes at different times. Then, using Eq. (1.4), one can approximate the Bogoliubov coefficient at very late times to be
This expression, in turn, permits one to evaluate the energy density of the created particles, which is arrived at by calculating the integral 
Let us now consider the case wherein there is a sharp transition from a phase of de Sitter inflation to a radiation dominated era. Let the transition take place at the conformal time, say, . In this scenario, the scalar curvature is non-zero, but constant (being related to the constant Hubble parameter during the de Sitter phase) for , while vanishes for (i.e. during the radiation dominated epoch). The integral (1.5) can be carried out explicitly in such a case and, one obtains that, , where is the incomplete Euler function [66, 67]. For large values of , one finds that , with the result that the corresponding energy density diverges logarithmically. However, as discussed in the original work , this conclusion is unphysical, and it is just an artifact of the abruptness of the transition from the de Sitter phase to the epoch of radiation domination. Indeed, if we now ‘regularize’ the transition, for instance, by smoothening out the quantity to be, say, , then the coefficient is found to be
In other words, one obtains an exponential cut-off in the spectrum, i.e. , of created particles (note that is negative), which occurs as a result of smoothening out the sharp transition. If we now calculate the corresponding energy density, then we arrive at a finite result, viz. . This unambiguously illustrates the point that the original logarithmic divergence was indeed an artifact and, upon modeling the transition more realistically, one obtains a result that is perfectly finite and physical.
In the same manner, the indefinite growth of the bi-spectrum at small scales in the Starobinsky model ought to be just an artifact and should be considered to be unphysical. In this work, focusing on the equilateral limit, we shall analytically investigate the effects of smoothening out the discontinuity in the derivative of the potential on the scalar bi-spectrum. We shall also compare the analytical results with the numerical results from the code Bi-spectra and Non-Gaussianity Operator or, simply, BINGO, which we had recently put together to compute the scalar bi-spectrum in inflationary models involving the canonical scalar field . As we shall illustrate, in the case of the bi-spectrum, smoothening out the discontinuity restores the scale invariance of the bi-spectrum at suitably large wavenumbers, depending on the extent of the smoothening. This allows us to conclude that the continued growth in the bi-spectrum at small scales, as was found earlier, can be attributed to the unrealistic assumption that the discontinuity in the derivative of the potential can be arbitrarily sharp.
The remainder of this paper is organized as follows. In the following two sections, we shall highlight a few essential aspects of the Starobinsky model and discuss the dominant contribution to the scalar bi-spectrum (in the equilateral limit) which arises due to the discontinuity in the first derivative of the potential in the model. In Sec. 4, we shall smoothen the discontinuity in a simple manner, which allows one to obtain the modes during the transition, and evaluate the corresponding contribution to the scalar bi-spectrum. We shall see that even the simplest of smoothening curtails the growth of the bi-spectrum on small scales. In Sec. 5, focusing on the limit of large wavenumbers, we shall discuss the effects of a more generic smoothening of the potential. We shall analytically illustrate that, if the potential is smoothened sufficiently, it ensures that the corresponding contributions to the bi-spectrum prove to be insignificant at suitably small scales. In Sec. 6, we shall compare the analytical expressions we obtain with the numerical results from BINGO. We shall conclude in Sec. 7 with a few general remarks.
Note that, we shall assume the background to be the spatially flat, FLRW line-element, which is described by the scale factor and the Hubble parameter . Also, we shall work with units such that , and we shall set . Moreover, shall denote the cosmic time coordinate, and we shall represent differentiation with respect to by an overdot. As we have already mentioned, represents the conformal time coordinate, while an overprime denotes differentiation with respect to . Further, shall denote the number of e-folds. Lastly, a plus sign, a zero or a minus sign in the sub-script or the super-script of any quantity shall denote its value or contribution before, during and after the field crosses the discontinuity in the derivative of the potential, respectively.
2 Essential aspects of the Starobinsky model
The Starobinsky model involves a canonical scalar field and it consists of a linear potential with a sudden change in its slope at a given point . The potential that describes the model can be written as follows:
Evidently, while the value of the scalar field where the slope, i.e. , changes abruptly is , the slope of the potential above and below are given by and , respectively. Moreover, the quantity denotes the value of the potential at . In this section, we shall highlight a few important points relating to the evolution of the background, in particular, the behavior of the slow roll parameters, in the Starobinsky model. We shall also discuss the behavior of the modes describing the curvature perturbation before and after the field crosses the point .
2.1 Evolution of the background
An important assumption of the Starobinsky model is that the value of is sufficiently large that it dominates the energy of the scalar field as it rolls down the potential across . As a result, the behavior of the scale factor proves to be essentially that of de Sitter. This, in turn, implies that the first slow parameter, viz. , remains much smaller than unity throughout the evolution, even as the field crosses the discontinuity in the potential. In fact, the first slow roll parameter before and after the transition, i.e. when the field crosses , can be shown to be 
respectively, where is a constant that is determined by the relation , while denotes the e-fold at the transition, and .
Before the transition, the second slow roll parameter, viz. , is determined by the slow roll approximation and is found to be . However, as the field crosses , the change in the slope causes a short period of deviation from slow roll. After the transition, the second slow roll parameter is found to be 
It is clear that turns large immediately after the transition and, when slow roll is restored eventually, one finds that , just as one would expect.
As we shall discuss in the following section, the dominant contribution to the scalar bi-spectrum arises due to the so-called fourth term in the Maldacena formalism (in this context, see, for instance, Refs. [39, 40, 57]). This contribution involves the time derivative of the second slow roll parameter . Upon using the background equations, one can show that can be written as [39, 62, 63]
where , and we should stress here that this expression is an exact one. In the case of the Starobinsky model, due to the discontinuity in the slope of the potential, clearly, the first term in the expression for above, which involves the second derivative of the potential, will lead to a Dirac delta function. The contribution to due to this specific term can then be written as
In fact, on large wavenumbers, as we shall soon discuss, it is this particular term that was found to lead to the dominant contribution to the scalar bi-spectrum [62, 63], if one works in the limit where the discontinuity in is infinitely sharp.
2.2 Evolution of the perturbations
Let us now turn to briefly discuss the behavior of the modes describing the scalar perturbations in the Starobinsky model.
Recall that, the Fourier modes of the curvature perturbations, say, , are governed by the differential equation 
where . In terms of the Mukhanov-Sasaki variable, , the above equation for reduces to
The ‘effective potential’ that appears in this differential equation can be written in terms of the slow roll parameters as follows:
where is the conformal Hubble parameter, while denotes the third slow roll parameter given by
Also, it should be emphasized that the above expression for is exact, and no approximation has been made in arriving at it.
In the Starobinsky model, due to certain cancellations that occur under the approximations of interest, one finds that the quantity reduces to before as well as after the transition [59, 39, 62, 63]. This basically corresponds to the de Sitter limit, which then implies that the Mukhanov-Sasaki variable during these regimes is essentially given by the conventional Bunch-Davies solutions . However, it should be clear from the expressions (2.9), (2.10) and (2.6) that, at the transition, it is the last term involving the quantity in above which will dominate. One finds that the corresponding effective potential is described by a Dirac delta function at the transition, and is given by 
where and denote the conformal time and the scale factor at the transition. We should clarify that, while the strictly de Sitter term, viz. , remains, it is the above term which a priori dominates at the transition.
Due to slow roll, before the transition, the modes can be described to a good approximation by following de Sitter solution:
Though slow roll is indeed restored at late times, due to the intervening epoch of fast roll, post-transition, the modes do not remain in the Bunch-Davies vacuum. Hence, after the transition, the solution to takes the general form
where and are the standard Bogoliubov coefficients. The expression (2.11) for then leads to the following matching conditions on the modes and their derivatives at the transition:
These conditions then allow us to determine the Bogoliubov coefficients and , which can be obtained to be
where corresponds to the mode that leaves the Hubble radius at the transition.
One can arrive at the corresponding expressions for the modes and the derivative before and after the transition from the above expressions for and its time derivative . Before the transition, the mode and the derivative are given by
Whereas, after the transition, one finds that
Note that, unlike the case of the Mukhanov-Sasaki equation (2.8), the governing equation (2.7) for involves only rather than . It should also be clear from the above arguments that will involve the Heaviside step function. This implies that the mode and its derivative are both continuous at the transition. As we shall discuss in the next section, the most significant contribution to the dominant term in the scalar bi-spectrum in the Starobinsky model shall depend on the mode and the derivative evaluated at the transition. Because of their simpler structure, it proves to be convenient to make use of the expressions (2.18) and (2.19) for the mode and before the transition. At the transition, these reduce to
where we have made use of the fact that to obtain the first expression, and have ignored the term involving , as is done in the slow roll approximation, to arrive at the second.
3 The dominant contribution to the scalar bi-spectrum
For simplicity, we shall focus on the equilateral limit in this work. It is well known that, when deviations from slow roll occur, it is the fourth term in the Maldacena formalism that leads to the dominant contribution to the bi-spectrum [40, 50, 51, 52, 53, 54, 55, 56]. In the equilateral limit of our interest, the fourth term, which we shall refer to as , is given by
where denotes the end of inflation. The quantity is described by the integral
where denotes a very early time, say, when the initial conditions are imposed on the perturbations.
Recall that, in a generic situation, the complete expression for the quantity is given by Eq. (2.5). As we have already discussed, in the Starobinsky model, the first term involving in the exact expression for leads to a delta function [cf. Eq. (2.6)]. It is then evident from the integral (3.2) that the corresponding contribution will be non-zero only at the transition. Actually, the contributions due to all the other terms, i.e. apart from the term involving in Eq. (2.5), can be evaluated analytically (in this context, see Ref. ). However, we shall focus here only on the specific contribution due to the term in , since it is this term that has been found to lead to the linear and indefinite growth on large wavenumbers in the bi-spectrum [62, 63]. We find that, with given by Eq. (2.6), the quantity can be written as
Towards the end of inflation, i.e. as , the mode simplifies to
where denotes the value of the first slow roll parameter at late times. Upon using the above two expressions for and in the expression (3.1) for , we find that we can write the contribution to the bi-spectrum due to the transition as follows:
where and the quantity is given by
As , we find that behaves as
while, in the limit , one obtains that
Note the linear growth with at large wavenumbers [57, 62, 63]. As we had discussed earlier, one physically expects the bi-spectrum to turn scale invariant for small scale modes that leave the Hubble radius at late times, when slow roll has been reestablished. However, one finds here that the bi-spectrum continues to grow indefinitely with the wavenumber. Evidently, this can be attributed to the fact that the potential contains an infinitely sharp transition, which can be considered to be unphysical, as discussed in the introductory section. As we shall illustrate in the following sections, the indefinite growth disappears as one smoothens the transition.
4 Effects of smoothening the discontinuity: A simple analytical treatment
In this section and the next, we shall analytically consider the effects of smoothening the discontinuity on the scalar bi-spectrum. We shall focus on the contribution due to the fourth term and we shall restrict ourselves to the specific term in (viz. the one involving ) that leads to the indefinite growth in the scalar bi-spectrum.
We shall first study the effects on the scalar bi-spectrum by smoothening the discontinuity in a specific fashion that permits a relatively complete analytical treatment of the problem. Essentially, we shall replace the delta function by one of its conventional representations. Let us write the delta function involved, viz. , in the following fashion:
where, for convenience, we have set
with being a small quantity (not to be confused with the slow roll parameters) that determines the width and the height of the transition. Obviously, the limit corresponds to the original sharp transition. In other words, instead of a function of infinite height and infinitesimal width, we shall alter the width and height suitably such that the area under the function is unity, as is required. In such a situation, in contrast to the infinitely sharp transition wherein there had existed just two domains, viz. the ones before and after the transition, there now exists a third domain corresponding to the period of the transition. It is then clear that, in the two original domains, i.e. when and , we have , just as we had before. Hence, the earlier solutions for , viz. (2.12) and (2.13) continue to remain valid during these domains. However, during the transition, i.e. when , we have
In order to be able to solve for the modes analytically corresponding to the above and also to be able to evaluate the integral describing the quantity [see Eq. (3.2)], we shall assume a few further points. Recall that, the delta function encountered in arises essentially due to its dependence on [cf. Eq. (2.11)]. Therefore, by altering the delta function, we have essentially modified the behavior of during the transition to be
In such a case, clearly, during the transition, we would have
where, for convenience, we have set . Actually, such a modification would also result in a change in the behavior of the first slow roll parameter and, needless to add, the scale factor as well. But, we shall assume that the scale factor continues to behave as that of de Sitter, and that the first slow roll parameter remains small and largely constant during the transition. As we shall see, these assumptions allow us to arrive at a complete analytical form for the resulting bi-spectrum, with the expected limit as .
Under the above assumptions, during the transition, the quantity is given by
and the corresponding solution to the Mukhanov-Sasaki equation can be written as
where and denote the Bogoliubov coefficients during the transition, while
It is important to stress that, since for the parameter values of our interest (in this context, see the caption of Fig. 1) and, as and are positive quantities, is a positive definite quantity. The corresponding mode and its derivative are given by
where and represent the first two slow roll parameters during the transition.
The expressions for the Bogoliubov coefficients during the transition, viz. and , are obtained by matching the modes and their derivatives on either side at . It should also be clear that the Bogoliubov coefficients after the transition, i.e. and , will no more be given by the original expressions [viz. Eqs. (2.16) and (2.17)], but will be modified. They are arrived at by matching the modes at . We find that the Bogoliubov coefficients and are given by
The Bogoliubov coefficients, say, and in the domain can be calculated to be
Recall that, our aim is to evaluate contribution to the bi-spectrum during the transition, when it has been smoothened. It is now a matter of substituting the mode (4.9) and the corresponding derivative (4.10) in the expression (3.2) and evaluating the integral involved from to . We find that, we can write as
where the quantities and are described by the integrals
We should mention that, in arriving at these integrals, we have ignored the term involving in the expression for [the one within the square brackets in Eq. (4.10)], and we have made use of the expressions (4.4) and (4.5) for and , respectively. If we also ignore the term involving [within the square brackets in Eq. (4.10)], we find that the above integrals can be trivially integrated to arrive at the following results111We should clarify a point here. We find that the final results and conclusions we have presented below remain largely unaffected, even if we retain the term involving .: