1 Introduction & summary

APCTP-Pre2014-009

KCL-PH-TH/2014-12


On degenerate models of cosmic inflation


Rhiannon Gwyn, Gonzalo A. Palma, Mairi Sakellariadou

and Spyros Sypsas


Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut

Am Mühlenberg 1, D-14476 Potsdam, Germany

Physics Department, FCFM, Universidad de Chile

Blanco Encalada 2008, Santiago, Chile

Department of Physics, King’s College London

Strand, London WC2R 2LS, U.K.

Asia Pacific Center for Theoretical Physics (APCTP)

Pohang 790-784, Republic of Korea


In this article we discuss the role of current and future CMB measurements in pinning down the model of inflation responsible for the generation of primordial curvature perturbations. By considering a parameterization of the effective field theory of inflation with a modified dispersion relation arising from heavy fields, we derive the dependence of cosmological observables on the scale of heavy physics . Specifically, we show how the non-linearity parameters are related to the phase velocity of curvature perturbations at horizon exit, which is parameterized by . Bicep2 and Planck findings are shown to be consistent with a value . However, we find a degeneracy in the parameter space of inflationary models that can only be resolved with a detailed knowledge of the shape of the non-Gaussian bispectrum.

1 Introduction & summary

Cosmic inflation [Guth:1980zm, Linde:1981mu, Albrecht:1982wi] successfully explains the origin of the primordial curvature perturbations needed to seed the observed large-scale structure of our universe and the cosmic microwave background anisotropies [Mukhanov:1981xt]. Its key predictions consist of a nearly Gaussian distribution of curvature perturbations characterized by a slightly red-tilted power spectrum, and the existence of primordial tensor modes. Cosmological observations have constrained various quantities, including the amplitude and spectral index of the power spectrum and, more recently, the tensor-to-scalar ratio [Komatsu:2010fb, Ade:2013ydc, Ade:2014xna], to a point where a large number of inflationary models have already been discarded. Despite this progress, it is clear that more data is required in order to gain insight into the nature of the fundamental theory hosting inflation. One of the most promising avenues for this is the study of the small departures from Gaussianity parameterized by the three-point correlation function (or bi-spectrum) of curvature perturbations [Linde:1996gt, Bartolo:2001cw, Bernardeau:2002jy, Maldacena:2002vr, Lyth:2002my, Seery:2005wm]. The amplitude and shape of this function are known to be sensitive to the self-interactions dictating the non-linear evolution of fluctuations, as well as to their interactions with other possible degrees of freedom relevant at the time of horizon exit [Bartolo:2004if].111Despite this, many degeneracies remain; see e.g. [Gwyn:2012pb, Gwyn:2012ey].

The recent development of the effective field theory (EFT) framework [Creminelli:2006xe, Cheung:2007st, Weinberg:2008hq, Senatore:2010wk, Khosravi:2012qg] to analyze the evolution of perturbations during inflation has been especially useful for discussing the potential existence of non-Gaussianity [Creminelli:2005hu, Senatore:2009gt]. Using general symmetry arguments on a Friedman-Lemaître-Robertson-Walker (FLRW) space-time, the authors of ref. [Cheung:2007st] were able to deduce the most general action describing curvature fluctuations generated by a single degree of freedom. This formulation has led to a model-independent parameterization of curvature modes’ self-interactions, exploiting the existence of non-linear relations among field operators of different orders in perturbation theory. In its simplest version, and up to cubic order, the EFT of inflation may be written in terms of a Goldstone boson field parametrizing fluctuations along the broken time translation symmetry direction of the background, often written as

(1)

where is the scale factor, is the Hubble expansion rate, is the usual slow roll parameter (terms sub-leading in the slow-roll parameters are omitted for convenience), and denotes the speed of sound at which the Goldstone mode propagates. This quantity may be expressed in terms of a mass scale used in the EFT expansion of ref. [Cheung:2007st] as

(2)

and will have a central role in our discussion. The other variable, , corresponds to a dimensionless quantity parametrizing non-linear interactions, and satisfies , where is the next to leading order mass parameter in the EFT expansion. In this formulation, the standard curvature perturbation is given in terms of the Goldstone boson by . The values of and characterize the cubic interactions, and are determined by the model being described. For instance, in single-field canonical models these two parameters take the values and , and the interactions are found to be suppressed with respect to the slow-roll parameters. In more exotic models, such as DBI inflation or multi-field inflation, the value of may vary in time, but with values restricted to be lower — or even much lower — than . In general, one expects the dimensionless parameter to be of order , which follows from naturalness arguments [Senatore:2009gt]. For instance, in the particular case of DBI inflation [Alishahiha:2004eh] one finds , in the case of two-scalar field canonical models with a heavy field one has  [Achucarro:2012sm], whereas in models with two or more heavy fields one finds the bound  [Cespedes:2013rda].

A suppressed value for the speed of sound changes the wavelength at which perturbations freeze, and increases the self-coupling between curvature perturbations, leading to the following formulas for the amplitude of the power spectrum , tensor-to-scalar ratio , and parameters (characterizing non-Gaussianity):

(3)

We see immediately that within this effective field theory parametrization is uniquely determined by and via

(4)

which implies, using recent observations [Ade:2013ydc, Ade:2014xna], a preferred value of GeV for the Hubble parameter during inflation. However, current observations cannot resolve the values of the slow roll parameter and the speed of sound . Determining these quantities requires better non-Gaussian constraints on the various parameters. The sensitivity of on has turned the speed of sound into a powerful parametrization of models beyond the single-field canonical paradigm. Current searches of non-Gaussianity [Ade:2013ydc] constrain the speed of sound to lie in the range .

More elaborate parameterizations of inflation are also possible within the EFT framework [Cheung:2007st]. For instance, it was argued on general grounds in ref. [Baumann:2011su] that, for short enough wavelengths of the curvature perturbations, the EFT could exhibit a non-trivial scaling of its field operators, enhanced by the broken time translation invariance of the background. For this to be possible, a new mass parameter needs to enter the EFT description, introducing a pivot scale at which this new scaling becomes operative. An example of such an EFT is obtained in the particular case where curvature perturbations interact with heavy scalar degrees of freedom, with masses such that . In this type of scenario, if the speed of sound and the Hubble scale satisfy , one obtains — after integrating out the heavy fields — an action of the form [Gwyn:2012mw]:

(5)

This action continues to describe a single degree of freedom, and therefore its cutoff energy scale is given by the mass of the heavy degrees of freedom [Achucarro:2012sm, Achucarro:2012yr, Gwyn:2012mw]. This version of the EFT may be seen as a non-trivial intermediate completion of the previous one shown in eq. (1), with Laplacians modifying the scaling of the operators affecting the evolution of perturbations. This scaling allows the EFT to display a smooth transition by remaining weakly coupled as it runs towards the ultraviolet (UV), where new degrees of freedom become operative. The energy range where this scaling becomes manifest is called the new physics regime [Baumann:2011su], a regime where linear perturbation theory is characterized by a dispersion relation, in Fourier space, of the form . Crucially, if curvature perturbations exit the horizon within this regime,222A good rule of thumb telling us the value of the wavelength at which curvature perturbations exit the horizon is given by the simple condition . Therefore, the freezing of the modes may happen during the new physics regime if the dispersion relation is of the form during horizon exit. then this time the amplitude of the power spectrum, tensor-to-scalar ratio, and parameters are found to be characterized respectively by:

(6)

These expressions may be compared with those of eq. (3): they have the same form but with replaced by . In particular, the dependence of both and on leads to the same equation (4) determining the Hubble parameter in terms of observables.

While it is not surprising that the new mass scale shows up in the observables, the fact that they lead to the same relation (4) suggests that and fulfil similar roles at linear perturbation level. Indeed, as we shall see, they both denote the phase velocity of the Goldstone mode at the moment of Hubble freezing in two different limits. As a result, the two EFT parameterizations are degenerate in the sense that they predict the same relations among observables involving the free field theory. On the other hand, one might have expected that self-interactions would break such a degeneracy by implying different non-Gaussian shapes for these models. We will show that this is not the case. A detailed analysis of the non-Gaussian shapes shows that both theories are indistinguishable for any practical purpose.

To judge the relevance of this situation, let us keep in mind that within the effective field theory framework it is of the utmost importance to understand how measurable — low-energy — quantities are related to the free parameters of the underlying theory. If one believes that single field canonical slow-roll inflation is only an effective description embedded in a more fundamental theory containing heavy degrees of freedom, then both (1) and (5) are equally natural parameterizations. This is because the UV physics responsible for the reduction in the speed of sound, parametrized by , may also contain heavy degrees of freedom, parametrized by . Adopting such a perspective, (6) implies that a non-Gaussian signal would provide information about the ratio (instead of ), while the recent results by Bicep2 [Ade:2014xna] would constrain the quantity (instead of ).

Let us examine this claim in the context of a well-studied UV inflationary model: D-brane inflation on a GKP background [Giddings:2001yu]. In such a scenario (see e.g. [Kachru:2003sx]), inflation appears because of the motion of a D-brane in a highly warped throat which is smoothed in the infrared (IR) by fluxes, and glued to a compact internal manifold in the UV. The fluxes are responsible for producing a non-trivial warp factor and for stabilizing the closed string moduli of the Calabi-Yau. The motion of the D-brane may be effectively described by the DBI action which contains higher-order kinetic terms resulting in a reduced propagation speed and a reduced sound horizon [Silverstein:2003hf, Alishahiha:2004eh, Chen:2004gc, Chen:2005ad]. These effects are parametrized by the coefficients of (1). However, as already mentioned, the presence of background fluxes also results in the stabilization of moduli. These massive scalars are parametrized by the parameter of (5). In the case where the length scale is small compared to the characteristic length of the perturbations, , the effect of these scalars is negligible. The action (5) becomes relevant in the opposite case.

Finally, let us stress that the action (5) is constructed entirely within the spirit of ref. [Cheung:2007st], where several operators were classified according to their compatibility with the symmetry of the low-energy theory. The operators involved in (5) satisfy this criterion and their physical interpretation is that they parametrize heavy degrees of freedom. Their relevance or not for CMB observations is a model-dependent question just as in the case of other sets of allowed operators like, for example, extrinsic curvature contributions [Cheung:2007st, Bartolo:2010bj, Bartolo:2010im, Anderson:2014mga], or Galilean operators [Creminelli:2010qf]. In the absence of a unique UV model, the best we can do is, as usual, parametrize our ignorance and constrain it through actual measurements.

The purpose of this article is to analyze the impact of future measurements — particularly related to non-Gaussianity — on discriminating between different models of inflation, described by effective field theories with drastically different parameterizations, such as those of eqs. (1) and (5). We will pay attention to the role of the non-Gaussianity shapes and show that new signatures are generated in the presence of heavy fields but they are degenerate with those of the low-derivative EFT, to a degree that renders the two descriptions indistinguishable from any practical perspective. What is important though is the precise connection of the observables to the dimensionful parameters of the underlying theory, and we will show how this occurs in our parametrization, so that recent results may constrain the scale of heavy physics directly — see [Assassi:2013gxa] for similar arguments. This will constitute one of our main results.

We have organized our work in the following way: In Section 2, we begin by explaining the dependence of the three-point amplitude on the scale of UV physics by showing that is related to the phase velocity of the Goldstone boson, which interpolates between the two predictions (3) and (6), depending on the value of the combination relative to . In Section 3, we calculate the three-point correlators and extract the precise dependence of on the parameters of the underlying intermediate EFT, which we then invert to obtain constraints, using Planck and Bicep2 results. In Section 4, we comment on the degeneracy of three-point functions of the two effective actions, while we conclude in Section 5.

2 Comments on the non-linearity parameters

It is well known that models of inflation with a speed of sound different from one are characterized by an enhancement of the equilateral shape of non-Gaussianity, with an amplitude of the order of . At perturbation level, the speed of sound is simply the phase velocity at which Goldstone boson modes propagate in the long wavelength limit , where is the comoving momentum of a given mode. Such models arise whenever non-trivial interactions modify the kinetic structure of the inflationary adiabatic curvature perturbations, which at low energies are well parametrized by the action (1).

However, as argued in the introduction, it is reasonable to expect that the interactions responsible for introducing a speed of sound may further modify the kinetic structure at short wavelengths. This is precisely the case for models of inflation where heavy fields interact with curvature perturbations [Tolley:2009fg, Achucarro:2010jv, Achucarro:2010da, Achucarro:2012sm, Achucarro:2012yr, Gwyn:2012mw]. Here, heavy fields may exchange energy with curvature perturbations producing a mixing between adiabatic and isocurvature modes, resulting in a non-trivial modification of their dispersion relations. In what follows we examine the EFT arising from having integrated out heavy fields that interact with curvature perturbations. For detailed discussions on how this EFT is deduced, see refs. [Achucarro:2012sm, Burgess:2012dz, Gwyn:2012mw, Cespedes:2013rda, Castillo:2013sfa, Noumi:2012vr]. For other discussions concerning the phenomenology of heavy fields during inflation, see refs. [Jackson:2010cw, Cremonini:2010sv, Jackson:2011qg, Shiu:2011qw, Cespedes:2012hu, Avgoustidis:2012yc, Gao:2012uq, Gao:2013ota, Gao:2013zga, Pi:2012gf, Achucarro:2012fd, Achucarro:2013cva, Achucarro:2014msa, Mizuno:2014jja, Battefeld:2014aea].

2.1 The effective action and free field dynamics

Integrating out a single333See [Cespedes:2013rda] and the appendix of [Gwyn:2012mw] for a more general case. heavy degree of freedom, one deduces the low-energy effective action for the adiabatic perturbation. This action reads[Gwyn:2012mw]

(7)

where , and where we have defined:

(8)

In these expressions represents a mass scale characterizing the heavy field sector that has been integrated out, while represents the speed of sound of the Goldstone boson modes in the long wavelength limit, given by (2). However, as already stressed in the introduction, the mass of the heavy degree of freedom corresponds to the combination , which may be much larger than if the speed of sound remains suppressed.

It may be seen that both (1) and (5) correspond to different limits of this action. More precisely, the action of eq. (1) is recovered in the limit , whereas the action of eq. (5) is recovered in the limit . In this sense, the action (7) may be thought of as an intermediate completion of the action (1) towards the cutoff scale , incorporating the non-trivial effects from heavy fields that cannot be encapsulated by (1) alone. The last interaction term in (7) arises from a cubic self-interaction of the heavy field with a dimensionful coupling , and was not considered in ref. [Gwyn:2012mw], since in this case the equation of motion for the heavy field is non-linear. However, such a term can be treated perturbatively in the interaction picture and we will thus include it in the present analysis. By first considering the action to quadratic order, one may derive the linear equation of motion:

(9)

where is given by the dispersion relation, deduced from the quadratic part of the action (7),

(10)

with , where denotes the comoving momentum. Assuming that all modes reach the Hubble scale () in the dispersive regime , or equivalently , the equation of motion (9) simplifies considerably and the solution for the curvature perturbation in the interaction picture is given by [Baumann:2011su]

(11)

where is the usual conformal time and denotes the Hankel function of the first kind. In the far IR limit the previous expression reads

(12)

and the amplitude of the power spectrum in eq. (6) is then recovered, i.e. .

2.2 The bispectrum amplitude

In order to understand what the three-point function amplitude probes, it is instructive to see how the operator , defined in (8), appears in the action. We will only consider momenta within the domain of validity of the effective field theory , where the dispersion relation (10) may be approximated by

(13)

omitting factors of . Let us now organize the cubic part of the Lagrangian (7) using the following notation:

(14)
(15)
(16)
(17)

Since we are interested in computing quantities around the freezing regime when all modes satisfy the horizon crossing condition , we are allowed to make the following replacements in these operators: and , where .444Note that this is not a recursive definition, as is determined uniquely by the condition and (13), and is a function of this . Rewriting the kinetic part of the Lagrangian (7) in terms of , we obtain

(18)

while the cubic operators may be written as

(19)
(20)
(21)
(22)

From (18) and (19)-(22), we see that the operator appears in the action in the same way that the coupling appears in the low derivative EFT (1), correlating — via symmetry — a low phase velocity with a large non-Gaussianity. We thus expect that the value of at the Hubble scale determines the amplitude of the three-point function. Indeed, taking the ratio of these expressions with (18), we immediately see that the , and operators lead to

(23)

up to numerical factors that we will include later. To further clarify this result, let us define a phase velocity from (13) as

(24)

The non-linearity parameters (23) may thus be written as

(25)

We may now use these relations to obtain a general expression for the amplitude of the three-point functions corresponding to these operators, for the full range of momenta . These expressions will depend on the ratio since the dispersive behaviour of the Goldstone boson at freezing depends on this quantity. The operator at the Hubble scale may be obtained using the dispersion relation at , which yields

(26)

Substituting these expressions into (23), we obtain

(27)

Taking the two limits and (or equivalently and ), we see that the momentum and the phase velocity (24) at the Hubble scale and the leading predictions for read555Recall that in the limit, the coefficient defined in (8) and consequently the non-linearity parameter vanish.

(28)

for the case , and

(29)

for the case . (Recall from 25 that is independent of ). These expressions are in accordance with the limit in which the EFT (7) flows to the EFT (1).

Therefore, the predictions (3) of the low-derivative EFT (1) generalise to the predictions (6) of the EFT (7), upon replacing the speed of sound (2) with the phase velocity (24). In both cases, the non-linearity parameter equals the inverse phase velocity squared. Depending on the value of the parameter , this phase velocity is related either to the ratio , or the ratio of the heavy physics scale to the Hubble scale, namely . Moreover, in [Gwyn:2012mw, Baumann:2011su] the symmetry breaking scale and the strong coupling scale were computed for the theory (7). In further support of our claim, let us point out that the same expressions for can be derived by taking the analogous expressions for the EFT (1) — see e.g. [Cheung:2007st] — and replacing with evaluated at the relevant energies (see Sec. 6.2 of [Sypsas:2014aua] for further details). In [Gwyn:2012mw] we proposed that the process of integrating out heavy physics may be thought of as the insertion of an effective UV medium through which the IR mode propagates. We see that encodes the “optical” properties of this medium, i.e. its refractive index.

3 Bispectra in the presence of heavy fields

Let us now compute the shapes of the bispectra in momentum space, defined as

corresponding to the cubic operators appearing in eqs. (14)-(17). These can be computed using the formalism [Maldacena:2002vr, Weinberg:2005vy], according to which the expectation value of an operator is evaluated using

with standing for time ordering and anti-ordering respectively, and with . Using the Baker-Campbell-Hausdorff formula one can expand the previous expression as

(30)

We will focus on the tree-level corrections consisting of the second term of (30), where the operator under consideration is . The field operator in Fourier space is defined by

where denotes the Fourier mode of the field with wavevector , and , and stand for the usual creation and annihilation operators obeying the canonical commutation relation:

From now on, we will focus on the part of the bispectrum induced by the operator of eq. (17), the computation of which we write in some detail, and simply quote the results for the other three operators appearing in eqs. (14)-(16). In the dispersive limit , where momentum dominates over the mass , the Hamiltonian in momentum space is given by

where , and from (30), the first tree-level correction to the three-point correlator reads

By expanding the commutator and performing the necessary contractions among the operators, we arrive at the final integral which is

(31)

with given by (11).

Let us first focus on the integral

Changing the integration variable from to (recall from eq. (29)) and using the solution (11), we obtain

where we have introduced the ratios and . Taking an analytic continuation , so that , with the modified Bessel function of the second kind, yields

(32)

We may now substitute (12) and (32) into (31) and obtain the three-point correlator for the operator .

In complete analogy, we may derive the expressions for the other operators in eqs. (14)-(16). Upon defining

and using the relation , the three-point functions for the Newtonian potential read

(33)

where is used to denote the shape function normalized at the equilateral limit , and the power spectrum is defined by , and may be computed using the late time solution (12). The non-linearity parameters read

(34)

with the phase velocity written in eq. (29). The shape functions are given by

(35)

and they are depicted in Fig. 1. Orthogonal and flattened shapes can be obtained from linear combinations of the three-point contributions in eq. (33) with appropriate values of and . For example, the combination with reproduces the orthogonal shape, while with it peaks for the flattened triangle. The same shapes can be obtained for similar values of by combining and .

Figure 1: The bispectra of the effective theory (7), normalized to one in the equilateral configuration. Clockwise from top left: . and are highly degenerate but evaluation at the flattened triangle reveals their difference.

In order to make contact with observation, it is necessary to project our predictions onto the templates actually used by experiments. Following [Babich:2004gb] and defining an inner product between two shapes and as

the projected non-linearity parameters can be computed using [Senatore:2009gt]

Using the templates [Creminelli:2005hu, Senatore:2009gt, Meerburg:2009ys, Ade:2013ydc]

we obtain

(36)

which can be inverted to yield

(37)

From this form one may proceed to input the Planck data [Ade:2013ydc] and derive constraints on the values of the dimensionful parameters of the underlying UV theory responsible for inflation. However, since the variables are correlated, one should use the covariance matrix to compute the error bars. Since such information is not available, what we can do is to examine if theoretically justified values of the parameters are within observational bounds.

In [Gwyn:2012mw], we argued that, naturally, the symmetry breaking and strong coupling scales of the EFT (7) should be of the order of , which implies, via scaling arguments (see Sec. 6.5.2 of [Sypsas:2014aua]), . Therefore, upon interpreting the Bicep2 results [Ade:2014xna] as fixing GeV, we are led to the value , which, interestingly, according to (37) can be achieved with . Such a number is consistent with a high tensor-to-scalar ratio, provided that the slow-roll parameter is in the range , compatible with the Planck bound [Ade:2013uln].

Constraints on and can be derived from the requirement that the mass parameter characterizing the heavy field satisfy . From and the values for and quoted above, we obtain , which666Notice that in the recent article [Baumann:2014cja] a new bound on was inferred by observing that the tensor-to-scalar ratio receives logarithmic contributions from the speed of sound . This result modifies the bounds discussed here (in the event that the value of turns out to be large) however it does not change our more general conclusions regarding the degeneracy between different classes of inflationary models. leads to . A value of the coupling close to the UV scale is consistent with our claim that the physics responsible for reducing the speed of sound also contains heavy degrees of freedom. Note also that a speed of sound of order is consistent with the requirements and , with the latter condition implying horizon exit in the dispersive regime. Upon assuming , it follows that which implies that the parameter should obey . An upper bound on cannot be derived due to the specific combination of mass scales appearing in — see eq. (8). The only information that can be extracted from this parameter is , which follows from and .

Finally, let us emphasize again that all these numbers must be taken with caution, since the Planck bounds on non-Gaussianity still leave a fairly large parameter space allowed, while the values for and