Bounds on non-adiabatic evolution in single-field inflation
We examine the regime of validity of -point spectra predictions of single field inflation models that invoke transient periods of non-adiabatic evolution. Such models generate oscillatory features in these spectra spanning frequencies up to the inverse time scale of the transient feature. To avoid strong coupling of fluctuations in these theories this scale must be at least of the Hubble time during inflation, where is the inflaton sound speed. We show that, in such models, the signal-to-noise ratio of the bispectrum is bounded from above by that of the power spectrum, implying that searches for features due to non-adiabatic evolution are best focussed first on the latter.
In this work we examine the regime of validity of models of single field inflation that invoke transient periods of non-adiabatic evolution to generate features in the spectra of curvature fluctuations. We clarify the limits that can be placed on the width of a feature in the inflationary potential or sound speed, and more generally, the shortest time scale or highest energy scale that can be probed by inflationary fluctuations while the effective theory remains weakly coupled. The characterization of the region of parameter space within which perturbation theory is under control has become more important since the release of the Planck data. These data now accurately probes a range of scales large enough to encompass modes that were on the horizon and those that were above the strong-coupling scale at a single epoch. Thus, the physical interpretation within the inflationary context of correlations between these scales must be treated with care.
Features in various aspects of the inflationary Lagrangian which generate a period of non-adiabatic evolution of the inflationary background are by now well established as a phenomenological way of introducing oscillatory signatures into the spectra of curvature fluctuations during inflation. Starobinsky first noted that a potential containing a singularity in its first derivative lead to oscillations in the resulting spectrum of curvature fluctuations Starobinsky (1992). More recently Starobinsky’s model has been extensively revisited Arroja et al. (2011); Martin and Sriramkumar (2012); Arroja and Sasaki (2012). A step in the potential of canonical inflation results similarly in oscillations of the power spectrum and bispectrum as shown both numerically Adams et al. (2001); Chen et al. (2007, 2008) and analytically Dvorkin and Hu (2010); Adshead et al. (2011, 2012); Adshead and Hu (2012); Miranda et al. (2012); Adshead et al. (2013); Miranda and Hu (2013). Oscillations in the potential lead to resonant logarithmic oscillations in these spectra Chen et al. (2008) and occur naturally in axion monodromy inflation Silverstein and Tong (2004); Flauger et al. (2010). Integrating out an orthogonal field in a multi-field inflationary model with a turning trajectory generates oscillatory signals Achucarro et al. (2012a, b, 2013a, 2013b). Further work on the effects of non-adiabatic evolution during slow roll inflation includes Chen (2010, 2011, 2012).
Observationally, features were first invoked to explain observed glitches in the angular spectrum of temperature fluctuations in the cosmic microwave background (CMB) in the year one analysis of the Wilkinson Microwave Anisotropy Probe (WMAP) data Peiris et al. (2003). Further studies were carried out fitting for these glitches Covi et al. (2006); Hamann et al. (2007). A hint of evidence for a high frequency oscillations due to a step feature in the inflationary potential was found in WMAP year 7 data by Adshead et al. (2012) and at a similar significance in the Planck temperature power spectrum Ade et al. (2013) with the same amplitude Miranda and Hu (2013). Evidence for oscillations due to a resonant enhancement of the fluctuations has also been reported in WMAP data by Peiris et al. (2013), however, the evidence weakened in the Planck data Ade et al. (2013); Meerburg and Spergel (2013); Easther and Flauger (2013). The Planck polarization power spectrum should soon provide a more definitive empirical check of these interpretations Mortonson et al. (2009).
In this paper, we examine theoretical limits both on these interpretations and on the effects of non-adiabatic evolution in general by requiring that perturbation theory remains valid throughout all aspects of their calculation. We detail how strong coupling of the fluctuations ultimately limit the shortest time scale associated with these phenomena and hence the parameters of any given model for inflaton features. These results parallel those found in the context of slow roll inflation Cheung et al. (2008a, b); Leblond and Shandera (2008); Shandera (2009); Baumann et al. (2011); Barnaby and Shandera (2012), here generalized for transient violations of slow-roll.
This work is organized as follows. In §II we study the breakdown of perturbation theory during inflation by considering the non-linearities of the equations of motion for the perturbations. In §III we relate this breakdown to the emergence of strong coupling from the perspective of the action for the fluctuations. In §IV we examine bounds on the maximal bispectrum consistent with perturbation theory, and its resulting signal to noise. In §V we collect some examples of non-adiabatic evolution during inflation and its impact on inflationary fluctuations. Finally we conclude in §VI.
Throughout, we work in natural units where the reduced Planck mass as well as .
Ii Perturbation Breakdown
In §II.1 we briefly review the calculation of -point observables using the linearized theory. In §II.2, we discuss nonlinearity in the kinetic part of the field equation and the energy density in fluctuations. We relate this nonlinearity to the strong coupling scale of the adiabatic vacuum in §II.3 and discuss nonlinearity due to non-adiabatic excitations in §II.4. The validity of the perturbative calculation requires that when the -point observables are formed, the modes in question lie below all scales associated with nonlinearity.
ii.1 Linear Perturbations
We begin by considering the theory described by the Lagrangian density
where the kinetic term of the inflaton field is
Note that this class contains canonical single field inflation where
as well as Dirac-Born-Infeld (DBI) inflation where
and is the warped brane tension. Varying the action with respect to the field yields the nonlinear field equation
This field equation enforces covariant conservation of the stress energy tensor , where
to all orders in field fluctuations. Note that
To calculate the -point observables, we expand either the action or the equation of motion around a homogeneous but time-dependent background
where solves both the homogeneous field equation (5) and the Friedmann equations
We then quantize the linear fluctuations, , deep within the horizon where the impact of the time dependence of the expanding background is weak and the fluctuations are assumed to obey the linearized field equation.
Expanding the field into modes, we obtain
where and are creation and annihilation operators satisfying the commutation relation
The modefunction, denoted here by , is the canonically normalized field defined as
while the sound speed is defined as
The adiabatic or Bunch-Davies vacuum state corresponds to the choice of boundary conditions such that the canonically normalized modefunction satisfies
where the sound horizon is
and denotes the end of inflation. The vacuum fluctuations are then evolved with the linear equations of motion or equivalently the quadratic action, including possible violations of the slow-roll assumptions, through sound horizon crossing where they freeze in. With , the linearized field equation is
Here characterizes deviations from de Sitter expansion and encodes the effect of slow-roll violations
The comoving curvature power spectrum defined as
is then given by
where and becomes independent of the evaluation point for . Higher order correlations such as the bispectrum
are calculated perturbatively at tree-level in the interaction picture from the modefunction and the higher order contributions to the Hamiltonian.
Sudden but transient violations of the slow roll assumptions, induced for example by rolling over features in , produce excitations and non-Gaussian correlations deep within the horizon. The validity of perturbative calculations of these effects critically relies on the ability to linearize fluctuations around the background and compute deviations from linearity within perturbation theory. In other words, they require that the theory of the fluctuations be weakly coupled across all scales of interest, including not only the horizon scale which the curvature perturbations freeze out, but also the scale at which they were excited as well as the scale at which the initial vacuum state before the excitation is defined. It is therefore useful to categorize the various checks of perturbation theory in terms of the various aspects of the field equation (5) which can become nonlinear.
Nonlinearity in can be recast as a comparison between the kinetic energy density of the fluctuations compared with the background whereas nonlinearity in is related to the sharpness of non-adiabatic features compared with the root-mean-square (rms) size of the field fluctuations.
Field equation nonlinearity is also distinguished by whether it arises from the adiabatic vacuum fluctuations themselves or the non-adiabatic excitations. Excitation nonlinearity places a bound on the amount of non-Gaussianity achievable within perturbation theory. Adiabatic nonlinearity means that the still linear excitations and the resulting non-Gaussianity may not be reliably calculated since the adiabatic vacuum state itself is strongly coupled.
ii.2 Quadratic Energy Density and Kinetic Nonlinearity
Nonlinearity in the kinetic part of the field equation (5) can be quantified by comparing the kinetic energy density in fluctuations to that of the background. Terms that are linear in vanish upon spatial or ensemble averaging the fluctuations leaving contributions that begin at second order in perturbation theory.
It is therefore useful to introduce the quadratic Lagrangian density,
Given that we are interested in subhorizon fluctuations, we ignore metric fluctuations. The represent terms that depend on derivatives with respect to ; their nonlinearity will be considered in §II.4. Here we consider nonlinearity during adiabatic evolution, e.g. well after a non-adiabatic event, where the omitted terms are negligible.
The quadratic Hamiltonian density associated with the theory gives the contribution to the energy density contained in the field fluctuations propagating on the background. It is constructed in the usual way from the canonical momenta
Now, note that in order that the theory is well defined, we require
The first condition ensures that the ground state energy is positive, while the second follows from the first after imposing that the fluctuations propagate subluminally.
We can compare the quadratic Hamiltonian to the change in the total energy in the presence of the field fluctuation. Again since we are evaluating the energy density for subhorizon fluctuations in an adiabatic regime, we ignore changes to the potential energy. Keeping quadratic contributions in the expansion of the kinetic terms in Eq. (7), we obtain
where we have introduced
For example, in DBI inflation
The term contains contributions that are quadratic in that are not included in . These terms are associated with the change in the background due to linear and quadratic terms in the fluctuations which then change the kinetic energy density carried by the fluctuations and background field respectively. Note that they involve terms that appear only at the cubic Lagrangian level.
On the other hand, there are analogous effects from the mean of the second order field that also renormalize the background. If we again compare fluctuations in a slow-roll sub-horizon regime where can be neglected, the field equation (5) is a conservation law for . Since this current is conserved exactly, it gives the second order contribution to the charge density from in terms of quadratic combinations of . For the spatially homogeneous component, the charge is conserved and so
This requires that and so
Thus in the energy density, the second order effects cancel the additional background renormalizing effects of the terms quadratic in the first order terms.
One condition for the validity of perturbation theory is that
so as not to disturb the background equation of motions (9). When this bound is violated due to excitations, one must take into account the effects of the backreaction of the fluctuations on the background through the Friedmann equation. A related and potentially stronger condition is that the field equation itself remains perturbative. Demanding that receives only small corrections from the terms quadratic in gives
By expanding to higher order in , one can show that this background condition is equivalent, up to numerical factors of order unity, to the condition that the equation of motion for the fluctuations, remains perturbative. Thus this condition is related to requiring that the fluctuations, with or without the non-adiabatic excitations, are not strongly coupled.
ii.3 Adiabatic Nonlinearity
We can apply the linearization conditions even in the absence of a non-adiabatic source of excitations. For this is the quadratic energy density contained in the Bunch-Davies vacuum fluctuations, which of course must be canceled off by counterterms. For , the kinetic linearization test on the equation of motion (32) determines the strong coupling scale of the vacuum fluctuations.
With the adiabatic vacuum fluctuations from Eq. (14), the energy to second order in fluctuations becomes
Integrating up to yields
This is the familiar zero point formula for a cutoff in momentum space. The additional factor of arises because energy is related to momentum as . This zero point energy is of course infinite if . We assume that it is renormalized away with appropriate counterterms. The direct energy comparison in Eq. (31) does not place a physical bound on the adiabatic vacuum fluctuations themselves.
Thus the more interesting comparison is to which tests whether the vacuum fluctuations can be treated using the linearized field equation. Here
Note that for , since the field equation is linear in the absence of whereas for small , there is a enhancement of . For example in DBI, and
places a limit on the for which we can reliably use linear perturbation theory. It is instructive to recast this bound using the power spectrum in the adiabatic limit
and compare the wavenumber to the sound horizon
For modes of higher frequency, the vacuum fluctuations no longer obey a linear equation and hence are strongly coupled. We call this the strong coupling scale. Notice that one can always tune in (36) to make vanish, in which case (41) diverges independently of . At the Lagrangian level (see Sec. III.1) this corresponds to canceling two cubic operators against each other. Since this is only possible at a fixed point in momentum space it does not apply to quartic order, and so there will still be strong coupling, however, it will arise from a different operator. For simplicity, we will assume that is such that away from .
Note that this is true even if the underlying theory is UV complete as in the DBI case. It represents not a breakdown of the theory itself but rather of the calculational tools of perturbation theory. Fortunately such a situation is not catastrophic for inflationary model building. The decoupling theorem implies that as long as frequencies redshift below the strong coupling scale in the adiabatic vacuum, the low-energy physics is decoupled from the physics above this scale. This means that, provided there remains a hierarchy between the Hubble scale, the scale of the cosmological experiment where observables are determined, and the strong coupling scale, there is no catastrophic consequence to this scale.
If on the other hand, slow-roll is interrupted by a non-adiabatic phase where the relevant timescale is much shorter than the Hubble time, then modes with frequencies associated with this scale are excited and can potentially be above the scale at which perturbation theory breaks down. We turn to this case next.
ii.4 Excitation Nonlinearity
If the inflaton transits a feature in much less than an efold, its fluctuations evolve non-adiabatically and the the observable impact of nonlinearity can be greatly enhanced. Let us consider the case where the field equation (5) gains a source from a feature in . In the linearized approach these lead to large contributions to the source of modefunction excitations in Eq. (16) as we shall see explicitly below. In addition to the considerations of the previous sections for nonlinearity in , the validity of this approach requires that can be approximated in perturbation theory.
The linearization involves approximating
For a sharp feature in field space of width this approximation will break down once the rms field fluctuation in the vacuum
exceeds it. In this expression is the maximum wave number we want to describe in our theory. This implies that perturbation theory can only be valid for all modes of interest if
If we wish to calculate out to the set by the feature width itself
This is the so-called symmetry breaking scale Baumann and Green (2011, 2012). Fluctuations of higher frequency no longer experience a sharp feature due to the presence of a perturbing background of field fluctuations, regardless of the amplitude of the feature. Above this scale it no longer makes sense to work with fluctuations about the classical background.
More specifically, if we consider a feature in of amplitude , then each successive term in the expansion in Eq. (42) will involve powers of so that
As , even becomes an uncontrolled expansion (see also §III.2). Note that, when the sound speed of the fluctuations , the scale at which the equations become non-linear due to this effect is above the strong coupling scale in Eq. (41) due to non-linearities in the kinetic term. Validity of perturbation theory requires that we calculate only below the lowest of the various scales. As we will see in the next section, violating the strong coupling bound while not violating the symmetry breaking scale still implies that the calculation of the -point functions breaks down.
Assuming the validity of the linearized field equation, we can compute the impact of non-adiabatic feature on the modefunctions and determine the excitations on top of the vacuum state. In the generalized-slow-roll (GSR) approximation Stewart (2002), one first defines the solution to Eq. (16) with and Bunch-Davies initial conditions
where and then replaces the RHS of Eq. (16) with . The solution is with
These expressions simplify in the limit of subhorizon fluctuations since and
and . We can then iterate to obtain
Note that only the positive frequency term of the second order field can contribute after averaging since the negative frequency term has no unperturbed part. Keeping only this term
we obtain the quadratic contributions as
This is the well known Boglioubov normalization relation derived perturbatively. In keeping with the Boglioubov language, we retain as representative only the piece of the modefunction change arising from particle excitations after the non-adiabatic feature. This is the negative frequency contribution, which reduces the total by a factor of 2.
Note that even after becomes a constant in time it still depends on in a manner that reflects the source of the excitation . Since we are interested in the piece of the source of Eq. (17) with the highest number of derivatives
and so it is useful to integrate by parts
Suppose now that the source has compact support around with some width . That finite width will introduce a cutoff due to the oscillatory integrand when . Thus
Given Eq. (20), this fractional change in the modefunction propagates into a change in the square root of the power spectrum as
To maintain generality for other types of excitations where does not necessarily have compact support, we use this notation from this point on.
With this in mind the second order energy density in the excitations
can be more instructively written by defining
which can be taken as a definition of and .
We then obtain an energy density of the same form as the zero point energy density in Eq. (34) at
where the cutoff of the effective theory is replaced by the cutoff imposed by the finite duration of the excitation. The energy density in excitations should not exceed the kinetic energy in the background
Note that the bound found by requiring is always weaker than the analogous bounds derived using from Eq. (46) for .
The analog of the strong coupling bound Eq. (41) for excitations becomes
In contrast to the adiabatic strong coupling scale, this bound weakens as . Violation of the bound implied by Eq. (41) but not (65) would indicate that the excitations are still linear around a vacuum state that is nonlinear. As such, the linearized calculation may be unreliable. On the other hand, violation of Eq. (65) means that the excitations themselves are strongly coupled and their interactions or non-Gaussianity can no longer be treated perturbatively. Thus saturation of this bound at gives the maximal level of non-Gaussianity that can be achieved by non-adiabatic excitations.
In summary, nonlinearity in the field equation provides four characteristic spatial scales relative to the sound horizon, or energy scale relative to the Hubble expansion rate. They are related by
and can be interpreted as , the strong coupling scale of vacuum fluctuations; the symmetry breaking scale beyond which features are blurred out by vacuum fluctuations; energy conservation violation scale; the strong coupling scale of the excitations in low sound speed models. The validity of perturbation theory requires that any non-adiabatic feature has a fractional temporal width , where is the smallest of the four scales. We loosely refer to this smallest of the four scales as the strong coupling scale of the theory for reasons that we make clear in the following section.
Iii Action Considerations
In this section we relate the four scales identified through nonlinearity of the field equations with the corresponding scales determined by expanding the action itself. We review the effective field theory method of identifying strong coupling of the adiabatic background Cheung et al. (2008a) and apply it to excitations. For canonical fields, this condition involves terms that break the exact shift symmetry and picks out the symmetry breaking scale for excitations. In §IV.1, we use the cubic action to derive a generic scaling relation for the maximal bispectrum due to a non-adiabatic feature.
iii.1 Effective Field Theory
The effective field theory of inflation Cheung et al. (2008a) provides a useful organizational structure for considering generic higher-order terms in the inflaton action. Given single field inflation, there is a preferred slicing called unitary gauge where . In this gauge the degrees of freedom are in the metric and one can write down all possible terms that obey unbroken spatial diffeomorphism invariance. We are primarily interested in models in this work, and so we will restrict our attention to the sector of the effective field theory which corresponds to this theory. This amounts to considering terms polynomial in , while disregarding higher derivative terms such as those that arise from curvature-squared terms, as well as powers of the extrinsic curvature Cheung et al. (2008a). Taylor expanding the resulting function around , we obtain the effective field theory action
Further demanding that its constant and linear terms satisfy the Friedmann equations gives
where we have explicitly kept to highlight the dimensions. We can restore temporal diffeomorphism invariance by using the Stückelberg trick which amounts to relating unitary time to an arbitrary slicing by introducing an auxiliary Goldstone field
and using the transformation rule for the metric
In the decoupling limit Cheung et al. (2008a) we can ignore the mixing due to metric fluctuations and set
Note that after reintroducing the Goldstone field and making use of Eqn. (70) the action at Eq. (67) is simply a change of variable of the action of a model to . In this case the theories can be matched by identifying
Let us begin with adiabatic assumptions where , and are taken to be approximately constant. Expanding the action in terms of and keeping quadratic and higher order terms, we have
where for even and for odd . Here the dimensionless parameters are Cheung et al. (2008a)
From the quadratic Lagrangian
we can further associate
For DBI the scale as expected at low
and vanish as .
iii.2 Strong Coupling
In general, one definition of strong coupling is the lowest scale for which one of the operators
for any . Using Eq. (III.1), and barring any cancellation between the -indexed contributions to we require for the smallest for which
To estimate the strong coupling scale given the power spectrum, we replace
Note that this differs from by the number of contractions but as we shall see in the following section, the same factors come into the calculation of Gaussian contributions to the -point functions.
Using the Stirling approximation, it is easy to show that for the DBI case the second term in Eq. (79) is and hence all operators becomes strongly coupled at the same scale Shandera (2009); Baumann et al. (2011)
as one might expect.
Moreover, this scales in the same way as that of Eq. (41). This is not surprising since strong coupling is simply a breakdown of perturbativity. As we discussed above, provided there is a sufficient hierarchy between the scales where the correlation functions are determined (e.g. sound horizon crossing) and this strong coupling scale there is no consequence of the existence of this scale.
Thus it is the interplay of strong coupling and non-adiabatic excitations that is important. While assuming an approximate shift symmetry for the inflationary history is technically natural in the EFT of inflation, it is in fact not required. Relaxing this assumption has two effects. For the operators included in Eq. (III.1), the modefunctions can gain a non-adiabatic excitation as described in §II.4. New operators also appear from allowing the coefficients in the unitary-gauge action Eq. (67) to vary.
We can estimate the effect of the former in a similar way to §II.4. Given a fractional change in the modefunction that takes place across a width in field space ,
If we want to calculate to the highest frequency set by the width of the feature then . Placing this scale into the comparison sets a bound on the width of the feature. If we assume that all operators become strongly coupled at the same scale without the excitation as in the DBI scaling of Eq. (77), then with the excitation we just introduce extra factors of from . If , the strongest constraint is from replacing a single modefunction with the excitation
such that for sufficiently high , independently of the value of . This fact merely reiterates that the strong coupling bound applies to any feature of such width or smaller. Once the adiabatic modes become strongly coupled it does not make sense to calculate even an infinitesimal excitation on top of them. For large the strongest condition comes from taking all factors to be excited and returns the excitation strong coupling scale .
The second change that non-adiabatic features make is to allow new operators associated with expanding the dependent coefficients in Eq. (67) in ,
where here and below the superscript on a quantity denotes , e.g. . For example, up to cubic order in fluctuations, the action for the effective field theory is Cheung et al. (2008b)
where here refer to terms higher order in powers of and and we have dropped total time derivative terms, which yield only slow roll suppressed contributions to the bispectrum Maldacena (2003). We have also dropped terms proportional to , such as the mass term for , consistent with taking the decoupling limit. In addition, we have introduced the slow-roll parameters
No slow roll expansion has been assumed, and at high energies, where decoupling is valid, the above action is a complete description of the interactions of , up to cubic order.
In order that the expansion of action remain valid, we require at least that
across all energy scales we wish to describe with our theory. Thus we lose perturbativity in the cubic operator beyond the highest energy scale for which the above is true. Note that this was misestimated in Ref. Adshead et al. (2012), where the perturbativity was only required near horizon crossing.
which is the energy bound . If we generalize these considerations to the interaction
(see Behbahani et al. (2012)) then the strong coupling condition contains terms for which
where we have integrated by parts times. Hence, in a fashion analogous to the low sound speed case of Eq. (83), as even as . Note that operators based on set the maximal scale if even for models where there is no non-adiabatic features whereas those in Eq. (III.1) would have strong coupling scales limited only by the Planck scale. The latter reflects the linearity of the kinetic term discussed in §II.2.
Iv Bispectrum and Signal to Noise
Limits on the amount of nonlinearity in the field equation or the action considered in the previous section directly translate into limits on non-Gaussianity, the observable impact of nonlinearity. In this section we consider the implications for the maximal bispectrum and the observability of features in the bispectrum vs. the power spectrum.
iv.1 Maximal Bispectrum
One advantage of working with the action is that, in addition to giving the scales where the perturbations break down, one can also easily read off estimates of the non-Gaussianity. When the correlation functions are determined at the sound horizon, a simple estimate is found from Cheung et al. (2008a); Baumann and Green (2011)
where is the relevant scale for the formation of the bispectrum. Thus the arguments above as to the scaling of this ratio directly give the desired result. For the adiabatic modes and , Eq. (79) for the minimal case of which must be present gives
with the appropriate understanding of what generates the excitation in the and limits. Note that the requirement that the scale set by the feature be below the strong coupling and symmetry breaking scales implies that non-Gaussianity remains small in all of the connected -pt correlation functions
by a factor of . A violation of the inequality in Eq. (96) clearly signifies that the total -point correlator has acquired strong non-Gaussianity from nonlinear terms whereas a violation of Eq. (97) does not necessarily.
iv.2 Signal to noise ratio
We want to determine whether or not the signal-to-noise in the bispectrum due to some sharp non-adiabatic evolution can exceed that in the 2-pt function in a regime where the theory is weakly coupled and all of the perturbativity criteria are met.
We can estimate the signal-to-noise ratio (SNR) in the 3-pt function in a finite volume by considering the minimum variance triangle weighting