Tensor to scalar ratio and large scale power suppression from pre-slow roll initial conditions
We study the corrections to the power spectra of curvature and tensor perturbations and the tensor-to-scalar ratio in single field slow roll inflation with standard kinetic term due to initial conditions imprinted by a “fast-roll” stage prior to slow roll. For a wide range of initial inflaton kinetic energy, this stage lasts only a few e-folds and merges smoothly with slow-roll thereby leading to non-Bunch-Davies initial conditions for modes that exit the Hubble radius during slow roll. We describe a program that yields the dynamics in the fast-roll stage while matching to the slow roll stage in a manner that is independent of the inflationary potentials. Corrections to the power spectra are encoded in a “transfer function” for initial conditions , , implying a modification of the “consistency condition” for the tensor to scalar ratio at a pivot scale : . We obtain to leading order in a Born approximation valid for modes of observational relevance today. A fit yields , with , and the Hubble scale during slow roll inflation, where curvature and tensor perturbations feature the same for a wide range of initial conditions. These corrections lead to both a suppression of the quadrupole and oscillatory features in both and with a period of the order of the Hubble scale during slow roll inflation. The results are quite general and independent of the specific inflationary potentials, depending solely on the ratio of kinetic to potential energy and the slow roll parameters to leading order in slow roll. For a wide range of and the values of corresponding to the upper bounds from Planck, we find that the low quadrupole is consistent with the results from Planck, and the oscillations in as a function of could be observable if the modes corresponding to the quadrupole and the pivot scale crossed the Hubble radius very few () e-folds after the onset of slow roll. We comment on possible impact on the recent BICEP2 results.
equationsection \affiliationDepartment of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260 \emailAddlal81@pitt.edu \emailAddboyan@pitt.edu \keywordsCMBR Theory, Initial Conditions and the Early Universe, Inflation, Physics of the Early Universe \arxivnumber
Inflation not only provides a solution to the horizon and flatness problems but also furnishes a mechanism for generating scalar (curvature) and tensor (gravitational wave) quantum fluctuations[1, 2, 3, 4]. These fluctuations seed the small temperature inhomogeneities in the cosmic microwave background (CMB) upon reentering the particle horizon during recombination. Most inflationary scenarios predict a nearly gaussian and nearly scale invariant power spectrum of adiabatic fluctuations [5, 9, 6, 7, 8]. These important predictions of inflationary cosmology are supported by observations of the cosmic microwave background[10, 11, 12, 13] which are beginning to discriminate among different scenarios.
Recent results from the Planck collaboration[12, 13, 14] have provided the most precise analysis of the (CMB) to date, confirming the main features of the inflationary paradigm, but at the same time highlighting perplexing large scale anomalies, some of them, such as a low quadrupole, dating back to the early observations of the Cosmic Background Explorer (COBE)[15, 16], confirmed with greater accuracy by WMAP and Planck[12, 13, 14]. Recently the BICEP2 collaboration  has provided the first measurement of primordial B-waves, possibly the first direct evidence of inflation.
The interpretation and statistical significance of these anomalies is a matter of much debate, but being associated with the largest scales, hence the most primordial aspects of the power spectrum, their observational evidence is not completely dismissed. The possible origin of the large scale anomalies is vigorously discussed, whether these are of primordial origin or a consequence of the statistical analysis (masking) or secondary anisotropies is still an open question. Some studies claim the removal of large scale anomalies (including the suppression of power of the low multipoles) after substraction of the integrated Sachs-Wolfe effect (ISW), however a different recent analysis finds that the low quadrupole becomes even more anomalous after subtraction of the (ISW) contribution, although some expansion histories may lead to an (ISW) suppression of the power spectrum. The most recent Planck[12, 13, 14] data still finds a statistically significant discrepancy at low multipoles, reporting a power deficit at with significance. This puzzling and persistent result stands out in an otherwise consistent picture of insofar as the (CMB) power spectrum is concerned. Recent analysis of this lack of power at low  and large angles, suggests that while limited by cosmic variance, the possibility of the primordial origin of the large scale anomalies cannot be dismissed and merits further study.
The simpler inflationary paradigm that successfully explains the cosmological data relies on the dynamics of a scalar field, the inflaton, evolving slowly during the inflationary stage with the dynamics determined by a fairly flat potential. This simple, yet observationally supported inflationary scenario is referred to as slow-roll inflation[9, 5, 6, 7, 8]. Within this scenario wave vectors of cosmological relevance cross the Hubble radius during inflation with nearly constant amplitude leading to a nearly scale invariant power spectrum. The quantization of the gaussian fluctuations (curvature and tensor) is carried out by imposing a set of initial conditions so that fluctuations with wavevectors deep inside the Hubble radius are described by Minkowski space-time free field mode functions. These are known as Bunch-Davies initial conditions (see for example[9, 5, 7, 8] and references therein).
The issue of modifications of these initial conditions and the potential impact on the inflationary power spectra[25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], enhancements to non-gaussianity[36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], and large scale structure have been discussed in the literature. Furthermore, arguments presented in refs.[34, 48] suggest that Bunch-Davies initial conditions are not the most natural ones to consider and may be unstable to small perturbations.
Whereas the recent results from Planck[12, 13, 14] provide tight constraints on primordial non-gaussianities including modifications from initial conditions, these constraints per se do not apply directly to the issue of initial conditions on other observational aspects.
Non-Bunch-Davies initial conditions arising from a pre-slow roll stage during which the (single) inflaton field features “fast-roll” dynamics have been proposed as a possible explanation of power suppression at large scales[49, 50, 51, 52, 53, 54, 55, 56]. This kinetically dominated fast roll stage has been considered previously using the Hamilton-Jacobi form of the Friedman equations in ref. with some of the consequences due to this type of scalar driven cosmology examined in detail in[58, 59, 60]. Alternative pre-slow-roll descriptions in terms of interpolating scale factors pre (and post) inflation have also been discussed in ref.. The influence of non-Bunch Davies initial conditions arising from a fast-roll stage just prior to slow roll on the infrared aspects of nearly massless scalar fields in de Sitter space time have been studied in ref.. Recent work  has shown that a kinetically dominated regime is in fact quite a generic feature under a very broad class of single field inflationary models providing further incentive for consideration of fast roll scenarios.
Motivations, goals and summary of results:
Inflationary scenarios predict the generation of primordial gravitational waves and their detection remains one of the very important goals of observational cosmology. Planck has placed constraints on the tensor to scalar ratio of while the BICEP  experiment has recently reported a measurement of . The BICEP value is much larger than many had expected and there exist models which can generate enhancements, refs  for example, which could explain the largeness of this value.
Suggestions of how to relieve the tension between the two experiments have been put forth where a possible solution invokes a running of the spectral index. Recently, in ref., a comparison between models featuring a running spectral index and models with a large scale power suppression has been made with the aim of determining which model relieved the tension most effectively. It was shown in this reference that a large scale power suppression of 35% yielded a considerably better fit to data than allowing a running of the spectral index, further improving the claims that the low l anomaly should be taken seriously.
The high amount of tension between these two experiments may be alleviated by future and forthcoming observations that will continue to constrain this important quantity, a quantity from which ultimately the scale of inflation may be extracted,
A distinct prediction of single field slow-roll inflationary models with a standard kinetic term is
with a (potential) slow roll parameter and is the index of the power spectrum of gravitational waves. The relation (2) is often quoted as a “consistency relation”. This relation is obtained by imposing Bunch-Davies initial conditions on tensor perturbations during the near de Sitter slow roll stage[9, 5, 7, 8].
Our goals in this article are the following:
Motivated by the recent results from the PLANCK collaboration[12, 13, 14] reporting the persistence of anomalies for small and large angular scales we study the modifications to the power spectra of curvature and tensor perturbations and the tensor to scalar ratio arising from non-Bunch Davies initial conditions imprinted from a pre-slow roll stage in which the dynamics of the scalar field is dominated by the kinetic term, namely a “fast-roll stage”. While previous studies of modifications of the scalar power spectrum from a fast roll stage focused on specific realizations of the inflationary potential, our goal is to extract the main corrections without resorting to a specific choice of the potential but by parametrizing the fast roll stage by the initial ratio of kinetic to potential energy of the inflaton, , and the potential slow roll parameters which have been constrained by Planck and WMAP-polarization (Planck+WP) to be .
To explore possible correlations between the suppression of the low multipoles in the temperature power spectrum, and features in the tensor to scalar ratio as a function of the pivot scale, and more generally, to the power spectrum of tensor perturbations, as a consequence of the fast roll stage.
To assess the scales and general aspects of features in the power spectra resulting from the modification of the initial conditions and their potential observability.
Brief summary of results: A fast roll stage prior to slow roll leads to non-Bunch-Davies conditions on the observationally relevant mode functions that cross the Hubble radius during slow roll. These modifications yield oscillatory corrections to the power spectra of curvature and tensor perturbations, with a period determined by the Hubble scale during slow roll inflation, and a modification of the consistency condition for the tensor to scalar ratio with oscillatory features as a function of the pivot scale. The results are general and do not depend on the specific form of the inflationary potential but to leading order in slow roll depend only on . We describe a systematic program that yields the solution interpolating between the fast and slow roll stages based on a derivative expansion and separation of scales, which is independent of the inflationary potentials provided these are monotonic and can be described in a derivative expansion characterized by slow roll parameters. The Non-Bunch Davies initial conditions from the fast roll stage lead to corrections to the power spectra of scalar and tensor perturbations in the form of oscillatory features with a typical frequency determined by the Hubble scale during slow roll. The corrections to the power spectrum for curvature perturbations lead to a suppression of the quadrupole that is correlated with the oscillatory features in the tensor to scalar ratio as a function of the pivot scale . The quadrupole suppression is consistent with the latest results from Planck and the oscillatory features in could be observable if the mode corresponding to the Hubble radius today crossed the Hubble radius a few e-folds from the beginning of slow roll.
2 Fast roll stage:
We consider a spatially flat Friedmann-Robertson-Walker (FRW) cosmology with
where and stand for cosmic and conformal time respectively and consider curvature and tensor perturbations. The dynamics of the scale factor in single field inflation is determined by Friedmann and covariant conservation equations
During the slow roll near de Sitter stage,
(here is the reduced Planck mass).
Instead, in this section we consider an initial stage dominated by the kinetic term, namely a fast roll stage, thereby neglecting the term in the equation of motion for the inflaton, (4) and consider the potential to be (nearly) constant and equal to the potential during the slow roll stage, namely . In the following section we relax this condition in a consistent expansion in .
The solution to (8) is given by
an initial value of the velocity damps out and the slow roll stage begins when . During the slow roll stage when it follows that
The dynamics enters the slow roll stage when as seen by (6). To a first approximation, we will assume that Eq.(9) holds not only for the kinetically dominated epoch, but also until the beginning of slow roll (). In Section 3, this approximation is justified and the error incurred from such an assumption is made explicit. The dynamics enters the slow roll stage at a value of the scale factor so that
We now use the freedom to rescale the scale factor to set
this normalization is particularly convenient to establish when a particular mode crosses the Hubble radius during slow roll, an important assessment in the analysis below.
In terms of these definitions and eqn. (11), we have that during the fast roll stage
Friedmann’s equation becomes
This equation for the scale factor can be readily integrated to yield the solution
where is an integration constant chosen to be
so that at long time . The slow roll stage begins when which corresponds to the value of given by
where to simplify notation later we defined
Introducing the dimensionless ratio of kinetic to potential contributions at the initial time
and assuming that the potential does not vary very much between the initial time and the onset of slow roll (this is quantified below), it follows from (11) that
Let us introduce
therefore , and
Before we continue with the analysis, it is important to establish the relative variation of the potential between the initial time and the onset of slow roll, assuming that the potential is monotonic and does not feature “bumps”, this is given by
with the result
For large it follows from (19) that , hence the logarithm of the term in brackets varies between for , therefore the relative change of the potential during the fast roll stage is for . This result will be used in the next section below to study a systematic expansion in to match with the slow roll results.
The acceleration equation written in terms of is given by
so that the inflationary stage begins when . At the initial time
hence, for the early stage of expansion is deccelerated and inflation begins when .
It proves convenient to introduce the variable
where is given by eqn. (19), and write in terms of this variable leading to
The number of e-folds between the initial time and a given time is given by
with a total number of e-folds between the beginning of the fast roll stage at and the onset of slow roll at given by
Fig. (1) shows as a function of for , inflation begins at and slow roll begins at . We find that this is the typical behavior for , namely for a wide range of fast roll initial conditions, the inflationary stage begins fairly soon and the fast roll stage lasts e-folds.
3 Matching to slow roll:
At the end of the fast roll stage the value of ), which is of the same order as the slow-roll solution of the equations of motion, and becomes smaller than the slow roll solution for for which . Therefore we must ensure a smooth matching to the slow roll stage. This is accomplished by recognizing that with the fast roll initial conditions there emerges a hierarchy of time scales as well as amplitudes for : during the fast roll stage the features a large amplitude and varies fast, while in the slow roll stage the amplitude is and varies slowly. Furthermore, in the previous section we have taken the potential to be (nearly) constant and recognized that the relative variation during the fast roll stage is .
In this section we treat the variation of the potential along with the slow roll corrections in a consistent perturbative formulation.
Therefore we write
where formally with being the fast roll solution (13) which is of amplitude during most of the fast roll stage. Furthermore, during the fast roll stage we assumed that the potential is nearly constant and equal to the potential during the slow roll stage, namely . We now relax this assumption by writing in the argument of the potential as in eqns. (26-29)
In writing this expression, we have assumed (self-consistently, see below) that at long time , hence adjusting an integration constant asymptotically at long time thereby extending the lower limit of the integral to . This assumption will be justified a posteriori from the solution.
Formally is of , is of , etc. Therefore
where is given by eqn. (32). From this we obtain
In the equation of motion
the right hand side is formally of order as can be seen from the definition of the slow roll variable (6). This suggests an expansion in powers of which leads to the following hierarchy of equations
Consistently with the slow roll approximation and to lowest order in slow roll we neglect in (53) as it can be shown (a posteriori) that , hence higher order in the slow roll expansion.
into eqn. (53) (neglecting ), we find
The function features the following asymptotic behavior,
Therefore for it follows that
With the results obtained above, it is straightforward to confirm that the second order correction is indeed of and further suppressed by a power of up to logarithmic terms in .
Therefore up to first order in slow roll
The second order contribution can be found by carrying out the integral in the second term in (41), this is achieved more efficiently by passing to the variable and expanding the function in a series in and integrating term by term. The result reveals that this correction is and suppressed by a power of as and is also subleading for .
The result (56) clearly shows that for
For these expressions reduce to the fast roll results of the previous section; however, for () we find
The correction to the scale factor is obtained by proposing a solution of the form
It is straightforward to find that asymptotically for
where a detailed calculation shows that the terms proportional to are subleading for . Therefore the improved fast roll solution (62) yields the correct long time behavior of the scale factor to leading order in slow roll; namely, for , the dynamics enters a near de Sitter stage
It is now clear from the solution (61) that for the fast roll, zeroth order solution () dominates but at ( and . Therefore, at , the solution is of order but off by a factor from the correct solution, resulting in an error of . In order to match to the correct slow roll solution the evolution must be continued past to a time at which the first order correction dominates. This “matching time”, , is determined by the error incurred in keeping the zeroth order term in the full solution. For example, requiring that the error be fixes so that
hence, at the “matching time”, we find that
The number of e-folds between the time , at which , and the matching time is
where the numerical result applies for . Therefore for , the total number of e-folds between the initial and the matching time is .
This quantity is a better indicator of the transition to slow roll, it features the following limits
with corrections of at the matching time .
Conformal time defined to vanish as is given by
where we integrated by parts and used the definition of given by eqn. (23). Adding and subtracting we find
The argument of the integrand in the second term in (76) vanishes to leading order in in the slow roll phase (when ). Therefore, during slow roll, .
The study in this section describes a systematic procedure to obtain a solution that is valid during the fast roll stage and that matches smoothly to the slow roll stage to any desired order in independently of the potential while under the assumption that the inflationary potential is monotonic and can be described by a derivative expansion characterized by slow roll parameters. The leading order solution is the fast roll solution (obtained in the previous section) and the above analysis shows that continuing this solution for time larger than incurs errors of order in the variable : at the zeroth-order and the improved solution differ by which in turn leads to corrections in the conformal time .
This analysis shows that the leading order corrections to the inflationary power spectra from a fast roll stage can be obtained by keeping only the fast-roll solution and integrating up to , at which point it matches to slow roll. Clearly keeping only the zeroth-order solution rather than the improved solution incurs errors of , which can (with numerical effort) be systematically improved upon by considering the corrections and improvements described in this section.
Having quantified the error incurred in keeping only the fast roll solution, we now proceed to obtain the corrections to the power spectra of scalar and tensor perturbations to leading order in the expansion in slow roll parameters, namely keeping only the fast roll solution.
4 Fast roll corrections to power spectra:
The analysis above clearly indicates that for a wide range of initial conditions dominated by the kinetic term of the inflaton potential, a fast roll stage merges with the slow roll stage within e-folds. Having quantified above the error incurred in keeping only the fast roll solution, we now proceed to obtain the corrections to the power spectra of scalar and tensor perturbations to leading order in the expansion in slow roll parameters, namely keeping only the fast roll solution. The results will yield the main corrections to the power spectra from the fast roll stage, with potential corrections of from the matching of scales. If the main features of the results obtained in leading order are supported observationally, this would justify a more thorough study that includes these corrections by implementing the systematic approach described in the previous section. Such a program would necessarily imply a larger numerical effort and would be justified if observational data suggest the presence of the main effects.
The observational constraint of nearly scale invariance suggests that wavelengths corresponding to observable quantities today crossed the Hubble radius during the slow roll era of inflation. Therefore our goal is to analyze the impact of the pre-slow roll dynamics upon perturbations with physical wavelengths that crossed the Hubble radius after the beginning of slow roll. As discussed in refs.[51, 52] and more recently in ref. the fast-roll stage prior to slow roll modifies the initial conditions on the mode functions from the usual Bunch-Davies case. The rapid dynamical evolution of the inflaton during the fast roll stage induces a correction to the potential in the equations of motion for the mode functions of curvature and tensor perturbations, which we now analyze in detail.
where stands for the derivative of the inflaton field with respect to the cosmic time .
The operators obey canonical commutation relations and the mode functions are solutions of the equation
where labels the two standard transverse and traceless polarizations and . The operators obey the usual canonical commutation relations, and are the two independent traceless-transverse tensors constructed from the two independent polarization vectors transverse to , chosen to be real and normalized such that .
The mode functions obey the differential equation of a massless minimally coupled scalar field, namely
In both these cases the mode functions obey an equation of the form,
This is a Schrödinger equation with playing the role of coordinate, the energy and a potential that depends on the coordinate . In the cases under consideration
During slow roll inflation the potential becomes
where to leading order in slow roll parameters