Covariant Predictions for Planck-Scale Features in
Primordial Power Spectra

Aidan Chatwin-Daviesa,b, Achim Kempf$\,{}^{c}$, and Petar Simidzijaa

aDepartment of Physics and Astronomy, University of British Columbia
6224 Agricultural Road, Vancouver, BC, V6T 1Z1, Canada
bInstitute for Theoretical Physics, KU Leuven
Celestijnenlaan 200D B-3001 Leuven, Belgium
cDepartment of Applied Mathematics, University of Waterloo
Waterloo, ON, N2L 3G1, Canada ††e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract

> In this companion to our letter (arXiv:2208.10514), we elaborate the full
> details of the predicted corrections to the primordial scalar and tensor
> power spectra that arise from quantum gravity-motivated, natural, covariant
> ultraviolet cutoffs. We implement these cutoffs by covariantly restricting
> the fields which are summed over in the path integrals for the primordial
> correlators, and we discuss in detail the functional analytic techniques
> necessary for evaluating such path integrals. Our prediction, which is given
> in terms of measured cosmological parameters and without assuming any
> particular inflationary potential, is that the corrections take the form of
> small oscillations which are superimposed on the conventional power spectra.
> The frequency of these oscillations only depends on the location of the
> cutoff scale and the first inflationary slow-roll parameter, while the
> amplitude and phase are also moderately sensitive to how smoothly the cutoff
> turns on. The specificity of the new predictions offers an opportunity to
> significantly enhance experimental sensitivity in observations of the cosmic
> microwave background and large-scale structure. This may be used to place
> ever higher bounds on the scale at which quantum gravity effects become
> important in quantum field theory or may even provide positive evidence for
> quantum gravity effects.

###### Contents

  1. 1 Introduction
  2. 2 Cosmological perturbations
  3. 3 Covariant ultraviolet cutoffs
  4. 4 Corrections to primordial power spectra
  5. 5 Discussion
  6. A Equivalence between path integrals and projectors
  7. B Bandlimited scalar fluctuations in de Sitter
  8. C Adiabatic approximation
  9. D Detailed features of the predicted signal

##  1 Introduction

The development of quantum gravity has been impeded by the lack of
experimental access to the Planck scale. For example, the peak energy of the
Large Hadron Collider is still about 15 orders of magnitude below the Planck
energy, and so the quantum gravitational regime remains well out of the reach
of accelerator experiments. The numbers are more favorable, however, in
cosmology. This is because, according to the standard model of cosmology, the
inhomogeneities in the Cosmic Microwave Background (CMB) originated in quantum
fluctuations of modes which froze when they exceeded the Hubble length during
inflation. Since the Hubble length at that time was only about 5 to 6 orders
of magnitude larger than the Planck length, Planck-scale effects in the CMB
should be correspondingly less suppressed and could perhaps even become
observable.

The question arises, therefore, as to what exact signature of potential
Planck-scale effects in the CMB to predict, so that experimental efforts can
be guided towards probing these predictions [1, 2]. The candidate theories for
quantum gravity differ strongly in their description of physics at the Planck
scale and therefore, in principle, each theory could yield its own predictions
[3, 4, 5, 6]. However, just below the Planck energy scale, each candidate
theory for quantum gravity must quite quickly reduce to quantum field theory
on curved spacetime in order to be consistent with the current standard model
of cosmology. This indicates that, at the Hubble scale during inflation, some
5 or 6 orders of magnitude from the Planck scale, only the most dominant
features of Planck-scale physics should have been able to leave an imprint in
the quantum fluctuations of the inflaton and metric modes that froze at that
time.

This leads to the question as to what is the most dominant impact of Planck-
scale physics on quantum field theory in curved spacetime at those scales
where quantum field theory is still a good enough model to describe inflation.
Since candidate theories of quantum gravity tend to predict the presence of
some form of natural ultraviolet (UV) cutoff [7, 8], it is natural to
conjecture that quantum field theory close to the Planck scale is modified
predominantly by the presence of an ultraviolet cutoff, which may be hard or
soft. The challenge is then to predict the impact that such a cutoff in
quantum field theory in curved spacetime would have on the predictions for the
CMB. The literature on this question has been mostly working with noncovariant
ultraviolet cutoffs and associated modified dispersion relations, see, e.g.,
[9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27,
28, 29, 30, 31, 32, 33]. For cutoff-free models, see, e.g., [34, 35, 36, 37].
There has also been significant interest in and progress made on this front
and related questions within the field of Loop Quantum Cosmology; see, e.g.,
[38, 39, 40, 41, 42, 43].

In previous work, we studied the case of a hard natural ultraviolet cutoff
that is covariant [44, 45]. That the cutoff is covariant is important to
ensure that the predictions arise only from the presence of the UV cutoff and
are uncontaminated by the breaking of symmetries. The cutoff is enacted on the
spectrum of the scalar field’s spacetime d’Alembertian, and it has an innate
information theoretic interpretation as a cutoff on the field’s density of
degrees of freedom in spacetime. We then presented a proof-of-principle
calculation to illustrate how the apparatus could be used to compute the
signature that a covariant UV cutoff would leave in the spectrum of
inflationary perturbations in the CMB [45].

In the present paper, we build on this ansatz and explicitly calculate the
correction that a natural covariant UV cutoff at (or near) the Planck scale
produces for inflationary primordial power spectra, assuming slow-roll
parameters that arise from observation. We focus on the scalar power spectrum,
i.e., the dimensionless power spectrum of the comoving curvature perturbation,
$\Delta_{\mathcal{R}}^{2}$, since it has already been characterized with ample
amounts of observational data [46, 47, 48], with more data from the CMB and
large scale structure surveys on the way [2]. A natural covariant UV cutoff
also produces a correction to the as-yet unobserved primordial spectrum of
tensor perturbations, $\Delta_{\mathcal{T}}^{2}$, which we also compute.

We assume that inflation is driven by a single inflaton field, but beyond this
assumption our calculation is model-independent, in the sense that we make no
assumptions about the inflaton’s potential. The only input required for the
calculation is the background Friedmann-Lemaître-Robertson-Walker (FLRW)
Hubble parameter which describes the inflationary phase, which we fix using
measured values of slow-roll parameters and the parameters that describe
$\Delta_{\mathcal{R}}^{2}$ (see Eq. (4.1)). We choose the Bunch-Davies vacuum
state for the cosmological perturbations so as to minimally diverge from the
standard inflationary calculation of $\Delta_{\mathcal{R}}^{2}$. It is
possible to accommodate a different choice of vacuum state within our
framework; however, the resulting predictions may change significantly as a
consequence.

We find that a natural covariant UV cutoff produces small oscillations which
are a function of the comoving momentum $k$, superimposed on top of the
uncorrected power spectrum, as illustrated in Fig. 2a. The predicted effect
depends only on two parameters: 1) the first slow-roll parameter
$\epsilon_{\star}\equiv\epsilon(k_{\star})$ at some fiducial momentum scale
$k_{\star}$, and 2) the dimensionless ratio $\sigma(k)$ of the cutoff length,
$\ell_{c}$, to the Hubble length, which varies with the comoving wave number
$k$ during the slow roll. In terms of observational parameters, $\sigma(k)$ is
given by

$\displaystyle\sigma(k)\equiv\left.\frac{H}{\Omega}\right|_{aH=k}=\frac{M_{\mathrm{Pl}}}{\Omega}\sqrt{\pi
A_{s}\epsilon_{\star}}\left(\frac{k}{k_{\star}}\right)^{-\epsilon_{\star}}.$
(1.1)

where $\Omega\equiv 1/\ell_{c}$ is the energy scale associated with the cutoff
and $A_{s}$ is the scalar perturbation amplitude. Our finding is that, for a
sharp cutoff, the relative change in the power spectrum as a function of the
comoving mode $k$ is given by

$\frac{\delta\Delta^{2}_{\mathcal{R}}}{\Delta^{2}_{\mathcal{R}}}=\mathcal{C}\frac{\sigma(k)^{3/2}}{\ln(\sigma(k)/2)}\sin\left(\omega(k)\,\sigma(k)\right),$
(1.2)

where $\mathcal{C}=0.8796...$ is a numerical constant and where we have
defined

$\displaystyle\omega(k)$
$\displaystyle\equiv\frac{1}{\sigma(k)^{2}}\ln\frac{\sigma(k)e}{2}.$ (1.3)

While the oscillations’ amplitude and phase depend mildly on fine-grained
details of the cutoff, such as how gradually it turns on, the oscillations’
frequency is a robust prediction which is essentially independent of the
hardness or softness of the cutoff. The only free parameters in this
prediction are the first slow-roll parameter, $\epsilon_{\star}$ (which is
constrained by CMB observations [49] to $0<\epsilon_{\star}\lesssim 0.004$),
and the precise energy scale of the UV cutoff, $\Omega$.

For a fixed value of $\epsilon_{\star}$, the smaller the value of $\Omega$,
the smaller is the peak oscillation frequency and the larger is the amplitude
of the oscillations, resulting in a larger imprint on the primordial power
spectrum; see Fig. 3 for illustration. Consequently, existing and future
measurements should be able to place bounds on $\Omega$, the scale at which
quantum gravity effects become important in inflation, in relation to
$\epsilon_{\star}$. In particular, an eventual measurement of
$\epsilon_{\star}$, such as through the amplitude of the CMB tensor modes,
would strenghthen the bound on $\Omega$. Furthermore, an explicit measurement
of the predicted oscillations would fix both the values of $\epsilon_{\star}$
and $\Omega$.

Since our calculations involve a variety of techniques, we have opted to be
liberal when covering background material and in the expositions. In Sec. 2,
we therefore begin by briefly reviewing relevant aspects of the theory of
cosmological perturbations and the computation of primordial power spectra
while establishing notation and definitions in preparation for our subsequent
calculations. Next, in Sec. 3, we review the definition and implementation of
the covariant natural UV cutoff which we consider. We demonstrate how such a
cutoff produces a correction to the power spectrum of a scalar field in de
Sitter spacetime and we explain how to extend the result to slowly rolling
FLRW spacetimes. In Sec. 4, we take the scalar field to be the comoving
curvature perturbation, and we compute the correction to its primordial power
spectrum assuming single field inflation and realistic slow-roll parameters.
We also discuss the correction to the tensor spectrum here. Finally, we end
with a summary and discussion in Sec. 5. The most intensive of calculational
details appear in the subsequent appendices.

##  2 Cosmological perturbations

The most remarkable success of the theory of inflation is its ability to
predict a primordial power spectrum which is in quantitative agreement with
observed large scale fluctuations in the universe, as seen, for example, in
the cosmic microwave background. Let us briefly review how this primordial
power spectrum is computed.

In the simplest model of inflation, which we consider here, the spacetime
metric $g_{\mu\nu}$ is coupled to a scalar field $\phi$, called the inflaton,
via the action

$\displaystyle
S=\frac{M_{\mathrm{Pl}}^{2}}{16\pi}\int\mathrm{d}^{4}x\sqrt{-g}R-\int\mathrm{d}^{4}x\sqrt{-g}\left[\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+V(\phi)\right],$
(2.1)

where $M_{\mathrm{Pl}}\equiv 1/\sqrt{G}$ is the Planck mass, and we work in
units $c=\hbar=1$. On the largest scales, the universe is nearly spatially
homogeneous and isotropic, and hence the metric and inflaton field can be
written as

$\displaystyle g_{\mu\nu}(\eta,\bm{x})$
$\displaystyle=a^{2}(\eta)\eta_{\mu\nu}+h_{\mu\nu}(\eta,\bm{x}),$ (2.2)
$\displaystyle\phi(\eta,\bm{x})$
$\displaystyle=\bar{\phi}(\eta)+\delta\phi(\eta,\bm{x}).$ (2.3)

The first terms describe a spatially flat FLRW cosmology with a scale factor
$a(\eta)$ and a spatially constant background field $\bar{\phi}(\eta)$, which
both depend only on the conformal time $\eta$, while the second terms allow
for deviations from spatial homogeneity and isotropy. We will assume that
these deviations are small and we will treat them quantum mechanically, while
the dominant background pieces will be treated classically.

#### Background

Substituting (2.2) and (2.3) into the action (2.1), and keeping only leading
order terms, one obtains the equations of motion for the background fields

$\displaystyle H^{2}$
$\displaystyle=\frac{8\pi}{3M_{\mathrm{Pl}}^{2}}\left(\frac{1}{2}\bar{\phi}^{\prime
2}+V(\bar{\phi})\right),$ (2.4) $\displaystyle\dot{H}$
$\displaystyle=-\frac{4\pi\bar{\phi}^{\prime 2}}{M_{\mathrm{Pl}}^{2}a^{2}},$
(2.5) $\displaystyle\bar{\phi}^{\prime\prime}$
$\displaystyle=-2aH\bar{\phi}^{\prime}-a^{2}V_{\phi}(\bar{\phi}),$ (2.6)

where $V_{\phi}\equiv dV/d\phi$, $H\equiv\dot{a}/a=a^{\prime}/a^{2}$ is the
Hubble parameter, primes denote derivatives with respect to conformal time,
$\eta$, and dots are derivatives with respect to cosmic time, $t$.

The theory of inflation postulates that in the very early universe, perhaps by
some abnormally large quantum fluctuation, the inflaton field found itself at
a value where the potential is large (close to the Planck scale) but the
gradient of the potential is given by a much lower scale. The equations of
motion (2.4) and (2.6) give the resulting background dynamics:
$\bar{\phi}(\eta)$ slowly rolls down the potential from its large value, while
$a(\eta)$ experiences a period of highly accelerated expansion, characterized
by a Hubble parameter $H(\eta)$ which is slowly decreasing in time. More
precisely, one can quantify the rate of change of $H$ via the slow-roll
parameters

$\displaystyle\epsilon\equiv-\frac{\dot{H}}{H^{2}},\quad\delta\equiv\frac{\ddot{H}}{2H\dot{H}}.$
(2.7)

Slow-roll inflation is characterized by the conditions $\epsilon\ll 1$ and
$\delta\ll 1$.

#### Perturbations

Now let us consider the fluctuations $\delta\phi$ and $h_{\mu\nu}$ on top of
this spatially homogeneous and isotropic classical background. To obtain the
dynamics of these fields, one again substitutes (2.2) and (2.3) into the
action (2.1), this time keeping terms up to second order in the perturbations;
hence, the perturbations are free fields. However, even in the absence of
interactions, there is a challenge in quantizing these fields due to the fact
that the theory is diffeomorphism-invariant, i.e. there is gauge freedom
associated with the choice of coordinates. The quantization of gauge theories
is complicated by the fact that for a gauge invariant action such as (2.1), it
is not manifest which fields constitute the physical degrees of freedom of the
theory and which fields can simply be gauged away. A careful analysis shows
that after gauge fixing, the only remaining degrees of freedom in the
perturbations are a scalar, $\mathcal{R}(x)$—the Mukhanov-Sasaki variable—and
two degrees of freedom associated with a transverse, traceless, symmetric
tensor, $h_{ij}(x)$. The gauge-fixed action is given by
$S=S_{\mathcal{R}}+S_{h}$, where [50]

$\displaystyle S_{\mathcal{R}}$
$\displaystyle=-\frac{1}{2}\int\mathrm{d}^{4}x\,z^{2}\eta^{\mu\nu}\partial_{\mu}\mathcal{R}\partial_{\nu}\mathcal{R},$
(2.8) $\displaystyle S_{h}$
$\displaystyle=-\frac{M_{\mathrm{Pl}}^{2}}{64\pi}\int\mathrm{d}^{4}x\,a^{2}\eta^{\mu\nu}\partial_{\mu}h_{ij}\partial_{\nu}h_{ij}.$
(2.9)

If we ignore the tensor structure, we see that the field $h_{ij}$—which
describes primordial gravitational waves with two polarizations—simply has the
dynamics of a massless scalar field in a FLRW spacetime with scale factor
$a(\eta)$. On the other hand, the Mukhanov-Sasaki variable experiences a
modified scale factor, $z(\eta)$, defined in terms of the background fields as

$\displaystyle
z\equiv\frac{a\dot{\bar{\phi}}}{H}=\frac{a^{2}\dot{\bar{\phi}}}{a^{\prime}}=\frac{a\bar{\phi}^{\prime}}{a^{\prime}}.$
(2.10)

Notice that the second Friedmann equation (2.5) implies
$z=(M_{\mathrm{Pl}}^{2}\epsilon/4\pi)^{1/2}a$. Hence, if the first slow-roll
parameter $\epsilon$ is constant, $z$ is simply proportional to $a$, and thus
the Hubble parameter $\dot{z}/z$ associated with the modified scale factor is
equal to the Hubble parameter $H=\dot{a}/a$ associated with the “true” scale
factor. More generally, $\epsilon$ is not constant, but rather varies as [51]

$\displaystyle\dot{\epsilon}=2H\epsilon(\epsilon+\delta),$ (2.11)

which implies

$\displaystyle\frac{\dot{z}}{z}=H(1+\epsilon+\delta).$ (2.12)

#### Canonical quantization

Let us now canonically quantize the scalar and tensor perturbations by
expanding them in terms of spatial Fourier modes. We obtain

$\displaystyle\mathcal{R}(x)$
$\displaystyle=\frac{1}{z(\eta)}\int\frac{\mathrm{d}^{3}\bm{k}}{(2\pi)^{3/2}}e^{i\bm{k}\cdot\bm{x}}u_{\bm{k}}(\eta)a_{\bm{k}}^{\dagger}+h.c.$
(2.13) $\displaystyle h_{ij}(x)$
$\displaystyle=\frac{\sqrt{16\pi}}{a(\eta)M_{\mathrm{Pl}}}\sum_{\lambda=1}^{2}\int\frac{\mathrm{d}^{3}\bm{k}}{(2\pi)^{3/2}}e^{i\bm{k}\cdot\bm{x}}\epsilon_{ij}(\hat{\bm{k}},\lambda)v_{\bm{k}}(\eta)b_{\bm{k},\lambda}^{\dagger}+h.c.$
(2.14)

Here, $a^{\dagger},b^{\dagger}$, and their adjoints are canonically commuting
creation and annihilation operators, and $\epsilon_{ij}(\hat{\bm{k}},1)$ and
$\epsilon_{ij}(\hat{\bm{k}},2)$ are two linearly independent, symmetric,
traceless, and transverse polarization tensors. The mode functions
$u_{\bm{k}}(\eta)$ and $v_{\bm{k}}(\eta)$ are harmonic oscillators with time-
dependent frequencies

$\displaystyle
u_{\bm{k}}^{\prime\prime}+\left(k^{2}-\frac{z^{\prime\prime}}{z}\right)u_{\bm{k}}$
$\displaystyle=0,$ (2.15) $\displaystyle
v_{\bm{k}}^{\prime\prime}+\left(k^{2}-\frac{a^{\prime\prime}}{a}\right)v_{\bm{k}}$
$\displaystyle=0.$ (2.16)

In the slow-roll regime, both $a(\eta)$ and $z(\eta)$ approach $1/(-\eta)$ for
large negative values of $\eta$, and hence in this limit both equations simply
reduce to that of a harmonic oscillator with frequency $k=|\bm{k}|$. A natural
choice of vacuum state is the Bunch-Davies vacuum, which corresponds to
setting the usual positive frequency initial conditions

$\displaystyle
u_{\bm{k}}(\eta)\rightarrow\frac{1}{\sqrt{2k}}e^{-i\omega\eta},\quad\text{and}\quad
v_{\bm{k}}(\eta)\rightarrow\frac{1}{\sqrt{2k}}e^{-i\omega\eta},$ (2.17)

at $\eta\rightarrow-\infty$. We denote this vacuum by $|0\rangle$ and assume
that the field starts out in this state.

The quantum fluctuations of the scalar and tensor perturbations can be
quantified in terms of their two-point correlators. The equal time correlators
are

$\displaystyle G_{\mathcal{R}}(\eta,\bm{x})$ $\displaystyle\equiv\langle
0|\mathcal{R}(\eta,\bm{x})\mathcal{R}(\eta,0)|0\rangle=\frac{1}{z(\eta)^{2}}\int\frac{\mathrm{d}^{3}\bm{k}}{(2\pi)^{3}}e^{-i\bm{k}\cdot\bm{x}}|u_{\bm{k}}(\eta)|^{2},$
(2.18) $\displaystyle G_{h}^{ij,kl}(\eta,\bm{x})$ $\displaystyle\equiv\langle
0|h_{ij}(\eta,\bm{x})h_{kl}(\eta,0)|0\rangle=\frac{16\pi}{a(\eta)^{2}M_{\mathrm{Pl}}^{2}}\int\frac{\mathrm{d}^{3}\bm{k}}{(2\pi)^{3}}e^{-i\bm{k}\cdot\bm{x}}|v_{\bm{k}}(\eta)|^{2}\Pi_{ij,kl}(\hat{\bm{k}}),$
(2.19)

where we define the quantity
$\Pi_{ij,kl}(\hat{\bm{k}})\equiv\sum_{\lambda}\epsilon_{ij}(\hat{\bm{k}},\lambda)\epsilon_{kl}^{*}(\hat{\bm{k}},\lambda)$.
The Fourier transforms of the two-point correlators are

$\displaystyle G_{\mathcal{R}}(\eta,\bm{k})$
$\displaystyle\equiv\frac{1}{(2\pi)^{3}}\int\mathrm{d}^{3}\bm{x}e^{i\bm{k}\cdot\bm{x}}G_{\mathcal{R}}(\eta,\bm{x})=\frac{|u_{\bm{k}}(\eta)|^{2}}{(2\pi)^{3}z(\eta)^{2}},$
(2.20) $\displaystyle G_{h}^{ij,kl}(\eta,\bm{k})$
$\displaystyle\equiv\frac{1}{(2\pi)^{3}}\int\mathrm{d}^{3}\bm{x}e^{i\bm{k}\cdot\bm{x}}G_{h}^{ij,kl}(\eta,\bm{x})=\frac{16\pi|v_{\bm{k}}(\eta)|^{2}}{(2\pi)^{3}a(\eta)^{2}M_{\mathrm{Pl}}^{2}}\Pi_{ij,kl}(\hat{\bm{k}}).$
(2.21)

Notice that the tensor structure of the quantity $G_{h}^{ij,kl}$ is purely
kinematic and that all of the dynamics is contained in the single function
$v_{\bm{k}}(\eta)$. This is simply the statement that the two linearly
independent tensor helicities have the same vacuum fluctuation amplitudes and
each fluctuate as a free scalar field. As shorthand, let $G_{h}$ denote
$G_{h}^{ij,kl}$ modulo its kinematic tensor structure, i.e.

$G_{h}(\eta,\bm{k})\equiv\frac{16\pi|v_{\bm{k}}(\eta)|^{2}}{(2\pi)^{3}a(\eta)^{2}M_{\mathrm{Pl}}^{2}}.$
(2.22)

#### Primordial power spectra

The scalar and tensor primordial power spectra are respectively defined as

$\displaystyle\Delta_{\mathcal{R}}^{2}(k)$
$\displaystyle\equiv\frac{k^{3}}{2\pi^{2}z(\eta_{k})^{2}}|u_{\bm{k}}(\eta_{k})|^{2},$
(2.23) $\displaystyle\Delta_{\mathcal{T}}^{2}(k)$
$\displaystyle\equiv\frac{k^{3}}{2\pi^{2}a(\eta_{k})^{2}}\frac{64\pi}{M_{\mathrm{Pl}}^{2}}|v_{\bm{k}}(\eta_{k})|^{2},$
(2.24)

where $\eta_{k}$ is the comoving time at which modes with comoving momentum of
magnitude $k=|{\bm{k}}|$ cross the Hubble horizon, i.e. it is defined as the
solution to the equation

$\displaystyle k=a(\eta_{k})H(\eta_{k}).$ (2.25)

We will make use of a few alternate expressions for the primordial power
spectra. In a slowly rolling spacetime, it is possible to obtain
approximations for the mode functions $u_{\bm{k}}$ and $v_{\bm{k}}$ (or
equivalently the two-point functions) at horizon crossing, i.e. at
$\eta=\eta_{k}$, in terms of the value of the Hubble parameter. This results
in the following expressions for the power spectra (see e.g. [52]):

$\displaystyle\Delta_{\mathcal{R}}^{2}(k)$
$\displaystyle=\frac{H^{2}}{\pi\epsilon
M_{\mathrm{Pl}}^{2}}\Big{|}_{\eta=\eta_{k}},$ (2.26)
$\displaystyle\Delta_{\mathcal{T}}^{2}(k)$ $\displaystyle=\frac{16H^{2}}{\pi
M_{\mathrm{Pl}}^{2}}\Big{|}_{\eta=\eta_{k}}.$ (2.27)

Notice that since the Hubble parameter and the slow-roll parameter $\epsilon$
vary slowly during inflation, we expect $\Delta_{\mathcal{R}}$ and
$\Delta_{\mathcal{T}}$ to vary only mildly with $k$—the primordial power
spectra should be roughly scale invariant. Indeed, observations of cosmic
perturbations indicate the scalar spectrum to be of the form

$\displaystyle\Delta_{\mathcal{R}}^{2}(k)=A_{s}\left(\frac{k}{k_{\star}}\right)^{n_{s}-1},$
(2.28)

where, at the pivot scale $k_{\star}=0.05~{}\mathrm{Mpc}^{-1}$, the amplitude
$A_{s}$ and spectral tilt $n_{s}$ are observed at the values [53]

$\displaystyle A_{s}$ $\displaystyle=(2.10\pm 0.03)\times 10^{-9},\quad$
(2.29) $\displaystyle n_{s}$ $\displaystyle=0.966\pm 0.004.$ (2.30)

Similarly, the tensor spectrum can be empirically fit to the curve

$\displaystyle\Delta_{\mathcal{T}}^{2}(k)=A_{t}\left(\frac{k}{k_{\star}}\right)^{n_{t}}.$
(2.31)

Since tensor fluctuations have not yet been observed, the values of $A_{t}$
and $n_{t}$ are unknown, but the tensor-to-scalar ratio $r\equiv A_{t}/A_{s}$
is constrained to $r<0.06$ [49].

Finally, note that using equations (2.20) and (2.22), we can write the power
spectra $\Delta_{\mathcal{R}}$ and $\Delta_{\mathcal{T}}$ in terms of two-
point correlators as

$\displaystyle\Delta_{\mathcal{R}}^{2}(k)$ $\displaystyle=4\pi
k^{3}G_{\mathcal{R}}(\eta_{k},k),$ (2.32)
$\displaystyle\Delta_{\mathcal{T}}^{2}(k)$ $\displaystyle=8\pi
k^{3}G_{h}(\eta_{k},k).$ (2.33)

(The extra factor of 2 in the tensor power spectrum comes from there being two
linearly independent gravitational wave polarizations [52].) The advantage of
this rewriting is that the two-point functions can be expressed in a
manifestly covariant manner using a path integral. The goal of this paper will
be to impose a covariant high energy cutoff on this path integral and to study
this as a simple model of the way in which Planck-scale physics might affect
inflationary power spectra.

##  3 Covariant ultraviolet cutoffs

In this section, we explain the machinery that we use to calculate predictions
for Planckian corrections to CMB power spectra. We begin by reviewing the
class of covariant natural UV cutoffs that we work with. We then give a
detailed description of how to impose such a cutoff on the fluctuation
spectrum of a quantized scalar field in de Sitter spacetime. Finally, we use
the de Sitter result to obtain the cutoff fluctuation spectrum for near-de
Sitter FLRW spacetimes.

The kinematics of covariant natural ultraviolet cutoffs are discussed in
detail in Refs.[54, 55, 56, 57, 44]. An outline of the de Sitter calculation
was given in Ref. [45], and here we elaborate the calculation in full while
making important refinements along the way.

#### Definition and implementation

Starting with a quantum field theory on a curved spacetime, our aim is to
model gravitational corrections to the effective theory via a natural UV
cutoff, and we wish to do so covariantly. Conceptually, our approach is to
suppress field configurations in the quantum field theoretic path integral
that lie beyond the Planck scale, which we parameterize in terms of
eigenfunctions and eigenvalues of the field’s d’Alembertian.

Concretely, let $(\mathcal{M},g)$ be a Lorentzian manifold $\mathcal{M}$ with
metric $g$ that supports a real scalar quantum field $\hat{\phi}$. Let $\Box$
denote the field’s d’Alembertian,

$\Box=\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\,\cdot\right),$
(3.1)

and let us suppose that any boundary conditions have been appropriately chosen
so that $\Box$ is self-adjoint. Given a non-negative function $f$, we define a
cutoff on a field configuration $\phi(x)$ via a linear combination of
projectors onto the eigenspaces of $\Box$:

$f(\Box)~{}:~{}\phi(x)~{}\mapsto~{}\sum_{\lambda\in\mathrm{spec}\,\Box}f(\lambda)\langle\psi_{\lambda},\,\phi\rangle\psi_{\lambda}(x)$
(3.2)

Here, $\psi_{\lambda}$ denotes the eigenfunction of $\Box$ with eigenvalue
$\lambda$, and $\langle\,\cdot\;,\,\cdot\,\rangle$ is the $L^{2}(\mathcal{M})$
inner product. For example, if

$f(\lambda)=\theta(\Omega^{2}-|\lambda|),$ (3.3)

where $\theta$ is the Heaviside step function, then $f(\Box)$ is a sharp
cutoff that projects fields onto the subspace spanned by eigenfunctions whose
eigenvalues’ magnitudes are less than $\Omega^{2}$. Cutoffs of the form (3.2)
are manifestly covariant; they do not depend on a choice of coordinates for
$\mathcal{M}$, as they are specified entirely in terms of the spectrum of
$\Box$, which is itself just a set of real numbers that depends on
$\mathcal{M}$ alone.

Flat spacetime is an illustrative example. Choosing
$\mathcal{M}=\mathbb{R}^{1,d}$ and usual Cartesian coordinates $(t,{\bm{x}})$,
the d’Alembertian is $\Box=-\partial_{t}^{2}+\partial_{j}\partial^{j}$ and its
eigenfunctions are plane waves,
$\psi_{k}(x)=e^{-ik^{0}t+i{\bm{k}}\cdot{\bm{x}}}$. The corresponding
eigenvalues are $(k^{0})^{2}-{\bm{k}}^{2}$, and so a sharp cutoff like (3.3)
removes Fourier contributions to a field configuration from plane waves whose
spacetime momentum-squared is greater than $\Omega^{2}$ or less than
$-\Omega^{2}$. For a given field configuration $\phi(x)$ with Fourier
transform $\tilde{\phi}(k)$, the action of

$P_{\Omega}\equiv\theta(\Omega^{2}-|\Box|)$ (3.4)

is therefore

$P_{\Omega}\phi(x)=\int_{|k_{\mu}k^{\mu}|\leq\Omega^{2}}\frac{\mathrm{d}^{d+1}k}{(2\pi)^{d+1}}~{}\tilde{\phi}(k)~{}e^{ik_{\nu}x^{\nu}}.$
(3.5)

The quantity $\Omega$ plays the role of a short-distance cutoff. Moreover,
$\Omega$ admits a natural information theoretic interpretation as a covariant
bandlimit on the density of degrees of freedom in spacetime, in the sense of
Nyquist-Shannon sampling theory [58, 59]. Given a Riemannian manifold, one can
view a _conventional_ bandlimit as a cutoff on the spectrum of the Laplacian,
$\bigtriangleup$. For example, for functions on $\mathbb{R}$, with
$\bigtriangleup=-\partial_{x}^{2}$ and eigenfunctions $e^{ikx}$, a cutoff
$\Lambda$ restricts $k^{2}\leq\Lambda^{2}$. _Bandlimited functions_ are then
functions whose Fourier transforms are compactly supported in a finite
interval $[-\Lambda,\Lambda]$, and the maximum Fourier frequency $\Lambda$ is
the _bandlimit_ , or equivalently here, the functions’ _bandwidth_. A sampling
theorem then applies: a bandlimited function can be perfectly reconstructed
everywhere on the real line knowing only its values at a discrete set of
sample points whose average density is greater than or equal to $\Lambda/\pi$
[60]. With appropriate modifications, versions of the sampling theorem above
generalize to $\mathbb{R}^{d}$ and to Riemannian manifolds [60, 61].

The covariant cutoff $\Omega$ on $\mathbb{R}^{1,d}$ is somewhat different,
mainly because the set of allowed eigenvalues
$|(k^{0})^{2}-{\bm{k}}^{2}|\leq\Omega^{2}$ is not compact. Nevertheless, each
partial Fourier transform of a covariantly bandlimited field such as (3.5)
enjoys a sampling theorem. That is, consider taking a partial Fourier
transform of $P_{\Omega}\phi$ with respect to $\bm{x}$ and holding $\bm{k}$
fixed:111We could equally have elected to take a partial Fourier transform
with respect to $t$ and hold $k^{0}$ fixed, but we will employ the former
choice in the calculations to come.

$P_{\Omega}\phi(t;{\bm{k}})=\int_{|(k^{0})^{2}-{\bm{k}}^{2}|\leq\Omega^{2}}\frac{\mathrm{d}k^{0}}{2\pi}\tilde{\phi}(k^{0},{\bm{k}})e^{-ik^{0}t}$
(3.6)

A conventional sampling theorem then applies to $P_{\Omega}\phi(t;{\bm{k}})$
because the allowed frequencies, $k^{0}$, form a compact set. In particular,
notice that arbitrarily large magnitudes of ${\bm{k}}$ are still allowed, but
the bandwidth in time for $P_{\Omega}\phi(t;{\bm{k}})$ falls to zero as
$|{\bm{k}}|\rightarrow\infty$; the spatial modes “freeze out” and their
density of degrees of freedom in time falls to zero. Furthermore, these
notions transform covariantly: a spatial mode that contracts under a Lorentz
transformation acquires a smaller bandwith in time, which is consistent with
the dilation of its degrees of freedom in time. See Ref. [44] for a more
extensive exposition.

Let us now return to a general Lorentzian manifold $\mathcal{M}$ and focus on
the sharp covariant cutoff $P_{\Omega}$. As a convenient piece of terminology,
we will say that $P_{\Omega}\phi(x)$ is a _covariantly bandlimited_ field.
Conceptually, we implement the covariant cutoff at the level of the quantum
field theoretic path integral by only integrating over covariantly bandlimited
fields. For example, consider the Feynman propagator, $G_{F}$, which can be
written in terms of a path integral as

$iG_{F}(x,x^{\prime})=\frac{\int\mathcal{D}\phi~{}\phi(x)\phi(x^{\prime})e^{iS[\phi]}}{\int\mathcal{D}\phi~{}e^{iS[\phi]}}.$
(3.7)

The covariantly bandlimited propagator, which we denote $G_{F}^{\Omega}$, is
then given by

$iG_{F}^{\Omega}(x,x^{\prime})=\frac{\int_{B_{\mathcal{M}}(\Omega)}\mathcal{D}\phi~{}\phi(x)\phi(x^{\prime})e^{iS[\phi]}}{\int_{B_{\mathcal{M}}(\Omega)}\mathcal{D}\phi~{}e^{iS[\phi]}},$
(3.8)

where
$B_{\mathcal{M}}(\Omega)\equiv\mathrm{span}\\{\psi_{\lambda}\,|\,\Box\psi_{\lambda}=\lambda\psi_{\lambda},|\lambda|\leq\Omega^{2}\\}$
denotes the space of covariantly bandlimited fields on $\mathcal{M}$.

In practice, we implement the covariant cutoff using the projectors
$P_{\Omega}$. One can view the propagator $G_{F}(x,x^{\prime})$ as the
integral kernel of an operator on $L^{2}(\mathcal{M})$ that is the right
inverse of the d’Alembert operator. The bandlimited propagator is then
obtained by projecting onto $B_{\mathcal{M}}(\Omega)$:

$G_{F}^{\Omega}=P_{\Omega}G_{F}P_{\Omega}$ (3.9)

This prescription is exactly equivalent to the path integral prescription
(3.8) when the scalar field’s action is of the form

$S[\phi]=\int\mathrm{d}^{d+1}x\sqrt{-g}~{}\phi F(\Box)\phi,$ (3.10)

as shown in App. A. In particular, this includes the case of a free scalar
field, with $F(\Box)=\Box-m^{2}$.

#### Cutoff fluctuation spectrum

Our next goal is to impose a covariant cutoff on the fluctuation spectrum of a
scalar field in a FLRW spacetime. We consider FLRW spacetimes with no spatial
curvature, for which we may write the line element

$\mathrm{d}s^{2}=a^{2}(\eta)(-\mathrm{d}\eta^{2}+\mathrm{d}x_{i}\mathrm{d}x^{i}).$
(3.11)

We choose Cartesian spatial coordinates $x^{i}$, and for an inflating
spacetime, the conformal time takes values $\eta\in(-\infty,0)$.

The fluctuation spectrum of a scalar field on such a spacetime was defined in
Eq. (2.23) in terms of its mode functions and in Eq. (2.32) in terms of its
two-point function. The two-point function coincides with the Feynman
propagator at equal times, so let us write

$\Delta^{2}_{\phi}(\eta,k)=\left.4\pi
k^{3}|G_{F}(\eta,\eta^{\prime};k)|\,\right|_{\eta=\eta^{\prime}}.$ (3.12)

$G_{F}(\eta,\eta^{\prime};k)$ denotes the spatial Fourier transform of $G_{F}$
with respect to ${\bm{x}}$, and it depends only on the magnitude
$k\equiv|{\bm{k}}|$ because of spherical symmetry. This motivates us to define
the (covariantly) cutoff fluctuation spectrum by

$(\Delta_{\phi}^{\Omega})^{2}(\eta,k)=\left.4\pi
k^{3}|G_{F}^{\Omega}(\eta,\eta^{\prime};k)|\,\right|_{\eta=\eta^{\prime}},$
(3.13)

and so we must compute the cutoff propagator $G_{F}^{\Omega}$.

In fact, what we are more interested in is the correction to the fluctuation
spectrum due to imposing a covariant cutoff. When we impose the covariant
cutoff, the Feynman propagator of course changes:

$G_{F}~{}\rightarrow~{}G_{F}^{\Omega}\equiv G_{F}+\delta G_{F}$ (3.14)

The change in the fluctuation spectrum is then

$\delta\Delta_{\phi}^{2}=4\pi k^{3}\left(|G_{F}+\delta G_{F}|-|G_{F}|\right).$
(3.15)

In practice, we will always have that $|\delta G_{F}|\ll|G_{F}|$, and so we
can compute the difference (3.15) via a Taylor series. One finds, to lowest
order in $\mathrm{Re}~{}\delta G_{F}$ and $\mathrm{Im}~{}\delta G_{F}$, that
the relative change in $\Delta_{\phi}^{2}$ is given by

$\frac{\delta\Delta_{\phi}^{2}}{\Delta_{\phi}^{2}}=\mathrm{Re}\left(\frac{\delta
G_{F}}{G_{F}}\right)+O(\delta G_{F}^{2}).$ (3.16)

In a similar vein, while the covariantly bandlimited propagator is formally
given by Eq. (3.9), we can access the correction to it directly by rewriting
(3.9) in terms of the complementary projector
$P_{\Omega}^{\perp}=I-P_{\Omega}$:

$G_{F}^{\Omega}=G_{F}+(P_{\Omega}^{\perp}G_{F}P_{\Omega}^{\perp}-P_{\Omega}^{\perp}G_{F}-G_{F}P_{\Omega}^{\perp}).$
(3.17)

The bracketed term is therefore the correction $\delta G_{F}$.

The calculation of $\delta G_{F}$ begins with writing down the Sturm-Liouville
eigenvalue problem: $\Box u(\eta,{\bm{x}})=\lambda u(\eta,{\bm{x}})$. Notice,
however, that a spatial Fourier transform with respect to ${\bm{x}}$ preserves
the spectrum of the d’Alembertian, i.e. if $\Box u(\eta,{\bm{x}})=\lambda
u(\eta,{\bm{x}})$, then $\Box_{\bm{k}}u(\eta,{\bm{k}})=\lambda
u(\eta,{\bm{k}})$. We may therefore impose the covariant cutoff on each
spatial mode individually, which will be practical since we are ultimately
interested in computing the correction to the spatial Fourier transform of the
propagator on a mode-by-mode basis. Next, we fix boundary conditions so that
$\Box_{\bm{k}}$ is self-adjoint, and we identify its spectrum. Then, with
eigenfunctions and eigenvalues in hand, for each $k$ we construct the
projector $P_{\Omega}^{\perp}$ and use it to obtain $\delta G_{F}$, and hence
$\delta\Delta^{2}_{\phi}/\Delta^{2}_{\phi}$.

#### Example: de Sitter inflation

As a both productive and necessary example, let us explicitly compute $\delta
G_{F}$ for a massless scalar field in de Sitter spacetime. This amounts to the
choice of scale factor $a(\eta)=(-H\eta)^{-1}$, and we also specialize to four
spacetime dimensions from now on. We will only describe the calculation in
broad strokes here, leaving a complete account of all the details to App. B.
In summary, the calculation essentially proceeds in six steps:

  1. 1.

Starting with the de Sitter $k$-d’Alembertian $\Box_{k}$, we write down the
two linearly independent solutions of the eigenvalue equation
$\Box_{k}u=\lambda u$ (Eqs. (B.4) and (B.5)). One solution is normalizable in
$L^{2}((-\infty,0),a^{4}(\eta)\,\mathrm{d}\eta)$ when $\lambda<9H^{2}/4$, and
so self-adjoint realizations of $\Box_{k}$ have point spectrum in this range.
In contrast, both solutions are non-normalizable for $\lambda\geq 9H^{2}/4$,
and so self-adjoint realizations of $\Box_{k}$ have continuous spectrum in
this range.

  2. 2.

We use an orthonormality relation (Eq. (B.7)) among point spectrum
eigenfunctions, $\psi_{n}$, to determine the different possible point spectra
corresponding to different self-adjoint realizations of $\Box_{k}$ as an
operator on $L^{2}((-\infty,0),a^{4}(\eta)\,\mathrm{d}\eta)$.

  3. 3.

We fix a particular choice of self-adjoint extension by requiring that
$\Box_{k}(G_{F}^{h}\psi_{n})=\psi_{n}$, where $G_{F}^{h}$ is the Hermitian
part of the Feynman propagator (Eq. (B.13)). The latter is calculated
according to canonical quantization with the Bunch-Davies vacuum chosen as the
field’s vacuum state (Eq. (B.12)). Choosing a different vacuum state would
therefore pick out a different self-adjoint extension of $\Box_{k}$.

  4. 4.

We then determine the continuous spectrum eigenfunctions, $\varphi_{q}$ (Eq.
(B.34)), by requiring that they be orthogonal to the $\psi_{n}$, as well as
mutually continuum-normalized.

  5. 5.

Next, we construct the projector $P_{\Omega}^{\perp}$ (Eq. (B.35)), which we
use to write down an expression for the change in the Feynman propagator,
$\delta G_{F}$, due to the covariant cutoff.

  6. 6.

Finally, we argue that we can neglect the point spectrum contribution to
$\delta G_{F}$, and we make several useful approximations for the continuous
spectrum contribution to arrive at a compact final expression for
$\delta\Delta^{2}_{\phi}/\Delta^{2}_{\phi}$.

We ultimately find that

$\frac{\delta\Delta^{2}_{\phi}}{\Delta^{2}_{\phi}}\approx\frac{2I(Q,x)\left[\left(\tfrac{2}{\pi}\right)^{3/2}Y_{3/2}(x)+\tfrac{4}{\pi^{3}}I(Q,x)\right]}{J_{3/2}(x)^{2}+Y_{3/2}(x)^{2}},$
(3.18)

where $I$ is the integral

$I(Q,x)=-\int_{0}^{\infty}\mathrm{d}b~{}e^{w(Q,b;2/(xe))}\sin\left(W[Q,b;2/(xe)]\right)$
(3.19)

and where we have defined $x=-k\eta$, $Q=\sqrt{\sigma^{-2}-9/4}$, and
$\sigma=H/\Omega$. The functions $w$ and $W$ are given by

$\displaystyle w(Q,b;2/(xe))$
$\displaystyle=-\frac{1}{2}\left(b+\frac{3}{2}\right)\ln\left(Q^{2}+b^{2}\right)-Q\arctan\left(\frac{b}{Q}\right)-b\ln\left(\frac{2}{xe}\right),$
(3.20) $\displaystyle W(Q,b;2/(xe))$
$\displaystyle=\frac{Q}{2}\ln\left(Q^{2}+b^{2}\right)-\left(b+\frac{3}{2}\right)\arctan\left(\frac{b}{Q}\right)+Q\ln\left(\frac{2}{xe}\right).$
(3.21)

Eq. (3.18) makes manifest that $\delta\Delta^{2}_{\phi}/\Delta^{2}_{\phi}$ is
only a function of two independent parameters: $x$, which characterizes when
we evaluate the fluctuation spectrum relative to horizon crossing, and
$\sigma$, the ratio of the cutoff and Hubble scales.

Although the integral (3.19) cannot be evaluated in closed form, note that for
$\sigma\ll 1$, i.e. $Q\gg 1$, the exponential function in the integrand decays
much more rapidly than the sine function oscillates. Hence, in this limit, the
integral can be well-approximated by expanding the functions $w$ and $W$ up to
second order in $b$ and analytically evaluating the resulting Gaussian
integral. This results in the asymptotic expansion222The next term in the
expansion for $I$ is
$\displaystyle\frac{\left(3\ln(2Q/x)+2\right)\cos\left(Q-Q\ln(2Q/x)\right)}{2Q^{5/2}\ln^{3}(2Q/x)},$
(3.22) which is smaller than the leading term by a factor of $1/Q$. Hence for
$Q\sim 1/\sigma\sim 10^{5}$, the leading approximation is quite accurate.

$\displaystyle I\sim\frac{\sin\left(Q-Q\ln(2Q/x)\right)}{Q^{3/2}\ln(2Q/x)}.$
(3.23)

We see that $I\ll 1$ for $Q\gg 1$, and thus to a good approximation we can
further neglect the $I^{2}$ term in the expression (3.18) for the relative
change in $\Delta_{\phi}$, which gives

$\frac{\delta\Delta^{2}_{\phi}}{\Delta^{2}_{\phi}}\approx-\frac{4(\cos x+x\sin
x)}{\pi(1+x^{2})}\frac{\sin(Q-Q\ln[2Q/x])}{Q^{3/2}\ln[2Q/x]}.$ (3.24)

It is expected that inflationary fluctuations freeze out (i.e. stop
fluctuating) as they cross the horizon, so from now on we will set $x=1$ to
reflect this expectation. Then, if we also approximate $Q\approx\sigma^{-1}$,
we arrive at the following compact expression for the relative change in
$\Delta_{\phi}$:

$\left.\frac{\delta\Delta^{2}_{\phi}}{\Delta^{2}_{\phi}}\right|_{x=1}\approx\frac{2(\cos
1+\sin 1)}{\pi}\frac{\sigma^{3/2}}{\ln(\sigma/2)}\sin(\omega(\sigma)\sigma).$
(3.25)

In the equation above, we defined the $\sigma$-dependent frequency

$\displaystyle\omega(\sigma)\equiv\frac{1}{\sigma^{2}}\ln\frac{\sigma e}{2}.$
(3.26)

These oscillations are a key feature of our predicted correction to the power
spectrum. A plot of the correction to $\Delta_{\phi}^{2}$ as a function of
$\sigma$, in which these characteristic oscillations can be seen, is shown in
Fig. 1.

Figure 1: Relative change in $\Delta^{2}_{\phi}$ due to a sharp covariant
cutoff as a function of $\sigma=H/\Omega$ with $x=1$ fixed to horizon
crossing. For the purposes of clearly illustrating the oscillations, here we
plot $\delta\Delta_{\phi}^{2}/\Delta_{\phi}^{2}$ for a fairly large value of
$\sigma\sim 10^{-2}$.

#### Predictions for slow-roll FLRW inflation

For the purpose of comparing with cosmological data, given a slowly-rolling
inflationary spacetime, the goal is to produce a prediction for
$\delta\Delta^{2}_{\phi}/\Delta^{2}_{\phi}$ evaluated at horizon crossing as a
function of the comoving mode $k$, namely

$\left.\frac{\delta\Delta^{2}_{\phi}}{\Delta^{2}_{\phi}}\right|_{aH=k}.$
(3.27)

For a de Sitter scale factor, horizon crossing happens at $x=-k\eta=1$ for all
modes, and so (3.27) is a constant correction for all $k$. This is just a
reflection of the fact that the proper size of the de Sitter horizon is
constant in time, and so all modes are the same proper size when they cross
the horizon. Consequently, the magnitude of their fluctuations is the same, as
well as the magnitude of the correction due to a covariant
cutoff.333Explicitly, here both $x=-k\eta$ and $\sigma=H/\Omega$ are constant
in using Eq. (3.18) to compute (3.27). On the other hand, in a slowly-rolling
inflationary spacetime, the proper size of the cosmological horizon is slowly
changing, and so different modes will have different proper wavelengths when
they cross the horizon. The prediction (3.27) will therefore be nontrivial as
a function of $k$.

The calculation that we carried out for de Sitter in Sec. 3 is intractable for
a generic FLRW spacetime, including even for a simple example such as a power
law scale factor. Our strategy will therefore be to approximate the prediction
(3.27) for a given slowly-rolling FLRW spacetime by a succession of
instantaneously de Sitter calculations. That is, we will use the de Sitter
result (3.18) to compute (3.27), but with a Hubble parameter $H$ that depends
on $k$. Intuitively, such an adiabatic approximation will be accurate provided
that the true time-dependent Hubble parameter of the FLRW spacetime,
$H(\eta)$, evolves sufficiently slowly, which is what we expect during slow-
roll inflation. We discuss this adiabatic, or “slow-roll” approximation and
additional supporting evidence for its validity in App. C.

In practice, for each given comoving mode $k$, we must fix the values of
$\eta$ and $H$, or equivalently the values of $x$ and $\sigma$, in Eq. (3.25).
Since we are interested in the correction to a mode’s fluctuations when it
crosses the cosmological horizon, and since Eq. (3.25) was derived for a de
Sitter background, we will set $\eta$ to be the de Sitter horizon-crossing
time; that is, we fix $x=-k\eta=1$ in Eq. (3.25). Then, to fix the value of
$H$, we use the horizon-crossing condition $a(\eta)H(\eta)=k$ for the slowly-
rolling $a(\eta)$ to determine the value of $H(\eta)$ at horizon crossing. We
show this calculation explicitly in the next section.

##  4 Corrections to primordial power spectra

In this section, we compute the correction to the primordial power spectrum
due to a covariant natural UV cutoff using realistic cosmological parameters.
We focus on the scalar spectrum, $\Delta_{\mathcal{R}}^{2}$, but the
calculation for the tensor spectrum, $\Delta_{\mathcal{T}}^{2}$, is completely
analogous.

As input to the prediction (3.27), we need the Hubble parameter seen by a mode
$k$ when it crosses the cosmological horizon. We compute this by comparing the
theoretical form of $\Delta_{\mathcal{R}}^{2}$ (without cutoff) given by Eq.
(2.26) with its observational parameterization (2.28).444Note that we are
justified in using the uncorrected value of $\Delta_{\mathcal{R}}^{2}$ to
compute $H$ because the error incurred in doing so is of second order, whereas
we are computing a first order correction. Equating the two, we obtain

$\left.\frac{H^{2}}{\epsilon}\right|_{aH=k}=M_{\mathrm{Pl}}^{2}\pi
A_{s}\left(\frac{k}{k_{\star}}\right)^{n_{s}-1}.$ (4.1)

In order to get $H^{2}$ as a function of $k$, we need to account for how the
first slow-roll parameter $\epsilon$ depends on $k$. To leading order in the
slow-roll expansion, this is given by

$\displaystyle\frac{d\ln\epsilon}{d\ln k}=2(\epsilon+\delta),$ (4.2)

where $\delta$ is the second slow-roll parameter, as defined in Eq. (2.7)
[52]. Taylor expanding $\ln\epsilon(k)$ about the pivot scale $k_{\star}$
therefore yields

$\displaystyle\epsilon(k)=\epsilon_{\star}\left(\frac{k}{k_{\star}}\right)^{2(\epsilon_{\star}+\delta_{\star})},$
(4.3)

where $\epsilon(k_{\star})\equiv\epsilon_{\star}$,
$\delta(k_{\star})\equiv\delta_{\star}$, and we suppress terms that are
quadratic or higher in $\epsilon_{\star}$ and $\delta_{\star}$. Similarly, to
leading order in the slow-roll expansion, the spectral tilt is given by

$\displaystyle n_{s}=1-4\epsilon_{\star}-2\delta_{\star}.$ (4.4)

Dividing by the cutoff scale $\Omega$, we therefore find that the ratio of the
Hubble and cutoff scales at the time when the mode $k$ crosses the horizon is
given by

$\displaystyle\sigma(k)\equiv\left.\frac{H}{\Omega}\right|_{aH=k}=\frac{M_{\mathrm{Pl}}}{\Omega}\sqrt{\pi
A_{s}\epsilon_{\star}}\left(\frac{k}{k_{\star}}\right)^{-\epsilon_{\star}}.$
(4.5)

From the expression above, we see that $\sigma(k)$ depends on two cosmological
parameters, $A_{s}$ and $\epsilon_{\star}$, as well as single new parameter,
namely, the covariant cutoff scale $\Omega$. Finally, we can use Eq. (4.5) in
Eq. (3.18) or Eq. (3.25) to obtain a prediction for the corrections
$\delta\Delta_{\mathcal{R}}^{2}/\Delta_{\mathcal{R}}^{2}$ to the scalar
primordial power spectrum, where we take the placeholder scalar field $\phi$
to be the Mukhanov-Sasaki variable, $\mathcal{R}$. The corrections are given
by

$\displaystyle\left.\frac{\delta\Delta^{2}_{\mathcal{R}}}{\Delta^{2}_{\mathcal{R}}}\right|_{aH=k}=\mathcal{C}\frac{\sigma(k)^{3/2}}{\ln(\sigma(k)/2)}\sin\left(\omega(k)\,\sigma(k)\right),$
(4.6)

where $\mathcal{C}=2(\cos 1+\sin 1)/\pi$,

$\displaystyle\omega(k)\equiv\omega(\sigma(k))=\frac{1}{\sigma(k)^{2}}\ln\frac{e\sigma(k)}{2},$
(4.7)

and where $\sigma(k)$ is given by Eq. (4.5). Both the amplitude and frequency
of the oscillations track the ratio $\sigma(k)$ of the Hubble and cutoff
scales at horizon crossing, and as a result both are functions of $k$.

Computing a concrete correction using Eq. (4.6) requires specifying values for
$A_{s}$, $\epsilon_{\star}$, and $\Omega$. The amplitude $A_{s}$ is tightly
constrained by CMB observations [49] and takes the value $A_{s}\approx
2.1\times 10^{-9}$. Assuming that the Hubble length increases during
inflation, as it must for inflation to ultimately end, $\epsilon_{\star}$ must
be positive. Additionally, CMB data gives an upper bound on $\epsilon_{\star}$
at around 0.004 for the pivot scale $k_{\star}=0.05\text{ Mpc}^{-1}$ [49].
Thus,

$\displaystyle 0\leq\epsilon_{\star}\lesssim 0.004.$ (4.8)

While $\Omega$ is a new parameter that is to be determined experimentally, it
is already subject to certain constraints out of principle. First, $\Omega$ is
presumably bounded from above by the Planck scale $M_{\mathrm{Pl}}$. There
should also be a lower bound on $\Omega$ to ensure that $\sigma(k)\leq 1$ for
all momenta $k$ corresponding to modes which we observe in the CMB, i.e. for
$k_{\text{min}}<k<k_{\text{max}}$, where $k_{\text{min}}=10^{-4}\text{
Mpc}^{-1}$ and $k_{\text{max}}=10\text{ Mpc}^{-1}$. This ensures that the
cutoff energy is above the Hubble energy for all visible modes, which should
be true because effective field theory (which should be valid only below the
scale of quantum gravity) seems to make accurate predictions for inflationary
perturbations. Because $\epsilon_{\star}>0$, $\sigma(k)$ is a decreasing
function of $k$, and so $\sigma(k)\leq 1$ is equivalent to
$\sigma(k_{\text{min}})\leq 1$. Hence, using Eq. (4.5), we expect that

$\displaystyle\sqrt{\pi
A_{s}\epsilon_{\star}}\left(\frac{k_{\text{min}}}{k_{\star}}\right)^{-\epsilon_{\star}}\leq\frac{\Omega}{M_{\mathrm{Pl}}}\leq
1.$ (4.9)

Given these constraints, the model (4.6) should be fit to observations of the
primordial power spectrum, which can be inferred from late time power spectra,
such as those of the CMB. We leave a detailed analysis of this fitting to
future work. Here, to get a qualitative sense of what our prediction looks
like, in Fig. 2 we plot the baseline PPS and its correction for particular
choices of $\epsilon_{\star}$ and $\Omega$. We find that a sharp covariant
natural UV cutoff causes small $k$-dependent oscillations which are
superimposed on the non-cutoff primordial power spectrum.

(a)

(b)

(c)

Figure 2: (a) The primordial scalar power spectrum, corrected due to a sharp
cutoff $M_{\mathrm{Pl}}/\Omega=1$, with $\epsilon_{\star}=0.003$. The inset
plot shows the correction itself, $\delta\Delta_{\mathcal{R}}^{2}$, over a
small interval of modes which resolves the oscillations. (b) The uncorrected
primordial power spectrum is shown in red, and the corrected spectrum due to a
sharp cutoff $M_{\mathrm{Pl}}/\Omega=2.1\times 10^{4}$ is shown in blue, also
with $\epsilon_{\star}=0.003$. (c) The same curves are plotted as in (b), but
here with $M_{\mathrm{Pl}}/\Omega=2.1\times 10^{4}$ and
$\epsilon_{\star}=0.004$. In all plots, other parameter values are
$A_{s}=2.1\times 10^{-9}$, $k_{\star}=0.05~{}\mathrm{Mpc}^{-1}$, and
$n_{s}=0.97$.

First we note that the oscillation frequency $\omega(\sigma(k))$ is the most
robust predicted feature of covariant natural UV cutoffs. So far, we have
focused on a sharp cutoff $P_{\Omega}=\theta(\Omega^{2}-|\Box|)$; however, the
class of cutoffs specified by Eq. (3.2) that we could consider is much larger.
For example, we could consider a cutoff $f(\Box)$ that smooths out the
Heaviside step function in $P_{\Omega}$. Doing so effectively introduces a
free functional parameter (which characterizes the degree of smoothness), yet
even so, the oscillation frequency is largely independent of such precise
details of the cutoff. This is in contrast to the oscillations’ amplitude and
phase, which depend more strongly on such details. These points are discussed
more thoroughly in App. D. In a nutshell, smoothing out the cutoff shifts the
oscillations’ phase and tends to reduce their amplitude. The former has little
practical consequences, but the latter makes the oscillations harder to
observe. With these findings in mind, here we continue to focus on the case of
a sharp cutoff, with the understanding that this corresponds to the strongest
possible signal.

Next, we would like to get a sense for how the predicted signal depends on
$\Omega$ and $\epsilon_{\star}$, which are the least constrained parameters of
the model. To this end, define the amplitude and number of oscillations in the
visible window $k_{\text{min}}<k<k_{\text{max}}$ as

$\displaystyle A(k)$
$\displaystyle\equiv\frac{\sigma(k)^{3/2}}{\ln(\sigma(k)/2)},$ (4.10)
$\displaystyle N_{\text{osc}}$
$\displaystyle\equiv\frac{\left|\omega(k_{\text{max}})\sigma(k_{\text{max}})-\omega(k_{\text{min}})\sigma(k_{\text{min}})\right|}{2\pi}.$
(4.11)

We plot $A(k_{\star})$ and $N_{\text{osc}}$ as functions of $\epsilon_{\star}$
and $\Omega$ in Fig. 3. For concreteness we plot the amplitude at the pivot
scale, but because the amplitude changes relatively little with $k$, the plot
would look qualitatively the same had we chosen any other scale between
$k_{\text{min}}$ and $k_{\text{max}}$.

(a) (b)

Figure 3: Density plots illustrating how (a) the amplitude, $A$, and (b) the
number of oscillations in the visible window of modes, $N_{\mathrm{osc}}$,
scale with the parameters $\epsilon_{\star}$ and $\Omega$. The white regions
correspond to disallowed parameter values for which $\sigma=H/\Omega>1$.

Let us consider the scaling of $A(k)$ and $N_{\text{osc}}$ with the parameters
$\Omega$ and $\epsilon_{\star}$. Since
$\sigma\sim\epsilon_{\star}^{1/2}M_{\mathrm{Pl}}/\Omega$, ignoring logarithmic
corrections we see that

$\displaystyle A$
$\displaystyle\sim\left(\frac{M_{\mathrm{Pl}}}{\Omega}\right)^{3/2}\epsilon_{\star}^{3/4},$
(4.12) $\displaystyle N_{\text{osc}}$
$\displaystyle\sim\frac{\Omega}{M_{\mathrm{Pl}}}\epsilon_{\star}^{1/2}.$
(4.13)

First, note that as the cutoff $\Omega$ increases, the predicted amplitude of
corrections to the PPS decreases. This makes sense, because the correction
should vanish in the limit that the cutoff is removed by taking
$\Omega\rightarrow\infty$. Interestingly, the scaling $\Omega^{-3/2}$ of the
amplitude exactly interpolates between previous findings in the literature
that the decay should be linear [17, 27] or quadratic [18, 31, 33] in the
ratio of cutoff and Hubble scales. That $N_{\text{osc}}$ increases with
increasing $\Omega$ is consistent with the intuition that the oscillations
arise as a sort of “interference” between the Hubble and cutoff scales [45].
Heuristically, peaks in $\delta\Delta_{\mathcal{R}}^{2}(k)$ occur for $k$ such
that an integer number of cutoffs lengths fit into $1/H(k)$, with similar
reasoning applying to troughs. Increasing $\Omega$ therefore accelerates this
interference pattern when scanning through a range of modes $k$.

We can understand the dependence of $A$ on $\epsilon_{\star}$ by noting that,
from the perspective of the uncorrected PPS (2.26), increasing
$\epsilon_{\star}$ may be compensated for by increasing $H$, which brings the
Hubble scale closer to the cutoff scale. To understand how $N_{\text{osc}}$
scales with $\epsilon_{\star}$, note that taking $\epsilon_{\star}\rightarrow
0$ corresponds (to leading order) to a static Hubble parameter for which the
correction $\delta\Delta_{\mathcal{R}}^{2}(k)$ is just a constant offset with
no oscillations. However, the $\epsilon_{\star}\rightarrow 0$ limit should be
taken with a grain of salt in the leading order calculations here, since the
Mukhanov-Sasaki variable is undefined if $\epsilon\rightarrow 0$, and
$\Delta_{\mathcal{R}}^{2}$ diverges.

The prediction for the primordial tensor power spectrum is computed similarly
to the scalar case. The only difference is that we write $\sigma$ in terms of
the tensor parameters as

$\left.\sigma\right|_{aH=k}=\frac{\sqrt{\pi
A_{t}}}{4\Omega}\left(\frac{k}{k_{\star}}\right)^{n_{t}}.$ (4.14)

Notice however, that unlike in the case of scalar perturbations, the slow roll
parameter $\epsilon_{\star}$ does not appear in this expression. Instead it is
replaced by the tensor spectral tilt index $n_{t}$. Should eventual
measurement of the tensor spectrum be sensitive enough to determine a value
for $n_{t}$, then $\Omega$ would be the only parameter left to constrain in
the tensor spectrum. Additionally, because the ratio of tensor to scalar
amplitudes, $A_{t}/A_{s}$, is equal to $16\epsilon_{\star}$, a measurement of
the tensor spectrum would also give a value to $\epsilon_{\star}$, in which
case $\Omega$ would be the only parameter left to constrain in the scalar PPS
as well.

##  5 Discussion

#### Summary

We calculated the signature that a generic, quantum gravity-motivated, natural
UV cutoff would leave in primordial inflationary power spectra.

The UV cutoff that we considered takes the form of a cutoff on the spectrum of
a scalar field’s d’Alembertian, i.e., it is a large eigenvalue cutoff for
scalar fields. As such, it covariantly generalizes the notion of a maximum
Fourier frequency to arbitrary Lorentzian manifolds. It also admits a natural
information theoretic interpretation as a cutoff on the density of field
degrees of freedom in spacetime, in the sense of Shannon sampling theory; see
[44, 45].

We implemented the natural UV cutoff in the language of path integrals by
restricting the space of fields integrated over in a path integral to only
those fields that are spanned by eigenfunctions of the d’Alembertian whose
eigenvalues are less than the cutoff scale. Conceptually, this can be thought
of as discarding, in a covariant way, the contributions to the path integral
of field configurations which fluctuate too far off shell. In practice, this
is equivalent to constructing projectors and using them to restrict operators
to the subspace defined by the cutoff. We illustrated this process by
calculating the covariantly cut-off Feynman propagator for a massless scalar
field in de Sitter spacetime and we explained how to generalize the result to
slowly-rolling FLRW spacetimes in Sec. 3.

Furthermore, we used the fact that the primordial scalar perturbation and each
polarization of the tensor perturbation are massless scalar fields that
propagate on a slowly-rolling FLRW background, and that the power spectra of
their fluctuations are straightforward to calculate in terms of the Feynman
propagator. This allowed us to use our results for the covariantly bandlimited
Feynman propagator to compute the correction that a covariant natural
ultraviolet cutoff produces on the primordial power spectra.

While our calculations for the tensor and scalar spectra are analogous, we
focused on the scalar spectrum, $\Delta^{2}_{\mathcal{R}}(k)$, due to its
better experimental prospects. We found that the correction induced by the
cutoff, $\delta\Delta^{2}_{\mathcal{R}}(k)$, takes the form of small
$k$-dependent oscillations;555Log-oscillations, i.e., corrections to the
primordial power spectrum that schematically oscillate like $\sin(\omega\ln
k)$, are not unique to our prediction; see, e.g., Ref. [27], or Ref. [34] and
Refs. [27-43] therein, and also Refs. [41, 40] for oscillatory predictions
from Loop Quantum Cosmology. To our knowledge, the chirping $k$-dependence of
$\omega(k)$ in Eq. (4.7), as well as the overall amplitude in Eq. (4.6) for
the case of a sharp cutoff, are specific to our prediction. see Eqs. (4.6) and
(4.7). The amplitude and phase of the oscillations depend moderately on the
precise details of the cutoff, namely, how smoothly and over how many Planck
lengths it turns on. However, the oscillation frequency is a particularly
robust characteristic of the prediction. The frequency, as a function of $k$,
tracks the ratio of the cutoff and Hubble scales when the mode $k$ crossed the
Hubble length during inflation, and its functional form is completely fixed
according to Eq. (4.7). The only free parameters of this prediction are the
first slow-roll parameter $\epsilon_{\star}$, and the location of the cutoff;
for fixed $\epsilon_{\star}$, the oscillation frequency decreases with
decreasing $\Omega$. Therefore, in the case that the cutoff scale is located
before the Planck scale, the predicted signature is stronger due to a larger
amplitude and lower frequency. This scenario could correspond to quantum
gravitational effects becoming important at, e.g., the string scale.

#### Effective field theory of inflation

Effective field theory has been used to systematically study the possible high
energy corrections to inflation [62]. As always in effective field theory, one
considers a Lagrangian with all possible local interaction terms that satisfy
the appropriate symmetries, and one provides a Wilsonian cutoff below which
the coupling coefficients to these terms can be deduced from experiments. By
integrating out higher energy contributions to the path integral, the
Wilsonian cutoff is lowered and the coupling coefficients flow to different
values. Of course, in the absence of experiments, effective field theory does
not pick out a particular theory (i.e. a particular set of coupling
coefficients) as the correct one.

What we are doing in this paper, on the other hand, is to study the free
theory of inflaton and linearized gravitational perturbations, with a
covariant cutoff imposed on the path integral. Unlike in the effective field
theory framework, we are studying, therefore, one particular theory; there are
no undetermined coefficients to fix using experiments. In particular, we
emphasize that we did not obtain our field theory by integrating out trans-
Planckian contributions to the path integral. Instead, we discard these
contributions from the quantum field theoretic path integral and study the
resulting modified theory.

Nevertheless, the formalism of effective field theory may well be broad enough
to encompass also the UV modification to quantum field theory that we are
considering here. To this end, a possible starting point is as follows. A path
integration over the space $B_{\mathcal{M}}(\Omega)$ of covariantly
bandlimited fields can be re-written as a path integral over the space of all
fields with an indicator function weighting the integration. Explicitly,

$\int_{B_{\mathcal{M}}(\Omega)}\mathcal{D}\phi~{}e^{iS[\phi]}=\int\mathcal{D}\phi~{}\Pi_{\Omega}[\phi]e^{iS[\phi]},$
(5.1)

where the indicator function $\Pi_{\Omega}[\phi]=1$ if $\phi\in
B_{\mathcal{M}}(\Omega)$ and vanishes otherwise, or is a more complicated
functional in the case of a soft cutoff. We may then exponentiate the
indicator to formally obtain

$\int\mathcal{D}\phi~{}e^{i(S[\phi]-i\ln\Pi_{\Omega}[\phi])}.$ (5.2)

One could then attempt to understand $i\ln\Pi_{\Omega}[\phi]$ as higher-order
corrections to the free field action $S[\phi]$. We leave these considerations
to future work.

### Choice of vacuum state

For our analysis, we chose the state of the quantum field to be the Bunch-
Davies state in order to deviate as minimally as possible from the standard
inflationary calculation. To put our motivation another way, because we are
modelling quantum gravitational physics by effectively discarding certain
modes’ contributions to the path integral, we did not want to conflate our
modelling of Planck-scale physics with effects that arise from altering the
vacuum states for these modes. Moreover, there is evidence that deviations
from the Bunch-Davies vacuum might interfere with gauge invariance and create
divergences in perturbation theory [63].

That said, given suitable motivation, it is certainly possible to make a
prediction, per our formalism, for a different choice of vacuum state. Within
our framework, changing the vacuum state can be accomplished by changing the
choice of self-adjoint extension of the d’Alembertian (Step 3 of the
calculation in Sec. 3). Such a change would eventually percolate to a
different prediction for $\delta\Delta_{\mathcal{R}}^{2}$, which could be
quite significant [38, 43], given that a different choice of vacuum state can
significantly alter the uncorrected primordial power spectrum itself.

#### Power at large comoving scales

In principle, our predictions are compatible with an apparent lack of power at
large scales in measured CMB power spectra; see, e.g., Sec. III.2.6 of Ref.
[64]. A covariant cutoff at a sufficiently low energy scale could produce a
long wavelength oscillation that amounts to a trough at large comoving scales
that would reduce the power in the primordial spectrum there. Such a
possibility is illustrated in Fig. 2c; whether or not this suppression happens
depends also on the value of the slow roll parameter $\epsilon_{\star}$.
Should this tension be confirmed to be more than a statistical anomaly, it may
help to further constrain the fit of our predicted template to observed data.

#### Observational prospects

A natural covariant UV cutoff, $\Omega$, is widely expected to exist somewhere
between the Hubble scale during inflation and the Planck scale. $\Omega$
should be no larger than the Planck scale, because this is the scale at which
we expect quantum gravitational effects to dominate, and hence the low-energy
path integral description to break down. Also, $\Omega$ is expected to be not
too close to the Hubble scale during inflation, in order to be consistent with
the standard quantum field theoretic description of cosmological
perturbations, which is supported by observations. These expectations are
encapsulated by the inequality (4.9).

Here, we calculated concrete predictions for the corrections to the primordial
scalar and tensor power spectra that arise from such a cutoff, as a function
of $\Omega$. The main prediction is that the presence of this UV cutoff
results in oscillations on top of the familiar, nearly-scale-invariant,
primordial power spectrum curve; see Fig. 2.

If the natural UV cutoff is located at what is presumably the upper limit of
its possible range, i.e., at the Planck scale with $\Omega=M_{\mathrm{Pl}}$,
then, as Fig. 2a shows for a sharp cutoff and a representative value of
$\epsilon_{\star}$, the predicted oscillations have an amplitude which is very
small (about 9 orders of magnitude smaller than the mean value of the power
spectrum) and their frequency is very high (there are roughly $10^{4}$
oscillations in the observable window
$10^{-4}~{}\mathrm{Mpc}^{-1}<k<10~{}\mathrm{Mpc}^{-1}$). These numbers would
seem to suggest that in order to measure this signal, measurements which are
about 9 orders of magnitude more sensitive than what is currently possible
would be needed. Moreover, it would seem necessary to measure at $\sim 10^{4}$
different $k$ values in order to fully resolve the oscillations. This seems
particularly challenging, given furthermore that fast oscillations run the
risk of being washed out in the processes of binning and computing derived
power spectra (such as primordial power spectra) from measured CMB power
spectra.

Fortunately, not quite as much accuracy should be needed to test the
predictions. In particular, it should be possible to place constraints on the
values of $\Omega$ and $\epsilon_{\star}$ even with existing data, given the
scaling of the predicted correction with these parameters.666In the absence of
an independent measurement of the first slow-roll parameter at the pivot
scale, $\epsilon_{\star}$, constraints on $\Omega$ will take the form of a
joint constraint on $\Omega$ and $\epsilon_{\star}$, since existing data
currently only provides an upper bound on $\epsilon_{\star}$. As illustrated
in Fig. 3, lowering the cutoff away from the Planck scale toward the Hubble
scale increases the oscillations’ amplitude and decreases their frequency,
with the effect becoming quite drastic as $\Omega$ approaches the Hubble
scale. Lower values of $\epsilon_{\star}$ result in smaller-amplitude
oscillations, which are unfavourable for detectability, but also slower
oscillation frequencies, which can counteract the accelerating effect of
having the cutoff approach the Planck scale. If the oscillations are too slow,
however, the correction $\delta\Delta_{\mathcal{R}}^{2}$ tends to a constant
offset in $\Delta_{\mathcal{R}}^{2}$, which is indistinguishable from a change
of the Hubble scale $H$ during inflation. Altogether and at the end of the
day, we hope to extract bounds on the possible values of a sharp cutoff
$\Omega$ and the slow-roll parameter $\epsilon_{\star}$ by comparing the
predictions to data.

To get a sense of what these bounds might look like, let us suppose that
oscillations whose amplitude is above some threshold and whose frequency lies
in some window are detectable. These ranges of amplitudes and frequencies
correspond to specific regions in the plots of Fig. 3. Superposing these plots
and taking the intersection of the aforementioned regions leads to a contour
such as the one shown in Fig. 4, which illustrates the region of the
$\Omega-\epsilon_{\star}$ parameter space which we could exclude. We emphasize
that the thresholds used to draw an exclusion region here are arbitrary and
meant only as illustration; a more careful statistical analysis is of course
needed in order to extract bounds, which we leave to future work.
Nevertheless, Fig. 4 illustrates what these bounds might look like. It is
worth mentioning that, in the best possible case in which the predicted
oscillations are resolved, this measurement would determine the values of both
$\Omega$ and $\epsilon_{\star}$.

Figure 4: The blue region consists of values of $\epsilon_{\star}$ and
$\Omega$ for which the amplitude of the oscillations satisfies $\log_{10}A>-8$
and for which the number of oscillations in the window of visible modes
satisfies $-1<\log_{10}N_{\text{osc}}<2$. The black contour lines match those
in Fig. 3. We emphasize that these amplitude and frequency thresholds have
been chosen arbitrarily and are only meant to illustrate the sorts of bounds
that may result from fits to real data.

In addition, observational prospects may be aided by the specificity of the
prediction. In particular, given the tightness of the constraints on $A_{s}$,
the frequency of the predicted oscillations in the primordial power spectra
depends essentially only on $\Omega$ and $\epsilon_{\star}$, even if the
cutoff is softened. If the cutoff is sharp, then even the phase and amplitude
of the predicted oscillations depend only on $\Omega$ and $\epsilon_{\star}$.
This means that the prediction can be thought of as a two-parameter family of
template waveforms, parameterized by $\Omega$ (or the ratio of the natural UV
cutoff scale to the Hubble scale) and $\epsilon_{\star}$. Experimentally then,
the search for these template waveforms should offer a significantly improved
signal to noise ratio as, in effect, template search allows one to filter out
the noise from that part of the function space that is orthogonal to the space
spanned by the template functions, similar to using a low-pass filter to
remove noise with frequencies above a signal’s frequencies. Template search
methods have of course recently been used to great effect in the detection of
gravitational waves, e.g., from black hole mergers. There, a three-parameter
family of template waveforms was successfully used to detect the passage of
gravitational waves with a strain of only $10^{-24}$ [65, 66].

It will be very interesting to compare the predictions here with present and
upcoming observation-based precision reconstructions of the scalar primordial
power spectrum. This will yield at least ever higher lower bounds on the
location of a natural UV cutoff scale $\Omega$ and it may eventually yield
positive evidence for the existence of a natural UV cutoff, a possible
experimental signal of quantum gravitational origin.

Acknowledgments

We thank Panos Betzios, François Bouchet, Richard Easther, Simon Foreman,
Lukas Hergt, Arjun Kar, Jorma Louko, Rob Martin, and Mark Van Raamsdonk for
helpful discussions during the preparation of this manuscript, as well as
Gianluca Calcagni and Albert Roura for comments on the first version. A.C.D.
acknowledges the support of the Natural Sciences and Engineering Research
Council of Canada (NSERC), [funding reference number PDF-545750-2020]. A.C.D.
was supported for a portion of this work as a postdoctoral fellow (Fundamental
Research) of the National Research Foundation – Flanders (FWO), Belgium. AK
acknowledges support through a Discovery Grant of the National Science and
Engineering Council of Canada (NSERC) and a Discovery Project grant of the
Australian Research Council (ARC). PS acknowledges support from the NSERC
CGS-D award.

##  Appendix A Equivalence between path integrals and projectors

Here, we show that the two expressions for $G_{F}^{\Omega}$ given in Eqs.
(3.8) and (3.9) are exactly equivalent for scalar field actions of the form
(3.10). Starting with Eq. (3.7), act on its left and right with projectors
$P_{\Omega}$:

$\displaystyle i(P_{\Omega}G_{F}P_{\Omega})(x,x^{\prime})$
$\displaystyle=i\iint\mathrm{d}y\,\mathrm{d}z~{}P_{\Omega}(x,y)G_{F}(y,z)P_{\Omega}(z,x^{\prime})$
$\displaystyle=\frac{1}{\mathcal{N}}\int\mathcal{D}\phi\left(\int\mathrm{d}y~{}P_{\Omega}(x,y)\phi(y)\right)\left(\int\mathrm{d}z~{}\phi(z)P_{\Omega}(z,x^{\prime})\right)e^{iS[\phi]}$

In the above, we denote by $\mathcal{N}$ the normalization

$\mathcal{N}=\int\mathcal{D}\phi~{}e^{iS[\phi]},$ (A.1)

and we abbreviate the spacetime integral measures, for example
$\mathrm{d}^{d+1}y\sqrt{-g(y)}$ by simply $\mathrm{d}y$. Next, because
$P_{\Omega}$ is symmetric, it follows that $P(z,x^{\prime})=P(x^{\prime},z)$,
and so we may write

$i(P_{\Omega}G_{F}P_{\Omega})(x,x^{\prime})=\frac{1}{\mathcal{N}}\int\mathcal{D}\phi~{}P_{\Omega}\phi(x)P_{\Omega}\phi(x^{\prime})e^{iS[\phi]}.$
(A.2)

Suppose now that $S[\phi]$ is of the form (3.10), and insert a resolution of
the identity $I=P_{\Omega}+P_{\Omega}^{\perp}$:

$\displaystyle S[(P_{\Omega}+P_{\Omega}^{\perp})\phi]$
$\displaystyle=\int\mathrm{d}x~{}(P_{\Omega}\phi)F(\Box)(P_{\Omega}\phi)+\int\mathrm{d}x~{}(P_{\Omega}^{\perp}\phi)F(\Box)(P_{\Omega}^{\perp}\phi)$
(A.3)
$\displaystyle\qquad+\int\mathrm{d}x~{}(P_{\Omega}\phi)F(\Box)(P_{\Omega}^{\perp}\phi)+\int\mathrm{d}x~{}(P_{\Omega}^{\perp}\phi)F(\Box)(P_{\Omega}\phi)$

Consider one of the cross terms in the second line of Eq. (A.3), and expand
the integrand in terms of eigenfunctions of the d’Alembertian,
$\psi_{\lambda}$. Explicitly, writing

$\phi(x)=\sum_{\lambda\,\in\,\mathrm{spec}\,\Box}\phi_{\lambda}\psi_{\lambda}(x),$
(A.4)

we find the following:

$\displaystyle\int\mathrm{d}x~{}(P_{\Omega}\phi)F(\Box)(P_{\Omega}^{\perp}\phi)$
$\displaystyle=\int\mathrm{d}x~{}\left(\sum_{|\lambda|\leq\Omega^{2}}\phi_{\lambda}\psi_{\lambda}(x)\right)F(\Box)\left(\sum_{|\lambda^{\prime}|>\Omega^{2}}\phi_{\lambda^{\prime}}\psi_{\lambda^{\prime}}(x)\right)$
$\displaystyle=\sum_{|\lambda|\leq\Omega^{2}}\sum_{|\lambda^{\prime}|>\Omega^{2}}\phi_{\lambda}\phi_{\lambda^{\prime}}F(\lambda^{\prime})\int\mathrm{d}x~{}\psi_{\lambda}(x)\psi_{\lambda^{\prime}}(x)$
$\displaystyle=\sum_{|\lambda|\leq\Omega^{2}}\sum_{|\lambda^{\prime}|>\Omega^{2}}\phi_{\lambda}\phi_{\lambda^{\prime}}F(\lambda^{\prime})\delta_{\lambda\lambda^{\prime}}$

Of course, the sums above are shorthand for a sum over spectrum and should be
replaced with integrals wherever $\Box$ has continuous spectrum. To go to the
final line, we used the fact that the d’Alembertian’s eigenfunctions are
orthonormal. However, notice that the values of $\lambda$ and
$\lambda^{\prime}$ that are summed over do not overlap, and so the cross-terms
in Eq. (A.3) vanish. We therefore have that
$S[\phi]=S[P_{\Omega}\phi]+S[P_{\Omega}^{\perp}\phi]$.

Since $P_{\Omega}\phi$ and $P_{\Omega}^{\perp}\phi$ are independent degrees of
freedom, we can path-integrate over them separately. Therefore, Eq. (A.2)
altogether reads

$i(P_{\Omega}G_{F}P_{\Omega})(x,x^{\prime})=\frac{1}{\mathcal{N}}\int\mathcal{D}(P_{\Omega}\phi)~{}P_{\Omega}\phi(x)P_{\Omega}\phi(x^{\prime})e^{iS[P_{\Omega}\phi]}\int\mathcal{D}(P_{\Omega}^{\perp}\phi)e^{iS[P_{\Omega}^{\perp}\phi]}.$
(A.5)

Similarly, it follows that

$\mathcal{N}=\int\mathcal{D}(P_{\Omega}\phi)~{}e^{iS[P_{\Omega}\phi]}\int\mathcal{D}(P_{\Omega}\phi^{\perp})~{}e^{iS[P_{\Omega}^{\perp}\phi]},$
(A.6)

and so we recover the path integral expression (3.8) for the covariantly
bandlimited Feynman propagator.

##  Appendix B Bandlimited scalar fluctuations in de Sitter

An important part of Sec. 3 was determining the relative change in the
fluctuation power spectrum of a massless scalar field, $\hat{\phi}$, in de
Sitter spacetime:

$\frac{\delta\Delta_{\phi}^{2}}{\Delta_{\phi}^{2}}=\mathrm{Re}\left(\frac{\delta
G_{F}}{G_{F}}\right).$ (B.1)

$G_{F}$ denotes the field’s Feynman propagator and $\delta
G_{F}=P_{\Omega}^{\perp}G_{F}P_{\Omega}^{\perp}-P_{\Omega}^{\perp}G_{F}-G_{F}P_{\Omega}^{\perp}$
is the correction induced by a covariant cutoff, $\Omega$. We only outlined
the six major steps of this calculation in the main text; here we go through
each of these steps in full detail. These steps are:

  1. 1.

Determining the spectrum of the $k$-d’Alembertian, $\Box_{k}$.

  2. 2.

Determining the self-adjoint realizations of $\Box_{k}$.

  3. 3.

Fixing the choice of self-adjoint realization.

  4. 4.

Determining the continuous spectrum eigenfunctions.

  5. 5.

Using the projector $P_{\Omega}^{\perp}$ to compute
$\delta\Delta_{\phi}^{2}/\Delta_{\phi}^{2}$.

  6. 6.

Making approximations for calculating $P_{\Omega}^{\perp}\chi_{0}$.

#### Step 1: Determining the spectrum of the $k$-d’Alembertian, $\Box_{k}$

Consider the flat slicing of de Sitter spacetime in four spacetime dimensions.
Explicitly, we take the line element to be

$\mathrm{d}s^{2}=a^{2}(\eta)\left[-\mathrm{d}\eta^{2}+\mathrm{d}x_{i}\mathrm{d}x^{i}\right]$
(B.2)

and the scale factor to be $a(\eta)=(-H\eta)^{-1}$, $\eta\in(-\infty,0)$. With
these choices, the Sturm-Liouville eigenvalue problem $\Box_{k}u=\lambda u$
reads

$(a^{2}u^{\prime})^{\prime}+k^{2}a^{2}u+\lambda a^{4}u=0,$ (B.3)

where ′ denotes a derivative with respect to the conformal time $\eta$ and
where we recall that $k=|{\bm{k}}|$ is the Fourier variable for $\bm{x}$. The
eigenfunction equation (B.3) has two linearly independent solutions,

$\displaystyle u_{J}(\eta)$
$\displaystyle=(-\eta)^{3/2}J_{p(\lambda)}(-k\eta)$ (B.4) $\displaystyle
u_{Y}(\eta)$ $\displaystyle=(-\eta)^{3/2}Y_{p(\lambda)}(-k\eta),$ (B.5)

where $J$ and $Y$ denote Bessel functions of the first and second kind,
respectively, and where

$p(\lambda)=\sqrt{\frac{9}{4}-\frac{\lambda}{H^{2}}}.$ (B.6)

Because we will ultimately be concerned with a realization of $\Box_{k}$ as a
self-adjoint operator and because the spectra of self-adjoint operators are
real, we need only consider real $\lambda$. Notice that when
$\lambda<9H^{2}/4$, $u_{J}$ is normalizable while $u_{Y}$ is not. Therefore,
self-adjoint realizations of $\Box_{k}$ will have point spectrum in this
range. Both solutions are non-normalizable for $\lambda\geq 9H^{2}/4$, and so
this range will be the continuous spectrum.

#### Step 2: Determining the self-adjoint realizations of $\Box_{k}$

A subtlety that we must address, however, is that the differential expression
$\Box_{k}$ has no unique realization as a self-adjoint operator on
$L^{2}((-\infty,0),a^{4}(\eta)\,\mathrm{d}\eta)$. As a symmetric operator, its
deficiency indices are $(1,1)$, meaning that $\Box_{k}$ has a one-parameter
family of self-adjoint extensions corresponding to different (generalized)
boundary conditions [67, 68, 69]. That the deficiency indices are $(1,1)$ can
be shown directly by inspecting the solutions of $\Box_{k}u=\pm iu$, i.e.,
Eqs. (B.4) and (B.5) with $\lambda=\pm i$. In both cases, $u_{J}$ is
normalizable while $u_{Y}$ is not; each deficiency space is therefore one-
dimensional and the deficiency indices are $(1,1)$.

A convenient way of parameterizing these self-adjoint extensions is by the
location of the largest eigenvalue in the point spectrum. In particular, a set
of square-integrable orthonormal eigenfunctions is given by

$\psi_{n}(\eta)=H^{2}\sqrt{2p_{n}}(-\eta)^{3/2}J_{p_{n}}(-k\eta),$ (B.7)

where $p_{n}=p_{0}+2n$, $n\in\mathbb{Z}$, $n\geq 0$, and $p_{0}\in(0,2]$. The
tower of discrete eigenvalues in the point spectrum is then

$\lambda_{n}=H^{2}\left(\frac{9}{4}-p_{n}^{2}\right),$ (B.8)

and so different choices of $p_{0}$ produce different point spectra that
correspond to the different self-adjoint extensions of $\Box_{k}$.
Orthonormality of the point spectrum eigenfunctions (B.7) follows immediately
from computing $\langle\psi_{m},\psi_{n}\rangle$, where the inner product is
given by

$\langle
u,v\rangle=\int_{-\infty}^{0}\mathrm{d}\eta\,a^{4}(\eta)~{}u^{*}(\eta)v(\eta)$
(B.9)

for $u,v\in L^{2}((-\infty,0),a^{4}(\eta)\mathrm{d}\eta)$, and from the
integral identity [70, Eq. 10.22.57]

$\int_{0}^{\infty}\frac{\mathrm{d}x}{x}J_{m}(x)J_{n}(x)=\frac{2\sin(\tfrac{\pi}{2}(m-n))}{\pi(m^{2}-n^{2})}.$
(B.10)

#### Step 3: Fixing the choice of self-adjoint realization

Having identified the different self-adjoint extensions of $\Box_{k}$, the
obvious question is which one should we use? We will choose the self-adjoint
extension that corresponds to choosing the Bunch-Davies vacuum state for the
scalar field, and we will determine the value of $p_{0}$ to which this
corresponds by examining the action of the full (unmodified) Feynman
propagator on a test eigenfunction.

Before getting into the above procedure and its meaning, we first need to
write down the Feynman propagator. According to canonical quantization, one
can express the Feynman propagator as a time-ordered expectation value,

$G_{F}(x,x^{\prime})=\langle
0|\mathcal{T}\hat{\phi}(x)\hat{\phi}(x^{\prime})|0\rangle.$ (B.11)

In the conformal coordinates of Eq. (3.11), making a Fourier transform with
respect to ${\bm{x}}-{\bm{x}}^{\prime}$, and choosing the Bunch-Davies vacuum
state for the mode functions of $\hat{\phi}$, one arrives at [71]

$\displaystyle
G_{F}(\eta,\eta^{\prime};k)=-\frac{i\pi}{4}\frac{\sqrt{\eta\eta^{\prime}}}{a(\eta)a(\eta^{\prime})}$
$\displaystyle\left[\theta(\eta-\eta^{\prime})H_{3/2}^{(1)}(-k\eta)H_{3/2}^{(2)}(-k\eta^{\prime})\right.$
(B.12)
$\displaystyle\left.\qquad+\theta(\eta^{\prime}-\eta)H_{3/2}^{(2)}(-k\eta)H_{3/2}^{(1)}(-k\eta^{\prime})\right],$

where $H^{(1)}$ and $H^{(2)}$ denote Hankel functions of the first and second
kind, respectively. For reasons that will soon become apparent, note that the
propagator above has both Hermitian and anti-Hermitian parts, given by

$G_{F}^{h}(\eta,\eta^{\prime};k)=\frac{1}{2}\left[G_{F}(\eta,\eta^{\prime};k)+G_{F}(\eta^{\prime},\eta;k)^{*}\right]$
(B.13)

and

$G_{F}^{ah}(\eta,\eta^{\prime};k)=\frac{1}{2}\left[G_{F}(\eta,\eta^{\prime};k)-G_{F}(\eta^{\prime},\eta;k)^{*}\right],$
(B.14)

respectively.

The Feynman propagator is a right inverse of the d’Alembertian, and so an
eigenfunction of the d’Alembertian with eigenvalue $\lambda\neq 0$ is also an
eigenfunction of the propagator, but with eigenvalue $\lambda^{-1}$. Given the
expression for $G_{F}$ in Eq. (B.12), we select a particular self-adjoint
extension by dialing the value of $p_{0}\in(0,2]$ (which parameterizes the
choice of self-adjoint extension) so that the action of $G_{F}$ on a test
eigenfunction of the form (B.7) is equal to the same eigenfunction, but
multiplied by $\lambda^{-1}$. In other words, the choice of the Bunch-Davies
vacuum in canonical quantization implies a particular choice of self-adjoint
extension in the functional analytic language that we are using here, and we
are deducing what that choice is.

An important subtlety is that (B.12) is not the integral kernel of an operator
on the Hilbert space $L^{2}((-\infty,0),a^{4}(\eta)\,\mathrm{d}\eta)$. As
operators, the d’Alembertian and Feynman propagator satisfy

$\hat{\Box}_{k}\hat{G}_{F(k)}=\hat{I},$ (B.15)

where we use a hat to explicitly indicate that this is an operator relation on
$L^{2}((-\infty,0),a^{4}(\eta)\,\mathrm{d}\eta)$. However, the integral kernel
$G_{F}(\eta,\eta^{\prime};k)$ in (B.12) does not define a good operator on
this Hilbert space. It is straightforward to check that its anti-Hermitian
part, Eq. (B.14), maps elements of
$L^{2}((-\infty,0),a^{4}(\eta)\,\mathrm{d}\eta)$ outside of the Hilbert space.
Rather, it is the Hermitian kernel (B.13) that defines an integral operator
whose domain and range are both contained in
$L^{2}((-\infty,0),a^{4}(\eta)\,\mathrm{d}\eta)$.

Once a self-adjoint extension has been fixed, each $k$-d’Alembertian can be
expressed in terms of its spectrum and eigenfunctions as

$\hat{\Box}_{k}=\sum_{\mathrm{spec}~{}\Box_{k}}\lambda~{}|\lambda\rangle\langle\lambda|,$
(B.16)

where we adopt bra-ket notation to denote vectors and their dual linear
functionals. As its right inverse, we may therefore write

$\hat{G}_{F(k)}=\sum_{\mathrm{spec}~{}\Box_{k}\setminus\\{0\\}}\frac{1}{\lambda}~{}|\lambda\rangle\langle\lambda|+(|0\rangle\langle
f|+|f\rangle\langle 0|).$ (B.17)

If zero is in the spectrum of the chosen self-adjoint extension of the
d’Alembertian, then the Feynman propagator’s range can have support on this
eigenspace as an operator since it is only a right-inverse of the
d’Alembertian (i.e. in this case the vector $|f\rangle$ is nonzero). This is
the role played by the bracketed term in Eq. (B.17) above. (Note that the term
is written out in a symmetric way, to make manifest the fact that
$\hat{G}_{F(k)}$ is Hermitian.) We will find that this is indeed the case.

Let $\psi_{n}(\eta)$ be a test eigenfunction as given by Eq. (B.7) with
$p_{0}$ not yet fixed. According to the spectral expansion (B.17), the action
of the (Hermitian part of the) Feynman propagator on $\psi_{n}(\eta)$ must
give

$(G_{F}^{h}\psi_{n})(\eta)\overset{!}{=}\frac{1}{\lambda_{n}}\psi_{n}(\eta)+\alpha(-\eta)^{3/2}J_{3/2}(-k\eta)$
(B.18)

where the constant $\alpha$ may not vanish if $\lambda=0$ is in the point
spectrum. Let us evaluate the left-hand side:

$\displaystyle(G_{F}^{h}\psi_{n})(\eta)$
$\displaystyle\equiv\int_{-\infty}^{0}a^{4}(\xi)\,\mathrm{d}\xi~{}G_{F}^{h}(\eta,\xi;k)\psi_{n}(\xi)$
$\displaystyle=-\frac{\pi}{4}\sqrt{2p_{n}}(-\eta)^{3/2}\left\\{J_{3/2}(-k\eta)\int_{0}^{\infty}\frac{\mathrm{d}x}{x}Y_{3/2}(x)J_{p_{n}}(x)\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad-
Y_{3/2}(-k\eta)\int_{0}^{\infty}\frac{\mathrm{d}x}{x}J_{3/2}(x)J_{p_{n}}(x)$
$\displaystyle\qquad\qquad\qquad\qquad-2J_{3/2}(-k\eta)\int_{0}^{-k\eta}\frac{\mathrm{d}x}{x}Y_{3/2}(x)J_{p_{n}}(x)$
$\displaystyle\left.\qquad\qquad\qquad\qquad\qquad-2Y_{3/2}(-k\eta)\int_{0}^{-k\eta}\frac{\mathrm{d}x}{x}J_{3/2}(x)J_{p_{n}}(x)\right\\}$

In the second equality, we changed the integration variable to $x=-k\xi$. The
following two antiderivatives are useful, assuming that $m^{2}\neq n^{2}$ [70,
Eq. 10.22.6]:

$\int\frac{\mathrm{d}x}{x}J_{m}(x)J_{n}(x)=\frac{x(J_{m-1}(x)J_{n}(x)-J_{m}(x)J_{n-1}(x))-(m-n)J_{m}(x)J_{n}(x)}{m^{2}-n^{2}}$
(B.19)
$\int\frac{\mathrm{d}x}{x}Y_{m}(x)J_{n}(x)=\frac{x(Y_{m-1}(x)J_{n}(x)-Y_{m}(x)J_{n-1}(x))-(m-n)Y_{m}(x)J_{n}(x)}{m^{2}-n^{2}}$
(B.20)

In particular, both antiderivatives vanish as $x\rightarrow 0^{+}$ if $m<n$.
Therefore, let us choose $p_{n}$ with $n\geq 1$ so that the $x=0$ endpoint of
the integrals above do not contribute. Making this choice, we arrive at

$\displaystyle(G_{F}^{h}\psi_{n})(\eta)$
$\displaystyle=\frac{1}{\lambda_{n}}\psi_{n}(\eta)+\frac{1}{2(\tfrac{9}{4}-p_{n}^{2})}\sqrt{2p_{n}}(-\eta)^{3/2}$
(B.21)
$\displaystyle\qquad\cdot\left\\{-J_{3/2}(-k\eta)\cos(\tfrac{\pi}{2}(\tfrac{3}{2}-p_{n}))+Y_{3/2}(-k\eta)\sin(\tfrac{\pi}{2}(\tfrac{3}{2}-p_{n}))\right\\}.$

The first term in the equation above is what we expect from inverse action,
and the second term is in the kernel of the d’Alembertian _as a differential
expression_. However, the $Y_{3/2}$ contribution is not normalizable, and so
it must vanish if we require the Hermitian part of the Feynman propagator _as
an operator_ to map into the Hilbert space. This happens when $p_{0}=3/2$,
giving $\sin(\tfrac{\pi}{2}(\tfrac{3}{2}-p_{n}))=0$. Altogether, we then
arrive at

$(G_{F}^{h}\psi_{n})(\eta)=\frac{1}{\lambda_{n}}\psi_{n}(\eta)-\frac{(-1)^{n}}{\lambda_{n}}\sqrt{\frac{p_{n}}{6}}\psi_{0}(\eta),$
(B.22)

and we have concluded that the self-adjoint extension implicit in the choice
of Bunch-Davies vacuum for the Feynman propagator is the one for which
$p_{0}=3/2$. Notice that $\lambda_{0}=0$ is therefore in the point spectrum.

#### Step 4: Determining the continuous spectrum eigenfunctions

Once we have fixed the self-adjoint extension of $\Box_{k}$ as well as the
orthonormal eigenvectors in the point spectrum, we can determine the
eigenfunctions for $\lambda>9H^{2}/4$ in the continuous spectrum.777We will
not need the edge case $\lambda=9H^{2}/4$, for which $\varphi_{0}(\eta)$ can
be expressed as a linear combination of $J_{0}(-k\eta)$ and $Y_{0}(-k\eta)$.
Let us take as an ansatz

$\varphi_{q}(\eta)=H^{2}(-\eta)^{3/2}N_{q}(a_{q}\,\mathrm{Re}\,J_{iq}(-k\eta)+b_{q}\,\mathrm{Im}\,J_{iq}(-k\eta)),$
(B.23)

where

$q\equiv q(\lambda)=\sqrt{\frac{\lambda}{H^{2}}-\frac{9}{4}},$ (B.24)

and the weights $a_{q}$, $b_{q}$ as well as the overall normalization $N_{q}$
are all real. $J_{iq}$ is a Bessel function of the first kind of purely
imaginary order, and we use its real and imaginary parts to form linearly
independent solutions of the eigenfunction equation.

First, we can determine $a_{q}$ and $b_{q}$ by demanding that the $\psi_{n}$
and $\varphi_{q}$ must be orthogonal:

$\displaystyle\langle\psi_{n},\varphi_{q}\rangle$
$\displaystyle=\int_{-\infty}^{0}a^{4}(\eta)\,\mathrm{d}\eta~{}H^{4}(-\eta)^{3}\sqrt{2p_{n}}N_{q}J_{p_{n}}(-k\eta)(a_{q}\,\mathrm{Re}\,J_{iq}(-k\eta)+b_{q}\,\mathrm{Im}\,J_{iq}(-k\eta))$
$\displaystyle\propto\int_{0}^{\infty}\frac{\mathrm{d}x}{x}J_{p_{n}}(x)(a_{q}\,\mathrm{Re}\,J_{iq}(x)+b_{q}\,\mathrm{Im}\,J_{iq}(x))$

In the second line we again defined $x=-k\eta$. According to [70, Eq.
10.22.57], we have that

$\displaystyle\int_{0}^{\infty}\frac{\mathrm{d}x}{x}J_{p_{n}}(x)J_{iq}(x)$
$\displaystyle=\frac{\Gamma(\tfrac{p_{n}}{2}+\tfrac{iq}{2})}{2\Gamma(1+\tfrac{p_{n}}{2}+\tfrac{iq}{2})\Gamma(1+\tfrac{p_{n}}{2}-\tfrac{iq}{2})\Gamma(1-\tfrac{p_{n}}{2}+\tfrac{iq}{2})}$
(B.25)
$\displaystyle=\frac{\sin\left(\tfrac{\pi}{2}(p_{n}-iq)\right)}{\tfrac{\pi}{2}(p_{n}^{2}+q^{2})}$
$\displaystyle=\frac{(-1)^{n}\sqrt{2}}{\pi(p_{n}^{2}+q^{2})}\left(\cosh(\tfrac{\pi}{2}q)+i\sinh(\tfrac{\pi}{2}q)\right).$

To go to the second line, we used gamma function identities, and in the third
line we used that $p_{n}=\tfrac{3}{2}+2n$. Consequently,

$\langle\psi_{n},\varphi_{q}\rangle\propto
a_{q}\cosh(\tfrac{\pi}{2}q)+b_{q}\sinh(\tfrac{\pi}{2}q).$ (B.26)

Motivated by hindsight, a convenient choice is
$a_{q}=\operatorname{sech}(\tfrac{\pi}{2}q)$ and
$b_{q}=-\operatorname{csch}(\tfrac{\pi}{2}q)$, giving

$\varphi_{q}(\eta)=H^{2}(-\eta)^{3/2}N_{q}\left[\operatorname{sech}(\tfrac{\pi}{2}q)\,\mathrm{Re}\,J_{iq}(-k\eta)-\operatorname{csch}(\tfrac{\pi}{2}q)\,\mathrm{Im}\,J_{iq}(-k\eta)\right].$
(B.27)

To fix the normalization, we will require that

$\langle\varphi_{q},\varphi_{q^{\prime}}\rangle=\delta(q-q^{\prime}).$ (B.28)

This amounts to setting $N_{q}=\sqrt{\tfrac{1}{2}q\tanh(\pi q)}$, which can be
verified by numerically integrating
$\langle\varphi_{q},\varphi_{q^{\prime}}\rangle$ around a small neighbourhood
of $q-q^{\prime}=0$ or via the following analytic argument.888We thank Jorma
Louko for pointing this out to us. On grounds of orthonormality, we know that
$\langle\varphi_{q},\varphi_{q^{\prime}}\rangle$ is proportional to
$\delta(q-q^{\prime})$. As such, when evaluating
$\langle\varphi_{q},\varphi_{q^{\prime}}\rangle$, we need only look for
contributions to its distributional part. By definition, we have that

$\langle\varphi_{q},\varphi_{q^{\prime}}\rangle=\int_{0}^{\infty}\frac{\mathrm{d}x}{x}\,\frac{1}{2}\left[\sqrt{q\tanh(\pi
q)}\left(\operatorname{sech}(\tfrac{\pi}{2}q)\,\mathrm{Re}\,J_{iq}(x)-\operatorname{csch}(\tfrac{\pi}{2}q)\,\mathrm{Im}\,J_{iq}(x)\right)\right]\cdot\left[q\rightarrow
q^{\prime}\right].$ (B.29)

For large $x$, $|J_{iq}(x)|/x\leq C/x^{3/2}$ for some constant $C$, and so the
integration over any interval $(\epsilon,\infty)$ with $\epsilon>0$ will not
give a distributional contribution. For small $x$,

$J_{iq}(x)=\left(\frac{x}{2}\right)^{iq}\frac{1}{\Gamma(1+iq)}(1+O(x^{2})).$
(B.30)

Therefore, we may replace $J_{iq}(x)$ with $(x/2)^{iq}/\Gamma(1+iq)$ to
compute the distributional contribution coming from the $x=0$ endpoint of
integration. Making this replacement, we find that

$\displaystyle\operatorname{sech}(\tfrac{\pi}{2}q)\,\mathrm{Re}\,J_{iq}(x)-\operatorname{csch}(\tfrac{\pi}{2}q)$
$\displaystyle\,\mathrm{Im}\,J_{iq}(x)$ (B.31)
$\displaystyle\rightarrow~{}\frac{1}{2}\left[(\operatorname{sech}(\tfrac{\pi}{2}q)+i\operatorname{csch}(\tfrac{\pi}{2}q))\left(\frac{x}{2}\right)^{iq}\frac{1}{\Gamma(1+iq)}\right.$
$\displaystyle\left.\qquad+(\operatorname{sech}(\tfrac{\pi}{2}q)-i\operatorname{csch}(\tfrac{\pi}{2}q))\left(\frac{x}{2}\right)^{-iq}\frac{1}{\Gamma(1-iq)}\right].$

The distributional part will therefore come from terms of the form

$\int_{0}^{\epsilon}\frac{\mathrm{d}x}{x}\left(\frac{x}{2}\right)^{i\gamma}=\int_{\tilde{\epsilon}}^{\infty}\mathrm{d}t~{}e^{-i\gamma
t}=\pi\delta(\gamma)+(\text{non-distributional}),$ (B.32)

where we let $x=2e^{-t}$ in the first equality. We can then read off the
coefficient of $\delta(q-q^{\prime})$ from Eqs. (B.29), (B.31), and (B.32); it
is obtained by collecting the coefficients of the cross-terms $(x/2)^{\pm
i(q-q^{\prime})}$, setting $q=q^{\prime}$, and multiplying by $\pi$, as we are
instructed to do by (B.32):

$\displaystyle\pi\cdot\frac{1}{2}q\tanh(\pi
q)\cdot\frac{1}{4}\left(2\frac{(\operatorname{sech}(\tfrac{\pi}{2}q)+i\operatorname{csch}(\tfrac{\pi}{2}q))(\operatorname{sech}(\tfrac{\pi}{2}q)-i\operatorname{csch}(\tfrac{\pi}{2}q))}{\Gamma(1+iq)\Gamma(1-iq)}\right)$
(B.33) $\displaystyle=\frac{\pi}{4}q\tanh(\pi
q)\left(\operatorname{sech}^{2}(\tfrac{\pi}{2}q)+\operatorname{csch}^{2}(\tfrac{\pi}{2}q)\right)\frac{\sinh(\pi
q)}{\pi q}$ $\displaystyle=1$

In going to the second and third lines, we used standard gamma function and
hyperbolic identities, respectively. Therefore, it follows that

$\varphi_{q}(\eta)=H^{2}(-\eta)^{3/2}\sqrt{\tfrac{1}{2}q\tanh(\pi
q)}\left[\operatorname{sech}(\tfrac{\pi}{2}q)\,\mathrm{Re}\,J_{iq}(-k\eta)-\operatorname{csch}(\tfrac{\pi}{2}q)\,\mathrm{Im}\,J_{iq}(-k\eta)\right]$
(B.34)

are indeed continuum-orthonormalized in the index $q$.

#### Step 5: Using the projector $P_{\Omega}^{\perp}$ to compute
$\delta\Delta_{\phi}^{2}/\Delta_{\phi}^{2}$

With all of the ingredients in hand, we can now construct the projector
$P_{\Omega}^{\perp}$. The (integral kernel of the) projector is given by

$P_{\Omega}^{\perp}(\eta,\eta^{\prime})=\int_{Q}^{\infty}\mathrm{d}q~{}\varphi_{q}(\eta)\varphi_{q}(\eta^{\prime})+\sum_{n\geq
N}\psi_{n}(\eta)\psi_{n}(\eta^{\prime}),$ (B.35)

where $Q=q(\Omega^{2})$, as given by Eq. (B.24), and
$N=\min\\{n:\lambda_{n}<-\Omega^{2}\\}$ (cf. Eq. (B.8)).

We must now compute $\delta\Delta^{2}_{\phi}/\Delta^{2}_{\phi}$, as given by
Eq. (3.16). First, note that this expression simplifies if we carefully
examine the structure of the Feynman propagator. The Feynman propagator
evaluated at equal times is a purely imaginary quantity,

$G_{F}(\eta=\eta^{\prime};k)=-\frac{i\pi}{4}(-\eta)^{3}H^{2}\left(J_{3/2}(-k\eta)^{2}+Y_{3/2}(-k\eta)^{2}\right).$
(B.36)

Moreover, the Hermitian (Eq. (B.13)) and anti-Hermitian (Eq. (B.14)) parts of
the Feynman propagator, and hence also the two correction terms $\delta
G_{F}^{h}$ and $\delta G_{F}^{ah}$, are purely real and purely imaginary,
respectively.999In fact, they are purely real and purely imaginary not just at
equal times, but for all times. Therefore, we have that

$\frac{\delta\Delta^{2}_{\phi}}{\Delta^{2}_{\phi}}\approx\left.\frac{\delta
G_{F}^{ah}}{G_{F}}\right|_{\eta=\eta^{\prime}},$ (B.37)

and so we only need to compute the correction to the anti-Hermitian part of
the propagator.

In analogy with the eigenfunction for the eigenvalue $\lambda=0$, let us
define the following function:

$\chi_{0}(\eta)=H^{2}\sqrt{3}(-\eta)^{3/2}Y_{3/2}(-k\eta)$ (B.38)

Then, we can write $G_{F}^{ah}(\eta,\eta^{\prime};k)$ as

$G_{F}^{ah}(\eta,\eta^{\prime};k)=-\frac{i\pi}{12H^{2}}\left[\psi_{0}(\eta)\psi_{0}(\eta^{\prime})+\chi_{0}(\eta)\chi_{0}(\eta^{\prime})\right],$
(B.39)

as well as its bandlimited version as

$\displaystyle G_{F}^{ah,\Omega}$
$\displaystyle=P_{\Omega}G_{F}^{ah}P_{\Omega}$ (B.40)
$\displaystyle=G_{F}^{ah}+(P_{\Omega}^{\perp}G_{F}^{ah}P_{\Omega}^{\perp}-P_{\Omega}^{\perp}G_{F}^{ah}-G_{F}^{ah}P_{\Omega}^{\perp})$
$\displaystyle\equiv G_{F}^{ah}+\delta G_{F}^{ah}.$

Since $\langle\psi_{\lambda},\psi_{0}\rangle=0$ for $\lambda\neq 0$, it is
straightforward to show that

$\delta
G_{F}^{ah}(\eta=\eta^{\prime})=\frac{i\pi}{6H^{2}}(P_{\Omega}^{\perp}\chi_{0})(\eta)\left[\chi_{0}(\eta)-\frac{1}{2}(P_{\Omega}^{\perp}\chi_{0})(\eta)\right],$
(B.41)

where we explicitly have that

$(P_{\Omega}^{\perp}\chi_{0})(\eta)=\int_{Q}^{\infty}\mathrm{d}q~{}\langle\varphi_{q},\chi_{0}\rangle\varphi_{q}(\eta)+\sum_{n\geq
N}\langle\psi_{n},\chi_{0}\rangle\psi_{n}(\eta).$ (B.42)

#### Step 6: Making approximations for calculating
$P_{\Omega}^{\perp}\chi_{0}$

Next we turn to computing $(P_{\Omega}^{\perp}\chi_{0})(\eta)$. Consider first
the contribution from the continuous spectrum, i.e., the first term on the
right side of Eq. (B.42). The inner product appearing in the integral is given
by

$\langle\varphi_{q},\chi_{0}\rangle=\frac{-2\sqrt{3q\tanh(\pi
q)}}{\pi(\tfrac{9}{4}+q^{2})},$ (B.43)

and so the integrand reads

$\langle\varphi_{q},\chi_{0}\rangle\psi_{q}(\eta)=-\frac{\sqrt{6}}{\pi}\frac{q\tanh(\pi
q)}{(\tfrac{9}{4}+q^{2})}H^{2}(-\eta)^{3/2}\left[\operatorname{sech}(\tfrac{\pi}{2}q)\mathrm{Re}J_{iq}(-k\eta)-\operatorname{csch}(\tfrac{\pi}{2}q)\mathrm{Im}J_{iq}(-k\eta)\right].$
(B.44)

Unfortunately, this cannot be integrated in closed form, and furthermore, it
is totally intractable numerically due to its oscillatory nature. Fortunately,
for large values of $q$, we can make several approximations to make this
integrand more tractable. We approximate

$\frac{q\tanh(\pi q)}{\tfrac{9}{4}+q^{2}}\approx\frac{1}{q},$ (B.45)

as well as

$\operatorname{sech}(\tfrac{\pi}{2}q)\approx\operatorname{csch}(\tfrac{\pi}{2}q)\approx
2e^{-\tfrac{\pi}{2}q}.$ (B.46)

Then,

$\langle\psi_{q},\chi_{0}\rangle\psi_{q}(\eta)\approx-\frac{2\sqrt{6}}{\pi}H^{2}(-\eta)^{3/2}\frac{1}{q}e^{-\tfrac{\pi}{2}q}\left[\mathrm{Re}J_{iq}(-k\eta)-\mathrm{Im}J_{iq}(-k\eta)\right].$
(B.47)

To make further progress, we can invoke an asymptotic expansion for Bessel
functions of large order:

$J_{\mu}(z)\approx\frac{1}{\sqrt{2\pi\mu}}\left(\frac{ez}{2\mu}\right)^{\mu}$
(B.48)

According to [70, Eq. 10.19.1] this expansion holds for large positive orders
$\mu$. However, by inspection, it seems to continue to hold if we analytically
continue $\mu=i\nu$.

After a bit of algebraic manipulation, we find that

$\mathrm{Re}J_{iq}(z)-\mathrm{Im}J_{iq}(z)\approx\frac{1}{\sqrt{\pi
q}}\mathrm{Re}\left[\left(\frac{ez}{2iq}\right)^{iq}\right].$ (B.49)

We further have that

$\displaystyle\left(\frac{ez}{2iq}\right)^{iq}$
$\displaystyle=\exp\left\\{iq\ln\left(\frac{ez}{2iq}\right)\right\\}$ (B.50)
$\displaystyle=\exp\left\\{\frac{\pi}{2}q+iq\ln\left(\frac{ez}{2q}\right)\right\\},$

where we made a branch cut for the complex logarithm. It therefore follows
that

$e^{-\tfrac{\pi}{2}q}(\mathrm{Re}J_{iq}(z)-\mathrm{Im}J_{iq}(z))\approx\frac{1}{\sqrt{\pi
q}}\cos\left(q\ln\left(\frac{ez}{2q}\right)\right).$ (B.51)

To get a sense of the goodness of the approximation, both sides of Eq. (B.51),
as well as their difference, is plotted in Fig. 5.

Figure 5: Left: Comparison of
$F_{1}(q)=e^{-(\pi/2)q}(\mathrm{Re}J_{iq}(1)-\mathrm{Im}J_{iq}(1))$ (blue) and
$F_{2}(q)=(\pi q)^{-1/2}\cos(q\ln(e/2q))$ (orange). Right: The relative
difference, $|F_{1}(q)-F_{2}(q)|/|F_{1}(q)|$. Beyond $q\sim O(10^{2})$,
$F_{1}(q)$ becomes numerically unstable.

Altogether, we arrive at a nice approximate but compact expression for the
integrand (B.44):

$\langle\psi_{q},\chi_{0}\rangle\psi_{q}(\eta)\approx-\sqrt{3}H^{2}\left(-\frac{2\eta}{\pi
q}\right)^{3/2}\cos\left(q\ln\left(\frac{2q}{-k\eta e}\right)\right)$ (B.52)

This is a vast improvement, but there is still no closed form expression for
the antiderivative

$\int\mathrm{d}q~{}q^{-3/2}\cos(q\ln(Aq)).$ (B.53)

Nevertheless, it converges when integrated over the interval $[Q,\infty)$, due
to the overall power of $q^{-3/2}$. A numerically efficient way of evaluating
this integral is to instead perform an equivalent integration along a contour
in the complex plane.

Define the integral

$\tilde{I}(Q,A)=\int_{Q}^{\infty}\mathrm{d}q~{}e^{iq\ln(Aq)-\tfrac{3}{2}\ln
q},$ (B.54)

so that its real part is the integral that we wish to compute. Denote the
argument of the exponential by

$\mathcal{I}(q,A)=iq\ln(Aq)-\frac{3}{2}\ln q.$ (B.55)

Its only singularities are at $q=0$ and $q\rightarrow\infty$, and so taking
the branch cut of the complex logarithm along the negative real axis, we can
safely deform the domain of integration to a contour which starts at $q=Q$ and
goes up vertically into the complex plane, to wit,

$\tilde{I}(Q,A)=\int_{q=Q}^{q=Q+i\infty}\mathrm{d}q~{}e^{\mathcal{I}(q,A)}=\int_{0}^{\infty}i\,\mathrm{d}b~{}e^{\mathcal{I}(Q+ib,A)}.$
(B.56)

Write $q=a+ib$ and split $\mathcal{I}$ into its real and imaginary parts by
writing

$\mathcal{I}(a+ib,A)=w(a,b;A)+iW(a,b;A).$ (B.57)

One then finds that

$\displaystyle w(a,b;A)$
$\displaystyle=-\frac{1}{2}\left(b+\frac{3}{2}\right)\ln\left(a^{2}+b^{2}\right)-a\arctan\left(\frac{b}{a}\right)-b\ln
A$ (B.58) $\displaystyle W(a,b;A)$
$\displaystyle=\frac{a}{2}\ln\left(a^{2}+b^{2}\right)-\left(b+\frac{3}{2}\right)\arctan\left(\frac{b}{a}\right)+a\ln
A.$ (B.59)

We only need the real part of $\tilde{I}$; therefore, we need only compute

$\displaystyle\mathrm{Re}\left[\tilde{I}(Q,2/(-k\eta e))\right]$
$\displaystyle=-\int_{0}^{\infty}\mathrm{d}b~{}e^{w(Q,b;2/(-k\eta
e))}\sin\left(W[Q,b;2/(-k\eta e)]\right)$ (B.60) $\displaystyle\equiv
I(Q,-k\eta).$

In particular, notice that even though there is an oscillatory component to
this integral, the $h$-dependent prefactor decays exponentially quickly with
$y$, and so this integral converges extremely rapidly. $I(Q,-k\eta)$ is
virtually indistinguishable from
$\int_{Q}^{\infty}\mathrm{d}q~{}q^{-3/2}\cos(q\ln[2q/(-k\eta e)])$, as shown
in Fig. 6, and the difference between the two is at the level of machine
precision.

Figure 6: $I(Q,-k\eta)$ and
$\int_{Q}^{\infty}\mathrm{d}q~{}q^{-3/2}\cos(q\ln[2q/(-k\eta e)])$ are
virtually indistinguishable, plotted here with $-k\eta=1$ and for a range of
$Q$ over which it is still possible to numerically integrate the latter. Their
relative difference is not shown, as it is at the level of machine precision.

Altogether, the projection $(P_{\Omega}^{\perp}\chi_{0})(\eta)$ therefore
contains a contribution from the continuous spectrum given by

$(P_{\Omega}^{\perp}\chi_{0})(\eta)\supset-\sqrt{3}H^{2}\left(-\frac{2\eta}{\pi}\right)^{3/2}I\left(Q,2/(-k\eta
e)\right).$ (B.61)

Next we consider the contribution from the point spectrum, i.e., the second
term on the right side of Eq. (B.42). Here, the inner product appearing in the
summand is given by

$\langle\psi_{n},\chi_{0}\rangle=-\sqrt{6p_{n}}\frac{(-1)^{n}}{\pi n(3+2n)},$
(B.62)

and so $(P_{\Omega}^{\perp}\chi_{0})(\eta)$ contains a contribution from the
point spectrum given by

$(P_{\Omega}^{\perp}\chi_{0})(\eta)\supset-\sqrt{3}H^{2}\left(-\frac{2\eta}{\pi}\right)^{3/2}\sum_{n\geq
N}\sqrt{\frac{\pi}{2}}\frac{(-1)^{n}}{n}\frac{(\tfrac{3}{2}+2n)}{(3+2n)}J_{p_{n}}(-k\eta).$
(B.63)

Because the sum is an alternating series, its magnitude is bounded from above
by

$\sqrt{\frac{\pi}{2}}\frac{1}{N}\frac{(\tfrac{3}{2}+2N)}{(3+2N)}|J_{p_{N}}(-k\eta)|\approx\frac{1}{(2N)^{3/2}}\left(\frac{-k\eta
e}{4N}\right)^{2N},$ (B.64)

where the approximation holds for large $N$. In particular, we approximated
$p_{N}\approx 2N$ and used the large-order approximation (B.48).

This bound is laughably tiny. For illustration, when $\Omega\gg H$, it follows
that $N\approx\Omega/(2H)$, and even for, e.g., $\Omega/H=100$, the bound
evaluates to $7\times 10^{-194}$ at horizon crossing (when $-k\eta=1$). The
value of $I(Q,2/e)$ is roughtly $1.1\times 10^{-4}$, and so we may safely
neglect the contribution from the point spectrum.

It is then a matter of straightforward algebraic manipulations to arrive at
the expression (3.18) for $\delta\Delta_{\phi}^{2}/\Delta_{\phi}^{2}$.

##  Appendix C Adiabatic approximation

Here we elaborate and further justify our use of the adiabatic, or “slow-roll”
approximation for nearly de Sitter spacetimes.

Suppose we wish to compute an inflationary observable $\mathcal{O}_{k}$ which
depends on the FLRW scale factor $a(\eta)$ and the time $\eta_{k}$ at which
the mode $k$ crosses the horizon. That is, $\eta_{k}$ is defined as the
solution of

$\displaystyle k=a(\eta_{k})H(\eta_{k}),$ (C.1)

where $H=a^{\prime}/a^{2}$. Hence, we can write
$\mathcal{O}_{k}=\mathcal{O}_{k}[a,\eta_{k}]$. The adiabatic approximation is
the statement that

$\displaystyle\mathcal{O}_{k}[a,\eta_{k}]\approx\mathcal{O}_{k}[\tilde{a},\tilde{\eta}_{k}],$
(C.2)

where $\tilde{a}(\eta)\equiv-1/\tilde{H}\eta$ is a de Sitter scale factor. We
define the de Sitter Hubble constant $\tilde{H}$ and the time
$\tilde{\eta}_{k}$ at which the mode $k$ crosses the de Sitter horizon via the
equations

$\displaystyle k$ $\displaystyle=\tilde{a}(\tilde{\eta}_{k})\tilde{H},$ (C.3)
$\displaystyle\tilde{a}(\tilde{\eta}_{k})$ $\displaystyle=a(\eta_{k}).$ (C.4)

Equation (C.3) is the usual horizon crossing condition for the mode $k$, while
(C.4) ensures that the de Sitter scale factor is equal to the “true” scale
factor at the mode crossing time. It is a generic feature of inflationary
cosmology that observables associated with a mode $k$ only depend on the FLRW
evolution near the time at which the mode $k$ crosses the horizon. Our
adiabatic approximation is simply making use of this fact to approximate an
observable in a generic FLRW spacetime with the same observable in a de Sitter
spacetime, but with the de Sitter spacetime tuned in such a way that near the
$k$-mode crossing time it looks similar to the true spacetime. The utility of
the approximation arises when it is difficult to compute the observable in the
true spacetime, but it is relatively easy to do so in the de Sitter spacetime.

For example, in this paper we were interested in the case where the observable
$\mathcal{O}_{k}$ is the covariantly bandlimited scalar or tensor power
spectrum. This quantity is unfeasible to compute for a generic FLRW spacetime,
but we are able to compute it in the de Sitter case. The adiabatic
approximation is therefore useful.

Let us now consider a simpler observable: the scalar power spectrum without a
cutoff. This is the quantity to which we are computing Planck-scale
corrections in this paper. For a FLRW spacetime with scale factor $a(\eta)$
and for a massless scalar field $\phi$, it is defined as

$\displaystyle\Delta^{2}_{\phi}(k)=\frac{k^{3}}{2\pi^{2}a^{2}(\eta_{k})}|v_{k}(\eta_{k})|^{2},$
(C.5)

where $v_{k}$ is the solution to the mode equation

$\displaystyle
v_{k}^{\prime\prime}+\left(k^{2}-\frac{a^{\prime\prime}}{a}\right)v_{k}=0,$
(C.6)

with initial condition $v_{k}\rightarrow\frac{1}{\sqrt{2k}}e^{-ik\eta}$ as
$\eta\rightarrow-\infty$ for the Bunch-Davies vacuum. This quantity can be
computed exactly for both power law and de Sitter spacetimes, allowing us to
use it to explicitly test the validity of the adiabatic approximation.

In other words, suppose that the “true” spacetime is a power law spacetime,
with a scale factor

$\displaystyle a=At^{c}=A[A(c-1)(-\eta)]^{c/(1-c)},$ (C.7)

where we take $c>2$ so that the spacetime is inflating. In terms of cosmic
time $t$, conformal time is given by

$\displaystyle\eta=-\frac{1}{A(c-1)t^{c-1}}.$ (C.8)

The Hubble parameter is

$\displaystyle H=\frac{c}{t}=c[A(c-1)(-\eta)]^{1/(c-1)}.$ (C.9)

Notice that the first slow-roll parameter

$\displaystyle\epsilon\equiv\frac{d}{dt}\left(\frac{1}{H}\right)=\frac{1}{c}$
(C.10)

vanishes in the limit $c\rightarrow\infty$. Thus, we might expect that in this
limit the power law spacetime behaves similarly to a de Sitter spacetime.
However, even at finite $c$, we will find that the adiabatic approximation can
accurately estimate the power spectrum $\Delta_{\phi}^{2}$ for a scalar field
in the power law spacetime.

To see this, let us first compute the exact answer for $\Delta_{\phi}^{2}$.
Taking the power law scale factor, the solution to the mode equation (C.6)
with the required initial conditions is

$\displaystyle
v_{k}(\eta)=-\frac{\sqrt{\pi}}{2}\sqrt{-\eta}H^{(1)}_{\nu_{c}}(-k\eta),$
(C.11)

where $\nu_{c}=\frac{3}{2}+\frac{1}{c-1}$. Hence, the exact power spectrum is

$\displaystyle\Delta_{\phi}^{2}(k)=\frac{c^{2}}{8\pi}\left(\frac{Ac}{k}\right)^{\frac{2}{c-1}}\frac{c}{c-1}\left|H^{(1)}_{\nu_{c}}\left(\frac{c}{c-1}\right)\right|^{2}.$
(C.12)

Let us now approximate the power spectrum using the adiabatic approximation.
To do so, we need the expression for $\Delta^{2}_{\phi}$ for a de Sitter
spacetime. In this case, the mode function solution of (C.6) can be written in
closed form as

$\displaystyle\tilde{v}_{k}(\tilde{\eta})=-\frac{\sqrt{\pi}}{2}\sqrt{-\tilde{\eta}}H^{(1)}_{\nu_{\infty}}(-k\tilde{\eta}),$
(C.13)

where $\nu_{\infty}=3/2$. The adiabatic approximation instructs us to set the
conformal time to $\eta_{k}$ and take the de Sitter Hubble constant as
$\tilde{H}$, where $\eta_{k}$ and $\tilde{H}$ are solutions to equations (C.3)
and (C.4). The solutions to these equations read

$\displaystyle\eta_{k}$ $\displaystyle=\frac{c}{(1-c)k},$ (C.14)
$\displaystyle\tilde{H}$
$\displaystyle=c\left(\frac{Ac}{k}\right)^{\frac{1}{c-1}},$ (C.15)

where we have also made use of the fact that the de Sitter horizon crossing
time is $\tilde{\eta}_{k}=-1/k$. The adiabatic approximation for the power
spectrum is thus

$\displaystyle\tilde{\Delta}_{\phi}^{2}(k)=\frac{c^{2}}{8\pi}\left(\frac{Ac}{k}\right)^{\frac{2}{c-1}}\left|H^{(1)}_{\nu_{\infty}}(1)\right|^{2}.$
(C.16)

Let us compare the approximate expression $\tilde{\Delta}_{\phi}^{2}$ in
equation (C.16) to the exact expression $\Delta_{\phi}^{2}$ in equation
(C.12). We find

$\displaystyle\left|\frac{\tilde{\Delta}_{\phi}(k)}{\Delta_{\phi}(k)}\right|=\sqrt{\frac{c-1}{c}}\left|\frac{H^{(1)}_{\nu_{\infty}}(1)}{H^{(1)}_{\nu_{c}}\left(\frac{c}{c-1}\right)}\right|=1-\frac{\alpha}{c}+\mathcal{O}\left(\frac{1}{c^{2}}\right),$
(C.17)

where $\alpha=0.10098...\,$. For large $c$, we can combine this with the
expression (C.10) for the first slow-roll parameter $\epsilon$ to obtain the
relative error due to the adiabatic approximation in this simple example:

$\displaystyle\text{relative error}\approx\alpha\epsilon.$ (C.18)

There are a couple of things to note regarding the results (C.17) and (C.18).
First, notice that (C.17) is independent of $k$. In this simple example of a
non-bandlimited power spectrum, the relative error incurred through the use of
the adiabatic approximation is the same for all modes. Extrapolating this
intuition to our covariantly bandlimited power spectra, this suggests that
although the use of the adiabatic approximation likely resulted in small
errors in the amplitude and phase of our predicted signal, it seems plausible
that our most universal prediction, the frequency of the signal, is unaffected
by the use of the adiabatic approximation.

Let us now look at the value of $\tilde{\Delta}_{\phi}/\Delta_{\phi}$ as a
function of the only variable on which this quantity depends: the power $c$ of
the power law expansion. The result is shown in Fig. 7. Although we expected
that the adiabatic approximation might do well in the $c\rightarrow\infty$
limit, we see that the adiabatic approximation in fact does much better than
anticipated. For example, for $c=2$, which is certainly far from a de Sitter
expansion, the error due to the adiabatic approximation is only 8%. Thus, for
the slowly rolling spacetimes which we consider in this paper—which are very
close to de Sitter—we expect that the adiabatic approximation is in fact very
accurate.

Figure 7: $|\tilde{\Delta}_{\phi}(k)/\Delta_{\phi}(k)|$ versus $c$.
$\Delta_{\phi}(k)$ is the exact value of the scalar power spectrum for a power
law spacetime with scale factor $a(t)=At^{c}$, while $\Delta_{\phi}(k)$ is the
adiabatic approximation to this quantity. The ratio is independent of $k$ for
a power law spacetime.

As a quantitative estimate of the accuracy of the adiabatic approximation,
note that observations put a constraint $\epsilon\ll 0.004$ on the first slow-
roll parameter [49]. From (C.10) we see that to get this small of a value for
$\epsilon$ via a power law expansion we require $c>250$. Equation (C.17), or
the large $c$ approximation (C.18), then give the relative error due to the
adiabatic approximation to be only $0.04\%$. Since there seems to be no reason
to expect that the adiabatic approximation should do worse when we introduce a
covariant cutoff, we estimate that the relative error incurred due to our use
of the adiabatic approximation in the covariantly bandlimited case is also
likely under one part in one thousand.

##  Appendix D Detailed features of the predicted signal

After establishing how a covariant natural UV cutoff, $f(\Box)$, is defined in
Eq. (3.2), we almost exclusively focused on the case of a sharp cutoff, for
which $f(\Box)=\theta(\Omega^{2}-|\Box|)$. The class of possible UV cutoffs
that one could consider is much larger, however, being parameterized by a
functional degree of freedom, $f$. In principle, $f$ need not even resemble a
cutoff; the definition goes through for any non-negative function. If we
insist that $f$ resemble what is normally thought of as a cutoff, however,
then we should have $f(\lambda)\approx 1$ for $|\lambda|\ll\Omega^{2}$,
$f(\lambda)\approx 0$ for $|\lambda|\gg\Omega^{2}$, and a monotonic transition
for $|\lambda|\sim\Omega^{2}$.

Here we investigate how the smoothness and the size of the interval over which
$f(\lambda)$ drops from 1 to 0 impacts
$\delta\Delta_{\phi}^{2}/\Delta_{\phi}^{2}$. While we of course cannot
characterize the full functional freedom in $f$, a convenient tool for our
investigations is the _smooth step function_ :

$S_{n}(x)=\left\\{\begin{array}[]{ll}0&x\leq 0\\\
x^{n+1}\sum_{k=0}^{n}{\binom{n+k}{k}}(1-x)^{k}&0<x<1\\\ 1&x\geq
1\end{array}\right.$ (D.1)

It has the property that its first $n$ derivatives are continuous at $x=0$ and
$x=1$, where $n\in\mathbb{Z}$ and $n\geq 0$, and its value increases from 0 to
1 over the interval $0<x<1$. Let us use it to define

$f(\lambda)=1-S_{n}\left(\frac{|\lambda|-\Omega^{2}}{\nu}+1\right).$ (D.2)

This profile is a $C^{n}$ function and it drops from 1 to 0 over the intervals
$|\lambda|\in(\Omega^{2}-\nu,\Omega^{2})$. The parameter $n$ therefore
characterizes the smoothness of the cutoff, and the parameter $\nu$
characterizes the size of the ramp over which the cutoff turns on.

If we replace the sharp profile $f(\lambda)=\theta(\Omega^{2}-|\lambda|)$ with
$f(\lambda)$ as defined in Eq. (D.2), after percolating through the steps of
the calculation, the final consequence is that the integral $I(Q,x)$ changes
in the final expression (3.18) for
$\delta\Delta_{\phi}^{2}/\Delta_{\phi}^{2}$. This is because this integral
came from $P_{\Omega}^{\perp}\chi_{0}$; see Eqs. (B.37) and (B.41). It is
convenient to reparameterize the width $\nu$ with a new parameter, $B$, in
term of which we replace $I(Q,x)$ with a new integral, $I_{n}(Q,x,B)$, given
by

$\displaystyle I_{n}(Q,x,B)$
$\displaystyle\equiv\int_{0}^{\infty}\mathrm{d}q~{}S_{n}\left(\frac{q-Q}{B}+1\right)q^{-3/2}\cos(q\ln(2q/xe))$
(D.3)
$\displaystyle=I(Q,x)+\int_{Q-B}^{Q}\mathrm{d}q~{}S_{n}\left(\frac{q-Q}{B}+1\right)q^{-3/2}\cos(q\ln(2q/xe)).$

In terms of the integration variable $q$, which is related to the eigenvalue
$\lambda$ via Eq. (B.24), the cutoff smoothly ramps up on an interval
$[Q-B,Q]$. Since $Q\approx\Omega/H=\sigma^{-1}$ when $\Omega\gg H$, it is
perhaps most meaningful to think of the ramp-up width $B$ as the number of
Planck lengths101010Or more generally cutoff lengths, if the cutoff is at some
scale other than the Planck scale. over which the cutoff turns on if we view
the cutoff scale $\Omega$ as being fixed.

Fig. 8 shows what happens to $\delta\Delta_{\phi}^{2}/\Delta_{\phi}^{2}$ as we
increase the width, $B$. The general trend is that increasing this width
decreases the amplitude of the oscillations and shifts their phase, but it
does not seem to change the frequency. Although increasing $B$ suppresses the
signal amplitude, it does not appear to be sending it to zero; see Fig. 9.
Note that we cannot let $B$ grow too large; otherwise, the approximations made
in arriving at the expression (B.52) for the integrand of $I_{n}(Q,x,B)$ break
down.

Figure 8: Plots of $\delta\Delta^{2}_{\phi}/\Delta^{2}_{\phi}$ as a function
of $\sigma$ with $x=1$ for various values of the smoothing width, $B$. We have
set the smoothness parameter to $n=2$. In this and the following plots, we
have that $\sigma\sim 10^{-2}$, which corresponds to a separation between the
Hubble and cutoff scales by only two orders of magnitude. Similarly to Fig. 1,
while this is not very realistic, it makes the features that we wish to
highlight easily visible. Figure 9: Plot of
$\delta\Delta^{2}_{\phi}/\Delta^{2}_{\phi}$ with $x=1$ and $\sigma=10^{-2}$ as
a function of the smoothing width parameter, $B$. We have set $n=2$.

Fig. 10 shows what happens when we adjust the smoothness by changing the
parameter $n$. Again, adjusting the smoothness changes the phase and
amplitude, but not the frequency of oscillations. It furthermore appears that
a smooth ramp-up suppresses the amplitude less as it is made smoother.

Figure 10: Plots of $\delta\Delta^{2}_{\phi}/\Delta^{2}_{\phi}$ as a function
of $\sigma$ with $x=1$ for various values of the smoothness parameter, $n$. We
have set $B=1$ in every case except for that with no smoothing ($B=0$).

Of course, precise details of how $\delta\Delta_{\phi}^{2}/\Delta_{\phi}^{2}$
changes depends on the choice of smoothing function; however, we expect that
the basic qualitative take-home lessons discussed here remain applicable in
general.

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