# An Étude on the Regularization and Renormalization of Divergences in
Primordial Observables

Anna Negro<EMAIL_ADDRESS>Subodh P. Patil
<EMAIL_ADDRESS>Instituut-Lorentz for Theoretical Physics, Leiden
University, 2333 CA Leiden, The Netherlands

(September 3, 2024; September 3, 2024)

###### Abstract

Many cosmological observables of interest derive from primordial vacuum
fluctuations evolved to late times. These observables represent statistical
draws from some underlying quantum or statistical field theoretic framework
where infinities arise and require regularization. After subtracting
divergences, renormalization conditions must be imposed by measurements or
observations at some scale, mindful of scheme and background dependence. We
review this process on backgrounds that transition from finite duration
inflation to radiation domination, and show how in spite of the ubiquity of
scaleless integrals, UV divergences can still be meaningfully extracted from
quantities that nominally vanish when dimensionally regularized. In this way,
one can contextualize calculations with hard cutoffs, distinguishing between
UV and IR scales corresponding to the beginning and end of inflation from UV
and IR scales corresponding the unknown completion of the theory and its
observables. This distinction has significance as observable quantities cannot
depend on the latter although they will certainly depend on the former. One
can also explicitly show the scheme independence of the coefficients of UV
divergent logarithms. Furthermore, certain IR divergences can be shown to be
an artifact of the de Sitter limit and are cured for finite duration
inflation. For gravitational wave observables, we stress the need to
regularize stress tensors that do not presume a prior scale separation in
their construction (as with the standard Isaacson form), deriving an improved
stress tensor fit to purpose. We conclude by highlighting the inextricable
connection between inferring $N_{\rm eff}$ bounds from vacuum tensor
perturbations and the process of background renormalization.

###### Contents

  1. I Introductory remarks
    1. I.1 Outline and scope
  2. II Preliminaries – regularization of stress tensors and correlation functions
    1. II.1 Regularization scheme (in)dependence
    2. II.2  UV and IR divergences in mass (in)dependent schemes
  3. III Non-interacting test scalar fields
    1. III.1  Divergences – (quasi) dS backgrounds
    2. III.2  Divergences – scalar fields and finite duration inflation
  4. IV Gravitational waves
    1. IV.1 Divergences – vacuum tensor modes and finite duration inflation
  5. V Renormalized stress tensors (and $N_{\rm eff}$ bounds)
  6. VI Concluding remarks
  7. A  Dimensional regularization of stress tensors in FRLW
  8. B  Oscillatory contributions to divergent integrals
  9. C  Point split comparison to physical cutoff regularization
  10. D  Improved stress tensor for gravitational waves

##  I Introductory remarks

Infinities often arise when computing physical observables in any quantum or
statistical field theoretic framework. A well defined prescription to
regularize and renormalize these divergences has been developed on flat
backgrounds111Among the many references one could consult on this topic, the
treatments in [1] and [2] in the context of renormalizable, and effective
field theories, respectively, provide comprehensive and pedagogical
introductions., and a transposition of this formalism to curved backgrounds is
also possible, albeit with numerous caveats and subtleties that one must be
mindful of depending on the context [3, 4, 5]222A number of practical
corollaries follow from the idealization of a Minkowski background that no
longer apply on curved backgrounds. These include the ability to define one
particle states as irreducible unitary representations of the background
symmetry group [6], the existence of a global timelike Killing vector through
which positive and negative frequency solutions can be defined [3, 4] and
energy scales cleanly identified to impose a separation of scales [2], in
addition to the existence of in and out states and asymptotic flatness that
premise the existence of an S-matrix [7]..

Divergences can generically be classed into two categories with distinct
physical interpretations. Short distance, or ultra-violet (UV) divergences are
unambiguously interpreted as an artifact of admitting arbitrarily high
energies or momenta in any given calculational setup as a mathematical
idealization. Once these infinities have been appropriately subtracted,
causality and locality conspire through decoupling [8, 9] to limit the
consequences of this idealization to a handful of relevant and marginal
operators in a parametrically controlled derivative expansion (see e.g. [1, 2,
10]).

Long wavelength, or infra-red (IR) divergences on the other hand require
additional care in their interpretation, as their implications can range from
the harmless to the severe. At the very least, they indicate that one has
additional work to do in order to arrive at a physical quantity. For instance,
IR divergences often feature in intermediate expressions in theories with
gapless excitations, and are rendered benign through cancellations when
computing physical observables (as is the case in QED333Where IR divergences
due to arbitrarily large numbers of soft quanta are compensated by phase space
factors in certain scattering processes [11].). IR divergences could also be
signaling the breakdown of a particular perturbative scheme, which can
nevertheless be tamed through resummation, as is the case when one studies
thermalization in an out of equilibrium context [12, 13, 14], or in QCD
scattering via the factorization properties of its soft and collinear limits
[15, 16]. In their most problematic incarnation, IR divergences are a
harbinger of the instability of the putative background one is attempting
perturbation theory around. Precisely which of the latter two possibilities
are signaled by certain IR divergences encountered on de Sitter (dS) or quasi
de Sitter backgrounds remains an open question (see e.g. [17, 18, 19, 20, 21,
22, 23, 24, 25, 26, 27, 29, 28, 30, 31]).

In operational terms, a class of IR divergences in cosmology trace to
complications that arise on time dependent backgrounds when standard
background field quantization methods can no longer be taken for granted.
Large quantum corrections to the background could invalidate any purported
classical background/fluctuation split, which is a limiting case of the
general complication of doing perturbation theory around a time evolving
background that must satisfy the tadpole condition at any given order in
$\hbar$. Although addressing these issues may seem to necessitate the full
regalia of non-equilibrium techniques, notable efforts within the context of
stochastic inflation and its background statistical field theory limit can be
found in [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48,
49, 50, 51, 52, 53], as well as within the dynamical renormalization group and
open effective field theory approaches [54, 55, 56].

Nevertheless, in situations where one can presume the validity of background
field quantization, dealing with divergences in a variety of cosmological
applications is a well understood prescription444See for instance [57, 58] for
a review on how to extract physically meaningful answers for well defined
observables in spite of the nominal presence of IR divergences in inflationary
cosmology.. So much so, that it is often taken for granted and elided in many
practical applications in the literature where the ability to meaningfully
extract physically observables quantities is taken as a given once divergences
have purportedly been tamed. However, as the adage goes, just because
something is infinite, does not mean it is zero555Unattributed quote recounted
in Ch. 1.3 of [6].. The purpose of this article is to review the details of
subtracting and renormalizing divergent quantities on backgrounds that should
be of practical interest to all cosmologists – one that transitions from an
arbitrary but finite duration of inflation through to a phase of radiation
domination. Doing so throws up qualitative differences and novelties relative
to the analogous procedure on backgrounds corresponding to a single epoch
(whether the inflationary, radiation, or matter dominated backgrounds) that
warrants a discussion even for the simplest possible scenario of non-
interacting, minimally coupled test scalar fields.

Foremost, we note the ubiquity of scaleless integrals for sufficiently light
(though not necessarily massless) fields even when one transitions between
epochs at a particular energy scale. Nevertheless, despite the fact that
scaleless integrals nominally vanish in mass independent regularization
schemes, it is still possible to cleanly separate UV and IR divergent
contributions (using techniques borrowed from matching calculations in non-
relativistic QCD and QED [2, 59]), and compare these to the contributions one
would have calculated in mass dependent regularization schemes, such as
through imposing hard cutoffs in physical momenta666The moral of which
seemingly inverts the previous adage: just because something is zero, doesn’t
mean it isn’t infinite.. This is useful, as UV and IR divergences have
distinct physical interpretations – whereas UV divergences are to be
subtracted with local counterterms, IR divergences cancel among themselves
once physical observables are computed777With the caveat that well defined
observables can be identified, which presume that some of the IR divergences
in question are not signaling the invalidity of the particular background
quantization.. Moreover, even as one finds the expected scheme dependence for
the coefficients of nominal power law divergences, one finds identical
coefficients for UV divergent logarithms whether one dimensionally
regularizes, point splits, or uses hard cutoffs in physical momenta.

Furthermore, one can demonstrate that certain IR divergences encountered in
inflationary cosmology are an artifact of the idealization of a past infinite
dS geometry and are absent in computations on backgrounds that transition in
and out of inflation, whereas unsurprisingly, UV divergences persist.
Specifically, we show how the end of inflation does not regulate UV
divergences, but instead parameterizes the coefficients of divergent
quantities through the ratio of the scales corresponding to the beginning and
end of inflation and must still be subtracted in the prescribed way. The
implications of this for vacuum contributions to measurable quantities with
blue spectra is discussed, as one must be cautious against including
contributions that could be nothing more than the Fourier transform of a UV
divergence that is to be subtracted on the way to computing observable
quantities.

We conclude with a discussion on following through on renormalizing
divergences after having subtracted them in the context of calculations that
attempt to compute quantum corrections to a given (tree level) background in
some form. In simple terms, it is to be stressed that all physical
observations are necessarily of fully dressed quantities, and the putative
classical background one performs background field quantization around is only
to be viewed as a calculational fiction that serves as bootstrap into
computing physical observables. Calculations of quantum corrections tell us
nothing of their absolute values. Once renormalization conditions are fixed by
measurements at a particular scale, all one can compute is how these
quantities change with scale. We discuss the implications of this point for
the interpretation of $N_{\rm eff}$ bounds from vacuum tensor perturbations in
the early universe, which can be boiled down to attempting to calculate a
corrected background (i.e. shifted tadpole condition) from an appropriately
derived stress tensor for gravitational waves. A definition of the latter that
is fit for the purposes of regularization forces us to work with a form that
does not assume a prior (time dependent) scale separation in its definition,
as is the case with the commonly used Isaacson form. We retrace the derivation
of the stress tensor for gravitational waves and derive an improved version
suitable for regularization. We comment in closing how can one in principle
compute the stress tensor for gravitational waves in a fully covariant
formalism, bypassing entirely the need to compute it via correlation functions
through Hadamard regularization techniques.

###  I.1 Outline and scope

The outline of this article is as follows, we first fix conventions and recap
how UV and IR divergences arise in computations of standard cosmological
observables, stressing that tracers corresponding to energy densities must be
extractable from regularized stress tensors or derivatives of two point
functions and their convolutions. For sufficiently light degrees of freedom,
we show the ubiquity of scaleless integrals, taking care to address the order
of various limits when one looks to extract divergences. Taking a cue from
non-relativistic QED and QCD, we then show how to split scaleless integrals
into UV and IR contributions that otherwise cancel in dimensional
regularization. Focusing only on the UV log-divergent parts, one can make
sense of calculations in the literature that impose hard cutoffs in physical
momenta, finding identical coefficients for the UV divergent logs no matter
the scheme. We then consider the process of regularization on backgrounds that
transition in and out of inflation to radiation domination. Although scales
corresponding to the beginning and end of inflation can be shown to merely
parameterize rather than regulate UV divergences, one finds that nominally IR
divergent contributions on a dS background are cured in the examples we
consider888Whether this generalizes to a wider class of IR divergences,
especially when higher point interactions are included is an important
question that demands to be followed up on.. In attempting to repeat this
process for divergences associated with vacuum tensor perturbations, one
immediately encounters the need to work with a form of the stress tensor that
does not assume a prior scale separation in its construction if one is to
integrate over all momenta in intermediate steps. We derive an improved stress
tensor for gravitational waves, highlighting its advantages relative to the
Isaacson form for applications to primordial observables, and address the
relevance of our observations for extracting various cosmological bounds,
before offering our concluding remarks. Various details are deferred to the
appendices.

This article is intended as a practical tour for the regularization and
renormalization of nominally divergent quantities on backgrounds of
cosmological interest using well established techniques, although applied to a
context where following through on the details is informative to the point of
novelty. For this reason, we focus on the simplest possible applications one
could think of given the richness of the problems already encountered there –
that of minimally coupled non-interacting test scalar fields, and vacuum
tensor perturbations sourced by the background expansion. Moreover, we work in
a formalism that should be familiar to most practicing cosmologists: two-point
functions, power spectra, and spectral densities, all constructed through the
intermediary of mode functions derived on a Friedmann-Robertson-Lemaître-
Walker (FRLW) slicing, as opposed to working in a fully covariant formalism
(see e.g. [3, 4] and references therein). The reason for this is as practical
as it is editorial – although the classic references exert a great deal of
effort in the study of the regularization of divergences, they are notably
sparse in detail when it comes to the process of following through on
renormalization though imposing renormalization conditions999This is perhaps a
historical corollary of the fact that many of the foundational references were
written in a time where measurements of the observables they were considering
(necessary for the imposition of renormalization conditions) seemed like a
distant prospect, which is no longer the case., which are typically imposed on
observables that convolve Fourier space quantities. In the interests of
accessibility for the target audience, full covariance is sacrificed for
clarity.

##  II Preliminaries – regularization of stress tensors and correlation
functions

All physical observations consist of some sort of energy or momentum transfer
between propagating degrees of freedom and some form of detector or tracer,
whether directly or indirectly. Although correlation functions serve as a
convenient calculational intermediary between any given (effective)
description valid during the primordial epoch and observables at late times,
they are not necessarily directly observable by themselves. They can, however,
be related to observations at later times by acting upon them with the
appropriate derivatives to extract energy and momentum, and subsequently
convolving them with transfer functions that encapsulate how they are
processed by the intervening cosmological evolution (see e.g. [60] for a
review). Correlation functions are typically computed in Fourier space,
whereas the observations they relate to are made at some fixed temporal and
spatial location and involve the coincident limits of bilinear or higher point
functions, necessitating subtraction of UV divergences associated with this
limit. Well defined coincident limits also implicitly depend on the IR
behavior of correlation functions in that any divergences encountered in the
latter must cancel between all contributions in convolution with the transfer
function for well defined physical observables in any self-consistent
calculational setup. In what follows, we elaborate on the process of
extracting and subtracting these divergences as appropriate on backgrounds
that transition through different epochs. We first focus upon the divergences
associated with vacuum two point correlation functions and the associated
energy momentum tensors of scalar fields, after which we shift our attention
to gravitational waves. In order to do this, we need to establish some
preliminary facts and conventions for the discussion that follows.

The energy momentum tensor for a non-interacting, minimally coupled test
scalar field $\phi$ on an FRLW background is given by

$T^{\mu}_{\leavevmode\nobreak\
\nu}=\partial^{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}\delta^{\mu}_{\nu}\left(g^{\lambda\beta}\partial_{\lambda}\phi\partial_{\beta}\phi+m^{2}\phi^{2}\right),$
(1)

from which we can extract

$-T^{0}{}_{0}:=\rho=\frac{\phi^{\prime
2}}{2a^{2}}+\frac{(\nabla\phi)^{2}}{2a^{2}}+\frac{m^{2}}{2}\phi^{2},$ (2)

where we work in conformal time, and primes denote derivatives with respect to
conformal time. The above is a local density that we can rewrite as the
coincident limit of a bilinear form as

$\rho:=\lim_{y\to x}\rho(\tau;x,y),$ (3)

where

$\rho(\tau;x,y):=\frac{1}{2a^{2}}\left[\phi^{\prime}(\tau,x)\phi^{\prime}(\tau,y)+\nabla_{x}\phi(\tau,x)\cdot\nabla_{y}\phi(\tau,y)+m^{2}a^{2}\phi(\tau,x)\phi(\tau,y)\right].$
(4)

Given the Fourier transform convention for the corresponding field operator
$\hat{\phi}$

$\hat{\phi}(\tau,x)=\int\frac{d^{3}k}{(2\pi)^{3}}\hat{\phi}(\tau,k)e^{ik\cdot
x},$ (5)

where the argument distinguishes the field operator in position space from its
Fourier component, and the definition101010Below and in the rest of what
follows, the equal time expectation values are short hand for in-in
correlation functions. For non-interacting test scalar fields evaluated in the
adiabatic vacuum, this reduces to Eq. 6 and the following.

$\frac{k^{3}}{2\pi^{2}}\langle\hat{\phi}(\tau,k)\hat{\phi}(\tau,k^{\prime})\rangle:=(2\pi)^{3}\delta^{3}(k+k^{\prime})\mathcal{P}_{\phi}(\tau,k),$
(6)

we can express the two point correlation function as

$\displaystyle\langle\hat{\phi}(\tau,x)\hat{\phi}(\tau,y)\rangle$
$\displaystyle=$
$\displaystyle\int\frac{d^{3}k}{4\pi}\frac{\mathcal{P}_{\phi}(\tau,k)}{k^{3}}e^{ik\cdot(x-y)},$
$\displaystyle=$
$\displaystyle\int\frac{dk}{k}\mathcal{P}_{\phi}(\tau,k)\frac{\sin(kr)}{kr},$

where we have assumed statistical isotropy to perform the angular integrals,
and $r:=|x-y|$, and will work with the adiabatic vacuum associated with a
given background in what follows. For inflationary spacetimes, this will be
the usual Bunch Davies vacuum state. The coincident limit of Eq. II can be
expressed as

$\langle\hat{\phi}(\tau,x)\hat{\phi}(\tau,x)\rangle=\int_{0}^{\infty}\frac{dk}{k}\mathcal{P}_{\phi}(\tau,k),$
(8)

and the coincident limit of Eq. 4 can similarly be expressed as

$\lim_{y\to
x}\rho(\tau;x,y)=\frac{1}{2a^{2}}\int_{0}^{\infty}\frac{dk}{k}\left[\left(k^{2}+a^{2}m^{2}\right)\mathcal{P}_{\phi}(\tau,k)+\mathcal{P}_{\phi^{\prime}}(\tau,k)\right],$
(9)

where we have used the shorthand

$\frac{k^{3}}{2\pi^{2}}\langle\hat{\phi}^{\prime}(\tau,k)\hat{\phi}^{\prime}(\tau,k^{\prime})\rangle:=(2\pi)^{3}\delta^{3}(k+k^{\prime})\mathcal{P}_{\phi^{\prime}}(\tau,k).$
(10)

In terms of canonically normalized mode functions $\phi_{k}(\tau)$ of the
appropriate adiabatic vacuum, the free field operator admits the expansion

$\hat{\phi}(\tau,k)=\hat{a}_{k}\phi_{k}(\tau)+\hat{a}^{\dagger}_{-k}\phi_{k}^{*}(\tau),$
(11)

where
$[\hat{a}_{k},\hat{a}^{\dagger}_{-k^{\prime}}]=(2\pi)^{3}\delta^{3}(k+k^{\prime})$,
so that

$\mathcal{P}_{\phi}(\tau,k)=\frac{k^{3}}{2\pi^{2}}|\phi_{k}(\tau)|^{2},$ (12)

and so that Eq. 9 can be expressed as

$\lim_{y\to
x}\rho(\tau;x,y)=\frac{1}{4\pi^{2}a^{2}}\int_{0}^{\infty}k^{2}\,dk\left[(k^{2}+a^{2}m^{2})|\phi_{k}(\tau)|^{2}+|\phi^{\prime}_{k}(\tau)|^{2}\right].$
(13)

On a purely dS background, the mode functions for a massless field are given
by

$|\phi_{k}(\tau)|^{2}=\frac{H^{2}}{2k^{3}}(1+k^{2}\tau^{2}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
|\phi^{\prime}_{k}(\tau)|^{2}=\frac{H^{2}}{2k^{3}}k^{4}\tau^{2},$ (14)

so that the late time power spectrum for a massless test scalar is exactly
scale invariant with amplitude determined from the Bunch Davies vacuum as
$\mathcal{P}_{\phi}=\left(\frac{H}{2\pi}\right)^{2}$, where $H$ is the Hubble
factor that defines the dS background, so that Eq. 8 reduces to

$\lim_{\tau\to
0}\langle\hat{\phi}(\tau,x)\hat{\phi}(\tau,x)\rangle=\left(\frac{H}{2\pi}\right)^{2}\int_{0}^{\infty}\frac{dk}{k}.$
(15)

This above is evidently divergent in both the UV and IR. Qualitatively similar
divergences are encountered when one constructs the corresponding energy
momentum tensor, as we attend to in the next section, warranting a detour on
the different approaches one might adopt to regularize expressions like the
above where important caveats immediately arise.

###  II.1 Regularization scheme (in)dependence

Given the absence of any scale in the integrand in Eq. 15, the integral
vanishes in mass independent regularization schemes such as dimensional or
zeta function regularization (see e.g. the relevant chapters of [1, 2]). One
might perhaps be wary of this and imagine instead regulating such an
expression by imposing hard cutoffs in both the UV and the IR. However, one is
immediately presented with a choice as to precisely how. For instance, [55]
adopts cutoffs in terms of physical momenta on the basis that since IR
divergences must cancel in all physical processes, the IR scales associated
with these must be expressed in terms of physical scales. On the other hand,
[61] argues that it is more practical to adopt cutoffs in terms of comoving
momenta for IR divergences given that the existence of a pre-inflationary
phase ought to serve as a natural regulator where the distinction becomes less
relevant (in the sense that it leads to only sub-leading
corrections)111111Logarithmic divergences are special in that they involve the
ratios of scales for some observable at some time, where the distinction
between comoving and physical cutoffs becomes irrelevant.. Although we
explicitly demonstrate the nature of a pre-inflationary phase as an IR
regulator in subsequent sections, we wish to stress that aside from the
expected scheme dependence that will drop out of physical quantities once
renormalization conditions are imposed consistently121212That is to say, a
suitable counterterm can be subtracted that doesn’t invalidate the background
field quantization prescription one has presumed., using hard cutoffs in
physical momenta works particularly straightfowardly for matching with the
logarithmic UV divergences one can extract from scaleless integrals, and
moreover leads to straightforward identification of the requisite
counterterms. Simply put, one can arrive at the same expressions for the
logarithmic running of physical quantities regardless of regularization scheme
with sufficient care, as is to be expected. The fact that UV and IR
divergences can be extracted and separated from scaleless integrals may come
as a surprise to those used to discarding them, so it behooves us to elaborate
on this in detail.

###  II.2  UV and IR divergences in mass (in)dependent schemes

Reconsider the scaleless integral Eq. 15, which can formally be rewritten as

$\int_{0}^{\infty}\frac{dk}{k}=\int_{0}^{\infty}\frac{k^{3}dk}{k^{4}}=\frac{1}{2\pi^{2}}\int_{-\infty}^{\infty}\frac{d^{4}k}{k^{4}},$
(16)

where the formal manipulations that result in the above render this integral
to be Euclidean by default. We see that Eq.16 is a specific case of the more
general form [2]

$I_{D}(m^{2})=\int\frac{d^{D}k}{(2\pi)^{D}}\frac{k^{2A}}{(k^{2}+m^{2})^{B}}.$
(17)

Formally, Eq. 17 evaluates to131313The integral evaluates to
$\big{|}\frac{x^{A+D/2}}{2A+D}{}_{2}F_{1}[B,\,A+D/2,\,1+A+D/2,\,-x]\big{|}^{\infty}_{0}$
with Eq. 18 coming from the upper limit along with additional divergent
contributions from the upper and lower limits if $A-B+D/2>0$ or if $A+D/2<0$,
which are subleading to the divergences in Eq. 18 for integer values of $A$
and $B$ as $D\to 4$.

$I_{D}(m^{2})=\frac{\Gamma\left(A+\frac{D}{2}\right)\Gamma\left(B-A-\frac{D}{2}\right)}{(4\pi)^{D/2}\Gamma\left(\frac{D}{2}\right)\Gamma(B)}\left(m^{2}\right)^{A-B+D/2},$
(18)

which permits analytic continuation to non-integer values of $D$. Re-
expressing the two point function Eq. 15 via Eq. 16 as141414Dimensional
regularization requires us to deform the dimension of every quantity that
depends on spacetime dimension, including the mode functions on the background
in question. This is especially crucial in the context of loop corrections in
the presence of interactions [62, 63, 64, 65], where spuriously large loop
corrections might otherwise be inferred. However, because we only consider
bubble diagrams of non-interacting fields in what follows (with no external
momenta), the result of doing so contributes only additional finite
contributions and will be elided in what follows.

$\lim_{\tau\to
0}\langle\hat{\phi}(\tau,x)\hat{\phi}(\tau,x)\rangle=\frac{H^{2}}{8\pi^{4}}\int_{-\infty}^{\infty}\frac{d^{4}k}{k^{4}}=\frac{H^{2}}{8\pi^{4}}\int_{-\infty}^{\infty}d^{4}k\left[\frac{1}{k^{2}(k^{2}+m^{2})}+\frac{m^{2}}{k^{4}(k^{2}+m^{2})}\right],$
(19)

we see via Eq. 18 that this evaluates in $D=4-\delta$ dimensions to two equal
and opposite contributions of the form

$\pm\frac{H^{2}}{4\pi^{2}}\left[\frac{1}{\delta}-\frac{1}{2}\left(\log\frac{m^{2}}{4\pi\mu^{2}}+\gamma_{E}-1\right)\right],$
(20)

where $\gamma_{E}$ is the Euler-Mascheroni constant, and $\mu$ is some
arbitrary mass scale necessitated by dimensional deformation. The sum of the
two contributions can be written as

$\lim_{\tau\to
0}\langle\hat{\phi}(\tau,x)\hat{\phi}(\tau,x)\rangle=\left(\frac{H}{2\pi}\right)^{2}\left[\frac{1}{\delta_{\rm
UV}}-\frac{1}{\delta_{\rm IR}}+\log\frac{\mu_{\rm UV}}{\mu_{\rm IR}}\right],$
(21)

where we have artificially given a separate label to the dimensional
deformation parameters $\delta$ and $\mu$ from the two terms to distinguish
the UV and IR divergent contributions in Eq. 19, illustrating the anatomy of
how dimensional regularization works in canceling two separately divergent
contributions by default.

It is informative to compare this result with the outcome of imposing hard
cutoffs. Whether we chose to do so in Eq. 15 in terms of physical momenta
(where $\Lambda_{\rm UV/IR}=a\,k_{\rm UV/IR}$) or in terms of comoving momenta
(where $\Lambda_{\rm UV/IR}=k_{\rm UV/IR}$) is immaterial in the context of
logarithmic divergences which only sees ratios of these scales:

$\lim_{\tau\to
0}\langle\hat{\phi}(\tau,x)\hat{\phi}(\tau,x)\rangle=\left(\frac{H}{2\pi}\right)^{2}\log\left(\frac{k_{\rm
UV}}{k_{\rm{IR}}}\right).$ (22)

From this, we see that in spite of the fact that scaleless integrals vanish
under dimensional regularization, factorizing them into a sum of scaleful
integrals allows one to match the coefficients of the UV divergent logarithms
in Eq. 21 with those in Eq. 22 obtained by imposing hard cutoffs in physical
momenta. As we show in subsequent sections, this will also be true no matter
the background.

##  III Non-interacting test scalar fields

In this section, we examine divergences in the two point correlation function
and energy momentum tensor for non-interacting, minimally coupled massless
test scalar fields on dS and quasi dS backgrounds, as well as that for
sufficiently light massive scalar fields on a dS background. All divergent
integrals turn out to be scaleless, a feature that persists even when one
considers a background that transitions from a pre-inflationary epoch through
to inflation and exiting to a terminal phase of radiation domination.
Nevertheless, one can extract UV divergences via the techniques elaborated in
section II.2 and compare with what one would have obtained with a hard cutoff
in physical momenta, recovering identical coefficients for the logarithmic
divergences.

###  III.1  Divergences – (quasi) dS backgrounds

On backgrounds that deviate from dS, one has to be careful to factor in the
difference in the scale factor from the pure dS form of $a(\tau)=-1/(H\tau)$.
For constant but non-zero $\epsilon:=-\dot{H}/H^{2}$, as is the case during
power law inflation for instance, one has $a(\tau)=1/[-\tau H_{0}]^{\nu-1/2}$,
where $\nu=\frac{3-\epsilon}{2(1-\epsilon)}$ (and where the integration
constant $H_{0}$ corresponds to the value of $H$ at $t=0$ in cosmological time
– cf. [60]). The net result for the coincident limit of the two point function
for a massless scalar field on a background corresponding to constant
$\epsilon$ is given by

$\lim_{\tau\to
0}\langle\hat{\phi}(\tau,x)\hat{\phi}(\tau,x)\rangle=\left(\frac{H_{0}}{2\pi}\right)^{2}\frac{\Gamma^{2}(\nu)}{\pi}2^{2\nu-1}\int_{0}^{\infty}\frac{dk}{k}\left(\frac{k}{H_{0}}\right)^{3-2\nu}\left[1+\frac{k^{2}\tau^{2}}{2(\nu-1)}+...\right],$
(23)

or more generally,

$\lim_{\tau\to
0}\langle\hat{\phi}(\tau,x)\hat{\phi}(\tau,x)\rangle\sim\left(\frac{H_{0}}{2\pi}\right)^{2}\int_{0}^{\infty}\frac{dk}{k}\left(\frac{k}{H_{0}}\right)^{n_{s}-1}\left[1+\left(\frac{k}{aH_{0}}\right)^{2}\frac{a^{n_{s}-1}}{2-n_{s}}+...\right]$
(24)

where the ellipses denote terms that vanish in the $\tau\to 0$ limit and the
$\sim$ indicates a numerical pre-factor that is close to unity for
$n_{s}-1\approx 1$ (and tends to it in the dS limit). In the dS limit, we find
the relatively simple expression

$\lim_{\tau\to
0}\langle\hat{\phi}(\tau,x)\hat{\phi}(\tau,x)\rangle=\left(\frac{H_{0}}{2\pi}\right)^{2}\int_{0}^{\infty}\frac{dk}{k}\left[1+\left(\frac{k}{aH_{0}}\right)^{2}\right],$
(25)

which supplements Eq. 15 with a sub-leading correction in the late time limit,
notable relative to the analogous expression on quasi dS Eq. 24 in that is Eq.
25 exact for all times. The scaleless nature of Eq. 24 implies that it
vanishes when regularized in any mass independent scheme. Instead, one might
contemplate imposing a hard UV or IR cutoff in physical momenta by setting
$k_{\rm UV/IR}=a\Lambda_{\rm UV/IR}$, and subtracting the UV divergence with
the necessary counterterm. Doing so for the expression Eq. 25, one finds (see
e.g. [55])

$\displaystyle\lim_{\tau\to
0}\langle\hat{\phi}(\tau,x)\hat{\phi}(\tau,x)\rangle$ $\displaystyle=$
$\displaystyle\left(\frac{H_{0}}{2\pi}\right)^{2}\left\\{\int_{a\Lambda_{\rm
IR}}^{a\Lambda_{\rm
UV}}\frac{dk}{k}\left[1+\left(\frac{k}{aH_{0}}\right)^{2}\right]-\int_{a\mu}^{a\Lambda_{\rm
UV}}\frac{dk}{k}\left[1+\left(\frac{k}{aH_{0}}\right)^{2}\right]\right\\},$
$\displaystyle=$
$\displaystyle\left(\frac{H_{0}}{2\pi}\right)^{2}\left\\{\log\left(\frac{\mu}{\Lambda_{\rm
IR}}\right)+\frac{1}{2H_{0}^{2}}\left(\mu^{2}-\Lambda_{\rm
IR}^{2}\right)\right\\},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ (n_{s}=1)$

where $\mu$ is some arbitrary renormalization scale. We note that the
counterterm that subtracts the UV divergence in Eq. III.1 is given by

${\rm
c.t.}=\left(\frac{H_{0}}{2\pi}\right)^{2}\left\\{\log\left(\frac{\mu}{\Lambda_{\rm
UV}}\right)+\frac{1}{2H_{0}^{2}}\left(\mu^{2}-\Lambda_{\rm
UV}^{2}\right)\right\\},$ (27)

corresponding to a scheme dependent renormalization of the cosmological
constant151515We note that imposing a comoving cutoff in the limits of Eq.
III.1 instead of a physical cutoff would have resulted in a time dependence in
the required counterterms (at odds with the dS invariance of the vacuum state)
and would have necessitated further gymnastics to arrive at any physically
meaningful quantity.. That these expressions are independent of time is a
consequence of having imposed a physical cutoff on a dS invariant background.
The remaining IR divergence is indicative of the fact that we have not yet
arrived at something that can be processed into an observable quantity. It
could end up being canceled by other contributions when we compute a well
defined observable, or it could herald the need for resummation in which case
it will also eventually get canceled. It could also indicate that the
background we are attempting quantization around is not what it seems [66],
and additional physical inputs are required in order to proceed. As we will
show explicitly in what follows, were we to abandon the assumption that
inflation was past dS eternal, then all IR divergences cancel among themselves
and are in effect regulated by the scale corresponding to the beginning of
inflation.

It is informative to compare what we would have obtained if we had
dimensionally regularized Eq. 25 instead. We first note that the scaleless
nature of the integral permits us to change variables to $q:=k/(aH_{0})$ for
any fixed time. Hence the integral we need to evaluate can be factorized as in
Eq. 19 as:

$\displaystyle\frac{H^{2}_{0}}{8\pi^{4}}\int_{-\infty}^{\infty}\frac{d^{4}q}{q^{4}}(1+q^{2})$
$\displaystyle=$
$\displaystyle\frac{H_{0}^{2}}{8\pi^{4}}\int_{-\infty}^{\infty}d^{4}q\left[\frac{1}{(q^{2}+\widetilde{m}^{2})}+\frac{1+\widetilde{m}^{2}}{q^{2}(q^{2}+\widetilde{m}^{2})}+\frac{\widetilde{m}^{2}}{q^{4}(q^{2}+\widetilde{m}^{2})}\right]$
(28)

where $\widetilde{m}:=\mu/H_{0}$ is some auxiliary dimensionless mass scale.
The first and second terms above are power law and log divergent in the UV,
respectively, whereas the third term is IR divergent. The UV divergences must
of course, be subtracted by an appropriate counterterm. One finds after
cancellations between the contributions from the first and second terms of Eq.
28, a remaining UV divergence

${\rm
c.t.}=\left(\frac{H_{0}}{2\pi}\right)^{2}\left\\{\log\left(\frac{\mu}{H_{0}}\right)-\frac{1}{\delta_{\rm
UV}}+\frac{1}{2}\left(\gamma_{E}-1-\log 4\pi\right)\right\\},$ (29)

which is to be compared to Eq. 27.

The situation for quasi dS is more complicated if one were to regularize it in
a mass dependent scheme, as is evident from Eq. 24. Nevertheless, it is still
worth elaborating upon provided we simplify matters by further specifying that
the background corresponds to that of eternal power-law inflation, defined by
a constant but non-zero $\epsilon$, so that
$n_{s}-1=-2\epsilon-2\epsilon^{2}+...$. We will address the more realistic
case of finite duration inflation next. One finds from inspection of the
integrand of Eq. 24:

$\int_{0}^{\infty}\frac{dk}{k}\left(\frac{k}{H_{0}}\right)^{n_{s}-1}\left[1+\left(\frac{k}{aH_{0}}\right)^{2}\frac{a^{n_{s}-1}}{2-n_{s}}\right]$
(30)

that for $n_{s}-1<0$, an IR divergent contribution results from the first
term:

$\int_{0}^{\infty}\frac{dk}{k}\left(\frac{k}{H_{0}}\right)^{n_{s}-1}=\frac{1}{|n_{s}-1|}\left(\frac{a\Lambda_{\rm
IR}}{H_{0}}\right)^{n_{s}-1},$ (31)

and a UV divergent contribution results from the second

$\int_{0}^{\infty}\frac{dk}{k}\left(\frac{k}{aH_{0}}\right)^{n_{s}+1}\frac{a^{2(n_{s}-1)}}{2-n_{s}}=\frac{a^{2(n_{s}-1)}}{(2-n_{s})(n_{s}-1)}\left(\frac{\Lambda_{\rm
UV}}{H_{0}}\right)^{n_{s}+1}.$ (32)

The interpretation of the IR divergences is as discussed above. The UV
divergence must be subtracted with a local counterterm if we are to obtain
physically meaningful quantities, and is given by

${\rm c.t.}=\lim_{\tau\to
0}\left(\frac{H_{0}}{2\pi}\right)^{2}\frac{a^{2(n_{s}-1)}}{(2-n_{s})(n_{s}-1)}\left[\left(\frac{\mu}{H_{0}}\right)^{n_{s}+1}-\left(\frac{\Lambda_{\rm
UV}}{H_{0}}\right)^{n_{s}+1}\right],$ (33)

where again $\mu$ is some renormalization scale. We first note that the
required counterterm is time dependent, which should not come as a surprise
given that the background is no longer maximally symmetric. It behooves us to
elaborate on the specific form of the local counterterm that could possibly
have this particular time dependence. Noting that on the specified power law
inflating background, $\dot{H},H^{2}\sim a^{n_{s}-1}$, so that curvature
squared invariants evaluated on this background have the required secular
dependence. That is, Eq. 33 derives from

${\rm c.t.}\subset\int
d^{4}x\sqrt{-g}\left[c_{1}(\mu)R^{2}+c_{2}(\mu)R_{\mu\nu}R^{\mu\nu}\right],$
(34)

which are the usual leading curvature squared counterterms encountered in the
effective field theory treatment of gravity [67, 68, 69, 70]. In the limit
that $\epsilon\to 0$, so that $n_{s}\to 1^{-}$, the counterterm Eq. 33
simplifies to

${\rm c.t.}=\lim_{\epsilon\to 0}\frac{a^{-4\epsilon}\Lambda^{2}_{\rm
UV}}{4\pi^{2}}\left\\{\frac{1}{2\epsilon}-\log\frac{\Lambda_{\rm
UV}}{H_{0}}+...\right\\},$ (35)

where given that $\Lambda_{\rm UV}\gg\mu$, we have neglected the sub-leading
dependence on $\mu$.

As before, we can retrace the same computation using dimensional
regularization. We first consider the separate contributions to the integrand
Eq. 30. The IR divergent contribution Eq. 31 can be re-expressed as

$\int_{0}^{\infty}\frac{dk}{k}\left(\frac{k}{H_{0}}\right)^{n_{s}-1}=\frac{1}{2\pi^{2}}\int_{-\infty}^{\infty}\frac{d^{4}q}{q^{4}}q^{n_{\rm
s}-1}=\frac{1}{2\pi^{2}}\int_{-\infty}^{\infty}d^{4}q\left[\frac{q^{n_{\rm
s}-1}}{q^{2}(q^{2}+\widetilde{m}^{2})}+\frac{\widetilde{m}^{2}q^{n_{s}-1}}{q^{4}(q^{2}+\widetilde{m}^{2})}\right],$
(36)

where we have switched to the dimensionless variable $q:=k/H_{0}$, and
similarly for $\widetilde{m}:=\mu_{\rm IR}/H_{0}$. The presence of the non-
integer exponents eliminates the $\delta$ poles161616Note that we are required
to take the limits in the order $\delta\to 0$ and then $\epsilon\to 0$. To do
the opposite would mean to have regularized on a dS background first and then
deformed the background in the hopes that no new divergences will have
appeared through this process. One can show that this is not justified on
general grounds [3], and is seen directly from the proliferation of higher
order $\delta$ poles encountered when taking the limits in the opposite
order., and one finds equal and opposite contributions from the two
constituent integrals, with the nominally IR divergent contribution coming
from the second term of the form

$\lim_{n_{s}\to 1^{-}}-\left\\{\frac{1}{n_{s}-1}+\log\frac{\mu_{\rm
IR}}{H_{0}}+(n_{\rm
s}-1)\left(\frac{\pi^{2}}{24}+\frac{1}{2}\log^{2}\frac{\mu_{\rm
IR}}{H_{0}}\right)+...\right\\},$ (37)

whose interpretation is as in the previous examples. Turning to the UV
divergent term Eq. 32:

$\frac{a^{2(n_{s}-1)}}{2-n_{s}}\int_{0}^{\infty}\frac{dk}{k}\left(\frac{k}{aH_{0}}\right)^{n_{s}+1},$
(38)

we see that we can again factor the scaleless integrand into the sum of two
terms as

$\int_{0}^{\infty}\frac{dk}{k}\left(\frac{k}{aH_{0}}\right)^{n_{s}+1}=\frac{1}{2\pi^{2}}\int_{-\infty}^{\infty}\frac{d^{4}q}{q^{4}}q^{n_{\rm
s}+1}=\frac{1}{2\pi^{2}}\int_{-\infty}^{\infty}d^{4}q\left[\frac{q^{n_{\rm
s}+1}}{q^{2}(q^{2}+\widetilde{m}^{2})}+\frac{\widetilde{m}^{2}q^{n_{s}+1}}{q^{4}(q^{2}+\widetilde{m}^{2})}\right],$
(39)

where just as we did on dS space, we work with dimensionless variables
$q:=k/aH_{0}$ and $\widetilde{m}:=\mu/H_{0}$. The first term above is UV
divergent, and necessitates the counterterm

$\displaystyle{\rm c.t.}$ $\displaystyle=$ $\displaystyle\lim_{\epsilon\to
0}-\frac{a^{2(n_{s}-1)}}{2-n_{s}}\left(\frac{H_{0}}{2\pi}\right)^{2}\frac{\mu^{2}}{H_{0}^{2}}\left\\{\frac{1}{n_{s}-1}+\log\frac{\mu}{H_{0}}\right\\}$
$\displaystyle=$ $\displaystyle\lim_{\epsilon\to
0}\frac{a^{-4\epsilon}\mu^{2}}{4\pi^{2}}\left\\{\frac{1}{2\epsilon}-\log\frac{\mu}{H_{0}}+...\right\\}$

which is to be compared to Eq. 35.

The generalization to massive fields is straightforward on dS backgrounds,
where analytic expressions for the mode functions can be obtained:

$|\phi_{k}(\tau)|^{2}=\frac{\pi}{4}H_{0}^{2}(-\tau)^{3}|H^{(1)}_{\nu_{m}}(-k\tau)|^{2},$
(41)

where

$\nu^{2}_{m}:=\frac{9}{4}-\frac{m^{2}}{H_{0}^{2}},$ (42)

and where $H^{(1)}_{\nu_{m}}$ is the corresponding Hankel function with degree
$\nu_{m}$. Presuming $0<m^{2}\leq 9H_{0}^{2}/4$ so that $\nu_{m}$ is still
real, the coincident limit of the two point function at late times can then be
computed as

$\lim_{\tau\to
0}\langle\hat{\phi}(\tau,x)\hat{\phi}(\tau,x)\rangle=\frac{H_{0}^{2}}{8\pi^{3}}2^{2\nu_{m}}\Gamma^{2}(\nu_{m})\int_{0}^{\infty}\frac{dk}{k}\left[\left(\frac{k}{aH_{0}}\right)^{3-2\nu_{m}}+\frac{1}{2\left(\nu_{m}-1\right)}\left(\frac{k}{aH_{0}}\right)^{5-2\nu_{m}}\right].$
(43)

It is notable that in spite of the presence of the additional mass scale $m$,
the integral remains scaleless171717This can be understood from the fact that
on dS, the quantity nominally labeled mass in Eq. 42 is constructed from
eigenvalues of the Casimir operators corresponding to conformal dimension and
spin expressed in units of the dS radius, which remains the only scale in the
problem. We thank Paolo Benincasa for pointing this out to us.. Here also, we
can make the change of variable to $q=k/(aH)$, so that the above can be recast
as

$\displaystyle\lim_{\tau\to
0}\langle\hat{\phi}(\tau,x)\hat{\phi}(\tau,x)\rangle$ $\displaystyle=$
$\displaystyle\frac{H_{0}^{2}}{16\pi^{5}}2^{2\nu_{m}}\Gamma^{2}(\nu_{m})\int_{-\infty}^{\infty}\frac{d^{4}q}{q^{4}}q^{3-2\nu_{m}}\left[1+\frac{q^{2}}{2(\nu_{m}-1)}\right],$
(44)

where the first thing to note is that the IR divergence encountered for a non-
interacting massless scalar on dS is eliminated given that
$\nu_{m}<3/2$181818A short-lived conclusion on dS backgrounds the minute any
interactions are incorporated.. Proceeding exactly as before, we can compare
the UV divergence obtained from imposing a hard cutoff in physical momenta to
that obtained by factorizing the scaleless integral in dimensional
regularization. In the limit $m^{2}/H_{0}^{2}\ll 1$, one finds that the
required counterterms are given by

${\rm
c.t.}=\frac{H_{0}^{2}}{8\pi^{2}}\frac{2^{2\nu_{m}}\Gamma^{2}(\nu_{m})}{\pi}\left[\frac{3H_{0}^{2}}{m^{2}}+\log\frac{\Lambda_{\rm
UV}}{H_{0}}+...\right]\leavevmode\nobreak\ \leavevmode\nobreak\ {(\rm
physical\leavevmode\nobreak\ cutoff)}$ (45)

and

${\rm
c.t.}=\frac{H_{0}^{2}}{8\pi^{2}}\frac{2^{2\nu_{m}}\Gamma^{2}(\nu_{m})}{\pi}\left[\frac{3H_{0}^{2}}{m^{2}}+\log\frac{\mu}{H_{0}}+...\right]\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ {(\rm dim\leavevmode\nobreak\ reg)}$
(46)

where again, one takes the limit $\delta\to 0$ to regularize the integrals
before taking the limit $m^{2}/H_{0}^{2}\to 0$. In both cases, the required
counterterm corresponds to a renormalization of the cosmological constant.

Having labored over the details of regularizing and renormalizing divergences
for the two point function of test scalar fields, repeating the exercise for
the associated stress tensor may seem a straightforward extension. However, we
immediately encounter a difficulty: it is not always possible to construct a
counterterm that subtracts the UV divergences encountered from background
geometric invariants if the regularization scheme itself does not respect this
symmetry. This is not to a concern to be casually dismissed191919An issue that
did not arise for the two-point function given that its nature as a scalar
bilinear., as persisting with this regularization scheme could lead us to the
erroneous conclusion that the background we attempted to quantize around is
transmuting into something else under renormalization if not worked through to
the end. It is here that dimensional and related zeta function regularization
techniques distinguish themselves in theories with general covariance, as they
effortlessly preserve the symmetries of the background even if the precise
mechanism by which this occurs can seem non-trivial. For this reason, it is
worth explicitly tracing through this process in detail202020Nevertheless, it
remains true that the coefficients of the UV divergent logarithms would match
regardless of whether one dimensionally regularizes or one imposes hard
cutoffs in physical momenta..

Consider a minimally coupled non-interacting test scalar field on dS space in
the Bunch Davies vacuum. The rotational invariance of the vacuum state allow
us to write the non-vanishing components as

$\rho=\frac{1}{4\pi^{2}a^{2}}\int_{0}^{\infty}k^{2}\,dk\left[\left(k^{2}+a^{2}m^{2}\right)|\phi_{k}(\tau)|^{2}+|\phi^{\prime}_{k}(\tau)|^{2}\right],$
(47)

and

$p=\frac{1}{4\pi^{2}a^{2}}\int_{0}^{\infty}k^{2}\,dk\left[-\left(\frac{k^{2}}{3}+a^{2}m^{2}\right)|\phi_{k}(\tau)|^{2}+|\phi^{\prime}_{k}(\tau)|^{2}\right],$
(48)

with all other components of the energy momentum tensor vanishing. From the
asymptotic forms of Eq. 41, one can infer that both integrals are UV
divergent. The energy density in the massless limit is straightforwardly
computed via Eq. 14, and found to be

$\rho=\frac{H^{4}}{8\pi^{2}}\int^{\infty}_{0}\frac{dk}{k}\left[\left(\frac{k}{aH}\right)^{2}+2\left(\frac{k}{aH}\right)^{4}\right],$
(49)

and the corresponding pressure is found to be

$p=\frac{H^{4}}{8\pi^{2}}\int^{\infty}_{0}\frac{dk}{k}\left[-\frac{1}{3}\left(\frac{k}{aH}\right)^{2}+2\left(\frac{k}{aH}\right)^{4}\right],$
(50)

Although one can regularize both components through imposing physical cutoffs,
one is immediately confronted by the fact that the required counterterm for
the leading divergences will not be proportional to the metric nor the Ricci
tensor on the presumed dS background (which would correspond either to
renormalizations of the cosmological and Newton’s constants). This is not the
case when one implements dimensional regularization, which may seem rather
non-trivial considering the forms that Eqs. 47 and 48 take. Although a fully
covariant formalism to regularize stress tensors using mass independent
schemes is the industry standard (as elaborated upon in [3, 4]), it is
nevertheless informative to trace through this process for the foliation
specific forms Eq. 47 and 48, as we do in Appendix A.

The important lesson to draw from this is that regularizing the divergence of
just one particular component of the stress tensor (such as the energy
density) is not enough. Computing physically sensible quantities derived from
the energy momentum tensor in a reliable manner necessitates constructing the
appropriate counterterm, without which renormalization conditions cannot be
consistently imposed. If the required counterterm cannot be constructed from
background quantities in a particular regularization scheme, then all
subsequent conclusions will necessarily be scheme dependent, and therefore
unphysical. For this reason, regularization schemes that preserve general
covariance such as dimensional or zeta function regularization are preferred
in this context. In spite of this caveat, however, one can still be forgiven
for using hard cutoffs in physical momenta when restricted to calculating
logarithmic divergences, as one can show the result to be identical to what
would have been obtained in a mass independent scheme. We will return to this
point in regards the regularization of the stress tensor for gravitational
waves in the next section.

###  III.2  Divergences – scalar fields and finite duration inflation

Having acquainted ourselves with the basics of regularizing and renormalizing
divergences for familiar quantities, one would like to generalize this to
backgrounds that realistically model the universe we inhabit. In particular,
one might wonder if the fact that we only obtained scaleless integrals in the
previous subsections is an artifact of considering the unrealistic scenario of
infinite duration inflation. We show that this situation persists even when we
consider a cosmology that transitions between different epochs.

In order for inflation to have finite duration by definition, we must be
considering an epoch before and after inflation, which we choose to be
radiation domination in both cases. We consider the scale factor to evolve as

$\displaystyle a(\tau)$ $\displaystyle=$ $\displaystyle a_{\rm
R}\left(2-\frac{\tau}{\tau_{\rm I}}\right)e^{-\mathcal{N}_{\rm
tot}}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \tau<\tau_{\rm I}$ $\displaystyle=$ $\displaystyle a_{\rm
R}\left(\frac{\tau_{\rm I}}{\tau}\right)e^{-\mathcal{N}_{\rm
tot}}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \tau_{\rm I}<\tau<\tau_{\rm R}$ $\displaystyle=$
$\displaystyle a_{\rm R}\left(2-\frac{\tau}{\tau_{\rm
R}}\right)\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \tau_{\rm
R}<\tau$

where $a_{\rm R}$ is the scale factor at reheating, and $\tau_{\rm I}$ and
$\tau_{\rm R}$ correspond to the (negative) conformal time at the start and
end of inflation212121Note that this puts the initial singularity at
$\tau=2\tau_{\rm I}$, however the only manner in which the pre-inflationary
phase bears on late time observables is through the choice to begin with the
adiabatic vacuum evolved to the start of the inflationary epoch at $\tau_{\rm
I}$. respectively, and

$\mathcal{N}_{\rm tot}=\log(a_{\rm R}/a_{\rm I})=\log(\tau_{\rm I}/\tau_{\rm
R})$ (52)

is the total amount e-folds of inflation. The domain of $\tau$ is
$(-\infty,\infty)$ with inflation occurring during negative conformal time.
With these definitions, the Hubble rate during inflation is given by

$H=-\frac{1}{a_{\rm R}\tau_{\rm R}}.$ (53)

In what follows, we keep the normalization $a_{\rm R}$ arbitrary, although one
can readily set $a_{\rm R}\equiv 1$ for convenience in what follows. Here
again, we focus on a massless, minimally coupled non-interacting scalar field
for illustrative purposes. The mode functions during the terminal stage of
radiation domination can be rewritten for the purposes of a matching to the
end of inflation as

$\displaystyle\phi^{\rm RD}_{k}$ $\displaystyle=$
$\displaystyle\frac{1}{a}\frac{1}{\sqrt{2k}}\left[\alpha_{k}^{\rm
R}e^{-ik\tau_{\rm R}\left(2-\frac{a}{a_{\rm R}}\right)}+\beta_{k}^{\rm
R}e^{ik\tau_{\rm R}\left(2-\frac{a}{a_{\rm R}}\right)}\right]$ (54)
$\displaystyle\phi^{\prime\rm RD}_{k}$ $\displaystyle=$
$\displaystyle\frac{a_{\rm R}}{a^{2}\tau_{\rm
R}}\frac{1}{\sqrt{2k}}\left[\alpha_{k}^{\rm R}e^{-ik\tau_{\rm
R}\left(2-\frac{a}{a_{\rm R}}\right)}\left(1-ik\tau_{\rm R}\frac{a}{a_{\rm
R}}\right)+\beta_{k}^{\rm R}e^{ik\tau_{\rm R}\left(2-\frac{a}{a_{\rm
R}}\right)}\left(1+ik\tau_{\rm R}\frac{a}{a_{\rm R}}\right)\right].$

The mode functions during inflation are given by

$\displaystyle\phi_{k}^{\rm I}$ $\displaystyle=$
$\displaystyle\frac{H}{\sqrt{2k^{3}}}\left[\alpha_{k}^{\rm
I}e^{i\frac{k}{aH}}\left(1-\frac{ik}{aH}\right)+\beta_{k}^{\rm
I}e^{-i\frac{k}{aH}}\left(1+\frac{ik}{aH}\right)\right]$ (55)
$\displaystyle\phi^{\prime\rm I}_{k}$ $\displaystyle=$
$\displaystyle-\frac{H}{\sqrt{2k^{3}}}\left[\alpha_{k}^{\rm
I}e^{i\frac{k}{aH}}\frac{k^{2}}{aH}+\beta_{k}^{\rm
I}e^{-i\frac{k}{aH}}\frac{k^{2}}{aH}\right],$

where the presence of non-trivial Bogoliubov coefficients come from having
matched to a radiation dominated pre-inflationary phase initiated in the
adiabatic vacuum, where the mode functions are given by

$\displaystyle\phi^{\rm PI}_{k}$ $\displaystyle=$
$\displaystyle\frac{1}{a}\frac{1}{\sqrt{2k}}e^{-ik\tau_{\rm
I}\left(2-\frac{a}{a_{\rm R}}e^{\mathcal{N}_{\rm tot}}\right)}$ (56)
$\displaystyle\phi^{\prime\rm PI}_{k}$ $\displaystyle=$
$\displaystyle-\frac{1}{a^{2}\sqrt{2k}}\frac{a_{\rm I}}{\tau_{\rm
I}}e^{-ik\tau_{\rm I}\left(2-\frac{a}{a_{\rm R}}e^{\mathcal{N}_{\rm
tot}}\right)}\left(-1+ik\tau_{\rm I}\frac{a}{a_{\rm I}}\right).$

One first obtains $\alpha^{\rm I}_{k}$ and $\beta_{k}^{\rm I}$ by matching to
the pre-inflationary radiation dominated epoch, so that

$\displaystyle\alpha_{k}^{\rm I}$ $\displaystyle=$
$\displaystyle\left(i-i\frac{a_{\rm I}^{2}H^{2}}{2k^{2}}+\frac{a_{\rm
I}H}{k}\right)$ (57) $\displaystyle\noindent\beta_{k}^{\rm I}$
$\displaystyle=$ $\displaystyle i\frac{a_{\rm
I}^{2}H^{2}}{2k^{2}}e^{2i\frac{k}{a_{\rm I}H}}.$

Similarly, after the matching between inflation and radiation dominated era,
one finds that $\alpha^{\rm R}_{k}$ and $\beta_{k}^{\rm R}$ are given by

$\displaystyle\alpha_{k}^{\rm R}$ $\displaystyle=$
$\displaystyle\alpha_{k}^{\rm I}\left(-i+\frac{a_{\rm R}H}{k}+i\frac{a_{\rm
R}^{2}H^{2}}{2k^{2}}\right)+i\beta_{k}^{\rm I}\frac{a_{\rm
R}^{2}H^{2}}{2k^{2}}e^{-2i\frac{k}{a_{\rm R}H}}$ $\displaystyle\beta_{k}^{\rm
R}$ $\displaystyle=$ $\displaystyle-i\alpha_{k}^{\rm I}\frac{a_{\rm
R}^{2}H^{2}}{2k^{2}}e^{2i\frac{k}{a_{\rm R}H}}+\beta_{k}^{\rm
I}\left(i+\frac{a_{\rm R}H}{k}-i\frac{a_{\rm R}^{2}H^{2}}{2k^{2}}\right).$
(58)

We now use these results to find expressions for the coincident limits of the
two point function and $\rho$ during the terminal stage of radiation
domination. The two point correlation function coincidence limit can be
expressed as

$\displaystyle\lim_{x\to y}\langle\hat{\phi}(\tau,x)\hat{\phi}(\tau,y)\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi^{2}a^{2}}\int_{0}^{\infty}\frac{dk}{k}\>\frac{k^{2}}{2}\left[1+2|\beta_{k}^{\rm
R}|^{2}+\alpha^{\rm R}_{k}\beta^{\rm R*}_{k}e^{\frac{2ik}{a_{\rm
R}H}\left(2-\frac{a}{a_{\rm R}}\right)}+\alpha^{\rm R*}_{k}\beta^{\rm
R}_{k}e^{-\frac{2ik}{a_{\rm R}H}\left(2-\frac{a}{a_{\rm R}}\right)}\right]$
(59) $\displaystyle=$
$\displaystyle\frac{1}{2\pi^{2}a^{2}}\int_{0}^{\infty}\frac{dk}{k}\>\frac{k^{2}}{2}\left[1+2|\beta_{k}^{\rm
R}|_{\rm power}^{2}+\\{{\rm osc}\\}\right]$

where the first line has used $|\alpha_{k}^{\rm I}|^{2}-|\beta_{k}^{\rm
I}|^{2}=|\alpha_{k}^{\rm R}|^{2}-|\beta_{k}^{\rm R}|^{2}=1$, and the second
line splits the integrand into strictly power law contributions and
oscillatory contributions. From Eqs. 57 and III.2, we find that

$\displaystyle|\beta^{\rm R}_{k}|_{\rm power}^{2}=\frac{a_{\rm
R}^{4}H^{4}+a_{\rm I}^{4}H^{4}}{4k^{4}}+\frac{a_{\rm I}^{4}a_{\rm
R}^{4}H^{8}}{8k^{8}},$ (60)

which nominally appears to have aggravated the IR divergence of the two point
function. However, as we demonstrate in Appendix B, although the oscillatory
terms can indeed be neglected for the purposes of extracting UV divergences,
the oscillations freeze in the IR and exactly cancel the contributions from
$|\beta^{\rm R}_{k}|_{\rm power}^{2}$. Specifically, one finds that

$\lim_{k\to 0}\\{{\rm osc}\\}=-\frac{a_{\rm R}^{4}H^{4}+a_{\rm
I}^{4}H^{4}}{2k^{4}}-\frac{a_{\rm I}^{4}a_{\rm
R}^{4}H^{8}}{4k^{8}}-1+\frac{(3a_{\rm I}^{3}a_{\rm R}+2a(a_{\rm R}^{3}-a_{\rm
I}^{3}))^{2}}{9a_{\rm I}^{2}a_{\rm R}^{6}},$ (61)

which cancels the contributions from $1+2|\beta^{\rm R}_{k}|_{\rm power}^{2}$,
leaving only the UV divergence resulting from the final term of Eq. 61 to
reckon with:

$\displaystyle\lim_{x\to
y}\langle\hat{\phi}(\tau,x)\hat{\phi}(\tau,y)\rangle_{\rm div}$
$\displaystyle=$ $\displaystyle e^{-4N_{\rm tot}}\left(3+\frac{2a}{a_{\rm
R}}\left[e^{3\mathcal{N}_{\rm
tot}-1}\right]\right)^{2}\frac{1}{36\pi^{2}a^{2}}\int_{0}^{\infty}dk\>k,$ (62)

where as before, $e^{-\mathcal{N}_{\rm tot}}=a_{\rm I}/a_{\rm R}$. Two things
are to be immediately noted. Firstly, that the physical UV and IR scales
$k_{\rm IR/UV}=a_{\rm I/R}H$ associated with the beginning and end of
inflation do not by themselves regulate the UV divergences. Instead, they
merely parameterize the divergence through their ratio $k_{\rm IR}/k_{\rm
UV}=e^{-\mathcal{N}_{\rm tot}}$ in the pre-factor above. Furthermore, the pre-
factor itself diverges as $\mathcal{N}_{\rm tot}\to\infty$, indicative of the
restoration of the pathologies of a past infinite dS space in that limit,
where moreover, a logarithmic IR divergence reappears. This is perhaps best
illustrated by Fig. 1, where we plot all contributions of the power spectrum
for finite duration inflation (identified through the logarithmic integrand of
Eq. 59) along with what would have resulted for a past-infinite dS cosmology
matched to a terminal phase of radiation domination222222The limit of a pre-
inflationary phase of infinite duration inflation can straightforwardly be
read off from inserting Eq. III.2 into Eq. 57 with $a_{\rm I}\equiv 0$..

(a) Power spectra evaluated at reheating $a=a_{\rm R}$, where $a_{\rm
I}=10^{-12}a_{\rm R}$ in units where $H$ is set to $2\pi$.

(b) Power spectra evaluated at different times during radiation domination.

Figure 1: Power spectrum comparison for a massless test scalar field with past
infinite vs finite duration dS inflation (dashed and bold lines, respectively)
matched to a terminal stage of radiation domination. Finite duration inflation
cures the would be IR divergences, leaving only UV divergences to attend to.

The UV divergence is unsurprisingly unchanged, and can be subtracted with a
cosmological constant counterterm were one to impose a cutoff in physical
momenta232323Where factorizing the divergence in dimensional regularization as
in the previous sub-section would instead see the UV contributions cancel
among themselves..

In computing the energy density $\rho$, we begin with

$\lim_{y\to
x}\rho(\tau;x,y)=\frac{1}{4\pi^{2}a^{2}}\int_{0}^{\infty}k^{2}\,dk\left[k^{2}\mathcal{|}\phi^{\rm
RD}_{k}(\tau)|^{2}+|\phi^{\rm RD\prime}_{k}(\tau)|^{2}\right]$ (63)

and use Eq. 54 to express the energy density during radiation domination as

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi^{2}a^{4}}\int_{0}^{\infty}\,\frac{dk}{k}\left[k^{4}\left(2+\frac{a_{\rm
R}^{4}H^{2}}{a^{2}k^{2}}\right)\left(1+2|\beta^{\rm R}_{k}|^{2}_{\rm
power}\right)+\\{{\rm osc}\\}\right],$ $\displaystyle:=$
$\displaystyle\int_{0}^{\infty}\,\frac{dk}{k}\left[\Omega^{\phi}_{\rm
power}(k)+\Omega^{\phi}_{\rm osc}(k)\right]$

where the integrated contribution of $\Omega^{\phi}_{\rm osc}$, defined as
$\rho_{\rm osc}$ is given in Eq. B, where it is explicitly shown not to
contribute to any UV divergences. As was the case with the two point function,
the contributions form Eq. 60 that nominally appear to aggravate IR
divergences in the context of Eq. III.2 are canceled by contributions from the
oscillatory terms that freeze out in the IR. One can expand the sum of all
contributions in the IR to find the IR safe scaling $\Omega^{\phi}_{\rm
tot}\propto k^{4}$, to be compared to what would have been obtained had there
been no pre-inflationary phase, where $\Omega^{\phi}_{\rm tot}\propto k^{2}$.
We illustrate this behavior, and the processing of the logarithmic spectra
power density as we go deeper into the radiation dominated regime in Fig. 2,
where the sub-horizon decay compensates for the $k^{2}$ scaling for the modes
that exited the Hubble horizon during inflation to produce a scale invariant
spectral density during radiation domination. We further note the additional
long wavelength suppression of the spectral density for a pre-inflationary
phase relative to past eternal dS.

(a)

(b)

(c)

(d)

Figure 2: Power spectral energy density for a massless test scalar field
comparing past infinite vs finite duration de Sitter inflation (dashed and
bold lines, respectively) matched to a terminal stage of radiation domination.
The panels are evaluated at subsequent times during radiation domination, with
$a_{\rm I}=10^{-10}a_{\rm R}$, and with $a/a_{\rm R}=1,10^{-8},10^{-10},$ and
$10^{-12}$, respectively, in units where $H=2\pi$.

Returning to Eq. III.2, we see that upon inserting Eq. 60, the only
divergences that need to be regulated are given by the contributions

$\rho_{\rm
div}=\frac{1}{8\pi^{4}a^{4}}\int_{-\infty}^{\infty}d^{4}k\left(1+\frac{H^{4}a_{\rm
R}^{4}+H^{4}a_{\rm
I}^{4}}{2k^{4}}\right)+\frac{1}{8\pi^{4}a^{6}}\int_{-\infty}^{\infty}d^{4}k\frac{a_{\rm
R}^{4}H^{2}}{2k^{2}},$ (65)

where again we recast the logarithmic integration into the formally equivalent
4D Euclidean form as per Eq. 16, and change integration variables to physical
momenta $q=k/(a_{\rm R}H)$. We stress that all factors of $H$ in the
expressions above correspond to the Hubble parameter during the intermediate
phase on inflation, which is presumed dS and therefore fixed as per Eq. 53.

One can proceed from here as we did in the previous subsection in comparing
the results from imposing cutoffs in physical momenta with that of
dimensionally regularizing factorized scaleless integrals for the energy
density, with the caveat that the counterterms are strictly speaking only to
be identified only through a scheme that preserves diffeomorphism invariance
as per the discussion at the end of the previous subsection. By imposing
cutoffs in physical momenta: $k=a\Lambda_{\rm UV}$, we see that the first term
in Eq. 65 corresponds to a divergence of the form

$\frac{1}{8\pi^{4}a^{4}}\int_{-\infty}^{\infty}\,d^{4}k=\frac{1}{4\pi^{2}a^{4}}\int_{0}^{a\Lambda_{\rm
UV}}\,k^{3}dk=\frac{\Lambda_{\rm UV}^{4}}{16\pi^{2}},$ (66)

and the third term corresponds to a divergence of the form

$\frac{1}{8\pi^{4}a^{6}}\int_{-\infty}^{\infty}\,d^{4}k\frac{a_{\rm
R}^{4}H^{2}}{2k^{2}}=\frac{a_{\rm
R}^{4}H^{2}}{8\pi^{2}a^{6}}\int_{0}^{a\Lambda_{\rm
UV}}\,kdk=\frac{\Lambda_{\rm UV}^{2}}{16\pi^{2}}\frac{H^{2}}{(a/a_{\rm
R})^{4}},$ (67)

which would nominally be subtracted by a counterterm proportional to $R$ were
we to insist on hard cutoff cutoff regularization. The former, when varied
with respect to the metric yields the contribution $R_{\mu\nu}\sim 1/a^{4}$
and thus corresponds to a renormalization of Newton’s constant. Finally, the
second term in Eq. 65 result in a divergence of the form

$\frac{1}{8\pi^{4}a^{4}}\int_{-\infty}^{\infty}\,d^{4}k\>\left(\frac{a_{\rm
R}^{4}H^{4}+a_{\rm
I}^{4}H^{4}}{2k^{4}}\right)=\frac{1}{8\pi^{2}a^{4}}\int_{a\Lambda_{\rm
IR}}^{a\Lambda_{\rm UV}}\,\frac{dk}{k}\left(a_{\rm R}^{4}H^{4}+a_{\rm
I}^{4}H^{4}\right)=\frac{H^{4}(1+e^{-4\mathcal{N}_{\rm
tot}})}{8\pi^{2}(a/a_{\rm R})^{4}}\log\frac{\Lambda_{\rm UV}}{\Lambda_{\rm
IR}},$ (68)

where the scale $\Lambda_{\rm IR}$ does not have the same significance as
before given the IR safe behavior of the spectral density, and where the above
also corresponds to a renormalization of Newton’s constant. We note that the
scales corresponding to the beginning and end of inflation appear in the
coefficient of the logarithm, whereas the UV scale corresponding to the
unknown completion the theory appears inside the logarithm.

Were we to now dimensionally regularize the divergences as advised, we can
perform a similar factorization of the scaleless integrals as in the previous
subsection. The nominally power law divergent parts can be isolated as

$\displaystyle\rho^{\rm UV}_{\rm div}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi^{4}a^{4}}\int_{-\infty}^{\infty}\,d^{4}k\left(1+\frac{a_{\rm
R}^{4}H^{2}}{2a^{2}k^{2}}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi^{4}a^{4}}\int_{-\infty}^{\infty}\,d^{4}k\left[\frac{k^{2}}{(k^{2}+m^{2})}+\frac{m^{2}}{(k^{2}+m^{2})}+\frac{a_{\rm
R}^{4}H^{2}}{2a^{2}}\left(\frac{1}{(k^{2}+m^{2})}+\frac{m^{2}}{k^{2}(k^{2}+m^{2})}\right)\right],$

where we see that power law UV divergences cancel among themselves, and so do
not necessitate any counterterms. On the other hand, the UV divergent
logarithmic term can be isolated and factorized as

$\displaystyle\rho^{\rm log}_{\rm div}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi^{4}a^{4}}\int_{-\infty}^{\infty}\,d^{4}k\>\left(\frac{a_{\rm
R}^{4}H^{4}+a_{\rm I}^{4}H^{4}}{2k^{4}}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi^{4}a^{4}}\int_{-\infty}^{\infty}\,d^{4}k\left(\frac{a_{\rm
R}^{4}H^{4}+a_{\rm
I}^{4}H^{4}}{2}\right)\left(\frac{1}{k^{2}(k^{2}+m^{2})}+\frac{m^{2}}{k^{4}(k^{2}+m^{2})}\right)$

By isolating the logarithmic UV poles, we find

$\rho^{\rm log}_{\rm div}=\frac{H^{4}(1+e^{-4\mathcal{N}_{\rm
tot}})}{8\pi^{2}(a/a_{\rm R})^{4}}\left[\frac{1}{\delta_{\rm
UV}}+1-\gamma_{E}+\log\left(\frac{\mu}{H}\right)\right]$ (71)

from which we read off an identical coefficient as computed in Eq. 68242424One
can also show that the analogous computation for the pressure component also
necessitates a counterterm corresponding to a renormalization of $G_{N}$ (i.e.
resulting in a divergent contribution proportional to the spatial components
of the Ricci tensor when varied with respect to the background metric)
resulting in a renormalized stress tensor that is traceless on a radiation
domination background. We do so explicitly for the case of tensor modes in
Appendix A.. For completeness, we show in Appendix C that the coefficient of
the logarithmic divergence obtained from point split regularization in the
FRLW slicing is also identical, with the same power law divergences identified
in physical cutoff regularization up to the expected scheme dependence in the
coefficients of the latter.

To summarize our findings in this subsection: the scales corresponding to the
beginning and end of inflation must in principle be separated from any UV and
IR scales parameterizing the unknown completion of our theory and well defined
observable quantities. Although physical quantities cannot depend on the
latter, they may certainly depend on the former. We found that all IR
divergences encountered in the simplified settings we worked in were regulated
by the existence of a pre-inflationary phase, and therefore an artifact of the
approximation of a past infinite dS phase. A highly pertinent question is
whether this persists once we move to incorporate higher point interactions
and couplings with other test fields. Unsurprisingly, UV divergences persist
for backgrounds corresponding to finite duration inflation, and the scale
corresponding to the end of inflation parameterizes their coefficients rather
than regulating them. Upon subtraction of these divergences with local
counterterms and the imposition of renormalization conditions, one is able to
draw meaningful physical conclusions. However, the two point function and
stress tensor of a test scalar field are of relatively academic interest given
the assumed negligibility of the scalar field background, and therefore
inability to fix renormalization conditions from observation. The analogous
set of questions for vacuum tensor perturbations presents a far more
interesting application given the assumption of an evolving background
gravitational field (that of FRLW cosmology), perturbations around which
represent gravitational waves, which is where we turn our attention towards
next.

##  IV Gravitational waves

Before delving into the substance of this section, it is informative to
contrast the results obtained in the previous section to analogous expressions
typically found in the literature in the context of gravitational waves (see
e.g. [71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84]), where imposing
hard cutoffs on the energy density of gravitational waves results in
expressions of the form:

$\rho_{\mathrm{GW}}\simeq\frac{A_{t}}{32\pi
G_{N}}\left(\frac{k_{\mathrm{UV}}}{k_{*}}\right)^{n_{t}}\frac{1}{2n_{t}}\frac{1}{a^{4}}\propto\frac{1}{a^{4}}\left[\frac{1}{n_{t}}+\log\frac{k_{\rm
UV}}{k_{*}}\right],$ (72)

where $k_{*}$ is some reference IR scale, and the approximation is only valid
when $n_{t}\to 0$ [71]. We deconstruct and rederive the energy density for
vacuum tensor perturbations in what follows, but before doing so, it is useful
to compare what one would have obtained in retracing the steps leading to Eq.
72 for a massless test scalar field, which would have resulted in the
expression

$\rho\simeq\lim_{n_{s}\to 1}\frac{A_{s}}{32\pi
G_{N}}\left(\frac{k_{\mathrm{UV}}}{k_{*}}\right)^{n_{\rm
s}-1}\frac{1}{2(n_{s}-1)}\frac{1}{a^{4}}\propto\frac{1}{a^{4}}\left[\frac{1}{n_{s}-1}+\log\frac{k_{\rm
UV}}{k_{*}}\right],$ (73)

which can directly be compared to the divergences computed in Eqs. 66 \- 68
from hard cutoffs in physical momenta and Eq. 71 from dimensional
regularization, all of which are understood to be intermediate expressions
that are to be subtracted and renormalized.

An immediate reservation one might express for expressions such as Eq. 72 is
the appearance of cutoffs in what should be a physical result. Although one
might argue that the effects of UV and IR modes are negligible relative to
what can be extracted from Eq. 72, its form nevertheless suggests that the
expression above is an intermediate result on the way to computing a physical
energy density. Moreover, one might be concerned in applying the formula Eq.
72 to the case of blue tilted spectra (i.e. for positive
$n_{t}\sim\mathcal{O}(1)$, so that $\rho_{\rm GW}\sim(k_{\rm
UV}/k_{*})^{n_{t}}$ as derived in e.g. [71]), that some part of this
expression is nothing other than the Fourier transform of a UV divergence that
gets subtracted in the usual way. The particular case of $n_{t}=3$ is
particularly interesting in this context, as part of it is exactly mimicked by
the familiar quadratic UV divergence from the perspective of cutoff
regularization, necessitating greater care in the interpretation of such power
spectra. In spite of this caution, it is both practical and straightforward to
follow through this process to the end to ensure that one is computing well
defined physical observables. In the specific context of vacuum tensor
perturbations, however, additional subtleties must be accounted for.

The formula for the stress energy tensor of gravitational waves was first
derived by Isaacson in the context astrophysically sourced gravitational waves
[85] (see also [86, 87]), which is nevertheless extensively referenced in
computations of the stress energy tensor of cosmological gravitational wave
backgrounds252525e.g. leading to Eq. 72. See also [88] and references
therein., and is given by262626Implicit in the expectation value of Eq. 74 is
an additional (Brill-Hartle) spatial and temporal averaging prescription. We
presume this as implicit when discussing the Isaacson form of the stress
tensor in what follows.

$\displaystyle\rho^{\rm Isc}_{\rm gw}=\frac{1}{32\pi a^{2}G_{N}}\left\langle
h^{\prime}_{ij}(\tau,k)h^{\prime ij}(\tau,k)\right\rangle.$ (74)

However, retracing the steps used in its derivation, it becomes immediately
clear that one needs to proceed with extra care for applications in a
cosmological context, in particular when considering questions of renormalized
stress tensors in prescriptions that involve integrations over all
momenta272727A feature that can be bypassed entirely in the context of
Hadamard regularization [89, 90].. This is because the Isaacson form presumes
a prior scale separation between fast and slow frequencies defined relative to
the curvature scale of the background (a criteria that is time dependent), and
repeatedly relies on this scale separation along with the implicit averaging
prescription to facilitate various approximations resulting in the simplified
final form Eq. 74. In the context of an expanding spacetime, with modes of
observational relevance that might have crossed any a priori defined scale
more than once over cosmic evolution, undoing the steps that relied on these
approximations is warranted. We do so in Appendix D, retracing the derivation
of the stress tensor for gravitational waves, finding the resulting expression
for the energy density282828From now on, even if we drop the hat in the
notation, $h_{ij}$ is an operator as defined in Eq. 76.:

$\displaystyle\rho^{\rm Imp}_{\rm gw}=\frac{1}{8\pi
G_{N}}\left\langle\frac{1}{8a^{2}}h_{ij}^{\prime}h^{\prime
ij}-\frac{3}{8a^{2}}\partial_{k}h_{ij}\partial^{k}h^{ij}-\frac{1}{2a^{2}}h_{ij}\partial_{k}\partial^{k}h^{ij}+\frac{1}{4a^{2}}\partial_{k}h_{ij}\partial^{j}h^{ik}+\frac{\mathcal{H}}{a^{2}}h_{j}{}^{i}h^{\prime}_{i}{}^{j}\right\rangle\>,$
(75)

where $\mathcal{H}:=a^{\prime}/a$. The derivation is straightforward, and
follows from the same starting point as the derivation leading to Eq. 74, but
with any steps that invoked a time averaging prescription or frequency
separation left undone (we elaborate on the details in Appendix D).

It is worth stressing that Eq. 75 is the temporal component of a covariantly
conserved tensor, and is under no obligation to conserved in isolation. This
is a corollary of the fact that on FRLW backgrounds, a locally conserved
energy cannot be defined. Moreover, it is also under no obligation to be
positive definite. In fact, were one to insist of constructing the spectral
density associated with the vacuum expectation value of Eq. 75, one can show
that it crosses zero at a comoving scale corresponding to the horizon scale at
any given time, consistent with the operational ambiguity of associating a
background/fluctuation split for wavelengths commensurate with the background
curvature. However, none of this is of any operational concern for our
purposes, as we stress that Eq. 75 is to be viewed as the components of a
covariantly conserved tensor corresponding to a massless spin two excitation,
whose role in renormalization of background quantities is well understood in
the covariant context, but obscured and widely conflated for a physical
contribution to the number of relativistic species in foliation specific
computations as we elaborate upon next. The specific computation we are
interested in performing is the renormalization of the graviton stress tensor
on a background corresponding to radiation domination preceded by a finite
period of inflation. In doing so, we will compare and contrast the results
obtained from both the Isaacson and improved forms of the stress tensor,
before revisiting the question of $N_{\rm eff}$ bounds from vacuum tensor
modes and concluding.

###  IV.1 Divergences – vacuum tensor modes and finite duration inflation

Working in transverse-traceless (TT) gauge (see Appendix D for the relevant
details and caveats), we expand the tensor perturbations as

$\hat{h}_{ij}(\tau,x)=\sum_{r=+,x}\int\frac{d^{3}k}{M_{\rm
pl}\left(2\pi\right)^{3}}e^{ix\cdot
k}\left[\epsilon_{ij}^{r}(k)\hat{a}_{k}\gamma_{k}(\tau)+\epsilon_{ij}^{r*}(-k)\hat{a}_{-k}^{{\dagger}}\gamma^{*}_{k}(\tau)\right]$
(76)

with polarization tensors normalized as
$\epsilon_{ij}^{s}\epsilon_{ij}^{r*}=4\delta^{rs}$,
$k^{i}\epsilon_{ij}^{r}=0$. The normalizations are chosen so that the relevant
mode functions during inflation and the pre and post-inflationary phases of
radiation domination are given for each polarization just as for massless
minimally coupled scalars292929The EOM for $h^{j}_{i}$ in TT gauge can be
derived from the first order expansion of Einstein equation:
$\hat{h}^{\prime\prime j}_{i}+2\mathcal{H}\hat{h}^{\prime
j}_{i}-\partial_{k}^{2}\hat{h}^{j}_{i}=0.$. These are:

$\displaystyle\gamma^{\rm PI}_{k}(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{a}\frac{1}{\sqrt{2k}}e^{-ik\tau_{\rm
I}\left(2-\frac{a}{a_{\rm R}}e^{\mathcal{N}}_{\rm tot}\right)}$ (77)
$\displaystyle\gamma^{\prime\rm PI}_{k}(\tau)$ $\displaystyle=$
$\displaystyle-\frac{1}{a^{2}\sqrt{2k}}\frac{a_{\rm I}}{\tau_{\rm
I}}e^{-ik\tau_{\rm I}\left(2-\frac{a}{a_{\rm R}}e^{\mathcal{N}}_{\rm
tot}\right)}\left(-1+ik\tau_{\rm I}\frac{a}{a_{\rm I}}\right),$

during the pre-inflationary phase,

$\displaystyle\gamma_{k}^{\rm I}(\tau)$ $\displaystyle=$
$\displaystyle\frac{H}{\sqrt{2k^{3}}}\left[\alpha_{k}^{\rm
I}e^{i\frac{k}{aH}}\left(1-\frac{ik}{aH}\right)+\beta_{k}^{\rm
I}e^{-i\frac{k}{aH}}\left(1+\frac{ik}{aH}\right)\right]$ (78)
$\displaystyle\gamma^{\prime\rm I}_{k}(\tau)$ $\displaystyle=$
$\displaystyle-\frac{H}{\sqrt{2k^{3}}}\left[\alpha_{k}^{\rm
I}e^{i\frac{k}{aH}}\frac{k^{2}}{aH}+\beta_{k}^{\rm
I}e^{-i\frac{k}{aH}}\frac{k^{2}}{aH}\right],$

during inflation, and

$\displaystyle\gamma^{\rm RD}_{k}(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{a}\frac{1}{\sqrt{2k}}\left[\alpha_{k}^{\rm
R}e^{-ik\tau_{\rm R}\left(2-\frac{a}{a_{\rm R}}\right)}+\beta_{k}^{\rm
R}e^{ik\tau_{\rm R}\left(2-\frac{a}{a_{\rm R}}\right)}\right]$ (79)
$\displaystyle\gamma^{\prime\rm RD}_{k}(\tau)$ $\displaystyle=$
$\displaystyle\frac{a_{\rm R}}{a^{2}\tau_{\rm
R}}\frac{1}{\sqrt{2k}}\left[\alpha_{k}^{\rm R}e^{-ik\tau_{\rm
R}\left(2-\frac{a}{a_{\rm R}}\right)}\left(1-ik\tau_{R}\frac{a}{a_{\rm
R}}\right)+\beta_{k}^{\rm R}e^{ik\tau_{\rm R}\left(2-\frac{a}{a_{\rm
R}}\right)}\left(1+ik\tau_{\rm R}\frac{a}{a_{\rm R}}\right)\right]$

during the terminal radiation dominated phase. The corresponding Bogoliubov
coefficients are the same as those given in Eq. 57 and Eq. III.2.

An immediate corollary of the above is that the two point correlation function
of each graviton polarization is identical to that of a massless, minimally
coupled scalar. Therefore, as illustrated in Figs. 1(a) and 1(b), one finds
that the IR divergences exhibited on a past infinite dS background are also
cured for the graviton two point function on a background corresponding to
finite duration inflation.

The energy density of gravitational waves presuming the Isaacson form of the
stress tensor Eq. 74 results in (with the reduced Planck mass defined as
$M^{2}_{\rm pl}=\frac{1}{8\pi G_{N}}$):

$\displaystyle\rho_{\rm gw}^{\rm Isc}$ $\displaystyle=$
$\displaystyle\lim_{y\to x}\rho_{\rm
gw}(\tau;x,y)=\frac{1}{\pi^{2}a^{2}}\int_{0}^{\infty}dk\
k^{2}\>\left[\gamma^{{}^{\prime}\rm RD}_{k}\gamma^{{}^{\prime}\rm
RD*}_{k}\right]$ (80)

as the tensor mode counterpart of Eq. 63, after having summed the
contributions from the two independent polarizations.

Figure 3: Power spectral density of the Isaacson stress tensor (dashed lines)
evaluated at $a=10^{9}a_{\rm R}$ (left) and $a=10^{13}a_{\rm R}$ (right), with
$a_{\rm I}=10^{-12}a_{\rm R}$ in units where $H$ is set to $2\pi$. The blue
line is the comparison to the spectral density of the improved stress tensor.
The gray shaded regions correspond to super-horizon scales, which are outside
the domain of validity of the Isaacson stress tensor (where we note that the
oscillations would not appear in its time averaged form), and also where the
spectral density for the improved stress tensor nominally becomes negative
(cf. discussion below Eq. 75).

Using the mode functions specified by Eqs. 79, the relations III.2, and
tracing through the steps of the previous section, we find

$\displaystyle\rho_{\rm gw}^{\rm Isc}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi^{2}a^{4}}\int_{0}^{\infty}\frac{dk}{k}\>k^{4}\left[\left(1+\frac{a_{\rm
R}^{4}H^{2}}{a^{2}k^{2}}\right)\left(1+2|\beta^{\rm R}_{k}|^{2}_{\rm
power}\right)+\\{{\rm osc}\\}\right],$ $\displaystyle:=$
$\displaystyle\int_{0}^{\infty}\,\frac{dk}{k}\left[\Omega^{\rm gw}_{\rm
power}(k,\tau)+\Omega^{\rm gw}_{\rm osc}(k,\tau)\right],$

where, as we did for the scalar case, we similarly separate the spectral
density into oscillatory and power law contributions, and all contributions to
$\rho_{\rm gw}$ can be found in Eq. B. Even though the Isaacson form of the
stress tensor is not strictly valid when incorporating wavelengths greater
than the Hubble scale at any given time, naïvely persisting with it for all
wavelengths would show an IR regular spectral density $\Omega^{\rm gw}_{\rm
total}\propto k^{2}$, with oscillatory contributions contributions giving
negligible contributions in the UV. Consequently, one only has the resulting
UV divergent contributions to regulate:

$\displaystyle\rho^{\rm Isc}_{\rm gw,div}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{4}a^{4}}\int_{-\infty}^{\infty}\,d^{4}k\>\left(1+\frac{a_{\rm
R}^{4}H^{4}+a_{\rm I}^{4}H^{4}}{2k^{4}}+\frac{a_{\rm
R}^{4}H^{2}}{a^{2}k^{2}}\right),$ (82)

which up to overall factors, is formally identical to the contributions Eq.
65, and whose subtraction proceeds along the same lines.

Were we to consider the improved stress tensor Eq. 75 (which does not invoke a
scale separation or time averaging prescription), one would obtain

$\displaystyle\rho_{\rm gw}^{\rm Imp}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi^{2}a^{2}}\int_{0}^{\infty}dk\
k^{2}\>\left[\gamma^{{}^{\prime}\rm RD}_{k}\gamma^{{}^{\prime}\rm
RD*}_{k}+k^{2}\gamma^{\rm RD}_{k}\gamma^{\rm
RD*}_{k}+4\mathcal{H}\left(\gamma^{{}^{\prime}\rm RD}_{k}\gamma^{\rm
RD*}_{k}+\gamma^{\rm RD}_{k}\gamma^{{}^{\prime}\rm RD*}_{k}\right)\right]$
(83)

as the counterpart of Eq. 80 that is now valid for for all wavelengths. Using
the mode functions and normalizations specified above, one finds

$\displaystyle\rho_{\rm gw}^{\rm Imp}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi^{2}a^{4}}\int_{0}^{\infty}\frac{dk}{k}\>k^{4}\left[\left(1-\frac{7a_{\rm
R}^{4}H^{2}}{2a^{2}k^{2}}\right)\left(1+2|\beta^{\rm R}_{k}|^{2}_{\rm
power}\right)+\\{{\rm osc}\\}\right],$ $\displaystyle:=$
$\displaystyle\int_{0}^{\infty}\,\frac{dk}{k}\left[\Omega^{\rm gw}_{\rm
power}(k,\tau)+\Omega^{\rm gw}_{\rm osc}(k,\tau)\right]$

where we can again separate the spectral density303030We stress that the
spectral density as defined in Eq. IV.1 is to be viewed as only a
calculational definition for the purposes of comparison to the literature, and
not be to viewed as the power spectral density of vacuum fluctuations without
additional caveats. into oscillatory and power law contributions, where all
contributions to $\rho^{\rm Imp}_{\rm gw}$ can be found in Eq. B.

One again finds that the spectral density summing all contributions in the IR
goes as $\Omega^{\rm gw}_{\rm total}\propto k^{2}$, albeit with a negative
overall coefficient (cf. the discussion below Eq. 75), with the following
divergent contributions in the UV that necessitate subtraction

$\displaystyle\rho_{\rm gw,div}^{\rm Imp}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{4}a^{4}}\int_{-\infty}^{\infty}\,d^{4}k\>\left(1+\frac{a_{\rm
R}^{4}H^{4}+a_{\rm I}^{4}H^{4}}{2k^{4}}-\frac{7\,a_{\rm
R}^{4}H^{2}}{2a^{2}k^{2}}\right).$ (85)

For illustrative purposes, we plot the power spectral densities of the (time
unaveraged) Isaacson and improved forms of the stress tensors in Fig. 3. For
sub-horizon modes, where a positive spectral density results, the improved
stress tensor evaluated at any given time does not exhibit
oscillations313131The oscillatory behavior of the time unaveraged Isaacson
stress tensor can be understood from the fact that the magnitude of the time
derivative squared of a linearized gravitational wave is by itself not a
constant of the linearized equations of motion (which requires restoring the
spatial derivative contributions), unlike for the improved form.. In
proceeding to regularizing divergences, we can identity the various UV
divergences is different schemes, finding identical coefficients for the log
divergences in all of them. By imposing cutoffs in physical momenta
($k=a\Lambda_{\rm UV}$) on the identifiably UV-divergent terms above, we
obtain

$\displaystyle\frac{1}{2\pi^{2}a^{4}}\int_{0}^{a\Lambda_{\rm
UV}}dk\>k^{3}-\frac{7}{2}\frac{a_{\rm
R}^{4}H^{2}}{2\pi^{2}a^{6}}\int_{0}^{a\Lambda_{\rm
UV}}dk\>k=\frac{\Lambda_{\rm UV}^{4}}{8\pi^{2}}-\frac{7H^{2}\Lambda_{\rm
UV}^{2}}{8\pi^{2}(a/a_{\rm R})^{4}}$ (86)

and

$\displaystyle\frac{1}{4\pi^{2}a^{4}}\int_{a\Lambda_{\rm IR}}^{a\Lambda_{\rm
UV}}\,\frac{dk}{k}\left(a_{\rm R}^{4}H^{4}+a_{\rm
I}^{4}H^{4}\right)=\frac{H^{4}\left(1+e^{-4\mathcal{N}_{\rm
tot}}\right)}{4\pi^{2}(a/a_{R})^{4}}\log\frac{\Lambda_{\rm UV}}{\Lambda_{\rm
IR}}.$ (87)

If instead we were to now isolate the power law UV divergent parts and
dimensionally regularize, we would obtain

$\displaystyle\rho^{\rm Imp}_{\rm gw,div}$ $\displaystyle\supset$
$\displaystyle\frac{1}{4\pi^{4}a^{4}}\int_{-\infty}^{\infty}\,d^{4}k-\frac{7a_{\rm
R}^{2}}{4\pi^{4}a^{6}}\int_{-\infty}^{\infty}\,d^{4}k\>\left(\frac{a_{\rm
R}^{2}H^{2}}{2k^{2}}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{4}a^{4}}\int_{-\infty}^{\infty}\,d^{4}k\left[\frac{k^{2}}{(k^{2}+m^{2})}+\frac{m^{2}}{(k^{2}+m^{2})}-\frac{7}{2}\frac{a_{\rm
R}^{4}H^{2}}{2a^{2}}\left(\frac{1}{(k^{2}+m^{2})}+\frac{m^{2}}{k^{2}(k^{2}+m^{2})}\right)\right]$

which, as for the dimensionally regularized scalar case Eq. III.2, has the
divergent contributions canceling among themselves and not necessitating any
counterterms. The logarithmic divergence can be expressed as

$\displaystyle\rho^{\rm Imp}_{\rm gw,div}$ $\displaystyle\supset$
$\displaystyle\frac{1}{4\pi^{4}a^{4}}\int_{-\infty}^{\infty}\,d^{4}k\>\left(\frac{a_{\rm
R}^{4}H^{4}+a_{\rm
I}^{4}H^{4}}{2k^{4}}\right)=\frac{H^{4}\left(1+e^{-4\mathcal{N}_{\rm
tot}}\right)}{4\pi^{2}(a/a_{\rm R})^{4}}\left[\frac{1}{\delta_{\rm
UV}}+1-\gamma_{E}+\log\left(\frac{\mu}{H}\right)\right].$ (89)

Lastly, one can point split regularize Eq. 82 by commuting the coincident
limit with the momentum integrations, so that one has to evaluate

$\displaystyle\rho_{\rm gw}^{\rm Imp}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}a^{4}r}\int_{0}^{\infty}\,k^{2}dk\,\sin
kr\>\left[\left(1-\frac{7a_{\rm
R}^{4}H^{2}}{2a^{2}k^{2}}\right)\left(1+2|\beta^{\rm R}_{k}|^{2}_{\rm
power}\right)+\\{{\rm osc}\\}\right],$ (90)

which can be shown to result in the UV divergent contributions

$\rho^{\rm Imp}_{\rm gw,div}=\lim_{\sigma\to
0}\left\\{-\frac{1}{\pi^{2}\sigma^{4}}-\frac{7H^{2}}{4\pi^{2}\sigma^{2}(a/a_{\rm
R}^{4})}+\frac{H^{4}\left(1+e^{-4\mathcal{N}_{\rm
tot}}\right)}{4\pi^{2}(a/a_{\rm
R})^{4}}\left[1-\gamma_{E}-\log{k_{p}\sigma}\right]\right\\},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ {\rm(point-
split)}$ (91)

where as before $\sigma=ar$ and $k_{p}:=k/a$ are physical quantities. We mote
that the coefficient of the logarithmic divergence is identical, with the
minus sign reflecting the fact that $\sigma\to 0$ inside the point split
logarithm, whereas $\Lambda_{\rm UV}\to\infty$ with hard cutoff
regularization, the results of which can be collected and summarized as

$\displaystyle\rho^{\rm Imp}_{\rm gw,div}$ $\displaystyle=$
$\displaystyle\lim_{\Lambda_{\rm
UV}\to\infty}\left\\{\frac{1}{2\pi^{2}}\frac{\Lambda_{\rm
UV}^{4}}{4}-\frac{7}{2}\frac{a_{\rm
R}^{4}H^{2}}{2\pi^{2}a^{4}}\frac{\Lambda_{\rm
UV}^{2}}{2}+\frac{H^{4}\left(1+e^{-4\mathcal{N}_{\rm
tot}}\right)}{4\pi^{2}(a/a_{\rm R})^{4}}\log\frac{\Lambda_{\rm
UV}}{\Lambda_{\rm IR}}\right\\},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm(cutoff)}$ (92)

and also compared to the answer obtained via dimensional regularization

$\displaystyle\rho^{\rm Imp}_{\rm gw,div}$ $\displaystyle=$
$\displaystyle\lim_{\delta_{\rm UV}\to
0}\left\\{\frac{H^{4}\left(1+e^{-4\mathcal{N}_{\rm
tot}}\right)}{4\pi^{2}(a/a_{R})^{4}}\left[\frac{1}{\delta_{\rm
UV}}+1-\gamma_{E}+\log\left(\frac{\mu}{H}\right)\right]\right\\}.\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ {\rm(dim-reg)}$ (93)

After having subtracting divergences with the appropriate counterterms, a
finite remainder remains that needs to be fixed by the imposition of
renormalization conditions by observations at a given scale. This in order to
have a predictive theory with which we can calculate the results for any
subsequent observations. As cautioned in the previous section, one may not be
able to do this reliably unless one uses regularization schemes that preserve
the symmetries of the background so that the required counterterms can be
consistently identified (see Appendix A for more details). For this reason, we
stick to dimensional regularization in what follows, although covariant point
splitting methods also offer a practical alternative323232The point split
regularization elaborated upon here is contingent on the FRLW slicing, which
permits a Fourier expansion, and so cannot be considered fully covariant. [3,
91, 92, 93, 94, 95, 96].

In order to understand which counterterms we need to add in order to reabsorb
the divergences in dimensional regularization, we need only to look at the
scale factor dependence of the divergences in Eq. 119. However, we immediately
notice that the divergences we have computed do not necessitate higher
derivative counterterms beyond those already in the Einstein Hilbert action.
In order to understand why this is, we trace through a treatment that uses
adiabatic regularization to renormalize the stress tensor of a non-minimally
coupled test scalar field [97], where mode functions are adiabatically
expanded in order to regularize divergences and identify counterterms. Higher
orders in the adiabatic expansion necessitate successively higher order
counterterms. The adiabatic solution of order zero is regularized by a
cosmological constant counterterm, the second order solution by a curvature
counterterm, with curvature squared counterterms needed to renormalize
divergences that appear only at fourth order in the adiabatic expansion. In
our case the adiabatic solution of order four is null and non-vanishing
curvature squared counterterms would only be necessitated were one to consider
couplings to other matter fields, or by incorporating the effects of loops and
higher order gravitational and matter coupling non-linearities.

##  V Renormalized stress tensors (and $N_{\rm eff}$ bounds)

We turn our attention in this section to the second and most consequential
step in the process of renormalization – that of extracting physical
observables after imposing renormalization conditions. We begin the discussion
with the relatively academic exercise of renormalizing the stress tensor of a
test scalar field on a background that transitions in and out of inflation. By
virtue of not having a classically evolving background and energy density, the
only effects of a test scalar will be in renormalizing background couplings in
the context of the effective theory of gravity [67, 68, 69, 70]. We return to
the more interesting and physically relevant case of primordial gravitational
waves next.

Our starting point is the bare matter and gravitational actions, along with
the requisite counterterms:

$S=S_{\rm EH}+S_{\rm bg}+S_{\phi}+S_{\rm ct}.$ (94)

In order to impose renormalization conditions after having regularized
divergences, we first consider the background equations of motion

$\frac{1}{8\pi G_{B}}\left(R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}\right)=T^{\rm
bg}_{\mu\nu}+T^{\phi}_{\mu\nu}+T^{\rm ct}_{\mu\nu},$ (95)

where we stress that the couplings that appear in Eq. 94 are to be understood
as bare couplings, and the presence of the counterterm shifts the tadpole
condition in a manner that we will shortly make precise. Given the consistency
of dimensional regularization with general covariance, we can proceed by
considering the above for any given component. For the 00 component, we have

$-\frac{R_{0}{}^{0}}{8\pi G_{B}}=\rho^{\rm cl}_{\rm bg}+\rho_{\phi}+\rho_{\rm
ct},$ (96)

where $\rho^{\rm cl}_{\rm bg}$ is the to be renormalized classical background
energy density that sources the expanding geometry around which we have
computed the stress tensor for the test scalar field, with corresponding
energy density $\rho_{\phi}$ defined as

$\rho_{\phi}:=\rho^{\rm cl}_{\phi}-\langle T^{0}{}_{0}\rangle\equiv-\langle
T^{0}{}_{0}\rangle,$ (97)

where $\rho_{\phi}^{\rm cl}\equiv 0$ for a test scalar by assumption. The
background satisfies

$-R_{0}{}^{0}=-\frac{3}{a^{2}}\left(\frac{a^{\prime}}{a}\right)^{\prime}=\frac{3H^{2}}{(a/a_{\rm
R})^{4}}$ (98)

via Eq. III.2 during the terminal radiation dominated phase, where it is to be
stressed again that $H$ is the Hubble constant during the intermediate phase
of inflation that enters the definition of the scale factor during radiation
domination via Eq. III.2. We need to identify the counterterm that absorbs the
divergence exhibited in Eq. 71:

$\frac{1}{8\pi G_{B}}\frac{3H^{2}}{(a/a_{R})^{4}}=\rho^{\rm cl}_{\rm
bg}+\frac{H^{4}(1+e^{-4\mathcal{N}_{\rm tot}})}{8\pi^{2}(a/a_{\rm
R})^{4}}\left[\frac{1}{\delta_{\rm
UV}}+1-\gamma_{E}+\log\left(\frac{\mu}{H}\right)\right]+\rho_{\rm
ct}+\rho_{\phi,\rm finite},$ (99)

where $\rho_{\rm finite}\equiv\rho_{\phi}-\rho_{\rm\phi,div}$. We are
immediately presented with a choice here – after subtracting the pole, do we
proceed to (multiplicatively) renormalize Newton’s constant, or (additively)
renormalize the background whose unshifted value is determined by $\rho^{\rm
cl}_{\rm bg}$? It turns out that this choice is rendered moot by the tadpole
condition that determines the background equations of motion at any given
order in $\hbar$, in that whichever choice we make will lead us to the same
shifted tadpole condition. We thus proceed by reabsorbing the pole by adding a
counterterm that multiplies the Ricci scalar in the Einstein Hilbert action
with coefficient $B$ defined as

$\rho_{\rm ct}=\frac{3H^{2}}{(a/a_{R})^{4}}\left(\frac{B_{-1}}{\delta_{\rm
UV}}+B_{0}\right).$ (100)

By assigning

$B_{-1}=-\frac{H^{2}(1+e^{-4\mathcal{N}_{\rm tot}})}{24\pi^{2}},$ (101)

one subtracts the pole contribution. By defining a scale dependent
gravitational coupling as

$\frac{1}{8\pi G_{N}(\mu)}=\frac{1}{8\pi
G_{B}}-B_{0}-\frac{H^{2}}{24\pi^{2}}(1+e^{-4\mathcal{N}_{\rm
tot}})\left\\{1-\gamma_{E}+\log\left(\frac{\mu}{H}\right)\right\\},$ (102)

we can rewrite Eq. 99 as

$\frac{1}{8\pi G_{N}(\mu)}\frac{3H^{2}}{(a/a_{R})^{4}}=\rho^{\rm cl}_{\rm
bg}+\rho_{\rm\phi,finite},$ (103)

where the finite remainder from the counterterm and $\rho_{\phi,\rm div}$ are
absorbed by the scale dependent gravitational coupling, allowing us to start
imposing renormalization conditions to fix the finite parts. We do so by
determining the Newtonian constant via a measurement at some energy scale
$\mu_{*}$, with which we can eliminate all reference to $G_{B}$ and $B_{0}$
via

$\frac{1}{8\pi G_{B}}=\frac{1}{8\pi
G_{N}(\mu_{*})}+B_{0}+\frac{H^{2}}{24\pi^{2}}(1+e^{-4\mathcal{N}_{\rm
tot}})\left\\{1-\gamma_{E}+\log\left(\frac{\mu_{*}}{H}\right)\right\\},$ (104)

and substituting the result into Eq. 102 to obtain

$\frac{1}{8\pi G_{N}(\mu)}=\frac{1}{8\pi
G_{N}(\mu_{*})}-\frac{H^{2}}{24\pi^{2}}(1+e^{-4\mathcal{N}_{\rm
tot}})\log\left(\frac{\mu}{\mu_{*}}\right).$ (105)

Picking $\mu_{*}$ to be some scale where we have determined Newton’s
constant333333Note that measuring the strength of the gravitational coupling
can only be done via a Cavendish type experiment, typically done at laboratory
scales where we have independent knowledge of the masses whose mutual
gravitational force we can determine. This yet another manner in which gravity
is distinguished among forces as the only force whose coupling strength we
measure in the UV (i.e. mm scale) and run into the IR, rather than the other
way around. to be $8\pi G_{N}(\mu_{*})=M_{\rm pl}^{-2}$ where $M_{\rm
pl}=2.435\times 10^{18}$ GeV, we can finally express $G_{N}(\mu)$ as

$8\pi G_{N}(\mu)=\frac{1}{M_{\rm
pl}^{2}}\left[1-\frac{H^{2}(1+e^{-4\mathcal{N}_{\rm tot}})}{24\pi^{2}M_{\rm
pl}^{2}}\log\left(\frac{\mu}{\mu_{*}}\right)\right]^{-1},$ (106)

which can be used to express Eq. 103 in its fully covariant form as

$G_{\mu\nu}=\frac{1}{M_{\rm pl}^{2}}T^{\rm
bg,shift}_{\mu\nu}\left[1-\frac{H^{2}(1+e^{-4\mathcal{N}_{\rm
tot}})}{24\pi^{2}M_{\rm
pl}^{2}}\log\left(\frac{\mu}{\mu_{*}}\right)\right]^{-1},$ (107)

where the shifted background stress tensor is defined as the sum of tree level
and finite contributions on the left hand side of Eq. 103. Several things are
to be immediately noted here – foremost is the minuscule nature of the scale
dependence of the gravitational coupling should we phrase it that way. We
could also simply view it as a multiplicative renormalization of the
background matter content that sources the expansion history343434This
interpretation is to be preferred if one would like keep the graviton to be
canonically normalized throughout all of cosmic history, something that is
implicitly taken for granted for most quoted observational results.. Given
that virtual effects from test scalar fields serve only to renormalize
background quantities353535Noting that given the vanishing background energy
density of the test scalar field, it contributes vanishingly to the curvature
perturbation. and impart scale dependence in observables associated to other
propagating degrees of freedom (and that too, in a highly suppressed manner
[63, 64, 65]), this is the furthest we can take this exercise. The situation
for gravitational waves is more interesting.

Retracing the steps above with the dimensionally regularized result for
gravitational waves on a finite duration background in Eq. 119, we end up with
the renormalized background

$G_{\mu\nu}=\frac{1}{M_{\rm pl}^{2}}T^{\rm
bg,shift}_{\mu\nu}\left[1-\frac{H^{2}(1+e^{-4\mathcal{N}_{\rm
tot}})}{12\pi^{2}M_{\rm
pl}^{2}}\log\left(\frac{\mu}{\mu_{*}}\right)\right]^{-1},$ (108)

which should be the starting point for determining any constraints on vacuum
sourced primordial gravitational waves from $N_{\rm eff}$ bounds. The latter
is in essence the question of how vacuum tensor perturbations renormalize the
background expansion through $1/a^{4}$ contributions that mimic additional
relativistic species. Unlike the case for virtual test scalars, however,
gravitational waves have a classically evolving background upon which they
represent perturbations – the background geometry itself. Therefore, we have
additional means to potentially measure the contributions from vacuum tensor
modes.

We first reconsider the spectral density for gravitational waves $\Omega^{\rm
gw}(k,\tau)$, whose amplitude on the scale invariant plateau for sub-horizon
modes is well defined, and given by

$\Omega^{\rm gw}(k,\tau)=\frac{H^{4}}{16\pi^{4}(a/a_{\rm
R})^{4}}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm(sub-horizon)}$ (109)

as plotted in Fig. 3. Let us presume that we have the means to determine the
ratio $H^{2}/M_{\rm pl}^{2}$ during inflation via measurement of the tensor to
scalar ratio at some pivot scale via B-mode anisotropy observations, or via
the measurement of the spectral density of the stochastic gravitational wave
background at some fixed scale via interferometeric means, or both363636In
practice, both B-mode polarization measurements and the detection of
stochastic backgrounds through interferometry measure momentum transfer of
gravitational waves and not the energy density associated with them.
Consequently the relevant quantity one can infer by such measurements would be
the analog of the Poynting vector associated with a flux of gravitational
waves, which would nevertheless still be sensitive to the ratio $H^{2}/M_{\rm
pl}^{2}$.. In the context of Eq. 108, which can now be re-expressed as

$\displaystyle\frac{3H^{2}}{(a/a_{R})^{4}}$ $\displaystyle=$
$\displaystyle\frac{1}{M_{\rm pl}^{2}}\left(\rho_{\rm bg}^{\rm cl}+\rho_{\rm
gw,finite}\right)\left[1-\frac{H^{2}(1+e^{-4\mathcal{N}_{\rm
tot}})}{12\pi^{2}M_{\rm
pl}^{2}}\log\left(\frac{\mu}{\mu_{*}}\right)\right]^{-1}$ $\displaystyle=$
$\displaystyle\frac{\rho_{\rm bg}^{\rm cl}}{M_{\rm pl}^{2}}\left(1+\delta_{\rm
r}\right)\left[1-\frac{H^{2}(1+e^{-4\mathcal{N}_{\rm tot}})}{12\pi^{2}M_{\rm
pl}^{2}}\log\left(\frac{\mu}{\mu_{*}}\right)\right]^{-1},$

where $\delta_{\rm r}$ is a constant since the putative tree level background
and the stochastic background of vacuum tensor modes both scale as
$\propto(a/a_{\rm R})^{4}$. Therefore, recalling that the quantities that
parameterized the background on the left hand side of Eq. 96 were also
implictly bare quantities, one finds that the shifted tadpole condition one
obtains upon renormalization is indistinguishable from a rescaling of the
scale factor normalization at reheating, or to simple shift the temperature
redshift relation in an otherwise unobservable manner373737We should also
stress that although we kept the scale dependent factor in Eq. V for
completeness, it can effectively be set to unity given the current upper
bounds on the tensor to scalar ratio $r_{*}$ or about $r_{*}\lesssim 3\times
10^{-2}$ [98, 99, 100], so that $H^{2}/(8\pi^{2}M_{\rm
pl}^{2})=\frac{r_{*}}{16}\Delta^{2}_{\mathcal{R}}\lesssim 10^{-12},$ in
combination with the fact that the log of the ratio between laboratory and
Hubble scales is no more than order $10^{2}$..

Therefore, vacuum tensor fluctuations by their very nature only serve to
renormalize background quantities, and do not enter as an additional effective
light species as registered by $N_{\rm eff}$ bounds. This should be
immediately apparent from the physical nature of $N_{\rm eff}$ bounds as
measuring the ratio of propagating light species that have undergone freeze-
out relative to the entropy density of the universe, which does not apply to
vacuum fluctuations. This is of course, not true for gravitons that are
physically produced by some mechanism in the early universe, however the
latter will also feature a bounded integrated spectral density.

##  VI Concluding remarks

Divergences in primordial observables are not something cosmologists have the
luxury of ignoring: every cosmological tracer corresponding to a density
fluctuation samples and convolves the coincident limit of a field bilinear,
necessitating subtraction. Merely regularizing, however, is not enough. The
process of arriving at a physical observable is incomplete unless one follows
through by constructing the requisite counterterms and fixing any finite
contributions that could accompany any subtraction via the imposition of
renormalization conditions. Failure to do so runs the risk of drawing
unphysical conclusions that include scheme (e.g. cutoff) dependence in
observables where there should not be any, or over-interpreting contributions
to physical observables that are absorbed or otherwise accounted for in the
process of renormalization.

Fortunately, the regularization and renormalization of divergences on
cosmological backgrounds is a well understood if often elided process. In
following through the details of this procedure at some length, we hope to
have guided the reader through how this works in practice in a foliation
specific formalism that should be familiar to most cosmologists, uncovering
various novelties along the way. Not only have we elaborated on how to make
sense of what one might obtain from putting hard cutoffs in physical momenta
relative to what one might have obtained from dimensional regularization, we
provided demonstrations of the scheme independence of logarithmic divergences
and explicitly tracked this process on backgrounds that have a beginning and
end to inflation. Certain nominally IR divergent quantities on backgrounds
that model inflation as past infinite dS space are cured when one considers
finite duration inflation. Whether this generalizes beyond the simple examples
elaborated upon this review to interacting theories seems like a highly
pertinent question to follow up on. We furthermore stressed the need to work
with an improved stress tensor for gravitational waves that does not presume a
prior scale separation in the context of cosmology, and demonstrated how the
process of attempting to extract $N_{\rm eff}$ bounds from vacuum tensor
fluctuations is inextricable from the process of background renormalization.

###### Acknowledgements.

We are grateful to Paolo Benincasa and Koenraad Schalm for many discussions
and comments on the draft. We are especially indebted to Cliff Burgess for
valuable discussions and insights that permeate this review, and to Alessandro
Bettini for his patience with the manuscript. AN is grateful to Dimitrios
Krommydas and Alice Garoffolo for the useful discussions, and the Casimir
Research School for travel funds to visit the Perimeter Institute, where some
of the work leading to this review was initiated.

##  Appendix A  Dimensional regularization of stress tensors in FRLW

In this appendix, we show how dimensional regularization renders covariant
counterterms for the divergences associated with the stress energy tensors for
scalar fields and tensor perturbations in a foliation specific formalism. We
first consider the example of a massive scalar field on an FRLW background,
and consider the analogs of Eqs. 47 and 48 in cosmic time where we furthermore
neglect the effects of background expansion. This is in order to be facilitate
working with transparent analytic expressions that moreover, are non-vanishing
(as would be the case for a massless scalar field), with the further
justification that as we are only interested in computing the relevant
counterterms to subtract UV divergences and can thus be forgiven for this
approximation for illustrative purposes. Details of how one dimensional
regularizes energy momentum tensors for more realistic examples in a fully
covariant approach can be found in e.g. [3, 4]. What follows below closely
tracks the treatment of [101, 102].

Working in $D=4-\delta$ dimensions, where the background metric is of the FRLW
form in cosmic time, one finds the following expressions for the vacuum
expectation values of the stress tensor:

$\displaystyle\left\langle 0\left|\hat{T}_{00}\right|0\right\rangle$
$\displaystyle=\frac{\mu^{4-D}}{2a^{D-1}(2\pi)^{D-1}}\int\mathrm{d}^{D-1}k\sqrt{m^{2}+\frac{k^{2}}{a^{2}}},$
(111) $\displaystyle\left\langle 0\left|\hat{T}_{ii}\right|0\right\rangle$
$\displaystyle=\frac{\mu^{4-D}}{2a^{D-1}(2\pi)^{D-1}}\frac{1}{D-1}\int\mathrm{d}^{D-1}k\frac{k^{2}}{\sqrt{m^{2}+\frac{k^{2}}{a^{2}}}},$

both components of which nominally have the same degree of divergence in the
UV, but with different coefficients. Imposing hard cutoffs in physical momenta
would necessitate a counterterm that cannot be constructed from background
geometric invariants. Instead, we proceed via Eq. 18, with $A=0$,
$B=-\frac{1}{2}$ in the first integral above, and $A=1$, $B=\frac{1}{2}$ for
the second integral, to find

$\displaystyle\int\mathrm{d}^{D-1}k\sqrt{m^{2}+\frac{k^{2}}{a^{2}}}=\frac{\left(ma\right)^{D}}{a}\frac{(2\pi)^{D-1}}{(4\pi)^{\frac{D-1}{2}}}\frac{\Gamma\left(\frac{D-1}{2}\right)\Gamma\left(-\frac{D}{2}\right)}{\Gamma\left(-\frac{1}{2}\right)\Gamma\left(\frac{D-1}{2}\right)},$
(112)
$\displaystyle\int\mathrm{d}^{D-1}k\frac{k^{2}}{\sqrt{m^{2}+\frac{k^{2}}{a^{2}}}}=a\left(ma\right)^{D}\frac{(2\pi)^{D-1}}{(4\pi)^{\frac{D-1}{2}}}\frac{\Gamma\left(\frac{1}{2}+\frac{D}{2}\right)\Gamma\left(-\frac{D}{2}\right)}{\Gamma\left(\frac{1}{2}\right)\Gamma\left(\frac{D-1}{2}\right)},$

so that the dimensionally regularized components of the stress energy tensor
become

$\displaystyle\left\langle 0\left|\hat{T}_{00}\right|0\right\rangle$
$\displaystyle=\frac{\mu^{4-D}}{2a^{D-1}(2\pi)^{D-1}}\frac{\left(ma\right)^{D}}{a}\frac{(2\pi)^{D-1}}{(4\pi)^{\frac{D-1}{2}}}\frac{\Gamma\left(\frac{D-1}{2}\right)\Gamma\left(-\frac{D}{2}\right)}{\Gamma\left(-\frac{1}{2}\right)\Gamma\left(\frac{D-1}{2}\right)}$
(113)
$\displaystyle=\frac{\mu^{4-D}}{2(4\pi)^{\frac{D-1}{2}}}\frac{\left(ma\right)^{D}}{a^{D}}\frac{\Gamma\left(-\frac{D}{2}\right)}{\Gamma\left(-\frac{1}{2}\right)}$
$\displaystyle=\frac{\mu^{4}}{2(4\pi)^{\frac{D-1}{2}}}\left(\frac{m}{\mu}\right)^{D}\frac{\Gamma\left(-\frac{D}{2}\right)}{\Gamma\left(-\frac{1}{2}\right)},$

for the energy density, and

$\displaystyle\left\langle 0\left|\hat{T}_{ii}\right|0\right\rangle$
$\displaystyle=\frac{\mu^{4-D}}{2a^{D-1}(2\pi)^{D-1}}\frac{1}{D-1}a\left(ma\right)^{D}\frac{(2\pi)^{D-1}}{(4\pi)^{\frac{D-1}{2}}}\frac{\Gamma\left(\frac{1}{2}+\frac{D}{2}\right)\Gamma\left(-\frac{D}{2}\right)}{\Gamma\left(\frac{1}{2}\right)\Gamma\left(\frac{D-1}{2}\right)}$
(114)
$\displaystyle=\frac{\mu^{4-D}}{2(4\pi)^{\frac{D-1}{2}}}\frac{1}{D-1}\frac{a\left(ma\right)^{D}}{a^{D-1}}\frac{\Gamma\left(\frac{D+1}{2}\right)\Gamma\left(-\frac{D}{2}\right)}{\Gamma\left(\frac{1}{2}\right)\Gamma\left(\frac{D-1}{2}\right)}$
$\displaystyle=-\frac{\mu^{4}a^{2}}{2(4\pi)^{\frac{D-1}{2}}}\left(\frac{m}{\mu}\right)^{D}\frac{\Gamma\left(-\frac{D}{2}\right)}{\Gamma\left(-\frac{1}{2}\right)},$

for the pressure components, where in the last equality we have used
$\Gamma(x+1)=x\Gamma(x)$ to re-express

$\frac{\Gamma\left(\frac{D+1}{2}\right)}{\Gamma\left(\frac{1}{2}\right)\Gamma\left(\frac{D-1}{2}\right)}=-\frac{D-1}{\Gamma\left(-\frac{1}{2}\right)}.$
(115)

From this, we see that the stress tensor has a divergence of the form

$\langle\hat{T}_{\mu\nu}\rangle_{\rm
div}=-g_{\mu\nu}\frac{m^{4}}{64\pi^{2}}\left\\{\frac{2}{\delta}-\log\left(\frac{m^{2}}{4\pi\mu^{2}}\right)+\gamma_{E}-\frac{3}{2}\right\\},$
(116)

which can straightforwardly be subtracted by a cosmological constant-like
counterterm.

Similarly, we follow through the same logic to study the divergencies
appearing in computing the energy density and pressure of gravitational waves.
We find that even if at the operator level this is not evident, once one
regularize using dimensional regualrization the equation of state
($P=\frac{1}{3}\rho$ in the case under analysis) is satisfied and the trace of
the energy tensor is proportional to the Ricci scalar ($T_{\mu}{}^{\mu}=R=0$
in the case under analysis), consistent with subtraction with a counterterm
that comes from varying the background curvature with respect to the metric
(and thus corresponds to a renormalization of $G_{N}$, cf. the discussion
following Eq. 93).

Following the procedure decribed in Appendix D, we derive the energy density
and pressure of gravitational waves during RD era that result respectivly

$\displaystyle\rho_{\rm gw}=-T_{0}{}^{0}=$ $\displaystyle\frac{1}{8\pi
a^{2}G_{N}}\left\langle\frac{1}{8}h^{\prime}_{j}{}^{i}h^{\prime}_{i}{}^{j}-\frac{3}{8}\partial_{k}h_{ij}\partial^{k}h^{ij}-\frac{1}{2}h_{ij}\partial_{k}\partial^{k}h^{ij}+\mathcal{H}h_{j}{}^{i}h^{\prime}_{i}{}^{j}+\frac{1}{4}\partial_{k}h_{ij}\partial^{j}h^{ik}\right\rangle,$
(117) $\displaystyle P_{\rm gw}=\frac{1}{3}T_{i}{}^{i}=$
$\displaystyle\frac{1}{3}\frac{-1}{8\pi
a^{2}G_{N}}\left\langle\frac{5}{8}h^{\prime}_{j}{}^{i}h^{\prime}_{i}{}^{j}-\frac{3}{8}\partial_{k}h_{ij}\partial^{k}h^{ij}-\frac{1}{2}h_{ij}\partial_{k}\partial^{k}h^{ij}+2\mathcal{H}h_{j}{}^{i}h^{\prime}_{i}{}^{j}+h_{j}{}^{i}h^{\prime\prime}_{i}{}^{j}\right\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{3}\frac{1}{8\pi
a^{2}G_{N}}\left\langle-\frac{5}{8}h^{\prime}_{j}{}^{i}h^{\prime}_{i}{}^{j}+\frac{3}{8}\partial_{k}h_{ij}\partial^{k}h^{ij}-\frac{1}{2}h_{ij}\partial_{k}\partial^{k}h^{ij}\right\rangle$

where in the second equality of the pressure we use the EOM. From the result
above it appears that the equation of state is not satisfied. However,
specifying the result in the case of finite inflation and following the
procedure in Section IV.1, we find that by using physical cutoff and
dimensional regularization respectively, the regularized pressure results

$\displaystyle P_{\rm gw,div}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\lim_{\Lambda_{\rm
UV}\to\infty}\left\\{\frac{1}{2\pi^{2}}\frac{\Lambda_{\rm
UV}^{4}}{4}-\frac{5}{2}\frac{a_{\rm
R}^{4}H^{2}}{2\pi^{2}a^{4}}\frac{\Lambda_{\rm
UV}^{2}}{2}+\frac{H^{4}\left(1+e^{-4\mathcal{N}_{\rm
tot}}\right)}{4\pi^{2}(a/a_{\rm R})^{4}}\log\frac{\Lambda_{\rm
UV}}{\Lambda_{\rm IR}}\right\\},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm(cutoff)}$ (118) $\displaystyle
P_{\rm gw,div}$ $\displaystyle=$ $\displaystyle\frac{1}{3}\lim_{\delta_{\rm
UV}\to 0}\left\\{\frac{H^{4}\left(1+e^{-4\mathcal{N}_{\rm
tot}}\right)}{4\pi^{2}(a/a_{R})^{4}}\left[\frac{1}{\delta_{\rm
UV}}+1-\gamma_{E}+\log\left(\frac{\mu}{H}\right)\right]\right\\}.\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ {\rm(dim-reg)}$ (119)

By recalling the result found in Section IV.1 for the energy density

$\displaystyle\rho^{\rm Imp}_{\rm gw,div}$ $\displaystyle=$
$\displaystyle\lim_{\Lambda_{\rm
UV}\to\infty}\left\\{\frac{1}{2\pi^{2}}\frac{\Lambda_{\rm
UV}^{4}}{4}-\frac{7}{2}\frac{a_{\rm
R}^{4}H^{2}}{2\pi^{2}a^{4}}\frac{\Lambda_{\rm
UV}^{2}}{2}+\frac{H^{4}\left(1+e^{-4\mathcal{N}_{\rm
tot}}\right)}{4\pi^{2}(a/a_{\rm R})^{4}}\log\frac{\Lambda_{\rm
UV}}{\Lambda_{\rm IR}}\right\\},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm(cutoff)}$ (120)
$\displaystyle\rho^{\rm Imp}_{\rm gw,div}$ $\displaystyle=$
$\displaystyle\lim_{\delta_{\rm UV}\to
0}\left\\{\frac{H^{4}\left(1+e^{-4\mathcal{N}_{\rm
tot}}\right)}{4\pi^{2}(a/a_{R})^{4}}\left[\frac{1}{\delta_{\rm
UV}}+1-\gamma_{E}+\log\left(\frac{\mu}{H}\right)\right]\right\\}.\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ {\rm(dim-reg)}$ (121)

we can see that only the result regularized using dimensional regularization
gives a traceless stress energy tensor and satisfies the equation of state for
radiation-like species.

##  Appendix B  Oscillatory contributions to divergent integrals

In deriving various expressions in Section III.2 and beyond, we neglected
oscillatory contributions coming from $\alpha^{\rm R*}_{k}\beta^{\rm R}_{k}$
as well as in computing the modulus square of $|\beta_{k}^{\rm R}|^{2}$ in
final expressions for the UV divergences we are left to regulate, such as Eqs.
62 and 65. In this appendix we show that this is justified focusing on the
details for the scalar case, with the extension to analogous computations for
gravitational waves a straightforward extension. We reconsider Eq. 65 but with
the neglected oscillatory contributions included:

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi^{4}a^{4}}\int_{-\infty}^{\infty}\,\frac{d^{4}k}{k}\left[k+\frac{a_{\rm
R}^{4}H^{2}}{a^{2}2k}\right]\left\\{1+\frac{a_{\rm
R}^{4}H^{4}}{2k^{4}}+2\left|\beta_{k}^{\rm I}\right|^{2}\left(1+\frac{a_{\rm
R}^{4}H^{4}}{2k^{4}}\right)\right\\}+\rho_{\rm osc},$ (122)

where $\rho_{\rm osc}$ is given by

$\displaystyle\rho_{\rm osc}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}a^{4}}\int_{0}^{\infty}\,dk\>k^{2}\left(k+\frac{a_{\rm
R}^{4}H^{2}}{a^{2}2k}\right)\left[-i\alpha_{k}^{\rm I}\beta_{k}^{\rm
I*}e^{2i\frac{k}{a_{\rm R}H}}\frac{a_{\rm
R}^{2}H^{2}}{k^{2}}\left(-i+\frac{a_{\rm R}H}{k}+i\frac{a_{\rm
R}^{2}H^{2}}{2k^{2}}\right)\right.$ $\displaystyle\left.+i\alpha_{k}^{\rm
I*}\beta_{k}^{\rm I}e^{-2i\frac{k}{a_{\rm R}H}}\frac{a_{\rm
R}^{2}H^{2}}{k^{2}}\left(i+\frac{a_{\rm R}H}{k}-i\frac{a_{\rm
R}^{2}H^{2}}{2k^{2}}\right)\right]$
$\displaystyle+\frac{1}{4\pi^{2}a^{4}}\int_{0}^{\infty}\,dk\>k^{2}\left[\alpha^{\rm
R}_{k}\beta^{\rm R*}_{k}e^{-2ik\tau_{\rm R}\left(2-\frac{a}{a_{\rm
R}}\right)}\left(\frac{k}{2}+\frac{a_{\rm
R}^{4}H^{2}}{a^{2}2k}\left(1+i\frac{ka}{a_{\rm
R}^{2}H}\right)^{2}\right)\right.$ $\displaystyle+\left.\alpha^{\rm
R*}_{k}\beta^{\rm R}e^{2ik\tau_{\rm R}\left(2-\frac{a}{a_{\rm
R}}\right)}\left(\frac{k}{2}+\frac{a_{\rm
R}^{4}H^{2}}{a^{2}2k}\left(1-i\frac{ka}{a_{\rm
R}^{2}H}\right)^{2}\right)\right].$

Eq. 65 presumes that $\rho_{\rm osc}$ renders vanishing contributions. Making
use of Eqs. 57 and III.2, one can integrate the various terms and expand the
result in the limit $k\to\infty$, upon which we obtain

$\displaystyle\rho_{\rm osc}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}a^{4}}\lim_{k\to\infty}\left[\left(e^{\left(-\frac{2ik(aa_{\rm
I}-2a_{\rm I}a_{\rm R}+a_{\rm R}^{2})}{Ha_{\rm I}a_{\rm
R}^{2}}\right)}+e^{\left(\frac{2ik(aa_{\rm I}2a_{\rm I}a_{\rm R}+a_{\rm
R}^{2})}{Ha_{\rm I}a_{\rm R}^{2}}\right)}\right)\left(\frac{H^{4}a_{\rm
I}^{3}a_{\rm R}^{4}}{4a\left(aa_{\rm I}-2a_{\rm I}a_{\rm R}+a_{\rm
R}^{2}\right)}\right)\right.$
$\displaystyle\left.+\left(e^{\left(-\frac{2ik(a_{\rm I}-a_{\rm R})}{Ha_{\rm
R}^{2}}\right)}+e^{\left(\frac{2ik(a_{\rm I}-a_{\rm R})}{Ha_{\rm
R}^{2}}\right)}\right)\left(-\frac{H^{4}a_{\rm R}^{6}}{4a^{2}-4aa_{\rm
R}}\right)+\mathcal{O}\left(\frac{1}{k}\right)\right].$

We note that this contribution is manifestly finite, and oscillatory in the
upper limit in a manner that would be eliminated via the analog of the
$i\epsilon$ prescription for the Minkowski space limit of all the two point
propagators in the in-in formalism. Similarly, for the tensor perturbations,
the full expression for the energy density including the oscillatory
contributions is given for the Isaacson form of the stress tensor Eq. 136, by

$\displaystyle\rho^{\rm Isc}_{\rm gw}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{4}a^{4}}\int_{-\infty}^{\infty}\,\frac{d^{4}k}{k}\left[k+\frac{a_{\rm
R}^{4}H^{2}}{a^{2}k}\right]\left\\{1+\frac{a_{\rm
R}^{4}H^{4}}{2k^{4}}+2\left|\beta_{k}^{\rm I}\right|^{2}\left(1+\frac{a_{\rm
R}^{4}H^{4}}{2k^{4}}\right)\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ ({\rm Isaacson})\right.$
$\displaystyle\left.-i\alpha_{k}^{\rm I}\beta_{k}^{\rm I*}e^{2i\frac{k}{a_{\rm
R}H}}\frac{a_{\rm R}^{2}H^{2}}{k^{2}}\left(-i+\frac{a_{\rm
R}H}{k}+i\frac{a_{\rm R}^{2}H^{2}}{2k^{2}}\right)+i\alpha_{k}^{\rm
I*}\beta_{k}^{\rm I}e^{-2i\frac{k}{a_{\rm R}H}}\frac{a_{\rm
R}^{2}H^{2}}{k^{2}}\left(i+\frac{a_{\rm R}H}{k}-i\frac{a_{\rm
R}^{2}H^{2}}{2k^{2}}\right)\right\\}$
$\displaystyle+\frac{1}{4\pi^{4}a^{4}}\int_{-\infty}^{\infty}\,\frac{d^{4}k}{k}\left(\frac{a_{\rm
R}^{4}H^{2}}{a^{2}k}\right)\left[\alpha^{\rm R}_{k}\beta^{\rm
R*}_{k}e^{-2ik\tau_{\rm R}\left(2-\frac{a}{a_{\rm
R}}\right)}\left(1+i\frac{ka}{a_{\rm R}^{2}H}\right)^{2}+\alpha^{\rm
R*}_{k}\beta^{\rm R}e^{2ik\tau_{\rm R}\left(2-\frac{a}{a_{\rm
R}}\right)}\left(1-i\frac{ka}{a_{\rm R}^{2}H}\right)^{2}\right],$

and for the improved form of the stress tensor Eq. 75, as:

$\displaystyle\rho_{\rm gw}^{\rm Imp}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{4}{a^{4}}}\int_{-\infty}^{\infty}\,\frac{d^{4}k}{k}\left[k-\frac{7a_{\rm
R}^{4}H^{2}}{a^{2}2k}\right]\left\\{1+\frac{a_{\rm
R}^{4}H^{4}}{2k^{4}}+2\left|\beta_{k}^{\rm I}\right|^{2}\left(1+\frac{a_{\rm
R}^{4}H^{4}}{2k^{4}}\right)\right.\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ ({\rm Improved})$ $\displaystyle\left.-i\alpha_{k}^{\rm
I}\beta_{k}^{\rm I*}e^{2i\frac{k}{a_{\rm R}H}}\frac{a_{\rm
R}^{2}H^{2}}{k^{2}}\left(-i+\frac{a_{\rm R}H}{k}+i\frac{a_{\rm
R}^{2}H^{2}}{2k^{2}}\right)+i\alpha_{k}^{\rm I*}\beta_{k}^{\rm
I}e^{-2i\frac{k}{a_{\rm R}H}}\frac{a_{\rm
R}^{2}H^{2}}{k^{2}}\left(i+\frac{a_{\rm R}H}{k}-i\frac{a_{\rm
R}^{2}H^{2}}{2k^{2}}\right)\right\\}$
$\displaystyle+\frac{1}{4\pi^{4}{a^{4}}}\int_{-\infty}^{\infty}\,\frac{d^{4}k}{k}\left[\alpha^{\rm
R}_{k}\beta^{\rm R*}_{k}e^{-2ik\tau_{\rm R}\left(2-\frac{a}{a_{\rm
R}}\right)}\left(\frac{k}{2}+\frac{a_{\rm
R}^{4}H^{2}}{a^{2}2k}\left(1+i\frac{ka}{a_{\rm
R}^{2}H}\right)^{2}-4\frac{a_{\rm
R}^{4}H^{2}}{a^{2}k}\left(1+i\frac{ka}{a_{\rm R}^{2}H}\right)\right)\right.$
$\displaystyle+\left.\alpha^{\rm R*}_{k}\beta^{\rm R}e^{2ik\tau_{\rm
R}\left(2-\frac{a}{a_{\rm R}}\right)}\left(\frac{k}{2}+\frac{a_{\rm
R}^{4}H^{2}}{a^{2}2k}\left(1-i\frac{ka}{a_{\rm
R}^{2}H}\right)^{2}-4\frac{a_{\rm
R}^{4}H^{2}}{a^{2}k}\left(1-i\frac{ka}{a_{\rm R}^{2}H}\right)\right)\right],$

which again results in oscillating but finite contributions in the UV-limit
that can similarly be discarded. Importantly, even though the oscillatory
contributions are negligible in the UV, they become relevant when the
oscillations freeze as $k\to 0$, also softening the IR behavior for the
spectral power density relative to what would have been on a past infinite dS
background.

##  Appendix C  Point split comparison to physical cutoff regularization

In what follows we compare the results of previous sections had we commuted
taking the coincidence limit before integrating over $\vec{k}$ to show
agreement between point split regularization and physical momentum cutoff
regularization. Recalling the steps that led up to Eq. 8, we are interested in
computing quantities that derive from the coincidence limit of the two point
function

$\displaystyle\lim_{x\to y}G(x,y)$ $\displaystyle=$ $\displaystyle\lim_{x\to
y}\langle\hat{\phi}(\tau,x)\hat{\phi}(\tau^{\prime},y)\rangle$
$\displaystyle=$ $\displaystyle\lim_{x\to
y}\int\frac{d^{3}k}{(2\pi)^{3}}\>\phi(\tau,k)\phi^{*}(\tau,k)e^{i{k}\cdot({x}-{y})},$

where in the second equality we use the Fourier transform convention of Eq. 5.
Similarly, in the derivation of the energy density for a massless test scalar
field, we are interested in computing the coincidence limit

$\displaystyle\rho(x)=\lim_{x\to y}\rho(x,y)$ $\displaystyle=$
$\displaystyle\lim_{x\to
y}\frac{1}{2a^{2}}\left[\langle\hat{\phi}^{\prime}(\tau,x)\hat{\phi}^{\prime}(\tau^{\prime},y)\rangle+\langle{\nabla}_{x}\hat{\phi}(\tau,x){\nabla}_{y}\hat{\phi}(\tau^{\prime},y)\rangle\right]$
$\displaystyle=$
$\displaystyle\lim_{{x}\to{y}}\frac{1}{2a^{2}}\int\frac{d^{3}k}{(2\pi)^{3}}\>\left[\phi^{\prime}(\tau,k)\phi^{*\prime}(\tau,k)+k^{2}\phi(\tau,k)\phi^{*}(\tau,k)\right]e^{i{k}\cdot({x}-{y})}.$

Defining $r:=|x-y|$, we now look to evaluate the following point split
quantities

$\displaystyle G(x,x)=\lim_{r\to
0}\int\frac{d^{3}k}{(2\pi)^{3}}\>\phi(\tau,k)\phi^{*}(\tau,k)e^{i{k}\cdot{r}}$
(129) $\displaystyle\rho(x)=\lim_{r\to
0}\frac{1}{2a^{2}}\int\frac{d^{3}k}{(2\pi)^{3}}\>\left[\phi^{\prime}(\tau,k)\phi^{*\prime}(\tau,k)+k^{2}\phi(\tau,k)\phi^{*}(\tau,k)\right]e^{i{k}\cdot{r}}.$

We note that in the derivations in the main body of this article, we freely
commuted the limit $\lim_{r\to 0}$ with the integral over $\vec{k}$. However,
this is strictly speaking only a well defined procedure if the integral is
absolutely convergent. In what follows, we compute the integral over $\vec{k}$
before taking the coincidence limit. We start by computing Eq. C for a non-
interacting, minimally coupled massless scalar field on a dS background,
obtaining

$\displaystyle G(x,x)$ $\displaystyle=$ $\displaystyle\lim_{r\to
0}\frac{1}{(2\pi)^{3}}\int_{0}^{\infty}dkk^{2}\int_{-1}^{1}d\cos\theta\int_{0}^{2\pi}d\varphi\>\frac{H^{2}}{2k^{3}}\left(1+k^{2}\tau^{2}\right)e^{ikr\cos\theta}$
$\displaystyle=$ $\displaystyle\lim_{r\to
0}\frac{H^{2}}{4\pi^{2}}\frac{1}{r}\int_{0}^{\infty}dk\>\left(\frac{1}{k^{2}}+\tau^{2}\right)\sin
kr$ $\displaystyle=$ $\displaystyle\lim_{r\to
0}\frac{H^{2}}{4\pi^{2}}\left(\frac{\tau^{2}}{r^{2}}-\left[\gamma_{E}-1+\log
kr\right]\right)$ $\displaystyle=$ $\displaystyle\lim_{\sigma\to
0}\frac{H^{2}}{4\pi^{2}}\left(\frac{1}{H^{2}\sigma^{2}}-\left[\gamma_{E}-1+\log
k_{p}\sigma\right]\right),$

where the third equality involves use of the principle value expression
$\int_{0}^{\infty}dx\sin x=1$, and the last equality is expressed in terms of
the physical distance $\sigma=ar$, with $k_{p}:=k/a$. As expected, Eq. C shows
that the two point function diverges in the coincident limit $\sigma\to 0$ in
the same manner as Eq. III.1 obtained by imposing hard momentum cutoffs. In a
similar manner, we can repeat the computation for the energy density given by
Eq. C, obtaining

$\displaystyle\rho_{\rm div}$ $\displaystyle=$ $\displaystyle\lim_{r\to
0}\frac{1}{(2\pi)^{3}}\frac{H^{2}}{4a^{2}}\int_{0}^{\infty}\frac{dk}{k}k^{2}\int_{-1}^{1}d\cos\theta\int_{0}^{2\pi}d\varphi\>\left(1+2k^{2}\tau^{2}\right)e^{ikr\cos\theta}$
$\displaystyle=$ $\displaystyle\lim_{r\to
0}\frac{1}{4\pi^{2}}\frac{H^{2}}{2a^{2}}\frac{1}{r}\int_{0}^{\infty}dk\>\left(1+2k^{2}\tau^{2}\right)\sin
kr$ $\displaystyle=$ $\displaystyle\lim_{r\to
0}\frac{H^{2}}{2a^{2}}\frac{1}{4\pi^{2}r^{2}}\left[1-\frac{4\tau^{2}}{r^{2}}\right]$
$\displaystyle=$ $\displaystyle\lim_{\sigma\to
0}\frac{H^{2}}{8\pi^{2}}\frac{1}{\sigma^{2}}\left[1-\frac{4}{H^{2}\sigma^{2}}\right],$

where going from the second to the third line involves integrating by parts
and again using the principal value scheme. Again, we reproduce the
divergences of Eq. 49 obtained by hard cutoffs in physical momenta, up to the
expected scheme dependence which as we will show shortly, which drops out when
computing the coefficients of logarithmic divergences. The comparison we are
most interested in exhibiting this, however, is for the the case of finite
duration of inflation. We exhibit the procedure for the energy density given
that the treatment for the two point function proceeds identically. Using the
results of Eqs. 54 and 65 in the context of Eq. 129, we find

$\displaystyle\rho_{\rm div}$ $\displaystyle=$ $\displaystyle\lim_{r\to
0}\frac{1}{4\pi^{2}a^{4}}\frac{1}{r}\int_{0}^{\infty}dkk^{2}\left(1+\frac{a_{\rm
R}^{4}H^{2}}{2a^{2}k^{2}}+\frac{a_{\rm R}^{4}H^{4}+a_{\rm
R}^{4}H^{4}}{2k^{4}}\right)\sin kr$ $\displaystyle=$ $\displaystyle\lim_{r\to
0}\left[\frac{1}{4\pi^{2}a^{4}}\frac{1}{r}\int_{0}^{\infty}dkk^{2}\sin
kr+\frac{a_{\rm R}^{4}H^{2}}{8\pi^{2}a^{6}}\frac{1}{r}\int_{0}^{\infty}dk\sin
kr+\frac{1}{4\pi^{2}a^{4}}\frac{1}{r}\int_{0}^{\infty}dk\left(\frac{a_{\rm
R}^{4}H^{4}+a_{\rm I}^{4}H^{4}}{2k^{2}}\right)\sin kr\right].$

Changing integration variables to $q=kr$, one can identify the terms that
survive in the $r\to 0$ limit, given by

$\displaystyle\rho_{\rm div}$ $\displaystyle=$ $\displaystyle\lim_{r\to
0}\left[\frac{1}{4\pi^{2}a^{4}}\frac{1}{r^{4}}\int_{0}^{\infty}dq\>q^{2}\sin
q+\frac{a_{\rm
R}^{4}H^{2}}{8\pi^{2}a^{6}}\frac{1}{r^{2}}\int_{0}^{\infty}dq\>\sin
q+\frac{a_{\rm R}^{4}H^{4}}{4\pi^{2}a^{4}}\left(1+\frac{a^{4}_{\rm
I}}{a^{4}_{\rm R}}\right)\int_{0}^{\infty}\frac{dq}{2q^{2}}\sin q\right]$
$\displaystyle=$ $\displaystyle\lim_{r\to
0}\left[-\frac{2}{4\pi^{2}a^{4}}\frac{1}{r^{4}}+\frac{a_{\rm
R}^{4}H^{2}}{8\pi^{2}a^{6}}\frac{1}{r^{2}}+\frac{a_{\rm