EFT for de Sitter Space Daniel Green Abstract The physics of de Sitter space is essential to our understanding of our cos-

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EFT for de Sitter Space

Daniel Green

Abstract The physics of de Sitter space is essential to our understanding of our cos-

mological past, present, and future. It forms the foundation for the statistical predic-

tions of inﬂation in terms of quantum vacuum ﬂuctuations that are being tested with

cosmic surveys. In addition, the current expansion of the universe is dominated by

an apparently constant vacuum energy and we again ﬁnd our universe described by a

de Sitter epoch. Despite the success of our predictions for cosmological observables,

conceptual questions of the nature of de Sitter abound and are exacerbated by tech-

nical challenges in quantum ﬁeld theory and perturbative quantum gravity in curved

backgrounds. In recent years, signiﬁcant process has been made using effective ﬁeld

theory techniques to tame these breakdowns of perturbation theory. We will discuss

how to understand the long-wavelength ﬂuctuations produced by accelerating cos-

mological backgrounds and how to resolve both the UV and IR obstacles that arise.

Divergences at long wavelengths are resummed by renormalization group (RG) ﬂow

in the EFT. For light scalar ﬁelds, the RG ﬂow manifests itself as the stochastic in-

ﬂation formalism. In single-ﬁeld inﬂation, long-wavelength metric ﬂuctuations are

conserved outside the horizon to all-loop order, which can be understood easily in

EFT terms from power counting and symmetries.

Keywords

Effective Field Theory, de Sitter space, Inﬂation, Cosmology

Daniel Green

UC San Diego, La Jolla, USA, e-mail: drgreen@physics.ucsd.edu

arXiv:2210.05820v1 [hep-th] 11 Oct 2022

2 Daniel Green

1 Introduction

The physics of de Sitter space forms an important pillar in our understanding of

the universe. Structure in the universe is widely believed to have originated in the

distance past during an approximately de Sitter phase, inﬂation, where the tools of

quantum ﬁeld theory (QFT) in curved space are essential for computing the statis-

tical predictions of inﬂation. While tree-level calculations are in precise agreement

with observational data, the theoretical foundations of these calculations are poorly

developed compared to their analogues in ﬂat space. These complications persist in

our in attempts to understand the universe at late-times. Observations of the expan-

sion of the universe at low redshifts are consistent with the existence of a new phase

de Sitter-like expansion. Our understanding of the universe in the current epoch is

limited by the same technical complications as inﬂation and by a number of open

questions about the origin and fate of the small non-zero vacuum energy.

In classical general relativity, de Sitter (dS) space presents few mysteries [89].

It is a maximally symmetric solution to Einstein’s equations, exhibiting a SO(d,1)

group of isometries in dspacetime dimensions. For understanding out own universe,

we often focus on the patch of de Sitter described in terms of an expanding FRW

solution,

ds2=gdS

µν dxµdxν=−dt2+a(t)2d~x2=a(τ)2(−dτ2+d~x2),(1)

where a(t) = eHt or a(τ)H=−1/τ.

The quantum nature of de Sitter space is signiﬁcantly more complicated, even

for non-interacting quantum ﬁelds. The litany of conceptual and technical chal-

lenges begin with the ultraviolet (UV) limit, where the so-called trans-Planckian

problem [22,91,23] suggests that even our choice of vacuum might be sensitive

to Planck-scale physics. The questions persist as we move to long distances, where

infrared (IR) divergences and secular growth challenge our notion of a perturbative

expansion [44,8,96,97,83,49,24,79,67,60,1]. This problem is compounded

at higher-loop order where our treatment of the UV and IR regimes manifests itself

as additional divergences in loop diagrams. The predictions for the scalar metric

ﬂuctuations that sourced structure in the universe are sensitive to how we treat these

regimes; therefore, our belief that inﬂation is consistent with observations is depen-

dent on a satisfactory resolution to these challenges.

Fortunately, effective ﬁeld theory (EFT) [77,47,65,30] provides a variety of

tools to understand both the short and long distance behavior of de Sitter space.

Fundamentally, the difﬁculty with cosmological backgrounds is that UV physics

evolves to the IR through the expansion of the universe. This process does not vi-

olate the fundamental principles of EFT, like the decoupling of scales, but it does

make their manifestation less transparent. Of course, we should expect these prin-

ciples to hold in dS, given that our universe is currently in a dS-like phase and we

use EFT successfully to describe the world around us. Yet, it remains challenging to

make sense of decoupling in contexts, like cosmological particle production, where

the curvature of spacetime is essential to the physical process. Our goal in this chap-

EFT for de Sitter Space 3

ter is to demonstrate that QFT and perturbative quantum gravity in dS can be recast

in the familiar language of EFT where the resolutions to many of these problems

have ready-made solutions.

The most basic insight that underlies the success of EFT is that the physics de

Sitter is characterized by a single energy scale, H, the Hubble scale. First and fore-

most, the blue-shifting of modes as we evolve backwards in time does not negatively

impact perturbation theory (in the Bunch-Davies vacuum, or a ﬁnite energy excita-

tion thereof) because physical processes do not occur at these high energies. In-

stead, the energy scale associated with particle production is H, such that Planckian

physics is exponentially suppressed when HMpl, where Mpl is the reduced Planck

mass. This observation is well-known, especially in inﬂationary phenomenology,

where the amplitude of primordial non-Gaussianity is usually determined by power

counting in H[29]. The implications are less obvious in loop diagrams, where one

encounters logarthmic divergences that need to be regulated. However, if one de-

ﬁnes the strength of couplings at the Hubble scale, the familiar RG from ﬂat space

is unnecessary and all such logarithmic terms vanish. Many of these observations

may even seem self-evident but can become obscured without effective regulators

in dS [84].

The second, and less obvious, outcome of the EFT approach to de Sitter [31], is

that the super-horizon evolution of ﬁelds and composite operators can be organized

by explicit power counting in powers of k/(aH)1. In the process, the degrees

of freedom are redeﬁned according to their scaling dimension in k, and time evolu-

tion is recast in the language of dynamical RG ﬂow, so that the powers of k/(aH)

that appear follow from dimensional analysis. The EFT does not contain any rele-

vant operators (in the RG sense), reproducing the long known result that corrections

grow at most logarithmically [104,105]. Logarithmic terms can be understood as

operator mixing, while irrelevant terms decouple at late times. For massless scalars,

an inﬁnite number of operators can mix, giving rise to the framework of stochas-

tic inﬂation as the master equation for this RG ﬂow. For metric ﬂuctuations, the

all-orders conservation of the adiabatic and tensor metric ﬂuctuations follow from

power counting, as the dimensions of these operators are ﬁxed by symmetries and

cannot be modiﬁed by RG.

The results in this chapter will be presented from the point of view of EFT, partic-

ularly Soft de Sitter Effective Theory (SdSET) [31], applied to (in-in) cosmological

correlators of scalar ﬁelds in ﬁxed dS and metric ﬂuctuations in single-ﬁeld inﬂa-

tion. Many of the key results have been or can be derived from different perspec-

tives, including conventional perturbation theory [16,17], the wavefunction of the

universe [50], and/or the physics of the static patch [69,70]. SdSET has the unique

advantage that many non-trivial results when explained in terms of diagrams of the

original theory, become simple observations about dimensional analysis within the

EFT. In addition, hard to interpret IR divergences in the full theory are traded for UV

divergences in the EFT where they have a standard interpretation in terms of RG.

Our emphasis on SdSET is similar to the role of the exact RG and EFT in Polchin-

ski’s proof of renormalizability of λ φ4in ﬂat space [76]; although one can reach

the main result by diagrammatic arguments [102], Polchinski’s exact RG makes the

4 Daniel Green

conceptual meaning of the result transparent, and generalizes it to other theories.

Our point of view in this chapter, as with many presentations of EFT, is that we

will only claim to have fully understood a phenomona when it can be explained by

symmetries and power counting.

This chapter will be organized as follows: In Section 2, we will discuss dS as an

EFT where the relevant energy scale is the expansion rate HMpl. Perturbation

theory will be controlled by the small size of the expansion rate in a predictable

way. We will speciﬁcally show how there is no trans-Planckian problem [22,91,23]

unless we give up the idea that short distance physics of de Sitter is similar to ﬂat

space (which would also contradict everyday experience). In Section 3, we discuss

how to understand inﬂationary and dS backgrounds on scales much larger than the

size of the cosmological horizon, H−1. These are the scales that give rise to IR di-

vergences and secular growth in traditional approaches to perturbation theory. We

will introduce the SdSET to handle this regime and we will see that the IR diver-

gences of the full theory are replaced with EFT UV divergences, so that they can be

resummed via renormalization group (RG) ﬂow, following the usual EFT playbook.

In Section 4, we apply these results to massless scalars and show how stochastic

inﬂation arises from operator mixing in SdSET. In Section 5, we then explain how

the all-orders conservation of the metric follows from power counting and discuss

some implications for slow-roll eternal inﬂation. We conclude in Section 6.

2 Effective Theory in de Sitter

All discussions of de Sitter space start from the central premise that the curvature

of dS is small in Planck units or HMpl. This ensures that our classical solution

for the background is under control and can be described geometrically, up to small

perturbations. Without such an assumption, there is no controlled background ge-

ometry in which to discuss quantum ﬁelds or metric ﬂuctuations. However, even

with this assumption, it is still not necessarily obvious that quantum gravitational

effects are always suppressed by at least (H/Mpl)2.

The most unambiguous way to deﬁne the energies relevant to a given process is to

calculate an observable quantity sensitive to the physical energy scale. To simplify

the discussion, we will mostly consider the case of a scalar ﬁeld φof mass mand

action (in the FRW slicing) ,

S=Zdtd3x a3(t)1

φ2−1

a(t)2∂iφ ∂ iφ−m2φ2,(2)

where the spatial indices are raised with the Kronecker δi j. Following standard

canonical quantization, we decompose the ﬁeld in terms of fourier modes, ~

k, ac-

cording to

EFT for de Sitter Space 5

φ(~x,τ) = Zd3k

(2π)3ei~

k·~x¯

φ∗~

k,τa†

k+¯

φ~

k,τa−~

k,(3)

where

φ~

k,τ=−iei(ν+1

2)π

2√π

2H(−τ)3/2H(1)

ν(−kτ),(4)

is a solution to the classical equations of motion, where ν=q9

4−m2

H2and H(1)

νis the

Hankel function of the ﬁrst kind. Next, we promote a†

kand a

kto quantum mechanical

operators that act on the vacuum state. The choice of vacuum is often presented as an

ambiguity unique to de Sitter; however, to be consistent with physical expectations

and experience in our own universe, we will require that when wavelength of the

modes is subhorizon, kaH, we reproduce the vacuum of ﬂat space. Speciﬁcally,

this means that as τ→−∞,φshould behave as a ﬁeld operator in ﬂat space, namely

that a†

kcreates a particle from the vacuum while a

kannihilates the vacuum. We can

see that Equation (4) is a negative frequency mode, as needed, by expanding the

Hankel function in τ→ −∞we ﬁnd

φ→ −iH(−τ)

√2kexp(−ikτ).(5)

This takes the form of a WKB solution for negative frequency mode if we identify

the physical (WKB) frequency as

ωphysical(t) = k

a(t)→Zt

dt0ω(t0) = Zt

dt0k

a(t0)=−k

a(t)H=kτ.(6)

In this sense, using the canonical commutation relation for a

kand a†

a†

k,a

k0= (2π)3δ~

k−~

k0a

k|0i=h0|a†

k=0,(7)

reproduced ﬂat space physics on short distances where kaH. We may choose

other states, corresponding to excitations of the ﬂat space vacuum. For most such

choices, the energy density also breaks the de Sitter symmetry and redshifts away

through the expansion of the universe. The exceptions are the α-vacua [4,71], which

are de Sitter invariant but correspond to inﬁnite energy conﬁgurations from the ﬂat

space perspective and may be ill-deﬁned when including interactions [12]. Arguably

the most important non-trivial excited state for the purpose of cosmology is a time-

dependent background for the scalar ﬁeld, φ(~x,t) = φ0(t). A case of particular in-

terest is inﬂation [13], where the energy density of the time evolution is well above

the Hubble scale, ˙

φ2

0H4.

For cosmological applications (see e.g. [13]), we are particularly interested in the

case of massless scalars, m2=0 (ν=3/2) where

φ~

k,τ→H

√2k3(1−ikτ)eikτ,(8)

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EFTfordeSitterSpaceDanielGreenAbstractThephysicsofdeSitterspaceisessentialtoourunderstandingofourcos-mologicalpast,present,andfuture.Itformsthefoundationforthestatisticalpredic-tionsofinationintermsofquantumvacuumuctuationsthatarebeingtestedwithcosmicsurveys.Inaddition,thecurrentexpansionoftheuniv...

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