1

23. Inflation

23. Inflation
Revised August 2023 by J. Ellis (King’s Coll. London; CERN) and D. Wands (Portsmouth U.).

23.1 Motivation and Introduction
The standard Big-Bang model of cosmology provides a successful framework in which to understand the thermal history of our Universe and the growth of cosmic structure, but it is essentially
incomplete. As described in Sec. 22.2.4, Big-Bang cosmology requires very specific initial conditions. It postulates a uniform cosmological background, described by a spatially-flat, homogeneousand isotropic Robertson-Walker (RW) metric (Eq. (22.1) in “Big Bang Cosmology” review), withscale factor R(t). Within this setting, it also requires an initial almost scale-invariant distribution ofprimordial density perturbations as seen, for example, in the cosmic microwave background (CMB)
radiation (described in Chap. 29, “Cosmic Microwave Background” review), on scales far larger
than the causal horizon at the time the CMB photons last scattered.
The Hubble expansion rate, H ≡ Ṙ/R, in a RW cosmology is given by the Friedmann constraint
equation (Eq. (22.8) in “Big Bang Cosmology” review)
H2 =

8πρ
k
Λ
+ − 2,
2
3
R
3MP(23.1)

where k/R2 is the intrinsic spatial curvature. We use natural units such that the speed of light−1/2
c = 1 and hence we have the Planck mass MP = GN ' 1019 GeV (see “Astrophysical Constants
and Parameters”). A cosmological constant, Λ, of the magnitude required to accelerate the Universe today (see Chap. 28, “Dark Energy” review) would have been completely negligible in theearly Universe where the energy density ρ  MP2 Λ ∼ 10−12 (eV)4 . The standard early Universe
cosmology, described in Sec. 22.1.5 in “Big Bang Cosmology” review, is thus dominated by nonrelativistic matter (pm  ρm ) or radiation (pr = ρr /3 for an isotropic distribution). This leads toa decelerating expansion with R̈ < 0.
The hypothesis of inflation [1,2] postulates a period of accelerated expansion, R̈ > 0, in the very
early Universe, preceding the standard radiation-dominated era, which offers a physical model forthe origin of these initial conditions, as reviewed in [3–7]. Such a period of accelerated expansion(i) drives a curved Robertson-Walker spacetime (with spherical or hyperbolic spatial geometry)
towards spatial flatness, and (ii) it also expands the causal horizon beyond the present Hubble
length, so as to encompass all the scales relevant to describe the large-scale structure observed in
our Universe today, via the following two mechanisms.
1. A spatially-flat Universe with vanishing spatial curvature, k = 0, has the dimensionless density
parameter Ωtot = 1, where we define (Eq. (22.13) in “Big Bang Cosmology” review; see Chap.
25, “Cosmological Parameters” review for more complete definitions)Ωtot ≡

8πρtot,3MP2 H 2(23.2)

with ρtot ≡ ρ + ΛMP2 /8π. If we re-write the Friedmann constraint (Eq. (23.1)) in terms ofΩtot we havek1 − Ωtot = − 2 .
(23.3)
Ṙ
Observations require |1 − Ωtot,0 | < 0.005 today [8], where the subscript 0 denotes the presentday value. Taking the time derivative of Eq. (23.3) we obtaindR̈(1 − Ωtot ) = −2 (1 − Ωtot ) .
dt
ṘS. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
31st May, 2024 10:15am(23.4)

2

23. Inflation
Thus in a decelerating expansion, Ṙ > 0 and R̈ < 0, any small initial deviation from spatial
flatness grows, (d/dt)|1 − Ωtot | > 0. A small value such as |1 − Ωtot,0 | < 0.005 today requires
an even smaller value at earlier times, e.g., |1−Ωtot | < 10−5 at the last scattering of the CMB,
which appears unlikely, unless for some reason space is exactly flat. However, an extended
period of accelerated expansion in the very early Universe, with Ṙ > 0 and R̈ > 0 and hence(d/dt)|1 − Ωtot | < 0, can drive Ωtot sufficiently close to unity, so that |1 − Ωtot,0 | remains
unobservable small today, even after the radiation- and matter-dominated eras, for a wide
range of initial values of Ωtot .
2. The comoving distance (the present-day proper distance) traversed by light between cosmic
time t1 and t2 in an expanding Universe can be written, (see Eq. (22.32) in “Big Bang
Cosmology” review), as
Z t2

D0 (t1 , t2 ) = R0
t1

dt
= R0
R(t)

Z ln R2ln R1d(ln R).
Ṙ

(23.5)In standard decelerated (radiation- or matter-dominated) cosmology the integrand, 1/Ṙ, decreases towards the past, and there is a finite comoving distance traversed by light (a particle
horizon) since the Big Bang (R1 → 0). For example, the comoving size of the particle
horizon at the CMB last-scattering surface (R2 = Rlss ) corresponds to D0 ∼ 100 Mpc, orapproximately 1◦ on the CMB sky today (see Sec. 22.2.4 in “Big Bang Cosmology” review).
However, during a period of inflation, 1/Ṙ increases towards the past, and hence the integral(Eq. (23.5)) diverges as R1 → 0, allowing an arbitrarily large causal horizon, dependent only
upon the duration of the accelerated expansion. Assuming that the Universe inflates with afinite Hubble rate H∗ at t1 = t∗ , ending with Hend < H∗ at t2 = tend , we haveD0 (t∗ , tend ) >
R0H∗−1 eN∗ − 1 ,Rend(23.6)

where N∗ ≡ ln(Rend /R∗ ) describes the duration of inflation, measured in terms of the logarithmic expansion (or “e-folds”) from t1 = t∗ up to the end of inflation at t2 = tend , and
R0 /Rend is the subsequent expansion from the end of inflation to the present day. If inflation−1
occurs above the TeV scale, the comoving Hubble scale at the end of inflation, (R0 /Rend )Hend,
11
is less than one astronomical unit (∼ 10 m), and a causally-connected patch can encompass
our entire observable Universe today, which has a size D0 > 30 Gpc, if there were more than
40 e-folds of inflation (N∗ > 40). If inflation occurs at the GUT scale (1015 GeV) then we
require more than 60 e-folds.
Producing an accelerated expansion in general relativity requires an energy-momentum tensor
with negative pressure, p < −ρ/3 (see Eq. (22.9) in “Big Bang Cosmology” review and Chap. 28,“Dark Energy” review), quite different from the hot dense plasma of relativistic particles in the hot
Big Bang. However a positive vacuum energy V > 0 does exert a negative pressure, pV = −ρV . The
work done by the cosmological expansion must be negative in this case so that the local vacuum
energy density remains constant in an expanding Universe, ρ̇V = −3H(ρV + pV ) = 0. Therefore,a false vacuum state can drive an exponential expansion, corresponding to a de Sitter spacetime
with a constant Hubble rate H 2 = 8πρV /3MP2 on spatially-flat hypersurfaces.
A constant vacuum energy V , equivalent to a cosmological constant Λ in the Friedmann equation
Eq. (23.1), cannot provide a complete description of inflation in the early Universe, since inflationmust necessarily have come to an end in order for the standard Big-Bang cosmology to follow. A
phase transition to the present true vacuum is required to release the false vacuum energy into theenergetic plasma of the hot Big Bang and produce the large total entropy of our observed Universe
31st May, 2024

23. Inflation

3

today. Thus, we must necessarily study dynamical models of inflation, where the time-invarianceof the false vacuum state is broken by a time-dependent field. A first-order phase transition would
produce a very inhomogeneous Universe [9] unless a time-dependent scalar field leads to a rapidlychanging percolation rate [10–12]. However, a second-order phase transition [13, 14], controlled bya slowly-rolling scalar field, can lead to a smooth classical exit from the vacuum-dominated phase.
As a spectacular bonus, quantum fluctuations in that scalar field could provide a source ofalmost scale-invariant density fluctuations [15, 16], as detected in the CMB (see Chap. 29), which
are thought to be the origin of the structures seen in the Universe today.
Accelerated expansion and primordial perturbations can also be produced in some modified
gravity theories (e.g., [1,17]), which introduce additional non-minimally coupled degrees of freedom.
Such inflation models can often be conveniently studied by transforming variables to an ‘Einsteinframe’ in which Einstein’s equations apply with minimally coupled scalar fields [18–20].
In the following we will review scalar field cosmology in general relativity and the spectra ofprimordial fluctuations produced during inflation, before studying selected inflation models.

23.2 Scalar Field Cosmology
The energy-momentum tensor for a canonical scalar field φ with self-interaction potential V (φ)
is given in Eq. (22.52) in “Big Bang Cosmology” review. In a homogeneous background thiscorresponds to a perfect fluid with density1
ρ = φ̇2 + V (φ) ,2(23.7)and isotropic pressure1
p = φ̇2 − V (φ) ,(23.8)
2
while the 4-velocity is proportional to the gradient of the field, uµ ∝ ∇µ φ.
A field with vanishing potential energy acts like a stiff fluid with p = ρ = φ̇2 /2, whereas if
the time-dependence vanishes we have p = −ρ = −V and the scalar field is uniform in time andspace. Thus a classical, potential-dominated scalar-field cosmology, with p ' −ρ, can naturally
drive a quasi-de Sitter expansion; the slow time-evolution of the energy density weakly breaks theexact O(1, 3) symmetry of four-dimensional de Sitter spacetime down to a Robertson-Walker (RW)
spacetime, where the scalar field plays the role of the cosmic time coordinate.
In a scalar-field RW cosmology the Friedmann constraint equation (Eq. (23.1)) reduces to
H2 =

8π
3MP2



1 2
φ̇ + V
2



−

k
,
R2

(23.9)

while energy conservation (Eq. (22.10) in “Big Bang Cosmology” review) for a homogeneous scalar
field reduces to the Klein-Gordon equation of motion (Eq. (22.54) in “Big Bang Cosmology” review)φ̈ = −3H φ̇ − V 0 (φ) .

(23.10)The evolution of the scalar field is thus driven by the potential gradient V 0 = dV /dφ, subject todamping by the Hubble expansion 3H φ̇.
If we define the Hubble slow-roll parameterH ≡ −

Ḣ
,H2(23.11)then we see that inflation (R̈ > 0 and hence Ḣ > −H 2 ) requires H < 1. In this case the spatial
curvature decreases relative to the scalar field energy density as the Universe expands. Hence, in
31st May, 2024

23. Inflation4

the following we drop the spatial curvature and consider a spatially-flat RW cosmology, assumingthat inflation has lasted sufficiently long that our observable Universe is very close to spatiallyflatness. However, we note that bubble nucleation, leading to a first-order phase transition during
inflation, can lead to homogeneous hypersurfaces with a hyperbolic (‘open’) geometry, effectively
resetting the spatial curvature inside the bubble [21]. This is the basis of so-called open inflation
models [22–24], where inflation inside the bubble has a finite duration, leaving a finite negative
spatial curvature.
In a scalar field-dominated cosmology (Eq. (23.11)) gives
H =

3φ̇2
,
2V + φ̇2(23.12)in which case we see that inflation requires a potential-dominated expansion, φ̇2 < V .
23.2.1 Slow-Roll Inflation
It is commonly assumed that the field acceleration term, φ̈, in (Eq. (23.10)) can be neglected,
in which case one can give an approximate solution for the inflationary attractor [25]. This slowroll approximation reduces the second-order Klein-Gordon equation (Eq. (23.10)) to a first-ordersystem, which is over-damped, with the potential gradient being approximately balanced againstto the Hubble damping:3H φ̇ ' −V 0 ,(23.13)and at the same time that the Hubble expansion (Eq. (23.9)) is dominated by the potential energy
H2 '8π
V (φ) ,
3MP2

(23.14)

corresponding to H  1.
A necessary condition for the validity of the slow-roll approximation is that the potential slowroll parameters M 2 V 00
M2 V 0 2≡ P, η≡ P,(23.15)16π V
8πVare small, i.e.,   1 and |η|  1, requiring the potential to be correspondingly flat. If we identify
V 00 with the effective mass of the field, we see that the slow-roll approximation requires that themass of the scalar field must be small compared with the Hubble scale. We note that the Hubble
slow-roll parameter (Eq. (23.11)) coincides with the potential slow-roll parameter, H ' , to leading
order in the slow-roll approximation.
The slow-roll approximation allows one to determine the Hubble expansion rate as a functionof the scalar field value, and vice versa. In particular, we can express, in terms of the scalar fieldvalue during inflation, the total logarithmic expansion, or number of “e-folds”:
RendN∗ ≡ lnR∗

Z tend=t∗

Hdt ' −Z φend rφ∗

4π dφ
for V 0 > 0 .
 MP(23.16)Given that the slow-roll parameters are approximately constant during slow-roll inflation, d/dN '2(η − 2) = O(2 ), we have4 ∆φN∗ ' √.
(23.17) MP
31st May, 2024

23. Inflation5

Since we require N > 40 to solve the flatness, horizon and entropy problems of the standard Big
Bang cosmology, we require either very slow roll,  < 0.01, or a large change in the value of thescalar field relative to the Planck scale, ∆φ > MP .
23.2.2 Reheating
Slow-roll inflation can lead to an exponentially large Universe, close to spatial flatness andhomogeneity, but the energy density is locked in the potential energy of the scalar field, and needs
to be converted to particles and thermalised to recover a hot Big Bang cosmology at the end of
inflation [26,27]. This process is usually referred to as reheating, although there was not necessarily
any preceding thermal era. Reheating can occur when the scalar field evolves towards the minimumof its potential, converting the potential energy first to kinetic energy. This can occur either throughthe breakdown of the slow-roll condition in single-field models, or due to an instability triggered by
the inflaton reaching a critical value, in multi-field models known as hybrid inflation models [28].
Close to a simple minimum, the scalar field potential can be described by a quadratic function,
V = m2 φ2 /2, where m is the mass of the field. We can obtain slow-roll inflation in such a potential
at large field values, φ  MP . However, for φ  MP the field approaches an oscillatory solution:
MP sin(mt)φ(t) ' √.
3π mt(23.18)

For |φ| < MP the Hubble rate drops below the inflaton mass, H < m, and the field oscillates manytimes over a Hubble time. Averaging over several oscillations, ∆t  m−1 , we find hφ̇2 /2i∆t 'hm2 φ2 /2i∆t and henceM2hρi∆t ' P2 , hpi∆t ' 0 .
(23.19)6πtThis coherent oscillating field corresponds to a condensate of non-relativistic massive inflaton particles, driving a matter-dominated era at the end of inflation, with scale factor R ∝ t2/3 .
The inflaton condensate can lose energy through perturbative decays due to terms in the interaction Lagrangian, such as
Lint ⊂ −λi σφχ2i − λj φψ̄j ψj(23.20)that couple the inflation to scalar fields χi or fermions ψj , where σ has dimensions of mass andthe λi are dimensionless couplings. When the mass of the inflaton is much larger than the decay
products, the decay rate is given by [29]Γi =λ2i σ 2
,
8πm

Γj =

λ2j m
.
8π

(23.21)

These decay products must in turn thermalise with Standard Model particles before we recoverconventional hot Big Bang cosmology. An upper limit on the reheating temperature after inflation
is given by [27]
100 1/4 p
Trh = 0.2
MP Γtot ,(23.22)g∗where g∗ is the effective number of degrees of freedom and Γtot is the total decay rate for the
inflaton, which is required to be less than m for perturbative decay.
The baryon asymmetry of the Universe must be generated after the main release of entropy
during inflation, which is an important constraint on possible models. Also, the fact that the
inflaton mass is much larger than the mass scale of the Standard Model opens up the possibility
that it may decay into massive stable or metastable particles that could be connected with dark
matter, constraining possible models. For example, in the context of supergravity models the reheat
31st May, 2024

6

23. Inflation

temperature is constrained by the requirement that gravitinos are not overproduced, potentiallydestroying the successes of Big Bang nucleosynthesis. For a range of gravitino masses one must
require Trh < 109 GeV [30, 31].
The process of inflaton decay and reheating can be significantly altered by interactions leading
to space-time dependences in the effective masses of the fields. In particular, parametric resonancecan lead to explosive, non-perturbative decay of the inflaton in some cases, a process often referred
to as preheating [26, 32]. For example, an interaction term of the form
Lint ⊂ −λ2 φ2 χ2 ,(23.23)

leads to a time-dependent effective mass for the χ field as the inflaton φ oscillates. This can leadto non-adiabatic particle production if the bare mass of the χ field is small for large couplings orfor rapid changes of the inflaton field. The process of preheating is highly model-dependent, but ithighlights the possible role of non-thermal particle production after and even during inflation.

23.3 Primordial Perturbations from Inflation
Although inflation was originally discussed as a solution to the problem of initial conditions
required for homogeneous and isotropic hot Big Bang cosmology, it was soon realised that inflation
also offered a mechanism to generate the inhomogeneous initial conditions required for the formationof large-scale structure [15–17, 33].
23.3.1 Metric Perturbations
In a homogeneous classical inflationary cosmology driven by a scalar field, the inflaton field
is uniform on constant-time hypersurfaces, φ = φ0 (t). However, quantum fluctuations inevitably
break the spatial symmetry leading to an inhomogeneous field:φ(t, xi ) = φ0 (t) + δφ(t, xi ) .

(23.24)At the same time, one should consider inhomogeneous perturbations of the RW spacetime metric(see, e.g., [34–36]):
ds2 = (1 + 2A)dt2 − 2RBi dtdxi− R2 [(1 + 2C)δij + ∂i ∂j E + hij ] dxi dxj ,(23.25)where A, B, E and C are scalar perturbations while hij represents transverse and tracefree, tensor
metric perturbations. Vector metric perturbations can be eliminated using Einstein constraint
equations in a scalar field cosmology.
The tensor perturbations remain invariant under a temporal gauge transformation t → t +δt(t, xi ), but both the scalar field and the scalar metric perturbations transform. For example, wehaveδφ → δφ − φ̇0 δt , C → C − Hδt .
(23.26)
However, there are gauge invariant combinations, such as [37]Q = δφ −φ̇0C,H(23.27)

which describes the scalar field perturbations on spatially-flat (C = 0) hypersurfaces. This is simply
related to the curvature perturbation on uniform-field (δφ = 0) hypersurfaces:
R=C−

H
H
δφ = − Q ,φ̇0φ̇031st May, 2024

(23.28)

23. Inflation

7

which coincides in slow-roll inflation, ρ ' ρ(φ), with the curvature perturbation on uniform-density
hypersurfaces [16]H
(23.29)ζ = C − δρ .
ρ̇0Thus scalar field and scalar metric perturbations are coupled by the evolution of the inflaton field.
23.3.2 Gravitational waves from inflation
The tensor metric perturbation, hij in Eq. (23.25), is gauge-invariant and decoupled from thescalar perturbations at first order. This represents the free excitations of the spacetime, i.e.,
gravitational waves, which are the simplest metric perturbations to study at linear order.
Each tensor mode, with wavevector ~k, has two linearly-independent transverse and trace-freepolarization states:hij (~k) = h~k qij + h̄~k q̄ij .
(23.30)The linearised Einstein equations then yield the same evolution equation for the amplitude as thatfor a massless field in RW spacetime:ḧ~k + 3H ḣ~k +k2
h~ = 0 ,R2 k(and similarly for h̄~k ). This can be re-written in terms of the conformal time, η =conformally rescaled field:
MP Rh~k.
u~k = √
32π

(23.31)R

dt/R, and the(23.32)This conformal field then obeys the wave equation for a canonical scalar field in Minkowski spacetime with a time-dependent mass:


u~00k + k 2 −R00
u~k = 0 .
R

(23.33)

During slow rollR00
' (2 − )R2 H 2 .
(23.34)
R
This makes it possible to quantise the linearised metric fluctuations, u~k → û~k , on sub-Hubble scales,
k 2 /R2  H 2 , where the background expansion can be neglected.
Crucially, in an inflationary expansion, where R̈ > 0, the comoving Hubble length H −1 /R = 1/Ṙdecreases with time. Thus, all modes start inside the Hubble horizon and it is possible to take theinitial field fluctuations to be in a vacuum state at early times or on small scales:ihu~k1 u~k2 i = (2π)3 δ (3) ~k1 + ~k2 .
2(23.35)In terms of the amplitude of the tensor metric perturbations, this corresponds to
hh~k1 h~k2 i =
1 Pt (k1 )(2π)3 δ (3) ~k1 + ~k2 ,32 4πk1(23.36)

where the factor 1/2 appears due to the two polarization states that contribute to the total tensor
power spectrum:


64π
k 2
Pt (k) = 2.
(23.37)MP 2πR
31st May, 2024

23. Inflation

Co
nc

av

e

0.10

0.15

TT,TE,EE+lowE+lensing
TT,TE,EE+lowE+lensing+BK15
TT,TE,EE+lowE+lensing+BK15+BAO
Natural inflation
Hilltop quartic modelα attractors
Power-law inflation
R2 inflation

Co
nv
ex

V ∝ φ2V ∝ φ4/3
V ∝φ

V ∝ φ2/3
Low scale SB SUSY
N∗ =50N∗ =600.05
0.00

Tensor-to-scalar ratio (r0.002 )

0.20

8

0.94

0.96
Primordial tilt (ns )

0.98

1.00

Figure 23.1: The marginalized joint 68 and 95% CL regions for the tilt in the scalar perturbation
spectrum, ns , and the relative magnitude of the tensor perturbations, r, obtained from the Planck2018 and lensing data alone, and their combinations with BICEP2/Keck Array (BK15) and (optionally) BAO data, confronted with the predictions of some of the inflationary models discussed
in this review. This figure is taken from [38].
On super-Hubble scales, k 2 /R2  H 2 , we have the growing mode solution to Eq. (23.33),
u~k ∝ R, corresponding to h~k → constant, i.e., tensor modes are frozen-in on super-Hubble scales,both during and after inflation. Thus, connecting the initial vacuum fluctuations on sub-Hubble
scales to the late-time power spectrum for tensor modes at Hubble exit during inflation, k = R∗ H∗ ,we obtain
64π H∗ 2
Pt (k) ' 2.
(23.38)MP 2π
In the de Sitter limit,  → 0, the Hubble rate becomes time-independent and the tensor spectrum
on super-Hubble scales becomes scale-invariant [39]. However slow-roll evolution leads to weak timedependence of H∗ and thus a scale-dependent spectrum on large scales, with a spectral tiltnt ≡d ln Pt' −2∗ .
d ln k(23.39)23.3.3 Density Perturbations from single-field inflation
The inflaton field fluctuations on spatially-flat hypersurfaces are coupled to scalar metric perturbations at first order, but these can be eliminated using the Einstein constraint equations toyield an evolution equation
"

k2
8π d
Q̈~k + 3H Q̇~k ++ V 00 − 2 3
2R
MP R dt

R3 φ̇2H!#

Q~k = 0 .

(23.40)Terms proportional to MP−2 represent the effect on the field fluctuations of gravity at first order.
As can be seen, this vanishes in the limit of a constant background field, and hence is suppressed
31st May, 2024

23. Inflation

9in the slow-roll limit, but it is of the same order as the effective mass, V 00 = 3ηH 2 , so must be
included if we wish to model deviations from exact de Sitter symmetry.
This wave equation can also be written in the canonical form for a free field in Minkowski
spacetime if we define [37]v~k ≡ RQ~k ,(23.41)to yield
v~k00 +



z 00
k −
v~k = 0 ,z
2

(23.42)where we define

Rφ̇
z 00
,
' (2 + 5 − 3η)R2 H 2 ,(23.43)Hzwhere the last approximate equality holds to leading order in the slow-roll approximation.
As previously done for gravitational waves, we quantise the linearised field fluctuations v~k → v̂~kon sub-Hubble scales, k 2 /R2  H 2 , where the background expansion can be neglected. Thus, we
imposeihv~k1 v~k0 i = δ (3) ~k1 + ~k2 .
(23.44)2
2
In terms of the field perturbations, this corresponds toz≡

hQ~k1 Q~k2 i =
PQ (k1 )
3 (3) ~
~k2 ,
(2π)
δ
k
+
1
4πk13

(23.45)where the power spectrum for vacuum field fluctuations on sub-Hubble scales, k 2 /R2  H 2 , issimplyk 2,(23.46)
PQ (k) =2πR
yielding the classic result for the vacuum fluctuations for a massless field in de Sitter at Hubble
exit, k = R∗ H∗ : 2
HPQ (k) '.
(23.47)2π ∗In practice there are slow-roll corrections due to the small but finite mass (η) and field evolution() [40].
Slow-roll corrections to the field fluctuations are small on sub-Hubble scales, but can become
significant as the field and its perturbations evolve over time on super-Hubble scales. Thus, it
is helpful to work instead with the curvature perturbation, ζ defined in equation (Eq. (23.29)),
which remains constant on super-Hubble scales for adiabatic density perturbations both during
and after inflation [16, 41]. Thus we have an expression for the primordial curvature perturbation
on super-Hubble scales produced by single-field slow-roll inflation:"Pζ (k) =

H
φ̇

2

#

4π
PQ (k) ' 2
MP
∗

" 

1


H
2π

2 #

.

(23.48)∗Comparing this with the primordial gravitational wave power spectrum (Eq. (23.38)) we obtain thetensor-to-scalar ratio
Pt' 16∗ .
(23.49)r≡Pζ
Note that the scalar amplitude is boosted by a factor 1/∗ during slow-roll inflation, because
small scalar field fluctuations can lead to relatively large curvature perturbations on hypersurfaces
31st May, 2024

23. Inflation

10

defined with respect to the density if the potential energy is only weakly dependent on the scalar
field, as in slow roll. Indeed, the de Sitter limit is singular, since the potential energy becomes
independent of the scalar field at first order,  → 0, and the curvature perturbation on uniformdensity hypersurfaces becomes ill-defined.
We note that in single-field inflation the tensor-to-scalar ratio and the tensor tilt (Eq. (23.39))
at the same scale are both determined by the first slow-roll parameter at Hubble exit, ∗ , giving
rise to an important consistency test for single-field slow-roll inflation:r
nt ' − .
8(23.50)This may be hard to verify if r is small, making any tensor tilt nt difficult to measure. On theother hand, it does offer a way to rule out single-field slow-roll inflation if either r or nt is large.
Given the relatively large scalar power spectrum, it has proved easier to measure the scalar tilt,
conventionally defined as ns − 1. Slow-roll corrections lead to slow time-dependence of both H∗and ∗ , giving a weak scale-dependence of the scalar power spectrum:ns − 1 ≡d ln Pζ' −6∗ + 2η∗ ,d ln k(23.51)and a running of this tilt at second-order in slow roll:dns' −8∗ (3∗ − 2η∗ ) − 2ξ∗2 ,d ln k(23.52)where the running introduces a new slow-roll parameter at second-order:
ξ2 =MP4 V 0 V 000.
64π 2 V 2

(23.53)23.3.4 Observational Bounds
The observed scale-dependence of the power spectrum makes it necessary to specify the comoving scale, k, at which quantities are constrained and hence the Hubble-exit time, k = a∗ H∗ , when
the corresponding theoretical quantities are calculated during inflation. This is usually expressedin terms of the number of e-folds from the end of inflation [42]:k
1N∗ (k) ' 67 − ln
+ ln
a0 H0
4
1
−
ln(g∗ ),
12




V∗2
MP4 ρend

!

ρrh
1
+
ln
12
ρend




(23.54)

where H0−1 /a0 is the present comoving Hubble length. Different models of reheating and and thus
different reheat temperatures and densities, ρrh in Eq. (23.54), lead to a range of possible values forN∗ corresponding to a fixed physical scale, and hence we have a range of observational predictions
for a given inflation model, as seen in Fig. 23.1.
The Planck 2018 temperature and polarization data (see Chap. 29, “Cosmic Microwave Background” review) are consistent with a smooth featureless power spectrum over a range of comoving
wavenumbers, 0.008 h−1 Mpc−1 ≤ k ≤ 0.1 h Mpc−1 . In the absence of running, the data measure
the spectral index to be [38]ns = 0.9649 ± 0.0042 ,(23.55)31st May, 2024

23. Inflation

110ln(V /Vpivot )0.05

inflation potential samples
inflation potential mean

−0.05

12 knots; p(convex) = 0.55
TT,TE,EE+lowE+lensing+BK15+BAO−0.5

0(φ − φpivot )/Mp

0.5

Figure 23.2: The result of reconstructing a single-field inflaton potential using a cubic-spline
power-spectrum mode expansion and the the full Planck, lensing, BK15 and BAO data set. This
figure is taken from [38].
corresponding to a deviation from scale-invariance exceeding the 7σ level. If running of the spectral
tilt is included in the model, this is constrained to be [38]dns
= −0.0045 ± 0.0067(23.56)d ln kat the 95% CL, assuming no running of the running. The most recent analysis [43] of the BICEP,Keck Array and Planck data places an upper bound on the tensor-to-scalar ratior < 0.036 ,(23.57)at the 95% CL.
These observational bounds can be converted into bounds on the slow-roll parameters and hence
the potential during slow-roll inflation. Setting higher-order slow-roll parameters (beyond secondorder in horizon-flow parameters [44]) to zero, the Planck collaboration obtain the following 95%
CL bounds when lensing and BK15 data are included [38]
 < 0.0044 ,(23.58)η = −0.015 ± 0.006 ,(23.59)= 0.0029+0.0073−0.0069 ,(23.60)ξ

2

31st May, 2024

1223. Inflation

which can be used to constrain models, as discussed in the next Section.
Fig. 23.1, which is taken from [38], compares observational CMB constraints on the tilt, ns ,in the spectrum of scalar perturbations and the ratio, r, between the magnitudes of tensor andscalar perturbations. Important rôles are played by data from the Planck satellite and on lensing,the BICEP2/Keck Array (BK15) and measurements of baryon acoustic oscillations (BAO). The
reader is referred to [38] for technical details. These experimental constraints are compared withthe predictions of some of the inflationary models discussed in this review. Generally speaking,models with a concave potential are favored over those with a convex potential, and models withpower-law inflation are now excluded, as opposed to models with de Sitter-like (quasi-)exponential
expansion.
There is no significant evidence for local features within the range of inflaton field values probed
by the data [38]. However, the data may be used to reconstruct partially the effective inflationary
potential over a range of inflaton field values, assuming that it is suitably smooth. The result of
one such exercise by the Planck collaboration [38] in the framework of a generic single-field inflaton
potential is shown in Fig. 23.2. This reconstruction assumes a cubic-spline power-spectrum mode
expansion and employs the full Planck, lensing, BK15 and BAO data set. The reader is again
referred to [38] for technical details. We see that the effective inflaton potential is relatively wellreconstructed over field values φ within ±0.5 of the chosen pivot value, but the potential is only
very weakly constrained for larger values of |φ−φpivot |, providing wide scope for inflationary modelbuilders.

23.4 Models
23.4.1 Pioneering Models
The paradigm of the inflationary Universe was proposed in [2], where it was pointed out thatan early period of (near-)exponential expansion, in addition to resolving the horizon and flatness
problems of conventional Big-Bang cosmology as discussed above (the possibility of a de Sitter
phase in the early history of the Universe was also proposed in the non-minimal gravity modelof [1], with the motivation of avoiding an initial singularity), would also dilute the prior abundanceof any unseen heavy, (meta-)stable particles, as exemplified by monopoles in grand unified theories(GUTs; see Chap. 93, “Grand Unified Theories” review). The original proposal was that thisinflationary expansion took place while the Universe was in a metastable state (a similar suggestion
was made in [45, 46], where in [45] it was also pointed out that such a mechanism could address
the horizon problem) and was terminated by a first-order transition due to tunnelling though apotential barrier. However, it was recognized already in [2] that this ‘old inflation’ scenario would
need modification if the transition to the post-inflationary Universe were to be completed smoothly
without generating unacceptable inhomogeneities.
This ‘graceful exit’ problem was addressed in the ‘new inflation’ model of [13] (see also [14]and footnote [39] of [2]), which studied models based on an SU(5) GUT with an effective potentialof the Coleman-Weinberg type (i.e., dominated by radiative corrections), in which inflation could
occur during the roll-down from the local maximum of the potential towards a global minimum.
However, it was realized that the Universe would evolve to a different minimum from the Standard
Model [47], and it was also recognized that density fluctuations would necessarily be too large [15],since they were related to the GUT coupling strength.
These early models of inflation assumed initial conditions enforced by thermal equilibrium inthe early Universe. However, this assumption was questionable: indeed, it was not made in themodel of [1], in which a higher-order gravitational curvature term was assumed to arise from quantum corrections, and the assumption of initial thermal equilibrium was jettisoned in the ‘chaotic’
inflationary model of [48]. These are the inspirations for much recent inflationary model building,
31st May, 2024

23. Inflation

13

so we now discuss them in more detail, before reviewing contemporary models.
In this section we will work in natural units where we set the reduced Planck mass to unity,i.e., 8π/MP2 = 1. All masses are thus relative to the reduced Planck scale.
23.4.2 R2 Inflation
R√
The first-order Einstein-Hilbert action, (1/2) d4 x −gR, where R is the Ricci scalar curvature,is the minimal possible theory consistent with general coordinate invariance. However, it is possiblethat there might be non-minimal corrections to this action, and the unique second-order possibility
is
!
Z
1
R2
4 √
S=
.
(23.61)d x −g R +2
6M 2
It was pointed out in [1] that an R2 term could be generated by quantum effects, and that
(Eq. (23.61)) could lead to de Sitter-like expansion of the Universe. Scalar density perturbations inthis model were calculated in [17]. Because the initial phase was (almost) de Sitter, these perturbations were (approximately) scale-invariant, with magnitude ∝ M . It was pointed out in [17] that
requiring the scalar density perturbations to lie in the range 10−3 to 10−5 , consistent with upper
limits at that time, would require M ∼ 10−3 to 10−5 in Planck units, and it was further suggested inthat these perturbations could lead to the observed large-scale structure of the Universe, including
the formation of galaxies.
Although the action (Eq. (23.61)) does not contain an explicit scalar field, [17] reduced thecalculation of density perturbations to that of fluctuations in the scalar curvature R, which could
be identified (up to a factor) with a scalar field of mass M . The formal equivalence of R2 gravity(Eq. (23.61)) to a theory of gravity with a massive scalar φ had been shown in [18], see also [19].
The effective scalar potential for what we would nowadays call the ‘inflaton’ [49] takes the form


Z
√
1
3 2
− 2/3φ 2
4 √2
S=)(23.62)d x −g R + (∂µ φ) − M (1 − e2
2when the action is written in the Einstein frame, and the potential is shown as the solid black line
in Fig. 23.3. Using (Eq. (23.48)), one finds that the amplitude of the scalar density perturbationsin this model is given by


3M 2
φ
4
√
∆R =
sinh
,
(23.63)8π 2
6The measured magnitude of the density fluctuations in the CMB requires M ' 1.3 × 10−5 inPlanck units (assuming N∗ ' 55), so one of the open questions in this model is why M is so small.
Obtaining N∗ ' 55 also requires an initial value of φ ' 5.5, i.e., a super-Planckian initial condition,
and another issue for this and many other models is how the form of the effective potential isprotected and remains valid at such large field values. Using Eq. (23.51) one finds that ns ' 0.965
for N∗ ' 55 and using (Eq. (23.49)) one finds that r ' 0.0035. These predictions are consistent
with the present data from Planck and other experiments, as seen in Fig. 23.1.
23.4.3 Chaotic Models with Power-Law Potentials
As has already been mentioned, a key innovation in inflationary model-building was the suggestion to abandon the questionable assumption of a thermal initial state, and consider ‘chaotic’
initial conditions with very general forms of potential [48]. (Indeed, the R2 model discussed above
can be regarded as a prototype of this approach.) The chaotic approach was first proposed in thecontext of a simple power-law potential of the form µ4−α φα , and the specific example of λφ4 wasstudied in [48]. Such models make the following predictions for the slow-roll parameters  and η:
1
 =
2

 2
α

φ

, η =

31st May, 2024α(α − 1),
φ2(23.64)

23. Inflation

14

Figure 23.3: The inflationary potential V in the R2 model (solid black line) compared with its
form in various no-scale models discussed in detail in [50] (dashed coloured lines).
leading to the predictions
4αα+2,ns − 1 ≈ −,(23.65)N∗
2N∗
which are shown in Fig. 23.1 for some illustrative values of α. We note that the prediction of theoriginal φ4 model lies out of the frame, with values of r that are too large and values of ns that are
too small. The φ3 model has similar problems, and would in any case require modification in orderto have a well-defined minimum. The simplest possibility is φ2 , but this is now also disfavored by
the data, at the 95% CL if only the Planck data are considered, and more strongly if other data
are included, as seen in Fig. 23.1. (For non-minimal models of quadratic inflation that avoid thisproblem, see, e.g., [51].)
Indeed, as can be seen in Fig. 23.1, all single-field models with a convex potential (i.e., one
curving upwards) are disfavored compared to models with a concave potential.
r ≈23.4.4 Hilltop Models
This preference for a concave potential motivates interest in ‘hilltop’ models [52], whose startingpoint is a potential of the formV (φ) = Λ4 1 −

 p

φ
µ



+ ... ,

(23.66)

where the . . . represent extra terms that yield a positive semi-definite potential. To first order inthe slow-roll parameters, when x ≡ φ/µ is small, one has
ns ' 1 − p(p − 1)µ−2

xp−2
3
− r,
p
(1 − x ) 8
31st May, 2024

r ' 8p2 µ−2x2p−2.
(1 − xp )2(23.67)

15

23. Inflation

As seen in Fig. 23.1, a hilltop model with p = 4 can be compatible with the Planck and othermeasurements, if µ  MP .
23.4.5 D-Brane Inflation
Many scenarios for inflation involving extra dimensions have been proposed, e.g., the possibility
that observable physics resides on a three-dimensional brane, and that there is an inflationary
potential that depends on the distance between our brane and an antibrane, with a potential of
the form [53]

 p

µ
4
V (φ) = Λ 1 −+ ... .
(23.68)φ
In this scenario the effective potential vanishes in the limit φ → ∞, corresponding to complete
separation between our brane and the antibrane. The predictions for ns and r in this model canbe obtained from (Eq. 23.67) by exchanging p ↔ −p, and are also consistent with the Planck andother data.
23.4.6 Natural Inflation
Also seen in Fig. 23.1 are the predictions of ‘natural inflation’ [54], in which one postulatesa non-perturbative shift symmetry that suppresses quantum corrections, so that a hierarchicallysmall scale of inflation, H  MP , is technically natural. In the simplest models, there is a periodic
potential of the form

 
φ
4
V (φ) = Λ 1 + cos,(23.69)f
where f is a dimensional parameter reminiscent of an axion decay constant (see the next subsection) [55], which must have a value > MP . Natural inflation can yield predictions similar toquadratic inflation (which are no longer favored, as already discussed), but can also yield an effective convex potential. Thus, it may lead to values of r that are acceptably small, but for values of
ns that are in tension with the data, as seen in Fig. 23.1.
23.4.7 Axion Monodromy Models
The effective potentials in stringy models [56, 57] motivated by axion monodromy may be of
the form
 pΛ
"
#
 
φ
φ φ pf +14−α α
4 −C φ0
V (φ) = µφ +Λ ecos γ +,
(23.70)f φ0
where µ, Λ, f and φ0 are parameters with the dimension of mass, and C, p, pΛ , pf and γ are dimensionless constants, generalizing the potential ( [54]) in the simplest models of natural inflation. The
oscillations in (Eq. 23.70) are associated with the axion field, and powers pΛ , pf 6= 0 may arise
from φ-dependent evolutions of string moduli. Since the exponential prefactor in (Eq. 23.70) is due
to non-perturbative effects that may be strongly suppressed, the oscillations may be unobservable
small. Specific string models having φα with α = 4/3, 1 or 2/3 have been constructed in [56, 57],
providing some motivation for the low-power models mentioned above.
As seen in Fig. 23.1, the simple axion monodromy models with the power α = 4/3 or 1 are nolonger compatible with the current CMB data at the 95% CL, while α = 2/3 is only marginallycompatible at 95% CL. The Planck Collaboration has also searched for characteristic effects associated with the second term in (Eq. (23.70)), such as a possible drift in the modulation amplitude(setting pΛ = C = 0), and a possible drifting frequency generated by pf 6= 0, without finding any
compelling evidence [38].
23.4.8 Higgs Inflation
Since the energy scale during inflation is commonly expected to lie between the Planck and TeV
scales, it may serve as a useful bridge with contacts both to string theory or some other quantum
31st May, 2024

23. Inflation

16

theory of gravity, on the one side, and particle physics on the other side. However, as the above
discussion shows, much of the activity in building models of inflation has been largely independentof specific connections with these subjects, though some examples of string-motivated models of
inflation were mentioned above.
The most economical scenario for inflation might be to use as inflaton the only establishedscalar field, namely the Higgs field (see Chap.11, “Status of Higgs boson physics” review). A
specific model assuming a non-minimal coupling of the Higgs field h to gravity was constructedin [58]. Its starting-point is the action
Z

S =

#

"

√
1
λ
M 2 + ξh2
d4 x −gR + ∂µ h∂ µ h − (h2 − v 2 )2 ,2
2
4

(23.71)

where v is the Higgs vacuum expectation value. The model requires ξ  1, in which case it can be
rewritten in the Einstein frame as
Z

S =

√
1
1
d x −g R + ∂µ χ∂ µ χ − U (χ) ,
2
2


4



(23.72)where the effective potential for the canonically-normalized inflaton field χ has the form
λ2χU (χ) =
1 + exp − √
2
4ξ
6MP




−2

,

(23.73)2which is similar to the effective potential√ of the R model at large field values. As such, the model
4
inflates successfully if ξ ' 5×10 mh /( 2v), with predictions for ns and r that are indistinguishable
from the predictions of the R2 model shown in Fig. 23.1.
This model is very appealing, but must confront several issues. One is to understand the valueof ξ, and another is the possibility of unitarity violation. However, a more fundamental issue iswhether the effective quartic Higgs coupling is positive at the scale of the Higgs field during inflation.
Extrapolations of the effective potential in the Standard Model using the measured values of themasses of the Higgs boson and the top quark indicate that probably λ < 0 at this scale [59], though
there are still significant uncertainties associated with the appropriate input value of the top massand the extrapolation to high renormalization scales.

23.4.9 Supersymmetric Models of Inflation
Supersymmetry [60] is widely considered to be a well-motivated possible extension of the Standard Model that might become apparent at the TeV scale. It is therefore natural to consider
supersymmetric models of inflation. These were originally proposed because of the problems of the
new inflationary theory [13, 14] based on the one-loop (Coleman-Weinberg) potential for breaking
SU(5). Several of these problems are related to the magnitude of the effective potential parameters:in any model of inflation based on an elementary scalar field, some parameter in the effective potential must be small in natural units, e.g., the quartic coupling λ in a chaotic model with a quartic
potential, or the mass parameter µ in a model of chaotic quadratic inflation. These parameters
are renormalized multiplicatively in a supersymmetric theory, so that the quantum corrections tosmall values would be under control. Hence, it was suggested that inflation cries out for supersymmetry [61], though non-supersymmetric resolutions of the problems of Coleman-Weinberg inflation
are also possible: see, e.g., Ref. [62].
In the Standard Model there is only one scalar field that could be a candidate for the inflaton,
namely the Higgs field discussed above, but even the minimal supersymmetric extension of theStandard Model (MSSM) contains many scalar fields. However, none of these is a promising
31st May, 2024

17

23. Inflation

Figure 23.4: The Bayes factors calculated in [63] for a large sample of inflationary models using
Planck 2015 data [64]. Those highlighted in yellow are featured in this review, according to thenumbers listed in the text.

candidate for the inflaton. The minimal extension of the MSSM that may contain a suitable
candidate is the supersymmetric version of the minimal seesaw model of neutrino masses, whichcontains the three supersymmetric partners of the heavy singlet (right-handed) neutrinos. One ofthese singlet sneutrinos ν̃ could be the inflaton [65]: it would have a quadratic potential, the masscoefficient required would be ∼ 1013 GeV, very much in the expected ball-park for singlet (righthanded) neutrino masses, and sneutrino inflaton decays also could give rise to the cosmologicalbaryon asymmetry via leptogenesis. However, as seen in Fig. 23.1 and already discussed, a purely
quadratic inflationary potential is no longer favored by the data. This difficulty could in principle
be resolved in models with multiple sneutrinos [66], or by postulating a trilinear sneutrino coupling
and hence a superpotential of Wess-Zumino type [67], which can yield successful inflation withpredictions intermediate between those of natural inflation and hilltop inflation in Fig. 23.1.
Finally, we note that it is also possible to obtain inflation via supersymmetry breaking, as in
the model [68] whose predictions are illustrated in Fig. 23.1.
23.4.10 Supergravity Models
Any model of early-Universe cosmology, and specifically inflation, must necessarily incorporate
gravity. In the context of supersymmetry this requires an embedding in some supergravity theory [69, 70]. An N = 1 supergravity theory is specified by three functions: a Hermitian function of
31st May, 2024

18

23. Inflation

the matter scalar fields φi , called the Kähler potential K, that describes its geometry, a holomorphic function of the superfields, called the superpotential W , which describes their interactions,and another holomorphic function fαβ , which describes their couplings to gauge fields Vα [71].
The simplest possibility is that the Kähler metric is flat:
K = φi φ∗i ,(23.74)

where the sum is over all scalar fields in the theory, and the simplest inflationary model in minimal
supergravity had the superpotential [72]W = m2 (1 − φ)2 ,(23.75)

Where φ is the inflaton. However, this model predicts a tilted scalar perturbation spectrum,
ns = 0.933, which is now in serious disagreement with the data from Planck and other experiments
shown in Fig. 23.1.
Moreover, there is a general problem that arises in any supergravity theory coupled to matter,namely that, since its effective scalar potential contains a factor of eK , scalars typically receive
squared masses ∝ H 2 ∼ V , where H is the Hubble parameter [73], an issue called the ‘η problem’.
The theory given by (Eq. (23.75)) avoids this η problem, but a generic supergravity inflationarymodel encounters this problem of a large inflaton mass. Moreover, there are additional challenges
for supergravity inflation associated with the spontaneous breaking of local supersymmetry [74–76].
Various approaches to the η problem in supergravity have been proposed, including the possibility of a shift symmetry [77], and one possibility that has attracted renewed attention recently is
no-scale supergravity [78, 79]. This is a form of supergravity with a Kähler potential that can be
written in the form [80]P i 2!
|φ |∗K = −3 ln T + T − i,(23.76)3which has the special property that it naturally has a flat potential, at the classical level andbefore specifying a non-trivial superpotential. As such, no-scale supergravity is well-suited forconstructing models of inflation. Adding to its attraction is the feature that compactifications ofstring theory to supersymmetric four-dimensional models yield effective supergravity theories of
the no-scale type [81]. There are many examples of superpotentials that yield effective inflationary
potentials for either the T field (which is akin to a modulus field in some string compactification)
or a φ field (generically representing matter) that are of the same form as the effective potentialof the R2 model (Eq. (23.62)) when the magnitude of the inflaton field  1 in Planck units, as
required to obtain sufficiently many e-folds of inflation, N∗ [82, 83]. This framework also offers thepossibility of using a suitable superpotential to construct models with effective potentials that are
similar, but not identical, to the R2 model, as shown by the dashed coloured lines in Fig. 23.3.
23.4.11 Other Exponential Potential Models
This framework also offers the possibility [82] of constructing models in which the asymptotic constant value of the potential at large inflaton field values is approached via a different
exponentially-suppressed term:hi

V (φ) = A 1 − δe−Bφ + O(e−2Bφ ) ,(23.77)

where the magnitude of the scalar density perturbations p
fixes A, but δ and B are regarded as freeparameters. In the case of R2 inflation δ = 2 and B = 2/3. In a model such as (Eq. (23.77)),
31st May, 2024

23. Inflation

19

one finds at leading order in the small quantity e−Bφ thatns = 1 − 2B 2 δe−Bφ ,r = 8B 2 δ 2 e−2Bφ ,1N∗ = 2 e+Bφ .
B δ(23.78)yielding the relations
ns = 1 −

2
,
N∗

r =

8
.
B 2 N∗2(23.79)These relations correspond to the class of models labeled as ‘α attractors’ [84] in Fig. 23.1. CMB
bounds on the tensor-to-scalar ratio translate into an upper bound on the parameter α = 2/3B 2
in α-attractor models, which determines the curvature of the Kähler manifold in supergravity
models [85]. There are generalizations of the simplest no-scale model√ (Eq. (23.76)) with prefactors
before the ln(. . . ) that are 1 or 2, leading to larger values of B = 2 or 1, respectively, and hence
smaller values of r than in the R2 model. CMB constraints on the scalar spectral index can also
place a lower bound on the magnitude of α [86], due to the implicit dependence of N∗ upon α [87].

23.5 Model Comparison
Given a particular inflationary model, one can obtain constraints on the model parameters,
informed by the likelihood, corresponding to the probability of the data given a particular choice ofparameters (see Sec. 40, “Statistics” review). In the light of the detailed constraints on the statistical
distribution of primordial perturbations now inferred from high-precision observations of the cosmic
microwave background, it is also possible to make quantitative comparison of the statistical evidence
for or against different inflationary models. This can be done either by comparing the logarithm of
the maximum likelihood that can be obtained for the data using each model, i.e., the minimum χ2(with some correction for the number of free parameters in each model), or by a Bayesian modelcomparison [88] (see also Sec. 40.3.4 in “Statistics” review).
In such a Bayesian model comparison one computes [7] the evidence, E(D|MA ) for a model,
MA , given the data D. This corresponds to the likelihood, L(θAj ) = p(D|θAj , MA ), integrated
over the assumed prior distribution, π(θAj |MA ), for all the model parameters θAj :E(D|MA ) =Z

L(θAj )π(θAj |MA )dθAj .

(23.80)The posterior probability of the model given the data follows from Bayes’ theoremp(MA |D) =E(D|MA )π(MA ),p(D)(23.81)

where the prior probability of the model is given by π(MA ). Assuming that all models are equally
likely a priori, π(MA ) = π(MB ), the relative probability of model A relative to a reference model,
in the light of the data, is thus given by the Bayes factor
BA,ref =E(D|MA ).
E(D|Mref )(23.82)Computation of the multi-dimensional integral (Eq. (23.80)) is a challenging numerical task. Even
using an efficient sampling algorithm requires hundreds of thousands of likelihood computations
for each model, though slow-roll approximations can be used to calculate rapidly the primordial
31st May, 2024

23. Inflation

20

power spectrum using the APSIC numerical library [7] for a large number of single-field, slow-roll
inflation models.
The change in χ2 for selected slow-roll models relative to the Starobinsky R2 inflationary model,
used as a reference, is given in Table 23.1 (taken from [38]). All the other inflation models require asubstantial amplitude of tensor modes, and so have an increased χ2 with respect to the Starobinsky
and other models with a scalar tilt but small tensor modes. Table 23.1 also shows the Bayesianevidence for (ln BA,ref > 0) or against (ln BA,ref < 0) a selection of inflation models using the Planck
analysis priors [38]. The Starobinsky R2 inflationary model may be chosen as a reference [38]that provides a good fit to current data. Higgs inflation [58] is indistinguishable using current
data, making the model comparison “inconclusive” on the Jeffrey’s scale (| ln BA,ref | < 1). (Recall,though, that this model is disfavored by the measured values of the Higgs and top quark masses [59].)
We note that although α-attractor models can provide a good fit to the data, they are disfavored
relative to the Starobinsky model due to their larger prior volume. There is now strong evidence(| ln BA,ref | > 5) against large-field models such as chaotic inflation with a quadratic or a quartic
potential. Indeed, over 30% of the slow-roll inflation models considered in Ref. [7] are strongly
disfavored by the Planck data.
Table 23.1: Observational evidence for and against selected inflation
models: ∆χ2 and the Bayes factors are calculated relative to the Starobinsky R2 inflationary model, which is treated as a reference. Results from
Planck 2018 analysis [38].
Model
R2 inflation
Power-law potential φ2/3
Power-law potential φ2Power-law potential φ4
Natural inflation
Hilltop quartic model

∆χ2
0
+4.0
+21.6
+75.3
+9.9
-0.3

ln BA,ref
0
−4.6
< −10
< −10
−6.6
−1.4

The Bayes factors for a wide selection of slow-roll inflationary models are displayed in Fig. 23.4,
which is adapted from Fig. 3 in [63], where more complete descriptions of the models and thecalculations of the Bayes factors using Planck 2015 data [64] are given. Models discussed in thisreview are highlighted in yellow, and numbered as follows: (1) R2 inflation (Sec. 23.4.2) and models
with similar predictions, such as Higgs inflation (Sec. 23.4.8) and no-scale supergravity inflation(Sec. 23.4.10); chaotic inflation models (2) with a φ2 potential; (3) with a φ4 potential; (4) witha φ2/3 potential, and (5) with a φp potential marginalising over p ∈ [0.2, 6] (Sec. 23.4.3); hilltop
inflation models (6) with p = 2; (7) with p = 4 and (8) marginalising over p (Sec. 23.4.4); (9) brane
inflation Sec. 23.4.5); (10) natural inflation (Sec. 23.4.6); (11) exponential potential models such asα-attractors (Sec. 23.4.11). As seen in Fig. 23.4 and discussed in the next Section, constraints onreheating are starting to provide additional information about models of inflation.

23.6 Constraints on Reheating
One connection between inflation and particle physics is provided by inflaton decay, whose
products are expected to have thermalized subsequently. As seen in Eq. (23.54), the number of
e-folds required during inflation depends on details of this reheating process, including the matter
density upon reheating, denoted by ρth , which depends in turn on the inflaton decay rate Γφ .
We see in Fig. 23.1 that, within any specific inflationary model, both ns and particularly r are
31st May, 2024

23. Inflation

21

sensitive to the value of N∗ . In particular, the uncertainty in the experimental measurement of ns
is comparable to the variation in many model predictions for N∗ ∈ [50, 60]. This implies that thedata start to constrain scenarios for inflaton decay in many models. For example, it is clear from
Fig. 23.1 that N∗ = 60 would be preferred over N∗ = 50 in a chaotic inflationary model with aquadratic potential.
As a specific example, let us consider α-attractor models that predict small values of r. As seenin Fig. 23.1, within these models the combination of Planck, BICEP2/Keck Array and BAO data
would require a limited range of ns , corresponding to a limited range of N∗ , as seen by comparing
the left and right vertical axes in Fig. 23.5:
N∗ & 52 (68% CL),N∗ & 44 (95% CL) .

(23.83)

Within any specific model for inflaton decay, these bounds can be translated into constraints on theeffective decay coupling. For example, if one postulates a two-body inflaton decay (Eq. (23.21)),the bounds (Eq. (23.83)) can be translated into bounds on y 2 ≡ 8πΓ/m. This is illustrated
in Fig. 23.5 [85], where any value of N0.05 (the number of e-folds for a scale 0.05 Mpc−1 , on
the left vertical axis), corresponding to the scalar tilt ns shown on the right vertical axis, when
projected onto the diagonal line representing one of the three illustrated choices of α-attractor
models, corresponds to a specific value of y (lower horizontal axis) and Treh (upper horizontalaxis). For example, for α = 1 one has
y & 10−10(68% CL),y & 10−17(95% CL) .

(23.84)

These bounds are not very constraining, but they can be expected to improve significantly in the
coming years and thereby provide significant information on the connections between inflation andparticle physics.

23.7 Beyond Single-Field Slow-Roll Inflation
There are numerous possible scenarios beyond the simplest single-field models of slow-roll inflation. These include models with features in the potential leading to deviations from slow roll,as well as theories in which non-canonical fields are considered, such as k-inflation [89] or DBI
inflation [90], and multiple-field models, such as the curvaton scenario [91]. As well as altering
the single-field slow-roll predictions for the primordial curvature power spectrum (Eq. (23.48)) andthe tensor-scalar ratio (Eq. (23.49)), they may introduce new quantities that vanish in single-fieldslow-roll models, such as isocurvature matter perturbations, corresponding to entropy fluctuationsin the photon-to-matter ratio, at first order:Sm =δnm δnγδρm 3 δργ
−
=
−
.
nm
nγ
ρm
4 ργ

(23.85)Another possibility is non-Gaussianity in the distribution of the primordial curvature perturbation(see Chap. 29, “Cosmic Microwave Background” review), encoded in higher-order correlators suchas the primordial bispectrum [92]hζ(k)ζ(k0 )ζ(k00 )i ≡ (2π)3 δ(k + k0 + k00 )Bζ (k, k 0 , k 00 ) ,(23.86)

which is often expressed in terms of a dimensionless non-linearity parameterfN L ∝ Bζ (k, k 0 , k 00 )/Pζ (k)Pζ (k 0 ).
The three-point function (Eq. (23.86)) can be thought of as defined on a triangle whose sides arek, k0 , k00 , of which only two are independent, since they sum to zero. Further assuming statistical
31st May, 2024

23. Inflation

22

isotropy ensures that the bispectrum depends only on the magnitudes of the three vectors, k, k 0
and k 00 . The search for fN L and other non-Gaussian effects was a prime objective of the Planck
data analysis [93, 94].
23.7.1 Enhanced perturbations on small scales
Although the observed primordial power spectrum on CMB scales is constrained to be almost
scale invariant, consistent with the simplest slow-roll inflation models, the form of the power spectrum is largely unconstrained on much smaller scales, corresponding to scales that would have left
the horizon closer to the end of inflation. Although free-streaming and other damping of perturbations in the primordial plasma are expected to have washed out small-scale fluctuations in thematter sector [95], gravitational relics such as primordial black holes [96,97] or induced gravitational
waves [98–101] produced by sufficiently large primordial density perturbations could have survived
and might be detectable in the present day. Rapid growth of the power spectrum on small scales
requires violation of slow roll in single-field models [102]. This could arise from features in thescalar field potential, such as an inflection point [103], leading to a transient phase of ultra-slow-roll
evolution [104, 105] which can give a k 4 -rise in the scalar power spectrum on small scales [106, 107].
The LIGO-Virgo-KAGRA collaboration has placed an upper limit on such a stochastic gravitational wave background (SGWB), conventionally expressed in terms of the effective density ofgravitational waves per logarithmic frequency interval, of ΩGW ≤ 5.8 × 10−9 for a scale-invariant
spectrum in the frequency range 20 − 77 Hz [108]. On the other hand, pulsar timing arrays havefound growing evidence for a low-frequency SGWB at nano-Hertz frequencies [109–112]. While
this could be associated with an astrophysical background due to supermassive black hole binary
systems, the NANOGrav collaboration report that a cosmological background provides a better fit
to their 15-year dataset [113]. This could be generated by large primordial density fluctuations on
the corresponding comoving scales, k ∼ 106 Mpc−1 , but these are constrained by the productionof primordial black holes of approximately solar mass [114] and possible one-loop or higher-ordercorrections to the primordial power spectrum on CMB scales.
23.7.2 Effective Field Theory of Inflation
Since slow-roll inflation is a phase of accelerated expansion with an almost constant Hubble
parameter, one may think of inflation in terms of an effective theory where the de Sitter spacetime
symmetry is spontaneously broken down to RW symmetry by the time-evolution of the Hubble
rate, Ḣ 6= 0. There is then a Goldstone boson, π, associated with the spontaneous breaking oftime-translation invariance, which can be used to study model-independent properties of inflation.
The Goldstone boson describes a spacetime-dependent shift of the time coordinate, corresponding
to an adiabatic perturbation of the matter fields:δφi (t, ~x) = φi (t + π(t, ~x)) − φi (t) .

(23.87)

Thus adiabatic field fluctuations can be absorbed into the spatial metric perturbation, R in
Eq. (23.28) at first order, in the comoving gauge:
R = −Hπ ,(23.88)where we define π on spatially-flat hypersurfaces. In terms of inflaton field fluctuations, we canidentify π ≡ δφ/φ̇, but in principle this analysis is not restricted to inflation driven by scalar fields.
The low-energy effective action for π can be obtained by writing down the most general Lorentzinvariant action and expanding in terms of π. The second-order effective action for the free-field
wave modes, πk , to leading order in slow roll is thenSπ(2) = −Z"#

√ M 2 Ḣ 2c2
d x −g P2π̇k − s2 (∇π)2 ,cs
R4

31st May, 2024

(23.89)23. Inflation

23

Figure 23.5: lllustration of the impact of the BICEP/Keck [43] and other constraints on the
inflaton decay coupling, y, and the number of e-folds at a scale 0.05 Mpc−1 , N0.05 , in α-attractor
models of inflation. The horizontal blue lines are 68% and 95% C.L. lower limits on ns . The left andright axes show the relation between N0.05 and ns and the top and bottom scales show the relation
between the inflaton coupling, y, and the reheating temperature, Treh . The diagonal dashed, solidand dotted black lines illustrate the correlations between these quantities for α-attractor models
with α = 0.1, 1, and 5. We also include the lower limit on y from Big Bang Nucleosynthesis (redline), the constraint that Treh be no smaller than the electroweak scale (gray line), and a constraint
from gravitino production (purple line) for α = 1, which strengthens for smaller α. Plot taken from
Ref. [85].
where H is the Hubble slow-roll parameter (Eq. (23.11)). We identify c2s with an effective soundspeed, generalising canonical slow-roll inflation, which is recovered in the limit c2s → 1.
The scalar power spectrum on super-Hubble scales (Eq. (23.48)) is enhanced for a reduced
sound speed, leading to a reduced tensor-scalar ratio (Eq. (23.49))
4π 1
Pζ (k) ' 2 2MP cs 



H
2π

2

,
∗

r ' 16(c2s )∗ .

(23.90)At third perturbative order and to lowest order in derivatives, one obtains [115]Sπ(3) =Z"#

M 2 (1 − c2 )Ḣ π̇(∇π)22 c̃3d x −g P 2 s− 1 + 2 π̇ 3 .
2
cs
R
3 cs
4

√

31st May, 2024



(23.91)

23. Inflation

24

Note that this expression vanishes for canonical fields with c2s = 1. For c2s 6= 1 the cubic action isdetermined by the sound speed and an additional parameter c̃3 . Both terms in the cubic action
give rise to primordial bispectra that are well approximated by equilateral bispectra. However, theshapes are not identical, so one can find a linear combination for which the equilateral bispectra ofeach term cancel, giving rise to a distinctive orthogonal-type bispectrum [115].
Analysis based on Planck 2018 temperature and polarization data has placed bounds on several
bispectrum shapes including equilateral and orthogonal shapes [94]:equilfN= −26 ± 47 ,L

orthogfN= −38 ± 24 (68% CL) .
L(23.92)For the simplest case of a constant sound speed, and marginalising over c̃3 , this provides a boundon the inflaton sound speed [94]cs ≥ 0.021 (95% CL) .
(23.93)For a specific model such as DBI inflation [90], corresponding to c̃3 = 3(1 − c2s )/2, one obtains atighter bound [94]:cDBI≥ 0.086 (95% CL) .
(23.94)s
The Planck team have analysed a wide range of non-Gaussian templates from different inflation models, including tests for deviations from an initial Bunch-Davies vacuum state, directiondependent non-Gaussianity, and feature models with oscillatory bispectra [94]. No individual feature or resonance is above the three-σ significance level after accounting for the look-elsewhere
effect. These results are consistent with the simplest canonical, slow-roll inflation models, but do
not rule out most alternative models; rather, bounds on primordial non-Gaussianity place important
constraints on the parameter space for non-canonical models.
23.7.3 Multi-Field Fluctuations
There is a very large literature on two- and multi-field models of inflation, most of which lies
beyond the scope of this review [116, 117]. However, two important general topics merit being
mentioned here, namely residual isocurvature perturbations and the possibility of non-Gaussian
effects in the primordial perturbations.
One might expect that other scalar fields besides the inflaton might have non-negligible values
that evolve and fluctuate in parallel with the inflaton, without necessarily making the dominant
contribution to the energy density during the inflationary epoch. However, the energy density insuch a field might persist beyond the end of inflation before decaying, at which point it might cometo dominate (or at least make a non-negligible contribution to) the total energy density. In sucha case, its perturbations could end up generating the density perturbations detected in the CMB.
This could occur due to a late-decaying scalar field [91] or a field fluctuation that modulates theend of inflation [118] or the inflaton decay [119].
23.7.3.1 Isocurvature Perturbations
Primordial perturbations arising in single-field slow-roll inflation are necessarily adiabatic, i.e.,they affect the overall density without changing the ratios of different contributions, such as thephoton-matter ratio, δ(nγ /nm )/(nγ /nm ). This is because inflaton perturbations represent a local
shift of the time, as described in section Sec. 23.7.2:
π=

δnγ
δnm
=
.
ṅγ
ṅm

(23.95)

However, any light scalar field (i.e., one with effective mass less than the Hubble scale) acquires aspectrum of nearly scale-invariant perturbations during inflation. Fluctuations orthogonal to the
31st May, 2024

25

23. Inflationinflaton in field space are decoupled from the inflaton at Hubble-exit, but can affect the subsequentevolution of the density perturbation. In particular, they can give rise to local variations in theequation of state (non-adabatic pressure perturbations) that can alter the primordial curvature
perturbation ζ on super-Hubble scales. Since these fluctuations are statistically independent of theinflaton perturbations at leading order in slow roll [117], non-adiabatic field fluctuations can only
increase the scalar power spectrum with respect to adiabatic perturbations at Hubble exit, whileleaving the tensor modes unaffected at first perturbative order. Thus, the single-field result for thetensor-scalar ratio (Eq. (23.49)) becomes an inequality [120]r ≤ 16∗ .

(23.96)

Hence, an observational upper bound on the tensor-scalar ratio does not bound the slow-roll parameter  in multi-field models.
If all the scalar fields present during inflation eventually decay completely into fully thermalizedradiation, these field fluctuations are converted fully into adiabatic perturbations in the primordial
plasma [121]. On the other hand, non-adiabatic field fluctuations can also leave behind primordial
isocurvature perturbations (Eq. (23.85)) after inflation. In multi-field inflation models it is thus possible for non-adiabatic field fluctuations to generate both curvature and isocurvature perturbations
leading to correlated primordial perturbations [122].
The amplitudes of any primordial isocurvature perturbations (Eq. (23.85)) are strongly constrained by the current CMB data, especially on large angular scales. Using temperature and low-`polarization data yields the following bound on the amplitude of cold dark matter isocurvature
perturbations at scale k = 0.002h−1 Mpc−1 (marginalising over the correlation angle and in theabsence of primordial tensor perturbations) [38]:PSm< 0.025 (95% CL) .
Pζ + PSm(23.97)

For fully (anti-)correlated isocurvature perturbations, corresponding to a single isocurvature fieldproviding a source for both the curvature and residual isocurvature perturbations, the bounds
become significantly tighter [38]:PSm< 0.0002 (95% CL), correlated ,Pζ + PSm
PSm< 0.003 (95% CL), anti-correlatedPζ + PSm(23.98)(23.99)23.7.3.2 Local-Type Non-Gaussianity
Since non-adiabatic field fluctuations in multi-field inflation may lead the to evolution of theprimordial curvature perturbation at all orders, it becomes possible to generate significant nonGaussianity in the primordial curvature perturbation. Non-linear evolution on super-Hubble scales
leads to local-type non-Gaussianity, where the local integrated expansion is a non-linear functionof the local field values during inflation, N (φi ). While the field fluctuations at Hubble exit, δφi∗ ,are Gaussian in the slow-roll limit, the curvature perturbation, ζ = δN , becomes a non-Gaussian
distribution [123]:X ∂N
1 X ∂2N
δφi +δφi δφj + . . .
(23.100)ζ=∂φi
2 i,j ∂φi ∂φjiwith non-vanishing bispectrum in the squeezed limit (k1 ≈ k2  k3 ):Bζ (k1 , k2 , k3 ) ≈12 local Pζ (k1 ) Pζ (k3 )f,
5 N L 4πk13 4πk33

31st May, 2024

(23.101)26

23. Inflation

where

∂2N

P

i,j ∂φ ∂φ
6 localf= P i 2j .
5 NL∂N(23.102)i ∂φiBoth equilateral and orthogonal bispectra, discussed above in the context of generalised single
field inflation, vanish in the squeezed limit, enabling the three types of non-Gaussianity to be
distinguished by observations, in principle.
local can often
Non-Gaussianity during multi-field inflation is highly model dependent, though fNL
be smaller than unity in multi-field slow-roll inflation [124]. Scenarios where a second light field
local , such as f local = −5/4
plays a role during or after inflation can make distinctive predictions for fN
L
NL
local
in some curvaton scenarios [123,125] or fN L = 5 in simple modulated reheating scenarios [119,126].
By contrast the constancy of ζ on super-Hubble scales in single-field slow-roll inflation leads to a
local =
very small non-Gaussianity [127, 128], and in the squeezed limit we have the simple result fNL
5(1 − nS )/12 [129, 130].
A combined analysis of the Planck 2018 temperature and polarization data [94] yields the
local defined in (Eq. (23.102)):
following range for fN
L
local
fN
L = −1 ± 5 (68% CL) .

(23.103)

This sensitivity is sufficient to rule out parameter regimes giving rise to relatively large nonlocal = O(), as expected in single-field models, or theGaussianity, but insufficient to probe fN
L
local = O(1) found in the simplest two-field models.
range fN
L
Local-type primordial non-Gaussianity can also give rise to a striking scale-dependent bias in
the distribution of collapsed dark matter halos and thus the galaxy distribution [131, 132]. Bounds
from high-redshift galaxy surveys are not currently competitive with the best CMB constraints.
An analysis based on the clustering of quasars in the final data release (DR16) of the extended
Baryon acoustic Oscillation Spectroscopic Survey (eBOSS) [133] yields
local
fN
L = −12 ± 21 (68% CL) .

(23.104)

A recent analysis based on the clustering of luminous red galaxies in imaging surveys from the Dark
local = 47+14 , suggesting that f local > 0
Energy Spectroscopic Instrument (DESI) [134] finds fN−11LNL
at more than 99.9% confidence, although the possibility that this could be due to unforeseensystematics cannot yet be excluded.

23.8 Initial Conditions and Fine-tuning
This review is based on the assumption that the inflationary paradigm is valid. However, itremains the object of many criticisms (see, e.g., [135]), many of them related to the perceived
unnaturalness of the required initial conditions.
Most work on inflation is done in the context of RW cosmology, which assumes a high degreeof symmetry, or small inhomogeneous perturbations (usually first order) about an RW cosmology.
The isotropic RW space-time is an attractor for many homogeneous but anisotropic cosmologies in
the presence of a false vacuum energy density [136], or a scalar field with suitable self-interaction
potential energy [137, 138]. However it is much harder to establish the range of highly inhomogeneous initial conditions that yield a successful RW Universe, with only limited studies initially(see, e.g., [139, 140]). A related open question is the general nature of the pre-inflationary state of
the inflaton and other fields that could have provided initial conditions suitable for inflation [135].
They would need to have satisfied non-trivial homogeneity and isotropy conditions, and one may
ask how these could have arisen, whether these are plausible, and whether there may be some
31st May, 2024

27

23. Inflation

observable signature of the pre-inflationary state. These and other criticisms of inflation were addressed in [141], which presented studies of the sensitivity of inflation to the initial conditions.
Complementing the studies reported in [141], there have been numerical relativity investigations ofhighly inhomogeneous initial conditions [142–144]. The general conclusion is that inflation is rather
robust with respect to inhomogeneities in the initial conditions in both the scalar field profile andthe extrinsic curvature, including large tensor perturbations.
To quantify the fine-tuning of initial conditions requires a measure in the space of possible
cosmologies [145], however it has been argued that some of the measures historically used to frame
this problem are formally invalid [146]. It is sometimes also objected that inflationary models
predict the existence of a multiverse, and potentially a loss of predictive power [147], if it undergoes
the process termed eternal inflation [148–150]. However, whether this is actually a bug or a featureremains a topic of debate [151, 152]. The existence of the multiverse is a purely philosophicalproblem, unless it has observable consequences, e.g., in the CMB.
One might expect signatures of any pre-inflationary state to appear at large angular scales, i.e.,
low multipoles `. Indeed, various anomalies have been noted in the large-scale CMB anisotropies,
as also discussed in Chap. 29, the “Cosmic Microwave Background” review, including a possible
suppression of the quadrupole and other very large-scale anisotropies, an apparent feature in therange ` ≈ 20 to 30, and a possible hemispheric asymmetry. However, none of these are highly
significant statistically, in view of the limitations due to cosmic variance [64]. They cannot yet be
regarded as signatures of initial conditions, the multiverse or some pre-inflationary dynamics, suchas might emerge from string theory.
A different kind of initial condition problem, called the trans-Planckian problem [153], is thatthe perturbations now seen in the CMB would have had wavelengths shorter than the Planck
length at the onset of inflation. However, under quite general and conservative assumptions theusual inflationary predictions would be quite robust [154], with the possibility of O((H/mP )n )corrections that might have interesting signatures in the CMB [155].
When inflation was first proposed [1] [2] there was no evidence for the existence of scalar fields orthe accelerated expansion of the Universe. The situation has changed dramatically in recent years
with the observational evidence that the cosmic expansion is currently accelerating and with thediscovery of a scalar particle, namely the Higgs boson (see Chap. 11, “Status of Higgs boson physics”
review). Combined with the lack of any widely accepted alternative model for the origin of cosmic
structure, these discoveries have lent support to the idea of a primordial accelerated expansion
driven by a scalar field, i.e., cosmological inflation. In parallel, successive CMB experiments havebeen consistent with generic predictions of inflationary models, although without yet providing
irrefutable evidence. It was concluded in [141] that the inflationary paradigm is not currently introuble. However, we note that inflation via a formally elementary scalar inflaton should probably
only be regarded as an effective field theory valid at energy densities hierarchically smaller than
the Planck scale. It should eventually be embedded in a suitable ultraviolet completion, on whichinflationary dynamics may be our clearest window.

23.9 Future Probes of Inflation
Prospective future CMB experiments, both ground- and space-based are reviewed in the separate PDG “Cosmic Microwave Background” review, Chap. 29. The main emphasis in CMB
experiments in the coming years will be on ground-based experiments providing improved measurements of B-mode polarization and greater sensitivity to the tensor-to-scalar ratio r, and more
precise measurements at higher ` that will constrain ns better. As is apparent from Fig. 23.1 andthe discussion of models such as R2 inflation, there is a strong incentive to reach a 5-σ sensitivity
to r ∼ 3 to 4 × 10−3 . This could be achieved with a moderately-sized space mission with large
31st May, 2024

28

23. Inflation

sky coverage [156], improvements in de-lensing and foreground measurements. The discussion in
Sec. 23.3 (see also Fig. 23.5), also brought out the importance of reducing the uncertainty in ns ,as a way to constrain post-inflationary reheating and the connection to particle physics. CMB
temperature anisotropies probe primordial density perturbations down to comoving scales of order
50 Mpc, beyond which scale secondary sources of anisotropy dominate. CMB spectral distortions
could potentially constrain the amplitude and shape of primordial density perturbations on comoving scales from Mpc to kpc due to distortions caused by the Silk damping of pressure waves in theradiation dominated era, before the last scattering of the CMB photons but after the plasma canbe fully thermalised [95].
Improved sensitivity to non-Gaussianities is also a priority. In addition to CMB measurements,
future large-scale structure surveys will also have roles to play as probes into models of inflation,
for which there are excellent prospects. Current surveys such as DESI may reach ∆fN L ∼ 4 [157]comparable with the Planck sensitivity. In the future, radio surveys such as SKA will measurelarge-scale structure out to redshift z ∼ 3 [158], initially through mapping the intensity of theneutral hydrogen 21-cm line, and eventually through radio galaxy surveys which will probe localtype non-Gaussianity to fN L ∼ 1.
Galaxy clustering using DESI and Euclid satellite data could also constrain the running of thescalar tilt to a precision of ∆αs ≈ 0.0028, a factor of 2 improvement on Planck constraints, or a
precision of 0.0016 using LSST data [157].
As an example of a proposed future satellite mission, SPHEREx [159] will use measurements ofthe galaxy power spectrum to target a measurement of the running of the scalar spectral index witha sensitivity ∆αs ∼ 10−3 and local-type primordial non-Gaussianity, ∆fN L ∼ 1. Including information from the galaxy bispectrum one might reduce the measurement error on non-Gaussianity to∆fN L ∼ 0.2, making it possible to distinguish between single-field slow-roll models and alternatives
such as the curvaton scenario for the origin of structure, which generate fN L ∼ 1.
Finally we note that future gravitational wave experiments such as the Laser InterferometerSpace Antenna (LISA) will be sensitive to a stochastic gravitational wave background, including
tensor perturbations produced during inflation or induced by large primordial density perturbations
on small scales after inflation [160], which could provide constraints on inflationary perturbations
on smaller scales.
Acknowledgements
The authors are grateful to Vincent Vennin for preparing Fig. 23.4 for this review. The work of
J.E. was supported in part by the UK STFC research grant ST/T000759/1. The work of D.W. was
supported in part by the UK STFC research grant ST/W001225/1.
References[1] A. A. Starobinsky, Phys. Lett. B91, 99 (1980), [Adv. Ser. Astrophys. Cosmol.3,130(1987)].
[2] A. H. Guth, Phys. Rev. D23, 347 (1981), [Adv. Ser. Astrophys. Cosmol.3,139(1987)].
[3] K. A. Olive, Phys. Rept. 190, 307 (1990).
[4] D. H. Lyth and A. Riotto, Phys. Rept. 314, 1 (1999), [hep-ph/9807278].
[5] A.R. Liddle and D.H. Lyth, Cosmological inflation and large-scale structure (Cambridge University Press, 2000).
[6] D. Baumann, in “Physics of the large and the small, TASI 09, proceedings of the Theoretical
Advanced Study Institute in Elementary Particle Physics, Boulder, Colorado, USA, 1-26 June
2009,” 523–686 (2011), [arXiv:0907.5424].
31st May, 2024

29

23. Inflation[7] J. Martin, C. Ringeval and V. Vennin, Phys. Dark Univ. 5-6, 75 (2014), [arXiv:1303.3787];
J. Martin et al., JCAP 1403, 039 (2014), [arXiv:1312.3529]; J. Martin, Astrophys. Space Sci.
Proc. 45, 41 (2016), [arXiv:1502.05733].
[8] P. A. R. Ade et al. (Planck), Astron. Astrophys. 594, A13 (2016), [arXiv:1502.01589].
[9] A. H. Guth and E. J. Weinberg, Nucl. Phys. B212, 321 (1983).
[10] D. La and P. J. Steinhardt, Phys. Rev. Lett. 62, 376 (1989), [Erratum: Phys. Rev.
Lett.62,1066(1989)].
[11] A. D. Linde, Phys. Lett. B249, 18 (1990).
[12] F. C. Adams and K. Freese, Phys. Rev. D43, 353 (1991), [hep-ph/0504135].
[13] A. D. Linde, Phys. Lett. 108B, 389 (1982), [Adv. Ser. Astrophys. Cosmol.3,149(1987)].
[14] A. Albrecht and P. J. Steinhardt, Phys. Rev. Lett. 48, 1220 (1982), [Adv. Ser. Astrophys.
Cosmol.3,158(1987)].
[15] W. H. Press, Phys. Scripta 21, 702 (1980); S. W. Hawking, Phys. Lett. 115B, 295 (1982);
A. A. Starobinsky, Phys. Lett. 117B, 175 (1982); A. H. Guth and S. Y. Pi, Phys. Rev. Lett.
49, 1110 (1982).
[16] J. M. Bardeen, P. J. Steinhardt and M. S. Turner, Phys. Rev. D28, 679 (1983).
[17] V. F. Mukhanov and G. V. Chibisov, JETP Lett. 33, 532 (1981), [Pisma Zh. Eksp. Teor.
Fiz.33,549(1981)].
[18] K. S. Stelle, Gen. Rel. Grav. 9, 353 (1978).
[19] B. Whitt, Phys. Lett. 145B, 176 (1984).
[20] D. Wands, Class. Quant. Grav. 11, 269 (1994), [arXiv:gr-qc/9307034].
[21] S. R. Coleman and F. De Luccia, Phys. Rev. D21, 3305 (1980).
[22] M. Sasaki et al., Phys. Lett. B317, 510 (1993).
[23] M. Bucher, A. S. Goldhaber and N. Turok, Phys. Rev. D52, 3314 (1995), [hep-ph/9411206].
[24] A. D. Linde and A. Mezhlumian, Phys. Rev. D52, 6789 (1995), [arXiv:astro-ph/9506017].
[25] A. R. Liddle, P. Parsons and J. D. Barrow, Phys. Rev. D50, 7222 (1994), [arXiv:astroph/9408015].
[26] L. Kofman, A. D. Linde and A. A. Starobinsky, Phys. Rev. D56, 3258 (1997), [hepph/9704452].
[27] B. A. Bassett, S. Tsujikawa and D. Wands, Rev. Mod. Phys. 78, 537 (2006), [arXiv:astroph/0507632].
[28] A. D. Linde, Phys. Rev. D49, 748 (1994), [arXiv:astro-ph/9307002].
[29] A. D. Dolgov and A. D. Linde, Phys. Lett. 116B, 329 (1982).
[30] J. R. Ellis et al., Nucl. Phys. B238, 453 (1984), [,223(1983)].
[31] M. Kawasaki and T. Moroi, Prog. Theor. Phys. 93, 879 (1995), [hep-ph/9403364].
[32] J. H. Traschen and R. H. Brandenberger, Phys. Rev. D42, 2491 (1990).
[33] G. V. Chibisov and V. F. Mukhanov, Mon. Not. Roy. Astron. Soc. 200, 535 (1982).
[34] H. Kodama and M. Sasaki, Prog. Theor. Phys. Suppl. 78, 1 (1984).
[35] V.F. Mukhanov, H.A. Feldman and R.H. Brandenberger, Phys. Rept. 215, 203 (1992).
[36] K. A. Malik and D. Wands, Phys. Rept. 475, 1 (2009), [arXiv:0809.4944].
[37] V. F. Mukhanov, Sov. Phys. JETP 67, 1297 (1988), [Zh. Eksp. Teor. Fiz.94N7,1(1988)].
31st May, 2024

30

23. Inflation[38] Y. Akrami et al. (Planck), Astron. Astrophys. 641, A10 (2020), [arXiv:1807.06211].
[39] A. A. Starobinsky, JETP Lett. 30, 682 (1979), [,767(1979)].
[40] E. D. Stewart and D. H. Lyth, Phys. Lett. B302, 171 (1993), [arXiv:gr-qc/9302019].
[41] D. Wands et al., Phys. Rev. D62, 043527 (2000), [arXiv:astro-ph/0003278].
[42] A. R. Liddle and S. M. Leach, Phys. Rev. D68, 103503 (2003), [arXiv:astro-ph/0305263].
[43] P. A. R. Ade et al. (BICEP/Keck), Phys. Rev. Lett. 127, 15, 151301 (2021),[arXiv:2110.00483].
[44] S.M. Leach, A.R. Liddle, J. Martin, and D.J. Schwarz, Phys. Rev. D66, 23515 (2002).
[45] D. Kazanas, Astrophys. J. 241, L59 (1980).
[46] K. Sato, Mon. Not. Roy. Astron. Soc. 195, 467 (1981).
[47] A. Billoire and K. Tamvakis, Nucl. Phys. B200, 329 (1982); J. D. Breit, S. Gupta and A. Zaks,
Phys. Rev. Lett. 51, 1007 (1983).
[48] A. D. Linde, Phys. Lett. 129B, 177 (1983).
[49] D. V. Nanopoulos, K. A. Olive and M. Srednicki, Phys. Lett. 127B, 30 (1983).
[50] J. Ellis, D. V. Nanopoulos and K. A. Olive, Phys. Rev. Lett. 111, 111301 (2013), [Erratum:
Phys. Rev. Lett.111,no.12,129902(2013)], [arXiv:1305.1247].
[51] T. S. Koivisto and F. R. Urban, JCAP 1503, 03, 003 (2015), [arXiv:1407.3445].
[52] L. Boubekeur and D. H. Lyth, JCAP 0507, 010 (2005), [hep-ph/0502047].
[53] G. R. Dvali, Q. Shafi and S. Solganik, in “4th European Meeting From the Planck Scaleto the Electroweak Scale (Planck 2001) La Londe les Maures, Toulon, France, May 11-16,
2001,” (2001), [hep-th/0105203]; J. Garcia-Bellido, R. Rabadan and F. Zamora, JHEP 01,
036 (2002), [hep-th/0112147]; S. Kachru et al., JCAP 0310, 013 (2003), [hep-th/0308055].
[54] F. C. Adams et al., Phys. Rev. D47, 426 (1993), [hep-ph/9207245].
[55] E. Pajer and M. Peloso, Class. Quant. Grav. 30, 214002 (2013), [arXiv:1305.3557].
[56] E. Silverstein and A. Westphal, Phys. Rev. D78, 106003 (2008), [arXiv:0803.3085].
[57] L. McAllister, E. Silverstein and A. Westphal, Phys. Rev. D82, 046003 (2010),[arXiv:0808.0706].
[58] F. L. Bezrukov and M. Shaposhnikov, Phys. Lett. B659, 703 (2008), [arXiv:0710.3755].
[59] D. Buttazzo et al., JHEP 12, 089 (2013), [arXiv:1307.3536].
[60] H. P. Nilles, Phys. Rept. 110, 1 (1984); H. E. Haber and G. L. Kane, Phys. Rept. 117, 75(1985).
[61] J.R. Ellis et al., Phys. Lett. 118B, 335 (1982).
[62] N. Okada and Q. Shafi (2013), [arXiv:1311.0921].
[63] J. Martin, C. Ringeval and V. Vennin, Phys. Rev. D93, 10, 103532 (2016), [arXiv:1603.02606].
[64] P. A. R. Ade et al. (Planck), Astron. Astrophys. 594, A20 (2016), [arXiv:1502.02114].
[65] H. Murayama et al., Phys. Rev. Lett. 70, 1912 (1993).
[66] J. Ellis, M. Fairbairn and M. Sueiro, JCAP 1402, 044 (2014), [arXiv:1312.1353].
[67] D. Croon, J. Ellis and N. E. Mavromatos, Phys. Lett. B724, 165 (2013), [arXiv:1303.6253].
[68] G. R. Dvali, Q. Shafi and R. K. Schaefer, Phys. Rev. Lett. 73, 1886 (1994), [hep-ph/9406319].
[69] D. V. Nanopoulos et al., Phys. Lett. 123B, 41 (1983).
[70] A. B. Goncharov and A. D. Linde, Phys. Lett. 139B, 27 (1984).
31st May, 2024

31

23. Inflation[71] E. Cremmer et al., Nucl. Phys. B212, 413 (1983), [,413(1982)].
[72] R. Holman, P. Ramond and G. G. Ross, Phys. Lett. 137B, 343 (1984).
[73] E. J. Copeland et al., Phys. Rev. D49, 6410 (1994), [arXiv:astro-ph/9401011]; E. D. Stewart,
Phys. Rev. D51, 6847 (1995), [hep-ph/9405389].
[74] G. D. Coughlan et al., Phys. Lett. 131B, 59 (1983); A. S. Goncharov, A. D. Linde and M. I.
Vysotsky, Phys. Lett. 147B, 279 (1984); T. Banks, D. B. Kaplan and A. E. Nelson, Phys.
Rev. D49, 779 (1994), [hep-ph/9308292]; B. de Carlos et al., Phys. Lett. B318, 447 (1993),[hep-ph/9308325]; M. Kawasaki, T. Moroi and T. Yanagida, Phys. Lett. B370, 52 (1996),[hep-ph/9509399].
[75] J. R. Ellis, D. V. Nanopoulos and M. Quiros, Phys. Lett. B174, 176 (1986).
[76] T. Moroi, M. Yamaguchi and T. Yanagida, Phys. Lett. B342, 105 (1995), [hep-ph/9409367].
[77] M. Kawasaki, M. Yamaguchi and T. Yanagida, Phys. Rev. Lett. 85, 3572 (2000), [hepph/0004243]; K. Nakayama, F. Takahashi and T. T. Yanagida, JCAP 1308, 038 (2013),[arXiv:1305.5099].
[78] E. Cremmer et al., Phys. Lett. 133B, 61 (1983).
[79] A. S. Goncharov and A. D. Linde, Class. Quant. Grav. 1, L75 (1984); C. Kounnas and
M. Quiros, Phys. Lett. 151B, 189 (1985).
[80] J. R. Ellis, C. Kounnas and D. V. Nanopoulos, Nucl. Phys. B247, 373 (1984).
[81] E. Witten, Phys. Lett. 155B, 151 (1985).
[82] J. Ellis, D. V. Nanopoulos and K. A. Olive, JCAP 1310, 009 (2013), [arXiv:1307.3537].
[83] J. Ellis et al., Class. Quant. Grav. 33, 9, 094001 (2016), [arXiv:1507.02308].
[84] R. Kallosh, A. Linde and D. Roest, JHEP 11, 198 (2013), [arXiv:1311.0472].
[85] J. Ellis et al., Phys. Rev. D 105, 4, 043504 (2022), [arXiv:2112.04466].
[86] L. Iacconi et al. (2023), [arXiv:2306.00918].
[87] J. Ellis et al., JCAP 1507, 07, 050 (2015), [arXiv:1505.06986].
[88] A. R. Liddle, Mon. Not. Roy. Astron. Soc. 377, L74 (2007), [arXiv:astro-ph/0701113].
[89] C. Armendariz-Picon, T. Damour and V. F. Mukhanov, Phys. Lett. B458, 209 (1999), [hepth/9904075].
[90] M. Alishahiha, E. Silverstein and D. Tong, Phys. Rev. D70, 123505 (2004), [hep-th/0404084].
[91] K. Enqvist and M. S. Sloth, Nucl. Phys. B626, 395 (2002), [hep-ph/0109214]; D. H. Lythand D. Wands, Phys. Lett. B524, 5 (2002), [hep-ph/0110002]; T. Moroi and T. Takahashi,
Phys. Lett. B522, 215 (2001), [Erratum: Phys. Lett.B539,303(2002)], [hep-ph/0110096].
[92] N. Bartolo et al., Phys. Rept. 402, 103 (2004), [arXiv:astro-ph/0406398].
[93] P. A. R. Ade et al. (Planck), Astron. Astrophys. 594, A17 (2016), [arXiv:1502.01592].
[94] Y. Akrami et al. (Planck), Astron. Astrophys. 641, A9 (2020), [arXiv:1905.05697].
[95] J. Chluba, J. Hamann and S. P. Patil, Int. J. Mod. Phys. D24, 10, 1530023 (2015),[arXiv:1505.01834].
[96] Y. B. Zel’dovich and I. D. Novikov, Soviet Astron. AJ (Engl. Transl. ), 10, 602 (1967).
[97] S. W. Hawking, Nature 248, 30 (1974).
[98] S. Matarrese, O. Pantano and D. Saez, Phys. Rev. Lett. 72, 320 (1994), [arXiv:astroph/9310036].
31st May, 2024

32

23. Inflation[99] S. Matarrese, S. Mollerach and M. Bruni, Phys. Rev. D 58, 043504 (1998), [arXiv:astroph/9707278].
[100] K. N. Ananda, C. Clarkson and D. Wands, Phys. Rev. D 75, 123518 (2007), [arXiv:grqc/0612013].
[101] D. Baumann et al., Phys. Rev. D 76, 084019 (2007), [hep-th/0703290].
[102] H. Motohashi and W. Hu, Phys. Rev. D 96, 6, 063503 (2017), [arXiv:1706.06784].
[103] J. Garcia-Bellido and E. Ruiz Morales, Phys. Dark Univ. 18, 47 (2017), [arXiv:1702.03901].
[104] S. Inoue and J. Yokoyama, Phys. Lett. B 524, 15 (2002), [hep-ph/0104083].
[105] W. H. Kinney, Phys. Rev. D 72, 023515 (2005), [arXiv:gr-qc/0503017].
[106] S. M. Leach et al., Phys. Rev. D 64, 023512 (2001), [arXiv:astro-ph/0101406].
[107] C. T. Byrnes, P. S. Cole and S. P. Patil, JCAP 06, 028 (2019), [arXiv:1811.11158].
[108] R. Abbott et al. (KAGRA, Virgo, LIGO Scientific), Phys. Rev. D 104, 2, 022004 (2021),[arXiv:2101.12130].
[109] G. Agazie et al. (NANOGrav), Astrophys. J. Lett. 951, 1, L8 (2023), [arXiv:2306.16213].
[110] J. Antoniadis et al. (EPTA) (2023), [arXiv:2306.16214].
[111] D. J. Reardon et al., Astrophys. J. Lett. 951, 1, L6 (2023), [arXiv:2306.16215].
[112] H. Xu et al., Res. Astron. Astrophys. 23, 7, 075024 (2023), [arXiv:2306.16216].
[113] A. Afzal et al. (NANOGrav), Astrophys. J. Lett. 951, 1, L11 (2023), [arXiv:2306.16219].
[114] B. Carr et al., Rept. Prog. Phys. 84, 11, 116902 (2021), [arXiv:2002.12778].
[115] L. Senatore, K. M. Smith and M. Zaldarriaga, JCAP 1001, 028 (2010), [arXiv:0905.3746].
[116] C. Gordon et al., Phys. Rev. D63, 023506 (2001), [arXiv:astro-ph/0009131]; R. Easther
et al., Phys. Rev. Lett. 112, 161302 (2014), [arXiv:1312.4035]; J. Ellis et al., JCAP 1501,
010 (2015), [arXiv:1409.8197]; S. Renaux-Petel and K. Turzynski, JCAP 1506, 06, 010 (2015),[arXiv:1405.6195].
[117] C. T. Byrnes and D. Wands, Phys. Rev. D74, 043529 (2006), [arXiv:astro-ph/0605679].
[118] D. H. Lyth, JCAP 0511, 006 (2005), [arXiv:astro-ph/0510443].
[119] G. Dvali, A. Gruzinov and M. Zaldarriaga, Phys. Rev. D69, 023505 (2004), [arXiv:astroph/0303591].
[120] D. Wands et al., Phys. Rev. D66, 043520 (2002), [arXiv:astro-ph/0205253].
[121] S. Weinberg, Phys. Rev. D67, 123504 (2003), [arXiv:astro-ph/0302326].
[122] D. Langlois, Phys. Rev. D59, 123512 (1999), [arXiv:astro-ph/9906080].
[123] D. H. Lyth and Y. Rodriguez, Phys. Rev. Lett. 95, 121302 (2005), [arXiv:astro-ph/0504045].
[124] F. Vernizzi and D. Wands, JCAP 0605, 019 (2006), [arXiv:astro-ph/0603799].
[125] M. Sasaki, J. Valiviita and D. Wands, Phys. Rev. D74, 103003 (2006), [arXiv:astroph/0607627].
[126] G. Dvali, A. Gruzinov and M. Zaldarriaga, Phys. Rev. D69, 083505 (2004), [arXiv:astroph/0305548].
[127] D. S. Salopek and J. R. Bond, Phys. Rev. D43, 1005 (1991).
[128] A. Gangui et al., Astrophys. J. 430, 447 (1994), [arXiv:astro-ph/9312033].
[129] J. M. Maldacena, JHEP 05, 013 (2003), [arXiv:astro-ph/0210603].
[130] V. Acquaviva et al., Nucl. Phys. B667, 119 (2003), [arXiv:astro-ph/0209156].
31st May, 2024

33

23. Inflation[131] N. Dalal et al., Phys. Rev. D77, 123514 (2008), [arXiv:0710.4560].
[132] S. Matarrese and L. Verde, Astrophys. J. 677, L77 (2008), [arXiv:0801.4826].
[133] E.-M. Mueller et al. (2021), [arXiv:2106.13725].
[134] M. Rezaie et al. (2023), [arXiv:2307.01753].
[135] A. Ijjas, P. J. Steinhardt and A. Loeb, Phys. Lett. B723, 261 (2013), [arXiv:1304.2785].
[136] R. M. Wald, Phys. Rev. D28, 2118 (1983).
[137] M. Heusler, Phys. Lett. B253, 33 (1991).
[138] Y. Kitada and K.-i. Maeda, Phys. Rev. D45, 1416 (1992).
[139] D. S. Goldwirth and T. Piran, Phys. Rept. 214, 223 (1992).
[140] T. Vachaspati and M. Trodden, Phys. Rev. D61, 023502 (1999), [arXiv:gr-qc/9811037].
[141] D. Chowdhury et al., Phys. Rev. D 100, 8, 083537 (2019), [arXiv:1902.03951].
[142] W. E. East et al., JCAP 1609, 09, 010 (2016), [arXiv:1511.05143].
[143] K. Clough et al., JCAP 1709, 09, 025 (2017), [arXiv:1608.04408].
[144] K. Clough, R. Flauger and E. A. Lim, JCAP 1805, 05, 065 (2018), [arXiv:1712.07352].
[145] G. W. Gibbons, S. W. Hawking and J. M. Stewart, Nucl. Phys. B281, 736 (1987).
[146] J. S. Schiffrin and R. M. Wald, Phys. Rev. D86, 023521 (2012), [arXiv:1202.1818].
[147] A. Ijjas, P. J. Steinhardt and A. Loeb, Phys. Lett. B736, 142 (2014), [arXiv:1402.6980].
[148] A. Vilenkin, Phys. Rev. D27, 2848 (1983).
[149] A. D. Linde, Phys. Lett. B175, 395 (1986).
[150] A. S. Goncharov, A. D. Linde and V. F. Mukhanov, Int. J. Mod. Phys. A2, 561 (1987).
[151] A. H. Guth, D. I. Kaiser and Y. Nomura, Phys. Lett. B733, 112 (2014), [arXiv:1312.7619].
[152] A. Linde, in “Proceedings, 100th Les Houches Summer School: Post-Planck Cosmology: Les
Houches, France, July 8 - August 2, 2013,” 231–316 (2015), [arXiv:1402.0526].
[153] J. Martin and R. H. Brandenberger, Phys. Rev. D63, 123501 (2001), [hep-th/0005209].
[154] R. H. Brandenberger and J. Martin, Mod. Phys. Lett. A16, 999 (2001), [arXiv:astroph/0005432].
[155] J. Martin and R. H. Brandenberger, in “Recent developments in theoretical and experimental
general relativity, gravitation and relativistic field theories. Proceedings, 9th Marcel Grossmann Meeting, MG’9, Rome, Italy, July 2-8, 2000. Pts. A-C,” 2001–2002 (2000), [arXiv:astroph/0012031].
[156] A. Kogut et al., JCAP 1107, 025 (2011), [arXiv:1105.2044].
[157] A. Font-Ribera et al., JCAP 1405, 023 (2014), [arXiv:1308.4164].
[158] R. Maartens et al. (SKA Cosmology SWG), PoS AASKA14, 016 (2015), [arXiv:1501.04076].
[159] O. Doré et al. (2014), [arXiv:1412.4872].
[160] P. Auclair et al. (LISA Cosmology Working Group) (2022), [arXiv:2204.05434].

31st May, 2024