The simplest models of inflation: recent
results and expected discoveries
Alexei A. Starobinsky
Landau Institute for Theoretical Physics RAS,
Moscow - Chernogolovka, Russia
20th Lomonosov Conference on Elementary
Particle Physics
Moscow State University, Moscow, 23.08.2021

Inflation and two new fundamental observational parameters
The simplest one-parametric inflationary models
The mixed R 2 -Higgs model
Post-inflationary behaviour of the mixed R 2 -Higgs model
Perspectives of future discoveries
Beyond the slow-roll approximation
Conclusions

Inflation
The inflationary hypothesis :Some part of the world which includes all its presently
observable part was as much symmetric as possible duringsome period in the past - both with respect to the geometricalbackground and to the state of all quantum fields (noparticles).
Non-universal (due to the specific initial condition) explanationof the cosmological arrow to time - chaos, entropy (in some
not well defined sense) can only grow after inflation.
Still this state is an intermediate attractor for a set of
pre-inflationary initial conditions with a non-zero measure.
Also it is not a unique one, there exists a class of such statesleading to the same observable predictions.

Successive realization of this idea is based on the two moredetailed and independent assumptions.
1. Existence of a metastable quasi-de Sitter stage in our
remote part which preceded the hot Big Bang. During it, theexpansion of the Universe was accelerated and close to the
exponential one, |Ḣ|  H 2 .
2. The origin of all inhomogeneities in the present Universe is
the effect of gravitational creation of pairs of particles antiparticles and field fluctuations during inflation from theadiabatic vacuum (no-particle) state for Fourier modes
covering all observable range of scales (and possibly somewhat
beyond).

Remark regarding these initial conditions for perturbations:they are not in the Bunch-Davies state in the rigorous sense,
since they may not be imposed for arbitrary large scales. As aconsequence, inflationary models typically does not predict
regular behaviour at spatial infinity both during and after
inflation (”multiverse”).
Existing analogies in other areas of physics.
1. The present dark energy, though the required degree ofmetastability for the primordial dark energy is much more thanis proved for the present one (more than 60 e-folds vs. ∼ 3).
2. Creation of electrons and positrons in an external electricfield.

Outcome of inflation
In the super-Hubble regime (k  aH) in the coordinate
representation:
ds 2 = dt 2 − a2 (t)(δlm + hlm )dx l dx m , l, m = 1, 2, 3
hlm = 2R(r)δlm +

2
X

(a)

g (a) (r) elm

a=1
l(a)

el

(a)

(a)

= 0, g ,l eml(a) = 0, elm e lm(a) = 1

R describes primordial scalar perturbations, g – primordial
tensor perturbations (primordial gravitational waves (GW)).
The most important quantities:
ns (k) − 1 ≡Pgd ln PR (k), r (k) ≡d ln kPR

In fact, metric perturbations hlm are quantum (operators inthe Heisenberg representation) and remain quantum up to thepresent time. But, after omitting of a very small part,
decaying with time, they become commuting and, thus,
equivalent to classical (c-number) stochastic quantities with
the Gaussian statistics (up to small terms quadratic in R, g ).
In particular:R̂k = Rk i(âk −âk† )+O(âk − âk† )2+...+O(10−100 )(âk +âk† )+, , ,The last term is time dependent, it is affected by physicaldecoherence and may become larger, but not as large as thesecond term.
Remaining quantum coherence: deterministic correlation
between k and −k modes - shows itself in the appearance of
acoustic oscillations (primordial oscillations in case of GW).

CMB temperature anisotropy
Planck-2015: P. A. R. Ade et al., arXiv:1502.01589

CMB temperature anisotropy multipoles
6000

D`T T [µK2 ]

5000
4000
3000
2000
1000

D`T T

0
600

60
30
0
-30
-60

300
0
-300
-600
2

10

30

500

1000

`

1500

2000

2500

CMB E-mode polarization multipoles
C`EE [10 5 µK2 ]

100
80
60
40
20

C`EE

0
4
0
-4
30

500

1000

`

1500

2000

New cosmological parameters relevant to inflation
Now we have numbers: N. Agranim et al., arXiv:1807.06209
The primordial spectrum of scalar perturbations has beenmeasured and its deviation from the flat spectrum ns = 1 in
the first order in |ns − 1| ∼ NH−1 has been discovered (using
the multipole range ` > 40):
 ns −1Z
PR (k)k2−9< R (r) >=dk, PR (k) = (2.10 ± 0.03)·10kk0k0 = 0.05 Mpc−1 , ns − 1 = −0.035 ± 0.004

Two fundamental observational constants of cosmology inaddition to the three known ones (baryon-to-photon ratio,
baryon-to-matter density and the cosmological constant).
Existing inflationary models can predict (and predicted, infact) one of them, namely ns − 1, relating it finally to
B Tγ
NH = ln k~H≈ 67.2. (note that (1 − ns )NH ∼ 2).
0

The simplest models producing the observed scalarslope
1. The R + R 2 model (Starobinsky 1980):
f (R)
R2
2
, MPl
= G −1, f (R) = R +
16πG
6M 2
 
55
−6
M = 2.6 × 10
MPl ≈ 3.1 × 1013 GeVNL=

ns − 1 = −212r≈ −0.036, r = 2 ≈ 0.004, nt = −
N
N
8

kf
Tγ
= ln
− O(10), HdS (N = 55) = 1.4 × 1014 GeVkk
2. The same prediction from a scalar field model with
4
V (φ) = λφ4 at large φ and strong non-minimal coupling togravity ξRφ2 with ξ < 0, |ξ|  1 (Spokoiny 1984), including
the Higgs inflationary model (Bezrukov & Shaposhnikov 2008).
N = lnThe simplest purely geometrical inflationary model
L=

R
N2
+
R 2 + (small rad. corr.)
16πG
288π 2 PR (k)R+ 5.1 × 108 R 2 + (small rad. corr.)
16πG
The quantum effect of creation of particles and field
fluctuations works twice in this model:a) at super-Hubble scales during inflation, to generate
space-time metric fluctuations;b) at small scales after inflation, to provide scalaron decay into
pairs of matter particles and antiparticles (AS, 1980, 1981).
=

Weak dependence of the time tr when the radiation dominatedstage begins:a0 H0 1 MPl 1
N(k) ≈ NH + ln
− ln
− ln(MPl tr )k3
M
6

Evolution of the R + R 2 model
1. During inflation (H  M):H=

M21
(tf − t) ++ ..., |Ḣ|  H 266(tf − t)(for the derivation of the second term in the rhs - see
A. S. Koshelev et al., JHEP 1611 (2016) 067).
2. After inflation (H  M):


2
2/3
a(t) ∝ t
1+
sin M(t − t1 )3Mt

The most effective decay channel: into minimally coupledscalars and the longitudinal mode of vector bosons withm  M. In the first case the formula
1 d √
R2
√
( −g ns ) =
−g dt
576π

(Ya. B. Zeldovich and A. A. Starobinsky, JETP Lett. 26, 252(1977)) can be used for simplicity, but the full
integral-differential system of equations for the Bogoliubov
αk , βk coefficients and the average EMT was in fact solved in
AS (1981). For this channel of the scalaron decay:Γ=

a0 H0 5 MPl
M3, N(k) ≈ NH + ln
− ln
2
192 MPlk
6
M

that gives N(k = 0.002 Mpc−1 ) ≈ 54. For the Higgs and themixed R 2 -Higgs models, N(k = 0.002 Mpc−1 ) ≈ 58, theincrease is mainly due to the large Higgs non-minimal coupling.
Scalaron decay into graviton pairs is suppressed(A. A. Starobinsky, JETP Lett. 34, 438 (1981)).

Possible microscopic origins of this phenomenological model.
1. Follow the purely geometrical approach and consider it as
the specific case of the fourth order gravity in 4D
L=

R+ AR 2 + BCαβγδ C αβγδ + (small rad. corr.)
16πG

for which A  1, A  |B|. Approximate scale (dilaton)
invariance and absence of ghosts in the curvature regime
A−2  (RR)/MP4  B −2 .
One-loop quantum-gravitational corrections are small (their
imaginary parts are just the predicted spectra of scalar andtensor perturbations), non-local and qualitatively have thesame structure modulo logarithmic dependence on curvature.

2. Another, completely different way:consider the R + R 2 model as an approximate description of
GR + a non-minimally coupled scalar field with a large
negative coupling ξ (ξconf = 16 ) in the gravity sector:
L=R
ξRφ2 1−+ φ,µ φ,µ − V (φ), ξ < 0, |ξ|  1 .
16πG
2
2

Geometrization of the scalar:for a generic family of solutions during inflation, the scalar
kinetic term can be neglected, so
ξRφ = −V 0 (φ) + O(|ξ|−1 ) .
No conformal transformation, we remain in the the physical(Jordan) frame!

These solutions are the same as for f (R) gravity with
L=

f (R)
ξRφ2 (R), f (R) = R −− V (φ(R)).
16πG
2

λ(φ2 −φ20 )2, this just produces
4
R2
1
f (R) = 16πG R + 6M 2 with M 2 = λ/24πξ 2 G and
2

For V (φ) = 

φ = |ξ|R/λ.

The same theorem is valid for a multi-component scalar field.
More generally, R 2 inflation (with an arbitrary ns , r ) serves as
an intermediate dynamical attractor for a large class of
scalar-tensor gravity models.

Inflation in the mixed R 2-Higgs model
M. He, A. A. Starobinsky and J. Yokoyama, JCAP 1805, 064(2018).
1
L=
16πG



4
ξRχ2 1
R2
,µ λχ
−
R+
+
χ
χ
−
, ξ < 0, |ξ|  1
,µ
6M 2
2
2
4

Can be conformally transformed to GR with two interacting
scalar fields in the Einstein frame. The effective two scalar
field potential for the dual model:
λ 4 M 2 αφ
2 2 2
−2αφ
U =e
χ + 2 e − 1 + ξκ χ
4
2α
r

2
α=
κ, κ2 = 8πG , R = 3M 2 e αφ − 1 + ξκ2 χ2
3

Attractiveness of the model from the quantum
field theory point of view
The mixed R 2 -Higgs model helps to remove some UV
problems of the Higgs inflationary model and may beconsidered as its UV-completion up to the Planck energy if
√

λ.

|ξ|M
.1
MPl

(see D. Gorbunov and A. Tokareva, Phys. Lett. B 788, 37(2019)).

Effective potential in the Einstein frameOne-field inflation in the attractor regime
In the attractor regime during inflation:


λ
|ξ|R
αφ
2
2
2
, e ≈ χ |ξ|κ +αφ  1, χ ≈λ3|ξ|M 2that directly follows from the geometrization of the Higgs
boson in the physical (Jordan) frame. Thus, we return to theR2f (R) = R + 6M
2 model with the renormalized scalaron mass
M → M̃:1
1
3ξ 2 κ2
= 2+
M
λ
M̃ 2
Double-field inflation reduces to the single (R + R 2 ) one forthe most of trajectories in the phase space. For λ = 0.01,|ξ| ≤ ξc ≈ 4400

Post-inflationary heating in the mixed R 2-Higgs
model through particle creation
The most effective channel of reheating though particle
creation: creation√ of longitudinal quanta of vector bosons withm  min(M, λMPl /|ξ|). More effective than in the pure R 2
model, but less effective than in the pure Higgs case.
The simplified variant - creation of NG (phase direction)
quanta of a complex Higgs-like scalar field: M. He, R. Jinno,
K. Kamada, S. C. Park, A. A. Starobinsky and J. Yokoyama,
Phys. Lett. B 791, 36 (2019) [arXiv:1812.10099]. Inflaton
decay is not instant and occurs after a large number ofscalaron oscillations. However, the Higgs field introducesanharmonic effects in these oscillations.

Post-inflationary heating in the mixed R 2-Higgs
model through tachyonic preheating
M. He, R. Jinno, K. Kamada, A. A. Starobinsky andJ. Yokoyama, JCAP 2101 (2021) 066 [arXiv:2007.10369]
Another mechanism of rapid creation and heating of matter
after inflation: tachyonic instability of the Higgs field leading
to formation of quasi-classical matter inhomogeneity. It arises
when the background Higgs field stays long near χ = 0 in theregime φ > 0.
Occurs not for all values of parameters M, ξ and requires
some fine-tuning of them to be efficient: at least < O(0.1) inthe deep Higgs-like regime with a large scalaron mass, whilemore severe fine-tuning ∼ O(10−4 ) − O(10−5 ) is needed in
the R 2 -like regime with a small non-minimal coupling.
Stochastic behaviour of φ, R and χ in this regime – stochastic
reheating.

0.6
φ/Mpl or 50h/Mpl

φ/Mpl or 50h/Mpl

0.6
0.4
0.2
0.0

0.4
0.2
0.0
-0.2

20

30

40

50

60

70

20

30

40

50

60

70

10-4 × Mpl t

0.03

0.03

0.02

0.02

φ/Mpl or 50h/Mpl

φ/Mpl or 50h/Mpl

10-4 × Mpl t

0.01
0.00
-0.01

0.01
0.00
-0.01
-0.02

34

35

36
10-4 × Mpl t

37

34

35

36

37

10-4 × Mpl t

Upper left: ξ = 4000. Upper right: ξ = 4100. Lower left:
ξ = 4435.759104801013. Lower right: ξ = 4435.7591048. All
the digits shown above are needed.

Perturbative reheating the mixed R 2-Higgs model
If the tachyonic preheating does not occur, the heating stage
ends in the perturbative regime, like in the R + R 2 model.
However, it is still sufficiently fast due to |ξ|  1. The
scalaron-Higgs decay rate is nowΓ=

M 3 (1 − 6ξ)2.
2
192 MPlAs a result, numerical calculations in M. He, JCAP 2105
(2021) 021 show that Treh = (1013 − 1014 ) GeV weakly
depending on ξ for |ξ|  1.

Perspectives of future discoveries
Primordial gravitational waves from inflation: r .
r . 8(1 − ns ) ≈ 0.3 (confirmed!) but may be much less.
However, under reasonable assumptions one may expectthat r & (ns − 1)2 ≈ 10−3 . The target prediction in thesimplest (one-parametric) models is
r = 3(ns − 1)2 ≈ 0.004.
I A more precise measurement of ns − 1 =⇒ duration oftransition from inflation to the radiation dominated stage=⇒ information on inflaton (scalaron) couplings to known
elementary particles at super-high energies E . 1013 Gev.
I Local non-smooth features in the scalar power spectrum
at cosmological scales (?).
I Local enhancement of the power spectrum at small scalesleading to a significant amount of primordial black holes(?).
I

Generating peaks and troughs in the primordialscalar spectrum
To obtain large peaks and troughs in PR , temporal breaking of
the slow-roll approximation during inflation is needed. The
simplest way: fast break in the first derivative of the inflaton
potential V (φ) (A. A. Starobinsky, JETP Lett. 55, 489(1992)). Leads to a step in PR with superimposed oscillations.
To obtain a peak, two such features with opposite signs, or afast break in the V (φ) itself are needed (so that an inflection
point appears in between). However, it is not sufficient tohave an inflection point only, it should be combined with astrong breaking of the slow-roll conditions.

Let V (φ) = V0 + A+ φ θ(φ − φ0 ) + A− φ θ(φ0 − φ) for φ close
to φ0 . Thenφ̇ = −A+
A− + (A+ − A− )e −3H0 t
θ(−t) −
θ(t)
3H0
3H0

The slow-roll spectrum PR is modulated by the multiplier


 

A−
2
1
2
D =1−3−1
1 − 2 sin 2y + cos 2y +
A+
y
y
2


A−
1
1
−1
1+ 2 ×
A+
y2
y




1
1
2
1 + 2 + 1 − 2 cos 2y − sin 2y ,
y
y
y
9
2



y=

k
A−
, D(0) =, D(∞) = 1
k0
A+Primordial

Linear power spectrum
0.1

2.5

P(k)/PP15(k) 1

Ps(k) (Mpc3)

3.0 1e 9

2.0

0.0
0.1

P15
WWIA
WWID
WWI'
P15+HFI

0.2
0.3

1.5
10 5

10 4

10 3 10 2
k (Mpc 1)

10 1

100

0.4 4
10

10 3

10 1
10 2
k (hMpc 1)100Non-scale-free features at cosmological scales
The most recent analysis of this type of spectra with power
suppression at large scales (D. K. Hazra, D. Paoletti,
I. Debono, A. Shafieloo, G. F. Smoot, A. A. Starobinsky,
arXiv:2107.09460) using the CMB temperature andpolarization data from the Planck 2018 data release shows
marginal (68% C.L.) preference of suppression from the large
scale temperature angular power spectrum. However, thelarge-scale E-mode likelihood does not support thissuppression and in the combined data the preference towards
the suppression becomes negligible. For models withoscillatory features along with the suppression, unbinned datafrom the recently released CamSpec v12.5 likelihood was usedwhich updates Planck 2018 results. Comparison of theBayesian evidences of the feature models with their baseline
slow-roll inflaton potentials showed that the latter aremoderately preferred against potentials with features.

PBHs and small-scale GWs in two-field models of
inflation
M. Braglia, D. K. Hazra, F. Fimelli, G. F. Smoot,L. Sriramkumar, A. A. Starobinsky, JCAP 2008 (2020) 001.
The simplest one-parameter inflationary models do not predict
PBHs, at least whose existing at present with M > 1015 g.
Previously known ways to obtain a large peak in the primordial
power spectrum of scalar adiabatic perturbations at small
scales:
1. A local feature in the inflaton potential V (φ) (a rapidchange of its slope or its amplitude, an inflection point with alarge V 000 (φ)).
2. A rapid turn of the inflaton trajectory in the field space inthe case of many-field models of inflation.
3. Phase transitions leading to large isocurvature
perturbations which transform to adiabatic ones afterwards.

Two-field inflation with large kinetic coupling
A novel mechanism: a two-field inflation with different inflaton
effective masses (that leads to two stages of inflation) and alarge non-standard kinetic coupling of the heavier field.
 2

Z
√
MPl
1
f (φ)
2
2
4
R − (∂φ) −(∂χ) − V (φ, χ)S(φ, χ) = d x −g222
V (φ, χ) = V0

mχ2 2
φ2
+
χ
φ20 + φ22Large kinetic coupling: f (φ) = exp(bφ), bMPl  1. The
peak in the spectrum arises when the heavier field goes out of
the slow-roll regime. It can lead to the formation of PBHs
with a wide range of masses and to the generation ofstochastic background of primordial gravitational waves
produced by second order scalar perturbations.

7.45

1.0

7.45

0.8

0.0

6.75

0.1

0.4

6.75

0.2

68
20

70 72
40
60

80

6.40

0.0

0

20

40

N

N

10

7.45

5
0

7.10
b1 [Mpl−1 ]

0

7.10

0.6

b [M −1 ]0.1²

b1 [Mpl−1 ]

7.10

η

φ and χ [Mpl ]

7
6
5
4
3
2
1
0

5
10

6.75

15
20

0

20

40

60

80

6.40

60

80

6.40

SKA

LISA

BBO

8.4

ET

10-5
10-8

7.4

P T ( k)

10-11
10-10
10-11
10-12 -610

b1 [Mpl−1 ]P R ( k)

10-2

10-3

100

103

106

109

1015

1012

1018

6.4

10-5
10-7
10-9
10-11
10-13
10-15
10-17
10-19
10-21
10-23 -10
10

100

10-3

PTA

SKA

10-6

10-9

M/M ¯

10-12

10-18

10-15

M100

10-21

LISA

ET aLVCEOGIMSBBO DEC

10-8

10-6

10-4

f [Hz]

10-2

100

102

8.4

7.4

6.4

b1 [Mpl−1 ]Ω GW h 2

k [Mpc −1 ]

1017

1020

1023

WD

MPBH [g]

1026

1029

1032

10-5

7.4

10-7

10-9

10-11 -19
10

10-16

10-13

10-10

10-7
M/M ¯

10-4

10-1

102

6.4

b1 [Mpl−1 ]γ EG bkg

CMB

HSC

fPBH (M)

8.4

UFD

EROS

10-1

10-3

1035

Kepler

Conclusions
The typical inflationary predictions that |ns − 1| is small
and of the order of NH−1 , and that r does not exceed∼ 8(1 − ns ) are confirmed. Typical consequences
following without assuming additional small parameters:
H55 ∼ 1014 GeV, minfl ∼ 1013 GeV.
I In f (R) gravity, the simplest R + R 2 model is
one-parametric and has the preferred values ns − 1 = − N2
and r = N122 = 3(ns − 1)2 ≈ 0.004. The first value
produces the best fit to present observational CMB data.
The same prediction follows for the Higgs and the mixed
R 2 -Higgs models, though actual values of N(k) areslightly different for these 3 cases.
I Inflation in f (R) gravity represents a dynamical attractor
for slow-rolling scalar fields strongly coupled to gravity .
As a result, double field inflation in the mixed R 2 -Higgs
model reduces to the single R 2 -like inflation for a dense
IThe mixed R 2 -Higgs model helps to remove some UV
problems of the Higgs inflationary model and may beconsidered as its UV-completion up to the Planck energy.
I The rate of post-inflationary heating though particle
creation in the mixed R 2 -Higgs model is intermediate
between those in the Higgs and R + R 2 models.
Generically inflaton (scalaron) decay is not instant andoccurs after a large number of its oscillations. However,
the reheating temperature is high, Treh = (1013 − 1014 )GeV, mainly due to a large non-minimal Higgs coupling togravity.
I In some fine-tuned sub-regions of parameters of themixed R 2 -Higgs model, more rapid preheating throughtachyonic instability of the Higgs field becomes possible.
IIn two-field inflationary models with a large non-standard
kinetic coupling of a heavier inflaton field, it is possible toproduce a large peak at small scales in the primordial
power spectrum of scalar adiabatic perturbations leading
to the formation of PBHs and to the peak in thestochastic background of primordial gravitational waves
produced by second order scalar perturbations.
I As for local non-scale-free features at cosmological scales,
the present CMB temperature anisotropy and polarization
data do not favor them, but are not able to exclude themcompletely.
I