Large and Small Field Inflation from Hyperbolic Sigma Models

Rolf<EMAIL_ADDRESS><EMAIL_ADDRESS>

Dept. of Physics

Indiana University South Bend

1700 Mishawaka Ave., South Bend, IN 466344

Abstract

> Long standing themes in inflation include the issue of large field vs. small
> field inflation as well as the question what fraction of phase space leads
> to sufficient inflation, and furthermore is compatible with the experimental
> data. In the present paper these issues are discussed in the context of
> modular inflation, a specialization of the framework of automorphic
> nonlinear $\sigma$-models associated to homogeneous spaces $G/K$ in which
> the continuous shift symmetry group $G$ is weakly broken to discrete
> subgroups $\Gamma$. The target spaces of these theories inherit a curved
> structure from the group $G$, which in the case of modular invariant
> inflation leads to a hyperbolic field space geometry. It is shown that in
> this class of models the symmetry structure leads to both large and small
> field inflationary trajectories within a single modular inflation model. The
> present paper analyzes the concrete model of $j$-inflation, a hyperbolic
> model with nontrivial inflaton interactions. It describes in some detail the
> structure of the initial conditions, including a systematic analysis of
> several phenomenological functions on the target space, leading to
> constraints on the curvature scalar of the field space by upcoming
> experiments, as well as a discussion of the scaling behavior of the spectral
> index, the finite volume fraction of the field space leading to sufficient
> inflation, the attractor behavior of $j$-inflation, and a comparison of
> inflaton trajectories vs. target space geodesics. The tensor-ratio analysis
> shows that $j$-inflation is an interesting target for upcoming ground and
> satellite experiments.

###### Contents

  1. 1 Introduction
  2. 2 Formulation of general multifield inflation
  3. 3 Structure of modular inflation on hyperbolic field space
  4. 4 Global structure of the potential surface of $j-$inflation
  5. 5 Large and small field inflation in hyperbolic $\sigma$-models
  6. 6 $j$-Inflation trajectories and hyperbolic geodesics
    1. 6.1 Hyperbolic geodesics and the effect of modular dynamics
    2. 6.2 Attractor-like behavior
  7. 7 Sufficient inflation
    1. 7.1 Initial conditions in $j$-inflation for sufficient inflation
    2. 7.2 A natural measure on the space of initial conditions
    3. 7.3 Finite measures of modular inflation and the swampland conjectures
  8. 8 Spectral index function $n_{{\cal R}{\cal R}}$ on the field space
    1. 8.1 $n_{{\cal R}{\cal R}}$ scans of the field space
  9. 9 Scaling behavior of the spectral index
  10. 10 The tensor-to-scalar ratio function $r$ on the field space
    1. 10.1 Large field inflation and small tensor-to-scalar ratios
    2. 10.2 Global $r$-scan
    3. 10.3 Effect of the field space curvature $R_{\rm fs}$ on the tensor ratio $r$
  11. 11 Effect of the $\mu$-variations
  12. 12 Conclusions

##  1 Introduction

The CMB experiments that have been conducted over the past decade have led to
dramatic constraints on the parameters of the models that make up the
extensive landscape of the inflationary theory space [1, 2, 3]. While most of
the progress has come from satellite probes, such experiments are currently
continued on the ground in several observatories around the globe, the results
of which will provide further insight into the theoretical parameter space [4,
5, 6]. While the observables measured or bounded by these experiments provide
constraints strong enough to exclude whole classes of models at high
confidence levels, the surviving theory space remains large and unstructured.
The amorphous nature of this space can be ameliorated somewhat by the
introduction of symmetry groups into multifield inflation that are motivated
by the fact that they provide an embedding of the shift symmetry. Shift
symmetries are often introduced as an ad-hoc device to ensure the absence of
terms that otherwise might affect the small parameters that enter the
observables of the theory. Embedding these symmetries into a proper discrete
group has the benefit of allowing a quantitative characterization of the model
space by leading to a foliation structure.

In the framework of automorphic inflation the groups that contain the shift
symmetry arise from nonlinear sigma models associated to homogeneous spaces
$G/K$ constructed from reductive groups $G$ and maximal compact subgroups $K$.
The resulting field space is curved with a target space metric that is
determined by the group $G$ [7, 8]. The discrete groups are obtained from
potentials that weakly break the continuous symmetry group $G({\mathbb{R}})$
to subgroups $\Gamma\subset G({\mathbb{Z}})$ defined over integers
${\mathbb{Z}}$. In the simplest case of modular inflation the discrete shift
symmetry results from the weak breaking of the continuous Möbius group ${\rm
SL}(2,{\mathbb{R}})$ to congruence subgroups $\Gamma$ of the modular group
${\rm SL}(2,{\mathbb{Z}})$. This approach was introduced for the case of level
one modularity in [9, 10] and for higher level in [11]. The field space of
modular inflation carries a nontrivial hyperbolic geometry that derives from
the homogeneous space $G/K={\rm SL}(2,{\mathbb{R}})/{\rm SO}(2,{\mathbb{R}})$
and it was shown in [9] that this metric has important implications for the
theory in that the geometry induces new terms in the observables, as compared
to a flat theory, that makes these observables quasi-modular, thereby
preserving the modularity, but not the underlying holomorphic nature. This
resolves a general issue of modular invariant inflation that was left open in
earlier work in the particular context of supergravity. The idea to embed the
shift symmetry into the modular group and to focus on the hyperbolic geometry
was emphasized after the appearance of [7] also in the papers [12, 13]. Recent
discussions concerned with the shift symmetry include [14, 15].

The concrete models briefly explored in earlier work include the model of
$j$-inflation at modular level one [7, 9] and the model of $h_{2}$-inflation
at level two [11]. It was shown in these papers that these models admit
inflaton trajectories that are compatible with the Planck data. The
compatibility of $j$-inflation with experiments thus represents the first
hyperbolic twofield inflation model consistent with the current cosmological
data. Work prior to [7] based on hyperbolic field space aimed at models with
singlefield potentials or effectively singlefield trajectories, for example
[16, 17]. Subsequent work with a focus on inflaton orbits often adopted
isomorphic versions such as the Poincaré disk or the Beltrami-Klein models.
This includes for example the references [18, 19] and [20, 21, 22, 23, 24, 25,
26, 27, 28]. Inflationary models based on negatively curved target spaces have
also been considered in other recent work [29, 30, 31, 32, 14, 33, 34] and
have played a role in the discussion of the conjectures concerned with the
swampland [11, 26, 35].

The goal of the present paper is to give a systematic analysis of modular
inflation by considering in some detail the slow-roll phase space of
$j$-inflation, relevant for the observable CMB regime. This analysis leads to
a number of observations that are of relevance beyond the specific model
considered here. One of these concerns the issue of large and small field
inflation. The distinction between inflaton fields with values larger or
smaller than the Planck mass has been a recurrent theme and has often been
used as a classification scheme to characterize different types of
inflationary models. In modular inflation it is possible to decompose the
field space into separate regions the union of which gives the total target
space. The structures that appear in these separate regions are then repeated
infinitely many times in the field space. The nontrivial hyperbolic geometry
of these individual regions is given by the metric considered originally by
Liouville, Riemann, Beltrami and Klein, hence is naturally called the Poincaré
metric in the literature. This metric leads to a stretching of the field space
as one approaches the boundary of the space, in particular when approaching
the origin of the space. This leads to trajectories with initial values that
can approach the origin arbitrarily close, thus leading to initial field
values that become small at any scale, in particular compared to the Planck
mass. Far away from the origin one or both of the inflaton components can be
large, leading to large field inflation. This shows that in modular inflation,
and more generally in some curved field space inflation, both small and large
field inflation can be realized in the same model.

Modular inflation is a field theory framework with curved target space
geometry, hence it is of interest to ask whether the dynamics of the theory
imposes constraints that lead to deviations from the geodesic structure
induced by the metric. It will be shown that while in $j$-inflation per se the
inflaton trajectories are strongly affected by the potential, they also show
attractor behavior in the sense that there are certain basins in the target
space to which the trajectories converge.

After the discussion of the modular symmetry and the comparison of
$j$-inflation trajectories with hyperbolic geodesics, the first
phenomenological focus of the present exploration will be to identify those
regions of the target space that lead to inflation with a number of e-folds in
some standard range. This first step is similar in spirit to earlier
investigations of other inflationary models such as hybrid inflation (see e.g.
[36, 37, 38, 39, 40, 41, 42, 43, 44, 45]) and other models [46], which focused
on the question of the existence of inflation per se, implemented as a lower
bound on the number of e-folds. In the context of considering the global scale
structure of the field space it is of interest that in modular inflation the
target space comes equipped with a finite measure that allows to quantify the
volume in phase space that lead to inflation. This is possible without the
need of a regularization of this measure, an issue that has afflicted many
discussions in this context [47, 48, 49, 50, 51, 52].

In the context of sufficient inflation the spectral index and the tensor ratio
are left unconstrained. The data obtained by extending the analysis to these
parameters can be used to address a question that concerns the possible
scaling of the spectral index and the tensor ratio. This was explored for
singlefield inflation in several papers [53, 54, 55], with the aim of
providing a priori constraints in the plane spanned by these two parameters.
The important question is what the behavior is of these two parameters in
multifield inflation. Finally, given the experimental data obtained from the
CMB satellites it is of interest to consider how these observational
constraints constrain the regions of viable initial conditions beyond those
obtained by the e-fold constraint alone.

The outline of this paper is as follows. In sections two and three multifield
inflation and modular inflation are briefly reviewed to establish the general
context and the notation used here. Section four describes the global
structure of the potential of $j$-inflation. Section five discusses the
modular structure of the critical points on the potential surface and explains
the associated existence structure of both large field and small field
inflation in the framework of modular invariant sigma models. Section six
shows that in the global target space there are broad basins of attraction and
compares the resulting inflaton trajectories with the geodesics of the
hyperbolic geometry. Section seven analyzes the behavior of the e-fold
function $N_{*}$ on the target space and shows that a finite volume fraction
of the target space leads to sufficient inflation, where the volume is
obtained from the nontrivial geometric measure on the field space. Sections
eight and nine consider the spectral index as a function on the target space,
including a discussion of the scaling behavior of the spectral index. Section
ten analyzes the behavior of the tensor-to-scalar ratio on the field space. It
is shown how the bounds of $r$ provided by the CMB probes constrain the
curvature scalar of the field space and that $j$-inflation presents an
interesting target for the upcoming CMB experiments such as the Simons
observatory, as well as LightBird and CMB-S4. This leads to a discussion of
large field inflation vs. small values of the tensor ratio, as well as of the
swampland conjectures. Section eleven analyzes how the variation of the mass
scale of $j$-inflation impacts the constraints of the CMB on the model.
Section twelve presents the conclusions.

##  2 Formulation of general multifield inflation

Automorphic inflation is a multifield inflation framework with fields
$\phi^{I},I=1,...,n$ and a nontrivial target space metric $G_{IJ}(\phi^{K})$
that leads to a nontrivial kinetic coupling action

${\cal A}~{}=~{}-\int
d^{4}x\sqrt{-g}\left(\frac{1}{2}G_{IJ}(\phi^{K})g^{\mu\nu}\partial_{\mu}\phi^{I}\partial_{\nu}\phi^{J}~{}+~{}V(\phi^{I})\right).$
(1)

The perturbations relevant in this general context can be parametrized as
$({\cal R},S^{IJ})$, where ${\cal R}$ is the comoving curvature perturbation
of Lukash and Bardeen [56, 57] (see also [58])

${\cal R}~{}=~{}H\delta u~{}-~{}\psi,$ (2)

in terms of the spacetime metric perturbation $\psi$, and $S^{IJ}$ is the
tensor of isocurvature perturbations [9]

$S^{IJ}~{}=~{}\frac{H}{{\dot{\sigma}}}\left(\sigma^{I}Q^{J}~{}-~{}\sigma^{J}Q^{I}\right).$
(3)

Here ${\dot{\sigma}}=(G_{IJ}{\dot{\phi}}^{I}{\dot{\phi}}^{J})^{1/2}$ is the
speed of the background inflaton and
$\sigma^{I}={\dot{\phi}}^{I}/{\dot{\sigma}}$ is the normalized inflaton
velocity. The closed form of the dynamics of the system $({\cal R},S^{IJ})$
was derived in [9] as

$\displaystyle{\dot{{\cal R}}}$ $\displaystyle=$
$\displaystyle-2H\eta_{IK}\sigma^{K}\sigma_{J}S^{IJ}$ $\displaystyle
D_{t}S^{IJ}$ $\displaystyle=$ $\displaystyle
2H\left(\eta_{KL}\sigma^{K}\sigma^{L}-\epsilon\right)S^{IJ}+H\left(\eta_{KL}G^{K[I}S^{J]L}-\frac{\epsilon}{3}M_{\rm
Pl}^{2}\sigma^{[I}R^{J]}_{KLM}\sigma^{K}S^{LM}\right),$ (4)

where

$\eta_{IJ}~{}=~{}M_{\rm Pl}^{2}\frac{V_{;IJ}}{V}$ (5)

and $V^{[I}W^{J]}=V^{I}W^{J}-V^{J}W^{I}$. The covariant derivative $D_{t}$
acts on the contravariant tensor $S^{IJ}$. The specialization to the twofield
case $({\cal R},{\cal S})$ is discussed in detail in [9].

The parameters considered in the present analysis include the spectral index
$n_{{\cal R}{\cal R}}$ of the scalar power spectrum ${\cal P}_{{\cal R}{\cal
R}}$ of ${\cal R}$, the tensor ratio and the number of e-folds

${\cal P}_{{\cal R}{\cal R}}~{}=~{}A_{{\cal R}{\cal
R}}\left(\frac{k}{k_{*}}\right)^{n_{{\cal R}{\cal R}}-1},~{}~{}~{}~{}{\cal
P}_{T}~{}=~{}A_{T}\left(\frac{k}{k_{*}}\right)^{n_{T}},~{}~{}~{}~{}r~{}=~{}\frac{{\cal
P}_{T}}{{\cal P}_{{\cal R}{\cal
R}}},~{}~{}~{}~{}~{}~{}~{}N_{*}=\int_{t_{*}}^{t_{e}}Hdt.$ (6)

The general expressions in multifield inflation with an inflaton multiplet
$\phi^{I}$ and a curved target space metric $G_{IJ}$ are given by

$\displaystyle n_{{\cal R}{\cal R}}$ $\displaystyle=$ $\displaystyle
1-3G^{IJ}\epsilon_{I}\epsilon_{J}~{}+~{}2\frac{\eta_{IJ}\epsilon^{I}\epsilon^{J}}{G^{KL}\epsilon_{K}\epsilon_{L}}$
$\displaystyle r$ $\displaystyle=$ $\displaystyle
8G^{IJ}\epsilon_{I}\epsilon_{J}$ $\displaystyle n_{T}$ $\displaystyle=$
$\displaystyle-\frac{r}{8}$ (7)

where the slow-roll parameters $\epsilon_{I}$ are defined as

$\epsilon_{I}~{}:=~{}M_{\rm Pl}\frac{V_{,I}}{V},$ (8)

and $H={\dot{a}}/a$ is the Hubble-Slipher parameter. The slow-roll dynamics
can be written in terms of the potential as

${\dot{\phi}}^{I}~{}=~{}-\sqrt{\frac{V}{3}}G^{IJ}\epsilon_{J}$ (9)

A systematic analysis of models in such a framework involves in the first
instance the variation of the energy scales that define the model, as well as
a scan of the phase space of the theory. In the context of the phase space
analysis the question of a measure arises, a problem that involves the
geometry of the target space. Other questions include the dichotomy of large
vs. small field inflation, an issue that also turns out to depend on the
geometry of the field space. Finally one can ask whether the scaling behavior
of the CMB observables observed in simple singlefield models has some counter
part in the multifield context.

##  3 Structure of modular inflation on hyperbolic field space

This section briefly outlines the essential structures that will be used in
the remainder of this paper. Modular inflation is a framework of twofield
inflation that specializes multifield theory of automorphic inflation [7, 8].
The latter is based on automorphic functions $F(\phi^{I})$, where the inflaton
multiplet defines the coordinates of quotient spaces $G/K$, defined in terms
of reductive groups $G$ and maximal compact subgroups $K$.

In the simplest case the doublet spans the curved target space geometry given
by the upper halfplane ${\cal H}$, which can be identified as the homogeneous
space

${\cal H}~{}:=~{}{\rm SL}(2,{\mathbb{R}})/{\rm SO}(2,{\mathbb{R}}).$ (10)

The geometry of this space is determined by the hyperbolic metric

$G_{IJ}~{}:=~{}\frac{\mu^{2}}{(\phi^{2})^{2}}\delta_{IJ},~{}~{}~{}I,J=1,2,$
(11)

where $\mu$ is an a priori arbitrary energy scale. This metric of the upper
halfplane leads to a space of constant curvature

$R~{}=~{}-\frac{2}{\mu^{2}}.$ (12)

Hence the energy scale $\mu$ can be thought of as a measure of the curvature
of the target space. In distinction to the energy scale $\Lambda$, the
parameter $\mu$ enters all the formulae that characterize $j$-inflation. As a
result the analyses below will show that the curvature scalar $R$ is
constrained by the CMB experiments. The hyperbolic modular inflation metric
admits an isometry group that contains the Möbius group and the transformation
$\tau\mapsto(-{\overline{\tau}})$.

In modular inflation it is convenient to complexify the inflaton doublet as
$\phi=\phi^{1}+i\phi^{2}$ and to introduce the dimensionless inflaton as
$\tau=\phi/\mu$. This recovers the so-called Poincaré metric of Liouville,
Beltrami and Klein as $ds^{2}=\mu^{2}d\tau d{\overline{\tau}}/({\rm
Im}~{}\tau)^{2}$. An important feature of hyperbolic geometry is that, while
the curvature is constant, the metric diverges as it approaches the boundary
of the space $\phi^{2}=0$, hence the target space in this parametrization is
the upper halfplane $\phi^{2}>0$. The form of the boundary can be transformed
into different shapes if different variables are used, such as in the Poincare
disk model which is obtained via the Cayley transform and which was considered
later in [12, 13]. Other models include the sinh-form of the metric considered
more recently in [20, 23, 31, 27]. These related models of the hyperbolic
geometry can be described in terms of homogeneous spaces based on quotients of
the groups ${\rm SU}(1,1)$ and ${\rm SO}(2,1)$ instead of ${\rm
SL}(2,{\mathbb{R}})$. This set-up can be compared to the metric sometimes
considered in axion-dilaton inflation with a radial field $\rho$ and an
angular field $\theta$ with a semi-flat metric that depends on the radial
field only. Writing the complex field as $\phi=\rho e^{i\theta}$ leads to the
modular inflation metric $G_{IJ}=\mu^{2}\delta_{IJ}/\rho^{2}\sin^{2}\theta$
involving both target space directions.

Given a hyperbolic metric on the target space with an associated symmetry
group it is natural to consider potentials that preserve at least some of
these symmetries. In modular inflation the idea is to break the continuous
Möbius group ${\rm SL}(2,{\mathbb{R}})$ to an infinite discrete group $\Gamma$
of the modular group ${\rm SL}(2,{\mathbb{Z}})$ that contains the shift
symmetry

${\rm SL}(2,{\mathbb{R}})~{}\stackrel{{\scriptstyle
V_{\Gamma}}}{{\longrightarrow}}~{}\Gamma\subset{\rm SL}(2,{\mathbb{Z}})$ (13)

where the potential

$V_{\Gamma}~{}=~{}\Lambda^{4}~{}|F|^{2}$ (14)

is defined in terms of a function $F$ that is invariant under the group
$\Gamma$. This allows for nontrivial potentials in which both components of
the inflaton doublet are coupled in a nontrivial way. Such modular invariant
inflation potentials can be constructed in different ways, but the most
convenient form for the phenomenological analysis is in terms of the
Eisenstein series $E_{4},E_{6}$, of weight four and six respectively, which
are defined as

$E_{w}(\tau)~{}=~{}1-\frac{2w}{B_{w}}\sum_{n\geq 1}\sigma_{w-1}(n)q^{n},$ (15)

where $q=e^{2\pi i\tau}$, the denominators $B_{w}$ are the Bernoulli numbers
and $\sigma_{w}(n)$ is the divisor function [7, 9].

In general modular inflation the models involve a modular invariant function
$F(\phi^{I}/\mu)$, hence depend in the formulation given here on the energy
scale $\mu$ that enters the potential. As a result the slow-roll parameters
$\epsilon_{I}$ depend on $\mu$ and can be expressed in the form [9]

$\epsilon_{I}~{}=~{}i^{I-1}\frac{M_{\rm
Pl}}{\mu}\left(\frac{F^{\prime}}{F}+(-1)^{I-1}\frac{{\overline{F}}^{\prime}}{{\overline{F}}}\right),$
(16)

leading to

$\epsilon_{V}~{}=~{}\frac{1}{2}G^{IJ}\epsilon_{I}\epsilon_{J}~{}=~{}2\frac{M_{\rm
Pl}^{2}}{\mu^{2}}({\rm Im}~{}\tau)^{2}\left|\frac{F^{\prime}}{F}\right|^{2}.$
(17)

The cosmological parameters considered above can be shown to be expressed in a
quasi-modular form as [9]

$\displaystyle n_{{\cal R}{\cal R}}$ $\displaystyle=$ $\displaystyle
1-4\frac{M_{\rm Pl}^{2}}{\mu^{2}}({\rm
Im}~{}\tau)^{2}\left[2\left|\frac{F^{\prime}}{F}\right|^{2}-{\rm
Re}\left(\frac{F^{\prime\prime}}{F^{\prime}}\frac{{\overline{F}}^{\prime}}{{\overline{F}}}\right)_{\rm
mod}+\frac{\pi}{3}{\rm
Im}\left({\widehat{E}}_{2}\frac{{\overline{F}}^{\prime}}{{\overline{F}}}\right)\right]$
$\displaystyle r$ $\displaystyle=$ $\displaystyle 32\frac{M_{\rm
Pl}^{2}}{\mu^{2}}({\rm
Im}~{}\tau)^{2}~{}\left|\frac{F^{\prime}}{F}\right|^{2}$ $\displaystyle N_{*}$
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{3}}\frac{\Lambda^{2}}{M_{\rm
Pl}}\int_{t_{*}}^{t_{e}}dt~{}|F(\tau(t)|,$ (18)

where the quasi-modular form ${\widehat{E}}_{2}$ is defined as

${\widehat{E}}_{2}(\tau)~{}=~{}E_{2}(\tau)-\frac{3}{\pi({\rm Im}~{}\tau)}.$
(19)

This shows that the spectral index is a modular invariant quantity, an issue
that was not addressed in the earlier literature, for example in the context
of supergravity theories.

The model of $j$-inflation is defined at level $N=1$, i.e. it is invariant
under the full modular group ${\rm SL}(2,{\mathbb{Z}})$ with $F(\tau)=j(\tau)$
. Here the absolute modular invariant $j(\tau)$-function is viewed as a
function of the dimensionless parameter $\tau=\phi/\mu$ in the complex upper
halfplane ${\cal H}$. The function $j(\tau)$ is invariant under the full
modular group ${\rm SL}(2,{\mathbb{Z}})$, a group that contains the shift
symmetry of the inflaton. The $j$-function is most naturally defined as the
quotient of two modular forms of weight twelve

$j(\tau)~{}:=~{}\frac{E_{4}^{3}(\tau)}{\Delta(\tau)},$ (20)

where $\Delta(\tau)=\eta^{24}(\tau)$ is the Ramanujan modular cusp form of the
full modular group, expressed in terms of the Dedekind eta function

$\eta(\tau)~{}=~{}q^{1/24}\prod_{n\geq 1}(1-q^{n}).$ (21)

The Ramanujan form can be written in terms of the generators $E_{4},E_{6}$ of
the space of all modular forms relative to the modular group ${\rm
SL}(2,{\mathbb{Z}})$ as $\Delta=(E_{4}^{3}-E_{6}^{2})/1728$.

The slow-roll parameters $\epsilon_{I}$ can now be expressed directly in terms
of undifferentiated modular forms as

$\epsilon_{I}~{}=~{}-2\pi i^{I}\frac{M_{\rm
Pl}}{\mu}\left(\frac{E_{6}}{E_{4}}+(-1)^{I}\frac{{\overline{E}}_{6}}{{\overline{E}}_{4}}\right)$
(22)

with

$\epsilon_{V}~{}=~{}8\pi^{2}\frac{M_{\rm Pl}^{2}}{\mu^{2}}({\rm
Im}~{}\tau)^{2}\left|\frac{E_{6}}{E_{4}}\right|^{2},$ (23)

and the above parameters are given directly in terms of the Eisenstein series
as [7, 8]

$\displaystyle n_{{\cal R}{\cal R}}$ $\displaystyle=$ $\displaystyle
1-\frac{8\pi^{2}}{3}\frac{M_{\rm Pl}^{2}}{\mu^{2}}({\rm
Im}~{}\tau)^{2}\left[8\left|\frac{E_{6}}{E_{4}}\right|^{2}-3{\rm
Re}\left(\frac{E_{4}^{2}}{E_{6}}\frac{{\overline{E}}_{6}}{{\overline{E}}_{4}}\right)~{}+~{}{\rm
Re}\left({\widehat{E}}_{2}\frac{{\overline{E}}_{6}}{{\overline{E}}_{4}}\right)\right]$
$\displaystyle r$ $\displaystyle=$ $\displaystyle 128\pi^{2}\frac{M_{\rm
Pl}^{2}}{\mu^{2}}~{}({\rm Im}~{}\tau)^{2}\left|\frac{E_{6}}{E_{4}}\right|^{2}$
$\displaystyle N_{*}$ $\displaystyle=$
$\displaystyle\frac{\Lambda^{2}}{\sqrt{3}M_{\rm Pl}}\int dt~{}|j|.$ (24)

The slow-roll dynamics takes for $j-$inflation the form

${\dot{\phi}}^{I}~{}=~{}\frac{2\pi i^{I}}{\sqrt{3}}\frac{M_{\rm
Pl}}{\mu}\Lambda^{2}\left({\rm
Im}~{}\tau\right)^{2}\left(\frac{E_{6}}{E_{4}}+(-1)^{I}\frac{{\overline{E}}_{6}}{{\overline{E}}_{4}}\right)~{}|j|.$
(25)

The energy scale $\Lambda$ does not enter $n_{{\cal R}{\cal R}}$ or $r$, but
it does appear in the differential equation and hence the number of e-folds
associated to inflaton trajectories. It can be expressed in terms of $\mu$ and
the amplitude of the primordial CMB amplitude $A_{{\cal R}{\cal R}}$ as

$\Lambda~{}=~{}M_{\rm Pl}\left(192\pi^{4}\frac{M_{\rm
Pl}^{2}}{\mu^{2}}A_{{\cal R}{\cal R}}({\rm
Im}~{}\tau)^{2}\left|\frac{E_{6}}{E_{4}}\right|^{2}\frac{1}{|j|^{2}}\right)^{1/4}.$
(26)

This leaves the parameter $\mu$, and hence the curvature scalar, to be
constrained by the cosmological observables. The phenomenological analysis now
aims to investigate the target space of $j$-inflation.

A more detailed foundational formulation of modular inflation at level one can
be found in [7, 9], and the generalization from the full modular group ${\rm
SL}(2,{\mathbb{Z}})=\Gamma_{0}(1)$ to Hecke congruence subgroups
$\Gamma_{0}(N)$ at higher level $N$ has been introduced in ref. [11].

##  4 Global structure of the potential surface of $j-$inflation

The aim of the preliminary phenomenological analysis of $j$-inflation
presented in ref. [9] was to establish the existence of inflaton trajectories
that are compatible with the known constraints for the spectral index of the
scalar power spectrum, the bound on the tensor power spectrum via the tensor
ratio, and the number of e-folds. Since the bound on the tensor ratio $r$
established by the CMB satellite probes is small, it is clear from the
$j$-inflation expression for $r$ given in eq. (24) that viable values for
$\phi_{*}^{I}$ can be found in the neighborhoods $U(\tau_{s})$ of the zeros of
the modular form $E_{6}$. One such zero is at $\tau_{s}=i$. A close-up view of
the neighborhood $U(\tau_{s})$ for $\tau_{s}=i$, together with the shape of a
typical trajectory, is shown in figure 4 of ref. [9]. The scale of the saddle
point at $\tau_{s}=i$ is given by the curvature matrix, which for the
$j$-inflation potential is given by

$V_{,IJ}(i)~{}=~{}\frac{3\cdot
432^{2}\Gamma\left(\frac{1}{4}\right)^{8}}{\pi^{4}}\frac{\Lambda^{4}}{\mu^{2}}\left(\matrix{-1&0\cr
0&1\cr}\right).$ (27)

This curvature matrix immediately gives the mass matrix of the inflaton
perturbations at the critical point. Away from the saddle point the mass
matrix receives contributions from terms that depend on the slow-roll
parameters. A recent discussion that aims at a characterization of critical
points in terms of discrete symmetries in a general context that includes
inflationary models can be found in [59].

Attempts have been made to classify singlefield inflation models by the
curvature $V^{\prime\prime}$ of the potential [60, 61, 62, 63, 64]. A natural
way to generalize these early discussions would be by keeping track of the
eigenvalue structure of the flat curvature matrix $V_{,IJ}$ of the potential,
or its covariant form $V_{;IJ}$. A simpler structure could alternatively be
obtained by considering the determinant $J(V)={\rm det}(V_{,IJ})$, or its
covariant form $J_{\rm cov}(V)$. A more phenomenological, but somewhat less
structural, characterization however might proceed by the generalization of
the singlefield slow-roll parameter $M_{\rm Pl}^{2}V^{\prime\prime}/V$. In
multifield inflation the analog of this parameter is
$\eta_{IJ}\epsilon^{I}\epsilon^{J}/(G^{KL}\epsilon_{K}\epsilon_{L})$, which is
not just a rescaled curvature matrix.

In the present paper the phenomenological considerations aim at a more global
view of the slow-roll phase space and the structure of the potential surface
away from the saddle point $\tau_{s}=i$ becomes of interest. One of the main
motivations for $j$-inflation is the fact that the shift symmetry, often
introduced as an ad hoc approximate invariance, is now part of a bona-fide
symmetry of the theory. Modular inflation at level one is invariant under the
full modular group ${\rm SL}(2,{\mathbb{Z}})$ of integral $2\times 2$ matrices
of determinant one. This infinite group can be generated by just two elements,
given by

$S~{}=~{}\left(\matrix{0&1\cr-1&0\cr}\right),~{}~{}~{}~{}T~{}=~{}\left(\matrix{1&1\cr
0&1\cr}\right).$ (28)

The generator $T$ shifts the inflaton multiplet, leading to a repetitive
pattern of the inflaton field space of the upper halfplane $\tau\in{\cal H}$
as $T$ shifts by one unit $\tau\longrightarrow\tau+1$. The investigation of
the behavior of modular inflation models can therefore be restricted without
loss of generality to bands of width one. In this paper the focus will be on
the vertical band bounded by $-1/2\leq\tau^{1}<1/2$.

The structure of the potential surface is simple for larger $\tau^{2}$, but
becomes more complicated as one approaches the region where
$\tau^{2}~{}{\rightarrow}~{}0$. Figure 1 illustrates this with a view that
shows a partial region of the embedded surface. This graph indicates a canyon
like structure that emerges as one approaches the boundary of the field space
given by the horizontal axis for $({\rm Im}~{}\tau)=0$, with saddle points
visible at $\tau_{s}=i$ and $\tau_{s}=(1+i)/2$. Further saddle points exist
closer to the boundary, but are obscured in this 3D picture. These will become
transparent in the contour graphs discussed further below. In coordinate space
the slopes steepen as ${\rm Im}~{}\tau~{}{\rightarrow}~{}0$, but the
hyperbolic metric stretches the physical distances in field space, thereby
flattening the potential surface. This will become important below in the
analysis of the behavior of the trajectories in $j$-inflation.

> Figure 1. A large scale view of the $j$-function surface $|j|^{2}$ (the
> energy scale parameter $\Lambda$ is determined by the CMB amplitude
> $A_{{\cal R}{\cal R}}$). The plot shows two of the saddle point regions, one
> of which has a mirror on the far side of the graph. The minimum of the
> potential is located at $\tau_{\rm min}=e^{2\pi i/6}$.

The minimum of the potential at $\tau_{\rm min}=e^{2\pi i/6}$, which is
determined by the zero of the Eisenstein series $E_{4}$ in the numerator of
the inflaton potential, combined with the fact that at the point $\tau_{\rm
min}$ the Ramanujan modular form $\Delta(\tau)$ that appears in the
denominator of $V$ does not vanish. As the component $\phi^{2}$ of the
inflaton doublet increases beyond the saddle point at $\tau_{s}=i$ the
$j$-inflation potential forms a wall and diverges. This can be seen explicitly
from the definition of the Eisenstein series given in eq. (15). As ${\rm
Im}~{}\tau~{}{\rightarrow}~{}\infty$ the variables $q=e^{2\pi i\tau}$ vanish
and the Eisenstein series approach unity in the normalization adopted here.
Since the Ramanujan form approaches $q$ in this limit the $j$-function
diverges.

The trajectories of the inflaton evolution considered in [9] all originated in
the neighborhood $U(\tau_{s})$ of the saddle point at $\tau_{s}=i$. The
evolution of those orbits was such that it lasts long enough to produce
sufficient inflation and is furthermore compatible with the observational
results of the CMB probes. Figure 1 shows that there are other saddle point
regions and that the potential surface of $j$-inflation has an intricate
structure that the large scale three-dimensional representation of the surface
does not display in full detail. Better insight into the potential surface can
be obtained by combining the 3D potential shown in figure 1 with the contour
plot in figure 2. This graph shows the saddle point region considered in [9],
as well as the structures in figure 1, and also shows a number of further
saddle points on both sides of the ridge of the saddle point $\tau_{s}=i$. A
zoom of the above contour plot in the right panel of figure 2 shows that the
potential becomes more complicated as one approaches the horizontal axis, the
boundary of the target space. This structure is reminiscent of a fractal
structure, but the picture suggested by this graph has to be complemented by
the behavior of the hyperbolic metric, which diverges as the saddles approach
the real axis.

> Figure 2. Contour lines of the potential surface modulo the energy scale.
> The higher the potential region, the lighter the color. This graph shows
> that there are further saddle point regions when $\tau^{2}$ approaches the
> boundary of the target space, determined by $\tau^{2}=0$. The ridge toward
> the top of the left panel shows the region around the saddle point
> $\tau_{s}=i$ which was the main focus of ref. [9].

The precise degree of detail obtained in the plots of the $j$-inflation
potential depends on the resolution and also on the accuracy with which the
potential is computed. This is relevant because the modular forms involved in
the definitions of the $j$-function are infinite series which have to be
truncated. An increase of the number of terms increases the amount of detail
that can be seen, thus the order to which the Eisenstein series are computed
needs to be adjusted depending on how close to the real axis the computation
should proceed.

##  5 Large and small field inflation in hyperbolic $\sigma$-models

One of the longstanding issues in inflation is the dichotomy between large
field and small field inflation, emphasized for example by Lyth in the context
of singlefield inflation [65]. These discussions revolve around the relation
between the distance covered by the inflaton in field space and the tensor
ratio, mediated by the number of e-folds during inflation. In multifield
inflation this relation is less direct because the distance in field space
involves not only changes of the fields $\Delta\phi^{I}$ but also depends on
the metric $G_{IJ}$ of the target space. It is therefore useful to separate
issues pertaining to the field values from those that depend on the distance
in field space, hence in general involve the metric.

Part of the current discussion that informs the planning and construction of
the next generation experiments aiming to discover the contribution of
inflationary tensor modes to the CMB is the formulation of significant targets
for the tensor ratio $r$, which will be discussed in detail in section ten.
Tensor mode targets that have been identified in recent years have been
motivated by distinguishing between models with inflaton values that are
either super- or sub-Planckian. This distinction has in particular led to the
formulation of classes of singlefield inflation models along these lines in
the hope that this leads to the identification of more generic characteristics
of inflationary models that might be testable with CMB data (early work in
this direction can be found in the refs. [60, 61, 62, 63, 64]). The
demarcation line between these two classes is not precise, but this lack of
precision is not important for the following. What is of interest is the
common association in the literature of large field inflation with large $r$
and small field inflation with smaller $r$. These discussions raise the
question whether a similar behavior can be identified in multifield inflation
in general.

In $j$-inflation there are two aspects that arise in this context. The first,
of relevance in principle for any multifield framework, is that the different
components of the inflaton multiplet can have rather different scales, making
the distinction of sub-Planckian vs. super-Planckian field values less a
feature of the field itself, even though it still retains its importance for
at least some components. This is the case in $j$-inflation in neighborhoods
$U(\tau_{s})$ of the saddle point $\tau_{s}=i$. For initial values $\tau_{*}$
in $U(i)$ at least one of the field values is super-Planckian for $\mu>M_{\rm
Pl}$. Hence the vertical component is large $\phi_{*}^{2}\cong\mu$, while the
horizontal component $\phi_{*}^{1}$ can be much smaller, depending on how
close to the saddle point $\phi_{s}$ the initial value $\phi_{*}$ is. In this
case the general philosophy can be applied by a slight change of formulation.

A second feature, specific to the modular inflation framework, is that modular
transformations connect an infinite number of different regions of the target
space. This also applies more generally to automorphic inflation. The contour
plots in figure 2 show sequences of critical points of the $j$-inflation
potential that form arcs oriented toward the origin of the $\tau$-plane. These
saddle point regions can be mapped with elements of the modular group ${\rm
SL}(2,{\mathbb{Z}}$), which is generated by the elements $S$ and $T$ in eq.
(28). The top arcs in the left panel of figure 2 can be reached for example
from the saddle point at $\tau_{s}=i$ with the sequences of group elements

$(TST)^{\pm n}(i)~{}=~{}\pm\frac{n}{n^{2}+1}+\frac{i}{n^{2}+1}.$ (29)

These representations of the primary arcs are not unique because $Si=i$, hence
these primary sequences can be obtained in a number of different group
elements.

The next sequences of saddle points, both illustrated in the left panel, but
more clearly visible in the zoom in the right panel, can be obtained via

$\displaystyle(TST)^{n}T(i)$ $\displaystyle=$
$\displaystyle\frac{2n+1}{2n^{2}+2n+1}~{}+~{}\frac{i}{2n^{2}+2n+1}$
$\displaystyle ST^{n}(TST)(i)$ $\displaystyle=$
$\displaystyle-\frac{2n+1}{2n^{2}+2n+1}~{}+~{}\frac{i}{2n^{2}+2n+1}.$ (30)

The general picture illustrated by these maps is that modular invariance of
the $j$-potential leads to saddle points $\tau_{n}$ obtained by maps
$\gamma_{n}$ obtained from elements in the modular group.

The main feature of these maps $\gamma_{n}$ is that they approach the origin
$\tau=0$ ever closer as $n$ increases, which is apparent for the images of the
$\tau_{s}$ in (29) and (30). For more general $\tau$ this can be illustrated
with the sequences $\gamma_{n}$ obtained for example by the iteration of a
generating element $\gamma$ as $\gamma_{n}=\gamma^{n}$ for the group element
$\gamma=TST$. These maps $\gamma_{n}$ send arbitrary points $\tau$ to images
of the form

$\gamma_{n}(\tau)~{}=~{}\frac{\tau^{1}+n|\tau|^{2}+i\tau^{2}}{1+2n\tau^{1}+n^{2}|\tau|^{2}},$
(31)

generalizing the sequence (29) of the saddle point $\tau_{s}=i$. For an
initial value $\tau_{*}$ in a neighborhood of $\tau_{s}=i$, this sequence
converges to the origin of the $(\tau^{1},\tau^{2})$-plane and in the process
sends both components $\tau^{I}_{*}$ to ever smaller field values. The
existence of such sequences that approach the origin of the field space shows
that it is possible to have viable $j-$inflation trajectories that start with
sub-Planckian field values.

The implication of these sequences of maps $\gamma_{n}$ is that large field
initial values $\tau_{*}$ in a neighborhood of the large field saddle point
$\tau_{s}=i$ with larger tensor ratios $r(\tau_{*})\geq 0.01$ (which satisfies
the Planck bound) are mapped into small field initial values
$\gamma_{n}(\tau_{*})$ with the same tensor ratio

$r(\gamma_{n}(\tau_{*}))~{}=~{}r(\tau_{*}).$ (32)

In section ten it will be shown that, depending on the parameters of the
model, $j$-inflation can lead to tensor ratios that reach the current bound
reported by the Planck collaboration, but can also reach down to the target
range of upcoming experiments, such as the Simons Observatory and the CMB-S4
experiment. Combined with the discussion above this shows that in
$j$-inflation, and eo ipso in multifield inflation, small field inflation can
lead to large tensor ratios and large field inflation can lead to small tensor
ratios.

##  6 $j$-Inflation trajectories and hyperbolic geodesics

The analysis discussed in the next section shows that sufficient inflation
takes place in a large volume of the field space of $j$-inflation. It is in
this context of interest to investigate in more detail the behavior of the
inflaton trajectories in different regions of the target space. The geometry
and the physics of the orbits depends of course on the potential surface above
the target space. In particular the structure of the regions with sufficient
inflation, or the existence of attractors, is dependent on the form of the
potential, hence necessitates the consideration of inflaton trajectories on
the potential surface. The inflaton behavior can also be considered in the
field space itself by projecting the inflaton paths down to the target space,
where they can be compared with the geodesics defined by the target space
metric, in the present case the hyperbolic metric of the upper halfplane. The
latter come in two types, straight vertical lines and semi-circles, and the
natural question is how the inflaton trajectories compare to the geodesic
paths. Naively one might expect the geodesic motion in the field space to be
irrelevant in slow-roll inflation because it is the gradient structure of the
potential that is important and the dynamics of the inflaton itself even along
approximately geodesic paths will depend on the potential. However, it will
become clear below that sufficient inflation can be achieved with a family of
inflaton orbits that interpolate between the two types of hyperbolic
geodesics.

###  6.1 Hyperbolic geodesics and the effect of modular dynamics

A natural question in the case of curved target spaces is how the physical
geometry of the field space differs from the purely geometric dynamics, i.e.
in what way the inflaton potentials affect the trajectories projected down to
the target space as compared to the geodesics. This is also of interest in the
context of how the swampland conjectures might relate to the observational
bounds on $r$ because the former are concerned in part with geodesic distances
while the latter impact the distances along inflaton trajectories. The
geodesic equations for the upper halfplane ${\cal H}$ lead to the coupled
system given by

$\displaystyle\tau^{1^{\prime\prime}}-\frac{2}{\tau^{2}}\tau^{1^{\prime}}\tau^{2^{\prime}}$
$\displaystyle=$ $\displaystyle 0$
$\displaystyle\tau^{2^{\prime\prime}}+\frac{1}{\tau^{2}}\left(\tau^{1^{\prime}})^{2}-(\tau^{2^{\prime}})^{2}\right)$
$\displaystyle=$ $\displaystyle 0,$ (33)

where the dimensionless variables $\tau^{I}$ are used and primes denote the
derivative relative to the dimensionless time. These equations can be solved
in terms of straight vertical lines $\tau^{1}={\rm const}$ as well as by semi-
circles that intersect the $\tau^{2}=0$ axis vertically, i.e. with centers
along the real axis. Parametric forms of these geodesics can be obtained in
two different ways, using either a group theoretic parametrization or a
formulation in terms of hyperbolic functions. More precisely, the geodesic
solutions can be given as a combination of a pair of maps of the point
$\tau=i$ defined by elements $\beta_{t},g$ in the Möbius group as

$\tau_{g}(t)~{}=~{}g\circ\beta_{t}(i),~{}~{}~{}~{}~{}\beta_{t}~{}:=~{}\left(\matrix{e^{t/2}&0\cr
0&e^{-t/2}\cr}\right),~{}~{}g~{}=~{}\left(\matrix{a&b\cr c&d\cr}\right)\in{\rm
SL}(2,{\mathbb{R}})$ (34)

where the action is given as usual by the fractional transformation. The
identity $g={\rm id}$ leads to vertical geodesics on the imaginary axis
$\tau=e^{t}i$ and the shift matrices generate other vertical lines. This shows
that the upper halfplane ${\cal H}$ is a geodesically complete manifold. The
group theoretic geodesics (34) can be mapped into the hyperbolic function
parametrization that provide an alternative parametrization of the semi-circle
geodesics.

The selection of $j$-inflation trajectories shown in figure 3 for several
initial values illustrates that for a wide range of initial values the
potential of $j$-inflation, and hence modular inflation in general, has an
important effect on the specifics of the dynamical solutions. Even with the
constraint imposed on the number of e-folds the $j$-inflation trajectories
form a continuous family that turns from the vertical line type of geodesics
into the semi-circle type of geodesics. For initial values with large ${\rm
Im}~{}\tau$ and ${\rm Re}~{}\tau$ closer to the boundary of the central
vertical band the trajectories approximate the vertical geodesics, while for
initial values closer to the saddle point $\tau_{s}=i$ they eventually
approximate the semi-circle geodesics. In the process of this interpolation
the family of trajectories inevitably contains paths that are far from being
of geodesic type.

If one imposes further phenomenological constraints such as the spectral index
and the tensor ratio the $j$-inflation initial values are forced to start
closer to the saddle points, as illustrated by a selection of different
trajectories in figure 3. These paths show that during the early phases of
$j$-inflation even those trajectories that are compatible with the Planck
constraints for the spectral index and the tensor ratio can have interesting
dynamical behavior that deviates from the geodesic. On the other hand there do
exist trajectories in $j$-inflation that approach the geodesic very quickly.
This comparison of course only concerns the geometric form of the paths
involved. Physically, the potential surface is still important even for
trajectories that are close to being geodesic because it is the potential that
determines the time scales of the inflaton dynamics.

> Figure 3. An illustration of $j$-inflation trajectories for different energy
> scales $\mu$, and the geodesic that connects the saddle point $\tau_{s}$
> with the minimum of the $j$-inflation potential. As before, these
> $j$-inflation trajectories are truncated at the end-of-inflation, but the
> geodesic is tracked to the minimum of the potential.

Figure 3 also provides another illustration of the attractor nature of the
inflationary trajectories on a more local scale. If one continues to follow
the dynamics along these orbits they all converge onto the geodesic path that
connects the saddle point $\tau_{s}=i$ with the minimum of the $j$-inflation
potential. This feature is enhanced for trajectories that originate close to
the saddle point.

###  6.2 Attractor-like behavior

In the context of the issue of fine-tuning of initial conditions it is natural
to ask whether inflationary models show features that are reminiscent of
attractor behavior for inflaton trajectories. The mathematical concept of an
attractor involves large time behavior [66], hence strictly speaking it is not
applicable in models in which inflation ends after a finite time, whatever the
mechanism. It is nevertheless of interest to ask whether an inflationary
potential has basins into which inflaton trajectories with quite different
initial conditions tend to converge and whether the trajectories that converge
to these basins have similar physical parameters. For singlefield inflation
there is a large amount of literature on this topic, and for a special class
of models this is discussed briefly in the recent reviews [67, 68] and
references therein. In the framework of multifield inflation attractors have
received less attention, but a recent discussion can be found in [69]. In
light of these discussions it is of interest to consider trajectories in
$j$-inflation.

The illustration of the potential in figure 1 shows that it increases and
forms a wall for large imaginary components of the inflaton doublet. One can
ask what the trajectories are that these initial conditions with sufficient
inflation lead to and figure 4 shows a selection of $j$-inflation initial
values and their associated trajectories. The paths in this graph all lead to
inflation within the canonical range for the number of e-folds
$N_{*}\in[50,70]$ adopted in the present paper, and they are tracked to the
end of inflation.

> Figure 4. $j$-Inflaton trajectories with e-folds in the standard range
> $N_{*}\in[50,70]$. The hyperbolic metric rescales the physical length of
> these paths drawn in the diagram.

Included in figure 4 are also trajectories that start in the neighborhood of
saddle points of the potential that are closer to the real axis. These
trajectories appear to be much shorter in the coordinate space variables of
this graph but because of the nontrivial hyperbolic metric of the field space
the physical length of these paths is larger since they are closer to the
horizontal axis. The factor $(1/{\rm Im}~{}\tau)$ in the length $ds$ leads to
an ever stronger stretching as ${\rm Im}~{}\tau$ approaches zero.

The trajectories in figure 4 show that sufficient inflation can be obtained in
$j$-inflation for a wide variety of different types of trajectories. Along
some of these orbits either of the two components $\tau^{I}$ of the inflaton
can be approximately constant, which is an assumption that is sometimes made
in the twofield inflation literature. There are however also trajectories for
which both of the components vary considerably, hence in $j$-inflation
sufficient inflation can be obtained along trajectories that represent true
twofield inflation.

The paths shown in figure 4 also illustrate how the trajectories that
originate close to the saddle point $\tau_{s}=i$ serve as an attractor basin
into which a continuum of trajectories merge that start away from the saddle
point. This behavior extends even to trajectories that start higher up the
wall region of the potential. Only orbits that approach the linear type
geodesics when projected down onto the field space approach the minimum in a
more direct way. Their structure approximates vertical geodesics, which are
briefly described above.

##  7 Sufficient inflation

The question whether a given field theoretic model might be useful in an
inflationary context is usually posed in a first iteration as a constraint on
the lower bound on the number of e-folds $N_{*}$. The original motivation for
inflation leads to a rough estimate for $N_{*}$ and a number of analyses in
different models involving an inflaton doublet have focused on an analysis of
the phase space that lead to sufficient inflation in this sense. A model that
has received particular attention over the years is hybrid inflation [36], and
papers that report results of such scans include [37, 38, 39, 40, 41, 42, 43].
Inflation must end eventually and in principle such scans of the phase space
should impose also an upper bound for $N_{*}$.

The duration of inflation has received much attention in the literature, not
only because it is the parameter that is most immediately relevant for the
puzzles that initially played an important role in the introduction of the
inflationary framework, but because it has also served as a focal point to
provide more precision to the question of how probable inflation is, for
example in the formulation of Gibbons and Turok [49]. It has turned out to be
difficult to constrain the number of e-folds precisely, in part because it
depends on the choices that have to be made to account for the post-
inflationary evolution. The main uncertainty in determining a definite range
is the reheating stage, which is poorly constrained observationally and which
is complicated because of its preheating phase during the early stages of this
process. Despite much work on issues related to preheating no clear estimates
for the number of e-folds and the reheat temperature have emerged.
Consequently, the range of e-folds that has been considered in the literature
covers a wide range, reflecting the lack of constraints on the energy scales
that are allowed in this context, reaching from a lower bound given by the
nucleosynthesis scale of some $(1-10){\rm MeV}$, to the GUT scale of
$10^{16}{\rm GeV}$. Given a choice for the accepted range of the number of
e-folds, i.e. a choice of the evolutionary scenario, a key issue to address is
whether the given model under discussion provides enough inflation in the
adopted framework. In the absence of better constraints on the post-
inflationary stages, a canonical range for the number of e-folds $N_{*}$
between horizon crossing and the end of inflation has emerged that posits that
50 to 60 is a reasonable minimal value for $N_{*}$. Such values were for
example imposed as a lower bound in the work on hybrid inflation. In the
present analysis of $j$-inflation more specific bounds for the number of
e-folds are implemented.

###  7.1 Initial conditions in $j$-inflation for sufficient inflation

In the present paper a conservative range for the number of e-folds is chosen
as $N_{*}\in[50,70]$ to define viable realizations. Using this as a constraint
on $N_{*}$, the existence of trajectories in $j$-inflation with enough
inflation was established for $j$-inflation in [7, 9]. (The generalization to
higher level models was considered in [11].) A systematic scan of the slow-
roll phase space is computationally expensive at high resolutions of the field
space, with most of the time spent on the computation of the number of e-folds
$N_{*}$. A balance thus has to be found between the resolution of the lattice
of the inflaton space as well as the time resolution used in the determination
of the end of inflation $t_{e}$. By choosing the inflaton resolution
$\delta\tau^{I}=\delta\phi^{I}/\mu$ low enough it is possible to scan the
central vertical band out to a range of ${\rm Im}~{}\tau$ where the e-folds
fall outside of the adopted range. A typical scan of the initial values leads
to a structure that is reminiscent of a butterfly, as shown in figure 5.

> Figure 5. The butterfly graph determined by the constraints on the number of
> e-folds $N_{*}$.

The feelers of the butterfly distribution reflect the intricate structure of
the potential surface as one approaches the horizontal axis $\phi^{2}=0$ that
defines the boundary of the modular inflation field space. This structure is
indicated in the 3D plot of figure 1 and can be seen more clearly in the
global contour plot of figure 2. A discussion of the behavior of the inflaton
trajectories corresponding to these initial values has been given in the
previous section. The structure of the 2D butterfly can be explained by
resolving the graph by adding the number of e-folds $N_{*}$ as a third
dimension. Figure 6 shows that the upper and lower boundaries of the butterfly
structure are determined by the bounds on the number of e-folds. They are
obtained because when moving away from of the thorax like structure that
appears close to the ridge of the potential along the imaginary axis the
e-folds produced by the potential are below the lower bound of the interval
$N_{*}\in[50,70]$ in the lower region of the target space and above the upper
bound in the upper region of the butterfly wings. Varying the bounds for the
e-folds therefore varies the wing range accordingly.

> Figure 6. The 3D function of the e-fold number $N_{*}$ on the dimensionless
> target space $(\tau^{1},\tau^{2})$ for e-folds in the range $[50,70]$. This
> graph shows that the upper and lower boundaries of the butterfly wings are
> determined by the upper and lower limits adopted for the number of e-folds.

###  7.2 A natural measure on the space of initial conditions

A question often raised in the context of inflation is how much of the space
of initial conditions actually leads to inflation. The answer to this question
involves some kind of measure that needs to be chosen. For general multifield
inflation with a curved field space the target space comes equipped with a
natural measure given by the volume measure $\omega=\sqrt{{\rm
det}~{}G_{IJ}}\prod_{K}d\phi^{K}$. Even though in general the volume of a
noncompact space will diverge in this measure, the situation is different in
modular inflation, where the metric measure provides a finite volume for the
fundamental domain of the target space. There are two features of modular
inflation that combine to allow the computation of a finite fraction of the
initial conditions that lead to sufficient inflation. The first is that the
modular symmetry constructs the full target space as an infinite number of
copies of the irreducible fundamental domain ${\cal F}$ of the upper halfplane
${\cal F}\subset{\cal H}$, given by

${\cal F}~{}=~{}{\cal H}/{\rm SL}(2,{\mathbb{Z}}),$ (35)

which extends infinitely high along the imaginary axis in the interval
$[-1/2,1/2)$. The second feature is that the metric is given by (11) and while
the euclidean area of ${\cal F}$ is infinite, the hyperbolic metric provides a
finite area for the fundamental domain, given by

${\rm vol}({\cal F})~{}=~{}\int_{\cal F}\sqrt{{\rm
det}~{}G}~{}d\phi^{1}d\phi^{2}~{}=~{}\frac{\pi}{3}\mu^{2}~{}=~{}-\frac{2\pi}{3}\frac{1}{R_{\rm
fs}}.$ (36)

It will become clear in the later sections that the CMB data constrains $\mu$
and hence the physical area of the fundamental domain.

Given that the volume of the fundamental domain ${\cal F}$ is finite, it is
natural to consider the quotient of the butterfly volume and the total volume
as a measure of the likelihood of $j$-inflation with a number of e-folds in
the canonical interval. This quotient can be interpreted as the probability
for $j$-inflation to produce $N_{*}\in[50,70]$ e-folds. It can also be
interpreted as the posterior probability by assuming that the prior
probability distribution is constant, which allows to cancel it in Bayes’
theorem. As a result modular inflation leads to the probability

$P(N_{*}\in{\cal I})~{}=~{}\frac{{\rm vol}(B_{\cal F})}{{\rm vol}({\cal F})}.$
(37)

A rough estimate of the volume of $B_{\cal F}$ shows that a large fraction of
the total field space leads to inflation in the range considered here. In the
literature, the probability of inflation is often discussed by setting a lower
cut-off for $N_{*}$, for example $N_{*}\geq 60$, see e.g. the papers [37, 38,
40, 41, 42, 44], where the space of initial conditions with $N_{*}\geq 60$ is
analyzed for hybrid inflation. The structure of $j$-inflation is such that if
one lets $N_{*}$ grow unbounded then the probability approaches one.

###  7.3 Finite measures of modular inflation and the swampland conjectures

In the past the question of how likely inflation is has often been phrased in
terms of measures that are not normalizable, leading to regularization issues
[47, 48, 50, 51]. It is worthwhile to note here that the conjectures that have
been formulated in the context of the swampland conjectures place modular
inflation in a larger framework in which the existence of a geometric measure
on the target space is guaranteed by fiat.

The swampland conjectures have been formulated as an attempt to formulate
criteria for effective field theories that can be embedded in a theory of
quantum gravity. For the most part it is assumed that this theory is string
theory, and much of the work aimed at providing evidence for the conjectures
has been done within this framework. A review of recent work can be found in
[70] and further work on the impact of these conjectures on inflation includes
[71]. However, the formulation of at least some of the criteria considered so
far takes place within the effective field theory and does not directly refer
to string theoretic features. Among the earliest constraints imposed were
conditions extrapolated by Ooguri and Vafa [72] from the behavior of string
theory moduli. These can be rephrased in terms of inflationary scalar fields,
independent of any moduli interpretation. The focus of these early
considerations was on the topological and geometric structure of target space
of the scalar fields, in general a multi-component field $\phi^{I},I=1,...,n$.
The topological conjecture states that the target space is non-compact in the
sense that there exist trajectories of infinite length. The precise length of
these trajectories depends on the form of the distance measure [73], but this
does not affect the basic picture of the topological conjecture. Figure 3
illustrates the existence of such paths as one increases the imaginary range
of the inflaton doublet indefinitely.

The geometric conjecture posits that despite the non-compactness the target
space has finite volume. What is assumed implicitly in such statements is that
the focus is on the fundamental domain. In this context $j-$inflation, and
more generally modular inflation, are examples of theories that satisfy the
finite volume conjecture [74, 75], as discussed above.

##  8 Spectral index function $n_{{\cal R}{\cal R}}$ on the field space

One of the fundamental parameters that has been determined with a dramatic
increase in precision by the post-COBE satellite experiments is the adiabatic
spectral index $n_{{\cal R}{\cal R}}$ of the curvature Lukash-Bardeen
perturbation ${\cal R}$. The specific value and uncertainty of the spectral
index depends on the details of the fits. For the most part the fits that have
been published by the collaborations are aimed at singlefield inflation and
the parameter space is correspondingly lower dimensional. Fits that are aimed
at twofield inflation usually adopt special types of parametrizations for the
scalar power spectrum and often make assumptions concerning the correlations
between the adiabatic and the isocurvature perturbation. Currently no
multifield specific fit of the final Planck data release has been performed,
but extended fits beyond the minimal parameter count have been obtained in
ref. [76]. In the present paper bounds are used that are close to the Planck
values.

In $j$-inflation the spectral index $n_{{\cal R}{\cal R}}(\tau^{I})$ is given
by the fairly complicated function in eq. (24) in terms of the Eisenstein
series. The analytic structure of this function is not immediately transparent
but its geometry can be determined numerically and is shown in figure 7. The
structure of this surface explains some of the features of the numerical scans
for viable initial conditions that satisfy not only the constraint on the
number of e-folds, but are also consistent with the results from the CMB
probes.

Figure 7. The spectral surface in a larger scale view.

###  8.1 $n_{{\cal R}{\cal R}}$ scans of the field space

It follows from the $j-$inflation expressions for the spectral index and the
tensor ratio in eq. (24) that the initial values in the neighborhood of the
saddle points $\tau_{s}$ are good candidates that might lead to viable
trajectories. The region around $\tau_{s}=i$ was the focus of the
phenomenological analysis of $j$-inflation of [9], which established the
existence of viable trajectories. In the present section the focus is on the
constraints imposed on the target space by the spectral index as the inflaton
ranges over the target space for a fixed energy scale $\mu$. The effect of the
variation of $\mu$ on the spectral index is considered later in this paper.
The results of a global scan of the target space is shown in figure 8
superimposed on the contour plot of the potential. This illustrates in some
detail the canyon-like structure of the field space close to the boundary.

Figure 8. A scan for viable spectral indices on the field space
$(\tau^{1},\tau^{2})$.

This global map indicates a chain like iterative structure of saddle points
which can be resolved only to a limited degree in the region very close to the
real axis. In order to obtain a more detailed view it is useful to construct a
zoom for smaller values of the vertical component of the inflaton. In figure 9
the focus is on the region to the right of the ridge, for which this higher
resolution graph shows a more detailed structure as one approaches the real
axis, in particular a double sequence of arcs made of saddle point regions
that range from the right boundary of the central vertical band toward the
origin of field space. These regions come in two different orientations, which
are roughly diagonal and off-diagonal. Three different arcs are clearly
visible and a fourth one is indicated. The amount of detail visible in such
runs depends on the size of the lattice used to scan the field space, as well
as the accuracy with which the Eisenstein series are computed.

Figure 9. A zoom of the right part of the central region of the inflaton space
$(\tau^{1},\tau^{2})$.

##  9 Scaling behavior of the spectral index

A question that has been raised many times in the inflation literature is
whether it is possible to derive constraints for the amplitude of the
gravitational contribution to the CMB background, given some plausible
assumptions. The idea has been to abstract some general behavior from simple
models that allows to deduce in particular a lower bound for the tensor-to-
scalar ratio $r$ considered by the CMB collaborations [1, 2, 3]. One strategy
that has been adopted in singlefield inflation is to assume some scaling
behavior of the equation of state parameter and to consider the implications
that result [53]. This translates immediately into an assumption for the
scaling behavior of the slow-roll parameters and hence the spectral index and
the tensor ratio. Given the CMB constraints on the spectral index this
translates into estimates for $r$ [53, 55].

In singlefield inflation such a scaling behavior is immediate in some simple
models. In the class of monomial inflation models [77], the spectral index

$n_{{\cal R}{\cal R}}~{}=~{}1-6\epsilon_{V}+2\eta_{V}$ (38)

simplifies because of the relation $\epsilon_{V}=2(p-1)\eta_{V}/p$, with $p$
the monomial exponent, as well as the fact that the slow-roll parameter
$\epsilon_{V}$ can be written in terms of the number of e-folds as
$\epsilon_{V}\cong 1/N$. This leads to the scaling relation

$n_{{\cal R}{\cal R}}~{}\cong~{}1-\frac{\alpha_{p}}{N},$ (39)

where $\alpha_{p}=(p+2)/p$. This relation also holds approximately for
different classes of models, for example the Starobinsky model [78]. For about
$N=60$ e-folds the resulting spectral index is consistent with the CMB
constraints that have been determined in recent years by the WMAP and Planck
satellites. The question can be raised whether this scaling holds for some
range of $N$ away from the specific value $N=60$, often adopted as the
canonical value.

As noted above, such scaling relations were assumed to hold in attempts to put
constraints on the undetermined, but bounded, tensor-to-scalar ratio $r$. It
was in particular observed by Mukhanov [53] that the observational bounds
obtained by the WMAP satellite for the spectral index [79], in combination
with a relation like (39), allow to determine lower bounds on the tensor-to-
scalar ratio $r$ (see also the papers [54, 55]). This raises the question
whether such a relation might be valid more generally in the framework of
multifield inflation. In this case there are in general no analytical formulae
for $N$ and hence there is no easy access to analytical scaling relations.
However, the analysis presented above of the behavior of $j-$inflation allows
to address this issue.

In order to test whether the above scaling relation for $n_{{\cal R}{\cal R}}$
holds more generally in multifield inflation it is useful to define the
function

$f_{\alpha}(n_{{\cal R}{\cal R}},N)~{}=~{}n_{{\cal R}{\cal
R}}-\left(1-\frac{\alpha}{N}\right)$ (40)

for some constant $\alpha$ and consider the correlation of this functions with
the spectral index. If the scaling relation above holds the function
$f_{\alpha}$ vanishes. The analysis of the scaling relation in $j$-inflation
shows that the values of the function $f_{\alpha}$ for $\alpha$ of order one
are small, of the order of a few percent. This is illustrated by the graph in
figure 10. While the function $f_{\alpha}$ is small, the behavior of the
tensor ratio $r$ further above shows that the bounds on $r$ obtained in the
framework of singlefield inflation are not valid in multifield inflation.

> Figure 10. Typical graph for the function $f_{\alpha}$ in correlation to the
> spectral index $n_{{\cal R}{\cal R}}$.

##  10 The tensor-to-scalar ratio function $r$ on the field space

One of the fundamental phenomenological parameters that constrain inflation is
the power amplitude of primordial gravitational waves. This is conventionally
quantified by the CMB collaborations in terms of the tensor-to-scalar ratio
$r$ defined in eq. (6). For $j$-inflation this parameter becomes a function of
the energy scale $\mu$ that determines the curvature of the target space and
the inflaton field $\phi^{I}$ as in eq. (24). This observable $r$ has not yet
been determined, but the bounds of the amplitude of the gravitational
perturbations have dramatically improved since COBE and further experiments to
constrain $r$ are currently under construction. The main point of these
efforts of course is to detect primordial gravitational waves, and in the
process determine some of the characteristic features of inflation. However,
while the scale of inflation is an important characteristic, it is not the
only information tied to $r$ and even in the absence of a detection the
results of future experiments are important because they provide essential
constraints for inflationary model building. While an improvement of an order
of magnitude in the recent past has not provided insight into the scale of
inflation [65], the experimental achievements have been strong enough to
exclude an infinite number of models, among them the class of monomial
inflation potentials.

###  10.1 Large field inflation and small tensor-to-scalar ratios

One of the issues in which the bounds on $r$ have been instrumental is the
question whether the experimental target values predicted to be reached in the
near future will be able to exclude certain types of inflationary models. As
noted above, this has already been achieved by the exclusion of certain
sequences of models. A more far reaching discussion that has been conducted in
a large number of papers, following [65], is concerned with the question
whether there are bounds that can exclude large field inflation altogether. In
the past strong bounds have been suggested based on restricting consideration
to special classes of inflationary models, and some of these bounds have been
used in proposals for new observatories. It was in particular noted early on
that a non-detection of $r$ at the level of $10^{-2}$ [80] or $10^{-3}$ [81]
would rule out large field inflation, see also [82]. This value was
subsequently reduced somewhat, depending on specific assumptions made in the
analysis. Slightly lower constraints than the milli-scale mentioned above were
obtained in [83, 84] in the context of singlefield inflation. For a special
class of models this bound was pushed down further in [85], where it was
argued that large field inflation is only possible if this ratio is larger
than $r\geq 2\times 10^{-5}$. In light of these discussions it is of interest
to analyze the tensor-to-scalar ratio $r$ in the more general context of
multifield inflation with curved target spaces.

The current experimental bound for $r$ is several magnitudes above the most
stringent values values just quoted. An often cited value is the result
reported by the Planck experiment, which is reported to be given by $r\leq
0.06$ [1, 2]. Such bounds depend on the type of fit adopted in the
experimental analysis and the above value was obtained in a fit by assuming a
minimal number of parameters, which is not appropriate for multifield
inflation. Extensions beyond the standard six parameter fits have been
considered, but at present no Planck based full analysis appropriate for
twofield inflation is available in the literature. In the following the
singlefield bound will therefore be adopted as the constraint. Ongoing
experiment such as CLASS [86, 6] aim to reduce the Planck bound to the level
of $10^{-2}$, while upcoming and future ground based experiments such as the
Simons Observatory, the BICEP Array, and the CMB-S4 project will be able to
reduce this by more than an order of magnitude [87, 4, 88, 5, 89, 90]. The
satellite experiment Light BIRD [91] is designed to reach down to $r=2\cdot
10^{-3}$ at 95% C.L., while the target value of the proposed satellite
experiment PICO [92, 93] reaches further down by almost another order of
magnitude to $r=5\cdot 10^{-4}$ at a $5\sigma$ confidence level, with
$\sigma(r)=1\cdot 10^{-4}$ at $r=0$. As noted already, even a non-detection of
the tensor modes at this level would have significant effects on the current
inflationary model landscape and it would provide a better perspective on
different ad hoc selection rules that have been used to provide a lower bound
on $r$ [94].

The goal in the present section is to analyze the tensor-to-scalar ratio of
$j$-inflation in light of the experimental and theoretical bounds discussed
above. The systematic results are of course limited by the resolution of the
scans, but even so it becomes clear from the analysis below that the parameter
space of $j$-inflation is constrained by the Planck bounds on $r$, hence
$j$-inflation presents a target for upcoming and planned experiments. It will
also become clear that there are regions in parameter space in which the lower
bound of $r$ can be pushed below the experimental floor of all upcoming
experiments. The $r$-range of $j$-inflation in particular reaches below the
boundary that has been identified in the literature as the dividing line for
large field inflation in the singlefield context. The fact that $j$-inflation
can reach below these proposed large field boundaries alleviates the concerns
expressed in a number of recent discussions [88, 89, 92, 93] that the
measurement of an experimental bound for the tensor ratio below $r\cong
10^{-3}$ would force a significant change in our understanding of the
primordial universe. The example of $j$-inflation shows that the framework of
twofield inflation is able to reach below these thresholds.

###  10.2 Global $r$-scan

As in all scans of the target space the specific results obtained for the
tensor ratio depend on the size of the fundamental lattice cell as well as the
parameter $\mu$ that is not fixed by the CMB amplitude in the formulation
considered here. What does not change is the global structure of the results.
As in the case of the spectral index, it is useful to consider the shape of
the analytical form of the tensor ratio $r(\tau^{I})$ of $j$-inflation as a
function on the target space via eq. (24). This is shown in figure 11.

Figure 11. Graph of the tensor-to-scalar ratio $r(\tau^{I})$ of eq. (24).

In figure 12 the results are given for a global scan of $r$along the central
vertical band. The resulting pattern of this band is repeated in the upper
halfplane to the left and right by the shift symmetry
$\tau~{}{\rightarrow}~{}\tau+1$ that is part of the modular group ${\rm
SL}(2,{\mathbb{Z}})$ that leaves the potential invariant. It is this
invariance that is part of the motivation of modular, and more generally
automorphic inflation [7, 8]. The structure of the central band changes in
dependence of the distance to the real axis. The areas closer to the boundary
of the target space given by ${\rm Im}~{}\tau=0$ are distorted in the plot of
figure 12 because it is drawn by using the euclidean metric. The hyperbolic
metric that describes the actual geometry of the field space increasingly
stretches these domains as the horizontal boundary is approached because of
the factor $1/({\rm Im}~{}\tau)^{2}$ that enters $ds^{2}$. This measure leads
to finite volumes of the target space in which the Planck bound on the tensor
ratio is satisfied.

> Figure 12. Global scan for viable initial values along the central vertical
> band consistent with the tensor ratio constraint. Close to the horizontal
> axis the effect of the hyperbolic metric leads to a stretching of the
> euclidean distances of this plot.

###  10.3 Effect of the field space curvature $R_{\rm fs}$ on the tensor ratio
$r$

Recall from the discussion above that once the energy scale $\mu$ is chosen,
the overall scale $\Lambda$ of the potential follows from the CMB amplitude,
making $\mu$ the only parameter that can be varied. This means that the only
free parameter in the model is the curvature scalar of the target space
$R_{\rm fs}=-2/\mu^{2}$ determined by the metric, eq. (12). The tensor ratio
in eq. (24) shows that $r$ scales like $1/\mu^{2}$, hence it scales with the
scalar curvature $R_{\rm fs}$ of the field space

$r~{}=~{}-64\pi^{2}M_{\rm Pl}^{2}~{}R_{\rm fs}~{}({\rm
Im}~{}\tau)^{2}\left|\frac{E_{6}}{E_{4}}\right|^{2}.$ (41)

The second ingredient that varies is the inflaton value $\phi_{*}^{I}$ at
horizon crossing and these two parameters determine the range of the $r$. The
precise boundary values of $\mu$, or $R_{\rm fs}$, and $\phi_{*}^{I}$ depend
on the range of the number of e-folds, and on the lattice resolution of the
scan.

To be concrete, for a fixed value of the target space curvature $R_{\rm fs}$
via a choice of $\mu$ the initial values $\tau^{I}_{*}$ that satisfy the
constraints adopted for $r,n_{{\cal R}{\cal R}}$ and $N_{*}$ sweep out a
finite band in the $(n_{{\cal R}{\cal R}},r)$-plane. As the target space
curvature is varied this band moves up and down along the axis defined by the
tensor ratio $r$. It is a priori not transparent how the imposed constraints
combine to determine the specific boundaries and the band in the $(n_{{\cal
R}{\cal R}},r)$-plane. This is illuminated by considering a three-dimensional
plot which adds the number of e-folds $N_{*}$ to this plane. Figure 13 shows
that doing so explains the boundaries as determined by the bounds adopted for
$N_{*}$. This lifting of the planar plot shows that the lower bound on $r$
arises from the upper bound of the number of e-folds. If the upper limit
$N_{*}$ were to be increased the lower bound on $r$ would decrease for fixed
$\mu$. The degeneracy of the sheet in figure 13 along the direction of the
spectral index shows that in $j$-inflation a scaling relation exists between
the tensor ratio $r_{*}$ and the number of e-folds $N_{*}$. Parametrizing the
tensor ratio as $r=\alpha N^{-\beta}$ one obtains for $\beta=1$ the monomial
class, for $\beta=2$ the Starobinsky type range of $r$, and for higher $\beta$
the range covered by $j$-inflation, with increasing suppression obtained for
larger target space curvature $R_{\rm fs}$.

> Figure 13. The $N_{*}$ dependence in the $(n_{{\cal R}{\cal R}},r,N)$ space
> at fixed $\mu$: as the number of e-folds grows the tensor ratio $r$
> decreases.

The issue of the upper limit for $N_{*}$ has a long history and the specific
bound $N_{*}\leq 70$, while often adopted, is not sharp. Discussions of the
allowed range of the number of e-folds during inflation can be found in [96,
97] and their implications for trans-Planckian physics has also been
extensively discussed. A review of these issues can be found in [67].

As noted above, the band swept out in the plane spanned by $n_{{\cal R}{\cal
R}}$ and $r$ in dependence of the CMB constraint on the spectral index and the
adopted range for the number of e-folds depend on the energy scale $\mu$ that
determines the field space curvature $R_{\rm fs}$. As the bounds obtained by
the Planck probe will be improved by the next generation of experiments
currently under construction the bound on the curvature scalar will become
stronger. This raises the question how much the tensor ratio is affected by a
variation of the target space curvature that remains consistent with the
Planck bounds on the spectral index and $N_{*}$. In figure 14 the results for
$R_{\rm fs}=1/\mu^{2}$ with $\mu=27M_{\rm Pl}$ and $\mu=35M_{\rm Pl}$ are
shown. This graph illustrates that the $r$-bands determined by the target
space curvature lie for both energy scales $\mu$ within the confidence region
of the Planck results. As the curvature $R_{\rm fs}$ increases the band moves
lower into the region targeted by the Simons observatory [88]. This shows that
the range of the tensor ratio obtained in $j-$inflation for different $\mu$
contains and extends the region of the $r$-value obtained within the class of
monomial inflation, as well as that of the Starobinsky model [78]. The latter
in particular is a prominent model that has motivated some experimental
proposals in the recent past.

> Figure 14. The results obtained for two scans of $j$-inflation at energy
> scales $\mu/M_{\rm Pl}=27,35$, with associated curvature $R_{\rm fs}$ of the
> target space, projected onto the Simons Observatory confidence regions [88].
> The results for the larger $\mu$-value (black, upper region of the data
> points) intersect with the Planck range, but not with the target range of
> the Simons telescope. For the smaller $\mu$-value (red, lower region) the
> distribution intersects with the region described by the Simons Array
> collaboration.

The analysis of this section shows that $j$-inflation provides an interesting
target for experiments that aim to discover the gravitational contribution of
the CMB signal, or at least to significantly improve the constraints on the
tensor ratio. This includes experiments that are scheduled to come online in
the near future, such as the Simons observatory, CMB-S4, as well as the
satellite experiment LightBird. The results from these observations will have
an impact on the size of the field space curvature of the hyperbolic field
space of modular inflation.

##  11 Effect of the $\mu$-variations

Modular inflation as described in section three is parametrized by the energy
scale $\Lambda$ and the inflaton scale $\mu$ needed to make the inflaton
dimensionless. The scale $\Lambda$ of the potential is as usual determined by
the amplitude of the CMB power spectrum measured by the satellite experiments.
This leaves the scale $\mu$ as an a priori undetermined parameter of the
theory, which is related to the field space scalar curvature via eq. (12).
This parameter is constrained however by the CMB results for the observables,
such as the spectral index and the bound on the tensor ratio. Any scan of the
target space is of course limited by the adopted lattice resolution and other
run parameters that are constrained by CPU resources. The scans discussed were
mostly performed with a fixed $\mu$ and the question arises how these change
when $\mu$ is varied. This is illustrated for the spectral constraint
$n_{{\cal R}{\cal R}}$ in figure 15, which represents a compilation of initial
inflaton values that satisfy the Planck constraint for the range of
$\mu\in[25,50]M_{\rm Pl}$. The modular symmetry of the $j$-inflation model
implies that the same structure is repeated along the modular sequences
described in sections four and five, repeating this pattern indefinitely.

> Figure 15. A compilation of results for the spectral scalar index for
> $\mu/M_{\rm Pl}$-variations between 25 and 50 in the region around the
> saddle point $\tau_{s}=i$.

Imposing not only the spectral index, but also the tensor-ratio bound of
Planck and the constraint on the number of e-folds $N_{*}$ selects from the
figure eight of the previous graph a subset of initial values that for the
same range of $\mu$-values and the same run-parameters produces a region in
the target space of the form of a nutcracker, shown in figure 16.

> Figure 16. The nutcracker of the constraints $n_{{\cal R}{\cal R}}\cap r\cap
> N_{*}$ for the runs in the previous graph, i.e. $\mu/M_{\rm
> Pl}\in[25,~{}50]$.

A natural question that can next be raised is how the tensor to scalar ratio
$r$ behaves as the energy scale is varied. In the context of the $(n_{{\cal
R}{\cal R}},r)$ plane usually considered by the CMB collaborations this has
been discussed in section ten and illustrated in figure 14. In figure 17 this
is addressed by considering the 3D-distribution of the parameters $(n_{{\cal
R}{\cal R}},r,N_{*})$ for three different values of $\mu$. This plot allows to
extrapolate the structure for the intermediate values and also how the pattern
extends beyond the values shown. The different bands in dependence of $\mu$
show that the scaling relation mentioned in the previous section between
$r_{*}$ and $N_{*}$ varies with $\mu$. Figure 17 shows that the scaling
relation mentioned already depends on the energy parameter $\mu$.

> Figure 17. Illustration of the $\mu$-dependence of the $(n_{{\cal R}{\cal
> R}},r,N_{*})$ distribution for $\mu~{}\in~{}\\{30,35,40\\}$. The sheets move
> from right to left with decreasing $\mu$.

##  12 Conclusions

The present paper has conducted an extensive systematic analysis of the
inflaton behavior in modular inflation with the potential given by the
$j$-function. In the process it has become apparent that the field space has
an interesting structure that originates from the underlying symmetry, which
allows to perform a quite complete scan of the total target space, due in part
to the shift-symmetry that is part of the modular invariance. The shift
symmetry in particular makes it possible to effectively scan completely the
part of the target space constrained by the Planck data because it leads to a
decomposition of the target space into an infinite number of vertical bands
that all have the same field theoretic behavior. Within each of these bands
the tessellation induced by the modular invariance in turn leads to an
iterative structure that maps large field regions to regions where the initial
field components take values that are arbitrarily small. This shows that
modular invariant inflation is a framework in which the distinction between
large field and small field inflation into separate classes no longer holds.
As a result the notion of large-field and small-field inflation is not a model
characteristic of general inflationary models. While this has been shown here
in the context of a particular model, the modularity admitted by this type of
geometry generalizes to other models in this class of theories. In the more
general context of automorphic inflation with an arbitrary number of inflaton
components a similar symmetry structure arises where the modular groups are
replaced by discrete groups $G({\mathbb{Z}})$ for reductive groups
$G({\mathbb{R}})$ [7, 8]. As a result the classification of inflation should
in general proceed along lines that do not reference the dichotomy of large
field vs. small field inflation. A further implication of the tessellation of
the field space in combination with the structure of the $j$-inflation
potential is that in the geometric measure provided by the target space metric
the fraction of the volume of the field space that leads to sufficient
inflation is finite.

A characteristic feature of the potential surface of $j$-inflation is a wall-
like structure of the potential along the vertical direction of the inflaton
doublet. The motion of the inflaton along this wall is of geodesic type, and
is restricted by the phenomenological constraints. Inflation trajectories that
are compatible with the Planck data exist in neighborhoods around the critical
points of the potential, leading to inflaton orbits that in general are
different from hyperbolic geodesics, but can approximate them as the initial
values get closer to these critical points. As a result, while the presence of
the hyperbolic metric in modular inflation has important implications for the
structure of the theory, the dynamics determined by the potential is crucial
for the behavior of the inflaton. In particular the attractor structure of the
model is determined by the potential, as expected. Nevertheless, there are
regions in the target space where the geometry of the inflaton trajectories
follows closely that of the different types of geodesics that characterize the
upper halfplane hyperbolic target space. In the context of the dynamical
structure the attractors in the potential surface project down to trajectories
that are close to the two different types of geodesics that are encountered in
hyperbolic geometry.

The phenomenological part of the analysis first shows that the region of the
target space with sufficient inflation in the canonical range has the shape of
a butterfly that covers a finite fraction the total field space. Imposing
further the CMB constraints on the spectral index and the tensor ratio leads
to initial values for which the spectral index admits an approximate scaling
behavior that is reminiscent of the scaling behavior of monomial and
Starobinsky inflation. Nevertheless the bounds on $r$ that have been discussed
in the literature in the context of singlefield inflation are not satisfied in
$j$-inflation. A detailed analysis of the tensor ratio on the space of initial
values instead shows that upcoming gravity wave experiments that can reach
down to $r\cong 10^{-4}$ and even smaller will not be able to distinguish
between large and small field inflation. These experiments are nevertheless
important even in the absence of a discovery because they provide essential
constraints for model building. They will in particular constrain the
parameter space for $j-$inflation, which presents a prime target for future
observations.

The CMB constraints that have been obtained since COBE have established the
amplitude of the scalar power spectrum, leading to a normalization of the
overall energy scale of inflationary models. In $j$-inflation as described in
the present paper this leaves the second energy parameter $\mu$, which is
constrained by the phenomenological parameters just discussed. This energy
scale uniquely determines the curvature scalar of the field space, hence these
observational constraints put bounds on how strongly curved this target space
can be.

Acknowledgement.
It is a pleasure to thank Monika Lynker and Deepu Sengupta for discussions.
This work was supported by a sabbatical leave of absence from Indiana
University South Bend.

## References

  * [1] N. Aghanim et al. (Planck Collab.), Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641 (2020) A6, arXiv: 1807.06209 [astro-ph.CO]
  * [2] Y. Akrami et al. (Planck Collab.), Planck 2018 results. X. Constraints on inflation, Astron. Astrophys. 641 (2020) A10, arXiv: 1807.06211 [astro-ph.CO]
  * [3] P. A. R. Ade et al., BICEP2 / Keck Array X: Constraints on Primordial Gravitational Waves using Planck, WMAP, and New BICEP2/Keck Observations through the 2015 Season, Phys. Rev. Lett. 121 (2018) 221301, arXiv: 1810.05216 [astro-ph.CO]
  * [4] P. Ade et al., The Simons Observatory: Science goals and forecasts, JCAP 02 (2019) 056 arXiv: 1808.07445
  * [5] K.N. Abazajian et al., CMB-S4 science case, reference design, and project plan, arXiv: 1907.04473 [astro-ph.IM]
  * [6] S. Dahal et al.(CLASS Collab.), Four-year cosmology large angular scale surveyor (CLASS) observations: On-sky receiver performance at 40, 90, 150 and 220 GHz frequency bands, arXiv: 2107.08022 [astro-ph.IM]
  * [7] R. Schimmrigk, Automorphic inflation, Phys. Lett. B748 (2015) 376 $-$ 379, arXiv: 1412.8537 [hep-th]
  * [8] R. Schimmrigk, A general framework of automorphic inflation, JHEP 05 (2016) 140, arXiv:1512.09082 [hep-th]
  * [9] R. Schimmrigk, Modular inflation observables and $j$-inflation phenomenology, JHEP 09 (2017) 043, arXiv: 1612.09559 [hep-th]
  * [10] R. Schimmrigk, Multifield reheating after modular j-Inflation, Phys. Lett. B782 (2018) 193 $-$ 197, arXiv: 1712.09961 [hep-th]
  * [11] M. Lynker and R. Schimmrigk, Modular inflation at higher level $N$, JCAP 06 (2019) 036, arXiv: 1902.04625 [astro-ph.CO]
  * [12] J.J.M. Carrasco, R. Kallosh and A. Linde, Hyperbolic geometry and cosmological attractors, Phys. Rev. D92 (2015) 04130, arXiv: 1504.05557 [hep-th]
  * [13] J.J.M. Carrasco, R. Kallosh and A. Linde, Cosmological attractors and initial conditions for inflation, Phys. Rev. D92 (2015) 063519, arXiv: 1506.00936 [hep-th]
  * [14] V. Aragam, S. Paban and R. Rosati, Multi-field inflation in high-slope potentials, JCAP 04 (2020) 022, arXiv: 1905.07495 [hep-th]
  * [15] A. Achucarro and Y. Welling, Orbital inflation: inflating along an angular isometry of field space, arXiv: [1907.02020 [hep-th]
  * [16] R. Kallosh and A. Linde, Multi-field conformal cosmological attractors, JCAP 12 (2013) 006, arXiv: 1309.2015
  * [17] R. Kallosh, A. Linde and D. Roest, Large field inflation and double $\alpha$-attractors, JHEP (2014), arXiv: 1405.3646
  * [18] R. Kallosh and A. Linde, Escher in the sky, Compt. Rend. Phys. 16 (2015) 914, arXiv: 1503.06785 [hep-th]
  * [19] A. Achucarro, R. Kallosh, A. Linde, D.-G. Wang and Y. Welling, Universality of multi-field $\alpha$-attractors, JCAP 04 (2018) 028, arXiv: 1711.09478 [hep-th]
  * [20] A. Brown, Hyperbolic inflation, Phys. Rev. Lett. 121 (2018) 251601, arXiv: 1705.03023 [hep-th]
  * [21] C.I. Lazaroiu and C.S. Shahbazi, Generalized two-field $\alpha$-attractor models from geometrically finite hyperbolic surfaces, Nucl. Phys. B936 (2018) 542 $-$ 596, arXiv:1702.06484 [hep-th]
  * [22] E.M. Babalic and C.I. Lazaroiu, Generalized $\alpha$-attractors from the hyperbolic triply-punctured sphere, Nucl. Phys. B937 (2018) 434 $-$ 477, arXiv:1703.06033 [hep-th]
  * [23] S. Mizuno and S. Mukohyama, Primordial perturbations from inflation with a hyperbolic field-space, Phys. Rev. D96 (2017) 103533, arXiv: 1707.05125 [hep-th]
  * [24] A. Linde, D.-G. Wang, Y. Welling and A. Achúcarro, Hypernatural inflation, JCAP 07 (2018) 035, arXiv: 1803.09911 [hep-th]
  * [25] L. Anguelova, E.M. Babalic and C.I. Lazaroiu, Hidden symmetries of two-field cosmological models, JHEP 09 (2019) 007, arXiv:1905.01611 [hep-th]
  * [26] S. Mizuno, S. Mukohyama, S. Pi and Y.-L. Zhang, Hyperbolic field space and swampland conjecture for DBI scalar, JCAP 09 (2019) 072, arXiv: 1905.10950 [hep-th]
  * [27] M. Bounakis, I.G. Moss and G. Rigoropoulos, Observational constraints on hyperinflation, arXiv: 2010.06461 [gr-qc]
  * [28] V. Aragam, S. Paban and R. Rosati, The multi-field, rapid-turn inflationary solution, JHEP 03 (2021) 09, arXiv: 2010.15933 [hep-th]
  * [29] X. Chen, G.A. Palma, W. Riquelme, B.S. Hitschfeld and S. Sypsas, Landscape tomography through primordial non-Gaussianity, Phys. Rev. D98 (2018) 083528, arXiv: 1804.07315
  * [30] T. Bjorkmo and M.C.D. Marsh, Hyperinflation generalised: from its attractor mechanism to its tension with the ‘swampland conditions’, JHEP 04 (2019) 172, arXiv: 1901.08603 [hep-th]
  * [31] P. Christodoulidis, D. Roest and E.I. Sfakianakis, Attractors, bifurcations and curvature in multi-field inflation, JCAP 08 (2020) 006, arXiv: 1903.03514 [gr-qc]
  * [32] O. Grocholski, M. Kalinowski, M. Kolanowski, S. Renaux-Petel, K. Turzynski and V. Vennin, On backreaction effects in geometrical destabilisation of inflation, JCAP 05 (2019) 008, arXiv: 1901.10468 [astro-ph.CO]
  * [33] D.-G. Wang, On the inflationary massive field with a curved field manifold, JCAP 01 (2020) 046, arXiv: 1911.04459 [astro-ph.CO]
  * [34] R. Kallosh and A. Linde, CMB targets after Planck, Phys. Rev. D100 (2019) 123523, arXiv: 1909.04687 [hep-th]
  * [35] R. Bravo, G.A. Palma and S. Riquelme M., A tip for landscape riders: multi-field inflation can fulfill the swampland distance conjecture, JCAP 02 (2020) 004, arXiv: 1906.05772 [hep-th]
  * [36] A. Linde, Hybrid inflation, Phys. Rev. D49 (1994) 748 $-$ 754, arXiv: astro-ph/9307002
  * [37] G. Lazarides, C. Panagiotakopoulos and N.D. Vlachos, Initial conditions for smooth hybrid inflation, Phys. Rev. D54 (1996) 1369 $-$ 1373, arXiv: hep-ph/9606297
  * [38] G. Lazarides and N.D. Vlachos, Initial conditions for supersymmetric inflation, Phys. Rev. D56 (1996) 4562 $-$ 4567, arXiv: hep-ph/9708296
  * [39] N. Tetradis, Fine tuning of the initial conditions for hybrid inflation, Phys. Rev. D57 (1998) 5997 $-$ 6002, arXiv: astro-ph/9707214
  * [40] L.E. Mendes and A.R. Liddle, Initial conditions for hybrid inflation, Phys. Rev. D62 (2000) 103511, arXiv: astro-ph/0006020
  * [41] S. Clesse and J. Rocher, Avoiding the blue spectrum and the fine-tuning of initial conditions in hybrid inflation, Phys. Rev. D79 (2009) 103507, arXiv: 0809.4355
  * [42] S. Clesse, C. Ringeval and J. Rocher, Fractal initial conditions and natural parameter values in hybrid inflation, Phys. Rev. D80 (2009) 123534, arXiv: 0909.0402
  * [43] S. Clesse, Hybrid inflation along waterfall trajectories, Phys. Rev. D83 (2011) 063518, arXiv: 1006.4522
  * [44] R. Easther and L.C. Price, Initial conditions and sampling for multifield inflation, JCAP 07 (2013) 027 arXiv: 1304.4244 [astro-ph.CO]
  * [45] S. Clesse and J. Rekier, Updated constraints on large field hybrid inflation, Phys. Rev. D90 (2014) 083527, arXiv: 1407.1984 [astro-ph.CO]
  * [46] P. Carrilho, D. Mulryne, J. Ronayne and T. Tenkanen, Attractor behavior in multifield inflation, JCAP 06 (2018) 032, arXiv: 1804.10489 [astro-ph.CO]
  * [47] G. Gibbons, S. Hawking and J. Stewart, A natural measure on the set of all universes, Nucl. Phys. B281 (1987) 736
  * [48] S. Hawking and D.N. Page, How probable is inflation?, Nucl. Phys. 298 (1988) 789 $-$ 809
  * [49] G. Gibbons and N. Turok, The measure problem in cosmology, Phys. Rev. D77 (2008) 063516, arXiv: hep-th/0609095
  * [50] S.M. Carroll and H. Tam, Unitary evolution and cosmological fine-tuning, arXiv: 1007.1417 [hep-th]
  * [51] J.S. Schiffrin and R.M. Wald, Measure and probability in cosmology, Phys. Rev. D86 (2012) 023521, arXiv: 1202.1818 [astro-ph.CO]
  * [52] G.N. Remmen and S.M. Carroll, Attractor solutions in scalar-field cosmology, Phys. Rev. D88 (2013) 083518, arXiv: 1309.2611 [gr-qc]
  * [53] V. Mukhanov, Quantum cosmological perturbations: predictions and observations, Eur. Phys. J. C73 (2013) 2486, arXiv: 1303.3925 [astro-ph.CO]
  * [54] D. Roest, Universality classes of inflation, JCAP 01 (2014) 007, arXiv: 1309.1285
  * [55] P. Creminelli, S. Dubovsky, D. Lopez Nacir, M. Simonovic, G. Trevisan, G. Villadoro and M. Zaldarriaga, Implications of the scalar tilt for the tensor-to-scalar ratio, Phys. Rev. D92 (2015) 123528, arXiv: 1412.0678 [astro-ph.CO]
  * [56] V.N. Lukash, Production of sound waves in the early universe, Zh. Eksp. Teor. Fiz. 31 (1980) 631 $-$ 635
  * [57] J.M. Bardeen, Gauge-invariant cosmological perturbations, Phys. Rev. D22 (1980) 1882 $-$ 1905
  * [58] S. Weinberg, Cosmology, Cambridge University Press, 2008
  * [59] R. Krishnan, Symmetries of stationary points of the $G$-invariant potential and the framework of the auxiliary group, Phys. Rev. D103 (2021) L051701, arXiv: 2011.11653 [hep-ph]
  * [60] S. Dodelson, W.H. Kinney and E.W. Kolb, Cosmic microwave background measurements can discriminate among inflation models, Phys. Rev. D56 (1997) 3207 $-$ 3215, arXiv: astro-ph/9702166
  * [61] W.H. Kinney, Constraining inflation with CMB polarization, Phys. Rev. D58 (1998) 123506, arXiv: astro-ph/9806259
  * [62] S. Hannestad, S.H. Hansen and F.L. Villante, Probing the power spectrum bend with recent CMB data, Astropart. Phys. 16 (2001) 137 $-$ 144, arXiv: astro-ph/0012009
  * [63] H.V. Peiris et al. (WMAP Collab.), First year WMAP observations: implications for inflation, Ap. J. Suppl. 148 (2003) 213 $-$ 231, arXiv: astro-ph/0302225
  * [64] W.H. Kinney, E.W. Kolb, A. Melchiorri and A. Riotto, WMAPping inflationary physics, Phys. Rev. D69 (2004) 103516, arXiv: hep-ph/0305130
  * [65] D.H. Lyth, What would we learn by detecting a gravitational wave signal in the cosmic microwave background anisotropy?, Phys. Rev. Lett. 78 (1997) 1861, arXiv: hep-ph/9606387
  * [66] J. Milnor, On the concept of attractors, Commun. Math. Phys. 99 (1985) 177 $-$ 195
  * [67] R. Brandenberger, Initial conditions for inflation $-$ a short review, Int. J. Mod. Phys. 26 (2016) 1740002, arXiv: 1601.01918 [hep-th]
  * [68] A. Linde, On the problem of initial conditions for inflation, Found. Phys. 48 (2018) 1246, arXiv: 1710.04278 [hep-th]
  * [69] K. Finn and S. Karamitsos, Finite measure for the initial conditions of inflation, Phys. Rev. D99 (2018) 063515 (Erratum: Phys.Rev. D99 (2019) no.10, 109901), arXiv:1812.07095 [gr-qc]
  * [70] E. Palti, The swampland: introduction and review, Fortsch. Phys. 67 (2019) 1900037, arXiv: 1903.06239 [hep-th]
  * [71] Z. Wang, R. Brandenberger and L. Heisenberg, Eternal inflation, entropy bounds and swampland, Eur. Phys. J. C80 (2020) 864, arXiv: 1907.08943 [hep-th]
  * [72] H. Ooguri and C. Vafa, On the Geometry of the String Landscape and the Swampland, Nucl. Phys. 766 (2006) 21$-$33, arXiv: hep-th/0605264
  * [73] R. Schimmrigk, The spectrum conjecture and the swampland, arXiv: 1810.11699 [hep-th],
  * [74] C. Vafa, The string landscape and the swampland, arXiv: hep-th/0509212 [hep-th]
  * [75] M.R. Douglas and Z. Lu, Finiteness of volume of moduli spaces, arXiv: hep-th/0509224 [hep-th]
  * [76] E. Di Valentino, A. Mechiorri and J. Silk Cosmological constraints in extended parameter space from the Planck 2018 legacy release, JCAP 01 (2020) 013, arXiv: 1908.01391 [astro-ph.CO]
  * [77] M.S. Turner, Coherent scalar-field oscillations in an expanding universe, Phys. Rev. D28 (1983) 1243 $-$ 1247
  * [78] A.A. Starobinsky, A new type of isotropic cosmological models without singularity, Phys. Lett. B91 (1980) 99 $-$ 102
  * [79] G. Hinshaw et al. (WMAP Collab.), Nine-year WMAP observations: cosmological parameter results, Astrophys. J. Suplp. 208 (2013) 19, arXiv: 1212.5226 [astro-ph.CO]
  * [80] D. Baumann et al. (CMBPol Study Team), CMBPol Mission concept study: A mission to map our origins, AIP Conf. Proc. 1141 (2009) 10, arXiv: 0811.3919 [astro-ph.CO]
  * [81] K.N. Abazajian et al., Inflation physics from the CMB and large scale structure, Astropart. Phys. 63 (2015) 55, arXiv:1309.5381 [astro-ph.CO]
  * [82] G. Efstathiou and S. Chongchitnan, The search for primordial tensor modes, Progr. Theor. Phys. Suppl. 163 (2006 204, arXiv: astro-ph/0603118
  * [83] J. Garcia-Bellido, D. Roest, M. Scalisi and I. Zavala, The Lyth bound of inflation with a tilt, Phys. Rev. D90 (2014) 123539, arXiv: 1408.6839 [hep-th]
  * [84] Q.-G. Huang, Lyth bound revisited, Phys. Rev. D91 (2015) 123532, arXiv: 1503.04513 [astro-ph.CO]
  * [85] A. Linde, Gravitational waves and large field inflation, JCAP 02 (2017) 006, arXiv: 1612.00020 [astro-ph.CO]
  * [86] J. Iuliano et al., The cosmology large angular scale surveyor receiver design, Proc. SPIE Int. Soc. Opt. Eng. 10708 (2018) 1070828, arXiv: 1807.04167 [astro-ph.IM]
  * [87] H. Hui et al., BICEP Array: a multi-frequency degree-scale CMB polarimeter, in Millimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for Astronomy  IX, ser. Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol. 10708, Jul. 2018, p. 1070807
  * [88] M.H. Abitbol et al., The Simons Observatory, Astro2020 APC Decadal Project White Paper , Bull. Am. Astron. Soc. 51 (2019) 147, arXiv: 1907.08284 [astro-ph.CO]
  * [89] K.N. Abazajian et al. CMB-S4 Decadal survey APC white paper, Bull. Am. Astron. Soc. 51 (2019) 209, arXiv: 1908.01062 [astro-ph.IM]
  * [90] K.N. Abazajian et al. CMB-S4: Forecasting constraints on primordial gravitational waves, arXiv: 2008.12619 [astro-ph.CO]
  * [91] M. Hazumi et al., (LiteBIRD Collab.), LiteBIRD: a satellite for the studies of B-mode polarization from cosmic background detection, J. Low. Temp. Phys. 194 (2019) 443
  * [92] S. Hanany et al., PICO: Probe of inflation and cosmic origins, arXiv: 1902.10541 [astro-ph.IM]
  * [93] M. Alvarez et al. PICO: Probe of inflation and cosmic origins, arXiv: 1908.07495 [astro-ph.IM]
  * [94] L. Boyle, P. Steinhardt and N. Turok, Inflationary predictions for scalar and tensor fluctuations reconsidered, Phys. Rev. Lett. 96 (2006) 111301, arXiv: astro-ph/0507455
  * [95] S. Shandera et al., Probing the origin of our universe through cosmic microwave background constraints on gravitational waves, Bull. Am. Astron. Soc. 51 (2019) 338, arXiv: 1903.04700 [astro-ph.CO]
  * [96] S. Dodelson and L. Hui, Horizon ratio bound for inflationary fluctuations, Phys. Rev. Lett. 91 (2003) 131301, arXiv: astro-ph/0305113
  * [97] A.R. Liddle and S.M. Leach, How long before the end of inflation were observable perturbations produced?, Phys. Rev. D68 (2003) 1003503, arXiv: astro-ph/0305263