Modular Inflation at Higher Level $N$

TL;DR

This paper extends modular inflation models to higher levels, analyzing their modular properties and proposing new models based on hauptmodul functions, with implications for CMB observations and quantum gravity conjectures.

Contribution

It generalizes modular inflation to higher levels and introduces a new class of models using hauptmodul functions, linking them to observable CMB signatures.

Findings

Modular structure ensures almost holomorphicity of inflation parameters.

Hauptmodul-based models provide testable CMB predictions.

Discussion of quantum gravity conjectures in the context of higher-level modular inflation.

Abstract

We introduce the framework of modular inflation with level structure, generalizing the level one theory considered previously to higher levels. We analyze the modular structure of CMB observables in this framework and show that the nontrivial geometry of the target space suffices to ensure the almost holomorphic modularity of the relevant parameters. We further introduce a concrete class of models based on hauptmodul functions that provide generators of the corresponding inflationary potentials at level $N>1$. The phenomenology of this class of models provides targets for ground-based CMB experiments in the immediate future. In the framework of our models we also discuss the status of the quantum gravity conjectures that have been formulated in the context of the swampland.

Figures

Equations

$$\mathcal{L}=-\frac{1}{2}{G}_{IJ}{g}^{\mu \nu }{\partial }_{\mu }{\varphi }^I{\partial }_{\nu }{\varphi }^J-V({\varphi }^I).$$

$$Q^I={\mathcal{Q}}^I+\frac{{\dot{\varphi }}^I}{H}\psi ,$$

$$\mathcal{R}=H\delta u-\psi =-\frac{H}{\dot{\sigma }}{\sigma }_I{Q}^I.$$

$$\dot{\sigma }:=\sqrt{G_{IJ}{\dot{\varphi }}^I{\dot{\varphi }}^J},$$

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

$$W_{IJ}={\sigma }_I{V}_{,J}-{\sigma }_J{V}_{,I}$$

$$\dot{\mathcal{R}}=\frac{H}{\dot{H}}\frac{1}{a^2}\Delta \psi +\frac{1}{\dot{\sigma }}{S}^{IJ}{W}_{IJ}$$

$${\alpha }^I=D_t{\sigma }^I=-\frac{P^{IJ}{V}_{,J}}{\dot{\sigma }}$$

$$\frac{1}{\dot{\sigma }}{S}^{IJ}{W}_{IJ}=-2\frac{H}{\dot{\sigma }}{\alpha }_I{Q}^I.$$

$$\mathcal{S}:=\frac{H}{\dot{\sigma }}{\nu }_I{Q}^I=-\frac{1}{2\alpha \dot{\sigma }}{S}^{IJ}{W}_{IJ}.$$

$$\dot{\mathcal{R}}$$

$$D_t{S}^{IJ}$$

$${ϵ}_I=M_{\mathrm{Pl}}\frac{V_{,I}}{V}$$

$${\eta }_{IJ}=M_{\mathrm{Pl}}^2\frac{V_{;IJ}}{V}.$$

$${\mathcal{P}}_{\mathcal{R}\mathcal{R}}=\frac{2{\pi }^2}{k^3}{P}_{\mathcal{R}\mathcal{R}}$$

$$⟨\mathcal{R}(\vec{k})\mathcal{R}({\vec{k}}^{\prime })⟩=(2\pi {)}^3{\delta }^{(3)}(\vec{k}+{\vec{k}}^{\prime })P_{\mathcal{R}\mathcal{R}}(k)$$

$${\mathcal{P}}_{\mathcal{R}\mathcal{R}}={\left(\frac{H}{2\pi }\right)}^2{\left(\frac{H}{\dot{\sigma }}\right)}^2=\frac{1}{12{\pi }^2{M}_{\mathrm{Pl}}^4}\frac{V}{G^{IJ}{ϵ}_I{ϵ}_J}.$$

$$n_{\mathcal{O}{\mathcal{O}}^{\prime }}=1+\frac{d\ln {\mathcal{P}}_{O{\mathcal{O}}^{\prime }}}{d\ln k}$$

$$n_{\mathcal{R}\mathcal{R}}=1-3G^{IJ}{ϵ}_I{ϵ}_J+2\frac{{\eta }_{IJ}{ϵ}^I{ϵ}^J}{G^{KL}{ϵ}_K{ϵ}_L}.$$

$$n_{\mathcal{S}\mathcal{S}}=n_{\mathcal{R}\mathcal{R}}$$

$${\gamma }_{\mathrm{corr}}=\frac{{\mathcal{P}}_{\mathcal{R}\mathcal{S}}}{\sqrt{{\mathcal{P}}_{\mathcal{R}\mathcal{R}}{\mathcal{P}}_{\mathcal{S}\mathcal{S}}}},$$

$${\beta }_{\mathrm{iso}}=\frac{{\mathcal{P}}_{\mathcal{S}\mathcal{S}}}{{\mathcal{P}}_{\mathcal{R}\mathcal{R}}+{\mathcal{P}}_{\mathcal{S}\mathcal{S}}}$$

$${\mathcal{P}}_T=\frac{2}{{\pi }^2}{\left(\frac{H}{M_{\mathrm{Pl}}}\right)}^2.$$

$$r=\frac{8}{M_{\mathrm{Pl}}^2}{\left(\frac{\dot{\sigma }}{H}\right)}^2=8G^{IJ}{ϵ}_I{ϵ}_J.$$

$$n_T=\frac{d\ln {\mathcal{P}}_T}{d\ln k},$$

$$n_T=-G^{IJ}{ϵ}_I{ϵ}_J=-\frac{r}{8},$$

$$N_k={\int }_{t_k}^{t_e}Hdt.$$

$$\gamma \equiv 1(\mathrm{mod}N),$$

$$\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$$

$$V={\Lambda }^4|F(f_i){|}^{2m}$$

$$f_i({\gamma }_i\tau )={\chi }_i(\gamma )(c\tau +d{)}^{w_i}f(\tau ),$$

Full text

1. 1 Introduction

2. 2 Multifield inflation

3. 3 CMB observables in MFI

4. 4 Modular inflation at higher level $N>1$

5. 4.1 Breaking the Möbius group to congruence subgroups

6. 4.2 Derivatives of the inflaton potential

7. 4.3 Geometry of higher level field spaces

8. 4.4 Modularity properties of CMB variables

9. 5 Geometric quantum gravity conjectures and modular inflation at higher level

10. 6 Transfer function dynamics

11. 7 $h_{N}$ Inflation

12. 8 CMB observables for $h_{N}$-inflation

13. 9 Phenomenology of $h_{N}$ inflation

14. 10 Conclusion

1 Introduction

Multifield inflation provides a useful perspective for the experimental results obtained over the past years with CMB probes. While singlefield inflation and low-dimensional fits suffice to account for the data collected by the satellite experiments COBE, WMAP and Planck [1, 2, 3, 4, 5, 6], such analyses do not give an accurate picture of what this data implies for the inflationary theory space. This is of particular importance because multifield theories predict effects that are absent or suppressed in single field models and thereby provide guidance for the phenomenological analysis. More work is necessary to properly constrain with the currently available data even the simplest extensions of inflationary models based on single fields.

The inflationary landscape has remained a somewhat amorphous object in its most general framework because of the lack of an organizing principle. Such a principle has been introduced in refs. [7, 8] for a subclass of theories based on automorphic symmetries. The automorphic structure endows the theory space with a foliation and the inflationary potentials are restricted because their building blocks are automorphic forms [9, 10]. Each of the leafs of the foliation is specified by numerical characteristics, and for each leaf only a finite number of fundamental building blocks exist. Furthermore, each of these functions is determined by a finite amount of information, hence can be identified uniquely by a finite number of computations. A further motivation for automorphic inflation arises from the fact that it provides a systematic framework in which the shift symmetry, often introduced as an ad hoc operation to provide protection of the inflaton potential against corrections, is an element of a bona fide discrete group that arises as the remnant of a weakly broken continuous group. This embedding of the shift symmetry allows to formulate the constraints on the model in an explicit and structured way.

The emergence of discrete symmetries makes the framework of automorphic inflation more compatible with conjectures concerning the nature of low energy theories that admit a UV completion. Inflationary models without symmetries will tend to be less compatible. These conjectures, initiated in the papers [11, 12] and continued in [13, 14, 15, 16, 17], aim at distinguishing potentially consistent theories from those that are not, thereby relegating the latter to the swampland introduced by Vafa. While the main purpose in this paper is not to add to the extensive discussions in the literature of these conjectures, we will comment on the status of some of the conjectures in the context of automorphic and modular inflation as we describe its structure in this paper.

The simplest realization of the above group theoretic formulation, leading to inflationary models for which a symmetry group provides protection against the aforementioned corrections, is provided by modular inflation. This represents the specialization to two fields of the more general theory of automorphic inflation, which is defined in terms of an arbitrary number of inflaton components and in which the discrete symmetry group is a subgroup of a general higher rank reductive continuous symmetry group [7, 8]. The idea to formulate automorphic inflation is analogous to the idea to embed gauge theoretic duality transformations of the coupling parameters into larger symmetry groups. In the context of the full modular group there exists a distinguished generator of all modular invariant functions, the absolute invariant $j$. It is therefore natural to formulate a model of $j$-inflation based on this function. The phenomenological analysis shows that this model is consistent with the data obtained from the CMB satellites [7, 18], and the reheating epoch after $j$-inflation was considered in [19]. Subsequent work emphasized the hyperbolic geometry underlying modular inflation in the context of $\alpha$-attractors [20, 21, 22, 23, 24], and hyperbolic inflation [25, 26, 27, 28], while the discussion of reheating after $j$-inflation was continued in ref. [29].

We will see below that the modular group is not the only discrete subgroup of the Möbius group that contains the shift symmetry. In the present work we introduce modular inflation at higher level congruence subgroups $\Gamma_{N}$ of the full modular group, generalizing the framework of modular inflation introduced in [7, 18] for the largest discrete subgroup of this type in the continuous Möbius group. The picture thus is that the Möbius group $G({\mathbb{R}})={\rm SL}(2,{\mathbb{R}})$ is weakly broken to some smaller subgroup $\Gamma_{N}\subset{\rm SL}(2,{\mathbb{Z}})$ at a scale whose value is determined by the amplitude of the adiabatic power spectrum. The advantage of considering smaller symmetry groups is that they provide weaker constraints on the theory and therefore extend the reach of the modular framework. Such discrete subgroups at higher level also appear in the context of duality in gauge theories [30, 31, 32], as well as in the microscopic description of black hole entropy [33, 34]. In the present paper we consider such groups in the context of inflationary models. In principle one might consider other classes of groups, such as Fuchsian groups for which a rich geometric theory exists that has been considered in discussions concerned with attractor theory in [22, 23, 24]. For such groups however the theory of modular forms and functions has not been developed sufficiently to lend itself to the framework of inflation with modular invariant potentials.

One of the issues that arise in inflationary models based on modular forms and functions is the modularity of the physical observables. Phenomenological variables such as the spectral index and the tensor-to-scalar ratio involve derivatives of the potential and hence derivatives of the modular forms that are the building blocks of modular invariant inflaton potentials. Derivatives of modular forms are not modular however, hence it is not a priori clear what the modularity properties are of the observables constructed from such derivatives. This problem has been addressed in the theory of modular inflation associated to the full modular group in ref. [18]. It was found there that the curved nature of the field space serves to restore modularity, albeit in a generalized sense, involving nonholomorphic modular forms. In the present paper we generalize the modularity analysis of [18] to inflation at higher levels and general potentials. We furthermore make our general framework concrete by introducing a class of inflationary modular invariant potentials given by hauptmodul functions that are based on modular forms relative to certain Hecke congruence subgroups. The groups considered here are distinguished by the fact that the associated space of modular invariant inflation potentials have a single generator, much like in the case of the full modular groups, where the $j$-function that defines $j$-inflation is the generator for modular invariant functions relative to ${\rm SL}(2,{\mathbb{Z}})$ [7, 18]. We further analyze the phenomenology of one of these models in more detail and establish that it leads to sufficient inflation with parameters that are compatible with the satellite constraints found by WMAP and Planck.

This paper is organized as follows. In $\S 2$ we briefly establish our multifield inflation notation and in $\S 3$ we formulate the multifield form of the CMB observables that we will focus on. In $\S$4 we introduce the general structure of higher level modular inflation, describe in particular the general form of the CMB observables computed later in the paper and establish their modularity in a generalized sense in terms of almost holomorphic modular forms. In $\S$5 we comment on the geometric part of the quantum gravity conjectures in the context of modular inflation. In $\S 6$ we briefly describe the transfer function evolution and in $\S 7$ we make our general framework concrete by introducing the class of hauptmodul inflation models, which we denote as $h_{N}$-inflation. In $\S 8,9$ we investigate this class of models in some detail and in $\S 10$ we make some concluding remarks.

2 Multifield inflation

In this section we briefly introduce our notation and recall a closed form for the multifield inflation with curved target spaces, the framework relevant for automorphic inflation in general and modular inflation in particular. In this context the field space carries a Riemannian geometry given by a nontrivial metric $G_{IJ}(\phi^{K})$ with the associated Levi-Civita connection determined by the Christoffel symbols $\Gamma_{IJ}^{K}(\phi^{L})$. The action is given in terms of the Lagrangian

[TABLE]

The perturbations $\delta\phi^{I}$ are encoded in terms of their covariant forms ${\cal Q}^{I}$ introduced in [35] via gauge invariant variables

[TABLE]

and projected to adiabatic and isocurvature directions. The adiabatic projection is chosen here to be the comoving curvature perturbation ${\cal R}$ of Lukash [36] and Bardeen [37] (see also [38]) which is given in the Newtonian gauge by

[TABLE]

Here $\psi$ is the spatial perturbation of the Friedman-Lemaitre metric with the convention $\delta g_{ij}=2a^{2}\psi\delta_{ij}$ and ${\dot{\sigma}}$ is a measure of the inflaton speed

[TABLE]

while $\sigma^{I}={\dot{\phi}}^{I}/{\dot{\sigma}}$ defines the normalized velocity vector. $H={\dot{a}}/a$ is the Hubble-Slipher parameter, first determined with Slipher’s redshift data [39, 40, 41] and Hubble’s distance data [42].

The isocurvature perturbations have been encoded in a number of different ways in the literature. The approach taken in [7, 18] is to use the structures that source the dynamics of the adiabatic perturbation ${\cal R}$ as it is obtained in the general case of multifield inflation with an arbitrary number of fields in the inflaton multiplet. This dynamics has been derived in closed form for an arbitrary number of fields in [18] in terms of the dimensionless contravariant fields

[TABLE]

and the covariant tensor

[TABLE]

as

[TABLE]

For large scales only the second term is relevant, hence this source term motivates the identification of the tensor $S^{IJ}$ as a measure for the isocurvature perturbations. The isocurvature projection $S^{IJ}W_{IJ}/{\dot{\sigma}}$, identified by the dynamics above as the large scale adiabatic source term, is related to the contraction of the variables $Q^{I}$ with respect to the turn-rate $\alpha^{I}$ often used in the multifield literature. Different notions of a turn-rate have been considered and the formal definitions that are in use are not all equivalent [43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53]. Defining $\alpha^{I}=D_{t}\sigma^{I}$ and using the exact relation

[TABLE]

in terms of the projection operator $P^{IJ}=G^{IJ}-\sigma^{I}\sigma^{J}$ we can relate our source term above to the projection $\alpha_{I}Q^{I}$ as

[TABLE]

Since $\alpha^{I}$ is orthogonal to the trajectory unit vectors $\sigma^{I}$ it is natural to rescale $\alpha^{I}$ into a unit vector $\nu^{I}=\alpha^{I}/\alpha$ with $\alpha=\left|D_{t}\sigma^{I}\right|$. This leads to the dimensionless isocurvature projection

[TABLE]

In the case of modular inflation considered below the isocurvature tensor has only one independent component and the relation (10) can be viewed as a rescaling.

In the slow-roll approximation the general multifield dynamics of the variables $({\cal R},S^{IJ})$ for curved target spaces with an arbitrary number of fields and associated adiabatic and isocurvature perturbations $S^{IJ}$ was obtained in ref. [18] in closed form as

[TABLE]

where $\epsilon=-\frac{{\dot{H}}}{H^{2}}$ is expressed in the slow-roll approximation in terms of slow-roll parameters

[TABLE]

as $\epsilon=G^{IJ}\epsilon_{I}\epsilon_{J}/2$ and $\eta_{IJ}$ are further slow-roll parameters, defined here as

[TABLE]

For the Levi-Civita connection this defines a symmetric tensor. The components $R^{I}_{JKL}$ of the curvature tensor are defined in terms of the Levi-Civita connection of the field space metric $G_{IJ}$. Here the antisymmetrization is defined as $\sigma^{[I}W^{J]}=\sigma^{I}W^{J}-\sigma^{J}W^{I}$. This generalizes the flat target twofield dynamics of ref. [44] to an arbitrary number of fields and curved field spaces.

3 CMB observables in MFI

The most important CMB observable for the experiments in the recent past has been the spectral index. In general, multifield inflation with curved target spaces this can be written in various ways. The dimensionless power spectrum ${\cal P}_{{\cal R}{\cal R}}$, defined as

[TABLE]

with

[TABLE]

can be written in terms of the Hubble-Slipher parameter and can be expressed in the slow roll approximation in terms of the potential as

[TABLE]

With the slow-roll parameters $\epsilon_{I}$ and $\eta_{IJ}$ the spectral index defined for perturbations ${\cal O},{\cal O}^{\prime}$ as

[TABLE]

takes for ${\cal O}={\cal R}={\cal O}^{\prime}$ the form

[TABLE]

Associated to the isocurvature perturbations $S^{IJ}$ are the power spectra ${\cal P}_{{\cal R}{\cal S}^{IJ}}$ and ${\cal P}_{{\cal S}^{IJ}{\cal S}^{KL}}$ and their spectral indices $n_{{\cal R}{\cal S}^{IJ}}$ and $n_{S^{IJ}S^{KL}}$, which can be explicitly computed as well. The power spectrum ${\cal P}_{{\cal S}{\cal S}}$ of the isocurvature projection (10) coincides with that of ${\cal P}_{{\cal R}{\cal R}}$ at horizon crossing

[TABLE]

and there is no cross correlation ${\cal P}_{{\cal R}{\cal S}}=0$. Hence at horizon crossing the correlation coefficient $\gamma_{\rm corr}$, defined by

[TABLE]

vanishes, and the isocurvature ratio $\beta_{\rm iso}$, defined as

[TABLE]

evaluates to $\beta_{\rm iso}=1/2$.

The primordial gravitational contribution of the power spectrum has not been observed directly. It is conventionally parametrized in terms of the tensor-to-scalar ratio $r$ defined as $r={\cal P}_{T}/{\cal P}_{{\cal R}{\cal R}}$, where the tensor power spectrum is given by [54]

[TABLE]

Both WMAP and Planck have strengthened the bounds on $r$ considerably over the past decade, with important ramifications for model selection. In multifield inflation the total scalar power spectrum includes contributions of the isocurvature perturbations, but we retain the above definition, which can be written in terms of the slow-roll parameters as

[TABLE]

The tensor spectral index, commonly defined without the shift of $(-1)$ conventionally adopted for the scalar power spectra as

[TABLE]

is not an independent parameter but is given in the slow-roll approximation by

[TABLE]

leading to a red-tilted spectrum. These relations are modified by the transfer functions considered further below.

The number of e-folds is a further key parameter that has to be considered. There are a number of different ways to encode this, with the most common being the number of e-folds $N_{k}$ between the horizon crossing of $k$ and the end of inflation

[TABLE]

In multifield inflation $N_{k}$ has to be evaluated numerically in general, after solving the dynamics. In the context of slow-roll inflation the first Friedman-Lemaitre equation transforms this into an integral that only involves the inflaton potential $V(\phi^{I}(t))$.

In twofield inflation there are then, in principle, at least 12 parameters of relevance for the experimental collaborations to fit, given by the amplitudes and indices of the power spectra $(A_{{\cal O}{\cal O}^{\prime}},n_{{\cal O},{\cal O}^{\prime}})$ for ${\cal O},{\cal O}^{\prime}={\cal R},{\cal S}$, the tensor observables $(A_{T},n_{T})$, and the canonical cosmological parameters given by the density parameters $\omega_{\rm CDM},\omega_{B},\omega_{\Lambda}$ and the optical depth $\tau$. In the slow-roll approximation these parameters reduce at horizon crossing according to the above relations.

Recent work in inflation has been concerned with conjectures initiated some time ago in refs. [11, 12, 13], and developed further in [14, 15, 17], to the effect that theories that can be extended in a UV consistent way should satisfy a number of criteria that can be made semi-quantitative. The most recent among these takes the form of a gradient conjecture, which posits that the relative dimensionless gradient of the potential should be bounded from below by a positive constant [14]. In the original conjecture, made in the context of a single scalar field with a flat metric, this constant is assumed to be of order one, in which case one obtains a conflict with the slow-roll condition of single field inflation. The above discussion shows how more generally the geometry of the multifield target space interferes when relating the gradient of the potential to the observables of the theory, such as the tensor-to-scalar ratio $r$. We will comment further below on the quantum gravity conjectures in the context of modular inflation.

4 Modular inflation at higher level $N>1$

Modular inflation is the simplest framework in which the shift symmetry is an element of a discrete group. The formulation introduced in [7, 18] is concerned with the modular group ${\rm SL}(2,{\mathbb{Z}})$ because it provides the strongest possible constraints in the standard theory of modular forms. A special feature of this framework is that the space of all modular invariant function is particularly simple because it is generated by a single function, the absolute invariant, denoted by $j$, in the sense that all others can be expressed in terms of $j$ by a quotient of two polynomials in $j$. The $j$-function thus is a distinguished element in this space and it is natural to use it to construct a modular invariant potential. It was shown in refs. [7, 18, 19] that the resulting inflationary model is compatible with the CMB observations of WMAP and Planck. Reheating aspects of this model were furthermore discussed in [19, 29]. In the present section we generalize the framework considered in [7, 18] to subgroups at higher levels $N>1$.

4.1 Breaking the Möbius group to congruence subgroups

The full modular group is not the only discrete subgroup of the Möbius group ${\rm SL}(2,{\mathbb{R}})$ that contains the shift symmetry and it is natural to explore whether the modular structure of $j$-inflation can be generalized to subgroups $\Gamma$ of ${\rm SL}(2,{\mathbb{Z}})$. In this more general context the continuous Möbius group is weakly broken to a discrete subgroup $\Gamma\subset{\rm SL}(2,{\mathbb{Z}})$, thereby realizing a special case of the symmetry breaking $G({\mathbb{R}})\longrightarrow G({\mathbb{Z}})$ outlined in [7, 8] for general reductive groups $G$. Smaller groups will provide weaker constraints, thereby enhancing the possibilities for modular invariant potentials. The goal in this paper is to focus on groups that have the same generating property as the modular group in the sense that their constraints are strong enough to identify a single generator of the associated space of modular invariant functions, leading to natural candidates for the inflationary potentials. For ${\rm SL}(2,{\mathbb{Z}})$ the existence of the special element $j(\tau)$ is explained by the fact that the compactified form of the quotient ${\cal H}/{\rm SL}(2,{\mathbb{Z}})$ of the complex upper halfplane ${\cal H}$ is a complex curve of genus zero, i.e. a sphere. While in general discrete subgroups $\Gamma$ of ${\rm SL}(2,{\mathbb{R}})$ lead to higher genus curves, in which case the modular invariant functions have more than one generator, there exist subgroups which contain the shift symmetry and which lead to the genus zero property.

Because we are interested in building blocks of inflationary potentials that are modular forms we focus here on the class of congruence subgroups, defined as those groups that contain a principal congruence subgroup $\Gamma[N]$ given by those elements $\gamma\in{\rm SL}(2,{\mathbb{Z}})$ that are congruent to the unit matrix ${\bf 1}$ mod $N$

[TABLE]

where the level $N$ is a positive integer. Among these congruence subgroups the most important type is given by the Hecke subgroups $\Gamma_{0}(N)$ of level $N$, defined by matrices

[TABLE]

such that $c$ is a multiple of $N$. These groups have been the main focus in the theory of modular forms for more than a century, leading to the most detailed structure for these objects. In the following we will focus on this framework. In the class of Hecke congruence subgroups there are a finite number of levels for which the complex curve associated to the group is of genus zero, leading to generators that we will denote by $h_{N}$. These generators are the natural generalization of the $j$-function and form the basis for our formulation of $h_{N}$-inflation introduced below. For Hecke groups $\Gamma_{0}(N)$ the specific levels that have genus zero property are given by $N=1,...,10$, 12, 13, 16, 18, 25 [55].

In a general modular inflation model we take the potential to be given by

[TABLE]

where $F$ is a dimensionless function of the modular forms $f_{i}$ associated to congruence groups $\Gamma_{i}\subset{\rm SL}(2,{\mathbb{Z}})$ of the full modular group and $m$ is an integer. Thus for $\gamma_{i}={\tiny\left(\begin{matrix}a&b\cr c&d\cr\end{matrix}\right)}\in\Gamma_{i}$ the $f_{i}$ transform as

[TABLE]

where $w_{i}$ is the weight of $f_{i}$ and $\chi_{i}$ is a possibly trivial character. The phenomenological analysis of modular inflation models involves derivatives of the potential and hence derivatives of the modular forms that define the model. Such derivatives in general involve nonmodular functions, which raises the question of the modularity status of inflationary observables. This issue was addressed first in [18] for modular inflation determined by modular forms relative to the full modular group. The fundamental picture in that case can be summarized by noting that the set of generators of the space of all modular forms has to be enlarged in the space of modular forms and their derivatives by including the nonmodular Eisenstein series $E_{2}$. This series transforms under a modular transformation $\gamma\in{\rm SL}(2,{\mathbb{Z}})$ of the full group as

[TABLE]

which makes it reminiscent to the inhomogeneous behavior of the Christoffel symbols. It was shown in [18] however that the CMB observables $n_{{\cal R}{\cal R}}$ and $r$ are modular invariant because the terms induced by the nontrivial geometry of the field space combine with the $E_{2}$-terms into modular invariant terms. These involve the modified Eisenstein series

[TABLE]

which does define a modular form, albeit a nonholomorphic one. In the more general case of modular forms with respect to congruence subgroups $\Gamma_{N}\subset{\rm SL}(2,{\mathbb{Z}})$ the symmetry constraints are weaker and the resulting structure is different. Since we will need to consider both level one and higher level modular forms for our models we briefly recall the pertinent points for the level one case from [18] before discussing the higher level generalization.

4.2 Derivatives of the inflaton potential

For the spectral index of the scalar power spectrum and the gravitational contribution of the total power spectrum, as measured by $r$, it is sufficient to know the first and second derivatives of the potential. Writing the modular function $F$ of the inflaton potential as a quotient

[TABLE]

of two modular forms $f,g$ with characteristics $(w_{f},N^{f},k_{f})$ and $(w_{g},N^{g},k_{g})$ respectively. Here the integer $w_{f}$ denotes the weight of the form, $N^{f}$ is its level, and $k_{f}$ is the lowest order in its Laurent expansion

[TABLE]

and similarly for $g$. If $w_{f}=w_{g}$ the function $F$ is a weight zero function.

For modular forms of weight $w$ and level $N$ the derivative furthermore involves higher level Eisenstein series, which we denote here by $E_{w}^{N}$, defined as

[TABLE]

where $E_{w}(\tau)$ is the standard Eisenstein series of weight $w$ given by

[TABLE]

where $q=e^{2\pi i\tau}$, the denominators $B_{w}$ are the Bernoulli numbers, and

[TABLE]

is the divisor function. For $w>2$ the Eisenstein series are modular forms of weight $w$ with respect to the full modular group, and the series defined by $E_{w}^{N}$ are modular forms with respect to a congruence subgroup of level $N$.

To determine the derivatives of modular forms relative to the Hecke congruence subgroups is somewhat involved and has only recently been completed for the groups of genus zero that are of interest in the present paper [56, 57, 58]. In the interest of simplicity we focus in the following on groups of prime level, even though the analysis can be generalized to the remaining groups at the cost of more complicated formulae. In the case of prime level the derivatives can be written as

[TABLE]

where ${\tilde{f}}$ is a modular form of weight $(w+2)$. The second term also defines a modular form of the same weight, as can be seen from the transformation behavior of $E_{2}^{N}$, which is given by

[TABLE]

The non-modular second term here is cancelled by the corresponding term of $E_{2}$, hence $NE_{2}^{N}-E_{2}$ defines a modular form of weight two for $\Gamma_{0}(N)$. Since this combination of $E_{2}^{N}$ and $E_{2}$ appears throughout our analysis, it is useful to introduce the abbreviation

[TABLE]

As noted above, the Eisenstein series $E_{2}$ is not modular, hence the final term of $f^{\prime}$ is not modular. We will return to this further below in the context of the CMB observables. An analogous expression is obtained for $g^{\prime}$. For modular forms at level $N=1$ this result specializes to

[TABLE]

for which the explicit construction of ${\tilde{f}}$ at level one has been described already in ref. [18].

The modular form of weight two ${\tilde{f}}$ is determined in the present case in terms of the quotient

[TABLE]

as

[TABLE]

Here $\eta(\tau)$ is the Dedekind eta-function, which can be written as an infinite product in terms of the variable $q=e^{2\pi i\tau}$ as

[TABLE]

Given these ingredients the form ${\tilde{f}}$ associated to $f$ is given by the following sum over the fundamental domain ${\cal F}_{p}$ of $\Gamma_{0}(p)$

[TABLE]

where ${\rm ord}_{\lambda}(f)$ is the vanishing order of $f$ at $\lambda$ and $o_{\lambda}^{p}$ is the order of isotropy groups at $\lambda$. More details about the structure of $H_{p}(\lambda,z)$ can be found in ref. [56].

With these ingredients we find that the first derivative is modular, and the quotient $F^{\prime}/F$ that appears in the CMB observables is of the form

[TABLE]

This quotient determines the slow roll parameters $\epsilon_{I}$ explicitly in terms of the modular forms $f,{\tilde{f}}$, the Eisenstein series $E_{2}^{N}$, and $E_{2}$. The fact that this expression shows the quotient $F^{\prime}/F$ to be a modular form of weight two will have implications for the general structure of inflationary observables discussed below.

For the spectral indices of the inflationary power spectra the second order derivatives are necessary. For higher level modular functions $F$ these second derivatives can be obtained by iteration in terms of modular forms ${\tilde{{\tilde{f}}}},{\tilde{\tilde{g}}}$ of weight $(w+4)$ that arise from the derivatives of ${\tilde{f}}$ and ${\tilde{g}}$. Introducing the abbreviations

[TABLE]

and similarly for $C_{p_{g}}$ we find for the second derivative the weight four expression

[TABLE]

This decomposes into a modular part $(F^{\prime\prime}/F)_{\rm mod}$ given by the first four terms and the nonmodular part given by the last term

[TABLE]

This will be enter below in the structure of the scalar spectral indices.

4.3 Geometry of higher level field spaces

As indicated earlier in the general multifield discussion, the observables in inflation depend on the geometry of the target field space, which in modular inflation is given by the Poincaré metric

[TABLE]

with the Kähler potential

[TABLE]

and the nonvanishing Christoffel symbols

[TABLE]

Also needed is the curvature tensor, which in two dimensions has only one independent component, given in modular inflation by

[TABLE]

The curvature scalar $R=G^{IJ}R_{IJ}$ of this curvature tensor is constant

[TABLE]

leading to the Gaussian curvature ${\cal K}=-1/\mu^{2}$. This shows that the energy scale $\mu$ in our formulation of modular inflation is a direct measure of the curvature of the field space. This scale, and hence the nontrivial target space geometry enters in the CMB observables of all modular inflation models.

While the metric of the field spaces is independent of the level $N$, the spaces themselves do depend on the level because the fundamental domain that defines the irreducible region of field space that generates the complete upper halfplane by motions via the discrete groups is given by the quotient

[TABLE]

The specific form of these regions can be obtained more explicitly from the fundamental region ${\cal F}={\cal F}_{1}$ of the full modular group ${\rm SL}(2,{\mathbb{Z}})$ via a finite union of images of ${\cal F}$ under maps constructed from the basic generators of the group.

4.4 Modularity properties of CMB variables

The quotient $F^{\prime}/F$ completely determines the tensor-to-scalar ratio and is part of the spectral index because the parameters $\epsilon_{I}$ are given for potentials of type (29) by

[TABLE]

which leads with eq. (23) to

[TABLE]

hence the tensor-to-scalar ratio is given in terms of the fundamental building blocks by

[TABLE]

The analysis of the subsection §4.2, in particular eq. (45), makes the modular invariance of $r$ in terms of standard holomorphic modular forms explicit.

The spectral index involves both the first derivative parameters $\epsilon_{I}$ and the second derivative slow-roll parameters $\eta_{IJ}$, which are defined in modular inflation via the covariant derivative because of the curved structure of the target field space. It is useful to decompose the parameters $\eta_{IJ}$ into a flat and a curved part as

[TABLE]

where the flat part is given by

[TABLE]

and the metric induced term $\eta_{IJ}^{\Gamma}$ is given by

[TABLE]

With these parameters the multifield spectral index (18) can be computed to be of the form

[TABLE]

where $F^{\prime}/F$ is as in eq. (46) and $F^{\prime\prime}/F^{\prime}$ is obtained from eq. (48) and eq. (46).

The results of the previous subsection in combination with the expression (62) make it possible to analyze the modularity of the spectral index in a concise way even though the detailed structure in terms of the fundamental building blocks at levels $N>1$ is somewhat involved. The above formula shows that the nonmodular term in $F^{\prime\prime}/F$ determined in eq. (49) contributes a nonmodular term to the spectral index that contains the factor

[TABLE]

This nonmodular part of $F^{\prime\prime}/F^{\prime}$ can be combined with the metric-induced final term of (62) to lead to the modified Eisenstein series ${\widehat{E}}_{2}$ defined in eq. (32). It follows from the transformation behavior of ${\rm Im}(\tau)$ given by

[TABLE]

that this modified Eisenstein series is a modular form of weight two. Thus, while ${\widehat{E}}_{2}$ is not holomorphic, it is modular. As a result the final two terms in (62) lead to a modular invariant spectral index at higher level $N$ given by

[TABLE]

We see from this that, as in the case of modular inflation at level one, the spectral index for higher level modular inflation is modular, although it is modular in a weaker sense because it involves an almost holomorphic modular forms. (More background and references concerning quasimodular and almost holomorphic modular forms can be found in ref. [18].) This generalized modularity is again obtained via the contribution of the curved target space that arises through the Christoffel term induced by the covariant derivative that enters the definition of the parameters $\eta_{IJ}$. This term combines with the quasimodular term in the spectral index into the almost modular form ${\widehat{E}}_{2}$. What changes at higher level is the specific form of the spectral index in terms of the fundamental variables given by the modular forms $f,g$ that define the inflaton potential, and the modular forms ${\tilde{f}},\tilde{{\tilde{f}}},{\tilde{g}},{\tilde{\tilde{g}}},E_{2,p},E_{4}$ that appear through the derivatives of the potential.

5 Geometric quantum gravity conjectures and modular inflation at higher level

We have noted above that in the past few years attempts have been made to formulate in concrete ways the issue whether inflation is constrained by quantum gravitational considerations in the sense that as an effective field theory it should be in principle be possible to embed it into a theory that admits a UV completion. Several conjectures have been suggested that formulate semi-quantitative expectations about what might distinguish theories that are UV completable from those that are not, the latter forming the swampland. Most of these constraints are concerned with moduli theory, hence are not directly relevant for inflation. It is however of interest to consider the geometry of the field theory space in a more general context, and in the present discussion we analyze how the geometric structure of modular inflation for arbitrary level $N$ relates to the geometry conjectured to be imposed by quantum gravity on the field space of scalar field theories. Perhaps the earliest conjecture in this direction is the volume conjecture for the field space, introduced in the discussion of the swampland by Vafa [11], and in a different context in [12]. In the general framework this constraint states that the $n$-dimensional volume obtained from the target space metric $G_{IJ}$ should be finite. A second conjecture formulated in the swampland discussions is the diameter conjecture. In the present section we consider these geometric conjectures for theories of modular inflation at higher level $N$.

For modular inflation at level one the area of the fundamental domain is given by $\pi/3$, but for higher $N$ the fundamental domain is larger because the group theoretic constraints are weaker. For modular inflation at arbitrary level $N$ for any discrete subgroup $\Gamma_{N}$ in ${\rm SL}(2,{\mathbb{Z}})$ the fundamental domain takes the form ${\cal F}_{N}={\cal H}/\Gamma_{N}$ and the volume conjecture takes for the complexified inflaton doublet $\tau$ the form

[TABLE]

One way to think about ${\cal F}_{N}$ is in terms of elements $\gamma_{i}$ in ${\rm SL}(2,{\mathbb{Z}})$ such that the combination of the sets $\gamma_{i}\Gamma_{N}$ recovers the modular group. With these elements the fundamental domain of $\Gamma_{N}$ can be constructed as

[TABLE]

where the index $\mu_{N}=[{\rm SL}:\Gamma_{N}]$ denotes the number of sets $\gamma_{i}\Gamma_{N}$ needed to obtain ${\rm SL}(2,{\mathbb{Z}})$. For the Hecke groups $\Gamma_{0}(N)$ this index can be obtained in terms of the prime divisors $p|N$ of the level $N$, leading to the volume of ${\cal F}_{N}$ given by

[TABLE]

This shows that the volume conjecture holds for all modular inflation models at arbitrary level $N$. It also shows that the discrete symmetry group motivated by the shift symmetry is necessary to obtain this constraint: without it the volume of the upper halfplane with the volume element given by the Poincaré metric would be infinite. In multifield inflation the target spaces that have been considered in the literature are infinite already in the special case of two fields. In such cases the most natural way to reduce the target space is by considering quotients of the field space by a discrete symmetry. If the resulting domain is unbounded in some directions then the quotient construction is still not enough if the target space is chosen to be flat, which is the case in most of the multifield literature. Compliance with the volume conjecture is facilitated by both the existence of a discrete group and a nontrivial metric, and inflationary theories without these ingredients are under pressure from the volume conjecture.

A second conjecture that has been formulated in the context of UV completable theories in ref. [13] is the diameter conjecture. In the context of moduli this conjecture claims that consistent theories are distinguished by the existence of paths in the target space that have infinite length in the metric $G_{IJ}$ (see also [59]). The above construction of the fundamental domain ${\cal F}_{N}$ shows that the domain ${\cal F}$ of the full modular group is part of ${\cal F}_{N}$, as expected from the fact that the Hecke subgroups $\Gamma_{0}(N)$ are strict subgroups. Since ${\cal F}$ contains paths of infinite lengths, in particular along the imaginary axis, so do the domains ${\cal F}_{N}$. While the Poincaré metric decreases in the direction of increasing imaginary values, the distance diverges logarithmically. It follows that the diameter conjecture is also satisfied in modular inflation at arbitrary levels $N$. For more general theories with potentials it is natural to consider the inflaton trajectories on the potential surface, in which case the target space metric $G_{IJ}$ is modified [60]. This modified metric still leads to paths of infinite length, hence the appropriate version of the diameter conjecture holds in the present framework.

A final geometric conjecture about the target space of low energy UV consistent theories is less of geometric but of topological nature. It has been posited in the work of Ooguri and Vafa [13] that the field space should be simply connected. The above construction of the fundamental domain in terms of the images of ${\cal F}$ under the elements that generate the left cosets can be used to show that the regions ${\cal F}_{N}$ are in fact simply connected.

We therefore see from this discussion that all three quantum gravity conjectures of geometric type whose violation would place a theory in the swampland are satisfied in all modular inflation theories at arbitrary level $N$. The essential features that bring modular inflation in compliance with these conjectures is the existence of an infinite discrete symmetry group and a curved field space.

The remaining conjectures that have been formulated in [11, 13, 14, 16, 17] are either automatically satisfied in a low-energy approach, or involve unknown parameters that make them less amenable to a quantitive check in an effective field theory context. The spectrum conjecture, introduced already in [13], is concerned with how the mass spectrum of the theory changes as the field multiplet evolution traverses the field space. This conjecture is however formulated in an asymptotic form and it contains an unknown parameter, hence there is no model independent estimate of how the spectrum should change in an inflationary theory. Finally, the gradient conjectures, introduced first in [14] and modified in [16, 17], impose constraints on the derivatives of the field theory potential that involves a further model dependent parameter whose value has been under discussion in the recent literature (see e.g. [61] and references therein). The goal of this conjecture is already achieved if the relevant parameter is strictly positive, if small, in which case compliance of slow-roll inflation with this conjecture is immediate.

6 Transfer function dynamics

In order to evolve the CMB observables given by the spectral indices and the tensor-to-scalar ratios it is useful to integrate the dynamics via transfer functions. Specializing the general adiabatic evolution equation of (11) to twofield inflation gives for conformally flat target spaces with metric factor $G$

[TABLE]

where $\eta_{\sigma s}$ is the curved target space slow-roll parameter contraction

[TABLE]

that results from the first eq. in (11) in terms of the parameters $\eta_{IJ}$ defined above. The isocurvature dynamics specializes to

[TABLE]

where $D_{t}$ is the covariant time derivative introduced above and ${\cal K}=-1/\mu^{2}$ is the Gaussian curvature which in the present case directly relates to the Riemann curvature tensor as

[TABLE]

The contraction $\eta_{ss}$ is obtained from the second equation of (11) as

[TABLE]

while $\eta_{\sigma\sigma}=\eta_{IJ}\sigma^{I}\sigma^{J}$.

Changing notation for the two-field system by writing ${\cal S}=S^{12}$ these dynamical equations have the general form of a coupled system

[TABLE]

with coefficient functions $A$ and $B^{\Gamma}$, that can be integrated to lead to

[TABLE]

where the transfer functions take the form

[TABLE]

In the context of a flat field space this was first considered in ref. [44].

With these transfer functions the evolution of the power spectra ${\cal P}_{{\cal O}{\cal O}^{\prime}}$ is given by

[TABLE]

and the spectral indices evolve as

[TABLE]

Furthermore the tensor-to-scalar ratio is suppressed by the transfer functions as

[TABLE]

We have seen above that the target space metric $G_{IJ}$ enters in $r_{*}$, hence the overall structure of the metric matters. This is relevant for the most recent of the quantum gravity conjectures, the gradient conjecture. In its original formulation this conjecture says that the slow-roll parameter is bounded from below by a constant $c$ whose value is not known. If this constant can be argued to be of order one then this obviously contradicts slow-roll inflation and the observational bounds of the tensor-to-scalar ratio. For the main purpose of the conjecture, to ensure the non-existence of de Sitter vacua [62], it is sufficient though to require $c>0$. In multifield inflation the natural generalization of the formulation in [14] is to postulate $M_{\rm Pl}\sqrt{G^{IJ}V_{,I}V_{,J}}\geq cV$ because this immediately relates to the slow-roll approximation $\epsilon=G^{IJ}\epsilon_{I}\epsilon_{J}/2$ of the standard parameter $\epsilon=-{\dot{H}}/H^{2}$. Even within a specific given target space geometry, such as the class of modular inflation models, the precise values at horizon crossing $G_{IJ}(\phi_{*}^{I})$ affect $\epsilon_{*}$, and hence $r_{*}$, as well as the spectral indices, because the metric can vary considerably for different models. It is in this context that the structure of the evolution of $r(t)$ becomes of interest because it can in principle enhance or suppress the slow-roll parameters and hence change the CMB parameters at horizon crossing. Furthermore, we see that the transfer functions provide another ingredient that can help suppress the physical observable affected by the gradient conjecture, in addition to the effect of the geometry of the target space. In general, multifield inflation with more than one isocurvature perturbation $S^{IJ}$ additional transfer functions will further suppress the tensor-to-scalar ratio.

For modular inflation at higher level the specific form the coefficient functions is somewhat involved. The expressions

[TABLE]

can be expressed in terms of the basic modular function $F$ because the slow-roll parameters are given by [18]

[TABLE]

The specific form of these parameters follows from the results (46) and (48) for the first and second derivatives of the modular invariant function $F$ at higher level $N>1$. These results allow to express the transfer functions in terms of the defining modular forms $f,g$ and the modular forms ${\tilde{f}},{\tilde{g}},{\tilde{{\tilde{f}}}},{\tilde{\tilde{g}}},E_{2,N}$, as well as the Eisenstein series $E_{2}$. Although somewhat complicated for the general case their structure simplifies in special cases, such as the model of $j$-inflation considered in [7, 18, 19], and the models of $h_{N}$ inflation, to which we turn in the next section.

7 $h_{N}$ Inflation

We have discussed earlier that a natural generalization of the $j$-function and the associated model of $j$-inflation is given by the class of generators of spaces of modular invariant functions associated to Hecke congruence groups $\Gamma_{0}(N)$. We define $h_{N}$-inflation, or $h$-inflation for short, in terms of a specific set of modular functions that belong to the class of hauptmodul functions $h_{N}$, which can be written in terms of the eta-function defined above as

[TABLE]

Inflaton potentials can be associated to these hauptmodul functions by considering dimensionless functions of the $h_{N}$, in combination with an energy scale $\Lambda$. A simple set of models is given by powers of the absolute value of $h_{N}$ as

[TABLE]

where $m$ is an arbitrary integer and the dimensionless variable $\tau=\tau^{1}+i\tau^{2}$ of the complex upper halfplane ${\cal H}$ again parametrizes the inflaton multiplet $\phi^{I}$ defined in terms of a second energy scale $\mu$ as $\tau^{I}=\phi^{I}/\mu$. In the present paper we focus on models with $m=2$:

[TABLE]

The integers $N$ are required to be such that the integers $(N-1)$ are divisors of $24$, which leads to a class of eight models which are all associated to genus zero Hecke congruence subgroups, hence the $h_{N}$ are of hauptmodul type and represent generators of the modular invariant functions for the respective groups $\Gamma_{0}(N)$. This defines the potentials $V_{N}$ in terms of modular forms $f,g$ of the same weight, given by $w_{N}=12/(N-1)$, but different levels $N$. The hauptmodul at $N=2$ in particular is given directly in terms of the Ramanujan form $\Delta=\eta^{24}$, which is a cusp form of weight twelve relative to the full modular group

[TABLE]

The form $\Delta$ appears in many different contexts because of its fundamental relation to the harmonic oscillator. For higher levels $N>2$ the underlying forms can be thought of as roots of the form $\Delta$ with rescaled arguments.

Similar to the $j$-function, the hauptmodul functions $h_{N}$ can be expanded as Laurent series whose leading term has order $(-1)$

[TABLE]

which again illustrates that the $h_{N}$ generalize the $j$-function in a natural way. Writing $j(\tau)=E_{4}^{3}/\eta(\tau)^{24}$ immediately shows that $j$ has the same leading order as the $h_{p}$ functions associated to the higher level groups of genus zero.

In the following we focus on the subclass associated to prime $N$ in order to simply the analysis. The primes $p$ such that $(p-1)$ divides 24 are the only primes for which $\Gamma_{0}(p)$ are of genus zero type. This can be seen as follows. The genus of the compact curve $C_{p}$ obtained from the quotient ${\cal H}/\Gamma_{0}(p)$ has for arbitrary $p$ the genus

[TABLE]

where $\nu_{\ell}(p)$ is the number of elements $\gamma$ in $\Gamma_{0}(p)$ of order $\ell$ such that ${\rm tr}(\gamma)<2$. Both $\nu_{\ell}$ are bounded by $\nu_{\ell}\leq 2$, hence it follows that for $p>13$ the genus of these curves is positive. For the primes $p\leq 13$ the computation of the $\nu_{\ell}$ shows that for these primes the genus vanishes.

The generators of the groups $\Gamma_{0}(p)$ include the shift symmetry. This follows from an analysis by Rademacher [63], who showed that these Hecke congruence subgroups have $2[p/12]+3$ generators, where $p>3$ and $[x]$ denotes the integer part of $x$. Thus for the models under consideration here the number of generators is at most five, while for $p=2,3$ there are two generators only. These generators include the translation $T$ of the full modular group, and hence the shift symmetry.

It is apparent from the definitions (82) and (44) that the potentials $V_{N}$ are intricate functions in which the two multiplet components $\phi^{I}$ interact in a highly nonlinear way. The structure of $V_{N}$ for $N=2,3$ and $N=5$ is illustrated in Fig. 1. This shows that the structure of the $V_{N}$ potentials increases in complexity as $N$, leading to an ever more intricate topography.

Fig. 1 *The structure of the $h_{N}$ inflation potentials $V/\Lambda^{4}=|h_{N}|^{2}$ for the modular levels $N=2,3,5$. The plots are not of the same scale. The size of the bounding box is changed as $N$ increases in order to make the structure of the individual potentials more transparent. These graphs illustrate that the topography of the potentials becomes more complex as the level $N$ increases. *

These different topographies of the potentials have an impact on the geometry relevant for the inflationary trajectories and illustrate how different potentials lead to different effects for the same target space metric. This in turn is of relevance for the quantum gravity conjectures mentioned in the context of the volume conjecture already. Recall that all modular inflation models have the same target space geometry given by the Poincaré metric. The different potentials of $h_{N}$-inflation, as well as $j$-inflation [7, 18, 19], however lead to different neighborhoods $U_{N}(\tau_{*}^{I})$ and $U_{j}(\tau_{*}^{I})$ of the domain in which the slow-roll parameters are small. Hence the degree of the deviation from the flat limit of the geometry is different for these models as $N$ varies. This affects the gradient conjecture because this immediately translates into a condition on the slow-roll parameter $\epsilon$ and hence the tensor-to-scalar ratio $r$. It turns out that $h_{N}-$inflation has a stronger metric dependence than $j-$inflation [7, 18].

8 CMB observables for $h_{N}$-inflation

We have outlined in the general discussion above that for the phenomenological analysis of the $h$-inflation models it is necessary to evaluate derivatives of the hauptmodul functions. These can be obtained by either specializing our results in $\S$4 on higher level inflation to the case $F=h_{N}$, or they can be computed directly by expressing the Dedekind eta function in terms of Eisenstein series via the relation of the Ramanujan modular form to the weight four and weight six Eisenstein series as

[TABLE]

and using Ramanujan’s results for the derivatives of $E_{w}$ [64] (a brief summary can also be found in [18]). With these results one obtains

[TABLE]

and

[TABLE]

This determines the slow-roll parameters $\epsilon_{I}$ as

[TABLE]

and leads to the tensor-to-scalar-ratio as

[TABLE]

Alternatively, this expression can be obtained from the general result above for $r$ for the case of prime levels. For arbitrary $N$ the modular invariance of $r$ arises from the fact that $E_{2,N}$ has weight two.

The spectral index involves first order and second order derivatives. For hauptmodul inflation the second order derivatives can be written in terms of the Eisenstein series $E_{w}$ as

[TABLE]

and

[TABLE]

where we introduce the abbreviation

[TABLE]

This leads to the adiabatic spectral index

[TABLE]

where ${\widehat{E}}_{2}$ is the nonholomorphic modular form defined in (32) and $E_{2,N}$ was defined in (40).

In order to determine the number of e-folds it is necessary to integrate the Klein-Gordon equation. In combination with the first Friedman-Lemaitre equation this leads to the coupled system of equations for $h_{N}$ inflation given by

[TABLE]

where we adopt $s=M_{\rm Pl}t$ as the dimensionless time variable.

Integrating this system allows us to evaluate the number of e-folds $N_{k}$ via

[TABLE]

once the energy scale $\Lambda$ has been determined in terms of the amplitude of the power spectrum. The number of e-folds is phenomenologically not precisely constrained and a wide range has been suggested in the literature. In the following we focus on the standard interval, which is given by $N_{k}\in[50,~{}70]$. We call trajectories with $N_{k}$ in this range as having enough inflation.

The slow-roll parameters (81) specialize to

[TABLE]

With these ingredients the evolution of the power spectra, the spectral indices, and the tensor-to-scalar ratio can be made explicit in $h_{N}$-inflation.

9 Phenomenology of $h_{N}$ inflation

For the phenomenological analysis of hauptmodul inflation we focus on the four main parameters that are commonly used to constrain models, the amplitude $A_{{\cal R}{\cal R}}$ and spectral index $n_{{\cal R}{\cal R}}$ of the adiabatic power spectrum, the tensor-to-scalar ratio $r$, and the number of e-folds $N_{k}$. These have to be matched with the two energy scales $(\Lambda,\mu)$ that define the parameters of the potentials in our framework. The expression of the spectral index and the ratio $r$ depend only on one of the two scales, $\mu$, which as noted above can be viewed as a proxy for the scalar curvature of the field space. The overall scale $\Lambda$ of $h_{N}$-inflation is determined by the amplitude ${\cal A}_{{\cal R}{\cal R}}$ of the power spectrum via the relation

[TABLE]

While this amplitude does not enter the spectral indices and the gravitational amplitude at horizon crossing, it does determine the evolution of these parameters, hence the number of e-folds. The precise values obtained for the observable parameters and their uncertainty in general, and for ${\cal A}_{{\cal R}{\cal R}}$ in particular, depend on the dimensionality of the phenomenological fit and on the precise combination of data sets used in the various analyses [5, 6, 65, 66, 67]. Since $\Lambda$ scales as ${\cal A}_{{\cal R}{\cal R}}^{1/4}$ the changes that result from such different choices are slight and do not affect the viability of realization of models located in the central regions of the relevant parameter intervals. Values for $\Lambda$ that are obtained for different trajectories range between $10^{14}{\rm GeV}$ and $10^{15}{\rm GeV}$.

This leaves the scale $\mu$ as a free parameter. Its range is constrained by the CMB data and the number of e-folds $N_{k}$ that are obtained by solving the dynamical equations (97) and integrating (98). As noted already, the value of $N_{k}$ is phenomenologically not well-constrained and a wide range of possible values has been discussed in the literature. In our analysis we focus on the standard range $N_{k}\in[50,~{}70]$. The reported experimental uncertainty of the spectral index is quite small for low-dimensional fits, leading for pure Planck data based on TT, TE, EE, lowE and lensing to a value of $0.9649\pm 0.0042$ in the six-parameter base model. Adding BAO data changes this to $0.9665\pm 0.0038$ [5]. Such low-dimensional fits however do not provide the most relevant constraints for the landscape of inflationary models in either the singlefield or multifield framework. In the absence of a multifield statistical analysis with a sufficiently extensive set of parameters we adopt for concreteness a parameter range centered around the value of the Planck collaboration $n_{{\cal R}{\cal R}}=0.96*$. The same uncertainties afflict the reported values for the tensor-to-scalar ratio $r$. The experimental value of this ratio is quite sensitive to what data set is being used and the pure Planck data mentioned above for the determination of the spectral index leads to a constraint that is higher than the Planck/BICEP2-Keck analysis [68]. The most stringent bounds have been obtained in the final data release of Planck as $r<0.65$ [5] and by the BICEP/Keck collaboration as $r<0.06$ at the pivot scale $k_{*}=0.05{\rm Mpc}^{-1}$. These are weakened if more realistic parameterizations are used. Adopting here the strongest constraint of these collaborations, keeping in mind the unrealistic nature of these fits, we set the upper bound for the tensor ratio $r$ as $r<0.06$.

The experimental constraints for $n_{{\cal R}{\cal R}}$ and $r$, in combination with their general multifield expressions (18) and (23), show that the slow-roll parameters should be small at horizon crossing. The $h_{N}$-inflation expressions (96) and (92) for the spectral index and the tensor ratio $r$ in turn show that the inflaton values $\tau_{*}^{I}$ should be in a region $U(\phi_{s}^{I})$ around a value of the inflaton multiplet for which the modified Eisenstein series $E_{2,N}$ is small. This suggests scanning the neighborhood $U(\tau_{s})$ for $\tau_{s}=(1+i)/2$. Fig. 2 shows a zoom of this region for the potential at level $N=2$, as well as a few trajectories. It illustrates that this point is a saddle point of the potential and that the inflaton can roll down from the ridge in either direction, depending on what precisely the value is of $\tau_{*}$.

Fig. 2. A zoom of the neighborhood $U(\tau_{s})$ of the potential surface of $V/\Lambda^{4}=|h_{2}|^{2}$, where $\tau_{s}$ is the zero of the function $E_{2,2}$. The curves on the surface illustrate a selection of inflationary trajectories that are compatible with the observational CMB constraints obtained in the recent literature [5, 69].

The detailed form of the inflationary path depends on whether the initial value is closer or farther from the open boundary of target space given by the upper halfplane given by the real axis. Depending on whether this point is on the near or the far side of the saddle point the trajectory will make a turn at the beginning of its evolution. This becomes more transparent in Figure 2, which shows trajectories with sufficient inflation and parameters compatible with the CMB constraint of the Planck and BICEP collaborations. Our numerical analysis of the inflaton target space for $h_{2}$ inflation with a step size of $\Delta\tau^{I}=10^{-3}$ produced values for the tensor ratio $r$ values in the range

[TABLE]

These values provide a target for experiments that are currently under constructions, such as the BICEP array and SPT-3G at the South Pole [69, 70], the CLASS and the Simons observatory in the Atacama desert [72, 73], and the Ali-CPT in Tibet [74], among others. The CLASS experiment aims to reach $r=0.01$, while the other experiments aim to reach target sensitivities $\sigma(r)=10^{-3}$ within the next few years, allowing them to nontrivially constrain the parameter space of $h$-inflation. In Figure 3 we place several of the $h_{2}$-models in the $(n_{{\cal R}{\cal R}},r)$ plane as determined by the Planck Collaboration [5]. In this plot the individual trajectories are obtained by fixing the initial value of the dimensionless inflaton and letting the energy scale $\mu$ run within an interval of $\Delta\mu/M_{\rm Pl}=\pm 2$ of the value identified by the phenomenological constraints, as described above.

Fig. 3  An illustration of the tensor-to-scalar ratio $r$ for $h_{2}$-inflation. The lines are $\mu$-trajectories in the plane spanned by the spectral index and $r$ as obtained in the final Planck data release. The detailed description of the various data sets can be found in [5].

In the discussion above of the general structure of modular inflation we have noted that the Poincaré metric is common to all such models, hence the geometric distortion determined by the metric in field space has a universal effect on the trajectories as they evolve. Since the magnitude of the metric coefficients increases as the trajectory approaches the open boundary of the field space, the impact of the nontrivial geometry becomes more pronounced close to this boundary. While this effect is universal, the details of this impact of the geometry will nevertheless be different for different potentials because the specific regions in the target space that give rise to enough inflation depend on the potential. In the case of $h$-inflation it turns out that it is the later part of the trajectory that is more affected by the metric than the earlier evolution. This is illustrated in Fig. 4 via an intensity map.

Fig. 4. An illustration of phenomenologically consistent trajectories for the inflationary model based on $h_{2}$. The intensity colors indicate the strength of the Poincaré metric, which increases as the imaginary component of the complex inflaton $\tau$ decreases.

For $N>2$ a similar analysis can be done. As shown in Fig. 1, the ridge structure of the potential surface becomes more intricate for higher $N$. The zeros of the functions $E_{2,N}$ that enter the slow-roll parameter $\epsilon$, and hence the CMB observables, vary with $N$, hence the neighborhoods $U(\tau_{*})$ are located in different regions of the target space. This also implies that the metric values will be different for these different higher level models, and hence the geometric effects on the observables will vary with $N$.

A more detailed analysis of the constraints implied by the satellite experiments necessitates the adoption of specific scenarios that parametrize the link between the field theoretic isocurvature perturbation and the isocurvature perturbations that are constrained by WMAP [2] and the Planck collaboration [3, 4, 5, 6]. The Planck group in particular has provided constraints for cold dark matter, neutrino density and neutrino velocity isocurvature perturbations, collectively denoted by ${\cal I}_{a}$ in [4, 5, 6]. These can be used to provide bounds on the corresponding fractions $U_{a}$ induced by the twofield isocurvature. A further extension of the work presented here can also include the period of reheating by generalizing the strategy developed in [19] for $j$-inflation to higher levels. This analysis will be left for future work.

10 Conclusion

In this paper we have introduced modular inflation at higher level $N>1$, which are theories with an underlying symmetry group $\Gamma_{N}$ embedded in the modular group ${\rm SL}(2,{\mathbb{Z}})$. For concreteness we have restricted our focus on the particular class given by Hecke congruence subgroups $\Gamma_{N}=\Gamma_{0}(N)$, because for these groups the theory of modular forms is most developed. This generalizes the framework introduced in ref. [18], where the discussion is restricted to level one. One of the key issues encountered at higher level is the modular nature of the observables because the phenomenological analysis introduces derivatives of modular forms that are not themselves modular. We have shown that the modularity structure found in [18] for the full modular group generalizes to higher level, although the modular building blocks are very different when $N>1$. As in the case of level one, the spectral indices as well as the tensor-to-scalar ratio are modular in a generalized sense in that modularity is more important than the holomorphic structure that the theory starts out with. While the holomorphic part of the observables is completely different from the structure at level one, it turns out that again the nonholomorphic modular form ${\widehat{E}}_{2}$ is sufficient to ensure the modularity of the phenomenological variables.

In order to make the framework of modular inflation at higher level $N>1$ concrete we have also introduced the class of $h_{N}$-inflation models based on hauptmodul functions, or principal modul functions. As in the case of $j$-inflation these functions are distinguished by the fact that they are generators of the space of inflation potentials at a given level. In the general discussion of the paper we have made the structure of modular inflation explicit for these theories. We have focused on the special case of prime levels because the analysis simplifies considerably, even though it remains more involved than at level one. It would be of interest to consider the explicit structure of the models introduced here at nonprime levels. For the specific $h_{N}$ model at $N=2$ we have analyzed the CMB observables $n_{{\cal R}{\cal R}},r,N_{k}$ and shown that in particular the tensor ratio $r$ provides target values that are accessible to ground-based experiments that are currently under construction and that are scheduled to reach their design sensitivity within the next few years.

In the course of the development of the general framework of modular inflation at higher level $N$, we have also considered features of these theories that are relevant for issues related to the possible UV completion of these models. In the context of this discussion a number of conjectures have been formulated under the heading of the swampland, introduced by Vafa, that pose restrictions on the topology and geometry of the target space and that also constrain the types of potentials. The status of these constraints remains under vigorous discussion. We have shown here that those conjectures that admit a precise formulation are satisfied in modular inflation while most of the multifield inflation theories considered in the literature are in conflict with the geometric constraints and hence belong to the swampland. Since the same ingredients that make modular inflation compatible with the quantum gravity conjectures operate in general automorphic inflation [7, 8] this more general multifield inflation framework will be in compliance with the constraints as well.

Acknowledgement.

This work was supported in part by a Faculty Research Grant of Indiana University South Bend.

References