# Encyclopædia Inflationaris: Opiparous Edition

Jérôme Martin, Christophe Ringeval, Vincent Vennin

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Jérôme Martin, Christophe Ringeval, Vincent Vennin. Encyclopædia Inflationaris: Opiparous Edition. Physics of the Dark Universe, 2024, 46, pp.101653. 10.1016/j.dark.2024.101653. insu-04851471

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Full length article

Encyclopædia Inflationaris

Opiparous Edition

Jérôme Martin<sup>a</sup>, Christophe Ringeval<sup>b,a,*</sup>, Vincent Vennin<sup>c,a</sup>

<figure><img src="image_3.png" /></figure>

ARTICLE INFO

Dedicated to Jean Le Rond d'Alembert (1717-1783) and Denis Diderot (1713-1784).

Keywords:

Cosmic inflation

Slow-roll

Reheating

Cosmic microwave background

Aspic

ABSTRACT

The current flow of high-accuracy astrophysical data, among which are the Cosmic Microwave Background (CMB) measurements by the Planck satellite, offers an unprecedented opportunity to constrain the inflationary theory. This is however a challenging project given the size of the inflationary landscape which contains hundreds of different scenarios. Given that there is currently no observational evidence for primordial non-Gaussianities, isocurvature perturbations or any other non-minimal extension of the inflationary paradigm, a reasonable approach is to consider the simplest models first, namely the slow-roll single-field models with minimal kinetic terms. This still leaves us with a very populated landscape, the exploration of which requires new and efficient strategies. It has been customary to tackle this problem by means of approximate model-independent methods while a more ambitious alternative is to study the inflationary scenarios one by one. We have developed the publicly available runtime library ASPIC<fnref n="1" /> to implement this last approach. The ASPIC code provides all routines needed to quickly derive reheating-consistent observable predictions within this class of scenarios. ASPIC has been designed as an evolutive code which presently supports 118 different models. In this paper, for each of the ASPIC models, we present and collect new results in a systematic manner, thereby constituting the first *Encyclopædia Inflationaris*.

0. Preface to the new edition

The first edition of *Encyclopædia Inflationaris* [1] compiles the single-field slow-roll models of inflation that were proposed prior to 2013. It contains accurate reheating-consistent slow-roll calculations of the dynamics of the background universe in these models, as well as of cosmological scalar and tensor perturbations. Although it is often seen as a review, it goes beyond the mere gathering of known results, since a fair fraction of these models had been studied only under rough approximations, which do not match the need for high accuracy from current and forthcoming cosmological data. The ambition of the encyclopaedic project is instead to provide an (almost) exact treatment of these models, using only the slow-roll approximation, and incorporating the effects of reheating on the scale correspondence in a consistent way. All these results are incorporated in a public library, ASPIC [2], standing for “Accurate Slow-roll Predictions for Inflationary Cosmology”, which has been used in a number of subsequent works, as for instance in the *Particle Data Group* [3]. Bayesian model comparison using the Cosmic Microwave Background (CMB) data has been presented in Refs. [4,5] while the constraints derived on the reheating epoch have been separately presented in Refs. [6–8].

Ten years later, inflation remains the most favored scenario of the early universe [9] but several developments call for the release of a new edition. First, at the theoretical level, new models have been proposed. Some of them boil down to one of the functional forms of the inflationary

<footnote><sup>a</sup> Institut d'Astrophysique de Paris, UMR 7095-CNRS, Université Pierre et Marie Curie, 98bis boulevard Arago, 75014 Paris, France</footnote>

<footnote><sup>b</sup> Cosmology, Universe and Relativity at Louvain, Institute of Mathematics and Physics, Louvain University, 2 Chemin du Cyclotron, 1348 Louvain-la-Neuve, Belgium</footnote>

<footnote><sup>c</sup> Laboratoire de Physique de l'École Normale Supérieure, ENS, CNRS, Université PSL, Sorbonne Université, Université Paris Cité, 75005 Paris, France</footnote>

<footnote>☆ Le but d'une encyclopédie est de rassembler les connaissances éparses sur la surface de la terre; d'en exposer le système général aux hommes avec qui nous vivons, et de les transmettre aux hommes qui viendront après nous; afin que les travaux des siècles passés n'aient pas été des travaux inutiles pour les siècles qui succéderont; que nos neveux, devenant plus instruits, deviennent en même temps plus vertueux et plus heureux, et que nous ne mourions pas sans avoir bien mérité du genre humain.</footnote>

<footnote>* Corresponding author at: Cosmology, Universe and Relativity at Louvain, Institute of Mathematics and Physics, Louvain University, 2 Chemin du Cyclotron, 1348 Louvain-la-Neuve, Belgium.</footnote>

<footnote>E-mail addresses: jmartin@iap.fr (J. Martin), christophe.ringeval@uclouvain.be (C. Ringeval), vincent.vennin@ens.fr (V. Vennin).</footnote>

<footnote n="1">¹ https://github.com/cosmicinflation/aspic</footnote>

<footnote n="2">² These models may nonetheless come with different values for the parameters describing the potential, i.e. different priors in the framework of a Bayesian analysis.</footnote>
potentials already encoded in ASPIC, in which case they have been added in the relevant sections.² Some other models give rise to new inflationary potentials, and therefore constitute new sections of *Encyclopædia Inflationaris*, as well as new entries in the ASPIC library. There are 24 such new potential functions in the new edition (here ordered alphabetically): Axion Hilltop Inflation (AHI), Cubically Corrected Starobinsky Inflation (CCSI), Double Exponential Inflation (DEI), Dual Inflation (DI), Fiber Inflation (FI), Generalized Double Well Inflation (GDWI), Hyperbolic Inflation (HBI), Hybrid Natural Inflation (HNI), Non-Renormalizable Corrected Loop Inflation (NCLI), N-Formalism Inflation (NFI), Non-Minimal Large Field Inflation (NMLFI), Pure Arctan Inflation (PAI), Radiatively Corrected Inflection Point Inflation (RCIPI), Radiatively Corrected Large Field Inflation (RCLFI), String Axion Inflation I (SAII), String Axion Inflation II (SAIII), Super-conformal Alpha Attractor A Inflation (SAAI), T-Model Inflation (TMI), Super-conformal Alpha Attractor B Inflation (SABI), Super-conformal Alpha Attractor T Inflation (SATI), Symmetry Breaking Kähler Inflation (SBKI), S-Dual Inflation (SDI), Smeared Higgs Inflation (SHI), Mukhanov Inflation (VFMI). The inclusion of these models allows the new edition to provide an up-to-date landscape of all single-field slow-roll inflationary models, bringing the number of models included in the ASPIC library to 118.

Second, at the observational level, the first edition compared the predictions of single-field models with the early release of the Planck 2013 data. Since then, additional data has been collected, and the second edition features second-order slow-roll constraints from the full Planck 2018 + Bicep-Keck data combination. The Bayesian evidence of all models has been re-assessed with the latest data sets, and the results are presented in a separate publication [8].

Third, as the accuracy of the recent data releases has kept improving, several projects are on their way that should deliver even more accurate cosmological data in the years to come. In particular, let us mention ground-based experiments that are currently operating such as BICEP3, Keck array [10,11] and SPT [12,13] in Antarctica, QUIJOTE [14] in the Canary islands, and CLASS [15,16], ACT [17,18] and POLARBEAR/SIMONS [19–22] in the Atacama desert. They will soon be joined by QUBIC [23]. In space, the EUCLID satellite [24,25] is taking data and will provide unprecedented measurements on the matter power spectrum, down to very small scales. The LiteBIRD satellite [26] is planned to be launched in 2028 and should allow us to further constrain the *B*-mode signal in the polarization of the CMB. These prospects of ever increasing precision confirm the relevance of the original *Encyclopædia Inflationaris* and ASPIC projects, namely the need for accurate predictions on a model-to-model basis. For this reason, we have continued to pay special attention to solve the inflationary dynamics exactly, without any other approximations than those contained in the slow-roll framework. This one has indeed been shown to be sufficiently accurate for the Planck CMB data [27,28] and can be extended to arbitrary precision if needed [29]. On the contrary, other commonly-employed approximations are now too imprecise to allow for a fair comparison with the data. As discussed in Ref. [28], this also applies to a number of “model-independent” approaches.

The role of reheating is also carefully taken care of, for two main reasons. First, since the reheating expansion history determines the part of the inflationary potential being probed by cosmological measurements, it can now substantially affect the preference shown by the data for a given model (in technical terms, the Bayesian evidence). This is one of the reasons why the Starobinsky model (SI) and Higgs Inflation (HI) are now treated as distinct models, since they come with different reheating histories. Moreover, even though they are often treated as sharing the same potential, this is only correct at leading order in an expansion with respect to the inverse of the non-minimal coupling of the field. Differences arise at next-to-leading order that we now account for in an exact manner. Second, taking the optimistic view that the data will narrow the number of viable models down to a few (which could be for instance happen as a result of a detection of the gravitational-wave background associated with inflation), the next question will clearly be to further constrain the parameters of the happy chosen models, among which are the reheating parameter. Coming back to the example of Starobinsky and Higgs inflation, once the energy scale of the potential has been set to match the amplitude of the CMB power spectrum, the only parameter of the model that is left describes the kinematics of reheating. Therefore, improved measurements of the primordial power spectra will allow us to directly constrain the reheating parameter, giving access to sectors of the theory describing the coupling between the inflaton and other degrees of freedom that we could not probe before. This again confirms the strategy adopted since the early days of *Encyclopædia Inflationaris* and ASPIC to derive reheating-consistent predictions.

Let us note that in its new edition, the format of *Encyclopædia Inflationaris* has been purposely kept similar to its original version. In particular, the introduction (Section 2) has been essentially left untouched in order to keep track of our original motivations, and of the main considerations that were discussed in the field at that time. We have nevertheless removed a section that was listing the new analytical results derived in the first 46 potentials, given that we think it has already become clear that *Encyclopædia Inflationaris* is more than a review indeed.

Finally, about the benefit of a second edition, let us quote Jean Le Rond d'Alembert (in a letter to Voltaire, June 23, 1766):

> Quant à l'ouvrage, il est maigre, mais il est aisé de lui donner de l'embonpoint dans une seconde édition.

We would like to thank all colleagues who wrote to us since 2013 to point out new models, ask for clarifications, and helped us to improve the content of both *Encyclopædia Inflationaris* and ASPIC. This clearly contributed to making these tools fully updated and best suited for analyzing current and forthcoming data.

Paris & Louvain-la-Neuve, May 2024.

J. Martin, C. Ringeval, V. Vennin.

# 1. List of potentials

The following table shows the acronym of the models contained in the current release of the ASPIC library. For each model, an hyperlink points to the adequate section in *Encyclopædia Inflationaris*, we give the number of free parameters characterizing the potential, the number of sub-models and the functional shape of the potential. The total number of models is 118 (see Table 1).
<table><caption>Table 1</caption><thead><tr><th>Name</th><th>Parameters</th><th>Sub-models</th><th>V(φ)</th></tr></thead><tbody><tr><td>SI</td><td>0</td><td>1</td><td>M<sup>4</sup>(1 - e<sup>-√2/3φ/M<sub>Pl</sub></sup>)<sup>2</sup></td></tr><tr><td>HI</td><td>0</td><td>1</td><td>M<sup>4</sup>(<sup>ħ<sup>2</sup>-ρ<sup>2</sup></sup>/<sup>1+ħ<sup>2</sup></sup>)<sup>2</sup><br>φ/M<sub>Pl</sub> = √6 + 1/ξ ln[√(1 + (1 + 6ξ)ħ<sup>2</sup>) + √(1 + 6ξ)ħ<sup>2</sup>]<br>+ √6 ln[√(1 + (1 + 6ξ)ħ<sup>2</sup>) + √6ξħ<sup>2</sup>]</td></tr><tr><td>RCHI</td><td>1</td><td>1</td><td>M<sup>4</sup>(1 - 2e<sup>-√2/3φ/M<sub>Pl</sub></sup> + <sup>A<sub>1</sub></sup>/<sub>16ξ</sub> <sup>φ</sup>/<sub>√6M<sub>Pl</sub></sup>)</td></tr><tr><td>LFI</td><td>1</td><td>1</td><td>M<sup>4</sup>(<sup>φ</sup>/<sub>M<sub>Pl</sub></sub>)<sup>2</sup></td></tr><tr><td>MLFI</td><td>1</td><td>1</td><td>M<sup>4</sup>(<sup>φ</sup>/<sub>M<sub>Pl</sub></sub>) (1 + α<sup>φ</sup>/<sub>M<sub>Pl</sub></sub>)</td></tr><tr><td>RCMI</td><td>1</td><td>1</td><td>M<sup>4</sup>(<sup>φ</sup>/<sub>M<sub>Pl</sub></sub>)<sup>2</sup> [1 - 2α<sup>φ</sup>/<sub>M<sub>Pl</sub></sub> ln(<sup>φ</sup>/<sub>M<sub>Pl</sub></sub>)]</td></tr><tr><td>RCQI</td><td>1</td><td>1</td><td>M<sup>4</sup>(<sup>φ</sup>/<sub>M<sub>Pl</sub></sub>)<sup>4</sup> [1 - α ln(<sup>φ</sup>/<sub>M<sub>Pl</sub></sub>)]</td></tr><tr><td>NI</td><td>1</td><td>1</td><td>M<sup>4</sup>[1 + cos(<sup>φ</sup>/<sub>φ</sub>)]</td></tr><tr><td>ESI</td><td>1</td><td>1</td><td>M<sup>4</sup>(1 - e<sup>-qφ/M<sub>Pl</sub></sup>)</td></tr><tr><td>PLI</td><td>1</td><td>1</td><td>M<sup>4</sup>e<sup>-αφ/M<sub>Pl</sub></sup></td></tr><tr><td>KMII</td><td>1</td><td>2</td><td>M<sup>4</sup>(1 - α<sup>φ</sup>/<sub>M<sub>Pl</sub></sub> e<sup>-φ/M<sub>Pl</sub></sup>)</td></tr><tr><td>HF1I</td><td>1</td><td>1</td><td>M<sup>4</sup>(1 + A<sub>1</sub><sup>φ</sup>/<sub>M<sub>Pl</sub></sub>)<sup>2</sup> [1 - <sup>2</sup>/<sub>3</sub>(<sup>A<sub>1</sub></sup>/<sub>1+A<sub>1</sub>φ/M<sub>Pl</sub></sub>)<sup>2</sup>]</td></tr><tr><td>CWI</td><td>1</td><td>1</td><td>M<sup>4</sup>[1 + α(<sup>φ</sup>/<sub>Q</sub>)<sup>4</sup> ln(<sup>φ</sup>/<sub>Q</sub>)]</td></tr><tr><td>LI</td><td>1</td><td>2</td><td>M<sup>4</sup>[1 + α ln(<sup>φ</sup>/<sub>M<sub>Pl</sub></sub>)]</td></tr><tr><td>RpI</td><td>1</td><td>3</td><td>M<sup>4</sup>e<sup>-2√2/3φ/M<sub>Pl</sub></sup> [e<sup>√2/3φ/M<sub>Pl</sub></sup> - 1]<sup>2p/(2p-1)</sup></td></tr><tr><td>DWI</td><td>1</td><td>1</td><td>M<sup>4</sup>[(<sup>φ</sup>/<sub>φ<sub>0</sub></sub>)<sup>2</sup> - 1]<sup>2</sup></td></tr><tr><td>MHI</td><td>1</td><td>1</td><td>M<sup>4</sup>[1 - sech(<sup>φ</sup>/<sub>μ</sub>)]</td></tr><tr><td>RGI</td><td>1</td><td>1</td><td>M<sup>4</sup>(<sup>φ</sup>/<sub>M<sub>Pl</sub></sub>)<sup>2</sup><br>α/(φ/M<sub>Pl</sub>)<sup>2</sup></td></tr><tr><td>MSSMI</td><td>1</td><td>1</td><td>M<sup>4</sup>[(<sup>φ</sup>/<sub>φ<sub>0</sub></sub>)<sup>2</sup> - <sup>2</sup>/<sub>3</sub>(<sup>φ</sup>/<sub>φ<sub>0</sub></sub>)<sup>6</sup> + <sup>1</sup>/<sub>5</sub>(<sup>φ</sup>/<sub>φ<sub>0</sub></sub>)<sup>10</sup>]</td></tr><tr><td>RIPI</td><td>1</td><td>1</td><td>M<sup>4</sup>[(<sup>φ</sup>/<sub>φ<sub>0</sub></sub>)<sup>2</sup> - <sup>4</sup>/<sub>3</sub>(<sup>φ</sup>/<sub>φ<sub>0</sub></sub>)<sup>3</sup> + <sup>1</sup>/<sub>2</sub>(<sup>φ</sup>/<sub>φ<sub>0</sub></sub>)<sup>4</sup>]</td></tr><tr><td>AI</td><td>1</td><td>1</td><td>M<sup>4</sup>[1 - <sup>2</sup>/<sub>π</sub> arctan(<sup>φ</sup>/<sub>μ</sub>)]</td></tr><tr><td>CNAI</td><td>1</td><td>1</td><td>M<sup>4</sup>[3 - (3 + α<sup>2</sup>) tanh<sup>2</sup>(<sup>α</sup>/<sub>√2</sub>M<sub>Pl</sub>)]</td></tr><tr><td>CNBI</td><td>1</td><td>1</td><td>M<sup>4</sup>[(3 - α<sup>2</sup>) tanh<sup>2</sup>(<sup>α</sup>/<sub>√2</sub>M<sub>Pl</sub>) - 3]</td></tr><tr><td>OSTI</td><td>1</td><td>1</td><td>-M<sup>4</sup>(<sup>φ</sup>/<sub>φ<sub>0</sub></sub>)<sup>2</sup> ln[(<sup>φ</sup>/<sub>φ<sub>0</sub></sub>)<sup>2</sup>]</td></tr><tr><td>WRI</td><td>1</td><td>1</td><td>M<sup>4</sup> ln(<sup>φ</sup>/<sub>φ<sub>0</sub></sub>)<sup>2</sup></td></tr><tr><td>DI</td><td>1</td><td>1</td><td>M<sup>4</sup>[1 + V<sub>0</sub>(f) - 2 <sup>K-E</sup>/<sub>mK</sub> - <sup>ν<sup>2</sup>θ(v)</sup>/<sub>mKK'</sub>]<br>with ν = 1 - <sup>8√2</sup>/<sub>π<sup>2</sup>f</sub> <sup>K</sup>/<sub>m<sup>1/2</sup>(E<sup>2</sup>-K<sup>2</sup>)</sub><br>and <sup>dφ</sup>/<sub>dm</sub> = -<sup>2√2</sup>/<sub>π</sub> <sup>√KK'</sup>/<sub>m<sup>3/2</sup></sub></td></tr></tbody></table>

(continued on next page)

## 2. Introduction

The theory of inflation [30–33] represents a cornerstone of the standard model of modern cosmology (the “hot Big-Bang model” of Lemaître and Friedmann) [34–37]. By definition, it is a phase of accelerated expansion which is supposed to take place in the very early universe, at very high energy, between Big-Bang Nucleosynthesis (BBN) and 10¹⁵ GeV. Inflation allows us to understand several puzzles that plagued the pre-inflationary standard model (before 1981) and that could not be understood otherwise. Without inflation, the standard model of cosmology would remain incomplete and highly unsatisfactory. The most spectacular achievement of inflation is that, combined with Quantum Mechanics, it provides a convincing mechanism for the origin of the cosmological fluctuations (the seeds of the galaxies and of the Cosmic Microwave Background - CMB - anisotropies) and predicts that their spectrum should be almost scale invariant (i.e. equal power on all spatial scales) [38–46] which is fully consistent with the observations. Let us notice in passing that this part of the scenario is particularly remarkable since it combines General Relativity and Quantum Mechanics [36,37,47–53]. Given all these spectacular successes and given the fact that, despite many efforts, inflation has not been
<table><caption>**Table 1 (continued).**</caption><thead><tr><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td>CCSI</td><td>1</td><td>3</td><td>M<sup>4</sup>(1 - e<sup>-√2/3φ/M<sub>Pl</sub></sup>)<sup>2</sup><br>× [1 + √(1 + 3α(e<sup>√2/3φ/M<sub>Pl</sub></sup> - 1)]<sup>-3</sup><br>× [1 + √(1 + 3α(e<sup>√2/3φ/M<sub>Pl</sub></sup> - 1) + 2α(e<sup>√2/3φ/M<sub>Pl</sub></sup> - 1)]</td></tr><tr><td>SBKI</td><td>1</td><td>1</td><td>M<sup>4</sup>(φ/M<sub>Pl</sub>)<sup>2</sup> exp[α(φ/M<sub>Pl</sub>)<sup>2</sup> + α<sup>2</sup>/6(φ/M<sub>Pl</sub>)<sup>4</sup>]</td></tr><tr><td>AHI</td><td>1</td><td>1</td><td>M<sup>4</sup>[ν<sub>0</sub> - 2 cos(φ/τ) + (π - φ/τ)sin(φ/τ)]</td></tr><tr><td>PAI</td><td>1</td><td>1</td><td>M<sup>4</sup> arctan(φ/μ)</td></tr><tr><td>SAAI</td><td>1</td><td>1</td><td>M<sup>4</sup>(1 - e<sup>-√2/3φ/M<sub>Pl</sub></sup>)<sup>2</sup></td></tr><tr><td>TMI</td><td>1</td><td>1</td><td>M<sup>4</sup>tanh<sup>2α</sup>(φ/M<sub>Pl</sub>√6)</td></tr><tr><td>SFI</td><td>2</td><td>1</td><td>M<sup>4</sup>[1 - (φ/μ)<sup>2</sup>]</td></tr><tr><td>II</td><td>2</td><td>1</td><td>M<sup>4</sup>(φ/φ<sub>0</sub>)<sup>-β</sup> - M<sup>4</sup>β<sup>2</sup>/6(φ/φ<sub>0</sub>)<sup>-β-2</sup></td></tr><tr><td>KMIII</td><td>2</td><td>1</td><td>M<sup>4</sup>(1 - αφ/M<sub>Pl</sub>)e<sup>-βφ/M<sub>Pl</sub></sup></td></tr><tr><td>LMI</td><td>2</td><td>2</td><td>M<sup>4</sup>(φ/M<sub>Pl</sub>)<sup>4(1-γ)</sup> exp[-β(φ/M<sub>Pl</sub>)<sup>γ</sup>]</td></tr><tr><td>TWI</td><td>2</td><td>1</td><td>M<sup>4</sup>[1 - A(φ/φ<sub>0</sub>)<sup>2</sup>e<sup>-φ/φ<sub>0</sub></sup>]</td></tr><tr><td>GMSSMI</td><td>2</td><td>2</td><td>M<sup>4</sup>[(φ/φ<sub>0</sub>)<sup>2</sup> - (2/3)α(φ/φ<sub>0</sub>)<sup>2</sup> + α/5(φ/φ<sub>0</sub>)<sup>10</sup>]</td></tr><tr><td>GRIP1</td><td>2</td><td>2</td><td>M<sup>4</sup>[(φ/φ<sub>0</sub>)<sup>2</sup> - (4/3)α(φ/φ<sub>0</sub>)<sup>3</sup> + α/2(φ/φ<sub>0</sub>)<sup>4</sup>]</td></tr><tr><td>BSUSYBI</td><td>2</td><td>1</td><td>M<sup>4</sup>(e<sup>√6φ/M<sub>Pl</sub></sup> + e<sup>√6φ/φ<sub>0</sub></sup>)</td></tr><tr><td>TI</td><td>2</td><td>3</td><td>M<sup>4</sup>(1 + cos φ/μ + α sin<sup>2</sup> φ/μ)</td></tr><tr><td>BEI</td><td>2</td><td>1</td><td>M<sup>4</sup>exp<sub>1-β</sub>(-βφ/M<sub>Pl</sub>)</td></tr><tr><td>PSNI</td><td>2</td><td>1</td><td>M<sup>4</sup>[1 + α ln(cos φ/τ)]</td></tr><tr><td>NCKI</td><td>2</td><td>2</td><td>M<sup>4</sup>[1 + α ln(φ/φ<sub>0</sub>) + β(φ/φ<sub>0</sub>)<sup>2</sup>]</td></tr><tr><td>CSI</td><td>2</td><td>1</td><td>M<sup>4</sup>(M<sup>2</sup>/[(1-αφ/φ<sub>0</sub>)<sup>2</sup>]</td></tr><tr><td>OI</td><td>2</td><td>1</td><td>M<sup>4</sup>(φ/φ<sub>0</sub>)<sup>4</sup> [(ln φ/φ<sub>0</sub>)<sup>2</sup> - α]</td></tr><tr><td>CNCI</td><td>2</td><td>1</td><td>M<sup>4</sup>[(3 + α<sup>2</sup>)coth<sup>2</sup>(α/√2φ/M<sub>Pl</sub>) - 3]</td></tr><tr><td>SBI</td><td>2</td><td>2</td><td>M<sup>4</sup>{1 + [-α + β ln(φ/φ<sub>0</sub>)](φ/φ<sub>0</sub>)<sup>4</sup>}</td></tr><tr><td>SSBI</td><td>2</td><td>6</td><td>M<sup>4</sup>[1 + α(φ/M<sub>Pl</sub>)<sup>2</sup> + β(φ/M<sub>Pl</sub>)<sup>4</sup>]</td></tr><tr><td>IMI</td><td>2</td><td>1</td><td>M<sup>4</sup>(φ/M<sub>Pl</sub>)<sup>-p</sup></td></tr><tr><td>BI</td><td>2</td><td>2</td><td>M<sup>4</sup>[1 - (φ/μ)<sup>-p</sup>]</td></tr><tr><td>SAII</td><td>2</td><td>2</td><td>M<sup>4</sup>[1 - cos(φ/μ) + αφ/μ sin(φ/μ)]</td></tr><tr><td>VFMI</td><td>2</td><td>1</td><td>M<sup>4</sup>{1 - β[2(1 + (2-α)/2√5βφ/M<sub>Pl</sub>)<sup>3/2-α</sup>]<sup>-1</sup><br>×exp{[3β/(1-α)[(1 + (2-α)/2√5βφ/M<sub>Pl</sub>)φ/φ<sub>0</sub>]<sup>3/2-α</sup> - 1]}<br>M<sup>4</sup>[(1 + 2/3δ)e<sup>-4/√3 M<sub>Pl</sub></sup><br>-4(1 + δ/6)e<sup>-1/√3 M<sub>Pl</sub></sup> + δ/(1+π)e<sup>3(1+α)/√3 M<sub>Pl</sub></sup> + 3 - δ/(1+α)]</td></tr><tr><td>FI</td><td>2</td><td>1</td><td>M<sup>4</sup>sinh<sup>α</sup>(φ/μ)</td></tr><tr><td>HBI</td><td>2</td><td>1</td><td>M<sup>4</sup>{[1 - (φ/φ<sub>0</sub>)<sup>2</sup>]<sup>2</sup> + α(φ/φ<sub>0</sub>)<sup>4</sup>[ln(φ/φ<sub>0</sub>) - 1/4] + α/4}</td></tr><tr><td>SHI</td><td>2</td><td>1</td><td>M<sup>4</sup>(e<sup>βφ/φ<sub>0</sub></sup> - β<sup>2</sup>e<sup>1/βφ/φ<sub>0</sub></sup>)</td></tr><tr><td>DEI</td><td>2</td><td>2</td><td>M<sup>4</sup>(M<sup>2</sup>/cosh<sup>(φ/μ)</sup>)</td></tr><tr><td>SDI</td><td>2</td><td>1</td><td>M<sup>4</sup>[(φ/φ<sub>0</sub>)<sup>2p</sup> - 1]<sup>2</sup></td></tr><tr><td>GDWI</td><td>2</td><td>1</td><td>M<sup>4</sup>[(φ/φ<sub>0</sub>)<sup>2p</sup> - 1]</td></tr></tbody></table>

(continued on next page)

superseded by its various challengers [54–82], this scenario has gradually become a crucial part of modern cosmology. As can be seen in Fig. 1, the number of papers devoted to this topic and published each year is inflating since the advent of inflation.

In order to produce a phase of inflation within General Relativity, the matter content of the universe has to be dominated by a fluid with negative pressure. At very high energy, matter is described by field theory, the prototypical example being a scalar field since it is compatible
<table><caption>**Table 1 (continued).**</caption><tr><td>NMLFI</td><td>2</td><td>4</td><td>M<sup>4</sup> (1+ħ<sup>2</sup>)/(1+ħ<sup>2</sup>)<br>φ/M<sub>pl</sub> = √(6+1/ξ) ln[√(1+(1+6ξ)ħ<sup>2</sup>) + √(1+(1+6ξ)ħ<sup>2</sup>)]<br>+ √6 ln[√(1+ħ<sup>2</sup>)/√(1+(1+6ξ)ħ<sup>2</sup>) + √6ξħ<sup>2</sup>]</td></tr><tr><td>SABI</td><td>2</td><td>1</td><td>M<sup>4</sup> (1 - e<sup>-√(ξ/2n)</sup>)</td></tr><tr><td>SATI</td><td>2</td><td>1</td><td>M<sup>4</sup> tanh<sup>2n</sup>(φ/M<sub>pl</sub>√6ξ)</td></tr><tr><td>RMI</td><td>3</td><td>4</td><td>M<sup>4</sup> [1 - (c/2)(-1/2 + ln φ/φ<sub>0</sub>)φ<sup>2</sup>/M<sub>pl</sub><sup>2</sup>]</td></tr><tr><td>VHI</td><td>3</td><td>1</td><td>M<sup>4</sup> [1 + (φ/μ)<sup>n</sup>]</td></tr><tr><td>DSI</td><td>3</td><td>1</td><td>M<sup>4</sup> [1 + (φ/μ)<sup>-n</sup>]</td></tr><tr><td>GMLFI</td><td>3</td><td>1</td><td>M<sup>4</sup> (φ/M<sub>pl</sub>)<sup>n</sup> [1 + α(φ/M<sub>pl</sub>)<sup>2n</sup>]</td></tr><tr><td>LPI</td><td>3</td><td>3</td><td>M<sup>4</sup> (φ/φ<sub>0</sub>)<sup>n</sup> (ln φ/φ<sub>0</sub>)</td></tr><tr><td>CNDI</td><td>3</td><td>3</td><td>M<sup>4</sup> (1 + β cos[α(φ/φ<sub>0</sub>)])<sup>2</sup></td></tr><tr><td>SAIII</td><td>3</td><td>6</td><td>M<sup>4</sup> {1 - cos(φ/μ) + α[φ/μ sin(φ/μ) + 1/2 β(φ/μ)<sup>2</sup>]}</td></tr><tr><td>RCLFI</td><td>3</td><td>4</td><td>M<sup>4</sup> [(φ/μ)<sup>n</sup> + α(φ/μ)<sup>4</sup> ln(φ/μ)]</td></tr><tr><td>NCLI</td><td>3</td><td>1</td><td>M<sup>4</sup> [1 + α ln(φ/M<sub>pl</sub>) + (φ/φ<sub>0</sub>)<sup>4+2n</sup>]</td></tr><tr><td>HNI</td><td>3</td><td>2</td><td>M<sup>4</sup> [1 + α cos(φ/M<sub>pl</sub>)]</td></tr><tr><td>NFI</td><td>3</td><td>4</td><td>M<sup>4</sup> exp[-α(φ/M<sub>pl</sub>)<sup>2</sup>]</td></tr><tr><td>RCIPI</td><td>3</td><td>2</td><td>M<sup>4</sup> (φ/M<sub>pl</sub>)<sup>n</sup> [1 + α ln(φ/M<sub>pl</sub>) + β ln<sup>2</sup>(φ/M<sub>pl</sub>)]</td></tr></table>

<figure><img src="image_4.png" /><figcaption>**Fig. 1.** Number of articles containing the word “inflation” and its variations (i.e. “inflating”, “inflationary”, etc...) in its title published each year since the advent of inflation. The total number exceeds 9200 papers.</figcaption></figure>

with the symmetries implied by the cosmological principle. Quite remarkably, if the potential of this scalar field is sufficiently flat (in fact, more precisely, its logarithm) so that the field moves slowly, then the corresponding pressure is negative. This is why it is believed that inflation is driven by one (or several) scalar field(s). For obvious reasons, this scalar field was given the name “inflaton”. However, the physical nature of the inflaton and its relation with the standard model of particle physics and its extensions remain elusive. Moreover the shape of its potential is not known except that it must be sufficiently flat. This is not so surprising since, as mentioned above, the inflationary mechanism is supposed to take place at very high energies in a regime where particle physics is not known and has not been tested in accelerators.

Another crucial aspect of the inflationary scenario is how it ends and how it is connected to the subsequent hot Big-Bang phase. It is believed that, after the slow-roll period, the field oscillates at the bottom of its potential, or undergoes tachyonic preheating, but finally decays into radiation. In this way, inflation is smoothly connected to the radiation-dominated epoch [83–92]. Unfortunately, very little is observationally known on this so-called reheating period. Let us stress that adiabatic initial conditions, as favored from the current CMB measurements, naturally stem from such a setup within single field models. Another hard bound is that the reheating energy, $T_{reh}$, must be higher than the nucleosynthesis scale (i.e. a few MeV). The very first constraints on $T_{reh}$ using CMB data (from the WMAP satellite) were derived in Ref. [93], but these were concerning a few models only.

We see that, despite the fact that it has become a cornerstone, the inflationary era is not as observationally known as the other parts of the standard model of Cosmology. However, there is now a flow of increasingly accurate astrophysical data which gives us a unique opportunity to learn more about inflation. In particular, the Planck satellite data [94–96] play a crucial role in this process. The mission complements and improves upon observations made by the NASA WMAP satellite [97,98] and is a major source of information relevant to several cosmological
issues including inflation [99–102]. As shown in Refs. [6–8], these data allow us to gain more than one bit of information on the reheating era when averaged over all the models presented in this paper. But the flow of new data does not only concern the CMB. The Supernovae projects [103–108] continue to measure the distances to the nearby exploding SN1 A stars while the large scale galaxy surveys such as the Sloan Digital Sky Survey (SDSS) [109,110] are providing an unprecedented picture of the structure of the universe. Galaxy surveys allow for measuring the so-called Baryonic Acoustic Oscillations (BAO) [111,112]. They are the red-shifted version of the acoustic oscillations observed in the CMB anisotropies which have been transferred to the galaxy power spectrum [113–116]. The “lever arm” in length scales between CMB and galaxy power spectra increases the sensitivity to the small deviations from scale invariance, and thus should be extremely powerful to constrain inflationary models. For this reason, the data from the Euclid satellite will be another step forward in our understanding of inflation [117–119]. Let us also mention the possibility of direct detection of the primordial gravitational waves for high energy inflationary models on large scales [120–126] and also on small scales [91,127].

The CMB small angular scales of Planck are already complemented by ground-based microwave telescopes such as the Atacama Cosmology Telescope (ACT) [17,128,129] or the South Pole Telescope (SPT) [130–134] while ultra-sensitive polarization dedicated experiments are on their way [135,136]. In a foreseeable future, the last bit of yet unexplored length scales are expected to be unveiled by the 21 cm cosmological telescopes. These ones will be sensitive to the red-shifted 21 cm line absorbed by hydrogen clouds before the formation of galaxies [137–143]. With such data, we will have a complete tomography of the universe history from the time of CMB emission at the surface of last scattering to the distribution of galaxies today.

The main goal of this article is to develop methods that will allow us to constrain the inflationary scenario at a level matching the accuracy of these new data. Since we have now entered the era of massive multi-data analysis, the project aims at a change of scale compared to previous approaches. In particular, one way to deal with this question is to perform systematic and “industrial” studies of this issue. Our ability to see through the inflationary window turns the early universe into a laboratory for ultra-high energy physics, at scales entirely inaccessible to conventional experimentation. In other words, this window offers a unique opportunity to learn about the very early universe and about physics in a regime that cannot be tested otherwise, even in accelerators such as the Large Hadron Collider (LHC) and its possible successors.

## 2.1. Methodology

Let us now discuss how, in practice, the above described goals can be reached. One issue often raised is that, since there are (literally) a few hundreds different scenarios, it is difficult to falsify inflation. This is, however, not a very convincing argument since different models belong to different classes and usually do differ in their observable predictions. They can thus be observationally distinguished. A natural way to proceed is therefore to test inflationary models step by step, starting with the simplest scenarios. This is consistent with the Occam’s razor point of view and the way inference is achieved within Bayesian statistics (see below). With this in mind, we can classify models in three different broad categories: single-field inflation (category I), multiple-field inflation (category II) and models where matter is not described by a scalar field as, for instance, vector inflation [144], chromo-natural inflation [145] and/or gauge-flation [146–148] (category III). Within each category, one could further identify various sub-categories. For example, within category I, the scalar field can possess a minimal kinetic term and a smooth potential (category IA), a minimal kinetic term and a potential with features (category IB), a non-minimal kinetic term with a smooth potential (category IC) or a non-minimal kinetic term and a potential with features (category ID, see for instance Ref. [149]) (a fifth category could be models of warm inflation [150–153]). The same four sub-categories can also be defined within category II [for instance, multiple Dirac Born Infeld (DBI) field inflation [154–156] belongs to category IIC] and so on. As already mentioned, each category leads to different predictions. For instance, all models of category IA predict a negligible level of non-Gaussianities, $f_{\text{NL}}^{\text{loc}} = 5(1 - n_s)/12 \approx 0.017$ [157–166] while, on the contrary, models of categories IB-ID yield non-negligible non-Gaussianities [167–182]; models belonging to IB and to IC, or II, may not predict exactly the same type of non-Gaussianities [183,184], etc... In this context, as already mentioned, a crucial step was the recent release of the Planck data [95,100,185–188]. Thanks to the polarization, lensing and BAO, they are compatible with a negligible (and slow-roll compatible) running $dn_s/d\ln k = 0.0011 \pm 0.0099$ and a negligible running of the running $d^2n_s/d\ln^2 k = 0.009 \pm 0.012$, with a pivot scale chosen at $k_* = 0.05 \text{ Mpc}^{-1}$. These data are also compatible with adiabaticity at 95% CL such that there is no evidence for isocurvature modes, although the analysis is done with one isocurvature mode at a time only. The Planck data do not find evidence for primordial non-Gaussianity, namely Ref. [189] reports $f_{\text{NL}}^{\text{loc}} = -0.9 \pm 5.1$, $f_{\text{NL}}^{\text{eq}} = -26 \pm 47$ and $f_{\text{NL}}^{\text{ortho}} = -38 \pm 24$. Therefore, at this stage, everything seems to be well described by simplest scenarios of inflation and, as consequence, a reasonable method is to start with the IA-models. Following category IA, if the present observational situation evolves in the future, one should then treat categories IB-ID, then category II and so on. In this way, one can falsify inflation step by step, in a Bayesian motivated fashion.

Bayesian inference for inflation requires some cosmological data that are sensitive to it, such as the ones enumerated above. For the purpose of illustration, let us consider the CMB angular power spectrum. Cosmological measurements give us a set of numbers, $C_{\ell}^{\text{meas}}$, that we are able to calculate theoretically within an inflationary model. This means that we know the functions $C_{\ell}^{\text{th}} \equiv C_{\ell}^{\text{th}}(\theta_{\text{stand}}, \theta_{\text{inf}})$, where $\theta_{\text{stand}}$ represents a set of parameters describing post-inflationary physics, i.e. $\theta_{\text{stand}} = (h, \Omega_A, \Omega_{\text{dm}}, \dots)$ and $\theta_{\text{inf}}$ a set of parameters describing inflationary physics. We are interested in constraining the values of those parameters, especially the $\theta_{\text{inf}}$’s. Within a given experiment, one is given a likelihood, or an effective chi-squared $\chi^2(\theta_{\text{stand}}, \theta_{\text{inf}})$, encoding all the underlying uncertainties. In a frequentist approach, the searched values of $\theta_{\text{stand}}$ and $\theta_{\text{inf}}$ would be chosen at the best fit, i.e. those verifying $\partial\chi^2/\partial\theta = 0$. In a Bayesian approach [190], we are interested in determining the posterior distributions of the parameters, using Bayes’s theorem

$$P(\theta_{\text{stand}}, \theta_{\text{inf}} | C_{\ell}^{\text{meas}}) = \frac{1}{N} \mathcal{L}(C_{\ell}^{\text{meas}} | \theta_{\text{stand}}, \theta_{\text{inf}}) \pi(\theta_{\text{stand}}, \theta_{\text{inf}}), \qquad (2.1)$$

where $\mathcal{L}(C_{\ell}^{\text{meas}} | \theta_{\text{stand}}, \theta_{\text{inf}}) = e^{-\chi^2(\theta_{\text{stand}}, \theta_{\text{inf}})/2}$ is the likelihood function, $\pi(\theta_{\text{stand}}, \theta_{\text{inf}})$ the prior distribution, describing our prejudices about the values of the parameters before our information is updated, and $N$ a normalization factor, also called Bayesian evidence. Because we are interested in the inflationary parameters, one has to integrate over the post-inflationary parameters in order to obtain the marginalized probability distribution $P(\theta_{\text{inf}} | C_{\ell}^{\text{meas}}) = \int P(\theta_{\text{stand}}, \theta_{\text{inf}} | C_{\ell}^{\text{meas}}) d\theta_{\text{stand}}$. CMB physics also tells us that the multipole moment $C_{\ell}^{\text{th}}$ can be written as

$$C_{\ell}^{\text{th}}(\theta_{\text{stand}}, \theta_{\text{inf}}) = \int_{0}^{+\infty} \frac{dk}{k} j_{\ell}(kr_{\ell ss}) T(k; \theta_{\text{stand}}) P_{\zeta}(k; \theta_{\text{inf}}), \qquad (2.2)$$
<figure><img src="image_5.png" /><figcaption>Fig. 2. Observational predictions for the LFI models, $V(\phi) \propto \phi^p$, in the plane $(n_s, r)$ (i.e. scalar spectral index and gravity wave contribution) compared to the Planck 2018 + Bicep-Keck data [95,99,100,185,186,209]. Each continuous line and each color represent a different value of $p$. Along each line, each point (i.e. each small “circle”) denotes a different reheating temperature compatible with the constraint $\rho_{\text{end}} > \rho_{\text{reh}} > \rho_{\text{nucl}}$. We see that the details of the reheating stage now matter: along a given line, some reheating temperatures are compatible with the observational constraints while others are not. This means that the CMB observations can now put constraints on $T_{\text{reh}}$. The mean equation of state parameter is defined in Eq. (3.39).</figcaption></figure>

where $j_\ell$ is a spherical Bessel function, $T(k; \theta_{\text{stand}})$ is the transfer function which describes the evolution of cosmological perturbations during the standard Friedmann-Lemaître eras and $P_\zeta$ is the inflationary power spectrum. As a result, the process of constraining inflation from the $C_\ell^{\text{meas}}$ reduces to the calculation of $P_\zeta$. The same lines of reasoning could be generalized to any other cosmological observables sourced during inflation, such as higher order correlation functions.

At this stage, there are, a priori, two possibilities (it is also worth noticing that yet another approach is the reconstruction program [191,192]). Either one uses a model-independent, necessarily approximate, shape for $P_\zeta$ or, on the contrary, one scans the inflationary landscape, model by model, and for each of them, calculates $P_\zeta$ exactly.

The advantage of working with a model-independent technique is obvious. However, it often requires an approximation scheme that may not be available for all models. In practice, an approximate method, the slow-roll approach, is known for the category IA and for the category IC, see the recent papers [29,193–198]. In this case, the set of inflationary parameters $\theta_{\text{inf}}$ becomes the Hubble flow functions: $\theta_{\text{inf}} = \{\epsilon_n\}$ where the $\epsilon_n$ are defined in Eq. (3.3) and the corresponding expression of $P_\zeta(k; \epsilon_n)$ is provided in Eqs. (3.18), (3.20), (3.21) and (3.22). Assuming some priors $\pi(\epsilon_n)$ on the Hubble flow functions, this method yields the posterior distributions $P(\epsilon_n|C_\ell^{\text{meas}})$ for the Hubble flow functions evaluated at the pivot scale. This approach has already been successfully implemented for the WMAP data in Refs. [93,199–202].

The second approach is more ambitious. It consists in treating exactly all the inflationary models that have been proposed so far and in a systematic manner. For each model, the power spectrum is determined exactly by means of a mode by mode numerical integration, for instance using the exact integration routines provided by the FieldInf code [203]. Such an approach can also be used with the higher correlation functions with, for instance, the recent release of the BINGO code calculating the inflationary bispectrum [204].

In this case, the set of parameters $\theta_{\text{inf}}$ differs according to the model considered. For instance, Large Field Inflation (LFI) for which $V(\phi) = M^4 (\phi/M_{\text{pl}})^p$, has $\theta_{\text{inf}} = (M, p)$ while Small Field Inflation (SFI) with $V(\phi) = M^4 [1 - (\phi/\mu)^p]$ has $\theta_{\text{inf}} = (M, p, \mu)$. From FieldInf one can then compute $P_\zeta(k; M, p)$ for LFI and $P_\zeta(k; M, p, \mu)$ for SFI without any other assumptions than linear perturbation theory and General Relativity. Starting from some priors on the model parameters, e.g. in the case of LFI, $\pi(M, p)$, this method allows us to determine the posterior distributions $P(M|C_\ell^{\text{meas}})$ and $P(p|C_\ell^{\text{meas}})$, thereby providing parameter inference about the corresponding inflationary model. This approach, which was successfully implemented for the first time in Refs. [200,205–207], and subsequently used in Ref. [208], has several advantages that we now discuss.

Firstly, the most obvious advantage is that the result is exact. The slow-roll method is an approximation and, for this reason, remains somehow limited. As mentioned before, there are plethora of models, such as single field models with features or multiple field scenarios, for which a numerical integration is mandatory.

A second reason is that a full numerical approach permits a new treatment of reheating. In the previous approaches, the influence of the reheating is only marginally taken into account. Any observable predictions depend on the number of e-folds associated with a reheating era. From the fact that the reheating must proceed after the end of inflation and before the Big-Bang Nucleosynthesis scale, one can put an order of magnitude bound on this number of e-folds [210]. This causes small uncertainties in the inflationary predictions that were not crucial in the past. However, with the accuracy of the present and future data this question now matters. This is illustrated in Fig. 2 which represents the slow-roll predictions of LFI for which $V(\phi) \propto \phi^p$. Each colored segment represents the range of observable predictions for a given value of $p$, each point within a segment corresponding to a given number of e-folds for the reheating or, equivalently, to a given reheating energy $T_{\text{reh}}$. We see that, for relatively small values of $p$, it is necessary to know the number of e-folds the Universe reheated to decide whether the model is compatible with the data or not. Conversely, the data are becoming so accurate that one can start constraining the reheating epoch [7]. Therefore, instead of viewing the reheating parameters as external source of uncertainties, it is more accurate to include them in the numerical approach and consider they are part of the inflationary model. In its simplest description, the reheating epoch can be modeled as a cosmological fluid with a mean equation of state $\bar{w}_{\text{reh}} > -1/3$. Notice that $w_{\text{reh}}$, the instantaneous equation of state parameter, does not need to be constant (see Section 3.2). For a simple quadratic potential, and a parametric reheating, one would have for instance $\bar{w}_{\text{reh}} = 0$. In this way, both $\bar{w}_{\text{reh}}$ and $T_{\text{reh}}$ are added to the inflationary
parameters, e.g. we now have $\theta_{inf} = (M, p, T_{reh}, \bar{w}_{reh})$ for LFI, and FieldInf computes $P_{\zeta}(k; M, p, T_{reh}, \bar{w}_{reh})$. Starting from some priors $\pi(T_{reh}, \bar{w}_{reh})$ one can then obtain the corresponding posterior distributions $P(T_{reh}|C_{\ell}^{meas})$ and $P(\bar{w}_{reh}|C_{\ell}^{meas})$. The feasibility of this method has already been demonstrated in Refs. [93,200,211] where constraints on the reheating temperature for LFI and SFI have been derived for the first time (see also Ref. [212] and later works of Refs. [6,7,213]). In view of the expected accuracy of the future data, the preheating/reheating era should become a compulsory element of inflationary model testing. This issue plays an important role in the proposal put forward in this article. In addition, let us also emphasize that a proper treatment of the reheating and preheating stages is mandatory in multiple field inflation because they can affect the evolution of $P_{\zeta}$ on large scales [214]. Only a numerical approach can presently deal with this problem.

A third advantage of the numerical approach is to address the question of the priors choice in a particularly well-defined way. A crucial aspect of Bayesian statistics is that the result depends on the choice of the priors. Therefore, these ones must be chosen and discussed carefully. In the slow-roll (approximated) approach described before, the priors are chosen on the slow-roll parameters themselves. For instance, a Jeffreys' prior is typically chosen on $\epsilon_1$ (i.e. uniform prior on $\log \epsilon_1$), as appropriate when the order of magnitude of a parameter is not known. However, from a physical point of view, it is better to choose the priors directly on the parameters of the model, e.g. the parameters entering the potential. For instance, several potentials that we will treat are the results of a one-loop calculation, namely a perturbative calculation with the coupling constant playing the role of the small parameter. It is clear that the prior must encode the fact that this parameter is small. With the numerical approach, this is very conveniently done since we directly compute the power spectrum from the potential itself. As another example, let us consider the case of LFI where $\epsilon_1 \approx p/(4\Delta N_* + p/4)$ ($\Delta N_*$ is the number of e-folds between Hubble exit and the end of inflation, see below). Owing to the non-trivial relation between the first slow-roll parameter and $p$, a Jeffreys' prior $\pi(\epsilon_1)$ on $\epsilon_1$ implies a complicated prior $\pi(p)$ on $p$ while a natural choice would be a flat prior. Again, implementing the priors directly on the parameters of the model is a more theoretically justified choice. Conversely, who could dispute that, beside the posterior $P(\epsilon_1|C_{\ell}^{meas})$, it is theoretically interesting to know the posterior distribution of $p$, i.e. $P(p|C_{\ell}^{meas})$. The exact numerical integration is a reliable technique to obtain such distributions.

The numerical approach, however, has also some disadvantages. Firstly, one needs to specify the inflationary scenarios explicitly and, therefore, the constraints obtained are not model-independent. Although this shortcoming can in fact never be avoided (we always need to make some assumptions even in the slow-roll approach) it may be partially overcome by scanning the complete inflationary landscape. Secondly, and more importantly, it is time consuming since the exact integration of the cosmological perturbations and of the corresponding correlation functions is heavy and can take up to a few minutes for complicated models. Finally, one should expect multiple degeneracies for models having a high number of inflationary parameters since the data have a limited sensitivity to the shape of the primordial observables.

Based on the previous considerations, we conclude that it would be very interesting to have an intermediate method that would allow us to get most of the results that can be derived using the exact numerical approach while being less time consuming and immune to the problems induced by high parameter degeneracies. This is what we suggest in the following. Our strategy is to use the slow-roll approximation in order to skip the numerical calculation of the power spectrum, while being combined with a systematic scan of the whole inflationary landscape and reheating properties. As argued before, the Planck data drive us towards testing inflation with the simplest models first and such a method would therefore need to be implemented for the class of scenarios IA only. More precisely, inferring the posterior distributions of the Hubble flow parameters $\epsilon_n$ becomes a first step, from which we take advantage of the fact that the $\epsilon_n$ can be computed in terms of the more fundamental parameters describing the reheating and $V(\phi)$. The degeneracies between the inflationary parameters being confined to the computation of the $\epsilon_n$, the data analysis used to determine the $\epsilon_n$'s posteriors remains immune to this issue. Then, in a second step, we use the knowledge of the functionals $\epsilon_n(\theta_{inf}, T_{reh}, \bar{w}_{reh})$ to fastly derive the posteriors for the inflationary and reheating parameters from the ones of the $\epsilon_n$. High degeneracies may still be present, but in the second step only. As shown in Ref. [27], for each model, the method permits a quick, efficient and accurate extraction of the posterior distributions of the inflationary and reheating parameters.

In our opinion, this third technique should not be viewed as a competitor of the two others mentioned earlier but rather as complementary and the corresponding results should be compared. Let us also notice that, if, in order to scan all the inflationary scenarios, the full exact numerical approach needs to be carried out at some point, this would by no means render the results derived in the present article useless. Indeed, the slow-roll approach is often a very useful guide of which kind of physics one should expect for a given model (initial conditions, range of the parameters, etc ...). In particular it allows us to understand any eventual parameter degeneracies within the primordial observables. In other words, the slow-roll method is an ideal tool to prepare a full numerical study.

At this point, it is worth making the following remark. The method put forward in this article uses an approximate shape for the power spectrum, namely $(k_*$ is the pivot scale)

$$P_{\zeta}(k) \propto a_0(\epsilon_n) + a_1(\epsilon_n) \ln\left(\frac{k}{k_*}\right) + \frac{1}{2}a_2(\epsilon_n) \ln^2\left(\frac{k}{k_*}\right) + \dots, \qquad (2.3)$$

in order to shortcut a numerical integration of $P_{\zeta}$ but is otherwise completely self-consistent. In other words, once the slow-roll approximation is accepted, no additional approximation should be made. This may still require some numerical calculations, however, in order to determine the coefficients $a_i$, or more precisely the explicit expression, at Hubble crossing, of $a_i = a_i[\epsilon_n(\theta_{inf}, T_{reh}, \bar{w}_{reh})]$. This is an important issue given the accuracy of the current data as it is illustrated in Fig. 3 (see also Ref. [200]). In this figure, we have represented the slow-roll predictions of a SFI model, $V(\phi) \propto 1 - (\phi/\mu)^4$. Each colored segment represents the exact slow-roll predictions of a model given the parameter $\mu$ and for different numbers of e-folds during the reheating. These predictions have been computed by solving numerically the slow-roll equations. But, in the same plot, there are also other segments, on the left, and represented in yellow only. They are predictions for different values of $\mu$ but based on widespread approximate slow-roll formulas used in the literature. We see that, given the accuracy of the data, the approximated formulas are no longer accurate enough: the approximate results would predict that models with $\mu/M_{pl} > 1$ are strongly disfavored while the correct slow-roll results show that they are still compatible with the data. Another textbook example is provided by Higgs inflation with radiative corrections (RCHI) and is presented in Fig. 4. This scenario is studied in detail in Section 5.1 and depends on one free parameter, $A_1$. The segments made of colored circles represent the exact predictions for different values of $A_1$ (see the color bar on the side of the plot). The colored “crosses” indicate predictions based on a commonly used approximate equation for the coefficients $a_i = a_i(\epsilon_n)$. We see that this is no longer sufficient as soon as $|A_1|$ becomes large. From these two examples, we conclude that it is safer to use the slow-roll approximation (which is usually extremely good) and nothing else, in particular no extra approximation on top of the slow-roll approximation. The fact that we may still need to use numerical calculations to establish the observational predictions of a model does not make our approach useless. Indeed, the numerics needed to estimate $a_i = a_i[\epsilon_n(\theta_{inf})]$ are, by far, much easier than those needed to exactly compute $P_{\zeta}$. Therefore, the gain in computational time mentioned above is huge and allows for a fast and reliable method to constrain the inflationary landscape.
<figure><img src="image_6.png" /><figcaption>Fig. 3. Exact slow-roll predictions for SFI models, $V(\phi) \propto 1 - (\phi/\mu)^4$, compared to the Planck 2018 + Bicep-Keck data [95,99,100,185,186,209]. Each colored segment represents a different value of $\mu$ and within each segment the color traces the reheating temperature. The segments made with “crosses”, systematically on the left, represent some extra approximations usually made in the literature on top of slow roll, valid for $\mu/M_p \ll 1$, see Eqs. (6.13). We see that both coincide at very small values of $r$ but differ already for $r \gtrsim 10^{-3}$ where the extra approximations become inaccurate. Moreover, these approximations would indicate that this class of models is disfavored while the correct slow-roll predictions show that, on the contrary, they remain compatible with the data.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 4. Predictions of the RCHI model in the plane $(n_s, r)$ together with Planck 2018 + Bicep-Keck data [95,99,100,185,186,209]. These predictions depend on one free parameter, $A_1$, for details see Section 5.1. The segments made of small circles represent the slow-roll predictions (same conventions as in Fig. 3), obtained when the coefficients $a_l = a_l[\epsilon_n(\theta_{inf}, T_{reh}, \tilde{w}_{reh})]$ are numerically evaluated. On the contrary, the segments made with crosses are obtained with some approximated predictions. We see that there is a significant difference at both negatively and positively large values of $A_1$.</figcaption></figure>

## 2.2. The ASPIC library

In order to implement the project described before, we have coded a public runtime library, named ASPIC [2] for “Accurate Slow-roll Predictions for Inflationary Cosmology”, which is meant to contain all the inflationary models that can be treated with the method described in Section 2.1. ASPIC already has 118 different inflationary scenarios, a number that should be compared to the three or four models that are usually considered. The ASPIC library is an open-source evolutive project and, although it already contains all the most popular inflationary scenarios, it aims at including more models. In this way, it will converge towards a situation where all the category IA models published since the advent of inflation are implemented thereby allowing us to exhaustively scan this part of the inflationary landscape. The list of the 118 ASPIC models, as well as their acronym, is presented in Section 1. If future cosmological data force us to move to more complicated scenarios, the ASPIC library will be upgraded accordingly. It can, moreover, already be interfaced with the exact field integration routines provided by FieldInf thereby allowing for a full numerical approach, if needed. This would be especially relevant for all the single field models with modified kinetic terms (category IB) such as DBI models, models with features (category IC) such as the Starobinsky model [215] or the multiple-field inflationary scenarios of category II, see for instance Refs. [216–226]. However, if the data continue to favor simple models, such as those producing negligible non-Gaussianities and isocurvature perturbations, the ASPIC library in its present form already contains the most relevant inflationary scenarios.

The ASPIC library contains all the necessary routines to compare the predictions of any of the 118 different models to high-accuracy data. It is programmed in modern fortran and contains an exhaustive documentation of its interfaces. The source files amount to more than 100 000 lines of code. They are publicly available and distributed under the GNU General Public License [227] at https://github.com/cosmicinflation/aspic
This paper presents the general architecture of the ASPIC project and all the calculations needed to understand and write these codes. For each model, we give the theoretical framework, the calculation of the three first slow-roll potential parameters, a discussion on how inflation ends, the calculation of the associated field value, a discussion of the priors, a calculation of the relevant range of variation of the reheating temperature and an exact integration of the slow-roll trajectory. Then, using our coding within the ASPIC library, we work out some basic reheating-consistent theoretical predictions in the planes $(\epsilon_1, \epsilon_2)$ and $(n_s, r)$. Let us stress again that, beside slow-roll, no other approximation is used in the numerical codes of ASPIC.

## 2.3. New results

Most of the ASPIC models have already been partially studied in the literature but let us emphasize that, for each of them, this paper contains new results, and, sometimes, corrects predictions that were not accurate enough for a proper use of the current data. In other words, it does not aim at being a mere review and, therefore, the presentation of already derived results has been kept to the minimal.

For all the models studied, this is the first time that their observational predictions are worked out when the constraints on the reheating phase are accurately taken into account. As explained in Ref. [93], and briefly reviewed in Section 3, it has become too inaccurate to derive the predictions of a model by simply assuming a fixed range for $\Delta N_*$.

For instance, this could lead to a reheating energy density larger than the energy density at the end of inflation, which is physically impossible (a concrete example for LFI being discussed in Ref. [93]). Therefore, the predictions have been re-worked in a consistent fashion (for the LFI and SFI models, reheating-consistent predictions were already derived in Ref. [93]). This already constitutes a significant result which goes beyond the current state-of-the-art. In the appendix, we present a series of plots which give some basic predictions of the various ASPIC models in the planes ($n_s, r$) and ($\epsilon_1, \epsilon_2$) for different values of the free parameters characterizing each potential and for different reheating energies. Most often, this is the first time that these predictions are worked out for such a wide range of parameters and, moreover, this is also the first time that these predictions are presented in this fashion. In some sense, our paper can be viewed as the first *Encyclopædia Inflationaris*.

## 2.4. Scope

Before concluding this introduction, let us stress that this article and the ASPIC library are the first step to carry out the final goal which consists in assessing how good a model is and in comparing the various inflationary models.

This very problem can be dealt with within Bayesian inference, using model comparison. It allows one to determine, in a statistically well-defined way, what “the best models of inflation” are given some data sets. For this purpose, one has to calculate, for each model, the global likelihood which is obtained by integrating the usual likelihood over all of the model parameter values, weighted by their respective prior probability distribution. The resulting quantity is a number associated with each model which gives the “evidence” that the model explains the data [this is the number $N$ in Eq. (2.1)]. Their respective ratios give the odds that one model explains all data compared to the others. Bayesian methods have the advantage to automatically incorporate the “Occam’s razor”: complicated inflationary models will be assigned large probability only if the complexity is required by the data. On the practical side, these two steps can be implemented by the use of Markov-Chains-Monte-Carlo (MCMC) methods, which are especially well suited with the approach advocated here.

The complete Bayesian data analysis and model comparison of all the models presented here have been achieved, using the ASPIC library, in various other works, and for different data sets, see Refs. [4–8,228]. For this reason, we will not discuss here what the favored models are and we refer, instead, the interested readers to these references. Nonetheless, because it is interesting to qualitatively assess if a model can have predictions compatible with the current cosmological data, we have reported, in all the figures of Appendix, the one- and two-sigma confidence contours in the planes $(\epsilon_1, \epsilon_2)$ and $(n_s, r)$. These confidence regions have been derived from a robust minimal set of cosmological data, Planck 2018 + Bicep-Keck, which concern CMB anisotropies (temperature, *E* and *B*-mode polarization), the derivation of which being detailed in Ref. [8].

This article is organized as follows. In the next section, Section 3, we briefly summarize slow-roll inflation and give the equations needed for the rest of this article. We also discuss the reheating stage and explains how it can be implemented. Then, in Section 4, we study inflationary models which, up to the potential normalization, do not contain any free parameter (concretely, at this stage, Starobinsky and Higgs inflation). In Sections 5–7, we analyze scenarios characterized by one, two and three free parameters, respectively. Finally, in Section 8, we present our conclusions and discuss future works. In Appendix, we give, in the planes $(n_s, r)$ and $(\epsilon_1, \epsilon_2)$, the reheating-consistent slow-roll predictions of all the 118 ASPIC models.

## 3. Basic equations

In this section, we very briefly recall the theoretical foundations of inflation and we present the main tools and equations that will be used in the rest of this paper. We start by reviewing the slow-roll phase, where the cosmological fluctuations are generated and, then, we describe how the end of inflation and the transition to the standard hot Big Bang phase can be modeled.
### 3.1. The slow-roll phase

Let us consider a single-field inflationary model with a minimal kinetic term and a potential $V(\phi)$. The behavior of the system is controlled by the Friedmann-Lemaître and Klein-Gordon equations, namely

$$H^2 = \frac{1}{3M_{\text{Pl}}^2} \left[ \frac{\phi^2}{2} + V(\phi) \right], \qquad (3.1)$$

$$\dot{\phi} + 3H\phi + V_{\phi} = 0, \qquad (3.2)$$

where $H \equiv \dot{a}/a$ denotes the Hubble parameter, $a(t)$ being the Friedmann-Lemaître-Robertson Walker (FLRW) scale factor and $\dot{a}$ its derivative with respect to cosmic time $t$. $M_{\text{Pl}} = 8\pi G$ denotes the reduced Planck mass. A subscript $\phi$ means a derivative with respect to the inflaton field. In order to describe the evolution of the background, it is convenient to introduce the Hubble flow functions $\epsilon_n$ defined by [229,230]

$$\epsilon_{n+1} \equiv \frac{d \ln |\epsilon_n|}{dN}, \quad n \ge 0, \qquad (3.3)$$

where $\epsilon_0 \equiv H_{\text{ini}}/H$ and $N \equiv \ln(a/a_{\text{ini}})$ is the number of e-folds. By definition, inflation is a phase of accelerated expansion, $\dot{a}/a > 0$, or, equivalently, $\epsilon_1 < 1$. As a consequence, the end of inflation is defined by the condition $\epsilon_1 = 1$. On the other hand, the slow-roll conditions (or slow-roll approximation) refer to a situation where all the $\epsilon_n$'s satisfy $\epsilon_n \ll 1$. If this is the case, then the parameters $\epsilon_n$ can also be expressed in terms of the successive derivatives of the potential, namely [46]

$$\epsilon_1 \approx \frac{M_{\text{Pl}}^2}{2} \left( \frac{V_{\phi}}{V} \right)^2, \qquad (3.4)$$

$$\epsilon_2 \approx 2M_{\text{Pl}}^2 \left[ \left( \frac{V_{\phi}}{V} \right)^2 - \frac{V_{\phi\phi}}{V} \right], \qquad (3.5)$$

$$\epsilon_2 \epsilon_3 \approx 2M_{\text{Pl}}^4 \left[ \frac{V_{\phi\phi\phi}V_{\phi}}{V^2} - 3\frac{V_{\phi\phi}}{V}\left(\frac{V_{\phi}}{V}\right)^2 + 2\left(\frac{V_{\phi}}{V}\right)^4 \right]. \qquad (3.6)$$

Therefore, a measurement of the $\epsilon_n$'s also provides information with regards to the shape of the inflationary potential. In terms of the number of e-folds, one can decouple Eqs. (3.1) and (3.2) to get the field evolution

$$\frac{1}{3-\epsilon_1} \frac{d^2\phi}{dN^2} + \frac{d\phi}{dN} = -M_{\text{Pl}}^2 \frac{d\ln V}{d\phi}, \qquad (3.7)$$

showing that the potential driving the field in FLRW spacetime is $\ln[V(\phi)]$. This equation can be further simplified by using the definition of $\epsilon_1$ and $\epsilon_2$ to get ride of the second order derivatives. From

$$\epsilon_1 = \frac{1}{2M_{\text{Pl}}^2} \left( \frac{d\phi}{dN} \right)^2, \qquad (3.8)$$

one gets

$$(1 + \frac{\epsilon_2}{6 - 2\epsilon_1}) \frac{d\phi}{dN} = -M_{\text{Pl}}^2 \frac{d\ln V}{d\phi}. \qquad (3.9)$$

As a result, in the slow-roll approximation, one has

$$\frac{d\phi}{dN} \approx -M_{\text{Pl}}^2 \frac{d\ln V}{d\phi}. \qquad (3.10)$$

This equation can be integrated to give an explicit expression of the classical trajectory. One arrives at

$$N - N_{\text{ini}} = - \frac{1}{M_{\text{Pl}}^2} \int_{\phi_{\text{ini}}}^{\phi} \frac{V(\chi)}{V_{\chi}(\chi)} d\chi. \qquad (3.11)$$

In this article, for each model, we provide the expressions of the first three Hubble flow parameters, a determination of $\phi_{\text{end}}$, the value of the field at which inflation comes to an end (and the corresponding discussion) and an explicit expression of the slow-roll trajectory Eq. (3.11).

Let us now consider the behavior of inflationary cosmological perturbations. The evolution of scalar (density) perturbations can be reduced to the study of a single variable, the so-called Mukhanov-Sasaki variable $v_k$. In Fourier space, its equation of motion can be expressed as [35–37,45]

$$v_k'' + \left[ k^2 - \frac{(a\sqrt{\epsilon_1})''}{a\sqrt{\epsilon_1}} \right] v_k = 0. \qquad (3.12)$$

Here, a prime denotes a derivative with respect to conformal time and the quantity $k$ is the comoving wave number modulus of the Fourier mode under consideration. This equation is the equation of a parametric oscillator, i.e. an oscillator with a time-dependent frequency. The time-dependence of the effective frequency is controlled by the dynamics of the background, more precisely by the scale factor and its derivatives (up to fourth order). The quantity $v_k$ is related to the curvature perturbation $\zeta_k$ through the following expression:

$$\zeta_k = \frac{1}{M_{\text{Pl}} a \sqrt{2\epsilon_1}}. \qquad (3.13)$$

The importance of $\zeta_k$ lies in the fact that it can be viewed as a “tracer” of the fluctuations on super-Hubble scales, i.e. for all $k\eta \ll 1$, where $\eta$ denotes the conformal time. Indeed, in the case of single-field inflation, this quantity becomes constant in this limit. Therefore, it can be used to “propagate” the perturbations from inflation to the subsequent cosmological eras. The statistical properties of the fluctuations can be characterized
by the n-point correlation functions of $\zeta_k$. In particular, the two-point correlation function can be written as an integral over wave numbers (in a logarithmic interval) of the power spectrum $P_\zeta(k)$, which can be expressed as

$$P_\zeta(k) \equiv \frac{k^3}{2\pi^2} |\zeta_k|^2 = \frac{k^3}{4\pi^2 M_{Pl}^2} \left| \frac{\nu_k}{a\sqrt{\epsilon_1}} \right|^2. \qquad (3.14)$$

In order to calculate $P_\zeta(k)$, one needs to integrate Eq. (3.12), which requires the knowledge of the initial conditions for the mode function $v_k$. Since, at the beginning of inflation, all the modes of cosmological interest today were much smaller than the Hubble radius, the initial conditions are chosen to be the Bunch-Davis vacuum which amounts to, up to a phase,

$$\lim_{k\eta \to -\infty} v_k = \frac{1}{\sqrt{2k}} e^{-ik\eta}, \qquad (3.15)$$

where $H = aH$ is the conformal Hubble parameter.

The evolution of tensor perturbations (or primordial gravity waves) can also be reduced to the study of a parametric oscillator. The amplitude of each transverse Fourier mode of the gravity wave, $\mu_k(\eta)$, obeys the following equation

$$\mu_k'' + \left( k^2 - \frac{a''}{a} \right) \mu_k = 0. \qquad (3.16)$$

We notice that the time-dependence of the effective frequency differs from that of the scalar case and now involves the derivative of the scale factor up to second order only. It is then straightforward to determine the resulting power spectrum. From a calculation of the two-point correlation function, one obtains

$$P_h(k) = \frac{2k^3}{\pi^2} \left| \frac{\mu_k}{a} \right|^2. \qquad (3.17)$$

In order to calculate this quantity, the equation of motion Eq. (3.16) needs to be solved. As it is the case for density perturbations, the initial state is chosen to be the Bunch-Davies vacuum.

The power spectra can be computed exactly by means of a mode by mode integration of Eqs. (3.12) and (3.16), which also requires an exact integration of the background, i.e. of Eqs. (3.1) and (3.2). As discussed in the introduction, this can be done with the help of publicly available codes such as FieldInf. We have seen above that the slow-roll approximation can be used to calculate the classical background trajectory. Quite remarkably, the same approximation also permits the derivation of the scalar and tensor power spectra. This involves a double expansion. The power spectra are expanded around a chosen pivot scale $k_*$ such that

$$\frac{P(k)}{P_0} = a_0 + a_1 \ln \left( \frac{k}{k_*} \right) + \frac{a_2}{2} \ln^2 \left( \frac{k}{k_*} \right) + \dots, \qquad (3.18)$$

where

$$P_{\zeta_0} = \frac{H^2}{8\pi^2 \epsilon_1 M_{Pl}^2}, \qquad P_{h_0} = \frac{2H^2}{\pi^2 M_{Pl}^2}, \qquad (3.19)$$

and, then, the coefficients $a_i$ are determined in terms of the Hubble flow functions. For scalar perturbations, one gets [193,194,230–235,235–237]

$$\begin{aligned} a_0^{(S)} = & 1 - 2(C+1)\epsilon_1 - C\epsilon_2 + \left(2C^2 + 2C + \frac{\pi^2}{2} - f\right)\epsilon_1^2 \\ & + \left(C^2 - C + \frac{7\pi^2}{12} - g\right)\epsilon_1\epsilon_2 + \left(\frac{1}{2}C^2 + \frac{\pi^2}{8} - 1\right)\epsilon_2^2 \\ & + \left(-\frac{1}{2}C^2 + \frac{\pi^2}{24}\right)\epsilon_2\epsilon_3, \end{aligned} \qquad (3.20)$$

$$a_1^{(S)} = -2\epsilon_1 - \epsilon_2 + 2(2C + 1)\epsilon_1^2 + (2C - 1)\epsilon_1\epsilon_2 + C\epsilon_2^2 - C\epsilon_2\epsilon_3, \qquad (3.21)$$

$$a_2^{(S)} = 4\epsilon_1^2 + 2\epsilon_1\epsilon_2 + \epsilon_2^2 - \epsilon_2\epsilon_3, \qquad (3.22)$$

where $C \equiv \gamma_E + \ln 2 - 2 \approx -0.7296$, $\gamma_E$ being the Euler constant, $f = 5$ and $g = 7$. For the gravitational waves, the coefficients $a_i$ read

$$\begin{aligned} a_0^{(T)} = & 1 - 2(C+1)\epsilon_1 + \left(2C^2 + 2C + \frac{\pi^2}{2} - f\right)\epsilon_1^2 \\ & + \left(-C^2 - 2C + \frac{\pi^2}{12} - 2\right)\epsilon_1\epsilon_2, \end{aligned} \qquad (3.23)$$

$$a_1^{(T)} = -2\epsilon_1 + 2(2C + 1)\epsilon_1^2 - 2(C + 1)\epsilon_1\epsilon_2, \qquad (3.24)$$

$$a_2^{(T)} = 4\epsilon_1^2 - 2\epsilon_1\epsilon_2. \qquad (3.25)$$

These expressions are actually known at one more order, namely third order in the Hubble flow functions, and can be found in Ref. [29]. The Hubble flow functions are time-dependent quantities such that in the above expression, it is understood that they should be evaluated at the time at which the pivot scale crosses the Hubble radius during inflation, i.e. at a time $\eta_*$ such that $k_* = H(\eta_*)$. Let us notice that setting the pivot at another time affects the previous expression. For instance, setting $\eta_*$ such that $k_*\eta_* = -1$ would set $f = 3$ and $g = 6$. We will see below that this introduces a dependence in the parameters describing the reheating stage.

The properties of the power spectra can also be characterized by the spectral indices and their “running”. They are defined by the coefficients of the Taylor expansions of the power spectra logarithm with respect to $\ln k$, evaluated at the pivot scale $k_*$. This gives

$$n_S - 1 \equiv \left. \frac{d \ln P_\zeta}{d \ln k} \right|_{k_*}, \qquad n_T \equiv \left. \frac{d \ln P_h}{d \ln k} \right|_{k_*}. \qquad (3.26)$$
For the runnings, one similarly has the two following expressions

$$ \alpha_s \equiv \left. \frac{d^2 \ln P_\zeta}{d(\ln k)^2} \right|_{k_*}, \qquad \alpha_T \equiv \left. \frac{d^2 \ln P_h}{d(\ln k)^2} \right|_{k_*}, \qquad (3.27) $$

and, in principle, we could also define the running of the running and so on. The slow-roll approximation allows us to calculate the quantities defined above. For instance, we have at first order in the Hubble flow parameters

$$ n_s = 1 - 2\epsilon_1 - \epsilon_2, \qquad n_T = -2\epsilon_1. \qquad (3.28) $$

Let us also notice that the tensor-to-scalar ratio at leading order can be expressed as

$$ r \equiv \frac{P_h}{P_\zeta} = 16\epsilon_1. \qquad (3.29) $$

In the rest of this article, we give the observational predictions of each inflationary model of the ASPIC library in the planes $(\epsilon_1, \epsilon_2)$ but also $(n_s, r)$.

Each inflationary model must also be CMB normalized, that is to say the amplitude of the power spectra, say at $k = k_*$, is completely fixed by the amplitude of the CMB anisotropies measured today. On the largest length scales, this is given to a good approximation by the CMB quadrupole $Q_{rms-PS}/T \equiv \sqrt{5C_2/(4\pi)} \approx 6 \times 10^{-6}$, where $T \approx 2.725$ K is the CMB blackbody temperature. This is achieved if $P_{\zeta_0} \approx 60 Q_{rms-PS}/T^2$. Using the slow-roll approximation of the Friedmann-Lemaître equation and writing the potential as $V(\phi) = M^4\nu(\phi)$, such that the mass scale M is singled out, one arrives at

$$ \left( \frac{M}{M_{\text{Pl}}} \right)^4 = 1440 \pi^2 \frac{\epsilon_1}{\nu(\phi_*)} \frac{Q_{\text{rms-PS}}^2}{T^2}. \qquad (3.30) $$

This is a model-depend expression (it depends on $\nu$) in which we have rendered explicit the dependence in the pivot time. On a more robust basis, CMB data are strongly constraining the value of $P_* \equiv P_\zeta(k_*)$, from the Planck 2018 + Bicep-Keck data one gets the one-sigma confidence interval

$$ \ln(10^{10} P_*) = 3.05 \pm 0.016, \qquad (3.31) $$

at $k_* = 0.05 \text{ Mpc}^{-1}$. This constraint and the one- and two-sigma contours in the planes $(\epsilon_1, \epsilon_2)$ and $(n_s, r)$ represented in all the figures have been obtained from a slow-roll analysis of the Planck 2018 + Bicep-Keck data. Since the analysis is in all point identical to the one of the WMAP seven years data performed in Ref. [93], we do not repeat it here. The interested reader can find all the details in appendix B of Ref. [93]. Moreover, in order to get a robust inference, we have used the second order expression for the power spectra. Therefore, all the results presented below are marginalized over the second order slow-roll parameters.

Since at leading order in the slow-roll expansion we have $P_* \approx H_*^2/(8\pi^2\epsilon_1 M_{\text{Pl}}^2)$, the Friedmann-Lemaître equation allows us to derive the relation

$$ \left( \frac{M}{M_{\text{Pl}}} \right)^4 = 24\pi^2 \frac{\epsilon_1}{\nu(\phi_*)} P_*, \qquad (3.32) $$

which is, as expected, formally identical to Eq. (3.30) with

$$ \frac{Q_{\text{rms-PS}}^2}{T^2} = \frac{P_*}{60}. \qquad (3.33) $$

It has however the advantage of using $P_*$ which is a well inferred quantity because it is fitted against all the $C_\ell$. In the following we will make no-distinction between the so-called COBE normalization and the CMB normalization, both being identical provided the above equation is used. For each inflationary model, these expressions will completely fix the allowed values for M.

We have shown how to calculate the two point correlation functions in the slow-roll approximation. The next logical step would be to determine the higher correlation functions. However, for the type of models considered here (i.e. category IA models), it is well-known that the corresponding signal is so small that it will stay out of reach for a while [157–161]. Therefore, we now consider the question of how to calculate the values of the $\epsilon_n$ when the pivot scale exits the Hubble radius and how this result depends on the details of the reheating period.

## 3.2. The reheating phase

In the last subsection, we have seen that the power spectrum in Eq. (3.18) can be calculated with the help of the slow-roll approximation and expressed in terms of the Hubble flow parameters evaluated at Hubble radius crossing. Here, we briefly explain how these Hubble flow parameters can be determined. It is easy to calculate $\epsilon_1$, $\epsilon_2$ and $\epsilon_3$ as a function of $\phi$ from Eqs. (3.4), (3.5) and (3.6). Then, from the trajectory in Eq. (3.11), one can calculate $N_{\text{end}}$, the total number of e-folds during inflation and $N_*$, the number of e-folds at the point when the pivot scale crosses the Hubble radius. If we denote by I the following primitive

$$ I(\phi) = \int^{\phi} \frac{V(\psi)}{V_{\psi}(\psi)} d\psi, \qquad (3.34) $$

which is also the slow-roll trajectory in Eq. (3.11), then we have

$$ N_{\text{end}} = -\frac{1}{M_{\text{Pl}}^2} [I(\phi_{\text{end}}) - I(\phi_{\text{ini}})], \qquad N_* = -\frac{1}{M_{\text{Pl}}^2} [I(\phi_*) - I(\phi_{\text{ini}})], \qquad (3.35) $$

where $\phi_*$ is the vacuum expectation value of the field, again evaluated when the pivot scale crosses the Hubble radius. From these two expressions, it follows that

$$ \phi_* = I^{-1} [I(\phi_{\text{end}}) + M_{\text{Pl}}^2 \Delta N_*], \qquad (3.36) $$

where

$$ \Delta N_* \equiv N_{\text{end}} - N_* . \qquad (3.37) $$
Inserting this formula into the expressions of the Hubble flow parameters allows us to find $\epsilon_{n*}$ and, therefore, $r$ and $n_s$.

However, in order to make the above-described calculation concrete, we need to say something about the quantity $\Delta N_*$, As was explained in details in Ref. [93], this requires to take into account the reheating stage. Let $\rho$ and $P$ be the energy density and pressure of the effective fluid dominating the Universe during reheating. Conservation of energy implies that

$$ \rho(N) = \rho_{\text{end}} \exp \left\{ -3 \int_{N_{\text{end}}}^{N} [1 + w_{\text{reh}}(n)] dn \right\}, \qquad (3.38) $$

where $w_{\text{reh}} \equiv P/\rho$ is the “instantaneous” equation of state during reheating. One can also define the mean equation of state parameter, $\bar{w}_{\text{reh}}$, by<fnref n="3" />

$$ \bar{w}_{\text{reh}} \equiv \frac{1}{\Delta N} \int_{N_{\text{end}}}^{N_{\text{reh}}} w_{\text{reh}}(n) dn, \qquad (3.39) $$

where

$$ \Delta N \equiv N_{\text{reh}} - N_{\text{end}}, \qquad (3.40) $$

is the total number of e-folds during reheating, $N_{\text{reh}}$ being the number of e-folds at which reheating is completed and the radiation dominated era begins. Then, one introduces a new parameter

$$ R_{\text{rad}} \equiv \frac{a_{\text{end}}}{a_{\text{reh}}} \left( \frac{\rho_{\text{end}}}{\rho_{\text{reh}}} \right)^{1/4}, \qquad (3.41) $$

where $\rho_{\text{reh}}$ has to be understood as the energy density at the end of the reheating era, i.e. $\rho(N_{\text{reh}})$. This definition shows that $R_{\text{rad}}$ encodes any deviations the reheating may have compared to a pure radiation era. In fact, $R_{\text{rad}}$ completely characterizes the reheating stage and can be expressed in terms of

$$ \ln R_{\text{rad}} \equiv \frac{\Delta N}{4} (-1 + 3 \bar{w}_{\text{reh}}), \qquad (3.42) $$

which renders explicit that if $\bar{w}_{\text{reh}} = 1/3$, i.e. the effective fluid during reheating is equivalent to radiation, then reheating cannot be distinguished from the subsequent radiation dominated era. In this case, one simply has $R_{\text{rad}} = 1$. Let us notice that it is also possible to express (or define) $\ln R_{\text{rad}}$ as

$$ \ln R_{\text{rad}} = \frac{1 - 3 \bar{w}_{\text{reh}}}{12(1 + \bar{w}_{\text{reh}})} \ln \left( \frac{\rho_{\text{reh}}}{\rho_{\text{end}}} \right). \qquad (3.43) $$

Using entropy conservation till the beginning of the radiation era, the redshift at which inflation ended can be expressed in terms of $R_{\text{rad}}$ as

$$ 1 + z_{\text{end}} = \frac{1}{R_{\text{rad}}} \left( \frac{\rho_{\text{end}}}{\bar{\rho}_{\gamma}} \right)^{1/4}, \qquad \bar{\rho}_{\gamma} \equiv Q_{\text{reh}} \rho_{\gamma}. \qquad (3.44) $$

The quantity $\rho_{\gamma} = 3H_0^2 M_{\text{pl}}^2 \Omega_{\gamma}$ is the total energy density of radiation today ($\Omega_{\gamma} \simeq 2.471 \times 10^{-5} h^{-2}$) while $Q_{\text{reh}} \equiv q_0^{4/3} g_{\text{reh}}/(q_{\text{reh}}^{4/3} g_0)$ is the measure of the change of relativistic degrees of freedom between the reheating epoch and today. In this expression $q$ and $g$ respectively denotes the number of entropy and energetic relativistic degrees of freedom. In view of the current CMB data, the precise value for $Q_{\text{reh}}$ is unimportant as this factor has only a minimal effect. At most it can shift the values of $\ln R_{\text{rad}}$ by a $\mathcal{O}(1)$ number.

Then, straightforward considerations [93,238] show that the quantities $\Delta N_*$ and $R_{\text{rad}}$ are related by

$$ \Delta N_* = \ln R_{\text{rad}} - N_0 - \frac{1}{4} \ln \left[ \frac{9}{\epsilon_{1*}(3 - \epsilon_{1\text{end}})} \frac{V_{\text{end}}}{V_*} \right] + \frac{1}{4} \ln(8\pi^2 P_*), \qquad (3.45) $$

where we have defined<fnref n="4" />

$$ N_0 \equiv \ln \left( \frac{k_* / a_0}{\bar{\rho}_\gamma^{1/4}} \right), \qquad (3.47) $$

which roughly measures the number of e-folds of deceleration of the Friedmann-Lemaître model. From Eq. (3.43), we see that the quantity $\ln R_{\text{rad}}$ is not arbitrary since $-1/3 < \bar{w}_{\text{reh}} < 1$ and $\rho_{\text{nuc}} < \rho_{\text{reh}} < \rho_{\text{end}}$. Notice that the range allowed for $\bar{w}_{\text{reh}}$ might be extended to smaller values if one allows a phase of acceleration to take place at lower energy than $\rho_{\text{end}}$, such as in thermal or multistage inflation [239,240]. The quantity $\Delta N_*$ is also constrained to vary in a given range, i.e. $\Delta N_* \in [\Delta N_{\text{nuc}}, \Delta N_{\text{end}}]$. Moreover, this range is model-dependent since $\rho_{\text{end}}$ or $V_{\text{end}}/V_*$ differ for different inflationary scenarios. In fact, for each allowed value of $\ln R_{\text{rad}}$, Eq. (3.45) must be viewed as an algebraic equation allowing us to determine the corresponding $\phi_*$. Explicitly, using Eq. (3.35), this equation reads

$$ \frac{1}{M_{\text{Pl}}^2} [I(\phi_*) - I(\phi_{\text{end}})] = \ln R_{\text{rad}} - N_0 - \frac{1}{4} \ln \left\{ \frac{9}{\epsilon_1(\phi_*)[3 - \epsilon_1(\phi_{\text{end}})]} \frac{V(\phi_{\text{end}})}{V(\phi_*)} \right\} + \frac{1}{4} \ln(8\pi^2 P_*). \qquad (3.48) $$

In general, this equation cannot be solved explicitly (except for LFI models, see Ref. [93]) and we have to rely on numerical calculations. Solving for each allowed value of $\ln R_{\text{rad}}$, one can determine the range of variation of $\phi_* \in [\phi_{\text{nuc}}, \phi_{\text{end}}]$ and, therefore, find the corresponding dispersion

$$ N_0 \rightarrow N_0 + \frac{1}{3} \ln \left[ 1 - \frac{\Omega_A \Omega_\gamma^3}{\Omega_{\text{dm}}^4} \left( \frac{g_{\text{eq}}}{g_0} \right)^3 \left( \frac{q_0}{q_{\text{eq}}} \right)^4 \right]. \qquad (3.46) $$

which is clearly a very tiny modification (the subscript “eq” denotes quantities at the equivalence time between radiation and matter).

<footnote n="3">³ In the figures, $\bar{w}_{\text{reh}}$ has been denoted by $w$ for simplicity.</footnote>

<footnote n="4">⁴ One may also wonder about the influence of the cosmological constant on this result. In fact, one can show that it leads to a negligible correction. Indeed, it simply amounts to redefining $N_0$ by</footnote>
in $r$ and $n_s$. In this paper, this task is carried out for all the models of the ASPIC library. Let us notice that it is compulsory to do so otherwise, assuming blindly say $\Delta N_* \in [40, 60]$, would lead to inconsistent reheating energy densities, either larger than $\rho_{end}$ or smaller than $\rho_{nuc}$. Clearly, this method also allows us to put model-dependent constraints on the reheating temperature. Indeed, for some values of $\rho_{rech}$, the corresponding $\epsilon_n(\phi_*)$ will turn out to be outside the $1\sigma$ or $2\sigma$ contours (depending on the criterion one wishes to adopt) thus signaling some tension with the data, see the discussion in the Introduction and Fig. 2.

Let us emphasize that the parametrization presented in this section is independent on the microphysics of reheating and we do not need to specify explicitly the couplings of the inflaton field with the rest of the world. In particular, preheating effects on the background evolution are already taken into account with the present framework. Furthermore, at the perturbed level, they cannot influence the shape of the large scale power spectrum for the class of models considered here [87].

Before closing this section, let us remind that, for each inflationary model, ASPIC gives the expression of the first three Hubble flow parameters, a discussion of the mechanism that ends inflation and the value of $\phi_{end}$, the classical trajectory $I(\phi)$, the CMB normalization $M/M_{Pl}$ and a determination of the exact range [$\phi_{nuc}^*$, $\phi_{end}^*$]. Then all these information are compared to CMB data in the planes $(\epsilon_1, \epsilon_2)$ and $(n_s, r)$. This provides a powerful tool to systematically derive the predictions for the ASPIC models and, therefore, to scan the inflationary landscape. In the next section, we start the systematic exploration of the category IA models that have been studied in the literature since the advent of inflation.

# 4. Zero parameter models

## 4.1. Starobinsky Inflation (SI)

### 4.1.1. Original theoretical justifications

One of the very first models of inflation was proposed by Alexei Starobinsky in 1980 in Ref. [241]. The idea is to generate inflation through a purely quantum-gravitational effect, by considering the case of a Friedmann–Lemaître–Robertson–Walker universe filled with massless conformally-covariant quantum fields. Because of conformal invariance, these massless fields do not undergo particle creation, so the stress–energy tensor is only made of terms that arise in the regularization process, i.e. from the interaction of quantum free matter fields with a classical gravitational field. Those terms are quadratic in the space–time curvature [242,243], and give rise to a non-vanishing expectation value for the stress–energy tensor, $\langle T_{\mu\nu} \rangle$, which, in the context of semi-classical gravity, sources the Einstein equations. It was then realized in Ref. [244] that the same stress–energy tensor can be obtained by varying the action

$$S = \frac{M_g^2}{2} \int d^4 x \sqrt{-g} f(R), \quad \text{where} \quad f(R) = R + \frac{R^2}{\mu^2}, \qquad (4.1)$$

where $\mu$ is a mass parameter that depends on the (conformal and massless) field content. The mass scale of gravity is denoted $M_g$ here. From the point of view of effective theories, Eq. (4.1) may be merely seen as the leading correction to General Relativity in the class of $f(R)$ theories. Indeed, at low energy, i.e. when $R$ is small, the leading term in a generic Taylor expansion of the $f(R)$ function dominates and one recovers an action looking like General Relativity and matching Newtonian gravity provided the numerical value of $M_g^2 \simeq (4/3)M_{Pl}^2$ (see Section 4.2.2 for more details on the scalar-tensor theories). Here one considers the first correction, that may play an important role at the high energies at which inflation proceeds. It is also worth mentioning that when that first correction is not quadratic but of a different order, one obtains $R + R^{2p}/\mu^{4p-2}$ Inflation (RPI), presented in Section 5.13, while, when the next-to-next-to-leading correction is included, i.e. when a term $R^3$ is also considered in the $f(R)$ function [namely, $f(R) = R + R^2/\mu^2 + \alpha R^3/\mu^4$], one obtains the Cubically Corrected Starobinsky Inflation model (CCSI), discussed in Section 5.25.

Let us first establish some general equations in the case where the action describing gravity is given by Eq. (4.1) to which we add a contribution representing matter fields, namely

$$S(\psi, g_{\mu\nu}) = \frac{M_g^2}{2} \int d^4 x \sqrt{-g} f(R) + \int d^4 x \mathcal{L}_{\text{mat}}(\psi, g_{\mu\nu}), \qquad (4.2)$$

where $\mathcal{L}_{\text{mat}}(\psi, g_{\mu\nu})$ is the Lagrangian of matter. The field $\psi$ being, “symbolically”, a matter field and we are implicitly assuming that $\mathcal{L}_{\text{mat}}$ contains the covariant volume factor $\sqrt{-g}$. Including the matter action in our considerations will be important when we deal with reheating. If viewed as exact, the above theory can be seen as a generalization of Einstein gravity. A maybe more realistic point of view, as already sketched above, is to interpret this framework as an effective theory of gravity taking into higher order operators into account, i.e. $f(R) = R + R^2/\mu^2 + \dots$. In this last point of view, however, one could also ask why other terms, such as $R_{\mu\nu}R^{\mu\nu}$, are not included (one could also add the contraction of the Riemann tensor but this can always be re-expressed in terms of the scalar curvature, the contraction of the Ricci tensor and the Gauss–Bonnet term, which is topological in four dimensions). Varying the above action, Eq. (4.2), with respect to the metric tensor lead to the following equations of motion

$$\Sigma_{\mu\nu} = F(R)R_{\mu\nu} - \frac{1}{2}f(R)g_{\mu\nu} - \nabla_{\mu}\nabla_{\nu}F(R) + g_{\mu\nu}\Box F(R) = \frac{1}{M_g^2}T_{\mu\nu}, \qquad (4.3)$$

where $F(R) = \partial f/\partial R$ and $T_{\mu\nu}$ is the stress–energy tensor of matter, namely

$$T_{\mu\nu} = -\frac{2}{\sqrt{-g}} \frac{\delta \mathcal{L}_{\text{mat}}}{\delta g^{\mu\nu}}. \qquad (4.4)$$

The tensor $T_{\mu\nu}$ is conserved and one can check that this is also the case for $\Sigma_{\mu\nu}$, $\nabla_{\mu}\Sigma^{\mu\nu} = 0$ which is evidently required for consistency of the equations of motion, see Eq. (4.3).

So far, we have worked in the so-called Jordan frame. However, as is well-known, see for instance Refs. [245,246], the above $f(R)$ theory can also be cast in different equivalent formulations. For instance, it is equivalent to the Brans–Dicke theory the action of which is given by

$$S_{\text{BD}}(\phi, \psi, g_{\mu\nu}) = M_g^2 \int d^4 x \sqrt{-g} \left[ \frac{1}{2} \phi R - \frac{\omega_{\text{BD}}}{\phi} \frac{1}{2} g^{\mu\nu} \partial_{\mu} \phi \partial_{\nu} \phi - V(\phi) \right] + \int d^4 x \mathcal{L}_{\text{mat}}(\psi, g_{\mu\nu}), \qquad (4.5)$$
where $\phi$ is a (dimensionless) scalar field, $\omega_{BD}$ the (dimensionless) Brans–Dicke parameter and $V(\phi)$ a (dimension 2) potential. In order to prove the equivalence between the $f(R)$ theory and the Brans–Dicke theory, let us consider the following action

$$S(\chi, \psi, g_{\mu\nu}) = \frac{M_g^2}{2} \int d^4x \sqrt{-g} \left[ f(\chi) + (R - \chi) \frac{\partial f}{\partial \chi} \right] + \int d^4x \mathcal{L}_{\text{mat}}(\psi, g_{\mu\nu}), \qquad (4.6)$$

where $\chi$ is a new field of dimension $M^2$. The function $f(\chi)$ is of same dimension since one can expand it as $f(\chi) = \chi + \cdots$. Varying this action with respect to $\chi$, one obtains

$$(R - \chi) \frac{\partial^2 f}{\partial \chi^2} = 0, \qquad (4.7)$$

which, provided $f''(\chi) \neq 0$ implies $\chi = R$ and Eq. (4.6) reduces to Eq. (4.1). Notice that for $f''(\chi) = 0$, one would simply recover General Relativity with a cosmological constant. The next step consists in introducing the dimensionless field $\phi$ defined by $\phi = \partial f / \partial \chi$ so that $\chi = \chi(\phi)$. Using this definition in Eq. (4.6), one arrives at

$$S_{\text{BD}}(\phi, \psi, g_{\mu\nu}) = M_g^2 \int d^4x \sqrt{-g} \left( \frac{1}{2}\phi R - \frac{1}{2}\{\chi(\phi)\phi - f[\chi(\phi)]\} \right) + \int d^4x \mathcal{L}_{\text{mat}}(\psi, g_{\mu\nu}), \qquad (4.8)$$

which is exactly of the Brans–Dicke form with $\omega_{BD} = 0$ and

$$V(\phi) = \frac{1}{2} \{\chi(\phi)\phi - f[\chi(\phi)]\} = \frac{\mu^2}{8} (\phi - 1)^2, \qquad (4.9)$$

the last equality holding for the Starobinsky model only.

One can also obtain another description of the same theory by using conformal transformations. For this purpose, let us rewrite the action (4.1) with the Lagrange multiplier introduced in Eq. (4.6). Defining

$$F(\chi) \equiv \frac{\partial f(\chi)}{\partial \chi}, \qquad (4.10)$$

one has

$$S(\psi, g_{\mu\nu}) = \int d^4x \sqrt{-g} \left\{ \frac{M_g^2}{2} F(\chi) R - \frac{M_g^2}{2} [F(\chi)\chi - f(\chi)] \right\} + \int d^4x \mathcal{L}_{\text{mat}}(\psi, g_{\mu\nu}). \qquad (4.11)$$

Let us now perform a conformal transformation induced by a dimensionless scalar field, say $\sigma$, and rewrite this action in the so-called Einstein frame of metric

$$\tilde{g}_{\mu\nu} = e^{-2\sigma} g_{\mu\nu}. \qquad (4.12)$$

In this section, quantities with a “tilde” will refer to the Einstein frame while quantities without are written in the Jordan frame. Under this conformal transformation, the scalar curvature changes according to

$$R = e^{-2\sigma} (\tilde{R} - 6\tilde{g}^{\mu\nu} \tilde{\nabla}_{\mu} \partial_{\nu} \sigma - 6\tilde{g}^{\mu\nu} \partial_{\mu} \sigma \partial_{\nu} \sigma). \qquad (4.13)$$

As a consequence, if we now express the action given by Eq. (4.11) in terms of quantities written in the Einstein frame, using the above transformation (4.13) for the scalar curvature, then one is led to the following expression

$$S(\sigma, \psi, \tilde{g}_{\mu\nu}) = \int d^4x \sqrt{-\tilde{g}} \left\{ e^{4\sigma} \frac{M_g^2}{2} F(\chi) e^{-2\sigma} (\tilde{R} - 6\tilde{g}^{\mu\nu} \tilde{\nabla}_\mu \partial_\nu \sigma - 6\tilde{g}^{\mu\nu} \partial_\mu \sigma \partial_\nu \sigma) \right. \\ \left. - e^{4\sigma} \frac{M_g^2}{2} [F(\chi)\chi - f(\chi)] \right\} + \int d^4x \mathcal{L}_{\text{mat}}(\psi, e^{2\sigma} \tilde{g}_{\mu\nu}). \qquad (4.14)$$

Since, by definition of the Einstein frame, we want a theory the action of which is linear in $\tilde{R}$, we see that one must choose $\sigma(\chi)$ such that $e^{-2\sigma} = F(\chi)$. Then, the term containing the second derivative of $\sigma$ is a total derivative and can be discarded. Indeed, for any metric tensor $\tilde{g}_{\alpha\beta}$, one has

$$\tilde{g}^{\alpha\beta} \nabla_{\alpha} \nabla_{\beta} \sigma = \frac{1}{\sqrt{-\tilde{g}}} \partial_{\alpha} \left( \sqrt{-\tilde{g}} \tilde{g}^{\alpha\beta} \partial_{\beta} \sigma \right). \qquad (4.15)$$

Moreover, if one defines the new scalar degree of freedom $\phi$ by

$$\frac{\phi}{M_g} \equiv \sqrt{\frac{3}{2}} \ln [F(\chi)] = -\sqrt{6} \sigma(\chi), \qquad (4.16)$$

then, one arrives at

$$S(\phi, \psi, \tilde{g}_{\mu\nu}) = \int d^4x \sqrt{-\tilde{g}} \left[ \frac{M_g^2}{2} \tilde{R} - \frac{1}{2} \tilde{g}^{\mu\nu} \partial_\mu \phi \partial_\nu \phi - V(\phi) \right] + \int d^4x \mathcal{L}_{\text{mat}}(\psi, e^{2\sigma} \tilde{g}_{\mu\nu}), \qquad (4.17)$$

with

$$V(\phi) = \frac{M_g^2 \chi F(\chi) - f(\chi)}{2 F^2(\chi)}. \qquad (4.18)$$

This potential is sometimes written in terms of $R$ instead of $\chi(\phi)$. Indeed, on shell, the Lagrange multiplier $\chi$ being the solution of Eq. (4.7), one has $\chi = R$. One therefore obtains, in the Einstein frame, Einstein gravity plus a canonically normalized scalar field $\phi$. This is an additional scalar mode propagating in the theory, and, an important ingredient for inflation, the coupling to matter is no longer universal due to the presence of the combination $e^{2\sigma} \tilde{g}_{\mu\nu}$ in the matter action. As already mentioned, this will play an important role for reheating. Let us mention that this additional
scalar degree of freedom may modify the measured gravitational constant and, in general, one cannot identify $M_g$ and $M_{Pl}$ [247,248]. However, if one assumes that, after inflation, this scalar degree of freedom relaxes to very small values (which is the case here since $\chi = R$), then, for all post-inflationary physics, $M_{Pl} \simeq M_g$.

Now, let us apply the previous considerations to the Starobinsky model. In that case, one has $F(\chi) \equiv \partial f / \partial \chi = 1+2\chi/\mu^2$, and the field $\phi$ evolves in the potential given by

$$V(\phi) = \frac{M_g^2}{2\mu^2} \frac{\chi^2}{\left(1 + 2\frac{\chi}{\mu^2}\right)^2}. \qquad (4.19)$$

Using the relationship (4.16) between the Lagrange multiplier $\chi$ and the field $\phi$, one gets

$$\chi = \frac{\mu^2}{2} \left( e^{\sqrt{2/3}\phi/M_g} - 1 \right), \qquad (4.20)$$

and the potential is explicitly given by

$$V(\phi) = M^4 \left( 1 - e^{-\sqrt{\frac{2}{3}}\frac{\phi}{M_g}} \right)^2, \qquad (4.21)$$

with $M^4 \equiv M_g^2 \mu^2/8$.

### 4.1.2. Other theoretical justifications

Many authors have tried to realize Starobinsky inflation in the framework of supersymmetry and supergravity [249–251]. One of the earliest attempt was based on models containing physical multiplets that are not chiral but vector or linear [252–256]. A great advantage of this type of approaches (compared to formulations using chiral multiplets) is that there is no need to stabilize additional scalar fields during inflation simply because there is none; indeed there is only one scalar field which is interpreted as the inflaton field. The extra fields are typically vector fields and they do not acquire a vacuum expectation value during inflation. The bosonic action obtained from the action of a massive vector field $V$ was derived in Ref. [252]. It reads

$$\mathcal{L} = -\frac{R}{2} - \frac{1}{4} F_{\mu\nu} F^{\mu\nu} + \frac{g^2}{2} J_{CC} B_{\mu} B^{\mu} + \frac{1}{2} J_{CC} \partial_{\mu} C \partial^{\mu} C - \frac{g^2}{2} J_C^2, \qquad (4.22)$$

where $C$ is the scalar field present in the vector multiplet, $B_\mu$ is the vector in the vector multiplet and the subscript “C” denotes a derivative with respect to the field $C$. The arbitrary function $J$ is written $J = 3/2 \ln \Phi$ where $\Phi$ is a function of $C$. Finally, the quantity $g$ is the gauge coupling.

As mentioned above, $B_\mu$ does not acquire a vacuum expectation value during inflation and, therefore, in Eq. (4.22), we are left with the action of gravity plus a non-canonical normalized scalar field $C$. If one chooses the function $\Phi$ such that

$$\Phi(C) = -Ce^C, \qquad (4.23)$$

and canonically normalize $C$ with $C = -e^{\sqrt{2/3}\phi/M_g}$, then the potential, which corresponds to the last term in Eq. (4.22), reads

$$V(\phi) = \frac{9g^2}{4} \left( 1 - e^{-\sqrt{2/3}\phi/M_g} \right)^2. \qquad (4.24)$$

One recognizes the Starobinsky potential already given in Eq. (4.21).

More recently, various other theoretical constructions have been proposed that also give a potential matching Eq. (4.21).

In Ref. [257], a supergravity realization of this model was presented that we now briefly review. The model is based on no-scale supergravity and has two fields, a modulus $T$ and the inflaton $\phi$. The Kähler and super-potential are given by

$$K = -3M_g^2 \ln \left( \frac{T}{M_g} + \frac{T^\dagger}{M_g} - \frac{|\phi|^2}{3M_g^2} \right), \qquad (4.25)$$

$$W = \hat{\mu}\phi^2 - \frac{\lambda}{3}\phi^3,$$

where $\hat{\mu}$ is of dimension 1 and $\lambda$ dimensionless (recall that the Kähler potential is of dimension 2 while the super-potential is of dimension 3), respectively. The quantities $\hat{\mu}$ and $\lambda$ are constants characterizing the model. It follows that the Kähler matrix and its inverse<fnref n="5" /> can be written as

$$K_{ij} = \frac{3}{\left[ T/M_g + T^\dagger/M_g - |\phi|^2/(3M_g^2) \right]^2} \begin{bmatrix} (T+T^\dagger)/(3M_g) & -\phi^\dagger/(3M_g) \\ -\phi/(3M_g) & 1 \end{bmatrix}, \qquad (4.26)$$

$$K^{kj} = \left( \frac{T}{M_g} + \frac{T^\dagger}{M_g} - \frac{|\phi|^2}{3M_g^2} \right) \begin{bmatrix} 1 & \phi/(3M_g) \\ \phi^\dagger/(3M_g) & (T+T^\dagger)/(3M_g) \end{bmatrix}. \qquad (4.27)$$

Then, assuming that the modulus is stabilized such $\langle T + T^\dagger \rangle = c M_g$ and $\langle T - T^\dagger \rangle = 0$, one obtains the effective Lagrangian

$$\mathcal{L}_{\text{eff}} = -\frac{c}{\Delta^2} \left| \partial_{\mu} \phi \right|^2 - \frac{1}{\Delta^2} \left| \frac{\partial W}{\partial \phi} \right|^2, \qquad (4.28)$$

where $\Delta \equiv c - |\phi|^2/(3M_g^2)$. The next step consists in introducing the fields $x$ and $y$ defined by

$$\frac{\phi}{M_g} \equiv \sqrt{3c} \tanh \left( \frac{x + iy}{M_g \sqrt{3}} \right). \qquad (4.29)$$

<footnote n="5">⁵ The inverse of the Kähler matrix is $K^{kj}$ so $K^{kj}$ is the transpose of the inverse</footnote>
Expressed in terms of these two fields, the previous Lagrangian takes the following form

$$ \mathcal{L}_{\text{eff}} = -\frac{1}{2 \cos^2[\sqrt{2/3}(y/M_g)]} [(\partial_\mu x)^2 + (\partial_\mu y)^2] \\ - \frac{\mu^2}{2} \frac{1}{2 \cos^2[\sqrt{2/3}(y/M_g)]} e^{-\sqrt{2/3}x} \left[ \cosh\left(\sqrt{\frac{2}{3}}\frac{x}{M_g}\right) - \cos\left(\sqrt{\frac{2}{3}}\frac{y}{M_g}\right) \right], \qquad (4.30) $$

where $\mu \equiv \hat{\mu}\sqrt{3/c}$. In order to obtain this formula, we have crucially assumed that

$$ \lambda = \frac{\mu}{3M_g}. \qquad (4.31) $$

The form of the effective Lagrangian has also been studied in Ref. [257] in the case where this relation is no longer valid. The last step consists in remarking that $y=0$ during inflation. If we expand the above Lagrangian about $y=0$, then the field $x$ is canonically normalized and the potential becomes precisely the one of Eq. (4.21). As such, it constitutes another scenario where this potential arises.

Let us also notice that other approaches based on superconformal D-term inflation also lead to the same potential [258]. Various multifield extensions have also been studied in which the inflationary phase can still be described by the one-field Higgs potential [259–261].

More recently, the Starobinsky model has also been derived from theories that are conformally invariant (with spontaneous symmetry breaking). Moreover, the supersymmetric version of these theories, superconformal theories (with spontaneous breaking of the superconformal symmetry) have also been shown to lead to the Starobinsky model, thus providing another supergravity description of this model. In the following, we present these considerations which are based on Ref. [262]. In order to understand the context in which these models have been developed, it is useful to first consider the action given by the following expression

$$ S(g_{\mu\nu}, \chi) = \frac{M_g^2}{2} \int d^4 x \sqrt{-g} \left( \frac{\chi^2}{6} R + g^{\mu\nu} \partial_\mu \chi \partial_\nu \chi - \frac{\lambda}{2} \chi^4 \right), \qquad (4.32) $$

where $\lambda$ is a coupling constant of dimension 2 (here, the field $\chi$ is dimensionless). It should be noticed that the sign of the kinetic term for the dimensionless field $\chi$ is the “wrong” one. Then, the fundamental remark is that the above action is invariant under the conformal transformation $\tilde{g}_{\mu\nu} = e^{-2\sigma} g_{\mu\nu}$ and $\tilde{\chi} = e^{\sigma}\chi$, where $\sigma$ is a dimensionless field. Indeed, if one inserts the previous transformation into the action (4.32), one obtains

$$ \begin{aligned} S(g_{\mu\nu}, \chi) &= \frac{M_g^2}{2} \int d^4 x e^{4\sigma} \sqrt{-\tilde{g}} \left\{ e^{-2\sigma} \frac{\tilde{\chi}^2}{6} e^{-2\sigma} \left[ \tilde{R} - 6\tilde{g}^{\mu\nu} \tilde{\nabla}_\mu \partial_\nu \sigma - 6\tilde{g}^{\mu\nu} \partial_\mu \sigma \partial_\nu \sigma \right] \right. \\ & \qquad \left. + e^{-2\sigma} \tilde{g}^{\mu\nu} \partial_\mu (e^{-\sigma} \tilde{\chi}) \partial_\nu (e^{-\sigma} \tilde{\chi}) - \frac{\lambda}{2} e^{-4\sigma} \tilde{\chi}^4 \right\} \\ &= \frac{M_g^2}{2} \int d^4 x \sqrt{-\tilde{g}} \left[ \frac{\tilde{\chi}^2}{6} \tilde{R} + \tilde{g}^{\mu\nu} \partial_\mu \tilde{\chi} \partial_\nu \tilde{\chi} - \frac{\lambda}{2} \tilde{\chi}^4 \right] - \frac{M_g^2}{2} \int d^4 x \sqrt{-\tilde{g}} \left[ 2\tilde{g}^{\mu\nu} \tilde{\chi} \partial_\mu \sigma \partial_\nu \tilde{\chi} \right. \\ & \qquad \left. + \tilde{\chi}^2 \tilde{g}^{\mu\nu} \tilde{\nabla}_\mu \partial_\nu \sigma \right], \end{aligned} \qquad (4.33) $$

where we have used that the transformation of the scalar curvature is given by Eq. (4.13). Using $\tilde{\chi}^2 \tilde{\nabla}_\mu \partial_\nu \sigma = \tilde{\nabla}_\mu (\tilde{\chi}^2 \partial_\nu \sigma) - \tilde{\nabla}_\mu (\tilde{\chi}^2) \partial_\nu \sigma$, the second term in the above expression reduces to a total derivative, thus showing that, indeed, the action is invariant, see Eq. (4.32).

The fact that the field $\chi$ has a kinetic term with a “wrong” sign is not problematic because, as explained in Ref. [262], it can be removed by fixing its value. If one takes $\chi = \sqrt{6}$, Eq. (4.32) reduces to

$$ S(g_{\mu\nu}) = \int d^4 x \sqrt{-g} \left( \frac{M_g^2}{2} R - 9\lambda M_g^2 \right). \qquad (4.34) $$

The field $\chi$ is called the “conformon” because it is used to break the conformal symmetry. This last equation is nothing but the action of GR with a cosmological constant. If this one is positive, the homogeneous and isotropic solution is de Sitter, the prototype of a Universe undergoing a phase of inflation.

The second step in Ref. [262] consists in introducing a two-field model, which is a generalization of the Eq. (4.32) and is described by the following expression

$$ S(g_{\mu\nu}, \chi, \phi) = \frac{M_g^2}{2} \int d^4 x \sqrt{-g} \left[ \frac{\chi^2}{6} R + g^{\mu\nu} \partial_\mu \chi \partial_\nu \chi - \frac{\phi^2}{6} R - g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi - \frac{\lambda}{2} (\phi^2 - \chi^2)^2 \right]. \qquad (4.35) $$

Obviously, this action resembles Eq. (4.32). The field $\chi$ is still a conformon since its kinetic term has the “wrong” sign but we notice that this is not the case for the field $\phi$. It is also clear that Eq. (4.35) is invariant under the conformal transformation $\tilde{g}_{\mu\nu} = e^{-2\sigma} g_{\mu\nu}$, $\tilde{\phi} = e^{\sigma}\phi$ and $\tilde{\chi} = e^{\sigma}\chi$. This action possesses an additional symmetry: it is invariant under global SO(1, 1) transformations in the fields $\phi$ and $\chi$. Let us recall that this group can be represented by the two-by-two matrices $M$ of the form

$$ M = \begin{pmatrix} a & b \\ b & a \end{pmatrix}, \qquad (4.36) $$

where $a$ and $b$ are real numbers such that $a^2 - b^2 = 1$. If

$$ \begin{pmatrix} \tilde{\phi} \\ \tilde{\chi} \end{pmatrix} = M \begin{pmatrix} \phi \\ \chi \end{pmatrix}, \qquad (4.37) $$

then $\phi^2 - \chi^2$ is a SO(1, 1)-invariant and this makes the invariance of the action (4.35) under SO(1, 1) explicit.
As before, the next step consists in fixing the conformal gauge. A first example is the so-called “rapidity” gauge defined by $\chi^2 - \phi^2 = 6$, which is SO(1, 1) invariant. Such a gauge condition does not completely fix the value of the conformon but only constrains its relationship with the field $\phi$. This constraint can also be enforced by introducing an additional field $\varphi$ and demanding that

$$ \chi = \sqrt{6} \cosh \left( \frac{\varphi}{\sqrt{6} M_g} \right), \quad \phi = \sqrt{6} \sinh \left( \frac{\varphi}{\sqrt{6} M_g} \right). \qquad (4.38) $$

Then, the Eq. (4.35) becomes

$$ S(g_{\mu\nu}, \varphi) = \int d^4 x \sqrt{-g} \left( \frac{M_g^2}{2} R - \frac{1}{2} g^{\mu\nu} \partial_\mu \varphi \partial_\nu \varphi - 9 \lambda M_g^2 \right). \qquad (4.39) $$

One recovers the action (4.34) but, this time, with one additional degree of freedom described by the field $\varphi$ which has a constant potential.

The idea to obtain a less trivial theory is to break the SO(1, 1) symmetry and to consider the potential $\lambda\phi^2(\phi - \chi)^2/4$ instead of $\lambda(\phi^2 - \chi^2)^2/2$ in Eq. (4.35). In the rapidity gauge, the potential of the field $\varphi$ now reads

$$ V(\varphi) = \frac{9 \lambda M_g^2}{4} \left( 1 - e^{-\sqrt{2/3}\varphi / M_g} \right)^2, \qquad (4.40) $$

which is exactly the Starobinsky model.

The above considerations were generalized to a supersymmetric framework (superconformal theory and supergravity) in Ref. [263]. As is well-known, standard supergravity depends on two functions, the Kähler and super potentials. The Kähler potential leads to the kinetic terms of the fields while the superpotential allows us to calculate the scalar potential of the theory. Standard supergravity implies that all the scalars in the theory are minimally coupled to gravity. However, supergravity can be reformulated, leading to conformal supergravity, in such a way that scalars can non-minimally couple to gravity and, in the following, we will be interested in this class of models. As it is the case for standard supergravity, conformal supergravity also depends on two functions, $N$, the embedding Kähler potential, and a superpotential $W$. The quantity $N$ which appears in front of the scalar curvature is also used to calculate the kinetic terms of the fields in the model, namely $G_{IJ} = \partial^2 N / (\partial X^I \partial \bar{X}^J)$. In this context, the Lagrangian of the theory can be written as

$$ \mathcal{L} = \sqrt{-g} \left( -\frac{N}{6} R - G_{I\bar{J}} \partial^\mu X^I \partial_\mu \bar{X}^{\bar{J}} - V \right). \qquad (4.41) $$

In order to implement the Starobinsky model, we use a version where there are three fields: the so-called compensator field $X^0$, the inflaton field $X^1 = \Phi$ and the so-called Goldstino superfield $X^2 = S$. The compensator field $X^0$ is also called the conformon field because the superconformal theory becomes supergravity after the conformal symmetry has been broken which can be achieved when the conformal field acquires a constant value, $X^0 = \bar{X}^0 = \sqrt{3} M_g$ (this particular value is chosen in order to correctly normalize gravity). Then, the corresponding $N = 1$ supergravity theory is described by the following Kähler and super potential

$$ \mathcal{N}(X^I, \bar{X}^I) \Big|_{X^0=\bar{X}^0=\sqrt{3}M_g} = -3 M_g^2 e^{-\frac{K(\Phi, \bar{\Phi}, S, \bar{S})}{3M_g^2}}, \\ W(X^I) \Big|_{X^0=\sqrt{3}M_g} = W(\Phi, S). \qquad (4.42) $$

The Starobinsky model can be obtained by assuming the following form for the potential of the embedding manifold and the superpotential

$$ \mathcal{N}(X^I, \bar{X}^I) = -|X^0|^2 \exp \left[ -\frac{|S|^2}{|X^0|^2} + \frac{1}{2} \left( \frac{\Phi}{X^0} - \frac{\bar{\Phi}}{\bar{X}^0} \right)^2 + \zeta \frac{|S|^4}{|X^0|^4} \right], \\ W(X^I) = \frac{M}{2\sqrt{3}M_g} S(X^0)^2 (1 - e^{-2\Phi/X^0}). \qquad (4.43) $$

where $M$ is a mass scale and $\zeta$ a dimensionless parameter. Using Eq. (4.42), after breaking the conformal symmetry, one finds that the corresponding Kähler and super potentials are given by

$$ K = |S|^2 - \frac{1}{2}(\Phi - \bar{\Phi})^2 - \frac{\zeta}{3} \frac{|S|^4}{M_g^2}, \\ W = \frac{M M_g \sqrt{3}}{2} S \left[ 1 - e^{-2\Phi/(\sqrt{3}M_g)} \right]. \qquad (4.44) $$

It is important to notice that the superpotential has the form $W = Sf(\Phi)$. This particular form will play an important role in the following, in particular in rendering the calculations much simpler.

From the above expression of the Kähler potential, one can calculate the Kähler matrix which reads

$$ G_{AB} = \frac{1}{M_g^2} \begin{pmatrix} 1 - \frac{4}{3}\zeta \frac{S\bar{S}}{M_g^2} & 0 \\ 0 & 1 \end{pmatrix}. \qquad (4.45) $$

Then, the F-term scalar potential can be inferred. It contains two terms corresponding to the two non-vanishing component of the Kähler matrix and can be expressed as

$$ V = \frac{e^{K/M_g^2}}{1 - \frac{4}{3}\zeta \frac{S\bar{S}}{M_g^2}} \left( \frac{1}{M_g^4} W W^\dagger \frac{\partial K}{\partial \bar{S}} \frac{\partial K}{\partial S} + \frac{1}{M_g^2} W^\dagger \frac{\partial K}{\partial \bar{S}} \frac{\partial W}{\partial S} + \frac{1}{M_g^2} W \frac{\partial K}{\partial S} \frac{\partial W^\dagger}{\partial \bar{S}} + \frac{\partial W}{\partial S} \frac{\partial W^\dagger}{\partial \bar{S}} \right) \\ + e^{K/M_g^2} \left( \frac{1}{M_g^4} W W^\dagger \frac{\partial K}{\partial \bar{\Phi}} \frac{\partial K}{\partial \Phi} + \frac{1}{M_g^2} W^\dagger \frac{\partial K}{\partial \bar{\Phi}} \frac{\partial W}{\partial \Phi} + \frac{1}{M_g^2} W \frac{\partial K}{\partial \Phi} \frac{\partial W^\dagger}{\partial \bar{\Phi}} + \frac{\partial W}{\partial \Phi} \frac{\partial W^\dagger}{\partial \bar{\Phi}} \right). \qquad (4.46) $$
Using the explicit form of the Kähler and super potentials, one finally gets

$$V = \frac{e^{K/M_g^2}}{1 - \frac{4}{3}\zeta \frac{S\bar{S}}{M_g^2}} \left[ \frac{1}{M_g^4} S\bar{S}f(\Phi)f(\bar{\Phi}) \left( S - \frac{2}{3}\zeta \frac{S^2\bar{S}}{M_g^2} \right) \left( \bar{S} - \frac{2}{3}\zeta \frac{S\bar{S}^2}{M_g^2} \right) \right. \\ + \frac{1}{M_g^2} \bar{S}f(\bar{\Phi}) \left( S - \frac{2}{3}\zeta \frac{S^2\bar{S}}{M_g^2} \right) f(\Phi) + \frac{1}{M_g^2} Sf(\Phi) \left( \bar{S} - \frac{2}{3}\zeta \frac{S\bar{S}^2}{M_g^2} \right) f(\Phi) + f(\Phi)f(\bar{\Phi}) \left. \right] \\ + e^{K/M_g^2} \left[ -\frac{1}{M_g^4} S\bar{S}f(\Phi)f(\bar{\Phi})(\Phi - \bar{\Phi})^2 + \frac{1}{M_g^2} \bar{S}f(\bar{\Phi})(\Phi - \bar{\Phi})S \frac{\partial f}{\partial \Phi} \right. \\ \left. - \frac{1}{M_g^4} Sf(\Phi)(\Phi - \bar{\Phi})\bar{S} \frac{\partial f}{\partial \bar{\Phi}} + S\bar{S} \frac{\partial f}{\partial \Phi} \frac{\partial f}{\partial \bar{\Phi}} \right].$$

Now if one considers the trajectory $S = 0$ and $\alpha = 0$, where $\Phi = (\varphi + i\alpha)/\sqrt{2}$, the potential reduces to

$$V(\varphi) = \frac{3}{4} M_g^2 M_g^2 \left(1 - e^{-\sqrt{2/3}\varphi/M_g}\right)^2, \qquad (4.47)$$

which is exactly the Starobinsky model. We notice that, in the full scalar potential, all the terms vanish thanks to $S = 0$, except $f(\Phi)f(\bar{\Phi})$, which gives rise the Starobinsky potential. As already mentioned, this is because the superpotential is of the form $W = Sf(\Phi)$.

Finally, it is important to notice that the inflationary trajectory considered above $S = 0$ is stable. In fact this is the whole purpose of introducing the term proportional to the parameter $\zeta$ in Eq. (4.43): it gives a positive mass to the field $S$ and renders the whole scenario consistent. More details on this class of models can be found in Ref. [263].

### 4.1.3. Slow-roll analysis

Let us move back to the original notation and denote by $\phi$ the inflaton field for the Starobinsky model in the Einstein frame. The potential is given by Eq. (4.21) and, defining $x \equiv \phi/M_g$, the first three slow-roll parameters are given by

$$\begin{align}
\epsilon_1 &= \frac{4}{3} \left(1 - e^{\sqrt{2/3}x}\right)^{-2}, & \epsilon_2 &= \frac{2}{3} \left[ \sinh\left(\frac{x}{\sqrt{6}}\right) \right]^{-2}, \nonumber \\
\epsilon_3 &= \frac{2}{3} \left[ \coth\left(\frac{x}{\sqrt{6}}\right) - 1 \right] \coth\left(\frac{x}{\sqrt{6}}\right). \tag{4.48}
\end{align}$$

Notice that Eqs. (3.4) to (3.6) are still applicable here with the formal replacement $M_{\text{Pl}} \to M_g$. These quantities are represented in Fig. 5 (left and right bottom panels) together with the potential and its logarithm. The minimum of the potential being at $x = 0$, after inflation the numerical value of $M_g \simeq M_{\text{Pl}}$.

In this model, as can be noticed on these plots, inflation stops by violation of the slow-roll conditions. The condition $\epsilon_1 = 1$ occurs for $x = x_{\text{end}}$ where $x_{\text{end}}$ can be expressed as

$$x_{\text{end}} = \sqrt{\frac{3}{2}} \ln \left( 1 + \frac{2}{\sqrt{3}} \right) \approx 0.94. \qquad (4.49)$$

In fact, before the end of inflation, the slow-roll approximation breaks down when $\epsilon_2$ becomes greater than one. This happens for $x = x_{\epsilon_2=1}$ where

$$x_{\epsilon_2=1} = \sqrt{6} \operatorname{arcsinh} \left( \sqrt{\frac{2}{3}} \right) \approx 1.83. \qquad (4.50)$$

The third slow-roll parameter $\epsilon_3$ also becomes greater than one before the end of inflation (but after the second slow-roll parameter has become unity). The corresponding vacuum expectation value can be written as

$$x_{\epsilon_3=1} = \sqrt{6} \operatorname{arctanh} \left( \frac{2}{1+\sqrt{7}} \right) \approx 1.51. \qquad (4.51)$$

We can calculate the slow-roll trajectory exactly. Using Eq. (4.21), it can be integrated and yields

$$N_{\text{end}} - N = \frac{1}{2} \sqrt{\frac{3}{2}} (x_{\text{end}} - x) + \frac{3}{4} \left( e^{\sqrt{\frac{2}{3}}x} - e^{\sqrt{\frac{2}{3}}x_{\text{end}}} \right). \qquad (4.52)$$

In the regime where $x \gg 1$, the last term is dominant. The trajectory can be inverted and expressed in term of the “-1-branch” of the Lambert function $W_{-1}$, leading to

$$x = \sqrt{\frac{3}{2}} \left\{ -\frac{4}{3}\Delta N + \sqrt{\frac{2}{3}}x_{\text{end}} - e^{\sqrt{\frac{2}{3}}x_{\text{end}}} - W_{-1} \left[ -\exp\left(-\frac{4}{3}\Delta N + \sqrt{\frac{2}{3}}x_{\text{end}} - e^{\sqrt{\frac{2}{3}}x_{\text{end}}}\right) \right] \right\}, \qquad (4.53)$$

where $\Delta N = N_{\text{end}} - N$. The fact that inflation proceeds on the “-1-branch of the Lambert function $W_{-1}$, as can be seen in Fig. 6, can be justified by the following considerations. When $\Delta N = 0$, the value taken by the Lambert function is $-\exp(\sqrt{2/3}x_{\text{end}})$, which is smaller than $-1$. On the other hand, if $x = 0$, the value given for $\Delta N$ by Eq. (4.52) can be inserted in Eq. (4.53) and one finds that the argument of the Lambert function is $-1$, i.e. the connection point between the “-1-branch and the 0 branch. Therefore inflation takes place between these two points.
<figure><img src="image_8.png" /><figcaption>Fig. 5. Starobinsky Inflation (SI). Top left panel: Starobinsky potential corresponding to Eq. (4.21). Top right panel: logarithm of the potential. It is clear from these two plots that inflation proceeds from the right to the left. Bottom left panel: slow-roll parameter $\epsilon_1$ as a function of the field $\phi$. Inflation ends when $\epsilon_1$ becomes larger than unity. Bottom right panel: slow-roll parameters $\epsilon_2$ (solid line) and $\epsilon_3$ (dotted line) for the same potential.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 6. Lambert functions $W_0(x)$ (dashed line) and $W_{-1}(x)$ (solid line). During Starobinsky inflation, inflation proceeds along the “-1” branch in the direction specified by the arrow in the figure.</figcaption></figure>

Finally, the value of the inflaton field, $x_*$, at which the pivot mode crossed the Hubble radius is related to the number $e$-folds before the end of inflation by

$$x_* = \sqrt{\frac{3}{2}} \left( -\frac{4}{3}\Delta N_* + \ln\left(1+\frac{2}{\sqrt{3}}\right) - \left(1+\frac{2}{\sqrt{3}}\right) - W_{-1}\left\{-\exp\left[-\frac{4}{3}\Delta N_* + \ln\left(1+\frac{2}{\sqrt{3}}\right) - \left(1+\frac{2}{\sqrt{3}}\right)\right]\right\} \right). \quad (4.54)$$

Assuming that $x_*$ is known, the energy scale of the potential is fixed by the CMB normalization and one obtains

$$\frac{M^4}{M_g^4} = 1920\pi^2 \left(1 - e^{\sqrt{\frac{2}{3}}x_*}\right)^{-4} e^{2\sqrt{\frac{2}{3}}x_*} \frac{Q_{\text{rms}-\text{PS}}^2}{T^2}. \quad (4.55)$$
Upon using the trajectory given by Eq. (4.54), for the fiducial value $\Delta N_*=55$, one gets $M \simeq 3.3 \times 10^{-3} M_g$, i.e., roughly speaking, inflation takes place at the GUT scale. This also implies that the mass scale $\mu$ is of the order $\mu \simeq 10^{-5} M_g \simeq 10^{13}$ GeV.

The actual values of $\Delta N_*$ and $x_*$ are obtained by solving the reheating equation. However, in the Einstein frame, this one is no longer given by Eq. (3.45). Indeed, in this frame, matter is not universally coupled to the metric tensor and, therefore, it is compulsory to re-consider the parametrization presented in Section 3.2. This is the subject of Section 4.1.4.

### 4.1.4. Reheating in the Einstein frame

In an Einstein frame of metric $g_{\mu\nu}$, the matter action is given by $S_{\text{mat}}[\psi, A^2(\phi)g_{\mu\nu}]$, where $\psi$ denotes some generic matter field and $g_{\mu\nu} \equiv F(\chi)\tilde{g}_{\mu\nu}$ with [248]

$$A \equiv \frac{1}{\Omega} = \frac{1}{\sqrt{F}}. \qquad (4.56)$$

Here, $\tilde{g}_{\mu\nu}$ denotes the metric in the Jordan frame. As most of the inflationary predictions are derived in the Einstein frame, in this section, we will be using the convenient convention that quantities in the Jordan frame have a “bar” whereas quantities in the Einstein frame are left untagged. Notice that this differs from the convention we have used for the theoretical motivations presented in Section 4.1.

In the Jordan frame, the energy density of a (conserved) fluid with a constant equation of state $w = \bar{\rho}/\bar{\rho}$ scales as $\bar{\rho} \propto \bar{a}^{-3(1+w)}$ while, in the Einstein frame, $\rho \propto A^4\bar{\rho} \propto A^{1-3w}a^{-3(1+w)}$ since the scale factors in the two frames are related by $\bar{a} = Aa$. As explained in Ref. [93] and briefly reviewed in Section 3.2, the dependence of the observational predictions on reheating originates from the gradient term $k/H$ present in the Mukhanov-Sasaki variable equation of motion. In order to evaluate concretely this term, one must relate the comoving wave-number $k$ during inflation with physical scales measured now. Clearly, this depends on the whole history of the universe and, therefore, explains why the final result depends on the reheating duration. In the Einstein frame, one can show that the gradient term takes the standard form, namely

$$\frac{k_*}{H} = \frac{e^{N_{\text{end}}-N_*} k_*}{H} \frac{k_*}{a_0} \left( \frac{\rho_{\text{end}}}{\bar{\rho}_\gamma} \right)^{1/4} \frac{1}{R_{\text{rad}}}, \qquad (4.57)$$

with

$$\ln R_{\text{rad}} = \frac{1 - 3w_{\text{reh}}}{12(1 + w_{\text{reh}})} \ln \left( \frac{\rho_{\text{reh}}}{\rho_{\text{end}}} \right) - \frac{1 - 3w_{\text{reh}}}{3(1 + w_{\text{reh}})} \ln \left( \frac{A_{\text{reh}}}{A_{\text{end}}} \right), \qquad (4.58)$$

where $w_{\text{reh}}$ is the equation of state of the effective dominant fluid during reheating. In the above expressions, it is important to emphasize that all the quantities are defined in the Einstein frame and that the non-standard scaling of the various energy densities (pressure-less matter and radiation) has been systematically taken into account. All the extra terms cancel out except in the expression of the parameter $R_{\text{rad}}$ where there is an additional term that involves the function $A$. Remarkably, this additional term is exactly such that the parameter $R_{\text{rad}}$ can be re-expressed in terms of the energy densities in the Jordan frame only, namely

$$\ln R_{\text{rad}} = \ln \bar{R}_{\text{rad}} \equiv \frac{1 - 3w_{\text{reh}}}{12(1 + w_{\text{reh}})} \ln \left( \frac{\bar{\rho}_{\text{reh}}}{\bar{\rho}_{\text{end}}} \right). \qquad (4.59)$$

In other words, this is exactly the parameter $\bar{R}_{\text{rad}}$ that one would have defined by looking only at energy densities in the Jordan frame. Let us stress again that the above equation has an unusual form: it is a quantity used in the Einstein frame but expressed in terms of quantities defined in the Jordan frame.

It is also important to notice an additional limitation compared to the standard case: in presence of non-minimal coupling to gravity, our parametrization of the reheating stage works only for a constant equation of state $w_{\text{reh}}$ while in Ref. [93] it was valid for any $w_{\text{reh}}$. We now explain the origin of this limitation. In the Einstein frame, the general expression of the parameter $R_{\text{rad}}$ is given by

$$\frac{1}{R_{\text{rad}}} = \left( \frac{\rho_{\text{reh}}}{\rho_{\text{end}}} \right)^{1/4} \frac{a_{\text{reh}}}{a_{\text{end}}}. \qquad (4.60)$$

In order to obtain Eq. (4.58) from that formula, one should express the Einstein frame scale factor in term of the energy density $\rho$. If the equation of state $w_{\text{reh}}$ is a constant, then $a \propto A^{(1-3w_{\text{reh}})/(3+3w_{\text{reh}})}a^{-1/(3+3w_{\text{reh}})}$. This is what has been used above and this led to Eqs. (4.58) and (4.59). But let us now assume that $w_{\text{reh}}$ is not a constant (notice that one always has $w = \bar{w}$ since, in the Einstein frame, the energy density and the pressure scale with the same power of the function $A$). Then, $\rho$ and $a$ are related by

$$\frac{d\rho}{\rho} = (1 - 3w_{\text{reh}}) \frac{dA}{A} - 3(1 + w_{\text{reh}}) \frac{da}{a}. \qquad (4.61)$$

If $A$ is a constant, one can always write [93]

$$\frac{a_{\text{reh}}}{a_{\text{end}}} = \left( \frac{\rho_{\text{reh}}}{\rho_{\text{end}}} \right)^{-1/(3+3\bar{w}_{\text{reh}})}, \qquad (4.62)$$

where $\bar{w}_{\text{reh}}$ is the mean equation of state during reheating, namely

$$\bar{w}_{\text{reh}} \equiv \frac{1}{N_{\text{reh}} - N_{\text{end}}} \int_{N_{\text{end}}}^{N_{\text{reh}}} w_{\text{reh}}(n)dn. \qquad (4.63)$$

If $A$ and $w_{\text{reh}}$, however, are not constant, it is no longer possible to express the final formula in terms of $\bar{w}_{\text{reh}}$. In particular, we do not obtain a term $A^{1-3\bar{w}_{\text{reh}}}$ as desired. Therefore, in what follows, we restrict our considerations to the case where the effective fluid dominating the matter content of the Universe has a constant equation of state.

Then, from Eqs. (4.57) and (4.59), at a given $\bar{R}_{\text{rad}}$, the remaining terms can be re-expressed in terms of quantities defined at Hubble radius crossing by using the Friedmann-Lemaître equations. In particular the energy density at the end of inflation in the Einstein frame reads

$$\frac{\rho_{\text{end}}}{M_{\text{Pl}}^4} = \frac{3H_*^2 V_{\text{end}}}{M_{\text{Pl}}^2 V_*} \frac{3 - \epsilon_{1*}}{3 - \epsilon_{1\text{end}}}, \qquad (4.64)$$
from which one obtains

$$ \Delta N_* = \ln \bar{R}_{\text{rad}} - \ln \left( \frac{k_* / a_0}{\bar{\rho}_\gamma^{1/4}} \right) + \frac{1}{4} \ln \left( \frac{H_*^2}{M_{\text{Pl}}^2 \epsilon_{1*}} \right) - \frac{1}{4} \ln \left( \frac{3}{\epsilon_{1*}} \frac{V_{\text{end}}}{V_*} \frac{3 - \epsilon_{1*}}{3 - \epsilon_{1\text{end}}} \right). \qquad (4.65) $$

In this last equation we have voluntarily made explicit the term in $H_*^2/(M_{\text{Pl}}^2 \epsilon_{1*}) = 8\pi^2 P_*$, the amplitude of the primordial power spectrum at the pivot scale, a well measured quantity. Of course, this equation resembles a lot Eq. (3.45) but one has to realize that it involves quantities defined both in the Einstein frame and in the Jordan frame. The term

$$ \frac{k_* / a_0}{\bar{\rho}_\gamma^{1/4}} = \frac{k_* / \bar{a}_0}{\bar{\rho}_\gamma^{1/4}} = e^{N_0}, \qquad (4.66) $$

and, therefore, its numerical value, is the standard one. The other quantities appearing in this equation are obtained using our standard procedures since they refer to the inflaton sector only.

However, the fact that $\ln \bar{R}_{\text{rad}}$, defined with energies in the Jordan frame, appears in Eq. (4.65), has various implications. For instance, the range of variation of $\Delta N_*$ in Eq. (4.65) is determined by putting limits on $\ln \bar{R}_{\text{rad}}$ coming from the fact that reheating must proceed between the end of inflation and BBN. This means that the physical value of the energy density at the end of reheating, that is to say $\bar{\rho}_{\text{reh}}$, must be such that $\bar{\rho}_{\text{nuc}} \equiv (10\text{MeV})^4 \le \bar{\rho}_{\text{reh}} \le \bar{\rho}_{\text{end}}$. We emphasize that physical limits must refer to quantities defined in the Jordan frame. The possible range for $\Delta N_*$ is $[\Delta N_*^{\text{nuc}}, \Delta N_*^{\text{end}}]$. The upper bound is obtained from the saturating value $\bar{\rho}_{\text{reh}} = \bar{\rho}_{\text{end}}$, which implies that $\ln \bar{R}_{\text{rad}} = 0$, and then

$$ \Delta N_*^{\text{end}} = -N_0 + \frac{1}{4} \ln (8\pi^2 P_*) - \frac{1}{4} \ln \left( \frac{3}{\epsilon_{1*}} \frac{V_{\text{end}}}{V_*} \frac{3 - \epsilon_{1*}}{3 - \epsilon_{1\text{end}}} \right). \qquad (4.67) $$

All the quantities in the above equation are calculated in the Einstein frame and are therefore unchanged compared to their standard value. The other limit is $\bar{\rho}_{\text{reh}} = \bar{\rho}_{\text{nuc}}$ and gives

$$ \Delta N_*^{\text{nuc}} = -N_0 + \frac{1}{4} \ln (8\pi^2 P_*) - \frac{1}{4} \ln \left( \frac{3}{\epsilon_{1*}} \frac{V_{\text{end}}}{V_*} \frac{3 - \epsilon_{1*}}{3 - \epsilon_{1\text{end}}} \right) + \frac{1 - 3w_{\text{reh}}}{12(1 + w_{\text{reh}})} \ln \left( \frac{\bar{\rho}_{\text{nuc}}}{M_{\text{Pl}}^4} \right) \\ - \frac{1 - 3w_{\text{reh}}}{12(1 + w_{\text{reh}})} \ln \left( \frac{\bar{\rho}_{\text{end}}}{M_{\text{Pl}}^4} \right). \qquad (4.68) $$

The quantity $\bar{\rho}_{\text{nuc}}$ is defined in the Jordan frame but its value is explicitly known, see above. On the other hand, we need to evaluate $\bar{\rho}_{\text{end}}$ since the Friedmann-Lemaître equations only determine $\rho_{\text{end}}$ by Eq. (4.64). By definition, we have

$$ \bar{\rho}_{\text{end}} = \frac{\rho_{\text{end}}}{A_{\text{end}}^4} = \Omega_{\text{end}}^4 \rho_{\text{end}}. \qquad (4.69) $$

Plugging this expression into Eq. (4.68) and making use of Eq. (4.64) one gets

$$ \Delta N_*^{\text{nuc}} = -N_0 + \frac{3w_{\text{reh}} + 1}{6(1 + w_{\text{reh}})} \ln (8\pi^2 P_*) - \frac{1}{3(1 + w_{\text{reh}})} \ln \left[ \frac{3}{\epsilon_{1*}^{(1+3w_{\text{reh}})/2}} \frac{V_{\text{end}}}{V_*} \frac{3 - \epsilon_{1*}}{3 - \epsilon_{1\text{end}}} \right] \\ + \frac{1 - 3w_{\text{reh}}}{12(1 + w_{\text{reh}})} \ln \left( \frac{\bar{\rho}_{\text{nuc}}}{M_{\text{Pl}}^4} \right) - \frac{1 - 3w_{\text{reh}}}{3(1 + w_{\text{reh}})} \ln |\Omega_{\text{end}}|. \qquad (4.70) $$

All terms but the last one are standard. The scalar-tensor effects appear in the term containing $\ln |\Omega_{\text{end}}|$. In most cases of interest, it is a very small correction which, for SI, amounts to

$$ \ln |\Omega_{\text{end}}| = \frac{1}{2} \ln |F(x_{\text{end}})| = \frac{x_{\text{end}}}{\sqrt{6}} = \frac{1}{2} \ln \left( 1 + \frac{2}{\sqrt{3}} \right) \approx 0.38. \qquad (4.71) $$

Even though it is a small effect, the scalar-tensor corrections on reheating are all included in the ASPIC library when the inflationary models are solved in the Einstein frame.

The reheating-consistent observational predictions of Starobinsky Inflation are represented in Fig. 125 where we have displayed their dependence on the reheating temperature.

## 4.2. Higgs Inflation (HI)

### 4.2.1. Non-minimal gravity

We start this section with some general considerations about non-minimal gravity or scalar-tensor theories. Non-minimal gravity plays an important role throughout this article for various reasons. Among them is the fact that the extra terms (compared to Einstein gravity) that characterize non-minimal gravity seem to be generated “automatically” by quantum corrections. From an effective field theory point of view, these models are therefore very well-motivated. Regarding inflation, as it will be discussed in details later on, non-minimal gravity can be used to “save” a model of inflation, that is to say a model can be ruled out when considered in the framework of Einstein gravity but compatible with the data when studied in a non-minimal setup. Several examples illustrating this claim will be studied in the following. Finally, in the inflationary context and contrary to what the name suggests, non-minimal gravity can be viewed as a framework which is as simple as Einstein gravity. Indeed, as we have already seen with Starobinsky inflation, non-minimal gravity alone has the same field content as Einstein gravity plus an additional scalar field. A simple inflationary model can thus be built only from the scalar-tensor action (of course, matter is needed when reheating is investigated but, again, the field content can be the same in both approaches). For all these reasons, inflationary scenarios based on scalar-tensor theories play an important role in the current efforts to understand the model building problem of inflation.
We have already discussed the $f(R)$ theory and how it is in fact equivalent either to the Brans–Dicke theory or to Einstein gravity plus a scalar field. Here, we discuss the same question by starting straightaway from a scalar-tensor theory, which can also be reduced to Einstein gravity plus a scalar field.

Let us consider the general action defining a scalar-tensor theory in a Jordan frame of metric $\tilde{g}_{\mu\nu}$. Here, in order to avoid any confusion, we will explicitly follow the conventions of Sections 4.1.1 and 4.1.4 and denote quantities in the Jordan frame with a “bar” and quantities in the Einstein frame with a “tilde”. Such an action reads

$$S(\tilde{g}^{\mu\nu}, \tilde{\phi}, \psi) = \frac{M_g^2}{2} \int d^4\mathbf{x} \sqrt{-\tilde{g}} [F(\tilde{\phi})\tilde{R} - Z(\tilde{\phi})\tilde{g}^{\mu\nu}\partial_{\mu}\tilde{\phi}\partial_{\nu}\tilde{\phi} - 2U(\tilde{\phi})] + \int d^4\mathbf{x} \mathcal{L}_{\text{mat}}(\psi, \tilde{g}_{\mu\nu}). \quad (4.72)$$

The gravity sector is characterized by three functions, $F(\tilde{\phi})$, $Z(\tilde{\phi})$ and $U(\tilde{\phi})$ and the mass scale $M_g$. Different representations can be used, for instance the Brans–Dicke representation where $F(\tilde{\phi}) = \tilde{\phi}$ and $Z(\tilde{\phi}) = \omega(\tilde{\phi})/\tilde{\phi}$ or the simple representation where, after having canonically renormalized the field $\tilde{\phi}$, one has $F(\tilde{\phi})$ arbitrary and $Z(\tilde{\phi}) = 1$. However, sometimes, this representation can be pathological and, in the most general situation, one has to keep the three functions.

Let us now consider the following conformal transformation, $\tilde{g}_{\mu\nu} = F(\tilde{\phi})\tilde{g}_{\mu\nu}$, where $\tilde{g}_{\mu\nu}$ is the metric tensor in the Einstein frame. Using Eq. (4.13), the action becomes

$$S(\tilde{g}^{\mu\nu}, \tilde{\phi}, \psi) = \int d^4\mathbf{x} \sqrt{-\tilde{g}} \left[ \frac{M_g^2}{2}\tilde{R} + \frac{M_g^2}{6}\tilde{g}^{\mu\nu}\tilde{\nabla}_\mu\tilde{\partial}_\nu(\ln F) - \frac{3M_g^2}{4F^2}\tilde{g}^{\mu\nu}\partial_\mu F\partial_\nu F \right. \\ \left. - \frac{M_g^2 Z(\tilde{\phi})}{2F(\tilde{\phi})}\tilde{g}^{\mu\nu}\partial_\mu\tilde{\phi}\tilde{\partial}_\nu\tilde{\phi} - M_g^2\frac{U(\tilde{\phi})}{F^2(\tilde{\phi})} \right] + \int d^4\mathbf{x} \mathcal{L}_{\text{mat}}[\psi, F^{-1}(\tilde{\phi})\tilde{g}_{\mu\nu}]. \quad (4.73)$$

The term which contains $\ln F$ is a total derivative and, therefore, can be discarded. If one introduces the field $\tilde{\phi}$ defined by the relation

$$\left(\frac{d\tilde{\phi}}{d\tilde{\phi}}\right)^2 = 2\left[\frac{3M_g^2}{4F^2(\tilde{\phi})}\left(\frac{dF}{d\tilde{\phi}}\right)^2 + \frac{M_g^2 Z(\tilde{\phi})}{2F(\tilde{\phi})}\right], \quad (4.74)$$

and if one defines a new potential $V = M_g^2 U/F^2$, then Eq. (4.72) takes the following form

$$S(\tilde{g}^{\mu\nu}, \tilde{\phi}, \psi) = \int d^4\mathbf{x} \sqrt{-\tilde{g}} \left[ \frac{M_g^2}{2}\tilde{R} - \frac{1}{2}\tilde{g}^{\mu\nu}\partial_\mu\tilde{\phi}\partial_\nu\tilde{\phi} - V(\tilde{\phi}) \right] + \int d^4\mathbf{x} \mathcal{L}_{\text{mat}}[\psi, F^{-1}(\tilde{\phi})\tilde{g}_{\mu\nu}]. \quad (4.75)$$

As announced before, one recognizes the action of Einstein gravity plus a scalar field with a minimal kinetic term. As it was the case for the $f(R)$ theory, one also notices that matter is no longer universally coupled to the metric tensor, as revealed by the term $F^{-1}(\tilde{\phi})\tilde{g}_{\mu\nu}$ in the matter action. As already mentioned, this has implications for reheating and these have been discussed in Section 4.1.4.

### 4.2.2. Theoretical justifications

Having briefly discussed non-minimal gravity, let us now apply it to Higgs inflation (the previous calculations will be useful in many other contexts throughout this article). This model has been proposed in Refs. [264–267] and postulates that the inflaton field is the Higgs field $\Sigma$ (discovered in 2012 at the Large Hadron Collider [268,269]) non-minimally coupled to gravity. Indeed, one can argue that, in curved spacetime, the simplest model compatible with our knowledge of particle physics is described by a Lagrangian which is the standard model Lagrangian plus an extra term of the form $\xi\Sigma^\dagger\Sigma R$. As already argued, this last term is natural since, in curved spacetime, it should be automatically generated by quantum corrections [270]. In the Jordan frame, the action of the model can be written as

$$S = \frac{M_g^2}{2} \int d^4\mathbf{x} \sqrt{-\tilde{g}} [F(h)\tilde{R} - Z(h)\tilde{g}^{\mu\nu}\partial_\mu h\partial_\nu h - 2U(h)], \quad (4.76)$$

where we have defined, in the unitary gauge,

$$\Sigma = \frac{M_g}{\sqrt{2}} \begin{pmatrix} 0 \\ h \end{pmatrix}, \quad (4.77)$$

and the quantity $M_g$ is a mass scale that, for the moment, is not identified with the Planck scale. As before, the tensor $\tilde{g}_{\mu\nu}$ denotes the metric in the Jordan frame. The three functions $F(h)$, $Z(h)$ and $U(h)$ completely characterize the model and are chosen to be

$$F(h) = 1 + \xi h^2, \quad Z(h) = 1, \quad U(h) = M_g^2 \frac{\lambda}{4} \left( h^2 - \frac{v^2}{M_g^2} \right)^2, \quad (4.78)$$

where $\xi$ is a new dimensionless parameter. The quantity $U(h)$ is the standard Higgs boson potential with $v \simeq 246$ GeV, the Higgs vacuum expectation value, and $\lambda \simeq 0.13$ the self-interacting coupling constant. Here, the field $h$ is dimensionless (as the functions $F$ and $Z$) while the potential $U$ is of dimension two. The effective gravitational constant (measured in Cavendish-type experiments) is affected by the new scalar degree of freedom and given by [248]

$$\frac{1}{M_{\text{Pl}}^2} = \frac{1}{M_g^2} \frac{1 + \xi h^2 + 8 \xi^2 h^2}{(1 + \xi h^2)(1 + \xi h^2 + 6 \xi^2 h^2)}. \quad (4.79)$$

Since, today, one has $h \simeq v/M_g \ll 1$, it follows that $M_g \simeq M_{\text{Pl}}$ at an accuracy far greater than the uncertainties associated with the measurements of the gravity coupling constant. As a result, from now on, we will take $M_{\text{Pl}}$ as the numerical value of $M_g$ in HI (see Section 6.28 for a model in which the equality is not always satisfied).

The above-described model can also be written in the Einstein frame where the corresponding slow-roll analysis is easier. For clarity, let us drop the “tilde” above Einstein frame quantities and denote in the following the metric tensor in this frame by $g_{\mu\nu}$. The action now takes the form

$$S = M_g^2 \int d^4\mathbf{x} \sqrt{-g} \left[ \frac{R}{2} - \frac{1}{2}g^{\mu\nu}\partial_\mu\chi\partial_\nu\chi - W(\chi) \right], \quad (4.80)$$
where the fields h and χ are related by

$$ \frac{d\chi}{dh} = \frac{\sqrt{1 + \xi(1 + 6\xi)h^2}}{1 + \xi h^2}, \qquad (4.81) $$

and the potential is given $V \equiv M_g^2 W = M_g^2 U/F^2$. Notice also that the canonically normalized field in the Einstein frame is simply given by $\phi \equiv M_g\chi$. It is also important to recall that, in the Einstein frame, matter is now explicitly coupled to the scalar field $\phi$. The consequences for reheating are discussed in Section 4.1.4 and Refs. [271–273]. The differential obtained in Eq. (4.81) can be integrated exactly and the result reads

$$ \chi = \sqrt{\frac{1 + 6\xi}{\xi}} \operatorname{arcsinh} \left[ h \sqrt{\xi(1 + 6\xi)} \right] - \sqrt{6} \operatorname{arctanh} \left[ \frac{\xi \sqrt{6}h}{\sqrt{1 + \xi(1 + 6\xi)h^2}} \right]. \qquad (4.82) $$

The inverse hyperbolic tangent is always well-defined since its argument is always smaller than one. This exact formula between the Einstein and Jordan frame fields was also derived in Ref. [271]. Using the identities

$$ \operatorname{arcsinh}(x) = \ln(x + \sqrt{1 + x^2}), \qquad \operatorname{arctanh}(x) = \frac{1}{2} \ln\left(\frac{1+x}{1-x}\right), \qquad (4.83) $$

and defining

$$ \bar{h} \equiv \sqrt{\xi} h, \qquad (4.84) $$

Eq. (4.82) can be further simplified as

$$ \chi = \sqrt{6 + \frac{1}{\xi}} \ln \left[ \sqrt{1 + (1 + 6\xi)\bar{h}^2} + \sqrt{(1 + 6\xi)\bar{h}^2} \right] + \sqrt{6} \ln \left[ \frac{\sqrt{1 + \bar{h}^2}}{\sqrt{1 + (1 + 6\xi)\bar{h}^2} + \sqrt{6\xi\bar{h}^2}} \right]. \qquad (4.85) $$

Higgs Inflation is usually considered in the large coupling limit $\xi \gg 1$ from which one gets

$$ \chi \approx \sqrt{6 \ln(2\bar{h}\sqrt{6\xi}) + \sqrt{6} \ln\left(\frac{\sqrt{1+\bar{h}^2}}{2\bar{h}\sqrt{6\xi}}\right)} = \sqrt{6} \ln\left(\sqrt{1+\bar{h}^2}\right) = \sqrt{\frac{3}{2}} \ln(1 + \xi h^2). \qquad (4.86) $$

The same expression can also be directly derived from Eq. (4.81) which, for $\xi \gg 1$, can be approximated as

$$ \frac{d\chi}{dh} \approx \frac{\sqrt{6\xi h}}{1 + \xi h^2}. \qquad (4.87) $$

The solution to this equation is exactly Eq. (4.86). The last step consists in inserting the approximate expression of h in terms of $\chi$ (and, therefore, in terms of $\phi$) into the definition of the potential V in the Einstein frame. This leads to the following expression

$$ V(\phi) \approx \frac{M_g^4 \lambda}{4\xi^2} \left(1 - e^{-\sqrt{2/3}\phi/M_g}\right)^2, \qquad (4.88) $$

i.e. one obtains the same potential as in Starobinsky Inflation, see Eq. (4.21). Interestingly enough, the parameters $\xi$ and $\lambda$ enter the approximate potential only through its overall amplitude. In the following, we define

$$ M_g^4 \equiv \frac{M_g^4 \lambda}{4\xi^2} \approx \frac{M_{\mathrm{Pl}}^4 \lambda}{4\xi^2}. \qquad (4.89) $$

In this sense, Higgs inflation is, as for Starobinsky inflation, a “zero-parameter model” since the scale M, and the parameter $\xi$, are entirely determined by the amplitude of the CMB anisotropies.

Let us stress, however, that the above potential is only approximate for Higgs inflation whereas Eq. (4.21) is exact for Starobinsky inflation. The two models match at leading order in $1/\xi$ only but are not strictly identical. In the slow-roll analysis below, we will be providing both a next-to-leading order and parametric exact treatment of Higgs inflation to quantity by how much the observable quantities between the two scenarios can differ.

Finally, let us also notice that other approaches based on superconformal D-term inflation also lead to the same potential [258]. Various multifield extensions have also been studied in which the inflationary phase can still be described by the one-field Higgs potential [259–261].

### 4.2.3. Next-to-leading order slow-roll analysis

As explained above, the leading-order potential of Higgs Inflation, Eq. (4.88), is the same as the one in Starobinsky Inflation, see Eq. (4.21), for which the slow-roll analysis was already performed in Section 4.1.3. Therefore, the results derived in Section 4.1.3 also apply to Higgs inflation at leading order.

There are however two differences between these models that we now further discuss. In Higgs inflation, Eq. (4.88) is only an approximation to the full potential, since the limit $\xi \gg 1$ was taken when inverting Eq. (4.82). In general, Higgs Inflation thus depends on two additional parameters, namely $\xi$ (which otherwise only appears in the overall normalization of the potential) and $\nu$ (the current vacuum expectation value of the Higgs field). It is displayed in Fig. 7 for $\xi = 1$ and $\nu = 246$ GeV, and compared with the approximation derived in Eq. (4.88). As we will see below, the value $\xi = 1$ is unrealistically small, but it allows one to distinguish between the two curves, otherwise the difference would not be visible by eye. One can check that, at large-field values, Eq. (4.88) indeed provides a good approximation to the full potential.

In order to better assess the reliability of this approximation, let us carry it out at next-to-leading order. Expanding Eq. (4.82) at order $\xi^{-1}$, thus at next-to-leading order compared to Eq. (4.88), one obtains for the inflationary potential

$$ V = \frac{\lambda M_g^4}{4\xi^2} \left(1 - e^{-\sqrt{2/3}x}\right)^2, \quad \text{with} \quad x = \frac{\phi/M_g}{1 + 1/(12\xi)} - \sqrt{\frac{3}{2}} \frac{1 + \ln(24\xi)}{1 + 12\xi}. \qquad (4.90) $$
<figure><img src="image_10.png" /><figcaption>Fig. 7. Higgs Inflation (HI). Left panel: full Higgs potential for $\xi = 1$ and $\nu = 246$ GeV (blue line) compared with a leading-order expansion in $\xi$ [green line, corresponding to Eq. (4.88)] and to a next-to-leading expansion in $\xi$ [red line, corresponding to Eq. (4.90)]. The value $\xi = 1$ is unrealistically small but it has been chosen to make the difference between the three curves visible. Right panel: first Hubble-flow parameter in the same situation as in the left panel.</figcaption></figure>

When $\xi \gg 1$, one has $x \simeq \phi/M_g$ so Eq. (4.88) is recovered. Let us note that we have neglected terms of order $\nu/M_g \simeq 10^{-16}$ when deriving this expression. It is displayed with the red solid line in Fig. 7 and one can see that already with $\xi = 1$, it provides an excellent approximation to the full potential in the inflating region. In the right panel of Fig. 7, the first Hubble-flow parameter is also displayed. At next-to-leading order, from Eq. (4.90), it is given by

$$ \epsilon_1 = \frac{4}{3} \left[ \frac{1 - e^{\sqrt{2/3}x}}{1 + 1/(12\xi)} \right]^2, \qquad (4.91) $$

where we recall that $x$ is defined in Eq. (4.90). One can see in Fig. 7 that, already with $\xi = 1$, this again provides an excellent approximation to the full first Hubble-flow parameter. This can be used to better estimate the error made when using Eq. (4.48) to compute the first Hubble-flow parameters. In the large-field regime where $\phi \gg M_g$, one indeed has

$$ \frac{\epsilon_1^{\text{nllo}}}{\epsilon_1^{\text{lo}}} \approx \frac{1}{(1 + 12\xi)^2} \exp \left\{ \frac{2}{1 + 12\xi} \left[ 1 + \ln(24\xi) + \sqrt{\frac{2}{3}} \frac{\phi}{M_g} \right] \right\}, \qquad (4.92) $$

where $\epsilon_1^{\text{lo}}$ stands for Eq. (4.48), and $\epsilon_1^{\text{nllo}}$ for Eq. (4.92). For this ratio to remain close to unity, hence for the relative error on the slow-roll parameters to remain small, one not only needs to impose $\xi \gg 1$ but also

$$ \xi \gg \frac{\phi}{M_g}. \qquad (4.93) $$

In particular, there is always a region far away in the plateau where the relative precision of the leading-order expressions breaks down. When $\xi$ is sufficiently large, this region is however removed out of the observable phase of the inflationary dynamics, and the leading-order expressions derived in Section 4.1.3 can be safely employed.

The reheating-consistent observational predictions of Higgs inflation are represented in Fig. 125 (there are almost the same as for Starobinsky Inflation) where we have displayed their dependence on the reheating energy defined in the Jordan frame by $T_{\text{reh}} = \rho_{\text{reh}}^{1/4}$. Notice that, a priori, the reheating temperature can be calculated exactly in Higgs inflation since all the couplings between the Higgs and the other fields in the standard model are known [271]. This gives a spectral index which is in good agreement with the data and a small contribution of gravitational waves. At this stage, in the Higgs case, the constraints on the parameter $\xi$ come from the amplitude of the CMB anisotropies, i.e. from Eq. (4.55). As explained below Eq. (4.55), for the fiducial value $\Delta N_*=55$, one gets $M \simeq 3.3 \times 10^{-3} M_g$, i.e., inflation takes place at the GUT scale. Then, using the expression of $M$, one obtains the following condition for the parameter $\xi$,

$$ \xi_* \approx 46000 \sqrt{\lambda}. \qquad (4.94) $$

The value of $\xi_*$ matching the amplitude of the CMB anisotropies thus depends on the self-interacting coupling constant $\lambda$ and, for $\lambda \simeq 0.13$, it satisfies Eq. (4.93) across the field range that is of observational relevance. These considerations are in agreement with the conclusions obtained in Refs. [264–266]. Notice that such a large value for the coupling constant $\xi$ is sometimes considered as problematic [274].

If we now consider the supergravity realization of the model described in Section 4.1.2, one obtains a constraint on the parameter $\hat{\lambda}$, hence, if one takes $c = 1$, one obtains a constraint on $\mu$ and $\lambda$, see Ref. [257].

### 4.2.4. Exact slow-roll and reheating analysis

We give in this section the exact slow-roll analysis of Higgs inflation as it is coded in the ASPIC library. Such a treatment is required for other non-minimal gravity models, such as Non-Minimal Large Field Inflation discussed in Section 6.28 and is analogous to models in which a non-canonical Kähler metric prevents the kinetic term of the inflaton to be explicitly normalized, as in Dual Inflation presented in Section 5.24.

Because Eq. (4.85) cannot be inverted to obtain an explicit potential $V(\phi)$, the analysis is parametric and uses as a proxy the dimensionless field $\bar{h} = \sqrt{\xi}h$. From Eq. (4.78), the parametric potential of Higgs inflation reads

$$ V(\bar{h}) = M^4 \left( \frac{\bar{h}^2 - \bar{v}^2}{1 + \bar{h}^2} \right)^2, \qquad (4.95) $$
where M is still defined by Eq. (4.89) and we have introduced the rescaled and dimensionless vacuum expectation value

$$ \bar{v} \equiv \sqrt{\xi} \frac{v}{M_g}. \qquad (4.96) $$

The canonically normalized field $\phi(\bar{h}) = M_g\chi(\bar{h})$ is explicit and given by Eq. (4.85).

The first Hubble flow function in the slow-roll approximation is given by

$$ \epsilon_1 = \frac{1}{2} \left( \frac{d \ln V}{d\chi} \right)^2 = \frac{\left( \frac{d \ln V}{d\bar{h}} \right)^2}{2 \left( \frac{d\chi}{d\bar{h}} \right)^2}. \qquad (4.97) $$

Using Eqs. (4.81) and (4.95) gives the explicit parametric expression

$$ \epsilon_1(\bar{h}) = \frac{8\xi\bar{h}^2(1 + \bar{v}^2)^2}{(\bar{h}^2 - \bar{v}^2)^2(1 + \bar{h}^2 + 6\xi\bar{h}^2)}. \qquad (4.98) $$

One can also obtain explicit expressions for the other Hubble-flow functions. The second Hubble-flow function reads

$$ \epsilon_2(\bar{h}) = \frac{8\xi(1 + \bar{h}^2)(1 + \bar{v}^2)[\bar{h}^2 + \bar{v}^2 + 2(1 + 6\xi)\bar{h}^4]}{(\bar{h}^2 - \bar{v}^2)^2[1 + (1 + 6\xi)\bar{h}^2]^2}, \qquad (4.99) $$

while the third one is given by

$$ \epsilon_3(\bar{h}) = \frac{8\xi\bar{h}^2(1 + \bar{v}^2)}{(\bar{h}^2 - \bar{v}^2)^2 [1 + (1 + 6\xi)\bar{h}^2]^2 [\bar{h}^2 + \bar{v}^2 + 2(1 + 6\xi)\bar{h}^4]} \times \left\{ \begin{aligned} & 3\bar{v}^2 - (1 + 12\xi)\bar{v}^4 + [1 + 2(5 + 21\xi)\bar{v}^2 - (1 + 6\xi)\bar{v}^4]\bar{h}^2 \\ & + 3(1 + 6\xi)(1 + 3\bar{v}^2)\bar{h}^4 + 2(1 + 6\xi)^2(2 + \bar{v}^2)\bar{h}^6 + 2(1 + 6\xi)^2\bar{h}^8 \end{aligned} \right\}. \qquad (4.100) $$

These expressions make explicit that the observable quantities of Higgs inflation, such as the spectral index and tensor-to-scalar ratio, depend on $\xi$. On the contrary, in Starobinsky Inflation, the same quantities do not depend on the new energy scale $\mu$. In other words, at same potential normalization and same reheating history, Higgs Inflation and Starobinsky Inflation predict a slightly different spectral index and tensor-to-scalar ratio. As detailed in the previous section, the differences in the observable range do not exceed $\mathcal{O}(1/\xi) \simeq 10^{-4}$.

The parametric field value $\bar{h}_{\text{end}}$ at which Higgs Inflation ends can also be determined analytically by solving $\epsilon_1(\bar{h}) = 1$. This is a cubic polynomial equation in $\bar{h}^2$ admitting a real root in the relevant domain $(\bar{h} > \bar{v})$

$$ \bar{h}_{\text{end}}^2 = \frac{1}{12(1 + 6\xi)} \left\{ \begin{aligned} & -4 + 8\bar{v}^2(1 + 6\xi) - 2i(i + \sqrt{3})[P(\xi, \bar{v})]^{1/3} - 2i(i - \sqrt{3})[P(\xi, \bar{v})]^{-1/3} \\ & \times [(1 + 12\xi)^2 + 2\bar{v}^2(1 + 6\xi)(1 + 24\xi)\bar{v}^4(1 + 6\xi)(1 + 30\xi)] \end{aligned} \right\}, \qquad (4.101) $$

where

$$ \begin{aligned} P(\xi, \bar{v}) = & 1 + 3\bar{v}^2 + 3\bar{v}^4 + \bar{v}^6 + 36\xi + 18\xi\bar{v}^2 - 72\xi\bar{v}^4 - 54\xi\bar{v}^6 + 216\xi^2 - 432\xi^2\bar{v}^2 - 1404\bar{v}^4\xi^2 \\ & - 756\xi^2\bar{v}^6 - 2592\xi^3\bar{v}^2 - 5184\xi^3\bar{v}^4 - 2376\xi^3\bar{v}^6 + 6\sqrt{6\xi}(1 + \bar{v}^2)(1 + 6\xi) \\ & \times [-\bar{v}^2(1 + \bar{v}^2)^3 - 2\xi(1 + \bar{v}^2)^2(1 + 20\bar{v}^2 + \bar{v}^4) + 4\xi^2(1 + \bar{v}^2) \\ & \times (-16 - 108\bar{v}^2 - 60\bar{v}^4 + 5\bar{v}^6) - 24\xi^3(4 + 8\bar{v}^2 + \bar{v}^4)^2]^{1/2} \end{aligned} \qquad (4.102) $$

The parametric slow-roll trajectory can be integrated analytically. Expressing Eq. (3.10) in terms of $\chi$ one gets

$$ \frac{d\chi}{dN} \approx -\frac{d\ln V}{d\chi} = -\frac{d\bar{h}}{d\chi} \frac{d\ln V}{d\bar{h}}, \qquad (4.103) $$

which can be used to have an explicit expression for

$$ \frac{d\bar{h}}{dN} = \frac{d\bar{h}}{d\chi} \frac{d\chi}{dN} \approx -\frac{1}{\left(\frac{d\chi}{d\bar{h}}\right)^2} \frac{d\ln V}{d\bar{h}} = -\frac{4\xi(1 + \bar{v}^2)\bar{h}(1 + \bar{h}^2)}{(\bar{h}^2 - \bar{v}^2)[1 + (1 + 6\xi)\bar{h}^2]}, \qquad (4.104) $$

where “$\simeq$” refers to the use of the slow-roll approximation when replacing $d\chi/dN \simeq -d\ln V/d\chi$. This equation can be analytically integrated to get the slow-roll trajectory as

$$ N_{\text{end}} - N = \frac{1}{8\xi(1 + \bar{v}^2)} \left[ (1 + 6\xi)(\bar{h}^2 - \bar{h}_{\text{end}}^2) - \bar{v}^2 \ln\left(\frac{\bar{h}^2}{\bar{h}_{\text{end}}^2}\right) - 6\xi(1 + \bar{v}^2) \ln\left(\frac{1 + \bar{h}^2}{1 + \bar{h}_{\text{end}}^2}\right) \right], \qquad (4.105) $$

where $\bar{h}_{\text{end}}$ is given in Eq. (4.101).

In order to determine the parametric field value $\bar{h}_*$ at which the pivot scale crosses the Hubble radius during inflation, one has to solve the reheating equation, in the Einstein frame, as we are in presence of a scalar-tensor inflaton. However, without making the approximations discussed in the previous sections, the slow-roll parameters, and the potential, depend on $\xi$. Its actual value, say $\xi_*$, being obtained from the amplitude of the CMB anisotropies, one ends up having a system of two coupled non-linear algebraic equations, the solution of which giving both $\bar{h}_*$ and $\xi_*$. The equation fixing the normalization of the potential is given by Eq. (3.32) and reads

$$ \frac{\lambda}{4\xi_*^2} = 24\pi^2 P_* \frac{\epsilon_1(\bar{h}_*, \xi_*)}{V(\bar{h}_*, \xi_*)/M^4}, \qquad (4.106) $$
<figure><img src="image_11.png" /><figcaption>Fig. 8. Top left panel: the solid blue line represents the radiatively corrected Higgs potential, see Eq. (5.11), with $A_1 = 5$. It is compared to the tree level potential given by Eq. (4.88) (dashed green line) and to Eq. (5.11) with $A_1 = 0$ (solid red line) which is supposed to be a good approximation of the tree level potential. It is obvious that this is indeed the case in the regime of interest, where the vev of the Higgs field is not too small. Top right panel: logarithm of potential, the three lines and the color code having the same meaning as in the top left panel. Bottom left panel: slow-roll parameter $\epsilon_1$ as a function of the field $\phi$, still with the same convention. As can be seen in this plot, even in presence of radiative corrections, the end of inflation occurs by violation of the slow-roll condition. Bottom right panel: slow-roll parameters $\epsilon_2$ (solid blue line) and $\epsilon_3$ (dashed blue line) for $A_1 = 5$ compared to their tree level counterparts (solid and dashed green lines, respectively).</figcaption></figure>

where $V(\tilde{h}_*, \xi_*)$ is given in Eq. (4.95) and $\epsilon_1(\tilde{h}_*, \xi_*)$ in Eq. (4.98). The reheating equation in the Einstein frame has been derived in Eq. (4.65) and reads

$$ \Delta N_{*} = \ln \bar{R}_{\text{rad}} - N_{0} + \frac{1}{4} \ln (8\pi^{2} P_{*}) - \frac{1}{4} \ln \left( \frac{3}{\epsilon_{1*}} \frac{V_{\text{end}}}{V_{*}} \frac{3 - \epsilon_{1*}}{3 - \epsilon_{\text{end}}} \right), \quad (4.107) $$

where it is understood that $V_{\text{end}} = V(\tilde{h}_{\text{end}}, \xi_*)$, $V_*=V(\tilde{h}_*, \xi_*)$, $\epsilon_{\text{end}} = 1$ and

$$ \Delta N_{*}(\tilde{h}_{*}, \xi_{*}) = N_{\text{end}} - N_{*}, \quad (4.108) $$

which is given in Eq. (4.105). Let us notice that from Eq. (4.101) one has $\tilde{h}_{\text{end}}(\xi_*)$ and that $\bar{\nu}$ is also an explicit function of $\xi_*$ given by Eq. (4.96). For $\nu \neq 0$, there is no analytical solution to the algebraic system made of Eqs. (4.106) and (4.107) and it is solved numerically in the ASPIC code. We do not plot the numerical solutions as they would be indistinguishable from the next-to-leading order analysis presented in Section 4.2.3.

The reheating-consistent observational predictions of Higgs Inflation are represented in Fig. 125 where we have displayed their dependence on the reheating temperature.

# 5. One parameter models

## 5.1. Radiatively Corrected Higgs Inflation (RCHI)

### 5.1.1. Theoretical justifications

Let us consider again the model given by Eq. (4.76). The three functions describing this action are modified when quantum corrections are taken into account. As a consequence, the potential which supports inflation is also modified and this leads to a new inflationary scenario that we call Radiatively Corrected Higgs Inflation (RCHI). This scenario has been studied in Refs. [275–280]. At first order, $M_g \simeq M_{\text{Pl}}$, the corrections to the function $Z(h)$ can be neglected while the corrections to $F(h)$ and to $U(h)$ read

$$ F(h) = 1 + \xi h^2 + \frac{C}{16\pi^2} h^2 \ln \left( \frac{M_{\text{Pl}}^2 h^2}{\mu^2} \right), \quad (5.1) $$

$$ U(h) = M_{\text{Pl}}^2 \frac{\lambda}{4} \left( h^2 - \frac{\nu^2}{M_{\text{Pl}}^2} \right)^2 + \frac{\lambda A}{128\pi^2} M_{\text{Pl}}^2 h^4 \ln \left( \frac{M_{\text{Pl}}^2 h^2}{\mu^2} \right), \quad (5.2) $$
where $\mu$ is the renormalization scale and A and C are two new constants given by

$$A = \frac{3}{8\lambda} [2g^4 + (g^2 + g'^2) - 16y_t^4] + 6\lambda + \mathcal{O}(\xi^{-2}), \qquad (5.3)$$

$$C = 3\xi\lambda + \mathcal{O}(\xi^0), \qquad (5.4)$$

$y_t$ being the Yukawa coupling of the top quark and g and g' the coupling constants of the SU(2)$_L$ and U(1)$_Y$ groups. The presence of quantum corrections modifies the relation between the Jordan and the Einstein frames and changes the shape of the potential in the Einstein frame. Assuming the smallness of $A/(32\pi^2) \ll 1$ and $C/(8\pi^2\xi) \ll 1$, which is necessary for the consistency of the one-loop calculation (the second condition is in fact equivalent to $C\lambda/(8\pi^2) \ll 1$ because C is proportional to $\xi$), one obtains the following expression

$$V \approx \frac{M_{Pl}^4 \lambda}{4\xi^2} \frac{\xi^2 h^4}{(1 + \xi h^2)^2} \left[ 1 - \frac{\xi h^2}{1 + \xi h^2} \frac{C}{8\pi^2 \xi} \ln \left( \frac{M_{Pl}^2 h^2}{\mu^2} \right) + \frac{A}{32\pi^2} \ln \left( \frac{M_{Pl}^2 h^2}{\mu^2} \right) \right]. \qquad (5.5)$$

Of course, if $A = C = 0$, one checks that this potential reduces to the potential of the previous section. Notice that, at this stage, we have not assumed that $\xi h^2 \gg 1$. If we further postulate that $\xi h^2 \gg 1$ and approximate $\xi^2 h^4 / (1 + \xi h^2)^2 \simeq 1 - 2/(\xi h^2)$, then the above formula reduces to

$$V \approx \frac{M_{Pl}^4 \lambda}{4\xi^2} \left[ 1 - \frac{2}{\xi h^2} + \frac{A_1}{16\pi^2} \ln \left( \frac{M_{Pl}h}{\mu} \right) \right], \qquad (5.6)$$

where $A_1 \equiv A - 12\lambda$ is the inflationary anomalous scaling. This formula coincides with Eq. (6) of Ref. [277] and Eq. (9) of Ref. [279]. Although the above formulas give $V$ in the Einstein frame, it is still expressed in term of $h$. The expression for the field in the Einstein frame, $\chi$, remains to be established. Assuming the smallness of the loop corrections, we obtain

$$\frac{d\chi}{dh} \approx \frac{\sqrt{3}h\xi}{(1 + \xi h^2)} \left[ 1 + \frac{C}{16\pi^2\xi} + \frac{C}{8\pi^2\xi} \frac{1}{1 + \xi h^2} \ln \left( \frac{M_{Pl}h}{\mu} \right) \right]. \qquad (5.7)$$

Notice that, in order to obtain this equation, we have neglected a term proportional to $1/(\xi h)^2 \ll 1$. Contrary to the assumption $\xi h^2 \gg 1$, the condition $(\xi h)^2 \gg 1$ was also used in Section 4.2. Then, the integration of this differential equation leads to

$$\chi \approx \frac{\sqrt{3}}{2} \ln(1 + \xi h^2) + \frac{\sqrt{3}C}{16\pi^2\xi} \left[ \ln h - \frac{1}{1 + \xi h^2} \ln \left( \frac{M_{Pl}h}{\mu} \right) \right]. \qquad (5.8)$$

Using only now the limit $\xi h^2 \gg 1$, this expression reduces to

$$\chi \approx \frac{\sqrt{3}}{2} \ln(\xi h^2) + \frac{\sqrt{3}C}{16\pi^2\xi} \ln h. \qquad (5.9)$$

As expected the relation between the Jordan frame field $h$ and the Einstein frame field $\chi$ is modified by the quantum corrections. Inverting the above formula gives

$$\xi^{1/2} h \approx e^{\chi/\sqrt{3}} - \frac{C}{16\pi^2\xi} e^{\chi/\sqrt{3}} \left( \frac{\chi}{\sqrt{3}} - \frac{1}{2} \ln \xi \right). \qquad (5.10)$$

This equation allows us to find the expression of the potential in the Einstein frame. Inserting Eq. (5.10) into Eq. (5.6) and introducing the canonically normalized field $\phi \equiv \sqrt{2}M_{Pl}\chi$, one obtains

$$V(\phi) \approx \frac{M_{Pl}^4 \lambda}{4\xi^2} \left[ 1 - 2e^{-2\phi/(\sqrt{6}M_{Pl})} - \frac{C}{4\pi^2\xi} e^{-2\phi/(\sqrt{6}M_{Pl})} \left( \frac{\phi}{\sqrt{6}M_{Pl}} - \frac{1}{2} \ln \xi \right) \right. \\ \left. + \frac{A_1}{16\pi^2} \ln \left( \frac{M_{Pl}}{\mu\sqrt{\xi}} \right) + \frac{A_1}{16\pi^2} \frac{\phi}{\sqrt{6}M_{Pl}} \right] \\ \approx \frac{M_{Pl}^4 \lambda}{4\xi^2} \left[ 1 - 2e^{-2\phi/(\sqrt{6}M_{Pl})} + \frac{A_1}{16\pi^2} \frac{\phi}{\sqrt{6}M_{Pl}} \right]. \qquad (5.11)$$

We see that we now deal with a “one parameter model”, $A_1$, since the mass scale $M^4 \equiv M_{Pl}^4 \lambda/(4\xi^2)$ is determined by the COBE normalization. In the case $A_1 = 0$, it is also interesting to compare the above potential with the one given by Eq. (4.88). We see that this corresponds to assuming that the exponential $e^{-2\phi/(\sqrt{6}M_{Pl})} \ll 1$ (or, equivalently, $\phi/M_{Pl} \gg 1$) and to expand the corresponding expression at first order in this small parameter. This leads to the following formula: $V \approx M^4 [1 - 2e^{-2\phi/(\sqrt{6}M_{Pl})}]$, i.e. exactly Eq. (5.11) for $A_1 = 0$. It is worth remarking that this approximation is not very good towards the end of inflation. Indeed, it is easy to show that (see below), for the potential (5.11) with $A_1 = 0$, $\phi_{end}/M_{Pl} = \sqrt{3/2} \ln(2 + 2/\sqrt{3}) \simeq 1.4$ which should be compared with Eq. (4.49) for the potential (4.88) according to which $\phi_{end}/M_{Pl} \simeq 0.94$.

### 5.1.2. Slow-roll analysis

Given Eq. (5.11), namely

$$V(\phi) = M^4 \left[ 1 - 2e^{-2\phi/(\sqrt{6}M_{Pl})} + \frac{A_1}{16\pi^2} \frac{\phi}{\sqrt{6}M_{Pl}} \right], \qquad (5.12)$$

we can now proceed to the slow-roll analysis. The potential (5.12) is represented and compared with its tree level counterpart in Fig. 8. Defining $x \equiv \phi/M_{Pl}$, the three first slow-roll parameters can be written as

$$\epsilon_1 = \frac{1}{12} \left[ \frac{4e^{-\sqrt{2/3}x} + A_1/(16\pi^2)}{1 - 2e^{-\sqrt{2/3}x} + A_1/(32\pi^2)\sqrt{2/3}x} \right]^2, \qquad (5.13)$$
$$ \epsilon_2 = \frac{8e^{-\sqrt{2/3}x} \left[ 1 + A_1/(16\pi^2) + A_1/(32\pi^2)\sqrt{2/3}x \right] + A_1^2/(256\pi^4)}{\left[ 1 - 2e^{-\sqrt{2/3}x} + A_1/(32\pi^2)\sqrt{2/3}x \right]^2}, \qquad (5.14) $$

and

$$ \epsilon_3 = 12 \left( 4 + \frac{A_1}{16\pi^2} e^{\sqrt{2/3}x} \right) \left\{ 48 + 8 \frac{A_1}{16\pi^2} \left( 9 + \sqrt{6x} \right) + 3 \frac{A_1^3}{4096\pi^6} e^{2\sqrt{2/3}x} \right. \\ \left. + 2e^{\sqrt{2/3}x} \left[ 12 + 18 \frac{A_1}{16\pi^2} \left( 1 + \frac{A_1}{16\pi^2} \right) + \sqrt{6} \frac{A_1}{16\pi^2} \left( 4 + 3 \frac{A_1}{16\pi^2} \right) x + 2 \frac{A_1^2}{256\pi^4} x^2 \right] \right\} \\ \times \left[ 24 + \frac{A_1}{16\pi^2} \left( 24 + 4\sqrt{6x} + 3 \frac{A_1}{16\pi^2} e^{\sqrt{2/3}x} \right) \right]^{-1} \left[ -12 + e^{\sqrt{2/3}x} \left( 6 + \sqrt{6} \frac{A_1}{16\pi^2} x \right) \right]^{-2}. \qquad (5.15) $$

These three slow-roll parameters are represented in Fig. 8 (bottom panels). It is interesting to compare these formulas with the expressions derived in Ref. [275] [see Eqs. (22) and (23) of that paper]. An approximate equation for the first slow-roll parameter is obtained by neglecting the second and third terms in the denominator of Eq. (5.13), which, as a matter of fact, consists in writing $V(\phi) \simeq M^4$. Then, it follows that

$$ \epsilon_1 \approx \frac{4}{3} e^{-2\sqrt{2/3}x} \left(1 + \frac{A_1}{64\pi^2} e^{\sqrt{2/3}x}\right)^2 \approx \frac{4}{3} \frac{1}{\xi^2 h^4} \left(1 + \frac{h^2}{h_1^2}\right)^2, \qquad (5.16) $$

where we have defined $h_1^2 \equiv 64\pi^2/(\xi A_1)$ in agreement with Ref. [275]. The same approximation is made for the second slow-roll parameter (except that Ref. [275] calculates $\hat{\eta} \equiv M_\phi^2 V_{\phi\phi}/V$ rather than $\epsilon_2$). The second field derivative of the potential can be written as $V_{\phi\phi} = -4M^4e^{-\sqrt{2/3}x}/(3M_{\phi}^2)$ and, therefore, if one considers that $V(\phi) \simeq M^4$, then $\hat{\eta} \simeq -4/(3\xi h^2)$. We conclude that our expressions of $\epsilon_1$ and $\epsilon_2$ reproduce Eqs. (22) and (23) of Ref. [275] in the limit where $V(\phi) \simeq M^4$.

Let us now study how inflation ends in this model. From Fig. 8, it is clear that this occurs by violation of the slow-roll conditions. Working out the condition $\epsilon_1 = 1$, it follows that

$$ x_{\text{end}} = \frac{1}{\sqrt{2}} - \sqrt{\frac{3}{2}} \frac{32\pi^2}{A_1} + \sqrt{\frac{3}{2}} W_{-1} \left[ \frac{64\pi^2}{A_1} \left( 1 + \frac{1}{\sqrt{3}} \right) e^{32\pi^2/A_1 - 1/\sqrt{3}} \right], \qquad (5.17) $$

where, if $A_1 > 0$, $W_0 = W_0$ while, if $A_1 < 0$, $W_0 = W_1$.

We now turn to the slow-roll trajectory. It can be integrated exactly and straightforward manipulations lead to the following expression

$$ N - N_{\text{ini}} = \sqrt{\frac{3}{2}} x - \frac{48\pi^2}{A_1} \left[ 1 + \frac{A_1}{32\pi^2} \left( 1 + \sqrt{\frac{2}{3}} x \right) \right] \ln \left( 1 + \frac{A_1}{64\pi^2} e^{\sqrt{2/3}x} \right) \\ - \frac{3}{2} \text{Li}_2 \left( -\frac{A_1}{64\pi^2} e^{\sqrt{2/3}x} \right) - \sqrt{\frac{3}{2}} x_{\text{ini}} + \frac{48\pi^2}{A_1} \left[ 1 + \frac{A_1}{32\pi^2} \left( 1 + \sqrt{\frac{2}{3}} x_{\text{ini}} \right) \right] \\ \times \ln \left( 1 + \frac{A_1}{64\pi^2} e^{\sqrt{2/3}x_{\text{ini}}} \right) + \frac{3}{2} \text{Li}_2 \left( -\frac{A_1}{64\pi^2} e^{\sqrt{2/3}x_{\text{ini}}} \right), \qquad (5.18) $$

where $\text{Li}_2$ denotes the dilogarithm function [281,282]. Let us also notice that if we use the approximation $V(\phi) \simeq M^4$ already discussed before, then one can obtain a much simpler formula, namely

$$ N - N_{\text{ini}} = -\frac{48\pi^2}{A_1} \ln \left( 1 + \frac{A_1}{64\pi^2} e^{\sqrt{2/3}x} \right) + \frac{48\pi^2}{A_1} \ln \left( 1 + \frac{A_1}{64\pi^2} e^{\sqrt{2/3}x_{\text{ini}}} \right). \qquad (5.19) $$

This expression is in agreement with Eq. (24) of Ref. [275]. In this case, the trajectory can even be inverted and the corresponding expression for the field $\phi$ reads

$$ x = \sqrt{\frac{3}{2}} \ln \left[ \left( \frac{64\pi^2}{A_1} + e^{\sqrt{2/3}x_{\text{ini}}} \right) e^{A_1(N-N_{\text{ini}})/(48\pi^2)} - \frac{64\pi^2}{A_1} \right]. \qquad (5.20) $$

We are now in a position where the predictions of the models can be calculated. They are presented in Fig. 125. We see that very negative values of $A_1$ are incompatible with the CMB while large values of $A_1$ remain close to the allowed contours. Of course $|A_1|$ cannot be too large since we have required $A_1/(64\pi^2) \ll 1$. We have chosen the upper bound in Fig. 125 to be $A_1 = 100$ for which $A_1/(64\pi^2) \simeq 0.16$, i.e. still a reasonable number. It is interesting to compare these findings with the existing literature. Using the approximate trajectory in Eq. (5.19) and neglecting the contribution originating from the end of inflation, one obtains

$$ x_* = \sqrt{\frac{3}{2}} \ln \left[ \frac{64\pi^2}{A_1} (e^{x_{\text{BKS}}} - 1) \right], \qquad (5.21) $$

where $x_{\text{BKS}} \equiv A_1 \Delta N_*/(48\pi^2)$ ($x_{\text{BKS}}$ is denoted $x$ in Ref. [275]). Then, from Eq. (5.16) and the fact that $\epsilon_2 = 4\epsilon_1 - 2\hat{\eta}$, it follows that

$$ \epsilon_1 = \frac{4}{3} \left( \frac{A_1}{64\pi^2} \right)^2 \left( \frac{e^{x_{\text{BKS}}}}{e^{x_{\text{BKS}}} - 1} \right)^2 = \frac{3}{4\Delta N_*^2} \left( \frac{x_{\text{BKS}}}{e^{x_{\text{BKS}}} - 1} \right)^2, \qquad (5.22) $$

$$ \epsilon_2 = 4\epsilon_1 + \frac{8}{3} \frac{A_1}{64\pi^2} \frac{1}{e^{x_{\text{BKS}}} - 1} = 4\epsilon_1 + \frac{2}{\Delta N_*} \frac{x_{\text{BKS}}}{e^{x_{\text{BKS}}} - 1}. \qquad (5.23) $$
<figure><img src="image_12.png" /><figcaption>Fig. 9. Predictions of the RCHI model in the plane $(n_s, r)$. The exact slow-roll predictions (colored circles starting in dark blue at the bottom/left part of the plot and ending in red slightly on the right of the allowed contours) are compared to various approximations represented by the second collection of segments made of “crosses”, by the orange thick dashed line and by the aquamarine dotted line, see the text for a detailed explanation. In the regime of large $|A_1|$, the exact predictions may significantly differ from the approximate ones.</figcaption></figure>

From these two expressions, one deduces that

$$n_s = 1 - \frac{2}{\Delta N_*} \frac{x_{\text{BKS}}}{e^{x_{\text{BKS}}} - 1}, \quad r = \frac{12}{\Delta N_*^2} \left( \frac{x_{\text{BKS}} e^{x_{\text{BKS}}}}{e^{x_{\text{BKS}}} - 1} \right)^2. \qquad (5.24)$$

Notice that, in the formula giving the spectral index, the contribution originating from $\epsilon_1$ has been neglected since it scales $\propto 1/\Delta N_*^2$. These approximate expressions match Eqs. (32) and (34) of Ref. [275]. For $\Delta N_* = 60$, they can be represented as a line $r = r(n_s)$ in the plane $(n_s, r)$, the parameter along the curve being $A_1$. This line has been plotted in Fig. 9 for $-30 < A_1 < 100$ (thick orange dashed line). Requiring $n_s$ to be within the $2\sigma$ Planck 2018 + Bicep-Keck suggests that $-8 \lesssim A_1 \lesssim 4$ (or $-12 \lesssim A_1 \lesssim 14$ with WMAP, again in agreement with Ref. [275]). These predictions are compared to the exact slow-roll predictions of Fig. 125. As before, the slow-roll predictions are represented by a collection of segments made of circles, each segment corresponding to different values of $A_1$ and each point of a given segment being in one-to-one correspondence with a given reheating temperature. The exact slow-roll predictions are such that, for $A_1 < 0$, the segments go to the bottom left side of the figure while for $A_1 \to 100$, the segments remain close to the allowed contours (see also Fig. 125). In the limit of “large” positive values of $A_1$, the exact slow-roll predictions and the predictions based on Eqs. (5.24) significantly differ.

Let us try to identify the origin of this discrepancy more precisely. In order to investigate this issue, we have also represented in Fig. 9, the predictions obtained when the approximate trajectory of Eq. (5.19), the approximate expression of the first slow-roll parameter in Eq. (5.16) and the relation $\epsilon_2 = 4\epsilon_1 - 2\hat{\eta}$ but, now, without neglecting $\epsilon_1$, are used together with an exact expression for $\phi_{\text{end}}$. They are represented by the second collections of segments made of crosses in Fig. 9. We see that for $A_1 \gtrsim 0$, they differ from the orange dashed thick line and bend towards the upper left part of the plot which is also the direction taken by the exact predictions. This suggests that neglecting the term $4\epsilon_1$ in the expression of $\epsilon_2$ causes a non-negligible error. This is confirmed if, instead of using Eq. (5.24) for $n_s$, we now take

$$n_s = 1 - \frac{9}{2\Delta N_*^2} \left( \frac{x_{\text{BKS}} e^{x_{\text{BKS}}}}{e^{x_{\text{BKS}}} - 1} \right)^2 - \frac{2}{\Delta N_*} \frac{x_{\text{BKS}}}{e^{x_{\text{BKS}}} - 1}, \qquad (5.25)$$

and plot again the line $r = r(n_s)$. This gives the orange dotted curve which follows the second collection of segments. If, however, we compare the red segments, namely those with $A_1$ “large”, corresponding the exact predictions to the approximate red ones, we see that including the term $4\epsilon_1$ is not sufficient. We conclude that RCHI represents a textbook case for ASPIC. It illustrates that, sometimes, “approximating the slow-roll approximation” can lead to too drastic conclusions, especially given the current accuracy of the data. It is an additional motivation to use the slow-roll method without any other scheme of approximations and this is the essence of the ASPIC project presented in this article.

A last word is in order concerning the constraints on the parameter $A_1$. Particle physics implies that $-48 \lesssim A_1 \lesssim -20$ and the previously discussed inaccuracies were concerning only a weaker upper limit on $A_1$. Therefore, when particle physics and cosmological data are simultaneously taken into account, the conclusions of Ref. [275] are unchanged and RCHI remains disfavored.

Finally, the scale $M$ can be determined from the CMB normalization and this leads to the following expression

$$\frac{M^4}{M_{\text{Pl}}^4} = 120\pi^2 \frac{Q^2}{T^2} \frac{[4e^{-\sqrt{2/3}x_*} + A_1/(16\pi^2)]^2}{[1 - 2e^{-\sqrt{2/3}x_*} + A_1/(32\pi^2)\sqrt{2/3}x_*]^3}. \qquad (5.26)$$

The knowledge of $\phi_*$ allows us to find the posterior distribution of $M$, that is to say of $\lambda/\xi^2$ or $\xi$, since the Higgs self coupling, $\lambda = m_H/v$, is now known.
<figure><img src="image_13.png" /><figcaption>Fig. 10. Large Field Inflation (LFI). Top left panel: large field potential for $p = 2$. Top right panel: logarithm of the potential for the same value of $p$. The required flatness of the potential becomes obvious on this plot. Bottom left panel: slow-roll parameter $\epsilon_1$ for a large field potential with $p = 2$. The shaded area indicates where acceleration stops. Bottom right panel: slow-roll parameters $\epsilon_2$ and $\epsilon_3$ for a large field potential with $p = 2$. Only one curve appears because $\epsilon_2 = \epsilon_3$. On this plot, the shaded region signals the breakdown of the slow-roll approximation, which is not necessarily the end of the accelerated phase.</figcaption></figure>

## 5.2. Large Field Inflation (LFI)

### 5.2.1. Theoretical justifications

Large fields models, also referred to as chaotic inflation [283], are characterized by the monomial potential [284–288] $V(\phi) \propto M^4\phi^p$. The number $p$ is the only model parameter, in addition to the normalization $M$ of the potential. The index $p$ is usually a positive integer (and it was recently realized in Ref. [289] that this type of scenario can emerge in the context of supergravity) but various models have been proposed in which it can also be a rational number [290–295]. It is interesting to briefly discuss concrete models where this is actually the case. Here, we follow Refs. [294,295]. These models are supergravity models where one assumes that the Kähler potential is invariant under a generalization of the shift symmetry (usually needed in order to avoid the so called $\eta$-problem). In the present case, the transformation is taken to be $\chi^n \to \chi^n + \alpha$ where $\alpha$ is a real number and $\chi$ a chiral superfield. This means that the Kähler potential should be a function of $\chi^n - \chi^{\dagger n}$ only. In addition, we allow the presence of a small breaking term in the Kähler potential of the form $b\chi\chi^\dagger$ where $b \ll 1$. We also assume that the superpotential breaks the generalized shift symmetry. Summarizing, we assume that

$$K = b\chi\chi^\dagger + c_1\kappa^{(n-1)/2}(\chi^n - \chi^{\dagger n}) - \frac{\kappa^{n-1}}{2}(\chi^n - \chi^{\dagger n})^2 + XX^\dagger, \quad (5.27)$$

$$W = \lambda X \chi^m, \quad (5.28)$$

where $X$ is another superfield and $\lambda$ and $c_1$ (notice that it is pure imaginary) are constant. The model is parametrized by the quantities $n$ and $m$ and $\kappa \equiv 1/M_{\text{Pl}}^2$. If, during inflation, $X$ acquires a large mass compared to the Hubble parameter and is stabilized at the origin, $\langle X \rangle = 0$, then it is not difficult to show that this supergravity model can be described by the following effective Lagrangian

$$\mathcal{L} = - \left[ b + n^2 \kappa^{n-1} (\chi \chi^\dagger)^{n-1} \right] \partial_\mu \chi \partial^\mu \chi^\dagger \\ - \exp \left[ b \kappa |\chi|^2 + c_1 \kappa^{n/2} (\chi^n - \chi^{\dagger n}) - \frac{\kappa^n}{2} (\chi^n - \chi^{\dagger n})^2 \right] \lambda^2 (\chi \chi^\dagger)^m. \quad (5.29)$$

Then, one can write the field $\chi$ in polar form, $\chi \equiv \alpha e^{i\beta}$ ($\alpha$ is of dimension one and $\beta$ dimensionless) and the above potential takes the form

$$V = \lambda^2 \alpha^{2m} \exp[b\kappa\alpha^2 + 2ic_1\kappa^{n/2}\alpha^n \sin(n\beta) + 2\kappa^n\alpha^{2n} \sin^2(n\beta)]. \quad (5.30)$$

Writing $\partial V / \partial \beta = 0$, one obtains the condition $2i\kappa^{n/2}\alpha^n \sin(n\beta) = -ic_1$ or $\kappa^{n/2} (\chi^n - \chi^{\dagger n}) = c_1$. It is thus natural to assume that the inflaton field rolls along that direction. As a consequence, the effective Lagrangian takes the form

$$\mathcal{L} = - \left[ b + n^2 \kappa^{n-1} (\chi \chi^\dagger)^{n-1} \right] \partial_\mu \chi \partial^\mu \chi^\dagger - e^{b\kappa |\chi|^2 + c_1^2/2} \lambda^2 (\chi \chi^\dagger)^m. \quad (5.31)$$
Now, in the regime $b\kappa|\chi|^2 \ll 1$, the exponential becomes essentially independent of the field $\chi$ and the coefficient $b$ in the kinetic term becomes negligible. It is therefore natural to define a new quantity $\theta \equiv \kappa^{(n-1)/2} \chi^n$ for which one obtains the Lagrangian of a canonically normalized field, namely

$$ \mathcal{L} = -\partial_{\mu}\theta\partial^{\mu}\theta^{\dagger} - e^{c_1^2/2}\lambda^2(\theta\theta^{\dagger})^{m/n}. \qquad (5.32) $$

Finally, we take the imaginary part of $\theta$ to be stabilized to $c_1$ in order to satisfy the condition discussed above and we define the real field $\phi$ by $\theta = \phi/\sqrt{2} + c_1/2$. As a consequence, it follows

$$ \mathcal{L} \approx -\frac{1}{2} \partial_{\mu} \phi \partial^{\mu} \phi^{\dagger} - e^{c_1^2/2} \lambda^2 \phi^{2m/n}. \qquad (5.33) $$

Therefore, we have obtained a LFI model with $p = 2m/n$ (neglecting a term $|c_1|^2$ in $V$). In Ref. [294], the case $n=2$ and $m=1$ was considered and we see that this leads to a linear potential. In Ref. [295], the generalized case considered before was introduced and studied. It is worth mentioning that, when the condition $b\kappa|\chi|^2 \ll 1$ is not satisfied, the potential remains of the LFI form but with a different $p$, see Ref. [295]. For instance, as shown in Ref. [294], if $n=2$ and $m=1$, the potential is in fact quadratic at the origin. This means that the standard relation between $p$ (in the inflationary regime) and the mean equation of state during reheating namely, $\bar{w}_{reh} = (p-2)/(p+2)$ [83], is no longer valid in that case.

### 5.2.2. Slow-roll analysis

Having studied how the LFI model can be implemented in high energy physics, we now turn to the inflationary analysis. In the following, we write $V(\phi)$ as

$$ V(\phi) = M^4 \left( \frac{\phi}{M_{\text{Pl}}} \right)^p. \qquad (5.34) $$

This potential is represented in Fig. 10 for $p=2$. The three Hubble flow functions are straightforwardly obtained from Eqs. (3.4), (3.5) and (3.6). Defining $x \equiv \phi/M_{\text{Pl}}$, one gets

$$ \epsilon_1 = \frac{p^2}{2x^2}, \qquad \epsilon_2 = \frac{2p}{x^2}, \qquad \epsilon_3 = \epsilon_2. \qquad (5.35) $$

These functions are represented in the two bottom panels of Fig. 10. They are monotonic decreasing functions of $\phi$. One can immediately deduce that, for a given $p$, the model in the plane $(\epsilon_1, \epsilon_2)$ is contained in the line $\epsilon_1 = (p/4)\epsilon_2$.

The slow-roll trajectory is completely explicit and obtained by quadrature from Eq. (3.11)

$$ N - N_{\text{end}} = -\frac{1}{M_{\text{Pl}}^2} \int_{\phi_{\text{end}}}^{\phi} \frac{V(\chi)}{V'(\chi)} d\chi = -\frac{1}{p} \int_{\phi_{\text{end}}/M_{\text{Pl}}}^{\phi/M_{\text{Pl}}} x dx = \frac{1}{2p} (x_{\text{end}}^2 - x^2). \qquad (5.36) $$

This expression can be inverted and reads

$$ x = \sqrt{x_{\text{end}}^2 - 2p(N - N_{\text{end}})}. \qquad (5.37) $$

For the large field models, inflation ends naturally when $\epsilon_1 = 1$ (see Section 2). Along the $\phi > 0$ branch of the potential, this leads to

$$ x_{\text{end}} = \frac{p}{\sqrt{2}}. \qquad (5.38) $$

This expression also allows us to obtain the total number of e-folds. Plugging Eq. (5.38) into Eq. (5.36), one arrives at

$$ N_{\text{end}} - N_{\text{ini}} = \frac{1}{2p} x_{\text{ini}}^2 - \frac{p}{4}, \qquad (5.39) $$

which can be very large if the initial field value is super-Planckian. Notice that this does not imply that the energy density is close to the Planck scale as this one is typically given by the potential and proportional to $M^4$. In fact, the model remains under control only if the initial energy density is smaller than $M_{\text{Pl}}^4$ and this imposes a constraint on both $\phi_{\text{ini}}$ and $M$ which reads

$$ x_{\text{ini}} = \frac{\phi_{\text{ini}}}{M_{\text{Pl}}} \lesssim \left( \frac{M_{\text{Pl}}}{M} \right)^{4/p}. \qquad (5.40) $$

Let us notice that, when the inflaton energy density approaches the Planck energy density, quantum effects become important. In this case, the stochastic inflation formalism must be used [296–302].

We now turn to the explicit determination of the slow-roll parameters. We have seen that the model is represented by the trajectory $\epsilon_1 = (p/4)\epsilon_2$ but observable models only lie in a limited portion of this straight line. Indeed, the Hubble flow parameters should be evaluated when the scales of astrophysical interest today left the Hubble radius during inflation. Following the discussion of Section 3.2, we assume the pivot mode crossed the Hubble radius for $\phi = \phi_*$ at the e-fold number $N_*$. From the trajectory, we have

$$ x_*^2 = 2p \left( \Delta N_* + \frac{p}{4} \right), \qquad (5.41) $$

and the slow-roll parameters read

$$ \epsilon_{1*} = \frac{p}{4(\Delta N_* + p/4)}, \qquad \epsilon_{2*} = \frac{1}{\Delta N_* + p/4}, \qquad \epsilon_{3*} = \epsilon_{2*}. \qquad (5.42) $$

Solving Eq. (3.48) for $\phi_*$ yields the slow-roll predictions represented in Fig. 127. As expected, the whole family lies in the region $\epsilon_2 > 0$ and verifies $\epsilon_1 = p/4\epsilon_2$. From Fig. 127, we see that all the models with $p \gtrsim 3$ lie outside the $2\sigma$ contour. The quadratic (or massive) model is under great pressure since it predicts quite a high contribution of gravitational waves, up to $r \simeq 15\%$ level.

Finally, the parameter $M$ can be determined from the amplitude of the CMB anisotropies, and one gets

$$ \frac{Q_{\text{rms-PS}}^2}{T^2} = \frac{1}{480\pi^2\epsilon_{1*}} \frac{H_*^2}{M_{\text{Pl}}^2} = \frac{1}{1440\pi^2\epsilon_{1*}} \frac{V_*}{M_{\text{Pl}}^4}. \qquad (5.43) $$
<figure><img src="image_14.png" /><figcaption>Fig. 11. Top left panel: mixed large field (MLFI) potential, see Eq. (5.45), for $\alpha = 0.05$. Top right panel: logarithm of the potential for the same value of $\alpha$. The dotted line indicates the potential $V(\phi) \simeq M^4\phi^2/M_{\text{Pl}}^2$ which is the limit of the MLFI potential in the regime $\phi/M_{\text{Pl}} \ll 1/\sqrt{\alpha}$ while the dashed line represents the expression $V(\phi) \simeq M^4\alpha\phi^3/M_{\text{Pl}}^3$, the limit of $V(\phi)$ when $\phi/M_{\text{Pl}} \gg 1/\sqrt{\alpha}$. For $\alpha = 0.05$ the two lines meet at the following value, $1/\sqrt{\alpha} \simeq 4.5$, as can be directly checked in the figure. The arrow in the top left and right panels indicate in which direction inflation proceeds. Bottom left panel: slow-roll parameter $\epsilon_1$ for a mixed large field potential with $\alpha = 0.05$. Bottom right panel: slow-roll parameters $\epsilon_2$ (solid line) and $\epsilon_3$ (dotted line) still for $\alpha = 0.05$.</figcaption></figure>

In the case of large fields model, this implies

$$ \left(\frac{M}{M_{\text{Pl}}}\right)^4 = \frac{720\pi^2 p^2 Q_{\text{rms-PS}}^2}{(x_*^2)^{p/2+1} T^2}, \qquad (5.44) $$

and given the constraints on $p$ and $\Delta N_*$, this leads to $M/M_{\text{Pl}} \simeq 3 \times 10^{-3}$. We recover the conclusion that, for large field models, inflation takes place close to the Grand Unified Theory (GUT) scale.

## 5.3. Mixed Large Field Inflation (MLFI)

This model is a generalization of the LFI model $V(\phi) \propto \phi^p$, see Section 5.2, where two monomials $\propto \phi^2$ and $\propto \phi^4$ are added. The MLFI potential reads

$$ V(\phi) = M^4 \frac{\phi^2}{M_{\text{Pl}}^2} \left( 1 + \alpha \frac{\phi^2}{M_{\text{Pl}}^2} \right), \qquad (5.45) $$

where $\alpha$ is a positive dimensionless parameter. If $\phi/M_{\text{Pl}} \ll 1/\sqrt{\alpha}$, then the potential is of the LFI type with $p=2$, i.e. $V(\phi) \simeq M^4\phi^2/M_{\text{Pl}}^2$, whereas if $\phi/M_{\text{Pl}} \gg 1/\sqrt{\alpha}$, the potential is of the LFI type with $p=4$, i.e. $V(\phi) \simeq M^4\alpha\phi^4/M_{\text{Pl}}^4$. Clearly, the interesting regime is when $\phi/M_{\text{Pl}} \simeq 1/\sqrt{\alpha}$, where the two terms are of equal importance. The potential and its logarithm are displayed in Fig. 11. We notice that $V(\phi)$ is an increasing function of the field $\text{vev}$ and, as a consequence, that inflation proceeds from the right to the left.

This model has been investigated in different contexts. Of course, the shape of the potential appears to be natural and well-motivated since it just represents a free theory (with particles of mass $2M^4/M_{\text{Pl}}^2$) corrected by the usual self-interacting quartic term. Therefore, it does not come as a surprise that this potential has been used in many different works. In Ref. [303], this model is studied in the case where a bulk scalar field is driving inflation in large extra dimensions. In Ref. [304], it is considered in a situation where inflation is driven by highly excited quantum states. In Refs. [305–307], the MLFI potential is utilized in the context of “fresh inflation”. The same potential was again considered in Ref. [308] where the role of inflaton is played by the Higgs triplet in a model where the type II seesaw mechanism is used to generate the small masses of left-handed neutrinos. Finally, it is also studied in Ref. [309] where supersymmetric hybrid inflation (in the framework of the Randall–Sundrum type II Braneworld model) is considered. The only constraint on the parameters of the model that is (sometimes) required is that the self-interacting term should be sub-dominant. This leads to the condition $\alpha M^4/M_{\text{Pl}}^4 \ll 1$. Given the typical values imposed by CMB normalization, i.e. $M/M_{\text{Pl}} \simeq 10^{-3}$ [see Eq. (5.44)], this is not very stringent and $\alpha$ can in fact vary in a quite large range of values.
<figure><img src="image_15.png" /><figcaption>Fig. 12. Lambert functions $W_0(x)$ (dashed line) and $W_{-1}(x)$ (solid line). During Mixed Large Field inflation, inflation proceeds along the “0” branch above the line $W = 1$ in the direction specified by the arrow.</figcaption></figure>

Defining $x \equiv \phi/M_{\text{Pl}}$, the three first slow-roll parameters can be expressed as

$$ \epsilon_1 = \frac{2}{x^2} \left( \frac{1 + 2\alpha x^2}{1 + \alpha x^2} \right)^2, \quad \epsilon_2 = \frac{4}{x^2} \frac{1 + \alpha x^2 + 2\alpha^2 x^4}{(1 + \alpha x^2)^2}, \qquad (5.46) $$

and

$$ \epsilon_3 = \frac{M_{\text{Pl}}^2}{x^2} \frac{1 + 2\alpha x^2 + 4 + 12\alpha x^2 + 8\alpha^3 x^6}{(1 + \alpha x^2)^2 + 1 + \alpha x^2 + 2\alpha^2 x^4}. \qquad (5.47) $$

They are displayed in Fig. 11. We see that the three slow-roll parameters are decreasing functions of the field $\text{vev}$, which means that they are all increasing functions during inflation. As a consequence, inflation can stop by violation of the slow-roll conditions at $x_{\text{end}}$ given by $\epsilon_1 = 1$ (see below). We also notice that $\epsilon_2$ and $\epsilon_3$ are larger than one at $x_{\text{end}}$. This means that the slow-roll approximation breaks down slightly before the end of inflation and that the last few e-folds of inflation may be not properly described by the slow-roll approximation.

Let us now study the slow-roll trajectory. It is given by

$$ N_{\text{end}} - N = -\frac{1}{8} \left[ x_{\text{end}}^2 + \frac{1}{2\alpha} \ln(1 + 2\alpha x_{\text{end}}^2) - x^2 - \frac{1}{2\alpha} \ln(1 + 2\alpha x^2) \right], \qquad (5.48) $$

where $N_{\text{end}}$ is the number of e-folds at the end of inflation. One can check that this expression is asymptotically correct. Indeed, when $\alpha \ll 1$, the slow-roll trajectory reduces to

$$ x_{\text{end}}^2 = x^2 - 4(N_{\text{end}} - N), \qquad (5.49) $$

which is the trajectory in the massive case, i.e. LFI with $p = 2$, see Eq. (5.36). On the other hand, in the limit $\alpha \to \infty$, one obtains

$$ x_{\text{end}}^2 = x^2 - 8(N_{\text{end}} - N), \qquad (5.50) $$

which is, as expected, the slow-roll trajectory in the quartic case, i.e. LFI with $p = 4$. In general, the trajectory can be inverted and expressed in terms of the Lambert function. Straightforward manipulations lead to

$$ x = \frac{1}{\sqrt{2\alpha}} \sqrt{-1 + W_0 \left[ e^{1+2\alpha x_{\text{end}}^2} (1 + 2\alpha x_{\text{end}}^2) e^{-16\alpha(N-N_{\text{end}})} \right]}. \qquad (5.51) $$

The corresponding Lambert function is displayed in Fig. 12, together with the region where inflation proceeds.

We have seen that, in MLFI, inflation stops by violation of the slow-roll condition. Let us therefore determine the corresponding $\text{vev}$ of the field. The condition $\epsilon_1 = 1$ leads to

$$ \alpha x_{\text{end}}^3 - 2\sqrt{2\alpha}x_{\text{end}}^2 + x_{\text{end}} - \sqrt{2} = 0. \qquad (5.52) $$

This is a cubic algebraic equation that can be solved exactly. In the limit $\alpha \gg 1$, the solution reads $x_{\text{end}} \approx 2\sqrt{2}$ which is indeed the solution for the quartic case, see Eq. (5.38). On the other hand, if $\alpha \ll 1$, then $x_{\text{end}} \approx \sqrt{2}$ which is also the correct result for the quadratic case. The general solution is

$$ x_{\text{end}} = \frac{2\sqrt{2}}{3} + \frac{1}{3\alpha} \left\{ \frac{1}{4\sqrt{2}} \left[ 4\alpha^2 (32\alpha + 9) + 2\alpha \sqrt{4\alpha^2 (32\alpha + 9)^2 - 8\alpha (8\alpha - 3)^3} \right] \right\}^{1/3} \\ + \frac{1}{3} (8\alpha - 3) \left\{ \frac{1}{4\sqrt{2}} \left[ 4\alpha^2 (32\alpha + 9) + 2\alpha \sqrt{4\alpha^2 (32\alpha + 9)^2 - 8\alpha (8\alpha - 3)^3} \right] \right\}^{-1/3}, \qquad (5.53) $$

which is the one used in the ASPIC library.
<figure><img src="image_16.png" /><figcaption>Fig. 13. Radiatively Corrected Massive Inflation (RCMI) for $\alpha = 0.01$. Top panels: potential (left) and logarithm of the potential (right). Bottom left panel: slow-roll parameter $\epsilon_1$ with respect to field values. The shaded area indicates where inflation stops. Bottom right panel: slow-roll parameters $\epsilon_2$ (solid line) and $\epsilon_3$ (dotted line).</figcaption></figure>

Finally, the parameter $M$ can be determined from the amplitude of the CMB anisotropies, and one gets

$$ \left(\frac{M}{M_{\text{Pl}}}\right)^4 = \frac{2880\pi^2 (1 + 2\alpha x_*^2)^2 Q_{\text{rms-Ps}}^2}{x^4 (1 + \alpha x_*^2)^3 T^2} \qquad (5.54) $$

Similarly to LFI (see Section 5.2), this gives rise to $M/M_{\text{Pl}} \approx 10^{-3}$. The reheating consistent slow-roll predictions for the MLFI models are displayed in Fig. 128. The reheating equation of state parameter $\bar{w}_{\text{reh}}$ has been taken to 0 which is consistent with the fact that the potential is quadratic close to its minimum. As expected, when $\alpha \ll 1$ the predictions of the model match those of LFI with $p = 2$ and are aligned along the $\epsilon_1 = \epsilon_2/2$ line. On the other hand, if $\alpha \gg 1$, then the predictions are consistent with those of LFI with $p = 4$ and are aligned along the $\epsilon_1 = \epsilon_2$ line. In the intermediate regime, it is interesting to notice that the MLFI predictions continuously interpolate between these two asymptotic solutions but do not remain inside the domain delimited by those two lines. Indeed, when $\alpha$ is larger than some value, one has $\epsilon_1 > \epsilon_2$. This means that, if one starts from a pure quartic potential (LFI with $p = 4$) and adds a small quadratic term, this extra term has the effect of increasing the “effective value” of $p$, which is quite counter intuitive. On the other hand, since the quadratic model fits better the data than the quartic one, small values for the parameter $\alpha$ are favored (all the models with $\alpha > 10^{-3}$ lie outside the $2\sigma$ contour of the Planck 2018 + Bicep-Keck data). High reheating temperatures are also preferred.

## 5.4. Radiatively Corrected Massive Inflation (RCMI)

This model is based on Ref. [310] and implements radiative corrections due to fermion couplings over the massive ($p = 2$) large field model (see Section 5.2). With an appropriate choice of the renormalization scale $\mu = gM_{\text{Pl}}$, $g$ denoting the Yukawa coupling, the potential is given by

$$ V(\phi) = \frac{1}{2}m^2\phi^2 - \frac{g^4}{16\pi^2}\phi^4\ln\left(\frac{\phi}{M_{\text{Pl}}}\right) = M^4\left(\frac{\phi}{M_{\text{Pl}}}\right)^2\left[1 - 2\alpha\frac{\phi^2}{M_{\text{Pl}}^2}\ln\left(\frac{\phi}{M_{\text{Pl}}}\right)\right], \qquad (5.55) $$

where

$$ M^4 \equiv \frac{1}{2}m^2 M_{\text{Pl}}^2, \qquad \alpha \equiv \frac{g^4 M_{\text{Pl}}^2}{16\pi^2 m^2}. \qquad (5.56) $$

This expression is obtained in the large field regime $\phi \gg m/g$ (this condition coming from the requirement that the fermion loop contribution dominates over the self-interaction loop contribution), i.e. assuming that the inflationary regime takes place under the condition

$$ \frac{\phi^4}{M_{\text{Pl}}^4} \gg \frac{1}{8\pi^2\alpha} \frac{M^4}{M_{\text{Pl}}^4}. \qquad (5.57) $$