The opiparous edition of Encyclopædia Inflationaris

We have released a new version of Encyclopædia Inflationaris and its associated public code ASPIC. This opiparous edition extends the original paper by 400 pages and features 46 new models of inflation that have been proposed in the last decade. Cosmic Inflation is ready to tackle the data challenges of tomorrow.

The first edition of Encyclopædia Inflationaris was published in Ref. [1] and authored by Jérôme Martin, Christophe and Vincent Vennin. It was built upon the slow-roll models of inflation proposed prior to 2013, the year of the first Planck data release. It contains accurate reheating-consistent slow-roll calculations of the background universe and of the expected scalar and tensor perturbations for all these models. Although it is now often seen as a review, it is a genuine paper as it goes beyond the mere gathering of known results. Indeed, a fair fraction of these models had been studied only under rough approximations, which were not matching the accuracy needed by the Cosmic Microwave Background (CMB) data. The ambition of the project was, and still is, to provide an (almost) exact treatment of single-field inflationary models using only the slow-roll approximation while incorporating the kinematic effects of reheating.

All these results are also incorporated in a public runtime library, ASPIC [2], which has been used in a number of subsequent works. For instance, Bayesian model comparison using the Planck CMB data has been presented in Refs. [3] and [4] while the constraints derived on the reheating epoch have been separately presented in Refs. [5] and [6].

Ten years later Cosmic Inflation remains the most favoured scenario of the early universe [7] but several developments have called 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 potentials already encoded in ASPIC, in which case they have been added in the relevant sections1. 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), Cubicly Corrected Starobinsky Inflation (CCSI),Double Exponential Inflation (DEI), Dual Inflation (DI), Fibre 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 Legacy + Bicep-Keck data combination. The Bayesian evidence of all models has also been re-computed with the new data set, and the results will be presented in a separate publication.

  • 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 and SPT in Antarctica, QUIJOTE in the Canary islands, and CLASS, ACT and Polarbear in the Atacama desert. They have been very recently joined by QUBIC. In space, the Euclid satellite is soon to be launched and will provide unprecedented measurements on the matter power spectrum, down to very small scales. The LiteBIRD satellite is planned to be launched in 2028 and should allow us to further constrain the \(B\)-mode signal in the polarisation 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 [8] and can be extended to arbitrary precision if needed [9]. On the contrary, as discussed in Ref. [10] other commonly-employed approximations are now too imprecise to allow for a fair comparison with the data.

Some of the new models are compatible with the present data, as one could have expected, but this statement is strongly reheating dependent. This can be illustrated by the following figure. It shows their predictions, namely spectral index \(n_\mathrm{S}\) and tensor-to-scalar ratio \(r\) as a function of the reheating energy scale \(E_{\mathrm{reh}}\) (assuming a matter-like reheating for illustration purposes). The contours are the \(68\%\) and \(95\%\) confidence intervals associated with the successive Planck data releases since 2013.


As this figure shows, getting reheating-consistent predictions has never been so crucial. 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, in the opirarous edition, 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.

The previous figure certainly confirms the strategy adopted since the early days of Encyclopædia Inflationaris to derive reheating-consistent predictions. Ignoring it can indeed be catastrophic. As an example, here are the reheating-consistent predictions of one of the new model proposed after the Planck data release. Not including reheating effects would imply selecting random points in this mess…


Let us note that in its new edition, the format of the paper 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 also not completed section 3 with a list of new analytical results derived in the 46 new potentials, given that we think it has already become clear that Encyclopædia Inflationaris is more than a review indeed. At the time of this post, the opiparous edition is made public only through arXiv:1303.3787v4 with a full open access CC BY-NC-SA 4.0 license.

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.

  1. 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. 


  • [1] Martin J, Ringeval C and Vennin V 2014 Encyclopædia Inflationaris Phys.Dark Univ. 5-6 75–235
    Abstract: arXiv:1303.3787v3
    Journal: 10.1016/j.dark.2014.01.003

  • [2] Martin J, Ringeval C and Vennin V 2018 ASPIC: Accurate Slow-roll Predictions for Inflationary Cosmology ascl:1806.031
    Ads: ascl:1806.031

  • [3] Martin J, Ringeval C, Trotta R and Vennin V 2014 The Best Inflationary Models After Planck JCAP 1403 039
    Abstract: arXiv:1312.3529
    Journal: 10.1088/1475-7516/2014/03/039

  • [4] Martin J, Ringeval C and Vennin V 2014 How Well Can Future CMB Missions Constrain Cosmic Inflation? JCAP 1410 038
    Abstract: arXiv:1407.4034
    Journal: 10.1088/1475-7516/2014/10/038

  • [5] Martin J, Ringeval C and Vennin V 2015 Observing Inflationary Reheating Phys. Rev. Lett. 114 081303
    Abstract: arXiv:1410.7958
    Journal: 10.1103/PhysRevLett.114.081303

  • [6] Martin J, Ringeval C and Vennin V 2016 Information Gain on Reheating: the One Bit Milestone Phys. Rev. D93 103532
    Abstract: arXiv:1603.02606
    Journal: 10.1103/PhysRevD.93.103532

  • [7] Chowdhury D, Martin J, Ringeval C and Vennin V 2019 Inflation after Planck: Judgment Day Phys. Rev. D100 083537
    Abstract: arXiv:1902.03951
    Journal: 10.1103/PhysRevD.100.083537

  • [8] Ringeval C 2014 Fast Bayesian inference for slow-roll inflation Mon.Not.Roy.Astron.Soc. 439 3253
    Abstract: arXiv:1312.2347
    Journal: 10.1093/mnras/stu109

  • [9] Auclair P and Ringeval C 2022 Slow-roll inflation at N3LO Phys. Rev. D 106 063512
    Abstract: arXiv:2205.12608
    Journal: 10.1103/PhysRevD.106.063512

  • [10] Martin J, Ringeval C and Vennin V 2016 Shortcomings of New Parametrizations of Inflation Phys. Rev. D94 123521
    Abstract: arXiv:1609.04739
    Journal: 10.1103/PhysRevD.94.123521