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 sections^{1}. 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.
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. ↩
At 2pm, room E349, Danièle Steer (APC), will be talking about
In this talk I will outline the main different methods on the market for measuring cosmological parameters (including modified gravity parameters) with GWs, and the main sources of errors in these measurements. Then I will highlight the current results on cosmological parameters obtained with the O3 run of LIGO-Virgo, as well as give predictions for future expectations. Finally I will go into more technical details and explain an analytical approach to estimate distance errors with GW observations (from any number of ground based interferometers placed at different positions on earth).
]]>The live stream can be watched there while a transcripted english translation has been published on the arXiv.
]]>At 2pm, room E349, Davide Dal Cin (SISSA), will be talking about
The radial direction of the Peccei-Quinn field can drive cosmic inflation, given a non-minimal coupling to gravity. This scenario has been considered to be capable of explaining inflation, the strong CP problem, and dark matter. We argue that Peccei-Quinn inflation is extremely sensitive to higher-dimensional operators. Further combining with the discussion on the axion quality required to solve the strong CP problem, we examine the validity of this scenario. We also show that a resonant amplification of the axion field is unavoidable after Peccei–Quinn inflation.
]]>Random Apollonian Packing (RAP) is inspired by the better-known Apollonian Gasket. In mathematics, an Apollonian gasket is a fractal generated by starting with a triple of circles, each tangent to the other two, and successively filling in more circles, each tangent to another three.
In Ref. (missing reference), Pierre examines a related mechanism in which \(d\)-dimensional spheres are randomly seeded in space, one at a time, within a finite-sized volume, and with the largest possible radius that avoids overlap.
The interest of the RAP mechanism is that it is thought to share universal properties with more general dynamic mechanisms, such as the ABK mechanism (named after Andrienko, Brilliantov and Krapivsky) in which bubbles grow linearly with time.
Pierre has built a model to explain at which rate Random Apollonian Packings grow after the insertion of one sphere after the other. The model’s prediction for the fractal properties of RAP are consistent with numerical simulations made in two, three and four dimensions.
If you want to explore a RAP in depth, feel free to zoom in that picture, it contains more than a million spheres.
]]>At 2pm, room E349, Nilanjan Bhaumik (IIS), will be talking about
Primordial black hole (PBH) is one of the most promising cold dark matter candidates in recent years. The signature of PBHs in the stochastic gravitational-wave background (SGWB) is natural as PBH formation requires a significant amplification in inflationary scalar curvature perturbation, which sources the tensor perturbation in second order and amplifies the SGWB. We explore this possibility to probe different possible reheating histories. We find that any change in reheating histories can shift the PBH mass range and the peak frequency of SGWB. Still, only the matter-dominated reheating phase can lead to an additional resonant amplification in SGWB at a very high frequency. Isocurvature perturbation from PBH distributions contributes to the adiabatic perturbation. It also leads to detectable amplification in the induced GW background and can contribute to Baryogenesis for ultra-low mass PBHs, which can dominate the universe for a small duration. Combining these two effects can help us probe primordial black hole scenarios and details of reheating histories, leaving diverse implications for cosmology and particle physics.
]]>Cosmic Inflation is an hypothetical early phase of accelerated expansion that has occurred before the first billionth of a second of existence of our Universe. It provides a natural mechanism to explain the observed flatness of our Universe today and naturally solves the so-called horizon problems of the Big-Bang model.
In a spectacular way, the quantum fluctuations that are inherently sourced during the inflationary era are exactly what is needed to explain the origin of the cosmological perturbations: the seeds of the galaxies of today.
These quantum fluctuations are deeply rooted in gravity and appear as both primordial gravitational waves \(h_{ij}\) and curvature perturbations \(\zeta\), with very peculiar correlation functions. In Ref. [1], we have pushed to third order the calculation of these correlation functions. They are completely determined by the Hubble parameter during inflation \(H(N)\) and its logarithmic derivatives, \(\epsilon_i(N) \equiv \mathrm{d}\ln H / \mathrm{d} \ln N\) (the so-called Hubble flow functions). Here \(N=\ln a\) is the logarithm of the scale factor \(a\).
Slow-roll inflation predicts the correlation functions to be given by these spectra:
They are expanded around an observable wavenumber \(k_*=0.05\,\mathrm{Mpc}^{-1}\) and readily testable with the incoming cosmological observations from the Euclid and LiteBird space telescopes, but also from the ground based CMB-S4 telescopes and Simons Observatory. Are we going to detect a non-vanishing \(\epsilon_{3*}\)?
]]>On display, you’ll find a black and white fractal image, extracted from Ref. [1].
Cosmic inflation is an hypothetic phase of accelerated expansion of the Universe, a tiny fraction of second after the Big-Bang, induced by the existence of a scalar field in the early Universe and suspected to be at the origin of the present structures of the Universe. During inflation, in about \(10^{-35}\) second, the distance between any two points in the Universe separated by one meter becomes larger than the distance to the farthest galaxies. This makes inflation one of the most fascinating phenomena in Science.
This picture shows the results of more than 4 millions numerical simulations of inflation in a model with two scalar fields, called hybrid inflation, for which inflation stops due a broken symmetry, similar to the Brout-Englert-Higgs mechanism. The \(x\) and \(y\) axis represent the initial value of each scalar field, a white dot indicate that the simulation led to enough expansion for being compatible with our Universe, whereas for black dots, there is no, or not enough, inflation.
Before these results were obtained in Ref. [1], at ULB and UCLouvain, it was thought that only the thin vertical white band could lead to inflation, suggesting a fine-tuning problem. This picture shows that, on the contrary, a large number of initial conditions are satisfactory and form a complex fractal structure. In analogy with anamorphosis, a phenomena consisting in the distortion of an image through an optical instrument, one can see this fractal structure as the image of the thin vertical band seen through the potential associated with the scalar fields, playing the role of the optical instrument.
Using the recently released public domain data DR3 from the GAIA satellite (ESA), we have reconstructed a view of our night sky, as you would see it with very sensitive eyes, eyes that would allow you to actually see daylight colors but in the dark.
This image represents the integrated light flux coming from all faint sources of light, of magnitude greater than \(10\), in human-visible colors (sRGB). These sources are mostly stars from our own Galaxy and have been measured by the Gaia satellite (ESA/Gaia/DPAC). There are more than 1.5 billion of stars accounted for in this image. The image making code is available here: gaialaxy.
Extracted from Ref. [2], a small piece of a computer generated all sky map of the Cosmic Microwave Background generated by Cosmic Strings.
These objects are line-like defects in the fabric of space-time and could have been formed in the earliest times of the Universe history. This picture is a small part of a full sky image computed using \(12000\) processors of one of the largest world computer at that time: the Cray XE6 Hopper at the NERSC. This image has been used to search for Cosmic Strings in the Planck satellite data (ESA).
]]>In Ref. [1], we have run new simulations of Nambu-Goto cosmic strings evolving during the radiation, the transition, and the matter eras to compute the unequal time correlators of the anisotropic stress tensor associated with the long strings. The following figure shows a snapshot of one of these simulation, the long strings have been represented in white whereas all the other objects are loops, see Ref. [2] and this post.
This correlator sources the gravitational waves and it allows us to solve for their creation, and propagation, all along the Universe history. By using the Green’s function method we can then predict the strain, \(k^2 \mathcal{P}_h\), and the energy density parameter \(\Omega_{\mathrm{GW}}^{\mathrm{mat}}\) of the gravitational waves that can be measured today. Their power spectra are represented below as a function of the wavenumber \(k\).
In these figures, the exact numerical result is represented in black while the blue and red curves show some semi-analytical approximations that we had proposed in a previous paper, calibrated using the amplitude found with the simulations, see Ref. [3]. The only significant deviations show up around \(k/\mathcal{H}_0 \simeq 100\), which corresponds to the transition between the radiation and matter era. An interesting point to notice is that most of strain signal is actually generated by the long cosmic strings in the matter era, i.e., close to us.
As we discuss in Ref. [1], this signal is quite small, but reachable by the LISA satellites. These ones will be sensitive to long strings that are undetectable today in the Cosmic Microwave Background.
]]>LISA can potentially detect GW signals from first order phase transitions occurring in the energy range \(100\,\mathrm{GeV}\) – \(1 \, \mathrm{TeV}\). There are several processes possibly leading to sizeable emission of GWs:
In Ref. [1], we propose a new model to the decay and decorrelation of a purely vortical velocity field and calculate the subsequent GW signal. To confirm the accuracy and relevance of the model, we have carried out a campaign of massively parallel direct numerical simulations of decaying turbulence. The figure below shows a slice across one of these simulations.
The following figure shows the initial conditions for the velocity field in real space
After some evolution, the velocity field decays and small scale structures develop. The next figure shows the velocity field after \(20\) eddy turnover times
Based on this modelling, we provide an approximate analytical form for the GW signal which could be readily used within the LISA collaboration to search for it. In the figure below, coloured lines show the results of the simulations, the black line the semi-analytical model and the grey line the analytical form usable for LISA.
Other new findings are presented in Ref. [1]. For instance, including an initial growth phase for the turbulent flow is shown to heavily influences the spectral shape of the SGWB. This highlights the importance of a complete understanding of the turbulence generation mechanism.
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