Single Field Inflationary Models Research Articles

Cosmic Inflation at the crossroads

The capability of Cosmic Inflation to explain the latest Cosmic Microwave Background and Baryonic Acoustic Oscillation data is assessed by performing Bayesian model comparison within the landscape of nearly three-hundred models of single-field slow-roll inflation. We present the first Bayesian data analysis based on the third-order slow-roll primordial power spectra. In particular, the fourth Hubble-flow function ε4 remains unbounded while the third function verifies, at two-sigma, ε3 ∈[-0.4,0.5], which is perfectly compatible with the slow-roll predictions for the running of the spectral index. We also observe some residual excess of B-modes within the BICEP/Keck data favoring, at a non-statistically significant level, non-vanishing primordial tensor modes: log(ε1) > -3.9, at 68% confidence level. Then, for 287 models of single-field inflation, we compute the Bayesian evidence, the Bayesian dimensionality and the marginalized posteriors of all the models' parameters, including the ones associated with the reheating era. The average information gain on the reheating parameter R reh reaches 1.3 ± 0.18 bits, which is more than a factor two improvement compared to the first Planck data release. As such, inflationary model predictions cannot meet data accuracy without specifying, or marginalizing over, the reheating kinematics. We also find that more than 40% of the scenarios are now strongly disfavored, which shows that the constraining power of cosmological data is winning against the increase of the number of proposed models. In addition, about 20% of all models have evidences within the most probable region and are all favored according to the Jeffreys' scale of Bayesian evidences.

The cosmological collider in R 2 inflation

Starobinsky's R 2 inflation manifests a best-fit scenario for the power spectrum of primordial density fluctuations. Observables derived from the slow-roll picture of the R 2 model in the Einstein frame relies on the conformal transformation of the metric, which inevitably induces a unique exponential-type couplings of the rolling scalaron with all matter fields during inflation. The “large-field” nature of the R 2 model further invokes non-negligible time and scale dependence to the matter sector through such an exponential coupling, modifying not only the dynamics of matter perturbations on superhorizon scales but also their decay rates. In this work, we identify the simplest observable of the cosmological collider physics built in the background of R 2 inflation, focusing on the so-called “quantum primordial clock” signals created by the non-local propagation of massive scalar perturbations. Our numerical formalism based on the unique conformal coupling can have extended applications to (quasi-)single-field inflationary models with non-trivial couplings to gravity or models that originated from the f(R) modification of gravity.

One consistency relation for all single-field inflationary models

In this paper, we present a non-Gaussianity consistency relation that enables the calculation of the squeezed limit bispectrum of the curvature perturbation in single-field inflationary models by carefully inspecting the background evolution and the linear perturbation theory. The consistency relation is more general than others in the literature since it does not require any specific symmetry, conservation of the curvature perturbation at large scales, attractor background evolution or canonical kinetic energy of the inflaton field. We demonstrate that all known examples of the squeezed limit bispectrum in single-field models of inflation can be reproduced within this framework.

Numerical 1-loop correction from a potential yielding ultra-slow-roll dynamics

Single-field models of inflation might lead to amplified scalar fluctuations on small scales due, for example, to a transient ultra-slow-roll phase. It was argued by Kristiano & Yokoyama in ref. [1] that the enhanced amplitude of the scalar power spectrum on small scales has the potential to induce a sizeable 1-loop correction to the spectrum at large scales. In this work, we repeat the calculation for the 1-loop correction presented in ref. [1]. We closely follow their assumptions but evaluate the loop numerically. This allows us to consider both instantaneous and smooth transitions between the slow-roll and ultra-slow-roll phases. In particular, we generate models featuring realistic, smooth evolution from an analytic inflationary potential. We find that, upon fixing the amplitude of the peak in the power spectrum at short scales, the resulting 1-loop correction is not significantly reduced by considering a smooth evolution. In particular, for a power spectrum with a tree-level peak amplitude potentially relevant for small-scale phenomenology, e.g. primordial black hole production, the 1-loop correction on large scales is a few percent of the tree-level power spectrum.

Spectral distortions from promising single and multifield inflationary models

Forthcoming missions probing the absolute intensity of the CMB are expected to be able to measurespectral distortions, which are deviations from its blackbody distribution. As cosmic inflation can induce spectraldistortions, these experiments offer a possibility to further test the various promising inflationaryproposals, whose predictions need to be carefully determined. After numerically fitting all inflationaryobservables to match current observations, we compute the predicted spectral distortions ofvarious promising single and multifield inflationary models. The predictions of single-fieldinflationary models display deviations between 0.5% and 20% with respect to the standard cosmologicalmodel in the observable window, where multi-natural and axion-monodromy inflation stand out in this respect.In the case of multifield inflation, we observe a richer structure of the power spectrum, which, inthe case of so-called hybrid attractors, yields spectral distortions about 100 times more intense thanthe standard signal. These observations open up questions about the relation among our resultsand other cosmological observables that are also to be probed soon, such as the production ofprimordial black holes and gravitational waves.

An exact model for enhancing/suppressing primordial fluctuations

Abstract Enhancements of primordial curvature fluctuations in single field inflation often involve departures from attractor trajectories in the phase space. We study enhancement/suppression of primordial fluctuations in one of the simplest models with exact background solutions for arbitrary initial conditions: a single field inflationary model with a piecewise exponential potential. We then present close to exact analytical solutions for primordial fluctuations in a general transition between two slow-roll attractors, valid whether the first slow parameter increases or decreases. The main features in the primordial spectrum are determined by the ratio of exponents of the potential. We also discuss the imprint of such features in the induced GW spectrum. Lastly, we apply the δN formalism to discuss non-Gaussianities and the tail of the probability distribution. We find that while non-Gaussianities are at most 𝒪(1) in the case of enhancement, they can be very large in the case of suppression. Our work can be easily generalized to multiple piecewise exponential potentials.

Effects of fragmentation on post-inflationary reheating

We consider the effects of fragmentation on the post-inflationary epoch of reheating.In simple single field models of inflation, an inflaton condensate undergoes an oscillatory phase once inflationary expansion ends. The equation of state of the condensate depends on the shape of the scalar potential, V(ϕ), about its minimum. Assuming V(ϕ) ∼ ϕk , the equation of state parameter is given by w = Pϕ /ρϕ = (k - 2)/(k + 2).The evolution of condensate and the reheating process depend on k. For k ≥ 4, inflaton self-interactions may lead to the fragmentation of the condensate and alter the reheating process. Indeed, these self-interactions lead to the production of a massless gas of inflaton particles as w relaxes to 1/3. If reheating occurs before fragmentation, the effects of fragmentation are harmless. We find, however, that the effects of fragmentation depend sensitively to the specific reheating process. Reheating through the decays to fermions is largely excluded since perturbative couplings would imply that fragmentation occurs before reheating and in fact could prevent reheating from completion. Reheating through the decays to boson is relatively unaffected by fragmentation and reheating through scatterings results in a lower reheating temperature.

Quantum loop effects on the power spectrum and constraints on primordial black holes

We present a detailed exposition on the prospects of the formation of Primordial Black Holes (PBHs) during Slow Roll (SR) to Ultra Slow Roll (USR) sharp transitions in the framework of single-field inflation. We use an effective field theory (EFT) approach in order to keep the analysis model-independent and applicable to both the canonical and non-canonical cases. We show in detail how renormalizing the power spectrum to one loop order in P(X,ϕ) theories severely limits the prospects for PBH formation in a single-field inflationary framework. We demonstrate that for the allowed range of effective sound speed, 1 < cs < 1.17, the consistency of one-loop corrected power spectrum leaves a small window for black hole masses, M PBH∼𝒪(102–103)gm to have sufficient e-foldings, Δ𝒩Total ∼ 𝒪(54–59) for inflation. We confirm that adding an SR regime after USR before the end of inflation does not significantly alter our conclusions. Our findings for sharp transition strictly rule out the possibility of generating large masses of PBHs from all possible models of single field inflation (canonical and non-canonical).Our results are at least valid for the situation where constraints from the loop effects are computed using either Late-Time (LT) or Adiabatic-Wave function(AF) scheme followed by Power Spectrum (PS) renormalization schemes.

Highly asymmetric probability distribution from a finite-width upward step during inflation

We study a single-field inflation model in which the inflaton potential has an upward step between two slow-roll regimes by taking into account the finite width of the step. We calculate the probability distribution function (PDF) of the curvature perturbation P[ℛ] using the δN formalism. The PDF has an exponential-tail only for positive ℛ whose slope depends on the step width. We find that the tail may have a significant impact on the estimation of the primordial black hole abundance. We also show that the PDF P[ℛ] becomes highly asymmetric on a particular scale exiting the horizon before the step, at which the curvature power spectrum has a dip. This asymmetric PDF may leave an interesting signature in the large scale structure such as voids.

Mimicking two field inflationary features with a single field

It is well known that damped superimposed oscillations at large scales in the primordial power spectrum can be generated in both single field and two field models. In single field inflationary models, these features typically arise due to deviations from the slow roll regime. On the other hand, in two field models, these features are generated due to a turn in the background trajectory in the field space. In this work, we demonstrate that both single field and two field models can produce identical features at large scales in the primordial power spectrum. To achieve this, we utilize the generalized slow roll approximation and successfully reconstruct single field models based on the featured power spectrum typically generated in two field models. To validate our methodology, we numerically calculate the power spectrum from the reconstructed potential and find a remarkable agreement with the power spectrum which is obtained from the two field model.

One-loop corrections in power spectrum in single field inflation

We revisit the one-loop correction in curvature perturbation power spectrum in models of single field inflation which undergo a phase of ultra slow-roll (USR) inflation. We include the contributions from both the cubic and quartic interaction Hamiltonians and calculate the one-loop corrections on the spectrum of the CMB scale modes from the small scale modes which leave the horizon during the USR phase. It is shown that the amplitude of one-loop corrections depends on the sharpness of the transition from the USR phase to the final slow-roll phase. For an arbitrarily sharp transition, the one-loop correction becomes arbitrarily large, invalidating the perturbative treatment of the analysis. We speculate that for a mild transition, the large one-loop corrections are washed out during the subsequent evolution after the USR phase. The implications for primordial black holes formation are briefly reviewed.

Tail diversity from inflation

The tail of the distribution of primordial fluctuations (corresponding to the likelihood of realization of large fluctuations) is of interest, from both theoretical and observational perspectives. In particular, it is relevant for the accurate evaluation of the primordial black hole (PBH) abundance. In this paper, we first analyze the non-perturbative δN formalism as a method to non-perturbatively estimate the probability distribution function (PDF) of primordial fluctuations, discuss its underlying assumptions and deal with several subtleties that may arise as a result of considering large fluctuations. Next, we employ the method to study several non-attractor single-field inflationary models as the simplest examples that may lead to the abundant production of PBHs. We conclude that the Gaussian extrapolation from linear perturbation theory may fail drastically to predict the likelihood of large fluctuations. Specifically, we show that a truncation of the tail, a power-law tail, a double-exponential tail, and a doubly peaked distribution can all be realized for the curvature perturbation in the single-field non-attractor models of inflation. We thus show that there is a diverse zoo of possible tails from inflation so that a model-dependent, non-perturbative study of the distribution of the primordial fluctuations seems inevitable concerning PBH abundance.

Primordial black holes and stochastic inflation beyond slow roll. Part I. Noise matrix elements

Primordial Black Holes (PBHs) may form in the early Universe, from the gravitational collapse of large density perturbations, generated by large quantum fluctuations during inflation. Since PBHs form from rare over-densities, their abundance is sensitive to the tail of the primordial probability distribution function (PDF) of the perturbations. It is therefore important to calculate the full PDF of the perturbations, which can be done non-perturbatively using the `stochastic inflation' framework. In single field inflation models generating large enough perturbations to produce an interesting abundance of PBHs requires violation of slow roll. It is therefore necessary to extend the stochastic inflation formalism beyond slow roll. A crucial ingredient for this are the stochastic noise matrix elements of the inflaton potential. We carry out analytical and numerical calculations of these matrix elements for a potential with a feature which violates slow roll and produces large, potentially PBH generating, perturbations. We find that the transition to an ultra slow-roll phase results in the momentum induced noise terms becoming larger than the field noise whilst each of them falls exponentially for a few e-folds. The noise terms then start rising with their original order restored, before approaching constant values which depend on the nature of the slow roll parameters in the post transition epoch. This will significantly impact the quantum diffusion of the coarse-grained inflaton field, and hence the PDF of the perturbations and the PBH mass fraction.

Primordial black holes from single-field inflation: a fine-tuning audit

All single-field inflationary models invoke varying degrees of tuning in order to account for cosmological observations. Mechanisms that generate primordial black holes (PBHs) from enhancement of primordial power at small scales posit inflationary potentials that transiently break scale invariance and possibly adiabaticity over a range of modes. This requires additional tuning on top of that required to account for observations at scales probed by cosmic microwave background (CMB) anisotropies. In this paper we study the parametric dependence of various single-field models of inflation that enhance power at small scales and quantify the degree to which coefficients in the model construction have to be tuned in order for certain observables to lie within specified ranges. We find significant tuning: changing the parameters of the potentials by between one part in a hundred and one part in 108 (depending on the model) is enough to change the power spectrum peak amplitude by an order one factor. The fine-tuning of the PBH abundance is larger still by 1–2 orders of magnitude. We highlight the challenges imposed by this tuning on any given model construction. Furthermore, polynomial potentials appear to require significant additional fine-tuning to also match the CMB observations.

Logarithmic Duality of the Curvature Perturbation.

We study the comoving curvature perturbation R in the single-field inflation models whose potential can be approximated by a piecewise quadratic potential V(φ) by using the δN formalism. We find a general formula for R(δφ,δπ), consisting of a sum of logarithmic functions of the field perturbation δφ and the velocity perturbation δπ at the point of interest, as well as of δπ_{*} at the boundaries of each quadratic piece, which are functions of (δφ,δπ) through the equation of motion. Each logarithmic expression has an equivalent dual expression, due to the second-order nature of the equation of motion for φ. We also clarify the condition under which R(δφ,δπ) reduces to a single logarithm, which yields either the renowned "exponential tail" of the probability distribution function of R or a Gumbel-distribution-like tail.

Production of Primordial Black Holes in Improved E-Models of Inflation

E-type α-attractor models of single-field inflation were generalized further in order to accommodate production of primordial black holes (PBHs) via adding a near-inflection point to the inflaton scalar potential at smaller scales, in good agreement with measurements of cosmic microwave background (CMB) radiation. A minimal number of new parameters were used but their fine-tuning was maximized in order to increase the possible masses of PBHs formed during an ultra-slow-roll phase, leading to a large enhancement in the power spectrum of scalar (curvature) perturbations by 6 or 7 orders of magnitude against the power spectrum of perturbations observed in CMB. It was found that extreme fine-tuning of the parameters in our models can lead to the formation of moon-sized PBHs, with masses of up to 1026 g, still in agreement with CMB observations. Quantum corrections are known to lead to the perturbative upper bound on the amplitude of large scalar perturbations responsible for PBH production. The quantum (one-loop) corrections in our models were found to be suppressed by one order of magnitude for PBHs with masses of approximately 1019 g, which may form the whole dark matter in the Universe.

Isocurvature constraints on scalar dark matter production from the inflaton

We investigate the production of a spectator scalar dark matter field that is directly coupled to the inflaton during inflation and reheating. We consider two specific inflationary potentials, namely the Starobinsky and T-model of inflation, which satisfy the constraints on the scalar tilt, $n_s$, and tensor-to-scalar ratio, $r$, measured by the Planck satellite. Excitation of light scalar dark matter during inflation may result in large isocurvature perturbations, which can be avoided by inducing a sizable effective dark matter mass during the inflationary phase. For purely gravitational production, the Planck isocurvature constraints require the dark matter mass to be larger than the Hubble scale at horizon exit, with $m_{\chi} \gtrsim 0.5H_*$. For small bare dark matter masses $m_{\chi} \ll H_*$, these constraints translate into a lower bound on the dark matter coupling to the inflaton. We argue that these constraints can be applied to a wide class of single-field slow-roll inflation models. We also derive isocurvature, dark matter abundance, and Lyman-$\alpha$ constraints on the direct coupling and bare dark matter mass.

Entanglement masquerading in the CMB

The simplest single-field inflation models capture all the relevant contributions to the patterns in the Cosmic Microwave Background (CMB) observed today. A key assumption in these models is that the quantum inflationary fluctuations that source such patterns are generated by a particular quantum state — the Bunch-Davies (BD) state. While this is a well-motivated choice from a theoretical perspective, the question arises of whether current data can rule out other, also well motivated, choices of states. In particular, as we previously demonstrated in [1], entanglement is naturally and inevitably dynamically generated during inflation given the presence of a “rolling” spectator scalar field — and the resulting entangled state will yield a primordial power spectrum with potentially measurable deviations compared to the canonical BD result. For this work we developed a perturbative framework to allow a systematic exploration of constraints on (or detection of) entangled states with Planck CMB data using Monte Carlo techniques. We have found that most entangled states accessible with our framework are consistent with the data. One would have to expand the framework to allow a greater variety of entangled states in order to saturate the Planck constraints and more systematically explore any preferences the data may have among the different possibilities.

Understanding large scale CMB anomalies with the generalized nonminimal derivative coupling during inflation

We study the observational implications of a class of inflationary models wherein the inflaton is coupled to the Einstein tensor through a generalized nonminimal derivative coupling (GNMDC). In particular, we explore whether these models can generate suitable features in the primordial spectrum of curvature perturbations as a possible explanation for the large-scale anomalies associated with the angular power spectrum of CMB temperature anisotropies. We derive model-independent constraints on the GNMDC function for such a scenario, considering both the scalar and tensor perturbations. We modify cosmomc to accommodate our GNMDC framework and investigate different classes of inflationary models using a fully consistent numerical approach. We find that the hilltop-quartic model with a specific choice of the GNMDC function provides a considerable improvement over the best-fit reference $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ model with a nearly scale-invariant power spectrum. While the large-scale structure observations should be able to provide independent constraints, future CMB experiments, such as CMB-S4 and CMB-Bharat, are expected to constrain further the parameter space of such beyond canonical single-field inflationary models.

Nonminimally assisted inflation: a general analysis

The effects of a scalar field, known as the “assistant field,” which nonminimally couples to gravity, on single-field inflationary models are studied. The analysis provides analytical expressions for inflationary observables such as the spectral index (ns ), the tensor-to-scalar ratio (r), and the local-type nonlinearity parameter (f NL (local)). The presence of the assistant field leads to a lowering of ns and r in most of the parameter space, compared to the original predictions. In some cases, ns may increase due to the assistant field. This revives compatibility between ruled-out single-field models and the latest observations by Planck-BICEP/Keck. The results are demonstrated using three example models: loop inflation, power-law inflation, and hybrid inflation.