We calculate the Bayesian evidences for a class of Ekpyrotic universe models, and compare with a model of single field inflation with a Higgs-type potential. Combining parsimony and observational constraints, this gives us a systematic way to evaluate the degree to which Ekpyrotic models are constrained by CMB data from Planck. We integrate the equations of motion numerically to define a likelihood using Planck 2018 data and sample this likelihood to obtain Bayesian evidences. Priors are justified and used to put Ekpyrotic models and inflation on equal footing. We find reasonable preference for one of the considered Ekpyrotic models over the others, but that even this one is disfavored compared with Higgs inflation.
Certain features in the primordial scalar power spectrum are known to improve the fit to the cosmological data. We examine whether bouncing scenarios can remain viable if future data confirm the presence of such features. In inflation, the fact that the trajectory is an attractor permits the generation of features. However, bouncing scenarios often require fine tuned initial conditions, and it is only the ekpyrotic models that allow attractors. We demonstrate, for the first time, that ekpyrotic scenarios can generate specific features that have been considered in the context of inflation.
The results from Planck2015, when combined with earlier observations from WMAP, ACT, SPT and other experiments, were the first observations to disfavor the "classic" inflationary paradigm. To satisfy the observational constraints, inflationary theorists have been forced to consider plateau-like inflaton potentials that introduce more parameters and more fine-tuning, problematic initial conditions, multiverse-unpredictability issues, and a new 'unlikeliness problem.' Some propose turning instead to a "postmodern" inflationary paradigm in which the cosmological properties in our observable unive…
The no-boundary proposal provides a compelling theory for the initial conditions of our universe. We study the implications of such initial conditions for ekpyrotic and cyclic cosmologies. These cosmologies allow for the existence of a new type of "ekpyrotic instanton", which describes the creation of a universe in the ekpyrotic contraction phase. Remarkably, we find that the ekpyrotic attractor can explain how the universe became classical. In a cyclic context, in addition to the ekpyrotic instantons there exist de Sitter-like instantons describing the emergence of the universe in the dark en…
The alternative scalar field dark sector of the Universe: Cosmological inflation, quintessence and ekpyrotic model in the framework of scalar-tensor cosmology
This paper investigates the manner in which classical universes are obtained in the no-boundary quantum state. In this context, universes can be characterised as classical (in a WKB sense) when the wavefunction is highly oscillatory, i.e. when the ratio of the change in the amplitude of the wavefunction becomes small compared to the change in the phase. In the presence of a positive or negative exponential potential, the WKB condition is satisfied in proportion to a factor $e^{-(ε- 3)N/(ε-1)},$ where $ε$ is the (constant) slow-roll/fast-roll parameter and $N$ designates the number of e-folds. …
We present a new class of complex instantons in the context of ekpyrotic cosmological theories. These instantons, which satisfy the "no-boundary" boundary conditions, describe the emergence of a classical, contracting universe out of nothing. The ekpyrotic attractor is essential in guaranteeing an evolution towards a real, Lorentzian history of the universe. In the context of the no-boundary proposal, the relative probability for such ekpyrotic histories compared to inflationary ones is very high -- in fact, assuming a bounce can be incorporated, these new instantons currently describe the mos…
Both inflationary and ekpyrotic scenarios can account for the origin of the large scale structure of the universe. It is often said that detecting primordial gravitational waves is the key to distinguish both scenarios. We show that this is not true if the gauge kinetic function is present in the ekpyrotic scenario. In fact, primordial gravitational waves sourced by the gauge field can be produced in an ekpyrotic universe. We also study scalar fluctuations sourced by the gauge field and show that it is negligible compared to primordial gravitational waves. This comes from the fact that the fas…
Ekpyrotic bouncing cosmologies have been proposed as alternatives to inflation. In these scenarios, the universe is smoothed and flattened during a period of slow contraction preceding the bounce while quantum fluctuations generate nearly scale-invariant super-horizon perturbations that seed structure in the post-bounce universe. An analysis by Tolley and Wesley (2007) showed that, for a wide range of ekpyrotic models, generating a scale-invariant spectrum of adiabatic or entropic fluctuations is only possible if the cosmological background is unstable, in which case the scenario is highly sen…
There is increasing evidence from string theory that effective field theories are only reliable over approximately sub-Planckian field excursions. The two most promising effective models for early universe cosmology, inflation and ekpyrosis, are mechanisms that, in order to address cosmological puzzles, must operate over vast expansion/energy ranges. This suggests that it might be appropriate to describe them using scaling laws. Here we combine these two ideas and demonstrate that they drive inflation and ekpyrosis to their extremes: inflation must start at ultra-slow-roll, and ekpyrosis at ul…
We derive all single-field cosmologies with unit sound speed that generate scale invariant curvature perturbations on a dynamical attractor background. We identify three distinct phases: slow-roll inflation; a slowly contracting adiabatic ekpyrotic phase, described by a rapidly-varying equation of state; and a novel adiabatic ekpyrotic phase on a slowly expanding background. All of these yield identical power spectra. The degeneracy is broken at the 3-point level: unlike the nearly gaussian spectrum of slow-roll inflation, adiabatic ekpyrosis predicts large non-gaussianities on small scales.
We consider models of accelerating Universe elaborated for Finsler like gravity theories constructed on tangent bundles to Lorentz manifolds. In the osculating approximation, certain locally anisotropic configurations are similar to those for f(R) gravity. This allows us to generalize a proposal (by Nojiri, Odintsov and Saez-Gomez, arXiv: 1108.0767) in order to reconstruct and compare two classes of Einstein-Finsler gravity, EFG, and f(R) gravity theories using modern cosmological data and realistic physical scenarios. We conclude that EFG provides inflation, acceleration and little rip evolut…
The scalar field is considered to have dominated the early Universe. One subtle yet crucial requirement of this assumption is that the solution must be highly stable, i.e., indifferent to any initial conditions because there are no favored ones. Inflation, which is now the most successful early Universe paradigm, answers most of the early Universe's problems, including the fact that it is mostly stable. In this article, in addition to the inflationary solution, we systematically investigate every possible early Universe solution in the presence of a barotropic fluid in the general non-minimal …
Cosmological models involving a bounce from a contracting to an expanding universe can address the standard cosmological puzzles and generate "primordial" density perturbations without the need for inflation. Some such models, in particular the ekpyrotic and cyclic models that we focus on, fit rather naturally into string theory. We discuss a number of topics related to these models: the reasoning that leads to the ekpyrotic phase, the predictions for upcoming observations, the differences between singular and non-singular models of the bounce as well as the predictive and explanatory power of…
Combining intervals of ekpyrotic (ultra-slow) contraction with a (non-singular) classical bounce naturally leads to a novel cyclic theory of the universe in which the Hubble parameter, energy density and temperature oscillate periodically, but the scale factor grows by an exponential factor from one cycle to the next. The resulting cosmology not only resolves the homogeneity, isotropy, flatness and monopole problems and generates a nearly scale invariant spectrum of density perturbations, but it also addresses a number of age-old cosmological issues that big bang inflationary cosmology does no…
By means of Bayesian techniques, we study how a premature ending of inflation, motivated by geometrical destabilization, affects the observational evidences of typical inflationary models. Large field models are worsened, and inflection point potentials are drastically improved for a specific range of the field-space curvature characterizing the geometrical destabilization. For other models we observe shifts in the preferred values of the model parameters. For quartic hilltop models for instance, contrary to the standard case, we find preference for theoretically natural sub-Planckian hill wid…
The current available CMB data show an anomalously low value of the CMB temperature fluctuations at large angular scales (l < 40). This lack of power is not explained by the minimal LCDM model, and one of the possible mechanisms explored in the literature to address this problem is the presence of features in the primordial power spectrum (PPS) motivated by the early universe physics. In this paper, we analyse a set of cutoff inflationary PPS models using a Bayesian model comparison approach in light of the latest Cosmic Microwave Background (CMB) data from the Planck Collaboration. Our result…
We present three non-parametric Bayesian primordial reconstructions using Planck 2018 polarization data: linear spline primordial power spectrum reconstructions, cubic spline inflationary potential reconstructions and sharp-featured primordial power spectrum reconstructions. All three methods conditionally show hints of an oscillatory feature in the primordial power spectrum in the multipole range $\ell\sim20$ to $\ell\sim50$, which is to some extent preserved upon marginalization. We find no evidence for deviations from a pure power law across a broad observable window ($50\lesssim\ell\lesssi…
We use the most recent cosmic microwave background (CMB) data to perform a Bayesian statistical analysis and discuss the observational viability of inflationary models with a non-minimal coupling,~$ξ$, between the inflaton field and the Ricci scalar. We particularize our analysis to two examples of small and large field inflationary models, namely, the Coleman-Weinberg and the chaotic quartic potentials. We find that (i) the $ξ$ parameter is closely correlated with the primordial amplitude; (ii) although improving the agreement with the CMB data in the $r - n_s$ plane, where $r$ is the tensor-…
We discuss the model selection problem for inflationary cosmology. We couple ModeCode, a publicly-available numerical solver for the primordial perturbation spectra, to the nested sampler MultiNest, in order to efficiently compute Bayesian evidence. Particular attention is paid to the specification of physically realistic priors, including the parametrization of the post-inflationary expansion and associated thermalization scale. It is confirmed that while present-day data tightly constrains the properties of the power spectrum, it cannot usefully distinguish between the members of a large cla…
In this paper we consider the issue of paradigm evaluation by applying Bayes' theorem along the following nested hierarchy of progressively more complex structures: i) parameter estimation (within a model), ii) model selection and comparison (within a paradigm), iii) paradigm evaluation. In such a hierarchy the Bayesian evidence works both as the posterior's normalization at a given level and as the likelihood function at the next level up. Whilst raising no objections to the standard application of the procedure at the two lowest levels, we argue that it should receive a considerable modifica…
We present and discuss a new approach increasing by orders of magnitude the speed of performing Bayesian inference and parameter estimation within the framework of slow-roll inflation. The method relies on the determination of an effective likelihood for inflation which is a function of the primordial amplitude of the scalar perturbations complemented with the necessary number of the so-called Hubble flow functions to reach the desired accuracy. Starting from any cosmological data set, the effective likelihood is obtained by marginalisation over the standard cosmological parameters, here viewe…
We present the first calculation of the Bayesian evidence for different prototypical single field inflationary scenarios, including representative classes of small field and large field models. This approach allows us to compare inflationary models in a well-defined statistical way and to determine the current "best model of inflation". The calculation is performed numerically by interfacing the inflationary code FieldInf with MultiNest. We find that small field models are currently preferred, while large field models having a self-interacting potential of power p>4 are strongly disfavoured. T…
We investigate the effects of non-zero spatial curvature on cosmic inflation in the light of cosmic microwave background (CMB) anisotropy measurements from the Planck 2018 legacy release and from the 2015 observing season of BICEP2 and the Keck Array. Even a small percentage of non-zero curvature today would significantly limit the total number of e-folds of the scale factor during inflation, rendering just-enough inflation scenarios with a kinetically dominated or fast-roll stage prior to slow-roll inflation more likely. Finite inflation leads to oscillations and a cutoff towards large scales…
Bayesian model comparison penalizes models with more free parameters that are allowed to vary over a wide range, and thus offers the most robust method to decide whether some given data require new parameters. In this paper, we ask a simple question: do current cosmological data require extensions of the simplest single-field inflation models? Specifically, we calculate the Bayesian evidence of a totally anti-correlated isocurvature perturbation and a running spectral index of the scalar curvature perturbation. These parameters are motivated by recent claims that the observed temperature aniso…
Modern cosmological data demand modern data analysis techniques. We introduce BayOp, a new likelihood sampling and maximisation method which is based on the Bayesian Optimisation algorithm and learns a function instead of randomly sampling from it. We apply BayOp to analyse Planck data for traces of inflationary features models with global periodic modulations of the primordial power spectrum. While we do not find any new evidence for features, we demonstrate that BayOp provides an extremely efficient way of sampling likelihoods over low-to-moderate-dimensional parameter spaces, even for very …
We compute the Bayesian evidence and complexity of 193 slow-roll single-field models of inflation using the Planck 2013 Cosmic Microwave Background data, with the aim of establishing which models are favoured from a Bayesian perspective. Our calculations employ a new numerical pipeline interfacing an inflationary effective likelihood with the slow-roll library ASPIC and the nested sampling algorithm MULTINEST. The models considered represent a complete and systematic scan of the entire landscape of inflationary scenarios proposed so far. Our analysis singles out the most probable models (from …
We perform a Bayesian analysis to study possible features in the primordial inflationary power spectrum of scalar perturbations. In particular, we analyse the possibility of detecting the imprint of these primordial features in the anisotropy temperature power spectrum of the Cosmic Microwave Background (CMB) and also in the matter power spectrum P (k). We use the most recent CMB data provided by the Planck Collaboration and P (k) measurements from the eleventh data release of the Sloan Digital Sky Survey. We focus our analysis on a class of potentials whose features are localised at different…
We implement Slow Roll Reconstruction -- an optimal solution to the inverse problem for inflationary cosmology -- within ModeCode, a publicly available solver for the inflationary dynamics. We obtain up-to-date constraints on the reconstructed inflationary potential, derived from the WMAP 7-year dataset and South Pole Telescope observations, combined with large scale structure data derived from SDSS Data Release 7. Using ModeCode in conjunction with the MultiNest sampler, we compute Bayesian evidence for the reconstructed potential at each order in the truncated slow roll hierarchy. We find th…
Future astrophysical datasets promise to strengthen constraints on models of inflation, and extracting these constraints requires methods and tools commensurate with the quality of the data. In this paper we describe ModeCode, a new, publicly available code that computes the primordial scalar and tensor power spectra for single field inflationary models. ModeCode solves the inflationary mode equations numerically, avoiding the slow roll approximation. It is interfaced with CAMB and CosmoMC to compute cosmic microwave background angular power spectra and perform likelihood analysis and paramete…
In this thesis we will primarily demonstrate how additional scalar degrees of freedom - which are motivated from many high-energy embeddings - open up new observational windows onto the physics of inflation. We construct a Bayesian framework to statistically compare models with additional fields given the current astronomical data. Putting inflation to the test, we perform our analysis on the quadratic curvaton accompanying a range of inflationary potentials, where we find that only one potential remains as a viable candidate. Furthermore, if the curvaton mechanism were to be confirmed by futu…
We constrain cosmological models where the primordial perturbations have both an adiabatic and a (possibly correlated) cold dark matter (CDM) or baryon isocurvature component. We use both a phenomenological approach, where the primordial power spectra are parametrized with amplitudes and spectral indices, and a slow-roll two-field inflation approach where slow-roll parameters are used as primary parameters. In the phenomenological case, with CMB data, the upper limit to the CDM isocurvature fraction is α<6.4% at k=0.002Mpc^{-1} and 15.4% at k=0.01Mpc^{-1}. The median 95% range for the non-adia…
We review the constraints that the recently released Cosmic Microwave Background (CMB) Planck data put on inflation and we argue that single field slow-roll inflationary scenarios (with minimal kinetic term) are favored. Then, within this class of models, by means of Bayesian inference, we show how one can rank the scenarios according to their performances, leading to the identification of ``the best models of inflation''.
In this short paper, we present constraints on the Galileon inflationary model from the CMB bispectrum. We employ a principal-component analysis of the independent degrees of freedom constrained by data and apply this to the WMAP 9-year data to constrain the free parameters of the model. A simple Bayesian comparison establishes that support for the Galileon model from bispectrum data is at best weak.
We investigate the implications for inflation of the detection of B-modes polarization in the Cosmic Microwave Background (CMB) by BICEP2. We show that the hypothesis of primordial origin of the measurement is only favored by the first four bandpowers, while the others would prefer unreasonably large values of the tensor-to-scalar ratio. Using only those four bandpowers, we carry out a complete analysis in the cosmological and inflationary slow-roll parameter space using the BICEP2 polarization measurements alone and extract the Bayesian evidences and complexities for all the Encyclopaedia Inf…
These lecture notes have been written for a short introductory course on the status of inflation after Planck and BICEP2, given at the Xth Modave School of Mathematical Physics. The first objective is to give an overview of the theory of inflation: motivations, homogeneous scalar field dynamics, slow-roll approximation, linear theory of cosmological perturbations, classification of single field potentials and their observable predictions. This includes a pedagogical derivation of the primordial scalar and tensor power spectra for any effective single field potential. The second goal is to pres…
How much more will we learn about single-field inflationary models in the future? We address this question in the context of Bayesian design and information theory. We develop a novel method to compute the expected utility of deciding between models and apply it to a set of futuristic measurements. This necessarily requires one to evaluate the Bayesian evidence many thousands of times over, which is numerically challenging. We show how this can be done using a number of simplifying assumptions and discuss their validity. We also modify the form of the expected utility, as previously introduced…
In a class of recently proposed models, the early universe is strongly coupled and described holographically by a three-dimensional, weakly coupled, super-renormalizable quantum field theory. This scenario leads to a power spectrum of scalar perturbations that differs from the usual empirical LCDM form and the predictions of generic models of single field, slow roll inflation. This spectrum is characterized by two parameters: an amplitude, and a parameter g related to the coupling constant of the dual theory. We estimate these parameters, using WMAP and other astrophysical data. We compute Bay…
We analyze a class of generalized inflationary models proposed in Ref. [1], known as $β$-exponential inflation. We show that this kind of potential can arise in the context of brane cosmology, where the field describing the size of the extra-dimension is interpreted as the inflaton. We discuss the observational viability of this class of model in light of the latest Cosmic Microwave Background (CMB) data from the Planck Collaboration through a Bayesian analysis, and impose tight constraints on the model parameters. We find that the CMB data alone prefer \textit{weakly} the minimal standard mod…
We study a model of inflation based on $\mathcal{N}=1$ supergravity essentially depending on one effective parameter. Under a field transformation we show that this model turns out to be equivalent to a previously studied supergravity model known to be ruled out with the original choice of the parameter. Such parameter measures the slope of the potential at observable scales. Through a Bayesian parameter estimation, it is shown how this model is compatible with recent CMB temperature measurements by {\it Planck 2018} giving rise to a simple, viable, single field model of inflation. The tensor …
We use WMAP 9-year bispectrum data to constrain the free parameters of an 'effective field theory' describing fluctuations in single-field inflation. The Lagrangian of the theory contains a finite number of operators associated with unknown mass scales. Each operator produces a fixed bispectrum shape, which we decompose into partial waves in order to construct a likelihood function. Based on this likelihood we are able to constrain four linearly independent combinations of the mass scales. As an example of our framework we specialize our results to the case of 'Dirac-Born-Infeld' and 'ghost' i…
We investigate the initial conditions problem for multifield inflation. In these scenarios the pre-inflationary dynamics can be chaotic, increasing the sensitivity of the onset of inflation to the initial data even in the homogeneous limit. To analyze physically equivalent scenarios we compare initial conditions at fixed energy. This ensures that each trajectory is counted once and only once, since the energy density decreases monotonically. We present a full analysis of hybrid inflation that reveals a greater degree of long range order in the set of "successful" initial conditions than was pr…
We constrain several models in Tachyonic Inflation derived from the large-$N$ formalism by considering theoretical aspects as well as the latest observational data. On the theoretical side, we assess the field range of our models by means of the excursion of the equivalent canonical field. On the observational side, we employ BK14+PLANCK+BAO data to perform a parameter estimation analysis as well as a Bayesian model selection to distinguish the most favoured models among all four classes here presented. We observe that the original potential $V \propto \textrm{sech}(T)$ is strongly disfavoured…
Bayesian analysis of $α$-Starobinsky model with Planck, ACT and DESI data
Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México（墨西哥国立自治大学物理科学研究所）；Instituto de Física, Universidad Nacional Autónoma de México（墨西哥国立自治大学物理研究所）；Facultad de Ciencias Físico-Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo（圣尼古拉斯德伊达尔戈米却肯大学理学院）
We study how the next generation of Cosmic Microwave Background (CMB) measurement missions (such as EPIC, LiteBIRD, PRISM and COrE) will be able to constrain the inflationary landscape in the hardest to disambiguate situation in which inflation is simply described by single-field slow-roll scenarios. Considering the proposed PRISM and LiteBIRD satellite designs, we simulate mock data corresponding to five different fiducial models having values of the tensor-to-scalar ratio ranging from $10^{-1}$ down to $10^{-7}$. We then compute the Bayesian evidences and complexities of all Encyclopaedia In…
Theory testing in the physical sciences has been revolutionized in recent decades by Bayesian approaches to probability theory. Here, I will consider Bayesian approaches to theory extensions, that is, theories like inflation which aim to provide a deeper explanation for some aspect of our models (in this case, the standard model of cosmology) that seem unnatural or fine-tuned. In particular, I will consider how cosmologists can test the multiverse using observations of this universe.
We perform a Bayesian model comparison for scenarios within the quadratic curvaton model, determining the degree to which both are disfavoured with respect to the $Λ$CDM concordance model and single-field quadratic inflation, using the recent \emph{Planck} data release. Despite having three additional model parameters, the simplest curvaton scenario is not disfavoured relative to single-field quadratic inflation, and it becomes favoured against this single-field model when we include the joint BICEP/Keck/\emph{Planck} analysis. In all cases we assume an instantaneous inflaton decay and no surv…
Bayesian inference techniques are used to investigate situations where an additional light scalar field is present during inflation and reheating. This includes (but is not limited to) curvaton-type models. We design a numerical pipeline where $\simeq 200$ inflaton setups $\times\, 10$ reheating scenarios $= 2000$ models are implemented and we present the results for a few prototypical potentials. We find that single-field models are remarkably robust under the introduction of light scalar degrees of freedom. Models that are ruled out at the single-field level are not improved in general, beca…
