Multi-field Inflation Models Research Articles

PyTransport: A Python package for the calculation of inflationary correlation functions

PyTransport constitutes a straightforward code written in C++ together with Python scripts which automatically edit, compile and run the C++ code as a Python module. It has been written for Unix-like systems (OS X and Linux). Primarily the module employs the transport approach to inflationary cosmology to calculate the tree-level power-spectrum and bispectrum of user specified models of multi-field inflation, accounting for all sub and super-horizon effects. The transport method we utilise means only coupled differential equations need to be solved, and the implementation presented here combines the speed of C++ with the functionality and convenience of Python. This document details the code and illustrates how to use it with a worked example. It has been updated to be a companion to the second version of the code, PyTransport 2.0, which includes functionality to deal with models of inflation with a curved field space metric.

Preheating after multifield inflation with nonminimal couplings. I. Covariant formalism and attractor behavior

This is the first of a three-part series of papers, in which we study the preheating phase for multifield models of inflation involving nonminimal couplings. In this paper, we study the single-field attractor behavior that these models exhibit during inflation and quantify its strength and parameter dependence. We further demonstrate that the strong single-field attractor behavior persists after the end of inflation. Preheating in such models therefore generically avoids the "de-phasing" that typically affects multifield models with minimally coupled fields, allowing efficient transfer of energy from the oscillating inflaton condensate(s) to coupled perturbations across large portions of parameter space. We develop a doubly-covariant formalism for studying the preheating phase in such models and identify several features specific to multifield models with nonminimal couplings, including effects that arise from the nontrivial field-space manifold. In papers II and III, we apply this formalism to study how the amplification of adiabatic and isocurvature perturbations varies with parameters, highlighting several distinct regimes depending on the magnitude of the nonminimal couplings $\xi_I$.

Large number limit of multifield inflation

We compute the tensor and scalar spectral index $n_t$, $n_s$, the tensor-to-scalar ratio $r$, the consistency relation $n_t/r$ in the general monomial multifield slow-roll inflation models with potentials $V \sim\sum_i\lambda_i \left|\phi_i\right|^{p_i}$. The general models give a novel relation that $n_t$, $n_s$ and $n_t/r$ are all proportional to the logarithm of the number of fields $N_f$ when $N_f$ is getting extremely large with the order of magnitude around $\mathcal{O}(10^{40})$. An upper bound $N_f\lesssim N_*e^{ZN_*}$ is given by requiring the slow variation parameter small enough where $N_*$ is the e-folding number and $Z$ is a function of distributions of $\lambda_i$ and $p_i$. Besides, $n_t/r$ differs from the single-field result $-1/8$ with substantial probability except for a few very special cases. Finally, we derive theoretical bounds $r>2/N_*$ ($r\gtrsim0.03$) and for $n_t$ which can be tested by observation in the near future.

Two-point correlators revisited: fast and slow scales in multifield models of inflation

We study the structure of two-point correlators of the inflationary field fluctuations in order to improve the accuracy and efficiency of the existing methods to calculate primordial spectra. We present a description motivated by the separation of the fast and slow evolving components of the spectrum which is based on Cholesky decomposition of the field correlator matrix. Our purpose is to rewrite all the relevant equations of motion in terms of slowly varying quantities. This is important in order to consider the contribution from high-frequency modes to the spectrum without affecting computational performance. The slow-roll approximation is not required to reproduce the main distinctive features in the power spectrum for each specific model of inflation.

Primordial black holes from single field models of inflation

Primordial black holes (PBH) have been shown to arise from high peaks in the matter power spectra of multi-field models of inflation. Here we show, with a simple toy model, that it is also possible to generate a peak in the curvature power spectrum of single-field inflation. We assume that the effective dynamics of the inflaton field presents a near-inflection point which slows down the field right before the end of inflation and gives rise to a prominent spike in the fluctuation power spectrum at scales much smaller than those probed by Cosmic Microwave Background (CMB) and Large Scale Structure (LSS) observations. This peak will give rise, upon reentry during the radiation era, to PBH via gravitational collapse. The mass and abundance of these PBH is such that they could constitute the totality of the Dark Matter today. We satisfy all CMB and LSS constraints and predict a very broad range of PBH masses. Some of these PBH are light enough that they will evaporate before structure formation, leaving behind a large curvature fluctuation on small scales. This broad mass distribution of PBH as Dark Matter will be tested in the future by AdvLIGO and LISA interferometers.

The field-space metric in spiral inflation and related models

Multi-field inflation models include a variety of scenarios for how inflation proceeds and ends. Models with the same potential but different kinetic terms are common in the literature. We compare spiral inflation and Dante's inferno-type models, which differ only in their field-space metric. We justify a single-field effective description in these models and relate the single-field description to a mass-matrix formalism. We note the effects of the nontrivial field-space metric on inflationary observables, and consequently on the viability of these models. We also note a duality between spiral inflation and Dante's inferno models with different potentials.

Effective theory of squeezed correlation functions

Various inflationary scenarios can often be distinguished from one another by looking at the squeezed limit behavior of correlation functions. Therefore, it is useful to have a framework designed to study this limit in a more systematic and efficient way. We propose using an expansion in terms of weakly coupled super-horizon degrees of freedom, which is argued to generically exist in a near de Sitter space-time. The modes have a simple factorized form which leads to factorization of the squeezed-limit correlation functions with power-law behavior in klong/kshort. This approach reproduces the known results in single-, quasi-single-, and multi-field inflationary models. However, it is applicable even if, unlike the above examples, the additional degrees of freedom are not weakly coupled at sub-horizon scales. Stronger results are derived in two-field (or sufficiently symmetric multi-field) inflationary models. We discuss the observability of the non-Gaussian 3-point function in the large-scale structure surveys, and argue that the squeezed limit behavior has a higher detectability chance than equilateral behavior when it scales as (klong/kshort)Δ with Δ < 1—where local non-Gaussianity corresponds to Δ = 0.

Computing observables in curved multifield models of inflation—A guide (with code) to the transport method

We describe how to apply the transport method to compute inflationary observables in a broad range of multiple-field models. The method is efficient and encompasses scenarios with curved field-space metrics, violations of slow-roll conditions and turns of the trajectory in field space. It can be used for an arbitrary mass spectrum, including massive modes and models with quasi-single-field dynamics.In this note we focus on practical issues. It is accompanied by a Mathematica code which can be used to explore suitable models, or as a basis for further development.

Multifield formulation of gravitational particle production after inflation

We study multi-field inflation models that contain a non-trivial field-space metric and a non-minimal coupling between the gravity and inflaton sectors. In such models it is known that even in the absence of explicit interaction terms the inflaton sector can decay into matter as a result of its non-minimal coupling to gravity, thereby reheating the Universe gravitationally. Using the Bogoliubov approach we evaluate the gravitational decay rates of the inflaton fields into both scalars and fermions, and analyse the reheating dynamics. We also discuss how the interpretation of the reheating dynamics differs in the so-called Jordan and Einstein frames, highlighting that the calculation of the Bogoliubov coefficients is independent of the frame in which one starts.

On reaching the adiabatic limit in multi-field inflation

We calculate the scalar spectral index ns and the tensor-to-scalar ratio r in a class of recently proposed two-field no-scale inflationary models in supergravity. We show that, in order to obtain correct predictions, it is crucial to take into account the coupling between the curvature and the isocurvature perturbations induced by the noncanonical form of the kinetic terms. This coupling enhances the curvature perturbation and suppresses the resulting tensor-to-scalar ratio to the per mille level even for values of the slow-roll parameter ϵ ∼ 0.01. Beyond these particular models, we emphasise that multifield models of inflation are a priori not predictive, unless one supplies a prescription for the post-inflationary era, or an adiabatic limit is reached before the end of inflation. We examine the conditions that enabled us to actually derive predictions in the models under study, by analysing the various contributions to the effective isocurvature mass in general two-field inflationary models. In particular, we point out a universal geometrical contribution that is important at the end of inflation, and which can be directly extracted from the inflationary Lagrangian, independently of a specific trajectory. Eventually, we point out that spectator fields can lead to oscillatory features in the time-dependent power spectra at the end of inflation. We demonstrate how these features can be model semi-analytically as well as the theoretical uncertainties they can entail.

MULTIMODECODE: an efficient numerical solver for multifield inflation

We present MULTIMODECODE, a Fortran 95/2000 package for the numerical exploration of multifield inflation models. This program facilitates efficient Monte Carlo sampling of prior probabilities for inflationary model parameters and initial conditions and is the first publicly available code that can efficiently generate large sample-sets for inflation models with \U0001d4aa fields. The code numerically solves the equations of motion for the background and first-order perturbations of multi-field inflation models with canonical kinetic terms and arbitrary potentials, providing the adiabatic, isocurvature, and tensor power spectra at the end of inflation. For models with sum-separable potentials MULTIMODECODE also computes the slow-roll prediction via the δ N formalism for easy model exploration and validation. We pay particular attention to the isocurvature perturbations as the system approaches the adiabatic limit, showing how to avoid numerical instabilities that affect some other approaches to this problem. We demonstrate the use of MULTIMODECODE by exploring a few toy models. Finally, we give a concise review of multifield perturbation theory and a user's manual for the program.

Using large scale structure to measurefNL,gNLandτNL

Primordial non-Gaussianity of local type is known to produce a scale-dependent contribution to the galaxy bias. Several classes of multifield inflationary models predict non-Gaussian bias which is stochastic, in the sense that dark matter and halos do not trace each other perfectly on large scales. In this work, we forecast the ability of next-generation large-scale structure surveys to constrain common types of primordial non-Gaussianity like ${f}_{\mathrm{NL}}$, ${g}_{\mathrm{NL}}$ and ${\ensuremath{\tau}}_{\mathrm{NL}}$ using halo bias, including stochastic contributions. We provide fitting functions for statistical errors on these parameters which can be used for rapid forecasting or survey optimization. A next-generation survey with volume $V=25{h}^{\ensuremath{-}3}\text{ }\text{ }{\mathrm{Gpc}}^{3}$, median redshift $z=0.7$ and mean bias ${b}_{g}=2.5$ can achieve $\ensuremath{\sigma}({f}_{\mathrm{NL}})=6$, $\ensuremath{\sigma}({g}_{\mathrm{NL}})={10}^{5}$ and $\ensuremath{\sigma}({\ensuremath{\tau}}_{\mathrm{NL}})={10}^{3}$ if no mass information is available. If halo masses are available, we show that optimally weighting the halo field in order to reduce sample variance can achieve $\ensuremath{\sigma}({f}_{\mathrm{NL}})=1.5$, $\ensuremath{\sigma}({g}_{\mathrm{NL}})={10}^{4}$ and $\ensuremath{\sigma}({\ensuremath{\tau}}_{\mathrm{NL}})=100$ if halos with mass down to ${M}_{\mathrm{min}}={10}^{11}{h}^{\ensuremath{-}1}{M}_{\ensuremath{\bigodot}}$ are resolved, outperforming Planck by a factor of 4 on ${f}_{\mathrm{NL}}$ and nearly an order of magnitude on ${g}_{\mathrm{NL}}$ and ${\ensuremath{\tau}}_{\mathrm{NL}}$. Finally, we study the effect of photometric redshift errors and discuss degeneracies between different non-Gaussian parameters, as well as the impact of marginalizing Gaussian bias and shot noise.

Gravitational wave consistency relations for multifield inflation.

We study the tensor spectral index n(t) and the tensor-to-scalar ratio r in the simplest multifield extension to single-field, slow-roll inflation models. We show that multifield models with potentials V∼[under ∑]iλ_{i}|ϕ_{i}|^{p} have different predictions for n(t)/r than single-field models, even when all the couplings are equal λ_{i}=λ_{j}, due to the probabilistic nature of the fields' initial values. We analyze well-motivated prior probabilities for the λ_{i} and initial conditions to make detailed predictions for the marginalized probability distribution of n(t)/r. With O(100) fields and p>3/4, we find that n(t)/r differs from the single-field result of n(t)/r=-1/8 at the 5σ level. This gives a novel and testable prediction for the simplest multifield inflation models.

Resilience of the Standard Predictions for Primordial Tensor Modes

We show that the prediction for the primordial tensor power spectrum cannot be modified at leading order in derivatives. Indeed, one can always set to unity the speed of propagation of gravitational waves during inflation by a suitable disformal transformation of the metric, while a conformal one can make the Planck mass time independent. Therefore, the tensor amplitude unambiguously fixes the energy scale of inflation. Using the effective field theory of inflation, we check that predictions are independent of the choice of frame, as expected. The first corrections to the standard prediction come from two parity violating operators with three derivatives. Also the correlator ⟨γγγ⟩ is standard and only receives higher derivative corrections. These results hold also in multifield models of inflation and in alternatives to inflation and make the connection between a (quasi-)scale-invariant tensor spectrum and inflation completely robust.

Non-Gaussianities in DBI inflation with angular motion

We study DBI spinflation models with angular potentials that are derived in string theory. We analyse the background dynamics with different parameter sets and study the impact of changing each parameter on inflationary dynamics. It is known that the conversion of the entropy perturbation into the curvature perturbation gives multi-field DBI inflation models a possibility of satisfying the observational constraints by relaxing the stringent microphysical constraint that disfavours single field DBI inflation models. we show that our model is excluded by the Planck satellite observations even with the conversion mechanism regardless of the parameter set.

Simple predictions from multifield inflationary models.

We explore whether multifield inflationary models make unambiguous predictions for fundamental cosmological observables. Focusing on N-quadratic inflation, we numerically evaluate the full perturbation equations for models with 2, 3, and O(100) fields, using several distinct methods for specifying the initial values of the background fields. All scenarios are highly predictive, with the probability distribution functions of the cosmological observables becoming more sharply peaked as N increases. For N=100 fields, 95% of our MonteCarlo samples fall in the ranges ns∈(0.9455,0.9534), α∈(-9.741,-7.047)×10-4, r∈(0.1445,0.1449), and riso∈(0.02137,3.510)×10-3 for the spectral index, running, tensor-to-scalar ratio, and isocurvature-to-adiabatic ratio, respectively. The expected amplitude of isocurvature perturbations grows with N, raising the possibility that many-field models may be sensitive to postinflationary physics and suggesting new avenues for testing these scenarios.

Multifield Inflation after Planck: The Case for Nonminimal Couplings

Multifield models of inflation with nonminimal couplings are in excellent agreement with the recent results from Planck. Across a broad range of couplings and initial conditions, such models evolve along an effectively single-field attractor solution and predict values of the primordial spectral index and its running, the tensor-to-scalar ratio, and non-Gaussianities squarely in the observationally most-favored region. Such models can also amplify isocurvature perturbations, which could account for the low power recently observed in the cosmic microwave background power spectrum at low multipoles. Future measurements of primordial isocurvature perturbations could distinguish between the currently viable possibilities.

Predictions in multifield models of inflation

This paper presents a method for obtaining an analytic expression for the density function of observables in multifield models of inflation with sum-separable potentials. The most striking result is that the density function in general possesses a sharp peak and the location of this peak is only mildly sensitive to the distribution of initial conditions. A simple argument is given for why this result holds for a more general class of models than just those with sum-separable potentials and why for such models, it is possible to obtain robust predictions for observable quantities. As an example, the joint density function of the spectral index and running in double quadratic inflation is computed. For scales leaving the horizon 55 e-folds before the end of inflation, the density function peaks at ns = 0.967 and α = 0.0006 for the spectral index and running respectively.

Multi-field conformal cosmological attractors

We describe a broad class of multi-field inflationary models with spontaneously broken conformal invariance. It generalizes the recently discovered class of cosmological attractors with a single inflaton field [1]. In the new multi-field theories, just as in the single-field models of [1], the moduli space has a boundary (Kähler cone) in terms of the original homogeneous conformal variables. Upon spontaneous breaking of the conformal invariance and switching to the Einstein frame, this boundary moves to infinity in terms of the canonically normalized inflaton field. This results in the exponential stretching and flattening of scalar potentials in the vicinity of the boundary of the moduli space, which makes even very steep potentials perfectly suitable for the slow-roll inflation. These theories, just like their single-field versions, typically lead to inflationary perturbations with ns = 1−2/N and r = 12/N2, where N is the number of e-foldings.

Non-linear curvature perturbation in multi-field inflation models with non-minimal coupling

Using the δN formalism we consider the non-linear curvatureperturbation in multi-field models of inflation with non-minimalcoupling. In particular, we focus on the relation between the δN formalism as applied in the conformally related Jordan and Einsteinframes. Exploiting results already known in the Einstein frame, wegive expressions for the power spectrum, spectral tilt and non-gaussianity associated with the Jordan frame curvature perturbation.In the case that an adiabatic limit has not been reached, we find thatin general these quantities differ from those associated with theEinstein frame curvature perturbation, and also confirm theirequivalence in the absence of isocurvature modes. We then proceed toconsider two analytically soluble examples, the first involving anon-minimally coupled `spectator' field and the second being anon-minimally coupled extension of the multi-brid inflation model. Inthe first model we find that predictions can easily be brought intoagreement with the recent Planck results, as the tensor-to-scalarratio is generally small, the spectral tilt tuneable and the non-gaussianitysuppressed. In the second model we find that predictions for all threeparameters can differ substantially from those predicted in theminimally coupled case, and that the recent Planck results forthe spectral tilt can be used to constrain the non-minimal couplingparameters.