Field-space Metric Research Articles

Analytic bounds on late-time axion-scalar cosmologies

The cosmological dynamics of multiple scalar/pseudoscalar fields are difficult to solve, especially when the field-space metric is curved. This presents a challenge in determining whether a given model can support cosmic acceleration, without solving for the on-shell solution. In this work, we present bounds on late-time FLRW-cosmologies in classes of theories that involve arbitrary numbers of scalar and pseudoscalar fields coupled both kinetically (leading to a curved field space metric) and through scalar potentials. Such bounds are proven analytically, independently of initial conditions, with no approximation in the field equations and without referring to explicit solutions. Besides their broad applications to cosmological model building, our bounds can be applied to studying asymptotic cosmologies of certain classes of string compactifications.

Dynamical consistency conditions for rapid turn inflation

We derive consistency conditions for sustained slow roll and rapid turn inflation in two-field cosmological models with oriented scalar field space, which imply that inflationary models with field-space trajectories of this type are non-generic. In particular, we show that third order adiabatic slow roll, together with large and slowly varying turn rate, requires the scalar potential of the model to satisfy a certain nonlinear second order PDE, whose coefficients depend on the scalar field metric. We also derive consistency conditions for slow roll inflationary solutions in the so called “rapid turn attractor” approximation, as well as study the consistency conditions for circular rapid turn trajectories with slow roll in two-field models with rotationally invariant field space metric. Finally, we argue that the rapid turn regime tends to have a natural exit after a limited number of e-folds.

Dynamical symmetries of homogeneous minisuperspace models

We investigate the phase space symmetries and conserved charges of homogeneous gravitational minisuperspaces. These ($0+1$)-dimensional reductions of general relativity are defined by spacetime metrics in which the dynamical variables depend only on a time coordinate and are formulated as mechanical systems with a nontrivial field space metric (or supermetric) and effective potential. We show how to extract conserved charges for those minisuperspaces from the homothetic Killing vectors of the field space metric. In the case of two-dimensional field spaces, we exhibit a universal eight-dimensional symmetry algebra $\mathcal{A}=(\mathfrak{sl}(2,\mathbb{R})\ensuremath{\bigoplus}\mathbb{R})⨭{\mathfrak{h}}_{2}$, based on the two-dimensional Heisenberg algebra ${\mathfrak{h}}_{2}\ensuremath{\simeq}{\mathbb{R}}^{4}$. We apply this to the systematic study of the Bianchi models for homogeneous cosmology. This extends the previous results on the $\mathfrak{sl}(2,\mathbb{R})$ algebra for Friedmann-Lema\^{\i}tre-Robertson-Walker cosmology and the Poincar\'e symmetry for Kantowski-Sachs metrics describing the black hole interior. The presence of this rich symmetry structure already in minisuperspace models opens new doors toward quantization and the study of solution generating mechanisms.

Dark energy from inspiraling in field space

We find an exact solution of the equations of motion of a two-field cosmological model, which realizes multi-field dark energy. The latter is characterized by field-space trajectories with turning rates that are always large. We study a class of two-field models and show that it is possible to have such trajectories, giving accelerated space-time expansion, even when the scalar potential preserves the rotational invariance of the field-space metric. For the case of Poincaré-disk field space, we derive the form of the scalar potential compatible with such background solutions and, furthermore, we find the exact solutions analytically. Their field-space trajectories are spirals inward, toward the center of the Poincaré disk. Interestingly, the functional form of the relevant scalar potential is compatible with a certain hidden symmetry, although the latter is broken by the presence of a constant term.

Multifield mimetic gravity

In this paper, we extend the mimetic gravity to the multifield setup with a curved field space manifold. The multifield mimetic scenario is realized via the singular limit of the conformal transformation between the auxiliary and the physical metrics. We look for the cosmological implications of the setup where it is shown that at the background level the mimetic energy density mimics the roles of dark matter. At the perturbation level, the scalar field perturbations are decomposed into the tangential and normal components with respect to the background field space trajectory. The adiabatic perturbation tangential to the background trajectory is frozen while the entropy mode perpendicular to the background trajectory propagates with the speed of unity. Whether or not the entropy perturbation is healthy directly depends on the signature of the field-space metric. We perform the full nonlinear Hamiltonian analysis of the system with the curved field space manifold and calculate the physical degrees of freedom verifying that the system is free from the Ostrogradsky-type ghost.

Absence of covariant singularities in pure gravity

The assumptions of the Hawking–Penrose singularity theorem are not covariant under field redefinitions. Following the works on the covariant formulation of quantum field theory initiated by Vilkovisky and DeWitt in the 80s, we propose to study singularities in field space, where the spacetime metric is treated as a coordinate along with the other fields in the theory. From this viewpoint, a spacetime singularity might be just a singularity in the field-space coordinates, analogously to the standard coordinate singularities in General Relativity. Objects invariant under field-space coordinate transformations can then reveal whether certain spacetime singularity is indeed singular. We recall that observables in quantum field theory are scalar functionals in field space. Therefore, in principle, spacetime singularities corresponding to regular field-space curvature invariants would not affect physical observables. In this paper, we show that the field-space Kretschmann scalar for a certain choice of the DeWitt field-space metric is everywhere finite. This fact could be interpreted as an indication that no singularities actually exist in pure gravity for any gravitational action. In particular, all vacuum singularities of General Relativity result from an unhappy choice of field variables. The extension to the case in which matter fields are present, as required by singularity theorems, is left for future development.

Generalizing the swampland: Embedding P(X,φ) inflationary theories in a curved multifield space

We study the general embedding of a $ P(X, \varphi) $ inflationary theory into a two-field theory with curved field space metric, which was proposed as a possible way to examine the relation between de Sitter Swampland conjecture and \textit{k}-inflation. We show that this embedding method fits into the special type of two-field model in which the heavy field can be integrated out at the full action level. However, this embedding is not exact due to the upper bound of the effective mass of the heavy field. We quantify the deviation between the speed of sound calculated via the $ P(X, \varphi) $ theory and the embedding two-field picture to next leading order terms. We especially focus on the first potential slow roll parameter defined in the two-field picture and obtain an upper bound on it.

A new class of non-canonical conformal attractors for multi-field inflation

We propose a new broad class of multi-field non-canonical inflationary models as an extension of multi-field conformal cosmological attractors. This also generalizes the recently discovered class of non-canonical conformal attractors for single field inflation. Kinetic terms of this class of models are phenomenologically arising from \U0001d4a9=1 supergravity and from \U0001d4a9=1 superconformal theory, with two conformal scalar compensator fields in the latter. We show that the inflationary dynamics and predictions of this class of models are stable with respect to the significant modification of both radial and angular part of the potential, but it is very sensitive to its minuscule modification in the geometry of the field space metric. We also show that our framework can pass the latest observational constraints set by Planck 2018.

Angular inflation in multi-field α-attractors

We explore the dynamics of multi-field models of inflation in which the field-space metric is a hyperbolic manifold of constant curvature. Such models are known as α-attractors and their single-field regimes have been extensively studied in the context of inflation and supergravity. We find a variety of multi-field inflationary trajectories in different regions of parameter space, which is spanned by the mass parameters and the hyperbolic curvature. Amongst these is a novel dynamical attractor along the boundary of the Poincare disc which we dub “angular inflation”. We calculate the evolution of adiabatic and isocurvature fluctuations during this regime and show that, while isocurvature modes decay during this phase, the duration of the angular inflation period can shift the single-field predictions of α-attractors. For highly curved field-space manifolds, this can lead to predictions that lie outside the current observational bounds.

Universality and scaling in multi-field α-attractor preheating

We explore preheating in multi-field models of inflation in which the field-space metric is a highly curved hyperbolic manifold. One broad family of such models is called α-attractors, whose single-field regimes have been extensively studied in the context of inflation and supergravity. We focus on a simple two-field generalization of the T-model, which has received renewed attention in the literature. Krajewski et al. concluded, using lattice simulations, that multi-field effects can dramatically speed-up preheating. We recover their results and further demonstrate that significant analytical progress can be made for preheating in these models using the WKB approximation and Floquet analysis. We find a simple scaling behavior of the Floquet exponents for large values of the field-space curvature, that enables a quick estimation of the T-model reheating efficiency for any large value of the field-space curvature. In this regime we further observe and explain universal preheating features that arise for different values of the potential steepness. In general preheating is faster for larger negative values of the field-space curvature and steeper potentials. For very highly curved field-space manifolds preheating is essentially instantaneous.

Preheating after Higgs inflation: Self-resonance and gauge boson production

We perform an extensive analysis of linear fluctuations during preheating in Higgs inflation in the Einstein frame, where the fields are minimally coupled to gravity, but the field-space metric is nontrivial. The self-resonance of the Higgs and the Higgsed gauge bosons are governed by effective masses that scale differently with the nonminimal couplings and evolve differently in time. Coupled metric perturbations enhance Higgs self-resonance and make it possible for Higgs inflation to preheat solely through this channel. For $\xi\gtrsim 100$ the total energy of the Higgs-inflaton condensate can be transferred to Higgs particles within $3$ $e$-folds after the end of inflation. For smaller values of the nonminimal coupling preheating takes longer, completely shutting off at around $\xi\simeq 30$. The production of gauge bosons is dominated by the gauge boson mass and the field space curvature. For large values of the nonminimal coupling $\xi \gtrsim 1000$, it is possible for the Higgs condensate to transfer the entirety of its energy into gauge fields within one oscillation. For smaller values of the nonminimal coupling gauge bosons decay very quickly into fermions, thereby shutting off Bose enhancement. Estimates of non-Abelian interactions indicate that they will not suppress preheating into gauge bosons for $\xi \gtrsim 1000$.

Squeezed bispectrum from multi-field inflation with curved field space metric

We investigate influences of the curved field-space metric of multi-field inflationary models on the squeezed bispectrum. The reduced bispectrum in squeezed limit is computed using the {\delta}N formalism. The calculation is performed under the slow-roll approximation and assumption that field derivative of the field-space metric is sufficiently small such that the contributions from Riemann tensor of the field-space can be approximately ignored. Based on these approximations, We compute the analytic expressions for the reduced bispectrum in squeezed limit, and find that, for such a nearly flat field-space metric, the field dependence of the metric can significantly alter both amplitude and shape of the reduced bispectrum. The reduced bispectrum from this nearly flat field-space metric can lead to spectral index of the halo bias which amplitude is 2 -- 4 times larger than that from the flat field-space model. This modification of the spectral index of the halo bias due to the curved field-space metric could leave observable imprints in future galaxy surveys.

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.

Numerically evaluating the bispectrum in curved field-space— with PyTransport 2.0

We extend the transport framework for numerically evaluating the power spectrum and bispectrum in multi-field inflation to the case of a curved field-space metric. This method naturally accounts for all sub- and super-horizon tree level effects, including those induced by the curvature of the field-space. We present an open source implementation of our equations in an extension of the publicly available PyTransport code. Finally we illustrate how our technique is applied to examples of inflationary models with a non-trivial field-space metric.

Strongly scale-dependent CMB dipolar asymmetry from super-curvature fluctuations

We reconsider the observed CMB dipolar asymmetry in the context of open inflation, where a supercurvature mode might survive the bubble nucleation. If such a supercurvature mode modulates the amplitude of the curvature power spectrum, it would easily produce an asymmetry in the power spectrum. We show that current observational data can be accommodated in a three-field model, with simple quadratic potentials and a non-trivial field-space metric. Despite the presence of three fields, we believe this model is so far the simplest that can match current observations. We are able to match the observed strong scale dependence of the dipolar asymmetry, without a fine tuning of initial conditions, breaking slow roll or adding a feature to the evolution of any field.

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.

Backreacted axion field ranges in string theory

String theory axions are interesting candidates for fields whose potential might be controllable over super-Planckian field ranges and therefore as possible candidates for inflatons in large field inflation. Axion monodromy scenarios are setups where the axion shift symmetry is broken by some effect such that the axion can traverse a large number of periods potentially leading to super-Planckian excursions. We study such scenarios in type IIA string theory where the axion shift symmetry is broken by background fluxes. In particular we calculate the backreaction of the energy density induced by the axion vacuum expectation value on its own field space metric. We find universal behaviour for all the compactifications studied where up to a certain critical axion value there is only a small backreaction effect. Beyond the critical value the backreaction is strong and implies that the proper field distance as measured by the backreacted metric increases at best logarithmically with the axion vev, thereby placing strong limitations on extending the field distance any further. The critical axion value can be made arbitrarily large by the choice of fluxes. However the backreaction of these fluxes on the axion field space metric ensures a precise cancellation such that the proper field distance up to the critical axion value is flux independent and remains sub-Planckian. We also study an axion alignment scenario for type IIA compactifications on a twisted torus with four fundamental axions mixing to leave an axion with an effective decay constant which is flux dependent. There is a choice of fluxes for which the alignment parameter is unconstrained by tadpoles and can in principle lead to a parametrically enhanced effective decay constant. However we show that these fluxes backreact on the fundamental decay constants so as to precisely cancel any enhancement.

General formula for the running of local [formula omitted

We compute the scale dependence of fNL for models of multi-field inflation, allowing for an arbitrary field space metric. We show that, in addition to multi-field effects and self-interactions, the curved field space metric provides another source of scale dependence, which arises from the field-space Riemann curvature tensor and its derivatives. The scale dependence may be detectable within the near future if the amplitude of fNL is not too far from the current observational bounds.

The inflationary bispectrum with curved field-space

We compute the covariant three-point function near horizon-crossing for a system of slowly-rolling scalar fields during an inflationary epoch, allowing for an arbitrary field-space metric. We show explicitly how to compute its subsequent evolution using acovariantized version of the separate universe or `δN' expansion,which must be augmented by terms measuring curvature of the field-space manifold, and give the nonlinear gauge transformation to the comoving curvature perturbation.Nonlinearities induced by the field-space curvature termsare a new and potentially significant source of non-Gaussianity.We show how inflationary models with non-minimal couplingto the spacetime Ricci scalar can be accommodated within this framework.This yields a simple toolkit allowingthe bispectrum to be computed inmodels with non-negligible field-space curvature.

A covariant approach to general field space metric in multi-field inflation

We present a covariant formalism for general multi-field system which enables us to obtain higher order action of cosmological perturbations easily and systematically. The effects of the field space geometry, described by the Riemann curvature tensor of the field space, are naturally incorporated. We explicitly calculate up to the cubic order action which is necessary to estimate non-Gaussianity and present those geometric terms which have not yet been known before.