In two-field inflation there are only a few kinematical and geometrical parame- ters that determine the evolution of linear perturbations. These are the radius of curvature of the inflationary trajectory in field space, theentropy mass (the mass of the isocurvature perturbations) and the Hubble slow-roll parameters. This motivates us to introduce Orbital Inflation, in which the radius of cur- vature is constant, and the remaining parameters are slowly varying. This is one of the simplest two-field extensions to single field inflation.
In§ 2.3we discussed the phenomenology of Orbital Inflation. In particular we focus on the regime of small entropy massµ2/H2≤0.2and we found that the predictions are substantially modified already for κ2/Mp2 .102. Further- more, in § 2.4 we showed how to explicitly construct exact models of Orbital Inflation. The key characteristic of these models is that inflation proceeds along an isometry direction of the field metric. We used a generalization of the Hamilton-Jacobi formalism to find the potential locally around the infla- tionary trajectory. Finally, we saw that an interesting limiting case is to have a vanishing entropy mass. This is reflected by a shift symmetry in the Hubble parameter rather than in the resulting potential. We investigate this particu- lar case in Chapter 3.
For convenience of the reader, we collect the relevant formulas and ele- ments that describe the two-field models we have reconstructed in § 2.4. The
Summary 71
procedure is more general, and can be used to reconstruct other multi-field models of inflation as well
• We specialized to a two-field model with two scalar fieldsθandρ descri- bed by the following action
S= 1 2 Z d4x√−g Mp2R−f(ρ)∂µθ∂µθ+∂µρ∂µρ−2V(θ, ρ) . (2.56)
with R the Ricci scalar of spacetime. The scalar kinetic term has an isometry in theθ direction.
• In order to achieve a constant radius of curvature, we force inflation to proceed exactly in the θ direction. This puts a restriction on the form of H(ρ, θ), namely that it has to be independent of θ on the trajectory ρ=ρ0.
• Next, we compute the entropy mass in terms ofH(ρ, θ)expanded around ρ=ρ0. We took into account that the entropy mass, defined inEq. 2.13,
receives geometrical and kinematical corrections as well. This allowed us to reconstruct the following potential, which admits an approximately constant entropy mass µ2/H2≈λup to slow-roll corrections
V = 3Mp4 W2(θ) +2M 2 pWθ2(θ) 3f(ρ) ! 1 + λ 12 (ρ−ρ0)2 M2 p +. . . 2 , (2.57)
The ellipses denote higher order terms in the expansion aroundρ=ρ0we
left unspecified. They determine higher order derivatives of the potential, such asVρρρ.
Hoofdstuk
3
Orbital Inflation with ultra-light fields
In this chapter we present a class of inflationary models with two light fields that have predictions similar to those of single field inflation. Inflation pro- ceeds along an ‘angular’ isometry direction in field space at arbitrary ‘radius’ and is a special case of Orbital Inflation discussed inChapter 2. More precisely, we study the Orbital Inflation in the limit of vanishing entropy mass1. We dub this ‘ultra-light Orbital Inflation’, because it realizes the shift symmetry de- scribed in [138]. If the field radius of curvature of the inflationary trajectory is sufficiently small, the amplitude of isocurvature perturbations and primordial non-Gaussianities are highly suppressed. Ultra-light Orbital Inflation mimics single field inflation, because only one degree of freedom is responsible for the observed perturbations.
We study a toy model of ultra-light Orbital Inflation in§ 3.2. This allows us to intuitively understand its interesting properties. In the successive sec- tions we make our intuitive arguments more precise. In § 3.3 we derive the family of two-field models which allow for ultra-light Orbital Inflation and give the corresponding exact solutions. We prove neutral stability of ultra-light Or- bital Inflation in§ 3.4. Then, in § 3.5we recap the definition of mass and the consequences of having massless isocurvature perturbations. Finally, we study the phenomenology of ultra-light Orbital Inflation in§ 3.6.
The results in this chapter are based on joint work with Ana Achúcarro, Ed- mund Copeland, Oksana Iargyina, Gonzalo Palma and Dong-Gang Wang.
1
The entropy mass is the effective mass of isocurvature perturbations. The definition of mass is non-trivial in a time-dependent inflationary background. By computing the normal modes of the coupled system of perturbations, we find a dispersion relation of the isocurva- ture perurbations corresponding to modes of massµas defined inEq. 3.18. Therefore it is
µthat we identify as the entropy mass. For more details, see§ 2.2.3.
74 Orbital Inflation with ultra-light fields
3.1 Introduction
The Planck data [115] reveal that inflationary perturbations are Gaussian and adiabatic to a high level of accuracy. A possible explanation for the observed simplicity is that the perturbations are generated by a single degree of freedom with small self-interactions. Do the observations therefore imply that, besi- des the inflaton, no other light fields are active during inflation? The answer is no. As pointed out in [138], in the limit that the other fields are massless, but coupled to the inflaton, the predictions mimic those of single field inflation.
Inflation with massless isocurvature modes behaves like single field infla- tion, because only one degree of freedom is relevant for the observed pertur- bations. The single field behavior is of dynamical origin. The key feature is that the isocurvature perturbations freeze out on superhorizon scales and constantly feed the curvature perturbations. Therefore, the isocurvature per- turbations generate the temperature fluctuations we observe in the sky.
In this work we provide a realization of a family of two-field inflationary models in which the isocurvature perturbations become exactly massless (see footnote 1). We dub this ultra-light Orbital Inflation. We use the two-field generalization of the Hamilton-Jacobi formalism [74,75,77,78] presented in Chapter 2 to derive the form of the potential. The key characteristic ofultra- light Orbital Inflation is that inflation proceeds along an isometry direction of the field metric at arbitrary radius. The resulting scalar field potential and kinetic term are given by
V = 3H2Mp2−2Mp4 H
2
θ
f(ρ), 2K=f(ρ)∂θ
2+∂ρ2 , (3.1)
where the fields are denoted by θ and ρ. Moreover, the Hubble parameter H is a function of θonly and f(ρ)>0.
In this chapter we point out several interesting properties of ultra-light Orbital Inflation:
• Each attractor is an exact solution to the highly non-linear system of field equations and Friedmann equation. This is ensured by using the Hamilton-Jacobi formalism.
• This system is neutrally stable. A small perturbation orthogonal to a given attractor solution will bring us to one of the neighboring attractors.
• Because isocurvature (= normal) perturbations move us freely between attractors, this implies that they are exactly massless. Thanks to the
A toy model with neutrally stable orbits 75
Hubble friction their velocity decays and therefore their amplitude freezes out.
• The quadratic action of perturbations has enhanced symmetry. On top of the usual shift symmetry of curvature perturbationsR → R+const, the masslessness of the isocurvature perturbation S implies a combined shift symmetry [138]
S → S +c, R →˙ R −˙ λc. (3.2)
We show that we can understand this as a result of the background dynamics. The symmetry transformation is related to a map of one background attractor to another, labeled by the continuous parameterc.
• For large enough λ this implies that curvature perturbations are dy- namically enhanced and the predictions of the power spectra coincide with those of single field inflation [138]. Moreover, the final curvature perturbations are completely determined by the initial isocurvature per- turbations. Therefore, these multi-field inflation scenarios mimic the predictions of single-field inflation.
We work in Planck units~=c= 1and the reduced Planck mass is given byMp = (8πG)−1/2.