Multi-field inflation and the field-space metric

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.


I. INTRODUCTION
Predictions for inflationary observables depend on both the field-space metric and potential of the fields responsible for the inflationary dynamics.Nontrivial kinetic terms which modify the field-space metric arise in many ways: from radiative corrections, from a higherdimensional origin of the fields, or simply from a field redefinition.Supersymmetric models of inflation typically include nontrivial Kahler potentials which modify the field-space metric, as in Ref. [1] and many of the models reviewed in Refs.[2].A covariant approach to analyzing fluctuations in an inflationary setting with nontrivial kinetic terms was developed in Ref. [3].Here we compare two classes of multi-field inflation models which differ only in their kinetic terms, and we discuss some of the lessons learned from these examples.We justify a single-field effective description of these models and derive a mass matrix appropriate for calculation of inflationary observables in these models.
Although the observation by the BICEP2 collaboration of B-modes in the polarization of microwave radiation [4] can be attributed to scattering off of galactic dust [5,6] as demonstrated by the Planck experiment [7], current and proposed experiments such as PIPER [8] remain sensitive to signatures of primordial gravitational waves produced during inflation.In slow-roll inflation models, the Lyth bound [9] implies that the inflaton field typically varies over super-Planckian values if sufficiently large power in gravitational waves is produced during inflation.This makes it difficult to describe such an inflationary scenario in terms of an effective field theory valid below the Planck scale.There are several ways to evade the Lyth bound, for example if the slow-roll parameter increases for some period during inflation, as happens in certain hybrid inflation models [10,11], or if the inflaton is embedded in a multi-field model in which one of the fields has a discrete shift symmetry, as in axion-monodromy models [12].Simplified models of the latter type were developed in Refs.[13,14].
Inflationary models based on one or more pseudo-Nambu-Goldstone bosons have a long history (for example, Refs.[15][16][17][18][19][20]). The Dante's inferno model, developed in Ref. [14], includes two axion fields which evolve along a trench in the potential during inflation, as in Fig. 1.The two axions r and θ in Dante's inferno have canonical kinetic terms, FIG. 1: The potential as a function of r and θ in Dante's Inferno with a quadratic shift-symmetrybreaking potential W (r) = 1 2 m 2 r 2 , as in Ref. [14].
The potential has the form where the discrete shift symmetry of the axion field r is broken by the term W (r) in the potential.A string-theoretic scenario which gives rise to the Dante's Inferno model was presented in Ref. [14], in which the shift-symmetry-breaking potential W (r) describes the axion on an NS5 brane wrapped on a 2-cycle belonging to a family of homologous 2-cycles which extend into a warped throat geometry.
We will consider a generalization of the potential Eq. (1.2) of the form, This class of potentials appears in models with a complex scalar field and a single anomalous U(1) symmetry, as in the axion inflation model of Ref. [21].In this case, the real fields r/ √ 2 and θ are the magnitude and phase, respectively, of a canonically normalized complex scalar field Φ = re iθ / √ 2, in which case we take f θ = 1.The trench spirals around the potential as in Fig. 2. The kinetic terms for the real scalars in these spiral inflation models are non-canonical, taking the form The additional factor of r 2 in the kinetic term for θ can have important consequences, even affecting the phenomenological viability of these models, as we will see.In this paper we compare the predictions for a number of two-field models with canonical and non-canonical kinetic terms of the form Eq. (1.1) and Eq.(1.4).These include models which are effectively either chaotic inflation or hybrid inflation models.Hybrid inflation models of this type include Dante's waterfall [22] and certain spiral inflation [23][24][25] models.In the case of spiral inflation we will take f θ = 1 so that the potential is periodic in θ → θ + 2π, while there is a monodromy in shifts of r.The qualitative difference between these models can be described in terms of the trajectories of the fields which evolve during inflation: In the Dante's inferno and Dante's waterfall scenarios the fields evolve along an approximately linear trajectory in the canonically normalized field space, whereas in spiral inflation models the fields evolve along a nearly circular trajectory.In a single-field effective description these are chaotic inflation models, but one must take care in the analysis of models with changing inflaton direction as in spiral inflation.
In Sec.II we describe the single-field effective description of these multi-field models, and derive a mass matrix whose smaller eigenvalue has the interpretation of the inflaton mass-squared.This mass matrix may be used in the calculation of inflationary observables.In Sec.III, we compare the predictions for inflationary observable in a variety of models which differ in their kinetic terms, most of which already appear in the literature.We conclude in Sec.IV.

II. SINGLE-FIELD EFFECTIVE DESCRIPTION
In this section we review the single-field description of spiral-inflation models with Lagrangian Eq. (1.4), and derive a mass-matrix description relevant for computation of inflationary observables.We first review the role of the field-space metric in the single-field effective description of these models.
A. From many fields to one Consider a model with real scalar fields φ a in a background spacetime described by the metric g µν .During inflation we assume the spacetime is given by the flat Friedmann-Robertson-Walker (FRW) metric g 00 = 1, g ij = −a 2 (t)δ ij , where i, j ∈ {1, 2, 3} and t ≡ x 0 , but for now we allow an arbitrary time-dependent metric.The Lagrangian for the theory is, where G ab ({φ c }) in the kinetic terms defines the field-space metric, which is taken to be symmetric in a ↔ b.Under a nonlinear field redefinition φ a → φa φ b , the Lagrangian transforms as, which defines the transformed field-space metric as In this sense, the field-space metric transforms as a tensor under field transformations.
Locally one can redefine the fields so that the field-space metric is flat, Gcd = δ cd , but this can be done globally only if the field-space metric originally describes a flat field space.
In order to compare with a single-field description we consider the equations of motion.
The equations of motion for the fields φ a are, We will be interested in spatially uniform solutions to the equations of motion, so that the fields φ a only have dependence on t.For these solutions, the equations of motion are where φa ≡ dφ a /dt.Now suppose that the trajectory describing a solution to the equations of motion is known, parametrized by a parameter I along the trajectory, so that along the given solution we have φ a (I).For such a solution, the equations of motion determine the time dependence of I. Multiplying Eq. (2.6) by φ a (I) gives, Now choose I to satisfy the field-space condition This condition makes the parameter I analogous to the invariant length, but in field space, and will give I the interpretation of a canonically normalized inflaton field, with kinetic term İ2 .A derivative of Eq. (2.8) with respect to I gives, Multiplying by İ2 , we have Using Eq. (2.10), the equations of motion Eq. (2.7) become, The first two terms in Eq. (2.11) combine to give a time derivative, or using Eq.(2.8), Together with the trajectory φ a (I) that solves the equations of motion, a solution to Eq. (2.13) then determines the time dependence of that trajectory.Consequently, Eq. ( 2.13) provides enough information to determine inflationary observables, as long as the fluctuations in the direction orthogonal to the trajectory are massive compared to H −1 so that they are not produced during inflation.
The field-space parameter I above plays the role of the inflaton in the single-field description of any model with Lagrangian of the form Eq. (2.1).The analysis above supposed that we knew the trajectory along a solution to the equations of motion.Now suppose that we had instead imposed as a constraint that the fields lie on the trajectory φ a (I).In Dante's inferno and spiral inflation models, the trajectory is approximately known due to the presence of a steep-walled trench in the potential.This is a holonomic constraint, as can be made explicit by inverting the expression for one of the fields, say φ 1 (I) to give I(φ 1 ).We assume that this inverse exists throughout the field trajectory.Then the remaining constraints are of the form φ a − φ a (I(φ 1 )) = 0.Such constraints can be imposed either by Lagrange multipliers in the Lagrangian, or by simply replacing φ a by φ a (I) in the Lagrangian.We are left with a description of the theory in terms of the single field I.
If we again choose I to satisfy the condition Eq. (2.8), then the Lagrangian Eq. (2.1) constrained to a field-space trajectory takes the canonical form, The equations of motion that follow from this singe-field effective description are the same as Eq.(2.13), which was derived in the multi-field description.This justifies the interpretation of the field I as the canonical inflaton in these models.Note that the only assumption in the analysis of this section was that we knew the trajectory taken by the fields φ a , which in many inflation models is known by the presence of a steep-walled trench in the potential.

B. Spiral Inflation Models and a Mass Matrix
At this stage we will focus on spiral inflation models, for which G rr = 1, G θθ = r 2 , and G rθ = G θr = 0.The condition Eq. (2.8) defining the canonical inflaton field can be written We suppose that the trajectory r(θ), approximately determined by the shape of the trench in the potential, is known.At a given time, the inflaton direction in field space is specified by the unit vector where and the unit vectors êr and êθ are the usual basis vectors in polar coordinates, which in a Cartesian coordinate system with x = r cos θ, y = r sin θ have components êr = cos θ êx + sin θ êy , êθ = − sin θ êx + cos θ êy .In spiral inflation models the field evolution is mostly in the êθ direction.In order to compare with a mass matrix description, as in Ref. [23], we make the approximation that the trajectory is nearly circular, and set to zero c r (θ), c θ (θ), which is a good approximation for typical parameter choices in these models as we will confirm numerically in Sec.III.
The slow-roll parameters, and consequently inflationary observables, depend on derivatives of the potential with respect to the canonically normalized inflaton field.In multi-field models this is a directional derivative (which for comparison with the previous section is simply the chain rule with Eq. (2.17)): where ∇V is the gradient in polar coordinates, ∇V = ∂ r V êr + 1/r ∂ θ V êθ .The derivative dV /dI determines the slow-roll parameter defined by where M * = 2.4 × 10 18 GeV is the reduced Planck mass.Noting that we have Eq. (2.24) can be simplified using yielding We can now identify the mass matrix appropriate for calculation of inflationary observables, In particular, the slow-roll parameter η is defined as, which may be calculated directly in the single-field effective description, or else (to good approximation) as the smaller eigenvalue of the mass matrix M 2 rθ .We note that the mass matrix M 2 rθ differs from the mass matrix of Refs.[23][24][25] in the off-diagonal terms, which explains differences in the results of this paper and those of some earlier papers. 1In particular, by identifying successive derivatives in the êr and êθ directions as ∂ r and ∂ θ /r, respectively, the mass matrix of Refs.[23][24][25] neglects the 1/(2r 2 )∂ θ V term 1 We are grateful to Gabriela Barenboim and Wan-Il Park for discussion on this point.
in the off-diagonal elements of Eq. (2.27).It is perhaps worthwhile therefore to discuss other mass matrices whose eigenvalues are not directly related to derivatives with respect to the inflaton in the single-field description.To that effect we will introduce some well motivated straw-man mass matrices in spiral inflation models, and describe their physical interpretation in relation to the inflaton dynamics.
Rather than begin with the field-space variables r and θ in spiral inflation models, which have noncanonical kinetic terms, one might have instead considered beginning with fieldspace variables x 1 ≡ r cos θ, x 2 ≡ r sin θ, in which case the kinetic terms are canonical and one can define the mass matrix (M 2 Cartesian where ∂ i ≡ ∂/∂x i .This mass matrix, evaluated at a point in field space, determines the quadratic terms in a Taylor expansion of the potential about that point.Then transforming to the polar variables in the neighborhood of that point, (dx, dy) T → (dr, r dθ) T = R(θ)(dx, dy) T , where R(θ) is the 2×2 rotation matrix with angle θ, gives the mass matrix M 2 Cartesian , where so that a Taylor expansion of the potential in Cartesian coordinates about a point (r 0 , θ 0 ) has quadratic part, where dr = (r − r 0 ), dθ = (θ − θ 0 ).The matrix M 2 Cartesian is also closely related to the matrix of covariant derivatives in polar coordinates, except that θ components have been rescaled by 1/r in M 2 Cartesian to transform to the basis (dr, r dθ) from (dr, dθ).Here, Γ c ab is the Christoffel symbol in field space, with nonvanishing components, Γ r θθ = −r, (2.32) The eigenvectors of the various mass matrices described above are numerically similar along the trench defined by ∂ r V = 0 in the models considered in this paper.The eigenvalues of the mass matrices, however are quite different.This is illustrated in Fig. 4 in a numerical example of Sec.III.
To summarize this section, with knowledge of the trajectory describing the evolution of fields constrained to follow a steep-walled trench during inflation, one can define a single-field effective description in terms of a potential V (I) in terms of a canonically normalized inflaton field I.The single-field description allows for straightforward computation of inflationary observables, and is the usual procedure for calculation of observables in multi-field models.
A mass matrix relating the single-field and multi-field descriptions may be constructed, and differs significantly from the mass matrix as usually defined if the direction of field evolution varies significantly during inflation, as in spiral inflation models.

III. RESULTS
We consider theories with both canonical and non-canonical kinetic terms in this section.We use units of the reduced Planck mass M * = 2.4 × 10 18 GeV throughout.Respectively, the Lagrangians are of the form Eq. (1.1) and Eq.(1.4), where V (r, θ) = W (r) + Λ 4 1 − cos ( r f ) n − θ .The inflaton field is defined so that along a trajectory (r(t), θ(t)) the field is canonically normalized.Recall that in the Dante's inferno-type model the fields r and θ are canonically normalized, and in spiral inflation models the fields are non-canonically normalized.In these cases, respectively, the inflaton field I(t) satisfies In both cases, the trajectory closely follows the bottom of the trench defined by We denote the trajectory by r(θ).Eq. (3.1) can be restated as The derivative of V with respect to I becomes We normally work in the region where r (θ) 1 in the canonical case, and and r (θ) r in the non-canonical case.Then, Eq. (3.4) can be approximated by The slow-roll parameters can now be calculated by The inflationary observables are then given by r = [16 ] where I i is the value of the inflaton field at the time when the observed inflationary perturbations were created, which in most models is 50-60 e-folds before the end of inflation, but is sensitive to the details of reheating after inflation.The observable r is the ratio of the tensor to scalar amplitude, where we use the unconventional tilde over r to distinguish the observable from the field r in these models.The other observables are the scalar tilt n s ; the scalar amplitude ∆ 2 R , also denoted A s ; and the running of the scalar tilt n r .Definitions in terms of the CMB spectrum are available in many places, for example in the Planck 2015 results papers [26].
The number of e-folds is given by In our numerical analysis we determine the initial point of inflation by fixing n s = 0.96 and ∆ 2 R = 2.2 × 10 −9 , close to the values measured by the Planck experiment [26], n s = 0.9655 ± 0.0062, ln(10 10 ∆ 2 R ) = 3.089 ± 0.036.The current experimental constraint on n r is based on the Planck measurement, n r = −0.003± 0.015 [26].The end of inflation occurs when either or when the potential reaches a hybrid-inflation-type instability as in the Dante's waterfall model.Two types of W (r) are studied in the following sections and their corresponding single-field approximations are compared with the full theory.

A. λr p
We first consider W (r) = λr p .The trench equation Eq. (3.2) becomes We consider the case that during inflation the magnitude of the right-hand side of Eq. (3.10) is 1, corresponding to a steep-walled trench, so that Eq. (3.10) can be solved by The single-field description of the potential in this approximation is therefore given by the potential, We work through the (p = 4, n = 1, 2) case for illustration.
First we show the predictions of the observables from the single-field approximation.
Using Eqs.(3.5)-(3.9),we analyze theories with both canonical and non-canonical kinetic terms, as earlier.For (p, n) = (4, 1), Eq. (3.11) is now θ = r f + 4λf Λ 4 r 3 .Assuming the second term on the right-hand-side is negligible, we get that the trench follows r(θ) ≈ f θ thus V (r(θ), θ) ≈ W (r(θ)) = λf 4 θ 4 .We determine the initial and final point of inflation in field space by fixing n s = 0.96 and [ ] θ=θ f = 1.Note that n s and are not sensitive to the overall scale in the potential while ∆ 2 R is, so ∆ 2 R can be controlled by rescaling the potential.Fixing observables this way, the model then predicts the number of e-folds during inflation and the ratio of tensor to scalar amplitudes r.The results are given in Table I.Working numerically in the complete two-field model, we find the following ex- ).Note that in the non-canonical case the coupling λ is driven to be nonperturbative and the perturbative analysis is not valid, but for the purpose of comparison with the single-field description we treat this case classically.The results match well with those from Table I, derived from the single-field approximation.The dynamical solutions to the equations of motion Eq. (2.6) are plotted in Fig 3 .Note that the nontrivial field-space metric in the non-canonical case has the consequence of reducing both the number of e-folds and r.However, this model is ruled out by the large values of r > 0.11 [26] and N e > 60 in the canonical case, and the large value of r in the non-canonical case.
For the non-canonical case, the eigenvalues and eigenvectors of the three different matrices mass matrix of Eq. (2.27), and the smaller eigenvalue of this matrix agrees with the second derivative of the potential along the inflaton direction.Hence, diagonalizing this mass matrix allows for calculation of observables that depend on that second derivative, although it is simpler to work with the single-field effective description.
Following the analysis of the previous section, the results are given in ).Note that in the canonical case the coupling λ is driven to be nonperturbative and the perturbative analysis is not valid, but for the purpose of comparison with the single-field description we treat this FIG.4: The solid blue line, dotted black line, and dashed red line correspond to our mass matrix Eq. (2.27), the Cartesian mass matrix Eq. (2.29), and the mass matrix of Refs.[23][24][25], respectively.
The lower eigenvalue of each matrix, indicated as m 2 in units of d 2 V /dI 2 , is plotted along the trench in the left graph.The corresponding eigenvector's slope is shown on the right, compared to that of the trench.
( 2 α 2 )   Λ 4 ) −1 .For a stable trench to exist, the trench equation Eq. (3.14) should be solvable; however, over a range of r a solution might not exist, depending on the model parameters [22,23].The canonical case is analyzed in Ref. [22], where a viable parameters space is found with inflation ending as in hybrid inflation.For the non-canonical case, with We note that in the Dante's waterfall model the ratio of tensor to scalar amplitudes r was found to be typically small with r < 0.03.The noncanonical kinetic term in the spiral inflation models above would predict larger values of r but smaller N e than in the Dante's waterfall model, and it is challenging to find a viable parameter space in this class of spiral inflation models.

FIG. 2 :
FIG.2:The potential as a function of r and θ in a spiral inflation model with a quadratic shiftsymmetry-breaking potential W (r) = 1 2 m 2 r 2 .The fields r and θ are represented in polar coordinates.

TABLE II :
Observables from the single-field approximation for the (p, n) = (4, 2) model, fixing n s = 0.96 and [ ] θ=θ f = 1.case classically.The results match relatively well with those from single-field approximations.Note that, again, the non-canonical kinetic term leads to a reduced r and N e .We also notice that the (4, 2) C model gives similar numerical predictions to the (4, 1) NC model.More generally, from Eq. (3.13) we see that the (p, n + 1) C model and (p, n) NC model have the same single-field approximation.This is a type of duality between inflation models.The dynamical solutions are plotted in Fig 5.