Slow-Twisting inflationary attractors Perseas Christodoulidis1and Robert Rosati2 1Department of Science Education Ewha Womans University Seoul 03760 Republic of Korea

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(Slow-)Twisting inﬂationary attractors

Perseas Christodoulidis1, ∗and Robert Rosati2, †

1Department of Science Education, Ewha Womans University, Seoul 03760, Republic of Korea

2NASA Postdoctoral Program Fellow, NASA Marshall Space Flight Center, Huntsville, AL 35812, USA

We explore in detail the dynamics of multi-ﬁeld inﬂationary models. We ﬁrst revisit the two-ﬁeld

case and rederive the coordinate independent expression for the attractor solution with either small

or large turn rate, emphasizing the role of isometries for the existence of rapid-turn solutions. Then,

for three ﬁelds in the slow-twist regime we provide elegant expressions for the attractor solution for

generic ﬁeld-space geometries and potentials and study the behaviour of ﬁrst order perturbations.

For generic N-ﬁeld models, our method quickly grows in algebraic complexity. We observe that

ﬁeld-space isometries are common in the literature and are able to obtain the attractor solutions

and deduce stability for some isometry classes of N-ﬁeld models. Finally, we apply our discussion

to concrete supergravity models. These analyses conclusively demonstrate the existence of N>2

dynamical attractors distinct from the two-ﬁeld case, and provide tools useful for future studies

of their phenomenology in the cosmic microwave background and stochastic gravitational wave

spectrum.

Contents

1. Introduction 2

2. Multi-ﬁeld Trajectories 3

3. Two-ﬁeld solutions 5

3.1. Coordinate-independent expression for the attractor solution 5

3.2. Three examples from the literature: hyperinﬂation, angular and sidetracked inﬂation 7

3.3. Two-ﬁeld metrics with isometries 9

3.4. Metrics without isometries and the stability conditions 10

4. Three-ﬁeld solutions 12

4.1. The zero-torsion case 12

4.2. Non-zero torsion 13

4.3. Examples from the literature 14

5. Speciﬁc N-ﬁeld cases 16

5.1. Metric with N − 1 isometries 17

5.2. Diagonal metric with one isometry 19

5.3. N-ﬁeld background stability 20

5.4. Remarks on the stability of inﬂationary backgrounds in supergravity models 22

6. First order perturbations 23

6.1. Three-ﬁeld perturbations 23

6.2. Two case examples 24

6.2.1. Model 1 24

6.2.2. Model 2 24

6.3. Many-ﬁeld simulations 26

7. Summary 31

Acknowledgments 31

∗Electronic address: perseas@ewha.ac.kr

†Electronic address: robert.j.rosati@nasa.gov

arXiv:2210.14900v3 [hep-th] 10 Nov 2023

A. Linking the Frenet-Serret equations with the equations of motion 31

B. Killing vectors of the hyperbolic space in two dimensions 32

C. Coordinate transformations and isometries 33

D. Eigenvalues of block matrices 34

E. Numerical solution for powerspectra 35

F. Initial condition search strategy 35

References 35

1. INTRODUCTION

The theory of cosmological inﬂation, originally introduced as a scenario that evades the ﬁne-tunings of the classical

FLRW universe [1–7], is currently the leading paradigm for the origin of structure formation [8]. In its simplest form,

the evolution of a scalar ﬁeld over ﬂat regions of its potential causes the accelerated expansion of the universe. One

might think that inﬂation is also subject to ﬁne-tuning problems of its own, because the onset of the accelerating

expansion is not guaranteed for all initial conditions. This turns out not to be the case due to the dissipative nature

of the cosmological equations in an expanding background which yields a notion of initial conditions independence

for models with a single scalar ﬁeld. This fact makes single-ﬁeld models of inﬂation both self-consistent and phe-

nomenologically successful in matching the CMB observations to a great accuracy. On a theoretical level this success

is eclipsed by obstacles in embedding the inﬂationary paradigm into high-energy physics.

More speciﬁcally, high-energy theories, such as string theory or supergravity, in general require consideration of the

dynamics of multiple scalar ﬁelds in the early universe, and through the swampland program, potentially limit their

allowed interactions (e.g. [9–17]). For a multi-ﬁeld model to be consistent with the general philosophy of inﬂation,

observables should have a weak dependence on initial ﬁeld conﬁgurations; this can only be achieved if any additional

degrees of freedom quickly become non-dynamical leaving just one ﬁeld to drive the evolution.1The truncation to one

ﬁeld is tricky and often times the dynamics of the eﬀective description is quite distinct from what one obtains from

purely single-ﬁeld models at the level of both the background and the perturbations (see e.g. [20–32]). In certain cases,

even though the background dynamics can be reduced to the dynamics of a single-ﬁeld, the behaviour of perturbations

can be drastically diﬀerent; the eﬀect of isocurvature perturbations can be absorbed into the deﬁnition of a nontrivial

speed of sound in the evolution of the curvature perturbation, leading to deviations from the single-ﬁeld predictions

(see for instance [33–39]).

Focusing on the background, it becomes apparent that consistent multi-ﬁeld models should display a strong attractor

behaviour, akin to single-ﬁeld models, that weakens their initial conditions dependence. The extra ﬁelds should remain

non-dynamical during the relevant evolution (at least 50-60 e-folds before the end of inﬂation) and, thus, the system

follows a speciﬁc trajectory in the ﬁeld space that we call the attractor solution. In general, the attractor solution is

not equivalent to the solution of a single-ﬁeld model because of the existence of turns in the trajectory; this happens

when the extra ﬁelds remain non-dynamical but are excited away from their respective minima of the potential. This

generic multi-ﬁeld dynamics at the background level has been explored recently in Ref. [40, 41], showing the existence

of rapid-turn solutions based on a generic calculation of the turn rate, and in Ref. [42] where analogous expressions for

the late-time solution were provided using an eﬀective potential that dictates dynamics at late-times. Moreover, this

class of models were shown to lie at the intersection of de-Sitter and scaling solutions where every “slow-roll”-type

approximation becomes exact [43]. It remains unclear whether two-ﬁeld studies capture all essential features of the

multi- or many-ﬁeld evolution, especially for concrete models derived from supergravity. Many studies of N>2

inﬂation exist in the literature, including [26, 44–55]. Many of these works have focused on speciﬁc types of slow-turn

inﬂationary models, the universality of the many-ﬁeld limit, or the eﬀective ﬁeld theory of the curvature perturbation

for N>2 models, overall leaving generic multi-ﬁeld background trajectories unexplored.

1For the simplest multi-ﬁeld models without any hierarchies between the model’s parameters (such as the masses of the ﬁelds or parameters

that quantify the strength of the ﬁeld space curvature) the non-uniqueness of observables is quite evident at the two-ﬁeld level, (see

e.g. [18]), and, moreover, it persists even in the many-ﬁeld limit [19].

In this work we study the model-independent attractor solutions of multi-ﬁeld inﬂation, providing analytic

expressions for three-ﬁeld models.2The expressions are particularly tractable in the slow-twist and rapid-turn

regime. We additionally explore the inﬂationary perturbations of these three-ﬁeld solutions, and recover an eﬀective

single-ﬁeld description. We also generalize our solutions to four or more ﬁelds by assuming ﬁeld spaces with

a suﬃciently high number of isometries. The paper is organized as follows: in Sec. 2 we derive the evolution

equations for the ﬁrst three basis vectors of the orthonormal Frenet-Serret system and calculate the set of higher

order bending parameters that parameterize how the ﬁeld-space trajectory bends in the N-dimensional space.

In Sec. 3 we perform a complete study of two-ﬁeld models under the assumption that the equations of motion

admit slow-roll-like solutions. Next, in Sec. 4 we argue that the complexity of the three-ﬁeld problem makes it

impossible to adopt a similar generic treatment and which forces us to consider certain simpliﬁcations, such as small

torsion. In Sec. 5 we focus on speciﬁc N-ﬁeld problems with isometries and discuss rapid-turn solutions and their sta-

bility. We analyse in more detail observables for three-ﬁeld models in Sec. 6. Finally, in Sec. 7 we oﬀer our conclusions.

Conventions: We will use Nfor the e-folding number, deﬁned from dN=Hdt, and Nfor the number of ﬁelds;

Gij denotes the ﬁeld-space metric; ti, ni, birepresent the components of the ﬁrst three basis vectors of the orthonormal

Frenet system. Since we will use multiple diﬀerent bases, to suppress notation we will use lower case Greek letters

(α, β, γ, ···) to denote indices belonging to the orthonormal kinematic basis while middle lower case Latin indices

(i, j, k, ···) represent general ﬁeld metric indices. We work in units with M2

Pl = 1.

2. MULTI-FIELD TRAJECTORIES

In this section we set the foundations for the rest of this work, and study inﬂationary trajectories in N>2 ﬁeld

space and compute the bending parameters as generically as possible. Our goal is to express these kinematic quantities

in terms of the ﬁeld space metric and potential using the equations of motion.

We study the evolution of Nscalar ﬁelds minimally coupled to gravity

S=Zd4x√−g1

2Re−1

2Gij (ϕk)∂µϕi∂µϕj−V(ϕk)(1)

where Reis the Ricci scalar associated with the spacetime metric and the ﬁelds interact via the ﬁeld-space metric

Gij (ϕk) and the potential V(ϕk).

As standard in the inﬂationary literature, we consider the ﬁelds spatially homogeneous at the classical level. The

classical equations of motion can be written

DN(ϕi)′+ (3 −ϵ)(ϕi)′+Gij V,j

H2= 0,(2)

where primes denote e-fold derivatives dN=Hdt, DN(ϕi)′≡(ϕj)′Dj(ϕi)′= (ϕi)′′ + Γi

jk(ϕj)′(ϕk)′, and the Γi

jk are

the connection components associated to the ﬁeld space metric Gij .

We deﬁne the slow-roll parameters

ϵ≡1

2(ϕi)′Gij (ϕj)′,(3)

η≡ϵ′/ϵ , (4)

which probe the kinetic energy and acceleration of the ﬁelds respectively.

We are interested in the turning parameters of the trajectory, which are deﬁned in the kinetic orthonormal basis

derived from the ﬁelds’ motion. This basis is formed from the velocity unit vector ti≡(ϕi)′/√2ϵand its derivatives.

The change of the velocity unit vector deﬁnes the turn rate of the inﬂationary trajectory, as well as the normal vector

to the trajectory

DNti≡Ωni.(5)

2During the preparation of this paper the preprint [56] was uploaded to the arXiv, which has as one of their conclusions that slow-roll,

rapid-turn behavior is short-lived or very rare in two-ﬁeld inﬂation. This claim seems to be in tension with previous literature, where

rapid-turn models can easily be constructed for highly curved spaces, and further work is necessary to reconcile them.

ϕ1

ϕ2

ϕ3

0< T < Ω, T =O(Ω)

FIG. 1: (Left) Pictorial representation of the orthonormal kinematic system on a three-dimensional curve. (Right) The helix

is the prototypical example of a curve with constant turn rate and torsion.

Similarly, the derivative of the normal vector deﬁnes the torsion or twist rate of the trajectory, as well as the binormal

vector

DNni≡ −Ωti+T bi.(6)

Further derivatives deﬁne higher-order bending parameters, for N − 1 total. This kinematic basis can be neatly

summarized in the Frenet-Serret system of equations [57]













=





0 Ω 0 0 ···

−Ω 0 T0···

0−T0T2···

0 0 −T20···

....



















,(7)

where bi

j, Tjwith j≥2 deﬁne additional bending parameters when N>3 (see Fig. 1 for examples).

The equations of motion are particularly simple in this kinetic basis, where (2) becomes

Ωni+3−ϵ+η

2ti+Gij V,j

√2ϵH2= 0.(8)

From here we can read that (on-shell) the potential gradient only has components along tiand ni, and also that the

turn rate Ω can be expressed

Ωni≡DNti=3−ϵ

√2ϵwσti−wi(9)

where we used the Friedmann equation V=H2(3 −ϵ) and deﬁned wi≡V,i/V and wσ≡witi.

For convenience, below we work with the trajectory’s arclength parameter σ, deﬁned so that DN≡√2ϵDσ.3

The equivalent bending parameters are deﬁned as k≡Ω/√2ϵ,τ≡T/√2ϵ,τj≡Tj/√2ϵ, etc. Through additional

3It is worth mentioning that the use of σas the independent “time” parameter is also physical relevant. Often the turning parameters in

(7) and ϵare analytically related, as in Sec. 5.2 and [23, 26]. In models without a known analytic relationship, we often ﬁnd numerically

that for either slow- or rapid-turn models the quantities that remain almost constant are the bending parameters kand τ.

derivatives of (9), we can ﬁnd expressions for the bending parameters in terms of only kinematic quantities and

covariant derivatives of the potential. We leave the details to Appendix A, but quote the torsion vector here

τbi="ln 3−ϵ

2ϵk ,σ

+3−ϵ

2ϵwσ#ni+3−ϵ

2ϵk wσσti−3−ϵ

2ϵk wi

σ.(10)

From these and similar expressions, it is possible to read oﬀ kinematic relationships for many of the potential

derivatives, which we leave to the appendices but use when applicable.

Below we describe the late-time solutions in terms of these kinematic quantities, assuming only slow-roll.

3. TWO-FIELD SOLUTIONS

3.1. Coordinate-independent expression for the attractor solution

In this section we present a neat way to ﬁnd the generic slow-roll two-ﬁeld attractor solution, before exploring higher

dimensional ﬁeld spaces. The solution is presented in a manifestly covariant expression when expressed in terms of

geometric objects constructed by the ﬁeld metric and the potential. The simplest objects we can construct include

the trace of products of the Hessian wij and projections of products of the Hessian along the gradient directions:

cn≡wiwj

i···wlm

| {z }

n times

wm,(11)

dn≡Tr 



wj

i···wlm

| {z }

n times





.(12)

With these objects to our disposal, the next step is to use an appropriate basis that relates the right hand side of

(11), (12) with the slow-roll parameter ϵand the turn rate and then form a system with a suﬃcient number of these

curvature invariants that allows us to solve back for ϵ.4We ﬁnd it easier to work with the usual kinematic frame in

a manner similar to Ref. [42] which, however, made use of a special coordinate system. We also avoid the potential

gradient-based orthonormal basis used in Refs. [40, 41].

We assume the attractor solution has a slowly-changing ϵ, so that ηis negligible in the equations of motion (8).

This assumption allows us to reduce the second order diﬀerential equation to an algebraic equation for the velocity,

which implies some sort of a late-time solution. From the equations of motion we ﬁnd the adiabatic component of the

gradient vector as of the equations of motion as

wσ=−√2ϵ3−ϵ+η/2

3−ϵ,(13)

and plugging this expression back into the deﬁnition of the turn rate we obtain

2wiwi≡ϵV=ϵ"1 + η

2(3 −ϵ)2

+Ω

3−ϵ2#.(14)

This is our ﬁrst expression that relates ϵwith the turn rate and the norm of the gradient vector. When the slow-roll

conditions are satisﬁed (ϵ, |η| ≪ 1) then Eq. (14) reduces to [58]

ϵV≈ϵ1 + 1

9Ω2,(15)

while ϵis approximated by

ϵ≈ϵad ≡1

2Vσ

V2

2w2

σ.(16)

4One could also consider contractions with objects such as V;ijk containing three or more covariant derivatives. However, in this case the

Frenet-Serret equations will constrain only a small number of them that inculde at least one σcomponent, leaving a larger number of

unknown parameters and hence a larger number of equations that have to be solved simultaneously for ϵ.

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(Slow-)TwistinginflationaryattractorsPerseasChristodoulidis1,∗andRobertRosati2,†1DepartmentofScienceEducation,EwhaWomansUniversity,Seoul03760,RepublicofKorea2NASAPostdoctoralProgramFellow,NASAMarshallSpaceFlightCenter,Huntsville,AL35812,USAWeexploreindetailthedynamicsofmulti-fieldinflationarymodels....

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Slow-Twisting inflationary attractors Perseas Christodoulidis1and Robert Rosati2 1Department of Science Education Ewha Womans University Seoul 03760 Republic of Korea

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