Dynamical consistency conditions for rapid turn inﬂation Lilia Anguelovaa1 Calin Iuliu Lazaroiub2

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Dynamical consistency conditions for

rapid turn inﬂation

Lilia Anguelovaa1, Calin Iuliu Lazaroiub2

aInstitute for Nuclear Research and Nuclear Energy,

Bulgarian Academy of Sciences, Soﬁa, Bulgaria

bHoria Hulubei National Institute for Physics and Nuclear

Engineering (IFIN-HH), Bucharest-Magurele, Romania

Abstract

We derive consistency conditions for sustained slow roll and rapid turn in-

ﬂation 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, to-

gether 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” approx-

imation, 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.

1anguelova@inrne.bas.bg

2lcalin@theory.nipne.ro

arXiv:2210.00031v3 [hep-th] 18 Apr 2023

Contents

1 Introduction 1

2 Two-ﬁeld cosmological models 4

2.1 Characteristics of an inﬂationary solution . . . . . . . . . . . . . . . . . . . 6

2.2 Slow roll and rapid turn regimes . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Some other useful parameters . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 The consistency condition for sustained slow roll with rapid turn 9

3.1 Theadaptedframe ............................... 9

3.2 Expressing ηkand η⊥in terms of θϕand c.................. 10

3.3 The condition for sustained rapid turn with third order slow roll . . . . . . 11

4 The case of rotationally invariant metrics 15

4.1 Slow roll consistency condition for circular trajectories . . . . . . . . . . . 16

5 A unifying approximation for rapid turn models 20

5.1 Equations of motion in adapted frame . . . . . . . . . . . . . . . . . . . . 20

5.2 Adapted frame parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5.3 Consistency conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6 The characteristic angle 24

7 Conclusions 27

A Derivatives of the adapted frame 29

B Example: Quasi-single ﬁeld inﬂation 30

1 Introduction

Modern observations have established – to a very good degree of accuracy – that the

present day universe is homogeneous and isotropic on large scales. This is naturally ex-

plained if one assumes that the early universe underwent a period of accelerated expansion

called inﬂation. This idea can be realized in models where the inﬂationary expansion is

driven by the potential energy of a number of real scalar ﬁelds called inﬂatons. The most

studied models of this type contain a single scalar ﬁeld. However, recent arguments related

to quantum gravity suggest that it is more natural, or may even be necessary [1–4] to have

more than one inﬂaton. This has generated renewed interest in multi-ﬁeld cosmological

models, which had previously attracted only limited attention.

Multiﬁeld cosmological models have richer phenomenology than single ﬁeld models

since they allow for solutions of the equations of motion whose ﬁeld-space trajectories are

not (reparameterized) geodesics. Such trajectories are characterized by a non-zero turn

rate. In the past it was thought that phenomenological viability requires small turn rate,

by analogy with the slow roll approximation used in the single-ﬁeld case. This assumption

leads to the celebrated slow-roll slow-turn (SRST) approximation of [5, 6]. However, in

recent years it was understood that rapid turn trajectories can also be (linearly) pertur-

batively stable [7, 8] and of phenomenological interest. For instance, a brief rapid turn

during slow-roll inﬂation can induce primordial black hole generation [9–12]; moreover,

trajectories with large and constant turn rate can correspond to solutions behaving as dark

energy [13, 14]. There is also a variety of proposals for full-ﬂedged rapid-turn inﬂation

models, relying on large turn rates during the entire inﬂationary period [15–22].

Finding inﬂationary solutions in multiﬁeld models is much harder than in the single-

ﬁeld case, because the background ﬁeld equations form a complicated coupled system

of nonlinear ODEs. Thus usually such models are either studied numerically or solved

only approximately.3Mathematically, this complicated coupled system is encoded by

the so-called cosmological equation, a nonlinear second order geometric ODE deﬁned on

the scalar ﬁeld space of the model. The latter is a connected paracompact manifold,

usually called the scalar manifold. In turn, the cosmological equation is equivalent with a

dissipative geometric dynamical system deﬁned on the tangent bundle of that manifold.

Little is known in general about this dynamical system, in particular because the scalar

manifold need not be simply-connected and – more importantly – because this manifold is

non-compact in most applications of physical interest and hence cosmological trajectories

can “escape to inﬁnity”. The resulting dynamics can be surprisingly involved4and hard to

3A notable exception is provided by models with hidden symmetry, which greatly facilitates the search

for exact solutions [23–27].

4It is sometimes claimed that the complexity of this dynamics could be ignored, because in “phe-

nomenologically relevant models” one should “expect that” all directions orthogonal to the physically

relevant scalar ﬁeld trajectory are heavy and hence can be integrated out, thus reducing the analysis

to that of a single-ﬁeld model. This argument is incorrect for a number of reasons. First, current phe-

nomenological data does not rule out multiﬁeld dynamics. Second, such a reduction to a one ﬁeld model

(even when possible) relies on knowledge of an appropriate cosmological trajectory, which itself must ﬁrst

be found by analyzing the dynamics of the multiﬁeld model.

analyze even by numerical methods (see [28–30] for some nontrivial examples in two-ﬁeld

models), though a conceptual approach to some aspects of that dynamics was recently

proposed in [31, 32].

A common approach to looking for cosmological trajectories with desirable properties

is to ﬁrst simplify the equations of motion by imposing various approximations (such

as slow-roll to a certain order, rapid-turn and/or other conditions). This leads to an ap-

proximate system of equations, obtained by neglecting certain terms in the original ODEs.

Then one attempts to solve the approximate system numerically or analytically. However,

there is apriori no guarantee that a solution of the approximate system is a good approx-

imant of a solution of the exact system for a suﬃciently long period of time. In general,

this will be the case only if the data which parameterizes the exact system (namely the

scalar ﬁeld metric and scalar potential of the model) satisﬁes appropriate consistency con-

ditions. Despite being of fundamental conceptual importance, such consistency conditions

have so far not been studied systematically in the literature.

In this paper, we investigate the problem of consistency conditions in two-ﬁeld models

with orientable scalar manifold for several commonly used approximations. First, we

consider third order slow-roll trajectories with large but slowly varying turn rate. In this

case, we show that compatibility with the equations of motion requires that the scalar

potential satisﬁes a certain nonlinear second order PDE whose coeﬃcients depend on the

scalar ﬁeld metric. This gives a nontrivial and previously unknown consistency condition

that must be satisﬁed in the ﬁeld-space regions where one can expect to ﬁnd sustained

rapid-turn trajectories allowing for slow roll inﬂation. Therefore, inﬂationary solutions

of this type are not easy to ﬁnd, implying that two-ﬁeld models with such families of

cosmological trajectories are non-generic. In particular, this shows that the diﬃculty in

ﬁnding such models which was noticed in [33] is not related to supergravity, but arises on

a more basic level.

We also discuss the case of rotationally invariant scalar ﬁeld-space metrics. In that

case, it is common to consider ﬁeld space trajectories which are nearly circular as can-

didates for sustained rapid turn inﬂation. Imposing the ﬁrst and second order slow roll

conditions in this context leads to a certain consistency condition for compatibility with

the equations of motion. This is again a PDE for the scalar potential with coeﬃcients

depending on the ﬁeld space metric, which does not seem to have been widely noticed in

the literature. We consider its implications for important examples in previous work.

Finally, we study in detail the consistency conditions for the approximation of [20],

which subsumes many prominent rapid turn models of inﬂation. We show that this

approximation is a special case of rapid turn with slow roll, instead of being equivalent to

it. We then derive conditions for compatibility of this approximation with the equations

of motion. Once again, these constrain the scalar potential and ﬁeld space metric.

Throughout the paper, we assume that the ﬁeld space (a.k.a. scalar manifold) of the

two-ﬁeld model is an oriented and connected paracompact surface. In our considerations,

a crucial role is played by a ﬁxed oriented frame (called the adapted frame) of vector

ﬁelds deﬁned on this surface which is determined by the scalar potential and ﬁeld-space

metric, instead of the moving oriented Frenet frame determined by the ﬁeld space trajec-

tory. The two frames are related to each other through a time-dependent rotation whose

time-dependent angle we call the characteristic angle. We conclude our investigations by

studying the time evolution of this angle. We show that, generically, this angle tends

rather fast to the value πmod 2π, which means that the tangent vector of the inﬂa-

tionary trajectory aligns with minus the gradient of the potential. This implies that the

rapid turn regime has a natural exit in the generic case. Our results also suggest that it

is diﬃcult to sustain this regime for a prolonged period.

The paper is organized as follows. Section 2 recalls some basic facts about two-

ﬁeld cosmological models and introduces various parameters which will be used later

on. Section 3 discusses the consistency condition for sustained rapid turn trajectories

with third order slow roll, which results from careful analysis of the compatibility of

the corresponding approximations with the equations of motion. Section 4 discusses the

consistency condition for circular trajectories in rotationally-invariant models, as well as

some implications for previous work on such inﬂationary trajectories. Section 5 discusses

the approximation of [20], showing how it diﬀers from rapid turn with second order slow

roll and extracts the relevant consistency conditions. Section 6 studies the time evolution

of the characteristic angle, while Section 7 presents our conclusions.

2 Two-ﬁeld cosmological models

The action for nreal scalar ﬁelds ϕI(xµ)minimally coupled to four-dimensional gravity

is:

S=Zd4xp−det gR(g)

2−1

2GIJ ({ϕI})∂µϕI∂µϕJ−V({ϕI}),(2.1)

where we took MPl = 1. Here gµν is the spacetime metric (which we take to have “mostly

plus” signature) and R(g)is its scalar curvature. The indices µ, ν run from from 0to

3and GIJ is the metric on the scalar manifold (a.k.a. “scalar ﬁeld space”) M, which is

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DynamicalconsistencyconditionsforrapidturninationLiliaAnguelovaa1,CalinIuliuLazaroiub2aInstituteforNuclearResearchandNuclearEnergy,BulgarianAcademyofSciences,Soa,BulgariabHoriaHulubeiNationalInstituteforPhysicsandNuclearEngineering(IFIN-HH),Bucharest-Magurele,RomaniaAbstractWederiveconsistencycond...

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Dynamical consistency conditions for rapid turn inﬂation Lilia Anguelovaa1 Calin Iuliu Lazaroiub2

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