††institutetext: Horia Hulubei National Institute of Physics and Nuclear
Engineering,
Reactorului 30, Bucharest-Magurele, 077125, Romania

# Dynamical renormalization and universality in classical multifield
cosmological models

Calin Iuliu Lazaroiu<EMAIL_ADDRESS>
###### Abstract

We study the scaling behavior of classical multifield cosmological models with
complete scalar manifold $(\mathcal{M},\mathcal{G})$ and positive smooth
scalar potential $\Phi$, introducing a dynamical renormalization group action
which relates their UV and IR limits. We show that the RG flow of such models
interpolates between a modification of the geodesic flow of
$(\mathcal{M},\mathcal{G})$ (obtained in the UV limit) and the gradient flow
of $(\mathcal{M},\mathcal{G},V)$ (obtained in the IR limit), where the
classical effective potential $V$ is proportional to $\sqrt{2\Phi}$. Using
this fact, we show that two-field models whose scalar manifold has constant
Gaussian curvature equal to $-1$, $0$ or $1$ are infrared universal in the
sense that they suffice to describe the first order IR approximants of
cosmological orbits for all two-field models with positive smooth scalar
potential.

## Introduction

The study of cosmological models with more than one real scalar field (known
as multifield cosmological models) is crucial for connecting fundamental
theories of gravity and matter to the early universe, since the effective
description of the generic string or M-theory compactification contains many
such fields. In particular, multifield models are important for cosmological
applications of the swampland program V ; OV  (see BCV ; BCMV  for reviews),
which drew attention to this often overlooked area of cosmology AP ; OOSV ; GK
.

Despite their importance, the current theory of such models is poorly
developed, especially as pertains to deeper conceptual and mathematical
aspects. In fact, most contributions to the area focus on analyzing the
leading orders in the formal cosmological perturbation expansion of such
models around a “God given” abstract cosmological curve SS ; NT ; GT ; GLM ;
EST ; Pinol ; Pinol2field  (often considering only two-field models and under
highly restrictive conditions such as assuming validity of the SRST
approximation of PT1 ; PT2 ) or on investigations of specific models and
cosmological curves aimed at illustrating the feasibility of various inflation
scenarios. Since the class of multifield models is continuously infinite, one
is left wondering if a more systematic approach is possible which could permit
fundamental progress in the field.

At the classical level (and before considering fluctuations) multifield models
are described by ODEs to which the powerful methods of the geometric theory of
dynamical systems Palis ; Katok ; Shub  could be applied. Even at this level,
the current literature remains largely concerned with local descriptions and
pays little attention to global aspects, thus hardly amounting to a geometric
theory. In particular, there were only a few attempts to develop an approach
to the subject that could place it on a firm mathematical foundation and hence
could serve as a stepping stone toward the program of connecting string theory
to the early universe in a systematic manner. The purpose of this paper is to
address some of these limitations by proposing a conceptual viewpoint on the
classical dynamics of such models.

As pointed out in genalpha , classical multifield cosmological models admit a
precise global description which leads to their formulation as geometric
dynamical systems. Mathematically, such a model is parameterized by the
reduced Planck mass $M$ (equivalently, by the rescaled Planck mass
$M_{0}\stackrel{{\scriptstyle{\rm def.}}}{{=}}M\sqrt{\frac{2}{3}}$) and by a
scalar triple $(\mathcal{M},\mathcal{G},\Phi)$, where $\mathcal{M}$ is a
(generally non-compact) connected manifold, $\mathcal{G}$ is a Riemannian
metric on $\mathcal{M}$ and $\Phi$ is a real-valued function defined on
$\mathcal{M}$; we assume that these three objects are smooth. Such data enters
the classical action of the scalar fields, which is described by a nonlinear
sigma model defined on spacetime, where:

  * •

The target space $\mathcal{M}$ is the space in which the scalar fields take
values, namely they are described as smooth maps from spacetime into this
manifold.

  * •

The scalar field metric $\mathcal{G}$ governs the kinetic energy of the scalar
fields.

  * •

The scalar potential $\Phi$ governs the potential energy of the scalar fields.

To ensure conservation of energy, one requires the metric $\mathcal{G}$ to be
complete on $\mathcal{M}$; this precludes existence of scalar field
configurations which “disappear into nothing”. To preclude global instability
of dynamics, one requires that $\Phi$ is bounded from below by zero; for
simplicity, we will assume throughout this paper that $\Phi$ is strictly
positive. The complete Riemannian manifold $(\mathcal{M},\mathcal{G})$ is
called the scalar manifold.

The classical cosmological model is obtained from this data by considering
gravity at rescaled Planck mass $M_{0}$ coupled to the sigma model
parameterized by $(\mathcal{M},\mathcal{G},\Phi)$. Before considering
perturbations, one takes the spacetime to be of FLRW form with cosmological
time $t$ and conformal factor $a$, while restricting the scalar fields to
depend only on $t$ and hence to be described by a curve
$\varphi:I\rightarrow\mathcal{M}$, where $I$ is a non-degenerate interval
(i.e. an interval on the real axis which is not empty or reduced to a point).
With these restrictions, the Einstein equations and scalar field equations of
motion reduce to a single second order ODE (known as the cosmological
equation) for $\varphi$, whose solutions we call cosmological curves. Since it
is geometric in the sense of Chern , the cosmological equation is equivalent
to a dynamical system defined on the total space of the tangent bundle
$T\mathcal{M}$ of the scalar manifold $\mathcal{M}$ by a certain vector field
$S$ which we call the cosmological semispray. The latter lives on
$T\mathcal{M}$ and is parameterized by $M_{0}$, by the scalar field metric
$\mathcal{G}$ and by the scalar potential $\Phi$. The equivalence between the
cosmological equation and the cosmological dynamical system follows from the
theory of geometric second order ODEs Kosambi ; Cartan ; Chern ; SLK ;
Bucataru  (see Nis  for a brief account). The flow defined by this dynamical
system on $T\mathcal{M}$ is called the cosmological flow of the scalar triple
$(\mathcal{M},\mathcal{G},\Phi)$ at rescaled Planck mass $M_{0}$.

Since the topology of $\mathcal{M}$ is arbitrary, a systematic approach to
such models requires the full force of the geometric theory of dynamical
systems as developed for example in Palis ; Katok ; Shub ; one cannot give a
satisfactory treatment merely by studying the description of this system in
local coordinates on the scalar manifold $\mathcal{M}$. For example,
$\mathcal{M}$ need not be simply connected so a maximal cosmological curve
need not be contractible. More importantly, $\mathcal{M}$ is generally non-
compact in physically interesting applications, so a cosmological curve can
“escape to infinity” in the sense that it can approach a Freudenthal end
Freudenthal ; Morita ; Isbell ; Peschke  of $\mathcal{M}$ for early or late
cosmological times depending on the behavior of $\Phi$ and $\mathcal{G}$ near
that end. Non-compactness of $\mathcal{M}$ complicates the classification of
past and future limit points of cosmological curves, which governs the early
and late time behavior of the model – an aspect which is of direct physical
interest. It also prevents the cosmological flow from being future complete
unless appropriate conditions are imposed on $\Phi$ and $\mathcal{G}$ in the
vicinity of the Freudenthal ends of $\mathcal{M}$. Such global aspects of
cosmological dynamics have direct implications for any attempt to construct
effective descriptions when one incorporates quantum effects (as envisaged for
example in cosmological applications of the swampland program). Indeed, any
effective description depends on having an a priori topological classification
of maximal cosmological curves since one cannot expect an effective
description to be the same in every topological class. Intuitively, the
topological classification of maximal cosmological curves partitions the
dynamics of the model into “phases” and every phase will have its own
effective description.

The global aspects mentioned above are already important for two-field
cosmological models i.e. when $\mathcal{M}$ is a surface, in which case we
denote it by $\Sigma$. The topological classification of borderless connected
paracompact surfaces Kerekjarto ; Stoilow ; Richards  through their
orientability, reduced genus and space of ends (together with its length two
chain of distinguished subspaces) shows that the global theory of cosmological
dynamical systems is already very rich in the two-field case. This is
especially true when the set of ends is infinite, in which case it can be for
example a Cantor space. The theory is quite rich even when one restricts to
oriented surfaces which are topologically finite in the sense that they have a
finitely-generated fundamental group and hence a finite number of Freudenthal
ends. The dynamical complexity of two-field models having such surfaces as
targets was illustrated in our previous work genalpha ; elem ; modular  (see
unif  for a brief review) when $\mathcal{G}$ has constant negative curvature;
this corresponds to two-field generalized $\alpha$-attractor models, which
form a very wide extension of the topologically trivial class of Poincaré disk
two-field models considered KLR .

Since the connected manifold $\mathcal{M}$, the complete Riemannian metric
$\mathcal{G}$ and the positive potential $\Phi$ are arbitrary, the task of
classifying multifield cosmological dynamics may seem hopeless at first sight.
From a dynamical systems perspective, one could attempt to classify
cosmological flows up to the appropriate notions of conjugation or
equivalence, a task which is nontrivial when $\mathcal{M}$ is non-compact and
was not yet attempted in the generality considered here.

In this paper, we propose a different approach to extract the essential
features of such models and to organize them into qualitatively distinct
dynamical classes. Our point of view is inspired by ideas akin to those used
in the theory of critical phenomena, with the points of $T\mathcal{M}$ playing
the role of microscopic states and $\mathcal{G}$ and $\Phi$ playing a role
similar to that of non-equilibrium thermodynamic observables. A state of the
system is a point $u(t)$ of $T\mathcal{M}$ which evolves in time according to
the cosmological flow and has “thermodynamic” parameters
$\frac{1}{2}||u(t)||_{\mathcal{G}}^{2}=\frac{1}{2}\mathcal{G}(\pi(u(t)))(u(t),u(t))$
and $\Phi(\pi(u(t)))$, where $\pi$ is the projection of $T\mathcal{M}$. As in
thermodynamics, knowing the values of these parameters does not determine the
“microscopic” state $u(t)$. Since the cosmological equation is autonomous, the
system is stationary in the sense that its flow is invariant under shifts of
the cosmological time by an arbitrary constant. Following the analogy with
critical phenomena, we consider the behavior of the model under scale
transformations $t\rightarrow t/\epsilon$ of the cosmological time (where
$\epsilon$ is a positive parameter), showing that such transformations induce
a renormalization group action on $M_{0}$, $\mathcal{G}$ and $\Phi$. The
scaling limits $\epsilon\rightarrow\infty$ and $\epsilon\rightarrow 0$ capture
the high frequency (or ultraviolet) and low frequency (or infrared) behavior
of cosmological curves in the sense that they “isolate” the high and low
frequency characteristic oscillations of such curves. We then show that taking
$\epsilon$ to be large or small corresponds respectively to replacing the
cosmological flow of $(\mathcal{M},\mathcal{G},\Phi)$ with a modification of
the geodesic flow of the scalar manifold $(\mathcal{M},\mathcal{G})$ or with
the gradient flow of the classical effective potential $V=M_{0}\sqrt{2\Phi}$
on this Riemannian manifold. The ultraviolet limit is scale invariant, while
the infrared limit (which we consider up to first order in the scale parameter
$\epsilon$) is invariant under Weyl transformations of $\mathcal{G}$ up to
reparameterization of the flow curves; in particular the first order IR
approximation of cosmological orbits is Weyl-invariant. This limit is
degenerate in that it confines the cosmological flow to the graph of the
vector field $-\mathrm{grad}_{\mathcal{G}}V$ inside $T\mathcal{M}$;
accordingly, the order of the cosmological equation drops by one in the
infrared limit. The scaling limits arise as the leading orders of systematic
asymptotic approximations (called the UV and IR expansions) of the
cosmological flow around the geodesic flow of $(\mathcal{M},\mathcal{G})$ or
around the gradient flow of $(\mathcal{M},\mathcal{G},V)$. Such expansions are
natural from a physics perspective. They are also mathematically natural since
geodesic and gradient flows are well-studied subjects in the theory of
dynamical systems – though the generic non-compactness of $\mathcal{M}$
complicates the analysis. Use of the scaling limits allows us to classify
multifield cosmological models into UV and IR universality classes whose study
relates to that of such classical flows.

The Weyl-invariance of infrared approximants to cosmological orbits has
striking implications for two-field models. In this case, the uniformization
theorem of Poincaré UnifT  states that the Weyl equivalence class of
$\mathcal{G}$ contains a unique complete metric $G$ (called the uniformizing
metric) which has constant Gaussian curvature $K$ equal to $-1$, $0$ or $+1$.
Thus the first order infrared approximants of cosmological orbits for any
cosmological two-field model with positive scalar potential $\Phi$ coincide
with those of the model obtained by replacing $\mathcal{G}$ with $G$ and with
the gradient flow orbits of the scalar triple $(\mathcal{M},G,V)$. In
particular, models whose scalar manifold metric has constant Gaussian
curvature provide distinguished representatives of the infrared universality
classes of all two-field models.

The generic case $K=-1$ arises whenever $\Sigma$ is a (compact or non-compact)
surface of general type, i.e. not diffeomorphic with a 2-plane, a 2-sphere, a
real projective plane, a 2-torus, a Klein bottle, an open 2-cylinder or an
open Möbius strip. In this situation, the uniformizing metric $G$ is
hyperbolic. When $\Sigma$ is diffeomorphic with $\mathrm{S}^{2}$ or
$\mathbb{R}\mathbb{P}^{2}$, the metric $G$ has Gaussian curvature $+1$, while
when $\Sigma$ is diffeomorphic with a torus or Klein bottle the uniformizing
metric is flat and complete. When $\Sigma$ is exceptional, i.e. diffeomorphic
with a plane, an open annulus or an open Möbius strip, the metric
$\mathcal{G}$ uniformizes to a complete flat metric or to a hyperbolic metric
depending on its conformal class. In this situation, a description of
universality classes requires considering both111For the three exceptional
surfaces, a hyperbolic metric on $\Sigma$ is conformally flat but it is not
conformally equivalent with a complete flat metric. models with complete flat
and hyperbolic scalar manifold metric. Notice that two-field models with
contractible target are of exceptional type and hence the Poincaré disk models
of KLR  cannot be IR universal among such. The qualitatively different
behavior of two-field models with distinct target space topology illustrates
the importance of global aspects in cosmological dynamics.

The paper is organized as follows. In Section 1, we recall the global
description of multifield cosmological models, outline its dynamical system
formulation and some of its basic properties and discuss a universal two-
parameter group of similarities of the cosmological equation. We also describe
two natural equivalence relations on multifield cosmological models which
arise from underlying groupoid structures and make some observations on the
early and late time behavior of cosmological curves for models with non-
compact target manifold. In Section 2, we discuss the scale transformations
and scaling limits of multifield cosmological models, showing that the UV and
IR limits recover respectively a modification of the geodesic flow of
$(\mathcal{M},\mathcal{G})$ and the gradient flow of
$(\mathcal{M},\mathcal{G},V)$, where $V=M_{0}\sqrt{2\Phi}$. We also derive
consistency conditions for the UV and IR approximations, showing that they
differ from other approximations commonly used in cosmology (such as the slow
roll and SRST approximations or the gradient flow approximation of genalpha ).
In Section 3, we introduce the dynamical renormalization group of such models.
We show invariance of first order IR approximant orbits under Weyl
transformations of the scalar manifold metric, define IR universality classes
and briefly discuss the late time infrared phase structure. In Section 4 we
prove IR universality of two-field models whose scalar manifold metric has
constant Gaussian curvature equal to $-1$, $0$ or $1$. Section 5 presents our
conclusions and some directions for further research. The Appendices contain
some technical notions and results used in the main text.

#### Notations and conventions.

All target manifolds $\mathcal{M}$ considered in this paper are connected,
smooth, Hausdorff and paracompact. If $V$ is a smooth real-valued function
defined on $\mathcal{M}$, we denote by:

$\mathrm{Crit}V\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\\{c\in\mathcal{M}|(\mathrm{d}V)(c)=0\\}$

the set of critical points of $V$. For any $c\in\mathrm{Crit}V$, we denote by
$\mathrm{Hess}(V)(c)\in\mathrm{Sym}^{2}(T^{\ast}_{c}\mathcal{M})$ the Hessian
of $V$ at $c$, which is a well-defined and coordinate independent symmetric
bilinear form defined on the tangent space $T_{c}\mathcal{M}$. Recall that a
critical point $c$ of $V$ is called nondegenerate if $\mathrm{Hess}(V)(c)$ is
a non-degenerate bilinear form. When $V$ is a Morse function (i.e. all of its
critical points are non-degenerate), the set $\mathrm{Crit}V$ is discrete.

We denote by $\hat{\mathcal{M}}$ the Freudenthal (a.k.a. end) compactification
of $\mathcal{M}$, which is a Hausdorff topological space containing
$\mathcal{M}$ as a dense subset (see Freudenthal ; Morita ; Isbell ). A metric
$\mathcal{G}$ on $\mathcal{M}$ is called hyperbolic if it is complete and of
constant sectional curvature equal to $-1$. In particular, a metric defined on
a surface is hyperbolic if it is complete and of Gaussian curvature $-1$.

##  1 Multifield cosmological models

Throughout this paper, a multifield cosmological model means a classical
cosmological model with a finite number $d>1$ of scalar fields, which is
derived from the following matter-gravity action on a spacetime with topology
$\mathbb{R}^{4}$:

$S[g,\varphi]=\int\mathrm{vol}_{g}\mathcal{L}[g,\varphi]~{}~{},$ (1)

where:

$\mathcal{L}[g,\varphi]=\frac{M^{2}}{2}\mathrm{R}(g)-\frac{1}{2}\mathrm{Tr}_{g}\varphi^{\ast}(\mathcal{G})-\Phi\circ\varphi~{}~{}.$
(2)

Here $M$ is the reduced Planck mass, $g$ is the spacetime metric on
$\mathbb{R}^{4}$ (taken to be of “mostly plus”) signature, while
$\mathrm{vol}_{g}$ and $\mathrm{R}(g)$ are the volume form and Ricci scalar of
$g$. The scalar fields are described by a smooth map
$\varphi:\mathbb{R}^{4}\rightarrow\mathcal{M}$, where $\mathcal{M}$ is a
(generally non-compact) connected, smooth and paracompact manifold of
dimension $d$ which is endowed with a smooth Riemannian metric $\mathcal{G}$,
while $\Phi:\mathcal{M}\rightarrow\mathbb{R}$ is a smooth function which plays
the role of potential for the scalar fields. As mentioned in the introduction,
we require that $\mathcal{G}$ is complete to ensure conservation of energy.
For simplicity, we will also assume that $\Phi$ is strictly positive on
$\mathcal{M}$. The quantity $\mathrm{Tr}_{g}\varphi^{\ast}(\mathcal{G})$ is
the trace of the $(1,1)$-tensor obtained by raising one of the indices of the
covariant 2-tensor $\varphi^{\ast}(\mathcal{G})$ with respect to the spacetime
metric $g$, while $\Phi\circ\varphi:\mathbb{R}^{4}\rightarrow\mathbb{R}$ is
the standard mathematical notation222This is commonly written as
$\Phi(\varphi)$ in the physics literature, though this notation is misleading
since $\varphi$ is not the argument of $\Phi$. for the real-valued function
defined on $\mathbb{R}^{4}$ which is obtained by composing $\Phi$ with
$\varphi$:

$(\Phi\circ\varphi)(x^{0},\ldots,x^{3})=\Phi(\varphi(x^{0},\ldots,x^{3}))~{}~{}.$

In local coordinates on $\mathcal{M}$, we have
$\varphi(x^{0},\ldots,x^{3})\\!=\\!(\varphi^{1}(x^{0},\ldots,x^{3}),\ldots,\varphi^{d}(x^{0},\ldots,x^{3}))$.
The second term in the Lagrangian above takes the familiar sigma model form if
one uses local coordinates on $\mathcal{M}$:

$\frac{1}{2}(\mathrm{Tr}_{g}\varphi^{\ast}(\mathcal{G}))(x)=\frac{1}{2}g^{\mu\nu}(x)\mathcal{G}_{\alpha\beta}(\varphi(x))\partial_{\mu}\varphi^{\alpha}\partial_{\nu}\varphi^{\beta}~{}~{}.$

Notice that the action (1) and its Lagrangian density (2) are manifestly
geometric, i.e. written in coordinate-free form and without making any
restrictive assumptions on the differential topology of $\mathcal{M}$, which
(except for being connected) can be arbitrary since any paracompact manifold
admits Riemannian metrics. Also notice that such a model is parameterized by
the quadruplet $\mathfrak{M}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(M_{0},\mathcal{M},\mathcal{G},\Phi)$.

###  1.1 The cosmological equation and cosmological dynamical system

The multifield cosmological model parameterized by
$(M_{0},\mathcal{M},\mathcal{G},\Phi)$ is obtained by assuming that $g$ is an
FLRW metric with flat spatial section:

$\mathrm{d}s^{2}_{g}=-\mathrm{d}t^{2}+a(t)^{2}\sum_{i=1}^{3}\mathrm{d}x_{i}^{2}$
(3)

(where $a(t)>0$) and that $\varphi$ depends only on
$t\stackrel{{\scriptstyle{\rm def.}}}{{=}}x^{0}$, which we call cosmological
time. In this case, the equations of motion derived from (1) (namely the
Einstein equations and the equation of motion for $\varphi$) amount to the
following system of coupled nonlinear ODEs:

$\displaystyle\nabla_{t}\dot{\varphi}+3H\dot{\varphi}+(\mathrm{grad}_{\mathcal{G}}\Phi)\circ\varphi$
$\displaystyle=$ $\displaystyle 0$
$\displaystyle\frac{1}{3}\dot{H}+H^{2}-\frac{\Phi\circ\varphi}{3M^{2}}$
$\displaystyle=$ $\displaystyle 0$ (4)
$\displaystyle\dot{H}+\frac{||\dot{\varphi}||_{\mathcal{G}}^{2}}{2M^{2}}$
$\displaystyle=$ $\displaystyle 0~{}~{},$

where the dot indicates derivation with respect to $t$ and
$H\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\frac{\dot{a}}{a}\in\mathcal{C}^{\infty}(\mathbb{R})$ is the
Hubble parameter. The last relation in the system above is called the
Friedmann equation. Notice that $a$ enters this system only through its
logarithmic derivative $H$.

#### The cosmological equation.

When $H$ is positive (which we assume throughout this paper), it can be
eliminated algebraically using last two equations, which give:

$H(t)=H_{\varphi}(t)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\frac{1}{3M_{0}}\left[||\dot{\varphi}(t)||_{\mathcal{G}}^{2}+2\Phi(\varphi(t))\right]^{1/2}~{}~{},$
(5)

where we defined the rescaled Planck mass $M_{0}$ though:

$M_{0}\stackrel{{\scriptstyle{\rm def.}}}{{=}}M\sqrt{\frac{2}{3}}~{}~{}.$ (6)

Eliminating $H$ though (5) allows one to reduce the system (1.1) to the
following autonomous geometric second order ODE, which we call the
cosmological equation:

$\nabla_{t}\dot{\varphi}(t)+\frac{1}{M_{0}}\left[||\dot{\varphi}(t)||_{\mathcal{G}}^{2}+2\Phi(\varphi(t))\right]^{1/2}\dot{\varphi}(t)+(\mathrm{grad}_{\mathcal{G}}\Phi)(\varphi(t))=0~{}~{}.$
(7)

Here $\nabla_{t}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\nabla_{\dot{\varphi}(t)}$. The cosmological equation is
equivalent with the system (1.1) when $H>0$. Since
$\mathrm{grad}_{\mathcal{G}}\Phi$ is invariant under transformations of the
form $(\mathcal{G},\Phi)\rightarrow(\lambda\mathcal{G},\lambda\Phi)$ with
$\lambda$ a positive constant, this equation can be written as:

$\nabla_{t}\dot{\varphi}(t)+\left[||\dot{\varphi}(t)||_{\mathcal{G}_{0}}^{2}+2\Phi_{0}(\varphi(t))\right]^{1/2}\dot{\varphi}(t)+(\mathrm{grad}_{\mathcal{G}_{0}}\Phi_{0})(\varphi(t))=0~{}~{},$

while (5) reads:

$H_{\varphi}(t)=\frac{1}{3}\left[||\dot{\varphi}(t)||_{\mathcal{G}_{0}}^{2}+2\Phi_{0}(\varphi(t))\right]^{1/2}~{}~{},$

where we defined the rescaled scalar field metric and rescaled scalar
potential through:

$\mathcal{G}_{0}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\frac{1}{M_{0}^{2}}\mathcal{G}~{}~{}\mathrm{and}~{}~{}\Phi_{0}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\ \frac{1}{M_{0}^{2}}\Phi~{}~{}.$

In particular, the cosmological equation depends only on the rescaled scalar
triple $(\mathcal{M},\mathcal{G}_{0},\Phi_{0})$. Given a solution
$\varphi:I\rightarrow\mathcal{M}$ of this equation, relation (5) determines
the scale factor $a(t)$ up to a multiplicative constant $C>0$:

$a(t)=Ce^{\int_{t_{0}}^{t}\mathrm{d}t^{\prime}H_{\varphi}(t^{\prime})}~{}~{},$

where $t_{0}\in I$ is chosen arbitrarily.

#### Cosmological curves and cosmological orbits.

The solutions $\varphi:I\rightarrow\mathcal{M}$ of (7) (where $I$ is a non-
degenerate interval) are called cosmological curves. The images $\varphi(I)$
of these curves in $\mathcal{M}$ will be called cosmological orbits. Notice
that a cosmological orbit need not be an immersed submanifold of
$\mathcal{M}$. It can be shown that the singular points of a cosmological
curve (i.e. those points where its tangent vector vanishes) form an at most
countable set whose complement in the corresponding orbit is a union of
mutually disjoint embedded submanifolds of $\mathcal{M}$. The cosmological
times corresponding to the singular points form a discrete subset of its
interval of definition.

#### The cosmological semispray.

Since the cosmological equation is manifestly geometric, it is also geometric
in the weaker sense that its local description in a coordinate system on
$\mathcal{M}$ is invariant under changes of local coordinates. It follows that
this equation is equivalent with the flow equation of a special type of vector
field (called a semispray or second order tangent vector field) defined on the
total space of the tangent bundle of $\mathcal{M}$. We refer the reader to SLK
; Bucataru  for an introduction to the well-developed theory of semisprays and
second order geometric ODEs; for completeness, let us recall the relevant
definitions and properties of such vector fields.

Let $J\in\mathrm{End}_{T\mathcal{M}}(TT\mathcal{M})$ be the canonical
endomorphism (a.k.a. tangent structure) of the double tangent bundle
$TT\mathcal{M}$ and $C\in\mathcal{X}(T\mathcal{M})$ be its Euler-Liouville
vector field. Let $\kappa:TT\mathcal{M}\rightarrow TT\mathcal{M}$ be the
canonical involution (a.k.a. flip) of $TT\mathcal{M}$,
$\pi:T\mathcal{M}\rightarrow\mathcal{M}$ be the bundle projection of
$T\mathcal{M}$ and $\pi_{T}:TT\mathcal{M}\rightarrow T\mathcal{M}$ be the
tangent bundle projection (a.k.a. first projection) of $TT\mathcal{M}$. Then
the map $\pi_{\ast}:TT\mathcal{M}\rightarrow T\mathcal{M}$ endows
$TT\mathcal{M}$ with a second vector bundle structure such that
$(TT\mathcal{M},\pi_{T},\pi_{\ast},\mathcal{M})$ is a double vector bundle. By
definition, a second order tangent vector to $\mathcal{M}$ is an element $w\in
TT\mathcal{M}$ such that $J(w)=C_{w}$. This condition is equivalent with
$\kappa(w)=w$ and also with $\pi_{\ast}(w)=\pi_{T}(w)$. We denote by
$T^{s}T\mathcal{M}$ the vector sub-bundle of $TT\mathcal{M}$ consisting of
second order tangent vectors. By definition, a smooth semispray on
$T\mathcal{M}$ is a smooth vector field $S\in\mathcal{X}(T\mathcal{M})$ which
is a section of this sub-bundle, i.e. which has the property that $S_{u}$ is a
double tangent vector to $\mathcal{M}$ for all $u\in T\mathcal{M}$. This
amounts to requiring that $S$ satisfies the equivalent conditions:

$J(S)=C\Longleftrightarrow\kappa(S)=S\Longleftrightarrow\pi_{\ast}(S)=\pi_{T}\circ
S~{}~{},$

where in the right hand side of the last equality $S$ is viewed as a map from
$T\mathcal{M}$ to $TT\mathcal{M}$.

The translation between the cosmological equation and the integral curve
equation of the cosmological semispray is performed by considering the
canonical lift of a cosmological curve $\varphi:I\rightarrow\mathcal{M}$,
which is defined as its first jet prolongation:

$c(\varphi)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}j^{1}(\varphi)=\dot{\varphi}:I\rightarrow T\mathcal{M}~{}~{},$

where $\dot{\varphi}(t)\in T_{\varphi(t)}\mathcal{M}$ is the tangent vector to
$\varphi$ at cosmological time $t$. The canonical lift gives an injective map:

$c:C(\mathcal{M})\rightarrow C(T\mathcal{M})$

from the set $C(\mathcal{M})$ of smooth curves in $\mathcal{M}$ to the set
$C(T\mathcal{M})$ of smooth curves in $T\mathcal{M}$. This is a right inverse
of the map $p:C(T\mathcal{M})\rightarrow C(\mathcal{M})$ defined through:

$p(\gamma)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\pi\circ\gamma:I\rightarrow\mathcal{M}$

for any smooth curve $\gamma:I\rightarrow T\mathcal{M}$. The relation $p\circ
c=\mathrm{id}_{C(\mathcal{M})}$ reflects the fact that $\dot{\varphi}(t)\in
T\mathcal{M}$ is a tangent vector to $\mathcal{M}$ at the point $\varphi(t)$.
Since the canonical lift $\varphi$ satisfies:

$\pi_{\ast}\circ c(c(\varphi))=\pi_{T}\circ c(\varphi)~{}~{},$

the image of the map $c$ coincides with the subset $C_{s}(T\mathcal{M})$ of
$C(T\mathcal{M})$ consisting of semispray curves in $T\mathcal{M}$, which are
defined as those smooth curves $\gamma:I\rightarrow T\mathcal{M}$ whose
tangent vector at any point is a double tangent vector to $\mathcal{M}$, i.e.
which satisfy $\dot{\gamma}(t)\in T^{s}_{\gamma(t)}T\mathcal{M}$ for all $t\in
I$. The inverse of the bijection $c:C(\mathcal{M})\rightarrow
C_{s}(T\mathcal{M})$ is the restriction of $p$ to $C_{s}(T\mathcal{M})$.

The canonical lifts of cosmological curves are called cosmological flow
curves. The cosmological semispray $S\in\mathcal{X}(T\mathcal{M})$ is defined
by the property that its integral curves coincide with the cosmological flow
curves. It can be shown that $S$ is given by:

$S=\mathbb{S}-Q~{}~{},$

where $\mathbb{S}$ is the geodesic spray of the scalar manifold
$(\mathcal{M},\mathcal{G})$ and the cosmological correction
$Q\in\mathcal{X}(T\mathcal{M})$ is the vertical vector field:

$Q=\mathcal{H}C+(\mathrm{grad}_{\mathcal{G}}\Phi)^{v}~{}~{}.$ (8)

Here the superscript $v$ denotes the vertical lift of vector fields from
$\mathcal{M}$ to $T\mathcal{M}$ and
$\mathcal{H}:T\mathcal{M}\rightarrow\mathbb{R}_{>0}$ is the reduced Hubble
function of $(\mathcal{M},\mathcal{G},\Phi)$, which is defined through:

$\mathcal{H}(u)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\frac{1}{M_{0}}\sqrt{||u||_{\mathcal{G}}^{2}+2\Phi(\pi(u))}~{}~{}\forall
u\in T\mathcal{M}~{}~{}.$ (9)

#### The cosmological dynamical system and cosmological flow.

The cosmological semispray $S$ defines an autonomous geometric dynamical
system $(T\mathcal{M},S)$ on the total space of the tangent bundle to
$\mathcal{M}$, whose flow $\Pi:\mathcal{D}\rightarrow T\mathcal{M}$ is called
the cosmological flow of the model parameterized by
$(M_{0},\mathcal{M},\mathcal{G},\Phi)$ (equivalently, of the rescaled scalar
triple $(\mathcal{M},\mathcal{G}_{0},\Phi_{0})$). Here
$\mathcal{D}\subset\mathbb{R}\times T\mathcal{M}$ is the maximal domain of
definition of the flow. Little is known about the global behavior of the
cosmological flow of a general non-compact scalar triple and especially about
its early and late time behavior.

#### Classical cosmological observables.

A basic local cosmological observable is a smooth function
$F:T\mathcal{M}\rightarrow\mathbb{R}$. The on-shell reduction of $F$ on a
cosmological flow curve $\gamma:I\rightarrow T\mathcal{M}$ is the function:

${\hat{F}}_{\gamma}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}F\circ\gamma:I\rightarrow\mathbb{R}~{}~{},$

while its on-shell reduction on a cosmological curve
$\varphi:I\rightarrow\mathcal{M}$ is the function:

$F_{\varphi}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}{\hat{F}}_{\dot{\varphi}}=F\circ\dot{\varphi}=F\circ
c(\varphi):I\rightarrow\mathbb{R}~{}~{}.$

One can also consider smooth functions
$F:J^{k}(\mathcal{M})\rightarrow\mathbb{R}$, where $k\geq 2$ and
$J^{k}(\mathcal{M})$ is the $k$-th jet bundle of $\mathcal{M}$. The reduction
of such a function on a cosmological curve $\varphi:I\rightarrow\mathcal{M}$
is defined though:

$F_{\varphi}\stackrel{{\scriptstyle{\rm def.}}}{{=}}F\circ
j^{k}(\varphi):I\rightarrow\mathbb{R}~{}~{},$

where $j^{k}(\varphi):I\rightarrow J^{k}(\mathcal{M})$ is the $k$-th jet
prolongation of $\varphi$. Repeated use of the cosmological equation allows
one to express $F_{\varphi}$ as:

$F_{\varphi}={\tilde{F}}_{\varphi}~{}~{},$

where ${\tilde{F}}\in\mathcal{C}^{\infty}(T\mathcal{M})$ is a basic local
observable constructed from $F$ and the jets $j^{k-1}(\mathcal{G})\in
J^{k-1}(\mathrm{Sym}^{2}(T\mathcal{M}),\mathbb{R})$ and $j^{k-1}(\Phi)\in
J^{k-1}(\mathcal{M},\mathbb{R})$ of $\mathcal{G}$ and $\Phi$.

#### Dissipativity and stationary points.

It is easy to check that the stationary points of the cosmological flow are
the images of the critical points of $\Phi$ through the zero section of
$T\mathcal{M}$. Accordingly, the stationary set of the cosmological model
coincides with the trivial lift:

$(\mathrm{Crit}\Phi)_{0}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\\{0_{c}|c\in\mathrm{Crit}\Phi\\}\subset T\mathcal{M}$

of the critical set:

$\mathrm{Crit}\Phi\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\\{c\in\mathcal{M}|(\mathrm{d}\Phi)(c)=0\\}$

of $\Phi$, whose complement:

$\mathcal{M}_{0}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathcal{M}\setminus\mathrm{Crit}\Phi\subset\mathcal{M}$

is an open submanifold of $\mathcal{M}$ called the noncritical set of the
model.

A cosmological flow curve is constant iff its orbit meets a point of the set
$(\mathrm{Crit}\Phi)_{0}$, in which case its orbit is reduced to that point.
Accordingly, a cosmological curve is constant iff its orbit meets a critical
point of $\Phi$ at zero speed, in which case the orbit coincides with that
critical point. When $\Phi$ is a Morse function, a straightforward computation
shows that the stationary points of the cosmological flow are hyperbolic.

It is also easy to see that the cosmological flow is dissipative. For this,
consider the total energy observable $E:T\mathcal{M}\rightarrow\mathbb{R}$
defined through:

$E(u)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\frac{1}{2}||u||_{\mathcal{G}}^{2}+\Phi(\pi(u))~{}~{}\forall u\in
T\mathcal{M}~{}~{},$

which is the sum of the kinetic and potential energies:

$E^{k}(u)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\frac{1}{2}||u||_{\mathcal{G}}^{2}~{}~{},~{}~{}E^{p}(u)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\Phi(\pi(u))$

and is related to the reduced Hubble function by
$\mathcal{H}=\frac{1}{M_{0}}\sqrt{2E}$. The last equation in (1.1) implies
that the total energy of a cosmological curve $\varphi$:

$E_{\varphi}(t)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}E(\dot{\varphi}(t))=\frac{1}{2}||\dot{\varphi}(t)||_{\mathcal{G}}^{2}+\Phi(\varphi(t))=\frac{9M_{0}^{2}}{2}H_{\varphi}(t)^{2}$
(10)

satisfies:

$\frac{\mathrm{d}E_{\varphi}(t)}{\mathrm{d}t}=-\frac{1}{M_{0}}\sqrt{2E_{\varphi}(t)}||\dot{\varphi}(t)||_{\mathcal{G}}^{2}$
(11)

and hence decreases strictly with time when $\varphi$ is non-constant. In
particular, any non-constant cosmological curve is aperiodic and without self-
intersections. Moreover, all non-constant cosmological flow curves are
embedded curves in $T\mathcal{M}$. As mentioned before, cosmological curves
need not be immersed in $\mathcal{M}$ but their singular times form an at most
countable discrete subset of their interval of definition. A cosmological
curve $\varphi:I\rightarrow\mathcal{M}$ is singular at cosmological time
$t_{0}\in I$ iff its complete lift $\gamma=c(\varphi)$ meets the zero section
of $T\mathcal{M}$ for $t=t_{0}$. Hence the embedded components of the orbit of
$\varphi$ are the $\pi$-projections of the set
$\gamma(I)\cap{\dot{T}}\mathcal{M}$, where ${\dot{T}}\mathcal{M}$ is the slit
tangent bundle of $\mathcal{M}$ (which is defined as the complement in
$T\mathcal{M}$ of the image of the zero section).

#### Future completeness when $\mathcal{M}$ is compact.

Since we assume that $\Phi$ is positive on $\mathcal{M}$, each energy sublevel
set:

$\mathcal{M}_{E}(C)\stackrel{{\scriptstyle{\rm def.}}}{{=}}\\{u\in
T\mathcal{M}~{}|~{}E(u)\leq C\\}~{}~{}(C>0)$

is contained in the tubular neighborhood of the zero section of $T\mathcal{M}$
defined by the inequality $||u||_{\mathcal{G}}\leq C\sqrt{2}$. Relation (11)
implies that the complete lift $\dot{\varphi}$ of a maximal cosmological curve
$\varphi$ is contained for $t\geq t_{0}$ within the tubular neighborhood with
$C=E_{\varphi}(t_{0})$. When $\mathcal{M}$ is compact, this tubular
neighborhood is compact and this observation together with the Escape Lemma
implies that $\varphi(t)$ is defined for all $t\geq t_{0}$, which shows that
in this case the cosmological flow is future-complete.

Notice, however, that $\mathcal{M}$ is non-compact in most applications. When
$\mathcal{M}$ is not compact, a maximal cosmological curve can “escape to
infinity” in the sense that its orbit for large times is not contained in any
compact subset of $\mathcal{M}$; this is equivalent with the statement that
the curve has a Freudenthal end of $\mathcal{M}$ among its limit points. In
this case, the late time behavior of those cosmological curves which escape to
infinity depends markedly on the asymptotic form of $\Phi$ and $\mathcal{G}$
near the Freudenthal ends of $\mathcal{M}$ and the cosmological flow need not
be future-complete.

###  1.2 The universal similarity group

Multifield cosmological models admit a universal two-parameter group of
similarities, which relate the cosmological curves of a model with those of
another model having the same target manifold $\mathcal{M}$ but different
parameters $(M_{0},\mathcal{G},\Phi)$. We first discuss scale transformations
of curves in $\mathcal{M}$, which enters the definition of this group action.

######  Definition 1.1.

Let $\epsilon>0$. The $\epsilon$-scale transform of a curve
$\varphi:I\rightarrow\mathcal{M}$ is the curve
$\varphi_{\epsilon}:I_{\epsilon}\rightarrow\mathcal{M}$ defined through:

$I_{\epsilon}\stackrel{{\scriptstyle{\rm def.}}}{{=}}\epsilon I=\\{\epsilon
t|t\in I\\}$

and:

$\varphi_{\epsilon}(t)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\varphi(t/\epsilon)~{}~{}\forall t\in I_{\epsilon}~{}~{}.$

The transformations $\varphi\rightarrow\varphi_{\epsilon}$ are called scale
transformations. They define an action of the multiplicative group
$\mathbb{R}_{>0}$ on the set:

$C(\mathcal{M})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\sqcup_{I\in\mathrm{Int}}\mathcal{C}^{\infty}(I,\mathcal{M})$

of all smooth curves in $\mathcal{M}$, where $\mathrm{Int}$ is the set of all
non-degenerate intervals on the real axis. Scale transformations are not
symmetries of the cosmological equation; the scale symmetry is “broken” by the
scalar potential $\Phi$, being restored in the limit $\Phi\rightarrow 0$
(which, as we will see below, corresponds to the UV limit).

We consider the following similarities of the cosmological equation (7):

  * •

The parameter homothety. The Lagrangian density $\mathcal{L}$ of (2) and the
action $S$ of (1) are homogeneous of degree one under the parameter homothety:

$\mathcal{G}\rightarrow\lambda\mathcal{G}~{}~{},~{}~{}\Phi\rightarrow\lambda\Phi~{}~{},~{}~{}M\rightarrow\lambda^{1/2}M~{}~{}(\mathrm{thus}~{}~{}M_{0}\rightarrow\lambda^{1/2}M_{0})~{}~{},$
(12)

where $\lambda$ is a positive constant. As a consequence, the equations of
motion (1.1) and the cosmological equation (7) are invariant under such
transformations333Notice that the Levi-Civita connection $\nabla$ of
$\mathcal{G}$ is invariant under constant rescalings of $\mathcal{G}$..

  * •

The scale similarity. The equations of motion (1.1) are invariant under the
transformations:

$t\rightarrow\epsilon
t~{}~{},~{}~{}\Phi\rightarrow\Phi_{\epsilon}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\Phi/\epsilon^{2}~{}~{}$

with $\epsilon>0$, which change $H=\frac{\dot{a}}{a}$ into
$\frac{1}{\epsilon}H$. Accordingly, the cosmological equation (7) is invariant
under:

$\varphi\rightarrow\varphi_{\epsilon}~{}~{},~{}~{}\Phi\rightarrow\Phi_{\epsilon}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\Phi/\epsilon^{2}~{}~{}(\epsilon>0)~{}~{}.$ (13)

Let $\mathrm{Met}(\mathcal{M})$ be the set of all Riemannian metrics defined
on $\mathcal{M}$ and $\mathrm{Pot}_{+}(\mathcal{M})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathcal{C}^{\infty}(\mathcal{M},\mathbb{R}_{>0})$ be the set of
all positive smooth functions defined on $\mathcal{M}$. The universal
cosmological similarity group is the multiplicative group
$T=\mathbb{R}_{>0}\times\mathbb{R}_{>0}$, where pairs of positive numbers
multiply componentwise:

$(\lambda_{1},\epsilon_{1})(\lambda_{2},\epsilon_{2})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(\lambda_{1}\lambda_{2},\epsilon_{1}\epsilon_{2})~{}~{}.$

The transformations above induce an action $\rho_{\mathrm{par}}$ of $T$ on the
set $\mathrm{Par}(\mathcal{M})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathbb{R}_{>0}\times\mathrm{Met}(\mathcal{M})\times\mathrm{Pot}_{+}(\mathcal{M})$
of parameters of models with fixed target space $\mathcal{M}$:

$\rho_{\mathrm{par}}(\lambda,\epsilon)(M_{0},\mathcal{G},\Phi)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(\lambda^{1/2}M_{0},\lambda\mathcal{G},\frac{\lambda}{\epsilon^{2}}\Phi)~{}~{}\forall(\lambda,\epsilon)\in
T~{}~{}\forall(M_{0},\mathcal{G},\Phi)\in\mathrm{Par}(\mathcal{M})~{}~{}.$
(14)

This action is free. On the other hand, scale transformations define an action
$\rho_{0}$ of $T$ on the space $C(\mathcal{M})$ of all smooth curves in
$\mathcal{M}$:

$\rho_{0}(\lambda,\epsilon)(\varphi)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\varphi_{\epsilon}~{}~{}\forall(\lambda,\epsilon)\in
T~{}~{}\forall\varphi\in C(\mathcal{M})~{}~{}.$

######  Definition 1.2.

The universal similarity action is the action $\rho$ of $T$ on the set
$C(\mathcal{M})\times\mathrm{Par}(\mathcal{M})$ given by:

$\rho(\lambda,\epsilon)(\varphi,M_{0},\mathcal{G},\Phi)=(\rho_{0}(\lambda,\epsilon)(\varphi),\rho_{\mathrm{par}}(\lambda,\epsilon)(M_{0},\mathcal{G},\Phi))=(\varphi_{\epsilon},\lambda^{1/2}M_{0},\lambda\mathcal{G},\frac{\lambda}{\epsilon^{2}}\Phi)~{}~{}.$

For any $(\lambda,\epsilon)\in T$, we have:

$\rho_{0}(\lambda,\epsilon)(\mathcal{S}^{M_{0},\mathcal{G},\Phi}(\mathcal{M}))=\mathcal{S}^{\rho_{\mathrm{par}}(\lambda,\epsilon)(M_{0},\mathcal{G},\Phi)}(\mathcal{M})~{}~{},$

where $\mathcal{S}^{M_{0},\mathcal{G},\Phi}(\mathcal{M})$ denotes the set of
cosmological curves of the model with target $\mathcal{M}$ and parameters
$(M_{0},\mathcal{G},\Phi)$. Thus $\varphi:I\rightarrow\mathcal{M}$ is a
cosmological curve for the model with parameters $(M_{0},\mathcal{G},\Phi)$
iff $\varphi_{\epsilon}:I_{\epsilon}\rightarrow\mathcal{M}$ is a cosmological
curve for the model with parameters
$(\lambda^{1/2}M_{0},\lambda\mathcal{G},\frac{\lambda}{\epsilon^{2}}\Phi)$.

The universal similarity action allows one to eliminate the overall scale of
$\Phi$ and another scale from the problem. For example, one can fix the
overall scales of $\mathcal{G}$ and $\Phi$, in which case one is left with the
single scale set by $M_{0}$. Equivalently, one can eliminate the rescaled
Planck mass by setting $M_{0}=1$ and the overall scale of $\Phi$, in which
case one is left with the single scale set by $\mathcal{G}$. Notice that one
cannot fix the scales of $M_{0}$ and $\mathcal{G}$ independently. Since $\Phi$
plays a distinguished role in this regard, it is natural to consider the
stabilizer $T_{\mathrm{ren}}\simeq\mathbb{R}_{>0}$ of $\Phi$ in $T$ with
respect to the universal similarity action:

$T_{\mathrm{ren}}=\mathrm{Stab}_{T}(\Phi)=\\{(\lambda,\epsilon)\in
T|\lambda=\epsilon^{2}\\}$ (15)

This is the subgroup of $T$ which can be used to rescale $M_{0}$ and
$\mathcal{G}$ once the scale of $\Phi$ has been fixed; it is the
renormalization group considered in Section 3.

###  1.3 Cosmological conjugation and equivalence

There exist two natural equivalence relations between cosmological models,
which are the isomorphism relations of two underlying groupoids. To describe
them, we first note some obvious properties of the canonical lift
$c:C(\mathcal{M})\rightarrow C_{s}(T\mathcal{M})$ of smooth curves from a
manifold $\mathcal{M}$ to its tangent bundle. Consider two manifolds
$\mathcal{M}_{i}$ with tangent bundle projections
$\pi_{i}:T\mathcal{M}_{i}\rightarrow\mathcal{M}_{i}$ ($i=1,2$) and canonical
curve lifts $c_{i}:C(\mathcal{M}_{i})\rightarrow C_{s}(T\mathcal{M}_{i})$. By
definition, a semispray map from $\mathcal{M}_{1}$ to $\mathcal{M}_{2}$ is a
smooth map $f:T\mathcal{M}_{1}\rightarrow T\mathcal{M}_{2}$ which satisfies
$f_{\ast}(T^{s}T\mathcal{M}_{1})=T^{s}T\mathcal{M}_{2}$. Precomposition with a
semispray map $f$ takes semispray curves in $T\mathcal{M}_{1}$ to semispray
curves in $T\mathcal{M}_{2}$, i.e. we have:

$f\circ(C_{s}(T\mathcal{M}_{1}))=C_{s}(T\mathcal{M}_{2})~{}~{}.$

Moreover, $f$ induces a map ${\hat{f}}:C(\mathcal{M}_{1})\rightarrow
C(\mathcal{M}_{2})$ defined through:

${\hat{f}}(\varphi_{1})\stackrel{{\scriptstyle{\rm def.}}}{{=}}\pi_{2}\circ
f\circ c_{1}(\varphi_{1})~{}~{}\forall\varphi_{1}\in
C(\mathcal{M}_{1})~{}~{}.$ (16)

This determines the precomposition of semispray curves with $f$ in the sense
that two smooth curves $\varphi_{i}:I\rightarrow\mathcal{M}_{i}$ ($i=1,2$)
satisfy $\varphi_{2}={\hat{f}}(\varphi_{1})$ iff $c_{2}(\varphi_{2})=f\circ
c_{1}(\varphi_{1})$.

Given a third manifold $\mathcal{M}_{3}$ with tangent bundle projection
$\pi_{3}:T\mathcal{M}_{3}\rightarrow\mathcal{M}_{3}$ and a second semispray
map $h:T\mathcal{M}_{2}\rightarrow T\mathcal{M}_{3}$, one easily checks the
relation:

$\widehat{h\circ f}={\hat{h}}\circ{\hat{f}}~{}~{}.$

Moreover, for any manifold $\mathcal{M}$ we have
$\widehat{\mathrm{id}_{T\mathcal{M}}}=\mathrm{id}_{C(\mathcal{M})}$. Hence the
maps $\mathcal{M}\rightarrow C(\mathcal{M})$ and $f\rightarrow{\hat{f}}$
define a functor from the category of manifolds and semispray maps to the
category of sets.

#### Cosmological conjugations.

Notice that a cosmological flow curve represents the time evolution of the
state of a multifield cosmological model, where the tangent bundle of
$\mathcal{M}$ is the space of classical states. Hence two cosmological models
parameterized by
$\mathfrak{M}_{i}=(M_{0i},\mathcal{M}_{i},\mathcal{G}_{i},\Phi_{i})$ ($i=1,2$)
and whose semisprays we denote by $S_{i}$ can be identified if there exists a
smooth semispray map $f:T\mathcal{M}_{1}\rightarrow T\mathcal{M}_{2}$ (called
smooth cosmological conjugation) which maps the cosmological flow curves of
the first model into those of the second, i.e. $\gamma_{1}:I\rightarrow
T\mathcal{M}_{1}$ is a cosmological flow curve for $\mathfrak{M}_{1}$ iff
$f\circ\gamma_{1}:I\rightarrow T\mathcal{M}_{2}$ is a cosmological flow curve
for $\mathfrak{M}_{2}$. This amounts to the requirement that $f$ is a smooth
topological conjugation between the cosmological flows of the two models (see
Appendix A), which in turn is equivalent with the condition:

$f_{\sharp}(S_{1})=S_{2}~{}~{},$

where $f_{\sharp}$ denotes the $f$-pushforward of vector fields. Since every
cosmological flow curve is the canonical lift of a cosmological curve, a
semispray map $f$ from $\mathcal{M}_{1}$ to $\mathcal{M}_{2}$ is a
cosmological conjugation iff the induced curve map ${\hat{f}}$ of (16) takes
cosmological curves of the model $\mathfrak{M}_{1}$ into those of the model
$\mathfrak{M}_{2}$. Since $f\rightarrow{\hat{f}}$ is a functor, it is clear
that cosmological models and smooth cosmological conjugations form a groupoid,
whose isomorphism classes we call smooth cosmological conjugacy classes.

When two models $\mathfrak{M}_{1}$ and $\mathfrak{M}_{2}$ are isomorphic in
this groupoid, we write $\mathfrak{M}_{1}\equiv\mathfrak{M}_{2}$ and say that
the models are smoothly conjugate. It is clear that the equivalence relation
$\equiv$ depends only on the rescaled scalar triples of the models, since so
do their cosmological equations. In particular, a parameter homothety with
parameter $\lambda$ corresponds to a conjugation with
$f=\mathrm{id}_{T\mathcal{M}}$ between the model parameterized by
$(M_{0},\mathcal{M},\mathcal{G},\Phi)$ and that parameterized by
$(\lambda^{1/2}M_{0},\mathcal{M},\lambda\mathcal{G},\lambda\Phi)$. If one also
requires $M_{01}=M_{02}$ then the conjugation is called strict. Any smooth
cosmological conjugation is the composite of a strict conjugation with a
parameter homothety.

A strict smooth cosmological conjugation preserves basic on-shell cosmological
observables in the sense that the pullback map
$f^{\ast}:\mathcal{C}^{\infty}(T\mathcal{M}_{2})\rightarrow\mathcal{C}^{\infty}(T\mathcal{M}_{1})$
satisfies:

$\widehat{(f^{\ast})(F_{2})}_{\gamma_{1}}=({\hat{F}}_{2})_{f\circ\gamma_{1}}$

for any basic observable $F_{2}\in\mathcal{C}^{\infty}(T\mathcal{M})$ of the
second model and any cosmological flow curve $\gamma_{1}$ of the first model.

######  Example 1.3.

A particularly simple class of strict smooth conjugations arises from
isomorphisms of scalar triples. We say that two scalar triples
$(\mathcal{M}_{1},\mathcal{G}_{1},\Phi_{1})$ and
$(\mathcal{M}_{2},\mathcal{G}_{2},\Phi_{2})$ are isomorphic if there exists an
isometry
$f_{0}:(\mathcal{M}_{1},\mathcal{G}_{1})\rightarrow(\mathcal{M}_{2},\mathcal{G}_{2})$
such that $\Phi_{1}=\Phi_{2}\circ f_{0}$; in this case, $f_{0}$ is called an
isomorphism of scalar triples and we write
$(\mathcal{M}_{1},\mathcal{G}_{1},\Phi_{1})\simeq(\mathcal{M}_{2},\mathcal{G}_{2},\Phi_{2})$.
We say that the models parameterized by $\mathfrak{M}_{1}$ and
$\mathfrak{M}_{2}$ are isomorphic and write
$\mathfrak{M}_{1}\simeq\mathfrak{M}_{2}$ if $M_{01}=M_{02}$ and
$(\mathcal{M}_{1},\mathcal{G}_{1},\Phi_{1})\simeq(\mathcal{M}_{2},\mathcal{G}_{2},\Phi_{2})$.
The differential $\mathrm{d}f_{0}$ of an isomorphism $f_{0}$ of cosmological
models is a strict smooth conjugation.

A (necessarily strict) smooth cosmological conjugation between a model
parameterized by $(M_{0},\mathcal{M},\mathcal{G}_{0},\Phi_{0})$ and itself is
called a cosmological symmetry of that model. Automorphisms of a cosmological
model are sometimes called visible symmetries while its remaining symmetries
are called hidden (see Noether1 ; Noether2 ; Tim19 ; Hesse ).

######  Remark 1.4.

The study of cosmological symmetries of multifield models in the mathematical
generality considered here has been limited. If one restricts attention to Lie
groups, this becomes the problem of determining the Lie symmetries of the
cosmological equation (7) and of classifying rescaled scalar triples whose
cosmological equation admits given Lie groups of symmetries. Various local
results about Lie symmetries of multifield models can be found in TP ;
Giacomini ; P ; Noether1 ; Mondal . The literature contains limited
information on the global geometry of the resulting scalar manifolds; see
Noether2 ; Hesse  for some results in that direction.

#### Cosmological equivalences.

An equivalence relation weaker than conjugation arises if one identifies
cosmological curves up to increasing reparameterization of the cosmological
time. We say that a smooth curve $\varphi:I\rightarrow\mathcal{M}$ is a pre-
cosmological curve of the model parameterized by
$\mathfrak{M}=(M_{0},\mathcal{M},\mathcal{G},\Phi)$ if there exists an
increasing reparameterization $\alpha:J\rightarrow I$ such that
$\varphi\circ\alpha$ is a cosmological curve of $\mathfrak{M}$. A smooth
semispray map $f:T\mathcal{M}_{1}\rightarrow T\mathcal{M}_{2}$ is called a
smooth cosmological equivalence between the models $\mathfrak{M}_{1}$ and
$\mathfrak{M}_{2}$ if the map ${\hat{f}}$ of (16) maps the pre-cosmological
curves of $\mathfrak{M}_{1}$ into those of $\mathfrak{M}_{2}$. This amounts to
the requirement that ${\hat{f}}$ identifies the oriented cosmological orbits
of the two models. Notice that cosmological equivalences between
$\mathfrak{M}_{1}$ and $\mathfrak{M}_{2}$ differ444Indeed, a cosmological
equivalence does not identify the cosmological flow curves of the two models
up to reparameterization, but up to reparameterization combined with a time-
dependent rescaling within the tangent bundle fibers. The notion of
topological equivalence of dynamical systems is recalled in Appendix A. from
smooth dynamical equivalences, which are defined by the requirement $f$ gives
a smooth topological equivalence between the cosmological flows of the models.

It is clear from the properties of ${\hat{f}}$ that cosmological models and
cosmological equivalence form a groupoid, whose isomorphism classes we call
smooth cosmological equivalence classes. When $\mathfrak{M}_{1}$ and
$\mathfrak{M}_{2}$ are isomorphic in this groupoid, we write
$\mathfrak{M}_{1}\sim\mathfrak{M}_{2}$ and say that the two models are
smoothly equivalent. This equivalence relation is weaker than cosmological
conjugation in the sense that $\mathfrak{M}_{1}\equiv\mathfrak{M}_{2}$ implies
$\mathfrak{M}_{1}\sim\mathfrak{M}_{2}$. Like cosmological conjugation,
cosmological equivalence depends only on the rescaled scalar triples; we say
that the equivalence is strict if $M_{01}=M_{02}$.

In Section 3, we will introduce other equivalence relations, which arise from
the study of UV and IR limits.

###  1.4 Some remarks on the early and late time behavior of cosmological
curves

Consider a maximal cosmological curve
$\varphi:(a_{-},a_{+})\rightarrow\mathcal{M}$, where
$a_{\pm}\in{\overline{\mathbb{R}}}$ can be chosen such that $a<0<b$ since the
cosmological equation is autonomous. Recall that the ordinary $\alpha$\- and
$\omega$\- limit sets of $\varphi$ are defined through Palis :

$\displaystyle\mathrm{Lim}_{\alpha}\varphi\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\\{m\in\mathcal{M}~{}|~{}\exists t_{n}\rightarrow
a_{+}:\lim_{n\rightarrow\infty}\varphi(t_{n})=m\\}\subset\mathcal{M}~{}~{}$
$\displaystyle\mathrm{Lim}_{\omega}\varphi\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\\{m\in\mathcal{M}~{}|~{}\exists t_{n}\rightarrow
a_{-}:\lim_{n\rightarrow\infty}\varphi(t_{n})=m\\}\subset\mathcal{M}~{}~{}.$

Let us assume that $\mathcal{M}$ is compact. Then for any $\omega$-limit point
$m$ of $\varphi$ and any sequence $t_{n}\in(a_{-},a_{+})$ with
$t_{n}\rightarrow a_{+}$ and $\varphi(t_{n})\rightarrow m$, the tangent
vectors $\dot{\varphi}(t_{n})$ are contained in a compact subset of
$T\mathcal{M}$ since $E_{\varphi}(t_{n})$ stays bounded by (11). Hence there
exists a subsequence $t^{\prime}_{n}$ of $t_{n}$ such that
$\dot{\varphi}(t^{\prime}_{n})$ converges to a point $u\in T\mathcal{M}$ with
$\pi(u)=m$. Since $u$ is a limit point of the integral curve
$\gamma=c(\varphi)$ of the vector field $S\in\mathcal{X}(T\mathcal{M})$, we
must have $\mathcal{S}(u)=0$ i.e. $u$ is a stationary point of the
cosmological flow. Hence $u$ belongs to the set $(\mathrm{Crit}\Phi)_{0}$ and
$m=\pi(u)$ belongs to $\mathrm{Crit}\Phi$. This shows that the ordinary
$\omega$-limit points of a cosmological curve are critical points of $\Phi$.
Moreover, the set
$\mathrm{Lim}_{\omega}\varphi=\pi(\mathrm{Lim}_{\omega}c(\varphi))$ is
nonempty, compact and connected since the map $\pi$ is continuous and the set
$\mathrm{Lim}_{\omega}c(\varphi)$ has these properties by (Palis, , Prop.
1.4).

When $\mathcal{M}$ is not compact, both ordinary limit sets of a maximal
cosmological curve $\varphi$ may be empty. Indeed, the orbit of $\varphi$ need
not be contained in any compact subset of $\mathcal{M}$, which means that
there there may exist a sequence of cosmological times $t_{n}\in(a_{-},a_{+})$
such that $\varphi(t_{n})$ approaches a Freudenthal end of $\mathcal{M}$ when
$t_{n}\rightarrow a_{-}$ or $t_{n}\rightarrow a_{+}$. To account for this, we
consider the end compactification $\hat{\mathcal{M}}$ of $\mathcal{M}$ (which
is a compact Hausdorff space containing $\mathcal{M}$ as a dense subset) and
view $\varphi$ as the continuous curve $\varphi_{e}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\iota\circ\varphi$ in the topological space $\hat{\mathcal{M}}$,
where $\iota:\mathcal{M}\hookrightarrow\hat{\mathcal{M}}$ is the inclusion. By
definition, the extended limit sets of $\varphi$ are the $\alpha$\- and
$\omega$-limit sets of $\varphi_{e}$ in the topological space
$\hat{\mathcal{M}}$:

$\displaystyle\mathrm{Lim}_{\alpha}^{e}\varphi\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{Lim}_{\alpha}\varphi_{e}=\\{{\hat{m}}\in\hat{\mathcal{M}}~{}|~{}\exists
t_{n}\rightarrow
a:\lim_{n\rightarrow\infty}\varphi(t_{n})={\hat{m}}\\}\subset\hat{\mathcal{M}}~{}~{}$
$\displaystyle\mathrm{Lim}_{\omega}^{e}\varphi\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{Lim}_{\omega}\varphi_{e}=\\{{\hat{m}}\in\hat{\mathcal{M}}~{}|~{}\exists
t_{n}\rightarrow
b:\lim_{n\rightarrow\infty}\varphi(t_{n})={\hat{m}}\\}\subset\hat{\mathcal{M}}~{}~{}.$

Let $\mathrm{Ends}(\mathcal{M})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\hat{\mathcal{M}}\setminus\mathcal{M}$ be the space of ends of
$\mathcal{M}$ (which is a totally disconnected topological space). We have:

$\mathrm{Lim}_{\alpha}^{e}\varphi=\mathrm{Lim}_{\alpha}\varphi\sqcup\Lambda_{\alpha}\varphi~{}~{},~{}~{}\mathrm{Lim}_{\omega}^{e}\varphi=\mathrm{Lim}_{\omega}\varphi\sqcup\Lambda_{\omega}\varphi~{}~{},$

where:

$\Lambda_{\alpha}\varphi\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{Lim}_{\alpha}\varphi_{e}\cap\mathrm{Ends}(\mathcal{M})~{}~{},~{}~{}\Lambda_{\omega}\varphi\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{Lim}_{\omega}\varphi_{e}\cap\mathrm{Ends}(\mathcal{M})$

are the sets of $\alpha$\- and $\omega$\- limit ends of $\varphi$. With these
definitions, $\varphi$ is contained in some compact subset of $\mathcal{M}$
iff $\Lambda_{\alpha}\varphi=\Lambda_{\omega}\varphi=\emptyset$. When the end
compactification $\hat{\mathcal{M}}$ is sequentially compact, an argument
similar to that of (Palis, , Prop. 1.4) shows that
$\mathrm{Lim}_{\alpha}^{e}\varphi$ and $\mathrm{Lim}_{\omega}^{e}\varphi$ are
nonempty, compact and connected subsets of $\hat{\mathcal{M}}$. The fact that
a Freudenthal end of $\mathcal{M}$ can act as a limit point for a cosmological
curve is a fundamental feature of models with non-compact target space.

In general, the continuous map
$\varphi_{e}:(a_{-},a_{+})\rightarrow\mathcal{M}$ need not have limits for
$t\rightarrow a_{-}$ or $t\rightarrow a_{+}$ in $\hat{\mathcal{M}}$. When the
corresponding limit exists, we call it the $\alpha$\- or $\omega$\- extended
limit of $\varphi$ and denote it by:

$\mathrm{lim}_{\alpha}\varphi\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\lim_{t\rightarrow
a_{-}}\varphi_{e}(t)\in\hat{\mathcal{M}}~{}~{}\mathrm{respectively}~{}~{}\mathrm{lim}_{\omega}\varphi\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\lim_{t\rightarrow
a_{+}}\varphi_{e}(t)\in\hat{\mathcal{M}}~{}~{}.$

##  2 Scaling limits and approximations

In this section, we study the UV and IR scaling limits of classical multifield
cosmological models with rescaled Planck mass $M_{0}$ and scalar triple
$(\mathcal{M},\mathcal{G},\Phi)$, showing that cosmological curves are
approximated in these limits by a reparameterization of the geodesic flow of
$(\mathcal{M},\mathcal{G})$ and by the gradient flow of
$(\mathcal{M},\mathcal{G},V)$, where $V=M_{0}\sqrt{2\Phi}$ is the classical
effective potential of the model. We also study the consistency conditions for
these approximations, showing that they differ from commonly used
approximations in cosmology, such as the slow roll approximation and its slow
roll-slow turn variant as well as from the gradient flow approximation of
genalpha .

###  2.1 The UV and IR limits

By definition, the behavior of the scale transform $\varphi_{\epsilon}$ at
time $t$ recovers the behavior of $\varphi$ at time $t/\epsilon$. Notice that
$\varphi$ can be recovered from its scale transform by setting $\epsilon=1$:

$\varphi=\varphi_{1}~{}~{}.$

A time interval $\Delta t$ is rescaled to $\frac{1}{\epsilon}\Delta t$ under
the scale transform by $\epsilon$. Hence cosmological time intervals are
compressed for large $\epsilon$ and expanded for small $\epsilon$.

Intuitively, the limits of small and large $\epsilon$ capture the low and high
frequency components of $\varphi$ since the large $\epsilon$ limit sharpens
the oscillations of $\varphi_{\epsilon}$ while the small $\epsilon$ limit
stretches them out. To make this quantitative, consider for simplicity the
case $\mathcal{M}=\mathbb{R}^{d}$ and a cosmological curve $\varphi$ which is
defined on $\mathbb{R}$. Then
$\varphi(t)=(\varphi^{1}(t),\ldots,\varphi^{d}(t))$, where
$\varphi^{i}:\mathbb{R}\rightarrow\mathbb{R}$ (for $i=1,\ldots,d$) are the
projections of $\varphi$ on the Cartesian coordinate axes. Since in this case
$\varphi$ is a vector-valued function, it has a Fourier decomposition:

$\varphi(t)=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\mathrm{d}\omega
e^{\mathbf{i}\omega t}\hat{\varphi}(\omega)~{}~{},$

where ${\hat{\varphi}}:\mathbb{R}\rightarrow\mathbb{R}^{d}$ is the Fourier
transform of $\varphi$:

${\hat{\varphi}}(\omega)=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\mathrm{d}te^{-\mathbf{i}\omega
t}\varphi(t)~{}~{}.$

We have:

$\varphi_{\epsilon}(t)=\varphi(t/\epsilon)=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\mathrm{d}\omega
e^{\mathbf{i}\omega
t/\epsilon}\hat{\varphi}(\omega)=\frac{\epsilon}{\sqrt{2\pi}}\int_{\mathbb{R}}\mathrm{d}\omega
e^{\mathbf{i}\omega t}\hat{\varphi}(\epsilon\omega)~{}~{},$

where we performed the change of variables $\omega\rightarrow\epsilon\omega$.
Hence the Fourier components of $\varphi_{\epsilon}$ are:

$\widehat{\varphi_{\epsilon}}(\omega)=\epsilon~{}\hat{\varphi}(\epsilon\omega)=\epsilon~{}({\hat{\varphi}})_{1/\epsilon}(\omega)$

and its Fourier transform is given by
$\widehat{\varphi_{\epsilon}}=\epsilon~{}({\hat{\varphi}})_{1/\epsilon}$. When
$\epsilon$ is very large or very small, the Fourier component of
$\varphi_{\epsilon}$ at frequency $\omega$ reproduces up to a rescaling of the
amplitude the Fourier component of $\varphi$ at the much higher (respectively
much lower) frequency $\epsilon\omega$. Hence the UV limit
$\epsilon\rightarrow\infty$ corresponds to the high frequency behavior of
$\varphi$ while the IR limit $\epsilon\rightarrow 0$ corresponds to its low
frequency behavior; these scaling limits correspond to the ultraviolet and
infrared limits of the oscillation spectrum of $\varphi$. Notice that the
kinetic energy of a plane wave component
${\hat{\varphi}}(\omega)e^{\mathbf{i}\omega t}$ of $\varphi$ equals
$E^{k}_{\varphi}(\omega)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\frac{1}{2}\omega^{2}||{\hat{\varphi}}(\omega)||_{\mathcal{G}}^{2}$
and we have:

$E^{k}_{\varphi_{\epsilon}}(\omega)=\frac{1}{2}\omega^{2}||\hat{\varphi_{\epsilon}}(\omega)||_{\mathcal{G}}^{2}=\frac{\epsilon^{2}}{2}\omega^{2}||{\hat{\varphi}}(\epsilon\omega)||_{\mathcal{G}}^{2}=\epsilon^{2}E^{k}_{\varphi}(\epsilon\omega)~{}~{}.$

######  Remark 2.1.

Recall that the generalized curvatures of a curve in a Riemannian manifold do
not depend on parameterization and hence are properties of the image of that
curve rather than of the curve itself. Hence the generalized curvatures of
$\varphi_{\epsilon}$ and $\varphi$ at every point on the orbit
$\mathrm{im}(\varphi_{\epsilon})=\mathrm{im}(\varphi)$ coincide. Since scale
transforms do not change the generalized curvatures of $\varphi$, the limits
of small and large $\epsilon$ have no connection with slow or fast turns of
the cosmological orbit, which form the focus of other approximation schemes in
cosmology.

###  2.2 The rescaled cosmological equation

For any $n\in\mathbb{N}_{>0}$, we have:

$\frac{\mathrm{d}^{n}\varphi_{\epsilon}}{\mathrm{d}t^{n}}=\frac{1}{\epsilon^{n}}\left(\frac{\mathrm{d}^{n}\varphi}{\mathrm{d}t^{n}}\right)_{\\!\epsilon}~{}~{}$
(17)

and:

$\nabla_{t}^{n}\frac{\mathrm{d}\varphi_{\epsilon}}{\mathrm{d}t}=\frac{1}{\epsilon^{n+1}}\left(\nabla_{t}^{n}\frac{\mathrm{d}\varphi}{\mathrm{d}t}\right)_{\\!\epsilon}~{}~{}.$
(18)

These relations imply that a curve $\varphi:I\rightarrow\mathcal{M}$ satisfies
the cosmological equation (7) iff its scale transform $\varphi_{\epsilon}$
satisfies the $\epsilon$-rescaled cosmological equation:

$\epsilon^{2}\nabla_{t}\frac{\mathrm{d}\varphi_{\epsilon}(t)}{\mathrm{d}t}\\!+\\!\frac{\epsilon}{M_{0}}\left[\epsilon^{2}\Big{\|}\frac{\mathrm{d}\varphi_{\epsilon}(t)}{\mathrm{d}t}\Big{\|}_{\mathcal{G}}^{2}+2\Phi(\varphi_{\epsilon}(t))\right]^{1/2}\\!\\!\frac{\mathrm{d}\varphi_{\epsilon}(t)}{\mathrm{d}t}+(\mathrm{grad}_{\mathcal{G}}\Phi)(\varphi_{\epsilon}(t))=0~{}~{}.$
(19)

Upon dividing by $\epsilon^{2}$, the latter is equivalent with:

$\nabla_{t}\frac{\mathrm{d}\varphi_{\epsilon}(t)}{\mathrm{d}t}+\frac{1}{M_{0}}\left[\Big{\|}\frac{\mathrm{d}\varphi_{\epsilon}(t)}{\mathrm{d}t}\Big{\|}_{\mathcal{G}}^{2}+2\Phi_{\epsilon}(\varphi_{\epsilon}(t))\right]^{1/2}\frac{\mathrm{d}\varphi_{\epsilon}(t)}{\mathrm{d}t}+(\mathrm{grad}_{\mathcal{G}}\Phi_{\epsilon})(\varphi_{\epsilon}(t))=0~{}~{},$
(20)

where $\Phi_{\epsilon}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\Phi/\epsilon^{2}$. Hence $\varphi$ satisfies the cosmological
equation of the scalar triple $(\mathcal{M},\mathcal{G},\Phi)$ iff
$\varphi_{\epsilon}$ satisfies the cosmological equation of the scalar triple
$(\mathcal{M},\mathcal{G},\Phi_{\epsilon})$. This also follows from the fact
that (13) is a symmetry of the cosmological equation. In particular, the UV
limit $\epsilon\rightarrow\infty$ amounts to taking the overall scale of
$\Phi$ to zero, while the IR limit $\epsilon\rightarrow 0$ amounts to taking
the overall scale of $\Phi$ to infinity.

The semispray defined by (20) is:

$S_{\epsilon}=\mathbb{S}-\mathcal{H}_{\epsilon}C-\frac{1}{\epsilon^{2}}(\mathrm{grad}_{\mathcal{G}}\Phi)^{v}~{}~{},$

where the function
$\mathcal{H}_{\epsilon}:T\mathcal{M}\rightarrow\mathbb{R}_{>0}$ is defined
through:

$\mathcal{H}_{\epsilon}(u)=\frac{1}{M_{0}}\left[||u||_{\mathcal{G}}^{2}+\frac{2\Phi(\pi(u))}{\epsilon^{2}}\right]^{1/2}~{}~{}.$

In the UV limit, $S_{\epsilon}$ reduces to the $N_{0}$-modification of the
geodesic spray of the rescaled scalar manifold
$(\mathcal{M},\mathcal{G}_{0})$:

$S_{\mathrm{UV}}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathbb{S}-N_{0}C~{}~{},$ (21)

where $N_{0}:T\mathcal{M}\rightarrow\mathbb{R}_{\geq 0}$ is the norm function
defined by $\mathcal{G}_{0}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\frac{1}{M_{0}^{2}}\mathcal{G}$ on $T\mathcal{M}$:

$N_{0}(u)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}||u||_{\mathcal{G}_{0}}=\frac{1}{M_{0}}||u||_{\mathcal{G}}~{}~{}.$

Notice that the geodesic spray $\mathbb{S}$ of $(\mathcal{M},\mathcal{G})$
coincides with that of $(\mathcal{M},\mathcal{G}_{0})$ since it is invariant
under constant rescalings of $\mathcal{G}$. Also notice that $N_{0}$ and the
spray (21) are continuous on $T\mathcal{M}$ but smooth only on the slit
tangent bundle $\dot{T}\mathcal{M}$. In the IR limit $\epsilon\rightarrow 0$,
$S_{\epsilon}$ is approximated by the vertical vector field:

$S^{\mathrm{IR}}_{\epsilon}=-\frac{1}{\epsilon}\frac{1}{M_{0}}\sqrt{2\Phi^{v}}C-\frac{1}{\epsilon^{2}}(\mathrm{grad}_{\mathcal{G}}\Phi)^{v}=-\frac{1}{\epsilon^{2}M_{0}}\sqrt{2\Phi^{v}}S_{\mathrm{IR},\epsilon}~{}~{},$

where:

$S_{\mathrm{IR},\epsilon}\stackrel{{\scriptstyle{\rm def.}}}{{=}}\epsilon
C-(\mathrm{grad}_{\mathcal{G}}V)^{v}\in\mathcal{X}(T\mathcal{M})$ (22)

and we defined the classical effective scalar potential $V$ through:

$V\stackrel{{\scriptstyle{\rm def.}}}{{=}}M_{0}\sqrt{2\Phi}~{}~{}.$ (23)

Notice that $J(S^{\mathrm{IR}}_{\epsilon})=0$, hence
$S^{\mathrm{IR}}_{\epsilon}$ is no longer a semispray. This signals that the
IR limit is degenerate. Here $\Phi^{v}$ is the vertical lift of $\Phi$, which
is defined through:

$\Phi^{v}(u)=\Phi(\pi(u))~{}~{}\forall u\in T\mathcal{M}~{}~{}.$

###  2.3 The UV and IR expansions

The rescaled cosmological equation can be used to construct UV and IR
expansions of cosmological curves and of the cosmological flow. When
$\epsilon$ is large, one can seek solutions $\varphi_{\epsilon}$ to (20) which
are expanded in positive powers of $\frac{1}{\epsilon^{2}}$; then
$\varphi(t)\stackrel{{\scriptstyle{\rm def.}}}{{=}}\varphi_{\epsilon}(\epsilon
t)$ is a solution of the cosmological equation (7) which is expanded in non-
negative powers of $\Phi$. This amounts to treating $\Phi$ as small, Taylor
expanding the reduced Hubble function (9) as:

$\mathcal{H}(u)=||u||_{\mathcal{G}}\left[1+\frac{2\Phi(\pi(u))}{||u||_{\mathcal{G}}^{2}}\right]^{1/2}=||u||_{\mathcal{G}}\left[1+\frac{\Phi(\pi(u))}{||u||_{\mathcal{G}}^{2}}-\frac{1}{2}\left(\frac{\Phi(\pi(u))}{||u||_{\mathcal{G}}^{2}}\right)^{2}+\ldots\right]$
(24)

and seeking solutions $\varphi$ of the cosmological equation expanded in
powers of $\Phi$. This produces the UV expansion of cosmological curves and a
corresponding expansion of the cosmological flow. Substituting (24) into (8)
gives an expansion of the cosmological semispray in powers of $\Phi$.

When $\epsilon$ is small, one can seek solutions $\varphi_{\epsilon}$ of (19)
which are expanded in powers of $\epsilon$; then
$\varphi(t)\stackrel{{\scriptstyle{\rm def.}}}{{=}}\varphi_{\epsilon}(\epsilon
t)$ is a solution of (7) expanded in powers of $\frac{1}{\sqrt{2\Phi}}$. This
amounts to treating $\Phi$ as large and expanding the reduced Hubble function
as:

$\mathcal{H}(u)=\sqrt{2\Phi(\pi(u))}\left[1+\left(\frac{||u||_{\mathcal{G}}}{\sqrt{2\Phi(\pi(u))}}\right)^{2}\right]^{1/2}=\sqrt{2\Phi(\pi(u))}\left[1+\frac{1}{2}\left(\frac{||u||_{\mathcal{G}}}{\sqrt{2\Phi(\pi(u))}}\right)^{2}-\frac{1}{8}\left(\frac{||u||_{\mathcal{G}}}{\sqrt{2\Phi(\pi(u))}}\right)^{4}+\ldots\right]$
(25)

Substituting (25) into (8) produces an expansion of the cosmological
semispray in powers of $\frac{1}{\sqrt{2\Phi}}$. Substituting (25) into the
cosmological equation and dividing both sides by $\sqrt{2\Phi(\varphi(t))}$
brings (7) to the form:

$\frac{1}{\sqrt{2\Phi(\varphi(t))}}\nabla_{t}\dot{\varphi}(t)+\frac{1}{M_{0}}\left[1+\left(\frac{||\dot{\varphi}(t)||_{\mathcal{G}}^{2}}{\sqrt{2\Phi(\varphi(t))}}\right)^{2}-\frac{1}{8}\left(\frac{||\dot{\varphi}(t)||_{\mathcal{G}}}{\sqrt{2\Phi(\varphi(t))}}\right)^{4}+\ldots\right]^{1/2}\dot{\varphi}(t)+(\mathrm{grad}_{\mathcal{G}}\sqrt{2\Phi})(\varphi(t))=0$
(26)

and one can seek solutions expanded in powers of $\frac{1}{\sqrt{2\Phi}}$.
This produces an asymptotic expansion of cosmological curves called the IR
expansion and a corresponding expansion of the cosmological flow. Notice that
the small expansion parameter multiplies the highest order term in (26). As a
consequence, the first order approximant $\varphi_{\mathrm{IR}}$ is obtained
by solving a first order ODE, which means that the corresponding approximant
$\Pi_{\mathrm{IR}}$ to the cosmological flow $\Pi$ is not defined on the whole
tangent bundle $T\mathcal{M}$ but on a closed submanifold of the latter. It is
only higher order approximants of the cosmological flow which are defined on
the entire tangent bundle of $\mathcal{M}$. In the next subsections, we
discuss the first order UV and IR approximants of cosmological curves and of
the cosmological flow.

######  Remark 2.2.

The UV and IR expansions admit a geometric description obtained by writing
$\varphi_{\epsilon}$ as:

$\varphi_{\epsilon}(t)=\exp_{\varphi_{\mathrm{UV}}(t/\epsilon)}\left(\sum_{n\geq
1}\frac{1}{\epsilon^{2n}}v_{n}(t)\right)~{}~{}\mathrm{with}~{}~{}v_{n}(t)\in
T_{\varphi_{\mathrm{UV}}(t/\epsilon)}\mathcal{M}$ (27)

respectively:

$\varphi_{\epsilon}(t)=\exp_{\varphi_{\mathrm{IR}}(t/\epsilon)}\left(\sum_{n\geq
1}\epsilon^{n}w_{n}(t)\right)~{}~{}\mathrm{with}~{}~{}w_{n}(t)\in
T_{\varphi_{\mathrm{IR}}(t/\epsilon)}\mathcal{M}~{}~{},$ (28)

where $\varphi_{\mathrm{UV}}$ and $\varphi_{\mathrm{IR}}$ are the first order
UV and IR approximants of $\varphi$ discussed below and
$\exp_{m}:T_{m}\mathcal{M}\rightarrow\mathcal{M}$ denotes the exponential map
of the Riemannian manifold $(\mathcal{M},\mathcal{G}_{0})$ at the point
$m\in\mathcal{M}$. The vector-valued functions $v_{n}(t)$ and $w_{n}(t)$ can
be determined by substituting (27) and (28) into the rescaled cosmological
equation and expanding respectively in powers of $1/\epsilon$ or $\epsilon$;
we will explain the technical details of this procedure in a different
publication. When $\epsilon$ is large (respectively small), the arguments of
the exponential maps in the expressions above tend to zero and hence the right
hand sides reduce to the UV and IR approximants.

###  2.4 The first order UV and IR approximants

Let $\varphi:I=(a_{-},a_{+})\rightarrow\mathcal{M}$ be a non-constant maximal
cosmological curve with $0\in(a_{-},a_{+})$, where
$a_{-}\in\mathbb{R}\cup\\{-\infty\\}$ and
$a_{+}\in\mathbb{R}\cup\\{+\infty\\}$.

#### The strict UV limit and first order UV approximant.

Let us fix $t\in I_{\epsilon}=(\epsilon a_{-},\epsilon a_{+})$ and set
$\lambda\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{sign}(t)\in\\{-1,0,1\\}$. In the strict UV limit
$\epsilon\rightarrow\infty$, we have
$\lim_{\epsilon\rightarrow\infty}I_{\epsilon}=\mathbb{R}$ and:

$\lim_{\epsilon\rightarrow\infty}\varphi_{\epsilon}(t)=\lim_{\epsilon\rightarrow\infty}\varphi(t/\epsilon)=\varphi(0)~{}~{},~{}~{}\lim_{\epsilon\rightarrow\infty}\frac{\mathrm{d}\varphi_{\epsilon}(t)}{\mathrm{d}t}=\lim_{\epsilon\rightarrow\infty}\frac{1}{\epsilon}\dot{\varphi}(t/\epsilon)=0~{}~{}.$

Hence the limiting curve $\varphi_{\infty}$ is the constant curve defined on
$I_{\infty}=\mathbb{R}$ whose image is the point $\varphi(0)$.

When $\epsilon$ is large but finite, the rescaled cosmological equation (20)
becomes:

$\nabla_{t}\frac{\mathrm{d}\varphi_{\epsilon}(t)}{\mathrm{d}t}+\frac{1}{M_{0}}\Big{\|}\frac{\mathrm{d}\varphi_{\epsilon}(t)}{\mathrm{d}t}\Big{\|}_{\mathcal{G}}\frac{\mathrm{d}\varphi_{\epsilon}(t)}{\mathrm{d}t}=\mathrm{O}(1/\epsilon^{2})$

and we have:

$\varphi_{\epsilon}(t)=\varphi(t/\epsilon)=\varphi(0)+\mathrm{O}(1/\epsilon)~{}~{},~{}~{}\frac{\mathrm{d}\varphi_{\epsilon}(t)}{\mathrm{d}t}=\frac{1}{\epsilon}\dot{\varphi}(t/\epsilon)=\frac{1}{\epsilon}\dot{\varphi}(0)+\mathrm{O}(1/\epsilon^{2})$

for $|t|\ll\epsilon$. Thus $\varphi_{\epsilon}$ is approximated up to first
order in $1/\epsilon$ by a curve
$t\rightarrow\varphi_{\mathrm{UV},\epsilon}(t)$ which satisfies:

$\nabla_{t}\frac{\mathrm{d}\varphi_{\mathrm{UV},\epsilon}(t)}{\mathrm{d}t}+\frac{1}{M_{0}}\Big{\|}\frac{\mathrm{d}\varphi_{\mathrm{UV},\epsilon}(t)}{\mathrm{d}t}\Big{\|}_{\mathcal{G}}\frac{\mathrm{d}\varphi_{\mathrm{UV},\epsilon}(t)}{\mathrm{d}t}=0$
(29)

and:

$\varphi_{\mathrm{UV},\epsilon}(0)=\varphi(0)~{}~{},~{}~{}\frac{\mathrm{d}\varphi_{\mathrm{UV},\epsilon}}{\mathrm{d}t}\Big{|}_{t=0}=\frac{1}{\epsilon}\dot{\varphi}(0)~{}~{}.$

Since (29) is scale-invariant, the curve
$\varphi_{\mathrm{UV}}:\mathbb{R}\rightarrow\mathcal{M}$ defined through
$\varphi_{\mathrm{UV}}(t)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\varphi_{\mathrm{UV},\epsilon}(\epsilon t)$ satisfies the same
equation:

$\nabla_{t}\frac{\mathrm{d}\varphi_{\mathrm{UV}}(t)}{\mathrm{d}t}+\frac{1}{M_{0}}\Big{\|}\frac{\mathrm{d}\varphi_{\mathrm{UV}}(t)}{\mathrm{d}t}\Big{\|}_{\mathcal{G}}\frac{\mathrm{d}\varphi_{\mathrm{UV}}(t)}{\mathrm{d}t}=0$
(30)

and the conditions:

$\varphi_{\mathrm{UV}}(0)=\varphi(0)~{}~{},~{}~{}\frac{\mathrm{d}\varphi_{\mathrm{UV}}}{\mathrm{d}t}\Big{|}_{t=0}=\dot{\varphi}(0)~{}~{}.$
(31)

It follows that $\varphi(t)=\varphi_{\epsilon}(\epsilon t)$ is approximated by
$\varphi_{\mathrm{UV}}(t)$ for $|t|\ll 1$. Notice that (21) is the spray
defined by equation (30). The complete lift of maximal solutions to (30)
defines the flow $\Pi_{\mathrm{UV}}$ of the $N_{0}$-modification
$S_{\mathrm{UV}}$ of the geodesic spray $\mathbb{S}$.

An increasing reparameterization $\sigma\rightarrow\sigma(t)$ shows that (30)
is equivalent with the geodesic equation:

$\nabla_{\sigma}\frac{\mathrm{d}\varphi_{\mathrm{UV}}}{\mathrm{d}\sigma}=0~{}~{},$
(32)

of $(\mathcal{M},\mathcal{G})$, where the affine parameter $\sigma$ is
obtained by solving the second order ODE:

$\ddot{\sigma}(t)+\frac{1}{M_{0}}||{\varphi}^{\prime}_{\mathrm{UV}}(\sigma)||_{\mathcal{G}}\dot{\sigma}(t)^{2}=0~{}~{}.$
(33)

Here the prime indicates derivation with respect to $\sigma$. Since the
Riemannian manifold $(\mathcal{M},\mathcal{G})$ is complete, its geodesic flow
is complete by the Hopf-Rinow theorem. As a consequence, the maximal geodesic
$\sigma\rightarrow\varphi_{\mathrm{UV}}(\sigma)$ is defined on the entire real
axis. Equation (33) can be written as:

$\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{1}{\dot{\sigma}}\right)=\frac{1}{M_{0}}||\varphi^{\prime}_{\mathrm{UV}}(\sigma)||_{\mathcal{G}}~{}~{}.$

Writing the left hand side as
$\dot{\sigma}\frac{\mathrm{d}}{\mathrm{d}\sigma}\left(\frac{\mathrm{d}t}{\mathrm{d}\sigma}\right)=\dot{\sigma}\frac{\mathrm{d}^{2}t}{\mathrm{d}\sigma^{2}}$,
this is equivalent with:

$\frac{\mathrm{d}^{2}t}{\mathrm{d}\sigma^{2}}=\frac{1}{M_{0}}||\varphi^{\prime}_{\mathrm{UV}}(\sigma)||_{\mathcal{G}}\frac{\mathrm{d}t}{\mathrm{d}\sigma}\Longleftrightarrow\frac{\mathrm{d}u}{\mathrm{d}\sigma}=\frac{1}{M_{0}}||\varphi^{\prime}_{\mathrm{UV}}(\sigma)||_{\mathcal{G}}u~{}~{},$

where we set $u\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\frac{\mathrm{d}t}{\mathrm{d}\sigma}$. Thus
$u(\sigma)=Be^{\frac{1}{M_{0}}\int^{\sigma}_{0}\mathrm{d}\sigma^{\prime}||\varphi^{\prime}_{\mathrm{UV}}(\sigma^{\prime})||_{\mathcal{G}}}$
and:

$t(\sigma)=A+B\int_{0}^{\sigma}\mathrm{d}\sigma^{\prime}e^{\frac{1}{M_{0}}\int_{0}^{\sigma^{\prime}}\mathrm{d}\sigma^{\prime\prime}||\varphi^{\prime}_{\mathrm{UV}}(\sigma^{\prime\prime})||_{\mathcal{G}}}~{}~{},$
(34)

where $A$ and $B$ are integration constants. Since $\sigma$ is an increasing
parameter for $\varphi_{\mathrm{UV}}$ (i.e. we require $\dot{\sigma}>0$ and
hence $\frac{\mathrm{d}t}{\mathrm{d}\sigma}>0$), we must take $B>0$. Notice
that (33) is invariant under affine transformations of $\sigma$, which
correspond to affine transformations of $t$, i.e. to changing the constants
$A$ and $B$ in (34). Since we require $\dot{\sigma}>0$, only affine
transformations of the form $\sigma\rightarrow\alpha+\beta\sigma$ with
$\beta>0$ are allowed. Using such transformations, we can set
$\sigma|_{t=0}=0$, which amounts to taking $A=0$ in (34). Suppose that
$\dot{\varphi}(0)\neq 0$, so that the maximal geodesic $\varphi_{\mathrm{UV}}$
is not constant. Then the remaining freedom of changing $B$ allows us to take
$\sigma$ to be the proper length parameter:

$s(t)=\int_{0}^{t}\mathrm{d}t^{\prime}||\dot{\varphi}_{\mathrm{UV}}(t^{\prime})||_{\mathcal{G}}~{}~{},$

in which case equation (34) (with $A=0$) gives:

$t(s)=B\int_{0}^{s}\mathrm{d}s^{\prime}e^{\frac{s^{\prime}}{M_{0}}}=B_{0}\left[e^{\frac{s}{M_{0}}}-1\right]~{}~{},$
(35)

where $B_{0}\stackrel{{\scriptstyle{\rm def.}}}{{=}}M_{0}B$ and we used the
relation
$\Big{\|}\frac{\mathrm{d}\varphi_{\mathrm{UV}}(s)}{\mathrm{d}s}\Big{\|}_{\mathcal{G}}=1$.
Moreover, (32) becomes:

$\nabla_{s}\frac{\mathrm{d}\varphi_{\mathrm{UV}}}{\mathrm{d}s}=0~{}~{}$ (36)

while conditions (31) require
$B_{0}=\frac{M_{0}}{||\dot{\varphi}(0)||_{\mathcal{G}}}$ and:

$\varphi_{\mathrm{UV}}|_{s=0}=\varphi(0)~{}~{},~{}~{}\frac{\mathrm{d}\varphi_{\mathrm{UV}}}{\mathrm{d}s}\big{|}_{s=0}=\frac{\dot{\varphi}(0)}{||\dot{\varphi}(0)||_{\mathcal{G}}}~{}~{}.$

To arrive at the last relations, we noticed that
$\frac{\varphi_{\mathrm{UV}}}{\mathrm{d}s}\big{|}_{s=0}=\dot{\varphi}_{\mathrm{UV}}(0)t^{\prime}(s)|_{s=0}=\frac{B_{0}}{M_{0}}\dot{\varphi}(0)$,
where we used the second of conditions (31). Since
$||\frac{\mathrm{d}\varphi_{\mathrm{UV}}}{\mathrm{d}s}||_{\mathcal{G}}=1$,
this requires $B_{0}=\frac{M_{0}}{||\dot{\varphi}(0)||_{\mathcal{G}}}$. In
particular, (35) reads:

$t(s)=\frac{M_{0}}{||\dot{\varphi}(0)||_{\mathcal{G}}}\left[e^{\frac{s}{M_{0}}}-1\right]~{}~{}.$
(37)

Notice that (37) requires
$t(s)>-\frac{M_{0}}{||\dot{\varphi}(0)||_{\mathcal{G}}}$, so the maximal curve
$t\rightarrow\varphi_{\mathrm{UV}}(t)$ is defined on the interval
$\left(-\frac{M_{0}}{||\dot{\varphi}(0)||_{\mathcal{G}}},+\infty\right)$. In
particular, the first order UV approximation must break down if $t\in I$ is
smaller than $-\frac{M_{0}}{||\dot{\varphi}(0)||_{\mathcal{G}}}$. In practice,
the approximation becomes inaccurate for negative times which are smaller in
absolute value since it is only appropriate for small $\frac{|s|}{M_{0}}$. We
have $t(s)\approx\frac{1}{||\dot{\varphi}(0)||_{\mathcal{G}}}s$ when
$\frac{|s|}{M_{0}}\ll 1$.

If $\dot{\varphi}(0)=0$, then $\varphi_{\mathrm{UV}}$ is the constant geodesic
at $\varphi(0)$ since it satisfies
$\varphi_{\mathrm{UV}}|_{\sigma=0}=\varphi(0)$ and
$\frac{\varphi_{\mathrm{UV}}}{\mathrm{d}\sigma}\Big{|}_{\sigma=0}=0$. In this
case, $\varphi_{\mathrm{UV}}$ satisfies (30) with respect to any parameter
$t$. Notice that relation (34) for $A=0$ formally gives $t=B\sigma$ with
$B>0$.

For any non-zero vector $u$ of $T\mathcal{M}$ such that
$m\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\pi(u)\in\mathcal{M}_{0}=\mathcal{M}\setminus\mathrm{Crit}V$,
there exists a unique maximal cosmological curve
$\varphi_{u}:I_{u}\rightarrow\mathcal{M}$ (defined on an open interval $I_{u}$
which contains zero) which satisfies $\varphi_{u}(0)=m$ and
$\dot{\varphi}_{u}(0)=u$. Setting $n\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\frac{u}{||u||_{\mathcal{G}}}$, this cosmological curve is
approximated in the IR limit by the reparameterization (37) of the unique
maximal normalized geodesic $\psi_{n}:\mathbb{R}\rightarrow\mathcal{M}$ of
$(\mathcal{M},\mathcal{G})$ which satisfies $\psi_{n}|_{s=0}=m$ and
$\frac{\mathrm{d}\psi_{n}}{\mathrm{d}s}\Big{|}_{s=0}=n$. Hence the
cosmological time along $\varphi_{u}$ is recovered in this approximation as:

$t(s)=\frac{M_{0}}{||u||_{\mathcal{G}}}\left[e^{\frac{s}{M_{0}}}-1\right]~{}~{}.$

Under the parameter homothety (12), the proper length parameter $s$ of
$\psi_{u}$ changes as $s\rightarrow\lambda^{1/2}s$ while the norm of $u$
changes as $||u||_{\mathcal{G}}\rightarrow\lambda^{1/2}||v||_{\mathcal{G}}$.
Thus we can use that similarity of the cosmological equation to set $M_{0}=1$;
this absorbs $M_{0}$ into $\mathcal{G}$ and changes the normalization of the
geodesic flow without changing the cosmological time $t$. Hence the first
order of the UV approximation of the cosmological flow of the model
parameterized by $(M_{0},\mathcal{M},\mathcal{G},\Phi)$ is determined by the
normalized geodesic flow of the rescaled scalar field metric
$\mathcal{G}_{0}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\frac{1}{M_{0}^{2}}\mathcal{G}$, which has proper length parameter
$s_{0}=\frac{1}{M_{0}}s$ and norm
$||~{}||_{\mathcal{G}_{0}}=\frac{1}{M_{0}}||~{}||_{\mathcal{G}}$. Summarizing
the discussion above gives the following:

######  Proposition 2.3.

Consider the cosmological model parameterized by
$(M_{0},\mathcal{M},\mathcal{G},\Phi)$ and set
$\mathcal{G}_{0}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\frac{1}{M_{0}^{2}}\mathcal{G}$. Let:

$\mathrm{o}(\mathcal{M})=\\{0_{m}|m\in\mathcal{M}\\}\subset T\mathcal{M}$

be the image of the zero section of $T\mathcal{M}$ and:

${\dot{T}}\mathcal{M}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}T\mathcal{M}\setminus\mathrm{o}(\mathcal{M})$

be the slit tangent bundle of $\mathcal{M}$. For each $u\in T\mathcal{M}$, let
$\varphi_{u}:I\rightarrow\mathcal{M}$ be the maximal cosmological curve of
this model which satisfies:

$\varphi_{u}(0)=\pi(u)~{}~{}\mathrm{and}~{}~{}\dot{\varphi}_{u}(0)=u~{}~{}.$

When $u\in{\dot{T}}\mathcal{M}$, this curve is approximated in the UV limit
(for $t\in I$ with $t>-\frac{1}{||u||_{\mathcal{G}_{0}}}$) by a
reparameterization of the maximal normalized geodesic
$\psi_{n}:\mathbb{R}\rightarrow\mathcal{M}$ of the rescaled scalar manifold
$(\mathcal{M},\mathcal{G}_{0})$ which satisfies:

$\psi_{n}|_{s_{0}=0}=\pi(u)~{}~{},~{}~{}\frac{\mathrm{d}\psi_{n}}{\mathrm{d}s_{0}}\Big{|}_{s_{0}=0}=n\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\frac{u}{~{}~{}||u||_{\mathcal{G}_{0}}}~{}~{},$

where the cosmological time $t$ is related to the proper length parameter
$s_{0}$ of $\psi_{n}$ through:

$t(s_{0})=\frac{e^{s_{0}}-1}{~{}~{}||u||_{\mathcal{G}_{0}}}~{}~{}.$

When $u\in\mathrm{o}(\mathcal{M})$, the cosmological curve $\varphi_{u}$ is
approximated in the UV limit by the constant geodesic of
$(\mathcal{M},\mathcal{G}_{0})$ at $\pi(u)$. Accordingly, the cosmological
flow $\Pi:\mathcal{D}\rightarrow T\mathcal{M}$ of the model parameterized by
$(M_{0},\mathcal{M},\mathcal{G},\Phi)$ is approximated in the UV limit by the
flow $\Pi_{\mathcal{G}_{0}}:\mathcal{D}_{\mathcal{G}_{0}}\rightarrow
T\mathcal{M}$ of the $N_{0}$-modification (21) of the geodesic spray of
$(\mathcal{M},\mathcal{G}_{0})$, which has maximal domain of definition:

$\mathcal{D}_{\mathcal{G}_{0}}=\Big{\\{}(t,u)\in\mathbb{R}\times
T\mathcal{M}\big{|}t>-\frac{1}{||u||_{\mathcal{G}_{0}}}\Big{\\}}~{}~{}.$

Notice that the approximations in the proposition are only asymptotic.

#### The strict IR limit and first order IR approximant.

When $\epsilon$ is small but not zero, the rescaled cosmological equation (19)
gives:

$\frac{1}{M_{0}}\epsilon\sqrt{2\Phi(\varphi_{\epsilon}(t))}\frac{\mathrm{d}\varphi_{\epsilon}(t)}{\mathrm{d}t}+(\mathrm{grad}_{\mathcal{G}}\Phi)(\varphi_{\epsilon}(t))=\mathrm{O}(\epsilon^{2})~{}~{}$
(38)

which is equivalent with:

$\epsilon\frac{\mathrm{d}\varphi_{\epsilon}(t)}{\mathrm{d}t}+(\mathrm{grad}_{\mathcal{G}}V)(\varphi_{\epsilon}(t))=\mathrm{O}(\epsilon^{2})~{}~{},$
(39)

where the classical effective scalar potential $V$ was defined in (23). Hence
$\varphi_{\epsilon}(t)$ is approximated to first order in $\epsilon$ by the
solution $\varphi_{\mathrm{IR},\epsilon}(t)$ of the equation:

$\epsilon\frac{\mathrm{d}\varphi_{\mathrm{IR},\epsilon}(t)}{\mathrm{d}t}+(\mathrm{grad}_{\mathcal{G}}V)(\varphi_{\mathrm{IR},\epsilon}(t))=0$
(40)

which satisfies:

$\varphi_{\mathrm{IR},\epsilon}(0)=\varphi(0)~{}~{}.$

It follows that $\varphi(t)=\varphi_{\epsilon}(\epsilon t)$ is approximated by
the solution $\varphi_{\mathrm{IR}}(t)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\varphi_{\mathrm{IR},\epsilon}(\epsilon t)$ of the equation:

$\frac{\mathrm{d}\varphi_{\mathrm{IR}}(t)}{\mathrm{d}t}+(\mathrm{grad}_{\mathcal{G}}V)(\varphi_{\mathrm{IR}}(t))=0$
(41)

which satisfies:

$\varphi_{\mathrm{IR}}(0)=\varphi(0)~{}~{}.$ (42)

Notice that the first order approximant $\varphi_{\mathrm{IR}}$ of $\varphi$
is entirely determined by $\varphi(0)$ and does not depend on
$\dot{\varphi}(0)$; one must consider higher orders of the IR expansion in
order to obtain an approximant of $\varphi$ which also depends on
$\dot{\varphi}(0)$. In the first order IR approximation, the initial speed
$\dot{\varphi}(0)\in T_{\varphi(0)}\mathcal{M}$ is approximated by the vector
$-(\mathrm{grad}_{\mathcal{G}}V)(\varphi(0))\in T_{\varphi(0)}\mathcal{M}$;
this approximation can only be accurate when
$||\dot{\varphi}(0)+(\mathrm{grad}_{\mathcal{G}}V)(\varphi(0))||_{\mathcal{G}}$
is small.

Equation (41) is equivalent with the condition:

$S_{\mathrm{IR}}(\dot{\varphi}_{\mathrm{IR}}(t))=0~{}~{},$ (43)

where:

$S_{\mathrm{IR}}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}-C-(\mathrm{grad}_{\mathcal{G}}V)^{v}\in\mathcal{X}(T\mathcal{M})$
(44)

is obtained from (22) for $\epsilon=1$. Condition (43) confines the canonical
lift of $\varphi_{\mathrm{IR}}$ to the gradient flow shell
$\mathrm{Grad}_{\mathcal{G}}V\subset T\mathcal{M}$ of the effective scalar
triple $(\mathcal{M},\mathcal{G},V)$, which is defined as the graph of the
vector field $-\mathrm{grad}_{\mathcal{G}}V$:

$\mathrm{Grad}_{\mathcal{G}}V\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{graph}(-\mathrm{grad}_{\mathcal{G}}V)=\\{u\in
T\mathcal{M}|u=-(\mathrm{grad}_{\mathcal{G}}V)(\pi(u))\\}=\\{u\in
T\mathcal{M}|S_{\mathrm{IR}}(u)=0\\}~{}~{}.$

To first order of the IR expansion, the cosmological flow curves of the model
degenerate to the canonical lifts of the gradient flow curves of
$(\mathcal{M},\mathcal{G},V)$, which lie within the gradient flow shell
$\mathrm{Grad}_{\mathcal{G}}V$. More precisely, consider a point $u\in
T\mathcal{M}$ and set $\pi(u)=m\in\mathcal{M}$. Then the maximal cosmological
curve $\varphi_{u}$ of this model which satisfies:

$\varphi_{u}(0)=m~{}~{}\mathrm{and}~{}~{}\dot{\varphi}_{u}(0)=u$

is approximated to first order of the IR expansion by the gradient flow curve
$\eta_{m}$ of $(\mathcal{M},\mathcal{G},V)$ which satisfies:

$\eta_{m}(0)=m$

This approximation is good for small $|t|$ only when
$||u+(\mathrm{grad}_{\mathcal{G}}V)(\pi(u))||_{\mathcal{G}}$ is small, i.e.
when $u$ is close the gradient flow shell $\mathrm{Grad}_{\mathcal{G}}V$ and
it is most precise when $u\in\mathrm{Grad}_{\mathcal{G}}V$.

In this approximation, a basic cosmological observable
$F:T\mathcal{M}\rightarrow\mathbb{R}$ reduces to the function:

$F_{\mathrm{IR}}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}F\circ(\mathrm{grad}_{\mathcal{G}}V)\in\mathcal{C}^{\infty}(\mathcal{M})$

defined on $\mathcal{M}$. For any cosmological curve $\varphi$, we have:

$F_{\varphi}\approx_{\mathrm{IR}}F_{\mathrm{IR}}\circ\varphi_{\mathrm{IR}}~{}~{}.$

Summarizing the discussion above gives the following:

######  Proposition 2.4.

Consider the cosmological model parameterized by
$\mathfrak{M}=(M_{0},\mathcal{M},\mathcal{G},\Phi)$ and define its classical
effective potential by $V\stackrel{{\scriptstyle{\rm
def.}}}{{=}}M_{0}\sqrt{2\Phi}$. For each $u\in T\mathcal{M}$ with
$\pi(u)=m\in\mathcal{M}$, the maximal cosmological curve $\varphi_{u}$ of the
model which satisfies:

$\varphi_{u}(0)=m~{}~{}\mathrm{and}~{}~{}\dot{\varphi}_{u}(0)=u$

is approximated to first order of the IR expansion by the gradient flow curve
$\eta_{m}$ of the effective scalar triple $(\mathcal{M},\mathcal{G},V)$ which
satisfies:

$\eta_{m}(0)=m~{}~{}.$

This approximation is optimal for small $|t|$ when
$u\in\mathrm{Grad}_{\mathcal{G}}V$. Accordingly, the cosmological flow
$\Pi:\mathcal{D}\rightarrow T\mathcal{M}$ of the model is approximated to
first order of the IR expansion by the map
$\Pi_{\mathrm{IR}}:{\hat{\mathcal{D}}}_{\mathrm{IR}}\rightarrow\mathrm{Grad}_{\mathcal{G}}V$
defined through:

$\Pi_{\mathrm{IR}}(t,u)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}-(\mathrm{grad}_{\mathcal{G}}V)(\Pi_{V}(t,\pi(u)))~{}~{}\forall(t,u)\in\mathcal{D}_{\mathrm{IR}}~{},$

where $\Pi_{\mathcal{G},V}:\mathcal{D}_{V}\rightarrow\mathcal{M}$ is the
gradient flow of the effective scalar triple $(\mathcal{M},\mathcal{G},V)$,
whose maximal domain of definition we denote by
$\mathcal{D}_{\mathcal{G},V}\subset\mathbb{R}\times\mathcal{M}$. Here:

${\hat{\mathcal{D}}}_{\mathrm{IR}}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\\{(t,u)\in
T\mathcal{M}~{}|~{}(t,\pi(u))\in\mathcal{D}_{\mathcal{G},V}\\}~{}~{}.$

As mentioned above, the first order IR approximation of a cosmological curve
$\varphi$ is optimal when $|t|\ll 1$ for those cosmological curves which
satisfy $\dot{\varphi(0)}\in\mathrm{Grad}_{\mathcal{G}}V$. This motivates the
following:

######  Definition 2.5.

A cosmological curve $\varphi$ of the model parameterized by
$\mathfrak{M}=(M_{0},\mathcal{M},\mathcal{G},\Phi)$ is called infrared optimal
if its orbit meets the gradient flow shell $\mathrm{Grad}_{\mathcal{G}}V$ of
the effective scalar triple $(\mathcal{M},\mathcal{G},V)$, where
$V\stackrel{{\scriptstyle{\rm def.}}}{{=}}M_{0}\sqrt{2\Phi}$.

Suppose that $\varphi:I\rightarrow\mathcal{M}$ is an infrared optimal
cosmological curve and let $t_{0}\in I$ be such that
$\dot{\varphi}(t_{0})=-(\mathrm{grad}_{\mathcal{G}}V)(\varphi(t_{0}))$.
Shifting $t$ by a constant we can assume that $t_{0}=0$ without loss of
generality. Then the first order IR approximant of $\varphi$ satisfies
$\varphi_{\mathrm{IR}}(0)=\varphi(0)$ and
$\dot{\varphi}_{\mathrm{IR}}(0)=\dot{\varphi}(0)$. Thus
$\varphi_{\mathrm{IR}}$ osculates in first order to $\varphi$ at $t=0$ and
hence approximates $\varphi$ to first order in $t$ for $|t|\ll 1$. Notice that
the covariant accelerations of $\varphi$ and $\varphi_{\mathrm{IR}}$ need not
agree at $t=0$ and hence the approximation need not hold to second order in
$t$.

######  Remark 2.6.

Equation (41) can also be written as:

$\frac{\mathrm{d}\varphi_{\mathrm{IR}}}{\mathrm{d}\tau}+(\mathrm{grad}\Phi)(\varphi_{\mathrm{IR}}(\tau))=0~{}~{},$

where $\tau$ is the increasing parameter defined though:

$\tau(t)=\tau_{0}+M_{0}\int_{t_{0}}^{t}\frac{\mathrm{d}t^{\prime}}{\sqrt{2\Phi(\varphi_{\mathrm{IR}}(t))}}~{}~{},$

with $\tau_{0}$ an arbitrary constant. Hence the cosmological curves of the
model parameterized by $\mathfrak{M}$ are approximated by the gradient flow
curves of the original scalar triple $(\mathcal{M},\mathcal{G},\Phi)$ up to
such a curve-dependent reparameterizations of the gradient flow curves. It is
more natural to work with the effective potential $V$ since its gradient flow
parameter coincides with the cosmological time $t$.

###  2.5 Consistency conditions for the UV and IR approximations

The UV and IR approximations discussed in the previous subsection are accurate
when the conditions given below are satisfied.

#### For the UV approximation.

Comparison of (30) with the cosmological equation (7) shows that the UV
approximation amounts to neglecting555This agrees with the fact that the UV
expansion is equivalent with an expansion in the overall scale of $\Phi$. the
contributions $2\Phi(\varphi(t))$ and
$(\mathrm{grad}_{\mathcal{G}}\Phi)(\varphi(t))$ in (7). Hence this
approximation is accurate when the following conditions are satisfied:

$\nu_{1\varphi}(t)\ll 1~{}~{}\mathrm{and}~{}~{}\nu_{2\varphi}(t)\ll 1~{}~{},$
(45)

where we defined the first and second UV parameters of $\varphi$ through:

$\nu_{1\varphi}(t)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\frac{2\Phi(\varphi(t))}{||\dot{\varphi}(t)||_{\mathcal{G}}^{2}}~{}~{}\mathrm{and}~{}~{}\nu_{2\varphi}(t)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}M_{0}\frac{||(\mathrm{d}\Phi)(\varphi(t))||_{\mathcal{G}}}{||\dot{\varphi}(t)||^{2}_{\mathcal{G}}}~{}~{}.$
(46)

Indeed, the cosmological equation (7) can be written as:

$\nabla_{t}\dot{\varphi}(t)+\frac{1}{M_{0}}[1+\nu_{1}(t)]^{1/2}||\dot{\varphi}(t)||_{\mathcal{G}}\dot{\varphi}(t)+(\mathrm{grad}_{\mathcal{G}}\Phi)(\varphi(t))=0~{}~{}.$

The first condition in (45) allows us to approximate this with:

$\nabla_{t}\dot{\varphi}(t)+\frac{1}{M_{0}}||\dot{\varphi}(t)||_{\mathcal{G}}\dot{\varphi}(t)+(\mathrm{grad}_{\mathcal{G}}\Phi)(\varphi(t))=0~{}~{},$
(47)

while the second condition allows us to approximate (47) with (30).

When conditions (45) are satisfied, the cosmological curve $\varphi$ is well-
approximated by the reparameterized geodesic $\varphi_{\mathrm{UV}}$ of
$(\mathcal{M},\mathcal{G})$ which satisfies (31). Using the proper length
parameter $s$ on the latter, equation (37) gives:

$t^{\prime}(s)=\frac{1}{||\dot{\varphi}(0)||_{\mathcal{G}}}e^{\frac{s}{M_{0}}}=\frac{1}{||\dot{\varphi}(0)||_{\mathcal{G}}}+\frac{1}{M_{0}}t(s)~{}~{}$
(48)

and we have
$\Big{\|}\frac{\mathrm{d}\varphi_{\mathrm{UV}}(s)}{\mathrm{d}s}\Big{\|}_{\mathcal{G}}=1$.
Thus:

$||\dot{\varphi}(t)||_{\mathcal{G}}\approx||\dot{\varphi}_{\mathrm{UV}}(t)||_{\mathcal{G}}=\frac{1}{t^{\prime}(s)}=||\dot{\varphi}(0)||_{\mathcal{G}}e^{-\frac{s}{M_{0}}}=\frac{M_{0}||\dot{\varphi}(0)||_{\mathcal{G}}}{M_{0}+t||\dot{\varphi}(0)||_{\mathcal{G}}}$

and:

$\nu_{1\varphi}(s)\approx\frac{2\Phi(\varphi(s))}{||\dot{\varphi}(0)||_{\mathcal{G}}^{2}}e^{\frac{2s}{M_{0}}}~{}~{}\mathrm{and}~{}~{}\nu_{2\varphi}(s)\approx\frac{M_{0}||(\mathrm{d}\Phi)(\varphi(s))||_{\mathcal{G}}}{||\dot{\varphi}(0)||_{\mathcal{G}}^{2}}e^{\frac{2s}{M_{0}}}~{}~{},$
(49)

i.e.:

$\displaystyle\nu_{1\varphi}(t)\approx\frac{2\Phi(\varphi(s))}{M_{0}^{2}||\dot{\varphi}(0)||_{\mathcal{G}}^{2}}(M_{0}+t||\dot{\varphi}(0)||_{\mathcal{G}})^{2}~{}~{}$
$\displaystyle\nu_{2\varphi}(t)\approx\frac{||(\mathrm{d}\Phi)(\varphi(s))||_{\mathcal{G}}}{||\dot{\varphi}(0)||_{\mathcal{G}}^{2}}(M_{0}+t||\dot{\varphi}(0)||_{\mathcal{G}})^{2}~{}~{}.$

For $t=0$, conditions (45) require:

$\frac{2\Phi(\varphi_{\mathrm{UV}}(0))}{||\dot{\varphi}(0)||_{\mathcal{G}}^{2}}\ll
1~{}~{}\mathrm{and}~{}~{}M_{0}\frac{||(\mathrm{d}\Phi)(\varphi_{\mathrm{UV}}(0))||_{\mathcal{G}}}{||\dot{\varphi}(0)||_{\mathcal{G}}^{2}}\ll
1~{}~{},$

while for $t\neq 0$ they constrain the cosmological time interval on which the
approximation is accurate. When these conditions are not satisfied, the
leading UV approximation has to be corrected by higher order terms in the UV
expansion.

#### For the IR approximation.

Comparison of (39) with the cosmological equation (7) shows that the IR
approximation amounts to neglecting the terms $\nabla_{t}\dot{\varphi}(t)$ and
$||\dot{\varphi}(t)||_{\mathcal{G}}^{2}$ in (7). Hence this approximation is
good when the cosmological curve $\varphi$ satisfies:

$\kappa_{1\varphi}(t)\ll 1~{}~{}\mathrm{and}~{}~{}\kappa_{2\varphi}(t)\ll
1~{}~{},$ (50)

where we defined the first and second IR parameters of $\varphi$ through:

$\kappa_{1\varphi}(t)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\frac{||\dot{\varphi}(t)||_{\mathcal{G}}^{2}}{2\Phi(\varphi(t))}=\frac{1}{\nu_{1\varphi}(t)}~{}~{}\mathrm{and}~{}~{}\kappa_{2\varphi}(t)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\frac{||\nabla_{t}\dot{\varphi}(t)||_{\mathcal{G}}}{||(\mathrm{d}\Phi)(\varphi(t))||_{\mathcal{G}}}~{}~{}.$
(51)

The first condition in (50) requires the kinetic energy of $\varphi$ to be
much smaller than its potential energy. Notice that:

$\kappa_{1\varphi}(t)=\kappa_{1}(\dot{\varphi}(t))~{}~{},$

where the function $\kappa_{1}:T\mathcal{M}\rightarrow\mathbb{R}_{>0}$ is
defined through:

$\kappa_{1}(u)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\frac{||u||_{\mathcal{G}}^{2}}{2\Phi(\pi(u)))}~{}~{}.$

The cosmological equation takes the form:

$\nabla_{t}\dot{\varphi}(t)=-\frac{1}{M_{0}}\sqrt{2\Phi(\varphi(t))}\left[\sqrt{1+\kappa_{1\varphi}(t)}\dot{\varphi}(t)+(\mathrm{grad}_{\mathcal{G}}V)(\varphi(t))\right]~{}~{}.$
(52)

Hence the second IR parameter of $\varphi$ can be written as:

$\kappa_{2\varphi}(t)=\frac{1}{M_{0}}\sqrt{2\Phi(\varphi(t))}\frac{||\sqrt{1+\kappa_{1\varphi}(t)}\dot{\varphi}(t)+(\mathrm{grad}_{\mathcal{G}}V)(\varphi(t))||_{\mathcal{G}}}{||(\mathrm{d}\Phi)(\varphi(t))||_{\mathcal{G}}}=\kappa_{2}(\dot{\varphi}(t))~{}~{},$
(53)

where the function $\kappa_{2}:T\mathcal{M}\rightarrow\mathbb{R}_{>0}$ is
defined through:

$\kappa_{2}(u)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\frac{1}{M_{0}}\sqrt{2\Phi(\pi(u))}\frac{||\sqrt{1+\kappa_{1}(u)}u+(\mathrm{grad}_{\mathcal{G}}V)(\pi(u))||_{\mathcal{G}}}{||(\mathrm{d}\Phi)(\pi(u))||_{\mathcal{G}}}~{}~{}.$
(54)

Let $\eta=\varphi_{\mathrm{IR}}$ be the first order IR approximant of
$\varphi$ at $t=0$. Since $\eta$ satisfies the gradient flow equation of
$(\mathcal{M},\mathcal{G},V)$, we have:

$\kappa_{1\eta}(t)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\kappa_{1}(\dot{\eta}(t))=M_{0}^{2}\frac{||(\mathrm{d}\Phi)(\eta(t))||_{\mathcal{G}}^{2}}{4\Phi(\eta(t))^{2}}={\hat{\kappa}}_{1}(\eta(t))~{}~{},$

where the function ${\hat{\kappa}}_{1}:\mathcal{M}\rightarrow\mathbb{R}_{>0}$
is defined through:

${\hat{\kappa}}_{1}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}M_{0}^{2}\frac{||\mathrm{d}\Phi||_{\mathcal{G}}^{2}}{4\Phi^{2}}~{}~{}.$

Since $\mathrm{grad}V=-\frac{M_{0}}{\sqrt{2\Phi}}\mathrm{grad}\Phi$, we
have666We temporarily denote $||~{}||_{\mathcal{G}}$ by $||~{}||$ and
$\mathrm{grad}_{\mathcal{G}}$ by $\mathrm{grad}$ to simplify notation.:

$\nabla_{t}\dot{\eta}(t)=\nabla_{\dot{\eta}(t)}\dot{\eta}(t)=(\nabla_{\mathrm{grad}V}\mathrm{grad}V)(\eta(t))=\left[\frac{M_{0}^{2}}{2\Phi}\nabla_{\mathrm{grad}\Phi}\mathrm{grad}\Phi-\kappa_{1}\mathrm{grad}\Phi\right]\Big{|}_{\eta(t)}$
(55)

because
$\nabla_{\mathrm{grad}\Phi}\Phi=(\mathrm{d}\Phi)(\mathrm{grad}\Phi)=||\mathrm{d}\Phi||^{2}$.
Thus:

$\kappa_{2\eta}(t)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\frac{||\nabla_{t}\dot{\eta}(t)||_{\mathcal{G}}}{||(\mathrm{d}\Phi)(\eta(t))||_{\mathcal{G}}}=\frac{M_{0}^{2}}{2\Phi}\\!\\!\left[\\!\frac{||\nabla_{\mathrm{grad}\Phi}\mathrm{grad}\Phi||^{2}}{||\mathrm{d}\Phi||^{2}}\\!\\!+\\!\frac{1}{4\Phi^{2}}||\mathrm{d}\Phi||^{4}\\!-\\!\frac{1}{2\Phi}\nabla_{\mathrm{grad}\Phi}||\mathrm{d}\Phi||^{2}\\!\right]^{1/2}\\!\\!\Big{|}_{\eta(t)}={\hat{\kappa}}_{2}(\eta(t))~{}~{},$

where we used the relation:

$\mathcal{G}(X,\nabla_{X}X)=\frac{1}{2}\nabla_{X}||X||^{2}~{}~{},$

which holds for any vector field $X\in\mathcal{X}(\mathcal{M})$ since
$\mathcal{G}$ is covariantly constant. Here the function
${\hat{\kappa}}_{2}:\mathcal{M}\rightarrow\mathbb{R}_{>0}$ is defined through:

$\displaystyle{\hat{\kappa}}_{2}$ $\displaystyle\stackrel{{\scriptstyle{\rm
def.}}}{{=}}$
$\displaystyle\frac{||\frac{M_{0}^{2}}{2\Phi}\nabla_{\mathrm{grad}\Phi}\mathrm{grad}\Phi-\kappa_{1}\mathrm{grad}\Phi||}{||\mathrm{d}\Phi||}$
$\displaystyle=$
$\displaystyle\frac{M_{0}^{2}}{2\Phi}\left[\frac{||\nabla_{\mathrm{grad}\Phi}\mathrm{grad}\Phi||^{2}}{||\mathrm{d}\Phi||^{2}}\\!\\!+\\!\frac{1}{4\Phi^{2}}||\mathrm{d}\Phi||^{4}-\frac{1}{2\Phi}\nabla_{\mathrm{grad}\Phi}||\mathrm{d}\Phi||^{2}\\!\right]^{1/2}$

On the other hand, we have::

$\kappa_{2}(\dot{\eta}(t))=\sqrt{1+\kappa_{1\eta}(t)}-1~{}~{},$ (56)

where we used (54) and the fact that $\eta$ satisfies the gradient flow
equation of $(\mathcal{M},\mathcal{G},V)$.

Using the first order IR approximation for $\varphi$ and its first time
derivative amounts to replace $\varphi(t)$ by $\eta(t)$ and $\dot{\varphi}(t)$
by $\dot{\eta}(t)$. Then $\kappa_{1\varphi}(t)$ is replaced by
$\kappa_{1\eta}(t)$ and $\kappa_{2\varphi}(t)=\kappa_{2}(\dot{\varphi}(t))$ is
replaced by $\kappa_{2}(\dot{\eta}(t))$. Accuracy of this approximation
requires $\kappa_{1\eta}(t)\ll 1$, which implies $\kappa_{2}(\dot{\eta}(t))\ll
1$ by (56). For the approximation to be accurate up to the second time
derivative of $\varphi$ and $\eta$, we must also have
$\kappa_{2\varphi}(t)\approx\kappa_{2\eta}(t)$, which requires
$\kappa_{2\eta}(t)\ll 1$.

######  Remark 2.7.

Suppose that $\varphi$ is an infrared optimal cosmological curve. In this
case, we have $\varphi(0)=\eta(0):=m\in\mathcal{M}$ and
$\dot{\varphi}(0)=\dot{\eta}(0)$, which gives:

$\kappa_{1\varphi}(0)=\kappa_{1\eta}(0)=\kappa_{1}(m)~{}~{}\mathrm{and}~{}~{}\kappa_{2\varphi}(0)=\kappa_{2}(\dot{\eta}(0))=\sqrt{1+{\hat{\kappa}}_{1}(m)}-1~{}~{}.$

The curves $\varphi$ and $\eta$ osculate in order two at the point $m$ iff
$\nabla_{t}\dot{\varphi}(0)=\nabla_{t}\dot{\eta}(0)$, i.e. (see (52) and
(55)):

$(\sqrt{1+{\hat{\kappa}}_{1}(m)}+{\hat{\kappa}}_{1}(m)-1)(\mathrm{grad}_{\mathcal{G}}\Phi)=\frac{M_{0}^{2}}{2\Phi(m)}(\nabla_{\mathrm{grad}\Phi}\mathrm{grad}\Phi)(m)~{}~{}.$
(57)

This requires $\kappa_{2\varphi}(0)=\kappa_{2\eta}(0)$, i.e.:

${\hat{\kappa}}_{2}(m)=\sqrt{1+{\hat{\kappa}}_{1}(m)}-1~{}~{}.$

When (57) is satisfied, $\varphi(t)$ is approximated by $\eta(t)$ to order two
in $t$ for $|t|\ll 1$.

###  2.6 Relation to some other approximations used in cosmology

Recall that the slow roll parameter of a cosmological curve $\varphi$ is
defined through:

$\bm{\varepsilon}_{\varphi}(t)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}-\frac{\dot{H_{\varphi}}(t)}{H_{\varphi}(t)^{2}}=-\frac{1}{H_{\varphi}(t)}\frac{\mathrm{d}}{\mathrm{d}t}\log
H_{\varphi}(t)~{}~{}.$

We have:

$\frac{\mathrm{d}}{\mathrm{d}t}\log
H_{\varphi}(t)=\frac{\mathcal{G}(\dot{\varphi}(t),\nabla_{t}\dot{\varphi}(t))+(\mathrm{d}\Phi)(\varphi(t))(\dot{\varphi}(t))}{||\dot{\varphi}(t)||_{\mathcal{G}}^{2}+2\Phi(\varphi(t))}=-\frac{3H_{\varphi}(t)||\dot{\varphi}(t)||_{\mathcal{G}}^{2}}{||\dot{\varphi}(t)||_{\mathcal{G}}^{2}+2\Phi(\varphi(t))}~{}~{},$

where in the last equality we used the cosmological equation and the relation:

$\mathcal{G}(\dot{\varphi}(t),(\mathrm{grad}_{\mathcal{G}}\Phi)(\varphi(t)))=(\mathrm{d}\Phi)(\varphi(t))(\dot{\varphi}(t))$

to simplify the numerator. Thus:

$\bm{\varepsilon}_{\varphi}(t)=\frac{3||\dot{\varphi}(t)||_{\mathcal{G}}^{2}}{||\dot{\varphi}(t)||_{\mathcal{G}}^{2}+2\Phi(\varphi(t))}=\frac{3}{1+\nu_{1\varphi}(t)}~{}~{},$

where
$\nu_{1\varphi}(t)=\frac{2\Phi(\varphi(t))}{||\dot{\varphi}(t)||_{\mathcal{G}}^{2}}=\frac{1}{\kappa_{1\varphi}(t)}$
is the first UV parameter and $\kappa_{1\varphi}(t)$ is the first IR parameter
of $\varphi$. Hence the slow roll condition $\bm{\varepsilon}_{\varphi}(t)\ll
1$ holds iff $\nu_{1\varphi}(t)\gg 1$ i.e. $\kappa_{1\varphi}(t)\ll 1$. In
particular, the slow roll approximation does not hold when the UV
approximation is accurate but it automatically applies when the first order IR
approximation is accurate up to first time derivatives. Notice, however, that
conditions (50) for the IR approximation to be accurate up to second time
derivatives are stronger (i.e. more restrictive) than the slow roll condition,
since they also require the parameter $\kappa_{2\varphi}(t)$ to be small. In
particular, the slow roll approximation by itself is not equivalent with the
first order IR approximation.

Define the scalar gradient flow parameter of $\varphi$ through:

$\bm{\eta}_{\varphi}(t)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\frac{1}{3}||\eta_{\varphi}(t)||_{\mathcal{G}}=\frac{1}{3H_{\varphi}(t)}\frac{||\nabla_{t}\dot{\varphi}(t)||_{\mathcal{G}}}{||\dot{\varphi}(t)||_{\mathcal{G}}}$

where $\eta_{\varphi}(t)$ is the vector gradient flow parameter of (genalpha,
, Sec. 1.5). Since
$||\nabla_{t}\dot{\varphi}(t)||_{\mathcal{G}}=\kappa_{2\varphi}(t)||(\mathrm{d}\Phi)(\varphi(t))||_{\mathcal{G}}$
and
$||\dot{\varphi}(t)||_{\mathcal{G}}=\sqrt{2\Phi(\varphi(t))\kappa_{1\varphi}(t)}$,
we have:

$H_{\varphi}(t)=\frac{1}{3M_{0}}\sqrt{2\Phi(\varphi(t))[1+\kappa_{1\varphi}(t)]}$

and:

$\bm{\eta}_{\varphi}(t)=\frac{M_{0}}{2}\frac{||(\mathrm{d}\Phi)(\varphi(t))||_{\mathcal{G}}}{\Phi(\varphi(t))}\frac{\kappa_{2\varphi}(t)}{\sqrt{\kappa_{1}(t)[1+\kappa_{1}(t)]}}~{}~{}.$

When the IR approximation is accurate to second order in time derivatives, we
have $\kappa_{1\varphi}(t),\kappa_{2\varphi}(t)\ll 1$ and the previous
relation gives:

$\bm{\eta}_{\varphi}(t)\approx\frac{M_{0}}{2}\frac{||(\mathrm{d}\Phi)(\varphi(t))||_{\mathcal{G}}}{\Phi(\varphi(t))}\frac{\kappa_{2}(t)}{\sqrt{\kappa_{1}(t)}}~{}~{}.$
(58)

Hence the gradient flow approximation of genalpha  applies within the IR
approximation only when this quantity is small. For two-field models, it was
shown in (genalpha, , Sec. 1.9) that the SRST approximation of PT1 ; PT2  is a
further specialization of the gradient flow approximation which applies only
when certain conditions on the Hessian of $\Phi$ are satisfied; these
conditions are much stronger than those for accuracy of the IR approximation.
Hence the SRST approximation can be applied within the IR approximation only
under very restrictive conditions. Thus the IR approximation is conceptually
and quantitatively different from the gradient flow approximation of genalpha
. It is markedly different from (and much less restrictive than) the SRST
approximation and its $n$-field variant.

##  3 Renormalization group and dynamical universality classes

In this section, we discuss the action of the subgroup
$T_{\mathrm{ren}}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{Stab}_{T}(\Phi)$ of the universal similarity group $T$ of
Subsection 1.2 on the parameters $(M_{0},\mathcal{G})$ of a multifield
cosmological model with target manifold $\mathcal{M}$, showing that it plays a
role similar to that of the renormalization group action in the theory of
critical phenomena. This action defines the RG flow of the model, which
proceeds by homothety transformations of the rescaled Planck mass $M_{0}$ and
of the scalar manifold metric $\mathcal{G}$. We also show that the first order
infrared approximation is invariant up to reparameterization under Weyl
transformations of $\mathcal{G}$ and define UV and IR universality classes of
classical cosmological models. After briefly discussing the infrared phase
structure which arises from the classification of limit points of cosmological
curves, we use these results and the uniformization theorem of Poincaré to
show that two-field models whose scalar manifold metric has constant curvature
are IR universal in the sense that they provide distinguished representatives
of the universality classes of all two-field models.

###  3.1 RG similarities and RG transformations

For fixed target space $\mathcal{M}$, the cosmological equation (7) is
invariant under the following action of the multiplicative group
$\mathbb{R}_{>0}$ on the curve $\varphi$ and on the parameters
$(M_{0},\mathcal{G},\Phi)$:

$\varphi\rightarrow\varphi_{\epsilon}~{}~{},~{}~{}M_{0}\rightarrow\epsilon
M_{0}~{}~{},~{}~{}\mathcal{G}\rightarrow\epsilon^{2}\mathcal{G}~{}~{},~{}~{}\Phi\rightarrow\Phi~{}~{}(\epsilon>0)~{}~{}.$
(59)

The RG similarity (59) is the composition of the parameter homothety (12) at
parameter $\lambda=\epsilon^{2}$ with the scale similarity (13) at parameter
$\epsilon$. These transformations give the restriction $\rho_{\mathrm{ren}}$
of the similarity action $\rho$ of the group $T=(\mathbb{R}_{>0})^{2}$ to the
renormalization subgroup
$T_{\mathrm{ren}}=\mathrm{Stab}_{T}(\Phi)\simeq\mathbb{R}_{>0}$ discussed in
Subsection 1.2:

$\rho_{\mathrm{ren}}(\epsilon)(\varphi,M_{0},\mathcal{G},\Phi)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\rho(\epsilon^{2},\epsilon)(\varphi,M_{0},\mathcal{G},\Phi)=(\varphi_{\epsilon},\epsilon
M_{0},\epsilon^{2}\mathcal{G},\Phi)~{}~{}(\epsilon>0)~{}~{}.$

The RG similarities (59) can be used to absorb $M_{0}$ into the overall scale
of $\mathcal{G}$ or viceversa without changing the scale of the scalar
potential $\Phi$. Notice that RG similarities are adiabatic in the sense that
they preserve the total energy
$E_{\varphi}^{\mathcal{G},\Phi}(t)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\frac{1}{2}||\dot{\varphi}(t)||_{\mathcal{G}}^{2}+\Phi(\varphi(t))$
of a cosmological curve $\varphi$. Namely, we have:

$E^{\epsilon^{2}\mathcal{G},\Phi}_{\varphi_{\epsilon}}(t)=E_{\varphi}^{\mathcal{G},\Phi}(t/\epsilon)~{}~{}.$

The time $t$ cosmological curves of the model with parameters $(\epsilon
M_{0},\epsilon^{2}\mathcal{G},\Phi)$ coincide with the time $t/\epsilon$
cosmological curves of the model with parameters $(M_{0},\mathcal{G},\Phi)$.
Hence the infrared and ultraviolet limits of the latter can be described
equivalently by taking $\epsilon$ to zero and infinity in the former. Due to
these properties, the RG transformations:

$M_{0}\rightarrow\epsilon
M_{0}~{}~{},~{}~{}\mathcal{G}\rightarrow\epsilon^{2}\mathcal{G}~{}~{},~{}~{}\Phi\rightarrow\Phi$
(60)

play a role akin to that familiar from the theory of critical phenomena. These
transformations give the restriction $\rho_{\mathrm{RG}}$ of the parameter
action $\rho_{\mathrm{par}}$ of $T$ (see (14)) to the renormalization group
$T_{\mathrm{ren}}$:

$\rho_{\mathrm{RG}}(\epsilon)(M_{0},\mathcal{G},\Phi)=\rho_{\mathrm{par}}(\epsilon^{2},\epsilon)(M_{0},\mathcal{G},\Phi)=(\epsilon
M_{0},\epsilon^{2}\mathcal{G},\Phi)~{}~{}(\epsilon>0)~{}~{}.$

Under RG transformations, the rescaled scalar field metric
$\mathcal{G}_{0}=\frac{1}{M_{0}^{2}}\mathcal{G}$ is invariant, while the
classical effective potential $V=M_{0}\sqrt{2\Phi}$ and its gradient with
respect to $\mathcal{G}$ change as:

$V\rightarrow\epsilon
V~{}~{},~{}~{}\mathrm{grad}_{\mathcal{G}}V\rightarrow\frac{1}{\epsilon}\mathrm{grad}_{\mathcal{G}}V~{}~{}.$

Hence $\mathrm{grad}_{\mathcal{G}}V$ tends to zero in the UV limit
$\epsilon\rightarrow\infty$, while it tends to a current supported on the non-
critical locus $\mathcal{M}_{0}=\mathcal{M}\setminus\mathrm{Crit}\Phi$ in the
strict IR limit $\epsilon\rightarrow 0$. Notice that the geodesic flow of
$(\mathcal{M},\mathcal{G}_{0})$ is invariant under RG transformations, since
the covariant derivative of $\mathcal{G}_{0}$ satisfies:

$\nabla^{\epsilon^{2}\mathcal{G}_{0}}=\nabla^{\mathcal{G}_{0}}~{}~{}.$

This also follows from the fact that the cosmological equation is invariant
under RG similarities while the geodesic equation is invariant under affine
reparameterizations. On the other hand, the gradient flow of
$(\mathcal{M},\mathcal{G},V)$ is invariant under RG transformations up to a
reparameterization
$\varphi_{\mathrm{IR}}\rightarrow\varphi_{\mathrm{IR},\epsilon}$ with positive
constant factor $1/\epsilon$; this also follows from invariance of the
cosmological equation under RG similarities.

###  3.2 The dynamical renormalization group flow

Let $\mathcal{M}$ be a smooth manifold and
$\mathcal{T}_{2}(\mathcal{M})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\Gamma(\mathrm{Sym}^{2}(T^{\ast}\mathcal{M}))$ be the infinite-
dimensional space of smooth symmetric covariant 2-tensor fields defined on
$\mathcal{M}$. A Riemannian homothety line on $\mathcal{M}$ is a one-
dimensional linear subspace $L\subset\mathcal{T}_{2}(\mathcal{M})$ which
contains a Riemannian metric defined on $\mathcal{M}$. In this case, all
elements of $L$ which are positively-homothetic with $\mathcal{G}$ are
Riemannian metrics on $\mathcal{M}$; such elements form an open half-line
$L_{+}$ contained in $L$ which satisfies $L=L_{+}\cup(-L_{+})\cup\\{0\\}$. The
cosmological homothety plane defined by $L$ is the linear space
$\Pi(\mathcal{M},L)\stackrel{{\scriptstyle{\rm def.}}}{{=}}\mathbb{R}\oplus
L\subset\mathbb{R}\times\mathcal{T}_{2}(\mathcal{M})$, which contains the
cosmological homothety cone
$\mathrm{C}(\mathcal{M},L)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathbb{R}_{>0}\oplus L_{+}$. The cosmological RG action on
$\mathrm{C}(\mathcal{M},L)$ is the action of the group
$T_{\mathrm{ren}}=\mathbb{R}_{>0}$ defined through:

$\rho_{\mathrm{RG}}(\epsilon)(M_{0},\mathcal{G})\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(\epsilon
M_{0},\epsilon^{2}\mathcal{G})~{}~{}\forall(M_{0},\mathcal{G})\in\mathrm{C}(\mathcal{M},L)~{}~{}\forall\epsilon>0~{}~{}.$

Setting $\epsilon=e^{\lambda}$ with $\lambda\in\mathbb{R}$, this action
describes the flow on the homothety cone of the Euler vector field $E_{L}$
defined through:

$E_{L}(M_{0},\mathcal{G})\stackrel{{\scriptstyle{\rm def.}}}{{=}}M_{0}\oplus
2\mathcal{G}\in\Pi(\mathcal{M},\mathcal{G})\equiv
T_{M_{0},\mathcal{G}}\mathrm{C}(\mathcal{M},L)~{}~{}\forall(M_{0},\mathcal{G})\in\mathrm{C}(\mathcal{M},L)~{}~{}.$

This flow is called the cosmological RG flow of $(\mathcal{M},L)$.

Any choice of a reference metric $\mathcal{G}_{\mathrm{ref}}\in L_{+}$ induces
coordinates $w_{1},w_{2}$ on $\mathrm{C}(\mathcal{M},L)$ given by:

$M_{0}=w_{1}~{}~{},~{}~{}\mathcal{G}=w_{2}\mathcal{G}_{\mathrm{ref}}~{}~{},$

which extend to coordinates on the homothety plane $\Pi(\mathcal{M},L)$ and
hence identify the latter with $\mathbb{R}^{2}$ and
$\mathrm{C}(\mathcal{M},L)$ with the first quadrant. Then $\rho_{\mathrm{RG}}$
identifies with the action:

$\rho_{\mathrm{RG}}(\epsilon)(w_{1},w_{2})=(\epsilon
w_{1},\epsilon^{2}w_{2})~{}~{}\forall\epsilon>0$

and $E_{L}$ identifies with the vector field $E$ on $\mathbb{R}_{>0}^{2}$
given by:

$E(w_{1},w_{2})=(w_{1},2w_{2})~{}~{}\forall w_{1},w_{2}>0.$

Moreover, the integral curves of the RG flow identify with the solutions of
the system:

$\displaystyle\frac{\mathrm{d}w_{1}(\lambda)}{\mathrm{d}\lambda}$
$\displaystyle=$ $\displaystyle
E_{1}(w_{1}(\lambda),w_{2}(\lambda))=w_{1}(\lambda)$
$\displaystyle\frac{\mathrm{d}w_{2}(\lambda)}{\mathrm{d}\lambda}$
$\displaystyle=$ $\displaystyle
E_{2}(w_{1}(\lambda),w_{2}(\lambda))=2w_{2}(\lambda)~{}~{},$ (61)

namely:

$w_{1}(\lambda)=e^{\lambda}w_{1}(0)~{}~{},~{}~{}w_{2}(\lambda)=e^{2\lambda}w_{2}(0)~{}~{}.$

The limit $\lambda\rightarrow+\infty$ recovers the UV limit
$\epsilon\rightarrow+\infty$ while $\lambda\rightarrow-\infty$ corresponds to
the IR limit $\epsilon\rightarrow 0$. These limits correspond to the fixed
points of the RG flow on the one-point compactification of the closure of the
homothety cone, which are the apex of the cone and the point at infinity (see
Figure 1).

(a) Integral curves of the RG flow on the homothety cone.

(b) Integral curves of the RG flow on the one-point compactification of the
homothety cone.

Figure 1: Integral curves of the RG flow on the homothety cone and on the one-
point compactification of its closure. In the second figure we identified the
Alexandroff compactification of the homothety plane with the unit sphere
through stereographic projection. Here $\phi\in[0,\pi]$ is the spherical
altitude angle on the unit sphere and $\theta\in[0,2\pi]$ is the
azimuth/longitude angle (which coincides with the polar angle in the homothety
plane), while $r=\sqrt{w_{1}^{2}+w_{2}^{2}}=\cot(\frac{\phi}{2})$ is the
distance from the origin in the homothety plane. The homothety cone identifies
with the region $(\phi,\theta)\in[0,\pi]\times[0,\frac{\pi}{2}]$ of the unit
sphere. The RG flow on the one-point compactification has fixed points at
$(\phi,\theta)=(\pi,0)$ (the red dot) and $(\phi,\theta)=(0,\frac{\pi}{2})$
(the blue dot), which correspond respectively to the apex of the homothety
cone and its point at infinity, i.e. to the south and north poles of the
sphere. These fixed points give the IR limit (the red dot) and the UV limit
(the blue dot).

Consider a model parameterized by $(M_{0},\mathcal{M},\mathcal{G},\Phi)$,
where $\mathcal{G}\in L_{+}$. The discussion of the previous subsection
implies that the cosmological RG flow curve of $(\mathcal{M},L)$ which passes
through the point $(M_{0},\mathcal{G})\in\mathrm{C}(\mathcal{M},L)$ induces a
curve in the space of all flows defined on $T\mathcal{M}$ which interpolates
when $\lambda$ runs from $-\infty$ to $+\infty$ between the gradient flow of
$(\mathcal{M},\mathcal{G},V)$ (where $V=M_{0}\sqrt{2\Phi}$) and a modification
of the geodesic flow of $(\mathcal{M},\mathcal{G}_{0})$ (where
$\mathcal{G}_{0}=\frac{1}{M_{0}^{2}}\mathcal{G}$).

###  3.3 Conformal invariance in the infrared

The gradient flow of a scalar potential $V$ defined on a Riemannian manifold
$(\mathcal{M},\mathcal{G})$ is invariant under Weyl rescalings of the metric
$\mathcal{G}$ up to reparameterization of the gradient flow curves. More
precisely, consider a solution $\eta:I\rightarrow\mathcal{M}$ of the gradient
flow equation:

$\frac{\mathrm{d}\eta(t)}{\mathrm{d}t}=-(\mathrm{grad}_{\mathcal{G}}V)(\eta(t))$
(62)

of $V$ with respect to the metric $\mathcal{G}$ and let $\Omega$ be a smooth
and everywhere-positive real-valued function defined on $\mathcal{M}$. Then
the $\eta$-dependent increasing reparameterization $t\rightarrow\tau$ defined
through:

$\tau(t)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\tau_{\eta}(t)=\int_{t_{0}}^{t}\mathrm{d}t^{\prime}\Omega(\eta(t^{\prime}))+C~{}~{},$

(where $t_{0}\in I$ and $C$ is an arbitrary constant) takes $\eta(t)$ into a
solution $\eta(\tau)$ of the gradient flow equation:

$\frac{\mathrm{d}\eta(\tau)}{\mathrm{d}\tau}=-(\mathrm{grad}_{\mathcal{G}^{\prime}}V)(\eta(\tau))$

of $V$ with respect to the metric
$\mathcal{G}^{\prime}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\Omega\mathcal{G}$. This implies that the gradient flow orbits of
$V$ with respect to the metrics $\mathcal{G}$ and $\Omega\mathcal{G}$ coincide
as oriented submanifolds of $\mathcal{M}$. In particular, these orbits depend
only on the Weyl-equivalence class of $\mathcal{G}$.

This observation can be generalized as follows. Recall that a smooth map
$f:(\mathcal{M}_{1},\mathcal{G}_{1})\rightarrow(\mathcal{M}_{2},\mathcal{G}_{2})$
between Riemannian manifolds $(\mathcal{M}_{1},\mathcal{G}_{1})$ and
$(\mathcal{M}_{2},\mathcal{G}_{2})$ is called a conformal diffeomorphism if
$f^{\ast}(\mathcal{G}_{2})=\Omega\mathcal{G}_{1}$ for some positive smooth
function $\Omega:\mathcal{M}_{1}\rightarrow\mathbb{R}_{>0}$.

######  Definition 3.1.

Two scalar triples $(\mathcal{M}_{1},\mathcal{G}_{1},V_{1})$ and
$(\mathcal{M}_{2},\mathcal{G}_{2},V_{2})$ are called smoothly conformally
equivalent if there exists a conformal diffeomorphism
$f:(\mathcal{M}_{1},\mathcal{G}_{1})\rightarrow(\mathcal{M}_{2},\mathcal{G}_{2})$
such that $V_{1}=V_{2}\circ f$. In this case, $f$ is called a (smooth)
conformal equivalence between the two triples. A conformal automorphism of a
scalar triple $(\mathcal{M},\mathcal{G},V)$ is a conformal equivalence from
$(\mathcal{M},\mathcal{G},V)$ to itself.

With this definition, we show in Appendix B that the gradient flows of
conformally-equivalent scalar triples are smoothly topologically equivalent.
In particular, the smooth topological equivalence class of the gradient flow
of a scalar triple is invariant under conformal automorphisms of that triple.
Since in the gradient flow of the function $V=M_{0}\sqrt{2\Phi}$ with respect
to $\mathcal{G}$ gives the first order IR approximant of cosmological curves
for the model parameterized by $(M_{0},\mathcal{M},\mathcal{G},\Phi)$, we
conclude that:

Up to topological equivalence, the first order IR approximant of the
cosmological flow of a multifield cosmological model parameterized by
$(M_{0},\mathcal{M},\mathcal{G},\Phi)$ depends only on the conformal
equivalence class of the effective scalar triple
$(\mathcal{M},\mathcal{G},V)$, where $V\stackrel{{\scriptstyle{\rm
def.}}}{{=}}M_{0}\sqrt{2\Phi}$. Up to curve-dependent increasing
reparameterization of the flow curves, this approximant depends only on
$\mathcal{M}$, $V$ and on the Weyl-equivalence class of $\mathcal{G}$.

###  3.4 Dynamical universality classes

Consider two cosmological models parameterized by
$\mathfrak{M}_{i}=(M_{0i},\\!\mathcal{M}_{i},\\!\mathcal{G}_{i},\\!\Phi_{i})$
($i=1,2$) whose tangent bundle projections and Levi-Civita connections we
denote by $\pi_{i}$ and $\nabla_{i}$. Let
$\mathcal{G}_{0i}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\frac{1}{M_{0i}^{2}}\mathcal{G}_{i}$ be the rescaled scalar
manifold metrics and $V_{i}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}M_{0i}\sqrt{2\Phi_{i}}$ be the effective scalar potentials of the
two models. Notice that $\nabla_{i}$ coincides with the Levi-Civita connection
of $\mathcal{G}_{0i}$.

#### UV conjugations and UV equivalences.

Recall the map (16) induced between the sets of smooth curves in two manifolds
by a smooth semispray map.

######  Definition 3.2.

A smooth geodesic conjugation between two connected Riemannian manifolds
$(\mathcal{M}_{1},\mathcal{G}_{01})$ and $(\mathcal{M}_{2},\mathcal{G}_{02})$
is a smooth semispray map $f:T\mathcal{M}_{1}\rightarrow T\mathcal{M}_{2}$
which restricts to a topological conjugation between their normalized geodesic
flows, i.e. $f$ restricts to a diffeomorphism
$f_{S}:ST\mathcal{M}_{1}\rightarrow ST\mathcal{M}_{2}$ between the unit sphere
bundles $ST\mathcal{M}_{1}$ and $ST\mathcal{M}_{2}$ and has the property that
${\hat{f}}(\psi_{1})$ is a normalized geodesic of
$(\mathcal{M}_{2},\mathcal{G}_{02})$ whenever $\psi_{1}$ is a normalized
geodesic of $(\mathcal{M}_{1},\mathcal{G}_{01})$.

######  Example 3.3.

Consider a diffeomorphism $f_{0}:\mathcal{M}_{1}\rightarrow\mathcal{M}_{2}$.
Then it is easy to see that $f_{0}$ maps normalized geodesics of
$(\mathcal{M}_{1},\mathcal{G}_{01})$ into normalized geodesics of
$(\mathcal{M}_{2},\mathcal{G}_{02})$ iff it is an isometry, which in turn is
equivalent with the condition that its differential $\mathrm{d}f_{0}$ maps
$ST\mathcal{M}_{1}$ into $ST\mathcal{M}_{2}$. In this case, the restriction
$\mathrm{d}f_{0}|_{ST\mathcal{M}_{1}}:ST\mathcal{M}_{1}\rightarrow
ST\mathcal{M}_{2}$ is a geodesic conjugation.

######  Remark 3.4.

Recall that an affine mapping from $(\mathcal{M}_{1},\mathcal{G}_{01})$ to
$(\mathcal{M}_{2},\mathcal{G}_{02})$ is a diffeomorphism
$f_{0}:\mathcal{M}_{1}\rightarrow\mathcal{M}_{2}$ such that
$f_{0}^{\ast}(\nabla_{2})=\nabla_{1}$. Such a map takes affinely parameterized
geodesics of $(\mathcal{M}_{1},\mathcal{G}_{01})$ into affinely parameterized
geodesics of $(\mathcal{M}_{2},\mathcal{G}_{02})$ but it need not take
normalized geodesics into normalized geodesics. An affine mapping from a
Riemannian manifold $(\mathcal{M},\mathcal{G}_{0})$ to itself is called an
affine transformation of $(\mathcal{M}_{0},\mathcal{G}_{0})$; such
transformations form a Lie group. An infinitesimal affine mapping is sometimes
called an affine collineation. When $(\mathcal{M},\mathcal{G}_{0})$ is
complete, irreducible and of dimension greater than one, any affine
transformation of $(\mathcal{M},\mathcal{G}_{0})$ is an isometry (see
Kobayashi ; IO ).

######  Definition 3.5.

A smooth geodesic equivalence between two Riemannian manifolds
$(\mathcal{M}_{1},\mathcal{G}_{01})$ and $(\mathcal{M}_{2},\mathcal{G}_{02})$
is a smooth semispray map $f:T\mathcal{M}_{1}\rightarrow T\mathcal{M}_{2}$
which identifies geodesic orbits, i.e.
$\psi_{1}:I_{1}\rightarrow\mathcal{M}_{1}$ is an arbitrarily parameterized
geodesic of $(\mathcal{M}_{1},\mathcal{G}_{01})$ iff ${\hat{f}}(\psi_{1})$ is
an arbitrary parameterized geodesic of $(\mathcal{M}_{2},\mathcal{G}_{02})$.

######  Example 3.6.

Consider a smooth geodesic (a.k.a. projective) mapping
$f_{0}:(\mathcal{M}_{1},\mathcal{G}_{01})\rightarrow(\mathcal{M}_{2},\mathcal{G}_{02})$,
i.e. a smooth diffeomorphism from $\mathcal{M}_{1}$ to $\mathcal{M}_{2}$ which
maps the geodesic orbits of $(\mathcal{M}_{1},\mathcal{G}_{01})$ into those of
$(\mathcal{M}_{2},\mathcal{G}_{02})$, a condition which amounts to the
requirement that the connections $\nabla_{1}$ and $f_{0}^{\ast}(\nabla_{2})$
be projectively-equivalent. Then $\mathrm{d}f_{0}:T\mathcal{M}_{1}\rightarrow
T\mathcal{M}_{2}$ is a smooth geodesic equivalence. The study of geodesic
mappings is a classical subject in Riemannian geometry (see, for example Mikes
) which is intimately connected (see CrampinSaunders ) to Cartan’s projective
differential geometry CartanProj .

The results of Section 2 motivate the following:

######  Definition 3.7.

Consider two classical cosmological models parameterized by
$\mathfrak{M}_{1}=(M_{01},\\!\mathcal{M}_{1},\\!\mathcal{G}_{1},\\!\Phi_{1})$
and
$\mathfrak{M}_{2}=(M_{02},\\!\mathcal{M}_{2},\\!\mathcal{G}_{2},\\!\Phi_{2})$
and let $V_{i}\stackrel{{\scriptstyle{\rm def.}}}{{=}}M_{0i}\sqrt{2\Phi_{i}}$
and $\mathcal{G}_{0i}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\frac{1}{M_{0i}^{2}}\mathcal{G}_{i}$ ($i=1,2$). We say that the
models are:

  * •

smoothly UV conjugate and write
$\mathfrak{M}_{1}\\!\equiv_{\mathrm{UV}}\mathfrak{M}_{2}$ if there exists a
smooth geodesic conjugation:

$f:(\mathcal{M},\mathcal{G}_{01})\rightarrow(\mathcal{M},\mathcal{G}_{02})~{}~{}.$

  * •

smoothly UV equivalent and write
$\mathfrak{M}_{1}\\!\sim_{\mathrm{UV}}\mathfrak{M}_{2}$ if there exists a
smooth geodesic equivalence:

$f:(\mathcal{M},\mathcal{G}_{01})\rightarrow(\mathcal{M},\mathcal{G}_{02})~{}~{}.$

In the situations above, $f$ is called respectively a smooth UV conjugation or
UV equivalence between $\mathfrak{M}_{1}$ and $\mathfrak{M}_{2}$.

#### IR conjugations and IR equivalences.

The results of Section 2 show that the IR limits of the two models can be
identified when the gradient flows of the scalar triples
$(\mathcal{M}_{1},\mathcal{G}_{1},V_{1})$ and
$(\mathcal{M}_{2},\mathcal{G}_{2},V_{2})$ are smoothly topologically
conjugate, i.e. related by a diffeomorphism
$f:\mathcal{M}_{1}\rightarrow\mathcal{M}_{2}$ (see Appendix A). This amounts
to the condition that the vector fields $\mathrm{grad}_{\mathcal{G}_{1}}V_{1}$
and $\mathrm{grad}_{\mathcal{G}_{2}}V_{2}$ be $f$-related, i.e.
$\mathrm{grad}_{\mathcal{G}_{2}}V_{2}=f_{\sharp}(\mathrm{grad}_{\mathcal{G}_{1}}V_{1})$,
where
$f_{\sharp}:\mathcal{X}(\mathcal{M}_{1})\rightarrow\mathcal{X}(\mathcal{M}_{2})$