Quintessential inflation:
A tale of emergent and broken symmetries

Dario Bettoni

Departamento de Física Fundamental and IUFFyM,
Universidad de Salamanca,
Plaza de la Merced S/N, E-37008 Salamanca, Spain

Javier Rubio

Centro de Astrofísica e Gravitação - CENTRA,
Departamento de Física, Instituto Superior Técnico - IST, Universidade de
Lisboa - UL,
Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal

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Abstract

Quintessential inflation provides a unified description of inflation and dark
energy in terms of a single scalar degree of freedom, the cosmon. We present
here a comprehensive overview of this appealing paradigm, highlighting its key
ingredients and keeping a reasonable and homogeneous level of details. After
summarizing the cosmological evolution in a simple canonical case, we discuss
how quintessential inflation can be embedded in a more general scalar-tensor
formulation and its relation to variable gravity scenarios. Particular
emphasis is placed on the role played by symmetries. In particular, we discuss
the evolution of the cosmon field in terms of ultraviolet and infrared fixed
points potentially appearing in quantum gravity formulations and leading to
the emergence of scale invariance in the early and late Universe. The second
part of the review is devoted to the exploration of the phenomenological
consequences of the paradigm. First, we discuss how direct couplings of the
cosmon field to matter may affect neutrinos masses and primordial structure
formation. Second, we describe how Ricci-mediated couplings to spectator
fields can trigger the spontaneous symmetry breaking of internal symmetries
such as, but not limited to, global U(1) or Z2 symmetries, and affect a large
variety of physical processes in the early Universe.

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###### Contents

  1. 1 Introduction
  2. 2 Cosmological evolution
    1. 2.1 Inflation
    2. 2.2 Kination
    3. 2.3 Heating
    4. 2.4 Hot big bang and dark energy
  3. 3 Field relativity
  4. 4 Symmetry principles
    1. 4.1 Ultraviolet regime
    2. 4.2 Crossover and infrared regimes
  5. 5 Phenomenology
    1. 5.1 Coupled quintessence
      1. 5.1.1 Growing neutrino masses
      2. 5.1.2 Primordial structure formation
    2. 5.2 Hubble-induced phase transitions
      1. 5.2.1 Spectator field dynamics
      2. 5.2.2 Applications
  6. 6 Discussion and outlook
  7. A Frame-invariant formulation of inflation
  8. B A workout example of crossover regime

##  1 Introduction

“Once upon a time the Universe sped up, out of emptiness. Then, there was
something all over the place. Lucky we were that it dawdled to hasten again,
so we can be here to observe”.

The current best-fitting picture of the statistical properties of the
Universe, from horizon scales ($15000$ Mpc) to typical galactic separations
($1$ Mpc), is the so-called $\Lambda$ Cold Dark Matter scenario
($\Lambda$CDM). This $6$-parameter model assumes the Universe to be composed
of photons, neutrinos, ordinary matter and a non-relativistic or cold dark
matter (CDM) component with gravitational interactions only. On top of that,
it presumes the existence of two accelerated expansion eras taking place in
the very early and the very late Universe: Inflation and Dark Energy.

  1. 1.

Inflation: This epoch provides a natural solution to the shortcomings of the
hot Big Bang (hBB) model, being able to generate, without finetunings, a flat,
homogeneous and isotropic Universe while removing unwanted relics such as
monopoles or other topological defects. More importantly, it provides a causal
mechanism for the generation of the primordial density perturbations seeding
structure formation. These fluctuations are unavoidably created out of the
quantum vacuum and stretched out to superhorizon scales, where they become
classical. In the simplest realizations, they are also adiabatic, Gaussian and
display a nearly scale-invariant spectrum at all observable scales. On top of
that, they remain frozen outside the horizon until they reenter it again long
after the end of inflation. When doing so, they generate the coherent set of
oscillations we observe in the Cosmic Microwave Background (CMB) and seed the
large-scale distribution of matter and galaxies in the Universe. Interestingly
enough, the results of three decades of observations are remarkably consistent
with the simplest incarnation of the inflationary paradigm: a canonical scalar
field slowly rolling in a featureless potential and eventually approaching a
global minimum, where oscillations are effectively damped by the quanta
production of the fields coupled to it.

Beyond temperature and density fluctuations, inflation predicts a so-far
unobserved background of tensor perturbations displaying also a quasi scale-
invariant spectrum. The main signature of this stochastic gravitational wave
background is a curl-like pattern in the CMB polarization. These $B$-modes are
the target of several ongoing and forthcoming initiatives, ranging from
ground-based experiments to stratospheric balloons and space missions [1]. Its
detection would constitute the first test of the quantum nature of spacetime,
while providing an indirect window to fundamental physics all the way up to
the unification scale.

  2. 2.

Dark energy: The discovery in the late 90’s of the current accelerated
expansion of the Universe revolutionized modern cosmology [2, 3]. Since then,
this result has been indirectly confirmed, among others, by CMB observations
[4, 5] and large scale structure surveys [6, 7, 8]. In the context of General
Relativity, the simplest mathematical explanation is a non-vanishing
cosmological constant $\Lambda$ leading to a time-independent energy density
of order $\rho_{\Lambda}\simeq(1\,{\rm meV})^{4}$. The inclusion of this
everlasting component in a Universe where all other matter species redshift
with expansion raises immediately two naturalness issues: i) why is the
cosmological constant so small as compared to all particle physics scales? and
ii) why did it start dominating precisely now? The lack of satisfactory
answers to these questions has motivated a plethora of alternative
explanations to the present accelerated expansion of the Universe, usually
coined under the generic name of dark energy models. Many of these proposals
involve a time-dependent (non-homogeneous) scalar field with an evolving
equation of state different from that of baryons, neutrinos, dark matter, or
radiation. This fifth or quintessential entity is generically designed to
mimic, with sufficient accuracy, a de Sitter-like behaviour at the present
cosmological epoch, in full analogy with the inflationary stage but at an
energy scale many orders of magnitude lower. Interestingly enough, some
quintessence scenarios involve appealing scaling solutions in which the dark
energy density remains comparable to the dominant radiation and matter
components for an extended period of time, eventually jumping out of this
behaviour to drive the late-time acceleration. Provided that the event
triggering this transition does not involve any additional tuning of times or
scales, this alleviates the aforementioned “why now” problem.

Since inflation and quintessence typically require a beyond the Standard Model
field to drive an almost de Sitter stage, it seems natural to seek for the
unification of these two stages into a common framework, potentially based on
an underlying symmetry principle. This is the main idea behind the so-called
Quintessential inflation paradigm, where the inflaton and dark energy fields
are identified as a single scalar component customarily dubbed cosmon [9].
This approach reduces the required number of degrees of freedom by one, with
the further advantage of eliminating the problem of setting the initial
conditions for the quintessence field, which are now determined by the
standard inflationary attractor. When written in terms of a canonical scalar
field, these scenarios involve a potential of runaway-type with two flat or
slow-roll regions connected by a steep transition spanning more than hundred
orders of magnitude. This large hierarchy of scales can be imposed by hand or
follow naturally from some underlying principle, as happen for instance in
quantum chromodynamics, where the existence of an ultraviolet (UV) fixed point
makes the infrared (IR) confinement scale exponentially smaller than the UV
cutoff of the theory. At the phenomenological level, the steep transition
leads generically to an intermediate cosmological epoch in which the energy
density of the Universe is dominated by the kinetic energy density of the
cosmon field. During this period, the global equation of state of the Universe
approaches unity and the total energy density decreases as the sixth power of
the scale factor. This peculiar scaling might allow to discriminate
quintessential inflation scenarios from standard oscillatory models. In fact,
as we will discuss in detail in this review, there are several interesting
phenomenological consequences associated to it. First, the fast dilution of
the cosmon energy density implies that any other form of matter will
eventually dominate the background evolution. This amplification effect has
advantages and drawbacks. On the one hand, it allows to recover the standard
hBB picture even if the cosmon field is only mildly depleted after inflation.
On the other hand, it induces a substantial growth of any stochastic
background of primordial gravitational waves at the corresponding scales,
leading to potential violations of Big Bang Nucleosynthesis (BBN) constraints
in scenarios involving very extended periods of kination.

Aim and spirit of this review: To make quantitative predictions within the
quintessential inflation paradigm one can follow two major strategies: perform
case-by-case studies involving specific choices of the cosmon potential or
follow a first-principles observational-oriented approach. The first point of
view has been widely explored in the literature [10, 11], from the early works
of Refs. [12, 13, 14, 15, 16, 17, 18] to the most recent developments [19, 20,
21, 22, 23, 24, 25, 26, 27]. In this review, we will follow the second
approach, trying to connect the behavior of the cosmon field to a minimum of
quantities and principles. In particular, we aim to provide an overview of
quintessential inflation for a wide audience, including both cosmologists and
particle physicists. To cope with this goal, we have focused on the generic
ingredients of the paradigm, keeping a reasonable and homogeneous level of
details throughout the whole manuscript and intentionally avoided the
exhaustive description of the myriad of models in the literature. Overall,
this approach brings clarity at the expense of disguising the technical
complexity of the issues we discuss, for which we refer the reader to Refs.
[10, 11]. For the sake of concreteness, we also exclude from this review a
number of theoretical topics, such as warm quintessential inflation [28],
curvaton-inspired scenarios [29], higher-dimensional settings [30] and non-
Riemannian approaches related to scale symmetry [31, 32].

This article is organized as follows. In Section 2, we overview the
cosmological history of a canonically normalized cosmon field minimally
coupled to gravity. After discussing the generalities of the inflationary
stage in Section 2.1, we concentrate in Section 2.2 on the unusual kinetic
dominated era, presenting the constraints on its duration imposed by BBN.
Special attention is then given to the unconventional heating stage, which we
describe in detail in Section 2.3, leaving for Section 2.4 the discussion of
the standard hBB evolution and the onset of scalar domination at late times.
The embedding of quintessential inflation in a more general variable gravity
framework is presented in Section 3, where we summarize the alternative
representations and equivalence classes in the literature. Section 4 discusses
the potential association of quintessential inflation with the emergence of
scale symmetry in the vicinity UV and IR fixed points, presenting the
conditions for the absence of fifth-forces and the temporal non-variation of
the Standard Model coupling constants. The last part of the review, Section 5,
presents an extensive discussion of the expected phenomenology in the presence
of interactions with a potential beyond the Standard Model sector. In
particular, Section 5.1 describes the coupled quintessence paradigm and its
applications in the context of growing neutrino quintessence and primordial
structure formation, while Section 5.2 discusses the impact of non-minimal
gravitational interactions and their effects on (re)heating, gravitational
waves’ production and baryogenesis. Finally, in Section 6, we comment on the
challenges and open avenues in this field of research.

Notation and conventions. We will work in natural units ($c=1$ and $\hbar=1$)
and use a mostly plus signature for the metric (-,+,+,+).

##  2 Cosmological evolution

In its simplest form, quintessential inflation can be formulated in terms of a
canonically-normalized scalar field minimally coupled to gravity. The
corresponding action takes the form

$S=\int
d^{4}x\sqrt{-g}\left[\frac{M_{P}^{2}}{2}R-\frac{1}{2}(\partial\phi)^{2}-V(\phi)\right]+S_{\rm
I}(g_{\mu\nu},\phi,\chi,A_{\mu},\psi)\,,$ (1)

with $M_{P}=(8\pi G)^{-1/2}=2.44\times 10^{18}$ GeV the reduced Planck mass,
$R$ the Ricci scalar and $(\partial\phi)^{2}$ a short hand notation for the
kinetic term $g^{\mu\nu}\partial_{\mu}\phi\,\partial_{\nu}\phi$, with
$g_{\mu\nu}$ the metric and $g={\rm det}\,(g_{\mu\nu})$ the metric
determinant. The cosmon potential $V(\phi)$ is assumed to be of a runaway form
and, as in any other dark energy-related models, it cannot innately explain
why the observed cosmological constant is so negligibly small, making
necessary the implementation of a symmetry principle or cancellation mechanism
in order to explain its value. For the purposes of this review, we will simply
assume that the cosmon potential vanishes asymptotically at large, positive
field values, $V(\phi\rightarrow\infty)\rightarrow 0$, albeit presenting in
Section 4 several arguments supporting this assumption. Finally, the action
$S_{\rm I}$ encodes the interactions of the cosmon field with scalar ($\chi$),
fermion ($\psi$) and vector ($A_{\mu}$) degrees of freedom in the Standard
Model sector and beyond.

On a flat Friedman-Lemaître-Robertson-Walker (FLRW) Universe with diagonal
metric $g_{\mu\nu}={\rm diag}(-1,a^{2}(t)\delta_{ij})$ and scale factor
$a(t)$, the Friedman and Klein-Gordon equations following from the variation
of the action (1) with respect to $g_{\mu\nu}$ and $\phi$ can be written as

$H^{2}=\frac{1}{3M_{P}^{2}}\sum_{i}\rho_{i}\,,\hskip
17.07164pt\dot{H}=-\frac{1}{2M_{P}^{2}}\sum_{i}(\rho_{i}+p_{i})\,,\hskip
17.07164pt\ddot{\phi}+3H\dot{\phi}-a^{-2}\nabla^{2}\phi+V_{,\phi}=Q(\phi,\chi,A_{\mu},\psi)\,,$
(2)

with the sum running over the cosmon field and all matter species in $S_{\rm
I}$ and the $Q$ term accounting for the potential interactions among these
components. Here $H\equiv\dot{a}/a$ is the Hubble expansion rate, the dots
denote derivatives with respect to the coordinate time $t$, $\rho_{i}$ and
$p_{i}$ stand for the energy densities and pressures of each component in $S$
and the subscript $\phi$ indicates a derivative with respect to the cosmon
field. Assuming the contribution of all matter fields in $S_{\rm I}$ to be or
become subdominant at very early and very late times, the dynamics of the
system of equations in (2) becomes controlled by the energy density and
pressure of the cosmon field at those epochs,

$\rho_{\phi}=\frac{1}{2}\dot{\phi}^{2}+\frac{1}{2a^{2}}|\nabla\phi|^{2}+V\,,\hskip
28.45274pt\textrm{and}\hskip
28.45274ptp_{\phi}=\frac{1}{2}\dot{\phi}^{2}-\frac{1}{6a^{2}}|\nabla\phi|^{2}-V\,,$
(3)

or equivalently by its energy density and the equation-of-state parameter

$w_{\phi}\equiv\frac{p_{\phi}}{\rho_{\phi}}\,.$ (4)

For a homogeneous scalar field, the latest quantity interpolates between a de
Sitter value $w_{\phi}\simeq-1$ for potential domination,
$V\gg\dot{\phi}^{2}$, and a stiff value $w_{\phi}\simeq+1$ for kinetic
domination, $\dot{\phi}^{2}\gg V$. Thus, as illustrated in Fig. 1, the
presence in the cosmon potential of two flat or slow-roll regions connected by
a steep transition is a priori sufficient to generate two periods of
exponential expansion at very different energy scales separated by a stage of
kinetic domination. Of course, the wealth of available observational data
ranging from CMB observations [4, 33] to BBN constraints [34] places several
requirements in this model building process [35, 36, 22, 37]. For instance, a
phenomenologically successful scenario should be able to generate the correct
amplitude and tilt for the primordial power spectrum of density fluctuations,
as well as to incorporate a stage of relativistic particle production giving
rise to the onset of the hBB era. On top of that, this ambitious program
requires the cosmon field to remain quiet and energetically subdominant for an
extended period of time after particle creation, such that the standard
radiation and matter dominated epochs have the sufficient duration.
Eventually, the cosmon field must also exit its quiescence and start behaving
as a quintessence component able to support the present accelerated expansion
of the Universe, as depicted in Figs. 2 and 3. In what follows, we will
provide a quick overview of the full cosmological evolution in this type of
scenarios, going through the various stages with a minimum amount of details
and referring to the extensive literature for more comprehensive analyses.

Figure 1: Schematic illustration of the quintessential inflation potential in
canonically-normalized scenarios. The presence of two flat or slow-roll
regions connected by a steep transition is a priori sufficient to generate two
periods of exponential expansion separated by a stage of kinetic domination.
Provided a sufficiently efficient creation of particles at the end of
inflation, this setting allows also for the standard hot big bang evolution
(radiation and matter domination). Figure 2: Cosmological evolution of the
background equation of state in quintessential inflation scenarios. The lack
of a potential minimum translates generically into the appearance of a kinetic
dominated era immediately after the end of inflation ($w\simeq-1$), during
which the global equation of state parameter is stiff ($w\simeq 1$). This
unusual expansion history extends till the onset of radiation ($w=1/3$) and
matter domination ($w=0$), where the cosmon field remains energetically
subdominant. Eventually, it starts dominating again, giving rise to the
current accelerated expansion of the Universe ($w\rightarrow-1$).  Figure 3:
Schematic description of the cosmological evolution of the energy densities in
quintessential inflation scenarios. The black, solid line corresponds to the
dominant component density. The dashed lines correspond to subdominant
radiation (yellow) and cosmon field (purple).

###  2.1 Inflation

The inflationary stage is assumed to start in a region of the Universe in
which the cosmon evolution is sufficiently slow and the energy density stored
in the potential term dominates the total energy budget, $V\gg
1/2\,\dot{\phi}^{2},\,1/2|\nabla\phi|^{2}$. As any other single-field
scenario, these models benefits from a late-time attractor solution that makes
the dynamics in phase space almost insensitive to the specific initial
conditions [38]. In order to sustain inflation for an extended period of time,
it is also necessary that the acceleration experienced by the cosmon is
sufficiently small as compared with its velocity per Hubble time,
$|\ddot{\phi}|\ll H|\dot{\phi}|$. Under these conditions, the first Friedman
and Klein-Gordon equations in (2) become respectively

$3M_{P}^{2}H^{2}\simeq V\,,\hskip 56.9055pt-V_{,\phi}\simeq 3H\dot{\phi}\,.$
(5)

This slow-roll regime can be supported by a variety of potential shapes around
the region of interest, including power-law forms [13, 14, 39], exponential
profiles [22] and plateau-like behaviours [40, 41]. The main requirement is
that the so-called slow-roll parameters

$\epsilon\equiv\frac{M_{P}^{2}}{2}\left(\frac{V_{,\phi}}{V}\right)^{2}\,,\hskip
56.9055pt\eta\equiv M^{2}_{P}\frac{V_{,\phi\phi}}{V}\,,$ (6)

encoding the local slope and curvature of the potential remain small for an
extended period of time. More precisely, the condition $\epsilon\ll 1$ is
required for the accelerated expansion to take place in the first place, while
the requirement $|\eta|\ll 1$ ensures that it lasts long enough to solve the
flatness and horizon problems. In the limit $\epsilon\approx 0$, the rate of
expansion is approximately constant and the scale factor grows exponentially
with time, $a\propto e^{Ht}$, leading to an almost de Sitter expansion. During
the inflationary stage, scalar (s) and tensor (t) perturbations are created
out of quantum vacuum fluctuations and spatially stretched to superhorizon
scales [42, 43, 44, 45, 46], where they become classical and inherit almost
scale-invariant spectra [42, 43, 44, 45, 46, 47, 48, 49, 50, 51]. At the
leading order in the slow-roll parameters, these spectra take the parametric
form

${\cal P}_{s}={\cal A}_{s}\left(\frac{k}{k_{\rm hc}}\right)^{n_{s}-1}\,,\hskip
28.45274pt{\cal P}_{t}={\cal A}_{t}\left(\frac{k}{k_{\rm
hc}}\right)^{n_{t}}\,,$ (7)

with the corresponding amplitudes and tilts,

$\displaystyle\mathcal{A}_{\,s}$ $\displaystyle=$
$\displaystyle\left.\frac{1}{2M_{P}^{2}\,\epsilon}\left(\frac{H}{2\pi}\right)^{2}\right|_{\rm
hc}\,,\hskip 56.9055ptn_{s}=1-6\epsilon_{\rm hc}+2\eta_{\rm hc}\,,$ (8)
$\displaystyle\mathcal{A}_{\,t}$ $\displaystyle=$
$\displaystyle\left.\frac{8}{M_{P}^{2}}\left(\frac{H}{2\pi}\right)^{2}\right|_{\rm
hc}\,,\hskip 71.13188ptn_{t}=-2\epsilon_{\rm hc}\,,$ (9)

depending only on the potential term and its derivatives through the relation
$H^{2}\simeq V/(3M_{P}^{2})$, and the subscript hc indicating the evaluation
of the corresponding quantities at the horizon exit of a given reference scale
$k^{-1}_{\rm hc}=(a_{\rm hc}H_{\rm hc})^{-1}$. Note that both ${\cal P}_{s}$
and ${\cal P}_{t}$ are time-independent as long as the associated fluctuations
are outside the horizon, meaning that their amplitudes at horizon re-entry,
and therefore the initial conditions for the subsequent causal evolution, are
the same as those at the first horizon crossing. The ratio of the spectral
amplitudes defines the so-called tensor-to-scalar ratio

$r\equiv\left.\frac{\mathcal{A}_{\,t}}{{\mathcal{A}}_{s}}\right|_{\rm
hc}=16\epsilon_{\rm hc}=-8n_{t}\,.$ (10)

The last equality can be understood as a consistency relation for single field
inflationary models. In particular, since we are working in Einstein gravity
and the cosmon field satisfies the null energy condition, the Hubble rate
decreases with time, leading to a red-tilted GW spectrum with negative tensor
spectral tilt.

The horizon crossing of a given momentum scale can be related to the number of
$e$-folds of inflation obtained by integrating the second equation in (5),
namely

$N\equiv\ln\frac{a_{\rm end}}{a}\simeq\frac{1}{M_{P}}\int_{\phi_{\rm
end}}^{\phi}\frac{d\phi}{\sqrt{2\epsilon}}\,,$ (11)

with $\phi_{\rm end}$ the value of the cosmon field following from the
violation of the slow-roll conditions, ${\rm max}[\epsilon(\phi_{\rm
end}),|\eta(\phi_{\rm end})|]=1$. In particular, for a scale $k^{-1}_{\rm
hc}=(a_{\rm hc}H_{\rm hc})^{-1}$ reentering the horizon at the present
cosmological epoch ($a_{\rm hc}H_{\rm hc}=a_{0}H_{0}$), we have

$N_{\rm hc}=-\ln\left(\frac{H_{\rm end}}{H_{\rm
hc}}\right)+\ln\left(\frac{a_{\rm end}H_{\rm end}}{a_{0}H_{0}}\right)\,.$ (12)

The first term in this expression depends implicitly on the specific cosmon
potential and therefore on the overall energy scale of inflation. The second
one accounts for the complete post-inflationary evolution and, in particular,
for any nonstandard expansion period, such as kinetic domination.

Recent observational constraints on the inflationary observables at a pivot
scale $k_{\rm hc}=0.05\,{\rm Mpc}^{-1}$ are provided by the Planck and
BICEP/Keck collaborations [4, 5],

$\displaystyle{\cal A}_{s}$ $\displaystyle=(2.099\pm 0.014)\times
10^{-9}\hskip 28.45274pt$ $\displaystyle\textrm{at 68\% CL}\,,$ (13)
$\displaystyle n_{s}$ $\displaystyle=0.9649\pm 0.0042\hskip 48.36967pt$
$\displaystyle\textrm{at 68\% CL}\,,$ (14) $\displaystyle r$
$\displaystyle<0.036\hskip 96.73936pt$ $\displaystyle\textrm{at 95\% CL}\,.$
(15)

Note that while the deviation of the scalar spectral tilt from unity is
statistically significant, the precise value of the tensor-to-scalar ratio
remains unknown, and with it the Hubble rate and the energy scale of inflation
at the pivot scale. In particular, combining Eqs. (8) and (10) with the
observed amplitude (13), we get

$\displaystyle H_{\rm hc}$ $\displaystyle=$ $\displaystyle 7.9\times
10^{12}\,{\rm GeV}\left(\frac{r}{0.001}\right)^{1/2}\lesssim 4.7\times
10^{13}\leavevmode\nobreak\ \rm GeV\,,$ (16) $\displaystyle V_{\rm hc}^{1/4}$
$\displaystyle=$ $\displaystyle 5.8\times 10^{15}\,{\rm
GeV}\left(\frac{r}{0.001}\right)^{1/4}\lesssim 1.4\times
10^{16}\leavevmode\nobreak\ \rm GeV\,,$ (17)

with the final bounds following directly from Eq. (15). The combination of
this upper limit with the consistency relation (10) implies also that the
tensor spectrum at CMB scales is almost exactly scale invariant, $-n_{t}\leq
0.0045\ll 1$, a condition that we will assume in what follows.

###  2.2 Kination

After the end of inflation, the cosmon field acquires a large kinetic energy
density at the expenses of its potential counterpart, which rapidly becomes
subdominant and irrelevant for the subsequent dynamics. This gives rise to a
kinetic dominated era dubbed deflation [12] or kination [52] where the global
equation of state of the Universe approaches the maximum value allowed by
causality, $w=1$. During this unusual cosmological epoch, the Ricci scalar is
negative [cf. Eq. (35)] and the first Friedmann and Klein-Gordon equations in
(2) take the approximate form

$3M_{P}^{2}H^{2}\simeq\frac{\dot{\phi}^{2}}{2}\,,\hskip
56.9055pt\ddot{\phi}+3H\dot{\phi}\simeq 0\,.$ (18)

This system of equations admits a solution

$\phi-\phi_{\rm kin}=\sqrt{6}\,M_{P}\,\Delta N_{\rm kin}\,,\hskip
56.9055ptH^{2}=H^{2}_{\rm kin}\left(\frac{a_{\rm kin}}{a}\right)^{6}\,,$ (19)

with

$\Delta N_{\rm kin}\equiv\log\left(\frac{a}{a_{\rm kin}}\right)$ (20)

the number of $e$-folds of kination. The quantitative differences among the
times, field values and energy scales in this expression (“kin”) and those at
the end of inflation (“end”) depend on the specific model under consideration.
However, for most practical purposes, they can be considered small and safely
neglected. This allows us to identify the end of inflation with the onset of
kination, i.e., $a_{\rm kin}=a_{\rm end}$, $\phi_{\rm kin}=\phi_{\rm end}$,
$H_{\rm kin}=H_{\rm end}$.

The scaling of the Hubble rate in (19) affects a plethora of particle physics
processes in the very early Universe, including, for instance, the generation
of the matter-antimatter asymmetry [52, 53, 54] or the abundance of dark
matter relics [55, 56, 57, 58, 59, 60, 61, 62, 63, 64]. Additionally, it
enhances any stochastic background of gravitational waves of primordial
origin, such as those generated by inflation [65, 66, 67, 39, 68, 69, 70, 71],
preheating [72, 73, 74, 75, 76, 77], cosmic strings [78, 79, 80, 81, 82, 83]
or phase transitions [84]; see Ref. [85] for a review. For the specific case
of inflationary perturbations, a global stiff equation of state translates
into an explicit breaking of scale invariance for modes reentering the horizon
during kinetic domination. To see this, let us consider the present density
parameter for GWs, customarily defined as the current GW energy-density
$\rho_{\rm GW}(\tau_{0},k)$ per logarithmic interval of comoving momentum $k$
normalized to the critical density $\rho_{\rm crit}=3M_{P}^{2}H^{2}$, [65, 66,
39, 68], i.e.

$\Omega_{\rm GW}^{(0)}(k)\equiv\dfrac{1}{\rho_{\text{crit}}}\dfrac{d\rho_{\rm
GW}(\tau_{0},k)}{d\ln
k}=\frac{1}{12}\left(\frac{k}{a_{0}H_{0}}\right)^{2}\Delta_{h}^{2}(\tau_{0},k)\,,$
(21)

with

$\Delta_{h}^{2}(\tau_{0},k)\equiv\frac{2k^{3}}{\pi^{2}}\left<|h_{k}(\tau_{0})|^{2}\right>$
(22)

the ensemble-averaged spectrum of tensor perturbations $h_{k}$ at present
time. These modes follow themselves from a Fourier decomposition

$h_{ij}(\tau,\mathbf{x})=\sum_{\lambda}\int\frac{d^{3}\textbf{k}}{(2\pi)^{3}}e^{i\mathbf{k}.\mathbf{x}}\epsilon_{ij}^{\lambda}(\mathbf{k})h^{\lambda}_{\mathbf{k}}(\tau)\,,$
(23)

with $h_{ij}$ a transverse-traceless metric perturbation
($\partial_{j}h_{ij}=h^{i}_{i}=0$) entering the linearized Einstein equations

$h_{ij}^{\prime\prime}+2\frac{a^{\prime}}{a}h_{ij}^{\prime}-\nabla^{2}h_{ij}=0\,.$
(24)

Here $\lambda=+,\times$ stand for the two physical graviton polarization
states and $\epsilon_{ij}^{\lambda}(\mathbf{k})$ denotes a basis of
polarization tensors, with
$\epsilon_{ij}^{\lambda}(\mathbf{k})=\epsilon_{ji}^{\lambda}(\mathbf{k})$,
$\epsilon_{ii}^{\lambda}(\mathbf{k})=0$,
$k_{i}\epsilon_{ij}^{\lambda}(\mathbf{k})=0$,
$\epsilon_{ij}^{\lambda}(\mathbf{k})=\epsilon_{ij}^{\lambda^{*}}(-\mathbf{k})$,
$\epsilon_{ij}^{\lambda}(\mathbf{k})\epsilon_{ij}^{\sigma^{*}}(\mathbf{k})=2\delta^{\lambda\sigma}$.
Assuming the tensor spectrum to be completely unpolarized and neglecting its
small tilt $n_{t}\ll 1$ [5], the present GW energy density parameter in a
Universe undergoing an epoch of kination between inflation and radiation
domination takes the form [70]

$\displaystyle\Omega_{\text{GW}}^{(0)}(k)\simeq\Omega_{\rm
GW}^{(0)}{\Big{|}}_{\rm no-kin}\times\left\\{\begin{array}[]{crl}1&\,,&\hskip
14.22636ptk\ll k_{\rm rad}\,,\vspace*{0.4cm}\\\
\frac{4}{\pi}\,\left(\frac{k}{k_{\rm rad}}\right)&\,,&\hskip 14.22636ptk\gg
k_{\rm rad}\,,\\\ \end{array}\right.$ (27)

with

$\Omega_{\rm GW}^{(0)}{\Big{|}}_{\rm no-kin}\simeq\leavevmode\nobreak\ 5\cdot
10^{-22}\left(\frac{H_{\rm kin}}{10^{11}\,{\rm GeV}}\right)^{2}$ (28)

the corresponding scale-invariant amplitude in the absence of kination and
$k_{\rm rad}$ the momentum associated with the horizon scale at the onset of
radiation domination.

The linear raising of (27) at high-momenta serves as an observational
signature of quintessential inflation. At the same time, it limits the
corresponding parameter space since the energy density of this relativistic
component affects directly the expansion rate of the Universe and with it the
abundance of light elements. In particular, BBN observations impose a limit on
the integrated GW contribution to the total energy density [86, 68], namely

$h^{2}\int_{k_{\text{BBN}}}^{k_{\rm kin}}\Omega^{(0)}_{\text{GW}}(k)d\ln k\leq
5.6\times 10^{-6}\,\Delta N_{\nu}\simeq 1.12\times 10^{-6}\,,$ (29)

with $h=0.678$ the reduced Hubble constant, $k_{\rm kin}$ and $k_{\rm BBN}$
the momenta associated to the horizon scale at the onset of kination and BBN,
and $\Delta N_{\nu}\lesssim 0.2$ an effective parameter encoding any extra
amount of radiation beyond the Standard Model content [87, 88]. This bound
applies to all sources of primordial gravitational waves and in particular to
the specific inflationary background in Eq. (27). Taking into account the
power law behaviour of this quantity, the integral in Eq. (29) can be
approximated by its upper end, getting a simpler expression
$h^{2}\Omega^{(0)}_{\text{GW}}(k\gg k_{\rm rad})\lesssim 1.12\times 10^{-6}$
that tends to exclude models with large inflationary scales and extended
periods of kination.

###  2.3 Heating

The measured abundance of light elements indicates that our Universe was
radiation dominated during BBN [34], thus setting a limit on the duration of
the kinetic dominated era discussed in the previous section [89],

$\Delta N_{\rm kin}\lesssim 27+\frac{1}{3}\ln\left(\frac{H_{\rm
kin}}{10^{11}{\rm GeV}}\right)\,.$ (30)

The end of this peculiar cosmological epoch will take place when the Universe
becomes dominated by the energy density of a relativistic component, created
potentially through the so-far irrelevant interaction terms in Eq. (1). Note,
however, that the generation of a thermal bath in quintessential inflation
cannot proceed as in the usual oscillatory models of inflation (for a review,
see e.g. Refs. [90, 91]). First, the runaway character of the potential
prevents the standard amplification of bosonic fields via parametric resonance
[92]. Second, the cosmon field cannot directly interact with the Standard
Model particles since that would induce the decay and destabilization of the
scalar potential tail. Hence, a successful heating mechanism must proceed
either through an indirect production of particles or involve a set of
sufficiently efficient couplings to an auxiliary matter sector quickly
shutting down the interaction and preventing the complete decay of the
inflaton condensate. In either case, the cosmon energy density must dilute
faster than the produced species or their decay products, in order to allow
for the onset of a radiation dominated epoch.

The background energy density during kination is initially dominated by the
cosmon field, which decays as $\rho_{\phi}\sim a^{-6}$. This rapid scaling
translates into a relative amplification of any radiation component
$\rho_{r}\sim a^{2}\rho_{\phi}$, making it eventually dominate the background
evolution [93]. A convenient way of quantifying the duration of this stage
while being agnostic on the particular model of heating is to introduce the
concept of heating efficiency [39]. The simplest definition of this quantity
involves the ratio between the energy density of the relativistic particles
produced at the onset of kination and the energy density of the cosmon field
at that time,

$\Theta\equiv\frac{\rho_{r}^{\rm kin}}{\rho_{\phi}^{\rm
kin}}=\left(\frac{a_{\rm kin}}{a_{\rm rad}}\right)^{2}\,.$ (31)

Albeit this definition assumes instant heating, the concept of heating
efficiency can be easily extended to non-instantaneous settings [39]. The
important point to keep in mind, however, is that the detailed description of
the process is not needed in practice. In other words, since the heating stage
in quintessential inflation requires almost compulsory an external sector with
new parameters, the introduction of a single one compressing that information
is a convenient choice. For example, taking into account the value of the
pivot scale $k_{\rm hc}=0.002\,{\rm Mpc}^{-1}=1.27\times 10^{-32}$ eV used in
the derivation of the combined Planck/BICEP2 results [94], the number of
relativistic degrees of freedom at matter-radiation equality $g_{\rm
mat}=3.36$ and the present value of the radiation temperature $T_{0}\simeq
2.73\,{\rm K}\simeq 2.35\times 10^{-4}$ eV, we can rewrite the required number
of e-folds of inflation in terms of the heating efficiency [39]

$N\simeq
63.3+\frac{1}{4}\ln\left(\frac{r}{0.001}\right)-\frac{1}{4}\ln\left(\frac{\Theta}{10^{-8}}\right)\,.$
(32)

Note that since comoving modes re-approach the Hubble scale faster during
kination than during radiation or matter domination, the required number of
$e$-folds needed to solve the flatness and horizon problems increases for
smaller heating efficiencies. Assuming a fiducial value $H_{\rm kin}\sim
10^{11}$ GeV and taking into account that the heating efficiency per degree of
freedom can vary between $\Theta\sim 10^{-19}$ and $\Theta\sim\mathcal{O}(1)$
depending on the specific heating mechanism [39] (see below), the minimal
number of $e$-folds in quintessential inflation scenarios is typically in the
range $60\lesssim N\lesssim 70$. Part of this window is, however, excluded by
Big Bang Nucleosynthesis constraints. In particular, the integrated bound in
Eq. (29) translates into a lower limit [39]

$\Theta\gtrsim 10^{-16}\left(\frac{H_{\rm kin}}{10^{11}{\rm
GeV}}\right)^{2}\,.$ (33)

Several heating mechanisms based on the exponential amplification of quantum
fluctuations via some kind of instability have been put forward over the years
[93, 12, 95, 96, 97, 98, 99, 29, 100, 101, 102, 103, 24, 27]. On general
grounds, they can be classified according to the presence or absence of direct
couplings to auxiliary matter sectors and to whether they produce particles
via adiabaticity violation or tachyonic instability. To illustrate this
explicitly, let us consider the following action for an exemplary and
energetically-subdominant scalar field $\chi$

$S=\int
d^{4}x\sqrt{-g}\left[-\frac{1}{2}(\partial\chi)^{2}-\frac{1}{2}(M^{2}(\phi)+\xi
R)\chi^{2}\right]\,.$ (34)

In this expression the inflaton field $\phi$ acts as a homogeneous background
field, $M(\phi)$ is a field-dependent mass potentially including a bare mass
contribution $m_{\chi}$, $\xi$ is a positive-definite non-minimal coupling and

$R=3H^{2}(1-3w)$ (35)

stands for the Ricci scalar in a FLRW background, written in terms the global
equation of state $w$. Note that we have intentionally restricted ourselves to
quadratic interactions in the scalar sector. This is indeed a good
approximation for the first stages of kination, where the auxiliary field
$\chi$ is close enough to the origin and particle production is most
efficient. However, as we will see in Section 5.2, higher-order operators can
play a crucial role at late times, especially in the presence of internal
symmetries.

The evolution of the field $Y$ is more easily described in the conformal
variables

$\chi=\chi_{*}\frac{a_{\rm kin}}{a}Y\,,\quad\quad dz=a_{\rm
kin}\chi_{*}d\tau\,,\quad\quad\vec{y}=a_{\rm
kin}\chi_{*}\vec{x}\,,\qquad\vec{\kappa}=\frac{\vec{k}}{a_{\rm
kin}\chi_{*}}\,,$ (36)

with $d\tau=dt/a(t)$ the conformal time and $\chi_{*}$ a potentially
convenient normalization. Introducing also the Fourier transform

$Y(\vec{y},z)=\int\frac{d^{3}\kappa}{(2\pi)^{3/2}}Y_{\vec{\kappa}}(z)e^{i\vec{\kappa}\cdot\vec{y}}\,,$
(37)

the equation of motion following from the variation of Eq. (34) with respect
to $\chi$ can be written as

$Y_{\vec{\kappa}}^{\prime\prime}+\omega^{2}_{\kappa}(z)Y_{\vec{\kappa}}=0\,,$
(38)

with

$\omega_{\kappa}^{2}(z)=\kappa^{2}+\frac{m_{\chi}^{2}+M^{2}_{\rm
eff}(\phi)}{\chi_{*}^{2}}\left(\frac{a}{a_{\rm
kin}}\right)^{2}-3\mathcal{H}^{2}(3w-1)\left(\xi-\frac{1}{6}\right)$ (39)

an effective frequency term, $\kappa=|\vec{\kappa}|$ and
$\mathcal{H}=a^{\prime}(z)/a(z)=a(z)H(z)$ the comoving Hubble rate.

The quantization of the scalar field $\chi$ proceeds according to the standard
procedure [104, 105]. The function $Y_{\vec{\kappa}}$ is promoted to a quantum
operator

$\hat{Y}_{\vec{\kappa}}=\hat{a}_{\vec{\kappa}}f_{\kappa}(z)+\hat{a}^{\dagger}_{-\vec{\kappa}}f^{*}_{\kappa}(z)$
(40)

satisfying the equal time commutation relations
$[\hat{Y}_{\vec{\kappa}}(z),\hat{\Pi}_{\vec{\kappa}^{\prime}}(z)]=i\delta(\vec{\kappa}+\vec{\kappa}^{\prime})$,
with $\hat{\Pi}_{\vec{\kappa}}=d\hat{Y}_{\vec{\kappa}}/dz$. Here,
$a_{\vec{\kappa}}$ and $a^{\dagger}_{\vec{\kappa}^{\prime}}$ are annihilation
and creation operators displaying equal-time commutation relations
$[a_{\vec{\kappa}},a^{\dagger}_{\vec{\kappa}^{\prime}}]=\delta({\vec{\kappa}}-{\vec{\kappa}^{\prime}})\,,[a_{\vec{\kappa}},a_{\vec{\kappa}^{\prime}}]=[a^{\dagger}_{\vec{\kappa}},a^{\dagger}_{\vec{\kappa}^{\prime}}]=0$
and $f_{\kappa}$ is a set of isotropic mode functions satisfying

$f_{\kappa}^{\prime\prime}+\omega^{2}_{\kappa}(z)f_{\kappa}=0\,,\hskip
56.9055pt2\,{\rm Im}(f^{\prime}_{\kappa}f^{*}_{\kappa})=1\,.$ (41)

Note that the choice of the basis $\\{f_{\kappa}(z),f^{*}_{\kappa}(z)\\}$ is
generically non unique, meaning that Eq. (40) should be understood as an
implicit definition of the operators $\hat{a}$ and $\hat{a}^{\dagger}$. For
constant positive square frequencies, a positive frequency mode
$f_{\kappa}\propto e^{-i\omega t}$ defines a unique vacuum state
$\hat{a}_{\vec{\kappa}}|0\rangle=0$. For non-constant non-positive square
frequencies, the notion of vacuum become ambiguous. In particular, particle
production can be achieved either because the adiabaticity condition
$|\omega_{\kappa}^{\prime}|/\omega_{\kappa}^{2}<1$ is violated or because the
frequency becomes tachyonic for a certain range of $\kappa$ modes,
$\omega_{\kappa}^{2}<0$ (i.e. $\omega_{\kappa}$ is imaginary). Given the
complications and extensive phenomenology of the latter scenario, we will
postpone its analysis to Section 5.2, focusing in what follows on the simplest
case $\omega^{2}_{\kappa}(z)>0$.

The amount of produced quanta for positive frequencies can be determined by
comparing the instantaneous adiabatic vacua at two different times. Since the
equation of motion (41) is second order, there must exist a linear Bogoliubov
transformation between two mode expansions $f_{\kappa}$ and $h_{\kappa}$
evaluated at the same time, namely

$f_{\kappa}(z)=\alpha_{\kappa}\,h_{\kappa}(z)+\beta_{\kappa}\,h^{*}_{\kappa}(z)\,,$
(42)

with the coefficients $\alpha_{\beta}$ and $\beta_{\kappa}$ satisfying the
normalization condition $|\alpha_{\kappa}|^{2}-|\beta_{\kappa}|^{2}=1$
following from the second expression in (41). As shown for instance in Ref.
[105], the occupation number of a mode $\kappa$ is given by square of the
second Bogoliubov coefficient $\beta_{\kappa}$,

$n_{\kappa}(z)=\left|\beta_{\kappa}(z)\right|^{2}\,,$ (43)

whose expression in terms of the mode expansion functions can be obtained by
solving the associated equation with proper “in” and “out” vacuum conditions.

Given the occupation numbers $n_{\kappa}(z)$ one can compute the total number
of $Y$ particles and the associated energy density

$n(z)=\frac{1}{(2\pi)^{3}}\int d^{3}\vec{\kappa}\,\,n_{\kappa}(z)\,,\hskip
42.67912pt\rho(z)=\frac{1}{(2\pi)^{3}}\int
d^{3}\vec{\kappa}\,\,\omega_{\kappa}(z)n_{\kappa}(z)\,.$ (44)

The resulting spectrum is not expected to be thermal. In fact, heating
mechanisms tend to amplify IR fluctuations below a given characteristic scale,
leading to a spectral shape initially peaked at low momenta. The subsequent
evolution of this overoccupied system proceeds typically through a rather slow
process of driven and free turbulence [106, 107, 103], for which thermal
techniques are not generically applicable. Still, it is sometimes convenient
to define an instantaneous temperature or energy scale

$T_{\rm r}(z)\equiv\left(\frac{30}{\pi^{2}g_{*}(z)}\,\rho_{\rm
r}(z)\right)^{1/4}\,,$ (45)

with $g_{*}(z)$ the effective number of relativistic degrees of freedom at a
given time $z$. The (re)heating temperature $T_{r}(z_{\rm rad})$ signaling the
onset of radiation domination is correspondingly defined by the condition
$\rho_{\phi}(z_{\rm rad})=\rho_{r}(z_{rad})$. Assuming exact kination ($w=1$,
$R=-6H^{2}$, $\rho_{\phi}\sim a^{-6}$), relativistic decay products
($\rho_{r}\sim a^{-4}$) and entropy conservation, this quantity can be related
to the maximum instantaneous temperature or energy scale at the onset of
kinetic domination, namely

$T_{r}(z_{\rm rad})=\left(\frac{g_{*,s}(z_{\rm kin})}{g_{*,s}(z_{\rm
rad})}\right)^{1/3}\Theta^{1/2}\,T_{r}(z_{\rm kin})\,,$ (46)

with $g_{*,s}(z_{\rm kin})$ and $g_{*,s}(z_{\rm rad})$ the entropic degrees of
freedom at the corresponding times. Accounting for all Standard Model degrees
of freedom, $g_{*}(z_{\rm kin})=g_{*,s}(z_{\rm kin})=g_{*,s}(z_{\rm
rad})=106.75$, we have

$\displaystyle T_{r}(z_{\rm rad})$ $\displaystyle\simeq$ $\displaystyle
2.7\times 10^{8}\leavevmode\nobreak\ {\rm
GeV}\left(\frac{\Theta}{10^{-8}}\right)^{\frac{3}{4}}\left(\frac{H_{\rm
kin}}{10^{11}\leavevmode\nobreak\ {\rm GeV}}\right)^{\frac{1}{2}}\,,$ (47)
$\displaystyle T_{r}(z_{\rm kin})$ $\displaystyle\simeq$ $\displaystyle
2.7\times 10^{12}\leavevmode\nobreak\ {\rm
GeV}\left(\frac{\Theta}{10^{-8}}\right)^{\frac{1}{4}}\left(\frac{H_{\rm
kin}}{10^{11}\leavevmode\nobreak\ {\rm GeV}}\right)^{\frac{1}{2}}\lesssim
6\times 10^{15}\leavevmode\nobreak\ {\rm GeV}\,\Theta^{1/4}\,,$ (48)

with the upper limit in the last expression following from (15) and the
restriction (16).

##### Specific cases:

Having presented the generalities of preheating via violations of the
adiabaticity condition $|\omega_{\kappa}^{\prime}|/\omega_{\kappa}^{2}<1$, let
us discuss some specific mechanisms connected with the choices of functions
and parameters in Eq. (39). As before, we assume exact kinetic domination
($w=1$, $R=-6H^{2}$).

  * •

Gravitational heating [93, 12, 95]

In this scenario, the production of particles is purely associated to the
expansion of the Universe, being essentially triggered by any rapid change in
the Ricci scalar $R$ [93, 108, 65]. This mechanism applies therefore to non-
conformally coupled scalar fields only ($\xi\neq 1/6$), since the evolution
equations of gauge bosons and chiral fermions in a conformally flat geometry
are invariant under Weyl rescalings.

For a light scalar field ($m_{\chi}\ll H_{\rm kin}$) in kinetic domination,
the effective frequency in Eq. (39) takes the form

$\omega_{\kappa}^{2}=\begin{cases}\kappa^{2}+\mathcal{H}^{2}\,\quad&{\rm
for}\quad\xi=0\,,\\\ \kappa^{2}+(1-6\xi)\mathcal{H}^{2}\,\quad&{\rm
for}\quad\xi<1/6\,,\end{cases}$ (49)

with the condition in the last expression ensuring that $\omega_{\kappa}^{2}$
is positive definite for all $\kappa$, such that the particle production is
only due to adiabaticity violation and the Bogoliubov formalism presented
above can be straightforwardly applied. The case with $\xi>1/6$ is analyzed in
detail in Section 5.2.

During a Hubble time in kinetic domination, the variation of the
(relativistic) energy density per spectator field is of the order
$\Delta\rho_{r}\sim\delta\times(\Delta H)^{4}\sim(\Delta a)^{-12}$, with
$\delta$ an efficiency parameter potentially depending on the non-minimal
coupling $\xi$ [12, 93] and typically much smaller that one. In consequence,
the production of particles is only efficient during the first stages of
kination [39], leading to a relatively small heating efficiency [39]

$\Theta=\frac{\delta\,{\cal N}}{1440\pi^{2}}\left(\frac{H_{\rm
kin}}{M_{P}}\right)^{2}=10^{-19}\,\delta\,{\cal N}\left(\frac{H_{\rm
kin}}{10^{11}{\rm GeV}}\right)^{2}\,,$ (50)

with ${\cal N}$ the number of light scalar degrees of freedom. Despite being
quite natural, this gravitational heating mechanism suffers from several
drawbacks. In particular, its low efficiency gives rise generically to a long-
lasting period of kination, which makes difficult to satisfy the GW bound (29)
unless a large number of species is introduced, ${\cal N}\gtrsim{\cal
O}(10^{2})\,\delta^{-1}$. However, this solution can lead to the generation of
sizeable isocurvature perturbations or secondary periods of inflation [97].
Another related possibility is to consider conformally-coupled particles with
heavy masses, $m_{\chi}>H$ eventually decaying into lighter ones to recover
the hBB era [109, 110, 111].

  * •

Matter preheating [96, 97, 98, 112, 39, 99, 64]

In this type of scenario the auxiliary field $\chi$ is directly coupled to the
cosmon via the effective mass function $M^{2}(\phi)$. A simple choice ensuring
decoupling at late times is given for instance by [112, 39, 64]

$M^{2}(\phi)=\begin{cases}g^{2}\phi^{2}&\quad\text{ for }\phi\leq 0\,,\\\
\tilde{M}^{2}&\quad\text{ for }\phi>0\,,\end{cases}$ (51)

with $g$ a dimensionless coupling constant and $\tilde{M}$ a constant mass
parameter. This unconventional behaviour appears naturally in quintessential
inflation models constructed on the basis of emergent scale symmetry in the
vicinity of UV and IR fixed points [112, 39], cf. Section 4. Alternative
choices smoothing the transition between negative and positive field values or
modifying its timing could be also considered without significantly changing
the discussion below, see e.g. Ref. [39].

Given the form (51), and omitting the subdominant gravitational contribution
in Eq. (49), the effective frequency (39) takes the form

$\omega_{\kappa}^{2}=\begin{cases}\kappa^{2}+g^{2}\left(\frac{a}{a_{\rm
kin}}\right)^{2}\frac{\phi^{2}(z)}{\chi_{*}^{2}}&\quad\text{ for }\phi\leq
0\,,\\\ \kappa^{2}+\left(\frac{a}{a_{\rm
kin}}\right)^{2}\frac{\tilde{M}^{2}}{\chi_{*}^{2}}&\quad\text{ for
}\phi>0\,.\end{cases}$ (52)

For small $\kappa$ values and $\phi\leq 0$, the condition for the violation of
the adiabaticity $|\omega_{\kappa}^{\prime}|/\omega_{\kappa}^{2}<1$ can be
safely approximated by $g|\phi^{\prime}_{0}|\gtrsim\phi^{2}$, with
$|\phi^{\prime}_{0}|$ the cosmon velocity at zero crossing. For sufficiently
large couplings, particle production takes place in a very narrow interval
$\Delta\phi\sim(|\phi^{\prime}_{0}|/g)^{1/2}$ around $\phi=0$, being the
process essentially instantaneous, $\Delta
z\sim\phi_{*}/|\phi^{\prime}_{0}|\sim(g|\phi^{\prime}_{0}|)^{-1/2}$, and
independent of the particle spin. The momentum of the created particles
follows directly from the uncertainty principle, $\Delta\kappa\sim(\Delta
z)^{-1}\sim(g|\phi^{\prime}_{0}|)^{1/2}$. Provided it to significantly exceed
the scale $\tilde{M}$ at positive cosmon values $\phi>0$,
($\Delta\kappa\gg\tilde{M}$), the produced particles are relativistic and can
trigger the onset of radiation domination. As shown in Ref. [64], the heating
efficiency in this particular example takes the form

$\Theta\simeq 2\times
10^{-8}\left(\frac{g}{0.02}\right)^{2}\left(\frac{10^{11}\,{\rm GeV}}{H_{\rm
kin}}\right)^{2}\left(\frac{|\phi^{\prime}_{0}|}{10^{-8}M_{P}^{2}}\right)^{2}\,.$
(53)

Alternative $Z_{2}$ symmetric-choices for the mass function $M^{2}(\phi)$
could be also considered [96, 97, 98, 64]. In this case, the energy transfer
happens through a combination of adiabaticity violation at zero crossing and a
rapid enhancement of the effective particle masses as the cosmon field evolves
towards large positive values. Since in this case the created particles become
rapidly non-relativistic, their energy density redshifts as matter,
$\rho_{\chi}\propto a^{-3}$, meaning that, in order to recover the hBB epoch,
they must eventually decay into light degrees of freedom. This decay can be
induced, for instance, by a direct Yukawa coupling $h\chi\bar{\psi}\psi$
between the spectator field $\chi$ and some fermionic species $\psi$. Since
the mass of $\chi$ grows with $\phi$ after zero crossing, so it does the
effective decay rate, $\Gamma\propto M(\phi)$, making this heating mechanism
very efficient.

###  2.4 Hot big bang and dark energy

Having discussed the production of matter fields coupled to the cosmon, let us
consider now the evolution of the system during radiation and matter
domination. The relevant equations can be conveniently written in terms of
cosmological observables, namely [113, 114, 35]

$\displaystyle\frac{w_{\phi}^{\prime}}{1-w_{\phi}}=-3(1+w_{\phi})+\lambda\sqrt{3(1+w_{\phi})\Omega_{\phi}}\,,$
(54)
$\displaystyle\,\Omega^{\prime}_{\phi}=-\Omega_{\phi}(1-\Omega_{\phi})\left(3(1+w_{\phi})-n\right)\,,$
(55)
$\displaystyle\lambda^{\prime}=-\sqrt{3(1+w_{\phi})\Omega_{\phi}}(\Gamma-1)\lambda^{2}\,,$
(56)

with the primes denoting derivatives with respect to the number of $e$-folds
and $n=4$ for radiation domination and $n=3$ for matter domination. The
parameters

$\lambda\equiv-M_{\rm P}\frac{V_{,\phi}}{V}\,,\hskip
56.9055pt\Gamma\equiv\frac{VV_{,\phi\phi}}{V_{,\phi}^{2}}\,,$ (57)

characterize, respectively, the local slope and curvature of the potential in
the corresponding regime. Although, there is not a first principle guide to
determine how the above system of equations will evolve during radiation and
matter domination or whether it can eventually support an accelerated
expansion era, a dynamical analysis is enough to single out a few options. In
fact, depending on the initial conditions and the specific shape of the cosmon
potential, we can distinguish three cases [115]:

  1. 1.

Thawing models: In this type of scenarios the Hubble friction freezes the
cosmon field evolution during radiation and early matter domination, being its
energy density completely dominated by the potential term. This corresponds to
the fixed point $w_{\phi}=-1$ in Eq. (54), which is, however, unstable for
$\lambda\neq 0$. In particular, as soon as the Hubble rate becomes smaller
than the mass of the scalar field, the cosmon will start rolling down the
potential, forcing the effective equation-of-state parameter to increase. The
growth of $w_{\phi}$ at late times can be analytically computed in some
specific limits. In particular, for approximately constant $\lambda$ and
$|1+w_{\phi}|\ll 1$, we have [116, 117, 118, 114, 119, 120, 121],

$1+w_{\phi}=\frac{\lambda^{2}}{3}F(\Omega_{\phi})\,,\hskip
56.9055pt\Omega_{\phi}=\frac{1}{1+\Delta_{0}\,a^{-3}}\,,$ (58)

with

$F(\Omega_{\phi})=\left[\frac{1}{\sqrt{\Omega_{\phi}}}-\Delta\tanh^{-1}\sqrt{\Omega_{\phi}}\right]^{2}$
(59)

a monotonically increasing function smoothly interpolating between $F(0)=0$ in
the deep radiation and matter dominated eras and $F(1)=1$ in the asymptotic
dark energy dominated regime and

$\Delta\equiv\frac{1-\Omega_{\phi}}{\Omega_{\phi}}\,,\hskip
56.9055pt\Delta_{0}\equiv\frac{1-\Omega_{\phi,0}}{\Omega_{\phi,0}}\,,$ (60)

with the subscript $0$ marking quantities evaluated today. The present value
of the dark-energy equation-of-state parameter follows directly from Eq. (58),

$\frac{1+w_{\phi,0}}{1+w_{\phi}(a)}=\frac{F(\Omega_{\phi,0})}{F(\Omega_{\phi})}.$
(61)

Note that these scenarios require a fine tuning of the local curvature and
amplitude of the potential in order to ensure that the scalar field starts
evolving at the correct time while reproducing the observed cosmological
constant energy density [114].

  2. 2.

Tracking freezing models: In this case the cosmon field is initially in slow-
roll motion and comes to a halt due the increasing flatness of the potential
after a certain critical value. This limit corresponds to a fixed point

$\Omega_{\phi}=\frac{3(1+w_{\phi})}{\lambda^{2}}\,,$ (62)

with constant $w_{\phi}$. Since this configuration is stable, the scalar field
trajectories will be dragged towards it for a large set of initial conditions,
meaning that we are dealing with a tracker solution. If then $\Gamma>1$ [122,
123], for which $\lambda$ approaches 0, the cosmon energy density will grow to
dominate the background evolution. Using the relation

$\frac{\Omega_{\phi}^{\prime}}{\Omega_{\phi}}=-2\frac{\lambda^{\prime}}{\lambda}$
(63)

following from Eq. (62) together with Eqs. (55) and (56) in the
$\Omega_{\phi}\ll 1$ limit,111Improved results accounting for $\Omega_{\phi}$
as a perturbation to the zeroth-order solution can be found in Ref. [124]. the
effective field equation of state along the tracker is given by

$w_{\phi}=-\frac{2(\Gamma-1)}{2\Gamma-1}\,.$ (64)

Note that, while the presence of the attractor alleviates the initial
condition problem in this class of models, a certain degree of tuning is still
required in order to recover the correct time at which the cosmon field starts
to dominate the energy budget.

  3. 3.

Scaling freezing models: In this scenario the cosmon field reaches as well a
tracking solution but this time with $\Omega_{\phi}={\rm const}$. A simple
inspection of Eqs. (54) and (55) reveals the existence of a stable fixed point
at [116, 125, 126, 117]

$\Omega_{\phi}=\frac{n}{\lambda}\,,\hskip 56.9055ptw_{\phi}=\frac{n-3}{3}\,,$
(65)

for constant $\lambda$. These conditions are verified for $\Gamma=1$ and
correspond to an exponential potential $V(\phi)=V_{0}e^{-\lambda\phi/M_{P}}$,
with $V_{0}$ a constant. Note that in this scenario the cosmon equation-of-
state parameter follows that of the background component during the radiation
and matter dominated eras, opening up a new line of attack to the coincidence
problem. The recovery of the late time acceleration of the Universe requires,
however, the introduction of an exit mechanism from this scaling regime. This
can be triggered by several mechanisms such as double exponential potentials
displaying two-late time attractors (one associated with the scaling solution
and one where the cosmon field dominates the energy density) [127, 117] or
growing neutrino masses. The first option does not provide, however, a
satisfactory solution to the “why now” problem since the amplitudes of the two
exponential potentials must be properly chosen to ensure that the transition
happens at the appropriate time. This is not the case of the growing neutrino
mass scenario, which we discuss in detail in Section 5.1.1.

As pointed out in the Introduction, observations reveal that the dark energy
equation-of-state parameter lies in a narrow strip around $w_{\phi}=-1$. For
an extensive review of these constraints and their impact of the above
scenarios, we refer the reader to Refs. [35, 128, 129] and references therein.

##  3 Field relativity

A canonical scalar field $\phi$ with a runaway potential is just one of the
many ways of generating an early and a late accelerated expansion of the
Universe using a single degree of freedom. A plethora of successful theories
can be constructed for instance within a scalar-tensor framework

$S[g_{\mu\nu},\varphi,f(\varphi),k(\varphi),V(\varphi)]=\int
d^{4}x\sqrt{-g}\,\left[\frac{f(\varphi)}{2}R-\frac{k(\varphi)}{2}\,g^{\mu\nu}(\nabla_{\mu}\varphi)(\nabla_{\nu}\varphi)-V(\varphi)\right]\,,$
(66)

with $f(\varphi)$, $k(\varphi)$ and $V(\varphi)$ arbitrary functions of a
scalar field $\varphi$. This parametrization leaves outside a large variety of
“safe” theories with second order derivatives of the scalar field, such as
Hordenski, beyond Horndeski and Degenerate Higher-Order Scalar-Tensor (DHOST)
theories [130], but it is still able to accommodate several modified gravity
scenarios such as $f(R)$ theories [131] and the low-energy limits of string
theories. On top of that, it can be easily extended to incorporate models
where the interactions with the matter sector play an important role, such as
the coupled quintessence settings discussed in Section 5.1 or those in Refs.
[23, 24, 27].

By specifying the theory defining functions in Eq. (66), one can construct an
equivalence class of theories related to each other by a frame transformation
[132, 133, 134], defined this as the combination of a Weyl rescaling of the
metric and a field reparametrization

$\tilde{g}_{\mu\nu}=\Omega^{2}(\varphi)\,g_{\mu\nu}\,,\hskip
56.9055pt\left(\frac{d\tilde{\varphi}}{d\varphi}\right)^{2}\equiv
K(\varphi)\,.$ (67)

In particular, taking into account the transformation of the Ricci scalar
under Weyl transformations,

$\tilde{R}=\Omega^{-2}\,R-6\,\Omega^{-3}\,g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\Omega\,,$
(68)

it is possible to rewrite the action (66) as

$S[\tilde{g}_{\mu\nu},\tilde{\varphi},\tilde{f}(\tilde{\varphi}),\tilde{k}(\tilde{\varphi}),\widetilde{V}(\tilde{\varphi})]=\int
d^{4}x\,\sqrt{-\tilde{g}}\,\left[\frac{\tilde{f}(\tilde{\varphi})}{2}{\tilde{R}}-\frac{\tilde{k}(\tilde{\varphi})}{2}\,\tilde{g}^{\mu\nu}(\nabla_{\mu}\tilde{\varphi})(\nabla_{\nu}\tilde{\varphi})-{\widetilde{V}(\tilde{\varphi})}\right]\,,$
(69)

with

$\displaystyle\tilde{f}(\tilde{\varphi})$ $\displaystyle=$
$\displaystyle\Omega^{-2}\,f\,,\hskip
14.22636pt\tilde{k}(\tilde{\varphi})=\frac{\Omega^{-2}}{K}\,\Big{(}k-6\,f\,\partial_{\varphi}\ln\Omega^{2}\>+6\,f_{,\varphi}\,\partial_{\varphi}\ln\Omega_{,\varphi}\Big{)}\,,\hskip
14.22636pt\tilde{V}(\tilde{\varphi})=\Omega^{-4}\,V\,.$

Note that even if the action $S$ is not invariant under Weyl rescalings and
field reparametrizations, its functional form remains invariant under these
combined transformations,

$S[\tilde{g}_{\mu\nu},\tilde{\varphi},\tilde{f}(\tilde{\varphi}),\tilde{k}(\tilde{\varphi}),\tilde{V}(\tilde{\varphi})]=S[g_{\mu\nu},\varphi,f(\varphi),k(\varphi),V(\varphi)]\,.$
(70)

In this sense, frame transformations are somehow analog to coordinates
transformations in classical mechanics. Different representations of a theory
may give rise to strikingly different pictures (such as expanding Universes
with constant particle masses or shrinking Universes with growing particles
masses), but they describe, however, the same physical reality.

In the so-called kinetial frame, the gravitational part of the action takes
the usual Einstein-Hilbert form, but the cosmon field displays a non-canonical
normalization,

$S=\int
d^{4}x\,\sqrt{-g}\,\left[\frac{M_{P}^{2}}{2}{R}-\frac{k(\varphi)}{2}\,g^{\mu\nu}(\nabla_{\mu}\varphi)(\nabla_{\nu}\varphi)-{V(\varphi)}\right]\,.$
(71)

Given the remaining freedom on the choice of the theory defining functions
$k(\varphi)$ and $V(\varphi)$, we can still distinguish some special cases:

  1. 1.

Exponential basis. This formulation makes use of a fixed exponential form for
the potential

$V(\varphi)=M^{4}\exp\left(-\frac{\varphi}{M}\right)\,,$ (72)

with $M$ a constant, leaving the detailed model information for the kinetial
$k(\varphi)$ [135, 136, 137, 138]. This choice is particularly useful for
model comparison since the relation between the scalar field value and the
potential energy density is universal. On top of that, all frame-covariant
inflationary observables in Appendix A become functions of $k(\varphi)$ only
[138].

  2. 2.

$\alpha$-attractor basis. This formulation assumes a singular function [41,
40, 99, 139],

$k(\varphi)=\frac{1}{1-\frac{1}{6\alpha}\frac{\varphi^{2}}{M^{2}_{P}}}\,,$
(73)

with $|\varphi|<\sqrt{6}M_{P}$. This choice can be motivated either by
invoking conformal symmetry or extended supergravity scenarios, where the
positive constant $\alpha$ describes the curvature of a hyperbolic Kähler
manifold. For describing quintessential inflation, the potential in this
setting is required to be asymmetric and non-singular, being the linear and
exponential choices commonly used in the literature. Remarkably, the
associated inflationary predictions are independent of the specific choice of
$V(\varphi)$, since this becomes exponentially stretched around the poles at
$|\varphi|=\sqrt{6}M_{P}$ when moving to canonically normalized variables. On
top of that, for moderate values of $\alpha$, and in contrast with the
canonical expectation (19), the field excursion during kination is ${\cal
O}(M_{P})$, as required by swamplamp distance conjecture [140] (in case you
take it seriously, of course). In spite of these positive aspects,
$\alpha$-attractor models cannot solve the cosmological constant problem
without invoking the inflationary landscape and anthropic considerations.

  3. 3.

Flattening basis. In this basis the theory defining functions are not
singular, but satisfy the relations

$k(\varphi)=\frac{1}{1+f(\varphi)}\,,\hskip
56.9055ptV(\varphi)=\frac{M^{4}f(\varphi)}{1+f(\varphi)}\,,$ (74)

with $M$ a constant with the dimension of mass and $f(\varphi)$ a
dimensionless runaway function of the cosmon field. This peculiar structure
appears naturally in Palatini scalar-tensor and modified gravity scenarios
involving Jordan-frame potentials proportional to the square of the non-
minimal coupling to gravity [141] or $R^{2}$ corrections [142]. Given
$f(\varphi)$, this setting generates an inflationary plateau for
$f(\varphi)\gg 1$, while reducing to the canonical form (1) for $f(\varphi)\ll
1$.

Note that, while being rather general, the above bases should be understood
just as convenient parametrizations for model building, not exhausting,
however, all quintessential scenarios proposed in the literature.

##  4 Symmetry principles

The naturalness of quintessential inflation is intimately related to that of
the theory defining functions driving the early- and late-time acceleration of
the Universe. In this regard, variable gravity scenarios involving the
realization of scale or dilatation symmetry constitute an interesting arena
for model building [136, 112, 39]. In this type of setting, inflation and dark
energy take place in the vicinity of UV and IR fixed points extending in
physical time to the infinite past and future. At these fixed points, any
information about intrinsic mass scales is completely lost and scale
invariance is manifestly realized even if broken in the underlying quantum
field theory. The anomalous breaking of dilatations plays, however, a key role
in the crossover among fixed points, which, depending on the model specifics,
can take place in one or several stages and involve physical scales separated
by many orders of magnitude, similarly to what happens in quantum
chromodynamics, where the confinement scale is naturally much smaller than a
given unification scale.

The relation of inflation and dark energy to scale invariance is more
naturally discussed in terms of a quantum effective action including all
fluctuation effects, since these are indeed needed for the emergence of fixed
points. In the lack of a first principles’ computation for the Standard Model
non-minimally coupled to gravity, several authors have adopted the following
parametrization for the graviscalar sector of the theory [136, 112, 19, 39],

$S_{c}=\int
d^{4}x\sqrt{-g}\,\left[\frac{\varphi^{2}}{2}R-\frac{1}{2}\left(B(\varphi/\mu)-6\right)g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi-\mu^{2}\varphi^{2}\right]\,.$
(75)

Here $\varphi$ stands for the cosmon field, $B(\varphi/\mu)$ is a positive
dimensionless function of the dimensionless ratio $\varphi/\mu$ and $\mu$ is a
mass parameter that can be associated with the dilatation anomaly. The absence
of a quartic potential term in this expression implies the asymptotic
vanishing of the observable cosmological constant when the theory is
transformed to the Einstein frame [116]. This can be motivated by functional
renormalization studies showing that the cosmon potential cannot increase as
$\varphi^{4}$ when $\varphi\to\infty$, since this would lead to a singularity
in the flow that tends to be avoided by strong gravity-induced renormalization
effects [143, 144]. Formally, this corresponds to requiring conformal
invariance ($B\to 0$) rather than scale symmetry ($B\to\textrm{constant}$) at
the infrared fixed point $\mu\to 0$ [145, 146]. In this conformal limit, the
cosmon is no longer a propagating degree of freedom, as can be easily seen by
transforming Eq. (75) to the Einstein frame.

The flow equation for $B(\varphi/\mu)$ encodes the relative change of the
spontaneous mass scales proportional to $\varphi$ as compared to the intrinsic
mass scales proportional to $\mu$. If all parameters with dimension of mass
are either proportional to $\mu$ or $\varphi$, the flow of $\varphi$ at fixed
$\mu$ can be related to the flow of $\mu$ at fixed $\varphi$, and viceversa,

$\varphi\frac{\partial B}{\partial\varphi}\Big{|}_{\mu}+\mu\frac{\partial
B}{\partial\mu}\Big{|}_{\varphi}=0\,.$ (76)

The UV fixed point $\mu\to\infty$ corresponds therefore to the $\varphi\to 0$
limit, while the IR fixed $\mu\to 0$ occurs for $\varphi\to\infty$. Due to the
divergence of the mass term in Eq. (75) as $\mu\to\infty$, the existence of an
UV fixed point requires the cosmon wave function renormalization to display an
anomalous dimension. The minimal realization of this condition follows from a
simple behaviour [112, 39],

$\mu\partial_{\mu}\ln B=\sigma\,,\hskip 28.45274pt\longrightarrow\hskip
28.45274ptB=\left(\frac{m}{\varphi}\right)^{\sigma}\,,$ (77)

with $\sigma>1$ and $m\equiv\mu\,\exp\,(c_{t})$ a crossover scale following
from the dimensional transmutation of an integration constant $c_{t}$
determining the particular trajectory on the flow. For this specific choice,
the effective action (75) at the UV point is invariant under the simultaneous
rescaling $g_{\mu\nu}\,\to\,\lambda^{2}g_{\mu\nu}$ and
$\varphi\,\to\,\alpha^{-\frac{2}{2-\sigma}}\varphi$, with $\lambda$ a
constant.

Albeit generically subdominant for the cosmological solutions considered here,
the graviscalar action (75) can be complemented with higher-order curvature
invariants such as Gauss-Bonnet or $R^{2}$ terms with coefficients depending
on the dimensionless ratio $\varphi/\mu$. These are expected to play an
important role at the UV fixed point [147], where the coefficient of the Ricci
scalar identically vanishes. On top of that, one should introduce a Standard
Model sector describing the usual matter and radiation components. The main
viability requirements are that i) the different renormalized dimensionless
couplings (gauge, Yukawa and Higgs self-interactions) become sufficiently
close to their IR fixed-point values before big bang nucleosynthesis and that
ii) all mass parameters in the theory (including the confinement and Fermi
scales) become proportional to $\varphi$ by that time [113], such that the
strict bounds on fifth forces [148] and the temporal variation of fundamental
constants [149] are trivially satisfied [113, 150, 151, 152, 153, 154]. This
restriction constitutes a key difference with Brans-Dicke scenarios and
similar scalar-tensor variants such as induced gravity theories [155]. It does
not apply, however, to beyond the Standard Model sectors, where the crossover
regime can take place even at the present cosmological epoch.

###  4.1 Ultraviolet regime

The field equations for the graviscalar action (75) admit an exact de Sitter
solution able to support an accelerated expansion era at very early times
[112],

$R_{\mu\nu}=\mu^{2}g_{\mu\nu}\,,\hskip 56.9055pt\varphi=0\,.$ (78)

However, the anomalous violation of dilatations in the vicinity of this UV
fixed point makes this situation unstable, forcing the cosmon field to evolve
slowly with time while providing a graceful exit from the inflationary state.
Once a specific form of $B$ is assumed, any subsequent analysis can be
performed by using the “classical field equations” obtained by varying the
quantum effective action (75). In particular, the inflationary observables
following from this expression can be straightforwardly computed using the
frame-covariant formalism in Appendix A, with $f(\varphi)=\varphi^{2}$,
$k(\varphi)\equiv B(\varphi/\mu)-6$ and $V(\varphi)=\mu^{2}\varphi^{2}$. The
spectral tilt and the tensor-to-scalar ration can be then written in a rather
suggestive way,

$1-n_{s}=\frac{r}{8}\left(1+\frac{1}{2}\,\beta(\mu)\right)\,,\hskip
56.9055ptr=\frac{32}{B(\mu)}\,,$ (79)

with

$\beta(\mu)\equiv\mu\,\partial_{\mu}\ln B=\frac{1}{2}\frac{\partial
B}{\partial N}$ (80)

the effective $\beta$ function for $B$, written in terms of the scale $\mu$
and the number of $e$-folds of inflation,

$N(\varphi)=\frac{1}{2}\int_{\varphi}^{\rm\varphi_{\rm
end}}\frac{d\varphi^{\prime}}{\varphi^{\prime}}\,B(\varphi^{\prime})\,,$ (81)

with $\varphi_{\text{end}}$ the value of $\varphi$ at the end of inflation.
The emergent dilatation symmetry at the UV fixed point is responsible for both
the flatness and amplitude of the primordial spectrum of density fluctuations.
A explicit illustration of this can be found in Refs. [112, 39], where the
fixed-point structure (77) was shown to generate a spectral amplitude ${\cal
A}_{s}\propto\mu^{2}/m^{2}$, leading naturally to tiny scalar fluctuations for
moderate values of the integration constant $c_{t}$ previously defined. The
tensor-to-scalar ratio following from this flow equation turns out to exceed,
however, the current observational constraints [5]. Smaller values of this
quantity can be easily obtained if the fixed point happens at some large but
finite $B_{*}$ value [135] or if the behavior of $B$ in the small field regime
is slightly modified. For instance, a suitable generalization

$\mu\,\partial_{\mu}\ln B=\sigma\,B^{1-\frac{1}{\alpha}}\,,\hskip
14.22636pt\longrightarrow\hskip 14.22636ptB(\bar{N})=B_{\rm
end}\frac{\bar{N}^{\alpha}}{c}\,,\hskip
28.45274ptB(\varphi)=\left[\frac{1-\alpha}{\alpha}\,\log\left(\frac{\tilde{m}}{\varphi}\right)^{\sigma}\right]^{\frac{\alpha}{1-\alpha}}\,,$
(82)

with $\alpha,\,\sigma\geq 1$ translate into a spectral tilt and a tensor-to-
scalar ratio

$1-n_{s}=\frac{\alpha}{\bar{N}}+\frac{r}{8}\,,\hskip
56.9055ptr=\frac{16\,c}{\bar{N}^{\alpha}}\,,$ (83)

with

$\bar{N}=N+c^{1/\alpha}\,,\hskip 42.67912ptc\equiv B_{\rm
end}\left(\frac{\alpha}{2\sigma}\right)^{\alpha}\,,\hskip
42.67912pt\tilde{m}\equiv\mu\,\exp\,(\tilde{c}_{t})\,,$ (84)

$B_{\rm end}\equiv B(N=0)=2$ the value of $B$ at the end of inflation, and
$\tilde{c}_{t}$ an integration constant determining the particular trajectory
on the flow. As shown in Fig. 4, values of these quantities compatible the
latest Planck/BICEP2 data are obtained for moderate $\alpha$ and $\sigma$
values.

Figure 4: Comparison between the inflationary predictions (83) for $N=60$ and
the Planck/BICEP2 data at $1\sigma$ and $2\sigma$ C.L.

###  4.2 Crossover and infrared regimes

Scale invariance is substantially violated when the coefficient of the kinetic
term in Eq. (75) changes from positive to negative values. During this
crossover regime rapid variations of the dimensionless couplings in the theory
are expected to occur as they transition from their UV to IR fixed-point
values. Among other effects, this can play a major role on the onset of the
standard hBB era. The minimal heating scenario involves the quantum effective
potential for Higgs field $H$,

$V_{H}=\frac{1}{2}\lambda(\varphi/\mu)\left(H^{\dagger}H-\alpha(\varphi/\mu)\varphi^{2}\right)^{2}\,,$
(85)

with the dimensionless couplings $\lambda(\varphi/\mu)$ and
$\alpha(\varphi/\mu)$ strongly violating the adibaticity condition after the
end of inflation and setting to their IR fixed-point values before big bang
nucleosynthesis. When written in the Einstein-frame this behaviour leads to
interactions of the runaway form (51), allowing to apply the techniques
presented in Section 2.3. Alternatively, one could consider a heating stage
involving the production of new particles, which should eventually decay into
the Standard Model content.

After heating and entropy production, the cosmon field starts to approach the
IR fixed point, with a characteristic time scale $\mu$. In the scaling frame
(75), the Universe shrinks, while both temperature and particle masses
increase with time and the size of atoms decrease [156]. For
$\mu^{-1}=10^{10}\,\textrm{yr}=1.2\times 10^{60}M_{P}^{-1}$, the current
expectation value of the cosmon field equals the observed reduced Planck mass,
$\varphi(t_{0})=2.48\times 10^{18}$ GeV, such that $\varphi/\mu=1.2\times
10^{60}$ [136, 112, 39]. This picture should be understood as equivalent to
the standard hBB picture of an expanding and cooling Universe, since the
observable dimensionless ratios still coincide with those in the usual
scenario. This can be seen explicitly by performing a Weyl redefinition of the
metric bringing the action (75) to a canonical form with constant Planck scale
and fixed particle masses [116]. This equivalence can be extended to other
relevant quantities for cosmology, such as the correlation functions of
primordial density fluctuations [157, 158].

Realistic scenarios involve, customarily, a constant or slowly varying form of
$B$ following from instance from an IR flow equation with anomalous dimension
$\sigma_{\rm IR}$ and small corrections proportional to $B$ [136, 116, 125,
112, 39],

$\mu\,\partial_{\mu}\ln B=\sigma_{\rm IR}+\kappa\,B\,.$ (86)

The dominant scaling violation at this time is associated with the potential
term $\mu^{2}\varphi^{2}$ in Eq. (75). In particular, the dimensionless ratio
of this quantity to the fourth power of the effective Planck mass $\varphi$
decreases with time, reaching very small values at present times. Note also
that the enhanced conformal symmetry for $B=\mu=0$ implies that the
$\beta$-function for $B$ must vanish at $B=0$. In this limit, $\sigma_{\rm
IR}=0$.

Approximate tracking solutions in which the dark energy evolution follows the
dominant radiation or matter components have been extensively studied in the
literature [116, 136, 125]. These scenarios include generically an additional
beyond the Standard Model sector with a late crossover regime able to stop the
tracking solution at the present cosmological era. A rather natural setup
appears in seesaw [159, 160, 161] and cascade [162, 163, 164] scenarios in
which the mass of heavy right-handed neutrinos or scalar triplets decreases
with the cosmon field $\varphi$, leading to an increasing neutrino-to-electron
mass ratio [165, 166]. In the Einstein frame with fixed electron mass, this
reduces to the growing neutrino mass mechanism described in Section (5.1.1).

##  5 Phenomenology

One of the most crucial aspects in the era of precision cosmology is to gain
knowledge on the fundamental mechanisms behind cosmological acceleration while
identifying new signatures potentially detectable with present or near future
observational campaigns.

In this second part of the review we will discuss the phenomenology associated
with quintessential inflation scenarios and the interplay between the cosmon
and other fields that might be present in the very early Universe.

###  5.1 Coupled quintessence

In this type of scenarios the cosmon field is allowed to exchange energy with
a given matter component, such as baryons, radiation, cold dark matter or
neutrinos. The interaction with baryons and photons is, however, strongly
restricted by local gravity measurements [167, 168, 169] and the non-variation
of fundamental constants [170],222Ways of avoiding these constraints include
symmetry principles as scale symmetry [151, 152, 153] and screening mechanisms
(see e.g. [171, 172, 173] and references therein). leaving the coupling to
dark matter [125, 174, 175, 176, 177, 178, 179, 180, 181, 182] and/or
neutrinos [183, 184, 165, 166, 185, 186, 187, 188, 189, 190, 191, 192, 193] as
the main viable possibility. This coupling can be formulated either at the
level of the equations of motion or at the level of the action.

Heuristic and phenomenological modifications of the equations of motion have
been extensively used in the literature, with typical interactions kernels
involving local and non-local functions of the relevant energy densities [178,
194, 195, 196, 197, 198], their derivatives [199, 200, 201, 198] and/or a
given temporal scale, such as the Hubble parameter [178, 194, 195, 196] or a
constant decay rate [196, 202]. These phenomenological approaches are
mathematically simple but in many cases physically ill-defined. On the one
hand, they usually lack a covariant form for the energy-momentum transfer able
to unambiguously describe realistic cosmologies involving perturbations. On
the other hand, there is no clear reason for the effective cosmon-matter
coupling to be dictated by global properties of the Universe, rather than by
purely local ones.

The above problems are avoided in field theory formulations. In particular,
cosmon-matter interactions appear naturally in variable and modified gravity
scenarios when written in the Einstein frame [203]. To show this, one can
consider an initial Jordan-frame action of the form (66) complemented with an
exemplary matter action for an uncoupled fermionic field $\psi$ with constant
mass $\bar{m}_{\psi}$, 333The fermionic nature of the matter field is chosen
purely for illustration purposes. For multifield generalizations and disformal
couplings to matter, see, for instance, Refs. [204, 205, 206].

$S_{\psi}=\int
d^{4}x\sqrt{-g}\left[\frac{i}{2}\left(\bar{\psi}\gamma^{\mu}\nabla_{\mu}\psi-\bar{\psi}\overleftarrow{\nabla}_{\mu}\gamma^{\mu}\psi\right)-\bar{m}_{\psi}\bar{\psi}\psi\right]\,.$
(87)

Transforming both sectors to the Einstein frame and performing a Weyl
rescaling of the spinor $\psi\to\Omega^{3/2}(\phi)\,\psi$ with

$\Omega^{2}=\frac{f(\phi)}{M_{P}^{2}}\,,\hskip
56.9055pt\left(\frac{d\phi}{d\varphi}\right)^{2}=\frac{3}{2}\left(\frac{\partial\ln
f}{\partial\varphi}\right)^{2}+\frac{k}{f}\,,$ (88)

we get

$S=\int{\rm d}^{4}x\sqrt{-g}\left[\frac{M^{2}_{\rm
P}}{2}R-\frac{1}{2}(\partial\phi)^{2}-V(\phi)+\frac{i}{2}\left(\bar{\psi}\gamma^{\mu}\nabla_{\mu}\psi-\bar{\psi}\overleftarrow{\nabla}_{\mu}\gamma^{\mu}\psi\right)-m_{\psi}(\phi)\bar{\psi}\psi\right]\,\,.$
(89)

In this representation, the matter field $\psi$ interacts with the cosmon via
the $\phi$-dependent mass term

$m_{\psi}(\phi)\equiv\frac{\bar{m}_{\psi}\,M_{P}}{\sqrt{f(\phi)}}\,.$ (90)

The variation of the action (89) with respect to the Einstein-frame metric
provides the usual Einstein’s equations. The conservation of stress-energy in
the presence of the aforementioned energy-momentum exchange can be then
written as

$\nabla_{\nu}T_{\phi}^{\mu\nu}=+\frac{\beta(\phi)}{M_{P}}T_{\psi}\partial^{\mu}\phi\,,\hskip
56.9055pt\nabla_{\nu}T_{\psi}^{\mu\nu}=-\frac{\beta(\phi)}{M_{P}}T_{\psi}\partial^{\mu}\phi\,,$
(91)

with

$\beta(\phi)\equiv-M_{P}\frac{\partial\ln m_{\psi}(\phi)}{\partial\phi}\,,$
(92)

and $T_{\psi}=T_{\psi}^{\mu\nu}g_{\mu\nu}$ the trace of the energy momentum
tensor for the $\psi$-field, which vanishes identically for relativistic
particles. As anticipated, these expressions are fully covariant and
determined by microphysics. The effective coupling $\beta$ modifies
equivalently the effective Klein-Gordon equation for the cosmon field

$\nabla_{\mu}\nabla^{\mu}\phi=V^{\prime}(\phi)+\beta(\phi)T_{\psi}\,,$ (93)

and the evolution of neutrinos on a classical path [188]

$\frac{\mathrm{d}u^{\mu}}{\mathrm{d}\tau}+\Gamma^{\mu}_{\rho\sigma}u^{\rho}u^{\sigma}=\beta(\phi)\,u^{\lambda}\partial_{\lambda}\phi\,u^{\mu}+\beta(\phi)\,\partial^{\mu}\phi\,,$
(94)

with $u^{\mu}$ the four-velocity, $\Gamma_{\mu\nu}$ the Christoffel symbols
associated to the metric $g_{\mu\nu}$ and $\tau$ the proper time. The right-
hand side of Eq. (94) plays the role of an additional fifth-force and consists
of a velocity-dependent piece
$\beta(\phi)\,u^{\lambda}\partial_{\lambda}\phi\,u^{\mu}$ ensuring momentum
conservation for particles moving in a varying cosmon field and a velocity-
independent piece $\beta(\phi)\,\partial^{\mu}\phi$ becoming dominant for
small velocities.

In the Einstein frame, a given model within the paradigm is specified by a
choice of the effective coupling $\beta(\phi)$. A simple possibility is to
consider an effective coupling $\beta(\phi)=-gM_{P}/(m_{0}+g\phi)$ following
from a renormalizable Yukawa interaction
$m_{\psi}(\phi)\bar{\psi}\psi=m_{0}\bar{\psi}\psi+g\phi\bar{\psi}\psi$, with
$m_{0}$ a mass parameter and $g$ a dimensionless coupling. Another option is
to consider a setup involving a constant $\beta$ coupling. This describes
dilatonic like interactions
$m_{\psi}(\phi)\bar{\psi}\psi=m_{0}\exp\left(-\beta\phi/M_{P}\right)\bar{\psi}\psi$
like those appearing in induced and modified gravity scenarios [155, 207,
208]. At the phenomenological level, these choices have been applied in a
variety of contexts, ranging from growing neutrino scenarios aiming to solve
the coincidence problem [183, 184, 165, 166, 185, 186, 187, 188, 189, 190,
191, 192, 193] to strongly coupled cosmologies [174, 209, 210, 211, 212, 213,
214, 215] inducing the formation of primordial black holes [216, 217, 218,
219] or similar compact objects [220] during radiation domination. In the
following sections, we describe these applications in detail.

####  5.1.1 Growing neutrino masses

Every quintessential inflation model must predict the correct amount of dark
energy, preferably without an excessive finetuning of the initial conditions.
In this regard, tracking or attractor solutions as those presented in Section
2.4 are particularly interesting, albeit they require an additional exit
mechanism able to initiate the current accelerated expansion of the Universe.
Growing neutrino quintessence explains the transition from a scaling solution
to a dark energy dominated era by identifying the fermion field $\psi$ in Eq.
(89) with the neutrino [183, 184, 165, 166, 185, 186, 187, 188, 189, 190, 191,
192, 193]. The special role of this particle can be motivated by a cross-over
regime in a beyond the Standard Model sector, manifesting itself in the
neutrino masses through a seesaw or cascade mechanism [112], cf. Section 4.

In growing neutrino quintessence, the “why now” problem becomes related to a
physical trigger event: the recent change of the effective neutrino equation
of state. In particular, when neutrinos become non-relativistic at redshift
$z\approx 5$ [185], a negative choice of $\beta$ induces a fixed point in the
Klein-Gordon equation (93) for the cosmon field,
$-\beta(\phi)\rho_{(\psi)}\simeq-V^{\prime}(\phi)$, effectively stopping its
evolution in the runaway potential $V(\phi)$ and pushing it away from the
scaling solution. For neutrino masses in the eV (or sub-eV) range, this leads
to an almost de Sitter equation of state [166, 165]

$1+w(t_{0})=\frac{m_{\psi}(t_{0})}{12\,{\rm eV}}\,,$ (95)

and an energy density roughly matching the observed value
$\rho_{\phi}(t_{0})^{1/4}=2\cdot 10^{-3}$ eV,

$\rho_{\phi}^{1/4}(t_{0})=1.27\left(\frac{\gamma_{\psi}\,m_{\psi}(t_{0})}{{\rm
eV}}\right)^{1/4}10^{-3}\,{\rm eV}\,,$ (96)

with $\gamma_{\psi}=\tilde{\gamma}(t_{0})$ a dimensionless ${\cal O}(1)$
parameter encoding the growth rate of the neutrino mass [166, 165].

Under some circumstances, the long-range interactions mediated by the cosmon
can give rise to the formation of dense non-linear neutrino lumps [185, 187,
221, 189, 222, 223, 188, 190, 191, 192, 193]. The formation of these objects
is conceptually similar to that of gravitational structures, being the main
differences associated to the field dependent mass $m(\phi)$, which varies
both in space and time. The impact of neutrino lumps on large scales depends
crucially on the presence or absence of a variation of the effective coupling
(92). In the constant coupling model, the instabilities are resolved by the
formation of sizable stable lumps typically exceeding the size of large galaxy
clusters. The associated backreaction effects lead to the suppression of the
(averaged) energy-momentum trace in Eq. (93), weakening the deceleration of
the cosmon field and making it difficult to obtain a realistic cosmology
[192]. This is due to two effects: i) the acceleration of neutrinos during
collapse, which tends to make them relativistic close to the center of the
lumps and ii ) the local negative value of the cosmon perturbation
$\delta\phi$ within the lumps, which leads to neutrino masses
$m_{\psi}(\bar{\phi}+\delta\phi)$ smaller than expected from the average field
$\bar{\phi}$ [189]. On the other hand, phenomenologically viable scenarios for
variable $\beta$ and small neutrino masses seem a priori plausible [193].

####  5.1.2 Primordial structure formation

The potential existence of an additional force stronger than gravity in
coupled quintessence scenarios opens the possibility of inducing structure
formation even in a radiation dominated epoch in which the standard
gravitational collapse is inefficient. To this end, the fermion field $\psi$
must be identified with a beyond the Standard Model particle playing the role
of dark matter and becoming non-relativistic prior to matter-radiation
equality. As shown in Refs. [125, 174, 224, 209, 210, 216], for constant444See
Refs. [217, 218, 219] for related settings with variable $\beta$. $\beta\gg 1$
and subleading fermion and cosmon components, this scenario admits a scaling
solution during radiation domination in which the mass of the fermion
decreases with time, $m_{\psi}\sim a^{-1}$, and

$\phi^{\prime}=\frac{M_{P}}{\beta}\,,\hskip
28.45274pt\Omega_{\psi}=\frac{1}{3\beta^{2}}\,,\hskip
28.45274pt\Omega_{\phi}=\frac{1}{6\beta^{2}}\,,\hskip
28.45274pt\Omega_{R}=1-\frac{1}{2\beta^{2}}\,,$ (97)

with $\Omega_{\psi},\Omega_{\phi},\Omega_{R}$ the energy density parameters of
the $\phi$, $\psi$ and radiation fluids. In the absence of shear or rotational
components in the initial velocity field, the density contrast of the $\psi$
field satisfies approximately [174, 209, 210, 216]

$\delta_{\psi}^{\prime\prime}-\delta_{\psi}^{\prime}-(1+\delta_{\psi})\delta_{\psi}-\frac{4}{3}\frac{\delta_{\psi}^{\prime
2}}{(1+\delta_{\psi})}=0\,.$ (98)

At early times, this differential equation admits a linearized solution [174,
224, 209, 210, 216]

$\delta_{\psi}=\delta_{\psi,\rm in}\left(\frac{a}{a_{\rm
in}}\right)^{p}\,,\hskip 56.9055ptp=\frac{1}{2}\left(1+\sqrt{5}\right)\,,$
(99)

with $a_{\rm in}$ the scale factor at the onset of the scaling regime and
$\delta_{\psi,\rm in}$ an initial value for the fluctuations which should be
determined by requiring compatibility with CMB observations [216].

Depending on the velocity distribution, the rapid growth (99) can lead to the
production of primordial black holes [216] or virialized halos which could
dominate the dark matter content of the Universe while passing the stringent
microlensing constraints and cosmic microwave background energy injection
bounds [220]. The formation process of primordial dark matter halos assumes
the onset of a screening mechanism at the time of formation. The analysis of
this stage is, however, complicated due to the highly non-linear and strong
coupling regime of the problem at hand. In particular, most of the equations
cannot be consistently linearized without leading to an unbounded growth of
perturbations that becomes unphysical at late times. On top of that, the
treatment implicitly neglects spatial density and temperature distributions
and the potential existence of smaller halos inside larger halos.
Consequently, the stability of the lumps can be most probably addressed via
$N$-body simulations only. If screening turns out to be inefficient, the long-
range forces could potentially cause the emission of scalar waves by
accelerated particles, dissipating energy from a halo and making it collapse
into black holes [218].

###  5.2 Hubble-induced phase transitions

In this section we present the phenomenology associated to the interplay
between non-minimally coupled spectator fields and a period of kination.
Before going into the details, it is worth to discuss the generality of this
scenario. On the one hand, non-minimal couplings to gravity are to be expected
from quantum field theory calculations on curved space-times [104]. On the
other, it is also natural to consider the spectator field to be massive and/or
self interacting, including potentially higher-order operators as those
generically expected in effective field theories not involving shift symmetry.
Additionally, the scalar field could display additional global or local
symmetries, being the former potentially broken by gravity [225]. As we shall
see, the combination of a long lasting period of kination with the ingredients
above leads unavoidably to the spontaneous breaking of the internal symmetry
of the spectator field when the background transitions from inflation to
kination. This Hubble-induced spontaneous symmetry breaking (SSB) has been
considered in the literature as a heating mechanism [226, 227, 101, 102, 103]
but also in the context of gravitational waves production [80], Affleck-Dine
baryogenesis [54] and dark matter [228, 229, 230].

####  5.2.1 Spectator field dynamics

To discuss the dynamics of Hubble-induced phase transitions, let us a consider
a canonical quintessential scenario like the one in Section 2 supplemented
with an exemplary action

$S_{\chi}=\int
d^{4}x\sqrt{-g}\left[-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\chi\partial_{\nu}\chi-\frac{1}{2}\left(m_{\chi}^{2}+\xi
R\right)\chi^{2}-V(\chi)\right]$ (100)

for an energetically-subdominant and non-minimally coupled spectator field
$\chi$ with bare mass $m_{\chi}$ and self-interaction potential

$V(\chi)=\frac{\lambda}{2n}\frac{\chi^{2n}}{\Lambda^{2n-4}}\,,$ (101)

with $\lambda$ a dimensionless coupling constant and $\Lambda$ an UV
suppression scale. For the sake of simplicity we have assumed the spectator
field to be real and imposed a discrete $Z_{2}$ symmetry. These restrictions
will not play, however, a key role in the following developments, being the
phenomenology presented below easily extendable to more general settings and
symmetry groups.

From the point of view of the spectator field, the only relevant aspect of the
cosmon evolution is the transition from inflation to kinetic domination, with
the correspondent change in the equation-of-state parameter from $w=-1$ to
$w=+1$. During inflation ($w=1$, $R=12H^{2}$) the effective square mass of the
$\chi$ field is large and positive for sufficiently big non-minimal couplings,
$\xi\gtrsim 1/12$, locking it at the origin potential and preventing the
generation of sizeable isocurvature perturbations in potential conflict with
CMB constraints [4]. However, as soon as the background evolution changes from
inflation to kination ($w=1$, $R=-6H^{2}$), the origin of the potential
becomes a local maximum for small bare masses $m^{2}_{\chi}<6\xi R$, forcing
the spectator field to evolve towards the new time-dependent minima appearing
at large field values555Notice that this holds even if the potential is a sum
of monomials with positive coefficients. The inclusion of negative
coefficients in the power series could give rise, however, to the appearance
of new local or global minima.

$|\chi_{\rm
min}|=H\left(\frac{6\xi-m^{2}_{\chi}/H^{2}}{\lambda}\right)^{\frac{1}{2(n-1)}}\left(\frac{\Lambda}{H}\right)^{\frac{n-2}{n-1}}\,.$
(102)

Figure 5: Schematic illustration of the evolution of the spectator field
fluctuations during inflation and kination. During inflation the field
distribution sits at the minimum of the effective potential. The internal
symmetry of the field is, however, spontaneously broken at the onset of
kination, forcing the field distribution to evolve towards two new minima
appearing at large field values. Due to the time-dependence of the potential,
the distribution is not trapped there, but rather oscillates with a bi-modal
pattern around the origin.

The transition between the inflationary and kinetic dominated epochs depends
on the details of the quintessential inflation potential. If it occurs
rapidly, the phase transition can be taken to be almost instantaneous. In
other words, the spectator field is not moving away from the inflationary
minimum while the potential is changing from the unbroken to the broken phase
form. This is typically the situation realized in quintessential inflationary
scenarios, where the cosmon potential at the end of inflation is steep.
Another important aspect to keep in mind is that the background evolution is
assumed to be homogeneous, and hence the transition occurs at the same time
everywhere. 666Alternative pictures involving first-order phase transitions
could appear in the presence of multiple higher-dimensional operators with
positive and negative coefficients. Then, at the time of symmetry breaking the
field has zero mean $\langle\chi\rangle=0$ and a certain variance
$\langle\chi^{2}\rangle$ dictated by vacuum fluctuations. During the broken
phase, the effective mass is negative and the system undergoes a stage of
tachyonic or spinodal instability, which rapidly amplifies the quantum
fluctuations while keeping their zero average value. In particular, the
symmetry breaking process is local and any vacua or direction to it is equally
probable. Then, the averaged value of the scalar field over scales much larger
than the correlation length must identically vanish. This crucial observation
implies that any description in terms of a homogeneous spectator field rolling
down a potential is inaccurate to start with [231, 232, 233, 103]. This has
two major implications. First, the dynamics of the scalar field leads to the
generation of temporary topological defects. Second, the energy in gradients
is non-negligible and the virialization of the system leads to an asymptotic
equation-of-state parameter $w=1/3$, regardless of the specific self-
interaction potential (101).

The tachyonic instability, which can be significantly long lasting since the
new minima of the effective potential can be quite displaced from the origin
and the kination period can last many $e$-folds, ends when the effective mass
of the $\chi$ field turns positive. This can occur at the onset of radiation
domination, where $R\simeq 0$, or be induced by the higher-order corrections
or the bare mass parameter in Eq. (100), provided that these overcome the
Ricci scalar contribution at a given time. At this point, the stability is
fully restored and the origin becomes again the true minimum of the potential.
However, as we shall see below, there is still the possibility that the
symmetry restoration is only effectively realized. This peculiar
characteristic of Hubble-induced phase transitions occurs when the energy lost
by the spectator field in one oscillation is smaller than the flattening of
the effective potential minimum, a quantity that can be easily inferred from
Eq. (102). In this case, the field distribution is able to pass back the
origin of the potential, giving rise to an interacting bimodal distribution
that oscillates around it. The other possibility is that the field radiates
enough energy as to get trapped in the new minima of the potential at a given
time, tracking their subsequent evolution and giving rise to more standard
topological defects with time-dependent tension and width [80]. This
interesting scenario could be realized, for instance, if the spectator field
is allowed to decay into other species or if the effective friction
experienced by it is somehow enhanced.

To quantify the above qualitative description, we can rewrite the action (100)
in terms of the coordinates and field variables in (36), getting

$S_{\chi}=\int d^{3}y\,dz\left[\frac{1}{2}(Y^{\prime})^{2}-\frac{1}{2}|\nabla
Y|^{2}+\frac{1}{2}M^{2}(z)Y^{2}-\frac{\lambda_{\rm eff}}{2n}Y^{2n}\right]\,,$
(103)

where the prime denotes derivative with respect to the conformal time $z$,

$M^{2}(z)=(4\nu^{2}-1)\mathcal{H}^{2}\,,\hskip 56.9055pt\lambda_{\rm
eff}=\left(\frac{\chi_{*}}{\Lambda}\frac{a_{\rm kin}}{a}\right)^{2n-4}\,,$
(104)

and

$\nu=\sqrt{3\xi/2}\,,\hskip
42.67912pt\mathcal{H}(z)=\frac{1}{2(z+\nu)}\,,\hskip
42.67912pt\mathcal{H}^{\prime}=-2\mathcal{H}^{2}\,,$ (105)

with the last relation holding during kination. At the onset of this expansion
epoch, the spectator field is close to the origin and the higher-order terms
in the potential can be safely neglected. This means that the early dynamics
of the system can be well described in terms of a free scalar field with a
time dependent mass $M^{2}(z)$ and, therefore, exactly solved, as we discussed
in Section 5.2.2. In fact, in terms of the dimensionless Fourier mode $\kappa$
in Eq. (36), the problem reduces to find the solutions of the Schrödinger-like
differential equation

$f^{\prime\prime}_{\kappa}+\omega^{2}_{\kappa}(z)f_{\kappa}=0\,,\hskip
56.9055pt\omega_{\kappa}^{2}=\kappa^{2}-M^{2}(z)\,,$ (106)

for the mode function $f_{\kappa}$ with standard vacuum initial conditions
$f_{\kappa}(0)=1/\sqrt{2\kappa}$ and
$f^{\prime}_{\kappa}(0)=-i\sqrt{\kappa/2}$. The detailed quantization of the
system follows closely the treatment of Refs. [234, 48, 49, 235] and it is
detailed for the present case in Ref. [233]. The tachyonic instability induced
by kinetic domination amplifies all subhorizon modes within the time-dependent
window $\kappa_{\rm min}(z)$ and $\kappa_{\rm max}(z)$, where

$\kappa_{\rm min}=\mathcal{H}(z)\,,\hskip 56.9055pt\kappa_{\rm
max}=(4\nu^{2}-1)^{1/2}\kappa_{\rm min}\,.$ (107)

The copious production of these modes translates eventually into a quantum-to-
classical transition [48, 47, 48, 49, 50, 51], meaning that the properties of
the system can be well-approximated by those of a classical random field [233,
103]. In particular, the above system can be solved in terms of Bessel’s
functions,

$f_{\kappa}(z)=\sqrt{z+\nu}\left[A_{\kappa}\mathcal{J}_{\nu}(\kappa(z+\
u))-B_{\kappa}\mathcal{Y}_{\nu}(\kappa(z+\nu))\right]\,,$ (108)

with $A_{\kappa}$ and $B_{\kappa}$ $\kappa$-dependent coefficients determined
by the aforementioned initial conditions. As shown in Fig. 6, the modes that
belongs to the unstable region are exponentially amplified. Moreover, one can
show that the Heisenberg uncertainty principle in Fourier space can be written
as [48, 47, 48, 49, 50, 51, 233, 103]

$\Delta
Y_{\kappa}^{2}\,\Delta\Pi_{\kappa}^{2}=|F_{\kappa}(z)|^{2}+\frac{1}{4}\geq{\frac{1}{4}}\,\Big{|}\langle[Y_{\kappa}(z),\
\Pi^{\dagger}_{\kappa}(z)]\rangle\Big{|}^{2}\,,$ (109)

with

$F_{\kappa}(z)={\rm Re}(f^{*}_{\kappa}f^{\prime}_{\kappa})\,,$ (110)

and the square brackets denoting the anticommutation of the corresponding
variables. It is then clear that for $|F_{\kappa}|\gg 1$, the ordering of the
operators becomes negligible, as expected for a classical system.

Figure 6: The functions $|f_{\kappa}|$ (left) and $|F_{\kappa}|$ (right)
evaluated at different conformal times $z$. The shaded region corresponds to
momenta outside the initial amplification band $\kappa>\kappa(z=0)$. The
dashed vertical lines corresponds to the time dependent amplification band
(107) at the given times.

The above description ceases to be valid as soon as non-linearities kick in.
At that moment, in order to properly account for the complete dynamics in the
evolution equation

$Y^{\prime\prime}-\nabla^{2}Y-M^{2}(z)Y+\lambda_{\rm eff}Y^{2n-1}=0\,,$ (111)

one needs to resort to numerical simulations. Such simulations have been
carried out in Ref. [103] for quartic ($n=4$, $\lambda_{\rm eff}=\lambda$) and
sextic potentials ($n=6$, $\lambda_{\rm eff}=\lambda(\chi_{*}/\Lambda\,a_{\rm
kin}/a)^{4}$). Despite the complexity of the non-linear physics involved, it
is still possible to picture the evolution during the broken phase in rather
simple terms. As soon as the relevant scales can be treated classically, the
field value moves locally towards the new minima, where locally means that the
field spatial distribution will look homogeneous on scales smaller than the
correlation length $\kappa_{*}=\sqrt{\nu+1}\mathcal{H}(z)$ [233]. This is
exactly what is shown in the first three panels of Fig. 7, where we display a
few two-dimensional snapshots of the simulation ran in Ref. [103] for a
quartic potential.777Movies covering the whole simulation are available at
this URL. The first box shows the initial classicalized configuration, with
the spectator field still very close to the vacuum initial conditions. The
second to fourth boxes display the displacement of the spectator field from
the origin, growing in amplitude and generating a rather inhomogeneous
distribution. Since we are considering a $Z_{2}$ symmetric field, this implies
the formation of regions of positive and negative field values separated by
domain wall configurations. It is worth noticing that the characteristic scale
of the inhomogeneity decreases with time. In particular, the profile of the
inhomogeneities is smoother when the field rolls towards the broken phase
minima, while it becomes more jagged in the subsequent evolution. This is due
to the fact that after the field distribution reaches a minimum and bounces
back, it is able to cross the origin towards the opposite minimum. Hence, the
spectator field is effectively oscillating in a quartic potential, permitting
energy exchange and making the oscillating defects fragment, while they
diminish in amplitude and size. Despite this, the energy stored in the
gradients of the spectator field is not decreasing with time and it represents
a non-negligible fraction of the total field energy density for the total
duration of the simulation. This confirms the idea that the Hubble-induced
symmetry breaking is an inherently inhomogeneous process.

Figure 7: Spectator field growth for the benchmark point $\nu=10$,
$\lambda=10^{-4}$. We display 2-dimensional sections of a fraction $V/8$ of
the 3-dimensional simulation volume $V$ at times $z=5,\,10,\,15,\,60$, with
the $z$-axis indicating the corresponding field value at each point in this
surface. The black segment in the second panel indicates the typical scale of
the highest peaks. (Images from [103])

####  5.2.2 Applications

In this last part of the manuscript, we will focus on the expected
phenomenology of the above scenario. In particular, we will describe the
consequences of inhomogeneities and review some new aspects to be expected
when changing the symmetry group or applying this mechanism to dark matter.

##### Heating

The fast amplification of the the spectator field fluctuations triggered by
the Hubble-induced SSB plays an important role when it comes to the heating of
the Universe. This can be understood by realizing that the effect of the non-
minimal coupling decreases with time, eventually becoming negligible in the
determination of the spectator field dynamics. From there on, the field
behaves almost as if it were minimally coupled to gravity. In particular, it
will tend to virialize, achieving an equation-of-state parameter
asymptotically close to the radiation value $w=1/3$. In fact, for monomial
potentials like the ones under consideration, the equations-of-state parameter
can be written as [236, 237]

$w_{\chi}=\frac{1}{3}+\frac{2}{3}\frac{(n-2)}{(n+1)+\langle(\nabla\chi/a)^{2}\rangle/\langle
V\rangle}\,,$ (112)

showing clearly that the field will behave as a radiation component provided
that the gradients dominate over the potential and independently of the
specific higher-order operators involved. For illustration purposes, we
display in Fig. 8 the time evolution of the equation-of-state parameter for
quartic and sextic potentials. It is worth stressing that if one were to
consider the spectator field as a homogeneous condensate the equation of state
would be that of radiation only for the first potential, taking the value
$w=1/2$ for the sextic one. This crucial difference remarks how crucial
inhomogeneities are in the context of SSB.

Figure 8:  (Left) Evolution of the volume and time averaged equation-of-state
parameter (112) for a quartic potential. (Right) Evolution of the same
quantity for a sextic potential. The red horizontal line corresponds to the
radiation value $1/3$ while the blue line corresponds to the value $1/2$.
(Images credit [103])

Another important aspect to keep in mind is that tachyonic instability only
amplifies IR modes thus generating an initially far-from-thermal spectrum, as
that displayed in Fig. 9. However, the subsequent turbulent evolution of the
system induces a slow shift of the spectral peak, which moves gradually
towards the UV, leading to eventually a close-to-thermal distribution [106,
107].

Figure 9: Time evolution of the occupation numbers for $\nu=10$ and
$\lambda=10^{-4}$. Different colors correspond to different times, as
indicated in the figure. (Image credit [103])

When combined with the fast dilution of the background component discussed in
Section 2, this implies that the spectator field will eventually become
dominant, starting an epoch of radiation domination and setting the conditions
for the onset of the hBB era. Of course, much still remains to be said about
this process. One might add, for instance, direct interactions between the
spectator field and fermions, like in other heating mechanisms [96, 97, 238,
239, 240, 241, 242], or consider this radiation component as an inert,
subdominant and extra radiation contribution (obviously satisfying all
observational bounds). Also, if a significant bare mass parameter is allowed,
the field could eventually oscillate around a quadratic potential, possibly
behaving as dark matter (see below).

##### Gravitational waves

Another interesting phenomenological aspect related to the presence of defects
and inhomogeneities is the associated emission of gravitational waves. Since
the distribution of the spectator field after the Hubble-induced phase
transition is highly inhomogeneous, the transverse traceless part of its
energy momentum tensor will generically be non-vanishing, hence providing a
source for GW emission. The production of these tensor modes due Kähler-
induced potentials has been considered in supersymmetric extensions of the
Standard Model [243] and applied to the present Ricci-based scenario in Ref.
[80] for a $U(1)$ symmetric spectator field. Although the generation of
gravitational waves from topological defects is a very well-known phenomenon
[244, 245, 246], the mechanism at hand possesses striking differences. First,
the defects will have a typical size proportional to the Hubble rate
$\mathcal{H}$, which makes them ‘fat’, as in the context of axion dark matter
[247, 248], and with a time-dependent tension and width, meaning that an exact
scaling regime [249, 250] is never achieved. Second, the energy density in GW
will be naturally amplified during kinetic domination. Although the precise
relevance of these phenomena will ultimately depend on the parameters of the
model, they represent nonetheless a generic feature of it, potentially within
the reach of forthcoming observational campaigns [251, 252], as shown
explicitly in Ref. [80]. Interestingly, the obtained GW spectrum displays also
a peaked structure, thus easing its discrimination from foreground signals.

##### Baryogenesis

The observed asymmetry between matter and anti-matter in the Universe remains
one of the biggest mysteries in particle physics. Even if the Standard Model
could a priori produce a baryonic asymmetry during the electroweak phase
transition [253], the resulting value turns out too small as compared with
observations. Traditionally, the amount of asymmetry is encoded in the baryon-
to-entropy ratio

$Y_{B}\equiv\frac{n_{B}}{s}=(0.861\pm 0.008)\times 10^{-10}\,,$ (113)

where $n_{B}=n_{b}-n_{\bar{b}}$ being $n_{b(\bar{b})}$ the number density of
(anti)baryons. One of the most successful proposals to explain this small
value of this quantity is the so-called Affleck-Dine mechanism [254, 255],
which makes use of a gravity-induced SSB during inflation. Interestingly
enough, Hubble-induced phase transitions are able to prompt a post-
inflationary Affleck-Dine asymmetry generation if the spectator field $\chi$
is charged under a weakly-broken $B-L$ symmetry [54]. In particular, the
change of sign of the Ricci scalar at the onset of kinetic domination breaks
spontaneously the $U(1)_{B-L}$ symmetry, making the complex spectator field
roll towards the new vacuum manifold of the potential with a certain phase
that, at each space point, takes a random value. The additional explicit
symmetry breaking requirement provides the charge non-conservation. In the
polar representation, $\chi=\tfrac{1}{\sqrt{2}}\rho e^{i\theta}$, the
associated symmetry-breaking operators adding to the potential888Notice that
since here the symmetry is $U(1)$ one needs to replace in the action (100) the
real scalar field by a complex one, i.e.
$\chi^{2}\rightarrow|\chi|^{2}=\chi\chi^{*}$, being $\chi^{*}$ the complex
conjugate of $\chi$. (101) can be written as

$\delta V=-\epsilon_{0}\Lambda^{4-2n}\rho^{2n}F(\theta,\theta_{0})\,,$ (114)

with $\epsilon_{0}$ and $\theta_{0}$ the amplitude and phase of a complex
symmetry-breaking coefficient $\epsilon=\epsilon_{0}e^{i\theta_{0}}$. A net
amount of particles over anti-particles can then be produced

$a^{3}n_{B-L}(t)=q\epsilon_{0}\Lambda^{4-2n}\int^{t}dt\,a^{3}(t)\rho^{2n}F^{\prime}(\theta,\theta_{0})\,,$
(115)

where the prime stands here for a derivative with respect to the phase
$\theta$ and $n_{B-L}=q\rho^{2}\dot{\theta}$ is the $B-L$ current, being $q$
the associated charge. The quantity $n_{B-L}$ keeps growing until the symmetry
is restored. In the case of a quartic operator, corresponding to $n=2$, the
produced asymmetry is estimated to be [54]

$\frac{Y_{B-L}}{10\,{\rm
GeV}}=\left(\frac{\Theta}{10^{-4}}\right)^{3/2}\frac{H_{\rm kin}}{10^{7}\,{\rm
GeV}}\,,$ (116)

with $\Theta$ the heating efficiency defined in Eq. (31). As this point, it is
important to emphasize a fundamental limitation of the procedure. The net
baryonic excess (116) is local, and, therefore, the cosmological evolution
will put in causal contact patches with different phase values, leading to a
net zero charge on large scales! Albeit not usually emphasized, this problem
is common to other post-inflationary mechanisms for the generation of baryon
asymmetry, as those considered, for instance, in Refs. [256, 257]. A possible
way out is to consider a biased minimum, so that the potential is tilted and a
unique global minimum exists, see e.g. [258, 259, 260]. Then, even if
initially all spectator phases are equally probable, the dynamical evolution
will generate a flux towards the true minimum, thus producing a net global
asymmetry [54].

##### Dark matter

The Hubble-induced SSB has also been explored in the context of primordial
dark matter production [261, 228, 229, 230]. In particular, Ref. [228] studied
the super-Hubble creation of particles999Notice that this does not violate
causality as long as it is sub-horizon, meaning by this that no modes smaller
than the inflationary particle horizon are considered [262]. when the post-
inflationary Ricci scalar can take negative values. This mechanism was also
combined there with a post-inflationary period of kination. By solving the
equation for the spectator field (111) at one loop precision, it is found a
dark matter abundance

$\Omega_{\chi}h^{2}=0.078\beta^{3/4}\left(\frac{m}{{\rm
GeV}}\right)\left(\frac{H_{\rm rad}}{10^{13}{\rm
GeV}}\right)^{3/2}\left(\frac{100}{g_{*,{\rm rad}}}\right)^{1/4}\,.$ (117)

Here $\beta$ is an $\mathcal{O}(1)$ parameter for $\xi\lesssim 10$
representing the energy density of the produced $\chi$ quanta in units of
$H_{\rm kin}$. $m$ is the dark matter bare mass parameter whose value is a
priory free. However, upper bounds on the cross section for dark matter
interactions obtained from galaxy clusters collisions impose the constraint

$\frac{m}{{\rm GeV}}>0.027\lambda^{2/3}\,.$ (118)

This bound is more constraining for larger values of $H_{\rm kin}$ and
$\lambda$. However, even when these restrictions are combined with the
requirement of avoiding dark matter overproduction, one still finds the very
broad range of allowed dark matter masses keV $<m<10^{4}$ GeV [228].

##  6 Discussion and outlook

Inflation and dark energy are usually understood as two independent epochs in
the expansion history of the Universe. In this review, we have described how
these accelerated expansion eras can be unified into a quintessential
inflation framework in which a single degree of freedom, the cosmon, plays
simultaneously the role of the inflaton and of the quintessence field.

The naturalness of quintessential inflation is intimately related to its
embedding in a fundamental framework able to produce a large hierarchy of
scales without involving unacceptable finetunings. Natural realizations might
appear, for instance, in variable gravity scenarios involving UV and IR fixed
points where scale symmetry is manifestly realized. On top of that, this
paradigm should provide a graceful solution to the “why now” problem,
potentially associated with some physical mechanism. In this respect, scaling
solutions are able to provide a dark energy density comparable to the dominant
matter content in the Universe for an extended period of time, reducing with
it the tuning on the initial conditions. A suitable clock signalling the end
of this scaling regime is naturally provided by neutrinos, which for sub-eV
masses becomes non-relativistic around the same time in which dark energy
should start to be dominant.

Probably, the most distinctive phenomenology of quintessential inflation is
associated to the unavoidable kinetic dominated regime. This unusual stage
induces a relative amplification of all radiation and matter components in the
Universe, including the products resulting from the heating stage and any
stochastic background of gravitational waves of primordial origin. This can be
used to put constraints on some of the theory defining parameters. For
instance, the compatibility of the primordial tensor perturbations with BBN
imposes bounds on the inflationary scale and the duration of kination, or
equivalently, on the efficiency of the heating stage. Additional constraints
on these quantities could follow from the spontaneous symmetry of internal
symmetries of fields non-minimally coupled to gravity and the associated
gravitational wave production. In this review we have discussed the Hubble-
induced symmetry breaking of two specific symmetries, namely a global U(1) and
a discrete Z2. However, it is worth stressing that this mechanism applies to
any symmetry group, provided the appearance of a time-dependent negative mass
parameter in the effective potential leading to new minima with a non-trivial
topological structure. Furthermore, the expansion rate may also affect a
plethora of particle physics processes in the very early Universe, from the
generation of the matter-antimatter asymmetry to the abundance of dark matter
relics.

## Acknowledgments

DB acknowledges support from Programa II: Contratos postdoctorales by
Salamanca University and from Project PGC2018-096038-B-I00 funded by the
Spanish “Ministerio de Ciencia e Innovación” and FEDER “A way of making
Europe”. JR thanks the Fundação para a Ciência e Tecnologia (FCT), Portugal,
for the financial support to the Center for Astrophysics and Gravitation-
CENTRA, Instituto Superior Técnico, Universidade de Lisboa, through the
Project No. UIDB $/00099/2020$ and for the financial support through the
CEECIND $/01091/2018$ grant.

##  Appendix A Frame-invariant formulation of inflation

In this Appendix, we summarize the frame-invariant formulation of inflation in
Refs. [263, 134], to which refer the reader for further details.

The variation of the action (75) with respect to the FLRW metric and the
scalar field $\varphi$ provides respectively the Friedmann equations [263],

$\displaystyle
H^{2}=\frac{fU}{3}\left(1-\frac{\epsilon_{H}}{3}+\frac{5\,\kappa_{H}}{3}+\frac{\theta_{H}}{3}\right)^{-1}\,,\hskip
42.67912pt\epsilon_{H}+\kappa_{H}-\theta_{H}=\frac{k\dot{\varphi}^{2}}{2H^{2}f}\,,$
(119)

and the Klein-Gordon equation

$\displaystyle{\ddot{\varphi}}+(3+\sigma_{H})H{\dot{\varphi}}+\frac{f^{3}}{E}U_{,\varphi}\
=0\,,$ (120)

with

$E\equiv kf+\frac{3}{2}f^{2}_{,\varphi}\,,\hskip
56.9055ptU\equiv\frac{V}{f^{2}}\,.$ (121)

The quantities

$\epsilon_{H}\equiv-\frac{\dot{H}}{H^{2}}\,,\hskip
42.67912pt\kappa_{H}\equiv\frac{1}{2}\frac{\dot{f}}{Hf}\,,\hskip
42.67912pt\sigma_{H}\equiv\frac{1}{2}\frac{\dot{E}}{HE}\,,\hskip
42.67912pt\theta_{H}\equiv\frac{1}{2}\frac{\ddot{f}}{H^{2}f}\,,$ (122)

in these expressions can be understood as a generalization of Hubble slow-roll
parameters. Assuming these to be small during the inflationary stage, we can
rewrite Eqs. (119) and (120) as

$\displaystyle H^{2}\simeq\frac{fU}{3}\,,\hskip
56.9055pt3H{\dot{\varphi}}\simeq-\frac{f^{3}}{E}U_{,\varphi}\,,$ (123)

and compute the scalar and tensor power spectra

$\displaystyle P_{s}$ $\displaystyle\approx$
$\displaystyle\frac{k^{3}}{24\pi^{2}}\,\frac{U}{\epsilon_{U}+\kappa_{U}}\,,\hskip
42.67912ptP_{t}\approx\frac{2k^{3}}{3\pi^{2}}\,U\,,$ (124)

and the associated spectral tilt and tensor-to-scalar ratio,

$\displaystyle 1-n_{s}$ $\displaystyle=$ $\displaystyle
4\epsilon_{U}+2(\sigma_{U}-\delta_{U}-\kappa_{U})\,,\hskip
42.67912ptr=16(\epsilon_{U}+\kappa_{U})\,,$ (125)

with

$\displaystyle\epsilon_{U}\ $ $\displaystyle\equiv$ $\displaystyle\
\frac{1}{2}\,\frac{fU_{,\varphi}(fU)_{,\varphi}}{EU^{2}}\ ,\hskip
28.45274pt\delta_{U}\equiv\frac{1}{2}\,\frac{fU_{,\varphi}(fU)_{,\varphi}}{EU^{2}}+\left(\frac{f^{2}U_{,\varphi}}{EU}\right)_{,\varphi}\,,$
(126) $\displaystyle\kappa_{U}$ $\displaystyle\equiv$
$\displaystyle-\frac{f_{,\varphi}}{2}\,\frac{fU_{,\varphi}}{EU}\,,\hskip
28.45274pt\sigma_{U}\equiv-\frac{1}{2}\,\frac{E_{,\varphi}}{E^{2}}\frac{f^{2}U_{,\varphi}}{U}\,,$
(127)

the generalization of the potential slow-roll parameters. These expressions
are intended to be evaluated at a number of $e$-folds $N=\int H\,dt$, computed
out of Eqs. (123),

$N(\varphi)\ =\
-\int_{\varphi}^{\varphi_{\text{end}}}d\varphi^{\prime}\frac{E(\varphi^{\prime})}{f(\varphi^{\prime})^{2}}\,\frac{U(\varphi^{\prime})}{U(\varphi^{\prime})_{,\varphi^{\prime}}}\,,$
(128)

with $\varphi_{\text{end}}$ the value of $\varphi$ at the end of inflation,
determined by a frame-invariant generalization of the standard condition
$\epsilon_{U}=1$,

$\max(\epsilon_{U}+\kappa_{U},|\epsilon_{U}+\delta_{U}+4\kappa_{U}-\sigma_{U}|)=1\,.$
(129)

Note that, at the leading order in the slow-roll approximation, the
inflationary observables are invariant under the separate action of conformal
transformations and field reparametrizations even if the slow-roll parameters
are themselves transformed, i.e.,
$\widetilde{n}_{s}(\tilde{\varphi})=n_{s}(\varphi)$,
$\widetilde{r}(\tilde{\varphi})=r(\varphi)$. Also, as written in Eq. (128),
the number of $e$-folds is invariant under frame transformations.
Consequently, we have $\widetilde{n}_{s}(\widetilde{N})=n_{s}(N)$ and
$\widetilde{r}(\widetilde{N})=r(N)$.

##  Appendix B A workout example of crossover regime

For illustration purposes, we present here a specific flow equation smoothly
interpolating between the fixed points considered in the main text and
allowing for explicit analytical solutions,

$\mu\,\partial_{\mu}\ln B=\frac{\kappa\,\sigma
B}{\sigma+\kappa\,B^{1/\alpha}}\,.$ (130)

Using Eq. (128) we get

$B(\bar{N})=\Big{\\{}\frac{\sigma}{\kappa}{\cal
W}\left[\frac{\kappa}{\sigma}\left(B_{\rm
end}\,e^{2\kappa\bar{N}}\right)^{p}\right]\Big{\\}}^{1/p}\,,\hskip 42.67912pt$
(131)

with

$\bar{N}=N+c^{p}\,,\hskip 42.67912ptc\equiv B_{\rm
end}\left(\frac{\alpha}{2\sigma}\right)^{\frac{1}{p}}\,.$ (132)

The above expression can be alternatively written as

$B(\bar{N})=B_{\rm
end}\Bigg{\\{}\left(\frac{1}{2\,\kappa\,p\,c^{p}}\right){\cal
W}\left[2\,\kappa\,p\,c^{p}\,e^{2\,\kappa\,p\,\bar{N}}\right]\Bigg{\\}}^{1/p}\,.$
(133)

The associated form $B(\varphi)$ can be trivially computed out of this
expression through the relation

$\frac{\partial\ln\varphi}{\partial N}=-\frac{2}{B}\,,$ (134)

getting

$\ln\left(\frac{\varphi}{m\,e^{-\frac{1}{\sigma}\frac{1}{1-q}}}\right)^{\sigma}=\frac{\sigma}{\kappa}(B^{-1})+\frac{(B^{-1})^{1-q}}{1-q}\,,$
(135)

where we have chosen the integration constant as
$C=m\,e^{-\frac{1}{\sigma}\frac{\alpha}{\alpha-1}}$. This can be alternatively
written as

$\ln\left(\frac{\varphi}{m}\right)^{\sigma}=\frac{\sigma}{\kappa}(B^{-1})+\ln_{q}B^{-1}\,,$
(136)

with

$\ln_{q}x=\begin{cases}\ln x\hskip 42.67912pt{\rm if}\hskip
14.22636ptx>0\,\,\&\,\,q=1\,,\\\ \frac{x^{1-q}-1}{1-q}\hskip 31.29802pt{\rm
if}\hskip 14.22636ptx>0\,\,\&\,\,q\neq 1\,,\\\ \textrm{undefined}\hskip
17.07164pt{\rm if}\hskip 14.22636ptx\leq 0\,,\end{cases}$ (137)

the so-called $q$-logarithm function used by Tsallis in non-extensive
statistical mechanics [264]. This function is related to the $q$-exponential

$e_{q}^{x}=\begin{cases}e^{x}\hskip 147.95433pt{\rm if}\hskip
14.22636ptq=1\,,\\\ (1+(1-q)x)^{1/(1-q)}\hskip 62.59605pt{\rm if}\hskip
14.22636ptq\neq 1\,,\,\,\&\,\,1+(1-q)x\geq 0\,,\\\ 0\hskip 153.64487pt{\rm
if}\hskip 14.22636ptq\neq 1\,,\,\,\&\,\,1+(1-q)x<0\,,\end{cases}$ (138)

(such that $\exp_{q}(\ln_{q}x)=x$ and $\ln_{q}(\exp_{q}x)=x$) and to the
Lambert-Tsallis function, defined as [264]

${\cal W}_{q}(x)e^{{\cal W}_{q}(x)}=x\,,$ (139)

and satisfying [264]

${\cal W}_{q}(x)=\ln_{q}x-_{q}\ln{\cal W}_{q}(x)\,,$ (140)

with

$a-_{q}b=\frac{a-b}{1+(1-q)b}\,.$ (141)

In general, Eq. (136) cannot be inverted. However, some solutions can be found
for specific values of $\alpha$, namely

$\displaystyle\alpha=1/2\,,\hskip
14.22636ptB(\varphi)=\frac{1}{2}\left(1-\log\left(\frac{\varphi}{m}\right)^{\sigma}\right)+\sqrt{\frac{\sigma}{\kappa}+\frac{1}{4}\left(1-\log\left(\frac{\varphi}{m}\right)^{\sigma}\right)^{2}}\,,$
(142) $\displaystyle\alpha=1\,,\hskip
14.22636ptB(\varphi)=\frac{\kappa}{\sigma}\,{\cal
W}^{-1}\left[\frac{\sigma}{\kappa}\left(\frac{\varphi}{m}\right)^{\sigma}\right]\,,$
(143) $\displaystyle\alpha=2\,,\hskip
14.22636ptB(\varphi)=\left(\frac{1+\sqrt{1+\frac{\sigma}{\kappa}(2+\ln\left(\frac{\varphi}{m}\right)^{\sigma})}}{2+\ln\left(\frac{\varphi}{m}\right)^{\sigma}}\right)^{2}\,.$
(144)

The cosmological evolution following from the $\alpha=1$ case has been
carefully analyzed in Ref. [39], where we refer the interested reader for
further details.

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