#  What is needed of a scalar field if it is to unify inflation and late time
acceleration?

Nur Jaman<EMAIL_ADDRESS>Centre for Cosmology and Science
Popularization(CCSP), SGT University, Gurugram 12006, India  Indian Institute
of Science Education and Research Kolkata, Mohanpur, WB 741246, India  M. Sami
<EMAIL_ADDRESS>Centre for Cosmology and Science Popularization(CCSP), SGT
University, Gurugram 12006, India  Center for Theoretical Physics, Eurasian
National University, Astana 010008, Kazakhstan.

###### Abstract

ymmsQuintessential inflation refers to scenarios in which a single scalar
field is used to describe inflation and late-time acceleration. This review is
dedicated to the framework of quintessential inflation, with a focus on the
building blocks of formalism. Consistent unification of inflation and late
time acceleration using a single scalar field asks for a shallow field
potential initially followed by steep behaviour thereafter and shallow again
around the present epoch. The requirement of non-interference of scalar field
with thermal history dictates steep nature of potential in the post-
inflationary era, with a further restriction that late time physics be
independent of initial conditions. We describe, in detail the scaling and
asymptotic scaling solutions and the mechanism of exit from the scaling regime
to late time acceleration. Review includes a fresh look at scaling solutions
which are central to the theme of unification of inflation and late time
acceleration. As for the exit mechanism, special attention is paid to the
coupling of massive neutrino matter to the scalar field, which builds up
dynamically and can give rise to late time acceleration. We present a detailed
analytical treatment of scalar field dynamics in the presence of coupling. We
briefly discuss the distinguished feature of quintessential inflation, namely,
the blue spectrum of gravity waves produced during the transition from
inflation to the kinetic regime.

##  I Introduction

Accelerated expansion is generic to cosmological dynamics Planck:2018vyg ,
Universe has gone through inflation Guth:1980zm ; Sato:1980yn ; Linde:1981mu ;
Albrecht:.1982wi ; Starobinsky:1980te ; Starobinsky:1982ee  (for review see,
Liddle:1999mq ; Tsujikawa:2003jp ; Martin:2013tda ; Vazquez:2018qdg ) at early
epochs and is accelerating at present SupernovaSearchTeam:1998fmf ;
SupernovaCosmologyProject:1998vns . Inflation addresses the logical
inconsistencies of hot big bang model and provides with a mechanism for
primordial perturbations believed to have grown via gravitational instability
into the structure we see today in the Universe Starobinsky:1982ee ; seed .
The age puzzle in hot big bang cries for late time acceleration Krauss:1995yb
; Turner:1997de  be due to dark energy $-$ cosmological constant, quintessence
or large scale modification of gravity. This phenomenon has been confirmed by
direct as well as indirect observations lateobs , which, however, elude
inflation. It is plausible to think that accelerated expansion is eternal, it
had gone into hiding at an early epoch (end of inflation) allowing the thermal
history to proceed as envisaged by the hot big bang and it reappears today to
account for the late time acceleration$-$re-incarnation of inflation. The term
“Quintessential Inflation” refers to a paradigm of unification of inflation
and late-time acceleration that does not disturb the thermal
historyPeebles:1998qn ; Sahni:2001qp ; Peebles:1987ek ; Huey:2001ae ;
Majumdar:2001mm ; Dimopoulos:2000md ; Sami:2003my ; Dimopoulos:2002hm ;
Dias:2010rg ; BasteroGil:2009eb ; Chun:2009yu ; Bento:2008yx ; Matsuda:2007ax
; Neupane:2007mu ; Dimopoulos:2007bp ; Rosenfeld:2006hs ; Gardner:2007ib ;
BuenoSanchez:2006ah ; Membiela:2006rj ; Cardenas:2006py ; Zhai:2005ub ; Da ;
Rosenfeld:2005mt ; Giovannini:2003jw ; Dimopoulos:2001qu ; Dimopoulos:2001ix ;
Yahiro:2001uh ; Kaganovich:2000fc ; Baccigalupi:1998mn ; Lee:2014bwa ;
Capozziello:2005tf ; Nojiri:2005pu ; Elizalde:2008yf ; Hossain:2014coa ;
Guendelman:2002js ; WaliHossain:2014usl ; Ahmad:2017itq ; deHaro:2021swo ;
Dimopoulos:2020pas ; Benisty:2020xqm ; Karciauskas:2021fdu ;
Capozziello:2003tk ; Sami:2004xk ; Dimopoulos:2017zvq ; Dimopoulos:2017tud ;
Bettoni:2021qfs ; Wetterich:2022brb ; Jaman:2018ucm ; Rosati:2003yw ;
Salati:2002md ; Akrami:2017cir ; Akrami:2020zxw ; Saba:2017xur ). In simple
cases, inflation is driven by a scalar field, which soon after the end of
inflation enters into an oscillatory phase and fast decays in particle species
giving rise to reheating/preheating of the Universe Albrecht:1982mp ;
Dolgov:1982th ; Abbott:1982hn ; Dolgov:1989us ; Ford:1986sy ; Traschen:1990sw
; Spokoiny:1993kt ; Shtanov:1994ce ; Kofman:1994rk ; Kofman:1997yn ; Garcia-
Bellido:1997hex ; Felder:1998vq ; Lyth:2001nq ; Feng:2002nb ; delCampo:2009yc
; Bassett:2005xm ; Hardwick:2016whe ; Campos:2002yk ; Allahverdi:2010xz ;
Amin:2014eta ; Garcia:2020eof ; Tambalo:2016eqr ; Dimopoulos:2017tud ;
Lopez:2021agu ; Saha:2021kez ; Pareek:2021lxz ; Bhattacharya:2019ryo ;
Dimopoulos:2019gpz ). Attempts have been made to achieve the aforementioned
goal by using a single scalar field as a source of both the phases of
accelerated expansion in the framework of modified theories of gravity,
particularly $f(R)$ theories Capozziello:2003tk ; Carroll:2003wy ;
DeFelice:2010aj ; Sotiriou:2008rp ; Gannouji:2012iy ; Cosmai:2013nva  . For
instance, putting the Starobinsky proposal for inflation and late time
acceleration together using $f(R)$ could suffice, see Ref.Nojiri:2007uq  on
the related theme. In these models, it is difficult to get rid of the strong
dependence on initial conditions. In this review, we shall focus on the scalar
field models where one has better control on this problem.

At the onset, it sounds fairly straightforward to carry the said unification
in a single scalar field using a runaway type of potential which is shallow
initially, suitable for slow roll, followed by steep behaviour for the entire
history of the Universe and turning shallow again at late times to account for
the observed acceleration. Though the presence of early and late time phases
of accelerated expansion is evident in the proposal, nonetheless, remarks
related to thermal history and the independence of late time evolution from
initial conditions are in order. Since the potential is of the runaway type,
the standard reheating mechanism is not operative in this framework. One needs
to employ an alternative framework such as gravitational particle production
Ford:1986sy ; Spokoiny:1993kt  (see also Chun:2009yu ), instant preheating
Felder:1998vq ; Sami:2004xk ; Campos:2002yk ; Dimopoulos:2017tud , curvaton
preheating Lyth:2001nq ; Feng:2002nb ; delCampo:2009yc ; Hardwick:2016whe ,
Ricci reheating Dimopoulos:2018wfg  (see also, Bettoni:2021zhq ;
Opferkuch:2019zbd ) and so on. Radiation density produced in these processes
at the end of inflation is smaller than the field energy density by several
orders of magnitude. Due to the steep nature of potential in the post-
inflationary era, scalar field enters the kinetic regime soon after inflation
ends ($\rho_{\phi}\sim a^{-6}$) Tashiro:2003qp  such that radiation domination
takes place after a long kinetic regime and field energy density becomes sub-
dominant. There are two requirements for the field dynamics before it emerges
from sub-dominance to dominance at late times: (1) It must adhere to the
nucleosynthesis constraint; and (2) Its emergence to late time acceleration
must be unaffected by initial conditions. The latter puts a restriction on the
class of field potentials that can be employed for quintessential inflation.

A comment on the duration of the kinetic regime is in order. The long kinetic
regime might be problematic for nucleosynthesis due to relic gravity waves.
Indeed, the energy density in relic gravity waves produced at the end of
inflation increases in comparison to $\rho_{\phi}$ during the kinetic regime
and might conflict with the nucleosynthesis constraint at the commencement of
the radiative regime. The nucleosynthesis constraint puts a lower bound on the
reheating temperature at the end of inflation and an upper bound on the
duration of the kinetic regime. One of the reheating alternatives is provided
by gravitational particle production which is a universal process that occurs
due to a non-adiabatic change in space-time geometry when inflation ends;
unlike other mechanisms, it does not require additional field(s); however,
nucleosynthesis bound is challenged. It should be noted that the presence of
the kinetic epoch soon after the end of inflation induces a blue tilt in the
gravitational waves spectrum. This feature enhances the prospects of detection
of gravitational waves in the forthcoming observational missions. Further, a
desirable feature of quintessential inflation should also be mentioned here,
namely, field should track the background in the post inflationary era, and
only at late times, it should exit to acceleration. Several exit mechanism
have bee analysed in the literature. We shall, however, focus on a particular
one due to coupling of scalar field to massive neutrino matter. Last but not
least, one should ask whether the unification scheme is an academic exercise
or has a distinguished observational signature. The presence of the kinetic
regime, a generic feature of quintessential inflation, induces a novel feature
in the spectrum of relic gravity waves, namely, the blue spectrum in the high-
frequency regime. This is a generic model-independent prediction of the
paradigm irrespective of its concrete realization, to be tested in future
observations (see Refs. Chun:2007np ; Kamada:2019ewe  on the related theme ).

This is a pedagogical review written specifically for young researchers.
Efforts have gone into highlighting the general requirements and the
corresponding building blocks of the paradigm rather than discussing the
technical details of concrete models of quintessential inflation. It goes
without saying that the issues raised in the review are equally important for
model building related to late time acceleration. We sincerely hope that the
review will be helpful for beginners interested in related issues. The review
should be read with footnotes, which include additional explanations and
clarifications. The organisation of the review is as follows. In Sec II, we
describe broad features of the paradigm: Suitable class of runaway potentials,
alternative reheating models and exit mechanisms. In Sec III, we discuss
cosmological dynamics of scalar field relevant to quintessential inflation.
This includes field dynamics during inflationary era, some model independent
feature of inflation, nucleosynthesis constraint, scaling and tracker
behaviour in models of quintessential inflation. In Sec. IV, we describe a
mechanism of exit from scaling regime to late time acceleration, which is
realized by invoking coupling to massive neutrino matter. In this section
prospects of detection of relic gravity waves are also discussed. Finally, we
summarise the review in Sec. IV. We use the metric signature, $(-,+,+,+)$ and
the notation for the reduced Planck mass, $M_{\textrm{Pl}}^{-2}=8\pi G$.

##  II  Quintessential inflation: Building blocks

In this section, we broadly indicate the basic features of model building for
quintessential inflation and list the building blocks of the model. Fig.1
shows the scalar field potential, flat at both the early and late time epochs
and steep in between, joining the two ends. One immediate implication of the
runaway type nature of the potential is the requirement for an alternative
reheating mechanism. There are a few alternatives to be mentioned in the
discussion to follow. It is important to note that in the context of warm
inflation Berera:1995wh ; Berera:1995ie  a separate reheating mechanism may
not be required as radiation is being produced during inflation itself due to
the coupling of the inflaton field to the radiation bath. The idea of “warm
inflation” also works in the context of quintessential inflation as well
Lima:2019yyv ; Dimopoulos:2019gpz ; Basak:2021cgk ; Levy:2020zfo ;
Gangopadhyay:2020bxn  We shall also mention specific field tracks that could
join the two phases of acceleration such that thermal history is left without
interference and the late-time physics is insensitive to initial conditions.

Figure 1: The figure displays a typical scalar field potential required to
realise quintessential inflation: potential shallow initially, followed by
steep behaviour thereafter, and shallow again around the present epoch.

####  II.0.1 Reheating through gravitational particle production

Gravitational particle production takes place at the end of inflation due to a
non-adiabatic change of space time geometry Ford:1986sy , (see also
Spokoiny:1993kt ). It is a universal process and does not require the presence
of additional field(s). Energy density of radiation produced in this process
is given by, $\rho_{r}\simeq 0.01\times g_{p}H^{4}_{\rm end}$, where
$g_{p}\sim 100$, the number of relativistic degrees produced in the process;
it involves $H_{\rm end}$, the natural scale available at the end of
inflation. In this case, typically, $\rho_{\phi}/\rho_{r}\simeq 10^{-10}$
Sahni:2001qp  and the process is inefficient and might challenge the
nucleosynthesis constraint due to over production of relic gravity waves.
Other alternatives include, instant preheating, curvaton reheating mechanism
and Ricci reheating.

####  II.0.2 Instant Preheating

Instant preheating Felder:1998vq ; Campos:2002yk  operates with an assumption
that the inflaton field $\phi$ interacts with another scalar field $\chi$
(with coupling $g$) whose mass depends upon $\phi$ such that as $\phi$, after
inflation, runs down its potential, $m_{\chi}$ increases and decays into
matter fields111In this framework, field $\phi$ is coupled to another scalar
field $\chi$ which in turn couples to matter fields:
$\mathcal{L}_{int}=-(1/2)g^{2}\phi^{2}\chi^{2}-h\chi\bar{\Psi}\Psi$. In this
case, $\chi$ does not bare mass, its effective mass depends upon $\phi$,
$m_{\chi}=g\phi$. After inflation, as $\phi$ runs down its potential fast,
$m_{\chi}$ changes in non-adiabatic fashion giving rise to production of
$\chi$ particles which might instantaneously decay into matter fields.
Assuming that energy of produced particles is instantaneously thermalized
(denoted by $\rho_{r}$), one finds at the end of inflation that
$(\rho_{\phi}/\rho_{r})_{end}=(2\pi)^{3}/g^{2}$. ($\chi$ couples to matter
fields with coupling $h$). By appropriately choosing couplings ($g,h<1$), one
can obtain instantaneous preheating at the end of inflation with the reheating
temperature that satisfies the bound implied by the nucleosynthesis
constraint.

####  II.0.3 Curvaton Reheating

The Lagrangian is supplemented with an extra field $\sigma$ with a quadratic
mass term of $m_{\sigma}^{2}\sigma^{2}$, such that $m_{\sigma}\ll H_{inf}$ and
the $\sigma$ field ceases to play any role during inflation Lyth:2001nq ;
Feng:2002nb ; delCampo:2009yc ; Hardwick:2016whe . After inflation ends, the
Hubble parameter decreases $m_{\sigma}\sim H$ and the $\sigma$ field begins to
oscillate and decay into matter fields, giving rise to reheating with desired
temperature at the end of inflation.

####  II.0.4 Ricci Reheating

The problem associated with gravitational reheating, that is over-production
of gravitational waves can be controlled by introducing another reheating
mechanism called Ricci Reheating Dimopoulos:2018wfg ; Bettoni:2021zhq ;
Opferkuch:2019zbd . In this mechanism a scalar field called reheaton is non-
minimally coupled directly with the Ricci scalar. The Ricci scalar changes
sign after the end of inflation when the kinetic energy takes over the
potential in the steep region of the potential. As a result the effective mass
of the reheaton changes sign and becomes tachyonic. Due the tachyonic nature,
reheaton filed grows as the filed rolls down towards new effective minima. The
rehating is realized by the transfer of energy from this reheaton to standard
model particles. It is also possible to identify reheaton with Higgs field
Opferkuch:2019zbd .

####  II.0.5 Independence of late time dynamics from initial conditions

The requirement of steep potential in the post inflationary era is dictated by
the need to achieve the commencement of radiation domination after which
scalar field remains sub-dominant. However, a specific type of steep behaviour
is asked for, which allows the scalar field to follow the background $-$
scaling solution which is an attractor of dynamical system. The
nucleosynthesis constraint, further, puts a lower bound on the slope of these
potentials. Scaling behaviour, for instance, is exhibited by a steep
exponential potential, ($V\sim Exp(-\alpha\phi/M_{Pl};\alpha^{2}>3$)
Ferreira:1997hj ; Copeland:1997et ; Ferreira:1997au ; SS . However, the
scaling solution is non-accelerating as , in this case,
$w_{\phi}=w_{b}=(0,1/3)$ for cold dark matter and radiation, respectively.
Hence, in this case, one needs to invoke an additional feature in the
potential that could trigger an exit to late time acceleration. Once the model
parameters are set to exit to acceleration at the present epoch, changing the
initial conditions within a reasonable range does not affect late-time physics
$\hat{\rm a}$ la a tracker Steinhardt:1999nw ; Chiba:2009gg .

####  II.0.6 Exit mechanisms

There are several ways to realize the exit from scaling regime to accelerated
expansion: (1) one might simply add a cosmological constant to scalar field
Lagrangian or if one is willing to go beyond, one might add to scalar field
potential, a term, $V_{1}\sim
Exp{(-\lambda_{1}\phi/M_{Pl})},\lambda_{1}^{2}<2$ Copeland:1997et ;
Barreiro:1999zs ; Haro:2019peq , such that it becomes operative only at late
times giving rise to slow roll. (2) One might add to Lagrangian a Gauss-Bonnet
term coupled to scalar field: $\xi(\phi)R^{2}_{GB}$ (
${R^{2}_{GB}}=R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}$
) which can induce minimum in the run away potential and might give rise to
late time acceleration by suitable choosing the couplingexitG . However, it
might be difficult to make this term invisible during inflation. (3a) It is
rather straight forward to induce minimum in the scalar field potential by
introducing coupling of scalar field to (cold) matter which is proportional to
trace of matter energy momentum tensor and vanishes identically during
radiation era Gumjudpai:2005ry . In this case, scalar field can settle in the
minimum of the effective potential giving rise to de Sitter like behaviour of
interest to late time cosmology. However, it happens soon after matter
domination is established where it is undesirable as matter phase should be
left intact. (3b) This problem can be circumvented by invoking coupling of
scalar field to massive neutrino matter which builds up dynamically only at
late times as massive neutrinos turn non-relativistic. In this paper, we shall
focus on the dynamics of this process.

####  II.0.7 Suitable class of scalar field potentials

It is difficult to have tractable analytical expression of a potential shallow
initially as required by inflation with a tracker like behaviour in the post
inflationary era. We often encounter the following class of potentials in the
literature. (a) Potentials steep initially with a desired behaviour
thereafter, for instance, inverse power law potentials. The known way to have
accelerated expansion in the initial phase in this case is provided by “Brane
Worlds” Sami:2004xk ; Sahni:2001qp ; Bento:2008yx ; Adhikari:2020xcg , where
high energy brane corrections can give rise to slow roll with a graceful exit
to deceleration as the field runs down its potential and brane damping
disappears. Unfortunately, this scenario gives rise to a large tensor to
scalar ratio of perturbations, which is ruled out by observation
Copeland:2000hn . (b) Class of potentials, flat initially, followed by a steep
behaviour. For instance, $V\sim
Exp(-\lambda\phi^{n}/M_{\textrm{Pl}}^{n}),n>1$Geng:2015fla . Potentials with
the said characteristic can also be constructed using non-canonical form of
scalar field Lagrangian. In this case one requires to invoke an exit mechanism
to late time acceleration. (c) If we do not adhere to tracker like behaviour
in the post inflationary era, it is possible to find a class of potentials
shallow at early and late times and steep in between with a thawing
realization Dimopoulos:2017zvq .

##  III Cosmological dynamics of Scalar field in a nut shell

Scalar field dynamics has played an important role in modern cosmology,
starting with inflation, which led to a paradigm shift resulting in the
integration of cosmology with high energy physics. In what follows, we present
the basic elements of scalar field dynamics with and without the presence of
background matter. Our discussion would revolve around scaling solutions and
slow roll attractors. Special attention would be paid to asymptotic scaling
solutions, one of the important building blocks of quintessential inflation.
However, since the scaling solutions are non-accelerating in nature, one
requires mechanisms of exit to late time acceleration. We discuss in detail
the exit caused by the coupling of the scalar field to massive neutrino
matter. The generic coupling is proportional to the trace of the energy
momentum tensor of massive neutrino matter, which vanishes as long as
neutrinos are relativistic. Coupling acquires non-zero value at late stages of
evolution when neutrinos become non-relativistic. This is a unique physical
process that occurred in the Universe, and neutrino mass is the only available
energy scale close to the mass scale associated with dark energy (see for
review on dark energy, Carroll:2003qq ; Copeland:2006wr ; Sahni:2006pa ;
Sahni:2004ai ; Li:2012dt ; Brax:2017idh ; Mortonson:2013zfa ; Sami:2013ssa ,
see alos Zhang:2021ygh ). The coupling of the scalar field to massive neutrino
matter invokes an important feature in the scalar field potential responsible
for late time acceleration. We will include details in the sub-section to
follow.

###  III.1 Field evolution in absence of background matter: Slow roll versus
fast roll

Scalar dynamics has been used extensively in the development of models for
both inflation and late-time acceleration.In this section, we shall focus on
dynamics of non-minimally coupled scalar field. Let us consider the action of
a minimally coupled scale field $\phi$ which populates the FLRW universe
Ratra:1987rm ; Copeland:2006wr ; samrev1 ; samrev2 ,

$\mathcal{S}=\int{d^{4}x\sqrt{-g}\left[\frac{M_{\textrm{Pl}}^{2}}{2}R+\mathcal{L}(\phi)\right]}\equiv\int{d^{4}x\sqrt{-g}\left[\frac{M_{\textrm{Pl}}^{2}}{2}R-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)\right]}$
(1)

The energy momentum tensor for the scalar field reads,

$T_{\mu\nu}(\phi)=\partial_{\mu}\phi\partial_{\nu}\phi+g_{\mu\nu}\mathcal{L}(\phi)$
(2)

such that the field energy density and pressure are given by,

$T^{0}{{}_{0}}\equiv-\rho_{\phi}=\frac{1}{2}\dot{\phi}^{2}+V(\phi);~{}~{}T^{i}{{}_{i}}\equiv
p_{\phi}=\frac{1}{2}\dot{\phi}^{2}-V(\phi)$ (3)

For FLRW background dominated by the scalar field energy density, we have,

$\displaystyle\left(\frac{\dot{a}}{a}\right)^{2}\equiv
H^{2}=\left(\frac{1}{3M_{\textrm{Pl}}^{2}}\right)\rho_{\phi}$ (4)
$\displaystyle-\frac{\dot{H}}{H^{2}}=\frac{3}{2}(1+w_{\phi})=3\frac{\dot{\phi}^{2}/{2}}{\rho_{\phi}}\Longleftrightarrow\frac{\ddot{a}}{a}=-\frac{1}{6M^{2}_{Pl}}\rho_{\phi}\left(1+3w_{\phi}\right)$
(5)

where the equation of state parameter for the field is defined as,

$w_{\phi}\equiv\frac{P_{\phi}}{\rho_{\phi}}=\frac{\frac{1}{2}\dot{\phi}^{2}-V(\phi)}{\frac{1}{2}\dot{\phi}^{2}+V(\phi)},$
(6)

which interpolates between $1$ and $-1$. The lower (upper) limit corresponds
to the potential (kinetic) energy dominated situation. The field configuration
with $w_{\phi}\simeq-1$ or $-\frac{\dot{H}}{H^{2}}\ll 1$ is referred to
(quasi) de Sitter for which, $H$ is approximately constant and $a(t)\sim
e^{Ht}$ 222Expansion has character of acceleration ($\ddot{a}/a>0$ for
$w_{\phi}<-1/3$), see Eq.(5).. In this case, field rolls slowly,

$\dot{\phi}^{2}/2\ll V(\phi)\Longrightarrow\epsilon\equiv
3\frac{\dot{\phi}^{2}/2}{\rho_{\phi}}\simeq\frac{3}{2}\frac{\dot{\phi}^{2}}{V}\ll
1$ (7)

where $\epsilon$ is one of the slow roll parameters. The (quasi) de Sitter
configuration is admitted by the scalar field dynamics as a possible fixed
point. Indeed, the scalar field equation that follows from (1) in the FLRW
background is,

$\ddot{\phi}+3H\dot{\phi}+V_{\phi}=0;~{}~{}V_{\phi}\equiv\frac{dV}{d\phi}$ (8)

where the second term with Hubble rate acts like a friction coefficient. Field
equation (8) is equivalent to energy conservation,

$\dot{\rho_{\phi}}+3H\rho_{\phi}(1+w_{\phi})=0$ (9)

which formally integrates to,

$\rho_{\phi}=\rho_{in}e^{-3\int{(1+w_{\phi})\frac{da}{a}}}\,,$ (10)

with $\rho_{in}$ as a integration constant. For potential energy dominated
situation, $\dot{\phi}^{2}/2\ll V$ ($w_{\phi}\simeq-1$), field energy density
is approximately constant where as in the opposite case $(\dot{\phi}/2\gg V)$,
dubbed kinetic regime, $\rho_{\phi}\sim 1/a^{6}$. Kinetic regime is realized
in case of a steep potential where field rolls down its potential fast making
the potential energy redundant giving rise to $\rho_{\phi}\sim a^{-6}$.

As mentioned before, we are interested in realizing the quasi de Sitter
configuration. To this effect, we notice that when, friction term in Eq.(8) is
large,

$\beta\equiv-\frac{\ddot{\phi}}{3H\dot{\phi}}\ll
1\Longrightarrow~{}~{}\text{Slow roll
regime}:~{}~{}3H\dot{\phi}+{V_{\phi}}\simeq
0;~{}~{}H^{2}\simeq\frac{1}{3M^{2}_{Pl}}V(\phi)$ (11)

where $\dot{\phi}$ is small and adheres to slow roll condition, namely,
$\epsilon\ll 1$, which in turn implies that the potential is nearly
flat,333The acceleration term, $\ddot{\phi}$ in (8) can no longer be be
dropped in case of a steep potential.

$\epsilon=\frac{M^{2}_{Pl}}{2}\left(\frac{V_{\phi}}{V}\right)^{2}$ (12)

In this case, even if we kick off with a huge acceleration, $\ddot{\phi}$, it
is taken care of by the friction term. Thus slow roll dynamics confirms the
existence of (quasi) de Sitter fixed point also known as slow roll solution.
We should , however, check the consistency of (11) with the field equation
(8). Taking the time derivative of slow roll equation (11) and using Eq.(5),
we have,

$\beta=-\frac{\ddot{\phi}}{3H\dot{\phi}}=+\frac{V_{\phi\phi}}{9H^{2}}-\frac{(1+w_{\phi})}{2};~{}~{}V_{\phi\phi}=\frac{d^{2}V}{d\phi^{2}}$
(13)

Keeping $\beta$ negligible requires,

$\displaystyle\epsilon=\frac{M^{2}_{Pl}}{2}\left(\frac{V_{\phi}}{V}\right)^{2}\ll
1$ (14) $\displaystyle\eta=M^{2}_{Pl}\frac{V_{\phi\phi}}{V}\ll 1,$ (15)

which express the self consistency of slow roll approximation with full scalar
field dynamics. Thus, we are hereby led to two necessary conditions for slow
roll. The first condition, tells us that the slope of the potential small and
potential is shallow ($M_{Pl}|V_{\phi}/V|\ll 1$) where as the second condition
is a statement about the smallness of the curvature which ensures that slow
roll can be sustained for a long duration (
$\dot{\epsilon}/H\epsilon=\mathcal{O}(\eta,\epsilon$ ) needed to collect
required number of e-foldings necessary to address the shortcomings of the hot
big bang model. Smallness of $\eta$ is also dictated by the flatness of
perturbation spectrum. Thus, the smallness of slow roll parameters ensures the
required number of e-foldings and the scale-independent spectrum of adiabatic
density perturbations produced during inflation.

###  III.2  Inflation: Model Independent Features

As mentioned before, we focus on generic features of the paradigm rather than
the concrete models. In what follows, we present the model independent
estimates of inflation to be used later. We recall that the tensor-to-scalar
ratio, $r$ is defined as the ratio of amplitudes of the tensor and scalar
perturbations,

$\displaystyle r\equiv\frac{P_{T}}{P_{S}}$ (16)

which is expressed through $\epsilon$ in the case of canonical scalar field as
$r=16\epsilon$. Observations Planck:2018jri  put a bound on the tensor to
scalar ratio of perturbations, namely, $r\lesssim 0.032$ Tristram:2021tvh
which translates into a restriction on the slope of the potential,
$\lambda_{s}\equiv-M_{\textrm{Pl}}\frac{V,\phi}{V}\lesssim 0.06$ which is a
more tighter bound on the slope than the one imposed by slow roll condition.
The scalar power spectrum,

$\displaystyle
P_{S}=\left.{\frac{1}{8\pi^{2}}\frac{1}{\epsilon}\frac{H^{2}}{M_{\textrm{Pl}}^{2}}}\right|_{k=aH}=\frac{2}{\pi}\,.\frac{1}{r}\frac{H_{\rm
inf}^{2}}{M_{\textrm{Pl}}^{2}}$ (17)

is bound due to CMB observations and adhering to recent findingsPlanck:2018jri
; Tristram:2021tvh , we have $H_{\rm inf}\simeq 5.8\times
10^{-5}\,r^{1/2}M_{\textrm{Pl}}=1.4\times 10^{14}r^{1/2}\rm GeV$ or
equivalently, $V_{\rm inf}^{1/4}=0.01\times r^{1/2}M_{\textrm{Pl}}$. At the
end of inflation, $\epsilon=1$ which implies that $\dot{\phi}_{\rm end}=V_{\rm
end}^{1/2}$ and we have the Hubble parameter at the end of inflation,

$\displaystyle H_{\rm end}=\frac{1}{\sqrt{2}M_{\textrm{Pl}}}V_{\rm
end}^{1/2}.$ (18)

Since inflation is a quasi de-Sitter phase of expansion it is expected H does
not change much from commencement to the end of inflation. The argument is
supported by the observational upper-bound on tensor-to-scalar ratio
$r\lesssim.032$ such that the field excursion during inflation is not much.
This is supported by numerical calculation for generalized exponential
potential which shows that not only $H_{inf}\&H_{end}$ but
$H_{inf},H_{end}\&H_{kin}$ are within the same order of magnitude
Ahmad:2017itq .

Using Eq. (18) we have the estimation for $V_{\rm end}$,

$\displaystyle V_{\rm end}\approx 2H_{\rm
inf}^{2}M_{\textrm{Pl}}^{2}\Rightarrow\rho_{\phi,\rm end}=\frac{3}{2}V_{\rm
end}\approx 3H_{\rm inf}^{2}M_{\textrm{Pl}}^{2}\,$ (19)

In the kinetic epoch followed by inflation, the energy densities of the filed
and radiation evolve as

$\displaystyle\rho_{\phi}$ $\displaystyle=$ $\displaystyle\rho_{\phi,\rm
end}\left(\frac{a_{\rm end}}{a}\right)^{6}$ (20) $\displaystyle\rho_{r}$
$\displaystyle=$ $\displaystyle\rho_{r,\rm end}\left(\frac{a_{\rm
end}}{a}\right)^{4}$ (21)

Thus, the ratio of the scale factors at the end of inflation to the
commencement of radiation epoch or that of the respective temperatures are
obtained by using $\rho_{\phi}(a_{r})=\rho_{r}(a_{r})$ which gives

$\displaystyle\frac{a_{r}}{a_{\rm end}}=\frac{T_{\rm
end}}{T_{r}}=\left[\frac{\rho_{\phi,\rm end}}{\rho_{r,\rm
end}}\right]^{1/2}\,.$ (22)

The length of kinetic regime depends upon the ratio of field energy density to
the radiation energy density at the end of inflation. Efficient reheating
process implies smaller value of this ratio corresponding to shorter kinetic
regime. Owing to the steep nature of scalar field potential in the post-
inflationary era, field runs down its potential fast after inflation ends and
in a short while, enters the kinetic. In fact, it is found numerically that
for the potential (69) ${H_{\rm inf}}~{},{H_{\rm end}}~{}\&~{}H_{\rm kin}$ are
within the same order of magnitude Ahmad:2017itq . We shall use, ${H_{\rm
inf}}\simeq{H_{\rm end}}\simeq~{}H_{\rm kin}$ in our estimations for model
independent predictions.

###  III.3 Evolution in presence of background matter: Scaling solution

Scalar field dynamics in presence of background matter (cold matter/radiation)
exhibits distinguished features. Under specific conditions scalar field might
mimic background matter $\hat{\rm a}$ la scaling solution which plays an
important in model building for dark energy and quintessential inflation. In
order to check for scaling behaviour of $\rho_{\phi},$, let us consider
evolution equations in the presence of background matterCopeland:2006wr ;
samrev1 ; Copeland:1997et ; Liddle:1998xm ; Skugoreva:2019blk

$\displaystyle H^{2}=\frac{1}{3M^{2}_{Pl}}\left(\rho_{b}+\rho_{\phi}\right)$
(23)
$\displaystyle\dot{H}=-\frac{1}{2M^{2}_{Pl}}\left((1+w_{B})\rho_{b}+\dot{\phi}^{2}\right)$
(24) $\displaystyle\dot{\rho}_{b}+3H\rho_{b}(1+w_{b})=0$ (25)

where $\rho_{b}$ and $w_{b}$ designate energy density and equation of state
parameter for background matter respectively. As for the scalar field
equation, background dependence is reflected there through Hubble parameter.
It follows from continuity equation for background matter that $\rho_{B}\sim
a^{-3(1+w_{b})}$. Scaling solution is defined requiring that scalar field
evolves exactly as the background matter,

$\text{ Scaling solution}:~{}~{}w_{\phi}=w_{b};~{}~{}\rho_{\phi}\sim
a^{-3(1+w_{b})}$ (26)

Scaling solution is a specific dynamical arrangement which preserves the ratio
of kinetic to potential energy of the field,

$w_{\phi}\equiv\frac{P_{\phi}}{\rho_{\phi}}=\frac{\frac{1}{2}\dot{\phi}^{2}-V(\phi)}{\frac{1}{2}\dot{\phi}^{2}+V(\phi)}=w_{b}\Rightarrow\dot{\phi}^{2}(1-w_{b})=2V(1+w_{b})$
(27)

We recall from earlier discussion that the interplay between the friction term
and the gradient of potential reflects the nature of the solution of the field
equation; the scaling solution should impose certain restrictions on their
ratio. Indeed, differentiating, (27) left right and using the field equation
(8), we have,

$(3H\dot{\phi}+V_{\phi})\dot{\phi}(1-w_{b})=-(1+w_{b})V_{\phi}\dot{\phi}\Rightarrow-3(1-w_{b})H\dot{\phi}^{2}=2V_{\phi}\dot{\phi}$
(28)

Eqs (27) and (28) yield two important properties of scaling solutions, namely
Skugoreva:2019blk ,

$\displaystyle X\equiv\frac{\dot{\phi}^{2}}{2V}=\frac{1+w_{b}}{1-w_{b}}$ (29)
$\displaystyle Y\equiv\frac{V_{\phi}}{\dot{\phi}H}=\frac{3}{2}(w-1)$ (30)

Let us also note that for scaling solution.

$\displaystyle-\frac{\dot{H}}{H^{2}}=\frac{3}{2}(1+w_{b})$ (31)
$\displaystyle\frac{\ddot{\phi}}{\dot{\phi}H}=-\frac{3}{2}(1+w_{b})\,.$ (32)

These equation readily yield the behaviour of scalar field in the scaling
regime Skugoreva:2019blk ; Haro:2019peq ,

$H(t)=\frac{1}{3(1+w_{b})t};~{}~{}\dot{\phi}\propto
t^{-1}\Rightarrow\phi(t)\propto\ln(t)$ (33)

So far, we did not need any information on the field potential that would
support the scaling behaviour. It is interesting to compute the
$\Omega_{\phi}$ for scaling solution that would tell us about the fraction of
total energy, $\rho_{\phi}$ constitutes and also the nature of the potential.
Using Eqs.(29) $\&$ and (30), we find,

$\Omega_{\phi}=\frac{\rho_{\phi}}{3M^{2}_{Pl}H^{2}}=\frac{1}{3M_{\textrm{Pl}}^{2}}\left(\frac{XY^{2}}{1-w_{b}}\right)\left(\frac{V}{V_{\phi}}\right)^{2}=\frac{3(1+w_{b})}{4\lambda^{2}_{s}};~{}~{}\lambda_{s}\equiv-
M_{Pl}\frac{V_{\phi}}{V}$ (34)

where $\lambda_{s}$ is the slope of the potential. Since $\rho_{\phi}$ scales
as the energy density of background matter in the scaling regime,
$\Omega_{\phi}=Const$, which in turn implies that the slope of the potential
should be constant and consequently, the exponential potential is singled out,

$V(\phi)=V_{0}e^{-\alpha\phi/M_{Pl}}~{}~{}~{};~{}\lambda_{s}=\alpha$ (35)

However, an important information is yet missing in our analysis. For
instance, if the slope is small, namely, $\alpha^{2}<2$, slow roll condition
applies and we have $w_{\phi}<-1/3\to\dot{\phi}^{2}/2<V/2$ and no scaling
solution in this case. This brings us to the question of the existence and
stability of the scaling solutions we are interested in, which is addressed by
dynamical analysis using the autonomous form of evolution equations. Using the
following notations Copeland:1997et 444It might look more natural to use the
variable $X\&Y$ as we know their values for scaling solution. However, in this
case, one requires one more variable $A$ which is directly linked to the field
$\phi$. But the equation for this variable decouples from the system and we
can analyse equations for $X\&Y$ without bothering about the third equation
for the variable $A$. These variables are useful in the analysis of asymptotic
scaling solutions which occur in case of steep potentials of variable slope
where we have system of three coupled equations for $X,Y$ and $A$.,

$x\equiv\frac{\dot{\phi}}{\sqrt{6}M_{Pl}H};~{}~{}~{}y\equiv\frac{\sqrt{V}}{\sqrt{3}M_{Pl}H},$
(36)

one can cast evolution equations, with exponential potential, in the
autonomous form,

$x^{\prime}=f(x,y);~{}~{}~{}y^{\prime}=g(x,y)$ (37)

along with the Friedman constraint equation,

$\frac{\rho_{B}}{3M^{2}_{Pl}H^{2}}+x^{2}+y^{2}=1$ (38)

where prime denotes the derivative with respect to $N\equiv\ln(a)$; $f$ and
$g$ are functions of $x$ $\&$ $y$ whose functional forms are not required
here. Fixed point of the dynamical system is the one for which,
$x^{\prime},y^{\prime}=0$ whose stability is checked looking at the sign of
eigen values of the perturbed matrix. Using this analysis, one finds
Copeland:1997et ; Liddle:1998xm ; Haro:2019peq ,

$\text{Scaling
solution}:~{}~{}~{}~{}~{}~{}w_{\phi}=w_{m};~{}~{}~{}\Omega_{\phi}=\frac{3(1+w_{b})}{\alpha^{2}}~{}~{}~{}\lambda^{2}_{s}=\alpha^{2}>3$
(39)

which is an attractor of the dynamical system and this completes the story. In
the scaling regime, $\rho_{\phi}$ constitutes a constant fraction of the total
energy density. The fraction is smaller for the larger value of the slope of
the potential. In fact, lower bound on the slope is fixed by the
nucleosynthesis constraint. Exponential potential with slope larger than
$\sqrt{3}$ is termed as steep otherwise shallow which also serves as yardstick
for an arbitrary potential. Let us note that field dominated solution
($\Omega_{\phi}=1$) is also an attractor for $\alpha^{2}<{3}$ that we are not
interested in here. The demarcation is clear: For $\alpha^{2}>3$, system
settles into a scaling regime, otherwise a field dominated solution, which is
accelerating if $\alpha^{2}<2$.

To summarize, scaling solution follows the background matter being sub-
dominant. During radiation era, field energy density scales as radiation
energy density ($\rho_{r}$), namely, $\rho_{\phi}\sim a^{-4}$ which adjusts
itself to $\rho_{m}$ (energy density of cold matter) after matter domination
is established and follows it for ever. Scaling regime is characterized by a
fixed kinetic to potential energy ratio, for instance, $\dot{\phi}^{2}/2=V$ in
radiation era whereas, $\dot{\phi}^{2}/2=2V$ for matter domination. Last but
not least, since scaling solutions are non-accelerating, one requires an
additional feature in the potential which would allow exit to late time
acceleration. In this framework, $\Gamma=1$ and $w_{\phi}=w_{b}$ for most of
the history of Universe and only at late times, transition to acceleration
would take place $\hat{\rm a}$ la a perfect tracker  In the forthcoming
discussion, we shall mention exit mechanism discussed in the literature.

###  III.4 Nucleosynthesis Constraint on Extra Species

One of the major successes of hot big bang includes the synthesis of light
elements, Deuterium (${}^{2}\rm{H}$), Tritium (${}^{3}\rm{H}$) and Helium
(${}^{4}\rm{He}$) in the early Universe. When temperature in the Universe was
around $1~{}\rm{TeV}$, all the standard model degrees of freedom were in
equilibrium. All of them were relativistic and contributed to radiation energy
density Kolb:1990vq ; Husdal:2016haj ,

$\displaystyle\rho_{r}=\frac{\pi^{2}}{30}g_{*}(T)T^{4}$ (40)

where $g_{*}(T)$ denotes the effective number of relativistic degrees of
freedom in the universe at temperature T. Relativistic species including both
the bosons and the fermions contribute to $g_{*}(T)$,

$g_{*}(T)=\sum_{Bosons}{g_{B}{\left(\frac{T_{i}}{T}\right)^{4}}}+\frac{7}{8}\sum_{Fermions}{g_{F}{\left(\frac{T_{i}}{T}\right)^{4}}},$
(41)

where $T$ is the photon gas temperature and $g_{B}~{}\&~{}g_{F}$ denote
bosonic and fermionic degrees of freedom with mass, $m_{i}\ll T$ ($m_{i}$ is
the mass of a particular species). The effective degrees of freedom is defined
relative to the photon gas. When temperature dropped to around $1$ $MeV$, the
relativistic species included $\gamma$, $e,e^{+}$ and three generations of
neutrinos $\nu$ such that, $g_{*}=10.75$. Weak processes such as,
$p+e^{-}\leftrightarrow n+\nu_{e}$ then froze out leaving behind a definite
ratio of $n/p$ densities. Freezing temperature can be estimated comparing the
reaction rate $\Gamma_{int}=G_{F}^{2}T^{5}_{f}$ with the Hubble parameter
$H(T)$ Cyburt:2015mya ,

$G_{F}^{2}T_{f}^{5}=\frac{g_{*}^{1/2}}{\sqrt{3}M_{\textrm{Pl}}}T_{f}^{2}\Rightarrow
T_{f}=\left(\frac{g_{*}^{1/2}}{\sqrt{3}G^{2}_{F}M_{\textrm{Pl}}}\right)^{1/3}\simeq
1~{}MeV$ (42)

where $G_{F}$ is Fermi constant and $T_{f}$ designates the freezing
temperature. It should be noted that the $n/p$ density ratio is temperature
sensitive, as $n/p\sim Exp(-Q/T_{f}$), where $Q$ denotes the neutron and
proton mass difference. Since $T_{f}$ is less than the binding energy of
Deuterium, its synthesis should naively begin. However, there is more than a
billion photons per proton ($\eta=n_{B}/n_{\gamma}\simeq 6\times 10^{-10}$,
$n_{B}$ designated baryon number density) in the universe such that the number
of photons whose energy exceeds $T_{f}$ is large and Deuterium does not form
due to photo-dissociation. This process is delayed and occurs at a lower
temperature, $T_{BBN}\simeq 0.1eV$, which then initiates the chain reaction
for Helium formation. Helium abundance in Universe has been measured
accurately; it is sensitive to $T_{f}$, for obvious reasons, it also depends
upon $\eta$. We should emphasize that $\eta$ is an important input here
required for a successful nucleosynthesis. It is further important to note
that bringing in any extra relativistic degree of freedom like a scalar field
would affect the numerical value of $T_{f}$ and consequently the Helium
abundance. The latter should, therefore,impose a constraint on any
relativistic degree of freedom in the Universe over and above the standard
model of particle physics $-$ nucleosynthesis constraint which, in particular,
applies to often used quintessence as well as relic gravity waves. Indeed, for
temperatures under consideration,

$g_{*}=2+\frac{7}{8}\left(4+2N_{\nu}\right)$ (43)

where the first term is due to photon, first term in the brackets is
attributed to electron and positron; $N_{\nu}=3$ in the standard model. Any
extra radiation present in the Universe, can be parametrized through
$N_{\nu}$Caprini:2018mtu

$\Delta\rho_{r}=\frac{\pi^{2}}{30}\frac{7}{4}\Delta N_{\nu}T^{4}$ (44)

where $\Delta N_{\nu}$ is anything over and above the standard model value of
$N_{\nu}$ ($N_{\nu}=3$) which should be constrained from the observed
primordial abundances and other data. Let us now assume an extra relativistic
degree of freedom in the Universe dubbed ”X” with energy density $\rho_{X}$
which should not exceed $\Delta\rho_{r}$ as a necessary requirement for not
disturbing the Helium abundance Caprini:2018mtu ,

$\left(\frac{\rho_{X}}{\rho_{\gamma}}\right)_{1~{}MeV}\leq\frac{7}{8}\Delta
N_{\nu}$ (45)

where $\rho_{\gamma}=(\pi^{2}/15)T^{4}$ is the energy density for photon
gas555Given that $\Delta N_{\nu}$ is constrained using primordial abundances
and other data, bound (45) is referred to as a ”nucleosynthesis” constraint or
”BBN” bound.. Let us note that Helium abundance depends upon two parameters,
$T_{f}$ and $\eta$ which can be used to constrain $\Delta N_{\nu}$. The
constraint can be improved considerably by considering, in addition, the
Deuterium abundance and CMB data Cyburt:2004yc . However, in this case, we
should estimate the the ratio (45) below the $e^{+}e^{-}$ annihilation
temperature, $T_{e^{+}e^{-}}$. After electron positron annihilation,
$T_{\nu}=(4/11)^{4/3}T$ such that,

$\left(\frac{\rho_{X}}{\rho_{\gamma}}\right)_{T_{e^{+}e^{-}}}\leq\frac{7}{8}\left(\frac{4}{11}\right)^{4/3}\Delta
N_{\nu}$ (46)

Observational constraint on $\Delta N_{\nu}$ should put a restriction on the
extra degree of freedom not the part of standard model. Remember that in
scalar field models with tracking behavior, $\rho_{\phi}\propto a^{4}$during
the radiation era, and the fraction of total energy that $\rho_{\phi}$
comprises is determined by the slope of the steep exponential potential
responsible for scaling behavior. Using combined data, gives $N_{\nu}<3.2$ at
$95\%$ confidence level which implies a bound on the slope of the
potentialCaprini:2018mtu ,

$\left(\frac{\rho_{X}}{\rho_{\gamma}}\right)_{T_{e^{+}e^{-}}}\simeq\Omega_{\phi}|_{T_{e^{+}e^{-}}}=\frac{4}{\alpha^{2}}\lesssim\frac{7}{8}\left(\frac{4}{11}\right)^{4/3}\times
0.2\Rightarrow\alpha\gtrsim 10.$ (47)

It may be noted that in case of thawing models, $\Omega_{\phi}\ll 1$ and the
$BBN$ constraint is trivially satisfied. Similarly, for relic gravity waves,
$\rho_{GW}$ should satisfy the following bound,

$\left(\frac{\rho_{GW}}{\rho_{\gamma}}\right)_{T_{e^{+}e^{-}}}\leq\frac{7}{8}\left(\frac{4}{11}\right)^{4/3}\Delta
N_{\nu}$ (48)

which can be translated into a bound on $\Omega_{GW,0}$,

$\Omega_{GW,0}=\frac{7}{8}\left(\frac{4}{11}\right)^{4/3}\Omega_{\gamma,0}\Delta
N_{\nu}\simeq 5.6\times 10^{-6}\Delta N_{\nu}$ (49)

where we used the fact that $\rho_{GW}\sim\rho_{\gamma}\sim a^{-4}$ and also
used the observed value of fractional energy density of radiation at the
present epoch, $\Omega_{\gamma,0}\simeq 2.47\times 10^{-5}$ Using the combined
abundances of Helium and Deuterium with the CMB data $(\Delta N_{\nu}\lesssim
0.2$), we finally have,

$\Omega_{GW,0}\lesssim 1.12\times 10^{-6}$ (50)

which puts an upper bound on the kinetic regime or the lower bound on
reheating temperature in models of quintessential inflation, see section V for
details.

###  III.5 Dynamics with general class of potentials and emergence of scaling
behaviour in the asymptotic regime

As discussed in the preceding sub-section, scaling solutions are specific to
exponential potential, which are either suited to field dominated situations
or scaling behaviour but not to both simultaneously. Thus, we might use a
potential suitable for inflation that subsequently changes to exponential form
soon after inflation ends or think of a non-exponential type of potential
capable of addressing the inflationary requirements at an early time and
dynamically mimicking the exponential behavior in the post inflationary era.

Exponential potential has a distinguished feature, its slope,
$\lambda_{s}\equiv-M_{Pl}V_{\phi}/V$, is constant. In this case, we have two
dynamical variable, $x$ and $y$ and correspondingly two autonomous equations.
For a potential which not an exponential, slope becomes a variable quantity
and one needs an additional dynamical equations for
$\lambda_{s}$Copeland:2006wr ; gammascale1 ; Nunes:2000yc ; Ng:2001hs ,

$\displaystyle
x^{\prime}=f(x,y,\lambda_{s});~{}~{}~{}y^{\prime}=g(x,y,\lambda_{s})$ (51)
$\displaystyle\lambda^{\prime}_{s}=-\sqrt{6}\lambda^{2}_{s}(\Gamma-1)x$ (52)
$\displaystyle\Gamma=\frac{V_{\phi\phi}V}{V^{2}_{\phi}}$ (53)

where $\Gamma$ is an important field construct which controls the dynamics of
the system with a general type of potential. For exponential function,
$\Gamma=1$, corresponding to constant slope of the potential making Eq.(52)
redundant. In this case, $\rho_{\phi}$ scales as background energy density
$\rho_{b}$ giving rise to scaling solution which is an attractor of dynamics.
Without the loss of generality, we may consider a class of potentials in which
field $\phi$ rolls down the potential from smaller to larger values
($\phi\to\infty$) such that $x>0$. In case of potentials with $\Gamma>1$, for
instance,

$V=V_{0}\frac{M^{n}_{Pl}}{\phi^{n}};~{}~{}\Gamma=1+1/n~{}(n>1),$ (54)

slope decreases with evolution, see Eq.(52), eventually, giving rise to slow
roll and forcing the field to evolve to quasi de Sitter
($\rho_{\phi}\simeq\textrm{const.},w_{\phi}\simeq-1$) which is an
attractorNunes:2000yc . While the field is rolling down the steep part of the
potential ($\lambda_{s}$ is large), $\rho_{\phi}$ closely follows the
background Steinhardt:1999nw ,

$w_{\phi}=\frac{2\Gamma}{2\Gamma-1}=w_{b}+\frac{2}{n}(1-w_{b})$ (55)

It should be noted that for generic values of $n$, $w_{\phi}\simeq w_{b}$ and
$\Gamma\simeq 1$666Notice that, $w_{\phi}=w_{b}$ for $n\to\infty$ which is not
surprising as power law corresponds to exponential in this limit. . However,
in this case ($\Gamma>1$), and slope ($\lambda_{s}=n/\phi$ ) decreases with
evolution such that the field enters the slow roll regime at late stages,

$w_{\phi}=-1+\frac{\lambda^{2}_{s}}{3}\to-1,~{}~{}\lambda_{s}\to 0$ (56)

and can account for late time acceleration. Scalar field solution, that mimics
the background in the post inflationary era and only at late stages overtakes
it to account for the observed late time acceleration, has been assigned a
nomenclature as tracker, see Fig.2(a). Obviously, the class of potentials with
$\Gamma>1$ are of interest and have been widely investigated in the
literature.

On the other hand, if $\Gamma<1$, slope of the potential increases with
evolution such that field runs down its potential faster and faster making the
potential energy redundant and naturally ends up in the kinetic regime with,
$\rho_{\phi}\sim a^{-6}$. In this case, it follows from (53) that slope
increases,

$\frac{1}{\lambda_{s}}=\int{x(\Gamma-1)}dN$ (57)

and in process of evolution as $\lambda_{s}$ becomes large and $\Gamma\to 1$.
In this case, system would eventually join the scaling track in the asymptotic
regime. We shall demonstrate it in an example. On the other hand, during
evolution, $\lambda_{s}\to 0$ if we begin from $\Gamma>1$ ( $\Gamma\neq
Constant$), and in this case $\Gamma$ increases towards infinity. In a sense,
”$\Gamma$” defines the type of steepness of the potential: Slope tells us
about the steepness of the potential, whereas $\Gamma$ expresses whether, with
evolution, steepness is unchanged ($\Gamma=1$), increases ($\Gamma<1$), or
decreases ($\Gamma>1$). Potentials with $\Gamma<1$ ($\Gamma>1)$ are sometimes
referred to in the literature as steeper (shallower) than the exponential
potential for brevity. It is interesting to think of slope $\lambda_{s}(\phi)$
as defined over the smooth set of functions $\\{V(\phi\\}$. Since,

$\frac{d\lambda_{s}}{d\phi}=-\lambda^{2}_{s}(\Gamma-1)=0\to\Gamma=1,$ (58)

slope, $\lambda_{s}$, is extremum for an exponential function. The second
derivative,

$\frac{d^{2}\lambda_{s}}{d\phi^{2}}=-\frac{1}{V^{4}_{\phi}}\left(V^{2}_{\phi}(V_{\phi\phi\phi}V+V_{\phi\phi}V_{\phi})-2V^{2}_{\phi\phi}V_{\phi}V\right)$
(59)

computed for exponential potentials also vanishes. Thus $\Gamma=1$ is saddle
point in the functional space which explains the mentioned features of the
dynamics.

It is desirable that for most of the history of Universe, field follows the
background (scaling behaviour) and only at late stages, it exits the scaling
regime, takes over the background to become the dominant component and might
mimics dark energy. In what follows, we discuss as how to incorporate this
feature in the dynamics.

####  III.5.1 Making a perfect tracker from scaling solutions:
$\Gamma=1~{}\&~{}\Gamma<1$

In view of the aforesaid, one needs a steep exponential type of behaviour in
the post inflationary era with a mechanism of late time exit to acceleration
$\hat{\rm a}$ la tracker not sensitive to initial conditions. In the framework
of quintessential inflation, one could design a potential that changes to
shallow exponential one suitable to inflation at early epochs and reduces to a
steep exponential function in the post inflationary era. It is easier to
accomplish this program using non-canonical scalar field actionGeng:2015fla ;
Wetterich:2013jsa ,

$\displaystyle\mathcal{S}=\int{d^{4}x\sqrt{-g}\left[\frac{M^{2}_{Pl}}{2}R-k^{2}(\phi)\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)\right]}$
(60) $\displaystyle
k^{2}(\phi)=\left(\frac{\alpha^{2}-\tilde{\alpha}^{2}}{\tilde{\alpha}^{2}}\right)\frac{1}{1+\beta^{2}e^{\alpha\phi/M_{Pl}}}+1$
(61) $\displaystyle V=M^{4}_{Pl}e^{-\alpha\phi/M_{Pl}}$ (62)

where, $\alpha,\tilde{\alpha}$ and $\beta$ are constants to be chosen by the
requirement of model building. Let us use the following transformation,

$\displaystyle\sigma=f(\phi)$ (63) $\displaystyle k^{2}=\frac{\partial
f}{\partial\phi}$ (64)

to change the action to canonical form,

$\mathcal{S}=\int{d^{4}x\sqrt{-g}\left[\frac{M^{2}_{Pl}}{2}R-\partial_{\mu}\sigma\partial^{\mu}\sigma-V(f^{-1}(\sigma))\right]}$
(65)

The effect of canonicalization is pushed into the potential which we need to
to work out in the small and large field limits to demonstrate the viability
of the underlying construction to early and late time dynamics. To this
effect, we need to express the canonical field $\sigma$ through field $\phi$
for $\tilde{\alpha}<1$ and $\alpha\gg\tilde{\alpha}$. In the small field limit
where we expect inflation, we have777We skip details here and refer the reader
to Geng:2015fla  for the same.,

$\displaystyle\phi\ll-2M_{Pl}\ln\beta/\alpha\Rightarrow
k^{2}(\phi)\simeq\frac{\alpha^{2}}{\tilde{\alpha}}$ (66)
$\displaystyle\sigma(\phi)\simeq\frac{\alpha}{\tilde{\alpha}}\phi\Rightarrow
V\propto e^{-\tilde{\alpha}\sigma/M_{Pl}}$ (67)

which can give rise to inflation for $\tilde{\alpha}<1$. In the large field
approximation,

$\phi\gg-2M_{Pl}\ln\beta/\alpha\to k^{2}(\phi)\simeq 1\Rightarrow V\propto
e^{-\alpha\sigma/M_{Pl}},$ (68)

with a desired scaling behaviour in the post inflationary era. It is therefore
not surprising that Lagrangian (62) can successfully describe inflation at
early times and with scaling behaviour in the post-inflationary era.

In the canonical formulation, an interesting possibility is provided by a
class of non-exponential type of potentials flat initially followed by steep
behaviour which is exponential like in the asymptotic regime, $\Gamma\to 1$
for large values of the field. To this effect, it is interesting to examine a
class of generalized exponential potentialsGeng:2015fla ,

$V(\phi)=V_{0}e^{-\alpha\phi^{n}/M_{Pl}^{n}};~{}~{}~{}n>1$ (69)

which can support slow roll ($\lambda_{s}=\alpha\phi^{n-1}$) at early times
when $\phi$ is small followed by a steep behaviour after inflation ends. In
this case, slope ($\lambda_{s}(\phi)$) is a dynamical variable whose evolution
is dictated by $\Gamma$,

$\Gamma(\phi)=1-\frac{n-1}{n\alpha}\left(\frac{M_{Pl}}{\phi}\right)^{n}$ (70)

which approaches one from below for large values of $\phi/M_{Pl}$. Before the
asymptotic region is reached, $\Gamma<1$ and (69) exhibits the properties of a
function steeper than the exponential potential pushing the field into kinetic
regime. However, for large values of the field (69) dynamically mimics the
behaviour of an exponential potential ($\Gamma\to 1,\phi\to\infty$) allowing
the field to finally catch up with the background. Does the field continue in
kinetic regime when $\Gamma<1$ ? In what follows, we describe a phenomenon
that occurs when the field $\phi$ runs down a steep potential in the presence
of background matter density. Thereafter, we shall be able to narrate the
complete story of field dynamics in a steep potential.

###  III.6 Slow roll in presence of background matter: Freezing regime

The formalism of slow roll parameters can be applied to quintessence, keeping
in mind that the Hubble rate is not solely defined by scalar field energy
density, background matter density also contributes to it.

(a) tracker

(b) Thawing

Figure 2:  Fig(a) shows evolution of field energy density $\rho_{\phi}$ and
background matter density (radiation/cold matter) versus $\ln(a)$ for a
potential shallower than the exponential potential ($\Gamma>1$), for instance,
inverse power law potentials, $V\sim\phi^{-n},n>0$ with $\Gamma=1+1/n$. After
the recovery from freezing regime, field yet on the steep part of potential
evolves with equation of state parameter close to that of the background. In
this case, slope of the potential gradually decreases and scalar field slowly
moves towards the background and overtakes it and joins the slow roll with
diminishing value of the slope $\hat{\rm a}$ la a tracker. Fig.(b) shows
schematic representation of thawing quintessence where the field is frozen on
a flat potential due to Hubble damping. When $\rho_{\phi}$ approaches the
background energy density at late times, the field resumes slow roll and
accounts for late time acceleration.

$\begin{array}[]{c@{\hspace{.1in}}cc}\includegraphics[width=202.35622pt,height=202.35622pt]{exposcale.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=202.35622pt,height=202.35622pt]{assymscaling.pdf}\\\
\mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)&\end{array}$

$\begin{array}[]{c@{\hspace{.1in}}cc}\includegraphics[width=202.35622pt,height=202.35622pt]{exitexpo.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=202.35622pt,height=202.35622pt]{exitassym.pdf}\\\
\mbox{(}c)\hfil\hskip 7.22743pt&\mbox{(}d)\end{array}$

Figure 3: Figure shows the qualitative behaviour of $\rho_{\phi}$ along with
the energy density of background matter (cold matter/radiation) versus $\ln a$
for a steep potential. As $\rho_{\phi}$ overshoot the background energy
density ($\rho_{\phi}\ll\rho_{b}$ $\rho_{b}$ designates background matter
energy density), field freezes on its potential due to Hubble damping. After
the recovery from freezing, field evolution crucially depends upon the nature
of steepness of potential. (a) In case of the exponential potential
($\Gamma=1$), field catches up with the background and tracks it for ever.
Fig.(b) exhibits the general feature of scalar field dynamics for a field
potential steeper than the exponential potential ($\Gamma<1$): after recovery
from freezing, $\rho_{\phi}$ evolves in steps (down and right) and eventually
catches up with the backgroundGeng:2015fla  $-$ asymptotic scaling solution.
Figs. (c) $\&$ (d) show exit from scaling and asymptotic scaling regimes to
late time acceleration at the present epoch $\hat{\rm a}$ la a perfect
tracker. Exit is triggered due a late time feature in the runaway potential
that makes it shallow at late stages of evolution.

To be fair, unlike in the case of inflation, slow roll parameters might be
useful here only for a broad perspective. An interesting feature of field
dynamics is noticed for field rolling down a steep potential when its energy
density is negligible compared to background energy density. In this case,
field freezes on its potential and waits there for conducive situation to
resume motion. Indeed, in case of slow roll888It should be noted that, in the
present situation, $-\dot{H}/H^{2}=(1+w_{b})/2$, is not related to slow roll
parameter due to the presence of background matter. In case of slowly rolling
quintessence, the friction term need not to be large and $\beta$ may not be
negligible. For tracker models, $|\beta|\ll 1$, however, for thawing
quintessence, $\beta=\mathcal{O}(1)$ and $\beta$ is nearly constant,
$\dot{\beta}/\beta H\ll 1$. Unlike the thawing case, the consistency of slow
roll gives rise to, $\eta=3(1+w_{b})/2=3/2$ in case of the trackers.

$\dot{\phi}^{2}/2\ll
V\Longrightarrow\epsilon=3\frac{\dot{\phi}^{2}/2}{\rho_{\phi}}\simeq\frac{3}{2}\frac{\dot{\phi}^{2}}{V}=\frac{1}{6}\frac{V^{2}_{\phi}}{VH^{2}},$
(71)

where we used the slow roll value of $\dot{\phi}$. In what follows, we shall
be interested in the case when $\rho_{\phi}\ll\rho_{b}$ such that,
$H^{2}\simeq\rho_{b}/3M^{2}_{Pl}$,

$\displaystyle\epsilon=\frac{M^{2}_{Pl}}{2}\left(\frac{V_{\phi}}{V}\right)^{2}\left(\frac{\rho_{\phi}}{\rho_{B}}\right)\equiv\epsilon_{0}\left(\frac{\rho_{\phi}}{\rho_{B}}\right)$
(72)
$\displaystyle\epsilon=\frac{\lambda^{2}_{s}}{2}\left(\frac{\rho_{\phi}}{\rho_{b}}\right)$
(73)

Since, we are dealing with a steep potential, $\epsilon_{0}>1$, but
interestingly, $\epsilon\ll 1$ as $\rho_{b}\gg V(\phi)$ $-$Hubble damping.
Thus, unlike the case of inflation, slow roll becomes operative here due to
Hubble damping. However, this is not useful for dark energy as
$\rho_{\phi}\ll\rho_{b}$ in this case, but it is certainly useful for the
understanding of scalar field evolution in a steep potential999Let us note
that this feature is central to thawing models where field is frozen on a
shallow potential such that field begins slow roll after it covers from Hubble
damping and accounts for late time acceleration. Initial conditions are set
specially or tuned allowing it to happen around the present epoch and model
parameters are chosen to comply with observation. As a result of Hubble
damping, field gets frozen on its potential and waits there till the ratio,
$\rho_{\phi}/\rho_{b}$ acquires a specific numerical value ($\rho_{b}\sim
a^{-3(1+w_{B})}$, $\rho_{\phi}=Const$) given by Eq.(73) when slow roll is
violated ($\epsilon=1$). Field evolution then commences but, hereafter, field
dynamics, crucially depends upon the type of steepness of the potential. We
shall come back to this point in the sub-section on post inflationary
evolution.

####  III.6.1 Recovery from freezing regime

For the sake of illustration, let us focus on a steep exponential potential.
After recovering from the freezing regime, which happens due to slow roll
violation, scalar field joins the scaling track. But when it happens,
crucially depends upon the numerical value of the slope of the potential.
Indeed, slow roll (72) is violated when,

$\epsilon\simeq
1\Rightarrow\left(\frac{\rho_{\phi}}{\rho_{b}}\right)\simeq\frac{2}{\alpha^{2}};~{}~{}\lambda^{2}_{s}=\alpha^{2}>3,$
(74)

which tells us that larger is the numerical value of $\alpha$, smaller is the
ratio $\rho_{\phi}/\rho_{b}$, when field resumes evolution. Actually, when
slow roll ends, scalar field begins running down the steep potential, its
kinetic energy gradually increases to scaling
value101010$\dot{\phi}^{2}/2=2V,V$ for radiation and matter domination
respectively. and it joins the scaling regime, it has no other option as
scaling solution is an attractor in this case. Hence, as the field recovers
from freezing and resumes evolution, it always finds itself on the entry to
scaling regime irrespective of the numerical value of the ratio,
$\rho_{\phi}/\rho_{b}$. As a result, even if the field energy density
constitutes a negligible fraction of the total energy, scalar field joins the
scaling track. In fact, this phenomenon has simple physical meaning, namely, a
particle can move in a medium despite large friction if force acting on it
(gradient of the force field potential) is large. This remark about the
interplay between the slope of potential and ratio of field to background
matter energy densities is specially important in relation to asymptotic
scaling solutions that we shortly discuss.

####  III.6.2 Slowly rolling quintessence

Though slow roll parameters do not play a similar roll for quintessence, they
are still helpful for a general perspective. The slow roll conditions should
be used with caution here because, unlike inflation, the Hubble parameter
receives contribution from background matter in this case. Quintessence models
can be classified as: Thawing models Caldwell:2005tm ; Linder:2006sv (see
also, Scherrer:2007pu ; Linder:2015zxa ) and Tracker models Steinhardt:1999nw
; Urena-Lopez:2020npg . In the first category, the scalar field is initially
frozen on a shallow potential due to the Hubble damping. Field recovers from
freezing regime ($w_{\phi}=-1$) at late stages and rolls slowly along a
shallow potential increasing the equation of state parameter to the level
consistent with observation, see Fig.2(b). As for the trackers, after recovery
from the freezing regime, the field (approximately) follows the background and
only at late times does it quit to join the slow roll due to the shallow
nature of the potential around the present epoch, see Fig.2(a). For thawing
models, $1/2<\beta<3/2$ and friction term is not operative here where as
friction term is large, $\beta\ll 1$, in case of trackersCaldwell:2005tm . We
shall assume that $\beta$ does not change much during slow roll in thawing
case. In terms of $\beta$, scalar field equation (8) can be cast as (see
Chiba:2009sj  for details),

$\dot{\phi}=-\frac{V_{\phi}}{3(1+\beta)H}$ (75)

We than ask for the consistence of $\beta$ being $\mathcal{O}(1)$ (thawing
models) or $\beta\simeq 0$ (trackers) with the field equation (8). Indeed,
differentiating Eq.(75) with respect to time, we have,

$-\frac{\beta
V_{\phi}}{1+\beta}=\ddot{\phi}\simeq-\frac{(1+w_{b})}{2(1+\beta)}+\frac{V_{\phi\phi}V_{\phi}}{9(1+\beta)^{2}H^{2}}\Rightarrow\beta\simeq\frac{(1+w_{b})}{2}-\frac{V_{\phi\phi}}{9(1+\beta)H^{2}}$
(76)

where we have ignored the term $\dot{\beta}/\beta H$ and used
$-\dot{H}/H^{2}\simeq 3(1+w_{b})/2$. In thawing case, $\beta$ is assumed to be
approximately constant and since the second term on right hand side of (76) is
time dependent, it is necessary for the equality to hold that,

$|\eta|\ll 1\Rightarrow\beta=\frac{3}{2}(1+w_{b})$ (77)

where we used the fact that $\beta=\mathcal{O}(1)$. We find that our
assumption that $\beta$ is constant is consistent with field equation. On the
other hand, $|\beta|\ll 1$ for trackers and we find the consistency condition,

$\eta=\frac{3}{2}(1+w_{b})$ (78)

which gives an order of magnitude estimate for the mass of quintessence,
namely, $m\simeq H_{0}\sim 10^{-33}~{}\rm{eV}$.

####  III.6.3 The asymptotic scaling solution

We argued that the class of potentials with $\Gamma<1$ mimic scaling behaviour
in the asymptotic regime. The generalized exponential potential (69), shallow
initially($\phi/M_{Pl}\ll 1$) with asymptotic features of an exponential
potential in the post inflationary region, is a suitable candidate for
quintessential inflation. As $\Gamma\to 1$ for large values of $\phi/M_{Pl}$,
we expect scaling regime in the asymptotic region. We need to set autonomous
system with suitable choice of variables, the choice for $X\&Y$ is obvious,
the third variable should explicitly include the information about asymptotic
nature of scaling solutionSkugoreva:2019blk ,

$\displaystyle
X\equiv\frac{\dot{\phi}^{2}}{2V};~{}~{}~{}Y\equiv\frac{V_{\phi}}{\dot{\phi}H}$
(79) $\displaystyle A\equiv\frac{1}{\phi/M_{Pl}+1}$ (80)

Let us cast the equation of motion in autonomous form in case of potential
(69) using variables defined in (80) $\&$ (79),

$\displaystyle\frac{dx}{dN}=-2X(3+Y+XY)$ (81)
$\displaystyle\frac{dY}{dN}=2XY^{2}\left(1-\frac{(n-1)A^{n}}{\alpha
n(1-A)^{n}}\right)+Y(9/2+Y+3w_{b}/2)-$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}XY^{3}A^{2n-2}(X(w_{b}-1)+w_{b}+1)\left[\frac{A-1}{\alpha
n(1-A)^{n}}\right]^{2}$ (82)
$\displaystyle\frac{dA}{dN}=\frac{2XYA^{n+1}(1-A)}{\alpha n(1-A)^{n}}$ (83)

Let us also express other quantities of interest through autonomous variables,

$\displaystyle\Gamma=1-\frac{(n-1)A^{n}}{\alpha n(1-A)^{n}}$ (84)
$\displaystyle w_{\phi}=\frac{X-1}{X+1}$ (85) $\displaystyle
w_{\phi}=\frac{2}{3}(X+1)XY^{2}A^{2n-2}\frac{1}{\alpha^{2}n^{2}(1-A)^{2n-2}}$
(86)

Since $\Gamma=1$ for $A=0$ ($\phi\to\infty$), system should mimic the
properties of exponential potential, thereby, we expect scaling behaviour in
the asymptotic regime. Indeed, we find the critical point,

$X=\frac{1+w_{b}}{1-w_{b}};~{}~{}Y=\frac{3}{2}(w_{b}-1):~{}~{}A=0$ (87)

as expected. It was demonstrated in reference Skugoreva:2019blk  that the
asymptotic scaling solution (87) is an attractor. It should be noted that in
case of $n=1$, the third Eq.(83) decouples from the other two equations which
can be solved without the reference to (83) which encodes the information
about asymptotic condition and we recover standard result corresponding to
exponential potential. It is important to note that unlike the case of
exponential potential, here, we can not infer time dependence of physical
quantities in scaling regime by the critical values in (87) as the fixed point
is reached in the limit, $t\to\infty;\phi\to\infty$. Substituting the critical
point in the expressions for $\Gamma,~{}w_{\phi}~{}\&~{}\Omega_{\phi}$, we
have

$\displaystyle\text{Asymptotic scaling solution}:~{}~{}~{}\Gamma\to
1;~{}~{}w_{\phi}\to w_{m}:~{}~{}\Omega_{\phi}\to 0$ (88)

In this case, passage to fixed point can be studied numerically. To
analytically understand the field dynamics, we need an anzats in the
asymptotic regime whose consistency with the field equation should be checked.
We have the following anzats in the asymptotic regime($t\to\infty$)staro  (see
Scherrer:2022umm  for comparison),

$\alpha\left(\frac{\phi(t)}{M_{Pl}}\right)^{n}=f_{0}\ln\left(\frac{t}{t_{1}}\right)\
+f_{1}\ln\left[\ln\left(\frac{t}{t_{1}}\right)\right]+..$ (89)

Substituting (89) in the field equation (8 and keeping in mind the asymptotic
behaviour, we find,

$f_{0}=2;~{}~{}f_{1}=\frac{2(n-1)}{2};~{}~{}t^{2}_{1}=\frac{M^{2}_{Pl}(1-w_{b})}{V_{0}n^{2}(1+w_{b})\alpha^{2/n}2^{(n-1)/n}},$
(90)

which allows us to compute important parameters in the asymptotic regime
showing passage to the critical point (87),

$\displaystyle
w_{\phi}=w_{b}+(w^{2}_{b}-1)\frac{(n-1)^{2}}{n^{2}}\frac{\ln\left[\ln\left(\frac{t}{t_{1}}\right)\right]}{\ln\left(\frac{t}{t_{1}}\right)}\to
w_{b},~{}t\to\infty$ (91)
$\displaystyle\Omega_{\phi}=\frac{3(1+w_{b})}{4n^{2}}\left(\frac{2}{\alpha}\right)^{2/n}\left[\ln\left(\frac{t}{t_{1}}\right)\right]^{2(1-n)/n}\to
0,~{}~{}t\to\infty~{}~{}(n>1)$ (92)

which obviously reduce to the standard results for $n=1$. We have, therefore
demonstrated that the scalar field system in case of the generalized
exponential potential (69) with $\Gamma<1$ eventually comes to the scaling
track in the asymptotic regime. We also analytically demonstrated the passage
to the critical point. We had argued that scalar field with an arbitrary steep
potential with $\Gamma<1$ should ultimately join the scaling track. In view of
quintessential inflation, it is desirable that $\Gamma\simeq 1$ for quite some
time while $\phi$ is rolling on steep part of the potential111111One should be
little careful here with regard to $\Gamma$ which is one for exponential
potential irrespective of its being steep or shallow. and $\rho_{\phi}$
follows the background energy density before exiting to acceleration to mimic
tracker like behaviour. Depending upon the underlying field potential,
approximate scaling regime might be invisible if overshoot of background
energy density is so deep that field freezes on its shallow part and after
recovery from Hubble damping it enters the slow roll mimic dark energy $-$
thawing realization, see Fig.4. Obviously, no exit mechanism is required in
this model but late time physics is sensitive to initial conditions. Let us
once again reiterate that in case of the generalized exponential potential
potential (69), with an exit mechanism, we have the desired situation to
realize a perfect tracker.

Using the above given technical information on scalar field dynamics, we would
be in position to narrate scalar field dynamics in the presence of background
in the post inflationary era. This would give us right perspective of
unification of inflation with late time acceleration without interfering with
the thermal history of Universe known to a good accuracy.

##  IV Post inflationary dynamics: The exit mechanism via coupling with
massive neutrino

In the preceding subsections, we described scalar field dynamics with and
without the presence of background energy density $\rho_{b}$. We argued that
the runaway potential generic to the underlying framework should be steep in
the post-inflationary era. The energy density of radiation produced at the end
of inflation $\rho^{end}_{r}$ in this framework is typically several orders of
magnitude lower than $\rho^{end}_{\phi}$, and the radiative regime takes a
long time to begin. After inflation ends, the field rolls down the steep
potential fast, making the potential energy redundant. As a result field
evolves in the kinetic regime ($\rho_{\phi}\sim a^{-6}$) and eventually
undershoots the background such that $\rho_{\phi}\ll\rho_{b}$. When this
happens, field enters the slow roll due to Hubble damping and freezes
($\rho_{\phi}=const$) on its potential (see, Eq.(72)) and waits there till the
background energy density ($\rho_{b}\sim a^{-m}$; m=4, 3 for radiation and
matter respectively) becomes comparable to $\lambda^{2}_{s}\rho_{\phi}/2$,
slow roll is then violated and evolution of field resumes. Hereafter, field
dynamics crucially depends upon the nature of steepness of the potential which
is dictated by $\Gamma$. For exponential potential $\Gamma=1$, scalar field
would follow the background as scaling solution is an attractor in this case,
see Fig.3(a). If $\Gamma<1$, the slope of the potential increases giving rise
to the increase in kinetic energy, field energy density would then move away
from the background and evolve in the kinetic regime ending up in the freezing
regime when $\rho_{\phi}\ll\rho_{b}$ and so on and so forth. However, slope
all the time increases during evolution and when it reaches large values,
$\Gamma$ then approaches its scaling value, $\Gamma=1$, see Eq.(57). As a
result, system eventually joins the scaling track, see Fig.3(b). In case of
generalized exponential potential (69), we demonstrated that scaling solution
is an attractor in the asymptotic regime. In this case,
$\lambda_{s}\sim(\ln(t))^{(1-1/n)}$, $\Omega_{\phi}\to 0$ and $w_{\phi}\to
w_{b}$ for $t\to\infty$. Obviously, in this case, there is no issue related to
nucleosynthesis. It should be noted that the nucleosynthesis constraint
imposes a lower bound on $\alpha$ for exponential potentials corresponding to
a small fraction of field energy density, which is automatically true for
asymptotic scaling solutions as $\Omega_{\phi}\to 0$ in this case.

If $\Gamma>1$ (inverse power law potentials), the slope of the potential
$\lambda_{s}$ decreases when the field rolls along the steep part of the
potential with a large value of $\lambda_{s}$, it evolves with an equation
state $w_{\phi}$, which is close to the equation of state of background matter
for generic values of $n$ $-$ approximate scaling behavior. However,
$\lambda_{s}$ decreases with evolution, causing the field energy density to
gradually move towards the background and eventually overtake it at late
times; the field then enters a slow roll with a diminishing slope
($w_{\phi}\to-1$) that accounts for late time acceleration, see Fig.2(a).
Setting the model parameters appropriately, one makes the framework consistent
with observation. The change of initial conditions related to scalar field
dynamics, within a broad range changes the time for field entry into the
(approximate) scaling regime, leaving the late time physics unchanged. A
remark about ”generic” initial conditions is in order. One can choose
extraordinary field initial conditions, giving rise to a deep overshoot,
allowing the field to recover from Hubble damping only at late times in the
region of slow roll, such that the observational consistency is guaranteed.
Late time physics, in this case, would become sensitive to initial conditions,
necessitating a re-adjustment of the model parameters to ensure observational
consistency$-$ thawing realization.

In the case of the generalised exponential potential, an interesting dynamical
feature is observed, where $\rho_{\phi}$ initially evolves in steps (down and
forward) and eventually catches up with the background in the asymptotic
region as $\Gamma\to 1$. Let us once again reiterate that the scaling, or at
least asymptotic scaling behaviour in the post-inflationary era, is required
for viable cosmological dynamics. Secondly, since the scaling solution is an
attractor which is non-accelerating, one needs a late time feature in the
potential to mimic slow roll allowing exit to late time acceleration dubbed a
perfect tracker, see Figs.3(c) $\&$ (d).

In what follows, we shall analyze an exit mechanism based upon scalar field
coupling to massive neutrino matter which modifies the scalar field potential.
The coupling is effective only at late stages of evolution, where the
effective potential has a minimum allowing the field to settle there at a late
times and an exit to a quasi de Sitter phase is realized.

Figure 4: Qualitative behaviour of $\rho_{\phi}$ for the potential used in
Ref.Dimopoulos:2017zvq  ($V\sim\exp\left[1-\tanh(\phi)\right]-1$) which is
shallow initially and at late stages and steep in between. In this case,
overshoot is deep enough that the field freezes at the shallow part of the
potential. After recovering from Hubble damping, it rolls slowly and adheres
to late time acceleration (figure is adapted from Ref. Dimopoulos:2017zvq ).

The late time exit for steep run away potential can be realized by non-minimal
coupling of the quintessence field to matter which modifies the runaway
potential such that the effective potential has minimum (see Refs.
Barreiro:1999zs ; Bartolo:1999sq ; Saridakis:2010mf  for other kind of exit
mechanism). The conformal coupling of field to matter under consideration is
proportional to the trace of energy momentum tensor of matter and vanishes
identically during radiation. Coupling becomes operative after matter
domination is established giving rise to effective potential that has minimum
which is right feature in the wrong time. In general, minimum in the potential
where field can settle down leads to de Sitter like behaviour and is unwanted
in this case, it can spoil the matter phase, it is more than desirable to
leave the matter phase intact. Problem finds its solution in the coupling to
massive neutrino matter which acquires non-vanishing value only at late times
when massive neutrinos become non-relativistic Wetterich:2013jsa ;
Wetterich:2013wza ; Hossain:2014xha ; Geng:2015fla .

Let us consider the action which describes coupling between massive neutrino
matter and scalar fieldSami:2021ufn ; Gumjudpai:2005ry ,

$\displaystyle\mathcal{S}=\int
d^{4}x\sqrt{-g}\left[\frac{M_{\textrm{Pl}}^{2}}{2}R-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)\right]+\mathcal{S}_{m}+\mathcal{S}_{r}+\mathcal{S}_{\nu}\left(A^{2}(\phi)g_{\alpha\beta},\Psi\nu\right)\,,$
(93)

where, $\mathcal{S}_{m}$ , $\mathcal{S}_{r}$ and $\mathcal{S}_{\nu}$ are
actions for cold matter, radiation and neutrino matter respectively.
$g_{\mu\nu}$ and $\tilde{g}_{\mu\nu}\equiv A^{2}(\phi)g_{\mu\nu}$ designate
the Einstein and Jordan metrics. It is important to note here that the metric
for the neutrino action is different from the rest (including matter and
gravity part). One should, therefore, be careful while deriving the relevant
quantities from this action. In this section we shall explicitly derive the
continuity equation using (93) for neutrino matter and scalar field in the
Einstein frame. Let us note that in the Jordan frame neutrino matter is
minimally coupled and its energy momentum tensor obeys the standard
conservation law. In the Jordan frame, energy momentum-tensor for neutrino
matter is given by,

$\displaystyle\tilde{T}_{\mu\nu}=\frac{-2}{\sqrt{-\tilde{g}}}\frac{\delta{\mathcal{S}}}{\delta{\tilde{g}_{\mu\nu}}}=\begin{pmatrix}\tilde{\rho}&0&0&0\\\
0&&&\\\ \,.0&&\tilde{a}^{2}\tilde{p}&\\\ 0&&&\end{pmatrix}$ (94)

Or equivalently,

$\displaystyle\tilde{T}^{\mu}~{}_{\nu}=\rm{diag}\left(-\tilde{\rho},\,\tilde{p},\,\tilde{p},\,\tilde{p}\right)\,,$
(95)

which satisfies the standard continuity equation,

$\displaystyle\tilde{\nabla}_{\mu}\tilde{T}^{\mu\nu}=0\,$ (96)
$\displaystyle\dot{\tilde{\rho}}+3\tilde{H}\left(\tilde{\rho}+\tilde{p}\right)=0\,.$
(97)

Since the Einstein and Jordan metrics are related via the conformal factor
$A^{2}(\phi)$, we have the following relations,

$\displaystyle g_{\mu\nu}$ $\displaystyle=$ $\displaystyle
A^{-2}\tilde{g}_{\mu\nu}$ $\displaystyle g^{\mu\nu}$ $\displaystyle=$
$\displaystyle A^{2}\tilde{g}^{\mu\nu}$ $\displaystyle\sqrt{-g}$
$\displaystyle=$ $\displaystyle A^{-4}\sqrt{\tilde{g}}$ (98) $\displaystyle a$
$\displaystyle=$ $\displaystyle A^{-1}\tilde{a}$ $\displaystyle dt$
$\displaystyle=$ $\displaystyle A^{-1}\tilde{dt}\,.$

Let us note that the energy-momentum tensor in the Einstein frame with metric
$g_{\mu\nu}$ is related to its counter part in the Jordan frame,

$\displaystyle T_{\mu\nu}=\frac{-2}{\sqrt{-g}}\frac{\delta\mathcal{S}}{\delta
g^{\mu\nu}}=\frac{-2}{A^{-4}\sqrt{\tilde{g}}}\frac{\delta\mathcal{S}_{\nu}}{\delta\tilde{g}_{\mu\nu}}=A^{2}\tilde{T}_{\mu\nu}\,.$
(99)

We can compare the individual components of stress-energy tensors of the two
frames by the following manipulation,

$\displaystyle
T^{\mu}~{}_{\nu}=g^{\mu\alpha}T_{\alpha\nu}=A^{2}\tilde{g}^{\mu\alpha}A^{2}\tilde{T}_{\alpha\nu}=A^{4}\tilde{T}^{\mu}~{}_{\nu}\Rightarrow\tilde{\rho}=A^{-4}\rho,~{}~{}\tilde{p}=A^{-4}p\,.$
(100)

The Hubble parameter for the two frames are related as,

$\displaystyle
H=\frac{1}{a}\frac{da}{dt}=\frac{A^{-1}\tilde{a}}{a}=\frac{1}{A^{-1}\tilde{a}}\left[\tilde{a}\frac{dA^{-1}}{dt}+\frac{d\tilde{a}}{d\tilde{t}}\right]=-\frac{\dot{A}}{A}+A\tilde{H}\,,$
(101)

where time deviate is taken with respect to $t$, as we want to express all
physical quantities in the Einstein frame without reference to Jordan frame.
We also have the following important relations,

$\displaystyle\dot{\tilde{\rho}}=\frac{d\tilde{\rho}}{d\tilde{t}}=A^{-1}\frac{1}{dt}\left(A^{-4}\rho_{\nu}\right)=-4A^{-5}\frac{\dot{A}}{A}\rho_{\nu}+A^{-5}\dot{\rho}_{\nu}$
(102)

Using Eq. (101) and (102), the continuity equation (97) gets transformed to
the Einstein frame,

$\displaystyle\dot{\rho}_{\nu}+3H\left(\rho_{\nu}+p_{\nu}\right)=\frac{\dot{A}}{A}\left(\rho_{\nu}-3p_{\nu}\right)$
$\displaystyle\rm{or~{}equivalently}$
$\displaystyle\dot{\rho}_{\nu}+3H\rho_{\nu}\left(1+w_{\nu}\right)=\frac{A,\phi}{A}\dot{\phi}\rho_{\nu}\left(1-3w_{\nu}\right)$
(103)

Figure 5: Schematic diagram showing the effective potential (green line); the
original runaway type of potential is depicted by blue line. Coupling of field
$\phi$ to massive neutrino matter manifests itself in the effective potential
(108) through $A(\phi)\hat{\rho}_{\nu}$
[$A(\phi)=Exp(\alpha\gamma\phi/M_{\textrm{Pl}})$] shown by dashed lines. The
graph shows that for smaller values of $\gamma$, the minimum of effective
potential hits larger values of the field.

In similar way, we derive the Klein-Gordon equation for scalar field,

$\displaystyle\ddot{\phi}+3H\dot{\phi}+V_{,\phi}=-\frac{A_{,\phi}}{A}(-\rho_{\nu}+3p_{\nu})\equiv-\frac{A_{,\phi}}{A}\rho_{\nu}(1-3w_{\nu}),$
(104)

which, after multiplying on both sides by $\dot{\phi}$, can be cast in the
following suggestive form,

$\displaystyle\dot{\rho_{\phi}}+3H\rho_{\phi}(1+w_{\phi})=-\frac{A_{,\phi}}{A}\dot{\phi}\rho_{\nu}(1-3w_{\nu}).$
(105)

It is evident from Eq. (103) and (105) that conservation law does not hold for
neutrinos and scalar field individually but the total energy density,
$\rho_{\phi}+\rho_{\nu}$ adheres to standard conservation as it should be.

Let us emphasize that neutrinos, due to their tiny masses
($m_{\nu}=\mathcal{O}(10^{-2})~{}eV$), remain relativistic for most of the
cosmic history with $w_{\nu}=\frac{1}{3}$ and their coupling to the scalar
field vanishes. Only at the late times, when they turn non-relativistic, they
start behaving like cold matter and the effect of the coupling comes into play
which has very interesting implications for late time dynamics. For a runaway
type potential of the form (69), we argue that coupling modifies the potential
such that the effective potential has a minimum around the present epoch. In
case of $n=1$, we shall demonstrate analytically that the field settles in the
minimum giving rise to a quasi de Sitter like behaviour which is exactly
followed by massive neutrino matter. Numerical investigations, confirm that
same conclusions hold for $n>1$.

Let us first begin from general considerations. Owing to non-relativistic
nature of neutrinos around the present epoch, we can approximate by
$w_{\nu}\approx 0$. It is further convenient for analytical estimation that we
choose, $\hat{\rho}_{\nu}\equiv A^{-1}\rho_{\nu}$, which is conserved in the
Einstein frame,

$\displaystyle\hat{\rho}_{\nu}+3H\hat{\rho}_{\nu}=0,$ (106)

and the field evolution equation in the approximation under consideration
acquires the following form,

$\ddot{\phi}+3H\dot{\phi}=-V_{,\phi}-A_{,\phi}\hat{\rho}_{\nu},$ (107)

It is indicative from Eq. (107), that we can write an effective potential, up
to an additive constant as,

$\displaystyle V_{\rm eff}=V(\phi)+A(\phi)\hat{\rho_{\nu}}\,.$ (108)

Let us now use a specific form of the conformal factor,

$A(\phi)=e^{\alpha\gamma\frac{\phi}{M_{\textrm{Pl}}}}.$ (109)

Although the original potential is runaway type with the chosen form of
$A(\phi)$, the effective potential do posses a minimum, see Fig. (5). It
should be noted here that Eqs. (106) and (107) are exact for coupling to
ordinary cold dark matter ($w=0$) whereas the equations hold only at late
times when $w_{\nu}\approx 0$ in case of coupling to massive neutrino matter.
To capture the behaviour of massive neutrino matter which is relativistic at
early times and non relativistic near the present (late time), we assume the
following antsatz,

$\displaystyle w_{\nu}=\frac{1}{6}\left\\{1+\tanh\left[\frac{\ln(1+z)-z_{\rm
eq}}{z_{\rm dur}}\right]\right\\},$ (110)

where $z$ is the red-shift and $z_{\rm dur}$ and $z_{\rm equal}$ are two
parameters to be fixed using fitting with observation. The aforesaid applies
to any runaway type of potential. However, in order to proceed further with
analytical estimates, we shall consider the exponential potential (35) or (69)
with $n=1$. In this case, we have the expression for the effective potential,

$\displaystyle V_{\rm
eff}=V_{0}e^{-\alpha\frac{\phi}{M_{\textrm{Pl}}}}+\hat{\rho_{\nu}}e^{\alpha\gamma\frac{\phi}{M_{\textrm{Pl}}}}\,$
(111)

Minimising the potential w.r.t $\phi$, we find,

$\displaystyle\phi_{\rm
min}=\log\left(\frac{V_{0}}{\gamma\hat{\rho_{\nu}}}\right)^{\frac{M_{\textrm{Pl}}}{\alpha(1+\gamma)}}\implies
V_{\rm
eff,min}=V_{0}\left(\frac{\gamma\hat{\rho_{\nu}}}{V_{0}}\right)^{\frac{1}{1+\gamma}}\,,\,,$
(112)

where the potential (111) is defined up to an additive constant and the above
relation is true for large values of $\gamma$ required by observation, (to be
discussed below). The question thus arises whether the minimum given by (112)
corresponds to the present dark energy density of the Universe. We will
demonstrate that it does.

Figure 6: Figure shows the plot of $\Gamma$ for effective potential given by
(111). $\Gamma$ increases fast during slow roll, which happens here around the
minimum which shifts towards larger values of $\phi$ for smaller values of
$\gamma$. The plot is for $\alpha=10$ and the potential is normalised by
$V_{0}$.

It is instructive to rewrite Eqs.(103 ) and ( 104) for the conformal coupling
(109),

$\displaystyle\dot{\rho}_{\nu}+3H\rho_{\nu}\left(1+w_{\nu}\right)=\frac{\alpha\gamma}{M_{\textrm{Pl}}}\dot{\phi}\rho_{\nu}\left(1-3w_{\nu}\right)$
(113)
$\displaystyle\ddot{\phi}+3H\dot{\phi}+V_{,\phi}=-\frac{\alpha\gamma}{M_{\textrm{Pl}}}\rho_{\nu}(1-3w_{\nu}),$
(114)

Using the effective potential picture, which is approximate in case of
coupling to massive neutrino matter, we have put forward a general perspective
to convince the reader that the coupling under consideration triggers transit
to late time acceleration. In what follows, we present the exact analytical
results based upon dynamical analysis. Indeed, in this case, the autonomous
system admits an attractor solution which corresponds toHossain:2014xha ,

$\displaystyle w_{\phi}$ $\displaystyle=$
$\displaystyle\frac{-3+\alpha^{2}(1+\gamma)}{\alpha^{2}(1+\gamma)^{2}}\,,$
(115) $\displaystyle\dot{\phi}$ $\displaystyle=$
$\displaystyle\frac{3HM_{\textrm{Pl}}}{\alpha(1+\gamma)}\,.$ (116)

It is interesting to check for the late time behaviour of neutrino matter
density. One might naively think that with the field approaching de Sitter:
$\dot{\phi}\to 0$ and $w_{\nu}\approx 0$ near the present epoch, neutrino
matter should decouple from scalar field and, $\rho_{\nu}\sim a^{-3}$, see
Eq.(113). On the contrary, $\rho_{\nu}$ follows the scalar field and behave
like dark energy which is counter intuitive. This behaviour can be understood,
recalling Eq. (103), and the fact that $p_{\nu}\approx 0$ around the present
epoch,. Indeed,

$\displaystyle\dot{\rho}_{\nu}+3H\rho_{\nu}\left(1-\frac{A,\phi}{A}\frac{\dot{\phi}}{3H}\right)=0\,.$
(117)

Taking into account that $A_{,\phi}/{A}=\alpha\gamma/M_{\textrm{Pl}}$, we
write the continuity equation in a suggestive form,

$\displaystyle\dot{\rho}_{\nu}+3H\left(1+w_{\nu\rm}^{\rm eff}\right)=0\
;~{}~{}w_{\nu}^{\rm
eff}\equiv-\frac{\alpha\gamma\dot{\phi}}{3HM_{\textrm{Pl}}}$ (118)

The behaviour of $w_{\nu}^{\rm eff}$ determines the late time evolution of
neutrino matter density. Indeed, plugging $\dot{\phi}$ from Eq. (116) in
Eq.(118), we get,

Figure 7: Evolution of $w_{\nu}^{\rm eff}$ versus the red-shift in case of the
generalized exponential potential (69) for $n=3$ and $\alpha\gamma=1000$,
compatible with observation Basak:2021cgk ; solid blue line corresponds
$w^{\rm eff}_{\nu}=-1$. Figure shows that $w_{\nu}^{\rm eff}$ oscillates and
eventually settles close to minus one.

$\displaystyle w_{\nu}^{\rm
eff}=-\frac{\alpha\gamma\dot{\phi}}{3HM_{\textrm{Pl}}}=-\frac{\gamma}{1+\gamma}.$
(119)

As mentioned before, nucleosynthesis constraint forces, $\alpha\gtrsim 10$,
using Eq.(115) then we have in the leading order in $\alpha$,

$w_{\phi}=-\frac{\gamma}{1+\gamma}+\mathcal{O}(\alpha^{-2})$ (120)

which is close to $-1$ for large values of $\gamma$, consistent with
observations on late time acceleration. This in turn implies that
$w_{\nu}^{\rm eff}\simeq-1$ for the fixed point and massive neutrino matter
follows the scalar field at late times.

Figure 8: Plot of energy densities versus the red-shift on the log scale for
generalized exponential potential (69). Evolution is shown from the end of
inflation. Field is shown to track the background with exit to de Sitter at
late times followed by the massive neutrino matter density. Figure corresponds
to numerical values of parameters: $\gamma=800$ , $\lambda=10^{-8}$ , $n=6$
$z_{eq}=2.55$ $\&$ $z_{dur}=3$.

In fact, numerical results confirm that same features are shared by the model
based upon the generalized exponential potential (69); see Figs. 7 $\&$ 8, and
Ref. Basak:2021cgk  for details. In terms of the effective potential picture,
it is clear that the minimum is time-dependent via $\hat{\rho}_{\nu}$, but the
system essentially settles around its attractor and matches observational
values of the dark energy paradigm at the present epoch by appropriately
choosing $``\gamma."$.

A comment on coupling of scalar field to dark matter is in order. In this
case, the exit to acceleration might happen soon after mater domination is
established. The only free parameter $\gamma$ should then be tuned to delay
the minimum to late stage which is accomplished by taking a $\gamma\ll 1$ (see
Figs. 5 $\&$ 6) ) but this defies acceleration as $w_{\phi}\to 0,\gamma\to 0$
and the system is back to scaling track. We have demonstrated that this issue
is successfully addressed by invoking coupling to massive neutrino matter.

Let us summarize our main findings in the subsection. Scalar field coupling to
massive neutrino matter, due to tiny neutrino masses, builds up dynamically
only at late stages giving rise to exit from scaling regime to de Sitter like
configuration around the present epoch. In case of exponential potential, we
supported our conclusions by analytical estimates. Indeed, in this case, as
$\gamma$ becomes larger and larger, $\dot{\phi}$ becomes smaller and smaller
($\dot{\phi}\to 0$) such that $\alpha\gamma\dot{\phi}\sim
H_{0}M_{\textrm{Pl}}\sim\rho^{1/2}_{cr}$ and the RHS of Eq.(113) is non-
vanishing as de Sitter is approached. As a result neutrino matter is never
decoupled from scalar field while de Sitter is approached. The latter forces
the neutrino matter to follow the scalar field at late times. Numerical
results confirms the same behaviour for $n>1$, see Fig.8.

##  V The distinguished features of quintessential inflation: Relic gravity
waves

The gravity waves (GW) are described by transverse trace-less component of the
metric perturbation over the background space time, say specially flat FLRW
space time,
$ds^{2}=-dt^{2}+a(t)^{2}\left(h_{ij}.+\delta_{ij}\right)dx^{i}dx^{j}$. The
perturbation satisfies the transverse trace-less condition
$h^{i}_{i}=0,\partial_{i}h^{j}=0,$. The Fourier decomposition for the the
perturbation for a mode with wave number k can be written as

$\displaystyle
h_{ij}(t,x)=\sum_{n=+,\times}\int\frac{d\textbf{k}}{(2\pi)^{2/3}}\varphi_{ij}^{n}(\textbf{k})h_{\textbf{k}}^{n}(k,t)e^{\textbf{k}\cdot\textbf{x}}\,,$
(121)

where the polarization tensor $\varphi_{ij}^{n}(\textbf{k})$ are symmetric and
satisfies transverse traceless condition. The Fourier component of the tensor
mode satisfies the wave equation Sahni:1990tx

$\displaystyle
h_{k}^{n^{\prime\prime}}(\tau)+2\frac{a^{\prime}}{a}h_{k}^{n^{\prime}}(\tau)+k^{2}h_{k}^{n}=0\,,$
(122)

where the ′ denotes derivative with respect to conformal time,
$d\tau\equiv\frac{dt}{a}$. Energy spectrum for the GWs with wave number $k$
are given by

$\displaystyle\Omega_{\rm GW}(k,\tau)\equiv\frac{1}{\rho_{\rm
cr}}\frac{d\rho_{\rm GW}}{d\ln k}\,.$ (123)

The GW energy density is given by

$\displaystyle\rho_{\rm
GW}=-T^{0}_{~{}0}=\frac{M_{\textrm{Pl}}^{2}}{4}\large\int\frac{d^{3}k}{(2\pi)^{3}}\frac{k^{2}}{a^{2}}2\sum_{n}|h_{k}^{n}|^{2}\,.$
(124)

The GW energy spectrum for the present time can be written as

$\displaystyle\Omega_{\rm
GW,0}=\frac{1}{12}\left(\frac{k^{2}}{a_{0}^{2}H_{0}^{2}}\right)P_{T}(k)T^{2}(k)\,\,,$
(125)

where, $P_{T}(k)=\left.\frac{2}{\pi^{2}}\frac{H_{\rm
inf}^{2}}{M_{\textrm{Pl}}^{2}}\right|_{k=aH}$ is the primordial tensor power
spectrum, defined at the horizon crossing and $T^{2}(k)$ is the transfer
function which describes time evolution of each mode after the end of
inflation. The transfer function today is given byBoyle:2005se ;
Watanabe:2006qe ; Kuroyanagi:2008ye .

Figure 9: The GW spectrum versus the frequency in Hz for the paradigm of
quintessential inflation. The blue tilt in the spectrum is visible on the high
frequency side due to the presence of long kinetic epoch. Instant preheating
is assumed as reheating mechanism with coupling constant $g$(consistent with
nucleosynthesis constraint), see subsection II.0.2 and Ref.Ahmad:2019jbm .

$\displaystyle T^{2}(k)=\frac{1}{2}\frac{a_{\rm hc}^{2}}{a_{0}^{2}}\,,$ (126)

where the subscript “$0$” represents value at present time and $\rm hc$ for
horizon crossing. Here it should be mentioned that in the super-horizon limit
$k\ll aH$, $h_{k}$ stays constant and vary as $h_{k}\propto\frac{1}{a}$ in the
sub-horizon limit, $k\gg aH$. Different mode of GW enter the horizon at
different epochs and to evaluate the spectrum today, we need to calculate the
Hubble parameter precisely,

$\displaystyle H\approx
H_{0}\sqrt{\Omega_{\phi}(a)+\Omega_{r0}\left(\frac{g_{*}}{g_{*0}}\right)\left(\frac{g_{*s}}{g_{*s0}}\right)^{-4/3}\left(\frac{a}{a_{0}}\right)^{-4}+\Omega_{m0}\left(\frac{a}{a_{0}}\right)^{-3}},$
(127)

where $g_{*}(s)$ denotes the effective number of relativistic degrees of
freedom at temperature $T$ contributing to entropy. It should mentioned that
the major contribution of scalar field to Hubble comes in two phases, namely,
kinetic regime which follows inflation and the dark energy era around the
present epoch. Let us put it in the following form,

$\displaystyle\Omega_{\phi}(a)=\Omega_{\phi K}+\Omega_{\Lambda 0}\,,$ (128)

where the latter is identified with the present value of dark energy. The
kinetic part, can be evaluated as,

$\displaystyle\Omega_{\phi,K}\equiv\frac{\rho_{\phi,K}}{3H_{0}^{2}M_{\textrm{Pl}}^{2}}=\frac{\rho_{\phi,\rm
end}}{3H_{0}M_{\textrm{Pl}}^{2}}\left(\frac{a}{a_{\rm
end}}\right)^{-6}=\frac{3H_{\rm
end}^{2}M_{\textrm{Pl}}^{2}}{3H_{0}^{2}M_{\textrm{Pl}}^{2}}\left(\frac{a}{a_{\rm
end}}\right)^{-6}=\frac{H_{\rm end}^{2}}{H_{0}^{2}}\left(\frac{a}{a_{\rm
end}}\right)^{-6}\,.$ (129)

Making use of the relation, $k=a_{\rm hc}H_{\rm hc}$ and Eq. (127), the scale
factor $a_{\rm hc}$, in different regimes, can be expressed as,

$\displaystyle a_{\rm hc,\rm MD}$ $\displaystyle=$
$\displaystyle\frac{H_{0}^{2}a_{0}^{3}}{k^{2}}\Omega_{\rm m0}$ (130)
$\displaystyle a_{\rm hc,\rm RD}$ $\displaystyle=$
$\displaystyle\frac{a_{0}^{2}H_{0}}{k}\left(\frac{g_{*}}{g_{*0}}\right)^{1/2}\left(\frac{g_{*s}}{g_{*s0}}\right)^{-2/3}\sqrt{\Omega_{r0}}$
(131)

The scale factor for those mode entering during kinetic epoch can be found by
using the fact that during kinetic epoch up-to the radiation commencement the
energy density goes as $\rho_{\rm tot}\approx\rho_{\phi}\propto a^{-6}$ (or
equivalently $H\propto a^{-3}$) and $k_{r}=H_{r}a_{r}$

$\displaystyle k$ $\displaystyle=$ $\displaystyle a_{\rm hc,\rm KD}H_{\rm
hc}=\frac{a_{\rm hc\rm KD}}{a_{r}}\frac{H_{\rm
hc}}{H_{r}}~{}~{}a_{r}H_{r}=\frac{a_{\rm hc\rm KD}}{a_{r}}\frac{H_{\rm
end}\left(\frac{a_{\rm end}}{a_{\rm hc,\rm KD}}\right)^{3}}{H_{\rm
end}\left(\frac{a_{\rm end}}{a_{r}}\right)^{3}}k_{r}$ (132) $\displaystyle=$
$\displaystyle\frac{a_{r}^{2}}{a_{\rm hc,\rm KD}^{2}}\implies\frac{a_{\rm
hc,\rm KD}}{a_{r}}=\left(\frac{k_{r}}{k}\right)^{1/2}\,.$

Substituting the values of the scale factors (130, 131) and (132), to the
transfer function (126), we obtain,

$\displaystyle\Omega_{\rm GW,0}^{(\rm MD)}$ $\displaystyle=$
$\displaystyle\frac{1}{6\pi^{2}}\Omega_{\rm m0}^{2}\frac{H_{\rm
inf}^{2}}{M_{\textrm{Pl}}^{2}}~{}\frac{a_{0}^{2}H_{0}^{2}}{k^{2}}~{}~{}~{}~{}~{}~{}~{}~{}{\rm
for}~{}~{}~{}(k_{0}<k<k_{\rm eq})$ (133) $\displaystyle\Omega_{\rm GW,0}^{\rm
RD}$ $\displaystyle=$ $\displaystyle\frac{1}{6\pi^{2}}\Omega_{\rm
r0}\frac{H_{\rm
inf}^{2}}{M_{\textrm{Pl}}^{2}}\left(\frac{g_{*}}{g_{*0}}\right)\left(\frac{g_{*,s}}{g_{*,s0}}\right)^{-4/3}~{}~{}{\rm
for}~{}~{}~{}(k_{\rm eq}<k<k_{\rm r})$ (134) $\displaystyle\Omega_{\rm
GW,0}^{\rm(KD)}$ $\displaystyle=$ $\displaystyle\Omega_{\rm
GW,0}^{\rm(RD)}\left(\frac{k}{k_{r}}\right)~{}~{}~{}~{}{\rm
for}~{}~{}~{}~{}~{}({k_{\rm end}<k<k_{r}})\,$ (135)

where $k_{0}$, $k_{\rm eq}$, $k_{\rm end}$, $k_{\rm r}$ represent values of
$k$ at present, matter-radiation equality, end of inflation and commencement
of radiation era respectively. The corresponding frequencies can easily be
evaluated using the relation, $f=\frac{aH}{2\pi}$

$\displaystyle f_{0}$ $\displaystyle=$
$\displaystyle\frac{a_{0}H_{0}}{2\pi}\sim 3\times 10^{-19}{\rm Hz}$ (136)
$\displaystyle f_{\rm eq}$ $\displaystyle=$ $\displaystyle\frac{a_{\rm
eq}H_{\rm eq}}{2\pi}\sim 10^{-17}\rm Hz$ (137) $\displaystyle f_{\rm end}$
$\displaystyle=$ $\displaystyle\frac{a_{\rm end}H_{\rm
end}}{2\pi}=\frac{T_{0}}{T_{\rm
end}}\left(\frac{43}{11g_{*s}}\right)^{1/3}\frac{H_{\rm end}}{2\pi}\gtrsim
10^{8}\rm Hz$ (138)

In the last step we have given an lower bound on frequency at end of inflation
in a model independent way by considering instantaneous energy transfer at
reheating.

Nucleosynthesis constraint on reheating temperature:  As discussed in section
III.4, the presence of any relativistic degree of freedom in the Universe,
over and above the standard model of particle physics, is subject to the
nucleosynthesis constraint. In particular, we have have a bound on relic
gravity waves given by (50). And since the major contribution to GW spectrum
comes from the blue-tilted spectrum due to kinetic regime ($k_{\rm
end}<k<k_{r}$), we have Cyburt:2015mya ; Figueroa:2018twl ,

$\displaystyle\Omega_{\rm GW,0}\simeq\Omega^{\rm KD}_{\rm
GW,0}\equiv\Omega^{(\rm max)}_{\rm GW,0}<1.12\times 10^{-6}$ (139)

Substituting the value of $\Omega_{\rm GW,0}^{\rm(KD)}$ from Eq. (135), and
using the fact that for highest frequency at kinetic epoch, $a=a_{\rm
kin}\approx a_{\rm end}$, we get,

$\displaystyle\Omega_{\rm GW,0}^{\rm max}=\frac{1}{6\pi^{2}}\Omega_{\rm
r0}~{}\frac{H_{\rm
inf}^{2}}{M_{\textrm{Pl}}^{2}}~{}\frac{g_{*}}{g_{*0}}~{}\left(\frac{g_{*s}}{g_{*s0}}\right)^{\frac{-4}{3}}\left(\frac{a_{r}}{a_{\rm
end}}\right)^{2}$ (140)

Putting the values, $\Omega_{\rm r0}=9.24\times 10^{-5}$, $g_{*}(a_{\rm
end})=g_{*s}(a_{\rm end})=106.75$, $g_{*0}=3.36$, $g_{*s0}=3.91$; $H_{\rm
inf}=1.4\times 10^{14}r^{\frac{1}{2}}~{}\rm GeV$ and using the upper bound on
tensor to scalar ratio of perturbations, $r\approx.05$, we obtain the upper
bound on the GW spectrum,

$\displaystyle\Omega_{\rm GW,0}^{\rm max}=3.8\times
10^{-16}\left(\frac{a_{r}}{a_{\rm end}}\right)^{2}=3.8\times
10^{-16}\left(\frac{T_{\rm end}}{T_{r}}\right)^{2}$ (141)
$\displaystyle\Omega_{\rm GW,0}^{\rm max}<1.12\times
10^{-6}\implies\left(\frac{T_{\rm end}}{T_{r}}\right)<5.5\times 10^{4},.$
(142)

where we have used the relation, ${a_{r}}/{a_{\rm end}}={T_{\rm
end}}/{T_{r}}$. This bound enables us to constrain the duration of kinetic
regime: from the end of inflation to the commencement of radiation era. Higher
is the radiation energy density produced at the end of inflation, shorter
would be the kinetic regime and the easier it would be to comply with the BBN
bound. Let us consider a natural mechanisms dubbed gravitational particle
production and check for its consistency with nucleosynthesis. In
gravitational reheating (see, II.0.1), the energy density produced after the
end of inflation is given by

$\displaystyle\rho_{r,\rm end}=10^{-2}g_{p}H_{\rm end}^{4}\approx 1.15\times
10^{-17}M_{\textrm{Pl}}^{4}r\,,$ (143)

where in the last step we used, $g_{p}=100$. From Eq. (19) we have
$\rho_{\phi,\rm end}\approx 1\times 10^{-8}M_{\textrm{Pl}}^{4}r^{2}$. Thus,
Eq. (22), yields,

$\displaystyle\left(\frac{a_{r}}{a_{\rm
end}}\right)^{2}=\left[\frac{\rho_{\phi,\rm end}}{\rho_{\rm
end}}\right]\approx 1.8\times 10^{10}\,.$ (144)

Finally, using Eq. (141), we arrive at the following estimate,

$\displaystyle\Omega_{\rm GW,0}^{\rm max}\approx 6\times 10^{-6}\,,$ (145)

which challenges the nucleosynthesis constraint (50). This is attributed to
inefficiency (smaller value of $\rho_{r,\rm end}$ and correspondingly larger
value of $(\rho_{\phi}/\rho_{r})_{end}$) of reheating process based upon
gravitational particle productionAhmad:2019jbm . In fact, other reheating
mechanisms can give rise to larger values of $\rho_{r,\rm end}$ circumventing
the problem due to relic gravity waves. For example in instant preheating
scenario, we have $(\rho_{\phi}/\rho_{r})_{end}=(2\pi)^{3}/g^{2}$, with $g$
being a coupling constant in the scenario. Constraining, $g$, one can easily
comply with the nucleosynthesis bound, see Ref.Ahmad:2019jbm  for details.

##  VI Summary

This review is a pedagogical exposition of scalar field dynamics in the FLRW
background applied to quintessential inflation. In this framework, two
specific roles are assigned to a scalar field, namely, a consistent
description of inflation and late time acceleration, with the understanding
that the field is invisible starting from the commencement of the radiative
regime till the beginning of late time acceleration. The other important
demand from scalar field dynamics is related to the insensitivity of late-time
physics concerning the initial (field) conditions. These requirements broadly
characterise the class of scalar field potentials suited to the framework
under consideration: potentials flat initially, followed by steep behaviour in
the post inflationary era, and again flat around the present epoch. Without
the loss of generality, one assumes that the runaway potential is such that
the field rolls away from the origin towards plus infinity. After inflation
ends, the field energy density $\rho_{\phi}$ is larger by several orders of
magnitude than the energy density of radiation assumed to be produced due to
some alternative mechanism. Consequently, the field evolves in the kinetic
regime for a long time before the commencement of radiation domination.
Following that, $\rho_{\phi}$ overshoots the background matter (radiation)
density such that $\rho_{\phi}\ll\rho_{r}$. Field then freezes on its
potential due to Hubble damping and waits till the radiation density redshifts
to the level that $\rho_{r}/\rho_{\phi}$ becomes comparable to
$\lambda_{s}^{2}/2$ ($\lambda_{s}$ is the slope of steep potential).
Hereafter, field dynamics are sensitive to the type of steepness of the
potential, which is quantified by the parameter $\Gamma$. If $\Gamma=1$
(exponential potential), the field exactly follows the background as the
scaling solution is an attractor in this case. In case, $\Gamma>1$, for
instance, inverse power law potentials ($V\sim\phi^{-n},\Gamma=1+1/n$), the
slope decreases such that the field nearly follows the background while
evolving along the steep part of the potential and at late times enters the
slow-roll regime. This class of potentials is suitable for tracker behaviour.
Slope, on the other hand, increases with evolution if $\Gamma<1$ and in this
case, $\Gamma$ eventually approaches its scaling value in the asymptotic
region. We demonstrated that the generalised exponential potential that falls
in this class gives rise to a scaling solution in the asymptotic regime, which
is an attractor. This class of potentials, shallow initially, is suitable for
inflation and, in the post-inflationary era, retains the characteristics of a
steep exponential potential. We further demonstrated that a perfect tracker
can be designed from the classes of potentials with $\Gamma\leq 1$ by invoking
a late time feature in the potentials that could facilitate exit to
acceleration. In case, $\Gamma=1$, we also need to invoke an early time
feature in the potential which, in particular, can be achieved using a non-
canonical kinetic term in the scalar field Lagrangian.

As for the exit to late time acceleration, we mentioned different ways to
realise it and presented detailed investigations of a mechanism based upon the
coupling of the field to massive neutrino matter. In this framework, the
distinguished physical process in the late Universe, namely, the turning of
massive neutrinos from relativistic to non-relativistic, can trigger the
desired exit.

The focus in this review was on the building blocks of quintessential
inflation rather than the concrete models. We emphasised the model-independent
and generic features of the paradigm. For instance, a long kinetic regime
after the end of inflation and before the commencement of a radiative regime
is an essential component of the framework, which distinguishes it from the
standard inflationary scenario. The latter induces a novel feature in the
spectrum of relic gravity waves, namely a blue tilted spectrum in the high-
frequency regime. To this effect, we have included a brief discussion on relic
gravity waves and presented the model-independent estimates to highlight the
distinguishing features of the paradigm.

Last but not least, our discussion, especially on scalar field dynamics in the
post-inflationary era, is pedagogical, and we hope the review will be helpful
for young researchers interested in topics related to quintessential inflation
and dark energy.

##  VII  Acknowledgements

We thank V. Sahni, R. Gannouji, Anzhong Wang, Shibesh Kumar, S. Capozziello,
Azam Mofazzal and Sang Pyo Kim for useful comments. We are indebted to
Konstantinos Dimopoulos for his invitation to write this review and for making
valuable comments that allowed us to improve the draft. MS is partially
supported by the Ministry of Education and Science of the Republic of
Kazakhstan, Grant No. 0118RK00935. NJ is supported by the National
Postdoctoral Fellowship of the Science and Engineering Research Board (SERB),
Department of Science and Technology (DST), Government of India, file No.
PDF/2021/004114. NJ thanks the Centre for Cosmology and Science Popularisation
(CCSP), SGT University for hospitality where the work was initiated.

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