# Unifying inflationary and reheating solution

Manjeet Kaur<EMAIL_ADDRESS>Debottam Nandi<EMAIL_ADDRESS>Sharath Raghavan B<EMAIL_ADDRESS>
Current address: Pranavam house, Kuttikol post, Kasargod, Kerala 671541
Department of Physics and Astrophysics, University of Delhi, Delhi 110007,
India

###### Abstract

The conventional background solution for the evolution of a single canonical
inflaton field performs admirably in extreme scenarios such as the slow-roll
phase (where the slow-roll parameter is much less than one) and deep reheating
era (where the Hubble parameter is much smaller than the effective mass of the
potential and the field oscillates around the minimum of the potential), but
fails to accurately depict the dynamics of the Universe near the end of
inflation and the initial oscillatory phases. This article proposes a single,
unified, model-independent analytical solution for such a model that bridges
the gap between these extremes, providing a comprehensive description of the
evolution of the Universe. This novel strategy has the potential to
substantially enhance both quantitative and qualitative cosmological
observational predictions.

††preprint: APS/123-QED

##  I Introduction

The inflationary paradigm [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,
16, 17, 18, 19, 20, 21, 22, 23, 24], a brief period of accelerated expansion
during the early Universe, not only overcomes the early Universe puzzles, such
as the horizon and the flatness problems, but also explains the observational
constraints [25, 26, 27]. Within the paradigm, the single canonical scalar
field-driven slow-roll inflationary models are the most successful ones, where
the inflaton (scalar) field slowly rolls down its potential, resulting in the
quasi-exponential expansion of the Universe. As the field approaches the
bottom of the potential, the inflationary stage ends, and the field smoothly
starts oscillating around the minimum of its effective potential. At this
stage, it couples to other (standard) particles and the inflaton field decays
into those elementary particles, resulting in the transfer of energy from the
inflaton field to those particles. This era is referred to as the reheating
epoch [21, 28, 29, 30, 31, 32, 33, 16, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43,
44, 45, 46, 47, 48, 49, 50, 51, 38, 52]. When the inflaton field completely
decays, the reheating era ends, and the Universe enters into the known
radiation-dominated era [21, 29, 31, 53].

For slow-roll inflation to occur, the magnitude of the slow-roll parameters
$\epsilon_{\rm{1}}$ and $\epsilon_{\rm{2}}$ must be extremely close to zero.
These conditions simply refer to the exceedingly slowly changing (decreasing)
Hubble parameter. This, in turn, leads to the quasi-exponential expansion of
the Universe. To better comprehend the dynamics, consider the following
potential chaotic inflation model [3]:

$V(\phi)=\frac{1}{2}m^{2}\phi^{2},$

where $m$ is the mass of the inflaton field. The slow-roll condition is met
for $|\phi|\gg 1$, and at this stage, the evolution of the Hubble parameter as
a function of cosmic time $t$ can be expressed as
$H(t)=H_{\rm{0}}-\frac{1}{3}m^{2}t$, where $H_{\rm{0}}$ is a constant (as will
be demonstrated later). It results in the scale factor solution
$a(t)\propto\exp{(H_{\rm{0}}t-\frac{1}{6}m^{2}t^{2})}$, i.e., a solution close
to exponential. As is evident, during the slow-roll era, $H\gg m$. The
inflationary era ends when $\epsilon_{\rm{1}}=1$, which implies $H\sim m$.
Shortly after the end of inflation, the field oscillates, and the reheating
epoch begins. During the reheating period, the scalar field $\phi$ oscillates
around the minimum, $\phi=0$, and as a consequence, the Hubble parameter
decreases as $H(t)\sim 2/(3t)$, indicating $H\ll m$.

These characteristics can be generalized to any inflationary potential: during
the slow-roll epoch, the magnitude of the Hubble parameter is considerably
higher than the effective mass of the potential and nearly constant. The
analytical solution, in this era, i.e., the slow-roll solution, is well-known.
Inflation ends approximately when the Hubble parameter equals the mass and,
during reheating, the Hubble parameter falls significantly over the mass and
decreases very quickly with time. In this period also, we can represent the
approximate analytical solution of the reheating epoch. However, both the
slow-roll inflationary solution as well as standard reheating solution fail to
address the smooth transition from the slow-roll regime to the oscillations,
and thus, the dynamics, as well as the complexities such as the study of the
perturbations at the end of inflation and the beginning of the reheating
process, are incomplete. This is because the slow-roll approximation fails
near the end of inflation, and hence, the slow-roll solution does not justify
the genuine solution near the end of inflation. Similarly, in solving the
dynamics in the reheating epoch, we use the approximation that the Hubble
parameter is subdominant over the effective mass of the potential (e.g., $H\ll
m$ in the case of chaotic inflation). Thus, the reheating solution only
accounts for the asymptotic oscillatory solution, while the solution near the
first and subsequent oscillations misses the true dynamics.

Such difficulties also affect the inflationary and reheating constraints. As
we know, there are primarily two ways to investigate the reheating era:
quantitative [39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 38, 52, 54]
and qualitative analysis [21, 28, 29, 30, 31, 32, 33, 16, 34, 35, 36, 37, 38].
In quantitative analysis, we presume the effective behavior of the Universe
during this stage and attempt to constrain both the reheating time
$N_{\rm{re}}$ (or the temperature $T_{\rm{re}}$ at the end of reheating) and
the effective equation of state $w_{\rm{re}}$ [55, 56, 45, 50] using the
perturbations generated during the inflationary regime. In the case of chaotic
inflation, for instance, the evolution during the preliminary stage of
reheating, i.e., the preheating stage, behaves like a matter-dominated
solution, so $w_{\rm{re}}=0$. In contrast, for qualitative analysis, we
approximate the evolution of the background and analyze the process of decay
of the inflaton field. In this scenario, parametric resonance serves as the
mechanism for the production of elementary particles. Thus, as dynamics near
the end and around the first few oscillations are analytically not known, the
qualitative and quantitative analysis of reheating are extremely difficult to
analyze analytically.

This paper addresses the analytical unification of the slow-roll inflationary
solution and the reheating solution into a single solution, which eventually
captures the true dynamics not only during the slow-roll and asymptotic
reheating epoch but also during the intermediate junction between the two
epochs, i.e., during the end of inflation and the beginning of reheating
epochs. Additionally, the suggested method is model-independent, meaning it
can account for both small and large field models. Suppose the full solution
can be achieved with the smooth transition from slow-roll to reheating. In
that case, it will not only provide a broader view of the unified early
Universe solution but also can provide better constraints on parameters using
both qualitative and quantitative methods of reheating. In other words, the
study can improve the accuracy of the $n_{\rm{s}}$ (scalar spectral index of
curvature perturbations) vs. $T_{\rm{re}}$ relation in the quantitative
picture of reheating [45, 50]. In contrast, qualitative analysis can also
account for the effects of the smooth transition from inflation to reheating
for perturbed variables. If improved, it can even be studied in special cases
of producing primordial black holes (PBHs), primordial gravitational waves
(PGWs), and other scenarios where using parametric resonance during reheating;
perturbed modes can be enhanced [57, 58, 59, 60, 61, 62]. It is crucial to
acknowledge that the work in this article only focuses on the background
dynamics of the Universe, and analysis of the perturbations is reserved for
future work as it is beyond the scope of this work.

For such analysis, in this article, we consider the single canonical scalar
field minimally coupled to gravity and provide a single analytical solution
for the early Universe. In doing so, rather than working in the phase space
consisting of $\\{\phi,\dot{\phi}\\}$, we conveniently choose $\theta$, which
represents the phase of the oscillatory solution, and the Hubble parameter
$H$. Since the particle production (resonance) occurs at the minimum of the
potential and it is difficult to be certain of the same using cosmic time $t$,
the introduction of the coordinate $\theta$ mitigates this issue. This method
systematically shows how to arise at different levels of Hubble parameter
values using $\theta$, i.e., $\theta=(2n+1)\pi/2,n\in N$ indicates the field
is at the bottom of the potential, whereas $\theta=n\pi$, $n\in N$ denotes the
field reaches the highest level of the potential, where the field velocity
$\dot{\phi}$ vanishes during reheating. Knowing this instances helps in
solving the system to achieve a detailed picture, as we will show in later
sections.

The following is how the article is written. The action responsible for the
early Universe dynamics is defined in Sec. II, and the generic background
equations are provided. Sec. III demonstrates how to obtain the usual slow-
roll and reheating solutions. In this part, we also present the phase space,
which comprises of $\theta$ and $H$, as well as the asymptotic reheating
solution, which is well-known. In Sec. IV, we extend the phase space solution,
which is used to get the reheating oscillatory solution, to the slow-roll
phase, and later we present our primary work in the next Sec. V. In this
section, we unify the inflationary and reheating solutions, i.e., the
complete, yet model-independent, solution of the homogeneous Universe
dominated by the canonical inflaton field, and demonstrate the result for the
chaotic inflationary model using a simple yet straightforward method. We
demonstrate that different values of $\theta$ represent different instances of
the early Universe. Thus, by providing the solution of $H$ (and other
background variables) in terms of $\theta$, we explicitly imply that, in each
of the instances listed, we know the value of the $H$ (and other background
variables), which clearly aids in understanding the dynamics. We also present
the solution of $\theta$ in terms of cosmic time $t$, completing the solution.
In Sec. VI, we extend our result and investigate several inflationary models,
demonstrating that our method brilliantly provides the entire background
solution of the dynamics, i.e., from slow-roll to reheating solution with a
smooth transition, and we study the observational consequences in Sec. VII.
Finally, in Sec. VIII, we conclude our work.

A few words about our conventions and notations are in order at this stage of
our discussion. In this work, we work with the natural units such that
$\hbar=c=1$, and we define the reduced Planck mass to be $M_{{}_{\rm
Pl}}\equiv(8\pi G)^{-1/2}=1$. We adopt the metric signature of $(-,+,+,+)$.
Also, we should mention that, while the Greek indices are contracted with the
metric tensor $g_{\mu\nu}$, the Latin indices are contracted with the
Kronecker delta $\delta_{ij}.$ Moreover, we shall denote the partial and the
covariant derivatives as $\partial$ and $\nabla$. The overdots and overprimes
denote derivatives with respect to the cosmic time $t$ and the conformal time
$\eta$ associated with the Friedmann-Lemaître-Robertson-Walker (FLRW) line-
element, respectively.

##  II General Equations

Let us first consider a single canonical scalar field $\phi$ minimally coupled
to the gravity with a potential $V(\phi)$, specified by the action

$\displaystyle S=\frac{1}{2}\int
d^{4}x\sqrt{-g}~{}\left(R-g^{\mu\nu}\partial_{\mu}\phi~{}\partial_{\nu}\phi-2V(\phi)\right),$
(1)

where $R$ is the Ricci scalar. The corresponding equations of motion, i.e.,
Einstein’s equations and the equation of the scalar field, can be written as

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R$ $\displaystyle=$
$\displaystyle T_{\mu\nu(\phi)},$ (2)
$\displaystyle\nabla_{\mu}T^{\mu\nu}_{{\rm{(\phi)}}}$ $\displaystyle=$
$\displaystyle 0,$ (3)

where $T^{\mu}_{\,\nu{\rm{(\phi)}}}$ is the stress-energy tensor corresponding
to the $\phi$ field:

$T_{\mu\nu{\rm{(\phi)}}}=\partial_{\mu}\phi\ \partial_{\nu}\phi-
g_{\mu\nu}\left(\frac{1}{2}\partial_{\lambda}\phi\
\partial^{\lambda}\phi+V(\phi)\right).$ (4)

Using the FLRW line element, describing the homogeneous and isotropic Universe
in cosmic time $t$:

$\displaystyle ds^{2}=-{\rm d}t^{2}+a^{2}(t){\rm d\bf x}^{2},$ (5)

where $a(t)$ is the scale factor, Eqs. (2) and (3) can be reduced to the
following forms:

$\displaystyle 3H^{2}=\frac{1}{2}\dot{\phi}^{2}+V(\phi),$ (6)
$\displaystyle\dot{H}=-\frac{1}{2}\dot{\phi}^{2},$ (7)
$\displaystyle\ddot{\phi}+3H\dot{\phi}+V_{,\phi}=0.$ (8)

where, $H\equiv\dot{a}/a$ is the Hubble parameter and $A_{,\phi}\equiv\partial
A/\partial{\phi}$. As one can see, the first one is a constrained equation,
and between the other two, one of them is independent, leaving the degrees of
the freedom of the system to one with a single evolutionary equation:

$\displaystyle\ddot{\phi}+\sqrt{\frac{3}{2}}\sqrt{\dot{\phi}^{2}+2V}+V_{,\phi}=0.$
(9)

The above equation is highly nonlinear; therefore, obtaining its general
solution is exceedingly challenging. Using certain approximations, Eq. (9) can
be solved under various conditions, as demonstrated in the following section.
The primary objective of this article is, contrary to the conventional method
of solving in various epochs (or conditions), to provide a complete solution
of the above equation for a variety of models, as will be demonstrated later.

Finally, we now can define the two slow-roll parameters $\epsilon_{\rm{1}}$
and $\epsilon_{\rm{2}}$ as

$\displaystyle\epsilon_{\rm{1}}\equiv-\frac{\dot{H}}{H^{2}},\qquad\epsilon_{\rm{2}}\equiv\frac{\dot{\epsilon_{\rm{1}}}}{H\epsilon_{\rm{1}}}.$
(10)

These slow-roll parameters play a crucial role in defining the dynamics in the
early Universe, mainly during slow-roll inflationary evolution. In the next
section, with the help of these parameters, we will establish the inflationary
as well as the reheating dynamics.

##  III Scalar field solutions in different regimes

Given that the generic background equations are now known and have already
been presented in the preceding section, one can obtain the evolution of the
Universe using those equations for a given potential as well as the initial
conditions. Let us first discuss the slow-roll inflation.

###  III.1 Slow-roll Equations

In order to achieve a slow-roll inflation in the early Universe, the above-
mentioned slow-roll parameters have to be extremely small, i.e.,

$\displaystyle\epsilon_{\rm{1}}\ll 1,\quad\epsilon_{\rm{2}}\ll 1.$ (11)

The first condition in the above equation leads to $\dot{\phi}^{2}\ll H^{2},$
meaning the field velocity is small compared to the potential, thus the name
slow-roll and the second condition leads to $\ddot{\phi}\ll\dot{\phi}H$,
implying that the field acceleration is extremely small, i.e., the first
condition stay relevant for a sufficient time, which constrains the Eqs. (6)
and (8) to,

$\displaystyle 3H^{2}\simeq V(\phi),\qquad 3H\dot{\phi}\simeq-V_{,\phi}.$ (12)

These equations define the dynamics corresponding to the slowly rolling scalar
fields. The two slow-roll conditions can then also be expressed directly in
terms of the shape of inflationary potential as

$\displaystyle\epsilon_{\rm{1}}\simeq\frac{1}{2}\left(\frac{V_{,\phi}}{V}\right)^{2}\qquad\epsilon_{\rm{2}}\simeq
2\left(\frac{V_{,\phi}^{2}}{V^{2}}-\frac{V_{,\phi\phi}}{V}\right),$ (13)

where, $A_{,xx}=\frac{\partial^{2}A}{\partial x^{2}}$. Given the potential as
well as the field value, if the aforementioned slow-roll parameters satisfy
the slow-roll conditions (11), then one can ensure the Universe is in the
slow-roll stage, and the specific dynamics can be obtained by solving the
slow-roll equations (12).

To illustrate the slow-roll inflationary scenario, consider the simplest model
with quadratic potential, i.e., the chaotic inflation model:

$\displaystyle V(\phi)=\frac{1}{2}m^{2}\phi^{2},$ (14)

where, $m$ is the mass of the scalar field $\phi$. Using Eqs. (13), the slow-
roll parameters are:

$\displaystyle{}\epsilon_{\rm{1}}\simeq\frac{2}{\phi^{2}},\quad\epsilon_{\rm{2}}\simeq\frac{4}{\phi^{2}},$
(15)

which implies that, only when $|\phi|\gg 1,$ the slow-roll conditions are met.
Only in this regime the slow-roll equations (12) can be used to obtain the
dynamics, and they are given as

$\displaystyle
3H^{2}\simeq\frac{1}{2}m^{2}\phi^{2},\qquad\dot{\phi}\simeq-m\sqrt{\frac{2}{3}}.$
(16)

As slow-roll can be achieved only during $|\phi|\gg 1,$ during this regime,
$H\gg m.$ As a result, the solution to these equations, i.e., the slow-roll
solutions, can be obtained as

$\displaystyle\phi\simeq\phi_{\rm{i}}-\sqrt{\frac{2}{3}}mt,$ (17)
$\displaystyle
H\simeq\left(\frac{1}{\sqrt{6}}m\phi_{\rm{i}}-\frac{1}{3}m^{2}t\right),$ (18)

and the solution of the scale factor during slow-roll can now be expressed as

$a(t)\simeq
a_{\rm{i}}\exp{\left(\frac{1}{\sqrt{6}}m\phi_{\rm{i}}t-\frac{1}{6}m^{2}t^{2}\right)}.$
(19)

$\phi_{\rm{i}}$ and $a_{\rm{i}}$ are the initial values of $\phi$ and $a$ at
$t=0$. Note that the inflation ends at $\epsilon_{\rm{1}}=1$, and by assuming
the slow-roll dynamics holds till the end of inflation, with the help of Eq.
(15), one can find the field value at the end of inflation, and it is
$|\phi_{\rm{end}}|\simeq\sqrt{2}$. In that case, we can also solve for cosmic
time $t_{\rm{end}}$ denoting the end of inflation as

$\displaystyle t_{\rm{end}}\simeq\sqrt{\frac{3}{2}}\ \frac{\phi_{\rm{i}}}{m}$
(20)

where $t=t_{\rm{end}}$ corresponds to the end of slow-roll inflation, and we
assume $|\phi_{\rm{i}}|\gg|\phi_{\rm{end}}|$ which is required for slow-roll
inflation. The Hubble parameter, then, at the end of inflation,
$H_{\rm{end}}$, is

$H_{\rm{end}}\simeq\frac{1}{\sqrt{3}}m.$ (21)

Please note that very close to the end of inflation, i.e., $|\phi|=\sqrt{2},$
the slow-roll parameters do not obey the slow-roll condition as
$\epsilon_{\rm{1}},\epsilon_{\rm{2}}\sim 1.$ In fact, $\epsilon_{\rm{2}}$
becomes one at $|\phi|\simeq 2,$ even before the end of inflation. Therefore,
these solutions do not represent the true solutions at the end of inflation
and thereafter.

###  III.2 Reheating

Deep within the slow-roll regime, the first slow-roll parameter is very close
to zero, i.e., $\epsilon_{\rm{1}}\ll 1$, by definition. Nonetheless, as the
field value decreases, $\epsilon_{\rm{1}}$ and $\epsilon_{\rm{2}}$ increase
(see, for instance, Eq. (15)), and inflation ceases when $\epsilon_{\rm{1}}$
equals 1. Consequently, during the slow-roll epoch, the potential energy
predominates over the kinetic energy, and as the inflation approaches the end
of it, the contribution to the kinetic energy increases while the contribution
to the potential energy decreases until they are almost equal at the end. The
field then begins to oscillate around the minimum of the potential, and the
reheating phase commences. To derive an analytical solution for this regime,
it is easier to work with the phase space orientation of the field, i.e.,
$\theta$ and $H$, as opposed to $\phi$ and $\dot{\phi}$ [15, 63, 64, 65, 66].
To illustrate this, let us define

$\displaystyle\frac{\dot{\phi}}{\sqrt{6}}\equiv-H\sin{\theta},\quad\sqrt{\frac{V}{3}}\equiv
H\cos{\theta},$ (22)

such that the energy equation (6) satisfies. Differentiating with respect to
the cosmic time $t$ and after re-arranging terms, we get,

$\displaystyle\dot{\theta}=\frac{V_{,\phi}}{\sqrt{2V}}-\frac{3}{2}H\sin{2\theta},$
(23) $\displaystyle\dot{H}=-3H^{2}\sin^{2}{\theta}.$ (24)

Such choice of orientation simply implies that, as $\theta\ll 1$, $\dot{\phi}$
is negative and significantly smaller than the Hubble parameter, indicating
the slow-roll regime. On the other hand,
$\theta=\theta_{\rm{end}}=\sin^{-1}\left(\frac{1}{\sqrt{3}}\right)$ defines
the exact epoch of end of inflation, for $\theta=(2n+1)\frac{\pi}{2},n\in N,$
the potential vanishes, and this corresponds to the bottom at the potential,
and for $\theta=n\pi,~{}n\in N,$ the field velocity is zero, and the field
reaches the peak of the potential. Unlike the slow-roll approximation, these
values are exact, which is one of the key reasons for using such a
formulation.

Therefore, instead of using the background equations (6), (7) and (8), here we
analyze the full solution of the system by solving the Eqs. (23) and (24).
Keeping that in mind, let us again consider the case of chaotic inflation.
Then, Eq. (23) can be rewritten as

$\displaystyle\dot{\theta}=m-\frac{3}{2}H\sin{2\theta}.$ (25)

As mentioned earlier, after the end of inflation, the Hubble parameter falls
significantly below effective mass, i.e., $H\ll m.$ As a consequence, during
reheating, the Eq. (23) can be approximated and solved as

$\displaystyle\dot{\theta}\simeq
m,\quad\theta=\theta_{\rm{0}}+m(t-t_{\rm{0}}),$ (26)

where $\theta(t=t_{\rm{0}})=\theta_{\rm{0}}$. Integrating Eq. (24), we can
write the solution of Hubble parameter $H$ as a function of $\theta$ as

$\displaystyle
H=\frac{H_{\rm{0}}}{1+\frac{3H_{\rm{0}}}{4m}\left(2(\theta-\theta_{\rm{0}})-(\sin
2\theta-\sin 2\theta_{\rm{0}})\right)},$ (27)

where $H=H_{\rm{0}}$ at $\theta=\theta_{\rm{0}}.$ In the deep oscillating
stage, $\theta\gg\theta_{\rm{0}}$ and $t\gg t_{\rm{0}}$, which can approximate
the above solution to

$\displaystyle H\simeq\frac{2m}{3\theta}\left(1+\frac{\sin
2\theta}{2\theta}\right),$ (28)

where $\theta_{\rm{0}}$ is chosen at any time at the bottom of the potential.
Substituting the solution of $\theta$ from Eq. (26) in above equation we get,

$\displaystyle H\simeq\frac{2}{3t}\left(1+\frac{\sin 2mt}{2mt}\right).$ (29)

Notice that the time average of the Hubble parameter behaves as $2/(3t)$,
i.e., like a dust-matter-dominated solution with the effective equation of
state $w_{\rm eff}=0$. The corresponding solution for the field $\phi$ in this
regime can be written as

$\displaystyle\phi\simeq\frac{2\sqrt{2}}{\sqrt{3}mt}\cos mt$ (30)
$\displaystyle\dot{\phi}\simeq-\frac{2\sqrt{2}}{\sqrt{3}t}\sin mt$ (31)

Also, the first and the second slow-roll parameters can be written as

$\displaystyle\epsilon_{\rm{1}}$ $\displaystyle=$ $\displaystyle
3\sin^{2}{mt},$ (32) $\displaystyle\epsilon_{\rm{2}}$ $\displaystyle=$
$\displaystyle 3mt\cot{mt}.$ (33)

This is the complete solution for the reheating era for the case of the
chaotic inflationary model.

Please note that the inflation ends at $\theta_{\rm{end}}$, and the field, for
the first time, reaches the bottom of the potential, making the first
oscillation at $\theta=\pi/2.$ At and around this stage, $H\sim m.$ Therefore,
the reheating solution (26) and the solutions thereafter cannot be trusted as
the above solutions are obtained using the approximation $H\ll m$. Only after
a few oscillations, when $H$ falls significantly below the mass of the
potential $m,$ the reheating solutions asymptotically merge with the solutions
given above.

To summarize, in this section, for the chaotic inflation model, we derive the
dynamics of the Universe in two distinct regimes. For $|\phi|\gg 1$, slow-roll
conditions are met, and using these conditions, we derive the slow-roll
dynamics, which leads to a quasi-exponential scale factor solution. For
$|\phi|\ll 1$, however, the field oscillates around the minimum of the
potential and decays into other particles, referred to as the reheating epoch,
and using $H\ll m$ approximations, we also obtain the asymptotic solution in
this epoch. The two approaches to achieving these two extreme solutions are
also entirely distinct. As previously stated, the solution when $H\sim m$ is
still not well understood, and the two solutions given above do not justify
around this regime. And because the methodologies are distinct, extrapolating
these two solutions into a single solution is also exceedingly challenging. In
the following section, we will demonstrate that this is, in fact, possible if
we contemplate a single method of solving these two regimes, which in our case
is identical to the method used to solve the reheating era, which is
characterized by the variable $\\{\theta,H\\}$.

##  IV extending the phase space solution method in slow-roll regime for
chaotic inflation

Let us now focus on the method to analyze the evolution of the Universe during
the slow roll. As mentioned earlier, during this epoch, $\theta$ is small, and
as a consequence, using Eq. (24), we can approximate the first slow-roll
parameter as

$\displaystyle\epsilon_{\rm{1}}\equiv-\frac{\dot{H}}{H^{2}}\simeq
3\theta^{2}.$ (34)

Using Eq. (13), one can immediately obtain the relation between the variable
$\theta$ and the scalar field $\phi$ as

$\displaystyle\theta\simeq\frac{V_{,\phi}}{\sqrt{6}V},$ (35)

which, in turn, leads to

$\displaystyle\dot{\theta}\simeq-\frac{1}{3\sqrt{2}}\left(\frac{V_{,\phi\phi}}{V}-\frac{V_{,\phi}^{2}}{V^{2}}\right)\frac{V_{,\phi}}{\sqrt{V}}.$
(36)

In the case of chaotic inflation, the above equations take the following form:

$\displaystyle\theta\simeq\sqrt{\frac{2}{3}}\frac{1}{\phi},\qquad\dot{\theta}\simeq
m\theta^{2}.$ (37)

Note that, $\phi\gg 1$ leads to $\theta\ll 1,$ and also $\dot{\theta}\ll 1,$
which make the above assumptions self-consistent. We now can integrate the
above equation as:

$\displaystyle\theta$ $\displaystyle\simeq$
$\displaystyle\frac{\sqrt{2}}{\sqrt{3}\phi_{\rm{i}}-\sqrt{2}mt},$ (38)

where $\phi(t=0)\equiv\phi_{\rm{i}}.$ We can also integrate Eq. (34) and
obtain the relation between $H$ and $\theta$ as

$\displaystyle H\simeq\frac{m}{3\theta},$ (39)

where we use the initial condition for chaotic inflation $H(\theta\rightarrow
0)\rightarrow\infty.$ It can now be seen that Eqs. (38) and (39) are in
agreement with the Eqs. (17) and (18).

Please note that since we now solve the system using the variables
$\\{\theta,H\\}$ even during slow-roll regime, the fundamental difference in
the dynamics of the chaotic inflationary model comes only in the expression
$\dot{\theta}$: i.e., during reheating, it was simply $m,$ a constant,
whereas, during slow-roll, it takes the form $m\theta^{2}.$ In the following
section, we will propose a method for obtaining the entire solution using this
information as an advantage.

##  V Proposed full solution for chaotic inflation

For chaotic inflation, we have discussed the dynamics of the Universe in two
different regimes, i.e., during the slow-roll and reheating era. Let us just
summarize the method in brief. Instead of Eqs. (6), (7), and (8), expressed in
$\phi$ and $\dot{\phi},$ we redefine these equations in terms of the variable
$\theta$ and $H,$ and equivalently, obtain two generalized equations (23) and
(24). Then, in the case of either inflation or reheating, we express
$\dot{\theta}$ as a function of $\theta$, i.e., during reheating epoch,
$\dot{\theta}$ is constant, whereas, during the slow-roll phase, it goes as
$\propto\theta^{2}$ (viz. Eqs. (26) and (37)):

$\displaystyle\dot{\theta}\simeq\left\\{\begin{array}[]{cc}{m\theta^{2}}&{~{}~{}~{}\text{Slow-
roll}}\\\ {m}&{~{}~{}~{}~{}\text{Reheating}}\\\ \end{array}\right.$ (42)

By solving these equations together with Equation (24), we obtain the dynamics
in these two distinct regimes. Please note, however, that, as previously
remarked, these approximations do not hold at the end of the inflation era and
the beginning of the reheating era, and the solution can only be completed if
we know how $\dot{\theta}$ behaves in this adjacent era.

Therefore, to obtain a complete solution from slow-roll inflation to
reheating, we require a solution in which $\dot{\theta}$ behaves as
$m\theta^{2}$ for $\theta\ll 1$ and as $m$ for $\theta\gg 1$, with a seamless
transition between these two solutions. Without worrying about the actual
solutions, one can make an intellectual conjecture as to the form of such
functions, and the possibilities are limitless. In this paper, we find one
basic yet effective form such that the solution can be solved analytically by
simple functions with the form:

$\displaystyle\dot{\theta}$ $\displaystyle=$
$\displaystyle\frac{m\theta^{2}}{1+\theta^{2}}.$ (43)

Note that inflation ends exactly at $\theta_{\rm{end}}\simeq 0.6$, and the
field, for the first time, reaches the bottom of the potential at
$\theta=\pi/2\sim 1.6.$ Therefore, one can verify that, for the inflationary
as well as the reheating solution, the above assumption depicts near-accurate
dynamics of $\dot{\theta}$ with the solution of $\theta$ as

$\displaystyle\theta=\left(\theta_{\rm{i}}^{2}+m\theta_{\rm{i}}t-1\right)+\sqrt{4\theta_{\rm{i}}^{2}+\left(\theta_{\rm{i}}^{2}+m\theta_{\rm{i}}t-1\right)^{2}}$
(44)

where,
$\theta(t=0)\equiv\theta_{\rm{i}}\equiv\sqrt{\frac{2}{3\phi_{\rm{i}}}}$, is
the initial condition, chosen during the deep slow-roll regime.

Figure 1: On the top, we plot the Hubble parameter $H$ as function of $\theta$
for chaotic inflation model. At the bottom, we also plot $\theta$ as a
function of cosmic time $t$.

Now that we know the solution of $\dot{\theta},$ one can again use the
equation related to the first slow-roll parameter, i.e., Eq. (24) and solve
the Hubble parameter and the subsequent dynamics. Similar to the previous
scenario, integrating Eq. (24) and using Eq. (43), evolution of the $H$ as
function of $\theta$ can be obtained for the above model as

$\displaystyle H=\frac{4m\theta}{\left(6(\theta^{2}-1)-3\theta\sin
2\theta+6\cos 2\theta+12\theta\ \text{Si}(2\theta)\right)},$ (45)

where, once again, we use the initial condition, $H(\theta\rightarrow
0)\rightarrow\infty$ for the chaotic inflation, and Si$(x)$ is Sine Integral
function. Note that $\text{Si}(x)\rightarrow x$ for $x\rightarrow 0$, and as a
result, under the limit $\theta\ll 1,$ one can verify that the above solution
coincides with the solution of slow-roll chaotic inflationary solution given
in Eq. (18). On the other extreme limit, $\text{Si}(x)\rightarrow\pi/2$ for
$x\gg 1,$ and thus, for $\theta\gg 1,$ the above solution coincides with the
reheating solution (28). Therefore, the above solution of the Hubble parameter
is consistent with both the slow-roll as well as the reheating solutions
discussed in the previous section. Similarly, the general solutions of $\phi$
and $\dot{\phi}$, by using Eq (22), can be written as:

$\displaystyle\phi$ $\displaystyle=$
$\displaystyle\frac{4\sqrt{6}\theta\cos\theta}{\left(6(\theta^{2}-1)-3\theta\sin
2\theta+6\cos 2\theta+12\theta\ \text{Si}(2\theta)\right)},$ (46)
$\displaystyle\dot{\phi}$ $\displaystyle=$
$\displaystyle-\frac{4\sqrt{6}m\theta\sin\theta}{\left(6(\theta^{2}-1)-3\theta\sin
2\theta+6\cos 2\theta+12\theta\ \text{Si}(2\theta)\right)},$

and the two slow-roll parameters can be expressed as:

$\displaystyle\epsilon_{\rm{1}}=3\sin^{2}\theta,$ (48)
$\displaystyle\epsilon_{\rm{2}}=\frac{\theta\cot\theta\left(6(\theta^{2}-1)-3\theta\sin
2\theta+6\cos 2\theta+12\theta\ \text{Si}(2\theta)\right)}{2(1+\theta^{2})}.$

Figure 2: We plot the Hubble parameter $H$ as function of $\theta$ for chaotic
inflation model as a function of cosmic time $t$ both numerically and
analytically and show that the analytical solution provides a good level of
accuracy of evaluating the background dynamics.  Figure 3: Plot of evolution
of first slow roll parameter $\epsilon_{\rm{1}}$ as function of cosmic time
$t$. Here blue(solid) line corresponds to the numerical solution obtained
after solving the equations of motion given by Eqs. (6), and (7) numerically.
The Red(dotted) line corresponds to the evolution of Hubble parameter $H$ and
the first slow roll parameter $\epsilon_{\rm{1}}$ obtained analytically by the
proposed model.

Figure 4: Plot of evolution scalar field $\phi$ as function of cosmic time
$t$. Here blue(solid) line corresponds to numerical solution obtained after
solving the equations of motion given by Eqs. (6), and (7) numerically. The
Red(dotted) line corresponds to the evolution of scalar field $\phi$ obtained
analytically by proposed model.

Please note that the effect of the Si$(x)$ function may appear to be
irrelevant. Nonetheless, it can be demonstrated that this function plays a
crucial role in the transition between the two epochs and provides greater
precision; therefore, it cannot be neglected.

Let us now discuss the impact of the full solution, which is demonstrated in
Fig. 1. As can be seen, since the expressions of all variables are now given
in terms of theta, we know the precise values of these variables in each
physical instance, as specified by the variable $\theta$. Consider, for
example, the variable Hubble parameter $H$ given in Eq. (45). At
$\theta=\sin^{-1}(1/\sqrt{3})$, inflation ends precisely, and the above
analytical solution yields $H_{\rm{end}}\simeq 0.503m$ at this level. In
contrast, using the slow-roll approximation, we previously obtained
$H_{\rm{end}}\simeq m/\sqrt{3}\simeq 0.577m$ using analytic techniques. Using
numerical simulations, we determine that $H_{\rm{end}}\simeq 0.504,$ which
demonstrates that our method provides a much higher degree of accuracy. On the
other hand, $\theta=\pi/2$ denotes when the field reaches the bottom of the
potential for the first time. Using our approach, it is now obvious that
$H\simeq 0.167m.$ For the second time, it reaches the bottom, $H\simeq
0.087m,$ for third time, it is $H\simeq 0.061m,$ and so on. On the other hand,
when the field reaches its first maxima, $H\simeq 0.112m,$ at the second
maxima, $H\simeq 0.072m,$ for the third, it is $0.053m,$ and so on. In Fig. 2,
we compare our result with the numerical simulations, and it can be seen that
our proposed method provides near-accurate results. In fact, using the above
expressions (44) and (45), once we know the initial conditions (i.e., $\phi$
and $\dot{\phi}$), we can easily evaluate $\theta,$ and subsequently, the
value of the Hubble parameter. Other variables such as $\epsilon_{\rm{1}}$
(Fig. 3) and $\phi$, $\dot{\phi}$ (see Fig. 4) are then straightforward to
evaluate. Thus, our analysis reflects, without the need for numerical
simulations, analytically and with great accuracy, how the Universe evolves
with time during any epoch, be it inflation or the reheating epoch (see Table
1).

$\hskip 28.45274pt\theta\hskip 28.45274pt$ | $\hskip 28.45274ptH\hskip 28.45274pt$ | $\hskip 28.45274pt\phi\hskip 28.45274pt$ | $\hskip 28.45274pt\dot{\phi}\hskip 28.45274pt$ | $\hskip 28.45274pt\epsilon_{\rm{1}}\hskip 28.45274pt$ | $\hskip 28.45274pt\epsilon_{\rm{2}}\hskip 28.45274pt$
---|---|---|---|---|---
$0$ | $\infty$ | $\infty$ | $0.816\ m$ | 0 | 0
$\sin^{-1}(1/\sqrt{3})$ | $0.503\ m$ | $\ 1.006$ | $-0.711\ m$ | $1$ | $1.544$
$\pi/2$ | $0.167\ m$ | $0$ | $-0.408\ m$ | $3$ | $0$
$\pi$ | $0.111\ m$ | $-0.273$ | $0$ | $0$ | $\infty$
$3\pi/2$ | $0.087\ m$ | $0$ | $0.213\ m$ | $3$ | $0$
$\pi$ | $0.072\ m$ | $0.176$ | $0$ | $0$ | $\infty$
Table 1: Statistics for Chaotic inflationary model for different values of
$\theta$.

This is one of the main results of this work.

##  VI Extended general solution

In this section, we will extend the solution for chaotic inflation to other
inflationary models. As we saw earlier, during the slow-roll stage, in the
case of chaotic inflation, $\dot{\theta}\propto\theta^{2}$, in the generalized
scenario, we can extend the exponent factor from two to any arbitrary real
positive number $n$. During reheating, however, we already know that if the
potential behaves nearly as $V(\phi)\propto\phi^{2}$, then $\dot{\theta}$
remains constant during this time period. For simplicity and to derive simple
analytical expressions, we also want to maintain this relationship in the
generalized scenario, i.e., near the bottom of the potential, we want it to
behave similarly to $\propto\phi^{2}$. A prime example, which will be
discussed in the following section, is the Starobinsky inflation, where the
potential is nearly flat during the slow-roll regime but behaves as $\phi^{2}$
near the bottom of the potential. Hence, for generalized inflationary models,
$\dot{\theta}$ can be expressed as

$\displaystyle\dot{\theta}\simeq\left\\{\begin{array}[]{cc}{\mu\theta^{n}}&{~{}~{}~{}\text{Slow-
roll}}\\\ {\nu}&{~{}~{}~{}~{}\text{Reheating}}\\\ \end{array}\right.$ (52)

where $\mu,~{}\nu$, and $n$ are all positive constants that can be correlated
with the model parameters, i.e., the inflationary potential. Here, we assume
the potential’s nature is simple and that the transition from the slow roll to
the reheating scenario is seamless. Any feature of the potential or deviation
from the slow-roll, such as ultra slow-roll, is not taken into account, as
$\dot{\theta}$ may differ from the above expression in such cases.

Similar to chaotic inflation discussed in the previous section, we now can
combine both cases and propose the general solution of $\dot{\theta}$ as a
function of $\theta$ as

$\displaystyle\dot{\theta}=\frac{\mu\theta^{n}}{1+\frac{\mu}{\nu}\theta^{n}}.$
(53)

The above equation can be integrated to get the solution of $\theta$ in terms
of the cosmic time as

$\displaystyle\mu(1-n)\theta+\nu\theta^{1-n}=C_{\rm{1}}+\mu\nu(1-n)t,$ (54)

where $C_{\rm{1}}\equiv(1-n)\mu\theta_{\rm{i}}+\nu\theta_{\rm{i}}^{1-n}$ is
the constant of integration, and $\theta(0)\equiv\theta_{\rm{i}}$. The
dependence of $\theta_{\rm{i}}$ on $\phi_{\rm{i}}$ depends on the form of
potential. Again, using Eq. (24) along with the above Eq. (53) and other
equations, we now can obtain the solution corresponding to $H$,
$\epsilon_{\rm{1}}$, $\epsilon_{\rm{2}}$, $\phi$, and $\dot{\phi}$ as a
function of $\theta$ as

$\displaystyle H=\frac{4\mu\nu(n-1)\theta^{n}}{\mu(n-1)(4\nu
C_{\rm{2}}+6\theta-\ 3\sin 2\theta)\
\theta^{n}+3\nu(n-1)\left(E_{\rm{n}}(2i\theta)+E_{\rm{n}}(-2i\theta)\right)\theta-6\nu\theta},$
(55) $\displaystyle\epsilon_{\rm{1}}=3\sin^{2}\theta,$ (61)
$\displaystyle\epsilon_{\rm{2}}=\frac{\mu(n-1)(4\nu C_{\rm{2}}+6\theta-\ 3\sin
2\theta)\
\theta^{n}+3\nu(n-1)\left(E_{\rm{n}}(2i\theta)+E_{\rm{n}}(-2i\theta)\right)\theta-6\nu\theta}{4(n-1)(\nu+\mu\theta^{n})}\cot\theta,$
$\displaystyle\dot{\phi}=-\frac{4\sqrt{6}\mu\nu(n-1)\theta^{n}\sin\theta}{\mu(n-1)(4\nu
C_{\rm{2}}+6\theta-\ 3\sin 2\theta)\
\theta^{n}+3\nu(n-1)\left(E_{\rm{n}}(2i\theta)+E_{\rm{n}}(-2i\theta)\right)\theta-6\nu\theta},$
$\displaystyle
V(\phi)=\frac{48\mu^{2}\nu^{2}(n-1)^{2}\theta^{2}n\cos^{2}\theta}{\left(\mu(n-1)(4\nu
C_{\rm{2}}+6\theta-\ 3\sin 2\theta)\
\theta^{n}+3\nu(n-1)\left(E_{\rm{n}}(2i\theta)+E_{\rm{n}}(-2i\theta)\right)\theta-6\nu\theta\right)^{2}},$

where $E_{\rm{n}}(x)$ is the exponential integral function, and $C_{\rm{2}}$
is the integration constant, which again depends on the shape of the
potential.

The background variable solutions mentioned above describe the evolution not
only during the slow-roll and reheating phases but also during the transition
phase. Additionally, it is model-independent, meaning that it may be used with
both small and large field models, and the values of $\mu~{},\nu$, and $n$ are
determined by the model parameters associated with the potential. As a result,
it represents the comprehensive, model-independent solution of all the
dynamical variables during the entire evolution from slow-roll to reheating,
which is the main outcome of this article.

Before proceeding into the next section, let us now discuss the parameters
$C_{\rm{2}}$ and $n.$ In order to determine the value or the range of it, let
us again consider the slow-roll regime. During slow-roll, i.e., for $\theta\ll
1,$ the Hubble and the slow-roll parameters take the form:

$\displaystyle H=\frac{1}{C_{\rm{2}}+\frac{3}{(3-n)\mu}\ \theta^{3-n}}.$ (62)
$\displaystyle\epsilon_{\rm{1}}=3\theta^{2}$ (63)
$\displaystyle\epsilon_{\rm{2}}=2\mu\theta^{n-1}\left(C_{\rm{2}}+\frac{3}{(3-n)\mu}\
\theta^{3-n}\right)$ (64)

Therefore, for $n<3,$ the Hubble parameter approaches the value $1/C_{\rm{2}}$
when $\theta$ approaches zero. On the other hand, if we set $C_{\rm{2}}=0,$
then the Hubble parameter $H\rightarrow\infty$, for $\theta\rightarrow 0$.
Therefore, in the case of large field models, where $H\rightarrow\infty$,
$C_{\rm{2}}$ equals zero. On the other hand, for small field models, as
$V(\phi)$ saturates to one value, say, $V_{\rm{0}}$ as $\theta\rightarrow 0,$
$H$ takes the form $\sqrt{V_{\rm{0}}/3}$. Therefore, $C_{\rm{2}}$ can be
associated with these values for large and small field models as

$\displaystyle C_{\rm{2}}=\left\\{\begin{aligned} &~{}~{}~{}0,\quad~{}~{}\
\text{large fields,}\\\ &\sqrt{\frac{3}{V_{\rm{0}}}},\hskip 10.0pt\text{small
fields.}\end{aligned}\right.$ (65)

The solution immediately translates to:

$\displaystyle\dot{\phi}=-\frac{\sqrt{6}\theta}{C_{\rm{2}}+\frac{3}{(3-n)\mu}\
\theta^{3-n}},$ (66)

and

$\displaystyle V(\phi)\simeq\left\\{\begin{aligned}
&\frac{\mu^{2}(3-n)^{2}}{3}\frac{1}{\theta^{6-2n}},\hskip 13.0pt\text{large
fields,}\\\
&\frac{3}{C_{\rm{2}}^{2}+\frac{6C_{\rm{2}}}{(3-n)\mu}\theta^{3-n}},\quad\text{small
fields.}\end{aligned}\right.$ (67)

where $n<3$ and $C_{\rm{2}}$ is given by Eq. (65). The above expression, based
on the functional form of the potential, leads to the functional dependence of
$\phi$ over $\theta.$ One can also immediately notice that $n>3$ is prohibited
as $H$ becomes negative. At the same time, $n>0$ is required as
$\dot{\theta}\rightarrow 0$ as $\theta\rightarrow 0.$ Therefore, the
constraint on $n$ is

$\displaystyle 0<n<3.$ (68)

Now that the generic solution has been provided, let’s examine various
inflationary models. There are typically two types of inflationary potential:
large field potentials and small field potentials. Observations have, however
already ruled out the possibility of large field inflationary potentials, such
as chaotic inflation. In contrast, among all small field models, we will
discuss two types of models and determine their complete solutions in this
paper.

###  VI.1 First kind of small field inflationary models

This kind of model, during inflation, for $\phi\gg 1$ can be expressed as
$V(\phi)\simeq A(1-B\phi^{-\alpha}),~{}\alpha>0.$ Again, we assume the
potential has a minimum at $\phi=0,$ and around it, it has a form
$V(\phi)\propto\phi^{2}$ for $\phi\ll 1.$ Therefore, the potential that we are
interested in can be expressed as:

$\displaystyle
V(\phi)\simeq\left\\{\begin{array}[]{cc}{A\left(1-B\phi^{-\alpha}\right)}&{~{}~{}~{}\text{Slow-
roll}}\\\ {\frac{1}{2}m^{2}\phi^{2}}&{~{}~{}~{}~{}\text{Reheating}}\\\
\end{array}\right.$ (71)

where $A$, $B$, $m$, and $\alpha$ are constants. This kind of model is called
the polynomial $\alpha$-attractor model [67, 68, 69, 70, 71, 72]. In this
case, one can relate the model parameter $\alpha$ to the exponent $n$, given
in Eq. (53) as

$\displaystyle n=\frac{3+2\alpha}{1+\alpha}.$ (72)

It is now obvious that, $\alpha>0$ implies:

$\displaystyle 2\leq n\leq 3.$ (73)

Similarly, $\mu$ and $\nu$ can also be expressed in terms of the models
parameters $A$ and $B$ as

$\displaystyle\mu=\frac{1}{3}\sqrt{\frac{A}{2}}B^{2}\alpha^{2}(1+\alpha)\left(\frac{\alpha
B}{\sqrt{6}}\right)^{-\frac{3+2\alpha}{1+\alpha}},\quad\nu=m.~{}~{}$ (74)

Using the above forms of $n,\mu$ and $\nu,$ along with Eqs. (54), (55), (61),
(61), (LABEL:eq:gen-phidot-theta), and (LABEL:eq:gen-pot-theta), we can obtain
the full solution of the dynamics using our proposed method. However, two
other pieces of information are needed to fully solve these equations. The
first one is how $\theta_{\rm{i}},$ i.e., the initial condition for $\theta$
depends on the initial condition of the field $\phi_{\rm{i}}$, such that (54)
can be properly solved. This can be obtained by using Eq. (35) as:

$\displaystyle\theta_{\rm{i}}=\frac{\alpha
B}{\sqrt{6}}\frac{1}{\phi_{\rm{i}}^{1+\alpha}}.$ (75)

Note that $\phi_{\rm{i}}>\phi_{*}$, where $\phi_{*}$ relates to the pivot
scale $k=0.05\ \text{Mpc}^{-1}.$ The other information needed is the constant
appearing in the general solution of the Hubble parameter in Eq. (55). Since
the model is categorized under small field models, as mentioned earlier,
$C_{\rm{2}}$ can then be expressed as

$\displaystyle C_{\rm{2}}=\sqrt{\frac{3}{A}}.$ (76)

The evolution corresponding to above model for a specific choice of $A$, $B$,
and $\alpha$ can be seen in Fig. 5.

Figure 5: Plot of evolution $\theta$ (Top) as function of cosmic time $t$ and
Hubble parameter $H$ (Bottom) as function of $\theta$ corresponding to the
potential given by Eq. (79) with parameters given by $A=10^{-10}$, $B=1$, and
$\alpha=12$.

###  VI.2 Second kind of small field inflationary models

In this case, the potential can be expressed as:

$\displaystyle
V(\phi)\simeq\left\\{\begin{array}[]{cc}{A\left(1-Be^{-\alpha\phi}\right)}&{~{}~{}~{}\text{Slow-
roll}}\\\ {\frac{1}{2}m^{2}\phi^{2}}&{~{}~{}~{}~{}\text{Reheating}}\\\
\end{array}\right.$ (79)

where $A$, $B$, $m$, and $\alpha$ are positive constants. This kind of model
can be categorized as $\alpha$-attractor model [67, 68, 69, 70, 71, 72].
where, again, we use the $Be^{-\alpha\phi}\ll 1.$ Similar to earlier case,
$n,\mu,\nu,\theta_{\rm{i}}$ and $C_{\rm{2}}$ can be expressed in terms of the
model parameters as

$\displaystyle n=2,\quad\mu=\alpha\sqrt{2}\sqrt{A},\quad\nu=m,$ (80)
$\displaystyle\theta_{\rm{i}}=\frac{\alpha\exp{(-\alpha\phi_{\rm{i}})}}{\sqrt{6}},\quad
C_{\rm{2}}=\sqrt{\frac{3}{A}}.$ (81)

One example of such a model is the famous Starobinsky model of inflation with
the potential given by

$\displaystyle
V(\phi)=\frac{3}{4}m^{2}\left(1-e^{-\sqrt{\frac{2}{3}}\phi}\right)^{2}.$ (82)

The evolution corresponding to above example can be seen in Fig. 6.

Figure 6: Plot of evolution $\theta$ (Top) as function of cosmic time $t$ and
Hubble parameter $H$ (Bottom) as function of $\theta$ corresponding to the
potential given by Eq. (82) with $m=10^{-5}$.

##  VII Observations

As mentioned earlier, during slow-roll, $\theta\ll 1$. During this epoch, Eqs.
(53) and (62) leads to

$\displaystyle\theta_{\rm{N}}\simeq\mu\theta^{n}\left(C_{\rm{2}}+\frac{3}{(3-n)\mu}\
\theta^{3-n}\right).$ (83)

Here, $N\equiv\ln a(t)$ is the e-folding number, $\theta_{\rm{N}}\equiv{\rm
d}\theta/{\rm d}N=\dot{\theta}/H$. Please note that, in the case of large
field models, $C_{\rm{2}}=0.$ On the other hand, for small field models,
$C_{\rm{2}}$ depends on the potential, as mentioned in the Eq. (65). Using
this feature of $C_{\rm{2}}$ during slow roll, Eq. (83) can rewritten as

$\displaystyle\theta_{\rm{N}}\simeq\left\\{\begin{aligned}
&\frac{3}{3-n}\theta^{3},\quad\text{Large fields,}\\\
&C_{\rm{2}}\mu\theta^{n}.\quad\quad\text{Small fields}.\end{aligned}\right.$
(84)

For these two separate cases, one can integrate the above equation, and also
by using the approximation $\theta\ll\theta_{\rm end}.$ one can then obtain
the relation between $\theta$ and $N$ as

$\displaystyle\theta\simeq\left\\{\begin{aligned}
&\sqrt{\frac{3-n}{6N}},&\quad\text{Large fields,}\\\
&\left(\frac{1}{C_{\rm{2}}\mu N}\right)^{\frac{1}{n-1}},&\hskip
13.0pt\text{Small fields.}\end{aligned}\right.$ (85)

Here $N$ denotes the duration e-folding number from $\theta$ to $\theta_{\rm
end}.$ This relation is needed to evaluate the perturbations for a specific
$k$ mode. The perturbations, observationally, can be characterized by mainly
four parameters: the scalar spectral index $n_{\rm{s}}$, the tensor spectral
index $n_{\rm{t}}$, tensor-to-scalar ratio $r$ and the scalar power spectrum
$\mathcal{P}_{\rm{s}}.$ For a single canonical scalar field minimally coupled
to gravity that leads to slow-roll inflation, these parameters can be written
in terms of potential, and the slow-roll parameters as

$\displaystyle n_{\rm{s}}\simeq 1-2\epsilon_{\rm{1}}-\epsilon_{\rm{2}},\quad
n_{\rm{t}}\simeq-2\epsilon_{\rm{1}},$ (86)
$\displaystyle\mathcal{P}_{\rm{s}}\simeq\frac{H^{2}}{8\pi^{2}\epsilon_{\rm{1}}},\quad\quad\quad\
r\simeq 16\epsilon_{\rm{1}}.$ (87)

Using Eqs. (62), (63) and (63), the above parameters can be expressed in terms
of $\theta$ as

$\displaystyle n_{\rm{s}}\simeq
1+\frac{6(4-n)\theta^{2}}{n-3}-2C_{\rm{2}}\mu\theta^{n-1},\ \quad
n_{\rm{t}}\simeq-6\theta^{2},$
$\displaystyle\mathcal{P}_{\rm{s}}\simeq\frac{1}{24\pi^{2}\theta^{2}\left(C_{\rm{2}}+\frac{3}{(3-n)\mu}\
\theta^{3-n}\right)^{2}},\quad r\simeq 48\theta^{2}.$

Observations (BICEP/Keck [26, 27] and PLANCK [25]) suggest that, at the pivot
scale ($k_{*}=0.05\,\text{Mpc}{}^{-1}$), the amplitude of the scalar power
spectrum is $\mathcal{P}_{\rm{s}}\simeq 2.101^{+0.031}_{-0.034}\times
10^{-9}(68\%\,\,\text{CL})$ with the scalar spectral index being
$n_{\rm{s}}=0.9649\pm 0.0042\,(68\%\,\,\text{CL})$, while the tensor-to-scalar
ratio $r$ is bounded from above by $r<0.028\,(95\%\,\,\text{CL})$. As of yet,
there is no bound on the tensor spectral index $n_{\rm{t}}.$ To evaluate Eq.
(VII), an additional requirement is necesaary: the relation between $\theta$
related to the pivot scale, i.e. $\theta_{*}$ and the e-folding number $N,$
given in Eq. (85). In general, the pivot scale leaves the Hubble horizon
$50-60$ e-folds before the end of inflation, i.e., $N_{*}\sim 50-60.$
Therefore, by using this information, one can evaluate the observational
parameters for any model of inflation.

Let us first discuss the large field models with
$V(\phi)\propto\phi^{\alpha},~{}\alpha>0$. In this case, one can verify that

$\displaystyle n=3-\frac{\alpha}{2}.$ (89)

It leads to $r=4\alpha/N_{*}$ and $n_{\rm{s}}=1-(2+\alpha)/(2N_{*}),$ which,
one can then quickly verify that for $N_{*}\sim 50-60,$ that these relations
do not obey the observational constraints, as is already mentioned in the
previous section. On the other hand, in the case of small field models, all
observational parameters can be expressed in terms of $C_{\rm{2}},\mu$ and $n$
as

$\displaystyle n_{\rm{s}}\simeq$ $\displaystyle 1-\frac{2}{(n-1)N_{*}},$ (90)
$\displaystyle n_{\rm{t}}\simeq$
$\displaystyle-\frac{6}{(C_{\rm{2}}\mu(n-1)N_{*})^{\frac{2}{n-1}}},$ (91)
$\displaystyle r\simeq$
$\displaystyle\frac{48}{(C_{\rm{2}}\mu(n-1)N_{*})^{\frac{2}{n-1}}},$ (92)
$\displaystyle\mathcal{P}_{\rm{s}}\simeq$
$\displaystyle\frac{(C_{\rm{2}}\mu(n-1)N_{*})^{\frac{2}{n-1}}}{24\pi^{2}C_{\rm{2}}^{2}}.$
(93)

Again, by using the observational constraints with $N_{*}\sim 50-60$, one can,
in general, obtain the constrained values of these parameters as

$\displaystyle 1.84<n<2.29,\quad C_{\rm{2}}>5.87\times 10^{4},$ (94)
$\displaystyle 0.83\times 10^{-5}<\mu<3.21\times 10^{-5}.$ (95)

These are the most general constraints on small-field inflationary models.

Let us now focus separately on the two different and special cases of small
field models that we discussed in the previous section. For the first kind of
small field models with $V(\phi)=A(1-B\phi^{-\alpha})$, all observational
parameters can be expressed as

$\displaystyle n_{\rm{s}}\simeq$ $\displaystyle
1-\frac{2(1+\alpha)}{(2+\alpha)N_{*}}$ (96) $\displaystyle n_{\rm{t}}\simeq$
$\displaystyle-\frac{(B\alpha)^{\frac{2}{2+\alpha}}}{((2+\alpha)N_{*})^{\frac{2(1+\alpha)}{2+\alpha}}}$
(97) $\displaystyle r\simeq$
$\displaystyle\frac{8(B\alpha)^{\frac{2}{2+\alpha}}}{((2+\alpha)N_{*})^{\frac{2(1+\alpha)}{2+\alpha}}}$
(98) $\displaystyle\mathcal{P}_{\rm{s}}\simeq$
$\displaystyle\frac{A}{12\pi^{2}}\frac{((2+\alpha)N_{*})^{\frac{2(1+\alpha)}{2+\alpha}}}{(B\alpha)^{\frac{2}{2+\alpha}}}.$
(99)

Please note that these expressions are obtained by using Eqs. (85) and (VII),
which, one can verify to be true by using $\phi$ and $N$ relations, as used in
general. It, therefore, shows that our method also provides consistent results
for the perturbations. Using these observational constraints, the constraint
on the model parameters, for this special case, can be obtained as:

$\displaystyle 2.40\leq\alpha\leq 55.14,\quad A\leq 8.70\times 10^{-10},$
(100)

and $B$ can be as large as $10^{126}$.

In the second special case with $V(\phi)=A(1-B\exp(-\alpha\phi))$, similarly,
the observable parameters can be written in terms of the e-folding number as

$\displaystyle n_{\rm{s}}\simeq$ $\displaystyle 1-\frac{2}{N_{*}},$ (101)
$\displaystyle n_{\rm{t}}\simeq$
$\displaystyle-\frac{1}{\alpha^{2}N_{*}^{2}},$ (102) $\displaystyle r\simeq$
$\displaystyle\frac{8}{\alpha^{2}N_{*}^{2}},$ (103)
$\displaystyle\mathcal{P}_{\rm{s}}\simeq$
$\displaystyle\frac{A\alpha^{2}N_{*}^{2}}{12\pi^{2}}.$ (104)

Using the observational constraints, the constraint on the model parameters
can be obtained as

$\displaystyle\alpha\geq 0.34,\quad A\leq 6.33\times 10^{-10}.$ (105)

It is important to note that the model parameter constraints mentioned above
do not account for the effect of reheating; therefore they do not justify the
actual limitations. To include the effect, we must analyze the effective
equation of state parameter during reheating $w_{\rm{re}}$, and the duration
of reheating $N_{\rm{re}}$, which is governed by the equation [49]:

$\displaystyle
N_{\rm{re}}=\frac{4}{3w_{\rm{re}}-1}\left(\log\left(\frac{k}{a_{\rm{0}}T_{\rm{0}}}\right)+N_{k}-\log(H_{k})+\right.$
$\displaystyle\left.\frac{1}{4}\log(\rho_{\rm{end}})+\frac{1}{3}\log\left(\frac{11g_{\rm{s,re}}}{43}\right)+\frac{1}{4}\log\left(\frac{30}{\pi^{2}g_{\rm{reh}}}\right)\right).$

Here, $\\{a_{\rm{0}},T_{\rm{0}}\\}$ are present values of the scale factor and
the temperature of the Universe, respectively. $H_{k}$ is the Hubble scale
when the mode leaves the Horizon; $\rho_{\rm{end}}$ is the energy density at
the end of inflation and $\\{g_{\rm{reh}},g_{\rm{s,re}}\\}$ are the effective
number of relativistic species upon thermalization and the effective number of
light species for entropy during reheating, respectively. As the reheating
epoch, as a result of the proposed new solution that modifies the smooth
transition from slow-roll to deep oscillations, deviates slightly different
than that previously expressed in the literature, we anticipate that the
constraints on the model parameters and the duration of reheating will also be
modified. It necessitates in-depth analysis, and thus, we will reserve it for
future work.

##  VIII Conclusions

We considered a single canonical scalar field model minimally coupled to
gravity with a potential $V(\phi)$ that leads to the evolution of the Universe
consisting of both slow-roll inflation and oscillatory behavior around the
potential’s minimum, also known as the reheating era. The complete solution
for the background field in this scenario remains elusive. Traditionally,
we’ve solved it using two discrete regimes, each with its own set of
conditions.

The first regime is characterized by the slow-roll condition, in which the
slow-roll parameters are considerably less than unity. In this regime, the
slow-roll solution is a well-established analytical solution. In contrast, the
second regime takes effect when the field enters the reheating phase, and the
Hubble parameters fall substantially below the effective mass of the
potential.

The difficulty arises, however, when attempting to bridge the distance between
these two regimes during a phase in which both slow-rolling and reheating
conditions fail. In order to resolve this dilemma, our work seeks to present a
model-independent, unified solution. With this objective in mind, we assume
the following:

  1. 1.

Simple potential: the nature of the potential is simple.

  2. 2.

Minimum of the potential potential has a minimum at $\phi=0$.

  3. 3.

Exact de-Sitter: $|\phi|\rightarrow\infty$ leads to $\epsilon_{\rm{1}}=0,$
implying the de-Sitter Universe.

  4. 4.

Slow-roll: the potential, for $|\phi|\gg 1$ leads to slow-roll inflation.

  5. 5.

Near-minimum behavior: In the vicinity of the minimum, the potential can be
approximated as $V(\phi)\propto\phi^{2}$ for simplicity.

To address this difficulty, we assume a new viewpoint by employing phase-space
variables $\\{\theta,H\\}$ rather than $\\{\phi,\dot{\phi}\\}$. This change is
advantageous because the variable $\theta$ corresponds directly to various
phases of cosmic evolution. For example, $\theta\ll 1$ (implies $|\phi|\gg 1$)
denotes a period of slow-roll inflationary evolution, whereas
$\theta=\sin^{-1}(1/\sqrt{3})$ (equivalent to $\epsilon_{\rm{1}}=1$) denotes
the precise end of the inflationary phase. In addition, $\theta=\pi/2$
corresponds to $\phi=0$, indicating that the field has reached the minimal
potential. This method is particularly significant because we know that
particle production, specifically resonance, occurs at the potential’s minimum
— a location that is difficult to pinpoint using cosmic time, $t$.

In our work, we first propose a unified solution for the Universe within the
chaotic inflation model, where the potential is $V(\phi)=1/2m^{2}\phi^{2}$. We
provide comprehensive solutions for critical parameters such as the Hubble
parameter $H$, slow-roll parameters $\epsilon_{\rm{1}}$ and
$\epsilon_{\rm{2}}$, the scalar field $\phi$, and its time derivative
$\dot{\phi}$ — all expressed in terms of $\theta$. To complete the dynamics,
we also furnish the solution for $\theta$ as a function of cosmic time, $t$.
We further extend our methods to broader models of inflation. This
accomplishment addresses three essential concerns:

  * (a)

Full Dynamics: We now possess the complete dynamical evolution of the
Universe, spanning from slow-roll inflation to reheating, including the
intermediate junction between these phases, rendered smoothly.

  * (b)

Intermediate identification: Second, as the full solution is now known, one
can immediately identify the evolution simply by knowing the correct value of
$\theta$ and analyzing the solution for the complete evolution of the
Universe.

  * (c)

Model Independence: Our solution is not tied to a specific model; instead, it
can be applied across a wide spectrum of inflationary models.

This integrated solution also provides insightful qualitative and quantitative
analysis of reheating. On the qualitative front, we can now incorporate the
effects of the end of inflation and the onset of reheating during the creation
of particles via parametric resonance, a process effectively described by the
Mathieu equation [35]. On the other hand, as stated previously, for
quantitative analysis, the equation that relates the reheating e-folding
number $N_{\rm{re}}$ (or the temperature at the end of reheating, i.e.,
$T_{\rm{re}}$) to the scalar spectral index $n_{\rm{s}}$, i.e., Eq. (VII), can
be modified by incorporating the proposed solution, which we believe can help
improve the observational constraints. In conclusion, by combining these two
analysis, the theoretical predictions can be substantially enhanced.

In conclusion, although our work focuses predominantly on simple models such
as chaotic inflation and small-field models like $\alpha$-attractors, we
recognize that more complex scenarios exist. These include models that deviate
from slow-roll during inflation, such as those that can generate primordial
black holes, as well as models with discrete behaviors around the minimum that
do not conform to $\phi^{2}$. Exploring these complex models and undertaking
in-depth perturbation analysis during the reheating period are promising
future research directions.

## Acknowledgements

DN is supported by the DST, Government of India through the DST-INSPIRE
Faculty fellowship (04/2020/002142). MK is supported by a DST-INSPIRE
Fellowship under the reference number: IF170808, DST, Government of India. DN
and MK are also very thankful to the Department of Physics and Astrophysics,
University of Delhi. MK and DN also acknowledge facilities provided by the
IUCAA Centre for Astronomy Research and Development (ICARD), University of
Delhi.

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