# Innovative Gain Reconfiguration for Active Fault-Tolerant Flight Control:
Balance of Stability and Agility

Ege C. ALTUNKAYA, Akin CATAK, Emre KOYUNCU, and Ibrahim OZKOL The authors are
with the Aerospace Research Center, Istanbul Technical University, Istanbul,
34469, Türkiye (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>ozkol@itu.edu.tr)(Corresponding author: Ege C.
ALTUNKAYA (e-mail: altunkaya16@itu.edu.tr))

###### Abstract

In this study, a distinct reconfigurable fault-tolerant flight control
strategy is addressed for mitigating one of the persistent safety-critical
issue, i.e. loss of control triggered by actuator faults. The attainable
acceleration set notion is taken a step further towards incremental attainable
acceleration set through a slight modification that enables instantaneous
controllability checks. The inner-loop gains are updated in case of a fault
using incremental attainable acceleration set and a tuning function, which is
in charge as a compensator of agility and robustness. Additionally, the outer-
loop gains are also such reconfigured that holding the bandwidth ratio of the
successive loops at a prudent level to ensure the closed-loop stability; for
this reason, an analytical outer-loop gain update law is derived based on the
inner-loop gains and bandwidth, actuator and command filter time constants.
Subsequently, the proposed architecture is assessed under a severe fault
scenario with a demanding maneuver mission. Noticeably, the proposed method
fulfills the expectations of stability and agility sufficiently, and surpasses
the fixed-gain approach.

###### Index Terms:

fault-tolerant flight control, reconfigurable flight control, agile aircraft
control, loss of control

##  I Introduction

Ensuring the safety of the aircraft under any circumstance is the main
objective of the entire design process, that’s why the civilian or military
certifications provide worthy guidelines to track during the design [1, 2].
However, unfortunately, regardless of how safely the aircraft is designed,
numerous fault-prone elements are mounted on an aircraft. Therefore, as one of
the safety assessments, the question of the likelihood of a fault occurrence
per flight can be partially answered at the system or whole aircraft level.
Since the impact of a fault can be dependent on the flight conditions,
estimating the potential effects on the aircraft is a tedious workload. At
this point, the necessity of an adaptiveness of the aircraft against any type
of fault under any circumstance arises.

Nevertheless, one of the disciplines which necessitates an adaptiveness
against presumable faults is the flight control, especially for an unstable
fighter aircraft. There exist numerous engineering and theoretical solutions
for accommodating these faults, such as robust flight control design or active
fault-tolerant control design [3, 4, 5]. Each technique, inherently, comes
with advantages, disadvantages, and even prerequisites, e.g. over-actuation
requires a control allocation method while robust flight control design may
compromise the agility of the aircraft [3]. However, the active fault-tolerant
control has attracted attention for a while instead of robust flight control
design for fault-tolerance [4].

In the open literature, there are plenty of active fault-tolerant flight
control proposals [6, 7, 8, 9, 10, 11, 12, 13]. Recently, the attention of
researchers has been drawn to reinforcement learning and artificial neural
network-based fault accommodation [14, 15]. Furthermore, control allocation-
based fault-tolerant control design methods continue to be proposed, such as
[16, 17]; however, it should not be confused that the existence of redundancy
in the control effectors implies that the system can cope with presumable
actuator faults under diverse circumstances. Therefore, care must be taken
even if the system is designed in an over-actuated suite. Some researchers
recognize the severity of the situation where an actuator fault during
maneuvers could trigger instability, emphasizing that fault-tolerant control
design cannot solely rely on handling model uncertainties [18].

Additionally, in [7], the author proposes dynamic control allocation depending
on actuator fault knowledge to accommodate various types of actuator faults,
such as loss of effectiveness, lock-in-place, and control surface damage.
However, the necessity of extensive and case-specific planning exists to
reconfigure the control allocation architecture. Furthermore, in [8], the
authors propose an active fault-tolerant flight control scheme by limiting the
achievable kinematics of the aircraft during a maneuver to prevent a loss of
control due to actuator faults, which is successfully simulated. Again,
however, offline and extensive planning is required to accommodate faults in-
flight. Vile et al. [11] propose a priority-weighted control allocation scheme
and sliding mode controller. While the proposed method performs
satisfactorily, it relies on offline prioritization, which is a conventional
approach. Also, since the flight control law is derived by leveraging sliding
mode control, passive fault tolerance, i.e., robust control characteristics,
may dominate the response instead of prioritization.

###  I-A Contributions

The main contributions of this study are as follows;

  * •

This study presents a unique perspective on active fault-tolerant flight
control by addressing the problem through aerospace notions. In this context,
the incremental attainable acceleration set (IAAS) is introduced and coherent
collaboration of nonlinear dynamic inversion and IAAS is proposed for fault
accommodation.

  * •

A new gain reconfiguration mechanism is developed that leverages both IAAS and
a tuning function which harmonizes the stability and agility of the aircraft,
which is one of the fundamental problems of the active fault-tolerant control.

  * •

Additionally, an analytical outer-loop gain update law is derived principally
based on inner-loop reconfiguration and successive loop bandwidth ratio
specification.

Consequently, the proposed method’s efficacy is examined through a coupled
maneuver mission.

##  II Preliminaries

The necessary background for the subsequent sections is established in this
section.

###  II-A Notations

Throughout the study, the time derivative of a $C^{1}$ function
$f:\mathbb{R}^{n}\xrightarrow[]{}\mathbb{R}$ is denoted by $\dot{f}$.
Moreover, a vector is denoted in the bold type, i.e. $\bm{v}$, and the vector
product of two vectors $\bm{x}$ and $\bm{y}$ is denoted by
$\bm{x}\times\bm{y}$. The operator of $\oslash$ denotes the Hadamard division,
and $sgn(*)$ is the signum function. Define $\gamma\in\mathbb{R}$, where
$\bm{\gamma}_{m}$ denotes the row vector of size $m$ with elements given by
$\gamma$. Furthermore, define a row vector of $\bm{\alpha}\in\mathbb{R}^{p}$
and a number of $\rho\in\mathbb{Z}_{>0}$, the notation of $\bm{\alpha}_{\rho}$
denotes the $\rho^{\text{th}}$ element of $\bm{\alpha}$. Similarly, if
$\mathbb{A}\in\mathbb{R}^{n\times n}$ is a diagonal matrix, the notation of
$\mathbb{A}_{\rho}$ denotes the $\rho^{\text{th}}$ diagonal element of
$\mathbb{A}$. Finally, a control affine system is described as given in
equation (1).

$\bm{\dot{x}}=\bm{f}(\bm{x})+\bm{g}(\bm{x})\bm{u}$ (1)

where $\bm{x}\in\mathbb{R}^{n}$ is the state vector, and
$\bm{u}\in\mathbb{R}^{m}$ is the control input vector. Nonlinear mappings of
$\bm{f}:\mathbb{R}^{n}\xrightarrow[]{}\mathbb{R}^{n}$ and
$\bm{g}:\mathbb{R}^{n}\xrightarrow[]{}\mathbb{R}^{n\times m}$ are locally
Lipschitz continuous functions.

###  II-B Flight Dynamics Model

The 6-degrees-of-freedom nonlinear translational and rotational dynamics of an
aircraft are given in a compact form in Eq. (2) and Eq. (3).

$\bm{\dot{V}}=m^{-1}[\bm{F}-\bm{\omega}\times m\bm{V}]$ (2)

$\bm{\dot{\omega}}=J^{-1}[\bm{M}-\bm{\omega}\times J\bm{\omega}]$ (3)

where $\bm{V}\in\mathbb{R}^{3\times 1}$ is the body velocity vector,
$\bm{\omega}\in\mathbb{R}^{3\times 1}$ is the angular rate vector,
$\bm{F}\in\mathbb{R}^{3\times 1}$ is the total body force vector,
$\bm{M}\in\mathbb{R}^{3\times 1}$ is the total moment vector,
$J\in\mathbb{R}^{3\times 3}$ is the inertia tensor, and $m$ is the mass of the
aircraft. Finally, the rotational kinematics are presented in equation (4).

$\bm{\dot{\Omega}}=\begin{bmatrix}1&\sin\phi\tan\theta&\cos\phi\tan\theta\\\
0&\cos\phi&-\sin\phi\\\
0&\sin\phi\sec\theta&\cos\phi\sec\theta\end{bmatrix}\bm{\omega}$ (4)

where $\bm{\Omega}$ is Euler’s angle vector.

Furthermore, the over-actuated aerodynamic modeling is derived from [19, 20],
where each coefficient is formulated as a polynomial function of the
corresponding states and control surface deflections. The formulation details
can be found in [19].

###  II-C Fault Detection and Isolation

The fault diagnosis modeling is not within the scope of the study; therefore,
it is assumed that such a fault detection and isolation methodology is
implemented, which detects and isolates the actuator jam fault within 0.1
seconds.

###  II-D Incremental Attainable Acceleration Set

The attainable acceleration set (AAS) is a distinct formation of the
attainable moment set (AMS) by leveraging the Euler’s equations of motion.
Since these sets give insight into the capabilities and controllable
boundaries of the aircraft, benefiting from AAS is reasonable in order to
assess whether the demanded commands are still bounded by AAS or not.

However, as a one step further, deriving the AAS by including the actuator
rate limits allows to investigate the instantaneous controllability. With this
rationale, primarily, the incremental attainable moment set notion is
introduced as given in equation (5).

$\Delta\mathbb{A_{M}}=\\{\Delta\bm{\tau}|\Delta\bm{\tau}=\Phi\Delta\bm{u},\hskip
4.26773pt\underline{\Delta\bm{u}}\leq\Delta\bm{u}\leq\overline{\Delta\bm{u}}\\}$
(5)

where $\Delta\bm{\tau}\in\mathbb{R}^{3\times 1}$ represents the vertices of
the convex-hull in question, while $\Delta\bm{u}$ denotes the control input in
the incremental form. Additionally, $\underline{\Delta\bm{u}}$ and
$\overline{\Delta\bm{u}}$ denotes the lower and upper limits of the control
inputs by considering the actuator constraints as defined in equation (6).

$\begin{split}\overline{\Delta\bm{u}}&=\textrm{min}(\dot{\bm{u}}_{max}\Delta
t_{s},\bm{u}_{max}-\bm{u}_{0})\\\
\underline{\Delta\bm{u}}&=\textrm{max}(-\dot{\bm{u}}_{max}\Delta
t_{s},\bm{u}_{min}-\bm{u}_{0})\end{split}$ (6)

in which $\dot{\bm{u}}_{max}$ is the maximum actuator rate limit, whereas
$\bm{u}_{max}$ and $\bm{u}_{min}$ are the maximum and minimum actuator
position limits, respectively. Also, $t_{s}$ is the fixed time step magnitude
and $u_{0}$ is the current actuator position. Based on the incremental AMS
derivation, the incremental AAS can be expressed as given in equation (7).

$\Delta\mathbb{A_{A}}=\\{\Delta\dot{\bm{\omega}}|\Delta\dot{\bm{\omega}}=J^{-1}[\Delta\bm{\tau}-\bm{\omega}\times
J\bm{\omega}],\hskip 4.26773pt\Delta\bm{\tau}\in\Delta\mathbb{A_{M}}\\}$ (7)

in which $\bm{\omega}=[p\hskip 4.26773ptq\hskip 4.26773ptr]^{T}$, representing
angular velocities. Consequently, the incremental AAS is derived to understand
if the aircraft is capable of performing the desired commands at that instant,
taking into account aerodynamic and mechanical characteristics, as well as
actuator constraints.

######  Assumption II.1 (Convex structure of the attainable sets)

The convex structure of the attainable acceleration set under healthy
circumstances may deteriorate after a fault occurrence. However, throughout
the study, it is assumed that the convex form is not significantly affected,
thus the convex-hull approach is still reasonable.

##  III Flight Control Architecture

###  III-A Control law design

The control law for the inner loop is designed using nonlinear dynamic
inversion. Since the aircraft is over-actuated, the inner-loop control law is
such derived that the output of the flight control module is to be the desired
control moment coefficient to feed the control allocation. Therefore, using
Euler’s equations of motion in equation (3), the necessary moment coefficients
can be expressed in the inverse form of equation (1) as given in equation (8).

$\begin{bmatrix}C_{l}\\\ C_{m}\\\
C_{n}\end{bmatrix}_{c}=\frac{J}{\bar{q}_{\infty}S}\begin{bmatrix}b&0&0\\\
0&\bar{c}&0\\\
0&0&b\end{bmatrix}^{-1}\Big{\\{}\bm{\dot{\omega}}_{c}+J^{-1}\Big{(}\bm{\omega}\times
J\bm{\omega}\Big{)}\Big{\\}}$ (8)

in which $b,\bar{c},S,\bar{q}_{\infty}$ are the wing span, mean aerodynamic
chord, wing area, and dynamic pressure, respectively. $[C_{l}\hskip
4.26773ptC_{m}\hskip 4.26773ptC_{n}]_{c}^{T}$ corresponds to the commanded
roll, pitch, and yaw moment coefficients. Also,
$\bm{\dot{\omega}}_{c}=[\dot{p}\hskip 4.26773pt\dot{q}\hskip
4.26773pt\dot{r}]_{c}^{T}$ is the virtual input as given in equation (9).

$\begin{bmatrix}\dot{p}\\\ \dot{q}\\\ \dot{r}\\\
\end{bmatrix}_{c}=\begin{bmatrix}\omega_{p}&&\\\ &\omega_{q}&\\\
&&\omega_{r}\\\ \end{bmatrix}\begin{bmatrix}p_{c}-p\\\ q_{c}-q\\\ r_{c}-r\\\
\end{bmatrix}$ (9)

where $\omega_{p},\omega_{q},\omega_{r}$ are control gains for $p$, $q$, and
$r$ channels, respectively. Consequently, if $\dot{\phi}=p$, $\dot{\theta}=q$,
and $\dot{\psi}=r$ are assumed while deriving the outer-loop control law for
the sake of simplicity, the following expression is obtained as given in
equation (10).

$\begin{bmatrix}p\\\ q\\\ r\end{bmatrix}_{c}=\begin{bmatrix}\omega_{\phi}&&\\\
&\omega_{\theta}&\\\ &&\omega_{\psi}\\\
\end{bmatrix}\begin{bmatrix}\phi_{c}-\phi\\\ \theta_{c}-\theta\\\
\psi_{c}-\psi\\\ \end{bmatrix}$ (10)

in which $\omega_{\phi},\omega_{\theta},\omega_{\psi}$ are control gains for
$\phi$, $\theta$, and $\psi$ channels, respectively. Deriving the control law
in this manner is preferred for creating an ease in the derivation of the
reconfiguration law, which will be introduced in the proceeding sections.

###  III-B Control allocation design

In order to capture the highly nonlinear aerodynamic characteristics of the
aircraft, the incremental nonlinear control allocation method is preferred as
the control allocation scheme. Therefore, the demanded control moment
coefficients vector is incrementally converted into the corresponding control
surface deflections. The formulation of the incremental nonlinear control
allocation in the general sense is given in equation (11).

$\Phi(x_{0},u_{0})\Delta\bm{u}=\Delta\bm{\tau_{c}}$ (11)

where $\Phi(x_{0},u_{0})\in\mathbb{R}^{3\times n}$ Jacobian matrix at that
instant, which $n$ is the number of control effectors. The Jacobian matrix
also refers to as the control effectivity matrix. Finally,
$\Delta\bm{\tau}_{c}$ is the difference between the demanded control moment
coefficient vector derived in the flight control module and the current moment
coefficient. Since the aircraft is over-actuated, the resulting control
effectivity matrix is not square; therefore, Moore-Penrose pseudo-inverse is
utilized in the weighted form given in equation (12).

$\displaystyle\Phi^{{\ddagger}}=W\Phi^{T}(\Phi W\Phi^{T})^{-1}$ (12)

where $\Phi^{{\ddagger}}$ is the pseudo-inverse of the control effectivity
matrix, and $W\in\mathbb{R}^{3\times 3}$ is the diagonal positive-definite
weight matrix. Consequently, the required control surface deflections are
calculated incrementally as given in equation (13).

$\displaystyle\Delta\bm{u}=\Phi^{{\ddagger}}\Delta\bm{\tau}_{c}$ (13)

At the end of the process, last actuator demand should be added to the
calculated incremental actuator demand, i.e. $\Delta\bm{u}=\bm{u}-\bm{u}_{0}$.
For the sake of clarity, entire flight control architecture is given in
diagram represented in Fig. 1.

Figure 1: Entire flight control architecture: nonlinear dynamic inversion
based control law derivation and weighted pseudo-inverse based incremental
nonlinear control allocation application to the nonlinear flight dynamics.

As expected, the corresponding state feedback are supplied from the flight
dynamics model. Also, as a note, $\phi_{c},\theta_{c},\psi_{c}$ and
$p_{c},q_{c},r_{c}$ commands are passed through a second-order and first-order
command filters, respectively.

##  IV Reconfigurable Fault-tolerant Flight Control

The reconfiguration of the flight control architecture depends on two notions:
incremental attainable acceleration set and bandwidth ratio, which are used
for inner-loop redesign and outer-loop redesign respectively. Moreover, the
processes introduced in the proceeding sections are activated only in the
event of a fault diagnosis; otherwise, the aircraft operates with the preset
flight control architecture.

###  IV-A Inner-loop Reconfiguration

The effect of the inner-loop on the aircraft is stabilizing it, meaning that
the inner-loop operates as a stability augmentation system. Therefore, the
controllability must be ensured in the inner-loop while reconfiguring the
corresponding gains. At this point, as a useful tool, the incremental
attainable aceleration set is benefited to create an adaptiveness of the
inner-loop for fault accommodation. Prior to introducing the reconfiguration
strategy, a representative illustration of the general framework is depicted
in Fig. 2.

Figure 2: Background for the reconfiguration strategy: A sample IAAS at an
arbitrary time step including commanded acceleration vector,
$\dot{\omega}_{c}$, which violates the boundary. Also, $\dot{\omega}_{c,r}$ is
the reconfigured acceleration command by reconfiguring $K_{\omega}$ to
$K_{\omega,r}$.

Obviously, it is possible to instantaneously stretch or shorten the desired
angular acceleration vector by comparing them to the incremental attainable
acceleration set at the corresponding time step. As a consequence, the desired
angular acceleration vectors can be guaranteed to be bounded by the attainable
set. In essence, this ensures controllability, as the boundaries of the
incremental attainable acceleration set determine the maximum or minimum
reachable accelerations without compromising controllability. However, there
are two presumable issues during the recalculation process: the chattering due
to the highly nonlinear aerodynamic characteristics and restricting the gain
alteration within a specific interval. Consequently, the following
reconfiguration strategy is established as given in equation (14) by
addressing the preceding issues.

$\begin{split}\underbrace{\begin{bmatrix}\omega_{p_{r}}&&\\\
&\omega_{q_{r}}&\\\ &&\omega_{r_{r}}\\\
\end{bmatrix}}_{\begin{subarray}{c}\text{reconfigured}\\\ \text{gain,
$K_{\omega,r}$}\end{subarray}}=\cdots\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\\\
\underbrace{\begin{bmatrix}\sigma_{p}&&\\\ &\sigma_{q}&\\\
&&\sigma_{r}\end{bmatrix}}_{\begin{subarray}{c}\text{tuning}\\\
\text{functions,
$\sigma_{\omega}$}\end{subarray}}\underbrace{\begin{bmatrix}\dot{p}_{\Delta\mathbb{A_{A}}}&&\\\
&\dot{q}_{\Delta\mathbb{A_{A}}}&\\\ &&\dot{r}_{\Delta\mathbb{A_{A}}}\\\
\end{bmatrix}}_{\begin{subarray}{c}\text{capability limits}\\\ \text{for each
channel,
$\Delta\mathbb{A_{A}}_{\omega}$}\end{subarray}}\oslash\underbrace{\begin{bmatrix}sgn(e_{p})(|e_{p}|+\epsilon)\\\
sgn(e_{q})(|e_{q}|+\epsilon)\\\
sgn(e_{r})(|e_{r}|+\epsilon)\end{bmatrix}}_{\begin{subarray}{c}\text{error for
each channel, $\bm{e}$}\end{subarray}}\end{split}$ (14)

where $\omega_{p_{r}},\omega_{q_{r}},\omega_{r_{r}}$ are the reconfigured
gains of the angular velocities and $\epsilon\in\mathbb{R}_{>0}$ is small
positive number to prevent the zero division. Additionally,
$\dot{p}_{\Delta\mathbb{A_{A}}}$, $\dot{q}_{\Delta\mathbb{A_{A}}}$,
$\dot{r}_{\Delta\mathbb{A_{A}}}\in\mathbb{R}$ are the local extremum
magnitudes of each angular accelerations over the incremental attainable
acceleration set, along the direction of the commanded acceleration vector.
Specifically, if the error for the corresponding control channel is positive,
the extremum value is positive; otherwise, vice versa holds as illustrated in
Fig. 2. Finally, one of the unique aspects of the formulation is the
utilization of tuning functions: $\sigma_{p},\sigma_{q}$, and $\sigma_{r}$
which will be discussed in the following section.

####  IV-A1 Tuning function design

In order to overcome the previously stated presumable issues, a sigmoid-like
tuning function is designed, and indeed, tuning function is uniquely and
separately designed for each inner-loop control channel. The formulation is
given in equation (15).

$\sigma_{\omega}=\sigma_{\omega_{l}}+\dfrac{\sigma_{\omega_{u}}-\sigma_{\omega_{l}}}{1+e^{-\beta(|e_{\omega}|-\alpha)}}$
(15)

where $\sigma_{{\omega}_{l}}$ and $\sigma_{{\omega}_{u}}$ are the lower and
upper limits of the tuning function for channel $\omega$, respectively. Also,
$\alpha\in\mathbb{R}_{>0}$ and $\beta\in\mathbb{R}_{>0}$ are hyper-parameters
to be tuned. The variation of the tuning function with the hyper-parameters of
$\alpha$ and $\beta$ is demonstrated in Fig. 3.

Figure 3: Tuning function behaviour with the change of hyper-parameters.

The purpose of the tuning function is to compensate for the adaptive gain’s
behavior concerning the error level. Therefore, it is designed as a function
of the error in the corresponding channel. For example, when the error level
is relatively low, the related gain tends to increase excessively due to the
division by error within the attainable set. However, at that moment, the
tuning function tends to decrease. As a result, the gain is prevented from
increasing excessively.

On the other hand, due to the nonlinear characteristics of aerodynamics, the
attainable set may exhibit chatter during unsteady flight. Therefore, the
tuning function is also responsible for ensuring that the gain alteration is
as smooth as possible. It’s important to note that the tuning function uses
hyper-parameters that need to be adjusted by the user. To achieve satisfactory
performance in terms of both tracking and stability, these hyper-parameters
should be carefully tuned, e.g. as $\alpha$ decreases, the reduction of the
tuning function magnitude takes place closer to zero error, whilst as $\beta$
decreases the gain gets more sluggish, or in other words smoother.

####  IV-A2 Inner-loop reconfiguration stability analysis

Since the tracking error stabilization is required to ensure the maneuver
stability, the Lyapunov candidate is constructed based on the tracking error,
as presented in Eq. (16).

$V(\bm{e})=\frac{1}{2}\bm{e}^{T}\bm{e}$ (16)

where $\bm{e}(t)=\bm{\omega}_{ref}(t)-\bm{\omega}(t)$, the difference between
the reference signal and obtained state. Obviously, $\bm{e}^{*}=\bm{0}_{n}$,
$V(\bm{e}^{*})=0$, and
$V(\bm{e})\in\mathbb{R}_{>0},\forall\bm{e}\in\mathbb{R}^{n}-\\{0\\}$. Simply,
the time derivative of the Lyapunov candidate is derived as in Eq. (17).

$\frac{dV(\bm{e})}{dt}=\frac{\partial
V}{\partial\bm{e}}\frac{\partial\bm{e}}{\partial t}=\bm{e}^{T}\dot{\bm{e}}$
(17)

Since $\dot{\bm{e}}=\dot{\bm{\omega}}_{ref}-\dot{\bm{\omega}}$, Eq. (17) is
rewritten in Eq. (18).

$\dot{V}=\bm{e}^{T}\dot{\bm{\omega}}_{ref}-\bm{e}^{T}[f(\bm{\omega})+g(\bm{\omega})\bm{u}]$
(18)

###### Assumption IV.1

The reference signal is known, $C^{1}$ class function.

Since the control input of $\bm{u}$ is derived using the nonlinear dynamic
inversion in equation (8), i.e.
$\bm{u}=\bm{g}(\bm{\omega})^{-1}[\bm{\dot{\omega}}_{c}-\bm{f}(\bm{\omega})]$,
the equation (18) turns into equation (19).

$\dot{V}=\bm{e}^{T}\dot{\bm{\omega}}_{ref}-\bm{e}^{T}\dot{\bm{\omega}}_{c}<0$
(19)

Recall that
$K_{\omega,r}=\sigma_{\omega}\Delta\mathbb{A_{A}}_{\omega}\oslash\bm{e}$ from
equation (14). Therefore, the following expression is obtained in equation
(20).

$\dot{V}=\bm{e}^{T}\dot{\bm{\omega}}_{ref}-\bm{e}^{T}[\sigma_{\omega}\Delta\mathbb{A_{A}}_{\omega}\oslash\bm{e}]\bm{e}<0$
(20)

For the asymptotically stable error dynamics, the following inequality in
equation (21) must be satisfied.

$\bm{e}^{T}\dot{\bm{\omega}}_{ref}<\bm{e}^{T}\sigma_{\omega}\Delta\mathbb{A_{A}}_{\omega}\bm{1}_{3}$
(21)

Consequently, the following relation in equation (22) concludes the stability
requirements.

$\lvert\dot{\bm{\omega}}_{ref,i}\rvert<\lvert\sigma_{\omega,i}\Delta\mathbb{A_{A}}_{\omega,i}\rvert\Rightarrow\dot{V}<0\\\
$ (22)

where $i={1,2,3}$ denotes the $p$, $q$, and $r$, respectively.

###### Corollary IV.1

The relation in equation (22) implies that there exist such a tuning function
of $\sigma_{\omega}$ that establishes a balance between the desired input and
capabilities of the aircraft. Furthermore, since the commanded acceleration
vector must be bounded by the IAAS, the maximum value of the tuning function
must be less than 1.

###  IV-B Outer-loop reconfiguration

The outer-loop serves as the control augmentation system, while the inner-loop
functions as the stability augmentation system. Consequently, once the inner-
loop achieves its stabilization objective, the main goal of outer-loop
reconfiguration should be to ensure overall closed-loop stability and achieve
good tracking performance. To satisfy these expectations, reconfiguring the
outer-loop gains through making them dependent on the inner-loop gains is
interiorized.

The bandwidth concept is highly suitable for achieving each of the
aforementioned goals. This is because keeping a certain and safe level of the
bandwidth ratio between the inner and outer loops can ensure overall closed-
loop stability. Additionally, bandwidth can be employed to adjust track
performance, as it is well-known that bandwidth is inversely proportional to
the damping ratio of the system. Consequently, it becomes possible to derive
an analytical and explicit expression for the outer-loop gains in relation to
the inner-loop gains. However, prior to delving into the derivation, let
introduce the equivalent linear representation of the overall system as shown
in Fig. 4.

Figure 4: The representation of the nonlinear closed-loop system in the
linearized form due to the nonlinear dynamic inversion: $\dot{\Omega}_{\nu}$
denotes the commanded virtual input, i.e. $\omega_{c}$, while
$\dot{\omega}_{\nu}$ also denotes the virtual input.

in which $K_{\Omega}$ are the outer-loops gains, $K_{\omega}$ are the inner-
loop gains, $A(s)$ is the actuator dynamics, $A(s)=1/(\tau_{a}s+1)$; $\tau$
and $\tau_{a}$ are the command filter and actuator time constants
respectively. Since the nonlinear dynamic inversion is employed in the inner-
loop control, the aircraft dynamics are transformed to a single integrator
system; however, the actuator dynamics and command filters still appear.
Moreover, since $\dot{\phi}=p$, $\dot{\theta}=q$, and $\dot{\psi}=r$ are
assumed, the representation provided above remains valid for investigating.
Subsequently, let derive the transfer functions for both inner-loop and
overall system. The inner loop transfer function can be easily obtained since
it is second-order simple system as given in equation (23) in a general sense.

$\dfrac{\omega}{\omega_{c}}=\dfrac{K_{\omega}}{\tau_{a}s^{2}+s+K_{\omega}}$
(23)

where the natural frequency is, $\omega_{n}=\sqrt{K_{\omega}/\tau_{a}}$, and
the damping ratio is, $\zeta=1/\sqrt{4K_{\omega}\tau_{a}}$. The bandwidth of a
conventional second-order system is given in equation (24).

$\nu=\omega_{n}\sqrt{1-2\zeta^{2}+\sqrt{2-4\zeta^{2}+4\zeta^{4}}}$ (24)

If the obtained natural frequency and damping ratio are plugged in equation
(24), the following expression for the inner-loop bandwidth is obtained as
given in equation (25).

$\begin{split}\nu_{\omega}=\sqrt{\dfrac{K_{\omega}}{\tau_{a}}}\sqrt{\sqrt{\dfrac{1}{4K_{\omega}^{2}\tau_{a}^{2}}-\dfrac{1}{K_{\omega}\tau_{a}}+2}-\dfrac{1}{2K_{\omega}\tau_{a}}+1}\end{split}$
(25)

Finally, the bandwidth of the inner-loop is expressed in terms of actuator
time constant and inner-loop gain of the corresponding channel. At this point,
let introduce the overall system transfer function to derive its bandwidth.
The overall system closed-loop transfer function is given in equation (26).

$\dfrac{\Omega}{\Omega_{c}}=\dfrac{K_{\Omega}K_{\omega}}{\tau_{a}\tau
s^{4}+(\tau_{a}+\tau)s^{3}+(1+K_{\omega}\tau)s^{2}+K_{\omega}s+K_{\Omega}K_{\omega}}$
(26)

Recall that the bandwidth frequency is the point where the gain of the system
is equal to the $1/\sqrt{2}$ of the gain at zero-frequency input, thus if the
equation of $|G(j\nu_{\Omega})|=\dfrac{1}{\sqrt{2}}|G(j0)|$ is set and
$s=j\nu_{\Omega}$ and $s=j0$ are plugged in the equation (26), the following
final expression is obtained as given in equation (27).

$\begin{split}2K_{\Omega}^{2}K_{\omega}^{2}-[\tau_{a}\tau\nu_{\Omega}^{4}-\nu_{\Omega}^{2}-K_{\omega}\tau\nu_{\Omega}^{2}+K_{\Omega}K_{\omega}]^{2}\cdots\\\
\cdots-[(-\tau_{a}-\tau)\nu_{\Omega}^{3}+K_{\omega}\nu_{\Omega}]^{2}=0\end{split}$
(27)

where $\nu_{\Omega}$ is the overall system bandwidth for the corresponding
channel.

If the equation (27) is solved for $\nu_{\Omega}$, the bandwidth can be
calculated; however, if the equation is solved for $K_{\Omega}$ in order to
obtain an explicit relationship between the inner and outer-loop gains, the
following non-trivial lengthy expression is achieved as given in equation
(28).

$\begin{split}K_{\Omega}=\dfrac{-\nu_{\Omega}^{2}+\nu_{\Omega}\sqrt{2K_{\omega}^{2}\tau^{2}\nu_{\Omega}^{2}+K_{\omega}^{2}-4K_{\omega}\tau_{a}\tau^{2}\nu_{\Omega}^{4}-2K_{\omega}\tau_{a}\nu_{\Omega}^{2}+2K_{\omega}\tau\nu_{\Omega}^{2}+2\tau_{a}^{2}\tau^{2}\nu_{\Omega}^{6}+\tau_{a}^{2}\nu_{\Omega}^{4}-2\tau_{a}\tau\nu_{\Omega}^{4}+\tau^{2}\nu_{\Omega}^{4}+2\nu_{\Omega}^{2}}}{K_{\omega}}\cdots\\\
\cdots+\dfrac{\tau_{a}\tau\nu_{\Omega}^{4}-K_{\omega}\tau\nu_{\Omega}^{2}}{K_{\omega}}\end{split}$
(28)

As a result, an analytical update law of the outer-loop gains based on the
inner-loop gains, actuator and command filter time constants, and overall
closed-loop system bandwidth could be achieved. Therefore, if the overall
closed-loop system bandwidth is specified while considering the bandwidth
ratio of the inner-loop and overall system, calculating the outer-loop gain
in-flight becomes a straightforward task.

###  IV-C Reconfiguration Strategy

Subsequent to diagnosing the actuator jam fault 0.1 seconds after the fault
occurrence by a hypothetical fault detection and isolation module, the preset
gains in the flight control unit are initialized to be reconfigured by
considering the proposed methods. Principally, the inner-loop reconfiguration
is achieved at each time step in an online manner by the incremental
attainable acceleration set and a compensator tuning function in order to
ensure the system stability to prevent a presumable loss of control due to the
actuator fault. Also, note that the inner-loop bandwidth is calculated at each
time step using equation (25).

Afterwards, in order to ensure the stability without sacrificing the agility,
the bandwidth ratio is restricted to a level of 15, i.e.
$\nu_{\omega}/\nu_{\Omega}=$ 15\. As a consequence, the overall system’s
bandwidth is determined in that manner and thanks to the prior knowledge of
the actuator and command filter time constants, the outer-loop gains are
calculated by plugging these obtained parameters in equation (28).

Finally, the inner-loop and outer-loop gains are reconfigured in-flight to
guarantee the controllability, stability, and satisfactory track performance.

##  V Results

The efficacy of the proposed method is demonstrated through an excessive test
case simulation, which encompasses both the right aileron and left horizontal
tail simultaneous lock-in-place fault injection at the 3 second of the
simulation111For the simulation videos of more test cases, please visit:
https://www.youtube.com/watch?v=7-TXah7Htbs. However, the aircraft begins
conducting a coupled commanded maneuver at 2 second of the simulation, meaning
that the lock-in-place occurs at a quite critical time instant. The
corresponding flight simulation state trajectories and control surface
deflection histories are illustrated in Fig. 5 and Fig. 6.

Figure 5: State history of the test maneuver: reconfigured gain and fixed
gain. Figure 6: Control deflection history and control effort of the test
maneuver: reconfigured gain and fixed gain.

Obviously, being conservative by increasing the bandwidth ratio is a prudent
step to ensure the system’s stability in the event of a fault. The bandwidth
ratio of the fixed-gain system is relatively low; therefore, the aircraft is
more responsive. As a consequence, the aircraft with fixed-gain approach has a
tendency to the loss of control. In contrast, the aircraft with reconfigured-
gain approach is in a trade-off between robustness and agility, since it is
bounded by the controllability, stability, and track performance metrics.
Furthermore, the updated inner and outer-loop gain histories are presented in
Fig. 7.

Figure 7: Gains and tuning function history of the corresponding channels.

Due to the ample volume of IAAS, it becomes possible to generate larger inner-
loop gains. Consequently, the gains in the angular rate channels rise to an
acceptable limit. In contrast, outer-loop gains typically decrease, often
falling below their initial values to achieve the desired bandwidth ratio. It
is noteworthy that the sigmoid-like tuning functions play a crucial role in
adjusting the inner-loop gains, taking into account the boundaries defined by
the IAAS, track performance expectations, and the need for smooth variations
over time. Finally, the IAAS variations with time for both approaches are
presented, afterwards the output trajectory of the flight is demonstrated in
Fig. 8 and Fig. 9, respectively.

---
(a) Fixed gain during maneuver

---
(b) Reconfigured gain during maneuver

Figure 8: Attainable acceleration set history: (a) fixed gain approach, (b)
reconfigured gain approach (red: if the control acceleration vector exceeds
the IAAS; green: if the control acceleration vector is bounded by IAAS).

Obviously, an unbounded control acceleration demand triggers a loss of
control; nevertheless, designing the inner-loop gains in such manner that the
generated control acceleration vector remains bounded by IAAS of the aircraft
can mitigate the risk of the loss of control subsequent to a fault. Also,
noticeably, the actuator fault causes IAAS to shrink and results in
disappearance of several vertices, i.e. controllable bounds are seriously
deteriorated.

Figure 9: 3D trajectory of the maneuver.

Consequently, it is observed that the proposed method is highly satisfactory
in terms of both track performance and stability.

##  VI Conclusion

Throughout this study, a unique active fault-tolerant flight control method is
thoroughly examined, designed to ensure the stability of the aircraft even
under demanding mission conditions. The inner-loop gains are updated using the
introduced notion, i.e. incremental attainable acceleration set, to ensure
that the requested control acceleration vectors remain bounded. Furthermore, a
more conservative formulation is derived for updating the outer-loop gains
based on the inner-loop gain, actuator and command filter time constants, and
a predetermined bandwidth ratio —specifically, $\nu_{\omega}/\nu_{\Omega}=$ 15
in this study. Finally, the proposed architecture is assessed using a high-
fidelity over-actuated F-16 nonlinear flight dynamics model running at 100Hz.
The result has demonstrated that the proposed method is capable of stabilizing
the aircraft even in the presence of extremely harsh fault and complex mission
requirement. As an extension of this work, we envision applying the same
approach to a real-world platform and conducting real-time experiments to
further validate its effectiveness.

##  VII Acknowledgement

The authors propose their gratitude to Dr. Mustafa Demir for his fruitful
contributions to the study.

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# The Exact Solution Approach to Warm Inflation

Oem Trivedi
School of Arts and Sciences
GICT Building, Central Campus, Ahmedabad University
Navrangpura, Ahmedabad, 380009
Gujarat,India
Email<EMAIL_ADDRESS>
###### Abstract

The theory of Cosmic Inflation has received a great amount of deserved
attention in recent years due to it’s stunning predictions about the Early
Universe. Alongside the usual Cold Inflation paradigm, Warm Inflation has
garnered a huge amount of interest in modern Inflationary Studies. It’s
peculiar features and specifically different predictions from Cold Inflation
have led to a substantial amount of literature about it. Various Modified
Cosmological Scenarios have also been studied in the Warm Inflationary regime.
In this work, we introduce the Exact Solution Approach for Warm Inflation.
This Approach allows one to directly study Warm Inflationary regime in a
variety of Modified Cosmological Scenarios. We begin by outlining our method
and show that it generalizes the Modified Friedmann Approach of Del Campo ,
and reduces to the well known Hamilton Jacobi Formalism for Inflation in
particular limits. We also find the Perturbation spectra for cosmological and
tensor perturbations in the Early Universe, and then apply our method to study
Warm Inflation in a Tsallis Entropy Modified Friedmann Universe. We end our
paper with some concluding remarks on the domain of applicability of our work.

###### Contents

  1. 1 Introduction
  2. 2 The Exact Solution Method
  3. 3  Perturbation Spectra Analysis
  4. 4 Warm Inflation in Tsallis Entropy Modified Friedmann Universe
  5. 5 Conclusions

##  1 Introduction

Some of the most captivating problems in modern cosmology concern the workings
of the very early Universe. A lot of the physics of how our Universe was
initially is not very well understood and this was evident ever since fine
tuning problems were discovered in the Traditional Big Bang Cosmology model
itself. The most prominent of these problems were the flatness, horizon and
monopole problems, which were very promisingly solved by the introduction of
Cosmic Inflation [1, 2, 3, 4]. Cosmic Inflation is the idea that the very
early Universe went through a period of rapid accelerated expansion. The
dynamics of the traditional Inflationary Models generally consider a real
scalar field, the ”Inflaton” field, as the predominant contributor to the
universal energy density at that time. While there are other Inflationary
Models which consider more exotic scenarios [5, 6, 7], there is still a very
high amount of evident interest in single real scalar field driven
Inflationary regimes [8, 9, 10, 11].

The conventional models of Inflation by a single scalar field consist of a
period of rapid accelerated expansion. The sub horizon scale quantum
fluctuations during this period of expansion, lead to cosmological
perturbations which give us a spectrum of scalar and tensor perturbations (
vector perturbations are usually ignored). This generates the large scale
structure of our Universe which we see today. Also , after the period of
expansion sets off a period of ” reheating” [2] , where the energy of the
Inflaton Field decays to form radiation energy densities. These models do not
take into account the dissipation of Inflaton energy to form radiation while
the Inflationary period itself is going on and hence, these are known
popularly now as ” Cold Inflation” scenarios. On the other hand, the class of
Inflationary Regimes which do take into account this dissipation effect are
known as ” Warm Inflation” regimes [12, 13, 14, 15]. Warm inflation includes
inflaton interactions with other fields throughout the inflationary epoch
instead of confining such interactions to a distinct reheating era.
Consequently, these inflationary models do not need a separate period of
reheating after Inflation to make the Universe radiation dominated, and hence
this provides a solution to the graceful exit problem of Inflation [16].

Recently a lot of interest has been weighed towards Cosmological models which
modify the structure of the usual Friedmann Equation

$H^{2}=\frac{8\pi}{3m_{p}^{2}}\rho$ (1)

where we have used the units $c=\hbar=1$ and $m_{p}^{2}$ is the usual reduced
planck mass. $\rho$ is the energy density in the Universe, which in the
context of usual Inflationary Models refers to the energy density of the
Inflaton Field. Various different cosmological scenarios bring about a change
in this usual Friedmann Equation . Superstring and M-Theory bring the
possibility of considering our universe as a domain wall embedded in a higher
dimensional space. In this scenario the standard model of particle is confined
to the brane, while gravitation propagate into the bulk spacetime. The effect
of extra dimensions induces a well known change in the Friedmann equation [17]
,

$H^{2}=\frac{8\pi}{3m_{p}^{2}}\rho(1+\frac{\rho}{2\lambda})$ (2)

where $\lambda$ is a measure of the brane tension.

It is also possible to consider quantum corrected Area Entropy relations and
study Universal Evolution in such scenarios. For example, [18] considered the
quantum corrected Entropy

$S=m_{p}^{2}\frac{A}{4}-\tilde{\alpha}\ln(m_{p}^{2}\frac{A}{4})$ (3)

where A is the Area of the apparent horizon and $\tilde{\alpha}$ is
dimensionless positive constant determined by conformal anolmaly of the
fields.This conformal anomaly is interpreted as a quantum correction to the
entropy of the apparent horizon. The resulting Friedmann Equation is of the
form

$H^{2}-\beta H^{4}=\frac{8\pi}{3m_{p}^{2}}\rho$ (4)

, where $\beta$ is some positive constant in the units of inverse mass
squared. This type of Friedmann Equation can also be arrived at by using a
Chern Simons modification of Gravity [19] . Finally, there has been a surge in
interest recently in the form of Black hole Entropy due to Tsallis and Cirto
[20] , where they argued that argued that the microscopic mathematical
expression of the thermodynamical entropy of a black hole does not obey the
area law and can be modified as

$S=\gamma A^{\kappa}$ (5)

, where $\gamma$ is an unknown constant , A is the Black Hole Horizon area and
$\kappa$ is a real parameter which quantifies the degree of nonextensivity,
known as the Tsallis Parameter. Considering the apparent horizon entropy of an
FLRW type Universe to be the Tsallis type, [21] was able to derive a Friedmann
Equation for the Universe while also taking into account and modifying
Padmanabhan’s Emergent Gravity proposal [22]. The concerned Friedmann Equation
is [21] ,

$(H^{2})^{2-\kappa}=\frac{8\pi}{3m_{p}^{2}}\rho$ (6)

.

Inflation has been studied in a lot of the popular Braneworld Scenarios , both
in the traditional supercooled regime [23, 24, 25] and in the warm Inflation
regime [26, 27, 28, 29]. [29] also studied Chern Simons type Modified Gravity
of [19] in the Cold Inflation regime. A crucial point , as noted in [29],
which emerges from the studies of Inflationary Studies in different
cosmological scenarios is that the concerned Friedmann Equation of
investigation can be often taken of the form

$F(H)=\frac{8\pi}{3m_{p}^{2}}\rho$ (7)

, where $F(H)$ is some function of the Hubble Parameter H. $\rho$ in the
context of usual Inflationary Models is the energy density of the real scalar
Inflaton Field , $\phi$ ,

$\rho=\rho_{\phi}$ (8)

.

A very useful approach to studying different cold Inflation scenarios was
discussed in [29], taking the lead of (7) . We would now like to study
different Cosmological Scenarios using (7) as the primary equation of
investigation. We hence structure our paper as follows. In section 2, we
outline the exact solution approach for Warm Inflation and describe the
dynamical aspects of the method while in Section 3 we discuss Perturbation
Spectra and derive crucial observational quantities like the scalar and tensor
spectral index in concern to the approach. In Section 4 we apply the method to
study Warm Inflation in the high dissipative regime in a Tsallis Entropy type
Universe of [21] . Then finally, we summarize our results in section 5 in the
Concluding remarks of the paper.

##  2 The Exact Solution Method

In Warm Inflationary approach, there is consideration given to both the
inflaton field and the radiation being created due to it’s dissipation during
Inflation itself. Hence, the usual Friedmann Equation of Warm Inflation is
[12] ,

$H^{2}=\frac{8\pi}{3m_{p}^{2}}(\rho_{\phi}+\rho_{r})$ (9)

whee the energy density $\rho$ consists of both inflaton and radiation energy
densities, $\rho_{\phi}$ and $\rho_{r}$ respectively. But, during Inflation ,
$\rho_{\phi}>>\rho_{r}$ [12] , hence (9) becomes ,

$H^{2}=\frac{8\pi}{3m_{p}^{2}}(\rho_{\phi})$ (10)

, where $\rho_{\phi}$ is given by ,

$\rho_{\phi}=\frac{\dot{\phi}^{2}}{2}+V(\phi)$ (11)

with $V(\phi)$ being the potential under which the Inflaton field is. The
Inflaton Field Equation , the inflaton energy density and radiation energy
density evolution equation is given by,

$\ddot{\phi}+(3H+\Gamma)\dot{\phi}+V^{\prime}(\phi)=0$ (12)
$\dot{\rho_{\phi}}+3H(\rho_{\phi}+p_{\phi})=-\Gamma\dot{\phi}^{2}$ (13)
$\dot{\rho_{r}}+4H\rho_{r}=\Gamma\dot{\phi}^{2}$ (14)

Here , $\Gamma$ is the dissipation coefficient of Inflaton Energy. Defining a
quantity $Q=\frac{\Gamma}{3H}$ , we can rewrite the above equations as

$\ddot{\phi}+3H(1+Q)\dot{\phi}+V\prime(\phi)=0$ (15)
$\dot{\rho_{\phi}}+3H(\rho_{\phi}+p_{\phi})=-3HQ\dot{\phi}^{2}$ (16)
$\dot{\rho_{r}}+4H\rho_{r}=3HQ\dot{\phi}^{2}$ (17)

During Inflation, we can take both $\dot{\rho_{r}}$ to be approximately zero
[12] . This gives us (17)

$\rho_{r}=\frac{3Q}{4}\dot{\phi}^{2}$ (18)

Now, this was just a small review of the basics of Warm Inflation which we
will now freely refer to throughout the entirety of paper.

Consider now the following general form of the Friedmann equation

$F(H)=\frac{8\pi}{3m_{p}^{2}}\rho_{\phi}=\frac{8\pi}{3m_{p}^{2}}\left(\frac{\dot{\phi}^{2}}{2}+V(\phi)\right)$
(19)

The famous slow roll conditions of cold Inflation [8] , hold well similarly in
Warm Inflation as well [12] . Hence during Warm Inflation,

$\dot{\phi}^{2}<<V(\phi)$ (20)

This gives us the energy density to be $\rho_{\phi}\approx V(\phi)$ Hence the
general Friedmann Equation (19) during Inflation, takes the form

$F(H)\approx\frac{8\pi}{3m_{p}^{2}}V(\phi)$ (21)

Now, it was shown in [30] that treating $\phi$ itself as the explicitly
dependent variable for H can lead to a very apt method of studying
Inflationary models, which is referred to as the ” Hamilton Jacobi Approach to
Inflation”. [31] studied the Hamilton Jacobi approach with respect to Warm
Inflation and found that treating the Hubble Parameter and the Disspiation
Coefficients in terms of $\phi$ is indeed very helpful for Warm Inflationary
regimes. Hence, now we treat the Hubble parameter ( and hence the scale factor
as well) primarily in terms of the field variable $\phi$. So $H=H(\phi)$.
Further the field variable will generally be time dependent, hence the Hubble
Parameter stays implicitly time dependent.

With these considerations, we take the derivative of (21) with respect to
$\phi$ ,

$F_{{}_{\prime}H}H^{\prime}(\phi)=\frac{8\pi}{3m_{p}^{2}}V^{\prime}(\phi)$
(22)

where , following the convention in [29], $F_{{}_{\prime}H}$ is the partial
derivative of F with respect to H , and the prime in the superscript denotes
derivative with respect to $\phi$ . In the slow roll approximation (12)
becomes,

$3H(1+Q)\dot{\phi}=-V^{\prime}(\phi)$ (23)

Using the expression for $V^{\prime}(\phi)$ from (22), we have $\dot{\phi}$ as
,

$\dot{\phi}=-\frac{m_{p}^{2}}{8\pi}\frac{F_{{}_{\prime}H}H^{\prime}}{H(1+Q)}$
(24)

Using (24) , we can get a different expression for $\rho_{r}$ by taking into
account (18) ,

$\rho_{r}=\frac{3Q}{4}\left(\frac{m_{p}^{2}}{8\pi}\frac{F_{{}_{\prime}H}H^{\prime}}{H(1+Q)}\right)^{2}$
(25)

Hence, we now have expression for the field velocity given $F(H)$ ,an ansatz
for H in terms of $\phi$ and a form of the dissipation coefficient. As
mentioned in [[32], [13] ] , dissipation coefficient is popularly used in
power laws in terms of both $\phi$ and temperature T. The temperature plays as
especially important role for high dissipative regimes, which are
characterized by

$Q>>1\implies\Gamma>>3H$

and temperature itself plays a very crucial in the generation of quantum
fluctuations during Inflation [[16] , [13]] .

Using (24) , we can now get a different expression for $\rho_{r}$ by taking
into account (18) ,

$\rho_{r}=\frac{3Q}{4}\left(\frac{m_{p}^{2}}{8\pi}\frac{F_{{}_{\prime}H}H^{\prime}}{H(1+Q)}\right)^{2}$
(26)

Using (25) , we can get an expression for $\rho_{r}$ in terms of
$\rho_{\phi}$. It’s easy to check that we’ll get

$\rho_{r}=\frac{Q\rho_{\phi}m_{p}^{2}}{32\pi
F}\left(\frac{F_{{}_{\prime}H}H^{\prime}}{H(1+Q)}\right)^{2}$ (27)

We also note that as $\rho_{r}$ is pure radiation , we can write

$\rho_{r}=\alpha T^{4}$ (28)

where T is the radiation temperature and $\alpha$ is the Stefan Boltzmann
Constant. Now, we have from (18) and (26),

$\alpha
T^{4}=\frac{3Q}{4}\left(\frac{m_{p}^{2}}{8\pi}\frac{F_{{}_{\prime}H}H^{\prime}}{H(1+Q)}\right)^{2}$
(29)

which gives us

$T=\left(\frac{3Q}{4\alpha}\right)^{1/4}\left(\frac{m_{p}^{2}}{8\pi}\frac{F_{{}_{\prime}H}H^{\prime}}{H(1+Q)}\right)^{1/2}$
(30)

We note from (19) that

$\frac{3m_{p}^{2}}{8\pi}F-\frac{1}{2}\dot{\phi}^{2}=V(\phi)$ (31)

Using the expression for $\dot{\phi}$ (24), we have the Potential given as

$V(\phi)=\frac{3m_{p}^{2}}{8\pi}F-\frac{1}{2}\left(\frac{m_{p}^{2}}{8\pi}\frac{F_{{}_{\prime}H}H^{\prime}}{H(1+Q)}\right)^{2}$
(32)

Also as

$\rho_{\phi}=\frac{3m_{p}^{2}}{8\pi}F$

, we can also write the potential in terms of the Inflaton energy density

$V(\phi)=\rho_{\phi}\left(1-\frac{m_{p}^{2}}{24\pi}\left(\frac{F_{{}_{\prime}H}H^{\prime}}{H(1+Q)}\right)^{2}\right)$
(33)

The number of e-folds is a very important quantity to measure the amount of
Inflation. The number of e-folds is [[8] ],

$N=\int_{\phi_{o}}^{\phi_{e}}\frac{H}{\dot{\phi}}d\phi$ (34)

where $\phi_{e}$ and $\phi$ are the field values at the end and beginning of
Inflation, respectively. Once again using (24), we write N as

$N=\int_{\phi_{e}}^{\phi}\frac{8\pi}{m_{p}^{2}}\frac{H^{2}(1+Q)}{F_{{}_{\prime}H}H^{\prime}}d\phi$
(35)

Now, by the definition , we have

$dN=\frac{da}{a}$ (36)

which gives us

$a(\phi)=a(\phi_{o})\exp\left(-\int_{\phi_{e}}^{\phi}\frac{8\pi}{m_{p}^{2}}\frac{H^{2}(1+Q)}{F_{{}_{\prime}H}H^{\prime}}d\phi\right)$
(37)

Another important quantity for Inflationary Models are the slow roll
Parameters. From the very basic definition of the $\epsilon$ slow roll
parameter , can write

$\frac{\dddot{a}}{a}=H^{2}+\dot{H}=H^{2}\left(1-\epsilon\right)$ (38)

where $\epsilon$ is the ”first” Hubble Slow Roll Parameter given by

$\epsilon=\frac{m_{p^{2}}}{8\pi}\frac{F_{{}_{\prime}H}}{H(1+Q)}\left(\frac{H^{\prime}}{H}\right)^{2}$
(39)

This can be arrived at directly by first realizing that

$\dot{H}=H^{\prime}\dot{\phi}=-\frac{m_{p}^{2}}{8\pi}\frac{F_{{}_{\prime}H}H^{\prime}}{H(1+Q)}$
(40)

This implies

$-\frac{\dot{H}}{H^{2}}=\frac{m_{p^{2}}}{8\pi}\frac{F_{{}_{\prime}H}}{H(1+Q)}\left(\frac{H^{\prime}}{H}\right)^{2}$
(41)

$\epsilon$ is also realized using the concurrent definition [33]

$\epsilon=-\frac{d\ln H}{d\ln
a}=\frac{m_{p^{2}}}{8\pi}\frac{F_{{}_{\prime}H}}{H(1+Q)}\left(\frac{H^{\prime}}{H}\right)^{2}$
(42)

In fact, the second Hubble Slow Roll Parameter, $\eta$ is similarly defined to
be

$\eta=-\frac{d\ln H^{\prime}}{d\ln
a}=\frac{m_{p^{2}}}{8\pi}\frac{F_{{}_{\prime}H}}{H(1+Q)}\frac{H^{\prime\prime}}{H}$
(43)

We now take a brief detour and discuss a bit of subtlety surrounding the
definition of slow roll parameters for Warm Inflation. While the $\epsilon$
and $\eta$ slow roll parameters above is derived as it comes from the basic
definition (40) and (41) , there still exists conflicting views in literature
about the appropriate definition for the parameter in the context of Warm
Inflation. While some authors [[34] , [35]] like to define the $\epsilon$ and
$\eta$ parameters in their usual supercooled Inflation forms , others [[28] ,
[31] ] like to ” absorb” the dependence of the damping function Q as it comes
into the slow roll parameters from the basic definitions (40) and (41) . The
proponents of the former approach feel that defining the first and second slow
roll parameters in the usual cold Inflation form allows them to have a more
relaxed constraint in Warm Inflationary Scenarios [34] . The author of this
paper is also in harmony with this line of thought, but however feels the
latter appraoch of defining the first two hubble parameters in Warm Inflation
allows for them to be evidently more general then their supercooled Inflation
form. This allows the reader to more smoothly see the transition of the
parameters from Warm to Cold Inflation in the extremely low dissipation
regime. But we are certainly of the opinion that while both approaches of
defining the parameters may appear different on the level of substance, they
carry virtually the same essence overall. So now we just note for the sake of
completeness that we can equivalently define the $\epsilon$ and $\eta$
parameters suitable for the approach [[34] , [35]] as (which we will call
$\epsilon_{a}$ and $\eta_{a}$)

$\epsilon_{a}=\frac{m_{p^{2}}}{8\pi}\frac{F_{{}_{\prime}H}}{H}\left(\frac{H^{\prime}}{H}\right)^{2}$
(44)
$\eta_{a}=\frac{m_{p^{2}}}{8\pi}\frac{F_{{}_{\prime}H}}{H}\frac{H^{\prime\prime}}{H}$
(45)

which is just the form of the parameters as shown in [29] . The second form of
the parameters , which is in line with the approach of [[28] , [31] ] , are
just the $\eta$ and $\epsilon$ parameters defined in (40) and (41). We readily
see that (42) and (43) are just (40) and (41) respectively, in the $Q<<1$
approximation.

Alongside the usual slow roll parameters $\eta$ and $\epsilon$ , [35] showed
that for appropriately studying Warm Inflationary Paradigms some other
parameters with the derivatives of $\Gamma$ would be very useful. These
parameters were derived by the using the slow roll conditions for Warm
Inflation. Hence we would now like to define some new slow roll parameters for
our model, in order to better cater to the needs of Warm Inflation scenarios.
One of the primary slow roll conditions reads

$\frac{-\ddot{H}}{\dot{H}H}<<1$ (46)

We would now evaluate the quantity $\frac{-\ddot{H}}{\dot{H}H}$ to arrive at
our new Slow Roll parameters taking lead from [31] . We begin by noting that

$\ddot{H}=\frac{d}{dt}\left(H^{\prime}\dot{\phi}\right)$ (47)

which can be realized using (38) . Now this allows us to write

$\ddot{H}=\dot{\phi}^{2}H^{\prime\prime}+\ddot{\phi}H^{\prime}$ (48)

To evaluate (46) , we need to first have an expression for $\ddot{\phi}$
appropriate for our use. This can be done by using (24) supplemented by the
fact that $\dot{F_{{}_{\prime}H}}=\dot{H}F_{{}_{\prime}HH}$ where
$\left(F_{{}_{\prime}HH}=\frac{\partial^{2}F}{{\partial H}^{2}}\right)$ . A
little bit of algebra leads us to

$\ddot{\phi}=\frac{-3m_{p}^{2}}{8\pi}\left(\frac{\left({H^{\prime}}^{2}F_{{}_{\prime}HH}+F_{{}_{\prime}H}H^{\prime\prime}\right)(3H+\Gamma)-(3H^{\prime}+\Gamma^{\prime})F_{{}_{\prime}H}H^{\prime}}{\left(3H+\Gamma\right)^{2}}\right)$
(49)

This allows us to write

$\frac{-\ddot{H}}{\dot{H}H}=-\eta-\frac{m_{p}^{2}}{8\pi
H^{2}}\left(\frac{\left({H^{\prime}}^{2}F_{{}_{\prime}HH}+F_{{}_{\prime}H}H^{\prime\prime}\right)(1+Q)-\frac{\left(3H^{\prime}+\Gamma^{\prime}\right)F_{{}_{\prime}H}H^{\prime}}{3H}}{\left(1+Q\right)^{2}}\right)$
(50)

And further we write

$\frac{-\ddot{H}}{\dot{H}H}=2\eta+\frac{m_{p}^{2}{H^{\prime}}^{2}}{8\pi
H^{2}}\frac{F_{{}_{\prime}HH}}{1+Q}-\frac{Q}{1+Q}\left(\frac{m_{p}^{2}}{8\pi}\frac{F_{{}_{\prime}H}}{H}\frac{\Gamma^{\prime}H^{\prime}}{\Gamma
H}+\frac{3m_{p}^{2}F_{{}_{\prime}H}}{8\pi H}\frac{H^{\prime}}{\Gamma
H(1+Q)}\right)$ (51)

this leads us to define $\beta$ parameter [[35] , [31] ] as ,

$\beta=\frac{m_{p}^{2}}{8\pi}\frac{F_{{}_{\prime}H}}{H}\frac{\Gamma^{\prime}H^{\prime}}{\Gamma
H}\frac{1}{1+Q}$ (52)

In addition to these slow roll parameters, we define other parameters which
will be helpful for our perturbation spectra studies. These are ,

$\chi=\frac{m_{p}^{2}F_{{}_{\prime}H}}{8\pi H}\frac{H^{\prime}}{\Gamma
H(1+Q)}$ (53)

and

$\gamma=\frac{m_{p}^{2}{H^{\prime}}^{2}}{8\pi
H^{2}}\frac{F_{{}_{\prime}HH}}{1+Q}$ (54)

Which now allows us to express (48) as

$\frac{-\ddot{H}}{\dot{H}H}=2\eta+\chi-\frac{Q}{1+Q}\left(\beta+3\gamma\right)$
(55)

In addition to these we define more parameters which will be helfpul for us in
our Perturbation spectra analysis

$\delta=\frac{m_{p}^{2}}{8\pi}\left(\frac{F_{{}_{\prime}H}}{(1+Q)H}\right)^{2}\frac{\Gamma^{\prime\prime}{H^{\prime}}^{2}}{\Gamma
H^{2}}$ (56)
$\sigma=\frac{m_{p}^{2}}{8\pi}\frac{1}{1+Q}\frac{F_{\prime}H}{H}\frac{H^{\prime\prime\prime}}{H^{\prime}}$
(57)
$\psi=\frac{m_{p}^{2}}{8\pi}\frac{F_{{}_{\prime}HHH}}{(1+Q)}\frac{{H^{\prime}}^{2}}{H}$
(58)

This completes the proper dynamical outline of the Exact Solution approach for
Warm Inflation. The following flowchart depicts how our work here reduces to
other related methods in appropriate units. [ [29] , [31] , [30] ].

Exact solution approach for Warm InflationFor $F(H)=H^{2}$ , the method of
[31] For $Q=0$ , the method of [29] For $F(H)=H^{2}$ and $Q=0$ , the method of
[30] Figure 1: Flowchart depicting how our approach relates with recent
studies

It is quick to check that this approach reduces to the usual Cold Inflation
approach for exact solutions [29] in the extremely low dissipation regime
$Q<<1$ . It also further reduces to the Hamilton Jacobi method for Warm
Inflation for $F(H)=H^{2}$ [31], and further to the usual cold Inflation
Hamilton Jacobi approach [30] in the extremely low dissipation regime for the
same $F(H)$.

With the dynamical aspects covered, we shall now explore the perturbation
spectra of Warm Inflation in this approach.

##  3  Perturbation Spectra Analysis

Cosmological density and gravitational wave perturbations in the inflationary
scenario arise as quantum fluctuations which redshift to long wavelengths due
to rapid cosmological expansion during Inflation [[8] , [36] , [37] , [38] ] .
In Warm Inflation, only the density perturbations couple strongly with the
thermal background and hence, the scalar density spectra is the one which
looks more evidently different from it’s usual supercooled Inflationary
counterpart [39] . Tensor perturbations do not couple strongly to the thermal
background and so gravitational waves are only generated by quantum
fluctuations, as in standard cold Inflation. In addition to the usual
Adiabatic Perturbations in cold Inflation, Isocurvature perturbations also are
generated in the warm inflationary era due to thermal fluctuations in the
radiation field. These perturbations can be characterised by fluctuations in
the entropy of the particle species undergoing thermal fluctuations relative
to the number density of photons. But in this paper, we will limit our focus
to the study of only adiabatic perturbations.

The square of the amplitude of adiabatic perturbations is calculated in a
similar way to Cold Inflation [[39] , [13] , [31]] ,

$P_{s}(k)=\frac{4}{25}\left(\frac{H}{\dot{|\phi|}}\right)^{2}{d\phi}^{2}$ (59)

where ${d\phi}^{2}$ for the high dissipative regime ($Q>>1$ )is given by [40]
,

${d\phi}^{2}=\frac{k_{F}T}{2\pi}$ (60)

where $k_{F}$ is the so called Freeze out number given by,

$k_{F}=\sqrt{\Gamma H}$ (61)

The definition of the freeze out number $k_{F}$ is not changed by considering
a general Friedmann Equation of the form (19). It is so because the definition
of the freeze out number stems primarily from the Field Equation of $\phi$
(12) , in particular from the evolution equation of the fluctuations
$\delta\phi(x,t)$ ( where $\phi$ (x,t) = $\phi_{o}(x,t)+\delta\phi(x,t)$ )
[[39] , [40] ].This remains unchanged by the consideration of (19) , takes the
usual form with the inclusion of the spatial Laplcian and an additional white
noise random force term ,

$\Gamma\frac{d\delta\phi(k,t)}{dt}=-[\left(k^{2}+V^{\prime\prime}(\phi_{o})\right)]\delta\phi(k,t)+\zeta(k,t)$
(62)

where we have Fourier transformed to the Momentum Space and $\zeta(k,t)$ is
the white noise term. In a similar way to [39] , we reach at
$k_{F}=\sqrt{\Gamma H}$. Now using the expression for T (28), and the
definition of $\epsilon$ (40) , we can write $P_{s}(k)$ as

${P_{s}(k)}^{2}=\frac{2}{25\pi}\sqrt{\Gamma
H}\left(\frac{3Q}{4\sigma}\right)^{\frac{1}{4}}\left(\frac{\epsilon
H^{2}}{H^{\prime}}\right)^{\frac{1}{2}}\left(\frac{H^{\prime}}{\epsilon
H}\right)^{2}$ (63)

which leads us to write

$P_{s}(k)=\left(\left(\frac{48}{1562500\sigma\pi^{4}}\right)^{\frac{1}{2}}\Gamma^{\frac{3}{2}}H^{-\frac{3}{2}}\epsilon^{-3}{H^{\prime}}^{3}\right)^{\frac{1}{2}}$
(64)

The scalar spectral index is defined by the well known equation

$n_{s}-1=\frac{d\ln{P_{s}(k)}^{2}}{d\ln k}$ (65)

where $d\ln k$ is given as the negative of the differential of the number of
e-folds

$d\ln
k=-dN=\frac{8\pi}{3m_{p}^{2}}\frac{H\Gamma}{F_{{}_{\prime}H}H^{\prime}}d\phi$
(66)

For proceeding further, we note that the derivative of the $\epsilon$
parameter is given by

$\epsilon^{\prime}=\frac{d\epsilon}{d\phi}=\frac{H^{\prime}}{H}\left(\gamma+2\eta-2\epsilon-\beta\right)$
(67)

This allows us to write

$\frac{d\epsilon}{d\ln k}=\epsilon(\gamma+2\eta-2\epsilon-\beta)$ (68)
$\frac{d\ln{P_{s}(k)}^{2}}{d\ln
k}=\frac{3\beta}{4}-\frac{3\epsilon}{4}+\frac{3\eta}{2}-\frac{3}{2}\frac{\epsilon^{\prime}}{\epsilon}\frac{3m_{p}^{2}}{8\pi}\frac{F_{{}_{\prime}H}H^{\prime}}{H(\Gamma)}$
(69) $\frac{d\ln{P_{s}(k)}^{2}}{d\ln
k}=\frac{9}{4}\beta+\frac{9}{4}\epsilon-\frac{3}{2}\gamma-\frac{3}{2}\eta$
(70)

This finally allows to us to express the scalar spectral index as

$n_{s}=1+\frac{9}{4}\beta+\frac{9}{4}\epsilon-\frac{3}{2}\gamma-\frac{3}{2}\eta$
(71)

This is the scalar spectral index for the high dissipative regime.

For the low dissipative regime ( $Q<<1$ ), the Amplitude of the scalar
perturbation spectra is still expressed by (54) , but $d\phi$ is given by

$d\phi=HT$ (72)

This allows us to write the power spectrum for the low dissipative regime as

${P_{s}^{\ast}(k)}^{2}=\frac{4}{25}\frac{{H^{\prime}}^{2}}{\epsilon^{2}}T^{2}$
(73)

Using the expression for the temperature derived above (28) , we have

${P_{s}^{\ast}(k)}^{2}=\left(\frac{4}{625\alpha}\left(H^{\prime}\right)^{2}H^{3}\Gamma\right)^{\frac{1}{2}}$
(74)

Again , the definition of the scalar spectral index in this case is the same
as for the high dissipation scenario

$n_{s}^{\ast}-1=\frac{d\ln{P_{s}^{\ast}(k)}^{2}}{d\ln k}$ (75)

Pursuing a similar analysis as for the previous case, we arrive at

$n_{s}^{\ast}=1+\frac{3}{2}\epsilon+\eta+\frac{\beta}{2}$ (76)

One of the more exciting findings of the observational data from the Planck,
WMAP and COBE experiments is that there is a significant running of the scalar
spectral index as well. Traditionally it was taken to be negligible but these
experimental findings make them a very important observational quantity.
Alongside the Running of the scalar spectral index and the index itself, the
tensor to scalar ratio is another important quantity of observational
relevance. We will now focus more on the high dissipative regime and calculate
the running of the scalar spectral index and the Tensor-to-Scalar Ratio in
that limit. We will not calculate the same in the low dissipative regime but
one can calculate them in that limit by pursuing a similar method as we do in
the following for the High Dissipative Regime.

The running of the Spectral Index is defined by it’s usual definition

$\alpha_{s}=\frac{dn_{s}}{d\ln k}$ (77)

In a similar way as we calculated $\frac{d\epsilon}{d\ln k}$ in (66), we
arrive at the following differentials

$\frac{d\eta}{d\ln k}=\eta\gamma+\sigma\epsilon-\eta\beta-\eta\epsilon$ (78)
$\frac{d\beta}{d\ln k}=\beta\gamma+\delta+\beta\eta-\beta\epsilon-2\beta^{2}$
(79) $\frac{d\gamma}{d\ln
k}=\psi\epsilon+\gamma\eta-\gamma\beta-\epsilon\gamma$ (80)

The above expressions allow us to write the running of scalar spectral index
using (69) as ,

$\alpha_{s}=\frac{15}{4}(\beta\gamma+\eta\beta+\gamma\epsilon+\eta\epsilon)-\frac{9}{2}(\epsilon\beta+\epsilon^{2}+\beta^{2}+\eta\gamma)-\frac{3}{2}(\psi\epsilon)+\frac{9}{4}\delta$
(81)

Further, the squared Tensor Perturbation power spectrum amplitude is defined
as [41]

${P_{T}(k)}^{2}=\frac{32}{75m_{p}^{4}}V(\phi)$ (82)

During Inflation, $V\approx\frac{3m_{p}^{2}F}{8\pi}$ which is clear by (21).
The definition of the tensor-to-scalar ratio is

$r=\frac{{P_{T}(k)}^{2}}{{P_{s}(k)}^{2}}$ (83)

Equation (80) now allows us to write the tensor-to-scalar ratio as

$r=\frac{2F}{m_{p}^{2}}\left(\frac{4\alpha}{3}\right)^{\frac{1}{4}}\left(\Gamma^{\frac{-3}{2}}{H^{\prime}}^{\frac{-3}{2}}\epsilon^{3}H^{3}\right)^{\frac{1}{2}}$
(84)

Again we remark that the above formulations of $r$ and $\alpha_{s}$ are
specifically for the high dissipative regime $Q>>1$ . One can easily formulate
the same for the low dissipative regime $Q<<1$ using the same procedure we
have shown above.
Now, we are completed with all the theoretical basis of our approach. We will
now apply it on a Tsallis Entropy Modified Friedmann Universe. We will study
aspects of Warm Inflation in this model in the high dissipative regime.

##  4 Warm Inflation in Tsallis Entropy Modified Friedmann Universe

A Tsallis Entropy Modified Friedmann Universe has the Friedmann Equation of
the form (6). This allows us to write $F(H)$ for this model as

$F(H)=H^{2(2-\kappa)}$ (85)

Moving forward we would need to ascertain two more quantities for studying
Warm Inflation in this Model, which are the Hubble Parameter $H$ and the
Dissipation Coefficient $\Gamma$ . Both of them will be taken as functions for
$\phi$ . While this statement can be understood in a straightforward way for
$H$ , for $\Gamma$ the answer could have been different. Usually, $\Gamma$ is
taken as a function of only $\phi$ or of both $\phi$ and the radiation
temperature $T$ [[31, 42, 43, 44] ] . This is because temperature plays a
crucial role in the dissipative scenario of Warm Inflation. However,
Temperature in general is written in terms of $\phi$ eventually and the
dissipation function in cases with $\Gamma=\Gamma(\phi,T)$ turns into a
function of only $\phi$ [[45, 32, 46, 47]] . So in our case we will treat both
H and $\Gamma$ as $H=H(\phi)$ and $\Gamma=\Gamma(\phi)$ . Further, we consider
both H and $\Gamma$ to be power law functions of $\phi$ , as

$H(\phi)=H_{o}\phi^{n}$ (86) $\Gamma(\phi)=\Gamma_{o}\phi^{m}$ (87)

where $H_{o}$ , $\Gamma_{o}$ are some constants. The powers n and m are left
undetermined here as we will use the Planck data to find out which power laws
best fit with our concerned model. The reason for choosing power law form for
the Hubbele Parameter is because they seem to be a good fit with the latest
Planck Data [48] , while the consideration of a supersymmetric interaction of
the authors [42, 44] lead to the dissipation coefficeint being a linear
function of $\phi$. While other authors from several distinct considerations
have been led to power law forms of the dissipation Coefficient [45, 47, 31].
This suggests to us that power law forms can indeed be very viable and general
forms of the dissipation coefficients. Hence, we take the coefficient to be in
a power law form of the field variable.

With all the preliminaries cleared up, we now move towards concrete
calculations. As stated previously, we will focus on Warm Inflation in the
High Dissipative regime for this case. To make progress, we would need the
form of the Field $\phi$ at the time of horizon exit and at the end of
Inflation. It is straightforward to get the latter by setting $\epsilon=1$ in
(40),

$1=\frac{3m_{p}^{2}2(2-\kappa)n^{2}H_{o}^{3-2\kappa}}{8\pi\Gamma_{o}}\phi_{e}^{(3-2\kappa)n-m-2}$
(88)
$\frac{3m_{p}^{2}2(2-\kappa)n^{2}H_{o}^{3-2\kappa}}{8\pi\Gamma_{o}}=\phi_{e}^{m+2-n(3-2\kappa)}$
(89)

In order to get the Field at the time of horizon exit, we take the help of the
Number of e-folds. For our model, it is given by

$N=\int_{\phi_{e}}^{\phi}\frac{8\pi}{3m_{p}^{2}}\frac{H_{o}\Gamma_{o}\phi^{n}\phi^{m}}{2(2-\kappa)H_{o}^{3-2\kappa}\phi^{n(3-2\kappa)nH_{o}\phi^{n-1}}}$
(90)

This leads to,

${3m_{p}^{2}2(2-\kappa)nH_{o}^{3-2\kappa}}{8\pi\Gamma_{o}}N=\frac{1}{m+2-n(3-2\kappa)}\left(\phi^{m+2-n(3-2\kappa)}-\phi_{e}^{m+2-n(3-2\kappa)N}\right)$
(91)

This finally allows us to write the field at the time of horizon exit as

$\phi^{m+2-n(3-2\kappa)}=\phi_{e}^{m+2-n(3-2\kappa)}\left(1+\frac{(m+2-n(3-2\kappa))N}{n}\right)$
(92)

Now, using (24) we can write

$\frac{d\phi}{dt}=-\frac{3m_{p}^{2}2(2-\kappa)H_{o}^{3-2\kappa}\phi^{n(3-2\kappa)}nH_{o}\phi^{n-1}}{8\pi\Gamma_{o}\phi^{m}}$
(93)

Integrating from $t_{o}$ to some t , we arrive at

$\phi^{m+2-n(4-2\kappa)}(t)=\frac{(4-2\kappa)n-m-2}{n}\phi_{e}^{m+2-n(3-2\kappa)}H_{o}(t-t_{o})+\\\
\left(\phi_{e}^{m+2-n(3-2\kappa)}\left(1+\frac{(m+2-n(3-2\kappa))N}{n}\right)\right)^{\frac{(m+2-n(4-2\kappa))}{m+2-n(3-2\kappa)}}$
(94)

This immediately allows us to write the Hubble Parameter and the dissipation
coefficient as functions of time

$H(t)=H_{o}\bigg{[}\frac{(4-2\kappa)n-m-2}{n}\phi_{e}^{m+2-n(3-2\kappa)}H_{o}(t-t_{o})+\\\
\left(\phi_{e}^{m+2-n(3-2\kappa)}\left(1+\frac{(m+2-n(3-2\kappa))N}{n}\right)\right)^{\frac{m+2-n(4-2\kappa)}{m+2-n(3-2\kappa)}}\bigg{]}^{\frac{n}{(m+2-n(4-2\kappa))}}$
(95)
$\Gamma(t)=\Gamma_{o}\bigg{[}\frac{(4-2\kappa)n-m-2}{n}\phi_{e}^{m+2-n(3-2\kappa)}H_{o}(t-t_{o})+\\\
\left(\phi_{e}^{m+2-n(3-2\kappa)}\left(1+\frac{(m+2-n(3-2\kappa))N}{n}\right)\right)^{\frac{m+2-n(4-2\kappa)}{m+2-n(3-2\kappa)}}\bigg{]}^{\frac{m}{(m+2-n(4-2\kappa))}}$
(96)

The Inflationary Potential corresponding to this Cosmology is given by (30)

$V(\phi)=\frac{3m_{p}^{2}H_{o}^{4-2\kappa}\phi^{4-2\kappa}}{8\pi}-\frac{1}{2}\left(\frac{3m_{p}^{2}2(2-\kappa)H_{o}^{4-2\kappa}n\phi^{(3-2\kappa)n-m-1}}{8\pi\Gamma_{o}}\right)^{2}$
(97)

By the definition of the Hubble Parameter we have ,

$Hdt=\frac{da}{a}$ (98)

Which gives us the scale factor as

$a(t)=a_{o}\exp(\int_{t_{o}}^{t}Hdt)$ (99)

The required time integral of the Hubble Parameter can be done quite easily to
get ,

$\int_{t_{o}}^{t}Hdt=\frac{nH_{o}\phi_{e}^{m+2-n(3-2\kappa)}}{(m+2-(3-2\kappa)n)N}\Bigg{[}\phi_{e}^{m+2-n(3-2\kappa)}\left(1+\frac{(m+2-n(3-2\kappa))N}{n}\right)-\\\
\bigg{[}\frac{(4-2\kappa)n-m-2}{n}\phi_{e}^{m+2-n(3-2\kappa)}H_{o}(t-t_{o})+\\\
\left(\phi_{e}^{m+2-n(3-2\kappa)}\left(1+\frac{(m+2-n(3-2\kappa))N}{n}\right)\right)^{\frac{m+2-n(4-2\kappa)}{m+2-n(3-2\kappa)}}\bigg{]}^{\frac{m+2-n(3-2\kappa)}{m+2-n(3-2\kappa)}}\Bigg{]}$
(100)

This gives us the scale factor as a function of time

$a(t)=a_{o}\exp\Bigg{(}\frac{nH_{o}\phi_{e}^{m+2-n(3-2\kappa)}}{(m+2-(3-2\kappa)n)N}\Bigg{[}\phi_{e}^{m+2-n(3-2\kappa)}\left(1+\frac{m+2-n(3-2\kappa)}{n}\right)-\\\
\bigg{[}\frac{(4-2\kappa)n-m-2}{n}\phi_{e}^{m+2-n(3-2\kappa)}H_{o}(t-t_{o})+\\\
\left(\phi_{e}^{m+2-n(3-2\kappa)}\left(1+\frac{(m+2-n(3-2\kappa))N}{n}\right)\right)^{\frac{m+2-n(4-2\kappa)}{m+2-n(3-2\kappa)}}\bigg{]}^{\frac{m+2-n(3-2\kappa)}{m+2-n(3-2\kappa)}}\Bigg{]}\Bigg{)}$
(101)

We can also get a relationship between the radiation energy density and the
inflaton energy density, as a function of $\phi$ and consequently of time.
Using (27) in the high dissipation regime we have,

$\rho_{r}(\phi)=\frac{3m_{p}^{2}\rho_{\phi}}{32\pi
F}\bigg{[}\frac{(F_{{}_{\prime}H}H^{\prime})^{2}}{\Gamma H}\bigg{]}$ (102)

Doing a little bit of algebra on this formula, we arrive at

$\rho_{r}(\phi)=\frac{m_{p}^{2}(2-\kappa)\phi_{e}^{m+2-n(3-2\kappa)}\rho_{\phi}}{16\pi\phi^{m+2-n(3-2\kappa)}}$
(103)

Using(94) , we can further write the above expression in terms of time ,

$\rho_{r}(t)=\frac{m_{p}^{2}(2-\kappa)\phi_{e}^{m+2-n(3-2\kappa)}\rho_{\phi}}{16\pi}\bigg{[}\frac{(4-2\kappa)n-m-2}{n}\phi_{e}^{m+2-n(3-2\kappa)}H_{o}(t-t_{o})+\\\
\left(\phi_{e}^{m+2-n(3-2\kappa)}\left(1+\frac{(m+2-n(3-2\kappa))N}{n}\right)\right)^{\frac{m+2-n(4-2\kappa)}{m+2-n(3-2\kappa)}}\bigg{]}^{\frac{n(3-2\kappa)-m-2}{m+2-n(4-2\kappa)}}$
(104)

We note further that at the time of horizon exit, (103) becomes

$\rho_{r}=\frac{m_{p}^{2}(2-\kappa)\rho_{\phi}}{16\pi}\left(\frac{n}{n+(m+2-n(3-2\kappa))N}\right)$
(105)

The above expression tells us that in the high dissipation regime at the time
of horizon exit, only the free parameters $\kappa$ , $m$ and $n$ and the
e-fold number N determines the relationship between $\rho_{\phi}$ and
$\rho_{r}$.

Now in order to fully get the details of Warm Inflation in the high
dissipation regime in this cosmology, we would like to have appropriate values
of the free parameters $\kappa$ , $n$ and $m$ which fit with the observational
data [[48] , [49]]. We would like to have expressions for important
observational quantities like the scalar spectral index and the running of the
scalar spectral index at the time of horizon exit. This can be done by
evaluating all the relevant slow roll and other cosmological parameters
defined previously at the time of horizon exit . Using (92) , we can evaluate
the slow roll parameters at the time of horizon exit. The $\epsilon$ slow roll
parameter in particular is

$\epsilon=\frac{n}{n+(m+2-n(3-2\kappa))N}$ (106)

We have emphasized about the $\epsilon$ parameter here because it is possible
to express all the other parameters which we have mentioned before in terms of
this parameter. Evaluating all the parameters at horizon exit, we get

$\beta=\frac{m}{n}\epsilon$ (107) $\eta=\frac{n-1}{n}\epsilon$ (108)
$\gamma=(3-2\kappa)\epsilon$ (109) $\sigma=\frac{(n-1)(n-2)}{n}\epsilon$ (110)
$\delta=\frac{(m)(m-1)}{n^{2}}\epsilon^{2}$ (111)
$\psi=(3-2\kappa)(2-2\kappa)\epsilon$ (112)

Using these definitions and (72), we arrive at the following expressions for
the scalar spectral index,

$n_{s}=1+\bigg{[}\frac{9m}{4n}+\frac{9}{4}+3\kappa-\frac{9}{2}+\frac{3}{2}\left(\frac{n-1}{n}\right)\bigg{]}\frac{n}{(m+2-n(2-2\kappa))N}$
(113)

And the running of the scalar spectral index is given by (82),

$\alpha_{s}=\Bigg{[}\frac{15}{4}\bigg{(}\frac{m}{n}\left(3-2\kappa\right)+\frac{(n-1)m}{n^{2}}+3-2\kappa+\frac{n-1}{n}\bigg{)}+\frac{9}{4}\frac{m(m-1)}{n^{2}}-\\\
\frac{9}{2}\bigg{(}\frac{m}{n}+1+\left(\frac{m}{n}\right)^{2}+\frac{n-1}{n}(3-2\kappa)\bigg{)}-3(3-2\kappa)(1-\kappa)\Bigg{]}\bigg{[}\frac{n}{n+(m+2-n(3-2\kappa))N}\bigg{]}^{2}$
(114)

With that, we are completed with the amount of material and detail we could
have reached building up from our theoretical groundwork. To get more insight
into the paradigm of Warm Inflation in this model, we will have to take note
of the Latest Observational Data Available from the Planck Satellite
Experiment [49] and see which values of the free parameters in this model most
suitably fit the data available on the spectral index and it’s running.
Another important quantity in this excursion of ours is the e-fold Number. For
Inflation to solve cosmological problems and contribute in large scale
structure formation, the e-fold number can conveniently between around 60 [8]
, so we will henceforth set $N=60$.

Using (72) and (82) , we find that a suitable choice of the free parameters
which satisfies the inflationary Requirements is $(m,n,\kappa)=(3,-5,1.4)$ .
For these values ,

$n_{s}\approx 0.964912$ (115) $\alpha_{s}\approx-0.003$ (116)

Which is in perfect agreement with the Planck 2018 data [48] of
$n_{s}=0.9649\pm 0.0042$ ( $68\%CL$ , Planck $TT,TE,EE+lowE+lensing$ ) and the
negligible running of the spectral index. Putting these values in the required
equations derived above would give one full details of Warm Inflation in this
cosmology.One can also further use the constraints on the tensor-to-scalar
ratio provided by the data to bound the constants $H_{o}$ and $\Gamma_{o}$ in
a similar way as done in [31] , but we do not pursue that here.

##  5 Conclusions

In this paper, we have introduced the exact solution approach for Warm
Inflation. We started off by showing how many modified Cosmological Scenarios
like Braneworld Cosmologies , Modified Gravity Cosmologies and various
Modified Entropy Cosmologies have a similar form of the Friedmann Equation
which can be used to consider a generalized Friedmann Equation with a General
Function of the Hubble Parameter . We then began the description of our method
with a light review of the basics dynamics of Warm Inflation. After that, we
described our approach and showed how various important quantities for Warm
Inflationary regimes like the Hubble Parameter, the Dissipation Coefficient,
the e-fold Number, the Inflaton Field Function etc. can be derived using this
approach. We further explored Scalar and Tensorial Inflationary Perturbations
in this method and derived important Inflationary Parameters like the Scalar
Spectral Index, it’s running, the tensor-to-scalar ratio etc. Finally, we
applied this method to study High Dissipation Warm Inflation in a Tsallis
Modified Entropy Friedmann Universe. We here point to one peculiarity of our
model which we have not yet touched upon. In obtaining the equation of the
inflaton field we have assumed that the matter, specified by the inflaton
scalar field, enters into the action Lagrangian in such a way that its
variation in a Friedmann-Robertson-Walker-Lemaitre background metric leads to
the Klein-Gordon equation, expressed by (12). Therefore our method is only
applicable to theories where the background metric alongside the
perturbations, are not modified.This means that Horava-Lifshitz theories of
gravity [50] or theories of similar plight are beyond the scope of our
approach.

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