# Bridging Logic and Learning: A Neural-Symbolic Approach for Enhanced
Reasoning in Neural Models (ASPER)

Fadi Al Machot,

Faculty of Science and Technology (REALTEK)
Norwegian University of Life Sciences
NMBU
1430 Ås, Norway
Corresponding author: Fadi Al Machot (e-mail<EMAIL_ADDRESS>
###### Abstract

Neural-symbolic learning, an intersection of neural networks and symbolic
reasoning, aims to blend neural networks’ learning capabilities with symbolic
AI’s interpretability and reasoning. This paper introduces an approach
designed to improve the performance of neural models in learning reasoning
tasks. It achieves this by integrating Answer Set Programming (ASP) solvers
and domain-specific expertise, which is an approach that diverges from
traditional complex neural-symbolic models. In this paper, a shallow
artificial neural network (ANN) is specifically trained to solve Sudoku
puzzles with minimal training data. The model has a unique loss function that
integrates losses calculated using the ASP solver outputs, effectively
enhancing its training efficiency. Most notably, the model shows a significant
improvement in solving Sudoku puzzles using only 12 puzzles for training and
testing without hyperparameter tuning. This advancement indicates that the
model’s enhanced reasoning capabilities have practical applications, extending
well beyond Sudoku puzzles to potentially include a variety of other domains.
The code can be found on GitHub: https://github.com/Fadi2200/ASPEN.

##  1 Introduction

Neural-symbolic learning aims to combine the adaptive learning abilities of
neural networks with symbolic AI’s clarity and logical capabilities. Thus,
research efforts have focused on creating hybrid models that combine Neural
Networks with Symbolic Learning, e.g., language translation [2], neural logic
machine [9], and logic synthesis from Input-Output examples [5].

This integration represents a step towards addressing some limitations in
training neural models, such as the need for large annotated datasets,
difficulty handling complex reasoning tasks, and the lack of interpretability
in model decisions [18]. While neural-symbolic AI is promising, it faces
significant challenges in lacking a generalizable framework that can be easily
adapted across different problem spaces. This limitation restricts the broader
application of these models and requires extensive customization for new
domains [3]. Addressing these limitations, this work introduces an approach
called ASPER that effectively integrates answer set programming (ASP) solvers
and domain-specific expertise into neural models.

This paper diverges from traditional neural-symbolic models by focusing on
straightforward, effective methods incorporating logical reasoning and expert
knowledge. The strength of the approach lies in its simplicity and
adaptability. The proposed approach does not require extensive datasets.
Instead, it leverages the logic in the problem domain to guide the learning
process. This aspect is crucial in fields where training data is limited or
expensive to collect. Additionally, the approach enhances the interpretability
of the model’s decisions. By incorporating rule-based logic into the training
process, the model’s decisions can be traced back to these underlying rules,
providing clearer explanations. This transparency is an advancement over
traditional ’black box’ models. Using a custom loss function designed to
integrate these logical constraints further represents the unique capabilities
of the ASPER. As a result, it ensures that the model not only learns from the
available data but also adheres to the logical structure of the problem,
resulting in more accurate and reliable outcomes.

##  2 Related works

Recent developments in neural-symbolic AI have revolutionized the integration
of low-level data with high-level reasoning capabilities. Central to this
advancement are approaches that combine data-driven neural modules with logic-
based symbolic systems [4, 10, 15, 8]. These combined systems have shown
remarkable proficiency in handling complex tasks such as visual question
answering (VQA), concept learning, and the development of explainable and
revisable models [25, 17, 7, 22].

Deep Probabilistic Programming Languages (DPPLs) [16] is a probabilistic logic
programming language that seamlessly integrates deep learning through neural
predicates. This pioneering framework uniquely combines general-purpose neural
networks with sophisticated probabilistic-logical reasoning, capable of both
symbolic and subsymbolic inference. Furthermore, NeurASP [24] is an innovative
extension of answer set programs that integrates neural networks for enhanced
computation. It treats neural network outputs as probability distributions for
atomic facts in answer set programs, offering a straightforward method to
merge sub-symbolic and symbolic processes. Moreover, Scallop [14] is a system
that combines deep learning with symbolic reasoning in a scalable way,
addressing limitations in DPPLs.

The previous approaches have introduced innovative ways to integrate neural
and symbolic modules, offering a scalable approach to AI systems. These
systems leverage neural predicates to enhance probabilistic reasoning, making
deep learning systems more flexible and efficient. Despite these advancements,
scalability remains a significant challenge in DPPLs. The primary issue lies
in the increasing computational complexity and demand for substantial memory
and processing power. Moreover, scaling up risks overfitting. Additionally, as
these systems grow in complexity, maintaining user-friendly interfaces that
allow efficient model building, training, and deployment without requiring
deep expertise in neural and symbolic AI domains is crucial for their
practical scalability.

In addition, recent studies have continued to explore various neural network
architectures to tackle the challenge of solving Sudoku puzzles. For instance,
a Neuro-Symbolic approach has been proposed, which blends deep reinforcement
learning techniques with symbolic learning, aiming to improve systematicity
and generalization across larger sets of inputs. This method emphasizes the
use of logic predicates to approach the problem systematically. It has shown a
promising success rate when solving the puzzles within a defined number of
steps [13]. However, the primary limitation noted is the consideration of only
3, 5, and 8 empty cells in their experiments.

Another innovative approach is the development of SATNet, a differentiable
maximum satisfiability solver incorporating CNNs. SATNet operates using a fast
coordinate descent approach to solve associated semidefinite programs, showing
potential in the field of deep learning for solving systematic problems like
Sudoku [23]. However, the researchers utilized a substantial dataset for
training, with their experiments based on 9,000 training examples and 1,000
test examples.

This work (ASPER) simplifies the integration of domain expert knowledge into
neural models. The empirical evaluations demonstrate the effectiveness of the
proposed approach in reducing the complexity of potential solutions and
facilitating end-to-end training of complex data structures by employing a
model that learns from both data and embedded logical constraints. This makes
the proposed method more scalable and adaptable to various problem domains
where reasoning and logic are essential.

##  3 Answer Set Programming (ASP)

Answer set programming (ASP), as outlined in [12] and [19], has gained
prominence in the field of artificial intelligence (AI) due to its robustness
in knowledge representation and reasoning. Notably recognized for its
expressiveness, ASP excels in handling scenarios with incomplete information.
An ASP program typically comprises two segments: a knowledge base with factual
information and a set of rules delineating the problem-solving approach. The
solutions in ASP are formulated as answer sets or models, essentially possible
solutions that comply with the given constraints.

In the language of AnsProlog, also known as A-Prolog, an ASP program is
constructed with rules like:

$a_{0}\leftarrow a_{1},\ldots,a_{m},\neg a_{m+1},\ldots,\neg a_{n},$ (1)

where $1\leq m\leq n$. Each $a_{i}$ denotes an atom in the propositional
language, with $\neg a_{i}$ signifying a negation-as-failure literal (naf-
literal). The rule is structured such that the head (left-hand side) and the
body (right-hand side) fulfil distinct roles. Notably, a rule can exist with
an empty head, termed a constraint, or with an empty body, known as a fact,
but cannot have both simultaneously absent.

Consider $X$ as a set of ground atoms in a given ASP program, forming its
Herbrand base. The body of a rule in the format of (1) is satisfied by $X$ if
the intersection of $\\{a_{m+1},\ldots,a_{n}\\}$ and $X$ is empty, and
$\\{a1,\ldots,a_{m}\\}$ is a subset of $X$. A rule with a non-empty head
satisfies $X$ if either $a0$ is in $X$ or its body is not satisfied by $X$. A
constraint is fulfilled by $X$ if its body is unsatisfied by $X$.

##  4 Methodology

The proposed approach, designed for adaptability and generalization across a
range of problem domains, seamlessly integrates three key components: the
Answer Set Programming (ASP) problem solver, a neural model, and a customized
loss function. The ASP solver is crucial for translating complex domain
knowledge into structured logic, making it adaptable to diverse scenarios
ranging from for example, puzzles, physics, and biology to strategic business
problems. The neural model, trained on empirical data and structured knowledge
from the ASP solver, is uniquely equipped for adaptability in various
environments, particularly where training data is limited. The core innovation
lies in the design of the loss function, which comprises overall data-driven
learning loss, constraints loss derived from ASP solver solutions for logical
consistency, and domain expertise constraints for aligning with expert
knowledge. This integrated loss function ensures a balanced learning process,
combining data, logical constraints, and expert insights.

###  4.1 Problem Definition

Board games, particularly puzzles, typically involve a grid-based structure
where each cell can contain an element from a predefined set. The objective
often revolves around arranging these elements according to specific rules.
Formally, such a grid can be represented as a matrix $G$ where $G_{i,j}$
represents the element in the $i^{th}$ row and $j^{th}$ column. The problem
can be defined as finding a matrix $G$ that satisfies a set of conditions
which are unique to each game but generally involve constraints on the
arrangement of elements. These conditions might include:

  1. 1.

Each cell $G_{i,j}$ must contain an element from a given set, e.g.,
$G_{i,j}\in\\{1,2,...,N\\}$ for a game with $N$ different elements.

  2. 2.

Certain elements may need to be aligned or arranged in specific patterns,
rows, columns, or subgrids, depending on the game’s rules.

  3. 3.

Additional constraints unique to the specific puzzle or game, dictating how
elements can be placed or moved within the grid.

The challenge in these games often lies in the logical arrangement or sequence
of moves required to achieve a final, ’solved’ state of the grid, adhering to
all specified rules and constraints.

###  4.2 Combined Loss

For any given puzzle, the puzzle should be represented in ASP. This involves
encoding the puzzle elements and constraints as logical rules. The general
form of these rules can be expressed as:

asp_rule(features, constraints, expert_insights) (2)

These rules ensure the ASP solver considers all possible configurations that
satisfy the puzzle’s constraints.

####  4.2.1 Standard loss

Parallel to the ASP formulation, a neural model should be employed. This model
uses a customized loss function that combines a standard loss with a puzzle-
specific constraints-based loss. The standard loss ($L_{standard}$) can be
generally defined using a squared error (MSE) or cross entropy as:

$L_{\text{standard}}=-\sum_{i=1}^{C}y_{i}\log(p_{i})$ (3)

This equation, $L_{standard}$ represents the Cross-Entropy loss for a
multiclass classification problem. Here, $C$ is the total number of classes,
$y_{i}$ is the binary indicator (0 or 1) if class label $i$ is the correct
classification for the observation, and $p_{i}$ is the predicted probability
of the observation being class $i$.

####  4.2.2 Constrained loss

The constraints-based loss $L_{constraints}$ for a generic puzzle is
formulated to penalize solutions that violate the puzzle’s specific rules and
constraints. It can be generalized as:

$L_{\text{constraints}}=\sum_{\text{all
constraints}}\text{penalty}(constraint_{violation})$ (4)

This loss ensures the model’s predictions adhere to the puzzle’s logical
structure.

####  4.2.3 Domain Expert Knowledge loss

Integrating domain expert knowledge, such as specialized strategies,
significantly enhances the interpretability and accuracy of machine learning
models. Answer Set Programming (ASP) solvers are key to this integration. The
results obtained from ASP solvers are particularly beneficial as they provide
structured, logical outcomes that can be leveraged for improved learning.

$L_{\text{expert}}=\sum_{\text{all expert
rules}}\text{penalty}(\text{deviation from expert rule})$ (5)

####  4.2.4 ASPER Approach - Combined Loss Function and Model Training

The combined loss function ($L_{combined}$) integrates both standard and
constraintsbased losses:

$L_{\text{combined}}=\alpha L_{\text{standard}}+\beta
L_{\text{constraints}}+\gamma L_{\text{expert}}$ (6)

Here, $\alpha$ and $\beta$ are weighting factors that balance the influence of
standard accuracy and adherence to constraints. The model is trained on data
generated by solving puzzles with ASP, to minimize $L_{combined}$. This
ensures that the model learns both the accuracy in puzzle element prediction
and adherence to the puzzle’s intrinsic rules.

Algorithm 1 ASPER - A General Methodology for Neural Learning with ASP Solver
Integration

1:Data, DomainKnowledge, Expertise

2:Trained Model

3:function TrainDeepLearningModel(Data, DomainKnowledge, Expertise)

4: ASP $\leftarrow$ Initialize ASP solver with domainKnowledge

5: Model $\leftarrow$ Initializethe Neural Model

6: $\alpha,\beta,\gamma\leftarrow$ Initialize weights for loss components

7: LossFunction $\leftarrow$ Create Custom Loss Function with Expertise
Knowledge

8: for epoch $\leftarrow$ 1 to Epochs do

9: for all puzzle in Data do

10: StructuredData $\leftarrow$ ASP.solve(puzzle)

11: Prediction $\leftarrow$ Model.predict(puzzle)

12: $L_{\text{standard}}\leftarrow$ CalculateStandardLoss(Prediction,
StructuredData)

13: $L_{\text{constraints}}\leftarrow$ CalculateConstraintsLoss(Prediction,
StructuredData)

14: $L_{\text{expert}}\leftarrow$ CalculateExpertLoss(Prediction,
StructuredData)

15: $L_{\text{combined}}\leftarrow\alpha L_{\text{standard}}+\beta
L_{\text{constraints}}+\gamma L_{\text{expert}}$

16: Model.update_weights($L_{\text{combined}}$)

17: end for

18: end for

19: return Model

20:end function

21:function SolvePuzzleWithModel(Puzzle, Model)

22: Prediction $\leftarrow$ Model.predict(Puzzle)

23: Solution $\leftarrow$ Post-process Prediction to satisfy game rules

24: return Solution

25:end function

Algorithms 1 shows the pseudocode details a methodical approach for
integrating neural models with Answer Set Programming (ASP), highlighting a
multi-faceted training process. It begins by initializing an ASP solver with
domain knowledge, ensuring that the model is attuned to the specific
constraints and logic of the problem domain. Consequently, the neural model is
set up to learn from empirical data. The idea lies in the custom loss
function, which combines standard data-driven learning loss, constraints-based
loss from the ASP solutions, and a domain expertise loss. This combination
ensures that the learning is data-centric, logically coherent, and aligned
with expert insights. The training loop iterates over the data, utilizing the
ASP solver for structured problem-solving and the deep learning model for
prediction, thus striking a balance between empirical learning and rule-based
reasoning. While demonstrated in the context of puzzles, this methodology has
a broad scope of applicability across various domains.

##  5 Use-Case - Suduku

In the realm of Sudoku, a puzzle is defined as a 9x9 grid where each cell may
contain a digit from 1 to 9. The challenge lies in filling the grid so that
each row, column, and 3x3 subgrid contains all digits from 1 to 9 exactly
once. Formally, the Sudoku grid can be represented as a matrix $S$ where
$S_{i,j}$ represents the value in the $i^{th}$ row and $j^{th}$ column. The
problem can be defined as finding a matrix $S$ that satisfies the following
conditions for all $i,j$ in the range 1 to 9:

  1. 1.

$S_{i,j}\in\\{1,2,...,9\\}$ for all $i,j$.

  2. 2.

Each number appears exactly once in each row: $\sum_{j=1}^{9}[S_{i,j}=k]=1$
for each $k$ and each $i$.

  3. 3.

Each number appears exactly once in each column: $\sum_{i=1}^{9}[S_{i,j}=k]=1$
for each $k$ and each $j$.

  4. 4.

Each number appears exactly once in each 3x3 subgrid.

In the proposed ASP-based approach for solving Sudoku, the puzzle’s
constraints are encoded as rules in an ASP program. This program uses the ASP
solver ‘clingo‘ to find solutions satisfying these constraints. The key
aspects of this implementation, based on the provided code, are (see Listing
1):

Listing 1: ASP code for Sudoku constraints

⬇

1 { cell(Row,Col,Val) : Val=1..9 } = 1 :- Row=1..9, Col=1..9.

2 :- cell(Row,Col1,Val), cell(Row,Col2,Val), Col1 != Col2.

3 :- cell(Row1,Col,Val), cell(Row2,Col,Val), Row1 != Row2.

4 :- cell(Row1,Col1,Val), cell(Row2,Col2,Val),

5 Row1 != Row2, Col1 != Col2, (Row1-1)/3 = (Row2-1)/3,

(Col1-1)/3 = (Col2-1)/3.

  1. 1.

Line 1 defines a rule that ensures each cell in the grid has exactly one
number between 1 and 9.

  2. 2.

Line 2 enforces that no two cells in the same row can have the same value.

  3. 3.

Line 3 ensures that no two cells in the same column have the same value.

  4. 4.

Line 4-5 imposes the constraint that no two cells in the same 3x3 subgrid have
the same value.

By leveraging ASP, we can efficiently solve Sudoku puzzles, regardless of
complexity. This declarative approach allows for a concise and clear puzzle
representation, facilitating intuitive and straightforward solution processes.

###  5.1 Loss functions of Suduku

ASPER integrates Answer Set Programming (ASP) with a deep learning model to
solve the defined Sudoku problem. In ASP, we express the constraints of Sudoku
as rules. For instance, the row constraint is encoded as:

$\sum_{j=1}^{9}\text{cell}(i,j,v)=1\quad\forall i,v$ (7)

Similar rules are constructed for columns and 3x3 subgrids, ensuring that the
ASP solver finds solutions that respect the Sudoku constraints.

The deep learning model, tailored to solve Sudoku, uses a loss function that
combines the standard cross-entropy with a constraints-based loss. The cross-
entropy, representing the standard loss ($L_{standard}$), is defined as:

$L_{\text{cross-
entropy}}=-\frac{1}{81}\sum_{i=1}^{9}\sum_{j=1}^{9}S_{i,j}\log(\hat{S_{i,j}})$
(8)

where $S_{i,j}$ represents the predicted value for the cell in the $i_{th}$
row and $j_{th}$ column.

The constraints-based loss ($L_{constraints}$) is formulated to penalize
solutions that violate Sudoku’s row, column, and subgrid rules. It can be
expressed as:

$\begin{split}L_{\text{constraints}}=\sum_{num=1}^{9}\Biggl{(}&\sum_{i=1}^{9}\left(1-\sum_{j=1}^{9}\text{is\\_empty}_{i,j}\times[y_{i,j}=num]\right)^{2}\\\
&+\sum_{j=1}^{9}\left(1-\sum_{i=1}^{9}\text{is\\_empty}_{i,j}\times[y_{i,j}=num]\right)^{2}\\\
&+\sum_{sub=1}^{9}\left(1-\sum_{(i,j)\in\text{sub}}\text{is\\_empty}_{i,j}\times[y_{i,j}=num]\right)^{2}\Biggr{)}\end{split}$
(9)

This loss function accounts for the frequency of each number in each row,
column, and subgrid, focusing on empty cells in the original puzzle.

Assuming that the domain knowledge expert loss calculates the difference in
the sum of predicted numbers and actual numbers in rows, columns, and
subgrids. It ensures that the total sum of numbers in each row, column, and
subgrid in the prediction matches that in the true solution. The loss is
defined as:

$\begin{split}L_{\text{expert}}=\sum_{i=1}^{9}\Biggl{(}&\left|\sum_{j=1}^{9}y_{\text{pred\\_row}_{i},j}-\sum_{j=1}^{9}y_{\text{true\\_row}_{i},j}\right|\\\
&+\left|\sum_{j=1}^{9}y_{\text{pred\\_col}_{i},j}-\sum_{j=1}^{9}y_{\text{true\\_col}_{i},j}\right|\\\
&+\sum_{j=1}^{9}\biggl{|}\sum_{(k,l)\in\text{subgrid}_{i,j}}y_{\text{pred\\_subgrid},k,l}\\\
&\quad-\sum_{(k,l)\in\text{subgrid}_{i,j}}y_{\text{true\\_subgrid},k,l}\biggr{|}\Biggr{)}\end{split}$
(10)

While the $L_{constraints}$ loss ensures each number appears exactly once per
row, column, and subgrid, the $L_{expert}$ specifically ensures that the sum
of the numbers in each row, column, and subgrid in the predicted solution
matches the sum in the true solution.

The constraints-based loss uses a squared difference to penalize deviations
from Sudoku rules. In contrast, $L_{expert}$ uses the absolute difference
between the sums of predicted and true values, which could be more sensitive
to deviations in number distribution. The $L_{expert}$ loss is particularly
effective in scenarios where maintaining the sum of numbers is crucial,
reflecting a deeper Sudoku-solving strategy often employed by experienced
players.

Combining these two losses with the neural model standard loss functions in a
neural network model for solving Sudoku puzzles enables the model to learn the
basic rules of Sudoku and incorporate more advanced solving strategies,
enhancing its problem-solving capabilities.

The combined loss function ($L_{combined}$) is a weighted sum of the standard
loss and the constraints loss:

$L_{\text{combined}}=\alpha L_{\text{standard}}+\beta
L_{\text{constraints}}+\gamma L_{\text{expert}}$ (11)

Training the model involves generating puzzles, solving them with ASP to
create training data, and then using this data to train the neural network to
minimize $L_{combined}$. This process ensures that the model not only learns
to predict the numbers in each cell accurately but also respects the
fundamental rules of Sudoku, thus effectively solving the puzzle.

##  6 Implementation and Evaluation

The Sudoku solving process employs a shallow neural network constructed using
Keras’ Sequential architecture [6]. This model starts with a Dense layer of 64
neurons and ReLU activation, processing a flattened 81-element array
representing the 9x9 Sudoku grid. It then includes a Dense layer (9x9x9) for
outputting potential cell values, reshaped into a 9x9x9 tensor and finalized
with a softmax activation layer. Optimization uses the Adam optimizer and
TensorFlow’s GradientTape for gradient computation [1]. Training involves
custom loss functions, including cross-entropy and Sudoku-specific
constraints, to ensure both accuracy and adherence to Sudoku rules. This
method is supported by Clingo [11], an ASP solver that verifies the
correctness of Sudoku puzzles generated for training, using KFold cross-
validation to enhance model robustness and unbiased accuracy.

The evaluation of accuracy is conducted through a KFold cross-validation
process. Each fold’s data is split into training and validation sets, and the
model is trained using custom loss functions. Post-training, the model’s
predictions on the validation set are compared to actual solutions. Accuracy
is calculated based on the proportion of correctly predicted cells across all
Sudoku puzzles in the test set. These accuracy metrics from each fold are then
aggregated to comprehensively assess the model’s overall performance in
solving the puzzles.

##  7 Results

# Puzzles |  difficulty |  Standard Only |  Standard + Expert |  Standard + Constraints |  All Combined
---|---|---|---|---|---
12 |  0.1 |  0.84 |  0.86 |  0.92 |  0.88
12 |  0.3 |  0.70 |  0.69 |  0.74 |  0.70
12 |  0.6 |  0.48 |  0.41 |  0.49 |  0.51
12 |  0.8 |  0.26 |  0.28 |  0.28 |  0.29
100 |  0.1 |  1.00 |  1.00 |  1.00 |  1.00
100 |  0.3 |  1.00 |  1.00 |  1.00 |  1.00
100 |  0.6 |  0.19 |  0.20 |  0.20 |  0.20
1000 |  0.8 |  0.11 |  0.11 |  0.11 |  0.11
Table 1: Comparative Evaluation of Various Loss Combinations on 12, 100, and
1000 Sudoku Puzzles Across Difficulty Levels 0.1, 0.3, 0.6, and 0.8. The data
presented reflects outcomes from a 3-fold cross-validation process, with the ±
symbol indicating the standard deviation observed across different folds. This
comprehensive analysis provides insights into the effectiveness of each loss
combination under varying puzzle complexities and sizes Figure 1: Evaluation
of Various Loss Combinations on 12, 100, and 1000 Sudoku Puzzles Across
Difficulty Levels 0.1, 0.3, 0.6, and 0.8. The data presented reflects outcomes
from a 3-fold cross-validation process.

The results detailed in Table 1 show the critical importance of selecting an
optimal loss function specifically tailored to the distinct characteristics of
a problem, such as the number of puzzles and the difficulty level,
particularly in the context of Sudoku puzzles. This study underscores the
power of employing a hybrid loss approach to tackle complex challenges. The
exploration involves various loss functions in training a deep learning model
for solving Sudoku puzzles, including the different loss combinations. The
analysis indicates that with higher difficulty levels in Sudoku puzzles, which
typically means less available information, there is a noticeable decrease in
accuracy across most loss combinations. This trend highlights deep learning
models’ challenges in effectively utilizing limited information. The analysis
further shows that the smaller the dataset used for training, the more is the
impact of ASPER on the overall performance. In addition, ASPER shows that
neural models can handle complex, less predictable puzzles with limited
training data under constrained training conditions. The All Combined approach
shows high performance, particularly in scenarios with lower difficulty levels
and smaller puzzle sets, suggesting its adeptness in handling complex
scenarios. Interestingly, in larger sets (100 puzzles), differences between
loss functions become less pronounced, while in smaller sets (12 puzzles), the
differences are significant, with the Combined loss showing its strength. The
study highlights the importance of constraint learning in deep learning, which
is crucial for model development in domains where rules and constraints play a
pivotal role. It also stresses the integration of domain expertise, which
enhances the model’s problem-solving capabilities, a principle extendable to
other fields for augmented AI performance. The role of cross-validation in
this context ensures that the model’s performance is unbiased, thereby
bolstering the reliability of the results. Importantly, the approach’s
potential extends beyond Sudoku, with applicability to other complex tasks
such as natural language processing and image recognition.

Figure 2: The image displays a set of four 9x9 Sudoku grids, each representing
a different stage of puzzle solving. The top left grid is the "Original
Puzzle" with all cells filled, indicating the starting complete Sudoku. The
top right grid is the "Masked Puzzle," showing the Sudoku with certain numbers
omitted, which presents the challenge to be solved. The bottom left grid is
the "Solution," providing the completed puzzle with all numbers correctly
placed. The bottom right grid is the "Predicted" solution, presumably produced
by a Sudoku-solving algorithm, with correct predictions highlighted in green
and incorrect ones in red. This visualization serves to compare the model’s
performance against the actual solution.

###  7.1 Discussion

ASPER reveals key insights into using different loss functions in neural-
symbolic AI, particularly in solving Sudoku puzzles. The finding shows that
using standard loss ($L_{standard}$) resulted in low accuracy when the number
of training data is small. This outcome underscores the insufficiency of using
a standard loss-based approach for complex problems like Sudoku. In contrast,
the incorporation of constraints-based loss ($L_{constraint}$) and domain
expert knowledge loss ($L_{exopert}$) led to an improvement when the number of
training data is small, considering different difficulty levels. This
highlights the role of integrating domain-specific knowledge and Sudoku-
specific rules into the learning process, illustrating the value of tailored
approaches in solving combinatorial problems. ASPER shows improved performance
across varied data samples and complexity levels, indicative of its
adaptability to diverse real-world scenarios. By using different datasets for
each set of conditions, we ensure the model’s effectiveness is not an artefact
of specific data characteristics, thereby reducing the risk of overfitting and
enhancing the reliability of ASPER findings. However, as the volume of
training data increases, the incremental benefits of ASPER become less
noticeable, aligning with expectations that extensive data can inherently
enhance deep learning model performance.

ASPER contributes to the evolving field of neuro-symbolic AI by showcasing the
successful integration of domain expertise into deep learning models. This
approach is particularly effective in complex reasoning tasks and scenarios
with limited data, combining neural networks’ pattern recognition capabilities
with symbolic AI’s structured, logical reasoning. This method reduces the
reliance on extensive datasets and enhances the model’s deductive reasoning
abilities. Moreover, ASPER addresses the crucial issue of AI interpretability
and transparency. Embedding logical rules into training provides more
traceable and understandable decision-making pathways. This is especially
important in fields where understanding the reasoning behind AI decisions is
critical. The methodology of embedding logical rules directly into the
training process can enhance the decision-making transparency in many AI
applications, a critical factor in fields such as healthcare, finance, and
commercial systems where understanding the rationale behind AI decisions is as
vital as the decisions themselves.

Method | Input | Number of Data for Training | Accuracy of Solution
---|---|---|---
(2018) [21] | Textual Sudoku (9x9) | 1 Million | 70.0%
(2018) [20] | Textual Sudoku (9x9) | 216,000 | 96.6%
(2019) [23] | Textual Sudoku (9x9) | 9000 | 98.3%
(2023) [13] | Textual Sudoku (9x9) | - | 100% (8 empty cells only)
Table 2: Comparison of different methods for solving Sudoku.

Table 2) shows that ASPER outperforms the benchmarks state-of-the-art
benchmarks in terms of handling different difficulty levels and achieving a
competitive accuracy a with a minimal dataset. It is also worth highlighting
that integrating expert knowledge and constraints into the learning process
has improved the accuracy of puzzles with difficulty levels of 0.1 and 0.3.
This integration appears to improve the model performance and provide the
ability to leverage additional information effectively, which is not solely
reliant on the data-driven patterns learned from the input data.

###  7.2 Conclusion

This paper proposes the ASPER approach to explore the integration of Answer
Set Programming (ASP) with neural models. The findings suggest that
incorporating ASP’s logical reasoning into the framework of neural models can
enhance their performance, particularly in combinatorial problems. This
integration appears to offer a way to embed domain-specific knowledge and
rules into AI models, an area of growing interest in machine learning. A key
observation from this paper is the potential improvement in model accuracy
when incorporating constraints-based loss. The ASPER approach demonstrates
that a combined loss methodology, which integrates various aspects of learning
and reasoning, may offer a balanced path forward in the development of more
sophisticated AI systems. Furthermore, the work addresses aspects of AI
interpretability and transparency by embedding logical rules directly into the
training process to enhance the decision-making transparency of AI models.
This approach could be particularly beneficial in sectors where understanding
the rationale behind AI decisions is required. Looking ahead, the ASPER
approach could be scaled and adapted to various applications, though this
would require careful consideration of each domain’s unique requirements and
constraints. The paper contributes to the ongoing discourse in the AI
community about balancing data-driven insights with rule-based logic,
particularly in the context of ethical AI development and deployment.

## References

  * [1] Martín Abadi, Paul Barham, Jianmin Chen, Zhifeng Chen, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Geoffrey Irving, Michael Isard, et al.  Tensorflow: A system for large-scale machine learning.  In 12th $\\{$USENIX$\\}$ Symposium on Operating Systems Design and Implementation ($\\{$OSDI$\\}$ 16), pages 265–283, 2016.
  * [2] Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio.  Neural machine translation by jointly learning to align and translate.  arXiv preprint arXiv:1409.0473, 2014.
  * [3] T. R. Besold, A. S. d’Avila Garcez, S. Bader, H. Bowman, P. Domingos, P. Hitzler, K. U. Kühnberger, L. C. Lamb, D. Lowd, P. M. V. Lima, L. de Penning, G. Pinkas, H. Poon, and G. Zaverucha.  Neural-symbolic learning and reasoning: A survey and interpretation.  arXiv preprint arXiv:1711.03902, 2017.
  * [4] Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al.  Language models are few-shot learners.  Advances in neural information processing systems, 33:1877–1901, 2020.
  * [5] Xinyun Chen, Chang Liu, and Dawn Song.  Towards synthesizing complex programs from input-output examples.  arXiv preprint arXiv:1706.01284, 2017.
  * [6] François Chollet et al.  Keras.  https://keras.io, 2015.
  * [7] Gabriele Ciravegna, Francesco Giannini, Marco Gori, Marco Maggini, and Stefano Melacci.  Human-driven fol explanations of deep learning.  In IJCAI, pages 2234–2240, 2020.
  * [8] Artur d’Avila Garcez and Luis C Lamb.  Neurosymbolic ai: the 3rd wave.  arXiv e-prints, pages arXiv–2012, 2020.
  * [9] Honghua Dong, Jiayuan Mao, Tian Lin, Chong Wang, Lihong Li, and Denny Zhou.  Neural logic machines.  arXiv preprint arXiv:1904.11694, 2019.
  * [10] Artur d’Avila Garcez, Marco Gori, Luis C Lamb, Luciano Serafini, Michael Spranger, and Son N Tran.  Neural-symbolic computing: An effective methodology for principled integration of machine learning and reasoning.  arXiv preprint arXiv:1905.06088, 2019.
  * [11] Martin Gebser, Roland Kaminski, Benjamin Kaufmann, and Torsten Schaub.  Multi-shot asp solving with clingo.  Theory and Practice of Logic Programming, 19(1):27–82, 2019.
  * [12] Michael Gelfond and Vladimir Lifschitz.  The stable model semantics for logic programming.  In ICLP/SLP, volume 88, pages 1070–1080. Cambridge, MA, 1988.
  * [13] Ashutosh Hathidara and Lalit Pandey.  Neuro-symbolic sudoku solver, 2023.
  * [14] Jiani Huang, Ziyang Li, Binghong Chen, Karan Samel, Mayur Naik, Le Song, and Xujie Si.  Scallop: From probabilistic deductive databases to scalable differentiable reasoning.  Advances in Neural Information Processing Systems, 34:25134–25145, 2021.
  * [15] Jindong Jiang and Sungjin Ahn.  Generative neurosymbolic machines.  Advances in Neural Information Processing Systems, 33:12572–12582, 2020.
  * [16] Robin Manhaeve, Sebastijan Dumancic, Angelika Kimmig, Thomas Demeester, and Luc De Raedt.  Deepproblog: Neural probabilistic logic programming.  Advances in neural information processing systems, 31, 2018.
  * [17] Jiayuan Mao, Chuang Gan, Pushmeet Kohli, Joshua B Tenenbaum, and Jiajun Wu.  The neuro-symbolic concept learner: Interpreting scenes, words, and sentences from natural supervision.  arXiv preprint arXiv:1904.12584, 2019.
  * [18] G. Marcus and E. Davis.  Rebooting ai: Building artificial intelligence we can trust.  Pantheon Books, 2019.
  * [19] Ilkka Niemelä.  Logic programs with stable model semantics as a constraint programming paradigm.  Annals of mathematics and Artificial Intelligence, 25:241–273, 1999.
  * [20] Rasmus Palm, Ulrich Paquet, and Ole Winther.  Recurrent relational networks.  In Proceedings of Advances in Neural Information Processing Systems, pages 3368–3378, 2018.
  * [21] Kyubyong Park.  Can convolutional neural networks crack sudoku puzzles?  https://github.com/Kyubyong/sudoku, 2018.  Accessed: [Insert Date Here].
  * [22] Wolfgang Stammer, Patrick Schramowski, and Kristian Kersting.  Right for the right concept: Revising neuro-symbolic concepts by interacting with their explanations.  In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 3619–3629, 2021.
  * [23] Po-Wei Wang, Priya Donti, Bryan Wilder, and Zico Kolter.  Satnet: Bridging deep learning and logical reasoning using a differentiable satisfiability solver.  In International Conference on Machine Learning, pages 6545–6554. PMLR, 2019.
  * [24] Zhun Yang, Adam Ishay, and Joohyung Lee.  Neurasp: Embracing neural networks into answer set programming.  arXiv preprint arXiv:2307.07700, 2023.
  * [25] Kexin Yi, Jiajun Wu, Chuang Gan, Antonio Torralba, Pushmeet Kohli, and Josh Tenenbaum.  Neural-symbolic vqa: Disentangling reasoning from vision and language understanding.  Advances in neural information processing systems, 31, 2018.


Observable $r$, Gravitino Dark Matter, and Non-thermal Leptogenesis in No-
Scale Supergravity

Waqas Ahmeda 111E-mail<EMAIL_ADDRESS> Muhammad Moosab 222E-mail:
<EMAIL_ADDRESS>Shoaib Munirc 333E-mail<EMAIL_ADDRESS>and
Umer Zubaird,e 444E-mail<EMAIL_ADDRESS>
a School of Mathematics and Physics, Hubei Polytechnic University,
Huangshi 435003, China
bDepartment of Physics, COMSATS University Islamabad, Islamabad 45550,
Pakistan
c East African Institute for Fundamental Research (ICTP-EAIFR),University of
Rwanda, Kigali, Rwanda
dDepartment of Physics, Saint Joseph’s University, Philadelphia, PA 19131, USA
eDivision of Science and Engineering,
Pennsylvania State University, Abington, PA 19001, USA

###### Abstract

We analyse the shifted hybrid inflation in a no-scale supersymmetric $SU(5)$
GUT model which naturally circumvents the monopole problem. The no-scale
framework is derivable as the effective field theory of the supersymmetric
(SUSY) compactifications of string theory, and yields a flat potential with no
anti-de Sitter vacua, resolving the $\eta$ problem. The model predicts a
scalar spectral tilt $n_{s}$ compatible with the most recent measurements by
the Planck satellite, while also accommodating large values of the tensor-to-
scalar ratio $r$ ($\sim 0.0015$), potentially measurable by the near-future
experiments. Moreover, the proton decay lifetime in the presence of the
dimension-5 operators is found to lie above the current limit imposed by the
Super-Kamiokande experiment. A realistic scenario of reheating and non-thermal
leptogenesis is employed, wherein the reheating temperature $T_{r}$ lies in
the $(2\times 10^{6}\lesssim T_{r}\lesssim 2\times 10^{9})$ GeV range, and at
the same time realizing gravitino as a viable dark matter (DM) candidate.

##  1 Introduction

Inflation provides a successful phenomenological framework for addressing
cosmological puzzles such as the size, the age, the homogeneity and the
(approximate) geometrical flatness of the Universe [1]. It also explains
large-scale structures, the smallness of the primordial density perturbations
measured in the cosmic microwave background (CMB), as well as the small tilt
($n_{s}=0.965$) in it and the inconsistency of this CMB with the Gaussian
white-noise spectrum [2]. The success of inflation has motivated several
attempts to relate it to the Standard Model (SM) of particle physics, and at
the same time to a candidate quantum theory of everything, including gravity,
such as string theory. The characteristic energy scale of inflation is
presumably intermediate between SM and quantum gravity, and inflationary
models may thus provide a welcome bridge between the two of them.

A connection between inflation and a viable quantum theory of gravity at some
very high scale and the SM at the electroweak (EW) scale makes models based on
no-scale supergravity (SUGRA) a particularly attractive choice [3, 4, 5]. No-
scale SUGRA appears generically in models with ultraviolet completion using
string theory compactifications [6]. It is known to be the general form of the
4-dimensional effective field theory derivable from string theory that
embodies low-energy supersymmetry. Moreover, no-scale SUGRA is an attractive
framework for constructing models of inflation [7, 8] because it naturally
yields a flat potential with no anti-de Sitter ‘holes’, resolving the so-
called $\eta$ problem. No-scale inflation also comfortably accommodates values
of the $r$ and $n_{s}$ perfectly compatible with the most recent measurements
of the Planck satellite, and potentially very similar to the values predicted
by the original Starobinsky model [9]. For more detailed studies in the
context of Starobinsky model, see Refs. [10, 11, 12, 13, 14, 15, 16]. The
simplest version of the Starobinsky proposal is equivalent to an inflationary
model where the scalar field couples non-minimally to gravity. A natural
choice for the inflaton field in the Grand Unified Theories (GUTs) with SUGRA
is the Higgs field responsible for breaking the GUT gauge group. (for more
details, see refs. [17, 18]).

Hybrid inflation is one of the most promising models of inflation that can be
naturally realized within the context of SUGRA theories [19, 20, 21, 22, 23,
24, 25]. In SUSY hybrid inflation, the scalar potential along the inflationary
track is completely flat at the tree level. The inclusion of radiative
corrections to this potential provides the necessary slope required to drive
the inflaton towards the SM vacuum. In such a scenario [19], the CMB
temperature anisotropy $\delta T/T$ is of the order $(M/m_{P})$, where $M$ is
the breaking scale of the parent gauge group $G$ of the Universe at the start
of inflation, and $m_{P}=2.4\times 10^{18}$ GeV is the reduced Planck mass. In
order for the self-consistency of the inflationary scenario to be preserved,
$M$ turns out to be comparable to the scale of grand unification,
$M_{\text{GUT}}\sim 10^{16}$ GeV, hinting that $G$ may be the GUT gauge group.

In the standard hybrid model of inflation, $G$ breaks spontaneously to its
subgroup $H$ at the end of the inflation [19, 26], which leads to topological
defects such as copious production of magnetic monopoles by the Kibble
mechanism [27]. These magnetic monopoles dominate the energy budget of the
Universe, contradicting the cosmological observations. In the shifted [28] or
smooth [30, 31, 32] variants of the hybrid inflation, $G$ is instead broken
during inflation, and in this way the disastrous monopoles are inflated away.

In this article, we study shifted hybrid inflation formulated in the framework
of no-scale SUGRA. In our model, the $SU(5)$ gauge group is spontaneously
broken down to the $G_{SM}$ by the vacuum expectation value (VEV) of the
$\Phi_{24}$ adjoint Higgs superfield. By generating a suitable shifted
inflationary track wherein the $SU(5)$ is broken during inflation, the
monopole density can be significantly diluted. The predictions of our model
are consistent with the Planck’s latest bounds [2] on $n_{s}$ and $r$.
Moreover, a wide range of the $T_{r}$ is obtained which naturally avoids the
gravitino problem. A model of non-thermal leptogenesis via right-handed
neutrinos is studied in order to explain the observed baryon asymmetry of the
Universe (BAU). As compared to the $SU(5)$ model with shifted hybrid inflation
studied outside the no-scale framework in [28], relatively large values of $r$
($\sim 0.0015$) are obtained here, potentially measurable by the future
experiments.

The layout of the paper is as follows. Sec. 2 presents the basic description
of the model including the field content, the superpotential and the global
minima of the potential. The inflationary trajectories and the dimension-5
proton decay are discussed in Sec. 3. The inflationary setup and the
theoretical details of reheating with non-thermal leptogensis are provided in
Sec. 4 and Sec. 5, respectively. The numerical analysis of the prospects of
observing primordial gravity waves, and of leptogenesis and gravitino
cosmology, is presented in Sec. 6. Finally, Sec. 7 summarizes our findings.

##  2 The Supersymmetric $SU(5)$ Model with $U(1)_{R}$ Symmetry

In the model we consider, the matter fields of the minimal supersymmetric
standard model (MSSM) reside in the following representations of the $SU(5)$
supermultiplets.

$\begin{split}F_{i}&\equiv
10_{i}=Q_{i}\left(3,2,1/6\right)+u_{i}^{c}\left(\overline{3},1,-2/3\right)+e^{c}\left(1,1,1\right),\\\
\overline{f}_{i}&\equiv\overline{5}_{i}=d_{i}^{c}\left(\overline{3},1,1/3\right)+\ell_{i}\left(1,2,-1/2\right),\\\
\nu_{i}&\equiv 1_{i}=\nu^{c}_{i}\left(1,1,0\right)\,,\end{split}$ (2.1)

where $i$ is the generation index $(i=1,2,3)$ and $\nu^{c}_{i}$ are right-
handed neutrnio superfields. The Higgs sector constitutes of a pair of 5-plet
superfields {$H\equiv H_{5},\,\overline{H}\equiv\overline{H}_{5}$} that
contain the colour Higgs triplets and the two doublets of the MSSM, a gauge-
singlet superfield $S$, and a 24-plet superfield $\Phi$ that belongs to the
adjoint representation. This superfield $\Phi$ is responsible for breaking the
$SU(5)$ gauge symmetry down to the SM gauge group by acquiring a non-zero VEV
in the hypercharge direction. These Higgs superfields are decomposed under the
$G_{SM}$ as

$\displaystyle H$ $\displaystyle=$
$\displaystyle{H_{T}(3,1,-1/3)}+H_{u}(1,2,1/2),$ $\displaystyle\overline{H}$
$\displaystyle=$
$\displaystyle\overline{H}_{T}(\overline{3},1,1/3)+H_{d}(1,2,-1/2),$
$\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\Phi_{24}(1,1,0)+W_{H}(1,3,0)+G_{H}(8,1,0)$ (2.2)
$\displaystyle+$ $\displaystyle
X_{H}(3,2,-5/6)+\overline{X}_{H}(\overline{3},2,5/6)\,.$

Following [28], the $U(1)_{R}$-charge assignments of the various superfields
are

$R\left(S,\Phi,H,\overline{H},F_{i},\overline{f}_{i},\nu_{i}\right)=\left(1,0,\frac{2}{5},\frac{3}{5},\frac{3}{10},\frac{1}{10},\frac{1}{2}\right)\,.$
(2.3)

The $U(1)_{R}$-symmetric superpotential of the $SU(5)$ group containing the
above superfields is written as

$\displaystyle W$ $\displaystyle=$ $\displaystyle\kappa
S\left(\mu^{2}-Tr(\Phi^{2})-\beta\frac{Tr(\Phi^{3})}{\kappa\,M_{*}}\right)+\widetilde{\gamma}\overline{H}\Phi
H+\widetilde{\delta}\overline{H}H$ (2.4) $\displaystyle+$ $\displaystyle
y_{ij}^{(u)}F_{i}F_{j}H+y_{ij}^{(d,e)}F_{i}\overline{f}_{j}\overline{H}+y_{ij}^{(\nu)}\nu_{i}\overline{f}_{j}H+m_{ij}\nu_{i}\nu_{j}+\lambda_{ij}\frac{Tr(\Phi^{2})\nu_{i}\nu_{j}}{M_{*}}\,,$

where $\kappa$, $\widetilde{\gamma}$, and $\lambda_{ij}$ are dimensionless
couplings, while $\widetilde{\delta}$ is a parameter with unit mass-dimension
and $M_{*}$ is a high cut-off scale (compactification scale in a string model
or the Planck scale) $M_{\text{GUT}}\lesssim M_{*}\lesssim m_{P}$. The Yukawa
couplings $y_{ij}^{(u)}$, $y_{ij}^{(d,e)}$ and $y_{ij}^{(\nu)}$ in the second
line above generate the quark and lepton masses after the EW symmetry-
breaking, whereas the last two terms generate the right-handed neutrino
masses, and are thus relevant for leptogenesis in the post inflationary era.

###  2.1 The Global SUSY Minima

The first line of the superpotential in Eq. (2.4) contains the terms
responsible for the shifted hybrid inflation, and can be rewritten in
component form as

$W\supset\kappa
S\left(\mu^{2}-\frac{1}{2}\sum_{i}\phi_{i}^{2}-\frac{\beta}{4\kappa\,M_{*}}d_{ijk}\phi_{i}\phi_{j}\phi_{k}\right)+\widetilde{\gamma}\,T_{ab}^{i}\phi^{i}\overline{H_{a}}H_{b}+\widetilde{\delta}\,\overline{H_{a}}H_{a}\,,$
(2.5)

where we have expressed the superfield $\Phi$ in the $SU(5)$ adjoint basis
$\Phi=\phi_{i}T^{i}$ with $\operatorname{Tr}(T_{i}T_{j})=\delta_{ij}/2$ and
$d_{ijk}=2\operatorname{Tr}(T_{i}\\{T_{j},T_{k}\\})$. Here the indices $i$,
$j$ and $k$ run from 1 to 24, whereas $a$ and $b$ run from 1 to 5. The global
$F$-term potential is obtained from $W$ as

$\displaystyle V_{F}\supset\sum_{i}\left|\frac{\partial W}{\partial
z_{i}}\right|^{2}$ $\displaystyle=$ $\displaystyle\sum_{i}\left|\kappa
S\phi_{i}+\frac{3\beta}{4M_{*}}d_{ijk}S\phi_{j}\phi_{k}-\widetilde{\gamma}T_{ab}^{i}\overline{H_{a}}H_{b}\right|^{2}+\sum_{b}\left|\widetilde{\gamma}T_{ab}^{i}\phi^{i}\overline{H_{a}}+\widetilde{\delta}\overline{H_{b}}\right|^{2}$
(2.6) $\displaystyle+$
$\displaystyle\kappa^{2}\left|\mu^{2}-\frac{1}{2}\sum_{i}\phi_{i}^{2}-\frac{\beta}{4M_{*}}d_{ijk}\phi_{i}\phi_{j}\phi_{k}\right|^{2}+\sum_{b}\left|\widetilde{\gamma}T_{ab}^{i}\phi^{i}H_{a}+\widetilde{\delta}H_{b}\right|^{2}\,,$

where $z_{i}$ are the scalar components of the Higgs superfields. The global
SUSY minimum of the above potential lies at the following VEVs of the fields

$\displaystyle\left<S\right>=\left<H_{a}\right>=\left<\overline{H_{a}}\right>=0,$
(2.7)

while $\langle\phi_{i}\rangle$ satisfy the condition

$\sum_{i=1}^{24}\left<\phi_{i}\right>^{2}+\frac{\beta}{2\kappa\,M_{*}}d_{ijk}\left<\phi_{i}\right>\left<\phi_{j}\right>\left<\phi_{k}\right>=2\mu^{2}\,.$
(2.8)

The VEV matrix $\left<\Phi_{i}\right>=\left<\phi_{i}\right>T^{i}$ can be
aligned in the hypercharge ($i=24$) direction using the $SU(5)$ transformation

$\left<\Phi_{24}\right>=\frac{\left<\phi_{24}\right>}{\sqrt{15}}\left(1,1,1,-3/2,-3/2\right)\,,$
(2.9)

such that $\left<\phi_{i}\right>=0,\,\forall\,i\neq 24$ and
$\left<\phi_{24}\right>\equiv\upsilon/\sqrt{2}$, where
$d_{24\,24\,24}=-1/\sqrt{15}$ and $\upsilon$ satisfies

$4\mu^{2}=\upsilon^{2}-\frac{\beta}{2\sqrt{30}\kappa\,M_{*}}\upsilon^{3}\,.$
(2.10)

The $D$-term contribution to the potential,

$V_{D}=\frac{g_{5}^{2}}{2}\sum_{i}\left(f^{ijk}\phi_{j}\phi_{k}^{\dagger}+T^{i}\left(\left|H_{a}\right|^{2}-\left|\overline{H}_{a}\right|^{2}\right)\right)^{2}\,,$
(2.11)

also vanishes for $\phi=\phi^{\ast}$ and $|\overline{H}_{a}|=|H_{a}|$ .

##  3 Inflationary Trajectories

(a) Standard ($\alpha=0$)

(b) Shifted ($\alpha=0.25$)

(c) Shifted ($\alpha=0.3$)

Figure 1: Normalized scalar potential
$\widetilde{V}(w,y)=V(w,y)/\kappa^{2}\mu^{4}$ as a function of dimensionless
variables $w$ and $y$, for different values of parameter $\alpha$. The
standard hybrid inflation potential is reproduced for $\alpha=0$ in panel (a).
For $\alpha\neq 0$, an additional shifted trajectory appears which lies higher
than the standard trajectory for $\alpha<\sqrt{2/27}$ and lower than the
standard trajectory for $\alpha>\sqrt{2/27}$. These shifted hybrid inflation
potentials are shown in panels (b) and (c) for $\alpha=0.25$ and $\alpha=0.3$,
respectively.

The scalar potential in Eq. (2.6) can be rewritten in terms of the
dimensionless variables

$y=\frac{\phi_{24}/\mu}{\sqrt{2}}~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}w=\frac{S/\mu}{\sqrt{2}}\,,$
(3.1)

as

$\widetilde{V}(w,y)=\frac{V(w,y)}{\kappa^{2}\mu^{4}}=\left(1-y^{2}+\alpha
y^{3}\right)^{2}+2w^{2}y^{2}\left(1-\frac{3\alpha y}{2}\right)^{2}\,,$ (3.2)

with $\alpha=\beta\mu/\sqrt{30}\,\kappa M_{*}$. This dimensionless potential
exhibits the following three extrema

$y_{1}=0,$ (3.3) $y_{2}=\frac{2}{3\alpha},$ (3.4)
$\displaystyle\text{and}\quad y_{3}=\frac{1}{3\alpha}$ $\displaystyle+$
$\displaystyle\frac{1}{3\sqrt[3]{2}\alpha}\Bigg{(}\sqrt[3]{2-27\alpha^{2}+\sqrt{\left(2-27\alpha^{2}\right)^{2}+4\left(9\alpha^{2}w^{2}-1\right)^{3}}}$
(3.5) $\displaystyle-$
$\displaystyle\sqrt[3]{-2+27\alpha^{2}+\sqrt{\left(2-27\alpha^{2}\right)^{2}+4\left(9\alpha^{2}w^{2}-1\right)^{3}}}\Bigg{)}\,,$

for any constant value of $w$. The dimensionless potential
$\widetilde{V}(w,y)$ is displayed in Fig. 1 for different values of $\alpha$.
The first extremum $y_{1}$ with $\alpha=0$ corresponds to the standard hybrid
inflation for which $\\{y=0$, $w>1\\}$ is the only inflationary trajectory
that evolves at $w=0$ into the global SUSY minimum at $y=\pm 1$ (panel (a)).
For $\alpha\neq 0$, a shifted trajectory appears at $y=y_{2}$, in addition to
the standard trajectory at $y=y_{1}=0$, which is a local maximum (minimum) for
$w<\sqrt{4/27\alpha^{2}-1}$ ($w>\sqrt{4/27\alpha^{2}-1}$). For
$\alpha<\sqrt{2/27}\simeq 0.27$, this shifted trajectory lies higher than the
standard trajectory (panel (b)). In order to have suitable initial conditions
for realizing inflation along the shifted track, we assume
$\alpha>\sqrt{2/27}$, for which the shifted trajectory lies lower than the
standard trajectory (panel (c)). Moreover, to ensure that the shifted
inflationary trajectory at $y_{2}$ can be realized before $w$ reaches zero, we
require $\alpha<\sqrt{4/27}\simeq 0.38$. Thus, for $0.27<\alpha<0.38$, while
the inflationary dynamics along the shifted track remain the same as for the
standard track, the $SU(5)$ gauge symmetry is broken during inflation, hence
alleviating the magnetic monopole problem. As the inflaton slowly rolls down
the inflationary valley and enters the waterfall regime at
$w=\sqrt{4/27\alpha^{2}-1}$, its fast rolling ends the inflation, and the
system starts oscillating about the vacuum at $w=0$ and $y=y_{3}$ .

###  3.1 Dimension-5 Proton Decay

Figure 2: The proton lifetime for the decay $p\rightarrow K^{+}\bar{\nu}$ as a
function of SUSY breaking scale $M_{\text{SUSY}}$ for different values of
$\tan\beta$. The curves are drawn for $SU(5)$ symmetry breaking scale
$M_{\alpha}$ fixed at $2\times 10^{16}$ GeV with $\alpha=0.3$. The black and
gray dashed lines represent the Super-Kamiokande ($\tau_{p}=5.9\times 10^{33}$
years) bounds and future Hyper-Kamiokande expected bounds ($\tau_{p}=3.2\times
10^{34}$ years) on proton lifetime, respectively.

Upon breaking of the $SU(5)$ symmetry, the last two terms in the $W$ in Eq.
(2.5) can be rewritten as,

$W_{H}\supset\left(\widetilde{\delta}-\frac{3\widetilde{\gamma}\phi_{24}^{0}}{2\sqrt{30}}\right){H}_{u}H_{d}+\left(\widetilde{\delta}+\frac{\widetilde{\gamma}\phi_{24}^{0}}{\sqrt{30}}\right)\overline{H}_{T}H_{T}\equiv\mu_{H}{H}_{u}H_{d}+M_{H_{T}}\overline{H}_{T}H_{T}\,,$
(3.6)

where $\mu_{H}$ is identified with the usual MSSM $\mu$-parameter, which is
taken to be of the order of TeV scale, and $M_{H_{T}}$ is the color-triplet
mass parameter given by

$\displaystyle
M_{H_{T}}\simeq\frac{10\widetilde{\gamma}M_{\alpha}}{3\sqrt{30\left(\frac{4}{27}-\alpha^{2}\right)}}\,.$
(3.7)

The above mass-splitting between the Higgs doublet and triplet can be
addressed by fine tuning of the parameters $\tilde{\delta}$ and
$\tilde{\gamma}$, such that

$\widetilde{\delta}\simeq\frac{3\widetilde{\gamma}\phi_{24}^{0}}{2\sqrt{30}}\,.$

However, the fermionic components of $H_{T}$, the color-triplet Higgsinos,
contribute to the proton decay via a dimension-5 operator, which typically
dominates the gauge boson mediated dimension-6 operators.

The proton lifetime for the decay $p\rightarrow K^{+}\bar{\nu}$ can be
approximated by the following formula [35],

$\tau_{p}\simeq 4\times
10^{31}\times\sin^{4}2\beta\left(\frac{M_{\text{SUSY}}}{1~{}\text{TeV}}\right)^{2}\left(\frac{M_{H_{T}}}{10^{16}~{}\text{GeV}}\right)^{2}\text{yrs}\,,$
(3.8)

The proton lifetime $\tau_{p}$ is shown in Fig. 2 as a function of
$M_{\text{SUSY}}$ for different values of $\tan\beta$, using Eq. (2.10), (3.7)
and (3.8). The curves are drawn for the $SU(5)$-breaking scale $M_{\alpha}$
fixed at $2\times 10^{16}$ GeV, with $\alpha=0.3$. It can be seen that the
proton lifetime is consistent with the experimental bound, $\tau_{p}>5.9\times
10^{33}$ years, from Super-Kamiokande [36], for $M_{\text{SUSY}}\gtrsim 12.5$
TeV and can be observed by Hyper-Kamiokande [37]. Our model thus remains safe
from proton decay even while adequately addressing the doublet-triplet
splitting problem.

##  4 No-Scale Shifted Hybrid Inflation

The Kähler potential with a no-scale structure, after including contributions
from the relevant fields in the model, takes the following form

$K=-3m_{P}^{2}\log\Delta\,,$ (4.1)

with

$\Delta=T+T^{\ast}-\frac{\operatorname{Tr}(\Phi^{2})+S^{\dagger}S+\overline{H}H+\nu^{c}\nu^{c}}{3m_{P}^{2}}+\gamma\frac{(S^{\dagger}S)^{2}}{3m_{P}^{4}}+\zeta\frac{\operatorname{Tr}(\Phi^{4})}{3m_{P}^{4}}+\sigma\frac{(S^{\dagger}S)^{3}}{3m_{P}^{6}}\,,$
(4.2)

where $\gamma$, $\zeta$ and $\sigma$ are dimensionless couplings, and $T$ and
$T^{*}$ are Kähler complex moduli fields given as $T=\left(u+i\,v\right)$, so
that $T+T^{\ast}=2u$ with $u=1/2$. The $F$-term SUGRA scalar potential is
given by

$V_{F}=e^{K/m_{P}^{2}}\left[\left(K_{i\bar{j}}\right)^{-1}\left(D_{z_{i}}W\right)\left(D_{z_{j}}W\right)^{*}-\frac{3\arrowvert
W\arrowvert^{2}}{m_{P}^{2}}\right]\,,$ (4.3)

where we have defined

$D_{z_{i}}W\equiv\frac{\partial W}{\partial z_{i}}+\frac{\partial K}{\partial
z_{i}}\frac{W}{m_{P}^{2}},\ \ \ \
K_{i\bar{j}}\equiv\frac{\partial^{2}K}{\partial z_{i}\partial z_{j}^{*}}\,,$
(4.4)

and $D_{z_{i}^{*}}W^{*}=\left(D_{z_{i}}W\right)^{*}.$

Since SUSY is temporarily broken along the inflationary trajectory, the
radiative corrections to the above $V_{F}$ along with the soft SUSY-breaking
potential $V_{\text{Soft}}$ can lift its flatness, while also providing the
necessary slope for driving inflation. For a detailed discussion on the mass
spectrum of the model, see Ref. [28]. The effective contribution of the one-
loop radiative corrections can be calculated using the Coleman-Weinberg
formula [38] as

$V_{1\text{-loop}}=\kappa^{2}M_{\alpha}^{4}\left(\frac{\kappa^{2}}{16\pi^{2}}[F(M_{\alpha}^{2},x^{2})+11\times
25F(5M_{\alpha}^{2},5x^{2})]\right)\,,$ (4.5)

where, for a non-canonically normalized field $x\equiv|S|/M_{\alpha}$,

$F(M_{\alpha}^{2},x^{2})=\frac{1}{4}\left((x^{4}+1)~{}\text{ln}\left(\frac{x^{4}-1}{x^{4}}\right)+2x^{2}~{}\text{ln}\frac{(x^{2}+1)}{(x^{2}-1)}+2~{}\text{ln}\left(\frac{\kappa^{2}M_{\alpha}^{2}x^{2}}{Q^{2}}\right)-3\right)\,,$
(4.6)

with $M_{\alpha}^{2}=\mu^{2}\left(\frac{4}{27\alpha^{2}}-1\right)$, and $Q$
being the renormalization scale.

As for the breaking of SUSY, in this study we consider the scenario wherein it
is communicated gravitationally from the hidden sector to the observable
sector. Following [39], the soft SUSY-breaking potential thus reads

$V_{\text{Soft}}\simeq am_{3/2}\kappa
M_{\alpha}^{3}x+M_{S}^{2}M_{\alpha}^{2}x^{2}+\frac{8M_{\phi}^{2}M_{\alpha}^{2}}{9\alpha^{2}\left(4/27\alpha^{2}-1\right)}\,,$
(4.7)

with

$a=2|A-2|\cos\left(\arg S+\arg|A-2|\right)\,.$

Here $m_{3/2}$ is the gravitino mass, $A$ is the complex coefficient of the
trilinear soft SUSY-breaking terms, $a$ and $M_{S}$ are the coefficients of
the soft linear and mass terms for $S$, respectively, while $M_{\phi}$ is the
soft mass parameter for the $\phi$ field. The complete effective scalar
potential during inflation is then given as

$\displaystyle V(x)$ $\displaystyle\simeq$ $\displaystyle
V_{F}+V_{1\text{-loop}}+V_{\text{Soft}}$ (4.8) $\displaystyle\simeq$
$\displaystyle\kappa^{2}M_{\alpha}^{4}\left[\left(1+2\left(\left(2\gamma-\frac{1}{3}\right)x^{2}-\frac{1}{\left(1-27\alpha^{2}/4\right)}\right)\left(\frac{M_{\alpha}}{m_{P}}\right)^{2}\right.\right.$
$\displaystyle+$
$\displaystyle\left.\left.\left(\frac{16\left(1-2\zeta\right)}{3\left(4-27\alpha^{2}\right)^{2}}+\frac{8x^{2}\left(1-12\gamma\right)}{3\left(4-27\alpha^{2}\right)}-x^{4}\left(16\gamma^{2}+\frac{14}{3}\gamma+\frac{9\sigma}{2}-\frac{5}{9}\right)\right)\left(\frac{M_{\alpha}}{m_{P}}\right)^{4}\right)\right.$
$\displaystyle+$
$\displaystyle\left.\frac{\kappa^{2}}{16\pi^{2}}\left(F(M_{\alpha}^{2},x^{2})+11\times
25\,F(5M_{\alpha}^{2},5\,x^{2})\right)\right]$ $\displaystyle+$ $\displaystyle
am_{3/2}\kappa
M_{\alpha}^{3}x+M_{S}^{2}M_{\alpha}^{2}x^{2}+\frac{8M_{\phi}^{2}M_{\alpha}^{2}}{9\alpha^{2}\left(4/27\alpha^{2}-1\right)}\,,$

where the $F$-term scalar potential along the shifted trajectory in the
$D$-flat direction has been obtained from Eq. (4.3). Finally, the action of
our model is given by

$\begin{split}\mathcal{A}=\int
dx^{4}\sqrt{-g}\left[\frac{m^{2}_{p}}{2}\mathcal{R}-K^{i}_{j}\partial_{\mu}x^{i}\partial^{\mu}x^{j}-V({x})\right]\,,\end{split}$
(4.9)

where $\mathcal{R}$ is the Ricci scalar. Introducing a canonically normalized
field $z$ satisfying

$\displaystyle\left(\frac{dz}{dx}\right)^{2}=\frac{\partial^{2}K}{\partial
S^{\dagger}\partial S}\quad\text{with}\quad
S^{\dagger}=S=xM_{\alpha}\quad\text{and}\quad\phi_{24}=\frac{\sqrt{2}M_{\alpha}y_{2}}{\left(4/27\alpha^{2}-1\right)}\,,$
(4.10)

requires modification of the slow-roll parameters, as shown later in section
6.

##  5 Reheating with Non-thermal Leptogenesis

A complete inflationary scenario should be followed by a successful reheating
that satisfies the constraint, $T_{r}<10^{9}$ GeV, from gravitino cosmology,
and generates the observed BAU. At the end of the inflationary epoch, the
system falls towards the SUSY vacuum and undergoes damped oscillations about
it. The inflaton (oscillating system) consists of two complex scalar fields
$S$ and $\theta=(\delta\phi+\delta\overline{\phi})/\sqrt{2}$. The canonical
normalized inflaton field can be defined as

$\delta\widetilde{\phi}=\langle J_{0}\rangle\delta\phi\,,$ (5.1)

with

$\delta\phi=\phi-\langle\phi_{24}\rangle_{y_{3}},\quad\text{and}\quad
J_{0}\equiv\frac{dz}{dx}\Big{|}_{\text{Minimum}}=\left(1-\frac{\langle\phi_{24}\rangle^{2}_{y_{3}}}{6m_{P}^{2}}+\frac{\zeta\langle\phi_{24}\rangle^{4}_{y_{3}}}{12m_{P}^{4}}\right)^{-1/2}\,,$
(5.2)

where

$\langle\phi_{24}\rangle_{y_{3}}\equiv\frac{\sqrt{2}M_{\alpha}y_{3}}{\left(4/27\alpha^{2}-1\right)}\,.$
(5.3)

The decay of the inflaton field, with its mass given by

$\displaystyle\widetilde{m}_{\text{inf}}^{2}$ $\displaystyle=$
$\displaystyle\frac{d^{2}V(\phi)}{dz^{2}}=\dfrac{1}{J_{0}}\left(\frac{d^{2}V(\phi)}{d\phi^{2}}\right)=\dfrac{m_{\text{inf}}^{2}}{J_{0}}=\frac{2\kappa^{2}M_{\alpha}^{2}y_{3}^{2}\left(1-3\alpha
y_{3}/2\right)^{2}}{J_{0}\left(4/27\alpha^{2}-1\right)}\,,$ (5.4)

gives rise to the radiation in the Universe. The inflaton decay into the
higginos and the right-handed neutrinos is induced by the superpotential
terms,

$W\supset\widetilde{\gamma}\overline{H}\Phi
H+\lambda_{ij}\frac{\operatorname{Tr}(\Phi^{2})\nu_{i}\nu_{j}}{M_{*}}\,.$
(5.5)

The Lagrangian terms relevant for inflaton decay into neutrinos are[40, 41],

$\displaystyle\mathcal{L}_{\text{inf}\rightarrow\nu_{i}^{c}\nu_{i}^{c}}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}e^{K/2m_{P}^{2}}\left(K_{\phi}W_{\phi,\nu_{i}^{c}\nu_{j}^{c}}+W_{\phi,\nu_{i}^{c}\nu_{j}^{c}}-2\Gamma^{\nu_{i}^{c}}_{\phi,\nu_{j}^{c}}W_{\phi,\nu_{i}^{c}\nu_{j}^{c}}\right)\,\phi\nu_{i}^{c}\nu_{i}^{c}+{\rm
h.c.}$ (5.6) $\displaystyle\to$ $\displaystyle-
g_{\nu_{ij}^{c}}\left(\delta\widetilde{\phi}\nu_{i}^{c}\nu_{j}^{c}+h.c\right)+...\,,$

where $W_{\phi,\nu^{c}_{i}\nu^{c}_{j}}$ is the second derivative of $W$ with
respect to $\nu^{c}$, and $g_{\nu_{ij}^{c}}$ is the effective inflaton-
neutrino coupling, defined as

$g_{\nu_{ij}^{c}}=\frac{\sqrt{2}\lambda_{ij}M_{\alpha}y_{3}J_{0}^{4}}{M_{*}\left(4/27\alpha^{2}-1\right)^{1/2}}\,.$
(5.7)

The decay width in the flavor-diagonal basis is thus given by [42]

$\displaystyle\Gamma_{\text{inf}\rightarrow\nu^{c}\nu^{c}}$ $\displaystyle=$
$\displaystyle\dfrac{1}{64\pi}g_{\nu_{i}^{c}}^{2}\widetilde{m}_{\text{inf}}\left(1-\frac{4M_{\nu_{i}^{c}}^{2}}{\widetilde{m}_{\text{inf}}^{2}}\right)^{3/2},$
(5.8) $\displaystyle=$
$\displaystyle\dfrac{J_{0}^{8}}{16\pi}\widetilde{m}_{\text{inf}}\left(\dfrac{M_{\nu_{i}^{c}}}{\quad\langle\phi_{24}\rangle_{y_{3}}}\right)^{2}\left(1-\frac{4M_{\nu_{i}^{c}}^{2}}{\widetilde{m}_{\text{inf}}^{2}}\right)^{3/2}\,,$

where $M_{\nu_{i}^{c}}$ represent the eigenvalues of the soft neutrino mass
matrix. Note that this term violates the lepton number by two units, $\Delta
L=2$. In addition, the Dirac mass terms for the neutrinos are obtained from
$W\supset
y_{ij}^{(\nu)}\,\nu_{i}^{c}\,\bar{f}_{j}\,H\to{m_{\nu_{D}}}_{ij}\nu_{i}\nu_{j}^{c}$
upon EW symmetry-breaking. The small neutrino masses, consistent with the
results from the neutrino oscillation experiments, are obtained by integrating
out the heavy right-handed neutrinos, so that

${m_{\nu_{D}}}_{\alpha\beta}=-\sum_{i}{y^{(\nu)}}_{i\alpha}{y^{(\nu)}}_{i\beta}\frac{v_{u}^{2}}{M_{\nu_{i}^{c}}}\,.$
(5.9)

The Dirac mass matrix above can be diagonalised by a unitary matrix $U_{\alpha
i}$ as ${m_{\nu_{D}}}_{\alpha\beta}=U_{\alpha i}U_{\beta i}m_{\nu_{D}}$, with
$m_{\nu_{D}}={\rm diag}(m_{\nu_{1}},m_{\nu_{2}},m_{\nu_{3}})$ .

The Lagrangian relevant for the inflaton decay to $H$ and $\overline{H}$ is

$\displaystyle\mathcal{L}_{\text{inf}\rightarrow H\overline{H}}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}e^{K/2m_{P}^{2}}\left(K_{\phi}W_{H\overline{H}}+W_{\phi
H\overline{H}}-2\Gamma^{H}_{\phi\overline{H}}W_{H\overline{H}}\right)^{*}m_{\text{inf}}\,\phi
H\overline{H}+{\rm h.c.}$ (5.10) $\displaystyle\to$ $\displaystyle-
g_{H}\widetilde{m}_{\text{inf}}\left(\delta\widetilde{\phi}H\overline{H}+h.c\right)+...\,,$

with the effective coupling

$g_{H}=\dfrac{\widetilde{\gamma}}{2}J_{0}^{3}\,.$ (5.11)

This leads to the partial decay width [42],

$\displaystyle\Gamma_{\text{inf}\rightarrow
H\overline{H}}=\frac{g_{H}^{2}}{8\pi}\widetilde{m}_{\text{inf}}\,.$ (5.12)

The reheating temperature of the Universe depends on the combined decay width
$\Gamma=\Gamma_{\text{inf}\rightarrow\nu_{i}^{c}\nu_{i}^{c}}+\Gamma_{\text{inf}\rightarrow\overline{H}H}$
as

$\displaystyle
T_{r}=\left(\frac{72}{5\pi^{2}g^{\ast}}\right)^{1/4}\sqrt{\Gamma m_{P}}\,.$
(5.13)

Assuming a standard thermal history, the number of $e$-folds, $N_{0}$, can be
written in terms of the $T_{r}$ as [43]

$N_{0}=54+\frac{1}{3}\ln\Big{(}\frac{T_{r}}{10^{9}\text{
GeV}}\Big{)}+\frac{2}{3}\ln\Big{(}\frac{V(x)^{1/4}}{10^{15}\text{
GeV}}\Big{)}\,.$ (5.14)

The ratio of the lepton number density to the entropy density in the limit
$T_{r}<M_{\nu_{1}^{c}}\leq m_{\text{inf}}/2\leq M_{\nu_{2,3}^{c}}$ is defined
as

$\frac{n_{L}}{s}\sim\frac{3}{2}\frac{\Gamma_{\text{inf}\rightarrow\nu_{1}^{c}\nu_{1}^{c}}}{\Gamma}\frac{T_{r}}{\widetilde{m}_{\text{inf}}}\epsilon_{cp}\,,$
(5.15)

where $\epsilon_{cp}$ is the CP-asymmetry factor, which is generated from the
out-of-equilibrium decay of the lightest right-handed neutrino. For a normal
hierarchical pattern of the light neutrino masses, this factor becomes [44]

$\epsilon_{cp}=\frac{3}{8\pi}\frac{M_{\nu_{1}^{c}}m_{\nu_{3}}}{v_{u}^{2}}\delta_{\rm
eff}\,,$ (5.16)

where $m_{\nu_{3}}$ is the mass of the heaviest light neutrino, $v_{u}$ is the
VEV of the Higgs doublet $H_{u}$, and $\delta_{\rm eff}$ is the CP-violating
phase.

The lepton asymmetry from experimental observations is [45],

$\displaystyle\mid n_{L}/s\mid\approx\left(2.67-3.02\right)\times 10^{-10}\,.$
(5.17)

In the numerical estimates discussed below, we take $m_{\nu_{3}}=0.05$ eV,
$|\delta_{\rm eff}|\leq 1$, $v_{u}=174$ GeV, while assuming large $\tan\beta$.
A successful baryogenesis is usually generated through the sphaleron process
[46, 47], where an initial lepton asymmetry, given by

$\frac{n_{L}}{s}\lesssim 3\times
10^{-10}\left(\frac{\Gamma_{\text{inf}\rightarrow\nu^{c}\nu^{c}}}{\Gamma}\right)\left(\frac{T_{r}}{\widetilde{m}_{\text{inf}}}\right)\left(\frac{M_{\nu_{1}^{c}}}{10^{6}\text{
GeV}}\right)\left(\frac{m_{\nu_{3}}}{0.05\text{ eV}}\right)\delta_{\rm eff},$
(5.18)

is partially converted into baryon asymmetry as $n_{B}/s=-0.35n_{L}/s$.

##  6 Numerical analysis

###  6.1 Inflationary Predictions

The inflationary slow-roll parameters can be expressed as

$\displaystyle\epsilon=\dfrac{1}{4}\left(\frac{m_{P}}{M_{\alpha}}\right)^{2}\left(\frac{V^{\prime}\left(x\right)}{V(x)z^{\prime}\left(x\right)}\right)^{2},,\quad\eta=\dfrac{1}{2}\left(\frac{m_{P}}{M_{\alpha}}\right)^{2}\left(\frac{V^{\prime\prime}\left(x\right)}{V(x)\left(z^{\prime}\left(x\right)\right)^{2}}-\frac{V^{\prime}\left(x\right)z^{\prime\prime}\left(x\right)}{V(x)\left(z^{\prime}\left(x\right)\right)^{3}}\right)\,,$
$\displaystyle\begin{aligned} \text{and}\quad
s^{2}=&~{}\dfrac{1}{4}\left(\frac{m_{P}}{M_{\alpha}}\right)^{4}\left(\frac{V^{\prime}\left(x\right)}{V(x)z^{\prime}\left(x\right)}\right)\Bigg{(}\frac{V^{\prime\prime\prime}\left(x\right)}{V(x)\left(z^{\prime}\left(x\right)\right)^{3}}-3\frac{V^{\prime\prime}\left(x\right)z^{\prime\prime}\left(x\right)}{V(x)\left(z^{\prime}\left(x\right)\right)^{4}}\\\
+&~{}3\frac{V^{\prime}\left(x\right)\left(z^{\prime\prime}\left(x\right)\right)^{2}}{V(x)\left(z^{\prime}\left(x\right)\right)^{5}}-\frac{V^{\prime}\left(x\right)z^{\prime\prime\prime}\left(x\right)}{V(x)\left(z^{\prime}\left(x\right)\right)^{4}}\Bigg{)}\,,\end{aligned}$
(6.1)

where a prime denotes a derivative with respect to $x$. In terms of these
parameters, we obtain

$r\simeq 16\epsilon,\quad n_{s}\simeq
1+2\eta-6\epsilon,\quad\text{and}\quad\frac{dn_{s}}{d\ln k}\simeq
16\epsilon\eta-24\epsilon^{2}+2s^{2}\,,$ (6.2)

where the last quantity gives the running of $n_{s}$. The number of $e$-folds
is given by

$N_{0}=2\left(\frac{M_{\alpha}}{m_{P}}\right)^{2}\int_{x_{e}}^{x_{0}}\left(\frac{V\left(x\right)z^{\prime}(x)^{2}}{V^{\prime}(x)}\right)dx\,,$
(6.3)

where $x_{0}$ is the field value at the pivot scale and $x_{e}$ is the field
value at the end of inflation (i.e., when $\epsilon=1$). Finally, the
amplitude of curvature perturbation $\Delta_{R}$ is obtained as

$\Delta_{R}^{2}=\frac{V\left(x\right)}{24\pi^{2}\epsilon\left(x\right)}\,.$
(6.4)

The results of our numerical calculations are displayed in Figs. 3 and 4,
which show the ranges of $\kappa$, $S_{0}$, and $T_{r}$ in the
$\tilde{\gamma}-\sigma$ plane. The color map in Fig. 3 corresponds to $r$ in
the top two panels and to the coupling $\gamma$ in the bottom panel. The
light-shaded region in all the panels implies that the inflaton predominantly
decays into the neutrinos, whereas the dark-shaded region represents a
Higgsino-dominant decay. In obtaining these results, we have used up to
second-order approximation on the slow-roll parameters, and have set
$\zeta=0$, $\alpha=0.3$, and $x_{e}=1$. Moreover, we have fixed the $SU(5)$
gauge symmetry-breaking scale $M_{\alpha}$ to $M_{\text{GUT}}=2\times 10^{16}$
GeV and $n_{s}$ to the central value (0.9655) of the bound from Planck’s data.
Finally, the soft masses have been fixed at $12.5$ TeV, in order to avoid the
dimension-5 proton decay. We restrict ourselves to the parameter region with
the largest possible values of $r$ observable by the near-future experiments
highlighted below.

In our analysis the soft SUSY contributions to the inflationary trajectory are
suppressed, while the radiative and SUGRA corrections, parametrised by
$\gamma$ and $\sigma$, play the dominant role. To keep the SUGRA expansion
under control we impose $S_{0}\leq m_{P}$. We further require
$\sigma\gtrsim-1$ and $\\{2\times 10^{6}\lesssim T_{r}\lesssim 2\times
10^{9}\\}$ GeV. These constraints appear in Figs. 3 and 4 as the boundaries of
the allowed region in the $\tilde{\gamma}-\sigma$ plane.

###  6.2 Observable Primordial Gravitational Waves

Figure 3: Variation of $\kappa$, $S_{0}$ and reheat temperature $T_{r}$ in
$\tilde{\gamma}-\sigma$ plane. The color map displays the range of tensor to
scalar ratio $r$ in the top two panels and $\gamma$ in the bottom panel. The
light shaded region corresponds to the neutrino dominant channel whereas, the
dark shaded region corresponds to Higgsino dominant channel.

The tensor-to-scalar ratio $r$ is the canonical measure of primordial
gravitational waves and the next-generation experiments are gearing up to
measure it. One of the highlights of PRISM [48] is to detect $r$ as low as
$5\times 10^{-4}$, and a major goal of LiteBIRD [49] is to attain a
measurement of $r$ within an uncertainty of $\delta r=0.001$. Furthermore, the
CORE [50] experiment is forecast to have sensitivity to $r$ as low as
$10^{-3}$, and PIXIE [51] aims to measure $r<10^{-3}$ at the 5$\sigma$ level.
Other future missions include CMB-S4 [52], which has the goal of detecting
$r\gtrsim 0.003$ at greater than 5$\sigma$ or, in the absence of a detection,
reaching an upper limit of $r<0.001$ at the 95$\%$ confidence level, and PICO
[53], which aims to detect $r=5\times 10^{-4}$ at 5$\sigma$.

The explicit dependence of $r$ on $\kappa$ and the $SU(5)$ symmetry-breaking
scale $M_{\alpha}$ is given by the following approximate relation obtained by
using the normalization constraint on $\Delta_{R}$

$r\simeq\left(\frac{2\kappa^{2}}{3\pi^{2}\Delta_{R}^{2}}\right)\left(\frac{M_{\alpha}}{m_{P}}\right)^{4}\,.$
(6.5)

One can see that, for fixed $M_{\alpha}$, large $r$ values should be obtained
with large $\kappa$, which is indeed the case. For $M_{\alpha}\simeq 2\times
10^{16}$ GeV, as used in our analysis, it can readily be checked that the
above equation gives $r\sim 0.0014$ for $\kappa\sim 0.1$, and $r\sim 6\times
10^{-7}$ for $\kappa\sim 0.002$. These approximate values are very close to
the actual values obtained in our numerical calculations, and the larger
tensor modes can be detected by future experiments.

Figure 4: Variation of reheat temperature $T_{r}$ in $\tilde{\gamma}-\sigma$
plane. The color map displays the range of right handed neutrino mass $M_{1}$
in the left panel and inflaton mass $m_{\text{inf}}$ in the right panel. The
light shaded region corresponds to the neutrino dominant channel whereas the
dark shaded region corresponds to Higgsino dominant channel.

In the leading order slow-roll approximation, the spectral index $n_{s}$ and
tensor to scalar ratio $r$ are given by

$\displaystyle n_{s}$ $\displaystyle\simeq$ $\displaystyle
1+\left(\frac{m_{P}}{M_{\alpha}}\right)^{2}\Bigg{[}-\frac{z^{{}^{\prime}}(x_{0})}{z(x_{0})^{3}}\bigg{(}4\left(\frac{M_{\alpha}}{m_{P}}\right)^{2}x_{0}\left(2\gamma-\frac{1}{3}\right)$
(6.6) $\displaystyle+$ $\displaystyle
4\left(\frac{M_{\alpha}}{m_{P}}\right)^{4}x_{0}^{3}\left(\frac{14}{3}\gamma+16\gamma^{2}-\frac{5}{9}+\frac{9}{2}\sigma\right)+\frac{275\kappa^{2}}{16\pi^{2}}F^{{}^{\prime}}(5x_{0})\bigg{)}$
$\displaystyle+$
$\displaystyle\frac{1}{z(x_{0})^{2}}\bigg{(}4\left(\frac{M_{\alpha}}{m_{P}}\right)^{2}\left(2\gamma-\frac{1}{3}\right)+12\left(\frac{M_{\alpha}}{m_{P}}\right)^{4}x_{0}^{2}\left(\frac{14}{3}\gamma+16\gamma^{2}-\frac{5}{9}+\frac{9}{2}\sigma\right)$
$\displaystyle+$
$\displaystyle\frac{275\kappa^{2}}{16\pi^{2}}F^{{}^{\prime\prime}}(5x_{0})\bigg{)}\Bigg{]}\Bigg{[}1+\left(\frac{m_{P}}{M_{\alpha}}\right)^{2}x_{0}^{2}\left(2\gamma-\frac{1}{3}\right)+\frac{275\kappa^{2}}{16\pi^{2}}F(5x_{0})\Bigg{]}^{-1}\,,$
$\displaystyle\text{and}\quad r$ $\displaystyle\simeq$ $\displaystyle
3\left(\frac{m_{P}}{M_{\alpha}}\right)^{2}\frac{1}{z(x_{0})^{2}}\Bigg{[}4\left(\frac{M_{\alpha}}{m_{P}}\right)^{4}x_{0}\left(2\gamma-\frac{1}{3}\right)$
(6.7) $\displaystyle+$ $\displaystyle
4\left(\frac{M_{\alpha}}{m_{P}}\right)^{4}x_{0}^{3}\left(\frac{14}{3}\gamma+16\gamma^{2}-\frac{5}{9}+\frac{9}{2}\sigma\right)+\frac{275\kappa^{2}}{16\pi^{2}}F^{{}^{\prime}}(5x_{0})\Bigg{]}\,,$

with

$z(x_{0})\simeq\sqrt{1+\left(\frac{M_{\alpha}}{m_{P}}\right)^{2}x_{0}^{2}}\,.$

Solving these two equations simultaneously, we obtain $\gamma\sim 0.17027$,
$\sigma\sim-0.15726$ for $S_{0}\simeq m_{P}$, $n_{s}\simeq 0.9655$, $r\simeq
0.0013$. Similarly, for $S_{0}\simeq 0.027m_{P}$, $n_{s}\simeq 0.9655$,
$r\simeq 7.5\times 10^{-7}$ we obtain $\gamma\sim 0.1683$, $\sigma\sim-1$.
Both these estimates are in good agreement with our numerical results
displayed in Figs. 3 and 4. Thus, for couplings $0.1678\lesssim\gamma\lesssim
0.1707$ and $-1\lesssim\sigma\lesssim-0.1525$, we obtain $n_{s}$ compatible
with the Planck constraints and $r$ in the $7.5\times 10^{-7}-1.5\times
10^{-3}$ range. At the same time, given our chosen values of $M_{\alpha}$ and
$M_{\text{SUSY}}$ that yield these results, the proton lifetime lies above the
lower bound from the Super-K experiment and should be testable by the future
Hyper-K experiment.

Fig. 4 is of particular importance in connection with the non-thermal
leptogenesis and shows the variation of $T_{r}$ in the $\tilde{\gamma}-\sigma$
plane. The color map displays the range of right handed neutrino mass
$(4.0\times 10^{13}\lesssim M_{\nu_{1}}\lesssim 1.6\times 10^{15})$ GeV in the
left panel and inflaton mass $(8.3\times
10^{13}\lesssim\widetilde{m}_{\text{inf}}\lesssim 3.3\times 10^{15})$ GeV in
the right panel. Imposing the kinematic condition,

$\frac{\widetilde{m}_{\text{inf}}}{M_{\nu_{1}}}\geq 2\,,$ (6.8)

and using $n_{L}/s=3\times 10^{-10}$, Eq. (5.18) can be written as,

$\frac{\Gamma_{\text{inf}\rightarrow\nu^{c}\nu^{c}}}{\Gamma_{\text{inf}\rightarrow\bar{H}H}}=\left(\frac{T_{r}}{2\times
10^{6}\text{ GeV}}\right)\,,$ (6.9)

which implies $T_{r}>4\times 10^{6}$ GeV for the Higgsino-dominant decay of
the inflaton, and $T_{r}<4\times 10^{6}$ GeV for the neutrino-dominant
channel.

###  6.3 BBN Constraints on Reheating Temperature and Gravitino Cosmology

Another important constraint on $T_{r}$ comes from gravitino cosmology, as it
depends on the SUSY-breaking mechanism and the gravitino mass. As noted in
[18, 54, 55, 56], one may consider the case of
$a)$ a stable gravitino as the LSP;
$b)$ an unstable long-lived ($\tilde{\tau}\gtrsim 1$ sec) gravitino with
$m_{3/2}<25$ TeV;
$c)$ an unstable short-lived ($\tilde{\tau}<1$ sec) gravitino with
$m_{3/2}>25$ TeV.

In models based on SUGRA, the relic abundance of a stable gravitino LSP is
given [57, 58, 59] by 555Taking into account only the dominant QCD
contributions to the gravitino production rate. In principle there are extra
contributions descending from the EW sector, as mentioned in [58] and recently
revised in [59].

$\Omega_{3/2}h^{2}=0.08\left(\frac{T_{r}}{10^{10}\;\text{GeV}}\right)\left(\frac{m_{3/2}}{1\;\text{TeV}}\right)\left(1+\frac{m_{\tilde{g}}^{2}}{3m_{3/2}^{2}}\right)~{},$
(6.10)

Figure 5: Possibility of the gravitino as the LSP for different values of its
mass, $m_{3/2}=(10,100)$ TeV, and the corresponding gluino masses.

where $m_{\tilde{g}}$ is the gluino mass, $h$ is the present Hubble parameter
in units of $100$ km $\text{sec}^{-1}\,\text{Mpc}^{-1}$, and
$\Omega_{3/2}=\rho_{3/2}/\rho_{c}$666$\rho_{3/2}$ and $\rho_{c}$ are the
gravitino energy density and the critical energy density of the present-day
Universe, respectively.. A stable LSP gravitino requires
$m_{\tilde{g}}>m_{3/2}$, while current LHC bounds on the gluino mass are
around $2.3$ TeV [60]. It follows from Eq. (6.10) that the overclosure limit,
$\Omega_{3/2}h^{2}<1$, puts a severe upper bound on $T_{r}$, depending on
$m_{3/2}$. Fig. 5 shows the gluino mass as a function of $T_{r}$ for
$m_{3/2}=(10,\,100)$ TeV, with the condition that $\Omega_{3/2}h^{2}$ does not
exceed the observed DM relic abundance, $\Omega h^{2}\leq 0.126$ [2], of the
Universe. We see that $(m_{\tilde{g}}>m_{3/2})$ is satisfied for the entire
range of the plotted parameter space, implying that the gravitino is
consistently realized as the LSP in the model, and acts as a viable DM
candidate.

When the gravitino is the next-to-LSP instead, the role of the LSP (and hence
the DM) can be played by the lightest neutralino, $\tilde{\chi}_{1}^{0}$,
which has two origins: thermal and non-thermal relic. Its thermal production
consists of the standard freeze-out mechanism of the weakly interacting
massive particles (WIMPs), whereas the non-thermal production proceeds via the
decay of the gravitino, itself produced during the reheating process [61, 62].
However, since the density of the thermal relic is strongly model-dependent,
we do not take into account its effect in calculating the density parameter
here.

An unstable gravitino can be long-lived or short-lived [63, 64]. For
$m_{3/2}<25$ TeV, a gravitino lifetime of $\tilde{\tau}\gtrsim 1\;\text{sec}$
[65, 66] can be sufficiently long to result in the cosmological gravitino
problem [63]. The fast decay of gravitino may affect the abundances of the
light nuclei, thereby ruining the success of the big-bang nucelosynthesis
(BBN) theory. To avoid this problem, one has to take into account the BBN
bounds on $T_{r}$, which are conditioned on $m_{3/2}$ in gravity mediated
SUSY-breaking as

$\begin{split}T_{r}&\lesssim 2\times 10^{9}\;{\rm GeV}\quad{\rm for}\quad
m_{3/2}\gtrsim 10\;{\rm TeV}\,.\end{split}$ (6.11)

A long-lived gravitino scenario is therefore consistent with the BBN bounds
(6.11) for the entire $(12.5\leq m_{3/2}\leq 25)$ TeV range, given the $T_{r}$
in our model.

For a short-lived gravitino, the above $T_{r}$ bounds from BBN do not apply,
and it can decay into the $\tilde{\chi}_{1}^{0}$ LSP, with the resultant
abundance given by

$\Omega_{\tilde{\chi}_{1}^{0}}h^{2}\simeq 2.8\times 10^{11}\times
Y_{3/2}\left(\frac{m_{\tilde{\chi}_{1}^{0}}}{1\;\text{TeV}}\right)\,,$ (6.12)

where the gravitino yield is defined as

$Y_{3/2}\simeq 2.3\times
10^{-12}\left(\frac{T_{r}}{10^{10}\;\text{GeV}}\right)\,.$ (6.13)

Requiring $\Omega_{\tilde{\chi}_{1}^{0}}h^{2}\lesssim 0.126$ leads to the
relation

$\displaystyle m_{\tilde{\chi}_{1}^{0}}\gtrsim
19.6\left(\frac{10^{11}\;\text{GeV}}{T_{r}}\right)~{}.$ (6.14)

The prediction of $\\{2\times 10^{6}\lesssim T_{r}\lesssim 2\times 10^{9}\\}$
GeV by our model with gravity-mediated SUSY-breaking easily satisfies the
$m_{\tilde{\chi}_{1}^{0}}\geq 18$ GeV [67] limit, thus making the short-lived
gravitino a viable scenario also.

##  7 Summary

To summarize, we have investigated various cosmological implications of a
generic model based on the $SU(5)$ gauge symmetry formulated in the framework
of no-scale supergravity, highlighting the issues of dimension-5 proton decay,
inflation, gravitino, as well as baryogenesis via non-thermal leptogenesis.
The breaking of $SU(5)$ gauge symmetry suffers from the magnetic monopole
problem. Employing the shifted extension of hybrid inflation the monopole
density is diluted and remain within the observable limit. The model yields
large values of the tensor-to-scalar ratio ($r\sim 0.0015$) that are
potentially measurable by future experiments and favors values of the scalar
tilt $n_{s}$ that are consistent with current constraints. Moreover, the model
avoids rapid proton decay via dimension-5 operators and also provides for
baryogenesis via non-thermal leptogenesis, with low reheating temperature
($2\times 10^{6}\lesssim T_{r}\lesssim 2\times 10^{9}$) GeV consistent with
gravitino cosmology.

## Acknowledgements

The authors are thankful to George K. Leonatris and Mansoor Ur Rehman for
helpful discussions. WA is partially supported by the Program for Excellent
Talents in Hubei Polytechnic University (21xjz22R, 21xjz21R, 21xjz20R) and SM
would like to acknowledge support from the ICTP through the Associates
Programme (2022-2027).

## References

  * [1] K. A. Olive, Phys. Rept. 190 (1990) 307; A. D. Linde, Particle Physics and Inflationary Cosmology (Harwood, Chur, Switzerland, 1990); D. H. Lyth and A. Riotto, Phys. Rep. 314 (1999) 1 [arXiv:hep-ph/9807278]; J. Martin, C. Ringeval and V. Vennin, Phys. Dark Univ. 5-6, 75-235 (2014) [arXiv:1303.3787 [astro-ph.CO]]; J. Martin, C. Ringeval, R. Trotta and V. Vennin, JCAP 1403 (2014) 039 [arXiv:1312.3529 [astro-ph.CO]]; J. Martin, Astrophys. Space Sci. Proc. 45, 41 (2016) [arXiv:1502.05733 [astro-ph.CO]].
  * [2] Y. Akrami et al. [Planck Collaboration], arXiv:1807.06211 [astro-ph.CO].
  * [3] E. Cremmer, S. Ferrara, C. Kounnas and D. V. Nanopoulos, Phys. Lett. B 133 (1983) 61.
  * [4] J. R. Ellis, A. B. Lahanas, D. V. Nanopoulos and K. Tamvakis, Phys. Lett. 134B, 429 (1984).
  * [5] A. B. Lahanas and D. V. Nanopoulos, Phys. Rept. 145 (1987) 1.
  * [6] E. Witten, Phys. Lett. 155B (1985) 151.
  * [7] J. Ellis, D. V. Nanopoulos and K. A. Olive, Phys. Rev. D 89 (2014) 4, 043502 [arXiv:1310.4770 [hep-ph]]; J. Ellis, M. A. G. García, D. V. Nanopoulos and K. A. Olive, JCAP 1510, 003 (2015) [arXiv:1503.08867 [hep-ph]]. S. F. King and E. Perdomo, JHEP 1905, 211 (2019) [arXiv:1903.08448 [hep-ph]]. J. Ellis, M. A. G. García, N. Nagata, D. V. Nanopoulos and K. A. Olive, JCAP 1611, no. 11, 018 (2016) [arXiv:1609.05849 [hep-ph]].
  * [8] J. Ellis, M. A. G. Garcia, N. Nagata, D. V. Nanopoulos and K. A. Olive, Phys. Lett. B 797, 134864 (2019) [arXiv:1906.08483 [hep-ph]]; J. Ellis, M. A. G. Garcia, N. Nagata, D. V. Nanopoulos and K. A. Olive, JCAP 2001, no. 01, 035 (2020) [arXiv:1910.11755 [hep-ph]]. E. Dudas, T. Gherghetta, Y. Mambrini and K. A. Olive, Phys. Rev. D 96, no. 11, 115032 (2017) [arXiv:1710.07341 [hep-ph]]; K. Kaneta, Y. Mambrini, K. A. Olive and S. Verner, Phys. Rev. D 101, no. 1, 015002 (2020) [arXiv:1911.02463 [hep-ph]].
  * [9] A. A. Starobinsky, Phys. Lett. B 91, 99-102 (1980)
  * [10] J. Ellis, D. V. Nanopoulos and K. A. Olive, Phys. Rev. Lett. 111, 111301 (2013) [Erratum-ibid. 111, no. 12, 129902 (2013)] [arXiv:1305.1247 [hep-th]].
  * [11] E. Cremmer, S. Ferrara, C. Kounnas and D. V. Nanopoulos, Phys. Lett. B133, 61 (1983).
  * [12] J. R. Ellis, C. Kounnas and D. V. Nanopoulos, Nucl. Phys.B 247, 373 (1984).
  * [13] M. B. Einhorn and D. R. T. Jones, JHEP 1003, 026 (2010) [arXiv:0912.2718 [hep-ph]].
  * [14] S. Ferrara, R. Kallosh, A. Linde, A. Marrani and A. Van Proeyen, Phys. Rev. D 83, 025008 (2011) [arXiv:1008.2942 [hep-th]].
  * [15] S. Ferrara, R. Kallosh, A. Linde, A. Marrani and A. Van Proeyen, Phys. Rev. D 82, 045003 (2010) [arXiv:1004.0712 [hep-th]].
  * [16] J. Ellis, M. A. G. Garcia, N. Nagata, N. D. V., K. A. Olive and S. Verner, Int. J. Mod. Phys. D 29 (2020) no.16, 2030011 [arXiv:2009.01709 [hep-ph]].
  * [17] W. Ahmed and A. Karozas, Phys. Rev. D 98, no.2, 023538 (2018) [arXiv:1804.04822 [hep-ph]].
  * [18] W. Ahmed, A. Karozas and G. K. Leontaris, Phys. Rev. D 104, no.5, 055025 (2021) [arXiv:2104.04328 [hep-ph]].
  * [19] G. R. Dvali, Q. Shafi and R. K. Schaefer, “Large scale structure and supersymmetric inflation without fine tuning,” Phys. Rev. Lett. 73, 1886-1889 (1994) [arXiv:hep-ph/9406319 [hep-ph]].
  * [20] E. J. Copeland, A. R. Liddle, D. H. Lyth, E. D. Stewart and D. Wands, “False vacuum inflation with Einstein gravity,” Phys. Rev. D 49, 6410-6433 (1994) [arXiv:astro-ph/9401011 [astro-ph]].
  * [21] A. D. Linde and A. Riotto, “Hybrid inflation in supergravity,” Phys. Rev. D 56, R1841-R1844 (1997) [arXiv:hep-ph/9703209 [hep-ph]].
  * [22] V. N. Senoguz and Q. Shafi, “Reheat temperature in supersymmetric hybrid inflation models,” Phys. Rev. D 71, 043514 (2005) [arXiv:hep-ph/0412102 [hep-ph]].
  * [23] M. U. Rehman, Q. Shafi and J. R. Wickman, “Supersymmetric Hybrid Inflation Redux,” Phys. Lett. B 683, 191-195 (2010) [arXiv:0908.3896 [hep-ph]].
  * [24] W. Buchmüller, V. Domcke, K. Kamada and K. Schmitz, “Hybrid Inflation in the Complex Plane,” JCAP 07, 054 (2014) [arXiv:1404.1832 [hep-ph]].
  * [25] M. M. A. Abid, M. Mehmood, M. U. Rehman and Q. Shafi, JCAP 10, 015 (2021) [arXiv:2107.05678 [hep-ph]].
  * [26] M. U. Rehman, Q. Shafi and U. Zubair, Phys. Rev. D 97, no.12, 123522 (2018) [arXiv:1804.02493 [hep-ph]].
  * [27] T. W. B. Kibble, J. Phys. A 9, 1387-1398 (1976)
  * [28] S. Khalil, M. U. Rehman, Q. Shafi and E. A. Zaakouk, Phys. Rev. D 83, 063522 (2011) [arXiv:1010.3657 [hep-ph]].
  * [29] M. A. Masoud, M. U. Rehman and Q. Shafi, JCAP 04, 041 (2020) [arXiv:1910.07554 [hep-ph]].
  * [30] M. U. Rehman and U. Zubair, Phys. Rev. D 91, 103523 (2015) [arXiv:1412.7619 [hep-ph]].
  * [31] M. U. Rehman and Q. Shafi, Phys. Rev. D 86, 027301 (2012) [arXiv:1202.0011 [hep-ph]].
  * [32] W. Ahmed, A. Karozas, G. K. Leontaris and U. Zubair, JCAP 06, no.06, 027 (2022)
  * [33] S. M. Barr, B. Kyae and Q. Shafi, [arXiv:hep-ph/0511097 [hep-ph]].
  * [34] M. Fallbacher, M. Ratz and P. K. S. Vaudrevange, Phys. Lett. B 705, 503-506 (2011)
  * [35] N. Nagata, “Proton Decay in High-scale Supersymmetry,” doi:10.15083/00006623
  * [36] K. Abe et al. [Super-Kamiokande], Phys. Rev. D 95, no.1, 012004 (2017) [arXiv:1610.03597 [hep-ex]].
  * [37] K. Abe et al. [Hyper-Kamiokande], [arXiv:1805.04163 [physics.ins-det]].
  * [38] S.R. Coleman and E.J. Weinberg, Phys. Rev. D 7, 1888 (1973) [arXiv:hep-ph/].
  * [39] H. P. Nilles, Phys. Rept. 110, 1-162 (1984)
  * [40] M. Endo, F. Takahashi and T. T. Yanagida, Phys. Rev. D 76, 083509 (2007) [arXiv:0706.0986 [hep-ph]].
  * [41] C. Pallis and N. Toumbas, JCAP 12, 002 (2011) [arXiv:1108.1771 [hep-ph]].
  * [42] G. Lazarides and N. D. Vlachos, Phys. Lett. B 441, 46-51 (1998) [arXiv:hep-ph/9807253 [hep-ph]].
  * [43] A. R. Liddle and S. M. Leach, Phys. Rev. D 68, 103503 (2003) [arXiv:astro-ph/0305263 [astro-ph]].
  * [44] K. Hamaguchi, [arXiv:hep-ph/0212305 [hep-ph]].
  * [45] N. Aghanim et al. [Planck], Astron. Astrophys. 641, A6 (2020) [erratum: Astron. Astrophys. 652, C4 (2021)] [arXiv:1807.06209 [astro-ph.CO]].
  * [46] V. A. Kuzmin, V. A. Rubakov and M. E. Shaposhnikov, Phys. Lett. B 155, 36 (1985)
  * [47] M. Fukugita and T. Yanagida, Phys. Lett. B 174, 45-47 (1986)
  * [48] P. Andre et al. [PRISM], [arXiv:1306.2259 [astro-ph.CO]].
  * [49] T. Matsumura et al. [Mission design of LiteBIRD], J. Low Temp. Phys. 176, 733 (2014) [arXiv:1311.2847 [astro-ph.IM]].
  * [50] F. Finelli et al. [CORE], JCAP 04, 016 (2018) [arXiv:1612.08270 [astro-ph.CO]].
  * [51] A. Kogut, D. J. Fixsen, D. T. Chuss, J. Dotson, E. Dwek, M. Halpern, G. F. Hinshaw, S. M. Meyer, S. H. Moseley, M. D. Seiffert, D. N. Spergel and E. J. Wollack, JCAP 07, 025 (2011) [arXiv:1105.2044 [astro-ph.CO]].
  * [52] K. Abazajian, G. Addison, P. Adshead, Z. Ahmed, S. W. Allen, D. Alonso, M. Alvarez, A. Anderson, K. S. Arnold and C. Baccigalupi, et al. [arXiv:1907.04473 [astro-ph.IM]].
  * [53] P. Ade et al. [Simons Observatory], JCAP 02, 056 (2019) [arXiv:1808.07445 [astro-ph.CO]].
  * [54] N. Okada and Q. Shafi, Phys. Lett. B 775, 348-351 (2017) [arXiv:1506.01410 [hep-ph]].
  * [55] M. U. Rehman, Q. Shafi and F. K. Vardag, Phys. Rev. D 96, no.6, 063527 (2017) [arXiv:1705.03693 [hep-ph]].
  * [56] W. Ahmed, M. Junaid, S. Nasri and U. Zubair, Phys. Rev. D 105, no.11, 115008 (2022) [arXiv:2202.06216 [hep-ph]].
  * [57] M. Bolz, A. Brandenburg and W. Buchmuller, Nucl. Phys. B 606, 518-544 (2001) [erratum: Nucl. Phys. B 790, 336-337 (2008)] [arXiv:hep-ph/0012052 [hep-ph]].
  * [58] J. Pradler and F. D. Steffen, Phys. Rev. D 75, 023509 (2007) [arXiv:hep-ph/0608344 [hep-ph]]. 022
  * [59] H. Eberl, I. D. Gialamas and V. C. Spanos, Phys. Rev. D 103, no.7, 075025 (2021) [arXiv:2010.14621 [hep-ph]].
  * [60] T. A. Vami [ATLAS and CMS], PoS LHCP2019, 168 (2019) [arXiv:1909.11753 [hep-ex]].
  * [61] M. Kawasaki, K. Kohri and T. Moroi, Phys. Rev. D 71, 083502 (2005) [arXiv:astro-ph/0408426 [astro-ph]].
  * [62] M. Kawasaki, K. Kohri, T. Moroi and Y. Takaesu, Phys. Rev. D 97, no.2, 023502 (2018) [arXiv:1709.01211 [hep-ph]].
  * [63] M. Y. Khlopov and A. D. Linde, Phys. Lett. B 138, 265-268 (1984)
  * [64] J. R. Ellis, J. E. Kim and D. V. Nanopoulos, Phys. Lett. B 145, 181-186 (1984)
  * [65] G. Lazarides, M. U. Rehman, Q. Shafi and F. K. Vardag, Phys. Rev. D 103, no.3, 035033 (2021) [arXiv:2007.01474 [hep-ph]].
  * [66] M. Kawasaki, K. Kohri, T. Moroi and A. Yotsuyanagi, Phys. Rev. D 78, 065011 (2008) [arXiv:0804.3745 [hep-ph]].
  * [67] D. Hooper and T. Plehn, Phys. Lett. B 562, 18-27 (2003) [arXiv:hep-ph/0212226 [hep-ph]].