Weak Cosmic Censorship and Weak Gravity Conjectures in CFT Thermodynamics
Jafar Sadeghi Saeed Noori Gashti Mohammad Reza Alipour Mohammad Ali S. Afshar



Weak Cosmic Censorship and Weak Gravity Conjectures
 in CFT Thermodynamics



Jafar Sadeghi^⋆[Email:   pouriya@ipm.ir],   Saeed Noori Gashti ^⋆,†[Email:   saeed.noorigashti@stu.umz.ac.ir],  Mohammad Reza Alipour^⋆[Email:   mr.alipour@stu.umz.ac.ir],  


Mohammad Ali S. Afshar ^⋆[Email:   m.a.s.afshar@gmail.com],


^⋆Department of Physics, Faculty of Basic
Sciences,
University of Mazandaran

P. O. Box 47416-95447, Babolsar, Iran


^†School of Physics, Damghan University, P. O. Box 3671641167, Damghan, Iran





In this paper, we explore the intriguing interplay between fundamental theoretical physics concepts within the context of charged black holes. Specifically, we focus on the consistency of the weak gravity conjecture (WGC) and weak cosmic censorship conjecture (WCCC) in the thermodynamics of conformal field theory (CFT), and restricted phase space (RPS) for AdS Reissner-Nordström black holes with a perfect fluid dark matter (RN-PFDM). The WGC ensures that gravity remains the weakest force in the system. Meanwhile, the WCCC addresses the cosmic censorship problem by preventing the violation of fundamental physical laws near the black hole singularity. First, we analyze the RN black hole's free energy in both spaces, revealing a distinctive swallowtail pattern indicative of a first-order phase transition when certain free parameter conditions are met. We explore the  WGC across different phase spaces, emphasizing the need for certain conditions in EPST, RPST, and CFT. We demonstrate that PFDM γ and the radius of AdS l have a vital role in proving the satisfaction of the WGC. Also, these values have a linear relation with the range compatibility of WGC. The range of compatibility for WGC in RPST and EPST is the same while for CFT, this range is larger than EPST, and RPST. It means somehow the WGC and CFT are more consistent. The WCCC was examined at the critical juncture, confirming its validity with f(M_c, Q_c, r_c) = 0. We conclude that the WGC is supported at the critical point of black holes, and the WCCC is also maintained, demonstrating the robustness of these conjectures within the critical ranges of black hole parameters.

Keywords: Weak Gravity Conjecture, Weak Cosmic Censorship Conjecture, Perfect Fluid Dark Matter, Restricted Phase Space, Conformal Field Theory




§ INTRODUCTION

The exploration of black hole thermodynamics, initially spurred by the work of Hawking and Bekenstein, delves into the thermal characteristics of these cosmic entities from a thermodynamic physics perspective<cit.>. This field posits a correlation between surface gravity and the event horizon's area with temperature and entropy, respectively, while often treating a black hole's mass as its internal energy. Despite the progress achieved, questions remain, particularly regarding the microscopic contributors to black hole entropy. The principles of holography and the AdS/CFT duality are instrumental in examining various black hole properties, where a state of an AdS black hole in the bulk corresponds to a thermal state in the dual field theory<cit.>. Additionally, the thermodynamics of black holes can be analyzed through the lens of different symmetries within a gravitational model<cit.>. Hawking and Page's advanced research on AdS black holes revealed a phase transition between such black holes accompanied by radiation and their thermal AdS counterparts<cit.>. The thermodynamic properties of black holes, especially within the framework of extended phase space thermodynamics, have been extensively discussed, where the negative cosmological constant is equated with pressure<cit.>. This perspective has led to the inclusion of other model parameters as novel thermodynamic variables<cit.>.


Furthermore, the use of P-V variables suggests that AdS black holes could function as heat engines<cit.>. A recent theoretical development has introduced central charge and chemical potential as a new pair of conjugate thermodynamic variables, enhancing our understanding of black hole thermodynamics. In the corresponding dual theory, the central charge is linked to the square of the number of colors<cit.>.
In the realm of AdS/CFT correspondence, a novel thermodynamic framework known as restricted phase space thermodynamics (RPST) has emerged, as introduced by Gao et al. This paradigm redefines black hole masses as a form of internal energy. Unlike the extended phase space thermodynamics (EPST) that utilizes pressure and volume, RPST employs a distinct set of conjugate thermodynamic variables: the chemical potential, denoted as µ, and the central charge, C. These variables represent color susceptibility and the degrees of freedom within the conformal field theory (CFT), respectively<cit.>.
The central charge, C, plays a role analogous to the particle count in the statistical physics of conventional matter. It is mathematically expressed in relation to Newton's constant, G, and the anti-de Sitter (AdS) curvature radius, through the formula C = l^d-2/G Here, d signifies the dimensionality of the AdS spacetime. The RPST framework has been applied to explore the homogeneity of the Smarr relation, various thermodynamic processes, and phase transitions across different black hole configurations such as Reissner-Nordström-AdS (RN-AdS), Kerr-AdS, Bañados-Teitelboim-Zanelli (BTZ), and Taub-NUT within the AdS context<cit.>.


The holographic principle has revolutionized our understanding of space and information, suggesting that the information contained within a volume of space can be represented on a lower-dimensional boundary. This concept has given rise to the field of holographic black hole chemistry. It posits a duality between gravitational dynamics in higher-dimensional Anti-de Sitter (AdS) space and quantum field theory on its boundary, a relationship known as the AdS/CFT correspondence. Through this lens, we can explore black hole thermodynamics by treating the cosmological constant as a variable akin to thermodynamic pressure in AdS space or a scaling factor on the boundary. This innovative approach has led to the development of black hole chemistry, which can define thermodynamic quantities like enthalpy, Gibbs free energy, and chemical potential for black holes<cit.>. It allows probing of the properties and phase transitions of black holes in AdS space through holographic duality. Notably, it provides insights into phenomena such as the Hawking-Page transition, which reflects a phase shift from thermal AdS space to a large AdS black hole, analogous to the confinement/deconfinement transition in conformal field theory. One of the intriguing outcomes is that AdS black holes adhere to a generalized Smarr relation, linking their mass, area, charge, angular momentum, and pressure. However, integrating the first law of black hole thermodynamics with the thermodynamics of its holographic dual presents challenges. This research is particularly focused on how the variation of the cosmological constant (Λ) correlates with changes in the central charge (C) and the volume (V) of the boundary CFT. This relationship necessitates a revision of the thermodynamic laws to account for the volume and its associated thermodynamic pressure<cit.>. The possibility of varying fundamental constants, like Newton's constant (G) and the cosmological constant, hints at more profound underlying theories. researchers addressed this by demonstrating that altering G alongside Λ maintains the constancy of the central charge C in the dual CFT. Consequently, the first law can be reformulated to include a new thermodynamic volume and chemical potential, bridging holographic and bulk thermodynamics through the central charge duality. Their work also uncovered phase structures with a critical central charge value, beyond which the phase diagram shows a swallowtail pattern indicative of a first-order phase transition. By treating G as a variable, we can incorporate the central charge into the first law and extend the thermodynamic discussion to various black holes. In essence, black hole chemistry reveals that AdS black holes experience phase transitions mirroring those found in conventional thermodynamic systems. Yet, the extended thermodynamics' holographic interpretation remained elusive for some time. Initial theories proposed that the volume-pressure term in AdS should correspond to a chemical potential-central charge term in the dual CFT, where the central charge is C and the chemical potential is its thermodynamic conjugate<cit.>.


The Swampland Program is an initiative in theoretical physics that seeks to identify effective low-energy theories which are not compatible with quantum gravity. This contrasts with the "string theory landscape," which includes theories known to be compatible with string theory, a consistent quantum theory of gravity. The Swampland encompasses theories that appear consistent but lack a consistent ultraviolet completion when gravity is added. The program was initiated by Cumrun Vafa and aims to delineate the boundaries of the quantum gravity landscape by identifying universal principles shared among all theories compatible with gravitational UV completion<cit.>. The  WGC posits that in any theory of quantum gravity, gauge forces must mediate interactions that are stronger than gravity for some particles. This conjecture has deep connections across various areas of physics and mathematics, suggesting that gravity should be the weakest force in any consistent theory of quantum gravity. It has been reviewed extensively and is supported by evidence from string theory examples. The Swampland Program or the WGC is significant in the ongoing effort to understand the fundamental principles that govern our universe at the quantum level. They have implications for particle physics, cosmology, general relativity, and mathematics, guiding researchers in exploring the limits and possibilities of theoretical physics. To study an example of applications of the swampland program in inflation, black holes, thermodynamics, and black brane, you can see in<cit.>.


In this exploration, we examine the nuanced interplay between the  WGC and the WCCC in the realm of charged black holes. Our focus is on the (RN) black holes that are encased within PFDM, considering diverse spatial frameworks like RPS and CFT. This analysis aims to deepen our understanding of the theoretical underpinnings that govern these cosmic entities.
The WCCC, introduced by Roger Penrose, is a pivotal concept in theoretical physics that addresses the potential issues arising from singularities within general relativity. The WCCC posits that singularities resulting from gravitational collapse are invariably concealed behind event horizons, thereby maintaining the causal structure and predictability of spacetime. This conjecture ensures that the "naked" singularities, which could expose the inner workings of black holes, do not disrupt the observable universe. In contrast, the  WGC suggests that in any consistent theory of quantum gravity, there must exist particles for which the electromagnetic force is stronger than their gravitational pull. This implies that black holes should be able to decay into these particles, preventing the formation of stable remnants and ensuring the discharge of black holes. The Reissner–Nordström (RN) black hole, a solution to the Einstein–Maxwell equations for a charged black hole, presents a unique challenge to these conjectures. If the charge Q of the black hole exceeds its mass M (Q > M), the RN solution would contravene the WCCC by revealing a naked singularity to distant observers. Conversely, when the RN black hole approaches an extremal state (Q = M), it suggests the existence of decay products with a charge-to-mass ratio exceeding unity. While this satisfies the WGC, it simultaneously breaches the WCCC, as these decay products cannot form black holes but are instead predicted to be elementary particles. This apparent contradiction between the WGC and the WCCC in the context of RN black holes is one of the intriguing puzzles at the intersection of quantum theory and gravitational physics<cit.>.


The  WGC and the WCCC are two distinct ideas in theoretical physics, each addressing different aspects of fundamental physics. While both conjectures involve the term "weak" and pertain to fundamental aspects of gravity and spacetime, they are distinct in their focus and implications. The WGC primarily concerns the relative strengths of forces in a quantum gravity framework, while the WCCC deals with the behavior of singularities and event horizons in gravitational collapse scenarios within classical general relativity. Combining these concepts, one might speculate about potential connections between the WGC and cosmic censorship, such as whether the WGC could have implications for cosmic censorship scenarios. Building on the concepts previously discussed, we delve into the intricate relationship between the  WGC and the WCCC within the framework of charged black holes, specifically the RN AdS black hole surrounded by PFDM. We examine this relationship from a thermodynamic standpoint, considering various theoretical spaces such as RPS and CFT by mentioning an important point that means.
The Mild WGC posits that gravity is the weakest force, yet it accommodates a wider spectrum of theoretical possibilities, making it less stringent than its counterpart, the Strong WGC. The latter is more narrowly focused on particles, asserting not just that gravity must be weaker but also stipulating the existence of particles that strictly adhere to this inequality. When considering black holes, the WCCC maintains that singularities should remain veiled behind event horizons, safeguarding the causal structure of spacetime. The interplay between the WGC and WCCC is subtle and complex. Our investigations have revealed that under specific conditions, such as the incorporation of PFDM, these conjectures can indeed coexist harmoniously. This concord is vital as it ensures that our theoretical models remain consistent and do not yield contradictory predictions about the thermodynamic behavior of black holes. By demonstrating compatibility under these conditions, we contribute to a more coherent understanding of the fundamental principles governing black hole physics. The paper is organized as follows.

In Section 2, we conduct a comprehensive study of the RN AdS black hole in the presence of  PFDM. Sections 3 and 4 are dedicated to examining the thermodynamics of the RN AdS black hole surrounded by PFDM within the frameworks of RPST and CFT, respectively. We also analyze the thermodynamic parameters in these two spaces. Section 5 explores the compatibility of the  WGC and the WCCC in both spaces, delving into the areas where these conjectures align. Finally, in Section 6, we briefly summarize our findings.


§ WEAK GRAVITY CONJECTURE

The Weak Gravity Conjecture  is a theoretical framework in the field of quantum gravity that suggests gravity should be the weakest force in any consistent theory of quantum gravity. It posits that for any gauge force, there must exist some particles for which the gauge force mediates interactions stronger than gravity. This conjecture has profound implications across various domains of physics and mathematics, and it has been a subject of significant interest and debate within the theoretical physics community. The WGC is not without its criticisms and potential weak points. Some of the criticisms include that the conjecture is based on qualitative arguments rather than a rigorous mathematical proof <cit.>. There are several variations of the conjecture, and it’s not clear which, if any, is the correct one. The conjecture has been challenged by certain string theory examples and other theoretical models that do not seem to satisfy its conditions.
The WGC has made several predictions, including:
The existence of particles with specific mass-to-charge ratios ensures gravity remains the weakest force.
Constraints on the properties of black holes, particularly regarding their charge and mass.
Implications for the stability of cosmic structures and the behavior of fundamental particles.

The WGC has interesting connections with thermodynamics, particularly in the context of black hole physics. It has been suggested that the conjecture could be derived from thermodynamic principles applied to black holes, such as entropy considerations and the behavior of quasinormal modes <cit.>. The implications of the WGC can extend to various spacetime geometries, including extended, restricted, and CFT spaces:
The conjecture has been explored in the context of theories with extra dimensions, such as string theory, where it places constraints on the geometry of the compactified dimensions.
In more constrained spacetime geometries, like those with a limited number of dimensions or symmetries, the WGC still implies the existence of particles that satisfy its mass-to-charge ratio condition.
In the AdS/CFT correspondence, the WGC translates into inequalities involving the dimension and charge of operators, as well as central charges in the dual CFTs. However, these translations are still speculative and may not apply universally across all CFTs <cit.>.
In summary, the WGC is a conjecture that aims to provide insights into the nature of gravity in quantum gravity theories. It has sparked a wide range of research, leading to discussions about its validity, implications, and connections to other physical principles. While it has faced criticism and challenges, it continues to be a pivotal concept in theoretical physics, driving exploration into the fundamental forces of nature. The WGC has far-reaching implications that extend beyond the realm of theoretical physics into cosmology, black hole thermodynamics, and the broader landscape of string theory. In cosmology, the WGC can influence our understanding of the early universe, particularly during the inflationary period. The conjecture suggests that inflationary models involving scalar fields should be coupled to gauge fields in a way that satisfies the WGC criteria <cit.>. This can lead to predictions about the spectrum of primordial fluctuations, potentially observable in the Cosmic Microwave Background (CMB) data. String theory provides a fertile ground for testing the WGC. The conjecture has been used to argue for the existence of certain particles in the string spectrum that have implications for the stability of extra dimensions and the nature of supersymmetry breaking. A proof of the WGC within the context of perturbative string theory has been proposed, which relates the black hole extremality bound to long-range forces calculated on the worldsheet <cit.>. The thermodynamics of black holes offers a unique testing ground for the WGC. The conjecture has been studied in the context of 4D Einstein-Maxwell-dilaton theory, where it places constraints on the Wilson coefficients of the theory, which are satisfied under certain assumption about the UV completion. This study bridges the gap between quantum field theory and gravitational phenomena, providing insights into the entropy and stability of black holes <cit.>.
The WGC may serve as a bridge between quantum mechanics and cosmic-scale physics. By examining the thermodynamics of black holes and their connection with the WGC, we can explore how quantum-level interactions manifest at cosmic scales. This could lead to a better understanding of the holographic principle and the nature of spacetime itself.
The phenomenological implications of the WGC are vast. It can affect predictions for particle accelerators, inform the search for dark matter candidates, and influence models of dark energy. The conjecture's requirement for the existence of certain particles could lead to observable signatures in high-energy physics experiments.
In summary, the WGC is not just a theoretical curiosity but a principle that can potentially offer solutions to existing problems in cosmology and string theory, provide a framework for understanding black hole thermodynamics, and open new avenues for research in the phenomenology of quantum gravity theories. The ongoing dialogue between theory and observation will continue to refine our understanding of the WGC and its implications across the cosmos. Considering all the mentioned concepts, we must now explore other applications of the WGC within various cosmological frameworks. This exploration aims to address potential issues and provide evidence supporting both the conjecture itself and string theory. By examining black hole thermodynamics and its relationship with the WGC, we can establish a link from quantum to cosmic scales. This not only allows for a review of existing concepts but also enables the results obtained to unveil new perspectives on the implications and phenomenology associated with the WGC.


§ THE MODEL

Reissner-Nordström  AdS black holes, which are enveloped by PFDM, carry an electrical charge and are affected by a theoretical type of matter that constitutes the majority of the universe's mass. These black holes exhibit intriguing thermodynamic behaviors and phase transition characteristics, which can be explored by broadening the phase space of the system. Additionally, some studies have investigated the WCCC in the context of these black holes. This conjecture posits that a black hole's singularity remains hidden from any external observer. The action under consideration pertains to a gravitational theory that interacts minimally with a gauge field amidst the presence of  PFDM<cit.>,

    S=∫ d^4x√(-g)[1/16π GR+1/4F^μνF_μν-Λ/8π G+ℒ_DM],

in the given context, g represents the determinant of the metric tensor, expressed as g = (g_ab), where g_ab are the components of the metric tensor. The scalar curvature is denoted by R, while G stands for the gravitational constant. The electromagnetic tensor F_μν is obtained from the gauge potential A_μ. The Lagrangian density L_DM characterizes the  PFDM. Utilizing the principle of least action allows for the derivation of the Einstein field equations, which can be articulated as follows,

    R_μν-1/2g_μνR=   -8π G(T^M_μν+T^DM_μν)≡-8π GT_μν,


    F^μν_ν=0,
       F^μν;α+F^να;μ+F^αμ;ν=0.

The energy-momentum tensor for ordinary matter is symbolized by T^M_μν, and the energy-momentum tensor for  PFDM is represented by T^DM_μν. These tensors are detailed in<cit.>. Consequently, the equations can be expressed as follows,

    T^μ_ν=g^μνT_μν,
       T^t_t=-ρ,T^r_r=T^θ_θ=T^ϕ_ϕ=P.

In the framework of the RN AdS metric influenced by  PFDM, it is posited that the components of the energy-momentum tensor satisfy T^r_r = T^θ_θ = T^ϕ_ϕ = T^t_t (1 - δ), with δ being a constant. This assumption is substantiated by <cit.>. The formulation of the RN AdS metric within the PFDM context is delineated as follows,

    ds^2=-f(r)dt^2+f(r)^-1dr^2+r^2(dθ^2+sin^2θ dϕ^2),

where,

    f(r)=1-2MG/r+GQ^2/r^2-Λ/3r^2+γ/rln(r/γ).

In this analysis, the parameters M, Q, and γ signify the mass, charge and the influence of (PFDM) on the black hole, respectively. Also, Λ = -3/ℓ^2
is the cosmological constant with ℓ being the AdS radius. In the absence of PFDM (γ = 0), the space-time metric simplifies to that of a standard RN AdS black hole.
The parameter γ can assume both positive and negative values, reflecting different PFDM scenarios. This study contemplates the implications of both positive and negative γ on the black hole's characteristics.


§ RPS THERMODYNAMICS

In the RPS formalism, the traditional variables of pressure P and volume V are eschewed due to the altered connotations they acquire within the holographic framework. Instead, two pivotal quantities, C and μ, assume significant roles within the realm of CFT. The central charge C is instrumental in ascertaining the microscopic degrees of freedom inherent in CFT, while its conjugate μ is regarded as a chemical potential, as referenced in literature<cit.>. Within the domain of extended phase space thermodynamics, the exploration of black hole thermodynamics necessitates the employment of a dynamic cosmological constant. This adjustment invariably leads to a transformation in the associated dynamical equations. To circumvent this modification, the thermodynamics of a constrained phase space is proposed, wherein Newton's constant is treated as a variable, supplanting the cosmological constant. This substitution preserves the original form of the field equations. Moreover, a correlation is established between the central charge and the cosmological and Newton's constants, expressed as C = ℓ^2/ G, where ℓ^2 = -3/Λ <cit.>. The first law of thermodynamics, as applied to a RN-AdS black hole surrounded by PFDM within the RPS framework, is articulated as follows,

    dM=T dS+ϕ̂ dQ̂+μ dC+Π dγ.

The observation aligns with an Euler-like relation, which is a fundamental principle in thermodynamics and fluid dynamics, often used to describe the relationship between various state functions. In the context of the RPS formalism, such relations are crucial for understanding the thermodynamic properties of black holes and their behavior in different spacetime geometries. By incorporating this Euler-like relation into the RPS framework, researchers can further elucidate the intricate dynamics of black holes within the holographic paradigm, offering deeper insights into the nature of spacetime and the fundamental forces that govern it,

    M=T S+ϕ̂Q̂+μ C+Πγ,

in the context of our black hole, the parameters M, T, and S represent the mass, temperature, and entropy, respectively. Moreover, these parameters can generally be characterized by corresponding quantities in the dual CFT,

    Q̂=Q ℓ/√(G)ϕ̂=ϕ√(G)/ℓ.

We will also have some parameters in RPS as follows,

    C=ℓ^2/G,    Q̂=Q√(C) ,     r_+=ℓ√(S/π  C).

Setting (f(r_+) = 0) and utilizing equations (6), (7), (9), and (10), we can derive some variables such as M, T, ϕ̂, Π, and μ of the black hole as follows,

    M=π  C S+π ^2 Q̂^2+S^2/2 π ^3/2ℓ√(C S)+C γ/2ℓ^2ln(√(S/π  C)ℓ/γ),



    T=(  ∂ M/∂ S)_Q̂,C,γ=π  C S-π ^2 Q̂^2+3 S^2/4 π ^3/2ℓ S √(C S)+C γ/4 ℓ^2 S



    ϕ̂=(  ∂ M/∂Q̂)_S,C,γ= √(π/SC)Q̂/ℓ,



    Π=(  ∂ M/∂γ)_S,Q̂,C=C/2ℓ^2[-1+ln(√(S/π  C)ℓ/γ) ],



    μ=(  ∂ M/∂γ)_S,Q̂,γ=π C S-π ^2 Q̂^2-S^2/4 π ^3/2 C ℓ√(C S) +γ/4ℓ^2[-1+2ln(√(S/π  C)ℓ/γ) ].

The T-S curve at a fixed value of Q̂ and C is a fundamental feature in thermodynamics. It captures the behavior of a system as temperature (T) and entropy (S) vary. Notably, this curve exhibits a first-order phase transition, which, intriguingly, becomes second-order at the critical point. To identify the critical point on the T-S curve (often referred to as the inflection point), we employ the following equation,

    (∂ T/∂ S)_Q̂,C,γ=0,     (∂^2 T/∂ S^2)_Q̂,C,γ=0.

Using Eqs. (12) and (16) for the critical parameters, we have,

    Y:=C ℓ^4/3( 81γ^2-ℓ^2+9ℓγ√(81γ^2-2ℓ^2))^1/3,      S_c=π (C ℓ^2+Y)^2/18 ℓ^2 Y,



    Q̂_c^2=-(C ℓ^2+Y)^2 (C^2 ℓ^4-4 C ℓ^2 Y+Y^2)/324 ℓ^4 Y^2+√(2 )γ  C^3/2(C ℓ^2+Y)/9√( Y)ℓ^2.

The critical values of T and M are calculated by,

    T_c=(C ℓ^2+Y) (-9 √(2) C^3/2γℓ^2 Y^3/2+C^3 ℓ^6+6 C^2 ℓ^4 Y+6 C ℓ^2 Y^2+Y^3)/3  π√(2C Y)ℓ^2 (C ℓ^2+Y)^3+9 γ C Y/2 π(C ℓ^2+Y)^2,



    M_c=√(C)(C ℓ^2+Y) (2 C ℓ^2+3 γ√(2 C Y)+2 Y)/9 ℓ^2 √(2 Y)(C ℓ^2+Y)+γ C   /2 ℓ^2ln(C ℓ^2+Y/γ√(18 C Y)).

When studying the thermodynamics of black holes using the Euclidean action, we often encounter a challenge known as redshift degeneracy in the action. To address this issue, we propose considering an appropriate counterterm derived from the AdS/CFT correspondence within the Euclidean action. By doing so, we can resolve this limitation and proceed to find the thermodynamic potential associated with both the canonical and grand canonical ensembles. As an illustrative example, let's refer to the Gibbs energy potential, denoted as G:
G = iβ where i represent the Euclidean action, and β is the inverse of the Hawking temperature. From this Gibbs energy potential, we can derive other essential thermodynamic quantities: Entropy (S): Given by S =- dG/dt, angular momentum (J): Given by J = -dG/dΩ, electric charge (Q): Given by Q =- dG/dϕ. This approach is equivalent to using the first law of thermodynamics to determine these quantities. Additionally, we can express these thermodynamic quantities in terms of the Helmholtz free energy (F) and its conjugates: Entropy (S): S = -dF/dt, angular momentum (J): J = -dF/dΩ, electric charge (Q): Q = -dF/dϕ. The connection between the Helmholtz free energy and the Euclidean action is given by F = iβ. Remarkably, the results obtained from the Helmholtz free energy remain consistent with those derived from the first law of thermodynamics. In the RPS formalism, Newton's constant is treated as a variable external to the Euclidean action. Consequently, it does not introduce any additional terms to the counterterm. By adjusting the counterterm appropriately, we can eliminate the redshift degeneracy for different types of black holes. Now, we delve into the Helmholtz free energy,

    F=M-TS=π  C S+3 π ^2 Q̂^2-S^2/4 π ^3/2ℓ√(C S)+γ C/4 ℓ^2[-1+2 ln(ℓ/γ√(S/π C))].

Figures 1 and 2 depict the evolution of free energy as a function of temperature for various free parameter values of the RN AdS black hole, analyzed within the framework of PFDM in RPS. When Q̂_c > Q̂ = 0.5, a distinctive swallowtail pattern emerges, indicative of a first-order phase transition. Also, for (Q̂_c = Q̂ ), we have a second-order phase transition. To enhance clarity, we have separated the figures. The specific free parameters pertinent to each figure are duly annotated. The separation of the figures isolates and emphasizes each parameter set's unique characteristics, facilitating a more detailed examination of the system's behavior under varying conditions. It is essential to recognize that the free parameters play a crucial role in shaping the physical properties of the black hole.







Figure 3 presents the Hawking temperature as a function of entropy for the AdS (RN) black hole amidst the presence of (PFDM) in the (RPS). We have set the constants C and ℓ to 1. For Figure 3a, the parameters are set to γ = 2 and Q̂_c= 0.93 , while for Figure 3b, γ = 1.5 and Q̂_c= 0.79 . Each subplot within the figure reveals certain zero points for varying free parameters within the range 0 < S < 1. Upon examining the entropy alongside the black hole's free parameter values, it is observed that for small values of S, the structural behavior of the black hole is markedly unique. Conversely, as the value of S increases, the figures tend to converge, indicating a unification in behavior for larger entropy values. This convergence suggests that the influence of the free parameters diminishes as the entropy grows, leading to a more predictable and uniform structure for the black hole.


§ CFT THERMODYNAMICS

To explore the thermodynamics of the charged AdS black hole with PFDM within the CFT framework, it's essential to understand the holographic links that connect the bulk and boundary quantities. We denote the boundary's curvature radius by R, which is distinct from the AdS radius ℓ in the bulk. The CFT metric, showcasing conformal scaling invariance, is given by,

    ds^2=ω^2(-dt^2+ℓ^2 dΩ_d-2^2).

Here, ω is the dimensionless conformal factor that varies freely, signifying the boundary theory's conformal symmetry. In a spherical scenario, dΩ_d-2^2 is the metric on a (d-2)-dimensional sphere. We assume ω is constant across the 
    𝒱=Ω_d-2 R^d-2,

where R = ωℓ signifies the manifold's variable curvature radius where the CFT resides. Notably, variations in the central charge C do not affect 𝒱. In Einstein's gravity context, C and ℓ exhibit a dual relationship, as indicated in a previous equation. We maintain a constant Newton's constant G while allowing ℓ to vary, which in turn alters C. The holographic dictionary facilitates the correlation of bulk parameters M, T, S, Φ, Q with their CFT counterparts E, T̃, S̃, Φ̃, Q̃, as follows,

    E=M/ω,     T̃=T/ω,     S̃=S,     Q̃=Q ℓ/√(G),     Φ̃=Φ√(G)/ωℓ.

Employing these relationships, we can reformulate the first law of thermodynamics for the charged AdS black hole in the CFT context:

    δ E= Φ̃δQ̃+ μδ C-pδ𝒱 +T̃δ S,

where the chemical potential μ and pressure p are defined as,

    μ=E-T̃S-Φ̃Q̃/C,


    p=E/(d-2)𝒱.

Furthermore, it's evident from the above equation that 𝒱 and C vary independently. This paper aims to delve into the consequences of this theoretical framework on the thermodynamics of RN AdS black holes in the presence of PFDM within the CFT paradigm. Our objective is to reformulate the bulk thermodynamic first law in a manner that incorporates the boundary central charge. To achieve this, we turn to the holographic dual relationship between key quantities in Einstein's gravity and the AdS scale, the central charge C, and Newton's constant G. In this context, we consider a setting where both the cosmological constant (Λ) and gravitational Newton constant (G) are allowed to vary within the bulk. By doing so, we can express the first law in a new form that involves both Λ (associated with thermodynamic pressure) and the central charge C from the dual CFT. These variables and their conjugates play a crucial role in understanding the thermodynamics of AdS black holes. This approach allows us to explore novel aspects of black hole behavior, including phase transitions, critical points, and universal scaling behavior, all while maintaining a fixed boundary central charge. The interplay between holography and bulk thermodynamics provides valuable insights into the nature of quantum gravity,

    C=Ω_d-2ℓ^d-2/16π G.

Here, we aim to derive the first law of thermodynamics within the context of the CFT mode for the RN-AdS black hole with PFDM. Our goal is to establish a meaningful correspondence between quantities in the bulk (associated with the black hole) and those on the boundary (related to the CFT). To achieve this, we introduce a scale transformation given by:E = M/ω where ω = R/ℓ represents the ratio of the AdS radius (R) to the characteristic length scale (ℓ). We focus on the case where the spacetime dimension is d=4. By analyzing the modified equations, we explore the interplay between the RN-AdS black hole, its thermodynamic properties, and the dual CFT description. This investigation sheds light on the intricate relationship between the bulk geometry and the boundary theory, providing valuable insights into the underlying physics,

    Ω_2=4π,       C=ℓ^2/4 G,

and

    δ( M/ω)= T/ωδ(A/4G)+(M/ω-TS/ω-QΦ/ω-Πγ/ω) δ ( ℓ^2/4G)/ℓ^2/4G
       -M/2ωδ (Ω_2 R^2)/(Ω_2 R^2)+Φ√(G)/ωℓδ(Qℓ/√(G))+Π√(G)/ωℓδ(γℓ/√(G)).

Based on the equation provided above, the correspondence between bulk and boundary quantities for the black hole can be established as follows,

    δ E= T̃δS̃+μδ C-p δ𝒱+ Φ̃δQ̃+  Π̃δγ̃,

where

    E=M/ω,   S̃=S,  T̃=T/ω,   Φ̃=Φ√(G)/ωℓ,  Q̃=Qℓ/√(G),   Π̃=Π√(G)/ωℓ,  γ̃=γℓ/√(G),

if β=γ̃/ℓ, we also calculate thermodynamic quantities for the CFT as follows,

    E=4 π  C S+π ^2 Q̃^2+S^2/2 π√(C S 𝒱)+2 β√(π  C)/√(𝒱)ln(√(S)/√(π)β),



    T̃=(  ∂ E/∂ S)_Q̃,C,β,𝒱=4 π  C S-π ^2 Q̃^2+3 S^2/4 π  S^3/2√(C 𝒱)+β√(π  C)/S √(𝒱),



    Φ̃=(  ∂ E/∂Q̃)_C,S,β,𝒱=πQ̃/√(C S 𝒱),



    μ=(  ∂ E/∂ C)_Q̃,S,β,𝒱=4 π  C S-π ^2 Q̃^2-S^2/4 π C √(C S 𝒱)+√(π)β/√(C 𝒱)ln(√(S)/√(π)β),



    Π̃=(  ∂ E/∂γ̃)_Q̃,S,C,𝒱=β√(π  C)/γ̃ S √(𝒱),



    -p=(  ∂ E/∂𝒱)_Q̃,S,C,β=-E/2𝒱.

Additionally, we can utilize the following relationships to identify the critical points,

    (∂T̃/∂ S)_Q̃,𝒱,C,β=0,     (∂^2 T̃/∂ S^2)_Q̃,𝒱,C,β=0.

By applying the connections between equations (<ref>) and (<ref>), we can determine the critical points as outlined below

    X:=(-4 C^3+81 β ^2 C^2+9β C^2 √(81 β ^2 -8  C))^1/3,      S_c=1/9( 2^5/3π  C^2/X+4 π  C+2^1/3π  X),



    Q̃_c^2=-2^10/3 C^4+2^11/3 C^3 X+24 C^2 X^2+2^7/3 C X^3-2^2/3 X^4/81 X^2+8/9β  C √(2^5/3 C^2/X+4 C+2^1/3 X),



    T̃_c=4 π  C S_c-π ^2 Q̃_c^2+3 S_c^2/4 π  S_c^3/2√(C 𝒱)+β√(π  C)/S_c √(𝒱),     
    E_c=4 π  C S_c+π ^2 Q̃_c^2+S_c^2/2 π√(C S_c 𝒱)+2 β√(π  C)/√(𝒱)ln(√(S_c)/√(π)β).

We utilize equations  (<ref>) and  (<ref>) to determine the Helmholtz energy.

    F=E-T̃S̃=12 π  C S+π ^2 Q̃^2+5 S^2/4 π√(C S𝒱)+β√(π  C)/√(𝒱)[-1+2 ln(√(S)/√(π)β)].

Figures 4 and 5 showcase the evolution of free energy as a function of temperature for diverse free parameter values of the RN AdS black hole, situated within the theoretical framework of PFDM in Conformal Field Theory. A notable swallowtail pattern is observed when Q̃_c > Q̃ = 1.5, signaling a first-order phase transition. Also, in Q̃_c=Q̃, we face with second phase transition. For enhanced visual clarity, we have separated the figures. Each figure is meticulously labeled with the relevant free parameters. By separating the figures, we aim to highlight and scrutinize the distinct features of each parameter set, thereby enabling a more granular analysis of the system's dynamics under different conditions. The role of free parameters is pivotal in determining the physical attributes of the black hole.




 





Figure 6 illustrates the Hawking temperature as a function of entropy for the RN AdS black hole in the context of  PFDM within CFT. We set C =𝒱=1. For Figure 6a, the parameters are chosen as β = 2 and Q̃_c = 2.52, whereas for Figure 6b, β = 1.5 and Q̃_c = 2.15 are selected. Each subplot in the figure discloses distinct zero points for varying free parameters across the entropy range 0 < S < 1. A detailed analysis of the entropy in relation to the black hole's free parameter values reveals a pronounced uniqueness in structural behavior for lower entropy values, denoted by S. This uniqueness is characterized by significant deviations in the thermal properties of the black hole. In contrast, as the entropy value S ascends, the subplots exhibit a tendency to align, signifying a behavioral unification at higher entropy levels.


§ WGC & WCCC

The primary objective of this work is to challenge the WGC and WCCC in different spaces. So first we investigate the thermodynamic properties of Reissner-Nordström black holes in the presence PFDM, utilizing the principles of RPS and CFT thermodynamics. Then we explore the special ranges where these conjectures are compatible with each other in the Q/M>√(G). We aim to explore the following key aspects:
This comprehensive study aims to deepen our understanding of black hole physics in the presence of dark matter and energy, contributing to the broader field of quantum gravity and cosmology.
This research objective is designed to be broad yet focused, allowing for a detailed exploration of the interconnected theories and their implications on the thermodynamics and stability of black holes in astrophysical scenarios influenced by dark matter.



To substantiate the WGC within various phase spaces, we must establish certain specific relations. In the EPST, the condition is H = Q^2/M^2 - 1 > 0, and for the RPST and CFT, the criterion is H = Q^2/M^2 - G > 0. Consequently, we can pursue several approaches, among which thermodynamics and the critical values of black hole parameters are particularly significant. In light of this, we integrate the critical points delineated in prior sections into each respective space under the conditions mentioned above. Given that the critical values related with two parameters, namely γ and l, it becomes evident that these parameters play pivotal roles in validating the WGC. As depicted in Figure 7, the ranges within which the  WGC is satisfied—specifically (H > 0) for EPST and RPST are the same. However, for CFT, this range is different and larger than other spaces.


    RPST=EPST →√(2/3)ℓ<γ<5ℓ
,


    CFT →√(3)ℓ<γ<10ℓ
.

Furthermore, through computational methods, we have approximated these intervals. Additionally, upon scrutinizing the WCCC at the critical juncture, we ascertain that it remains valid viz f(M_c, Q_c, r_c)=0. Thus, it can be concluded that even though the WGC is upheld for the black hole at the critical point, the WCCC is also concurrently sustained. This dual validity underscores the robustness of the conjectures within the critical framework of black hole parameters.


The  WGC in CFT thermodynamic space is considered to yield a better result and is within a more acceptable range for several reasons and the coordination of the WGC and CFT can be checked in different works<cit.>


In summary, the WGC in CFT thermodynamic space is favored because it integrates well with established principles of quantum field theory and quantum gravity, offers a stable thermodynamic interpretation, and avoids theoretical inconsistencies, making it a more reliable and acceptable result within the theoretical physics community.


§ CONCLUSIONS

In this paper, we explore the intriguing interplay between fundamental theoretical physics concepts within the context of charged black holes. Specifically, we focus on the consistency of the  WGC and WCCC in CFT, and  RPS for RN AdS-PFDM. First, we have meticulously explored the thermodynamic properties of RN AdS black holes in the presence of  PFDM within the realm of RPS and CFT. Our research has been underpinned by a series of detailed figures that depict the evolution of free energy and Hawking temperature as functions of temperature and entropy, respectively. Our findings reveal a remarkable swallowtail pattern in the free energy of RN AdS black holes, which emerges under specific free parameter values and signifies a first-order phase transition. We found that despite the differences in EPST, RPST and CFT spaces, similar phase behavior is observed in all these spaces. In our study, we delve into the intricacies of the " WGC", which is a theoretical framework within the field of quantum gravity. The WGC posits that for any gauge force to be consistent with quantum gravity, it must exert forces that are stronger than gravitational forces on some particles. This conjecture has profound implications across various phase spaces, and we specifically focus on its application in "EPST", " RPST, and CFT" spaces. Our investigation highlights the critical role that thermodynamic principles and black hole parameters play in substantiating the WGC. Through rigorous exploration, we delineate the regions within these phase spaces where the WGC holds met. Interestingly, we observe that the parameter space accommodating the WGC is more expansive in the context of CFT, surpassing the confines observed in EPST and RPST spaces. Moreover, we pinpoint specific intervals for the parameters γ and l that lend credence to the WGC. By employing computational techniques, we not only bolster the WGC but also affirm the "WCCC". The WCCC, which is a fundamental principle in the study of black holes, asserts that singularities arising within black holes should be concealed from external observers, thereby preserving the predictability of physical laws. Hence, for a charged black hole, when (Q/M>√(G)), we will face a naked singularity, which violates WCCC, and therefore is not compatible with WGC, but in this case, adding PFDM will solve the mentioned problem and the compatibility of these two conjectures appears. In a pivotal examination at the critical juncture of black hole parameters, we scrutinize the WCCC and establish its validity. The critical condition f(M_c, Q_c, r_c) = 0 serves as a testament to the conjecture's soundness. Our findings lead us to conclude that both the WGC and WCCC receive substantial support at the critical points of black holes. This not only demonstrates the robustness of these conjectures but also underscores their significance within the critical areas of black hole physics.
11
1
Bekenstein, Jacob D. "Black holes and the second law." JACOB BEKENSTEIN: The Conservative Revolutionary. 2020. 303-306.
2
Bekenstein, Jacob D. "Black holes and entropy." Physical Review D 7.8 (1973): 2333.
3
Bardeen, James M., Brandon Carter, and Stephen W. Hawking. "The four laws of black hole mechanics." Communications in mathematical physics 31 (1973): 161-170.
4
Susskind, Leonard. "The world as a hologram." Journal of Mathematical Physics 36.11 (1995): 6377-6396.
5
Maldacena, Juan. "The large-N limit of superconformal field theories and supergravity." International journal of theoretical physics 38.4 (1999): 1113-1133.
6
Wald, Robert M. "Black hole entropy is the Noether charge." Physical Review D 48.8 (1993): R3427.
7
Hawking, Stephen W., and Don N. Page. "Thermodynamics of black holes in anti-de Sitter space." Communications in Mathematical Physics 87 (1983): 577-588.
8
Dolan, Brian P. "Compressibility of rotating black holes." Physical Review D 84.12 (2011): 127503.
9
Kubiznak, David, and Robert B. Mann. "P-V criticality of charged AdS black holes." Journal of High Energy Physics 2012.7 (2012): 1-25.
10
Cai, Rong-Gen, et al. "PV criticality in the extended phase space of Gauss-Bonnet black holes in AdS space." Journal of High Energy Physics 2013.9 (2013): 1-22.
11
Xu, Wei, and Liu Zhao. "Critical phenomena of static charged AdS black holes in conformal gravity." Physics Letters B 736 (2014): 214-220.
12
Xu, Wei, Hao Xu, and Liu Zhao. "Gauss–Bonnet coupling constant as a free thermodynamical variable and the associated criticality." The European Physical Journal C 74.7 (2014): 2970.
13
Yang, Si-Jiang, et al. "Weak cosmic censorship conjecture for a Kerr-Taub-NUT black hole with a test scalar field and particle." Physical Review D 101.6 (2020): 064048.
14
Cong, Wan, David Kubizňák, and Robert B. Mann. "Thermodynamics of AdS black holes: critical behavior of the central charge." Physical Review Letters 127.9 (2021): 091301.
15
Johnson, Clifford V. "Holographic heat engines." Classical and Quantum Gravity 31.20 (2014): 205002.
16
Ciambelli, Luca, et al. "Topological terms and the Misner string entropy." Physical Review D 103.2 (2021): 024052.
17
Karch, Andreas, and Brandon Robinson. "Holographic black hole chemistry." Journal of High Energy Physics 2015.12 (2015): 1-15.
18
Wei, Shao-Wen, Bin Liang, and Yu-Xiao Liu. "Critical phenomena and chemical potential of a charged AdS black hole." Physical Review D 96.12 (2017): 124018.
19
Rafiee, Morteza, et al. "Universal criticality of thermodynamic geometry for boundary conformal field theories in gauge/gravity duality." Physical Review D 105.2 (2022): 024058.
20
Sadeghi, Jafar, et al. "RPS thermodynamics of Taub–NUT AdS black holes in the presence of central charge and the weak gravity conjecture." General Relativity and Gravitation 54.10 (2022): 129.
21
Gao, Zeyuan, Xiangqing Kong, and Liu Zhao. "Thermodynamics of Kerr-AdS black holes in the restricted phase space." The European Physical Journal C 82.2 (2022): 112.
22
Bai, Yan-Ying, et al. "Revisit on thermodynamics of BTZ black hole with variable Newton constant." arXiv preprint arXiv:2208.11859 (2022).
23
Kong, Xiangqing, et al. "Restricted phased space thermodynamics for black holes in higher dimensions and higher curvature gravities." Entropy 24.8 (2022): 1131.
24
Ladghami, Y., et al. "4D-EGB black holes in RPS thermodynamics." Physics of the Dark Universe 41 (2023): 101261.
25
Ladghami, Yahya, et al. "Barrow entropy and AdS black holes in RPS thermodynamics." Physics of the Dark Universe (2024): 101470.
26
Cong, Wan, David Kubizňák, and Robert B. Mann. "Thermodynamics of AdS black holes: critical behavior of the central charge." Physical Review Letters 127.9 (2021): 091301.
27
Ahmed, Moaathe Belhaj, et al. "Holographic dual of extended black hole thermodynamics." Physical Review Letters 130.18 (2023): 181401.
28
Gong, Ting-Feng, Jie Jiang, and Ming Zhang. "Holographic thermodynamics of rotating black holes." Journal of High Energy Physics 2023.6 (2023): 1-22.
29
Ladghami, Yahya, and Taoufik Ouali. "Black holes thermodynamics with CFT re-scaling." arXiv preprint arXiv:2402.15913 (2024).
30
Palti, Eran. "The swampland: introduction and review." Fortschritte der Physik 67.6 (2019): 1900037.
31
van Beest, Marieke, et al. "Lectures on the swampland program in string compactifications." Physics Reports 989 (2022): 1-50.
32
Vafa, Cumrun. "The String landscape and the swampland." arXiv preprint hep-th/0509212 (2005).
33
Ooguri, Hirosi, and Cumrun Vafa. "On the Geometry of the String Landscape and the Swampland." Nuclear physics B 766.1-3 (2007): 21-33.
34
Arkani-Hamed, Nima, et al. "The string landscape, black holes and gravity as the weakest force." Journal of High Energy Physics 2007.06 (2007): 060.
35
Heidenreich, Ben, Matthew Reece, and Tom Rudelius. "Evidence for a sublattice weak gravity conjecture." Journal of High Energy Physics 2017.8 (2017): 1-40.
36
Palti, Eran. "The weak gravity conjecture and scalar fields." Journal of High Energy Physics 2017.8 (2017): 1-26.
37
Odintsov, Sergei D., and Vasilis K. Oikonomou. "Swampland implications of GW170817-compatible Einstein-Gauss-Bonnet gravity." Physics Letters B 805 (2020): 135437.
38
Ooguri, Hirosi, and Cumrun Vafa. "On the Geometry of the String Landscape and the Swampland." Nuclear physics B 766.1-3 (2007): 21-33.
c
Liu, Yang. "Higgs inflation and its extensions and the further refining dS swampland conjecture." The European Physical Journal C 81.12 (2021): 1122.
d
Liu, Yang. "Higgs inflation and scalar weak gravity conjecture." The European Physical Journal C 82.11 (2022): 1052.
e
Liu, Yang. "Higgs inflation and its extensions and the further refining dS swampland conjecture." The European Physical Journal C 81.12 (2021): 1122.
f
Arkani-Hamed, Nima, et al. "The string landscape, black holes and gravity as the weakest force." Journal of High Energy Physics 2007.06 (2007): 060.
g
Gashti, S. Noori, and J. Sadeghi. "Refined swampland conjecture in warm vector hybrid inflationary scenario." The European Physical Journal Plus 137.6 (2022): 1-13.
h
Alipour, Mohammad Reza, Jafar Sadeghi, and Mehdi Shokri. "WGC and WCC for charged black holes with quintessence and cloud of strings." The European Physical Journal C 83.7 (2023): 1-7.
i
Alipour, Mohammad Reza, Jafar Sadeghi, and Mehdi Shokri. "WGC and WCCC of black holes with quintessence and cloud strings in RPS space." Nuclear Physics B 990 (2023): 116184.
j
Sadeghi, Jafar, Mohammad Reza Alipour, and Saeed Noori Gashti. "Emerging WGC from the Dirac particle around black holes." Modern Physics Letters A 38.26n27 (2023): 2350122.
k
Schöneberg, Nils, et al. "News from the Swampland—constraining string theory with astrophysics and cosmology." Journal of Cosmology and Astroparticle Physics 2023.10 (2023): 039.
l
Kadota, Kenji, et al. "Trans-Planckian censorship and single-field inflaton potential." Journal of Cosmology and Astroparticle Physics 2020.01 (2020): 008.
m
Oikonomou, V. K. "Rescaled Einstein-Hilbert gravity from f (R) gravity: Inflation, dark energy, and the swampland criteria." Physical Review D 103.12 (2021): 124028.
n
Sadeghi, J., et al. "de Sitter Swampland Conjecture in String Field Inflation." The European Physical Journal C 83 (2023) : (635).
o
Harlow, Daniel, et al. "Weak gravity conjecture." Reviews of Modern Physics 95.3 (2023): 035003.
oooo
Cho, Minseok, et al. "Supersymmetric Cardy formula and the Weak Gravity Conjecture in AdS/CFT." Journal of High Energy Physics 2023.11 (2023): 1-26.
ooooo
Nakayama, Yu, and Yasunori Nomura. "Weak gravity conjecture in the AdS/CFT correspondence." Physical Review D 92.12 (2015): 126006.
oooooo
Aalsma, Lars, et al. "A new spin on the weak gravity conjecture." Journal of High Energy Physics 2021.3 (2021): 1-39.
aaaaa
Heidenreich, Ben, Matthew Reece, and Tom Rudelius. "The weak gravity conjecture and axion strings." Journal of High Energy Physics 2021.11 (2021): 1-24.
aaaa
Loges, Gregory J., Toshifumi Noumi, and Gary Shiu. "Thermodynamics of 4D dilatonic black holes and the weak gravity conjecture." Physical Review D 102.4 (2020): 046010.
p
Capozziello, Salvatore, et al. "Hydrostatic equilibrium and stellar structure in f (R) gravity." Physical Review D 83.6 (2011): 064004.
q
Gashti, S. Noori, et al. "Swampland dS conjecture in mimetic f (R, T) gravity." Communications in Theoretical Physics 74.8 (2022): 085402.
r
Das, Suratna. "Distance, de Sitter and Trans-Planckian Censorship conjectures: the status quo of Warm Inflation." Physics of the Dark Universe 27 (2020): 100432.
s
Yuennan, Jureeporn, and Phongpichit Channuie. "Further Refining Swampland Conjecture on Inflation in General Scalar‐Tensor Theories of Gravity." Fortschritte der Physik 70.6 (2022): 2200024.
t
Bedroya, Alek, and Cumrun Vafa. "Trans-Planckian censorship and the swampland." Journal of High Energy Physics 2020.9 (2020): 1-34.
u
Sadeghi, Jafar, et al. "Can black holes cause cosmic expansion?." arXiv preprint arXiv:2305.12545 (2023).
v
Mohammadi, Abolhassan, Tayeb Golanbari, and Jamil Enayati. "Brane inflation and Trans-Planckian censorship conjecture." Physical Review D 104.12 (2021): 123515.
w
Sadeghi, Jafar, Mohammad Reza Alipour, and Saeed Noori Gashti. "Scalar Weak Gravity Conjecture in Super Yang-Mills Inflationary Model." Universe 8.12 (2022): 621.
x
Kallosh, Renata, et al. "dS Vacua and the Swampland." Journal of High Energy Physics 2019.3 (2019): 1-18.
y
Guleryuz, Omer. "On the Trans-Planckian Censorship Conjecture and the generalized non-minimal coupling." Journal of Cosmology and Astroparticle Physics 2021.11 (2021): 043.
z
Osses, Constanza, Nelson Videla, and Grigoris Panotopoulos. "Reheating in small-field inflation on the brane: the swampland criteria and observational constraints in light of the PLANCK 2018 results." The European Physical Journal C 81 (2021): 1-29.
aa
Sadeghi, J., S. Noori Gashti, and M. R. Alipour. "Notes on further refining de Sitter swampland conjecture with inflationary models." Chinese Journal of Physics 79 (2022): 490-502.
bb
Brahma, Suddhasattwa. "Trans-Planckian censorship, inflation, and excited initial states for perturbations." Physical Review D 101.2 (2020): 023526.
cc
Brandenberger, Robert. "Trans-Planckian censorship conjecture and early universe cosmology." arXiv preprint arXiv:2102.09641 (2021).
dd
Sadeghi, J., S. Noori Gashti, and F. Darabi. "Swampland conjectures in hybrid metric-Palatini gravity." Physics of the Dark Universe (2022): 101090.
ee
Geng, Hao, Sebastian Grieninger, and Andreas Karch. "Entropy, entanglement and swampland bounds in DS/dS." Journal of High Energy Physics 2019.6 (2019): 1-16.
ff
Gashti, S. Noori, J. Sadeghi, and B. Pourhassan. "Pleasant behavior of swampland conjectures in the face of specific inflationary models." Astroparticle Physics 139 (2022): 102703.
gg
Sadeghi, J., E. Naghd Mezerji, and S. Noori Gashti. "Study of some cosmological parameters in logarithmic corrected f (R) gravitational model with swampland conjectures." Modern Physics Letters A 36.05 (2021): 2150027.
hh
Sadeghi, J., S. Noori Gashti, and E. Naghd Mezerji. "The investigation of universal relation between corrections to entropy and extremality bounds with verification WGC." Physics of the Dark Universe 30 (2020): 100626.
ii
Agrawal, Prateek, et al. "On the cosmological implications of the string swampland." Physics Letters B 784 (2018): 271-276.
jj
Odintsov, Sergei D., and Vasilis K. Oikonomou. "Swampland implications of GW170817-compatible Einstein-Gauss-Bonnet gravity." Physics Letters B 805 (2020): 135437.
kk
Sadeghi, J., and S. Noori Gashti. "Investigating the logarithmic form of f (R) gravity model from brane perspective and swampland criteria." Pramana 95 (2021): 1-8.
ll
Sharma, Umesh Kumar. "Reconstruction of quintessence field for the THDE with swampland correspondence in f (R, T) gravity." International Journal of Geometric Methods in Modern Physics 18.02 (2021): 2150031.
mm
Sadeghi, Jafar, Mohammad Reza Alipour, and Saeed Noori Gashti. "Strong cosmic censorship in light of weak gravity conjecture for charged black holes." Journal of High Energy Physics 2023.2 (2023): 1-14.
nn
Odintsov, S. D., V. K. Oikonomou, and L. Sebastiani. "Unification of constant-roll inflation and dark energy with logarithmic R2-corrected and exponential F (R) gravity." Nuclear Physics B 923 (2017): 608-632.
oo
Sadeghi, J., et al. "de Sitter Swampland Conjecture in String Field Inflation."  The European Physical Journal C 83 (2023): (635).
pp
Shokri, Mehdi, Jafar Sadeghi, and Saeed Noori Gashti. "Quintessential constant-roll inflation." Physics of the Dark Universe 35 (2022): 100923.
qq
Sadeghi, J., et al. "Swampland conjecture and inflation model from brane perspective." Physica Scripta 96.12 (2021): 125317.
rr
Shokri, Mehdi, et al. "Warm inflation with bulk viscous pressure for different solutions of an anisotropic universe." arXiv preprint arXiv:2112.12309 (2021).
ss
Sadeghi, J., et al. "The emergence of universal relations in the AdS black holes thermodynamics." Physica Scripta 98.2 (2023): 025305.
tt
Yuennan, Jureeporn, and Phongpichit Channuie. "Composite Inflation and further refining dS swampland conjecture." Nuclear Physics B 986 (2023): 116033.
uu
Gashti, S. Noori, and J. Sadeghi. "Refined swampland conjecture in warm vector hybrid inflationary scenario." The European Physical Journal Plus 137.6 (2022): 1-13.
vv
Kinney, William H. "Eternal inflation and the refined swampland conjecture." Physical review letters 122.8 (2019): 081302.
ww
Kinney, William H. "The swampland conjecture bound conjecture." arXiv preprint arXiv:2103.16583 (2021).
xx
Yu, Ten-Yeh, and Wen-Yu Wen. "Cosmic censorship and weak gravity conjecture in the Einstein–Maxwell-dilaton theory." Physics Letters B 781 (2018): 713-718.
yy
Gashti, S. Noori. "Two-field inflationary model and swampland de Sitter conjecture." Journal of Holography Applications in Physics 2 (1) (2022): 13-24.
zz
Vafa, Cumrun. "The String landscape and the swampland." arXiv preprint hep-th/0509212 (2005).
aaa
Sadeghi, Jafar, et al. "Weak gravity conjecture from conformal field theory: a challenge from hyperscaling violating and Kerr-Newman-AdS black holes." Chinese Physics C 47.1 (2023): 015103.
bbb
van Beest, Marieke, et al. "Lectures on the swampland program in string compactifications." Physics Reports 989 (2022): 1-50.
ccc
Sadeghi, J., et al. "Weak gravity conjecture, black branes and violations of universal thermodynamics relation." Annals of Physics 447 (2022): 169168.
ddd
Kolb, Edward W., Andrew J. Long, and Evan McDonough. "Gravitino swampland conjecture." Physical Review Letters 127.13 (2021): 131603.
eee
Yuennan, Jureeporn, and Phongpichit Channuie. "Further Refining Swampland Conjecture on Inflation in General Scalar‐Tensor Theories of Gravity." Fortschritte der Physik 70.6 (2022): 2200024.
fff
Palti, Eran. "The swampland: introduction and review." Fortschritte der Physik 67.6 (2019): 1900037.
ggg
Noori Gashti, S., J. Sadeghi, and M. R. Alipour. "Further refining swampland dS conjecture in mimetic f (G) gravity." International Journal of Modern Physics D 32.03 (2023): 2350011.
hhh
Sadeghi, Jafar, et al. "Weak gravity conjecture from conformal field theory: a challenge from hyperscaling violating and Kerr-Newman-AdS black holes." Chinese Physics C 47.1 (2023): 015103.
iii
Sadeghi, Jafar, et al. "Cosmic evolution of the logarithmic f (R) model and the dS swampland conjecture." Universe 8.12 (2022): 623.
39
Penrose, Roger. "Gravitational collapse and space-time singularities." Physical Review Letters 14.3 (1965): 57.
40
Harlow, Daniel, et al. "The weak gravity conjecture: a review." arXiv preprint arXiv:2201.08380 (2022).
41
Shaymatov, Sanjar, Bobomurat Ahmedov, and Mubasher Jamil. "Testing the weak cosmic censorship conjecture for a Reissner–Nordström–de Sitter black hole surrounded by perfect fluid dark matter." The European Physical Journal C 81 (2021): 1-11.
42
Kiselev, VV1966820. "Quintessence and black holes." Classical and Quantum Gravity 20.6 (2003): 1187.
43
Li, Ming-Hsun, and Kwei-Chou Yang. "Galactic dark matter in the phantom field." Physical Review D 86.12 (2012): 123015.
44
Kiselev, V. V. "Quintessential solution of dark matter rotation curves and its simulation by extra dimensions." arXiv preprint gr-qc/0303031 (2003).
45
Xu, Zhaoyi, Xian Hou, and Jiancheng Wang. "Kerr–anti-de Sitter/de Sitter black hole in perfect fluid dark matter background." Classical and Quantum Gravity 35.11 (2018): 115003.
455
Aharony, Ofer, and Eran Palti. "Convexity of charged operators in CFTs and the weak gravity conjecture." Physical Review D 104.12 (2021): 126005.
4555
Urbano, A. "Towards a proof of the weak gravity conjecture (2018)." arXiv preprint arXiv:1810.05621.
45555
Sadeghi, Jafar, and Saeed Noori Gashti. "Reissner-Nordström black holes surrounded by perfect fluid dark matter: testing the viability of weak gravity conjecture and weak cosmic censorship conjecture simultaneously." Physics Letters B (2024): 138651.
455555
Moitra, Upamanyu. "Newton vs Coulomb in AdS/CFT and the weak gravity conjecture." Physical Review D 109.4 (2024): L041903.

