We investigate the phenomenon of gravitational baryogenesis within the context of a specific modified theory of gravity namely energymomentum squared gravity or fR TmunuTmunu gravity. In this framework the gravitational Lagrangian is formulated as a general function of the Ricci scalar R and the selfcontraction of the energymomentum tensor mathcalT2 equiv TmunuTmunu. This approach extends the conventional paradigm of gravitational baryogenesis by introducing new dependencies that allow for a more comprehensive exploration of the baryon asymmetry problem. Our analysis aims to elucidate the role of these gravitational modifications in the generation of baryon asymmetry a critical issue in cosmology that remains unresolved within the Standard Model of particle physics. By incorporating mathcalT2 into the gravitational action we propose that these modifications can significantly influence the dynamics of the early universe thereby altering the conditions under which baryogenesis occurs. This study not only provides a novel depiction of gravitational baryogenesis but also offers insights into how modified gravity theories can address the longstanding question of baryon asymmetry. The implications of our findings suggest that fR TmunuTmunu gravity could play a crucial role in understanding the fundamental processes that led to the matterantimatter imbalance observed in the universe today.
['General Relativity and Quantum Cosmology', 'Astrophysics - High Energy Astrophysical Phenomena', 'High Energy Physics - Theory']
Gravitational baryogenesis in energymomentum squared gravity
{'Gravitational baryogenesis in energy-momentum squared gravity': "David S. Pereira, 1, 2, ∗ Francisco S.N. Lobo, 1, 2, † and Jos'e Pedro Mimoso 1, 2, ‡  \n1 Departamento de F'ısica, Faculdade de Ciˆencias da Universidade de Lisboa,  \nCampo Grande, Edif'ıcio C8, P-1749-016 Lisbon, Portugal  \n2 Instituto de Astrof'ısica e Ciˆencias do Espa¸co, Faculdade de Ciˆencias da Universidade de Lisboa, Campo Grande, Edif'ıcio C8, P-1749-016 Lisbon, Portugal  \n(Dated: September 10, 2024)  \nWe investigate the phenomenon of gravitational baryogenesis within the context of a specific modified theory of gravity, namely, energy-momentum squared gravity or f ( R,T µν T µν ) gravity. In this framework, the gravitational Lagrangian is formulated as a general function of the Ricci scalar R and the self-contraction of the energy-momentum tensor, T 2 ≡ T µν T µν . This approach extends the conventional paradigm of gravitational baryogenesis by introducing new dependencies that allow for a more comprehensive exploration of the baryon asymmetry problem. Our analysis aims to elucidate the role of these gravitational modifications in the generation of baryon asymmetry, a critical issue in cosmology that remains unresolved within the Standard Model of particle physics. By incorporating T 2 into the gravitational action, we propose that these modifications can significantly influence the dynamics of the early universe, thereby altering the conditions under which baryogenesis occurs. This study not only provides a novel depiction of gravitational baryogenesis but also offers insights into how modified gravity theories can address the longstanding question of baryon asymmetry. The implications of our findings suggest that f ( R,T µν T µν ) gravity could play a crucial role in understanding the fundamental processes that led to the matter-antimatter imbalance observed in the universe today.", 'I. INTRODUCTION': 'Among the open questions concerning the primordial universe, one of the most intriguing is the observed asymmetry between matter and antimatter [1], as indicated by cosmological observations [2-4]. This asymmetry is characterized by the baryon-to-entropy ratio, η s = n b /s , where n b represents the net baryon number density, the difference between the number density of baryons ( n B ) and the number density of anti-baryons ( n ¯ B ), and s denotes the radiation entropy density. Measurements of the Cosmic Microwave Background anisotropies and data from Big Bang Nucleosynthesis suggest that the magnitude of the baryon asymmetry is approximately [5-8] η s ≲ 9 . 2 × 10 -11 . Indeed, in 1967, Sakharov [9] proposed that the present baryon density might not stem from unnatural initial conditions but could instead be understood through fundamental microphysical laws. These laws describe how an initially symmetric universe could evolve to produce the observed asymmetry. Sakharov identified three key conditions necessary for generating this asymmetry: (i) baryon number violation; (ii) violation of C (charge conjugation symmetry) and CP (the combined symmetry of charge conjugation and parity); and (iii) a departure from thermal equilibrium.  \nThe first two conditions are relatively straightforward: a baryon number-violating process is logically required to generate an excess of baryons over anti-baryons, and the violation of C (charge conjugation) and CP (charge  \nconjugation and parity) symmetries is necessary to favor matter over antimatter in baryon number-violating interactions. If these symmetries were preserved, such interactions would produce baryons and anti-baryons at equal rates, resulting in a net baryon number of zero. Such processes, like sphalerons, are present in the Standard Model within electroweak theory, leading to the wellknown mechanism of electroweak baryogenesis [10]. However, this mechanism encounters significant challenges, such as the insufficient amount of CP violation within the Standard Model and the nature of the electroweak phase transition, which is a crossover rather than a firstorder transition. For further details, see [10-13].  \nThe third condition, unlike the others, is not as straightforward to justify. The underlying principle here is that in thermal equilibrium, any process generating baryon asymmetry will have a corresponding reverse process. More specifically, if a process X → Y + b occurs in thermal equilibrium, it will be counterbalanced by the reverse process Y + b → X , resulting in no net change in baryon number. Statistically, when the Hamiltonian commutes with CPT symmetry, [H , CPT] = 0, the distribution of particles and anti-particles remains identical, leading to n B = n ¯ B . Thus, a departure from equilibrium appears necessary to achieve a net baryon asymmetry. However, these conditions are not strictly indispensable [14-18]. For example, the first condition can be circumvented by considering the existence of a hidden or sterile baryon sector [18]. In this scenario, the baryon asymmetry in the visible sector would be exactly balanced by an asymmetry in the hidden sector composed of these sterile baryons, thus conserving the overall baryon number.  \nRegarding the out-of-equilibrium criterion, it has been  \n̸  \nshown in [14] that baryon asymmetry can be generated even within a state of thermodynamic equilibrium, without requiring a departure from thermal equilibrium. This finding significantly advances our understanding of baryogenesis. The key result is that in an expanding Universe at finite temperature, CPT symmetry may not be preserved, meaning that CPT can be (spontaneously) violated. Under CPT invariance, the baryon number must be generated out of thermal equilibrium, as CPT symmetry ensures that the thermal distributions of baryons and anti-baryons are identical. However, if CPT is spontaneously violated within a theoretical framework, this condition is disrupted, leading to an altered baryonantibaryon spectrum. This disparity results in a nonzero baryon asymmetry, n B -n ¯ B = 0. A CPT violation enables the generation of baryon asymmetry during a period when interactions that violate baryon and lepton number conservation persist in thermal equilibrium. This asymmetry becomes fixed at the decoupling temperature T D , marking the point where these interactions cease to be in equilibrium. The value of T D can be determined by equating the rate of these processes, denoted by Γ, to the expansion rate of the Universe, characterized by the Hubble constant H . In other words, this occurs when H ( T D ) ≃ Γ( T D ).  \nIn the context of an expanding Universe, the generation of baryon asymmetry in thermal equilibrium proceeds as follows: During epochs when the expansion rate H significantly exceeds the interaction rate Γ, i.e., H ≫ Γ, corresponding to T > T D , baryon asymmetry is generated by processes that violate baryon number (B) or lepton number (L) while maintaining thermal equilibrium. As the temperature decreases to T = T D , where H ≈ Γ, these processes decouple. Subsequently, when H < Γ, or T < T D , the baryon asymmetry becomes effectively frozen. This sequence is schematically illustrated in Fig. 1. This phenomenon has been extensively studied in the context of spontaneous baryogenesis [14, 19], which requires only baryon number (B) violation to occur while maintaining thermal equilibrium, and is generally more efficient under these conditions.  \nSince the formulation of the Sakharov conditions [9], numerous theories have emerged that propose mechanisms responsible for generating baryon asymmetry, collectively referred to as baryogenesis mechanisms. These theories currently represent the leading explanations for the observed matter-antimatter imbalance in the universe (see [20-23] for comprehensive reviews). Typically, these mechanisms are grounded in particle physics, utilizing fundamental interactions and processes to produce the necessary asymmetry. Gravitational effects, such as the expansion rate of the universe, are often considered secondary influences in these frameworks. However, in a significant departure from this conventional approach, Davoudiasl et al. [24] proposed a novel mechanism known as gravitational baryogenesis. This mechanism considers that gravitational interactions, specifically those involving the derivative of the Ricci scalar curvature and  \nFIG. 1. Schematic representation of the generation of baryon asymmetry in the regime of thermal equilibrium.  \n<!-- image -->  \nthe baryon-number current, can play a critical role in generating baryon asymmetry. This idea introduces a new perspective on how gravity can play a crucial role in baryogenesis.  \n̸  \nFollowing this groundbreaking work, a number of subsequent studies have investigated gravitational baryogenesis within the framework of modified theories of gravity. These studies not only altered the traditional descriptions of gravitational interactions but also introduced new theoretical constructs that incorporate similar forms of interaction to those proposed by Davoudiasl et al . For instance, gravitational baryogenesis was modified in f ( R ) gravity [25], where it was shown that gravitational Lagrangians of the form f ( R ) ∼ R n , where n = 1 (where n = 1 reduces to General Relativity) only for tiny deviations of a few percent, are consistent with the current bounds on the observed baryon asymmetry. This led to the emergence of studies on gravitational baryogenesis within the context of modified gravity theories [26-31], where the concept of introducing alternative CPviolating terms, in place of the conventional term described in [24] (see section II below), became a recurring approach. These explorations have expanded our understanding of how modified theories of gravity might impact the baryogenesis process, potentially offering new insights into the fundamental mechanisms that govern the matter-antimatter asymmetry in the universe.  \nIn this paper, we propose a detailed depiction of modified gravitational baryogenesis in the context of EnergyMomentum-Squared Gravity (EMSG), i.e., f ( R,T µν T µν )  \ngravity [32, 33] (see Ref. [34] for a recent review). In this framework, R denotes the Ricci scalar, while T µν represents the energy-momentum tensor. EMSG introduces quadratic terms involving the energy-momentum tensor components into the gravitational action, which has significant and intriguing implications for cosmology, especially during the early stages of the universe. The incorporation of these quadratic terms modifies the gravitational dynamics under high-energy conditions, offering potential solutions to several longstanding issues in General Relativity (GR). Notably, EMSG could provide insights into the nature of the initial singularity and address specific challenges related to Big Bang Nucleosynthesis.  \nA distinctive feature of EMSG is its explicit nonminimal coupling between matter and geometry, which leads to the non-conservation of the energy-momentum tensor. This non-conservation requires the application of irreversible thermodynamics to effectively analyze cosmological scenarios within the EMSG framework. Through this formalism, the energy balance equations in EMSG can be interpreted from a thermodynamic perspective, describing them as irreversible matter creation processes. Several related theories also exhibit this nonminimal coupling, including f ( R, L m ) [35, 36], f ( R,T ) [37, 38], and f ( R,T,R µν T µν ) [39, 40] theories of gravity. Here, R represents the Ricci scalar, R µν denotes the Ricci tensor, L m is the matter Lagrangian density, and T is the trace of the energy-momentum tensor T µν . These theories modify the standard Einstein-Hilbert action by introducing additional terms that couple the spacetime geometry with matter in various ways, leading to new dynamics and effects that extend beyond the predictions of GR.  \nWhile EMSG reduces to f ( R ) gravity [41] (and to GR for f ( R ) = R ) in the absence of matter, deviations become pronounced in the presence of an energymomentum distribution, especially in high-curvature regions. Thus, in the context of modified gravitational baryogenesis, we will examine these interactions within f ( R,T µν T µν ) gravity. This framework allows us to explore how the inclusion of quadratic terms involving the energy-momentum tensor affects gravitational baryogenesis. By analyzing this specific theoretical setup, we aim to uncover new insights into how modifications to gravitational theories can influence the generation of baryon asymmetry and address some of the outstanding questions in cosmology and gravitational physics.  \nThis paper is organized in the following manner: In Sec. II, we provide a theoretical introduction to gravitational baryogenesis and review the underlying mechanisms of this process. In Sec. III, we introduce the concept of modified gravitational baryogenesis, dividing the discussion into two potential approaches based on modified gravity theories. Section IV presents the theoretical framework of f ( R,T µν T µν ) gravity and the cosmological implications of the theory. In Sec. V, we explore the modified gravitational baryogenesis framework within the context of EMSG. Finally, Sec. VI provides a  \nsummary and discussion of our results.  \nThroughout this paper, we consider natural units where ℏ = c = k B = 1, and define the Planck mass to be M Pl = (8 πG ) -2 , with G being the gravitational constant. For notational simplicity, we also consider the definition T 2 = T µν T µν .', 'A. Formalism': 'Gravitational baryogenesis [24] is a mechanism that generates baryon asymmetry while maintaining thermal equilibrium, through the interaction term:  \nL int = 1 M 2 ∗ ∂ µ ( R ) J µ B . (1)  \nIn an expanding universe, this term leads to CP and CPT violating interactions by directly coupling the derivative of the Ricci scalar, R , with the baryon number (B) current J µ B . This interaction can also be applied to a lepton (L) current or any current that results in a net B -L charge in equilibrium [24]. The parameter M 2 ∗ represents the cutoff scale of the effective theory, and it is hypothesized that this interaction originates from a higherdimensional operator in supergravity theories [42, 43].  \nFurthermore, as suggested in [24], such an operator may emerge within a low-energy effective field theory of quantum gravity, particularly if the fundamental scale M ∗ is of the order of the reduced Planck scale M Pl = (8 πG N ) -1 / 2 ≈ 2 . 4 × 10 18 GeV. This mechanism requires baryon number-violating processes to be in thermal equilibrium to produce the asymmetry. The key feature of this mechanism is that CP violation arises from a gravitational interaction [24]. As the universe expands, microscopic CP violation is amplified by a shift in baryon energy, leading to a dynamic violation of CPT that alters the relative energies of particles and antiparticles. As a result, a CP-conserving but baryon number-violating interaction in equilibrium can generate the asymmetry, which is enhanced by the term (1) and becomes fixed when the baryon number-violating interaction decouples.  \nHence, the asymmetry resulting from (1) is derived as follows: in a flat FLRW Universe, for a spatially constant R , the interaction (1) induces different energy contributions for particles and antiparticles, thereby altering their thermal equilibrium distributions, in the following way  \n1 M 2 ∗ ( ∂ µ R ) J µ B = 1 M 2 ∗ ˙ R ( n B -n ¯ B ) . (2)  \nHere, we use J 0 B = n B -n ¯ B , where n B and n ¯ B represent the baryon and anti-baryon number densities, respectively. This contribution introduces a shift in the energy of baryons, approximately 2 ˙ R/M 2 ∗ relative to anti-baryons, leading to CPT symmetry violation. As a result, the chemical potential for baryons is given by  \nµ B = -µ ¯ B = -˙ R/M 2 ∗ . Under thermal equilibrium conditions, in the regime where T ≫ m , the net baryon number density is expressed as [44]:  \nn b = g b 2 π 2 ∫ d 3 p [ 1 e ( p -µ B ) T +1 -1 e ( p + µ B ) T +1 ] , (3)  \nwhere g b is the number of internal degrees of freedom of baryons. In an homogeneous and isotropic Universe, this reduces to  \nn b = g b T 3 6 π 2 [ π 2 ( µ B T ) + ( µ B T ) 3 ] . (4)  \nThe conservation of entropy allows the number of a given particle species within a comoving volume to be expressed as the ratio of the particle number density to the entropy density. Formally, this relationship can be written as  \nN i ≡ n i s , (5)  \nwhere  \ns = 2 π 2 45 g ∗ s ( T ) T 3 , (6)  \nand g ∗ s ( T ) is given by  \ng ∗ s ( T ) = ∑ i =bosons g i ( T i T ) 3 + 7 8 ∑ i =fermions g i ( T i T ) 3 . (7)  \nBy considering that the asymmetry is generated in thermal equilibrium, we can reasonably assume that g ∗ s = g ∗ , where g ∗ denotes the total effective number of relativistic degrees of freedom [44]. Therefore, if particles are neither produced nor destroyed, as is the case after baryogenesis has concluded, the number density n i scales with the inverse cube of the scale factor, n i ∝ a -3 , and N i constant. This provides a well-defined parameter for observational purposes.  \nIn an expanding Universe, considering the relativistic regime ( T ≫ m ) and T ≫ µ B , using Eq.(3) and Eq.(6) the asymmetry produced by the interaction given in Eq.(1) at the temperature where the baryon number violation processes decouple, T D , is given by [24]  \nn b s ≃ -15 g b 4 π 2 g ∗ ˙ R M 2 ∗ T ∣ ∣ ∣ ∣ ∣ T D , (8)  \nwhere ˙ R is determined based on the cosmological model used. In essence, the coupling term in (1) modifies the thermal equilibrium distribution and the chemical potential, driving the universe toward a nonzero equilibrium B-asymmetry (or L-asymmetry) through B-violating (or L-violating) interactions. This mechanism bears a resemblance with spontaneous baryogenesis [14, 19], wherein gravitational baryogenesis involves the Ricci scalar of  \na gravitational background assuming the role typically played by an axion in spontaneous baryogenesis.  \nIn spontaneous baryogenesis, the underlying symmetry of the theory, which ensures the conservation of total baryon number in the unbroken phase, is spontaneously broken. When this symmetry breaks, the Lagrangian density acquires the following term [14, 19, 45]  \nL = ( ∂ µ ϕ ) f J µ B , (9)  \nwhere ϕ is typically a pseudo-Nambu-Goldstone boson associated with an approximate symmetry related to the baryon number and f is a parameter associated with the symmetry breaking scale. Assuming a spatially homogeneous field, ϕ ≡ ϕ ( t ), in the same way as done in gravitational baryogenesis one has  \nL = ˙ ϕ f J 0 B = ˙ ϕ f ( n B -n ¯ B ) . (10)  \nHowever, in spite of the similarity between the spontaneous baryogenesis and gravitational baryogenesis, the distinctions between the two primarily lie in the behavior of both entities, the scalar field ϕ and the Ricci scalar. For spontaneous baryogenesis to occur, it is imperative that the scalar field ϕ engaged in the process of spontaneous symmetry breaking undergoes a uniform evolution. This evolution must be such that it progresses homogeneously in one direction while contrasting with the evolution in the other direction to bring about the desired asymmetry. Moreover, a spatially consistent representation is also essential. These requirements are fulfilled inherently in gravitational baryogenesis due to the homogeneity of the Universe.  \nOne challenge with the scalar field in spontaneous baryogenesis is that its minimum occurs where the average of ˙ ϕ equals zero, driving the generated asymmetry to zero and complicating perturbations around this minimum. In contrast, the Ricci scalar, with a nonzero mean value, ˙ R ∼ H 3 , avoids this issue. Additionally, as with gravitational baryogenesis, it is possible to identify ˙ ϕ/f as a chemical potential in Eq.(10). However, as noted in [46, 47], this interpretation is complex due to the specific representation chosen for the fermionic fields. Therefore, the assumption that including the term ( ˙ ϕ/f ) J 0 B ∼ ( ˙ ϕ/f ) n b in the Lagrangian density of spontaneous baryogenesis leads to an energy difference between particles and antiparticles, effectively acting as a chemical potential that generates baryon asymmetry, should be approached with caution and careful consideration.', 'B. Instabilities': 'While gravitational baryogenesis offers significant advantages, potential instabilities must be carefully considered. As noted in [18] and [48], certain models incorporating the CPT-violating term (1) in the gravitational  \naction and a matter Lagrangian that includes bosonic or fermionic components can lead to instabilities, rendering the cosmological model untenable. Recently, the authors who identified these instabilities proposed a possible solution in [49]. Their solution involves modifying the theory of gravity by adding a quadratic Ricci scalar term to the gravitational action. Such modifications can prevent these instabilities, supporting the argument for introducing modified gravity in the context of gravitational baryogenesis. Although these models exhibit instabilities, their relevance becomes questionable when considering the scales involved in the cosmological description.  \nBy employing a thermodynamic approach to L m , a stable cosmological model can be achieved without the presence of such instabilities. The inclusion of the CPviolating term (1) in the gravitational action, however, lacks proper justification and has been introduced primarily to investigate possible back-reaction effects of the interaction considered. Given that this term is expected to arise from quantum gravity and supergravity theories as a higher-dimensional operator in the Kahler potential, the construction of a classical theory of gravity that includes both GR and the asymmetry-generating term is questionable. Therefore, this issue will not be addressed in this work.  \nIn this study, gravitational baryogenesis is described without incorporating the specific matter sectors containing bosons or fermions done in [18, 48]. Instead, a macroscopic description of the matter Lagrangian is used. The gravitational action will not include the interaction term from Eq. (1), which will be treated separately.', 'III. MODIFIED GRAVITATIONAL BARYOGENESIS': 'The concept of modifying gravitational baryogenesis emerged soon after the original formulation was introduced [50]. The first modification, also known as generalized gravitational baryogenesis, involved coupling ∂ µ f ( R ) with J µ , where f ( R ) ∼ ln( R ), leading to the interaction term  \nL int = -α ∂ µ R R J µ . (11)  \nThis operator is of a type expected to arise when heavy particles or extra dimensions are integrated out [51, 52], with α representing the strength of this new interaction in the effective theory.  \nBuilding on this work, a modification to gravitational baryogenesis was explored by employing f ( R ) gravity to describe the gravitational interaction, which subsequently influenced the outcomes of gravitational baryogenesis [25]. These developments sparked an emerging interest in studying gravitational baryogenesis within the context of modified theories of gravity, where alternative CP-violating terms, beyond the standard term (1), were  \nfrequently considered. In this work, we propose the introduction of specific modified terms within the framework of these theories, categorizing them into two distinct classes.  \nFrom a phenomenological standpoint, modifying gravitational baryogenesis can address several challenges inherent in the original formulation. One key issue is generating a non-vanishing asymmetry during the radiationdominated epoch characterized by w = 1 / 3, although this concern can be circumvented by considering w ≈ 1 / 3 [24]. Furthermore, scenarios that produce either an insufficient or excessive asymmetry may benefit from modified gravitational baryogenesis. This approach may also help resolve inherent issues, such as the aforementioned instabilities. In this context, we propose two approaches to modify and expand gravitational baryogenesis 1 , which can be applied separately or in combination.', 'A. Modified gravitational theories': 'The first approach involves adopting alternative descriptions of gravity, leading to different field equations, cosmological models, or frameworks. These alternatives result in distinct outcomes for the original gravitational baryogenesis scenario. This is exemplified by the modified f ( R ) gravity case presented in [25], and further explored in well-constructed examples of cosmological framework modifications [57-59].  \nThe motivation for altering the gravitational and cosmological framework can be linked to addressing current issues, such as the lack of a solid theoretical foundation for the accelerated expansion of the universe, cosmological tensions, singularities in GR (e.g., in black hole interiors and cosmological solutions), the lithium problem [60], and even the study of quantum gravity theories. Employing such frameworks enables a more comprehensive exploration of these alternative approaches.', 'B. Different couplings with J µ': "The second approach involves generalizing the CPviolating interaction itself. Rather than coupling only the derivative of the Ricci scalar with the baryon current, other quantities can be considered for this coupling. This modification is linked to the specific modified theory of gravity being utilized and is introduced in a similar manner to spontaneous baryogenesis [14, 19] or quintessential baryogenesis [61]. The initial generalization in this context is to consider f ( R ) instead of R , leading to a term  \nof the form:  \n1 M 2 ∗ ∫ d 4 x √ -g ∂ µ ( f ( R )) J µ B . (12)  \nThe first class of modified terms we consider involves altering the term (1) by substituting the Ricci scalar with an alternative geometric quantity. This category includes modified gravity theories that utilize different geometric constructs to describe gravity. In these frameworks, it is natural to modify the term (1) to incorporate the specific geometric quantities relevant to the particular theory, as the Ricci scalar may not be the central term in the geometric description of gravity. This approach is classified under geometric modifications. For example, in extended teleparallel theories of gravity [62, 63], the CP-violating term can be constructed as follows:  \n1 M 2 ∗ ∫ d 4 x √ -g ∂ µ ( -T ) J µ B , (13)  \nwhere T is the torsion scalar leading to the general term  \n1 M 2 ∗ ∫ d 4 x √ -g ∂ µ ( -f ( T )) J µ B , (14)  \nThis can analogously be applied to f ( Q ) gravity, where Q is the non-metricity.  \nOther robust and high order theories of gravity like Gauss-Bonnet gravity [65, 66], generalized Einsteinian cubic gravity [67, 68] are also a possible path to modified gravitational baryogenesis [27, 69].  \nWe also include in this group special cases that are done in the context of black holes [70, 71] that typically evolve a term with the form  \n∂ α ( R µνρσ R µνρσ ) J α , (15)  \nwhere R µνρσ is the Riemann tensor and R µνρσ R µνρσ is the Kretschmann scalar.  \nThe second class of modifications involves theories of gravity that incorporate matter sectors into the gravitational action with a nonminimal coupling (NMC) to geometry. Examples of such theories include f ( R, L m ) gravity [36] and f ( R,T ) gravity [37] 2 . In these theories, we propose that analogous terms to (1) can be constructed using the matter components L m , T , and others. This idea is supported by the fact that theories with nonminimal couplings between matter and geometry can exhibit gravitationally-induced particle production, as described in [76-80] 3 .  \nThe possibility of gravitationally-induced particle production provides a strong rationale for exploring the direct coupling of matter-related quantities with the baryonic current. However, it is important to clarify that here gravitational particle production is not directly used to generate baryonic asymmetry. Instead, this mechanism, which arises from the non-conservation of the energymomentum tensor in theories with geometry-matter couplings, can be integrated with gravitational baryogenesis to achieve baryonic asymmetry. This integration requires careful consideration, such as ensuring entropy conservation after the baryogenesis epoch and developing a method to produce more particles than antiparticles through some B-violating process, which may or may not be gravitationally created.  \nIn gravitational baryogenesis, despite the term 'gravitational', gravity does not directly cause the B (or BL) violation. Instead, gravity influences the net baryon current through the background, and the baryon asymmetry results from an existing B-violating process. The interaction term from Eq.(1) enhances the baryon asymmetry to its observed value, which remains fixed once these B-violating processes decouple. Therefore, a mechanism that uses gravitational particle production to produce baryonic asymmetry, coupled with a B-violation process potentially originating from gravitational effects, and further amplifies this asymmetry through gravitational baryogenesis, represents a primarily gravitational means of achieving baryogenesis. This holds true even when supplementary mechanisms are considered for generating the asymmetry.  \nA similar concept was explored in [82], though this work used GR for the gravitational description, leading to a different justification for gravitational particle production compared to the approach described here. More recently, in [83], a comparable mechanism was proposed, integrating Cosmological Gravitational Particle Production (CGPP) [84], which combines cosmic inflation, quantum field theory, general relativity, and particle properties to produce particles, with baryogenesis from GrandUnification Theories (GUT).  \nWith this clarified, in this work, we consider a modification based on the geometry-matter coupling theory, namely, f ( R, T 2 ) gravity [32], where T 2 ≡ T µν T µν , that recently has been studied in the context of gravitational induced particle production [85]. We will explore the consequences for gravitational baryogenesis when Eq.(1) is built with the scalar T 2 instead of R . These modifications can elucidate the consequences of the geometrymatter coupling and implications of these type of couplings for baryon asymmetry shedding light on the intricate interplay between fundamental scalar fields and baryonic processes in the early universe. Furthermore, it opens new avenues for exploring the interplay between gravity and particle physics. Therefore, considering the derivative of the scalar T 2 as the term that couples to  \nJ u B leads to  \n1 M 2 ∗ ∫ d 4 x √ -g ∂ µ ( T 2 ) J µ B , (16)  \nand consequently yields the asymmetry  \nn b s ≃ -15 g b 4 π 2 g ∗ ˙ T 2 M 2 ∗ T ∣ ∣ ∣ ∣ ∣ T D . (17)  \nThus, for all future expressions derived from it, the dimensional quantities involved will be adjusted by dividing by the appropriate units to ensure consistency in the units of the expression  \nIt is also possible to admit an interplay of the Ricci scalar and the T 2 scalar in the asymmetry generator term leading to  \n1 M 2 ∗ ∫ d 4 x √ -g ∂ µ ( R + T 2 ) J µ B , (18)  \nthat, in the same way as done for f ( R ) gravity, can be generalized to  \n1 M 2 ∗ ∫ d 4 x √ -g ∂ µ ( f ( R, T 2 )) J µ B , (19)  \nleading to the asymmetry  \nn b s ≃ -15 g b 4 π 2 g ∗ ˙ Rf ,R + ˙ T 2 f , T 2 M 2 ∗ T ∣ ∣ ∣ ∣ ∣ T D . (20)  \nTo calculate the asymmetry using Eqs. (8), (17), and (20), a cosmological model that enables the computation of ˙ R and ˙ T 2 is required. With this framework established, we will first investigate the implications of f ( R, T 2 ) gravity for gravitational baryogenesis, specifically focusing on the term (1). Next, we will examine the effects of the modified terms (16) and (19) on baryogenesis.", 'IV. f ( R, T 2 ) GRAVITY': "One of the main reasons for exploring alternative theories of gravity beyond GR is the observed accelerated expansion of the universe. The most common explanation for this acceleration is dark energy, which is thought to drive the expansion by exerting a negative pressure that counteracts the gravitational attraction, resulting in an accelerated expansion of the universe. The leading candidate for dark energy is the cosmological constant, which represents a constant energy density added to Einstein's equations through the term 2Λ. This term is included in the Einstein-Hilbert action [86] to produce the observed acceleration. However, this solution faces several challenges. The theoretical origin of dark energy remains unclear, and observational data do not perfectly align with the theoretical description of the cosmological  \nconstant as vacuum energy. These issues have led to the search for more sophisticated solutions to the problem of the late-time acceleration.  \nOne notable alternative is the f ( R ) theory of gravity, first introduced by A.H. Buchdahl [87], where f is a function of the Ricci scalar R . This theory has been thoroughly examined, with detailed discussions available in sources like [41, 88]. In addition, more complex gravitational theories have emerged and gained interest, such as Teleparallel Gravity [92], Horndeski/Galileon theories [93], and f ( R, L m ) gravity [36], f ( R,T ) gravity [37], and f ( R, T 2 ) gravity [32]. These theories were initially developed to address the late-time acceleration problem, but they also show promise in tackling issues related to spacetime singularities that occur within the framework of GR. Since GR alone is insufficient due to the anticipated effects of quantum gravity, theories like f ( R, T 2 ) provide a compelling alternative.", 'A. Action and field equations': "The action for f ( R, T 2 ) gravity reads  \nS = 1 2 κ ∫ d 4 x √ -gf ( R, T 2 ) + ∫ d 4 x √ -g L m , (21)  \nwhere κ = 8 πG = M -2 Pl and f ( R, T 2 ) is a well behaved function of the Ricci scalar and T 2 .  \nWe define the energy-momentum tensor of the matter fields as  \nT µν = -2 √ -g ∂ ∂g µν ( √ -g L m ) , (22)  \nand imposing that L m depends solely on the metric components and not on their derivatives we obtain  \nT µν = g µν L m -∂ L m ∂g µν . (23)  \nVarying the action (21) with respect to the metric yields the field equations  \nf ,R R µν -1 2 g µν f + ( g µν ∇ α ∇ α -∇ µ ∇ ν ) f ,R = κ ( T µν -1 κ f T 2 θ µν ) , (24)  \nwhere subscripts denote differentiation with respect to the respective quantity and θ µν is defined as  \nθ µν ≡ δT αβ T αβ δg µν = -2 L m ( T µν -1 2 g µν T ) -TT µν +2 T α µ T να -4 T αβ ∂ 2 L m ∂g µν ∂g αβ . (25)  \nRelative to the matter sector, we assume that it is described by a perfect fluid  \nT µν = ( ρ + p ) u µ u ν + pg µν , (26)  \nwhere ρ is the energy density and p the isotropic pressure. For this description, the trace of the energy-momentum tensor and the scalar T 2 are given by  \nT = 3 p -ρ, (27)  \nand  \nT 2 = ρ 2 +3 p 2 , (28)  \nrespectively.  \nFor the matter Lagrangian, we choose L m = p [32, 35, 40, 94], as it effectively represents the energy characteristics of the primordial universe during the baryogenesis epoch. This choice provides an accurate thermodynamic description of the early universe and allows for a thorough analysis of the theory's implications for gravitational baryogenesis. In the ultra-relativistic regime, this approach avoids issues associated with the choice of L m when p = 0, which is typical for dust. Consequently, we derive the expression for the tensor θ µν as follows  \nθ µν = -( ρ 2 +4 pρ +3 p 2 ) u µ u ν . (29)  \nFor the f ( R, T 2 ) form, we considered a particular class of models given by  \nf ( R, T 2 ) = R + ηM 2 -8 n Pl ( T 2 ) n , (30)  \nthat is dubbed as energy-momentum powered gravity, which is a special case of f ( R, T 2 ) gravity. In this context, η is a dimensionless constant that is responsible for determining the coupling strength of the scalar T 2 and n is the parameter that specifies the model. The introduction of M 2 -8 n Pl serves to adjust the units of the new terms introduced by the scalar T 2 in the cosmological equations and for simplicity we defined η ' = ηM 2 -8 n Pl . For n = 1, the model corresponds to EMSG [33] and n = 1 / 2 and n = 1 / 4 are models that were studied in detail in the literature [32].  \nFor this type of models, the field equations read  \nR µν -1 2 g µν R = κT eff µν , (31)  \nwhere T eff µν is given by  \nT eff µν = T µν + η ' κ ( T 2 ) n -1 [ 1 2 ( T 2 ) g µν -nθ µν ] . (32)  \nTaking the trace of the field equations leads to  \n2 ( T 2 ) n -n ( T 2 ) n -1 θ = -1 η ' ( κT + R ) , (33)  \nwith θ being the trace of θ µν and T the trace of the energy-momentum tensor. This equation allows to evaluate T 2 depending on the choice of n .", 'B. Cosmology in EMSG': "To develop the foundational framework of gravitational baryogenesis, it is essential to provide a cosmological description of the primordial universe, particularly focusing on the radiation-dominated era, during which the baryon asymmetry is considered to be generated. For this purpose, we assume a flat FLRW (Friedmann-LemaˆıtreRobertson-Walker) metric, given by  \nd s 2 = -d t 2 + a 2 ( t )d V 2 , (34)  \nwhere t is the comoving proper time, a ( t ) is the expansion scale factor and d V is the volume element in comoving coordinates.  \nUsing this metric, the modified Friedmann equations are  \nH 2 = κ ρ 3 + η ' 3 ( ρ 2 +3 p 2 ) n -1 [ ( n -1 2 )( ρ 2 +3 p 2 ) + 4 nρp ] , (35)  \nwhere H ≡ ˙ a/a is the Hubble parameter, and the acceleration equation  \n˙ H + H 2 = -κ ρ +3 p 6 -η ' 3 ( ρ 2 +3 p 2 ) n -1 × × [ n +1 2 ( ρ 2 +3 p 2 ) + 2 nρp ] , (36)  \nrespectively.  \nAssuming that the matter fields obey a barotropic equation of state, p = wρ , where w is a constant, the additional terms introduced by the scalar T 2 are all of the form ρ 2 n , each multiplied by a constant. Therefore the cosmological equations can be recast as [95]  \nH 2 = κ ρ 3 + η ' ρ 2 n 3 C Frd ( n, w ) , (37)  \nwhere C Frd is a function that depends on the choice of n and w and is given by  \nC Frd ( n, w ) ≡ (1 + 3 w 2 ) n -1 [ ( n -1 2 )(1 + 3 w 2 ) + 4 nw ] , (38)  \nand the acceleration equation becomes  \n˙ H + H 2 = -κ 1 + 3 w 6 ρ -η ' ρ 2 n 3 C Acc ( n, w ) , (39)  \nwhere C Acc provides a constant based on the choice for n and w , and is given by  \nC Acc ≡ (1 + 3 w 2 ) n -1 [ n +1 2 (1 + 3 w 2 ) + 2 nw ] . (40)  \nThe term ρ 2 n present in both equations resembles quantum geometry effects in loop quantum gravity [96] and in braneworlds [97].  \nFinally, the modified continuity equation is obtained by direct derivation from the Friedmann equation (35) leading to  \n˙ ρ = -3 Hρ (1 + w ) C con ( n, w ) , (41)  \nwhere C con ( n, w ) is the extra term originated from T 2 , and is given by  \nC con ( n, w ) ≡ [ κ + η ' ρ 2 n -1 n (1 + 3 w )(1 + 3 w 2 ) n -1 κ +2 η ' ρ 2 n -1 nC Frd ( n, w ) ] . (42)  \nAn analysis of the cosmological equations reveals that the cosmological description is largely determined by the three functions under consideration and the chosen value of n . For the baryogenesis epoch, we focus on the radiation-dominated era following inflation thus avoiding the issue of asymmetry washout due to inflation. Typically, when choosing to use this era as the stage for baryogenesis, one sets w = 1 / 3 as this choice for w corresponds to radiation. Note, however, that from this choice, Eq. (41) does not reduce to the typical form for the energy density, ρ = Ca -4 , expected for the radiation era. In a first instance, this subtle point can be addressed by imposing  \nC con ( n, w ) = 1 , (43)  \nas done in [95]. This condition, although able to recreate the standard GR continuity equation, restricts unnecessarily n and w . If one only wishes to obtain the usual energy density form expected for a Universe dominated by radiation, a softer constraint can be built, by imposing  \n(1 + w ) C con ( n, w ) = 4 3 . (44)  \nSolving this condition in order to ρ 2 n -1 gives rise to  \nρ 2 n -1 = 1 -3 w ηM 4 -8 n Pl (3 nA ( w ) -8 nC Frd ) , (45)  \nwith A ( w )  \nA ( w ) ≡ (1 + w )(1 + 3 w )(1 + 3 w 2 ) n -1 . (46)  \nAnalysing Eq.(45), the right-hand corresponds to a constant, implying that for the equation to yield a physically meaningful result within a dynamical cosmological framework, it is necessary to consider n = 1 / 2, as a constant ρ does not present meaningful value. This result indicates that within the context of energy-momentum powered gravity, the n = 1 / 2 model stands out as a favourite model being able to reproduce ρ = Ca -4 for specific values of η and w . Imposing n = 1 / 2, the condition (45) can be written in a way that establishes a relation between η and w leading to  \nη ∗ ≡ η ( w ) = 1 -3 w 3 2 A ( w ) -4 C Frd ( 1 2 , w ) . (47)  \nThis relation not only provides the desired energy density form but also acts as a constraint on η . The value of w determines the η needed to satisfy this condition, leading to new scenarios for studying baryogenesis. An intriguing result is that for w = 1 / 3, the required η is zero, effectively reducing the theory to GR. This result will be explored with more details in the following section.  \nWith this framework established, w = 1 / 3 will be used for all models, while for the n = 1 / 2 model, additional values of w will be explored using the condition in (47). In this context, the new terms associated with each cosmological equation depend only on the specific model chosen, particularly the power of T 2 , except for the special case of n = 1 / 2, where they also depend on w . This highlights the importance of the model in shaping the cosmological framework and its parameters. Additionally, the coupling constant η is expected to play a key role in cosmological dynamics, significantly influencing the asymmetry produced through gravitational baryogenesis. Since the generated asymmetry largely depends on this parameter, η can be constrained by imposing observational limits on the resulting asymmetry.", 'V. MODIFIED GRAVITATIONAL BARYOGENESIS IN EMSG': 'In this section, we examine the effects of f ( R, T 2 ) gravity in the context of modified gravitational baryogenesis and constrain η using the observational limit on baryonic asymmetry, specifically η s ≲ 9 . 2 × 10 -11 [5-8]. We consider various values of n , focusing on three cases: n = 1 / 4, n = 1 / 2, and n = 1, analyzing their influence on modified gravitational baryogenesis as discussed in Sec. III. Additionally, we will explore the extreme cases where n < 1 / 4 and n > 1.  \nThe first step in this study involves carefully selecting the values of M ∗ and T D , as these are critical parameters requiring close attention. In the original gravitational baryogenesis framework, as noted in [24], M ∗ does not necessarily need to reach the Planck scale. For instance, in scenarios where B-violation is characterized by the Majorana mass M R of the right-handed neutrino, which represents a soft B-violation, the term (1) avoids unitarity violation up to the Planck mass, even when M 2 ∗ = M R M Pl [24]. As a result, the cutoff scale for the ∂ µ R coupling can be lowered if necessary to support baryogenesis. For the modified couplings introduced in an ad hoc manner, the cutoff scale will depend on various scenarios and numerical outcomes that lead to successful baryogenesis. Once a successful baryogenesis scenario is established, these new interactions can be tested within the LHC energy scale, providing a way to rule out such terms if M ∗ is below the LHC threshold, as B-violating interactions would have been detected up to this scale.  \nIt is generally accepted in the literature that during the radiation-dominated epoch following inflation, the decou-  \npling temperature, T D , is approximately equal to the inflationary scale, M I , where M I ≃ 1 . 6 × 10 16 GeV, representing the upper bound based on tensor mode fluctuation constraints [98]. While this approximation is widely used, our study aims to examine this assumption more critically. During slow-roll inflation, as inflation ends, the scalar field begins oscillating around the potential minimum, initiating the reheating phase essential for the thermalization of the Universe. Therefore, if baryogenesis is assumed to occur during the radiation era postinflation, the decoupling temperature should satisfy the condition T D < T RD < M I , where T RD is the temperature at which the Universe becomes radiation-dominated, or equivalently, the reheating temperature following inflation. The specific value of T RD depends on the chosen inflationary model (see comprehensive reviews and analyses in [99-102]), and while it can theoretically reach up to ∼ 10 16 GeV, similar to the GUT scale, this is often considered unrealistic. Consequently, the assumption T D ≈ M I is problematic, as it would require T RD to be nearly equal to M I , a scenario typically not realized.  \nTo maintain a general discussion, we avoid assuming any specific inflationary model. Instead, we consider the upper bound M I alongside the small temperature fluctuations observed in the CMB and adopt T RD ≲ 10 9 -10 14 GeV [44, 98, 102-104] as a conservative estimate. This approach, combined with the condition T D ≈ T RD , allows for a broader range of potential values for T D compared to the case where T D ≃ M I . Determining the lower bound of T D is more complex. Within the temperature range of 100 to 10 12 GeV, gauge interactions and sphaleron processes maintain equilibrium [23]. Additionally, the lower bound of T D is tied to the chosen B-violation mechanism, which is crucial for gravitational baryogenesis. A detailed analysis of this aspect will be addressed in the final section.  \nIn this section, we will explore the impact of f ( R, T 2 ) gravity on gravitational baryogenesis from an effective perspective, without specifying any particular Bviolation process. To maintain a general description, we assume baryogenesis occurs in the early stages of the radiation epoch with T D ≳ 10 7 GeV, leading to the interval 10 7 ≲ T D ≲ 10 14 GeV. Although this lower bound includes a selective group of B-violation mechanisms, such as certain leptogenesis models [105], it excludes others that occur at lower energy scales.', '(i) Model n = 1 / 4:': 'Combining the results from Sections V A and V B, the expression from the ∂ µ f ( R,T ) for the n = 1 / 4 model is given by  \nn b s ≃ -g b πηT D 2(192) 1 / 4 g 1 / 2 ∗ M Pl M 2 ∗ [ ( √ 3 -2(192) 1 / 4 ) × × ( 3 η √ 10( 3 64 ) 1 / 4 M 2 Pl +2 πg 1 / 2 ∗ T 2 D )] . (111)  \nAs observed in the case where n = 1 for the ∂ µ T 2 coupling, by setting M ∗ = M Pl and performing computational calculations, we found an inverse relationship between the decoupling temperature T D and the parameter η . Specifically, as T D increases, the value of η required to satisfy the observational constraints on baryonic asymmetry decreases. This effect arises due to the interaction between the Ricci scalar and the scalar term T 2 , which allows for the generation of an acceptable asymmetry. This result differs from previous cases involving the ∂ µ R and the ∂ µ T 2 couplings. For instance, considering T D = 10 8 GeV and M ∗ = M Pl , the plot of the baryon-to-entropy ratio n b s ( η ) in Fig. 4 shows that the asymmetry increases rapidly with η , indicating  \nFIG. 4. Plot of n b s vs η for the asymmetry generated from the general coupling. The green line shows the evolution of n b s ( η ), the dashed red line indicates the observation constraint for the asymmetry and the red dot marks the point where n b s ( η crit ) ≈ 9 . 1988327 × 10 -11 with η crit = 1 . 4615892317846357.  \n<!-- image -->  \na strong dependence on the coupling parameter.', '(ii) Model n = 1 / 2:': 'For n = 1 / 2, using the results from Sections V A and VB the asymmetry is given by  \nn b s ≃ 15 g b g 1 / 2 ∗ πT 5 D M 2 ∗ [ 3 α ( η )(2 α ( η ) -1) (30 ρ 0 ) 3 / 2 -η 10 1 / 2 ( ρ 0 ) 1 / 2 30 M 2 Pl ] . (112)  \nThis case clearly demonstrates the interplay between the asymmetry originating from the Ricci scalar, represented by the first term in the parentheses, and the asymmetry from the scalar T 2 , associated with the second term in the parentheses, where the decoupling temperature plays a significant role. As discussed in Section VA, the term related to R is negative for positive values of η . Similarly, the term arising from T 2 also remains negative when η > 0. Consequently, selecting η > 0 results in a negative asymmetry, which indicates an imbalance favoring anti-matter. In the n = 1 / 2 model for the ∂ µ R coupling, we found scenarios that produce an asymmetry consistent with observational data when M ∗ is below the LHC energy scale and for reasonable values of η . These results are observed for decoupling temperatures below 10 12 GeV. For example, with T D = 10 11 GeV and M ∗ = 9 × 10 3 GeV, the critical value of η was calculated to be η crit = -0 . 000852. However, these scenarios have been previously deemed unrealistic.  \nFor T D = 1 × 10 14 GeV and M ∗ = 1 × 10 12 GeV, the evolution of the baryon-to-entropy ratio n b s ( η ) is graphically depicted in Fig. 5. Once again, successful baryo-  \nFIG. 5. Graphical representation of n b s vs η derived from the general coupling term f ( R, T 2 ) for the case n = 1 / 2. The blue line represents n b s ( η ), the dashed red line indicates the observation constraint for n b s and the red dot marks the point where n b s ( η crit ) ≈ 9 . 195741 × 10 -11 with η crit = -0 . 01058.  \n<!-- image -->  \nis is achieved with larger values of η . This outcome is driven by the interaction between the Ricci scalar and the T 2 scalar. If the contribution came solely from the term associated with R , the result would match what was obtained in Section V A. However, the inclusion of the T 2 scalar term significantly increases the value of η . In other words, the scalar T 2 enhances the value of η , highlighting its crucial role in the baryogenesis scenario. To support this conclusion, the results from Section V B show that the ∂ µ T 2 coupling alone leads to an excessive increase in baryonic asymmetry. In the general case of f ( R, T 2 ), however, the Ricci scalar and the additional terms derived from f T 2 help balance the over-contribution of the scalar T 2 on its own.  \nRegarding the case with the constraint, the expression for the generated asymmetry is  \nn b s ≃ 15 g b γ π 2 g ∗ M 2 ∗ [ 3(2 γ -1)( π 2 g ∗ 30 ) 3 / 4 γ T 3 /γ -1 D p 3 / 4 γ c + η ∗ (1 + 3 w 2 ) 1 / 2 ( π 2 g ∗ 30 ) 1+1 / 4 γ T 3+1 /γ D p 1 / 4 γ c M 2 Pl ] . (113)  \nOnce more, as expected, there exists an interplay between both contributions from the different scalars. For some subspaces of the model parameter space the asymmetry is negative rendering these cases non-viable. Considering T D = 1 × 10 13 GeV and M ∗ = 1 × 10 12 GeV two successful cases where for, one for w = 0 . 3 with ρ c = 7 . 5 × 10 32 giving n B /s ≃ 9 . 06400 × 10 -11 and for w = -1 / 3 with ρ c = 8 . 8 × 10 32 resulting in n B /s ≃ 9 . 043081 × 10 -11 . These results appear less  \nfavorable when compared to those obtained from each individual coupling separately. This outcome arises from the interplay between the different couplings, where both contributions exhibit limitations that often invalidate the resulting asymmetry. For instance, in the case of w = 1, the parameter η ∗ is negative rendering the contribution of f T 2 negative, which necessitates a compensating contribution from the sector associated with the derivative of the Ricci scalar to counterbalance this negative value. However, for w = 1, the contribution from the derivative of the Ricci scalar is also negative, thereby failing to provide the required compensation.', '(iii) Model n = 1:': "For n = 1, there is no contribution to the asymmetry from the Ricci scalar because, in this case, R = 0. By applying Eq.(106) and the decoupling time from Eq.(109), the asymmetry in this scenario is given by  \nn b s ≃ (8 / 3) 3 / 2 g b g 2 ∗ π 4 η 3 / 2 7200 M 9 Pl M 2 ∗ T 11 D . (114)  \nThis case closely resembles the scenario for n = 1 in the ∂ µ T 2 coupling with J µ , differing only by a factor of η '-1 . As a result, the behavior in this case will be identical to that described for n = 1 in Section V B. However, for T D = 10 7 GeV with M ∗ = M Pl , the values of η required to produce an acceptable asymmetry are too large to be considered viable, rendering this case nonviable.", 'B. Couplings between ∂ µ ( T 2 ) and J µ': '(i) Model n = 1 / 4: Using Eq. (28) for this case one has  \nT 2 = 12 M 4 Pl ( z e zt -2 ) 4 , (95)  \nand taking into account Eqs. (60) and (61) provides the following expression for ˙ T 2  \n˙ T 2 = -48 π 4 g 2 ∗ T 8 D [ 3( 3 64 ) 1 / 4 η √ 10 M 2 Pl +2 πg 1 / 2 ∗ T 2 D 90 5 / 2 M Pl ] , (96)  \nresulting in the asymmetry  \nn b s ≃ g b g ∗ π 3 T 7 D 130 √ 10 M 2 ∗ M Pl [ 3 ( 3 64 ) 1 / 4 η √ 10 M 2 Pl + 2 πg 1 / 2 ∗ T 2 D ] . (97)  \nThe expression for this scenario fails to generate an acceptable asymmetry, as it consistently suppresses the observable value for asymmetry. The contribution from T 2 excessively enhances the asymmetry. Thus, this case is not viable for achieving successful baryogenesis.', 'D. Models with n < 1 / 4 and n > 1': "In addition to the three models previously analyzed, it is possible to explore the extreme cases where n < 1 / 4 and n > 1 within the context of gravitational baryogenesis. However, we did not consider these scenarios because they involve coupling constants with unnatural dimensions, which is why we refer to them as extreme cases. Nonetheless, we will provide a potential theoretical framework for both cases to be used in the context of gravitational baryogenesis.  \nStarting with the modified Friedmann equations with w = 1 / 3 one has  \nH 2 = ρ 3 M P l 2 + η ' ρ 2 n 3 C Frd ( n, 1 3 ) , (115)  \n˙ H + H 2 = -ρ 3 M P l 2 -η ' ρ 2 n 3 C Acc ( n, 1 3 ) , (116)  \nrespectively.  \nConsidering the energy scale of the very early universe ( ρ →∞ ), two distinct approximations can be made based on whether n < 1 / 4 or n > 1. For n < 1 / 4, the ρ term is expected to dominate over the ρ 2 n term because the exponent associated with ρ 2 n is less than 1. This allows the modified cosmological equations to be recast as those  \nof GR. In the case of n > 1, a similar reasoning applies, but here the dominant term is the quadratic term, ρ 2 n , leading to modified Friedmann equations  \nH 2 = η ' ρ 2 n 3 C Frd ( n, 1 3 ) , (117)  \n˙ H + H 2 = -η ' ρ 2 n 3 C Acc ( n, 1 3 ) . (118)  \nIt is important to emphasize that these results are valid only for sufficiently large values of the energy density. If this condition is not met, the parameter η ' invalidates both sets of results. Additionally, a key point to consider is the applicability of the result derived for n > 1 to the specific case of n = 1. In this study, we did not consider n = 1 as an extreme case due to the unique properties that this model exhibits. Supporting this reasoning, the range of decoupling temperatures considered, which occur after inflation and the era of grand unification, do not satisfy the required energy density condition. We considered that for n = 1 the approximation considered only holds when one takes into account pre-inflation or inflation as the epochs of study.", 'VI. SUMMARY AND DISCUSSION': "In this paper, we introduced the concept of modified gravitational baryogenesis through two distinct approaches, which can be applied either independently or in combination. Within the second approach, we further propose two rationales that give rise to two different categories. The first category is based on geometric modifications of the original interaction term, given by Eq. (1), while the second category involves the ad hoc introduction of scalar fields that emerge from modified theories of gravity, featuring a non-minimal coupling between geometry and matter. These scalar fields, such as the trace of the energy-momentum tensor or the scalar T 2 , which are incorporated into the gravitational action, interact with the baryonic current in a manner similar to the coupling with ∂ µ R .  \nWith this framework established, we explored its applications within the context of f ( R, T 2 ) gravity. Specifically, we examined the class of models presented in Eq. (30), where the value of n determines the specific model. We focused on three models: n = 1 / 4, n = 1 / 2, and n = 1, and also proposed approaches to address the extreme cases of n < 1 / 4 and n > 1. In addition to studying the implications of f ( R, T 2 ) for gravitational baryogenesis, we used the observed value of baryonic asymmetry to constrain the η parameter within this framework. We also introduced a method to rule out certain positive results for baryogenesis, allowing for a cut-off scale lower than the LHC energy scale.  \nFor n = 1 / 4, the only coupling capable of producing an acceptable asymmetry was the general coupling in Eq.  \n(19). The result from the ∂ µ T 2 coupling showed that while the scalar T 2 contributes to asymmetry, its impact is too strong, leading to excessive matter production. In the general case, the interplay between the Ricci scalar and the new scalar helps balance the extreme enhancement caused by T 2 . This pattern of overproduction and suppression was observed across all three models.  \nThe n = 1 / 2 case is particularly interesting because here, the contribution of T 2 to the cosmological equation manifests as an interplay between the gravitational constant and η . However, the constraint on η derived from Eq. (8) is incompatible with the one obtained in [32] for BBN. A potential solution could involve developing a model with a variable η that, after asymmetry production, adjusts the parameter to preserve the BBN results. For the general coupling, we again found a successful baryogenesis scenario where the interplay between the Ricci scalar and the T 2 part led to an increase in η . This model also displays the capability for specific η and w to describe the behaviour ρ ∝ a -4 for the radiation-epoch while also achieving successful baryogenesis.  \nFinally, in the n = 1 model, an acceptable asymmetry was only achieved with the ∂ µ T 2 coupling. However, the resulting value of η contradicts the conditions necessary for an accelerated late-time expansion predicted by this model.  \nThese results lead to two important conclusions. Firstly, they highlight the different cosmological equations arising from the three distinct models examined. For the n = 1 / 2 model using the coupling (1), the contribution from T 2 to the cosmological equations can facilitate a successful baryogenesis scenario. However, as discussed in Sec.V A, only contributions that align with the standard cosmological equations, specifically those involving ρ terms with the same exponent as in GR, result in successful asymmetry generation. The n = 1 / 4 model introduces an additional energy-density term that is insufficient to achieve the desired asymmetry, leading to the opposite effect. Meanwhile, the n = 1 model demonstrates a notable characteristic: despite the inclusion of an extra quadratic term in the Friedmann equations, the Ricci scalar remains zero in a radiation-dominated universe, resulting in zero asymmetry.  \nSecondly, the results address the specific contribution from the ∂ µ T 2 coupling. In most scenarios, this scalar field significantly amplifies the asymmetry, leading to unacceptably high values. In the framework of f ( R, T 2 ) gravity, where the new scalar is proportional to ρ 2 , these substantial contributions are too large to effectively resolve the asymmetry issue. An intriguing alternative could involve coupling with ρ in a linear rather than quadratic form. Additionally, the successful outcome observed for the n = 1 case provides strong evidence that integrating matter-related quantities into the gravitational baryogenesis framework can yield favorable results. This finding highlights the potential effectiveness of coupling matter within the context of gravitational baryogenesis to generate baryon asymmetry.  \nFor the extreme cases, a theoretical framework was developed by considering the energy scale of the very early universe, including during inflation. This framework is based on the behavior of the energy density parameter, ρ , in regimes of very large values. Specifically, for n < 1 / 4, the ρ term dominates over the ρ 2 n term. Conversely, for n > 1, the ρ 2 n term becomes dominant over ρ . As a result, these conditions yield the standard Friedmann equations of GR for n < 1 / 4, and quadratic modified Friedmann equations for n > 1.  \nTo fully realize gravitational baryogenesis, it is essential to consider a baryon-violating (B-violating) process that generates the necessary asymmetry. Various Bviolating processes can be analyzed, but to keep the discussion general and within the scope of this paper, we focus on a B-violating interaction represented by an operator O B . This operator has a mass dimension of 4+ m , where m> 0. For this interaction, the rate of B-violation is given by:  \nΓ B = T 2 m +1 M 2 m B , (119)  \nwhere M B denotes the mass scale associated with O B . As discussed earlier, B-violation processes cease when the Hubble parameter H ( T ) approximately equals the interaction rate Γ B ( T ). This condition allows us to determine the decoupling temperature by setting the expansion rate equal to the interaction rate.  \nAn interaction of this type may either be part of an extension to the Standard Model of particle physics (SM) or arise as a nonrenormalizable interaction within an effective field theory. Additionally, from the viewpoint that gravitational baryogenesis is derived from supergravity theories, such theories are expected to include B-violating processes. This establishes a coherent link between the interaction term that enhances the asymmetry and the process responsible for generating it. Consequently, to adopt this perspective, constraints on the reheating temperature, T RD , must be applied [110].  \nIn supersymmetric models, particularly those involving warm inflation [111], additional constraints on T RD arise due to gravitino production and associated challenges. These constraints are as follows: First, to protect the results of Big Bang Nucleosynthesis (BBN) from being disrupted by late gravitino decays. Second, to prevent the overclosure of the universe due to an excessive abundance of gravitinos. Since parameters like the gravitino mass m 3 / 2 and the gravitino decay processes are model-dependent, we will use a conservative upper limit for T RD , specifically T RD < 10 9 GeV [108, 112, 113].  \nFor the ∂ µ R coupling with n = 1 / 2 the case without the imposed form for ρ and the two first cases in Table I have a the temperature constraints that do not permit the B-violation process to originate from supergravity. For the not constrained case, using Eq.(119), Eq.(66), and Eq.(76) with η = η crit and a D = 7 operator, we find M B ≈ 1 . 96 × 10 14 GeV. This value is below the typical mass scale for Grand Unified Theories (GUTs)  \nand could be associated with a GUT Higgs boson [114]. In the same way, using η = η ∗ , the constrained cases have M B ( w = 0 . 3) ≈ 1 . 996 × 10 14 GeV, M B ( w = -1 / 3) ≈ 1 . 932 × 10 14 GeV, M B ( w = -2 / 3) ≈ 1 . 915 × 10 14 GeV and M B ( w = -0 . 7) ≈ 1 . 9120 × 10 14 GeV.  \nDoing the same calculations for the ∂ µ T 2 coupling with n = 1 / 2, the three cases of the constrained condition gives M B ( w = 1) ≈ 7 . 90 × 10 7 M B ( w = 0 . 3) ≈ 7 . 65 × 10 7 GeV and M B ( w = -1 / 3) ≈ 7 . 40 × 10 7 . For n = 1, a similar calculation with D = 7 yields M B = 160 . 96 GeV. This result compared to the previous one is much lower and such value for M B is ruled out by observations as this mass scale would already have been observed. Therefore, the B-violation can not be described by the general description considered. For the general coupling ∂ µ f ( R, T 2 ) with D = 6, the model with n = 1 / 4 results in M B ≈ 6 . 92 × 10 8 GeV, while the model with n = 1 / 2 gives M B ≈ 2 . 19 × 10 14 GeV, M B ( w = 0 . 3) ≈ 2 . 42 × 10 13 Gev and M B ( w = -1 / 3) ≈ 2 . 34 × 10 13 GeV.  \nIn conclusion, modifying the existing gravitational and cosmological frameworks and incorporating new interactions based on these modifications offer a compelling approach for advancing our understanding of gravitational baryogenesis. By revisiting and extending the traditional theories of gravity and cosmology, we can explore new mechanisms that may account for the observed baryon asymmetry in the universe. Specifically, the study of f ( R, T 2 ) gravity reveals its potential to generate an acceptable level of asymmetry through gravitational baryogenesis. This approach leverages the interplay between  \n- [1] F. Wilczek, 'The cosmic asymmetry between matter and antimatter,' Scientific American . 243 , 82-91 (1980)\n- [2] S. 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