Baryogenesis in f(Q, T ) Gravity

The article communicates exploration of gravitational baryogenesis in presence of f(Q, T ) gravity where Q denote the nonmetricity and T the trace of the energy momentum tensor. We study various baryogenesis interactions proportional to Q̇ and Q̇fQ for the f(Q, T ) gravity model f(Q, T ) = αQ + βT , where α, β and n are model parameters. Additionally we report the viable parameter spaces for which an observationally consistent baryon-to-entropy can be generated. Our results indicate that f(Q, T ) gravity can contribute significantly and consistently to the phenomenon of gravitational baryognesis.


Introduction
Our universe favors matter over antimatter for some mysterious reasons.
Observations from Cosmic Microwave Background [1], coupled with successful predictions from the Big Bang Nucleosynthesis [2], recommend an overwhelming supremacy of matter over antimatter.
Cosmological theories that aim at resolving this fundamental issue falls under the domain of Baryogenesis. Theories such as GUT baryogenesis, Thermal Baryogenesis, Affleck-Dine Baryogenesis, Electroweak Baryogenesis, Black hole evaporation Baryogenesis and Spontaneous Baryogenesis propose interactions which goes beyond the standard model to explain this profound dominance of matter in the universe [3]. These mechanisms were further developed in [4] Gravitational Baryogenesis is one such theory proposed in [5] and further developed and extended to many modified gravity theories [6]. This particular theory employs one of the Sakharov criterion [7] which assures a baryon-antibaryon asymmetry from the existence of a CP-violating interaction, which reads where M * is the mass parameter of the underlying effective theory, g, J i and R denote respectively the metric scalar, baryon current and Ricci scalar. Hence, for a flat FLRW background, the baryon to entropy ratio η B /s is proportional to time derivative of Ricci scalarṘ. For a radiation dominated universe with EoS parameter ω = p/ρ = 1/3, the net baryon asymmetry produced by (1) is zero.
The paper aims at investigating gravitational Baryogenesis through other curvature invariants and specifically through the nonmetricity Q. For the f (Q, T ) gravity, the CP-violating interaction is given by where T denote the trace of energy momentum tensor and the nonmetricity Q is defined as [8] where H(t) represents Hubble parameter and N (t) the lapse function. A remarkable difference between (1) with (2) is that the latter yields a nonzero baryon asymmetry even for a radiation dominated universe (ω = 1/3). We shall investigate here in detail the baryogenesis terms proportional to ∂ i Q and ∂ i f (Q) and compare our results with cosmological observations. The paper is organized as follows: In Section 2 we provide a summary of f (Q, T ) gravity and obtain the field equations. In Section 3 we explain in detail the gravitational baryogenesis in f (Q, T ) gravity and infer the viability of a f (Q, T ) gravity model in producing observationally acceptable baryon to entropy ratio and finally in Section 4 we present our conclusions.

Field Equations in f (Q, T ) Gravity
The action in f (Q, T ) gravity is given as [8] S where g ≡ det(g ij ) denote the metric scalar.
Variation of action (4) with respect to metric tensor components yields the field equations in f (Q, T ) gravity as [8] 8πT and P α ij is called superpotential and is defined as [8] where We now consider a flat FLRW spacetime of the form where a(t) represent the scale factor and the lapse function N (t) = 1 for a flat spacetime.
Employing (9) in (5), we finally obtain the modified Friedmann equations with where According to cosmological observations such as CMB [1] and BBN [2], the observed baryon to entropy ratio reads Sakharov reported three conditions for a net baryon asymmetry to occur through baryon number violation, C and CP violation and processes occurring outside of thermal equilibrium [7].
When the temperature T falls below a critical value T D through the evolution of the Universe, the baryon to entropy ratio can be written as [5] η where g b represent the total number of intrinsic degrees of freedom of the baryons, g * s represent the total number of degrees of freedom of the massless particles and the critical temperature T D is the temperature of the cosmos when all the interactions producing baryon asymmetry comes to a halt.
We shall presume that a thermal equilibrium prevails with energy density being associated with temperature T as Hence, for a CP violating interaction of (2), the resulting baryon to entropy We shall assume cosmological pressure and density obeys a barotropic equation of state of the form p = (γ − 1)ρ, where γ is a constant and 1 ≤ γ ≤ 2. For relativistic matter p = ρ/3 and hence T = 0. Thus, the baryon to entropy ratio in f (Q, T ) gravity for radiation dominated universe reduces to We shall now compute the baryon-to-entropy ratio for a CP-violating interaction proportional to the nonmetricity Q for two cases: First, with the Universe comprising predominantly of a perfect fluid with cosmological pressure p and matter density ρ following a barotropic equation of state and second, when the Universe is filled with a perfect fluid and the cosmic dynamics is governed by the f (Q, T ) theory of gravity.

The perfect fluid Case
For a perfect fluid following barotropic equation of state, the Ricci scalar Thus, for a radiation dominated universe, γ = 4/3 and hence R = 0 which further implies 14 acquire a null value. Nonetheless, we shall show that when the baryon-to-entropy ratio is proportional to ∂ i Q (Eq. 2), the resultant baryon to entropy ratio is non-zero even for γ = 4/3. Assuming a flat FLRW background with (−,+,+,+) metric signature with the scale factor a(t) = a 0 t 1/2 and therefore density ρ ∼ ρ 0 t −2 , the baryon to entropy ratio (17) in a radiation dominated universe reads where we have usedQ = 12HḢ and decoupling time t D is written in terms of critical temperature T D as Substituting g * s = 106, g b ∼ 1, ρ 0 = 10 −6 GeV ,T D M * = 2 × 10 16 GeV and a 0 7.8 × 10 −13 , the resultant baryon to entropy ratio reads which is close to the observational value (13). Thus, for the perfect fluid case, (17) produces non zero and observationally consistent baryon to entropy ratio.

The perfect fluid with f (Q, T ) gravity case
We shall now compute baryon to entropy ratio for the case when the Universe is filled with a perfect fluid and the evolution of the Universe is governed by the f (Q, T ) theory of gravity.
Interestingly, in Fig. (1b), the baryon to entropy ratio becomes negative for β −8π which is unphysical as it implies an overabundance of antimatter over ordinary matter. Also note that for n 2.2, η B /s = 0 which indicate no asymmetry between antimatter and matter and therefore not acceptable.
We shall now define a more complete and generalized baryogenesis interaction proportional to ∂ i f (Q, T ). The CP-violating interaction then reads For (28), the resulting baryon to entropy ratio reads As discussed in the previous section that for a radiation dominated universe Substituting (23), (25) and (26) in (30), the baryon to entropy ratio for γ = 4/3 (radiation dominated universe) then reads where Substituting M * , g * s , g b , T D as before, α = 0.004, β = 8.1π and n = −2.4 the resultant baryon to entropy ratio reads η B /s =∼ 8.7 × 10 −11 which is very close to observational constraints. In Fig. (2a), (2b) and (2c) we show η B /s for the generalized baryogenesis interaction as a function of α, β and n respectively.  (2c) shows η B /s as a function of n. We choose g * s = 106, g b ∼ 1 and T D M * = 2 × 10 16 GeV .

Conclusions
The article presented a thorough investigation of gravitational baryogenesis interactions in the framework of f (Q, T ) gravity where Q denote the nonmetricity and T the trace of the energy momentum tensor. For this type of modified gravity we find the baryon-to-entropy ratio to be proportional toQ, since for a radiation dominated universe T = 0. We ascertained the baryon-to-entropy ratio proportional to nonmetricity Q two different scenarios, First, with the Universe comprising predominantly of a perfect fluid with cosmological pressure p and matter density ρ following a barotropic equation of state and second, when the Universe is filled with a perfect fluid and the cosmic dynamics is governed by the f (Q, T ) theory of gravity. We choose the functional form of f (Q, T ) gravity to be f (Q, T ) = αQ n+1 + βT , where α, β and n are model parameters. For the perfect fluid case, the obtained baryon-to-entropy ratio η B /s 8 × 10 −11 while for the f (Q, T ) gravity model we obtained η B /s ∼ 6.17 × 10 −11 , both of which are in excellent agreement with observational value of 9 × 10 −11 . Next, for the f (Q, T ) gravity model, we explored a more complete and generalized baryogenesis interaction proportional toQf Q . For this baryogenesis interaction, we found the baryon-to-entropy ratio η B /s ∼ 8.7 × 10 −11 which is very close to the observational value.
As a final note we add that the values of the free parameters obtained from the analysis satisfies only one cosmological observation. More research is needed to further pin point the values which ultimately would enable us to understand whether this modified gravity is viable in representing the current state of the Universe.