Gravitational baryogenesis in non-minimal kinetic coupling model

In this work, we consider the gravitational baryogenesis in the framework of non-minimal derivative coupling model. A mechanism to generate the baryon asymmetry based on the coupling between the derivative of the Ricci scalar curvature and the baryon current in context of non-minimal derivative coupling model is investigated. We show that, in this model, the temperature increases during the reheating periods to the end of reheating period or beginning of radiation dominated era. Therefore the reheating temperature is larger then decoupling temperature. It can be demonstrated that, the evaluation of baryon asymmetry is not depends on coupling constant. In this model we can generate baryon asymmetry at low and high reheating temperature, by considering the high friction constraint.


Introduction
One of the greatest puzzles in the standard model of cosmology and astro particle physics is the dominance of matter against anti-matter.In the other words the number of baryons in the universe is larger then the number of anti baryons.The cosmic microwave background radiation [1] and the abundance of the primordial light elements from big bang nucleosynthesis (BBN) [2] shows that the ratio of baryon number density n s to the entropy density s is As A.Sakharov in Ref. [3] showed that baryon asymmetry may be dynamically generated from the following conditions.(i) processes that violate baryon number; (ii)violation of charge (C) and charge parity (CP) symmetry; (iii) out of the thermal equilibrium.This three assumptions are now known as Sakharov conditions.
Several interesting and applicable mechanisms for generation of baryon asymmetry are proposed.The first suggestion for this topic used on the out of equilibrium decay of a massive particle such as a super heavy GUT gauge of Higgs boson where dubbed GUT baryogenesis [4].The other mechanism involving the decay of flat directions in super symmetric models is known as the Affleck-Dine scenario [5].Also the possibility of generating the baryon asymmetry at the electro-weak scale has been considered, these interactions conserve the sum of baryon and lepton number, which is converted to a baryon asymmetry at the electro-weak scale This mechanism is known as lepto-baryogenesis [6].The spontaneous baryogenesis has been proposed with the characteristic generation of baryon asymmetry in thermal equilibrium without the necessity of C and CP violation [7,8].
Davoudiasl et al in Supergravity have proposed a mechanism for generating of the baryon asymmetry on the basis of spontaneous baryogenesis during the expansion of the universe [9].This mechanism is known as gravitational baryogensis.In this approach they introduced an interaction between derivative of the Ricci scalar curvature and baryon current J µ ∂ µ R which dynamically violate CPT and CP symmetries in expanding universe.During the last years, other scenarios to extend this coupling, has been receiving a great amount of attention from the many authors.In [10] the effect of time dependence of the equation of state parameter on gravitational baryogenesis has been considered.Gravitational baryogenesis in f (R) theories in [11] has been considered.Also gravitational baryogenesis in f (T ) theory of gravity has been considered where T is the torsion scalar [12].Some variant forms of gravitational baryogenesis containing the partial derivative of Gauss-Bonnet scalar coupled to baryon current are investigated in [13].Generalized gravitational baryogenesis of f (R, T ), f (Q, τ ), f (T, T G ) and f (T, B) where τ denote the trace of energy-momentum tensor, Q is the nonmetricity, T G is the teleparallel equivalent to the Gauss-Bonnet term, B denotes boundary term between torsion and Ricci scalar are discussed in [14][15][16][17][18].In [19] the baryon asymmetry is generated dynamically during an inflationary epoch powered by ultra-relativistic particle production.In [20][21][22] has been suggested that anisotropy of the universe can enhance the generation of the baryon asymmetry.The authors of [22] clarified, if we into account the gravitino problem (T RD < 10 9 GeV ), gravitational baryogenesis [9] is incapable to explain generation of sufficient baryon asymmetry.They show that if there exists a huge shear in the radiation dominated era there is a little possibility of gravitational baryogenesis.
In this paper, we examine gravitational baryogenesis in context of nonminimal derivative coupling model and the effect of "gravitationally enhanced friction" on the evolution of the scalar field on the baryon asymmetry has been considered.
In this coupling the inflaton field evolves more slowly relative to the case of standard inflation due to a gravitationally enhanced friction which its capacity to explain the Higgs inflation.
An emphasize feature of the non-minimal derivative coupling with the Einstein tensor is that the mechanism of the gravitationally enhanced friction during inflation, by which even steep potentials with theoretically natural model parameters can drive cosmic acceleration [26,27].Thus it is well motivated to propose the gravitational baryogenesis in context of non-minimal derivative coupling model.The oscillatory inflation, reheating process after the slow roll and warm inflation in the presence of a non-minimal kinetic coupling, was studied in [28][29][30][31][32].
In the present work, inspired by the above mentioned models, we will consider gravitational baryogenesis mechanism in non-minimal derivative coupling model.The paper is organized as follows: In the section 2 we briefly introduce non-minimal derivative coupling model.In the section 3 we examine conditions for oscillatory inflation and study the reheating phase in this model and the temperature at the end of warm inflation is calculated.In the section 4 we discuss gravitational baryogenesis scenarios during oscillatory inflaton dominated in the context of non-minimal derivative coupling model and also we briefly investigate gravitational baryogenesis scenarios during radiation dominated phase.In the section 5 we consider the qualitative implication of non-minimal derivative coupling by calculating the corresponding baryon asymmetry and compare our results with the observation.In the last section we conclude our results.
We use units = c = 1 through the paper.

The model
In this section, we will introduce reheating of universe after inflation in nonminimal kinetic coupling model where rapid oscillatory inflaton decaying to radiation.Let us consider the total action of non-minimal derivative coupling model [25,26] where Rg µν is the Einstein tensor, M is a coupling constant with mass dimension, M P = 2.4 × 10 18 GeV is the reduced Planck mass, S r is the radiation action and S int describes the interaction of the scalar field with radiation.In order to describe gravitational baryogenesis, we define action S B by interaction between derivative of Ricci scalar curvature ∂ µ R and baryon current J µ as [9] where M * is the cutoff scale of the effective theory.We can obtain energy momentum tensor by variation of the action (1) with respect to the metric, Where T (r) µν is the energy momentum tensor for radiation described as u µ is the four-velocity of the radiation and T (ϕ) µν is the energy momentum tensor for minimal and non-minimal coupling counterparts of scalar field as follows Energy transfer between the scalar field and radiation is assumed to be [33][34][35][36][37][38] Γ, is decay rate of scalar field, where in general, is a function of ϕ and temperature [34,39].The covariant derivative of energy momentum tensor becomes The equation of motion, for the spatially flat FLRW Universe, and in presence of the dissipative term becomes where H = ȧ/a is the Hubble parameter, overdot sign is derivative with respect to cosmic time t, prime is derivative with respect to the scalar field ϕ, Γ φ is the friction term which describes the decay of the scalar field to radiation.The Friedmann equations are given by Where ρ r and P r are the energy density and the pressure of radiation, respectively.In the non-minimal derivative coupling model, the energy density ρ ϕ and the pressure of inflaton field P ϕ in FRW metric can be expressed as [25,26] We can write energy density of radiation as a function of temperature T and entropy density s, ρ r = (3/4)T s [34].Using the equation of state parameter for radiation ω r = 1/3, we obtain the rate of radiation production as 3 Reheating after inflation In this section we will consider reheating of the Universe after the end of slow-roll inflation.At the end of inflation, oscillation of the scalar field about the bottom of potential begins.We assume that the potential is even, V (−ϕ) = V (ϕ), and consider rapid oscillating solution to equation ( 9) around ϕ = 0.The inflaton energy density may estimated as ρ ϕ = V (Φ(t)), where Φ(t) is the amplitude of inflaton oscillation.In this epoch ρ ϕ and H change insignificantly during a period of oscillation [29,30].
By averaging the continuity equation, we obtain [32] We can simply derive the average of energy density of scalar field at the high friction limit and ΓM 2 ≪ 3H 3 constrain as M from equation (48).In the case that decoupling take place during the reheating era for quadratic inflationary potential (q = 2), dimension-6 B-violating interaction (n = 2) and By relation (19) and Friedmann equation, in the scalar field dominated era, we can obtain Therefore the Hubble parameter in the energy density of scalar field dominated era, can be estimated as H ≈ 2/(3γt).In the rapid oscillation phase and with the power law potential (16), we can write the amplitude of the oscillation as We have seen at high friction limit φ2 ≈ γM 2 P M 2 is nearly constant.Therefore from equations ( 14) and ( 16) we can calculate evolution of radiation and scalar field energy density as where t o is the beginning of scalar field rapid oscillation, which ρ r (t = t o ) = 0 and ρ o = ρ ϕ (t o ).In the rapid oscillation phase, energy density of radiation increases slowly, so that at the time t RD the energy density of radiation becomes equal to the scalar field ρ r (t RD ) ≈ ρ ϕ (t RD ).From equations (22) and (10) we can calculate t RD as [30,32]  ).The solid black curve correspond to "decoupling of B-violating processes" with M B = 10 −4 M p .In the case that decoupling take place during the reheating era for quadratic inflationary potential (q = 2) and dimension-6 B-violating interaction (n = 2).The admissible regions for decoupling temperature and reheating temperature are displayed with a brown color spectrum, from high friction condition.Intersection of the dashed curves and solid curve are the points where defines, T RD and T D .
We can write energy density of radiation as a function of reheating temperature T RD as [33] where g ⋆ is the number of degree of freedom at the reheating temperature and T RD is the temperature of radiation at the beginning of radiation dominated era.Therefore the temperature of the universe at the beginning of radiation dominated universe becomes The solid black curve correspond to "decoupling of B-violating processes" with M B = 10 −12 M p .In the case that decoupling take place during the reheating era for quadratic inflationary potential (q = 2) and dimension-6 B-violating interaction (n 2).The admissible regions for decoupling temperature and reheating temperature are displayed with a brown color spectrum, from high friction condition.Intersection of the dashed curves and solid curve are the points where defines, T RD and T D .

Reheating period
During the reheating era, the scalar field oscillates about minimum of potential and decays to ultra relativistic particles.In this period the energy density of oscillatory scalar field is dominated and the Universe expansion is accelerated.The equation ( 20) is equivalent to where a RD is scale factor at the beginning of radiation dominated era.The evolution of the energy density of radiation during the rapid oscillation of scalar field from relation (14) becomes Therefore we can write the energy density of radiation as a function of scale factor as From relation (19) the energy density of scalar field is given by From relation (29) and Fridmann equation (10) we can calculate evolution of scale factor as a function of temperature, during reheating period as This relation show that in the non-minimal derivative coupling model, the temperature of Universe increases by expansion of the Universe until beginning of radiation dominated period.While in the standard thermal history of the Universe, temperature had opposite evolution.By replacement of relation (31) into equation ( 30) the energy density of inflaton field becomes If there exist B-violating interaction in thermal equilibrium then it can generate net baryon asymmetry.In the expanding Universe From action (3) we have [9,22] where g b = −gb denotes the number of intrinsic degree of freedom of baryons.n b and nb are the number densities of baryon and antibaryon respectively.An effective chemical potential follow as ⋆ , the entropy density of the Universe is given by s = 2π 2 g ⋆ T 3 /45, the baryon number density, in thermal equilibrium, becomes n B = (g b n b +gbnb) = −g b µ b T 2 /6 [40].As a result, we can write the baryon to entropy ratio (baryon asymmetry) in an accelerating universe as where temperature T D is the temperature of the Universe at which the baron current violation decouples.In the spatially flat FLRW metric, Ricci scalar curvature R is equal to by using equation (31) for the scale factor, easily we can calculate Ricci scalar curvature as a function of cosmic time as Now, with the time derivative of Ricci scalar curvature (36) and Fridmann equation H 2 ≈ ρ ϕ /3M 2 p in the scalar field domination period, we have Finally, by substituting Ṙ and ρ ϕ from equations ( 37) and ( 32) into equation (34), we obtain baryon asymmetry Y B as a function of Universe temperature We continue with a brief mention of the origin of the B-violating interaction that is indispensable for any baryogenesis scenario.We assume that B-violating interactions, which are given by an operator O B of mass dimension D = 4 + n [9,22].We need n > 0 for the B-violating interaction.In the B-violating interactions, coupling constants are proportional to M −n B , where M B is the mass scale, the rate of generation of B-violating interaction in thermal equilibrium with the temperature T can be cast in the form [9] Decoupling of B-violating processes occurs at T = T D , when Γ falls below H = 2/3γt.Therefore we can obtain temperature of decoupling from equation (32) as Therefore by substituting of relation (40) into equation (38) we have ).In the case that decoupling take place during the reheating era for quadratic inflationary potential (q = 2) and dimension-6 B-violating interaction (n = 2).The admissible regions for decoupling temperature and reheating temperature are displayed with a brown color spectrum, from high friction condition.Intersection of the dashed curves and solid curve are the points where defines, T RD and T D .
From high friction condition H 2 /M 2 ≫ 1 we can constrain the decoupling temperatures T D as Contrary to minimal coupling model, we have seen during the oscillatory scalar field dominated universe for γ = 1/3 the temperature of the universe increases as T ≈ T RD (a(t)/a RD ) (1/8) .Therefore, the decoupling temperature T D is smaller than the reheating temperature T RD , in pleasant accordance with the condition (42).

Radiation dominated period
In this section we consider the gravitational baryogenesis during the radiation dominated era after the reheating period.The challenge of gravitational ).In the case that decoupling take place during the reheating era for quadratic inflationary potential (q = 2) and dimension-6 B-violating interaction (n = 2).The admissible regions for decoupling temperature and reheating temperature are displayed with a brown color spectrum, from high friction condition.Intersection of the dashed curves and solid curve are the points where defines, T RD and T D .baryogenesis during radiation dominated universe is characterised by equation of state ω ≈ 1/3.If ω were equal to 1/3 exactly, then R = 3(1 − 3ω)H 2 would vanish and Y B = 0. Then the baryon asymmetry effect would never be generated during the radiation dominated era.If we look more closely at the problem, it is not so serious, so that if we take into account the effect of interactions among massless particles lead to trace anomaly that make T µ µ = 0. Therefore the equation of state is given by 1 − 3ω ∼ 10 −2 − 10 −1 [9,22].
In the radiation dominated era a ∝ √ t, ρ r ∝ a −4 , H ≈ 1/2t and by relation ρ r = (π 2 g ⋆ /30)T 4 we arrive at We can calculate the time of decoupling by equality Γ = H, as , and the temperature of decoupling becomes By using Fridmann equation during of radiation dominated era H 2 ≈ ρ r /3M 2 p we have R = 3(1 − 3ω)(1/4t 2 ), therefore baryon asymmetry reads We assume that reheating temperature is T RD = 10 −9 M p and the golden region show the Hence,by choosing dimension-6 B-violating interaction (n = 2) and g ⋆ = 106 we have similarly, by choosing dimension-5 B-violating interaction (n = 1) we have

numerical analyses
As we have shown in the previous section, the value of reheating temperature T RD is depends on coupling constant M. On the other hand, high friction condition, constrain reheating temperature.To display high friction condition H 2 /M 2 ≫ 1 in our plots, we can parameterize relation (42) as As we can see, this constrain does not depend on q and n parameters, but it depends on coupling constant M. Therefore, we must chose T D and T RD where satisfy in this condition, for any value of M. In all of the plots Figure 1-Figure 6, decoupling take place during the reheating era for quadratic inflationary potential (q = 2 or γ = 1/3).Also, we assume that dimension-6 B-violating interaction (n = 2), the ultra relativistic degrees of freedom at the electroweak energy scale g ⋆ = 106.75, the number of intrinsic degree of freedom of baryons g b ≈ O(1).
In Figure 1 the admissible region of decoupling temperature T D and reheating temperature T RD from high friction condition, in the case that decoupling take place during the reheating era for quadratic inflationary potential (q = 2), dimension-6 B-violating interaction (n = 2),coupling constant M = 10 −8 M p , and different values of C has been depicted.As a reference, the acceptable regions are C > 10 where displayed by the brown color.So the region C < 1 is out of bounds and the values of reheating temperature and decoupling temperature in this region violate the high friction condition.The bright colors represent low values while the dark colors represent high values for C.
We can rewrite relation ( 38), ( 40) and ( 41) for γ = 1/3, n = 2, in the form of In Figure 2 high friction condition (48) for different values of coupling constant M has been depicted.Clearly, as the coupling constant M is decreased, the acceptable region for T RD and T D associated with high temperature.
Figure 3 shows decoupling temperature T D in terms of reheating temperature T RD , for coupling constant M = 10 −8 M p .The blue, green and red dashed curves correspond to different values of cutoff scale M ⋆ , to explain the observed baryon asymmetry (Y B = 8.64 × 10 −11 ) in relation (49).The solid black curve correspond to "decoupling of B-violating processes" with M B = 10 −4 M p in relation (50).The admissible regions for decoupling temperature and reheating temperature are displayed with a brown color spectrum, from high friction condition.Intersection of the dashed curves and solid curve are the points where defines, T RD and T D .We have seen, intersection points take place in admissible regions C > 100 and C > 1000.Therefore, if we want to explain, generation of baryon asymmetry in high reheating temperature, then we have to choose higher values of M ⋆ and M B .
Therefore more appropriate choices, are larger cutoff scale M ⋆ which fall into the dark eras.
The Figure 4 is similar to Figure 3, but we drawing this plot for low reheating temperature and different parameters values.We have seen, if we want to explain, generation of baryon asymmetry in low reheating temperature, then we have to choose smaller values of M ⋆ , M B and M.
The Figure 6 shows decoupling temperature T D in terms of reheating temperature T RD , for coupling constant M = 10 −8 M p and high reheating temperature.The blue and red dashed curves correspond to different values of M B in "decoupling of B-violating processes" in relation (50).The solid black curve correspond to cutoff scale M ⋆ = 10 −1 M p , to explain the observed baryon asymmetry (Y B = 8.64 × 10 −11 ) in relation (49).The admissible regions for decoupling temperature and reheating temperature are displayed with a brown color spectrum, from high friction condition.Intersection of the dashed curves and solid curve are the points where defines, T RD and T D where satisfy in relations (49), (50) and high friction condition.We have seen, intersection points take place in admissible regions C > 100.
The Figure 6 is similar to Figure 6, but we drawing this plot for low reheating temperature and different parameters values.We have seen, if we want to explain, generation of baryon asymmetry in low reheating temperature, then we have to choose smaller values of M ⋆ , M B and M.
The Figure 7, the acceptable range for M ⋆ and M B to explain the observed baryon asymmetry (Y B = 8.64 × 10 −11 ) has been depicted.The solid blue curve, correspond to dimension-6 B-violating interaction (n = 2) and red dashed curve dimension-5 B-violating interaction (n = 1).We assume that reheating temperature is T RD = 10 −9 M p and the golden region show the T RD < 10 −9 M p .
We conclude that in non-minimal derivative coupling model, sufficient baryon asymmetry has been generated in low and high reheating temperature during reheating phase.

Conclusion
In this paper, We investigated the gravitational baryogenesis mechanism in the non-minimal derivative coupling model in high friction regime.We used coupling between derivative of Ricci scalar curvature and baryon current to describe baryon asymmetry.In this model, inflaton begins a coherent rapid oscillation, after the slow roll inflation.During this stage, inflaton decays to radiation and reheats the Universe.We calculated the baryon to entropy ratio in the case that reheating period described by coherent rapid oscillation in the non-minimal derivative coupling model.As we demonstrated, in contrast to the standard gravitation baryogenesis where could not explain baryon asymmetry in low reheating temperature, in the non-minimal derivative coupling model we can describe baron asymmetry in low and high reheating temperature.

Figure 1 :
Figure 1: The admissible region of decoupling temperature T D and reheating temperature T RD from high friction condition, in the case that decoupling take place during the reheating era for quadratic inflationary potential (q = 2) and dimension-6 B-violating interaction (n = 2).We assumes coupling constant M = 10 −8 M p and different values of C. The allowed regions (C > 10) are restricted to the dark brown.The bright colors of the region represent low values while the dark colors represent high values of C.

Figure 2 :
Figure 2: Comparison high friction condition for different values of coupling constant

Figure 3 :
Figure 3: decoupling temperature T D in terms of reheating temperature T RD , for coupling constant M = 10 −8 M p and high reheating temperature.The blue, green and red dashed curves correspond to different values of cutoff scale M ⋆ , to explain the observed baryon asymmetry (Y B = 8.64 × 10 −11).The solid black curve correspond to "decoupling of B-violating processes" with M B = 10 −4 M p .In the case that decoupling take place during the reheating era for quadratic inflationary potential (q = 2) and dimension-6 B-violating interaction (n = 2).The admissible regions for decoupling temperature and reheating temperature are displayed with a brown color spectrum, from high friction condition.Intersection of the dashed curves and solid curve are the points where defines, T RD and T D .

Figure 4 :
Figure 4: Decoupling temperature T D in terms of reheating temperature T RD , for coupling constant M = 10 −14 M p and low reheating temperature.The blue, green and red dashed curves correspond to different values of cutoff scale M ⋆ , to explain the observed baryon asymmetry (Y B = 8.64 × 10 −11 ) in relation (49).The solid black curve correspond to "decoupling of B-violating processes" with M B = 10 −12 M p .In the case that decoupling take place during the reheating era for quadratic inflationary potential (q = 2) and dimension-6 B-violating interaction (n 2).The admissible regions for decoupling temperature and reheating temperature are displayed with a brown color spectrum, from high friction condition.Intersection of the dashed curves and solid curve are the points where defines, T RD and T D .

Figure 5 :
Figure 5: decoupling temperature T D in terms of reheating temperature T RD , for coupling constant M = 10 −8 M p and high reheating temperature.The blue and red dashed curves correspond to different values of M B in "decoupling of B-violating processes".The solid black curve correspond to cutoff scale M ⋆ = 10 −1 M p , to explain the observed baryon asymmetry (Y B = 8.64 × 10 −11).In the case that decoupling take place during the reheating era for quadratic inflationary potential (q = 2) and dimension-6 B-violating interaction (n = 2).The admissible regions for decoupling temperature and reheating temperature are displayed with a brown color spectrum, from high friction condition.Intersection of the dashed curves and solid curve are the points where defines, T RD and T D .

Figure 6 :
Figure 6: decoupling temperature T D in terms of reheating temperature T RD , for coupling constant M = 10 −14 M p and low reheating temperature.The blue and red dashed curves correspond to different values of M B in "decoupling of B-violating processes".The solid black curve correspond to cutoff scale M ⋆ = 10 −9 M p , to explain the observed baryon asymmetry (Y B = 8.64 × 10 −11).In the case that decoupling take place during the reheating era for quadratic inflationary potential (q = 2) and dimension-6 B-violating interaction (n = 2).The admissible regions for decoupling temperature and reheating temperature are displayed with a brown color spectrum, from high friction condition.Intersection of the dashed curves and solid curve are the points where defines, T RD and T D .

Figure 7 :
Figure 7: The acceptable range for M ⋆ and M B to explain the observed baryon asymmetry (Y B = 8.64 × 10 −11).The solid blue curve, correspond to dimension-6 B-violating interaction (n = 2) and red dashed curve dimension-5 B-violating interaction (n = 1).We assume that reheating temperature is T RD = 10 −9 M p and the golden region show the T RD < 10 −9 M p .