*f*(*R*, *T*) models applied to baryogenesis

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## Abstract

This paper is devoted to the reproduction of the gravitational baryogenesis epoch in the context of *f*(*R*, *T*) theory of gravity, where *R* and *T* are respectively the curvature scalar and the trace of the energy-momentum tensor, respectively. It is assumed a minimal coupling between matter and gravity. In particular we consider the following two models, \(f(R,T) = R +\alpha T + \beta T^2\) and \(f(R,T) = R+ \mu R^2 + \lambda T\), with the assumption that the universe is filled by dark energy and perfect fluid where the baryon to entropy ratio during a radiation domination era is non-zero. We constrain the models with the cosmological gravitational baryogenesis scenario, highlighting the appropriate values of model’s parameters compatible with the observation data of the baryon-entropy ratio.

## 1 Introduction

^{1}Thus the possibility arises of processes which preferentially produce matter rather than antimatter (although our present theoretical understanding doesn’t allow us to deduce this directly from the observed CP violation). However, even if this is the case, the ratio of particles and antiparticles will be very close to unity providing they are in equilibrium, as will be the case when the universe was very hot. Only as it cools and the equilibrium is removed will the tiny asymmetry in the particle interactions be amplified to an actual asymmetry in number densities. These requirements to produce matter-antimatter asymmetry, namely, (a) non-conservation of baryon number, (b) CP violation and (c) non-equilibrium are known as the Sakharov conditions [16]. In order to connect to dark energy, the authors [17, 18] have studied a class of models of spontaneous baryo(lepto)genesis by introducing a interaction between the dynamical dark energy scalars and the ordinary matter. Recently, Davoudiasl et al. [19] have proposed a mechanism for generating the baryon number asymmetry in thermal equilibrium during the expansion of the Universe by means of a dynamical breaking of CP. The interaction responsible for CP violation is given by a coupling between the derivative of the Ricci scalar

*R*and the baryon current \(J^{\mu }\) of the form

*g*and

*R*being respectively, the metric determinant and the Rurvature scalar. Other scenario to extend this well known theory by using a similar couplaging between the Ricci scalar and the baryonic current has been discussed by many authors. This scenario extends the well known theory that uses a similar coupling between the Ricci scalar and the baryonic current. In [20],

*f*(

*R*) theories of gravity are reviewed in the context of the so called gravitational baryogenesis. Some variant forms of gravitational baryogenesis by using higher order terms containing the partial derivative of the Gauss-Bonnet scalar coupled to the baryonic current are discussed in [21] whereas in [22], the gravitational baryogenesis scenario, generated by an

*f*(

*T*) theory of gravity where

*T*is the torsion scalar are proposed.

The purpose of this paper is to investigate the gravitational baryogenesis mechanism in *f*(*R*, *T*) modified theory of gravity, a theory in which matter and geometry are minimally coupled and well known as generalization of General Theory of Relativity. This Theory was firstly introduced by the authors of [23] and several works with interesting results have been found in [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42].

The paper is organized as follows: A brief review in *f*(*R*, *T*) gravity is performed in Sect. 2. We investigate the essential features of baryogenesis in *f*(*R*, *T*) gravity by calculating the corresponding baryon to entropy ratio in universe containing a dark energy and the perfect fluid with constant equation of state parameter in Sect. 3. Some conclusions are presented in the last section.

## 2 Brief review in *f*(*R*, *T*) gravity

*f*(

*R*,

*T*) gravity by

*R*,

*T*are the curvature scalar and the trace of the energy-momentum tensor, respectively,

*G*being the gravitation constant.

*f*(

*R*,

*T*) with respect to

*R*and

*T*, respectively. The field equations (4) are reduced to Einstein field equations when \(f(R,T) \equiv R\).

*R*and the trace

*T*of the energy momentum tensor

*a*(

*t*) is the scale factor and the matter content of the universe is a perfect fluid for which the matter Lagrangian density can be taken as \(\mathcal {L}_m = -p \). For this, the Eqs. (4) and (7) become

*t*and \(H=\frac{\dot{a}}{a}\), the Hubble parameter. In the above equations, \(\rho \) is the matter density,

*p*the matter pressure and the trace \(T= \rho -3p\).

## 3 *f*(*R*, *T*) baryogenesis

*f*(

*R*,

*T*) gravity where we take account a minimal coupling between matter and geometry, we consider a CP-violating interaction term generating by the baryon asymmetry of the universe of the form,

*R*and the trace \(T= \rho (1-3w) \) of the energy-momentum tensor of matter are related as

*f*(

*R*,

*T*) theories of gravity. To do, we focus on attention on two particulars

*f*(

*R*,

*T*) models namely \(f(R,T) = R +\alpha T + \beta T^2\) and \(f(R,T) =R+ \mu R^2 + \lambda T\) to describe how we can recover the baryogenesis epoch with these models. We calculate the baryon to entropy ratio for each model by considering a universe filled by the dark energy and perfect fluid with constant equation of state parameter \( w= \frac{p}{\rho }\) and assuming that the scale factor evolve as power-law \(a(t) = B t^{\gamma }\) where

*B*is a constant parameter.

### 3.1 \(f(R,T) = R +\alpha T + \beta T^2\) cases

*f*(

*R*,

*T*) particular model

**various**values of the parameter \(\beta \).

Some values of baryon to entropy ratio for \(\gamma =0.4\) and \(\alpha = 10^{-20}\)

\(\beta \) | \(-10^{-9}\) | \(\mathbf{-2\times 10^{-9}}\) | \(\mathbf{-3 \times 10^{-9}}\) | \(\mathbf{-4\times 10^{-9}}\) | \(\mathbf{-5\times 10^{-9}}\) | \(\mathbf{-6\times 10^{-9}}\) |
---|---|---|---|---|---|---|

\(\frac{ n_{B}}{s}\) | \( {1.12\times 10^{-11} }\) | \({3.18\times 10^{-11}}\) | \( {5.85\times 10^{-11}} \) | \({9.01\times 10^{-11}}\) | \({1.25\times 10^{-10}}\) | \({1.66\times 10^{-10}}\) |

According to the results of this table, we observe that for \(\beta = \mathbf{-4\times 10^{-9}}\), \(n_{B}/s\) \(\mathbf{= 9.01\times 10^{-11}}\), which is very agreement with observations and practically equal to the observed value (\(n_{B}/s \mathbf{\simeq 9.42\times 10^{-11}}\)) whereas when \(\beta > \mathbf{-4\times 10^{-9}}\), we denote a significantly small values.

### 3.2 \(f(R,T) = R+ \mu R^2 + \lambda T\) cases

*f*(

*R*,

*T*) model considered. Notice that for \(\gamma = 0.3\) and \(\mu = \lambda = 10^{-5}\), the baryon to entropy ratio \(n_{B}/s = \mathbf{8.28\times 10^{-11}}\), which is a compatible with the observational value. In the same way, we plot in Fig. 3, the evolution of baryon to entropy ratio (23) versus \(\gamma \) for \(M_{*} = 10^{12}\) GeV and \({\mathcal {T_D}} = \mathbf{2\times 10^{16}}\) GeV in comparison with the curve traducing the observational value.

In Figs. 2 and 4, we have noticed that how we can constrain each model to see the observational value of baryon to entropy ratio in the frame of the modified theories of gravity. The different values of the parameters that can constrain these models in baryogenesis era are denoted by the intersection of each curve for specific parameter with the observational curve of baryon to entropy ratio.

## 4 Conclusion

The paper is devoted to the study of gravitational baryogenesis mechanism in the context of *f*(*R*, *T*) theories. According that the CP-violating interaction that will generate the baryon asymmetry of the Universe and considering that the matter content of the universe as perfect fluid with constant equation of state parameter, we evaluate the baryon to entropy ratio for two particulars *f*(*R*, *T*) models. In contrast with GR, we show for both models that the baryon to entropy ratio is non-zero in radiation dominated epoch if the parameter \(\gamma \ne \frac{1}{2}\) and the baryogenesis epoch can be reproduced in such theory.

## Footnotes

- 1.
One is the charge conjugation symmetry (

*C*-symmetry) and the other is the parity symmetry (*P*– symmetry). The combined symmetry of the two is called,*CP*– symmetry.

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