# Jerk Parameter and Modified Theory

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## Abstract

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## 1. Introduction

## 2. Field Equations of $\mathit{f}(\mathit{R},\mathit{T})$ Gravity

#### 2.1. Field Equations for the Non-Minimal Coupling (NMC) Form

#### 2.2. Field Equations for the Simplest Minimal Coupling (SMC) Form

## 3. Modified Field Equations in the Flat FLRW Background

#### 3.1. The NMC Form

#### 3.2. The SMC Form

## 4. Solution of the Field Equations

#### 4.1. Decelerating Model for NMC Form

#### 4.2. Accelerating Model for NMC Form

#### 4.3. Transit Model for NMC Form

#### 4.4. Decelerating Model for SMC Form

#### 4.5. Accelerating Model for SMC Form

#### 4.6. Transit Model for SMC Form

## 5. Wec Analysis

## 6. Conclusions

- This study entails a much deeper analysis of the concept of the implications of a constant jerk parameter for modified gravity.
- Most studies of $f(R,T)$ theory are based on the linear form of $f(R,T)$. In this work, we considered the nonlinear form $f(R,T)=R+RT$. This is much more difficult to analyze.
- The complete solution to the differential Equation (39) for $j=1$ has been presented for the first time here as far as we are aware.
- Since these equations are fairly complicated, and some of them are complex (and therefore imaginary), we selected one of the solutions. We simplified it to a manageable form that could be analyzed more easily, and we provided a complete analysis of this solution (the transit solution).
- Using observations, we calculated the values of the parameters that occur in the transit solution. Then, we calculated the Hubble parameter at the present time, the deceleration parameter at present and at transition, and the transition time from deceleration to acceleration. The transit solution appears to be viable, but we plan a more detailed analysis of its implications in the future, including observations.
- The transit solution was compared with the two other solutions (power law and exponential). Detailed plots were provided for all three solutions to illustrate the results more clearly.
- Energy conditions were analyzed in this paper; these were not addressed in the Phys. Sc. Forum paper. These conditions were analyzed in detail for all three types of models, and detailed comparisons were made amongst all three.
- This work has 23 figures illustrating our results here, as compared to 2 in the previous work.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

GR | General Relativity |

DE | Dark Energy |

$\mathsf{\Lambda}$CDM | $\mathsf{\Lambda}$ Cold Dark Matter |

WEC | Weak Energy Condition |

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**Figure 1.**Evolution of the energy density of matter $\rho $ against time t for decelerating model in NMC form.

**Figure 2.**Evolution of the total effective energy density ${\rho}^{tot}$ against time t for decelerating model in NMC form.

**Figure 3.**Evolution of the total effective pressure ${p}^{tot}$ against time t for decelerating model in NMC form.

**Figure 4.**Evolution of the energy density $\rho $ against time t for accelerating model in NMC form.

**Figure 5.**Evolution of the total effective energy density ${\rho}^{tot}$ against time t for accelerating model in NMC form.

**Figure 6.**Evolution of the total effective pressure ${p}^{tot}$ against time t for accelerating model in NMC form.

**Figure 10.**Evolution of the total effective energy density ${\rho}^{tot}$ against time t for transit model in NMC form.

**Figure 11.**Evolution of the total effective pressure ${p}^{tot}$ against time t for transit model in NMC form.

**Figure 13.**Evolution of the energy density ${\rho}^{tot}$ against time t for transit model in SMC form.

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**MDPI and ACS Style**

Sofuoğlu, D.; Beesham, A.
Jerk Parameter and Modified Theory. *Universe* **2023**, *9*, 430.
https://doi.org/10.3390/universe9100430

**AMA Style**

Sofuoğlu D, Beesham A.
Jerk Parameter and Modified Theory. *Universe*. 2023; 9(10):430.
https://doi.org/10.3390/universe9100430

**Chicago/Turabian Style**

Sofuoğlu, Değer, and Aroonkumar Beesham.
2023. "Jerk Parameter and Modified Theory" *Universe* 9, no. 10: 430.
https://doi.org/10.3390/universe9100430