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Research Objective

To explore the LRS Bianchi Type-I space-time filled with a perfect fluid within the framework of $f(R,T)$ gravity, analyzing two specific models to achieve an exact field solution leading to an accelerated universe, characterized by a time-dependent deceleration parameter.

Methodology

Theoretical analysis within the framework of $f(R,T)$ gravity, where $R$ is the Ricci scalar and $T$ is the trace of the stress-energy momentum tensor. Two specific $f(R,T)$ models (Model-I: $f(R,T) = 1R + 2T + 4T^2$ and Model-II: $f(R,T) = R - 2T$) are examined using the LRS Bianchi Type-I line element ($ds^2 = dt^2 - A^2dx^2 - B^2(dy^2 + dz^2)$) filled with a perfect fluid ($T_{ij} = (\rho)u_i u_j - p g_{ij}$). Solutions are derived by assuming a specific relation for the scale factors ($A = B^n$) and examining the resulting cosmological parameters.

Methodology Flowchart

                        graph TD; A[Define LRS Bianchi Type-I Metric & Perfect Fluid $T_ij$] --> B[Select $fR,T$ Model]; B --> C[Derive Modified Einstein Field Equations]; C --> D[Apply Specific Constraints/Assumptions e.g., $A=B^n$]; D --> E[Solve Field Equations for Scale Factors At, Bt]; E --> F[Calculate Cosmological Parameters H, q, $\rho$, p]; F --> G[Analyze Physical InterpretationGraphs/State Finder]; G --> H[Conclusion on Accelerated Expansion];                    

Discussion

The study confirms that modifying General Relativity via $f(R,T)$ gravity, using a perfect fluid source, can naturally explain the observed accelerated expansion without explicitly invoking dark energy components, provided a time-dependent deceleration parameter is assumed. The physical interpretation of the derived graphs (Figs 1-4) supports the accelerating phase indicated by negative pressure.

Key Findings

Both Model-I and Model-II yield solutions where the deceleration parameter ($q$) is negative across the evolution, confirming an accelerated expansion of the universe. For Model-I, energy density decreases towards a constant value as $T \to \infty$, while pressure is negative and approaches zero. For Model-II, energy density is a decaying function approaching a constant value, and pressure is negative, approaching zero.

Conclusion

The analysis of LRS Bianchi Type-I models in $f(R,T)$ gravity, using a perfect fluid, provides exact solutions demonstrating continuous accelerated expansion, as evidenced by the consistently negative deceleration parameter across the explored models.

Fact Check

1. Dark energy occupies 73% of the matter in the universe according to WMAP satellite experiment. (Claimed in text)
2. Model-I parameters used: $m=1, n=0, 4=13, p=1, q=0$. (Claimed in text)
3. Model-II parameters used: $\lambda=1, m=1, n=0, \beta=0$. (Claimed in text)

Mind Map