Cosmic String Isotropic Model in Modified Theory of Gravity

Abstract

The spatially homogeneous Bianchi type-I in Kasner form have been considered in modified theory of gravitation. The exact solutions of the field equations in f(R,T) gravity theory for Bianchi type- I(kasner form) have been obtained. The physical parameters of the models are also mentioned.

Introduction

The standard model of cosmology, based on General Relativity (GR), successfully explains much of the universe's large-scale structure. However, observations like accelerated cosmic expansion suggest that GR may be incomplete. This has motivated the study of modified gravity theories, such as f(R,T)f(R,T)f(R,T) gravity—a framework introduced by Harko that extends GR by incorporating a function of the Ricci scalar (R) and the trace of the energy-momentum tensor (T).

This theory introduces matter-geometry coupling, potentially modeling phenomena like quantum effects or interactions between matter and curvature without requiring dark energy.

2. Model: Bianchi Type-I in Kasner Form

• The study focuses on an anisotropic cosmological model: Bianchi type-I, specifically in the Kasner form—a generalization of flat FLRW space-time allowing different expansion rates along each axis.

• The Kasner metric:

  ds2=dt2−t2p1dx2−t2p2dy2−t2p3dz2ds^2 = dt^2 - t^{2p_1} dx^2 - t^{2p_2} dy^2 - t^{2p_3} dz^2ds2=dt2−t2p1?dx2−t2p2?dy2−t2p3?dz2

  with constants p1,p2,p3p_1, p_2, p_3p1?,p2?,p3? satisfying:

  p1+p2+p3=s,p12+p22+p32=θp_1 + p_2 + p_3 = s, \quad p_1^2 + p_2^2 + p_3^2 = \thetap1?+p2?+p3?=s,p12?+p22?+p32?=θ

3. Field Equations in f(R,T)f(R,T)f(R,T) Gravity

• The field equations are derived using:

  • The Kasner metric.

  • A cloud string energy-momentum tensor.

  • The function f(T)=μTf(T) = \mu Tf(T)=μT, which leads to modified field equations incorporating the matter-geometry coupling.

• Solving these equations yields:

  p1=p2=p3=P⇒Isotropic evolutionp_1 = p_2 = p_3 = P \quad \Rightarrow \text{Isotropic evolution}p1?=p2?=p3?=P⇒Isotropic evolution

  So, the metric becomes:

  ds2=dt2−t2P(dx2+dy2+dz2)ds^2 = dt^2 - t^{2P}(dx^2 + dy^2 + dz^2)ds2=dt2−t2P(dx2+dy2+dz2)

  reducing to an isotropic model resembling the flat FLRW Universe.

4. Key Physical Quantities Derived

• Volume scale factor: V=ts=t3PV = t^s = t^{3P}V=ts=t3P

• Mean scale factor: a(t)=ts/3a(t) = t^{s/3}a(t)=ts/3

• Hubble parameter: H=a?a=PtH = \frac{\dot{a}}{a} = \frac{P}{t}H=aa??=tP?

• Expansion scalar: θ=3H\theta = 3Hθ=3H

• Shear scalar σ\sigmaσ and anisotropy parameter Δ\DeltaΔ: Both are zero, indicating isotropy.

• Pressure and energy density expressions:

  p(t)=−12t2+4π(1+Λ)+32p(t) = -\frac{1}{2t^2} + 4\pi(1 + \Lambda) + \frac{3}{2}p(t)=−2t21?+4π(1+Λ)+23? ρ(t)=8π(1+Λ)+24(4π+1)−14(4π+1)t2\rho(t) = \frac{8\pi(1 + \Lambda) + 2}{4(4\pi + 1)} - \frac{1}{4(4\pi + 1)t^2}ρ(t)=4(4π+1)8π(1+Λ)+2?−4(4π+1)t21?

5. Results & Physical Implications

• Isotropy Achieved: Despite starting with an anisotropic model, the solution leads to isotropic evolution due to equal directional Hubble rates (H1=H2=H3H_1 = H_2 = H_3H1?=H2?=H3?).

• Shear Vanishes: With σ=0\sigma = 0σ=0, the universe evolves toward an FRW-like behavior.

• Late-Time Universe:

  • As t→∞t \to \inftyt→∞, both ρ(t)\rho(t)ρ(t) and p(t)p(t)p(t) approach constants.

  • The universe becomes Λ-dominated and isotropic, consistent with current observations.

• Consistency Conditions:

  • Time-dependent pressure and density must match isotropy assumptions.

Conclusion

We have considered Bianchi type-I metric (Kasner form) in the presence of cosmic string in modified theory of gravity. It is interesting to know that our model is isotropic and Shear free. For the present cosmological model the anisotropy parameter ?=0 implies? H?_1=H_2=H_3=H ; hence the directional Hubble rates coincide. Consequently the shear scalar ?^2=0 which is also vanishes. The vanishing of ? and ? therefore indicates that the model is kinematically isotropic in effect the Bianchi type–I solutions reduce to an FRW like behaviour. All directional pressures become equal, shear-driven terms drop out of the field equations, and the late-time dynamics are dominated by the constant ? term (consistent with the numerical plots which show ?(t) and p(t) approaching constant values).

References

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Copyright

Copyright © 2025 V. M. Raut, A. S. Mankar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.