# Stability Analysis of the Inhomogeneous Perturbed Einstein Universe in Energy–Momentum Squared Gravity

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Einstein Universe

## 3. Inhomogeneous Scalar Perturbations

## 4. Stability Analysis

#### 4.1. Case I

#### 4.2. Case II

## 5. Final Remarks

- A unique expression of ${f}_{2}\left({T}^{2}\right)$ for the conserved EMT case was developed that satisfies the conservation equation. We investigated the stable modes of the ESU against ${\mathrm{b}}_{1}$ for distinct values of $\lambda $. It was found that stability of the ESU existed for all values of ${\mathrm{b}}_{1}$ corresponding to closed and open cosmic models.
- We assumed a particular type of ${f}_{2}\left({T}^{2}\right)$ in the non-conserved case and analyzed the stability of the ESU for different values of $\eta $. We found that stable modes existed for entire values of $\omega $. These stable regions became more smooth as the model parameter increased in the open universe model and decreased in the closed universe model. It is worthwhile to mention here that our solutions reduced to homogeneous perturbations for $\lambda =0$.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Stable modes of the ESU for $\lambda =2$ (blue) and $\lambda =15$ (orange), corresponding to closed universe model (

**right**), and for ${\lambda}^{2}=2$ (blue) and ${\lambda}^{2}=15$ (orange), corresponding to the open universe model (

**left**).

**Figure 2.**Stable regions of the ESU for $K=-1$, $\lambda =2$ (blue) and $\lambda =15$ (orange), corresponding to $\eta =1$ (

**left**) and $\eta =-1$ (

**right**).

**Figure 3.**Stable regions of the ESU for $K=-1$, $\lambda =2$ (blue) and $\lambda =15$ (orange), corresponding to $\eta =7$ (

**left**) and $\eta =-7$ (

**right**).

**Figure 4.**Stable regions of the ESU for $K=1$, ${\lambda}^{2}=2$ (blue) and ${\lambda}^{2}=15$ (orange), corresponding to $\eta =1$ (

**left**) and $\eta =-1$ (

**right**).

**Figure 5.**Stable regions of the ESU for $K=1$, ${\lambda}^{2}=2$ (blue) and ${\lambda}^{2}=15$ (orange), corresponding to $\eta =7$ (

**left**); $\eta =-7$ (

**right**).

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**MDPI and ACS Style**

Sharif, M.; Gul, M.Z.
Stability Analysis of the Inhomogeneous Perturbed Einstein Universe in Energy–Momentum Squared Gravity. *Universe* **2023**, *9*, 145.
https://doi.org/10.3390/universe9030145

**AMA Style**

Sharif M, Gul MZ.
Stability Analysis of the Inhomogeneous Perturbed Einstein Universe in Energy–Momentum Squared Gravity. *Universe*. 2023; 9(3):145.
https://doi.org/10.3390/universe9030145

**Chicago/Turabian Style**

Sharif, Muhammad, and Muhammad Zeeshan Gul.
2023. "Stability Analysis of the Inhomogeneous Perturbed Einstein Universe in Energy–Momentum Squared Gravity" *Universe* 9, no. 3: 145.
https://doi.org/10.3390/universe9030145