# A Critical Examination of the Standard Cosmological Model: Toward a Modified Framework for Explaining Cosmic Structure Formation and Evolution

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{−1}Mpc [19]. This finding was recently challenged through a homogeneity test for the matter distribution based on the Baryon Oscillation Spectroscopic Survey Data Release 12 CMASS galaxy sample [20]. It was found that the observed distribution of matter is statistically unlikely to be a random arrangement up to a radius of 300 h

^{−1}Mpc, which is approximately the largest statistically available scale [18].

## 2. Background Model Formulation

#### 2.1. Parametric Model

#### 2.2. Non-Parametric Model

#### 2.3. The Friedmann Model

#### 2.4. Modified Redshift Model

## 3. Analytical Solutions

#### 3.1. Light Intensity–Modified Redshift

**,**it is easy to show that:

#### 3.2. Number Density–Modified Redshift

## 4. MATLAB Graphical Simulations

^{−27}kgm

^{−3}to ρ(to) = 8.78 × 10

^{−25}kgm

^{−3}, speed of light c = 3 × 10

^{8}m/s, cosmic scale factor R(to) = 9 × 10

^{25}m (modifiable as needed), gravitational constant G = 6.67 × 10

^{−11}m

^{3}kg

^{−1}s

^{−2}, and the geometric curvature of the universe, where κ = 0 signifies a flat universe, κ = +1 designates a closed universe, and κ = −1 represents an open universe.

#### 4.1. Light Intensity–Redshift Graphs

#### 4.2. Number Density–Redshift Graphs

## 5. Discussion

#### 5.1. Light Intensity of Galaxy Distribution

#### 5.2. Number Density of Galaxy Formation

#### 5.3. Comparing the Standard Redshift and the Modified Redshift Friedmann Cosmological Models

#### 5.4. Transition from Decelerating to Accelerating Expanding Universe

#### 5.5. Meaning of Our Results

## 6. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Simulation result for log (I) against redshift $z$ for $z=0$ to $z=5$. The solid curves represent the standard redshift while dotted curves represent the modified redshift $f\left(z\right)={\alpha}_{1}z+{\alpha}_{2}{z}^{2}$ with ${\alpha}_{1}=2.005$ and ${\alpha}_{2}=0.005.$ Both models without λ.

**Figure 2.**Simulation result for log (I) against redshift $z$ for $z=0$ to $z=5$. The solid curves represent the standard redshift while dotted curves represent the modified redshift $f\left(z\right)$=$z+\gamma {\left(z\right)}^{2}$, where $\gamma \left(z\right)$ is a free function of $z$ with $\gamma =0.45$. Both models without λ.

**Figure 3.**Simulation result for log (I) against redshift $z$ for $z=0$ to $z=5$. The solid curves represent the standard redshift while dotted curves represent the modified redshift $f\left(z\right)=\frac{z}{\epsilon}$ with $\epsilon =0.45$. Both model without λ.

**Figure 4.**Simulation result for log (I) against redshift $z$ for $z=0$ to $z=5$. The solid curves represent log (I) against standard redshift $z$ with λ while dotted curves represent log (I) against modified redshift $f\left(z\right)={\alpha}_{1}z+{\alpha}_{2}{z}^{2}$ with ${\alpha}_{1}=2.005$ and ${\alpha}_{2}=0.005$ without λ.

**Figure 5.**Simulation result for log (I) against redshift $z$ for $z=0$ to $z=5$. The solid curves represent log (I) against standard redshift $z$ with λ while dotted curves represent log (I) against modified redshift $f\left(z\right)$=$z+\gamma {\left(z\right)}^{2}$, where $\gamma \left(z\right)$ is a free function of $z$ with $\gamma =0.45$ without λ.

**Figure 6.**Simulation result for log (I) against redshift $z$ for $z=0$ to $z=5$. The solid curves represent log (I) against standard redshift$z$ with λ while dotted curves represent log (I) against modified redshift $f\left(z\right)=\frac{z}{\epsilon}$ with $\epsilon =0.45$ without λ.

**Figure 7.**Simulation result for log (I) against redshift $z$ for $z=0$ to $z=5$. The solid curves represent log (I) against standard redshift $z$ with

**λ**while dotted curves represent log (I) against standard redshift $z$ without λ.

**Figure 8.**Simulation result for log (n) against redshift $z$ for $z=0$ to $z=5$. The solid curves represent log (n) against standard redshift $z$ while dotted curves represent log (n) against modified redshift $f\left(z\right)={\alpha}_{1}z+{\alpha}_{2}{z}^{2}$ with ${\alpha}_{1}=2.005$ and ${\alpha}_{2}=0.005$. Both models without λ.

**Figure 9.**Simulation result for log (n) against redshift $z$ for $z=0$ to $z=5$. The solid curves represent the standard redshift while dotted curves represent the modified redshift $f\left(z\right)$ =$z+\gamma ({z)}^{2}$, where $\gamma \left(z\right)$ is a free function of $z$ and $\gamma =0.45$. Both models without λ.

**Figure 10.**Simulation result for log (n) against redshift $z$ for $z=0$ to $z=5$. The solid curves represent the standard redshift while dotted curves represent the modified redshift $f\left(z\right)=\frac{z}{\epsilon}$ with $\epsilon =0.45$. Both models without λ.

**Figure 11.**Simulation result for log (n) against redshift z for $z=0$ to $z=5$. The solid curves represent the standard redshift while dotted curves represent the modified redshift $f\left(z\right)={\alpha}_{1}z+{\alpha}_{2}{z}^{2}$ with λ = 0, where${\alpha}_{1}=2.005$ and ${\alpha}_{2}=0.005$.

**Figure 12.**Simulation result for log (n) against redshift z for $z=0$ to $z=5$. The solid curves represent the standard redshift with λ while dotted curves represent t the modified redshift $f\left(z\right)$ = $z+\gamma ({z)}^{2}$ with λ = 0, where $\gamma \left(z\right)$ is a free function of $z$ and $\gamma $ = 0.45.

**Figure 13.**Simulation result for log (n) against redshift z for $z=0$ to $=5$. The solid curves represent the standard redshift with λ while dotted curves represent the modified redshift $f\left(z\right)=\frac{z}{\epsilon}$ with λ = 0, where $\epsilon =0.45$.

**Figure 14.**Simulation result for log (n) against redshift z for $z=0$ to $z=5$ The solid curves represent the standard redshift with λ while dotted curves represent the standard redshift with a vanishing λ (λ = 0).

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**MDPI and ACS Style**

Nyagisera, R.N.; Wamalwa, D.; Rapando, B.; Awino, C.; Mageto, M.
A Critical Examination of the Standard Cosmological Model: Toward a Modified Framework for Explaining Cosmic Structure Formation and Evolution. *Astronomy* **2024**, *3*, 43-67.
https://doi.org/10.3390/astronomy3010005

**AMA Style**

Nyagisera RN, Wamalwa D, Rapando B, Awino C, Mageto M.
A Critical Examination of the Standard Cosmological Model: Toward a Modified Framework for Explaining Cosmic Structure Formation and Evolution. *Astronomy*. 2024; 3(1):43-67.
https://doi.org/10.3390/astronomy3010005

**Chicago/Turabian Style**

Nyagisera, Robert Nyakundi, Dismas Wamalwa, Bernard Rapando, Celline Awino, and Maxwell Mageto.
2024. "A Critical Examination of the Standard Cosmological Model: Toward a Modified Framework for Explaining Cosmic Structure Formation and Evolution" *Astronomy* 3, no. 1: 43-67.
https://doi.org/10.3390/astronomy3010005