# Cosmological Observations in a Modified Theory of Gravity (MOG)

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## Abstract

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## 1. Introduction

## 2. Modified Gravity Theory

#### 2.1. Scalar-Tensor-Vector Gravity

#### 2.2. Point Particles in a Spherically Symmetric Field

#### 2.3. The MOG Poisson Equation

## 3. MOG and the Matter Power Spectrum

#### 3.1. Density Fluctuations in Newtonian Gravity

#### 3.1.1. Newtonian Theory of Small Fluctuations

#### 3.1.2. Analytical Approximation

#### 3.2. Density Fluctuations in Modified Gravity

**Figure 1.**The matter power spectrum. Three models are compared against five data sets (see text): Λ-cold dark matter (Λ-CDM) (dashed blue line, ${\Omega}_{b}=0.035$, ${\Omega}_{c}=0.245$, ${\Omega}_{\Lambda}=0.72$, $H=71$ km/s/Mpc), a baryon-only model (dotted green line, ${\Omega}_{b}=0.035$, $H=71$ km/s/Mpc) and modified gravity (MOG) (solid red line, $\alpha =19$, $\mu =5$ h Mpc${}^{-1}$, ${\Omega}_{b}=0.035$, $H=71$ km/s/Mpc), Data points are colored light blue [Sloan Digital Sky Survey (SDSS) 2006], gold (SDSS 2004), pink [Two-degree-Field (2dF)], light green [UK Schmidt Telescope (UKST)] and dark blue (CfA).

#### 3.3. Discussion

**Figure 2.**The effect of window functions on the power spectrum is demonstrated by applying the SDSS luminous red galaxy survey window functions to the MOG prediction. Baryonic oscillations are greatly dampened in the resulting curve (solid red line). A normalized linear $\Lambda -$CDM estimate is also shown (thin blue line) for comparison.

## 4. MOG and the CMB

`CMBFAST`[22]. Unfortunately, such software packages cannot easily be adapted for use with MOG. Instead, at the present time, we opt to use the excellent semi-analytical approximation developed by [23]. While not as accurate as numerical software, it lends itself more easily to nontrivial modifications, as the physics remain evident in the equations.

#### 4.1. Semi-Analytical Estimation of CMB Anisotropies

#### 4.2. The MOG CMB Spectrum

**Figure 3.**MOG and the acoustic power spectrum. Calculated using ${\Omega}_{M}=0.3$, ${\Omega}_{b}=0.035$, ${H}_{0}=71$ km/s/Mpc. Also shown are the raw Wilkinson Microwave Anisotropy Probe (WMAP) three-year data set (light blue), binned averages with horizontal and vertical error bars provided by the WMAP project (red) and data from the Boomerang experiment (green).

#### 4.3. Discussion

## 5. Conclusions

`CMBFAST`[22] and its derivatives, are ill suited for this investigation, as it is difficult to disentangle the use of quantities proportional to $G\rho $ in gravitational vs. nongravitational contexts. Before embarking on what seems to be a formidable task, we turned to a semi-analytical approximation [23]. While many of the approximations employed by [23] are not physically motivated, but numerical fitting formulae, nonetheless, the role played by quantities proportional to $G\rho $ can be clearly discerned, and the formulae can be suitably adapted. While we recognize that this is not a conclusive result, we find it nonetheless encouraging that the CMB acoustic power spectrum was faithfully reproduced.

## Acknowledgments

## Conflict of Interest

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**MDPI and ACS Style**

Moffat, J.W.; Toth, V.T. Cosmological Observations in a Modified Theory of Gravity (MOG). *Galaxies* **2013**, *1*, 65-82.
https://doi.org/10.3390/galaxies1010065

**AMA Style**

Moffat JW, Toth VT. Cosmological Observations in a Modified Theory of Gravity (MOG). *Galaxies*. 2013; 1(1):65-82.
https://doi.org/10.3390/galaxies1010065

**Chicago/Turabian Style**

Moffat, John. W., and Viktor T. Toth. 2013. "Cosmological Observations in a Modified Theory of Gravity (MOG)" *Galaxies* 1, no. 1: 65-82.
https://doi.org/10.3390/galaxies1010065