# Varying Constants Entropic-ΛCDM Cosmology

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## Abstract

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## 1. Introduction

## 2. Entropic Force Field Equations and Varying Constants

## 3. Gravitational Thermodynamics and Varying Constants

## 4. Gravitational Thermodynamics—Cosmological Solutions

#### 4.1. G Varying Models Only: $G\left(t\right)={G}_{0}{a}^{q};$ $q,{G}_{0}=const.$, $\dot{c}\left(t\right)=0$.

#### 4.2. c Varying Models Only: $c\left(t\right)={c}_{0}{a}^{n};$ ${c}_{0},n=const.$, $\dot{G}\left(t\right)=0$

## 5. Entropic Pressure Modified Equations

#### 5.1. G Varying Models Only: $\dot{G}\left(t\right)\ne 0$ and $\dot{c}\left(t\right)=0$; $q\ne 0,n=0$.

#### 5.2. c Varying Models Only: $\dot{G}\left(t\right)=0$ and $\dot{c}\left(t\right)\ne 0$; $q=0,n\ne 0$

## 6. Gravitational Thermodynamics—Horizon Heat Flow

## 7. Observational Parameters

## 8. Data Analysis

#### 8.1. Type Ia Supernovae

**θ**will include ${\mathit{\theta}}_{c}$ and the other fitting parameters, which in this case are: α and β, which characterize the stretch-luminosity and color-luminosity relationships; and the nuisance parameter ${\mathcal{M}}_{B}$, expressed as a step function of two more parameters, ${\mathcal{M}}_{B}^{1}$ and ${\Delta}_{m}$:

#### 8.2. Baryon Acoustic Oscillations

#### 8.3. Cosmic Microwave Background

#### 8.4. Results

- the entropic scenario plus varying c and/or G is quite indistinguishable from a pure-ΛCDM model, that is why we call it an entropic-ΛCDM model. Present data is still unable to differentiate between the two scenarios;
- the best fit for the value of the Hawking temperature coefficient γ is quite different from the theoretical values used in literature, i.e., $\gamma =3/\left(2\pi \right)$ or $1/2$; it should be pointed out that other considered entropic scenarios have the values of $O\left(1\right)$ (e.g., [25]);
- the value for γ is compatible with zero since we were able to put only an upper limit to it. This would mean that the Hawking temperature was zero for the models under study;
- it is also clear that we still have the deceleration-acceleration transition, as we show in the plot of the relation for $q\left(z\right)$ and also for $q\left(a\right)$ in Figure 2, where our models are compared with a standard ΛCDM resulting in being, as said above, barely distinguishable.

## 9. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**(

**Left**) Varying G scenario: $68\%$ and $95\%$ confidence levels for q and γ; (

**Right**) Varying c scenario: $68\%$ and $95\%$ confidence levels for n and γ.

**Figure 2.**Transition from deceleration to acceleration for the model Equation (94) as function of redshift (

**Left**) and scale factor (

**Right**). Solid line is for standard ΛCDM model; dashed line for varying–G–entropic–ΛCDM model; dotted line is for varying–c–entropic–ΛCDM model. The values of the parameters are taken from Table I and from the Planck data (ΛCDM).

$\mathrm{id}.$ | ${\Omega}_{m}$ | ${\Omega}_{b}$ | h | $q/n$ | γ | α | β | ${\mathcal{M}}_{B}^{1}$ | ${\Delta}_{m}$ |
---|---|---|---|---|---|---|---|---|---|

$G={G}_{0}\phantom{\rule{0.277778em}{0ex}}{a}^{q}$ | $0.{314}_{-0.008}^{+0.009}$ | $0.{0453}_{-0.0009}^{+0.0009}$ | $0.{698}_{-0.007}^{+0.007}$ | $0.{048}_{-0.033}^{+0.042}$ | $<0.022$ | $0.{141}_{-0.006}^{+0.007}$ | $3.{106}_{-0.087}^{+0.077}$ | $-19.{044}_{-0.019}^{+0.018}$ | $-0.{071}_{-0.023}^{+0.023}$ |

$c={c}_{0}\phantom{\rule{0.277778em}{0ex}}{a}^{n}$ | $0.{311}_{-0.007}^{+0.007}$ | $0.{046}_{-0.001}^{+0.001}$ | $0.{696}_{-0.007}^{+0.007}$ | $0.{00049}_{-0.00053}^{+0.00049}$ | $<0.0007$ | $0.{141}_{-0.007}^{+0.007}$ | $3.{100}_{-0.080}^{+0.080}$ | $-19.{043}_{-0.018}^{+0.018}$ | $-0.{070}_{-0.022}^{+0.023}$ |

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Da̧browski, M.P.; Gohar, H.; Salzano, V.
Varying Constants Entropic-ΛCDM Cosmology. *Entropy* **2016**, *18*, 60.
https://doi.org/10.3390/e18020060

**AMA Style**

Da̧browski MP, Gohar H, Salzano V.
Varying Constants Entropic-ΛCDM Cosmology. *Entropy*. 2016; 18(2):60.
https://doi.org/10.3390/e18020060

**Chicago/Turabian Style**

Da̧browski, Mariusz P., Hussain Gohar, and Vincenzo Salzano.
2016. "Varying Constants Entropic-ΛCDM Cosmology" *Entropy* 18, no. 2: 60.
https://doi.org/10.3390/e18020060