# Cosmological Solutions from a Multi-Measure Model with Inflaton Field

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## Abstract

**:**

## 1. Introduction

## 2. Overview of the Multi-Measures Model

## 3. Numerical Results

- (a) only deceleration;
- (b) deceleration followed by acceleration. The acceleration can be (i) very strong, i.e., we have inflation, in which case the deceleration epoch is extremely short, or (ii) very slow, i.e., corresponding to acceleration with the cosmological constant as given in Equation (21). Which of these options is realized depends on the asymptotic value of $\varphi $.
- We also observe a (c) “physically realistic” Universe with four epochs: a short first deceleration epoch (FD), early inflation (EI), a second deceleration (SD), which we interpret as radiation and matter determined epochs together, and finally, an infinite slow accelerating expansion (AE). Some examples of this type of “physically realistic” evolution are shown in Figure 1, Figure 2, Figure 3 and Figure 4, and hereafter, we consider only this type of evolution.

- At ${t}_{0}=0$ we observe the EOS of ultra-relativistic matter with $w=1/3$. The existence of this phase does not contradict the observations, because currently, we have information solely from the time after the initial inflation;
- Initial inflation with EOS of dark energy $w\to -1$;
- Matter domination stage where $w>-1/3$;
- Accelerated expansion with $w<-1/3$.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The Universe’s evolution for $({\chi}_{2}=1,\phantom{\rule{0.277778em}{0ex}}{M}_{0}=0.0001,\phantom{\rule{0.277778em}{0ex}}{M}_{1}=0.08,\phantom{\rule{0.277778em}{0ex}}{M}_{2}=0.001)$. The parameters $\{\alpha ,{b}_{0},{p}_{u},{f}_{1},{f}_{2}\}$ are $\{1.1,0.00119,1.5\times {10}^{-7},20,{10}^{-3}\}$. On the panels are the evolution of: (

**a**) the scale factor $a(t)$, (

**b**) the second derivative of scale factor $\ddot{a}(t)$, (

**c**) the inflaton field $\varphi (t)$, and (

**d**) the equation of state $w(t)=p(t)/\rho (t)$.

**Figure 2.**(

**a**) The beginning of the matter dominated epoch ${t}_{\mathrm{SD}}$ as a function of the parameter $\alpha $ for ${\chi}_{2}=1,\phantom{\rule{0.277778em}{0ex}}{M}_{0}=-1,\phantom{\rule{0.277778em}{0ex}}{M}_{1}=4,\phantom{\rule{0.277778em}{0ex}}{M}_{2}=0.001,{p}_{u}={10}^{-12},{f}_{2}=0.001$. (

**b**) The ${f}_{1}$ parameters used. (

**c**) The ${b}_{0}$ parameters used.

**Figure 3.**(

**a**) The beginning of the matter dominated epoch ${t}_{\mathrm{SD}}$ as a function of the parameter ${f}_{2}$ for ${\chi}_{2}=1,\phantom{\rule{0.277778em}{0ex}}{M}_{0}=-1,\phantom{\rule{0.277778em}{0ex}}{M}_{1}=4,\phantom{\rule{0.277778em}{0ex}}{M}_{2}=0.001,{p}_{u}={10}^{-12},\alpha =1.1$. (

**b**) The ${f}_{1}$ parameters used. (

**c**) The ${b}_{0}$ parameters used.

**Figure 4.**(

**a**) The beginning of the matter dominated epoch ${t}_{\mathrm{SD}}$ as a function of the parameter ${p}_{u}$ for ${\chi}_{2}=1,\phantom{\rule{0.277778em}{0ex}}{M}_{0}=-1,\phantom{\rule{0.277778em}{0ex}}{M}_{1}=4,\phantom{\rule{0.277778em}{0ex}}{M}_{2}=0.001,{p}_{u}={10}^{-12},\alpha =1.1$. (

**b**) The ${f}_{1}$ parameters used. (

**c**) The ${b}_{0}$ parameters used.

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Staicova, D.; Stoilov, M.
Cosmological Solutions from a Multi-Measure Model with Inflaton Field. *Symmetry* **2019**, *11*, 1387.
https://doi.org/10.3390/sym11111387

**AMA Style**

Staicova D, Stoilov M.
Cosmological Solutions from a Multi-Measure Model with Inflaton Field. *Symmetry*. 2019; 11(11):1387.
https://doi.org/10.3390/sym11111387

**Chicago/Turabian Style**

Staicova, Denitsa, and Michail Stoilov.
2019. "Cosmological Solutions from a Multi-Measure Model with Inflaton Field" *Symmetry* 11, no. 11: 1387.
https://doi.org/10.3390/sym11111387