# Dynamically Generated Inflationary ΛCDM

^{1}

^{2}

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^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. A Simple Model of Unification of Dark Energy and Dark Matter

- The first term in (10) is the standard Einstein–Hilbert action with $R\left(g\right)$ denoting the scalar curvature with respect to metric ${g}_{\mu \nu}$ in the second order (metric) formalism;
- ${\Phi}_{0}\left(A\right)$ is particular representative of a $D=4$ non-Riemannian volume-element density (6):$${\Phi}_{0}\left(A\right)=\frac{1}{3!}{\epsilon}^{\mu \nu \kappa \lambda}{\partial}_{\mu}{A}_{\nu \kappa \lambda}\phantom{\rule{0.277778em}{0ex}}.$$
- $L(\phi ,X)$ is general-coordinate invariant Lagrangian of a single scalar field $\phi \left(x\right)$:$$L(\phi ,X)=X-V\left(\phi \right),\phantom{\rule{1.em}{0ex}}X\equiv -\frac{1}{2}{g}^{\mu \nu}{\partial}_{\mu}\phi {\partial}_{\nu}\phi \phantom{\rule{0.277778em}{0ex}}.$$

**Remark**

**1.**

- A dynamically generated dark matter part given by the first term in (19), where ${p}_{\mathrm{DM}}=0$, ${\rho}_{\mathrm{DM}}={\rho}_{0}$ with ${\rho}_{0}$ as in (20), which in fact according to (21) and (22) describes a dust fluid with fluid density ${\rho}_{0}$ flowing along geodesics. Thus, we will refer to the $\phi $ scalar field by the alias “darkon.”

## 3. Inflation and Unified Dark Energy and Dark Matter

- (i) ${U}_{\mathrm{eff}}\left(u\right)$ (37) has an almost flat region for large positive u: ${U}_{\mathrm{eff}}\left(u\right)\simeq 2{\Lambda}_{0}$ for large u. This almost flat region corresponds to "early universe" inflationary evolution with energy scale $2{\Lambda}_{0}$, as will be evident from the autonomous dynamical system analysis of the cosmological dynamics in Section 4.
- (ii) ${U}_{\mathrm{eff}}\left(u\right)$ (37) has a stable minimum for a small finite value $u={u}_{*}$: $\frac{\partial {U}_{\mathrm{eff}}}{\partial u}=0\phantom{\rule{0.277778em}{0ex}}$ for $u\equiv {u}_{*}$, where:$$exp\left(-\frac{{u}_{*}}{\sqrt{3}}\right)=\frac{{M}_{1}}{4{M}_{0}}\phantom{\rule{1.em}{0ex}},\frac{{\partial}^{2}{U}_{\mathrm{eff}}}{\partial {u}^{2}}{|}_{u={u}_{*}}=\frac{{M}_{1}^{2}}{12{M}_{0}}>0\phantom{\rule{0.277778em}{0ex}}.$$
- (iii) As it will be explicitly exhibited in the dynamical system analysis in Section 4, the region of u around the stable minimum at $u={u}_{*}$ (41) corresponds to the late-time de Sitter expansion of the universe with a slightly varied late-time Hubble parameter (dark energy dominated epoch), wherein the minimum value of the potential:$${U}_{\mathrm{eff}}\left({u}_{*}\right)=2{\Lambda}_{0}-\frac{{M}_{1}^{2}}{8{M}_{0}}\equiv 2{\Lambda}_{\mathrm{DE}}$$

## 4. Cosmological Implications

**Remark**

**2.**

- (A) Stable critical point:$${x}_{*}=0\phantom{\rule{1.em}{0ex}},\phantom{\rule{1.em}{0ex}}{y}_{*}=0\phantom{\rule{1.em}{0ex}},\phantom{\rule{1.em}{0ex}}{H}_{*}=\sqrt{\frac{{\Lambda}_{\mathrm{DE}}}{3}}\phantom{\rule{0.277778em}{0ex}},$$
- (B) Unstable critical point:$${x}_{**}=0\phantom{\rule{1.em}{0ex}},\phantom{\rule{1.em}{0ex}}{y}_{**}=\sqrt{1-\frac{{\Lambda}_{\mathrm{DE}}}{{\Lambda}_{0}}}=\frac{{M}_{1}}{4\sqrt{{M}_{0}{\Lambda}_{0}}}\phantom{\rule{1.em}{0ex}},\phantom{\rule{1.em}{0ex}}{H}_{**}=\sqrt{\frac{{\Lambda}_{0}}{3}}\phantom{\rule{0.277778em}{0ex}},$$$$\begin{array}{c}\hfill \u03f5=-\frac{\stackrel{.}{H}}{{H}^{2}}\approx {\left(\frac{\frac{\partial {U}_{\mathrm{eff}}}{\partial u}-\frac{1}{2\sqrt{3}}\phantom{\rule{0.166667em}{0ex}}{\rho}_{\mathrm{DM}}}{{U}_{\mathrm{eff}}+{\rho}_{\mathrm{DM}}}\right)}^{2}+\frac{3}{2}\frac{{\rho}_{\mathrm{DM}}}{{U}_{\mathrm{eff}}+{\rho}_{\mathrm{DM}}}\phantom{\rule{0.277778em}{0ex}},\end{array}$$$$\begin{array}{c}\hfill \eta =-\frac{\stackrel{.}{H}}{{H}^{2}}-\frac{\stackrel{..}{H}}{2H\stackrel{.}{H}}\approx -2\frac{\frac{{\partial}^{2}{U}_{\mathrm{eff}}}{\partial {u}^{2}}+\frac{1}{12}\phantom{\rule{0.166667em}{0ex}}{\rho}_{\mathrm{DM}}}{{U}_{\mathrm{eff}}+{\rho}_{\mathrm{DM}}}+\mathrm{O}\left({\rho}_{\mathrm{DM}}\right)\phantom{\rule{0.277778em}{0ex}}.\end{array}$$

## 5. Numerical Solutions

## 6. Conclusions and Outlook

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Shape of the effective potential ${U}_{\mathrm{eff}}\left(u\right)$ in the Einstein-frame (37). The physical unit for u is ${M}_{Pl}/\sqrt{2}$.

**Figure 2.**Numerical shape of the evolution of $u\left(t\right)$. The physical unit for u is ${M}_{Pl}/\sqrt{2}$.

**Figure 3.**Slow-roll parameters $\u03f5$ and $\eta $ before and around end of inflation. When $\u03f5=1$ the inflation ends.

**Figure 4.**Numerical shape of the evolution of $H\left(t\right)$. Here ${H}_{**}\equiv \sqrt{{\Lambda}_{0}/3}$ as in (58).

**Figure 5.**In the left panel—blown-up portion of the plot on Figure 2 around and after end of inflation depicting the oscillations of $u\left(t\right)$ after end of inflation. In the right panel—oscillations of $\stackrel{.}{u}\left(t\right)$ after end of inflation.

**Figure 6.**Evolution of w parameter of the equation of state with sharp growth above $w\approx -1$ for a short time interval after end of inflation—matter domination.

**Figure 7.**The scalar to tensor ratio r and the scalar spectral index ${n}_{s}$ vs. the number of e-folds for different values of the initial conditions. The sampling of the latter is done with a normal distribution ${\Lambda}_{0}=50\pm 10$, ${M}_{1}=20\pm 10$.

**Figure 8.**The relation between the scalar to tensor ratio r and the scalar spectral index ${n}_{s}$ via sampled initial conditions with a normal distribution ${\Lambda}_{0}=50\pm 10$, ${M}_{1}=20\pm 10$. All of the sampled values fall well inside the Planck data constraint (67).

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Benisty, D.; Guendelman, E.I.; Nissimov, E.; Pacheva, S. Dynamically Generated Inflationary ΛCDM. *Symmetry* **2020**, *12*, 481.
https://doi.org/10.3390/sym12030481

**AMA Style**

Benisty D, Guendelman EI, Nissimov E, Pacheva S. Dynamically Generated Inflationary ΛCDM. *Symmetry*. 2020; 12(3):481.
https://doi.org/10.3390/sym12030481

**Chicago/Turabian Style**

Benisty, David, Eduardo I. Guendelman, Emil Nissimov, and Svetlana Pacheva. 2020. "Dynamically Generated Inflationary ΛCDM" *Symmetry* 12, no. 3: 481.
https://doi.org/10.3390/sym12030481