# Dark Energy and Dark Matter Interaction: Kernels of Volterra Type and Coincidence Problem

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## Abstract

**:**

## 1. Introduction

## 2. Phenomenological Approach to the Problem of Interactions in the Dark Sector of the Universe

#### 2.1. Two-Fluid Model in the Einstein Theory of Gravity

#### 2.2. Description of the DE/DM Coupling in the Framework of an Isotropic Homogeneous Cosmological Model

## 3. Rheological-Type Model of the DE/DM Coupling

#### 3.1. Reconstruction of the Kernel $Q\left(t\right)$

- (i)
- The function $Q\left(t\right)$ is presented by the integral operator of the Volterra type:$$Q\left(t\right)={\int}_{{t}_{0}}^{t}d\xi K(t,\xi )[E\left(\xi \right)-W\left(\xi \right)]\phantom{\rule{0.166667em}{0ex}}.$$
- (ii)
- The Volterra integral contains the difference of the energy density scalars $E\left(\xi \right)$ and $W\left(\xi \right)$.
- (iii)
- The kernel of the Volterra integral $K(t,\xi )$ has a specific multiplicative form$$K(t,\xi )={K}_{0}H\left(t\right)H\left(\xi \right){\left[\frac{a\left(\xi \right)}{a\left(t\right)}\right]}^{\nu}\phantom{\rule{0.166667em}{0ex}}.$$

- (1)
- In the context of rheological approach we assume that the state of a fluid system at the present time moment t is predetermined by whole prehistory of its evolution from the starting moment ${t}_{0}$ till to the moment t. More than century ago it was shown, that the mathematical formalism appropriate for description of this idea can be based on the theory of linear integral Volterra operators, which have found numerous applications to the theory of media with memory. We also use this fruitful idea.
- (2)
- Our ansatz is that the interaction between two constituents of the Dark Fluid vanishes, if the DE energy density coincides identically with the DM energy density, $W\equiv E$. When $W\ne E$ the integral mechanism of self-regulation inside the Dark Fluid switches on. For instance, during the cosmological epochs with DE domination, i.e., when $W>E$, the corresponding contribution into the interaction term Q is negative, the rates $\dot{W}$ and $\dot{E}$ obtain negative and positive contributions, respectively (see (10) and (11)); when $W<E$, the inverse process starts thus regulating the ratio between DE and DM energy densities.
- (3)
- For classical models of fading memory the kernel of the Volterra operator is known to be of exponential form $K(t,\xi )=\mathcal{K}exp\frac{(\xi -t)}{{T}_{0}}$, where the parameter ${T}_{0}$ describes the typical time of memory fading, and the quantity $\mathcal{K}$ has the dimensionality ${\left[\mathrm{time}\right]}^{-2}$. When we work with the de Sitter scale factor $a\left(t\right)=a\left({t}_{0}\right)exp{H}_{0}t$, we can rewrite the kernel of the Volterra operator as follows:$$K(t,\xi )=\mathcal{K}exp\left[\frac{{H}_{0}(\xi -t)}{{H}_{0}{T}_{0}}\right]={K}_{0}{H}_{0}^{2}{\left[\frac{a\left(\xi \right)}{a\left(t\right)}\right]}^{\frac{1}{{H}_{0}{T}_{0}}}\phantom{\rule{0.166667em}{0ex}},$$

#### 3.2. Key Equation of the Model

## 4. Classification of Solutions

#### 4.1. The Scheme of Classification

#### 4.2. Solutions Corresponding to the Negative Discriminant, $\Delta <0$

#### 4.2.1. The Structure of the Exact Solution

#### 4.2.2. Two Auxiliary Characteristics of the Model and a Scheme of Estimation of the Kernel Parameters

- (1)
- The acceleration parameter q

- (2)
- The DM/DE energy density ratio ω

#### 4.2.3. Admissible Asymptotic Regimes, and Constraints on the Model Parameters

- (i)
- If the maximal real root, say ${\sigma}_{1}$, is positive and the set of initial data is general, we see that $W\to \infty $, $E\to \infty $ and $H\to \infty $, when $x\to \infty $. The integral in (17) converges at $a\left(t\right)\to \infty $, and the scale factor $a\left(t\right)$ follows the law $a\left(t\right)={a}_{*}{({t}_{*}-t)}^{-\frac{2}{{\sigma}_{1}}}$, and reaches infinity at $t={t}_{*}$. We deal in this case with the so-called Big Rip asymptotic regime, and the Universe follows the catastrophic scenario [11,19]. In particular, when ${\sigma}_{1}>0$ and ${\sigma}_{2}<0$, ${\sigma}_{3}<0$, according to the Viète theorem, we can definitely say only that ${\sigma}_{1}{\sigma}_{2}{\sigma}_{3}=-D>0$, i.e., ${K}_{0}(\Gamma +\gamma )+3\nu \Gamma \gamma <0$. The asymptotic value of the acceleration parameter is equal to $-q(\infty )=1+\frac{{\sigma}_{1}}{2}$. The final ratio between the DM and DE energy densities$$\omega (\infty )=\frac{{\sigma}_{1}^{2}+{\sigma}_{1}(\nu +3\Gamma )+{K}_{0}+3\nu \Gamma}{{K}_{0}}$$
- (ii)
- If the maximal real root, say ${\sigma}_{1}$, is equal to zero, we see that $D=0$, and thus$${K}_{0}(\Gamma +\gamma )+3\nu \Gamma \gamma =0\phantom{\rule{0.166667em}{0ex}}.$$In this case, the Hubble function tends asymptotically to constant ${H}_{\infty}$, given by$${H}_{\infty}=\sqrt{\frac{\Lambda}{3}+\frac{\kappa \left(2{K}_{0}+3\nu \Gamma \right)}{3{K}_{0}{\sigma}_{2}{\sigma}_{3}}\left\{W\left(1\right)\left[{\sigma}_{2}{\sigma}_{3}+3\Gamma ({\sigma}_{2}+{\sigma}_{3})+9{\Gamma}^{2}-{K}_{0}\right]+{K}_{0}E\left(1\right)\right\}}\phantom{\rule{0.166667em}{0ex}},$$$$A=-({\sigma}_{2}+{\sigma}_{3})>0\phantom{\rule{4pt}{0ex}}\to \nu +3(\Gamma +\gamma )>0\phantom{\rule{0.166667em}{0ex}},$$$$B-A={\sigma}_{2}{\sigma}_{3}>0\phantom{\rule{4pt}{0ex}}\to 2{K}_{0}+3\nu (\Gamma +\gamma )+9\Gamma \gamma >0\phantom{\rule{0.166667em}{0ex}}.$$These requirements restrict the choice of model parameters.
- (iii)
- If all the roots are negative, we see that $H\to {H}_{0}\equiv \sqrt{\frac{\Lambda}{3}}$, when $x\to \infty $, thus we obtain the classical de Sitter asymptote with $-q(\infty )=1$. When $\Lambda =0$, all the roots are negative, and, say, ${\sigma}_{1}$ is the maximal among them, we see that $W\to 0$, $E\to 0$ at $x\to \infty $. The scale factor behaves asymptotically as the power-law function $a\left(t\right)\propto {t}^{\frac{2}{|{\sigma}_{1}|}}$; the acceleration parameter $-q(\infty )=1-\frac{|{\sigma}_{1}|}{2}$ is positive, when $|{\sigma}_{1}|<2$. In particular, when ${\sigma}_{1}<0$, ${\sigma}_{2}<0$, ${\sigma}_{3}<0$, we see that, first, ${\sigma}_{1}{\sigma}_{2}{\sigma}_{3}=-D<0$, i.e., ${K}_{0}(\Gamma +\gamma )+3\nu \Gamma \gamma >0$; second, $A=-({\sigma}_{1}+{\sigma}_{2}+{\sigma}_{3})>0$; third, $B-A={\sigma}_{1}{\sigma}_{2}+{\sigma}_{1}{\sigma}_{3}+{\sigma}_{3}{\sigma}_{2}>0$.There are also cases related to the special choice of initial data $W\left(1\right)$, $E\left(1\right)$, as well as, of the choice of parameters ${K}_{0}$, $\nu $, $\Gamma $, $\gamma $. For instance, if we deal with the situation indicated as (i) but now ${C}_{1}=0$ due to specific choice of $W\left(1\right)$, $E\left(1\right)$, (see (40)), we obtain the situation (ii) or (iii).

#### 4.3. Solutions Corresponding to the Positive Discriminant, $\Delta >0$

#### Admissible Asymptotic Regimes

- (i)
- The first new regime can be indicated as a quasi-periodic expansion; it can be realized when ${\sigma}_{1}=0$, $\alpha $ is negative, and ${H}_{\infty}^{2}>\left|h\right|$. The square of the Hubble function can be now rewritten as follows:$${H}^{2}\to {H}_{\infty}^{2}+h{x}^{-\left|\alpha \right|}sin[\beta logx+\psi ]\phantom{\rule{0.166667em}{0ex}}.$$Asymptotically, the Universe expansion tends to the Pseudo Rip regime; however, this process has quasi-periodic features.
- (ii)
- The second new regime relates to ${\sigma}_{1}=0$, $\alpha =0$ and ${H}_{\infty}^{2}>\left|h\right|$. The square of the Hubble function, the DE and DM energy densities become now periodic functions (see, e.g., (63) with $\alpha =0$).
- (iii)
- The third regime is characterized by the following specific feature: ${H}^{2}$ takes zero value at finite $x={x}_{*}$. This regime can be effectively realized in two cases: first, when ${\sigma}_{1}=0$, $\alpha <0$ and ${H}_{\infty}^{2}<\left|h\right|$; second, when ${\sigma}_{1}=0$, $\alpha >0$. In both cases the size of the Universe is fixed by the specific value of the scale factor ${a}^{*}=a\left({t}_{0}\right){x}_{*}$.

#### 4.4. Solutions Corresponding to the Vanishing Discriminant, $\Delta =0$

#### 4.4.1. Two Roots Coincide, $q\ne 0$

#### 4.4.2. Three Roots Coincide, $q=0$

#### 4.4.3. Admissible Asymptotic Regimes

## 5. Three Examples of Explicit Model Analysis

#### 5.1. First Explicit Submodel, $\Delta <0$, $q=0$ and $\Lambda =0$; How Do the Initial Data Correct the Universe Destiny?

- (i)
- When $\omega \left(1\right)=2+\sqrt{3}$, i.e., ${C}_{1}=0$, and the growing mode is deactivated, the DE energy density, DM energy density take, respectively, the form$$W\left(x\right)=\frac{W\left(1\right)}{\sqrt{3}}{x}^{-\frac{3}{2}\gamma}\left[(\sqrt{3}+1)-{x}^{-\frac{3\sqrt{3}}{2}\gamma}\right]\ge 0\phantom{\rule{0.166667em}{0ex}},$$$$E\left(x\right)=\frac{E\left(1\right)}{\sqrt{3}}{x}^{-\frac{3}{2}\gamma}\left[(\sqrt{3}-1)+{x}^{-\frac{3\sqrt{3}}{2}\gamma}\right]\ge 0\phantom{\rule{0.166667em}{0ex}}.$$The function $\omega \left(x\right)=\frac{E\left(x\right)}{W\left(x\right)}$, which is given by$$\omega \left(x\right)=(2+\sqrt{3})\frac{\left[(\sqrt{3}-1)+{x}^{-\frac{3\sqrt{3}}{2}\gamma}\right]}{\left[(\sqrt{3}+1)-{x}^{-\frac{3\sqrt{3}}{2}\gamma}\right]}\phantom{\rule{0.166667em}{0ex}},$$$${H}^{2}\left(x\right)=\frac{\kappa W\left(1\right)(\sqrt{3}+1)}{3\sqrt{3}}{x}^{-\frac{3}{2}\gamma}\left(2+{x}^{-\frac{3\sqrt{3}}{2}\gamma}\right)\ge 0\phantom{\rule{0.166667em}{0ex}}.$$The scale factor $a\left(t\right)$ can be found from the quadrature:$$\sqrt{\frac{2\kappa W\left(1\right)(\sqrt{3}+1)}{3\sqrt{3}}}(t-{t}_{0})={\int}_{1}^{\frac{a\left(t\right)}{a\left({t}_{0}\right)}}\frac{dx{x}^{\frac{3\gamma}{4}-1}}{\sqrt{1+\frac{1}{2}{x}^{-\frac{3\sqrt{3}}{2}\gamma}}}\phantom{\rule{0.166667em}{0ex}}.$$In the asymptotic regime the scale factor behaves as $a\left(t\right)\propto {t}^{\frac{4}{3\gamma}}$, and the Hubble function $H\left(t\right)$ tends to zero as $H\left(t\right)\simeq \frac{4}{3\gamma t}$.
- (ii)
- When $0<\omega \left(1\right)<2+\sqrt{3}$, i.e., ${C}_{1}>0$, the integral ${\int}_{1}^{\infty}\frac{dx}{xH\left(x\right)}$ converges, so the scale factor $a\left(t\right)$ reaches infinite value at finite value of the cosmological time. The growing mode, which relates to the positive root ${\sigma}_{1}$, become the leading mode, and we obtain the model of the Big Rip type.
- (iii)
- When $\omega \left(1\right)>2+\sqrt{3}$, i.e., ${C}_{1}<0$, we obtain the model in which the square of the Hubble function takes zero value at some finite time moment. In other words, the Universe expansion stops, the Universe volume becomes finite.

#### 5.2. Second and Third Explicit Submodels: The Case $\Delta =0$ and $q=0$

#### 5.2.1. The Case $W\left(1\right)+E\left(1\right)=0$: Solution of the Bounce Type

#### 5.2.2. The Case $W\left(1\right)+E\left(1\right)\ne 0$: Super-Inflationary Solution

## 6. Discussion

- The model of kernel of the DE/DM interaction, which possesses two extra parameters, ${K}_{0}$ and $\nu $, is able to describe many known interesting cosmic scenaria: Big Rip, Little Rip, Pseudo Rip, de Sitter-type expansion; the late-time accelerated expansion of the Universe is the typical feature of the presented model.
- When $2{K}_{0}+3\nu \Gamma \ne 0$, the solution of a new type appears, which is associated with the so-called Effective Cosmological Constant. Indeed, if the standard cosmological constant vanishes, $\Lambda =0$, we obtain according to (52) that the parameter ${H}_{\infty}\ne 0$ plays the role of an effective Hubble constant. It appears as the result of integration over the whole time interval; it can be associated with the memory effect produced by the DE/DM interaction; we can introduce the effective cosmological constant ${\Lambda}_{*}\equiv 3{H}_{\infty}^{2}$, which appears just due to the interaction in the Dark sector of the Universe.
- The model of the DE/DM coupling based on the Volterra-type interaction kernel can solve the Coincidence problem. Indeed, the asymptotic value $\omega (\infty )$ of the function $\omega \left(x\right)=\frac{E\left(x\right)}{W\left(x\right)}$ is predetermined by the choice of parameters ${K}_{0}$ and $\nu $ entering the integral kernel (13), (14). Even if the initial value $E\left(1\right)$ of the Dark Matter energy density is vanishing, the final value $E(\infty )$ is of the order of the final value $W(\infty )$ due to the integral procedure of energy redistribution, which is described by the Volterra operator (see, e.g., the example (50)). In other words, the DE component of the Dark Fluid transmits the energy to the DM components during the whole evolution time interval, and this action "is remembering" by the Dark Fluid.
- Optimization of the model parameters ${K}_{0}$, $\nu $, $\Gamma $, $\gamma $ using the observational data is the goal of our next work. However, some qualitative comments concerning the ways to distinguish the models of DE/DM interactions can be done based on the presented work. For instance, when one deals with the standard $\Lambda $CDM model, the profile of the energy density associated with the Dark Energy is considered to be the horizontal straight line; the DM energy density profile decreases monotonically, thus providing the existence of some cross-point at some finite time moment. For this model the time derivative $\dot{W}\left(t\right)$ vanishes, so that $\dot{W}\left(t\right)=0$ and $\dot{E}\left(t\right)\ne 0$ never coincide. In the model under discussion, the profiles $E\left(t\right)$ and $W\left(t\right)$ do not cross; these quantities tend to one another asymptotically. As for the rates of evolution, the quantities $\dot{W}\left(t\right)$ and $\dot{E}\left(t\right)$ can coincide identically (see, e.g., (96)), or can tend to one another asymptotically. In other words, one can distinguish the models of DE/DM interaction if to analyze and compare the rates of evolution of the DE and DM energy density scalars.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DE | Dark Energy |

DM | Dark Matter |

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**MDPI and ACS Style**

Balakin, A.B.; Ilin, A.S.
Dark Energy and Dark Matter Interaction: Kernels of Volterra Type and Coincidence Problem. *Symmetry* **2018**, *10*, 411.
https://doi.org/10.3390/sym10090411

**AMA Style**

Balakin AB, Ilin AS.
Dark Energy and Dark Matter Interaction: Kernels of Volterra Type and Coincidence Problem. *Symmetry*. 2018; 10(9):411.
https://doi.org/10.3390/sym10090411

**Chicago/Turabian Style**

Balakin, Alexander B., and Alexei S. Ilin.
2018. "Dark Energy and Dark Matter Interaction: Kernels of Volterra Type and Coincidence Problem" *Symmetry* 10, no. 9: 411.
https://doi.org/10.3390/sym10090411