Difference between revisions of "Phenomenology of interacting models"

The system which describes the interacting dark components can be transformed into the standard form that corresponds to non-interacting components by redefining the parameters $w_{de}$ and $w_{dm}$ \[\dot\rho_i+3H(1+w_{(i)eff})\rho_i=0,\quad i=\{de,dm\}.\] where in the case of cold DM ($w_{dm}=0$) \[w_{(de)eff}=w_{de}+\frac{Q}{3H\rho_{de}},\quad w_{(dm)eff} = -\frac{Q}{3H\rho_{dm}}.\] If one "turns off" the interaction ($Q=0$), the original parameters are recovered: \[w_{(de)eff}=w_{de}, \quad w_{(dm)eff}=0.\]

If $Q>0$: dark energy $\to$ dark matter, $w_{(dm)eff}<0$ and dark matter redshifts slower than $a^{-3}$, $w_{(de)eff}>w_{de}$ and dark energy has less accelerating power.
If $Q<0$: dark matter $\to$ dark energy, $w_{(dm)eff}>0$ and dark matter redshifts faster than $a^{-3}$, $w_{(de)eff}>w_{de}$ and dark energy has more accelerating power.

For the warm dark matter \[w_{(de)eff}=w_{de}+\frac{Q}{3H\rho_{de}},\quad w_{(dm)eff} = w_{dm}-\frac{Q}{3H\rho_{dm}}.\] Note that the behavior of $w_{(dm)eff}$ can be quite different if $w_{dm}\ne0$. With the usual assumption of cold DM ($w_{dm}=0$, $Q>0$), which implies that energy is transferred from DE to DM, some kind of exotic dark matter with a negative EoS parameter is driven, assuming, of course, that we are in an expanding Universe ($H>0$). For warm dark matter we would have, depending on the type of interaction considered and the strength of the coupling constant $Q$ appearing, a possibility to change the sign of the effective parameter during the cosmic evolution.
For $Q>0$ the expression for $w_{(de)eff}$ indicates, even for $w_{de}=-1$ (the cosmological constant), that the DE fluid will behave as a quintessence field. Thus an effective phantom behavior can only be obtained if $w_{de}<-1$.

Use the identity \[\frac{d}{dt} = \frac{d}{dz}\frac{dz}{da}\frac{da}{dt} = -(1+z)H(z)\frac{d}{dz}\] and transform the basic equations \[\dot\rho_{dm} + 3H\rho_{dm} = Q,\] \[\dot\rho_{de} + 3H\rho_{de}(1+w_{de}) = -Q\] to the form \[\frac{d\rho_{dm}}{dz}-\frac{3}{1+z}\rho_{dm} = -\frac{Q(z)}{(1+z)H(z)},\] \[\frac{d\rho_{de}}{dz}-\frac{3}{1+z}(1+w_{de})\rho_{de} = \frac{Q(z)}{(1+z)H(z)}.\]

We assume for simplicity some two-component fluid with effective pressure and energy \[p=p_{de},\quad \rho=\rho_{de}+\rho_{dm}.\] The conservation condition can be rewritten to the form \begin{equation}\label{IDE_9_0_1} \frac1{a^3}\frac d{dt} (\rho_{dm}a^3)+\frac1{a^{3(1+w_{de})}}\frac d{dt}\left(\rho_{de}a^{3(1+w_{de})}\right)=0. \end{equation} Relation (\ref{IDE_9_0_1}) can be written as \begin{equation}\label{IDE_9_0_2} \frac1{a^3}\frac d{dt} (\rho_{dm}a^3) = \gamma(t),\quad \frac1{a^{3(1+w_{de})}}\frac d{dt}\left(\rho_{de}a^{3(1+w_{de})}\right)=-\gamma(t). \end{equation} The function $\gamma(t)$ describes the interaction between two dark components. Integration of (\ref{IDE_9_0_2}) gives \begin{align} \nonumber \rho_{dm}a^3= & \rho_{dm0}a_0^3 + \int\limits_{t_0}^t\gamma(t)a^3dt,\\ \label{IDE_9_0_3} \rho_{de}a^{3(1+w_{de})}= & \rho_{de0}a_0^{3(1+w_{de})} + \int\limits_{t_0}^t\gamma(t)a^{3(1+w_{de})}dt. \end{align} Let us represent the first Friedman equation in the form of energy conservation law for a particle moving in the one-dimensional potential \begin{equation}\label{IDE_9_0_4} \frac{\dot a^2}2 +V(a)=\frac k2,\quad V(a) = -\frac{a^2}6(\rho_{dm}+\rho_{de}). \end{equation} In presence of the interaction the potential function is explicitly time dependent and now takes the following form \begin{equation}\label{IDE_9_0_5} V(a) = -\frac12\left[\frac{A(t)} a+ \frac{B(t)}{a^1+3w_{de}}\right], \end{equation} where \begin{equation}\label{IDE_9_0_6} A(t)=\frac{\rho_{dm}}3,\quad B(t)=\frac{\rho_{de}a^{3(1+w_{de})}}3. \end{equation} In the SCM model both matter and the cosmological constant are treated separately without the interaction, so both functions $A(t)$ and $B(t)$ are constant. The presence of the interaction manifests in the model by appearing the time dependence of the potential function (\ref{IDE_9_0_5}).

\[\dot\rho_{dm} + 3H(\rho_{dm} + \Pi_{dm} = 0,\] \[\dot\rho_{de} + 3H(\rho_{dm} + p_{de} + \Pi_{de} = 0,\] In this case, the conservation equations formally look as those for two independent fluids. A coupling between them has been mapped into the relation $\Pi_{de}=-\Pi_{dm}$.

Use the standard definition for the dark matter density \begin{equation}\label{IDE_10_1} \rho_{dm} = m_{dm}(\varphi) n_{dm} \end{equation} The number density $n_{dm}$ satisfies the equation \begin{equation}\label{IDE_10_2} \dot n_{dm} + 3Hn_{dm} = 0 \end{equation} Taking time derivatives of (\ref{IDE_10_1}) and using (\ref{IDE_10_2}), one obtains \[\dot\rho_{dm} + 3H\rho_{dm} = \frac{1}{m_{dm}(\varphi)}\frac{d m_{dm}(\varphi)}{d\varphi}\dot\varphi\rho_{dm}.\] In the case of mass independent from $\varphi$ one recovers the usual equation for the energy density of dark matter particles \[\dot\rho_{dm} + 3H\rho_{dm} = 0\] As general covariance lead to conservation law for total energy of dark matter and the scalar field, one finally gets \[\dot\rho_{de} + 3H(\rho_{de} + p_{de}) = -\frac{1}{m_{dm}(\varphi)}\frac{d m_{dm}(\varphi)}{d\varphi}\dot\varphi\rho_{dm}.\]

Using the result of the previous problem one finds \[Q = \frac{1}{m_{dm}(\varphi)}\frac{d m_{dm}(\varphi)}{d\varphi}\dot\varphi\rho_{dm}= -\lambda\dot\varphi\rho_{dm}.\]

Substitute of standard definitions of energy density and pressure of the scalar field into the conservation equation to obtain \[\ddot\varphi + 3H\dot\varphi + \frac{dV}{d\varphi} = -\frac{1}{m_{dm}(\varphi)}\frac{d m_{dm}(\varphi)}{d\varphi}\rho_{dm}.\]

\[\rho' = -\left(1+\frac{w_{de}}{1+r}\right)\rho,\] \[r' = r\left[w_{de} -\frac{(1+r)^2}{r\rho}\Pi\right],\] where \[\Pi\equiv-\frac{Q}{3H}\] is effective pressure and the prime denote the derivatives with respect to $\ln a^3$.

\[\rho' = -\left(1+\frac{w_{dm} r + w_{de}}{1+r}\right)\rho,\] \[r' = r\left[w_{de} -w_{dm} -\frac{(1+r)^2}{r\rho}\Pi\right].\]

Using the result of problem \ref{IDE_12n} and the relation \[\frac d{du}=\frac1{3H}\frac d{dt},\] where $u\equiv\ln a^3,$ one obtains \[\dot r=3Hr\left[w_{de}+\frac Q{9H^3}\frac{(r+1)^2}{r}\right].\] To calculate $dr/dH$ use the relations \[\dot r=\dot H\frac{dr}{dH},\quad \dot H=-\frac12(\rho_{dm}+\rho_{de}+w_{de}\rho_{de}).\] Substitute $\rho_{de}+\rho_{de}= 3H^2$ and \[\rho_{de}=\frac{3H^2}{1+r}\] to obtain \[\dot H=-\frac32\frac{1+w_{de}+r}{1+r}H^2.\] Ultimately the derivative $dr/dH$ takes the form \[\frac{dr}{dH}=\frac I H,\quad I\equiv-2r\frac{1+r}{1+w_{de}+r}\left[w_{de}+\frac Q{9H^3}\frac{(r+1)^2}{r}\right].\]

A combination of field equations for a spatially flat homogeneous and isotropic Universe \begin{align} \nonumber H^2 & =\frac13\rho_{tot},\\ \nonumber \dot H & =-\frac12(\rho_{tot}+p_{tot}), \end{align} yields \[2\frac{\dot H}{H}=-3H\left(1+\frac{p_{tot}}{\rho_{tot}}\right).\] Using \[\frac{\dot\rho_{de}}{\rho_{de}}=-3H(1+w_{de})-\Gamma,\] we obtain \[\frac{\dot\rho_{de}}{\rho_{de}} - 2\frac{\dot H}{H}=-\left(3Hw_{de}\frac r{1+r}+\Gamma\right).\] Comparing this equation with \[\dot r=(1+r)\left[3Hw_{de}\frac{r}{1+r}+\Gamma\right],\] we find that the dynamics of the ratio $r$ is governed by \[\dot r =-(1+r)\left(\frac{\dot\rho_{de}}{\rho_{de}} - 2\frac{\dot H}{H}\right).\]

A combination of field equations for a spatially flat homogeneous and isotropic Universe \begin{align} \nonumber H^2 & =\frac13\rho_{tot} -\frac k{a^2},\\ \nonumber \dot H & =-\frac12(\rho_{tot}+p_{tot}) + \frac k{a^2}, \end{align} yields \[2\frac{\dot H}{H}=-3H\left(1+\frac{p_{tot}}{\rho_{tot}}\right) -\frac{k}{a^2H}\left(1+3\frac{p_{tot}}{\rho_{tot}}\right).\] In this case \[\frac{\dot\rho_{de}}{\rho_{de}} - 2\frac{\dot H}{H}=-\left(3Hw_{de}\frac r{1+r}+\Gamma\right) + \frac{k}{a^2H}\left(1+\frac{3w}{1+r}\right).\] Comparing this equation with \[\dot r=(1+r)\left[3Hw_{de}\frac{r}{1+r}+\Gamma\right],\] we find that the dynamics of the ratio $r$ is governed by \[\dot r =-(1+r)\left[\frac{\dot\rho_{de}}{\rho_{de}} - 2\frac{\dot H}{H}-\frac{k}{a^2H}\left(1+\frac{3w}{1+r}\right)\right].\]

The critical points of the equation \[\rho' = -\left(1+\frac{w_{de}}{1+r}\right)\rho,\] are determined by the condition $\rho'=0$. The relevant critical point is \[r_c=-(1+w_{de}).\] Consequently, for positive values of $r$, the existence of a critical point requires $w_{de}<-1$, i.e., dark energy of the phantom type. This conclusion does not depend on the interaction.

The critical points of the equation \[\rho' = -\left(1+\frac{w_{dm} r + w_{de}}{1+r}\right)\rho,\] are determined by the condition $\rho'=0$. The relevant critical point is \[r_c=-\frac{1+w_{de}}{1+w_{dm}}\] If $w_{dm}>0$ (warm dark matter) and since $r$ must be positive, it follows that $w_{de}<-1$, which corresponds to a phantom DE.

The condition $\rho'=r'=0$ provides \[\rho_c = -\frac{w_{de}}{1+w_{de}}\Pi_c,\quad \Pi_c\equiv \Pi(\rho_c,r_c)\] Since $w_{de}<-1$, a positive stationary energy density $\rho_c$ requires $\Pi_c<0$, which is equivalent to $Q_c>0$. Consequently, the existence of the critical points $\rho_c$ and $r_c$ requires a transfer from dark energy to dark matter. Note that $\rho_c$ remains undetermined for a linear dependence of $\Pi$ on $\rho$.

For this case the expression for $\rho_c$ becomes \[\rho_c=\frac{w_{de}-w_{dm}}{(1+w_{de})(1+w_{dm})}\frac{Q}{3H}.\] The condition $w_{de}<-1$ leads in this case to the same result for the sign of $Q$ as in the previous problem. This result also holds for $-1<w_{de}<0.$.

Consider the time evolution of the ratio $r$, \[\dot r = r\left(\frac{\dot\rho_{dm}}{\rho_{dm}} - \frac{\dot\rho_{de}}{\rho_{de}}\right) = 3Hr \left( w_{de} \frac{1+r}{\rho_{dm}} \frac{Q}{3H}\right).\] For $r\propto a^{-\xi}$ \[\frac{Q}{3H\rho_{dm}}=-\frac{w_{de}+\xi/3}{1+r}.\] The obtained result demonstrates that by choosing a suitable interaction between dark components, one can produce any desired scaling behavior of the energy densities. The uncoupled case, corresponding to $Q=0$, is given by $w_{de} + \xi/3=0$. The SCM model (the special uncoupled case) corresponds to $w_{de}=-1$, $\xi=3$. Generally, interacting models are parameterized by deviations from $\xi=-3w_{de}$.

For $\xi>0$, the interaction \[Q=-3H\rho_{dm}\frac{w_{de}+xi/3}{1+r}\] (see previous problem) becomes very small for $a\ll1$. Consequently, the interaction is not relevant at high redshifts. This guarantees the existence of an early matter-dominated epoch.

\[\dot\rho_{dm}=\dot\rho_{de}f+\rho_{de}f'\dot a,\] where the prime denotes derivative with respect to the scale factor. Substitute this expression into the conservation equation for the dark matter to obtain the following: \[\dot\rho_{de}f+\rho_{de}f'\dot a + 3H\rho_{de} f = Q.\] Insert into the latter the expression for $\dot\rho_{de}$, \[\dot\rho_{de}=-Q-3H\rho_{de}(1+w_{de})\] to finally obtain \[Q=-3H\rho_{de}\frac{f}{1+f}\left(w_{de}-\frac13\frac{f'a}{f}\right).\] For $f\propto a^{-\xi}$ this result coincides with the one obtained in the previous problem.

Represent the Friedman equation with the conservation law in the following form \begin{align} \nonumber H^2 &=\frac13\rho_{tot},\\ \nonumber \frac{\dot H}{H^2} &=\frac32\left(1+\frac{p_{tot}}{\rho_{tot}}\right),\\ \nonumber \dot\rho_{dm}+3H\rho_{dm} &=\frac{\dot f}{f}\rho_{dm},\\ \nonumber \dot\rho_{dm}+3H(1+w_{de})\rho_{de} &=-\frac{\dot f}{f}\rho_{dm}. \end{align} From the conservation equations one easily finds \[w_{de}=-\frac{\dot f}{3Hf}\left(1+\frac{\rho_{dm}}{\rho_{de}}\right)= -\frac{\dot f}{3Hf}(1+r).\] The deceleration parameter is \[q=\frac12\left(3\frac{p_{tot}}{\rho_{tot}+1}\right)=\frac12\left(3\frac{w_{de}}{1+r}+1\right)=\frac12\left(1-\frac{\dot f}{Hf}\right).\]

Let us analyze the result, obtained in the previous problem. The sign of $q$ is defined by the ratio \[\frac{\dot f}{Hf}.\] For \[\frac{\dot f}{Hf}<1\] we have $q>0$, i.e., decelerated expansion. For \[\frac{\dot f}{Hf}>1\] we have $q<0$---accelerated expansion. If, in particular, $f$ is such that the ratio $\dot f/f$ changes from $1$ to $-1$, this corresponds to a transition from decelerated to accelerated expansion. Consequently, this transition occurs solely due to interaction.