# Physical mechanism of energy exchange

### Problem 1

Models with interaction between DM and the DE field can be realized if we make just an obvious assumption: the mass of the cold DM particles is a function of the DE field. Let the dark matter particles will be collisionless and nonrelativisic. Hence, the pressure of this fluid and its energy density are \(p_{dm}=0\) and \(\rho_{dm}=nm\) respectively, where $m$ is the rest mass and $n$ is the number density of DM particles. We define $m=\lambda\varphi$ where $\varphi$ is a scalar field and $\lambda$ is a dimensionless constant. Show how such assumption affects the scalar field dynamics (after [1]). [2]

The energy density associated with the considered fluid is \[\rho_{dm}=\lambda n\varphi.\] We will assume that the comoving number density of DM particles is constant during the epochs of interest, i.e the particles are neither created nor destroyed. Thus, the number density is only a function of physical volume and $n\propto a^{-3}$. Since the energy density of the dark matter particles depends on $\varphi$, the scalar field feels an additional effective potential when it is in a "bath" of DM particles. Taking this effect into account, the equation of motion for the scalar field becomes \[\ddot\varphi+3H\dot\varphi+\frac{dV_{eff}}{d\varphi}=0\quad V_{eff}\equiv V(\varphi)+\lambda n\varphi.\] Consequently, \[\ddot\varphi+3H\dot\varphi+\frac{dV}{d\varphi}=-\lambda n.\] The only difference between this equation and a noninteracting scalar field is the term on the right hand side, which accounts for the interaction.

### Problem 2

Show that the DM on a inhomogeneous vacuum background can be treated as as interacting DE and DM. (after [3].)

If one treats the dark energy as the energy of vacuum then its energy-momentum tensor must be proportional to the metric tensor \[T^\nu_{\mu de}= -Vg_\mu^\nu.\] By analogy with the energy-momentum tensor of a perfect fluid one can identify the vacuum energy density and pressure with $\rho_{de}=-p_{de}=V$. The vacuum energy that is homogeneous throughout the spacetime--$\nabla_\mu V=0$-- is equivalent to the cosmological constant in Einstein's gravity \(\Lambda=8\pi GV\). Let us consider the possibility of a time and/or space dependent vacuum energy, \[\nabla_\mu T^\nu_{\mu de}= F_\mu,\quad F_\mu\equiv -\nabla_\mu V.\] One can therefore identify an inhomogeneous vacuum with the interacting one: $F_\mu\ne0$. Indeed, the conservation of the total energy-momentum (including matter fields and the vacuum energy \[\nabla_\mu(T^\nu_{\mu de}+T^\nu_{\mu dm})= 0\] implies that the vacuum transfers energy-momentum to or from the matter fields \[\nabla_\mu T^\nu_{\mu dm}= -F_\mu.\]

### Problem 3

Obtain general equations of motion for DE interacting with DM (after [4]).

We assume a Universe formed by only dark matter and dark energy. The equations of motion that describe the dynamics of the Universe as a whole are the Einstein field equations \begin{equation}\label{int22} {R_{\mu \nu }} - \frac{1}{2}R{g_{\mu \nu }} = 8\pi G\left( {{T_{(de)}}_{\mu \nu } + {T_{(dm)}}_{\mu \nu }} \right), \end{equation} whereas the conservation equation for each component are \begin{equation}\label{int23} \begin{gathered} {\nabla ^\nu }{T_{(de)}}_{\mu \nu } = {F_\mu }, \\ {\nabla ^\nu }{T_{(dm)}}_{\mu \nu } = - {F_\mu }. \\ \end{gathered} \end{equation} where the respective energy momentum tensor for the component $i\;\left( {i = dm,de} \right)$ is \begin{equation}\label{int24} {T_{\left( i \right)\mu \nu }} = \left( {{\rho_i} + {p_i}} \right){u_\mu }{u_\nu } - {p_i}{g_{\mu \nu }}, \end{equation} where ${u_\mu }$ is the velocity of the fluid (the same for each one) and ${\rho_i}$ and ${p_i}$ are respectively the density and pressure of the fluid $i$ measured by an observer with the velocity ${u_\mu }$ . ${F_\mu }$ is the 4-vector of interaction between dark components and its form is not known a priori. Equations \eqref{int23} can be projected on the time or on the space direction of the comoving observer. We project these equations in a part parallel to the velocity ${u^\mu }$ \begin{equation}\label{int25} \begin{gathered} {u^\mu }{\nabla ^\nu }{T_{\left( {dm} \right)\mu \nu }} = - {u^\mu }{F_\mu }, \\ {u^\mu }{\nabla ^\nu }{T_{\left( {de} \right)\mu \nu }} = {u^\mu }{F_\mu }, \\ \end{gathered} \end{equation} and in other part orthogonal to the velocity using the projector ${h_{\beta \mu }} = {g_{\beta \mu }} - {u_\beta }{u_\mu }$ \begin{equation} \label{GrindEQ__2_21_} \begin{array}{l} {\; \; h^{\mu \beta } \nabla ^{\nu } T_{\left(dm\right)\mu \nu } =-h^{\mu \beta } F_{\mu } ,} \\ {\; h^{\mu \beta } \nabla ^{\nu } \nabla ^{\nu } T_{\left(de\right)\mu \nu } =h^{\mu \beta } F_{\mu } }. \end{array} \end{equation} Using \eqref{int24} in \eqref{int25} we obtain the energy conservation equations for each dark component \begin{equation} \label{GrindEQ__2_22_} \begin{array}{l} {u^{\mu } \nabla_{\mu } \rho_{dm} +\left(\rho_{dm} +p_{dm} \right)\nabla_{\mu } u^{\mu } =u^{\mu } F_{\mu } ,} \\ {u^{\mu } \nabla_{\mu } \rho_{de} +\left(\rho_{de} +p_{de} \right)\nabla_{\mu } u^{\mu } =-u^{\mu } F_{\mu } }. \end{array} \end{equation} On the other hand, substitution of \eqref{int24} into \eqref{GrindEQ__2_22_} leads to the Euler equations for each component, \begin{equation} \label{GrindEQ__2_23_} \begin{array}{l} {h^{\mu \beta } \nabla_{\mu } p_{dm} +\left(\rho_{dm} +p_{dm} \right)u^{\mu } \nabla_{\mu } u^{\beta } =-h^{\mu \beta } F_{\mu } ,} \\ {h^{\mu \beta } \nabla_{\mu } p_{de} +\left(\rho_{de} +p_{de} \right)u^{\mu } \nabla_{\mu } u^{\beta } =h^{\mu \beta } F_{\mu } } \end{array}. \end{equation} We assumed that the background metric is described by the flat FLRW metric . In the comoving coordinates we choose $u^{\mu } =\left(1,0,0,0\right)$. With this choice \begin{equation} \label{GrindEQ__2_24_} \begin{array}{l} {\nabla_{\mu } u^{\mu } =3H,} \\ {u^{\mu } \nabla_{\mu } u^{\nu } =0} \end{array}. \end{equation} Using the notation $u^{\mu } F_{\mu } =Q(a)$, we transform the equations \eqref{GrindEQ__2_22_} to their final form \begin{equation} \label{GrindEQ__2_25_} \begin{array}{l} {\dot{\rho }_{dm} +3H\rho_{dm} =Q,} \\ {\dot{\rho }_{de} +3H(\rho_{de} +p_{de} )=-Q} .\\ {} \end{array} \end{equation} The function Q(a) is known as the interaction function, and depends on the scale factor. We note that the equations \eqref{GrindEQ__2_21_} are satisfied identically (taking into account the condition $h^{\mu \nu } F_{\nu } =0$ ) and do not produce any new equations.