Peculiarity of dynamics of scalar field coupled to dark matter

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Interacting quintessence model

Given that the quintessence field and the dark matter have unknown physical natures, there seem to be no a priori reasons to exclude a coupling between the two components. Let us consider a two-component system (scalar field $\varphi$ + dark matter) with the energy density and pressure \[\rho=\rho_\varphi+\rho_{dm},\quad p=p_\varphi+p_{dm}\] (we do not exclude the possibility of warm DM ($p_{dm}\ne0$).) If some interaction exists between the scalar field and DM, then \[\dot\rho_{dm}+3H(\rho_{dm}+p_{dm})=Q\] \[\dot\rho_\varphi+3H(\rho_\varphi+p_\varphi)=-Q.\] Using the effective pressures $\Pi_\varphi$ and $\Pi_{dm}$, \[Q=-3H\Pi_{dm}=3H\Pi_\varphi\] one can transit to the system \begin{align} \nonumber \dot\rho_{dm}+3H(\rho_{dm}+p_{dm}+\Pi_{dm}) & =0,\\ \nonumber \dot\rho_\varphi+3H(\rho_\varphi+p_\varphi+\Pi_\varphi) & =0. \end{align}


Problem 1

Obtain the modified Klein-Gordon equation for the scalar field interacting with the dark matter.


Problem 2

Consider a quintessence scalar field $\varphi$ which couples to the dark matter via, e.g., a Yukawa-like interaction $f(\varphi/M_{Pl})\bar\psi\psi$, where $f$ is an arbitrary function of $\varphi$ and $\psi$ is a dark matter Dirac spinor. Obtain the modified Klein-Gordon equation for the scalar field interacting with the dark matter in such way. (after [1])


Problem 3

(The problems #IDE_62-#IDE_66 are inspired by [2].)

Show that the Friedman equation with interacting scalar field and dark matter allow existence of stationary solution for the ratio $r\equiv\rho_{dm}/\rho_\varphi$.


Problem 4

Find the form of interaction $Q$ which provides the stationary relation $r$ for interacting cold dark matter and quintessence in spatially flat Universe.


Problem 5

For the interaction $Q$ which provides the stationary relation $r$ for interacting cold dark matter and quintessence in spatially flat Universe (see the previous problem), find the dependence of $\rho_{dm}$ and $\rho_\varphi$ on the scale factor.


Problem 6

Show that in the case of interaction $Q$ obtained in the problem #IDE_63, the scalar field $\varphi$ evolves logarithmically with time.


Problem 7

Reconstruct the potential $V(\varphi)$, which realizes the solution $r=const$, obtained in the problem #IDE_63.


Problem 8

(After [3])

Let the DM particle's mass $M$ depend exponentially on the DE scalar field as $M=M_*e^{-\lambda\varphi}$, where $\lambda$ is positive constant and the scalar field potential is \[V(\varphi)=V_* e^{\eta\varphi}.\] Obtain the modified Klein-Gordon equation for this case.


Problem 9

Let the DM particle's mass $M$ depend exponentially on the DE scalar field as $M=M_*e^{-\alpha}$, and the scalar field potential is \[V(\varphi)=V_* e^{\beta},\] where $\alpha,\beta>0$. Obtain the modified Klein-Gordon equation for this case.


Interacting Phantom

Let the Universe contain only noninteracting cold dark matter ($w_{dm}=0$) and a phantom field ($w_{de}<-1$). The densities of these components evolve separately: $\rho_{dm}\propto a^{-3}$ and $\rho_{de}\propto a^{-3(1+w_{de})}$. If matter domination ends at $t_m$, then at the moment of time \[t_{BR}=\frac{w_{de}}{1+w_{de}}t_m\] the scale factor, as well as a series of other cosmological characteristics of the Universe become infinite. This catastrophe has earned the name "Big Rip". One of the way to avoid the unwanted big rip singularity is to allow for a suitable interaction between the phantom energy and the background dark matter.


Problem 10

Show that through a special choice of interaction, one can mitigate the rise of the phantom component and make it so that components decrease with time if there is a transfer of energy from the phantom field to the dark matter. Consider case of $Q=\delta(a)H\rho_{dm}$ and $w_{de}=const$.


Problem 11

Calculate the deceleration parameter for the model considered in the previous problem.


Problem 12

(After [4].)

Let the interaction $Q$ of phantom field $\varphi$ with DM provide constant relation $r=\rho_{dm}/\rho_\varphi$. Assuming that $w_\varphi=const$, find $\rho_\varphi(a)$, $\rho_\varphi(\varphi)$ and $a(\varphi)$ for the case of cold dark matter (CDM).


Problem 13

Construct the scalar field potential, which realizes the given relation $r$ for the model considered in the previous problem.}


Tachyonic Interacting Scalar Field

Let us consider a flat Friedmann Universe filled with a spatially homogeneous tachyon field $T$ evolving according to the Lagrangian \[L=-V(T)\sqrt{1-g_{00}\dot T^2}.\] The energy density and the pressure of this field are, respectively \[\rho_T=\frac{V(T)}{\sqrt{1-\dot T^2}}\] and \[p_T=-V(T)\sqrt{1-\dot T^2}.\] The equation of motion for the tachyon is \[\frac{\ddot T}{1-\dot T^2}+3H\dot T+\frac{1}{V(T)}\frac{dV}{dT}.\] (Problems #IDE_73 - #IDE_77 are after [5].)


Problem 14

Find interaction of tachyon field with cold dark matter (CDM), which results in $r\equiv\rho_{dm}/\rho_T=const$.


Problem 15

Show that the stationary solution $\dot r=0$ exists only when the energy of the tachyon field is transferred to the dark matter.


Problem 16

Find the modified Klein-Gordon equation for arbitrary interaction $Q$ of tachyon scalar field with dark matter.


Problem 17

Find the modified Klein-Gordon equation for the interaction $Q$ obtained in the problem \ref{IDE_73} and obtain its solutions for the case $\dot\varphi=const$.


Problem 18

Show that sufficiently small values of tachyon field provide the accelerated expansion of Universe.