Chapter 11 · Section 7 of 11
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}
Obtain the modified Klein-Gordon equation for the scalar field interacting with the dark matter.
\[\ddot\varphi+3H\dot\varphi+\frac{dV}{d\varphi}=-\Gamma,\quad \Gamma\equiv\frac{Q}{\dot\varphi}.\] Here $Q$ is the constant of interaction between dark energy and dark matter.
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])
In the presence of this dark sector interaction, the energy density in the dark matter no longer redshifts as $a^{-3}$ but instead scales as \[\rho_{dm}\propto\frac{\rho_{dm0}}{a^3}f(\varphi/M_{Pl}).\] The conservation equation in this case transforms into \[\dot\rho_{dm}+3H\rho_{dm}=\frac{\rho_{dm0}}{a^3}\frac{df/d\varphi}{f_0},\] where $f_0=f(\varphi/M_{Pl})$ with $\varphi_0$ is the field value today. Consequently the usual Klein-Gordon equation for quintessence models \[\ddot\varphi+3H\dot\varphi+\frac{dV}{d\varphi}=0\] is transformed into \[\ddot\varphi+3H\dot\varphi+\frac{dV}{d\varphi}=-\frac{\rho_{dm0}}{a^3}\frac{df/d\varphi}{f_0}.\]
(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 $r\equiv\rho_{dm}/\rho_\varphi$.
Evolution of the ratio $r\equiv\rho_{dm}/\rho_\varphi$ is described by the equation \[\dot r=r\left(\frac{\dot\rho_{dm}}{\rho_{dm}}-\frac{\dot\rho_\varphi}{\rho\varphi}\right).\] Introducing the barotropic index $\gamma_i=w_i+1$, $(i=dm,\varphi)$, one obtains \[\dot r =3Hr\left(\gamma_{dm}-\gamma_\varphi+\frac{1+r}{\rho_\varphi r}\Pi_{dm}\right).\] The existence of a stationary solution $\dot r=0$ is guaranteed by the condition \[\Pi_{dm}=(\gamma_\varphi-\gamma_{dm})\frac{r}{1+r}\rho_\varphi.\]
Find the form of interaction $Q$ which provides the stationary relation $r$ for interacting cold dark matter and quintessence in spatially flat Universe.
For cold dark matter, $\gamma_{dm}\approx1$, and for dark energy as quintessence, $\gamma_\varphi=\dot\varphi^2/\rho_\varphi$. The coupling term $Q$ in this case is \[Q=-3H(\gamma\varphi-1)\frac{r}{1+r}\rho_\varphi.\] In a spatially flat Universe \[H^2=\frac13(\rho_\varphi+\rho_{dm}),\] and consequently \[Q=-\sqrt3(\gamma_\varphi-1)\frac{\rho_\varphi\rho_{dm}}{\sqrt{\rho_\varphi+\rho_{dm}}}.\]
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.
For the interaction under consideration, the conservation equation for $\rho_{dm}$ and $\rho_\varphi$ transform into \[\dot\rho_{dm}+3H\rho_{dm} =-3H(\gamma_\varphi-1)\frac{r}{1+r}\rho_{dm},\] \[\dot\rho_\varphi+3H\gamma_\varphi\rho_\varphi= -3H(\gamma_\varphi-1)\frac{r}{1+r}\rho_\varphi.\] Assuming $\gamma_\varphi$ to be constant, one obtains \[\rho_\varphi\propto a^{-\nu},\quad \rho_{dm}\propto a^{-\nu},\quad \nu=3\frac{\gamma_\varphi+r}{1+r}.\] Both energy densities happen to redshift at the same rate because we have chosen $Q$ to correspond to the $r=const$.
Show that in the case of interaction $Q$ obtained in the problem #IDE_63, the scalar field $\varphi$ evolves logarithmically with time.
With $\rho\propto a^{-\nu}$ (see the previous problem) one can solve the first Friedmann equation to find $a\propto t^{2/\nu}$. Using \[\Omega_\varphi=\frac{1}{1+r}=\frac{8\pi G}{3}\frac{\nu^2}{4}t^2\rho_\varphi,\quad \nu=3\frac{\gamma_\varphi+r}{1+r},\] one finds \[\rho_\varphi=\frac{1}{6\pi G}\frac{1+r}{(\gamma_\varphi+r)^2}\frac{1}{t^2}.\] Taking into account that \[\gamma_\varphi=\frac{\dot\varphi^2}{\rho_\varphi},\] one finds \[\dot\varphi=\sqrt{\frac{\gamma_\varphi(1+r)}{6\pi G}}\frac{1}{\gamma_\varphi+r}\frac1t,\] i.e., $\varphi$ evolves logarithmically with time.
Reconstruct the potential $V(\varphi)$, which realizes the solution $r=const$, obtained in the problem #IDE_63.
With the help of \[\rho_\varphi= \frac{\dot\varphi^2}2+V(\varphi)\] and \[\rho_\varphi= \frac{\dot\varphi^2}{\gamma_\varphi},\] one finds \[V(\varphi)=\frac{\dot\varphi^2}{\gamma_\varphi}\left(1-\frac{\gamma_\varphi}2\right).\] Using the expression for $\dot\varphi$, obtained in the previous problem, one obtains \[V(\varphi)=\frac{1}{6\pi G}\left(1-\frac{\gamma_\varphi}2\right)\frac{1+r}{(\gamma_\varphi+r)^2}\frac1{t^2}.\] Since \[\frac{dV}{d\varphi}\dot\varphi=\frac{dV}{dt}=-2\frac V t,\] then one obtains the following equation for the scalar field potential \[\frac{dV(\varphi)}{d\varphi}=-\lambda V(\varphi),\quad \lambda\equiv\sqrt{\frac{24\pi G}{\gamma_\varphi(1+r)}}(\gamma_\varphi+r),\] and, consequently, \[V(\varphi)=V_0\exp\left[-\lambda(\varphi-\varphi_0)\right].\]
(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.
Transit to derivatives of the scalar field with respect to $u=\ln a$ in the Klein-Gordon equation and take into account that $Q=-\lambda\dot\varphi\rho_{dm}$ in the considered case to obtain \[H^2\varphi''+(\dot H+3H^2)\varphi'=\lambda\rho_{dm}-\eta V.\] Substitute \[H^2=\frac13\frac{\rho_{dm}+V}{1-\frac16\varphi''}\] and \[\dot H=-\frac12(\rho_{tot}+p_{tot})\] to find \[\frac13\frac{\rho_{dm}+V}{1-\frac16\varphi''}\varphi''+\frac12(\rho_{dm}+2V)\varphi'=\lambda\rho_{dm}-\eta V.\]
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.
Like in the previous problem, transit to derivatives of the scalar field with respect to $u=\ln a$ in the Klein-Gordon equation and take into account that \[Q=-\alpha\frac{\dot\varphi}{\varphi}\rho_{dm}\] in the considered case to obtain \[\frac13\frac{\rho_{dm}+V}{1-\frac16\varphi''}\varphi''+\frac12(\rho_{dm}+2V)\varphi'=\frac{\alpha}{\varphi} \rho_{dm}-\frac{\beta}{\varphi} V.\]
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.
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 $w_{de}=const$.
If only $\delta$ depends on the scale factor, then \[\rho_{dm}=\rho_{dm0}a^{-3}e^{\int\delta(a)d\ln a}.\] One more time use the assumption \[r\equiv\frac{\rho_{dm}}{\rho_{de}}=\frac{\rho_{dm0}}{\rho_{de0}}a^{-\xi}=A^{-1}a^{-\xi},\quad A\equiv\frac{\rho_{dm0}}{\rho_{de0}}=\frac{\Omega_{dm0}}{\Omega_{de0}}.\] Consequently, \[\rho_{de}=\frac{Aa^\xi}{1+Aa^\xi}\rho_{tot},\quad \rho_{dm}=\frac{1}{1+Aa^\xi}\rho_{tot}.\] Then the total energy density satisfies the equation \[\frac{d\rho_{tot}}{da}+\frac3a\frac{1+(1+w_{de})Aa^\xi}{1+Aa^\xi}\rho_{tot}=0.\] Integrate this expression to obtain \[\rho_{tot}=\rho_{tot0}a^{-3}\left[1-\Omega_{de0}(1-a^\xi)\right]^{-3w_{de}/\xi},\quad \rho_{tot0}=\rho_{de0}+\rho_{dm0}.\] Consequently the first Friedmann equation can be written as \[H^2=H_0^2 a^{-3}\left[1-\Omega_{de0}(1-a^\xi)\right]^{-3w_{de}/\xi}.\] Using the conservation equation for DM one can get the coupling function \[\delta=3+\frac1H\frac{\dot\rho_{dm}}{\rho_{dm}}=-\frac{(3+w_{de})Aa^\xi}{1+Aa^\xi}= -(3+w_{de}) \frac{\rho_{de}}{\rho_{tot}}.\] This relation can be expressed as \[\delta=\frac{\delta_0}{\Omega_{de0}+(1-\Omega_{de0})a^{-\xi}}, \quad \delta_0\equiv-\Omega_{de0}(\xi+3w_{de}).\] Let us analyze the obtained expression. Asymptote of the interaction $\delta$ represents a constant, $\delta(a\to\infty)=\delta_0/\Omega_{de0}$. Therefore, if the expansion dynamics is such that $\xi>-3w_{de}$, then $\delta<0$, which implies that the energy flow is from the dark matter to dark energy. On the contrary, when $0<\xi<-3w_{de}$, the energy flow is from the phantom dark energy to dark matter. Further, we can see that there is no coupling between the dark energy and dark matter at $\xi=-3w_{de}$. Specifically, there is no coupling in SCM, for which $\xi=3$, $w_{de}=-1$.
Calculate the deceleration parameter for the model considered in the previous problem.
\[q=-\frac{\ddot a}{aH^2}=-1+\frac{\dot H}{H^2}=-1+\frac32\frac{1-\Omega_{de0}+(1+w_{de})\Omega_{de0}a^\xi}{1-\Omega_{de0}(1-a^\xi)}.\] Note that the deceleration parameter values $q(a\to1)$ and $q(a\to\infty)$ are negative, as expected in the dark energy dominated epoch.
(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).
Use the result of problem #IDE_63 \[Q=-3H(\gamma\varphi-1)\frac{r}{1+r}\rho_\varphi\] to find that the conservation equation for the fantom field takes on the form \[\dot\rho_\varphi+3\frac{\gamma_\varphi+r}{1+r}H\rho_\varphi=0.\] It then follows that \[\rho_\varphi\propto a^{-3\nu},\quad \nu\equiv\gamma_\varphi\frac{r}{1+r}.\] If the expansion is matter dominated until the time $t_m$, then we can write the scale factor as \[a(t)=a(t_m)\left(1-\nu+\nu\frac t {t_m}\right)^{\frac2{3\nu}}.\] Using the definitions of $r$ and $\gamma_varphi$, the Friedmann equation gives \[a\propto\exp\left(\frac{\sqrt{1+r}}{\sqrt{-3(1+w_\varphi)}}\kappa\varphi\right),\quad \kappa^2\equiv 8\pi G.\] In terms of $\varphi$, $\rho_\varphi$ becomes \[\rho_\varphi\propto\exp\left(\frac{\sqrt{3(1+r)}\nu}{\sqrt{-(1+w_\varphi)}}\kappa\varphi\right).\]
Construct the scalar field potential, which realizes the given relation $r$ for the model considered in the previous problem.}
For the scalar field $V(\varphi)\propto\rho_\varphi-p_\varphi$, and for $w_\varphi=const$ one has $V(\varphi)\propto\rho_\varphi$. It follows that \[V(\varphi)=V_0 \exp\left(\frac{\sqrt{3(1+r)}\nu}{\sqrt{-(1+w_\varphi)}}\kappa\varphi\right).\]
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].)
Find interaction of tachyon field with cold dark matter (CDM), which results in $r\equiv\rho_{dm}/\rho_T=const$.
Proceeding like in solution of analogous problem 4 for quintessence, one obtains that the condition $\dot r=0$ is realized for the interaction \[Q=3H\frac{r}{(1+r)^2}(1-\dot T^2)(\rho_T + \rho_{dm}).\]
Show that the stationary solution $\dot r=0$ exists only when the energy of the tachyon field is transferred to the dark matter.
Since $\dot T^2<1$, we have according to previous problem $Q > 0$. Therefore, the stationary solution exists only when the energy of the tachyon field is transferred to the dark matter.
Find the modified Klein-Gordon equation for arbitrary interaction $Q$ of tachyon scalar field with dark matter.
Use the relations \[\rho_T=\frac{V(T)}{\sqrt{1-\dot T^2}}\] and \[p_T=-V(T)\sqrt{1-\dot T^2}.\] to obtain the following transformations \[\dot\rho_t+3H(\rho_T+p_T)=-Q\quad\to\] \[\dot\rho_t+3H\dot\varphi^2\rho_T=-Q\quad\to\] \[\frac{\ddot\varphi}{1-\dot\varphi^2}+3H\dot\varphi+\frac{V'}{V}=-\frac{\sqrt{1-\dot\varphi^2}}{V\dot\varphi}Q,\quad V'\equiv\frac{dV}{d\varphi}.\]
Find the modified Klein-Gordon equation for the interaction $Q$ obtained in the problem 14 and obtain its solutions for the case $\dot\varphi=const$.
The modified Klein-Gordon equation or the tachyon field in the case $\dot r=const$ reads \[\frac{\ddot\varphi}{1-\dot\varphi^2}+3H\dot\varphi+3H\frac{1-\dot\varphi^2}{\varphi}\frac{r}{r+1}+\frac{V'}{V}=0.\] When $\dot\varphi=const$ \[\rho_{dm},\rho_T,V\propto a^{-3\frac{r+\dot\varphi^2}{r+1}}.\] It corresponds to a power law expansion \[a(t)\propto t^n,\quad n=\frac23\frac{r+1}{r+\dot\varphi^2}=const\] and \[\rho_{dm}, \rho_T\propto t^{-2}.\]
Show that sufficiently small values of tachyon field provide the accelerated expansion of Universe.
For $\rho\propto t^n$ the accelerated expansion $\ddot a>0$ is guaranteed by the condition $n>1$, which is equivalent \[\dot\varphi^2<\frac{2-r}3,\] i.e. $\dot\varphi^2$ has to be sufficiently small.