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Problem 1

problem id: G-interact-1

Let us consider a very simple model with the following conservation equations: \begin{equation}\label{conserv_Qm} \dot{\rho_{dm}}+3H\rho_{dm}=Q, \end{equation} \begin{equation}\label{conserv_Qe} \dot{\rho_{de}}+3H\rho_{de}(1+w)=-Q, \end{equation} where $Q=3H\delta\rho_{dm}$ is the source of interaction and $w$ is the state equation parameter for dark energy. The positive small coupling constant $\delta$ will eventually characterize how evolution of the matter energy density \begin{equation}\label{rho-m1} \rho_{dm}= \rho_{dm0}a^{-3(1-\delta)}, \end{equation} will deviate from the standard case ($\rho_{m0}$ is the present value of $\rho_m$).

For this model find dependence of the dark energy density on the scale factor and using the method described in the problem New_problems#1301_38 express Friedmann equation in terms of the cosmographic parameters $q, j, l, s$.

Problem 2

problem id: G-interact-2

Make the same calculations as in the previous problem for model with interaction term $Q=3H\delta\rho_{de}$.

Problem 3

problem id: G-interact-3

Consider Model with a Decaying Vacuum Energy: \begin{equation} \label{eq:1} 3 \frac{\dot{a}^{2}}{a^{2}} = \rho_{\text{m}} + \rho_{\text{vac}} = \rho_{\text{m}}+ \Lambda_{\text{bare}} + \frac{3\beta}{t^2} \end{equation} where matter density fulfils the conservation condition \begin{equation} \bar{T}^{\alpha \beta}_{\quad ;\beta} = T^{\alpha \beta}_{\text{m}} + \Lambda(t) g^{\alpha \beta}= 0. \end{equation} This condition for the flat homogeneous and isotropic Universe has the form \begin{equation} \label{eq:5} \dot{\rho}_{\text{m}} + 3 H \rho_{\text{m}} = - \dot{\Lambda}(t), \end{equation} where $\Lambda(t) = \Lambda_{\text{bare}} + \frac{3\beta}{t^{2}}$.

For the model with vacuum decaying \begin{equation} \label{eq:9} \left( \frac{H}{H_0} \right)^2 = \Omega_{\text{m}} + \Omega_{\Lambda_{\text{bare}}} + \frac{\beta}{H_{0}^{2}} T(z)^{-2} \end{equation} where \[ T(z) = - \int_{\infty}^{z} \frac{dz}{(1+z)H(z)} \] is the age of the Universe up to the redshift $z$. For estimation of the model parameters use such parametrization of $H(z)$ in which $T(z) \propto H^{-1}$. Then \begin{equation} \left( \frac{H}{H_0} \right)^2 = \Omega_{\text{m}} + \Omega_{\Lambda_{\text{bare}}} + \frac{\beta }{H_{0}^{2}} H^{2}. \end{equation} The acceleration equation has the form \begin{equation} \dot{H} = \frac{\Lambda_{\text{bare}}}{2} - \frac{3}{2} (1-\beta) H^2. \end{equation} After changing the variable $t \to a$ and solving this equation we will find \begin{equation} \left( \frac{H}{H_0} \right)^2 = \frac{\Omega_{\text{m}} a^{-3(1-\beta)}}{1-\beta} + \frac{\Omega_{\Lambda_{\text{bare}}}}{1-\beta}. \end{equation}

\[ \Omega_{\text{m}0} + \Omega_{\Lambda_{\text{bare}0}} +\beta =1. \] As we can see when $\beta = 0$ the model reduces to the standard $\Lambda$CDM model.

For this model express Friedmann equation in terms of the cosmographic parameters $q, j, l, s$

Problem 4

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Problem 5

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Problem 6

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