Difference between revisions of "New oleg"
(4 intermediate revisions by the same user not shown) | |||
Line 18: | Line 18: | ||
will deviate from the standard case ($\rho_{m0}$ is the present value of $\rho_m$). | 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 $q, j, l, s$ | + | 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 $q, j, l, s$. |
<div class="NavFrame collapsed"> | <div class="NavFrame collapsed"> | ||
<div class="NavHead">solution</div> | <div class="NavHead">solution</div> | ||
Line 50: | Line 50: | ||
− | <div id="G-interact- | + | <div id="G-interact-2"></div> |
<div style="border: 1px solid #AAA; padding:5px;"> | <div style="border: 1px solid #AAA; padding:5px;"> | ||
'''Problem 2''' | '''Problem 2''' | ||
− | <p style= "color: #999;font-size: 11px">problem id: G-interact- | + | <p style= "color: #999;font-size: 11px">problem id: G-interact-2</p> |
Make the same calculations as in the previous problem for model with interaction term $Q=3H\delta\rho_{de}$. | Make the same calculations as in the previous problem for model with interaction term $Q=3H\delta\rho_{de}$. | ||
<div class="NavFrame collapsed"> | <div class="NavFrame collapsed"> | ||
Line 60: | Line 60: | ||
<p style="text-align: left;">For model with interaction term $Q=3H\delta\rho_{de}$ the equivalent Friedmann equation expressed in cosmographic parameters has the form: | <p style="text-align: left;">For model with interaction term $Q=3H\delta\rho_{de}$ the equivalent Friedmann equation expressed in cosmographic parameters has the form: | ||
\begin{equation}\label{fridman_Q_rde} | \begin{equation}\label{fridman_Q_rde} | ||
− | 1+3(w+\delta)=-2(j-3q(1+w+\delta)) | + | 1+3(w+\delta)=-2(j-3q(1+w+\delta)). |
\end{equation} | \end{equation} | ||
− | |||
And coupling constant $\delta$ | And coupling constant $\delta$ | ||
\begin{equation}\label{delta-rde} | \begin{equation}\label{delta-rde} | ||
Line 76: | Line 75: | ||
− | <div id=""></div> | + | <div id="G-interact-3"></div> |
<div style="border: 1px solid #AAA; padding:5px;"> | <div style="border: 1px solid #AAA; padding:5px;"> | ||
'''Problem 3''' | '''Problem 3''' | ||
− | <p style= "color: #999;font-size: 11px">problem id: </p> | + | <p style= "color: #999;font-size: 11px">problem id: G-interact-3</p> |
+ | 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 $q, j, l, s$ | ||
<div class="NavFrame collapsed"> | <div class="NavFrame collapsed"> | ||
<div class="NavHead">solution</div> | <div class="NavHead">solution</div> | ||
<div style="width:100%;" class="NavContent"> | <div style="width:100%;" class="NavContent"> | ||
− | <p style="text-align: left;"></p> | + | <p style="text-align: left;">We can obtain equation which is expressed only in cosmographical parameters |
+ | \begin{equation}\label{dr1} | ||
+ | s+(2-3\beta)(q+j)+qj=0. | ||
+ | \end{equation} | ||
+ | And $\beta$ for this model has the form | ||
+ | \begin{equation}\label{delta-qua} | ||
+ | \beta=\frac{qj+2(q+j)+s}{3(q+j)}. | ||
+ | \end{equation} | ||
+ | </p> | ||
</div> | </div> | ||
</div></div> | </div></div> |
Latest revision as of 22:05, 9 April 2015
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 $q, j, l, s$.
The dark energy density in the consider model model takes on the form: \begin{equation}\label{rhode} \rho_{de}=\rho_{de0} a^{-3(1+w)}-\frac{a^{-3(1-\delta)}\delta\rho_{dm0}}{w+\delta}, \end{equation} where the integration constant $\rho_{de0}$ equals to the present value of the dark energy density. Substituting the energy density dependence of the scale factor into the Friedmann equation, we get: \begin{equation}\label{habl} H^2=\frac 13\left(\rho_{de0} a^{-3(1+w)}-\frac{a^{-3(1-\delta)}\delta\rho_{dm0}}{w+\delta}+\rho_{dm0}a^{-3(1-\delta)} \right). \end{equation} Note that here we consider a flat ($k=0$) Universe. Using the method described in the problem New_problems#1301_38 we can write the equivalent Friedmann equation expressed in cosmographic parameters: \begin{equation}\label{fridman_Q_rm} (1+3w)(3\delta-1)=2(j-3q(1+w-\delta)). \end{equation} In order to express the coupling constant $\delta$ in terms of the cosmographic parameters we must know the second, the third and the fourth derivative of the scale factor, because we need three independent equations to resolve three independent constants. After the straightforward calculations we obtain \begin{equation}\label{delta-first11} -2q(2+3w)(3\delta-2)+j(-5+3(\delta-w))-qj-s=0, \end{equation} or \begin{equation}\label{delta-first} \delta=\frac{-2q(2+3w)+j(5+3w)+jq+s}{3(j-q(2+3w))}. \end{equation}
Problem 2
problem id: G-interact-2
Make the same calculations as in the previous problem for model with interaction term $Q=3H\delta\rho_{de}$.
For model with interaction term $Q=3H\delta\rho_{de}$ the equivalent Friedmann equation expressed in cosmographic parameters has the form: \begin{equation}\label{fridman_Q_rde} 1+3(w+\delta)=-2(j-3q(1+w+\delta)). \end{equation} And coupling constant $\delta$ \begin{equation}\label{delta-rde} -2q(2+3(w+\delta))+j(2+3(w+\delta))+qj+s=0, \end{equation} or \begin{equation}\label{delta-rde} \delta=\frac{2q(2+3w)-j(5+3w)-qj-s}{3(j-2q)}. \end{equation}
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 $q, j, l, s$
We can obtain equation which is expressed only in cosmographical parameters \begin{equation}\label{dr1} s+(2-3\beta)(q+j)+qj=0. \end{equation} And $\beta$ for this model has the form \begin{equation}\label{delta-qua} \beta=\frac{qj+2(q+j)+s}{3(q+j)}. \end{equation}
Problem 4
problem id:
Problem 5
problem id:
Problem 6
problem id: