Skip to content
Universe in Problems Interactions in the Dark Sector Log in
Classic view
Page tools
Chapter 11
Section 11 of 11

Chapter 11 · Section 11 of 11

Transient acceleration

6 problems




11.11.1

Problem 11.11.1

Consider a simple model of transient acceleration with decaying cosmological constant \begin{equation} \label{ec} \dot{\rho}_{m} + 3\frac{\dot{a}}{a}\rho_{m} = - \dot{\rho}_\Lambda\;, \end{equation} where $\rho_{m}$ and $\rho_\Lambda$ energy density DE and cosmological constant $\Lambda$. At the early stages of the expansion of the Universe, when $\rho_\Lambda$ is quite small, such a decay does not influence cosmological evolution in any way. At later stages, as the DE contribution increases, its decay has an ever increasing effect on the standard dependence of the DM energy density $\rho_{m} \propto a^{-3}$ on the scale factor $a$. We consider the deviation to be described by a function of the scale factor - $\epsilon(a)$. \begin{equation} \label{dm} \rho_{m} = \rho_{m, 0}a^{-3 + \epsilon(a)}\;, \end{equation} where $a_0 = 1$ in the present epoch. Other fields of matter (radiation, baryons) evolve independently and are conserved. Hence, the DE density has the form \begin{equation}\label{decayv} \rho_{\Lambda} = \rho_{m0} \int\limits_{a}^{1}\frac{\epsilon(\tilde{a}) + \tilde{a}\epsilon' \ln(\tilde{a})}{\tilde{a}^{4 - \epsilon(\tilde a)}} d\tilde{a} + {\rm{X}}\;, %\tilde\rho_{\Lambda 0} + \end{equation} where the prime denotes the derivative with respect to the scale factor, and ${\rm{X}}$ is the integration constant. If radiation is neglected, the first Friedmann equation takes the form \begin{equation} \label{friedmann} {H}= H_0\left[\Omega_{b,0}{a}^{-3} + \Omega_{m0}\varphi(a) + {\Omega}_{\rm{X,0}}\right]^{1/2}, \end{equation} Using the assumption that the function $\epsilon (a)$ has the following simple form \begin{equation} \label{Parametrization_a} \epsilon(a) = \epsilon_0a^\xi\ = \epsilon_0(1+z)^{-\xi}, \end{equation} where $\epsilon_0$ and $\xi$ can take both positive and negative values, find function $\varphi(a)$ and relative energy density $\Omega_b(a)$, $\Omega_{m}(a)$ and $\Omega_{\Lambda}(a)$.



11.11.2

Problem 11.11.2

Using the results of the previous problem, find the deceleration parameter for this model is $q(a)$. Draw the graph deceleration parameter as a function of $\log(a)$ for various values of $\epsilon_0$ and $\xi$: $\xi = 1.0$ and $\epsilon_0 = 0.1$, $\xi = -1.0$ and $\epsilon_0 = 0.1$, $\xi = 0.8$ and $\epsilon_0 = 0.5$, $\xi = -0.5$ and $\epsilon_0 = -0.1.$.



11.11.3

Problem 11.11.3

Consider the possibility of an accelerating transient regime within the interacting scalar field model using potential of the form \begin{equation} \label{V_fransient} V(\phi)=\rho_{\phi\,0}[1-\frac{\lambda}{6}(1+\alpha \sqrt{\sigma}\phi)^2)] \exp{[-\lambda\sqrt{\sigma}(\phi+ \frac{\alpha \sqrt{\sigma}}{2}\phi^2)]}, \end{equation} where $\rho_{\phi\,0}$ is a constant energy density, $\sigma=8\pi G/\lambda$, and $\alpha$ and $\lambda$ are two dimensionless, positive parameters of the model, that the deceleration parameter is non-monotonically dependent on the scale factor. Plot the deceleration parameter as a function of the scale factor.



11.11.4

Problem 11.11.4

Consider a simple parameterization: \begin{equation} \label{Qsimple} Q=3\beta(a)H \rho_{de} \end{equation} with a simple power-law ansatz for $\beta(a)$, namely: \begin{equation} \label{cas32} \beta(a)=\beta_0 a^\xi. \end{equation} Substituting this interaction form into conservation equations for DM and DE: \begin{eqnarray}\label{eom1} \dot{\rho}_{dm}+3H\rho_{dm}=Q, \end{eqnarray} \begin{eqnarray}\label{eom2} \dot{\rho}_{de}+3H(\rho_{de}+p_{de})=-Q, \end{eqnarray} we get \begin{equation} \label{rhophi2} \rho_{de}=\rho_{de0}\, a^{-3(1+w_0)}\cdot \exp{\left[\frac{3\beta_0(1-a^\xi)}{\xi}\right]}, \end{equation} where the integration constant $\rho_{de0}$ is value of the dark energy at present, and the dark energy EoS parameter $w\equiv p_{de}/\rho_{de}$ is a constant-$w_0$. Substituting Eq. (\ref{rhophi2}) into Eq. (\ref{eom2}), we get the dark matter energy density, \begin{equation}\label{rhom2} \rho_{dm}=f(a)\rho_{dm0}, \end{equation} where \begin{equation} \label{f1} f(a)\equiv \frac{1}{a^3}\left\{1-\frac{\Omega_{de0}}{\Omega_{dm0}}\frac{3\beta_0 a^{-3w_0}e^{\frac{3\beta_0}{\xi}}}{\xi}\cdot\left[a^\xi E_{\frac{3w_0}{\xi}}\left(\frac{3\beta_0 a^\xi}{\xi}\right)-a^{3w_0} E_{\frac{3w_0}{\xi}} \left(\frac{3\beta_0}{\xi}\right)\right]\right\}, \end{equation} where $\rho_{dm0}$ is dark matter density at present day, and $E_n(z)=\int_1^\infty t^{-n}e^{-xt}dt$ the usual exponential integral function. Note however that Eq. (\ref{rhom2}) is an analytical expression, while in the corresponding expressions were left as integrals and were calculated numerically. Obviously, in the case of non-interaction (that is, for $\beta_0=0$), Eq. (\ref{rhom2}) recovers the standard result $\rho_{dm}=\rho_{dm0}/a^3$. For the special case $\xi=0$ find dimensionless Hubble parameter $E^2(z)\equiv \frac{H^{2}}{H^{2}_0}$, the evolution of the density parameters $\Omega_b(a)$, $\Omega_{dm}(a)$ and $\Omega_{de}(a)$ and $q(a)$. For what values of $\beta_0$ the cosmic acceleration is transient?



11.11.5

Problem 11.11.5

Consider the flat FLRW cosmology with two coupled homogeneous scalar fields $\Phi$ and $\Psi$: \begin{eqnarray} \dot{\rho_b} &=& -3 H \gamma_b \rho_b \\ \ddot{\Phi} &=& -3H\dot{\Phi} - \partial_\Phi V \\ \ddot{\Psi} &=& -3H\dot{\Psi} - \partial_\Psi V \\ \dot{H} &=& -4\pi G(\gamma_m\rho_m+\gamma_r\rho_r+\gamma_Q\rho_Q)~, \end{eqnarray} \begin{equation} H^2 = \frac{8\pi G}{3}(\rho_m+\rho_r+\rho_Q)-\frac{k}{a^2}~. \end{equation} Here a dot denotes a derivative with respect to the cosmic time $t$, the subscript $b$ refers to the dominant background quantity, either dust (m) or radiation (r) while $Q$ refers to the Dark Energy sector, here the two quintessence scalar fields.

The quintessence fields with potential $V$ have the following energy density and pressure: \begin{eqnarray} \rho_Q =\frac{1}{2}\dot{\Phi}^2 + \frac{1}{2}\dot{\Psi}^2 + V(\Phi,\Psi)\\ p_Q =\frac{1}{2}\dot{\Phi}^2 + \frac{1}{2}\dot{\Psi}^2 - V(\Phi,\Psi) \end{eqnarray} with $p_Q = (\gamma_Q-1)\rho_Q$. It is convenient to define the following new variables: \begin{equation} X_{\Phi}=\sqrt{\frac{8\pi G}{3H^2}}~\frac{\dot{\Phi}}{\sqrt{2}}, X_{\Psi}=\sqrt{\frac{8\pi G}{3H^2}}~\frac{\dot{\Psi}}{\sqrt{2}}, X_V=\sqrt{\frac{8\pi G}{3H^2}}~\sqrt{V}. \end{equation} Find expressions for the $\Phi'$, $\Psi'$, $X_{\Phi}'$, $X_{\Psi}'$, $X_V'$ where a prime denotes a derivative with respect to the quantity $N$, the number of e-folds with respect to the present time, \begin{equation} N\equiv {\rm ln} \frac{a}{a_0}~,\label{N} \end{equation} and we have also $H=\dot{N}$.



11.11.6

Problem 11.11.6

Using result from the previous problem find the relative energy density for matter, radiation and quintessence, $\Omega_m$, $\Omega_r$ and $\Omega_Q$, the deceleration parameter $q$, the Hubble-parameter-free luminosity distance $D_L$ and the age of the Universe $t_0$.


Retrieved from "https://universeinproblems.com/index.php?title=Transient_acceleration&oldid=2401"
Chapter 11 · Interactions in the Dark Sector
All 18 chapters · Book order
  1. 01Cosmo warm-up9 sections
  2. 02Dynamics of the Expanding Universe9 sections
  3. 03Dynamics of the Universe in the Big Bang Model9 sections
  4. 04Black Holes5 sections
  5. 05Cosmic Microwave Background (CMB)6 sections
  6. 06Thermodynamics of Universe8 sections
  7. 07Perturbation theory in cosmology5 sections
  8. 08Inflation3 sections
  9. 09Dark Energy7 sections
  10. 10Dark Matter7 sections
  11. 11Interactions in the Dark Sector11 sections
  12. 12Standard Cosmological Model2 sections
  13. 13Weak field limit and gravitational waves10 sections
  14. 14Observational cosmology2 sections
  15. 15Holographic Universe1 section
  16. 16Horizons9 sections
  17. 17Deceleration Parameter74 problems
  18. 18Quantum Cosmology27 problems