Statefinder parameters for interacting dark energy and cold dark matter

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Problem 1

(after [1])

Show that in flat Universe both the Hubble parameter and deceleration parameter do not depend on whether or not dark components are interacting. Become convinced the second derivative $\ddot H$ does depend on the interaction between the components.

Problem 2

Find statefinder parameters for interacting dark energy and cold dark matter.

Problem 3

Show that the statefinder parameter $r$ is generally necessary to characterize any variation in the overall equation of state of the cosmic medium.

Problem 4

Find relation between the statefinder parameters in the flat Universe.

Problem 5

Express the statefinder parameters in terms of effective state parameter $w_{(de)eff}$, for which \[\dot\rho_{de}+3H(1+w_{(de)eff})\rho_{de}=0.\]

Problem 6

Find the statefinder parameters for $Q=3\delta H\rho_{dm}$, assuming that $w_{de}=const$.

Problem 7

Find statefinder parameters for the case $\rho_{dm}/\rho_{de}=a^{-\xi}$, where $\xi$ is a constant parameter in the range $[0,3]$ and $w_{de}=const$.

Problem 8

Show that in the case $\rho_{dm}/\rho_{de}=a^{-\xi}$ the current value of the statefinder parameter $s=s_0$ can be used to measure the deviation of cosmological models from the SCM.

Problem 9

Find how the statefinder parameters enter the expression for the luminosity distance.