# Time-dependent Cosmological Constant

(Inspired by I.Shapiro,J.Sola, H.Stefancic, hep-ph/0410095)

### Problem 1

Obtain the analogue of the conservation equation $\dot\rho+3H(\rho+p)=0$ for the case when the gravitation constant $G$ and the cosmological "constant" $\Lambda$ depend on time.

### Problem 2

Show that when $G$ is constant, $\Lambda$ is also a constant if and only if the ordinary energy-momentum tensor $T_{\mu\nu}$ is also conserved.

### Problem 3

Show that in case the cosmological constant depends on time, the energy density related to the latter can be converted into matter.

### Problem 4

Derive the time dependence of the scale factor for a flat Universe with the $\Lambda$--dynamical constant $\Lambda=\Lambda_0(1+\alpha t)$.

### Problem 5

Construct the dynamics of the Universe in the cosmological model with $\Lambda=\sigma H ,\; \sigma>0.$

### Problem 6

Construct the dynamics of the Universe in the cosmological model with $\Lambda(H)=\sigma H +3\beta H^2$ and $\rho=\rho_\Lambda+\rho_m$.