# Lost and Found

### Problem 1

Consider the case of spatially flat Universe dominated by non-relativistic matter and spatially homogeneous scalar complex field $\Phi$ and obtain the equations to describe the dynamics of such a Universe.

### Problem 2

Consider the case of the Universe composed of non-relativistic matter and quintessence and relate the quantities $\varphi ,\,\rho _\varphi ,\,H,\,V(\varphi )$ with the redshift-dependent state equation parameter $w(z)$.

### Problem 3

When solving the equation for the scalar field one assumes that the time dependence of the scalar factor, which is necessary to calculate the Hubble parameter in the equation, is determined by the dominant component. Such approximation becomes invalid at some time moment, because the energy density for the scalar field decays slower than that of matter or radiation. Determine the value of the scalar field at that time moment for the potential of the problem