According to data, collected by astronomers at the end of XXs century and the beginning of the XXIs centery, shining objects in the observable universe move away from each other with increasing velocity during last $2.5-4~\text{Myr}$ ($\text{Myr}=10^{6}~\text{yr}$). This observation haven't being explained by cosmological models, which assume that the universe consists only of ordinary matter. At the same time, it doesn't matter whether this matter visible (it radiates or scatters EM waves) or dark(it doesn't interact with EM waves) - the only important thing is that it normally gravitationally interacts with other objects. Therefore in the modern cosmology we add to the model various vacuum-like forms of matter with unusual properties. These forms of matter are called "dark matter". Certainly, scientific models of "quintessence" or "dark matter" are being developed with Metric-affine gravitation theory, Quantum field theory and cutting-edge math. However, several interesting phenomena could be studied with quite simple models. We will work with a similar one, which we will call the Toy Cosmological Model or TCM. In the TCM not only an ordinary matter (which behaves according to Newton's law of gravity) is represented, but also the quintessence. When quintessence is mixed with an ordinary matter, it makes a negative contribution to the gravitational mass of that matter. Also, the quintessence creates a pressure $p_q$ that acts on the ordinary matter and this pressure is determined by the mass density $\rho_q = M_q/V_q<0$ of the quintessence: $p_q = A (-\rho_q)^{5/3},$ where $A$ is a constant. Note, that $p_q$ is also an internal energy density and in order to move or deform a quintessence you must do work on it. The total mass of the ordinary matter is $M_s>0$ and the total mass of the quintessence is $M_q<0$. Also $M_s+M_q>0$. Moreover, we assume that the universe at any given time $t$ is a ball with radius $R(t)$, homogeneously filled with a mixture of ordinary matter and quintessence. The motion of ordinary matter obeys Newton's laws and quintessence has no inertial mass, so its kinetic energy is equal to zero. Remark: if the mass of system increases from $m$ to $m + \Delta m$ without any changing the relative mass distribution, the gravitational potential energy increases by $\Delta E_G = \int\limits_0^{\Delta m} \varphi \, dm,$ where $\varphi$ is a gravitational potential at the point of mass $dm$. Also $\varphi$ is chosen so that $\varphi=0$ on infinity.

A1 Find the total energy $E$ of the universe as a function of its radius $R$ in the form
$E = -\frac{\alpha}{R} + \frac{\beta}{R^2} + E_K,$
where $E_K$ is the kinetic energy of the universe.

A2 For given $M_s$, $M_q$, $A$ and $E$ of the universe find a range of radii which it can have. Write the expression in terms of $\alpha$, $\beta$ and $E$.

A3 Could the universe described by TCM be stable, i.e., have a constant radius? If so, describe it quantitatively.

Let's describe a particular realization of the universe in TCM. Let the total energy $E$ of the universe be equal to zero and its history start from zero expansion velocity with the minimum radius. We also assume that in this universe the Hubble's law holds: at any given time, the distribution of velocities $v(r,t)$ of the ordinary matter as a function of distance $r$ from the center is given by
$v(r,t) = \frac{r}{R} v(R,t),$
where $R$ is the radius of the universe.

A4 How long will the universe continue to expand at a positive acceleration?