After Isaac Newton's discovery of the Law of Universal Gravitation, Pierre-Simón Laplace attempted to describe the behavior of the Universe as a cloud of moving matter. Laplace assumed that, in description of motion of this cloud, out of all interactions only gravity plays significant role at large distances between bodies. However, he lacked experimental data to accomplish construction of a realistic model.

Only in the 20th century, astronomical observations made it possible to establish that in the observed part of the Universe matter is distributed almost uniformly and isotropically (i.e. in the areas containing many galaxies the average density of matter is practically the same). In addition, Edwin Hubble discovered a law according to which distant objects move away from us along the line of sight with velocities $V_R$ proportional to distance $R$: $V_R = H \cdot R$, where $H$ is called the Hubble constant; it was found to be independent of distance and approximately equal to $7 \cdot 10^{-11}~\text{years}^{-1}$. George Gamow suggested that such velocity distribution is due to the Big Bang, an explosion that occurred in a small area of space, and then the matter flew in different directions at different speeds. Therefore, the particles, which flew faster, have by now gone farther away from the explosion area

In theoretical physics, Einstein's general relativity equations are used to describe the expansion of the Universe after the Big Bang. However, it's interesting to find the conclusions Laplace would have drawn if he used Hubble law and information about the homogeneity of the Universe in the model based on Newton's laws. To find such conclusions, let us consider the expansion of the so-called Newtonian Universe (NU). The NU is a homogeneous ball of total mass $M=10^{55}~\text{kg}$, in which particles of matter interact due to the Newton’s law of gravity with gravitational constant $G=6.7\cdot 10^{-11}~\text{m}^3/(\text{kg}\cdot \text{s}^2)$, and velocities of matter are distributed according to Hubble law, in which the "constant" $H$ is actually a function of time $H = H(t)$.

*Remark:* if the mass of the 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.

It is clear that in the process of further expansion (over time $t$) the matter in the NU will be decelerated by gravity. Let's assume that at time $t$, which is counted from the Big Bang, some "residents" of the NU measured the average density $\rho(t)$ of matter in it and the Hubble constant $H(t)$.

\[E = -\frac{2}{15} Mc^2,\]

where $c$ is the speed of light in vacuum. Find the maximum radius of the NU in the process of expansion. Write down the formula and get a numerical answer in parsecs ($1 ~\text{parsec}$ is approximately equal to $3.2~\text{light years}$, or $3\cdot10^{16}~\text{m}$).

*Remark:* $$ \int\limits_{0}^{1} \frac{\sqrt{x}}{\sqrt{1-x}} d x=(y=\sqrt{1-x})=2 \int\limits_{0}^{1} \sqrt{1-y^{2}} d y $$