The continuity equation takes the form: \[\frac{\partial \rho}{dt} +\mbox{div}\left(\vec v\rho\right) = 0.\] It follows from the homogeneity of the Universe that $\rho$ can depend on time but it is independent on coordinates, i.e. $\rho = \rho (t)$. Then in accordance with the Hubble's law $\vec v = H\vec r$ one obtains \[\text{div}(\rho\vec v) = \rho\text{div}(\vec v) =\rho\text{div}(H\vec r) = 3\rho H.\] Therefore, if the distribution $\rho$ was coordinate independent in some reference frame, then the expansion law $\vec v = H\vec r$ preserves the homogeneity of $\rho$ in all subsequent moments of time. Thus the initial homogeneity is conserved forever.

Constant Hubble's parameter provides exponential expansion of the Universe with $a = {a_0}{e^{Ht}}$. Consider a region of space with initial volume $V_0$ and density $\rho_0$. Then at time $t$ the density in the chosen region is \[\rho = \frac{V_{0}\rho _0}{V} = \frac{V_{0}\rho_{0}} {\frac{4}{3}\pi a_0^3{e^{3Ht}}} = {\rho _0}{e^{ - 3Ht}},\] and one obtains $\dot \rho = - 3H\rho$. Then to preserve the constant density of the Universe one has to introduce additional density growth rate equal to $3H\rho$. Note that $ H \approx 2 \cdot {10^{-18}\;s^{- 1}}\approx 6\cdot 10^{ - 11} \;\mbox{year}^{-1}$ and \[3H\rho \approx 3H\rho _{cr} \approx 2 \cdot 10^{ - 39}\mbox{ g} \cdot \mbox{cm}^{-3} \cdot \mbox{ yr}^{-1} \approx m_p\,\mbox{km}^{- 3} \cdot \mbox{ yr}^{ - 1},\] where ${m_p} = 1.67 \cdot {10^{ - 24}}\;\mbox{ g}$. Thus the constancy of the Hubble's parameter along with constant density requires creation "from nothing" of on average one proton in cubic kilometer per year. So, for example, our Earth during its lifetime would have "gained weight" of about $\sim 10^{-2} {\rm g}$. Paul Dirac was one of active supporters of cosmological models with matter creation. In regard to his and Hoyle's models he wrote the following: "... Hoyle assumed that the Universe is in a homogeneous and isotropic state, and continuous creation of matter is aimed to supply the substance which leaves the observable zone due to the expansion. The Hoyle's theory sets the quantity $G$ constant, while in mine $G$ changes with time, and it makes the main distinction from the Hoyle's theory. I propose a theory where continuous creation of matter is combined with variable $G$. Both assumptions result from the large numbers' hypothesis".

From the condition $\frac{V_p}{V_H} \ll 1$ one gets $R \gg \frac{V_p}{H_0} \approx 1.5~\mbox{Mpc}$.

We assume that the Universe expanded in the past with constant velocity equal to its modern value. Then the physical distance between any two cosmological objects equals to $R = V_Ht_H = H_0Rt_H,$ and $t_H = H_0^{-1} \approx 4 \cdot 10^{17} ~\mbox{ s} \approx 14\cdot10^9~\mbox{years.}$

A star with absolute luminosity $L$, situated at distance $r$, has the apparent luminosity equal to $4\pi r^2L$. If the number density of the stars is constant and equals to $n$, then the number of stars in a spherical layer between $r$ and $r+dr$ equals to $4\pi n r^2dr$, and the total density of radiation energy from all the stars is equal to \[\rho _s = \int\limits_0^{R_H} \frac L {4\pi r^2} 4\pi r^2 dr = Ln\int\limits_0^{R_H} dr = LnR_H.\] The solution of the Olbers' paradox is thus possible due to the fact that an observer registers radiation from the stars inside of the Hubble's sphere only, rather than from the infinite Universe.