Difference between revisions of "Homogeneous Universe"

A system with constant magnetic field is homogeneous but not isotropic, because the direction along the field and perpendicular to it are not equivalent. On the contrary, a spherically symmetric distribution of electric charges is by construction isotropic, but not, in general, homogeneous.

Due to isotropy the distribution must coincide in any two points, because they can be transformed to each other by rotation by $180^\circ$ around the middle of the interval connecting them. The opposite is not true ,see the problem on homogeniety vs isotropy.

There are three cases: three-dimensional plane (zero curvature), sphere (positive curvature) and hyperboloid (negative curvature).

The main qualitative difference from the usual explosion lies in the fact that the explosive charge is usually surrounded by atmospheric air. The expansion is then caused by the difference between the huge pressure of the gaseous products of the explosion and comparatively small pressure of the surrounding air. But when considering the expanding Universe, one assumes that the pressure (according to the cosmological principle) is uniformly distributed too. Therefore there are neither pressure gradients nor forces that could cause or even affect the expansion. The expansion of the Universe itself is the result of initial velocity distribution.

Suppose the Hubble's law holds in the reference frame $K$, in which matter is at rest at the origin. Consider another frame $K'$, with origin at some point $A$, moving with velocity $\boldsymbol{v}_{A}$ relative to $K$. The quantities measured in the latter frame are denoted by primes. Then in $K'$ one has $\boldsymbol r' = \boldsymbol r - {\boldsymbol r_A}$ and \[\boldsymbol v' = \boldsymbol v - {\boldsymbol v_A} = H\boldsymbol r' = H\boldsymbol r - H{\boldsymbol r_A} = H\left( {\boldsymbol r - {{\boldsymbol r}_A}} \right) = H\boldsymbol r'.\] Therefore the distribution law for velocities in the new frame has the same form with the same value of Hubble's parameter as in the original one.

The continuity equation takes the form: \[\frac{\partial \rho}{dt} +\mbox{div}\left(\boldsymbol v\rho\right) = 0.\] 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 $\boldsymbol v = H\boldsymbol r$ one obtains \[\text{div}(\rho\boldsymbol v) = \rho\text{div}(\boldsymbol v) =\rho\text{div}(H\boldsymbol r) = 3\rho H.\] Therefore, if the distribution $\rho$ was coordinate independent in some reference frame, then the expansion law $\boldsymbol v = H\boldsymbol r$ 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.