Homogeneous Universe
Homogeneous and isotropic Universe, Hubble’s law
Problem 1.
Most cosmological models are based on the assumption that the Universe is spatially homogeneous and isotropic. Give examples to show that the two properties do not automatically follow one from the other.
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.
Problem 2.
Show that if some spatial distribution is everywhere isotropic then it is also homogeneous. Is the opposite true?.
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 problem.
Problem 3.
What three-dimensional geometrical objects are both homogeneous and isotropic?
There are three cases: three-dimensional plane (zero curvature), sphere (positive curvature) and hyperboloid (negative curvature).
Problem 4.
Why the notion of Big Bang regarding the early evolution of the Universe should not be treated too literally?
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.
Problem 5.
Show that the Hubble's law is invariant with respect to Galilean transformations.
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 $\vec{v}_{A}$ relative to $K$. The quantities measured in the latter frame are denoted by primes. Then in $K'$ one has $\vec r' = \vec r - {\vec r_A}$ and \[\vec v' = \vec v - {\vec v_A} = H\vec r' = H\vec r - H{\vec r_A} = H\left( {\vec r - {{\vec r}_A}} \right) = H\vec r'.\] 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.
Problem 6.
Show that the Hubble's law represents the only form of expansion compatible with homogeneity and isotropy of the Universe.
Problem 7.
Show that if expansion of the Universe obeys the Hubble's law then the initial homogeneity is conserved for all its subsequent evolution.
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.
Problem 8.
In the 1940-ties Bondi, Gold and Hoyle proposed a stationary model of the Universe basing on the generalized cosmological principle, according to which there is no privileged position either in space or in time. The model describes a Universe, in which all global properties and characteristics (density, Hubble parameter and others) remain constant in time. Estimate the rate of matter creation in this model.
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".
Problem 9.
Galaxies typically have peculiar (individual) velocities of the order of $V_p \approx 100~\mbox{km/s}.$ Estimate how distant a galaxy should be for its peculiar velocity to be negligible compared to the velocity of Hubble flow $V_H=H_{0}R$.
From the condition $\frac{V_p}{V_H} \ll 1$ one gets $R \gg \frac{V_p}{H_0} \approx 1.5~\mbox{Mpc}$.
Problem 10.
Estimate the age of the Universe basing on the observed value of the Hubble's constant (the Hubble time $t_H$).
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 $t_H = H_0^{-1} \approx 4 \cdot 10^{17} ~\mbox{ s} \approx 14\cdot10^9~\mbox{years.}$
Problem 11.
Show that the model of the expanding Universe allows one to eliminate the Olbers' paradox.
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.
